Research ArticlePricing of Vehicle-to-Grid Services in a Microgrid byNash Bargaining Theory

Mohammad Hossein Sarparandeh and Mehdi Ehsan

Department of Electrical Engineering Sharif University of Technology Tehran Iran

Correspondence should be addressed to Mohammad Hossein Sarparandeh sarparandeheesharifedu

Received 2 September 2016 Accepted 27 December 2016 Published 19 January 2017

Academic Editor Linni Jian

Copyright copy 2017 Mohammad Hossein Sarparandeh and Mehdi Ehsan This is an open access article distributed under theCreative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium providedthe original work is properly cited

Owners of electric vehicles (EVs) can offer the storage capacity of their batteries to the operator of a microgrid as a servicecalled vehicle-to-grid (V2G) to hold the balance between supply and demand of electricity particularly when the microgrid hasintermittent renewable energy sources Literature review implies that V2G has economic benefits for both microgrid operator andEV owners but it is unclear how these benefits are divided between themThe challenge grows when the policy makers rely on theV2G revenue as an incentive for expanding the penetration of EVs in the automotive market This paper models the interactionbetween microgrid operator and EV owners as a bargaining game to determine how the benefits of V2G should be divided Themethod has been implemented on a hybrid power system with high wind penetration in addition to diesel generators in ManjilIran The results indicate that in addition to V2G benefits government subsidies are necessary to promote the use of EVs

1 Introduction

Among the advantages which policymakers have enumeratedfor applying electric vehicles (EVs) in transportation areemission reduction and energy diversity of primary sourcesthat generate electricity as a fuel for these vehicles [1 2]These superiorities over the conventional gasoline-poweredvehiclemdashwhose fuel is dependent on finite oil reservesmdashhave caused some experts to predict a bright future for EVmarket shares [3ndash6] Nevertheless some have argued that theadditional cost of EVs is an obstacle to the adoption of thistechnology [1 6 7]

The first solution to encourage large-scale utilization ofEVs by customers might be subsidies A subsidy of up to$7500 by the US government is an example [7] In additionelectric vehicle owners could earn money from providing aservice called vehicle-to-grid (V2G) in which the electricalenergy stored in their EV battery is injected to the gridwhenever the grid operator calls on the vehicle owners forthe service

Since it is not feasible for a bulk power system operator toconnect with a huge number of cars with insignificant power

capacity several papers have proposed an aggregator to bea middleman entity between vehicle owners and the systemoperator [8ndash15]mdashalthough the function of the aggregator inthese papers is dissimilar In this manner it would be possiblefor dispersed vehicles to participate in the electricity marketbut the required communication infrastructure would becostly and widespread implementation of such a projectwould be a long-term program

A more readily available application of V2G is to use thestorage capacity of EV batteries to mitigate the intermittencyof renewable sources of electricity generation in a microgridespecially with high penetration of renewable generation(eg wind or solar) as shown in [16ndash21] In the short term it iscertainly more possible to provide the essentials for initiatingV2G inmicrogrids because amicrogrid is a low extent powernetworkwhose communication and control system can easilydeal with dispersed energy storage systems

Although technical feasibility and economic profitabilityof V2G have been addressed in several studies such as[10 20ndash27] the division of benefits between the entity whoutilizes V2G services and the vehicle owners is a point thathas been missed in the literature Although some studies

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 1840140 11 pageshttpsdoiorg10115520171840140

2 Mathematical Problems in Engineering

like Lee and Park [28] tried to model interaction betweenEVs and a microgrid like a shopping mall but indeed thereis as of yet no proposed model on how the players of thisbilateral agreement compromise Since planners considervehicle-to-grid as an incentive to promote EV penetration inthe automotive market it is vital to know how much impactV2G will have on each side of the contract This paper offersa principle to be regarded in subsequent cost-benefit analysesof V2G

The rest of the article is organized as follows The advan-tages of incorporating the EV batteries storage capacity ina microgrid and its challenges are introduced in Section 2Section 3 describes the Nash bargaining theorem and itsapplication for division of added value to themicrogrid due toV2G Section 4 is devoted to a case study inwhich the benefitssharing model is utilized for a microgrid in Manjil IranThe subsequent results are presented in Section 5 Finallysome remarks and directions for future work are concludedin Section 6

2 Problem Description

A microgrid is part of a local utility distribution networkthat uses distributed energy resources to supply electricityfor associated local loads and can operate in either grid-connected mode or islanded mode The microgrid operatorcontrols the interconnection switch loads and distributedenergy resources [29] According to [30] distributed energyresources include distributed generation and distributedstorage and often both are used to provide energy within themicrogrid Microgrids enhance power quality reliability andefficiency while lowering greenhouse gas emissions per unitof final energy consumed [31 32]

Although high penetration of renewable energy sources(eg wind turbines and photovoltaic panels) as distributedgeneration in a microgrid brings environmental benefitsbut it may also jeopardize the balance between load andgeneration during islanded operation A microgrid mustbe designed so that in islanded mode distributed energyresources can follow the load Therefore renewable sourcesof energy are usually installed in combination with dieselgenerators [33] Distributed storage is considered as analternative technology to reduce diesel generator size as wellas improving efficiency When the renewable generation ishigh but the load is low the excess power can be stored ina distributed storage system instead of being wasted as heatin dump loads However as [34] points out connection ofsuch storage systems to the grid requires smart control andadvanced power electronic converters

EV batteries are potential dispersed energy storageresources that may be viewed as an opportunity for micro-grids with a high penetration of intermittent power gener-ation sources From a technical point of view this requirescommunication and control infrastructure [9 13 35] andfrom a commercial point of view both microgrid operatorand vehicle owner should derive a benefit In terms of themicrogrid operator V2Gdecreases investment cost and oper-ating cost due to a reduction in diesel generator size or otherdistributed energy resources required for load followingThe

additional costs of associated infrastructure should also beconsidered

With respect to the vehicle owner replacing a con-ventional car with an electric one has its costs (althoughgovernment subsidies reduce the price gap) Moreover V2Gdecreases the battery lifetime and accelerates battery replace-ment [3 36 37] So the microgrid operator should pay thevehicle owners for the service and access the EV batteries sothat V2G does not limit the vehicle functionality as a meansof transport

A model determining the amount of payment to vehicleowners bymicrogrid operator is described in the next sectionThe model must take into account the interests of both sidesand investigate if the payment would be a sufficient incentivefor the public to use the green technology of electric vehicles

3 V2G Benefits Sharing Model

If the microgrid operator wants to contract with vehicleowners to use their EV batteries energy storage capabilitiesit must reach an agreement on V2G service characteristicsand price The situation can be modeled by game theory as abargaining game with the microgrid operator and all vehicleowners as players It has specifications of a bargaining gamebecause the players benefit from cooperation as discussed inprevious section but at the same time themicrogrid operatorwants to pay less to vehicle owners while the vehicle ownerswant to gain more fee There are many vehicles in this gameand this increases the complexity but it is possible to considera single representative as it is considered in [38ndash40] for awage bargaining game In a wage bargaining game a firmand a unionmdashthat is representative of all workersmdashbargainover the division of profits between them In our problemall vehicle owners have similar interests and the interactionof their community with the microgrid operator affects theprice and specifications of the V2G service contracts sotheir community can be assumed as a single player againstmicrogrid operator as the other player Before describingthe model proposed in this paper we review the theory ofbargaining and Nash bargaining solution

31 Nash Bargaining Theory A bargaining problem withtwo players describes a situation in which both playersare motivated to cooperate although there is a conflictof interest about agreement terms Bargaining theory wasformally introduced by John Nash in 1950 by an axiomatic(or cooperative) approach and followed by the strategic(or noncooperative) framework with alternating offers byRubinstein in 1982 [38]

The axiomatic approach presented by Nash sets a seriesof axioms (ie properties that characterize the solution)and the solution must satisfy these axioms In the strategicapproach players offer alternating proposals to each otherSo the bargaining is modeled as a sequential game withan infinite bargaining horizon where delays in agreementlead to reduced payoffs According to [38] the Rubinsteinmodel is suitable for strategic relationships arising in dynamic(ie over time) negotiations in which there is no foreseeabledeadline In contrast the Nash solution explains a situation

Mathematical Problems in Engineering 3

where bargainers sign a binding contract once an agreementis struck

The Nash solution of a bargaining game is a mutuallybeneficial agreement that maximizes the product of playersrsquoexcess payoffs due to agreement In other words considering1198751 and 1198752 as the payoffs of players when there is no agreement1198751 and 1198752 will be the Nash solution of the bargaining game ifand only if they maximize the term in

max11987511198752(1198751 minus 1198751) times (1198752 minus 1198752) (1)

The Nash solution is Pareto optimal so there is no agree-ment in which the utilities of both players are simultaneouslymore than 1198751 and 119875232 The Two-Stage Model Formulation If we want to applyNash bargaining theory to the V2G pricing problem in amicrogrid we have to determine the payoffs of both micro-grid operator and vehicle owners union with and withoutV2G Payoffs can be expressed by monetary unit Time valueof money should be taken into consideration Costs as wellas revenues form a cash flow and are discounted using aninterest rate for an analysis period

The formula for Nash bargaining is presented in

max119873119901(MGCbase minusMGCV2G minus 119873 times 119901)

times (119873 times CVC minus 119873 times EVC + 119873 times 119901) (2)

where MGCbase is the net present cost of investment andoperation of the microgrid without V2G MGCV2G is the netpresent cost of investment and operation of the microgridutilizing V2G 119901 is the present value of payment to the vehicleowners for V2G service by the microgrid operator 119873 is thenumber of electric vehicles that participate in V2G CVC isthe net present cost for buying and keeping a conventionalgasoline-powered vehicle (including fuel cost) and EVC isthe net present cost for buying and keeping an electric vehicle(including fuel cost)

In (2) 119873 and 119901 are decision variables Therefore thenumber of vehicles and V2G price should be determinedsimultaneously to maximize the Nash product Alternativelydecision variables can be determined sequentially by a two-stage model in which the optimum number of electric vehi-cles for V2G is appointed first through minimizing the termdenoted in (3) Then the Nash product in (2) is maximizedwith respect to 119901 while 119873 is assumed to be fixed It can beproved that both alternativemethods have similar results (seeAppendix) In (3) the total cost of themicrogrid operator andelectric vehicle union for replacing vehicles and utilizingV2Gis minimizedThis means that players should consider globalutility to set119873 and then bargain over 119901

min119873(MGCV2G (119873) + 119873 times (EVC minus CVC)) (3)

By solving (2)with respect to119901 the payment to the vehicleowners for V2G service will be specified as presented in

119901= 12 (MGCbase minusMGCV2G + 119873 times EVC minus 119873 times CVC) (4)

Hence the proper V2G price will be revealed dependent onthe pricing scheme and regarding 119901 as the present value oftotal payment for V2G to each electric vehicle

4 Case Study A Microgrid in Manjil Iran

Manjil a city located in the north of Iran has wind farmsthat deliver electricity to the national grid Commonly facingpower interruptions feasibility studies have been conductedsince 2010 to investigate if the Manjil distribution networkcould supply its local critical load when it is isolated fromthe national grid in rare events that may occur a few hours ayear Local wind turbines were presumed as one of the powergeneration sources in the incoming microgrid Intermittencyand variability of wind power oblige the utilization of acomplementary diesel generator to maintain a steady balanceof supply and demand Determining the size of the dieselgenerator for the microgrid in Manjil was a multiobjectivedecision-making problemwhose goalwas finding the optimalMW rating that ensures the reliability of the microgrid withminimum cost The problem has been discussed in [41] indetail

As mentioned in the previous sections V2G can be con-sidered as an alternative energy source which will decreasethe diesel generator size and consequently the investmentcost of the microgrid If the Manjil microgrid wants V2Gto be applied in the island mode of operation the paymentto vehicle owners should be agreed so that it is mutuallybeneficial Hence the previously introduced two-stagemodelhas been utilized to compute the V2G price It should benoted that the pricing scheme proposed in this paper can beapplied to various cases other than Manjil too

According to the first stage the optimum number ofelectric vehicles participating in V2G must be calculated atthe beginningThe reducedMWrating of the diesel generatoris derived in the same stage to be compared with the initialrating Therefore two cases have been analyzed for planningof Manjil microgrid in island mode of operation Both casesuse two 660 kW Vestas wind turbines Case 1 is the basecase in which a diesel generator adjoins when the microgridis isolated from grid The size of the diesel generator is theonly decision variable in the design of the base case In Case2 electric vehicles service the microgrid thus the decisionvariables in this case are the number of vehicles and themodified size of the diesel generator Figure 1 demonstratesthe configuration of the Manjil microgrid

Wind speed data for Manjil has been recorded in 10-minute time steps in 2009 The time series for wind powergeneration is derived from wind speed data and the powercurve of the turbine generatorsTheyearly peak load inManjilis about 5MW but in island modemdashwhich may last up to 6

4 Mathematical Problems in Engineering

Circuit breaker

Grid

Microgridoperator

Diesel generatorWind turbine 1 Wind turbine 2

Critical loadElectric vehicle 1Electric vehicle 2

Monitoring amp control

Electric vehicle N

Figure 1 Configuration of Manjil microgrid

hoursmdashonly the critical load approximately 20 of the totalload must be supplied

Sizing of microgrid components is dealt with as a mul-tiobjective problem by the NSGA-II algorithm in MATLABReliability and cost are the two objectives that form the axis ofPareto front (a set of solutions with no other solution whichcan improve at least one of the objectives without degradingany other objective [42]) as output of the program ldquoSystemminutesrdquo is selected as the reliability index The operatingcost of the diesel generator is negligible since the dieselgenerator is only operated in island mode and isolation ofManjil from the grid is infrequent and of a short durationSo its operating cost is much less than its investment costand can be neglected Operating cost of the wind turbines isnegligible too The investment cost of wind turbines is notconsidered in the planning problem because they are presentin the Manjil wind farm now and are operated all year longAll other costs are accounted and transferred in cash flow tothe present value

The optimum value for decision variables in each casehas been found using a hybrid method which is a mixture ofsimulation andmultiobjective optimizationThe advantage ofsimulation as stated in [43] is studying the details that cannotbe easily considered by other methods If the alternatives orpotential solutions are limited the most appropriate way tofind the best solution is to simulate all of them But when thesolution space is extended it would be too time consumingto simulate all possible solutions In such circumstances aheuristic search of the solution space is beneficial

One of the methods that can intelligently search thesolution space is the genetic algorithm which is compatiblewith multiobjective optimization NSGA-II is a controlledelitist genetic algorithm As it is elitist it prefers solutionswith better fitness values As it is controlled it also favorsthe solutions that increase the diversity of the population

Diversity of population helps the algorithm converge to anoptimal Pareto front

Thus the steps of the algorithm for finding the sizeparameters can be summarized as follows

(1) Random generating of an initial population for deci-sion variables (initial values for size of diesel generatorand number of electric vehicles)

(2) Simulating the operation of the microgrid withdefined decision variables and evaluating the fitnessof solutions according to prescribed criteria (reliabil-ity and cost)

(3) Investigating the termination criteria and moving toStep (5) if they have been satisfied

(4) Intelligent selection of next generation of solutionswith genetic algorithm and returning to Step (2)

(5) Plotting the optimal Pareto front

Simulation of each solution which is equivalent to aspecific design is performed with 10-minute time steps fora 6-hour interval in which the microgrid is operated in islandmode Due to the random nature of wind speed and itsdifferent patterns in various months of the year four 6-hourintervals are randomly picked for each month and the fitnessvalues would be eventually obtained from the average of 48present scenariosThe algorithm termination criteria includea combination of the maximum number of generations timeconstraints and lack of significant improvement in fitnessvalues The output of the program is a Pareto front for eachcase whose horizontal axis denotes cost and whose verticalaxis denotes the reliability index All points on the Paretofront may be viewed as an optimal point For final decisionmaking one can set the desired reliability index and comparethe present values of total cost associated with each case

Mathematical Problems in Engineering 5

To simulate the operation of the microgrid as the secondstep in the algorithm used to find the optimum numberof electric vehicles it is necessary to model the microgridcomponents (ie diesel generator wind turbines electricvehicles and critical load) and the relationships betweenthese components As mentioned before for the powergenerated by wind turbines and the power consumed bycritical load a simulated time series vector has been utilizedFor electric vehicles their batteries are considered as energystorage systems that can be charged with at most 3 kW powerfrom the grid and deliver at most the same power to the gridwhen necessary The rating of vehicle batteriesrsquo energy sizeis assumed to be 12 kWh Both charging efficiency (120578119888) anddischarging efficiency (120578119889) are considered to be 095 whichresults in a round-trip efficiency of approximately 09

Regarding the interaction between electric vehicles andmicrogrid operator it is also accepted thatV2Gwould only beused when themicrogrid switches to islandmode Hence theconvenience of electric vehicle owners will not be influencedexcept for a limited number of days when the microgrid isdisconnected from the bulk power grid During the periodof disconnection the battery chargedischarge control ofthe electric vehicles that have contracted to participate inV2G and are available to the grid will be performed by themicrogrid operator Available vehicles are those which areparked and connected to the grid The availability indexwhich has been defined as the ratio of available vehicles tothe total vehicles under contract is 70 percent In [11 1644] some statistical information has been provided abouthow many hours the cars are parked in a day The requiredcommunication and control infrastructure for monitoringdata acquisition and sending control commands to electricvehicles should be provided by the microgrid operator

