Co-Optimization Strategy for VPPs Integrating Generalized Energy Storage Based on Asymmetric Nash Bargaining

Abstract

With the in-depth construction of the new power system, the importance of demand-side resources is becoming more and more prominent. The virtual power plant (VPP) has become a powerful means to explore the potential value of distributed resources. However, the differentiated resources between different VPPs are not reasonably deployed, and the problem of realizing the sharing of resources and the distribution of revenues among multi-VPP needs to be urgently solved. A cooperative operation optimization strategy for multi-VPP to participate in the energy and reserve capacity markets is proposed, and the potential risks associated with uncertainty in distributed generators (DGs) output are quantitatively assessed using conditional value-at-risk (CVaR). Firstly, due to the good adjustable performance of electric vehicles (EVs) and thermostatically controlled loads (TCLs), their virtual energy storage (VES) models are established to participate in VPP scheduling. Secondly, based on the asymmetric Nash negotiation theory, a P2P trading method between VPPs in a multi-marketed environment is proposed, which is decomposed into a virtual power plant alliance (VPPA) benefit maximization subproblem and a cooperative revenue distribution subproblem. The alternating direction multiplier method is chosen to solve the model, which protects the privacy of each subject. Simulation results show that the proposed multi-VPP cooperative operation optimization strategy can effectively quantify the uncertainty risk, maximize the alliance benefit, and reasonably allocate the cooperative benefit based on the contribution size of each VPP.

1. Introduction

The energy crisis and global warming are becoming increasingly severe [1] and have attracted widespread attention. Although access to many renewable energy sources has effectively alleviated the energy shortage problem, their volatility and intermittency have brought new challenges to the power grid. However, more than the regulation capability of the power side is needed to match the volatility of renewable energy, and the load side needs to be involved in the adjustment to increase the system’s flexibility. Based on advanced control technology, VPP can combine all kinds of distributed resources into a whole that can be managed in a unified way [2] and constructed as a power plant with the characteristics of distributed resources. Through the VPP, various demand-side resources in the distribution network can be effectively and uniformly scheduled [3], which is crucial for balancing supply and demand in the power grid [4].

Among the adjustable loads, air conditioners [5] and EVs [6,7] can shift or cut the load for a certain period, which has substantial adjustable potential. EVs [8] and TCLs [9] also have energy storage properties and can be constructed as VES models. The charging load of EVs based on V2G technology can be flexible and adjustable [10], which allows EVs to participate in demand response as an adjustable resource [11]. However, EV charging stations are not constructed to be considered as a whole and are not explored as an adjustable resource for BES. Literature [12] modeled an inverter air conditioning system as a VES, proposed a polymeric thermal battery model, and compared it with lithium-ion batteries. Literature [13] established a unified state-space model for inverter air conditioning and EV cluster to achieve joint frequency regulation. Based on the data-driven methodology, an EV was modeled as a BES, and the TCL was modeled as a VES system, which can be effectively dispatched and controlled [14]. Literature [15] establishes a battery energy storage system model with a second-order equivalent thermal parameter model and describes the TCL as a dynamic energy storage model with limitations on the output power and energy capacity. This approach effectively exploits the regulation of TCLs [16]. EVs and TCLs are important resources for VPPs, and their effective scheduling through VPPs after constructing them as VES requires further research.

VPPs aggregate demand-side distributed resources with specific adjustable capabilities and can participate in the electricity market [17], including the electric energy [18] and auxiliary service [19] markets. The independent operation of VPPs will face the problems of low economic efficiency and uncertainty of new energy output, which will affect the safe and stable operation of VPPs [20]. Multi-VPP’s coordinated and optimized operation has become a popular research topic [21]. The coordinated operation of multi-VPP can improve stability and increase the alliance revenue [22]. Literature [23] takes the VPP with surplus energy as the supplier and the VPP lacking energy as the consumer. It improves the overall benefit of the VPPA through the matching of supply and demand. Reference [24] developed a cooperative game and market bidding-based model for P2P electricity trading among multiple VPPs. However, the above studies primarily focus on electricity trading and overlook VPP participation in the reserve market.

However, in the cooperative game of VPPs, the conflict of interest between VPPs is complex to avoid [25]. Using cooperative game theory, cooperative benefits can be allocated using an allocation mechanism based on the Shapley value [26] and the nucleolus [27]. However, the Shapley value cannot always guarantee a stable benefit distribution. The nucleolus incentivizes consumer-producers with higher performance than the Shapley value, but the computational complexity of the nucleolus increases exponentially with the number of participants. Literature [28] establishes a decentralized bilateral transaction mechanism and proposes a distributed interaction algorithm based on the alternating direction of multipliers method. However, it does not allocate the revenue rationally to incentivize the participants. Literature [29] uses asymmetric Nash bargaining to quantify and reward the contribution of microgrids and uses differentiated bargaining power to distribute the benefits. The analysis of differences in allocation mechanisms is shown in

Table 1. Analysis of differences in allocation mechanisms.

Allocation Mechanism	Fairness	Stability	Computational Efficiency
Standard Nash negotiation	☒	☑	☑
Asymmetric	☑	☑	☑
Nash negotiation
Shapley value	☑	☒	☑
nucleolus	☑	☑	☒

. Most of the studies mentioned above use the ADMM algorithm. VPPs need to form alliances with each other to compete together in the market, but the fine-grained modeling of the resources within VPPs needs to be improved. Investigating the differentiation of resources within VPPs with energy storage properties and between different VPPs is important.

VPPs face several uncertainties in the process of exploring distributed resources. The intermittent nature of renewable energy sources can lead to mismatches between supply and demand within the VPP. To reduce the risks and improve the control performance of VPPs, research has been conducted on flexible resources [30], control mechanisms [31], and market participation [32]. The CVaR theory has been used by many scholars to quantify the risk loss caused by uncertainty in the system and is widely used in power systems [33]. In reference [34], CVaR is used to quantify the uncertainty of new energy sources, and a two-layer stochastic optimization model is established. References [35,36] propose optimal bidding strategies for VPPs to participate in the combined energy and reserve capacity markets, considering uncertainty and using CVaR theory to avoid risk losses caused by uncertainty.

