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Two-stage Mechanism for Massive Electric VehicleCharging Involving Renewable Energy

Ran Wang, Ping Wang, Senior Member, IEEE, and Gaoxi Xiao, Member, IEEE

Abstract—Integrating massive electric vehicles (EVs) into thepower grid requires the charging to be coordinated to reducethe energy cost and peak to average ratio (PAR) of the system.The coordination becomes more challenging when the highlyfluctuant renewable energies constitute a significant portion ofthe power resources. To tackle this problem, a novel two-stageEV charging mechanism is designed in this paper, which mainlyincludes three parts as follows. At the first stage, based on theprediction of future energy requests and considering the elasticcharging property of EVs, an offline optimal energy generationscheduling problem is formulated and solved in a day-aheadmanner to determine the energy generation in each time slotnext day. Then at the second stage, based on the planned energygeneration day-ahead, an adaptive real-time charging strategyis developed to determine the charging rate of each vehicle in adynamic manner. Finally, we develop a charging rate compression(CRC) algorithm which tremendously reduces the complexityof the problem solving. The fast algorithm supports real-timeoperations and enables the large-scale small-step scheduling moreefficiently. Simulation results indicate that the proposed schemecan help effectively save the energy cost and reduce the systemPAR. Detailed evaluations on the impact of renewable energyuncertainties show that our proposed approach achieves a goodperformance in enhancing the system fault tolerance againstuncertainties and the noises of real-time data. We further extendthe mechanism to track a given load profile and handle thescenario where EVs only have several discrete charging rates. Asa universal methodology, the proposed scheme is not restrictedto any specific data traces and can be easily applied to manyother cases as well.

Index Terms—Electric vehicles (EVs), charging mechanismdesign, renewable energy, energy generation scheduling, powerregulation, peak shaving.

I. INTRODUCTION

IN the world today, fossil fuels are the dominant energysources for both transportation sector and electricity gener-

ation industry. Statistics show that transportation and electric-ity generation account for over 60% of global primary energydemands [1]. The future solution for the fossil fuels scarcity,as well as the growing environmental problems associatedwith their wide usage, will most likely involve an extensiveuse of electric vehicles (EVs) and adopting renewable energysources for electric energy production [2]–[4]. Under such

Copyright c⃝ 2015 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

Ran Wang and Ping Wang are with the School of Computer Engi-neering, Nanyang Technological University, Singapore (e-mail: {wangran,wangping}@ntu.edu.sg).

Gaoxi Xiao is with the School of Electrical and Electronic Engineering,Nanyang Technological University, Singapore (e-mail: egxxiao@ntu.edu.sg;Phone: (+65) 6790-4552; Fax: (+65) 6793-3318).

cases, renewable energy supplied EV charging is becominga popular approach for greener and more efficient energyusage. Since EVs have controllable charging rate, they can beconsidered as flexible loads in grid system which can benefitthe grid system with demand response or load following.Accordingly, charging scheduling of EVs in the presence ofrenewable energy becomes a practical and important researchproblem [5], [6].

A number of technical and regulatory issues, however, haveto be resolved before renewable energy supplied EV chargingbecomes a commonplace. The arrival of EVs and their requiredenergy amount may appear to be random, which increasesthe demand side uncertainties. In addition, while renewableenergy offers a cheaper and cleaner energy supply, it imposesgreat challenges to the stability and safety of the chargingsystem because of its high inter-temporal variation and lim-ited predictability. Therefore, the stochastic characteristics ofboth EVs and renewable energy sources should be carefullyconsidered. Stand-by generators, back-up energy suppliers orbulk energy storage systems may be necessary to alleviatethe unbalancing issue caused by renewable energy fluctuation,which results in extra cost. In order to minimize the costfor obtaining extra energy and to increase energy efficiency,a flexible and efficient EV charging mechanism has to beproperly designed to dynamically coordinate the renewableenergy generation and energy demands of EVs.

A. Related Work

The existing EV charging control mechanisms roughlyfall into two categories: centralized charging strategies anddecentralized charging strategies. The main idea of central-ized control is utilizing centralized infrastructure to collectinformation from all EVs and centrally optimize EV chargingconsidering the grid technical constraints. In such a strategy,the master controller decides on the rate and duration foreach EV charge. References [7]–[10] develop various cen-tralized charging strategies to achieve different optimizationobjectives, including saving system cost, minimizing powerloss, adjusting power frequency and satisfying EV owners.Optimization methods and heuristic algorithms are adoptedby researchers to solve such problems. In [11], a hierarchicalcontrol scheme is proposed for EVs’ charging station loads in adistribution network while minimizing energy cost and abidingby substation supply constraints. The scheduling is based onthe forecast load information. Recent literature [12]–[14] alladopt receding horizon control based techniques to tackle theuncertainties in the dynamic systems. Jin et al. [15] study EV

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charging scheduling problems from a customer’s perspectiveby jointly considering the aggregator’s revenue and customers’demands and costs. Different from previous papers, bothstatic and dynamic charging scenarios are considered in [15].Though the centralized charging strategy is straightforward,the size of the centralized optimization increases with thenumber of EVs. Accurate information collection from a largenumber of EVs may also impose a challenge. Designing aneffective centralized EV charging strategy therefore remainsas a difficult problem.

In contrast, the vehicle owners directly control their EVs’charging patterns employing the decentralized charging strate-gies [16]–[27]. Gan et al. [16] propose a decentralized algo-rithm to schedule EV charging to fill the electric load valley.This charging control strategy iteratively solves an optimalcontrol problem in which the charging rate of each vehicle canvary continuously within its upper and lower bounds. In eachiteration, each EV updates its own charging profile accordingto the control signal broadcast by the utility and the utilitycompany alters the control signal to guide their updates. In[17], [20]–[25], various decentralized charging frameworks tocoordinate charging demand of EVs are implemented basedon game theory concepts. Considering the selfish nature ofpeople, authors of [18] define some weighting factors inthe objective function of EV charging management problemaiming at modeling users’ convenience in the presented opti-mization procedure. In the decentralized charging strategy, theglobal optimization is achieved through the influence of priceor control signals over the EVs. However, the last decision istaken by each EV, indicating that there are some uncertaintiesin the final results. Also, simultaneous reactions may happen,that is, a huge number of EVs can change their charge rateat the same moment in response to a significant fall/rise ofelectricity prices [26]. In [19], an decentralized online valleyfilling algorithm for EV charging is proposed. An optimalpower flow (OPF) framework is adopted to model the networkconstraint that rises from charging EVs at different locations.In [27], the authors formulate the EV charging problem as aconvex optimization problem and then propose a decentralizedwater-filling-based algorithm to solve it. A receding horizonapproach (similar to that in [12]–[14]) is utilized to handlethe random arrival of EVs and the inaccuracy of the forecastnon-EV load.

