d3.5 – report on driving range prediction and extension ... · thesis and 1 whitepaper....

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Electric Vehicle Enhanced Range, Lifetime And Safety Through INGenious battery management

D3.5 – Report on driving range prediction and extension algorithm

February 2020

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 713771


Author: Paul Padilla (TU/e)

EVERLASTING - Grant Agreement 71377 (Call: H2020-GV8-2015) Electric Vehicle Enhanced Range, Lifetime And Safety Through INGenious battery management

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PROJECT SHEET

Project Acronym EVERLASTING

Project Full Title Electric Vehicle Enhanced Range, Lifetime And Safety Through INGenious battery management

Grant Agreement 713771

Call Identifier H2020-GV8-2015

Topic GV-8-2015: Electric vehicles’ enhanced performance and integration into the transport system and the grid

Type of Action Research and Innovation action

Project Duration 48 months (01/09/2016 – 31/08/2020)

Coordinator VLAAMSE INSTELLING VOOR TECHNOLOGISCH ONDERZOEK NV (BE) - VITO

Consortium Partners

COMMISSARIAT A L ENERGIE ATOMIQUE ET AUX ENERGIES ALTERNATIVES (FR) - CEA SIEMENS INDUSTRY SOFTWARE SAS (FR) - Siemens PLM TECHNISCHE UNIVERSITAET MUENCHEN (DE) - TUM TUV SUD BATTERY TESTING GMBH (DE) - TUV SUD ALGOLION LTD (IL) - ALGOLION LTD RHEINISCH-WESTFAELISCHE TECHNISCHE HOCHSCHULE AACHEN (DE) - RWTH AACHEN LION SMART GMBH (DE) - LION SMART TECHNISCHE UNIVERSITEIT EINDHOVEN (NL) - TU/E VOLTIA AS (SK) - VOLTIA VDL ENABLING TRANSPORT SOLUTIONS (NL) – VDL ETS

Website www.everlasting-project.eu




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DELIVERABLE SHEET

Title D3.5 – Report on driving range prediction and extension algorithm

Related WP WP3 (Extended driving range)

Lead Beneficiary TUe

Author(s) Paul Padilla (TU/e) Tijs Donkers (TU/e)

Reviewer(s) Will Hendrix (TUE) Roshni Digumoorthi (VDL) Juan Carlos Flores (VDL)

Type Report

Dissemination level PUBLIC

Due Date M42

Submission date January 27, 2020

Status and Version Draft, version 0.4




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REVISION HISTORY

Version Date Author/Reviewer Notes

V0.1 20/01/2020 Paul Padilla (TUE) Tijs Donkers (TUE)

First draft

V0.2 24/01/2020 Will Hendrix (TUE)

V0.3 30/01/2020 Roshni Digumoorthi (VDL) Juan Carlos Flores (VDL)

V0.4 04/02/2020 Carlo Mol (VITO)

Quality check

V1.0 DD/MM/YYYY Carlo Mol (VITO)Coordinator

Submission to the EC




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DISCLAIMER The opinion stated in this report reflects the opinion of the authors and not the opinion of the European Commission. All intellectual property rights are owned by the EVERLASTING consortium members and are protected by the applicable laws. Except where otherwise specified, all document contents are: “© EVERLASTING Project - All rights reserved”. Reproduction is not authorised without prior written agreement. The commercial use of any information contained in this document may require a license from the owner of that information. All EVERLASTING consortium members are committed to publish accurate information and take the greatest care to do so. However, the EVERLASTING consortium members cannot accept liability for any inaccuracies or omissions nor do they accept liability for any direct, indirect, special, consequential or other losses or damages of any kind arising out of the use of this information. ACKNOWLEDGEMENT This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 713771




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EXECUTIVE SUMMARY

Range extension of electric vehicles has become a prominent objective to achieve in order to mitigate range anxiety and consequently, accelerate the adoption of electric vehicles in the market. In fact, range anxiety mitigation by vehicle range optimization has become one of the main objectives in the EVERLASTING project. Range optimization can be achieved by using vehicle energy management strategies, which aim to optimize the total energy consumption of the vehicle. In most of the cases, it is simple and economical compared to other approaches to extend the vehicle range, e.g., energy dense batteries, extended charging infrastructure, light weight vehicle parts, etc. For this reason, this report documents the work done in Task 3.5, where the objective has been to optimize and to predict the driving range in terms of energy. The compendium of publications obtained in this project ([1]–[11]) are included as appendixes of this report. In particular, the work carried out in Task 3.5 has lead to 2 journal publications, 7 refereed conference publications, 1 MSc thesis and 1 whitepaper. Furthermore, a preliminary experimental study shows that optimization of the vehicle energy consumption can lead to a reduction in energy consumption of 6.94%, which leads to a range extension of 7.45%, thereby laying the basis for achieving the objective of range extension of 5% as described in the project’s description of action (DoA).




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TABLE OF CONTENTS

EXECUTIVE SUMMARY ........................................................................................................ 6

TABLE OF CONTENTS .......................................................................................................... 7

LIST OF ABBREVIATIONS AND ACRONYMS ......................................................................... 8

1 INTRODUCTION .......................................................................................................... 10

2 CONCLUSION .............................................................................................................. 12

3 LIST OF PUBLICATIONS .............................................................................................. 12

APPENDIX COMPLETE VEHICLE ENERGY MANAGEMENT FOR ELECTROMOBILITY…………………………………………………………..14

RANGE ESTIMATION FOR ELECTRIC VEHICLE……………………………………………………………………………………………27

A GLOBAL OPTIMAL SOLUTION TO THE ECO-DRIVING PROBLEM…………………………………………………………………41

A SHRINKING HORIZON APPROACH TO ECO-DRIVING FOR ELECTRIC CITY BUSES.………………………………………47

A DISTRIBUTED OPTIMIZATION APPROACH FOR COMPLETE VEHICLE ENERGY MANAGEMENT.………………………53

EFFECTS OF BATTERY CHARGE ACCEPTANCE AND BATTERY AGING IN COMPLETE VEHICLE ENERGY MANAGEMENT……………………………………………………………………………………………………………………………………….67

GLOBAL SOLUTIONS TO THE COMPLETE VEHICLE ENERGY MANAGEMENT PROBLEM VIA FORWARD-BACKWARD OPERATOR SPLITTING……………………………………………………………………………………………………………………………74

A PORT-HAMILTONIAN APPROACH TO COMPLETE VEHICLE ENERGY MANAGEMENT………………………………………80

TRAFFIC-AWARE VEHICLE ENERGY MANAGEMENT STRATEGIES VIA SCENARIO-BASED OPTIMIZATION………………………………………………………………………………………………….…………………………………88

VEHICLE ENERGY MANAGEMENT WITH ECO-DRIVING……………………………………………………………………………….96

ECO-DRIVING FOR ENERGY EFFICIENT CORNERING OF ELECTRIC VEHICLES IN URBAN SCENARIO………………104




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LIST OF ABBREVIATIONS AND ACRONYMS

ACRONYM DEFINITION DoA Description of activityCVEM Complete Vehicle Energy Management HVAC Heat Ventilation and Air Conditioning System FWD Forward Wheel Drive RWD Rear Wheel Drive SQP Sequential Quadratic Programming BMS Battery Management System




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1 INTRODUCTION

This report documents the work done in Task 3.5. It contains a description of the main objectives and contributions of the work. The publications that have resulted from the work done in Task 3.5 are included in the appendix. This chapter connects the objectives of Task 3.5 to these publications. An extensive literature review of the state-of-the-art exploration of the research paths on complete vehicle energy management for electromobility applications is presented in [1]. This exploration is directed to the modelling, optimization techniques and prediction in view of exploitation of preview information aspects to predict power request that arise from the inclusion of velocity as a decision variable in energy management problems. Thus, aiming to include ’anticipation’ and ’reaction’ as way of predictions as new features of the complete vehicle energy management concept (CVEM). In this work, energy management strategies are divided in 2 groups. First, CVEM that aims to maximize the range of the vehicle by minimizing the total power consumption using an optimal splitting of energy among all the interconnected subsystems in the vehicle for a given power request. Second, Eco-driving strategies, which extends the driving range of a vehicle by the selection of an energy optimal velocity profile. A first contribution to a power request and range predictor has been described in [2]. Accurate range estimation in electric vehicles is a crucial challenge to increase customer trust in the use of batteries as an alternative source of power in mobility, thereby mitigating the effect of range anxiety for drivers. This work presents a novel approach to estimate the range of an electric vehicle with a low computational load by estimating the battery pack state and forecasting the future power demand using available vehicle information. The results show that the driving range of an electric bus can be accurately estimated with a maximum range estimation error of less than 10%. Range prediction can also be used as a tool to predict future power request in a vehicle, which is important aspect to consider during the implementation of CVEM strategies. In fact, the efficiency achieved by CVEM strategies depends on the quality of the predictions for future power request. Further explorations for more accurate predictions of the vehicle are performed in Task 3.3, whose results are reported in Deliverable 3.3. In [5], an energy management strategy to improve the operational efficiency of a vehicle, and thus , extending the driving range is proposed as a distributed optimization approach is proposed to solve the CVEM problem of a vehicle with several controllable auxiliaries. This approach uses a dual decomposition, which allows the underlying optimal control problem to be solved for every subsystem separately. Simulation results show that the energy consumption can be reduced up to 0.52% on a heavy-duty truck, by including smart auxiliaries in the energy management problem. More interestingly, the computation time is reduced by a factor of 64 up to 1825, compared with solving a centralized convex optimization problem. In [6], the methodology presented in [5] has been extended to include charge acceptance limitations and battery aging effects, showing that battery aging constraints can be partly compensated by smart control of other energy buffers, which shows the true benefit of CVEM. The existence of only global optimal solutions for the CVEM problem was demonstrated in [7]. As a result, a distributed optimization algorithm for non-convex optimization problems using operator splitting techniques was proposed. The relevance of this contribution is that it opens the door for online implementations of CVEM using a decentralized architecture. In [8], a different exploration of CVEM is performed using port-based models. Specifically, Port-Hamiltonian theory is used to formulate a decomposable optimal control problem for CVEM applications. The advantage of this alternative modeling framework with respect to the power-based approach used in [5],[6],[7],[9], is that it is able to capture physically relevant variables of the systems and describe energy loses in an intuitive way, which is relevant for this type of energy




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management problems. A simulation case study for an electric city bus with heating, ventilation and air conditioning (HVAC) system was presented., where it was possible to see that the HVAC system can be used as an additional energy buffer. The simulation results showed that in specific cases an extension of the driving range up to 25% could be possible. Real driving conditions are subject to uncertainty, which directly affect power request predictions, thereby making the real-time optimization of the energy consumption of a vehicle to be a challenging problem. In order to deal with this situation, in [9] the current framework of CVEM was employed in a receding horizon fashion in which random constraints representing realizations of the disturbance, i.e., the uncertain driving conditions, are considered. Additionally, several methods for power request prediction are described and compared, e.g., using average traffic flow information, a method based on Gaussian process regression, and a method that combines both. The proposed strategies are tested with real traffic data as a case study where the use of a Gaussian process regression and the average traffic speed achieves near optimal fuel consumption. The driving range of a vehicle can also be extended by the selection of an energy optimal velocity profile, which is referred as the eco-driving problem. Eco-driving is an energy management method that aims maximize the range of a vehicle by means of finding energy optimal velocity profiles. In [3], a detailed analysis on the global optimal solution of the Eco-driving problem was presented. This result was exploited to propose a static non-linear optimization algorithm that can be used for an online implementation of this method. As a consequence of the results in [4], an online implementation of the eco-driving algorithm for electric city buses was presented. The optimization algorithm was used to formulate a shrinking horizon model predictive control that provides real time feedback to the driver about the optimal velocity profile needed to arrive the next bus stop on a given time. In order to validate this approach, experiments on an electric city bus were performed obtaining as a result average energy savings of 6.94%. It should be noted that energy savings of 6.94% leads to an extension of the driving range of 7.45%. Namely, it holds that

𝑉𝑒ℎ𝑖𝑐𝑙𝑒 𝑟𝑎𝑛𝑔𝑒 𝑘𝑚 𝐸𝑛𝑒𝑟𝑔𝑦 𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 𝑘𝑊ℎ 𝑣𝑒ℎ𝑖𝑐𝑙𝑒 𝑠𝑝𝑒𝑒𝑑

𝐸𝑛𝑒𝑟𝑔𝑦 𝑐𝑜𝑛𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛 𝑘𝑊

And a reduction in energy consumption by a factor (1-0.0694) leads to an extension of the range by a factor 1/(1-0.0694)=1.0745. In [11], the work proposed in [3] is extended by considering a model for eco-driving that captures cornering effects, which is especially relevant in urban scenarios. The proposed model purely relies on the geometric configuration of the vehicle and road. Moreover, it can be applied for vehicles with front wheel drive (FWD) or rear wheel drive (RWD). Simulation results for an electric vehicle executing cornering maneuvers showed and approximated improvement of 8% in energy savings with respect to traditional eco-driving strategies, especially in trajectories with large curvatures. Finally, in [8] the CVEM and eco-driving strategies (presented in [3] and [5]) are connected and extended to incorporate velocity profiles as part of the CVEM problem. It is shown that a Sequential Quadratic Programming (SQP) algorithm, which uses a convex subproblem with Thikhonov regularization, can be embedded in a distributed optimization approach, allowing it to be used for CVEM, incorporating optimal control of the vehicles auxiliary systems, in combination with eco-driving.




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2 CONCLUSION

In the Description of Activities (DoA), objective 6 states:

Objective 6: To extend the driving range the EVERLASTING project will develop BMS technology that will increase driving range up to 15% without increasing the physical battery size. It will also decrease the charging time by 30%.

To reach the 15% increase of driving range several topics have been listed in the DoA to be subject of research, where the results of this separate research activities all contribute to reach this 15% target. This Deliverable focusses on the topic:

Optimizes the driving behavior, resulting in an increase of the driving range by 5%. The above discussed publications have explored range prediction, which crucially depends on the vehicle’s ability to predicts the vehicle’s power request. This is considered in more detail in Task 3.3 and Task 3.4, with corresponding Deliverables 3.3 and 3.4. Furthermore, algorithms that can be used to extending the driving range have been studied extensively and first experimental results show that an extension of the driving range of more than 5% is expected, as was intended as one of the project’s overall objectives.

3 LIST OF PUBLICATIONS

[1] G.P. Padilla, and M.C.F. Donkers, “Complete Vehicle Energy Management for Electromobility”, White paper, Eindhoven University of Technology, 2017

[2] Francisco Ozuna, “Range Estimation for Electric Vehicles”, Msc. Thesis, Eindhoven University of Technology, 2017.

[3] G.P. Padilla, S. Weiland, and M.C.F. Donkers, “A Global Optimal Solution to the Eco-Driving Problem”, in IEEE Control Systems Letters, 2018.

[4] J.C. Flores, G.P. Padilla, & M.C.F. Donkers. “A shrinking horizon approach to eco-driving for electric city buses: implementation and experimental results”, in IFAC International Symposium on Advances on Automotive Control, 2019.

[5] T.C.J. Romijn, M.C.F. Donkers, John T. B. A. Kessels, and Siep Weiland, “A Distributed Optimization Approach for Complete Vehicle Energy Management”, in IEEE Transactions on Control Systems Technology, 2018.

[6] Z. Khalik T.C.J. Romijn M.C.F. Donkers and S. Weiland, “Effects of Battery Charge Acceptance and Battery Aging in Complete Vehicle Energy Management”, in IFAC World Congress, 2017.

[7] G.P. Padilla, G. Belgioioso and M.C.F. Donkers, “Global Optimal Solutions to Complete Vehicle Energy Management Problem via Forward-Backward Operator Splitting”, in IEEE Control and Decision Conference, 2019.




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[8] J.C. Flores, G.P. Padilla, and M.C.F. Donkers, “A Port-Hamiltonian Approach to Complete Vehicle Energy Management for an Electric Bus with an HVAC System”, in IEEE American Control Conference, 2020.

[9] A. Wulf, G.P. Padilla, and M.C.F. Donkers, “Traffic-Aware Vehicle Energy Management Strategies via Scenario-Based Optimization”, in IFAC world congress, 2020. (Submitted)

[10] Z. Khalik, G.P. Padilla, T.C.J. Romijn and M.C.F. Donkers, “Vehicle Energy Management with Ecodriving: A Sequential Quadratic Programming Approach with Dual Decomposition”, in IEEE American Control Conference, 2018.

[11] G.P. Padilla, C. Pelosi, C.J.J. Beckers and M.C.F. Donkers, “Eco-Driving for Energy Efficient Cornering of Electric Vehicles in Urban Scenarios”, in IFAC world congress, 2020. (Submitted)

EINDHOVEN UNIVERSITY OF TECHNOLOGY, PHD FIRST YEAR EVALUATION, NOVEMBER 2017 1

Complete Vehicle Energy Management for ElectromobilityPaul Padilla

Abstract— This paper presents an exploration of the fu-ture research on complete vehicle energy management forelectromobility applications. This exploration is directed tothe modelling, optimization techniques and exploitation ofpreview information aspects that arise from the inclusion ofvelocity as a decision variable in energy management problems.Thus aiming to include ’anticipation’ and ’reaction’ as newfeatures of the complete vehicle energy management concept.Preliminary results that show the potential of this researchtopic are presented and concrete plans for future extensionsare discussed.

I. INTRODUCTION

Transportation has deeply influenced the evolution of ourcivilization. The formation of modern urban centers havebeen shaped by the development of transportation systems,which is also directly connected to the economic welfare;e.g., transporting raw materials and finalized goods over longdistances is an essential activity for trading. On the otherhand, the environmental role of transportation has a negativeimpact for the modern society. For instance, the transportsector was responsible for 14% of the global CO2 emissionsfrom fossil fuel consumption in 2014 [1]. Additionally, in2016, it was estimated that 95% of the global transportationenergy was based in petroleum fuels [2]. This oil dependencyis also a sensitive factor that influences the geopoliticalstability since petroleum is a finite resource.

Despite of those negative factors, transportation still en-compasses crucial aspects in our society. Thus, in the lastyears, the electrification of transport systems has raised asa promising approach to mitigate effects caused by theenvironmental role of transportation and to face the imminentdepletion of fossil fuels. Assuming that the electricity con-sumed by electric transportation systems is produced fromrenewable energy sources, the impact on the reduction ofCO2 emissions is direct; under the same assumption, it isexpected that electric transportation drastically reduce thehigh oil dependency of the modern society [3].

The electrification in automotive sector has been studiedas electromobility in [4], which can be defined as vehiclesthat are propelled by electricity. This is a simple yet broaddefinition due to the fact that not only encircles hybridelectric vehicles (HEV) or battery electric vehicles (EV),this definition also includes vehicles that obtain energy formthe grid or converts other energy sources to electricity,i.e., biomass, hydrogen, etc. Hence, electromobility impliesflexibility for the electrified vehicles to have different energysources [4].

The evolution of the EV market in the recent years haveopened the door for optimistic predictions about the futuremarket for electromobility. For instance, in 2016. the EV

made up 29% of the market in Norway, while in China,the number of EV units increased from 100.000 in 2014 to650.000 in 2016 [5]. This tendency indicates that by 2030electromobility might reach a stage of mass market adoption.This global optimism is also supported by the novel researchannouncements related to batteries, charging infrastructureand energy efficiency in vehicles.

However, the goal of a complete acceptance of electro-mobility in the market is still far and several problems needto be solved. It has been identified that one of the mainreasons that limits the penetration rate of EV in the marketis range anxiety [6], [7]; which is defined as the concern(experienced by users) that the vehicle has insufficient energyto reach the next charging station [5]. There are severalfactors that influence range anxiety, e.g, reduced capacityof batteries to store energy, weather and traffic conditionsthat affects drastically the prediction of the remaining drivingrange [8], limited number of charging stations and the time-consuming charge process [9]. As a response to these factors,the mitigation of the anxiety range can be achieved by acombination of the next approaches:

1) Energy management strategies.2) Light weight and energy dense batteries.3) Fast charging.4) Extended charging infrastructure.The use of light-weight and energy dense batteries is the

most logical choice to reduce the energy anxiety effect byextending the driving range of the EV. On the other hand, theextension of charging infrastructure and the development offast charging technologies aim to reduce the range anxiety byoffering the user the possibility to always have a reachablecharging station where the charging process is performed ina reduced time, which is resembles the functionality of thecurrent gas stations. The last proposed approach to mitigaterange anxiety is related to energy management strategies,which aims at extending the driving range of the vehicleby maximizing the energy efficiency during its operation.In particular, energy efficient operation of vehicles can bedescribed as the solution to an optimal control problem thataims to obtain the optimal conversion of energy between theenergy consumers in the power network of the vehicle. Themain advantage for this approach is that energy managementstrategies can (for most of the cases) be applied directly assoftware in the vehicle. Hence, the implementation cost ismarginal. For this reason, energy management strategies havebeen a topic of intensive research in the last years and definesthe global scope of the contents treated in this manuscript.

This paper aims to present a future view of energy man-agement strategies for electromobility.

The content in this document is organized as follows. In




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Section II, an overview of architectures and basic modelingfor energy management applications are presented. The stateof the art for energy management systems is discussed inSection III. In Section IV, the main research question isposed, several problems are formulated and possible method-ologies to solve those problems are discussed. Preliminaryresults are presented in Section V. Finally, the future workand conclusions are discussed in Section VI and Section VIIrespectively.

II. OVERVIEW OF ENERGY MANAGEMENT CONCEPTS

In this section, the concept of complete vehicle en-ergy management (CVEM) is presented. Additionally, anoverview of the power network concepts is presented as ageneral network architecture applied to electromobility.

A. Complete Vehicle Energy ManagementThe CVEM concept was originally introduced in [10]

as an holistic approach to energy management in vehicles.Later, methodologies to solve the CVEM problem in severalautomotive applications have been reported in [11]–[13]. Theindividual optimization of energy consumption in all thesubsystems of a vehicle is incapable to produce a globalenergy efficient operation. This limitation is consequence ofneglecting the iteration that exists between all the energyconsumers inside a vehicle. Hence, an holistic approachthat optimizes energy consumption of all the interconnectedsubsystems in the vehicle is needed, which is the basis forthe CVEM concept.

The energy in the vehicle is transformed between differentphysical domains, e.g., chemical energy that is stored asfuel can transformed into electrical energy by an electricgenerator unit; later, the produced electricity can be eithertransformed into mechanical energy to propel the vehicle ortransformed into thermal energy modify the temperature ofthe cabin. For this reason, considering power networks todescribe the power iteration between the energy consumersis a useful tool in the CVEM framework; in Section II-B,a generalized power network architecture for electromobilityis detailed.

The main features of CVEM are consequences of itsinterconnected nature and are summarized next:• Globally Optimal: By considering the iteration of all

the subsystems in the power network, the design ofenergy management strategies that determine a globallyefficienct operation of the vehicle is feasible.

• Scalable: The complexity of the energy managementsystem should not be sensitive to the introduction ofadditional subsystems in the power network.

• Reconfigurable / Flexible: Updating or completelyremoving subsystems or its parameters should not implya major reconfiguration of the energy managementstrategy.

B. Power Network Architecture for ElectromobilityDue to the broad nature of electromobility it is important

to have a general architecture that can describe a large varietyof vehicle topologies.

Fig. 1: General topology for electromobility.

The idea of general power network architecture is pre-sented in Fig. 1 and describes the conversion and interactionof power flows between different energy domains [10]. Thepower network depicted in this figure is composed by energybuffers and converters, which are represented by rhombusand rectangles respectively. The subsystems in the networkare constituted by a single converter or by an energy bufferand a converter that are directly connected (grey dashedline). In Fig. 1, the blue colour is used to represent elementsassociated to electric power, while the back colour is usedto describe other non-electric energy domain power flows.

The subsystems located in the left hand side of Fig. 1describe energy sources. For instance, energy source A canrepresent a super capacitor that stores electric energy, andthe converter block describes the efficiency of this element.Energy source B can be used to describe batteries or fuelcells; in this case, the chemical energy stored in the bufferis converted to electric energy. Finally, energy source C canrepresent an internal combustion engine, where the energy istransformed from the chemical to mechanical domain.

It is interesting to note that the vehicle auxiliaries can bealso considered as energy storing subsystems. For example,at low environment temperatures, the heat ventilation and airconditioning (HVAC) system can be used to store the excessof recovered energy form braking as heat. Similarly, theinertia subsystem stores kinetic and potential energy of thevehicle depending on the current velocity and road elevationof the vehicle; in some applications, where the trajectoryof the vehicle is known in advance, this buffer can also beexploited by the energy management strategies.

The proposed network has a general configuration thatallows to easily describe specific electromobility cases. Forinstance, the topology for hybrid parallel vehicle can bedescribed by removing energy source A from the generalarchitecture. Removing the clutch, energy source A and Cthe topology obtained can represent an EV; if an additionalenergy source B is considered, the same configuration canrepresent the power network topology for an hydrogen fuelcell vehicle.

C. CVEM as an Optimal Control Problem

The CVEM problem can be described as an optimalcontrol problem. The formulation of the problem depends




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on the application goals; however, it is possible to identifya general structure that CVEM follows and is given by

minξ(τ)

J(ξ(τ)) (1a)

subject to D(ξ(τ)) = 0, (1b)H(ξ(τ)) = 0, (1c)N (ξ(τ)) = 0, (1d)

¯b(ξ(τ)) 6 B(ξ(τ)) 6 b(ξ(τ)), (1e)

where ξ(τ) is a time or space dependent vector of decisionvariables that normally represent power, forces or evenvelocities in the vehicle. The cost function of the optimalcontrol problem (1) is given by J(ξ(τ)), which it is normallydescribed as the cumulative power consumed from the energysources in the vehicle over a driving cycle. The problem issubject to the dynamics of all the subsystem (1b), the powerdefinition for each converter in the network (1c) and to (1d)that describes the power balance for all the nodes in thepower network. Additionally, the optimal control problem(1) is bounded by (1e), which often represents power, torqueand even velocity bounds.

III. STATE OF THE ART

A large amount of energy management strategies (EMS)have been reported in the last two decades. As can beseen in Fig. 2, the number of papers published in 2017 isapproximately two times larger than the amount of paperspublished in 2010; thus, showing the growing interest ofEMS the automotive industry. In order to describe the stateof the art in this field, in this paper, the research literatureavailable is divided in four relevant groups that can bedirectly connected to electromobility.

A. Classic EMS

Traditionally, energy management problems are focusedon controlling the power split between the combustion engine

2000 2002 2004 2006 2008 2010 2012 2014 2016

Year

0

10

20

30

40

50

Num

ber

of P

aper

s

Fig. 2: Research trend in EMS (number of papers peryear published in IEEE that contain the key words “energymanagement” and “automotive”).

and the electric machine of a hybrid electric vehicle. Bystoring regenerative braking energy and shifting the operatingpoints of the combustion engine, a significant amount of fuelcan be saved. In [14], [15, Ch. 4] optimization techniques forenergy control on hybrid vehicles are summarized; addition-ally, the books [16], [17] present a complete introduction ofthis research area.

Classic EMS literature can be classified in an ad hocsolutions and optimal control approaches. Additionally, theoptimal control approaches can be subdivided into off-lineand on-line control problems.

1) Ad-hoc solutions:The set of EMS that are included in this classification

are normally characterised to be computationally fast, whichmake them suitable candidates for on-line implementations.Historically, this type of approaches were the first to ap-pear in EMS literature [18], [19]. Fuzzy logic [20] andneuronal networks [21] are popular strategies among thisclassification. The disadvantage of this strategies is thatthe performance of the systems is sensitive to changes inoperation, and the glocal optimality of the solution cannotbe guaranteed. Nevertheless, some authors are currentlyproposing solutions to properly recalibrate this strategiesusing dynamic programming (DP) [22], thus obtaining anacceptable approximation of the global solution.

2) Off-line optimal control problems:EMS under this category are normally used as benchmark

solutions for specific applications or configurations of theoptimal control problem due to the fact that that globaloptimal solutions are typically achieved.

A representative optimization technique in this categoryis Dynamic Programming (DP) [23]; for instance in [24],applying DP forces the convergence to the global optimumof the energy management problem proposed. However, DPhas the inherent disadvantage that the computational burdenincreases with the number of states.

Optimization methods based on the Pontryagin’s Max-imum Principle (PMP), see, e.g., [25], [26] can handlecomputational complexity of multi-state energy managementproblems. In PMP, the problem is reduced to solving a two-point boundary value problem, which can be difficult to solvein the presence of state constraints. Moreover, the globaloptimality of the solutions obtained can only be guaranteedif the formulation of the optimal control problem is convex.

Similarly, static optimization methods can guarantee aglobal optimality only for convex approximations of the en-ergy management problems, e.g., see [27] and the referencestherein.

Finally, it is important to remark that the EMS thatbelong to this classification need to have a priori completeinformation related to driving cycle, i.e., velocity and roadgradient. This is the main reason that limits the use of thisEMS approach for on-line implementation.

3) On-line optimal control problems:In this case, algorithms that requires a low computational

effort are described; besides, the requirement of a priori




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information related to the velocity and road profile is relaxed,which normally leads ot a suboptimal solutions.

In [28], [29] stochastic DP is used to performance an off-line calculation of the optimal EMS, which is given by anstochastic policy that is implemented as a look-up table in alow computational power embedded system.

A fast computational approach that uses PMP is known asthe equivalent consumption minimization strategy (ECMS)[30]. In this strategy, the energy consumption in the batteryis transformed into an equivalent fuel consumption, whichis represented by a co-state function related to the batteryenergy. The co-state can be estimated at every time instant,thus eliminating the necessity for the complete driving profileinformation in advance. Consequently, ECMS can be imple-mented on-line; however, updating the co-state is a delicatetask that often produce to suboptimal solutions.

Model predictive control (MPC) is also used to implementon-line EMS [31], [32]. A finite-time horizon predictionof the future the energy consumption is used to calculateoptimal control strategy, from which only the first controldecision is implemented. A successive execution of thisprocedure obtains a suboptimal energy management policythat depending on the quality of the predictions can be veryapproximated to the optimal solution.

B. CVEM Approaches

In general, the classic EMS approaches consider onlya reduced set of subsystems, i.e., the power interactionbetween the the internal combustion engine, electric machineand batteries. This motivated the emergence of the CVEMconcept, as it was already discussed in Section II-A.

CEVM aims to extend classic EMS approaches to in-corporate more subsystems in the optimal control problem.The goal is to increase scalability of the static optimizationproblem to allow for a large number of auxiliary systems.The extra degrees of freedom added to the problem, makesCVEM appealing to be used in fully electric vehicles,where the power split between the electric machine and thecombustion engine has disappeared.

CVEM was first proposed in [10] and a deep discussionabout the difficulties using classic EMS techniques to solvethe CVEM problem was also presented. For instance, DPcan provide global optimal solutions to CVEM problems,however, scalability becomes an issue due to the curse ofdimensionality observed in this technique. Similarly, theinherit set of state constraints presented in the CVEMproblem makes PMP approaches difficult to be used for thisapplication. As a consequence, centralized and decentralizedoptimization approaches have been proposed to satisfy therequirements imposed by the CVEM concept.

The research conducted within [11] has made some earlyattempts to incorporate auxiliaries into the classic EMS ap-proaches. Due to the use of a centralized optimization meth-ods and convexification techniques the optimal control for-mulations ’global optimality’ and ’scalability’ requirementsfor CVEM are satisfied. However, the approaches reportedare not flexible, i.e., the EMS is not easily reconfigurable

when new subsystems are introduced in the optimizationproblem due to the convexification procedure. Additionally,some of the results presented in [11] have included thevehicle inertia in the energy management problem, whichwas solved by proposing a convex reformulation of theproblem in space domain; in this case, the lack of flexibilityis also an unsolved issue.

The research results presented in [12] uses game-theoreticapproach to solve the CVEM problems in a decentralizedmanner, where all the subsystems share a limited amountof information and are able to take some decisions au-tonomously. On-line implementations were obtained withinthis approach, where predicted information is not considered,thus limiting the performance of the strategy. In this research,the inclusion of battery wear and charging acceptance arepromising research directions that were not addressed. More-over, an exploration of algorithms that converge to a Nash’sequilibrium are still an open task in in this study.

A distributed optimization approach for CVEM is ex-plored in [13] obtaining both on-line and off-line EMSimplementations that are scalable and flexible for a largenumber of subsystems (including on/off auxiliaries) in thevehicle. This approach uses a dual decomposition to splita large and complex optimal control problem into severalsimple problems related to the subsystems in the vehicle.These sub-problems iteratively share information with acentral coordinator in order to achieve an equilibrium that isrepresented by the optimal solution of the CVEM problem.The off-line formulations of the distributed optimizationproblem presented in [13] are able to drastically reducethe simulation time compared to classic EMS; however,this approach requires a priori information of the drivingprofile and power consumption. Furthermore, there are openquestions related to the numerical aspects of the algorithmsproposed to solve the distributed optimization problem, i.e.,the convergence of the algorithms proposed has not beenformally proved. On the other hand, the distributed opti-mization on-line formulation of the CVEM problem usesfinite-time horizon predictions in an MPC fashion to obtaina sub-optimal solution. A disadvantage of the distributedoptimization approach to CVEM is that the global optimalityof the solution requires the components to be describedby linear and (convex) quadratic models. This renders thedistributed optimization approach not usable for the ap-plications where non-linear models are required. However,particular extensions of the distributed optimization approachthat include non-linear models have already been explored.For instance, in [33], a non-linear battery ageing model hasbeen introduced to the CVEM problem, while in [34], theinertia of the vehicle is considered as an energy buffer andconnected to CVEM (see Section V-C for more details).In both cases, the potential of the distributed optimizationapproach to manage non-linear models has showed promis-ing results; in that sense, a possible extension to [13] isthe generalization of the distributed optimization frameworktowards non-linear formulations.




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C. Eco-driving

In the aforementioned approaches, the vehicle velocity(and thereby the power needed to propel the vehicle) is oftenassumed to be completely given a priori. Still, the vehicleinertia, which is the largest energy buffer in the vehicle canhave a large impact in energy savings and consequently inthe extension of the driving range. For instance, in [35] ithas been reported that changes in driving behavior couldimprove the energetic performance of the vehicle more than30%. These promising improvements in energy efficiencyhave contributed to the emergence of the eco-driving concept,which aims to increase the energy efficiency of a vehicleby means of a convenient selection of driving strategies;i.e. laws, technological implementations or simply changesin the drivers behavior. Hence, it is clear that eco-drivingis a broad definition where government, manufacturers andusers participate [36]. The problem of optimizing the velocityprofile (in isolation form the other subsystems in the vehicle)to achieve optimal energy consumption has been consideredfor conventional vehicles in [37]–[39], for hybrid electricvehicles by [40], [41], and for electric vehicles by [42]–[46].

To solve the eco-driving problem, standard techniquesused in optimal control have been adopted. In [47], [48]dynamic programming (DP) has been used. Alternatively,Pontryagin’s minimum principle (PMP) has been used in[43], [49], [50] and [42] to solve the optimal control prob-lems presented. The main disadvantage is that PMP onlyprovides a necessary condition for optimality and cannotincorporate state constraints easily. Therefore, in [39], [40],[46] static non-linear optimization techniques are used tosolve the problem in presence of state constraints. In orderto guarantee that these static optimization techniques and themethods based on PMP find global optimal solutions, it isimportant to understand convexity of the eco-driving optimalcontrol problem. Unfortunately, the literature related to thistopic is scarce. The noticeable exception is [40], where thecontinuous-time optimal control problem is approximated toa convex formulation. However, the possible loss of convexityin the discretisation step (to arrive at a finite dimensionaloptimization problem) has not been considered in [40].As consequence of this limitation, in [51], conditions forconvexity of the eco-driving control problem are presented(see Section V-B).

Eco-driving solutions have been widely implemented inEco-Driving Assistance Systems (EDAS). Depending on themethod used to influence the driving profile, the implemen-tation of eco-driving solutions can be seen as an advisorysystem, where the driver receives suggestion to adjust thedriving style to save energy consumption [52] or as adaptivecruise control (ACC) system, where the vehicle takes controlon the velocity profile [53].

D. Preview Information

The energy performance for real time implementationsof classic EMS and CVEM strategies strongly depends onthe quality of the predictions that are used to solve theoptimal control problem. In the case of eco-driving, complete

information related to the driving profile is not necessary;however, this approach assume no obstacles in the path, e.g.,traffic lights, intersections, vehicles, etc. This implies thatthe performance of the real time eco-driving solutions can beimproved using accurate predictions of the the environmentalconditions that surrounds the vehicle. Therefore, severalmethods to improve the preview information used in real timevehicle energy management applications have been reportedin literature.

In [39], traffic information is integrated into the energymanagement problem as time and space varying velocityconstraints that are updated every five minutes by the Free-way Performance Measurement System [54]. Similarly, in[55], the eco-driving problem is solved considering trafficconditions; the authors propose a method to automaticallycreate a velocity corridor from statistical real truck operationdata that is used to represent varying traffic flow. For both ofthe previous approaches, the results presented are promising;however, these methods are mainly applicable in highwayswhere the influence of vehicles, traffic lights of intersectionsare not significant for the overall performance.

In the last years, energy management in urban trafficenvironments has become a relevant research branch. Forthis case, the preview information is mainly obtained bypredictions based on data from sensors and communicationnetworks, e.g, vehicle to vehicle (V2V) or vehicle to infras-tructure (V2I). Approaches to solve the eco-driving problemfor a road segment with several traffic lights have beenreported in [56], [57]. In [58], [59] energy optimal adaptivecruise controllers are proposed, this approaches uses sensorinformation and communication V2V to design energy opti-mal velocity profiles constrained by the interaction with othervehicles in urban environments.

Interestingly, most of cases previously described onlyfocus its attention to optimize velocity profile neglectingthe influence of other subsystems in the energy managementproblem.

IV. RESEARCH OUTLOOK

In this section, the main research question of this workis stated and the related implications to this question arediscussed as problems and possible solution paths to follow.

The state of the art previously discussed shows that a largeportion of the research conducted on energy management forautomotive applications have been devoted to optimize thepower split among the the subsystems in the vehicle, i.e.,classic EMS and CVEM strategies; while the other largesector has been oriented to optimize the velocity profilesof the vehicle. However, the literature that links both ofthese large research fields is still scarce, albeit the significantimprovements in performance and implementability that thisresearch direction promises.

It is possible to consider the classic EMS as a subset ofCVEM strategies. The main disadvantage in this case is thestrong dependence on a priori information of the drivingcycle, i.e., the velocity profile for a known trajectory inspecific time interval. This driving cycle can be directly




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obtained as a solution of the eco-driving problem, thusovercoming in some degree the main disadvantage present inCVEM strategies. As a consequence, the following researchquestion is posed:

“What are the potential advantages ofincluding the optimization of velocity tra-jectories in the CVEM framework for ap-plications in electromobility?”

The implications that arise from the exploration of thisquestion are discussed in the following Sections.

A. General framework for modeling and optimal problemformulation.

Problem Overview:Obtaining optimal velocity profiles in the CVEM strategy

directly implies that velocity becomes a decision variable inthe energy management optimal control problem. The currentmodeling framework used in CVEM applications describesa network of subsystems where the iteration between themis based on energy flows. In that sense, the subsystems aremodelled as energy buffers or energy converters and theformulation of the optimal control problem uses as decisionvariables the power produced / consumed by the subsystems.Therefore, understanding what is the most beneficial way toinclude velocity in the CVEM optimal control problem formu-lation becomes a priority task. For instance, [11] proposesa formulation where space is selected as the independentvariable in order to avoid the non convexity introduced by theinclusion of velocity in the optimization problem; however,this produces non-linear and possibly non convex modelsfor other subsystem that are normally linear and convex informulations where time is considered as the independentvariable.

The current modeling process tends to simplify the non-linearities in the models and mostly aims obtain to con-vex approximations that are convenient in view of theoptimization methods performance. It is clear to note thatthese approximations introduce errors that can constraint theperformance of the CVEM strategies. Therefore, it is relevantto investigate what are the problems and potential solutionsthat appear as a consequence of increasing the complexityof the models used for CVEM. An early exploration in thisdirection was already reported in [33], observing that theincluding ageing in the battery model changed the behaviorof the CVEM strategy, i.e., increasing the complexity ofthe models to capture additional physical phenomena in theCVEM can lead to an improved performance of the strategy.

Research Paths:In order to address these open topics on modeling, the first

necessary step it to formalize and generalize a modellingframework for CVEM applications on electromobility. Ageneralized framework facilitates the design of systematicmethods to obtain CVEM strategies, hence accelerating the

study of the benefits for different formulations of the energymanagement problem. An early idea of the features expectedin this modeling framework are already presented in SectionV-A. In this case, non-linear models are considered andthe states of the energy buffers are explicitly expressed inthe network; furthermore, the use of time or space as theindependent variable in the model is also envisioned. Thusallowing a the use of more detailed models to improve theperformance of the CVEM strategy.

The implications of the level of accuracy required in themodels is other topic that needs to be studied. Physicallyaccurate models impose a higher computational effort to findthe optimal solution of the energy management problem.The trade off between accuracy and complexity of themodels is not a trivial task. One of the main issues is thatthe complexity of the models introduces non-convexity inthe optimal control problem formulation. This problem canbe addressed by the application of lossless convexificationtechniques. However, it is not always possible to successfullyapply this approach; therefore, a detailed analysis of the con-vexity properties of the optimization problem is useful. Thisanalysis helps to identify which physical properties can berelaxed or even removed from the formulation producing theminimum impact in the performance of the EMS. The idea isto systematically qualify and quantify the errors introducedin the formulation and use this insight to make decisionsabout the convexification techniques that can be applied tothe CVEM problem. In [51], a first attempt in this researchline has been presented. The convexity of the eco-drivingproblem in isolation was analyzed, thus identifying realisticassumptions and conditions that permit the formulation ofa convex eco-driving optimization problem. These ideas canbe extended to more complex formulations where additionalnon-convex subsystems are included in the CVEM problem.

B. Numerical Optimization Methods for CVEM

Problem Overview:As it was already stated in the previous section, the use

of richer models can lead to non linear and non convexformulations of the CVEM problem. This issue could bepartially addressed by the use of convexification techniquesin the formulation of the problem. However, for some casesthose techniques fail to obtain formulations where convexoptimization techniques can be used to efficiently solvethe CVEM problem. This issue directly affects the realtime implementation of CVEM strategies. In that sense, itis important to explore what optimization methods can besuitable to efficiently solve the CVEM problem under nonlinear and non convex formulations.

Research Paths:Since the distributed optimization approach for CVEM

studied in [13] has reported satisfactory results in terms ofcomputational performance for off-line and on-line appli-cations, it can be considered a promising method includethe optimization of velocity profiles in the CVEM prob-lem. However, the convergence in several of the algorithmsproposed has not been guaranteed. In order to improve




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the reliability of the distributed optimization approach itbecomes relevant to explore convergence properties of thealgorithms proposed in [13].

Additionally, in [13], alternating direction method of mul-tipliers (ADMM) was used to split the horizon of the CVEMproblem into small sub-problems that could be solved ina more efficient manner. The results reported still showa considerable computational effort to obtain the optimalsolution of the problem. Thus, research should be conductedon this topic in order to adapt or obtain faster algorithms andalso explore how these methods can exploit the structure ofthe problem in order to be used in non-convex formulations.

Algorithms that can tackle possibly non-convex formula-tions of the CVEM problem have been recently explored.For instance, sequential quadratic programming (SQP) andregularization techniques have presented promising resultsin [34]. This research line could be extended to achieve fastoptimization algorithms that can be implemented in real timeapplications.

C. Preview Information using Sensor and CommunicationNetworks

Problem Overview:Considering velocity as a decision variable for the CVEM

problem relaxes the requirement of a priori informationrelated to the driving cycle. The additional freedom providedto the optimal control problem implies that a reduced setof information from the trajectory is needed. For instance,the eco-driving problem normally requires the position andvelocities at the starting and final points of the trajectoryand the complete road grade. Hence, it is clear to see thatproviding this information is less complicated than providinga detailed velocity profile, which in most of the cases issub-optimal. However, this approach assumes ideal trafficconditions in the trajectory, e.g., a road without traffic lights,intersections and vehicles. Clearly, traffic conditions limitsthe performance of the energy management strategies, there-fore it is relevant to investigate what preview information canbe used to improve the performance of the CVEM strategiesunder traffic conditions and how this information can be usedin the CVEM framework.

Research Paths:It can be argued that for specific cases, e.g., a vehicle driv-

ing in a highway, the assumption of ideal traffic conditionsholds. However, it is difficult to predict when this idealizedcase can be expected for an specific road. On the other hand,for urban traffic applications this idealized assumption isvirtually impossible to satisfy.

In Section III-D, it was reported that several attempts havealready been made to obtain and use preview informationrelated to the traffic conditions. A large majority of thecurrent work has been oriented to design cruise controllersthat react to the environment conditions; unfortunately, onlya reduced amount of these approaches explicitly considersenergy consumption in the problem. Nevertheless, some ofthe concepts can be adapted to be used in energy manage-ment applications.

The use of statistical information to predict a velocitycorridor that simulates traffic conditions for an specific tra-jectory was presented in [55]. In this case, the preview infor-mation is encoded as position dependent velocity boundariesthat allows to reduce the energy consumed for braking. Thisapproach can be extended to consider data form operation,but also climate conditions and time, which are parametersthat influence the traffic flow.

Additionally, it can be noted that in the previous approachtraffic lights or intersections are not considered. However,this information is available in cases where the vehicle usescommunication networks I2V and I2I. This preview infor-mation can be used to obtain efficient management strategieswhen approaching to traffic lights or intersections. Moreover,the use of sensors in the vehicle give the possibility to defineenergy efficient cruise controllers that react to the behaviorof the vehicle in front.

The applications previously mentioned are currently beingexplored by several authors; unfortunately, many of theseapproaches are disconnected from the energy managementperspective. Including these ideas to the CVEM frameworkis not a trivial task. The main challenges are to investigatehow the formulation of the CVEM problem has to beadapted and what optimization techniques can lead to theexpected performance. Additionally, this implies that theCVEM concept can consider additional features, thus the listpresented in Section II-A can be updated as:• Globally Optimal.• Scalable.• Reconfigurable / Flexible.• Anticipative.• Reactive.

These the new features are related to the way in whichthe preview information is used. An ’Anticipative’ energymanagement strategy implies the exploitation of the directinformation received from the communication networks andthe predictions obtained form statistical models to react inadvance to traffic conditions. The ’Reactive’ property refersto the capacity to react to events that were not predicted orcommunicated in advance, e.g., vehicles that are driving infront without communication capabilities.

V. PRELIMINARY RESULTS

A. Power Network Modeling

The formulation of optimal energy control problems re-quires a proper description of the power flows among thesubsystems that operate in the vehicle.

From the energetic perspective, the basic functions of asubsystem are to store and convert (and/or dissipate) energybetween different physical domains [10]. For instance, afuel cell uses the chemical energy stored as hydrogen toobtain electricity by means of a chemical reaction thatoccurs in the cell; similarly electric energy is transformedinto mechanical energy by an electric machine. Hence, asubsystem m ∈ M, is represented by an energy bufferconnected to a power converter cm as can be observed in Fig.




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Fig. 3: Basic elements for modeling the vehicle powernetwork.

3 [60]. The dynamical behavior of the buffer is representedby

xm(t) = fm(xm(t), um(t)),

zm(t) = gm(xm(t), um(t))(2)

where the mappings fm : Rnx × Rnu → Rnx , gm : Rnx ×Rnu → R. The is the energy stored in the subsystem is zm(t)and xm(t), um(t) represents the states and the inputs of thebuffer respectively. The energy conversion in cm is given by

ym(t) = hm(um(t)), (3)

where hm : Rp → R, ym(t) is the power of subsystemm flowing in the vehicle power network. Note that (3)describes the efficiency of subsystem m to convert energy.It is important to remark that it is also possible to havesubsystems without energy buffers, for those cases hm :R→ R defines a mapping between power in different energydomains.

The iteration of power flows can be observed in the nodesof the network; thus, for all the nodes j ∈ J it is possibleto see that

∑

pj∈Pj

ypj = 0 (4)

where Pj ⊆ M is the subset of power flows interacting inthe node j. Note that by convection ypj is positive when isreaching the node; for cases where the flow is bidirectionalthe positive flow direction can be selected arbitrarily.

B. Convexity of the Eco-driving Problem

This section summarizes the work presented in [51], wherea detailed view of the convexity issues of the eco-drivingoptimal control problem is described and a method to refor-mulate and discretize the problem preserving its convexity isproposed. Furthermore, physically realistic assumptions andconditions that guarantee the convexity of the eco-drivingproblem are given and a Sequential Quadratic Programmingalgorithm that exploits these results is proposed.

1) Optimal Eco-driving Control Problem:Eco-driving aims at obtaining an optimal traction force

u(t) and velocity profile v(t) that minimizes the total powerP (v, u) consumed by a vehicle while traveling during a giventime interval [to, tf ] over a given trajectory s(t) ∈ [so, sf ]with known geographical characteristics, i.e., with a givenroad grade α : [so, sf ] → [−π2 , π2 ], where α(s) is the gradeat position s; while being subject to longitudinal vehicledynamics, non-negative velocity bounds v(t) ∈ [

¯v, v], and

boundary conditions on position and velocity. This can bestated in the form of the following optimal control problem:

mins(t),v(t),u(t)

∫ tf

to

P (v(t), u(t))dt (5a)

subject to u = mdvdt σdv

2 +mg(crγr(s) + γg(s)), (5b)dsdt = v, (5c)

s(to) = so, s(tf ) = sf (5d)v(to) = vo, v(tf ) = vf (5e)

¯v 6 v 6 v, (5f)

where

γg(s) = sin(α(s)) and γr(s) = cos(α(s)), (6)

In (5b), m represents the combined mass of the vehicleand the inertia of the driveline, g is the gravitational constant,cr > 0 describes the rolling force coefficient and σd =12cdρaAf with cd > 0 is the drag coefficient, in which ρadenotes the air density and Af is the frontal area of thevehicle.

The consumed power P (v, u) can be obtained from differ-ent modeling approaches that capture the energy consump-tion in the powertrain. In this paper, it is assumed to be aquadratic function of the form

P (v(t), u(t)) = β0v2 + β1vu+ β2u

2, (7)

for some non-negative parameters β0, β1 and β2. Equation(7) is a physically realistic approximation, e.g., for electricmotors due to the fact that the friction and Ohmic losses arecaptured by the terms β0v2 and β2u2, respectively.

In general, (5) is a non-linear optimal control problem thatmight be nonconvex due to specific features of the vehiclemodel and road profile, see (5b). Furthermore, it has beenproved that a direct discretization of the eco-driving canlead to a nonconvex optimization problem even for specificcases where the continuous time problem is convex. Thisimplies that direct optimization approaches and methodsbased on PMP only provide candidate minima, which mightnot correspond to the global solution to problem (5).

2) Assumptions and Conditions for Convexity:It is convenient to obtain an equivalent optimization

problem that can be discretized using a forward Eulermethod while preserving its convexity properties. This goalis achieved by the substitution of (5b) into (5a) to obtain

mina(t),s(t),v(t)

∫ tf

to

PR(a, s, v) dt, (8a)

subject to (5c)-(5f) anddvdt = a; (8b)

where a(t) is a new decision variable, which represents thevehicle acceleration. In this case the cost function is definedby

PR(a, s, v) =β0v2 + β1v(σdv

2 + crmgγr(s)) + β2(ma)2

+β2(mgγg(s)+σdv2+crmgγr(s))

2

+2β2m2ga(γg(s)+crγr(s)). (9)




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The reduced eco-driving problem (8) is convex under acondition imposed on the lower velocity bound. To presentthis result, we need the following assumption.

Assumption 1: The given road grade α(s) and the corre-sponding elevation profile h(s) satisfies

dγg(s)ds =

d2γg(s)ds2 = 0 and dγr(s)

ds = d2γr(s)ds2 = 0. (10)

where

γg(s) = dhds . (11)

Under this assumption, a sufficient condition for globaloptimality of (8) can be derived, which are presented in thefollowing theorem.

Theorem 1: Suppose that the optimization problem (8)satisfies Assumption 1. Then, the problem (8) is convex ifthe lower bound on the velocity

¯v satisfies

¯v ≥

¯vc(s) = min

vv |H33(s, v) > 0 (12)

for all s ∈ [so, sf ], where

H33(s, v) = 2β0 + 6β1σdv + 12β2σ2dv

2

+ 4β2σdmg(γg(s) + crγr(s)). (13)

The result from Theorem 1 imposes¯vc(s) as the lower

bound on the velocity that the vehicle has while drivingdownhill to guarantee a convex formulation of the optimiza-tion problem (8). This condition is satisfied for the majorityof practical cases. For instance, in Fig. 4a, it is possible toobserve the minimum velocity imposed by this condition fora vehicle with parameters presented in Table I. Recallingthat γg(s) = sin(α(s)), it can be observed that the vehiclecould drive downhill with no velocity limitations for slopeslarger than −20°. Even for roads with pronounced decreasingslopes, the minimum velocities allowed are still modest,which means that the condition stated in Theorem 1 is nothighly restrictive for real-world applications.

Finally, it should be noted that Assumption 1 is an ac-ceptable approximation. Indeed, for realistic roads, the rate ofchange of the elevation profile is relatively constant for smallintervals. This can be observed in Figs. 4b and 4c, wherethe first and second derivatives of γg and γr are depicted fora road with slopes α(s) ∈ [−60°, 40°], which is an extremecase. In both cases, the derivatives are virtually zero, meaningthat Assumption 1 holds for a numerical solution to theoptimal control problem (8). We will exploit this propertyin the numerical scheme that we will present below.

3) Numerical Example:A Sequential Quadratic Programming (SQP) method that

takes advantage of results from Theorem 1 has been proposedto solve the finite dimensional version of the eco-driving

TABLE I: Vehicle parameters.

m = 18000[kg] g = 9.8[m/s2]σd= 3.1246 cr= 0.3β0= 0.1564 β1= 1.005β2= 0.0002652

problem (8). Specifically, this formulation uses QP sub-problem that represents a convex second-order approxima-tion of (8). The effectiveness of this methods is presented ina numerical example where a conventional driver behavioris compared with an eco-driving strategy for a heavy-dutyvehicle. The settings used in this example this example aresummarized in Table II and the elevation profile is definedby

h(s) = 50 cos( 2π21000s+ π

4 ) + 50 [m], (14)

which is depicted in Fig. 6 as a green surface.

TABLE II: Eco-driving problem parameters for a heavy-dutyvehicle.

β0= 0.292 v(to)= 0[m/s] m= 1595[kg] to= 0[s]β1= 1.005 v(tf )= 0[m/s] g= 9.81[m/s2] tf= 1080[s]β2= 2.652×10−4 s(to)= 0[km] cr= 0.1 v= 100[km/h]τ= 5[s] s(tf )= 21[km] σd= 3.1246

¯v= 0[km/h]

The convexity of the optimization problem can be verifiedusing (12). In Fig. 5, the satisfaction of this condition isdepicted for all s ∈ [0, 21] [km]. Thus, form Theorem 1 itcan be concluded that the problem in this example is convex.

The presumed conventional driver behavior is describedby a solid line in Fig. 6. In this case, the total energyconsumed by the vehicle is 6.218 × 104[kJ ]. On the otherhand, the dashed line presented in Fig. 6 describes thevelocity profile obtained as a solution of the eco-drivingoptimal control problem studied in this case. For this strategythe acceleration and deceleration stages are performed atreduced time intervals with larger magnitudes. Consequently,the average velocity achieved during the trip is considerablylower than the average velocity for the conventional case.The total energy consumed by the vehicle under this strategyis 5.165 × 104[kJ ], which is approximately 16.92% lowerthan the energy consumed by the vehicle under conventionaldriving behavior.

Recalling that the optimization problem was certified tobe convex a priori, it can be stated that the reported solutionis an accurate approximation of the global solution of theeco-driving problem analysed in this example.

C. Complete Vehicle Energy Management and Eco-driving

The main result presented in [34] is briefly summarized inthis section. Here, a first attempt to connect CVEM with eco-driving is proposed. Based on the distributed optimizationapproach of [61], the optimal velocity profile and power splitof energy is obtained. The optimization sub-problem relatedto eco-driving is solved by the application of Thiakonovregularization to an SQP formulation.

1) Problem Formulation:As a case study for CVEM with eco-driving, we consider a

series-hybrid electric vehicle, consisting of an Electric Motor(EM), Engine-Generator Unit (EGU) and a High-VoltageBattery (HVB). The topology is shown in Fig. 7, in whichyegu and uegu denote the ICE’s fuel and mechanical power,respectively, yhvb and uhvb the battery’s electrical and storedchemical power, ybr is an artificial braking power exerted at




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-90 -80 -70 -60 -50 -40 -30 -200

10

20

30

40

50

60

70

Lo

wer

bo

un

d v

c [K

m/h

]

(s )[deg]Road grade

(a) Minimum downhill velocity required to guar-antee convexity of the problem.

