Trading strategy for virtual power plant clusters based on a multi-subject game model

Abstract

With the rapid development of renewable energy and the continuous growth of new loads, VPP has become an important form of smart grid and energy internet due to its flexible and effective management of distributed energy. During the operation of Virtual power plant, there is a game relationship between the system operator and VPP, and they are in a non-complete information environment. However, most of the current game optimization modeling is under the condition of complete information, and the game model based on complete information cannot solve this problem. This article focuses on the VPP cluster trading problem based on non-complete information game theory, constructs a Bayesian game model for multiple VPPs with multiple subjects under the master-slave game framework by introducing the Bayesian concept to optimize the cluster transactions within VPPs, and verifies the effectiveness of the model through simulation experiments. The experimental results show that the multi-VPP multi-subject Bayesian game model established in the study can guarantee the privacy of each subject and effectively reduce PAR, thus ensuring the security and stability of the VPP network and reducing cost expenditures, which has practicality in actual VPP cluster transactions.

Keywords

Multi-subject game non-complete information virtual power plant cluster trading strategy

1. Introduction

The consumption of traditional fossil energy sources has aggravated environmental pollution and led to the development of new energy sources. However, the rapid development of new energy sources has led to an increase in the penetration of distributed energy sources in the distribution network, which poses many challenges for the power system [1]. The emergence of Virtual Power Plant (VPP) has reduced the pressure of high proportion of distributed energy and become an effective way of distributed energy management [2]. Based on this, a wide range of domestic and international scholars have conducted in-depth research on this topic. Gough et al. developed a technology VPP model to optimize the dispatch of distributed energy in the former energy source [3]. Li et al. analyzed in detail the role of VPP as a service provider in the distribution system and built a flexible market framework to effectively ensure the safe operation of the grid [4]. Sharma and Mishra constructed a VPP model as an aid to effectively reduce the penetration of distributed power sources such as solar PV [5]. In response to the issue of the impact of large-scale grid connection of renewable energy on the supply and demand balance of the power system, Yan et al. proposed a VPP energy management method for optimal energy and operating reserve (OR) scheduling [6]. Considering noise and communication delay, Naina et al. proposed a consensus based distributed robust algorithm to suppress the adverse effects of communication delay and noise [7]. Aiming at the temporary faults in VPPs, Majumder et al. proposed a VPP pre emergency scheduling strategy considering chance constraints [8]. At present, VPP has a relatively mature operation optimization technology in the physical level. However, research on transaction problems considering multi-agent games is still immature. In this context, the study introduces Bayesian (Bayesian) under non-complete information conditions and constructs a multi-subject Bayesian game model under the master-slave game framework, aiming to protect the private information of multiple subjects within the VPP while ensuring the safe and smooth operation of the grid by reducing the PAR value and reducing the cost of electricity consumption of multiple subjects in the multi-VPP.

2. Literature review

In recent years, with the rapid development of smart grid, there are more and more participants in various aspects of power generation and distribution, and the application of game theory is a scenario analysis of multiple stakeholders for interactive decision making, thus providing a new way of thinking for optimal decision making of multiple subjects [9]. At present, a large number of domestic and foreign scholars have conducted a lot of research on the application of game theory to electric power systems. Zhang et al. constructed a two-strategy evolutionary game model with three groups, and used it to simulate the electricity market data, thus helping to develop reasonable rules for electricity trading [10]. Chowdhury has analyzed the role of game theory in resource allocation, and provided suggestions for the allocation of resources in radio networks in electric power systems. Resource allocation in the power system [11]. Panda et al. provide a detailed analysis of VPP modeling, management, etc., thus providing help in ensuring the quality and efficiency of electric power [12]. Ali et al. present a detailed description of point-to-point trading in grid-connected multi-microgrid systems, thus using game theory to effectively improve the efficiency and profitability of power trading [13]. Kapoor et al. present centralized and distributed pricing schemes, and coordinated the EV load using game theory, thus improving the stability of the distribution network system [14].

