Solving PEV Charging Strategies with an Asynchronous Distributed Generalized Nash Game Algorithm in Energy Management System

Abstract

As plug-in electric vehicles (PEVs) become more and more popular, there is a growing interest in the management of their charging power. Many models exist nowadays to manage the charging of plug-in electric vehicles, and it is important that these models are implemented in a better way. This paper investigates a price-driven charging management model in which all plug-in electric vehicles are informed of the charging strategies of neighboring plug-in electric vehicles and adjust their own strategies to minimize the cost, while an aggregator determines the unit price based on overall electricity consumption to coordinate the charging strategies of the plug-in electric vehicles. In this article, we used an asynchronous distributed generalized Nash game algorithm to investigate a charging management model for plug-in electric vehicles in a smart charging station (SCS). In a charging management model, we need to consider constraints on the charge and discharge rates of plug-in electric vehicles, the battery capacity, the amount of charge per plug-in electric vehicle, and the maximum electrical load that the whole system can allow. Meeting the constraints of plug-in electric vehicles and smart charging stations, the model coordinates the charging strategy of each plug-in electric vehicle to ultimately reduce the cost of smart charging stations, which is the cost that the smart charging station should pay to the higher-level power supply facility. To the best of our knowledge, this algorithm used in this paper has not been used to solve this model, and it has better performance than the generalized Nash equilibria (GNE) seeking algorithm originally used for this model, which is called a fast alternating direction multiplier method (Fast-ADMM). In the simulation results, the asynchronous algorithm we used showed a correlation error of 0.0076 at the 713th iteration, compared to 0.0087 for the synchronous algorithm used for comparison, and the cost of the smart charging station was reduced to USD 4800.951 after coordination using the asynchronous algorithm, which was also satisfactory. We used an asynchronous algorithm to better implement a plug-in electric vehicle charging management model; this also demonstrates the potential advantages of using an asynchronous algorithm for solving the charging management model for plug-in electric vehicles.

1. Introduction

PEVs have gotten a lot of attention from the automotive industry and the government due to the advantages of energy saving and emission reduction [1]. With the gradual increase in the penetration of PEVs, charging peaks may occur because PEVs need to be charged frequently [2]. When a large number of PEVs are being charged at the same time resulting in an excessive charging peak, the stability and effectiveness of the SCS will be greatly affected [3], and the grid may even collapse [4]. As a result, the charging of PEVs must be managed.

Recently, the charging problem of PEVs has been extensively studied [5,6,7,8,9,10]. In these studies, the electrical-energy-related aspects of the PEV charging problem are managed through different aspects. Smith et al. [5] investigated the effect of variable-rate charging on reducing overload in charging PEVs. In [6], the problem of transformer overload due to PEV charging was investigated by considering different cases of assigning different available transformer capacities to PEV chargers. In [7], the charging demand of PEVs was assessed by considering the category of PEVs, charging strategy, battery capacity, and other factors, and then a corresponding power supply scheme was developed based on the obtained demand. In [8], Egbue et al. investigated the future development of PEVs by considering the resources (e.g., lithium) available for the batteries of PEVs. Betancur et al. [9] modeled the impact of different types of loads on the grid charged by PEVs in preparation for the future electric vehicle transition. In [10], the use of photovoltaic systems to supply power in the PEV charging problem was studied.

As can be seen from the above, the PEV charging problem is currently being studied more and more intensively. A number of algorithms are also currently available for solving the proposed charging management model for PEVs. In [11], a real-time charging scheme was used to coordinate the PEV charging and accommodate demand response programs in the parking station by charging schedule. In [12], a centralized optimization technique based on learning particle swarm optimization was used to optimize the charging schedule and costs of PEVs. In [13], a centralized approach is proposed to optimize the combined operation of PEVs and photovoltaic units in residential distribution networks. In the above-mentioned works, the optimization is performed in the centralized case. In this case, however, calculating the optimal policy is computationally burdensome and does not better protect the user’s private information [14].

As a result, distributed approaches have received a great deal of attention. In the distributed case, there are a large number of game-theoretic approaches that can be used to deal with the problem. In [15], Li et al. used a non-cooperative game technique to minimize the total charging cost of each PEV in a price-driven charging model. To achieve charging management of PEVs in an SCS considering time anxiety and user behavior, a non-cooperative game model was used in [16]. Li et al. [17] investigated data-driven charging strategies for PEV-based taxis using a stochastic game. To solve the charging control problem, Shamshirband et al. [18] used a novel stochastic cost emission-based approach under the realistic constraints of PEVs. However, the above algorithms are all considered for the synchronous case. When communication delays, packet loss, and changes in the topology of the communication network [19] occur, all PEVs cannot update their policies synchronously, which cannot be solved efficiently by using the synchronization algorithm. Then, it is better to use asynchronous algorithms for processing.

