Applied Energy 178 (2016) 155–163
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Applied Energy journal homepage: www.elsevier.com/locate/apenergy
Pricing mechanisms design for guiding electric vehicle charging to fill load valley Zechun Hu a,⇑, Kaiqiao Zhan b, Hongcai Zhang a, Yonghua Song a a b
Department of Electrical Engineering, Tsinghua University, Beijing, People’s Republic of China Electric Power Research Institute, China Southern Grid, Guangzhou, People’s Republic of China
h i g h l i g h t s Pricing mechanisms to guide EV charging behaviors for load valley filling are investigated. Sufficient and necessary conditions of the valley-filling pricing mechanisms are derived. Coordinated charging strategies are proposed for non-cooperative and cooperative scenarios. Simulation results show effectiveness of the proposed pricing mechanisms.
a r t i c l e
i n f o
Article history: Received 3 April 2016 Received in revised form 29 May 2016 Accepted 11 June 2016
Keywords: Electric vehicles Load valley filling Pricing mechanism Coordinated charging Nash equilibrium
a b s t r a c t The uncoordinated charging load of large-scale electric vehicles (EVs) may increase the gap between peak load and valley load of future power grids. By designing proper charging pricing mechanisms for EVs to guide their charging behaviors, the negative effect caused by uncoordinated EV charging can be alleviated and the flexible EV charging load can even help achieve valley filling for the power grid and therefore increases the social welfare. This paper designs two valley-filling pricing mechanisms under non-cooperative and cooperative scenarios respectively that can incent EV owners to shift their charging schedules for flattening the power load profiles. In the non-cooperative scenario, each EV schedules its own charging power without cooperation with the other EVs, while in the cooperative scenario, all the EVs are controlled by an aggregator. Sufficient and necessary conditions of the valley-filling pricing mechanisms are derived for both the non-cooperative and cooperative scenarios. And the corresponding coordinated charging strategies are proposed for the two scenarios, respectively. Simulation results show that under the proposed pricing mechanisms, cost-minimizing charging schedules of self-interested EVs can also fill the load valley effectively. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Because of the growing concerns about fossil energy shortage and environmental pollution [1,2], electric vehicles (EVs) have drawn much attention around the world. Battery modeling [3], EV energy management [4] and vehicle-to-grid (V2G) [5] technology have been research hotspots over recent years. On one hand, without proper control, high penetrations of EVs will increase the gap between peak load and valley load of the power grid and may overload distribution lines and transformers, resulting in higher network losses and shorter equipment lifetime [6–8]. On the other hand, EVs can also be viewed as interruptible loads or even distributed energy storage devices and they can ⇑ Corresponding author. E-mail address:
[email protected] (Z. Hu). http://dx.doi.org/10.1016/j.apenergy.2016.06.025 0306-2619/Ó 2016 Elsevier Ltd. All rights reserved.
support the optimal operation of the power grid, e.g., decreasing the network losses [9–12], balancing renewable energy fluctuations [13,14], participating frequency regulation [15,16] and so forth. One important application of coordinated charging of EVs is load valley filling. With a better valley-filling performance, the utilization of electrical equipment can be improved. And therefore, the electric power companies can serve power load with lower costs. According to [17], most EVs are used only 4% of the time for transportation. Thus, EV owners have the flexibility to select during which periods they will charge their vehicles, making the EV charging load an ideal resource to fill the load valley. Coordinated EV charging strategies in previous published papers can mainly be divided into three categories: centralized charging strategies [18–23], decentralized charging strategies [24–30], hierarchical charging strategies [31,32].
