The structure and energy flow of the IES analyzed in this research are depicted in Figure 1, which has three modules: energy supply, energy conversion, and load demand. The system’s electrical load is met by the grid, photovoltaic (PV), gas turbine (GT), and electric storage (ES). The electrical load is met via a heat recovery steam generator (HRSG), gas boilers (GBs), electric boilers (EBs), and heat storage (HS). The electrical load is met via an absorption chiller (AC), electric chiller (EC), ground source heat pump (GSHP), and cooling storage (CS). The gas network supplies the gas load; GT, GB, and AC combine to generate combined cooling, heating, and power (CCHP).

This paper considers price-based electricity–gas load demand response, and a price-based demand response model based on the load price elasticity coefficient matrix is developed to guide customers in responding to changes in energy prices in order to adjust load demand. The change in load is represented by a price elasticity matrix based on the price elasticity coefficient while employing time-sharing electricity and gas pricing (Tan et al., 2020). The elasticity coefficient is defined as the ratio of the rate of change in load demand to the rate of change in price and is calculated as follows: ε i j x = Δ Q j x Q j x / Δ P i x P i x , (8) where ε i j x is the price elasticity coefficient; Q j x and Δ Q j x are the load demand and load change, respectively; ε i j x is the self-elasticity coefficient when i = j; ε i j x is the cross-elasticity coefficient when i ≠ j; and x = 1 for electric load and x = 2 for gas load. The load change reflected by the price elasticity matrix can be calculated using the price elasticity coefficient: Δ Q 1 x / Q 1 x Δ Q 2 x / Q 2 x ⋮ Δ Q n x / Q n x = ε 11 x ε 12 x ⋯ ε 1 n x ε 21 x ε 22 x ⋯ ε 2 n x ⋮ ⋮ ⋮ ε n 1 x ε n 2 x ⋯ ε n n x Δ P 1 x / P 1 x Δ P 2 x / P 2 x ⋮ Δ P n x / P n x . (9) In order to find the price elasticity matrix consisting of the price elasticity coefficients in Equation 9, assuming that for a given time interval l, the energy price at any moment i has the same effect on the consumption of the same type of energy at moments i + l, the following assumptions are made in this paper: ε 11 x = ε 22 x = ⋯ = ε n n x = ε 0 ε 12 x = ε 23 x = ⋯ = ε n − 1 n x = ε 1 x ε 13 x = ε 24 x = ⋯ = ε n − 2 n x = ε 2 x ε 21 x = ε 32 x = ⋯ = ε n n − 1 x = ε − 1 x ε 31 x = ε 12 x = ⋯ = ε n n − 2 x = ε − 2 x ⋯ . (10) Therefore, Equation 9 can be simplified as follows: Δ Q i x Q i x = ∑ l = − m + m ε l Δ P i + l x P i + l x , (11) where i = 1, 2, …, n and P i + l x are the energy prices at neighboring moment i at a distance l; ε _l is the response elasticity coefficient; and m is the range of moments that have an effect on energy prices at moment i. The specific procedure for finding the price elasticity matrix is as follows:

Step 1. Based on the historical energy use data of the studied IES (June–July 2022), corresponding to time period i, the average value of energy prices p _i and the average value of energy use q _i are calculated as follows: p i = R f i + R g i + R p i q f i + q g i + q p i , (12) q i x = q f i x + q g i x + q p i x , (13) where R is the energy price; q is the energy usage; and the subscripts f, g, and p are the peak, valley, and normal periods, respectively.

Step 2. Outlier handling

The 3σ criterion is chosen to remove outliers from the sample. | v k | = X k − X ̄ > 3 σ . (14) If the residual satisfies Equation 14, the value is considered an outlier. Since the standard deviation σ is usually unknown, the experimental standard deviation difference s (X _k ) is used instead of σ. The formula for s (X _k ) is as follows: s X k = 1 n − 1 ∑ k = 1 n X k − X ̄ 2 , (15) where X _k is the value of the first k samples; X ̄ is the mean value of the samples.

Step 3. The rate of change of energy and energy prices is calculated. The rate of change of energy and energy prices in Equation 9 is calculated using Equations 15, 16, respectively: Δ Q i x Q i x = Q i 2 x − Q i 1 x Q i 1 x , (16) Δ P i x P i x = P i 2 x − P i 1 x P i 1 x , (17) where P i 2 x is the average value of energy price in July at i period; P i 1 x is the average value of energy price in June at i period; Q i 2 x is the energy usage in July at i period; and Q i 1 x is the energy usage in June at i period.

