Optimal Scheduling for Charging and Discharging
of Electric Vehicles
Yifeng He,Member,IEEE,Bala Venkatesh,Senior Member,IEEE,and Ling Guan,Fellow,IEEE
Abstract—The vehicle electrification will have a significant im-pact on the power grid due to the increa in electricity consump-tion.It is important to perform intelligent scheduling for charging and discharging of electric vehicles(EVs).However,there are two major challenges in the scheduling problem.First,it is challenging tofind the globally optimal scheduling solution which can mini-mize the total cost.Second,it is difficult tofind a distributed sched-uling scheme which can handle a large population and the random arrivals of the EVs.In this paper,we propo a globally optimal scheduling scheme and a locally optimal scheduling scheme for EV charging and discharging.Wefirst formulate a global sched-uling optimization problem,in which the charging powers are opti-mized to minimize the total cost of all EVs which perform charging and discharging during the day.The globally optimal solution pro-vides the globally minimal total cost.However,the globally optimal scheduling scheme is impractical since it requires the information on the future ba loads and the arrival times and the charging pe-riods of the EVs that will arrive in the future time of the day.To develop a practical scheduli
ng scheme,we then formulate a local scheduling optimization problem,which aims to minimize the total cost of the EVs in the current ongoing EV t in the local group.The locally optimal scheduling scheme is not only scalable to a large EV population but also resilient to the dynamic EV arrivals.Through simulations,we demonstrate that the locally optimal scheduling scheme can achieve a clo performance compared to the globally optimal scheduling scheme.
Index Terms—Charging and discharging,convex optimization, distributed solution,electric vehicle,optimal scheduling,smart grid,vehicle-to-grid(V2G).
N OMENCLATURE
Interval t.
Set of electric vehicles(EVs),
Charging-only EV t.
Vehicle-to-grid(V2G)EV t.
Charging power of EV in interval.
Charging period of EV.
Length of an interval
Initial energy of EV.
Battery capacity of EV.
Final energy of EV.
Manuscript received February21,2011;revid June20,2011,September 12,2011;accepted October17,2011.Date of publication July19,2012;date of current version August20,2012.Paper no.TSG-00027-2011.
The authors are with the Department of Electrical and Computer Engi-neering,Ryerson University,Toronto,Ontario,M5B2K3,Canada(e-mail: son.ca;bala@ryerson.ca;son.ca).
叮当猫乐队Color versions of one or more of thefigures in this paper are available online at ieeexplore.ieee.
Digital Object Identifier10.1109/TSG.2011.2173507
Maximum charging power.
无论如何的英文Final energy ratio of EV.
Charging-interval matrix.
Total load in interval.
Real ba load in interval.
Forecasted ba load in interval.
Charging load in interval.
Intercept in the real-time pricing model.asdfg
Slope in the real-time pricing model.
Cost for EV charging in interval.
Previous-interval t of interval.
Group t.2010年6月四级真题
Ongoing EV t at the beginning of interval in
group.
Sliding window at the beginning of interval in
group.
Charging-only EV t at the beginning of interval
in group.
V2G EV t at the beginning of interval in
group.
Arrival time of EV.
Departure time of EV.
Start time of the charging period of EV.
End time of the charging period of EV.
I.I NTRODUCTION
T HE automotive industry is heavily investing in plug-in hy-brid electric vehicles(PHEVs)and fully electric vehicles (EVs)mainly in order to reduce the CO emissions and oil de-pendency of current automotive technology.The vehicle elec-trification will have significant impacts on the power grid due to the increa in electricity consumption.
