The integrated energy system can integrate various energy resources to solve the complexity of purchasing and managing energy resources separately. At the same time, under unified scheduling and management, it reduces energy loss and waste, thereby reducing the operating costs of enterprises. Integrated Energy Systems (IES) can simultaneously integrate electricity, heat, cooling, and gas. The system is equipped with energy storage devices to assist in balancing the supply and demand of these four types of energy, and includes energy conversion devices to enable flexible conversion between the four types of energy, maximizing the utilization of renewable energy. During system operation, the four types of energy units develop output plans based on their own operational characteristics, energy demands, and the price differences between various energies. IES uses energy conversion devices to achieve mutual conversion between various types of energy, enhancing the penetration rate of renewable energy. Unlike traditional power systems, IES includes multi-energy flow coupling. Figure 1 shows the IES architecture, which involves multiple energies such as electricity, heat, cooling, and gas. IES consists of an energy supply network, energy exchange links, and end use energy units.

The IES energy supply equipment includes photovoltaic (PV), wind power (WT), and gas boilers (GB). The CCHP system typically consists of a gas turbine (GT), a heat recovery boiler (HRB), and an ice storage air conditioning system (ISAC). Energy storage equipment is composed of batteries (BT) and thermal storage tanks (HST).

The goal of carbon trading policies is to stimulate the initiative of traditional units to reduce carbon emissions by adjusting carbon trading costs, thereby increasing the competitiveness of wind and solar power in the electricity market (Ruitian et al., 2022). A common method is to obtain carbon trading costs through a fixed carbon price, which is relatively convenient in terms of operation and settlement (Yongli et al., 2023). However, the carbon trading costs and operational decisions of most entities are affected by the carbon price, and the fixed carbon price model has limited incentives for entities to actively reduce emissions.

Unlike the usual carbon trading mechanism, the stepped carbon pricing mechanism divides the purchasing range into different intervals. As the volume of carbon emissions traded increases, the carbon price in the interval also rises. The stepped carbon trading cost C c t is as following Formula 39: C c t = λ E t   E t ≤ L λ 1 + α E t − L + λ L   L ≤ E t ≤ 2 L λ 1 + 2 α E t − 2 L + λ 2 + α L   2 L ≤ E t ≤ 3 L λ 1 + 3 α E t − 3 L + λ 3 + 3 α L   3 L ≤ E t ≤ 4 L λ 1 + 4 α E t − 4 L + λ 4 + 6 α L   E t ≥ 4 L (39)

Where λ represents the baseline carbon trading price; L represents the length of the carbon emission interval; α represents the price increase; E t represents the total amount of carbon emission rights traded in IES.

The integrated energy system constructed in this article includes thermal power units, wind power units, and solar photovoltaic power units. Referring to the “Management Measures for Carbon Emission Trading”, carbon quotas are allocated to each subject of integrated energy system. The calculation formula is obtained by multiplying the unit carbon quota allocation coefficient with the power generation of the units, as following Equation 40: Q t = ∑ f = 1 F Q f , t + ∑ w = 1 W Q w , t + ∑ p = 1 P Q p , t (40)

In the formula Q f , t , Q w , t , Q p , t represent the carbon quotas of thermal power units, wind power units, and solar photovoltaic units during the t period respectively.

The entities in the integrated energy system can purchase excess carbon emission rights in the carbon market, avoid paying fines for excess carbon emissions, and also sell excess carbon quotas to obtain profits. The formula for calculating carbon trading costs is as following Equations 41–44: M t C = M t f + M t w + M t p (41) M t f = ∑ f = 1 F P C O 2 E f , t − E f , 0 (42) M t w = ∑ f = 1 W P C O 2 E w , t − E w , 0 (43) M t p = ∑ p = 1 P P C O 2 E p , t − E p , 0 (44)

In the formula E f , t , E w , t , E p , t represent the carbon emissions generated by thermal power units, wind power units, and solar photovoltaic units during time period t respectively.

Game theory focuses on the interaction between multiple entities, exploring strategies to maximize the interests of each entity through modeling and quantifying information exchanges, constraints, etc., Among them. In games, any action by one party will affect the other parties to varying degrees, potentially leading them to change their strategies. The actions of one party influence the interests of the other parties, prompting strategy adjustments, which in turn impact the original party’s interests, resulting in conflicts of interest.