Step 4. The price elasticity coefficient matrix based on Equation 11 is calculated. A multiple regression algorithm is applied to calculate the price elasticity coefficient ɛ _l , l = −m, − m + 1, …, + m.

After using the time-sharing electricity–gas price, the following formula may be derived based on Equation 9: Q 1 * x Q 2 * x ⋮ Q n * x = Q 1 x Q 2 x ⋮ Q n x + Q 1 x 0 ⋯ 0 0 Q 2 x ⋯ 0 ⋮ ⋮ ⋮ 0 0 ⋯ Q n x Δ Q 1 x / Q 1 x Δ Q 2 x / Q 2 x ⋮ Δ Q n x / Q n x , (18) where Q n * x is the electricity–gas load demand after the introduction of time-of-use pricing.

Adjustable cooling load demand response modifies the cooling load within the user’s temperature comfort range based on the ambiguity of the user’s temperature demand and transforms the cooling load curve into a cooling load interval, increasing system scheduling flexibility. A building heat balance model is a physical description of the building heat exchange process that can analyze the principle and process of temperature change in the building, describe the process of heat gain and loss in the building, and obtain the connection between temperature and cooling power in the building (Triolo et al., 2023). The following heat balance equation can be used to describe the temperature change inside the building: c g a s m g a s + c s m s d T b d t = q l o a d t − q c t , (19) where c _gas and c _s are the specific heat capacity of air and the specific heat capacity of the envelope, respectively; m _gas and m _s are the air mass and the envelope mass, respectively; dT _b /dt is the differentiation of room temperature with respect to time; q _c(t) is the cooling power; and q _load(t) is the building load.

The building load q _load(t) consists of the following components: q l o a d t = q 1 t + q 2 t + q 3 t , (20) where q ₁(t) is the heat exchange between the maintenance structure and the environment; q ₂(t) is the heat transfer from the outside air; and q ₃(t) is the heat dissipation inside the building. q ₁(t), q ₂(t), and q ₃(t), respectively, can be expressed as q 1 t = K F T i n t − T o u t t q 2 t = T i n t − T o u t t c g a s ρ g a s V q 3 t = A e t + p e t (21) where K is the average heat transfer coefficient of the maintenance structure; F is the heat transfer area of the maintenance structure; T _in(t) is the indoor temperature; T _out(t) is the outdoor temperature; ρ _gas is the air density; V is the air exchange volume; A is the floor area; e(t) is the amount of heat dissipated by the equipment per unit area; and P _e(t) is the heat dissipation power per unit area. The discrete treatment of the building heat balance equation is as follows: q c t =   − T i n t + 1 − T i n t c g a s m g a s + c s m s Δ t +   q l o a d t . (22) According to the aforementioned calculation, the user’s cooling load demand is proportional to the indoor temperature and equals the cooling power under supply–demand balance. Thus, the adjustment interval of the cold power supplied by the IES may be estimated based on the user’s temperature comfort range, reducing the burden of delivering energy during the peak cooling period.

The daily operating cost of the system includes the purchase cost of electricity and gas f _buy and the operation and maintenance cost of each equipment f _om. F 1 = f b u y + f o m , (24) f b u y = ∑ t = 1 24 P g r i d t C g r i d t + ∑ t = 1 24 G g r i d t C g a s t , (25) where P _grid(t) is the purchased power at time t, C _grid(t) is the price of electricity at time t, G _grid(t) is the purchased gas volume at time t, and C _gas(t) is the price of gas at time t. f o m = ∑ i = 1 n ∑ t = 1 24 P i t η i , (26) where P _i (t) is the power output of the ith device at time t; η _i is the operation and maintenance cost of the ith equipment output.

Carbon emissions from power and gas purchases are included in the IES carbon emissions. The entire system’s carbon emissions are calculated using the unit carbon emission factor as follows: F 2 = ∑ t = 1 24 P g r i d t λ g r i d t + ∑ t = 1 24 G g r i d t λ g a s , (27) where λ _grid is the purchased power carbon emission factor; λ _gas is the purchased gas carbon emission factor.

