Secondary Frequency Regulation Strategy for Battery Swapping Stations Considering the Behavioral Model of Electric Vehicles

Abstract

The development of vehicle-to-grid (V2G) technique and the growth of battery swapping stations are expected to enhance the resilience of power networks. However, V2G battery swapping stations exhibit inconsistencies among internal battery packs, where the power capacity is significantly affected by the battery swapping behavior of electric vehicle (EV) users. To address this issue, this paper proposes a secondary frequency control strategy for V2G battery swapping stations that accounts for battery pack heterogeneity. First, a user behavioral model is developed through quantitative analysis of key factors such as economic incentives, time costs, and battery degradation, which is then used to optimize the operation of V2G battery swapping stations. Moreover, active balancing of EV battery energy levels is achieved by incorporating penalty terms into the objective function. Finally, a distributed secondary frequency control strategy based on the consensus algorithm is established to minimize total frequency control loss. Simulation results demonstrate that the proposed strategy effectively meets the secondary frequency control requirements of the power grid.

1. Introduction

In recent years, the electric vehicle (EV) industry has experienced rapid growth due to its environmental benefits, energy efficiency, and sustainability. A robust charging infrastructure is essential for the widespread adoption of EVs. Battery swapping stations effectively address EV users’ range concerns by providing instant recharging services similar to traditional gasoline stations and are strongly supported by national policies [1,2]. In 2024, the number of battery swapping stations in China reached 4039, an increase of 819 compared to 2023, a year-on-year growth of 25.43%.

Battery swapping stations present new opportunities for enhancing the resilience of urban distribution grids [3]. Specifically, with the prevalence of vehicle-to-grid (V2G) technology, bidirectional energy flows between the power grid and EVs can significantly improve the flexibility of EVs. Similar to battery storage systems, V2G battery swapping stations have the advantages of fast regulation speed and high precision, demonstrating great potential in frequency regulation [4,5]. Moreover, leveraging V2G battery swapping stations can effectively reduce the cost of establishing large-scale energy storage stations in power systems while improving the flexibility regulation capability of the power grid [6].

Unlike fuel stations, battery swapping stations do not always guarantee that users will receive a fully charged battery. Instead, their services are highly flexible, allowing users to charge or discharge based on their needs. The amount of charge or discharge, as well as the timing of battery replacement, can be adjusted according to grid demand and user preferences. Depicting EV users’ willingness to participate in frequency control is essential for developing effective frequency regulation strategies for V2G battery swapping stations [7,8,9]. The authors in [10] propose a frequency regulation strategy for battery swapping stations considering factors such as time and state of charge. They establish a two-layer frequency regulation model by integrating the ability of electric vehicles to charge and discharge. Wu et al. [11] introduce a two-stage scheduling model for the day-ahead energy and frequency regulation market based on historical data. This model aims to analyze users’ frequency regulation demand and investigate uncertainties faced by EV battery swapping stations when engaging in frequency control and energy trading markets. While the distribution of EV demand can be predicted using an uncertainty model, there is a lack of research on classifying and assessing the willingness of EV users. Current studies on user behavior primarily concentrate on EV load forecasting. The study in [12] presents an EV charging load forecasting model based on the regret theory to analyze the influence of driving time, queuing time, and electricity price on users’ decision-making processes, aiding in predicting the spatial and temporal distribution of EV loads. Furthermore, the authors in [13] depict the heterogeneity and incomplete rationality of EV users during decision-making, employing a discrete choice model to illustrate users’ preferences for charging stations and path selection.

In a V2G battery swapping station, the challenge of inconsistent battery clusters is significant due to variations in brands, models, and battery ages across different vehicles, as well as diverse driving habits and charging patterns among users. Disparities in the state of charge within battery packs at these stations can result in detrimental outcomes such as battery overcharging or over-discharging, heightening the risks of battery damage and fire hazards [14]. Moreover, this imbalance diminishes the operational efficiency of the battery swapping station and, in severe instances, may cause the malfunction or even explosion of the power exchange equipment, posing a substantial threat to the safe and stable operation of the station [2].

Researchers have proposed various control strategies to address the issue of battery inconsistency [15]. The study by Li et al. [16] employs a dynamic feedback mechanism, adjusting the battery energy storage system (BESS) in each storage unit to balance the state of charge (SOC) among the battery cells. Moreover, the research in [17] achieves active balance in the BESS by configuring the equalization factor and adaptive parameters to effectively regulate the charging and discharging processes. However, these approaches exhibit high computational complexity and neglect the economic impact of battery degradation, which is challenging for practical implementation. Wang et al. [18] employ a hierarchical control approach to improve SOC consistency through vector control and an enhanced allocation strategy. Nonetheless, the complexity of the strategy may elevate the risks of equipment overloads, while communication delays could compromise the effectiveness of frequency regulation efforts.

To accommodate the scalable and distributed nature of battery packs in V2G battery swapping stations and fulfill EV users’ plug-and-play charging and discharging requirements, distributed algorithms have gained increasing attention in frequency control strategies due to their high flexibility and resilience [19,20,21,22]. The work in [23] proposes a distributed control strategy based on the iterative gradient method to improve the tracking accuracy of grid frequency control in time-varying distributed power networks by minimizing the quadratic cost function. The research in [24] utilizes a distributed model predictive control strategy for dynamic rolling optimization, which effectively coordinates the outputs of individual generating units to achieve precise control of system frequency stability. Zhao et al. [25] propose a secondary frequency regulation control strategy for battery storage based on a consensus algorithm. This approach successfully alleviates the burden on central dispatch and improves system reliability.

Comparative analysis was conducted with the related works, as shown in

Table 1. Comparison with state-of-the-art secondary frequency regulation strategies.

