The electricity supply of IESs should consider the income from selling energy F _sell, trading costs with external electricity grids F _util, and various energy supply costs. The energy supply costs include the natural gas fuel cost F _f and the equipment maintenance cost F _om. The optimal electricity supply model is as follows: max ⁡ F = F sell − F f + F util + F om . (1)

The IESs can earn profits by selling the produced electricity and heat to SMIPs. The price and power of the sold energy are obtained based on the electricity market clearing. Then, the income from selling energy F _sell is given as follows: F sell = ∑ t = 1 T P e l t C e t + ∑ t = 1 T Q h l t C h t , (2)

where P _el (t) and Q _hl (t) represent the electricity power and heat power, respectively, at time t under the response to the demand of the SMIPs. C _e (t) and C _h (t) represent the electricity prices and heat prices, respectively, at time t cleared by the EM. T represents the optimization time.

IESs can arbitrage through trading with external grids. When the electricity price of the power grid is low, the IESs purchase the electricity and sell electricity when its price is high, and then the profits can be obtained. The trading costs with the external electricity grid F _util is given as follows: F util = ∑ t = 1 T P u t i l t C u t i l t , (3)

where P _util (t) is the interactive power between the IESs and the external electricity grid at time t. When P _util (t) > 0, it means that the IESs purchase electricity from the external electricity grid. When P _util (t) < 0, it means that the IESs sell electricity from the external electricity grid. C _util (t) is the interactive electricity price between the IESs and the external electricity grid at time t.

The energy provided by IESs is divided into two categories. The first category is wind turbines (WTs) and photovoltaic (PV) power generation, and this type of energy does not need to be purchased. The second category is gas turbines and gas boilers, which burn natural gas to generate electricity and heat, respectively. For this category, the purchasing cost of natural gas needs to be included in the cost, which is called fuel cost. The fuel cost F _f is given as follows: F f = C g a s L ⋅ ∑ t = 1 T P G T t η G T + Q G B t η G B , (4)

where C _gas is the price of natural gas and L is the low calorific value of natural gas, which represents the heat released by burning a certain volume of natural gas. P _GT (t) and Q _GB (t) represent the output powers of the gas turbine and gas boiler at time t, respectively. η _GT and η _GB represent the efficiencies of the gas turbine and gas boiler, respectively.

The distributed energy equipment in the system needs maintenance, and the equipment maintenance cost F _om is given as follows: F o m = ∑ i = 1 I ∑ t = 1 T ξ i S P D G i t , (5)

where I is the number of power generation equipment. ξ i S is the cost coefficient of the ith power generation equipment. P _DGi (t) represents the supply power of the ith power generation equipment at time t. In the IES optimization model, the constraints of various parameters are given as follows:

(1) Constraints on electricity and heat power balance

Formulas 6, 7 are the constraints on electricity power balance, and Formulas 8, 9 are the constraints on heat power balance. L _e (t) and L _h (t) represent the original electricity and heat loads of SMIPs at time t, respectively. Q(t) is the heat power recovered from the gas boiler at time t. P _EDR (t) and Q _eHDR (t) represent the electricity and heat load responding to the demand of SMIPs at time t, respectively.

(2) Constraints on the output of distributed energy generation equipment

where P _{i min}(t) and P _{i max}(t) represent the upper and lower limits of the power of the ith distributed energy equipment, respectively.

(3) Constraints on the power exchange with the external electricity grid

where P max s e l l and P max b u y are the upper limits of the power sold to and bought from the external electricity grid by the IESs, respectively.

(4) Constraints on the power of energy equipment participating in the electricity market clearing

where P ^' _k,e (t) and Q ^' _s,h (t) represent the powers cleared in the EM for each electric and thermal unit at time t, respectively. P _k,e (t) and Q _s,h (t) represent the actual power generated by each electric and thermal unit at time t, respectively.

The energy utilization optimization model of SMIPs takes minimizing the energy purchase cost as the objective function. In addition to paying the energy purchase fee F _sell to the IESs, SMIPs can also obtain income compensation F _DR by reducing a certain amount of load through demand response. Therefore, the objective function is established as follows: min ⁡ F u s e r = F s e l l − F D R , (14) F s e l l = ∑ t = 1 T L e t + P E D R t ⋅ C e t + ∑ t = 1 T L h t + Q H D R t ⋅ C h t , (15) F D R = ∑ t = 1 T v e P E D R t + ∑ t = 1 T v h Q H D R t , (16)

where v _e and v _h represent the reduction compensation coefficients for the electricity and heat demand response of SMIPs, respectively. For the demand response to SMIPs, various constraints need to be considered as follows:

(1) Constraints on the income of SMIPs

The benefits of users after demand response F u s e r D R should be greater than the benefits before the response F ₀ . The mathematical formula is F u s e r D R ≥ F 0 . (17)

