1. Introduction
More people are expected to purchase electric vehicles (EVs) because of their advantages for energy conservation and emission reductions. In the United States, 18.7 million electric vehicles are anticipated to be on the road by 2030, making up 7% of the estimated total number of vehicles [
1]. The growing number of EVs will have a substantial influence on the electrical system. Through strategic charging and discharging, EVs can serve as system-wide temporary energy sources in addition to being a flexible load with controllable charging features [
2,
3]. Therefore, an operator such as a battery charging station (BCS) which polymerizes the charging and discharging behavior of EVs is needed. However, the charging process of electric vehicles at a BCS requires more time than refueling non-electric vehicles at a gas station, which hinders the promotion of EVs [
4]. Different from BCS, battery swapping stations (BSS) have the advantages of high energy supplement efficiency, electricity savings, and prevention of excessive battery charging and discharging. By participating in the market and offering services such as demand response and energy storage, BSS aim to maximize earnings. However, the price of swapping batteries has become a concern for most people. The traditional modeling methods of BSS consider the interaction either between the distribution system operator (DSO) and BSS or between BSS and EVs. The combined interactions of DSO, BSS, and EVs have always been neglected. Thus, it is necessary to propose a model considering the three as a whole.
The development of Battery-to-Grid mode (B2G) and Grid-to-Battery mode (G2B) technology enables BSS to provide more service types [
5] and also leads to greater requirements for flexibility in the power grid electricity price. Recently, a coordinated and distributed bi-level peer-to-peer transaction energy management framework has been proposed in [
6] which uses distribution locational marginal price (DLMP) and allows producers and sellers to participate in the local power market through iterative distributed energy pricing. The five parts of DLMPs—marginal costs for active power, reactive power, congestion, voltage support, and loss—offer price signals to encourage market participants to help control congestion and provide voltage support [
7]. In [
8], the generation cost is integrated into the BSS operation optimization problem, and a BSS battery charging scheme for different power exchange scenarios is proposed. There needs to be more DSO regulation and supervision of the relatively disordered load. Reference [
9] integrates the grid-connected operation of electric buses into a dynamic market framework and uses the allocation location marginal price algorithm for load congestion management. Therefore, it is necessary to introduce DLMP to calculate the DSO market clearing process and improve BSSs’ charging and discharging behavior.
Although the construction of BSS is relatively slow, the research on the operation and business model of BSS has made some preliminary progress. Considering the charging and discharging state of a BSS battery, binary variables are introduced, and a BSS operation model has been developed using a large-size mixed integer nonlinear programming (MINLP) technique. In this model, the configuration of rechargeable, replaced, and standby batteries and the different battery rental costs of BSS are analyzed as a whole to meet the users’ battery exchange needs and improve the profitability of BSS [
10]. Considering the scheduling flow of BSS in the past and using G2B and B2G services to obtain market profits, a complete BSS operation model is established in [
11]. In another model, the impacts of battery degradation, market price uncertainty, and battery demand uncertainty are analyzed to determine the economic benefits. To determine the BSS batteries’ charging procedure, the number of batteries withdrawn from stock to fulfill all swap orders from incoming EVs is calculated [
12]. However, the influence of EV behavior on BSS operation is not considered, and the pricing decision of BSS is not involved. Hence, in order to model the BSS pricing process and achieve more adaptable service costs, a dynamic battery replacement price calculation method should be used.
The traditional EV models consider the travel characteristics of electric vehicles, but only a Monte Carlo simulation is carried out in these existing models [
13], which is quite different from the real situation. Reference [
14] uses an uncertainty modeling approach with stochastic intervals for the uncertainty of PV generation and electric energy prices and iteratively solves a stochastic model for EV charging demand. Another bi-level model of BSS and EV operation optimization based on microgrids aims to minimize EV operation cost and maximize BSS profit and adopts incentive-based pricing rules. That is, the price of the power exchange service is positively related to the proportion of electric vehicle load [
15]. Reference [
16] considers a network flow model in a bi-level autonomous mobility-on-demand system to maximize the system’s profit. In [
17], a battery charging optimization model based on the Markov process is proposed to minimize PV charging station power and user cost while maximizing renewable energy consumption. As a result, it is crucial to take into account EV travel demand and travel patterns. With knowledge of potential EV travel routes, BSS can employ dynamic electricity rates to increase market revenues.