In addition, Li et al. used game theory related theories and models to help coordinate the load of the system with there [15]. Yang et al. used deep learning to construct a power data-driven decision making method to effectively improve the stability of the power system [16]. Ghadimi et al. developed a stochastic bookstore expansion plan for wind farms to improve the reliability of the wind power system while reducing costs. Power system reliability and N-1 contingency is taken into consideration in [17] to proposed a stochastic transmission expansion planning of wind farms. Lei et al. propose an intelligent algorithm to optimize power distribution and reduce power consumption in the context of game theory [18]. Literature [19] proposed a comprehensive evaluation method based on game theory to identify the key transmission lines of the power system. A multi-agent game strategy considering integrated demand response is proposed in [20] to optimal the operation of integrated energy system. In order to improve the economic efficiency of VPP operation, Liu et al. analyzed the game relationship among multiple VPPs, and proposed the optimal operation strategy considering multiple VPP games [21].

From the research of domestic and foreign scholars, it can be found that most of the current research on game theory for power systems is conducted under complete information conditions. In the actual power dispatch, each subject is often unwilling to share private information and make the information incomplete. Therefore, the research study introduces Bayesian and puts the constructed multi-VPP multi-subject Bayesian game model under the framework of master-slave game, which has high practicality and innovation.

3. A cluster trading strategy for virtual power plants based on a multi-subject game model

3.1 Analysis of trading mechanism and master-slave game framework

In order to balance the interests of different VPP subjects, the study optimizes the scheduling of multi-VPP based on game theory and constructs a multi-subject game model. In the concept of game theory, a complete game generally includes game parties, strategy sets and interests. The game can be classified as a complete information game and an incomplete information game based on the information that a game party has about the other game participants [22]. For the actual VPP internally distributed power supply, it is full of uncertainty and has very limited ability to regulate itself. At the same time, its power generation and consumption per unit time are not necessarily balanced, thus presenting a surplus phenomenon to the outside. On this basis, the less-powered VPP will make necessary power purchases from the distribution grid, while the more-powered VPP will sell the surplus power to the distribution grid. Therefore, the study reduces the pressure on the grid by introducing Distribution System Operator (DSO) and using it as an agent that exists between VPP and the electricity market to guide the purchase and sale of electricity from VPP by determining the price of VPP purchase and sale, and increasing the sharing of electricity consumption among VPPs. The schematic diagram of the trading mechanism of DSO and VPP proposed in the study is shown in Fig. 1.

Figure 1.

Schematic diagram of transaction mechanism of DSO and VPP.

In Fig. 1, the red solid arrows indicate power purchases and the purple dashed arrows indicate power sales. As seen in Fig. 1, the multi-power VPP sells excess power to the DSO based on the sale price set by the DSO, and purchases surplus power from the power purchased by the DSO using the purchase price set by the DSO; the DSO does this by interacting between the VPPs through the network and grid prices set in the power market. At the same time, DSOs are profitable through differences in the pricing of power purchases and power sales. The higher the proportion of electricity allocated in the power system, the higher the efficiency of the power system. It is important to note that the DSO must not sell electricity at a price lower than the feed-in price and must not purchase electricity at a price higher than the electricity market in order to ensure that the VPPs are motivated to trade with the DSO.

In Fig. 1, there are two main interests, DSO and VPP. Considering that each interest is rational and the interests of each interest are maximum, the policies they develop are influenced by each other. In addition, DSO’s pricing will affect the generation plan and cost in the electricity market, while VPP’s purchase and sale will directly affect DSO’s pricing and profit. Therefore, there is a game relationship between DSOs and VPPs. Moreover, in the transactions between DSO and VPP, the price responses of the two are sequential and in different positions, with DSO being the price setter and the manager of VPP with priority decision making power. Therefore, the study considers the owners of both as the participants of the game, and the master-slave game framework constructed in this way is shown in Fig. 2.

Figure 2.

Schematic diagram of master-slave game framework.

In Fig. 2, the DSO, as the lead party, is responsible for collecting the procurement and sales of each VPP and combining them with the grid and feed-in tariffs as a way to maximize its own profit by determining the trading tariff of each VPP based on the pricing response of the VPP. At the same time, each VPP is considered as a tracking object, accepts the trading price determined by DSO, and makes a reasonable allocation of its internal DER to minimize its operating costs. If the costs are not minimized, the steps need to be repeated until the DSO determines the tariff and the VPP’s generation strategy does not change, finally achieving a dynamic balance between the DSO’s dynamic pricing and the VPP’s price response, thus obtaining the traded tariff and the equilibrium point of the electricity market. A multi-VPP multi-subject Bayesian (Bayesian) game consisting of leaders and followers, in which each VPP makes decisions at the same time, thus forming a noncooperative game.

