A Non-Cooperative Game-Based Retail Pricing Model for Electricity Retailers Considering Low-Carbon Incentives and Multi-Player Competition

Abstract

This paper addresses the retail pricing problem for electricity retailers who also act as virtual power plant (VPP) operators, aggregating distributed energy resources (DERs). In future power markets where multiple such retailers compete for customers, a key challenge is to design pricing strategies that balance economic profitability with low-carbon objectives. Existing research often overlooks the impact of retailers’ heterogeneous resource portfolios, particularly the share of low-carbon resources like photovoltaics (PVs), on their competitive advantage and pricing decisions. To bridge this gap, we propose a novel retail pricing model that integrates a non-cooperative game framework with Markov Decision Processes (MDPs). The model enables each retailer to formulate optimal real-time pricing strategies by anticipating competitors’ actions and customer responses, ultimately reaching a Nash equilibrium. A distinctive feature of our approach is the incorporation of spatially differentiated carbon emission factors, which are adjusted based on each retailer’s share of PV generation. This creates a tangible low-carbon incentive, allowing retailers with greener resource mixes to leverage their environmental advantage. The proposed framework is validated on a modified IEEE 30-bus system with six competing retailers. Simulation results demonstrate that our method effectively incentivizes optimal load distribution, alleviates network congestion, and improves branch loading indices. Critically, retailers with a higher share of PV resources achieved significantly higher profits, directly translating their low-carbon advantage into economic value. Notably, the Branch Load Index (BLI) was reduced by 12% and node voltage deviations were improved by 1.32% at Bus 12, demonstrating the model’s effectiveness in integrating economic and low-carbon objectives.

1. Introduction

1.1. Background and Motivation

The liberalization of electricity markets and the proliferation of distributed energy resources (DERs) are fostering the emergence of new entities, such as virtual power plants (VPPs). These entities increasingly act as electricity retailers, procuring electricity from wholesale markets and selling it to end-use customers. A key competitive lever for these retailer VPPs is the formulation of dynamic retail pricing strategies to attract and retain customers [1,2]. As this business model matures, it is anticipated that multiple such retailer VPPs will coexist and compete within the same geographical region [3,4]. Crucially, these competitors possess heterogeneous portfolios of aggregated DERs. Their resource mix—encompassing conventional distributed generators (DGs), energy storage systems (ESS), flexible loads, and renewable sources like distributed photovoltaics (DPV)—varies significantly [5,6,7,8,9,10,11]. This heterogeneity leads to fundamental differences not only in their operational costs but also in their environmental footprint, particularly their carbon emission intensity [12]. The evolution of VPPs is driven by several key factors: the liberalization of electricity markets, which creates opportunities for new market entrants; the rapid proliferation of DERs, which provides the physical foundation for aggregation; and pressing global decarbonization goals, which incentivize the integration of renewable energy sources. Initially conceptualized primarily as aggregators of DERs for grid services, the role of VPPs is rapidly evolving. They are increasingly acting as competitive electricity retailers, actively participating in both wholesale and retail markets. This transition marks a significant shift in the energy landscape, moving from a passive support function to a proactive, market-driven role.

This context creates a complex, multi-faceted competitive landscape. While all retailer VPPs share the broad goals of profit maximization and reliable service, they compete fiercely for customers. Traditionally, price is the primary differentiator. However, the global push for decarbonization, exemplified by carbon pricing mechanisms in regions like the European Union and China, introduces a new dimension of competition. A retailer-VPP with a higher share of DPV inherently possesses a lower carbon emission factor for its electricity supply. This constitutes a potential low-carbon competitive advantage that is not adequately captured in existing market frameworks, which often rely on broad, regional-average carbon emission factors [13]. This gap leads to our core research question: How can competing retailer VPPs develop optimal retail pricing strategies that effectively leverage their unique, spatially differentiated low-carbon attributes to maximize profitability, while also contributing to the stability and efficiency of the distribution network?

While prior research has made significant strides in incorporating low-carbon objectives into VPP operations, a specific gap remains when these objectives are applied to a competitive retail pricing context. Existing low-carbon VPP models typically operate in isolation or under coordination with the DSO [14], and often rely on regional-average carbon emission factors. Consequently, they fail to capture the strategic interactions among multiple retailer VPPs and cannot translate a VPP’s unique, spatially differentiated low-carbon advantage (stemming from its specific resource portfolio) into a competitive edge in retail pricing. In other words, prior work has not modeled the tripartite interaction among (a) non-cooperative multi-retailer competition, (b) spatially differentiated carbon accounting, and (c) dynamic retail pricing strategy formulation within a unified game-theoretic framework.

1.2. Literature Review

1.2.1. VPP Operations and Coordination with DSO

Regarding the impact of internal DER dispatching by VPPs on retail electricity pricing, this area has drawn increasing academic interest in recent years. By designing electricity retail rates, VPPs aim to attract end-users to participate in their electricity trading and dispatching, thereby improving the overall economic efficiency and low carbon emissions of the distribution system [15]. For instance, reference [16] designed a hierarchical deep reinforcement learning Hierarchical-TD3 algorithm to realize the online coordinated scheduling of VPPs and distribution networks. Reference [17] proposed an energy-reserve coordinated optimization model led by the distribution system operator that involves the participation of both multi-VPPs and shared energy storage. Reference [18] proposed hierarchical robust day-ahead coordination of a VPP and distribution system operator based on the local market to improve the voltage stability of the distribution network by considering the uncertainty of DERs. Reference [19] formulated a novel virtual power plant framework based on the unique resource endowment of rural areas. Reference [20] constructed a novel virtual power plant system incorporating waste incineration plants and electric waste trucks. Reference [21] hypothesized that a residential area could form a VPP capable of providing power-balancing services to a national power system. Their simulation results demonstrated that all power surpluses within the system were effectively utilized by the integrated heat pumps. However, much of the existing research primarily focuses on the economic aspect of transaction mechanisms, often overlooking other critical factors such as carbon emissions. Despite the fruitful outcomes achieved in multi-VPP regulation strategies and profit distribution, the main objective lies in maximizing profits for an independent VPP, often neglecting changes in regional grid load characteristic indicators during the optimization process. When each VPP aims to maximize profits, it can lead to deterioration in regional load characteristic indicators.

1.2.2. Incorporating Low-Carbon Considerations

Furthermore, although the aforementioned solutions are economically viable, they may not be environmentally friendly and might even violate carbon emission limits. Many countries and regions, such as the European Union and China, have implemented stringent policies and regulations regarding carbon emissions, requiring relevant entities within the electric power system to ensure sufficient carbon allowances for power generation and consumption activities. Scholars have attempted to incorporate carbon emissions into VPP regulation models [22]. For example, Reference [23] proposed the concept of a cleanness value of distributed energy storage and constructed an optimal low-carbon dispatch method for a VPP with aggregated distributed energy storage, wherein energy value and cleanness value were both considered. Reference [24] proposed a low-carbon economic operation scheduling optimization method for VPPs considering multiple heterogeneous resource aggregation. Reference [25] proposed a low-carbon operation mode of a VPP considering emission arbitrage. Reference [26] proposed an optimal allocation method for a low-carbon economy for VPPs involving a Load Aggregator (LA) joined by EV. Reference [27] focused on the information asymmetry between VPPs and clean Power to Hydrogen (P2H) and proposed a decentralized coordinated operation method of a VPP-P2H combined system considering the profits of each player. Reference [28] investigated the low-carbon capacity of integrated hydrogen energy utilization within a system that includes natural gas and hydrogen doping. Reference [29] proposed a low-carbon operation strategy incorporating a progressive demand response mechanism. The study concluded that this approach not only enhances the feasibility and effectiveness of market supervision strategies but also offers valuable insights for strategy selection by both system operators and demand-side users. Reference [30] introduced a low-carbon-oriented tri-stage collaborative energy management framework. The efficacy of the proposed method was validated through extensive experiments based on the case of the Guangdong electricity market. Reference [31] focused on a specific design for park-level integrated energy systems with near-zero emissions. However, these studies often rely on regional-level carbon emission factors, which fail to capture the spatial heterogeneity introduced by distributed renewable generation, and lack a corresponding retail electricity pricing method that can reflect the environmental characteristics of individual VPPs.

1.2.3. Pricing Strategies for VPPs

Some research has been conducted to investigate VPP pricing strategies. For instance, ref. [32] developed a Deep Deterministic Policy Gradient (DDPG) algorithm relying on prioritized experience replay to formulate a real-time electricity price plan. Reference [33] formulated a non-cooperative pricing game modeling the competitive behaviors between the two VPPs. Reference [34] pointed out that a VPP operator will be able to optimize their procurement expenditures by incentivizing flexible demand in proportion to different electricity tariffs. Reference [35] introduced a framework that optimizes bidding strategies and maximizes a VPP’s profit on day-ahead and real-time bases by pricing. Summarizing the aforementioned methods, the pricing strategy of VPPs for retail users mainly includes game theory-based models and Reinforcement Learning (RL). Game theory-based models are suitable for solving multi-player decision-making problems but have limited capability in handling dynamic and uncertain environments. In contrast, RL based on Markov Decision Processes (MDPs) is an effective approach for decision-making under conditions of uncertainty. The MDP framework’s advantages include adaptability to dynamic environments, holistic consideration of temporal dynamics, and reduced reliance on historical data. However, the application of RL also faces challenges, such as the need for large samples for training and limited generalization capabilities. Therefore, how to leverage the strengths of existing methods while overcoming their weaknesses to address the VPP pricing issue remains an unexplored research area. Given this, we propose a non-cooperative game algorithm based on the MDP framework to model the pricing decision problem in scenarios involving multiple VPPs. This approach allows each VPP to make optimal pricing decisions considering potential responses from competitors, thereby reaching a Nash equilibrium.

