Gradient Revision Method for Demand Response Stimulus Parameters of the Integrated Energy System

Abstract

Integrated Demand Response (IDR) enhances the operational flexibility of Integrated Energy Systems (IES) and promotes renewable energy integration. However, limited interaction between the Integrated Energy Operator (IEO) and users during actual energy transactions can lead to biases in IDR planning, compromising user response effectiveness. To address this, this paper proposes a method for revising IDR stimulus parameters in IES based on gradient descent within a Stackelberg game framework. First, an IDR model based on consumer psychology principles is constructed to establish an IES Stackelberg game, in which the IEO acts as the leader and the load aggregator acts as the follower. Subsequently, during the game, the IEO utilizes users’ energy consumption strategies to adjust the stimulus threshold parameters of the dead zone and saturation zone along the negative gradient direction, thereby updating its decision for the next round. Furthermore, a response adjustment mechanism is designed to refine the IDR plan, enhancing its feasibility. Finally, comparative analyses across diverse scenarios demonstrate that the proposed method reduces deviations in planned IDR, thereby enhancing the low-carbon performance and renewable energy integration capacity of IES.

1. Introduction

Driven by the imperative to address climate change and advance the global energy transition, energy systems are evolving toward clean, low-carbon, secure, and efficient operation. The development of multi-energy coupling technologies and the deep exploration of flexible demand-side resources are crucial for supply–demand regulation [1,2]. Integrated energy systems (IES) overcome the limitations of traditional single-energy systems by leveraging the complementary and mutual support among multiple energy sources, thereby enhancing overall energy utilization efficiency and system flexibility [3,4]. Integrated demand response (IDR), as a key demand-side management tool, promotes IES supply–demand balance by steering users to modify their energy consumption behaviors and unlocking adjustable load resources [5]. However, implementing IDR often impacts users’ energy comfort and experience. To mitigate this conflict, integrated energy operators (IEOs) develop targeted pricing and incentive strategies based on users’ distinct energy consumption characteristics and preferences, thereby enhancing the dispatchability of demand-side resources and load response effectiveness. Therefore, considering the bidirectional interaction between IEO and users, exploring how IES can accurately capture and utilize user energy characteristics to enhance IDR implementation effectiveness is crucial for improving IES economics and market-oriented development [6].

Currently, many scholars simulate the supply–demand interaction in energy markets by constructing Stackelberg game models. In reference [7], a bidirectional Stackelberg game model incorporating shared energy storage systems was proposed, employing the bisection method for solution. In reference [8], a Stackelberg game model was proposed to optimize the efficiency of integrated energy systems. This model utilizes the exergy properties of electrical and thermal energy and employs genetic algorithms to achieve multi-objective optimization. However, Stackelberg game models solved using iterative algorithms implicitly require users to possess high-frequency autonomous decision-making capabilities [9], which is not conducive to user participation in demand response and the market-oriented operation of demand response. Furthermore, the computational complexity and number of iterations required by traditional iterative algorithms to solve bilevel models are typically significantly higher than those for single-layer problems. Considering the actual supply–demand interactions and to reduce computational complexity for the solution, some scholars have transformed the two-layer model of principal-agent games into a single-layer model using Karush-Kuhn-Tucker (KKT) conditions [10,11]. However, solving the model using centralized algorithms requires complete information from the user side, which is similarly challenging in practice. Moreover, computational burden increases as the number of market participants grows. Consequently, some researchers have begun exploring how to construct more refined and complex user-side models. These models simulate real user feedback to assist operators in independently making precise decisions. In reference [12], a Stackelberg game model was developed based on demand response to simultaneously coordinate logistics operations, port energy scheduling, and vessel power requirements. In reference [13], a unified user satisfaction metric was proposed for cooling, heating, and electricity loads, based on a comprehensive consideration of their demand response. In reference [14], an EV charging/discharging incentive model that accounts for battery degradation was established, with the electric vehicle load aggregator positioned as a follower, to achieve refined energy management. While these studies examine the construction and solution of Stackelberg game models from various perspectives, they encounter challenges in simultaneously balancing the reduction in solution iterations and minimizing the requirement for user information or detailed user-side model construction.

Currently, IDR models that integrate users’ energy consumption characteristics can support decision-making for operators and electricity retailers, which has led scholars to conduct in-depth research on such models and classify them into price-based demand response (PBDR), incentive-based demand response (IBDR), and substitution-based demand response (SBDR) [15]. If the IEO can capture users’ energy consumption characteristics, it can design pricing strategies to steer IDR effectively. However, parameters reflecting users’ inherent response characteristics are difficult to acquire directly and thus require estimation using historical IDR data. In reference [16], support vector machines were applied to assess the potential of demand response. In reference [17], an LSTM-based method was proposed to recognize and predict user response behavior. In reference [18], a reinforcement learning model was presented to derive optimal incentive strategies based on user characteristics and historical data from the preceding week. Although these methods accurately estimate IDR parameters, IDR data is limited due to the ongoing development of services. This data scarcity makes it difficult for sample-dependent approaches to effectively estimate key parameters [19].

Consequently, researchers have turned to traditional statistical methods to estimate parameters related to user energy consumption characteristics in IDR models. In reference [20], a linear regression algorithm was utilized to estimate the parameters of the user response model, based on historical data within the IBDR framework. In reference [21], the least squares method was used to estimate user risk-sensitivity parameters for day-ahead demand response. In reference [22], a moving-window linear regression method utilizing maximum likelihood estimation was introduced for determining the unknown parameters of the IDR model. Existing traditional statistical methods have achieved success in improving the IDR model parameter estimation accuracy. However, these methods rely solely on historical IDR data. Consequently, they fail to reflect dynamic changes in user energy consumption characteristics during actual game interactions. This limitation often leads to systematic biases in IDR model parameters and affects the applicability of the model.

To address the problems mentioned as in

Table 1. Comparison of the research in this paper with related works.

Refs.	Solution Algorithm		Parameter Estimation		Parameter Revision During Optimization
	Centralized	Decentralized	Data-Driven	Traditional Statistical	Yes	No
[7,8,12,13,14]		√
[10,11]	√
[16,17,18]			√
[20,21,22]				√		√
This paper		√		√	√

, this study proposes a gradient descent-based integrated response stimulus threshold parameters revision method (GD-DRPR) for adjusting stimulus threshold parameters of the dead zone and saturation zone of the IDR model based on the principle of consumer psychology during the IES Stackelberg game. Compared to the KKT-based method, GD-DRPR does not require the IEO to have complete information about the load aggregator (LA). Unlike traditional distributed algorithms for solving Stackelberg game models, which often require numerous rounds of information exchange between the IEO and LA, GD-DRPR incorporates the demand response model as an approximate reaction function of the LA. This enables the IEO to make decisions by anticipating the LA’s responses, thereby improving the efficiency of transactions. Note that in previous studies, key parameters of the IDR model are usually estimated using historical data prior to the optimization or game process and remain fixed during optimization. GD-DRPR is not intended to improve the method of parameter estimation but focuses on revising original parameters during the game process based on user feedback.

The main contributions of this paper are summarized as follows:

• This paper establishes an IES optimization model based on the Stackelberg game. As the leader, the IEO sets the energy selling price and demand response incentive compensation price according to the estimated response of the LA from the IDR model to maximize profits. As a follower, LA optimizes energy consumption strategies by integrating demand response resources to minimize costs.
• This paper constructs a demand response model based on the principle of consumer psychology. It proposes the GD-DRPR by establishing an objective function based on the difference between expected demand response rates of the IEO and the LA. This approach improves the IEO’s decision-making within the IDR model and enables precise mobilization of LA resources. Furthermore, it reduces the number of interactions between the IEO and the LA, increases transaction efficiency, and lowers the barrier for user participation in IDR.
• A response adjustment mechanism is designed. After the Stackelberg game concludes, the IEO verifies and adjusts the IDR plan with the LA to ensure its practical feasibility.

The remainder of this paper is organized as follows: Section 2 introduces the Stackelberg game framework. Section 3 presents the demand response model and GD-DRPR. Section 4 describes the IEO and LA models, along with the solution process for the Stackelberg game under GD-DRPR. In Section 5, a case study is conducted, and the results are discussed. Finally, Section 6 provides the conclusion.

2. Structure of IES and Stackelberg Game Framework

2.1. Structure of IES

The IES structure is shown in Figure 1. This system uses an IEO and an LA to coordinate transactions of electricity, gas, heat, and cooling between users and the upper-level energy network. The IEO is equipped with photovoltaic (PV), wind turbine (WT), combined heat and power (CHP), gas boiler (GB), electric boiler (EB), absorption chiller (AC), electric chiller (EC), electrical storage system (ESS), and heat storage system (HSS). The LA aggregates electrical, heat, and cooling loads. The IEO is located between the higher-level energy network and the LA, which manages the energy equipment within the IES and participates in market transactions. The IEO sets both the energy purchase price for LA and the incentive compensation price for demand response. Once LA accepts these prices, it aggregates flexible demand-side resources to provide IDR services. If the IEO cannot meet the LA’s energy demand with its own resources, it may purchase energy from the upper-level power grid or gas network. In this paper, the IEO purchases electricity from the upper-level power grid at time-of-use grid tariffs and natural gas from the upper-level gas network at a fixed price, while LA only purchases energy from the IEO.

