prof. moura optimal charging of a vehicle-to-grid ... · an aggregator needs to deal with many evs...

CE290-002 • Prof. Moura • Spring 2014

Optimal charging of a Vehicle-to-Gridaggregator under power supply constraint

Francois Debaillon-Vesque, Caroline Le Floch∗

University of California, Berkeley

May 2014

Abstract

This paper develops a modeling and controlling paradigm to optimize the aggregated charging dynamics of afleet of plug-in electric vehicles (PEV) under the constraint of supplying a predetermined amount of powerback to the grid. A key feature of this model is the use of coupled partial differential equations (PDE) tocapture the dynamics of the energy levels of the aggregated fleet of vehicles. The optimal control policyobtained is then validated against a Monte Carlo simulator.

I. Introduction

A. Motivation and Background

As electrification of automobile transportation is soar-ing and the share of renewable energies in the elec-tricity mix is increasing, the smart charge of Plug-inElectric Vehicles (PEV) is becoming increasingly crit-ical. Hundreds of thousands of multi-kWh batteriesconnect to the grid on a daily basis, and charge ina straight-forward pattern regardless of the demandfor electricity on the grid. Not only can these cur-rently sub-optimal charging habits be improved bytaking advantage of off peak hours, but aggregatedPEV batteries potentially represent a great means ofenergy regulation for the grid.

Currently, frequency regulation is provided by gen-erators on a day-ahead market at a MW level. Conse-quently, if PEVs, with battery powers usually in the10-20 kW range, are to play a role in regulation, itis necessary to combine their power by using an ag-gregator. Such an aggregator would enable a fleet ofvehicles to meet the minimum threshold to enter themarket while at the same time enabling a more effi-cient charge of vehicles.

Load control of PEVs can be challenging becauseof uncertainties and specificities associated with eachdriver. In this work, we consider a PEV aggrega-tor, which stocks, charges PEVs and dispatch themto customers. Optimal PEV charging schedule has

often addressed to solve the single-vehicle problem;however, for the purpose of interacting with the grid,an aggregator needs to deal with many EVs and needsa highly scalable optimization tool. In this work, wedesign a scalable optimization framework to optimizethe charging schedule of a Vehicle-To-Grid fleet.Section II will develop a transport-based method tomodel the aggregate charging dynamics of a Vehicle-To-Grid fleet; Section III will develop a Linear Pro-gram to solve the optimal charging problem.

B. Relevant literatureIn the recent literature, a number of modeling frame-works and optimization methods have been used todeal with the problem of optimal charging patterns ofPEVs. Most of the studies have considered the prob-lem for a single vehicle.

S. Bashash develops in [1] a Qradratic Program-ming framework to find the optimal cost charging pat-tern of one vehicle with varying electricity prices. Thismethod gives a computationally efficient way to solvethe problem for a single vehicle.

Soohe Han [2] has faced the problem from an ag-gregation point of view and has developed a dynamicprogramming solver to maximize the revenue of theaggregator. The model framework is very close fromthe individual vehicle point of view in term of Costformulation and solving methods; it gives interestinginsight into the aggregation problem and proposes a

∗[email protected]

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solution based on ’utility weights’, which depend onthe state of charge (SOC) of vehicles, to adapt charg-ing patterns to vehicles. However, the latter modelconsiders that the aggregator is allowed to supply anyamount of power to the regulation and that the userdrive-cycles are deterministic.

Bashash and Fathy in [3] have developed a first-order bilinear transport PDE, representing the charg-ing dynamics a G2V fleet; the authors developed anoptimal control algorithm for the use of ”intelligent”charging stations, which can adapt the charging powerthat is uniformly provided to plugged-in vehicles;therefore charging pattern of the fleet is unique andthe optimal control is solved for a unique variable.