In simulating the microgrid operation critical load levelandwind speed have been considered as exogenous variablesWhat the microgrid operator has under his control includesthe amount of power generated by the diesel generatorand the chargedischarge rate of power tofrom the vehiclebatteries The strategy applied in this paper to adjust chargelevels of vehicle batteries is intended to maximize the energyreserved in the batteries Therefore in every time step inwhich the vehicle batteries are not fully charged and totaloutput power of the wind turbines plus maximum power ofthe diesel generator exceeds the critical load level batterieswill be charged This helps sufficient energy to be availablein the batteries for the next time steps with a probableshortage in wind power Hence a new parameter has beendefined called the power absorption capability (PAC) ofvehicle batteries whose quantity in each time step is obtainedfrom

PAC (119905) = 119873sum119894=1

min(119875119887119894 (1 minus SOC119894 (119905)) times 119864119887119894120578119888 times 119879 ) (5)

where119873 is the number of electric vehicles 119875119887119894 is the nominalpower of the 119894th battery converter in kW SOC119894(119905) is thestate of charge for 119894th vehicle battery at time step 119905 119864119887119894 isthe capacity of 119894th vehicle battery in kWh 120578119888 is the charging

efficiency of 119894th vehicle battery and119879 is the time step durationin ℎ

At this stage if total power generation of wind turbines isgreater than critical load plus PAC(119905) the wind turbines willbe disconnected one by one until the consumption surpassesthe generationThen the rate of power thatmust be generatedby the diesel generator can be obtained from

119875119889 (119905) = min(119875119889max 119871 (119905) minus 119873119908sum

119894=1

119875119908119894 (119905) + PAC (119905)) (6)

where 119875119889(119905) is the power generated by the diesel generator attime step 119905 in kW 119875119889max

is the maximum power of the dieselgenerator in kW 119871(119905) is the amount of critical load at timestep 119905 in kW 119873119908 is the number of wind turbines that arestill connected and 119875119908119894(119905) is the power generated by 119894th windturbine at time step 119905 in kW

Afterwards the microgrid operator will determine thepower which should be exchanged with the electric vehiclesunion from

119875V2G (119905) = 119871 (119905) minus119873119908sum119894=1

119875119908119894 (119905) + 119875119889 (119905) (7)

When 119875V2G(119905) is positive it means that the power generatedby the diesel generator and wind turbines is not sufficient forsupplying load so the microgrid operator orders the electricvehicles to deliver electricity to the microgrid Neverthelessit is possible that some part of the load has to be interruptedThe amount of interrupted load may be calculated from(8) When 119875V2G(119905) is zero the batteries of electric vehiclesconnected to the grid are fully charged and when 119875V2G(119905)is negative it means that vehicle batteries can be chargedbecause of adequate power supply An allocation scheme for119875V2G(119905) among electric vehicles is beyond the scope of thispaper

ENS (119905) = 119871 (119905) minus 119873119908sum119894=1

119875119908119894 (119905) + 119875119889 (119905) + 119875V2G (119905) (8)

In the above equation ENS(119905) is the energy not suppliedat time step 119905 in kWhThe total energy not supplied during alltime spans for 48 simulation scenarios forms the reliabilityindex as one of the design criteria Total cost the seconddecision criterion is obtained from

TC = 119875119889maxtimes 120587119889 + 119873 times 120587EV (9)

where TC is the total cost in $ 120587119889 is the diesel generatorinvestment unit price which is supposed to be $800kW and120587EV is the surplus cost of an electric vehicle with respect toa conventional gasoline-powered vehicle in $ with regard togovernment subsidies The reasons for not considering otherparameters in determining the total cost had been discussedabove The required cost for installing the communicationand control infrastructure to implement V2G has beenaccounted in 120587EV

To compare the cash flows of conventional and electricvehicles it should be noted that the lifetime of an automobile

6 Mathematical Problems in Engineering

in Iran is about 20 years The lifetime of the diesel generatorand analysis time span is assumed to be 20 years too Anelectric vehicle is almost $6000 more expensive to buy andneeds battery replacement in the tenth year It is assumedthat the cost of battery replacement is $6000 and the cost ofestablishing the infrastructure for V2G is $1000 per vehicleOn the other hand a typical PHEV consumes approximately005 literkm In comparison with a 008 literkm fuel con-sumption rate for a conventional vehicle and assuming anaverage distance traveled of 30000 kilometers per year therewill be a 900-liter-per-year differenceThe gas price in Iran isabout $03liter which gives a $270 per year cost reductionby replacing a conventional vehicle with an electric oneTherefore 120587EV is equivalent to the present value of the cashflow as shown in the following equation

120587EV = 120587119904 + 120587119894 + 120587119887 times 1(1 + 119894)10 minus 120587119891 times(1 + 119894)20 minus 1119894 times (1 + 119894)20 (10)

where 120587119904 is the price difference between an electric vehicleand a conventional one 120587119894 is the V2G infrastructure cost pervehicle 120587119887 is the battery replacement cost 120587119891 is the annualcost reduction due to fuel saving of an electric vehicle and 119894 isthe discount rate in Iran thatmay reasonably be approximatedwith 20 percent in 2014 Hence 120587EV in this case is about$6654

5 Results

The results of the analysis for Manjil show that utilizationof electric vehicles with the current price difference betweenconventional cars and electric carsmdashwhich is mainly dueto battery costsmdashis not economically attractive even whenconsidering the benefits of V2G One can only hope thatwidespread use of electric vehicles in the short term maybe a possibility if the government provides a subsidy forbuying electric vehicles in order to protect the environmentand promote green technologies Unfortunately accordingto calculations made in this study any subsidy less than$5000 per electric vehicle would not encourage the publicto buy an electric car Figure 2 demonstrates the optimalPareto fronts of microgrid sizing plans for various scenariosThe horizontal axis in this figure illustrates the equivalentpresent value of the plan while the vertical axis representsthe reliability index expressed in system minutes

Figure 2 shows four scenarios in the first one vehiclebatteries are not utilized and the Pareto front has beenextracted from various designs with different diesel generatorpower sizes In other scenarios vehicle batteries are used tosupply load along with a diesel generator as the complementsto wind turbines In one of these three cases there is nogovernment subsidy but in the other two scenarios thesubsidies equal to $5000 and $5500 have been assumedIn Figure 3 a scenario with a $6000 government subsidyis added to the previous figure If the government subsidywas more than $6654 buying an electric vehicle becameeconomically attractive even without receiving V2G incomeThe results of analyzing this case indicate that considering

No V2GV2G no subsidy

V2G $5000 subsidyV2G $5500 subsidy

0

50

100

150

200

250

300

Syste

m (m

inut

es)

750 760 770 780 790 800 810 820 830740Total cost (times$1000)

Figure 2 Optimal Pareto fronts of microgrid sizing plan for variousscenarios

No V2GV2G no subsidyV2G $5000 subsidy

V2G $5500 subsidyV2G $6000 subsidy

500 550 600 650 700 750 800 850450Total cost (times$1000)

0

50

100

150

200

250

300Sy

stem

(min

utes

)

Figure 3 Optimal Pareto fronts with a new scenario $6000 subsidy

the proceeds of V2G a $6000 subsidy is an appropriatedecision

One way to attain a certain design for sizing of microgridcomponents among the optimal options on Pareto front is tospecify a minimum requirement for reliability In this paperthe system minutes index is supposed to be less than 30minutes in a year as a decision criterion Accordingly Table 1has been obtained for various scenarios of the problem

As it is shown in Table 1 a subsidy of $6000 has alarge impact on the optimal design of the system and theinvestment cost needed to meet the load In the rest ofthis paper it has been assumed that this amount of subsidywill be granted from the government to everyone who buysan electric vehicle Therefore for determining the price ofV2G services via the Nash bargaining theory two cases havebeen studied and compared first operation of the microgridwithout V2G and second operation of the microgrid with

Mathematical Problems in Engineering 7

Table 1 Size cost and reliability indices for various scenarios

Scenario No V2G V2G (no subsidy) V2G ($5000 subsidy) V2G ($5500 subsidy) V2G ($6000 subsidy)System minutes 292 298 290 290 299Microgrid inv cost ($) 801100 801000 663096 635281 44543Total cost ($) 801100 801000 797070 749527 461795Diesel generator size (MW) 9750 9749 8050 7708 533Number of PHEVs 0 0 81 99 638

Table 2 Results for V2G service pricing

Equivalent present value of total V2G service price $5869045Equivalent present value of V2G service price pervehicle

$919913

Equivalent annual payment to each vehicle for V2Gservice

$18891

Excess payoff of microgrid operator due to V2G $1696525

Excess payoff of each vehicle owner due to V2G $265913

Payment to each vehicle per hour available $449786

V2G and the government subsidy ($6000) In Figures 4(a)ndash4(d) the results of simulating the system during a six-hourinterval of isolation from grid for both cases in a windycondition are depicted Figures 5(a)ndash5(d) display the resultsof the simulation for both cases in a low wind six-hour timeinterval

As can be seen in Figures 4 and 5 when only thediesel generator is used to follow the difference between thepower generated by wind turbines and the power consumedby load a high capacity of diesel generator is needed tobe installed When chargedischarge power of the vehiclebatteries follows the load a low capacity of diesel generatoris required and it will work with an almost smooth outputpower When the electric power generated by wind turbinesis high the batteriesrsquo state of charge will be approximatelyinvariant However when the wind speed drops the vehiclebatteries will gradually discharge The possibility of storingextra power generated by wind turbines in vehicle batterieseliminates cutting of wind turbines due to excess powerin island mode This enhances the energy efficiency in themicrogrid

Using (4) and the results listed in Table 1 the price ofvehicle-to-grid service revenues and profits of themicrogridoperator and electric vehicle owners from V2G agreement isshown in Table 2

In Table 2 the first row represents the equivalent presentvalue of the total payment by the microgrid operator to allvehicle owners during a 20-year time interval for participat-ing in the vehicle-to-grid programThe second row indicatesthe same parameter per vehicle while the price in the thirdrow is the equivalent annual payment by the microgridoperator to each vehicle during the 20-year period given adiscount rate equal to 20 Excess payoffs of the microgrid

operator and vehicle owners due to V2G implementation arelisted as equivalent present values In the last row it has beenassumed that the pricing scheme is based on the hours thatan electric vehicle is available for V2G when it is called forPercentage of available vehicles is supposed to be 70 and itis assumed that isolation from grid takes place for on average6 hours a year Hence a vehicle owner would be paid about$45 for one-hour availability

6 Conclusion

The utilization of batteries of electric vehicles as energystorage systems can help microgrids in supplying load whenthey become isolated from the grid This service which isknown as V2G requires the active participation of electricvehicle owners To develop and maintain such partnershipsthe interests of both microgrid operators and vehicle ownershould be considered Otherwise the participants wouldnot have sufficient motivation This paper has proposed amodel to determine how to divide the proceeds of V2Gamong its contributors based on Nash bargaining theoryTheoutput of themodel specifies the optimumnumber of electricvehicles and the amount of money that must be paid by themicrogrid operator to the electric vehicle owners which hasbeen interpreted as the V2G service price Moreover thismodel can be used to determine the appropriate rate forsubsidies granted by the government to promote the purchaseof electric vehicles

The results of implementation of the proposed model inthe Manjil case in Iran indicate that to encourage the publicto buy electric vehicles it is essential that the governmentcompensate much of the cost difference between conven-tional and electric vehicles in the form of subsidies Theremaining difference can be compensated by the revenues ofthe electric vehicle owner from V2G service The microgridoperator will earn a profit from a reduction in the amountof investment necessary for supplying critical load by dieselgenerator in emergencies This profit will be shared with thevehicle owners

This paper has analyzed the V2G service pricing problemfor a case in which the vehicle batteries are utilized onlywhenever the microgrid goes into island mode Since such asituation rarely happens in a year vehicle owners will not beinconvenienced and battery depreciation cost due to frequentchargedischarge cycles will be negligible For future studyone could also develop a model for V2G service pricing in

8 Mathematical Problems in Engineering

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

700Cr

itica

l loa

d (k

W)

(a)

0

100

200

300

400

500

600

700

Tota

l win

d po

wer

(kW

)

1 2 3 4 5 60Time (hour)

(b)

No V2GV2G

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

Die

sel g

ener

ator

pow

er (k

W)

(c)

1 2 3 4 5 60Time (hour)

minus400

minus200

0

200

400

600

V2G

pow

er (k

W)

(d)

Figure 4 Simulation results for a windy isolation period

the case where V2G is applied every day of the year Prior tofinding amodel that considers the interests of all players of thegame policymakers cannot count on the success of vehicle-to-grid idea in such a case

Appendix

This appendix proves that the two-stage model proposedin this paper is equivalent to simultaneous maximizationof Nash product with respect to both variables The termspresented in (2) are expressed in other words in

max119873119901

MGEP (119873 119901) times EVEP (119873 119901) (A1)

where MGEP(119873 119901) and EVEP(119873 119901) denote the excess pay-offs of the microgrid operator and electric vehicles unionrespectively and can be written as follows

MGEP (119873 119901) = MGCbase minusMGCV2G minus 119873 times 119901EVEP (119873 119901) = 119873 times CVC minus 119873 times EVC + 119873 times 119901 (A2)

To maximize the Nash product presented in (A1) itspartial derivatives with respect to both119873 and 119901must be zerosimultaneously So we have

120597 (MGEP (119873 119901) times EVEP (119873 119901))120597119901 = 0 (A3)

120597 (MGEP (119873 119901) times EVEP (119873 119901))120597119873 = 0 (A4)

From (A3) we have

minus 119873 timesMGEP (119873 119901) times +119873 times EVEP (119873 119901) = 0 997904rArrMGEP (119873 119901) = EVEP (119873 119901) (A5)

And from (A4)

120597MGEP (119873 119901)120597119873 times EVEP (119873 119901) + 120597EVEP (119873 119901)120597119873timesMGEP (119873 119901) = 0 997904rArr(minus120597MGCV2G (119873)120597119873 minus 119901) times EVEP (119873 119901)+ (CVC minus EVC + 119901) timesMGEP (119873 119901) = 0

(A6)

Mathematical Problems in Engineering 9

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600Cr

itica

l loa

d (k

W)

(a)

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

Tota

l win

d po

wer

(kW

)

(b)

No V2GV2G

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

700

Die

sel g

ener

ator

pow

er (k

W)

(c)

1 2 3 4 5 60Time (hour)

minus600

minus400

minus200

0

200

400

600

V2G

pow

er (k

W)

(d)

Figure 5 Simulation results for a low wind isolation period

According to (A5) in optimum of the Nash product func-tion the product terms are equal Hence we conclude

minus120597MGCV2G (119873)120597119873 minus 119901 + CVC minus EVC + 119901 = 0 997904rArr120597MGCV2G (119873)120597119873 minus CVC + EVC = 0

(A7)

The above equation is equivalent to (3) that had been pro-posed as the two-stage model Furthermore by substituting(A2) into (A5) the payment to the vehicles for V2Gservicemdashwhen the number of vehicles is optimalmdashwill be thesame as (4)

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan and A Kushler ldquoPlug-in electric vehicles challengesand opportunities American council for an energy-efficienteconomyrdquo 2013 httpwwwaceeeorgresearch-reportt133

[2] M Duvall and E Knipping Environmental Assessment of Plug-In Hybrid Electric Electric Power Research Institute (EPRI)2007

[3] A Bandyopadhyay L Wang V K Devabhaktuni and R CGreen ldquoAggregator analysis for efficient day-time charging ofPlug-in Hybrid Electric Vehiclesrdquo in Proceedings of the IEEEPower and Energy Society General Meeting pp 1ndash8 IEEEDetroit Mich USA July 2011

[4] K Clement E Haesen and J Driesen ldquoCoordinated chargingof multiple plug-in hybrid electric vehicles in residential dis-tribution gridsrdquo in Proceedings of the IEEEPES Power SystemsConference and Exposition (PSCE rsquo09) pp 1ndash7 IEEE SeattleWash USA March 2009

[5] International Energy Agency World Energy Outlook 2011httpwwwieaorgpublicationsfreepublicationspublicationWEO2011 WEBpdf

[6] International Energy Agency Global EV Outlook Understand-ing the Electric Vehicle Landscape to 2020 2013 httpwwwieaorgpublicationsfreepublicationspublicationGlobalEV-Outlook 2013pdf

[7] H Lee and G Lovellette ldquoWill electric cars transform the USvehicle market An analysis of the key determinantsrdquo Discus-sion Paper 2011-08 Belfer Center for Science and InternationalAffairs Cambridge Mass USA 2011

[8] C Hay M Togeby N C Bang C Sondergren and L HHansen Introducing Electric Vehicles into the Current ElectricityMarkets EDISON Consortium 2010

Mathematical Problems in Engineering 3

where bargainers sign a binding contract once an agreementis struck

The Nash solution of a bargaining game is a mutuallybeneficial agreement that maximizes the product of playersrsquoexcess payoffs due to agreement In other words considering1198751 and 1198752 as the payoffs of players when there is no agreement1198751 and 1198752 will be the Nash solution of the bargaining game ifand only if they maximize the term in

max11987511198752(1198751 minus 1198751) times (1198752 minus 1198752) (1)

