Open AccessArticle

A Stackelberg Game Model for the Energy–Carbon Co-Optimization of Multiple Virtual Power Plants

Dayong Xu

^* and

Mengjie Li

Huizhou Electric Power Supply Bureau of Guangdong Power Grid Corporation, Huizhou 516003, China

Author to whom correspondence should be addressed.

Inventions 2025, 10(1), 16; https://doi.org/10.3390/inventions10010016

Submission received: 21 December 2024 / Revised: 24 January 2025 / Accepted: 27 January 2025 / Published: 8 February 2025

(This article belongs to the Section Inventions and Innovation in Electrical Engineering/Energy/Communications)

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Browse Figures

Figure 1
The trading structure of the DSO and VPP. "> Figure 2
Framework of Stackelberg game model. "> Figure 3
Forecast load for three VPPs. "> Figure 4
Forecast wind for three VPPs. "> Figure 5
Trading electricity prices. "> Figure 6
Sum of VPP power exchange with DSO. "> Figure 7
Sharing power of VPPs. "> Figure 8
Optimal results of power for VPPs. (a) Optimal results of power for VPP1 in Case 1. (b) Optimal results of power for VPP1 in Case 3. (c) Optimal results of power for VPP2 in Case 1. (d) Optimal results of power for VPP2 in Case 3. (e) Optimal results of power for VPP3 in Case 1. (f) Optimal results of power for VPP3 in Case 3. "> Figure 9
Trading carbon prices. "> Figure 10
Sum of VPP carbon allowance exchange with DSO. "> Figure 11
Sharing carbon allowance of VPPs. ">

Versions Notes

Abstract

As energy and carbon markets evolve, it has emerged as a prevalent trend for multiple virtual power plants (VPPs) to engage in market trading through coordinated operation. Given that these VPPs belong to diverse stakeholders, a competitive dynamic is shaping up. To strike a balance between the interests of the distribution system operator (DSO) and VPPs, this paper introduces a bi-level energy–carbon coordination model based on the Stackelberg game framework, which consists of an upper-level optimal pricing model for the DSO and a lower-level optimal energy scheduling model for each VPP. Subsequently, the Karush-Kuhn-Tucker (KKT) conditions and the duality theorem of linear programming are applied to transform the bi-level Stackelberg game model into a mixed-integer linear program, allowing for the computation of the model’s global optimal solution using commercial solvers. Finally, a case study is conducted to demonstrate the effectiveness of the proposed model. The simulation results show that the proposed game model effectively optimizes energy and carbon pricing, encourages the active participation of VPPs in electricity and carbon allowance sharing, increases the profitability of DSOs, and reduces the operational costs of VPPs.

Keywords:

virtual power plants; Stackelberg game; pricing; energy-carbon coordination; Karush-Kuhn-Tucker conditions

1. Introduction

In the context of global energy shortages and severe environmental pollution [1,2], renewable energy resources, including photovoltaics and wind power, have emerged as the mainstream trend in energy development [3,4,5]. However, their large-scale integration into the power grid poses significant challenges to the grid’s safe operation [6,7]. The virtual power plant (VPP), serving as a key solution to the grid connection challenges posed by multiple distributed energy resources (DERs) [8,9], effectively aggregates distributed generation, energy storage, controllable loads, and other components into a virtual entity [10] to participate in various markets, as well as the operation and dispatch of the power grid [11,12]. It will become a crucial aspect in driving the development of smart grids.

Currently, numerous studies have studied the optimal scheduling of the VPP from an independent stakeholder perspective [13]. In [14], a two-level optimization framework was introduced for determining the optimal trading strategies of a price-maker VPP in the day-ahead market. In [15], a combined approach of interval and deterministic optimization was utilized to address the dispatch problem of a VPP with uncertainties. In [16], a two-stage optimal operational framework was introduced for a VPP considering reserve uncertainty. In [17], a unique holistic peer-to-peer market framework hosted in a VPP environment was presented, aimed at rewarding VPP participants financially and optimizing both network and market operations concurrently.

Given the heterogeneity of VPPs, research emphasis has shifted from focusing on individual VPPs to coordinating energy management across multiple VPPs, aiming to achieve complementary resource utilization [8,18,19,20,21,22,23,24]. Each VPP is affiliated with different interest stakeholders. Driven by individual rationality, these entities pursue the maximization of their own interests, making traditional optimization scheduling methods for a single entity difficult to apply [25]. Game theory is a prevalent approach for analyzing the equilibrium of interests among multiple agents and is extensively applied in the economic domain [26]. Currently, game theory is employed to examine the equilibrium of interests among and within VPPs [27]. In [28], a one-leader multi-follower game model was constructed to investigate the dynamic pricing behavior of a distribution system operator (DSO) and the corresponding price response of VPPs. In [29], a bi-level equilibrium problem with equilibrium constraints was used to model the joint operation of energy and reserve market to boost

μ

VPP entry in the retail electricity market.

As the carbon market evolves, carbon trading is recognized as a crucial tool for balancing environmental and economic concerns [30]. As the carbon trading mechanism improves, the focus on achieving low-carbon operation for VPPs has intensified, becoming a prominent issue [31]. In [32], a Stackelberg game-based electricity—heat trading model was introduced for the operator and multiple VPPs, taking into account a ladder-type carbon pricing mechanism. In [33], an optimal dispatching model of the VPP considering carbon emission trading was presented. In [34], a flexible carbon emission mechanism for a VPP was proposed to reduce carbon emissions without reducing profits. In [35], an optimal scheduling strategy using multi-level games was proposed for multiple VPPs considering carbon emission flow and carbon trade. However, the existing research either only considers carbon allowance cost or energy trading, without any consideration of sharing from the perspective of energy—carbon coupling.

To fill the research gap, this paper presents a Stackelberg game model for the energy–carbon co-optimization of multiple virtual power plants. The main contributions of this paper are outlined as follows:

Designing an energy–carbon allowance trading mechanism for DSOs to guide VPPs from the perspective of electricity–carbon coupling.
Developing an optimal energy–carbon pricing model for DSOs and energy management models for multiple VPPs based on a Stackelberg game considering the different benefits requirements for DSOs and VPPs.