Considering the problems above, this paper proposes a three-level pricing model of BSS in the electricity market. This new model contains a mathematical formulation and a market strategy and is built on a framework including DSO, BSS, and EVs. Firstly, DLMP for BSS to trade electricity is proposed to explain the market-clearing process of DSO. Secondly, the BSS operation model is established to determine the optimal time-varying swapping price. Then, the EV driving mode is considered in EV running costs. Finally, by introducing a penalty function, an iteration method is used to solve the three-level model and calculate the optimal swapping price of BSS and the driving schedule of EVs, converting the three-level BSS business model into an iterative BSS optimization search process. The main contributions are summarized as follows:
- (1)
A business framework for BSS is proposed which includes bidding in the electricity market and pricing for EVs.
- (2)
A method for solving the Nash equilibrium solution of the three-level model is proposed.
- (3)
Driving demand for EVs is analyzed to fit the situation closely.
The rest of this paper is organized as follows. The details of the BSS operation mechanism are presented in
Section 2. The business model of BSS is examined in
Section 3. The three-level business model’s reformulation and relaxation strategies are provided in
Section 4. The case study is elaborated on in
Section 5, and the conclusion and suggestions for future research are given in
Section 6.
3. Mathematical Formulation of the BSS Business Model
3.1. DSO Market Schedule
In the proposed three-level pricing model, the DSO is responsible for the management of branch currents and node voltages of the entire transmission network in a centralized form. To reduce overall energy costs, DSO makes separate system-level trading plans for energy services. The market clearing model is listed in (1)–(8), where the objective function is formulated in (1), day-ahead domains are restricted by (2)–(8), and the location and time indices of variables are neglected for brevity.
DLMP depends on factors such as the load size, electrical location of nodes, line blockage, etc., so DLMP differs in time and space. In this paper, the marginal cost of electricity consumption at each node of the distribution network at different times is obtained through the second-order cone AC Optimal power flow (SOC-ACOPF) model, and a linearized power flow is established to obtain the DLMP, including the blocking cost.
Target function:
where
is the resistance of branch
; and
is the current of branch
at time
.
Constraints:
- (1)
Power flow balance constraints:
- (2)
Node voltage constraints:
- (3)
Second-order cone constraints on the variables and :
where
and
are the active and reactive power of the first end of branch
at time
;
is the reactance of branch
;
and
are the minimum and maximum value of voltage of node
at time
; and
and
are the second power of node
voltage and branch
current at time
. The variables
,
and
at the end of the constraints are their Lagrange multipliers, and
is the DLMP of bus
at time slot
.
The constraint space corresponding to Equation (8) is a quadratic cone, and the problem is transformed into second-order cone programming (SOCP). The optimal global solution is obtained quickly by common commercial solvers such as YALMIP [
20].
To write the dual-cone model, the cone constraint is transformed to obtain the dual problem.
Solving the second-order cone current dual problem yields the DLMP for each node at time .
3.2. BSS Pricing Model
The BSS buys electricity from the DSO and charge its batteries. The BSS obtains its main source of income by swapping batteries for EVs. Meanwhile, the BSS can sell excess electricity from the reserve batteries to the DSO for arbitrage. The BSS swaps EV batteries to maximize the operation’s total profits. Considering the life cycle of the battery, each charging and discharging process has an impact on the battery. The battery loss is included in the BSS optimization objective function as a factor for the BSS to formulate the electricity price and charging and discharging plan [
21]. The BSS operation model is listed in (17)–(26), where the objective function is formulated in (17). Numerous rounds of charging and discharging the batteries in the BSS shorten their lifespan and lower their maximum capacity. As seen in the third formula of this article, the battery properties are significantly dependent on the number of cycles. The charge/discharge constraints and battery SOC constraints are restricted by (18)–(23) and (24) and (25).
The objective of the BSS-level model is to minimize its operation cost, which depends on the number of batteries for swapping, the electricity price, and the degradation cost of charging and discharging.
Target function:
where
is the swap price determined by BSS at time
;
is a binary variable representing whether the EV is charged at time
;
and
are the charge and discharge price in the electricity market at time
;
and
are the total charge and discharge power of BSS at time
;
is the battery degradation coefficient;
and
are the charge and discharge power of the battery
at time
; and
is the capacity of a single battery.
Constraints:
- (1)
Battery charge/discharge constraints:
- (2)
Swapping battery constraints:
- (3)
Swap price constraints:
where
is a binary variable to ensure that the charging process and discharge process cannot be carried out at the same time;
is the SOC of battery
at time
;
and
are the minimum and maximum value of the SOC of a battery;
and
are the charging and discharge efficiency;
is the time interval;
is the time of battery
to be swapped;
and
are the minimum and maximum values of the swap price.
