research article a predictive distribution model for...

Research ArticleA Predictive Distribution Model for Cooperative BrakingSystem of an Electric Vehicle

Hongqiang Guo,1 Hongwen He,1 and Xuelian Xiao2

1 National Engineering Laboratory for Electric Vehicles, Beijing Institute of Technology, Beijing 100081, China2 China North Vehicle Research Institute, Beijing 100072, China

Correspondence should be addressed to Hongwen He; [email protected]

Received 10 October 2013; Accepted 19 December 2013; Published 9 February 2014

Academic Editor: Wudhichai Assawinchaichote

Copyright © 2014 Hongqiang Guo et al.This is an open access article distributed under theCreative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A predictive distribution model for a series cooperative braking system of an electric vehicle is proposed, which can solve thereal-time problem of the optimum braking force distribution. To get the predictive distribution model, firstly three disciplinesof the maximum regenerative energy recovery capability, the maximum generating efficiency and the optimum braking stabilityare considered, then an off-line process optimization stream is designed, particularly the optimal Latin hypercube design (OptLHD) method and radial basis function neural network (RBFNN) are utilized. In order to decouple the variables between differentdisciplines, a concurrent subspace design (CSD) algorithm is suggested. The established predictive distribution model is verifiedin a dynamic simulation. The off-line optimization results show that the proposed process optimization stream can improve theregenerative energy recovery efficiency, and optimize the braking stability simultaneously. Further simulation tests demonstrate thatthe predictive distribution model can achieve high prediction accuracy and is very beneficial for the cooperative braking system.

1. Introduction

As a core technology in electric vehicles, cooperative brakingwith regenerative braking and mechanical braking can notonly improve the fuel efficiency but also maintain a satisfyingbraking stability, as studied in the literature [1–3]. For acooperative braking system, two braking scenarios are pre-sently studied. The first one is normal deceleration process,withinwhich the braking force distribution strategy is usuallystudied.The second one is emergency braking process, withinwhich the antilock braking system (ABS) is required to worktogether with the regenerative braking simultaneously [4].

Generally, when the vehicle is decelerating, the normaldeceleration process happens in most cases, which indicatethat the braking force distribution strategy is vital for thecooperative braking system. With respect to this, how tomaximize the regenerative energy recovery efficiency underthe premise of braking stability is the focus of traditionalstudy. Many researchers have made many efforts to developdifferent distribution strategies. Based on the braking theory,

the references [5–11] presented a distribution strategy, inwhich the braking stability was taken as a precondition, andthe maximum regenerative energy recovery efficiency wasregarded as the target. If the generatorsmet the required brak-ing force, the braking force was afforded by generators, oth-erwise, hydraulic brakes compensated the rest braking force.The reference [11] considered the low generating efficiency ofgenerators, and a similar distribution strategy was proposedaccording to the ECE braking regulation. The references[12, 13] proposed an optimization method in which brakingstability was regarded as the constraint and the maximumregenerative energy recovery efficiency was designed as thetarget. Generally, the strategies described above focusedmainly on the objective of the regenerative energy recoveryefficiency. However, the braking stability is also very impor-tant. Braking strategies which can maximize the regenerativeenergy recovery efficiency and optimize the braking stabilityobjectives simultaneously are a key technology in cooperativebraking systems. Additionally, the rapidity of a real-time con-trol may affect the performance of the cooperative braking

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 828269, 11 pageshttp://dx.doi.org/10.1155/2014/828269

2 Mathematical Problems in Engineering

Power coupling

equipment

Motor/generator 1

Motor/generator 2 Rear axle

Front axle

Motor/generator integration controller

Cooperative braking controller

Vehicle Breaking severity (z)

Battery SoC

Hydraulic brakes integration controller

Battery

speed (�)

Figure 1: The cooperative braking system of the electric vehicle.

systems, so good performance of the real-time control is alsoimportant.

The cooperative braking systems can be classified intoparallel and series types [14]. With the parallel system,motor/generators add an additional braking force on thehydraulic braking force, and there will be no coordinatebetween motor/generators and mechanical hydraulic brakes.Considering the braking stability requirement, the regener-ative energy recovery efficiency is usually constrained, butthe traditional braking system does not need to be reformed[15]. With the series system, a real-time coordinate betweenmotor/generators and hydraulic brakes is considered. Inaddition, the total output braking force will follow therequired braking force. Since series systems can achieve betterregenerative energy recovery efficiency and better brakingstability simultaneously, it has been in the spotlight [16–19].