(2) Constraints on the power of load transfer

During the optimization process, the SMIPs can participate in price-based demand response. Thus, the power of electricity and heat load transfer cannot exceed the limit value P E D R max and Q H D R max . The mathematical formula is P E D R t ≤ P E D R max , (18) Q H D R t ≤ Q H D R max . (19)

The EM aims at maximizing the social surplus profit, which reflects the balance of benefits between the supply and demand. The maximum social surplus profit in this paper consists of the profit surplus of IESs and profit surplus of the users. The objective function of the model is established as follows: max ⁡ H = ∑ t = 1 T H e m o t + H u s e r t , (20) H e m o t = ∑ k = 1 K   C e t − C e , ⁡ min t P k , e ′ t + ∑ s = 1 S C h t − C h , ⁡ min t Q s , h ′ t , (21)

where H _emo (t) and H _user (t) represent the surplus profits of the IESs and the surplus profits of SMIPs at time t, respectively. K represents the number of electricity power generation equipment. S represents the number of heat power generation equipment. C _e,min(t) and C _h,min(t) represent the lower limits of the electricity and heat bidding prices of the IESs, respectively. C _e,max(t) and C _h,max(t) represent the upper limits of electricity and heat bidding prices of SMIPs, respectively.

The price and power of the electricity and heat cleared by the EM need to be constrained to ensure that the clearing data are within a reasonable range. The formulas are as follows: P e , ⁡ min ′   t ≤ P e ′ t ≤ P e , ⁡ max ′   t , (22) Q h , ⁡ min ′   t ≤ Q h ′ t ≤ Q h , ⁡ max ′   t , (23) C e , ⁡ min t ≤ C e t ≤ C e , ⁡ max t , (24) C h , ⁡ min t ≤ C h t ≤ C h , ⁡ max t , (25)

where P′ _k.e (t) represents the electricity power cleared by the EM. Q′ _s.h (t) represents the heat power cleared by the EM. P′ _e,min(t) represents the essential need of electricity load. Q′ _s.h (t) represents the essential need of heat load. P′ _e,max(t) represents the electricity load after the demand response. Q′ _h,max(t) represents the heat load after the demand response. To solve the proposed optimal electricity supply model for IESs, the energy utilization optimization model for the SMIPs, and the market clearing optimization model for the EM, an improved particle swarm optimization algorithm (Xiao et al., 2017) is utilized to conduct the optimization.

To characterize the benefit relationship among multiple stakeholders, a leader–follower game model based on the Stackelberg game is proposed in this paper to maximize benefits for multiple stakeholders. In the proposed leader–follower game mode, there are three stakeholders: the EM, SMIPs, and the IESs. The EM is the leader in the game, while the SMIPs and IESs are followers. The SMIPs and the IESs respond to the decisions of the EM and adjust the strategy according to their objective functions. The EM aims to maximize social surplus profits, and the strategy set includes clearing electricity power, heat power, and energy prices to the SMIPs and the IESs. The IESs aim to maximize operating benefits, and the strategy set includes energy equipment output, external grid trading amount, and energy storage management. The SMIPs aim to minimize energy procurement costs, and the strategy set includes the electricity loads for the demand response and heat loads for the demand response. The interactive framework of the leader–follower game is shown in Figure 2.

For the Stackelberg equilibrium of a non-cooperative game, when Eq. 26 is satisfied, it indicates that the game has reached equilibrium. At this point, the followers make the best response according to the strategy of the leader. In addition, each stakeholder cannot obtain more profits by changing their own strategy set. E m a r I m a r , F e m o * , L u s e r * ≤ E m a r I m a r * , F e m o * , L u s e r * E u s e r I m a r * , F e m o , L u s e r * ≤ E u s e r I m a r * , F e m o * , L u s e r * E e m o I m a r * , F e m o * , F e m o ≤ E e m o I m a r * , F e m o * , F e m o * , (26)

where E _mar represents the profits of the EM. E _user represents the profits of the SMIPs. E _emo represents the profits of the IESs. L _user , F _emo , and I _mar represent the strategy sets for the SMIPs, the IESs, and the EM, respectively. L*_user , F ^* _emo , and I*_mar represent the optimal strategy sets for the SMIPs, the IESs, and the EM, respectively.

To solve the proposed method, an iteration search method proposed by Chuang et al. (2001) is employed to find the Nash equilibrium point. The solution of the proposed scheduling model is summarized as follows:

Step 1

The strategies of all the stakeholders are initialized as D ^old = [I ⁰ _mar , L ⁰ _user , F ⁰ _emo ], and the profits of all stakeholders are calculated as F ^old = [E ⁰ _mar , E ⁰ _user , E ⁰ _emo ].

Step 2

The scheduling model for each stakeholder is solved based on the exchanged game strategies from other stakeholders.

Step 3

The game strategy of each stakeholder is updated as D ^new = [I ¹ _mar , L ¹ _user , F ¹ _emo ], and the operating cost of all stakeholders is calculated as F ^new = [E ¹ _mar , E ¹ _user , E ¹ _emo ].

