A Stackelberg Game-Based Model of Distribution Network-Distributed Energy Storage Systems Considering Demand Response

Abstract

In the context of national efforts to promote country-wide distributed photovoltaics (DPVs), the installation of distributed energy storage systems (DESSs) can solve the current problems of DPV consumption, peak shaving, and valley filling, as well as operation optimization faced by medium-voltage distribution networks (DN). In this paper, firstly, a price elasticity matrix based on the peak and valley tariff mechanism is introduced to establish a master–slave game framework for DN-DESSs under the DPV multi-point access environment. Secondly, the main model optimizes the pricing strategy of peak and valley tariffs with the objective of the lowest annual operating cost of the DN, and the slave model establishes a two-layer optimization model of DESSs with the objective of the maximum investment return of the DESSs and the lowest daily operating costs and call the CPLEX solver and particle swarm optimization algorithm for solving. Finally, the IEEE33 node system is used as a prototype for simulation verification. The results show that the proposed model can not only effectively reduce the operating cost of the distribution network but also play a role in improving the energy storage revenue and DPV consumption capacity, which has a certain degree of rationality and practicality.

1. Introduction

1.1. Motivation

With the depletion of traditional energy sources, the development of new energy generation has become a favored choice for various countries. However, the power generation of photovoltaics is greatly affected by meteorological factors, and the volatility and randomness of its output will impact the safe and stable operation of the distribution network. Large-scale DPV grid connection will bring problems such as tidal current fluctuation, rising node voltage deviation, and increasing line loss to the distribution network [1].

In the distribution grid system containing a high percentage of grid-connected DPVs, reasonable access to the energy storage system can better solve the above problems [2,3], and the user side of the DESS configuration, because of its fast response and ease of control, can be adjusted in both directions and so on [4], and gradually become a popular choice of distribution grid to deal with the access of large-scale distributed DPVs, but the energy storage access to the distribution grid must be taken into account in the premise of the security of distribution grid. Inappropriate energy storage capacity configuration or charging and discharging operation curve will add uncertainty factors but are not conducive to the safe and stable operation of the DN [5,6].

1.2. Review

There are many factors affecting the configuration of DESSs. Literature [7] takes a lead-acid battery storage system as an example to model the battery aging model and optimize its control strategy. Literature [8] provides an energy storage allocation method based on vulnerability analysis, which takes into account the factors of sudden grid faults as well as the probabilistic output of DPVs in order to facilitate the quick determination of the optimal allocation of DPVs and energy storage in unexpected conditions. Literature [9] introduces the power demand and non-black start (NB-S) generating unit capacity accessibility concept and proposes an optimal selection and siting scheme for battery storage and DPV. Literature [10] proposed a battery energy storage system (BESS) charging and discharging algorithm for the possible over-voltage problem associated with grid-connected DPVs and verified its effectiveness in a 10kV-level grid-connected DPV system. Literature [11] proposed, from the viewpoint of the distribution network operator, a bounded uncertainty robust optimization (BURO) framework. Literature [12] proposed an effective strategy to optimize the DESSs in distribution networks using the ABC meta-heuristic optimization technique, which solves and mitigates the key problems of voltage deviation and power loss in distribution networks.

In addition to the grid side, many researchers have also tried to consider the configuration scheme of DESSs from the user side to achieve the overall scheduling optimization of the microgrids [13,14]. Literature [15] proposed a two-stage optimization method for reducing the long-term operating costs of user-side DESSs based on the existing two-part tariff mechanism. Literature [16] evaluated the economic benefits of photovoltaic (DPV) energy storage systems and optimized a combined battery and thermal storage system in smart buildings, and Reference [17] optimized the short-term performance of smart microgrids by optimizing the short-term performance of smart microgrids in order to minimize the operating costs and emissions. Literature [18] integrates wind power plant (WPP), photovoltaic (PV) generators, conventional gas turbines (CGT), energy storage systems (ESS), and demand resource providers (DRP) into a virtual power plant (VPP), and establishes a two-layer optimization model considering uncertainty and robustness, which is used to achieve dispatch optimization for WPP and PV; literature [19] considers the important roles played by ESSs and renewable energy sources, and provides a model for the Combined Heat and Power (CHP)-District Heat (DH) system, which provides an efficient modeling method and verifies the validity of the model and optimization method with an arithmetic example. In addition, this paper compiles some of the highly cited papers on energy storage planning in distribution networks in recent years, and the results are shown in the

Table 1. The Distribution Network revenue optimization results.

References	WT	PV	MT	Diesel Power Generation	CHP	Solution Methods
[20]	√	√		√		Grasshopper Optimization Algorithm (GOA)
[21]	√	√	√			Modified Bat Algorithm (MBA)
[22]		√			√	CPLEX solver
[23]	√					Hybrid TS/PSO algorithm
[24]	√	√				Improved bat algorithm
[25]	√				√	CPLEX solver
[26]	√	√		√		Genetic Algorithm (GA)
[27]	√	√	√			Jaya-IPM
[28]		√		√		X-press
[29]		√			√	Improved Differential Evolution Algorithm (IDEA)
[30]	√	√			√	CPLEX solver

:

In today’s environment of power market reform, the traditional model of energy storage invested by the grid operator has gradually revealed its shortcomings, replaced by a third-party investor-funded investment, jointly considering the interests of the distribution network and users of the energy storage configuration model. This model is closer to the market competition model and can take into account the interests of multiple parties [31]. Wang et al. [32] defined the basic concept of demand response (DR), and the literature [33] proposed a multi-stage stochastic planning considering load demand response under the framework of energy trading for determining the coalition composition of multiple microgrids. Overall, demand response is proposed to describe well the responses made by users in the face of changes in the market environment and has become an important reference factor when optimizing energy storage allocation [34,35].