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Ref. [18] proposed an aggregated EV charging model and developed a centralized charging strategy to reduce the power fluctuations. Ref. [19] developed a strategy for peak load shaving and valley filling using V2G technology. The influences of the EV population and target curve were analyzed quantitatively. Ref. [20] proposed a centralized strategy to optimize the battery swapping behaviors of EVs. In [21], a centralized double-layer smart charging management algorithm was designed which not only optimizes the charging power of EVs but also navigates the EVs to proper charging stations. In [22], the authors developed an optimization technique to coordinate V2G power to deal with the intermittency in renewable power generation. Ref. [23] proposed a joint power scheduling strategy for EVs and wind generation utilities participating in the day-ahead energy, balancing, and regulation markets. In [18–23], centralized controllers are used to determine the charging schedules for all EVs. Centralized control strategies can guarantee optimal controlling results. However, for a large population of EVs, optimizing and controlling EV charging loads utilizing centralized strategies will be computationally expensive. Ref. [24] proposed a decentralized EV control method in which each EV updates its charging schedule according to the broadcasted control signals and the aggregated EV load will converge to an optimal valley-filling charging profile with asynchronous computation. Decentralized charging control strategies were developed in [25,26] to schedule autonomous EV charging loads for valley filling based on non-cooperative games. In [27,28], decentralized EV charging control strategies that optimize charging of EV fleets according to price-based signals were proposed, which require iterative interaction between EVs and the market operator. In [29], a decentralized control strategy was established and implemented in a multi-agent system framework, which requires information exchanges among neighboring agents. Although decentralized strategies require little computational resources, some of them require higher communication burden and due to lack of global information, it is hard for distributed control strategies to reach global optimal solutions. In [30], a data-driven approach was developed to identify EV user patterns so as to provide guidance for utilities to issue incentive price signals. Refs. [31,32] proposed hierarchical control strategies for large-scale EV coordinated charging control, which combine the advantages of both centralized control strategies and decentralized control strategies. In [25,26], different control strategies were proposed to make individual cost-minimizing charging schedules converge to the valley-filling schedules under load-based pricing mechanisms, which shows that proper pricing mechanisms can incent self-interested EV owners to shift their charging loads for load valley filling. However, it is not clear what properties that the valley-filling pricing mechanisms should have so that the individual goals of EVs (minimizing charging costs) are also the social goal (valley filling), which is the focus of this paper. Two charging scenarios are considered, i.e., non-cooperative scenario and cooperative scenario. In the non-cooperative scenario, each EV schedules its own charging power without cooperation with the others, while in the cooperative scenario, all the EVs are controlled by an aggregator. Compared with previous researches, the contributions of this paper are as follows: (1) The sufficient and necessary conditions of valley-filling pricing mechanisms in both non-cooperative and cooperative scenarios are derived, which can provide sufficient guidance for charging pricing mechanism design. To the best of our knowledge, this is the first work that has provided these derivations.
(2) Based on mathematic derivations, two charging pricing mechanisms and the corresponding coordinated charging strategies for non-cooperative and cooperative scenarios are respectively proposed, which can guarantee quick convergence and require low communication burden. (3) Numerical experiments are conducted to validate the proposed valley-filling pricing mechanisms and the corresponding coordinated charging strategies. The rest of this paper is organized as follows. Section 2 describes the objectives of the valley-filling pricing mechanisms. Sufficient and necessary conditions of valley-filling pricing mechanisms in non-cooperative and cooperative scenarios are derived in Section 3. In Sections 4 and 5, the implementation of the valley-filling mechanisms for the non-cooperative and cooperative scenarios are studied and simulated, respectively. Section 6 concludes the paper. 2. Objectives of the valley-filling pricing mechanisms In this paper, the pricing mechanism design is for load valley filling, which aims at maximizing the benefit of the whole society. The optimization problem of valley filling can be formulated as minimization of the load variance:
0 !2 1 X X 2 min @ ðPt þ Dt Þ =T ðPt þ Dt Þ=T A; t2s
ð1Þ
t2s
where s denotes the set f1; 2; . . . ; Tg, Pt denotes the total EV charging load at time t, and Dt denotes the predicted non-deferrable load at time t. Assuming that the total required EV charging energy, i.e., P t2s P t , is fixed, (1) can be simplified as:
min
X ðP t þ Dt Þ2 :
ð2Þ
t2s
It is assumed that EV owners are self-interested and their individual goals are to minimize their charging costs. Two different scenarios, i.e., non-cooperative and cooperative scenarios, are respectively considered. In the non-cooperative scenario, EV owners make charging decisions independently based on the electricity price signals from the market without cooperating with the other EV owners [33–35]. The goal of the ith EV is to minimize its individual charging cost:
min
X ct Pi;t ;
ð3Þ
t2s
where ct denotes the electricity price at time t, and Pi;t denotes the charging power of the ith EV at time t.