3.2 Collaborative optimization analysis of multiple VPP subjects in non-complete information

In the actual operation of VPP, there is a state of multiple subjects such as distributed power sources. Therefore, the study analyzes the cluster trading strategy, specifies multi-VPP to power-related trading, and focuses on the problem of incomplete information game in the optimal scheduling of day-ahead energy in grid operation based on multi-VPP. The specific scenario of incomplete information optimization constructed by the study is shown in Fig. 3.

Figure 3.

Specific scenarios of incomplete information optimization.

In Fig. 3, the optimization scenario constructed by the study contains a grid company, multiple grid-operated virtual power plants, etc. Among them, the power system is operated by the grid company, which aggregates distributed PV generation and multiple VPPs, and guides the VPPs to adjust the flexible load of the main body by signing a contract with the VPPs. And participate in demand response, thus realizing the absorption of renewable energy and also reducing the total load on the demand side. In addition, by participating in the demand response of multiple VPPs, a single VPP is able to reduce the electricity expenses of internal subjects. Among them, the VPP is the service unit of the power system, and it can use the communication network to collect the basic information of each subject within each VPP. Based on the base information provided by the VPP, the power system analyzes the historical responses of each VPP and enters into different demand response subsidy agreements with the VPP to increase user participation. Therefore, it contains the benefit model of VPP-related operators as well as subjects within it. In the benefit model of multiple VPP-related operators, the formula for calculating the electric load demand of all VPPs at a certain time period is shown in Eq. (1).

$\displaystyle W^{h}=\sum\nolimits_{i=1}^{I}\sum\nolimits_{j=1}^{J_{i}}ew_{ij}^% {h}=\sum\nolimits_{j=1}^{J_{i}}({fw_{ij}^{h}+sw_{ij}^{h}})$ (1)

In Eq. (1), $W$ denotes total load demand; $e w$ denotes the load demand of $i$ vpp in $j$ subject; $h$ denotes time period; $i$ denotes VPP; $j$ denotes subject; $J_{i}$ denotes number of subjects; $f w$ denotes fixed load; and $s w$ denotes levelable load. The study constructs a multi-VPP scenario in which the grid-operated VPP aggregates distributed PV, demand-side loads for day-ahead optimal scheduling, aiming to shift load demand from peak to grid, so as to ensure stable grid operation. Based on this, Peak to Average Ratio (PAR) is introduced to quantitatively study the role of VPP in collaborative optimal scheduling of multiple entities. The formula for calculating the Peak to Average Ratio is shown in Eq. (2).

$\displaystyle\textit{PAR}=\frac{W_{\textit{peak}}}{W_{\textit{avg}}}=H\frac{% \mathop{\max}\limits_{h\in H}W^{h}}{\sum\nolimits_{h\in H}{W^{h}}}$ (2)

In Eq. (2), $W_{\textit{peak}}$ denotes the maximum load throughout the day of the dispatch day; $W_{\textit{avg}}$ denotes the average load throughout the day of the dispatch day; and $H$ denotes the set of time periods $h$ . In addition, the associated benefit model of the multi-VPP includes the power purchase cost and the subsidy revenue in response to the demand in the power market. In order to enable the subjects within the VPP to reasonably determine the electricity sales price and to improve the optimal dispatch of each subject participating in the VPP day before, the grid-operated VPP releases the electricity sales price pricing mechanism to the VPP subject to the corresponding conditions. First, the power procurement cost function of each VPP is a strictly increasing function at any time period and is related to the total power demand of the VPP. Second, the power purchase cost function must be continuously derivable and at least continuously derivable in the VPP. Finally, the cost of electricity purchased by the subject of the VPP is related to the time period of use, in addition to its own electricity consumption, where the cost of electricity purchased by the consumer is higher during peak hours than during low peak hours. Among them, the strictly increasing function is often used as a cost function for electricity purchase due to its strict convexity, and its corresponding expression is shown in Eq. (3).

$\displaystyle C^{h}({W^{h}})=a^{h}({W^{h}})^{2}+b^{h}({W^{h}})$ (3)

In Eq. (3), $C^{h}$ denotes the cost of purchasing electricity; $a^{h}$ and $b^{h}$ denote the tariff coefficients. From Eq. (3), the cost of purchasing a unit of electricity can be derived, and its calculation formula is shown in Eq. (4).