There has been some research into the use of asynchronous algorithms to deal with the PEV charging management problem. In [19], a cutting-plane-based distributed algorithm was used to solve an equivalent surrogate model which has potential advantages in coping with the asynchronous situation when coordinating the charging of PEVs. Wang et al. [20] solved the optimal power control problem in microgrids by an asynchronous algorithm. In [21], an asynchronous multiagent optimization algorithm is proposed for solving a convex optimization problem in a multiagent network asynchronously. These have demonstrated the potential of using asynchronous algorithms to deal with the PEV charging management problem.

We used an asynchronous distributed generalized Nash game algorithm proposed in [22] to solve the PEV charging management model proposed in [23]. Compared to the model in [24,25,26], the model chosen in this paper also takes into account the potential exchange between all PEVs; that is, after obtaining the price decided by the aggregator embedded in an SCS, each PEV chooses its optimal policy based on the policies of the other PEVs and then summarizes the total load information to the aggregator. Compared to the algorithms in [19,20,21], the algorithm used in this paper is able to deal with a GNE game model that is subject to inequality constraints, which is closer to the reality of a charging management model for PEVs in an SCS.

A large number of models exist today for the charging management of PEVs. For a good model, it is important to have a good algorithm to implement it. For the charge management model chosen in this paper, results were obtained using synchronous algorithms but not yet with asynchronous algorithms. It is clear that asynchronous algorithms are better able to simulate the actual charging/discharging of PEVs than synchronous algorithms, since in practice, it is not guaranteed that all PEVs will have access to the information they need at the same time if a policy update is to be made. Therefore, in this paper, we implemented a model using an asynchronous algorithm to confirm the superiority of the asynchronous algorithm.

In light of the above discussion, our main contributions are summarized as follows:

(1)

Preferably, the model we chose is more in line with the situation in which it is shown to exist. Compared to [15,16,17], a new communication mechanism is used in the model used in this paper, which takes into account the potential communication between PEVs, where each PEV can exchange information directly with each other, instead of only individuals with the aggregator. This facilitates the implementation of distributed algorithms and is more in line with real-life situations.

(2)

Secondly, we chose an asynchronous algorithm to solve this model. The final simulation results show that the asynchronous algorithm has a good convergence effect and can also better coordinate the charging and discharging strategies between PEVs. Moreover, compared to the algorithm used in [23] to solve the model, the algorithm used in this paper performs better based on the convergence rate and the coordinated charging of PEVs. This further demonstrates the superiority of using asynchronous algorithms to solve charging management models.

2. Notation

$\mathbb{N}$ denotes the natural number set. $ℝ$ denotes the positive number set. The transpose of the square matrix $Q\in {ℝ}^{n\times n}$ is $Q^T$, the $i$th row is ${[Q]}_i$, and the element, which is in the $i$th row and $j$th column, is ${[Q]}_{ij}$. The Kronecker product of the matrices $A$ and $B$ is $A\otimes B$. The vector/matrix with only 0 (resp.1) element is $\mathbf{0}$ (resp.$\mathbf{1}$). The identity matrix is $I_n\in {ℝ}^{n\times n}$. Given $x_1,\dots ,x_N\in {ℝ}^n$, $\mathit{x}=\mathrm{col}({(x_i)}_{i\in (1,\dots ,N)})={[x_1^T,\dots ,x_N^T]}^T$.

3. System Model and Problem Formulation

By establishing the time-varying price, a PEV aggregator in the SCS controls how a group $\mathcal{N}=\{1,2,\dots ,N\}$ of PEVs charge across a number of charging intervals $\mathcal{T}=\{1,2,\dots ,T\}$. For an SCS, there is a finite number of PEVs that can be accommodated in the same period, and we set its maximum limit to $N_{max}$. In other words, the SCS is only available when $N\le N_{max}$. We define the charging/discharging strategy of each PEV $n$ at time slot $t\in \mathcal{T}$ as $x_{n,t}$, where $x_{n,t}>0$ indicates charging, $x_{n,t}<0$ indicates discharging, and $x_{n,t}=0$ indicates that PEV $n$ is in idle time. At all times, the charging file of each PEV $n$ is ${\mathit{x}}_n={(x_{n,1},x_{n,2},\dots ,x_{n,T})}^T$, and the charging file of all PEVs is $\mathit{x}={({\mathit{x}}_1^T,{\mathit{x}}_2^T,\dots ,{\mathit{x}}_N^T)}^T$.