X Pi;t ¼ Pt ;
ð4Þ
i2m
where m denotes the set f1; 2; . . . ; Ng. In the cooperative scenario, participants collaborate to achieve the maximum collective interest. In this paper, we assume the EVs are controlled by an aggregator. And the aggregator is responsible to optimize the EV fleet’s total charging load so that the overall charging costs are minimized. The objective of the aggregator is:
min
X ct P t :
ð5Þ
t2s
Well-designed pricing mechanisms for EV charging should be able to guarantee the social welfare, i.e., load valley-filling, to be achieved while the market participants, i.e., individual EVs or the aggregator, pursue their own interests, i.e., minimizing charging
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costs. In other words, under a well-designed price function ct which is called a valley-filling price in this paper, the optimal solutions of (2) and (3) in the non-cooperative scenario should be the same while the optimal solutions of (2) and (5) in the cooperative scenario should be the same. 3. Sufficient and necessary conditions of valley-filling pricing mechanisms From [25,36], it is shown that static electricity price-based strategies have difficulty in filling the night-time valley effectively. For example, with a TOU pricing mechanism, self-interested EV owners will charge their EVs during the predetermined low-price period, which is the non-deferrable load valley period. However, when the EV population is large, a new charging load peak may arise during the predetermined low-price period since there may be many EVs getting recharged simultaneously. A dynamic loadbased pricing strategy, which can react to the dynamic change of the total load, may avoid this situation. The price function for EV charging, i.e., ct , should be a two-variable function of the charging load, i.e., P t , and the non-deferrable load, i.e., Dt , as follows:
ct ¼ cðPt ; Dt Þ:
ð6Þ
In this section, the requirements of the valley-filling price functions, i.e., cðP t ; Dt Þ, will be derived in both non-cooperative and cooperative scenarios. 3.1. Valley-filling pricing mechanism for the non-cooperative scenario In the non-cooperative scenario, EV owners optimize their charging schedules to minimize their individual charging costs. According to the Game Theory, the final cost-minimizing charging schedule of each EV, ai;t , will constitute a Nash equilibrium [37]. When the Nash equilibrium is reached, no EV owner can gain extra profits by changing his/her own charging schedule unilaterally. Therefore, the cost-minimizing schedule of the ith EV at time t, ai;t , satisfies:
X ai;t ¼ arg min ct ai;t P0 ai;t
t2s
ai;t
t2s
X ¼ arg min cðai;t P0 þ ai;t P0 ; Dt Þai;t P0 :
ð7Þ
Note that EVs are assumed to be charged in the on–off mode, because compared to modulating charging power of each EV, simply turning charging on or off may be more practical [38]. In (7), P 0 denotes the rated charging power of EVs, ai;t denotes the on/off charging status of the ith EV at time t and ai;t denotes the sum of the other EVs’ cost-minimizing charging schedules at time t excluding the ith EV:
ai;t ¼ 0 or 1; ai;t ¼
t 2 s;
ð8Þ ð9Þ
j2m
X ai;t P0 Dt ¼ Ei
g
ð10Þ
t2s
where g is the coefficient of charging efficiency, Dt is the time interval duration, and Ei denotes the required energy of the ith EV. In the non-cooperative scenario, valley-filling charging schedules can be obtained by solving:
i2m
ai;t2 ¼ a0i;t1 ¼ 0; ai;t ¼ a0i;t ;
!2 :
ð12Þ
t 2 s; t – t 1 ; t – t 2 :
Since a0i;t is not a valley-filling solution, the objective applying a0i;t is higher than the optimal value:
X X ai;t P0 þ Dt t2s
!2
Pt2 þ Dt2 ;
where D is an extremely small positive variable. According to (21), we have:
ðPt1 þ Dt1 Þ þ ðPt2 þ Dt2 Þ < ðPt1 þ D þ Dt1 Þ þ ðPt2 D þ Dt2 Þ ; 2
2
2
2
ð22Þ X 0 X 2 2 ðPt þ Dt Þ < ðPt þ Dt Þ : t2s
ð23Þ
t2s
Therefore, P0t is not a valley-filling solution. If the price function, i.e., cðP; DÞ, is a valley-filling function, P0t should not be a cost-minimizing solution. Therefore, we have:
X X ðcðPt ; Dt ÞPt Þ < ðcðP0t ; Dt ÞPt Þ: t2s
ð24Þ
t2s
By simplifying (24), we have:
cðP t1 ; Dt1 ÞPt1 þ cðPt2 ; Dt2 ÞPt2 < cðPt1 þ D; Dt1 ÞðPt1 þ DÞ þ cðPt2 D; Dt2 ÞðPt2 DÞ:
ð25Þ
Assuming that cðP; DÞ is differentiable, Expansion, (25) can be rewritten as:
utilizing
cðPt1 ; Dt1 Þ þ c0P ðP t1 ; Dt1 ÞD þ oðD2 Þ ðPt1 þ DÞ þ cðPt2 ; Dt2 Þ c0P ðPt2 ; Dt2 ÞD þ oðD2 Þ ðPt2 DÞ > c P t1 ; Dt1 Pt1 þ c P t2 ; Dt2 Pt2