The remaining sections of this paper are organized as follows. Section 2 establishes a Stackelberg game model for DSOs and VPPs considering energy–carbon coor-dinated optimization. Section 3 proposes the solving method for the proposed Stackelberg game model. Section 4 presents a case study to verify the effectiveness of the proposed Stackelberg game model. Finally, Section 5 draws the conclusions of the paper.

2. Stackelberg Game Model of DSO and VPPs

Each VPP exhibits distinct characteristics in terms of its functionality, energy equipment setup, and methodologies for energy production and consumption. The internal energy supply and demand within each VPP frequently experience imbalances, leading to either shortages or surpluses of energy. Therefore, this paper proposed a electricity-carbon sharing mechanism for VPPs.

The trading structure of the DSO and VPP is depicted in Figure 1. As shown in Figure 1, the DSO is responsible for coordinating the sharing and trading of electricity and carbon allowance and for establishing the shared electricity prices and carbon prices under the electricity-carbon sharing business model. Under the energy sharing mechanism, VPPs buy/sell electricity to/from the DSO based on DSO-set prices and their own needs. When the electricity produced by a VPP exceeds its consumption, the VPP sells surplus electricity to the DSO for profit. When the electricity produced is less than its consumption, the VPP purchases electricity from the DSO to maintain normal industrial production. Similarly, under the carbon sharing mechanism, VPPs trade carbon allowance with the DSO, selling excess carbon allowance for profit and buying additional carbon allowance when needed. The DSO serves as the intersection hub for energy and carbon flows between VPPs, the distribution grid, and the external carbon market. After each round of sharing, it aggregates electricity and carbon allowance information from VPPs and participates in external electricity and carbon markets to achieve internal energy and carbon allowance balance among multiple VPPs, profiting from the price differential between market and set prices.

Given the different statuses and objectives of the DSO and VPPs, this paper considers the leader of the DSO and VPP as the followers of the game to construct the framework of a Stackelberg game model as illustrated in Figure 2. In the upper-level model, the DSO maximizes the own profit by setting the optimal trading electricity price and carbon allowance price for VPPs. In the lower-level model, each VPP minimizes its energy cost by deciding the optimal amount energy–carbon allowance to trade with the DSO and the operation of each DER.

2.1. Upper-Level Model: Benefit Maximization of DSO

2.1.1. Objective Function of Upper-Level Model

The objective function of the DSO includes both the costs and benefits related to buying and selling electricity/carbon allowance with the electricity/carbon market and VPPs. The details are as follows:

max U^{DSO} = \sum_{t = 1}^{T} (\begin{matrix} λ_{t}^{grid, s} P_{t}^{DSO, s} - λ_{t}^{grid, b} P_{t}^{DSO, b} + \\ λ_{t}^{e, b} \sum_{i = 1}^{I} P_{i, t}^{VPP, b} - λ_{t}^{e, s} \sum_{i = 1}^{I} P_{i, t}^{VPP, s} \end{matrix}) + \sum_{t = 1}^{T} (\begin{matrix} λ_{t}^{CM, s} M_{t}^{DSO, s} - λ_{t}^{CM, b} M_{t}^{DSO, b} + \\ λ_{t}^{c, b} \sum_{i = 1}^{I} M_{i, t}^{VPP, b} - λ_{t}^{c, s} \sum_{i = 1}^{I} M_{i, t}^{VPP, s} \end{matrix})

(1a)

P_{t}^{DSO} = \sum_{i \in I} (P_{i, t}^{VPP, b} - P_{i, t}^{VPP, s})

(1b)

P_{t}^{DSO, b} = \{\begin{matrix} P_{t}^{DSO}, & P_{t}^{DSO} \geq 0 \\ 0, & P_{t}^{DSO} < 0 \end{matrix}

(1c)

P_{t}^{DSO, s} = \{\begin{matrix} - P_{t}^{DSO}, & P_{t}^{DSO} < 0 \\ 0, & P_{t}^{DSO} \geq 0 \end{matrix}

(1d)

M_{t}^{DSO} = \sum_{i \in I} (M_{i, t}^{VPP, b} - M_{i, t}^{VPP, s})

(1e)

M_{t}^{DSO, b} = \{\begin{matrix} M_{t}^{DSO}, & M_{t}^{DSO} \geq 0 \\ 0, & M_{t}^{DSO} < 0 \end{matrix}

(1f)

M_{t}^{DSO, s} = \{\begin{matrix} - M_{t}^{DSO}, & M_{t}^{DSO} < 0 \\ 0, & M_{t}^{DSO} \geq 0 \end{matrix}

(1g)

where

U^{DSO}

denotes the profit of the DSO.

λ_{t}^{grid, s}

and

λ_{t}^{grid, b}

denote the selling and buying electricity prices of the grid, respectively.

λ_{t}^{CM, s}

and

λ_{t}^{CM, b}

denote the selling and buying prices of carbon allowance in the carbon market, respectively.

λ_{t}^{e, s}

and

λ_{t}^{e, b}

denote the selling and buying electricity prices set by the DSO, respectively.

λ_{t}^{c, s}

and

λ_{t}^{c, b}

denote the selling and buying prices of carbon allowance set by the DSO, respectively.

P_{t}^{DSO, b}

and

P_{t}^{DSO, s}

denote the electric power purchased and sold by the DSO in the grid.

P_{i, t}^{VPP, b}

and

P_{i, t}^{VPP, s}

denote the electric power purchased and sold by the VPP in the DSO.

M_{i, t}^{DSO, b}

and

M_{i, t}^{DSO, s}

denote the carbon allowance purchased and sold by the DSO in the carbon market.

M_{i, t}^{VPP, b}

and

M_{i, t}^{VPP, s}

denote the carbon allowance purchased and sold by the VPP in the DSO.