The Nash solution is Pareto optimal so there is no agree-ment in which the utilities of both players are simultaneouslymore than 1198751 and 119875232 The Two-Stage Model Formulation If we want to applyNash bargaining theory to the V2G pricing problem in amicrogrid we have to determine the payoffs of both micro-grid operator and vehicle owners union with and withoutV2G Payoffs can be expressed by monetary unit Time valueof money should be taken into consideration Costs as wellas revenues form a cash flow and are discounted using aninterest rate for an analysis period

The formula for Nash bargaining is presented in

max119873119901(MGCbase minusMGCV2G minus 119873 times 119901)

times (119873 times CVC minus 119873 times EVC + 119873 times 119901) (2)

where MGCbase is the net present cost of investment andoperation of the microgrid without V2G MGCV2G is the netpresent cost of investment and operation of the microgridutilizing V2G 119901 is the present value of payment to the vehicleowners for V2G service by the microgrid operator 119873 is thenumber of electric vehicles that participate in V2G CVC isthe net present cost for buying and keeping a conventionalgasoline-powered vehicle (including fuel cost) and EVC isthe net present cost for buying and keeping an electric vehicle(including fuel cost)

In (2) 119873 and 119901 are decision variables Therefore thenumber of vehicles and V2G price should be determinedsimultaneously to maximize the Nash product Alternativelydecision variables can be determined sequentially by a two-stage model in which the optimum number of electric vehi-cles for V2G is appointed first through minimizing the termdenoted in (3) Then the Nash product in (2) is maximizedwith respect to 119901 while 119873 is assumed to be fixed It can beproved that both alternativemethods have similar results (seeAppendix) In (3) the total cost of themicrogrid operator andelectric vehicle union for replacing vehicles and utilizingV2Gis minimizedThis means that players should consider globalutility to set119873 and then bargain over 119901

min119873(MGCV2G (119873) + 119873 times (EVC minus CVC)) (3)

By solving (2)with respect to119901 the payment to the vehicleowners for V2G service will be specified as presented in

119901= 12 (MGCbase minusMGCV2G + 119873 times EVC minus 119873 times CVC) (4)

Hence the proper V2G price will be revealed dependent onthe pricing scheme and regarding 119901 as the present value oftotal payment for V2G to each electric vehicle

4 Case Study A Microgrid in Manjil Iran

Manjil a city located in the north of Iran has wind farmsthat deliver electricity to the national grid Commonly facingpower interruptions feasibility studies have been conductedsince 2010 to investigate if the Manjil distribution networkcould supply its local critical load when it is isolated fromthe national grid in rare events that may occur a few hours ayear Local wind turbines were presumed as one of the powergeneration sources in the incoming microgrid Intermittencyand variability of wind power oblige the utilization of acomplementary diesel generator to maintain a steady balanceof supply and demand Determining the size of the dieselgenerator for the microgrid in Manjil was a multiobjectivedecision-making problemwhose goalwas finding the optimalMW rating that ensures the reliability of the microgrid withminimum cost The problem has been discussed in [41] indetail

As mentioned in the previous sections V2G can be con-sidered as an alternative energy source which will decreasethe diesel generator size and consequently the investmentcost of the microgrid If the Manjil microgrid wants V2Gto be applied in the island mode of operation the paymentto vehicle owners should be agreed so that it is mutuallybeneficial Hence the previously introduced two-stagemodelhas been utilized to compute the V2G price It should benoted that the pricing scheme proposed in this paper can beapplied to various cases other than Manjil too

According to the first stage the optimum number ofelectric vehicles participating in V2G must be calculated atthe beginningThe reducedMWrating of the diesel generatoris derived in the same stage to be compared with the initialrating Therefore two cases have been analyzed for planningof Manjil microgrid in island mode of operation Both casesuse two 660 kW Vestas wind turbines Case 1 is the basecase in which a diesel generator adjoins when the microgridis isolated from grid The size of the diesel generator is theonly decision variable in the design of the base case In Case2 electric vehicles service the microgrid thus the decisionvariables in this case are the number of vehicles and themodified size of the diesel generator Figure 1 demonstratesthe configuration of the Manjil microgrid

Wind speed data for Manjil has been recorded in 10-minute time steps in 2009 The time series for wind powergeneration is derived from wind speed data and the powercurve of the turbine generatorsTheyearly peak load inManjilis about 5MW but in island modemdashwhich may last up to 6

4 Mathematical Problems in Engineering

Circuit breaker

Grid

Microgridoperator

Diesel generatorWind turbine 1 Wind turbine 2

Critical loadElectric vehicle 1Electric vehicle 2

Monitoring amp control

Electric vehicle N

Figure 1 Configuration of Manjil microgrid

hoursmdashonly the critical load approximately 20 of the totalload must be supplied

Sizing of microgrid components is dealt with as a mul-tiobjective problem by the NSGA-II algorithm in MATLABReliability and cost are the two objectives that form the axis ofPareto front (a set of solutions with no other solution whichcan improve at least one of the objectives without degradingany other objective [42]) as output of the program ldquoSystemminutesrdquo is selected as the reliability index The operatingcost of the diesel generator is negligible since the dieselgenerator is only operated in island mode and isolation ofManjil from the grid is infrequent and of a short durationSo its operating cost is much less than its investment costand can be neglected Operating cost of the wind turbines isnegligible too The investment cost of wind turbines is notconsidered in the planning problem because they are presentin the Manjil wind farm now and are operated all year longAll other costs are accounted and transferred in cash flow tothe present value

The optimum value for decision variables in each casehas been found using a hybrid method which is a mixture ofsimulation andmultiobjective optimizationThe advantage ofsimulation as stated in [43] is studying the details that cannotbe easily considered by other methods If the alternatives orpotential solutions are limited the most appropriate way tofind the best solution is to simulate all of them But when thesolution space is extended it would be too time consumingto simulate all possible solutions In such circumstances aheuristic search of the solution space is beneficial

One of the methods that can intelligently search thesolution space is the genetic algorithm which is compatiblewith multiobjective optimization NSGA-II is a controlledelitist genetic algorithm As it is elitist it prefers solutionswith better fitness values As it is controlled it also favorsthe solutions that increase the diversity of the population

Diversity of population helps the algorithm converge to anoptimal Pareto front

Thus the steps of the algorithm for finding the sizeparameters can be summarized as follows

(1) Random generating of an initial population for deci-sion variables (initial values for size of diesel generatorand number of electric vehicles)

(2) Simulating the operation of the microgrid withdefined decision variables and evaluating the fitnessof solutions according to prescribed criteria (reliabil-ity and cost)

(3) Investigating the termination criteria and moving toStep (5) if they have been satisfied

(4) Intelligent selection of next generation of solutionswith genetic algorithm and returning to Step (2)

(5) Plotting the optimal Pareto front

Simulation of each solution which is equivalent to aspecific design is performed with 10-minute time steps fora 6-hour interval in which the microgrid is operated in islandmode Due to the random nature of wind speed and itsdifferent patterns in various months of the year four 6-hourintervals are randomly picked for each month and the fitnessvalues would be eventually obtained from the average of 48present scenariosThe algorithm termination criteria includea combination of the maximum number of generations timeconstraints and lack of significant improvement in fitnessvalues The output of the program is a Pareto front for eachcase whose horizontal axis denotes cost and whose verticalaxis denotes the reliability index All points on the Paretofront may be viewed as an optimal point For final decisionmaking one can set the desired reliability index and comparethe present values of total cost associated with each case

Mathematical Problems in Engineering 5

To simulate the operation of the microgrid as the secondstep in the algorithm used to find the optimum numberof electric vehicles it is necessary to model the microgridcomponents (ie diesel generator wind turbines electricvehicles and critical load) and the relationships betweenthese components As mentioned before for the powergenerated by wind turbines and the power consumed bycritical load a simulated time series vector has been utilizedFor electric vehicles their batteries are considered as energystorage systems that can be charged with at most 3 kW powerfrom the grid and deliver at most the same power to the gridwhen necessary The rating of vehicle batteriesrsquo energy sizeis assumed to be 12 kWh Both charging efficiency (120578119888) anddischarging efficiency (120578119889) are considered to be 095 whichresults in a round-trip efficiency of approximately 09

Regarding the interaction between electric vehicles andmicrogrid operator it is also accepted thatV2Gwould only beused when themicrogrid switches to islandmode Hence theconvenience of electric vehicle owners will not be influencedexcept for a limited number of days when the microgrid isdisconnected from the bulk power grid During the periodof disconnection the battery chargedischarge control ofthe electric vehicles that have contracted to participate inV2G and are available to the grid will be performed by themicrogrid operator Available vehicles are those which areparked and connected to the grid The availability indexwhich has been defined as the ratio of available vehicles tothe total vehicles under contract is 70 percent In [11 1644] some statistical information has been provided abouthow many hours the cars are parked in a day The requiredcommunication and control infrastructure for monitoringdata acquisition and sending control commands to electricvehicles should be provided by the microgrid operator

In simulating the microgrid operation critical load levelandwind speed have been considered as exogenous variablesWhat the microgrid operator has under his control includesthe amount of power generated by the diesel generatorand the chargedischarge rate of power tofrom the vehiclebatteries The strategy applied in this paper to adjust chargelevels of vehicle batteries is intended to maximize the energyreserved in the batteries Therefore in every time step inwhich the vehicle batteries are not fully charged and totaloutput power of the wind turbines plus maximum power ofthe diesel generator exceeds the critical load level batterieswill be charged This helps sufficient energy to be availablein the batteries for the next time steps with a probableshortage in wind power Hence a new parameter has beendefined called the power absorption capability (PAC) ofvehicle batteries whose quantity in each time step is obtainedfrom

PAC (119905) = 119873sum119894=1

min(119875119887119894 (1 minus SOC119894 (119905)) times 119864119887119894120578119888 times 119879 ) (5)

where119873 is the number of electric vehicles 119875119887119894 is the nominalpower of the 119894th battery converter in kW SOC119894(119905) is thestate of charge for 119894th vehicle battery at time step 119905 119864119887119894 isthe capacity of 119894th vehicle battery in kWh 120578119888 is the charging

efficiency of 119894th vehicle battery and119879 is the time step durationin ℎ

At this stage if total power generation of wind turbines isgreater than critical load plus PAC(119905) the wind turbines willbe disconnected one by one until the consumption surpassesthe generationThen the rate of power thatmust be generatedby the diesel generator can be obtained from

119875119889 (119905) = min(119875119889max 119871 (119905) minus 119873119908sum

119894=1

119875119908119894 (119905) + PAC (119905)) (6)

where 119875119889(119905) is the power generated by the diesel generator attime step 119905 in kW 119875119889max

is the maximum power of the dieselgenerator in kW 119871(119905) is the amount of critical load at timestep 119905 in kW 119873119908 is the number of wind turbines that arestill connected and 119875119908119894(119905) is the power generated by 119894th windturbine at time step 119905 in kW

Afterwards the microgrid operator will determine thepower which should be exchanged with the electric vehiclesunion from

119875V2G (119905) = 119871 (119905) minus119873119908sum119894=1

119875119908119894 (119905) + 119875119889 (119905) (7)

When 119875V2G(119905) is positive it means that the power generatedby the diesel generator and wind turbines is not sufficient forsupplying load so the microgrid operator orders the electricvehicles to deliver electricity to the microgrid Neverthelessit is possible that some part of the load has to be interruptedThe amount of interrupted load may be calculated from(8) When 119875V2G(119905) is zero the batteries of electric vehiclesconnected to the grid are fully charged and when 119875V2G(119905)is negative it means that vehicle batteries can be chargedbecause of adequate power supply An allocation scheme for119875V2G(119905) among electric vehicles is beyond the scope of thispaper

ENS (119905) = 119871 (119905) minus 119873119908sum119894=1

119875119908119894 (119905) + 119875119889 (119905) + 119875V2G (119905) (8)

In the above equation ENS(119905) is the energy not suppliedat time step 119905 in kWhThe total energy not supplied during alltime spans for 48 simulation scenarios forms the reliabilityindex as one of the design criteria Total cost the seconddecision criterion is obtained from

TC = 119875119889maxtimes 120587119889 + 119873 times 120587EV (9)

where TC is the total cost in $ 120587119889 is the diesel generatorinvestment unit price which is supposed to be $800kW and120587EV is the surplus cost of an electric vehicle with respect toa conventional gasoline-powered vehicle in $ with regard togovernment subsidies The reasons for not considering otherparameters in determining the total cost had been discussedabove The required cost for installing the communicationand control infrastructure to implement V2G has beenaccounted in 120587EV

To compare the cash flows of conventional and electricvehicles it should be noted that the lifetime of an automobile

6 Mathematical Problems in Engineering

in Iran is about 20 years The lifetime of the diesel generatorand analysis time span is assumed to be 20 years too Anelectric vehicle is almost $6000 more expensive to buy andneeds battery replacement in the tenth year It is assumedthat the cost of battery replacement is $6000 and the cost ofestablishing the infrastructure for V2G is $1000 per vehicleOn the other hand a typical PHEV consumes approximately005 literkm In comparison with a 008 literkm fuel con-sumption rate for a conventional vehicle and assuming anaverage distance traveled of 30000 kilometers per year therewill be a 900-liter-per-year differenceThe gas price in Iran isabout $03liter which gives a $270 per year cost reductionby replacing a conventional vehicle with an electric oneTherefore 120587EV is equivalent to the present value of the cashflow as shown in the following equation

120587EV = 120587119904 + 120587119894 + 120587119887 times 1(1 + 119894)10 minus 120587119891 times(1 + 119894)20 minus 1119894 times (1 + 119894)20 (10)

where 120587119904 is the price difference between an electric vehicleand a conventional one 120587119894 is the V2G infrastructure cost pervehicle 120587119887 is the battery replacement cost 120587119891 is the annualcost reduction due to fuel saving of an electric vehicle and 119894 isthe discount rate in Iran thatmay reasonably be approximatedwith 20 percent in 2014 Hence 120587EV in this case is about$6654

5 Results

The results of the analysis for Manjil show that utilizationof electric vehicles with the current price difference betweenconventional cars and electric carsmdashwhich is mainly dueto battery costsmdashis not economically attractive even whenconsidering the benefits of V2G One can only hope thatwidespread use of electric vehicles in the short term maybe a possibility if the government provides a subsidy forbuying electric vehicles in order to protect the environmentand promote green technologies Unfortunately accordingto calculations made in this study any subsidy less than$5000 per electric vehicle would not encourage the publicto buy an electric car Figure 2 demonstrates the optimalPareto fronts of microgrid sizing plans for various scenariosThe horizontal axis in this figure illustrates the equivalentpresent value of the plan while the vertical axis representsthe reliability index expressed in system minutes

Figure 2 shows four scenarios in the first one vehiclebatteries are not utilized and the Pareto front has beenextracted from various designs with different diesel generatorpower sizes In other scenarios vehicle batteries are used tosupply load along with a diesel generator as the complementsto wind turbines In one of these three cases there is nogovernment subsidy but in the other two scenarios thesubsidies equal to $5000 and $5500 have been assumedIn Figure 3 a scenario with a $6000 government subsidyis added to the previous figure If the government subsidywas more than $6654 buying an electric vehicle becameeconomically attractive even without receiving V2G incomeThe results of analyzing this case indicate that considering

No V2GV2G no subsidy

V2G $5000 subsidyV2G $5500 subsidy

0

50

100

150

200

250

300

Syste

m (m

inut

es)

750 760 770 780 790 800 810 820 830740Total cost (times$1000)

Figure 2 Optimal Pareto fronts of microgrid sizing plan for variousscenarios

No V2GV2G no subsidyV2G $5000 subsidy

V2G $5500 subsidyV2G $6000 subsidy

500 550 600 650 700 750 800 850450Total cost (times$1000)

0

50

100

150

200

250

300Sy

stem

(min

utes

)

Figure 3 Optimal Pareto fronts with a new scenario $6000 subsidy

the proceeds of V2G a $6000 subsidy is an appropriatedecision

One way to attain a certain design for sizing of microgridcomponents among the optimal options on Pareto front is tospecify a minimum requirement for reliability In this paperthe system minutes index is supposed to be less than 30minutes in a year as a decision criterion Accordingly Table 1has been obtained for various scenarios of the problem

As it is shown in Table 1 a subsidy of $6000 has alarge impact on the optimal design of the system and theinvestment cost needed to meet the load In the rest ofthis paper it has been assumed that this amount of subsidywill be granted from the government to everyone who buysan electric vehicle Therefore for determining the price ofV2G services via the Nash bargaining theory two cases havebeen studied and compared first operation of the microgridwithout V2G and second operation of the microgrid with

Mathematical Problems in Engineering 7

Table 1 Size cost and reliability indices for various scenarios

Scenario No V2G V2G (no subsidy) V2G ($5000 subsidy) V2G ($5500 subsidy) V2G ($6000 subsidy)System minutes 292 298 290 290 299Microgrid inv cost ($) 801100 801000 663096 635281 44543Total cost ($) 801100 801000 797070 749527 461795Diesel generator size (MW) 9750 9749 8050 7708 533Number of PHEVs 0 0 81 99 638

Table 2 Results for V2G service pricing

Equivalent present value of total V2G service price $5869045Equivalent present value of V2G service price pervehicle