The overall load profile of electric system will be changed due to the introduction of EV charging and discharging.The charging of a large population of EVs has a significant impact on the power grid.It have been estimated that the total charging load of the EVs in US can reach18%of the U.S.summer peak at the EV penetration level of30%[1].On the other hand,an EV can also provide energy to the power grid by discharging
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the battery,which is known as vehicle-to-grid(V2G)[2].An intelligent scheduling scheme can optimally schedule the EV charging patterns such that the load profile of the electric system can be effectivelyflattened.This will reduce potential capital costs and minimize operational costs.Intelligent scheduling for EV charging and discharging has become a vital step towards smart grid implementation[3],[4].The esntial principle in intelligent scheduling is to reshape the load profile by charging the EV battery from the grid at the time when the demand is low and discharging the EV battery to the grid when the demand is high.However,it is challenging to schedule the patterns of EV charging and discharging in an optimal way.First,it is difficult tofind the globally optimal scheduling solution which can min-imize the overall charging cost,especially in the prence of a large EV population.Second,the scheduling scheme is required to have the capacity to efficiently handle the random arrivals of the EVs.
In the recent literature,a number of scheduling schemes for EV charging and discharging have been propod[5]–[8].How-ever,the scheduling schemes in[5],[6]only dealt with battery charging without V2G function.Though the existing work on V2G scheduling[7],[8]tried to optimize the charging and dis-charging powers to minimize the cost,their methods are esn-tially centralized algorithms,which may not be suitable for the EV charging and discharging systems with a large population and dynamic arrivals.
现在进行时表将来
In this paper,we propo a globally optimal scheduling scheme and a locally optimal scheduling scheme for EV charging and discharging.Our contributions are summarized as follows.
•We formulate a global scheduling optimization problem, which aims to minimize the total cost for charging all EVs within the day.The optimization problem is a convex optimization problem,which can be solved efficiently.
The globally optimal scheduling scheme determines the optimal charging powers for all EVs for all intervals by solving a single global scheduling optimization problem, thus obtaining the globally minimal total cost.
•We formulate a local scheduling optimization problem for the EVs in the local group.Bad on the local scheduling optimization problem,we develop a locally optimal sched-uling scheme,which is performed in an independent and distributed way.The locally optimal scheduling scheme is very appropriate for the EV charging and discharging systems with a large population and dynamic arrivals.The performance of the locally optimal scheduling scheme is lower than but very clo to that of the globally optimal scheduling scheme.
The globally optimal scheduling scheme provides the glob-ally minimal total cost.However,the globall
y optimal sched-uling scheme is impractical since it requires the information on the future ba loads and the arrival times and the charging pe-riods of the EVs that will arrive in the future time of the day. Though the locally optimal scheduling scheme performs a little wor than the globally optimal scheduling scheme,it it is a practical scheme which can efficiently handle a large EV popu-lation and dynamic EV arrivals.Therefore,the locally optimal scheduling scheme is thefinal solution suggested in the paper.With the globally minimal total cost provided by the globally optimal scheduling scheme,we canfind out the optimality gap between the two schemes.
The remainder of the paper is organized as follows.Section II discuss the related work.In Section III,we formulate and solve the global scheduling optimization problem.In Section IV,we formulate and solve the local scheduling op-timization problem.The simulation results are prented in Section V,and the conclusions are drawn in Section VI.
II.R ELATED W ORK
Depending on the direction of energyflow,existing work on EV charging scheduling can be classified into two class: 1)scheduling for charging only,and2)scheduling for both charging and discharging.
In charging-only scheduling,the scheduler tries to optimize the energyflow from the grid to the battery cfa培训
of the EV.In[5], Shrestha et al.optimized the EV battery charging during the low-cost off-peak period to minimize the charging cost in the context of Singapore.The paper in[9]examined the problem of optimizing the charge trajectory of a PHEV,defined as the time and the rate with which the PHEV obtains electricity from the power grid.In[1],a decentralized charging control algorithm was propod to schedule charging for large populations of EVs. The paper in[10]optimized EV battery charging behavior to minimize charging costs,achieving satisfactory state-of-energy levels,and optimal power balancing.Mets et al.in[6]prented smart energy control strategies for charging residential PHEVs, aiming to minimize the peak load andflatten the overall load profile.The impact of different battery charging rates of EVs on the power quality of smart grid distribution systems was studied in[11].In[12],Clement et al.propod coordinated charging with stochastic programming,which was introduced to repre-nt the error in the load forecasting.