C. Focus of this studyThe main contribution of this paper is to model theoptimization problem of charging an aggregated fleetof vehicles under V2G constraints, by a system of cou-pled partial differential equations. These equationscapture the dynamics of the aggregated energy levelof the PEV batteries. We seek to find the optimalcharge and discharge policy in order to reach the low-est cost of charge while supplying a predeterminedamount of power to the grid.

In this study we assume the price of electricity isknown prior to the time interval. This assumptionis relevant since in reality prices are known 24 hoursahead of time. We also consider deterministic drivingcycles. In other words, the time when drivers leavewith their vehicle and return is known in advance.

D. NomenclatureIn the following work, we will use the nomenclature:w Distribution of charging carv Distribution of idle carsγ Distribution of discharging carsσi→c Flow of vehicles from Idle to Chargeσi→d Flow of vehicles from Idle to Dischargeq Instantaneous charging powerXmax Storage capacity of batteriesXdep Min allowed charge for cars to departXDmin Minimum allowed charged for

discharging and idle carsXCmax Max allowed charge for charging carsNmin Min number of cars required to

be charged enough at TmaxCelec Price of electricityP desgrid Power required by the grid

II. Continuous V2G modeldevelopment

A. Continuous Transport-Based modelThis section details the use of a hyperbolic partialdifferential equation (PDE) to model the dynamics ofthe aggregated charge of the PEVs. We assume thatall the vehicles that are charging do so at a constantrate qc and that the operator can control which of theplugged vehicles are charging. The total power de-mand for charging these PEVs is proportional to thenumber of vehicles that are actually charging.

The charging PEVs can be considered as dynamicloads diffusing towards higher SOC at the chargingrate qc.

The distribution of the number of cars that arecharging at their docking station in function of thestate of charge is approximated by a continuous func-tion, as represented in figure 1.

Figure 1: Distribution of the number of vehicles fora given state of charge. Since PEVs are charging thiscurve diffuses towards higher SOCs

To describe with a mathematical expression thetransport process from low SOC to higher SOC, weconsider the number of vehicles charging at time twith a state of charge x: w(x, t).These vehicles are charging at a speed qc which cor-responds to the transport rate.We then consider an infinitesimal slice of vehicleswhose SOC is comprised between x and x+ dx. Thevariation of the number of vehicles in that SOC rangebetween time t and t+dt corresponds to the differencebetween the entering flux and the exiting flux, respec-tively w(x− qcdt, t)dt and w(x, t)dt. The variation ofloads inside the control volume also depends on theflux of vehicles that plug into the grid at a moment

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between t and t+ dt. This source term correspondingto the algebraic number of vehicles that start chargingat time t with a SOC of x is noted σi→c(x, t).

Figure 2: Algebraic flows in the control volume ofcharging vehicles

Consequently, a conservation law applied to thenumber of charging vehicles yields

w(x, t+dt)dx = w(x, t)dx+w(x−qcdt, t)dt−w(x, t)dt

+σi→c(x, t)dxdt

This equation can be rewritten as

∂w

∂t(x, t) = −qc

∂w

∂x(x, t) + σi→c(x, t) (1)

Equation (1) is the governing PDE of the dynam-ics of the distribution of charging cars.

To define the boundary conditions of (1) we as-sume all the vehicles have a SOC greater then Xmin

and lesser than Xmax. In other words, the numberand flux of cars for x ≤ Xmin is null.

w(x, t) = 0 ∀x ≤ Xmin

B. Dynamics of a V2G fleetThree-state model To complete the derivation ofour continuum level charging dynamics of a V2G fleet,we must consider three possible states for the vehicles.PEVs can either be• Charging, retrieving power from the grid and

increasing SOC.• Discharging, supplying power back to the grid

(V2G mode).• Idle, plugged into the grid but with no interac-

tion.There is also a fourth distinct state, which the ag-

gregator does not control:

• Vehicles on the road or not connected to theircharging station

In this model, the aggregator has control only overvehicles that are plugged into their charging stationand not over the vehicles that are on the road. Thecontrol policy therefore applies transfers between thefirst three states. The global dynamics are mathemat-ically described by coupled PDEs.