2.2. Stackelberg Game Framework for IES Under GD-DRPR

The Stackelberg game model framework constructed in this paper is shown in Figure 2. The upper-layer model is the operational planning and pricing strategy decision-making model for the leader IEO, which determines the energy purchasing plan, equipment output plan, energy selling price, and incentive compensation price. The lower-layer model is the energy consumption decision-making model for the follower LA, which determines the energy purchasing plan and the expected IDR.

Unlike traditional IES Stackelberg game models, the interaction mechanism is enhanced by the GD-DRPR. In this framework, the lower-layer LA decision model optimizes its energy consumption strategy based on current prices and transmits this expected response plan to the IEO. Upon receiving the LA’s feedback, the IEO does not directly input this load data into the upper-layer pricing model. Instead, the IEO utilizes the LA’s energy consumption strategy as a reference to execute the GD-DRPR process, thereby revising the stimulus threshold parameters of the dead zone and saturation zone. Subsequently, these revised parameters are incorporated into the upper-layer IEO decision model to generate more precise pricing and incentive strategies for the next iteration. After the iterative game process converges, the IEO receives the confirmed response level from the LA and adjusts the energy procurement plan and equipment output plan based on the LA’s expected response.

3. IDR Model and Its Threshold Parameters Revision Method

3.1. A Demand Response Model Based on the Principle of Consumer Psychology

According to the principle of consumer psychology, consumer responses to economic stimulus exhibit significant differences when reaching distinct psychological thresholds. A demand response model was constructed based on this principle, shown in Figure 3. According to users’ demand response characteristics at different stimulus levels, this model categorizes the response process into three stages, including the dead zone, linear zone, and saturation zone [23]. IDR can be categorized into PBDR, SBDR, and IBDR, and aligns with the fundamental assumptions of the principle of consumer psychology. Therefore, the model is extended for application within the IDR framework [23,24]. The relationship between demand response stimulus levels $r_t^{x,y}$ and demand response rates ${\lambda }_t^{x,y}$ can be expressed as follows:

$$0<r_{t,\min }^{x,y}<r_{t,\max }^{x,y}$$

(1)

$${\lambda }_t^{x,y}=\left\{\begin{array}{l}\begin{array}{cc}0,& 0\le r_{t,\min }^{x,y}\le r_{t,\max }^{x,y}\end{array}\\ \begin{array}{cc}{\displaystyle \frac{{\lambda }_{\max }^{x,y}(r_t^{x,y}-r_{t,\min }^{x,y})}{r_{t,\max }^{x,y}-r_{t,\min }^{x,y}}},& r_{t,\min }^{x,y}\le r_t^{x,y}\le r_{t,\max }^{x,y}\end{array}\\ \begin{array}{cc}{\lambda }_{\max }^{x,y},& r_{t,\max }^{x,y}\le r_t^{x,y}\end{array}\end{array}\right.$$

(2)

where $r_{t,\min }^{x,y}$ and $r_{t,\max }^{x,y}$ are the dead zone and saturation zone thresholds of demand response stimulus level at time t, respectively; ${\lambda }_{\max }^{x,y}$ is the maximum demand response rate; the superscript x denotes the type of responsive load, where $x\in \left\{\mathrm{TL},\mathrm{SL},\mathrm{CL}\right\}$ with TL, SL, and CL represent transferable load, substitutable load, and curtailable load, respectively; and the superscript y denotes the energy type of load, where $y\in \left\{\mathrm{e},\mathrm{h},\mathrm{c}\right\}$ with e, h, and c represent electrical, heat, and cooling loads, respectively.

3.2. IDR Model

3.2.1. PBDR Model

This paper temporarily disregards the transfer behavior of heat and cooling loads, focusing only on the transfer of electrical loads to construct the PBDR model. The IEO guides LA participation in peak shaving and valley filling by setting electricity prices, with the difference between the current price and the historical average price serving as the stimulus level for PBDR. Combining Equations (1) and (2), the PBDR model is expressed as follows:

$$\left\{\begin{array}{l}{P}_t^{\mathrm{TL},\mathrm{e}}=P_t^{\mathrm{L},\mathrm{e}}{\lambda }_t^{\mathrm{TL},\mathrm{e}}\\ {\displaystyle \sum _t^T{P}_t^{\mathrm{TL},\mathrm{e}}=0}\end{array}\right.$$

(3)

$$\left\{\begin{array}{l}{\lambda }_t^{\mathrm{TL},\mathrm{e}}={\displaystyle \frac{{\lambda }_{\max }^{\mathrm{TL},\mathrm{e}}(U_t^{\mathrm{TL},\mathrm{e},\mathrm{out}}-U_t^{\mathrm{TL},\mathrm{e},\mathrm{in}})(r_t^{\mathrm{TL},\mathrm{e}}-r_{t,\min }^{\mathrm{TL},\mathrm{e}})}{r_{t,\max }^{\mathrm{TL},\mathrm{e}}-r_{t,\min }^{\mathrm{TL},\mathrm{e}}}}\\ r_t^{\mathrm{TL},\mathrm{e}}=(U_t^{\mathrm{TL},\mathrm{e},\mathrm{out}}-U_t^{\mathrm{TL},\mathrm{e},\mathrm{in}})(c_t^{\mathrm{e}}-\overline{c^{\mathrm{e}}})\\ r_{t,\max }^{\mathrm{TL},\mathrm{e}}\le r_t^{\mathrm{TL},\mathrm{e}}\le r_{t,\min }^{\mathrm{TL},\mathrm{e}}\\ U_t^{\mathrm{TL},\mathrm{out}}{U}_t^{\mathrm{TL},\mathrm{in}}=0\end{array}\right.$$

(4)

where $P_t^{\mathrm{L},\mathrm{e}}$ is the baseline electric load at time t; $P_t^{\mathrm{TL},\mathrm{e}}$ is the shiftable electric load at time t; T is the total scheduling period; ${\lambda }_t^{\mathrm{TL},\mathrm{e}}$ is the rate of the PBDR for electric energy at time t; ${\lambda }_{\max }^{\mathrm{TL},\mathrm{e}}$ is the maximum rate of the PBDR for electric energy; $r_{t,\max }^{\mathrm{TL},\mathrm{e}}$ and $r_{t,\min }^{\mathrm{TL},\mathrm{e}}$ are the dead zone and saturation zone thresholds for the stimulus level of the PBDR for electric energy at time t, respectively; $c_t^{\mathrm{e}}$ is the electricity selling price of the IEO at time t; $\overline{c^{\mathrm{e}}}$ is the historical average electricity selling price; $U_t^{\mathrm{TL},\mathrm{e},\mathrm{out}}$ is a binary variable indicating the state of shifting electric load from time t to other periods, taking a value of 1 if the load is shifted and 0 otherwise; and $U_t^{\mathrm{TL},\mathrm{e},\mathrm{in}}$ is a binary variable indicating the state of shifting electric load from other periods to time t, taking a value of 1 if the load is shifted and 0 otherwise.

3.2.2. SBDR Model

IEO enables substitutable loads to participate in SBDR by setting different energy prices, which steer the LA to adjust its energy consumption structure. The model considers only the energy substitution between electricity–heat and electricity–cooling loads, establishing SBDR stimulus levels based on the price differentials between electricity and heat, and between electricity and cooling [23]. By combining Equations (1) and (2), the SBDR model is as follows:

$$\left\{\begin{array}{l}{P}_t^{\mathrm{SL},\mathrm{e}}={\eta }_{\mathrm{hte}}{P}_t^{\mathrm{SL},\mathrm{h}}+{\eta }_{\mathrm{cte}}{P}_t^{\mathrm{SL},\mathrm{c}}\\ -w_{\max }^{\mathrm{SL},\mathrm{e}}{P}_t^{\mathrm{L},\mathrm{e}}\le P_{\mathrm{SL},\mathrm{e}}^t\le w_{\max }^{\mathrm{SL},\mathrm{e}}{P}_t^{\mathrm{L},\mathrm{e}}\end{array}\right.$$

(5)

$$\left\{\begin{array}{l}{P}_t^{\mathrm{SL},\mathrm{h}}=(U_{t,\mathrm{eth}}^{\mathrm{SL}}-U_{t,\mathrm{hte}}^{\mathrm{SL}})P_t^{\mathrm{L},\mathrm{h}}{\lambda }_t^{\mathrm{SL},\mathrm{h}}\\ r_t^{\mathrm{SL},\mathrm{h}}=(U_{t,\mathrm{eth}}^{\mathrm{SL}}-U_{t,\mathrm{hte}}^{\mathrm{SL}})(c_t^{\mathrm{h}}-c_t^{\mathrm{e}})\\ r_{t,\min }^{\mathrm{SL},\mathrm{h}}\le r_t^{\mathrm{SL},\mathrm{h}}\le r_{t,\max }^{\mathrm{SL},\mathrm{h}}\\ {\lambda }_t^{\mathrm{SL},\mathrm{h}}={\displaystyle \frac{{\lambda }_{\max }^{\mathrm{SL},\mathrm{h}}{r}_t^{\mathrm{SL},\mathrm{h}}}{r_{t,\max }^{\mathrm{SL},\mathrm{h}}-r_{t,\min }^{\mathrm{SL},\mathrm{h}}}}\\ U_{t,\mathrm{hte}}^{\mathrm{SL}}{U}_{t,\mathrm{eth}}^{\mathrm{SL}}=0\end{array}\right.$$