In this paper, a series cooperative braking system withtwo motor/generators is studied for the normal decelerationprocess. Different from the traditional methods, a brakingforce distribution strategy based on a predictive distributionmodel is proposed. The predictive distribution model is con-stituted by a predictive model and an additional condition.To get the predictivemodel, some key technologies have beenstudied. Firstly, a general mathematical model is establishedby three disciplines and a braking force analysis. Secondly,an off-line process optimization stream which is constitutedby the optimal Latin hypercube design (Opt LHD) and aconcurrent subspace design (CSD) method is designed toget an off-line optimization data. Thirdly, a predictive model

based on a radial basis function neural network (RBFNN) bythe off-line optimization data and the general mathematicalmodel is presented. Finally, the predictive distribution modelis verified in a dynamic simulation framework.

2. The Cooperative Braking Structure

Figure 1 shows the cooperative braking system of an electricvehicle. The cooperative braking system is constructed byfour hydraulic brakes and two motor/generators. The twomotor/generators are connected to a power coupler. Four-wheel drive, front-wheel drive, and rear-wheel drive canbe realized according to different working modes of thepower coupler. This paper only focuses on the front-wheeldrive mode. For motor/generator 1, the coupling ratio is1.442; for motor/generator 2, the coupling ratio is 2.225.The vehicle speed is V, the breaking severity is 𝑧, and thebattery SoC are considered as inputs, and the cooperativebraking predictive distribution model is implemented in thecooperative braking controller. When the vehicle decelerates,the motor/generators work as generators, the cooperativebraking controller can coordinate the braking force distribu-tion between hydraulic brakes and generators. In addition,based on the charging performance of the battery, SoC isconstrained by 0.1–0.8. Considering the constraint of the cou-pler ration, V is constrained by 10–50 km/h. Since a normaldeceleration process is studied, 𝑧 is constrained by 0.01–0.4.The scopes of SoC, V, and 𝑧 constitute the continuous designspace in this paper.

Mathematical Problems in Engineering 3

�𝜔f 𝜔r

Nxf

Nzf

Fxf

Fzf

Tm

rwThf

mfg

Nxr

Nzf

Fxr

Fzr

rwThr

mrg

Figure 2: The force diagram of the cooperative braking system.

3. The Cooperative BrakingMathematical Model

3.1. General Mathematical Model. A force diagram of thecooperative braking system is shown in Figure 2.

In Figure 2, 𝑇𝑚

denotes the total regenerative brakingtorque, which is enforced on the front wheels. 𝑇

ℎ𝑓, 𝑇ℎ𝑟

arehydraulic braking torques which are applied the on front andrear wheels, respectively; 𝐹

𝑥𝑓, 𝐹𝑥𝑟

denote the road brakingforces the on front and rear wheels, respectively; 𝐹

𝑧𝑓, 𝐹𝑧𝑟

denote the road normal reaction forces the on front and rearwheels, respectively; 𝑟

𝑤denotes the radius of the wheels; 𝜔

𝑓,

𝜔𝑟denote the angular velocities of the front and rear wheels,

respectively;𝑁𝑥𝑓,𝑁𝑥𝑟denote the thrusts of the front and rear

wheel axles, respectively; 𝑚𝑓𝑔, 𝑚𝑟𝑔 denote the gravities of

the front and rear wheel axles, respectively.A general cooperative braking mathematical model is

established to induce 𝑇𝑚, 𝑇ℎ𝑓, and 𝑇

ℎ𝑟. The sum of the

hydraulic braking torques and the total regenerative brakingtorque should be equal to the required braking torque.Consider the following:

𝑇ℎ𝑓

+ 𝑇𝑚+ 𝑇ℎ𝑟

= 𝑇𝑟𝑚, (1)

where 𝑇𝑟𝑚

is the required braking torque, which is given by(2).

𝑇𝑟𝑚

= 𝑚𝑔𝑟𝑤𝑧, (2)

where𝑚 is the total mass of the vehicle.When the vehicle decelerates, the generators will work

simultaneously.

𝑇𝑚1

+ 𝑇𝑚2

= 𝑇𝑚, (3)

where𝑇𝑚1,𝑇𝑚2

denote the regenerative braking torques of thegenerator 1 and the generator 2, respectively.

According to (1)–(3), 𝑇𝑟𝑚

can be viewed as a knownvariable, however, 𝑇

𝑚, 𝑇ℎ𝑓, 𝑇ℎ𝑟, 𝑇𝑚1, and 𝑇

𝑚2are unknown

variables, and only two equations which are (1) and (3)are presented, so another three balance equation should berequired.

Define𝑇𝑚1

𝑇𝑚

= 𝜆,

𝑇ℎ𝑓

+ 𝑇𝑚

𝑇𝑟𝑚

= 𝛽,

𝑇ℎ𝑓

𝑇ℎ𝑓

+ 𝑇ℎ𝑟

= 𝛼,

(4)

where 𝜆 is the regenerative braking torque distributioncoefficient.

