Optimal Regulation Strategy of Distribution Network with Photovoltaic-Powered Charging Stations Under Multiple Uncertainties: A Bi-Level Stochastic Optimization Approach

Abstract

In order to consider the impact of multiple uncertainties on the interaction between the distribution network operator (DNO) and photovoltaic powered charging stations (PVCSs), this paper proposes a regulation strategy for a distribution network with a PVCS based on bi-level stochastic optimization. First, the interaction framework between the DNO and PVCS is established to address the energy management and trading problems of different subjects in the system. Second, considering the uncertainties in the electricity price and PV output, a bi-level stochastic model is constructed with the DNO and PVCS targeting their respective interests. Furthermore, the conditional value-at-risk (CVaR) is introduced to measure the relationship between the DNO’s operational strategy and the uncertain risks. Next, the Karush–Kuhn–Tucker (KKT) conditions and duality theorem are utilized to tackle the challenging bi-level problem, resulting in a mixed-integer second-order cone programming (MISCOP) model. Finally, the effectiveness of the proposed regulation strategy is validated on the modified IEEE 33-bus system.

1. Introduction

With the advancement of green, environmentally friendly and low-carbon energy consumption patterns, electric vehicles (EV), which are synonymous with clean energy, have gained widespread attention [1]. EV ownership has continued to rise over the last decade, reaching 18 million EVs worldwide by the end of 2022 [2]. However, charging stations that serve a large number of EVs can have serious impacts on the distribution network if left unsupervised, such as voltage violation, peak load stacking, and high energy costs [3]. In addition, the interaction between the distribution network and charging stations is subject to multiple uncertainties, which complicates the energy management of the distribution network and poses risks to the system’s operation [4]. Therefore, it is important to study the optimal regulation strategy of the distribution network containing charging stations, considering uncertainties.

Existing studies on the interaction strategy between charging stations and the distribution network are mainly divided into two categories: multi-objective optimization and bi-level optimization. In the field of multi-objective optimization, reference [5] regulates resources such as EVs and micro gas turbines (MTs), proposing an optimization strategy aimed at minimizing the peak–valley difference, minimizing the operating costs of the DNO, and maximizing the utilization rate of renewable energy, thereby effectively ensuring the economic and safe operation of the distribution network. Reference [6] not only focuses on the peak–valley difference in the distribution network but also minimizes the energy cost of charging stations, taking into account the benefits of both. Reference [7] considers the EV charging costs and the carbon trading costs for generators, ensuring both the economic efficiency and low-carbon performance of the system. However, the aforementioned references are based on the principle of overall optimization, treating charging stations as price takers and guiding them with a pre-determined electricity price. Unlike other controllable resources, charging stations, as third-party entities, are not directly dispatched by the DNO. Therefore, a more reasonable pricing strategy is needed to guide charging stations in their electricity purchase and sale behaviors, as well as their charging and discharging behaviors, in order to explore their adjustable capacity. Within the bi-level optimization framework, the transaction electricity price between the PVCS and the DNO is treated as a decision variable, determined through optimization rather than being fixed in advance. Reference [8] constructs a bi-level model between the distribution network and charging stations, where the distribution network aims to minimize the cost of utilizing various resources, and the charging stations aim to minimize energy costs. This model utilizes the transaction electricity price derived from optimization to facilitate the coordinated output of various resources by the DNO and the charging stations. Reference [9] constructs a Stackelberg game model between the distribution network and charging stations, which formulates power interaction strategies and game prices, fully leveraging the adjustable potential of EVs while reflecting both parties’ pursuit of their respective interests. It can be seen that in order to improve the operating revenue of the DNO, the coordination between the DNO issuing scheduling instructions to controllable resources and pricing PVCSs is of great significance.

The impact of uncertainty on the interaction strategy between charging stations and the distribution network has been extensively studied. Reference [10] discusses the impact of multiple uncertainties such as the electricity price and load demand on energy management in a distribution network. Reference [11] takes into account the PV uncertainty and proposes a scheduling strategy that utilizes EV resources to improve the flexibility of the distribution network. Reference [12] proposes a pricing strategy for an EV-containing distribution network considering the uncertainties of new energy. The uncertainty analysis in the above references focuses on the side of the distribution network. However, an increasing number of charging stations are now equipped with renewable energy sources, becoming prosumers participating in market transactions. References [13,14] indicate that charging stations equipped with PVs can increase their own benefits. The inherent uncertainty of the PV output poses challenges to the power balance within the charging stations, which can further impact the pricing strategy and power interaction strategy of the distribution network. To measure the risk posed by uncertainties in a DNO’s operation, theories such as variance, shortfall probability, expected shortage, value-at-risk (VaR), and CVaR have been applied to DNO risk management [15]. Among them, the CVaR is widely used because, besides being a coherent risk measure, it can be expressed using a linear formulation. Reference [16] introduces CVaR to address the uncertainty risks from renewable energy sources, thereby enhancing the risk management level of the distribution network. Reference [17] considers multiple types of uncertainties faced by AC-DC networks and uses CVaR to achieve a balance between the risk and cost.

Therefore, this paper proposes an optimal regulation strategy for a distribution network with a PVCS under multiple uncertainties based on bi-level stochastic optimization, taking into account the uncertainties faced by different subjects. The main contributions of this paper are as follows:

• A bi-level stochastic model is constructed. The upper level is a two-stage stochastic model of the distribution network, where the first stage targets the PVCS that is not directly dispatched by the DNO and decides the electricity price associated with the PVCS. The second stage targets the resources that are directly dispatched by the DNO and decides their active power. The lower level is the PVCS’s energy cost model. Regarding the issue of EV constraints at the lower level, the Minkowski Sum is employed to aggregate the EV clusters in order to reduce the model dimensionality.
• The CVaR theory is introduced to measure the impact of the uncertainty risk on the regulation strategy for electricity price uncertainty on the DNO side and PV uncertainty on the PVCS side.
• It is demonstrated that the proposed model’s complementary constraints for electricity purchasing and selling in PVCSs, as well as for charging and discharging EVs, are redundant. The KKT conditions and duality theorem are utilized to tackle the challenging bi-level problem, resulting in a MISCOP model.

The rest of this paper is organized as follows: In Section 2, the PVCS–DNO game interaction framework based on bi-level stochastic optimization is proposed. In Section 3, a bi-level stochastic model considering uncertainties is constructed. Section 4 provides proof of the redundancy of some constraints and transforms the model for a solution using KKT conditions, the duality theorem, and the Big-M method. Section 5 verifies the effectiveness of the bi-level stochastic model in the modified IEEE 33-bus distribution network. Finally, Section 6 summarizes the conclusions of this paper.