With the surge in distributed resources in the power system, the number of EVs, TCLs, and DGs has increased dramatically. This would substantially increase the ability of the VPP to adjust its load, but it also entails a greater risk of loss. Proper coordination of resources within VPPs and energy sharing between VPPs is a promising but challenging approach to better utilize demand-side resources and reduce operating costs. VPPs enable them to meet the criteria for market participation by aggregating distributed resources. No studies have examined the operation and economics of VPPs participating in primary market transactions and P2P transactions and considering their internal DGs, EVs, TCLs, and battery energy storages (BESs). However, this is an urgent problem for future power grids with more and more distributed resources. In order to investigate this problem, this paper proposes an optimization model for the cooperative operation of multi-VPP based on asymmetric Nash bargaining. The main contributions of this paper are as follows:

• A trading framework for VPPs’ participation in the main and P2P markets is constructed and utilized to study the interactions between VPPs and the main market and within the VPP consortium.
• Aiming at the uncertainty of DG output and the huge variable space of distributed resources, we construct a generalized energy storage model for EVs and TCLs and quantify the risky loss using CVaR theory. We also analyze the impact of the risk aversion coefficient on the operating cost of VPPs.
• The validity of the mixed integer linear programming model proposed in this paper is verified through simulation. The impact of whether or not a VPP participates in energy sharing on the operating cost of a VPP is analyzed. The rationality of the asymmetric Nash bargaining method on distributing cooperative benefits is also analyzed.

The rest of the paper is organized as follows: Section 2 presents a framework for multi-VPP to participate in primary market trading and peer-to-peer trading. Section 3 constructs a generalized energy storage model for VPP component units. Section 4 constructs a CVaR-based cooperative operation model for multi-VPP. Section 5 presents the solution method based on the ADMM algorithm. Section 6 presents a case study, and Section 7 concludes the paper.

2. Trading Architecture Among Multi-VPP

There are multiple VPPs in the market with internal differences in their resources, so building a market architecture that can satisfy the cooperative operation of multiple VPPs and P2P trading is necessary. The VPPs form a coalition in the primary market for P2P trading. Since P2P energy trading exists only between a small number of VPPs, and all members of the alliance gain after P2P trading, P2P trading is also called energy sharing in this paper. The multi-VPP trading architecture is shown in Figure 1. The VPP aggregates DGs, BESs, EVs, TCLs, and base loads. The VPPs can participate in electricity trading and reserve capacity trading in the primary market, while only electricity is traded in P2P trading. When internal power imbalance occurs within a VPP, priority is given to energy exchange with other VPPs. Any remaining imbalance after energy sharing is resolved through transactions with the grid. In P2P electricity-sharing transactions, the VPPs negotiate among themselves to determine the amount of electricity to be traded and the price of electricity.

Based on a VPPA electricity sharing architecture for P2P electricity trading, a cooperative optimization model for multi-VPP electricity sharing was constructed and decomposed into an energy sharing subproblem and a revenue distribution subproblem. In the energy sharing subproblem, the interaction power flow between VPPs is determined based on the objective of maximizing the benefits of the VPPA. In the revenue distribution subproblem, revenue is allocated according to the contributions of VPPs in energy sharing, based on asymmetric Nash equilibrium theory.

3. Generalized Energy Storage Model

The demand-side resources with energy storage attributes can be equated to the energy storage model and constructed as generalized energy storage. Due to the variety of demand-side resources, it is difficult to carry out effective regulation. However, the generalized energy storage model can effectively characterize demand-side resources’ flexibility and adjustable capacity. Generalized energy storage is divided into narrow energy storage and VES. Narrowly defined energy storage refers to BES, specifically those equipped with lithium-ion energy storage modules. VES refers to all demand-side resources with energy storage capabilities. This section constructs VES models for EVs and TCLs.

3.1. Virtual Energy Storage Model for EVs

Based on V2G technology, EVs in slow charging mode have load shifting and reverse power supply capabilities, which provide a theoretical basis for their power adjustment. A charging station is a natural aggregator. Based on the energy consumption data and vehicle data of the EVs in the station, the charging station can be constructed as a VES for overall regulation. The charging station records historical EV data, including the EV’s arrival time, departure time, battery level on arrival, battery level on departure, battery capacity limit, and maximum charge and discharge power, as shown in Formula (1).

$$E_{Vn}=\left\{t_n^{arr},t_n^{lea},e_n^{arr},e_n^{lea},{\underset{¯}{e}}_n^{EV},{\overline{e}}_n^{EV},{\overline{P}}_n^{EV,c},{\overline{P}}_n^{EV,d}\right\},\forall n$$

(1)

Based on Formula (1), the dispatchable potential of the EV cluster can be derived and used as an equipment parameter for the VES of the charging station. The specific derivation procedure can be found in Appendix A. The derived equipment parameter for the VES of the EVs is shown in Formula (2):

$${\mathrm{\Lambda }}_V=\left\{{\overline{P}}_{it}^{EVs,c},{\overline{P}}_{it}^{EVs,d},{\underset{¯}{e}}_{it}^{EVs},{\overline{e}}_{it}^{EVs},\Delta e_{it}^{EVs}\right\},\forall t$$

(2)

By means of the Minkowski summation, the EVs in the station are aggregated into a VES, as shown in Constraints (3) to (9). The specific derivation procedure can be found in Constraints (A1) to (A19) of Appendix A.

$$0\le P_{ist}^{EVs,c}\le {\overline{P}}_{it}^{EVs,c},\forall i,\forall s,\forall t$$

(3)

$$0\le P_{ist}^{EVs,d}\le {\overline{P}}_{it}^{EVs,d},\forall i,\forall s,\forall t$$

(4)

$$s_{ist}^{EVs}=\left(e_{is(t-1)}^{EVs}+\Delta e_{it}^{EVs}+{\eta }^{EV}{P}_{ist}^{EVs,c}\Delta t-{\displaystyle \frac{P_{ist}^{EVs,d}}{{\eta }^{EV}}}\Delta t\right)/{\overline{e}}_{it}^{EVs},\forall i,\forall s,\forall t$$

(5)

$${\underset{¯}{e}}_{it}^{EVs}/{\overline{e}}_{it}^{EVs}\le s_{ist}^{EVs}\le 1,\forall i,\forall s,\forall t$$