Usually, 𝜆 is regarded as a known variable; however, in theoptimization mathematic model, it is an unknown variable,and will be varied from 0 to 1. If 𝜆 = 0, it means thatthe whole regenerative braking torque is only provided bygenerator 2; similarly, if 𝜆 = 1, it means that the wholeregenerative braking torque is only provided by generator 1;additionally, if 𝜆 ∈ (0, 1), it means that the whole regenerativebraking torque is provided by generator 1 and generator2 simultaneously. It is significant to define this parameter,for generators; the generating efficiency will be differentunder different total regenerative braking torques and motorspeeds, so it is necessary to find an optimum distributioncoefficient tomaximize the generating efficiency between twogenerators. 𝛽 is the braking torque distribution coefficientbetween front and rear wheels. The same as 𝜆, herein, it isa known variable, but in optimizationmathematic model it isan unknown variable andwill be varied by the requirement ofthe braking regulation and braking stability. 𝛼 is the brakingtorque distribution coefficient between hydraulic brakes. Itcan coordinate the hydraulic brakes to meet the objectivesof optimization, and varies from 0 to 1 in optimizationmathematical model. If 𝛼 = 0, it means that only the frontaxle hydraulic brakes work; if 𝛼 = 1 it means that only rearaxle hydraulic brakes work; if 𝛼 ∈ (0, 1), it means that all thehydraulic brakes will work simultaneously.

According to the above equations, 𝑇𝑚, 𝑇ℎ𝑓, 𝑇ℎ𝑟, 𝑇𝑚1

and𝑇𝑚2

can be formulated as follows:

𝑇𝑚=

𝛽 − 𝛼

1 − 𝛼

𝑇𝑟𝑚,

𝑇ℎ𝑓

=

𝛼 (1 − 𝛽)

1 − 𝛼

𝑇𝑟𝑚,

𝑇ℎ𝑟

= (1 − 𝛽) 𝑇𝑟𝑚,

𝑇𝑚1

= 𝜆𝑇𝑚,

𝑇𝑚2

= (1 − 𝜆) 𝑇𝑚.

(5)

3.2. Optimization Mathematical Model. Three disciplines areproposed in this study. The first one is the maximum regen-erative energy recovery capability discipline, which the totalregenerative braking torque should be close to the optimumcharging torque. The second one is the maximum generatingefficiency discipline, in which the charging torque efficiencywill be maximized under the given total regenerative brakingtorque of the first discipline. The third one is the optimum


braking stability discipline, in which can be classified intofour levels as follows.

(1) No wheels are locked, each wheel maintains rollingand sliding state simultaneously, with this state, ABSis usually used. In this study, no ABS is equipped.

(2) No wheels are locked earlier, if locked, all the wheelswill be locked simultaneously. In this case, vehicle willkeep the optimum braking stability. It is also the opti-mum braking stability objective in this study, whichthe braking force distribution coefficient should beclose to it.

(3) Front wheels will be locked earlier than rear wheels.In this case, the steering capability will be lost butstill is a stability state. In this study, it is viewed as aprecondition for braking stability.

(4) Rear wheels will be locked earlier than front wheels.In this case, rear wheels will spin and will be in aninstability state. In this study, it is considered as aninfeasible region.

A concurrent subspace design (CSD) is performed hereand its optimization mathematical model can be set up asfollows.

(a) The Maximum Regenerative Energy Recovery CapabilityDiscipline. Consider the following:

Find: 𝛼, 𝛽

Minimize: 𝑓1(𝛼, 𝛽) = (

𝑇𝑚− 𝑇opt

𝑇opt)

2

Satisfy: 𝑇𝑚+ 𝑇ℎ𝑓

< 𝑇𝑓,

𝑇ℎ𝑟

< 𝑇𝑟,

𝑐 = 𝛽 − 𝛼 > 0,

𝑏1< 𝛽 < 𝑏

2,

0 < 𝛼 < 1,

𝑇𝑚< 𝑇opt

Output: 𝑇𝑚,

(6)

where 𝑐 is an auxiliary constraint, which can ensure 𝑇𝑚

> 0;𝑇𝑓, 𝑇𝑟are the maximum braking torques which are subjected

to a dry pavement; and 𝑇opt denotes the optimum chargingtorque, which can be expressed as

𝑇opt = min [max (𝑇mot brake) ,max (𝑇battery charge)] , (7)

where max (𝑇mot brake) is the maximum regenerative brak-ing torque; max (𝑇battery charge) is the maximum rechargeable

torque; and 𝑏1, 𝑏2denote the lower and upper boundary con-

ditions of the braking stability, respectively, which are definedaccording to the braking regulation of the ZBT24007-1989 in(8) and the braking stability precondition in (9) as follows:

𝛽 ≥

𝑏 + 𝑧ℎ𝑔

𝐿

, 0 ≤ 𝑧 ≤ 0.3,

(𝑧 − 0.08) (𝑏 + 𝑧ℎ𝑔)

𝑧𝐿

< 𝛽 ≤

(𝑧 + 0.08) (𝑏 + 𝑧ℎ𝑔)

𝑧𝐿

,

0.15 ≤ 𝑧 ≤ 0.3,

𝛽 ≥ 1 −

(𝑧 + 0.25) 𝐿

0.74 (𝑎 − 𝑧ℎ𝑔) 𝑧

, 0.3 ≤ 𝑧 ≤ 0.8,

1 −

(𝑎 − 𝑧ℎ𝑔) ((𝑧 − 0.1) /0.85 + 0.2)

𝐿𝑧

≤ 𝛽 ≤

(𝑏 + 𝑧ℎ𝑔) ((𝑧 − 0.1) /0.85 + 0.2)

𝐿𝑧

, 0.2 ≤ 𝑧 ≤ 0.8,

(8)

𝛽 ≥

𝑏 + 𝑧ℎ𝑔

𝐿

, 0 ≤ 𝑧 ≤ 0.8, (9)

where 𝑎, 𝑏, 𝐿, and ℎ𝑔denote the front wheelbase, rear

wheelbase, wheelbase, and the height of vehicle body masscenter respectively.

(b) The Maximum Generating Efficiency Discipline. Considerthe following:

Given: 𝑇𝑚, 𝑛

Find: 𝜆

Minimize: 𝑓2(𝜆) =

1

𝑓generator1 efficiency𝑓generator2 efficiency,

Satisfy: 0 ≤ 𝜆 ≤ 1,

𝑇𝑚1

≤ 𝑇out1,

𝑇𝑚2

≤ 𝑇out2

Output: 𝑓2,

(10)

where 𝑛 is the generator’s speed; 𝑓generator1 efficiency and𝑓generator2 efficiency denote the generator’s efficiencies, which canbe obtained by an efficiency map in Figure 3; 𝑇out1, 𝑇out2 arethe maximum output regenerative torques which can be gotby Figure 4.

(c)The Optimum Braking Stability Discipline. Generally, if theadhesions between front and rear wheels are equal under anybraking severity, the vehicle can get the best braking stability[7]; in this state, the braking force distribution curve is theideal curve (𝐼 curve).Thus the optimum braking distributioncoefficient can be realized as follows:

𝛽opt =𝑏 + 𝑧ℎ

𝑔

𝐿

. (11)


0 100200 300

0200040006000

8000

80

90

100

Effici

ency

(%)

78808284868890929496

T (Nm)n (rpm)

Figure 3: The efficiency map of the generators.

0 1000 2000 3000 4000 5000 6000 7000 8000 900050

100

150

200

250

300

T(N

m)

n (rpm)

Figure 4: The regenerative torque versus speed curve.

Additionally, if𝛽 > (𝑏+𝑧ℎ𝑔)/𝐿, itmeans that the adhesion

of the front wheels is bigger than the rear wheels, and it is alsoa stability state.

The optimization mathematical model can be expressedas follows:

Find: 𝛽

Minimize: 𝑓3(𝛽) = (

𝛽 − 𝛽opt

𝛽opt)

2

Satisfy: 𝑏1< 𝛽 < 𝑏

2.

(12)

3.3. Coupling Relationship between Different Disciplines. Ascan be seen in Figure 5, 𝛼, 𝛽, and 𝜆 are system variables; 𝛽is the system coupling variable, which can affect discipline 1and discipline 3 simultaneously; 𝑇

𝑚is the discipline coupling

variable, which is both the output variable of discipline 1 andthe input variable of discipline 2; and 𝑓

1, 𝑓2, and 𝑓

3are the

output variables which are passed to the system.

3.4. Off-Line Process Optimization Stream. As shown inFigure 6, design of experiment (DOE) 1 and another threemodules are defined in the stream. DOE 1 is used to discretethe continuous input design space with the optimal Latinhypercube design (Opt LHD)method. For the threemodules,we have the following.