References	Battery Energy Storage System	Traditional Generating Units	Algorithms (Distributed/ Centralized)	Balancing of Battery Energy Levels	EV Users’ Behavioral Model
[26,27,28]	✓	×	Distributed	✓	×
[29]	✓	✓	Centralized	×	×
[30]	✓	✓	Distributed	×	×
[31]	✓	×	Centralized	✓	✓
Ours	✓	✓	Distributed	✓	✓

. The proposed model incorporates user behavior modeling, battery energy level balancing, coordination between BESS and traditional generators, and a distributed control framework, thereby significantly enhancing system performance and facilitating user participation.

To address the inherent randomness of EV user behavior and weak consistency in battery packs, this paper proposes a Secondary Frequency Regulation Strategy for Battery Swapping Stations Considering EV Users’ Behavioral Model. In Section 2, a user willingness model is constructed by analyzing key factors such as economic incentive, time cost, and battery loss. Then, the loss function of the battery swapping station is proposed, and the active balance of the battery storage is obtained by setting the penalty term of the objective function. In Section 3, a distributed algorithm based on the consensus algorithm is developed to achieve the minimization of the total frequency regulation loss. The simulation results and conclusion are presented in Section 4 and Section 5, respectively.

2. Secondary Frequency Control Model

As shown in Figure 1, this paper proposes a secondary frequency control strategy for V2G battery swapping stations. First, the regulation cost functions for both BESS and conventional generation units are established. In particular, EV users’ willingness to participate, quantified using a behavioral model, is incorporated as an input parameter. Next, the power system operator allocates the regulation demand between V2G swapping stations and conventional units based on a consensus algorithm. During this process, battery packs exchange consensus variables, which gradually converge through information interactions, ensuring state-of-charge (SOC) balancing among different battery clusters. Finally, each frequency regulation unit delivers its power output to maintain system frequency stability. This process is repeated at each time step, involving both information flow and power flow, thereby enhancing system robustness.

2.1. The Behavioral Model of EV Agents

Since EV drivers may not directly own the battery packs associated with their vehicles, their behavior is influenced by the pricing set by service providers, who determine these prices based on total operational costs—including those related to battery degradation during charging and discharging cycles. Hence, we introduce the concept of an EV agent to capture the overall impact of these costs on individual users. An EV agent refers to either a group of EV users who share access to a pool of battery packs or fleet operators who own both the vehicles and the batteries.

Figure 1. Secondary frequency regulation control strategy for V2G battery swapping station based on distributed algorithm.

Figure 1. Secondary frequency regulation control strategy for V2G battery swapping station based on distributed algorithm.

The willingness of an EV agent to participate in V2G services is primarily driven by two factors: time anxiety and battery degradation. Time anxiety represents the opportunity cost of time spent engaging in V2G activities, which is significantly affected by the EV agent’s travel requirements and scheduling constraints.

Additionally, battery degradation leads to increased maintenance costs, which are typically passed on to consumers through higher service fees. It may also reduce vehicle range, thus affecting the overall experience of the EV agent. Frequent charging and discharging cycles further accelerate battery wear, thereby increasing operational costs [31].

In a V2G battery swapping station, the aggregate anxiety cost of an EV agent, $U_a$, is defined as a function of the battery loss anxiety cost, $U_{loss}$, and the time anxiety cost, $U_T$, to quantify the EV agent’s willingness to participate in V2G services. This relationship is expressed as follows:

$$U_a=\epsilon {\displaystyle \frac{K_r-U_{loss}}{U_T^{max}}}$$

(1)

where $K_r$ represents the expected reserve revenue for EV agents participating in V2G services, and $U_T^{max}$ denotes the maximum time anxiety cost associated with V2G participation. The net expected benefit for EV agents, after accounting for additional battery losses, is given by $K_r-U_{loss}$. The parameter $\epsilon$, known as the conversion factor, adjusts the EV agent’s sensitivity to economic incentives, ensuring a balance between reserve revenue and anxiety costs when evaluating willingness. A larger $\epsilon$ indicates that EV agents prioritize economic benefits, leading to higher participation in regulation. However, this may also result in overcommitment, increasing uncertainty in actual execution. Conversely, a smaller $\epsilon$ suggests that EV agents focus more on personal costs, such as battery degradation and reduced vehicle flexibility, leading to lower participation but improved reliability in regulation execution. As such, $U_a$ represents the trade-off between the incentive for EV agents to participate in V2G services for financial benefits and the associated time-related costs. A higher $U_a$ signifies a stronger willingness to engage in V2G services, whereas a lower value indicates reduced participation willingness.

Moreover, we characterize the relationship between the willingness of EV agents and the aggregate anxiety cost $U_a$ through a membership function, categorizing EV agents into three clusters: ${\mathcal{D}}_1$, ${\mathcal{D}}_2$, and ${\mathcal{D}}_3$, representing strong, medium, and weak participation willingness, respectively [32].

Given that the willingness of EV agents may fluctuate in response to external environmental or personal factors, it is not definitive to assume that an agent, even if classified into a cluster with a strong willingness to participate, will unequivocally engage in V2G services during the simulation of user willingness. Conversely, an agent’s classification into a cluster with a weak willingness to participate does not conclusively preclude their engagement in V2G services. Therefore, to mitigate the absolute error in the user intention determination process, a binomial distribution is incorporated for each cluster. The method for determining the willingness of EV agents to participate in V2G services is as follows:

$$\begin{array}{cc}\hfill & U_i=\left\{\begin{array}{c}W(1,A),U_aϵ{\mathcal{D}}_1\hfill \\ W(1,B),U_aϵ{\mathcal{D}}_1\hfill \\ W(1,C),U_aϵ{\mathcal{D}}_3\hfill \end{array}\right.\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & 0<C<B<A<1\hfill \end{array}$$

(3)

where $U_i$ is a binary variable, indicating the individual EV agent’s intention, and A, B, and C are probability factors used to reflect the strength of agent participation willingness within each battery cluster of the battery swapping station.