In the above mentioned literature, the charging energy issupplied purely from power grid, largely generated by conven-tional units. The main goal of introducing EVs, namely reduc-ing the pollution and green house gas of transportation sectoris consequently greatly abated, as the pollution is transferredfrom vehicle itself to the conventional energy units. Renewableenergy should play a role as significantly as possible to achievethe real environmental advantage. Renewable energy based EVcharging hence becomes a practical and critical problem.

Though the topic has not been well investigated in literature,a few related works can still be found dealing with thecharging scheduling of EVs with renewable energy integration.Moeini et al. [28] propose a charging management frameworkconsidering multiple criteria including total loss of distribu-tion networks, rescheduling cost and wind energy utilization.

In [28], it is assumed that the energy demand of EVs isknown by the controller. In [29], a price-incentive model isutilized to generate the management strategy to coordinatethe charging of EVs and battery swapping station (BSS). In[30], mathematical models are built for both smart chargingand V2G operation with distribution grid constraints. Authorsof both [29] and [30] assume that the EVs are static andalways available to be charged/discharged. In [31], a stochasticoptimization algorithm is presented to coordinate the chargingof electric-drive vehicles (EDVs) in order to maximize theutilization of renewable energy in transportation. Due to thestochastic nature of the transportation patterns, the MonteCarlo simulation is applied to model uncertainties presentedby numerous scenarios. In [32], the charging problem isformulated as a stochastic semi-Markov decision process withthe objective of maximizing the energy utilization. In recentwork [33], the uncertainties of the EV arrival and renewableenergy are described as independent Markov processes. In[34] and [35], the authors tackle the EV charging schedulingproblem adopting Lyapunov optimization techniques, such thatstatistics of the underlying processes does not need to beknown in prior.

Compared with what has been proposed in the past, ourEV charging mechanism mainly has the following severaladvantages: 1) renewable energies can be effectively utilizedby the EVs; 2) compared with the online scheduling schemes,the proposed mechanism incorporates useful estimated infor-mation day-ahead to help reduce the uncertainties in the real-time scheduling stage; 3) compared with the offline schedulingschemes, our mechanism is fairly flexible such that it caneffectively respond to real-time incidents; 4) A fast computingalgorithm is designed which can easily tackle a large numberof EVs, i.e., one weakness of the centralized charging strate-gies is overcome.

B. Main Contributions

In this paper, we consider charging scheduling of a largenumber of EVs at a charging station which is equipped withrenewable energy generation devices. The charging stationcan also obtain energy through controllable generators orbuying energy from outside power grid. Stimulated by thefact that in practical scenario, EV arrival and renewableenergy may not follow Markov process yet obtaining somestatistical information of future EVs’ arrivals (departures) ispossible, we propose a novel two-stage EV charging mech-anism to reduce the cost and efficiently utilize renewableenergy. Several uncertain quantities such as the arrival anddeparture time of the EVs, their charging requirements andavailable renewable energies are all taken into account. Inaddition, the mechanism allows more information of EVarrivals (departures) and renewable energy generation to beeffectively incorporated into the charging mechanism whensuch information is available. The main contributions of thispaper can be briefly summarized as follows:

• A day-ahead cost minimization problem is formulatedand solved, based on the prediction of future renewableenergy generation and EVs’ arrivals (departures) to de-

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termine the amount of energy generated or imported in aday ahead manner.

• We propose a real-time EV charging and power regulationscheme based on the planned energy generation day-ahead to determine the charging rate of each vehicleand power output adjustments in a dynamic and flexiblemanner.

• We develop a fast charging rate compression (CRC)algorithm which significantly reduces the complexity ofsolving the real-time EV charging scheduling problem.The proposed algorithm supports real-time operationsand enables the large-scale small-step scheduling moreefficiently.

• We further extend our mechanism to be applicable to twopractical scenarios: 1) the charging station needs to tracka given load profile; and 2) the EVs only have discretecharging rates.

Simulation results indicate that our proposed two-stage EVcharging mechanism can effectively reduce the system expen-diture and peak to average ratio (PAR). Moreover, the proposedmechanism enhances the system fault tolerance against re-newable energy uncertainties and the noises of real-time data.Note that the proposed charging scheme adopts a universalmethodology which is not restricted to the specific data tracesused in the paper: as long as the renewable energy generationdata and EVs pattern data (including EVs battery level, desiredcharging amount, charging speed and arrival/departure times)can be obtained, the proposed EV charging scheduling schemecan be implemented with virtually no change.

The remainder of this paper is organized as follows: SectionII introduces the problem formulation and two-stage decisionmaking process. In Section III, we present the fast chargingcompression algorithm. The simulation results and discussionsare presented in Section IV. An extension of the proposedcharging mechanism is discussed in Section V. Finally, SectionVI concludes the paper.

II. TWO-STAGE DECISION MAKING MODEL ANDPROBLEM FORMULATION

A. Two-stage Decision-making Model

As shown in Fig. 1, we consider a charging park where anintelligent controller is responsible for the charging schedulingof a large number of EVs. To meet the EVs’ energy demands,the intelligent controller 1) acquires electricity from eithercontrollable energy plants (a dedicated power supply [2]) orcentral power grid; and 2) harvests the renewable energy fromlocal solar panels or wind turbines. Considering the practiceof energy acquisition from controllable generators or powergrid and the limited predictability of renewable energy, wepropose a two-stage model for decision making as shown inFig. 2. Specifically, at the first stage, we divide time intodiscrete time slots with equal length. The preliminary energyacquisition profile Ec(h) and energy transfer factor α(h) aredetermined day-ahead before dispatch based on the estimatedEV energy demand Ev(h) and renewable energy generationEr(h), where h ∈ H is the time slot index and H is the setof time slots in day-ahead scale. Note that Ev(h) is computed

Controllable

Energy Plants

...EV Charging

Station

Intelligent Controller

Solar Panels

Fig. 1: The architecture of the EV charging station.

Fig. 2: Illustration of two-stage decision making model. Firststage (day-ahead): the decision variables are acquisition profileEc(h) and energy transfer factor α(h). Second stage (real-time): the decision variables are the charging speeds of EVsVi(t).

through the EVs arriving and departing pattern predictions. Onthe other hand, the supply of renewable power Pr(t) and EVsreal power demand Pv(t) at time t can only be known in realtime, which requires the real-time control to balance the powersupply and demand at the second stage (real-time stage) ifnecessary. Hence during the real-time EV charging scheduling,we try to obtain the proper EVs charging rates Vi(t) and real-time power acquisition Pc(t) given the real-time renewablepower generation Pr(t), EVs real-time parking profiles andday-ahead dispatched acquired power Pc(t) (determined inthe first stage). Note that for the first stage, the decisionmaking is done one time day-ahead. For the second stage,it is done more frequently in real time, i.e., as long as therenewable power generation or the parking states change, theEVs charging decision coordinates accordingly. Table I liststhe main notations to be used in the rest of this paper.