Some researchers pay more attention to the mutual competitive behavior that exists between multiple subjects under the power market mechanism [36]. In terms of energy storage configuration, the grid operator hopes to act as the dominant player in energy storage configuration and minimize the cost of maintaining the normal operation of the grid under the premise of ensuring its stable operation. The energy storage investment and construction company hope to construct the relevant energy storage device and obtain economic benefits through the reasonable charging and discharging control of the energy storage batteries. And the power user participates in the actual operation process of the energy storage and hopes to configure the energy storage device with a certain capacity to minimize their electricity expenses [37]. References [38,39] consider the different operation strategies of multiple subjects and realize the co-optimization of real-time tariff and storage operation curves. Literature [40] considers renewable energy power generation cost, storage depreciation cost, and two-way energy trading, establishing a model combining the optimization of energy consumption and storage capacity and finally adopting a distributed algorithm to find its Nash equilibrium solution. Literature [41] designed a mechanism based on auction trading to determine the internal trading price of renewable energy through the Stackelberg game.

In response to the above literature, some important issues must be noted:

(1)

Most of the literature does not consider the joint role of demand response and game structure. In the existing Stackelberg game structure, customers’ electricity consumption behavior does not change due to changes in the cost of electricity consumption, which is almost impossible to happen in practice;

(2)

In the existing references, various intelligent algorithms have been widely used to solve all kinds of complex problems, but all algorithms will inevitably have the situation that they easily fall into the local optimum and cannot find the optimal solution. At this time, it is necessary to consider the excellent performance of the CPLEX solver in finding the optimal.

1.3. Contributions

At present, domestic residential users still hold a more conservative attitude towards the installation of distributed photovoltaics (DPVs) and related energy storage devices, and the installation of distributed DPVs is still dominated by industrial-type users. In the face of the medium-voltage distribution network operation optimization problem where large-scale industrial and commercial users installing DPVs coexist with small and medium-sized users not installing DPVs, this paper proposes a co-optimization method of distribution network-DESSs that takes demand response mechanism into account and realizes the dual optimization of distribution network tariff mechanism and energy storage allocation scheme by establishing a master–slave game model between the two. Finally, the simulation example is verified in the IEEE 33-node system. The optimization results show that the optimized energy storage allocation model proposed in this paper can effectively improve the security and economy of distribution network operations. The main contributions of this paper are as follows:

(1)

An optimization method for DESSs for commercial and industrial users installing DPV is proposed;

(2)

Considering the dynamic impact of demand response, a three-layer game model of distribution network-DESSs is developed;

(3)

The synergistic optimization of the daily operation of the distribution network and the DESSs allocation scheme is achieved through the distribution network tariff adjustment and the rational allocation of DESSs.

1.4. Organization

The rest of the paper is organized as follows. Section 2 describes the demand response mechanism under peak and valley tariffs and the related modeling methodology. Section 3 presents the main framework of the game model developed in this paper. Section 4 and Section 5 describe in detail the objective functions and constraints of the distribution network model and the BESS model. Section 6 describes the solution methods used in this paper. Section 7 performs an arithmetic simulation using the IEEE 33-node system as a prototype and presents the analytical results. Section 8 summarizes the whole paper.

2. Demand Response Mechanism Based on Peak and Valley Tariffs

Demand response (DR) in the power market refers to the short-term behavior of power users to promote the balance of power supply and demand and guarantee the stable operation of the power grid by adjusting their own power consumption behavior and power consumption plan according to the price or incentives when the price of the power market is significantly higher (lower) or when there is a risk to the security and reliability of the system [42].

The implementation of the peak and valley tariff mechanism is an effective means of DR. The peak and valley tariff mechanism refers to the division of a day into three periods of electricity consumption, peak, flat, and valley, according to the different amounts of electricity consumed by users during each time period, with different prices for the sale of electricity during different periods of electricity consumption. The implementation of the peak and valley tariff mechanism can appropriately guide local users to reduce the consumption of electricity during peak hours, transfer that part of the consumption to the consumption of electricity in the valley, encourage users to carry out the consumption of low tariff hours, not only to reduce the user’s electricity expenditure but also to achieve the purpose of the power system to cut the peaks and fill the valleys [43].

Electricity is a special commodity, and according to the principle of economics, changes in the price of a commodity will, to varying degrees, cause changes in the demand for that commodity. Under the peak and valley tariff mechanism, the user’s behavioral response of electricity consumption to the change of tariff in this time period, on the one hand, is shown in the reduction or increase of the electricity load in this time period, and on the other hand, is shown in the load change generated by the transfer of this time period to the other time periods of electricity consumption, which defines the ratio of the change in the tariff to the change in the amount of electricity load as the coefficient of elasticity of demand [44]. The coefficients are calculated as follows:

$$e_{XX}=\frac{\Delta Q_X/Q_X}{\Delta P_X/P_X}$$

(1)

$$e_{XY}=\frac{\Delta Q_X/Q_X}{\Delta P_Y/P_Y}$$

(2)