2.1.2. Constraints of Upper-Level Model

In order to ensure that VPPs are willing to trade with the DSO, the buying and selling prices set by the DSO should satisfy the following constraints:

λ_{t}^{grid, s} \leq λ_{t}^{e, s} \leq λ_{t}^{e, b} \leq λ_{t}^{grid, b}

(2)

λ_{t}^{CM, s} \leq λ_{t}^{c, s} \leq λ_{t}^{c, b} \leq λ_{t}^{CM, b}

(3)

In Equations (2) and (3), the DSO sets the buying prices lower than the market prices and the selling price higher than the feed-in tariff. Consequently, the VPPs will opt to trade with the DSO in order to maximize their own interests.

2.2. Lower-Level Model: Cost Minimization of VPP

2.2.1. Objective Function of Lower-Level Model

The objective function of VPPi includes electricity/carbon allowance purchase costs, operational cost of micro turbines (MT)

C_{i, t}^{MT}

, operational cost of energy storages (ES)

C_{i, t}^{ES}

and operational cost of interruptible load (IL)

C_{i, t}^{IL}

, which is shown as follows:

min C_{i}^{VPP} = \sum_{t = 1}^{T} (λ_{t}^{e, b} P_{i, t}^{VPP, b} - λ_{t}^{e, s} P_{i, t}^{VPP, s} + λ_{t}^{c, b} M_{i, t}^{VPP, b} - λ_{t}^{c, s} M_{i, t}^{VPP, s} + C_{i, t}^{MT} + C_{i, t}^{ES} + C_{i, t}^{IL})

(4a)

C_{i, t}^{MT} = a_{i} {(P_{i, t}^{MT})}^{2} + b_{i} P_{i, t}^{MT} + c_{i}

(4b)

C_{i, t}^{ES} = λ_{i}^{ES} {(P_{i, t}^{ES})}^{2}

(4c)

C_{i, t}^{IL} = λ_{i}^{IL} P_{i, t}^{IL}

(4d)

where

C_{i}^{VPP}

denotes the cost of VPPs.

a_{i}

b_{i}

and

c_{i}

are the cost coefficients of MT.

λ_{i}^{ES}

and

λ_{i}^{IL}

denote the scheduling cost coefficients of ES and IL, respectively.

P_{i, t}^{MT}

P_{i, t}^{ES}

and

P_{i, t}^{IL}

denote the power of MT, ES and IL, respectively.

2.2.2. Constraints of Lower-Level Model

The VPP is required to satisfy the power/carbon balance constraints and the operational constraints of each DER when responding to the price, which can be expressed as (5)–(16).

P_{i, t}^{VPP, b} + P_{i, t}^{MT} + P_{i, t}^{ES} + P_{i, t}^{IL} + P_{i, t}^{W} = P_{i, t}^{D} + P_{i, t}^{VPP, s} : γ_{i, t}^{e}

(5)

0 \leq P_{i, t}^{VPP, b} \leq {\bar{P}}_{i}^{VPP} : {\underset{̲}{v}}_{i, t}, {\bar{v}}_{i, t}

(6)

0 \leq P_{i, t}^{VPP, s} \leq {\bar{P}}_{i}^{VPP} : {\underset{̲}{u}}_{i, t}, {\bar{u}}_{i, t}

(7)

0 \leq P_{i, t}^{MT} \leq {\bar{P}}_{i}^{MT} : {\underset{̲}{ε}}_{i, t}^{MT}, {\bar{ε}}_{i, t}^{MT}

(8)

R_{i}^{MT, dn} \leq P_{i, t}^{MT} - P_{i, t - 1}^{MT} \leq R_{i}^{MT, up} : w_{i, t}^{dn}, w_{i, t}^{up}

(9)

{\underset{̲}{P}}_{i}^{ES} \leq P_{i, t}^{ES} \leq {\bar{P}}_{i}^{ES} : {\underset{̲}{ε}}_{i, t}^{ES}, {\bar{ε}}_{i, t}^{ES}

(10)

S_{i, t}^{ES} = S_{i, t - 1}^{ES} - \frac{P_{i, t}^{ES}}{E_{i}} Δ t : μ_{i, t}

(11)

{\underset{̲}{S}}_{i}^{ES} \leq S_{i, t}^{ES} \leq {\bar{S}}_{i}^{ES} : {\underset{̲}{η}}_{i, t}, {\bar{η}}_{i, t}

(12)

S_{i, 0}^{ES} = S_{i, T}^{ES} : σ_{i}

(13)

0 \leq P_{i, t}^{IL} \leq {\bar{P}}_{i}^{IL} : {\underset{̲}{ε}}_{i, t}^{IL}, {\bar{ε}}_{i, t}^{IL}

(14)

0 \leq P_{i, t}^{W} \leq {\bar{P}}_{i}^{W} : {\underset{̲}{ε}}_{i, t}^{W}, {\bar{ε}}_{i, t}^{W}

(15)

M_{i, t}^{VPP, b} - M_{i, t}^{VPP, s} = [δ^{MT} P_{i, t}^{MT} + δ^{g} (P_{i, t}^{VPP, b} - P_{i, t}^{VPP, s})] - [χ^{MT} P_{i, t}^{MT} + χ^{g} (P_{i, t}^{VPP, b} - P_{i, t}^{VPP, s})] : γ_{i, t}^{c}

(16)

where

P_{i, t}^{W}

denotes the output of wind turbines.

P_{i, t}^{D}

denotes the forecast value of the load.

S_{i, t}^{ES}

denotes the state of charge (SOC) of ES.

{\bar{P}}_{i}^{VPP}

denotes the upper bound of the electric power purchased and sold by the VPP.

{\bar{P}}_{i}^{MT}

{\bar{P}}_{i}^{IL}

, and

{\bar{P}}_{i}^{W}

denote the maximum power of the MT, IL and output of wind turbines, respectively.

R_{i}^{MT, dn}

and

R_{i}^{MT, up}

denote the upward and downward hourly ramping rate limits of MT, respectively.

{\underset{̲}{P}}_{i}^{ES}

and

{\bar{P}}_{i}^{ES}

denote the lower and upper bounds of the charging/discharging power of ES, respectively.