The analysis on the willingness of the EV agent to participate in frequency control is the basis for obtaining the power limit of the battery clusters in the V2G battery swapping station, which will be used as input parameters for frequency control strategies in the following.

Notably, the behavioral model of EV agents proposed in this study considers not only economic benefits but also the trade-off between reserve revenue and anxiety costs. This approach ensures that the modeling of user willingness more accurately reflects real-world behavior, enabling precise predictions of users’ willingness to participate in V2G frequency regulation. Furthermore, it provides theoretical support for assessing the predictability of battery swapping stations as frequency regulation resources, effectively mitigating shortages or surpluses in regulation capacity caused by agent behavior uncertainty, thereby enhancing the reliability of battery swapping stations in frequency regulation scheduling.

2.2. Virtual Frequency Regulation Loss Function

In this paper, we consider the frequency control model containing conventional generation units and V2G battery swapping stations, as shown in Figure 2. It contains I conventional generation units and J clusters of battery clusters in the battery swapping station, with a total of $N(N=I+J)$ units. In Figure 2, $\Delta P_{tie}$ is the exchange power of the interconnection grid contact line, B is the modulation coefficient of secondary frequency control, $K_I$ is the integration coefficient, $\Delta f$ is the frequency deviation of the system, $P^{AGC}$ is the secondary frequency control command signal, namely, the total of the secondary frequency control outputs that need to be supplied by all the frequency control units in the region, $P_{i,k}(i=1,2,\cdots ,I)$ is the secondary frequency control output of the ith traditional unit, $P_{j,k}(j=1,2,\cdots ,J)$ is the secondary frequency control output of the jth battery group in the battery swapping station, $T_g$ is the time constant of the thermal unit governor, $T_{ch}$ is the response time of the control valve control of the reheat unit, $T_{rh}$ is the reheater time constant, $F_{hp}$ is the reheater gain, $T_{wj}$ the time constant of the j-th battery cluster, $P_L$ is the net load fluctuation of the system, R is the modulation index of the primary frequency control, M is the inertia time constant of the regional power grid, and D is the load damping coefficient. The key to the secondary frequency control strategy for the V2G battery swapping station is the optimal allocation of the frequency control signal $P^{\mathrm{AGC}}$ according to the characteristics of different battery clusters in the battery swapping station.

To assess the cost of traditional generator units and battery swapping stations for frequency control, this paper develops a loss function that accounts for various types of frequency control sources. The function is constructed based on the cost function of traditional generation units, serving as a basis for secondary frequency control power allocation. The operating cost of traditional units has a quadratic relationship with the output power. During secondary frequency control, the change in operating cost due to alterations in the power output of traditional generators is the loss incurred by these units for frequency control.

$$C_{i,k}^I={\alpha }_i{P}_{i,k}^2$$

(4)

where $C_{i,k}^I$ is the frequency control loss of the i-th conventional unit at time k, $P_{i,k}^2$ is the frequency control output of the i-th conventional unit at time k, i.e., the power value of the unit changed due to frequency control need at time k, and ${\alpha }_i$ is the weighting factor.

The battery cluster of the V2G station can be regarded as a voltage source, analogous to the frequency control loss function of conventional generating units. When the battery cluster of the battery swapping station bears large amplitude frequency modulation signal components or is in a deep charging and discharging state, it increases the unit power frequency modulation cost. Consequently, this paper introduces a virtual frequency control loss function tailored for battery clusters in a battery swapping station, specifically,

$$C_{j,k}^I=a_j\left[{\left(P_{j,k}^c\right)}^2+{\left(P_{j,k}^d\right)}^2\right]+b_j{S}_j^2{\left(So{C}_{j,k}-So{C}_j^{\mathrm{ref}}\right)}^2$$

(5)

where $C_{j,k}^I$ is the frequency control loss of the j-th battery cluster at the moment k, $P_{j,k}^c$ and $P_{j,k}^d$ are the charging and discharging powers, i.e., frequency control output of the j-th battery cluster at time k, respectively, $So{C}_{j,k}^{ref}$ represents the charge condition of the j-th battery cluster battery array at the k-th moment, i.e., the proportion of the remaining capacity of the battery to its nominal capacity ($W·h$), $So{C}_j^{ref}$ is the optimal SOC value desired for battery j in frequency control, $S_j$ is the rated capacity of the j-th battery cluster ($W·h$), and $a_j$ and $b_j$ are the weighting coefficients. Equation (5) describes the power deviation cost and state of charge offset cost for battery swapping station. It is worth noting that by adding the cost term for SOC deviation in the objective function, the algorithm will prefer the strategy with the smallest SOC deviation, thus enhancing the accuracy of the frequency control results and achieving the consistency of battery clusters.