2. Interactive Framework for Distribution Network with PVCS

This section proposes an interactive framework for a distribution network with a PVCS based on bi-level stochastic optimization, as shown in Figure 1. The upper-level model addresses the problem of maximizing the operational revenue of the distribution network, which depends on the electricity price from the main grid and the interaction with the PVCS. The lower-level model addresses the regulation and energy usage of the PVCS for the EV clusters, which is influenced by the EV clusters’ travel behavior and electricity price traded with the DNO. The uncertainties faced by the proposed strategy in this paper are modeled using a scenario-based approach. To more clearly explain the proposed bi-level stochastic model optimization framework, two uncertainty sets, $W$ and $O$, as well as the scenario indices $o$ and $w$, are defined according to the problem associated with the uncertainty parameters. Set $O$ consists of uncertainties related to the upper-level model, specifically, the electricity price from the main grid for the distribution network. Set $W$ consists of uncertainties related to the lower-level model, specifically, the PV output within the PVCS.

As mentioned earlier, the PVCS is different from other controllable resources in that it is not directly dispatched by the DNO. Additionally, there are pricing issues between the PVCS and the DNO. Considering the differences between the PVCS and other resources, we construct a two-stage model in the upper level. In the first stage of the model, the DNO aims to maximize the revenue from transactions with the PVCS and makes decisions regarding the pricing issues and power interaction strategy with the PVCS. The variables in this stage are referred to as “here-and-now” variables [18]. Based on the optimization results of this stage, a corresponding transaction pricing curve between the DNO and PVCS can be established. In the second stage of the model, the DNO aims to maximize the revenue from resource scheduling, deciding on the electricity purchase from the main grid, MT output, and ESS output. This stage primarily concerns resources directly managed by the distribution network, and the variables are referred to as “wait-and-see” variables [19]. The lower-level model is based on the perspective of the PVCS, accounting for discharge losses and aiming to minimize energy costs. It interacts in a strategic game with the distribution network’s first-stage decision-making. The interaction results of the PVCS also influence the second-stage scheduling strategy of the distribution network.

Regarding uncertainty scenarios, this paper generates prediction error values based on forecasted values using the Latin hypercube sampling method to form an uncertainty scenario set. Furthermore, a fast-forward scenario reduction technique based on the probability distance is employed to obtain a set of representative scenarios [20].

3. The Model of DNO with PVCS Based on Bi-Level Stochastic Optimization

3.1. Upper-Level Model: DNO’s Operational Revenue Maximization Problem

3.1.1. Objective Function of the Upper-Level Model

The upper-level model’s objective is to maximize the expected operational revenue of the DNO, with the objective function established as

$$\begin{array}{l}\max F_a=F_1+F_2\\ \left\{\begin{array}{l}{F}_1={\displaystyle \sum _{w=1}^W{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^I{\rho }_w(c_{w,i,t}^{\mathrm{buy}}{P}_{w,i,t}^{\mathrm{buy}}-c_{w,i,t}^{\mathrm{sell}}{P}_{w,i,t}^{\mathrm{sell}}})}}\\ F_2={\displaystyle \sum _{w=1}^W{\displaystyle \sum _{o=1}^O{\displaystyle \sum _{t=1}^T{\rho }_w{\rho }_o\left[\begin{array}{l}{c}_t^{\mathrm{LOAD}}{\displaystyle \sum _{j=1}^J{P}_{j,t}^{\mathrm{LOAD}}}-c_{o,t}^{\mathrm{DNO}}{P}_{wo,t}^{\mathrm{DNO}}-\\ {\displaystyle \sum _{m=1}^M(a_m^{\mathrm{MT}}{P}_{wo,m,t}^{\mathrm{MT}}+b_m^{\mathrm{MT}})-{\displaystyle \sum _{n=1}^N{c}_n^{\mathrm{ESS}}(P_{wo,n,t}^{\mathrm{ESSch}}+P_{wo,n,t}^{\mathrm{ESSdis}})}}\end{array}\right]}}}\end{array}\right.\end{array}$$

(1)

where $F_1$ represents the expected revenue from transactions between the DNO and PVCS, corresponding to the first-stage revenue. $F_2$ represents the expected revenue from the scheduling of other resources, which corresponds to the second-stage revenue and includes revenue from conventional loads, the cost of purchasing electricity from the main grid, the cost of MT generation, and the cost of ESS operation. T represents the scheduling period. I, M, N and J represent the number of PVCSs, MTs, and ESSs and the conventional load, respectively. ${\rho }_w$ and ${\rho }_o$ represent the probability of scenarios $w$ and $o$, respectively. $c_{w,i,t}^{\mathrm{buy}}$, $c_{w,i,t}^{\mathrm{sell}}$, $P_{w,i,t}^{\mathrm{buy}}$, and $P_{w,i,t}^{\mathrm{sell}}$ represent the price of electricity purchasing and selling and the amount of electricity purchasing and selling, respectively. $c_t^{\mathrm{LOAD}}$ and $P_{j,t}^{\mathrm{LOAD}}$ represent the price of electricity consumption and active power for conventional loads, respectively. $c_{o,t}^{\mathrm{DNO}}$ and $P_{wo,t}^{\mathrm{DNO}}$ represent the electricity purchase price and the electricity purchase quantity for the DNO from the main grid, respectively. $P_{wo,m,t}^{\mathrm{MT}}$ represents the output power of the MT, while $a_m^{\mathrm{MT}}$ and $b_m^{\mathrm{MT}}$ represent the cost coefficients. $c_n^{\mathrm{ESS}}$, $P_{wo,n,t}^{\mathrm{ESSch}}$, and $P_{wo,n,t}^{\mathrm{ESSdis}}$ represent the cost coefficient, charging power, and discharging power of the ESS, respectively. In particular, a formula with subscript containing “$w$” means that it is related to the lower-level uncertainty scenario; a formula with subscript containing “$o$” means that it is related to the upper-level uncertainty scenario, and a formula with subscript containing “$wo$” means that it is related to both the lower-level and upper-level uncertainty scenarios. In addition, a subscript without “$w$” or “$o$” means that it is not affected by the scenario. This will not be repeated later.

3.1.2. Risk Management

Uncertainty in the electricity price of the main grid can directly affect the distribution grid’s decisions on power purchasing and the dispatch of various resources. Furthermore, uncertainty in the PV output will affect the power balance in the charging station, which in turn will indirectly affect the formulation of a regulation strategy for the distribution grid. Therefore, the decision-making processes of both the distribution network and PVCS must fully account for the risks arising from both types of uncertainties.

This paper introduces the CVaR theory for the risk management of the DNO’s regulation strategy, allowing risks from both types of uncertainties to be specifically characterized as risks of reduced operational revenue for the DNO. As a risk management approach derived from VaR, CVaR can control the risk beyond the VaR threshold, thus better reflecting the potential value of risk [21].

The value of CVaR typically represents the expected maximum loss of a portfolio at a given confidence level. This risk measurement method considers the tail risk beyond the VaR quantile, providing a more comprehensive assessment of the portfolio’s risk [22]. The calculation is shown in the following equation:

$$\mathrm{CVaR}={\displaystyle \frac{1}{1-\alpha }}{\displaystyle \underset{f(x,y)<=VaR\alpha }{\int }f(x,y)\rho }(y)dy$$

(2)

where x represents the decision variable. $f(x,y)$ is the loss function. $\rho (y)$ is the probability density function of the random variable $y$. $\alpha$ is the confidence level. $VaR\alpha$ is the expected maximum loss at the confidence level.