(1)Module 1 is the original approximationmodelmodule.Since different sampling data correspond to different inputs,

Discipline 1 the maximum energy

recovery capability

Discipline 2 the maximum generating

efficiency

System

Discipline 3 the optimum braking

stability

f1

f2

f3

Tm

𝛽

𝛼, 𝛽

𝜆

Figure 5: The coupling relationship between different disciplines.

in order to ensure the generated sampling data correspondingto the given input parameter, deleting the existing sam-pling data firstly is necessary in this module (delete theexisting sampling point data component). 𝑇opt, 𝛽opt, 𝑇𝑟𝑚,𝑇out1, 𝑇out2, 𝑏1, 𝑏2, 𝑇

𝑓, and 𝑇

𝑟can be calculated by the

calculation program component. DOE 2 component is usedto discrete the design variables with the Opt LHD method.An approximation model with respect to

𝑇𝑚,𝑇𝑚1, 𝑇𝑚2, 𝑇ℎ𝑓,

𝑇ℎ𝑟, 𝑓1, 𝑓2, and

𝑓3can be built by the approximation model

component, which 𝑇𝑚, 𝑇𝑚1, 𝑇𝑚2, 𝑇ℎ𝑓, 𝑇ℎ𝑟, 𝑓1, 𝑓2, and

𝑓3are

the approximate values of 𝑇𝑚, 𝑇𝑚1, 𝑇𝑚2, 𝑇ℎ𝑓, 𝑇ℎ𝑟, 𝑓1, 𝑓2and

𝑓3.(2) Module 2 is the CSD algorithm module, which

is constituted by a system optimization layer and threediscipline layers. For the discipline layers, no optimizationwill be carried out according to the CSD algorithm; the onlyrole for them is to add three new sets of sampling pointsto update the sampling point component in every cycle andaccelerate the convergence of the approximationmodel. Aim-ing at a specific discipline layer, the approximation values ofother disciplines will be provided with the approximationmodel and the real values of the specific discipline will beobtained with the real mathematic model. According to theupdate of the sampling points component, a higher-precisionapproximation model will be obtained by the update approx-imation model component, based on which the system opti-mization layer can carry out the optimization task; addition-ally, the optimization mathematic model is shown in (13).With the iterative calculation of the optimization, the approx-imation model precision will be higher and higher and theoptimization results will be converged sooner.

(3) Module 3 is the predictive model module. The mostimportant task for it is to set up a high-precision predictivemodel. The input parameters of SoC, V, and 𝑧 are viewedas inputs, and the optimization values of 𝛼, 𝛽, and 𝜆 whichare denoted by 𝛼optimization, 𝛽optimization, and 𝜆optimization areviewed as outputs. A radial basis function neural network(RBFNN) will be generated based on the inputs and outputs.Additionally, a feedback cycle-control strategy is designed.If the precision of the RBFNN meets the requirement, then,end; otherwise, additional sampling points will be added tothe DOE 1 component.


DOE 1Delete the existing sampling point

data

DOE 2 Approximation model

System optimization layer

Discipline 1 layer The maximum energy

recovery capability

Approximation model

Update the sampling

points

Update the approximation

model

System objective

and constraints

Discipline 2 layer The maximum energy

recovery efficiency

Approximation model

Discipline 3 layer The optimum

braking stability

Approximation model

Predictive model

Calculation program

Begin End

Prediction accuracy

YesNo

Add new sampling points

Module 1

Module 2

Module 3

(SoC, �, z)

(SoC, �, z)

(𝛼, 𝛽, 𝜆)

(𝛼, 𝛽, 𝜆) Tm, ��m1, Tm2, Thf, Thr, f1, f2, f3(𝛼optimization , 𝛽optimization , 𝜆optimization)

f1, f3, Thf, Thr, Tm

f2, f3, Tm1, Tm2

f3

f1, f2, Tm1, Tm2, Thf, Thr, Tm

f1, Thf, Thr, Tm

Trm, Tout1, Tout2, b1, b2, Tf, Tr

f2, Tm1, Tm2

Topt , 𝛽opt ,

Figure 6: The off-line process optimization stream.

Then, the optimization mathematic model is

𝑓1= (

𝑇𝑚− 𝑇opt

𝑇opt)

2

,

𝑓2=

1

𝑓generator1 efficiency


,

𝑓3= (

𝛽 − 𝛽opt

𝛽opt)

2

,

Find 𝛼, 𝛽, 𝜆

Minimize 𝐹 (𝛼, 𝛽, 𝜆)

= 𝑤1(

𝑇𝑚− 𝑇opt

𝑇opt)

2

+

𝑤2



+ 𝑤3(

𝛽 − 𝛽opt

𝛽opt)

2

Satisfy 𝑇𝑚+𝑇ℎ𝑓

< 𝑇𝑓,

𝑇ℎ𝑟

< 𝑇𝑟,

𝑐 = 𝛽 − 𝛼 > 0,

𝑏1< 𝛽 < 𝑏

2,

0 < 𝛼 < 1,

𝑇𝑚< 𝑇opt,

0 ≤ 𝜆 ≤ 1,


b2 − b1

b 2−b 1 0.5

0.4

0.3

0.2

0.1

00 500 1000 1500 2000 2500

Sampling point

𝛽op

timiz

atio

n−b 1

𝛽optimization − b1

and

Figure 7: The braking force distribution coefficient.