1.3. Contributions

To summarize, while significant progress has been made in multi-VPP management models, several research gaps still exist, particularly concerning the integrated economic–environmental optimization of retail pricing and its impact on distribution network operations. To fill these gaps, this paper proposes an MDP-based non-cooperative game VPP retail pricing model and constructs a collaborative energy management framework aimed at scenarios where multiple VPPs coexist. The contributions can be summarized as follows:

(1)

We develop an MDP-based non-cooperative game model for electricity retailers. This model integrates a User Equilibrium (UE) model to capture price-sensitive customer choices, enabling retailers to dynamically optimize their retail prices in a competitive environment while considering the operational constraints and costs of their internal DERs.

(2)

We design and incorporate a low-carbon incentive mechanism into the pricing model. This mechanism utilizes spatially differentiated carbon emission factors, which are dynamically adjusted for each retailer based on the real-time proportion of its PV generation to total electricity sales. This directly quantifies and monetizes the low-carbon attributes of a retailer’s energy mix, transforming it into a competitive edge within the pricing game.

(3)

Through case studies on a modified IEEE 30-bus system, we demonstrate the practical efficacy of the proposed framework. The results validate that our model not only ensures profitability for retailers but also contributes to the technical security of the distribution network by alleviating congestion and mitigating voltage violations.

1.4. Structure of the Paper

The concrete framework of this paper is organized as follows: Section 2 mainly introduces the framework. Section 3 provides the technical details of the proposed research. Section 4 conducts case studies to verify the effectiveness of the proposed research. Section 5 summarizes this research and highlights the conclusions.

2. Problem Analysis

2.1. Multi-VPP Architecture in Electricity and Carbon Markets

To serve the power system dispatch of multi-VPP access, a dispatching architecture associated with the VPP serving entities is proposed. The architecture of the VPPs is designed to operate within wholesale energy and retail transactions, taking into account carbon emission controls and penalties. Six types of entities are involved, including (1) Distribution Network Operators (DNOs); (2) Carbon Market Operators (CMOs); (3) virtual power plant operators (VPPOs); (4) VPPs; (5) distributed energy resources (DERs); and (6) customers. The architecture of the coupled networks with the flow of energy, information, and money is shown as Figure 1.

The energy flow reflects the energy from the network to end-users via different entities, which is the physical aspect of the price-making model. The money flow describes the cost or revenue of each entity, which is the part of the business model that supports the interactions among these five entities and maintains the pricing scheme operating economically and sustainably. The information flow represents the key parameters exchanged between each entity in the information network, which is the essential element of the VPP pricing model.

2.2. Connections and Business Model of Related Entities

Virtual power plant operators (VPPO) connect Distribution Network (DN) Operators, Carbon Market (CM) Operators, VPPs, and customers through an information network, creating complex couplings among the power grid, users, regulation resources, and carbon emission systems. To ensure a VPP’s pricing strategy maintains a smooth flow across financial, energy, and information chains, an analysis of relevant stakeholders was conducted, leading to the proposal of a novel business model. As shown by Arrow 1, the DNO transmits electricity generated from centralized power plants through the distribution network to the VPP while adhering to constraints such as distribution network line power flow and voltage limits. The VPP, acting as a price taker, pays the DNO corresponding fees based on the uniform marginal clearing price of the wholesale electricity market, as indicated by Arrow 2.

The VPPO, serving as the VPP’s manager, is responsible for standardizing grid connection testing for VPP access and manages all the operational information of the VPP. It reports power-related information to DNOs while simultaneously sharing the uniform marginal clearing price of the wholesale electricity market and grid regulation requirements (e.g., line congestion, node voltage violations) with VPPs, as shown by Arrow 3. It is important to note that, according to technical access requirements or regulatory mandates, multi-VPP systems must have a power system operator or government-mandated central management platform—the VPPO.

The VPPO provides VPPs with critical parameters including distribution network load forecasts, predictions of the uniform marginal clearing price, and customer price preferences. Upon receiving these parameters, each VPP formulates optimal retail pricing strategies and DER (distributed energy resource) regulation strategies based on its aggregated resources and available information. These strategies, along with DER regulation results, are reported to the VPPO, as shown by Arrow 4.

Based on the retail electricity prices set by various VPPs (Arrow 6), customers select their preferred VPP as their electricity supplier while paying for their consumed electricity (Arrow 5). VPPs develop DER regulation strategies (Arrow 11) considering contracted customers’ forecasted load curves, their own DERs’ (including photovoltaics (PVs), Distributed Generation (DG), energy storage system (ESS), and flexible loads) operational costs and availability, and compensate DER participants accordingly (Arrow 10).

Given the growing number of carbon market transactions internationally, user electricity consumption has become a major source of power sector emissions. Anticipating potential inclusion of users in carbon compliance frameworks, this model envisions VPPOs participating in carbon markets as an aggregated entity. CMOs issue uniform carbon emission factors (CEFs) to VPPs. VPPOs report renewable generation data (primarily from DPV and wind) and adjusted carbon emission factors to CMOs (Arrow 7). Customers derive their electricity’s low-carbon profile based on their chosen VPP’s emission factors, while CMOs can provide low-carbon consumption recommendations and certifications (Arrow 9).

Customers can access a VPPO’s aggregated information on VPP pricing and low-carbon metrics to aid decision-making, and VPPOs also offer value-added services like energy consumption advice (Arrow 8). This integrated framework enables multi-dimensional coordination across energy, finance, and carbon emission management systems.

2.3. Carbon Emission Factor

From an electricity generation standpoint, it is evident that a significant portion of carbon emissions stems from the generator sector due to fossil fuel combustion, while minimal emissions originate from the transmission and consumption sectors. However, electricity generation is fundamentally driven by consumer demand. Therefore, accurately quantifying carbon emissions at each stage, spanning from generation to consumption, becomes crucial.

The average carbon emission factor (CEF) represents the carbon emissions per kilowatt-hour (kWh) of electricity consumed by the load, calculated by converting the emissions associated with the power source unit commitment and actual output. This implies that, without considering the carbon emissions from resources internal to a virtual power plant (VPP), the carbon emission per unit of electricity consumption is considered consistent across different VPPs. By tracking and quantifying the carbon emissions from all source-side generators within the power system and integrating this data with power system flow analysis, Branch CEFs and Node CEFs (expressed in kgCO₂/kWh) can be computed, as detailed in [36]. While the methodology for calculating CEFs is not the primary focus of this paper, it is important to note that the current spatial resolution of CEF calculations is limited to the 10 kV voltage level. Consequently, the average CEF cannot effectively guide low-carbon scheduling of resources within a VPP.

This paper focuses on how a virtual power plant operator (VPPO), after obtaining the unified average CEF, can determine a more refined, corrected carbon emission factor based on the total electricity consumption and the proportion of green electricity generated by distributed resources within the VPP. This corrected CEF is then applied to enable low-carbon scheduling.

3. Methodology and Mathematical Models

This paper considers VPPs as electricity retailers who purchase power from wholesale markets and determine retail electricity prices for end-users. First, VPPs need to optimize the formulation of retail electricity prices by comprehensively considering factors such as wholesale electricity procurement costs, carbon market emission costs, and DER control costs in order to attract electric consumers and enhance sales profits. Second, it is necessary to optimize the control of various types of DERs and adjust the amount of electricity purchased from the wholesale market to reduce VPPs’ electricity selling costs. The framework of the proposed methodology is shown in Figure 2, with the work of each step summarized as follows:

Step 1: The control strategy of a single VPP considers distribution network power flow and voltage constraints. Assuming the controllable resources within VPP include DG units (i.e., diesel units, gas turbines), ESS, controllable loads (i.e., load shifting and load curtailment via demand response), and RPCE, the control model of a single VPP is established by incorporating resource control constraints and distribution network control constraints, aiming to achieve optimal economic efficiency and low-carbon performance. This serves as the foundation for competitive game analysis among multiple VPPs.

Step 2: Carbon emission factor correction. Based on the proportion of renewable energy generation in the total electricity output of each VPP, the uniform carbon emission factor at the distribution network level is adjusted to refine the spatial granularity of carbon emission accounting for each VPP. This correction also provides guidance for electricity consumers in selecting retail tariff packages.

Step 3: Non-cooperative game-based VPP pricing model using MDP. Based on the control model and carbon emission model established in Step 1 and Step 2, the pool state model, pool state transition probability, and VPPs’ utility are constructed. Through non-cooperative game theory, the optimal retail electricity prices for multiple VPPs are derived.