0 1 2 3 4 5 6 7 8

Distance [km]

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

[1/m]

[1/m ]

(b) Derivatives of γg(s) with respect to the displace-ment s.

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

[1/m]

[1/m ]

Distance [km]

(c) Derivatives of γr(s) with respect to the dis-placement s.

Fig. 4: Example illustrating Assumption 1 and Theorem 1.

0 5 10 15 20

Distance s [km]

-2.5

-2

-1.5

-1

-0.5

0

Vel

oci

ty [

km

/h]

vc

v

Fig. 5: Certification of convexity.

Fig. 6: Conventional vs. eco-driving profiles.

the interconnection of the subsystems, uem and yem the EM’smechanical force and electrical power, respectively, xhvbdenotes the battery state of energy and xem = [ vs ], denotesthe states speed v and distance travelled s of the vehicle.In CVEM, the main goal is to minimize fuel consumption,given by

∑

k∈Kτyegu,k, (15a)

subject to the dynamics and input-output behaviour of theconverters in Fig 7, and the power balance at interconnectionof the subsystems, i.e.,

yem,k − uegu,k − yhvb,k = ybr 6 0, (15b)

for all k ∈ K. We use the constraint (15b) and energy balanceconstraint of the HVB, i.e.,

∑

k∈Kτuhvb = xhvb,K − xhvb,0 = 0 (15c)

Fig. 7: Topology.

to rewrite (15a) as a ‘sum of losses’, i.e.,

τ∑

k∈Kyegu,k−uegu,k+uhvb,k−yhvb,k+yem,k−ybr,k. (15d)

By substituting (15b) and (15c) into (15d), we retrieve theoriginal objective function (15a). The goal is to solve theCVEM problem by formulating it as a convex SQP problemand then apply dual decomposition as presented in [61].

2) Numerical Example:In this simulation study, we consider the Series-Hybrid

Electric Vehicle (SEHV) case study presented in V-C.1.We will refer to the ‘forward’ optimization as solving thevehicle energy management problem with eco-driving, andrefer to the ‘backward’ optimization as solving an energymanagement problem without eco-driving, where the vehicletrajectory information, i.e., speed vk and distance sk for allk ∈ K, are given.

We base our case study on the work done in [11], inwhich a convex optimization approach is taken to solve theSEHV case study, where it is formulated a second-order coneprogram. Due to the chosen problem formulation, the authorsin [11] have opted for a piece-wise linear EM model, linearEGU model and a quadratic HVB model. Furthermore, theauthors in [11] define the optimization problem in the spacedomain, for which the physical interpretation of some partsof their problem formulation is not easy to understand. As wehave defined the SEHV case study in the time domain, we donot face this issue. The parameters used for the simulationstudy are shown in Table III. We remark here that the upperand lower bounds for the state and input variables specifiedin Table III are defined for all k ∈ K.

The simulations are done for 1080 s over a distance of 21km with a step size τ = 5 s, which gives an optimizationhorizon K = 216. In ‘backward’ optimization, the speed is




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TABLE III: Series Hybrid Electric Vehicle Parameters

γem0 = 0.5856 m = 15950 [kg] sK = 21000 [m]γem1 = 1.005 g = 9.81 [m/s2] xhvb,0 = 11988 [kJ]γem2 = 5.052×10−4 cd = 0.7 xhvb,K = xhvb,0γegu0

= 38 cr = 0.0007 vk = 22.22 [m/s]γegu1

= 2.52 Af = 7.54 [m2] vk = 16.67 [m/s]γegu2

= 2×10−5 ρA = 1.184 kg/m2 xhvb,k = 22680 [kJ]γhvb0

= 0 v0 = 19.44 [m/s] xhvb,k = 7560 [kJ]γhvb1

= −1 vK = v0 uhvb,k = 92.4 [kW]γhvb2

= 1.671×10−3 s0 = 0 [m] uhvb,k = −92.4 [kW]

given by a constant speed of vk = 70 km/h for all k ∈ K,and in ‘forward’ optimization the initial and final velocity aregiven, while over the trajectory the speed is allowed to varyby 70 ± 10 km/h. As initial conditions for the ‘forward’optimization, we choose the solutions of the ‘backward’optimization.

In Fig. 8, the simulation results for the ‘backward’ and‘forward’ optimization method are given. In the ‘forward’optimization results, we see that between 0 and 2 km,where the slope becomes negative, the vehicle reaches thelower speed bound. This action allows the available potentialenergy from the road profile, between 2 and 13 km, to bemaximally converted to kinetic energy; the vehicle speed ismaximized in this interval. After 13 km, the road gradientbecomes positive and speed is minimized such that the finalspeed constraint is met. Therefore, we can see that as aresult of having the speed as a decision variable, in the‘forward’ case, the EGU may provide less power over thecourse of the trajectory, and noticeably less braking poweris applied, when it is compared to the ‘backward’ case. Thefuel consumption of the ‘backward’ and ‘forward’ simulationcases are 23.41 l/100 km and 22.31 l/100 km respectively.Thus, by including the eco-driving problem into the vehicleenergy management problem, approximately 4.7 % decreasein fuel consumption is achieved. This is comparable to thefuel consumption obtained in [11], which are 24.35 l/100 kmand 23.98 l/100 km for the ‘backward’ and ‘forward’ caserespectively. This is a 1.54 % decrease in fuel consumption

0

100

200

ue

gu

[k

W]

Backward

Forward distributed

-200

0

200

ue

m [

kW

]

-40

-20

0

yb

r [

kW

]

2

4

6

xh

vb

[k

Wh

]

60

70

80

v [

km

/h

]

0 2 4 6 8 10 12 14 16 18 20

Distance [km]

0

100

200

300

Ro

ad

alt

itu

de

[m

]

Fig. 8: Backward vs forward optimization. The dotted linesrepresent minimum and maximum state constraints.

between the ‘backward’ and ‘forward’ case. We may explainthis difference in fuel consumption savings largely due to thedifferent models used.

VI. FUTURE WORK

In this section, concrete plans for the immediate futureresearch are presented. Specific details for the extensions ofthe preliminary results presented in Section V are given anda short discussion of a case study for CVEM on electricbuses is summarized.

The next step in this research is to extend the resultsobtained in [51] to include force and power constraints in theconvexity analysis. It has been observed that the same set ofassumptions used in [51] can be used to obtain conditionsfor the convexity of the eco-driving problem under this newset of constraints. Additionally, guaranteeing global solutionsin cases where the convexity condition (12) is satisfied onlyfor a a subset of the feasible se is an interesting researchextension. This could be achieved by proving the uniquenessof the critical point in optimization the problem, which seemsfeasible under the same assumptions presented in [51]. Theseadditional results on the convexity and global solutions of theeco-driving problem can be connected to the work presentedin [34]. Consequently, a convex formulation for the CVEMproblem with velocity optimization could be proposed, thusalso a reduced computational effort needed is expected.

A case study to design CVEM strategies for electric busesis an research line that will be addressed. The main goal isto reduce the range anxiety of electric buses by improvingthe operational performance of the bus. There are two energybuffers that must be considered, i.e., the vehicle inertia andHVAC system. Bus manufactures have reported that theHVAC system normally is the first energy consumer, whichis expected due to the large volume of the passenger cabinand the substantial exchange of energy with the externalenvironment at every bus stop. Therefore, a CVEM strategywith velocity optimization that takes advantage of previewinformation of the next stop can be designed in order toexploit the capabilities of the energy buffers in the bus, hencemaximizing the driving range of the vehicle. Additionally,a potential on-line implementation of the strategy is beingconsidered, therefore the research activities will be alignedto this goal.

VII. CONCLUSIONS

In this paper, a view of the the future research on CVEMon electromobility has been presented. The implications ofincluding velocity optimization in the CVEM frameworkhave been discussed, thereby concluding that modeling,optimization methods and the exploitation fo preview infor-mation are feasible research lines to follow. Consequently,’anticipation’ and ’reaction’ were proposed as additionalfeatures of the complete vehicle energy management concept.Moreover, preliminary results that shows the potential ofoptimizing velocity profiles in the CVEM problem have beensummarized. Finally, concrete plans for the extension ofthose results have been proposed and the design of a casestudy for electric buses was also discussed.




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1

Range Estimation for Electric VehiclesFrancisco Ozuna

Abstract—Accurate range estimation in electric vehiclesis a crucial challenge to increase customer trust in the useof batteries as an alternative source of power in mobility.The extractable energy in battery packs is determined bythe amount of energy in the cells and the future energyconsumption of the vehicle. Furthermore, a battery packconsists of hundreds of cells and having a state estimatorfor each cell is challenging for on-line implementation.Therefore, a low computational load algorithm is required toestimate the states of the pack and consequently the electricrange. This paper presents a novel approach to estimatethe range of an electric vehicle with a low computationalload by estimating the battery pack state and forecasting thefuture power demand using available vehicle information.The results show that the driving range of an electric bus canbe accurately estimated with a maximum range estimationerror of less than 10%.

Keywords—Range estimation, Battery pack, Electric Vehi-cles.

I. INTRODUCTIONElectric vehicles (EVs) research is playing an important

role in the development of alternative transport meansto reduce the environmental impact of fossil fuels [1].Nevertheless, there are some factors that discouragepotential costumers to buy EVs. Among these factors,driving range stands as one of the major issues. Most ofcurrent EVs have driving ranges around 400 km which isaround 40% of similar sized internal combustion enginevehicles (ICEV) as can be seen in Fig. 1. Furthermore,the EV convenience usage is limited by the EV charginginfrastructure. In the U.S. the number of public stationsis estimated to be around 16,000 which is low comparedto the estimated 112,000 gasoline stations. This meansa reduced possibility to recharge an EV compared toan ICEV. Moreover, the gas refueling takes a couple ofminutes to provide a range of approximately 300 milesfor an average ICEV. Meanwhile, EVs require on averagetens of minutes to hours to recharge [2].

For these reasons, people are still reluctant to acquireEVs due to the concern of how far they can travel beforethe battery depletes. For electric buses this is relevant aswell. A service needs to be provided and it is importantto know how far electric bus can travel and when theyneed to be recharged. The estimation of the distancewhich an EV can travel before the battery depletes, ishenceforth referred to as range estimation.

The remaining range estimation of any electric vehiclecan be computed using

Range[km] =ES [J ]

EC [J/km](1)

0 200 400 600 800 1000

Range [km]

0

500

1000

1500

2000

2500

Curb

weig

ht [k

g]

Opel Ampera-E

Renault Zoe

Tesla S

Nissan leaf

BMW i3

Chevrolet Spark

Renault Twingo

Nissan Cube

Kia Soul

EV

ICEV

Fig. 1. Comparison between EVs and ICEVs driving ranges (Vehiclemodels 2017).

where ES is the remaining available energy stored in thebattery and EC is the future energy consumption that thevehicle will require to complete a driving cycle.

There is much research on estimating the batteryavailable energy. The research in [3],[4] review over350 sources of scientific literature where the methodsrelated to this topic are reviewed. In [5], the remainingdischarge (RD) energy in the battery pack is defined asthe cumulative energy from the present condition to thestate when one of the cells in the pack reaches the lowercut-off voltage and can be represented as

ERD(t) =N∑

i=1

∫ t

0

V oc,i(SoC)Qidt (2)

where ERD(t) is the maximum RD energy at time t,N is the number of cells in the battery pack connectedin series, V oc,i(SoC) is the open-circuit-voltage (OCV)which is function of the state-of-charge (SoC) and Qi

is the maximum available charge of the ith cell. In thisresearch, the remaining range is calculated consideringthe battery pack inconsistency and predefined drivingcycles. The battery pack parameters are identified witha recursive least square (RLS) and the SoC is estimatedusing an unscented Kalman filter (UKF). As we can note,available energy in EVs is strongly dependent on thebattery pack SoC. To estimate it, various approacheshave been proposed [6]–[8]. In [9], Plett proposes the




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2

use of a method for pack state estimation called bar-delta filtering. An average SoC estimate at pack levelis provided by an estimator called bar filter and thedifference between each cell SoC and the average is givenby an estimator called delta filter. The computationaltime required to estimate the battery pack SoC is reducedby executing the delta filter at lower rate than the barfilter.

There is some research focused on predicting EC inEVs [10], [11]. Wenjia Wang et. al. [12] introduces amethod to estimate how far an EV can travel by cal-culating the possible routes it can take from its actuallocation to the surroundings with a rough and preciseestimation considering different factors that influence thefuture energy consumption, which is defined as

EC = f(EC,cruise, EC,RG, EC,acc,

EC,AR, EC,ED, EC,DS)(3)

where EC,cruise refers to the energy consumption due tovehicle speed, , EC,RG is the energy consumption relatedto road grade, EC,acc is the energy consumption due toacceleration, EC,AR is the aerodynamic resistance energyconsumption, EC,ED is the use of energy for electricdevices and EC,DS is the driving style which affects theenergy consumption. These factors are determined by theuse of a telematics center where the possible routes andvehicle status are calculated. Finally, range is computedusing (1). However, this research considers the battery asa single cell; thus, ignoring potential imbalances withinthe pack and the battery efficiency losses are not consid-ered. In [13], Liu et. al. integrate the energy in the batteryand future energy consumption to estimate the remain-ing discharge energy in a cell under certain dynamicoperating profiles using predictive control theory andadaptive model prediction. This method consists of theuse of a predetermined future current profile, estimationof battery present states and future variables such as SoC,voltage and battery parameters. This algorithm proves tobe accurate to estimate the future available energy withan error below 2% of an EV for cell level with a givenfuture current profile.

Although some results exist to estimate the remainingrange for EVs, such methods do not consider jointlyenergy estimation in the battery and future energy con-sumption for battery packs. In [5], the available energyis estimated considering a battery pack with multiplecells with state estimation. However, the future energyconsumption is not addressed in this research. In con-trast, [12] predicts only the future energy consumptionwithout considering battery state and energy losses dueto current profiles. A research done in [13], estimatesthe available energy in a battery but at cell level witha predetermined future current profile. Battery packstypically consist of multiple cells connected in series andparallel. To estimate battery pack energy, a same numberof algorithms would be required to estimate the battery

pack state, which would represent a significant computa-tional burden for the Battery Management System (BMS)requiring increased memory allocation.

In this paper, the remaining range of an electric vehicleis estimated using a novel approach with low computa-tional complexity. Cells within battery packs may presentimbalances; therefore, the state in the pack is estimatedto obtain its energy. Meanwhile the forthcoming drivingconditions are forecast to compute the future energyconsumption by reading available on-line informationfrom the EV. This research provides for the first timean integration of battery pack state estimation for avail-able energy considering imbalances in the battery packand prediction of energy consumption to estimate theremaining range of an EV. The energy in the batterypack is estimated by using the bar-delta filter principlefrom [9] and the future energy consumption is forecastby a low pass filter using past driving conditions. Theremaining range accuracy is tested in an electric busapplication and is computed using 3 algorithms to showimprovement of range estimation with different energyapproaches. The first approach calculates the remainingrange without considering battery energy inefficiency,whereas the second approach considers the availableenergy in the battery via a forecast in current. Finally,the third approach estimates the battery energy via apredicted power demand.

This paper is organized as follows. Section II presentsthe SoC estimation for the battery pack to compute itsenergy. Section III describes the methodology to calculatethe future energy consumption. Section IV introduces the3 approaches to estimate the remaining range. SectionV presents the implementation of the approaches andshows the results of range estimation for a model of anelectric bus. Finally, Section VI contains the conclusionsof this research.

II. BATTERY PACK STATE ESTIMATIONAs presented in the introduction of this paper, SoC

is an important state to estimate the available energyin the battery, and as consequence, relevant to computethe remaining range of any EV. Battery packs in EVs arecomprised normally of hundreds of cells connected inseries and parallel. These may present non-uniformitiescausing state and parameter differences. For this reason,it is important to track the state of each cell withinthe pack. However, having one estimator for each cellwould represent a high computational burden for theBattery Management System (BMS). Therefore, this sec-tion introduces the process for estimating the state ofeach cell in the battery pack for multiple cells with lowcomputational load for range estimation.

A. Cell Equivalent Circuit ModelingTo estimate the SoC, a cell model is needed. Since the

low computational load implementation is important, arelatively simple Equivalent Circuit Model (ECM) is used




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3

Fig. 2. ECM model with one RC block

020406080100

SoC [%]

0

0.1

0.2

0.3

Resis

tance [

Ω]

0

1

2

3

C1 [m

F]

R0 Discharge

R0 Charge

R1

C1

Fig. 3. ECM parameters

as shown in Fig. 2 [14], [15]. Then, the model is definedby

d

dtχsoc =

1

CbattIbatt(t)

d

dtV C

1

= − V C1

R1C1+

1

C1Ibatt(t)

V batt = V oc(χsoc) + V C1

+R0(χsoc)Ibatt(t),

(4)

where χsoc[%] is the SoC, V oc[V] is the open circuitvoltage (OCV), Cbatt [Ah] is the cell capacity, R1[Ω]and C1 [F] are the resistance and capacitance of theoverpotential, R0[Ω] is the cell’s series resistance andIbatt [A] is the cell current. Battery charging occurs whenIbatt > 0 and discharging when Ibatt < 0. The batterymodel parameters and OCV corresponding to a Li-ioncell are presented in Figures 3 and 4, respectively.

Li-ion batteries operate between well defined voltageboundaries. During discharge, when the battery reachesa predefined voltage, the operation stops. These bound-aries are defined as

020406080100

SoC [%]

3

3.5

4

4.5

Vo

c [

V]

Fig. 4. Battery Open Circuit Voltage

V batt < V batt < V batt (5)

where V batt is the minimum discharge voltage called cut-off voltage and V batt is the maximum charge voltage.

By defining the battery states as x = [χsoc V C1

]T , (4)can be rewritten as the state space model

d

dtx =

[0 0

0 − 1R1C1

]x+

[1

Cbatt

1C1

]Ibatt

V batt = V oc([1 0]x) + [0 1]x+R0Ibatt.

(6)

B. PACK SoC ESTIMATION WITH BAR-DELTA FILTER

A typical approach for state estimation is to use anExtended Kalman Filter (EKF), which is a generalizationof the Kalman Filter (KF) towards a non-linear system. Todevelop the SoC estimation based on the EKF, the system(4) is discretized at xk = x(kδt), k ∈ N with sample timeδt > 0, using a forward Euler discretization, leading to

xk+1 =

[1 0

0 1− δtR1C1

]

︸︷︷︸:=A

xk +

[δt

Cbatt

δtC1

]

︸︷︷︸:=B

Ibattk

V battk = V oc([1 0]xk) + [0 1]xk +R0Ibattk .

(7)

The introduced discrete time ECM can be used to es-timate the SoC in the battery. EKF uses first orderapproximations of the nonlinear dynamics in the errorcovariance propagation and consists of a prediction step

xk+1|k = Axk|k +BIbattk

Φk+1|k = AΦk|kAT +Qk,

(8)




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4

and correction step

xk+1|k+1 = xk+1|k +Kk(V battk − V battk )

Φk+1|k+1 = Φk+1|k −KkCΦk+1|k,(9)

in which

Kk = ΦkCTk (CkΦkC

Tk +Rk)−1, (10)

and

Ck =∂V batt

∂xk

∣∣∣∣xk

, (11)

where A and B represent the matrices from (7) andQk and Rk are the process and measurement noisecovariance matrices, respectively.

Considering a high voltage EV battery pack, N cellsare connected in series to supply the voltage demand.For instance, the trend in electric vehicles is towardsbattery pack voltages between 400-500 [V], which wouldrequire around 125 cells of 4 [V] connected in series.These cells may present non-uniformities causing stateand parameter differences within the pack; therefore, allthe cell states must be estimated. Assuming all cells aredifferent and the difference is unknown, each cell modelsatisfies

xik+1 = Axik +BIbattk

V batt,ik = V oc([1 0]xik) + [0 1]xik −R0,iIbattk .(12)

Nevertheless, having one estimator for each cell wouldrepresent a high computational burden to the BatteryManagement System (BMS). Instead of having N EKFsin parallel to estimate each cell state, the average SoCcan be tracked by

xk =1

N

N∑

i=1

xik, (13)

which leads to

xk+1 =1

N

N∑

i=1

xik+1 = Axk +BIbattk

V battk =1

N

N∑

i=1

V batt,ik ≈ V oc([1 0]xk) + [0 1]xk −R0Ibattk .

(14)

The second equation in (14) holds approximately, be-cause

1

N

N∑

i=1

V oc([1 0]xik) ≈ V oc([1 0]1

N

N∑

i=1

xik). (15)

and the smoothness of the OCV from Fig. 4. Furthermore,the differences between each cell and the average aretracked by

x∆i

k = xik − xk, (16)

which leads to

x∆i

k+1 = Ax∆i

k

V batt,∆i

k ≈ V oc([1 0]x∆i

k ) + [0 1]x∆i

k .(17)

EKFs on (14) and (17) can be designed using (7). Abar filter estimates the average state in the pack and adelta filter estimates the individual state differences ofevery cell. To estimate the battery pack state, a bar filterand N delta filters are used. It would appear that thebattery pack state estimation computational complexityhas been increased from N to N + 1, but this is not thecase. An EKF is used to estimate the average state and asimpler EKF without coulomb counting and voltage dropcalculation due to battery current is employed to estimateeach state difference. Moreover, as the differences inthe individual states change at a lower speed than theaverage state themselves, each cell ∆state is estimatedin a lower rate than the average pack state. In this case,the bar delta filter is estimated every 0.1 seconds and thedelta filters are estimated every 100 seconds.

C. Bar-delta filter performanceIn this subsection, the performance of the bar-delta

filter is verified by first comparing the SoC of a mod-eled battery pack to an estimated SoC performed in asimulation environment with the bar-delta filter algo-rithm. Then the estimated SoC of a real battery pack iscompared later to a true SoC of the same real batterypack obtained using coulomb counting with an accuratecurrent sensor.

First, to obtain the SoC of a modeled battery pack, adriving cycle shown in Fig. 5 is simulated in an electricbus vehicle model. As shown in Fig. 6, the electric busvehicle model, which is developed by TNO, simulates,among other subsystems, the powertrain of an electricbus. The powertrain is a forward facing model thattransforms a driving cycle speed profile to an electricbus battery pack power demand. For practical datavalidation, the electric bus power demand is scaled toa power demand for a battery pack of N cells by

P pack,N =P pack,EBusNpack

NEBus, (18)

where P pack,N is the scaled power demand correspond-ing to a battery pack of Npack number of cells, P pack,EBusis the original power demand in the electric bus andNEBus is the total amount of number of cells in a batterypack of the electric bus. A battery pack model withpower as input, is simulated with P pack,N to obtaincurrent in the battery pack Ipack, terminal voltage andthe SoC of every cell. The battery pack model consists ofNpack cells connected in series with the ECM model asshown in Fig. 7.

With the input conditions shown in Table I, the electricbus model is simulated to obtain the pack current, the




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0 0.5 1 1.5 2 2.5 3 3.5

Time [h]

0

10

20

30

40

50

60

70

80

Speed(t

) [k

m/h

]

Fig. 5. Bus driving cycle corresponding to the trajectory of a real busin a city in the Netherlands.

Longitudinal vehicle

Driving cycle [m /s ]

Chargingmode

Passengers

T amb[°C ]

HVACmode

I pack [A ]

V batt ,i[V ]

v [m /s ]

s [m ]

Ppack , EBus[W ]

dynamicsmodel

#

Ppack , N[W ] Battery pack

N packmodel[10 ] xk

i[%]

N pack/N EBus

Fig. 6. Electric bus model.

SoC and the terminal voltage of every cell. The estimatedSoC for each cell is obtained by simulating the bar-deltafilter approach with the electric bus model. For visualclarity, Fig. 8 shows the comparison between the SoC ofthe model and the SoC estimation from 3 cells of the packusing the bar-delta filter. The 3 cells are referred as cellA, B and C and correspond to the highest and lowestSoC in the pack and one cell in-between, respectively.Fig. 9 shows the estimation for these cells. It can be seenthat the bar-delta filter is able to track the difference inSoC for each cell with low estimation error for the entiredriving cycle. Furthermore, when the simulation stopswith the conditions from (5), the cell in the pack with thelowest SoC is at 4% and the highest is at 7.5%, indicatingthat not all energy in all cells could be extracted.

To obtain estimated SoC of every cell from a realbattery pack, a current profile is run in a pack comprised

+

-V pack

V oc ,1

R01I pack

C1,1 R11

R0,1

R1,1

V oc , N

R0, N

C1, N R1, N

V batt ,1+

-

V batt , N+

-

Fig. 7. Battery pack model with N cells connected in series

TABLE I. ELECTRIC BUS MODEL INPUT CONDITIONS

Input ConfigurationDriving cycle Repeated driving cycle from Figure 5

until stop conditions are reached:V batt,i = V cf

Charging mode Disbled# Passengers 50 passengers during the

entire driving cycleTamb 23CHVAC Heating-Ventilation-Air Conditioning system

is turned off while vehicle is stopped

TABLE II. COMPUTATIONAL LOAD BATTERY PACK BAR-DELTAFILTER ALGORITHM

Sim time Speedup Sim time SpeedupTest (12 cells) (12 cells) (96 cells) (96 cells)One EKF per cell 6.02 [s] 1.0 29.28 [s] 1.0One pack bar filter 3.88 [s] 1.55 3.88 [s] 7.54One pack bar, N delta filters 4.04 [s] 1.49 7.41 [s] 3.95

of 12 Molicel IHR18650A cells connected in series. Thebar-delta filter estimates the state of each cell during theentire current cycle profile. To validate the algorithm, alaboratory test is performed to determine a posteriori thetrue SoC of every cell in the pack.

A Battery Test Setup is used to apply the current profileto the real battery pack. A diagram of this setup is shownin Fig. 10. The setup consists of 6 main components: apower supply (PS), electronic load (EL) and relay boardwhich provide a real current based on current profile. Adata-acquisition device reads CAN data from the batterypack BMS to provide measured terminal voltages ofevery cell and measured pack current to a PC runninga Simulink R© model which controls the tests and runsthe bar-delta (BD) filter algorithm on-line to obtain theestimated SoC.

The battery pack contains 11 cells with an initial 90%SoC and one cell at 78% SoC, which is a realistic scenarioif a pack is charged only with fast charging at constantcurrent or a cell is substantially aged. The battery pack isdischarged using the current profile without cell passive




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6

1 2 3 4 5 6

Time [h]

10

20

30

40

50

60

70

80

90

SoC

[%

]

Estimated SoC cell A

Modeled SoC cell A

Estimated SoC cell B

Modeled SoC cell B

Estimated SoC cell C

Modeled SoC cell C

3.6 3.7 3.846

47

48

49

50

51

Fig. 8. Bar-Delta filter SoC estimation for every cell compared to SoCobtained from a battery model during an entire driving cycle.

1 2 3 4 5 6

Time [h]

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Err

or

[%]

Cell A

Cell B

Cell C

Fig. 9. Error in SoC estimation for every cell in the pack during theentire driving cycle.

balancing. This allows to observe the voltage differencesin the cells due to non-uniformity in the battery pack. Forvisual clarity, Fig. 11 shows the bar-delta filter estimationcomparison of the average true SoC and the SoC of 2cells in the pack. These 2 cells correspond to the highestand lowest SoC estimated by the delta filter and arereferred as cells A and B, respectively. Fig. 12 shows theestimation error. It can be seen that the bar-delta filter iscapable to track the SoC of a real battery with highererror than the simulated pack; however, the accuracydecreases due to impreciseness from the current andvoltage measurement. Nevertheless, the SoC estimationis considered accurate since the maximum error is lessthan 2%.

Finally, the speed performance of the bar-delta filter

PS

EL

RelayBoard

Dataacquisition

I pack , profile

CANV batt ,i

I pack ,real

I pack ,real<0

I pack ,real>0PC

simulinkBD

filter

I pack ,measured

BMS

BatteryPack

I pack , profile

[on/ off ]

Fig. 10. Battery Test Setup diagram.

0.5 1 1.5 2 2.5 3 3.5 4

Time [h]

0

10

20

30

40

50

60

70

80

90

SoC

[%

]

SoC estimation cell A

True SoC cell A

SoC estimation cell B

True SoC cell B

Average SoC estimation

Average true SoC

1.8 2 2.245

50

55

60

65

Fig. 11. Bar-delta filter SoC estimation for every cell in the packcompared to a true SoC during an entire driving cycle.

is tested. The purpose of using this filter is to have anaccurate and fast algorithm that estimates the state ofeach cell within the battery pack and therefore, a lowcomputational load range estimation algorithm. Table IIshows the computational load comparison between dif-ferent bar-delta filter configurations for a battery pack of12 and 96 cells. In all cases, the algorithm was simulatedin Simulink R© with voltage and current profiles from afull driving cycle. The time to run all the algorithm ismeasured using a computer running Matlab R© functioncputime with an Intel R© Core i7 @ 2.40GHz processor and7.7 GB of RAM. It can be seen from this table that themore cells the battery pack contains, the more benefit ofcomputational load is obtained using the bar-delta filteralgorithm.

The tests presented in this section show that the bar-delta filter algorithm is capable of tracking on-line state




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0 0.5 1 1.5 2 2.5 3 3.5 4

Time [h]

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Err

or

[%]

Cell A

Cell B

Average SoC

Fig. 12. Error in SoC estimation during the entire driving cycle froma real battery pack.

variations in the pack for a real electric bus driving cyclewith low computational load, which is relevant for therange estimation in EVs. Estimating the lowest SoC inthe pack is important to estimate the ES in any EV as itdetermines the maximum extractable energy in the pack.

III. FUTURE ENERGY PREDICTIONTo predict the future energy consumption, the forecast

of driving conditions is an important factor. This sectionintroduces the methodology to predict the future cycleconditions of vehicle speed, pack current and power inthe battery pack of an EV. The driving cycle forecast is achallenging task to solve since the driver can modify thedriving behavior at any time and the initial conditionsof any driving cycle will not define the total cycle out-put. However, past information of the drive cycle mayprovide data about future behavior.

A. Low pass filter and cut-off frequency selectionTo forecast the driving conditions, the low frequencies

of vehicle speed, pack current and power in the batterypack are projected into the future using a low-pass filterin real-time. The filters are given by

vvehk+1 = (1− δtf c)vvehk + (δtfc)vvehk

Ipackk+1 = (1− δtf c)Ipackk + (δtfc)Ipackk

P packk+1 = (1− δtf c)P packk + (δtfc)P packk ,

(19)

where vvehk+1 is the vehicle speed prediction, Ipackk+1 andP packk+1 are the battery pack current and power predic-tion, respectively. A crucial decision when selecting thismethod is defining the low cut-off frequency f c. To

10 -5 10 -4 10 -3 10 -2 10 -1

Frequency [Hz]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

vve

h(f

)*

Electric bus Cycle 1

Electric bus Cycle 2

USA CITY II

USA FTP 75

Electric bus complete

Fig. 13. PSD for electric bus, USA CITY II and USA FTP 75 drivingcycles.

select this cut-off frequency, the Power Spectral Density(PSD) of the speed from different driving cycles need tobe analyzed to check which frequencies have the mostinformation about a driving cycle.

Fig. 13 shows the PSD of different driving cycles thatlast 1.5 hours each, including the electric bus speedprofile from Fig. 5 and the standardized USA CITY II andUSA FTP driving cycles. A complete driving cycle fromthe electric bus corresponding from full to empty batteryis called electric bus complete, which lasts 6 hours.The speed profile from each driving cycle is shown asfunction of frequency and is normalized from 0 to 1 as

vveh(f)∗ =vveh(f)

|vveh(f)| . (20)

The electric bus driving cycle contains 2 main sub-cycles of approximately 1.5 hours each, which representshort trips the bus takes from the bus station to a finaldestination. As the low bounds of the PSD are givenby the signal length, most of the cycle information forelectric bus cycles 1 and 2 is contained within the fre-quencies above 1.8 · 10−4 Hz. Standardized USA CITY IIand USA FTP 75 driving cycles are repeated to completea 1.5 hours speed profile to emulate the driving profileof a bus inside a city. Since these cycles have the samecycle duration of 1.5 hours, the cycle information isalso contained within the frequencies above 1.8 · 10−4

Hz. Meanwhile, the complete electric bus driving cycleshows to contain most of the information above 10−5 Hz.This means that the cut-off frequency represents the timeframe of past information that is filtered by (19).

In EVs, the available energy in the battery changesduring the driving cycle; therefore, to predict the drivingconditions, the span of time which is looked into the pasthas to be selected according to the level of energy avail-able in the battery. For the previous reason, to predict




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the future driving conditions, the cutoff frequency of thefirst order low pass filter is set as

f c =

4.5 · 10−5 for 75% < SoC ≤ 100%

2.5 · 10−4 for 30% < SoC ≤ 75%

1.5 · 10−3 for 0% ≤ SoC ≤ 30%

(21)

where f c is the cut-off frequency of the low passfilter expressed in [Hz]. For high SoC values, the cut-off frequency considers a time frame of 6 hours ofpast information to predict the future. For SoC valuesbetween 30% and 75%, the time frame corresponds to 1.1hours. Finally, for SoC below 30%, the last 11 minutes ofvehicle operation are used for prediction.

To predict future driving conditions, the filtered dataof the speed, current and power is projected to the futureusing (19) and cut-off frequency conditions from (21) ateach time step.

B. Energy Prediction performanceTo measure the performance of the forecast algorithm,

a real average is calculated a posteriori by

vvehk =1

M

M∑

i=k

vvehi

Ipackk =1

M

M∑

i=k

Ipacki

P packk =1

M

M∑

i=k

P packi ,

(22)

where M is the number of points from time step k untilthe end of cycle.

Fig. 14 shows the comparison between speed, current,power prediction obtained from predicting the futureaverage signals using past information using (19) andthe real averages calculated from (22) during the entiredriving cycle from full to empty battery. It can be seenthat the prediction is overestimated and underestimatedfor all driving conditions during different time steps.For instance, when speed is over estimated in the firsthour of the driving cycle, current and power are alsoover estimated. Furthermore, at the beginning of thedriving cycle, the prediction has a smooth pattern sincethe driving conditions are being filtered with the lowestcut-off frequency. This means that the time frame of pastinformation used for prediction corresponds to a fullcycle of 6 hours. As the driving cycle ends, the timeframe of past information used for prediction is reducedto 10 minutes using the cut-off frequency of 1.5·10−3 Hz.

the low pass filter cut-off frequency increases to lookfor past driving conditions from a short period of timein the past.

has a delayed following pattern of the real futureaverage which

0 1 2 3 4 5 6

Time [h]

0

20

40

60

Speed [km

/h]

(a)

0 1 2 3 4 5 6

Time [h]

0

20

40

Speed [km

/h]

(b)

Prediction

Real average

0 1 2 3 4 5 6

Time [h]

-0.8

-0.6

-0.4

-0.2

Curr

ent [A

]

(c)

0 1 2 3 4 5 6

Time [h]

-60

-40

-20

0

Pow

er

[kW

]

(d)

0 1 2 3 4 5 6

Time [h]

0

50

100

Err

or

[%]

(e)

Speed

Current

Power

Fig. 14. (a) Electric bus driving cycle corresponding from a pointwith a battery pack fully charged until cut-off voltage is reached.(b) Comparison between real average vehicle speed and predictionthroughout the entire driving cycle. (c) Comparison between realaverage battery pack current and prediction throughout the entiredriving cycle. (d) Comparison between real battery pack power andprediction throughout the entire driving cycle. (e) Error in predictionfor vehicle speed, battery pack current and power.

TABLE III. FORECAST ERROR FOR DIFFERENT TIME STEPS IN THEDRIVING CYCLE

Driving condition Maximum Error Average errorvvehk 108% 23.37%

Ipackk 65.55% 18.57%]Ppack

k 84.17% 20.30%

becomes more sensitive when the battery gets closerto empty due to the conditions from (21).

Table III shows the maximum and average errors forspeed, current and power for the entire driving cyclewithout considering prediction error when the electricbus is stopped since the remaining range of the vehicledoes not change and is not necessary to forecast drivingconditions. Current prediction has the lowest maximumand average error from the 3 driving conditions. Mean-while, speed has the highest maximum error and itoccurs when the cut-off frequency of the low pass filteris changed to predict short drivng cycles. Before thistime step, the prediction error is maintained below 50%;however, the prediction for low SoC has to become moresensitive since any change in driving conditions willaffect more the real remaining range.

This section introduced the forecast of energy con-sumption using a first order low pass filter, which pro-vides a low computational effort algorithm. The filtered




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9

values of speed, pack current and power are projectedin the future to predict future driving conditions at eachtime step. The results reflect that the filter is able topredict driving conditions with an average error below24%.

IV. RANGE ESTIMATIONThis section introduces the estimation the remaining

range of an electric vehicle employing the SoC estimatedby the bar-delta filter from Section II and the predictionof vehicle driving conditions using the low pass filterfrom Section III.

From (2), range estimation can be expressed as

R =ESkECk

= vvehk temptyk (23)

where ESk [J] is the energy estimation in the source, ECk[J/km] is the future energy consumption, vvehk [km/h] isa predicted speed and temptyk [h] is the time the batteryterminal voltage will take to reach cut-off voltage, whichcan be expressed as

temptyk =ESkP battk

=Qbattk V battk

Ibattk V battk

, (24)

where P battk and Ibattk are predicted power and currentin the battery expressed in [J/h] and [A], respectively.Qbattk is the estimated extractable charge in the batteryexpressed in [Ah] and V battk is the predicted terminalvoltage in the battery.

The extractable charge in a battery is directly relatedto the energy which can be extracted from the battery.From the battery model in (7), when current is appliedto the battery, power losses occur in R0 and R1 causingbattery voltage to stop at cut-off voltage even if energyis still available in the battery, making the extractableenergy (ES,ext) less than the total energy (ES,total) in thebattery, hence, providing less extractable charge. Fig. 15shows a representation of this energy difference, wherethe extractable charge is noted as Qbatt,ext(Ibatt) andV oc,cf is the open circuit voltage when cut-off voltageis reached.

The battery model in (7), consists of current as inputand voltage as output. However, automotive applicationsrequire a constant power demand which needs to bedelivered by the battery. Power in a battery can beexpressed as

P battk = Ibattk V battk . (25)

By substituting (7) in (25), P batt can be described in termsof battery parameters as

P battk = Ibattk (V oc(χsock ) + V C1

k +R0Ibattk ). (26)

The difference in battery behavior is evaluated byapplying a constant current and constant power to the

V batt[V ]

Qbatt , ext(I batt

)Qbatt

[Ah]

ES , total

ES , ext

V oc(xsoc

)

V batt , I batt<0

V oc , cf

Qbatt ,max0

V batt

Fig. 15. Comparison between total available energy ES,total andextractable energy ES,ext when Ibatt < 0.

0102030405060708090100

SoC [%]

3

3.5

4

4.5

Vbatt

[V

]

(a)

Constant current

Constant power

0102030405060708090100

SoC [%]

-0.6

-0.4

Ibatt

[A

]

(b)

0102030405060708090100

SoC [%]

-2.5

-2

-1.5

Pbatt

[W

]

(c)

Fig. 16. Comparison between applying constant Ibatt and constantP batt in a battery. (a) V batt with constant current and power demands.(b) Constant current demand Ibatt at 0.25 C-rate and Ibatt behaviorwith a constant power demand. (c) Constant power demand in thebattery corresponding to the same C-rate and nominal battery voltagecompared to power behavior with constant current demand.

battery model. A simulation is performed with constantcurrent using (7) and another simulation is run withconstant power using (26). This comparison is shownin Fig. 16. It can be clearly seen that power decreaseswith a constant current demand profile, which is notrepresentative for automotive applications. Furthermore,in order to keep a constant power demand, current needsto increase as battery voltage becomes lower, providingless extractable charge since the power losses are biggerwith a higher current.

As it can be seen, any current demand will makethe battery to deliver less energy than the total energystored in the battery. Furthermore, the battery behavesdifferently with a power demand. In this view, the re-




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10

maining range estimation accuracy is benchmarked with3 approaches:

1) Charge Based (CB) approach. All energy in thebattery is considered extractable; therefore it doesnot depend on power or current prediction. Thetime to reach cut-off voltage is calculated by using(24) considering a constant battery voltage.

2) Voltage Based (VB) approach. It considers thatextractable energy is less than the total batteryenergy. The extractable charge depends on a pre-dicted current demand and the time to reach cut-off voltage is calculated using (24) with a constantbattery voltage.

3) Power Based (PB) approach. It considers that ex-tractable energy is less than the total battery en-ergy. The extractable energy depends on a powerprediction. The time to reach cut-off voltage iscalculated using (24).

All approaches calculate the time that it will takethe voltage of the cell with lowest SoC from presentconditions to reach cut-off voltage with the steady statecondition

d

dtV C

1

= 0. (27)

A. Charge based approachThe CB approach is a simple solution that estimates

the remaining range without modeling the energy lossesby considering that all charge is extractable from the cellwith the lowest SoC.

The extractable charge in the battery is estimated by

Qk = χsoc,lowk Cbatt, (28)

where Qk is the estimated extractable charge in a batteryand is expressed in [Ah], χsoc,lowk is the lowest SoC fromthe pack estimated with the bar-delta filter. Then, thetime that the battery will take to reach cut-off voltage iscalculated with (24) using the current prediction in thebattery. Considering a battery pack arrangement whereall cells in parallel are equally balanced, current in thebattery will be

Ibattk ≈ Ipackk

Nparallel(29)

where Ipackk is the pack current prediction calculatedfrom Section III and Nparallel is the number of cellsin parallel in the pack. Finally, the remaining range isestimated with (23).

B. Voltage Based approachThe VB approach estimates the remaining range by

considering the extractable charge which depends on acurrent prediction. The time to reach cut-off voltage iscalculated from the cell with lowest SoC.

The extractable charge in a battery can be estimatedby

Qk = (χsoc,lowk − χsoc,cfk (Ibattk+1))Cbatt (30)

where χsoc,cfk (Ipackk+1 ) is the SoC when cut-off voltage isreached in the same cell with lowest estimated SoCand is function of the current prediction. Consideringthe steady state condition from (27), the SoC at cut-offvoltage can be calculated by

V oc,cfk (χsoc,cfk ) = V batt − Ibattk+1[R0,cfk +R1,cf

k ] (31)

where V oc,cfk is the OCV at cut-off voltage, R0,cf is theseries resistance and R1,cf is the resistance of the overpotential when cut-off voltage is reached. Using the OCVfrom (31), SoC at the cut-off voltage is obtained fromthe OCV-SoC relation shown in Fig. 4. Then, the time toreach cut-off voltage is calculated with (24). Finally, theremaining range is estimated using (23).

C. Power Based approachThe PB approach estimates the remaining range by

considering an extractable energy as a function of thepower prediction in the battery with lowest SoC in thepack.

The extractable energy in a battery as a function ofpower prediction can be calculated by

ESk = V batt(χsoc,lowk − χsoc,cfk (P battk ))Cbatt, (32)

where ESk is the extractable energy in a battery and isexpressed in [Wh], χsoc,cfk (P battk ) is the SoC when cut-offvoltage is reached in the same cell with lowest estimatedSoC and depends on the power prediction. Consideringthe steady state condition (27), current in the battery canbe written as

Ibattk =V battk − V oc(χsoc)

R0 +R1. (33)

OCV at cut-off voltage can be calculated by substituting(33) in (26) by

V oc,cf (χsoc,cfk ) =P battk (R0,cf +R1,cf )

V batt+ V batt. (34)

Considering that power prediction in a battery from thepack can be described as

P battk+1 =P packk+1

NseriesNparallel, (35)

where Nseries and Nparallel are the number of cells inseries and parallel in the pack, respectively.

To estimate the remaining range, first OCV at the cut-off voltage is calculated from (34), then SoC at the cut-offvoltage is obtained through the OCV-SoC relation shown




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11

in Fig. 4. Then, the extractable energy in the battery withlowest SoC is estimated with (32) and the time to reachcut-off voltage is calculated with (24). Finally, from (23),the remaining range is estimated.

V. RESULTS AND DISCUSSION

This section presents the remaining range estimationresults from the approaches introduced in Section IV.As explained in Section II-C, a real driving cycle in 2scenarios is simulated with an electric bus vehicle modelto obtain speed, traveled distance, cell terminal volt-age, cell SoC and battery pack current and power. Thefirst scenario considers a balanced battery pack and thesecond one considers an unbalanced battery pack. Thetraveled distance in the vehicle model is the referencefor the remaining range estimation in the 3 approaches:Charge Based, Voltage Based and Power Based. Theremaining range estimation is compared to the referencetraveled distance and the accuracy of the results arepresented in both scenarios. Finally, a sensitivity analysisis performed to analyze and discuss the research results..

A. Remaining Range estimation resultsThe real range is obtained from simulating the electric

bus vehicle model with the conditions presented in TableI in 2 scenarios:

1) Scenario 1. A balanced battery pack model of 12cells is included in the electric bus vehicle model.All cells start at 92% SoC and have the sameparameters shown in Fig. 3. The capacity of eachcell is different and is assigned between 2.01 and2.10 Ah.

2) Scenario 2. An battery pack model of 11 balancedcells and 1 unbalanced cell is used in the electricbus vehicle model. The 11 balanced cells start at95% SoC with different capacities between 2.01and 2.10 Ah and the unbalanced cell starts at 85%SoC with a capacity of 1.8 Ah. As described inSection II-C, having this unbalanced situation isa possible real scenario for battery pack which isfast charged with only constant current or with asubstantially aged cell.

For each scenario, a repeated real driving cycle fromFig. 5 is simulated in the electric bus vehicle model untilone of the cells in the battery pack reaches cut-off voltage.The SoC of each cell in the battery pack for scenario 1and 2 are shown in Figures 17 and 18, respectively.

The electric bus speed, current and power in the packare recorded from the vehicle model and then simulatedwith each range estimation algorithm. The range calcu-lation is started in the moment the electric bus startsmoving and the estimation algorithm is stopped whilethe electric bus is not moving. Figures 19 and 20 showthe remaining range estimation results for each algorithmfor scenarios 1 and 2, respectively.

0 1 2 3 4 5 6

Time [h]

0

10

20

30

40

50

60

70

80

90

SoC

[%

]

Fig. 17. SoC of the cells in the balanced batery pack

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time [h]

0

10

20

30

40

50

60

70

80

90

SoC

[%

]

Fig. 18. SoC of the cells in the unbalanced batery pack

For scenario 1, it can be clearly seen that ChargeBased approach is overestimating the range for the entiredriving cycle with an average error of 10.70 km and amaximum error of 20.38 km. On the other hand, Voltageand Power Based approach remaining range estimationsare closer to the range reference with an average error of3.02 km and 3.43 km, respectively, with errors below 10%.For scenario 2, the Charge Based approach has a biggerrange overestimation with an average and maximumerrors of 12.36 km and 22.74 km, however, Voltage andPower Based approaches keep a low range estimationerror with average and maximum errors close to 4 kmand 11 km, respectively.

A sensitivity analysis is done to analyze the effect ofthe prediction of the driving conditions in the range esti-mation for each algorithm. The range estimation is tested




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12

0 1 2 3 4 5 6

Time [h]

0

20

40

60

80

100

Range [km

]

(a)

Charge Based approach

Voltage Based approach

Power Based approach

Real Range EBus

0 1 2 3 4 5 6

Time [h]

-20

-10

0

10

20

Err

or

[%]

(b)




Fig. 19. Range estimation results with 3 range estimation approaches.(a) Range estimation for each approach for the entire driving cyclecompared to a true electric bus range. (b) Range estimation error foreach approach during the entire driving cycle.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time [h]

0

20

40

60

80

100

Range [km

]

(a)




Real Range EBus

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time [h]

-30

-20

-10

0

10

20

30

Err

or

[%]

(b)

Fig. 20. Range estimation results using an unbalanced battery packwith 3 range estimation approaches.(a) Range estimation for eachapproach for the entire driving cycle compared to a true electric busrange. (b) Range estimation error for each approach during the entiredriving cycle.

in 2 different scenarios of predicted driving conditions:1) Predicted Average. Range estimation is calculated

with predicted driving conditions from (19).2) Real Average. A real predicted average of each

driving condition from (22) is used as an inputfor each range estimation algorithm.

Furthermore, the influence of the bar-delta filter algo-rithm in the range estimation is tested by using the

lowest or average SoC in the balanced and unbalancedpacks. This SoC is used as an input for estimatingextractable energy in (32) for the Power Based approach,the extractable charge in (28) and (30) for Charge Basedand Voltage Based approaches, respectively. Finally, abattery model from (7) is used to benchmark the rangeestimation algorithms accuracy. The predicted and av-eraged currents are used to measure the time that ittakes the cell with lowest SoC to reach cut-off voltageby simulating the battery model. The range is estimatedusing (23). Table IV shows the results of this sensitiv-ity analysis including the testing of predicted and realaverage driving conditions.

As expected, by using the real average speed, currentand power in the pack from (22), the range estimationbecomes more accurate than using the predicted con-ditions form the low pass filter. Furthermore, Voltageand Power Based approaches show a similar accuracywhile estimating the range. Both of them have similarmaximum and average errors for a balanced pack andfor an unbalanced pack.

Contrary to expectations, using the bar filter providesa slight better range estimation in a balanced batterypack for Voltage and Power Based approaches, wherethe average range accuracy is improved around 0.5 km.On the other hand, the bar filter worsens Charge Basedrange estimation accuracy. The estimation error for thisapproach is 10.70 km for the delta filter and 11.53 km forthe bar filter.

Meanwhile, the range estimation in an unbalancedbattery pack shows different results. The delta filterimproves the range estimation in all the cases. Themaximum and average errors using the delta filter arelower than the bar filter.

Using a battery model provides a range estimationerror below 5 km for all average errors. The maximumerrors are maintained under 10 km in all cases exceptwhen using the predicted average in an unbalanced pack.This indicates that using a battery model provides arange estimation close to the real range without highvariation along the driving cycle.

B. Remaining Range estimation discussionAs it can be seen, the future energy prediction has a

significant influence on the remaining range estimation.From Fig. 19, it can be noticed that the range estimationin the voltage and power approaches follow similar pat-terns as the results from the low pass filter shown in Fig.14. When the prediction is overestimated, more speed,current and power are predicted. Therefore, the rangeis underestimated since the model forecasts that moreenergy will be consumed in the future, and ultimately,less remaining range is available. On the other hand,when the prediction is under estimated the oppositeeffect occurs.

Even though the forecast has a significant influencein range estimation, it is still important to consider




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13

TABLE IV. RANGE ESTIMATION SENSITIVITY ANALYSIS

Balanced pack Unbalanced packRange algorithm Real average Predicted average Real average Predicted average

Approach Bar-delta filter Max error Avg error Max error Avg error Max error Avg error Max error Avg errorCharge Based Bar filter 4.57 km 4.03 km 20.92 km 11.53 km 20.99 km 19.37 km 33.49 km 26.05 km

Delta filter 4.01 km 3.28 km 20.38 km 10.70 km 16.63 km 6.56 km 22.74 km 12.36 kmVoltage Based Bar filter 7.95 km 4.60 km 12.91 km 2.77 km 10.48 km 8.85 km 21.05 km 15.48 km

Delta filter 9.38 km 5.25 km 12.38 km 3.02 km 11.76 km 4.50 km 10.76 km 4.78 kmPower Based Bar filter 8.67 km 3.22 km 9.34 km 2.98 km 10.48 km 8.65 km 17.28 km 12.50 km

Delta filter 8.8 km 3.90 km 9.42 km 3.43 km 11.43 km 4.39 km 11.25 km 3.64 kmBattery Model 6.53 km 3.96 km 6.34 km 4.23 km 9.97 km 3.36 km 14.57 km 4.68 km

the energy that can be extracted from the battery. Inthe balanced and unbalanced packs, the Charge Basedapproach has a bad range estimation performance com-pared to the Voltage Based approach despite the factthat the same speed and current prediction were usedfor both algorithms. The Voltage Based approach had abetter accuracy than the Charge Based algorithm since itestimates the extractable energy.

In the balanced pack, the delta filter improved therange estimation only for the Charge Based approachsince this algorithm is initially overestimating a range.Therefore, the delta filter helps to lower the range estima-tion since less energy available is calculated. On the otherhand, the the bar filter makes the range estimation worsesince it considers more extractable energy. With respectto the Voltage and Power approaches, using the forecastof driving conditions and estimating the extractable en-ergy with the bar filter is enough to estimate the rangewith low estimation error. As a consequence, the deltafilter worsens the range estimation as the future energyconsumption is overestimated for most part of the cycle.

The opposite effect happens when using an unbal-anced battery pack. The range estimation is improved byusing the delta filter in all cases. The extractable energyin the pack is more accurately estimated than the barfilter since the lowest SoC is estimated. The dynamics ofthe cell with the lowest SoC determine the extractableenergy in the pack, hence it is the limiting factor howfar an EV can travel. In the case of the unbalancedpack, the difference between the lowest SoC and theSoC of the rest of the pack is significant; therefore, thedelta filter estimates the extractable energy more accurateproviding a better range estimation than the bar filter,which overestimates the extractable energy.

The purpose of benchmarking the accuracy of rangeestimation with the Voltage and Power approaches is totest the difference of using a current and power demandprofiles as shown in Fig 16. The use of predicted currentand power demand in the algorithms did not yield asignificant difference in the range estimation for both,balanced and unbalanced battery packs. For instance, inthe balanced pack, the Voltage Based approach had abetter accuracy for 0.5 km in the average error through-out the entire driving cycle; however, the Power Basedapproach had less peak errors than the Voltage approachfor less than 4 km.

VI. CONCLUSIONS AND FUTURE WORK

This paper introduced the remaining range estimationfor any EV based on the integration of states estimationin a battery pack using a bar-delta filter and the predicteddriving conditions using a low pass filter with a lowcomputational effort. The results show that range can beestimated with an error of less than 5% for a balancedpack and less than 7% for an unbalanced pack in asimulation study of an electric bus.

Contrary to expectations, the delta filter showed tohave a low influence in range estimation for a balancedbattery pack. Using the cell with the lowest SoC in thepack to calculate the time to reach cut-off voltage didnot have a significant influence in the range estimationsince the cell with the lowest SoC of the pack has not aconsiderable difference compared to the SoC of the restof the cells. We assume that in the case of balanced packs,using a delta filter for this purpose is not as critical asforecasting accurately the driving conditions. However,results indicate that for an unbalanced battery pack, thedelta filter provides a better range estimation than usingthe bar filter. We conclude that predicting the drivingconditions has more influence in the range estimationwhen the battery pack is balanced. Nevertheless, asthe battery pack ages, unbalances might become moresignificant. Furthermore, a fast charging session withonly a constant current might lead to have unbalances inthe pack as well. Therefore, the bar-delta filter algorithmtakes more relevance as the battery ages and it shouldbe considered to estimate the range of an EV.

Using past information to predict the future demon-strated to be helpful to estimate the remaining range.The predictions in this research had an average errorof around 20%, which still provided range estimationresults below 10% error. Having a precise prediction offuture driving conditions is a challenging task. However,there can be alternatives to improve the accuracy ofthe forecast. The prediction of driving conditions canbe improved by having an adaptive and self learninglow pass filter according to present driving conditions.Furthermore, incorporating more information such asGPS route and traffic data might increase the accuracy ofthe forecast. In the case of electric bus applications, theenergy consumption can be linked to the time of the dayof the driving cycle. For certain times of the day, differentenergy consumptions can be predicted by using peak andoff-peak hour information such as number of passengers




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14

and traffic. The repetitiveness of the driving cycles forpublic transport can be an opportunity to conduct afurther research on the use of filters to estimate futuredriving conditions.

VII. REFERENCESREFERENCES

[1] E. A. Nanaki and C. J. Koroneos, “Climate changemitigation and deployment of electric vehicles inurban areas,” Renewable Energy, 8–2016.

[2] M. Fuller, “Wireless charging in california: Range,recharge, and vehicle electrification,” Transportationresearch, 3–2016.

[3] W. Waag, C. Fleischer, and D. U. Sauer, “Critical re-view of the methods for monitoring of lithium-ionbatteries in electric and hybrid vehicles,” Journal ofPower Sources, 3–2014.

[4] A. Farmann and D. U. Sauer, “A comprehensive re-view of on-board state-of-available-power predic-tion techniques for lithium-ion batteries in electricvehicles,” Journal of Power Sources, 8–2016.

[5] X. Zhang, Y. Wang, C. Liu, and Z. Chen, “A novelapproach of remaining discharge energy predictionfor large format lithium-ion battery pack,” Journalof Power Sources, 1–2017.

[6] M. Dubarry and B. Y. Liaw, “From li-ion single cellmodel to battery pack simulation,” Journal of PowerSources, 10–2008.

[7] C. Truchot, M. Dubarry, and B. Y. Liaw, “State-ofcharge estimation and uncertainty for lithium-ion battery strings,” Applied Energy, 1–2014.

[8] X. Zhang, Y. Wang, D. Yang, and Z. Chen, “Anon-line estimation of battery pack parameters andstate-of-charge using dual filters based on packmodel,” Energy, 9–2016.

[9] G. L. Plett, “Efficient battery pack state estimationusing bar-delta filtering,” 5–2009.