$\displaystyle G_{\textit{sell}}^{h}({W^{h}})={C^{h}({W^{h}})}/{W^{h}=a^{h}W^{h% }+b^{h}}$ (4)

In Eq. (4), $G_{\textit{sell}}^{h}$ denotes the electricity sales price. Based on this, the total purchased cost of electricity can be calculated based on Eqs (1) and (4). The formula is shown in Eq. (5).

$\displaystyle C_{i}({sw_{i}})=\sum\nolimits_{h=1}^{H}\left({G_{\textit{sell}}^% {h}({W^{h}})\sum\nolimits_{j=1}^{J_{i}}{({fw_{ij}^{h}+sw_{ij}^{h}})}}\right)$ (5)

In Eq. (5), $sw_{i}$ denotes the set of load adjustment strategies that can be panned by the virtual power plant $i$ during a day. Therefore, the expression for setting the power prediction value of the distributed PV plant in 24 hours is shown in Eq. (6).

$\displaystyle pv=[{pv^{1},pv^{2},\ldots,pv^{H}}]$ (6)

In Eq. (6), $p v$ denotes the power prediction value. The study developed different subsidy coefficients for different VPP response positivity and also gave the subsidy unit price considering the power prediction value, which is calculated as shown in Eq. (7).

$\displaystyle\gamma_{i}=\frac{pv}{pv\max}bt_{i}=\left[{\frac{pv^{1}}{pv\max}bt% _{i},\frac{pv^{2}}{pv\max}bt_{i},\ldots\frac{pv^{H}}{pv\max}bt_{i}}\right]$ (7)

In Eq. (7), $\gamma$ denotes the subsidy unit price; $b t$ denotes the subsidy coefficient. According to Eq. (7), the actual subsidy can be derived, and the calculation formula is shown in Eq. (8).

$\displaystyle B_{i}({sw_{i}})=\sum\nolimits_{h=1}^{H}{\left({\gamma_{i}\sum% \nolimits_{j=1}^{J_{i}}{({fw_{ij}^{h}+sw_{ij}^{h}})}}\right)}=\sum\nolimits_{h% =1}^{H}\left({\frac{pv^{h}}{pv\max}bt_{i}\sum\nolimits_{j=1}^{J_{i}}{({fw_{ij}% ^{h}+sw_{ij}^{h}})}}\right)$ (8)

In Eq. (8), $B$ denotes the actual subsidy obtained in response to the relevant demand participating in the VPP. Based on this, the benefit function associated with optimal scheduling can be derived from Eqs (5) and (8).

$\displaystyle U_{i}({sw_{i}})=B_{i}({sw_{i}})-C_{i}({sw_{i}})=\sum\nolimits_{h% =1}^{H}\left[{\left({\left({\frac{pv^{h}}{pv\max}bt_{i}-G_{sell}^{h}({w_{i}})}% \right)\sum\nolimits_{j=1}^{J_{i}}{({fw_{ij}^{h}+sw_{ij}^{h}})}}\right)}\right]$ (9)

In Eq. (9), $U_{i}({sw_{i}})$ denotes the benefit function involved in the optimal scheduling associated with the VPP day before. Therefore, it is assumed that there is a certain proportional relationship between this benefit function and the load demand, and the definition of this proportional correlation coefficient is given as shown in expression (10).

$\displaystyle o_{ij}=\frac{\sum\nolimits_{h=1}^{H}{({fw_{ij}^{h}+sw_{ij}^{h}})% }}{\sum\nolimits_{j=1}^{J_{i}}{\sum\nolimits_{h=1}^{H}{({fw_{ij}^{h}+sw_{ij}^{% h}})}}}$ (10)

In Eq. (10), $o_{ij}$ represents the scale factor. Based on equation (10), the benefit function related to the subject $i j$ in the virtual power plant $i$ can be calculated as shown in Eq. (11).

$\displaystyle U_{ij}=o_{ij}U_{i}({sw_{i}})$ (11)

In Eq. (11), $U$ represents the benefit function; $s$ represents a set of translatable load adjustment strategies for a smart community in a day. Taken together, the VPP will maximize the relevant benefits as the goal, and thus participate in the day-ahead scheduling of the VPP, and then the DSO can allocate the relevant benefits according to Eqs (10) and (11).