(6)

$$(P_{ist}^{EVs,d}+P_{ist}^{EVs,rm})\Delta t/{\eta }^{EV}\le e_{ist}^{EVs},\forall i,\forall s,\forall t$$

(7)

$$P_{ist}^{EVs,rm}-P_{ist}^{EVs,c}{\eta }^{EV}+P_{ist}^{EVs,d}/{\eta }^{EV}\le {\overline{P}}_{ist}^{EVs,d},\forall i,\forall s,\forall t$$

(8)

$$0\le P_{ist}^{EVs,rm}\le {\overline{P}}_{it}^{EVs,c},\forall i,\forall s,\forall t$$

(9)

The discrete EV data is aggregated into VES through the method of Minkowski summation. Essentially, the variable space of individual EVs is projected onto a hypercube space while retaining the Constraint relationship. The hypercube contains all possible behavioral decisions of the charging station. This method can effectively dispatch the power of the EV cluster and fully tap its adjustable potential.

3.2. Virtual Energy Storage Model for TCLs

Variable-frequency air conditioners can adjust power consumption by controlling the compressor’s frequency, enabling load variations within a certain range. By altering power consumption, the indoor temperature deviates from the set point, resulting in changes to indoor energy dynamics. Based on this principle, TCLs can be modeled as VES, providing flexibility in VPP.

Equivalent thermal parameters (ETP) can describe the thermal dynamic process within TCLs with inverter air conditioners. This paper simplifies the model by ignoring the effect of solar radiation on indoor temperature. The ETP model is shown in Formula (10):

$$C_a{\displaystyle \frac{d{T}_{ismt}}{dt}}={\displaystyle \frac{T_t^{out}-T_{ismt}}{R}}+Q^{in}-Q_{ismt}^{AC},\forall i,\forall s,\forall m,\forall t$$

(10)

Unlike fixed-frequency air conditioners, inverter air conditioners can adjust the thermoelectric conversion efficiency by changing the frequency of the compressor. The air conditioner’s power and cooling power can be expressed as a first-degree function of the air conditioner frequency, as shown in Formulas (11) and (12):

$$P_{ismt}^{AC}=k_1{f}_{ismt}^{AC}+l_1,\forall i,\forall s,\forall m,\forall t$$

(11)

$$Q_{ismt}^{AC}=k_2{f}_{ismt}^{AC}+l_2,\forall i,\forall s,\forall m,\forall t$$

(12)

The connection between the cooling power and the electric power of an inverter air conditioner can be derived from Formulas (11) and (12), as shown in Formula (13):

$$Q_{ismt}^{AC}={\displaystyle \frac{k_2}{k_1}}{P}_{ismt}^{AC}+{\displaystyle \frac{k_1{l}_2-k_2{l}_1}{k_1}},\forall i,\forall s,\forall m,\forall t$$

(13)

The electrical power consumed by the air conditioner when operating at the set temperature is defined as the baseline load of the air conditioner, as shown in Formula (14):

$$P_{imt}^{AC,baseline}={\displaystyle \frac{k_1(T_t^{out}-T^{set})}{k_{2}R}}+{\displaystyle \frac{k_1{Q}^{in}+k_2{l}_1-k_1{l}_2}{k_2}},\forall i,\forall m,\forall t$$

(14)

Due to the frequency limit of the air conditioning compressor, its power meets certain constraints, as shown in Constraint (15):

$${\underset{¯}{P}}_{im}^{AC}\le P_{ismt}^{AC}\le {\overline{P}}_{im}^{AC},\forall i,\forall s,\forall m,\forall t$$

(15)

After the air conditioner receives external instructions, the power changes and deviates from the baseline load. The resulting power deviation is defined as the charging and discharging power of the VES for TCL, as shown in Formula (16):

$$P_{ismt}^{AC,ves}=P_{ismt}^{AC}-P_{imt}^{AC,baseline},\forall i,\forall s,\forall m,\forall t$$

(16)

The user-defined acceptable temperature fluctuation range determines the upper and lower temperature limits for temperature adjustment of the TCL. The virtual power level for TCLs is equivalently expressed using the state of change in room temperature over a range of temperature fluctuations, as shown in Formula (17).

$$s_{ismt}^{AC}={\displaystyle \frac{\overline{T}-T_{ismt}}{\overline{T}-\underset{¯}{T}}},\forall i,\forall s,\forall m,\forall t$$

(17)

Based on the thermodynamic principles of the inverter air conditioning system [12], the energy constraints of the VES of the TCL can be expressed as Constraints (18) to (20). Constraint (18) specifies the energy levels of the TCLs in the VPP for each period, and these energy levels receive the limitations of Constraint (20).

$$s_{ism(t+1)}^{AC}=\alpha s_{ismt}^{AC}+\beta P_{ismt}^{AC,ves}\Delta t+\gamma ,\forall i,\forall s,\forall m,\forall t$$

(18)

$$\left\{\begin{array}{c}\alpha =e^{-{\displaystyle \frac{\Delta t}{R{C}_a}}}\\ \beta ={\displaystyle \frac{k_{2}R}{k_1(\overline{T}-\underset{¯}{T})}}(1-\alpha )\\ \gamma =(1-\alpha )s_{ism(t=0)}^{AC}\end{array}\right.,\forall i,\forall s,\forall m$$

(19)

$$0\le s_{ismt}^{AC}\le 1,\forall i,\forall s,\forall m,\forall t$$

(20)

In summary, a VES model for TCLs has been established, and its power and energy constraints have a similar expression form as that of battery energy storage.