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TABLE I: Notations used in this paper

Symbol Defination

H Set of time slots in day-ahead scale, |H| = Hh Element in H, time slot index in day-ahead schedulingt Time index in the real-time scheduling

Ec(h) Predetermined energy acquisition at time slot hEv(h) Estimated EV energy demand at time slot hPv(t) Estimated EV power demand at time t

Er(h) Estimated renewable energy generation at time slot hT Length of one time slot

α(h) Energy transfer factor at time slot hM(t) The number of EVs in the charging park at time twi(t) Priority factor of EV i at time tVimax The maximum charging rate of EV iVimin The minimum charging rate of EV iVi(t) Charging rate of vehicle i at time tVd(t) The desired total charging demand at time tPr(t) Renewable power realization at time tPc(t) Power generated or imported in real-timeΓ The set of charging tasks whose charging rates can varyΓS The set of charging tasks whose charging rates are fixed

to the maximumτi Charging task of EV i.

B. Modeling System Uncertainties

It can be noticed that the intelligent charging operationinvolves several uncertain quantities including power availablefrom the renewable energy system, the EVs’ arrival and depar-ture time, and their required charging amount. These quantitiesare crucial parameters for managing the energy generationand consumption of the system. Although these quantities arerandom, there are good reasons to expect that some statisticalinformation may be obtained through accumulation of histori-cal records. For example, the average energy generated by therenewable energy sources at each time slot can be estimatedin a day-ahead manner based on the historical data and theweather forecast, inspecting a large number of samples of EVs’arrival and departure time, a probability distribution trend canbe envisioned. We assume that the parking lot can roughlyestimate the following parameters day-ahead: EVs arrival timedistribution fA(x), departure time distribution fD(x), the totalnumber of EVs being charged in a day N , and the averagecharging rate of an EV µv . In this case, the estimated power(energy density) demand at time t can be expressed as:

Pv(t) =

∫ t

0

(fA(x)− fD(x)

)dx · N · µv, (1)

and the estimated energy demand during time slot h is:

Ev(h) =

∫ h

h−1

Pv(t)dt, ∀h ∈ H. (2)

C. Day-ahead Energy Acquisition Scheduling

The intelligent controller will firstly decide how muchenergy needs to be generated or imported in a day-aheadmanner to minimize the expected energy acquisition costwhile fulfilling the energy demand of EV charging station.The day-ahead energy acquisition scheduling problem can be

formulated as:

minEc(h),α(h)

H∑h=1

Ch(Ec(h)

)(3)

s.t. Ec(h) + Er(h) ≥ Ev(h) · α(h) (4)H∑

h=1

Ev(h) · α(h) =H∑

h=1

Ev(h) (5)

αL ≤ α(h) ≤ αU ,∀h ∈ H, (6)

where Ch(·) is the cost function of the electricity acquisitionfor the charging station, which is assumed to be an increasingconvex function. The convex property reflects the fact thateach additional unit of power needed to serve the demandsis provided at a non-decreasing cost. Example cases includethe quadratic cost function [36] [37] and the piecewise linearcost function [38] [39], etc. Without loss of generality, weconsider quadratic cost function throughout this paper. As tothe renewable energy cost, for typical renewable energies (e.g.solar and wind energy), capital cost dominates. The operationand maintenance costs are typically very low or even negligible[40] [41]. In this paper, it is assumed that the renewable energygenerators such as solar panels and wind turbines have alreadybeen installed, and the marginal cost of renewable energy canbe neglected, leading to its omission in the objective function[42]. Due to the flexibility of EVs’ charging tasks, it is possibleto shift some energy demand to other time slots to achieve thedemand response target and reduce the total cost. α(h) > 0 isan energy transfer factor and 1−α(h) controls the portion ofdemand at time slot h shifted to other time slots. If α(h) > 1,energy demand from other time slots is transferred to time sloth, whereas if α(h) < 1, the energy demand in time slot h isshifted to other time slots. Note that α(h) can vary within itslower bound αL and upper bound αU . Constraint (4) is theload balance constraint, simply indicating that energy in eachtime slot should be balanced. Constraint (5) reveals the factthat the total energy required from EVs during a day remainsunchanged, i.e., demand only transfers between time slots.

D. Real-time Power Regulation and Elastic EV Charging

It is assumed that a two-way communication infrastructure(e.g., a local area network (LAN)) is available between theintelligent controller and vehicles. When an EV plugs in,it informs the intelligent controller its unplug time, desiredcharging amount, maximum and minimum allowable chargingrates. Also, it is assumed that the EV owners are rational sothat the desired charging amount won’t exceed the maximumcharging capacity of vehicle during its parking period. Inother words, if the vehicle is charged at its maximum speedduring the entire parking period, it can definitely reach thepre-set desired battery level. For the real-time operation,the intelligent controller has two tasks. First, given the realrenewable generation and EVs’ charging requirements, it hasto determine a proper charging rate for each EV to achieve theoptimal utilization of renewable energy and finish the charg-ing tasks before EVs’ departures. Second, the total acquiredpower should be properly regulated around the predetermined

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generation profile in real-time to match the fluctuant powerdemand, i.e., demand and supply should be balanced at anytime instance.

From the standpoint of EV owners, it is desirable toreduce their EVs’ charging time. For example, decreasing thecharging time provides more flexibility for the owners to leavethe charging station earlier. This objective can be captured bythe constrained optimization problem as follows:

minVi(t)

∑τi∈Γ

wi(t)(Vimax − Vi(t)

)2, (7)

s.t.∑τi∈Γ

Vi(t) +∑

τi∈ΓS

Vimax ≤ Vd(t), (8)

Vi(t) ≥ Vimin∀τi ∈ Γ, (9)

Vi(t) ≤ Vimax ∀τi ∈ Γ. (10)

In (7), decision variable is Vi(t) which is the charging speed ofEV i to be determined at time t. τi represents the charging taskof vehicle i. Parameter wi(t) ≥ 0 is a priority factor whichreflects the urgent degree of a charging task. More urgent taskswould have larger wi(t). Without loss of generality, wi(t) canbe determined dynamically according to the state of the EV,which is defined as follows:

wi(t) =Er

i

T di − t

, ∀τi ∈ Γ, (11)

where Eri is the amount of remaining requested energy for

charging and T di is EV i’s departure time. Equation (11)

indicates that urgent charging tasks will have a higher priorityfactor so as to be charged faster. This is to ensure that EVsdepart with desired battery level. wi also denotes the averagecharging rate EV i needs to finish the charging task τi on time.Vimax is the maximum charging rate (i.e., the desired chargingrate) of EV i. Vimin is the minimum allowable charging rate ofEV i. At any time t, the charging tasks can be first classifiedinto two categories: Γ is the set of charging tasks whosecharging rate can vary, i.e., Γ = {τi| wi(t) < Vimax

}. ΓS

denotes the set of charging tasks whose charging rates haveto be fixed at the maximum charging rates because of theurgent charging time, i.e., ΓS = {τi| wi(t) = Vimax}. Notethat elements in Γ and ΓS may vary with time and for τi ∈ Γ,Vi ≤ Vimax , for τi ∈ ΓS , Vi = Vimax . This EV classificationapproach ensures that all the EVs depart with satisfactorycharging amount. Vd(t) is the desired total charging demandat time t. The way to set Vd(t) will be introduced later.

