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Presentation Material for Rinko 25 June 2010

Development of Vehicle-to-grid aggregator on Smart Grid

Graduate School of IST, The University of Tokyo D2 Sezaki Lab. Han, Sekyung

Abstract—For Vehicle-to-Grid (V2G) frequency regulationservices, we propose an aggregator that makes efficient useof the distributed power of electric vehicles to produce thedesired grid-scale power. The cost arising from the batterycharging and the revenue obtained by providing the regulationare investigated and represented mathematically. Some designconsiderations of the aggregator are also discussed togetherwith practical constraints such as the energy restriction ofthe batteries. The cost function with constraints enables us toconstruct an optimization problem. Based on the developedoptimization problem, we apply the dynamic programmingalgorithm to compute the optimal charging control for eachvehicle. Finally, simulations are provided to illustrate theoptimality of the proposed charging control strategy withvariations of parameters.

Index Terms—Vehicle-to-Grid, V2G, Aggregator, Battery,PHEV, Regulation, Electric Vehicle, Dynamic Programming

I. I NTRODUCTION

W ITH the growing acceptance of global climate changeas a critical environmental problem, re-electrification

of automobile transportation, referred to as Vehicle-to-Gridor V2G, is getting the spotlight. So far, V2G researchers havebeen mainly focused on how to connect the vehicles batteriesto the power grid [11], [14], [16]. Much attention has alsobeen paid to prove the validity of the V2G [15], identifyits feasible services [16], and pioneer its new markets [19],[17]. Especially, extremely fast charging rate of the batterymakes the frequency regulation acknowledged as one of themost promising and practical services with V2G [11], [13],[19].

Currently, most of the frequency regulation is provided bygenerators that bid into the market. Such frequency regula-tion is carried out on MW basis between a grid operator andplants that operate single or a few generators [18]. Since atypical single vehicle battery could provide only 10kW to20kW of the power capacity [20], an intermediate system,called an aggregator, is necessary to deal with small-scalepower of vehicles while providing the regulation service onlarge-scale power. This aggregator would play an importantrole in charging the batteries of more than hundreds orthousands of vehicles. The vehicles pertaining to the ag-gregator are charged or discharged alternatively to meet therequested power from the grid operator and fulfill chargingtheir batteries. Since the conditions of each vehicle suchas aimed or current state-of-charge, and expected durationof being plugged in differ from each other, it would bemeaningful to design an efficient aggregator that providesthe regulation service in an optimal way as well as chargeeach vehicle within a specific time.

Recently, for optimal operation of the V2G, some effortshave been made to extend an existing method, called unitcommitment (UC), that schedules available generating unitsto operate a power system efficiently [10]. However, sincethe UC involves entire units in a grid, it requires muchtime-consuming computation and complicated numerical al-gorithms. The extended UC with V2G makes the problemeven more complicated, which naturally leads to stochasticmethods such as the particle swarm optimization (PSO). Inaddition, vehicles are considered just as generating unitsin the extended UC and are assumed to be charged fromrenewable sources, which is unrealistic to be applied to entirevehicles. The optimality is pursued only from the perspectiveof efficient grid operation rather than that of each vehicle.Thus, it is intractable to attract the vehicle owners to join theV2G voluntarily. Moreover, when it comes to the regulation,the decision strategy should be entirely revised as the pricingmechanism of regulation is based on the available powercapacity, not the generation cost.

In this paper, we approach the V2G regulation from astrategic perspective for the first time. The goal of thispaper is to design an optimal aggregator with respect tothe frequency regulation. Instead of considering the entirepower system, we merely focus on the clustered vehiclesunder supervision of the aggregator. Unlike other conven-tional generating units, vehicle batteries have totally differentfeatures. Vehicle batteries neither have a start-up cost nora shut-down cost while typical generators do. In addition,generation cost which is the most important factor in the UCis trivial in frequency regulation since the long-term meanof regulation request is almost zero [13]. Nevertheless, theenergy constraint is still a matter of concern for the frequencyregulation. Furthermore, since a vehicle has both aspects ofenergy consumer and regulation provider, the market pricesfor those are considered simultaneously.