In addition, the reserve capacity constraint for TCL can be expressed as Constraint (21):

$$0\le P_{ismt}^{AC,rm}\le P_{ismt}^{AC}-{\underset{¯}{P}}_{im}^{AC},\forall i,\forall s,\forall m,\forall t$$

(21)

3.3. Battery Energy Storage Model

Constraint (22) specifies the energy levels of the BES in the VPP during the dispatch cycle, and these energy levels are limited by Constraint (23). Constraints (24) and (25) limit the charging and discharging power of the BES. Constraint (26) specifies the non-simultaneity of the charging and discharging states of the BES system. Constraints (27) to (29) limit the reserve capacity of the BES. Constraint (27) specifies the limitations related to the remaining power that the reserve capacity should fulfill, and Constraints (28) and (29) specify the limitations related to the maximum power level that the reserve capacity should fulfill.

$$s_{is(t+1)}^{BES}=(e_{ist}^{BES}+{\eta }^{BES}{P}_{ist}^{BES,c}\Delta t-{\displaystyle \frac{P_{ist}^{BES,d}}{{\eta }^{BES}}}\Delta t)/{\overline{e}}_i^{BES},\forall i,\forall s,\forall t$$

(22)

$$0.1\le s_{is(t+1)}^{BES}\le 0.9,\forall i,\forall s,\forall t$$

(23)

$$0\le P_{ist}^{BES,c}\le {\overline{P}}_i^{BES}{u}_{ist}^{BES,c},\forall i,\forall s,\forall t$$

(24)

$$0\le P_{ist}^{BES,d}\le {\overline{P}}_i^{BES}{u}_{ist}^{BES,d},\forall i,\forall s,\forall t$$

(25)

$$u_{ist}^{BES,c}+u_{ist}^{BES,d}\le 1,u_{ist}^{BES,c},u_{ist}^{BES,d}\in \{0,1\},\forall i,\forall s,\forall t$$

(26)

$$(P_{ist}^{BES,d}+P_{ist}^{BES,rm})\Delta t/{\eta }^{BES}\le e_{ist}^{BES},\forall i,\forall s,\forall t$$

(27)

$$P_{ist}^{BES,rm}-P_{ist}^{BES,c}{\eta }^{BES}+P_{ist}^{BES,d}/{\eta }^{BES}\le {\overline{P}}_i^{BES},\forall i,\forall s,\forall t$$

(28)

$$0\le P_{ist}^{BES,rm}\le {\overline{P}}_i^{BES},\forall i,\forall s,\forall t$$

(29)

4. Multi-VPP Cooperative Operation Model

This section constructs a cooperative optimization operation model for multi-VPP considering risk.

4.1. Objective Function

The objective function (30) aims to minimize the VPPA’s expected operating costs. Additionally, it utilizes CVaR theory to measure the risk of loss due to uncertainty in DG output.

$$\begin{array}{l}\min \left(1-{\beta }^{CVaR}\right){\displaystyle \sum _{i\in I}{\displaystyle \sum _{s\in S}{\pi }_{i,s}}}{\displaystyle \sum _{t\in T}{C}_{ist}^{VPP}}+{\beta }^{CVaR}{\displaystyle \sum _{i\in I}{\delta }_i}\\ =\left(1-{\beta }^{CVaR}\right){\displaystyle \sum _{i\in I}{\displaystyle \sum _{s\in S}{\pi }_{i,s}}}{\displaystyle \sum _{t\in T}\left(C_{it}^{EN}-C_{it}^{RM}+C_{it}^{P2P}+C_{ist}^{BES}+C_{ist}^{TCL}+C_{ist}^{EVs}+C_{ist}^{DG}\right)+{\beta }^{VaR}{\displaystyle \sum _{i\in I}{\delta }_i}}\end{array}$$

(30)

The objective function (30) consists of two terms: the expected cost of the VPP operation and the CVaR.

Constraint (31) represents the cost of the VPP’s participation in energy trading in the primary market. Constraint (32) represents the cost of a VPP’s participation in standby capacity trading in the primary market. Constraint (33) represents the transaction cost of P2P trading between VPPs.

$$C_{it}^{EN}=P_{it}^{buy}{C}_t^{buy}-P_{it}^{sell}{C}_t^{sell}$$

(31)

$$C_{it}^{RM}=P_{it}^{VPP,rm}{c}_t^{rm}$$

(32)

$$C_{it}^{P2P}=c_{\left(i-j\right)t}^{P2P}{P}_{\left(i-j\right)t}^{P2P}$$

(33)

Constraint (34) represents the cost of calling for BES. Constraint (35) represents the cost of invocation of TCL. Constraint (36) represents the cost of EV invocation and charging revenue. The charging tariff for EV users consists of the price of electricity and the service charge.

$$C_{ist}^{BES}={\mu }^{BES}\left(P_{ist}^{BES,c}+P_{ist}^{BES,d}\right)$$

(34)

$$C_{ist}^{TCL}={\displaystyle \sum _{m\in M}{u}^T}\left|T_{ismt}-T^{set}\right|$$

(35)

$$C_{ist}^{EVs}={\mu }^P{P}_{ist}^{EVs,d}+{\mu }^E\left(1-s_{ist}^{EVs}\right)-\left(c_t^{buy}+c^{ser}\right)P_{ist}^{EVs,c}$$

(36)

Constraint (37) represents the cost of curtailing new energy sources. A cost is incurred when the new energy output is not fully utilized.

$$C_{ist}^{DG}={\mu }^{DG}\left({\overline{P}}_{ist}^{DG}-P_{ist}^{DG}\right)$$

(37)

Constraint (38) represents the CVaR cost of the VPP. Risks in VPP operation arise from uncertainty in the output of DGs.

$${\delta }_i={\xi }_i+{\displaystyle \frac{1}{1-\alpha }}{\displaystyle \sum _{s\in S}{\pi }_{is}{\eta }_{is}}$$

(38)

4.2. Economic and Technical Constraints

4.2.1. CVaR Risk Measure

The last term of expression (30) considers only the scenarios where the cost is higher than the value-at-risk. In contrast, the rest of the scenarios are assigned a zero value according to Constraints (39) and (40). Constraint (39) allows us to determine the value of the auxiliary variable ${\eta }_{is}$ equal to the difference between ${\xi }_i$ and the cost of the corresponding scenario; otherwise, Constraint (40) assigns ${\eta }_{is}$ is defined as a non-negative variable with a value of 0 for scenarios with costs inferior than ${\xi }_i$. The trade-off between expected costs and CVaR of the VPP is modeled through the parameter ${\xi }_i$. The decision maker can adopt different risk aversion strategies by choosing different values of ${\xi }_i$.