3.1. System Model

3.1.1. Feasible Charging Profiles

Electrical energy stored in the battery of each PEV $n$ at the time slot $t\in \mathcal{T}$ is ${\pi }_{n,t}$, and PEVs can be charged or discharged as appropriate. Then, the electricity corresponding to the next time is

$${\pi }_{n,t+1}={\pi }_{n,t}+x_{n,t},\forall t\in \mathcal{T},\forall n\in \mathcal{N}.$$

(1)

Moreover, at each time slot $t\in \mathcal{T}$, the electricity of each PEV $n\in \mathcal{N}$ needs to be within its battery capacity, i.e.,

$${\pi }_n^{\min }\le {\pi }_n^t\le {\pi }_n^{\max },\forall t\in \mathcal{T},\forall n\in \mathcal{N},$$

(2)

where ${\pi }_n^{\min }$ is the lower limit of the battery storage capacity, and ${\pi }_n^{\max }$ is the upper limit. The charge/discharge rate of each PEV $n$ is bounded by

$$x_n^{\min }\le x_{n,t}\le x_n^{\max },\forall t\in \mathcal{T},\forall n\in \mathcal{N},$$

(3)

where $x_n^{\min }$ is the lower limit of charge rate, and $x_n^{\max }$ is the upper limit. In addition, assuming that the beginning and the final electricity of PEV $n$ are ${\pi }_n^0$ and ${\pi }_n^T$, respectively, the required total electricity is

$$R_n={\pi }_n^T-{\pi }_n^0,\forall n\in \mathcal{N}.$$

(4)

For each PEV $n$, the total electrical energy obtained in the end should reach its charging requirement, i.e.,

$${\displaystyle \sum _{t=1}^T{x}_{n,t}=R_n,\forall n\in \mathcal{N}}.$$

(5)

Moreover, the charging demand of all PEVs at each time could not exceed the SCS’s maximum power supply, which is the maximum amount of power that the SCS can supply to PEVs while ensuring the normal operation of the system, i.e.,

$${\displaystyle \sum _{n=1}^N{x}_{n,t}\le C_{\max },\forall t\in \mathcal{T}}.$$

(6)

Then, for each PEV $n$, the feasible charging profile is

$${\mathcal{X}}_n=\left\{{\mathit{x}}_n\right|(2)(3)(5)(6)\}.$$

(7)

For all PEVs, we have

$$\mathcal{X}=\{({\mathit{x}}_1,{\mathit{x}}_2,\dots ,{\mathit{x}}_N)|{\mathit{x}}_n\in {\mathcal{X}}_n,\forall n\in \mathcal{N}\}.$$

(8)

3.1.2. Electricity Price

The electricity price $p_t$ is defined by the following linear function [27]:

$$p_t={\kappa }_t{L}_t+{\beta }_t,\forall t\in \mathcal{T},$$

(9)

where ${\kappa }_t$ represents the tariff premium factor at time slot $t$, ${\beta }_t$ represents the initial tariff at time slot $t$, which is proportional to the distance from the power station to the smart charging station, and $L_t={\displaystyle {\sum }_{n=1}^N{x}_{n,t}}$ represents the total load of the SCS at time slot $t$. When there is a huge number of PEVs being charged at the same time, it will lead to an increase in the total load, which will further lead to an increase in the tariff. To minimize charging expenses, PEVs that do not require urgent charging will adjust their charging times, i.e., choose other periods when the tariff is lower.

3.1.3. Cost of SCS

We specify the cost function $C_t(L_t)$ as the payment of the SCS to the higher-level power supply facility at time slot $t$, which needs to satisfy the following assumptions:

Assumption 1. 

At each time slot$t\in \mathcal{T}$, the function$C_t(L_t)$is assumed to be a non-decreasing function.

Then, the cost model of the SCS can be described as a quadratic function [28] as follows:

$$C_t(L_t)=p_t{L}_t={\kappa }_t{L}_t^2+{\beta }_t{L}_t,\forall t\in \mathcal{T}.$$

(10)

Therefore, the total cost during the whole charging period is

$$C_{tol}={\displaystyle \sum _{t=1}^T{C}_t(L_t).}$$

(11)

3.2. Problem Formulation

The purpose of charging coordination is to reduce the cost to the SCS of purchasing power from a higher level of power resource provider by determining the optimal charging strategies for all PEVs without violating the PEV and SCS constraints; hence, the problem may be written as

$$\{\begin{array}{c}{\min }_{x\in \mathcal{X}}{\mathrm{C}}_{tol},\hfill \\ s.t.\left(7\right),\forall n\in \mathcal{N}.\hfill \end{array}$$

(12)

The optimization problem in (12) is strictly convex and has a unique optimal solution as the function $C_t(L_t)$ in (10) is strictly convex and the set ${\mathcal{X}}_n$ in (7) is nonempty, compact, and convex [29].