We begin the article with an overview of the regula-tion scheme followed by several design considerations forthe V2G regulation. Section 3 constructs a mathematicalmodel with appropriate assumptions and incorporate severalfactors for the V2G regulation into performance criteria.Optimal solutions are then obtained through the dynamicprogramming. The solutions are represented in terms of thecurrent state of each vehicle and hence lead to a closed loopsystem guaranteeing the optimality. In section 4, we providesimulation results proving the optimality of the proposedmethod and demonstrate the balancing mechanism betweendesired departure state-of-charge level and acquired profit by

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providing regulation power.

II. V2G AGGREGATOR FOR FREQUENCY REGULATION

A. System Overview

An aggregator should be able to provide the regulationin desired scale by organizing electric-drive vehicles. Itmakes a contract with each vehicle owner. Based on thetotal contracted amount of the vehicle power, the aggregatorthen makes another contract with a grid operator. Whenthe contract is established, an energy management system(EMS), placed at the grid operator, dispatches appropriateregulation signal within the contracted boundary to theaggregator based on its own algorithm. That is, aggregatorcan concentrate on its own performance without caring aboutthe grid side benefit as long as it can respond to the regulationsignal.

As a typical performance measure, revenue of the ag-gregator can be considered. To make the revenue model,related markets should be analyzed first. Generally, dispatchalgorithm of the EMS forces the power fluctuations aboveand below zero to average out to approximately zero netenergy over time. Thus, almost all payment of the frequencyregulation is based on the available power capacity ratherthan the actually dispatched amount of power. This meansa generator sitting idle with the ability of providing theregulation is almost paid the same amount as the generatorthat was actually called upon to provide the regulation. Ifwe replace it with V2G terminology, a vehicle being idlewhile connected to the grid could be paid for its availablepower capacity. Meanwhile, the primary purpose of a vehicleto plug in to the grid is to charge its battery to serve thenext driving. With the conventional concept, the vehiclewould start to charge upon plugging in at its maximum rateuntil the desired target state-of-charge (SOC) is reached. Toprovide the regulation service, however, the vehicle shouldnot operate by its own decision. Instead, an aggregator shouldcontrol the sequence, duration and the rate of charging foreach vehicle based on a performance measure that maxi-mizes revenue of the aggregator. Typically, the performancemeasure should include electricity price since it varies overtime in the most power market. Fig. 1 is an actual LocationalMarginal Pricing (LMP) from PJM, a regional transmissionorganization in North America. It actually reflects the valueof the energy at the specific location and time it has de-livered during September 1st - 7th, 2009. Although thereare some deviations depending on the day, the daily patternis similar throughout the seven days, and thus a day-aheadprice model could be a significant factor in scheduling thevehicle charging. Secondly, the performance measure shouldreflect regulation price as well. As mentioned previously,each vehicle being idle under the control of aggregator ispotentially providing the regulation service for its availablepower capacity. Meanwhile, regulation price varies on timeand daily curves show similar pattern as in the wholesaleelectricity market. Thus, it is important to determine theidle time throughout the connection. Fig. 2 is a Regulation

Market Clearing Price (RMCP) during September 2nd - 8th,2008 from PJM. As in Fig. 1, the curves show similar patternwith each other throughout the investigated days.

Fig. 1. PJM Locational Marginal Pricing (LMP) during September 1st -7th, 2009.

Fig. 2. PJM Regulation Market Clearing Price (RMCP) during September2nd - 8th, 2008

Hence, it can be concluded that a vehicle under the controlof an aggregator has two aspects. One is a consumer sideaspect, and the other is a supplier side aspect. These twoaspects are strongly related. That is, whenever a vehiclebecomes idle, it can be considered as providing the fre-quency regulation and thus the vehicle should be paid forits available power capacity. On the other hand, if a vehiclestarts charging, the payment direction should be reversed.Therefore, these two price factors have to be reflected as acombined form in the performance measure. What shouldnot be confused with is that the charging operation initiatedby the command of regulation down is different from theoperation triggered to actually fill up the charge. Althoughboth operations are initiated by the aggregator and haveno difference in the physical aspect, they are completely