3.1. Dispatching Model for a Single VPP

3.1.1. Objective Function

The objective of a single VPP is to maximize its operational profit by optimizing electricity retail prices while minimizing costs through purchasing electricity at the right time on the wholesale market and coordinated control of internal DERs. The profit of a single VPP is defined as (1) and (2).

$$F_i(t)=\max \left(\underset{R_i^{retail}(t)}{\underbrace{{\lambda }_i^{retail}(t)r_i^{retail}(t)}}-\underset{C_i^{wholesale}(t)}{\underbrace{{\lambda }^{wholesale}(t)p_i^{grid}(t)}}-C_i^{DER}(t)-C_i^{CEF}(t)\right)$$

(1)

$$C_i^{DER}(t)=C_i^{DG}(t)+C_i^{PV}(t)+C_i^{FL}(t)+C_i^{ESS}(t)$$

(2)

where $F_i(t)$ is the utility function for the $i$ th VPP, USD. $R_i^{retail}(t)$ is the VPP’s revenue as an electricity retailer, which is equal to the product of the electricity selling price ${\lambda }_i^{retail}(t)$ set by the VPP and the amount of electricity $r_i^{retail}(t)$ sold at each time point, kW. The electricity selling price ${\lambda }_i^{retail}(t)$ is the output of the non-cooperative game model in Section 3.3, USD/kW. $C_i^{wholesale}(t)$ is the cost of purchasing electricity in wholesale markets, which is equal to the product of wholesale market electricity prices and the electricity purchased by the VPP from the wholesale market, USD. ${\lambda }^{wholesale}(t)$ is the wholesale electricity purchase price for VPPs, which is the same for all VPPs, USD/kW. $p_i^{grid}(t)$ is the wholesale electricity purchase volume for the VPP $i$, kW. $C_i^{DER}(t)$ is the sum of various DER adjustment costs, including DG, PV, FL, and ESS, USD. $C_i^{CEF}(t)$ is the carbon emission cost of the VPP, which is equal to the product of the carbon emission factor, total electricity consumption, and carbon market price, USD.

3.1.2. Constraints

The constraints mainly include the overall power balance of the VPP and the operational constraints of each DER, including generation-type, load-type, and storage-type DERs.

• Overall power balance of the VPP

The VPP’s overall power balance should be satisfied, which requires that, for each VPP, the retail electricity sales equal the sum of the electricity purchased from the wholesale market and the adjusted output from DERs, as specified in (3).

$$r_i^{retail}(t)=p_i^{grid}(t)+p_i^{DG}(t)+p_i^{PV}(t)+p_i^{LS}(t)-p_i^{lsto}(t)+p_i^{LC}(t)+p_i^{dis}(t)-p_i^{chr}(t)$$

(3)

where $p_i^{DG}(t)$ and $p_i^{PV}(t)$ are the generation power of DG and PV in VPP i, respectively, kW. $p_i^{LS}(t)$ and $p_i^{lsto}(t)$ represent the load shift-out power and shift-in power of shiftable flexible load (LS-FL) at time t, respectively, kW. These two states are mutually exclusive at any given time t. $p_i^{LC}(t)$ denotes the load curtailment of curtailable flexible load (LC-FL) at time t, kW. $p_i^{dis}(t)$ and $p_i^{chr}(t)$ indicate the discharge power and charge power of ESS at time t, respectively, kW. ESS operates in only one mode at a time (charge or discharge), making these two variables mutually exclusive at time t.

2.

Generation-type resources

Distributed Generators: The output constraints of the distributed generators can be given by (4)–(7). Equation (4) indicates the unit output limit. Equation (5) indicates the unit ramp rate limit. The cost of the distributed generators can be formulated by (6) and (7).

$$P^{DG,\min }\le p_i^{DG}(t)\le P^{DG,\max }$$

(4)

$$-R^D\cdot \mathsf{\Delta }t\le p_i^{DG}(t)-p_i^{DG}(t-1)\le R^U\cdot \mathsf{\Delta }t$$

(5)

$$C_i^{DG}(t)={\alpha }_i^{DG}(t)C^{DG,SU}+{\beta }_i^{DG}(t)f^{DG}\left(p_i^{DG}\right)+{\gamma }_i^{DG}(t)C^{DG,SD}$$

(6)

$$C_i^{DG}(t)={\alpha }_i^{DG}(t)C^{DG,SU}+{\beta }_i^{DG}(t)f^{DG}\left(p_i^{DG}\right)+{\gamma }_i^{DG}(t)C^{DG,SD}$$

(7)

where $P^{DG,\min }$ and $P^{DG,\max }$ are the minimum and maximum generation power of the DG, respectively, kW. $R^D$ and $R^U$ represent the maximum downward and upward ramp rates, respectively, kW/h. ${\alpha }_i^{DG}(t)$, ${\beta }_i^{DG}(t)$, and ${\gamma }_i^{DG}(t)$ are binary variables indicating the startup, operation, and shutdown states of the DG at time t, respectively. $C^{DG,SU}$ and $C^{DG,SD}$ denote the startup cost and shutdown costs of the DG, respectively. $f^{DG}\left(p_i^{DG}\right)$ is the operational cost function, which is a quadratic function of the generation output at current time step t. $a^{DG}$, $b^{DG}$, and $c^{DG}$ are the cost coefficients of the DG’s operational cost function.

Distributed Photovoltaics: The forecasted DPV power and the cost of abandoned power which was not fully consumed are mainly considered in the regulation model. The forecasted PV output is taken as the maximum dispatchable output in (8), and the cost of PVs is shown in (9).

$$0\le p_i^{PV}(t)\le P_i^{PV,\max }(t)$$

(8)

$$C_i^{PV}(t)=c^{WPV}(P_i^{PV,\max }(t)-p_i^{PV}(t))$$

(9)

where $P_i^{PV,\max }(t)$ is the maximum available power of DPVs in VPP i at time t, kW. $c^{WPV}$ denotes the curtailment cost coefficient for DPV, USD.

3.

Load-type resources

The flexible load of customers can be dispatched by participating in demand response (DR) implemented by VPP and can obtain corresponding compensation. Before explaining the mechanism of dispatching loads in detail, it is necessary to introduce the concept of Customer Baseline Load (CBL) [37]. The CBL means that if the customers do not participate in the DR program, the electricity that they would have consumed. For example, when the power system is suffering from a power imbalance, the VPP will respond to the instruction from the DSO and dispatch the customers’ flexible load via load shifting (LS) or load curtailment (LC) contracts [38].

Load shifting (LS): The VPP can shift the customers’ flexible load from one timeslot to another timeslot with the predefined quantity in the LS quantity. The dispatching process of LS load and relevant constraints can be formulated by (10)–(13).

$$\left\{\begin{array}{l}{p}_{i,c}^{\mathrm{base}}(t)\le p_{i,c}^{lsto}(t)\le p_{i,c}^{\max }(t),u_{i,c}^{\mathrm{LS}}(t)=0,\forall c\in {\mathit{C}}_i^{LS}\\ p_{i,c}^{\mathrm{LS}}(t)=p_{i,c}^{\mathrm{base}}(t)-q_{i,c}^{\mathrm{LS}}(t),u_{i,c}^{\mathrm{LS}}(t)=1,\forall c\in {\mathit{C}}_i^{LS}\end{array}\right.$$

(10)

$${\displaystyle \sum _{t\in \mathit{T}}{u}_{i,c}^{\mathrm{LS}}(t)=N_{i,c,\max }^{\mathrm{LS}}},\forall c\in {\mathit{C}}_i^{LS}$$

(11)

$${\displaystyle \sum _{t\in \mathrm{T}}{p}_{i,c}^{\mathrm{LS}}(t)={\displaystyle \sum _{t\in T}{p}_{i,c}^{\mathrm{base}}(t)}},\forall c\in {\mathit{C}}_i^{LS}$$

(12)

$$p_i^{LS}(t)={\displaystyle \sum _{c\in {\mathit{C}}_i^{LS}}{p}_{i,c}^{\mathrm{LS}}(t)},p_i^{lsto}(t)={\displaystyle \sum _{c\in {\mathit{C}}_i^{LS}}{p}_{i,c}^{lsto}(t)}$$

(13)

where ${\mathit{C}}_i^{LS}$ is the set of shiftable flexible loads (LS-FL) in VPP $i$. $p_{i,c}^{\mathrm{base}}(t)$ and $p_{i,c}^{\max }(t)$ are the baseline load and maximum load of the $c$-th customer, respectively, kW. $p_{i,c}^{\mathrm{LS}}(t)$ and $p_{i,c}^{lsto}(t)$ denote the load shift-out power and shift-in power of the $c$-th LS-FL user in VPP $i$ at time $t$, respectively, kW. These two states are mutually exclusive at any given time $t$. $q_{i,c}^{\mathrm{LS}}(t)$ is the net load shift amount of the $c$-th LS-FL user in VPP $i$ at time $t$. $u_{i,c}^{\mathrm{LS}}(t)$ is a binary variable indicating whether load shifting is activated for the $c$-th LS-FL user in VPP $i$ at time $t$. $N_{i,c,\max }^{\mathrm{LS}}$ is the maximum number of time slots allowed for load shifting by the $c$-th LS-FL user in VPP $i$.

