Robust Operation of Electric–Heat–Gas Integrated Energy Systems Considering Multiple Uncertainties and Hydrogen Energy System Heat Recovery

Abstract

Due to the high cost of hydrogen utilization and the uncertainties in renewable energy generation and load demand, significant challenges are posed for the operation optimization of hydrogen-containing integrated energy systems (IESs). In this study, a robust operational model for an electric–heat–gas IES (EHG-IES) is proposed, considering the hydrogen energy system heat recovery (HESHR) and multiple uncertainties. Firstly, a heat recovery model for the hydrogen system is established based on thermodynamic equations and reaction principles; secondly, through the constructed adjustable robust optimization (ARO) model, the optimal solution of the system under the worst-case scenario is obtained; lastly, the original problem is decomposed based on the column and constraint generation method and strong duality theory, resulting in the formulation of a master problem and subproblem with mixed-integer linear characteristics. These problems are solved through alternating iterations, ultimately obtaining the corresponding optimal scheduling scheme. The simulation results demonstrate that our model and method can effectively reduce the operation and maintenance costs of HESHR-EHG-IES while being resilient to uncertainties on both the supply and demand sides. In summary, this study provides a novel approach for the diversified utilization and flexible operation of energy in HESHR-EHG-IES, contributing to the safe, controllable, and economically efficient development of the energy market. It holds significant value for engineering practice.

1. Introduction

With the proposal of the “dual carbon” goal, countries around the world have launched various explorations around establishing a new energy system that is economically efficient, safe, controllable, clean, and low-carbon [1]. Hydrogen, as a new type of clean energy, is widely regarded as a key pathway for replacing traditional energy sources and facilitating the construction of a new energy system due to its high energy density, zero-carbon emissions during production or utilization, and versatile applications [2,3]. However, the high costs associated with processes such as production, storage, and utilization, as well as inadequate supporting infrastructure, have significantly hindered the development of hydrogen energy. Therefore, there has been a growing interest among scholars in the diversified utilization and efficient collaborative operation planning of hydrogen energy [4].

EHG-IES, as a typical type of IES, has attracted extensive attention from domestic and international scholars [5,6]. Currently, research on EHG-IES is mainly focused on multi-energy coupling technology, capacity planning, energy efficiency improvement, and operational optimization scheduling [7,8]. With the gradual development and maturation of hydrogen energy utilization technologies [9,10], scholars have successively started to explore the introduction of hydrogen energy and related equipment in IES [11,12], aiming to further increase the proportion of new energy sources in power supply [13,14]. Tostado et al. developed a comprehensive model of an electric hydrogen gas microgrid, incorporating various types of energy charging stations for electricity, gas, and hydrogen [15]. Through summer and winter scenario simulations, the model accurately reflects the operational performance of the electric hydrogen gas microgrid under multiple operating conditions. Ozturk et al. introduced hydrogen energy into an electricity-to-gas micro energy network, which effectively enhances the integration of renewable energy and promotes coordinated operation of the electricity-to-gas conversion system [16]. Shi et al. introduced hydrogen energy into the concept of shared energy storage, proposing a strategy to optimize microgrid clusters through shared hydrogen storage [17]. Case studies were conducted to validate the complementary advantages of microgrids and hydrogen energy storage, which reduced the dependency of microgrids on the main power grid and facilitated the friendly supply of renewable energy sources. Clearly, a multi-energy complementary IES with hydrogen can effectively coordinate various energy supply sources, improve economic efficiency, and reduce carbon emissions. However, there is limited research on heat recovery from hydrogen equipment in the aforementioned literature. Some studies focus only on the heat recovery from hydrogen production processes, while neglecting the investigation of heat recovery from methane reforming equipment or hydrogen fuel cells. Therefore, there is an urgent need for systematic research on the utilization of waste heat recovery in hydrogen energy systems.

Regarding IES involving hydrogen energy, researchers have conducted studies on capacity configuration, dispatch optimization, and other related issues. Khalil et al. integrated hydrogen production with the integration of renewable energy sources, effectively promoting renewable energy supply while utilizing surplus electricity for hydrogen production [18]. Case studies demonstrate that this system can significantly reduce or eliminate reliance on fossil fuels. Zhang et al. applied hydrogen storage technology to a port IES, achieving a zero-carbon emission hydrogen energy port. Numerous numerical experiments demonstrate that the hydrogen energy port IES can promote the consumption of new energy sources such as wind and PV power while achieving zero-carbon emissions [19]. Yun et al. established an EHG-IES coupled with oxygen rich combustion and hydrogen doped gas equipment to reduce carbon emissions [20]. Simulation results demonstrate that this system has a wind abandonment rate of 0% and achieves actual carbon emissions of only 35.8% of the quota. It effectively reduces both carbon emissions and economic costs. Tanumoy et al. studied the integration strategies of liquid air energy storage and hydrogen energy storage technologies in subcritical coal-fired power plants. The simulation results indicate that this solution enhances energy utilization efficiency and advances the development of renewable energy storage technologies [21]. In most studies on hydrogen-integrated IES, including but not limited to the aforementioned literature, there are factors that are difficult to accurately predict. On the source side, there are variables such as PV radiation intensity and wind speed. On the demand side, various types of load requirements exist, including electricity, gas, heat, and hydrogen. The high uncertainty of these factors inevitably disrupts the scheduling and operation of the system. However, these uncertain factors are not considered in the aforementioned models. The proposed models are obtained using deterministic optimization methods based on known source-load data. As a result, the resulting scheduling plans and planning strategies carry certain risks.

In summary, we can observe the following:

(1) At present, there is a lack of systematic research on the heat recovery and utilization of hydrogen energy systems, which has resulted in the inability to fully exploit the diversified utilization of hydrogen energy. This has led to long-term high production costs and low energy utilization efficiency. In other words, the energy utilization potential of hydrogen energy systems has not been fully tapped.

(2) Currently, in addressing the multiple uncertainties in IES, compared to RO, methods such as reinforcement learning and digital twins face challenges in parameter setting, making it difficult to achieve the desired results. On the other hand, DRO relies heavily on historical data, making ARO more suitable for solving optimization problems in uncertain energy systems. To date, there has been no research on using ARO to address the scheduling and planning problems of HESHR-EHG-IES involving hydrogen heat recovery.

This study constructs an EHG-IES with diversified hydrogen utilization by incorporating devices such as water electrolysis equipment for hydrogen production, methanation equipment, and hydrogen fuel cells. Throughout the various stages of the hydrogen energy system, significant amounts of heat energy are released during reactions. Research on HESHR can effectively enhance energy utilization efficiency. However, the contradiction between economic costs and system stability, as well as the interference from multiple uncertainties, severely hinder the stable, efficient, and low-carbon operation of EHG-IES. In light of this, this study proposes a robust operational model for EHG-IES considering both HESHR and multiple uncertainties. Based on the column and constraint generation (C&CG) algorithm and strong duality theory, the original problem is decomposed and transformed. Subsequently, the master problem with mixed-integer linear characteristics and subproblems are obtained, which are solved iteratively through alternating iterations, ultimately obtaining the corresponding optimal solutions. The specific contributions of this study are outlined below:

(1) By fully leveraging the advantages of diversified hydrogen utilization, this study constructs an EHG-IES for hydrogen production, storage, and utilization; taps into the potential of heat recovery in the hydrogen energy system; and further establishes the HESHR-EHG-IES. The impact of the hydrogen energy system with heat recovery on energy exchange among the different components in EHG-IES is investigated, opening up new avenues for exploring diversified hydrogen utilization.

(2) Considering the multiple uncertainties in HESHR-EHG-IES, a two-stage robust model is proposed based on ARO, which achieves optimal scheduling of system operation under the “worst-case” scenario. Compared with methods such as RO and SO, this approach introduces uncertain adjustment parameters and maximum allowable deviation to flexibly adjust the conservatism of the scheduling scheme obtained by the robust model, helping decision-makers at higher levels make reasonable choices between economic efficiency and stability.

(3) Through case studies, the following can be verified: (1) Considering the HESHR can effectively promote the coordinated operation of electricity and heat, and it has certain advantages in reducing overall system costs and improving energy utilization efficiency. (2) By adjusting the maximum allowable deviation and uncertainty parameters, it is possible to flexibly control the conservatism of the scheduling scheme in the ARO model. (3) The day-ahead scheduling scheme under the ARO model exhibits stronger robustness and the ability to withstand price fluctuations in various energy markets, including electricity, heat, hydrogen, and gas, in real-time.

(4) The effective utilization of waste heat recovery from the hydrogen energy system enhances the system’s economic performance and improves its overall thermal efficiency. Meanwhile, by comparing different uncertainty handling methods, after considering both source and load uncertainties in the system, the solution obtained through ARO ensures 100% feasibility in random scenarios, avoiding the conservatism of decision-making.

2. HESHR-EHG-IES Model

The HESHR-EHG-IES constructed in this paper is a system that integrates multiple energy and energy storage technologies, aiming to provide a reliable and efficient solution for optimizing the IES operation. The HESHR-EHG-IES structure is depicted in Figure 1.

The HESHR-EHG-IES mainly includes the hydrogen energy system (hydrogen production through water electrolysis, methane production, hydrogen fuel cells, hydrogen storage tanks), as well as the power generation system (wind and PV power generation, micro gas turbines), combined heat and power (CHP), and energy storage systems. It also considers the comprehensive loads, including electricity load, heat load, gas load, and hydrogen load. A traditional hot water heating system is used, with a supply water temperature of 100~130 °C and a return water temperature of 40~70 °C. In addition to using electricity to produce hydrogen and producing electricity and methane using hydrogen, the hydrogen energy system is also coupled with the hot water return pipeline. By preheating the return water, it provides heat support for the heat energy system, achieving heat recovery and utilization.

2.1. HESHR Model

The efficiency of hydrogen production through water electrolysis generally ranges from 51% to 70%, with heat losses accounting for 20% to 30%. The overall efficiency of the electricity-to-gas-to-electricity conversion process does not exceed 40%. The efficiency of hydrogen fuel cells is approximately 60%, while the efficiency of methane production from hydrogen is around 75% to 80% [22]. Throughout the entire process, a significant amount of heat energy is released. If this heat energy can be effectively recovered and utilized, it will greatly enhance the energy utilization efficiency of the system.