Global dynamics The equation detailed in partA models the dynamics of charging vehicles. Simi-larly, the dynamics for the vehicles that are discharg-ing (providing power to the grid) are modeled by atransport equation. For discharging cars, the trans-port rate is −qd: in this case the distribution of vehi-cles diffuses towards the left from high SOCs to lowerSOCs as vehicles use battery energy to provide powerto the grid.

∂γ

∂t(x, t) = qd

∂γ

∂x(x, t) + σi→d(x, t) (2)

Finally, the number of idle vehicles v(x, t) correspondsto the remaining vehicles, which are plugged into theircharging station, but are neither charging nor dis-charging. Here we must also consider vehicles comingfrom a third source term σi→or(x, t) = Dep(x, t) −Arr(x, t). This represents the algebraic number of ve-hicles plugging into their charging station: a negativenumber means more vehicles are leaving the garagethan vehicles are returning. This input is uncontrol-lable since we can not decide when drivers start usingtheir vehicles or stop using them. The variation ofthe number of idle vehicles is therefore the differencebetween the three transfer terms.

vt(x, t) = (Arr(x, t)−Dep(x, t))−σi→c(x, t)−σi→d(x, t)

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Figure 3: Conservation of the number of vehicles andtransfers between different states

Figure 3 represents the three possible states forthe plugged vehicles as well as the three transfers be-tween them. At this stage it is important to note thatthe control variables are σi→c and σi→d, the transferbetween states w,γ and v. and that, σi→or(x, t) istaken as an uncontrollable input.

The system of coupled PDEs mathematically rep-resents the dynamics of the aggregated load of charg-ing, discharging and idle vehicles.

∂w∂t (x, t) = −qc ∂w∂x (x, t) + σi→c(x, t)

∂γ∂t (x, t) = qd

∂γ∂x (x, t) + σi→d(x, t)

∂v∂t (x, t) = −σi→or(x, t)− σi→c(x, t)− σi→d(x, t)

The appropriate boundary conditions and initial con-ditions are developed in the next section.

III. Optimal control of loads

A. System modeling and constraintsThe objective here is to design the best distributionof vehicles between charging, discharging and idlestates over time. The aggregator minimizes the costof charging vehicles over the time period Tmax. Attime t, the aggregator only charges vehicles that arein the charging state W (t) =

∫Xmax

0 w(x, t)dx. Thenthe total cost over time is given by:

C =∫ Tmax

t=0Celec(t)

∫ Xmax

0w(x, t)qdxdt (3)

To ensure physical meaning of the system, we im-pose boundaries on SOC values for each category: forx > XCmax, cars cannot charge any more and forx < XDmin, cars are forced to charge:

w(x, t) = 0 ∀x ≥ XCmax

γ(x, t) = 0 ∀x ≤ XDmin (4)v(x, t) = 0 ∀x ≤ XDmin

The system is faced with two different types of con-straints: following the power demand from the gridand meeting the demand of vehicles from customers.

Power supply constraint We consider that theV2G aggregator sells electricity in a day ahead mar-ket: the amount of power P desgrid(t) is known one day inadvance. This condition sets the number of discharg-ing cars Γ(t) =

∫Xmax

0 γ(x, t)dx over time:

Psupply(t) =∫ Xmax

0γ(x, t)qdx = P desgrid(t) ∀t (5)

Drivers’ demand constraint We consider thatthe demand and arrivals of cars are known one dayin advance. This could be enabled by a reservation-based system; in practice these functions can repre-sent expected arrivals and demands. The arrival ofcars Arr(x, t) is known for all time and all SOC val-ues. Similarly the total demand of vehicles over timeDem(t) is known one day in advance. It remains tothe aggregator system to optimize the distribution ofleaving cars Dep(x, t) over SOC values. We formu-late this problem by fixing a minimum required chargelevelXdep to allow a car to be taken by a driver. Then,Dep(x, t) becomes a controllable input, which satisfies

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the following constraint:∫ Xmax

Xdep

Dep(x, t)dx = Dem(t) ∀t (6)

Time horizon and final condition The problemhere is a finite-horizon optimization: minimizing costover time period Tmax. To ensure continuity of thesystem after time period Tmax, we impose that thesystem contains a minimum number of vehicles Nminthat are able to depart after Tmax.