𝑇𝑚1

≤ 𝑇out1,

𝑇𝑚2

≤ 𝑇out2,

(13)

where 𝑤1, 𝑤2, and 𝑤

3are weighting coefficients, in which 𝑤

1

and 𝑤3are 1 and 𝑤

2is 0.01.

4. Results and Discussions

4.1. Off-Line Optimization Results. As shown in Figure 7,𝛽optimization should be bigger than or equal to 𝛽opt, accordingto the braking stability requirement. Herein, 𝛽optimization −

𝑏1is close to 0 but bigger than 0, and lower than 𝑏

2− 𝑏1,

whichmeans that the vehicle can always keep a better brakingstability.

Define

𝜀 =

𝑇𝑚 optimization − 𝑇opt

𝑇opt100%,

𝜉 =

𝑇𝑚 optimization𝑓2 − 𝑇opt

𝑇opt100%,

(14)

where 𝜀 denotes the deviate error between the regenerativebraking torque and the ideal charging torque; 𝜁 denotes thedeviate error between the charging torque between actual andideal; and 𝑇

𝑚 optimization denotes the optimization value of 𝑇𝑚.

It can be seen from Figure 8 that the value of 𝜀 is verysmall and below zero, whichmeans that the total regenerativebraking torque always follows 𝑇opt. Two generators cancoordinate themselves to maximize the generating efficiency.For that, the average value of 𝜁 is only 20%, and the averagegenerating efficiency of the generators is nearly 80%.

As shown in Figure 9, 𝛼optimization is close to 0 at somesampling points, whichmeans that only the hydraulic brakingtorque of rear wheels is needed to realized the optimumbraking force distribution. Apart from those sampling points,the value of 𝛼optimization is between 0 and 1, which means thatall the hydraulic brakes will be worked. Similarly, with respectto 𝜆optimization, at some sampling points, 𝜆optimization is closeto 1, which means that with only generator 1 offering the

𝜀an

d𝜁(%

)

𝜀

𝜁

Effici

ency

(%)

0

−20

−400 500 1000 1500 2000 2500

Sampling point

90

80

70

Figure 8: 𝜀, 𝜁, and generating efficiency.

𝛼op

timiz

atio

n

𝜆op

timiz

atio

n

Sampling point

1

0.5

0

1

0.5

00 500 1000 1500 2000 2500

Figure 9: 𝛼optimization and 𝜆optimization.

total regenerative braking torque, the generating efficiencycan be maximized; on the contrary, when 𝜆optimization is closeto 0, it means that with only generator 2 offerring the totalregenerative braking torque, the generating efficiency can bemaximized. Additionally, when 𝜆optimization is between 0 and1, it means that the generating efficiency can be maximizedwhen the two generators work simultaneously.

The above optimization results show that the optimiza-tion model with the CSD method can get good results, andlay a good foundation for a further high-presicison predictivemodel.

4.2. Predictive Distribution Model. Considering the poorreal-time control of the optimization, a predictive distri-bution model together with a predictive model and anadditional condition is presented. For the predictive model,the multiple correlation coefficients (𝑅2) of ��,

𝛽, and 𝜆 are

0.95, 0.999, and 0.96045, respectively. Generally, although ��,𝛽, and

𝜆 have a relatively high predictive precision, predictiveerrors still exist; a relatively small predictive error may causethe predictive parameters to fall into the infeasible region.

To ensure that the predictive distribution model meetsthe cooperative braking requirement, some key predictiveparameters’ scope should be verified as follows.

(1) As stated above, 𝑇𝑚

> 0 is the precondition torealize the regenerative braking energy recovery. So,�� should be lower than

𝛽 based on (5). Figure 10shows the predictive results of �� −

𝛽 which is denoted

by 𝛼predictive value − 𝛽predictive value. It can be seen that

8 Mathematical Problems in Engineering𝛼

pred

ictiv

eval

ue−𝛽

pred

ictiv

eval

ue

Sampling point

0.2

0

−0.2

−0.4

−0.6

0 500 1000 1500 2000 2500

0.00576820.0177131

Infeasiblepoints

Figure 10: The predictive results of 𝛼predictive value − 𝛽predictive value.

𝛽pr

edic

tive

valu

e−𝛽

opt

Sampling point

0.03

0.025

0.02

0.015

0.01

0.005

00 500 1000 1500 2000 2500

Figure 11: The predictive results of 𝛽predictive value − 𝛽opt.

Sampling point

0.6

0.4

0.2

0

0 500 1000 1500 2000 2500

−0.027 −0.0103 −0.0113 −0.0371 −0.01146

𝛼pr

edic

tivev

alue

Figure 12: The predictive results of 𝛼predictive.

an overwhelming majority of the sampling pointscan meet the requirement of the cooperative brakingsystem except two sampling points.