In charging and discharging scheduling,the scheduler tries to optimize the bidirectional energyflows:from the grid to the EV battery and from the EV battery to the grid.Binary particle swarm methods were employed to optimize the V2G scheduling in a parking lot to maximize the profit[7],[8],[13].Sortomme et al.propod an unidirectional regulation at the aggregator, in which veral smart charging algorithms were examined to t the point about which the rate of charge varies while per-f
orming regulation[14].The paper in[16]developed an ag-gregator for V2G frequency regulation with the optimal control strategy,which aims to maximize the revenue.Jang et al.pro-pod a method for an analytic estimation of the probability dis-tribution of the procured power capacity(PPC),bad on which the optimal contract size was decided[17].The paper in[18] prented a real-time model of afleet of plug-in vehicles per-forming V2G power transactions.In[19],Singh et al.demon-strated that the coordinated charging and discharging of EVs can improve the voltage profile and reduce the power transmission loss.The paper in[15]discusd the vehicle to grid integration and described the vehicle-to-grid communication interface.
III.G LOBAL S CHEDULING O PTIMIZATION
In this ction,we formulate a global scheduling optimization for EV charging and discharging bad on a real-time pricing
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Fig.1.Charging period of EV.
model.The solution to the optimization problem provides a globally optimal scheduling scheme which minimizes the total cost.
A.System Models
We study the battery charging and discharging of EVs during a day,which is evenly divided into a t of intervals.The in-terval t is denoted by.The length of an interval is denoted by.We assume that the charging or discharging power in an interval is kept unchanged.In this paper,we divide the day into 24intervals such that the interval length is given by h. The t of the EVs,which perform charging
and discharging during the day,is denoted by.The EV t consists of two ts:1)the charging-only EV t,which includes the EVs that only charge their battery and do not provide the battery energy to the grid,and2)the V2G EV t, which includes the EVs that perform both battery charging and battery discharging.We have.The charging or discharging power of EV in interval is denoted by.In order to unify the notation, we just call the charging power of EV in interval.If ,it means that EV charges its battery in interval.If
,it means that EV discharges its battery in interval .The EVs in the charging-only t always satisfy since they do not discharge their battery at any time. On the other hand,the EVs in the V2G t may have a positive,zero,or negative charging power in interval
since they have bidirectional energyflows between the battery and the power grid.
The arrival time of EV,denoted by,is the time when EV is plugged into the charging station.The departure time of EV,denoted by,is the time when EV is plugged out of the charging station.The charging period of EV,de-noted by,is the period in which EV charges or discharges its battery.Since we divide the time into multiple intervals,we define the charging period of EV as the t of continuous intervals that fall between the arrival time and the departure time of EV,as illustrated in Fi
g.1.The initial energy of EV,denoted by,is defined as the battery energy at the arrival time.The battery capacity of EV is denoted by.Thefinal energy of EV,denoted by,is defined as the battery energy at the departure time.Thefinal energy is no larger than the battery capacity.We define afinal energy ratio of EV as where. The charging station can automatically detect the arrival time, the initial energy and the battery capacity of EV when the EV is connected to the charging station.The departure time and thefinal energy ratio of EV are provided to the charging station by the ur of EV before charging is started.The charging station can determine the charging period of EV from the parameters and.EV performs charging and discharging activities during the charging period.To reprent the relationship between the charging/discharging ac-tivities and the intervals,we define a charging-interval matrix
where and denote the number of elements in the t and the t,respectively.The elements of are defined as
(1) In this paper,we consider the scheduling of EV charging and discharging in a small geographic area.In our real-time pricing model,we make two assumptions:1)the loss between nodes are small and thus negligible,and2)there is no congestion in transmission.The two assumptions allow us to neglect the spa-tial variation of the electricity prices.The electricity price at a time instant is the same
regardless of the charging location.The optimizations of EV charging bad on only temporal variation but not spatial variation of the price have be en in[1],[6]. The electricity price is modeled as a linear function of the in-stant load[1],which is given as follows:
(2)
where is the intercept and is the slope,which are both non-negative real number,and is the total load at time. The total load in interval consists of two parts:1)the ba load,which reprents the load of all electricity consump-tions in interval except EV charging,and2)the charging load ,which reprents the load of EV charging in interval.We assume that the ba load remains constant in interval.The charging load in interval is given by. If the load from the grid to the batteries of the is greater than that from the batteries of the EVs to the grid in interval ,the charging load is positive.Otherwi,it is negative. The total load in interval is given by
.Since both the ba load and the charging power remain constant in interval, the total load is constant in interval.