{\underset{̲}{S}}_{i}^{ES}

and

{\bar{S}}_{i}^{ES}

denote the minimum and maximum values of the SOC, respectively.

δ^{MT}

and

δ^{g}

denote the carbon emission coefficients of MT and unit electricity purchased from the grid, respectively.

χ^{MT}

and

χ^{g}

denote the carbon allowance coefficients of MT and unit electricity purchased from the grid, respectively.

γ_{i, t}^{e}

{\underset{̲}{v}}_{i, t}

{\bar{v}}_{i, t}

{\underset{̲}{u}}_{i, t}

{\bar{u}}_{i, t}

{\underset{̲}{ε}}_{i, t}^{MT}

{\bar{ε}}_{i, t}^{MT}

w_{i, t}^{dn}

w_{i, t}^{up}

{\underset{̲}{ε}}_{i, t}^{ES}

{\bar{ε}}_{i, t}^{ES}

μ_{i, t}

{\underset{̲}{η}}_{i, t}

{\bar{η}}_{i, t}

σ_{i}

{\underset{̲}{ε}}_{i, t}^{IL}

{\bar{ε}}_{i, t}^{IL}

{\underset{̲}{ε}}_{i, t}^{W}

{\bar{ε}}_{i, t}^{W}

and

γ_{i, t}^{c}

denote dual variables of the corresponding constraints, respectively.

Specifically, (5) represents the power balance constraint. (6) and (7) represent the power limits of the electric power purchased and sold by the VPP, respectively. (8) and (9) represent the electric power limit and ramping rate limit of MT, respectively. (10) represents the electric power limit of ES. (11) and (12) represent the upper and lower bounds of stored energy in ES. (13) represents the target energy of ES, which should be identical to the energy stored in ES at the initial time. (14) and (15) represent the electric power limits of interruptible load and wind turbine output, respectively. (16) represents the carbon balance constraint.

2.3. Stackelberg Game Model

Based on Section 2.1 and Section 2.2, the Stackelberg game is a decision-making process where the DSO and VPPs pursue their optimal objectives, respectively. The Stackelberg game model, which takes the DSO as the leader and VPPs as followers, is shown as below.

\begin{matrix} max_{λ} U^{DSO} (λ, P) \\ s . t . \{\begin{matrix} λ \in Ω^{DSO} \\ P_{i} = arg min_{P_{i}} C_{i}^{VPP} (λ, P_{i}), \forall i \\ P_{i} \in Ω_{i}^{VPP}, \forall i \end{matrix} \end{matrix}

(17)

where

λ = {(λ_{t}^{e, b}, λ_{t}^{e, s}, λ_{t}^{c, b} λ_{t}^{c, s})}^{T}

, P denotes the decision variable set of VPP i.

Ω^{DSO}

and

Ω_{i}^{VPP}

denote the feasible regions of the DSO and VPP i, respectively.

In (17), the DSO and the VPP formulate their strategies with the objectives of maximizing profit and minimizing operating costs, respectively. The profit of the DSO is related to the set prices and the amount of electricity/carbon allowance traded with VPPs. The larger the difference between the trading prices and the amount of electricity/carbon allowance shared by the VPPs, the larger the profit of the DSO. However, the behavior of VPPs in response to the prices also influences the DSO’s profit. Higher buying prices lead to fewer electricity/carbon allowance purchases by the VPPs, while lower selling prices result in less electricity/carbon allowance sold by them, thereby decreasing the total electricity/carbon allowance exchanged among the VPPs. It can be seen that there is a strategic game between the DSO and VPPs. In order to maximize its own profit, the DSO must consider how the VPPs will respond to changes in price, and find the Nash equilibrium solution as the best tariff strategy.

3. Stackelberg Game Model Solution Method

Since the trading electricity/carbon prices are given when VPPs make decisions, the lower-level model is linear programming. However, when the DSO makes decisions, it must consider both trading electricity/carbon prices and the VPPs’ responses to the prices. Therefore, the Stackelberg game between the DSO and VPPs is neither linear nor convex.

In this section, the game is recast as a mathematical problem with equilibrium constraints (MPEC). The Karush-Kuhn-Tucker (KKT) optimality conditions [36,37,38] and the duality theorem of linear programming are used to solve the double-level Stackelberg game model established in Section 2.

3.1. MPEC Formulation of DSO and VPP

Since the trading electricity/carbon prices are decided by the DSO upper-level model, they can be considered as parameters in the VPP lower-level problem. This makes the lower-level model linear and thus convex. Therefore, the KKT optimality conditions can be used to replace the lower-level model (5)–(16). The KKT optimality conditions for each VPP are calculated as follows:

λ_{t}^{e, b} - γ_{i, t}^{e} - {\underset{̲}{v}}_{i, t} + {\bar{v}}_{i, t} - γ_{i, t}^{c} δ^{g} + γ_{i, t}^{c} χ^{g} = 0

(18)

- λ_{t}^{e, s} + γ_{i, t}^{e} - {\underset{̲}{u}}_{i, t} + {\bar{u}}_{i, t} + γ_{i, t}^{c} δ^{g} - γ_{i, t}^{c} χ^{g} = 0

(19)

2 a_{i} P_{i, t}^{MT} + b_{i} - γ_{i, t}^{e} - {\underset{̲}{ε}}_{i, t}^{MT} + {\bar{ε}}_{i, t}^{MT} - γ_{i, t}^{c} δ^{MT} + γ_{i, t}^{c} χ^{MT} - w_{i, t}^{dn} + w_{i, t}^{up} + w_{i, t + 1}^{dn} - w_{i, t + 1}^{up} = 0, t < T

(20)

2 a_{i} P_{i, T}^{MT} + b_{i} - γ_{i, T}^{e} - {\underset{̲}{ε}}_{i, T}^{MT} + {\bar{ε}}_{i, T}^{MT} - γ_{i, T}^{c} δ^{MT} + γ_{i, T}^{c} χ^{MT} - w_{i, T}^{dn} + w_{i, T}^{up} = 0, t = T