Notice that constraint (8) simply states the schedulabilitycondition, and the rest of the constraints bound the charg-ing rates. Due to EVs’ arrivals and departures, the systemis dynamic and the number of vehicles and their chargingrequirements will change over time. Therefore, the intelligentcontroller can solve problem (7)-(10) to obtain the chargingrate for each EV at time t. When the renewable powerrealization changes, or an EV’s status changes (τi changesfrom Γ to ΓS) or a vehicle enters or departs the system, theintelligent controller will update Γ, ΓS and Vd(t) in real timeand then re-do the calculation. Next, we will show how todetermine Vd(t) to optimally utilize the renewable energy.

Let Pc(t) = Ec(h)T denote the dispatched acquired power

(i.e., the day-ahead pre-scheduled power generation) at time

t, where T is the length of a time slot, and Pr(t) denote therenewable generation realization at time t. Then, Vd(t) can bedefined as follows:

• If∑

τi∈Γ Vimin +∑

τi∈ΓSVimax > Pc(t) + Pr(t), then

Vd(t) =∑

τi∈Γ Vimin +∑

τi∈ΓSVimax , Pc(t) = Vd(t)−

Pr(t), Pc(t) is the acquired power in real time. Thisis for the case where the renewable energy generationis very low, i.e., even though all the controllable EVs(EVs that belong to set Γ) charge at their minimumallowable charging rates, the demand is still higher thanthe available supply. Therefore, up regulation is requiredto guarantee the power balancing, i.e., more energy hasto be imported, either by raising up the output level offast response generators or buying more electricity fromancillary service markets.

• If∑

τi∈Γ Vimin +∑

τi∈ΓSVimax ≤ Pc(t) + Pr(t) ≤∑

τi∈Γ∪ΓSVimax , then Vd(t) = Pc(t)+Pr(t) and Pc(t) =

Pc(t). This investigates the scenario where the renewableenergy generation deviates not far from the previousprediction, i.e., the power demand of EVs can be adjustedto match the available supply. This represents the mostcommon situation the charging system encounters. Undersuch case, the power demand of controllable EVs canbe adjusted to match the supply, thus power acquisitionprofile does not need to be changed and is equal to thedispatched load determined day-ahead.

• If∑

τi∈Γ∪ΓSVimax < Pr(t) + Pc(t), then Vd(t) =∑

τi∈Γ∪ΓSVimax

and Pc(t) =∑

τi∈Γ∪ΓSVimax

− Pr(t).This corresponds to the case where the renewable energygeneration is plenty enough that even the highest chargingdemand can be satisfied, i.e., although all the EVs chargeat the maximum charging rates, available power stillexceeds. In this case, down regulation is required to makesure that power is balanced, i.e., the intelligent controllercan reduce the acquired power level or sell the extrapower out and only compensate the mismatch betweenthe maximum charging demand and the renewable energyoutput.

Remark: In day-ahead energy acquisition scheduling, theintelligent controller aims at minimizing the expected cost ofthe charging park given the estimated renewable energy supplyEr(h) and EVs’ energy demand Ev(h), h ∈ H. Decisionvariable Ec(h) is the scheduled electricity to be broughtfrom day-ahead energy market or generated by base-loadplants. In real-time power regulation, system reliability andEVs’ charging requirements become the main concerns. Theaforementioned up/down regulation is provided by ancillaryservice markets or fast response generators [43].

III. THE CHARGING RATE COMPRESSION ALGORITHM

The problem (7)-(10) belongs to the category of convexquadratic programs and can be solved in polynomial time.Many commercial optimization solvers including CPLEX,Mosek, FortMP and Gurobi, etc., can be utilized to solve suchproblems. However solving such a problem using quadraticprogram solver during run time can be still too costly, espe-cially when the number of EVs is large and the response time

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has to be very short so as to quickly respond to EVs. Whatmakes the above formulation attractive is that a charging ratecompression (CRC) algorithm can be proposed such that theproblem solving can be extremely fast. We first develop theCRC algorithm and then introduce a lemma and a theorem toprove that it can solve the problem (7)-(10).

At each time instance t, the set Γ of charging tasks canbe further divided into two subsets: a set Γf of chargingtasks with the minimum charging rate and a set Γv ofcharging tasks whose charging rate can still be compressed.Let V0 =

∑i∈Γ Vimax be the maximum power level of the

charging task set Γ, Vv0 be the sum of maximum charging ratesof charging tasks in Γv , and Vf be the sum of the chargingrates of charging tasks in Γf . To achieve a desired powerlevel Vd(t) < V0 +

∑i∈ΓS

Vimax , each charging task has tobe compressed up to the following charging rate:

∀τi ∈ Γv, Vi = Vimax − (Vv0 − Vm(t) + Vf )Wv

wi, (12)

where

Vm(t) = Vd(t)−∑

τi∈ΓS

Vimax (13)

Vv0=

∑τi∈Γv

Vimax(14)

Vf =∑

τi∈Γf

Vimin (15)

Wv =1∑

τ∈Γv

1wi

. (16)

If there exist charging tasks where Vi < Vimin , then thecharging rates of these vehicles have to be fixed at theirminimum value Vimin . Sets Γf and Γv have to be updated(Therefore Vf , Vv0 and Wv have to be recomputed) and (12) isapplied again to the charging tasks in Γv . If a feasible solutionexists, i.e., the desired power level of the system is higherthan or equal to the minimum power level

∑M(t)i=1 Vimin , the

iterative process ends until each value computed by (12) isgreater than or equal to its corresponding minimum Vimin .The algorithm for compressing the charging rate of a set Γof EVs to a desired charging power level Vd(t) is shown inAlgorithm 1.