In this paper,we define the charging cost in interval,denoted as,as the total amount of the money that the customers pay for charging and discharging of their EVs in interval.Bad on the pricing model,t
he charging cost in interval
is given by
manhub com
(3) As shown in(3),the charging cost can be positive or nega-tive.If the charging load,given by,in interval is positive,the charging cost is positive.Otherwi,it is negative.
B.Problem Formulation and Solution
In order tofind a globally optimal scheduling scheme for the EVs that perform charging and discharging during the day,we
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make the following assumptions:1)the arrival time and the de-parture time of each EV in the EV t are known(this is real-istic in the ca where each EV ur signs the charging contract and bring in the EV at a designated time);2)the initial energy and thefinal energy of the battery for each EV in the EV t are known;3)the ba load in each interval of the day is known; and4)a central controller collects all the information and then performs the scheduling optimization.
The total cost is defined as the sum of the charging costs over the interval t.The total cost is then given by
(4) The global scheduling optimization problem can be stated as to minimize the total cost of the EVs which perform charging and discharging during the day,by optimizing the total load in interval and the charging power
,subject to the relationship between the total load in an interval and the charging power of an individual EV,the instant energy constraints,thefinal energy constraints,and the lower bound and the upper bound of the charging power.Mathematically,the optimization problem can be formulated as follows:
(5a)
(5b)
(5c)
(5d)
(5e)
(5f) In the optimization problem(5),the objective function(5a) to be minimized is the total cost of the EVs which perform charging and discharging during the day.Constraints(5b)rep-rent the relationship between the total load in an interval and the charging power of an individual EV.Constraints(5c)are the instant energy constraints,which require the energy of EV
at the end of interval,given by
,to be no less than0and no larger than the battery capacity of EV.Constraints(5d) are thefinal energy constraints,which require thefinal energy of EV,given by, to be no less than the specified energy level,is given by .Constraints(5e)specify the lower bound0and the upper bound of the charging power for the EVs in the charging-only t.Constraints(5f)specify the lower bound and the upper bound of the charging power for the EVs in the V2G t.
In the optimization problem(5),the objective function(5a) is convex,and all the constraint functions are linear.Therefore the optimization problem(5)is a convex optimization problem, which can be solved efficiently with the interior point methods [20].The solution to the optimization problem(5)provides the globally optimal scheduling scheme for EV charging and dis-charging during the day.
IV.L OCAL S CHEDULING O PTIMIZATION
The globally optimal scheduling scheme gives the globally minimal total cost.However,the globally optimal scheduling scheme is impractical due to the following reasons.First,the EVs that will arrive in the future time of the day are unknown at the current moment.Second,the ba load in the future time of the day is unknown at the current moment.Third,it is not scalable for a centralized scheduling scheme in which the cen-tral controller may be overrun by a large number of EVs.
In this ction,we formulate a local scheduling optimiza-tion problem,which relaxes the assumptions ud in the global scheduling optimization problem(5).The solution to the local scheduling optimization problem is a locally optimal scheduling scheme,which can achieve the performance clo to that in the globally optimal scheduling scheme.Compared to the glob-ally optimal scheduling scheme,the locally optimal scheduling scheme is practical and scalable.