(6)

$$\left\{\begin{array}{l}{P}_t^{\mathrm{SL},\mathrm{c}}=(U_{t,\mathrm{etc}}^{\mathrm{SL}}-U_{t,\mathrm{c}\mathrm{t}\mathrm{e}}^{\mathrm{SL}})P_t^{\mathrm{L},\mathrm{c}}{\lambda }_t^{\mathrm{SL},\mathrm{c}}\\ r_t^{\mathrm{SL},\mathrm{c}}=(U_{t,\mathrm{etc}}^{\mathrm{SL}}-U_{t,\mathrm{c}\mathrm{t}\mathrm{e}}^{\mathrm{SL}})(c_t^{\mathrm{c}}-c_t^{\mathrm{e}})\\ r_{t,\min }^{\mathrm{SL},\mathrm{c}}\le r_t^{\mathrm{SL},\mathrm{c}}\le r_{t,\max }^{\mathrm{SL},\mathrm{c}}\\ {\lambda }_t^{\mathrm{SL},\mathrm{c}}={\displaystyle \frac{{\lambda }_{\max }^{\mathrm{SL},\mathrm{c}}{r}_t^{\mathrm{SL},\mathrm{c}}}{r_{t,\max }^{\mathrm{SL},\mathrm{c}}-r_{t,\min }^{\mathrm{SL},\mathrm{c}}}}\\ U_{t,\mathrm{c}\mathrm{t}\mathrm{e}}^{\mathrm{SL}}{U}_{t,\mathrm{etc}}^{\mathrm{SL}}=0\end{array}\right.$$

(7)

where $w_{\max }^{\mathrm{SL},\mathrm{e}}$ is the upper limit proportion of the substitutable electric load; ${\eta }_{\mathrm{hte}}$ and ${\eta }_{\mathrm{cte}}$ are the energy substitution coefficients for heat-to-electrical and cooling-to-electrical, respectively; $P_t^{\mathrm{SL},\mathrm{e}}$, $P_t^{\mathrm{SL},\mathrm{h}}$ and $P_t^{\mathrm{SL},\mathrm{c}}$ are the substitutable electrical, heat and cooling loads at time t, respectively; $P_t^{\mathrm{L},\mathrm{h}}$ and $P_t^{\mathrm{L},\mathrm{c}}$ are the baseline heat load and baseline cooling load at time t, respectively; $U_{t,\mathrm{hte}}^{\mathrm{SL}}$, $U_{t,\mathrm{eth}}^{\mathrm{SL}}$, $U_{t,\mathrm{c}\mathrm{t}\mathrm{e}}^{\mathrm{SL}}$ and $U_{t,\mathrm{etc}}^{\mathrm{SL}}$ are binary variables denote the states of heat-to-electrical, electrical-to-heat, cooling-to-electrical, and electrical-to-cooling load substitution at time t, taking a value of 1 if substitution occurs and 0 otherwise; $r_t^{\mathrm{SL},\mathrm{h}}$, $r_{t,\min }^{\mathrm{SL},\mathrm{h}}$ and $r_{t,\max }^{\mathrm{SL},\mathrm{h}}$ are the stimulus level of the heat energy SBDR at time t and its dead zone and saturation zone thresholds, respectively; ${\lambda }_t^{\mathrm{SL},\mathrm{h}}$ and ${\lambda }_{\max }^{\mathrm{SL},\mathrm{h}}$ are the rate of SBDR for heat energy at time t and its upper limit, respectively; $r_t^{\mathrm{SL},\mathrm{c}}$, $r_{t,\min }^{\mathrm{SL},\mathrm{c}}$ and $r_{t,\max }^{\mathrm{SL},\mathrm{c}}$ are the stimulus level of the cooling energy SBDR at time t and its dead zone and saturation zone thresholds, respectively; ${\lambda }_t^{\mathrm{SL},\mathrm{c}}$ and ${\lambda }_{\max }^{\mathrm{SL},\mathrm{c}}$ are the rate of SBDR for cooling energy at time t and its upper limit, respectively; $c_t^{\mathrm{h}}$ and $c_t^{\mathrm{c}}$ are the heating and cooling selling prices of the IEO at time t, respectively.

3.2.3. IBDR Model

IEO incentivizes curtailable loads to participate in IBDR by contracting with LA and setting the stimulus level as $r_t^{\mathrm{CL},y}=c_t^{\mathrm{CL},y}$, where $c_t^{\mathrm{CL},y}$ is the incentive compensation price at time t. Combining Equations (1) and (2), the IBDR model is as follows:

$$P_t^{\mathrm{CL},y}=P_t^{\mathrm{L},y}{\lambda }_t^{\mathrm{CL},y}$$

(8)

$$\left\{\begin{array}{l}{\lambda }_t^{\mathrm{CL},y}={\displaystyle \frac{{\lambda }_{\max }^{\mathrm{CL},y}{r}_t^{\mathrm{CL},y}}{r_{t,\max }^{\mathrm{CL},y}-r_{t,\min }^{\mathrm{CL},y}}}\\ U_t^{\mathrm{CL},y}{r}_{\min ,t}^{\mathrm{CL},y}\le r_t^{\mathrm{CL},y}\le U_t^{\mathrm{CL},y}{r}_{t,\max }^{\mathrm{CL},y}\end{array}\right.$$

(9)

where $P_t^{\mathrm{CL},y}$ is the curtailable load of y class energy at time t; ${\lambda }_{\max }^{\mathrm{CL},y}$ denotes the upper limit of the IBDR rate for energy y; $r_{t,\max }^{\mathrm{CL},y}$ and $r_{t,\min }^{\mathrm{CL},y}$ denote the dead zone and saturation zone threshold values, respectively, for the stimulus level of IBDR at time t; and $U_t^{\mathrm{CL},y}$ is a 0–1 variable indicating the load curtailment status at time t, where 1 signifies load curtailment, and 0 signifies no curtailment.

3.3. Revision Method for IDR Stimulus Threshold Parameters

3.3.1. GD-DRPR Process

The $K_t^{x,y}={\lambda }_{\max }^{x,y}/(r_{t,\max }^{x,y}-r_{t,\min }^{x,y})$ is the slope of the linear region in Figure 3 and indicates user sensitivity to demand response stimulus level, while $r_{t,\max }^{x,y}$ and $r_{t,\min }^{x,y}$ are key parameters derived from historical data [19]. This historical derivation is often inaccurate because it fails to capture recent dynamics, including behavioral shifts, equipment updates, or schedule changes, and because the IEO has incomplete user information. To improve accuracy, $r_{t,\max }^{x,y}$ and $r_{t,\min }^{x,y}$ are revised by ${\lambda }_{t,n}^{\mathrm{IEO},x,y}$ and ${\lambda }_{t,n}^{\mathrm{LA},x,y}$, where ${\lambda }_{t,n}^{\mathrm{IEO},x,y}$ and ${\lambda }_{t,n}^{\mathrm{LA},x,y}$ are the expected demand response rates of the IEO and the LA at time t. The objective function $J$ is then formulated based on these parameters, Equation (2), and the least squares method:

$$\min J(r_{t,n,\min }^{x,y},r_{t,n,\max }^{x,y})={[{\lambda }_{\max }^{x,y}(r_{t,n}^{x,y}-r_{t,n-1,\min }^{x,y})-{\lambda }_{t,n}^{\mathrm{LA},x,y}(r_{t,n-1,\max }^{x,y}-r_{t,n-1,\min }^{x,y})]}^2$$

(10)

where n is the iteration step count for the inner iteration within GD-DRPR.

Furthermore, considering that the model depicted in Figure 3 provides only an approximate description of actual response behavior, an excessively rapid decrease in the objective function value during the incentive threshold parameters adjustment process may cause these parameters to deviate from the true response characteristics. Therefore, the GD-DRPR method is proposed based on the gradient descent approach.

This paper compares the expected post-response load from the IEO and the LA. The mean absolute percentage error (MAPE) is utilized as the convergence indicator for the GD-DRPR algorithm. To evaluate the effectiveness of revision for IDR stimulus threshold parameters, both MAPE and the root mean square error (RMSE) are employed as performance metrics. The expressions for these metrics are as follows:

$$D_{\mathrm{MAPE}}(n)={\displaystyle \frac{100\%}{3T}}{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{y\in \left\{\mathrm{e},\mathrm{g},\mathrm{h}\right\}}\left|\frac{P_{t,n}^{\mathrm{IEO},y}-P_{t,n}^{\mathrm{LA},y}}{P_{t,n}^{\mathrm{LA},y}}\right|}}$$

(11)

$$R_{\mathrm{MSE}}(n)=\sqrt{{\displaystyle \frac{1}{3T}}{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{y\in \left\{\mathrm{e},\mathrm{g},\mathrm{h}\right\}}{(P_{t,n}^{\mathrm{IEO},y}-P_{t,n}^{\mathrm{LA},y})}^2}}}$$

(12)

where $P_{t,n}^{\mathrm{IEO},y}$ and $P_{t,n}^{\mathrm{LA},y}$ are the expected post-response load of y class energy at time t during iteration n for IEO and LA, respectively.