4. Problem Solution

4.1. Problem Formulation

We assume that all PEVs in an SCS are rational and self-interested, and their goal is to minimize their total cost. Since the total cost of each PEV $n$ depends on the total load at the corresponding moment, it follows that their total cost depends on the charging strategies taken by the other PEVs in the same SCS. Let us define $f_n$ as the total cost of PEV $n$ when charging is completed, then it can be defined as follows:

$$f_n({\mathit{x}}_n,{\mathit{x}}_{-n})={\gamma }_n{\displaystyle \sum _{t=1}^T[{\kappa }_t({\displaystyle \sum _{n=1}^N{x}_{n,t}{)}^2+{\beta }_t{\displaystyle \sum _{n=1}^N{x}_{n,t}]}}},$$

(13)

where ${\mathit{x}}_n\in {\mathcal{X}}_n$ and ${\mathit{x}}_{-n}=({\mathit{x}}_1,\dots {\mathit{x}}_{n-1},{\mathit{x}}_{n+1},\dots {\mathit{x}}_N)$ is the charging strategies of all PEVs except PEV $n$. Let $\delta =\frac{{\displaystyle {\sum }_{n=1}^N{f}_n}}{{\displaystyle {\sum }_{t=1}^T{C}_t(L_t)}}$ denote whether the SCS is profitable or not: when $\delta =1$, the SCS breaks even; when $\delta >1$, the SCS can make a profit; and when $\delta <1$, the SCS loses money. Without the loss of generality, the $\delta$ is unchanged, and we need to ensure $\delta \ge 1$. Let ${\xi }_n=\frac{{\displaystyle {\sum }_{t=1}^T{x}_{n,t}}}{{\displaystyle {\sum }_{k\in \mathcal{N}}{\displaystyle {\sum }_{t=1}^T{x}_{k,t}}}}$ as the total energy consumed by PEV $n$ in proportion to the total energy provided by the SCS. Then, we can derive that ${\gamma }_n=\delta {\xi }_n$. Since each PEV $n$ must reach its charging demand at the end of charging, the total charge of all PEVs and the total charge of each PEV $n$ are constant, which means that ${\xi }_n$ is deterministic for each PEV $n$ in the SCS. Then, the ${\gamma }_n$ is fixed for each PEV $n$.

Based on (13), we have

$$f_n({\mathit{x}}_n,{\mathit{x}}_{-n})={\gamma }_n{\displaystyle \sum _{t=1}^T[{\kappa }_t{(L_t)}^2+{\beta }_t(L_t)]=}{\gamma }_n{\displaystyle \sum _{t=1}^T{C}_t(L_t)}={\gamma }_n{C}_{tol}.$$

(14)

According to (14), we can say that each PEV $n$ is also trying to minimize $C_{tol}$ when it minimizes its own total cost $f_n$. Since each PEV $n$ chooses its optimal strategy for minimizing $C_{tol}$, the non-cooperative game model $\mathcal{G}=\{\mathcal{N},{\left\{{\mathcal{X}}_n\right\}}_{n\in \mathcal{N}},{\left\{f_n\right\}}_{n\in \mathcal{N}}\}$ is defined, and it consists of the following elements:

• Players: the PEVs in set $\mathcal{N}$;
• Cost function: $f_n({\mathit{x}}_n,{\mathit{x}}_{-n})$ for each PEV $n\in \mathcal{N}$;
• Strategy set: ${\mathcal{X}}_n$ for each PEV $n\in \mathcal{N}$ is nonempty, compact, and convex.

For this model, we adopt GNE as the strategic choice, and the total spending of each PEV $n$ is influenced by the other PEVs. For each PEV $n\in \mathcal{N}$, we assume that there are $M$ constraints.

Definition 1. 

For the game model$\mathcal{G}$, a vector${\mathit{x}}^{*}=({\mathit{x}}_n^{*},{\mathit{x}}_{-n}^{*})\in \mathcal{X}$is a GNE if, for each PEV$n$, it holds

$$f_n({\mathit{x}}_n^{*},{\mathit{x}}_{-n}^{*})\le f_n({\mathit{x}}_n,{\mathit{x}}_{-n}^{*}),\forall ({\mathit{x}}_n,{\mathit{x}}_{-n}^{*})\in \mathcal{X},\forall n\in \mathcal{N}.$$

(15)

That is, when all PEVs choose the strategies in ${\mathit{x}}^{*}$, no PEV $n$ can minimize its expense by unilaterally changing its charging strategy.

Next, the pseudo-gradient mapping is defined as:

$$F(\mathit{x})=\mathrm{col}({({\nabla }_{x_n}{f}_n({\mathit{x}}_n,{\mathit{x}}_{-n}))}_{n\in \mathcal{N}}),$$

(16)

which collects the gradients of (13) each with respect to (w.r.t) its strategy into a collective vector. $F(\mathit{x})$ represents the possible reduction in cost for each PEV $n$ if it unilaterally changes its strategy without the other PEVs changing theirs. For the mapping $F(\mathit{x})$, we make the following assumptions.

Assumption 2. 

The mapping$F(\mathit{x})$in (16) is$\alpha$-strong monotone and$\ell$-Lipschitz continuous, for$\alpha ,\ell >0$.