[10] J. Hong, S. Park, and N. Chang, “Accurate remain-ing range estimation for electric vehicles,” 3–2016.

[11] P. Ondrúška and I. Posner, “Probabilistic attainabil-ity maps: Efficiently predicting driver-specific elec-tric vehicle range,” IEEE Intelligent Vehicles Sympo-sium (IV), 7–2014.

[12] W. Wang, Y. Kobayashi, and K. Shirai, “Remain-ing driving range estimation of electric vehicle,”4–2012.

[13] G. Liu, M. Ouyang, L. Languang, L. Jianqiu,and H. Jianfeng, “A highly accurate predictive-adaptive method for lithium-ion battery remainingdischarge energy prediction in electric vehicle ap-plications,” Applied Energy, 7–2015.

[14] H. Rahimi and M.-Y. Chow, “Adaptive online bat-tery parameters/soc/capacity co-estimation,” 8–2013.

[15] H. Zhang and M.-Y. Chow, “Comprehensive dy-namic battery modeling for phev applications,”IEEE PES General Meeting, 7–2010.




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A Global Optimal Solution to the Eco-DrivingProblem

G.P. Padilla, S. Weiland, M.C.F. Donkers

Abstract—Eco-driving aims at minimizing the energy con-sumption of a vehicle by adjusting the vehicle’s velocity. Thiscan be formulated as an optimal control problem and thispaper provides a detailed view on the global optimal solutionto this problem. A method to reformulate and discretize theproblem avoiding the introduction of additional nonconvex termsis presented. Furthermore, physically realistic conditions aregiven that guarantee the existence of the global optimal solutionto the eco-driving problem. Subsequently, a sequential quadraticprogramming algorithm is provided that allows finding theglobal optimal solution. Finally, two numerical examples are usedto illustrate how solutions of the eco-driving problem can beobtained.

Index Terms—Automotive control, Optimal control, Optimiza-tion.

I. INTRODUCTION

IMPROVING energy efficiency of vehicles is an importanttopic of research for the automotive industry. High energy

efficiency is important for reducing fuel consumption andmeeting emission legislations. Moreover, energy efficiencyis also supported by the functional argument of mitigatingrange anxiety of electric vehicles, i.e., giving a sufficientlylarge driving range for an electrical vehicle. The problem ofreducing the energy consumption of a vehicle over a certaindrive cycle can be formulated as an optimal control problemand its solution is often referred to as an energy managementstrategy. Most of these energy management problems arefocused on controlling the power split between the combustionengine and the electric machine of a hybrid electric vehicle[1], [2]. By storing regenerative braking energy and shiftingthe operating points of the combustion engine, a significantamount of fuel can be saved. A recent trend is to extend thisenergy management system to incorporate more subsystemsof the vehicle [3] or to consider emission constraints in theoptimal control problem [4].

In the aforementioned approaches, the vehicle velocity (andthereby the power needed to propel the vehicle) is assumed tobe fixed. Nevertheless, the vehicle inertia, which is the largestenergy buffer in the vehicle, can have a large impact in energysavings and consequently in the extension of the driving range.For instance, in [5] it has been reported that changes indriving behavior could improve the energetic performance of

This work has received financial support from the Horizon 2020 pro-gramme of the European Union under the grant ‘Electric Vehicle En-hanced Range, Lifetime And Safety Through INGenious battery management’(EVERLASTING-713771).

Paul Padilla, Tijs Donkers and Siep Weiland are with the Depart-ment of Electrical Engineering of Eindhoven University of Technol-ogy, Netherlands. E-mail: g.p.padilla.cazar, m.c.f.donkers,[email protected]

the vehicle more than 30%. The promising improvements inenergy efficiency have contributed to the emergence of the eco-driving concept, which aims to increase the energy efficiencyof a vehicle by means of a convenient selection of drivingstrategies; i.e. legal regulations, technological implementationsor simply changes in the driver behavior. Hence, it is clear thateco-driving is a broad concept where government, manufac-turers and users participate [6].

To solve the eco-driving problem, standard techniques usedin optimal control have been adopted, see [7] for a detailedoverview of the recent literature. In [8], [9], dynamic program-ming (DP) has been used to find a global solution to this prob-lem. Alternatively, Pontryagin’s minimum principle (PMP) hasbeen used in [10]–[13]. The main disadvantages PMP are thatit only provides a necessary condition for optimality and thatincorporating state constraints is not a simple task. Therefore,in [14]–[16] static nonlinear optimization techniques are usedto solve the problem in the presence of state constraints. Itremains unclear from the papers that use static optimizationtechniques or PMP whether the solutions are, in fact, globallyoptimal. Unfortunately, the literature related to this topic isscarce. The noticeable exception is [15], where the continuous-time optimal control problem is certified to be convex, whichguarantees that the obtained solution is globally optimal.However, [15] does not discuss the possible loss of convexitydue to the discretization process. This might occur, as it isdemonstrated in this manuscript.

This paper aims to expose a detailed view of the globaloptimal solution to the eco-driving problem. The results of thispaper can be used to certify optimality of the results presentedin the existing literature and in future works. The maincontributions presented in this paper are threefold: Firstly, amethod to reformulate and discretize the problem is presented.This is initially done for a simplified case to illustrate thenonconvexity of the problem and subsequently extended to thecomplete eco-driving problem. Secondly, a detailed analysis ofthe uniqueness of the solution to the reformulated problem isused to obtain a set of mild conditions that guarantee the globaloptimality of the solution. Thirdly, a sequential quadraticprogramming (SQP) method is employed to efficiently solvethe eco-driving problem.

II. CONTINUOUS-TIME PROBLEM FORMULATION

In this section, a continuous-time formulation of the eco-driving concept as an optimal control problem is provided.Eco-driving aims at obtaining an optimal control force u(t)and velocity profile v(t) that minimizes the integral of thepower P (v, u) consumed by a vehicle while traveling duringa given time interval [to, tf ] over a given trajectory s(t) ∈

This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/LCSYS.2018.2846182

Copyright (c) 2018 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].




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[so, sf ] with known geographical characteristics, i.e., with agiven road grade α : [so, sf ] → [−π2 ,

π2 ], where α(s) is

the grade at position s; while being subject to longitudinalvehicle dynamics, non-negative velocity bounds v(t) ∈ [

¯v, v],

and boundary conditions on position and velocity. This can bestated in the form of the following optimal control problem:

mins(t),v(t),u(t)

∫ tf

to

P (v(t), u(t))dt (1a)

subject to mdvdt = u(t)− f(v(t), s(t)), (1b)dsdt = v(t), (1c)

s(to) = so, s(tf ) = sf (1d)v(to) = vo, v(tf ) = vf (1e)

¯v ≤ v(t) ≤ v, (1f)

where (1b) represents the longitudinal vehicle dynamics, inwhich u(t) is the control force and f(v, s) describes theaerodynamic drag, rolling resistance and gravity forces as

f(v, s) = σdv2 + crmg cos(α(s)) +mg sin(α(s)), (2)

In (1b) and (2), m represents the combined mass of thevehicle and the inertia of the driveline, g ≈ 9.81[m/s2] isthe gravitational constant, cr > 0 describes the rolling forcecoefficient and σd = 1

2cdρaAf with cd > 0 is the dragcoefficient, in which ρa denotes the air density and Af isthe frontal area of the vehicle.

The consumed power P (v, u) can be obtained from differentmodeling approaches that capture the energy consumption inthe powertrain. In this paper, it is assumed to be a quadraticfunction of the form

P (v, u) = β0v2 + β1vu+ β2u

2, (3)for some non-negative parameters β0, β1 and β2. Equation(3) is a physically realistic approximation, e.g., for electricmotors due to the fact that the friction and Ohmic lossesare captured by the terms β0v2 and β2u

2, respectively. Theoptimal control problem (1) does not consider constraints onthe control force u(t) nor constraints on propulsion powerP (v, u), as in (3). However, the results of this paper can beextended to include such constraints. Alternatively, constraintson propulsion power can be incorporated by connecting theeco-driving to vehicle energy management, as was done in[17], [15], where power constraints of powertrain componentsare present.

In general, (1) is a nonlinear optimal control problem thatmight have multiple local solutions due to specific features ofthe vehicle model and road profile, see (2). This implies thatdirect optimization methods or methods based on PMP onlyprovide candidate minima, which might not correspond to theglobal solution to problem (1).

III. CONVEXITY IN RELATION TO DISCRETIZATION

To illustrate that the solution methods for the controlproblem (1) can introduce nonconvexity, which might com-plicate finding the global minimum, a simplified version ofproblem (1) is considered in this section. Using this simplifiedexample, we will illustrate a possible reformulation of theproblem as a convex optimal control problem, which impliesthe existence of a unique solution. Because of the presence

of state constraints in (1), we will focus on discrete-timeapproximations of the optimal control, which might by itselfintroduce nonconvexity, even for specific cases where thecontinuous-time problem (1) is convex.

For a simplified version of problem (1) used in this section,consider β0 = 0, β1 = 1 and β2 = 0 in (3), which correspondsto an electric motor with perfect energy conversion and nofriction and Ohmic losses. Constant velocity bounds are alsoconsidered. Moreover, assume a flat road, meaning that α(s) =0 so that the rolling friction is constant, i.e. cos(α(s)) = 1,and the gravitational force has no effect on the longitudinalvehicle dynamics as sin(α(s)) = 0. Under these assumptions,(1)-(3) reduces to

mins(t),v(t),u(t)

∫ tf

to

u(t) v(t) dt. (4a)

subject to (1c)-(1f) andmdv

dt = u(t)− σdv(t)2 − crmg. (4b)The above optimization problem is nonconvex due to (4b),meaning that application of PMP or finite dimensional opti-mization methods cannot guarantee a global optimal solution.We will show how this problem can be reformulated as aconvex optimization problem, where we focus on discrete-time approximations so that static optimization methods canbe applied.

A. Direct Discretization

In order to illustrate that caution should be taken whendiscretizing the optimal control problem (4) with (1c)-(1f),we show that direct discretization leads to a nonconvex op-timization problem. In an attempt to assess convexity of (4)with (1c)-(1f), we eliminate u(t) by substituting the equalityconstraints (4b) into the objective function (4a), thereby pro-ducing an equivalent optimization problem. This procedure isa useful tool to analyse the convexity of the problem in thefeasible set, see, e.g., [18], and leads to

mins(t),v(t)

∫ tf

to

mdvdt v(t) + crmgv(t) + σdv(t)3 dt, (5)

subject to (1c)-(1f).In order to solve (5) subject to (1c)-(1f), using direct

optimization methods, we can approximate the integral inthe objective function using a forward Euler discretizationmethod at vk = v(tk) and sk = s(tk), i.e., at instances tk,k ∈ 0, . . . , N, for some N ∈ N, where tk+1 > tk, t0 = to,tN = tf and step size τk = tk+1 − tk > 0. This leads to aforward Euler discretization of the objective function in (5),given by∑N−1

k=0 τk

(m (vk+1−vk)

τkvk + σdv

3k + crmgvk

)(6)

which can be rewritten as

12m(v2f−v2o)+

N−1∑k=0

τk(σdv3k+crmgvk)− 1

2m(vk+1 − vk)2, (7)

using (1e). The last term in this expression is nonconvex,which is a direct consequence of the forward Euler discretiza-tion. It should be noted that this result is independent of thestep size τk and a similar conclusion can be drawn from theapplication of a backward Euler method. This shows the loss of






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convexity when applying discrete-time approximations of theeco-driving problem (1). Higher-order discrete approximationsof (5) possibly lead to a convex optimization problem, albeitat the cost of increased complexity. Instead, we will herereformulate the optimal control problem (5) subject to (1c)-(1f) in a way that introduction of nonconvex terms is avoided,as will be shown below.

B. Continuous-Time Reformulation

The reformulation of the simplified eco-driving problemis based on the observation that, instead of discretizing theobjective function directly, the first two terms in (5) can beintegrated over the boundary conditions (1d) and (1e) to obtain

mins(t),v(t)

12m(v2f−v2o) + crmg(sf−s0) +

∫ tf

to

σdv(t)3 dt, (8)

subject to (1c)-(1f). The two terms in this objective functiondescribe the change in kinetic energy of the vehicle over thecomplete trajectory, and the loss due to rolling resistance,respectively. Since vo, so, vf and sf are known, these terms donot influence the optimal solution. Interestingly, the remainingterm describes the energy losses due to aerodynamic drag.Thus, the reduced expression is discretized using a forwardEuler approach, leading to

minsk,vk

∑N−1k=0 τkσdv

3k, (9)

subject to sk+1 = sk + τkvk, (1d)-(1f). The resulting discreteobjective function (9) is convex for vk > 0, which correspondsto the vehicle driving in forward direction.

Hence, this simplified case shows that a direct discretizationof the eco-driving can lead to a nonconvex optimizationproblem, while a reformulated problem yields a convex opti-mization problem after discretization. The ideas in this sectionwill be extended in the next section towards the complete eco-driving problem (1) to prove the existence of a unique solutionof the optimal control problem.

IV. A GLOBAL SOLUTION TO THE ECO-DRIVINGPROBLEM

In this section, the ideas presented in Section III will beapplied to problem (1), without making the simplificationsconsidered in Section III. This allows an equivalent opti-mization problem of reduced complexity to be formulated,which can be discretized using a forward Euler methodwithout introducing additional nonconvex terms. The structureof the reduced discrete-time optimal control problem will beexploited to prove that a unique global optimal solution existsunder mild and realistic conditions.

A. Reduction of the Continuous-Time Problem

Nonconvex optimization problems can show multiple localminima, therefore finding a global solution of the problemcould be a cumbersome task. Nonconvexity is not only aconsequence of a discretization action, it can also be related toother parameters of the nonlinear optimal control problem (1).For instance, (1a) could be a nonconvex objective function,which occurs for particular values of β0, β1 and β2 thatmake (3) a nonconvex function. Moreover, a realistic road

Fig. 1: Geometry of the road profile.

grade α(s) might also introduce nonconvexity in the equalityconstraint (1b). As done in the previous section, an equivalentoptimal control problem will be obtained by substituting theequality constraint (1b) into the objective function.

Adopting the basic ideas presented in Section III, thecontinuous-time optimal control problem (1) is reformulatedinto a convenient form given by

mina(t),s(t),v(t)

∫ tf

to

P (v(t),ma(t) + f(v(t), s(t))) dt (10a)

subject to (1c)-(1f) anddvdt = a(t), (10b)

where a(t) is a new decision variable, which represents thevehicle acceleration, and f(v, s) is given by (2). It is importantto remark that (10a) is obtained from the substitution of (1b)into (1a), and its integrand is given byP (v,ma+f(v,s)) = β2(mgγg(s)+cdv

2+crmgγr(s)+ma)2

+β1v(ma+cdv2+mgγg(s)+crmgγr(s))+β0v

2 (11)in which γg(s) = sin(α(s)) and γr(s) = cos(α(s)).The relevance of (11) is that it contains information of thelongitudinal vehicle dynamics, where the majority of thenonlinearities of the problem are embedded. This informationand the structure of the problem (10) with (1c)-(1f) can beexploited to obtain a reduced, yet equivalent, optimal controlproblem. The first step to achieve this goal is to observe that

γg(s) = dhds and γr(s) = ds′

ds (12)where h(s) is the given elevation profile and s′(t) is thehorizontal projection of s(t). The validity of these expressionscan be explained using Fig. 1, where the geometry in thisphysical configuration shows that for any specific point in theroad, the change in elevation and horizontal projection withrespect to the displacement are given by dh

ds = sin(α(s)), andds′

ds = cos(α(s)), respectively.The next step is to remove the terms in the objective

function (10a) that can be solved in advance and have nocontribution to the optimal control problem. In particular,consider the following terms from (10a):

EG =

∫ tf

to

β1m(a+ gγg(s) + crgγr(s))v + 2β2mσdv2a dt

=

∫ tf

to

β1m(v dv

dt +g dhds

dsdt +gcr

ds′

dsdsdt

)+2β2mσdv

2 dvdt dt

=12β1m(v2f−v2o)+β1mg(hf−ho)+β1mgcr(s

′f−s′o)

+ 23β2mσd(v

3f−v3o). (13)






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The first equality in (13) is obtained by substitution of (1c),(10b) and (12), and the second equality by solving the integralover the boundary conditions of the problem. The first twoterms shows that the total kinetic and potential energy of thevehicle depend only on the velocities and elevations at theboundaries. In a similar way, the last two terms demonstratethat part of the energy consumed by the drag and rollingforces, respectively, are defined by the velocity and horizontaldisplacement at the boundaries.

Since the value of (13) is given by the boundary conditionsof the optimization problem (10), it is possible to rewrite (10a)as ∫ tf

to

P (v,ma+ f(v, s)) dt =

∫ tf

to

PR(a, v, s)dt+ EG, (14)

wherePR(a, s, v) = β0v

2 + β1σdv3 +2β2m

2ga(γg(s)+crγr(s))

+ β2(ma)2+β2(mgγg(s)+σdv2+crmgγr(s))

2. (15)Removing the constant term EG from the objective function,thereby changing (11) to (15) in the optimal control problem,does not change its optimal solution.

B. Unique Solution to the Discrete-Time Optimal ControlProblem

The reduced continuous-time optimal control problem ob-tained in the previous section can be discretized in order tomake it solvable using static optimization methods. In thissection, it will be shown that the discrete-time problem hasa unique solution under realistic conditions and an efficientmethod to obtain this global minimum will be presented inSection V.

In order to discretize the problem, we consider again aforward Euler discretization method and define ak = a(tk),vk = v(tk) and sk = s(tk) at instances tk = kτ + to,k ∈ K = 0, . . . , N, with fixed step size τ =

tf−toN , for

some N ∈ N. This leads tomin

ak,sk,vk

∑N−1k=0 τPR(ak, sk, vk) (16a)

subject to sk+1 = sk + τvk, (16b)vk+1 = vk + τak, (16c)s0 = so, sN = sf , (16d)v0 = vo, vN = vf , (16e)

¯v ≤ vk ≤ v (16f)

The theorem below is the main results of this section andprovides conditions under which the optimal control problem(16) has a unique global minimum.

Theorem 1. Suppose optimization problem (16) is feasible.If β2 > 0 and if gτ2

(dγg(sk)ds + cr

dγr(sk)ds

)6= −1 for all k ∈

0, . . . , N−1, optimization problem (16) has a unique globalminimum.

Proof. The first-order necessary conditions for optimality of(16) are given by the so-called Karuhn-Kuhn-Tucker (KKT)conditions, see, e.g, [18]. Instrumental in these KKT condi-

tions is the LagrangianL(ak, sk, vk, κk, λk, µk, νk) =∑N−1

k=0 β0τv2k + β1τσdv

3k + β2τ(mak)2

+β2τ(mgγg(sk)+σdv2k+crmgγr(sk))2

+ 2β2τm2ga(γg(sk)+crγr(sk)) + κk+1(sk+1−sk−τvk)

+ λk+1(vk+1−vk−τak)+µk(vk − v)+νk(v − vk). (17)Since all constraints are linear, a critical point, or stationarypoint, of (16) is characterized by∂L∂ak

=2β2m2τ(ak+gγg(sk)+crgγr(sk))−λk+1τ=0 (18a)

∂L∂sk

= 2β2τmg(mak +mgγg(sk) + σdv2k + crmgγr(sk))

×(dγg(sk)

ds + crdγr(sk)

ds

)+κk − κk+1 = 0 (18b)

∂L∂vk

= 2β0τvk + 3β1τσdv2k +4β2τσdvk

(mgγg(sk)+σdv

2k

+crmgγr(sk))−τκk+1+λk−λk+1+(µk−νk)=0 (18c)

and ∂L∂κk

, ∂L∂λk

described by (16b) and (16c) respectively forsome Lagrange multipliers κk, λk and µk > 0, νk > 0, k ∈0, . . . , N, such that

µk(vk − v) = 0 and νk(v − vk) = 0, (18d)and κ0, λ0, κN , and λN chosen such a way that (16d) and(16e) are satisfied. Now rewriting (18a) as

ak = 12β2m2λk+1 − gγg(sk)− crgγr(sk)), (19)

and substituting this into (18b) yields

κk=κk+1−2β2τmg( 12mβ2

λk+1+σdv2k)(dγg(sk)

ds +crdγr(sk)

ds

).

(20)Substituting (19) into (16c) leads to a characterization of acritical point of (16), given by[

sk+1

vk+1

]=[

sk + τvkvk − τgφ(sk) + τ

2β2m2 λk+1

](21a)

and[κkλk

]=

[1 −τg dφ

dskτ 1

][κk+1

λk+1

]+[

01

](νk−µk)− τΦ(sk, vk) (21b)

with

Φ(s, v)=

[2β2mg2σdv

2 dφdsk

2β0v+3β1σdv2+4β2σdv

(σdv

2k +mgφ(sk)

)] (21c)

and φ(s) = γg(s) + crγr(s). For this characterization, µk ≥ 0and νk ≥ 0 can always be uniquely chosen so that λk has thevalue needed to ensure v ≤ vk ≤ v, while satisfying (18d).

Furthermore, since the matrix[

1 −τg dφdsk

τ 1

]is full rank for

every k ∈ 0, . . . , N − 1 by the hypothesis of the theorem,every pair (κN , λN ) leads to a unique trajectory of κk and λk, meaning that they can be uniquely chosen such that (16d)and (16e) are satisfied if (16) is feasible.

The KKT conditions provide necessary conditions for opti-mality in the sense that they characterize points that satisfy

∇F (x) d > 0 (22)for all feasible directions d at x. In the above equation∇F (x) is the gradient of the objective function (16a), wherex = [a0, . . . , aN−1, s0, . . . , sN−1, v0, . . . , vN−1]>, and thefeasible directions d at x ∈ F are all vectors that satisfyx+αd ∈ F for some α > 0 and where F = x|(16b)−(16f),which is a compact set. Points satisfying (22) can in principlebe minima, maxima or saddle points. Because of uniqueness






44 / 110

of the solutions to (21), the obtained critical point has to bea minimum. Indeed, suppose the critical point would be amaximum or a saddle point, there would exist at least oneother x that would satisfy the necessary conditions. In otherwords, if x is a saddle point or a maximum, it is possible tomove away from this x and lower the value of the objectivefunction until the boundary of the feasible set F would bereached. At this point, we would find another point x thatsatisfies (22). This contradicts the fact that there is only onecritical point. Therefore, this unique critical point has to bethe global minimum, which completes the proof.

Remark 1. The conditions presented in Theorem 1 aresatisfied for many realistic cases. For instance, β2 > 0 isalways satisfied if Ohmic losses of the electric motor areconsidered in the objective function. Moreover, standard roaddesign guidelines, e.g, [19], suggest curvatures that yield∣∣∣dγg(sk)ds

∣∣∣ 1 and∣∣∣dγr(sk)ds

∣∣∣ 1.

V. SOLUTION TO THE ECO-DRIVING PROBLEM

In this section, we will propose an efficient numericalmethod for solving the optimal control problem (16). The solu-tion method we propose uses Sequential Quadratic Program-ming (SQP). The effectiveness of SQP to solve constrainednonlinear optimization problems is the main reason for thelarge acceptance of this method. The scent of this approachis to iteratively solve linearly constrained quadratic (QP) sub-problems, that are an approximation of the original problemevaluated in a previous solution, until some convergencecriterion is achieved [20]. The uniqueness of the solutionguarantees that the SQP algorithm finds the global minimum,provided that the algorithm converges.

Since the objective function in (16) is nonconvex even afterthe reformulation and reduction presneted in Section IV, weemploy the results of [21] to formulate a convex QP sub-problem by making an approximation of the Hessian matrixof (15), which is given by

H=

∂2PR∂a2

∂2PR∂a ∂s

∂2PR∂a ∂v

∂2PR∂a ∂s

∂2PR∂s2

∂2PR∂s ∂v

∂2PR∂a ∂v

∂2PR∂s ∂v

∂2PR∂v2

=

[h11 h12(s) 0h12(s) h22(a,s,v) h23(s,v)

0 h23(s, v) h33(s,v)

], (23)

in which we have omitted the arguments of the functionPR(a, s, v) for compactness of notation, and where

h11 =2β2m2, (24a)

h12(s)=2β2m2g(dγg(s)

ds + crdγr(s)

ds

), (24b)

h22(a, s, v)=2β2mg(mg(

dγg(s)ds +cr

dγr(s)ds )2

+(ma+mgγg(s)+σdv2+mgcrγr(s))

×(d2γg(s)

ds2 +crd2γr(s)

ds2 )), (24c)

h23(s,v)=4mgβ2σdv(dγg(s)

ds +crdγr(s)

ds ), (24d)h33(s, v)= 4β2σdmg(γg(s)+crγr(s))+ 2β0+6β1σdv

+ 12β2σ2dv

2. (24e)Since h22(a, s, v) and h33(s, v) can become negative, wepropose in this paper to useH(s, v) = diag

(2β2m

2, ε22, max(h33(s, v), ε33)), (25)

as a positive definite approximate Hessian in the SQP algo-rithm presented below, in which ε22, ε33 are small positivenumbers and h33(s, v) is given by (24e).

The approximated Hessian matrix allow us to solve the eco-driving problem by sequentially solving the following convexsecond-order approximation of (15) as:ai+1k , si+1

k , vi+1k k∈K =

argminak,sk,vk

∑N−1k=0 τPQP (ak, sk, vk, a

ik, s

ik, v

ik), (26)

subject to (16b)-(16f), wherePQP (ak, sk, vk, a

ik, s

ik, v

ik) =

12

[akskvk

]>H(sik, v

ik)[akskvk

]+

∇PR(aik, sik, v

ik)−

[aiksikvik

]TH(sik, v

ik)

[akskvk

].

(27)

In (27), ∇PR(aik, sik, v

ik) is the gradient of (15), and H(sik, v

ik)

the positive definite approximated Hessian matrix given by(24), both evaluated at (aik, s

ik, v

ik), where i indicates the

iteration of the SQP algorithm. Convergence of the proposedSQP approach is guaranteed due to the fact that the Hessianmatrix (25) satisfies the conditions for convergence presentedin [21, Section 3.2].

VI. NUMERICAL EXAMPLES

In this section, the SQP algorithm proposed in Section V isused to find the optimal solution of two numerical examplesof the eco-driving problem.

A. Benchmark Problem for Electric Vehicles

In this example, we revisit the numerical benchmark prob-lem for eco-driving that have been introduced in [10] anduses PMP to solve it. The energy consumption in this modelconsiders a frictionless electric motor, i.e., β0 = 0. The effectsof the rolling force in the vehicle are assumed to be constant,i.e., γr = 1, while the effects of the gravitational force aredescribed by

γ(s) = gγg(s) = p0 + p1s+ p2s2 + p3s

3. (28)It is important to note that the polynomial function γ(s)also embeds information related to the road profile used inthis example. In Table I, the parameters presented in [10]are translated to the eco-driving formulation proposed in thispaper. Since the example of [10] is a numerical example, theparameters in Table I and the optimal solution do not have aphysical interpretation. The method proposed in Section Vreturns an optimal solution that is depicted in Fig. 2a. Thissolution shows the same features reported in [10]. In fact,the final cost obtained using SQP is 1.2295 × 106, whichdiffers only 0.0767% from the results reported in [10], whilehaving a similar computation time. Considering that in thiscase β2 > 0, from Theorem 1, it is posible to conclude that the

TABLE I: EV parameters in [10].

p0: 3 v(to): 0 m: 1 to: 0p1: 0.4 v(tf ): 0 g: 1 tf : 1p2: −1 s(to): 0 cr : 0.1 τ : 0.001p3: 0.1 s(tf ): 10 σd: 10−3 v0(t): 0






45 / 110

0 0.2 0.4 0.6 0.8 1-10

-5

0

5

10

15

Time [s]

s

v

u/

γ

(a) Original formulation.

Time [s]0 0.2 0.4 0.6 0.8 1

-10

-5

0

5

10

15

s

v

u/

γ

(b) Including velocity contraints.

Fig. 2: Global solution to the eco-driving problem [10].

0 5 10 15 20 s [km]

60

70

80

v [k

m/h

]

0

500

h [m

]

Constant speed drivingEco-driving

0 5 10 15 20 s [Km]

-20

0

20

u [K

N]

0

500

h [m

]

Fig. 3: Global solution to the eco-driving problem.

solution is a global minimum of the problem. Using the SQPapproach presented in this paper, velocity constraints can beeasily added, as shown in Fig. 2b, which is not straightforwardusing the PMP-based approach of [10].

B. Hybrid Electric Heavy-Duty Vehicle

In this example, a hybrid electric heavy-duty vehicle drivingat a constant speed is compared with an eco-driving strategy,where the velocity is allowed to change between given bounds.The settings of the eco-driving problem used in this exampleare summarized in Table II and the elevation profile is de-fined by

h(s) = 225 cos( 3π21000s+ π

4 ) + 225 [m], (29)which is depicted in Fig. 3 as a green surface.

The constant-speed driving profile is described by a solidline in Fig. 3. In this case, the total energy consumed by thevehicle is 3.0719 × 104[kJ ]. On the other hand, the dashedline presented in Fig. 3 describes the optimal control force uand velocity profile v obtained as the global solution to theeco-driving optimal control problem studied in this case. Thetotal energy consumed by the vehicle under this strategy is2.843×104[kJ ], which is approximately 7.44% lower than theenergy consumed by the vehicle driving at constant velocity.

TABLE II: Parameters for the heavy-duty vehicle example.

β0: 0.292 v(to): 70[km/h] m: 15950[kg] to: 0[s]β1: 1.005 v(tf ): 70[km/h] g: 9.81[m/s2] tf : 1080[s]β2: 2.652×10−4 s(to): 0[km] cr : 0.1 v: 80[km/h]τ : 5[s] s(tf ): 21[km] σd: 3.1246

¯v: 60[km/h]

VII. CONCLUSIONS

In this paper, a detailed study has been conducted on theglobal optimality of the eco-driving optimal control problem.We have proposed to reformulate and discretize the problemand have subsequently derived conditions that guarantee theexistence of the global solution to the eco-driving problem.Taking advantage of these results, a SQP algorithm that effi-ciently solves the eco-driving problem has been proposed. Themethodologies and results were illustrated in two numericalexamples.

REFERENCES

[1] B. Egardt, N. Murgovski, M. Pourabdollah, and L. J. Mardh, “Electro-mobility Studies Based on Convex Optimization: Design and ControlIssues Regarding Vehicle Electrification,” IEEE Control Sys Mag, 2014.

[2] J. Kessels, J. H. M. Martens, P. P. J. van den Bosch, and W. H. A.Hendrix, “Smart vehicle powernet enabling complete vehicle energymanagement,” in Proc. Vehicle Propulsion & Power Control , 2012.

[3] T. C. J. Romijn, M. C. F. Donkers, J. T. Kessels, and S. Weiland, “A dualdecomposition approach to complete energy management for a heavy-duty vehicle,” IEEE Trans on Control Syst Technol, 2018.

[4] M. Donkers, J. van Schijndel, W. Heemels, and F. Willems, “Optimalcontrol for integrated emission management in diesel engines,” ControlEngineering Pract., 2017.

[5] C. Bingham, C. Walsh, and S. Carroll, “Impact of driving characteristicson electric vehicle energy consumption and range,” IET Intell. TransportSyst., 2012.

[6] B. Seraens, “Optimal Control Based Eco-Driving: Theoretical Approachand Practical Applications,” Ph.D. dissertation, Katholieke UniversiteitLeuven, 2012.

[7] A. Sciarretta, G. D. Nunzio, and L. L. Ojeda, “Optimal EcodrivingControl: Energy-Efficient Driving of Road Vehicles as an OptimalControl Problem,” IEEE Control Systems, 2015.

[8] E. Hellstrom, J. Aslund, and L. Nielsen, “Management of kinetic andelectric energy in heavy trucks,” SAE Journal of Engines, 2010.

[9] E. Ozatay, S. Onori, J. Wollaeger, U. Ozguner, G. Rizzoni, D. Filev,J. Michelini, and S. D. Cairano, “Cloud-Based Velocity Profile Optimiza-tion for Everyday Driving: A Dynamic-Programming-Based Solution,”IEEE Trans on Intell Transport Syst, 2014.

[10] N. Petit and A. Sciarretta, “Optimal drive of electric vehicles usingan inversion-based trajectory generation approach,” Proc. IFAC WorldCongress, 2011.

[11] W. Dib, A. Chasse, P. Moulin, A. Sciarretta, and G. Corde, “Optimalenergy management for an electric vehicle in eco-driving applications,”Control Eng Pract, 2014.

[12] E. Ozatay, U. Ozguner, and D. Filev, “Velocity profile optimization ofon road vehicles: Pontryagin’s Maximum Principle based approach,”Control Eng Pract, 2017.

[13] M. A. S. Kamal, M. Mukai, J. Murata, and T. Kawabe, “EcologicalVehicle Control on Roads With Up-Down Slopes,” IEEE Transactionson Intelligent Transportation Systems, vol. 12, 2011.

[14] N. J. Kohut, J. K. Hedrick, and F. Borrelli, “Integrating traffic data andmodel predictive control to improve fuel economy,” Proc. IFAC WorldCongress, 2009.

[15] N. Murgovski, L. Johannesson, X. Hu, B. Egardt, and J. Sjoberg,“Convex relaxations in the optimal control of electrified vehicles,” inProc. American Control Conf, 2015.

[16] M. Vajedi and N. L. Azad, “Ecological Adaptive Cruise Controllerfor Plug-In Hybrid Electric Vehicles Using Nonlinear Model PredictiveControl,” IEEE Trans on Intell Transport Syst, 2016.

[17] Z. Khalik, G. P. Padilla, T. C. J. Romijn, and M. C. F. Donkers,“Vehicle Energy Management with Ecodriving: A Sequential QuadraticProgramming Approach with Dual Decomposition,” in Proc AmericanControl Conference, 2018.

[18] S. P. Boyd and L. Vandenberghe, Convex optimization, CambridgeUniversity Press New York, 2004.

[19] A. A. of State Highway and T. O. (AASHTO), “A policy on geometricdesign of highways and streets,” Tech. Rep.

[20] J. Nocedal and S. J. Wright, Numerical Optimization. Springer, 2006.[21] P. T. Boggs and J. W. Tolle, “Sequential quadratic programming,” Acta

numerica, 1995.






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A Shrinking Horizon Approach toEco-driving for Electric City Buses:

Implementation and Experimental Results ?

J.C. Flores Paredes, G.P. Padilla Cazar, M.C.F. Donkers

Dept Electrical Eng, Eindhoven University of Technology, Netherlands,(e-mail: [email protected], [email protected],

[email protected])

Abstract: Eco-driving aims at minimizing the energy consumption of a vehicle by changing thevehicle’s velocity. This can be formulated as an optimal control problem and this paper presentsan efficient shrinking horizon implementation to solve this problem. The efficient implementationis demonstrated on a case study of a fully electric bus, driving on an inner-city public transportroute. Because the bus drives on designated bus lanes, meaning that it has little interactionwith other traffic, and because it has frequent and predictable stops, this case will have a goodenergy consumption savings potential. An energy consumption reduction of 11.43% is achievedon a simulation study for the case that the vehicle is fully autonomous and a reduction of 6.94%is achieved experimentally for the case that the driver is ‘coached’ using a driver assistancesystem.

Keywords: Automotive Control, Advanced Driver Assistance Systems, Eco-driving

1. INTRODUCTION

Electrification of vehicles, and particularly the deploymentof fully electric vehicles, is considered as a way to mitigatethe production of CO2 emissions from fossil fuels (Grauerset al., 2012). However, electric vehicles are known to havelimited range and to charge slowly, when compared totraditional vehicles, leading to range anxiety, which isthe user’s concern to not reach the vehicle’s destination(Nilsson, 2011). There are several factors that influencerange anxiety, e.g., the limited energy storage capacityof batteries, the ability to accurately predict the vehicle’senergy consumption and the total energy consumption ofthe vehicle (including its auxiliaries). The last item men-tioned can be improved using vehicle energy management(de Jager et al., 2013; Onori et al., 2016) or using eco-driving strategies.Eco-driving intends to change the vehicle’s speed betweendeparture and arrival, either by assuming full automationof the vehicle or by changing the driver’s behavior (Bing-ham et al., 2012; Sciarretta et al., 2015). This problem canbe formulated as an optimal control problem (OCP). Tosolve this problem, standard techniques from optimal con-trol are often adopted. In (Ozatay et al., 2014), DynamicProgramming (DP) has been used to solve the eco-drivingproblem. Alternatively, Pontryagin’s Minimum Principle(PMP) has been used in (Dib et al., 2014; Ozatay et al.,2017; Kamal et al., 2011). The main disadvantage of DP isthat it is computationally intensive and the main disadvan-tage of PMP is that incorporating state constraints is not a

? This work has received financial support from the Horizon 2020program of the European Union under the grant ‘Electric VehicleEnhanced Range, Lifetime And Safety Through INGenious batterymanagement’ (EVERLASTING-713771).

simple task. Therefore, static nonlinear optimization tech-niques are used in (Padilla et al., 2018; Romijn et al., 2018;Murgovski et al., 2015; Vajedi and Azad, 2016) to solve theproblem in the presence of state constraints. These staticoptimization problems lead to an optimization problemthat is often seen in model-predictive control and allowsfor fast online implementations.While many of the existing results on eco-driving presentonly simulation results (Kamal et al., 2013; Ojeda et al.,2017; Vajedi and Azad, 2016) and/or present a one-shotoptimal solution when the full drive trajectory is known(Padilla et al., 2018; Ozatay et al., 2014; Murgovski et al.,2015), this paper presents an online shrinking horizon im-plementation of eco-driving, and experimental results thatshows its potential to reduce the energy consumption. As acase study, a fully electric city-bus is used on an inner-citypublic transport route. The considered city bus drives ondesignated lanes, meaning that it has limited interactionwith other traffic, and has frequent and predictable stops.Therefore, a good energy savings potential is expected.First, an efficient online implementable algorithm of thesolution strategy of (Padilla et al., 2018) is presented.In this efficient algorithm, the solution to the state-spacemodel is substituted into the objective function and con-straints, as is often done in model-predictive control, lead-ing to a significant reduction of the number of decisionvariables, when compared to (Padilla et al., 2018). Thisimplementation can also be used in a so-called shrinkinghorizon fashion. A shrinking horizon is needed because atevery time instant, the optimal control input is recom-puted for a shorter time horizon, due to the fact thatthe time and distance to the next bus stop is smallerthan during the previous time instant. We will presenttwo results: the first being a simulation study for the case




47 / 110

that the vehicle speed is exactly controlled (i.e., assuminga fully automated vehicle), and the second being an ex-perimental study for the case that the driver is ‘coached’using a driver assistance system that interacts with thedriver through a human-machine interface. Both studieswill show that the energy consumption of the bus can bereduced significantly.

2. ECO-DRIVING CONTROL PROBLEM

In this section, we introduce the continuous-time eco-driving problem and the discrete-time solution approachpresented in (Padilla et al., 2018). This solution approachwill serve as a basis for our proposed efficient real-timeimplementation.

2.1 Continuous-Time Formulation

The eco-driving problem considered in (Padilla et al.,2018) aims at minimizing the total energy consumed by avehicle over a given time interval [t0, tf ] and trajectory s(t)∈ [s0, sf ] for a road grade α(s)∈

[−π2 , π2

], which depends

on the position s. This can be formulated as a continuous-time Optimal Control Problem (OCP) given by

mins(t),v(t),u(t)

∫ tf

t0

P (v(t), u(t))dt (1a)

subject to mdvdt = u(t)− σdv2 −mgγ(s), (1b)

dsdt = v(t), dv

dt = a(t), (1c)

s(t0) = s0, v(t0) = v0, (1d)

s(tf ) = sf , v(tf ) = vf , (1e)

¯v ≤ v(t) ≤ v, (1f)

¯a ≤ a(t) ≤ a, (1g)

where (1b) represents the longitudinal vehicle dynamicsof a vehicle with mass m, aerodynamic drag coefficient σdand rolling resistance and gravity forces, with gravitationalconstant g, where the latter two are combined in

γ(s) = cr cos(α(s)) + sin(α(s)), (2)

where cr describes the rolling force coefficient. Further-more, the non-negative velocity is bounded by (1f). In(1g), we have also included bounds on the acceleration,which can be done without compromising the uniquenessof the solution claimed by (Padilla et al., 2018). Finally,the consumed power in the driveline is assumed to be aquadratic function given by

P (v, u) = β0v2 + β1vu+ β2u

2, (3)

where β0, β1 and β2 are non-negative parameters thatdescribe the contribution of Ohmic and mechanical frictionloses in the power consumption.

2.2 Discretization

In (Padilla et al., 2018), the OCP (1) is discretized attimes tk = kτ+t0, k∈K = 0, . . . , N, with time steps τ =tf−t0N , for some N ∈ N using a forward Euler discretization

method. Additionally, the non-linear equality constraint(1b) is included in the objective function (1a), resulting aQuadratic Programming (QP) problem, given by

minak,sk,vkk∈K

N−1∑

k=0

τP (ak, sk, vk) (4a)

subject to sk+1 = sk + τvk, (4b)

vk+1 = vk + τak, (4c)

s0 = so, v0 = vo, (4d)

sN = sf , vN = vf , (4e)

¯v ≤ vk ≤ v, (4f)

¯a ≤ ak ≤ a, (4g)

where ak = a(tk), vk = v(tk) and sk = s(tk), and

P (ak, sk, vk) =β0v2k + β1vk

(mak + σdv

2k +mgγ(sk)

)

+ β2

(mak + σdv

2k +mgγ(sk)

)2(5)

represents the driveline power consumption, now depen-dent on the acceleration, position and velocity of thevehicle.

2.3 Sequential Quadratic Programming for Eco-driving

The uniqueness and global optimality of the solution to thediscrete-time OCP (4) has been formally shown by (Padillaet al., 2018). Moreover, it was suggested that SequentialQuadratic Programming (SQP) is a suitable approach toobtain the solution.In SQP, a nonlinear program is solved by iterativelysolving QP approximations of the nonlinear program. Forthe OCP (4), this leads to

ai+1, xi+1 = argmina, x

∑

k∈Kτ PQP (a,x,ai,xi) (6)

subject to (4b)− (4g),

for all k∈K, with PQP (a,x,ai,xi), a second-order approx-imation of (5), given by

PQP (a,x,ai,xi)= 12

[ax

]>H

[ax

]+

[Gi

a

Gix

]>−[ai

xi

]>H

[ax

],

(7)

where the superscript i refers to the solution of the i-thiteration of the SQP algorithm, and

a =[a0, . . . , aN−1

]> ∈ RN , (8)

x =[s0, v0, . . . , sN−1, vN−1

]> ∈ R2N , (9)

H = diag(Ha, Hx

)∈ R3N×3N , (10)

Ga =[∂P (a0,s0,v0)

∂a0, . . . , ∂P (aN−1,sN−1,vN−1)

∂aN−1

]>∈ RN , (11)

Gx =[∂P (a0,s0,v0)

∂s0, ∂P (a0,s0,v0)

∂v0, . . . , ∂P (aN−1,sN−1,vN−1)

∂sN−1,

∂P (aN−1,sN−1,vN−1)∂vN−1

]>∈ R2N . (12)

In (10), the Hessian matrix is approximated to a diagonalpositive definite matrix (Padilla et al., 2018), given by

Ha = 2β2m2IN ∈ RN×N , (13)

Hx = diag(εs,max(hv0 , εv), . . . ,

. . . , εs,max(hvN−1, εv)

)∈ R2N×2N , (14)

in which In ∈ Rn×n is a n-dimensional identity matrix,εs and εv are small positive numbers that guarantee thepositive definiteness of the Hessian matrix, and




48 / 110

hvk = 2β0 + 6β1σdvk + 4β2σd

(mgγ(sk) + 3σdv

2k

). (15)

The gradients of the power consumption (5) are consideredin (11) and (12), where∂P (ak,sk,vk)

∂ak=2β2m

2(gγ(sk)+ak

), (16)

∂P (ak,sk,vk)∂sk

=2β2mg∂γ(sk)∂sk

(mak+mgγ(sk)+σdv

2k

), (17)

∂P (ak,sk,vk)∂vk

=2β0vk+3β1σdv2k+4β2σdvk

(mgγ(sk)+σdv

2k

).

(18)

The QP iterations (6) repeat until its solution converges.We consider the algorithm to have converged if∥∥∥∥∥∥

[ai

xi

]−[ai−1

xi−1

]∥∥∥∥∥∥2

≤ ρ, (19)

where,‖·‖2 is the 2-norm operator, i and i−1 represent thecurrent and previous QP solutions, respectively, and ρ ∈ Ris a given non-negative convergence tolerance.This approach provides a tractable solution to the eco-driving OCP (4). In the next section, we will modify thismethod to obtain a real-time implementation for the eco-driving problem.

3. REAL-TIME IMPLEMENTATION

To arrive at a real-time implementable algorithm, thecomputation time needs to be low, such that solutions canbe obtained in a shorter time than the sampling periodused in the embedded system. In this section, we willmodify the SQP method presented in the previous sectionto reduce the computation time by means of a reductionon the number of decision variables and the application ofa shrinking horizon approach that uses a previous solutionas warm-start for the next optimization problem. Thisreduction of the number of decision variables does notaffect the solution (and its optimality).

3.1 Improving Computational Efficiency

In this section, we will reduce the number of decisionvariables by eliminating the state x in (6). This leads to asignificant reduction of the computation time, because thecomputation time of the QP subproblems scales cubicallywith the number of decision variables (Diehl et al., 2009).By eliminating the state xk = [sk vk]> from the QPproblems an improvement of the computation time isexpected by a factor of 33 = 27, when compared to (Padillaet al., 2018).In order to reduce the number of decision variables,the longitudinal motion of the vehicle described by theequality constraints (4b)-(4d) can be recast as a predictionmodel for instants k∈K as

x = Φx0 + Γa, (20)

where Φ∈R2N×2 and Γ∈R2N×N are defined by

Φ =

I2A

A2

.

.

.

AN−1

, Γ =

02 02 . . . 02

B 02 . . . 02

AB B . . . 02

.

.

....

. . ....

AN−2

B AN−3

B . . . 02

(21)

with

A =[1 τ

0 1

], B =

[0

τ

], (22)

and 0n a n-dimensional column vector with all entries zero.For k = N , the final state given by (4e) can be expressedas the following equality constraint[

AN−1B AN−2B . . . B]a = xN −ANx0. (23)

By substituting (10) and (20) into (7), we remove thedependency on the state x in (7). After removing theconstant terms (which do not change the solution), weobtain

PQP(a,ai) = 12a>(Hi

a + Γ>HixΓ)a

+(Gi>

a − ai>Hi

a +(Gi>

x − ai>Γ>Hix

)Γ)a, (24)

where Hia and Hi

x are the approximated Hessian respectto acceleration and state, respectively, evaluated in the i-th iteration of the SQP algorithm.Using the above notation, we obtain the following QPproblem with acceleration a as the only decision variable

ai+1 = argmina

∑

k∈Kτ PQP(a,ai) (25a)

subject to (23),

¯v − ΞΦx0 ≤ ΞΓa ≤ v − ΞΦx0, (25b)

1N¯a ≤ a ≤ 1N a, (25c)

where the constraint (25b) is obtained by substituting (20)in (4f) for all k ∈ K and considering v = Ξx, with aselection matrix Ξ = IN ⊗

[0 1]∈ RN×2N , in which the

operator ⊗ denotes the Kronecker product. Besides, theconstraint (25c) is directly obtained from (4g) for all k∈Kand considering 1n∈Rn, a n-dimensional column vectorwith all entries one.The QP subproblem (25) is sequentially solved until itconverges, thus finding the global optimal solution to thediscrete optimal control eco-driving problem (4), with thecorresponding convergence criteria defined by

||ai − ai−1||2 ≤ ρ. (26)

In practice, a smaller number of decision variables letthe optimization problem converges with fewer iterations,which might lead to an additional reduction in computa-tion time.

3.2 Considerations for Real-time Implementation

In order to achieve further improvements in terms ofcomputation time, we use an efficient QP solver withwarm-start and a shrinking horizon approach. Shrinkinghorizon works in a similar way as receding horizon, alsoknown as model-predictive control. At every time instantk, the full optimization problem (25) is solved over thehorizon N using the actual state xk as initial value. Thisleads to a sequence of optimal control inputs a, whereonly the first a0 is implemented at time instant k. Now attime tk+1 the process repeats, but in contrary to recedinghorizon control, the control horizon N now becomes 1 timestep shorter. The use of shrinking horizon for eco-drivingof buses is motivated by the fact that we only need tooptimize the vehicle speed until the next bus stop andat every recomputation of the optimal control inputs, thenext bus stop becomes nearer.The fact that the control horizon shrinks makes theoptimization problem time dependant. In particular, attime instant k, the eco-driving problem has to be solved




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over the discrete-time set Kk = k, . . . , N. This causesthe QP subproblem at time k to become

ai+1 = argmina

∑

k∈Kk

τ PQP(a,ai) (27)

subject to (23), (25b), (25c)

where N has become N−k in (23) and the matrices in(25b) and (25c) are modified similarly. The Hessian andgradient matrices in the SQP algorithm also need to bemodified mutatis mutandis. The shrinking of the horizonis repeated until N−k < Nmin for some Nmin to avoidunnecessary computations in the last iterations, where notmuch energy consumption improvement is expected.The QP problem (27) is implemented in Simulink, with thesolver mpcqpsolver.m, from the Matlab/Simulink ModelPredictive Control Toolbox. This function allows to com-pile the Simulink model as a system function that canembed the eco-driving problem and provide the solutionin real-time. This solver can also make a so-called warm-start to reduce the computation time of the solution byproviding the optimal solution of the previous iteration ofthe algorithm as an inital guess for the next iteration.

4. SIMULATION STUDY FOR ELECTRIC CITYBUSES

Inner-city public transportation buses, sometimes drivethrough exclusive lanes to arrive and depart from stopsat well defined times. This makes this scenario to havehigh predictability, which makes it highly suitable for theimplementation of eco-driving approaches.In this section, we will analyze the performance of thesolution to eco-driving problem, presented in Section 3.1,in the context of electric city buses. The first part ofthis section shows an identification approach to obtain acontrol-oriented model for the power consumption of thedriveline in an electric bus, which can be used in the real-time optimal ecodriving problem of Section 3. Later, thismodel is used to obtain an optimal velocity profile for anspecific driving route and finally, ideal energy savings arecalculated with respect to a benchmark velocity profile,that has been obtained from real data logged from a busdriving on a segment of 2.5[km] between two bus stops,where the elevation between the highest and the lowestpoint is less than one meter. Hence, the grade α(s) isconsidered zero for all s.

4.1 Identification of the Power Consumption Model

The OCP (4) considers a simplified control-oriented powerconsumption model defined by (5). For the identificationof this model, several realizations over the 2.5[km] routementioned above are used. This leads to a set of velocityprofiles vjk, where j ∈ J = 1, . . . , J represents thej-th realization of the velocity profile. These velocityprofiles are then simulated on a high-fidelity (hi-fi) vehicle

model to obtain power consumption profiles P jk for eachgiven velocity profile. This data can be used to find theparameters β0, β1 and β2 in (5) by solving the followingconstrained least-squares problem

β0, β1, β2 = argmin∑

j∈J

∑

k∈K

∥∥∥P (ajk, sjk, v

jk)− P jk

∥∥∥2

2

subject to β0 > 0, β1 > 0 and β2 > 0, (28)

where P (ajk, sjk, v

jk) is defined in (5). It is important to

remark that the hi-fi vehicle model is a validated model.Fig. 1(b) depicts the normalized vehicle power consump-tion, based on a real velocity profile that is shown inFig. 1(a). The power consumption described by the hi-fi model is represented by the dotted red line, while theblue line is obtained by identifying the three unknownparameters on the simplified consumption model (5).Although the identified model shows a considerable errorwith respect to hi-fi model, it is capable to describe ageneral trend in its behavior. In the last part of thissection, we will show that this feature is enough to produceaccurate solutions in term of energy savings.

4.2 Computation Time

The identified model obtained in the previous section isused to obtain the optimal velocity profile for an electricbus driving on the 2.5[km] flat route. The optimal velocityprofile is obtained by the method proposed in (Padillaet al., 2018) and the approach proposed in this paper. Thecomputation time for both cases is compared in Table 1.It should be noted that the computation time presentedin (Padilla et al., 2018) is obtained using a different QPsolver and cannot be compared to the results presented inTable 1, where we have compared the method presentedin this paper with (Padilla et al., 2018) using the same QPsolver.As it was stated in Section 3.1, the improved computationtime obtained by the approach proposed in this paperis mainly produced by the reduced number of decisionvariables. The results shown in Table 1 demonstrate thatthe proposed implementation can be used in real-timeapplications.

0 50 100 150 200

time [s]

0

20

40

60

Ve

locity [km

/h]

real velocity

bounds

(a) Real velocity profile.

0 50 100 150 200time [s]

-50

0

50

Con

sum

ed p

ower

[% ]

hi-fi modelControl model

(b) Normalized power consumption.

Fig. 1. High fidelity and identified control models.




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Table 1. Computation time

ApproachNumber of Computation

decision variables time

Padilla et al. (2018) 636 282.5[s]Problem in (25) 211 0.2[s]

4.3 Energy Savings

In order to analyze the performance of the method pro-posed in this paper, in terms of energy savings, a real veloc-ity profile is used as benchmark. In Fig. 2, the blue dashedline depicts a real velocity profile obtained from loggeddata of an electric bus, while the red solid line representsthe optimal velocity profile obtained as solution to (27). Byusing the hi-fi model we obtain 11.43% of energy savings,while using the simplified function (5), we save 12.21% ofenergy. This shows that the simplified power consumptionmodel (5) is an adequate approximation that enables anefficient calculation of velocity profiles and provides a goodapproximation of the expected energy savings.The power profile of this benchmark is represented as a reddashed line on Fig. 1(b). Here, it is possible to see the on-off action created by the real behavior of a driver. Althoughthe velocity profile generated by the driver seems to beclose to the optimal velocity profile presented in Fig. 2,the constant on-off control action created by the driverproduces energy loses that explains the energy savingswhen applying the optimal velocity profile.

5. ASSISTED DRIVING EXPERIMENTAL RESULTS

In this section, we will discuss the experimental results ob-tained by using a driving assistance system that ‘coaches’the driver to follow the optimal velocity profile obtainedas a solution to the eco-driving problem (27). First, weprovide details about the case study analysed in this sec-tion. Second, the Human Machine Interface (HMI) usedto provide visual feedback to the driver is described; andfinally, experimental results for the case study are providedand discussed.

5.1 Case Study

The experiments have been conducted in the previouslyselected road, with a distance of 2.5[km] that shouldbe covered in 200[s], considering a real speed limit of60[km/h]. During the experiments, it was registered windvelocity of 30[km/h], approximately matching the roaddirection.The baseline for energy savings is generated by the drivercovering the selected road, with the knowledge of velocity

0 50 100 150 200

time [s]

0

20

40

60

Ve

locity [km

/h]

baseline

eco-driving solution

bounds

Fig. 2. Velocity profile for simulated power consumption

(a) Increase velocity (b) Correct velocity (c) Decrease velocity

Fig. 3. Human-Machine Interface for velocity feedback

bounds and initial and final states. This is compared tothe energy consumed while providing visual feedback tothe driver and expressed as a percentage ratio of energysavings.Two drivers have performed the experiments during oneday, driving the same bus in both directions. The powerconsumption and velocity profiles have been recorded witha sample rate of 1[Hz] for data post-processing.

5.2 Human-Machine Interface

The Human-Machine Interface (HMI) provides visual feed-back to the driver at a specific frequency, based on thesolution to the real-time implementation of the eco-drivingproblem (27). The HMI updating frequency should takeinto account the speed of reaction of the driver. In practice,this means that the update frequency presented to thedriver is considerably smaller than the one used for thereal-time implementation of (27).In this case, an external computer manages the HMI withan update frequency of 0.1[Hz]. It has been developedin Vector CANoe (Vector Informatik GmbH, 2016) anddepicted in Fig. 3. The HMI sends the request to increase(Fig. 3(a)), hold (Fig. 3(b)) or decrease (Fig. 3(c)) thecurrent velocity, in order to follow the optimal velocityprofile within a tolerance of ±1[km/h].