Taylor
ð26Þ
where c0P ðP; DÞ denotes @cðP; DÞ=@P. By ignoring the quadratic and higher terms of D, we have:
cðP t1 ; Dt1 Þ þ c0P ðPt1 ; Dt1 ÞP t1 cðPt2 ; Dt2 Þ c0P ðPt2 ; Dt2 ÞPt2 > 0:
ð27Þ
Pt1 þ Dt1 > Pt2 þ Dt2 ; f ðP t1 ; Dt1 Þ > f ðPt2 ; Dt2 Þ:
Z
gðP þ DÞdP ¼ Z ¼
Z
cðP; DÞdP þ
ð28Þ
@cðP; DÞ dP @P
dðcðP; DÞPÞ ¼ cðP; DÞP:
Pt
t2s
Pt
t2s
ð32Þ
P According to [24], min t2s GðPt þ Dt Þ is equivalent to P min t2s ðPt þ Dt Þ2 . Therefore, the cost-minimizing schedule is equivalent to the valley filling schedule. Thus, we can conclude that in the cooperative scenario a price function is valley-filling if and only if the price function satisfies (31). 4. Implementation and case studies for the non-cooperative scenario In this section, a charging pricing mechanism for the non-cooperative scenario is designed based on the analysis in Section 3.1 and a charging price signal based charging-guide method is proposed to help EVs to determine their cost-minimizing schedules. Then cost-minimizing charging behaviors of all EVs are simulated under the proposed pricing mechanism to evaluate its valley-filling performance. 4.1. Charging-guide method to get Nash equilibrium solutions In the non-cooperative scenario, the cost-minimizing charging schedules of EVs will constitute a Nash equilibrium, which generally requires repeated interactions. To help each EV determine its cost-minimizing schedule in a short time without repeated interactions, a charging-guide method, as shown in Fig. 1, is developed. At the first step, the grid operator predicts the non-deferrable load, i.e., Dt , and selects a strictly increasing price function, i.e., cðxÞ. The initial price curve, i.e., c0;t , equals to cðDt Þ. Then, each EV is numbered according to its arrival time. The price curve ci1;t is calculated according to charging schedules of the already arrived i 1 EVs and the non-deferrable load Dt , using
! i1 X an;t P0 þ Dt :
ð33Þ
n¼1
According to the derivation presented in Appendix A, a price function, i.e., f ðP; DÞ, which always satisfies (28), should be equivalent to a strictly increasing function gðP þ DÞ. According to the definition of f ðP; DÞ, we have
Z
t2s
X ¼ arg min GðPt þ Dt Þ
ci1;t ¼ c
Define f ðP; DÞ ¼ cðP; DÞ þ c0P ðP; DÞP. Then we have:
ð31Þ
X ðcðPt ; Dt ÞPt Þ
X ¼ arg min ðGðPt þ Dt Þ þ HðDt ÞÞ :
ð21Þ
t 2 s; t – t 1 ; t – t 2 ;
GðxÞ is strictly convex:
If the price function, i.e., cðP; DÞ, is a valley-filling one for the cooperative scenario, it should satisfy (31). We have proven that in the cooperative scenario, Proposition S3 , the price function is valley-filling, implies Proposition S4 , the price function satisfies (31). Now we prove that S4 also implies S3 . If S4 holds, the cost-minimizing charging schedule satisfies: Pt
D;
ð30Þ
where HðDÞ is a random function of D. Thus we have:
cðP; DÞ ¼ ðGðP þ DÞ þ HðDÞÞ=P;
subject to (18) and (19). If the price function, i.e., cðP; DÞ, is a valley-filling one, there is a charging schedule, i.e., Pt , which is the optimal solution of both (17) and (20). Similar to the non-cooperative scenario, let us consider a suboptimal charging schedule P0t that is slightly different from P t at only two time intervals t1 and t2 . The selected P0t , t 1 and t2 should satisfy:
P0t1 ¼ P t2 Pt ¼ P0t ; Pt1 þ Dt1
Z GðP þ DÞ þ HðDÞ ¼
ð29Þ
Let us consider a function GðxÞ that satisfies G0 ðxÞ ¼ gðxÞ. Because gðxÞ is strictly increasing, GðxÞ is strictly convex. Thus we have
Then ci1;t is sent to the ith EV that has been newly connected to the grid. The cost-minimizing charging schedule of the ith EV, i.e., ai;t , can be obtained by solving:
min
X ci1;t ai;t P0 ;
ð34Þ
t2s
subject to (8) and (10). Then the ith EV submits its charging schedule ai;t to the grid operator. According to ai;t , the operator updates the load curve and the price curve. It can be proven that ai;t is also the solution of the optimization problem:
min
X cN1;t si;t P0 ; t2s
ð35Þ
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Z. Hu et al. / Applied Energy 178 (2016) 155–163
5
Initialization
10
4
4.5
i=1
4
The i EV is connected to the grid
i=i+1
Price curve is sent to the ith EV. The ith EV makes its cost-minimizing charging schedule
Total Load (kW)
3.5 th
3 2.5 2 1.5
The charging schedule of the ith EV is sent to the operator. The load and price curves are updated.