A cluster of batteries in a V2G battery swapping station will only be in one state of charging or discharging at the same time. Therefore, when the battery cluster is in the charging state, there exists only the charging power $P_{j,k}^c$, and the discharging power $P_{j,k}^d=0$. Similarly, when the battery cluster is in the discharging state, there exists only the discharging power $P_{j,k}^d$, and the charging power $P_{j,k}^c=0$. Thus, we have $P_{j,k}^c·P_{j,k}^d=0$ at any moment. Therefore, the charging and discharging power $P_{j,k}$ can be defined as follows:

$$P_{j,k}=\left\{\begin{array}{c}{P}_{j,k}^c,P_{j,k}<0\hfill \\ P_{j,k}^d,P_{j,k}>0\hfill \end{array}\right.$$

(6)

A simplification of the frequency control loss function for the battery cluster of the swapping station, i.e.,

$$C_{j,k}^I={\alpha }_j{P}_{j,k}^2+{\beta }_{j,k}{P}_{j,k}+{\gamma }_{j,k}$$

(7)

When $P_{j,k}<0$, we have

$$\left\{\begin{array}{cc}\hfill {\alpha }_j& =a_j+b_j{\left({\eta }_j^c\Delta t\right)}^2\hfill \\ \hfill {\beta }_{j,k}& =-2b_j{S}_j\left(So{C}_{j,k-1}-So{C}_j^{\mathrm{ref}}\right){\eta }_j^c\Delta t\hfill \\ \hfill {\gamma }_{j,k}& =b_j{S}_j^2{\left(So{C}_{j,k-1}-So{C}_j^{\mathrm{ref}}\right)}^2\hfill \end{array}\right.$$

(8)

When $P_{j,k}>0$, we have

$$\left\{\begin{array}{cc}\hfill {\alpha }_j& =a_j+b_j{(\Delta t/{\eta }_j^d)}^2\hfill \\ \hfill {\beta }_{j,k}& =-2b_j{S}_j\left(So{C}_{j,k-1}-So{C}_j^{\mathrm{ref}}\right)\Delta t/{\eta }_j^d\hfill \\ \hfill {\gamma }_{j,k}& =b_j{S}_j^2{\left(So{C}_{j,k-1}-So{C}_j^{\mathrm{ref}}\right)}^2\hfill \end{array}\right.$$

(9)

When the net load frequency fluctuation is slow and of large amplitude, the battery frequency control loss at the swapping station is significant, which contradicts the objective of minimizing such loss. Consequently, the traditional generation units assume this task, benefiting from their low gradient rate, thereby mitigating mechanical wear and tear associated with frequent gradient changes. This arrangement leverages the strengths of both systems. Conversely, when the net load frequency fluctuation is rapid and of small amplitude, the battery cluster’s frequency control loss is minimal, and the swapping station assumes priority, capitalizing on the battery’s ability to respond swiftly and accurately to frequency adjustments. Furthermore, the values of $a_j$ and $b_j$ reflect the weighting of the battery cluster’s frequency control loss affected by power transfer capabilities and battery charge levels, with $a_j$ and $b_j$ parameters calibrated based on the battery’s rated power and capacity.

In the battery frequency control loss function Equation (5), $So{C}_{j,k}$ can be further expressed as a functional form related to the variables $P_{j,k}^c$ and $P_{j,k}^d$, i.e.,

$$So{C}_{j,k}=So{C}_{j,k-1}-\left({\eta }_j^c{P}_{j,k}^c+P_{j,k}^d/{\eta }_j^d\right)\Delta t/S_j$$

(10)

where $So{C}_{j,k-1}$ denotes the charging state of the battery cluster of the swapping station at the moment $k-1$, ${\eta }_j^c$ and ${\eta }_j^d$ are the charging and discharging efficiencies of the battery cluster of the station, respectively, and $\Delta t$ represents the duration between successive measurement instances.

2.3. Quadratic Frequency Regulation Control Model

2.3.1. Objective Function

The total frequency control loss $C_k^I$ for the secondary frequency control containing the conventional generation unit and the battery of the V2G swapping station is

$$\left\{\begin{array}{cc}\hfill C_k^I& =\sum _{iϵJ}{C}_{i,k}^I+\sum _{jϵJ}{C}_{j,k}^I\hfill \\ \hfill C_{i,k}^I& ={\alpha }_i{P}_{i,k}^2\hfill \\ \hfill C_{j,k}^I& ={\alpha }_j{P}_{j,k}^2+{\beta }_{j,k}{P}_{j,k}+{\gamma }_{j,k}\hfill \end{array}\right.$$

(11)

where ${\alpha }_i$, ${\beta }_{j,k}$, and ${\gamma }_{j,k}$ are the coefficients of the j-th battery cluster at the k-th moment. ${\alpha }_i$ is a constant, and ${\beta }_{j,k}$ and ${\gamma }_{j,k}$ are determined by the charging state at k-th moment.

2.3.2. Frequency Regulation Constraints

For traditional generator sets, their frequency control capability is mainly limited by two aspects: One is the limitation of output power, and the other is the rate of power variation, i.e., the rate of climb.

Conventional units provide the frequency control output based on their frequency control capacity in a range of objective constraints, specifically reflected in two aspects, the frequency control power backup and climbing rate. The frequency control capability constraints of conventional units can be defined as

$$\left\{\begin{array}{c}{P}_i^{min}\le P_{i,k}\le P_i^{max}\hfill \\ R_i^{min}\le {\displaystyle \frac{P_{i,k}-P_{i,k-1}}{\Delta t}}\le R_i^{max}\hfill \end{array}\right.$$

(12)

where $P_i^{min}$ and $P_i^{max}$ are the minimum and maximum values of the conventional unit’s frequency control output, respectively, $R_i^{min}$ and $R_i^{max}$ are the minimum and maximum values of the climb rate, respectively, and $P_{i,k-1}$ is the power level of the conventional unit at the k-1st frequency control moment.