$$\begin{array}{l}[{\displaystyle \sum _{t\in T}\left[\left(P_{it}^{buy}{c}_t^{buy}-P_{it}^{sell}{c}_t^{sell}\right)-P_{it}^{VPP,rm}{c}_t^{rm}+{\displaystyle \sum _{i\in I,j\ne i}{c}_{\left(i-j\right)t}^{P2P}{P}_{\left(i-j\right)t}^{P2p}}\right]}\\ +{\displaystyle \sum _{t\in T}[{\mu }^{BES}}\left(P_{ist}^{BES,c}+P_{ist}^{BES,d}\right)+{\displaystyle \sum _{m\in M}{\mu }^T}\left|T_{ismt}-T^{set}\right|+{\mu }^P{P}_{ist}^{EVs,d}\\ +{\mu }^E\left(1-s_{ist}^{EVs}\right)-\left(c_t^{buy}+c^{ser}\right)P_{ist}^{EVs,c}+{\mu }^{DG}\left({\overline{P}}_{ist}^{DG}-P_{ist}^{DG}\right)]]-{\xi }_i\le {\eta }_{is},\forall i,\forall s\end{array}$$

(39)

$$0\le {\eta }_{is},\forall i,\forall s$$

(40)

4.2.2. VPP Energy and Reserve Capacity Balancing

Constraint (41) represents the power balance within the VPP, considering the assets in the VPP and the energy interaction between the VPP and the external market. This constraint stipulates that the power consumed by the distributed resources within the VPP must be balanced by the power that the VPP interacts with the external market. Constraint (42) represents the balance of reserve capacity within the VPP. This constraint stipulates that the reserve capacity sold by the VPP to the grid is equal to the sum of the reserve capacities provided by the on-site distributed resources within the VPP.

$$\begin{array}{l}{P}_{ist}^{load}+P_{ist}^{EVs,c}+P_{ist}^{BES,c}+{\displaystyle \sum _{m\in M}{P}_{ismt}^{AC}}+P_{it}^{sell}+{\displaystyle \sum _{j\in I,j\ne i}{P}_{\left(i-j\right)t}^{P2P}}\\ =P_{it}^{buy}+P_{ist}^{DG}+P_{ist}^{EVs,d}+P_{ist}^{BES,d},\forall i,\forall s,\forall t\end{array}$$

(41)

$$P_{it}^{VPP,rm}=P_{ist}^{BES,rm}+P_{ist}^{EVs,rm}+{\displaystyle \sum _{m\in M}{P}_{ismt}^{AC,rm}+P_{ist}^{DG,rm}},\forall i,\forall s,\forall t$$

(42)

4.2.3. P2P Transaction

Constraints (43) and (44) represent the P2P transaction constraints between VPPs. The constraints indicate the transaction energy and payment amount constraints that need to be satisfied.

$${\displaystyle \sum _{i,j\in I,i\ne j}{P}_{\left(i-j\right)t}^{P2P}}=0,\forall t$$

(43)

$${\displaystyle \sum _{i,j\in I,i\ne j}{c}_{\left(i-j\right)t}^{P2P}{P}_{\left(i-j\right)t}^{P2P}}=0,\forall t$$

(44)

4.2.4. DGs Operation

Constraints (45) and (46) specify the production limits of the distributed generation in the VPP. The inequality indicates that the output power of the distributed generation and the reserved reserve capacity should be less than its maximum output.

$$P_{ist}^{DG}+P_{ist}^{DG,rm}\le {\overline{P}}_{ist}^{DG},\forall i,\forall s,\forall t$$

(45)

$$0\le P_{ist}^{DG},0\le P_{ist}^{DG,rm},\forall i,\forall s,\forall t$$

(46)

5. Solution Method

5.1. ADMM-Based Subproblem for Minimizing Coalition Cost

ADMM is a method based on augmented generalized Lagrangian, commonly used to solve optimization problems with linearly divisible constraints and objectives. It can be effective for the distributed solution of the energy-sharing problem of a VPPA, and the alliance cost minimization problem can be formulated as Formulas (47) and (48):

$$\begin{array}{l}\min (1-{\beta }^{CVaR})\left[{\displaystyle \sum _{i\in I}{\displaystyle \sum _{t\in T}\left[\left(P_{it}^{buy}{c}_t^{buy}-P_{it}^{sell}{c}_t^{sell}\right)-P_{it}^{VPP,rm}{c}_t^{rm}+{\displaystyle \sum _{i\in I,j\ne i}{c}_{\left(i-j\right)t}^{P2P}{P}_{\left(i-j\right)t}^{P2p}}\right]}}\right.\\ +{\displaystyle \sum _{i\in I}{\displaystyle \sum _{s\in S}{\pi }_{i,s}{\displaystyle \sum _{t\in T}\left[{\mu }^{BES}\left(P_{ist}^{BES,c}+P_{ist}^{BES,d}\right)+{\displaystyle \sum _{m\in M}{\mu }^T}\left|T_{ismt}-T^{set}\right|+{\mu }^P{P}_{ist}^{EVs,d}+{\mu }^E\left(1-s_{ist}^{EVs}\right)\right.}}}\\ -\left(c_t^{buy}+c^{ser}\right)P_{ist}^{EVs,c}+{\mu }^{DG}\left.\left({\overline{P}}_{ist}^{DG}-P_{ist}^{DG}\right)\right]+{\beta }^{CVaR}{\displaystyle \sum _{i\in I}\left[{\xi }_i+\frac{1}{1-\alpha }{\displaystyle \sum _{s\in S}{\pi }_{is}{\eta }_{is}}\right]}\end{array}$$

(47)

s.t.

Constraints (3)–(9), and (14)–(29).