The master-slave game, a type of game theory, is dynamic. The entity with leadership advantages occupies a favorable position in the game, making decisions first, which serve as constraints for the followers. The followers then make decisions based on the leader’s actions, and the leader updates their decision based on the followers’ responses.

The master-slave game relationship among IES entities proposed in this paper is shown in Figure 2. The leader, Energy Management Operator (EMO), and followers, Energy Generation Operator (EGO), Energy Storage Operator (ESO), and users, aim to maximize their respective benefits while considering environmental benefits in their trading strategies. This improves their operational states, achieving economic and efficient energy supply and rational and scientific energy use. EMO plays a leading and coordinating role in the energy market, balancing power sources, loads, and storage, acting as an energy pool with input and output energy flows. EMO typically purchases electricity from EGO to supply to users. If the purchase quantity cannot meet the demand, EMO must pay more to purchase electricity externally and bear the additional carbon emission costs from the grid purchase. As the leader, EMO’s advantage lies in utilizing pricing strategies to enhance the flexibility of distributed energy sources, making them more competitive in the energy market compared to traditional energy sources, and effectively incentivizing users to reasonably adjust their energy use. The core of EGO is CCHP, optimizing the output of each unit considering profits, fuel costs, and carbon trading costs. ESO formulates buying and selling strategies based on price information, which influences EGO’s energy sales strategy and users’ demand response strategies through EMO. Conversely, adjustments by EGO and users also affect the strategies of ESO and users. On the user side, shiftable loads and reducible thermal (cooling) loads are introduced, allowing users to adjust their energy demand based on energy costs, comfort, and other factors. Changes in user energy use will impact the interests and strategies of other entities.

The master-slave game described in this paper depicts how EGO, ESO, and users, as followers, optimize their actions based on the leader EMO’s price decisions to achieve their optimal objectives. The Stackelberg model is as following formula: G = { E M O ;   E G O , E S O , U S E R ;   ρ E M O ;   σ E G O , σ E S O , σ U S E R ;   F E M O ;   F E G O , F E S O , F U S E R }

The master-slave game model consists of three elements: participants, decision variables, and utility. Participant refers to every decision-maker involved in the game. Decision variables refer to the actions taken by participants in the game process. Utility is used to measure the level of utility obtained by participants in a game. The model constructed in this paper includes four participants: one leader EMO and three followers EGO, ESO, and users. The decision variables are: buying and selling prices ρ E M O will influence the decisions of each follower; output power of energy supply equipment δ E G O determines the energy supply; storage power of batteries and thermal storage tanks δ E S O determines the energy storage; shiftable electricity, heating, and cooling loads δ U S E R from users determine the variable energy demand. The utility for each participant corresponds to the aforementioned objective functions. If none of the participants can unilaterally change their strategies to achieve higher profits, the game reaches equilibrium (Yuanyuan et al., 2022), referred to as the equilibrium solution ρ E M O * , δ E G O * , δ E S O * , δ U S E R * , satisfying the following Formula 45: F E M O ρ E M O * , σ E G O * , σ E S O * , σ U S E R *   ≥ F E M O ρ E M O , σ E G O * , σ E S O * , σ U S E R *   F E M O ρ E M O * , σ E G O * , σ E S O * , σ U S E R *   ≥ F E M O ρ E M O * , σ E G O , σ E S O * , σ U S E R *   F E M O ρ E M O * , σ E G O * , σ E S O * , σ U S E R *   ≥ F E M O ρ E M O * , σ E G O * , σ E S O , σ U S E R *   F E M O ρ E M O * , σ E G O * , σ E S O * , σ U S E R *   ≥ F E M O ρ E M O * , σ E G O * , σ E S O * , σ U S E R     (45)

Formula 5 is a nonlinear formula. This article linearizes the max and min functions by introducing two non negative auxiliary variables and using appropriate constraints to simulate them. z 1 = max L e − P e g , 0 z 2 = min L e − P e g , 0

The rewritten equations are shown in Formulas 46–49: z 1 = L e − P e g   L e − P e g ≥ 0 (46) z 1 = 0   L e − P e g < 0 (47) z 2 = L e − P e g   L e − P e g ≤ 0 (48) z 2 = 0   L e − P e g > 0 (49)

To simulate these conditions, we can use binary variables δ and corresponding linear inequalities and equations to replace these conditions. Assuming that δ is a binary variable, where δ = 1 when L e − P e g ≥ 0 , otherwise δ = 0 . M is a sufficiently large number.