GD-DRPR is a dual-layer iterative process. The outer layer models the game-theoretic interaction between IEO and LA. In the inner layer, IEO iteratively revises the IDR stimulus threshold parameter. The process proceeds as follows:

1.

Set the initial parameters as n = 0, m = 0, $D_{\mathrm{MAPE}}(n)=100\%$ and $D(m)=100\%$, where m is the count of interactions between the IEO and LA, and $D(m)$ is the MAPE calculated following each interaction. The IEO performs its initial decision-making based on the initial IDR stimulus threshold.

2.

Let m = m + 1. The LA executes a decision-making round according to the current price, subsequently providing the expected response rate ${\lambda }_{t,n}^{\mathrm{LA},x,y}$ to the IEO.

3.

Let n = n + 1. The gradient of the current objective function is then calculated with the following formula:

$${\displaystyle \frac{\mathit{\partial }J}{\partial r_{t,n-1,\min }^{x,y}}}=M({\lambda }_{t,n}^{\mathrm{LA},x,y}-{\lambda }_{\max }^{x,y})$$

(13)

$${\displaystyle \frac{\partial J}{\partial r_{t,n-1,\max }^{x,y}}}=-M{\lambda }_{t,n}^{\mathrm{LA},x,y}$$

(14)

$$M={\lambda }_{\max }^{x,y}(r_{t,n}^{x,y}-r_{t,n-1,\min }^{x,y})-{\lambda }^{\mathrm{LA},x,y}(r_{t,n-1,\max }^{x,y}-r_{t,n-1,\min }^{x,y})$$

(15)

4.

The IDR stimulus threshold parameters are updated along the negative gradient using the following formula:

$$r_{t,n,\min }^{x,y}=r_{t,n-1,\min }^{x,y}+\alpha M({\lambda }_{\max }^{x,y}-{\lambda }_{t,n}^{\mathrm{LA},x,y})$$

(16)

$$r_{t,n,\max }^{x,y}=r_{t,n-1,\max }^{x,y}+\alpha M{\lambda }_{t,n}^{\mathrm{LA},x,y}$$

(17)

where $\alpha$ is the iterative step size.

5.

The IEO determines its pricing strategies based on the current IDR stimulus threshold parameters.

6.

Calculate $D_{\mathrm{MAPE}}(n)$ for iteration n. If $D_{\mathrm{MAPE}}(n)\ge D_{\mathrm{MAPE}}(n-1)$, let $r_{t,n,\min }^{x,y}=r_{t,n-1,\min }^{x,y}$, $r_{t,n,\max }^{x,y}=r_{t,n-1,\max }^{x,y}$, and $D(m)=D_{\mathrm{MAPE}}(n)$. Otherwise, return to Step 3 to continue the inner loop.

7.

If $\left|D(m)-D(m-1)\right|<0.01\%$, the algorithm concludes. Otherwise, pass the current pricing strategy to LA and return to Step 2 to begin a new outer loop.

3.3.2. Proof of Convergence for GD-DRPR

The GD-DRPR mentioned in this text involves inner and outer iterations. The outer iteration corresponds to traditional Stackelberg games of IES, while the inner iteration involves the adjustment of stimulus threshold parameters, thus requiring proof of the convergence of the inner iteration. This section utilizes the Lyapunov function to prove the convergence of the GD-DRPR inner iteration.

Theorem 1. 

Global convergence to the minimum point of the objective function $J(\mathit{r})$.

Proof of Theorem 1. 

Let $\mathit{r}={[r_{t,\min }^{x,y},r_{t,\max }^{x,y}]}^{\mathrm{T}}$, and let $w={[{\lambda }_t^{\mathrm{LA},x,y}-{\lambda }_{\max }^{x,y},-{\lambda }_t^{\mathrm{LA},x,y}]}^{\mathrm{T}}$. Choose the Lyapunov function as $V(\mathit{r})=J(\mathit{r})$, and when $V(\mathit{r})=0$, it corresponds to the global minimum point, then:

$$\Delta V_n=V_{n+1}({\mathit{r}}_n)-V_n({\mathit{r}}_{n-1})$$

(18)

Based on updated Formulas (13)–(15):

$$\Delta V_n=V_n({\mathit{r}}_{n-1})[{(1-2\alpha {‖{\mathit{w}}_n‖}^2)}^2-1]$$

(19)

To ensure $\Delta V_n\le 0$, it is necessary that:

$$1-2\alpha {‖{\mathit{w}}_n‖}^2{)}^2-1\le 0$$

(20)

Solving this inequality gives:

$$0\le \alpha \le {\displaystyle \frac{1}{{‖{\mathit{w}}_n‖}^2}}$$

(21)

Under this condition, $\Delta V_n\le 0$. Since $w_n$ does not change during each interaction between IEO and LA, when $V(\mathit{r})>0$ and $0\le \alpha \le 1/{‖{\mathit{w}}_n‖}^2$, $\Delta V_n\le 0$ and $J(\mathit{r})$ globally converges to the minimum point of the objective function. ☐

4. IES Stackelberg Game Model

4.1. IEO Pricing Decision Model

4.1.1. Objective Function

The IEO steers the LA’s response by setting energy selling prices and incentive compensation. The objective of optimization is to maximize the IEO’s daily revenue, formulated as follows:

$$\max C^{\mathrm{IEO}}=C^{\mathrm{sell}}-C^{\mathrm{buy}}-C^{\mathrm{om}}-C^{\mathrm{idr}}-C^{\mathrm{CO}2}$$

(22)

$$C^{\mathrm{sell}}={\displaystyle \sum _{t=1}^T(c_t^{\mathrm{e}}{P}_t^{\mathrm{IEO},\mathrm{e}}+c_t^{\mathrm{h}}{P}_t^{\mathrm{IEO},\mathrm{h}}+c_t^{\mathrm{c}}{P}_t^{\mathrm{IEO},\mathrm{c}})}$$

(23)

$$C^{\mathrm{buy}}={\displaystyle \sum _{t=1}^T(P_t^{\mathrm{GRID}}{p}_t^{\mathrm{GRID}}+G_t^{\mathrm{GAS}}{p}_t^{\mathrm{GAS}})}$$

(24)

$$C^{\mathrm{om}}={\displaystyle \sum _{t=1}^T\left[\begin{array}{l}{c}^{\mathrm{CHP}}{P}_t^{\mathrm{CHP}}+c^{\mathrm{GB}}{H}_t^{\mathrm{GB}}+c^{\mathrm{EB}}{H}_t^{\mathrm{EB}}+c^{\mathrm{AC}}{Q}_t^{\mathrm{AC}}+c^{\mathrm{EC}}{Q}_t^{\mathrm{EC}}+\\ c^{\mathrm{ESS}}(P_{t,\mathrm{dis}}^{\mathrm{ESS}}-P_{t,\mathrm{char}}^{\mathrm{ESS}})+c^{\mathrm{HSS}}(P_{t,\mathrm{dis}}^{\mathrm{HSS}}-P_{t,\mathrm{char}}^{\mathrm{HSS}})\end{array}\right]}$$

(25)

where $C^{\mathrm{IEO}}$ is the daily operational revenue of the IEO; $C^{\mathrm{sell}}$ is the energy sales revenue of the IEO; and $C^{\mathrm{buy}}$, $C^{\mathrm{om}}$, $C^{\mathrm{idr}}$, and $C^{\mathrm{CO}2}$ are the energy purchase cost, equipment maintenance cost, IDR incentive compensation cost, and carbon emission cost of the IEO, respectively. $P_t^{\mathrm{GRID}}$ and $G_t^{\mathrm{GAS}}$ are the electricity and gas purchased by IEO at time t, respectively; $p_t^{\mathrm{GRID}}$ and $p_t^{\mathrm{GAS}}$ are the electricity and gas purchase prices from the upper-level energy network, respectively. $P_t^{\mathrm{CHP}}$ is the electric energy output of the CHP at time t; $H_t^{\mathrm{GB}}$ and $H_t^{\mathrm{EB}}$ are the heat energy output of GB and EB at time t, respectively; $Q_t^{\mathrm{EC}}$ and $Q_t^{\mathrm{AC}}$ are the cooling energy output of EC and AC at time t, respectively; $P_{t,\mathrm{char}}^{\mathrm{ESS}}$ and $P_{t,\mathrm{dis}}^{\mathrm{ESS}}$ are the charging and discharging electric energy of the ESS at time t, respectively; $P_{t,\mathrm{char}}^{\mathrm{HSS}}$ and $P_{t,\mathrm{dis}}^{\mathrm{HSS}}$ are the charging and discharging heat energy of the HSS at time t, respectively; and $c^{\mathrm{CHP}}$, $c^{\mathrm{GB}}$, $c^{\mathrm{EB}}$, $c^{\mathrm{AC}}$, $c^{\mathrm{EC}}$, $c^{\mathrm{ESS}}$, and $c^{\mathrm{HSS}}$ are the operating maintenance costs per unit output power for the CHP, GB, EB, AC, EC, ESS, and HSS, respectively.