Xmax∫Xdep

(w + γ + v)(x, Tmax)dx ≥ Nmin (7)

B. Formulation of the optimization problemEventually, we derive the resulting optimization prob-lem:

minσi→d,σi→c,Dep

C =Tmax∫t=0

Celec(t)xmax∫0

w(x, t)qdxdt

subject to

wt(x, t) = −qcwx(x, t) + σi→c(x, t)γt(x, t) = qdγx(x, t) + σi→d(x, t)vt(x, t) = −[σi→c(x, t) + σi→d(x, t)]

+Arr(x, t)−Dep(x, t)

σi→d(x, t) ∈ [−γ(x, t), v(x, t)]σi→c(x, t) ∈ [−w(x, t), v(x, t)]

w(., 0) = w0, γ(., 0) = γ0, v(., 0) = v0

(4), (5), (6), (7)

We can remark here that all the functions and con-straints are linear regarding the variables w, v and γ.This is a key property of the model, which allows useof Linear Programing.

Discretization We note n ∈ [0, N ] the index fordiscretization in time with time step ∆t, Tmax =N∆t. We note j ∈ [0, J ] the discretization index forState of Charge with step ∆x. The function variablesare discretized into vectors: fnj = f(j∆x, n∆t).

We choose a Lax-Wendroff numerical scheme todiscretize the PDEs. This numerical scheme hasgood properties when applied to hyperbolic PDEs.The main condition of this numerical scheme is theCourant-Friedrichs-Lewy condition q∆x

∆t ≤ 1 where qis the celerity. Appendices A and B give more detailsabout discretization choices.Boundary conditions are necessary to ensure that thePDE problem is well formulated. For transport PDEs,the necessary point depends on the sign of the speedpropagation. For charging cars, we need a boundarycondition in x = 0, for discharging cars we need aboundary condition in x = Xmax. For charging cars,the boundary condition is the absence of a source termin x = 0−: this means wn0 = 0; we have the sameboundary condition in x = Xmax for discharging cars.

Linear Program We note Mc and Md the transi-tion matrices derived from the discretization of PDEs(see Appendix A). The dynamics of charging and dis-charging cars reads

wn+1 = Mcwn + σn+1

c

γn+1 = Mdγn + σn+1

d

After discretization of all the functions, we solve theoptimization problem for the optimal values of w, v,γ and Dep.The coupled PDE constraints result in the transitionconstraint:[w + γ + v]n+1 + Depn+1

∆x = Mcwn + Mdγ

n + Arrn+1

∆x

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Then, the final Linear Problem reads:

minw,v,γ,Dep

∆t∆xq∑∑

Cnelecwnj

subject to

[w + γ + v]n+1 + Depn+1

∆x = Mcwn +Mdγ

n + Arrn+1

∆xwnj = 0, ∀j > Jc

γnj = 0, ∀j < Jd

vnj = 0, ∀j < Jd

w0 = winit, γ0 = γinit, v

0 = vinit

w, v, γ,Dep ≥ 0

q∆x∑j

γnj ≥ Pn

Jmax∑j=Jdep

Depnj = Demn

∆xJmax∑j=Jdep

wNj + vNj + γNj ≥ Nminfinal

IV. Results and validation

In this section, we first examine the performance ofthe model to control the charging schedule of a fleet of500 vehicles. Then, we derive discrete integer controlsand validate the result with a Monte Carlo simulator.