A.Problem Formulation and Solution
In the globally optimal scheduling scheme,since we assume that we have the global knowledge of the information about the EVs and the ba load within the day,we canfind the optimal charging powers at each interval by solving the global sched-uling optimization problem(5)only once.In the loc
ally op-timal scheduling scheme,we do not know the information of the future load and the future EVs.We propo a locally op-timal scheduling scheme tofind the optimal charging powers in the next interval for the local EVs by using a sliding window mechanism.
In the locally optimal scheduling scheme,we perform the scheduling optimization bad on groups.A group of EVs in-cludes the EVs in one location or multiple nearby locations.For example,the EVs which perform charging and discharging in a parking lot can be classified into a group,and the EVs in a residential garage can be classified into another group.There is a local controller(LC)for each group.The communications and controls in the locally optimal scheduling scheme are il-lustrated in Fig.2.The local controller establishes communica-tion connections with the central controller located in the utility company and the charging stations at the local site.The local controller receives the forecasted loads for the day from the central controller.The local controller communicates with each charging station in real time to collect the EV information,bad on which it performs scheduling optimization and then instructs each local EV to charge or discharge the battery with the optimal charging powers.
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Fig.2.Illustration of communications and controls in the locally optimal scheduling
scheme.
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Fig.3.Illustration of the ongoing EV t and the sliding window in the locally optimal scheduling scheme.
We denote the group t by .Since each local controller per-forms scheduling independently,we will just study the sched-uling optimization in group .The local controller does not know the future arrivals of the EVs in the group.There-fore,we propo to update the charging powers at the begin-ning of each interval by using a sliding window.At the begin-ning of interval ,we need to first determine the
current ongoing EV t
and the current sliding window .Let the current time be the beginning of interval
.Each EV has a charging period.The start time and
the end time of the charging period of EV is denoted by
and ,respectively.If EV satis fies and
,we say that EV belongs to the current ongoing
EV t .The current sliding window at the beginning of interval is de fined as the t of the concutive intervals be-tween the start time and the end time of the sliding window.The start time of the sliding window is always given by ,and the end time of the sliding window is de fined by .Fig.3illustrates the ongoing EV t and the sliding window at the beginning of interval 2.As shown in Fig.3,EV 1has completed charging since and .EVs 2,3,and 4satisfy
and .Therefore the current ongoing
EV t is given by
,and the current sliding window is given by
.EV
performs charging and discharging activities during its charging period.At the beginning of interval ,we de fine a charging-interval matrix
who elements are given by
(6)
宾语从句pptIn order to determine the charging powers in the current sliding window,we need to know the ba loads in the sliding window ,which can be forecasted using similar-day approach,regression methods or time-ries methods [21].In this paper,we adopt the similar-day approach [21],in which the ba load in each interval of the sliding window is estimated by averaging the ba loads of the same interval of the recent days with similar weather conditions.The forecasted ba load is denoted by for .
Bad on the current ongoing EV t and the current sliding window ,we formulate the local scheduling opti-mization problem for the current moment in group .The opti-mization problem can be stated as to minimize the total cost of
the EVs in the current ongoing EV t
during the current sliding window
,by optimizing the total load in interval and the charging power
,subject to the relationship between the total load in an interval and the charging power of an individual EV ,the instant energy constraints,the final energy constraints,and the lower bound and the upper bound of the charging power.Mathemati-cally,the optimization problem can be formulated as follows.
(7a)
(7b)
(7c)(7d)(7e)(7f)
the kings speech
In the local scheduling optimization problem (7),the objec-tive function (7a)to be minimized is the total cost of the EVs in the current ongoing EV t during the current sliding
window
.
Constraints (7b)reprent the relationship be-tween the total load and the charging power of an individual EV in an interval of the current sliding window .Con-straints (7c)are the instant energy constraints,which require the energy of EV at the end of interval
,given by ,
to be no less than 0and no larger than the battery capacity
of EV .In Constraints (7c),denotes the energy at the beginning of interval denotes the current pre-vious-interval t,de fined as the t of intervals that belong to the current sliding window but are no later than interval .Constraints (7d)are the final energy constraints,which re-quire the final energy of EV
to be no less than .Constraints (7e)specify the lower bound 0and