Since Equation (2) contains VaR, which makes it difficult to find CVaR directly, it is necessary to transform Equation (2) into a mathematical optimization problem with continuously differentiable convex functions by introducing a transformation function. In this optimization problem, the CVaR value is used as the objective function. The transformation function is defined as

$$F_{\alpha }(x,\beta )=\beta +{\displaystyle \frac{1}{(1-\alpha )}}{{\displaystyle \int \left[f(x,y)-\alpha \right]}}^{+}\rho (y)dy$$

(3)

where ${\left[f(x,y)-\alpha \right]}^{+}$ represents $\max \left\{f(x,y)-\alpha ,0\right\}$. $\beta$ is the value of VaR at the confidence level $\alpha$. Considering the need to minimize the maximum loss, we have $\mathrm{CVaR}=\min F_{\alpha }(x,\beta )$.

However, the objective function in this paper is a revenue function, not a loss function. The DNO faces uncertainties in terms of the electricity price and PV output, which means that the DNO will bear corresponding uncertainty risks when operating each unit, affecting the DNO’s revenue. Therefore, in this paper, VaR represents the minimum revenue boundary for the DNO when facing uncertainties at a certain confidence level, while CVaR represents the expected value of the tail (1−$\alpha$) of the DNO’s operational revenue distribution. Unlike the loss function, when the object of risk quantification is a revenue function, the optimization goal is to maximize the minimum revenue. Let the revenue function be $f_1(x,y)$; then, Equation (3) can be rewritten as

$$F_{\alpha }(x,\beta )=\beta -{\displaystyle \frac{1}{(1-\alpha )}}{{\displaystyle \int \left[\alpha -f_1(x,y)\right]}}^{+}\rho (y)dy$$

(4)

where ${\left[\alpha -f_1(x,y)\right]}^{+}$ represents $\max \left\{0,\alpha -f_1(x,y)\right\}$. At this point, we have $\mathrm{CVaR}=\max F_{\alpha }(x,\beta )$. Since the integral term in the transformation function is complex to compute, it typically needs to be linearized. Considering the revenue function in this model, the linear form of CVaR can be expressed as

$$\mathrm{CVaR}=\beta -{\displaystyle \frac{1}{1-\alpha }}{\displaystyle \sum _{w=1}^W{\displaystyle \sum _{o=1}^O{\rho }_w}}{\rho }_o{\delta }_{wo}$$

(5)

$$\begin{array}{l}\left\{\begin{array}{l}{\displaystyle \sum _{w=1}^W{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^I(c_{w,i,t}^{\mathrm{buy}}{P}_{w,i,t}^{\mathrm{buy}}-c_{w,i,t}^{\mathrm{sell}}{P}_{w,i,t}^{\mathrm{sell}}})}}+\\ {\displaystyle \sum _{w=1}^W{\displaystyle \sum _{o=1}^O{\displaystyle \sum _{t=1}^T[c_t^{load}{P}_t^{load}-c_{o,t}^{\mathrm{DNO}}{P}_{wo,t}^{\mathrm{DNO}}}}}\\ -{\displaystyle \sum _{m=1}^M(a_m^{\mathrm{MT}}{P}_{wo,m,t}^{\mathrm{MT}}+b_m^{\mathrm{MT}})-}{\displaystyle \sum _{n=1}^N{c}_n^{\mathrm{ESS}}(P_{wo,n,t}^{\mathrm{ESSch}}+P_{wo,n,t}^{\mathrm{ESSdis}})}]\end{array}\right\}\ge \beta -{\delta }_{wo}\\ \end{array}$$

(6)

$$\delta wo\ge 0$$

(7)

where ${\delta }_{wo}$ is a non-negative auxiliary variable, specifically representing the amount by which the DNO’s operational revenue falls below $\beta$ under various scenarios.

Furthermore, the objective function of the upper-level model, accounting for risk, can be expressed as

$$\max F^U=(1-\gamma )F_a+\gamma \mathrm{CVaR}$$

(8)

where $\gamma$ is the risk aversion coefficient.

3.1.3. Constraints of the Upper-Level Model

(1) Power flow constraints

The conventional distribution network has a radial structure and uses the linear DistFlow equations to describe the power flow constraints [23]:

$${\displaystyle \sum _{q\in h(j)}{P}_{wo,jq,t}-}{\displaystyle \sum _{i\in e(j)}(P_{wo,ij,t}+r_{ij}{I}_{wo,ij,t})=}{P}_{wo,j,t}^{\mathrm{MT}}+P_{wo,j,t}^{\mathrm{DNO}}-P_{j,t}^{\mathrm{LOAD}}-P_{wo,j,t}^{\mathrm{ESSch}}+P_{wo,j,t}^{\mathrm{ESSdis}}-P_{w,j,t}^{\mathrm{buy}}+P_{w,j,t}^{\mathrm{sell}}$$

(9)

$${\displaystyle \sum _{q\in h(j)}{Q}_{wo,jq,t}-}{\displaystyle \sum _{i\in w(j)}(Q_{wo,ij,t}-}{x}_{ij}{I}_{wo,ij,t})=Q_{wo,j,t}^{\mathrm{MT}}+Q_{wo,j,t}^{\mathrm{DNO}}-Q_{j,t}^{\mathrm{LOAD}}$$

(10)

$$U_{wo,j,t}=U_{wo,i,t}-2(r_{ij}{P}_{wo,ij,t}+x_{ij}{Q}_{wo,ij,t})+(r_{ij}^2+x_{ij}^2)I_{wo,ij,t}$$

(11)

$${‖\begin{array}{c}2P_{wo,ij,t}\\ 2Q_{wo,ij,t}\\ I_{wo,ij,t}-U_{wo,i,t}\end{array}‖}_2\le I_{wo,ij,t}+U_{wo,i,t}$$

(12)

where $h(j)$ and $e(j)$ denote the sets of terminal nodes at the end and start of branches with node j as the respective end. $P_{wo,ij,t}$ and $Q_{wo,ij,t}$ denote the active and reactive power flows from node i to node j on branch ij. $r_{ij}$ and $x_{ij}$ are the resistance and reactance of branch ij, respectively. $Q_{wo,j,t}^{\mathrm{DNO}}$ and $Q_{wo,j,t}^{\mathrm{MT}}$ represent the reactive power injected by the main grid and the reactive power of MT, respectively. $Q_{j,t}^{\mathrm{LOAD}}$ denotes the reactive power of the conventional load.

(2) Safety operation constraints

$$U_i^{\min }\le U_{wo,i,t}\le U_i^{\max }$$

(13)

$$0\le I_{wo,ij,t}\le I_{ij,t}^{\max }$$

(14)

where $U_i^{\max }$ and $U_i^{\min }$ represent the limits of the squared node voltage, respectively, and $I_{ij,t}^{\max }$ denotes the square of the maximum safe current for branch ij.