A. Case study

The initial fleet contains 500 vehicles; the arrivals anddepartures of cars are shown figure 4. The total num-ber of cars which are plugged-in vary from 213 to 525.This distribution is chosen to fit with a real case sce-nario where many cars leave during the morning andcome back during evening or night.

Figure 4: Arrivals and departure of cars

Figure 5: Number of cars in the system

The price of electricity and power required by thegrid are shown figure 6 and 7. The parameters of thebattery are chosen as follows:• Instantaneous charging power: q=2kW• Battery storage capacity : Xmax = 38kWh• Xdep = 28.5 kWh• XCmax = 36.1 kWh, XDmin = 7.6kWh• Nmin = 80

B. Control implementation

The discretization steps are ∆t = 30min and ∆x =1kWh. We apply the Linear Program to the system toget the best distribution of cars over time and Stateof Charge.The total number of cars in each category (idle, dis-charge and charge) is shown figure 6.

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Figure 6: Optimal distribution of cars over time andPrice of electricity

Figure 7: Power required by the grid and power gen-erated by V2G

The controller is able to force enough cars in thedischarging category to meet the power supply de-mand (figure 7); then the remaining plugged-in carsare managed between charge and idle in order to meetdemand from drivers and to minimize the overall cost.Figure 6 shows flows from charge to idle during peakhours; in this case study, cars mainly charge from 1am to 6 am, when a lot of cars are still plugged-in andthe price of electricity is low.A second aspect of the controller is the optimizationof the distribution of cars along SOC values. Figure8 and 9 show how the controller tends to aggregatecars around the typical SOC values of the system. At

the end of the optimization period, cars are mainly inthe idle category and are charged at the minimum re-quired value XDmin = 7.6kWh, a second peak of carsis aggregated at Xdep = 28.5kWh, which correspondsto the final constraint of having a minimum numberof cars Nmin = 80 charged for the second day. Thecontroller also optimizes at which SOC level cars de-part with respect to the condition x ≥ Xdep. Figure 9shows that almost all cars depart with the minimumrequired charge value.

Figure 8: Distribution of cars along SOC values atTmax

Figure 9: Distribution of cars that departed alongSOC values

C. Discrete control and validationIn this section we explain how the control can be ap-plied to real fleets of vehicles. The continuous modelgives an optimal continuous control over SOC distri-butions. However, in a real case scenario, only dis-

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crete values of SOC and rounded number of cars canbe controlled.We validate our model using a Monte-Carlo-style sim-ulation: the control obtained form the LP formulationis transformed in rounded flows of cars. The initialfleet is modeled with a discrete approach : each car ismodeled one by one and can be assigned to 4 differentstates : charging, discharging, idle or absent. For eachof these states, the dynamic of the battery charge ismodeled as following: if the state is ’Charge’, the bat-tery charges at rate q; if the state is ’Discharge’, thebattery discharges at rate q; if the state is ’Idle’, thecharge remains constant; if the state is ’Absent’, thebattery charge is not tracked. The discrete control isapplied to this fleet of vehicles and the evolution ofthe system is simulated during N time steps, whichcorrespond to the same Tmax = 24h. Figure 10 sum-marizes the steps of the simulation.

Figure 10: Global scheme of controller

The distributions of cars resulting from the dis-crete controller are presented figure 11 and 12. Thegraphs show that the passage of the controller fromcontinuous to discrete cases has very satisfying re-sults. The resulting distributions are very close whenwe look at the aggregated level.

Figure 11: Distribution of cars along SOC values atTmax

Figure 12: Power supplied by V2G with discrete con-trol

However, for the purpose of examining the suit-ability of the continuous model, we have taken verysimilar evolutions of charge between the Monte Carlosimulator and the continuous simulator. To get moreinsight into the robustness of the model, one couldadd more complexity to the Monte Carlo simulatorand test the resulting aggregation controller.