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different from the semantic perspective. While the formeris a supplier side operation, the latter is a consumer sideoperation and hence the payment direction is opposite. Itmay seem reasonable that a vehicle being charged for itsown shake could be interrupted by a regulation command tobe used for the regulation. Thus, even the time for chargingthe battery could be accounted for providing the regulation.However, when a vehicle battery is charged for its own sake,it plays a role as a load to the grid necessarily affectingthe required regulation amount. Consequently, if the loadsuddenly disappears (or is adjusted) when the regulationcontrol is trying to accommodate the load, it would distractthe control loop of balancing between supply and demand.Moreover, as the scale becomes significant by being simulta-neously operated by an aggregator, it would cause a seriousoscillation to the generation amount. Therefore, a regulationrequest should not interrupt the vehicles which are undercharging operation for their own sake.

B. Design Considerations

Other than the price issues, there are several topics relatedto the design of optimal aggregator. We will discuss threeimportant design considerations in this section.

1) System Wide Optimality vs Single vehicle Optimality:As discussed previously, the main goal of an aggregator isto maximize its revenue, which is the payment from a gridoperator by providing the power capacity for the frequencyregulation. Therefore, the aggregator should provide as muchpower capacity as possible whenever the regulation price isexpensive in order to maximize its revenue. Since the powerof aggregator is just a summation of each vehicle power,revenue of the aggregator could be maximized by performinga charging control so that each vehicle remains idle wheneverthe regulation price is expensive.

Meanwhile, since vehicles have no freedom for chargingcontrol, aggregator is ultimately responsible for purchasingcheap power from the grid for the vehicle battery charging.In this case, the revenue of aggregator is affected by the totalexpense of purchasing the charging power. Nevertheless,aggregator still can concentrate on the charging control ofeach vehicle as long as each control does not affect theothers. Basically, the influence of each vehicle charging is anoise on the grid-scale, and hence does not affect the priceof the power. After all, optimal charging control to maximizerevenue of each vehicle naturally leads to the maximumrevenue of the aggregator.

2) Energy Constraint :The main reason why the regu-lation service is paid mostly by its power capacity ratherthan the actually dispatched energy is that the fluctuationsof power changes between positive and negative are almostevenly distributed [13]. That is, the energy delivered andabsorbed is almost equal over a long term regulation.

Conventionally, the regulation service is provided by agenerator that is contracted to provide a nominal power withthe capability of adjusting the power level on the regulation

signal from the grid operator. In this case, only the adjustablepower level and the response time are of concerns. On theother hand, if an aggregator joins as a regulation providerwith V2G, it would have to consider one more aspect, anenergy constraint. Although power flow of vehicle battery ismostly restricted by its surrounding devices such as cables,relays and chemical characteristics of the battery, SOC isanother important factor that limits the power. Inherently,the charge and discharge are disallowed at top and bottomof the SOC respectively. Moreover, the power capacity isalso diminished to protect the battery itself by a BatteryManagement System (BMS) as the SOC approaches eitherlimit [2]. As a result, this energy constraint should beappropriately reflected in the performance measure of theaggregator.

3) User Behavior: Basically, it is a precondition thatthe aggregator is notified with the expected departure timewhenever a vehicle is plugged in. Unlike other vehicleinformation such as SOC or power limit, a departure timeis not quantitative information that could be measured orcalculated. Nevertheless, it is essential information neededfor optimal charging control. Therefore, it is mandatory thatdrivers actively notify the expected departure time uponplugging in. A driver would sign on a contract that he orshe would keep the vehicle connected to the grid for certainamount of time in return of incentives such as a life timebattery warranty [14]. However, it could be easily inferredthat not all drivers would sincerely abide by the contract.There may be occasions when drivers drive away beforethe pre-notified departure time. In this case, the battery maynot be charged enough even after being plugged in for longenough time, which the drivers have to accept. Therefore, theaggregator may fail to reach the pre-calculated optimal resultdue to this breach of contract and as a result it will fail toprovide the contracted regulation power. If it deals with onlya few vehicles, then an unexpected early departure wouldcause a serious failure in providing the contracted amount ofpower. As the number of vehicles increases, however, the rateof early departure would be constant, and hence predictable.Therefore, the contract between grid operator and aggrega-tor would be made considering the portion of unexpectedearly departures obtained experimentally. In particular, sincegrid operators would not deal with a small-scale regulationprovider with less than a MW, a practical aggregator shouldhave contracted with hundreds or thousands of vehicles, atleast, considering the typical small KW range of vehiclepower to provide a reasonable scale regulation. That is, whenthe aggregator is launched in the world, the early departurewould no more be a problem.