Step 4

The operating cost difference of F ^new and F ^old is calculated. If the cost difference is smaller than its threshold, the procedure is terminated, and the new strategies D ^new is output. Otherwise, D ^old is reset as D ^new, and Step 2 onwards is repeated.

This paper conducted a simulation analysis of the day-ahead EM clearing with SMIPs as an example. The WT and PV forecast data, as well as the electricity and heat load power of users, are shown in Figure 3. Table 1 shows the time-of-use energy prices of the electricity and gas grids. Table 2 shows the parameters for IESs and SMIPs.

In order to analyze the advantages of the proposed method, three scenarios are set up. In scenario 1, the objective function of the operation is maximizing the profits of IESs, electricity and heat prices are fixed, and users do not participate in the demand response. In scenario 2, the objective function of the operation is still maximizing the profits of IESs, but the user side will actively respond according to the change in energy prices. In scenario 3, the EM, IES, and SMIP carry out the operation with the proposed strategy of this paper, and the power and price of energy are determined by market clearing through market-side quoting and bidding.

Table 3 shows the operation results of the three scenarios. It shows that under the proposed method, there have been varying degrees of improvement in social surplus profits, profits of the IESs, and profits of the SMIPs, with the most notable increase in social surplus profits.

Compared to scenario 1, in scenario 2, the social surplus increase in the profits of the IES decrease slightly, and the energy purchase costs for SMIPs decrease significantly. This is because through price-based demand response, the user side can peak-shave and valley-fill to smooth the electricity and heat load curves, thereby obtaining extra compensation benefits. As the load demand curve becomes smoother, the arbitrage space obtained by IESs through energy storage will be correspondingly reduced, leading to a decrease in profits. As the load demand becomes flat, the marginal cost of the energy equipment output is reduced, and the social surplus profits increase. In scenario 3, the profits of all stakeholders increase significantly, and the energy purchase costs of SMIPs have been reduced. This is because under the guidance of the EM mechanism, the energy trading price between IESs and SMIPs is determined based on the clearing of the supply and demand relationship at each moment. Thus, the price can better reflect the degree of energy surplus or scarcity within the SMIPs. The proposed method can also increase efficiency in energy storage arbitrage and trading with the external electricity grid, as well as improve the precision of user demand response.

Figure 4 shows the convergence diagram of profits for each stakeholder in scenario 3. It well reflects the game process between various stakeholders throughout the entire iteration process, and finally, equilibrium is achieved at about 80 iterations, which takes 16.3 min. In the game process, the declared energy power and prices from other stakeholders are constantly cleared by the EM. In addition, the EM, as the leader in the entire game process, shows a gradual upward trend in its profits. The SMIPs and IESs adjust their own strategy sets continuously and rationally based on the clearing results of the EM. The SMIPs and IESs, as followers, also engage in game interaction at the same time and finally reach convergence. When the leader and two followers reach the Stackelberg equilibrium, their strategy sets no longer change.

The energy clearing prices are shown in Figure 5. The electricity clearing price in the EM peaks from 18:00 to 22:00. During this period, the PV output is low, and the demand for electricity load is high. To meet the demand, the IES will utilize more gas turbines. At the same time, the gas turbines have the highest marginal cost among all units, so their prices are the highest. Meanwhile, the demand response from the SMIPs is low, and the electric load demand curve is flat. Therefore, the EM will clear the electricity generated by gas turbines at high prices. The heat clearing price in the EM shows a significant peak from 00:00 to 04:00. Therefore, during this period, the output of gas boilers needs to be scheduled to meet the heat needs of the SMIPs. The cost of gas boilers is higher than that of other heat sources, so the heat clearing price in the EM shows a short-term peak.

Figure 6 shows the comparison of the before and after demand response of the SMIPs under the proposed strategy. It indicates that the fluctuations in electricity and heat loads have been significantly reduced, and the effects of peak-shaving and valley-filling are obvious. SMIPs respond more accurately to the energy price in the EM clearing, which brings a lot of hidden benefits to the DN. After the demand response, the electricity and heat load curves are smoothed within an appropriate range, and the energy purchase costs on the user side are reduced. At the same time, the energy supply pressure of the IESs is eased.

The final optimized electricity and heat power of the IESs are shown in Figures 7, 8. According to the following optimization results, several key time periods with obvious characteristics are analyzed:

1) From 0:00 to 9:00, the electric load demand is low. During this period, the electricity clearing price of the EM is low, so the SMIPs follow the price-based demand response and increase electricity usage when the electricity price is low. The output of the WT is relatively high, and only a minimal amount of gas turbine output is needed to supplement the electricity supply alongside the WT. At the same time, the energy prices of the upper-layer electricity and gas grids are relatively low. Therefore, with the complete consumption of the WT, the IESs can use a small amount of gas turbine power while purchasing electricity from the external electricity grid. Then, the electrical energy is stored under the premise of meeting the electrical load demand.