Lemma 1. Given the constraint optimization problem asspecified in (7)-(10) and

∑τi∈Γ Vimax > Vm(t), any solution,

V ∗i (t), to the problem must satisfy

∑τi∈Γ V

∗i (t) = Vm(t) and

V ∗i (t) = Vimax , for all τi ∈ Γ.Theorem 1. Given the optimization problem as specified in

(7)-(10),∑

τi∈Γ Vimax > Vm(t), and∑

τi∈Γ Vimin < Vm(t),let V (t) =

∑V ∗i (t)=Vimin

Vimax+∑

V ∗i (t)=Vimin

Vimin. A

solution is optimal if and only if

V ∗i (t) = Vimax

−1

wi(t)(V (t)− Vm(t))∑

V ∗j (t) =Vjmin

(1/wj), (17)

for V (t) > Vm(t) and V ∗i (t) > Vimin , and V ∗

i (t) = Vimin

otherwise.The proofs of Lemma 1 and Theorem 1 are given in the

Appendix. Based on the previous lemma and theorem, we candraw the conclusion as follows:

Algorithm 1 Algorithm for compressing the charging rate fora charging task set of Γ at time t.

Input: Vd(t), Vimin , Vimax , wi, ∀τi ∈ Γ.Output: Vi, ∀τi ∈ Γ.

1: Begin2: V0 =

∑τi∈Γ Vimax ;

3: Vmin =∑

τi∈Γ Vimin ;4: Vm(t) = Vd(t)−

∑τi∈ΓS

Vimax ;5: if (Vm(t) < Vmin)6: Return INFEASIBLE;7: else8: do {9: Γf = {τi|Vi = Vimin};

10: Γv = Γ− Γf ;11: Vv0 =

∑τi∈Γv

Vimax ;12: Vf =

∑τi∈Γf

Vimin ;13: Wv = 1∑

τ∈Γv1wi

;14: OK= 1;15: for (each τi ∈ Γv)16: Vi = Vimax

− (Vv0 − Vm(t) + Vf )Wv

wi;

17: if (Vi < Vimin)18: Vi = Vimin

;19: OK= 0;20: end if21: end for22: } while (OK== 0);23: return FEASIBLE;24: end if25: End

Corollary 1. Consider the charging tasks of |Γ ∪ ΓS | EVs,where Vi(t) is the charging rate of the ith vehicle. Let Vimax

denote the initial desired charging rate of charging task τi ∈Γ∪ΓS and wi(t) be the set of priority factors. Let Vd(t) be thedesired power level of the system and

∑τi∈Γ Vimax > Vm(t).

The charging rate Vi, τi ∈ Γ, obtained from Algorithm 1minimizes ∑

τi∈Γ

wi(t)(Vimax − Vi(t)

)2subject to the inequality constraints

∑τi∈Γ Vi(t) +∑

τi∈ΓSVimax ≤ Vd(t), Vi(t) ≥ Vimin , and Vi(t) ≤ Vimax for

τi ∈ Γ.Remark: Through analysis, the time complexity of Algo-

rithm 1 is O(n2), where n is the number of tasks in Γ.

IV. SIMULATION RESULTS AND DISCUSSIONS

In this section, we present simulation results based on realworld traces for assessing the performance of the proposedtwo-stage EV charging scheme.

A. Parameters and Settings

We assume there are solar panels providing renewableenergy for the charging station. The area of the solar panelsin the system is set to be 6.25 × 104 m2. The energy

7

6 8 10 12 14 16 180

100

200

300

400

500

600

Time (h)

Solarirradiance

(W/m

2)

real−time solar irradiancepredicted solar irradiance

Fig. 3: Solar irradiance in a day

conversion efficiency is 0.4. The solar radiation intensitystatistic is adopted from [44], from which we employ the solarradiation data of a typical day in winter (17/01/2013). Thedata utilized for the day-ahead energy acquisition schedulingand real-time EV charging are depicted in Fig. 3. Note thatthe predicted average solar radiance utilized in the day-aheadenergy generation scheduling is plotted in the blue circledline, and the actual real-time solar radiance adopted in thereal-time charging is shown by the red curve. We envisionthe scenario that the charging station is located at a workplace (e.g., a campus) that is active from 6:00 AM to 6:00PM. Vehicles arrive earlier than 6:00 AM start to charge at6:00 AM while those depart later than 6:00 PM finish theircharging before 6:00 PM. We simulate the operation processof a large scale charging station which serves totally 3000EVs arriving and departing independently in a typical day.It is assumed that the arrival time distribution and departuretime distribution are all Gaussian with parameters shown inTable II (similar assumptions can be found in many papers,e.g., [34] [45]). EVs are active for charging during theirparking time and discharging is not permitted. The amountof energy needed for the EVs are evenly distributed between20 KWh and 50 KWh. The maximum allowable charging rateof an EV is 62.5 KW (e.g., high-voltage (up to 500 VDC)high-current (125 A) automotive fast charging [46]) and theminimum charging rate of an EV is 0 KW. The cost functionof the electricity acquisition is Ch

(Ec(h)

)= ah · Ec(h)

2 andah = 150 $ · (MWh)−2.

TABLE II: Parameters of the arrival and departure timeprobability distribution

Time parameter Arrival Departure

mean: µr 10 14standard deviation: σr 1.2 1.3

6 8 10 12 14 16 180

2

4

6

8

10

12

Time

Pow

ersupply

from

conventional

generators(MW)

scheme 1scheme 2

Fig. 4: Energy supply from conventional generators underdifferent charging schemes

B. Results and Discussions

The simulation process contains two parts. First, given theestimated solar energy in each time slot (in the simulation, onetime slot is set as one hour), we solve the day-ahead energyacquisition scheduling problem (3)-(6) and obtain Ec(h) andα(h) for h = 1, ...,H . The upper bound and lower boundof energy transfer parameter α(h) is set to be 2 and 0.5respectively. Once the dispatched energy acquisition in eachtime slot is obtained, we are ready to simulate the chargingprocess of EVs based on the real time renewable powergeneration and EVs’ real time arrival (departure) patterns.Adopting the data previously mentioned, all the simulationsare conducted on an Intel workstation with 6 processorsclocking at 3.2 GHZ and 16 GB of RAM. We repeated thesimulation for 10 times. All the 3000 EVs complete chargingwith required amount before their departures. By utilizing theCRC algorithm introduced in Section III, the simulation timeis reduced from 1005.1 s to 101.2 s, showing that the proposedCRC algorithm can significantly reduce the complexity ofthe problem solving. Note that our CRC algorithm does notsacrifice the problem solving accuracy and we obtain exactlythe same results when adopting quadratic programming solversand our CRC algorithm.