III. PROBLEM FORMULATION

If we assume that the target SOC is always above theinitial charge and thus considers charging as the only controlfactor, it is obvious that the aggregator has three controlvariables - charging sequence, charging duration and the

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Fig. 3. Example of a charging rate control with a vehicle connected for12 hours.

charging rate. However, if the target SOC is given as a pointrather than a range, the charging duration becomes just afunction of two other variables. For example, if the chargingrate is fixed at 0.1 C-rate and the gap between the initialand target SOC is 50%, total duration to fill up the requiredcharge can be easily calculated as 5 hours. (1 C-rate chargingfills up the SOC as much as 100% during an hour) Most gridoperators contract for power (and regulation as well) on adaily and hourly basis. Therefore, the minimum control unitshould be an hour. Fig. 3 shows an example of chargingcontrol. In this paper, we assume that target SOC is alwaysgiven as a point value and hence solve the control problemwith respect to charging sequence and charging rate only.

A. Analysis on Charging Rate Control

As discussed in section II-A, a vehicle is considered asproviding regulation service when being idle and the vehicleis paid from the grid operator. On the other hand, when avehicle is charging, it has to pay for purchasing the powerfrom the grid. Thus, a revenue functionR could be definedas follows:

R(Tc, r(t))△=

∫T−TC

PR(t) dt − M

∫TC

r(t)PC(t) dt (1)

subject to:

M

∫TC

r(t) dt = Q (2)

whereT : Expected plug-in durationTC : Required amount of time for target SOC chargePR(t): Regulation price scaled for each vehiclePC(t): Unit price for purchasing power from gridM : maximum possible charging rater(t): Charging Rate ,0 ≤ r(t) ≤ 1Q: Required energy for target SOC charge

Rewriting above equation with respect to the controlvariablesr(t) andPC(t) yields:

R(Tc, r(t))△=

∫T

PR(t) dt

−∫

TC

[Mr(t)PC(t) + PR(t)] dt (3)

In equation (3), sincePR(t) andPC(t) are given, the firstintegral term is a constant. Thus, we only need to minimizethe second integral term to acquire maximum revenue.

Assuming that the prices are given as hourly data, thesecond integral term in a discrete form yields:

MinN−1∑k=0

H(k, r(k)) (4)

subject to:

N−1∑k=0

r(k) =Q

Mand 0 ≤ r(k) ≤ 1 (5)

where H(k, r(k)) = Mr(k)PC(k) + PR(k) and N is thenumber of increment in expected plug-in duration T. Hereall k, and hencer(k) must be chosen to minimize each ofH(k, r(k)) to satisfy (4).

Now, let’s assume that there exist optimalr(k1) andr(k2)such that0 < r(k1), r(k2) < 1. It follows then that we have,

H(k1, r(k1)) + H(k2, r(k2))= r(k1)PC(k1) + PR(k1) + r(k2)PC(k2) + PR(k2)= r(k1) [PC(k1) − PC(k2)]

+SPC(k2) + PR(k1) + PR(k2) (6)

whereS△= r(k1) + r(k2). If we assume S to be a constant,

then (6) becomes a first-order equation with respect tor(k1),the extrema occur only at both ends of the range ofr(k1).That is, (6) has a minimum at either ofr(k1) = 0 or r(k1) =1 from (5), and thus it contradicts the assumption that0 <r(k1), r(k2) < 1. Therefore,r(k) should be either 0 or 1 tosatisfy the minimizing condition (4).

The implication of this result is that charging controlshould be on or off at maximum charging rate to maximizethe revenue. Thus, we no more need to be concerned aboutthe charging rate. Instead, we focus on how to determine thecharging sequence. Fig. 4 is revised example of Fig. 3. Sinceall charging rate is set as 1, only charging sequence (whento turn on the charging) is considered.