In the above equations, $e_{XX}$ and $e_{XY}$ represent the self-elasticity coefficient of the electricity demand of time period X to the tariff of time period X and the cross-elasticity coefficient of the electricity demand of time period X to the tariff of time period Y. $Q_X$ and $\Delta Q_X$ represent the total electricity demand of time period X and its variation. $P_X$ and $\Delta P_X$ represent the tariff of time period X and its variation, respectively. Accordingly, the price elasticity matrix of each time period under the peak and valley time-sharing tariff mechanism can be established as follows:

$$E_d=\left[\begin{array}{ccc}{e}_{11}& e_{12}& e_{13}\\ e_{21}& e_{22}& e_{23}\\ e_{31}& e_{32}& e_{33}\end{array}\right]$$

(3)

The electricity consumption in the three time periods under peak and valley tariffs can be expressed as:

$$Q=Q_0+\left[\begin{array}{ccc}{Q}_{1,0}& 0& 0\\ 0& Q_{2,0}& 0\\ 0& 0& Q_{3,0}\end{array}\right]\times E_d\times \left[\begin{array}{l}\Delta p_1/p_0\\ \Delta p_2/p_0\\ \Delta p_3/p_0\end{array}\right]$$

(4)

In the above equations, $Q_0$ and $Q$ denote the demand for electricity before and after the implementation of peak and valley tariffs, respectively, and $Q_0={\left[\begin{array}{ccc}{Q}_{1,0}& Q_{2,0}& Q_{3,0}\end{array}\right]}^T$, $Q={\left[\begin{array}{ccc}{Q}_1& Q_2& Q_3\end{array}\right]}^T$, $\Delta p_1$, $\Delta p_2$ and $\Delta p_3$ represent the difference between the tariffs of each time period before and after the implementation of peak/valley tariffs and the original tariff of $P_0$, respectively.

In general, an increase in tariff in one time period leads to a decrease in electricity load in that time period, which in turn leads to an increase in electricity demand in other time periods. Therefore, the auto-elasticity coefficients located on the diagonal of the matrix are negative, and the cross-elasticity coefficients are positive. The exact values of the coefficients of the matrix are usually known from social surveys or various regression methods [45].

3. A Stackelberg Game Framework for DN-DESSs

The Stackelberg game belongs to one kind of non-cooperative game model. In the Stackelberg game framework, the leader occupies a dominant position and is able to dynamically adjust its own strategy according to the actions of the followers. The followers cannot foresee the leader’s behavior and can only formulate the most suitable strategy for themselves according to the leader’s decision [46]. In this paper, the distribution grid operator occupies a dominant position and has the ability to plan and decide the grid operation strategy. The energy storage investor adjusts its own configuration and operation strategy according to the environment set by the grid operator in order to maximize the benefits of the energy storage system [47]. Specifically, the DN operator wants to configure the energy storage capacity as large as possible to fully address the range of operational issues associated with the DPV access. The energy storage investor, on the other hand, is responsible for the investment and operation of the DESSs and wants to minimize the capacity of the DESSs allocation to reduce the investment cost while profiting from the energy storage allocation. It can be seen that there are two subjects of interest in the optimal allocation of DESSs, and different subjects have different profit goals, strategies, and constraints, which form a master–slave game structure in which the grid operator is the subject of the game, and the energy storage investor is the slave of the game. The specific game framework is shown in the following Figure 1.

The upper-layer model represents the grid operator, which can set the tariff, earn economic benefits by selling electricity to users in the region, and guide the users and the energy storage system to adjust their own power curves. The middle and lower models represent the energy storage investor; the energy storage system can respond to the grid tariff in combination with its own needs and realize the “low storage and high generation” of electric energy by optimizing its own capacity allocation and operation strategy to obtain economic benefits, and then transmit its own configuration and operation strategy to the grid operator of the upper model, and the power grid operator will adjust the original tariff again in combination with the operating cost of the distribution network, and issue it to the energy storage investor. Follow the above process for multiple iterations until grid operators and energy storage investors find the optimal peak-to-valley tariff scheme and energy storage operation strategy.

4. The DN Operation Optimization Model

4.1. Objective Function

The distribution network layer model mainly considers the operation optimization problem of the distribution network after the connection of photovoltaic and energy storage. The objective function is composed of the annual income from electricity sales and the electricity cost loss caused by network loss, as follows:

$$\max \hspace{0.17em}{C}_{ALL}=C_{sales}-C_{loss}$$

(5)

where $C_{sales}$ denotes the annual electricity sales revenue of the distribution network and $C_{loss}$ denotes the annual network loss cost of the distribution network.

$$C_{loss}={\displaystyle \sum _{d=1}^{365}{\displaystyle \sum _{l=1}^L{\displaystyle \sum _{t=1}^{24}{P}_l^{loss}(t)\times p_t}}}$$

(6)

$$C_{sales}={\displaystyle \sum _{d=1}^{365}{\displaystyle \sum _{n=1}^N{\displaystyle \sum _{t=1}^{24}{P}_i^{sales}(t)\times p_t}}}$$

(7)

where $P_l^{loss}(t)$ denotes the network loss power of the l-th branch of the distribution network at the moment t, $P_i^{sales}(t)$ denotes the net power of the i-th node of the DN at the moment t, $p_t$ is the tariff of the distribution network at the moment t, N denotes the total number of nodes of the distribution network, and L denotes the total number of lines of the distribution network.