$919913

Equivalent annual payment to each vehicle for V2Gservice

$18891

Excess payoff of microgrid operator due to V2G $1696525

Excess payoff of each vehicle owner due to V2G $265913

Payment to each vehicle per hour available $449786

V2G and the government subsidy ($6000) In Figures 4(a)ndash4(d) the results of simulating the system during a six-hourinterval of isolation from grid for both cases in a windycondition are depicted Figures 5(a)ndash5(d) display the resultsof the simulation for both cases in a low wind six-hour timeinterval

As can be seen in Figures 4 and 5 when only thediesel generator is used to follow the difference between thepower generated by wind turbines and the power consumedby load a high capacity of diesel generator is needed tobe installed When chargedischarge power of the vehiclebatteries follows the load a low capacity of diesel generatoris required and it will work with an almost smooth outputpower When the electric power generated by wind turbinesis high the batteriesrsquo state of charge will be approximatelyinvariant However when the wind speed drops the vehiclebatteries will gradually discharge The possibility of storingextra power generated by wind turbines in vehicle batterieseliminates cutting of wind turbines due to excess powerin island mode This enhances the energy efficiency in themicrogrid

Using (4) and the results listed in Table 1 the price ofvehicle-to-grid service revenues and profits of themicrogridoperator and electric vehicle owners from V2G agreement isshown in Table 2

In Table 2 the first row represents the equivalent presentvalue of the total payment by the microgrid operator to allvehicle owners during a 20-year time interval for participat-ing in the vehicle-to-grid programThe second row indicatesthe same parameter per vehicle while the price in the thirdrow is the equivalent annual payment by the microgridoperator to each vehicle during the 20-year period given adiscount rate equal to 20 Excess payoffs of the microgrid

operator and vehicle owners due to V2G implementation arelisted as equivalent present values In the last row it has beenassumed that the pricing scheme is based on the hours thatan electric vehicle is available for V2G when it is called forPercentage of available vehicles is supposed to be 70 and itis assumed that isolation from grid takes place for on average6 hours a year Hence a vehicle owner would be paid about$45 for one-hour availability

6 Conclusion

The utilization of batteries of electric vehicles as energystorage systems can help microgrids in supplying load whenthey become isolated from the grid This service which isknown as V2G requires the active participation of electricvehicle owners To develop and maintain such partnershipsthe interests of both microgrid operators and vehicle ownershould be considered Otherwise the participants wouldnot have sufficient motivation This paper has proposed amodel to determine how to divide the proceeds of V2Gamong its contributors based on Nash bargaining theoryTheoutput of themodel specifies the optimumnumber of electricvehicles and the amount of money that must be paid by themicrogrid operator to the electric vehicle owners which hasbeen interpreted as the V2G service price Moreover thismodel can be used to determine the appropriate rate forsubsidies granted by the government to promote the purchaseof electric vehicles

The results of implementation of the proposed model inthe Manjil case in Iran indicate that to encourage the publicto buy electric vehicles it is essential that the governmentcompensate much of the cost difference between conven-tional and electric vehicles in the form of subsidies Theremaining difference can be compensated by the revenues ofthe electric vehicle owner from V2G service The microgridoperator will earn a profit from a reduction in the amountof investment necessary for supplying critical load by dieselgenerator in emergencies This profit will be shared with thevehicle owners

This paper has analyzed the V2G service pricing problemfor a case in which the vehicle batteries are utilized onlywhenever the microgrid goes into island mode Since such asituation rarely happens in a year vehicle owners will not beinconvenienced and battery depreciation cost due to frequentchargedischarge cycles will be negligible For future studyone could also develop a model for V2G service pricing in

8 Mathematical Problems in Engineering

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

700Cr

itica

l loa

d (k

W)

(a)

0

100

200

300

400

500

600

700

Tota

l win

d po

wer

(kW

)

1 2 3 4 5 60Time (hour)

(b)

No V2GV2G

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

Die

sel g

ener

ator

pow

er (k

W)

(c)

1 2 3 4 5 60Time (hour)

minus400

minus200

0

200

400

600

V2G

pow

er (k

W)

(d)

Figure 4 Simulation results for a windy isolation period

the case where V2G is applied every day of the year Prior tofinding amodel that considers the interests of all players of thegame policymakers cannot count on the success of vehicle-to-grid idea in such a case

Appendix

This appendix proves that the two-stage model proposedin this paper is equivalent to simultaneous maximizationof Nash product with respect to both variables The termspresented in (2) are expressed in other words in

max119873119901

MGEP (119873 119901) times EVEP (119873 119901) (A1)

where MGEP(119873 119901) and EVEP(119873 119901) denote the excess pay-offs of the microgrid operator and electric vehicles unionrespectively and can be written as follows

MGEP (119873 119901) = MGCbase minusMGCV2G minus 119873 times 119901EVEP (119873 119901) = 119873 times CVC minus 119873 times EVC + 119873 times 119901 (A2)

To maximize the Nash product presented in (A1) itspartial derivatives with respect to both119873 and 119901must be zerosimultaneously So we have

120597 (MGEP (119873 119901) times EVEP (119873 119901))120597119901 = 0 (A3)

120597 (MGEP (119873 119901) times EVEP (119873 119901))120597119873 = 0 (A4)

From (A3) we have

minus 119873 timesMGEP (119873 119901) times +119873 times EVEP (119873 119901) = 0 997904rArrMGEP (119873 119901) = EVEP (119873 119901) (A5)

And from (A4)

120597MGEP (119873 119901)120597119873 times EVEP (119873 119901) + 120597EVEP (119873 119901)120597119873timesMGEP (119873 119901) = 0 997904rArr(minus120597MGCV2G (119873)120597119873 minus 119901) times EVEP (119873 119901)+ (CVC minus EVC + 119901) timesMGEP (119873 119901) = 0

(A6)

Mathematical Problems in Engineering 9

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600Cr

itica

l loa

d (k

W)

(a)

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

Tota

l win

d po

wer

(kW

)

(b)

No V2GV2G

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

700

Die

sel g

ener

ator

pow

er (k

W)

(c)

1 2 3 4 5 60Time (hour)

minus600

minus400

minus200

0

200

400

600

V2G

pow

er (k

W)

(d)

Figure 5 Simulation results for a low wind isolation period

According to (A5) in optimum of the Nash product func-tion the product terms are equal Hence we conclude

minus120597MGCV2G (119873)120597119873 minus 119901 + CVC minus EVC + 119901 = 0 997904rArr120597MGCV2G (119873)120597119873 minus CVC + EVC = 0

(A7)

The above equation is equivalent to (3) that had been pro-posed as the two-stage model Furthermore by substituting(A2) into (A5) the payment to the vehicles for V2Gservicemdashwhen the number of vehicles is optimalmdashwill be thesame as (4)

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan and A Kushler ldquoPlug-in electric vehicles challengesand opportunities American council for an energy-efficienteconomyrdquo 2013 httpwwwaceeeorgresearch-reportt133

[2] M Duvall and E Knipping Environmental Assessment of Plug-In Hybrid Electric Electric Power Research Institute (EPRI)2007

[3] A Bandyopadhyay L Wang V K Devabhaktuni and R CGreen ldquoAggregator analysis for efficient day-time charging ofPlug-in Hybrid Electric Vehiclesrdquo in Proceedings of the IEEEPower and Energy Society General Meeting pp 1ndash8 IEEEDetroit Mich USA July 2011

[4] K Clement E Haesen and J Driesen ldquoCoordinated chargingof multiple plug-in hybrid electric vehicles in residential dis-tribution gridsrdquo in Proceedings of the IEEEPES Power SystemsConference and Exposition (PSCE rsquo09) pp 1ndash7 IEEE SeattleWash USA March 2009

[5] International Energy Agency World Energy Outlook 2011httpwwwieaorgpublicationsfreepublicationspublicationWEO2011 WEBpdf

[6] International Energy Agency Global EV Outlook Understand-ing the Electric Vehicle Landscape to 2020 2013 httpwwwieaorgpublicationsfreepublicationspublicationGlobalEV-Outlook 2013pdf

[7] H Lee and G Lovellette ldquoWill electric cars transform the USvehicle market An analysis of the key determinantsrdquo Discus-sion Paper 2011-08 Belfer Center for Science and InternationalAffairs Cambridge Mass USA 2011

[8] C Hay M Togeby N C Bang C Sondergren and L HHansen Introducing Electric Vehicles into the Current ElectricityMarkets EDISON Consortium 2010

4 Mathematical Problems in Engineering

Circuit breaker

Grid

Microgridoperator

Diesel generatorWind turbine 1 Wind turbine 2

Critical loadElectric vehicle 1Electric vehicle 2

Monitoring amp control

Electric vehicle N

Figure 1 Configuration of Manjil microgrid

hoursmdashonly the critical load approximately 20 of the totalload must be supplied

Sizing of microgrid components is dealt with as a mul-tiobjective problem by the NSGA-II algorithm in MATLABReliability and cost are the two objectives that form the axis ofPareto front (a set of solutions with no other solution whichcan improve at least one of the objectives without degradingany other objective [42]) as output of the program ldquoSystemminutesrdquo is selected as the reliability index The operatingcost of the diesel generator is negligible since the dieselgenerator is only operated in island mode and isolation ofManjil from the grid is infrequent and of a short durationSo its operating cost is much less than its investment costand can be neglected Operating cost of the wind turbines isnegligible too The investment cost of wind turbines is notconsidered in the planning problem because they are presentin the Manjil wind farm now and are operated all year longAll other costs are accounted and transferred in cash flow tothe present value

The optimum value for decision variables in each casehas been found using a hybrid method which is a mixture ofsimulation andmultiobjective optimizationThe advantage ofsimulation as stated in [43] is studying the details that cannotbe easily considered by other methods If the alternatives orpotential solutions are limited the most appropriate way tofind the best solution is to simulate all of them But when thesolution space is extended it would be too time consumingto simulate all possible solutions In such circumstances aheuristic search of the solution space is beneficial

One of the methods that can intelligently search thesolution space is the genetic algorithm which is compatiblewith multiobjective optimization NSGA-II is a controlledelitist genetic algorithm As it is elitist it prefers solutionswith better fitness values As it is controlled it also favorsthe solutions that increase the diversity of the population

Diversity of population helps the algorithm converge to anoptimal Pareto front

Thus the steps of the algorithm for finding the sizeparameters can be summarized as follows

(1) Random generating of an initial population for deci-sion variables (initial values for size of diesel generatorand number of electric vehicles)

(2) Simulating the operation of the microgrid withdefined decision variables and evaluating the fitnessof solutions according to prescribed criteria (reliabil-ity and cost)

(3) Investigating the termination criteria and moving toStep (5) if they have been satisfied

(4) Intelligent selection of next generation of solutionswith genetic algorithm and returning to Step (2)

(5) Plotting the optimal Pareto front

Simulation of each solution which is equivalent to aspecific design is performed with 10-minute time steps fora 6-hour interval in which the microgrid is operated in islandmode Due to the random nature of wind speed and itsdifferent patterns in various months of the year four 6-hourintervals are randomly picked for each month and the fitnessvalues would be eventually obtained from the average of 48present scenariosThe algorithm termination criteria includea combination of the maximum number of generations timeconstraints and lack of significant improvement in fitnessvalues The output of the program is a Pareto front for eachcase whose horizontal axis denotes cost and whose verticalaxis denotes the reliability index All points on the Paretofront may be viewed as an optimal point For final decisionmaking one can set the desired reliability index and comparethe present values of total cost associated with each case

Mathematical Problems in Engineering 5

To simulate the operation of the microgrid as the secondstep in the algorithm used to find the optimum numberof electric vehicles it is necessary to model the microgridcomponents (ie diesel generator wind turbines electricvehicles and critical load) and the relationships betweenthese components As mentioned before for the powergenerated by wind turbines and the power consumed bycritical load a simulated time series vector has been utilizedFor electric vehicles their batteries are considered as energystorage systems that can be charged with at most 3 kW powerfrom the grid and deliver at most the same power to the gridwhen necessary The rating of vehicle batteriesrsquo energy sizeis assumed to be 12 kWh Both charging efficiency (120578119888) anddischarging efficiency (120578119889) are considered to be 095 whichresults in a round-trip efficiency of approximately 09

Regarding the interaction between electric vehicles andmicrogrid operator it is also accepted thatV2Gwould only beused when themicrogrid switches to islandmode Hence theconvenience of electric vehicle owners will not be influencedexcept for a limited number of days when the microgrid isdisconnected from the bulk power grid During the periodof disconnection the battery chargedischarge control ofthe electric vehicles that have contracted to participate inV2G and are available to the grid will be performed by themicrogrid operator Available vehicles are those which areparked and connected to the grid The availability indexwhich has been defined as the ratio of available vehicles tothe total vehicles under contract is 70 percent In [11 1644] some statistical information has been provided abouthow many hours the cars are parked in a day The requiredcommunication and control infrastructure for monitoringdata acquisition and sending control commands to electricvehicles should be provided by the microgrid operator

In simulating the microgrid operation critical load levelandwind speed have been considered as exogenous variablesWhat the microgrid operator has under his control includesthe amount of power generated by the diesel generatorand the chargedischarge rate of power tofrom the vehiclebatteries The strategy applied in this paper to adjust chargelevels of vehicle batteries is intended to maximize the energyreserved in the batteries Therefore in every time step inwhich the vehicle batteries are not fully charged and totaloutput power of the wind turbines plus maximum power ofthe diesel generator exceeds the critical load level batterieswill be charged This helps sufficient energy to be availablein the batteries for the next time steps with a probableshortage in wind power Hence a new parameter has beendefined called the power absorption capability (PAC) ofvehicle batteries whose quantity in each time step is obtainedfrom

PAC (119905) = 119873sum119894=1

min(119875119887119894 (1 minus SOC119894 (119905)) times 119864119887119894120578119888 times 119879 ) (5)

where119873 is the number of electric vehicles 119875119887119894 is the nominalpower of the 119894th battery converter in kW SOC119894(119905) is thestate of charge for 119894th vehicle battery at time step 119905 119864119887119894 isthe capacity of 119894th vehicle battery in kWh 120578119888 is the charging

efficiency of 119894th vehicle battery and119879 is the time step durationin ℎ

At this stage if total power generation of wind turbines isgreater than critical load plus PAC(119905) the wind turbines willbe disconnected one by one until the consumption surpassesthe generationThen the rate of power thatmust be generatedby the diesel generator can be obtained from

119875119889 (119905) = min(119875119889max 119871 (119905) minus 119873119908sum

119894=1

119875119908119894 (119905) + PAC (119905)) (6)

where 119875119889(119905) is the power generated by the diesel generator attime step 119905 in kW 119875119889max

is the maximum power of the dieselgenerator in kW 119871(119905) is the amount of critical load at timestep 119905 in kW 119873119908 is the number of wind turbines that arestill connected and 119875119908119894(119905) is the power generated by 119894th windturbine at time step 119905 in kW

Afterwards the microgrid operator will determine thepower which should be exchanged with the electric vehiclesunion from

119875V2G (119905) = 119871 (119905) minus119873119908sum119894=1

119875119908119894 (119905) + 119875119889 (119905) (7)

When 119875V2G(119905) is positive it means that the power generatedby the diesel generator and wind turbines is not sufficient forsupplying load so the microgrid operator orders the electricvehicles to deliver electricity to the microgrid Neverthelessit is possible that some part of the load has to be interruptedThe amount of interrupted load may be calculated from(8) When 119875V2G(119905) is zero the batteries of electric vehiclesconnected to the grid are fully charged and when 119875V2G(119905)is negative it means that vehicle batteries can be chargedbecause of adequate power supply An allocation scheme for119875V2G(119905) among electric vehicles is beyond the scope of thispaper

ENS (119905) = 119871 (119905) minus 119873119908sum119894=1

119875119908119894 (119905) + 119875119889 (119905) + 119875V2G (119905) (8)

In the above equation ENS(119905) is the energy not suppliedat time step 119905 in kWhThe total energy not supplied during alltime spans for 48 simulation scenarios forms the reliabilityindex as one of the design criteria Total cost the seconddecision criterion is obtained from

TC = 119875119889maxtimes 120587119889 + 119873 times 120587EV (9)

where TC is the total cost in $ 120587119889 is the diesel generatorinvestment unit price which is supposed to be $800kW and120587EV is the surplus cost of an electric vehicle with respect toa conventional gasoline-powered vehicle in $ with regard togovernment subsidies The reasons for not considering otherparameters in determining the total cost had been discussedabove The required cost for installing the communicationand control infrastructure to implement V2G has beenaccounted in 120587EV

To compare the cash flows of conventional and electricvehicles it should be noted that the lifetime of an automobile

6 Mathematical Problems in Engineering

in Iran is about 20 years The lifetime of the diesel generatorand analysis time span is assumed to be 20 years too Anelectric vehicle is almost $6000 more expensive to buy andneeds battery replacement in the tenth year It is assumedthat the cost of battery replacement is $6000 and the cost ofestablishing the infrastructure for V2G is $1000 per vehicleOn the other hand a typical PHEV consumes approximately005 literkm In comparison with a 008 literkm fuel con-sumption rate for a conventional vehicle and assuming anaverage distance traveled of 30000 kilometers per year therewill be a 900-liter-per-year differenceThe gas price in Iran isabout $03liter which gives a $270 per year cost reductionby replacing a conventional vehicle with an electric oneTherefore 120587EV is equivalent to the present value of the cashflow as shown in the following equation