3.3 Multi-VPP multi-subject Bayesian game study

Based on the game theory under imperfect information, the study uses Bayesian games to describe the cluster transactions of multi-VPP, i.e., delineated to the cooperative optimization of subjects among multi-VPP. In the Bayesian game, the main analytical method is the Harsanyi transformation. The basic idea is to introduce the concept of “nature” into the game, converting the uncertain information before the start of the game into a random outcome after the game, so that the problem of incomplete information can be solved by probability theory, that is, by using probability theory to maximize the expected return of each participant [23]. Before that, the study first constructs the VPP multi-subject complete information game model under the framework of master-slave game. And based on this, the Bayesian game model of multi-VPP is constructed. Under the condition of complete information, each VPP has the relevant revenue functions of other VPPs, so it can optimize its own strategy through the optimal scheduling before the day, and thus obtain the highest revenue [24]. Therefore, the expression describing the VPP complete information multi-subject game model is shown in Eq. (12).

$\displaystyle L_{V}=\{c_{1},\ldots,c_{I};sw_{1},\ldots,sw_{I};q_{1},\ldots,q_{% I}\}$ (12)

In Eq. (12), $L_{V}$ denotes the VPP game multi-subject game in complete information. It is obvious from Eq. that the VPP complete information multi-subject game model contains three parts, which are the game parties, the relevant strategies of the game parties, and the relevant revenue functions of the game parties. Upon receiving the real-time electricity price from VPP, according to the strategies of other game parties, VPP will continuously adjust its strategy to maximize its profit according to the strategies of other VPPs until none of the game parties are willing to change their electricity consumption strategies, thus forming a Nash equilibrium [25]. However, in the actual environment, it is difficult for VPPs to know the revenue functions of other VPPs completely, so the study constructs a multi-subject game model of VPP with non-complete information based on this model, i.e., multi-subject Bayesian game model. Each VPP within the VPP can formulate a revenue maximization strategy among VPPs based on this model, i.e., collaborative optimization. the Bayesian game model can be transformed into a multi-subject Bayesian game model based on The Bayesian game model can be transformed into a complete information game model based on the Harsanyi transformation, so the payoff function of VPP is equivalent to the expected value in the payoff function of the complete information game, and its calculation formula is shown in Eq. (13).

$\displaystyle Eq_{i}({p_{i}})=\sum\nolimits_{p_{-i}\in P_{-i}}{q_{i}}({p_{i},% sw_{i}({p_{i}}),sw_{-i}({p_{-i}})})q_{i}(p_{-i}|{p_{i}})$ (13)

In Eq. (13), $sw_{i}({p_{i}})$ denotes the levelable load adjustment strategy of VPP $i$ when the type is. $p_{i}sw_{-i}({p_{-i}})$ denotes the combination of the reviewable load adjustment strategies for VPPs other than the VPP of type combination $p_{-i}$ . Among them, the Nash equilibrium formed by the study can also be called Bayesian Nash equilibrium, and its state expression formula is shown in Eq. (14).

$\displaystyle Eq_{i}({sw^{*}_{i}({p_{i}}),sw^{*}_{-i}({p_{-i}})})\geqslant Eq_% {i}({sw_{i}({p_{i}}),sw^{*}_{-i}({p_{-i}})})$ (14)

In Eq. (18), $sw^{*}_{-i}({p_{-i}})$ denotes the equilibrium strategy at all time points. Since there are multiple types of VPP, the Bayesian game can be transformed when it satisfies three conditions. The conditions are that the types of the game parties are limited, the combination of the game party types conforms to the joint probability distribution, and the probability of any type of the game party is greater than zero. On this basis, the transformed expression is shown in Eq. (15).

$\displaystyle\overline{Eq}_{i}=\sum\nolimits_{p_{-i}\in P_{-i}}{q_{i}}({p_{i},% sw_{i}({p_{i}}),sw_{-i}({p_{-i}})})q(t)$ (15)

Based on Eq. (15), the descriptive expression of the VPP multi-subject Bayesian game model can be given, as shown in Eq. (16).

$\displaystyle L_{F}=\{c_{1},\ldots,c_{I};P_{i},\ldots,P_{I};sw_{1}({p_{1}}),% \ldots,sw_{I}({p_{I}});\overline{Eq}_{i},\ldots,\overline{Eq}_{I}\}$ (16)

In Eq. (16), $L_{F}$ denotes the VPP multi-subject Bayesian game. In addition, in solving the Bayesian Nash equilibrium optimization problem, the study enters an optimized iterative algorithm for the solution, whose algorithmic flow is shown in Fig. 4.

Figure 4.

Nash equilibrium iterative optimization algorithm process.

