Centralized Bi-level Spatial-Temporal Coordination ... - IEEE Xplore

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CSEE JOURNAL OF POWER AND ENERGY SYSTEMS, VOL. 1, NO. 4, DECEMBER 2015

Centralized Bi-level Spatial-Temporal Coordination Charging Strategy for Area Electric Vehicles Lei Yu, Tianyang Zhao, Qifang Chen, and Jianhua Zhang, Member, CSEE

Abstract—Increased penetration of electric vehicles (EVs) is expected to impact power system performance in adverse ways, e.g., overloading, uncertain power quality, and increased voltage fluctuation, particularly at the distribution level. Most EV charging control strategies that have been proposed only benefit the grid or EV users. A centralized EV charging strategy based on bilevel optimization is proposed in this paper with the objectives of deriving benefits for the grid and EV users simultaneously. The proposed strategy involves distributing the EV charging load more beneficially across both spatial and temporal levels. In the spatial problem, the whole fleet of EVs is controlled to minimize load variance as spatial coordination, with total charging rate and energy needed as the constraint. While in the temporal problem, EVs in each aggregator are controlled to minimize the charging cost or maximize the EV user’s degree of satisfaction with each aggregator’s charging rate and energy needed as the constraint. The proposed bi-level charging strategy is transformed to a single-stage optimization problem and solved using the classical optimization method. The impacts of uncontrolled charging on the grid and EV users are studied using the Monte Carlo Simulation (MCS) method. Then, the effectiveness of the proposed charging strategy is demonstrated via results obtained in the MCS. Index Terms—Bi-level optimization, centralized charging, electric vehicles, spatial temporal coordination.

I. I NTRODUCTION S energy security needs and climate change measures are viewed globally as pathways to reduce dependence on petroleum resources, electric vehicles (EVs) have drawn attention in the new energy automobile market for their high efficiency, low noise, zero emissions, and zero fossil energy use. EVs represent an important breakthrough for the automobile industry to curb CO2 emissions. A case in point is a study in Denmark that has demonstrated how integration of the electric power and transportation sectors can lead to reduced transportation related CO2 emissions by 85% [1]. An analysis conducted for three regions in China suggests that CO2 reductions occur under all scenarios, even in regions that

A

Manuscript received April 21, 2015; revised August 18 and October 26, 2015; accepted October 29, 2015. Date of publication December 30, 2015; date of current version December 7, 2015. This work was supported by the Fundamental Research Funds for the Central Universities (No. 2014XS09). L. Yu, T. Y. Zhao (corresponding author), Q. F. Chen, and J. H. Zhang are with State Key Laboratory of Alternate Power System with Renewable Energy Resources, North China Electric Power University, Beijing 102206, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.17775/CSEEJPES.2015.00050

rely heavily on coal [2]. The Chinese government has stated the goal to attain the accumulated production and sales of EVs to 0.5 million by 2015 and 5 million by 2020 [3]. It has been well recognized that the widespread adoption of EVs connected to the grid along with uncontrolled charging will cause adverse impacts on the planning and operation of the power system from different perspectives [4]–[10]. In fact, in [4], researchers have shown that peak load will increase under a simple charging strategy, requiring extra investment in generation and transmission capacity. Based on real data of the driving behavior of light duty vehicles and data from the Ontario Electric Power system [5], it is estimated that 133,548 plug-in EVs could be introduced without adversely affecting the electric grid in Ontario, Canada, by 2025. In the U.S., the existing grid capacity would allow for 73% of the light-duty fleet converted to PHEVs [6]. Reference [7] is the earliest literature that investigates the impacts of EVs on distribution networks. In [8], it is suggested that EV penetrations over 20% will bring 35.8% load growth to the distribution network. EV charging load impacting voltage profiles of distribution networks is described in [9]. Based on the integration of power systems and transportation analysis, a spatial-temporal model of a plug-in EV charging load is formulated in [10]. This work analyzes the effect of largescale deployment of EVs on urban distribution networks. At the same time, the wide adoption of EVs presents opportunities for power system operation and EV users. An important feature of EVs is that they can be charged for relatively long periods, especially during low price and low demand times [10], [11]. Based on this, researchers have developed various EV charging strategies, e.g., delayed charging, nighttime charging, and smart charging [11]. The drawbacks and advantages of each charging strategy have been well surveyed in [12] from the perspectives of implementation, information, communication technology (ICT) functions, and so on. Since smart charging allows EV users and network operators to pursue technical and economic benefits by scheduling EV charging plans, smart charging has become a very promising option for coordinating EV charging within the network. As a first step to implementing the smart charging strategy, the controllability factor of EVs needs to be studied. Vehicle technology, driving patterns, and charging behavior are the three main factors affecting the energy use of EVs [13]. To this end, in [14], researchers have surveyed battery charger topologies, charging power levels, and charging infrastructure for plug-in EVs. To the best of our knowledge, the first attempt