1
normal load (without EV charging load) total load (under valley-filling pricing mechanism, 247 EVs)
0.5
N
total load (under valley-filling pricing mechanism, 2476 EVs) total load (uncoordinated charging, 2476 EVs)
Stop?
0 12:00
Y
18:00
24:00
6:00
12:00
Time of Day
End Fig. 2. Load profiles of different EV numbers in the non-cooperative scenario. Fig. 1. Flow chart of the charging-guide method in the non-cooperative scenario.
4.2. Simulation results In this case, a strictly increasing function,
cðP þ DÞ ¼ 3:2065 104 ðP þ DÞ 3:1073
ð36Þ
is chosen to be the price function, in cent/kWh. Units of P and D are kW. It should be noted that, besides (36), any functions which are strictly increasing could be chosen as the price function. EV parameters such as arrival time, expected departure time, state of charge (SOC) are randomly generated according to the distributions given in Table 1. The battery capacity of each EV is assumed to be 32 kW h and the rated charging power is 7 kW [39] and the charging efficiency is 0.90. Then optimal charging scheduling of all the EVs are simulated using the charging-guide method and the results are shown in Fig. 2. The peak-to-valley ratios, i.e., ep=v , are calculated in Table 2 to evaluate the load valley-filling performance of the pricing function (36). The smaller ep=v is, the better the valley-filling performance is. As shown in Fig. 2, without proper guidance, EV owners will charge their EVs as soon as they get home, which increases the peak load by 34.65% than that of the no-EV case at about 8:00 p. m. Table 2 also shows that ep=v of the uncoordinated charging case is greater than ep=v of the no-EV case. On the other hand, with the guide of the price function (36), the valley of the non-deferrable
Table 2 Peak-to-valley ratios of different load profiles (non-cooperative scenario with 2476 EVs). Load profiles
Without EV charging loads
Uncoordinated charging
Under valley-filling pricing mechanism
Peak-to-valley ratio
2.02
2.72
1.28
load is filled by EV charging loads and ep=v decreases from 2.72 to 1.68. In addition, the valley-filling performance becomes better when the number of EVs increases from 247 to 2476. Fig. 3 shows the valley-filling price curve in the 2476-EV case. The price curve is flat during the EV charging period so that no EV owner can decrease his/her charging cost by changing the charging schedule unilaterally. 5. Implementation and case studies for the cooperative scenario In this section, a rolling optimization method for the aggregator is proposed to determine the cost-minimizing EV charging schedule. A charging pricing mechanism for the cooperative scenario is designed based on the analysis in Section 3.2 and benchmarked with two other pricing mechanisms. Then cost-minimizing charging behaviors of the aggregator are simulated under designed pricing mechanisms to evaluate the valley filling performances of different pricing mechanisms.
9 8
Price (cent/kW)
which means the current cost-minimizing charging schedule ai;t is exactly the final cost-minimizing charging schedule considering all EVs (see Appendix C for detailed derivations). The equilibrium solutions, i.e., ai;t , are obtained without multiple bargaining between the EV and the grid operator. The advantages of the charging-guide method include: (1) each EV can obtain a cost-minimizing charging schedule without repeated interactions with the grid operator, and (2) although the price curve changes when new EVs arrive, the cost-minimizing charging schedules for the previous-arrived EVs will not change.
7 6 5 4 3 2
the valley-filling price curve (2476 EVs)
1
Table 1 Distributions of the EV parameters.