Fig. 4. Revised example of Fig. 3. The charging rate is fixed to maximum,and hence only the charging sequence is considered.

B. Analysis on Charging Sequence Control

Now the problem can be formulated simply as when toturn on or turn off the charging operation. Therefore, the

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continuous variabler(t) can be replaced by a step functionC(t), which is only 1 or 0 for the entire duration. In addition,since the price can vary depending on the absolute time, thetotal charging durationTc is indicated by the integration ofC(t) between the explicit start and the end time. Then therevenue function (3) is revised as:

R(t1, t2, C(t))△=

∫ t2

t1

[PR(t) − C(t) (PR(t) + PC(t))] dt (7)

subject to: ∫ t2

t1

C(t) dt = N (8)

wheret1: Plug in timet2: Expected plug out timeC(t): Step function indicating charging on or off (1 or 0)N : Minimum duration required to fill up the gap betweeninitial SOC and final SOC

Since PR(t), PC(t) are given values, andC(t) is theonly control variable, an optimal control sequenceC(t)could be obtained simply by turning on (making it 1) fromthe minimal point of (PR(t) + PC(t)) until the chargingduration constraint (8) is satisfied.

However, the revenue function (7) does not reflect theenergy constraint mentioned in section II-B2. For example,from the moment SOC reaches 100%, regulation ‘down’,which corresponds to charging of the battery, cannot beperformed at all. In order to reflect the energy constraintto the cost function, we define a weight function for each ofregulation up and down. This function indicates a normalizedcapacity that a vehicle battery can provide at a given SOC.Since the power fluctuations caused by regulation are fairlydistributed so that no, or at least only a small deviation ismade from zero. Thus, the weight function would be 1 (one)indicating full rate service as long as current SOC has moremargin than the maximum deviation caused by providingregulation. On the other hand, charging or discharging wouldbe restricted proportionally as the SOC exceeds marginalpoint MPdown or MPup that leaves less capacity than themaximum deviation. That is, if SOC is at the middle ofthe marginal pointMPdown and the top SOC (100%), itis expected that the regulation down could last only half ofthe average time. This is because the deviation would causethe battery to charge with two times more energy than it iscurrently capable of. Therefore, it is reasonable to assumethat the value created by providing regulation down at thatSOC is half of the full value. In that manner, the degradationof the weight could be set as linearly proportional to theremaining energy capacity for each direction. Fig. 5 depictstwo weight functions for each of the regulation up and theregulation down.

Consequently, the regulation pricePR(t) in (7) should bemodified to reflect the energy constraint as follows:

PR(t, x(t)) = PRU (t)WU (x(t)) + PRD(t)WD(x(t)) (9)

Fig. 5. A linearly degrading weight function

wherePRU : Scaled price of regulation up for each vehiclePRD: Scaled price of regulation down for each vehicleWU : Weight function for regulation upWD: Weight function for regulation downx:SOC of the vehicle battery

As shown in Fig. 1 and Fig. 2, both price patterns arerepeated every 24 hours. Moreover, they are contracted onhourly basis at most energy market [18]. It means that therevenue function (7) could be discretized with a unity stepand thus the control sequence ofC(t) yields a discrete stepfunction which changes only at the beginning of every hour.

C. Optimal Control with Dynamic Programming

To determine the control sequenceC(t) regarding therevenue function (7) with the constraint (8), we employeda numerical method, the dynamic programming. In orderto perform the dynamic programming, the revenue functionwas employed as a performance measure with a slightmodification to reflect the desired final SOC control as acost. Thus, the performance measure is defined as:

M(t1, t2, C(t))△=

∫ t2

t1

[PR(t, x(t))

−C(t) (PR(t, x(t)) + PC(t))] dt

−α (x(t2) − xT )2 (10)

where x(t2) is an actual SOC at the end of the control,xT is a desired final SOC, andα is a proportional factorwhich reflects the relative importance of the desire to drivethe system to the final SOC,xT .

In the system, the only state variable affected by thecontrol is SOC, which can be estimated by integrating thecharging amount over the control duration. Thus, the requiredsystem dynamic can be described as follows:

x(t) = x(t1) + K

∫ t

t1

C(t) dt (11)

wherex(t1) is an initial SOC andK is a maximum chargingrate for each vehicle. Since the initial SOC is assumed to be

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notified upon plugging in, current SOCx(t) can be estimatedas the control sequenceC(t) is determined.