5.3 Experimental Results

In this section, we will experimentally determine theamount of energy that eco-driving is able to save. Sub-sequently, we will compare this results with simulationsin the hi-fi model with the velocity profiles generatedexperimentally.While performing the experiments, a wind velocity of30[km/h] was registered, with a direction approximatelymatching the selected road. Hence, the experimental eco-driving energy savings are separated for headwind andtailwind driving, and shown in Table 2. In average, weobtained 6.94% of energy savings.Note that the percentages in Table 2 were obtained from20 experiments performed by the two drivers, i.e., four ex-periments for each wind direction, plus the correspondingbaselines. The velocity profiles of two experiments with its

Table 2. Experimental energy savings

Wind direction Driver Energy savings Average

HeadwindDriver 1 1.27%

3.18%Driver 2 5.09%

TailwindDriver 1 12.70%

10.70%Driver 2 8.70%

Total Average 6.94%




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Table 3. Simulated energy savings

Wind direction Driver Energy savings Average

HeadwindDriver 1 6.48%

6.89%Driver 2 7.29%

TailwindDriver 1 17.50%

15.96%Driver 2 14.42%

Total Average 11.43%

baselines are depicted in Fig. 4, one for each driver withthe specified wind direction. Here, the blue dotted linerepresents the baseline and the red line shows the driverbehavior when receiving eco-driving feedback.Tailwind driving saves more energy than headwind. In thiscase, the difference is approximately a factor of three (seeTable 2). The difference in energy savings between thedrivers can be explained by their own driving style andexperience, as well as random disturbances, e.g., changesin intensity and direction of the wind, temperature varia-tions, traffic conditions, etc.The experimental velocity profiles were logged by VectorCANoe and used as input to the hi-fi vehicle model. Usingthe same baseline for each driver and wind direction, thesimulated energy savings are shown in Table 3. Tables 2and 3 show correlation between experimental and simu-lated energy savings, verifying that the hi-fi vehicle modelhas been validated. Note that in both cases the eco-drivingsolution to (27) reduces the energy consumption.

6. CONCLUSIONS

This paper has presented an efficient shrinking horizonimplementation to solve the eco-driving OCP. The imple-mentation has been demonstrated on a case study of afully electric bus, driving on an inner-city public transportroute. We have shown that the computational performancecan be reduced significantly using the proposed implemen-tation. Furthermore, a energy consumption reduction of11.43% is achieved on a simulation study for the casethat the vehicle is fully autonomous and a reduction of6.94% is achieved experimentally for the case that thedriver is ‘coached’ using a driver assistance system. Theobtained results are promising in terms of energy savingsby providing feedback to the driver and, at the same time,it has become evident that better results can be achieved

0 20 40 60 80 100 120 140 160 180 200

time [s]

0

20

40

60

Ve

locity [

km

/h]

Driver 1 - headwind

baseline

with feedback

bounds

0 20 40 60 80 100 120 140 160 180

time [s]

0

20

40

60

Ve

locity [

km

/h]

Driver 2 - tailwind

baseline

with feedback

bounds

Fig. 4. Experimental velocity profiles

by improving the accuracy when following the eco-drivingfeedback.

REFERENCES

Bingham, C., Walsh, C., and Carroll, S. (2012). Impactof driving characteristics on electric vehicle energy con-sumption and range. IET Intell. Transport Syst.

de Jager, B., van Keulen, T., and Kessels, J. (2013).Optimal Control of Hybrid Vehicles. Springer.

Dib, W., Chasse, A., Moulin, P., Sciarretta, A., and Corde,G. (2014). Optimal energy management for an electricvehicle in eco-driving applications. Control Eng Pract.

Diehl, M., Ferreau, H.J., and Haverbeke, N. (2009). Effi-cient numerical methods for nonlinear mpc and movinghorizon estimation.

Grauers, A., Sarasini, S., and Karlstrom, M. (2012). Whyelectromobility and what is it? In B. Sanden andP. Wallgren (eds.), Syst. perspective on Electromobility.

Kamal, M.A.S., Mukai, M., Murata, J., and Kawabe, T.(2011). Ecological Vehicle Control on Roads With Up-Down Slopes. IEEE Trans Intell Transport Syst.

Kamal, M., Mukai, M., Murata, J., and Kawabe, T. (2013).Model predictive control of vehicles on urban roads forimproved fuel economy. IEEE Trans Control Syst Techn.

Murgovski, N., Johannesson, L., Hu, X., Egardt, B., andSjoberg, J. (2015). Convex relaxations in the optimalcontrol of electrified vehicles. In Proc. Am Control Conf.

Nilsson, M. (2011). Electric vehicles: the phenomenon ofrange anxiety. Technical report, Lindholmen SciencePark, Sweden.

Ojeda, L.L., Han, J., Sciarretta, A., Nunzio, G.D., andThibault, L. (2017). A real-time eco-driving strategy forautomated electric vehicles. Proc. IEEE Conf. Decisionand Control.

Onori, S., Serrao, L., and Rizzoni, G. (2016). Hybrid elec-tric vehicles: Energy management strategies. Springer.

Ozatay, E., Onori, S., Wollaeger, J., Ozguner, U., Rizzoni,G., Filev, D., Michelini, J., and Cairano, S.D. (2014).Cloud-Based Velocity Profile Optimization for Every-day Driving: A Dynamic-Programming-Based Solution.IEEE Trans on Intell Transport Syst.

Ozatay, E., Ozguner, U., and Filev, D. (2017). Velocityprofile optimization of on road vehicles: Pontryagin’sMaximum Principle based approach. Control Eng Pract.

Padilla, G.P., Weiland, S., and Donkers, M.C.F. (2018). AGlobal Optimal Solution to the Eco-Driving Problem.IEEE Control Systems Letters.

Romijn, T.C.J., Donkers, M.C.F., Kessels, J.T., and Wei-land, S. (2018). A dual decomposition approach tocomplete energy management for a heavy-duty vehicle.IEEE Trans Control Syst Techn.

Sciarretta, A., Nunzio, G.D., and Ojeda, L.L. (2015).Optimal Ecodriving Control: Energy-Efficient Drivingof Road Vehicles as an Optimal Control Problem. IEEEControl Syst.

Vajedi, M. and Azad, N.L. (2016). Ecological AdaptiveCruise Controller for Plug-In Hybrid Electric VehiclesUsing Nonlinear Model Predictive Control. IEEE Transon Intell Transport Syst.

Vector Informatik GmbH (2016). CANoe.J1939.LIN.Eth-ernet.XCP.




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TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 1

A Distributed Optimization Approach forComplete Vehicle Energy Management

T.C.J. Romijn, M.C.F. Donkers, J.T.B.A. Kessels, and S. Weiland

Abstract—In this paper, a distributed optimization approachis proposed to solve the complete vehicle energy managementproblem of a hybrid truck with several controllable auxilia ries.The first part of the approach is a dual decomposition, whichallows the underlying optimal control problem to be solvedfor every subsystem separately. For the second part of theapproach, the optimal control problem for every subsystem isfurther decomposed by splitting the control horizon into severalsmaller horizons. Two methods for splitting the control horizonare used; the first method uses Alternating Direction MethodofMultipliers and divides the horizon a priori, while the secondmethod divides the horizon iteratively by solving unconstrainedoptimization problems analytically. We demonstrate the approachby solving the complete vehicle energy management problem ofa hybrid truck with a refrigerated semi-trailer, an air supp lysystem, an alternator, a DCDC converter, a low-voltage batteryand a climate control system. Simulation results show that thefuel consumption can be reduced up to 0.52 % by includingsmart auxiliaries in the energy management problem. Moreinterestingly, the computation time is reduced by a factor of 64 upto 1825, compared with solving a centralized convex optimizationproblem.

Index Terms—Energy Management, Optimal Control, Dis-tributed Optimization, Hybrid Vehicles, Smart Auxiliarie s

I. I NTRODUCTION

REDUCING fuel consumption has been one of the toppriorities of the automotive industry in the last decades.

A low fuel consumption is important for a sustainable societyand is also one of the largest competitive factors for sales.Over the last decades, several new technologies have beenintroduced in vehicles to improve fuel efficiency. A majorimprovement is obtained with the introduction of hybridtechnology. By adding an electric motor with a high-voltagebattery to the powertrain, braking energy can be recuperatedand the internal combustion engine can be controlled at a moreefficient operating point. The energy management strategy inthis case is restricted to determining the (optimal) powersplitbetween the electric motor and the internal combustion engine.The energy consumption is, however, not only related tothe driving functionality of the vehicle, especially for heavy-duty vehicles, as part of the energy is used for auxiliaries,e.g., an air supply system, a refrigerated semi-trailer and

This work has received financial support from the Horizon 2020 pro-gramme of the European Union under the grant ‘Electric Vehicle En-hanced Range, Lifetime And Safety Through INGenious battery man-agement’ (EVERLASTING-713771). Constantijn Romijn, TijsDonkersand Siep Weiland are with the Department of Electrical Engineering ofEindhoven University of Technology, the Netherlands, and John Kesselsis with the Vehicle Control group of DAF Trucks NV. E-mail ad-dresses: t.c.j.romijn,m.c.f.donkers,[email protected],[email protected].

a climate control system. These auxiliaries have a potentialenergy buffer that can be utilized by the energy managementsystem to schedule energy flows and thereby further improvingthe vehicle energy efficiency, i.e., Complete Vehicle EnergyManagement (CVEM, see [1]). At the same time, heavy-dutyvehicles are augmented with many different type of auxiliaries,which requires the energy management system to be flexibleand scalable with low complexity to reduce development timeand costs. Moreover, as heavy-duty vehicles typically drivelong distances, it is needed to solve the CVEM problem overlarge horizons to demonstrate and benchmark the benefit ofCVEM.

Many different solution strategies for solving the energymanagement problem have been proposed over the pastdecades. The proposed solution strategies can be dividedinto so-called online and offline solution strategies [2], [3].Online solution strategies are real-time implementable andcan be further divided in rule-based strategies (see, e.g.,[4]),equivalent consumption minimization strategies (ECMS, see,e.g., [5]) or model predictive control (MPC, see, e.g., [6]).The online solution strategies only rely on feedback and/orpredictions and can therefore not guarantee the global optimalsolution. To verify the performance of the online solutionstrategies and to analyze different configurations, so-calledoffline solution strategies have been developed based on, e.g.,dynamic programming (DP, see, e.g., [7]–[9]), Pontryagin’sminimum principle (PMP, see, e.g., [10]–[12]) or convex opti-mization (see, e.g., [13], [14]). The offline solution strategiesrequire all disturbances to be known (e.g., the driving cycle)so that the global optimal solution can be computed andcan therefore not be implemented in real-time. Still, they doprovide a benchmark for online solution methods and aretherefore valuable tools. We focus on offline solution strategiesin this paper.

While online optimization methods based on ECMS mightbe able to handle the complexity of the CVEM problem,the aforementioned offline optimal control methods cannot.Itshould be noted that multi-state energy management problems,e.g., including battery state-of-health [15], battery aging [16],thermal management [17]–[19] and the control of a wasteheat recovery system [20] are all based on ECMS, meaningthat global optimality of the solution cannot be guaranteed.For the offline optimization methods, scalability is poor asthe computational complexity of DP increases exponentiallywith the number of states and solving the two-point boundaryvalue problem resulting from PMP is difficult, particularlywhen state constraints are present, see, e.g., [17] in the contextof thermal dynamics. Finally, a convex approximation of the




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energy management problem can lead to a globally optimalsolution, but still requires a large-scale optimization problemto be solved.

For this reason, distributed solutions for energy managementstart to appear. In [21], [22], an online implementable game-theoretic approach to CVEM is shown. In [23], scalabilityis obtained by using the Alternating Direction Method ofMultipliers (ADMM) while ideas based on ECMS are usedto calculate the equivalent costs at a supervisory level. Still,these distributed solutions are all online solution methods forwhich the global optimal solution is not guaranteed.

In this paper, we propose to use methods from distributedoptimization to solve the convex approximation of the CVEMproblem and obtain the global optimal solution which can beused to verify the performance of online solution methodsand to analyze the potential of different auxiliaries for energymanagement. In particular, we use the dual decompositionapproach for CVEM that we first introduced in [24] in com-bination with efficient algorithms to solve the dual functionsthat we first introduced in [25]. Dual decomposition has beenused in large-scale optimization since the early 1960s [26]and have since then been applied to problems with large scaledynamical systems (see, e.g., [27]–[29]).

With the dual decomposition approach [24], the optimalcontrol problem is decomposed into smaller dual problems.Each of the dual problems is then either solved explicitly, withan ADMM algorithm or solved with a Lagrangian Method[25]. Both papers showed good performance on a simplifiedCVEM problem which considered only an internal combus-tion engine, an electric motor, a high-voltage battery and arefrigerated semi-trailer while real scalability of the methodwith the number of components has not been demonstrated.Contrary to [24] and [25], we present the general CVEMproblem as a quadratically constrained linear program (QCLP),where we apply a relaxation to ensure convexity of theCVEM optimal control problem. This allows the method tobe applied to various vehicle configurations. Moreover, thedual decomposition approach is extended with a Newton dualupdate algorithm to improve convergence speed. The ADMMalgorithm is extended for multi-state dynamical systems andthe Lagrangian Method is extended to handle systems withsaturated states on the lower or upper bound. Finally, to fully

Converter

Storage

Converter

Converter

ConverterStoragex

m,kConverter Storage

Storage

Converter

ym,k

um,k

subsystem m

Converter

v,k

v,k

v,k

Fig. 1: Power net with energy storage devices and energyconverters.

demonstrate the performance of the distributed optimizationapproach, the optimal control of a hybrid truck with aninternal combustion engine, an electric motor, a high-voltagebattery, a refrigerated semi-trailer, an air supply system, analternator, a DCDC converter, a low-voltage battery and aclimate control system is solved. Moreover, the proposedalgorithm is compared with the state-of-the-art solver CPLEX[30].

The remainder of this paper is organized as follows. Thegeneral optimal control problem and the application of thedual decomposition is given in Section II. In Section III,solution methods are presented to solve the dual functions thatresulted from the dual decomposition. The CVEM problem iscasted as an optimal control problem in Section IV and, finally,simulation results are presented in Section V.

II. D ISTRIBUTED OPTIMIZATION OF POWER NETS

In this section, we consider the optimal control of theenergy flows in a power net, which is illustrated in Fig. 1.The power net consist of energy storage devices, e.g., ahigh-voltage battery, and energy converters, e.g., an electricmachine. The storage devices are connected to the converters,while the outputsym,k and inputsum,k of the converters areconnected to each other via nodes according to a specifictopology, i.e., energy can be exchanged directly between con-verters but not directly between storage devices. At each noden ∈ 1, . . . , N, there is also a known exogenous load signalvn,k given for each time instantk. Subsystems are composedof a combination of a converter, possibly with an energystorage device. The goal of the power net is to minimizethe cumulative energy losses of all subsystems, while meetingconstraints on the inputs, outputs and states in each subsystem.In this section, we will introduce the optimal control problemfor this power net and we will give a dual decompositionapproach to solve the optimal control problem. In Section IVwe will show that the Complete Vehicle Energy Management(CVEM) problem can be represented as a power net, whereminimizing the energy losses is equivalent to minimizing thefuel consumption.

A. Optimal Control Problem

The optimal control problem for the power net is given by

minum,k,ym,k

∑

m∈M

∑

k∈Kcmum,k − dmym,k, (1a)

where um,k ∈ R and ym,k ∈ R are the (scalar) inputsand outputs of the converter in subsystemm ∈ M =1, . . . , M with M the number of subsystems and at timeinstant k ∈ K = 0, 1, . . . , K − 1, with K the horizonlength. In (1a),cm ∈ R and dm ∈ R+ are coefficients thatdefine the energy losses in converterm. Moreover, we usethe notationum,k, ym,k to indicateum,k, ym,km∈M,k∈K.This notation will be used throughout the paper for minimizingover a set. The optimization problem (1a) is to be solvedsubject to a quadratic equality constraint describing the input-output behavior of each converter, i.e.,

12qm,ku2

m,k + fm,kum,k + em,k + ym,k = 0, (1b)




54 / 110


with qm,k ∈ R+, fm,k ∈ R and em,k ∈ R being efficiencycoefficients of the converterm ∈ M at time instantk ∈ K,and subject to linear system dynamics of the storage devicein subsystemm ∈ M, i.e.,

xm,k+1 = Amxm,k + Bm,wwm,k + Bm,uum,k, (1c)

for all k ∈ K, where the initial statexm,0 and final statexm,K

of the storage device are assumed to be given,wm,k is a knownload signal at every time instantk and the inputum,k ∈ R issubject to linear inequality constraints, i.e.,

um,k 6 um,k 6 um,k, (1d)

for all k ∈ K, m ∈ M, and the statexm,k is subject to linearinequality constraints, i.e.,

xm,k 6 xm,k 6 xm,k, (1e)

for all k ∈ K andm ∈ M. Finally, the optimization problemis solved subject to a linear equality constraint describing theinterconnection of the subsystems, i.e.,

∑

m∈MAmum,k + Bmym,k + 1

M vk = 0, (1f)

for all k ∈ K, whereAm ∈ RN andBm ∈ RN are vectorswith the n-th element being−1 if the power flow to noden is positive, 0 if there is no power flow to noden and1 if the power flow to noden is negative. Here,N is thenumber of nodes in the topology where power is aggregated.Furthermore, the load signalvk = [v1,k . . . vN,k]

T ∈ RN isassumed to be known at each time instantk ∈ K. We definethe primal optimal solutionas the solutionu∗

m,k, y∗m,k that

satisfies (1) andp∗ as the primal optimal value of (1). Note thatquadratic behavior (1b) is a good assumption for a wide rangeof converters as most converters have quadratic power losses,e.g., the internal combustion engine or the electric machine,which will be shown in Section IV.

B. Dual Decomposition and Convex Relaxation

The optimization problem (1) can be a large-scale problem(when K and M are large), which is not convex due to thequadratic equality constraint (1b). We propose in this paper tosolve (1) by decomposing it into several smaller problems andto relax (1b). In doing so, we can solve (1) efficiently withoutsacrificing optimality of the solution as we will show below.Problem (1a) subject to (1b) - (1f) cannot be separated due tothe complicating constraint (1f). Therefore, we decomposetheproblem via dual decomposition by introducing the followingso-calledpartial Lagrangian

L(um,k, ym,k, µk) =∑

m∈M

∑

k∈Kcmum,k − dmym,k

+ µTk (Amum,k + Bmym,k + 1

M vk), (2)

where µk ∈ RN is a Lagrange multiplier, subject to (1b)-(1e). Indeed, the partial Lagrangian is obtained by adding thecomplicating constraints (the constraints that act on morethan

one subsystem) to the objective function in (1a). The partialLagrange dual function is now given by

g(µk) = minum,k,ym,k


m∈Mgm(µk),

(3a)with

gm(µk) = minum,k,ym,k

∑



M vk), (3b)

subject to (1b)-(1e). Note that each of the Lagrange dualfunctions (3b) subject to (1b)-(1e) is related to one of the sub-systems and can be solved independently. The dual problemis given by

maxµk

g(µk) = d∗, (4)

subject to (1b)-(1e) whered∗ is defined as the dual optimalvalue. The dual problem (4) gives a lower bound on the primaloptimal valuep∗ of problem (1), i.e.,

d∗ 6 p∗. (5)

The dual optimal valued∗ will be equal to the primal optimalvalue p∗, i.e., p∗ = d∗, if problem (1) is strictly convexand the constraints satisfy Slater’s constraint qualifications[31]. Problem (1), however, is not strictly convex due to thequadratic equality constraint (1b). By relaxing this constraintto an inequality constraint, i.e.,

ym,k + 12qm,ku2

m,k + fm,kum,k + em,k 6 0, (1b’)

for m ∈ M, k ∈ K, the optimization problem becomes strictlyconvex. We define the dual optimal solution as the solutionu∗

m,k, y∗m,k, µ∗

k that satisfies (4) withg(µk) defined in (3)with the minimum taken subject to (1c)-(1e) and (1b’) insteadof (1b). The following theorem provides a condition for whichthe dual optimal solution leads to the primal optimal solution.

Theorem 1. The dual optimal solutionu∗m,k, y

∗m,k, µ∗

k tothe dual problem(4) when (1b) is replaced by (1b’), solvesthe primal optimization problem(1) with u∗

m,k, y∗m,k (and

(1b) instead of (1b’)) if

• the dual optimal solution satisfiesdm − BTmµ∗

k > 0 forall m ∈ M and for all k ∈ K,

• there exist a feasible pointum,k, ym,k for all m ∈ Mand for all k ∈ K satisfying(1c) - (1e) and (1b’) withstrict inequality.

Proof. Because the second condition in the hypothesis of thetheorem implies that Slater’s constraint qualification (see e.g.,[31]) holds and that we have strong duality, we only need toshow that the solution to (4) subject to (1c) - (1e) and (1b’)is achieved with equality ifdm − BT

mµ∗k > 0.

To show thatdm − BTmµ∗

k > 0 for all m ∈ M, k ∈ Kimplies that the dual optimal solution yields the primal optimalsolution that satisfies (1b), consider the partial Lagrangedualfunction of problem (1) subject to (1b’) instead of (1b) whichis given by




55 / 110



m∈M

∑



M vk)

+ νm,k(ym,k + 12qm,ku2

m,k + fm,kum,k + em,k), (6)

subject to (1c)-(1e) and whereνm,k > 0 is the (scalar) La-grange multiplier associated with the (scalar-valued) quadraticinequality constraint (1b’). The derivative with respect to ym,k

at u∗m,k, y∗

m,k, µ∗k of this partial Lagrangian (6) (one of the

necessary conditions for optimality, see, e.g., [31]) is given by

BTmµ∗

k − dm + ν∗m,k = 0. (7)

Sincedm − BTmµ∗

k > 0 by the hypothesis of the theorem, itfollows that ν∗

m,k > 0. The positivity of νm,k ensures thatinequality (1b’) is satisfied as an equality by complementarityslackness [31] which completes the proof.

The first condition of this theorem is in general not mild butprovides an a posteriori check for the dual optimal solutionu∗

v,k, y∗v,k, µ∗

k that satisfies (4) withg(µk) defined in (3)with the minimum taken over (1c)-(1e) and (1b’) insteadof (1b), to be equal to the primal optimal solution to (1).Moreover, for the objective function (1a) this condition isintuitively satisfied as the energy losses in each subsystemm ∈ M are defined ascmum,k−dmym,k for each time instantk ∈ K, so that the quadratic inequality constraint (1b’) at theoptimal solution implies

cmu∗m,k − dmy∗

m,k >dm

(12qm,k(u

∗m,k)

2 + fm,ku∗m,k + em,k

)

+ cmu∗m,k. (8)

for all m ∈ M and k ∈ K. If the quadratic inequalityconstraint holds with equality at the optimal solution forall m ∈ M and k ∈ K, then the energy losses in eachsubsystemm ∈ M are minimal, which is intuitively neededfor the solution to be optimal. The second condition in thehypothesis of this theorem is relatively mild (Slater’s constraintqualification, see, e.g., [31]) and can be satisfied for thenumerical example given in Section IV. The dual problem (4)can be solved efficiently using a subgradient method as willbe shown below.

C. Maximizing the Lagrange Dual Function

Maximizing the Lagrange dual function (3) overµk can bedone with a ‘steepest ascent’ method, i.e.,

µs+1k = µs

k + αsk

( ∑

m∈MAmus

m,k + Bmysm,k + 1

M vk

), (9)

for all k ∈ K where αk is a suitably chosen matrix ands ∈ N is the iteration counter. In [24], a diagonal matrixwith sufficiently small positive constant step sizes on itsdiagonal was chosen such that the Lagrange dual problemwill always converge but convergence tended to be slow. Abetter convergence rate is achieved with a Newton scheme(see e.g., [32]). We will derive this scheme intuitively from aprimal feasibility perspective. The idea is to update the dual

variablesµk such that for the next iteration primal feasibilityfor the complicating constraints holds, i.e.,

∑

m∈MAmus+1

m,k + Bmys+1m,k + 1

M vk = 0. (10)

Furthermore, we can approximate the value forus+1m,k andys+1

m,k

linearly by

us+1m,k ≈ us

m,k +

(∂us

m,k

∂µk

)T

(µs+1k − µs

k), (11a)

ys+1m,k ≈ ys

m,k +

(∂ys

m,k

∂µk

)T

(µs+1k − µs

k), (11b)

where∂us

m,k

∂µkis a vector with the approximations of the

first-order derivatives ofusm,k(µk) with respect to the dual

variablesµk at iterations. Similarly,∂ys

m,k

∂µkis a vector with the

approximations of the first-order derivatives ofysm,k(µk) with

respect to the dual variablesµk at iterations. By substituting(11) into (10) and solving forµs+1

k we obtain (9) with

αsk =

( ∑

m∈M−Am

(∂us

m,k

∂µk

)T

− Bm

(∂ys

m,k

∂µk

)T)−1

, (12)

which can be obtained by calculating the vector with deriva-tives for each subproblem in a distributed fashion. Note thatcalculating the derivatives with respect toµk in (11) canbe hard and might not even exist due to the presence ofconstraints. Instead, the derivatives can be approximatedbyneglecting the state constraints (1e). This might cause that(9) does not converge. In this case, sufficiently small constantstep sizes can be chosen as was done in [24]. Convergencespeed might be significantly slower in this case, as will beshown in the simulation study in Section V. Finally, the dualdecomposition algorithm consists of iteratively solving (3) toobtainus

m,k, ysm,k and updating the Lagrange multipliers by

solving (9) to obtainµs+1k . In the Section below, we will

provide methods to efficiently solve the dual functions (3b).

III. E VALUATING THE DUAL FUNCTIONS

Each of the Lagrange dual functions (3b) related to one ofthe subsystems can be solved separately and can be written asalinearly constrained quadratic program (LCQP) by substituting(1b) into (3b), which gives

gm(µk)= minum,k

∑

k∈K

12Hm,ku2

m,k+Fm,kum,k+Em,k, (13a)

with

Hm,k = (dm − BTmµk)qm,k, (13b)

Fm,k = cm +ATmµk + (dm − BT

mµk)fm,k (13c)

Em,k = µTk

1M vk + (dm − BT

mµk)em,k (13d)

and subject to (1c) - (1e). Note that for strict convexity of(13a), it is required thatdm − BT

mµ∗k > 0 for all k ∈ K,

m ∈ M, at every iteration of the dual decomposition algorithmwhich is a more restrictive condition than in Theorem 1 forthe dual optimal solution to be equal to the primal optimalsolution. For the simulation study presented in section IV,




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this condition is satisfied, as will be shown with the numericalexample in Section V.

As a result, the dual decomposition allows solving thequadratically constrained quadratic program by solving mul-tiple LCQPs iteratively. However, solving a LCQP for alarge horizon lengthK is still very inefficient. Therefore,we introduce two solution methods to solve the optimizationproblem (13a), related to each of the subsystems, efficiently.Both methods use the principle of splitting the horizonK intomultiple smaller intervals, where each interval is defined asKℓ = Kℓ−1, . . . , Kℓ − 1 with 0 = K0 < K1 < . . . <KL = K and whereℓ ∈ L = 1, . . . , L with L the numberof intervals. To decompose the constraints in the optimizationproblem (13a) into smaller optimization problems, we recallthat a solution to (1c) satisfies

xm,k+1 = Ak+1−Kℓ−1m xm,Kℓ−1

+∑

i∈Kℓ−1,...,kAk−i

m (Bm,wwm,i + Bm,uum,i), (14a)

for all k ∈ Kℓ, where the local initial conditionxm,Kℓ−1at

each intervalℓ is equal to the final condition at intervalℓ− 1,i.e.,

xm,Kℓ−1= xm,Kℓ−1

, (14b)

for ℓ ∈ L and xm,0 = xm,0 and xm,L = xm,K which followfrom the initial and final condition of the full horizon. Usingthese constraints, we can write the dual function (13a) as

gm= minum,k,xm,Kℓ−1

∑

ℓ∈L

∑

k∈Kℓ

12Hm,ku2

m,k+Fm,kum,k+Em,k,

(15)subject to (1d), (1e) and (14). Note that the problem (15)subject to (1d), (1e) and (14) is only coupled by (14b).

We will introduce two solution methods that can be usedto select the intervalsKℓ and the initial state at each intervalxm,Kℓ−1

. In the first solution method, based on AlternatingDirection Method of Multipliers (ADMM), the horizon is splita priori in a fixed number of intervals. For each interval,an optimization problem is solved that takes the initial statexm,Kℓ−1

as decision variable. ADMM is a suitable method asthe resulting decomposed problem is not strictly convex, yetstill convex, as we will show below. In the second solutionmethod, based on the Lagrangian Method, the horizon issplit iteratively and the initial state is fixed on the lower orupper state constraint depending on the solution of the state-unconstrained optimization problem. The Lagrangian Methodis only applicable to systems with scalar states, while theADMM method is applicable to systems with multiple states.

A. Horizon Splitting with ADMM

For this method, we define a priori the setsKℓ =Kℓ−1, . . . , Kℓ − 1, ℓ ∈ L = 1, . . . , L. This method issimilar to the method proposed in [33] where intervals thatcontain only one time instant, i.e.,Kℓ = ℓ− 1 are used forsolving the problem over a short horizon. Contrary to [33],we use intervals containing multiple time instants, therebymaking it more applicable for solving the problem over a longhorizon as will be demonstrated with the numerical example

in Section V. The objective function in (15) is separable invariables related to each interval but is not strictly convexdue to the minimization over the local initial statexm,Kℓ−1

,which is an essential assumption for the dual decompositionapproach taken in the previous section. Lagrangian methodsas used in Section II.B, however, assume convexity of theobjective function rather than strict convexity. Instead,thepartial augmented Lagrangian for problem (15) can be definedas

L(um,k, xm,Kℓ−1, νm,ℓ) =

∑

ℓ∈L

∑

k∈Kℓ

12Hm,ku2

m,k

+ Fm,kum,k + Em,k + νTm,ℓ−1(xm,Kℓ−1

− xm,Kℓ−1)

+ 12

(xm,Kℓ−1

− xm,Kℓ−1

)TR(xm,Kℓ−1

− xm,Kℓ−1

), (16)

in which νm,ℓ ∈ Rdim(xm,k) are Lagrange multipliers andwhereR ≻ 0 is a diagonal matrix with penalty parameterson its diagonal. In this expression, we temporary omit theconstraints that are acting only within one interval, i.e.,(1d),(1e) and (14a) and will reintroduce them later in the decom-posed problem. The partial augmented Lagrange dual functionis given by

gm(νm,ℓ) = minum,k,xm,Kℓ−1

L(um,k, xm,Kℓ−1

, νm,ℓ)

=∑

ℓ∈Lgm,ℓ(νm,ℓ−1, νm,ℓ), (17a)

with

gm,ℓ(νm,ℓ−1, νm,ℓ) = minum,k,xm,Kℓ−1

∑

k∈Kℓ

12Hm,ku2

m,k

+ 12 xT

m,Kℓ−1Rxm,Kℓ−1

+Gxm,Kℓ−1+Fum,k+E, (17b)

in which

G = νTm,ℓ−1 − xT

m,Kℓ−1R − νT

m,ℓAKℓ−Kℓ−1m , (17c)

F = Fm,k − νTm,ℓA

Kℓ−1−km Bm,u, (17d)

E = Em,k − νTm,ℓA

Kℓ−1−km Bm,wwm,k, (17e)

for ℓ ∈ L, with νm,L = 0 and is to be solved subject to(1d), (1e) and (14a). Expressions (17c,d,e) are obtained bysubstituting (14a) fork = Kℓ−1 into (16), only for the linearpart of the equation. This gives a more desirable expressionfor(17c), i.e., the Hessian matrix is diagonal. However, as a result,(17c) is not separable due to the termxm,Kℓ−1

in (17c). Byminimizing (17) sequentially from intervalℓ = 1 to intervalℓ = L, as part of the ADMM algorithm (see, e.g., [31]), theminimization problem (17), can still be solved efficiently.

To maximize the partial augmented Lagrange dual function,we use a ‘steepest ascent’ method, i.e.,

νt+1m,ℓ−1 = νt

m,ℓ−1 + R(xtm,Kℓ−1

− xtm,Kℓ−1

) (18)

with t ∈ N the iteration number and for some given initialcondition ν0

m,ℓ−1 for ℓ ∈ L. Finally, the ADMM algorithmconsists of iteratively solving (17) subject to (1c) - (1e) toobtainut

m,k, xtm,Kℓ−1

for ℓ ∈ L, k ∈ K and solving (14a)for k = Kℓ−1 to obtainxt

m,Kℓfor ℓ ∈ L, k ∈ K, followed by

an update of the Lagrange multipliers through (18) to obtainνt+1

m,ℓ−1 for ℓ ∈ L.




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B. Horizon Splitting with the Lagrangian Method

Fixing the intervalKℓ a priori and use ADMM to solve(15) results in a general solution method to solve the LCQP(13). We will also develop an iterative procedure that involvessplitting the intervals based on solving (15) for the particularcase wherexm,k ∈ R, i.e., the energy storage device insubsystemm is a scalar-state system. In Section IV, it willbe shown that many components in the CVEM problem canbe represented by a scalar-state system and in Section V wewill show that a tailored solution method for these componentsis more favorable with respect to computation time, whichalso emphasizes the advantage of using the dual decompositionapproach to CVEM where each of the dual functions can besolved with the most suistable solution method.

For this method, we initially take only one interval, i.e. thefull horizon, so thatKℓ = Kℓ−1, . . . , Kℓ−1 = 0, . . . , K−1 and ℓ ∈ L = 1 and solve (13a) subject to (1d) and(14) without considering the state constraints (1e). The mainreason for this is that the problem without (1e) is much easierto solve. Depending on the solution of a state-unconstrainedoptimization, extra intervals will be added as will be shownlater in this section.

First, we define the state unconstrained problem for subsys-tem m, which is given by (15) subject to (1d) and (14). TheLagrangian of this problem is given by

L(um,k, λℓ, νk, νk) =∑

ℓ∈L

∑

k∈Kℓ

12Hm,ku2

m,k + Fm,kum,k

+ Em,k + νk(um,k − um,k) + νk(um,k − um,k)

+ λℓ

(AKℓ−Kℓ−1

m xm,Kℓ−1− xm,Kℓ

+∑

i∈Kℓ

AKℓ−1−im (Bm,wwm,i + Bm,uum,i)

), (19)

with λℓ ∈ R, the Lagrange multiplier associated with theconstraint (14),νk ∈ R andνk ∈ R, the Lagrange multipliersassociated with the upper and lower input constraints (1d),respectively. The Karush-Kuhn-Tucker conditions [31] forminimizing the Lagrangian in (19) are given by the first-ordernecessary optimality condition

∂L(um,k,λℓ,νk,νk)∂um,k

= Hm,kum,k+Fm,k+νk−νk

+ λℓ

∑

i∈Kℓ

AKℓ−1−im Bm,u= 0, (20)

for all k ∈ Kℓ, ℓ ∈ L, feasibility of the constraint (14)and the complementary slackness conditions for the inequalityconstraints

νk(um,k − um,k) = 0, νk(um,k − um,k) = 0, (21)

for all k ∈ Kℓ, ℓ ∈ L with νk > 0 and νk > 0. Finding asolution for (20) and (21) simultaneously is difficult and oftenthe solution is found with a shooting method and a bisectionalgorithm overλℓ, leading to the optimal solution

u∗m,k=−H−1

m,k(Fm,k+λℓ

∑

i∈Kℓ

AKℓ−1−im Bm,u+νk−νk), (22)

for all k ∈ Kℓ, ℓ ∈ L for a givenλℓ, νk and νk. Instead,we propose a procedure that aims, for each intervalℓ ∈ L, at

solving

λt+1ℓ = (1− γ)λt

ℓ + γH−1

(AKℓ−Kℓ−1


+

∑

i∈Kℓ

AKℓ−1−im

(Bm,wwm,i−Bm,uH−1

m,i(Fm,i+νti−νt

i)))

,

(23a)

with relaxation parameterγ ∈ (0, 1] and with

H=∑

i∈Kℓ

AKℓ−1−im Bm,uH−1

m,i

∑

i∈Kℓ

AKℓ−1−im Bm,u, (23b)

and

νt+1k =max

0,−Hm,kum,k−Fm,k−λt+1

ℓ

∑

i∈Kℓ

AKℓ−1−im Bm,u

,

(23c)

νt+1k =max

0, Hm,kum,k+Fm,k+λt+1

ℓ

∑

i∈Kℓ

AKℓ−1−im Bm,u

,

(23d)

with t ∈ N the iteration index and forλ0 = 0 andν0k = ν0

k =0 for all k ∈ K, until (14) is satisfied within some desiredtolerance. The expressions in (23) are obtained by substituting(22) into (14) and (1d). IfHm,k is strictly positive and if thereexists an optimal solutionu∗

m,k for which (1d) and (14) aresatisfied, then the solution of (23) will converge to the solutionof (15) subject to (1d) and (14) for a well chosen relaxationparameterγ ∈ (0, 1].

For solving the state-constrained optimization problem (13),we use an idea proposed in [10]. In [10] it is proven thatwhen the relation betweenλℓ and the final statexm,Kℓ

ismonotonic (note that it is linear in this paper), the time instantat which the state constrained is violated mostKℓ is a contactpoint of the state-constrained solution, i.e.,xm,Kℓ

= xm,k or

xm,Kℓ= xm,k. This allows us to addKℓ to the set of splitting

instancesKℓℓ∈L and we fixxm,kℓat eitherxm,kℓ

or xm,kℓ

depending on whether the upper or lower bound was violatedmost and solve smaller optimal control problems subject toinitial and terminal constraints. Note that due to the particularprojection on the state constraints, this method is only suitablefor scalar-state systems, i.e.,xm,k ∈ R.

As a result of splitting the time horizonK into segmentsusing the method outlined before, we might have a largenumber of ‘contact points’, which makes the method ineffi-cient. We can prove that the optimal state trajectory equalseither xm,k = xm,k or xm,k = xm,k for all k ∈ Kℓ undercertain conditions, i.e., it is saturated on the upper or lowerbound. This might occur when eitherxm,Kℓ−1

= xm,Kℓ−1and

xm,Kℓ= xm,Kℓ

or xm,Kℓ−1= xm,Kℓ−1

andxm,Kℓ= xm,Kℓ

.The following theorem provides conditions, that can be evalu-ateda priori, for which the optimal state trajectory is saturatedon the upper or lower bound.

Theorem 2. The optimal state trajectoryxm,k satisfiesxm,k = xm,k for all k ∈ Kℓ and the corresponding controlinput satisfies

u∗m,k = 1

Bm,u

((1− Am)xm,k + Bm,wwm,k

)(24a)




58 / 110


if xm,Kℓ−1= xm,Kℓ−1

andxm,Kℓ= xm,Kℓ

, and if either oneof the following conditions holds for allk ∈ Kℓ:

• Am > 0, Bm,u < 0, Hm,k+1u∗m,k+1 − Hm,ku∗

m,k < 0,Fm,k+1 − Fm,k < 0

• Am > 0, Bm,u > 0, Hm,k+1u∗m,k+1 − Hm,ku∗

m,k > 0,Fm,k+1 − Fm,k > 0.

Similarly, the optimal state trajectoryxm,k satisfiesxm,k =xm,k for all k ∈ Kℓ and the corresponding control inputsatisfies

u∗m,k = 1

Bm,u

((1− Am)xm,k + Bm,wwm,k

)(24b)

if xm,Kℓ−1= xm,Kℓ−1

andxm,Kℓ= xm,Kℓ

, and if either oneof the following conditions holds for allk ∈ Kℓ:

• Am > 0, Bm,u < 0, Hm,k+1u∗m,k+1 − Hm,ku∗

m,k >0,Fm,k+1 − Fm,k > 0

• Am > 0, Bm,u > 0, Hm,k+1u∗m,k+1 − Hm,ku∗

m,k <0,Fm,k+1 − Fm,k < 0.

Proof. We only prove the case thatxm,k = xm,k for all k ∈Kℓ and the proof forxm,k = xm,k follows mutatis mutandis.

For u∗m,k to be a feasible solution it needs to satisfy (1d)

by definition, such that the Lagrangian of (15) subject to (14)and (1e) on intervalℓ is given by

L(u∗m,k, λℓ, υk) =

∑

k∈Kℓ

12Hm,k(u

∗m,k)

2 + Fm,ku∗m,k + Em,k

+ υk(xm,k − xm,k) + λℓ

(AKℓ−Kℓ−1


+∑

i∈Kℓ

AKℓ−1−im (Bm,wwm,i + Bm,uum,i)

), (25)

wherexm,k is given by (14a) for allk ∈ Kℓ. In (25), υk ∈ Ris the Lagrange multiplier associated with the lower stateconstraint (1e). We will show that the lower state is activefor all k ∈ Kℓ which means that the upper state is inactiveand can be left out of the Lagrangian. The Karush-Kuhn-Tucker conditions [31] for minimizing the Lagrangian in (25)are given by the first-order necessary optimality condition

∂L(u∗m,k,λℓ,υk)∂u∗

m,k= Hm,ku∗

m,k + Fm,k

+ λℓ

∑

i∈Kℓ

AKℓ−1−im Bm,u − υk

k∑

i=Kℓ−1

Ak−im Bm,u= 0, (26)

for all k ∈ Kℓ, feasibility of the constraint (14) and the com-plementary slackness conditions for the inequality constraint

υk(xm,k − xm,k) = 0, (27)

for all k ∈ Kℓ, for a given υk > 0. Asλℓ

∑i∈Kℓ

AKℓ−1−im Bm,u is a constant in (26), we can derive

the following relation between the Lagrange multipliersυk attime instantk + 1 andk

υk+1 =

k+1∑

i=Kℓ−1

Ak−im Bm,u

−1 (υk

k∑

i=Kℓ−1

Ak−im Bm,u

+ Hm,k+1u∗m,k+1−Hm,ku

∗m,k + Fm,k+1 − Fm,k

), (28)

for k ∈ Kℓ and whereυKℓ−1> 0 if xm,Kℓ−1

= xm,Kℓ−1, such

that if Am > 0, Bm,u < 0, Hm,k+1u∗m,k+1 − Hm,ku∗

m,k < 0

and Fm,k+1 − Fm,k < 0 or if Am > 0, Bm,u > 0,Hm,k+1u

∗m,k+1 − Hm,ku∗

m,k > 0 and Fm,k+1 − Fm,k > 0then there exist aυk+1 > 0 for all k ∈ Kℓ, such that the first-order optimality conditions and the complementary slacknessconditions are satisfied andu∗

m,k is optimal for all k ∈ Kℓ.This completes the proof forxm,k = xm,k.

This theorem provides a priori verifiable conditions whenthe optimal state trajectory is saturated at the lower boundorupper bound for allk ∈ Kℓ−1, . . . , Kℓ − 1, respectively.The three preceding results, i.e., i) the solution of the optimalcontrol problem without considering the state constraints, ii)the iterative method for splitting the control problem intosmaller ones to incorporate state constraints and iii) conditionsfor which the optimal solution satisfiesxm,k = xm,k orxm,k = xm,k for all k ∈ Kℓ allow us to propose the followingalgorithm for solving (13).

Algorithm 1. TakeK1 = 0, . . . , K − 1, L = 1 and letxm,0 and xm,K be given.

• For each interval ℓ ∈ L, check if the conditions ofTheorem 2 are satisfied.

– If the conditions of Theorem 2 are satisfied, theoptimal solution satisfies (24a) or (24b).

– If the conditions of Theorem 2 are not satisfied,compute the input constrained solution using(23).Then verify

εℓ = maxk∈Kℓ

xm,k − xm,k, (29a)

εℓ = maxk∈Kℓ

xm,k − xm,k. (29b)

∗ If εℓ > 0 and εℓ > εℓ, the lower state constraintis violated more than the upper state constraintand

Kℓ = arg maxk∈Kℓ

xm,k − xm,k, (29c)

is added to the set of contact pointsKℓℓ∈Land re-ordered, i.e.,0 = K0 6 . . . 6 Kℓ−1 6Kℓ 6 Kℓ 6 KL to define new subsetsKℓ =Kℓ−1, . . . , Kℓ − 1 and xm,Kℓ

= xm,Kℓ.

∗ If εℓ > 0 and εℓ > εℓ, the upper state constraintis violated more than the lower state constraintand

Kℓ = arg maxk∈Kℓ

xm,k − xm,k, (29d)

is added to the set of contact pointsKℓℓ∈Land re-ordered, i.e.,0 = K0 6 . . . 6 Kℓ−1 6Kℓ 6 Kℓ 6 KL to define new subsetsKℓ =Kℓ−1, . . . , Kℓ − 1 and xm,Kℓ

= xm,Kℓ.

∗ If both (29b) and (29a) are nonpositive, theℓ-thinterval does not have to be further divided.

• Repeat untilmaxxm,k−xm,k, xm,k−xm,k 6 0 for allk ∈ K.

Similarly as the dual decomposition allows the large-scaleoptimal control problem to be solved by solving smalleroptimal control problems on subsystem level, Algorithm 1and the ADMM algorithm allow the optimal control problemover a large horizon to be solved through multiple optimal




59 / 110


ICE

Gearbox

Electric

machine

HV Battery

Refrigerated

semi-trailer

xice

DC/DCAir supply

LV Battery

Alternator

CCS xccs

xlvb

xhvb

xrst

Brakes xbr

yice

uice

ygb

ybr

uem

yem

yhvb

yrst

uhvb

urst

yas

udc

ua

yccs

ylvb

yay

dc ulvb

xas

uas

uccs

ugb

ubr

v,k

v,k

v,k

Fig. 2: Vehicle topology where the arrows indicate the direc-tion of a positive power flow.

control problems over a smaller horizon. Note that, to ensureconvergence of the solution to the dual problem (4), thesolution to each dual function obtained with Algorithm 1 orADMM (18) needs to be converged before proceeding withthe maximization in (9). Still, by combining these solutionmethods, scalability is significantly improved, which willbedemonstrated on the complete vehicle energy managementproblem that we will introduce in the next section.

IV. A PPLICATION TO A VEHICLE POWER NET

The distributed optimization approach presented in theprevious sections will be used to find the optimal solution forenergy management of a vehicle with an internal combustionengine (ICE), an electric machine, a high-voltage battery,arefrigerated semi-trailer, an air supply system, an alternator, aDCDC converter, a low-voltage battery and a climate controlsystem (CCS). The topology is shown in Fig. 2. This topologyhas three exogenous load signals, i.e.,vk = [v1,k v2,k v3,k]

T ∈R3, which are the power required to drive a certain drivecycle, the power required for uncontrolled high-voltage aux-iliaries and the power required for uncontrolled low-voltageauxiliaries, respectively. These three signals are assumed to beknown for every time instantk ∈ K. Furthermore, we assumethat the gearshift strategy is fixed such that the rotationalvelocity of the drive lineωk is known for every time instantk ∈ K. We also assume that the power losses in the gearboxare negligible, i.e.,ygb,k = ugb,k, such that nodes connectingthe subsystems are given by

v1,k − ybr,k − yice,k + uem,k + ualt,k − yccs,k = 0, (30a)

v2,k − yem,k − yhvb,k − yrst,k − yas,k + udc,k = 0, (30b)

v3,k − yalt,k − ylvb,k − ydc,k = 0, (30c)

which we can write in the form of (1f) with

Aice = [0 0 0 ]T , Bice = [−1 0 0 ]T , (31a)

Aem = [1 0 0 ]T , Bem = [ 0 −1 0 ]T , (31b)

Aalt = [1 0 0 ]T , Balt = [ 0 0 −1]T , (31c)

Adc = [0 1 0 ]T , Bdc = [ 0 0 −1]T , (31d)

Abr = [0 0 0 ]T , Bbr = [−1 0 0 ]T , (31e)

Ahvb = [0 0 0 ]T , Bhvb = [ 0 −1 0 ]T , (31f)

Alvb = [0 0 0 ]T , Blvb = [ 0 0 −1]T , (31g)

Arst = [0 0 0 ]T , Brst = [ 0 −1 0 ]T , (31h)

Aas = [0 0 0 ]T , Bas = [ 0 −1 0 ]T , (31i)

Accs = [0 0 0 ]T , Bccs = [−1 0 0 ]T . (31j)

Note that we have replaced the setM = 1, . . . , Min the general optimization problem (1) by the setM = ice, em, hvb, rst, as, ccs, dc, lvb, alt, br, to betterindicate the physical origin of the power flows.

A. Objective Function

The objective in energy management is to minimize the fuelconsumption, which is equivalent to minimizing the equivalentfuel energy, i.e.,

minuice,k

∑

k∈Kτuice,k (32)

where the equivalent fuel power is given byuice,k = H0mf,k

with H0 the lower heating value of the fuel andmf,k the fuelconsumption rate at time instantk. This objective function isonly defined in variables related to the internal combustionengine. However, we can obtain an objective function that isdefined in variables related to every subsystem in the network,i.e., in the form of (1a), which is equivalent to (32) for a spe-cific cm anddm. In particular, this holds if we choosecm anddm for every subsystem such that

∑k∈K cmum,k − dmym,k

represents the energy losses in the subsystem. The energylosses in each converter are given by the energy flowing intothe converter minus the energy flowing out of the converterwhere the energy flowing into the converter is indicated withthe arrow in Fig. 2. According to this topology, the energylosses in the internal combustion engine, the electric motor, thealternator, the DCDC converter, the high-voltage battery andthe low-voltage battery are given by the difference betweenthe input and output power, i.e.,

cm = τ, dm = τ, (33a)

for m ∈ ice, em, alt, dc, hvb, lvb. For the refrigerated semi-trailer, the air supply system, the climate control system andthe mechanical brakes, all energy flowing into the subsystemis eventually lost and therefore the energy losses are givenfor

cm = 0, dm = τ, (33b)

for m ∈ rst, as, ccs, br. By substituting (33) into (1a), weobtain

minum,k,ym,k

∑

m∈M

∑

k∈Kcmum,k − dmym,k =

minum,k,ym,k

τ(uice,k − yice,k + uem,k+ualt,k−yccs,k−ybr,k

−yem,k−yhvb,k−yrst,k−yas,k+udc,k−yalt,k−ylvb,k−ydc,k

+uhvb,k+ulvb,k

). (34a)

By using the power balance constraints (30), we can reducethis equation to

minuice,k,uhvb,k,ulvb,k

∑

k∈Kτuice,k − τ (v1,k + v2,k + v3,k)

+ τ(uhvb,k + ulvb,k). (34b)

Moreover, as will be explained in more detail in the next sub-section, the high-voltage battery and low-voltage batterysat-isfy integrator dynamics so that we can write

∑k∈K τum,k =




60 / 110


xm,0 − xm,K for m ∈ hvb, lvb and further reduce (34b) to

minuice,k

∑

k∈Kτuice,k − τ (v1,k + v2,k + v3,k)

+ xhvb,0 − xhvb,K + xlvb,0 − xlvb,K . (34c)

As the load signals are known for allk ∈ K and the initialand final statesxm,0 andxm,K for m ∈ hvb, lvb are fixed,the optimal value foruice,k for (34c) is equivalent to theoptimal valueuice,k for (32) for all k ∈ K. Hence, the optimalcontrol problem (1) with objective function (1a) andcm anddm given by (33) provides the optimal solution for which thefuel consumption is minimized over allk ∈ K as is indicatedin (32).

B. Subsystem Modeling

The models for the internal combustion engine, the electricmachine, the alternator, the DCDC converter, the mechanicalbrakes the high-voltage battery, the low-voltage battery,therefrigerated semi-trailer, the air supply system and the climatecontrol system will be presented below.

1) Internal combustion engine, electric machine and alter-nator: The input-output behavior of the internal combustionengine, the electric machine and alternator can be describedby the quadratic function (1b) form ∈ ice, em, alt wherethe efficiency coefficients depend on speed, i.e.,

qm,k = qm(ωk), fm,k = fm(ωk), em,k = em(ωk), (35)

for m ∈ ice, em, alt whereqm(ωk), fm(ωk) andem(ωk) arefunctions parameterizing the efficiency coefficients as functionof drive line speedωk. The input power of the converters arebounded by (1d) form ∈ ice, em, alt where the upper andlower bound depend on speed, i.e.,

um,k = um(ωk), um,k = um(ωk). (36)

for m ∈ ice, em, alt whereum(ωk) and um(ωk) are func-tions parameterizing the lower and upper bound as functionof drive line speedωk. The input-output behavior is shown inFig. 3 where the power losses, i.e.,um,k − ym,k are given forthe internal combustion engine, electric machine and alternatorfor two different speeds. In this figure, we show measuredefficiencies of typical components in a truck and the accuracyof the quadratic approximations. We normalized the powerlosses to avoid sharing confidential information. Still, thefigure shows that the quadratic behavior, the lower and upperbound strongly depend on the driveline speedωk. Furthermore,the quadratic assumption on the behavior of converters holdswell as the models are close to the measurement data. To beprecise, the average root mean square error over all drive linespeeds is 2.32 kW, 0.18 kW and 0.05 kW for the internalcombustion engine, the electric machine and the alternator,respectively. The internal combustion engine, the electric ma-chine and the alternator are subsystems without a (constrained)energy storage. Therefore, constraints on the system dynamics(1c) and state constraints (1e) do not have to be taken intoaccount in these subsystems.

Po

wer

lo

sses

[-]

y ice

uic

e-

Input power [kW]uice

Po

wer

lo

sses

[-]

y em

uem

-

Input power uem

Po

wer

lo

sses

[-]

y alt

ualt-

Input power [kW]ualt

0 200 400 600 8000

0.5

1

1000 rpm 2000 rpm Data

−150 −100 −50 0 50 100 1500

0.5

1

[kW]

0 1 2 3 4 5 6 70

0.5

1

Fig. 3: Quadratic approximation of input-output behavior forthe internal combustion engine, electric machine and alterna-tor.

2) DCDC converter and mechanical brakes:The input-output behavior of the DCDC converter and mechanical brakescan be described by the quadratic function (1b) form ∈dc, br with efficiency coefficientsqm,k ∈ R+, fm,k ∈ Rand em,k ∈ R for m ∈ dc, br which do not depend onspeed. The input power is bounded by (1d) form ∈ dc, br.The DCDC converter and the mechanical brakes are alsosubsystems without a (constrained) energy storage. Therefore,constraints on the system dynamics (1c) and state constraints(1e) do not have to be taken into account in these subsystems.

3) High- and low-voltage battery:The models of the high-and low-voltage battery are derived from a battery equivalentcircuit model, i.e., an open circuit voltageUm,oc for m ∈hvb, lvb in series with a resistanceRm,i for m ∈ hvb, lvb(see, e.g., [2]), which lead to an input-output behavior ofthe converter that can be described by the quadratic function(1b) for m ∈ hvb, lvb with qm,k =

Rm,i

U2m,oc

, fm,k = −1

and em,k = 0 for m ∈ hvb, lvb. The input power ofthe high-voltage and low-voltage battery are bounded by (1d)for m ∈ hvb, lvb. The dynamics are given by (1c) form ∈ hvb, lvb with Am = 1, Bm,w = 0 and Bm,u = −τwith τ being the sample time. Here, the statexm,k representsthe energy in the battery at time instantk.

4) Refrigerated semi-trailer:The input-output behavior ofthe converter in the refrigerated semi-trailer can be describedby the quadratic function (1b) form ∈ rst and the inputpower of the converter is bounded by (1d) form ∈ rst.The dynamics of the refrigerated semi-trailer are assumed tosatisfy a thermal energy balance (see, e.g., [34]) given by

CrstddtTrst =

(urst − h(Trst − αTamb)

), (37)

whereCrst is the heat capacity of the refrigerated semi-trailerand its contents,urst is the thermal power where negative




61 / 110


values indicate cooling,h is the heat transfer coefficientbetween the refrigerated semi-trailer and the environment, αis an insulation coefficient,Trst is the temperature inside therefrigerated semi-trailer andTamb is the ambient temperature(which is assumed to be constant). Similar to the battery, wecan represent the refrigerated semi-trailer model in termsofstored energy by defining the thermal energy relative to theambient temperaturexrst = Crst(αTamb − Trst). By doing soand by making a forward Euler approximation of (37), thedynamics can be represented by (1c) form ∈ rst withArst = 1− τh

Crst, Brst,w = 0 andBrst,u = −τ for all k ∈ K.

5) Air supply system:The input-output behavior of theconverter in the air supply system can be described by thequadratic function (1b) form ∈ as and the input power ofthe converter is bounded by (1d) form ∈ as. The dynamicsof the air supply system are assumed to satisfy a mass energybalance (see, e.g., [22]) given by

V ddtPas = R(Tinmin − Toutmout), (38)

whereR is the specific gas constant for air,V is the lumpedvolume of the air vessels,min is the mass flow into the airvessels with air temperatureTin and mout is the mass flowout of the air vessels with air temperatureTout. Similar tothe battery, we can represent the air supply system modelin terms of stored energy by defining the pneumatic energyrelative to the ambient pressurexas = (Pas−Pamb)V

γ−1 whereγ = cp/cv is the ratio of specific heats (approximately 1.4for air). Furthermore, we define the pneumatic power byuas =

RTinmin

γ−1 . By doing so and by making a forward Eulerapproximation of (38), the dynamics can be represented by(1c) for m ∈ as with Aas = 1, Bas,w = −τ , Bas,u = τ andwrst,k is the power, i.e,RToutmout

γ−1 released to the environmentat time instantk.