c 2015 CSEE 2096-0042

YU et al.: CENTRALIZED BI-LEVEL SPATIAL-TEMPORAL COORDINATION CHARGING STRATEGY FOR AREA ELECTRIC VEHICLES

to study EV charging controllability appears in [15]. Traditionally, controllability is only modeled from a temporal perspective [12]. In [16], however, the spatial distribution of EVs is seen as also playing an important role in the smart charging strategy. To coordinate EV charging with other entities, the smart charging scheme should have various types of objective functions, e.g., charging cost minimization, frequency regulation, and power loss reduction [11]. According to the distinction of implication structures, these strategies can be classified as decentralized control, centralized control, and a mixture [17]– [22]. In decentralized/agent-based control, the EV aggregator will locally optimize the EV charging in its control area and share limited information with the external aggregators. In centralized control, all information is sent to the central controller and the charging plan is made only by the central controller, while in the mixed control, both aggregators and central controllers can decide how EVs should be charged. In [17], a decentralized charging strategy for the utility company and the EV aggregators is formulated as a no-cooperative Stackelberg game to realize the demand response of EVs. First, the utility company publishes the charging price as the leader, and then the EV aggregators act according to the given price as followers. An efficient iterative distributed solving method based on the Karush-Kuhn-Tucker (KKT) conditions for the EV aggregators’ decision making is also proposed. In [18], the authors propose both centralized and decentralized charging strategies for the utility company by optimizing the EVs’ charging and discharging rates to minimize the total electricity bill. In the decentralized charging strategy, the consumers only need to report their aggregated demand to the utility company. Since decentralized control does not always guarantee that a global optimum charging plan can be obtained, a centralized control for EVs is called for where each EV charging is controlled by a centralized controller. In [20], a centralized charging strategy based on maximum sensitivity selection optimization is proposed to minimize power losses and enhance the voltage profile. To reduce the power loss, the changing frequency of the on-load tap changer is minimized, which smoothens the distribution of daily load profiles and improves EV users’ degree of satisfaction (DOS) simultaneously. In [21] a centralized charging optimization model is proposed in which charging and discharging behaviors of EVs connected to one specific node are optimized. In a centralized control strategy, each EV’s charging/discharging rate needs to be optimized, which in turn is not feasible for controlling EVs on a large scale. Thus a mixed charging strategy is proposed in [21], consisting of three steps: aggregation, centralized energy optimization, and decentralized power control. Moreover, since EV charging causes geographically mobile demand rather than stationary loads [11], it is necessary to also take into consideration the spatial perspective. This is especially relevant in urban distribution power supply zones, where EV charging can pose challenges in terms of expanding the distribution facilities, e.g., cables and distribution transformers. Most smart charging strategies proposed in [17]–[22] pursue a specific objective, i.e., since EVs are owned by the