Here, we are concerned with vGNE, which is a subset of GNE, and we can seek vGNE by solving the variational inequality (VI). Using the Hessian matrix, we can prove that $f_n$ is a convex function [23]. Moreover, it is clear that $f_n$ is continuously differentiable. Then, we can obtain that $F(\mathit{x})$ is a single-valued function, and the vGNE of game $\mathcal{G}$ corresponds to the solution of $VI(F(\mathit{x}),\mathcal{X})$. The solution of $VI(F(\mathit{x}),\mathcal{X})$ exists while $F(\mathit{x})$ is continuous and $\mathcal{X}$ is compact, and furthermore, there is one and only one solution for $VI(F(\mathit{x}),\mathcal{X})$ while $F(\mathit{x})$ is strongly monotonic; that is, game $\mathcal{G}$ has a unique vGNE. At this point, the cost function $f_n$ of each PEV $n$ takes on a minimum value. According to the previous discussion, the strictly convex function $C_{tol}$ takes on the unique optimal solution.

4.2. Communication Network

We use an undirected and connected graph ${\mathcal{G}}^c=({\mathcal{N}}^c,{ℰ}^c)$ to describe the communication network between PEVs in the same SCS in game $\mathcal{G}$ where ${\mathcal{N}}^c$ represents the set of PEVs, and ${ℰ}^c\subseteq {\mathcal{N}}^c\times {\mathcal{N}}^c$ represents the set of edges. Given two PEVs $i,j\in {\mathcal{N}}^c$ and ${\mathcal{N}}_i^c$, which is the neighbor set of PEV $i$, if PEV $i$ and PEV $j$ share information with each other directly, then PEV $j$ is a neighbor of PEV $i$, i.e., $j\in {\mathcal{N}}_i^c$, and the couple (i,j) belongs to ${ℰ}^c$. The number of edges is $E=\left|{ℰ}^c\right|$, and the number of PEVs is $N=\left|{\mathcal{N}}^c\right|$. Then, we label the edges as $e_l$, for $l\in \{1,\dots ,E\}$. For the incidence matrix $V\in {ℝ}^{E\times N}$, we define ${[V]}_{li}=1$ if $e_l=(i,\cdot )$, ${[V]}_{li}=-1$ if $e_l=(\cdot ,i)$, and ${[V]}_{li}=0$ otherwise. By construction, $V{\mathit{1}}_N={\mathit{0}}_N$. We define the node Laplacian $L\in {ℝ}^{N\times N}$ as $L=V^{T}V$, which is a symmetric matrix; then, we have $L{\mathit{1}}_N={\mathit{0}}_N$. Next, we define ${ℰ}_i^{out}$(resp. ${ℰ}_i^{in}$) as the set of all the index $l$ of the edges $e_l$, which start from (resp. end in) PEV $i$, and hence, ${ℰ}_i^c={ℰ}_i^{out}\cup {ℰ}_i^{in}$.

4.3. vGNE Seeking

We define $\mathit{x}=\mathrm{col}({(x_i)}_{i\in {\mathcal{N}}^c})$, $\mathit{\lambda }=\mathrm{col}({({\lambda }_i)}_{i\in {\mathcal{N}}^c})$, $\mathit{z}=\mathrm{col}({(z_l)}_{l\in \{1,\dots ,E\}})$, where $x_i$ is the strategy of PEV $i$, ${\lambda }_i$ is the dual variable of PEV $i$, which can be used to express the fairness of the constraints each PEV is subject to when deciding on its own best response, and $z_l$ is the auxiliary variable of $l$th edge, which is used to enforce the consensus among the dual variables and ultimately ensure that all PEVs choose their best response within a fair constraint. Considering time delays due to communication delays, etc., we define ${\phi }_i(k)$ as the delay of PEV $i$ at iteration $k$. We have ${\tilde{x}}_i=x_i(k-{\phi }_i(k))$, ${\tilde{\lambda }}_i={\lambda }_i(k-{\phi }_i(k))$, and ${\tilde{z}}_l=z_l(k-{\phi }_i(k))$ for all $l\in {ℰ}_i^{out}$, and furthermore, $\tilde{\mathit{x}}=\mathrm{col}({({\tilde{x}}_i)}_{i\in {\mathcal{N}}^c})$, $\tilde{\mathit{\lambda }}=\mathrm{col}({({\tilde{\lambda }}_i)}_{i\in {\mathcal{N}}^c})$, and $\tilde{\mathit{z}}=\mathrm{col}({({\tilde{z}}_l)}_{l\in \{1,\dots ,E\}})$. Let us define $\varpi =\mathrm{col}(\mathit{x},\mathit{z},\mathit{\lambda })$. Then, to simplify the symbolic representation, we define $\varpi =\varpi (k)$, $\tilde{\varpi }=\tilde{\varpi }(k)$, $\widehat{\varpi }=\widehat{\varpi }(k)$, and ${\varpi }^{\oplus }=\varpi (k+1)$.