(21)

2 λ_{i}^{ES} P_{i, t}^{ES} - γ_{i, t}^{e} - {\underset{̲}{ε}}_{i, t}^{ES} + {\bar{ε}}_{i, t}^{ES} - μ_{i, t} Δ t = 0

(22)

- {\underset{̲}{η}}_{i, t} + {\bar{η}}_{i, t} - μ_{i, t} + μ_{i, t + 1} = 0, t < T

(23)

- {\underset{̲}{η}}_{i, T} + {\bar{η}}_{i, T} - μ_{i, T} - σ_{i} = 0, t = T

(24)

λ_{i}^{IL} - γ_{i, t}^{e} - {\underset{̲}{ε}}_{i, t}^{IL} + {\bar{ε}}_{i, t}^{IL} = 0

(25)

- γ_{i, t}^{e} - {\underset{̲}{ε}}_{i, t}^{W} + {\bar{ε}}_{i, t}^{W} = 0

(26)

λ_{t}^{c, b} - γ_{i, t}^{c} = 0

(27)

- λ_{t}^{c, s} + γ_{i, t}^{c} = 0

(28)

0 \leq P_{i, t}^{VPP, b} ⊥ {\underset{̲}{v}}_{i, t} \leq 0

(29)

0 \leq ({\bar{P}}_{i}^{VPP} - P_{i, t}^{VPP, b}) ⊥ {\bar{v}}_{i, t} \leq 0

(30)

0 \leq P_{i, t}^{VPP, s} ⊥ {\underset{̲}{u}}_{i, t} \leq 0

(31)

0 \leq ({\bar{P}}_{i}^{VPP} - P_{i, t}^{VPP, s}) ⊥ {\bar{u}}_{i, t} \leq 0

(32)

0 \leq P_{i, t}^{MT} ⊥ {\underset{̲}{ε}}_{i, t}^{MT} \leq 0

(33)

0 \leq ({\bar{P}}_{i}^{MT} - P_{i, t}^{MT}) ⊥ {\bar{ε}}_{i, t}^{MT} \leq 0

(34)

0 \leq (P_{i, t}^{MT} - P_{i, t - 1}^{MT} - R_{i}^{MT, dn}) ⊥ w_{i, t}^{dn} \leq 0

(35)

0 \leq (R_{i}^{MT, up} - P_{i, t}^{MT} + P_{i, t - 1}^{MT}) ⊥ w_{i, t}^{up} \leq 0

(36)

0 \leq (P_{i, t}^{ES} - {\underset{̲}{ε}}_{i}^{ES}) ⊥_{i, t}^{ES} \leq 0

(37)

0 \leq ({\bar{P}}_{i}^{ES} - P_{i, t}^{ES}) ⊥ {\bar{ε}}_{i, t}^{ES} \leq 0

(38)

0 \leq (S_{i, t}^{ES} - {\underset{̲}{S}}_{i}^{ES}) ⊥ {\underset{̲}{η}}_{i, t} \leq 0

(39)

0 \leq ({\bar{S}}_{i}^{ES} - S_{i, t}^{ES}) ⊥ {\bar{η}}_{i, t} \leq 0

(40)

0 \leq P_{i, t}^{IL} ⊥ {\underset{̲}{ε}}_{i, t}^{IL} \leq 0

(41)

0 \leq ({\bar{P}}_{i}^{IL} - P_{i, t}^{IL}) ⊥ {\bar{ε}}_{i, t}^{IL} \leq 0

(42)

0 \leq P_{i, t}^{W} ⊥ ε_{i, t}^{W} \leq 0

(43)

0 \leq ({\bar{P}}_{i, t}^{W} - P_{i, t}^{W}) ⊥ {\bar{ε}}_{i, t}^{W} \leq 0

(44)

3.2. Linearized MPEC Formulation

As the term

\sum_{t = 1}^{T} (λ_{t}^{e, b} P_{i, t}^{VPP, b} - λ_{t}^{e, s} P_{i, t}^{VPP, s} + λ_{t}^{c, b} M_{i, t}^{VPP, b} - λ_{t}^{c, s} M_{i, t}^{VPP, s})

in Equation (1) and the complementarity conditions (38)–(53) are non-linear, some linearization techniques are applied to transform the non-linear MPEC formulation into the mixed-integer linear problem (MILP).

The strong duality equality shows that the objective function values of the primal problem and the dual problem are equal at the optimal solution. Therefore, the non-linear term

\sum_{t = 1}^{T} (λ_{t}^{e, b} P_{i, t}^{VPP, b} - λ_{t}^{e, s} P_{i, t}^{VPP, s} + λ_{t}^{c, b} M_{i, t}^{VPP, b} - λ_{t}^{c, s} M_{i, t}^{VPP, s})

is equivalent to (45).

\begin{matrix} \sum_{t = 1}^{T} (λ_{t}^{e, b} P_{i, t}^{VPP, b} - λ_{t}^{e, s} P_{i, t}^{VPP, s} + λ_{t}^{c, b} M_{i, t}^{VPP, b} - λ_{t}^{c, s} M_{i, t}^{VPP, s}) \\ = & \sum_{t = 1}^{T} (\begin{matrix} - γ_{i, t}^{e} P_{i, t}^{D} - {\bar{v}}_{i, t} {\bar{P}}_{i}^{VPP} - {\bar{u}}_{i, t} {\bar{P}}_{i}^{VPP} - {\bar{ε}}_{i, t}^{MT} {\bar{P}}_{i}^{MT} \\ - {\bar{ε}}_{i, t}^{ES} {\bar{P}}_{i}^{ES} + {\underset{̲}{ε}}_{i, t}^{ES} {\underset{̲}{P}}_{i}^{ES} - {\bar{η}}_{i, t} {\bar{S}}_{i}^{ES} + {\underset{̲}{η}}_{i, t} {\underset{̲}{S}}_{i}^{ES} \\ - {\bar{ε}}_{i, t}^{IL} {\bar{P}}_{i}^{IL} - {\bar{ε}}_{i, t}^{W} {\bar{P}}_{i, t}^{W} - (C_{i, t}^{MT} + C_{i, t}^{ES} + C_{i, t}^{IL}) \end{matrix}) + \sum_{t = 1}^{T - 1} (w_{i, t}^{dn} R_{i}^{MT, dn} - w_{i, t}^{up} R_{i}^{MT, up}) \end{matrix}

(45)

Moreover, to linearized the complementarity conditions (18)–(44), the big-M method is utilized. The complementarity condition, which has the form

0 \leq x ⊥ ζ \leq 0

, is equivalent to

x \geq 0

ζ \geq 0

and

x ζ = 0

and can be reformulated by the following MILP equations.