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users and distribution networks are owned by the utilities, they tend to pursue conflicting objectives under different market frameworks, e.g., charging cost minimization for the users and flattened load profile for the grid. In this paper, based on our previous work on batteryswap mode EV charging [23], we propose an agent-based centralized spatial-temporal coordination charging strategy for the plug-in EVs within a given area, e.g., a distribution power supply zone. The strategy takes both the spatial and temporal characteristics of EV charging controllability into consideration while allowing the EV charging controller to pursue different objectives from specific levels, e.g., load flattening and charging cost minimization. This paper is organized as follows. Section II describes the energy model of EVs, followed by a discussion of the energy model of EV aggregator and introduction of our centralized spatial-temporal coordination charging framework. In section III, a centralized spatial-temporal coordination charging model is proposed for the EV aggregators as a bi-level optimization problem. The solving method is proposed in section IV. In section V, the impact of uncontrolled charging on the grid and EV users is described, as well as two case studies that verify the effectiveness of the proposed charging strategy. Conclusions of this work are presented in section VI. II. S YSTEM M ODEL In this work, the EVs within a given subarea are controlled by a specific aggregator. The aggregator predicts when the EV will be plugged in and when it leaves the charging pilot, as well as the energy needed by each EV. The aggregator then updates its charging capability to a central control center (CCC). The controller then constructs the charging plan for all aggregators for the next day. The control period is divided into T equals time slots, and the control step ∆h is set to 1 hour. A. Electric Vehicle Model Each EV i in aggregator k is specified by its plug-in time tin,k,i , departure time tdep,k,i , energy needed Ereq,k,i , and maximum charging rate pmax,k,i . In our research, it is necessary to assume (tdep,k,i − tin,k,i ) pmax,k,i ≥ Ereq,k,i , which means the EV should be integrated to the charging pilot before its energy needs can be fulfilled. Similar to [22], the maximum and minimum cumulative energy needed at each time interval t during its integration period Ti is depicted as: t Emax,k,i = min{pmax,k,i (t − tin,k,i ), Ereq,k,i }

∀t ∈ Ti , i ∈ Ik , k ∈ K

(1)

t Emin,k,i = min{pmax,k,i (tdep,k,i − t),

max{Ereq,k,i − pmax,k,i (tdep,k,i − t), 0}}

(2)

∀t ∈ Ti , i ∈ Ik , k ∈ K where i is the ith EV; k is the k th aggregator; t is the tth t t time interval; Emax,k,i and Emin,k,i are the maximum and minimum cumulative energy needed by the ith EV within the k th aggregator at time interval t during the integration period Ti , MWh; tin,k,i and tdep,k,i are the plug-in and departure

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CSEE JOURNAL OF POWER AND ENERGY SYSTEMS, VOL. 1, NO. 4, DECEMBER 2015

time of the ith EV within the k th aggregator, hour; Ereq,k,i is the energy needed by the ith EV within the k th aggregator when plug-in, MWh; Ti is the set of integration period of the ith EV within the k th aggregator, [tin,k,i , tdep,k,i ]; Ik is the set of all EVs within the k th aggregator; and K is the set of aggregators. As presented in (1), the cumulative energy EV i charged from the pilot at time interval t will not exceed the energy needed. Further, as shown in (2), the cumulative energy EV i charged from the pilot at time interval t is limited by both the energy needed and charging pilot’s charging capacity.

charging on the grid, and this can benefit EV users, who will be able to adjust the charging rate during the integration period of their EV. In the control scheme designed for this work, there is more than one aggregator. How to coordinate all aggregators’ charging behavior to benefit the grid is defined as the spatial coordination, and the coordination of each aggregator’s charging behavior to benefit the EV user is defined as the temporal coordination. Central charging center

The grid

B. Electric Vehicle Aggregator Model Aggregator 1

At time interval t, the cumulative maximum and minimum energy needed by aggregator k can be represented as follows: X t t Emax,k = {(t ∈ Ti ) & Emax,k,i }, ∀t ∈ T, k ∈ K (3) i∈K t Emin,k =