(3) Electricity purchase constraints

$$0\le P_{wo,j,t}^{\mathrm{DNO}}\le P^{\mathrm{DNO}\max }$$

(15)

$$0\le Q_{wo,j,t}^{\mathrm{DNO}}\le Q^{\mathrm{DNO}\max }$$

(16)

where $P^{\mathrm{DNO}\max }$ and $Q^{\mathrm{DNO}\max }$ denote the maximum active and reactive electricity purchase quantities, respectively.

(4) MT constraints

$$0\le P_{wo,j,t}^{\mathrm{MT}}\le P^{\mathrm{MT}\max }$$

(17)

$$0\le Q_{wo,j,t}^{\mathrm{MT}}\le Q^{\mathrm{MT}\max }$$

(18)

where $P^{\mathrm{MT}\max }$ and $Q^{\mathrm{MT}\max }$ represent the maximum active power and maximum reactive power of the MT, respectively.

(5) ESS constraints

$$\left\{\begin{array}{l}0<=P_{wo,j,t}^{\mathrm{ESSch}}<=P_j^{\mathrm{ESSchmax}}\hfill \\ 0<=P_{wo,j,t}^{\mathrm{ESSdis}}<=P_j^{\mathrm{ESSdismax}}\hfill \\ P_{wo,j,t}^{\mathrm{ESSch}}{P}_{wo,j,t}^{\mathrm{ESSdis}}=0\hfill \\ S_{wo,j,t}^{ESS}=S_{wo,j,t-1}^{ESS}+{\eta }^{\mathrm{ech}}{P}_{wo,j,t}^{\mathrm{ESSch}}\Delta t-P_{wo,j,t}^{\mathrm{ESSdis}}\Delta t/{\eta }^{\mathrm{edis}}\hfill \\ S_j^{ESS\min }<=S_{wo,j,t}^{ESS}<=S_j^{ESS\max }\hfill \\ S_{wo,j,1}^{ESS}=S_{wo,j,T}^{ESS}=S_{wo,j,0}^{ESS}\hfill \end{array}\right.$$

(19)

where $P_j^{\mathrm{ESSchmax}}$ and $P_j^{\mathrm{ESSdismax}}$ represent the maximum charging and discharging capacities of the ESS, respectively. ${\eta }^{\mathrm{ech}}$ and ${\eta }^{\mathrm{edis}}$ represent the charging efficiency and discharging efficiency of the ESS, respectively. $S_j^{\mathrm{ESSmin}}$ and $S_j^{\mathrm{ESSma}x}$ indicate the safe boundary limits for the ESS’s battery charge. $S_{wo,j,1}^{\mathrm{ESS}}$ and $S_{wo,j,T}^{\mathrm{ESS}}$ represent the battery charge after the first scheduling period and after the final scheduling period. $S_{wo,j,0}^{\mathrm{ESS}}$ represents the initial battery charge.

(6) Pricing constraints for transactions between DNO and PVCS

The DNO needs to establish the pricing for transactions with the PVCS and cannot arbitrarily lower the price at which the PVCS sells electricity to the DNO. Equation (22) ensures that the price at which the PVCS sells electricity to the DNO is lower than the price at which it purchases electricity.

$$c_{i,t}^{\mathrm{EVsmin}}\le c_{w,i,t}^{\mathrm{sell}}\le c_{i,t}^{\mathrm{EVs}\max }$$

(20)

$$c_{i,t}^{\mathrm{EVb}\min }\le c_{w,i,t}^{\mathrm{buy}}\le c_{i,t}^{\mathrm{EVb}\max }$$

(21)

$$c_{i,t}^{\mathrm{EVs}\max }<c_{i,t}^{\mathrm{EVb}\min }$$

(22)

where $c_{i,t}^{\mathrm{EVs}\max }$ and $c_{i,t}^{\mathrm{EVsmin}}$ represent the limits of the price at which the PVCS sells electricity to the DNO. $c_{i,t}^{\mathrm{EVb}\max }$ and $c_{i,t}^{\mathrm{EVb}\min }$ represent the limits of the price at which the PVCS purchases electricity from the DNO.

3.2. Lower-Level Model: Minimizing the Energy Costs for PVCS

In the lower-level model, the PVCS faces uncertainties related to the PV output and aims to reduce its energy costs by adjusting its electricity purchasing and selling strategy as well as the charging and discharging strategy for EVs.

3.2.1. Objective Function of the Lower-Level Model

The PVCS, while meeting users’ demand, adjusts its electricity purchasing and selling strategy with the DNO based on the electricity price to minimize the electricity cost. This objective function includes the cost of purchasing electricity from the DNO, the revenue from selling electricity, and the cost of discharging losses. For any scenario $w$, the objective function is as follows:

$$\min F_w^{\mathrm{L}}={\displaystyle \sum _{i=1}^I{\displaystyle \sum _{t=1}^T(c_{w,i,t}^{\mathrm{buy}}{P}_{w,i,t}^{\mathrm{buy}}-c_{w,i,t}^{\mathrm{sell}}{P}_{w,i,t}^{\mathrm{sell}}+\kappa P_{w,i,t}^{\mathrm{dis}})}}$$

(23)

where $\kappa$ is the compensation coefficient for EV discharging [24].

3.2.2. Constraints of the Lower-Level Model

(1) EV Cluster Constraints

To avoid the high dimensionality that arises from modeling individual EVs, this paper employs an EV aggregation method based on the Minkowski Sum [25]. This approach projects the variable space of individual EVs into a hypercube space, effectively compressing the EV cluster into a generalized energy storage device, thereby significantly reducing the model’s dimensionality. The constraints for the EV cluster based on the Minkowski Sum are as follows:

$$0\le P_{w,i,t}^{\mathrm{ch}}\le P_{i,t}^{\mathrm{chmax}}:{\mu }_{w,i,t}^{1L},{\mu }_{w,i,t}^{1U}\ge 0$$

(24)

$$0\le P_{w,i,t}^{\mathrm{dis}}\le P_{i,t}^{\mathrm{dismax}}:{\mu }_{w,i,t}^{2L},{\mu }_{w,i,t}^{2U}\ge 0$$

(25)

$$S_{w,i,t}=\left\{\begin{array}{ll}{\eta }^{\mathrm{ch}}{P}_{w,i,t}^{\mathrm{ch}}\Delta t-P_{w,i,t}^{\mathrm{dis}}\Delta t/{\eta }^{\mathrm{dis}}+d{S}_{w,i,t},\hfill & t=1\hfill \\ S_{w,i,t-1}+{\eta }^{\mathrm{ch}}{P}_{w,i,t}^{\mathrm{ch}}\Delta t-P_{w,i,t}^{\mathrm{dis}}\Delta t/{\eta }^{\mathrm{dis}}+d{S}_{w,i,t},\hfill & t\in [2,T]\hfill \end{array}\right.:{\lambda }_{w,i,t}^1$$