The IES contains several types of energy with varying characteristics. Exergy can represent the “quantity” as well as the “quality” of energy. We introduce an energy quality system to measure the losses in each energy source’s conversion process and determine the utilization of each energy source, i.e., exergy efficiency (Chen et al., 2022). The inverse of exergy efficiency is used as the optimization objective in the following equation: F 3 = ∑ t = 1 24 P l d t λ e + H l d t λ h + C l d t λ c + G l d t λ g ∑ t = 1 24 P g r i d t λ e + P p v t λ p v + G g r i d t λ g , (28) where P _ld(t), H _ld(t), C _ld(t), and G _ld(t) are the electrical, thermal, cooling, and gas loads of the system at time t, respectively; λ _e, λ _h, λ _c, λ _g, and λ _pv are the energy quality coefficients of electrical, thermal, cooling, and PV, respectively; and P _pv(t) is the PV power generated at time t.

The IES is subjected to power balance constraints and conditions when solving optimal scheduling. The electrical power balance constraint is shown as follows: P g t t + P p v t +   P d i s 1 t + P g r i d t = P e c t + P e b t + P g p t + P l d t +   P c h a r 1 t . (29)

The thermal power balance constraint is H g b t + H e b t + H h g t + P d i s 3 t = H l d t + P c h a r 3 t . (30)

The cooling power balance constraint is C a c t + C e c t + C g p t + P d i s 2 t = C l d t + P c h a r 2 t . (31) The gas power balance constraint is G g r i d t = G g t t + G g b + G l d t . (32)

The system power and gas purchase constraints are 0 ≤ G g r i d t ≤ G b u y m a x 0 ≤ P g r i d t ≤ P g r i d m a x (33) where G _buymax is the upper limit of purchased gas power; P _gridmax is the upper limit of purchased electricity power.

According to the “Office Building Design Standards” in China, the indoor temperature of office buildings in summer should be in the range of 25°C–28°C. The temperature comfort constraint is 25 ≤ T t ≤ 28 . (34)

In this paper, NSGA-II based on Tent mapping chaos optimization is used to solve the proposed multi-objective optimization model, due to its advantages of large search space and fast convergence (Verma et al., 2021). The specific solving process is depicted in Figure 2. After obtaining the Pareto optimal frontier solution set using NSGA-II based on Tent mapping chaos optimization, the optimal scheduling scheme is selected using the VIKOR decision method, and the specific process is as follows (Zheng and Wang, 2020):

Step 1. Let A = {A ₁, A ₂, …, A _n } be the set of force solutions to be selected, n = 100; G = {G ₁, G ₂, …, G _m } be the set of objectives, m = 3; and W = {w ₁, w ₂, …, w _m } be the set of objective weights.

Step 2. The raw data x _ij = {i = 1, 2, …, n; j = 1, 2, ⋯m} are normalized. v i j = x i j ∑ i = 1 n x i j 2 , (35) where v _ij is the normative value of the j objective of the i scenario.

Step 3. The positive and negative ideal solutions are obtained. r + = max x i 1 , max x i 2 , … , max x i m , r − = min x i 1 , min x i 2 , … , min x i m (36) where r ⁺ and r ⁻ are the positive and negative ideal solutions, respectively.

Step 4. The group benefit values and individual regret values are calculated. S i = ∑ w i j r j + − v i j r j + − r j − , (37) R i = max w i j r j + − v i j r j + − r j − (38) where S _i and R _i are the group effect value and individual regret value of the i decision option, respectively.

Step 5. The decision index value Q _i based on S _i , R _i is calculated. Q i = v S i − S − S + − S − + 1 − v R i − R − R + − R − , (39) where Q _i is the value of the decision indicator; S ⁺ and S ⁻ are the maximum and minimum values of group effect, respectively; and R ⁺ and R ⁻ are the maximum and minimum values of individual regret, respectively.

Step 6. According to the decision index values of each solution, the solution corresponding to the first ranked index value is the optimal solution.

The Sino-German Ecological Park in Qingdao, China, is chosen as the research object to investigate the applicability and effectiveness of the proposed IES multi-objective optimal dispatch model incorporating exergy efficiency and demand responsiveness (DR). The multi-objective optimal scheduling analysis is performed in this example based on the electricity, gas, cooling, and heating loads in typical summer days. The global scheduling time is 24 h, and the unit scheduling time is 1 h. Figure 3 depicts the system’s projection for power, gas, cooling, and heating loads and solar output. The price information of natural gas and electricity used in this case study is listed in Table 1. Table 2 lists the economic parameters of the equipment. Table 3 lists the parameters of the modeling. Among them, IES internal equipment parameters, energy storage equipment parameters, and energy time-of-use prices are derived from Wang et al. (2022). The following NSGA-II settings are considered: the population size is 50, the maximum number of iterations is 300, the crossover percentage is 0.7, and the mutation percentage is 0.3. The average summer days of IES are chosen for this paper’s energy supply study. Because the heat load is small and stable, it is excluded in demand response.