To mathematically elaborate the update mechanism of the algorithm, we introduce $N$ diagonal matrices ${\mathit{H}}_i$, which represents the update of PEV $i$. ${[{\mathit{H}}_i]}_{jj}$ is 1 if the $j$th element of $\mathrm{col}(\mathit{x},\mathit{z},\mathit{\lambda })$ is an element of $\mathrm{col}(x_i,{\left\{z_l\right\}}_{l\in {ℰ}_i^{out}},{\lambda }_i)$; ${[{\mathit{H}}_i]}_{jj}$ is 0, otherwise. The selection of which PEV to update at the $k$th iteration is determined by an identically distributed random variable $\zeta (k)$, which takes values in $\mathit{H}={\left\{{\mathit{H}}_i\right\}}_{i\in {\mathcal{N}}^c}$. Given a discrete probability distribution $(p_1,\dots ,p_N)$, let $\mathbb{P}[\zeta (k)={\mathit{H}}_i]=p_i$, for each PEV $i\in {\mathcal{N}}^c$. That is, each PEV $i$ updates the best response at the $k$th iteration with probability $p_i$, thus simulating situations such as communication delays.

The delay time of each PEV $i$ is bounded, as formalized next.

Assumption 3. 

The delays are all upper limited in the same way; that is, there exits$\overline{\phi }>0$such that$su{p}_{k\in {\mathbb{N}}_{\ge 0}}ma{x}_{i\in {\mathcal{N}}^c}\{{\phi }_i(k)\}\le \overline{\phi }\le +\infty$.

Theorem 1 [22]. 

For each iteration, the step size of each PEV$i\in {\mathcal{N}}^c$, and$\vartheta \in ℝ$such that

$${\tau }_i\le {(\Vert A_i\Vert +\vartheta )}^{-1},$$

(17a)

$${\upsilon }_i\le {(2\rho +\vartheta )}^{-1},$$

(17b)

$${\epsilon }_i\le {(\rho \left|{\mathcal{N}}_i^c\right|+\Vert A_i\Vert +\vartheta )}^{-1},$$

(17c)

where$\vartheta >\frac{1}{2\nu }$and$\nu =\min \{\frac{\alpha }{{\ell }^2},{\lambda }_{max}{\left\{L\right\}}^{-1}\}$.

Based on all the above definitions, the algorithm we use to solve the game $\mathcal{G}$, which means finding the strategy of each PEV $n$ that minimizes $f_n$, is as Algorithm 1. Its convergence was proved in [22].

Algorithm 1. Asynchronous distributed algorithm with edge variables (AD-GEED) [22]

Initialization:$k=0,{\mathit{x}}^0\in {ℝ}^{T\times N},{\mathit{\lambda }}^0\in {ℝ}^{M\times N}$ and ${\mathit{z}}^0={\mathbf{0}}_{M\times E}$, choose ${\upsilon }_i,{\mathit{\epsilon }}_i$, and ${\mathit{\tau }}_i$ satisfying (17) and relaxation factor $\eta \in (0,1)$ Iteration k: Select PEV $i_k$ with probability $\mathbb{P}[\zeta (k)={\mathit{H}}_{i_k}]=p_{i_k}$. Reading: PEV $i_k$ reads the current values which are in the public memory (i.e., ${\tilde{\mathit{x}}}_j,$${\tilde{\lambda }}_j,\forall j\in {\mathcal{N}}_{i_k}^c$ and ${\tilde{z}}_l,\forall l\in {ℰ}_{i_k}^{in}$ and $l\in {ℰ}_j^{out}$) into its private memory. Update: ${\widehat{x}}_{i_k}=pro{j}_{{\Omega }_{i_k}}(x_{i_k}-{\tau }_{i_k}({\nabla }_{i_k}(f_{i_k}(x_{i_k},{\tilde{\mathit{x}}}_{-i_k})+A_{i_k}^T{\lambda }_{i_k}))$ ${\widehat{z}}_l=z_l+{\upsilon }_{i_k}\rho ({[V]}_l\otimes I_m)\tilde{\mathit{\lambda }},\forall l\in {ℰ}_{i_k}^{out}$ ${\widehat{\lambda }}_{i_k}=pro{j}_{{ℝ}_{\ge 0}^m}({\lambda }_{i_k}+{\epsilon }_{i_k}(A_{i_k}(2{\widehat{x}}_{i_k}-x_{i_k})-b_{i_k}-\rho ({[V^T]}_{i_k}\otimes I_m)\tilde{\mathit{z}}-(2{\upsilon }_{i_k}{\rho }^2+1){\displaystyle {\sum }_{j\in {\mathcal{N}}_{i_k}^c}({\lambda }_{i_k}}-\tilde{\lambda })$ $x_{i_k}^{\oplus }=x_{i_k}+\eta ({\widehat{x}}_{i_k}-x_{i_k})$ $z_l^{\oplus }=z_l+\eta ({\widehat{z}}_l-z_l),\forall l\in {ℰ}_{i_k}^{out}$ ${\lambda }_{i_k}^{\oplus }={\lambda }_{i_k}+\eta ({\lambda }_{i_k}-{\lambda }_{i_k})$ Writing: PEV $i_k$ writes the new values to the public memories of PEV $j\in {\mathcal{N}}_{i_k}^c$ $(x_{i_k},{\lambda }_{i_k})\leftarrow (x_{i_k}^{\oplus },{\lambda }_{i_k}^{\oplus })$ ${\left\{z_l\right\}}_{l\in {ℰ}_{i_k}^{out}}\leftarrow {\left\{z_l^{\oplus }\right\}}_{l\in {ℰ}_{i_k}^{out}}$ $k\leftarrow k+1$