120587EV = 120587119904 + 120587119894 + 120587119887 times 1(1 + 119894)10 minus 120587119891 times(1 + 119894)20 minus 1119894 times (1 + 119894)20 (10)

where 120587119904 is the price difference between an electric vehicleand a conventional one 120587119894 is the V2G infrastructure cost pervehicle 120587119887 is the battery replacement cost 120587119891 is the annualcost reduction due to fuel saving of an electric vehicle and 119894 isthe discount rate in Iran thatmay reasonably be approximatedwith 20 percent in 2014 Hence 120587EV in this case is about$6654

5 Results

The results of the analysis for Manjil show that utilizationof electric vehicles with the current price difference betweenconventional cars and electric carsmdashwhich is mainly dueto battery costsmdashis not economically attractive even whenconsidering the benefits of V2G One can only hope thatwidespread use of electric vehicles in the short term maybe a possibility if the government provides a subsidy forbuying electric vehicles in order to protect the environmentand promote green technologies Unfortunately accordingto calculations made in this study any subsidy less than$5000 per electric vehicle would not encourage the publicto buy an electric car Figure 2 demonstrates the optimalPareto fronts of microgrid sizing plans for various scenariosThe horizontal axis in this figure illustrates the equivalentpresent value of the plan while the vertical axis representsthe reliability index expressed in system minutes

Figure 2 shows four scenarios in the first one vehiclebatteries are not utilized and the Pareto front has beenextracted from various designs with different diesel generatorpower sizes In other scenarios vehicle batteries are used tosupply load along with a diesel generator as the complementsto wind turbines In one of these three cases there is nogovernment subsidy but in the other two scenarios thesubsidies equal to $5000 and $5500 have been assumedIn Figure 3 a scenario with a $6000 government subsidyis added to the previous figure If the government subsidywas more than $6654 buying an electric vehicle becameeconomically attractive even without receiving V2G incomeThe results of analyzing this case indicate that considering

No V2GV2G no subsidy

V2G $5000 subsidyV2G $5500 subsidy

0

50

100

150

200

250

300

Syste

m (m

inut

es)

750 760 770 780 790 800 810 820 830740Total cost (times$1000)

Figure 2 Optimal Pareto fronts of microgrid sizing plan for variousscenarios

No V2GV2G no subsidyV2G $5000 subsidy

V2G $5500 subsidyV2G $6000 subsidy

500 550 600 650 700 750 800 850450Total cost (times$1000)

0

50

100

150

200

250

300Sy

stem

(min

utes

)

Figure 3 Optimal Pareto fronts with a new scenario $6000 subsidy

the proceeds of V2G a $6000 subsidy is an appropriatedecision

One way to attain a certain design for sizing of microgridcomponents among the optimal options on Pareto front is tospecify a minimum requirement for reliability In this paperthe system minutes index is supposed to be less than 30minutes in a year as a decision criterion Accordingly Table 1has been obtained for various scenarios of the problem

As it is shown in Table 1 a subsidy of $6000 has alarge impact on the optimal design of the system and theinvestment cost needed to meet the load In the rest ofthis paper it has been assumed that this amount of subsidywill be granted from the government to everyone who buysan electric vehicle Therefore for determining the price ofV2G services via the Nash bargaining theory two cases havebeen studied and compared first operation of the microgridwithout V2G and second operation of the microgrid with

Mathematical Problems in Engineering 7

Table 1 Size cost and reliability indices for various scenarios

Scenario No V2G V2G (no subsidy) V2G ($5000 subsidy) V2G ($5500 subsidy) V2G ($6000 subsidy)System minutes 292 298 290 290 299Microgrid inv cost ($) 801100 801000 663096 635281 44543Total cost ($) 801100 801000 797070 749527 461795Diesel generator size (MW) 9750 9749 8050 7708 533Number of PHEVs 0 0 81 99 638

Table 2 Results for V2G service pricing

Equivalent present value of total V2G service price $5869045Equivalent present value of V2G service price pervehicle

$919913

Equivalent annual payment to each vehicle for V2Gservice

$18891

Excess payoff of microgrid operator due to V2G $1696525

Excess payoff of each vehicle owner due to V2G $265913

Payment to each vehicle per hour available $449786

V2G and the government subsidy ($6000) In Figures 4(a)ndash4(d) the results of simulating the system during a six-hourinterval of isolation from grid for both cases in a windycondition are depicted Figures 5(a)ndash5(d) display the resultsof the simulation for both cases in a low wind six-hour timeinterval

As can be seen in Figures 4 and 5 when only thediesel generator is used to follow the difference between thepower generated by wind turbines and the power consumedby load a high capacity of diesel generator is needed tobe installed When chargedischarge power of the vehiclebatteries follows the load a low capacity of diesel generatoris required and it will work with an almost smooth outputpower When the electric power generated by wind turbinesis high the batteriesrsquo state of charge will be approximatelyinvariant However when the wind speed drops the vehiclebatteries will gradually discharge The possibility of storingextra power generated by wind turbines in vehicle batterieseliminates cutting of wind turbines due to excess powerin island mode This enhances the energy efficiency in themicrogrid

Using (4) and the results listed in Table 1 the price ofvehicle-to-grid service revenues and profits of themicrogridoperator and electric vehicle owners from V2G agreement isshown in Table 2

In Table 2 the first row represents the equivalent presentvalue of the total payment by the microgrid operator to allvehicle owners during a 20-year time interval for participat-ing in the vehicle-to-grid programThe second row indicatesthe same parameter per vehicle while the price in the thirdrow is the equivalent annual payment by the microgridoperator to each vehicle during the 20-year period given adiscount rate equal to 20 Excess payoffs of the microgrid

operator and vehicle owners due to V2G implementation arelisted as equivalent present values In the last row it has beenassumed that the pricing scheme is based on the hours thatan electric vehicle is available for V2G when it is called forPercentage of available vehicles is supposed to be 70 and itis assumed that isolation from grid takes place for on average6 hours a year Hence a vehicle owner would be paid about$45 for one-hour availability

6 Conclusion

The utilization of batteries of electric vehicles as energystorage systems can help microgrids in supplying load whenthey become isolated from the grid This service which isknown as V2G requires the active participation of electricvehicle owners To develop and maintain such partnershipsthe interests of both microgrid operators and vehicle ownershould be considered Otherwise the participants wouldnot have sufficient motivation This paper has proposed amodel to determine how to divide the proceeds of V2Gamong its contributors based on Nash bargaining theoryTheoutput of themodel specifies the optimumnumber of electricvehicles and the amount of money that must be paid by themicrogrid operator to the electric vehicle owners which hasbeen interpreted as the V2G service price Moreover thismodel can be used to determine the appropriate rate forsubsidies granted by the government to promote the purchaseof electric vehicles

The results of implementation of the proposed model inthe Manjil case in Iran indicate that to encourage the publicto buy electric vehicles it is essential that the governmentcompensate much of the cost difference between conven-tional and electric vehicles in the form of subsidies Theremaining difference can be compensated by the revenues ofthe electric vehicle owner from V2G service The microgridoperator will earn a profit from a reduction in the amountof investment necessary for supplying critical load by dieselgenerator in emergencies This profit will be shared with thevehicle owners

This paper has analyzed the V2G service pricing problemfor a case in which the vehicle batteries are utilized onlywhenever the microgrid goes into island mode Since such asituation rarely happens in a year vehicle owners will not beinconvenienced and battery depreciation cost due to frequentchargedischarge cycles will be negligible For future studyone could also develop a model for V2G service pricing in

8 Mathematical Problems in Engineering

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

700Cr

itica

l loa

d (k

W)

(a)

0

100

200

300

400

500

600

700

Tota

l win

d po

wer

(kW

)

1 2 3 4 5 60Time (hour)

(b)

No V2GV2G

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

Die

sel g

ener

ator

pow

er (k

W)

(c)

1 2 3 4 5 60Time (hour)

minus400

minus200

0

200

400

600

V2G

pow

er (k

W)

(d)

Figure 4 Simulation results for a windy isolation period

the case where V2G is applied every day of the year Prior tofinding amodel that considers the interests of all players of thegame policymakers cannot count on the success of vehicle-to-grid idea in such a case

Appendix

This appendix proves that the two-stage model proposedin this paper is equivalent to simultaneous maximizationof Nash product with respect to both variables The termspresented in (2) are expressed in other words in

max119873119901

MGEP (119873 119901) times EVEP (119873 119901) (A1)

where MGEP(119873 119901) and EVEP(119873 119901) denote the excess pay-offs of the microgrid operator and electric vehicles unionrespectively and can be written as follows

MGEP (119873 119901) = MGCbase minusMGCV2G minus 119873 times 119901EVEP (119873 119901) = 119873 times CVC minus 119873 times EVC + 119873 times 119901 (A2)

To maximize the Nash product presented in (A1) itspartial derivatives with respect to both119873 and 119901must be zerosimultaneously So we have

120597 (MGEP (119873 119901) times EVEP (119873 119901))120597119901 = 0 (A3)

120597 (MGEP (119873 119901) times EVEP (119873 119901))120597119873 = 0 (A4)

From (A3) we have

minus 119873 timesMGEP (119873 119901) times +119873 times EVEP (119873 119901) = 0 997904rArrMGEP (119873 119901) = EVEP (119873 119901) (A5)

And from (A4)

120597MGEP (119873 119901)120597119873 times EVEP (119873 119901) + 120597EVEP (119873 119901)120597119873timesMGEP (119873 119901) = 0 997904rArr(minus120597MGCV2G (119873)120597119873 minus 119901) times EVEP (119873 119901)+ (CVC minus EVC + 119901) timesMGEP (119873 119901) = 0

(A6)

Mathematical Problems in Engineering 9

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600Cr

itica

l loa

d (k

W)

(a)

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

Tota

l win

d po

wer

(kW

)

(b)

No V2GV2G

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

700

Die

sel g

ener

ator

pow

er (k

W)

(c)

1 2 3 4 5 60Time (hour)

minus600

minus400

minus200

0

200

400

600

V2G

pow

er (k

W)

(d)

Figure 5 Simulation results for a low wind isolation period

According to (A5) in optimum of the Nash product func-tion the product terms are equal Hence we conclude

minus120597MGCV2G (119873)120597119873 minus 119901 + CVC minus EVC + 119901 = 0 997904rArr120597MGCV2G (119873)120597119873 minus CVC + EVC = 0

(A7)

The above equation is equivalent to (3) that had been pro-posed as the two-stage model Furthermore by substituting(A2) into (A5) the payment to the vehicles for V2Gservicemdashwhen the number of vehicles is optimalmdashwill be thesame as (4)

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan and A Kushler ldquoPlug-in electric vehicles challengesand opportunities American council for an energy-efficienteconomyrdquo 2013 httpwwwaceeeorgresearch-reportt133

[2] M Duvall and E Knipping Environmental Assessment of Plug-In Hybrid Electric Electric Power Research Institute (EPRI)2007

[3] A Bandyopadhyay L Wang V K Devabhaktuni and R CGreen ldquoAggregator analysis for efficient day-time charging ofPlug-in Hybrid Electric Vehiclesrdquo in Proceedings of the IEEEPower and Energy Society General Meeting pp 1ndash8 IEEEDetroit Mich USA July 2011

[4] K Clement E Haesen and J Driesen ldquoCoordinated chargingof multiple plug-in hybrid electric vehicles in residential dis-tribution gridsrdquo in Proceedings of the IEEEPES Power SystemsConference and Exposition (PSCE rsquo09) pp 1ndash7 IEEE SeattleWash USA March 2009

[5] International Energy Agency World Energy Outlook 2011httpwwwieaorgpublicationsfreepublicationspublicationWEO2011 WEBpdf

[6] International Energy Agency Global EV Outlook Understand-ing the Electric Vehicle Landscape to 2020 2013 httpwwwieaorgpublicationsfreepublicationspublicationGlobalEV-Outlook 2013pdf

[7] H Lee and G Lovellette ldquoWill electric cars transform the USvehicle market An analysis of the key determinantsrdquo Discus-sion Paper 2011-08 Belfer Center for Science and InternationalAffairs Cambridge Mass USA 2011

[8] C Hay M Togeby N C Bang C Sondergren and L HHansen Introducing Electric Vehicles into the Current ElectricityMarkets EDISON Consortium 2010

Mathematical Problems in Engineering 5

To simulate the operation of the microgrid as the secondstep in the algorithm used to find the optimum numberof electric vehicles it is necessary to model the microgridcomponents (ie diesel generator wind turbines electricvehicles and critical load) and the relationships betweenthese components As mentioned before for the powergenerated by wind turbines and the power consumed bycritical load a simulated time series vector has been utilizedFor electric vehicles their batteries are considered as energystorage systems that can be charged with at most 3 kW powerfrom the grid and deliver at most the same power to the gridwhen necessary The rating of vehicle batteriesrsquo energy sizeis assumed to be 12 kWh Both charging efficiency (120578119888) anddischarging efficiency (120578119889) are considered to be 095 whichresults in a round-trip efficiency of approximately 09

Regarding the interaction between electric vehicles andmicrogrid operator it is also accepted thatV2Gwould only beused when themicrogrid switches to islandmode Hence theconvenience of electric vehicle owners will not be influencedexcept for a limited number of days when the microgrid isdisconnected from the bulk power grid During the periodof disconnection the battery chargedischarge control ofthe electric vehicles that have contracted to participate inV2G and are available to the grid will be performed by themicrogrid operator Available vehicles are those which areparked and connected to the grid The availability indexwhich has been defined as the ratio of available vehicles tothe total vehicles under contract is 70 percent In [11 1644] some statistical information has been provided abouthow many hours the cars are parked in a day The requiredcommunication and control infrastructure for monitoringdata acquisition and sending control commands to electricvehicles should be provided by the microgrid operator

In simulating the microgrid operation critical load levelandwind speed have been considered as exogenous variablesWhat the microgrid operator has under his control includesthe amount of power generated by the diesel generatorand the chargedischarge rate of power tofrom the vehiclebatteries The strategy applied in this paper to adjust chargelevels of vehicle batteries is intended to maximize the energyreserved in the batteries Therefore in every time step inwhich the vehicle batteries are not fully charged and totaloutput power of the wind turbines plus maximum power ofthe diesel generator exceeds the critical load level batterieswill be charged This helps sufficient energy to be availablein the batteries for the next time steps with a probableshortage in wind power Hence a new parameter has beendefined called the power absorption capability (PAC) ofvehicle batteries whose quantity in each time step is obtainedfrom

PAC (119905) = 119873sum119894=1

min(119875119887119894 (1 minus SOC119894 (119905)) times 119864119887119894120578119888 times 119879 ) (5)

where119873 is the number of electric vehicles 119875119887119894 is the nominalpower of the 119894th battery converter in kW SOC119894(119905) is thestate of charge for 119894th vehicle battery at time step 119905 119864119887119894 isthe capacity of 119894th vehicle battery in kWh 120578119888 is the charging

efficiency of 119894th vehicle battery and119879 is the time step durationin ℎ

At this stage if total power generation of wind turbines isgreater than critical load plus PAC(119905) the wind turbines willbe disconnected one by one until the consumption surpassesthe generationThen the rate of power thatmust be generatedby the diesel generator can be obtained from

119875119889 (119905) = min(119875119889max 119871 (119905) minus 119873119908sum

119894=1

119875119908119894 (119905) + PAC (119905)) (6)

where 119875119889(119905) is the power generated by the diesel generator attime step 119905 in kW 119875119889max

is the maximum power of the dieselgenerator in kW 119871(119905) is the amount of critical load at timestep 119905 in kW 119873119908 is the number of wind turbines that arestill connected and 119875119908119894(119905) is the power generated by 119894th windturbine at time step 119905 in kW

Afterwards the microgrid operator will determine thepower which should be exchanged with the electric vehiclesunion from

119875V2G (119905) = 119871 (119905) minus119873119908sum119894=1

119875119908119894 (119905) + 119875119889 (119905) (7)

When 119875V2G(119905) is positive it means that the power generatedby the diesel generator and wind turbines is not sufficient forsupplying load so the microgrid operator orders the electricvehicles to deliver electricity to the microgrid Neverthelessit is possible that some part of the load has to be interruptedThe amount of interrupted load may be calculated from(8) When 119875V2G(119905) is zero the batteries of electric vehiclesconnected to the grid are fully charged and when 119875V2G(119905)is negative it means that vehicle batteries can be chargedbecause of adequate power supply An allocation scheme for119875V2G(119905) among electric vehicles is beyond the scope of thispaper

ENS (119905) = 119871 (119905) minus 119873119908sum119894=1

119875119908119894 (119905) + 119875119889 (119905) + 119875V2G (119905) (8)

In the above equation ENS(119905) is the energy not suppliedat time step 119905 in kWhThe total energy not supplied during alltime spans for 48 simulation scenarios forms the reliabilityindex as one of the design criteria Total cost the seconddecision criterion is obtained from

TC = 119875119889maxtimes 120587119889 + 119873 times 120587EV (9)

where TC is the total cost in $ 120587119889 is the diesel generatorinvestment unit price which is supposed to be $800kW and120587EV is the surplus cost of an electric vehicle with respect toa conventional gasoline-powered vehicle in $ with regard togovernment subsidies The reasons for not considering otherparameters in determining the total cost had been discussedabove The required cost for installing the communicationand control infrastructure to implement V2G has beenaccounted in 120587EV