(2) With respect to 𝛽, it should be bigger than or equal to

𝛽opt according to the requirement of braking stability.Figure 11 shows the predictive results of 𝛽−𝛽opt whichis denoted by 𝛽predictive value − 𝛽opt. Obviously, due tothe high predictive precision of 𝛽, it can always meetsthe requirement of braking stability.

(3) With regard to �� and 𝜆. �� should not be lower than

0; 𝜆 should not be lower than 0 and bigger than 1according to the requirement in (4). Figures 12 and13 show the predictive values of �� and

𝜆, which aredenoted by𝛼predictive value and𝜆predictive value. Both of the

𝜆pr

edic

tive v

alue

Sampling point

1

0.8

0.6

0.4

0.2

0

0 500 1000 1500 2000 2500−0.034

1.0095 1.0472 1.0173 1.02113 1.0272

Figure 13: The predictive results of 𝜆predictive value.

predictive parameters have some infeasible samplingpoints.

Given the above analysis, an additional condition shouldbe added to the predictive distribution model. For the addi-tional condition, the basic principle is that, if the predictiveparameters fall into the infeasible region, the braking torquewill be only provided by hydraulic brakes. The predictivedistribution model is shown as follows:

𝑇𝑚=

𝛽 − ��

1 − ��

𝑇𝑟𝑚,

𝑇ℎ𝑓

=

�� (1 −𝛽)

1 − ��

𝑇𝑟𝑚,

𝑇ℎ𝑟

= (1 −𝛽)𝑇𝑟𝑚,

𝑇𝑚1

=𝜆𝑇𝑚,

𝑇𝑚2

= (1 −𝜆)𝑇𝑟𝑚,

Define: 𝜂 = 2.5

If �� < 0 ‖𝜆 > 1 ‖

𝜆 < 0 ‖ �� −

𝛽 > 0,

𝑇𝑚1

= 0,

𝑇𝑚2

= 0,

𝑇ℎ𝑓

=

𝜂𝑇𝑟𝑚

(1 + 𝜂)

,

𝑇ℎ𝑟

=

𝑇𝑟𝑚

(1 + 𝜂)

,

End,

(15)

where ��, 𝛽, and

𝜆 are the predictive values of 𝛼, 𝛽, and 𝜆; 𝑇𝑚,

𝑇ℎ𝑓, 𝑇ℎ𝑟, 𝑇𝑚1, 𝑇𝑚2

are predictive values which are obtained by��, 𝛽, and

𝜆.

4.3. Dynamic Simulation. Generally, for the series system,the optimum braking stability object can be realized eas-ily through the coordination between the generators and


(a)

(b)

Time (s)

Time (s)

0.2

0.1

0

0.5004

0.5002

0.5

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

SoC

z

With the predictive distribution modelWith the regenerative braking torquedistribution strategy of 1 : 1

SoCincreasing rate = 0.0689%

SoCincreasing rate = 0.0564%

t0 t0t1 t2

Figure 14: The simulation results of SoC.

the hydraulic brakes. So, with regard to the predictive distri-butionmodel, the only advantage is embodied in the regener-ative braking energy recovery efficiency. To verify the advan-tages of the predictive distribution model, two simulationmodels of the vehicle are established in MATLAB/Simulink.One is established with the predictive distribution model,the other is based on the ideal braking force distributionstrategy which is also striving to maximize the regenerativebraking energy recovery and optimize the braking stabilitysimultaneously. Additionally, no optimizationmethod is usedin the other simulation model and the distribution of theregenerative braking torque is based on 1 : 1.

The other braking force distribution strategy can beexpressed as follows.

First, based on the requirement of the ideal braking forcedistribution, the required braking torque of the front and rearwheels under a required braking torque can be obtained asfollows:

𝑇front require + 𝑇rear require = 𝑇𝑟𝑚,

𝑇front require

𝑇𝑟𝑚

= 𝛽opt,(16)

where 𝑇front require, 𝑇rear require denote the required brakingtorque of the front and rear wheels, respectively.

Then, the optimumcharging torque𝑇opt can be calculatedby (7), if 𝑇opt is bigger than 𝑇front require, then make 𝑇

𝑚=

𝑇front require; 𝑇ℎ𝑓 = 0; 𝑇ℎ𝑟

= 𝑇rear require. If 𝑇opt is lower than orequal to𝑇front require, thenmake𝑇

𝑚= 𝑇opt;𝑇ℎ𝑓 = 𝑇front require−

𝑇opt; 𝑇ℎ𝑟 = 𝑇rear require.Finally, the distribution of 𝑇

𝑚between two generators is

1 : 1.In the simulation models, the initial vehicle speed is

48 km/h. The battery SoC is 0.5. The road is assumed to be adry pavement. Two cooperative braking processes are definedin Figures 14(a)–16(a); the value of 𝑡

0is 1.54, which is the

interval time between two braking processes, 𝑡1is the first

braking time, and 𝑡2is the second braking time.