6) Climate control system:The input-output behavior of theconverter in the climate control system can be described bythe quadratic function (1b) form ∈ ccs where the efficiencycoefficients are speed dependent, i.e.,

qccs,k = qccs(ωk), fccs,k = fccs(ωk), eccs,k = eccs(ωk),(39)

whereqccs(ωk), fccs(ωk) andeccs(ωk) are functions parame-terizing the efficiency coefficients as function of the drivelinespeedωk. The input power of the climate control system isbounded by (1d) form ∈ ccs where the bounds depend onspeed, i.e.,

uccs,k = uccs(ωk), uccs,k = uccs(ωk), (40)

whereuccs(ωk) anduccs(ωk) are functions parameterizing thelower and upper bound as function of drive line speedωk. Thedynamics of the climate control system are assumed to satisfya coupled thermal energy balance (see, e.g., [35]) given by

CrddtTr = hi(Tw − Tr) + Qc, (41a)

CwddtTw = Ql + ho(Tamb − Tw) + hi(Tr − Tw), (41b)

where Cr and Cw are the heat capacities of the refrigerantand walls of the evaporator, respectively,Tr and Tw are thetemperatures of the refrigerant and walls of the evaporator,

respectively,Qc is the cooling power from the compressor,Tamb is the ambient temperature,hi and ho are the heattransfer coefficients between the inner and outer walls ofthe evaporator, respectively andQl is the latent heat. Similarto the battery, we can represent the climate control systemmodel in terms of stored energy by defining the thermalenergy in the wall and refrigerant relative to the ambienttemperature, i.e,xccs = [Cr(Tamb − Tr), Cw(Tamb − Tw)]

T .By doing so and by making a forward Euler approximation of(41), the dynamics can be represented by (1c) form ∈ ccs

with Accs =

[1− τhi

Cr

τhi

Cwτhi

Cr1− τ(hi+ho)

Cw

], Bccs,w =

[0−τ

],

Bccs,u =

[−τ0

]and wrst,k is the latent heatQl at time

instantk.The CVEM problem for a vehicle with an internal com-

bustion engine, an electric machine, a high-voltage battery, arefrigerated semi-trailer, an air supply system, an alternator, aDCDC converter, a low-voltage battery and a climate controlsystem is now fully described by the optimal control problemdefined in Section II. In particular, the topology of the vehicleis described through (1f) for a givenAm andBm, the objectivefunction, i.e., minimizing fuel consumption, is describedby(1a) by choosingcm and dm appropriately and the behaviorof each subsystem is fully described by (1b) - (1e) by choosingthe efficiency coefficientsqm,k, fm,k andem,k, the state-spacematricesAm, Bm,u, Bm,w describing the dynamics of thesubsystems, the upper and lower bound on the inputsum,k

and um,k, respectively, and the upper and lower bounds onthe statesxm,k andxm,k, respectively. This allows the CVEMproblem to be solved with the solution methods proposed inSection II and Section III as will be demonstrated in the nextsection.

V. SIMULATION RESULTS

In this section, we will demonstrate the distributed opti-mization approach to complete vehicle energy management(CVEM) by using a simulation study. First, we will givethe exogenous signals that we used for the simulation studyfollowed by the results that will be discussed in three subsec-tions. In the first subsection, we will analyze the computationalperformance and comparing it with the state-of-the art solverCPLEX [30]. In the second subsection, we will discuss theoptimal power flows and state trajectories and, in the lastsubsection, the fuel consumption reduction for CVEM willbe discussed.

A. Exogenous Inputs

The driving cycle is commonly described by a velocity pro-file over time. If the gear shift strategy is assumed to be known,the velocity profile can be converted to a power required atthe wheels and the engine speed. This set of data is derivedfor a PAN European driving cycle and shown in Fig. 4. It canbe seen that the brake power can reach -1000 kW. However,only a small part of the total braking power can be recoveredby the subsystems. Therefore, the power request used as loadsignalv1,k is limited to the maximum braking power that can




62 / 110


0 1 2 3 4 5−1000

−500

0

500

0 1 2 3 4 50

1000

2000

3000

Original Limited

Time [s] x 104

Power request

[ kW]

Engine speed

[rpm]

v

Fig. 4: PAN European driving cycle.

be recovered with all subsystems combined. Furthermore, theuncontrolled high-voltage auxiliaries are assumed to be absentsuch thatv2,k = 0 kW for all k ∈ K and the power requiredfrom uncontrolled low-voltage auxiliaries is assumed to beconstant, i.e.,v3,k = 1.5 kW for all k ∈ K. A quasi-staticapproach is generally sufficient for energy management (see,e.g., [3]) such that the sample time is chosen to be 1 second,i.e., τ = 1, which is smaller than the time constants of thedynamics in the subsystems.

B. Computational Performance

The local optimization problem related to each componentdefined in (13a) can be solved via different solution methodsintroduced in Section III. In particular, the performance of theADMM solution method depends on the penalty parameterRin (18) and the interval lengthKℓ − Kℓ−1 for ℓ ∈ L. Forsimplicity we assume that the interval length is equal for allintervalsℓ ∈ L. The maximum, average and minimum time tocompute the solution of the local optimization problem (13a)are given in Table I for the high-voltage battery (HVB), therefrigerated semi-trailer (RST), the air supply system (AS) andthe climate control system (CCS) for different values ofKℓ −Kℓ−1 and two horizon lengthsK. The value of the penaltyparameterR in (18) is manually tuned for eachKℓ − Kℓ−1

but is kept constant for different lengthsK of the drive cycle.The computation time required to solve the optimization

problem strongly depends on the amount of iterationst,as in (18), required for the ADMM method to converge,which depends on the initial guess of the dual variables.We use the dual variables from the previous iterationsin the dual decomposition, see (9), as an initial guess andtherefore the amount of ADMM iterations reduces as the dualdecomposition converges. As a consequence, the maximumtime is significantly larger than the minimum time. The mainconclusion drawn from this table is that the optimalKℓ−Kℓ−1

differs per component and it should be chosen neither toosmall nor too large. Moreover, it does not seem to dependon K. For example, the high-voltage battery has the highestperformance forKℓ−Kℓ−1 = 200 which is the optimal trade-off between the size and number of QPs.

With the results of Table I, we chooseKℓ−Kℓ−1 = 200 forthe high-voltage battery,Kℓ −Kℓ−1 = 50 for the refrigeratedsemi-trailer,Kℓ − Kℓ−1 = 200 for the air supply system

TABLE I: Computation time in seconds for ADMM methodwith different interval lengths

K1000 5000

Kℓ −Kℓ−1 min av max min av max

HVB

25 0.09 1.40 4.38 2.6 12.3 23.750 0.08 0.69 2.04 1.7 6.15 11.7100 0.14 0.69 1.97 2.46 6.62 11.4200 0.16 0.41 0.99 0.93 3.94 7.6500 0.71 1.73 5.0 4.64 28.0 14.5

RST

25 0.09 1.16 3.7 0.45 6.3 19.850 0.09 0.72 1.94 0.46 4.3 18.3100 0.17 0.71 1.54 0.93 5.79 24.3

AS

25 0.09 1.04 8.7 0.42 10.34 46.850 0.09 1.07 8.4 0.40 5.35 41.8100 0.15 0.6 7.5 0.74 5.81 62.9200 0.18 0.44 4.5 0.85 3.78 36.8500 0.99 1.78 5.8 4.5 11.18 87.5

CCS

25 0.1 0.32 1.8 0.49 2.37 20.4550 0.1 0.28 1.54 0.5 1.85 12.7100 0.12 0.33 1.91 0.61 1.88 12.2200 0.14 0.39 2.35 0.7 2.13 13.7

and Kℓ − Kℓ−1 = 50 for the climate control system. Theseresults are compared with other solution methods in Table II.This table shows the average computation time to solve thelocal optimization problem (13a) over all iterationss in thedual decomposition. Here, QP indicates the computation timefor solving the local optimization problem (13a) with theQP solver CPLEX [30] directly, ADMM corresponds withsection III.A and LM corresponds with the Lagrangian Methodintroduced in Section III.B. This table shows that ADMMoffers a large improvement compared to QP, especially forlarge horizonsK. The LM method (with relaxation parameterγ = 1) reduces the computation time even further and dependson the number of intervalsL required for the LM method. ForK = 3000, the amount of intervals are 2, 6 and 53 for thehigh-voltage battery, the air supply system and the refrigeratedsemi-trailer, respectively. Note that this method cannot be usedfor the climate control system as this method is only suitablefor scalar-state systems, i.e.,xm,k ∈ R. Due to the absenceof state constraints, the optimization problem for the internalcombustion engine, electric machine, alternator, DCDC con-verter and mechanical brakes can be solved explicitly and arenot shown in the table. Since the dynamics of the low-voltagebattery are similar to the dynamics of the high-voltage battery,we conjecture that LM is also best for the low-voltage battery.

To assess the computational performance of solving theenergy management problem for different vehicle configura-tions, we define six case studies with increasing complexity.These case studies are introduced in Table III. To demonstratethat the conditions in Theorem 1 hold, we show in Table IVthe minimum value of the dual variablesmink∈K(µ

(i)k ), over

all iterations in the dual decomposition algorithm where(i)denotes the i-th element of the vectorµk ∈ R3. Note that withBm = −1 for all m ∈ M and dm = τ for all m ∈ M, thecondition in Theorem 1 is satisfied if and only ifµ

(i)k > −τ for

all k ∈ K andi ∈ 1, 2, 3, which is always satisfied as shown




63 / 110


TABLE II: Average computation time in seconds per compo-nent, per dual decomposition iteration

K1000 2000 3000 4000 5000

HVBQP 2.81 23.0 93.0 253.8 527.6

ADMM 0.41 1.22 2.37 3.32 3.94LM 2.5e-04 0.066 0.063 0.07 0.09

RSTQP 3.93 29.4 118.4 291.1 562.3

ADMM 0.72 1.38 1.94 2.60 4.3LM 0.045 0.21 0.38 0.74 1.29

ASQP 4.98 33.9 130.7 366.3 602.6

ADMM 0.44 0.59 1.15 3.09 3.78LM 0.012 0.034 0.046 0.071 0.061

CCSQP 3.50 40.2 89.6 136.9 159

ADMM 0.28 0.7 0.93 1.23 1.85

in Table IV forτ = 1. Theorem 1 is satisfied for all simulationsand is not further demonstrated in this section. Moreover, thistable also shows the reduced iterations of the dual Newtonupdate strategy compared with an update strategy with fixedstep sizes, i.e., withαs

k being a constant. The Newton strategyalways converged which implies that the derivatives in (12)are sufficiently well approximated.

The computation times are given in Table V for eachconfiguration. For these simulations, the optimal control prob-lems related to each subsystem are solved in series whichis more straightforward to implement and moreover, thisresults already in sufficient computational benefits for offlineenergy management. The computation time of the DistributedOptimization (DO) method introduced in this paper are com-pared with the computation time of the QCQP solver CPLEX[30]. The CPLEX solver cannot handle quadratic constraintswritten in vector format and every quadratic constraint needsto be programmed separately. This requires a large amountof assembly time which is not used for solving the actualoptimization problem. Therefore, the computation times ofCPLEX with and without assembling the optimization problemare given. If we compare only the time required to solvethe optimization problem, DO is still 1825 times faster forCase 1 withK = 5000 and 64 times faster for Case 6 withK = 3000. Scalability of DO in the horizon lengthK issuperior compared with CPLEX. Scalability in the number ofcomponents is not always better with DO but only in the rarecase with smallK and many more components, CPLEX couldbe better than DO. The flexibility of adding and removingcomponents with CPLEX remains poor though.

TABLE III: Case studies with problem size defined in numberof inputs, states and quadratic constraints.

quadr.inputs states constr.

Case 1 Truck with ICE, EM and a HVB 4K K K

Case 2 Case 1 with a RST 5K 2K K

Case 3 Case 2 with an AS 6K 3K K

Case 4 Case 3 with a CCS 7K 5K 2K

Case 5 Case 4 with an ALT and a LVB 9K 6K 3K

Case 6 Case 5 with a DCDC converter 10K 6K 3K

TABLE IV: Number of iterations and minimum of the dualvariables over all iterations forK = 5000

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6

iter. (αks constant) 105 122 119 228 221 384

iter. (Newton) 26 30 31 131 134 117

mink∈K(µ(1)k ) 0.416 0.430 0.431 −0.03 −0.03 −0.03

mink∈K(µ(2)k ) 0.621 0.622 0.626 0.616 0.623 0.612

mink∈K(µ(3)k ) N.A. N.A. N.A. N.A. 0.522 0.628

TABLE V: Computation times of DO and CPLEX in seconds

K1000 2000 3000 4000 5000

Case 1CPLEX1 8.1 80 300 727 1259

CPLEX2 46 389 1405 3403 6509

DO 0.035 0.24 0.21 0.74 0.69

Case 2CPLEX1 20.3 369 1186 2355 6464

CPLEX2 90 924 3085 6946 15422

DO 0.14 0.99 1.19 1.96 3.25

Case 3CPLEX1 45 334 2014 6650 > 104

CPLEX2 146 1177 4863 13528 > 104

DO 0.34 1.72 1.69 2.74 7.0

Case 4CPLEX1 71 454 3112 > 104 > 104

CPLEX2 357 2721 10770 > 104 > 104

DO 4.35 10.5 18.6 24.2 40.1

Case 5CPLEX1 76 958 2834 > 103 > 103

CPLEX2 791 6835 22213 ≫ 104 ≫ 104

DO 9.6 23.1 43.8 60.9 120.3

Case 6CPLEX1 79 965 3285 > 104 > 104

CPLEX2 1003 8304 28202 ≫ 104 ≫ 104

DO 11.3 31.5 51.4 73.2 150.71 Computation times without assembly time2 Computation times with assembly time

C. Optimal Input and State Trajectories

The optimal power flows as function of time are shownin Fig. 5 for the complete vehicle and with a drive cyclelength of onlyK = 3000 for clarity. Both, the results fromDO, as well as the results from solving the optimizationproblem with CPLEX are shown. This figure demonstratesthat both methods converge to the same solution (within adesired tolerance). Moreover, the fuel consumption of DO is0.019 % smaller compared with CPLEX, which is negligible.Two important observations can be made from this figure,1) all auxiliaries are used to store (brake) energy and 2) theDCDC converter is generally used to supply the low-voltageauxiliaries, except when free brake energy is available, thenthe alternator supplies the low-voltage auxiliaries and chargesthe low-voltage battery. Observation 1 can also be seen fromthe state trajectories given in Fig. 6 wherexhvb = xhvb

Ehvb

is the high-voltage battery energy normalized with respectto the maximum battery capacityEhvb, xlvb = xlvb

Elvbis the

low-voltage battery energy normalized with respect to themaximum battery capacityElvb, Trst is the air temperaturein the refrigerated trailer,Pas is the air pressure in the airsupply system andTccs = [Tw Tr]

T is the wall and refrigeranttemperature in the climate control system. This figure showsthat all state constraints are met, where for the climate controlsystem, only the constraint on the wall temperature is shown.




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−500

0

500

uic

e

−200

0

200

uem

−200

0

200

uhv

b

−4

−2

0

urs

t

−4

−2

0

uas

−4

−2

0

ual

t

−4

−2

0

ud

c

−2

0

2

ulv

b

0 500 1000 1500 2000 2500 3000−4

−2

0

ucc

s

Time [s]

DO CPLEX

Fig. 5: Optimal power flows (in kW) for DO and CPLEX

D. Fuel Consumption Reduction

To analyze the fuel consumption for different parts of thecomplete drive cycle, the drive cycle is split into three parts.The first part is given byk ∈ 0, . . . , 19999, the secondpart by k ∈ 20000, . . . , 39999 and the third part byk ∈40000, . . . , 55579. The fuel consumption reduction for eachof the cases and for each of these drive cycles are given inTable VI. For the first case, the baseline is a non-hybrid truckwith the air temperature in the refrigerated semi-trailer keptat its upper bound, the air pressure in the air supply systemkept at its lower bound and the temperature in the climatecontrol system kept at its upper bound. For the next cases,the baseline is the previous case to emphasize the potential

TABLE VI: Fuel consumption reduction results.

Fuel ConsumptionCase Part 1 Part 2 Part 3 Complete Part 1 (switched)

1 −6.00% −10.93% −2.33% −6.64% −6.00%

2 −0.08% −0.12% −0.03% −0.07% −0.42%

3 −0.02% −0.03% −0.01% −0.02% −0.04%

4 −0.01% −0.04% −0.02% −0.03% −0.21%

5 −0.07% −0.11% −0.02% −0.06% −0.07%

6 −0.34% −0.32% −0.25% −0.34% −0.34%

0

0.5

1

x

hv

b [−

]

4

5

6

Trs

t [°C

]

10

15

Pas

[ b

ar ]

0.8

0.85

xlv

b

500 1000 1500 2000 2500 3000

0

10

20

[°

C ]

Time [s]

Wall Refrigerant

[−

]T

ccs

~~

Fig. 6: Optimal state trajectories for DO.

of each auxiliary. The DCDC converter (Case 6) is the mostpotential auxiliary for reducing fuel with 0.34 %.

Although, the current trend in automotive applications isto electrify the auxiliaries in the vehicle which do allow forcontinuous control (see, e.g., [36]), still the refrigerated semi-trailer, the air supply system and the climate control system areoften attached to the engine via a clutch and, as such, cannotbecontinuous controlled. These auxiliaries are switched betweenan on and off state. Therefore, comparing with a baselinecontinuous controller gives not the full potential of CVEM.The last column in Table VI is therefore added which uses abaseline controller where the auxiliary is turned on when thestate hits the lower bound and turned off when the state hitsthe upper bound. The fuel reduction for the refrigerated semi-trailer and climate control system with the switched baselineis much higher, i.e., 0.42 % and 0.21%, respectively. However,this requires a switched controller with many more switchescompared with the baseline. More switches will reduce thelife time of the auxiliaries and an optimal trade-off must befound between the number of switches and the fuel reduction.

The fuel reduction for some auxiliaries, e.g., the air supplysystem and the climate control system, is very low and integra-tion of those auxiliaries into the energy management strategymight not outweigh the additional cost and complexity. How-ever, the main result of the distributed optimization approachpresented in this paper is that the energy management problemis decomposed into smaller energy management problemsrelated to each subsystem. Each of the energy managementproblems on subsystem level is much easier to solve and can besolved with different algorithms, e.g., an ADMM method or aLagrangian Method. Extensions to more sophisticated models,e.g., by including battery aging or battery thermal dynamics,will be easier and part of future work. Moreover, developingoptimal control algorithms for different subsystems can bedone in parallel, e.g., thermal management of the internalcombustion engine can be included in the internal combustionengine optimization problem, while at the same time thermal




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management of the high-voltage battery can be included in thehigh-voltage battery optimization problem.

VI. CONCLUSIONS

In this paper, a distributed optimization approach has beenproposed to solve the complete vehicle energy managementproblem of a hybrid truck with auxiliaries. A dual decomposi-tion is applied first to the optimal control problem such thattheproblem related to each subsystem can be solved separately.Then, either an ADMM method or a Lagrangian Method hasbeen used to efficiently solve the optimal control problem forevery subsystem in the vehicle. The proposed approach hasbeen demonstrated by solving the complete vehicle energymanagement problem of a hybrid truck with a refrigeratedsemi-trailer, an air supply system, an alternator, a DCDCconverter, a low-voltage battery and a climate control system.Simulation results have shown that the computation time isreduced by a factor of 64 up to 1825, compared to solving theproblem with the CPLEX solver, depending on the vehicleconfiguration and driving conditions. The fuel consumptioncan be reduced up to 0.52 % by including auxiliaries in theenergy management problem, assuming that the auxiliariesare continuous controlled. Fuel consumption can be furtherreduced up to 1.08 % when compared to a baseline with on/offcontrol, but the amount of switches need to increase signifi-cantly. An interesting extension amounts to finding the optimaltrade-off between the maximum allowable number of switchesand fuel reduction. Moreover, the presented approach can bemodified to be used for real-time control, i.e., the optimalcontrol problem can be defined over a shorter horizon. Afterdual decomposition, this results in small Linearly ConstrainedQuadratic Programs for each of the subsystems which can besolved in real-time.

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[2] B. de Jager, T. van Keulen, and J. Kessels,Optimal Control of HybridVehicles. Springer, 2013.

[3] L. Guzzella, A. Sciarrettaet al., Vehicle propulsion systems. Springer,2007, vol. 1.

[4] T. Hofman, M. Steinbuch, R. Van Druten, and A. Serrarens,“Rule-based energy management strategies for hybrid vehicles,”InternationalJournal of Electric and Hybrid Vehicles, vol. 1, no. 1, pp. 71–94, 2007.

[5] G. Paganelli, S. Delprat, T.-M. Guerra, J. Rimaux, and J.-J. Santin,“Equivalent consumption minimization strategy for parallel hybrid pow-ertrains,” in IEEE 55th Vehicular Technology Conf., 2002.

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[11] R. Cipollone and A. Sciarretta, “Analysis of the potential performanceof a combined hybrid vehicle with optimal supervisory control,” in IEEEInt. Conf. on Control Applications, 2006, pp. 2802–2807.

[12] L. Serrao and G. Rizzoni, “Optimal control of power split for a hybridelectric refuse vehicle,” inProc. of the American Control Conf., 2008,pp. 4498–4503.

[13] N. Murgovski, L. Johannesson, J. Sjoberg, and B. Egardt, “Componentsizing of a plug-in hybrid electric powertrain via convex optimization,”Mechatronics, vol. 22, no. 1, pp. 106–120, 2012.

[14] P. Elbert, T. Nuesch, A. Ritter, N. Murgovski, and L. Guzzella, “Engineon/off control for the energy management of a serial hybrid electric busvia convex optimization,”IEEE Trans. Veh. Tech., 2014.

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[16] L. Serrao, S. Onori, A. Sciarretta, Y. Guezennec, and G.Rizzoni,“Optimal energy management of hybrid electric vehicles includingbattery aging,” inProc. American Control Conf., 2011.

[17] F. Merz, A. Sciarretta, J. Dabadie, and L. Serrao, “On the optimalthermal management of hybrid-electric vehicles with heat recoverysystem,”Oil & Gas Science and Tech., 2012.

[18] J. Lescot, A. Sciarretta, Y. Chamaillard, and A. Charlet, “On theintegration of optimal energy management and thermal management ofhybrid electric vehicles,” inProc. IEEE Vehicle Power and PropulsionConf., 2010.

[19] T. Pham, J. Kessels, P. van den Bosch, R. Huisman, and R. Nevels,“On-line energy and battery thermal management for hybrid electricheavy-duty truck.” inProc. American Control Conf., 2013.

[20] F. Willems, M. Donkers, and F. Kupper,Optimal Control of DieselEngines with Waste Heat Recovery System. Springer, 2014.

[21] H. Chen, J. Kessels, M. Donkers, and S. Weiland, “Game-theoreticapproach for complete vehicle energy management,” inProc. IEEEVehicle Power and Propulsion Conf., 2014.

[22] K. D. Nguyen, E. Bideaux, M. T. Pham, and P. Le Brusq, “Gametheoretic approach for electrified auxiliary management inhigh voltagenetwork of hev/phev,” inIEEE International Electric Vehicle Confer-ence, 2014, pp. 1–8.

[23] M. Nilsson, L. Johannesson, and M. Askerdal, “ADMM applied toenergy management of ancilliary systems in trucks,” inProc. AmericanControl Conf., 2015.

[24] T. Romijn, M. Donkers, J. Kessels, and S. Weiland, “A dual decomposi-tion approach to complete energy management for a heavy-duty vehicle,”in Proc. 53rd Annu. Conf. on Decision and Control, 2014.

[25] ——, “Complete vehicle energy management with large horizon opti-mization,” in Proc. 54rd Annu. Conf. on Decision and Control, 2015.

[26] G. Danzig and P. Wolfe, “The decomposition algorithm for linearprogramming,”Econometrica, vol. 4, pp. 767–778, 1961.

[27] G. Cohen, “Auxiliary problem principle and decomposition of opti-mization problems,”Journal of optimization Theory and Applications,vol. 32, no. 3, pp. 277–305, 1980.

[28] A. Rantzer, “Dynamic dual decomposition for distributed control,” inProc. American Control Conf., 2009, pp. 884 – 888.

[29] R. Scattolini, “Architectures for distributed and hierarchical modelpredictive control–a review,”Journal of Process Control, vol. 19, no. 5,pp. 723–731, 2009.

[30] IBM Corp: IBM ILOG CPLEX V12.1, Users Manual for CPLEX(2009).

[31] S. Boyd and L. Vandenberghe,Convex Optimization. CambridgeUniversity Press, Cambridge, 2004.

[32] A. Kozma, J. V. Frasch, and M. Diehl, “A distributed method for convexquadratic programming problems arising in optimal controlof distributedsystems,” inIEEE 52nd Ann. Conf. on Decision and Control, 2013.

[33] G. Stathopoulos, T. Keviczky, and Y. Wang, “A hierarchical time-splitting approach for solving finite-time optimal controlproblems,” inEuropean Control Conf. IEEE, 2013, pp. 3089–3094.

[34] D. van Leeuwen, “A model-based simulation tool for optimal compo-nent sizing for heavy-duty hybrid trucks,” Master’s thesis, EindhovenUniversity of Technology, 2011.

[35] N. van der Sanden, “Model identification of dafs euro-6 air-conditioningsystem applied in an optimal control study,” Master’s thesis, EindhovenUniversity of Technology, 2014.

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Effects of Battery Charge Acceptance andBattery Aging in Complete Vehicle Energy

Management ?

Z. Khalik T.C.J. Romijn M.C.F. Donkers S. Weiland

Dept. of Elect. Eng., Eindhoven University of Technology, Netherlands

Abstract: In this paper, we propose a solution to the complete vehicle energy managementproblem with battery charge acceptance limitations and battery aging limitations. The problemis solved using distributed optimization for a case study of a hybrid heavy-duty vehicle, equippedwith a refrigerated semi-trailer for two different battery models. The first battery model takescharge acceptance into account by adding an additional energy state to the optimizationproblem. The second battery model includes battery aging for which a novel iterative algorithm isproposed. Simulation results show that charge acceptance limitations only have a minor effect onthe solution to the energy management problem, while battery aging leads to a trade-off betweenbattery capacity loss and fuel consumption reduction. In particular, a decrease in capacity lossby 260%, leads to a drop in fuel consumption reduction from 9.40% (without battery aging)to 8.61% (with battery aging), both when compared to a conventional vehicle. This is causedby the fact that aging limitations cause less energy to be stored in the high-voltage battery.Still, because energy can also be stored in the refrigerated semi-trailer, smart control of thisrefrigerated semi-trailer leads to an additional fuel reduction 0.53% in the case where batteryaging is incorporated, while it was only 0.09% when battery aging was not considered. In otherwords, the drop in the fuel consumption reduction caused by battery aging constraints can bepartly compensated by smart control of other energy buffers, which shows the true benefit ofcomplete vehicle energy management.

Keywords: Energy Management, Smart Auxiliaries, Charge Acceptance, Battery Aging

1. INTRODUCTION

Modern vehicles are subjected to strict emission regula-tions, forcing car manufacturers to find ways to reduceemissions and fuel consumption. The fuel consumption ofhybrid electric vehicles is reduced by adding an electricmotor and a high-voltage battery to the power train andoptimally controlling the energy flow between the electricmotor and combustion engine. However, auxiliaries, espe-cially in heavy-duty vehicles, add to the fuel consumptionas well. Examples of such auxiliaries are an air supplysystem, a refrigerated semi-trailer and a climate controlsystem. These auxiliaries provide an opportunity to beused as a buffer to temporarily store excess energy insituations where it is not optimal to store energy in thehigh-voltage battery system. Therefore, it is important toconsider these additional energy flows in the global energymanagement problem, which is referred to as complete ve-hicle energy management (CVEM), (Kessels et al., 2012).

Typically, the global optimal solution to the energy man-agement problem is found by Dynamic Programming(Perez et al., 2006). This method however, suffers from anexponentially increasing computation time with the num-ber of states. Optimization methods based on Pontryagins

? This work has received financial support from the Horizon 2020programme of the European Union under the grant ‘Electric VehicleEnhanced Range, Lifetime And Safety Through INGenious batterymanagement’ (EVERLASTING-713771).

Minimum Principle (Van Keulen et al., 2014) can handlecomputational complexity of multi-state energy manage-ment problems, e.g., battery state-of-health is included in(Ebbesen et al., 2012) and battery thermal managementin (Pham et al., 2013). The solution is found by solving atwo-point boundary value problem, which can be difficultin the presence of state constraints.

For this reason, solutions that use static optimization forenergy management start to appear. As CVEM typicallyleads to a large-scale optimization problem, a distributedoptimization approach to CVEM is proposed in (Romijnet al., 2014, 2015). This allows handling a large numberof components and a large control horizon in the CVEMproblem and results in the global optimal solution underthe condition that the CVEM problem is strictly convex.The battery model used in (Romijn et al., 2014, 2015),however, is an equivalent circuit model without the dy-namics of charge acceptance and restrictions on batteryaging. These models are typically used in optimal controlproblems due to its simplicity. However, the capability ofstoring energy in the high-voltage battery system is inher-ently overestimated, which might lead to a conservativeusage of the energy buffers in the auxiliaries.

Charge acceptance, i.e., the resistance of the batteryagainst storing energy for a longer period, has, to ourbest knowledge, never been included in the energy man-agement problem. Battery aging or state-of-health has




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been included in the energy management problem (see,e.g., (Ebbesen et al., 2012), (Serrao et al., 2011), (Tangand Rizzoni, 2016)) and introduces a trade-off betweenbattery life and fuel consumption. The aforementionedpapers however, do not consider auxiliaries in the energymanagement problem.

Therefore, we will extend the CVEM problem of (Romijnet al., 2014) with the dynamics of charge acceptance andbattery aging. The case study consists of a hybrid-electricheavy-duty vehicle with a refrigerated semi-trailer. Thedynamics of charge acceptance can easily be integratedinto the approach of (Romijn et al., 2014) as it only addsan additional state to the CVEM problem. Battery aging isincluded using a convex approximation of a semi-empiricalbattery aging model proposed in (Wang et al., 2011).This requires a nonlinear constraint to be added to theCVEM problem which is solved by iteratively solving aquadratically constrained quadratic program.

2. COMPLETE VEHICLE ENERGY MANAGEMENT

In this paper, we consider the case study of (Romijn et al.,2014), consisting of a heavy-duty vehicle that includesan internal combustion engine (ICE), an electric machine(EM), a high-voltage battery (HVB) and a refrigeratedsemi-trailer (RST). The topology is schematically shownin Fig. 1, in which Pice1 and Pice2 denote the ICE’s fueland mechanical power, respectively, Pem1 and Pem2 theEM’s mechanical and electrical power, respectively, Phvb2

and Phvb1 the battery’s electrical and stored chemicalpower, respectively, Prst2 and Prst1 the refrigerated semi-trailer’s electrical and thermal power, respectively, Pbr isthe mechanical braking power, Pr is the power required fordriving, Ehvb denotes the battery state of energy and Erst

denotes the thermal energy in the refrigerated semi-trailer.No power losses are assumed in the gearbox (GB) whichimplies that Pgb1

= Pgb2. In this section, we will present

the associated optimal control problem for minimizing thetotal fuel consumption based on the work presented in(Romijn et al., 2014).

2.1 Optimal Control Problem

The optimal control problem of CVEM is given by

minPm1,k,Pm2,k

∑m∈M

∑k∈K

cmPm1,k + dmPm2,k (1a)

where Pm1,k ∈ R and Pm2,k ∈ R are the (scalar) inputsand outputs of the converter in subsystem m ∈ M =ice, em,hvb, rst at time instant k ∈ K = 0, 1, . . . ,K −1, with K the horizon length. In (1a) we use the notationPm1,k, Pm2,k to indicate Pm1,k, Pm2,km∈M,k∈K. Thisnotation will be used throughout the paper for minimizing

ICE

GB

EM

HVB

RST

Ehvb

Erst

BR

PicePice

PgbPbr

PemPem

Phvb

Prst

Phvb

Prst

Pgb

Pr

Fig. 1. Vehicle topology with the ICE, EM, HVB and RST

over a set. The coefficients cm and dm in (1a) can bechosen such that (1a) minimizes the energy losses in thesystem which is equivalent to minimizing fuel consumption(Romijn et al., 2014). The optimization problem (1a) isto be solved subject to a quadratic equality constraintdescribing the input-output behavior of each converter,i.e.,

Pm2,k= 12qm,kP

2m1,k+fm,kPm1,k+am,kEm,kPm1,k+em,k,

(1b)

for all k ∈ K, m ∈ M, and subject to linear systemdynamics of the energy storage device in subsystem m ∈M, i.e.,

Em,k+1 = AmEm,k +BmPm1,k, (1c)

for all k ∈ K where the initial state Em,0 and final stateEm,K of the energy storage device are assumed to begiven, and the input Pm1,k is subject to linear inequalityconstraints, i.e.,

Pm1,k6 Pm1,k 6 Pm1,k, (1d)

for all k ∈ K, m ∈ M and the state Em,k is subject tolinear inequality constraints, i.e.,

Em,k 6 Em,k 6 Em,k, (1e)

for all k ∈ K and m ∈ M. Finally, the optimizationproblem is solved subject to linear equality constraintsdescribing the interconnection of the subsystems, i.e.,

Pr,k − Pbr2,k − Pice2,k − Pem1,k = 0, (1f)

Pem2,k − Phvb2,k − Prst2,k = 0, (1g)

which can be compactly written as

C(Pr,k − Pbr,k) +∑m∈M

AmPm1,k + BmPm2,k = 0, (1h)

for all k ∈ K, where Am ∈ R2, Bm ∈ R2 and C = [1 0]T .This topology has one exogenous disturbance signal, i.e.,the power required to drive a certain drive cycle Pr,k whichis assumed to be known for every time instant k ∈ K.

In this paper, we extend the optimal control problem withan additional constraint representing battery aging, i.e.,∑

k∈K

αk(Phvb1,k)P 2hvb1,kQ

z−1z

k 6 QK −Q0, (1i)

where αk is a HVB power dependent aging factor, 0 <z6 1is a battery constant, QK is the maximum battery agingallowed at the end of a drive cycle with length K and Qkis the battery aging at time instant k given by

Qk+1 = Qk + αk(Phvb1,k)P 2hvb1,kQ

z−1z

k (1j)

Finally, we define the primal optimal solution as thesolution P ∗m1,k

, P ∗m2,k that satisfies (1) and p∗ as the

primal optimal value of (1).

2.2 Dual Decomposition

Problem (1a) subject to (1b) - (1h) cannot be separateddue to the complicating constraint (1h). Therefore, wedecompose the problem via dual decomposition by intro-ducing the following so-called partial Lagrangian

L(Pm1,k, Pm2,k, µk)=∑k∈K

C(Pr,k − Pbr,k)+∑m∈M

cmPm1,k

+ dmPm2,k + µTk (AmPm1,k + BmPm2,k), (2)

where µk ∈ RN is a Lagrange multiplier, subject to (1b)-(1e). Indeed, the partial Lagrangian is obtained by adding




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the complicating constraints (the constraints that act onmore than one subsystem) to the objective function in(1a). The partial Lagrange dual function is now given by

g(µk) = minPm1,k,Pm2,k

L(Pm1,k, Pm2,k, µk)

= µTk C(Pr,k − Pbr,k)) +∑m∈M

gm(µk), (3a)

with

gm(µk) = minPm1,k,Pm2,k

∑k∈K

cmPm1,k + dmPm2,k

+ µTk (AmPm1,k + BmPm2,k), (3b)

subject to (1b)-(1j) but not subject to (1h). In (Romijnet al., 2014), it is shown that minimizing over the mechan-ical braking power Pbr,k is not necessary as it is given by(1f) for the first element of µk, which is smaller than zero.Note that each of the Lagrange dual functions (3b) subjectto (1b)-(1j), but not subject to (1h) is related to one ofthe subsystems and can be solved independently. The dualproblem is given by

maxµk

g(µk) = d∗, (4)

subject to (1b)-(1j) except (1h) where d∗ is defined as thedual optimal value. It has been shown in (Romijn et al.,2016) that under very mild and verifiable conditions, theprimal and dual solution are the same, i.e., d∗ = p∗.

Maximizing the Lagrange dual function (3) over µk canbe done with a steepest ascent method. This results in analgorithm for solving the CVEM problem, given by

P sm1,k, Psm2,k = argmin

Pm1,k,Pm2,kL(Pm1,k, Pm2,k, µ

sk) (5a)

µs+1k =µsk+Ssk

(∑m∈M

AmP sm1,k+BmP sm2,k+C(Pr,k−Pbr,k)),

(5b)

subject to (1b)-(1j) but not subject to (1h) for all k ∈ Kwhere Sk is a suitably chosen matrix and s ∈ N is theiteration counter.

2.3 Dual Functions

Each of the Lagrange dual functions (3b) related to oneof the subsystems can be solved separately and can bewritten as a quadratic program by substituting (1b) into(3b), which gives

gm(µk)

= minPm1,k

∑k∈K

12

[Pm1,k

1

]T[Hm,k FTm,kFm,k 2Gm,k

][Pm1,k

1

]T, (6a)

with

Hm,k=(dm + BTmµk)qm,k, (6b)

Fm,k=cm+ATmµk+(dm+BTmµk)(fm,k+am,kEm,k), (6c)

Gm,k=(dm + BTmµk)em,k (6d)

subject to (1c) - (1e) and for the HVB subject to (1i)and (1j) as well. Note that for a nonzero matrix am,k in(6c), the quadratic program becomes more difficult to solvedue to the cross products between Pm1,k for k ∈ K. Still,efficient solution methods for these optimization problemswithout (1i) and (1j), i.e., an ADMM solution methodand a Lagrangian solution method, are given in (Romijn

et al., 2015). The battery aging constraint (1i), however, isa nonlinear constraint for which the methods proposed in(Romijn et al., 2015) cannot be used. In this paper, we willpropose a modified Lagrangian solution method that takesinto account the nonlinear constraint for the case thatam,k = 0. In Section 3, we will show that if the dynamics ofcharge acceptance are not taken into account, am,k is zero.Hence, the battery optimization problem cannot be solvedwith this method for the combination of charge acceptanceand battery aging. In Section 4, we will show that this isnot a significant restriction as battery aging is dominantover charge acceptance.

2.4 Lagrangian Solution Method

The battery optimization problem (6) for m ∈ hvbsubject to the nonlinear constraint (1i) is difficult to solvedue to the nonlinear constraint. However, by fixing thebattery aging parameters αk for all k ∈ K and batteryaging values Qk for all k ∈ K, the nonlinear constraint(1i) is quadratic for which we can solve the optimizationproblem via a modified Lagrangian Method of (Romijnet al., 2014). This fixation basically implies that we havea (not optimal) solution Phvb1,k for all k ∈ K for whichwe can calculate αk and Qk for all k ∈ K as in (1i). Byiteratively updating αk and Qk for all k ∈ K in (1i) withthe solution of the optimization problem, the solution ofthe optimization problem converges to the global optimumsolution.

To show this, let us define first the state-unconstrainedproblem for the HVB, which is given by (6) subject to(1d) and (1c) with given initial state Ehvb,0 and final stateEhvb,K . The Lagrangian of this problem is given by

L(Phvb1,k, λ1, λ2, νk, νk) =(∑k∈K

12

[Phvb1,k

1

]T [Hhvb,k FThvb,kFhvb,k 2Ghvb,k

] [Phvb1,k

1

]T(7)

+

[νkνk

]T [Phvb1,k − P hvb1,k

P hvb1,k − Phvb1,k

])

+

[λ1λ2

]T[AKhvbEhvb,0−Ehvb,K+

∑k∈KA

K−1−khvb BhvbPhvb1,k

Q0 −QK +∑k∈K

12γkP

2hvb1,k

]with λ1 ∈ R, the Lagrange multiplier associated with thefinal state constraint

Ehvb,K = AKhvbEhvb,0 +∑k∈K

AK−1−khvb BhvbPhvb1,k, (8)

and λ2 ∈ R the Lagrange multiplier associated withthe battery aging constraint (1i). Furthermore, we have

introduced γk = 2αk(Phvb1,k)Qz−1z

k which is fixed for agiven Phvb1,k for all k ∈ K.

The Karush-Kuhn-Tucker conditions (Boyd and Vanden-berghe, 2004) for minimizing the Lagrangian in (7) aregiven by the first-order necessary optimality condition

∂L(Phvb1,k,λ1,λ2,νk,νk)

∂Phvb1,k= Hhvb,kPhvb1,k + Fhvb,k

+ νk − νk + λ1Bhvb = 0, (9)

for all k ∈ K with Hhvb,k = Hhvb,k + λ2γk and Bhvb =∑k∈KA

K−1−khvb Bhvb, feasibility of the constraints (8) and




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(1i) and the complementary slackness conditions for theinequality constraints

νk(Phvb1,k − P hvb1,k) = 0, (10a)

νk(P hvb1,k− Phvb1,k) = 0, (10b)

λ2(Q0 −QK +∑k∈K

12γkP

2hvb1,k) = 0 (10c)

for all k ∈ K with νk > 0, νk > 0 and λ2 > 0. The optimalsolution to (9) and (10) is given by

P ∗hvb,k =− H−1hvb,k(Fhvb,k+λ1Bhvb + νk−νk), (11)

for all k ∈ K for a given λ1, λ2, νk and νk. To find theoptimal solution, we propose an algorithm for which wefirst fix λ2 and γk and solve

λt+11 =H−1hvb

(AKhvbEhvb,0−Ehvb,K

+∑k∈K

AK−1−khvb BhvbH−1hvb,i(Fhvb,k+νtk−νtk)

), (12a)

with Hhvb =∑k∈KA

K−1−khvb BhvbH

−1hvb,kBhvb and

νt+1k =max

0,−Hhvb,kP hvb,k−Fhvb,k−λt+1

1 Bhvb

(12b)

νt+1k =max

0, Hhvb,kP hvb,k + Fhvb,k + λt+1

1 Bhvb

(12c)

with t ∈ N the iteration index and for ν0k = ν0k = 0 for allk ∈ K, until (8) is satisfied within some desired tolerance.The expressions in (12) are obtained by substituting (11)

into (8) and (1d). If Hhvb,k is strictly positive definite forall k ∈ K and if there exists an optimal solution P ∗hvb,kfor which (1d) is satisfied, then the solution of (12) willconverge to the solution of (6) subject to (1d) and (8) butnot (1i).

To satisfy (1i), γk is updated after convergence of thealgorithm in (12) with the optimal solution (11) by

γs+1k = (1− ρ1)γsk + ρ1(2αk(P ∗hvb1,k)Q∗k

z−1z ) (13)

with s ∈ N the iteration index, ρ1 ∈ (0, 1] is a relaxationparameter and until γs+1

k ≈ γsk for all k ∈ K. Thisalgorithm will converge under the conditions that ρ1 issufficiently small and if γk is a convex function of thebattery power Phvb1,k. Still, with γs+1

k ≈ γsk, equation (1i)is not satisfied as λ2 needs to be converged to satisfy (1i).We propose to update λ2 every time γk is converged byusing a ‘steepest ascent’ direction, i.e.,

λr+12 = λr2 + ρ2(Q0 +

∑k∈K

12γkP

2hvb1,k −QK) (14)

with r ∈ N the iteration index and where ρ2 > 0 is thestep size.

The above presented procedure still only provides a solu-tion for the state-unconstrained problem. For solving thestate-constrained dual function, we use a property firstintroduced in (Van Keulen et al., 2014). In particular, wemake use of the monotonic relation between the Lagrangemultiplier λ1 and the final state Ehvb,K . If this relationholds, the time instant at which the unconstrained stateexceeds its constraints the most, is a contact point of thestate-constrained solution, and, therefore, fixes a part ofthe optimal solution to the optimization problem. As aresult, the optimization problem can be split into two newunconstrained optimal control problems with known initialand terminal states. This property is used in (Van Keulenet al., 2014) to calculate the optimal state constrained

solution by using Pontryagin’s Minimum Principle. Here,we used the same property to calculate the optimal state-constrained solution in combination with the Lagrangianmethod. For details, we refer to (Van Keulen et al., 2014).

3. MODELING OF COMPONENTS

After presenting the optimal control problem related toCVEM in the previous section, we now explain the modelsused in the optimal control problem. The models forthe ICE, EM, RST will only be briefly discussed. Themodels for the HVB will be discussed more extensively,and extended compared to previous work in (Romijn et al.,2014). In particular, we extend in this paper the HVBmodel to incorporate charge acceptance and battery aging.

3.1 ICE, EM and RST

The input-output behavior of the ICE and EM can bedescribed by the quadratic function (1b) form ∈ ice, em,where the efficiency coefficients for the ICE and the EMare speed dependent, i.e.,

qm,k = qm(ωk), fm,k = fm(ωk), em,k = em(ωk), (15)

for m ∈ ice, em where qm(ωk), fm(ωk) and em(ωk)are functions parameterizing the efficiency coefficients asfunction of drive line speed ωk. The input power of the ICEand EM are bounded by (1d) for m ∈ ice, em where theupper and lower bound for the ICE and the EM are speeddependent, i.e.,

Pm1,k= Pm1

(ωk), Pm1,k = Pm1(ωk). (16)

for m ∈ ice, em where Pm1(ωk) and Pm1

(ωk) arefunctions parameterizing the lower and upper bound asfunction of drive line speed ωk. Constraints on the systemdynamics (1c) and state constraints (1e) do not have tobe taken into account as the ICE and EM do not have a(constrained) energy storage.

The input-output behavior of the RST can be describedby the quadratic function (1b) for m ∈ rst. The inputpower of the RST is bounded by (1d) for m ∈ rst. Thedynamics of the RST are assumed to satisfy a thermalenergy balance given by

CrstddtTrst = (Prst1 − β1(Trst − β2Tamb)), (17)

where Crst is the heat capacity of the RST and its contents,Prst1 is the thermal power where negative values indicatecooling, β1 is the heat transfer coefficient between theRST and the environment, β2 is an insulation coefficient,Trst is the temperature inside the RST and Tamb is theambient temperature (which is assumed to be constant).We can represent the RST model in terms of stored energyby defining the thermal energy relative to the ambienttemperature Erst = Crst(β2Tamb − Trst). By doing so andby making a forward Euler approximation of (17) withstep size τ > 0, the dynamics can be represented by (1c)

for m ∈ rst with Arst = 1 − τβ1

Crstand Brst = −τ for all

k ∈ K.

3.2 HVB Equivalent Circuit Model

The HVB dynamics are modeled with a battery equivalentcircuit model shown in Fig. 2. We consider the so-calledRint model as well as the so-called Thevenin model (He




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R

R

C

Uoc

Ub

I

Ub

Rint Thevenin

Fig. 2. Equivalent circuit models

et al., 2012). The Rint model consists of a circuit witha constant voltage source and a resistor. This model isgenerally used for energy management due to its simplicity.However, it is also the least accurate HVB model (Heet al., 2012). The Thevenin model is a Rint model withadditionally a parallel RC network connected to it, therebyincluding the dynamics of charge acceptance. With theThevenin model, the HVB undergoes a low resistancewhen the capacitor C1 is not charged and the resistanceincreases as the charge increases. The dynamics of theHVB with the Rint model are given by

q = −I, (18a)

with q the HVB charge and I the HVB current and for theThevenin model given by (18a) and

U1 = 1C1

(I − U1

R1), (18b)

where U1 is the voltage over capacitor C1. The HVBterminal voltage is given by

Ub = Uoc −R0I − U1 (18c)

for the Thevenin model and for the Rint model given by(18c) with U1 = 0.

We can represent the HVB model in terms of stored energyby defining the energy in the HVB by Ehvb = qUoc, theenergy related to the capacitor by E1 = C1UocU1 and thebattery power Phvb11 = UocI. By doing so and by making aforward Euler approximation of (18) with step size τ > 0,the dynamics can be represented by (1c) for m ∈ hvbwith Ahvb = 1, Bhvb = −τ for all k ∈ K for the Rint

model and with Ahvb =

[1 00 1− τ

C1R1

], Bhvb = [−τ τ ]

T

for all k ∈ K for the Thevenin model. The input-outputbehavior of the HVB can be described by the quadraticfunction (1b) for m ∈ hvb with fm,k = R0

U2oc

for both

the Rint as well as the Thevenin model and ahvb,k = 0 for

the Rint model and ahvb,k =[0 −1C1U2

oc

]for the Thevenin

model.

3.3 HVB model with aging

In (Wang et al., 2011), the following semi-empirical batteryaging model is given for graphite-LiFePO4 battery cells

Q = B · exp(−Ea

RT )(Ah)z (19a)

where Q is the battery capacity loss in percentages, R isthe gas constant, T is the battery temperature which isassumed to be constant through active cooling, Ah is thetotal Ah-throughput, Ea is the activation energy given by

Table 1. Pre-exponential factor B.

Crate C/2 2C 6C 10CB 31630 21681 12934 15512

-100 -50 0 50 1000

2

4× 10 -4

-100 -50 0 50 100

Phvb

[kW]

0

1

2

3× 10 -6

Battery power1

erro

r

h(Phvb

)

(Phvb

)Phvb

2α

1

1 1

[% /

s]1 z

Agi

ng

fact

or[%

/s]

1 z

Fig. 3. Convex approximation of the aging factor.

Ea = 31700− 370Crate (19b)

and B is a pre-exponential factor measured at differentC-rates in (Wang et al., 2011) and given in Table 1.If we assume that the energy and power are uniformlydistributed over the cells in the HVB system, then theC-rate can be expressed as

Crate =|Phvb1

|nUcellCmax

(19c)

where Phvb1 is the HVB storage power, n the total numberof cells in the HVB system, Ucell is the cell voltage andCmax the total battery capacity. The total Ah-throughputis given as:

Ah = 13600nUcell

(E0 +

∫ t

0

|Phvb1|dt) (19d)

where E0 is the initial total energy throughput of the HVBsystem and t is the time horizon over which the energythroughput is evaluated. In (Pham, 2015), a quasi-staticapproach is utilized where the rate of change of the Crate isneglected in order to derive the incremental capacity loss

dQdt = h(Phvb1

)Qz−1z , (20a)

with

h(Phvb1) = [B · exp(−Ea

RT )]1z z3600npnsUcell

|Phvb1|. (20b)

It should be noted that (19) is the solution to (20), aswas shown in (Pham, 2015). By making a forward Eulerapproximation of (20a) with step size τ > 0, the batterycapacity loss in discrete time is given by

Qk+1 = Qk + τh(Phvb1,k)Qz−1z

k . (21)

Note that h(·) is not necessarily a convex function de-pending on the parameterizing of B as function of theC-rate. Therefore, the battery capacity loss (21) can beapproximated with (1i) by taking the convex function

α(Phvb1,k) = p1·exp(p2·Phvb1,k)+p3·exp(p4·Phvb1,k) (22)

such that α(Phvb1,k)P 2hvb1,k

≈ h(Phvb1,k). The parametersp1, p2, p3, p4 are obtained with a least-squares fittingtechnique. The results are shown in Fig. 3 where h(Phvb1

)and α(Phvb1

)P 2hvb1

are shown together with the absoluteerror between the two functions.

4. SIMULATION RESULTS

In this section, we analyze the effect of using the HVBmodel with the dynamics of charge acceptance and the




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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

× 10 4

-200

0

200

400

Pow

er r

eques

t P r

[kW

]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time [s] × 10 4

500

1000

1500

2000

Engi

ne

spee

d [rp

m]

Fig. 4. The Pan-European driving cycle.

HVB aging constraint on the CVEM solution by meansof a simulation study. To do so, the optimization problemgiven in (1) is solved for part of a Pan-European drivingcycle with length K = 20000 s. The power request at thewheels and the engine speed for this driving cycle are givenin Fig. 4.

4.1 Case studies

To study the effect of different battery models on CVEM,we define the following cases with increasing complexity

(1) In this case, the battery is not used, i.e., Phvb2 =Phvb1 = 0. Moreover, the temperature in the refrig-erated semi-trailer is fixed at a constant temperature(namely, its upper bound) requiring a constant ther-mal power of Prst1 = β1(T rst − β2Tamb).

(2) In this case, one decision variable is added withrespect to Case 1, namely the battery storage powerPhvb1 .The temperature in the refrigerated semi-traileris still fixed at the upper bound. This case has onestate, being the battery state-of-energy Ehvb.

(3) In this scenario, another decision variable is addedwith respect to Case 2, namely the thermal powerPrst1 . The thermal energy in the refrigerated semi-trailer is no longer fixed but constrained in an inter-val. This case has two energy states Ehvb and Erst.

Note that only for Case 2 and Case 3 the HVB is usedso that we can solve the associated optimization problemwith the HVB Rint model or the HVB Thevenin modelintroduced in Section 3.2. Moreover, with the solutionmethod proposed in Section 2.4, the optimization problemfor Case 2 and Case 3 can be solved with the HVBRint model together with the constraint on battery aging(1i). To avoid disclosing confidential information, (19) isscaled with a constant factor but within a realistic range.Furthermore, the battery parameters Uoc, R0, R1 and C1

are optimized to give the best approximation of a high-fidelity simulation model where, in general, the resistanceR0 is higher for the Rint model compared to the Theveninmodel due to its reduced accuracy.

4.2 Results

In Fig. 5, the HVB power and RST power for Case 3 andfor each of the three battery models, i.e., the Rint model,the Thevenin model and the Rint model with a constraint

0 100 200 300 400 500-150

-100

-50

0

50

Phvb

[kW

]

∆ = 5.3 kW

0 100 200 300 400 500

Time [s]

-15

-10

-5

0

Prs

t [kW

]

Rint model

Thevenin model

Rint model with aging constraint

22

Fig. 5. HVB and RST power for the three different HVBmodels, for a small part of the driving cycle.

0 0.5 1 1.5 2

Time [s] × 10 4

1

1.1

1.2

1.3

Capac

ity los

s Q

[%

]

No aging consideration

Aging consideration

Fig. 6. Capacity loss comparison over time in Case 3.

on aging are shown for a small part of the driving cycle.This figure clearly shows the charge acceptance dynamicsin the Thevenin model. At short peaks, the Theveninmodel can have a larger HVB power than the Rint model,i.e., the internal resistance R0 is lower and the influence ofR1 is small when the capacitor is not charged. However,at longer positive/negative power intervals, the capacitorC1 is charged such that the Thevenin model effectivelyhas a higher resistance than the Rint model, resulting inlower HVB powers. As less energy (power) is stored in thebattery, more energy (power) is stored in the RST. Theeffect of charge acceptance on the HVB power and theRST power is still very limited though.

The effect of the constraint on battery aging is morepronounced. This is due to the fact that the batterylifetime without a constraint on aging is significantlyshorter than with a constraint on aging, i.e., for thisdriving cycle, 0.11 % of battery capacity loss is allowedwhile without aging constraint 1 , the battery capacity losswas 0.29 %. This is also shown in Fig. 6 where Q0 = 1%.Here, the black line visualizes the imposed constraint onbattery aging. This results in the battery power beinglimited in order to preserve battery life as is shown inFig. 5. It can also be noted that large battery powersare reduced much more in magnitude than smaller batterypowers, which is explained by Fig. 3, where it can be seenthat larger battery powers have an exponentially larger

1 The constraint of 0.11% is based on the desire to complete thedrive cycle of Fig. 4 1000 times before reaching a capacity loss of 20%,while not changing the solution of the optimal control problem. Itshould be noted that for the same energy management strategy, thecapacity decreases more initially than during the 1000nd repetitionof the drive cycle.




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0 0.5 1 1.5 2

× 10 4

5

5.5

Trs

t [o

C]

0 0.5 1 1.5 2

Time [s] × 10 4

0

50

100

SoC

[%

]

Rint model Thevenin model Rint model with aging constraint

Fig. 7. RST temperature and SoC for the three differentbattery models.

effect on battery aging. This larger difference in batterypower results in a larger difference in RST power as well,as seen in Fig. 5.

Finally, in Fig. 7, the state-of-charge (SoC) and RSTtemperature are shown for the complete driving cycle. Thisfigure clearly shows that with battery aging, substantiallymore energy is put into the RST and less energy is drawnfrom the battery compared to the battery models withoutconstraint on aging. This also has a significant effect onthe fuel reduction given in Table 2 for Case 2 with respectto Case 1 and Case 3 with respect to Case 2. For Case 2,the fuel reduction with the Rint model is not significantlydifferent than with the Thevenin model. The fuel reductionby taking into account the constraint on battery aging ishowever, significantly lower, i.e., 8.61% instead of 9.4 %.This is caused by the fact that aging limitations cause lessenergy to be stored in the high-voltage battery. For Case3 however, energy can also be stored in the refrigeratedsemi-trailer, smart control of this refrigerated semi-trailerleads to an additional fuel reduction 0.53% in the casewhere battery aging is incorporated, while it was only0.09% when battery aging was not considered. In otherwords, the drop in the fuel consumption reduction causedby battery aging constraints can be partly compensatedby smart control of other energy buffers, which shows thetrue benefit of complete vehicle energy management.

5. CONCLUSION

In this paper, we added battery charge acceptance lim-itations and battery aging limitations to the completevehicle energy management problem. The problem hasbeen solved using a distributed optimization approach fora case study of a hybrid heavy-duty vehicle, equippedwith a refrigerated semi-trailer for two different batterymodels. The first battery model takes charge acceptanceinto account by adding an additional energy state to theoptimization problem. The second battery model includesbattery aging for which a novel iterative algorithm hasbeen proposed. Simulation results showed that chargeacceptance limitations only have a minor effect on the

Table 2. Fuel consumption reduction.