For IBDR, IEO must simultaneously establish incentive compensation prices and IBDR plans, where the IDR compensation cost is as follows:

$$C^{\mathrm{idr}}={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{y\in \left\{\mathrm{e},\mathrm{g},\mathrm{h}\right\}}{c}_t^{\mathrm{CL},y}{P}_t^{\mathrm{CL},y}}}$$

(26)

Carbon emission allowances are allocated based on the baseline method. If emissions exceed the allocated allowance, additional allowances must be purchased [25]. The resulting carbon emission cost is calculated as follows:

$$C^{\mathrm{CO}2}={\displaystyle \sum _{t=1}^T{c}^{\mathrm{re}}\left[e^{\mathrm{GRID}}{P}_t^{\mathrm{GRID}}+P_t^{\mathrm{CHP}}(e^{\mathrm{CHP}}-{\delta }^{\mathrm{CHP}})+H_t^{\mathrm{GB}}(e^{\mathrm{GB}}-{\delta }^{\mathrm{GB}})\right]}$$

(27)

where $e^{\mathrm{GRID}}$ is the carbon dioxide emission factor for the power grid; $e^{\mathrm{CHP}}$ is the carbon dioxide emission factor for CHP; $e^{\mathrm{GB}}$ is the carbon dioxide emission factor for GB; $c^{\mathrm{re}}$ signifies the unit price of carbon allowances; ${\delta }^{\mathrm{CHP}}$ is the free carbon emission allowance coefficient for CHP; and ${\delta }^{\mathrm{GB}}$ is the carbon emission allowance coefficient for GB.

4.1.2. Constraints

The constraints of the IEO upper-layer pricing decision model must satisfy power balance constraints, operational constraints for equipment, price constraints, and demand response constraints as defined in Equations (3)–(9). Its power balance constraints are as follows:

$$\left\{\begin{array}{l}{P}_t^{\mathrm{GRID}}+P_t^{\mathrm{WT}}+P_t^{\mathrm{PV}}+P_t^{\mathrm{CHP}}+P_{t,\mathrm{dis}}^{\mathrm{ESS}}-P_t^{\mathrm{cut}}-P_{t,\mathrm{char}}^{\mathrm{ESS}}-P_t^{\mathrm{EC}}-P_t^{\mathrm{EB}}=P_t^{\mathrm{IEO},\mathrm{e}}\\ G_t^{\mathrm{GAS}}=G_t^{\mathrm{CHP}}+G_t^{\mathrm{GB}}\\ H_t^{\mathrm{EB}}+H_t^{\mathrm{GB}}+H_t^{\mathrm{CHP}}+P_{t,\mathrm{dis}}^{\mathrm{HSS}}-P_{t,\mathrm{char}}^{\mathrm{HSS}}-H_t^{\mathrm{AC}}=P_t^{\mathrm{IEO},\mathrm{h}}\\ Q_t^{\mathrm{AC}}+Q_t^{\mathrm{EC}}=P_t^{\mathrm{IEO},\mathrm{c}}\end{array}\right.$$

(28)

$$\left\{\begin{array}{l}{P}_t^{\mathrm{IEO},\mathrm{e}}+P_t^{\mathrm{TL},\mathrm{e}}+P_t^{\mathrm{CL},\mathrm{e}}-P_t^{\mathrm{SL},\mathrm{e}}=P_t^{\mathrm{L},\mathrm{e}}\\ P_t^{\mathrm{IEO},\mathrm{h}}+P_t^{\mathrm{CL},\mathrm{h}}+P_t^{\mathrm{SL},\mathrm{h}}=P_t^{\mathrm{L},\mathrm{h}}\\ P_t^{\mathrm{IEO},\mathrm{c}}+P_t^{\mathrm{CL},\mathrm{c}}+P_t^{\mathrm{SL},\mathrm{c}}=P_t^{\mathrm{L},\mathrm{c}}\end{array}\right.$$

(29)

where $P_t^{\mathrm{EB}}$ and $P_t^{\mathrm{EC}}$ are the electric power inputs to EB and EC at time t; $P_t^{\mathrm{WT}}$ and $P_t^{\mathrm{PV}}$ are the output powers of WT and PV at time t, respectively; $P_t^{\mathrm{cut}}$ is the electric energy curtailed during at time t; $H_t^{\mathrm{CHP}}$ is the heat energy output from CHP at time t; $H_t^{\mathrm{AC}}$ is the heat energy input to AC at time t; and $G_t^{\mathrm{CHP}}$ and $G_t^{\mathrm{GB}}$ are the natural gas inputs to CHP and GB, respectively.

Due to the similarities in their modeling, a unified model is adopted for both EES and HSSs, as presented in Equation (30).

$$\left\{\begin{array}{l}{S}_t^B=(1-{\sigma }_B)S_{t-1}^B+({\eta }_{B,\mathrm{char}}{P}_{t,\mathrm{char}}^B-{\displaystyle \frac{P_{t,\mathrm{dis}}^B}{{\eta }_{B,\mathrm{dis}}}})\\ 0\le P_{t,\mathrm{c}}^B\le U_{t,\mathrm{char}}^B{P}_{\mathrm{char},\max }^B\\ 0\le P_{t,\mathrm{dis}}^B\le U_{t,\mathrm{dis}}^B{P}_{\mathrm{dis},\max }^B\\ U_{t,\mathrm{char}}^B+U_{t,\mathrm{dis}}^B=0\\ {\mu }_{\min }^B{S}_{\max }^B\le S_t^B\le {\mu }_{\max }^B{S}_{\max }^B\\ S_0^B=S_T^B={\mu }_{\min }^B{S}_{\max }^B\\ {\displaystyle \sum _{t=1}^T(P_{t,\mathrm{char}}^B-P_{t,\mathrm{dis}}^B)}=0\end{array}\right.$$

(30)

where $S_t^B$ and $S_{t-1}^B$ are the stored energies of the energy storage equipment at time t and time t−1, respectively; ${\sigma }_B$ is the loss coefficient of the B-class storage equipment; ${\eta }_{B,\mathrm{char}}$ and ${\eta }_{B,\mathrm{dis}}$ are the B-class storage equipment charging and discharging efficiency, respectively; and B is the type of energy storage equipment, where $B\in \left\{\mathrm{ESS},\mathrm{HSS}\right\}$. $U_{t,\mathrm{char}}^B$ and $U_{t,\mathrm{dis}}^B$ are 0–1 variables representing the charging and discharging states of the B-class storage equipment at time t, respectively; $P_{\mathrm{char},\max }^B$ and $P_{\mathrm{dis},\max }^B$ are the B-class storage equipment charging and discharging maximum power, respectively; $S_{\max }^B$ is the maximum capacity of the B-class storage equipment; ${\mu }_{\min }^B$ and ${\mu }_{\max }^B$ are the minimum and maximum capacity coefficients of the B class storage equipment, respectively; and $S_0^B$ and $S_T^B$ are the energy stored at the beginning and end of the scheduling cycle, respectively.

The operation constraints of the energy conversion equipment, including CHP, GB, EB, EC, and AC, are shown in Equations (31) and (32).

$$\left\{\begin{array}{l}{P}_t^{\mathrm{CHP}}={\eta }_{\mathrm{CHP},\mathrm{e}}{G}_t^{\mathrm{CHP}}\\ H_t^{\mathrm{CHP}}={\eta }_{\mathrm{CHP},\mathrm{h}}(1-{\eta }_{\mathrm{CHP},\mathrm{e}})G_t^{\mathrm{CHP}}\\ H_t^{\mathrm{GB}}={\eta }_{\mathrm{GB}}{G}_t^{\mathrm{GB}}\\ H_t^{\mathrm{EB}}={\eta }_{\mathrm{EB}}{P}_t^{\mathrm{EB}}\\ Q_t^{\mathrm{AC}}={\eta }_{\mathrm{AC}}{H}_t^{\mathrm{AC}}\\ Q_t^{\mathrm{EC}}={\eta }_{\mathrm{EC}}{P}_t^{\mathrm{EC}}\end{array}\right.$$

(31)

$$\left\{\begin{array}{l}0\le H_t^{\mathrm{CHP}}\le H_{\max }^{\mathrm{CHP}}\\ 0\le H_t^{\mathrm{GB}}\le H_{\max }^{\mathrm{GB}}\\ 0\le H_t^{\mathrm{EB}}\le H_{\max }^{\mathrm{EB}}\\ 0\le Q_t^{\mathrm{AC}}\le Q_{\max }^{\mathrm{AC}}\\ 0\le Q_t^{\mathrm{E}\mathrm{C}}\le Q_{\max }^{\mathrm{EC}}\end{array}\right.$$

(32)

where ${\eta }_{\mathrm{CHP},\mathrm{e}}$ and ${\eta }_{\mathrm{CHP},\mathrm{h}}$ are the electrical and heat energy conversion efficiency of CHP; ${\eta }_{\mathrm{GB}}$, ${\eta }_{\mathrm{EB}}$, ${\eta }_{\mathrm{AC}}$, and ${\eta }_{\mathrm{EC}}$ are the energy conversion efficiencies of GB, EB, AC, EC, respectively; and $H_{\max }^{\mathrm{CHP}}$, $H_{\max }^{\mathrm{GB}}$, $H_{\max }^{\mathrm{EB}}$, $Q_{\max }^{\mathrm{AC}}$ and $Q_{\max }^{\mathrm{EC}}$ are the upper power limits of CHP, GB, EB, AC, and EC.