X

t {(t ∈ Ti ) & Emin,k,i }, ∀t ∈ T, k ∈ K

(4)

i∈K t t where Emax,k and Emin,k are the maximum and minimum cumulative energy needed by the EVs within the k th aggregator at time interval t, MWh; T is the set of control period. As shown in (3), EVs within aggregator k are aggregated from the spatial perspective, which means the aggregator will gather each EV’s maximum and minimum energy needed at each time interval. The EV aggregator will charge the EVs according to the CCC’s charging plan expressed in the next section, and the CCC will not directly control each EV’s charging process. Since our concern is about the energy flow for the next day, real-time power control is not included in this paper. Due to the spatial distribution of the charging pilots, the maximum EV charging power is not only limited by the energy needed by the group of EVs within the aggregator k, but is also limited by the EV charging pilot’s charging capacity. The maximum EV charging power can be depicted as:

0≤

≤

X

pmax,k,i , ∀t ∈ T, k ∈ K

t Emin,k ≤

. Pev,l,1i

. Pev,l,1Nk

l,2 Pev,1

.

. Pev,l,2Nk

l,3 Pev,1

.

. Pev,l,3N k

l,4 Pev,1

.

. Pev,l,4Nk

Pevu,1 Pevu,2 u,3 Pev,1 u,4 Pev,1

. l, j Pev,1

. . l, j . Pev, i

. . . l, j . Pev, Nk

Pevu, j

.

.

. . . l,T . Pev,i

. . . l,T . Pev, N k

.

l,T Pev,1

.

Lower optimization

Pevu,T Upper optimization

Energy flow Information flow

Spatial coordination

(5)

i=1 t X

l,1 Pev,1

Fig. 1. Schematic diagram of the control mode.

NOI,k t Pev,k

Temporal coordination

Aggregator N k

Aggregator i

j t Pev,k ∆h ≤ Emax,k , ∀t ∈ T, k ∈ K

(6)

j=1 t where Pev,k is the charging load of the k th aggregator at time interval t, MW; NOI,k is the number of the charging pilots within aggregator k.

C. Spatial and Temporal Coordination of EV Charging After the formulation of each aggregator’s charging capability, as described in section II B., all aggregators are controlled by a CCC, as shown in Fig. 1. The CCC will make the charging plan for all aggregators along the control period, e.g., from 15:00 to 08:00 when most EVs arrive at home and remain stationary according to the survey carried out in [24]. Since the CCC is the bridge between the grid and users, it will function to reduce the adverse impacts of uncontrolled

By coordinating aggregator charging behavior as a way to minimize real power losses or load shifting along the control horizon, the CCC will be able to solve the spatial coordination optimization problem (upper level optimization), from which u,t can be obtained the spatial coordination charging plan {Pev , t ∈ T } (as shown in Fig. 1). The charging plan will then be set as the reference charging point for all aggregators. In the temporal coordination procedure (lower level optimization), each aggregator decides its temporal coordination l,t charging plan {Pev,k , k ∈ K, t ∈ T } (as shown in Fig. 1) as a means to minimize the charging cost or to improve EV users’ DOS. The spatial coordination plan, thus, functions as the soft constraint. In other words, the spatial coordination could be treated as the unique leader that determines the quota for all aggregators at each time interval, while all aggregators would compete with each other in the temporal coordination procedure to obtain the maximum amount of products to maximize benefits. Therefore, the spatial and temporal coor-

YU et al.: CENTRALIZED BI-LEVEL SPATIAL-TEMPORAL COORDINATION CHARGING STRATEGY FOR AREA ELECTRIC VEHICLES

dination are separated in pursuing objective functions, and yet are integrated in controlled objects. It is for this reason then that we model the spatial and temporal coordination charging strategy as a hierarchical optimization problem, as shown in the next section.