We first investigate the effectiveness of our proposed EVcharging mechanism. Specifically, two charging schemes arecompared. In the first scheme, EVs are kept charging duringtheir parking time and the charge speeds are the averagerates that they need to fulfill the charging tasks. Conventionalgenerators generate electricity for the unbalanced power de-mand in an on-demand manner. While in the second scheme,the charging station charges EVs’ batteries according to themechanism we proposed, and electricity is generated basedon the day-ahead scheduling and real-time adjustment. Thesimulation results concerning the power supply curves, totalsystem cost and peak to average ratio (PAR) under these twoschemes are given in Fig. 4 and Fig. 5, respectively. As wementioned previously, quadratic cost functions are adopted to

8

case 1 case 2 case 1 case 21.5

2

2.5

3

3.5

4

Different cases

PAR

scheme 1scheme 2

scheme 1 scheme 2

1.5

2

2.5

3

3.5

4

x 104

Different charging strategies

Cost

($)

Fig. 5: Cost and PAR comparisons of different chargingschemes

compute the system expenditures for both schemes.In Fig. 4, it is shown that by optimally controlling the charg-

ing rates of EVs, our proposed charging strategy successfullytransfers the peak demand to the off-peak hours, which canhelp stabilize the operations of the charging system and reducethe energy cost. As shown in Fig. 5, the total expenditure of thecharging station decreases from $4.1×104 per day in scheme1 to $1.8 × 104 per day in our proposed scheme, achievinga cost saving of 56.1%. Therefore, one of the aims of thedeveloped charging strategy, which is reducing the expenditureof the system, is achieved. To investigate the variation of PAR,we study two cases: 1) PAR of the aggregated supply (i.e., thesupply from controllable generators plus the supply from solarpanels); and 2) PAR of the controllable generators’ output. Aswe observe in Fig. 5, with scheme 1, the PAR of the aggregatedsupply and the PAR of controllable generators’ output are 2.69and 3.95, respectively. By adopting the proposed chargingscheme, these two PAR values reduce to 2.02 and 1.78(decrease about 25% and 55%), respectively. The proposedEV charging strategy presents much better PAR performanceduring the 12-hour operation. An interesting observation isthat in scheme 1, the PAR of controllable generators’ output ismuch higher than that of the aggregated power supply, howeverthe situation is exactly opposite in our proposed scheme. Inother words, under normal circumstances, utilizing renewableenergy will make the output of controllable generators morefluctuant, whereas EVs can help solve this problem by properlyvarying their charging speeds, i.e., charging quickly whenrenewable energy is sufficient and reducing the rate when notenough renewable energy is available.

In our scheme, the first-stage day-ahead energy genera-tion scheduling is based on the estimated renewable energygeneration in next day. Normally the real renewable energygeneration might be different from the estimated one. Next, weinvestigate the cost sensitivity with respect to this deviation.The simulation results are depicted in Fig. 6. Specifically,we conduct the experiment as follows. In the first step, theday-ahead energy generation scheduling is done based on

−10% −8% −6% −4% −2% 0 2% 4% 6% 8% 10%

1.5

2

2.5

3

3.5

4

4.5

5

5.5x 10

4

∆m/m

Costofthechargingpark

($)

scheme 1scheme 2

Fig. 6: System cost with respect to the real-time renewablegeneration deviation (∆m represents the deviation of real solarirradiance from the estimated one, and m is the actual datatrace)

the estimated solar irradiance and EVs’ arriving (departing)patterns. Then, for the real-time charging, we vary the solarirradiance data based on the real world trace to representdifferent estimation error levels. As it is observed in Fig. 6,system cost is much more sensitive in scheme 1 than thatin our scheme when the deviation varies. The reason is thatby applying our charging strategy, deviations of the solarpower can be distributed to the whole time horizon. Howeverin scheme 1, the situation that solar power is excessiveduring some time periods and insufficient in some other timebecomes more severe. Under such case, solar energy utilizationefficiency fluctuates more extensively when deviation levelincreases, and accordingly, system cost varies more violently.Hence, our charging mechanism can effectively reduce thefinancial risks caused by the estimation error of the renewableenergy generation.

Figure 7 illustrates how system cost varies under differentfluctuation levels of solar energy. In this experiment, we add 0-mean Gaussian noise to the real-time solar irradiance data andthen evaluate its impact on the system cost. Different standarddeviations of the noise reflect different fluctuation levels ofsolar energy. It appears that the fluctuation of renewableenergy has less impact on the system cost when adoptingour proposed scheduling scheme. This observation is intuitivesince by properly altering their charging rates, EVs act as anenergy storage which may to a certain extent alleviate theuncertainty problem. However in scheme 1, the controllablegenerators have to compensate the solar power fluctuationduring the entire time horizon. In this case, the system cost willbe affected more extensively when fluctuation level increases.Note that this experiment also simulates the scenario thatsystem data is affected by noises. Thus, we claim that the ourproposed EV charging mechanism shows good performancesin dealing with uncertainties of renewable energy and noisesof real-time data.

9

0 0.02 0.04 0.06 0.08 0.1

1.5

2

2.5

3

3.5

4

4.5

5

5.5x 10

4

σ/m

Costofthechargingpark

($)

scheme 1scheme 2

Fig. 7: System cost with respect to the different fluctuationlevel of renewable energy (m represents the actual data traceand σ represents the standard deviation of noise).

V. EXTENSIONS

A. Tracking a Given Load Profile

The electricity utilized for EV charging can be provided bya utility company. The objective of the utility company maybe to flatten the total load profile. The utility company mayalso need to buy electricity in day-ahead electricity market andsupply the electricity to the charging parking as well as otherenergy consumers in real-time. Under such case, the utilitycompany may want the charging station to properly schedulethe charging of EVs so that the demand can track the electricityprofile it brought in the day-ahead electricity market. Denotethe load profile that the charging park tracks as L(t). Ourcharging scheme can be extended to track L(t) by solving thefollowing constraint optimization problem:

minVi(t)

∑τi∈Γ

wi(t)(Vimax − Vi(t)

)2, (18)

s.t.∑τi∈Γ

Vi(t) +∑

τi∈ΓS

Vimax ≤ L(t), (19)

Vi(t) ≥ Vimin ∀τi ∈ Γ, (20)Vi(t) ≤ Vimax ∀τi ∈ Γ. (21)

Figure 8 shows the simulation results of tracking giventarget load profiles. The intelligent controller is in charge ofmanaging 3000 EVs in a day on their charging schedules.These vehicles plug in uniformly distributed between 6 : 00and 14 : 00, with deadlines uniformly distributed between10 : 00 and 18 : 00. The amount of energies needed to chargeare evenly distributed between 20 KWh and 50 KWh. Twotestings are conducted to show the load tracking results withdifferent target profiles. The target profiles are representedby the blue dot-circled curves. The red dash curves andgreen solid curves correspond to the aggregated charging ratesobtained from our EV charging mechanism and scheme 1,respectively. We observe that the aggregated charging demand

6 8 10 12 14 16 180

5

10

15

Time of day

Load

(MW)

Test 1

Target profileScheme 1Our mechanism

6 8 10 12 14 16 180

5

10

15

Time of day

Load(M

W)

Test 2

Target profileScheme 1Our mechanism

Fig. 8: Tracking given target load profiles

can closely follow the target load profiles when adopting ourproposed charging scheme. There are only small discrepanciesaround 18 : 00 due to the early or late departures of EVs.