Constraint condition is shown in Formulas 50–54: z 1 ≥ L e − P e g (50) z 1 ≤ M δ (51) z 2 ≤ L e − P e g (52) z 2 ≥ − M 1 − δ (53) δ ∈ 0 , 1 (54)

In summary, the following linearized model is obtained is shown in Equation 55: f e t = z 1 p r g b + z 2 p r g s (55)

Traditional optimization methods require collecting various data from participants, such as equipment parameters, user preferences, and environmental factors, but some information in the electricity market is not publicly transparent (Yang et al., 2025). Therefore, this paper adopts the Differential Evolution (DE) algorithm and the Cplex solver to solve the proposed master-slaver game model. The DE algorithm is an algorithm based on genetic algorithms (GA) that can solve multi-objective optimization problems. It is a heuristic random search algorithm based on group differences. Compared with other algorithms (GA, SA), DE, it has fewer parameters and is simpler to compute, making it commonly used to solve optimization scheduling problems in the power sector. The approximation effect of differential evolution algorithm is more significant compared to genetic algorithm. Solving the model proposed in this article belongs to high-dimensional problems. Other algorithms have slow or even difficult convergence speed for high-dimensional problems.

The solving process includes five steps: initializing the population, population mutation, crossover to form a new population, selection, and termination (Wanxing et al., 2023). As shown in the Figure 3, DE algorithm solution, initial data and parameters are first input in the upper-level, then an initial population a is generated, with the iteration count K = 0. The lower-level uses the Cplex solver to calculate the optimal results for the followers, and the leader updates its decision based on the results, calculating its own utility F1. The population a forms a new population b through mutation and crossover, and the lower-level repeats the followers’ solution and recalculates its own utility F2. A selection process follows: if F2 is greater than F1, b is assigned to a and F2 to F1, otherwise, it is retained. Finally, judgment is made. If the iteration count K meets the requirement, the final result is output; otherwise, the mutation, crossover, and other steps are repeated.

This paper conducts a case study analysis based on a typical region in China. The predicted curves for PV, wind power, and electric, thermal, and cooling loads in this region are shown in Figure 4. The PV generation period is from 6:00 to 18:00, mainly generating power from 9:00 to 14:00, while wind power generation is mostly from 19:00 to 24:00. The peak electricity demand occurs between 10:00 and 12:00 and 17:00 and 19:00, the peak thermal load demand between 20:00 and 22:00, and the cooling load peak between 11:00 and 16:00. The user preference coefficients for electricity, cooling, and heating are: Ve = 1.5,ae = 0.0009,Vh = 1.1,ah = 0.0011. Shiftable loads account for 20% of the total electric load, and adjustable cooling and heating loads account for 10% of the total cooling and heating loads. Other relevant IES parameters are listed in Table 1.

The purchase and sale pricing strategy of the upper-level EMO is shown in Figure 5. The selling price of electricity is similar to the time of use price. The heating purchase prices are higher than the lower price limit. In Figure 5A, the red and green dashed lines represent the time-of-use electricity price and the on-grid electricity price, respectively. EMO determines the pricing strategy within this range to provide EGO and users with more favorable prices than the grid. The electricity purchase price and load demand vary similarly at different times. The electricity prices are higher between 12:00–13:00 and 22:00–23:00, reaching 0.60 yuan/kWh and 0.63 yuan/kWh, respectively. During these periods, PV and wind power generation are higher, and the higher purchase price encourages more renewable energy generation, reducing the need to purchase electricity from the grid to increase profits. The trend in EMO’s electricity sales prices is similar to the grid’s time-of-use electricity prices, with higher prices between 11:00–14:00 and 17:00–21:00, reaching 1.25 yuan/kWh. In Figure 5B, the red and green dashed lines represent the upper and lower limits of heating prices in North China, respectively. EMO similarly determines purchase and sale prices within this envelope. The heating sale prices reach the upper limit during multiple periods, such as 3:00–4:00, 6:00–9:00, 12:00–14:00, 15:00–16:00, and 20:00–21:00. The heating purchase prices peak at 7:00–8:00, 11:00–12:00, and 22:00–23:00.