Before the numerical procedure of the dynamic program-ming, aforementioned equations should be approximated bydiscrete forms to utilize a digital computer. Deferring theconversion procedure till Appendix A, the discrete form of(11) can be represented as:

x(n + 1) = KC(n) + x(n) (12)

where n is a discrete step during the time interval(t1 ≤ t ≤ t2) . Similarly, the performance measure (10)could be discretized as:

M(k1, k2, C(k)) =k2−1∑n=k1

[PR(n, x(n))

−C(n) (PR(n, x(n)) + PC(n))]−α (x(k2) − xT )2 (13)

wherek1 andk2 are the nearest discrete time instances fort1 and t2, respectively.

IV. SIMULATIONS AND PERFORMANCEANALYSIS

In order to verify the optimality and to assess the controlresults, a simulator which performs the dynamic program-ming for the designed system was developed. We employedactual day-ahead LMP and RMCP from PJM Interconnectionfor the battery charging cost and regulation price, respec-tively. The price was scaled so that it could be used fora single vehicle control. Regarding the vehicle parameters,we assumed 20KWh battery with maximum charging anddischarging rate of 0.1C, which in turn is 2KW for eachdirection. If departure SOC should be met at whatever cost,the proportional factorα in (13) must be set to a big enoughvalue. In our first simulation, it was set as 0.01 which isacquired experimentally. The vehicle was considered with10% initial SOC, and the desired final SOC was set to 90%.The total plug-in duration was set to 12 hours. With themaximum charging rate of 0.1C, the battery can be charged10% in every hour. Thus, the aggregator should control thecharging to be on for 8 hours to transfer the SOC to 90%.In this case, there is a variety of12C8 = 495 ways for thepossible charging sequence. Fig. 6 depicts expected revenuesfor all 495 sequences, and they are sorted ascending in termsof the revenue.

Fig. 7 depicts a set of optimal control sequences derivedthrough the simulation varying the initial SOC from 0%to 100% by 10%. From Fig. 6 and Fig. 7, we can seethe revenue derived from the simulation result with 10%initial SOC matches exactly the maximum revenue amongall possible 495 control ways.

As expected, all control sequences successfully transferthe SOC to 90%, the designated departure SOC, regardlessof the initial SOC. Since discharging is not considered here,control sequences with more than 90% of initial SOC do notaffect the SOC throughout the entire connection duration.

Fig. 6. Revenues for all possible 495(=12 C8) cases with 10% initialSOC

Fig. 7. Optimal control sequences for each SOC withα = 0.01

However, as the importance of the final SOC control isreduced by decreasing the proportional factorα, not allcontrols transfer the SOC to the designated one. Fig. 8 showsa set of the optimal control sequences withα = 0.0005which is much smaller than that of in Fig. 7. In contrast withthe result in Fig. 7, some controls do not meet the final SOCcondition exactly. It happens when the aggregator judges thatthe expected revenue is great enough to sacrifice the exactcontrol of the final SOC. Since this relative importance isexpressed byα, it is more likely to happen asα is decreased.As a proof, we can find out that controls which failed totransfer SOC to 90%, the designated departure value, resultin more revenue than those in Fig. 7 where they alwayssucceed to finish at 90%.

Also, it is noted that the revenue obtained with 90% initialSOC is bigger than that with 100% initial SOC due to theenergy constraint mentioned in section II-B2.

The entire control sequence could be obtained imme-diately as soon as a vehicle is plugged in. However, thesequence needs to be adaptively recalculated due to SOCfluctuations caused by the regulation. Although the long termfluctuations would end up with the zero mean, a certainamount of deviation could still be incurred over a short term.Especially, as the expected departure time is approaching,this deviation could cause a failure of the final SOC control.Therefore, the decision for the optimal control should be

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Fig. 8. Optimal control sequences for each SOC withα = 0.0005

made every moment before an actual control is performed.Fortunately, a table for the entire optimal control sequencescan be obtained through the dynamic programming. Uponthe plugging in of a vehicle, this table is built for the entireSOC range and every control step. If we assume that theSOC is decreased no more than a certain amount from theinitial SOC, the effort for building this table could be savedas much as the SOC range which is out of the concern.The control strategy, of course, would vary depending on therelative importance of the final SOC control,α, as well. Fig.9 illustrates two optimal control tables regarding different‘α’s.