Load Curtailment (LC): During the DR period, the VPP can reduce the load of the customers under the predefined quantity set in the LC contract. For the other periods, the customers’ flexible load is equal to the CBL. The customer can turn off unnecessary devices to reduce the power. The mathematical model of the LC contract is presented by (14)–(16).

$$\left\{\begin{array}{l}{p}_{i,c}^{\mathrm{LC}}(t)=p_{i,c}^{\mathrm{base}}(t),\forall c\in {\mathit{C}}_i^{LC},\forall t\in \mathit{T}\backslash {\mathit{T}}_{\mathit{w}}\\ p_{i,c}^{\mathrm{LC}}(t)=p_{i,c}^{\mathrm{base}}(t)-F_{i,c}^{\mathrm{LC}}(t),\forall c\in {\mathit{C}}_i^{LC},\forall t\in {\mathit{T}}_{\mathit{w}}\\ p_{i,c}^{\min }(t)\le p_{i,c}^{\mathrm{LC}}(t)\le p_{i,c}^{\mathrm{base}}(t),\forall c\in {\mathit{C}}_i^{LC},\forall t\in \mathit{T}\end{array}\right.$$

(14)

$$\left\{\begin{array}{l}{F}_{i,c}^{\mathrm{LC}}(t)=0,u_{i,c}^{\mathrm{LC}}(t)=0,\forall c\in {\mathit{C}}_i^{LC},\forall t\in {\mathit{T}}_{\mathit{w}}\\ F_{i,c}^{\mathrm{LC}}(t)\in \{q_{i,c}^{\mathrm{LC},\mathrm{deg}1},q_{i,c}^{\mathrm{LC},\mathrm{deg}2},q_{i,c}^{\mathrm{LC},\mathrm{deg}3}\},u_{i,c}^{\mathrm{LC}}(t)=1,\forall c\in {\mathit{C}}_i^{LC},\forall t\in {\mathit{T}}_{\mathit{w}}\end{array}\right.$$

(15)

$$p_t^{\mathrm{LC}}(t)={\displaystyle \sum _{c\in {\mathit{C}}_i^{LC}}{p}_{i,c}^{\mathrm{LC}}(t)},\forall t\in \mathit{T}$$

(16)

where ${\mathit{C}}_i^{LC}$ is the set of curtailable flexible load (LC-FL) in VPP $i$. ${\mathit{T}}_{\mathit{w}}$ is the set of time slots during which load curtailment is permitted. $p_{i,c}^{\min }(t)$ is the minimum load of the $c$-th customer, kW. $p_{i,c}^{\mathrm{LC}}(t)$ denotes the load curtailment power of the $c$-th LC-FL user in VPP $i$ at time $t$, kW. $F_{i,c}^{\mathrm{LC}}(t)$ represents the actual curtailed load amount, kW. This paper considers residential air conditioning as the primary source for load curtailment, with discrete reduction levels corresponding to temperature adjustments of 1 °C, 2 °C, and 3 °C, quantified by parameters $q_{i,c}^{\mathrm{LC},\mathrm{deg}1}$ (kW), $q_{i,c}^{\mathrm{LC},\mathrm{deg}2}$ (kW), and $q_{i,c}^{\mathrm{LC},\mathrm{deg}3}$ (kW), respectively.

After the DR, the VPP will compensate the customers according to their performance. In this paper, the compensation is modeled as (17) and (18). The compensation to the customers is the main source of costs in the scheduling process.

$${\mathsf{\Pi }}_c^{comp}(t)=(1+{\displaystyle \frac{|p_{i,c}^{\mathrm{base}}(t)-p_{i,c}^{\mathsf{\Omega }}(t)|}{p_{i,c}^{\mathrm{base}}(t)}})\cdot {\mathsf{\Pi }}^{average},\forall c\in {\mathit{C}}_i^{\mathsf{\Omega }},\mathsf{\Omega }\in \{LS,LC\}$$

(17)

$$C_i^{FL}(t)={\displaystyle \sum _{c\in {\mathit{C}}_i^{LS}}{\mathsf{\Pi }}_c^{comp}(t)\cdot q_{i,c}^{LS}(t)}+{\displaystyle \sum _{c\in {\mathit{C}}_i^{LC}}{\mathsf{\Pi }}_c^{comp}(t)\cdot F_{i,c}^{\mathrm{LC}}(t)}$$

(18)

where ${\mathsf{\Pi }}^{average}$ is the traditional compensation coefficient for DR, typically uniform across all customers, USD. This paper uses a contribution based compensation coefficient to modify the traditional compensation coefficient, that is, the larger the change compared to baseline load, the larger the compensation coefficient, in order to motivate users to actively respond to demand response. ${\mathsf{\Pi }}_c^{comp}(t)$ denotes the modified compensation coefficient, USD.

4.

Storage-type resources

The key operation features of the ESS can be concluded by the state of charge (SOC), the rated power, the efficiency, and the degradation rate. The SOC can reflect the percentage of the remaining capacity of the ESS. Because of the degradation, the ESS shouldn’t be charged or discharged at a high frequency so the number of charge or discharge times should be limited. The process of charging and discharging can be formulated by (19)–(23).

$$e_i^{ESS}(t)=e_i^{ESS}(t-1)+{\eta }_i^{chr}{p}_i^{chr}(t)\mathsf{\Delta }t-{\displaystyle \frac{1}{{\eta }_i^{dis}}}{p}_i^{dis}(t)\mathsf{\Delta }t$$

(19)

$$0\le e_i^{ESS}(t)\le E_i^{ESS,rated}$$

(20)

$$0\le p_i^{dis}(t)\le p_i^{dis,\max }$$

(21)

$$0\le p_i^{chr}(t)\le p_i^{chr,\max }$$

(22)

$$C_i^{ESS}(t)={\kappa }_i^{\mathrm{deg}}(|p_i^{chr}(t)|+|p_i^{dis}(t)|)\mathsf{\Delta }t$$

(23)

where $e_i^{ESS}(t)$ represents the actual energy level of the ESS at time t, kWh. ${\eta }^{chr}$ and ${\eta }^{dis}$ denote the charging efficiency and discharging efficiency, respectively. $p_i^{chr,\max }$ and $p_i^{dis,\max }$ are the maximum charging power and maximum discharging power, respectively, kW. ${\kappa }_i^{\mathrm{deg}}$ is the cycle degradation cost coefficient.

3.2. Carbon Emission Factor Correction

To refine the spatial granularity of CEF accounting for VPPs, this study proposes a dynamic correction mechanism for the initial uniform CEF. The correction adjusts the CEF for each VPP based on its renewable energy generation proportion and total electricity sales, ensuring that the total system-wide carbon emissions remain constant. This approach promotes fairness in carbon accounting and incentivizes VPPs to integrate more renewable energy. While total emissions remain constant (zero-sum redistribution), the mechanism creates real competitive incentives through relative cost differentials among VPPs. The CMO allows this internal adjustment to leverage the VPPO’s local information advantages while maintaining control over the aggregate carbon budget. The specific procedure for this correction and its application is illustrated in Figure 3.

The VPPO first receives the uniform CEF issued by the CMO. Subsequently, the CEF for each VPP is adjusted based on the proportion of DPV generation to total electricity sales at each time period for each VPP, as shown in (24a), $k\in \left\{PV,wind,hydro\right\}$ represents the set of renewable energy types integrated into VPP $i$ The equivalent renewable energy generation is calculated by weighting each renewable energy type according to its life cycle carbon performance, as shown in Equation (24b), $P_i^k(t)$ is the actual generation from renewable energy type $k$; The weight coefficient $w_k$ for each renewable energy type is determined based on its LCA carbon intensity using a tiered approach defined in Equation (24c). This tiered structure reflects the climate targets outlined in IPCC assessments, where electricity systems must achieve carbon intensities below 15 gCO₂-eq/kWh by 2050 to meet 2 °C warming goals [39,40].

At the same time, it is required that the total carbon emissions within the VPPO’s jurisdiction remain unchanged before and after the correction, as expressed in (25).

$$c_i^{adj}(t)=\left\{\begin{array}{l}\beta (t)\cdot c^0(t)\cdot (1-{\displaystyle \frac{P_i^{RE,weighted}{p}_i^{PV}(t)}{r_i^{retail}(t)}}),r_i^{retail}(t)\ge 0\\ c^0(t),r_i^{retail}(t)=0\end{array}\right.$$

(24a)

$$P_i^{RE,weighted}(t)={\displaystyle \sum _{k\in \left\{PV,wind,hydro\right\}}{w}_k\cdot P_i^k(t)}$$

(24b)

$$\left\{\begin{array}{c}{\omega }_k=1.0,LCA<10 \mathrm{g}{\mathrm{CO}}_2\text{-}\mathrm{eq}/\mathrm{kWh}\\ {\omega }_k=0.9,10\le LCA<30 \mathrm{g}{\mathrm{CO}}_2\text{-}\mathrm{eq}/\mathrm{kWh}\\ {\omega }_k=0.7,30\le LCA<100 \mathrm{g}{\mathrm{CO}}_2\text{-}\mathrm{eq}/\mathrm{kWh}\\ {\omega }_k=0.5,100\le LCA<200 \mathrm{g}{\mathrm{CO}}_2\text{-}\mathrm{eq}/\mathrm{kWh}\end{array}\right.$$

(24c)

$${\displaystyle \sum _{i,t}(c_i^{adj}(t)\cdot r_i^{retail}(t))}={\displaystyle \sum _{i,t}(c^0(t)\cdot r_i^0(t))}$$

(25)

$$C_i^{CEF}(t)={\lambda }^{CEF}(t)\cdot c_i^{adj}(t)\cdot r_i^{retail}(t)$$

(26)

where $c^0(t)$ is the initial unified CEF, kgCO₂/kWh. $c_i^{adj}(t)$ is the revised CEF, kgCO₂/kWh. A correction factor $\beta (t)$ is introduced to ensure the feasibility of the solution. ${\lambda }^{CEF}(t)$ is the price of carbon emissions in the carbon market, USD/kgCO₂.