4.2. Constraints

The constraints mainly consider the indicators during the normal operation of the distribution network. They are described below:

(1)

Node voltage constraint

During normal operation of the DN, it must be ensured that the voltage at each node is not out of bounds. The voltage constraints are as follows:

$$U_i^{\min }\le U_i(t)\le U_i^{\max }$$

(8)

where $U_i(t)$ denotes the voltage of the i-th node in the distribution system at the t-th moment, and $U_i^{\min }$ and $U_i^{\max }$ denote the upper and lower limit constraints of the distribution node voltage, respectively.

(2)

Distribution network current constraint

The DN must satisfy the branch current equation constraints during normal operation. It is expressed as follows:

$$P_i(t)=U_i(t){\displaystyle \sum _{j=1}^N{U}_j(t)[G_{ij}\cos {\delta }_{ij}(t)+B_{ij}\sin {\delta }_{ij}(t)]}$$

(9)

$$Q_i(t)=U_i(t){\displaystyle \sum _{j=1}^N{U}_j(t)[G_{ij}\sin {\delta }_{ij}(t)-B_{ij}\cos {\delta }_{ij}(t)]}$$

(10)

In the above equation, $P_i(t)$ and $Q_i(t)$ denote the active and reactive power of the i-th node of the distribution network at time t respectively, N is the total number of nodes in the distribution network, $U_i(t)$ and $U_j(t)$ denote the nodal voltage values of node i and node j at time t, respectively, $G_{ij}$ and $B_{ij}$ denote the conductance and conductance between nodes i and j, respectively, and ${\delta }_{ij}$ is the phase angle between nodes i and j.

(3)

Tariff Constraint

When setting tariffs for distribution grids, certain tariff constraints must be observed [48]. In the model established in this paper, the grid operator and the energy storage investor are different interest subjects, and the tariff should be set to satisfy the interests of both parties. If the tariff is too low, users will increase the amount of power purchased from the grid, harming the interests of the energy storage investor. When the tariff is too high, users will reduce the power purchased from the grid, which is also detrimental to the stable operation of the grid. Therefore, the tariff set by the distribution network must be kept within a reasonable range, namely:

$$p_t^{\min }\le p_t\le p_t^{\max }$$

(11)

In the formula, $p_t^{\min }$ and $p_t^{\min }$ represent the maximum and minimum values of the tariff in time period t, respectively.

5. The DESSs Two-Tier Optimized Configuration Model

In this paper, a two-layer planning model is used to optimize the configuration of DESSs in the medium-voltage DN. The upper-layer model considers the planning problem for solving the optimal capacity configuration of the energy storage system, and the lower-layer model considers the operation problem for optimizing the daily charging and discharging strategies of the DESS system.

5.1. The Battery Storage System Model

The battery storage system is the core research object of this paper. It can both absorb excess power from the grid for first PV consumption and emit power to solve the problem of insufficient power supply capacity of the grid. The state of charge of the battery can be calculated by the following Equation:

$$SOC(t+1)=(1-\sigma )\times SOC(t)+P_{Bess,ch}(t)\times {\eta }_{ch}/E_{Bess}-P_{Bess,dis}(t)/(E_{Bess}\times {\eta }_{dis})$$

(12)

where $SOC(t)$ denotes the state of charge of the energy storage at time t, $\sigma$ denotes the self-discharge coefficient of the battery. $E_{bess}$ indicates the rated capacity of the battery. $P_{Bess,ch}$ denotes the input power of the BESS from the grid and $P_{Bess,dis}$ denotes the discharge power of the BESS to the load. ${\eta }_{ch}$ and ${\eta }_{dis}$ denote the charging and discharging efficiencies of the BESS, respectively. In addition, other constraints related to the batteries are listed below:

$$SO{C}_{\min }\le SOC(t)\le SO{C}_{\max }$$

(13)

$$-P_{i,t}^{\max }\le P_{Bess,ch},P_{Bess,dis}\le P_{i,t}^{\max }$$

(14)

Equation (13) indicates that the state of charge of the BESS at any point in time cannot exceed its minimum and maximum capacity levels, and Equation (14) indicates that the operating power of the BESS cannot exceed its rated power.

5.2. The Upper-Layer Planning Model

5.2.1. Objective Function

The upper-layer model aims to maximize the profit after energy storage allocation, considering the tariff arbitrage gains, government tariff subsidy gains, and various investment costs obtained by the energy storage investor during the whole life cycle of the DESSs, and the decision variable is the rated capacity of the DESSs. In regions with peak and valley tariffs, peak-time tariffs are usually much higher than valley-time tariffs. Therefore, by purchasing and storing electricity in the valley time, and then releasing it in the peak time, the energy storage system utilizes the difference in tariff between different time periods in order to achieve profitability, and this revenue model can reduce the expenditure on electricity and alleviate the pressure of power supply of the grid during peak load to a certain extent. The objective function of the upper layer model is:

$$\max \hspace{0.17em}\{{\displaystyle \sum _{y=1}^{Year}[{\displaystyle \sum _{d=1}^{365}(C_{arbitrage}+C_{subsidy}-C_{op})}]\times \frac{\gamma {(1+\gamma )}^y}{{(1+\gamma )}^y-1}}\}-C_{bess}$$

(15)

$$C_{arbitrage}={\displaystyle \sum _{n=1}^N{\displaystyle \sum _{t=1}^{24}{P}_{Bess}(t)\times p_t}}$$

(16)

where $C_{arbitrage}$ denotes the revenue generated by DESSs through the electricity consumption differential, $P_{Bess}(t)$ denotes the operating power of the energy storage system at the t-th moment, $p_t$ denotes the tariff at the t-th moment, $\gamma$ is the discount rate, and $Year$ is the life span of the configured energy storage.