To compare the cash flows of conventional and electricvehicles it should be noted that the lifetime of an automobile

6 Mathematical Problems in Engineering

in Iran is about 20 years The lifetime of the diesel generatorand analysis time span is assumed to be 20 years too Anelectric vehicle is almost $6000 more expensive to buy andneeds battery replacement in the tenth year It is assumedthat the cost of battery replacement is $6000 and the cost ofestablishing the infrastructure for V2G is $1000 per vehicleOn the other hand a typical PHEV consumes approximately005 literkm In comparison with a 008 literkm fuel con-sumption rate for a conventional vehicle and assuming anaverage distance traveled of 30000 kilometers per year therewill be a 900-liter-per-year differenceThe gas price in Iran isabout $03liter which gives a $270 per year cost reductionby replacing a conventional vehicle with an electric oneTherefore 120587EV is equivalent to the present value of the cashflow as shown in the following equation

120587EV = 120587119904 + 120587119894 + 120587119887 times 1(1 + 119894)10 minus 120587119891 times(1 + 119894)20 minus 1119894 times (1 + 119894)20 (10)

where 120587119904 is the price difference between an electric vehicleand a conventional one 120587119894 is the V2G infrastructure cost pervehicle 120587119887 is the battery replacement cost 120587119891 is the annualcost reduction due to fuel saving of an electric vehicle and 119894 isthe discount rate in Iran thatmay reasonably be approximatedwith 20 percent in 2014 Hence 120587EV in this case is about$6654

5 Results

The results of the analysis for Manjil show that utilizationof electric vehicles with the current price difference betweenconventional cars and electric carsmdashwhich is mainly dueto battery costsmdashis not economically attractive even whenconsidering the benefits of V2G One can only hope thatwidespread use of electric vehicles in the short term maybe a possibility if the government provides a subsidy forbuying electric vehicles in order to protect the environmentand promote green technologies Unfortunately accordingto calculations made in this study any subsidy less than$5000 per electric vehicle would not encourage the publicto buy an electric car Figure 2 demonstrates the optimalPareto fronts of microgrid sizing plans for various scenariosThe horizontal axis in this figure illustrates the equivalentpresent value of the plan while the vertical axis representsthe reliability index expressed in system minutes

Figure 2 shows four scenarios in the first one vehiclebatteries are not utilized and the Pareto front has beenextracted from various designs with different diesel generatorpower sizes In other scenarios vehicle batteries are used tosupply load along with a diesel generator as the complementsto wind turbines In one of these three cases there is nogovernment subsidy but in the other two scenarios thesubsidies equal to $5000 and $5500 have been assumedIn Figure 3 a scenario with a $6000 government subsidyis added to the previous figure If the government subsidywas more than $6654 buying an electric vehicle becameeconomically attractive even without receiving V2G incomeThe results of analyzing this case indicate that considering

No V2GV2G no subsidy

V2G $5000 subsidyV2G $5500 subsidy

0

50

100

150

200

250

300

Syste

m (m

inut

es)

750 760 770 780 790 800 810 820 830740Total cost (times$1000)

Figure 2 Optimal Pareto fronts of microgrid sizing plan for variousscenarios

No V2GV2G no subsidyV2G $5000 subsidy

V2G $5500 subsidyV2G $6000 subsidy

500 550 600 650 700 750 800 850450Total cost (times$1000)

0

50

100

150

200

250

300Sy

stem

(min

utes

)

Figure 3 Optimal Pareto fronts with a new scenario $6000 subsidy

the proceeds of V2G a $6000 subsidy is an appropriatedecision

One way to attain a certain design for sizing of microgridcomponents among the optimal options on Pareto front is tospecify a minimum requirement for reliability In this paperthe system minutes index is supposed to be less than 30minutes in a year as a decision criterion Accordingly Table 1has been obtained for various scenarios of the problem

As it is shown in Table 1 a subsidy of $6000 has alarge impact on the optimal design of the system and theinvestment cost needed to meet the load In the rest ofthis paper it has been assumed that this amount of subsidywill be granted from the government to everyone who buysan electric vehicle Therefore for determining the price ofV2G services via the Nash bargaining theory two cases havebeen studied and compared first operation of the microgridwithout V2G and second operation of the microgrid with

Mathematical Problems in Engineering 7

Table 1 Size cost and reliability indices for various scenarios

Scenario No V2G V2G (no subsidy) V2G ($5000 subsidy) V2G ($5500 subsidy) V2G ($6000 subsidy)System minutes 292 298 290 290 299Microgrid inv cost ($) 801100 801000 663096 635281 44543Total cost ($) 801100 801000 797070 749527 461795Diesel generator size (MW) 9750 9749 8050 7708 533Number of PHEVs 0 0 81 99 638

Table 2 Results for V2G service pricing

Equivalent present value of total V2G service price $5869045Equivalent present value of V2G service price pervehicle

$919913

Equivalent annual payment to each vehicle for V2Gservice

$18891

Excess payoff of microgrid operator due to V2G $1696525

Excess payoff of each vehicle owner due to V2G $265913

Payment to each vehicle per hour available $449786

V2G and the government subsidy ($6000) In Figures 4(a)ndash4(d) the results of simulating the system during a six-hourinterval of isolation from grid for both cases in a windycondition are depicted Figures 5(a)ndash5(d) display the resultsof the simulation for both cases in a low wind six-hour timeinterval

As can be seen in Figures 4 and 5 when only thediesel generator is used to follow the difference between thepower generated by wind turbines and the power consumedby load a high capacity of diesel generator is needed tobe installed When chargedischarge power of the vehiclebatteries follows the load a low capacity of diesel generatoris required and it will work with an almost smooth outputpower When the electric power generated by wind turbinesis high the batteriesrsquo state of charge will be approximatelyinvariant However when the wind speed drops the vehiclebatteries will gradually discharge The possibility of storingextra power generated by wind turbines in vehicle batterieseliminates cutting of wind turbines due to excess powerin island mode This enhances the energy efficiency in themicrogrid

Using (4) and the results listed in Table 1 the price ofvehicle-to-grid service revenues and profits of themicrogridoperator and electric vehicle owners from V2G agreement isshown in Table 2

In Table 2 the first row represents the equivalent presentvalue of the total payment by the microgrid operator to allvehicle owners during a 20-year time interval for participat-ing in the vehicle-to-grid programThe second row indicatesthe same parameter per vehicle while the price in the thirdrow is the equivalent annual payment by the microgridoperator to each vehicle during the 20-year period given adiscount rate equal to 20 Excess payoffs of the microgrid

operator and vehicle owners due to V2G implementation arelisted as equivalent present values In the last row it has beenassumed that the pricing scheme is based on the hours thatan electric vehicle is available for V2G when it is called forPercentage of available vehicles is supposed to be 70 and itis assumed that isolation from grid takes place for on average6 hours a year Hence a vehicle owner would be paid about$45 for one-hour availability

6 Conclusion

The utilization of batteries of electric vehicles as energystorage systems can help microgrids in supplying load whenthey become isolated from the grid This service which isknown as V2G requires the active participation of electricvehicle owners To develop and maintain such partnershipsthe interests of both microgrid operators and vehicle ownershould be considered Otherwise the participants wouldnot have sufficient motivation This paper has proposed amodel to determine how to divide the proceeds of V2Gamong its contributors based on Nash bargaining theoryTheoutput of themodel specifies the optimumnumber of electricvehicles and the amount of money that must be paid by themicrogrid operator to the electric vehicle owners which hasbeen interpreted as the V2G service price Moreover thismodel can be used to determine the appropriate rate forsubsidies granted by the government to promote the purchaseof electric vehicles

The results of implementation of the proposed model inthe Manjil case in Iran indicate that to encourage the publicto buy electric vehicles it is essential that the governmentcompensate much of the cost difference between conven-tional and electric vehicles in the form of subsidies Theremaining difference can be compensated by the revenues ofthe electric vehicle owner from V2G service The microgridoperator will earn a profit from a reduction in the amountof investment necessary for supplying critical load by dieselgenerator in emergencies This profit will be shared with thevehicle owners

This paper has analyzed the V2G service pricing problemfor a case in which the vehicle batteries are utilized onlywhenever the microgrid goes into island mode Since such asituation rarely happens in a year vehicle owners will not beinconvenienced and battery depreciation cost due to frequentchargedischarge cycles will be negligible For future studyone could also develop a model for V2G service pricing in

8 Mathematical Problems in Engineering

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

700Cr

itica

l loa

d (k

W)

(a)

0

100

200

300

400

500

600

700

Tota

l win

d po

wer

(kW

)

1 2 3 4 5 60Time (hour)

(b)

No V2GV2G

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

Die

sel g

ener

ator

pow

er (k

W)

(c)

1 2 3 4 5 60Time (hour)

minus400

minus200

0

200

400

600

V2G

pow

er (k

W)

(d)

Figure 4 Simulation results for a windy isolation period

the case where V2G is applied every day of the year Prior tofinding amodel that considers the interests of all players of thegame policymakers cannot count on the success of vehicle-to-grid idea in such a case

Appendix

This appendix proves that the two-stage model proposedin this paper is equivalent to simultaneous maximizationof Nash product with respect to both variables The termspresented in (2) are expressed in other words in

max119873119901

MGEP (119873 119901) times EVEP (119873 119901) (A1)

where MGEP(119873 119901) and EVEP(119873 119901) denote the excess pay-offs of the microgrid operator and electric vehicles unionrespectively and can be written as follows

MGEP (119873 119901) = MGCbase minusMGCV2G minus 119873 times 119901EVEP (119873 119901) = 119873 times CVC minus 119873 times EVC + 119873 times 119901 (A2)

To maximize the Nash product presented in (A1) itspartial derivatives with respect to both119873 and 119901must be zerosimultaneously So we have

120597 (MGEP (119873 119901) times EVEP (119873 119901))120597119901 = 0 (A3)

120597 (MGEP (119873 119901) times EVEP (119873 119901))120597119873 = 0 (A4)

From (A3) we have

minus 119873 timesMGEP (119873 119901) times +119873 times EVEP (119873 119901) = 0 997904rArrMGEP (119873 119901) = EVEP (119873 119901) (A5)

And from (A4)

120597MGEP (119873 119901)120597119873 times EVEP (119873 119901) + 120597EVEP (119873 119901)120597119873timesMGEP (119873 119901) = 0 997904rArr(minus120597MGCV2G (119873)120597119873 minus 119901) times EVEP (119873 119901)+ (CVC minus EVC + 119901) timesMGEP (119873 119901) = 0

(A6)

Mathematical Problems in Engineering 9

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600Cr

itica

l loa

d (k

W)

(a)

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

Tota

l win

d po

wer

(kW

)

(b)

No V2GV2G

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

700

Die

sel g

ener

ator

pow

er (k

W)

(c)

1 2 3 4 5 60Time (hour)

minus600

minus400

minus200

0

200

400

600

V2G

pow

er (k

W)

(d)

Figure 5 Simulation results for a low wind isolation period

According to (A5) in optimum of the Nash product func-tion the product terms are equal Hence we conclude

minus120597MGCV2G (119873)120597119873 minus 119901 + CVC minus EVC + 119901 = 0 997904rArr120597MGCV2G (119873)120597119873 minus CVC + EVC = 0

(A7)

The above equation is equivalent to (3) that had been pro-posed as the two-stage model Furthermore by substituting(A2) into (A5) the payment to the vehicles for V2Gservicemdashwhen the number of vehicles is optimalmdashwill be thesame as (4)

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan and A Kushler ldquoPlug-in electric vehicles challengesand opportunities American council for an energy-efficienteconomyrdquo 2013 httpwwwaceeeorgresearch-reportt133

[2] M Duvall and E Knipping Environmental Assessment of Plug-In Hybrid Electric Electric Power Research Institute (EPRI)2007

[3] A Bandyopadhyay L Wang V K Devabhaktuni and R CGreen ldquoAggregator analysis for efficient day-time charging ofPlug-in Hybrid Electric Vehiclesrdquo in Proceedings of the IEEEPower and Energy Society General Meeting pp 1ndash8 IEEEDetroit Mich USA July 2011

[4] K Clement E Haesen and J Driesen ldquoCoordinated chargingof multiple plug-in hybrid electric vehicles in residential dis-tribution gridsrdquo in Proceedings of the IEEEPES Power SystemsConference and Exposition (PSCE rsquo09) pp 1ndash7 IEEE SeattleWash USA March 2009

[5] International Energy Agency World Energy Outlook 2011httpwwwieaorgpublicationsfreepublicationspublicationWEO2011 WEBpdf

[6] International Energy Agency Global EV Outlook Understand-ing the Electric Vehicle Landscape to 2020 2013 httpwwwieaorgpublicationsfreepublicationspublicationGlobalEV-Outlook 2013pdf

[7] H Lee and G Lovellette ldquoWill electric cars transform the USvehicle market An analysis of the key determinantsrdquo Discus-sion Paper 2011-08 Belfer Center for Science and InternationalAffairs Cambridge Mass USA 2011

[8] C Hay M Togeby N C Bang C Sondergren and L HHansen Introducing Electric Vehicles into the Current ElectricityMarkets EDISON Consortium 2010

6 Mathematical Problems in Engineering

in Iran is about 20 years The lifetime of the diesel generatorand analysis time span is assumed to be 20 years too Anelectric vehicle is almost $6000 more expensive to buy andneeds battery replacement in the tenth year It is assumedthat the cost of battery replacement is $6000 and the cost ofestablishing the infrastructure for V2G is $1000 per vehicleOn the other hand a typical PHEV consumes approximately005 literkm In comparison with a 008 literkm fuel con-sumption rate for a conventional vehicle and assuming anaverage distance traveled of 30000 kilometers per year therewill be a 900-liter-per-year differenceThe gas price in Iran isabout $03liter which gives a $270 per year cost reductionby replacing a conventional vehicle with an electric oneTherefore 120587EV is equivalent to the present value of the cashflow as shown in the following equation

120587EV = 120587119904 + 120587119894 + 120587119887 times 1(1 + 119894)10 minus 120587119891 times(1 + 119894)20 minus 1119894 times (1 + 119894)20 (10)

where 120587119904 is the price difference between an electric vehicleand a conventional one 120587119894 is the V2G infrastructure cost pervehicle 120587119887 is the battery replacement cost 120587119891 is the annualcost reduction due to fuel saving of an electric vehicle and 119894 isthe discount rate in Iran thatmay reasonably be approximatedwith 20 percent in 2014 Hence 120587EV in this case is about$6654

5 Results

The results of the analysis for Manjil show that utilizationof electric vehicles with the current price difference betweenconventional cars and electric carsmdashwhich is mainly dueto battery costsmdashis not economically attractive even whenconsidering the benefits of V2G One can only hope thatwidespread use of electric vehicles in the short term maybe a possibility if the government provides a subsidy forbuying electric vehicles in order to protect the environmentand promote green technologies Unfortunately accordingto calculations made in this study any subsidy less than$5000 per electric vehicle would not encourage the publicto buy an electric car Figure 2 demonstrates the optimalPareto fronts of microgrid sizing plans for various scenariosThe horizontal axis in this figure illustrates the equivalentpresent value of the plan while the vertical axis representsthe reliability index expressed in system minutes

Figure 2 shows four scenarios in the first one vehiclebatteries are not utilized and the Pareto front has beenextracted from various designs with different diesel generatorpower sizes In other scenarios vehicle batteries are used tosupply load along with a diesel generator as the complementsto wind turbines In one of these three cases there is nogovernment subsidy but in the other two scenarios thesubsidies equal to $5000 and $5500 have been assumedIn Figure 3 a scenario with a $6000 government subsidyis added to the previous figure If the government subsidywas more than $6654 buying an electric vehicle becameeconomically attractive even without receiving V2G incomeThe results of analyzing this case indicate that considering

No V2GV2G no subsidy

V2G $5000 subsidyV2G $5500 subsidy

0

50

100

150

200

250

300

Syste

m (m

inut

es)

750 760 770 780 790 800 810 820 830740Total cost (times$1000)

Figure 2 Optimal Pareto fronts of microgrid sizing plan for variousscenarios

No V2GV2G no subsidyV2G $5000 subsidy

V2G $5500 subsidyV2G $6000 subsidy

500 550 600 650 700 750 800 850450Total cost (times$1000)

0

50

100

150

200

250

300Sy

stem

(min

utes

)

Figure 3 Optimal Pareto fronts with a new scenario $6000 subsidy

the proceeds of V2G a $6000 subsidy is an appropriatedecision

One way to attain a certain design for sizing of microgridcomponents among the optimal options on Pareto front is tospecify a minimum requirement for reliability In this paperthe system minutes index is supposed to be less than 30minutes in a year as a decision criterion Accordingly Table 1has been obtained for various scenarios of the problem

As it is shown in Table 1 a subsidy of $6000 has alarge impact on the optimal design of the system and theinvestment cost needed to meet the load In the rest ofthis paper it has been assumed that this amount of subsidywill be granted from the government to everyone who buysan electric vehicle Therefore for determining the price ofV2G services via the Nash bargaining theory two cases havebeen studied and compared first operation of the microgridwithout V2G and second operation of the microgrid with