To enhance the model’s general applicability, we have conducted an analysis of its extensibility to multiple renewable energy types. While this study focuses on distributed PV generation—the most prevalent renewable technology in user-side VPPs—the proposed CEF correction framework is designed to accommodate multiple renewable energy types. As shown in Figure 3, when VPPs integrate distributed wind, small hydropower, or other renewable sources, the calculation logic remains consistent across all renewable energy types: each technology contributes to the CEF reduction proportionally to both its generation output and its life cycle carbon performance.

The framework’s modular design requires only two steps for integration of additional renewable technologies:

• Adding the corresponding generation term $P_i^k(t)$ to the summation in Equation (24b).
• Calibrating the weight coefficient $w_k$ based on technology-specific LCA data.

This ensures seamless scalability to emerging distributed energy technologies. For example, distributed wind turbines with LCA values of 13.5–225 gCO₂/kWh [41,42] would receive ${\omega }_{wind}=0.85-0.95$, while small run-of-river hydropower achieving 10–20 gCO₂/kWh [43] would receive ${\omega }_{hydro}=0.95$. Small reservoir-based hydropower with significantly higher carbon intensity (200–500 gCO₂/kWh) [44] would receive ${\omega }_{hydro}=0.60-0.75$.

The current focus on distributed PV is justified by several factors beyond its high penetration and predictability. First, our research specifically examines demand-side virtual power plants, where distributed PV accounts for over 70% of distributed renewable capacity in many urban and commercial settings, while distributed wind and small hydropower have substantially lower penetration rates in user-side applications. Second, distributed PV installations benefit from well-established monitoring infrastructure, standardized performance metrics, and consistent LCA estimates (6–48 gCO₂/kWh range) [45]. In contrast, small wind and hydropower exhibit highly site-specific performance with capacity factors varying dramatically, greater measurement uncertainties, and wider LCA variability (10–500+ gCO₂/kWh range) [46]. Third, by validating the core CEF correction mechanism with the most reliable and well-characterized technology first, we establish a solid methodological foundation before extending to more complex or variable cases, which aligns with established practices in energy systems modeling. Future extensions will integrate distributed wind data from pilot projects (Phase 1), incorporate small run-of-river hydropower suitable for user-side applications (Phase 2), and develop a comprehensive regional LCA database for dynamic weight coefficient calibration (Phase 3).

3.3. Non-Cooperative Game Between VPPs Based on MDP

Assuming that VPPs are behavioral rational, the pricing mechanism of VPP can change the trading behavior of customers, thus affecting the branch power flow and node voltage of distribution network. The schematic diagram of the game model is shown in Figure 4.

3.3.1. VPP Behavioral Modeling

The selling price matrix of VPPs is ${\mathbf{\lambda }}^{\mathit{retail}}=[{\mathbf{\lambda }}_1,{\mathbf{\lambda }}_2,\cdots ,{\mathbf{\lambda }}_{\mathit{N}}]$, where ${\mathbf{\lambda }}_{\mathit{i}}$ is the price vector of the VPP $i\in [1,\cdots ,N]$. Assuming that there are $K$ customers, the load demand matrix is $\mathbf{R}=[{\mathbf{r}}_1,{\mathbf{r}}_2,\cdots ,{\mathbf{r}}_K]$, where $r_k={[r_k(1),r_k(2),\cdots ,r_k(T)]}^T$ is the load demand vector of customer $k\in [1,\cdots ,K]$, and $r_k(t)$ is the power consumption of the customer in period $t\in [1,\cdots ,\mathrm{T}]$. Each customer has a budget constraint $b_k$, as shown in (27)

$$c_k={\displaystyle \sum _{t\in \mathit{T}}{\displaystyle \sum _{i\in \mathit{N}}{r}_k(t)}{\lambda }_i^{retail}(t)}\le b_k,\forall k\in [1,\cdots ,K]$$

(27)

Customers can only choose to sign an agreement with one VPP within a cycle. Customers compare the prices of all VPP and calculate the utility obtained, and then decide which VPP provides the power providing service. The discrete choice model (DCM) is used to describe the choice behavior of customers, and the probability of customers choosing an VPP is obtained according to the utility maximization criterion. Specifically, this paper describes the customer selection behavior based on the Multi-nominal Logit model. The utility of customer $k$ choosing VPP $i$ is shown in (28)–(30).

$$u_{ki}={\alpha }_k-c_{ki}+{\eta }_{ki}={\upsilon }_{ki}+{\eta }_{ki}$$

(28)

$${\alpha }_k={\displaystyle \sum _{t\in \mathit{T}}{r}_k(t){\gamma }_k(t)}$$

(29)

$$c_{ki}={\displaystyle \sum _{t\in \mathit{T}}{r}_k(t){\lambda }_i^{retail}(t)}$$

(30)

where ${\gamma }_k(t)$ is the unit utility for using electricity at time $t$. ${\alpha }_k$ is the total utility of customer $k$, and $c_{ki}$ is the power consumption cost of customer $k$. ${\upsilon }_{ki}$ is observable utility, ${\eta }_{ki}$ is the preference of customer $k$ for VPP $i$, determined by the TAP-UE model [47].

Assume that variable ${\eta }_{ki}$ satisfies independent and identically distributed (i.i.d.), obeys Gumbel distribution (i.e., extremum type I distribution), the probability density function of ${\eta }_{ki}$ is shown in (31), and the cumulative probability density distribution is shown in (32).

$$\mathrm{f}({\eta }_{ki})=e^{-{\eta }_{ki}}{e}^{e^{-{\eta }_{ki}}}$$

(31)

$$\mathrm{F}({\eta }_{ki})=e^{e^{-{\eta }_{ki}}}$$

(32)

The probability of customer k choosing VPP i is shown in (33).

$$\begin{array}{ll}{P}_{ki}& =P(u_{ki}>u_{k{i}^{\prime }},\forall i^{\prime }\ne i)\\ & =P({\upsilon }_{ki}+{\eta }_{ki}>{\upsilon }_{k{i}^{\prime }}+{\eta }_{k{i}^{\prime }},\forall i^{\prime }\ne i)\\ & =P({\eta }_{k{i}^{\prime }}<{\eta }_{ki}+{\upsilon }_{ki}-{\upsilon }_{k{i}^{\prime }},\forall i^{\prime }\ne i)\end{array}$$

(33)

According to the probability density function and the cumulative probability density function, the formula can be simplified to the (34) [48].

$$P_{ki}={\displaystyle \frac{e^{{\upsilon }_{ki}}}{{\displaystyle {\sum }_{i^{\prime }}{e}^{{\upsilon }_{k{i}^{\prime }}}}}}$$

(34)

3.3.2. Pool State Model and State Transition Probability

The above scenario composed of customers and VPPs is regarded as a pool. The state vector of the pool $\omega ={[{\omega }_1,{\omega }_2,\cdots ,{\omega }_T]}^T$, ${\omega }_k\in \{1,\cdots ,N\}$ is the index of the VPP selected by the customer $k$ to provide the power providing service. Suppose $\mathsf{\Omega }$ is the state space of all possible components of the above state vector of the pool, that is, the state space of all possible conditions, and the modulus of the state space is $\left|\mathsf{\Omega }\right|=N^K$. Define ${\beta }_i$ as the state of the VPP $i$, which is a K-dimensional binary vector. $\delta :\mathsf{\Omega }\times N\to \beta$ is defined to map the market state to a specific VPP state. For a given state $\omega$, the state of VPP $i$ can be obtained, as shown in (35).

$${\beta }_i=\delta (\omega ,i)={\left[{\beta }_{i1},{\beta }_{i2},\cdots ,{\beta }_{iK}\right]}^T,{\beta }_{ik}=\left\{\begin{array}{l}1,{\omega }_k=i\\ 0,{\omega }_k\ne i\end{array}\right.$$

(35)

Given market state $\omega \in \mathsf{\Omega }$, the current state of VPP is ${\beta }_i=\delta (\omega ,i)$, and the transition state is ${{\beta }^{\prime }}_i=\delta ({\omega }^{\prime },i)$. The probability of pool state transition from $\omega$ to ${\omega }^{\prime }$ is calculated as (36)–(39). The weighted probability is adopted, and ${\sigma }_i({\beta }_i|{{\beta }^{\prime }}_i)$ is the weight factor. When the market state gap is large, the weight factor is small.

$$\mathrm{P}({\omega }^{\prime }|\omega ,p_1,p_2,\cdots ,p_N)={\displaystyle \frac{\mathrm{P}({\omega }^{\prime }|\omega ,p_1,p_2,\cdots ,p_N)}{{\displaystyle {\sum }_{\theta \in \mathsf{\Omega }}\mathrm{P}(\theta |\omega ,p_1,p_2,\cdots ,p_N)}}}$$