D. Smoothing of controlsThe impact of frequent controls on battery degrada-tion remains unclear. The continuous model allows toshift the global control policy towards smoother andless frequent controls. For this purpose, we can addto the cost function a penalization for high values of

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controls. We derive the following penalization:

Pen = λ(||(σni→d)n||1 + ||(σni→c)n||1)

= λ(||(wn+1 −Mcw

n)n||1

+||(γn+1 −Mdγn)n||1

)This is very similar to adding a L1 penalization onthe derivative of the state function.It is worthy to note that||(wn+1−Mcw

n)n||1 + ||(γn+1−Mdγn)n||1 is a convex

function with respect to the variables (w, γ, v,Dep).Hence, we use the optimization solver CVX [7] to solvethe problem with the penalized cost. The resultingoptimal distribution of cars is shown figure 13

Figure 13: Aggregate distribution of cars after penal-ization L1 with λ = max(Celec)

1.4

Figure 14: Final distribution of cars along SOC valuesafter penalization L1

Figure 13 shows that the penalized optimal prob-

lems drives to more ’global’ policies. As in the previ-ous case, the controller forces the number of discharg-ing cars to meet the power demand; then the fleettends to be either completely idle during peak hours,or completely charging when prices of electricity arelow. Figure 14 shows that the cars are more equallydistributed across SOC values, even though XminD isstill an aggregation point.

V. Conclusion and Discussion

In this report we developed a modeling and controlprogram to optimize the charging and dischargingschedule of a fleet of PEVs, which supply energy tothe grid on a day ahead market. We modeled a sys-tem where vehicles can be assigned to one of the threestates: charging, discharging and idle. We formulatedthe collective charging dynamics of each category us-ing transport-based modeling principles, and coupledthe resulting PDEs to model the global evolution ofthe system. We examined different finite differencemethods to discretize the system of PDEs and geta stable numerical scheme. We developed a LinearProgram to solve the charge and discharge schedule inthe discretized spaces. The optimization gives at eachtime step the optimal flows of cars along SOC val-ues; results show that the optimization program canboth meet grid and drivers’ requirements and controlfor the cost of charging vehicles. Controlling for bothtime and SOC distributions gives large freedom to thecontroller. From the LP results, we derived discrete,integer controls to be applied the fleet of vehicles.We modeled the fleet of vehicles with a Monte Carlosimulator which takes the same assumptions for pa-rameters than the continuous model. Results showthat the performance of the integer controller is veryclose to the continuous’.

This work has developed an innovative frame-work to model and control fleets of vehicles. Thisframework allows to solve optimal charging and dis-patching problems in both time and SOC spaces in ahighly scalable way. The technique mainly relies onTransport-Based models with transfers between dif-ferent states, which represent charge, discharge andidle. This framework is very suitable to describe thedynamics of most loads. Hence, this modeling tech-nique is very promising to address future optimizationchallenges of dealing with demand-Side managementand high numbers of loads.

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A. Appendix I. Discretization

In this section we describe the discretization methodapplied to the PDEs of the form

wt(x, t) = −qcwx(x, t) + σi→c(x, t).

Two particularly interesting characteristics of theLax-Wendroff schemes for a transport equation are• Second order accuracy in time and space

achieved with a relatively low computational in-tensity.• Stability for transport equations under the CFL

condition.The numerical method based on explicit time in-

tegration is detailed below. We note n ∈ [0, N ] theindex for discretization in time with time step ∆t,andj ∈ [0, J ] the discretization index for State of Chargewith step ∆x. Define

wnj = w(xj , tn) = w(j∆x, n∆t)

with the Taylor series

w(x, t+∆t) = w(x, t)+∆twt(x, t)+ ∆t2

2 wtt+O(∆t3)

Since

wt(x, t) = −qcwx(x, t) + σi→c(x, t)

which gives

w(x, t+ ∆t) = w(x, t)− qc∆twx + q2c

2 wxx + ∆tσi→c

−qc∆t2

2 (σi→c)x + ∆t2

2 (σi→c)t

Replacing the derivatives by their approximate

wx(x, t) ≈ w(xj + ∆x, t)− w(xj , t)∆x

≈wnj+1 − wnj

∆xWe obtain

wn+1j = M∗c [wnj−1, w

nj , w

nj+1]T + σi→c)n+1

j

Where vector M∗c is defined as

M∗c =(C2 + C2

2 1− C2

2 −C2 + C2

2

)Matrix Mc can then be written

Mc =

M0c 0 · · · · · · 0

0 M∗c. . .

......