Due to communication delays, packet losses, communication network topology changes, etc., the time it takes for each PEV in an SCS to get new information from the other PEVs will vary. When we execute the Algorithm 1, in the synchronous mechanism, the next iteration can only be performed when all PEVs have finished iterating, whereas, in the asynchronous mechanism, each PEV does not need to wait until the other PEVs have finished iterating before iterating but can complete its next iteration based on the data that have not been updated by the other PEVs. In this way, each PEV can iterate asynchronously without having to wait for the other PEVs to update together synchronously. Updating asynchronously is suitable for more general situations.

In Algorithm 1, we use edge auxiliary variables ${\left\{z_l\right\}}_{l\in \left\{1\dots E\right\}}$ to compute vGNE asynchronously, where the implementation of asynchrony is performed using the “Arock” framework [22]. All PEVs in the algorithm only exchange information with their neighboring PEVs, and all PEVs can acquire information and update their policies at different times, and it can be shown that the algorithm still converges relatively quickly under these conditions [22].

In Algorithm 1, we randomly select a PEV $n$ for the best response follow-up at each iteration, which is used to represent the asynchronous update of PEVs due to various reasons. At each iteration, we first find the optimal response of the PEV $n$ by using the gradient descent and dual ascend structure. This corresponds to the optimal decision of the PEV $n$ given that the decisions of the other PEVs remain the same, i.e., to choose to charge when the electricity price is cheaper as far as possible under certain constraints. Among other things, we can note that the mapping $F(x)$ in the game $\mathcal{G}$ is strongly monotonic, and it also means that each PEV $n$ chooses the optimal strategy each time based on reducing its expenditure. To obtain the vGNE, we then update $x$ by using preconditioned forward–backward splitting. As the optimal strategies of all PEVs which eventually minimize $C_{tol}$ in the game $\mathcal{G}$ correspond to their best responses when computed with AD-GEED, the best response set of all PEVs at the end of AD-GEED is also equivalent to the vGNE of the game $\mathcal{G}$. At this time, the total cost $f_n$ of each PEV $n$ under certain constraints is minimized, and in turn, the total cost $C_{tol}$ of the SCS is minimized.

5. Simulation

We carried out simulation experiments with MATLAB 2021a. From the numerical results of the simulations, we can see that the AD-GEED algorithm can effectively tackle the charging management of PEVs, and the AD-GEED algorithm shows better performance compared to the numerical results obtained by the Fast-ADMM algorithm originally used in the model.

In an SCS with a capacity of 10 PEVs (i.e., $N_{max}=10$), we consider the case where 10 PEVs come to charge. The communication network between them is shown in Figure 1, where each node represents a PEV $n$, and the two PEVs connected by each edge can communicate directly. After the aggregator has obtained the total load information, it then determines the time-varying price information. For comparison purposes, the parameters of this model for simulation are set identically to the corresponding parameters of the model in [23]. The initial charge, total charge required, upper/lower limit of battery capacity, upper/lower limit of charge/discharge rate, and the time of arrival and departure from the SCS of each PEV $n$ are shown in

Table 1. PEV parameter settings.

PEV	$${\mathit{\pi }}_{\mathit{n}}^0 \left(\mathbf{kW}\right)$$	$${\mathit{R}}_{\mathit{n}} \left(\mathbf{kW}\right)$$	$${\mathit{\pi }}_{\mathit{n}}^{\mathbf{max}} \left(\mathbf{kW}\right)$$	$${\mathit{\pi }}_{\mathit{n}}^{\mathbf{min}} \left(\mathbf{kW}\right)$$	$${\mathit{x}}_{\mathit{n}}^{\mathbf{max}} (\mathbf{kW}/\mathbf{h})$$	$${\mathit{x}}_{\mathit{n}}^{\mathbf{min}} (\mathbf{kW}/\mathbf{h})$$	AT	DT
1	7.5	67.5	75	5	15	−15	17:00	22:00
2	6	72	80	5	15	−15	18:00	23:00
3	7	67.5	75	5	10	−10	19:00	6:00
4	5.6	63	70	5	8	−8	17:00	7:00
5	6.7	58.5	65	5	10	−10	18:00	8:00
6	7.5	67.5	75	5	12	−12	12:00	18:00
7	8.1	58.5	65	5	10	−10	11:00	23:00
8	9	72	80	5	10	−10	12:00	6:00
9	7.2	63	70	5	8	−8	13:00	7:00
10	7.5	67.5	75	5	15	−15	14:00	20:00