Define ${\mathsf{\Omega }}_{i,k}$ as the feasible domain of the i-th conventional unit output power at moment k:

$${\mathsf{\Omega }}_{i,k}=\left\{P_{i,k}\right|max(P_i^{min},R_i^{min}\Delta t+P_{i,k-1})\le P_{i,k}\le min(P_i^{max},R_i^{max}\Delta t+P_{i,k-1})\}$$

(13)

The frequency control capability of the battery of a swapping station is mainly constrained by power limitations, climbing rate limitations, and battery charge aspects. The SOC of the battery’s charging state can vary between 0 and 1, but in order to avoid the life loss caused by overcharging and over-discharging of the battery, the SOC of the battery usually needs to be kept within a certain range. At the same time, in order to achieve the equalization of the battery charge, the constraints on the SOC value of the battery at the off-grid moment are set. The frequency control capability constraint of the battery bin cluster can be defined as

$$\left\{\begin{array}{c}{P}_j^{min}\le P_{j,k}\le P_j^{max}\hfill \\ R_j^{min}\le {\displaystyle \frac{P_{j,k}-P_{j,k-1}}{\Delta t}}\le R_j^{max}\hfill \\ SO{C}_j^{min}\le SO{C}_{j,k}\le SO{C}_j^{max}\hfill \\ SO{C}_{j,k}\ge SO{C}_j^{exp}\hfill \end{array}\right.$$

(14)

In the above equation, $P_j^{min}$ and $P_j^{max}$ are the minimum and maximum values of the frequency control output power of the battery clusters, respectively, $R_j^{min}$ and $R_j^{max}$ are the minimum and maximum values of the climbing rate of the battery cluster, respectively, $SO{C}_j^{min}$ and $SO{C}_j^{max}$ are the minimum and maximum values of the load state of the battery cluster, respectively, and $SO{C}_j^{exp}$ denotes the expected SOC value of the battery cluster when it is off-grid.

${\mathsf{\Omega }}_{j,k}$ represents the viable energy dispatch interval for the j-th battery cluster at time k.

$${\mathsf{\Omega }}_{j,k}=\left\{P_{j,k}|P_{j,k}^{\mathrm{L}}\le P_{j,k}\le P_{j,k}^{\mathrm{U}}\right\}$$

(15)

$$\begin{array}{c}\hfill P_{j,k}^{\mathrm{L}}=\left\{\begin{array}{cc}max\{P_j^{min},R_j^{min}\Delta t+P_{j,k-1},\left(So{C}_{j,k-1}-So{C}_j^{max}\right)S_j/\Delta \mathrm{t}{\eta }_j^c\},\hfill & P_{j,k}\le 0\hfill \\ max(P_j^{min},R_j^{min}\Delta \mathrm{t}+P_{j,k-1},0),\hfill & P_{j,k}>0\hfill \end{array}\right.\end{array}$$

(16)

$$\begin{array}{ccc}\hfill & P_{j,k}^{\mathrm{U}}\hfill & \hfill =\left\{\begin{array}{c}min\left\{P_j^{max},R_j^{max}\Delta t+P_{j,k-1},0,\left({SoC}_{j,k-1}-So{C}_j^{exp}\right)S_j/\Delta {\mathrm{t}}_j^c\right\},P_{j,k}\le 0\\ min\left\{\begin{array}{c}{P}_j^{max},R_j^{max}\Delta \mathrm{t}+P_{j,k-1},{\displaystyle \frac{\left(So{C}_{j,k-1}-So{C}_j^{min}\right)S_j{\eta }_j^d}{\Delta \mathrm{t}}},\\ \left(So{C}_{j,k-1}-So{C}_j^{exp}\right)S_j{\eta }_j^d/\Delta \mathrm{t},\end{array}\right\},P_{j,k}>0\end{array}\right.\end{array}$$

(17)

In the above equation,$P_{j,k}^L$ and $P_{j,k}^U$ are the lower and upper limits of the output power of the j-th battery cluster at moment k, respectively.

2.3.3. Frequency Regulation Control Model

The grid dispatch center assigns the frequency control responsibility to the conventional units and the battery clusters of the swapping station, taking advantage of the respective frequency control advantages of both. The system’s secondary frequency control demand $P_k^{AGC}$ is equal to the active output of all frequency control power sources, i.e.,

$$P_k^{\mathrm{AGC}}=\sum _{i\in I}{P}_{i,k}+\sum _{j\in J}{P}_{j,k}$$

(18)

In the above equation, $P_k^{AGC}$ is the secondary frequency control command in the control area at moment k, and ${\sum }_{i\in I}{P}_{i,k}$ and ${\sum }_{j\in J}{P}_{j,k}$ are the total active outputs of the conventional unit and the battery of the swapping station, respectively.

In summary, the mathematical model of secondary frequency control considering the battery of the converter station is as follows:

$$min{C}_k^I=min\left(\sum _{iϵI}{D}_{i,k}^I+\sum _{jϵJ}{D}_{j,k}^I\right)$$

(19)

$$\mathrm{subject}\mathrm{to}{P}_k^{\mathrm{AGC}}=\sum _{i\in I}{P}_{i,k}+\sum _{j\in J}{P}_{j,k}$$

(20)

$$\left\{\begin{array}{c}{\mathsf{\Omega }}_{i,k}=\left\{P_{i,k}|\begin{array}{c}max\left(P_i^{min},R_i^{min}\Delta t+P_{i,k-1}\right)\le P_{i,k}\le min\left(P_i^{max},R_i^{max}\Delta t+P_{i,k-1}\right)\end{array}\right\}\hfill \\ {\mathsf{\Omega }}_{j,k}=\left\{P_{j,k}|P_{j,k}^{\mathrm{L}}\le P_{j,k}\le P_{j,k}^{\mathrm{U}}\right\}\hfill \end{array}\right.$$

(21)