To validate the feasibility and effectiveness of the IES optimal scheduling model developed in this paper, five scenarios are set up for comparative analysis, and the settings for each scenario are shown in Table 4. The optimization outcomes under each scenario are solved after optimization computation, and the system operation cost, carbon emission, and energy efficiency are compared, and the optimum scheduling results are shown in Table 5. As shown in Table 5, the performance of individual indexes in Scenario 4 is slightly worse than that in Scenarios 1, 2, and 3; however, it can take into account the three indices of system economy, environmental protection, and energy efficiency at the same time. Compared to Scenario 1, exergy efficiency improves by 5.1%; compared to Scenario 2, carbon emissions are decreased by 3.5%; and compared to Scenario 3, operation costs are lowered by 5.2%. As a result, the model suggested in this study can meet IES scheduling criteria for numerous purposes, such as economy, environmental protection, and exergy efficiency.

The load variations following demand response are depicted in Figure 4 and Figure 5. When combined with Table 5, it can be observed that all indicators of Scenario 5 are better than those of Scenario 4, in which energy efficiency is increased by 6.4%, carbon emissions are decreased by 1.3%, and operation costs are lowered by 1.7%. This is primarily due to the fact that after demand response, IES can appropriately cut the peak values of electric load, gas load, and cooling load while satisfying the basic load demand and maintaining a comfortable temperature and, lowering the cost of purchasing high-priced energy during the peak period; at the same time, it can shift the peak load, allowing for multi-objective optimization to reduce carbon emissions and improve exergy efficiency.

The peak–valley difference of electric load after demand response decreases by 21.9% and the peak–valley difference of gas load after demand response decreases by 17.8% compared to those before demand response. The load fluctuation is leveled off, which is due to the impact of price response on users’ energy consumption habits, and users’ loads are shifted downward and upward during peak and valley hours, respectively, as shown in Figure 4 and Figure 5. The efficiency of price-based demand response for optimizing electricity–gas load is demonstrated. During the peak cold load period, the cold load is properly reduced to ensure the user’s comfort, relieving supply pressure during the peak energy consumption period. This is due to the human body’s ambiguity for temperature changes within a certain range, and adjusting the cold load to change the temperature within this range has no effect on the comfortable temperature, so the cold load curve is transformed into a cold load interval, transforming the cold load into a flexible adjustable load to participate in demand response to enhance system scheduling flexibility. The electric, gas, and cooling load curves can be efficiently smoothed using demand response, ensuring improved system operating economy, environmental protection, and exergy efficiency.

The Pareto solution set of Scenario 5 is illustrated in Figure 6, demonstrating that the low operating cost, low carbon emission, and high exergy efficiency of IES are three competing optimization objectives, and obtaining the optimal solution simultaneously is difficult. As a result, the model developed in this paper can satisfy IES’s optimal scheduling requirements in terms of economy, environmental protection, and exergy efficiency. Decision-makers in the multi-objective optimal decision method used in this study can alter the VIKOR decision weight parameters based on the real needs of the project to acquire the appropriate optimal scheduling solution.

The optimal dispatching result of Scenario 5 is shown in Figure 7, Figure 8, Figure 9, and Figure 10. Due to the cost-effectiveness, minimal carbon footprint, and high exergy efficiency of PV, it is completely consumed during the dispatching procedure. During 23:00–06:00, the system purchases low-cost electricity from the grid as its primary supply and from GT as its supplementary supply, while the storage apparatus stores electricity. At this time, the cold load is low, with EC and GSHP as the primary supply method and AC as the supplementary supply; during 07:00–10:00, the electric load and cold load continue to increase, the price of electricity is moderate, and the joint power supply equipment can meet the needs of a variety of loads; during 11:00–13:00, the electric load and cold load reach their peak, the price of electricity is at its highest, and GT generation is more advantageous. AC as the main refrigeration equipment, which can use low-grade thermal energy to cool, has a more balanced exergy efficiency and economy compared to the consumption of high-grade electrical energy to cool the electric refrigeration equipment; during 14:00–17:00, the cold load demand is greater, the electricity price is flat, and the system is suited to increase the output of the electric refrigeration equipment; and during 18:00–22:00, which are the peak electricity price hours, the output of the electric refrigeration equipment should be decreased.