$C_{subsidy}$ indicates the government’s additional subsidized revenue for the discharge of energy storage devices. As the configuration cost of the current energy storage system is still high, in order to promote the development of the energy storage industry, China has introduced a series of price subsidy policies for energy storage devices, and there are two main forms of subsidies: a construction investment subsidy, and a tariff subsidy. This paper adopts the form of tariff subsidy; the specific calculation formula is as follows:

$$C_{subsidy}={\displaystyle \sum _{n=1}^N{\displaystyle \sum _{t=1}^{24}{P}_{dis}(t)\times p_{sub}}}$$

(17)

where $P_{dis}(t)$ denotes the power discharged by the storage at the t-th moment and $p_{sub}$ denotes the government-subsidized tariff for discharging DESSs.

$C_{op}$ denotes the operation and maintenance cost of the DESS system in one day, and the operation and maintenance cost of the energy storage is mainly related to the charging and discharging power of the energy storage. The specific calculation equation is as follows:

$$C_{op}={\displaystyle \sum _{n=1}^N{\displaystyle \sum _{t=1}^{24}{P}_{Bess}(t)\times K_{op}}}$$

(18)

where $K_{op}$ denotes the unit charge/discharge cost factor for energy storage.

$C_{bess}$ is the fixed capacity investment cost for DESSs, expressed as follows:

$$C_{bess}=S_{bess}\times K_S$$

(19)

where $S_{bess}$ denotes the total capacity configured by the DESSs and $K_S$ is the unit capacity cost factor.

5.2.2. The Upper-Layer Model Constraints

(1)

Charge/Discharge Continuity Constraint

Every 24 h of operation of the energy storage is regarded as the completion of a complete operation cycle. The constructed energy storage operation model must ensure the continuity of the charging and discharging strategies, i.e., the energy storage operation states before and after completing an operation cycle must be equal, as follows:

$$SO{C}_{i,1}=SO{C}_{i,24}$$

(20)

where $SO{C}_{i,1}$ denotes the initial charge state of the i-th energy storage in a day, and $SO{C}_{i,24}$ denotes the charge state of the energy storage at the last moment of a day.

(2)

Energy Storage Installation Capacity Constraint

The capacity configuration of the energy storage must meet the maximum access capacity limit for each node as follows:

$$E_{Bess,i}\le E_{Bess,i}^{\max }$$

(21)

where $E_{Bess,i}$ denotes the configured energy storage capacity of the i-th node, and $E_{Bess,i}^{\max }$ denotes the maximum access capacity limit for energy storage of the current node.

(3)

Energy Storage Charge State Constraint

In order to ensure the life cycle of the energy storage system, the energy storage system is usually not allowed to operate at full charge and discharge. The state of charge of the energy storage system at any moment must satisfy certain constraints, as follows:

$$SO{C}_{\min }\le SO{C}_{i,t}\le SO{C}_{\max }$$

(22)

where $SO{C}_{i,t}$ denotes the charging state of the i-th energy storage at time t, and $SO{C}_{\min }$ and $SO{C}_{\max }$ denote the minimum and maximum values of the charging state of the energy storage, respectively.

5.3. The Lower-Layer Planning Model

5.3.1. Objective Function

The lower-layer model is the daily operation optimization model for the DESSs. The main consideration is the daily running cost after configuring DESSs. The objective function of the model is:

$$\min \hspace{0.17em}{\displaystyle \sum _{d=1}^{365}{C}_{op}}$$

(23)

The relevant parameters have been given in the previous section and will not be repeated here.

5.3.2. Constraints

The lower model is an operational optimization model for DESSs, and the constraints to be satisfied are given in Equations (12)–(14) and will not be repeated.

6. Methods of Solving the Model

6.1. The Particle Swarm Optimization Algorithm

Particle Swarm Optimization (PSO) is an intelligence optimization algorithm whose basic idea is to find the optimal solution through the collaboration and information sharing of individuals in the swarm. The PSO algorithm optimizes its own position and velocity parameters by designing a swarm of particles possessing two attributes: position and velocity, which simulate the foraging behavior of a flock of birds in the natural world [49]. After several rounds of iterations, the particles gradually converge and finally find the optimal solution of the model. Due to the simplicity of the algorithmic process and fewer parameter settings, the PSO algorithm is often used to solve optimization problems in power system planning and operation. In this paper, the PSO algorithm is used for iterative optimization of the pricing scheme for peak and valley tariffs as well as the optimal allocation capacity of energy storage.

6.2. The CPLEX Solver

The CPLEX solver, developed by IBM, is a general-purpose mathematical modeling solver that is often used to solve multi-constraint linear programming problems and has been widely used in the study of energy storage configuration in power systems. In the two-layer planning model established in this paper, considering that the decision variables include the capacity of energy storage configuration, the charging and discharging power of energy storage at each moment, and the constraints are mostly inequality constraints containing upper and lower bounds, this paper decides to use the swarm intelligent optimization algorithm and the CPLEX solver to carry out a hybrid solution, in which the CPLEX solver is used in the lower-layer model to solve for the optimal operation strategy of the energy storage system.