Rint Thevenin Rint model andmodel model aging constraint

Case 2 vs. Case 1 9.40 9.36 8.61

Case 3 vs. Case 2 0.089 0.091 0.53

solution to the energy management problem, while batteryaging leads to a trade-off between battery capacity lossand fuel consumption reduction. The drop in the fuelconsumption reduction caused by battery limitations canbe partly compensated by smart control of other energybuffers such as the refrigerated semi-trailer.

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He, H., Xiong, R., Guo, H., and Li, S. (2012). Comparisonstudy on the battery models used for the energy manage-ment of batteries in electric vehicles. Energy Conversionand Management.

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Perez, L.V., Bossio, G.R., Moitre, D., and Garcıa, G.O.(2006). Optimization of power management in an hybridelectric vehicle using dynamic programming. Math.Comput. Simulation.

Pham, H.T. (2015). Integrated energy and battery life man-agement for hybrid vehicles. Ph.D. thesis, TechnischeUniversiteit Eindhoven.

Pham, T., Kessels, J., van den Bosch, P., Huisman, R., andNevels, R. (2013). On-line energy and battery thermalmanagement for hybrid electric heavy-duty truck. InProc. American Control Conf.

Romijn, T., Donkers, M., Kessels, J., and Weiland, S.(2014). A dual decomposition approach to completeenergy management for a heavy-duty vehicle. In Proc.IEEE Conf. on Decision and Control.

Romijn, T., Donkers, M., Kessels, J., and Weiland, S.(2015). Complete vehicle energy management withlarge horizon optimization. In in Proc. IEEE. Conf.on Decision and Control.

Romijn, T., Donkers, M., Kessels, J., and Weiland, S.(2016). A distributed optimization approach for com-plete vehicle energy management. Submitted for journalpublication.

Serrao, L., Onori, S., Sciarretta, A., Guezennec, Y., andRizzoni, G. (2011). Optimal energy management ofhybrid electric vehicles including battery aging. In Proc.American Control Conf.

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Global Solutions to the Complete Vehicle Energy Management Problemvia Forward-Backward Operator Splitting

G.P. Padilla, G. Belgioioso, M.C.F. Donkers

Abstract— Complete Vehicle energy Management (CVEM)aims to minimize the energy consumption of all subsystemsin a vehicle. We consider the case where the subsystemsconsist of energy buffers with linear dynamics and/or energyconverters with quadratic power losses. In this paper, weshow the existence of only global solutions for the CVEMoptimal control problem and propose a reformulation of thisproblem so that it can be solved using a Forward-Backwardsplitting algorithm for nonconvex optimization problems. Theregularization properties inherent to this algorithm allow us tosolve CVEM cases that were difficult using other approachesas dual decomposition.

I. INTRODUCTION

Improving energy efficiency of vehicles is an importanttopic of research for the automotive industry. While fortraditional and hybrid vehicles, a better energy efficiencyleads to lower emissions, the main motivation to improveefficiency of electric vehicles is that it will extend the rangeof the vehicle and thereby mitigating range anxiety. Rangeanxiety is the concern experienced by the user to be unableto reach a final destination with the current energy [1]. Anenergy management strategy (EMS) aims at reducing theenergy consumption by an optimal distribution of poweramong the powertrain components (to provide the requiredpower to the wheels) and the auxiliary systems. In essence,the EMS is the solution to an optimal control problemand optimization techniques for energy control on hybridvehicles have been discussed in [2], [3]. A recent trend inthis research area is to consider all the energy consumersinside the vehicle and this concept is known as CompleteVehicle Energy Management (CVEM) [4]. CVEM requiresthe optimal control problem to be scalable, as it requires alarger number of subsystems to be connected to the powernetwork compared to earlier solutions for EMS.

Static optimization techniques have emerged as tractableapproaches to tackle the CVEM problem, see, e.g., [5]–[8].In particular, [6] has incorporated the control of auxiliarysystems into the EMS and [5] has used convex relaxationsto guarantee optimality of the solution. However, due to theuse of centralized optimization methods, those approachesare not flexible in the sense that subsystems cannot be easilyadded or removed. The research presented in [7] uses agame-theoretic approach to solve the CVEM problems in

This work has received financial support from the Horizon 2020 pro-gramme of the European Union under the grant ‘Electric Vehicle EnhancedRange, Lifetime And Safety Through INGenious battery management’(EVERLASTING-713771).

The authors are with the Dep. of Electrical Eng. of Eindhoven Uni-versity of Technology, Netherlands. E-mail: g.p.padilla.cazar,g.belgioioso, [email protected]

a decentralized manner, where all the subsystems share alimited amount of information and are able to take somedecisions autonomously. Global optimality of the centralizedoptimal control problem is not guaranteed.

A static distributed optimization approach for CVEM ispresented in [8]. The method proposed in these papersare scalable and flexible. The main idea is to use dualdecomposition to split the original optimal control probleminto several simple problems related to interconnected sub-systems. As a result, the computational time is drasticallyreduced and adding and removing subsystems becomes easy.However, several open questions exist related to numericalaspects of the algorithms proposed to solve the distributedoptimization problem. For instance, [8] proposes a second-order dual update for which the convergence of the algorithmhas not been formally proven. Additionally, the approachrequires components to be described with linear dynamicsand quadratic energy conversion models. In case the energyconversion is almost linear, the distributed optimization ap-proach of [8] becomes ill conditioned, causing the algorithmto have difficulties to converge.

In this paper, we design an efficient algorithm withconvergence guarantee for the CVEM problem, based onthe Forward-Backward (FB) operator splitting method [9,§5.1]. First, we show that the non-convex CVEM prob-lem only has global optimal solutions. Then, by takingadvantage of the globality of the solutions, we massage theFB splitting method onto the CVEM problem to obtain aparallelizable static optimization algorithm. The convergenceanalysis of the proposed algorithm is based on a recentresult on projected-gradient dynamics for non-convex op-timization problems [9, §5.3]. Remarkably, the use of thisprojection method introduces regularization that prevents ill-conditioning of the optimization problem. This means thatconvergence is possible even if linear models are used todescribe the power consumption of the vehicle subsystems.

Nomenclature: R denotes the set of real numbers, R+ theset of non-negative real numbers, and R := R ∪ +∞ theset of extended real numbers. For a, b ∈ R, a significantstrict inequality that is denoted by a b implies that ais much less than b. The matrices 0 and 1 denote matriceswith all elements equal to 0 and 1, respectively, and thematrix I denotes the identity matrix. To improve clarity, wesometimes add the dimension of these matrices as subscript.Furthermore, diag(A1, · · · , AN ) denotes a block-diagonalmatrix with A1, · · · , AN as diagonal blocks. Given N vec-tors x1, . . . , xN ∈ Rn, we denote col(xii∈1,...,N) =[x>1 , . . . , x

>N ]>. The short hand notation xp,q is used to




74 / 110

denote xp,qp∈P,q∈Q.Given a set S ⊆ Rn, the mapping ıS : Rn → 0, +∞

denotes the indicator function satisfying ıS(x) = 0 if x ∈S and ıS(x) = ∞ if x /∈ S, and set-valued mappingNS : Rn ⇒ Rn denotes the normal cone operator satisfyingNS(x) = v ∈ Rn | supz∈S v

>(z − x) ≤ 0 if x ∈ Sand NS(x) = ∅ if x /∈ S. The mapping projS : Rn → Sfor a closed set S ⊆ Rn denotes the projection onto S, i.e.,projS(x) = argminy∈S ‖y − x‖2.

For a function ψ : Rn → R with dom(ψ) := x ∈Rn | ψ(x) < ∞, the subdifferential set-valued mapping∂ψ : dom(ψ) ⇒ Rn is defined as ∂ψ(x) := v ∈ Rn |ψ(z) ≥ ψ(x) + v>(z − x) for all z ∈ dom(ψ). In case ψis continuously differentiable, the subgradient is equal to itsgradient, i.e., ∂ψ(x) = ∇ψ(x).

II. CVEM AS OPTIMAL CONTROL PROBLEM

In this section, we first discuss a mathematical descriptionof a power network model for CVEM. Later, we willformulate the discrete optimal control problem for CVEM,which will turn out to be non-convex, and we will show thatall the solutions to this non-convex optimal control problemare global optimal solutions. Finally, we will propose aconvenient equivalent formulation of the problem that willbe exploited in Section III to find an optimization algorithmbased on operator splitting techniques.

A. Power Network Model for CVEMThe CVEM problem aims to minimize the

energy consumption for a network of subsystemsm ∈ M := 1, . . . ,M, where M is the totalnumber of subsystems. These subsystems are composedof an energy converter possibly combined with an energybuffer. For instance, considering a battery, the capacityof storing energy is modeled as an energy buffer, whilethe power losses produced during the transformation ofchemical energy into electrical energy in the battery arerepresented by an energy converter.

A power network for CVEM is schematically depicted inFig. 1. It can be noted that the energy buffers are alwaysconnected to energy converters by the power input um,k. Onthe other hand, the converters are connected to each otheraccording to a specific topology via the network nodes j ∈J := 1, . . . , J, where J is the total number of nodes in thenetwork. In this case, power outputs ym,k and inputs um,k ofthe converters could be directly connected to a network node.Every node j ∈ J can have a known exogenous load signalwj,k given for each time instant k. The following assumptionon the network topology is considered in this paper.

Assumption 1: The power network has a tree structuretopology, i.e., every subsystem is connected to only onenode, and two consecutive nodes are always bridged by anindividual power converter. Moreover, only converters canbe connected directly to a network node. Note that, albeit of its simplicity, the network presentedin Fig. 1 contains the essential features described in As-sumption 1, which can be found in more complicated powernetworks.

metsysbuS m

x k,mum,k ym,k

retrevnoC

Buffer

ConverterBuffer

Converter

retrevnoC Buffer

Converter Buffer

wj,k

edoN j

Fig. 1: Power network for CVEM.

B. Optimal Control ProblemThe CVEM problem is an optimal control problem that

aims to minimize the total aggregated energy consumption ofthe components over a time horizon K := 0, 1, . . . ,K−1,while considering the interaction of all the componentsinterconnected in the power network. Thus, the optimalcontrol problem is given by

minym,k,um,k,xm,k

∑

m∈M

∑

k∈Kamym,k + bmum,k, (1a)

where xm,k ∈ Rnm are the states, um,k ∈ R are the (scalar)inputs and ym,k ∈ R are the (scalar) outputs of the converterin subsystem m ∈ M, while am ∈ R+ and bm ∈ Rare coefficients to define the objective function. A typicalobjective in CVEM is to minimize the fuel consumption, inwhich case assume the subsystem m = 1 corresponds to thecombustion engine and y1,k denotes the chemical fuel powerflow at time k ∈ K. In this case, only a1 = 1, while all otheram and bm are zero.

The objective function (1a) is to be solved subject to aquadratic equality constraint that describes the input-outputbehavior of each converter, i.e.,

ym,k = 12q2,mu

2m,k + q1,mum,k + q0,m, (1b)

where q0,m ∈ R, q1,m ∈ R and q2,m ∈ R+ are efficiencycoefficients of the converter m ∈ M, and subject to thelinear system dynamics of the energy buffer

xm,k+1 = Amxm,k +Bmum,k, (1c)

for all k ∈ K with Am ∈ Rnm×nm and Bm ∈ Rnm×1. Theinitial state xm,0 of the storage device is assumed to be givenand the inputs and states are subject to

xm ≤ xm,k ≤ xm and um ≤ um,k ≤ um, (1d)

where for all m ∈ M the given state and input boundsare respectively xm, xm ∈ Rnm and um, um ∈ R. In [8],it has been shown that quadratic static models for energyconverters and linear dynamical models for buffers are anadequate approximation to describe typical components inCVEM.

The interaction between the subsystems in the powernetwork is given by the power balance at each node j ∈ J .We distinguish between energy conserving nodes, given by

∑

m∈Mcj,mym,k + dj,mum,k + wj,k = 0, (1e)

for j ∈ Jc = 1, 2, . . . , Jc and energy dissipating nodes∑

m∈Mcj,mym,k + dj,mum,k + wj,k ≤ 0, (1f)




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for j ∈ Jd = Jc + 1, Jc + 2, . . . , J. Observe that Jc ∩Jd = ∅ and Jc∪Jd = J . In these expressions, cj,m is 1 ifthe correspondent power signal ym,k is connected to the nodej and 0 otherwise. The constant dj,m is −1 if the respectivepower signal um,k, flows into node j,it is 1 if the power flowsout of node j, and 0 if the respective signal is not connectedto the node. It should be noted that dissipating nodes (1f)exist, e.g., because mechanical braking can be modeled as apower dissipation in a node. Finally, due to Assumption 1,we have that for every m ∈ M, cj,m = 1 for only a singlej ∈ J .

C. Global Solutions to the Nonconvex CVEM Problem

The optimal control problem (1) is non-convex due to (1b).This might cause solvers to get stuck in a local minimum.In this section, we show that all (local) solutions to (1) areglobal solutions under very mild conditions.

To show that (1) has only global solutions, we relax theequality constraint (1b) to an inequality constraint, i.e.,

12q2,mu

2m,k + q1,mum,k + q0,m − ym,k ≤ 0 (1b’)

for m ∈ M and k ∈ K, allowing us to define a relaxedoptimal control problem

minym,k,um,k,xm,k

∑m∈M

∑k∈K amym,k+bmum,k

s.t. (1b’), (1c)−(1f).(2)

The discrete optimal control problem (2) is convex, thushas only global solutions. The optimal value of the originalproblem (1) and of the convex relaxation (2) satisfy

pCR ≤ pNC, (3)

where pCR and pNC denote the optimal value of the convexrelaxation (2) and the non-convex optimal control prob-lem (1), respectively, because the convex relaxation has alarger feasible set due to (1b’).

The next Theorem states that for this particular optimalcontrol problem and its relaxation, it holds that, pCR = pNC

and that (2) always has a solution that satisfies (1b’) withequality, thus it satisfies (1b) and solves (1).

Theorem 1: Assume that the bounds umm∈M andumm∈M are finite and that there exists a feasible pointum,k, ym,k, xm,km∈M,k∈K for (2) with strict inequali-ties (1b’) and (1d). Then, an optimal solution to (2) existsthat satisfies (1b’) with equality, pCR = pNC, and (1) onlyhas global optimal solutions.

D. An Equivalent Formulation of the CVEM Problem

In this subsection, we will reformulate the optimal controlproblem (1) into an equivalent form that will later be recastas a static optimization problem that is suitable to be solvedwith the operator splitting technique presented in Section III.The reformulation of (1) considers three main steps that aredescribed and justified as follows:

1) Substitution of the quadratic equality constraint (1b)into (1a), (1e), (1f). This substitution aims to reducethe number of decision variables by the elimination ofym,km∈M,k∈K from the optimal control problem.

2) Conversion of the inequality constraint (1f) intoan equality constraint using slack variableszj,kj∈J ,k∈K, of which some are constrainedto zj,k = 0 for all j ∈ Jc and k ∈ K. This yields thatthe Lagrangian of the optimal control problem (4) isa mapping from R to R, which is necessary for thealgorithm that will be proposed Section III, see [9,§5].

3) Introduction of quadratic penalty terms associated tothe equality constraints (1e) and (1f). This is donewithout removing these constraints. The resulting La-grangian function of the optimal control problem canbe seen as an augmented Lagrangian function. Theadvantage of this formulation is that it introducesregularization to the optimization procedure, therebyimproving the convergence properties of the algorithm[10, §3.2.1, §4.2].

The equivalent optimal control problem obtained as aresult of the above steps is given by

minum,k,xm,kzj,k

∑

k∈K

( ∑

m∈Mpm(um,k)+

∑

j∈J

σj

2

(∑

m∈Mrj,m(um,k)+ 1

2z2j,k

)2)

(4a)

s.t. (1c), (1d)∑

m∈Mrj,m(um,k) + 1

2z2j,k = 0 for all j ∈ J ,

(4b)zj,k = 0 for all j ∈ Jc, (4c)

where σj ∈ R+ are coefficients that weight the penalty termsin (4a). Furthermore,

pm(um,k)=am( 12q2,mu

2m,k+q1,mum,k+q0,m)+bmum,k,

(5)

rj,m(um,k)=cj,m( 12q2,mu

2m,k+q1,mum,k+q0,m)+

dj,mum,k + wj . (6)

It should be noted that constraint (4c) acts only on thenon dissipating nodes Jc, which means that the equalityconstraint (1e) is embedded in (4b) for this formulation.

The equivalence of the discrete time optimal controlproblems (1) and (4) is the main result of this section. Inorder to present this result, we will indicate a constraintqualification (CQ) that (4) satisfies. For a complete surveyof CQ for non-linear programming see [11]. In the followinglemma, we present a condition to guarantee that (4) satisfiesa linear independence CQ (LICQ).

Lemma 1: The feasible set of the discrete optimal controlproblem (4) satisfies LICQ, if for all m ∈M the bounds onum,k in (1d) satisfy

¯um > − q1,mq2,m

or um < − q1,mq2,m. (7)

The condition to guarantee LICQ of discrete optimalcontrol problem (4) presented in this lemma can be tested apriori via a simple inspection of the power bounds for all thesubsystems. For realistic applications, it typically holds that




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|q2,m| |q1,m|, which suggests that (7) is a mild condition.The satisfaction of LICQ by the optimal control problem (4)indicates that its critical points are regular. This will beexploited in the following Theorem to show the equivalencebetween the problem formulations (1) and (4) in terms ofthe Karush-Kuhn-Tucker (KKT) conditions for optimality.

Theorem 2: The optimization problems (1) and (4) havethe same necessary conditions for optimality, and sameglobal minimizers, if the conditions in Lemma 1 are satisfied.

III. A FORWARD-BACKWARD ALGORITHM FOR CVEMIn the previous section, we showed that the CVEM

problem (1) only has global solutions, thus solvers cannotget stuck in local minima. Moreover, it was demonstratedthat (4) is an equivalent formulation to (1). In this section,we will take advantage of the previous results to proposea method to solve the CVEM problem (1) using an oper-ator splitting approach. To achieve this we will recast theequivalent optimal control problem (4) into a non-convexstatic optimization problem. Then, we will split the costfunction of this static optimization problem as the sum ofa non-convex function and a convex set-valued function.This allows the forward-backward splitting method for non-convex optimization problems to be applied.

A. Reformulation as a Static Optimization Problem

To reformulate the optimal control problem (4) as a staticoptimization problem, we define the following vectors

um = col(um,kk∈K) ∈ RK , (8a)

xm = col(xm,kk∈K) ∈ RKnm , (8b)

zj = col(zj,kk∈K) ∈ RK , (8c)

wj = col(wj,kk∈K) ∈ RK , (8d)

for m ∈M and j ∈ J . This notation allows us to write thesolutions to (1c) in the compact form

xm = Φmxm,0 + Γmum, (9)

with matrices Φm ∈ RKnm×nm and Γm ∈ RKnm×K , andwrite (1d) as

¯xm1 ≤ xm ≤ xm1 and

¯um1 ≤ um ≤ um1. (10)

Now, by exploiting (8) - (10), we rewrite the discrete optimalcontrol problem (4) as the following static optimizationproblem:

minum,zj

∑

m∈MPm(um) +

∑

j∈J

σj

2

∥∥∥∑

m∈MRm,j(um) + 1

2z2j

∥∥∥2

2

(11a)s.t.

∑

m∈MRm,j(um) + 1

2z2j = 0, for all j∈J (11b)

zj = 0, for all j∈Jc (11c)um ∈ Ωm, for all m∈M,(11d)

where

Pm(um)= 1TK (amym(um)+bmum) , (12a)Rm,j(um)= cj,mym(um) + dj,mum + wj , (12b)

Ωm=um∈RK |1K¯xm≤Φmxm,0+Γmum≤1K xm,

1K¯um≤um≤1K um, (12c)

in which

ym(um)= col( 12q2,mu

2m,k+ q1,mum,k+ q0,mk∈K). (13)

This convenient reformulation shows a non-convex static op-timization problem that will be used to design a parallelizablealgorithm to solve the CVEM problem (1), based on the FBoperator splitting method [9, §5.1].

B. KKT Conditions

Operator splitting methods [12, §26] and [13] can beused to find zeros of (set-valued) mappings. In constrainedoptimization theory, these splitting methods are used to findthe points that satisfy the KKT conditions. In particular, con-straints (11d) will be embedded in the optimization problemusing indicator functions, which will make the optimizationproblem non-smooth. In doing so, the KKT conditions of thestatic optimization problem (11) can be represented as a set-valued mapping, which is shown in this subsection, so that(candidate) minimizers can be found using operator splittingmethods.

Let us first introduce vectors u = col(umm∈M),z = col(zjj∈J ), λ = col(λjj∈J ), and the Lagrangianfunction of (11), which is given by

L(u,z,λ) = h(u,z,λ) +∑

m∈MıΩm

(um) +∑

j∈Jc

ı0(zj), (14)

where ı0(zj) and ıΩm(um) are the indicator functionscorresponding to (11c) and (11d), respectively. Moreover,

h(u, z,λ)=∑

m∈MPm(um)+

∑

j∈J

(σj

2

∥∥∥∑

m∈MRm,j(um)+ 1

2z2j

∥∥∥2

2+

λ>j( ∑

m∈MRm,j(um) + 1

2z2j

)), (15)

with Lagrange multipliers λj ∈ RK , j ∈ J . The specificdesign of the Lagrangian function (14) is connected to thesplitting algorithm that will be presented in Section III-C. Namely, the linear part of the feasible set is expressedas indicator functions while the non-linear and possiblynon-convex part of the problem (11) is embedded in thecontinuously differentiable function h. The linear part of thefeasible set is also separable per m ∈ M, which will makethe algorithm parallelizable.

The KKT conditions of the augmented optimization prob-lem (11) can be expressed in terms of the subdifferential ofthe Lagrangian function (14) as 0 ∈ ∂L(u, z,λ), with

∂L(u, z,λ) =

col(Fm(u,z,λ)m∈M)col(Gj(u,zj ,λj)j∈Jc )col(Gj(u,zj ,λj)j∈Jd

)col(−Λj(u, zj)j∈J )

︸︷︷︸:=∇h

+

col(NΩm (um)m∈M)col(N0(zj)j∈Jc )

00

︸︷︷︸:=N

Ω×R(Jd+J)K

, (16)




77 / 110

where Ω = (∏m∈M Ωm)× 0JC and

Fm(u,z,λ)=∇Pm(um)+∑

j∈Jc

∇Rm(um)(λj+σjΛj(u,zj)

),

(17a)

Gj(u,zj ,λj)=diag(zj)(λj + σjΛj(u, zj)

), (17b)

Λj(u,zj)=∑

m∈MRm(um) + 1

2z2j , (17c)

in which

∇Pm(um) = am1K∇ym(um)+bm1K , (18a)∇Rm(um) = cj,m∇ym(um)+dj,mIK , (18b)∇ym(um) = diag(q2,mum,k + q1,mk∈K). (18c)

In (16), we characterized the KKT conditions related tothe static optimization problem (11). Under certain regu-larity conditions, the points u∗, z∗,λ∗ satisfying 0 ∈∂L(u∗, z∗,λ∗) provide candidate minima of the optimiza-tion problem (11), see [10, §3.3.1] and [14] for a detaileddiscussion on this topic. In the theorem below, we formallystate that 0 ∈ ∂L(u∗, z∗,λ∗) leads to global minimizers tothe discrete time optimal control problem (1).

Theorem 3: Suppose conditions of Lemma 1 are satisfied,and sets Ωm for all m ∈ M are compact. Then, globalsolutions y∗m,k, u∗m,k, x∗m,k to the discrete time optimalcontrol problem (1) can be obtained from points u∗, z∗,λ∗satisfying 0 ∈ ∂L(u∗, z∗,λ∗).

C. Forward-Backward (FB) Splitting AlgorithmIn this section, we apply the FB splitting method presented

in [9, §5.1], which will be used to find points col(u∗, z∗,λ∗)satisfying 0 ∈ ∂L(u∗, z∗,λ∗), thereby finding global min-imizers of the optimal control problem in (1). It should benoted that for convex optimization problems, FB splittingleads to the widely used proximal gradient method (see [12,§26.5]), while recent extensions of the FB splitting showapplication to solve non-convex problems [9], [15].

To see how FB splitting is applied here, note that the right-hand side of (16) consists of two terms: ∇h, which is thegradient of the continuously differentiable function (15), andNΩ×R(Jd+J)K , which is the operator that contains the normalcones of the linear feasible set defined by constraints (11c)and (11d). Now let Φ be a preconditioning matrix, given by

Φ := diag(αmm∈M, θjj∈J), , γjj∈J (19)

where αm, γj and θj are positive real constants that definethe step sizes of the proposed algorithm. The FB operator

ICEyice uice

HBVxhvb

w b

ybr

EM

yem

uem

uhvb yhvb w a

Electric powerNon-electric power

a

b

Fig. 2: Power network topology for a parallel-hybrid vehicle.

splitting method [9, §5.1], applied on the mapping in (16)with preconditioning Φ reads as

ωi+1 = projΩ×R(Jd+J)K (ωi −Φ∇h(ωi)), (20)

with ω = col(u, z,λ) and superscript i ∈ 1, 2, . . .denoting the i-th iteration. It should be noted that both thefunction ∇h as well as the projection are highly structured.This allows for a paralellizable algorithm. Furthermore, theprojections on Ωm can be implemented efficiently as aconstrained least squares problem. The final algorithm ispresented below.

ALGORITHM 1: Forward-Backward Splitting

Initialization:Select step-sizes αm, m ∈ M, γj , j ∈ J and θj , j ∈ Jd;select regularization weights σj , j ∈ J and select initialvalues u0

m ∈ Ωm for all m ∈M, zij = 0K×1 for all j ∈ Jcand all i ∈ N, z0

j ∈ RK for all j ∈ Jd, and λ0j ∈ RK for all

j ∈ J .

Iterate until convergence:

ui+1m = projΩm

(uim−αmFm(ui, zi,λi)), for all m ∈Mzi+1j = zij − θjGj(u

i, zij ,λij), for all j ∈ Jd

λi+1j = λij + γjΛj(u

i, zij), for all j ∈ J

It remains to show how to choose the step sizes inAlgorithm 1 so that the algorithm converges. In [9, Th. 5.3],it is shown that if αm, γj and θj are chosen sufficientlysmall, the algorithm converges. For more details, the readeris referred to [9].

IV. NUMERICAL EXAMPLE

In this section, we will use the method presented inSection III-C to solve the CVEM optimal control problem fora parallel-hybrid electric vehicle, consisting of and internalcombustion engine (ICE), electric machine (EM) and a high-voltage battery (HVB). For notational convenience, we usethe set M = ice, em, hvb, instead of M = 1, 2, 3, andthe set J = a, b, instead of J = 1, 2.

The power network for this study case is presented inFig. 2. There exists two nodes J that describe the mechanicaland electrical power balance in the vehicle. These nodes arebridged by the EG that acts as a single power converterto transform power between the electrical and mechanicaldomains. The power network also contains exogenous signalsthat represent the power consumed by auxiliary subsystemswa = 1.5[kw] and the traction power request wb = yreq,which is presented in Fig.3a. The presence of the mechanicalbraking power ybr ≤ 0 in node b implies that it is adissipative node, thus Jc = a and Jd = b. Theparameters that describe the subsystems are given in Table Iand by selecting, db,ice = da,em = −1, ca,hvb = cb,em = 1and all other coefficients as zero. Thus, the power networkis described by

yhvb,k + yem,k + wa,k = 0, (21)−uice,k − uem,k + wb,k = ybr,k ≤ 0. (22)




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0 10 20s [km]

60

65

70

75

80

v [k

m/h

]

0

50

100

150

200

250

300

0 10 20s [km]

-200

-100

0

100

200

300

y req [k

w]

0

50

100

150

200

250

300

Elev

atio

n pr

ofile

h [m

]

(a) Driving cycle and power request provided.

0 10 20s [Km]

0

50

100

150

200

250

y ice [k

w]

Solution 1Solution 2

0 10 20s [km]

-80

-60

-40

-20

0

20

y hvb [k

w]


(b) Optimal power of the internal combustionengine and the high voltage battery.

0 10 20s [km]

-100

-50

0

y br [k

w]


0 10 20s [km]

3

4

5

6

x hvb [k

wh]


(c) Optimal breaking power and stored energy onthe battery.

Fig. 3: Numerical Example

TABLE I: Subsystem Parameters

ICEqice,2 = 0 qice,1 = 1 qice,0 = 0

¯uice = 310[Kw] uice = 0[kw]

EMqem,2 = 2× 10−5 qem,1 = 2.53 qem,0 = 19

¯uem = 210[kw] uem = −210[kw]

HVBqhvb,2 = 0.001671 qhvb,1 = −1 qhvb,0 = 0

¯uhvb = 92.4[kw] uhvb = −92.4[kw]

¯xhvb = 210[kw] xhvb = −210[kw]xhvb,0 = 11988[kJ ] xhvb,K = 11988[kJ ]Ahvb = 1 Bhvb = −5

For this case, the CVEM is formulated to minimize theenergy consumed by the ICE, which implies that aice = 1while the other am and all bm are zero. In order to obtainresults that reflect the energy consumption only related tofuel, constraints on the initial and final charge states of thebattery are imposed, i.e., xhvb,0 = xhvb,K . It is important toremark that this constraint on the final state can be easilyincluded in the feasible set (12c) for m = hvb, withoutaffecting the main results and the methodology presented inthis paper. Finally, it is important to mention that the problemhas been discretized using a sampling time τ = 5[s], whichleads to Ahvb = 1 and Bhvb = −5.

Given the fact that condition (7) of Lemma 1 can be easilyverified for this example, we have that Theorem 3 states thatAlgorithm 1 gives a global solution. For this example, twopossible solutions are presented in Fig. 3b and Fig. 3c, whichare obtained using the same initial conditions ( u0

m = 0,z0j = 0 and λ0

j = 0 for all m ∈ M and j ∈ J ) forAlgorithm 1. but using different step sizes, see Table II.The energy consumed in both cases is 9.889551[l]/100[km],which shows that both solutions are indeed global minimizersof the optimal control problem. The convergence of Solu-tion 1 was obtained after 365 iterations of the algorithm,while Solution 2 converged after 432 iterations. This is aremarkable result due to the fact that the use of the dualdecomposition approach proposed in [8], does not convergefor this numerical example. The observed improvement inconvergence is related to the regularization properties intro-duced by the projections performed in Algorithm 1.

V. CONCLUSIONS

In this paper, we have proven that the non-convex CVEMproblem only has global optimal solutions under mild condi-

TABLE II: Parameters used for Algorithm 1.

Solution 1 Solution 2αice 100.9 80.9αem 2.1 2.1αhvb 62.1 62.1γa 0.001 0.001γb 0.01 0.001θa 0.04 0.1σa 0.008 0.008σb 0.01 0.01

tions. Moreover, we exploited this result in order to proposea reformulation of the CVEM problem that can be solvedusing a FB splitting algorithm. The guaranteed convergenceand the regularization properties of the algorithm allowed usto solve a CVEM problem for a parallel hybrid vehicle thatpresents almost linear power losses, which was not possiblewith approaches like dual decomposition.

REFERENCES

[1] “Global EV Outlook 2017,” Int. Energy Agency, Tech. Rep., 2017.[2] B. de Jager, T. van Keulen, and J. Kessels, Optimal Control of Hybrid

Vehicles, ser. Advances in Industrial Control, 2013.[3] L. Guzzella and A. Sciarretta, Vehicle Propulsion Systems. Springer,

2013.[4] J. Kessels, J. H. M. Martens, P. P. J. Van den Bosch, and W. H. A.

Hendrix, “Smart vehicle powernet enabling complete vehicle energymanagement,” in Proc. IEEE Vehicle Power and Propuls Conf., 2012.

[5] N. Murgovski, L. Johannesson, X. Hu, B. Egardt, and J. Sjoberg,“Convex relaxations in the optimal control of electrified vehicles,” inProc. IEEE American Control Conf., 2015.

[6] E. Feru, N. Murgovski, B. de Jager, and F. Willems, “Supervisorycontrol of a heavy-duty diesel engine with an electrified waste heatrecovery system,” Control Eng. Pract., 2016.

[7] H. Chen, J. T. B. A. Kessels, M. C. F. Donkers, and S. Weiland,“Game-theoretic approach for complete vehicle energy management,”in Proc. IEEE Vehicle Power and Propulsion Conf., 2014.

[8] T. C. J. Romijn, M. C. F. Donkers, J. T. B. A. Kessels, and S. Weiland,“A distributed optimization approach for complete vehicle energymanagement,” IEEE Trans. on Control Syst. Technol., 2018.

[9] H. Attouch, J. Bolte, and B. F. Svaiter, “Convergence of descentmethods for semi-algebraic and tame problems: proximal algorithms,forward–backward splitting, and regularized gauss–seidel methods,”Mathematical Programming, vol. 137, no. 1-2, pp. 91–129, 2013.

[10] D. P. Bertsekas, Nonlinear Programming. Athena Scientific, 1999.[11] J. V. Burke, “Constraint qualifications for nonlinear programming -

Numerical Optimization. Course Notes, AMath/Math 516,” 2012.[12] H. H. Bauschke, P. L. Combettes et al., Convex analysis and monotone

operator theory in Hilbert spaces. Springer, 2017.[13] E. K. Ryu and S. Boyd, “A Primer on Monotone Operator Methods,”

Appl. Comput. Math., 2016.[14] F. J. Gould and J. W. Tolle, “A Necessary and Sufficient Qualification

for Constrained Optimization,” SIAM J. Appl. Math, 1971.[15] A. Themelis, L. Stella, and P. Patrinos, “Forward-backward envelope

for the sum of two nonconvex functions: Further properties and non-monotone line-search algorithms,” arXiv:1606.06256 [math], 2016.




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A Port-Hamiltonian Approach to Complete Vehicle Energy Management:A Battery Electric Vehicle Case Study

G.P. Padilla, J.C. Flores Paredes, M.C.F. Donkers

Abstract— In this paper, we present a modelling approach tovehicle energy management based on Port-Hamiltonian systemsrepresentations. We consider a network of interconnected port-Hamiltonian systems that describes the powertrain componentsand auxiliaries in the vehicle. This description is suitable toobtain a systematic approach to formulate a decomposableoptimal control problem for Complete Vehicle Energy Manage-ment. A cost function that describe total energy consumptionof the vehicle is proposed in terms of internal energy andlosses of each system connected system in the network, whichhas provided an insightful physical interpretation. Takingadvantage of the modularity of the formulation proposed, wepresent a distributed optimization algorithm to find solutionsto the energy management problem. To illustrate this modellingmethodology, a case study to optimize the energy consumptionof a battery electric vehicle is proposed.

I. INTRODUCTION

Electrification of vehicles, and particularly the deploymentof fully electric vehicles, is considered a way to mitigateCO2 emissions related to the transport sector, which wasresponsible for 14% of the global CO2 emissions in 2014 [1],with 95% of the transportation energy based on fossil fuels,estimated in 2016 [2]. However, electric vehicles are knownto have a limited range and a time-consuming chargingprocess, leading to range anxiety, which is the user’s concernto be unable to reach the destination with the current batterycharge [3]. This concern has a negative impact for theintroduction of electric vehicles in the market [4], [5].

In order to mitigate range anxiety, it is possible to useEnergy Management Strategies (EMS) [6], [7], which aimto minimize energy consumption during operation, thereforeextending the driving range. These strategies can most ofthe time be implemented as software in the vehicle, whichimplies limited cost for production. A specific branch ofEMS that considers influence of all auxiliary subsystems inthe vehicle is referred as Complete Vehicle Energy Man-agement (CVEM) [8]. Specifically, CVEM strategies aresolutions to optimal control problems (OCP) that aim tominimize the power consumption of the vehicle subject tothe dynamical behavior of interconnected subsystems in anetwork, e.g., internal combustion engine, battery, electricmotor, climate system, etc. Obtaining as a result an energyefficient operation of the vehicle.


Paul Padilla, Juan Flores and Tijs Donkers are with the Department ofElectrical Engineering of Eindhoven University of Technology, Netherlands.E-mail: g.p.padilla.cazar, [email protected]

In the current literature, the modelling and the OCPformulation are mainly obtained by applying power-basedapproaches [9]–[13]. The main idea of these approaches isto describe the behavior of the interconnected subsystemsnetwork in term of power interactions. In other words,each subsystem is normally represented by an energy bufferconnected to a power converter. The buffer stores energyand it is modeled as a dynamic linear system, which oftenresembles an integrator. The power converter representsenergy consumption and is modeled as a static non-linearity,normally approximated to a quadratic function from datameasurements, which often lacks a physical interpretation.By considering only power and energy to model the subsys-tems, the number of decision variables in the OCP problem isreduced. Moreover, the models show low complexity whichmakes the CVEM problem tractable.

However, power-based approaches are difficult to usewhen the optimization variables cannot be easily describedin terms of power. For instance, in [14], [15] attempts tounify CVEM and eco-driving (finding energy optimal veloc-ity profiles) are presented. In this case, velocity becomesa decision variable and describing it in terms of poweris a non trivial task. Other disadvantage of power-basedapproaches is that it is difficult to constrain physical variablesthat describe power in the model. For instance, imposingconstraints on the battery power do not necessarily implythat the solution of the OCP is physically realizable. Power inthe battery is described as the product of voltage and current,thus an optimal solution that satisfies power constraints notnecessarily satisfies the bounds on voltage and currents.Additionally, power-based approaches do not always allowfor an intuitive formulation of decomposable OCP, which areuseful when distributed optimization technique are applied toapproximate solutions. For instance, in [10] the concept of‘sum of losses’ is introduced to formulate separable OCP forCVEM. Unfortunately, this concept is not always simple toapply.

In the paper, we propose an alternative modelling frame-work based on interconnected port-Hamiltonian (pH) sys-tems [16], [17] aiming to overcome the previously discussedlimitations of power-based approaches. In this modellingapproach, each subsystem is represented as a dissipativedynamical system that inherently describes energy lossesand changes of internal energy in the subsystem. The maincontributions of this work are threefold. First, we proposeto use pH representations to model networks for CVEMapplications, thus unifying the power based concepts ofenergy buffers and power converters in a single dynamical




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model, i.e., in terms of internal energy and power losses.Second, a systematic approach is proposed to formulatea decomposable OCP for CVEM, which is useful whendistributed static optimization techniques are considered tosolve the CVEM OPC. Third, an adaptation of the distributedoptimization algorithm of [13] is presented to find solutionsto the proposed pH CVEM OCP.

The remainder of this paper is organized as follows. InSection II, we briefly introduce the pH modeling theoret-ical background and use these concepts to model CVEMnetworks, which in Section III, allow us us to formulatedecomposable OCP for CVEM with pH representations. InSection IV, a distributed static optimization problem to solvepH OCP for CVEM is summarized. The pH methodologyfor CVEM is applied to a case study presented in SectionV. Finally, the conclusions are drawn in Section VI.

II. A PORT-HAMILTONIAN MODELLING APPROACH FORCVEM

The main objective of this section is to present the pHmodeling framework for CVEM and the formulation ofdecomposable OCP. Furthermore, we adapt the pH formalismto the holistic philosophy of CVEM presented in [8]. Toachieve this, we briefly discuss the main concepts of pHrepresentations. Later, the main features of networks ofinterconnected subsystems for CVEM are translated into acompatible description with pH models.

A. Port-Hamiltonian Representation

From a modeling perspective, a pH representation isa port-based modelling approach that considers a pair ofconjugated variables in each port, which product representspower. This has been depicted in Fig. 1, where a genericgraphical representation of a two-port pH system is pre-sented. Although in literature there is a large amount ofpossible pH representations, we will only focus on linearinput-state-output models with direct feedthrough term in thispaper.

Consider m ∈M := 1, . . . ,M linear subsystem, whereM is the total number of interconnected subsystems in thenetwork. The internal energy of the subsystems is expressedas a quadratic function given by

Hm(xm) = 12x>mQmxm, (1)

where xm ∈Rn represents the state vector and Qm ∈Rn×nis a symmetric positive semi-definite matrix, which is calledenergy matrix. This energy function is used in an input-state-output with feedthrough term pH representation [17], [16, §4]

yinm Σm

uinm

uoutm

youtm P outmP in

m Subsystem

Fig. 1: Port-Hamiltonian (sub)system Σm

of the m subsystem, and is given by

xm=(Jm−Rm

)Qmxm+Bmum (2a)

ym=(Bm + 2Pm)>Qmxm + (Mm + Sm)um (2b)

where xm∈Rn is the state vector and um, ym∈R2 are thecontrol input and output, respectively, given by

um =

[uinm

uoutm

]and ym =

[yinm

youtm

], (3)

From the port-Hamiltonian formalism in (2), we candescribe the following power balance equation [16, §4]

∂Hm∂t

(xm) = −[xmum

]>Lm

[xmum

]+ y>mum. (4)

in which

Lm :=

[QmRmQm QmPmP>mQm Sm

]. (5)

In case Lm 0, passivity of the system is guaranteed[18, §7.1]. Furthermore, an energy balance can be describedby reorganizing and integrating both sides of (4) over theinterval [t0, tf ], hence obtaining

∆Em = Einm + Eout

m = ∆Hm + Lm (6)

with

∆Em =

∫ tf

t0

y>mum dt =

∫ tf

t0

yinmu

inm + yout

m uoutm dt

︸︷︷︸Einm+Eout

m

(7a)

∆Hm = Hm(xm(tf ))−Hm(xm(t0)) (7b)

Lm =

∫ tf

t0

[xmum

]>Lm

[xmum

]dt (7c)

where ∆Em represents the net supplied energy, ∆Hm is thechange on internal energy and Lm describes the total energylosses. Moreover, let us define the set of source subsystems

S := m ∈M|Einm = 0, (8a)

the set consumersC :=M\ S, (8b)

and the set of terminal subsystems

T := m ∈M|Eoutm = 0. (8c)

A physical example of a source subsystem, which only hasan output port, is chemical energy in a battery. On the otherhand, mechanical brake is a physical example of a terminalsystem, which has only an input port.

B. Port-based Network Topology

The interconnections of pH systems have been studiedin [19], where a methodology to describe networks of pHsystems with the final goal to perform stability analysis hasbeen proposed. Although in this paper we consider someideas from the work presented in [19], we will deviate fromthis approach to propose a specialized description of network




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Subsystem m

Junction j

youtm

uoutm

uinm

yinm

Equalitynode

nodeAdditive

Powerjunction

Fig. 2: Port-based topology for interconnected port-Hamiltonian systems

topologies that capture specific features found in CVEMapplications.

A tree structure port-based network topology is presentedin Fig. 2, where each subsystem m ∈ M is depicted by agray square. The interconnection between subsystems takesplace through junctions j ∈ J := 1, . . . , J, where J ∈ Nis the total number junctions in the network. These junctionsare power preserving interconnections that are described interms of the conjugated input output variables as follows∑

m∈Maj,mu

inmy

inm + bj,mu

outm y

outm = 0, for all j ∈ J , (9)

where aj,m = 1 if the input port of subsystem m is connectedto junction j, and aj,m = 0 otherwise. Similarly, bj,m = 1if the output port of subsystem m is connected to junctionj, and bj,m = 0 otherwise.

In CVEM networks, there is a type of junction thatcan be decomposed into an additive node and an equalitynode. A physical interpretation of these type of nodes isobserved in electrical interconnections between subsystems,e.g., Kirchhoff’s law of currents can be seen as an additivenode, while Kirchhoff’s law of voltages is an equality node.Let us define the set of all the decomposable nodes in thenetwork Jn := 1, . . . , Jn, where Jn ≤ J is the totalnumber of these type of junctions in the network. Thus, thepower preserving equation (9) can be rewritten as∑

m∈M[aj,m 0]um + [0 bj,m]ym = 0, (10a)

[aj,m 0]ym − [0 bj,m]um = 0 for all m ∈M, (10b)

for all j ∈ Jn. Note that (10a) and (10b) are referred asadditive node and an equality nodes, respectively.

Another type of power preserving interconnections ob-served in CVEM networks are power junctions. The powerjunction artifact introduced in this framework obeys thenecessity to resemble the functionality of DC-DC convertersin CVEM networks. It is well known that DC-DC converterscan be represented as non-linear pH systems, e.g., see [20].In fact, from steady-state approximations of the modelsproposed in [20] under the assumption that the converter islossless, it is possible to prove that DC/DC converters satisfy

(9). In order to keep consistency in the notation, it is possibleto rewrite (9) as

∑

m∈My>m

[aj,m 0

0 bj,m

]um= 0, for all j∈ Jp; (11)

where Jp := Jn+1, . . . , J is the set of all power junctionsin the network.

III. OPTIMAL CONTROL PROBLEM FOR PH CVEM

In the first part of section, we first provide a generalformulation for a continuous-time optimal control problem(OCP) for the pH CVEM modelling approach described inthe previous section. Subsequently, we propose a physicallyinsightful cost function that enables the decomposability ofthe proposed pH OCP for CVEM. Finally, we present adiscrete-time OCP for pH CVEM.

The main objective of CVEM is to minimize the totalenergy consumed by a network of interconnected subsystemsover a fixed time interval, subject to the dynamical behaviorof each subsystem and bounds on the respective inputs,outputs, and states. From a mathematical perspective, thisgoal can described as an OCP. Hence, a general formulationof the pH OCP for CVEM is given by

minxm,um,ymm∈M

J(xm, um, ymm∈M

)(12a)

subject to: (2), (10), (11),

¯xm ≤ xm ≤ xm, for all m ∈M (12b)

¯um ≤ um ≤ um, for all m ∈M (12c)

¯ym ≤ ym ≤ ym, for all m ∈M (12d)

Note that (12a) represents a general cost function that aimsto describe the total energy consumed by the subsystems inthe network. We will propose a physically insightful, anddecomposable cost function for the OCP (12) below.

A. Power Consumption in pH CVEM

The physical interpretations provided by pH models interms of internal energy and energy losses of the subsystemscan be exploited to formulate a sensitive description ofthe energy consumed in the network. Thus, we could aimto maximize the change of internal energy in each sourcesubsystem, which implies minimizing consumption, i.e., see(7b). This can be described by (12) if the cost function isdefined as

J =∑

m∈S−∆Hm. (13)

Note that the minus sign in (13) is necessary because (12)is defined as a minimization problem.

Interestingly, by decomposing the cost function (13) itis possible to obtain an equivalent expression that can beinterpreted as minimizing the aggregated internal energy ofall the consumers and the energy losses of all the subsystems.In order to see this, let us recall that power preservinginterconnections are considered for all the junctions j ∈ J




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in the network. Therefore, by integrating (9) for a given timeinterval we obtain the following energy balance

∑

m∈Maj,mE

inm + bj,mE

outm = 0, (14)

for all j ∈ J . Hence, it is possible to follow a recursiveprocedure where we substitute (6) and (14) into (13) to obtainthe partially expanded function

J =∑

m∈SLm − Eout

m

=∑

m∈S

(Lm + bm,j

∑

n∈Can,jE

inn

)

=∑

m∈S

(Lm+bm,j

∑

n∈Can,j

(∆Hn + Ln−Eout

n

)). (15)

The recursive expansion of this cost function ends withterminal subsystems in each branch of the tree network.Hence, the complete expansion of (13) is given by

J =∑

m∈MLm +

∑

n∈C∆Hn, (16)

which implies that maximization of internal energy is equiv-alent to minimizing the losses of all subsystems and theinternal energy of only the consumers in the network.

These results not only show the importance of pH for-mulations to obtain physically insightful interpretations ofcost functions, but also open the door to obtain possibleequivalent cost functions that could be beneficial for specificsolution methods used to solve the CVEM OCP (12), suchas the method presented in Section IV

B. DiscretizationThe use of a static optimization techniques to approxi-

mate solutions of (12) in a finite time horizon requires thediscretization of this problem. It is important to remark thatwe are aware that there are no conclusive results aboutpreservation of the discrete-time energy balance equationunder interconnection of discrete-time pH systems. Althoughpromising results can be found in literature, e.g., [21], [22],we will neglect this issue in this work.

To arrive to discrete-time OCP, (12) is discretized at timestk = kδt+ t0, k ∈ K = 0, . . . ,K−1, with time step δt =tf−t0K for some K ∈ N using the forward Euler discretization

method. This lead to

minxm,k,um,k,ym,km∈M

k∈K

∑

m∈Mem∆Hm + Lm (17a)

subject to the discrete-time dynamical model

xm,k+1 =(I + δt(Jm−Rm

)Qm)xm,k+δtBmum,k, (17b)

ym,k=(Bm + 2Pm)>Qmxm,k + (Mm + Sm)um,k, (17c)

and discrete-time versions of (10), (11), andxm,k, um,k, ym,k ∈ Ωm for all m ∈ M k ∈ K,where

Ωm :=xm,k ∈ Rn, ym,k ∈ Rp, um,k ∈ Rp|satisfy (12b), (12c) and (12d). (17d)

Note that (17a) is an equivalent formulation of the costfunction (16), where em = 1 if m ∈ C and em = 0 otherwise.This indicates that the cost function (17a) penalizes changesof internal energy only for the set of energy consumers. Thediscretized versions of internal energy and losses are givenrespectively by

∆Hm = 12x>m,KQmxm,K − 1

2x>m,0Qmxm,0, (18a)

Lm = δt∑

k∈K

[xm,kum,k

]>Lm

[xm,kum,k

]. (18b)

IV. A DISTRIBUTED OPTIMIZATION APPROACH TO PHCVEM

In the previous section we proposed a complete frameworkto formulate OCP for CVEM applications using pH models.An advantage of this approach it is that is possible toformulate decomposable problems by expanding the costfunction in terms of internal energy and losses of the system.In this section, we use this advantage to adapt the Forward-Backward Operator Splitting (FBS) method presented in[13] to find solutions to the pH CVEM problem (17).The aforementioned FBS method is a parallelizable staticoptimization technique that takes advantage of the modularstructure of the OCP (17).

The FBS method is based on the augmented Lagrangianfunction

L = F +∑

k∈K

∑

m∈MıΩm(xm,k, um,k, ym,k) (19)

where the dependencies in functions L and F have beenomitted to improve readability. In (19), ıZ(z) is an indicatorfunction such that ıZ(z) = 0 if z ∈ Z and ıZ(z) = +∞otherwise. Furthermore, we define

F (xm,k, um,k, ym,k, λj,k, µj,k, κj,k) =

=∑

m∈M

(em∆Hm + Lm

)

+∑

k∈K

( ∑

j∈Jnλi,k

( ∑

m∈M[aj,m 0]um,k + [0 bj,m]ym,k

)

+∑

j∈Jnηj

∥∥∥∥∥∑

m∈M[aj,m 0]um,k + [0 bj,m]ym,k

∥∥∥∥∥

2

2

+∑

j∈Jnµ>j,k

( ∑

m∈M[aj,m 0]ym,k − [0 bj,m]um,k

)

+∑

j∈Jnρj

∥∥∥∥∥∑

m∈M[aj,m 0]ym,k − [0 bj,m]um,k

∥∥∥∥∥

2

2

+∑

j∈Jp

∑

m∈Mκ>j,k

(y>m,k

[aj,m 0

0 bj,m

]um,k

)

+∑

j∈Jp

∑

m∈Mσj

∥∥∥∥y>m,k[aj,m 0

0 bj,m

]um,k

∥∥∥∥2

2

), (20)

In (20), λi,k ∈ R2, µi,k ∈ R2, κi,m,k ∈ R, representdual variables related to the discrete-time versions of the




83 / 110

Vhvb

Ihvb

Battery

Freq

vreq

vbr

Fbr

vem

Fem

Electric

High Voltage

Machine

n

Power

Junction

Mechanical

Junction

DC/DC

Vem

Iem

MechanicalBrake

Driveline

Power Request

ElectricCircuit

Source

Voc

Ihvb

Fig. 3: CVEM port-based configuration for a battery electric vehicle.

junction constraints (10) and (11) respectively, and the non-negative constants ηi, ρi, σj,m are weights to the respectiveregularization terms of the aforementioned constraints. Now,by defining the following vectors

xm =[x1 . . . xK

]> ∈ RnK (21a)

um =[u0 . . . uK−1

]> ∈ R2K (21b)

ym =[y1 . . . yK

]> ∈ R2K (21c)

λj =[λj,1 . . . λj,K

]> ∈ R2K (21d)

µj =[µ>j,1 . . . µ>j,K

]> ∈ R2K (21e)

κj =[κ>j,1 . . . κ>j,K

]> ∈ RK (21f)

wm =[x>m u>m y>m

]> ∈ R(4+n)K (21g)

it is possible to rewrite (20) and (17d) in vector form toobtain F(wm,λj ,µj ,κj) and Ωm(wm) respectively. Thus,by applying the operator splitting method presented [13], wecan be obtain the following static optimization algorithm tofind solutions to the pH CVEM OCP (17).

ALGORITHM 1: Forward-Backward Splitting

Initialization:Select step-sizes γm, m ∈ M, φj , j ∈ Jn, θj , j ∈ Jn andυj , j ∈ Jn; select regularization weights ηj , j ∈ Jn, ρj ,j ∈ Jn, σj , j ∈ Jp and select initial values w0

m ∈ Ωm forall m ∈ M, λ0

j ∈ RpK for all j ∈ Jn, µ0j ∈ RpK for all

j ∈ Jn and κ0j ∈ RK for all j ∈ Jp; , and all the iterations

i ∈ N.

Iterate until convergence:

wi+1m = projΩm

(wim−γm∇wm

F), for all m ∈M

λi+1j = λj + φj∇λjF, for all j ∈ Jn

µi+1j = µj + θj∇µjF, for all j ∈ Jn

κi+1j = κj + υj∇κjF, for all j ∈ Jp

The superscript i ∈ 1, 2, . . . denotes the i-th iteration, thegradient of a continuously differentiable function G(·) withrespect the variable z is represented by ∇zG(·), and themapping projZ : RZ → Z for a closed set Z ⊆ RZ denotesthe projection onto Z , i.e., projZ(z) = argminv∈Z ‖v−z‖2.

Algorithm 1 takes advantage of the modularity of CVEMproblem by the use of parallel projections that can be

efficiently solved using linearly constrained least-squares.solvers. Additionally, note that the procedure to obtainAlgorithm 1 has been briefly described in this section asa derivation of the work presented in [13]. Although it ispossible to rigorously prove that the solutions provided byAlgorithm 1 corresponds to solutions of the OPC (12) byfollowing steps similar to the ones described in [13], weconsider that this discussion is outside of the scope of thispaper and we will leave it for future extensions of this work.

V. CASE STUDY: BATTERY ELECTRIC VEHICLE

The pH CVEM framework described in Section II andthe associated OCP of Section III are illustrated in a casestudy presented in this section. We consider a Battery ElectricVehicle (BEV), whose topology is depicted in Fig. 3. A high-voltage battery (HVB) is connected via a DC/DC converterto an electric machine (EM). The mechanical part in thisdriveline yields a power balance between the requestedpropulsion power and the mechanical braking power. Eventhough this configuration is simple, it allows to illustrate theproposed modelling approach for CVEM and the solutionmethod.

In the remainder of this section, we provide details formodelling the topology presented in Fig. 3 and formulate anOCP for CVEM. Later, we discuss the connections betweenpower-based and pH OCP formulations highlighting the ad-vantages of the the approach proposed in this paper. Finally,results of simulations for this case study are presented andanalyzed.

A. Modelling

In the topology presented in Fig. 3 is possible to observethe physical conjugated variables that describe the powerpreserving interconnection between subsystems. We will usethese variables to define the input-output ports for eachsubsystem. Additionally, the pH models for the subsystemswill be obtained from first-principle models and described interms of the pH representation given in Section II-A.

The set M is the set of interconnected subsystems forthe BEV considered in this case. For the sake of readability,we use M = vs, ec, em, br, req instead of a enumeratedset to denote battery voltage source, battery electric circuit,electric machine, mechanical brake and power request, re-spectively. Note that vs is a source subsystem, while br and




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req are terminal subsystems. Similarly, Jn = 1 representsthe mechanical node in the topology, while Jp = 2 isthe power junction of the network which corresponds to theDC/DC converter.

1) High-Voltage Battery: We consider an equivalent cir-cuit model with constant open circuit voltage Voc and internalresistance Rhvb in this paper. The voltage and current in theterminals are denoted as Vhvb and Ihvb respectively. The pHformulation of the HVB constituted of subsystems vs andec, which represent an ideal lossless voltage source and theinternal resistance of the battery, respectively. The input andoutput ports and the states of the aforementioned subsystemsare defined as

uvs=

[0

−Ihvb

], yvs=

[0Voc

], (22a)

uec=

[VocIhvb

], xec=SoC, yec=

[Ihvb−Vhvb

], (22b)

where SoC denotes the HVB state-of-charge. Note thatthese definitions indicate that the interconnection betweenthe voltage source vs and the electric circuit ec is powerpreserving, i.e., yout

vs uoutvs + yin

ecuinec = 0. Therefore, the pH

models of the subsystems in the HVB are given by

Qec= Jec=Rec= 0, Pec=[0 0

], Bec =

[0 − 1

qhvb

]

Mec=

[0 1−1 0

], Sec=

[0 00 Rhvb

], (22c)

where qhvb denotes the HVB capacity. Numerical values forthe parameters associated to the HVB are given in Table I.