(26)

$$S_{w,i,T}=-d{S}_{i,T+1}:{\lambda }_{w,i}^2$$

(27)

$$S_{i,t}^{\min }\le S_{w,i,t}\le S_{i,t}^{\max }:{\mu }_{w,i,t}^{3L},{\mu }_{w,i,t}^{3U}\ge 0$$

(28)

where $P_{w,i,t}^{\mathrm{ch}}$, $P_{w,i,t}^{\mathrm{dis}}$, $P_{i,t}^{\mathrm{chmax}}$, and $P_{i,t}^{\mathrm{dismax}}$ represent the charging and discharging power of the i-th EV cluster and their respective upper limits. $S_{w,i,t}$, $S_{i,t}^{\min }$, and $S_{i,t}^{\max }$ denote the capacity of the EV cluster and its lower/upper limits. ${\eta }^{\mathrm{ch}}$ and ${\eta }^{\mathrm{dis}}$ are the charging and discharging efficiency coefficient. $d{S}_{w,i,t}$ is the step change in energy caused by the connection or disconnection of individual EVs to the grid. Equation (27) ensures the constraint that the EV clusters must meet the required energy level at the time of disconnection from the grid. Additionally, the variables ${\mu }_{w,i,t}^{1L}$, ${\mu }_{w,i,t}^{1U}$, ${\mu }_{w,i,t}^{2L}$, ${\mu }_{w,i,t}^{2U}$, ${\mu }_{w,i,t}^{3L}$, ${\mu }_{w,i,t}^{3U}$, ${\lambda }_{w,i,t}^1$, and ${\lambda }_{w,i}^2$ are all dual variables, each defined after their respective constraints (with the inequality constraints indicated by λ, and the equality constraints indicated by μ). The same definitions apply to the dual variables ${\mu }_{w,i,t}^{4L}$, ${\mu }_{w,i,t}^{4U}$, ${\mu }_{w,i,t}^{5L}$, ${\mu }_{w,i,t}^{5U}$, ${\mu }_{w,i,t}^6$, ${\mu }_{w,i,t}^7$, and ${\lambda }_{w,i,t}^3$ in the subsequent sections.

(2) Power Purchasing and Selling Constraints

$$0\le P_{w,i,t}^{\mathrm{buy}}\le P_{i,t}^{\mathrm{buymax}}:{\mu }_{w,i,t}^{4L},{\mu }_{w,i,t}^{4U}\ge 0$$

(29)

$$0\le P_{w,i,t}^{\mathrm{sell}}\le P_{i,t}^{\mathrm{sellmax}}:{\mu }_{w,i,t}^{5L},{\mu }_{w,i,t}^{5U}\ge 0$$

(30)

where $P_{i,t}^{\mathrm{buymax}}$ and $P_{i,t}^{\mathrm{sellmax}}$ denote the upper limits for the PVCS’s electricity purchasing and selling, respectively.

(3) Power Balance Constraint

$$P_{w,i,t}^{\mathrm{sell}}-P_{w,i,t}^{\mathrm{buy}}+P_{w,i,t}^{\mathrm{ch}}-P_{w,i,t}^{\mathrm{dis}}=P_{w,i,t}^{\mathrm{pv}}:{\lambda }_{w,i,t}^3$$

(31)

where $P_{w,i,t}^{\mathrm{pv}}$ represents the active power of the PV.

(4) Charger Power Constraint

$$P_{w,i,t}^{\mathrm{ch}}\le P_i^{\mathrm{cmax}}:{\mu }_{w,i,t}^6\ge 0$$

(32)

$$P_{w,i,t}^{\mathrm{dis}}\le P_i^{\mathrm{dmax}}:{\mu }_{w,i,t}^7\ge 0$$

(33)

where $P_i^{\mathrm{cmax}}$ and $P_i^{\mathrm{dmax}}$ represent the maximum charging and discharging power limits of the station’s chargers, respectively.

In practical applications, the PVCS typically does not engage in both purchasing and selling electricity simultaneously, nor does it charge and discharge at the same time. However, the lower-level model presented in this paper does not include complementary constraints for PVCS electricity trading ($P_{w,i,t}^{\mathrm{buy}}{P}_{w,i,t}^{\mathrm{sell}}=0$) and EV charging/discharging ($P_{w,i,t}^{\mathrm{ch}}{P}_{w,i,t}^{\mathrm{dis}}=0$). This is because these constraints are redundant in this study, as will be demonstrated in Section 4.1.

4. Model Proof and Solution

4.1. Proof of Redundancy of Complementary Constraints

In the lower-level model, the nonlinearity of complementary constraints can make the optimization problem difficult to solve [26]. By combining the KKT conditions, it can be demonstrated that the complementary constraints for purchasing and selling electricity, as well as for charging and discharging, are redundant in this optimization model.

(1) $P_{w,i,t}^{\mathrm{buy}}{P}_{w,i,t}^{\mathrm{sell}}=0$

According to the complementary slackness conditions, the dual variables satisfy the following:

$${\mu }_{w,i,t}^{4L}{P}_{w,i,t}^{\mathrm{buy}}=0$$

(34)

$${\mu }_{w,i,t}^{5L}{P}_{w,i,t}^{\mathrm{sell}}=0$$

(35)

Assuming that there exists a simultaneous purchase and sale of electricity, i.e., $P_{w,i,t}^{\mathrm{buy}}>0$ and $P_{w,i,t}^{\mathrm{sell}}>0$, we have ${\mu }_{w,i,t}^{4L}=0$ and ${\mu }_{w,i,t}^{5L}=0$. Further, assuming that $L_w$ is the Lagrangian function of the lower-level model (see Appendix A (A1)), we have

$${\displaystyle \frac{\partial L_w}{\partial P_{w,i,t}^{\mathrm{buy}}}}=c_{w,i,t}^{\mathrm{buy}}-{\mu }_{w,i,t}^{4L}+{\mu }_{w,i,t}^{4U}+{\lambda }_{w,i,t}^3=0$$

(36)

$${\displaystyle \frac{\partial L_w}{\partial P_{w,i,t}^{\mathrm{sell}}}}=-c_{w,i,t}^{\mathrm{sell}}-{\mu }_{w,i,t}^{5L}+{\mu }_{w,i,t}^{5U}-{\lambda }_{w,i,t}^3=0$$

(37)

Next, Equation (36) is added to Equation (37):

$$c_{w,i,t}^{\mathrm{buy}}-c_{w,i,t}^{\mathrm{sell}}+{\mu }_{w,i,t}^{4U}+{\mu }_{w,i,t}^{5U}=0$$

(38)

Since the dual variables satisfy ${\mu }_{w,i,t}^{4U}\ge 0$ and ${\mu }_{i,t}^{5U}\ge 0$, and $c_{w,i,t}^{\mathrm{buy}}>c_{w,i,t}^{\mathrm{sell}}$ from constraints (20)–(22), Equation (38) does not hold.

Therefore, the PVCS does not simultaneously purchase and sell electricity, meaning that $P_{w,i,t}^{\mathrm{buy}}{P}_{w,i,t}^{\mathrm{sell}}=0$ in this model is redundant.