During the operation of hydrogen energy systems, a significant amount of heat energy is released during the energy conversion processes of water electrolysis, methane production, and hydrogen fuel cells. To construct an accurate HESHR model, it is necessary to establish a comprehensive heat energy recovery model for each module of the hydrogen energy system based on the principles of the system’s equipment and thermodynamic equation. This is a prerequisite for the utilization of waste heat in hydrogen energy systems and provides a theoretical foundation for the further study of constructing HESHR-EHG-IES under multiple uncertainties.

2.1.1. Heat Energy Recovery Model for Water Electrolysis

(1)

Electrolysis water energy conversion equation.

According to the principle of hydrogen production by electrolysis of water, the mathematical model of electric energy conversion is shown in Equations (1)–(3).

$$m_{ED,H}\left(t\right)={\displaystyle \frac{{\eta }_{ED,H}}{{\mathrm{LHV}}_{\mathrm{H}}}}{P}_{ED,el}\left(t\right)$$

(1)

$$P_{ED,el}^{\min }\le P_{ED,el}\left(t\right)\le P_{ED,el}^{\max }$$

(2)

$$C_{WE}\left(t\right)=c_{ED}^{om}{P}_{ED,el}\left(t\right)$$

(3)

where $P_{ED,el}(t)$ represents the total energy consumption for hydrogen production and heat generation during water electrolysis at time $t$. $m_{ED,H}(t)$ represents the hydrogen production rate at time $t$ during water electrolysis. $LH{V}_H$ represents the lower heating value of hydrogen. ${\eta }_{ED,H}$ represents the energy conversion efficiency. $P_{ED,el}^{\min }$ and $P_{ED,el}^{\max }$ represent the minimum and maximum input power of the water electrolysis system, respectively. $C_{WE}\left(t\right)$ represents the operation and maintenance cost of hydrogen production from water electrolysis at time $t$. $c_{ED}^{om}$ represents the unit operation and maintenance cost of hydrogen production from water electrolysis.

(2)

Thermodynamic equation for electrolytic water.

Water electrolysis technology can be primarily categorized into low-temperature electrolysis and high-temperature electrolysis. Low-temperature electrolysis includes alkaline electrolysis and proton exchange membrane electrolysis, while high-temperature electrolysis is represented by high-temperature solid oxide electrolysis [23,24]. Both types of electrolysis technologies have the potential for heat recovery and utilization. Research indicates that low-temperature electrolysis has greater value in terms of heat recovery. In this study, we adopt the low-temperature electrolysis mode for hydrogen production.

Based on the principle of hydrogen production through low-temperature electrolysis, the steady-state heat model for the hydrogen production reaction can be expressed as shown in Equation (4).

$$T_{LED}^{\min }\le T_{LED}\left(0\right)+{\displaystyle \frac{1-{\eta }_{ED,H}}{C_{LED}}}{\displaystyle \sum _{t^{\prime }=1}^t[P_{LED,el}\left(t\right)}\mathsf{\Delta }t-{\displaystyle \frac{1}{C_{LED}}}{\displaystyle \sum _{t^{\prime }=1}^t{P}_{LED,rec}\left(t\right)}\mathsf{\Delta }t-{\displaystyle \frac{1}{C_{LED}}}{\displaystyle \sum _{t^{\prime }=1}^t{P}_{LED,loss}\left(t\right)}\mathsf{\Delta }t]\le T_{LED}^{\max }$$

(4)

where $T_{LED}\left(0\right)$ represents the operating temperature during the initial period of low-temperature water electrolysis. $C_{LED}$ represents the equivalent heat capacity of low-temperature water electrolysis. $T_{LED}^{\min }$ and $T_{LED}^{\max }$ represent the minimum and maximum values of the hydrogen production operating temperature, respectively. $P_{LED,el}\left(t\right)$ represents the total energy consumption for hydrogen production and heat generation at time t during low-temperature water electrolysis. $P_{LED,rec}\left(t\right)$ represents the recoverable thermal energy at time t during low-temperature water electrolysis. $P_{LED,loss}(t)$ represents the thermal loss at time $t$ during low-temperature water electrolysis.

According to Equation (4), it can be inferred that the heat energy storage capacity of water electrolysis is primarily determined by the size of the equivalent capacitance, indicating a certain level of heat energy storage capability. The heat loss power consumption of low-temperature electrolysis can be represented as Equation (5).

$$P_{ED,loss}\left(t\right)={\displaystyle \frac{1}{R_{ED}}}\left[T_{LED}\left(0\right)+{\displaystyle \frac{1-{\eta }_{ED,H}}{C_{LED}}}{\displaystyle \sum _{t^{\prime }=1}^t\begin{array}{l}\left[P_{LED,el}\left(t\right)\mathsf{\Delta }t\right]-\frac{1}{C_{LED}}{\displaystyle \sum _{t^{\prime }=1}^t\left[P_{LED,rec}\left(t\right)\mathsf{\Delta }t\right]}\\ -\frac{1}{C_{LED}}{\displaystyle \sum _{t^{\prime }=1}^t\left[P_{LED,loss}\left(t\right)\mathsf{\Delta }t\right]}\end{array}}-T_a\right]$$

(5)

where $R_{ED}$ represents the equivalent thermal resistance of water electrolysis, and $T_a$ denotes the ambient temperature. Constrained by the characteristics of heat recovery equipment and electrolytic hydrogen operation, the recoverable heat energy is subject to the constraint described by Equation (6).

$$P_{LED,rec}^{\min }\le P_{LED,rec}\left(t\right)\le P_{LED,rec}^{\max }$$

(6)

(3)

Heat recovery model.

Based on the principles of water electrolysis and thermodynamics, a low-temperature heat recovery model is established, considering the operational characteristics under multiple conditions, as shown in Equation (7).

$$Q_{LED}\left(t\right)={\eta }_{LED,he}{P}_{LED,rec}\left(t\right)$$

(7)

where $Q_{LED}\left(t\right)$ represents the actual recoverable thermal energy at time $t$ during low-temperature water electrolysis. ${\eta }_{LED,he}$ represents the equivalent heat exchange efficiency of the water electrolysis heat exchanger.

2.1.2. Heat Energy Recovery Model for Hydrogen Fuel Cells

Based on the principles of hydrogen fuel cells and thermodynamics, the heat recovery model is given by Equations (8)–(10).

$$\left\{\begin{array}{l}{P}_{HFC,e}\left(t\right)={\eta }_{HFC,e}{m}_{HFC}\left(t\right)LH{V}_H\\ P_{HFC,h}\left(t\right)={\eta }_{HFC,h}{m}_{HFC}\left(t\right)LH{V}_H\end{array}\right.$$

(8)

$$0\le P_{HFC,e}\left(t\right)\le P_{HFC,e}^{\max }$$

(9)

$$C_{HFC}\left(t\right)=c_{HFC}^{om}LH{V}_H\times m_{HFC}(t)$$

(10)

where $m_{HFC}\left(t\right)$ represents the hydrogen consumption at time $t$ for the hydrogen fuel cell. $P_{HFC,e}(t)$ and $P_{HFC,h}\left(t\right)$ represent the power output and heat generation power of the hydrogen fuel cell at time t, respectively. ${\eta }_{HFC,e}$ and ${\eta }_{HFC,h}$ represent the discharge efficiency and heat generation efficiency of the hydrogen fuel cell, respectively. $P_{HFC,e}^{\max }$ represents the maximum discharge power of the hydrogen fuel cell. $C_{HFC}\left(t\right)$ represents the operational and maintenance cost of the hydrogen fuel cell at time $t$. $c_{HFC}^{om}$ represents the unit operational and maintenance cost of the hydrogen fuel cell.

2.1.3. Heat Energy Recovery Model for Methane Production Equipment

According to the chemical reaction equations of H₂-CH₄, the reaction not only produces CH₄ but also releases a significant amount of heat energy. However, the release of such a large amount of heat energy can severely hinder the progress of the chemical reaction. To ensure the safe operation of methane production equipment, temperature management systems are generally required. In most cases, cascading devices are used to complete the reaction in multiple reactors to produce methane. Heat exchange is carried out in different reactors, and heat recovery steps can be added to achieve cooling during the process.

As indicated by the thermochemical Equations of methane production equipment, it is necessary to achieve a mass balance between the products and reactants during the reaction process. The specific relationship constraints can be described as shown in Equations (11)–(13).

$${\displaystyle \frac{m_{C{O}_2}\left(t\right)}{M_{C{O}_2}}}={\displaystyle \frac{m_{MR,H}\left(t\right)}{n_{C{H}_4 - h}{M}_H}}={\displaystyle \frac{m_{C{H}_4}\left(t\right)}{M_{C{H}_4}}}=n_{C{H}_4}\left(t\right)$$

(11)

$$P_{MR,heat}\left(t\right)={\displaystyle \frac{{\eta }_{MR}{m}_{MR,H}\left(t\right)\mathsf{\Delta }H}{n_{C{H}_4 - H}{M}_H}}={\eta }_{MR}{n}_{C{H}_4}\left(t\right)\mathsf{\Delta }H$$

(12)

$$Q_{MR}\left(t\right)={\eta }_{\mathrm{MR},\mathrm{heat}}{P}_{\mathrm{MR},\mathrm{heat}}\left(t\right)$$

(13)

where $m_{C{O}_2}\left(t\right)$ and $m_{MR,H}\left(t\right)$ represent the consumption rates of CO₂ and H₂ by the methanation equipment at time $t$, respectively. $m_{C{H}_4}\left(t\right)$ and $n_{C{H}_4}\left(t\right)$ represent the production rate of CH₄ and the molar rate of the methanation equipment at time $t$, respectively. $n_{C{H}_4-H_2}$ represents the molar ratio of CH₄ production to H₂ consumption by the methanation equipment. $M_{C{O}_2}$, $M_H$, and $M_{C{H}_4}$ represent the relative molecular masses of CO₂, H₂, and CH₄, respectively. $P_{MR,heat}\left(t\right)$ represents the heat power generated by the methanation process at time $t$. ${\eta }_{MR}$ represents the operational efficiency of the methanation equipment. $\mathsf{\Delta }H$ represents the heat energy generated per mole of CH₄ produced, approximately 165 J/mol. $Q_{MR}(t)$ represents the heat energy recovered by the methanation equipment at time $t$. ${\eta }_{MR,heat}$ represents the heat recovery efficiency of the methanation equipment.