(a)

(b)

Time (s)

Time (s)

0.2

0.1

0

1

0.8

0.6

0.40 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

𝛽

z

𝛽ideal braking distribution strategy

𝛽predictive distribution modelb2

b1

t0 t0t1 t2

Figure 15: The simulation results of 𝛽.

Define the increasing rate of SoC as follows:

SoCincreasing rate =SoCend − SoCinitial

SoCinitial, (17)

where SoCincreasing rate denotes the increasing rate of SoC; andSoCend denotes the end state of SoC; and SoCinitial denotes theinitial state of SoC.

As shown in Figure 14, during the braking time of 𝑡0,

SoC will remain unchanged for the two models. During thebraking time of 𝑡

1, SoC will increase linearly and the advan-

tage of the predictive distribution model begins to manifest,which is that SoCincreasing rate is bigger than the ideal brakingforce distribution strategy. During the braking time of 𝑡

2, the

increasing tendency is similar to 𝑡1, and the only difference

is that the advantage of SoCincreasing rate is highlighted in thisbraking process. The reason can be interpreted to the factthat the bigger 𝑧 is, the more regenerative braking energy willbe obtained and the advantage of the predictive distributionmodel will be more obvious. At the end of braking, thesimulation model with the predictive distribution modelSoCincreasing rate is 0.0689%, the other is 0.0564%.

It can be seen from Figure 15, both of the braking forcedistribution of the models are close to 𝑏

1, and fall into the

scope between 𝑏1and 𝑏2, which means that the vehicle can

avoid any wheel be locked earlier, in addition, if the wheelsbe locked, all the wheels will be locked at the same time forthe both simulation models. So the vehicle can maintain theoptimum braking stability.

Figure 16 shows the simulation results of the predictivedistribution model. If 𝑧 is 0, which is corresponding to thebraking time of 𝑡

0, V is almost not changed. 𝛼predictive value and

𝑇 are equal to 0, and no braking torque will be generated bythe braking apparatuses. 𝜆predictive value is equal to 1, actually,whatever the value of 𝜆 is, the ultimate output torque forevery braking apparatus will be unaffected, the reason is that𝑧 is equal to 0 which is the precondition for the outputbraking torques. Additionally, the values of 𝛼predictive value and𝜆predictive value make no sense during this braking time and the


(a)

(b)

(c)

(a)

(d)

(e)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

0.2

0.1

0

0.2

0.1

0

60

40

20

0.4

0.3

0.2

0.1

00 2 4 6 8

0 2 4 6 8

0 2 4 6 80 2 4 6 8

Time (s)0 2 4 6 8

0 2 4 6 8

z z

�(k

m/h

)𝛼

pred

ictiv

eva

lue

Tpr

edic

tive

valu

e(N

m)

𝜆pr

edic

tive

valu

e

0.9

0.8

0.7

6000

4000

2000

0

TmTm1

Tm2

ThfThrTrm

t0 t0t1 t2 t0 t0t1 t2

Figure 16: The simulation results of the predictive distribution model.

values are wrong as a result of the design space of 𝑧 (0.01–0.4). During the braking process of 𝑡

1, 𝑧 is 0.1, which is a rel-

atively low braking severity, so the vehicle decelerates slowly.𝛼predictive value and 𝜆predictive value are changed and they meetthe cooperative braking objects’ requirements. 𝛼predictive valuechanges linearly and 𝜆predictive value changes nonlinearly. Sincethe value of 𝜆predictive value is close to 1, generator 1 will affordthe mostly charging torque. During the braking time of 𝑡

2,

the changing trends of variables are similar to 𝑡1. The only

difference is that the descending slope of V is bigger than 𝑡1

due to the higher braking severity.

5. Conclusions

This paper carries out a systematic study for a predictive dis-tributionmodel of a series cooperative braking system.Threedisciplines of the maximum regenerative energy recoverycapability, the maximum generating efficiency, and the opti-mum braking stability are considered with an off-line opti-mizationmethod. In consideration of the poor real-time per-formance of optimization, a predictive model which is basedon the off-line optimization data is presented. Finally, a pre-dictive distributionmodel which is constituted by the predic-tive model and an additional condition is proposed. The off-line optimization data proves that the optimization methodcan meet every discipline and improve the cooperative

braking performance. The dynamic simulation results showthat the predictive distribution model is reasonable for real-time control.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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