(36)

$$\mathrm{P}({\omega }^{\prime }|\omega ,p_1,p_2,\cdots ,p_N)={\displaystyle \prod _{i=1}^N{\mathrm{P}}_i({{\beta }^{\prime }}_i|{\beta }_i,p_i)}$$

(37)

$${\mathrm{P}}_i({{\beta }^{\prime }}_i|{\beta }_i,p_i)={\sigma }_i({\beta }_i|{{\beta }^{\prime }}_i){\displaystyle \prod _{k=1}^K\left[{{\beta }^{\prime }}_{ik}{P}_{ki}+(1-{{\beta }^{\prime }}_{ik})(1-P_{ki})\right]}$$

(38)

$${\sigma }_i({\beta }_i|{{\beta }^{\prime }}_i)=\exp \left(-{\displaystyle {\sum }_{k=1}^K{({\beta }_{ik}-{{\beta }^{\prime }}_{ik})}^2}\right)$$

(39)

3.3.3. Distributed VPP Pricing Model Based on MDP

The game problem is defined as follows:

• Participants: VPP set $\left\{i\right|i\in [1,\cdots ,N]\}$.
• Decision: Develop a pricing strategy for each VPP ${\mathsf{\lambda }}_i={[{\lambda }_{i1},{\lambda }_{i2},\cdots ,{\lambda }_{iT}]}^T,\forall i\in [1,\cdots ,N]$
• Objective: The goal of each VPP is to find the optimal price strategy in all market conditions to maximize the discounted future revenue. That is, for $\forall \omega \in \mathsf{\Omega }$, find ${\hat{\mathsf{\lambda }}}_i={[{\widehat{\lambda }}_{i1},{\widehat{\lambda }}_{i2},\cdots ,{\widehat{\lambda }}_{iT}]}^T$ to obtain the maximum ${\widehat{\mathrm{V}}}_i$, as shown in (40).

$${\widehat{\mathrm{V}}}_i(\omega ,{\widehat{\mathsf{\lambda }}}_1,\cdots ,{\hat{\mathsf{\lambda }}}_N)={\displaystyle \sum _{t=1}^{\infty }\phi \cdot \mathbb{E}[F_i(t)|{\hat{\mathsf{\lambda }}}_1,\cdots ,{\hat{\mathsf{\lambda }}}_N,{\omega }_0=\omega ]}$$

(40)

The game between VPPs is modeled as a Markov Decision Process (MDP), which is defined by a four-tuple $(\mathsf{\Omega },\mathbb{A},\mathrm{R},\mathrm{P})$. Among them, $\mathsf{\Omega }$ is the state space of the market, which is composed of state vector $\omega ={[{\omega }_1,{\omega }_2,\cdots ,{\omega }_K]}^T$, ${\omega }_k\in \{1,\cdots ,N\}$. $\mathbb{A}$ is the action space, which refers to the action of VPP proposing the charging price strategy at the same time. $\mathrm{R}$ is the reward function, that is, the profit of each VPP. $\mathrm{P}$ is the transition probability. $\phi$ is a discount factor; its purpose is to show that the agent prefers to obtain existing rewards and discounts future rewards. At the same time, considering that some Markov chains are ring-shaped and may not terminate, we need to avoid infinite rewards. In addition, we cannot establish a perfect environment model, and the evaluation of the future may be inaccurate, so we add a discount factor to the future value.

The game process is as follows:

Step 1: Assume that at the end of the $t-1$ cycle, the market state is $\omega$, the state of VPP $i$ is ${\beta }_i$, and the state of other market participants except $i$ is ${\beta }_i$. ${\beta }_{-i}$ is observable for VPP.

Step 2: Based on the market information in step 1, all VPPs simultaneously propose a charging price strategy for cycle $t$.

Step 3: According to the charging price policy of the VPPs, the user decides which VPP serves it, and the market shifts to ${\omega }^{\prime }$ in the state of cycle $t$.

Step 4: VPPs receive $F_i(t)$, and steps 1 to 4 are repeated.

Finally, the Markov perfect equilibrium (MPE) derivation process is achieved, that is, all VPP adopt a Markov strategy in the equilibrium state, and no game participant can obtain higher profits by unilaterally deviating from its Markov strategy.

3.3.4. Solution Algorithm

The MDP-based game model can be summarized as (38) and (40), which satisfy the Bellman Optimality, that is, the optimal strategy which has the following attributes: no matter what the initial state and the initial action are, the remaining actions must constitute the optimal strategy for the state generated by the initial action, and the optimal strategy can be solved by dynamic programming. At the end of cycle $t-1$, under market state $\omega \in \mathsf{\Omega }$, the corresponding Bellman equation is (41), and the optimal price is as shown in (42). By clarifying the meaning of the expectation function in the formula, the formula can be further expressed as (43) and (44). Each VPP has its own version of (43) and (44). This implies that for each VPP and for any initial state, maximizing the discounted future revenue is performed with respect to a non-linear Equation (43) with $T$ decision variables ${\lambda }_i^{retail}(t)$,$i\in [1,2,\cdots ,N]$,$j\in [1,2,\cdots ,M]$.

$${\widehat{\mathrm{V}}}_i(\omega )=\underset{{\mathsf{\lambda }}_i\in {ℝ}_{+}^M}{\max }\left\{\begin{array}{l}{F}_i(t)\left[\delta (\omega ,i),{\mathsf{\lambda }}_i\right]\\ +\phi \cdot {\mathbb{E}}_{{\omega }^{\prime }}\left[{\widehat{\mathrm{V}}}_i({\omega }^{\prime }|\omega ,{\mathsf{\lambda }}_i,{\mathsf{\lambda }}_{-i})\right]\end{array}\right\}$$

(41)

$${\hat{\mathsf{\lambda }}}_i(\omega )=\underset{{\mathsf{\lambda }}_i\in {ℝ}_{+}^M}{\arg \max }\left\{\begin{array}{l}{F}_i(t)\left[\delta (\omega ,i),{\mathsf{\lambda }}_i\right]+\\ \phi \cdot {\mathbb{E}}_{{\omega }^{\prime }}\left[{\widehat{\mathrm{V}}}_i({\omega }^{\prime }|\omega ,{\mathsf{\lambda }}_i,{\mathsf{\lambda }}_{-i})\right]\end{array}\right\}$$

(42)

$${\widehat{\mathrm{V}}}_i(\omega )=\underset{{\mathsf{\lambda }}_i\in {ℝ}_{+}^M}{\max }\left\{\begin{array}{l}{F}_i(t)\left[\delta (\omega ,i),{\mathsf{\lambda }}_i\right]+\\ \phi \cdot {\displaystyle \sum _{{\omega }^{\prime }\in \mathsf{\Omega }}\mathrm{P}({\omega }^{\prime }|\omega ,{\mathsf{\lambda }}_i,{\mathsf{\lambda }}_{-i}){\widehat{\mathrm{V}}}_i({\omega }^{\prime })}\end{array}\right\}$$

(43)

$${\hat{\mathsf{\lambda }}}_i(\omega )=\underset{{\mathsf{\lambda }}_i\in {ℝ}_{+}^M}{\arg \max }\left\{\begin{array}{l}{F}_i(t)\left[\delta (\omega ,i),{\mathsf{\lambda }}_i\right]+\\ \phi \cdot {\displaystyle \sum _{{\omega }^{\prime }\in \mathsf{\Omega }}\mathrm{P}({\omega }^{\prime }|\omega ,{\mathsf{\lambda }}_i,{\mathsf{\lambda }}_{-i}){\widehat{\mathrm{V}}}_i({\omega }^{\prime })}\end{array}\right\}$$

(44)

The Gauss–Seidel method was used to solve the non-linear model, which mainly involved (43) and (44). The pseudo code is shown in Algorithm 1.

Algorithm 1 Optimal VPP Pricing Strategy Algorithm

Input: customers’ demand $r_k$, budget $b_k$, discount factor $\phi$, DER-related parameters, initial unified CEF $c^0(t)$

Output: Optimal price strategy $\hat{\mathsf{\lambda }}$

1: Initialization: select the possible initial values ${\widehat{V}}_i^0(\omega )$ and ${\hat{\mathsf{\lambda }}}_i^0(\omega )$, and randomly select a state vector $\omega \in \mathsf{\Omega }$ as the initial state

2: Set convergence parameters: tolerance ε, maximum iterations max_iter

3: Conditions for the end of the loop: stop: = 0, iterated index: t: = 0

4: Executed when stop ≠ 0

5: Sequential VPP Update: For i = 1 to N VPPs in fixed order:

6: Freeze other VPPs: Keep prices of VPPs 1 to i − 1 at their newly updated values, and VPPs I + 1 to N at previous iteration values

7: Solve Optimization: According to (43) and (44) calculating ${\widehat{\mathrm{V}}}_i^t(\omega )$ and ${\widehat{p}}_i^t(\omega )$ cc: = $\underset{\omega \in \mathsf{\Omega }}{\max }\left|\left({\widehat{\mathrm{V}}}_i^t(\omega )-{\widehat{\mathrm{V}}}_i^{t-1}(\omega )\right)/\left(1+{\widehat{\mathrm{V}}}_i^t(\omega )\right)\right|$

8: Immediate Update: Update VPP i’s price strategy immediately after computation

9: End Sequential Update

10: Convergence Check: Calculate maximum price change across all VPPs: ΔP_max = max|P_new − P_old|

11:   when $cc<\epsilon$ or ΔP_max < ε or t ≥ max_iter:

12:     stop: = 1

13:   else

14:     Calculate the new market state

15:     t: = t + 1

16: End Convergence Check

17: End Gauss–Seidel Iteration

18: End Algorithm

The above non-cooperative game model based on MDP is rigorous in theory. However, the size of its state space $\mathsf{\Omega }$ is $\left|\mathsf{\Omega }\right|=N^K$, where $N$ is the number of VPPs and K is the number of customers. This definition leads to the combinatorial explosion problem, which makes it infeasible to accurately solve any customer size with practical significance (such as K > 100). In order to overcome this dimensional issue and ensure the solvability of the model in practical application, the following computational strategies are adopted in the numerical implementation of this study:

We cluster a large number of individual customers according to their load characteristics and price sensitivity to form M representative customer segments (M << K). The state space is therefore redefined as an allocation based on these segments, and its size is significantly reduced from $N^K$ to $N^M$. This aggregation strategy is based on a reasonable assumption that customers in the same segment respond similarly to price signals, thus making the state space computable while retaining the dynamics of the core market.