(2) $P_{w,i,t}^{\mathrm{ch}}{P}_{w,i,t}^{\mathrm{dis}}=0$

Taking the situation of the PVCS purchasing power and EV supplemental power within a single time period as an example for analysis, the PVCS has two options: Option 1 involves EV cluster-only charging, while Option 2 involves both charging and discharging (with the charging amount exceeding the discharging amount). In both options, the EV cluster receives the same amount of energy within the time period; therefore,

$$P_{w,i,t}^{\mathrm{ch}1}{\eta }^{\mathrm{ch}}=P_{w,i,t}^{\mathrm{ch}2}{\eta }^{\mathrm{ch}}-P_{w,i,t}^{\mathrm{dis}2}/{\eta }^{\mathrm{dis}}$$

(39)

where $P_{w,i,t}^{\mathrm{ch}1}$ represents the charging power in Option 1. $P_{w,i,t}^{\mathrm{ch}2}$ and $P_{w,i,t}^{\mathrm{dis}2}$ represent the charging and discharging power in Option 2, respectively.

Since the PVCS only purchases electricity, the selling power is zero. Combined with Equation (31), the equations related to the purchased electricity for Option 1 and Option 2 can be derived as follows:

$$P_{w,i,t}^{\mathrm{ch}1}=P_{w,i,t}^{\mathrm{buy}1}+P_{w,i,t}^{\mathrm{pv}}$$

(40)

$$P_{w,i,t}^{\mathrm{ch}2}-P_{w,i,t}^{\mathrm{dis}2}=P_{w,i,t}^{\mathrm{buy}2}+P_{w,i,t}^{\mathrm{pv}}$$

(41)

where $P_{w,i,t}^{\mathrm{buy}1}$ and $P_{w,i,t}^{\mathrm{buy}2}$ represent the amount of electricity purchased in Option 1 and Option 2, respectively.

Multiplying both Equations (40) and (41) by ${\eta }^{\mathrm{ch}}$, we have

$$P_{w,i,t}^{\mathrm{ch}1}{\eta }^{\mathrm{ch}}=\left(P_{w,i,t}^{\mathrm{buy}1}+P_{w,i,t}^{\mathrm{pv}}\right){\eta }^{\mathrm{ch}}$$

(42)

$$\left(P_{w,i,t}^{\mathrm{ch}2}-P_{w,i,t}^{\mathrm{dis}2}\right){\eta }^{\mathrm{ch}}=\left(P_{w,i,t}^{\mathrm{buy}2}+P_{w,i,t}^{\mathrm{pv}}\right){\eta }^{\mathrm{ch}}$$

(43)

Combining Equation (42) with Equation (39) and eliminating $P_{w,i,t}^{\mathrm{ch}1}$, we have

$$P_{w,i,t}^{\mathrm{ch}2}{\eta }^{\mathrm{ch}}-P_{w,i,t}^{\mathrm{dis}2}/{\eta }^{\mathrm{dis}}=\left(P_{w,i,t}^{\mathrm{buy}1}+P_{w,i,t}^{\mathrm{pv}}\right){\eta }^{\mathrm{ch}}$$

(44)

Subtracting Equation (43) from Equation (44) and eliminating $P_{w,i,t}^{\mathrm{ch}2}$ as well as $P_{w,i,t}^{\mathrm{pv}}$, we have

$$-P_{w,i,t}^{\mathrm{dis}2}{\eta }^{\mathrm{ch}}+P_{w,i,t}^{\mathrm{dis}2}/{\eta }^{\mathrm{dis}}=(P_{w,i,t}^{\mathrm{buy}2}-P_{w,i,t}^{\mathrm{buy}1}){\eta }^{\mathrm{ch}}$$

(45)

Since ${\eta }^{\mathrm{dis}}$ and ${\eta }^{\mathrm{ch}}$ are less than 1, Equation (45) simplifies to

$$\left(P_{w,i,t}^{\mathrm{buy}2}-P_{w,i,t}^{\mathrm{buy}1}\right)=P_{w,i,t}^{\mathrm{dis}2}({\displaystyle \frac{1}{{\eta }^{\mathrm{dis}}{\eta }^{\mathrm{ch}}}}-1)>0$$

(46)

Equation (46) implies more electricity needs to be purchased in Option 2, indicating that Option 2 is uneconomical. Consequently, according to the objective function (23), Option 2 is not the optimal solution. Thus, the PVCS does not simultaneously charge and discharge, making the corresponding constraint redundant in this model. The proof for other situations such as electricity selling and discharging follows similarly and is omitted here. As a result, the lower-level model proposed in this paper is a linear convex optimization problem, which allows the bi-level model to be transformed into a single-level model using KKT conditions.

4.2. Model Solution

4.2.1. KKT Conditions

The above linear convex optimization problem can be replaced by KKT optimality conditions, which include Equations (24)–(33) along with other dual variable constraints and complementary slackness constraints (see Appendix A for details). Thus, the proposed bi-level stochastic model can be transformed into a single-level stochastic model.

4.2.2. Linearization of the Objective Function

However, the single-level model still contains nonlinear terms that are challenging to solve directly using a solver and thus require further linearization. The nonlinear components include the objective function and the complementary constraints listed in Appendix A (A7)–(A18).

(1) The nonlinear terms in the objective function are linearized by applying the strong duality theorem.

$$\begin{array}{l}{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{t=1}^T(c_{i,t}^{\mathrm{buy}}{P}_{i,t}^{\mathrm{buy}}-c_{i,t}^{\mathrm{sell}}{P}_{i,t}^{\mathrm{sell}})}}=\\ {\displaystyle \sum _{i=1}^I{\displaystyle \sum _{t=1}^T\left(-\kappa P_{i,t}^{\mathrm{dis}}-{\mu }_{w,i,t}^{1U}{P}_{i,t}^{\mathrm{chmax}}-{\mu }_{w,i,t}^{2U}{P}_{i,t}^{\mathrm{dismax}}+{\mu }_{w,i,t}^{3L}{S}_{i,t}^{\min }-{\mu }_{w,i,t}^{3U}{S}_{i,t}^{\max }+{\lambda }_{w,i,t}^{1}d{S}_{w,i,t}\right.}}\\ \left.-{\mu }_{w,i,t}^{4U}{P}_{w,i,t}^{\mathrm{buymax}}-{\mu }_{w,i,t}^{5U}{P}_{w,i,t}^{\mathrm{sellmax}}+{\lambda }_{w,i,t}^3{P}_{w,i,t}^{\mathrm{pv}}-{\mu }_{w,i,t}^6{P}_{i,t}^{\mathrm{cmax}}-{\mu }_{w,i,t}^7{P}_{i,t}^{\mathrm{dmax}}\right)-{\displaystyle \sum _{i=1}^I{\lambda }_{w,i}^{2}d{S}_{i,T+1}}\end{array}$$

(47)

(2) The complementary slackness constraints in Appendix A (A7)–(A18) are linearized using the Big-M method.