In Fig. 4, m is the type. It can be seen from the figure that in the process of incomplete information game, each VPP must continuously adjust its strategy until it reaches Bayesian Nash equilibrium. Moreover, these dynamic adjustment processes are carried out by the collaborative control centers of each VPP, and an equilibrium solution is available for each VPP under the operation of the collaborative control centers of each VPP. At the same time, the Bayesian Nash equilibrium solution exists for the Bayesian game among the VPPs within the VPP, so the equilibrium solutions obtained by the operations of the collaborative control centers of each VPP are the same.

4. Analysis of VPP cluster transaction calculus under multi-subject bayesian game model

To verify the practicality and effectiveness of VPP cluster trading under the multi-subject Bayesian game model constructed under the master-slave game framework, the study conducted an experimental analysis of its co-optimization algorithm. Before the experiments, the relevant parameters of the multi-VPP were first set up, which are shown in Fig. 5.

Figure 5.

Parameter settings related to multi VPP.

In Fig. 5, a total of five VPPs were selected for the study, which contain typical daily predicted load curves within the VPPs as well as the as-predicted PV output curves of the PV plants. As can be seen from the figure, the predicted loads of the five VPPs are maintained within the range of 0–1000 kw, and the overall trend is one of first decreasing and then increasing. Overall the predicted load is lower after 22:00 and before 6:00 and higher between 6 and 22:00. In addition, the predicted PV output curve ebbs and flows from 8:00–18:00, and is 0 kw for the rest of the time. At the same time, the subsidy coefficients in the case are 0.1 yuan/kWh and 0.2 yuan/kWh, respectively, with type sets of 1 and 2, which correspond to the subsidy coefficients. There are 32 types of combinations in the type space, and the number of households in the community is set to 100; The probability of Joint probability distribution is set to 1/32.

Based on this, the study verifies the validity of the VPP multi-subject Bayesian game model in three aspects in total. The first is the solution of Bayesian Nash equilibrium. The study assumes the existence of 10 panning load adjustment strategies for 5 VPPs and assumes category 1 for VPP2, category 2 for VPP3 and category 1 for VPP5. Therefore, the panning load adjustment strategies for VPP2 under category 1 and the comparison results of load profiles before and after applying the multi-VPP subject Bayesian game model are shown in Fig. 6.

Figure 6.

The translatable load adjustment strategy of VPP2 under category 1 and the load curve before and after adjustment.

In Fig. 6a, positive values indicate the increase in electricity consumption and negative values indicate the decrease in electricity consumption. It is obvious from the figure that the increase in electricity consumption of VPP2 under category 1 is concentrated at points 1 to 17, when the electricity price is relatively low. In addition, the VPP load is significantly higher at points 17–24 before the game optimization, and there is a significant reduction after the optimization, showing a better optimization effect. In addition, the relevant data of VPP5 under the same category are shown in Fig. 7.

Figure 7.

Related data of VPP5 under type 1.

In Fig. 7, the electricity consumption abatement of VPP5 under category 1 is also concentrated at 17–24 points. And there is a significant increase from 0:00 to 3:00, indicating that it is in a low electricity price state at this time. Compared with VPP2, the increase in electricity consumption of VPP5 from 13 to 16 points is significantly lower than that of VPP2. In addition, the relevant data of VPP3 under type 2 are shown in Fig. 8.

Figure 8.

Related data of VPP3 under type 2.

In Fig. 8, the difference between the electricity consumption of VPP3 and the other two VPPs is not significant. It is worth noting that, according to the definition of Eq. (11), the unit price of VPP subsidy corresponding to category 2 is significantly higher than that of category 1 in the period from 13:00 to 16:00, so the increase of VPP in this period has a more obvious change. The comparison results before and after the optimization of the three VPP games are comprehensive, and under the Bayesian Nash equilibrium condition, each VPP shifts most of the load at the peak tariff time to 13:00–16:00, so as to obtain a larger demand response subsidy, and at the same time, the load curve of each VPP fluctuates more smoothly relative to that before the optimization, so as to ensure the stability of the total load and avoid spikes in real-time electricity prices. On this basis, the study analyzes the benefits of each subject, including the benefits of VPP and the subject VPP benefits. In Eq. (6) the study introduces the peak-to-average ratio to describe the benefits after the Bayesian game, according to which it can be clearly calculated that the PAR value before the game is 1.858 and after the game is 1.487, which has a significant decrease. Therefore, the study compared the total load before and after the game for both category combinations, and the results are shown in Fig. 9.