As in the spatial coordination, since the CCC controls EV charging from the spatial perspective, it is necessary to take the following constraints into consideration: 1) Total charging load will not exceed charging capacity of all charging pilots, and vehicle to grid (V2G) will not be considered, as expressed in (8):

III. B I - LEVEL S PATIAL AND T EMPORAL C OORDINATION C HARGING S TRATEGY

u,t 0 ≤ Pev ≤

In this section, the spatial temporal coordination charging strategy will be formulated as a bi-level optimization problem (BLP), which is essentially a class of hierarchical mathematical programing [25]. BLP has origins both in mathematics and game theory [26]. In terms of the former, BLP was formulated by Jerome Bracken and James T. McGill in 1973, and defined as a mathematical program with an optimization problem in its constraints [27]. BLP has also been well defined from a game theory perspective since its application to the Stackelberg games [25], [26]. BLP’s multi-level programming advantage has led to its wide deployment across many fields, such as revenue management, congestion management, origindestination matrix estimation, and network design [25]. In a BLP problem, two or more mathematical programs are formulated into a single two-stage instance, with one being part of the constraints of the other. Because of the existence of this hierarchical relationship, there is always an upper problem, which makes the decision in the first stage, and then a lower problem, which reacts to this decision in the second stage [28]. In our work, as stated in section II C., the spatial coordination is formulated as the upper problem, and the temporal coordination is formulated as the lower problem. B. Upper Level Optimization Taking into account the regulated power market policy in China, the main purpose behind spatial coordination in the day-ahead market is to minimize the real power losses. In reference [31] it has been shown that feeder loss minimization and load variance reductions are equivalent. Then, the role of the upper level optimization (spatial coordination) is to coordinate EV charging load with a given load profile predicted in the day-ahead market. The objective function can be described as follows:

min Pvar u,t l,t Pev ,Pev,k ,∀t∈T

=

T t u,t X {(Plocal +Pev − t=1

+C(

Nk X

k=1

i=1

t u,t (Plocal +Pev )

T

Nk NX OI,k X

pmax,k,i , ∀t ∈ T.

(8)

k=1 i=1

A. Brief Review of Bi-level Optimization Problem

T P

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)

2

l,t u,t 2 Pev,k − Pev ) }

(7) where Pvar is the load variance during control period MW; u,t Pev is the charging load at time interval t in the upper l,t optimization problem MW; Pev,k is the charging load of the th k aggregator at time interval t in the lower optimization t problem MW; Nk is the number of aggregators; Plocal is the given load profile at time interval t within the given area MW; and C is the penalty factor.

2) Cumulative energy obtained from the grid will not exceed maximum energy needed by all EVs, and will be larger than the minimum energy needed by all EVs, as expressed in (9). Nk X

t Emin,k ≤

t X

u,j Pev ∆h ≤

j=1

k=1

Nk X

t Emax,k , ∀t ∈ T. (9)

k=1

From (7)–(9), the spatial coordination charging is formulated as a linear constrained quadratic programing problem (LCQP) and is the upper optimization problem. Since each EV aggregator’s charging load is unknown, the upper problem, therefore, cannot be settled until the temporal coordination is solved. C. Lower Level Optimization To realize the temporal coordination of EVs within a given subarea, the lower level optimization problem for each aggregator can be formulated as: min fc,k = l,t Pev,k ,∀t∈T

T X

l,t t Pev,k cev,k ∆h

(10)

t=1

where fc,k is the charging cost for the k th aggregator, yuan; ctev,k is the charging cost coefficient for the k th aggregator at time t, yuan/MWh. From an EV user’s perspective, EVs should be well charged when they are needed. Therefore, if each aggregator wants to improve EV users’ DOS, it would be feasible to minimize the gap between the cumulative energy obtained from the grid and cumulative maximum energy needed, as shown in (11): min Dos,k =

T t X X l,j t ( Pev,k ∆h − Emax,k )2

l,j Pev,k ,∀t∈T

t=1 j=1

(11)

where Dos,k is EV users’ DOS within the k th aggregator, further Dos represents for all EV users’ DOS. Similar to (8) and (9), constraints for each aggregator can be represented as follows: NOI,k

0≤

l,t Pev,k

≤

X

pmax,k,i , ∀t ∈ T, k ∈ K

(12)

i=1 t Emin,k ≤

t X

l,t t Pev,k ∆h ≤ Emax,k , ∀t ∈ T, k ∈ K.