2.1.4. HESHR Model

Hydrogen energy systems are connected to external heating networks and coupled with external heating systems through heat exchangers and heat pumps. Simultaneously, the internal heat energy management system of the hydrogen energy system achieves efficient waste heat recovery and utilization by controlling the cooling mass flow rates and temperatures of water electrolysis devices, hydrogen fuel cells, and methane production equipment. This enhances energy conversion efficiency. In this study, based on the principle of energy conservation, a HESHR is established as shown in Equation (14).

$$Q_{out}\left(t\right)=Q_{\mathrm{ED}}\left(t\right)+Q_{\mathrm{MR}}\left(t\right)+P_{\mathrm{HFC},\mathrm{h}}\left(t\right)$$

(14)

where $Q_{out}\left(t\right)$ represents the heat energy recovered by the hydrogen energy system at time t, consisting of heat recovery from each stage of the water electrolysis device, hydrogen fuel cell, and methanation equipment.

2.2. Micro Gas Turbine Model

The HESHR-EHG-IES power system primarily consists of wind and solar power generation, micro gas turbines, and energy storage batteries, with the generation cost represented by a linear function. The cost function of the micro gas turbine is given by Equation (15):

$$C_G\left(t\right)=\left[a{P}_G\left(t\right)+b\right]\mathsf{\Delta }t$$

(15)

$$P_{\mathrm{G}}^{\min }\le P_{\mathrm{G}}(t)\le P_{\mathrm{G}}^{\max }$$

(16)

where $C_G\left(t\right)$ and $P_G\left(t\right)$ represent the generation cost and output power of the micro gas turbine at time t, respectively. $a$ and $b$ are the cost coefficients. This study considers a micro gas turbine with a relatively fast power response rate in relation to an hourly scheduling period and thus neglects its ramp-up rate constraints. Only the maximum and minimum output power constraints are considered, as shown in Equation (16). $P_G^{\max }$ and $P_G^{\min }$ represent the upper and lower limits of the output power of the micro gas turbine, respectively.

2.3. Storage Model

The cost of the energy storage system consists of its one-time purchase cost and operation and maintenance costs. The integrated charging and discharging cost $C_S\left(t\right)$ of the energy storage system during the operational period at time $t$ is given by Equation (17).

$$C_{\mathrm{S}}(t)=K_{\mathrm{S}}\left[P_{\mathrm{S}}^{dis}(t)/{\eta }_S+P_{\mathrm{S}}^{ch}(t){\eta }_S\right]\mathsf{\Delta }t$$

(17)

$$0\le P_S^{ch}(t)\le [1-U_S(t)]P_S^{max}$$

(18)

$$0\le P_S^{d\mathrm{i}s}(t)\le U_S(t)P_S^{\max }$$

(19)

$${\eta }_S{\displaystyle \sum _{t=1}^{N_T}\left[P_S^{ch}\left(t\right)\mathsf{\Delta }t\right]}-{\displaystyle \frac{1}{{\eta }_S}}{\displaystyle \sum _{t=1}^{N_T}\left[P_S^{dis}\left(t\right)\mathsf{\Delta }t\right]}=0$$

(20)

$$E_S^{\min }\le E_S\left(0\right)+{\eta }_S{\displaystyle \sum _{t^{\prime }=1}^t\left[P_S^{ch}\left(t^{\prime }\right)\mathsf{\Delta }t\right]}-{\displaystyle \frac{1}{{\eta }_S}}{\displaystyle \sum _{t^{\prime }=1}^t\left[P_S^{dis}\left(t^{\prime }\right)\mathsf{\Delta }t\right]}\le E_S^{\max }$$

(21)

where $P_S^{dis}\left(t\right)$ and $P_S^{ch}\left(t\right)$ represent the charging and discharging power of the energy storage system at time $t$, respectively. $\eta$ represents the efficiency of the energy storage system. $K_S$ represents the unit charging and discharging conversion cost of the energy storage system. Additionally, the energy storage system must satisfy the following constraint Equations (18)–(21) during operation. $P_S^{\max }$ represents the maximum allowable charging and discharging power. $E_S\left(0\right)$ represents the capacity at the initial time of the scheduling period. $E_S^{\max }$ and $E_S^{\min }$ represent the maximum and minimum capacities of the energy storage system within the scheduling period, respectively, to prevent the system from experiencing overcharging or overdischarging. ${\eta }_S$ represents the energy storage charging and discharging efficiency.

2.4. Combined Heat and Power Unit Model

According to the mathematical model of CHP units, the conversion relationships between electricity, heat, and gas, as well as the operational costs, can be expressed as shown in Equations (22)–(24).

$$\left\{\begin{array}{l}{P}_{CHP,e}\left(t\right)=G_{gas}^{CHP}\left(t\right)LH{V}_{gas}{\eta }_{CHP,e}\\ P_{CHP,h}\left(t\right)=G_{gas}^{CHP}\left(t\right)LH{V}_{gas}{\eta }_{CHP,h}\end{array}\right.$$

(22)

$$0\le P_{CHP,e}\left(t\right)\le P_{CHP,e}^{\max }$$

(23)

$$C_{CHP}(t)=c_{CHP}^{om}{G}_{gas}^{CHP}\left(t\right)$$

(24)

$P_{CHP,e}\left(t\right)$ and $P_{CHP,h}\left(t\right)$ represent the electricity and heat generation power of the CHP, respectively. $G_{gas}^{CHP}\left(t\right)$ represents the gas consumption of the CHP at time $t$, while ${\eta }_{CHP,h}$ and ${\eta }_{CHP,e}$ represent the heat and electricity generation efficiencies of the CHP, respectively. $C_{CHP}\left(t\right)$ represents the operation and maintenance cost of the CHP during the time period $t$. $c_{CHP}^{om}$ represents the unit operation and maintenance cost of the CHP, while $LH{V}_{gas}$ represents the lower heating value of natural gas.

2.5. Hydrogen Storage Ttank Model

According to the operational principles and mathematical model of the hydrogen storage tank [25], it is necessary to satisfy the constraints expressed in Equations (25)–(30).

$$p_{\mathrm{HS}}\left(t\right)V_{\mathrm{HS}}={\displaystyle \frac{m_{\mathrm{HS}}\left(t\right)}{M_H}}R{T}_H$$

(25)

$$p_{HS}^{\min }\le p_{HS}\left(t\right)\le p_{HS}^{\max }$$

(26)

$${\eta }_{\mathrm{HS},\mathrm{in}}{\displaystyle \sum _{t^{\prime }=1}^{Nt}\left[m_{\mathrm{HS},\mathrm{in}}\left(t\right)\mathsf{\Delta }t\right]}-{\displaystyle \frac{1}{{\eta }_{\mathrm{HS},\mathrm{out}}}}{\displaystyle \sum _{t^{\prime }=1}^{Nt}\left[m_{\mathrm{HS},out}\left(t\right)\mathsf{\Delta }t\right]}=0$$

(27)

$$m_{HS}^{\min }\le m_{\mathrm{HS}}\left(0\right)+{\eta }_{\mathrm{HS},\mathrm{in}}{\displaystyle \sum _{t^{\prime }=1}^t{m}_{\mathrm{HS},\mathrm{in}}\left(t\right)}\mathsf{\Delta }t-{\displaystyle \frac{1}{{\eta }_{\mathrm{HS},\mathrm{out}}}}{\displaystyle \sum _{t^{\prime }=1}^t{m}_{\mathrm{HS},out}\left(t\right)}\mathsf{\Delta }t\le m_{HS}^{\max }$$

(28)

$$0\le m_{HS,out}\left(t\right)\le U_{HS}\left(t\right)m_{HS,in/out}^{\max }$$

(29)

$$0\le m_{HS,in}\left(t\right)\le \left[1-U_{HS}\left(t\right)\right]m_{HS,in/out}^{\max }$$

(30)

Equation (25) represents the relationship between the pressure and mass of the ideal gas in the hydrogen storage tank. $P_{HS}\left(t\right)$ denotes the pressure within the tank at time $t$, $m_{HS}\left(t\right)$ represents the hydrogen mass stored in the tank during the time period $t$, $R$ is the ideal gas constant, $T_H$ is the gas temperature within the storage tank, and $V_{HS}$ is the volume of the storage tank. Equation (26) represents the upper and lower bound constraints of the pressure within the hydrogen storage tank. $p_{HS}^{\min }$ and $p_{HS}^{\max }$ denote the minimum and maximum pressure within the tank, respectively. Equation (28) describes the relationship between the hydrogen mass in the tank at adjacent time intervals. $m_{HS}\left(0\right)$ represents the initial hydrogen mass in the tank, while ${\eta }_{HS,in}$ and ${\eta }_{HS,out}$ correspond to the hydrogen charging and discharging rates within the storage tank. $m_{HS,in}\left(t\right)$ and $m_{HS,out}\left(t\right)$ represent the hydrogen storage and discharge amounts in the tank at time $t$. Equations (29) and (30) represent the constraints on the hydrogen charging and discharging state variables in the storage tank. It is not allowed to charge and discharge hydrogen simultaneously, and both charging and discharging rates must not exceed or fall below their respective limits. $m_{HS,in\backslash out}^{\max }$ denotes the maximum allowable hydrogen charging and discharging rates within the storage tank, respectively. $U_{HS}\left(t\right)$ represents the 0–1 state variable for hydrogen charging and discharging, where a value of 1 indicates the tank is in the discharging state, and 0 indicates it is in the charging state.