0 \leq x \leq α Q

(46)

0 \leq ζ \leq (1 - α) Q

(47)

α \in \{0, 1\}

(48)

where

α

denotes an auxiliary binary variable and Q is a large positive constant. The linearization of conditions (29)–(44) is specified in the Appendix A.

In summary, the double-level game model can be reformulated as the following MILP:

\begin{matrix} max_{λ} \sum_{t = 1}^{T} (λ_{t}^{g, s} P_{t}^{DSO, s} - λ_{t}^{g, b} P_{t}^{DSO, b} + λ_{t}^{u, s} M_{t}^{DSO, s} - λ_{t}^{u, b} M_{t}^{DSO, b}) + \\ \sum_{t = 1}^{T} (\begin{matrix} - γ_{i, t}^{e} P_{i, t}^{D} - {\bar{v}}_{i, t} {\bar{P}}_{i}^{VPP} - {\bar{u}}_{i, t} {\bar{P}}_{i}^{VPP} - {\bar{ε}}_{i, t}^{MT} {\bar{P}}_{i}^{MT} \\ - {\bar{ε}}_{i, t}^{ES} {\bar{P}}_{i}^{ES} + {\underset{̲}{ε}}_{i, t}^{ES} {\underset{̲}{P}}_{i}^{ES} - {\bar{η}}_{i, t} {\bar{S}}_{i}^{ES} + {\underset{̲}{η}}_{i, t} {\underset{̲}{S}}_{i}^{ES} \\ - {\bar{ε}}_{i, t}^{IL} {\bar{P}}_{i}^{IL} - {\bar{ε}}_{i, t}^{W} {\bar{P}}_{i, t}^{W} - (C_{i, t}^{MT} + C_{i, t}^{ES} + C_{i, t}^{IL}) \end{matrix}) + \sum_{t = 1}^{T - 1} (w_{i, t}^{dn} R_{i}^{MT, dn} - w_{i, t}^{up} R_{i}^{MT, up}) \end{matrix}

(49)

s.t. (2) and (3), (18)–(28), (A1)–(A32).

4. Case Study

Numerical simulations are presented to demonstrate the performance of the proposed model in this section. The optimization models are all formulated with YALMIP and solved in Matlab R2022b equipped with a Gurobi 10.0.0 solver on a desktop computer with an Intel(R) Core(TM) i7-9700 CPU.

4.1. Data Description

The case study is built on a multi-virtual power plant group composed of three virtual power plants, namely VPP1, VPP2 and VPP3. The detailed parameters of MT and ES for the three VPPs are listed in Table 1 and Table 2, respectively. The maximum wind output and electric load are illustrated in Figure 3 and Figure 4, respectively [28].

The carbon emission coefficients of MT and unit electricity purchased from grid are 0.736 kg/kWh and 1.08 kg/kWh, respectively [39]. The carbon allowance coefficients of MT and unit electricity purchased from grid are 0.7 kg/kWh and 0.728 kg/kWh, respectively. The buying and selling prices of the carbon allowance are 0.3 CNY/kg and 0.15 CNY/kg, respectively [40].

To validate the effectiveness of the proposed scheduling model, two cases are conducted as follows:

Case 1: the reference scheduling model, where the Stackelberg game is not considered. In this case, DSO does not optimize electricity and carbon prices, instead utilizing external electricity and carbon markets as the prices for electricity and carbon trading. Each VPP optimizes its operation with the goal of minimizing its own cost.

Case 2: the comparison scheduling model, where the Stackelberg game is only considered for energy sharing. In this case, DSO only optimizes electricity prices.

Case 3: the proposed scheduling model, where the Stackelberg game is considered for both energy and carbon allowance sharing. In this case, DSO optimizes both electricity and carbon prices.

4.2. Analysis of Optimization Results

Comparing Case 1 with Case 3, the pricing strategy set by the DSO for electricity is depicted in Figure 5. The amount of power trading between VPPs and the DSO is shown in Figure 6, where the positive and negative bars of the power trading represent the buying and selling power, respectively. The sharing power between the VPPs is shown in Figure 7. Figure 8 shows the optimal results of power for VPPs in Case 1 and Case 3. Combining Figure 5, Figure 6, Figure 7 and Figure 8, the following observations can be made based on the mutual influence between trading prices and VPP energy management:

During 9:00–10:00, all VPPs sell electricity to the DSO, resulting in no energy sharing among the VPPs. To guarantee profitability, the DSO sets the selling prices equivalent to the grid feed-in tariff. Similarly, during 17:00–18:00, all VPPs buy electricity from the DSO. The DSO sets the buying prices as the grid tariff to maintain the cost.
During 12:00–14:00, the overall aim of the three VPPs is to sell electricity to the DSO as there remains a surplus even after the internal energy sharing. During these periods, it is advisable to reduce the purchase price of electricity to incentivize VPPs to consume more electricity. For instance, VPP3 purchases more electricity compared to Case 1 due to the lower electricity buying price of Case 3 within the period of 12:00–14:00.
During 19:00–22:00, the overall aim of the three VPPs is to buy electricity from the DSO as internal energy sharing is insufficient to meet the electricity demand. Consequently, the selling price can be increased to guide the VPPs to sell more power to meet the demand. For example, in the period of 19:00–22:00, VPP2 needs to buy electricity from the DSO in Case 1. However, in Case 3, VPP2 transitions from being a power buyer to a power seller by optimizing its own energy equipment outputs in response to the increased selling price of electricity.