(13)

j=1

With (12) and (13), the temporal coordination charging is formulated as a linear programming (LP) problem with objective function (10) defined as the lower price coordination

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method. Further, when the objective function is replaced by objective function (11) formulating an LCQP, it is defined as the lower DOS coordination method. The lower price coordination method or lower DOS coordination method are integrated into the spatial coordination as the lower level optimization problem. IV. S OLVING M ETHOD Since the BLP is a relatively complex approach, researchers have deployed complementary pivoting, descent methods, penalty function methods and so on, to solve the BLP [28]. As shown in section III, the decision variables in our coordination strategy are the EV’s charging plan during the next day, continuous variables with linear constraints, and quadratic objective functions. Moreover, it is feasible to transform the BLP into a single stage optimization problem based on first order optimal conditions for non-linear problems, such as KKT conditions or Fritz John (FJ) (necessary) conditions [24]. In this section, the lower level optimization of LCQP is taken as an example, as the LP can be transferred in a similar way. According to the method proposed in [29], the core idea is to translate the lower problem into a series of linear or non-linear constraints integrated into the upper problem. In our case, the KKT conditions are deployed to transform the lower level optimization problem into a series of nonlinear constraints, as presented in section III C. In order to obtain these constraints, the Lagrangian relaxation for each aggregator is first introduced: min

l,j Pev,k ,λt1,k ,λt2,k ,λt3,k ,λt4,k ,∀t∈T

Lk =

t T X X l,j t Pev,k ∆h − Emax,k )2 ( t=1 j=1

− −

T X t=1 T X t=1

−

T X

l,j λt1,k Pev,k −

T X

NOI,k

λt2,k (

t=1

X

l,j pmax,k,i − Pev,k )

(14)

i=1

t X l,j t λt3,k ( Pev,k ∆h − Emin,k )

t=1

t X

l,j Pev,k ∆h)

j=1

where λt1,k , λt2,k , λt3,k , λt4,k are the Lagrangian multipliers for the constraints of the k th aggregator in the lower optimization problem. After the KKT approach is deployed, the lower optimization problem is represented as follows: l,t t 2Pev,k ∆h2 −2Emax,k − λt1,k +

λt2,k −

t X

λj3,k ∆h +

j=1 l,t t λ1,k Pev,k =

t X

λj4,k ∆h = 0

X i=1

l,t t Pev,k ∆h − Emin,k )=0

j=1 t λt4,k (Emax,k −

t X

l,t Pev,k ∆h) = 0

j=1

λt1,k , λt2,k , λt3,k , λt4,k ≥ 0, ∀t ∈ T, k ∈ K. As shown in (15), a series of constraints are formulated to replace the lower/temporal optimization problem. The complementarity constraints in (15) are bilinear; it is possible to transform these bilinear constraints into mixed-integer linear constraints, according to the method deployed in [30]. t 0 ≤ λt1,k ≤ M1,k M0 l,t t Pev,k ≤ (1 − M1,k )M0 t 0 ≤ λt2,k ≤ M2,k M0 NOI,k

X

l,t t pmax,k,i − Pev,k )M0 ≤ (1 − M2,k

i=1 t 0 ≤ λt3,k ≤ M3,k M0 t X

(16)

l,t t t Pev,k ∆h − Emin,k ≤ (1 − M3,k )M0

j=1 t 0 ≤ λt4,k ≤ M4,k M0 t Emax,k −

t X

l,t t Pev,k ∆h ≤ (1 − M4,k )M0 , ∀t ∈ T, k ∈ K

j=1 t t t t where M1,k , M2,k , M3,k , M4,k are auxiliary binary vectors of Lagrangian multipliers in the lower optimization problem; M0 is a big scalar that should include the range of the Lagrangian multipliers and is set to 100,000 in this paper. After the reformulation proposed above, the bi-level spatialtemporal-coordination optimization problem is transformed into a single stage mixed-integer LCQP program, which can be solved using classical optimization techniques.