. . . . . . . . ....

.... . . M∗c 0

0 · · · · · · 0 MKc

The parameter C = qc∆x∆t often called the Courant

number is particularly important. Indeed, the schemeis stable if C ≤ 1, which creates a constraint on thechoice of the time steps and space steps. It is impor-tant to note however that the Courant Friedrich Lewycondition is not a sufficient condition for the approx-imate solution to converge to the analytic solution.

B. Appendix II. Discretizationaccuracy

A. Analytical solution

In order to test the validity of our numerical method,we solve the wave equation analytically in some simplecases and confront them to the numerical results.

The analytical solution is derived by using themethod of characteristics. In the case of a simple hy-perbolic equation the characteristics are simply lineswith a slope corresponding to the velocity qc writ-ten as c in this section. Depending on where thecharacteristics originate from, we either choose theboundary condition or the initial condition. A simplecase where the input is constant over time is presentedbelow.

Initial condition: w(x, t = 0) = w0(x) ∀xBoundary condition: w(x = 0, t) = 0 ∀tSource term: σi→c(x, t) = α+ βx ∀x, t

As a result

w(x ≥ ct, t) = w0(x− ct) + αt+ β

2c (2xct− c2t2)

w(x ≤ ct, t) = αxc + βx2

2c

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B. Comparison between numeric and ana-lytic results

We illustrate here the importance of the choice of theCourant number which influences significantly the ac-curacy of the result.

Because the source term is not smooth and in mostcases contains important discontinuities, if qc∆x

∆t < 1,artifacts similar to the Gibbs phenomenon with un-wanted oscillations, are created. Figure 15 illustratesthis for C = 0.5.

Figure 15: Example of numerical solving of the waveequation with C = 0.5

However, if the time step is chosen more wiselyin order to have C = 1 the match between numericresults and the real solution is largely improved andthe relative error is at maximum of the order of 10−3.

Figure 16: Example of numerical solving of the waveequation with C = 1

References

[1] Bashash, S., & Fathy, H. K. (2013, June),Optimizing demand response of plug-in hybridelectric vehicles using quadratic programming, InAmerican Control Conference (ACC), 2013 (pp.716-721). IEEE.

[2] Han, S., Han, S., & Sezaki, K. (2010) Devel-opment of an optimal vehicle-to-grid aggregatorfor frequency regulation. , Smart Grid, IEEETransactions on, 1(1), 65-72.

[3] Bashash, S., & Fathy, H. K. (2012) Transport-based load modeling and sliding mode controlof plug-in electric vehicles for robust renewablepower tracking. , Smart Grid, IEEE Transactionson, 3(1), 526-534.

[4] Bashash S., Moura S., Forman J & Fathy H.K. (2010) Plug-in hybrid electric vehicle chargepattern optimization for energy cost and batterylongevity Control Optimization Laboratory,Department of Mechanical Engineering, TheUniversity of Michigan, Ann Arbor, MI 48109,USA

[5] John C. Strikwerda Finite difference schemes andpartial differential equations , Wadsworth Publ.Co. Belmont, CA, USA 1989, ISBN:0-534-09984-X

[6] Randall J. LeVeque, Finite Volume Methodsfor Hyperbolic Problems , Cambridge UniversityPress, 2002. ISBN 0-521-81087-6

[7] Michael Grant and Stephen Boyd. CVX: Matlabsoftware for disciplined convex programming, ver-sion 2.0 beta http://cvxr.com/cvx

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