As shown in Equation (33), the operation of each energy conversion equipment must also satisfy ramping constraints.

$$\left\{\begin{array}{l}\left|P_t^{\mathrm{CHP}}-P_{t-1}^{\mathrm{CHP}}\right|\le P_{\max ,\mathrm{R}}^{\mathrm{CHP}}\\ \left|H_t^{\mathrm{GB}}-H_t^{\mathrm{GB}}\right|\le H_{\max ,\mathrm{R}}^{\mathrm{GB}}\\ \left|H_t^{\mathrm{EB}}-H_{t-1}^{\mathrm{EB}}\right|\le H_{\max ,\mathrm{R}}^{\mathrm{EB}}\\ \left|Q_t^{\mathrm{AC}}-Q_{t-1}^{\mathrm{AC}}\right|\le Q_{\max ,\mathrm{R}}^{\mathrm{AC}}\\ \left|Q_t^{\mathrm{EC}}-Q_{t-1}^{\mathrm{EC}}\right|\le Q_{\max ,\mathrm{R}}^{\mathrm{EC}}\end{array}\right.$$

(33)

where $P_{\max ,\mathrm{R}}^{\mathrm{CHP}}$, $H_{\max ,\mathrm{R}}^{\mathrm{GB}}$, $H_{\max ,\mathrm{R}}^{\mathrm{EB}}$, $Q_{\max ,\mathrm{R}}^{\mathrm{AC}}$, and $Q_{\max ,\mathrm{R}}^{\mathrm{EC}}$ are the maximum ramp rate of CHP, GB, EB, AC, and EC.

As shown in Equation (34), the energy selling price and incentive compensation price established by the IEO must satisfy the upper and lower limits, as well as the average selling price constraint. This ensures that both energy selling prices and incentive compensation prices remain within a reasonable range, preventing extreme price scenarios [26].

$$\left\{\begin{array}{l}{c}_{\min }^y\le c_t^y\le c_{\max }^y\\ {\displaystyle \sum _{t=1}^T{c}_t^y}\le T{c}_{\mathrm{av}}^y\\ c_{\min }^{\mathrm{CL},y}\le c_t^{\mathrm{CL}.y}\le c_{\max }^{\mathrm{CL},y}\end{array}\right.$$

(34)

where $c_{\min }^y$ and $c_{\min }^{\mathrm{CL},y}$ are the minimum energy selling price and minimum incentive compensation price for the y class energy at time t, respectively; and $c_{\mathrm{av}}^y$ is the maximum average selling price for y class energy at time t. $c_{\max }^y$ and $c_{\max }^{\mathrm{CL},y}$ are the maximum energy selling price and maximum incentive compensation price for y class energy at time t, respectively.

4.2. LA Energy Consumption Decision Model

4.2.1. Objective Function

Following the modeling approach in [27], the satisfaction loss incurred by the LA due to demand response deviations from baseline load is quantified using the following function:

$$f_t^{\mathrm{LA}}={\displaystyle \sum (a^{x,y}}{P}_t^{x,y}+b^{x,y}{P}_t^{x,y2})$$

(35)

where $f_t^{\mathrm{LA}}$ is the satisfaction loss cost at time t; and $a^{x,y}$ and $b^{x,y}$ denote the satisfaction loss parameters for the x-class demand response of y energy.

4.2.2. Constraints

LA adjusts electricity, heating, and cooling demand based on the energy sales and incentive compensation prices established by the IEO, aiming to minimize the comprehensive cost of LA’s economic efficiency and satisfaction. This cost, representing the sum of LA’s energy procurement cost, satisfaction loss cost, and IDR compensation, can be expressed as follows:

$$\min C^{\mathrm{LA}}={\displaystyle \sum _{t=1}^T(f_t^{\mathrm{LA}}+c_t^e{P}_t^{\mathrm{LA},\mathrm{e}}+c_t^h{P}_t^{\mathrm{LA},\mathrm{h}}+c_t^c{P}_t^{\mathrm{LA},\mathrm{c}}-C^{\mathrm{idr}})}$$

(36)

$$\left\{\begin{array}{l}{P}_t^{\mathrm{LA},\mathrm{e}}+P_t^{\mathrm{TL},\mathrm{e}}+P_t^{\mathrm{CL},\mathrm{e}}-P_t^{\mathrm{SL},\mathrm{e}}=P_t^{\mathrm{L},\mathrm{e}}\\ P_t^{\mathrm{LA},\mathrm{h}}+P_t^{\mathrm{CL},\mathrm{h}}+P_t^{\mathrm{SL},\mathrm{h}}=P_t^{\mathrm{L},\mathrm{h}}\\ P_t^{\mathrm{LA},\mathrm{c}}+P_t^{\mathrm{CL},\mathrm{c}}+P_t^{\mathrm{SL},\mathrm{c}}=P_t^{\mathrm{L},\mathrm{c}}\end{array}\right.$$

(37)

Unlike the demand response constraints in the IEO pricing decision mode, the demand response constraints in the LA decision model only consider the power balance constraints of IBDR and SBDR, as well as the maximum response rate constraint.

$$\left\{\begin{array}{l}{\displaystyle \sum _t^T{P}_t^{\mathrm{TL},\mathrm{e}}}=0\\ P_t^{\mathrm{SL},\mathrm{e}}={\eta }_{\mathrm{hte}}{P}_t^{\mathrm{SL},\mathrm{h}}+{\eta }_{\mathrm{cte}}{P}_t^{\mathrm{SL},\mathrm{c}}\\ 0<P_t^{\mathrm{x},\mathrm{y}}<{\lambda }_{\max }^{x,y}{P}_t^{\mathrm{L},y}\end{array}\right.$$

(38)

4.3. Rescheduling Model for IEO Under a Response Adjustment Mechanism

In this paper, since the IDR model serves as a linear approximation of the user’s actual response characteristics, it does have certain limitations and may not fully reflect the actual user response patterns. Inaccuracies can become more pronounced near the thresholds of the dead zone and saturation regions. Consequently, following the conclusion of the game, the IEO is required to verify and adjust its IDR plan in consultation with the LA to ensure its feasibility. This process involves the IEO modifying its equipment output and energy procurement schedules based on the determined energy selling price, incentive compensation price, and the confirmed expected response load from LA. The objective function of this response adjustment model is presented in Equation (22), subject to the constraints defined in Equations (28)–(33).

4.4. Model Solution

In the constructed IES Stackelberg game model, the upper-layer IEO decision model is a mixed-integer nonlinear model.

Due to the presence of 0–1 variables in Equations (3), (6) and (7), the big-M method is employed for linearization. After the Stackelberg game model converges, the response adjustment model also be solved. The model solution process is illustrated in Figure 4.

5. Case Study

5.1. Basic Data

This section validates the rationality and effectiveness of the IDR and the proposed GD-DRPR method through case study simulations. The output prediction curves for PV and WT, along with the LA energy consumption load curve, are shown in Figure 5 [28]. The time-of-use tariffs and natural gas for the upper-level grid are listed in

Table 2. The time-of-use tariffs and natural gas price.

Energy Type	Time	Price (CNY/kWh)
Electricity	13:00–16:00, 19:00–22:00	1.1398
	08:00–13:00, 16:00–19:00, 22:00–24:00	0.7112
	24:00–08:00	0.3815
Natural gas	Full day	0.35

. The energy trading limits are detailed in

Table 3. Energy trading limits.

Energy Type	Price Floor (CNY/kWh)	Price Cap (CNY/kWh)	Average Price Cap (CNY/kWh)
Electricity	0.2	1.24	0.8
Heat	0.2	0.6	0.5
Cooling	0.2	0.6	0.46

. The parameters of each piece of equipment in IES are detailed in

Table 4. Equipment parameters for IES.

Parameters	Numerical Value	Parameters	Numerical Value
$S_{\max }^{\mathrm{ESS}}$/(kWh)	800	${\eta }_{\mathrm{EC}}$	3
$S_{\max }^{\mathrm{HSS}}$/(kWh)	1000	$P_{\mathrm{dis},\max }^{\mathrm{ESS}}$, $P_{\mathrm{char},\max }^{\mathrm{ESS}}$/(kW)	50, 50
${\sigma }_{\mathrm{ESS}}$	0.0025	$P_{\mathrm{dis},\max }^{\mathrm{HSS}}$, $P_{\mathrm{char},\max }^{\mathrm{HSS}}$/(kW)	50, 50
${\sigma }_{\mathrm{HSS}}$	0.0017	$Q_{\max }^{\mathrm{CHP}}$/(kW)	80, 800
${\mu }_{\min }^{\mathrm{ESS}}$, ${\mu }_{\max }^{\mathrm{ESS}}$	0.1, 0.9	$P_{\max ,\mathrm{R}}^{\mathrm{CHP}}$/(kW/h)	25
${\mu }_{\min }^{\mathrm{HSS}}$, ${\mu }_{\max }^{\mathrm{HSS}}$	0.1, 0.9	$Q_{\min }^{\mathrm{GB}}$/(kW)	80, 800
${\eta }_{\mathrm{ESS},\mathrm{char}}$, ${\eta }_{\mathrm{ESS},\mathrm{dis}}$/(kW)	0.95, 0.95	$Q_{\max ,\mathrm{R}}^{\mathrm{GB}}$/(kW/h)	33.3
${\eta }_{\mathrm{HSS},\mathrm{char}}$, ${\eta }_{\mathrm{HSS},\mathrm{dis}}$/(kW)	0.9, 0.9	$Q_{\max }^{\mathrm{EB}}$/(kW)	450
${\eta }_{\mathrm{CHP},\mathrm{e}}$	0.33	$Q_{\max ,\mathrm{R}}^{\mathrm{EB}}$/(kW/h)	100
${\eta }_{\mathrm{CHP},\mathrm{h}}$	0.51	$C_{\max }^{\mathrm{AC}}$/(kW)	150
${\eta }_{\mathrm{GB}}$	0.9	$C_{\max ,\mathrm{R}}^{\mathrm{AC}}$/(kW/h)	33.3
${\eta }_{\mathrm{EB}}$	0.95	$C_{\max }^{\mathrm{EC}}$/(kW)	250
${\eta }_{AC}$	0.8	$C_{\max ,\mathrm{R}}^{\mathrm{EC}}$/(kW/h)	100

. All parameters are configured according to the Reference [29]. In the proposed model, the LA reports its demand response capability to the IEO. The maximum demand response rates and the threshold parameters of the dead zone and saturation zone are set with reference to [30], where the maximum demand response rates for PBDR, SBDR, and IBDR are 10%, 6%, and 5%, respectively. The simulation was performed in MATLAB (R2023a) using the Yalmip toolbox to call the commercial solver Gurobi 11.0.