α = 0.01 α = 0.0005SOC\ HR 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11

100 x x x x x x x x x x x x x x x x x x x x x x x x95 x x x x x x x x x x x x x x x x x x x x x x x x90 x x x x x x x x x x x x x x x x x x x x x x x x85 x x x x x x x x x x x x x x x x x x x x x x x x80 x x x x x x x x x x x o x x x x x x x x x x x x75 x x x x x x x x x x x o x x x x x x x x x x x x70 x x x x x x x x x x o o x x x x x x x x x x o o65 x x x x x x x x x x o o x x x x x x x x x x o o60 x x x x x x x x o o o o x x x x x x x x o x o o55 x x x x x x x x o o o o x x x x x x x x o x o o50 x x x x x x x o o o o o x x x x x x x x o x o o45 x x x x x x x o o o o o x x x x x x x x o o o o40 x x x x x x x o o o o o x x x x x x x x o o o o35 x x x x x x x o o o o o x x x x x x x o o o o o30 x x x x o o o o o o o o x x x x x x x o o o o o25 o x x o o o o o o o o o o x x o o o x o o o o o20 o o o o o o o o o o o o o o o o o o x o o o o o15 o o o o o o o o o o o o o o o o o o x o o o o o10 o o o o o o o o o o o o o o o o o o o o o o o o5 o o o o o o o o o o o o o o o o o o o o o o o o0 o o o o o o o o o o o o o o o o o o o o o o o o

Fig. 9. Optimal control tables for each ofα = 0.01 andα = 0.0005. ‘o’means charging while ‘x’ means idle.

Although the control decision is made only at specificSOC instances, a linear interpolation can be made for theinterim values and thus the optimal decision can be madeat any moment with any SOC value. For example, assumethat the table is built withα = 0.01 and the current SOC is30%, then an optimal control at 5th hour step can be madeas ‘ON’ according to the left table in Fig. 9.

It is also noted that the table withα = 0.01 has more ‘o’sthan that withα = 0.0005 indicating more effort to transfer

SOC to the designated departure SOC.

V. CONCLUSION

This paper developed an aggregator for V2G frequencyregulation regarding the optimal control strategy for thefirst time. A performance measure was mathematically for-mulated to maximize the revenue. During the formulation,energy capacity of the battery was considered as an impor-tant factor, and weight functions were employed to reflectthe energy constraint. By using dynamic programming, anoptimal charging control was pursued for each vehicle andthe optimality of the results was verified by simulations.Moreover, a mechanism that effectively adjusts the relativeimportance between the final SOC control and the revenuewas provided as well. Currently, the departure SOC isassumed to be given as a point rather than a range. However,the vehicle owners would not stick to an exact point as longas the SOC is high enough for one driving. In this case,the revenue could be increased by widening the range ofthe departure SOC as shown in Fig. 8 under the customer’sagreement. Finally, this study is a first strategic approach forthe V2G aggregator. Moreover, the proposed method couldbe applied not only to the frequency regulation but also toother V2G applications with some modifications.

APPENDIX ADISCRETIZATION OF THE SYSTEM DYNAMICS

Differentiating equation (11) yields:

dx(t)dt

= KC(t) (14)

Above differential eqaution can be discretized most con-veniently by dividing the control intervalt1 ≤ t ≤ t2 intoN equal increments,∆t yielding to:

x(t + ∆t) − x(t)∆t

≈ KC(t), (15)

or

x(t + ∆t) = KC(t)∆t + x(t) (16)

If we assume the control changes only at the instantst =0, ∆t, ..., (N − 1)∆t;thus, fort = k∆t,

x(n + 1) = KC(n)∆t + x(n) (17)

Since the electric power is hourly priced in most energymarkets, it is possible to put∆t as 1 yielding above equationto (12). In a similar way, the performance measure (10) canbe converted to (13) easily.

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