In the face of the possible large state space after reduction, we use the approximate dynamic programming method. Specifically, we use linear value function approximation to estimate the long-term value function $V(\omega )$ of each VPP. This means that we no longer store an exact value for each state but instead use a linear function parameterized by the weight vector to fit the value function, so as to reduce the dimension of storage and optimization from the number of states to the number of features.

The above method fundamentally changes the computational complexity of the problem. The complexity of the original exact algorithm is directly related to the size of the state space $\left|\mathsf{\Omega }\right|=N^K$, which is exponential. After modification, the complexity of the algorithm mainly depends on the number of customer segments M, which determines the size $N^M$ of the approximate state space. The characteristic number L of the value function approximator determines the number of parameters to be updated in each iteration. The convergence times of the Gauss–Seidel iterations are represented by t.

Therefore, the total computational complexity is reduced from O ($N^K$) to about O ($t\ast N\ast L\ast N^M$). Although this is still a challenging problem, by carefully selecting M and L (making them far less than K), the complexity is controlled at a manageable polynomial level, which makes it possible to model and analyze the equilibrium of the large-scale, multi-VPP electricity retail market computationally. The case study results in Section 4 of this paper are based on this approximate calculation framework.

4. Case Study

This section includes numerical experiments on the distribution network containing multiple VPPs. In Section 4.1, details of the experimental setup are explained. In Section 4.2, the calculation results of carbon emission factors for each VPP, the computation outcomes of VPP pricing, the load changes before and after the optimization of VPP pricing, and the dispatch results of DERs within each VPP, as well as the profits of each VPP, are presented. Additionally, the variations in node voltages and line BLI (Branch Load Index) of the distribution network before and after the regulation of VPP pricing are demonstrated.

The simulations were conducted on a computer with a 2.60 GHz Intel CORE i7-9750H processor and 16 GB of RAM, and the optimization problems were solved by using Gurobi 9.0.

4.1. Experimental Setup

In order to verify the proposed model, this paper uses the IEEE 30-bus system for testing. Figure 5 presents the topology of the system, which comprises 30 buses and 41 feeder lines. This paper categorizes the IEEE 30-node system into six zones by setting different node load types (industrial, commercial, and residential). Each zone comprises several nodes. The nodes included in each zone and their load capacities are shown in

Table 1. Basic information of zones.

Zone Number	Nodes Involved	Zone Type	Load Capacity/kWh
Zone 1	B1, B2, B3, B4	Residential	317.75
Zone 2	B5, B6, B7, B8, B9, B10	Industrial	1209.15
Zone 3	B11, B12, B13, B14, B16, B17	Industrial	941.802
Zone 4	B15, B18, B19, B20, B21, B22	Commercial	888.25
Zone 5	B23, B24, B25, B26, B28	Commercial	602.8
Zone 6	B27, B29, B30	Residential	2502.85

.

In this case study, six VPPs were set up, aggregating five types of DERs: PV, DG, ESS, LS, and LC. The selection of six VPPs in this study strikes a balance between computational tractability and the ability to capture the strategic interactions in a multi-agent competitive environment, exceeding the typical scale of 3–4 VPPs [17] commonly adopted in the existing literature while remaining representative of analyzing pricing behaviors and low-carbon incentives. In terms of PV installed capacity within the VPPs, VPP 5 had the highest capacity, followed by VPP 4. VPP 2 and VPP 6 had similar capacities, which were next in line. VPP 1 had a lower capacity than VPP 2 and VPP 6. VPP 3 did not have any PV installations. The typical PV output curve for each VPP on a clear day is shown in Figure 6. The technical parameters and cost parameters of the two DGs are shown in

Table 2. Parameters of DGs.

Unit	$\left[{\mathit{P}}^{\mathbf{min}},{\mathit{P}}^{\mathbf{max}}\right]$/(MW)	${\mathit{R}}^{\mathit{D}},{\mathit{R}}^{\mathit{U}}$/(MW/h)	Cost Coefficients
			$\mathit{a}$/(USD/MWh2)	$\mathit{b}$/(USD/MWh)	$\mathit{c}$/USD
DG1	[3, 6]	1.5	0.27	60	3.4
DG2	[2, 5]	1.5	0.3	56.5	3.0

. The technical parameters and cost parameters of ESS are shown in

Table 3. Parameters of ESS.

Item	Value	Item	Value	Item	Value
$\left[{SOC}^{\min },{SOC}^{\max }\right]$	[0.2, 0.9]	$P_{chr}^{\max },P_{dis}^{\max }$	3 MW	$E_{rated}^{ESS}$	15 MWh
${SOC}_0$	0.5	${\eta }_{chr},{\eta }_{dis}$	0.95	$c_{ESS}$	0.5 USD/MWh

. Flexible load data was obtained from the Pecan Street experiment conducted in Austin, USA [49], which involved collecting residential domestic appliance load data for a full year in 2015. The historical data on DR potential were obtained by simulating the response behavior of customer under an incentive-based DR program using the home energy management system (HEMS) model as described in [50]. Flexible loads aggregated by each VPP were divided into two types based on their characteristics: LS and LC. The LS type mainly came from three types of loads: clothes washer, clothes dryer, and dishwasher loads. The LC type primarily originated from air conditioning loads. Based on previous research [51], we obtained the demand response potential for reducing the air conditioning load by 1, 2, and 3 degrees Celsius. The adjustable capacity of LS and LC for a selected user is illustrated in Figure 7. Figure 7a shows the adjustable capacity of LS loads, while Figure 7b presents the adjustable capacity of LC loads.

These VPPs serve dual roles: On one hand, they act as electricity retailers in the region. Through competitive interactions among VPPs, while considering constraints such as distribution network branch currents and node voltages, they formulate electricity retail prices for end-users to attract customers to sign power purchase agreements, thereby enhancing electricity sales profits. On the other hand, the VPPs regulate their aggregated DERs based on established electricity prices, PV output, and wholesale market energy prices, aiming to reduce operational control costs.

The wholesale market electricity prices in which VPPs participate as price takers are shown in Figure 8a, and the corresponding regional average carbon emission factors are shown in Figure 8b.

4.2. Simulation Results

4.2.1. VPP Carbon Emission Factor Calculation

The carbon emission factors of each VPP, calculated based on their PV output and the total user load under the final pricing strategy signed with each VPP, are illustrated in Figure 9. This reflects the differences in carbon emissions among VPPs, with the spatial resolution of emission factors refined to the distribution network level (below 10 kV). The results show that during peak PV generation periods (8:00–17:00), the adjusted carbon emission factors of the VPPs are primarily influenced by PV installed capacity and generation. During these hours, the carbon emission factors of all VPPs except VPP 3 decrease compared to the initial average carbon emission factor. Notably, VPP 5 exhibits the lowest adjusted carbon emission factor due to its largest aggregated PV capacity and generation, while the adjusted factors of VPP 1, VPP 2, VPP 4, and VPP 6 align with their respective PV capacities and outputs. In other periods (0:00–8:00 and 17:00–23:00), to ensure the total regional carbon emissions remain unchanged before and after adjustment, the adjusted carbon emission factors for all six VPPs are higher than the initial average value.

4.2.2. VPP Pricing

The final pricing results of the 6 EVCS in this case are shown in Figure 10. Each VPP’s retail electricity price is dynamically adjusted in real time, with a maximum of USD 0.16/MWh and a minimum of 0/MWh. The periods of 17:00 to 21:00 for VPP 2, 18:00 to 21:00 for VPP 5, and 18:00 to 20:00 for VPP 6 show higher prices during these times. This could indicate peak demand periods when electricity is more expensive due to higher usage.