2.6. Wind and PV Power Generation Model

In HESHR-EHG-IES, the operational costs of wind power and photovoltaic power generation can be represented by Equation (31).

$$C_{WT}^{PV}\left(t\right)=P_{WT}\left(t\right)\times c_{WT}^{om}+P_{PV}\left(t\right)\times c_{WT}^{PV}$$

(31)

where $C_{WT}^{PV}$ represents the operation and maintenance cost of wind and solar power generation during time period $t$. $P_{WT}\left(t\right)$ and $P_{PV}\left(t\right)$ denote the output power of wind power generation and photovoltaic power generation at time $t$, respectively. $c_{WT}^{om}$ and $c_{WT}^{PV}$ represent the unit operation and maintenance costs of wind power generation and photovoltaic power generation, respectively.

2.7. Model of Energy Transmission Between Integrated Energy System and Higher-Level Network

In the HESHR-EHG-IES, there are various conversions and dispatches between heterogeneous energies, including electricity, heat, gas, and hydrogen. The synergistic scheduling of these different energy sources improves the flexibility and diversity of the system, and the model is more inclined towards the multi-energy architecture composition of future IES networks. In order to depict the energy interactions between the different heterogeneous energy sources within the system and the power distribution network, heat network, gas network, and hydrogen energy company, it is necessary to construct separate models for the energy interactions between the upper-level network and the electricity, gas, heat, and hydrogen energy within the IES.

2.7.1. Distribution Network Energy Interactive Model

In HESHR-EHG-IES, when the total power generation of the system is unable to meet the demand for electricity, it is necessary to purchase electricity from the power distribution network. Conversely, excess electricity can be sold to the power grid for profit. The constraints on the interactive power between the IES and the power distribution network can be expressed as shown in Equations (32)–(35).

$$\begin{array}{l}{P}_M^{buy}(t)-P_M^{sell}(t)=P_S^{ch}(t)+P_L(t)+P_{ED,el}\left(t\right)-P_{CHP,e}\left(t\right)-P_G(t)-P_S^{dis}(t)\\ -P_{PV}(t)-P_{WT}(t)-P_{HFC,e}\left(t\right)\end{array}$$

(32)

$$0\le P_M^{buy}\left(t\right)\le U_M\left(t\right)P_M^{\max }$$

(33)

$$0\le P_M^{sell}\left(t\right)\le \left[1-U_M\left(t\right)\right]P_M^{\max }$$

(34)

$$C_M\left(t\right)=\lambda \left(t\right)\left[P_M^{buy}\left(t\right)-P_M^{sell}\left(t\right)\right]\mathsf{\Delta }t$$

(35)

where $P_M^{buy}\left(t\right)$ and $P_M^{sell}\left(t\right)$ represent the power purchased from and sold to the distribution grid by the integrated energy system at time $t$, respectively. $P_L\left(t\right)$ represents the load demand power at time $t$. The interaction power constraints between HESHR-EHG-IES and the power grid are given by Equations (33) and (34), where $P_M^{\max }$ represents the maximum value of the interaction power between the system and the distribution grid. $U_M\left(t\right)$ represents the power purchase and sale status of the HESHR-EHG-IES with the distribution grid. A $U_M\left(t\right)$ value of 1 indicates power purchase, while a value of 0 indicates power sale to the distribution grid. Equation (35) represents the interaction cost between the integrated energy system and the distribution grid, denoted by $C_M\left(t\right)$. $\lambda \left(t\right)$ represents the day-ahead electricity trading price of the distribution grid.

2.7.2. Natural Gas Network Interaction Energy Model

In HESHR-EHG-IES, the gas consumption of the CHP unit, gas load, transaction volume with the upper-level gas network, and methane gas production must satisfy the gas flow balance constraint expressed as Equations (36) and (37).

$$G_{gas}\left(t\right)+{\displaystyle \raisebox{1ex}{$m_{C{H}_4}\left(t\right)$}\!\left/ \!\raisebox{-1ex}{${\rho }_{C{H}_4}$}\right.}={\displaystyle \raisebox{1ex}{$P_{gas,L}\left(t\right)$}\!\left/ \!\raisebox{-1ex}{$LH{V}_{gas}$}\right.}+G_{gas}^{CHP}\left(t\right)$$

(36)

$$C_{gas}\left(t\right)=c_{gas,buy}{G}_{gas}\left(t\right)$$

(37)

Here, ${\rho }_{C{H}_4}$ represents the methane density; $G_{gas}\left(t\right)$ denotes the natural gas purchase quantity at time $t$; $P_{gas,L}\left(t\right)$ is the gas load at time $t$; $C_{gas}\left(t\right)$ is the natural gas purchase cost at time $t$; and $c_{gas,buy}$ is the unit purchase cost of natural gas at time $t$.

2.7.3. Heat Network Interaction Energy Model

$$Q_{heat,buy}\left(t\right)+Q_{out}\left(t\right)+P_{CHP,h}\left(t\right)=P_{heat,L}(t)$$

(38)

$$C_{heat}\left(t\right)=c_{heat,buy}{Q}_{heat,buy}\left(t\right)$$

(39)

Here, $Q_{heat,buy}\left(t\right)$ and $P_{heat,L}\left(t\right)$ represent the heating power purchase and the heat load power at time $t$, respectively. $c_{heat}\left(t\right)$ is the interaction cost between the system and the heating network at time $t$, and $c_{heat,buy}$ is the unit cost of heating power purchase.

2.7.4. Hydrogen Energy Company Interaction Energy Model

$${\displaystyle \raisebox{1ex}{$P_{HL}\left(t\right)$}\!\left/ \!\raisebox{-1ex}{$LH{V}_H$}\right.}+m_{HFC}\left(t\right)+m_{H,sell}\left(t\right)+m_{MR,H}\left(t\right)+m_{HS,in}\left(t\right)=m_{ED,H}\left(t\right)+m_{HS,out}\left(t\right)$$

(40)

$$C_H\left(t\right)=-c_{H,sell}{m}_{H,sell}\left(t\right)$$

(41)

Here, $P_{HL}\left(t\right)$ represents the hydrogen load at time $t$; $m_{H,sell}\left(t\right)$ denotes the hydrogen sale quantity to the hydrogen company at time $t$; $C_H\left(t\right)$ is the interaction cost between the system and the hydrogen company at time $t$; $c_{H,sell}$ is the unit cost of hydrogen sale. $LH{V}_H$ is the lower heating value of hydrogen.

3. HESHR-EHG-IES Deterministic Model

In HESHR-EHG-IES, the wind and PV power output and various loads of electric heat gas and hydrogen have strong uncertainty and volatility. With the continuous improvement of source-load forecasting technology, it provides some assistance to the dispatch of the IES. However, there still exists a gap of 10% to 20% between the predicted values and the actual values, which indicates a certain deviation from the idealized scheduling of the IES. Therefore, it is necessary to consider the optimization problem of source-load uncertainty, which can be viewed as solving a two-stage robust problem of min–max–min. This enhances the sophistication of the system’s coordinated scheduling strategy.

The objective function is defined as minimizing the daily operational cost, as represented by Equation (42). This cost includes the maintenance costs of various system components and the costs associated with interaction with the upper-level energy network. It is composed of Equations (3), (10), (15), (17), (24), (31), (35), (37), (39) and (41).

$$\min {\displaystyle \sum _{t=1}^{N_T}\left[\begin{array}{l}{C}_{WE}\left(t\right)+C_{HFC}\left(t\right)+C_G\left(t\right)+C_S\left(t\right)+C_{CHP}\left(t\right)+\\ C_{WT}^{PV}\left(t\right)+C_M\left(t\right)+C_{gas}\left(t\right)+C_{\mathrm{heat}}\left(t\right)+C_H\left(t\right)\end{array}\right]}$$

(42)

When the uncertainties of wind and PV power output and load fluctuations are not considered, the economic optimization scheduling model of HESHR-EHG-IES can be derived. The general form of the deterministic solution model is expressed as shown in Equation (43).

$$\left\{\begin{array}{c}\underset{x,y}{\min } c^{T}y\\ \mathrm{s}.\mathrm{t}. Dy\ge d\\ Ky=k\\ Fx+Gy\ge h\\ I_{u}y=\stackrel{⌢}{u}\end{array}\right.$$

(43)

In this context, $x$ and $y$ represent the optimization variables, and their specific meanings are defined as shown in Equation (44).

$$\left\{\begin{array}{l}x={[U_S\left(t\right),U_{HS}\left(t\right),U_M\left(t\right)]}^T\\ y={\left[\begin{array}{l}{m}_{ED,H}\left(t\right),P_{ED,el}\left(t\right),P_{LED,rec}\left(t\right),P_{LED,loss}\left(t\right),Q_{LED}\left(t\right),P_{HFC,e}\left(t\right),\\ P_{HFC,h}\left(t\right),m_{HFC}\left(t\right),m_{c{o}_2}\left(t\right),m_{MR,H}\left(t\right),m_{C{H}_4}\left(t\right),n_{C{H}_4}\left(t\right),P_{MR,heat}\left(t\right),\\ Q_{MR}\left(t\right),Q_{out}\left(t\right),P_G\left(t\right),P_S^{dis}\left(t\right),P_S^{ch}\left(t\right),P_{CHP,e}\left(t\right),P_{CHP,h}\left(t\right),G_{gas}^{CHP}\left(t\right),\\ p_{HS}(t),m_{HS}(t),m_{HS,in}\left(t\right),m_{HS,out}\left(t\right),P_{WT}\left(t\right),P_{PV}\left(t\right),P_M^{buy}\left(t\right),P_M^{sell}\left(t\right),\\ P_L\left(t\right),G_{gas}\left(t\right),P_{gas,L}\left(t\right),Q_{heat,buy}\left(t\right),P_{heat,L}\left(t\right),P_{HL}\left(t\right),m_{H,sell}\left(t\right)\end{array}\right]}^{\mathrm{T}}\\ t=\left(1,2,3\cdots N_T\right)\end{array}\right.$$

(44)

In Equation (43), $c$ represents the coefficient matrix vector corresponding to the objective function (42); $D$, $K$, $F$, $G$, and $I_u$ are coefficient matrix vectors corresponding to the constraint conditions; $d$, $k$, and $h$ are constant column vectors of variables. Equations (2), (4), (6), (16), (21), (23), (26) and (28) represent the inequality constraint Equations in the first row of the deterministic model general form (43) for HESHR-EHG-IES; Equations (1), (5), (7), (8), (11)–(14), (20), (22), (25), (27), (32), (36), (38) and (40) represent the equality constraint Equations in the second row of the constraint conditions for the model; Equations (18), (19), (29), (30), (33) and (34) represent the inequality constraint Equations in the third row of the constraint conditions, which are related to the variables $x$ and $y/x$.