Remark: In order to ensure that the electricity demand ofEVs can closely follow the target load profile, load profile L(t)should not go beyond the variation limits of EVs’ chargingrates, i.e.:

L(t) ≥∑τi∈Γ

Vimin +∑

τi∈ΓS

Vimax (22)

and

L(t) ≤∑τi∈Γ

Vimax +∑

τi∈ΓS

Vimax (23)

B. Discrete Charging Rates

In our proposed charging scheme, we assume that thecharging rate can vary continuously within the EV’s maximumand minimum allowable rates, determined by the charger.Similar assumptions can be found in many literature including[7] [47] and [16]. However in some circumstances, if onlya few discrete charging speeds are allowed, the proposedEV charging scheme can be easily extended to handle suchcase. Let Vi denote the set of allowable charging rates ofvehicle i. To capture the discrete charging rate case, we replaceconstraints (9), (10) with the following constraint in real-timeEV charging:

Vi(t) ∈ Vi ∀τi ∈ Γ. (24)

Although the CRC algorithm is only suitable for the continu-ous charging rate case, simulations show that with discreteallowable charging rates, the proposed two-stage chargingmechanism still has an acceptable computation-time perfor-mance. In the simulation, each EV has 4 allowable chargingspeeds, i.e., Vi = {0 KW, 20 KW, 40 KW and 62.5 KW},∀τi ∈ Γ [2]. The number of EVs served in a day is still3000. The simulation results comparison with the continuouscharging rate case is summarized in the first row of Table III.

10

TABLE III: Simulation results under continuous charging rate case and discrete charging rate case (all results are 10 timesaverage).

charging park size power level type charging cost ($) cost growthlarge scale continuous charging rate case 17993.1 -(3000 EVs) discrete charging rate case 18028.8 0.2%

medium scale continuous charging rate case 497.1 -(500 EVs) discrete charging rate case 511.2 2.8%small scale continuous charging rate case 22.2 -(100 EVs) discrete charging rate case 27.9 25.7%

Note that the simulations under both cases are conducted 10times and results in Table III are the average.

As it is shown in Table III, two main observations can befound as follows:

• For the discrete charging rate case, though the simulationtime is much longer for the continuous charging rate case,our two-stage EV charging mechanism still performsacceptably for the real-time scheduling since computationtime for updating the charging rates of active vehicles isabout 0.25 s on average. Note that this is the updatingtime running on the computer whose configuration isspecified in the previous subsection.

• The system cost increases slightly (about 0.2%) whenonly several discrete charging rates are allowed. Thisobservation is intuitive since with discrete charging rates,the scheduling flexibility is abated and mechanism perfor-mance gets worse. In other words, when EVs’ chargingrates can vary continuously, the power demand can followthe desired power supply more closely and thus utilize therenewable energy in a more efficient manner. However,since the number of EVs is large, discrepancy between thecombinations of EVs’ discrete charging rates and desiredenergy supply level is not significant. Thus, the cost onlyincreases slightly.

We further reduce the simulation scale to medium size (e.g.,500 vehicles) and small size (e.g., 100 vehicles) to investigatehow the size of the charging park impacts the performancesof the proposed scheme. Besides the charging park size,simulation process and system parameters are exactly thesame to those in the previous subsection. The area of thesolar panels varies proportionally with the charging park size.We also simulate 10 times and the results data are depictedin Table III. It appears that for large-, medium- and small-scale charging parks, system costs in discrete charging ratecase are 0.2%, 2.8% and 25.7% higher than those in thecontinuous charging rate case, respectively. In other words,system cost is more sensitive to the discrete charging ratecondition when the scale of charging park shrinks. The reasonfor this phenomenon is that when the number of connectedvehicles gets small and only several discrete charging ratesare allowed, the flexibility of the system deteriorates. Therewill be a higher probability that aggregated charging demandcannot match the available power. For instance, when thereare only 30 KW power available and 2 vehicles are active at agiven time, for the continuous charging rate case, EVs are ableto follow the supply closely. Whereas for the discrete charging

rate case, either 10 KW power is wasted or conventionalunits have to generate 10 KW more so that discrete demandcan be matched. Therefore, power utilization becomes lessefficient and conventional generators have to produce moreelectricity to ensure that charging tasks can be finished intime. As we mentioned previously, when the number of EVs islarge, discrepancy between the combinations of EVs’ discretecharging rates and desired energy supply level becomes lesssignificant, leading to only marginal increase in cost. Theproposed EV charging scheme favors reasonably for a largecharging park when only discrete charging rates are allowed.

VI. CONCLUSIONS

In this paper, we investigated the cost-effective schedulingapproach of EV charging at a renewable energy aided charg-ing station. We designed a two-stage EV charging schemeto determine energy generation and charging rates of EVs.Specifically, at the first stage, based on the EV pattern andrenewable energy generation estimation, a cost minimizationproblem was formulated and solved to obtain a preliminaryenergy generation or importation scheduling in a day-aheadmanner. Then at the second stage, a real-time EV charging andpower regulation scheme was proposed. Such a scheme allowsconvenient handling of volatile renewable energy and indeter-minate EV patterns. We also developed an efficient chargingcompression algorithm to further lower the complexity of theproblem solving. Simulation results indicate the satisfactoryefficiency of the proposed EV charging mechanism and thecost benefits obtained from it. Moreover, the impacts ofrenewable energy uncertainties have been carefully evaluated.The results show that the proposed EV charging scheme hasa good performance in enhancing the system fault toleranceagainst uncertainties and the noises of real-time data. Suchevaluations, as we believe, reveal that the proposed chargingmechanism is suitable for the case with a large number ofEVs and unstable renewable energy. Furthermore, we extendthe mechanism to track a given load profile and handle thescenario that EVs only have discrete charging rates. As auniversal methodology, the proposed scheme is not restrictedto any specific data traces and can be easily applied to manyother cases as well.