(48)

When solving the VPPA cost minimization problem, the revenue from electricity trading between entities, i.e., $C_{it}^{P2P}$, is ignored. A vector of auxiliary variables, $P_{(i-j)t}^{P2P}$, is introduced to transform the multiple coupling model constructed by the original problem into a double coupling model, as shown in Constraint (49). If Formula (49) holds, a consensus on electricity trading has been reached among the subjects.

$$P_{(i-j)t}^{P2P}+P_{(j-i)t}^{P2P}=0,\forall i,\forall t$$

(49)

After the decoupling of multiple couplings is complete, the Formula constraints are introduced into the objective function using ADMM to obtain the augmented Lagrangian form, as shown in Formula (50):

$$L_i^e={\displaystyle \sum _{s\in S}{\pi }_{is}}{\displaystyle \sum _{t\in T}{C}_{ist}^{VPP}}+{\displaystyle \sum _{j\in I}{\displaystyle \sum _{t\in T}{\lambda }_{i-j}^e}}(P_{(i-j)t}^{P2P}+P_{(j-i)t}^{P2P})+{\displaystyle \sum _{j\in I}\frac{{\rho }^e}{2}{\displaystyle \sum _{t\in T}{‖P_{(i-j)t}^{P2P}+P_{(j-i)t}^{P2P}‖}_2^2}}$$

(50)

The iterative process of the ADMM algorithm is shown in Formulas (51) and (52):

$$\begin{array}{cc}{P}_{(i-j)t}^{P2P}(k+1)& =\arg \min \left[{\displaystyle \sum _{s\in S}{\pi }_{is}}{\displaystyle \sum _{t\in T}{C}_{ist}^{VPP}}+{\displaystyle \sum _{j\in I}{\displaystyle \sum _{t\in T}{\lambda }_{i-j}^e}}(k)\left(P_{(i-j)t}^{P2P}(k)+P_{(j-i)t}^{P2P}(k)\right)\right.\hfill \\ & \left.+{\displaystyle \sum _{j\in I}\frac{{\rho }^e}{2}{\displaystyle \sum _{t\in T}{‖P_{(i-j)t}^{P2P}(k)+P_{(j-i)t}^{P2P}(k)‖}_2^2}}\right]\hfill \end{array}$$

(51)

$$\begin{array}{cc}{P}_{(j-i)t}^{P2P}(k+1)& =\arg \min \left[{\displaystyle \sum _{s\in S}{\pi }_{is}}{\displaystyle \sum _{t\in T}{C}_{ist}^{VPP}}+{\displaystyle \sum _{j\in I}{\displaystyle \sum _{t\in T}{\lambda }_{j-i}^e}}(k)\left(P_{(j-i)t}^{P2P}(k)+P_{(i-j)t}^{P2P}(k+1)\right)\right.\hfill \\ & \left.+{\displaystyle \sum _{j\in I}\frac{{\rho }^e}{2}{\displaystyle \sum _{t\in T}{‖P_{(j-i)t}^{P2P}(k)+P_{(i-j)t}^{P2P}(k+1)‖}_2^2}}\right]\hfill \end{array}$$

(52)

The Lagrange multiplier update rule is shown in Formula (53):

$${\lambda }_{i-j}^e(k+1)={\lambda }_{i-j}^e(k)+{\rho }^e\left(P_{(i-j)t}^{P2P}(k+1)+P_{(j-i)t}^{P2P}(k+1)\right)$$

(53)

The condition for the ADMM algorithm to stop iterating is shown in Constraint (54):

$${\displaystyle \sum _{i\in I}{\displaystyle \sum _{t\in T}{‖P_{(i-j)t}^{P2P}(k+1)+P_{(j-i)t}^{P2P}(k)‖}_2}}\le {\delta }_e$$

(54)

5.2. Revenue Allocation Problem Based on Asymmetric Nash Games

After the VPPA shares energy with the goal of minimizing operating costs, this section solves the problem of revenue distribution based on the basic theory of asymmetric Nash bargaining. After the VPP cooperates, its dependence on the grid decreases, operating costs are reduced, and output and received power contribute to the alliance. However, it is generally considered that the contribution of output energy is more significant than received energy.

In this paper, a nonlinear function based on the natural logarithm is proposed to quantify each entity’s contribution to energy sharing. The asymmetric bargaining factor is calculated based on the amount of electricity provided and received by the VPP in energy sharing. The asymmetric bargaining factor quantifies the contribution, and its magnitude measures the VPP’s bargaining power. The VPP with stronger bargaining power can allocate more benefits in the Nash negotiation. The electricity price for P2P transactions is determined through Nash bargaining, and the energy-sharing revenue is then distributed. The bargaining power of the VPP is shown in Formulas (55) and (56):

$$d_i=e^{E_i^s/{\overline{E}}^s}-e^{-E_i^r/{\overline{E}}^r},\forall i$$

(55)

$$\left\{\begin{array}{c}{E}_i^s={\displaystyle {\sum }_{t\in T}\max (0,P_{(i-j)t}^{P2P})}\\ {\overline{E}}^s=\max (E_i^s)\\ E_i^r=-{\displaystyle {\sum }_{t\in T}\min (0,P_{(i-j)t}^{P2P})}\\ {\overline{E}}^r=\min (E_i^r)\end{array}\right.,\forall i$$

(56)

Based on Nash bargaining theory and asymmetric bargaining factors, the problem of revenue distribution in a VPPA can be formulated as an asymmetric Nash game. This model can distribute revenue unequally based on the size of the Nash bargaining factor. The constructed asymmetric Nash game model is expressed as Formulas (57) and (58):

$$\left\{\begin{array}{c}\max {{\displaystyle {\prod }_{i\in I}\left[C_i^0-{\displaystyle {\sum }_{t\in T}{C}_{it}^{VPP}+{\displaystyle {\sum }_{t\in T}{C}_{it}^{P2P}}}\right]}}^{d_i}\\ \mathrm{s}.\mathrm{t}.(44)\end{array}\right.$$

(57)

$$C_i^0-{\displaystyle \sum _{t\in T}{C}_{it}^{VPP}}+{\displaystyle \sum _{t\in T}{C}_{it}^{P2P}}>0,\forall i$$

(58)

There is an exponential function in Formula (57), which is converted to facilitate model solving. The converted objective function is shown in Formula (59):

$$\min {\displaystyle \sum _{i\in I}-d_i}\ln \left[C_i^o-{\displaystyle \sum _{t\in T}{C}_{it}^{VPP}+{\displaystyle \sum _{t\in T}{C}_{it}^{P2P}}}\right]$$

(59)

The asymmetric Nash bargaining model is solved using the ADMM algorithm, and the specific steps of the solution are described in detail in Constraints (A20) to (A24) of Appendix A.

6. Case Study

6.1. Data

In this paper, we construct a simulated case study involving three VPPs to investigate P2P energy trading among them.