. The table shows that each PEV owner chooses his or her own charging time and amount of charge according to his or her circumstances. The battery capacity and charging/discharging rates of each PEV $n$ depend on the specific configuration for different PEVs. The maximum power supplied by the SCS is 50 kWh. Moreover, ${\kappa }_t=0.3\$/\mathrm{k}\mathrm{W}\mathrm{h}$ during the peak charging period between 18:00 and 06:00 and ${\kappa }_t=0.2\$/\mathrm{k}\mathrm{W}\mathrm{h}$ during the off-peak charging period between 7:00 and 17:00. In all periods, ${\beta }_t=0.8\$/\mathrm{k}\mathrm{W}\mathrm{h}$. Let us consider the scenario of the SCS breaking even (i.e., $\delta$ = 1), and in AD-GEED, for every PEV $n\in {\mathcal{N}}^c$, let $\eta$ = 0.5, $\overline{\phi }$ = 4, and $\mathbb{P}[\zeta (k)={\mathit{H}}_n]=\frac{1}{10}$.

The convergence results are shown in Figure 2. The smaller the correlation error, the closer the strategies chosen by all PEVs at this point are to the optimal strategies at the Nash equilibrium point of the game $\mathcal{G}$, and the closer the total cost of the SCS is to its minimum value. It can be seen that the AD-GEED algorithm converges at a fast rate, with the relative error reduced to ${10}^{-5}$ by the 10000th iteration. A comparison of the convergence results of the AD-GEED and Fast-ADMM algorithms at the 713th iteration is shown in

Table 2. Comparison of the two algorithms.

Algorithms	Relative Error at 713th Iteration		Total Cost of SCS
	Synchronous Situation	Asynchronous Situation
AD-GEED	0.0076	0.0298	USD 4800.951
Fast-ADMM	0.0087	-	USD 4670.042

. It shows that the Fast-ADMM algorithm has a relative error of 0.0087, while the AD-GEED algorithm has a relative error of 0.0298. The difference between the two relative errors is small. The reason why the relative error of the AD-GEED algorithm is larger is that this paper considers the asynchronous case, whereas the Fast-ADMM algorithm only considers the synchronous case. When the AD-GEED algorithm is used to deal with the case of synchronization, the relative error is 0.0076. It can be seen that the AD-GEED algorithm achieves the desired results faster for processing this model. Figure 3 shows the optimal charging/discharging strategies for all PEVs at each moment when game $\mathcal{G}$ achieved GNE. We can see that for each PEV $n$, its charging/discharging strategy is constantly adapted to the specific circumstances of each moment. During the period from 17:00 to 22:00, three PEVs successively supply power to the other PEVs by discharging, which allows the SCS to supply less power to meet the demand of all PEVs and also ensures that the total load of the SCS at the corresponding moment does not exceed its maximum value, which is shown in Figure 4. Moreover, in Figure 5 [23], the total load of the SCS reaches its maximum at about 17:00, whereas, as shown in Figure 4, the total SCS load is below the maximum at any point in this paper, which also guarantees a certain level of safety. The total cost of the SCS when no charge coordination is performed is USD 6298.702. What we can see from Table 2 is that the Fast-ADMM algorithm reduced the total cost of the SCS to USD 4670.042, while the AD-GEED algorithm reduced it to USD 4800.951. It can be seen that the AD-GEED algorithm also performs well in terms of reducing the total cost of the SCS. In Figure 6, we show the change in charging/discharging strategies for all PEVs throughout the iteration at 17:00. In each iteration of the AD-GEED algorithm, PEV $n$ reselects its own optimal strategy for the current situation based on the strategies of the other PEVs. Eventually, at the GNE, all PEVs determine their optimal strategy choice and do not change it again, as no PEV’s strategy change will result in a lower cost to itself at this point. This again illustrates that the AD-GEED algorithm is capable of efficiently resolving the charge issue in the SCS.

6. Conclusions

In this article, for a plug-in electric vehicle charging management model, we use an asynchronous distributed algorithm with an edge variables algorithm for solving. The asynchronous algorithm we used is better suited to deal with communication delays, packet losses, and communication network topology changes than the synchronous algorithm fast alternating direction multiplier method originally used in this model. This is more applicable to real-life situations. In the simulation results, the use of the asynchronous distributed algorithm with an edge variables algorithm converges faster in the asynchronous case, and in the final coordination results, the total load in all time slots is below the maximum that the system can sustain. It is clear from the simulation results that the asynchronous distributed algorithm with an edge variables asynchronous algorithm outperforms the fast alternating direction multiplier method algorithm in terms of convergence rate and coordination results for plug-in electric vehicles. This also demonstrates the potential advantages of using the asynchronous algorithm used in this paper to coordinate the charging of plug-in electric vehicles. However, for the same number of iterations, the convergence rate of the asynchronous algorithm is a little slower due to the difference between asynchronous and synchronous, which could also be used as a direction for future improvement.