3. Distributed Control Strategy for Secondary Frequency Regulation of Battery Swapping Station

3.1. Distributed Consensus Algorithm

Compared with the centralized algorithm, the distributed algorithm can significantly improve the computational efficiency and has higher reliability and stability, which is suitable for the battery system of the swapping station with plug-and-play demand. The distributed control algorithm used in this paper is based on the consensus algorithm, the core of which is the iterative process of the consensus variable. The distributed algorithm control strategy for the battery of the swapping station is shown in Figure 3. The gradient component with respect to the frequency control loss function with respect to the frequency control power is defined as the consensus variable $\lambda$. The expression of the consensus variable for the conventional unit and the battery of the swapping station is shown in Equation (22):

$$\left\{\begin{array}{c}{\lambda }_{i,k}={\displaystyle \frac{\partial C_{n,k}^I}{\partial P_{i,k}}}=2{\alpha }_i{P}_{i,k}\hfill \\ {\lambda }_{j,k}={\displaystyle \frac{\partial C_{n,k}^I}{\partial P_{j,k}}}=2{\alpha }_j{P}_{j,k}+{\beta }_{j,k}\hfill \\ {\beta }_{j,k}=-2b_j{S}_j\left(So{C}_{j,k-1}-So{C}_j^{\mathrm{ref}}\right){\eta }_j^c\Delta t,P_{j,k}<0\hfill \\ {\beta }_{j,k}=-2b_j{S}_j\left(So{C}_{j,k-1}-So{C}_j^{\mathrm{ref}}\right)\Delta t/{\eta }_j^d,P_{j,k}>0\hfill \end{array}\right.$$

(22)

From the perspective of frequency control allocation, the secondary frequency control active demand should be undertaken by the frequency control power supply with the smallest unit frequency control power loss, that is, the frequency control power supply with the smallest consensus variable $\lambda$; however, over time, changes in the frequency control power and real-time charging state of the frequency control power supply may lead to an increase in the value of $\lambda$, which in turn generates a new frequency control power supply with the smallest value of $\lambda$ to undertake the frequency control task. Therefore, instead of assigning all secondary frequency control active demands to the same frequency control power supply, the frequency control scheme is determined by an optimization calculation: when the $\lambda$ values of all frequency modulation power sources tend to be consistent, the total frequency modulation loss is minimized.

In power system networks, the connectivity of distributed algorithms is an important parameter to describe the topological characteristics of the network. In this paper, the frequency control power sources that can be communicatively connected are referred to as neighboring power sources. The connectivity degree is defined as the number of neighboring power sources for each frequency control power source. For example, a schematic diagram of the connectivity degree of six frequency control power supplies is illustrated in Figure 4. The connectivity of frequency regulation unit 2 is 3, and the connectivity of frequency control power supply 3 is 2. If the connectivity of all the power supplies is added and divided by the total number of power supplies, 9, the average connectivity of the entire regional power grid can be obtained as 2.

3.2. Secondary Frequency Regulation Distributed Control Strategy

(1) Initialization of frequency control power

After starting the secondary frequency control control, the initial value of the frequency control power of each frequency control power supply should be set first. The $P^{AGC}$ secondary frequency control command signal can be equally distributed to the conventional unit and the battery cluster of the swapping station, as shown in Equation (23):

$$P_{n,k}^0={\displaystyle \frac{P^{\mathrm{AGC}}}{N}}$$

(23)

In turn, the initial value of each power supply consensus variable is calculated as shown in Equation (24):

$$\left\{\begin{array}{cc}{\lambda }_{i,k}^0=2{\alpha }_i{P}_{i,k}^0,\hfill & n\in I\hfill \\ {\lambda }_{j,k}^0=2{\alpha }_j{P}_{j,k}^0+{\beta }_{j,k},\hfill & n\in J\hfill \end{array}\right.$$

(24)

(2) Secondary frequency control allocation optimization strategy

The secondary frequency control output optimization strategy for each frequency control power source at moment k is shown in Figure 5.

Each distributed frequency regulation unit assesses the variation in the consensus variable $\lambda$ value compared to adjacent frequency regulation units and determines if the variation surpasses a threshold.

$$\sum _{m\in \delta \left(n\right)}^N({\lambda }_{m,k}^{q-1}-{\lambda }_{n,k}^{q-1})<\xi$$

(25)

where the adjacent node set of node n is denoted by $\delta \left(n\right)$, and $m\in \delta \left(n\right)$ indicates that m is an adjacent node of n. This implies that frequency control power supply m is adjacent to frequency control power supply n. The variables ${\lambda }_{m,k}^{q-1}$ and ${\lambda }_{n,k}^{q-1}$ represent the consensus variables of frequency control power supplies m and n after $q-1$ iterations, respectively. The parameter $\xi$ is typically set as a small constant close to zero. Furthermore, N denotes the n-th frequency control power source, and q represents the number of iterations.

Each frequency control power supply updates the consensus variable by adjusting its own secondary frequency control output, so that its consensus variable is approximately the same as that of the neighboring power supplies, and the iterative equations [19] are shown in Equations (26)–(30).

$${\overline{\lambda }}_{n,k}^q={\lambda }_{n,k}^{q-1}-{\sigma }_1\sum _{t=1}^{q-1}\left(\sum _{m\leftrightarrow n}^N\left({\lambda }_{m,k}^{q-t}-{\lambda }_{n,k}^{q-t}\right)\right)+{\sigma }_2\left(P_{n,k}^0-P_{n,k-1}\right)$$

(26)

$${\lambda }_{n,k}^q={\left[{\overline{\lambda }}_{n,k}^q\right]}_{\sigma }$$

(27)

$$\left\{\begin{array}{cc}{\overline{P}}_{n,k}^q={\displaystyle \frac{{\lambda }_{n,k}^q}{2\alpha }},\hfill & nϵI\hfill \\ {\overline{P}}_{n,k}^q={\displaystyle \frac{{\lambda }_{n,k}^q-{\beta }_{n,k}^q}{2\alpha }},\hfill & nϵJ\hfill \end{array}\right.$$