2) Electric Machine: The HVB is connected to an EMthrough an ideal DC/DC convertor. The DC/DC converteris considered a power junction in this is approach, whosedescription will be provided later in this section. The pHformulation for the EM is defined using

uem=

[VemFem

], xem=

[LIem

grrwImvem

], yem=

[Iem−vem

], (23a)

where Vem, Iem denotes the voltage and current in theEM respectively, Fem is the propulsion force and vem thelongitudinal velocity. Additionally, the positive constants L,Im, rw and gr denote the electric inductance, moment ofinertia, wheel radius and total gear ratio, respectively. Thesedefinitions allow the pH model of the EM to be defined using

TABLE I: High Voltage Battery parameters

Variable Value Definition ObservationsVoc 658 open circuit voltage [V ]Rhvb 0.33 series resistance [Ω]qhvb 180 capacity of the battery [Ah]

¯Ihvb -800 minimum current [A]Ihvb 800 maximum current [A]

¯Vhvb 400 minimum voltage [V ]Vhvb 1000 maximum voltage [V ]

TABLE II: Driveline parameters

Variable Value Definition ObservationsRe 1 equivalent resistance [Ω]L 2 equivalent inductance [H]Ie 99.68 equivalent axle inertia [kgm2]β 1.1e-03 friction coefficient (rotational) [Nm s/rad)]κτ 4.2175 torque constant [Nm/A]κV 4.27 back-electromotive constant [V s/rad]rw 0.5715 wheel radius [m]gr 6.1 gear ratio [−]

¯Iem -600 minimum current [A]Iem 600 maximum current [A]

¯Vem -1000 minimum voltage [V ]Vem 1000 maximum voltage [V ]

Qem =

[1L 00 1

Im

], Jem =

[0 κV +κτ

2−κV +κτ

2 0

], (23b)

Rem =

[Re

κV −κτ2

κV −κτ2 β

], Bem =

[1 00 − rwgr

], (23c)

Pem = Mem = Sem =

[0 00 0

], (23d)

where Re, κV , κτ and β are positive constants that denotethe equivalent electric resistance, back-electromotive con-stant, torque constant and damping coefficient, respectively.In Table II, numerical values of the driveline parameters usedfor this case study.

3) Mechanical Braking: The function of this subsystemis to dissipate all the power provided to it. It is consideredas a terminal subsystem and its ports are defined as

ubr =

[vbr0

], ybr =

[Fbr0

], (24)

where Fbr ≥ 0 is the mechanical braking force and vbrdenotes the longitudinal velocity of the vehicle. Note, thata complete description of the subsystem is not necessarybecause we are only interested in its consumption. Thus,the terms that describe internal energy and losses for themechanical braking subsystem in the cost function (16) canbe replaced by the total energy consumed by the subsystem,i.e.,

∆Hbr + Lbr =

∫ tf

t0

y>brkubrdt. (25)

4) Power Request: This subsystem describes the powerrequired by the vehicle to travel a certain period of time witha given longitudinal velocity profile vreq and traction forceFreq. It is considered a terminal subsystem, whose ports aredefined as

ureq =

[vreq

0

], yreq =

[Freq

0

]. (26)

The requested force and velocity profiles used in this casestudy are shown in Fig. 4. In this case, a complete descriptionof the subsystem is not provided because its behavior isalready given. However, to preserve consistency in the costfunction formulation (16), the terms that describe internal




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energy and losses are replaced by the total power consumedby the subsystem, i.e.,

∆Hreq + Lreq =

∫ tf

t0

y>requreqdt. (27)

5) Interconnections: The set J = Jn ∪ Jp = 1, 2contains the two junctions in network topology depicted inFig. 3. This junctions are described by

bem,1 = abr,1 = areq,1 = 1, (28a)bec,2 = aem,2 = 1, (28b)

and zero for the remaining coefficients.

B. Optimal Control Problem

After describing pH representations for all the drivelinecomponents and its interconnections, the procedure to for-mulate a CVEM OCP for this case study is reduced to asimple substitution. In particular, a continuous-time CVEMOCP for this case study is obtained by substituting (22)-(28)into (12) and (16). Similarly, substituting (22)-(28) into (17)and (18) generates a discrete-time OCP.

Interestingly, for this case study it is also possible tofind and equivalent OCP using a power-based approach. Forinstance, let us consider the modelling framework proposedin [10]. In this approach, the source subsystem is representedas linear system that describes the accumulation of energyand the conversion of power in all subsystems is defined bya quadratic function of the form

Pout = 12γ2P

2in + γ1Pin + γ0, (29)

where Pout, Pin represent the power in the input and outputport of the subsystem, respectively. To illustrate the connec-tion between the power-based approach in [10] with the pHframework proposed in this paper. Let us consider the EMmodel described by (2) and (23) in steady state. After somealgebraic manipulations and by defining Pout = VemIem andPin = FemVem, it is possible to obtain

γ2 =Rer

2w

κ2V g

2rv

2em

, γ1 =2Reβ

κ2v

+κτκV

,

γ0 =

(Reβ

2

κ2v

+κτβ

κV

)g2r

r2w

v2em. (30)

Hence, considering that the longitudinal velocity vem isgiven, equation (29) describes only a static relation between

0 200 400 600 800 1000

Time [s]

0

20

40

60

80

Vole

city

[km

/h]

-40

-20

0

20

40

Tra

ctio

n F

orc

e [k

N]

Fig. 4: Requested traction force and longitudinal velocity.

input and output powers. Performing a similar procedure inthe rest of the subsystems of this example, it allows us toobtain an equivalent OCP that has power in the input-outputports of the subsystems as decision variables.

The advantage of this equivalent formulation is that ifstatic optimization methods are used to obtain solutions, thenumber of decision variables is lower than in the port-basedcase. However, this framework fails to describe problems ofhigher complexity. For instance, see [14] where an attempt tounify the optimization of velocity profiles, i.e., eco-driving,and CVEM for the subsystems of a vehicle is presented.Using the pH framework for CVEM presented in this paperthe integration of eco-driving and CVEM is natural. In fact,it is equivalent to include the longitudinal vehicle dynamicsas an additional subsystem in the port-based network.

C. Simulation Results

We use Algorithm 1 to find a solution to the discrete-time OCP obtained from the substitution of (22)-(28) into(17) and (18). In this case, we use a sampling time δt =1[s] and a horizon of length K = 990. The results obtainedare compared to a benchmark case where all the mechanicalbraking power is zero, which means that the energy producedby deceleration of the vehicle is completely stored in theHVB. In Fig. 5, the SoC and the mechanical braking forcecan be observed as function of time for both the benchmarkas well as the optimal solution. The final SoC correspondingto the optimal solution is 0.0717% higher than the benchmarksolution, which implies energy savings of 4.896% for thegiven power request. Interestingly, the optimal solution stillrequires some mechanical braking which clearly shows thatthe optimal solution selects efficient operational points forthe subsystems instead of only trying to recover as muchmechanical energy as possible.

Physical insight about the HVB operation for both cases isgiven Fig. 7 and Fig. 6, where the behavior of battery voltageand current, respectively, is depicted for both solutions.Note that regions where the voltage in the battery terminalsincreases correspond to negative currents in the HVB. Thisimplies charging the battery, which is also observed in Fig.5. An advantage of the pH CVEM framework proposed inthis paper is that is possible to give independent treatmentto each conjugated variable in the port of the subsystems,i.e., setting independent constraints to each variable. For

0 200 400 600 800 1000Time [s]

48

48.5

49

49.5

50

SoC

[%]

0

5

10

15

20

Bra

king

For

ce [k

N]Optimal Solution

Benchmark Solution

Fig. 5: State of charge and braking force.




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0 200 400 600 800 1000

Time [s]

-200

0

200

400

600

800

I hvb [A

]

Optimal SolutionBenchmark Solution

Fig. 6: Battery current.

0 200 400 600 800 1000Time [s]

400

500

600

700

Vhv

b [V]

Optimal SolutionBenchmark Solution

Fig. 7: Battery voltage.

this study case, the constrains on the voltage and currentof HVB aim to capture the charge acceptance phenomenonin the battery. This is not directly possible in power-basedapproaches because voltage and current are combined aspower.

VI. CONCLUSIONS

We have proposed a systematic modelling approach todescribe CVEM networks in terms of interconnected pHsystems representations. This is an alternative to powerbased approaches for CVEM that has allowed us to unifythe concepts of energy storage and losses directly in thedynamical models of the subsystems. We have highlightedthe relevance of the pH framework in terms of formulationof physically insightful decomposable OCP. Moreover, theextra freedom obtained by describing power in terms ofconjugated variables has open the door to include additionalphysical phenomena in the as constraints in the OCP. Finally,we have adapted the distributed optimization algorithm forpower-based CVEM proposed in [13] to find solutions to thepH CVEM problem.

REFERENCES

[1] C. W. Team, R. Pachauri, and L. Meyer, Eds., Climate Change 2014:Synthesis Report. Contribution of Working Groups I, II and III to the

Fifth Assessment Report of the Intergovernmental Panel on ClimateChange. Geneva, Switzerland: IPCC, 2014.

[2] O. US EPA, “Global Greenhouse Gas Emissions Data,” Jan.2016. [Online]. Available: https://www.epa.gov/ghgemissions/global-greenhouse-gas-emissions-data

[3] M. Nilsson, “Electric vehicles: the phenomenon of range anxiety,”Lindholmen Science Park, Sweden, Tech. Rep., 2011.

[4] T. R. International Energy Agency, “Global EV Outlook 2017,” 2017.[5] T. R. European Energy Agency, “Electric vehicles in Europe,” 2016.[6] B. Egardt, N. Murgovski, M. Pourabdollah, and L. J. Mardh, “Electro-

mobility Studies Based on Convex Optimization: Design and ControlIssues Regarding Vehicle Electrification,” IEEE Control Sys Mag,2014.

[7] B. de Jager, T. van Keulen, and J. Kessels, Optimal Control of HybridVehicles. Springer, 2013.

[8] J. T. B. A. Kessels, J. H. M. Martens, P. P. J. van den Bosch, andW. H. A. Hendrix, “Smart vehicle powernet enabling complete vehicleenergy management,” Proc. Vehicle Propulsion & Power Control,2012.

[9] S. Onori, L. Serrao, and G. Rizzoni, Hybrid electric vehicles: Energymanagement strategies. Springer, 2016.

[10] T. C. J. Romijn, M. C. F. Donkers, J. T. B. A. Kessels, and S. Weiland,“A Distributed Optimization Approach for Complete Vehicle EnergyManagement,” IEEE Transactions on Control Systems Technology,2018.

[11] H. Chen, “Game-theoretic solution concept for complete vehicleenergy management,” Ph.D, dissertation, Eindhoven University ofTechnology, 2016.

[12] T. C. J. Romijn, M. C. F. Donkers, J. T. B. A. Kessels, and S. Weiland,“Receding horizon control for distributed energy management of a hy-brid heavy-duty vehicle with auxiliaries,” IFAC-PapersOnLine, 2015,4th IFAC Workshop on Engine and Powertrain Control, Simulationand Modeling E-COSM 2015.

[13] G. P. Padilla, G. Belgioioso, and M. C. F. Donkers, “Global Solutionsto the Complete Vehicle Energy Management Problem via Forward-Backward Operator Splitting,” IEEE Conference on Decision andControl, 2019.

[14] Z. Khalik, G. P. Padilla, T. C. J. Romijn, and M. C. F. Donkers, “Ve-hicle Energy Management with Ecodriving: A Sequential QuadraticProgramming Approach with Dual Decomposition,” Proc AmericanControl Conference, 2018.

[15] N. Murgovski, L. Johannesson, X. Hu, B. Egardt, and J. Sjoberg,“Convex relaxations in the optimal control of electrified vehicles,” inProc. American Control Conf, 2015.

[16] A. van der Schaft and D. Jeltsema, Port-Hamiltonian Systems Theory:An Introductory Overview. now Publishers Inc., 2014.

[17] A. van der Schaft, “Interconnections of input-output Hamiltoniansystems with dissipation,” Johann Bernoulli Institute for Mathematicsand Computer Science, 2016.

[18] V. Duindam, A. Macchelli, S. Stramigioli, and H. Bruyninckx, Model-ing and Control of Complex Physical Systems: The Port-HamiltonianApproach. Springer Berlin Heidelberg, 2009.

[19] A. van der Schaft, “Interconnections of input-output hamiltoniansystems with dissipation.”

[20] O. D. M. Giraldo, A. G. Ruiz, I. O. Velazquez, and G. R. E. Perez,“Passivity-based control for battery charging/discharging applicationsby using a buck-boost DC-DC converter,” in 2018 IEEE GreenTechnologies Conference (GreenTech).

[21] E. Celledoni and E. H. Hoiseth, “Energy-Preserving and Passivity-Consistent Numerical Discretization of Port-Hamiltonian Systems,”IEEE Transactions on Automatic Control, 2017.

[22] Y. Yalcin, G. Sumer, and K. S., “Discrete-time modeling of Hamilto-nian systems,” Turkish Journal of Electrical Engineering & ComputerSciences, 2013.




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Traffic-Aware Vehicle Energy ManagementStrategies via Scenario-Based

Optimization ?

L.A. Wulf Ribelles G.P. Padilla M.C.F. Donkers

Dept. Electrical Eng, Eindhoven University of Technology, Netherlands(e-mail:[email protected],g.p.padilla.cazar,[email protected])

Abstract: This paper explores the development of traffic-aware energy management strategiesby means of scenario-based optimization. This is motivated by that fact that real drivingconditions are subject to uncertainty, thereby making the real-time optimization of the energyconsumption of a vehicle to be a challenging problem. In order to deal with this situation, weemploy the current framework of complete vehicle energy management in a receding horizonfashion in which we consider random constraints representing realizations of the disturbance, i.e.,the uncertain driving conditions. Additionally, we study three methods for velocity predictionin energy management strategies, i.e., a method based on (average) traffic flow information, amethod based on Gaussian process regression, and a method that combines both. The proposedstrategy is tested with real traffic data using a case study of the power split in a series-hybridelectric vehicle. The behavior of the battery, control inputs and fuel consumption generated fromthe randomized strategy are compared against the optimal solution from an offline benchmarkand a situation with perfect prediction of the future, where the use of a Gaussian processregression and the average traffic speed achieves near optimal fuel consumption.

Keywords: Vehicle Energy Management,Model Predictive Control, Scenario Optimization,Power request predictions

1. INTRODUCTION

Nowadays, vehicle efficiency has become more relevantdue to the need to mitigate the environmental impactof fossil fuels and meet the CO2 emission targets setfor 2030 as expressed in the ”Global EV Outlook 2018”and ”IEA New Policies Scenario” (noa, 2018). In fact,several countries are proposing regulations to stop thecommercialization and use of non-electrified vehicles in thecity centers between 2025 and 2040. However, e-mobilityfaces a crucial problem to experience a total incorporationin the transportation market denominated range anxiety,i.e., users are concerned about not having enough energyto reach their final destination (Melliger et al., 2017). Fromthis perspective, the development and implementationof Energy Management Strategies (EMSs) becomes arelevant research topic in the automotive industry, becauseits implementation does not require substantial hardwaremodifications to achieve longer traveling distances usingonly a reduced amount of energy.

Basically, EMSs determine how to optimize the energyconsumption of a vehicle by establishing an appropriatedivision of the energy used by its components and subsys-tems, which in a broader sense is referred as Complete Ve-hicle Energy Management (CVEM) (Kessels et al., 2012).Generally, the EMS literature can be divided into online


and offline methods. Offline methods are typically based onDynamic Programming (Perez et al., 2006), Pontryagin’sMinimum Principle (Serrao et al., 2009) or static optimiza-tion techniques (Khalik et al., 2018; Padilla et al., 2019),and require the drive cycle to be known a priori. Thesemethods are not real-time implementable and neglect thepresence of uncertain driving conditions, e.g., traffic con-gestion, varying speed limits and different driving styles.Alternatively, different online methods explored in theliterature are given by rule-based techniques (Jalil et al.,1997), expressing the energy required by the subsystemsthrough Equivalent Consumption Minimization Strategies(ECMS) (Sciarretta et al., 2004) or implementing ModelPredictive Control (MPC) methods with online predic-tions of the driving mission (Romijn et al., 2017).

Despite being online implementable, all these methodsrequired tuning, which often relies on offline solutions,or assume exact predictions of the power request, lim-iting their use under real-life situations. Alternatively,stochastic optimal control methods provide noticeable ex-tensions for Traffic-aware Energy Management Strategies(TaEMS), i.e., strategies that take into account the uncer-tainty present in real traffic conditions. These strategiesare typically obtained using Stochastic Dynamic Program-ming (SDP) (Moura et al., 2011; Johannesson et al., 2007),which suffers from scalability problems known as ”Curseof Dimensionality”, or Stochastic MPC (Di Cairano et al.,2014), which might become computationally demandingwhen the number of subsystems considered in the controlproblem increases.




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In this paper, we use the recent developments of scenario-based optimization (Campi and Garatti, 2018; Schildbachet al., 2013, 2014) to extend the current framework ofCVEM as in Romijn et al. (2017). This will lead to atractable method for traffic-aware complete vehicle en-ergy management, in which in intuitive tradeoff can bemade between computational complexity and robustnessdepending on the number of scenarios considered. Fur-thermore, the proposed method has the potential of usingdistributed optimization techniques to improve its imple-mentation capabilities (although this will not be addressedin this paper). In addition, we propose the use of GaussianProcesses Regression for this TaEMS to generate multiplepredictions of the future driving situations, i.e., samplerandom scenarios, which are combined with traffic flowinformation to provide long-term speed predictions.

This paper is organized as follows: In Section 2, thegeneral vehicle energy management problem formulationis presented and extended as an uncertain optimal controlproblem. A description of the prediction methods fortraffic-aware vehicle energy management is included inSection 3. Section 4 presents the implementation detailsand simulation results obtained on a case study. Finally,conclusions and future work are presented in Section 5.

2. TRAFFIC-AWARE VEHICLE ENERGYMANAGEMENT

In this section, we present the mathematical formulationdescribing the optimal control problem arising from theCVEM framework in a receding horizon fashion. Addition-ally, an extension of the resulting Receding-Horizon Opti-mal Control Problem (RHOCP) in the context of scenario-based optimization is introduced to account for uncertaindisturbances affecting a vehicle, i.e., the uncertain powerrequest caused by, e.g., unknown driving conditions.

2.1 Receding Horizon Optimal Control Problem

In general, the CVEM problem aims to define the opti-mal energy flows between the subsystems in the powernetwork of a vehicle over a prediction horizon k ∈ K =0, 1, . . . ,K−1 given the measurements at time step t ∈ Nrepresented by

minum,k|t,ym,k|t

∑

m∈M

∑

k∈Kam,kym,k|t + bm,kum,k|t (1a)

where um,k|t ∈ R are scalar inputs, ym,k|t ∈ R arescalar outputs of the converter of subsystem m ∈ M =1, . . . ,M, and the coefficients am,k, bm,k ∈ R define thedesired cost metric based on the energy consumed by eachsubsystem at time instant k+ t. The minimization of (1a)is subject to a set of constraints describing the behaviourof the vehicle’s power network and the exchanges of powerin it (Fig. 1 shows a general network structure). First,we consider quadratic equality constraints that define theinput-output behaviour of the converter in each subsystem

ym,k|t = 12γ2,mu

2m,k|t + γ1,mum,k|t + γ0,m (1b)

with γ2,m ∈ R+, γ1,m ∈ R and γ0,m ∈ R being coefficientsthat define the efficiency of converter m ∈ M predictedat time k + t given information of time t ∈ N, k ∈ K.Furthermore, the network presents different states that are

Fig. 1. Power network structure

being controlled, imposing constraints based on the linearsystem dynamics of the energy buffers

xm,k+1|t = Am,kxm,k|t +Bm,kum,k|t (1c)

in which xm,k|t ∈ Rnm and um,k|t ∈ R denote the predictedstate and input, respectively, of subsystem m ∈M at timek + t, given information at time t ∈ N and k ∈ K. Theadmissible states and inputs are subject to constraints,i.e.,

xm,k|t ∈ Xm and um,k|t ∈ Um (1d)

Moreover, the interconnections of subsystems are de-scribed by J = 1, . . . , J nodes and no direct interactionsbetween them are considered, i.e., each subsystem can beconnected only to a node, resulting in the power balances

gj(ym,k|t, um,k|t, wj,k|t) ≤ 0 (1e)

with

gj(ym,k|t,um,k|t,wj,k|t)=∑

m∈Mcj,mym,k|t+dj,mum,k|t+wj,k|t

(1f)for all j ∈ J , k ∈ K and t ∈ N.

In (1e), wj,k|t are disturbances acting on each node, e.g.,the power request from the driver or the auxiliaries. Gen-erally for EMS, it is assumed that these disturbances areknown in advance or can be perfectly predicted. However,this might not be always true, as they are generated by theenvironment or external factors, e.g., the driver. Therefore,we can consider that these unknown disturbances have astochastic nature, turning the CVEM problem (1) into aStochastic RHOCP. Even though different methods canbe used to solve this problem, we make use of the scenarioapproach (Campi and Garatti, 2018) to solve the resultinguncertain RHOCP in a computationally advantageous wayas presented in the remainder of this section.

2.2 Stochastic RHOCP

Before presenting the scenario approach, let us considera stochastic extensions to problem (1) accounting for theunknown disturbances in nodes j ∈ J . In particular, wecan post the resulting CVEM problem as the followingchance-constrained RHOCP

minum,k|t,ym,k|t

∑

m∈M

∑


subject to (1b) - (1d), and

Prgj(ym,k|t, um,k|t, wj,k|t) ≤ 0 ≥ 1− εj (2b)

where the parameters εj are acceptable infeasibility levelsand the functions gj are defined as in (1e).




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The main restriction in (2) is the need to guarantee thatthe chance-constraints (2b) will hold for any realization ofthe uncertain disturbances wj,k|t, since the distribution ofthe uncertainty might be unknown and, even if it is known,the solution could lead to a more conservative solution andundesired performance.

2.3 Scenario-Based Traffic-Aware Energy Management

In order to deal with the restrictive characteristics of thechance-constrained formulation in the previous section, wemake use of the scenario approach (Campi and Garatti,2018) instead. This methodology for data-driven opti-mization aims to solve the resulting chance-constrainedRHOCP by means of a deterministic approximation thatconsiders only a finite number of realizations of the distur-bances wj,k|t, thereby providing a tuning knob to balancerobustness versus performance of the scenario solution.This allows to achieve a computationally tractable prob-lem when multiple subsystems are considered, in compari-son to classic stochastic EMS based on SDP (Moura et al.,2011; Johannesson et al., 2007). With this in mind, thepower balances (2b) in problem (2) can be replaced by a setof randomly sampled constraints, leading to what we referto as scenario-based Traffic-aware Energy ManagementStrategy (TaEMS).

Now, following the scenario approach and the resultsin (Schildbach et al., 2013, 2014), we introduce somedefinitions and assumptions required for the scenario-based RHOCP:

(1) The uncertainties w[ι]j,k|t of each node are contained in

a single variable w[ι]k|t = [w1,k|t, . . . , wJ,k|t]ᵀ which is a

random variable with (maybe unknown) probabilitymeasure Pr and support set W.

(2) A sequence of variables w[ι]0|t, . . . , w

[ι]K−1|t is a re-

alization of the uncertainty w[ι]k|t over the prediction

horizon defining the scenario w[ι]t .

(3) Enough independent and identically distributed sam-

ples w[ι]t can be obtain at every time instant, giving

a set of scenarios I = 1, . . . , I.(4) The scenario-based RHOCP problem has a feasible

solution for almost any w[ι]t .

With this definitions, the resulting scenario-based TaEMSproblem is given by

minum,k|t,ym,k|t

∑


subject to

ym,k|t = 12γ2,mu

2m,k|t + γ1,mum,k|t + γ0,m (3b)

xm,k+1|t = Am,kxm,k|t +Bm,kum,k|t (3c)

xm,k|t ∈ Xm and um,k|t ∈ Um (3d)

andgj(ym,k|t, um,k|t, w

[ι]j,k|t) ≤ 0, (3e)

for all ι ∈ I and j ∈ J , and k ∈ K, m ∈M at time t ∈ N.

From the above formulation, we make use of the resultsin Schildbach et al. (2013) to address the selection of thenumber of scenarios required for a particular feasibility

level. To this end, let us define the probability of constraintviolation

Vj,k(y∗m,k|t, u∗m,k|t) = Prgj(y∗m,k|t, u∗m,k|t, w

[ι]j,k|t) > 0

(4)

where u∗m,k|t, y∗m,k|t refer to the scenario solution. It has

been shown in Campi and Garatti (2018) and Schildbachet al. (2013) that Vj,k(y∗m,k|t, u

∗m,k|t) is bounded by a Beta

distribution B(ρj , I − ρj + 1), such that

PrIVj,k(y∗m,k|t, u∗m,k|t) > εj ≤ B(ρj , I − ρj + 1) (5)

where ρj is the support rank of the constraint in node j.Here, we make use of the results in Schildbach et al. (2013)instead of considering the number of decision variables asin the classic scenario approach presented in Campi andGaratti (2018). This is favorable as only a reduced numberof the decision variables in problem (3) is affected by (3e)at every step on the prediction horizon k regardless of thenumber of sampled scenarios.

From this formulation, we aim to find the minimum num-ber of samples required to satisfy the original chanceconstraint, as the more samples are drawn, the moreconservative the solution becomes. Nevertheless, giventhat new samples are drawn at each time step, weconsider a bound on the expected violation probabilityEI [Vj,k(y∗m,k|t, u

∗m,k|t)] ≤ εj , which leads to a sample size

of εj ≤ ρj/(I+ 1). This result follows from the integrationof (5), which can be interpreted as the probability thatthe I + 1 sample becomes a support constraint, i.e., thesolution obtained with the scenarios I does not satisfythe power balances gj(·) ≤ 0 (see Campi and Garatti(2018); Schildbach et al. (2013, 2014) for further detailsand proofs).

3. SCENARIO GENERATORS

In order to make predictions of the unknown traffic con-ditions, and thus the disturbance wj,k, we present thethree velocity prediction methods used in this work, e.g.,predictions based on GPS/eHorizon data, predictions us-ing a Gaussian Process Regression (GPR) model, and amixed generator that combines both the GPS and theGPR model.

3.1 GPS / eHorizon Methods

First, we consider that the vehicle has access to trafficinformation through a Global Positioning System (GPS)or an electronic horizon (eHorizon), which are devisesavailable in today’s vehicles. Here, the average traffic speedis calculated based on the traffic flow through a particularsection of the road as follows:

vηavg,tw = 1Zη

Zη∑

z=1

vztw (6)

where η are the road sections and Zη is number of vehiclespassing through a particular road section during a timewindow, e.g., an update frequency between 1 to 5 minutes,as is usually done in mapping and traffic managementsystems (AhmedAl-Sobky and Mousa, 2015).




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3.2 Gaussian Process Regression

Since the average traffic speed only provides a deter-ministic estimate of the traffic situation, a probabilisticmodel to forecast the future speed of the vehicle is pro-posed in this section. In particular, we propose to use aGaussian Process Regression (GPR) model (Rasmussenand Williams, 2006). The selection of this non-parametricmodel is motivated by the remarkable prediction capa-bilities achieved with machine learning methods in (Sunet al., 2015; Lefevre et al., 2014; Liu et al., 2019) and, atthe same time, its particular ability to provide a directmeasure for the uncertainty of the predictions. For ourapplication, the GPR model is defined as a NonlinearAuto-Regressive Model (NAR-GP), which results in re-gressing a function yk = f(xk) + ek that maps vectorinputs xk ∈ Rp to scalar outputs yk ∈ R, where we feeda feature vector xk = vk−p|t, . . . , vk|t and predict the

future speed yk = vk+1|t. Additionally, ek ∼ N (0, σ2n)

is a noise term acting on the output of the function andf ∼ GP(µ, ker), which is a GPR defined with a priordistribution with µ = 0, a covariance/kernel functionker, e.g., squared exponential, Matern, rational quadraticexponential, etc, and a set of hyper-parameters Θ. Further-more, the definition of these hyper-parameters is done byminimizing the negative log-likelihood function on trainingdata D = (xn, yn) | n = 1, . . . , N, see (Rasmussen andWilliams, 2006) for further details.

Once the NAR-GP is fully defined, we obtain a model thatallows us to generate random samples from the posteriordistribution, such that

vk+1|t ∼ Pr(y∗|D, x∗) = N (fpost, kerpost + σ2n; Θ) (7a)

in which

fpost =kerᵀ(x∗)(ker + σ2nI)−1y (7b)

kerpost =ker(x∗, x∗)− kerᵀ(x∗)(ker+σ2nI)−1ker(x∗) (7c)

with y the output training data, I the identity matrixand points x∗, y∗ referring to a test input and output,respectively.

Although predicting over multiple time steps ahead withthe NAR-GP results in predictions at uncertain inputs,i.e., regressing the prediction of vk+1|t that is a randomvariable, we only consider a naive approach to generatethe predictions over the prediction horizon K, since itsimplifies the prediction process and avoids large speedchanges which might result in intractable predictions.

3.3 Mixed Generator Approach

Given that long prediction horizons lead to a better per-formance of the EMS (Romijn et al., 2017), a combinationof the velocity prediction methods is considered in thiswork in order to exploit the benefits of each method, e.g.,account for the uncertainty in a short-term and preservethe preview of the traffic situation given by the averagetraffic speed. This combination is motivated by the factthat most of the maneuvers in car following or trafficsituations require a very short time (see Lefevre et al.(2014) and references therein) and the mismatch presentbetween the traffic speed and the individual speed profiles.At the same time, it is known that machine learningmethods tend to incur in large prediction errors when

Fig. 2. Powertrain topology

the prediction horizon length increases, as these methodsare not able to account for long-term traffic dependencies(Liu et al., 2019). In fact, most of the methods present inthe literature are restricted to predictions of 10 secondsin the future and, therefore, the integration of externalinformation could lead to substantial fuel savings whilegenerating more robust solutions against the actions ofthe driver.

4. CASE STUDY

In this section, we first define the case study consideredto evaluate the potential of the proposed scenario formu-lation. The case study is based on TaEMS for a serieshybrid vehicle. We will start by presenting the RHOCPformulation, followed by the selection of the sample sizeand the explanation of the power request determination.

4.1 Receding Horizon Optimal Control Problem

The case study in this paper is based on the series-hybridelectric vehicle (SHEV) presented in (Khalik et al., 2018).In particular, the powertrain topology of the vehicle isrepresented as the network of energy buffers depicted inFig. 2. In this figure, EGU stands for Engine GeneratorUnit, with yegu being the fuel consumption and ueguthe power supplied by the EGU to the power network.Furthermore, uhvb and yhvb define the electric powercoming from the High-Voltage Battery (HVB) and xhvbrepresents the stored energy in the battery. Besides thissubsystems, the Electric Motor (EM) provides yem, whichrepresents the (unknown) power request defined by thedriver. Note that yem propels the vehicle when being apositive magnitude and negative values denote brakingactions.

The node interconnecting the elements in the networkdefines a power balance as in (1e), where the parameterschvb = −1, degu = −1 and cem = 1 are specified accordingto the flow direction of the power for each subsystemand all the others are set to zero. On top of this, anexternal braking signal ybr is introduced to account for

Table 1. Powertain model coefficients

EGU

γ2,egu = 2 · 10−5 γ1,egu = 2.52 γ0,egu = 19uegu = 210 [kW] uegu = 0 [kW]

HVB

γ2,hvb = 1.671 · 10−3 γ1,hvb = 1 γ0,hvb = 0uhvb = 92.4 [kW] uhvb = −92.4 [kW] xhvb,0 = 11988 [kJ]xhbv = 22680 [kJ] xhbv = 7560 [kJ] xhvb,K = 11988 [kJ]




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the mechanical braking dissipating the excess of energy inthe powertrain.

According to the problem formulation in Section 2.1,the task of reducing the fuel consumed by the SHEV isdescribed by cost function with aegu,k|t = τk and all theremaining parameters am,k = bm,k = 0, as they do notcontribute to the objective of the problem. In this case,yem,k|t is considered to be an uncertain consequence ofthe driver actions and only the power request yem,0|t isknown at the current time step t, which is a realisticassumption given the on-board sensors in today’s vehicles.Furthermore, Table 1 presents the parameters defining thepowertrain in this case study.

After some substitutions, the problem can be reformu-lated as a Quadratically Constrained Quadratic Program(QCQP) which can be efficiently solved with specializedsolvers, e.g., CPLEX CPL (2019), resulting in

minum,k|t

∑

k∈Kτk( 1

2γ2,eguu2egu,k|t + γ1,eguuegu,k|t + γ0,egu)

(8a)

subject to12γ2,hvbu

2hvb,k|t−γ1,hvbuhvb,k|t − uegu,k|t︸︷︷︸

gqcqp

−γ0hvb ≤ −y[ι]em,k|t

(8b)

for all ι ∈ I and

xhvb,k+1|t = xhvb,k|t − τkuhvb,k|t (8c)

xhvb,k|t ∈ Xhvb and um,k|t ∈ Um (8d)

with k ∈ K, m ∈ M and t ∈ N. Note that this problem isconvex due as γ2,m > 0, which leads to a direct definitionof the samples for the scenario-based RHOCP.

4.2 Sample Size for Scenario-based RHOCP

Since the power balance (8b) is present for every predictionof time k+ t, given information at time t, the problem hasK scenario constraints and affect only the particular in-puts at that stage in horizon K. Therefore, it is straightfor-ward to define the support rank of the scenario constraintand specify the required sample size. In this case, thesupport rank of (8b) is given by ρ = d−L = 2, leading toan expected violation probability EI [Vj,k(u∗m,k|t)] ≤ 2

I+1 ,

where Vj,k(u∗m,k|t) = Prgqcqp − (γ0hvb − y[ι]em,k|t) > 0. In

this case study, we have defined a sample size of I = 119implying a theoretical bound of ε ≤ 2/120 ≈ 1.66%.

4.3 Power Request Definition

Given that the scenarios forecast possible velocity profiles,the power request is calculated using a longitudinal vehicledynamics model, such that:

yem,k|t =vk|tσu

(vk+1|t−vk|t

τk+σvv

2k|t+σr+g sin(θ(sk|t))) (9)

where σr = gcr, σu = 1m , σv = 1

2mcdρaAf and θ define therolling resistance, inverse of the mass, aerodynamic dragand the road slope, respectively. The definition and valuesof these coefficients for the vehicle considered in this casestudy can be found in Table 2. Moreover, a constant roadslope θ = 0 is considered based on the driving cycles andavailable information.

Fig. 3. Spatio-temporal vehicle trajectories

5. SIMULATION RESULTS

In order to analyse the performance of the scenario gen-erators and the TaEMS for the power split problem, wefirst describe the particular characteristics considered inthe simulations and later, we present results obtainedwith each method. Here, the solutions of the TaEMS arecompared to the optimal performance given by an offlinebenchmark and an online solution with perfect predictionof the future driving cycle. Additionally, we assess thebenefits of hybridization under uncertain predictions by in-cluding the fuel consumption generated by a conventionalvehicle, where the power request is directly covered by thecombustion engine.

For this case study, we used the traffic data set from theMobile Century field experiment (Herrera et al., 2010),which was a project carried out to evaluate the use ofGPS-enabled smartphones for accurate traffic informationsystems. The data set consists of position, velocity andtime information of individual vehicles driving over a 16km section of the Interstate 880 highway in Californiafrom 10 a.m. to 6 p.m., Fig. 3 shows the different vehicletrajectories in the northbound direction (see Herrera et al.(2010) for descriptions). Particularly, we consider differentdriving cycles recorded between 10 a.m. and 12 noon dueto the presence of a traffic congestion event around 10:30a.m., which allows for a more complete evaluation of theproposed TaEMS. The different driving cycles selected toevaluate the performance of the TaCVEM are shown inFig. 4.

For implementation, we use a moving average filter witha Gaussian window to smooth intractable speed changespresent in the predictions due to the piece-wise constantprofiles obtained from the traffic speed or large noiserealizations in the NAR-GP samples, as this providedsimilar results compared to other smoothing methodsin the literature (Thorsell, 2018), while keeping moreinformation of the NAR-GP samples.

Table 2. Vehicle coefficients

Parameter symbol Value

Frontal drag area Af 7.5400 [m2]

Drag coefficient cd 0.7

Rolling resistance cr 0.007

Air density ρa 1.1840

Mass m 15950 [kg]

Gravitational acceleration g 9.81 [m/s2]




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200 400 600 800 1000 1200 1400 1600 1800 2000Time [s]

5

10

15

20

25

30

Spe

ed [m

/s]

Test Driving CyclesTest Cycle 1Test Cycle 2Test Cycle 3

Fig. 4. Test driving cycles considered for simulations

5.1 GPS / eHorizon Method

In order to replicate the information supplied by a GPS,we divided the road in 100 segments and considered a timewindow tw = 300 seconds according to the update fre-quency in (Herrera et al., 2010). A density map depictingthe traffic speed obtained with (6) is presented in Fig. 5.For implementation purposes, we consider that the vehiclehas access to the average speed relative to the currenttime of the driving cycle, e.g., at 10:43am the informationobtained at 10:40am is known and an update is availableat 10:45am.

5.2 Gaussian Process Regression

For this probabilistic velocity prediction method, thetraining data D was composed by real driving cycles takenfrom the Mobile Century data set starting before 10:30a.m., and the HWFET, LA92 short and EPA standarddriving cycles in order to provide more dynamic data tothe model. Furthermore, the covariance function used inthis work is the Matern 5

2 function, see, e.g., (Rasmussenand Williams, 2006). This selection was motivated by thefact that real driving cycles present long braking patterns,which were not properly captured when using a squaredexponential function due to its smoothness characteristics.Additionally, the number of lags in the NAR-GP was set top = 5, since it was observed that longer dependencies didnot provide a substantial improvement of the predictions.

As an example, the mean predictions generated with theNAR-GP for Test Cycle 1 are shown in Fig. 8, where thetop plot presents the prediction accuracy with differentprediction horizons and the trajectories for 10 seconds ofprediction are shown in the bottom plot. As it can beseen, when the predictions are made for short periods

Fig. 5. Average traffic speed during congestion event

Fuel Consumption vs. Prediction Horizon

21.522

22.523 K=20

K=30K=60K=120No Hybrid

2525.5

26

Fue

l Con

sum

ptio

n [l]

/100

[km

]

Perfect Prediction NAR-GP GPS20

21

22

a)

b)

c)

Fig. 6. Fuel consumption vs. prediction horizon length.a) Test cycle 1. b) Test cycle 2. c) Test cycle 3

of time, e.g., 10 seconds, 95% of the errors are smallerthan 1 m/s, which is acceptable for the development ofenergy management strategies. Nevertheless, these errorspresent an increasing trend when generating a velocityforecast for longer horizon lengths. Besides this, the NAR-GP occasionally generates wrong braking predictions, e.g.,predictions around second 1100, but taking multiple ran-dom samples results in a more cautious use of the battery.

5.3 Prediction Horizon Length

In order to establish an appropriate length of the pre-diction horizon, the fuel consumption obtained with theaverage traffic speed and the NAR-GP predictions wereevaluated. Fig. 6 shows the the fuel consumption with aperfect prediction, the scenario solution with the NAR-GPand the average traffic speed (GPS), considering the offlinebenchmark as the baseline for three test cycles. From thisfigure, it can be seen that a longer prediction horizonresulted in a lower fuel consumption since it allows a largerdeviation from the final state constraint imposed in theproblem. For test cycle 2, the traffic speed captures the fastdeceleration accurately, leading to an appropriate use ofthe battery compared to the NAR-GP, where the batteryis mainly used after the braking event starts, see Fig. 7.Regarding the third test cycle, the use of the average trafficspeed incurs in a higher fuel consumption after one minuteof prediction due to the mismatch between the traffic flowand the actions of the driver. In the same way, the NAR-GP leads to better fuel economy since the braking event inthis driving cycle presents a softer deceleration comparedto the other tests and sampling different profiles accountsfor the fast accelerations at the end of the cycle.

50 100 150 200 250 300 350 400 450Time [s]

2.5

3

3.5

4

4.5

5

5.5

6

x hvb [K

Wh]

Battery State Trajectory for Test Cycle 2

OfflinePerfect PredictionNAR-GPGPS

Fig. 7. Trajectory of the energy stored in the battery fortest cycle 2




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Fig. 8. NAR-GP predictions for test cycle 1. Top: Errorof the predicted mean vs. prediction horizon length.Bottom: Predicted mean speed with horizon of 10seconds.

5.4 Mixed Scenario Generator

As shown before, the NAR-GP is capable of producingaccurate predictions when a short horizon is specified, butfails to anticipate longer braking events. For this reason,a horizon length of 10 seconds was selected to generatethe speed profiles and to define a reference point to othervelocity prediction models studied in the literature. Afterthis predictions are made, we use the average traffic speedto complete the remaining part of the prediction horizonin order to keep the preview of the future traffic conditionsand provide more freedom for the usage of the battery.

Finally, since longer horizons have a large impact in thecomputation time due to the increment of constraints anddecision variables in the RHOCP, we incorporate the vari-able step-size approach proposed in (Romijn et al., 2017)for the predictions of the average traffic speed since, asindicated by the authors, coarser predictions of the futurehave minor impacts in the fuel savings while noticeablyreducing the computation complexity of the problem. Weconsider (τ1, . . . , τK) = (1, 1, 2, 4, 6, 8, 10, 12, 16, 20, 40) asthe step-size sequence used to generate long-term predic-tions, where the specific values of the sequence are definedas suggested in (Romijn et al., 2017), such that the totallength of the horizon obtained is 120 seconds.

The resulting fuel savings obtained with the mixed sce-nario generator are reported in Table 3, where we presentthe fuel consumed with ‘full mixed’ scenario generators(i.e., with constant step sizes τk), the ‘variable mixed’scenario generators (i.e., with variable step sizes, as ex-plained above) and the results when the variable stepsize is incorporated and only the average traffic speed isconsidered while using the same sampling time sequence,i.e., ‘variable GPS’. Moreover, the state trajectory andcontrol inputs for Test cycle 2 are shown in Fig. 9.

From Table 3, it can be seen that the proposed EMSpresents a slight improvement when the full predictions ofthe mixed scenario generator are considered in comparisonthe GPS-based predictions (see also Fig. 6). Nevertheless,the main advantage is observed when the decision variablesare reduced by means of the variable sampling time τk. Inthis case, we see that the fuel savings decrease as shownin (Romijn et al., 2017) but the incorporation of multiplepredictions keeps the usage of fuel closer to their optimalvalues and reaching up to 99.3% when full predictions are

100 200 300 400Time [s]

4

6

x hvb [k

Wh]

Offline Perfect Prediction GPS Mixed SG

100 200 300 400Time [s]

0

100

200

u egu [k

W]

100 200 300 400Time [s]

-50

0

50

y hvb [k

W]

100 200 300 400Time [s]

-150-100

-500

y br [k

W]

Fig. 9. State trajectory and power balance signals fromthe GPS and mixed scenario generator in test cycle 2using sampling sequence τk.

Table 3. Fuel consumption of TaCVEM withmixed scenarios and GPS information

TestCycle

MethodFuel Consumption

Optimality[l]/100[km]

1Full Mixed 21.7 97.2 %

Variable Mixed 21.9 96.1 %Variable GPS 22.2 94.7 %

2Full Mixed 24.8 99.3 %


3Full Mixed 20.8 97.6 %


used and 98.2% with the sapling sequence τk. On the otherhand, it can be noticed there exists a negative effect offaulty predictions, as the consumption increases due tothe mismatch in the long-term prediction information inTest Cycle 3.

6. CONCLUSIONS

In this paper, a traffic-aware energy management strategyhas been developed that accounts for uncertain distur-bances. This strategy is based on the solution to a scenario-based optimal control problem that is used in a recedinghorizon fashion. The scenario-based approach reduces theprobability of running out of energy during a driving mis-sion. Different alternatives to include traffic informationfor the generation of scenarios have been defined andevaluated, where up to 99.27 % of the optimality in termsof fuel consumption was achieved with a suitable mix ofthe available information and 98.24 % with variable step-size predictions.

Future work involves the improvement of the scenariogenerator to increase the accuracy of the predictions whilehandling more complex driving conditions. Besides, exten-sions to distributed optimization are suggested by decom-posing the problems in terms of subsystems, scenarios andhorizon so as to reduce the computational time.

REFERENCES

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IBM ILOG CPLEX V12.9 Users Manual for CPLEX,2019.




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M. C. Campi and S. Garatti. Introduction to the ScenarioApproach, Volume 26 of MOS-SIAM Series on Opti-mization. SIAM, 2018.

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J.C. Herrera, D. Work, R. Herring, X. Ban, Q. Jacobson,and A.M. Bayen. Evaluation of traffic data obtainedvia gps-enabled mobile phones: The mobile century fieldexperiment. Transportation Research Part C: EmergingTechnologies, 2010.

N. Jalil, N. Kheir, and M. Salman. A Rule-based EnergyManagement Strategy for a Series Hybrid Vehicle. InProc American Control Conf, 1997.

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Z. Khalik, G.P. Padilla, T.C.J. Romijn, and M.C.F.Donkers. Vehicle Energy Management with Ecodriving:A Sequential Quadratic Programmming Approach withDual Decomposition. In Proc American Control Conf,2018.

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K. Liu, Z. Asher, X. Gong, M. Huang, and I. Kolmanovsky.Vehicle Velocity Prediction and Energy ManagementStrategy Part 1: Deterministic and Stochastic VehicleVelocity Prediction Using Machine Learning. SAETechnical Paper, 2019.

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J.S. Moura, H.K. Fathy, D.S. Callaway, and J.L. Stein. AStochastic Optimal Control Approach for Power Man-agement in Plug-In Hybrid Electric Vehicles. IEEETrans on Control Syst Techn, 2011.

G.P. Padilla, G. Belgioioso, and M.C.F. Donkers. Globalsolutions to the complete vehicle energy managementproblem via forward-backward operator splitting. InProc Conf on Decision and Control, 2019.

L.V. Perez, G.R. Bossio, D. Moitre, and G.O. Garcıa.Optimization of power management in an hybrid electricvehicle using dynamic programming. Mathematics andComputers in Simulation, 2006.

C. E. Rasmussen and C. K. I. Williams. Gaussian Pro-cesses for Machine Learning. The MIT Press, 2006.

T.C.J. Romijn, M.C.F. Donkers, J.T.B.A. Kessels, andS. Weiland. Real-time distributed economic model pre-dictive control for complete vehicle energy management.Energies, 2017.

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G. Schildbach, L. Fagiano, C. Frei, and M. Morari. Thescenario approach for stochastic model predictive con-trol with bounds on closed-loop constraint violations.Automatica, 2014.

A. Sciarretta, M. Back, and L. Guzzella. Optimal Controlof Parallel Hybrid Vehicles. IEEE Trans on Control SystTechn, 2004.

L. Serrao, S. Onori, and G. Rizzoni. Ecms as a Realizationof Ponrtyagin’s Minimum Principle for HEV Control. InProc American Control Conf, 2009.

C. Sun, X. Hu, S.J. Moura, and F. Sun. Velocity predictorsfor predictive energy management in hybrid electricvehicles. IEEE Trans on Control Syst Techn, 2015.

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Vehicle Energy Management with Ecodriving: A Sequential QuadraticProgramming Approach with Dual Decomposition

Z. Khalik, G.P. Padilla, T.C.J. Romijn, M.C.F. Donkers

Abstract— In this paper, we propose to solve the ecodrivingproblem using a Sequential Quadratic Programming (SQP)algorithm. We formulate the ecodriving problem as discrete-time (possibly nonconvex) nonlinear optimal control problemand solve this using SQP, in which we form a convex SQPsubproblem using Thikhonov regularization. We will furthershow that the SQP algorithm can be embedded in a distributedoptimization approach, allowing it to be used for CompleteVehicle Energy Management (CVEM), incorporating optimalcontrol of the vehicles auxiliary systems, in combination withecodriving. We consider two case studies for the ecodrivingproblem. The first case study concerns the optimal control of aFull-Eletric Vehicle, which has one control input and two statesand is solved with the SQP algorithm. The second case studylays a foundation for CVEM with ecodriving, where we willsolve an energy management problem with ecodriving for aSeries-Hybrid Electric Vehicle, which has three control inputsand three states, using the aforementioned SQP algorithm anddual decomposition.

I. INTRODUCTION

Hybrid electric vehicles offer the potential to reduce fuelconsumption of a vehicle by adding an electric motor with ahigh-voltage battery to the powertrain. This allows brakingenergy to be recuperated and allows the combustion engineto work at a more efficient operating point. In energymanagement, supervisory control is used to determine theoptimal power flow between the electric machine and thecombustion engine. A recent trend is to extend the energymanagement problem to incorporate more auxiliary devices,such as a refrigerated semitrailer or a climate control sys-tem [1], engine thermal management [2], battery thermalmanagement [3], battery ageing [4], [5]. Incorporating allthe vehicle energy consumers into the energy managementproblem is referred to as Complete Vehicle Energy Manage-ment (CVEM) in [6]. The rationale behind this is that theauxiliaries also consume a considerable amount of energy.However, in all these vehicle energy management problems,the vehicle speed is assumed to be given, while most of thepower generated by the powertrain is used for propelling thevehicle. Therefore, optimizing the speed of a vehicle over acertain trajectory, thereby allowing for an optimal conversionof potential energy from the road profile into kinetic energyof the vehicle, can lead to a considerable energy consumption


Zuan Khalik and Tijs Donkers are with the Department of ElectricalEngineering of Eindhoven University of Technology, Netherlands. E-mail:z.khalik, [email protected]

reduction. In this paper, we refer to this latter problem as theecodriving problem.

In Vehicle Energy Management (VEM), global optimalsolutions are typically achieved using Dynamic Program-ming (DP) [7], see e.g., [8]. However, DP has the inherentdisadvantage that the computational burden increases withthe number of states. Optimization methods based on thePontryagins Maximum Principle (PMP), see, e.g., [4], [9]can handle computational complexity of multi-state energymanagement problems. In PMP, the problem is reduced tosolving a two-point boundary value problem, which can bedifficult to solve in the presence of state constraints. Staticoptimization methods guarantee a global optimal solution forconvex approximations of the energy management problems,e.g., [10]. To increase scalability of the static optimizationproblem to allow for a large number of auxiliary systems,distributed optimization approaches have been proposed in[1] for complete vehicle energy management. Using a dualdecomposition, a large optimal control problem is splitup into several smaller optimal control problems. A dis-advantage of the convex optimization approach to vehicleenergy management of [1] is that it requires the powertraincomponents to be described by (convex) quadratic models.This renders the distributed optimization approach not usablefor the ecodriving problem, as the longitudinal vehicle dy-namics is nonlinear. However, a particular extension of thedistributed optimization approach of [1] towards a nonlinearbattery ageing model has been made in [5], showing itsability to handle nonlinear convex models as well.

Some of the above mentioned optimization methods havebeen applied to the ecodriving problem, e.g. PMP in [11],[12], DP in [13], [14], and static optimization in [15]. Theapproaches based on PMP and DP suffer from their inherentdifficulties, i.e., incorporating state constraints in PMP andmore dynamics in DP, while the approach based on (convex)static optimization [15] applies an Euler discretization to acontinuous time optimal control problem. As the problemis formulated as a second-order cone program, some of thepowertrain components can only be modeled using (piece-wise) linear functions. Furthermore, convexity might be lostafter applying the Euler discretization.

In this paper, we will propose to solve the nonlinear(and possibly nonconvex) ecodriving problem using a staticoptimization approach. To handle the nonlinearity and non-convexity of the resulting optimal control problem, we em-ploy a Sequential Quadratic programming (SQP) algorithm[16], [17]. The SQP algorithm is similar in nature to thealgorithm in [18], where the hessians are approximated to




96 / 110

accelerate convergence, however we take advantage of theproblem structure and further eliminate the state variables asis often done in MPC. Furthermore, we will show that thepresented SQP algorithm can be embedded in the distributedoptimization approach of [1], allowing it to be used forcomplete vehicle energy management, incorporating optimalcontrol of the vehicles auxiliary systems, in combination withecodriving. We will benchmark our solution strategy on aFull Electric Vehicle (FEV) problem presented in [12], whichsolves the problem using PMP, and solve the problem usingSQP. We show the advantage of SQP over the PMP approachused in [12], as state constraints can be very easily addedin our proposed approach. Furthermore, we demonstrate thecombined ecodriving problem with the powersplit control ofa series hybrid powertrain, as was also considered in [15].

II. ECODRIVING PROBLEM FORMULATION

In this section, we formulate the Ecodriving problem.Moreover, we will propose a discrete-time formulation ofthe problem and we will show that this leads to a nonconvexoptimization problem. We define the Ecodriving problem asminimizing traction power over a certain trajectory:

minv(t),s(t),u(t)

∫ tf

t0

Ptrac(v(t), u(t))dt, (1a)

where v is speed, u is the mechanical force, t0 and tf are theinitial and final time respectively. In this paper, we assume

Ptrac(v, u) = 12γ2u

2 + γ1uv + 12γ0v

2, (1b)

where γ2, γ1 and γ0 are parameters. The model (1b) is areasonable model for electric motors, because v2 is relatedto friction losses and u2 is related to dynamic losses of thetraction motor. The objective function in (1a) is minimizedsubject to (1b) and the longitudinal dynamics of the vehicle,given byv(t) = σuu(t)− σvv(t)2 − σr − g sin(α(s(t))),

s(t) = v(t),(1c)

for t ∈ [t0, tf ], and to lower and upper bounds on v and u,i.e.,

v(t) 6 v(t) 6 v(t), u(t) 6 u(t) 6 u(t), (1d)

for t ∈ [t0, tf ], and v(t0), s(t0), v(tf ) and s(tf ) given.In (1c), α is the road slope, σr = gcr, σu= 1

m and σv =1

2mcdρaAf . The coefficients m, g, cd, ρa, Af and cr arethe vehicle’s mass, gravitational acceleration, aerodynamicdrag coefficient, air density, frontal drag area and rollingresistance coefficient, respectively.

To arrive at a finite dimensional optimization problem, wediscretize (1) using a forward Euler discretization, and arriveat a discrete-time nonlinear optimal control problem of theform:

minxk,uk

∑

k∈K

12 [ xkuk

]>Hk[ xk

uk] + F>k [ xk

uk], (2a)

subject to state dynamics,

xk+1 =f(xk, uk), (2b)

and state constraints and input constraints,

xk 6 xk 6 xk, uk 6 uk 6 uk, (2c)

for k ∈ K = 0, 1, . . . ,K − 1, where K =tf−t0τ is the

optimization horizon with x0, xK given and τ > 0, whichis chosen such that τtf ∈ N, is the step size. To arrive at adiscrete-time approximation of (1), we choose

xk = [ vksk ], Hk = τ[ γ0 0 γ1

0 0 0γ1 0 γ2

], Fk =

[000

], (3a)

and

f(xk, uk) =

[vk+τ

(σuuk−σvv2k −σr−g sin(α(sk))

)

sk + τvk

].

(3b)

An electric machine, represented by (1b), typically hasa close to linear input-output behavior, such that γ1 ≈ 1,γ0 1 and γ2 1, as the power losses are small generally.This might cause (2) with (3) to be a non-convex optim-ization problem, as Hk is note positive definite. However,(2) may still be convex in the feasible domain, i.e., wherethe constraints (2b) - (2c) are satisfied. We refer to [Paul’spaper] where it is shown that due to the discretization stepto arrive at (2), the discretized problem is nonconvex evenin the feasible domain for any sampling time τ > 0. As aresult, the method presented in this paper will give only alocal minimum, which might not be the global minimum. Amethod to arrive at a convex formulation of the ecodrivingproblem (1) can be found in [Paul’s paper].