Mathematical Problems in Engineering 7

Table 1 Size cost and reliability indices for various scenarios

Scenario No V2G V2G (no subsidy) V2G ($5000 subsidy) V2G ($5500 subsidy) V2G ($6000 subsidy)System minutes 292 298 290 290 299Microgrid inv cost ($) 801100 801000 663096 635281 44543Total cost ($) 801100 801000 797070 749527 461795Diesel generator size (MW) 9750 9749 8050 7708 533Number of PHEVs 0 0 81 99 638

Table 2 Results for V2G service pricing

Equivalent present value of total V2G service price $5869045Equivalent present value of V2G service price pervehicle

$919913

Equivalent annual payment to each vehicle for V2Gservice

$18891

Excess payoff of microgrid operator due to V2G $1696525

Excess payoff of each vehicle owner due to V2G $265913

Payment to each vehicle per hour available $449786

V2G and the government subsidy ($6000) In Figures 4(a)ndash4(d) the results of simulating the system during a six-hourinterval of isolation from grid for both cases in a windycondition are depicted Figures 5(a)ndash5(d) display the resultsof the simulation for both cases in a low wind six-hour timeinterval

As can be seen in Figures 4 and 5 when only thediesel generator is used to follow the difference between thepower generated by wind turbines and the power consumedby load a high capacity of diesel generator is needed tobe installed When chargedischarge power of the vehiclebatteries follows the load a low capacity of diesel generatoris required and it will work with an almost smooth outputpower When the electric power generated by wind turbinesis high the batteriesrsquo state of charge will be approximatelyinvariant However when the wind speed drops the vehiclebatteries will gradually discharge The possibility of storingextra power generated by wind turbines in vehicle batterieseliminates cutting of wind turbines due to excess powerin island mode This enhances the energy efficiency in themicrogrid

Using (4) and the results listed in Table 1 the price ofvehicle-to-grid service revenues and profits of themicrogridoperator and electric vehicle owners from V2G agreement isshown in Table 2

In Table 2 the first row represents the equivalent presentvalue of the total payment by the microgrid operator to allvehicle owners during a 20-year time interval for participat-ing in the vehicle-to-grid programThe second row indicatesthe same parameter per vehicle while the price in the thirdrow is the equivalent annual payment by the microgridoperator to each vehicle during the 20-year period given adiscount rate equal to 20 Excess payoffs of the microgrid

operator and vehicle owners due to V2G implementation arelisted as equivalent present values In the last row it has beenassumed that the pricing scheme is based on the hours thatan electric vehicle is available for V2G when it is called forPercentage of available vehicles is supposed to be 70 and itis assumed that isolation from grid takes place for on average6 hours a year Hence a vehicle owner would be paid about$45 for one-hour availability

6 Conclusion

The utilization of batteries of electric vehicles as energystorage systems can help microgrids in supplying load whenthey become isolated from the grid This service which isknown as V2G requires the active participation of electricvehicle owners To develop and maintain such partnershipsthe interests of both microgrid operators and vehicle ownershould be considered Otherwise the participants wouldnot have sufficient motivation This paper has proposed amodel to determine how to divide the proceeds of V2Gamong its contributors based on Nash bargaining theoryTheoutput of themodel specifies the optimumnumber of electricvehicles and the amount of money that must be paid by themicrogrid operator to the electric vehicle owners which hasbeen interpreted as the V2G service price Moreover thismodel can be used to determine the appropriate rate forsubsidies granted by the government to promote the purchaseof electric vehicles

The results of implementation of the proposed model inthe Manjil case in Iran indicate that to encourage the publicto buy electric vehicles it is essential that the governmentcompensate much of the cost difference between conven-tional and electric vehicles in the form of subsidies Theremaining difference can be compensated by the revenues ofthe electric vehicle owner from V2G service The microgridoperator will earn a profit from a reduction in the amountof investment necessary for supplying critical load by dieselgenerator in emergencies This profit will be shared with thevehicle owners

This paper has analyzed the V2G service pricing problemfor a case in which the vehicle batteries are utilized onlywhenever the microgrid goes into island mode Since such asituation rarely happens in a year vehicle owners will not beinconvenienced and battery depreciation cost due to frequentchargedischarge cycles will be negligible For future studyone could also develop a model for V2G service pricing in

8 Mathematical Problems in Engineering

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

700Cr

itica

l loa

d (k

W)

(a)

0

100

200

300

400

500

600

700

Tota

l win

d po

wer

(kW

)

1 2 3 4 5 60Time (hour)

(b)

No V2GV2G

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

Die

sel g

ener

ator

pow

er (k

W)

(c)

1 2 3 4 5 60Time (hour)

minus400

minus200

0

200

400

600

V2G

pow

er (k

W)

(d)

Figure 4 Simulation results for a windy isolation period

the case where V2G is applied every day of the year Prior tofinding amodel that considers the interests of all players of thegame policymakers cannot count on the success of vehicle-to-grid idea in such a case

Appendix

This appendix proves that the two-stage model proposedin this paper is equivalent to simultaneous maximizationof Nash product with respect to both variables The termspresented in (2) are expressed in other words in

max119873119901

MGEP (119873 119901) times EVEP (119873 119901) (A1)

where MGEP(119873 119901) and EVEP(119873 119901) denote the excess pay-offs of the microgrid operator and electric vehicles unionrespectively and can be written as follows

MGEP (119873 119901) = MGCbase minusMGCV2G minus 119873 times 119901EVEP (119873 119901) = 119873 times CVC minus 119873 times EVC + 119873 times 119901 (A2)

To maximize the Nash product presented in (A1) itspartial derivatives with respect to both119873 and 119901must be zerosimultaneously So we have

120597 (MGEP (119873 119901) times EVEP (119873 119901))120597119901 = 0 (A3)

120597 (MGEP (119873 119901) times EVEP (119873 119901))120597119873 = 0 (A4)

From (A3) we have

minus 119873 timesMGEP (119873 119901) times +119873 times EVEP (119873 119901) = 0 997904rArrMGEP (119873 119901) = EVEP (119873 119901) (A5)

And from (A4)

120597MGEP (119873 119901)120597119873 times EVEP (119873 119901) + 120597EVEP (119873 119901)120597119873timesMGEP (119873 119901) = 0 997904rArr(minus120597MGCV2G (119873)120597119873 minus 119901) times EVEP (119873 119901)+ (CVC minus EVC + 119901) timesMGEP (119873 119901) = 0

(A6)

Mathematical Problems in Engineering 9

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600Cr

itica

l loa

d (k

W)

(a)

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

Tota

l win

d po

wer

(kW

)

(b)

No V2GV2G

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

700

Die

sel g

ener

ator

pow

er (k

W)

(c)

1 2 3 4 5 60Time (hour)

minus600

minus400

minus200

0

200

400

600

V2G

pow

er (k

W)

(d)

Figure 5 Simulation results for a low wind isolation period

According to (A5) in optimum of the Nash product func-tion the product terms are equal Hence we conclude

minus120597MGCV2G (119873)120597119873 minus 119901 + CVC minus EVC + 119901 = 0 997904rArr120597MGCV2G (119873)120597119873 minus CVC + EVC = 0

(A7)

The above equation is equivalent to (3) that had been pro-posed as the two-stage model Furthermore by substituting(A2) into (A5) the payment to the vehicles for V2Gservicemdashwhen the number of vehicles is optimalmdashwill be thesame as (4)

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan and A Kushler ldquoPlug-in electric vehicles challengesand opportunities American council for an energy-efficienteconomyrdquo 2013 httpwwwaceeeorgresearch-reportt133

[2] M Duvall and E Knipping Environmental Assessment of Plug-In Hybrid Electric Electric Power Research Institute (EPRI)2007

[3] A Bandyopadhyay L Wang V K Devabhaktuni and R CGreen ldquoAggregator analysis for efficient day-time charging ofPlug-in Hybrid Electric Vehiclesrdquo in Proceedings of the IEEEPower and Energy Society General Meeting pp 1ndash8 IEEEDetroit Mich USA July 2011

[4] K Clement E Haesen and J Driesen ldquoCoordinated chargingof multiple plug-in hybrid electric vehicles in residential dis-tribution gridsrdquo in Proceedings of the IEEEPES Power SystemsConference and Exposition (PSCE rsquo09) pp 1ndash7 IEEE SeattleWash USA March 2009

[5] International Energy Agency World Energy Outlook 2011httpwwwieaorgpublicationsfreepublicationspublicationWEO2011 WEBpdf

[6] International Energy Agency Global EV Outlook Understand-ing the Electric Vehicle Landscape to 2020 2013 httpwwwieaorgpublicationsfreepublicationspublicationGlobalEV-Outlook 2013pdf

[7] H Lee and G Lovellette ldquoWill electric cars transform the USvehicle market An analysis of the key determinantsrdquo Discus-sion Paper 2011-08 Belfer Center for Science and InternationalAffairs Cambridge Mass USA 2011

[8] C Hay M Togeby N C Bang C Sondergren and L HHansen Introducing Electric Vehicles into the Current ElectricityMarkets EDISON Consortium 2010

Mathematical Problems in Engineering 7

Table 1 Size cost and reliability indices for various scenarios

Scenario No V2G V2G (no subsidy) V2G ($5000 subsidy) V2G ($5500 subsidy) V2G ($6000 subsidy)System minutes 292 298 290 290 299Microgrid inv cost ($) 801100 801000 663096 635281 44543Total cost ($) 801100 801000 797070 749527 461795Diesel generator size (MW) 9750 9749 8050 7708 533Number of PHEVs 0 0 81 99 638

Table 2 Results for V2G service pricing

Equivalent present value of total V2G service price $5869045Equivalent present value of V2G service price pervehicle

$919913

Equivalent annual payment to each vehicle for V2Gservice

$18891

Excess payoff of microgrid operator due to V2G $1696525

Excess payoff of each vehicle owner due to V2G $265913

Payment to each vehicle per hour available $449786

V2G and the government subsidy ($6000) In Figures 4(a)ndash4(d) the results of simulating the system during a six-hourinterval of isolation from grid for both cases in a windycondition are depicted Figures 5(a)ndash5(d) display the resultsof the simulation for both cases in a low wind six-hour timeinterval

As can be seen in Figures 4 and 5 when only thediesel generator is used to follow the difference between thepower generated by wind turbines and the power consumedby load a high capacity of diesel generator is needed tobe installed When chargedischarge power of the vehiclebatteries follows the load a low capacity of diesel generatoris required and it will work with an almost smooth outputpower When the electric power generated by wind turbinesis high the batteriesrsquo state of charge will be approximatelyinvariant However when the wind speed drops the vehiclebatteries will gradually discharge The possibility of storingextra power generated by wind turbines in vehicle batterieseliminates cutting of wind turbines due to excess powerin island mode This enhances the energy efficiency in themicrogrid

Using (4) and the results listed in Table 1 the price ofvehicle-to-grid service revenues and profits of themicrogridoperator and electric vehicle owners from V2G agreement isshown in Table 2

In Table 2 the first row represents the equivalent presentvalue of the total payment by the microgrid operator to allvehicle owners during a 20-year time interval for participat-ing in the vehicle-to-grid programThe second row indicatesthe same parameter per vehicle while the price in the thirdrow is the equivalent annual payment by the microgridoperator to each vehicle during the 20-year period given adiscount rate equal to 20 Excess payoffs of the microgrid

operator and vehicle owners due to V2G implementation arelisted as equivalent present values In the last row it has beenassumed that the pricing scheme is based on the hours thatan electric vehicle is available for V2G when it is called forPercentage of available vehicles is supposed to be 70 and itis assumed that isolation from grid takes place for on average6 hours a year Hence a vehicle owner would be paid about$45 for one-hour availability

6 Conclusion

The utilization of batteries of electric vehicles as energystorage systems can help microgrids in supplying load whenthey become isolated from the grid This service which isknown as V2G requires the active participation of electricvehicle owners To develop and maintain such partnershipsthe interests of both microgrid operators and vehicle ownershould be considered Otherwise the participants wouldnot have sufficient motivation This paper has proposed amodel to determine how to divide the proceeds of V2Gamong its contributors based on Nash bargaining theoryTheoutput of themodel specifies the optimumnumber of electricvehicles and the amount of money that must be paid by themicrogrid operator to the electric vehicle owners which hasbeen interpreted as the V2G service price Moreover thismodel can be used to determine the appropriate rate forsubsidies granted by the government to promote the purchaseof electric vehicles

The results of implementation of the proposed model inthe Manjil case in Iran indicate that to encourage the publicto buy electric vehicles it is essential that the governmentcompensate much of the cost difference between conven-tional and electric vehicles in the form of subsidies Theremaining difference can be compensated by the revenues ofthe electric vehicle owner from V2G service The microgridoperator will earn a profit from a reduction in the amountof investment necessary for supplying critical load by dieselgenerator in emergencies This profit will be shared with thevehicle owners

This paper has analyzed the V2G service pricing problemfor a case in which the vehicle batteries are utilized onlywhenever the microgrid goes into island mode Since such asituation rarely happens in a year vehicle owners will not beinconvenienced and battery depreciation cost due to frequentchargedischarge cycles will be negligible For future studyone could also develop a model for V2G service pricing in

8 Mathematical Problems in Engineering

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

700Cr

itica

l loa

d (k

W)

(a)

0

100

200

300

400

500

600

700

Tota

l win

d po

wer

(kW

)

1 2 3 4 5 60Time (hour)

(b)

No V2GV2G

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

Die

sel g

ener

ator

pow

er (k

W)

(c)

1 2 3 4 5 60Time (hour)

minus400

minus200

0

200

400

600

V2G

pow

er (k

W)

(d)

Figure 4 Simulation results for a windy isolation period

the case where V2G is applied every day of the year Prior tofinding amodel that considers the interests of all players of thegame policymakers cannot count on the success of vehicle-to-grid idea in such a case

Appendix

This appendix proves that the two-stage model proposedin this paper is equivalent to simultaneous maximizationof Nash product with respect to both variables The termspresented in (2) are expressed in other words in

max119873119901

MGEP (119873 119901) times EVEP (119873 119901) (A1)

where MGEP(119873 119901) and EVEP(119873 119901) denote the excess pay-offs of the microgrid operator and electric vehicles unionrespectively and can be written as follows

MGEP (119873 119901) = MGCbase minusMGCV2G minus 119873 times 119901EVEP (119873 119901) = 119873 times CVC minus 119873 times EVC + 119873 times 119901 (A2)

To maximize the Nash product presented in (A1) itspartial derivatives with respect to both119873 and 119901must be zerosimultaneously So we have

120597 (MGEP (119873 119901) times EVEP (119873 119901))120597119901 = 0 (A3)

120597 (MGEP (119873 119901) times EVEP (119873 119901))120597119873 = 0 (A4)

From (A3) we have

minus 119873 timesMGEP (119873 119901) times +119873 times EVEP (119873 119901) = 0 997904rArrMGEP (119873 119901) = EVEP (119873 119901) (A5)

And from (A4)

120597MGEP (119873 119901)120597119873 times EVEP (119873 119901) + 120597EVEP (119873 119901)120597119873timesMGEP (119873 119901) = 0 997904rArr(minus120597MGCV2G (119873)120597119873 minus 119901) times EVEP (119873 119901)+ (CVC minus EVC + 119901) timesMGEP (119873 119901) = 0

(A6)

Mathematical Problems in Engineering 9

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600Cr

itica

l loa

d (k

W)

(a)

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

Tota

l win

d po

wer

(kW

)

(b)

No V2GV2G

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

700

Die

sel g

ener

ator

pow

er (k

W)

(c)

1 2 3 4 5 60Time (hour)

minus600

minus400

minus200

0

200

400

600

V2G

pow

er (k

W)

(d)

Figure 5 Simulation results for a low wind isolation period

According to (A5) in optimum of the Nash product func-tion the product terms are equal Hence we conclude

minus120597MGCV2G (119873)120597119873 minus 119901 + CVC minus EVC + 119901 = 0 997904rArr120597MGCV2G (119873)120597119873 minus CVC + EVC = 0

(A7)

The above equation is equivalent to (3) that had been pro-posed as the two-stage model Furthermore by substituting(A2) into (A5) the payment to the vehicles for V2Gservicemdashwhen the number of vehicles is optimalmdashwill be thesame as (4)

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan and A Kushler ldquoPlug-in electric vehicles challengesand opportunities American council for an energy-efficienteconomyrdquo 2013 httpwwwaceeeorgresearch-reportt133

[2] M Duvall and E Knipping Environmental Assessment of Plug-In Hybrid Electric Electric Power Research Institute (EPRI)2007

[3] A Bandyopadhyay L Wang V K Devabhaktuni and R CGreen ldquoAggregator analysis for efficient day-time charging ofPlug-in Hybrid Electric Vehiclesrdquo in Proceedings of the IEEEPower and Energy Society General Meeting pp 1ndash8 IEEEDetroit Mich USA July 2011

[4] K Clement E Haesen and J Driesen ldquoCoordinated chargingof multiple plug-in hybrid electric vehicles in residential dis-tribution gridsrdquo in Proceedings of the IEEEPES Power SystemsConference and Exposition (PSCE rsquo09) pp 1ndash7 IEEE SeattleWash USA March 2009

[5] International Energy Agency World Energy Outlook 2011httpwwwieaorgpublicationsfreepublicationspublicationWEO2011 WEBpdf

[6] International Energy Agency Global EV Outlook Understand-ing the Electric Vehicle Landscape to 2020 2013 httpwwwieaorgpublicationsfreepublicationspublicationGlobalEV-Outlook 2013pdf