To analyze the effectiveness of the GD-DRPR proposed in this article, nine scenarios were set up as shown in

Table 5. Information for scenarios.

Scenario	IDR	Stackelberg Game	Response Adjustment Mechanism	Objective		Solution Method
				IEO Revenue	IEO Cost	GD-DRPR	KKT-Based	AGA
1				√
2	√			√
3		√		√		√
4	√				√
5	√		√		√
6	√	√			√	√
7	√	√	√		√	√
8	√	√		√			√
9	√	√		√				√

. Scenario 3 is the method proposed in this paper.

5.2. Analysis of IDR and IES Optimization Results

To analyze the optimization results of IDR and IES, this section compares Scenarios 1, 2, and 3. The optimization results for Scenarios 1, 2, and 3 are shown in

Table 6. Comparison of results for Scenarios 1, 2, and 3.

Scenario	Carbon Emissions/kg	${\mathit{C}}^{\mathbf{sell}}$/CNY	${\mathit{C}}^{\mathbf{IEO}}$/CNY	${\mathit{C}}^{\mathbf{LA}}$/CNY
1	5538	34,491	8659	34,247
2	5388	36,877	12,308	37,730
3	4899	37,898	13,459	38,890

. The IDR and pricing results for Scenario 3 are illustrated in Figure 6 and Figure 7.

As shown in Table 6, Scenario 2 demonstrates significant improvements following the implementation of IDR. Total IEO revenue increases by 42.14% relative to Scenario 1, and carbon emissions decrease by 2.71%. These results indicate that IDR effectively enhances both the economic efficiency and the low-carbon performance of integrated energy system operations.

Further analysis reveals that Scenario 3 delivers additional benefits. Building on Scenario 1, the operational revenue of the IEO rises by 13.29%, and carbon emissions are reduced by 8.83% compared to Scenario 2. This finding suggests that GD-DRPR can further amplify the advantages brought by IDR.

It should also be noted that, in Scenario 3 compared to Scenario 1, the IEO’s energy sales revenue increases by 3407 CNY. This increment is less than the 4643 CNY rise in LA costs. The difference is primarily attributed to the increase in the cost associated with LA satisfaction loss. This outcome reflects the IEO’s ability to steer LA participation in IDR and modify energy consumption behavior through its pricing mechanism. While IDR enhances system efficiency, it may impose burdens on users through diminished satisfaction, potentially deterring their long-term participation. If users disengage due to these costs, the market sustainability of the IDR mechanism could be severely compromised. Long-term contracts between IEO and LA should be established. Such contracts enable IEO to face energy market risks on behalf of LA, thereby cultivating trust and ensuring stable involvement. Concurrently, these contracts can regulate IEO pricing strategies to prevent excessive cost burdens on LAs, ensuring that economic incentives are aligned with sustainable system operations.

As shown in Figure 6, after implementing IDR, the peak-to-valley difference in electrical load decreased from 1134.79 kW to 872.56 kW, demonstrating a significant smoothing effect on the load curve. Combined with the optimized electricity price curve for Scenario 3 in Figure 7b, it can be observed that the electricity price exhibits a time-of-use pattern with peak-valley-flat characteristics. This means that PBDR can encourage users to shift part of their load from peak periods to off-peak periods, thereby achieving peak shaving and valley filling.

A comparison of Figure 6b and Figure 7a reveals a significant decline in heat load during the period from 23:00 to 8:00. During this interval, the heat selling price exceeds the electricity selling price. Under the SBDR, part of the heat load is shifted to electrical load, thereby realizing load substitution and reducing the difference between peak and valley values for both electrical and heat loads. This process further enhances the system’s capacity to integrate renewable energy. In the period from 10:00 to 22:00, the heat selling price is lower than the electricity selling price, which in turn induces a shift in electricity load to heat load as a result of price signals. Further analysis of Figure 6c and Figure 7a indicates that, from 10:00 to 22:00, the cooling selling price is also lower than the electricity selling price. This price relationship leads to a corresponding shift in electricity load to cooling load. During other time periods, the price difference between cooling and electricity sales remains relatively small, resulting in minimal load substitution toward cooling. In summary, the price differential among various energy sources plays a pivotal role in influencing the magnitude of response under substitution-based demand response. When the price difference is limited, the effect of load substitution remains insignificant. In contrast, a larger price differential strengthens the guiding effect of SBDR and results in more notable substitution effects.

Furthermore, as shown in Figure 7b, the incentive compensation price for electricity remains relatively high during periods when the electrical load increases following the response, while it is nearly zero during other intervals. Additionally, Figure 6 further demonstrates that the heat load and cooling load exhibit similar variation patterns. This is because the IEO possesses stronger profit motivation during peak load periods and, therefore, is less inclined to guide users to reduce load through incentive compensation when load demand can be met. This observation indicates that the IEO primarily opts for PBDR and SBDR, rather than IBDR.

5.3. Analysis of GD-DRPR and Response Adjustment Mechanism

On one hand, IBDR reduces the energy purchased by the LA from the IEO; on the other hand, the IEO must compensate the IBDR for the reduced energy. When the IEO considers energy sales revenue and aims to maximize daily operational income, it tends to sell more energy to increase profits, which discourages the IEO from formulating IBDR plans. Therefore, when the IEO prioritizes energy sales revenue, IBDR becomes less active, making it difficult to analyze. To facilitate a simultaneous analysis of the effects of GD-DRPR and response adjustment mechanisms on PBDR, SBDR, and IBDR, this section compares Scenarios 4, 5, 6, and 7, in which the IEO considers only operational costs.

The optimization results for Scenarios 4, 5, 6, and 7 are shown in

Table 7. Comparison of Results for Scenarios 4, 5, 6, and 7.

Scenario	Carbon Emissions/kg	IEO Cost/CNY	MAPE	RSME
4	4788	23,805	2.43%	32.18
5	4859	24,597	0%	0
6	4689	24,505	0.18%	2.34
7	4691	24,513	0%	0

. The adjustment amounts for the scheduling plans between Scenarios 5 and 7 compared to Scenarios 4 and 6, are illustrated in Figure 8. The IDR plan deviations for IEO in Scenarios 4 and 6 are depicted in Figure 9.

As shown in Table 7, both carbon emissions and IEO costs increase in Scenario 5 and Scenario 7 compared to Scenario 4 and Scenario 6. This is because the proposed method requires the IEO to estimate the LA’s response using the IDR model, rather than making decisions directly based on the LA’s energy consumption plan. As a result, the IEO must adjust the response plan through the response adjustment mechanism before rescheduling. Figure 8 shows that the maximum power adjustment in the rescheduling plan for Scenario 7 is 22.7 kW, which is significantly lower than the 293.79 kW observed in Scenario 5. Furthermore, Scenario 5 exhibits a 71 kg increase in carbon emissions and a 792 CNY increase in IEO costs compared to Scenario 4. Scenario 7, on the other hand, shows only a 2 kg increase in carbon emissions and an 8 CNY increase in IEO costs relative to Scenario 6. These results indicate that the rescheduling outcomes for Scenario 7 are closer to the original scheduling outcomes than those for Scenario 5.

In addition, it can be observed that smaller adjustments to the dispatch plan result in post-rescheduling outcomes that are more closely aligned with the pre-rescheduling results. Moreover, compared to Scenario 5, Scenario 7 reduces carbon emissions and IEO costs by 3.46% and 0.34%, respectively. This demonstrates that smaller adjustments to the dispatch plan yield more favorable results in terms of low-carbon performance and economic efficiency after rescheduling. In Figure 9, a response deviation greater than 0 indicates that the post-response load in the IDR plan is higher than the expected post-response load by LA. From 24:00 to 8:00, Scenario 4 shows a higher planned post-response electrical load and a lower planned post-response heat load, which is reflected in Scenario 5 as reduced electricity purchases in Figure 8a.