3.3. EV Behavior
From the EVs’ perspective, EV owners should not only consider the cost of battery replacement, but also the satisfaction of their own travel needs. Because the electricity price offered by the BSS varies during the day, the swapping schedule should be optimized to minimize the electricity cost over the entire swapping duration.
The objective function is set up in terms of a monetary value to find the best exchanging time and driving schedule. The cost for serving the EV swapping orders is as follows. Equation (27) is the target function minimizing the daily vehicle maintenance cost, which is composed of the swap cost and driving dissatisfaction cost. Equations (28) and (29) are EV discharge constraints, and Equations (30)–(34) are EV discharge and swap status constraints.
Target function:
where
is the discharge power of EV
at time
;
is the primal driving demand of EV
at time
;
is the expectation punishment coefficient of EV
at time
.
Constraints:
- (1)
EV discharging constraints:
- (2)
Discharge and swap status constraints:
where
is a binary variable representing whether the EV is discharged at time
.
4. Approximations and Relaxations of the BSS Business Model
Considering the nonlinear problem caused by the absolute value in the BSS objective function, set
, and
and
are binary variables.
By introducing linear, binary, and relaxation variables, the nonlinear objective function of BSS is converted into linear form, improving the calculation speed.
Considering the relationship between layers of the three-layer model, in order to solve the optimal BSS pricing strategy, we adopt the iterative solution method to decouple the interaction between the DSO, BSS and EVs. Iteration is the most typical mathematical method for solving multi-layer models. The logic of iteration is illustrated in Algorithm 1.
Algorithm 1. DSO–BSS–EV iteration method. |
Input: |
Number of network nodes, branches, network topology, line parameters and load output. |
The iteration parameters include the convergence error
and the maximum number of iterations . |
Output: the optimal price of BSS and optimal driving schedule of EV. |
1. Initialization: Set iteration number =1. Set initial network load and charging and discharging power of BSS. |
2. repeat: |
3. DSO: According to the charging and discharging power of the BSS, the network load is redefined, the market is cleared and the DLMP is calculated. |
4. Algorithm 2 BSS pricing iteration method: Calculate the equilibrium of BSS and EV objective functions and . |
5. Calculate . |
6. then |
7. break |
8. end if |
9. until |
The logic of the BSS pricing iteration method is illustrated in Algorithm 2.
Algorithm 2. BSS pricing iteration method. |
Input: |
EV driving demand . |
The iteration parameters include the convergence error and the maximum number of iterations. |
Output: the optimal price of BSS |
1. Initialization: Set iteration number =1. Set initial EV driving schedule. |
2. repeat: |
3. BSS: According to the EV driving schedule, the BSS operator optimizes the swapping price and calculates the optimal profit. |
4. EV: According to the swapping price , EV owners calculate the optimal driving schedule . |
5. Calculate . |
6. if then |
7. break |
8. end if |
In order to improve the convergence speed, the iterative objective is added to the optimization objective function of BSS as a penalty term.
5. Case Studies
In this section, the optimization model and algorithm are verified on a modified IEEE 33-bus distribution network [
22] shown in
Figure 2. The reference voltage and capacity values are 12.66 kV and 1 MW, respectively. The NREL Database is used to produce the data on solar radiation and wind speed [
23]. The data for BSS and EV in
Table 1 are from Reference [
24]. All numerical simulations are carried out through Gurobi 9.5.1 [
25] and YALMIP within MATLAB on a desktop computer with an Intel
® Core™ i5 of a 1.6 GHz CPU and 16 GB of memory.
We determined the daily time-series EV profiles for typical EV demand [
26]. Combined with the BSS data of Li-ion batteries, the raw EV power of discharge shown in
Figure 3 is designed to characterize the EV owners’ daily activities.
5.1. Designed Cases and Arrangements
The following cases investigate the benefits brought by the market clearing process and EV behavior simulation.
Case 1: Optimal BSS scheduling considering bidding without pricing. In this case, the BSS provides EVs with a swapping service at a fixed price.
Case 2: Optimal BSS scheduling considering pricing without bidding. In this case, the pricing process of the BSS ignores the market clearing process, and the time-of-use electricity price is used to represent the grid electricity price.
Case 3: Optimal BSS scheduling considering pricing and bidding. In this case, the BSS swapping pricing optimization model considering DSO market clearing and EV driving demand proposed in this paper is adopted.
Case 4: On the basis of Case 3, the data scale is increased with a primal algorithm, specifically increasing the number of BSS batteries and the number of EVs in the distribution network. In this case, the BSS swapping pricing optimization model considering DSO market clearing and EV driving demand proposed in this paper is adopted. This case requires no model reconstruction, with 100 batteries in the BSS and 60 EVs.