Figure 5. Trading electricity prices.

Figure 6. Sum of VPP power exchange with DSO.

Figure 7. Sharing power of VPPs.

Figure 8. Optimal results of power for VPPs. (a) Optimal results of power for VPP1 in Case 1. (b) Optimal results of power for VPP1 in Case 3. (c) Optimal results of power for VPP2 in Case 1. (d) Optimal results of power for VPP2 in Case 3. (e) Optimal results of power for VPP3 in Case 1. (f) Optimal results of power for VPP3 in Case 3.

Figure 9 shows the pricing strategy set by the DSO for carbon allowance. Figure 10 shows the amount of carbon allowance trading between VPPs and the DSO. Figure 11 shows the sharing carbon allowance among the VPPs. As shown in Figure 9, it can be seen that the carbon prices exhibit less variation compared to the electricity prices as the electricity market is more flexible than carbon market. Combining Figure 10 and Figure 11, it is obvious that Case 3 demonstrates a substantial increase in carbon allowance sharing among VPPs and a significant rise in the number of sharing time slots after considering the Stackelberg game. Consequently, this leads to a reduction in the volume of carbon allowances traded between the DSO and the carbon market, ultimately increasing the DSO’s profit.

Table 3 shows the amount of sharing electricity and carbon allowance. It can be observed that the total amount of sharing electricity and carbon allowance among VPPs increases significantly in Case 2. Further consideration of electricity-carbon sharing has contributed to an additional increase in this total amount. Table 4 shows the costs of VPPs and the profit of the DSO under three cases. It can be seen that the initial profit without any sharing is 10,073 yuan. After considering electricity sharing, the profit of the the DSO increases to 11,720 yuan. Besides, in Case 3, with both energy and carbon allowance sharing in place, the profit of the DSO further rises to 11,767 yuan, which has increased by 16.82% compared to Case 1. This is because the increased internal electricity and carbon allowance exchange decreases the trading volume from the grid and carbon market. Consequently, this leads to a decrease in the purchase cost of electricity and carbon allowances, and an increase in the profit from their sale. In addition, the optimal electricity and carbon allowance trading prices set by the DSO are more beneficial than the market prices, resulting in reduced costs for VPPs.

5. Conclusions

This paper proposed a novel energy-carbon coordinated optimization model based on Stackelberg game for multiple VPPs. Firstly, a bi-level a Stackelberg game model involving DSO and VPPs was formulated. Then, the KKT conditions and the duality theorem are utilized to solve the game model. Finally, a case study was carried out to verify the superiority of the proposed model. The main conclusions are summarized as follows.

The optimized traded electricity and carbon allowances of VPPs will affect the DSO’s determination of electricity and carbon prices, while the DSO can guide the VPPs in making decisions through dynamic pricing.
In the established Stackelberg game model, the DSO promotes the sharing of electricity and carbon allowance among VPPs through the optimization of electricity and carbon prices, and it can not only can increase its own profit, but also can reduce the operating costs of VPPs.

The electricity and carbon trading markets considered in this paper are relatively simple, potentially failing to capture the full complexity of real-world scenarios. In future research, the external electricity and carbon market models, transaction matching and allocation rules, can be refined to foster a more comprehensive and efficient development of the VPPs electricity-carbon coordination model.

Author Contributions

D.X.: conceptualization, methodology, formal analysis, writing—original draft; M.L.: supervision, resources, project administration. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by China Southern Power Grid Company Limited Technology Project Funding (No. 031300KC23070006).

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Conflicts of Interest

All Authors were employed by the Huizhou Electric Power Supply Bureau of Guangdong Power Grid Corporation. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Appendix A

The linearization of conditions (29)–(44) is as follows:

0 \leq P_{i, t}^{VPP, b} \leq α_{i, t}^{(1)} Q^{(1)}

(A1)

0 \leq {\underset{̲}{v}}_{i, t} \leq (1 - α_{i, t}^{(1)}) Q^{(1)}

(A2)

0 \leq {\bar{P}}_{i}^{VPP} - P_{i, t}^{VPP, b} \leq α_{i, t}^{(2)} Q^{(2)}

(A3)

0 \leq {\bar{v}}_{i, t} \leq (1 - α_{i, t}^{(2)}) Q^{(2)}

(A4)

0 \leq P_{i, t}^{VPP, s} \leq α_{i, t}^{(3)} Q^{(3)}

(A5)

0 \leq {\underset{̲}{u}}_{i, t} \leq (1 - α_{i, t}^{(3)}) Q^{(3)}

(A6)

0 \leq {\bar{P}}_{i}^{VPP} - P_{i, t}^{VPP, s} \leq α_{i, t}^{(4)} Q^{(4)}

(A7)

0 \leq {\bar{u}}_{i, t} \leq (1 - α_{i, t}^{(4)}) Q^{(4)}

(A8)

0 \leq P_{i, t}^{MT} \leq α_{i, t}^{(5)} Q^{(5)}

(A9)

0 \leq {\underset{̲}{ε}}_{i, t}^{MT} \leq (1 - α_{i, t}^{(5)}) Q^{(5)}

(A10)

0 \leq {\bar{P}}_{i}^{MT} - P_{i, t}^{MT} \leq α_{i, t}^{(6)} Q^{(6)}

(A11)

0 \leq {\bar{ε}}_{i, t}^{MT} \leq (1 - α_{i, t}^{(6)}) Q^{(6)}

(A12)

0 \leq P_{i, t}^{MT} - P_{i, t - 1}^{MT} - R_{i}^{MT, dn} \leq α_{i, t}^{(7)} Q^{(7)}

(A13)

0 \leq w_{i, t}^{dn} \leq (1 - α_{i, t}^{(7)}) Q^{(7)}

(A14)

0 \leq R_{i}^{MT, up} - P_{i, t}^{MT} + P_{i, t - 1}^{MT} \leq α_{i, t}^{(8)} Q^{(8)}

(A15)