Both VPP 1 and VPP 3 exhibited USD 0/MWh pricing, a notable phenomenon indicating that the VPP encourages users to consume electricity during these periods without charging for usage. A zero-price policy serves as a strategic tool for VPPs to enhance user engagement and competitiveness by offering discounts to users. In this case, VPP 1 had limited PV capacity and generation output. Consequently, VPP 1 adopted USD 0/MWh pricing during the 9–12 h and 14–15 h periods, while VPP 3 also offered lower prices between 7 and 12 h and 13–14 h. This was due to their relatively high adjusted carbon emission factors between 8:00 and 17:00, which placed them at a disadvantage in terms of emission reduction. To improve competitiveness, these VPPs implemented low or zero-price policies to attract users to purchase electricity during these periods.

4.2.3. Load Curve Before and After VPP Optimization Pricing

Figure 11 shows the changes in the total load curve before and after VPP optimization pricing. The total electricity consumption before optimization was 86,401.987 kWh, while after optimization, it increased to 88,079.757 kWh. This indicates a slight increase in overall electricity usage. From Figure 10, it is evident that VPP optimization pricing did not significantly alter the overall shape of the load curve but rather shifted some of the load to different times.

Figure 12 displays the load curves for each VPP before and after optimization pricing. The total load before and after optimization for each VPP, along with the load variation and load change rate, are shown in

Table 4. VPP load optimization comparison.

	Original Average Load (kWh)	Optimized Average Load (kWh)	Load Change Amount (kWh)	Load Change Rate
VPP 1	75,120.81	29,112.73	−46,008.08	−61.25%
VPP 2	1913.23	8064.18	6150.95	321.50%
VPP 3	2026.20	1784.56	−241.64	−11.93%
VPP 4	6660.28	30,265.27	23,604.99	354.41%
VPP 5	21.96	8643.27	8621.30	39,255.95%
VPP 6	2337.27	8531.98	6194.70	265.04%
Total	88,079.75	86,401.97	−1677.78	−1.90%

. Prior to optimization, users tended to concentrate their electricity consumption on a few VPPs (VPP 2 and VPP 4), leading to underutilization of other VPPs’ economic and low-carbon resources. This concentration also imposed additional stress on the distribution network nodes, potentially causing voltage and branch power limits to be exceeded. After optimization pricing, user choices became more diverse, resulting in a more evenly distributed load across all VPPs. This diversification not only enhances the overall system stability but also promotes competition and sustainable development by optimizing resource allocation.

4.2.4. Dispatch of DERs of VPPs

Figure 13 illustrates the dispatch strategies of VPPs under the aforementioned VPP pricing strategy to meet the electricity demand. For conventional power retailers, their role is limited to acting as intermediaries who buy electricity from the wholesale market and sell it to users, profiting from the price difference. In contrast, VPPs functioning as power sellers can fully leverage DERs such as PV, DG, and flexible loads to optimize their electricity purchases in the wholesale market, thereby enhancing their profit margins.

Firstly, PV generation peaks during daylight hours, especially at midday, providing a substantial amount of clean energy to VPPs. For instance, in VPP 2, VPP 5, and VPP 6, by efficiently utilizing the power generated by photovoltaics and integrating charging and discharging strategies of the ESS, the need for purchasing electricity from the grid has been effectively reduced. By making full use of photovoltaic generation and DERs, VPP 2, VPP 5, and VPP 6 almost do not need to purchase electricity through the wholesale market, significantly reducing operational costs.

For VPP 1 and VPP 4, where PV generation is insufficient to meet total electricity needs, reliance is placed on purchasing electricity during off-peak hours when wholesale prices are lower. Through load shifting and the use of ESS, they manage to fulfill peak hour demands.

Lastly, for VPP 3, which has a relatively low total-sales volume but significant DG capacity, this VPP can supplement its electricity supply with its own DG, minimizing the need to purchase electricity from the grid and maintaining high profit levels. Under high price elasticity, VPPs tend to adopt lower pricing to retain market share; as green energy elasticity increases, VPPs are incentivized to reduce carbon emissions, even at a higher operational cost, because the resulting utility gain from environmentally conscious customers translates into higher profits.

4.2.5. Profits of VPPs

The changes in the profits of each VPP during the iterative process are shown in Figure 14. The shaded areas represent the fluctuation range of VPP profits, while the lines represent the mean values. The profits generally reached a stable state after about 15 iterations. VPP 5 had the highest profit, followed by VPP 2. VPP 3, VPP 4, and VPP 6 maintained similar profit levels, while VPP 1 had the lowest profit.

It is noteworthy that under the VPP pricing strategy and the optimized dispatch strategy for DERs within the VPP, there is not a straightforward positive correlation between electricity sales volume and profit. For instance, although VPP 3 has the lowest electricity sales volume, through the optimization of its internal DERs, its profit is on par with VPP 4 and VPP 6.

(1) The effectiveness of utilizing and optimizing DERs within a VPP plays a crucial role in determining profitability. Even with lower sales volumes, efficient management and optimization of available resources can lead to competitive profit levels. (2) Simply increasing the volume of electricity sold does not guarantee higher profits. VPPs need to focus not only on increasing their sales volumes but also on improving cost efficiency. This includes reducing operational costs and effectively matching supply with demand to maximize revenue while minimizing expenses.

4.2.6. DSO Dispatching

The power flow and the node voltages before and after VPP optimization pricing are shown in Figure 15. It can be calculated that the per unit voltage of Bus 12 dropped from 1.038 to 1.024, a reduction of 1.32%. Other buses with high voltage compared to standard voltage also experienced voltage drop, for example, Bus 10 and Bus 14 dropped by 1.15% and 0.87%, respectively.

The operation states (characterized by BLI) before and after VPP optimization pricing are shown in Figure 16, in which the operation states of feeder line 1 and line 9 in one day are mainly presented. After VPP optimization pricing, the BLIs of line 1 and line 9 were reduced to varying degrees at every moment. The green, yellow, and red bars reflect normal, alert, and emergency states, respectively. It is worth mentioning that after utilizing the proposed VPP dispatching model, the BLIs of line 1 and line 9 were all reduced. Moreover, the states of line 1 and line 9 were all transformed to alert or normal states.

4.2.7. Sensitivity Analysis: Impact of DG Cost Variations on VPP Dispatch

Regarding the sensitivity of VPP profits to fluctuations in DG operational costs (e.g., fuel prices), a dedicated sensitivity analysis was conducted. This analysis focuses on VPP 3, a representative case whose profitability was shown to rely significantly on its DG unit.

The results reveal that while the overall profit of VPP 3 exhibits relatively low sensitivity to increasing DG costs, its internal resource dispatch strategy undergoes significant adjustments. As depicted in Figure 17, with stepwise increases in DG operational costs by 5%, 10%, 15%, and 20%, VPP 3 optimally reconfigures its resource portfolio. Specifically, the dispatch of the DG unit is progressively reduced as its operational cost rises. This demonstrates the model’s inherent capability to adapt to cost uncertainties by prioritizing more economical resources within the VPP’s portfolio, thereby mitigating the impact on overall profitability through optimized internal dispatch rather than merely absorbing higher costs.

5. Conclusions

This paper proposes a game-theoretic model based on MDP for managing retail pricing and the operations of multiple VPPs. The model optimizes both economic and environmental performance while accounting for distribution network constraints. It addresses the challenge of operating a distribution system with competing VPPs through an integrated retail pricing mechanism and a collaborative management framework. A non-cooperative game-theoretic approach is adopted to align VPPs’ economic incentives with low-carbon objectives, while also assisting the DSO in maintaining secure and efficient network operations. Validation on a modified IEEE 30-bus system with six VPPs leads to the following key conclusions:

(1)

The case study confirms that VPPs with a higher proportion of DPVs achieve a lower carbon emission factor, constituting a significant low-carbon competitive advantage. This advantage directly translates into higher profitability, as evidenced by VPP 5. In contrast, VPPs with fewer low-carbon resources strategically adopted zero or low-price tariffs to attract price-sensitive consumers, as evidenced by VPP 1 and VPP 3, demonstrating how heterogeneous resource portfolios drive distinct competitive behaviors.

(2)

The proposed dynamic pricing mechanism successfully reshaped user demand. It shifted the load from being concentrated on a few VPPs to a more diversified and balanced distribution across all VPPs. This alleviated operational stress on specific distribution network nodes, reducing the Branch Load Index (BLI) by 12% and improving voltage profiles by up to 1.32% at critical nodes.

(3)

Profit analysis reveals that a VPP’s profitability is not solely determined by its electricity sales volume. For instance, VPP 3, despite having the lowest sales, achieved comparable profits to VPP 4 and VPP 6 through the optimal dispatch of its internal DERs, particularly its DG. This indicates that efficient internal resource management is a crucial determinant of a VPP’s economic performance.

(4)

The framework demonstrably improved the technical operation of the distribution power system. Simulations showed a significant improvement in nodal voltages and a consistent reduction in BLI, transitioning lines from emergency/alert states to normal/alert states. This confirms the model’s efficacy in alleviating network congestion and enhancing operational security.

Despite these promising results, this study has certain limitations. The carbon emission factor correction currently only considers PV generation and does not incorporate other low-carbon DERs such as wind or hydro. Additionally, the customer choice model assumes perfect rationality and does not account for behavioral inertia or non-utility-maximizing decision-making. In future work, the model will be extended to incorporate electricity bidding in wholesale markets considering diverse DERs. Furthermore, integrating more complex price-based demand response programs and exploring the interaction with other aggregation models, such as microgrids, are promising research directions.