$$\left\{\begin{array}{l}0\le Y\le X^{dual}M\\ 0\le \Delta P\le \left(1-X^{dual}\right)M\end{array}\right.$$

(48)

$$Y={[{\mu }_{w,i,t}^{1L},{\mu }_{w,i,t}^{1U},{\mu }_{w,i,t}^{2L},{\mu }_{w,i,t}^{2U},{\mu }_{w,i,t}^{3L},{\mu }_{w,i,t}^{3U},{\mu }_{w,i,t}^{4L},{\mu }_{w,i,t}^{4U},{\mu }_{w,i,t}^{5L},{\mu }_{w,i,t}^{5U},{\mu }_{w,i,t}^6,{\mu }_{w,i,t}^7]}^{\mathrm{T}}$$

(49)

$$\begin{array}{l}\Delta P=[P_{w,i,t}^{\mathrm{ch}},P_{i,t}^{\mathrm{chmax}}-P_{w,i,t}^{\mathrm{ch}},P_{w,i,t}^{\mathrm{dis}},P_{i,t}^{\mathrm{dismax}}-P_{w,i,t}^{\mathrm{dis}},S_{w,i,t}-S_{i,t}^{\min },S_{i,t}^{\max }-S_{w,i,t},\\ P_{w,i,t}^{\mathrm{buy}},P_{i,t}^{\mathrm{buymax}}-P_{w,i,t}^{\mathrm{buy}},P_{w,i,t}^{\mathrm{sell}},P_{i,t}^{\mathrm{sellmax}}-P_{w,i,t}^{\mathrm{sell}},P_{i,t}^{\mathrm{cmax}}-P_{w,i,t}^{\mathrm{ch}},P_{i,t}^{\mathrm{dmax}}-P_{w,i,t}^{\mathrm{dis}}{]}^{\mathrm{T}}\end{array}$$

(50)

where $Y$, $X^{dual}$, and $\Delta P$ are all vectors, and the elements of $X^{dual}$ are binary variables (0–1). $M$ is a relatively big constant.

At this point, we obtain a MISCOP problem. The objective function of the model includes (1) and (8). The constraints include (1) CVaR constraints (5)–(7); (2) the DNO’s operational constraints (9)–(22); (3) KKT constraints (26)–(27), (31), (A2)–(A6) and (48)–(50).

4.2.3. Procedures of the Regulation Strategy

In the interaction between the DNO and PVCS, the uncertainty of the electricity price and PV output affects the system. The regulation strategy in this paper aims to promote the coordinated operation of the DNO and PVCS while controlling the impact of uncertainties. The implementation process of the regulation strategy is shown in Figure 2 and includes the following main steps: First, scenario generation and reduction techniques are applied to generate typical scenarios for the PV output and electricity price. Second, a bi-level optimization model is constructed, with the upper level focusing on the distribution network and the lower level focusing on the PVCS. Next, the CVaR theory is introduced to control the uncertainty and manage the risks. Then, the duality theory, KKT conditions, and the Big-M method are used to transform the bi-level model into a single-level model that is easier to solve. Finally, the optimal scheduling solution is obtained by solving the transformed model with the solver.

5. Case Analysis

This paper uses the modified IEEE 33-bus distribution network system to validate the effectiveness of the proposed bi-level model and its risk management capability. All simulation tests are implemented using the YALMIP toolbox in MATLAB R2019a for modeling and are solved with the GUROBI solver.

5.1. Parameter Setting

The modified IEEE 33-bus distribution network system is shown in Figure 3. This distribution network includes three PVCSs (located at nodes 15, 21, and 31), two MTs (located at nodes 11 and 29), and two ESSs (located at nodes 7 and 22). The EVs, MTs, ESSs and other parameters are detailed in Appendix B. Additionally, 5 electricity price scenarios o and 4 PV scenarios w are generated, resulting in a total of 20 scenarios. Other parameters are the same as the standard IEEE 33-bus system. Since some EV charging and discharging behaviors will be carried out at night, if the simulation scheduling time starts from 0:00, the complete scheduling cycle of the EV cannot be obtained. Considering that EV charging at night usually ends at around 8:00 AM, and daytime charging users usually connect to the grid around 8:00 AM, the scheduling start time in this paper is set to 8:00 AM.

5.2. Analysis of Optimization Results for the Bi-Level Model

This section analyzes the effectiveness of the proposed bi-level model without considering risk aversion in the DNO’s energy management strategy (i.e., $\gamma =0$). In terms of the probability of each scenario (Appendix B,

Table A5. Probability of typical sets of scenarios.

	o = 1	o = 2	o = 3	o = 4	o = 5
Electricity price from main grid (Set O)	0.1920	0.0860	0.2090	0.1050	0.4080
	w = 1	w = 2	w = 3	w = 4
PV output (Set W)	0.1264	0.1747	0.4758	0.2231

), the probability of “o = 5 and w = 3” is the largest, so this paper shows the specific optimization results under this scenario. Figure 4 illustrates the internal power results of the distribution network and the electricity price from the main grid, while Figure 5 shows the internal power strategy of the three PVCSs and the pricing results from their interaction with the DNO. In addition, the figure displaying the voltage distribution is provided in Appendix C.

Figure 4 indicates that during periods of high electricity prices from the main grid, the DNO tends to actively schedule the output of the MTs, such as during hours 8–11 and 19–23. Conversely, during periods of low electricity prices, the DNO prefers to purchase a large amount of electricity from the main grid, such as during hours 16–18 and 3–7 the following day. In other words, without considering risk aversion, the DNOs maximize the use of low-cost sources of electricity as a way to increase their own revenue. Combined with the time-of-use electricity price (Appendix B,

Table A6. Time-of-use electricity price.

Periods	The Time Periods	Electricity Price (Yuan/kWh)
Peak period	11:00–13:00 17:00–19:00	1
Normal period	8:00–11:00 13:00–17:00 19:00–24:00	0.5
Valley period	0:00–8:00	0.3

), it can be seen that the PVCS’s electricity sales behavior is concentrated in the peak tariff hours, in the instantaneous segments 12–13 and 19–20. In the peak hours, the PVCS can game for a higher selling price and sell electricity as much as possible, thus reducing its energy costs. In other periods, the selling price is not high enough for the PVCS to sell electricity, and the PVCS mainly purchases electricity more economically according to the guidance of the purchasing price.

Regarding the PVCS section, Figure 5 shows that the electricity purchasing and selling price, the purchasing and selling strategy, and the EV clusters’ scheduling strategy for each PVCS exhibit similarities. This is due to the fact that all three PVCSs are influenced by the same DNO’s strategic interactions. Specifically, the peak periods for the PVCS’s sale and EV clusters’ discharging coincide with the peak periods of the selling price, which are from 12–13 to 19–20. Each PVCS, while meeting the EV clusters’ energy needs, aims to maximize EV discharging during peak periods to increase electricity sales. Notably, during the time periods 12 and 20, the game price are the highest within the adjacent periods, with the EV discharging power reaching its maximum. For example, PVCS1 has discharging powers of 0.67 MW and 0.68 MW during these two periods, PVCS2 has discharging powers of 0.65 MW and 0.81 MW, and PVCS3 has discharging powers of 0.78 MW and 0.85 MW. In other periods, the discharge price obtained through the game is not high, and considering the loss costs associated with discharging, the EV reduces its discharging behavior during this phase. Additionally, while the game price and power strategy of each PVCS have similarities, they also exhibit differences due to variations in the on-site PV output, the number of EVs connected/disconnected, and their power requirements. Furthermore, as seen in Figure 4 and Figure 5, there are no instances of PVCSs simultaneously purchasing and selling electricity or EVs simultaneously charging and discharging, which indirectly confirms the effectiveness of the redundancy proof provided in Section 4.1.