In this section, the performance of the proposed spatial and temporal coordination charging strategy is evaluated using real EV user traveling data based on a survey conducted in Beijing [24]. The impacts of uncontrolled charging on the grid and EV users are carried out in section V B. The uncontrolled charging refers to that EVs would be charged immediately when being integrated to the charging pilots and would depart from the charging pilots when being well charged. Then the proposed strategies, lower price coordination and lower DOS coordination (see section III C.) are deployed to mitigate the impacts of uncontrolled charging on the grid and guarantee EV users’ DOS.

j=1

0

A. Case Description

NOI,k

λt2,k (

t X

V. P ERFORMANCE E VALUATION

j=1 t λt4,k (Emax,k −

λt3,k (

l,t pmax,k,i − Pev,k )=0

(15)

As the discussion in section II A. illustrates, three factors define an EV’s traveling characteristics, i.e., plug-in time,

YU et al.: CENTRALIZED BI-LEVEL SPATIAL-TEMPORAL COORDINATION CHARGING STRATEGY FOR AREA ELECTRIC VEHICLES

departure time, and energy needed. These data are obtained from the work carried out in [24]. ft,in =

1 0.75π[1 + 3(tin,k,i − 17.2)2 ]

0 ≤ tin,k,i ≤ 24

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There are five aggregators in this study. The CCC gives all five the reference-charging plan, according to the spatial coordination strategy presented in section III B. The given load profile is shown in Fig. 3.

(17)

15

ft,dep

No-EV Load (MW)

2

(t −18.2) 1 − dep,k,i 2×0.82 e =√ 2π × 0.8

0 ≤ tdep,k,i ≤ 24 (18)

fds,k,i

x 1 − b5 = a da−1 s,k,i e b (a − 1)!

ds,k,i ≥ 0

(19)

where ft,in is the probability distribution function (pdf) of the starting time of EVs; ft,dep is the pdf of the departure time of EVs; fds,k,i is the pdf of the travel distance per time of EVs; ds,k,i travel distance per time of the ith EV within the k th aggregator, km. Then the energy needed by each EV can be depicted as: Ereq,k,i = ds,k,i eav /ηc /1000, ∀i ∈ Ik , k ∈ K

(20)

where eav is the energy needed per kilo meter, and is set to 0.15 kWh/km; ηc is the charging efficiency, and is set to 0.9. Since the EV users’ driving behavior is stochastic, the Monte-Carlo Simulation (MCS) method is used in this paper to verify the effectiveness of our proposed strategy under most conditions. The simulation process could be depicted by Fig. 2, where Iter is the counter in MCS, and Itermax is the maximum simulation times. Pdf pdf of of EVs EVs plug plug in in time time

Pdf pdf of of EVs EVs departure departure time time

Pdf pdf of of EVs EVs travel travel distance distance

Real Real data data according according to to the the survey survey [19] [19]

EVs’ travel EVsÿ travel distance distance

EVs’ EVs departure departure time time

EVs EVs driving driving behavior behavior

Impacts Impacts on on the the grid grid Impacts Impacts on on the the users users Lower Lower price price coordination coordination

Impacts Impacts on on the the grid grid Impacts Impacts on on the the users users Uncontrolled Uncontrolled charging charging

Impacts Impacts on on the the grid grid Impacts Impacts on on the the users users Lower Lower DOS DOS coordination coordination

Proposed Proposed strategy Strategy Lower Lower the the impacts impacts on on the the grid? grid?

Users’ charging Usersÿcharging cost cost decreases? decreases?

Is Is the the proposed proposed strategy strategy effective? effective? IIter ter