Figure 6 shows the electric, thermal, and cooling load curves on the user side before and after demand response. The EL curve presents an M-shape and CL curve presents an N-shape. The TL curve shows a trend of low in the middle and high on both sides. From Figure 6A, under the influence of electricity price incentives, the electric load curve before and after demand response shows a “peak shaving and valley filling” change, reducing the total electricity cost. Before demand response, the peak electric load occurred between 18:00 and 19:00, reaching 2050 kW, with other peaks at 12:00–13:00 and 22:00–23:00, reaching 1857 KW and 1867 kW, respectively. The electric load valleys occurred between 0:00–7:00 and 23:00–24:00, with the lowest value being 847 KW. After demand response, the peak value dropped to 1874 kW, and the lowest value increased to 980 kW, with a noticeable load shift. In Figure 6B, the thermal load curve also achieved “peak shaving and valley filling”. Before demand response, peaks occurred at 2:00–3:00 and 21:00–22:00, reaching 1584 KW and 1496 kW, respectively. The period between 10:00–17:00 showed a downward trend before rising, with a valley value of 1008 KW. After optimization, the peak value dropped to 1433 kW, and the valley value increased to 1087 KW. Figure 6C shows the cooling load curve, also exhibiting “peak shaving and valley filling”. Cooling demand was high between 8:00–18:00 and low between 0:00–6:00 and 23:00–24:00. Before optimization, the peak cooling load occurred at 14:00–15:00, reaching 1042 kW, with a low demand of about 135 KW between 0:00–4:00 and 23:00–24:00. Through demand response, the peak cooling load dropped to 922 kW, and the valley value increased to 251 KW. This proves that the optimization model proposed in this paper can reduce energy costs and improve energy economy while ensuring user comfort.

Figure 7 shows the optimized scheduling of electricity, thermal, and cooling in the IES. Wind and PV power generation are low-carbon, so EGO prioritizes selling renewable energy to EMO. When the supply is insufficient, other gas-fired units are activated to supplement wind and PV shortfalls. For example, between 23:00–6:00, the period of off-peak electricity prices is mainly covered by wind power. During periods of high electricity demand such as 10:00–13:00 and 17:00–22:00, wind and PV power are fully utilized, and gas-fired units increase output to make up for the missing energy, while surplus energy is stored in ESO’s BT. During peak electricity demand periods such as 17:00–19:00, EGO limits the output of gas-fired units to some extent after analyzing energy prices and operational costs. If EGO cannot meet the load demand, EMO will purchase electricity from the grid to make up for the shortfall. Gas turbines generate heat during power generation, which is proportional to the electricity generated. Utilizing waste heat can reduce heating costs. EMO encourages the gas boiler’s output through pricing policies, and coordinating with the heat recovery boiler can effectively avoid penalties for heating interruptions, ensuring a stable heat supply. When the heat supply cannot be met, ESO compensates by releasing stored heat from HST during high-price periods. Cooling demand is relatively low, and ice storage air conditioners provide cooling through both air conditioning and ice melting modes, with air conditioning as the primary method. During peak cooling periods such as 8:00–20:00, the ice storage air conditioning system will additionally activate the ice melting mode, and ESO’s storage devices will supplement the cooling demand.

After optimization, EMO reduces the amount of external electricity purchased, and ESO mitigates the output pressure on EGO units by charging during low-price periods and discharging during high-price periods, providing users with more affordable energy prices. This helps reduce EMO’s external electricity purchase costs and users’ electricity costs. Table 2 compares the profits of each IES entity and total carbon emissions before and after optimization. The introduction of stepped carbon trading has reduced IES carbon emissions by 11%. Higher carbon emissions will result in paying more expensive fees. It can be calculated from the Table 2 that profits of EMO increase 34%. After optimization through the master-slave game model, energy supply and demand have reached a balance. There is no need to purchase electricity from external sources at high prices during peak periods, nor is there a need to sell surplus electricity at low prices during low periods. Profits of EGO increase 46%. After optimization EGO obtains more benefits through reasonable planning of power generation. Profits of ESO increase 31%. ESO purchases excess wind and photovoltaic power during low periods and sell it during peak periods to seek more profits. Profits of users increase 7%. By introducing a demand response mechanism, users have changed their electricity usage habits, resulting in increased revenue.

Figure 8 shows the comparison of power output of thermal power units before and after considering carbon trading. Without considering carbon trading, thermal power units output between 8:00–23:00. After considering carbon trading, output is reduced between 2:00–7:00. This is because the addition of carbon trading mechanisms requires thermal power units in the integrated energy system to pay more expensive fees for carbon emissions.