The fourth row of the constraint conditions states that, in HESHR-EHG-IES, for the deterministic model, the values of wind and PV power output, as well as electricity, gas, heat, and hydrogen load, are the predicted values for each time period, as shown in Equation (45).

$$\stackrel{⌢}{u}={\left[{\stackrel{⌢}{u}}_{PV}\left(t\right),{\stackrel{⌢}{u}}_{WT}\left(t\right),{\stackrel{⌢}{u}}_L\left(t\right),{\stackrel{⌢}{u}}_{gas}\left(t\right),{\stackrel{⌢}{u}}_{H_2}\left(t\right),{\stackrel{⌢}{u}}_{heat}\left(t\right)\right]}^T,t=\left(1,2\cdots N_T\right)$$

(45)

where ${\stackrel{⌢}{u}}_{PV}\left(t\right),{\stackrel{⌢}{u}}_{WT}\left(t\right),{\stackrel{⌢}{u}}_L\left(t\right),{\stackrel{⌢}{u}}_{gas}\left(t\right),{\stackrel{⌢}{u}}_{heat}\left(t\right),{\stackrel{⌢}{u}}_{H_2}\left(t\right)$ represent the predicted power values of photovoltaic output, wind power output, electricity load, gas load, heat load, and hydrogen load, respectively, at time period $t$.

In summary, with the objective function of minimizing daily operation and maintenance costs, a HESHR EHG-IES deterministic model is constructed as above. The optimization problem of the deterministic model mentioned above is a mixed-integer linear programming problem, which can be solved using intelligent algorithms or solvers. However, the economic and advanced nature of obtaining the optimal results often depends on the accuracy of the predicted values for photovoltaic output, wind power output, electricity load, gas load, heat load, and hydrogen load. The resulting solution may be overly optimistic, and there are certain risks in adopting this solution in practical cases. Hence, it is necessary to analyze factors such as strong uncertainties in output and load demands within the system. Taking into account the impact of uncertainties in HESHR-EHG-IES, the uncertain model of HESHR-EHG-IES needs to be constructed and solved to obtain the optimal solution under the uncertainty model.

4. HESHR-EHG-IES Uncertainty Model

4.1. Two-Stage Robust Uncertainty Model

Due to the strong uncertainties at the source and load ends within the system, the solution obtained from the deterministic model ignores these factors and carries certain risks. Therefore, it is necessary to consider the uncertainties of the source and load and construct the HESHR-EHG-IES model. The solution of the deterministic model heavily relies on the accuracy of the source and load predictions. Although prediction technologies have greatly improved, there are still errors between predicted values and actual values due to climate conditions and human factors. It is necessary to incorporate the impact of the error between the predicted and actual values in the model to obtain accurate results. Therefore, the prerequisite for constructing the uncertain model of HESHR-EHG-IES is to establish a set of elements representing system uncertainties.

Considering the errors between predicted and actual values of wind power output, photovoltaic output, electricity load, gas load, heat load, and hydrogen load, a box uncertainty set $U$ is constructed, as shown in Equation (46).

$$U:=\left\{\begin{array}{l}u={[u_{PV}\left(t\right),u_{WT}\left(t\right),u_L\left(t\right),u_{gas}\left(t\right),u_{H_2}\left(t\right),u_{heat}\left(t\right)]}^T\in {ℝ}^{N_T\times 6}\\ t=\left(1,2\cdots N_T\right)\\ u_{PV}\left(t\right)\in \left[{\stackrel{⌢}{u}}_{PV}\left(t\right)-\mathsf{\Delta }{u}_{PV}^{\max }\left(t\right),{\stackrel{⌢}{u}}_{PV}\left(t\right)+\mathsf{\Delta }{u}_{PV}^{\max }\left(t\right)\right]\\ u_{WT}\left(t\right)\in \left[{\stackrel{⌢}{u}}_{WT}\left(t\right)-\mathsf{\Delta }{u}_{WT}^{\max }\left(t\right),{\stackrel{⌢}{u}}_{WT}\left(t\right)+\mathsf{\Delta }{u}_{WT}^{\max }\left(t\right)\right]\\ u_L\left(t\right)\in \left[{\stackrel{⌢}{u}}_L\left(t\right)-\mathsf{\Delta }{u}_L^{\max }\left(t\right),{\stackrel{⌢}{u}}_L\left(t\right)+\mathsf{\Delta }{u}_L^{\max }\left(t\right)\right]\\ u_{gas}\left(t\right)\in \left[{\stackrel{⌢}{u}}_{gas}\left(t\right)-\mathsf{\Delta }{u}_{gas}^{\max }\left(t\right),{\stackrel{⌢}{u}}_{gas}\left(t\right)+\mathsf{\Delta }{u}_{gas}^{\max }\left(t\right)\right]\\ u_{H_2}\left(t\right)\in \left[{\stackrel{⌢}{u}}_{H_2}\left(t\right)-\mathsf{\Delta }{u}_{H_2}^{\max }\left(t\right),{\stackrel{⌢}{u}}_{H_2}\left(t\right)+\mathsf{\Delta }{u}_{H_2}^{\max }\left(t\right)\right]\\ u_{heat}\left(t\right)\in \left[{\stackrel{⌢}{u}}_{heat}\left(t\right)-\mathsf{\Delta }{u}_{heat}^{\max }\left(t\right),{\stackrel{⌢}{u}}_{heat}\left(t\right)+\mathsf{\Delta }{u}_{heat}^{\max }\left(t\right)\right]\end{array}\right.$$

(46)

where $u_{PV}\left(t\right)$, $u_{WT}\left(t\right)$, $u_L\left(t\right)$, $u_{gas}\left(t\right)$, $u_{H_2}\left(t\right)$, and $u_{heat}\left(t\right)$ represent the sets of uncertainty variables for wind and PV power output and load power, taking into account the uncertainties on both sides of the system source and load; $\mathsf{\Delta }{u}_{PV}^{\max }\left(t\right)$, $\mathsf{\Delta }{u}_{WT}^{\max }\left(t\right)$, $\mathsf{\Delta }{u}_L^{\max }\left(t\right)$, $\mathsf{\Delta }{u}_{gas}^{\max }\left(t\right)$, $\mathsf{\Delta }{u}_{H_2}^{\max }\left(t\right)$, and $\mathsf{\Delta }{u}_{heat}^{\max }\left(t\right)$ represent the maximum allowable error values between the actual and predicted values for wind and PV power output and load power, all of which are greater than zero.

The purpose of this study is to construct a ARO model to discover the economically optimal coordinated scheduling solution when the uncertain variable $u$ changes towards the worst-case scenario within the uncertainty set $U$. The problem described in Equations (43)–(46) can be viewed as a two-stage robust planning model, and the problem-solving process can be transformed into a mathematical model represented by the min–max–min two-stage optimization problem, as shown in Equation (47).

$$\left\{\mathrm{s}.\mathrm{t}\begin{array}{c}\underset{x}{\min }\left\{\left.\underset{u\in U}{\max }\underset{y\in \Omega \left(x,u\right)}{\min }{c}^{T}y\right\}\right.\\ x={\left(x_1,x_2,\cdots ,x_{3\times N_T}\right)}^T\\ x_i\in \left\{0,1\right\},\forall i\in \left(1,2,3,\cdots ,3\times N_T\right)\\ \begin{array}{l}y={(y_1,y_2,y_3,\cdots ,y_{34\times N_T})}^T\\ y_i\in \left\{0,+\infty \right\},\forall i\in \left(1,2,3,\cdots ,34\times N_T\right)\end{array}\end{array}\right.$$

(47)

The outer minimization is the first-stage problem, with x as the optimization variable. The inner problem consists of a minimization followed by a maximization, involving two variables $u$ and $y$, where $\min c^{T}y$ is equivalent to the objective function of solving the deterministic model described in Equation (43), representing the minimization of daily operating cost of the system. The optimization variables x and y are defined as shown in Equation (44), where x is a binary variable and y is a decimal variable. $\mathsf{\Omega }\left(x,u\right)$ represents the feasible domain of the optimization variable y, given a set of variables $\left(x,u\right)$, and is expressed specifically as shown in Equation (48).

$$\Omega \left(x,u\right):=\left\{\begin{array}{c}\left.y\right|\\ Dy\ge d:\gamma \\ Ky=k:\lambda \\ Fx+Gy\ge h:v\\ I_{u}y=u:\pi \end{array}\right\}$$

(48)

where $\gamma$, $\lambda$, $v$, and $\pi$ correspond to the dual variables associated with the constraints of the minimization subproblems in the second-stage min–max problem.

For each given set of uncertain variables $u$, it can be transformed into a deterministic model as shown in Equation (43) through Equation (48). The purpose of solving the maximization problem in the second-stage robust uncertainty model is to find the worst-case scenario that causes the maximum daily operating cost under uncertain conditions.

4.2. Solving Algorithm

In consideration of uncertainty and hydrogen energy heat recovery, a two-stage robust model, namely HESHR-EHG-IES, is constructed in this study. The C&CG algorithm is employed for solving the model [26]. Similar to the Benders decomposition algorithm, the C&CG algorithm decomposes a complex two-stage problem into an outer master problem and inner subproblems, which are iteratively solved to obtain the optimal solution of the original problem. The main difference lies in the fact that the C&CG algorithm gradually introduces constraints and variables related to the subproblems into the computation process of the master problem. As a result, it can obtain a more compact lower bound of the original objective function value, effectively reducing the number of iterations.