6.3. The Solution Flow

The optimization model for DN-DESSs constructed in this paper is a three-layer structure, with the upper layer being the DN model, which is responsible for optimizing and issuing the tariffs, and the middle and lower layers being the DESSs two-layer model, which is able to optimize its own allocation scheme according to the tariffs. The overall solution process of the model is shown in Figure 2:

6.3.1. The DN Layer Model

The objective function of the DN model is shown in the previous section, and its decision variable is the tariff in each time period. In this paper, the PSO algorithm is used to solve the DN layer model iteratively. The solution process is shown in the following Figure 3:

The steps for solving the problem are as follows:

(1)

Input the power of the distributed DPV, the line parameters of the distribution network, and the load data of each node;

(2)

Initialize the PSO algorithm to generate tariff particles;

(3)

Calculate the value of the fitness function (i.e., the cost of operating the distribution grid) for all tariff particles in conjunction with a two-layer optimization model for energy storage;

(4)

Update the positions and velocities of all particles, recalculate the fitness function values of each particle, and obtain the optimal solution for this round of iteration;

(5)

Determine whether the obtained optimal solution satisfies the termination condition. If it is satisfied, output the optimal function value and the optimal tariff particle. If not, return to step (4) to continue the iteration.

6.3.2. The DESSs Two-Layer Optimization Model

When the DESS model is undergoing the optimization process, the tariffs for each time period are issued by the distribution network operator and are known quantities at the time of optimization. In the two-layer model developed in this paper, the upper-layer model has the objective of maximizing the revenue after the DESS configuration, and the decision variable is the rated capacity of the DESSs. The lower-layer model has the objective of minimizing the daily operating cost of the DESSs, and the decision variable is the power of the storage operation at each moment. The iterative flowchart of the two-tier model is shown in the Figure 4:

This two-layer model is solved using a particle swarm algorithm together with the CPLEX solver. The specific steps are as follows:

(1)

Enter the initial tariff as well as the DPV output and load information for each node;

(2)

The upper-layer model randomly generates particles and initializes the position (representing the rated capacity of the DESSs) and velocity of the particle swarm;

(3)

The lower-layer model solves for the optimal operation policy corresponding to each particle using a CPLEX solver based on the rated capacity represented by each particle;

(4)

The upper-layer model calculates the fitness function value of each particle and updates the positions and velocities of all particles based on the obtained energy storage capacity and the operation strategy of the lower layer and obtains the global optimal solution for the current iteration round;

(5)

Determine whether the current optimal solution satisfies the iteration termination condition. If satisfied, output the rated capacity and daily operating power of the DESS. If not, update the position and velocity of each particle and return to step (3) to continue the iteration.

7. Example Analysis

7.1. Parameterization of the Algorithm

In this paper, the IEEE33 node system model [50] is used for simulation verification. As shown in the following Figure, in the IEEE33 node system, a total of nine nodes access the distributed DPV power generation system, and the specific system structure diagram is shown in the Figure 5. The storage access node is known; its unit capacity cost factor and unit charging and discharging power cost factor are taken as ¥667 per kW·h and ¥0.045 per kW, respectively; the government subsidized tariff is taken as ¥0.1 per kW, the initial SOC is taken as 0.5, the discount rate is 6.79%, and the life span of the storage is 20 years. For the existing 365-day load and DPV processing data of a real distribution network, this paper adopts the k-means clustering method to generate a total of nine scenarios. In order to verify the effectiveness of the proposed model, this paper selects the scenario with the smallest net load (i.e., the largest storage allocation demand) as the typical day scenario, and the relevant curves are shown in Figure 6 and

Table 2. DPV access nodes and capacity.

The DPV Access Node	6	9	16	24	30
DPV access capacity (kW·h)	400	500	400	450	400

.

According to the results of related literature [51], the elasticity matrix of demand response of tariff is shown below:

$$E_d=\left[\begin{array}{ccc}-0.0870& 0.0502& 0.0201\\ 0.0912& -0.0526& 0.0211\\ 0.0915& 0.0528& -0.0211\end{array}\right]$$

7.2. The Comparison of the Algorithms’ Precision

In this paper, the PSO algorithm [52] combined with the CPLEX solver [53] is adopted as the solution method. In order to demonstrate the accuracy of the adopted solving methods, this paper takes the objective function of the upper layer model as an example and shows the iterative process of the three methods respectively, and the evaluation indexes adopted as well as the comparison results are as follows:

$$\frac{f(x_{optimal})-f(x_i)}{f(x_{optimal})-f(x_0)}$$

(24)

where $f(x_{optimal})$ denotes the optimal solution that can be obtained from the objective function, $f(x_i)$ denotes the value of the objective function obtained in the i-th round of iteration, and $f(x_0)$ denotes the initial value of the iteration.The specific results are shown in Figure 7.

From the above Figure, it can be seen that the PSO-CPLEX solver is more efficient and accurate than the remaining two methods. This is because the upper model of this paper is a mixed integer nonlinear programming (MINLP) problem, which is suitable to be solved by using intelligent optimization algorithms, while the lower model is a purely linear problem, in which case the CPLEX solver has a better solving performance compared to intelligent optimization algorithms. Therefore, it can be expected that a mixture of the two solvers is more effective than a single solver.

7.3. The Optimization Results of the DESSs

The

Table 3. Energy storage configuration capacity.