Figure 9.

Total load results before and after two types of combination games.

In Fig. 9, the total load demand of VPPs significantly decreases after participating in the game optimization when the grid load demand is high from 17:00–24:00 (which is also the period of high tariff), and they all shift to the time period of 0:00–6:00 and the time period of 13:00–16:00 when the PV forecast output is higher (higher demand response subsidy unit price). Taken together, after the Bayesian game, the load curve changes for each VPP level off, with a significant drop in peak load near 22:00, which is the most important factor contributing to the decrease in PAR. Considering the natural advantage of solar energy, 12–14 o’clock is the time when PV generation is the largest, and the load at this time will match the generation characteristics of PV, thus effectively promoting the consumption of PV. In addition, the results of the main VPP gain are shown in Fig. 10.

Figure 10.

Benefit result of residents’ smart community.

In Fig. 10, the total expected cost of energy consumption for each VPP is significantly reduced after the Bayesian game optimization. Taking VPP3 as an example, the total cost of energy consumption is $8995 before Bayesian game optimization, and after game optimization, the energy consumption expectation is reduced to $7057, which is about 21.5% lower. By calculating the expected energy consumption of each VPP and the expected load of the subjects in the VPP, the expected cost can be derived, and this is used to allocate to the different subjects in the VPP according to Eqs (10) and (11), which leads to a reduction in the cost of energy consumption desired by the subjects during the day as well. Finally, the study analyzed the number of VPPs and the sensitivity of the subsidy coefficient, the results of which are shown in Fig. 11.

Figure 11.

Sensitivity analysis results of the number of VPPs and subsidy coefficient.

From Fig. 11, an increase in the number of VPPs causes an increase in the running time of the optimized iterative algorithm when the Bayesian game model reaches equilibrium. The optimal number of VPPs to maximize the cluster transaction sensitivity of the Bayesian game model is 5, at which time the algorithm consumes 22.028 s. In addition, the increase in the subsidy factor shows a trend of decreasing and then increasing PAR values, indicating that the VPPs will regulate most of the load to the peak PV output rip-off time when the subsidy factor is too high, so as to obtain more subsidy revenue. Therefore, VPP should set a reasonable response subsidy factor in order to avoid adverse effects on the grid.

5. Conclusion

To achieve smooth operation of the power grid and to reduce the cost of electricity consumption of multiple VPPs within the grid, the study constructs a multi-VPP multi-subject Bayesian game model under the framework of non-complete information conditions and master-slave game, and conducts simulation experiments with the help of actual VPP cases to verify the effectiveness of the model and the iterative algorithm. The experimental results show that in the Bayesian Nash equilibrium experiment, the three VPPs concentrate on the decrease of electricity consumption from 17:00 to 24:00 and the increase of electricity price at the trough. In the subject benefit analysis, there is a significant decrease in total load demand after Bayesian game optimization, at which time the PAR value shifts from 1.858 before optimization to 1.487 after the decrease, a decrease of about 20%; and the subject cost expenditure also decreases significantly, by about 21.5%. In the sensitivity analysis, the Bayesian game model cluster transaction sensitivity is optimal when the VPP is 5, at which time the iterative algorithm consumes 22.028 s. The increase of the subsidy coefficient makes the PAR value show a trend of first decreasing and then increasing, indicating that if the subsidy coefficient is too high, the VPP will adjust most of the load to the maximum power, so as to obtain more subsidies. Taken together, the VPP multi-subject Bayesian game model constructed by the study can work under non-complete information conditions, i.e., it can ensure the smooth operation of the VPP grid and significantly reduce the energy cost of each subject by reducing the PAR value while protecting the privacy of each subject. However, the game model built in the study only considers the game of multiple VPPs for the time being, and the impact of energy storage devices needs to be analyzed later.

Authors’ contributions

Ximing Wan contributed to the motivation, and wrote the first draft. Bihong Tang provided the interpretation of the methods, the data analysis and results. Xuan Wen provided the revised versions, references. Qinfei Sun provided the data and results, the revised versions and references. All authors reviewed the manuscript.

Funding

The work was financially supported by Science and Technology Projects from State Grid Corporation of China, (Research and Application of Key Technologies for Economic Operation of Virtual Power Plant Considering Real-time Carbon Flow, 5108-202218280A-2-383-XG).

Footnotes

Conflict of interest

The authors declare no conflict of interest.

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