The main problem and subproblems solved in this paper have the same objective function, which is the minimization of daily operating cost, and both are in linear form. Therefore, by decomposing Equation (47), we can obtain the general form of the objective function and constraint conditions for the main problem as shown in Equation (49).

$$\left\{\begin{array}{c}\underset{x}{\min }\alpha \\ \mathrm{s}.\mathrm{t}. \alpha \ge c^T{y}_l,D{y}_l\ge d,Fx+G{y}_l\ge h\\ K{y}_l=k,I_u{y}_l=u_l^{*}\end{array}\right.;\forall l\ge J$$

(49)

Here, $J$ denotes the current iteration number. $y_l$ represents the solution of the subproblem after the l-th iteration. $u_l^{*}$ corresponds to the values of the uncertain variables $u$ generated by the subproblem under the worst-case scenario after the l-th iteration.

After decomposition, the subproblem is expressed as follows:

$$\underset{u\in U}{\max }\underset{y\in \Omega \left(x,u\right)}{\min c^{T}y}$$

(50)

From Equations (47) to (49), it can be seen that, given $\left(x,u\right)$, the inner $\min c^{T}y$ in Equation (50) can be equivalently transformed into a linear problem. Referring to the strong duality theory based on the corresponding relationship in Equation (48) [27], the inner minimization problem can be converted into a maximization form and combined with the outer maximization problem to obtain the specific dual problem expressed as Equation (51).

$$\left\{\mathrm{s}.\mathrm{t}\begin{array}{c}\underset{u\in U,\gamma ,\lambda ,v,\pi }{\max }{d}^T\gamma +k^T\lambda +{\left(h-Fx\right)}^{T}v+u^T\pi \\ D^T\gamma +K^T\lambda +G^{T}v+I_u^T\pi \le c\\ \gamma \ge 0,v\ge 0\end{array}\right.$$

(51)

As shown in Equation (46), $u^T\pi$ is a bilinear term. According to an analysis of the literature, the optimal solution $u^{*}$ of the dual problem described by Equation (51) is a vertex of the uncertainty set $U$. That is, when Equation (51) reaches its maximum value, the value of the uncertain variable $u$ should be the boundary of the source-load fluctuation interval described by Equation (46).

In summary, analyzing the proposed HESHR-EHG-IES model in this paper reveals that in Equation (46), when the wind and PV power outputs reach the minimum value within the predicted power range, and various types of loads reach their maximum values within the predicted power range, the operating cost of the integrated energy system is higher, which aligns with the definition of the “worst-case” scenario. Therefore, Equation (46) can be rewritten as follows in a more concise form Equation (52).

$$U:=\left\{\begin{array}{l}u={[u_{PV}\left(t\right),u_{WT}\left(t\right),u_L\left(t\right),u_{gas}\left(t\right),u_{H_2}\left(t\right),u_{heat}\left(t\right)]}^T\in {ℝ}^{N_T\times 6}\\ \mathrm{Source} \mathrm{side}:\\ u_{PV}\left(t\right)={\widehat{u}}_{PV}\left(t\right)-B_{PV}\left(t\right)\mathsf{\Delta }{u}_{PV}^{\max }\left(t\right),u_{WT}\left(t\right)={\widehat{u}}_{WT}\left(t\right)-B_{WT}\left(t\right)\mathsf{\Delta }{u}_{WT}^{\max }\left(t\right)\\ {\displaystyle \sum _{t=1}^{N_T}{B}_{PV}\left(t\right)\le {\mathsf{\Gamma }}_{PV}},{\displaystyle \sum _{t=1}^{N_T}{B}_{WT}\left(t\right)\le {\mathsf{\Gamma }}_{WT}}\\ \mathrm{Load} \mathrm{side}:\\ u_L\left(t\right)={\widehat{u}}_L\left(t\right)+B_L\left(t\right)\mathsf{\Delta }{u}_L^{\max }\left(t\right),u_{gas}\left(t\right)={\widehat{u}}_{gas}\left(t\right)+B_{gas}\left(t\right)\mathsf{\Delta }{u}_{gas}^{\max }\left(t\right)\\ u_{H_2}\left(t\right)={\widehat{u}}_{H_2}\left(t\right)+B_{H_2}\left(t\right)\mathsf{\Delta }{u}_{H_2}^{\max }\left(t\right),u_{heat}\left(t\right)={\widehat{u}}_{heat}\left(t\right)+B_{heat}\left(t\right)\mathsf{\Delta }{u}_{heat}^{\max }\left(t\right)\\ {\displaystyle \sum _{t=1}^{N_T}{B}_L\left(t\right)\le {\mathsf{\Gamma }}_L},{\displaystyle \sum _{t=1}^{N_T}{B}_{gas}\left(t\right)\le {\mathsf{\Gamma }}_{gas}},{\displaystyle \sum _{t=1}^{N_T}{B}_{H_2}\left(t\right)\le {\mathsf{\Gamma }}_{H_2}},{\displaystyle \sum _{t=1}^{N_T}{B}_{heat}\left(t\right)\le {\mathsf{\Gamma }}_{heat}}\end{array}\right.$$

(52)

The variable $B={\left[B_{PV}\left(t\right),B_{WT}\left(t\right),B_L\left(t\right),B_{gas}\left(t\right),B_{H_2}\left(t\right),B_{heat}\left(t\right)\right]}^T$ is binary; ${\mathsf{\Gamma }}_{PV}$ and ${\mathsf{\Gamma }}_{WT}$ are uncertainty adjustment parameters for the source-side photovoltaic and wind power outputs, respectively. ${\mathsf{\Gamma }}_L,{\mathsf{\Gamma }}_{gas},{\mathsf{\Gamma }}_{H_2},{\mathsf{\Gamma }}_{heat}$ are uncertainty adjustment parameters for the load-side electrical load, gas load, hydrogen load, and heat load, respectively. The uncertainty adjustment parameters on the source and load sides take integer values within the range of $\left[0,N_T\right]$. These values represent the total number of time periods within one scheduling cycle where the fluctuation intervals described by Equation (52) for the source-side wind and PV power outputs, as well as the various load demands on the electrical, gas, hydrogen, and heat load sides, reach their maximum or minimum values. The purpose is to allow for adjustable uncertainty factors on both the source and load sides during the scheduling period, thus adjusting the conservatism of the optimal solution. Smaller parameter values lead to a more “adventurous” approach in obtaining the solution, while larger values result in a more “conservative” approach.

After substituting Equation (51) into Equation (52), continuous variable product terms and binary variable forms emerge, which are not conducive to solving the optimization model. Based on the big M method [28], auxiliary variables and related constraint conditions are introduced to linearize the obtained model, resulting in Equation (53).

$$\left\{\begin{array}{l}\underset{B,B^{\prime },\gamma ,\lambda ,v,\pi }{\max }{d}^T\gamma +k^T\lambda +{\left(h-Fx\right)}^{T}v+{\widehat{u}}^T\pi +\mathsf{\Delta }{u}^T{B}^{\prime }\\ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{c}{D}^T\gamma +K^T\lambda +G^{T}v+I_u^T\pi \le c\\ -MB\le B^{\prime }\le MB\\ \pi -M\left(1-B\right)\le B^{\prime }\le \pi +M\left(1-B\right)\\ {\displaystyle \sum _{t=1}^{N_T}{B}_{PV}\left(t\right)\le {\mathsf{\Gamma }}_{PV}},{\displaystyle \sum _{t=1}^{N_T}{B}_{WT}\left(t\right)\le {\mathsf{\Gamma }}_{WT}},{\displaystyle \sum _{t=1}^{N_T}{B}_L\left(t\right)\le {\mathsf{\Gamma }}_L},\\ {\displaystyle \sum _{t=1}^{N_T}{B}_{gas}\left(t\right)\le {\mathsf{\Gamma }}_{gas}},{\displaystyle \sum _{t=1}^{N_T}{B}_{H_2}\left(t\right)\le {\mathsf{\Gamma }}_{H_2}},{\displaystyle \sum _{t=1}^{N_T}{B}_{heat}\left(t\right)\le {\mathsf{\Gamma }}_{heat}}\end{array}\right.\end{array}\right.$$

(53)

where $\mathsf{\Delta }u$ and $B^{\prime }$ are introduced as continuous auxiliary variables, with their specific expressions given by Equation (54); M represents the upper bound of the dual variable, which is typically chosen as a sufficiently large positive real number.

$$\begin{array}{l}\mathsf{\Delta }u={[\mathsf{\Delta }{u}_{PV}^{\max }\left(t\right),\mathsf{\Delta }{u}_{WT}^{\max }\left(t\right),\mathsf{\Delta }{u}_L^{\max }\left(t\right),\mathsf{\Delta }{u}_{gas}^{\max }\left(t\right),\mathsf{\Delta }{u}_{H_2}^{\max }\left(t\right),\mathsf{\Delta }{u}_{heat}^{\max }\left(t\right)]}^T\\ B^{\prime }={[B_{PV}^{\prime }\left(t\right),B_{WT}^{\prime }\left(t\right),B_L^{\prime }\left(t\right),B_{gas}^{\prime }\left(t\right),B_{H_2}^{\prime }\left(t\right),B_{heat}^{\prime }\left(t\right)]}^T\end{array}$$

(54)

Based on Equations (42) to (54), a two-stage robust model for the uncertainty and heat recovery in hydrogen energy systems can be considered, with both the master problem and subproblem decoupled into mixed-integer linear forms. The master problem and subproblem are given by (49) and (53), respectively, and can be solved using the C&CG algorithm. The specific solution steps are as follows.

Step 1: For the iteration with $J=1$, the initial worst-case scenario is given by the predicted values of uncertain variables such as PV power output, electric load, gas load, hydrogen load, and heat load. The upper and lower bounds for the daily operational cost objective function are set as $+\infty$ and $-\infty$, respectively, representing the target range for system dispatch.

Step 2: Solving the master problem using the worst-case scenario $u_l^{*}$ by substituting it into Equation (49), we calculate the optimal solution $\left(x_k^{*},{\alpha }_k^{*},y^{1*},\cdots ,y^{J*}\right)$. The objective function value obtained from solving the master problem under this scenario is set as the new lower bound $LB={\alpha }_J^{*}$.

Step 3: Substituting the solution $x_J^{*}$ of the master problem into subproblem (53), we calculate the corresponding optimal solution $f_J^{*}\left(x_J^{*}\right)$ for the subproblem’s objective function and obtain the values $u_{J+1}^{*}$ for the uncertain variable $u$ under the worst-case scenario. We then update the upper bound to $UB=\min \left\{UB,f_J^{*}\left(x_J^{*}\right)\right\}$.