	Node 15 (DESS 1)	Node 23 (DESS 2)
The DESSs’ configuration capacity (kW·h)	1179.3	394.1

and the Figure 8 show the operating power of the DESSs as well as the SOC at each moment, where a positive power indicates that the DESS is discharging, and a negative power indicates that the DESS is charging. From the operating power curves of the DESS, it can be seen that the operating curves of the two configured energy storage have basically the same trend. The 0:00–6:00 a.m. period belongs to the trough period of electricity consumption; the power supply capacity of the DN can completely meet the user’s demand. At this time, the two energy storage systems are charged to varying degrees. After 6:00 a.m., the user load ushered in the first peak period of the day, but due to the distributed DPV being out of the power and load curve of the complementary nature of the DESS 2 is not discharged, instead of a certain degree of charging operation, to smooth out the DPV power generation, and to reduce the power consumption. After 14:00, due to the gradual weakening of the DPV generation power, its generation power is no longer able to meet the demand of the DPV-load complementarity, the load curve enters into the second peak of the day, and the two storage systems in the grid basically carry out the discharging operation at the same time to alleviate the problem of node overloading. After 22:00, the consumer load, as well as the tariff level, are into the low valley period, and the DESSs start charging until a complete operation cycle is completed, which can not only solve the voltage over the upper limit problem that may be caused by the low user load but also carry out the power reserve in the low tariff period.

7.4. The DN Benefits Analysis

The initial tariffs used in the model are based on the peak and valley tariff system for commercial and industrial users, with the values shown below. The low valley hours are from 1:00–8:00 daily, the peak hours are from 15:00–17:00 and 20:00–22:00, and the rest of the time is the usual period. The comparison of the final tariff optimization results of the upper distribution model is shown in the

Table 4. Peak and valley tariff optimization results.

Title 1	Valley Period	Flat Period	Peak Period
Initial tariff (¥/kW·h)	0.390	0.780	1.290
Final tariff (¥/kW·h)	0.379	0.766	1.301

below.

Compared to the pre-optimization period, the optimized tariff strategy adjusts the peak hour tariff upwards while slightly lowering the usual and valley hour tariffs. The purpose of the distribution network is to guide the users in reducing the consumption of electricity during peak hours and reducing the peak-valley difference in the load of the grid. The following Figure 9 shows the change in the load factor of the distribution network at each moment after the tariff guidance (expressed as “current load/maximum load before optimization × 100%”).

The above Figure shows the adjustment of electricity consumption made by users according to the demand response mechanism after the tariff guidance. It can be seen that the DN upwardly adjusted the price of electricity in the peak time and reduced the price of electricity in the usual time period and the low time period, and the users shifted part of the load in the peak time to the low time and the flat time period based on the goal of minimizing the cost of electricity consumption. After reasonable tariff guidance, the peak-valley difference of electricity load in the distribution network has been significantly reduced, realizing the goal of load shaving and valley filling using tariff adjustment.

The network loss diagram of the distribution network before and after optimization are shown in Figure 10.

As can be seen from the network loss comparison graph of the distribution network before and after optimization, the network loss peaks of the distribution network before optimization mainly appeared in the nodes close to the first end, and the time was concentrated in the two peak periods of the customer load. After the tariff-driven effect and the optimized allocation of energy storage, the network loss problem of the distribution network has been significantly improved, and its network loss peak is significantly reduced, which is specifically shown before and after the two peak periods of 8:30–10:30 and 15:00–18:00 every day, the adjustment of electricity price affects the charging and discharging strategy of DESSs, and energy storage is discharged during the peak electricity consumption period, which not only optimizes the network loss distribution of the distribution network, but also alleviates the problem of electricity consumption tension. The results are shown in

Table 5. The Distribution Network revenue optimization results.

	Daily Network Loss Cost (¥)	Daily Revenue from Electricity Sales (¥)	Daily Revenue (¥)
pre-optimization	896.3	403,659.7	402,763.4
post-optimization	828.6	415,578.2	414,749.6

.

The Table 5 shows the revenue of the DN’s operator before and after tariff optimization. As can be seen from the above table, after configuring DESSs, the DN’s daily network loss cost decreased by ¥67.7, the revenue from electricity sales increased by ¥11,918.5, and finally, the DN’s overall revenue increased by about 3.0% compared with the previous one. Overall, according to the above results, through the configuration of DESSs and reasonable tariff adjustment, the DN realizes the peak shaving of the load curve and the increase in the revenue from electricity sales, and it improves the economy and security of the grid operation.

7.5. The Demand Response Sensitivity Analysis

In this paper, the user’s demand response mechanism to changes in tariff is considered in the process of optimal allocation of energy storage. When demand response is not considered, the user’s electricity consumption behavior is only related to their own electricity demand and is not affected by changes in tariff. The comparative optimization results, with or without considering demand response, are shown in

Table 6. The Sensitivity comparison results for demand response mechanisms.

	Tariff (¥/kW·h)	Daily Net Loss Cost (¥)	DESS 1’s Capacity (kW·h)	DESS 2’s Capacity (kW·h)	Total Capacity (kW·h)
pre-optimization	1.290	896.3	-	-	-
	0.780
	0.390
No Demand Response	1.298	891.7	1101.0	322.7	1427.3
	0.791
	0.390
Demand Response	1.301	845.8	1179.3	394.1	1573.4
	0.766
	0.379

and

Table 7. The Optimization results of the DESSs revenue before and after considering demand response (kW·h).

	Total Construction Cost of the DESSs (¥)	Daily Operating Costs (¥)	Annual Arbitrage Gains from Energy Storage (¥)	Annual Subsidized Energy Storage Benefits (¥)	Return on Investment in Energy Storage
No Demand Response	121.5 × 104	128.6	62,506.8	104,434.9	6.53%
Demand Response	141.8 × 104	133.5	74,580.0	108,270.0	7.87%

.