0 12:00
Parameter
Arrival time
Expected departure time
SOC
Distribution
N(19:00, 12)
N(7:00, 12)
N(0.3, 0.12)
18:00
24:00
6:00
12:00
Time of Day Fig. 3. Valley-filling price curve (2476 EVs) in the non-cooperative scenario.
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Z. Hu et al. / Applied Energy 178 (2016) 155–163 10
Initialization
9
k=1
8
Price (cent/kW)
7
k=k+1
Set the optimization period to [Tk,Tk+24)
Charge EVs as planned N
Any newly arrived EV at time Tk ? Y
6 5 4 3 Price Curve 1 (TOU) Price Curve 2 (linear) Price Curve 3 (valley-filling)
2 1
Solve (37) and update charging schedules of each EV
0 12:00
18:00
min
X ðcðPt ; Dt ÞPt Þ t2s
X0 s:t:g Pi;t Dt ¼ Ei ; t2si
X Pi;t ¼ Pt ;
4
i2m
Pi;t ¼ 0; t 2 s0 ;
Total Load (kW)
4 3.5
2.5 2 load curve without EV charging load load curve with electricity price function 1 load curve with electricity price function 2 load curve wtih electricity price function 3
0.5 0 12:00
18:00
24:00
6:00
ð37Þ t 2 si ; i 2 m ;
t R si ; i 2 m;
where s0 denotes time periods of the next 24 h, ½T k ; T k þ 24Þ; T k means the current time; si stands for the feasible charging period of the ith EV, ½T start;i ; T end;i Þ, where T start;i is the greater value between T k and the arrival time of the ith EV and T end;i is the expected departure time of the ith EV; Ei denotes the required energy of the ith EV at current time.
3
1
i 2 m;
t 2 s0 ;
Pi;t ¼ 0 or Pi;t ¼ P0 ;
4.5
1.5
12:00
Fig. 6. Comparison of different price curves in the cooperative scenario (2476 EVs).
Fig. 4. Flow chart of the rolling control method for an aggregator in the cooperative scenario.
x 10
6:00
Stop? Y End
5
24:00
Time of Day
N
5.2. Simulation results 12:00
Time of Day Fig. 5. Load profiles with three different price functions in the cooperative scenario (2476 EVs).
5.1. Rolling optimization method for a EV aggregator In the cooperative scenario, charging schedules of EVs are made by an aggregator to minimize the total charging costs. Fig. 4 shows a rolling optimization method designed for the aggregator. First, the price function and the predicted non-deferrable load curve are sent to the aggregator for calculating an optimal charging schedule. It should be noted that being informed of the price function is necessary for the aggregator to make an optimal charging schedule, because the change of the aggregator’s charging schedule can affect the price. Then the aggregator checks every 15 min for newly arrived EVs that need to get charged. If no EV arrives, the already connected EVs will be charged as planned. Otherwise, the aggregator will update information including the SOCs, arrival time and expected departure time of newly arrived EVs and solve the following optimization problem to obtain an updated charging schedule for each EV:
The parameters of EVs and the non-deferrable load are the same as those used in the non-cooperative scenario. Cost-minimizing charging behaviors of the aggregator are simulated using the rolling optimization method to evaluate the valley-filling performances of three different pricing functions. Price function 1 is a TOU price (cent/kWh) represented by
8 > < 6:0 t 2 ½0 : 00;6 : 00Þ [ ½23 : 00; 24 : 00Þ c1 ¼ 7:5 t 2 ½6 : 00; 8 : 00Þ [ ½11 : 30; 16 : 30Þ [ ½21 : 00;23 : 00Þ : > : 9:0 t 2 ½8 : 00; 11 : 30Þ [ ½16 : 30; 21 : 00Þ
ð38Þ Price function 2 is the same as (36),
c2 ¼ cðP þ DÞ ¼ 3:2065 104 ðP þ DÞ 3:1073:
ð39Þ
Price function 3 is designed according to the sufficient and necessary conditions of valley-filling pricing mechanisms in the cooperative scenario. According to (31), we assume that:
GðP þ DÞ ¼ 1:2133 104 ðP þ DÞ2 ;
ð40Þ
HðDÞ ¼ 1:2133 104 D2 :
ð41Þ
Table 3 Peak-to-valley ratios under different price functions (Cooperative scenario with 2476 EVs). Cases
Price function 1 (TOU price)
Price function 2 (linear price)
Price function 3 (valley filling price)
Without EV charging load
Peak-to-valley ratio
2.25
1.38
1.28
2.02
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Z. Hu et al. / Applied Energy 178 (2016) 155–163 Table 4 Costs and peak load shaving performance analyses under cooperative and non-cooperative scenarios. Price function
c1 (TOU price) c2 (Linear price) c3
Total charging costs of EVs/ $
Peak-to-valley ratio
Non-cooperative scenario
Cooperative scenario
Non-cooperative scenario
Cooperative scenario
370.75 448.60 479.62
370.75 356.38 370.93
2.25 1.28 1.47
2.25 1.38 1.28
Obviously, GðxÞ is a strictly convex function of the total load. Thus we have:
c3 ¼ ðGðP þ DÞ þ HðDÞÞ=P ¼ 1:2133 104 ðP þ 2DÞ:
ð42Þ
Then cost-minimizing charging schedules of the aggregator are simulated using the rolling optimization method and the results are shown in Fig. 5. Peak-to-valley ratios are calculated in Table 3 to evaluate the valley-filling performances of the three different pricing functions. As shown in Fig. 5, price function 1 (TOU price) will result in forming a new load peak at the beginning of the valley price period because all EV charging loads will be scheduled to the lowest price period. Since the TOU prices are issued beforehand and cannot be updated with the change of the real-time load, it cannot achieve satisfactory valley-filling performances and may result in unexpected load peaks during valley price periods. And the valley-filling performance of price function 2 in the cooperative scenario is not as good as that in the non-cooperative scenario. Price function 3, which is designed according to the sufficient and necessary conditions of valley-filling pricing mechanisms, has the best valley-filling performance. This result verifies our conclusion that in a cooperative scenario a price function which satisfies (31) is a valley-filling one. From Table 3, we can see that ep=v of the TOU price case is greater than that of the no-EV case. Both price function 2 (linear price function) and price function 3 (valley-filling price function) can decrease the load peak-valley difference. Price function 3 has the best valley-filling performance among three price functions. Fig. 6 shows the three price curves in the cooperative scenario. Unlike the price curve in the non-cooperative scenario, the prices (price curves 2 and 3) are not flat during the EV-charging period, which means parts of EVs can decrease their charging costs by changing their own charging schedules unilaterally. However, this behavior will increase the total charging costs. The costs and peak load shaving performance of different price functions under cooperative and non-cooperative scenarios are compared in Table 4. It can be seen that total charging costs under cooperative scenario are lower than those under non-cooperative scenario, which indicates the benefits of coordinated control by an EV aggregator. With price function c1 , since EVs all tend to charge during low price period, the total costs under cooperative and non-cooperative scenarios are the same; while with price function c2 , the total cost under cooperative scenario is lower than that under non-cooperative scenario by 20.6% and with price function c3 , the former is lower than the latter by 22.7%. Regarding peak load shaving performance, price function c2 outperforms the other two price functions under non-cooperative scenario, while c3 is the best peak-load-shaving price function under cooperative scenario. With TOU price (price function c1 ), charging power of large-scale EVs may increase the peak-to-valley ratio. 6. Conclusions
valley filling are investigated. The sufficient and necessary conditions of the valley-filling pricing mechanisms are derived for the non-cooperative and cooperative scenarios, respectively. It is concluded that in the non-cooperative scenario a valley-filling price function should be a strictly increasing function of the total load while in the cooperative scenario it should satisfy (31). In the non-cooperative scenario, a charging-guide method is developed to help EV owners make their cost-minimizing charging schedules, avoiding frequent interaction between EVs and the grid operator. In the cooperative scenario, a rolling optimization strategy is designed for the aggregator to schedule EV charging power based on the charging price functions. Cost-minimizing charging schedules are simulated in both scenarios to verify the valley-filling performances of the pricing mechanisms and the simulation results indicate that the proposed pricing mechanisms can effectively incent EV owners or aggregator to shift the charging loads for load valley filling. With the increase of EV population, the charging load of EVs may lead to the congestion in distribution systems, how to design nodal price mechanisms for EV charging will be an interesting area. The incentive pricing mechanism to promote vehicle-to-grid applications will also be our future work. Acknowledgment This work is supported by the National Natural Science Foundation of China (No. 51477082).