Table 2. Flexibility resources within the VPPs.

Type of Resource	VPP1	VPP2	VPP3
Base load	☑	☑	☑
Photovoltaic	☑	☒	☒
Wind power	☒	☑	☑
TCL	☑	☑	☑
EV	☑	☑	☑
BES	☑	☑	☑

displays the flexibility resources within the VPPs. Employ the Latin hypercube generation and reduction method to create ten DG output scenarios. The parameters of the equipment inside the VPP are shown in

Table A1. System parameter.

Parameter Name	Parameter Value	Parameter Name	Parameter Value
${\overline{e}}_i^{BES}$	0.9 × 800 kWh	$k_1,k_2,l_1,l_2$	0.04, −0.4, 0.06, −0.3
${\underset{¯}{e}}_i^{BES}$	0.1 × 800 kWh	$T^{set}$	24 °C
$P_i^{BES}$	400 kW	${\overline{P}}_{im}^{AC}$	5 kW
${\eta }^{BES}$	0.95	${\underset{¯}{P}}_{im}^{AC}$	1 kW
${\overline{e}}_n^{EV}$	0.9 × 50 kWh	$\overline{T}$	26 °C
${\underset{¯}{e}}_n^{EV}$	0.1 × 50 kWh	$\underset{¯}{T}$	22 °C
${\overline{P}}_n^{EV,c}$	7 kW	${\mu }^{BES}$	0.2 CNY/kWh
${\overline{P}}_n^{EV,d}$	7 kW	${\mu }^T$	0.5 CNY/°C
${\eta }^{EV}$	0.95	${\mu }^P$	0.5 CNY/kWh
$C_a$	2 kJ/°C	${\mu }^E$	0.01 CNY
$R$	2 °C/kW	${\mu }^{DG}$	0.6 CNY/kWh
$Q^{in}$	0.3 kJ	$c^{ser}$	0.2 CNY/kW

of Appendix B. We categorize EVs into three types based on their travel patterns, and the sampling parameters for these types are listed in

Table A2. Sampling parameters for three types of EVs.

Types of EVs	${\mathit{T}}_{\mathit{n}}^{\mathit{a}\mathit{r}\mathit{r}}$	${\mathit{T}}_{\mathit{n}}^{\mathit{l}\mathit{e}\mathit{a}}$	${\mathit{e}}_{\mathit{n}}^{\mathit{a}\mathit{r}\mathit{r}}$ (kWh)
Group I vehicles	N (18:00, 4:00)	N (4:00, 8:00)	U (25, 35)
Group II vehicles	N (21:00, 1:00)	N (1:00, 7:00)	U (15, 25)
Group III vehicles	N (9:00, 2:00)	N (17:00, 2:00)	U (20, 30)

of Appendix B. We use the Monte Carlo simulation method to model the day-ahead adjustable potential of the charging station.

Table A3. Number of EVs in VPPs.

Types of EVs	Group I Vehicles	Group II Vehicles	Group III Vehicles
VPP1	U (170, 210)	U (180, 220)	U (20, 40)
VPP2	U (170, 200)	U (80, 120)	U (340, 380)
VPP3	U (10, 30)	U (80, 100)	U (380, 420)

and

Table A4. Number of TCLs in VPPs.

Types of EVs	VPP1	VPP2	VPP3
Number of TCLs	350	300	400

of Appendix B present the counts of EVs and TCLs within different VPPs. The VPPs purchase electricity from the electricity market at time-of-day tariffs. To discourage arbitrage in the electricity market by the VPPs, we set the electricity purchase price at 1.2 times the electricity sale price. Figure A1 of Appendix B illustrates the power purchase, sale, and reserve capacity prices.

To solve the proposed model, this paper decomposes it into two problems. First, the energy sharing problem among VPPs is solved, followed by the revenue sharing problem among VPPs. Based on the aforementioned assumptions, this study will use purely numerical simulations to analyze the proposed method to provide case studies and guidance for the operation of virtual power plant alliances in electricity markets.

6.2. VPPA Risk Analysis

The risk aversion coefficient reflects decision makers’ degree of risk loss aversion. To verify the guiding effect of CVaR theory on the risk aversion during the operation of VPPs, the operating cost of VPPs under different risk aversion coefficients is analyzed. The running cost and CVaR of the VPP coalition under different risk aversion coefficients are shown in Figure 2. It can be seen that regardless of whether VPPs form alliances or not, as the risk aversion coefficient rises, VPPs make more conservative decisions. The increased operating costs and the decrease in CVaR manifest this. However, the effective frontier curve shifts forward after VPPs ally. This indicates that energy sharing can improve VPPs’ operational efficiency and risk resistance. Through the effective frontier curve, the decision makers of VPPs can determine their risk appetite according to their psychological expectations. Decision makers can increase the risk aversion coefficient during periods of heightened risk aversion sentiment, thereby reducing the potential risks associated with VPP operations.

6.3. Operation of the VPPA

This section analyses the optimization of the VPPA’s cooperative operation. It employs a risk aversion coefficient of 0.6.

6.3.1. Energy Sharing Results

The optimization results of the P2P power interactions of the VPPA are shown in Figure 3. From 09:00 to 16:00, VPP 1 has surplus PV power and transmits the surplus power to VPP 2 and VPP 3, which are short of power. During the rest of the time, the increase in wind power output of VPP 2 creates a power surplus. It starts to transmit the power to the remaining two VPPs. VPP1 starts to purchase power from the other VPPs after 23:00 because of the scarcity of power due to the charging of EVs and photovoltaic power reduction. VPP 3’s dependence on the remaining VPPs decreases during this period due to load drop and increase in wind power output. Through energy sharing, energy is complementary to resources within the VPPA.

Figure 4 shows the electricity traded with the grid before and after participating in energy sharing. Before participating in energy sharing, each VPP must trade electricity with the grid to maintain internal energy balance when there is a surplus or lack of internal electricity. The power interaction between the VPPA and the grid is 45,250.60 kWh. After the VPPs participate in energy sharing, they prioritize getting or giving power to other VPPs in the alliance to consume power internally first. If the power remains unbalanced within the alliance, then power is traded with the grid. This leads to a decrease in the amount of power interaction between the alliance and the grid. The power interaction between the VPPA and the grid drops to 22,881.21 kWh, a decrease of 49.4%. Through energy sharing, VPP operators can sell surplus electricity at higher prices or purchase electricity at lower prices, thereby reducing operational costs. The alliance’s dependence on the grid is significantly reduced.