Multi-objective optimization: The model proposed in this paper effectively balances multiple objectives, including system operating costs, carbon emissions, and exergy. By providing a Pareto optimal frontier solution set, it enables decision-makers to weigh the trade-offs and make informed choices that are in line with their priorities. In this paper, we use NSGA-II to achieve efficient solution space exploration and obtain diverse sets of Pareto optimal solutions. This feature enables decision-makers to identify a range of feasible and desirable solutions. Integrated demand response: The introduction of a demand response mechanism allows dynamic load regulation, promotes demand-side management and energy conservation, and helps improve the overall economic and environmental performance of the IES. The model takes into account user’s comfort when optimizing energy scheduling, ensuring user’s comfort, enhancing user’s satisfaction, and increasing user’s participation in demand response programs.

Scalability and complexity: Although the model in this paper considers different types of energy devices and demand response scenarios, it may face challenges when applied to larger and more complex IESs. Future research should address scalability issues for a wide range of practical applications. Data requirements and model generalization: Models are highly dependent on accurate and real-time data, and obtaining and managing these data can be resource-intensive and may present challenges in practice; as with any optimization model, the generalizability of the approach presented in this paper to different types of IESs requires further study. The unique characteristics and constraints of a particular system may require customization of the model.

In the study of this paper, a multi-objective optimal dispatch model is proposed which takes into account energy efficiency and demand response. This model may have an impact on the intensity of communication between participants in an IES. In the real world, different participants in the energy system need to exchange information frequently in terms of supply and demand matching and energy trading. Our model may introduce new information exchange needs, e.g., sharing of energy supply plans during demand response adjustments to ensure smooth operation of the system. Such information exchange may affect aspects such as cooperation patterns, data sharing, and decision coordination among participants. In addition, different types of energy equipment and systems may involve different modes of information exchange. For example, there may be differences in the way information is exchanged between electric, thermal, and gas energy systems because of their different characteristics and operational needs. The model in this paper may introduce closer information exchange between different systems for cross-system coordination and optimization. This diversity of information exchange models requires adequate technical support to ensure the safety and reliability of data transmission. In this paper, we pay special attention to the modeling of information exchange and technological diversity in order to better understand the impact of our model on the intensity of communication between participants in an IES. In IESs, effective modeling of information exchange is essential to achieve efficient system operation. Górski (2023) emphasized the importance of information exchange in software applications, especially in the communication between different systems, introducing integration services and business views. The multi-objective optimal scheduling model in this paper also aims to create an integrated view between various energy devices. Such a view helps reveal the information needs and communication patterns among the energy system participants. In Zhao et al. (2023), the role of emerging information and communication technologies in driving energy system transformation is emphasized. The model developed in this paper is based on an IES that leverages information exchange and technological diversity to optimize energy scheduling. In the model proposed in this paper, different types of energy devices and demand response mechanisms are considered, and frequent data and information exchanges are required among the various players. Therefore, the modeling of information exchange needs to be fully considered in the model, including data transmission methods, communication protocols, and the time cost of information processing and delivery. In an IES, different energy devices and technologies are usually diverse, such as smart meters, sensors, and communication networks. For example, in our model, we introduce a price elasticity matrix to simulate the exchange of information in the energy market to support electricity–gas load demand response. This has similarities to the idea of using message flow modeling discussed in the paper, where information exchange is used to achieve coordination among system components. The diversity of these technologies not only provides broader possibilities for information exchange but also brings a number of challenges, such as data compatibility and information security. Taken together, the multi-objective optimal scheduling model in this paper has far-reaching implications for the modeling of information exchange and technological diversity in IESs. In order to further improve the model, appropriate modeling of information exchange needs to be embedded in the model to ensure efficient communication among participants. At the same time, different information and communication technologies should be actively applied to accommodate the technological diversity in IESs and to address the associated technological challenges. Through such explorations and improvements, the model proposed in this paper will be better adapted to practical applications and make positive contributions to the intelligent and sustainable development of IESs.