(28)

$${\overline{\overline{P}}}_{n,k}^q=P_{n,k-1}^{q-1}+{\sigma }_3\left({\overline{P}}_{n,k}^q-P_{n,k}^{q-1}\right)$$

(29)

$$P_{n,k}^q={\left[{\overline{\overline{P}}}_{n,k}^q\right]}_{\mathsf{\Omega }}$$

(30)

Equation (26) defines the update conditions for the virtual consensus variables for the qth iteration by correcting the consensus variables and power constraints for the q-1st iteration. ${\overline{\lambda }}_{n,k}^q$ is corrected in two parts: one part is to sum the difference between ${\lambda }_{n,k}^{q-1}$ of this frequency control power supply and ${\lambda }_{x,k}^{q-1}$ of the neighboring power supplies, respectively, and the other part is to sum the power constraints on the consensus variables, and ${\sigma }_1$ and ${\sigma }_2$ are both correction coefficients.

Equation (27) represents the constraints on the virtual consensus variable ${\overline{\lambda }}_{n,k}^q$ to ensure that the running results are in line with the actual situation and guarantee the convergence of the algorithm. The value range of the consensus variable ${\lambda }_{n,k}^q$ is set as $\lambda ϵ{({\lambda }_{min},{\lambda }_{max})}_{\delta }$ in the practical situation. When the virtual consensus variable ${\overline{\lambda }}_{n,k}^q$ of frequency control power supply exceeds the range, it takes the boundary value ${\lambda }_{min}$ or ${\lambda }_{max}$; when it does not exceed the boundary value, it remains unchanged.

Equation (28) can be obtained by inverting Equation (24) for the consensus variable $\lambda$. Equation (29) indicates that the virtual power offset is equal to the result after correcting the frequency control output; at this time, no constraints have been set on it, and ${\sigma }_3$ is the correction coefficient.

We apply Equation (30) and then the virtual power offset ${\overline{\overline{P}}}_{n,k}^q$ constraints to obtain the power offset $P_{n,k}^q$. Judging the type of frequency control power supply, according to the above-mentioned constraints, $P_{n,k}$ can be $\mathsf{\Omega }\left(P_{i,k}ϵ{\mathsf{\Omega }}_{i,k},P_{j,k}ϵ{\mathsf{\Omega }}_{j,k}\right)$, and the constraints ${\overline{\overline{P}}}_{n,k}^q$. The constraint process is similar to the constraint of consensus variables.

(3) Determine whether the timer upper limit is reached.

If the timer reaches the preset value, the result of the last iteration is used as the secondary frequency control output command; otherwise, we return to step (1). Since, in the secondary frequency control, it is necessary to calculate the size of the secondary frequency control output of the device at this moment in time $\Delta t$, the timer is generally set to a fixed duration $\Delta t$, namely, the signal sampling time interval.

(4) Update the charging state of the battery of the swapping station.

The updated formula is given in the following equation:

$$So{C}_{j,k}=\left\{\begin{array}{c}So{C}_{j,k-1}-{\displaystyle \frac{{\eta }_j^c{P}_{j,k}^c\Delta t}{S_j}}\hfill \\ \\ So{C}_{j,k-1}-{\displaystyle \frac{P_{j,k}^d\Delta t}{{\eta }_j^d{S}_j}}\hfill \end{array}\right.$$

(31)

At the end of the secondary frequency control configuration at moment $k-1$, the state of charge of the battery clusters is updated according to their actual outputs, which are used in the calculation of the secondary frequency control output allocation at moment k.

4. Case Study

In this paper, the distributed control strategy for secondary frequency control of battery swapping stations was simulated and verified in MATLAB/Simulink. The simulation example consisted of a conventional unit and four battery storage clusters in the V2G battery swapping station, and it was assumed that the conventional unit participates in both primary and secondary frequency control, and the battery storage clusters participate in the secondary frequency control process only. The total capacity of the regional grid was 5000 MW, and the values of the simulation system parameters M and D were 5 and 1, respectively, the unit control power of the primary frequency control was set to 0.04, the values of the secondary frequency control coefficient B and the integration coefficient $K_I$ were 26 and −0.15, respectively, the frequency control period was set to 4 s, and the entire frequency control period was set to 7200 s. The parameter values were calibrated based on real-world systems and referenced from [33]. The net load fluctuation of the local electricity network is depicted in Figure 6, and there is a step-up and a step-down of the grid load near 300 s and 4500 s, respectively, and the rest of the time is a small fluctuation.

Figure 7 shows the curve of grid frequency deviation with time during the participation of the battery swapping station in secondary frequency control. It can be seen that the described control strategy can ensure accurate tracking of the signal and control the grid frequency deviation $\Delta f$ within a very small range. Specifically, the maximum frequency deviation in the extreme condition of the regional power grid is only $1.116\times {10}^{-3}\mathrm{Hz}$, and the fluctuation in the general condition is around $1.946\times {10}^{-7}\mathrm{Hz}$, which can meet the requirement of frequency fluctuation of the regional power grid.

Figure 8 shows the error curve between the common active output of the conventional unit and the battery swapping station’s battery cluster and the secondary frequency control command signal $P^{AGC}$. It can be seen that the described control strategy can achieve accurate tracking of the secondary frequency control signal. Except for the instantaneous tracking errors of $2.632\times {10}^{-3}$ MW and $-3.067\times {10}^{-3}$ MW near 305 s and 4504 s, respectively, the tracking effect for the $P^{AGC}$ signal is relatively good for the rest of the time period, and the errors are all within $-9.802\times {10}^{-6}$ MW.