[7] H Lee and G Lovellette ldquoWill electric cars transform the USvehicle market An analysis of the key determinantsrdquo Discus-sion Paper 2011-08 Belfer Center for Science and InternationalAffairs Cambridge Mass USA 2011

[8] C Hay M Togeby N C Bang C Sondergren and L HHansen Introducing Electric Vehicles into the Current ElectricityMarkets EDISON Consortium 2010

8 Mathematical Problems in Engineering

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

700Cr

itica

l loa

d (k

W)

(a)

0

100

200

300

400

500

600

700

Tota

l win

d po

wer

(kW

)

1 2 3 4 5 60Time (hour)

(b)

No V2GV2G

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

Die

sel g

ener

ator

pow

er (k

W)

(c)

1 2 3 4 5 60Time (hour)

minus400

minus200

0

200

400

600

V2G

pow

er (k

W)

(d)

Figure 4 Simulation results for a windy isolation period

the case where V2G is applied every day of the year Prior tofinding amodel that considers the interests of all players of thegame policymakers cannot count on the success of vehicle-to-grid idea in such a case

Appendix

This appendix proves that the two-stage model proposedin this paper is equivalent to simultaneous maximizationof Nash product with respect to both variables The termspresented in (2) are expressed in other words in

max119873119901

MGEP (119873 119901) times EVEP (119873 119901) (A1)

where MGEP(119873 119901) and EVEP(119873 119901) denote the excess pay-offs of the microgrid operator and electric vehicles unionrespectively and can be written as follows

MGEP (119873 119901) = MGCbase minusMGCV2G minus 119873 times 119901EVEP (119873 119901) = 119873 times CVC minus 119873 times EVC + 119873 times 119901 (A2)

To maximize the Nash product presented in (A1) itspartial derivatives with respect to both119873 and 119901must be zerosimultaneously So we have

120597 (MGEP (119873 119901) times EVEP (119873 119901))120597119901 = 0 (A3)

120597 (MGEP (119873 119901) times EVEP (119873 119901))120597119873 = 0 (A4)

From (A3) we have

minus 119873 timesMGEP (119873 119901) times +119873 times EVEP (119873 119901) = 0 997904rArrMGEP (119873 119901) = EVEP (119873 119901) (A5)

And from (A4)

120597MGEP (119873 119901)120597119873 times EVEP (119873 119901) + 120597EVEP (119873 119901)120597119873timesMGEP (119873 119901) = 0 997904rArr(minus120597MGCV2G (119873)120597119873 minus 119901) times EVEP (119873 119901)+ (CVC minus EVC + 119901) timesMGEP (119873 119901) = 0

(A6)

Mathematical Problems in Engineering 9

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600Cr

itica

l loa

d (k

W)

(a)

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

Tota

l win

d po

wer

(kW

)

(b)

No V2GV2G

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

700

Die

sel g

ener

ator

pow

er (k

W)

(c)

1 2 3 4 5 60Time (hour)

minus600

minus400

minus200

0

200

400

600

V2G

pow

er (k

W)

(d)

Figure 5 Simulation results for a low wind isolation period

According to (A5) in optimum of the Nash product func-tion the product terms are equal Hence we conclude

minus120597MGCV2G (119873)120597119873 minus 119901 + CVC minus EVC + 119901 = 0 997904rArr120597MGCV2G (119873)120597119873 minus CVC + EVC = 0

(A7)

The above equation is equivalent to (3) that had been pro-posed as the two-stage model Furthermore by substituting(A2) into (A5) the payment to the vehicles for V2Gservicemdashwhen the number of vehicles is optimalmdashwill be thesame as (4)

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan and A Kushler ldquoPlug-in electric vehicles challengesand opportunities American council for an energy-efficienteconomyrdquo 2013 httpwwwaceeeorgresearch-reportt133

[2] M Duvall and E Knipping Environmental Assessment of Plug-In Hybrid Electric Electric Power Research Institute (EPRI)2007

[3] A Bandyopadhyay L Wang V K Devabhaktuni and R CGreen ldquoAggregator analysis for efficient day-time charging ofPlug-in Hybrid Electric Vehiclesrdquo in Proceedings of the IEEEPower and Energy Society General Meeting pp 1ndash8 IEEEDetroit Mich USA July 2011

[4] K Clement E Haesen and J Driesen ldquoCoordinated chargingof multiple plug-in hybrid electric vehicles in residential dis-tribution gridsrdquo in Proceedings of the IEEEPES Power SystemsConference and Exposition (PSCE rsquo09) pp 1ndash7 IEEE SeattleWash USA March 2009

[5] International Energy Agency World Energy Outlook 2011httpwwwieaorgpublicationsfreepublicationspublicationWEO2011 WEBpdf

[6] International Energy Agency Global EV Outlook Understand-ing the Electric Vehicle Landscape to 2020 2013 httpwwwieaorgpublicationsfreepublicationspublicationGlobalEV-Outlook 2013pdf

[7] H Lee and G Lovellette ldquoWill electric cars transform the USvehicle market An analysis of the key determinantsrdquo Discus-sion Paper 2011-08 Belfer Center for Science and InternationalAffairs Cambridge Mass USA 2011

[8] C Hay M Togeby N C Bang C Sondergren and L HHansen Introducing Electric Vehicles into the Current ElectricityMarkets EDISON Consortium 2010

Mathematical Problems in Engineering 9

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600Cr

itica

l loa

d (k

W)

(a)

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

Tota

l win

d po

wer

(kW

)

(b)

No V2GV2G

1 2 3 4 5 60Time (hour)

0

100

200

300

400

500

600

700

Die

sel g

ener

ator

pow

er (k

W)

(c)

1 2 3 4 5 60Time (hour)

minus600

minus400

minus200

0

200

400

600

V2G

pow

er (k

W)

(d)

Figure 5 Simulation results for a low wind isolation period

According to (A5) in optimum of the Nash product func-tion the product terms are equal Hence we conclude

minus120597MGCV2G (119873)120597119873 minus 119901 + CVC minus EVC + 119901 = 0 997904rArr120597MGCV2G (119873)120597119873 minus CVC + EVC = 0

(A7)

The above equation is equivalent to (3) that had been pro-posed as the two-stage model Furthermore by substituting(A2) into (A5) the payment to the vehicles for V2Gservicemdashwhen the number of vehicles is optimalmdashwill be thesame as (4)

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan and A Kushler ldquoPlug-in electric vehicles challengesand opportunities American council for an energy-efficienteconomyrdquo 2013 httpwwwaceeeorgresearch-reportt133

[2] M Duvall and E Knipping Environmental Assessment of Plug-In Hybrid Electric Electric Power Research Institute (EPRI)2007

[3] A Bandyopadhyay L Wang V K Devabhaktuni and R CGreen ldquoAggregator analysis for efficient day-time charging ofPlug-in Hybrid Electric Vehiclesrdquo in Proceedings of the IEEEPower and Energy Society General Meeting pp 1ndash8 IEEEDetroit Mich USA July 2011

[4] K Clement E Haesen and J Driesen ldquoCoordinated chargingof multiple plug-in hybrid electric vehicles in residential dis-tribution gridsrdquo in Proceedings of the IEEEPES Power SystemsConference and Exposition (PSCE rsquo09) pp 1ndash7 IEEE SeattleWash USA March 2009

[5] International Energy Agency World Energy Outlook 2011httpwwwieaorgpublicationsfreepublicationspublicationWEO2011 WEBpdf

[6] International Energy Agency Global EV Outlook Understand-ing the Electric Vehicle Landscape to 2020 2013 httpwwwieaorgpublicationsfreepublicationspublicationGlobalEV-Outlook 2013pdf

[7] H Lee and G Lovellette ldquoWill electric cars transform the USvehicle market An analysis of the key determinantsrdquo Discus-sion Paper 2011-08 Belfer Center for Science and InternationalAffairs Cambridge Mass USA 2011

[8] C Hay M Togeby N C Bang C Sondergren and L HHansen Introducing Electric Vehicles into the Current ElectricityMarkets EDISON Consortium 2010

10 Mathematical Problems in Engineering

[9] C Quinn D Zimmerle and T H Bradley ldquoThe effect of com-munication architecture on the availability reliability andeconomics of plug-in hybrid electric vehicle-to-grid ancillaryservicesrdquo Journal of Power Sources vol 195 no 5 pp 1500ndash15092010

[10] C Guille and G Gross ldquoA conceptual framework for thevehicle-to-grid (V2G) implementationrdquo Energy Policy vol 37no 11 pp 4379ndash4390 2009

[11] C Guille and G Gross ldquoDesign of a conceptual framework forthe V2G implementationrdquo in Proceedings of the IEEE Energy2030 Conference (ENERGY rsquo08) pp 1ndash3 November 2008

[12] S Han S Jang K Sezaki and S Han ldquoQuantitative modelingof an energy constraint regarding V2G aggregator for frequencyregulationrdquo in Proceedings of the 9th International Conference onEnvironment and Electrical Engineering (EEEIC rsquo10) pp 114ndash116IEEE Prague Czech Republic May 2010

[13] N Matta R Rahim-Amoud L Merghem-Boulahia and AJrad ldquoA cooperative aggregation-based architecture for vehicle-to-grid communicationsrdquo in Proceedings of the Global Infor-mation Infrastructure Symposium (GIIS rsquo11) pp 1ndash6 Da NangVietnam August 2011

[14] D Wu C Liu and S Gao ldquoCoordinated control on a vehicle-to-grid systemrdquo in Proceedings of the International Conferenceon ElectricalMachines and Systems (ICEMS rsquo11) pp 1ndash6 BeijingChina August 2011

[15] J Xu and V W S Wong ldquoAn approximate dynamic program-ming approach for coordinated charging control at vehicle-to-grid aggregatorrdquo in Proceedings of the IEEE 2nd InternationalConference on Smart Grid Communications (SmartGridCommrsquo11) pp 279ndash284 IEEE Gaithersburg Md USA October 2011

[16] M El ChehalyO Saadeh CMartinez andG Joos ldquoAdvantagesand applications of vehicle to grid mode of operation in plug-in hybrid electric vehiclesrdquo in Proceedings of the IEEE ElectricalPower and Energy Conference (EPEC rsquo09) pp 1ndash6 MontrealCanada October 2009

[17] H Lund and W Kempton ldquoIntegration of renewable energyinto the transport and electricity sectors through V2Grdquo EnergyPolicy vol 36 no 9 pp 3578ndash3587 2008

[18] Y Ota H Taniguchi T Nakajima and K M Liyanage ldquoAuto-nomous distributed V2G (vehicle-to-grid) considering charg-ing request and battery conditionrdquo in Proceedings of the IEEEPES Innovative Smart Grid Technologies Conference Europe(ISGT Europe) pp 1ndash6 Gothenberg Sweden October 2010

[19] R Walawalkar J Apt and R Mancini ldquoEconomics of electricenergy storage for energy arbitrage and regulation inNewYorkrdquoEnergy Policy vol 35 no 4 pp 2558ndash2568 2007

[20] W Kempton J Tomic S Letendre A Brooks and T LipmanVehicle-to-Grid Power Battery Hybrid and Fuel Cell Vehiclesas Resources for Distributed Institute of Transportation Studies(UCD) 2001

[21] W Kempton and J Tomic ldquoVehicle-to-grid power imple-mentation from stabilizing the grid to supporting large-scalerenewable energyrdquo Journal of Power Sources vol 144 no 1 pp280ndash294 2005

[22] Y Wang B Wang C-C Chu H Pota and R Gadh ldquoEnergymanagement for a commercial buildingmicrogrid with station-ary and mobile battery storagerdquo Energy and Buildings vol 116pp 141ndash150 2016

[23] W Shi and V W S Wong ldquoReal-time vehicle-to-grid controlalgorithm under price uncertaintyrdquo in Proceedings of the IEEE2nd International Conference on Smart Grid Communications(SmartGridComm rsquo11) pp 261ndash266 IEEE Brussels BelgiumOctober 2011

[24] E Sortomme andM A El-Sharkawi ldquoOptimal charging strate-gies for unidirectional vehicle-to-gridrdquo IEEE Transactions onSmart Grid vol 2 no 1 pp 131ndash138 2011

[25] M A Ortega-Vazquez ldquoOptimal scheduling of electric vehiclecharging and vehicle-to-grid services at household level includ-ing battery degradation and price uncertaintyrdquo IET GenerationTransmission amp Distribution vol 8 no 6 pp 1007ndash1016 2014

[26] W Kempton and J Tomic ldquoVehicle-to-grid power fundamen-tals calculating capacity and net revenuerdquo Journal of PowerSources vol 144 no 1 pp 268ndash279 2005

[27] J Donadee andM Ilic ldquoStochastic co-optimization of chargingand frequency regulation by electric vehiclesrdquo in Proceedingsof the North American Power Symposium (NAPS rsquo12) pp 1ndash6Champaign Ill USA September 2012

[28] J Lee and G-L Park ldquoA heuristic-based electricity tradecoordination for microgrid-level V2G servicesrdquo InternationalJournal of Vehicle Design vol 69 no 1-4 pp 208ndash223 2015

[29] C M Colson M H Nehrir and C Wang ldquoAnt colony opti-mization for microgrid multi-objective power managementrdquoin Proceedings of the IEEEPES Power Systems Conference andExposition (PSCE rsquo09) pp 1ndash7 Seattle Wash USA March 2009

[30] B Kroposki T Basso and R DeBlasio ldquoMicrogrid standardsand technologiesrdquo in Proceedings of the IEEE Power and EnergySociety General Meeting Conversion and Delivery of ElectricalEnergy in the 21st Century (PES rsquo08) pp 1ndash4 IEEE PittsburghPa USA July 2008

[31] A D Hawkes and M A Leach ldquoModelling high level systemdesign and unit commitment for a microgridrdquo Applied Energyvol 86 no 7-8 pp 1253ndash1265 2009

[32] C AHernandez-Aramburo T CGreen andNMugniot ldquoFuelconsumption minimization of a microgridrdquo IEEE Transactionson Industry Applications vol 41 no 3 pp 673ndash681 2005

[33] M T Lawder V Viswanathan and V R Subramanian ldquoBalanc-ing autonomy and utilization of solar power and battery storagefor demand based microgridsrdquo Journal of Power Sources vol279 pp 645ndash655 2015

[34] A Mohd E Ortjohann A Schmelter N Hamsic and DMorton ldquoChallenges in integrating distributed energy storagesystems into future smart gridrdquo in Proceedings of the IEEEInternational Symposium on Industrial Electronics (ISIE rsquo08) pp1627ndash1632 Cambridge UK June 2008

[35] P Han J Wang Y Han and Y Li ldquoResident Plug-In ElectricVehicle charging modeling and scheduling mechanism in thesmart gridrdquo Mathematical Problems in Engineering vol 2014Article ID 540624 8 pages 2014

[36] P Denholm and R Sioshansi ldquoThe value of plug-in hybridelectric vehicles as grid resourcesrdquo in Proceedings of the 34thIAEE International Conference Stockholm Sweden 2011

[37] C Quinn D Zimmerle and T H Bradley ldquoAn evaluation ofstate-of-charge limitations and actuation signal energy contenton plug-in hybrid electric vehicle vehicle-to-grid reliability andeconomicsrdquo IEEE Transactions on Smart Grid vol 3 no 1 pp483ndash491 2012

[38] P Manzini ldquoGame theoretic models of wage bargainingrdquo Jour-nal of Economic Surveys vol 12 no 1 pp 1ndash41 1998

[39] A Muthoo Bargaining Theory with Applications CambridgeUniversity Press Cambridge UK 1999

[40] A E Roth Game-Theoretic Models of Bargaining CambridgeUniversity Press Cambridge UK 2005

[41] M H Sarparandeh M Moeini-Aghtaie P Dehghanian I Har-sini and A Haghani ldquoFeasibility study of operating anautonomous power system in presence of wind turbines A

Mathematical Problems in Engineering 11

practical experience in Manjil Iranrdquo in Proceedings of the11th International Conference on Environment and ElectricalEngineering (EEEIC rsquo12) pp 1011ndash1016 Venice Italy May 2012

[42] K Y Lee andMA El-SharkawiModernHeuristic OptimizationTechniques Theory and Applications to Power Systems Wiley-IEEE Press 2008

[43] H L Willis and W G Scott Distributed Power Generation Pla-nning and Evaluation Marcel Dekker New York NY USA2000

[44] K Qian C Zhou M Allan and Y Yuan ldquoModeling of loaddemand due to EV battery charging in distribution systemsrdquoIEEE Transactions on Power Systems vol 26 no 2 pp 802ndash8102011

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

practical experience in Manjil Iranrdquo in Proceedings of the11th International Conference on Environment and ElectricalEngineering (EEEIC rsquo12) pp 1011ndash1016 Venice Italy May 2012

[42] K Y Lee andMA El-SharkawiModernHeuristic OptimizationTechniques Theory and Applications to Power Systems Wiley-IEEE Press 2008

[43] H L Willis and W G Scott Distributed Power Generation Pla-nning and Evaluation Marcel Dekker New York NY USA2000

[44] K Qian C Zhou M Allan and Y Yuan ldquoModeling of loaddemand due to EV battery charging in distribution systemsrdquoIEEE Transactions on Power Systems vol 26 no 2 pp 802ndash8102011

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Stochastic AnalysisInternational Journal of