5.3. Comparison and Analysis of Interaction Strategy

Furthermore, the superiority of the proposed interaction strategy between the PVCS and DNO is discussed. For this purpose, two comparison strategies are established. In both comparison strategies, the PVCS acts as a price taker and performs optimization based on fixed electricity prices (with a purchase electricity price set at 1.2 times the time-of-use price and a sale electricity price at 0.8 times the time-of-use price).

Strategy 1: Single-objective optimization. First, optimize the PVCS with the goal of minimizing energy costs, and then optimize the DNO with the goal of maximizing the operational revenue.

Strategy 2: Multi-objective optimization using the weighted coefficient method [27], where the PVCS optimizes with the goal of minimizing energy costs, while the DNO optimizes with the goal of minimizing the operational revenue.

Strategy 3: Using the strategy proposed in this paper, the PVCS’s purchasing and selling electricity price is optimized by different subjects’ games.

Table 1. DNO’s expected revenue and PVCS’s expected cost under different strategies.

		DNO			PVCS
	F1 (Yuan)	F2 (Yuan)	Total Expected Revenue (Yuan)	Cost of Purchase and Sale of Electricity (Yuan)	Cost of Discharging Losses (Yuan)	Total Expected Cost (Yuan)
Strategy 1	258.5	646.0	904.5	258.5	622.0	880.5
Strategy 2	1166.1	147.6	1313.7	1166.1	211.5	1377.7
Strategy 3	758.7	741.7	1500.4	758.7	510.5	1269.2

presents the expected revenue of the DNO and the expected cost of the PVCS under different strategies. In fact, Strategy 1 prioritizes the optimization of the PVCS’s interests, which submits to the DNO a plan for purchasing and selling power that is designed to minimize its own costs, and then the DNO schedules resources to maximize its profits. As a result, the energy costs for the PVCS under Strategy 1 are minimized, amounting to only 880.5 yuan. Strategy 2 is a multi-objective optimization, which takes into account the interests of the DNO compared to Strategy 1. Compared to Strategy 1, the DNO’s revenue in Strategy 2 increased by 409.2 yuan. But this is accompanied by a 497.2 yuan increase in the cost of the PVCS, and overall, the result obtained from this strategy is not economical. The strategy proposed in this paper considers the pursuit of self-interest by the DNO and PVCS, and the result is the equilibrium solution of the game. As shown in Table 1, compared to Strategy 1, the proposed strategy increases the total revenue of the DNO by 595.9 yuan while only increasing the total cost for the PVCS by 388.7 yuan. Compared to Strategy 2, the proposed strategy enhances the distribution network’s revenue by 186.7 yuan (a 14.2% increase) and reduces the total cost for the PVCS by 108.5 yuan (a 7.9% reduction), demonstrating the effectiveness of the proposed strategy in this paper. Additionally, it can be seen that the core of the proposed strategy is that the DNO, through the electricity price, guides the PVCS toward carrying out more economic purchasing and sales of electricity and charging and discharging behaviors, so as to reduce the purchase of electricity in the period of the higher price of the main grid, which in turn improves the expected revenue of the second stage (F₂), to achieve the optimization results that are acceptable to both sides. Both the single-objective optimization strategy and the multi-objective optimization strategy are optimized based on a fixed electricity price, which makes it difficult to reflect the game–theoretic interactions between different stakeholders. This can lead to situations where one party sacrifices the other’s interests to increase its own benefit. Therefore, compared with the optimization strategy of a fixed electricity price, the strategy in this paper makes full use of the interactive relationship between the two sides to maximize their own interests, so that the transaction price shows dynamic changes, which better motivates both parties.

5.4. Risk Analysis

This section analyses the effect of risk management on the DNO’s expected revenue. Setting the confidence level at 0.8 and changing the risk aversion coefficient $\gamma$, the expected revenue and CVaR of the DNO under different risk aversion coefficients are obtained, as shown in Figure 6. $\gamma$ reflects the trade-off between the risk and revenue, and the smaller the value of $\gamma$ is, the smaller the degree of the DNO’s aversion to risk is, and the same rule applies the larger the value of $\gamma$ is. It is evident that the strategy’s revenue differs significantly under varying risk aversion coefficients. The difference is due to the following reason: when $\gamma =0$, the DNO prioritizes dispatching low-cost electricity sources (as discussed in Section 5.2, representing a risk-neutral scenario), whereas when $\gamma >0$, the DNO incorporates risk aversion into its operations, which allows for the inclusion of high-cost electricity sources in the dispatch process. As $\gamma$ increases, a greater proportion of high-cost electricity sources are utilized. Additionally, as shown in Figure 6, with the increase in $\gamma$, the DNO’s expected revenue decreases while CVaR increases. Specifically, when $\gamma$ increases from 0.1 to 0.9, the DNO’s expected revenue decreases from 1503.8 yuan to 1389.5 yuan, a decrease of 7.6%; and the CVAR increases from 293.7 yuan to 407 yuan, an increase of 38.58%. It can be seen that as the $\gamma$ increases, the decision maker tends to choose a more conservative operating strategy, i.e., the decision maker reduces the potential uncertainty risk by sacrificing profits.

6. Conclusions

To account for the impact of multiple uncertainties on the interaction and game between the DNO and PVCS, this paper proposes a distribution network regulation strategy with a PVCS based on bi-level stochastic optimization. After systematically transforming the problem using the KKT conditions, duality theorem, and Big-M method, the strategy is solved using a commercial solver, and a simulation analysis is conducted. On the one hand, the strategy proposed in this paper guides the PVCS to formulate more reasonable electricity purchases and sale strategies as well as EV cluster charging and discharging strategies, in the form of game pricing. Compared to the multi-objective optimization strategy, it can increase the DNO’s revenue by 14.2% and reduce the PVCS’s energy costs by 7.9%. This strategy helps to balance the interests of all parties. On the other hand, the proposed risk management strategy can effectively address the uncertainties of the electricity price on the DNO side and PV output on the PVCS side. When the risk aversion coefficient increases from 0.1 to 0.9, the DNO’s expected revenue decreases by 7.6%, while the CVaR increases by 38.58%.

This paper considers uncertainties related to the electricity price and PV output. In practice, EVs themselves also present significant uncertainties. Aggregating and scheduling EVs with uncertainties will be a key focus of future research in this study.