Case 5: On the basis of Case 3, the data scale is increased with reformulation of the algorithm, specifically by increasing the number of BSS batteries and the number of EVs in the distribution network. In this case, the BSS swapping pricing optimization model considering DSO market clearing and EV driving demand proposed in this paper is adopted. This case requires model reconstruction, with 100 batteries in the BSS and 60 EVs.
The case study is performed as follows. Firstly, the benefits of the dynamic pricing approach are examined, and profit fluctuations in BSS are also evaluated, using Cases 1 and 3. Secondly, the profit resulting from BSS participation in market bidding is analyzed by comparing the results obtained from Case 2 and Case 3. Thirdly, the practicability of the proposed pricing approach is verified by testing a large-scale distribution system. The computational performance and application are finally explored.
5.2. Pricing Strategy Analysis
The pricing strategy analysis is performed by comparing the traditional EV demand considered as an uncertain variable and fixed-price pricing based on the experience method (i.e., Case 1), modeling EV behavior but ignoring the market impact method (i.e., Case 2) and the proposed three-level pricing method (i.e., Case 3).
As can be seen from
Table 2, Case 1 which ignores the dynamic pricing of the BSS and Case 2 which ignores the impact of the DSO results in an artificially high daily operating profit for the BSS. Neither Case 1 or Case 2 can meet the daily driving needs of EV owners to the maximum, which is not conducive to the long-term operation and development of the BSS. Only in Case 3, and with full consideration of the impact of market clearing and EV dynamics, can the optimal pricing and operation scheme of the BSS be realized.
As shown in
Figure 4, BSS affects the market clearing process by adjusting strategies, effectively reducing DLMP. The DLMP reduction during peak and valley periods in the figure is more prominent, corresponding to the peak charge and discharge of the BSS, which is conducive to reducing BSS electricity costs.
Figure 5a represents the time-varying electricity price and charging/discharging power of the BSS in Case 2. The electricity price is in line with the supply and demand relationship with electricity consumption and decreases with the load reduction in the morning and late at night. The BSS charges when the electricity price is low (5:00–7:00) and discharges larger amounts when the electricity price is high (17:00–19:00) to reduce the cost of electricity purchases. Due to the existence of standby batteries, BSS uses the spare batteries for low storage and high-power generation with the power grid to realize arbitrage and obtain additional income beyond the power exchange service.
Figure 5b represents the swapping price and driving demand of EVs in Case 2. The power exchange price of BSS is affected by the power exchange plan of BSS and users, reaching the maximum level from 19:00 to 24:00. On the premise of meeting the driving demand, the EV adjusts its driving behavior according to the impact of the BSS electricity exchange price to reduce the cost of using the EV.
5.3. Calculation Speed and Convergence Analysis
The convergence rates of the above three cases are analyzed and listed in
Table 3. Due to the consideration of market clearing and in Case 1 and Case 3, the number of iterations required for calculation increases, and the convergence speed slows down. The main reason is that the equilibrium of the BSS and EVs is easy to solve through generations. However, this iterative result is challenging to converge when iterating with the settlement of the upper-level DSO market clearing.
To compare the computation and convergence speed of the algorithm in a large-scale distribution network, we designed application scenarios with a BSS containing more batteries and a more significant number of EVs.
Compared with the model before modification, which is Case 4, the model of Case 5 has better convergence and faster convergence speed, which is shown in
Table 4. Modifying the objective function of the BSS and introducing a convergence penalty term is beneficial when applying the model proposed in this paper in large-scale power distribution systems.
6. Conclusions
This paper proposes a BSS swapping pricing optimization model considering DSO market clearing and EV driving demand. The establishment of the dynamic swap price affects the income of the BSS. In addition, the revenue of the BSS is under the influence of the charge/discharge electricity prices. Here, DLMP is used to express the impact of the power exchange with the DSO. The results show that the BSS pricing scheme considering the synergy of DSO and EVs, can effectively improve the operating income of the BSS, reduce the electricity cost to EV owners, and facilitate the regulation and adjustment of real-time electricity prices to market behavior.
The random variables in the model are yet to be modeled. Future studies should simulate should this uncertainty fully. In addition, the spatiotemporal coupling characteristics of EVs are well worth studying. Distinguished from EV demand, the space–time coupling problem combines the actual actions of EVs in the grid for analysis and consideration, which is more appropriate to the actual EV operation mode in practice.