At the same time, the output of the CHP, which can simultaneously generate electricity and supply heat, mainly replaces the output of the GB, which can only supply heat. From 9:00 to 23:00, Scenario 4 exhibits a lower planned post-response electrical load and a higher planned post-response heat load, which is reflected in Scenario 5 as increased electricity purchases and reduced GB output in Figure 8a. Additionally, the output of the EC replaces that of the AC systems. Furthermore, in Scenario 4, the planned response-adjusted cooling load is significantly lower only during the periods from 7:00 to 8:00 and from 22:00 to 24:00, which is reflected in Scenario 5 as increased EC and AC outputs in Figure 8a. According to Table 1, in Scenario 4, the IEO increases electricity purchases during periods when the upper-level grid’s time-of-use price is higher, and decreases purchases when the price is lower, thereby increasing the IEO’s overall costs.

In Figure 9b, at 1:00, 7:00, and from 19:00 to 23:00, the planned response in Scenario 6 leads to a higher heat load, resulting in a reduction in EB or GB output in Scenario 7 that exceeds the corresponding increase in CHP output. At 8:00, the planned response results in lower heat load, leading to the replacement of GB output with EB and CHP. At 9:00, due to the lower planned post-response electrical load, the ESS in Scenario 7 charges to store excess electricity. Additionally, since deviations in other IDR planned values are relatively small, the subsequent changes in electricity purchases and equipment output are comparatively balanced, leading to only minimal differences between the results of Scenario 7 and Scenario 6.

Additionally, without introducing a response adjustment mechanism, the planned post-response load error results for Scenarios 4 and 6 are shown in Table 7. As indicated in Table 7, GD-DRPR reduced MAPE and RSME by 92.45% and 92.73%, respectively. In summary, GD-DRPR effectively corrects the IDR incentive threshold parameter, aligning the IDR plan formulated by the IEO more closely with actual user responses. This reduces the need for schedule adjustments caused by IDR plan deviations, thereby lowering system operating costs and validating the method’s effectiveness in enhancing the economic efficiency of the IES.

5.4. The Effect of the Satisfaction Loss Parameters on GD-DRPR

To evaluate the adaptability of GD-DRPR under varying degrees of user energy consumption characteristics, this section compares the parameter revision effects of IEO and LA during a single interaction process. The revised IDR stimulus threshold parameter from Scenario 6 is set as the initial value, while the user satisfaction loss parameters are allowed to fluctuate randomly around the initial value. Its maximum fluctuation rate gradually increased from 10% to 50%.

Figure 10 illustrates the effect of satisfaction loss parameters on GD-DRPR. The initial MAPE increases as the volatility of the user satisfaction loss parameters rises, demonstrating a clear positive correlation between the two. This is attributed to the gradient descent mechanism employed by GD-DRPR. Larger variations in user energy consumption characteristics lead to significant deviations in the IDR incentive threshold parameters, thereby necessitating more rounds of parameter correction to achieve convergence. Although the MAPE decreases significantly during the initial iterations, the rate of decline slows down after the third iteration. Furthermore, the MAPE exhibits peaks near specific fluctuation rates, indicating that user energy consumption characteristics are more sensitive within these ranges, exerting a more pronounced influence on the parameter adjustment process.

5.5. Convergence Analysis of GD-DRPRs

Through experiments, this paper finds that the proposed GD-DRPR algorithm performs well with a step size of 0.001; therefore $\alpha$ = 0.001 is set.

This section sets the step sizes to 0.3 $\alpha$, 0.7 $\alpha$, $\alpha$, 1.5 $\alpha$, 2 $\alpha$, and 10 $\alpha$ to analyze the impact of different step sizes on algorithm convergence. Figure 11 shows the convergence curves for different step sizes, where the number of iterations refers to the inner iteration step n. It can be observed that the algorithm is sensitive to the step size. When the step size is large, the algorithm is prone to stopping early or even causes the objective function value to increase. When the step size is small, the algorithm converges slowly. When the step size is 0.001, the algorithm can converge relatively quickly.

5.6. Comparative Analysis of GD-DRPR and Other Methods

This section compares the differences between GD-DRPR and other traditional solution methods in terms of final optimization results and the number of game interactions. Traditional methods primarily fall into two categories: first, those that equivalently transform the two-layer model into a single-layer model for solution based on KKT conditions; second, those that employ iterative optimization algorithms for solution, with the adaptive genetic algorithm (AGA) serving as a representative example in this paper. Therefore, this section conducts a comparative analysis of Scenarios 3, 8, and 9, examining the solution methods for different Stackelberg game models.

To evaluate the adaptability of GD-DRPR under varying wind and solar penetration rates, the predicted wind and solar outputs were scaled proportionally, with penetration rates set at 40%, 50%, and 60%. The results for Scenarios 3, 8, and 9 under each penetration rate are compared in

Table 8. Comparison of GD-DRPR with other methods.

Renewable Energy Penetration	Scenario	Interaction Count	Runtime/s	Carbon Emissions/kg	Curtailment Rate	${\mathit{C}}^{\mathbf{IEO}}$/CNY
40%	3	4	488	4134	0%	19,929
	8	1	56	4206	0%	20,112
	9	43	774	4460	0%	19,905
50%	3	4	501	2532	1.91%	23,537
	8	1	57	2590	2.28%	23,626
	9	44	793	2671	2.40%	23,105
60%	3	4	497	1901	4.72%	25,748
	8	1	56	1938	5.16%	25,914
	9	47	830	2030	5.37%	25,589

.

The KKT-based solution method requires the IEO to have complete information, including user energy consumption characteristics. Therefore, under different penetration rates, the revenue of the IEO for Scenario 8 in Table 8 consistently remains the highest. In both GD-DRPR and AGA, the IEO operates without complete information regarding the LA, which more accurately reflects real scenarios. As Table 8 illustrates, under these same conditions of incomplete information, the number of interactions in Scenario 9 is significantly higher than in Scenario 3. This demonstrates that GD-DRPR effectively minimizes the need for repeated gaming between the two parties and thereby enhances transaction efficiency. Moreover, the solving time in Scenario 3 is shorter than in Scenario 9, demonstrating that, despite the increased model complexity introduced by GD-DRPR, it still maintains a computational efficiency advantage over AGA. Meanwhile, the optimization results for Scenario 3 are generally closer to those of Scenario 8. This is primarily attributed to the fact that AGA is more prone to getting trapped in local optima during the search process. Considering that Scenario 8 represents the ideal case where the IEO possesses full information, its results serve as the theoretical upper bound. Therefore, the fact that Scenario 3 achieves performance close to this upper bound under conditions of incomplete information demonstrates the effectiveness and practicality of GD-DRPR.

As illustrated in Figure 11, the GD-DRPR algorithm converges at 17 inner iterations when an appropriate step size is applied. In conjunction with Table 8, it is evident that the IEO and LA models exchange information only four times, meaning the Stackelberg game process concludes in exactly three outer iterations. Since the inner iterations of the proposed approach are not reset at the onset of a new outer iteration cycle, the algorithm only needs to solve the IEO model approximately a dozen times and the LA model a few times. In contrast, the AGA method requires solving the IEO model several dozen times per iteration, depending on the configured population size. Consequently, when applied to large-scale complex systems and extended to multi-tier games, the proposed method not only shows lower computational complexity than conventional distributed algorithms but also effectively reduces the overhead communication among different agents.

Notably, when wind and photovoltaic penetration reach 50% and 60%, Scenario 3 exhibits superior performance regarding carbon emissions and power curtailment. This demonstrates that GD-DRPR enables the IEO to derive pricing strategies that align closely with real-world user behavior. Consequently, this effectively mobilizes demand-side resources, enhancing both the low-carbon operation and the renewable energy integration capability of the IES.

6. Conclusions

This paper addresses the issue that IEO struggles to formulate accurate IDR plans due to incomplete user information, leading to deviations in user demand response. It proposes the GD-DRPR and response adjustment mechanism, then establishes an IES Stackelberg game model based on this approach. Through case study verification and analysis, the following conclusions are drawn:

• The proposed GD-DRPR reduces the MAPE of planned IDR loads by 92.45% and the RMSE by 92.73%, enabling rapid revision of stimulus threshold parameters in the dead zone and saturation zone of the IDR model.
• The IES Stackelberg game model proposed in this paper, the IEO employs the IDR model to support decision-making. Compared to scenarios without GD-DRPR, carbon emissions were reduced by 13.29%, while the IEO’s total revenue increased by 8.83%, achieving synergistic improvements in both the economic efficiency and low-carbon performance of IES operations.
• Compared to traditional Stackelberg game methods, GD-DRPR reduces the interaction frequency between IEO and LA, thereby improving transaction efficiency. This reduction in interactions also alleviates the decision-making burden on the LA, effectively lowering the market entry barriers for both users and LAs. The improved transaction efficiency facilitates the IEO’s participation in more complex and large-scale market environments.

The method proposed in this paper can serve as a reference for revising the demand response model and solving the Stackelberg game model. However, many issues still need to be addressed before its practical application. These mainly include:

• The initial stimulus threshold parameters and the upper limits of users’ demand response are merely set within a reasonable range based on previous studies. The adaptability of GD-DRPR in complex real-world environments remains to be verified.
• This study only considers a deterministic model. The actual performance of GD-DRPR under uncertain conditions needs to be validated.

In future work, we will conduct further research to address these two issues for further optimization.