6.3.2. VPPA Revenue Analysis

Table 3. Primary market trading results.

VPP Number	Cost of Electricity (CNY)		Revenue from Spare Capacity (CNY)
	Before Cooperation	After Cooperation	Before Cooperation	After Cooperation
VPP1	18,144.08	12,354.53	712.14	716.42
VPP2	−4362.64	4251.42	870.62	871.23
VPP3	13,683.00	4466.82	489.92	493.16
VPPA	27,464.44	21,072.77	2072.68	2080.81

gives the cost of electric energy and revenue from the reserve capacity of VPPs before and after cooperation. Before the cooperation, the VPPA’s electric energy cost was 27,464.44 CNY. After the cooperation, energy sharing is done among the VPPs, and the electric energy demand to the grid decreases. The alliance’s electric energy cost decreased to 21,072.77 CNY, 23.3% lower. After participating in energy sharing, the electricity consumption of adjustable loads rises due to the ability to obtain lower-cost electricity at other VPPs. This means that adjustable loads have more space for load reduction, VPPs can sell more reserve capacity to the grid, and VPP revenues in the reserve capacity market have increased.

The graph of the electricity trading price of VPPA energy sharing after asymmetric Nash bargaining is shown in Figure 5. The analysis shows that because the VPPs can directly trade power purchases and sales with the grid, the electricity transaction price between VPPs will be between the purchase and sale prices. Therefore, the VPPs can sell electricity to other VPPs through P2P transactions at a price higher than the price of electricity sales and can also buy electricity from other VPPs through P2P transactions at a price lower than the price of electricity purchases. Through P2P transactions, VPPs reduce energy costs and improve operational efficiency.

The results of the asymmetric bargaining factors are given in

Table 4. Distribution of gains from asymmetric Nash bargaining.

VPP Number	Electricity Contribution (kWh)	Asymmetric Bargaining Factor
VPP1	−10,592.70	1.09
VPP2	16,683.90	1.94
VPP3	−6091.23	0.59

. Before asymmetric Nash bargaining, each VPP obtained different asymmetric bargaining factors according to the size of their respective contributions to energy sharing. VPP2 has the most significant energy contribution value, has a more prominent asymmetric bargaining factor, and can allocate more cooperative benefits in Nash bargaining. VPP1 and VPP3 receive energy or provide less energy in energy sharing, get smaller asymmetric bargaining factors, and allocate less cooperative benefits in Nash bargaining. The asymmetric bargaining factor realizes a non-equal non-allocation of cooperative gains.

Table 5. Distribution of gains from asymmetric Nash bargaining.

Negotiation Modalities	VPP Number	Operating Cost (CNY)			Benefits of Cooperation (CNY)
		Before Cooperation	After Energy Sharing	After Income Distribution
Standard Nash negotiation Standard	VPP1	2963.09	−49.69	2182.11	780.98
	VPP2	−23,767.56	−14,248.90	−24,548.70	781.14
	VPP3	−148.18	−8995.37	−929.79	781.61
Asymmetric Nash negotiation	VPP1	2963.09	−49.69	2256.39	706.70
	VPP2	−23,767.56	−14,248.90	−25,025.50	1257.98
	VPP3	−148.18	−8995.37	−528.214	380.04

shows the VPPA’s operating costs and revenue distribution. The total operating cost of the VPPA is −20,952.65 CNY before the P2P transaction. Through the P2P transaction, the overall revenue of the VPPA is improved by 2341.31 CNY, and the revenue of the consortium is improved by 11.2% due to the decrease in the energy cost of the VPPs. When the standard Nash negotiation is used to distribute the revenue, the revenue distribution of each VPP is nearly equal, which is 781 CNY. However, the producer and seller identities and contribution sizes of each VPP are different, and dividing the revenue equally is unfair. It is necessary to distribute the revenue unequally. After the asymmetric allocation of the revenue through asymmetric Nash bargaining based on the contribution size, each VPP’s revenue enhancement of each VPP is 706.70 CNY, 1257.99 CNY, and 380.04 CNY, respectively. Due to differing contributions to energy sharing, VPP2’s revenue increased by 61.0%, while VPP3’s revenue decreased by 51.3%. Each VPP receives the revenue according to the size of its contribution to energy sharing through the P2P transaction.

7. Conclusions

This paper investigates and analyzes cooperative operations among VPPs. An optimization model for the cooperative operation of an alliance consisting of multiple VPPs is proposed in a multi-market environment. The paper quantifies the risky losses associated with DGs using CVaR theory. Energy is shared between VPPs through P2P transactions. Energy sharing improves the operational efficiency of VPPs and reduces energy dependence on the grid. Meanwhile, the benefits are distributed using the asymmetric Nash theory. The VPPA will distribute the cooperative revenue based on the energy contribution value. Finally, the model is solved using the ADMM algorithm.

VPPs have many types of internal resources with different characteristics, so it is difficult to explore their adjustable potential. In the process of participating in the energy market and standby capacity market, a unified model of generalized energy storage can be established by extracting its characteristic parameters, such as charging and discharging power, power level, etc., which can effectively characterize its operating characteristics and explore the flexibility of demand-side resources. Multiple VPPs establish an alliance to share energy to replace power trading with the grid. After energy sharing, the amount of power interaction with the grid is reduced by 49%, reducing the power purchase cost of the VPPs and reducing dependence on the grid. In addition, it reduces the risk of losses in the operation of the VPP. The asymmetric Nash bargaining method can better characterize the degree of contribution of each member in the alliance to the cooperative alliance. The revenue sharing ratio changes from equal sharing in standard Nash bargaining to about 3:5:2. This study provides an effective VPP cooperation strategy by examining the participation of multiple VPPs in the primary market. This also plays a positive role in improving the market mechanism and accelerating the construction of modern power systems. However, this paper only considered the uncertainty of distributed power sources and did not consider the uncertainty of generalized energy storage. The uncertainty of generalized energy storage devices and temperature can be further studied in future research.