0 \leq w_{i, t}^{up} \leq (1 - α_{i, t}^{(8)}) Q^{(8)}

(A16)

0 \leq (P_{i, t}^{ES} - {\underset{̲}{P}}_{i}^{ES}) \leq α_{i, t}^{(9)} Q^{(9)}

(A17)

0 \leq {\underset{̲}{ε}}_{i, t}^{ES} \leq (1 - α_{i, t}^{(9)}) Q^{(9)}

(A18)

0 \leq {\bar{P}}_{i}^{ES} - P_{i, t}^{ES} \leq α_{i, t}^{(10)} Q^{(10)}

(A19)

0 \leq {\bar{ε}}_{i, t}^{ES} \leq (1 - α_{i, t}^{(10)}) Q^{(10)}

(A20)

0 \leq S_{i, t}^{ES} - {\underset{̲}{S}}_{i}^{ES} \leq α_{i, t}^{(11)} Q^{(11)}

(A21)

0 \leq {\underset{̲}{η}}_{i, t} \leq (1 - α_{i, t}^{(11)}) Q^{(11)}

(A22)

0 \leq {\bar{S}}_{i}^{ES} - S_{i, t}^{ES} \leq α_{i, t}^{(12)} Q^{(12)}

(A23)

0 \leq {\bar{η}}_{i, t} \leq (1 - α_{i, t}^{(12)}) Q^{(12)}

(A24)

0 \leq P_{i, t}^{IL} \leq α_{i, t}^{(13)} Q^{(13)}

(A25)

0 \leq {\underset{̲}{ε}}_{i, t}^{IL} \leq (1 - α_{i, t}^{(13)}) Q^{(13)}

(A26)

0 \leq {\bar{P}}_{i, t}^{IL} - P_{i, t}^{IL} \leq α_{i, t}^{(14)} Q^{(14)}

(A27)

0 \leq {\bar{ε}}_{i, t}^{IL} \leq (1 - α_{i, t}^{(14)}) Q^{(14)}

(A28)

0 \leq P_{i, t}^{W} \leq α_{i, t}^{(15)} Q^{(15)}

(A29)

0 \leq {\underset{̲}{ε}}_{i, t}^{W} \leq (1 - α_{i, t}^{(15)}) Q^{(15)}

(A30)

0 \leq {\bar{P}}_{i, t}^{W} - P_{i, t}^{W} \leq α_{i, t}^{(16)} Q^{(16)}

(A31)

0 \leq {\bar{ε}}_{i, t}^{W} \leq (1 - α_{i, t}^{(16)}) Q^{(16)}

(A32)

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Figure 1. The trading structure of the DSO and VPP.

Figure 2. Framework of Stackelberg game model.

Figure 3. Forecast load for three VPPs.

Figure 4. Forecast wind for three VPPs.

Figure 9. Trading carbon prices.

Figure 10. Sum of VPP carbon allowance exchange with DSO.

Figure 11. Sharing carbon allowance of VPPs.

Table 1. Parameter setting of MT.

VPP	$a_{i}$	$b_{i}$	$c_{i}$	${\bar{P}}_{i}^{MT}$	$R_{i}^{MT, dn}$	$R_{i}^{MT, up}$
1	0.08	0.9	1.2	6 MW	−3.5 MW	3.5 MW
2	0.1	0.6	1	5 MW	−3 MW	3 MW
3	0.15	0.5	0.8	4 MW	−2 MW	2 MW

Table 2. Parameter setting of ES.

VPP	$E_{i}$	${\underset{̲}{P}}_{i}^{ES}$	${\bar{P}}_{i}^{ES}$	${\underset{̲}{S}}_{i}^{ES}$	${\bar{S}}_{i}^{ES}$
1	1 MWh	−0.6 MW	0.6 MW	0.2	0.9
2	1 MWh	−0.6 MW	0.6 MW	0.2	0.9
3	2 MWh	−1.2 MW	1.2 MW	0.2	0.9

Table 3. Optimal scheduling results of energy and carbon allowance sharing.

Case	Electricity (MWh)	Carbon Allowance (t)
Case 1	22.17	8.03
Case 2	28.43	9.61
Case 3	28.51	9.67

Table 4. VPPs’ costs and DSO’s profit in different cases (CNY

\times 10^{3}

Table 4. VPPs’ costs and DSO’s profit in different cases (CNY

\times 10^{3}

Case	DSO Profit	VPP1 Cost	VPP2 Cost	VPP3 Cost
Case 1	10.073	71.434	34.448	60.158
Case 2	11.720	71.429	34.105	59.859
Case 3	11.767	71.417	34.097	59.849

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Xu, D.; Li, M. A Stackelberg Game Model for the Energy–Carbon Co-Optimization of Multiple Virtual Power Plants. Inventions 2025, 10, 16. https://doi.org/10.3390/inventions10010016

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Xu D, Li M. A Stackelberg Game Model for the Energy–Carbon Co-Optimization of Multiple Virtual Power Plants. Inventions. 2025; 10(1):16. https://doi.org/10.3390/inventions10010016

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Xu, D., & Li, M. (2025). A Stackelberg Game Model for the Energy–Carbon Co-Optimization of Multiple Virtual Power Plants. Inventions, 10(1), 16. https://doi.org/10.3390/inventions10010016

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A Stackelberg Game Model for the Energy–Carbon Co-Optimization of Multiple Virtual Power Plants

Abstract

1. Introduction

2. Stackelberg Game Model of DSO and VPPs

2.1. Upper-Level Model: Benefit Maximization of DSO

2.1.1. Objective Function of Upper-Level Model

2.1.2. Constraints of Upper-Level Model

2.2. Lower-Level Model: Cost Minimization of VPP

2.2.1. Objective Function of Lower-Level Model

2.2.2. Constraints of Lower-Level Model

2.3. Stackelberg Game Model

3. Stackelberg Game Model Solution Method

3.1. MPEC Formulation of DSO and VPP

3.2. Linearized MPEC Formulation

4. Case Study

4.1. Data Description

4.2. Analysis of Optimization Results

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

Appendix A

References

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