As can be seen from the table above:

(1)

When Demand Response mechanisms are not considered, the results show that the DN’s operator increases the tariffs for all time periods, but the final optimization effect is not obvious, and the allocation capacity of the energy storage is too small, which not only fails to achieve the purpose of distribution network operation optimization but also fails to give full play to the arbitrage potential of the low storage and high generation of the energy storage during the peak tariff period. The procedure, after considering the demand response, adjusts the peak tariffs upwards and adjusts the usual tariffs and valley tariffs downwards, which provides proper guidance to the users’ electricity consumption behavior is appropriately guided. It can be seen that the operation indexes and economic indexes of DN are optimized substantially compared with those without considering demand response, which indicates that the full consideration of users’ demand response to tariff can bring practical references to the decision-making of the distribution network;

(2)

The network loss cost of the distribution network depends on the net load power of each node. As the DESSs are able to provide part of the power to the users to realize arbitrage during the peak period, it can not only alleviate the power supply problem of the distribution network at the peak moment but also reduce the network loss cost of the distribution network. After considering demand response, the users are able to transfer the loads to the low valley hours spontaneously, and the daily network loss cost of the distribution network has been reduced by ¥45.9, which is about 5.1% lower compared with the result when the demand response is not taken into account;

(3)

From the model comparison, it can be seen that the consideration of the demand response mechanism, as well as the reasonable allocation of the DESSs, are both conducive to the improvement of the energy storage arbitrage revenue and subsidy revenue and, at the same time, is conducive to the peak shaving of loads by the distribution grid. Users shift their loads during peak hours, which reduces electricity expenditure. The DESSs can better absorb the DPV output that cannot be absorbed because of users shifting their own loads, which improves the revenue of the DESSs’ operators. The DN operators maximize the revenue by manipulating the peak and valley tariff strategy and guiding the behavior of users and energy storage in the game process as well.

(4)

In the model established in this paper, the peak period of the daily load curve of the user occurs around 10:00 and 15:00 every day, and part of the DPV power can be realized to be locally consumed. However, due to the existence of the demand response mechanism, the user will spontaneously transfer the load to the valley time, reducing the amount of DPV consumption during the peak load period, thus leading to a reduction in the rate of DPV consumption in order to achieve the same amount of DPV consumption with the pre-demand response, it is necessary to appropriately increase the capacity allocation of the DESSs. As can be seen from the above table, the total capacity of energy storage in the distribution network after the demand response and the total construction cost increased by ¥179,000 compared to the pre-demand response. However, overall, the increase in the cost of energy storage capacity configuration is still favorable to the DPV consumption in the distribution grid as well as the economic operation of the system.

8. Conclusions

This paper fully considers the role of the peak and valley tariff mechanism as a link between the DN and the DESSs and proposes a method for the optimal configuration of the DN-DESSs under the demand response mechanism based on the demand response mechanism of the peak and valley tariff. In the DN model, the DN operator considers its own operation demand and sets the tariff. In the DESS planning model, this paper considers the load-storage synergy and optimizes the capacity configuration and daily operation strategy of the energy storage system. In the arithmetic simulation, the following conclusions can be drawn:

(1)

Most of the existing master–slave game models do not consider the impact of tariff changes on users’ electricity consumption. This paper introduces the price leverage of demand response control and establishes a tariff elasticity matrix, which can better reflect the users’ response to tariff changes, and the game model in the framework of distribution grid-energy storage can more accurately study the operation of the network. The model, after considering demand response, reduces the cost of energy storage allocation and distribution network operation and improves the coordination of distribution network-energy storage operation.

(2)

This stage is still in the preliminary stage of the whole country’s DPV promotion; distributed DPV-DESS economic benefits are not obvious relative to the huge distribution network, which appears to be too small. The model optimization process, in order to reflect the optimization effect of the model, must give more weight to the storage of economic benefits. However, with the future large-scale distributed DPV projects and landing, DPV energy storage commercial investment potential will be further enhanced. At the same time, with the gradual progress of the power market reform, the user’s response to changes in the market mechanism as well as the distribution grid—third-party energy storage investment company game processes must be considered. The model proposed in this paper investigates the distribution grid-energy storage operation under the peak and valley tariff mechanism, which provides some reference value for the investment decision of the future user-side configuration of the energy storage system;

(3)

In this paper, a deterministic distributed DPV power scenario is used in the case analysis, and the uncertainty of DPV power and the diversity of load scenarios are not considered for the time being. In addition, this paper only verifies the validity of the model under the peak and valley tariff mechanism. However, at present, there is a hybrid tariff form of “peak and valley tariff + ladder tariff” in China, and the simple peak and valley tariff mechanism cannot completely simulate the real tariff fluctuation and user behavior. The subsequent optimization of the model will take into account different tariff modes and the diversity of DPV and load scenarios in order to obtain a more realistic optimization model to promote the coordinated operation of the DN-DESSs in the context of DPV in the whole country;

(4)

From the results of the arithmetic example, it is necessary to consider the dynamic impact of price factors on user behavior in power system planning. Most of the current literature treats users’ electricity consumption behavior as a known quantity without considering the impact of changes in users’ behavior on the game results, which can lead to insufficient accuracy in the optimization results of planning projects. The Stackelberg game model proposed in this paper takes this into account and is able to better describe the behavioral changes of the user as the subject of electricity consumption, which can be applied to different regions and time-of-use tariffs with flexibility and portability.