Step 4: While iteratively updating the upper bound UB and lower bound LB of the objective function in the model, we set $\epsilon$ as the convergence threshold for the algorithm’s bounds. If $UB-LB\le \epsilon$, indicating that the bounds have converged, the iteration is immediately stopped, and the optimal solutions $x_J^{*}$ and $y_J^{*}$ are returned. Otherwise, the algorithm proceeds to the next iteration $J=J+1$, jumping back to step 2 and introducing additional variables and constraints Equation (55).

$$\left\{\mathrm{s}.\mathrm{t}.\begin{array}{c}\alpha \ge c^T{y}^{J+1}\\ D{y}^{J+1}\ge d,Fx+G{y}^{J+1}\ge h\\ K{y}^{J+1}=k,I_u{y}^{J+1}=u_{J+1}^{*}\end{array}\right.$$

(55)

Step 5: Solve Equation (55) and sequentially perform the steps until the algorithm converges to $UB-LB\le \epsilon$.

Therefore, the HESHR-EHG-IES two-stage robust model considering uncertainty proposed in this article, the flowchart of the specific solution steps is shown in Figure 2.

5. Simulation Studies

This study uses the HESHR-EHG-IES (Figure 1) as a case study and constructs a two-stage RO model considering source-load multiple uncertainty and HESHR. To validate the effectiveness of the proposed model and solution algorithm, simulation analyses are conducted to discuss three aspects: the impact of heat recovery on system performance, analysis of flexible source and load uncertainty, and comparison of optimization models. Based on relevant literature [29,30,31], the operating parameters of the HESHR-EHG-IES are presented in

Table 1. Parameters of the HESHR-EHG-IES.

Equipment Type	Parameters
Electrolytic water heat recovery model	$\begin{array}{l}{P}_{ED,el}^{\min }=750 \mathrm{kW}, P_{ED,el}^{\max }=3000 \mathrm{kW}, {\eta }_{ED,H}=0.7, {\eta }_{LED,he}=0.9, \\ C_{ED}=20 \mathrm{kW}/°\mathrm{C},R_{ED}=4.36\times {10}^3 °\mathrm{C}/\mathrm{kW}, T_{LED}^{\min }=70 °\mathrm{C}, \\ T_{LED}^{\max }=90 °\mathrm{C},T_a=25 °\mathrm{C},T_0=80 °\mathrm{C},c_{LED}^{om}=0.07 \yen /\mathrm{kW}, \\ P_{LED,rec}^{\min }=0 \mathrm{kW}, P_{LED,rec}^{\max }=700 \mathrm{kW}\end{array}$
Hydrogen fuel cell	$\begin{array}{l}{P}_{HFC,e}^{\max }=1000 \mathrm{kW}, P_{HFC,e}^{\min }=0 \mathrm{kW}, \\ {\mathrm{c}}_{HFC}^{om}=0.07 \yen /\mathrm{kW},{\eta }_{HFC,e}=0.5, {\eta }_{HFC,h}=0.3\end{array}$
Methanation equipment	$\begin{array}{l}{M}_{C{O}_2}=0.044 \mathrm{kg}/\mathrm{mol}, M_H=0.002 \mathrm{kg}/\mathrm{mol}, \\ M_{C{H}_2}=0.016 \mathrm{kg}/\mathrm{mol}, {\eta }_{MR}=0.6,{\eta }_{MR,heat}=0.3, \\ n_{C{H}_4-H_2}=2, \mathsf{\Delta }H=0.165 \mathrm{kJ}/\mathrm{mol}, {\rho }_{C{H}_4}=0.66{ \mathrm{kg}/\mathrm{m}}^3\end{array}$
Micro gas turbine	$P_G^{\max }=4000 \mathrm{kW}, P_G^{\min }=500 \mathrm{kW}, a=0.67$
Storage energy system	$\begin{array}{l}{P}_S^{\max }=500 \mathrm{kW}, E_S^{\max }=1800 \mathrm{kW}\cdot \mathrm{h}, E_S^{\min }=400 \mathrm{kW}\cdot \mathrm{h}\\ K_S\left(0\right)=1000 \mathrm{kW}\cdot \mathrm{h}, K_S=0.38 \mathrm{kW}\cdot \mathrm{h}, {\eta }_S=0.95\end{array}$
CHP	$\begin{array}{l}{\eta }_{CHP,e}=0.4, {\eta }_{CHP,h}=0.6, P_{CHP,e}^{\max }=800 \mathrm{kW}, \\ {\mathrm{c}}_{CHP}^{om}=0.08 \yen /\mathrm{kW}, LHVgas=9.7{ \mathrm{kWh}/\mathrm{m}}^3\end{array}$
HST	$\begin{array}{l}{V}_{HS}=60{ \mathrm{m}}^3, \mathrm{R}=0.008314 \mathrm{kJ}\cdot {\mathrm{mol}}^{-1}\cdot {\mathrm{K}}^{-1},T_H=293 \mathrm{K},\\ {\eta }_{HS,loss}=0.03,{\eta }_{HS,in}=0.95,{\eta }_{HS,out}=0.95,m_{HS,in/out}^{\max }=320 \mathrm{kg}/\mathrm{h},\\ m_{HS}^{\max }=985.22 \mathrm{kg}/\mathrm{h},m_{HS}^{\min }=246.30 \mathrm{kg}/\mathrm{h},p_{HS}^{\min }=5000 \mathrm{kPa},\\ p_{HS}^{\max }=\mathrm{20,000} \mathrm{kPa},m_{HS}\left(0\right)=492.5 \mathrm{kg}\end{array}$
Energy interaction	$\begin{array}{l}{c}_{WT}^{om}=0.09 \yen /\mathrm{kW},c_{PV}^{om}=0.07 \yen /\mathrm{kW},P_M^{\max }=4000 \mathrm{kW}\\ {\mathrm{c}}_{gas,buy}=4.5 \yen /{\mathrm{m}}^3,{\mathrm{c}}_{heat,buy}=2.5 \yen /\mathrm{kW},{\mathrm{c}}_{H,sell}=35 \yen /\mathrm{kg}\end{array}$

. All simulations were performed on a desktop computer running Windows 10, equipped with an Intel(R) Core (TM) i7-14700KF 3.40 GHz processor. MATLAB R2022a software and the Gurobi 10.0.2 solver were used for solving the optimization problems.

5.1. Two-Stage Robust Uncertainty Model

The day-ahead trading electricity price of the distribution network is shown in Figure 3.

5.2. Impact of HESHR on Operational Performance of IES

The uncertainty adjustment parameters $\left({\mathsf{\Gamma }}_{PV},{\mathsf{\Gamma }}_{WT},{\mathsf{\Gamma }}_L,{\mathsf{\Gamma }}_{gas},{\mathsf{\Gamma }}_{H_2},{\mathsf{\Gamma }}_{heat}\right)$ for both the source and load sides are set to 6, indicating that within one scheduling cycle, the wind and PV power outputs on the source side can each take up to 6 values at their respective minimum values in the power prediction space. Similarly, the electrical, gas, hydrogen, and heat loads on the load side can each take up to six values at their respective maximum values in their power prediction spaces. For all other time periods, the wind and PV power outputs and various load demands are equal to their predicted values.

In engineering practice, the maximum allowable deviation fluctuation of wind and PV power outputs and various load power demands in IES can be determined based on historical prediction deviation data. In this study, based on the historical data of source and load prediction deviations in IES, the maximum allowable deviation for wind power output and photovoltaic power output on the source side is set to 15% to 20% of the predicted value. The comparison between the predicted and actual power curves for wind and PV power outputs is shown in Figure 4a,b. On the load side, the maximum allowable deviation for electrical, gas, hydrogen, and heat loads is set to 10% of their respective predicted values. The comparison between the predicted and actual power demand curves for electrical, gas, hydrogen, and heat loads is shown in Figure 4c–f.

The optimization scheduling results of HESHR are shown in Figure 5. Figure 5 represents the optimized operation results of electrolysis water and heat recovery, including the released heat energy during the electrolysis process, recoverable heat energy, actual recovered heat energy, and variation in operating temperature. Figure 6 illustrates the energy balance of the HESHR-EHG-IES, where Figure 6a–d represent the energy interaction status of each subsystem in terms of electricity, gas, hydrogen, and heat energy, respectively.

In Figure 5, the heat recovery from hydrogen production via water electrolysis is related to the power input, heat release, and working temperature. The system’s heat load and hydrogen demand have a significant impact on the operation of the hydrogen energy system. Within the time interval of 1–24 h, when the heat release from hydrogen production is less than the recoverable heat energy, the working temperature decreases. Conversely, if the heat release exceeds the recoverable heat energy, the working temperature increases. If the two values are equal, the working temperature remains unchanged and is equal to the working temperature at the end of the previous time interval. This satisfies the heat recovery equations for electrolytic hydrogen production.

The water electrolysis device operates for 1–4 h, during which the hydrogen demand is basically zero and the heat demand is at its peak. The reaction temperature of the electrolysis cell is maintained at 70–72 °C to reduce the release of hydrogen and improve the heat recovery efficiency. The hydrogen generated during this period is mainly used for the operation, storage, and trading of the methanation equipment. From 5–7 h, the working temperature increases mainly due to the overall upward trend of hydrogen demand, but the demand is relatively small compared to the high heat demand. The hydrogen produced during this period is mainly used for storage and trading to reserve hydrogen energy for the peak demand and achieve hydrogen energy peak shaving. From 8–9 h, the working temperature is maintained at around 89 °C to meet the peak demand of hydrogen load and increase hydrogen production. At 10 h, both the hydrogen and heat loads decrease, and the working temperature is lowered to 76–70 °C to reduce hydrogen production and heat release. From 11–14 h, both the hydrogen and heat loads are in a low-demand period of the scheduling cycle, so the reaction temperature is maintained at 70 °C to operate the system at the minimum hydrogen production temperature and meet the minimum hydrogen load fluctuations. From 15–17 h, the working temperature increases as the demand for hydrogen load continues to grow and approach the second peak value while the heat demand is still in a low period. The working temperature is 74–89 °C, and the heat recovery value decreases. The reaction temperature needs to be increased to increase hydrogen production and meet the growing demand for hydrogen load. At 18–19 h, the hydrogen load is at its peak energy consumption period, while the heat demand is rapidly increasing. The reaction temperature of the electrolysis cell is maintained at 90 °C to meet the hydrogen load demand while achieving maximum heat recovery with heat release values that are almost equal. From 20 to 21 h, the hydrogen load gradually decreases while the heat demand continues to grow. The reaction temperature is lowered to 89–74 °C, and the heat recovery value is greater than the heat release value. This optimizes heat recovery and helps to reduce the temperature of the electrolysis cell, reducing hydrogen production to meet the hydrogen load demand. From 22 to 24 h, the hydrogen demand is approximately zero, while the heat demand is at its peak. The reaction temperature is maintained at 70 °C, and the generated hydrogen is used for storage or trading with hydrogen energy companies.