Figure 9 shows the active output curves for each of the four battery clusters of the conventional unit and the swapping station. It can be seen that the four battery clusters of the conventional unit and the converter station change with the fluctuation of the net load of the regional grid, and the more drastic the fluctuation of the net load is, the faster the change of the active output is. Specifically, at 0–300 s, the net load fluctuates in a small range near 0. At 300 s, the net load increases in a stepwise manner, and each frequency control power source starts to increase its output in response to $P^{AGC}$. The climbing rate of the battery clusters is much larger than that of the conventional units, so, at the beginning of the stepwise change, the battery clusters have a faster frequency control speed and a larger active output, and since the real-time SOC value of battery cluster 1 is larger than that of battery cluster 2, the battery clusters with a larger capacity have a larger output accordingly. Conventional units have a smaller rate of climb, and the units produce power smoothly with a smooth curve. The net load tends to be stable in the middle of frequency control; at this time, the traditional unit mainly undertakes frequency control independently, and the battery cluster maintains the power. At 4500 s, the net load decreases stepwise, and the battery bank cluster responds quickly to track the $P^{AGC}$ signal. The unit is smoothly out of power in the late frequency control period.

Figure 10 shows the variation curves of the consensus variables for the four battery clusters of the conventional unit and the swapping station. The consensus variables of the four battery clusters of the traditional unit and the swapping station are approximately the same throughout the frequency control process, and only in the vicinity of 300 s and 4500 s; due to the step change of the frequency control demand, the consensus variables of the frequency control power sources undergo a short adjustment and converge to the approximate value rapidly, reflecting the good convergence performance of the distributed algorithm.

Figure 11 shows the SOC variation curves of the four battery clusters of the swapping station. It can be seen that the four battery clusters of the swapping station change with the fluctuation of the net load of the regional grid: when the net load increases, the SOC of the four battery clusters all decrease; when the net load decreases, the SOC of the batteries all increase. Specifically, at the beginning of frequency control, the net load increases stepwise, the frequency control units increase the active output in response to the frequency control signal, and the SOC of all four battery clusters gradually decreases; the battery cluster with the largest initial SOC decreases the most, and the battery cluster with the smallest initial SOC decreases the least; at the middle of the frequency control, the frequency control is mainly undertaken by the conventional units independently, and the four battery clusters keep the power, and the SOC is basically unchanged. In the later stages of frequency control, after the traditional units are able to follow the frequency control signals, the battery clusters are charged in a timely manner. As a result, the SOC of all four battery clusters gradually increases, slowly returning to the optimal SOC range, but due to the smallest capacity of battery cluster 4’s batteries, the responsibility for frequency control is correspondingly smaller, so the value of the upward rise of the SOC is even less.

Therefore, this frequency control control strategy is more reasonable for the active output distribution of the frequency control power supply, which can make the battery cluster of the swapping station meet the system frequency control needs while maintaining the SOC in the vicinity of the expected SOC value of the battery (0.5) to ensure the power balance of the battery cluster, and realize the goal of minimizing the total frequency control loss.

To evaluate the effectiveness of the proposed frequency control methods, two widely used control models were selected as benchmark methods. The first, i.e., Mode 1, excludes the participation of the swapping station in frequency regulation. Only traditional units participate in the primary and secondary frequency control. The second, i.e., Mode 2, disregards the behavioral modeling of EV users within the swapping station. The traditional unit participates in the primary and secondary frequency control, and all EV users agree to participate in the secondary frequency control. In comparison, our model assumes that the traditional unit participates in the first and second frequency control, and takes into account the EV users’ willingness to participate in the second frequency control. The results are presented in Figure 12, Figure 13, Figure 14 and Figure 15.

As can be seen from Figure 12, under Mode 1, the frequency deviation of the regional grid is controlled within a small range as a whole. However, due to the limitation of the climbing rate of the conventional unit, a large grid frequency deviation occurs when the net load of the regional grid undergoes a step-up (300 s) and a step-down (4500 s). As can be seen in Figure 13, in Mode 2, the frequency deviation of the regional grid is significantly smaller than that of Mode 1. The fast response characteristics of the battery clusters of the swapping station significantly improve the grid frequency deviation. Mode 2 is extremely similar to the frequency deviation of the strategy proposed in this paper (Figure 7), demonstrating the superiority of the battery storage of the swapping station to participate in the grid frequency control.

From Figure 14, it can be seen that, in Mode 1, the tracking effect for the $P^{AGC}$ signal is not good, and near 300 s and 4500 s, respectively. A large instantaneous tracking error (namely, $P_{error}$) occurs near 300 s and 4500 s, respectively, and the average error for the remainder of the time frame is $-4.332\times {10}^{-4}$ MW, which is much larger than the average error of the control strategy proposed in this paper (Figure 8). From Figure 15, it is observable that, in Mode 2, the tracking effect for the $P^{AGC}$ signal is a little better than that of Mode 1, but there is still a big gap compared with the control strategy proposed in this paper. A large instantaneous tracking error occurs near 300 s and 4500 s, respectively, and the average error for the rest of the time period is $2.351\times {10}^{-4}$ MW.

5. Conclusions

This paper proposes a secondary frequency regulation strategy for V2G battery swapping stations that considers user behavior. By analyzing key factors such as economic incentives, time costs, and battery degradation, a user behavioral model is developed. The loss function for the frequency control in the swapping station is formulated, and the active balance of battery packs is achieved by incorporating a penalty term into the objective function. A distributed control strategy for V2G swapping stations, based on the consensus algorithm, is established to minimize total frequency control loss while meeting the secondary frequency control requirements through iterative updates of consensus variables. Simulation results demonstrate that the proposed strategy not only meets the secondary frequency control requirements of the grid and optimizes the frequency control capabilities of battery clusters but also ensures power balance within the clusters, enhancing grid stability and reliability. Furthermore, the strategy improves system response speed and scalability while reducing communication costs.