APPENDIX

A. Proof for Lemma 1

Proof: We prove the lemma by adopting the Karush-Kuhn-Tucker (KKT) optimality conditions for the solution to

11

the given problem. The Lagrangian function for the problem(7)-(10) is:

L(V, λ, ν)

=∑τi∈Γ

wi(Vimax − Vi)2 + λ0

(∑τi∈Γ

Vi − Vm

)(25)

+∑τi∈Γ

λi(Vimin − Vi) +∑τi∈Γ

νi(Vi − Vimax),

where λ0 ≥ 0, λi ≥ 0 and νi ≥ 0 for τi ∈ Γ areLagrangian multipliers associated with constraints (8), (9) and(10). Through Slater’s condition, strong duality holds for thisproblem. In such case, the sufficient and necessary conditionsfor the existence of a minimum value at V ∗

i are, for all τi ∈ Γ

∂L

∂V ∗i

= −2wi(Vimax − V ∗i ) + λ0 − λi + νi = 0, (26)

λ0

(∑τi∈Γ

Vi − Vm(t)

)= 0, (27)∑

τi∈Γ

λi(Vimin − Vi) = 0, (28)∑τi∈Γ

νi(Vi − Vimax) = 0. (29)

First assume that (8) is inactive, which means that∑τi∈Γ V

∗i − Vm(t) < 0 and λ0 = 0. In this case, at least

one constraint in (9) or (10) must be active. Let’s assumethat the kth constraint in (9) is active, i.e., V ∗

k = Vkmin andλk ≥ 0. Then, the kth constraint in (10) must be inactive, thatis V ∗

k − Vkmax < 0 and νk = 0. From (26), we then obtain

λk = −2wi(Vimax − V ∗i ) < 0, (30)

which contradicts the assumption that λk ≥ 0. Hence, we havethe conclusion that if any V ∗

k = Vkmin , constraint (8) have tobe active.

Similarly, if at least one constraint in (10) is active whileothers are inactive, i.e., V ∗

h = Vhmax (active) and Vlmin <V ∗l < Vlmax (inactive), then we can obtain that λh = 0, νh ≥

0, λk = 0 and νk = 0. Based on (26), we obtain the followingtwo equations:

λ0 = 2wi(Vhmax − V ∗h ) + λh − νh = −νh ≤ 0, (31)

λ0 = 2wi(Vkmax − V ∗k ) + λk − νk

= 2wi(Vkmax − V ∗k ) > 0. (32)

Note that the above equations (31) and (32) cannot be satisfiedsimultaneously, which means that all the constraints in (10)can either be active or inactive. Under such cases, if all theconstraints in (10) are active, we have∑

τi∈Γ

V ∗i =

∑τi∈Γ

Vimax > Vm(t), (33)

which contradicts the constraint (8) that the charging task isschedulable. If all the constraints in (10) are inactive, then from(29) we have λ0 = 0 from (32), which means that

∑M(t)i=1 V ∗

i −Vm(t) = 0 given (27). This again contradicts the assumptionthat (8) is inactive. Therefore, we have the conclusion that forany solution to the optimization problem (7)-(10), constraint

(8) is active, i.e.,∑

τi∈Γ V∗i (t) = Vm(t) and V ∗

i (t) = Vimax ,for τi ∈ Γ. Hence, Lemma 1 is proved.

B. Proof for Theorem 1

Proof: Consider the KKT optimality condition in (26)-(29). We have proved in Lemma 1 that any solution, V ∗

i (t) tothe optimization problem must satisfy

∑τi∈Γ V

∗i (t) = Vm(t)

and V ∗i (t) = Vimax , for τi ∈ Γ. Therefore, we only need to

consider the condition that νi = 0, for τi ∈ Γ. Suppose thatthe hth constraint in (9) is active, i.e., V ∗

h = Vhmin and

λh = λ0 + ν0 − 2wh(Vhmax − V ∗h ) (34)

= λ0 − 2wh(Vhmax − Vhmin).

For other constraints that are inactive, we have λk = 0 basedon (28). Based on (26), we have:

λ0

wi=

λi

wi+ 2(Vimax

− V ∗i ). (35)

By summing up the above equation for all i that satisfy V ∗i =

Viminwe can get:

λ0

∑V ∗i =Vimin

1

wi= 2

∑V ∗i =Vimin

(Vimax − V ∗i ), (36)

which is equivalent to

λ0

∑V ∗i =Vimin

1

wi(37)

= 2

( ∑V ∗i =Vimin

Vimax +∑

V ∗i =Vimin

Vimin

−∑

V ∗i =Vimin

Vimin −∑

V ∗i =Vimin

V ∗i

)

= 2(V (t)− Vm(t)

),

and thus:

λ0 =2(V (t)− Vm(t)

)∑

V ∗i =Vimin

(1/wi)(38)

as long as V (t) > Vm(t), λ0 > 0, λi ≥ 0 and constraint (10)are satisfied. Under such case, the optimal charging rate V ∗

i

either satisfies V ∗i = Vimin or

V ∗i = Vimax − λ0

2wi(39)

= Vimax −1wi

(V (t)− Vm(t)

)∑

V ∗i =Vimin

(1/wi).

Since Slater condition holds for problem (7)-(10), the KKTconditions provide necessary and sufficient condition for op-timality. Theorem 1 is proven.

12

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13

Ran Wang is currently working as a Project Officerat the School of Electrical & Electronic Engineering,Nanyang Technological University (NTU), Singa-pore. He received his B.E. in Electronic and Infor-mation Engineering from Honors School, Harbin In-stitute of Technology (HIT), P.R. China in July 2011and Ph.D. in Computer Engineering from NanyangTechnological University (NTU), Singapore in Jan-uary 2016. His current research interests includeintelligent management and control in Smart Gridand evolution of complex networks.

Ping Wang (M’08, SM’15) received the Ph.D.degree in electrical engineering from University ofWaterloo, Canada, in 2008. Currently she is anAssociate Professor in the School of Computer Engi-neering, Nanyang Technological University (NTU),Singapore. Her current research interests includeresource allocation in multimedia wireless networks,cloud computing, and smart grid. She was a core-cipient of the Best Paper Award from IEEE Wire-less Communications and Networking Conference(WCNC) 2012 and IEEE International Conference

on Communications (ICC) 2007. She is an Editor of IEEE Transactions onWireless Communications, EURASIP Journal on Wireless Communicationsand Networking, and International Journal of Ultra Wideband Communica-tions and Systems.

Gaoxi Xiao (M’99) received the B.S. and M.S.degrees in applied mathematics from Xidian Uni-versity, Xi’an, China, in 1991 and 1994 respectively.He was an Assistant Lecturer in Xidian Universityin 1994-1995. In 1998, he received the Ph.D. de-gree in computing from the Hong Kong PolytechnicUniversity. He was a Postdoctoral Research Fellowin Polytechnic University, Brooklyn, New York in1999; and a Visiting Scientist in the University ofTexas at Dallas in 1999-2001. He joined the Schoolof Electrical and Electronic Engineering, Nanyang

Technological University, Singapore, in 2001, where he is now an AssociateProfessor. His research interests include complex systems and networks,optical and wireless networking, smart grid, system resilience and Internettechnologies. He has published more than 130 papers in refereed journals andconferences.