III. SQP APPROACH TO ECODRIVING

In this section, we will present a Sequential QuadraticProgramming (SQP) algorithm to solve the nonlinear optimalcontrol problem (2). SQP aims at solving a nonlinear optim-ization problem by sequentially solving linearly constrainedquadratic programs (LCQP), which are formed, e.g., by ap-proximating the objective function with a quadratic equationand linearizing the constraints. We will further prove thatsolving the SQP algorithm yields a solution to (2), providedthat the solution converges. In particular, we will solve (2)by recursively solving the SQP subproblem

xi+1k , ui+1

k k∈K =

argminxk,uk

∑

k∈K

12

[xk−xi

k

uk−uik

]>Rk

[xk−xi

k

uk−uik

]+(Hk

[xik

uik

]+Fk

)>[xkuk

],

(4a)

subject to linearized state dynamics,

xk+1 =f(xik, uik) +∇f(xik, u

ik)[xk−xi

k

uk−uik

], (4b)

and state constraints and input constraints,

xk 6 xk 6 xk, uk 6 uk 6 uk, (4c)

for all k ∈ K and i ∈ N, and for given x0, xK aswell as some suitably well chosen x0k, u0kk∈K, such that




97 / 110

a feasible solution exists for the next iteration of the SQPsubproblem (4). Thus, we have formed the SQP subproblem(4) by linearizing the state equations (2b), and linearizing theobjective function (2a) and adding an Rk 0 to ensure aconvex objective function in (4a), which can be regarded asa proximal or Thikonov regularization. The matrix Rk canbe chosen to warrant that the SQP subproblem (4) is strictlyconvex in its control variables ukk∈K and converges. Wenote that by choosing Rk = Hk, the SQP objective function(4a) becomes exactly the original objective function (2a).The SQP algorithm can be terminated when, e.g.,

|J i+1 − J i| 6 ∆tol, (5a)

in which ∆tol is a certain specified tolerance, and

J i=∑

k∈K

12

[xik

uik

]>Hk

[xik

uik

]+F>k

[xik

uik

]+ν∑

k∈K|xik+1−f(xik, u

ik)|

(5b)

is the optimal cost at iteration i, which can be considered asa merit function for the SQP approach (4). In (5b) ν > 0 ischosen such that infeasible solutions to the SQP subproblem(4) at iteration i return a higher cost than optimal feasiblesolutions. Note that in this SQP approach, we allow infeas-ible solutions at iteration i, and as the algorithm converges,i.e., |J i+1 − J i| → 0, feasibility is obtained in the limit.Also note that as the state and input constraints (4c) areautomatically satisfied with a solution to the SQP (4), andthus there is no contribution from these inequality constraintsto the merit function (5b).

We can prove that when the SQP problem has converged,i.e. xi+1

k = xik and ui+1k = uik for all k ∈ K, the first-

order necessary conditions for optimality, i.e., the so-calledKarush-Kuhn-Tucker (KKT) conditions [19], for (4) areidentical to the KKT conditions of (2). The KKT conditionsuse the notion of a Lagrangian, which for (2) is given by,

L(xk, uk, λk, µk) =∑

k∈K

12 [xkuk

]>Hk[xk

uk]+Fk[xk

uk]

+∑

k∈Kµ>k+1

(xk+1−f(xk, uk)

)+λ>kh(xk, uk), (6)

where h(x, u) = [ (x−x)> (x−x)> (u−u)> (u−u)> ]>. For a

local optimal solution x∗k, u∗kk∈K, there exist λ∗k, µ∗kk∈Kthat satisfy stationarity of the Lagrangian, i.e.,

Hk

[x∗ku∗k

]+ Fk +

[µ∗k0

]−∇f(x∗k, u

∗k)>µ∗k+1

+∇h(x∗k, u∗k)>λ∗k = 0, (7a)

and primal feasibility and complementarity slackness of theconstraints, i.e.,

x∗k+1 − f(x∗k, u∗k) = 0 (7b)

0 6 λ∗k ⊥ −h(x∗k, u∗k) > 0, (7c)

where the notation 0 6 a ⊥ b > 0 indicates a, b > 0,a>b = 0. Similarly, the necessary conditions for optimality

of (4) are given by a stationarity condition, i.e.,

Rk

[x∗k−xi

k

u∗k−uik

]+Hk

[xik

uik

]+Fk +

[µ∗k0

]

−∇f(xik, uik)>µ∗k+1 +∇h(x∗k, u

∗k)>λ∗k = 0, (8a)

and primal feasibility and complementary slackness of theconstraints, i.e.,

x∗k+1 − f(xik, uik) +∇f(xik, u

ik)[x∗k−xi

k

u∗k−uik

]= 0, (8b)

0 6 λ∗k ⊥ −h(x∗k, u∗k) > 0. (8c)

We observe that when x∗k = xi+1k = xik and u∗k = ui+1

k = uikthe KKT conditions (8) and (7) are equal. Thus, we can find(local) solutions of (2) by solving the SQP (4), providedthat the iterates converge, in the sense that xi+1

k → xik andui+1k → uik for all k ∈ K when i→∞.We remark that the state variables may be eliminated in the

SQP problem (4), by rewriting the linearized state equations(4b) in a prediction form, where the state variables are givenby a set of prediction matrices and the inputs, i.e.

x =

[x0...xK

]= Φ(xi,ui) + Γ(xi,ui)u, (9)

where u = [ u0 ··· uK−1 ]> and Φ, Γ are some matrices

such that (9) represents (4b). By substituting (9) into theobjective function (4a) and state constraints (4c), as isoften done in model-predictive control, we can arrive at anSQP subproblem with only the input variables as decisionvariables, and the state variables can be updated with (9).

IV. ENERGY MANAGEMENT WITH ECODRIVING

As a case study for Vehicle Energy Management (VEM)with ecodriving, we consider a series-hybrid electric vehicle,consisting of an electric motor (EM), engine-generator unit(EGU) and a high-voltage battery (HVB). The topology isshown in Fig. 1, in which yegu and uegu denote the ICE’sfuel and mechanical power, respectively, yhvb and uhvb thebattery’s electrical and stored chemical power, ybr is anartificial braking power exerted at the interconnection of thesubsystems, uem and yem the EM’s mechanical force andelectrical power, respectively, xhvb denotes the battery stateof energy and xem = [ vs ], denotes the states speed v anddistance traveled s of the vehicle. In VEM, the main goal isto minimize fuel consumption, given by

∑

k∈Kτyegu,k, (10a)

EGU

xhvb HVB

EM xem

yegu

uhvb

yem uem

uegu

yhvb ybr

Fig. 1: Topology




98 / 110

subject to the dynamics and input-output behavior of theconverters in Fig 1, and the power balance at interconnectionof the subsystems, i.e.,

yem,k − uegu,k − yhvb,k = ybr 6 0, (10b)

for all k ∈ K. We use the constraint (10b) and energy balanceconstraint of the HVB, i.e.,

∑

k∈Kτuhvb = xhvb,K − xhvb,0 = 0 (10c)

to rewrite (10a) as a ‘sum of losses’, i.e.,

τ∑

k∈Kyegu,k−uegu,k+uhvb,k−yhvb,k+yem,k−ybr,k. (10d)

By substituting (10b) and (10c) into (10d), we retrieve theoriginal objective function (10a). In this section, we solve theVEM problem by formulating it as a convex SQP problemand then apply dual decomposition as presented in [1].

A. Optimal Control Problem

The objective is to minimize fuel consumption, for whichwe use the equivalent fuel consumption (10d), which maybe written as

minum,k,ym,k

∑

m∈M

∑

k∈Kamum,k + bmym,k (11a)

where um,k ∈ R and ym,k ∈ R are the (scalar) inputsand outputs of the converter in subsystem m ∈ M =em, egu,hvb at time instant k ∈ K. The optimizationproblem (11a) is to be solved subject to an equality con-straint describing the quadratic input-output behavior of eachconverter, i.e.,

ym,k = 12

[ xm,kum,k

]>Hm

[ xm,kum,k

]+ F>m

[ xm,kum,k

]+ em (11b)

for all k ∈ K, m ∈M, and subject to the system dynamicsof the states in subsystem m ∈M, i.e.,

xm,k+1 = fm(xm,k, um,k) (11c)

for all k ∈ K where the initial state xm,0 and final statexm,K of the energy storage device are assumed to be given,and the input um,k is subject to linear inequality constraints,i.e.,

um,k 6 um,k 6 um,k, (11d)

for all k ∈ K, m ∈M and the state xm,k is subject to linearinequality constraints, i.e.,

xm,k 6 xm,k 6 xm,k, (11e)

for all k ∈ K and m ∈M. Finally, the optimization problemis solved subject to a linear inequality constraint describingthe interconnection of the subsystems given by (10b), whichwe write as

∑

m∈Mcmum,k + dmym,k = ybr,k 6 0, (11f)

for all k ∈ K, where cm, dm ∈ R. Note that we have leftout the term −ybr,k in (11a), as ybr,k for all k ∈ K is

already given by the inequality (11f), which renders ybr,k asan implicit decision variable. To have that (11a) correspondsto (10d), we choose

aegu =bhvb =−τ, ahvb =begu =bem =τ, aem =0, (12a)

and to have that (11f) corresponds to (10b), we choose

cegu =dhvb =−1, dem =1, chvb =cem =degu =0. (12b)

Finally, we assume in this paper that the coefficients for theinput-output behavior of the converters in (11b) are given by

Hem =

[γem0 0 γem1

0 0 0γem1 0 γem2

], Fem =

[000

], eem =0,

Hhvb =[0 00 γhvb2

], Fhvb =

[0

γhvb1

], ehvb =γhvb0

,Hegu =γegu2

, Fegu =γegu1, eegu =γegu0

,(12c)

for some parameters γm2, γm1

, γm0, m ∈ M, the state

dynamics for the EM are given by

f(xem,k, uem,k) =[vk+τ

(σuuem,k−σvv2k −σr−g sin(α(sk))

)

sk + τvk

], (12d)

and the battery state of energy is given by

fhvb(xhvb,k, uhvb,k)=xhvb,k−τuhvb,k. (12e)

B. SQP Formulation

To form a convex SQP formulation, we propose to re-lax the (nonconvex) quadratic input-output behavior of theconverters (11b) to a convex quadratic approximation bylinearizing (11b) around [(xi

m,k)> (ui

m,k)>]> and adding a

convex quadratic part, i.e.,

ym,k≈ 12

([xkuk

]−[xik

uik

])>Rm([xkuk

]−[xik

uik

])

+(Hm

[xik

uik

]+Fm)>[xk

uk]+em, (13)

for all k ∈ K and m ∈ M, where the matrix Rm 0 ischosen such that (13) is convex. We note that the approxima-tion error disappears when xm,k → xim,k, um,k → uim,k, andby choosing Rm = Hm, we retrieve the original quadraticequation (11b). By substituting (13) into (11a) and (11f) andlinearizing the dynamics (11c), we arrive at the convex SQPsubproblem

xi+1m,k, u

i+1m,kk∈K,m∈M=

argminxm,k,um,k

∑

m∈M

∑

k∈K

12bm

([xkuk

]-[xik

uik

])>Rm([xkuk

]-[xik

uik

])

+(bmHm

[xik

uik

]+bmFm +

[0am

])>[xkuk

] + bmem, (14a)

subject to the linearized state dynamics, i.e.,

xm,k+1 − fm(xim,k, uim,k)

−∇fm(xim,k, uim,k)

([xkuk

]−[xik

uik

])= 0, (14b)

for all k ∈ K, m ∈ M, and subject to linear inequalityconstraints, i.e.,

xm,k 6 xm,k 6 xm,k, um,k 6 um,k 6 um,k (14c)




99 / 110

for all k ∈ K, m ∈ M, and further subject to the convexquadratic inequality constraint specifying the interconnectionof the subsystems∑

m∈M

12dm

([xkuk

]−[xik

uik

])>Rm([xkuk

]−[xik

uik

])

+(dmHm

[xik

uik

]+dmFm +

[0cm

])>[xkuk

]+dmem 6 0, (14d)

for all m ∈M.

C. Dual Decomposition

Note that (14a) subject to (14b) and (14c) is entirelyseparable, and the only complicating constraint is (14d),which is the constraint that acts on all components m ∈M. Therefore, we propose to decompose (14) via dualdecomposition by augmenting the objective function withthe constraint (14d), which results in the so-called ’partialLagrangian’:

L(xm,k, um,k,λk)=∑

k∈K

∑

m∈M

12

([xkuk

]-[xik

uik

])>Rm,k

([xkuk

]-[xik

uik

])

+(Hm,k

[xik

uik

]+Fm,k

)>[xkuk

]+Em,k, (15a)

in which,

Rm,k = (bm+dmλk)Rm,

Hm,k = (bm+dmλk)Hm,

Fm,k = (bm+dmλk)Fm+(am+cmλk)[ 01 ],

Em,k = (bm+dmλk)em, (15b)

where λk > 0 ∈ RN is a Lagrange multiplier. The partialLagrange dual function is then given by

g(λk)= minxm,k,um,k

L(xm,k, um,k, λk)

=∑

m∈Mgm(λk) + Em,k, (16a)

subject to (14b) and (14c), with

gm(λk)= minxm,k,um,k

∑

k∈K

12

([xkuk

]-[xik

uik

])>Rm,k

([xkuk

]-[xik

uik

])

+(Hm,k

[xik

uik

]+Fm,k

)>[xkuk

], (16b)

subject to (14b) and (14c), the dual problem for eachcomponent. Note that for the components whose dynamicsfm(xm,k, um,k) are linear and Rm = Hm yields a convexdual problem (16b), SQP is not needed to solve the dualproblem, as they can be solved as a QP problem. Typically,gegu and ghvb are both QP problems and gem is an SQPproblem that can be solved through the approach given inSection III. However, it may still be beneficial to applyregularization to the other components, since it may improveconvergence properties, as we will show in Section V. Thedual problem is given by

maxλk

g(λk) = d∗, (17)

subject to (14b) and (14c), where d∗ is defined as the dualoptimal solution. The dual problem (17) gives a lower boundon the primal optimal value p∗ of problem (14), i.e.,

d∗ 6 p∗. (18)

The dual problem equals the primal problem, i.e. d∗ = p∗,if problem (14) is convex and the constraints satisfy Slater’sconstraint qualifications [19]. As we have formed a convexSQP subproblem (14) and assume that the Slater’s constraintqualifications are satisfied for this problem formulation, wehave that d∗ = p∗ in our SQP and dual decompositionapproach. We maximize the dual problem (17) with a steepestascend method, i.e.,

λi+1k = max

0, λik+ρk

( ∑

m∈Mcmu

i+1m,k+dmy

i+1m,k

), (19)

for all k ∈ K, where ρk > 0 is chosen small enough such thatthe dual problem converges. In our approach, note that in 1iteration, we solve both the SQP subproblem (14) as well asupdate the Lagrangian multiplier λk for all k ∈ K. We havefound that as long as Rm for all m ∈ M are well chosen,this is the fastest way to let the dual problem converge. Thedual problem (17) is terminated similarly to the SQP (4), i.e,it is terminated under the condition that

|J i+1 − J i| 6 ∆tol, (20a)

in which ∆tol is a certain specified tolerance, and

J i=∑

k∈Kτyegu,k+ν1

∑

k∈Kmax

0,∑

m∈Mcmu

im,k+dmy

im,k

+ν2∑

m∈M

∑

k∈K|xim,k+1−f(xim,k, u

im,k)|,

(20b)

where ν1 > 0 and ν2 > 0 are penalty parameters. Themerit function (20b) has a cost term that defines the fuelconsumption, a penalty term related to the violation of thestate equality constraints (11c), and a penalty term relatedto the augmented inequality constraint (11f). Indeed, thesetwo constraints are the only constraints that may be violatedat iteration i of the dual problem (17), and thus have acontribution in the merit function (20b). This concludesthe SQP and dual decomposition approach presented inthis section, where by formulating (11) as a convex SQPsubproblem (14), we could form the dual problem (17),which solves the vehicle energy management problem (11).

V. RESULTS

In this section, we show the results of two simulationstudies where the Ecodriving problem is present. In the firstsimulation study, we will use the SQP approach presentedin Section III to replicate the results of the Ecodrivingproblem for a full-electric vehicle (FEV) detailed in [12],using the SQP approach presented in Section II. In thesecond simulation study, we consider an energy managementproblem for a series-hybrid electric vehicle, and solve thiswith the approach presented in Section IV. The results of thissimulation are comparable to the results presented in [15],




100 / 110

0 0.2 0.4 0.6 0.8 1

Time [s]

-10

-5

0

5

10

15

distance [m]speed [m/s]gsin(α(s)) [m/s]u/10 [%]

Fig. 2: Results of the FEV case study.

yet not exactly the same, due to slightly different models thatare used.

A. Full Electric Vehicle

The full electric vehicle simulation study as presented in[12] has a powertrain consisting of an electric motor (EM)and a high-voltage battery (HVB). However, in this casestudy, the HVB is not considered, i.e., it is assumed thatthe HVB has infinite energy storage and no power limita-tions, which makes it a simple but representative examplefor ecodriving. Furthermore, the optimal control problemgiven in [12] does not consider input and state constraints.This follows from the fact that [12] applies Pontryagin’sMaximum Principle, in which it is hard to introduce state andinput constraints. We solve the ecodriving problem presentedin [12] using the discretization approach as presented inSection II, with τ = 0.0005, and SQP, as presented inSection III. We also show that with the SQP approach, addingstate and input constraints becomes trivial.

The optimal control problem defined in [12] is given bythe ecodriving problem (1) as presented in Section II, with

g sin(α(s)) = p0 + p1s+ p2s2 + p3s

3. (21)

The parameters used for the FEV simulation study are givenin Table I. We note that in this formulation, the control inputu is a torque percentage defined as u = T

Tmax× 100%,

in which T is the EM mechanical torque and Tmax is themaximum EM torque. It is important to remark that thediscretized objective function (2a) is not convex with thechosen parameters; however, choosing Rk = Hk in theSQP subproblem (4a) results in a convex objective functionafter the substitution of the state variables (9) into the SQPsubproblem. To select initial conditions for the input andstates, we choose a constant velocity profile vk = 0 m/sfor all k ∈ K, and determine sk and uk for all k ∈ Kfrom the longitudinal vehicle dynamics (2b). We set the

TABLE I: Full Electric Vehicle Parameters

p0 = 3 v(t0) = 0 σr = 0.1 γ0 = 0p1 = 0.4 v(tf ) = 0 σv = 10−3 γ1 = 103

p2 = −1 s(t0) = 0 σu = 1 γ2 = 2×103p3 = 0.1 s(tf ) = 10 t0 = 0 tf = 1

0 0.2 0.4 0.6 0.8 1

Time [s]

-10

-5

0

5

10

15

distance [m]speed [m/s]gsin(α(s)) [m/s]u/10 [%]

Fig. 3: FEV case study with state constraints. The blackdotted line represents the maximum speed constraint.

tolerance to terminate the SQP algorithm defined in (5a) to∆tol = 1 and select for the penalty parameter ν = 2 × 104

in (5b). In Fig. 2, the results of the FEV simulation study,without state and input constraints are shown, in which wesee that the initial and final state constraint are satisfied.We further note that the speed curve is not symmetrical,due to the road profile; close to where the slope is steepest,speed is maximized, as is expected. The cost as defined bythe discrete-time objective function by the merit function(5b) is J = 1227247.8, which differs by 0.1% to the costobtained in [12]. This small difference may be caused due tonumerical inaccuracies presented in both this work, as wellas in [12], and the finite sampling time. It is interesting tonote that although global optimality is guaranteed neither inthe approach used in this paper nor in the approach usedin [12], the solutions obtained are practically identical. Weremark that global optimality is guaranteed in neither ourapproach in this paper, nor the approach based on PMPin [12], because both approaches only consider necessaryconditions for optimality. As an illustration to the ease ofadding state and input constraints, in Fig. 3, the results of thesimulation study with a maximum speed constraint of 12 m/sare shown. We observe that with a cost of J = 1511796.5, byadding this constraint, the cost becomes higher than withoutstate constraints, as is expected.

B. Series-Hybrid Electric Vehicle

In this simulation study, we consider the Series-HybridElectric Vehicle (SEHV) case study presented in Section IV.We solve the case study in a ‘forward’ and ‘backward’ sim-ulation using the approach presented in Section IV. The term‘forward’ and ‘backward’ refer to way in which the longitud-inal vehicle dynamics differential equations (1c) are treated.That is, in ‘backward’ simulation, the differential equationsare treated as ‘quasi-static’ equations, i.e., the states v and sare given a priori and in ‘forward’ simulation, the differentialequations are treated ‘dynamically’, i.e., the states remaindecision variables. We will refer to ‘forward’ optimizationas solving the vehicle energy management problem withecodriving (11), and refer to ‘backward’ optimization assolving (11) as an energy management problem withoutecodriving, where the vehicle trajectory information, i.e.




101 / 110

speed vk and distance sk for all k ∈ K, are given. We furthersolve the ‘forward’ optimization problem with the SQP (14),although without applying dual decomposition, which leadsto a Quadratically Constrained Quadratic Program (QCQP),which we will refer to as the ‘direct’ method, to validatethe dual decomposition method. We refer to the SQP anddual decomposition approach presented in Section IV as the‘distributed’ method. To show additional capabilities of theSQP approach, we show an example where we introducetime-varying minimum and maximum speed constraints.

We base our case study on the work done in [15], inwhich a convex optimization approach is taken to solve theSEHV case study, where it is formulated a second-order coneprogram. Due to the chosen problem formulation, the authorsin [15] have opted for a piece-wise linear EM model, linearEGU model and a quadratic HVB model. Furthermore, theauthors in [15] define the optimization problem in the spacedomain, for which the physical interpretation of some partsof their problem formulation is not easy to understand. Aswe have defined the SEHV case study in the time domain,we do not face this issue. To have somewhat comparableresults, we fit the parameters of our quadratic EM modelusing a least-squares fitting tool. We further approximate thelinear EGU model with a quadratic EGU model by choosingthe quadratic coefficient γegu2

in (12c) sufficiently small.The parameters used for the simulation study are shown inTable II. We remark here that the upper and lower bounds forthe state and input variables specified in Table II are definedfor all k ∈ K.

The simulations are done for 1080 s over a distance of21km with a step size τ = 5 s, which gives an optimizationhorizon K = 216. In ‘backward’ optimization, the speed isgiven by a constant speed of vk = 70 km/h for all k ∈ K,and in ‘forward’ optimization the initial and final velocity aregiven, while over the trajectory the speed is allowed to varyby 70 ± 10 km/h. We choose Rhvb = Hhvb for the HVBdual problem ghvb, as it has a convex quadratic objectivefunction and linear constraints. As the EGU dual problemgegu also has a convex quadratic objective function and linearconstraint, it may be solved as a QP problem, and thus wemay choose Regu = Hegu. However, we will show that,because the quadratic term Hegu is small in magnitude forthe dual problem gegu, it is beneficial to tune Regu 0,such that convergence of the dual problem (17) is improve.Specifically, we will vary Regu as follows

Regu = (1− αR)Hegu + αR, (22)

TABLE II: Series Hybrid Electric Vehicle Parameters

γem0 = 0.5856 m = 15950 kg sK = 21000 mγem1 = 1.005 g = 9.81 m/s2 xhvb,0 = 11988 kJγem2 = 5.052×10−4 cd = 0.7 xhvb,K = xhvb,0γegu0

= 38 cr = 0.0007 vk = 22.22 m/sγegu1

= 2.52 Af = 7.54 m2 vk = 16.67 m/sγegu2

= 2×10−5 ρA = 1.184 kg/m2 xhvb,k = 22680 kJγhvb0

= 0 v0 = 19.44 m/s xhvb,k = 7560 kJγhvb1

= −1 vK = v0 uhvb,k = 92.4 kWγhvb2

= 1.671×10−3 s0 = 0 m uhvb,k = −92.4 kW

0

100

200

ue

gu [k

W]

BackwardForward distributedForward direct

-200

0

200

ue

m [k

W]

-40

-20

0

y br [k

W]

2

4

6

x hvb

[kW

h]

60

70

80

v [k

m/h

]0 2 4 6 8 10 12 14 16 18 20

Distance [km]

0

100

200

300

Roa

d al

titud

e [m

]

Fig. 4: Backward vs forward optimization. The dotted linesrepresent minimum and maximum state constraints.

where αR is a tuning parameter and we note that αR = 0gives Regu = Hegu. The EM dual problem gem has a noncon-vex quadratic objective function and nonlinear constraints,and thus we are restricted to choose an Rem 0. Spe-cifically, we choose Rem = 0.02[ 0 0

0 I ]. As initial conditionsfor the ‘forward’ optimization, we choose the solutions ofthe ‘backward’ optimization. For the termination of the dualproblem (17), we choose δtol = 10−3 in (20a) and ν1 = 103,ν2 = 104 for the penalty parameters of the merit function. InFig. 4, the simulation results for ‘backward’ and ‘forward’optimization using the ‘distributed’ method, and ‘forward’optimization using the ‘direct’ method are given. It can beseen that the results for ‘forward’ optimization using the‘direct’ method and ‘distributed’ method are almost the same,which validates the dual decomposition approach. In the‘forward’ optimization results, we see that between 0 and 2km, where the slope becomes negative, the vehicle reachesthe lower speed bound. This action allows the availablepotential energy from the road profile, between 2 and 13km, to be maximally converted to kinetic energy; the vehiclespeed is maximized in this interval. After 13 km, the roadgradient becomes positive and speed is minimized such thatthe final speed constraint is met. Therefore, we can see thatas a result of having the speed as a decision variable, in the‘forward’ case, the EGU may provide less power over thecourse of the trajectory, and noticeably less braking poweris applied, when it is compared to the ‘backward’ case. Thefuel consumption of the ‘backward’ and ‘forward’ simulationcases are 23.41 l/100 km and 22.31 l/100 km respectively.Thus, by including the ecodriving problem into the vehicleenergy management problem, approximately 4.7 % decrease




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100 101 102 103 104

Iteration i

106

107

108

Cos

t J

αR=0

αR=0.00005

αR=0.0005

αR=0.002

Fig. 5: Convergence of the SQP and dual decompositionapproach.

in fuel consumption is achieved. This is compared to thefuel consumption obtained in [15], which are 24.35 l/100 kmand 23.98 l/100 km for the ‘backward’ and ‘forward’ caserespectively. This is a 1.54 % decrease in fuel consumptionbetween the ‘backward’ and ‘forward’ case. We may explainthis difference in fuel consumption savings largely due to thedifferent models used.

Finally, the convergence of the SQP + dual decompositionapproach presented in section IV is shown in Fig 5. Wesee that the number of iterations for the dual problem (17)is greatly reduced by tuning Regu, even though Regu =Hegu = 2× 10−5 yields a convex problem. Specifically,The dual problem (17) converges after 1003 iterations withαR = 0.002 and converges after around 21000 iterationswith αR = 0. We refer to convergence as the point where thedual problem may be terminated according to the tolerancespecified by (20a), although we have not terminated thedual problem at these specified iterations for the purpose ofillustration. This shows the additional advantage of regular-ization, which is the potential to improve converge, whereaswe have used regularization for the EM dual problem gegu toenforce convexity. However, from Fig. 5 we also see that wesacrifice monotonicity for speed of convergence by choosingαR > 0, where the graphs shows more oscillatory behavioras αR increases. However, this is not an issue as long as therequired tolerances are reached for the termination of thedual problem.

VI. CONCLUSIONIn this paper, we have solved the ecodriving problem using

a Sequential Quadratic Programming (SQP) algorithm. Wehave formulated the ecodriving problem as a discrete-timenonlinear optimal control problem, and have shown that dueto the discretization in this approach, the formulation may benonconvex. In the SQP algorithm, we have formed a convexSQP subproblem using Thikonov regularization. To show-case the scalability of the distributed optimization approachpresented in [1], we have embedded the SQP algorithm intothis distributed optimization approach, which allows it to beused for Complete Vehicle Energy Management (CVEM) incombination with ecodriving. We have done so by formulat-ing a vehicle energy management with ecodriving problemas a convex SQP problem and applying dual decomposition

to solve a dual problem. We have considered two case studiesfor the ecodriving problem. In the first case study, we havesolved the ecodriving problem for a Full Electric Vehicleusing the SQP algorithm and have shown that the algorithmyields a very similar result as the benchmark problem setin [12], and shown that adding state constraints is trivialusing SQP. In the second case study, we have solved anenergy management problem with ecodriving for a Series-Hybrid Electric Vehicle with the SQP algorithm and dualdecomposition. We have shown that using our approach,we have had the opportunity to use more representablepowertrain models than in [15], and as a result have thepotential to save more fuel by incorporating ecodriving tothe energy management problem.

REFERENCES

[1] T. Romijn, M. Donkers, J. Kessels, and S. Weiland, “A dual decom-position approach to complete energy management for a heavy-dutyvehicle,” in Proc. IEEE Conf. on Decision and Control, 2014.

[2] M. Donkers, J. Van Schijndel, W. Heemels, and F. Willems, “Optimalcontrol for integrated emission management in diesel engines,” ControlEngineering Practice, 2017.

[3] T. M. Padovani, M. Debert, G. Colin, and Y. Chamaillard, “Optimalenergy management strategy including battery health through thermalmanagement for hybrid vehicles,” Proc. of the IFAC World Congress,2013.

[4] S. Ebbesen, P. Elbert, and L. Guzzella, “Battery state-of-health per-ceptive energy management for hybrid electric vehicles,” IEEE Trans.Control Syst. Tech., 2012.

[5] Z. Khalik, T. Romijn, M. Donkers, and S. Weiland, “Effects ofbattery charge acceptance and battery aging in complete vehicle energymanagement,” Proc. of the IFAC World Congress, 2017.

[6] J. Kessels, J. Martens, P. van den Bosch, and W. Hendrix, “Smartvehicle powernet enabling complete vehicle energy management,” inProc. IEEE Vehicle Power and Propulsion Conf., 2012.

[7] D. P. Bertsekas, Dynamic programming and optimal control. AthenaScientific Belmont, MA, 1995, vol. 1, no. 2.

[8] L. V. Perez, G. R. Bossio, D. Moitre, and G. O. Garcıa, “Optimizationof power management in an hybrid electric vehicle using dynamicprogramming,” Math. Comput. Simulation, 2006.

[9] T. Pham, J. Kessels, P. van den Bosch, R. Huisman, and R. Nevels,“On-line energy and battery thermal management for hybrid electricheavy-duty truck.” in Proc. American Control Conf., 2013.

[10] N. Murgovski, L. Johannesson, J. Sjoberg, and B. Egardt, “Componentsizing of a plug-in hybrid electric powertrain via convex optimization,”Mechatronics, 2012.

[11] T. van Keulen, B. de Jager, and M. Steinbuch, “Optimal trajectoriesfor vehicles with energy recovery options,” Proc. of the IFAC WorldCongress, 2011.

[12] N. Petit and A. Sciarretta, “Optimal drive of electric vehicles usingan inversion-based trajectory generation approach,” IFAC World Con-gress, 2011.

[13] G. Heppeler, M. Sonntag, U. Wohlhaupter, and O. Sawodny, “Predict-ive planning of optimal velocity and state of charge trajectories forhybrid electric vehicles,” Control Engineering Practice, 2017.

[14] E. Hellstrom, J. Aslund, and L. Nielsen, “Design of an efficientalgorithm for fuel-optimal look-ahead control,” Control EngineeringPractice, 2010.

[15] N. Murgovski, L. Johannesson, X. Hu, B. Egardt, and J. Sjoberg,“Convex relaxations in the optimal control of electrified vehicles,” inAmerican Control Conference (ACC), 2015. IEEE, 2015.

[16] J. Nocedal and S. J. Wright, Sequential quadratic programming.Springer, 2006.

[17] P. T. Boggs and J. W. Tolle, “Sequential quadratic programming,” Actanumerica, 1995.

[18] L. Etman, A. A. Groenwold, and J. Rooda, “First-order sequentialconvex programming using approximate diagonal qp subproblems,”Structural and Multidisciplinary Optimization, 2012.

[19] S. Boyd and L. Vandenberghe, Convex Optimization. CambridgeUniversity Press, Cambridge, 2004.




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Eco-Driving for Energy Efficient Corneringof Electric Vehicles in Urban Scenarios ?

G.P. Padilla C. Pelosi C.J.J. Beckers M.C.F. Donkers

Eindhoven University of Technology, Netherlands(e-mail: g.p.padilla.cazar,c.j.j.beckers,[email protected], [email protected])

Abstract: In this paper, we propose a model for eco-driving that considers cornering effects.The proposed model purely relies on the geometric configuration of the vehicle and road.Consequently, we propose and eco-driving optimal control problem formulation that is suitablefor both straight and curved trajectories in urban scenarios. Moreover, it can be applied forvehicles with front wheel drive (FWD) or rear wheel drive (RWD). We use a case studyfor an electric vehicle executing cornering maneuvers to validate the proposed approach andhighlight its advantages with respect to traditional eco-driving formulations. Results show anapproximated improvement of 8% in energy savings with respect to traditional eco-drivingstrategies, especially in trajectories with large curvatures.

Keywords: Energy management system, optimal control, velocity control, electric vehicles,vehicle dynamics.

1. INTRODUCTION

In recent years, electrification of transport systems orelectromobility (Grauers et al., 2013), has been raised asa promising approach to mitigate environmental effectscaused by CO2 emissions and to face the imminent deple-tion of fossil fuels reserves. However, the goal of completeacceptance of electromobility in the market is still farand several problems like range anxiety still have to besolved. Range anxiety is defined as the fear experienced bydrivers of having insufficient energy to arrive to the nextcharging station (AccessScience Editors, 2014). In general,energy management strategies are a direct method to re-duce range anxiety. In particular, the capability to extenddriving range by reducing power demand demonstrated byeco-driving techniques has positioned these approaches asstrong tools alleviate effects of range anxiety, e.g., Floreset al. (2019).

As a general concept, eco-driving is a selection of energyefficient driving strategies (Bingham et al., 2012), whichcan be obtained by solving an optimal control problem(OCP) that aims to find optimized velocity profiles thatreduce the total consumption of the vehicle, e.g., seeSciarretta et al. (2015). This concept has been studiedin terms of the energy source of the vehicle, e.g., forconventional vehicles in Maamria et al. (2017), for hybridelectric vehicles by Khalik et al. (2018), and for electricvehicles by Petit and Sciarretta (2011). Eco-driving hasalso been analysed in terms of the surrounding operationalconditions. For urban scenario cases, Nunzio et al. (2013)have included traffic light information into the eco-drivingformulation, Henriksson et al. (2017) use traffic statistic


data to create velocity corridors where solutions the eco-driving OCP lay, and Han et al. (2018) consider theeffects of the preceding vehicle in the problem formulation.Interestingly, the literature about eco-driving for urbantrajectories where cornering is considered is scarce.

In (Beckers et al., 2019), a high-fidelity model is presentedto calculate additional energy losses during cornering ofthe vehicle. The results presented in this work have shownthat energy losses during cornering maneuvers should notbe neglected. Cornering losses become relevant duringurban scenarios where the vehicle might follow routeswhere several turning maneuvers are performed. Often,eco-driving formulations that consider the road curvatureinclude a constraint linked to the centripetal acceleration,which indirectly limits the maximum velocity during cor-nering, e.g., Polterauer et al. (2019). In this case, thecornering losses are not directly considered in the for-mulation. As a result, the solutions obtained are unlikelyto be energy optimal during the cornering maneuver. Aninteresting case is presented in Ikezawa et al. (2017), wherea dynamical vehicle model is used to describe a simplifiedmodel that can be used to formulate a eco-driving OCP.Unfortunately, this description depends on cornering stiff-ness of the tires, which can show significant variationsbetween vehicles and road conditions. Other approachesaim to achieve energy efficient cornering by applying en-ergy efficient torque vectoring to the vehicle (Edren et al.,2019). Unfortunately, these approaches partially neglectthe eco-driving concept by only focusing on the corneringmaneuver itself.

In this paper, we aim to bridge the gap observed in liter-ature by extending the current eco-driving OCP formula-tion of Padilla et al. (2018) to consider cornering effects.In particular, our main contributions are the use of a kine-matic bicycle model to approximate the dissipative forcesproduced during cornering in the longitudinal direction of




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the vehicle. This model purely depends on the geometryof the road and the vehicle, which makes it suitable to bedeployed in real applications where the tire properties areoften unknown. Thus, we propose a trajectory dependentmodel for eco-driving that can be used for vehicles withfront wheel drive (FWD) and rear wheel drive (RWD).This allow us to formulate generalized eco-driving OCPthat is suitable to be used in urban routes.

The remainder of this paper is organized as follows. InSection 2, the cornering effects are approximated in alow complexity model for eco-driving applications. Section3, proposes and OCP formulation that takes advantagesof the modeling choice proposed in this paper. Later, acase study for an electric vehicle performing a corneringmaneuver in a urban intersection is presented in Section4. Here we will validate our approach using a high-fidelity model to calculate losses during cornering and wewill highlight the advantages of the proposed approach.Finally, we will draw conclusions in Section 5.

2. A TRAJECTORY-DEPENDENT MODEL FORECO-DRIVING

In this section, we aim to provide a general vehicle dynam-ical model to be used in the eco-driving optimal controlproblem proposed in this paper. To this end, we analysethe representations used for straight trajectories, whichcoincides with traditional models used for eco-driving inthe current literature. Later, we propose a generalizeddescription of the dissipative forces for curved trajectories,which are often present in urban scenarios. Finally, weshow that the generalized models obtained for curvedtrajectories can easily describe the dynamics for straighttrajectories as well.

The main objective of eco-driving strategies is to obtainenergy optimal velocity profiles. In general, the dynamicalmodels considered in traditional eco-driving approachesrepresent the longitudinal vehicle dynamics by the inter-action between the traction force in the longitudinal axisFl(t) and dissipative forces Fd(t), i.e.,

ma = Fl − Fd, (1)

where m represents the equivalent mass of the vehicle, anda(t) = dv

dt is the vehicle acceleration. The definition ofFd(t) in (1) can vary depending on the trajectory that thevehicle is describing. Specifically, we consider the cases forstraight and curved trajectories below.

2.1 Straight Trajectories

A traditional assumption for eco-driving problem formu-lations is that the vehicle moves in a straight trajectorysuch that the dissipative force is given by

Fd(v, s)=σdv2

︸︷︷︸Fair

+mg(cr cos(α(s))+sin(α(s)))︸︷︷︸Froll+grav

, (2)

where v(t) represents the vehicle velocity, s(t) describesdisplacement, α(s(t)) is the road grade, g is the gravi-tational constant, cr > 0 is the rolling resistance, andσd = 1

2cdρaAf with the aerodynamical drag coefficientcd > 0, air density ρa and frontal area of the vehicle Af .In the right-hand side of (2), the term Fair representsthe force produced by aerodynamical drag, and the term

rlfl

R

ICR

β−δ

v,acF

δββ−2π

clF

airF+rollF

s(t)CG

lFairF+rollF

Curved trajectory

lF

Straigth trajectory

v,a

CG

β

s(t)

Fig. 1. Forces on the londitudinal axis for both straightand curved trajectories.

Froll+grav is the force connected to rolling resistance andgravity. These forces are depicted in Fig. 1 for a straighttrajectory case. Finally, by defining the total traction forceprovided by the electric motor Fu(t), we see that

Fl = Fu (3)

Note that this relationship is independent of the driveconfiguration of the vehicle, i.e., (3) holds for both RWDand FWD configurations.

The assumption of a straight trajectory is widely adoptedin literature, see, e.g., Sciarretta et al. (2015) and the ref-erences therein. It is intuitive to validate this assumptionfor vehicles driving in highways, where trajectories withlarge curvature are not often present. However, corneringactions are often needed during a typical urban trajectory.Moreover, the curvatures of these trajectories are signifi-cantly higher than in highways, which is translated into alarger energy consumption that should not be neglected,e.g., (Beckers et al., 2019; Ikezawa et al., 2017).

2.2 Curved Trajectories

The main contribution of this section is the inclusion ofcornering effects in the model used for eco-driving OCPin urban scenarios, i.e., assuming curved trajectories. Theeffect of cornering in terms of energy consumption hasbeen discussed in Beckers et al. (2019), where a highlydetailed non-linear model has been developed to describeadditional tire slip losses during cornering. The resultspresented in the aforementioned work show that the energyconsumed as consequence of cornering is significant. In thissection, we follow a simplified approach to approximate theeffects of cornering into a low-complexity model that willbe used to formulate the eco-driving OCP in Section 3.

Let us consider the curved trajectory observed in Fig. 1,which is characterized by a position-dependent curvaturefunction K(s), where the curvature is defined as thereciprocal of the radius R(s), i.e.,

K(s) = 1R(s) . (4)

The radius R is measured from the instantaneous center ofrotation (ICR) to the vehicle center of gravity (CG). We




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assume that the CG shown in Fig. 1 is located at a distancelr from the rear wheel axle and a distance lf form the frontwheel axle. Moreover, the vehicle depicted in Fig. 1 has afront wheel steering angle δ(s) ∈ [−π2 , π2 ].

In order to analyse the interaction of traction and dissipa-tive forces in this case, we use a kinematic bicycle model.Despite the simplicity of a kinematic bicycle model, it canachieve similar results as a dynamical bicycle model forvehicle control purposes (Kong et al., 2015). The kine-matic bicycle model considers that the two front and rearwheels are respectively lumped into one single front andrear wheel, which in Fig. 1 are represented in red color.Even thought this is strictly true only for low lateralvehicle accelerations, applying this model to high-velocitycorners will result is an over-estimate of the corneringenergy, thereby making the model quantitatively conser-vative. Moreover, we assume vehicles with front-wheels-only steering systems, implying that the rear wheel willbe aligned with the longitudinal axis of the vehicle for theentire route. For the sake of simplicity, we assume thatthe dissipative forces Froll and Fair are aligned with thelongitudinal axis of the vehicle. Note that the velocity ofthe vehicle v(t) is tangential to the trajectory and showsan angle β(s(t)) ∈ [−π2 , π2 ] with respect to the longitudinalaxis, which is given by

β(s) = arcsin (lrK(s)) (5)

The total centripetal force applied at CG and its projec-tion into the longitudinal vehicle axis are given by

Fc(v, s) = mv2K(s), (6)

and

Fcl(v, s) = Fc(v, s) cos(π2 − β(s(t))) = mlrv2K(s)2, (7)

respectively.

An analysis of the forces acting on the longitudinal axisof the vehicle shows that for curved trajectories the totaldissipative force in that direction is given by

Fd(v, s) = Froll+grav + Fair + Fcl. (8)

Note that projection of the centripetal force into thelongitudinal axis (7) can be physically interpreted as alumped approximation in the longitudinal direction of thelateral forces generated on the tires during cornering. From(2) and (7), it is possible to rewrite (8) as

Fd(v, s) =

mg(crcos(α(s))+sin(α(s))) + (σd+mlrK(s)2)v2. (9)

Unlike to the approach of Ikezawa et al. (2017), the dissi-pating force introduced by cornering Fcl is an approxima-tion that only depends on the geometric configuration ofthe vehicle and the road, which comes as a consequence ofthe use of a kinematic model. In Section 4, we will use anumerical example to show that albeit the simplicity of theobtained approximation, it properly captures the behaviorof the vehicle during cornering for control purposes.

The traction force on the longitudinal direction Fl de-pends on the drive configuration of the vehicle, i.e., in aRWD configuration the total motor traction force Fu(t)is directly applied directly to the longitudinal directionof the vehicle, while for FWD case only a fraction ofFu(t) is applied in the longitudinal direction. This ideais depicted in Fig. 2, where the traction forces for the two

FWDRWD

uFδ

CG

lF uF=uF

δ

CGlF

uF

Fig. 2. Traction force for RWD and FWD configurations.

drive configurations are shown. It is clear to see that thetraction force for a vehicle with RWD can be described by(3). On the other had, the traction force in the longitudinaldirection for a FWD vehicle depends on the steering angleas

Fl(Fu, s) = Fu cos(δ(s)), (10)

where according to the kinematic bicycle model

δ(s) = arctan

(lf + lrlr

tan (β(s))

). (11)

After substituting (5) into (11) and applying a compositionof trigonometric and inverse trigonometric functions, thetraction force into the longitudinal direction (12) can bereformulated as

Fl(Fu, s) =Fu√

1 +(lf+lr)2K2(s)

1–l2rK2(s)

. (12)

The projection of the centripetal force and the tractionforce in the direction of motion allows us to generalizethe longitudinal vehicle dynamics (1) model for curvedtrajectories as

ma = Fl(Fu, s)− Fd(v, s), (13)

where a(t) = dvdt is the acceleration in the tangential

direction to the trajectory. Considering that the forcesare analyzed in the longitudinal axis of the vehicle, theuse of a(t) instead of the longitudinal acceleration al(t)in (13) is justified by a small angle approximation suchthat al(t) = a(t) cos(β(s)) ≈ a(t). Additionally in (13),Fd(v, s) is given by (9), and Fl(Fu, s) is described by (3)and (12) for vehicles with FWD and RWD, respectively.As a final observation, it should be noted that takingK(s) = 0 implies that the vehicle moves on a straighttrajectory. For this specific case, (9) is equivalent to(2), which represent the dissipating forces for a straighttrajectory. Similarly, for FWD vehicles (12) is equivalent to(3), which describes the traction force in the longitudinaldirection for vehicles driving in straight trajectories. Theseobservations show the generality of the proposed models.

3. OPTIMAL CONTROL PROBLEM

A continuous-time optimal control problem (OCP) formu-lation that represents the eco-driving problem for urbancity scenarios is provided in this section. In particular, wewill extend the eco-driving formulation proposed in Padillaet al. (2018) to include the cornering effects captured bythe trajectory dependent models presented in the previoussection. Additionally, we discuss the differences betweenthe OCP proposed in this paper and common approachesof eco-driving for cornering available in literature.




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3.1 Problem Formulation

For a route with given curvature K(s) and road gradeα(s), eco-driving aims to minimize the aggregative powerP (v(t), Fu(t)) over a fixed period of time [to, tf ] requiredby a vehicle driving a trajectory s(t) ∈ [so, sf ], while beingsubject to position dependent velocity and accelerationbounds v(t) ∈ [v(s), v(s)], a(t) ∈ [a(s), a(s)], respectively.Moreover, the vehicle longitudinal dynamics and initialfinal conditions for position and velocity are considered.A mathematical formulation of the eco-driving problem asan OCP is given by

mins(t),v(t),a(t),Fu(t)

∫ tf

to

P (v(t), Fu(t))dt (14a)

subject to ma(t)=Fl(Fu(t), s(t))−Fd(v(t), s(t)), (14b)ddts(t) = v(t), (14c)ddtv(t) = a(t), (14d)

s(to) = so, s(tf) = sf (14e)

v(to) = vo, v(tf) = vf (14f)

a(t)2 + v(t)4K(s(t))2 ≤ (µsg)2, (14g)

v(s(t)) ≤ v(t) ≤ v(s(t)), (14h)

a(s(t)) ≤ a(t) ≤ a(s(t)), (14i)

where the power consumed by the electric motor anddriveline at a given time instant is

P (v, u) = β2F2u + β1vFu + β0v

2 (15)

with positive coefficients β2 to penalize the Ohmic losses,β1 to describe effective power consumed, and β1 thatpenalizes the friction losses in the electric motor. Thelongitudinal vehicle dynamics are described by (14b) withthe definitions provided for (13), the time evolution ofacceleration, velocity and position are described by (14d)and (14c), and boundaries for position of velocity arerepresented by (14e) and (14f), respectively. Finally, (14g)represents a constraint imposed on the total acceleration ofthe vehicle, where µs > 0 is a friction coefficient dependenton the characteristics of the tires and the road conditions.In order to give a physical justification to (14g), let us notethat the maximum friction force between the road and tiresis given by Ffric = mµsg. For safety reasons, is required toavoid the vehicle to slip, which implies that the total forceapplied to the vehicle is lower than the maximum frictionforce during normal operation, i.e.,

(ma(t))2 + (mv(t)2(K(s(t))))2 ≤ (mµsg)2, (16)

which can be simplified into (14g). The first term in theleft hand side of (16) represents the resultant force appliedto the vehicle in the tangential direction of the trajectory,and the second term is the centripetal force. Note that(14g) is mainly relevant during cornering. During straighttrajectories this constraint is inactive because upper accel-eration bound in (14i) is expected to satisfy a(s) ≤ µsg.

3.2 Effects of Cornering

From the scarce literature about eco-driving approachesthat consider cornering effects, it is possible to note thatoften the OCP formulation includes a constraint linkedto the centripetal acceleration of the vehicle, see, e.g.,Polterauer et al. (2019), which is similar to (14g). Thistype of hard constraint implicitly imposes a limit to

the maximum velocity during cornering. In the approachpresented in this paper, we improve the representation ofcornering effects by including those effects in longitudinalvehicle dynamics, i.e., (14b). This specific modeling choicecan be seen as a soft constraint that highly penalizesthe velocity while cornering. To this end, we will use asimplified example that, without losing generality, allowus to observe the effects cornering in the OCP (14).

Let us consider a RWD vehicle, driving in a circulartrajectory on a flat road, i.e., K(s) = 1

R and α(s) = 0.It is possible to find an equivalent formulation fo the OCP(14) by substituting (14b) into (14a) (see , e.g., (Boyd andVandenberghe, 2004, §4.1.3)), leading to

mins(t),v(t),a(t)

∫ tf

to

P (v(t),ma(t) + Fd(v(t), s(t)))dt (17)

subject to (14c)-(14i),

in which

P (v,ma+Fd(v, s)) = β2(m(a+gcr)+(σd+mlrK

2)v2)2

+β1v(m(a+gcr)+(σd+mlrK

2)v2)+β0v

2. (18)

Thus, we can note the the term mlrK2 drastically penal-

izes velocity in the new cost function. This penalization,depends quadratically on the road curvature, which im-plies that the optimal solution might show lower deceler-ations during cornering maneuvers.

4. CASE STUDY

In this section, we study an electric vehicle executing acornering maneuver in an urban environment with differ-ent curvatures. The advantages of the model and OPCformulation presented in this paper are highlighted inthis example. To this end, we will contrast the resultswith traditional eco-driving approaches. Moreover, we willvalidate the proposed approach using a high-fidelity modelto calculate the instantaneous power and total energyconsumption produced during the cornering maneuversanalyzed.

Let us consider an electric vehicle with RWD and param-eters listed in Table 1. The vehicle executes a corneringmaneuver on a city intersection as is depicted in Fig. 3,which curvature is defined as

K(s) =

1

Rfor sco ≤ s ≤ scf ,

0 otherwise;(19)

Table 1. Vehicle and OCP parameters

Variable Value Units

m 15000 [kg]σd 3.24625 [Ns2/m2]cr 0.007 −µs 0.35 −β0 0.292 [ws2/m2]β1 1.005 −β2 2.652e-4 [w/N2]s0 0 [m]sf 150 [m]v0 30 [m/s]vf 35 [m/s]v 60 [m/s]v 0 [m/s]a 0.2g [m/s2]a -0.2g [m/s2]




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ocsos

fcs

ov

fs fv

R

Fig. 3. Urban intersection considered in the case study.

where sco is the position where the corner begins and scfis the final corner position. The curvature (19), will bespecified for three different scenarios that are detailed inTable 2.

The OCP (14) is formulated using the parameters in Table1. To find the solutions of this OCP, we first dicretize theproblem (14) and later apply a static optimization tech-nique, i.e., sequential quadratic programing (Boggs andTolle (1995)). The optimal solutions to (14) are depicted inFig. 4 as velocity and acceleration profiles for each scenarioconsidered in the case study. Note that the vertical linesdepicted in both profiles indicates the initial and finalpositions of the curved section of the trajectory. It can beobserved that the vehicle decelerates before entering thecurved section of the trajectory. When the vehicle entersthe curved section of the trajectory the deceleration rateis reduced and approximately at half of the curved sectionthe vehicle begins accelerating. As soon as the vehicleleaves the curved section of the trajectory the accelera-tion is immediately increased. Interestingly, the differencesbetween scenarios considered is clearly observed in the ve-locity profiles, where at higher curvatures lower velocitiesare observed in the curved section of the trajectory. Thisresults is expected because velocity is highly penalized forhigh curvatures in the road. This effect has been discussedin Section 3.2.

In order to highlight the advantages of our general ap-proach, we compare the optimal solution to the OCP (14)with a traditional eco-driving OCP formulation. Specifi-cally, a traditional OCP considers the constraint (14g), butthe cornering effects in (14b) are neglected. In Fig. 5, theoptimal velocity and acceleration profiles for traditionaland general eco-driving formulations are presented for thespecific scenario where R = 14 [m]. Note that the velocityprofile of the traditional approach shows constant velocityduring the curved section of the road, which also indicateszero acceleration during that section. This is expectedsince traditional eco-driving strategies avoids changes in

Table 2. Parameters for different scenarios.

R sco scf12 [m] 70 [m] 70+Rπ/2 [m]14 [m] 70 [m] 70+Rπ/2 [m]17 [m] 70 [m] 70+Rπ/2 [m]

acceleration to reduce energy consumption. As a conse-quence, the vehicle crosses the curved section with themaximum possible velocity, which is defined by constrain(14g). On the other hand, the optimal strategy obtained

20 40 60 80 100 120 140s(t) [m]

-0.5

0

0.5

1

a(t)

[m

/s]

R = 12[m]R = 14[m]R = 17[m]

(a) Acceleration profiles.

20 40 60 80 100 120 140s(t) [m]

20

25

30

35

v(t)

[km

/h]

R = 12[m]R = 14[m]R = 17[m]

(b) Velocity profiles.

Fig. 4. Optimal solutions to the OCP (14) for differentcurvatures.

20 40 60 80 100 120 140s(t) [m]

-1

-0.5

0

0.5

1

a(t)

[m

/s]

Gen. Eco-driv.Trad. Eco-driv.corner location

(a) Acceleration profiles.

20 40 60 80 100 120 140s(t) [m]

24

26

28

30

32

34

36

v(t)

[km

/h]

Gen. Eco-driv.Trad. Eco-driv.corner location

(b) Velocity profiles.

Fig. 5. Comparison for OCP formulations for corneringscenario with R = 14 [m].




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0 50 100 150s(t) [m]

-100

0

100

200

Cum

ulat

ive

pow

er [

kW]

20

25

30

35

40

v(t)

[km

/h]

inertiaaero.rol. res.e-mach. losscorn. res.velocity

(a) Tradditional eco-driving aprpoach.

0 50 100 150s(t) [m]

-100

0

100

200

Cum

ulat

ive

pow

er [

kW]

20

25

30

35

40

v(t)

[km

/h]

inertiaaero.rol. res.e-mach. losscorn. res.velocity

(b) Eco-driving for for general trajec-tories.

Fig. 6. Velocity and energy loses for an cornering scenariowith R = 14[m].

from the general approach proposed in this paper showslower velocities during the curved section, which is con-nected to the velocity being penalized by the curvature ofthe road. This causes non-zero longitudinal accelerationin the curved section. The effects of both strategies interms of power consumption can be observed in in Fig.6, where the instantaneous cumulative power consumptionproduced by these two strategies is depicted. Here, we usethe high-fidelity model from Beckers et al. (2019) to cal-culate the cornering losses, in combination with instanta-neous power consumed by the inertia, aerodynamical drag,rolling resistance, electric machine losses. As expected, themain difference between both strategies are observed inthe curved section of the trajectory. For the traditionalstrategy, the power loses due to cornering and electric ma-chine losses are constant during the curved section, whilethe general strategy indicates virtually no electric machinelosses during the first half of the curved trajectory. Thisshows that the cornering maneuver obtained form the gen-eral strategy is more energy efficient that the traditionalcounterpart. This observation is reinforced by simulationsperformed on the rest of the scenarios considered in thiscase study.

In Fig. 7, we present the cumulative energy obtained byusing the traditional and general strategies for scenarioswith three different curvatures, and in Table 3, we sum-marize the total energy for each case and strategy. Thetendency observed from these results indicates that thegeneral strategy proposed in this paper is able to improvethe energy efficiency of the vehicle for all the consideredscenarios. In fact, the proposed eco-driving strategy savesa larger amount of energy for scenarios with larger cur-

R = 12[m] R = 14[m] R = 17[m]0

200

400

600

800

1000

Tota

l E

ner

gy [

kJ]

inertiaaero.rol. res.e-mach. loss.corn. res.

Tra

dit

ional

Gen

eral

Tra

dit

ional

Gen

eral

Tra

dit

ional

Gen

eral

Fig. 7. Cumulative energy consumption for different sce-narios and eco-driving strategies.

Table 3. Comparison of energy consumptionbetween strategies.

Scenario Tot. Energy Tot. Energy Diff.(Traditional) (General) %

R = 12 [m] 820.2 [kJ ] 757.6 [kJ ] 8.26R = 14 [m] 687.5 [kJ ] 658.2 [kJ ] 4.45R = 17 [m] 574.2 [kJ ] 573.4 [kJ ] 0.14

vatures, while for roads with small curvature the totalconsumption of the general strategy tends to approximateto the traditional approach. This is a consistent result sinceour OCP formulation resembles traditional eco-drivingformulations when roads with no curvature are considered,as has been discussed in Section 2.2. Remarkably, in Fig.7, it is possible to see that a higher energy efficiency of theproposed strategy mainly comes from a reduction of theenergy losses in the electric machine.

5. CONCLUSIONS

A model that approximates cornering forces into the longi-tudinal axis of the vehicle has been proposed in this paper.The simplicity of this model relies on the geometry of thevehicle and the road, which makes it easy to use for a broadspectrum of cases without the necessity of identificationof specific tire parameters. Based on this model, we haveproposed a general eco-driving optimal control problemformulation that can be used for both straight and curvedtrajectories. Moreover, it can be used for front wheel driveand rear wheel drive configurations of the vehicle. Theadvantages of the proposed OCP have been analysed ona case study, where the use of a high-fidelity model haveallowed not only to validate the proposed model and OCPformulation, but also to show the improvements of thisapproach with respect to traditional approaches found incurrent literature. The results have shown that the useof the general OCP formulation proposed in this paperyields to an improvement energy savings for corneringmaneuvers in trajectories with large curvatures, which areapproximately up to 8% larger than traditional eco-drivingstrategies.

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