Appendix A Prove that if cðx; yÞ is a continuous function that satisfies
8x1 ; y1 ; x2 ; y2 P 0; x1 þ y1 < x2 þ y2 ;
then cðx; yÞ is equivalent to a strictly increasing function hðx þ yÞ. Proof. Set
e > 0. According to (A.1), we have 8x; y P 0;
cðx; yÞ < cðx þ y; e=2Þ < cðx þ e; yÞ: Let
e ! 0, we have
cðx; yÞ 6 lim cðx þ y; e=2Þ 6 lim cðx þ e; yÞ; e!0
e!0
cðx; yÞ 6 cðx þ y; 0Þ 6 cðx; yÞ; cðx; yÞ ¼ cðx þ y; 0Þ: Let
hðx þ yÞ cðx þ y; 0Þ: Hence
In this paper, valley-filling pricing mechanisms that can incent self-interested EV owners to shift their charging loads for load
ðA:1Þ
cðx1 ; y1 Þ < cðx2 ; y2 Þ;
hðx þ yÞ ¼ cðx; yÞ:
162
Z. Hu et al. / Applied Energy 178 (2016) 155–163
Therefore we can find i1 > i0 , which satisfies
Rewriting (A.1), we have
8x1 ; y1 ; x2 ; y2 P 0;
ai1 ;t1
x1 þ y 1 < x2 þ y 2 ;
i0 1 i1 i0 1 i1 X X X X aj;t1 P0 þ aj;t1 P0 þ Dt1 > aj;t2 P0 þ aj;t2 P0 þ Dt2 :
hðx1 þ y1 Þ < hðx2 þ y2 Þ:
j¼1
j¼i0 þ1
ðC:5Þ Note that
Appendix B
ai0 ;t1 ¼ ai1 ;t1 ¼ 1;
ðC:6Þ
ai0 ;t2 ¼ ai1 ;t2 ¼ 0:
Prove S2 implies S1 . Proof. Let ai;t denote cost-minimizing charging schedule of the ith EV at time t.
(C.5) can be rewritten as
ai1 ;t1
X min hðai;t P 0 þ Dt þ ai;t P0 Þai;t ;
ðB:1Þ
t2s
where hðxÞ is a strictly increasing function. Because ai;t is binary, (B.1) is equivalent to
X min hðai;t P 0 þ Dt Þai;t :
¼ 1; ai1 ;t2 ¼ 0;
i1 1 i1 1 X X aj;t1 P0 þ Dt1 > aj;t2 P0 þ Dt2 j¼1
ðC:7Þ
j¼1
which contradicts (C.3). The proof is complete. h
ðB:2Þ
t2s
References
According to the Rearrangement Inequality, (B.2) is equivalent to
X min ai;t ðai;t P0 þ Dt Þ;
ðB:3Þ
t2s
which is equivalent to
X ðai;t P0 þ ai;t P 0 þ Dt Þ2
ðB:4Þ
t2s
It is easy to know that the valley filling schedules is the optimal solution of (B.4). That is to say, if hðxÞ is a strictly increasing function, the costminimizing schedule ai;t is also the valley-filling charging schedule. h Appendix C For any i 2 ½1; 2; . . . ; n,
! ! i1 X X ai;t ¼ arg min c aj;t P0 þ ai;t P0 þ Dt ai;t P0 ; ai;t
t2s
ðC:1Þ
j¼1
where cðxÞ is a strictly increasing function. Prove
X ai;t ¼ arg min cðai;t P 0 þ ai;t P0 þ Dt Þai;t P 0 : ai;t
ðC:2Þ
t2s
Proof. Since ai;t is the solution of (C.1), using the Rearrangement Inequality we can simply prove
8ai;t1 ¼ 1; ai;t2 ¼ 0; i1 i1 X X aj;t1 P0 þ Dt1 6 aj;t2 P0 þ Dt2 : j¼1
ðC:3Þ
j¼1
for any i 2 ½1; 2; . . . ; n. Assuming there is ai0 ;t that is not the solution of (C.2), similar to the proof of (C.3), we have
9ai0 ;t1 ¼ 1; ai0 ;t2 ¼ 0; i0 1 i0 1 n n X X X X aj;t1 P0 þ aj;t1 P0 þ Dt1 > aj;t2 P0 þ aj;t2 P0 þ Dt2 : j¼1
j¼i0 þ1
j¼1
Thus, hðxÞ is strictly increasing. h
min
¼ 1; ai1 ;t2 ¼ 0;
j¼i0 þ1
j¼1
j¼i0 þ1
ðC:4Þ
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