In summary, the electrolysis water hydrogen production system with heat recovery has the ability to flexibly adjust and store heat energy. It can control the storage and regulation of heat energy according to the system’s hydrogen and heat load requirements, thereby improving the flexibility of hydrogen energy system operation, promoting the coordinated operation of electric–heat–hydrogen multi-heterogeneous energy sources and achieving efficient utilization of energy.

According to Figure 6, the energy generated by the system is positive, while the energy consumed is negative. The electric, natural gas, hydrogen, and heat loads are all represented by dashed lines on the positive axis. Throughout one scheduling cycle, the outputs of electricity, gas, hydrogen, and heat energy vary in accordance with the changes in the loads, maintaining the energy balance of the system.

In Figure 6c, during the periods of 0–7 h, 10–15 h, and 22–24 h, although there are fluctuations in hydrogen load demand, the demand is essentially zero during these time periods. During this time, the generated hydrogen is stored in hydrogen storage tanks and consumed by the methanation equipment to produce methane, and any excess hydrogen is sold to hydrogen energy companies for profit. During the periods of 8–9 h and 16–18 h, it is the peak period of hydrogen load demand. Hydrogen is produced by water electrolysis and released from the hydrogen storage tank to meet the hydrogen load demand during the 0–7 h period. From 19–21 h, the hydrogen load decreases. During this period, the hydrogen produced by electrolysis water meets the hydrogen load demand, and the remaining hydrogen is consumed by the methanation equipment to produce methane, which is supplied to the CHP unit to meet the high heat load demand during this period.

In Figure 6d, during a 4 h period, the heat load reaches its peak and is jointly supported by the HESHR, CHP, and heat purchased from the heating network. During the time periods of 1–3 h and 5–24 h, the heat load demand is solely met by the HESHR and CHP. Throughout the 1–24 h period, the output power of the CHP unit remains relatively stable over one scheduling period, while the heat power output of the hydrogen energy system fluctuates greatly, thereby reducing the pressure on the heat energy output of the CHP unit.

Therefore, it can be observed that after the involvement of the HESHR in heat load scheduling, it effectively promotes the coordinated operation of electricity and heat, alleviates the operational pressure of CHP units in terms of electricity and heat coupling, achieves efficient utilization of hydrogen and heat energy in the hydrogen energy system, and improves the energy efficiency of the system.

5.3. Robustness Analysis

To validate the effectiveness of the proposed HESHR-EHG-IES model, which considers uncertainties on both the source and load sides, an analysis will be conducted on the convergence of the solution algorithm and the robustness of the obtained solutions. The uncertain parameters $\mathsf{\Gamma }$ are set to 6 and 12, the robustness level $\mathsf{\Delta }u$ to 0.1, and the convergence gap to 1 × 10⁻⁶. The convergence effect of the HESHR-EHG-IES robust operating model is shown in Figure 7.

Analysis of Figure 7 reveals that the robust HESHR-EHG-IES operational model converges by the second iteration, with the convergence performance satisfying the specified gap. This indicates that the C&CG algorithm demonstrates good convergence characteristics when solving the robust HESHR-EHG-IES model. The solution times are 0.502 s and 0.508 s, respectively, indicating high computational efficiency, which fully meets the requirements for day-ahead system scheduling decisions. To validate the robustness of the decision-making solution, the uncertain parameter $\mathsf{\Gamma }=12$ and the robustness level $\mathsf{\Delta }u=0.2$ were fixed, and the corresponding operational results are presented in Figure 8.

Analysis of Figure 8 shows that when the value remains constant, the economic cost of HESHR-EHG-IES increases progressively with higher robustness levels. This relationship is largely positive and linear, indicating that as investors are willing to tolerate higher operational costs, the robustness of the operational decision-making solution also improves correspondingly. Corresponding to Figure 9, when $\mathsf{\Delta }u$ remains constant, the economic cost of HESHR-EHG-IES increases as $\mathsf{\Gamma }$ increases. This simultaneously indicates that the system’s ability to withstand uncertainties on both the supply and demand sides is also enhanced. This aligns with the principles of robust optimization, where higher operational costs accepted by investors lead to a higher level of system robustness. Consequently, the conservativeness of the system also increases.

By setting the uncertainty parameter $\mathsf{\Gamma }=15$ and the robustness level $\mathsf{\Delta }u=0.2$, the worst-case operational scenario derived from the HESHR-EHG-IES robust planning model is illustrated in Figure 10.

Analysis of Figure 10 reveals that the worst-case scenarios for wind and solar output are positioned at the lower boundaries of the uncertainty set. This indicates that when the actual wind and solar output is less than the forecasted values, the system shifts towards the worst-case scenario. When the output reaches the lower boundary extreme, wind and solar generation are at their worst-case levels during that period. Primarily, under the premise of fixed switching of system equipment in the first-stage problem, the second-stage decision cost, specifically the cost of purchasing electricity from the grid, far exceeds the operational and maintenance costs of wind and solar power. Therefore, the worst-case scenario on the supply side is at the lower boundary of the wind and solar uncertainty set, which is similar to the extreme scenario optimization theory.

The worst-case scenarios for various types of electric, thermal, natural gas, and hydrogen loads are all located at the upper boundary of the uncertainty set. This is understandable from the perspective of the lower-level second-stage objective: higher load demands will result in greater operational and maintenance costs. Therefore, the worst-case scenarios for load demands are at the upper boundary of the uncertainty set.

It is important to note that there is a distinction between solving the source-side or load-side uncertainty problems independently and solving them simultaneously. When solved independently, the interaction between source and load does not need to be considered while identifying the worst-case scenario for each side. However, in solving the worst-case scenarios for both source and load sides simultaneously, the coupling relationship between the worst-case scenarios of both sides is taken into account. In the operational model presented in this study, the theoretical worst-case scenario typically occurs when wind and solar output are at their minimum while load demand is at its maximum, which is more aligned with actual engineering conditions.

5.4. Comparative Analysis of Optimization Models

To compare the robust operation models of the EHG-IES considering multiple uncertainties and HESHR, four typical models were constructed from the perspectives of flexibility in uncertainty model and economic feasibility of heat recovery.

Model 1: Source-load deterministic model, without considering HESHR;

Model 2: Source-load deterministic model, considering HESHR;

Model 3: Source-load uncertainty model, without considering HESHR;

Model 4: Source-load uncertainty model, considering HESHR.

The total cost of formulating the day-ahead dispatch plan in the HESHR-EHG-IES for the four models are compared and analyzed, as shown in

Table 2. Comparison of model operating costs.

Models	Maintenance Costs (CNY)	Energy Interaction Cost (CNY)	Total Cost (CNY)
1	58,291.6	13,591.6	71,883.2
2	58,443.5	−7725.3	50,718.3
3	58,291.6	13,591.6	71,883.2
4	58,443.5	−7725.3	50,718.3

. By setting the uncertainty adjustment parameter values on both sides of the source and load to 0 for models 3 and 4, the operational costs, energy interaction costs, and total costs of the four models are obtained, as shown in Table 2. The energy interaction cost is positive, indicating that the system purchases energy from the higher-level energy network, while a negative value indicates selling energy to the higher-level energy network.

When the uncertainty adjustment parameters in models 3 and 4 are both set to 0, it indicates that the output and demand on both sides of the source and load are equal to the predicted values, which means that the prediction is 100% accurate, and the uncertain model should be equivalent to the deterministic model. As shown in Table 2, models 1 and 3 as well as models 2 and 4 have completely equal operational costs, energy interaction costs, and total costs. Therefore, model 3 is equivalent to model 1, and model 4 is equivalent to model 2 under these conditions.

By setting the uncertainty adjustment parameter values on both sides of the source and load in models 3 and 4, considering that photovoltaic power generation forecasts have higher reliability, and setting the maximum allowable deviation between actual values and predicted values according to case 4 in Table 2, the hydrogen consumption rates of the hydrogen equipment for the four models are obtained as shown in

Table 3. Hydrogen consumption of each equipment under different models. (kg).

Models	Hydrogen Production Capacity	Hydrogen Fuel Cell	Methanation Equipment	Hydrogen Sales Volume	Hydrogen Load
1	669.14	0	0	441.49	217.56
2	695.88	0	175.28	282.64	217.56
3	669.12	0	0	422.02	235.22
4	695.90	52.23	240.47	135.71	235.22

.

From Table 3, it can be observed that models 2 and 4, compared to models 1 and 3, show a 4% increase in hydrogen production when considering the HESHR model. This is mainly due to the inclusion of heat generation capabilities in the hydrogen energy system in models 2 and 4, which alleviates the heat generation pressure on the CHP units and allows the electricity to be used for hydrogen production. The hydrogen consumption for methane conversion equipment accounts for 0% of the total hydrogen production in models 1 and 3, while it accounts for 25.2% and 34.5% in models 2 and 4, respectively. This is because the hydrogen energy system in models 2 and 4 has the attribute of electricity–hydrogen–heat–gas cogeneration. The consumption of hydrogen by the methane conversion equipment for cogeneration reduces the costs of purchasing heat and gas for the system compared to direct hydrogen sales, resulting in lower costs. Therefore, the hydrogen sales volume in models 2 and 4 is significantly reduced.

According to the analysis in

Table 4. Comparison of costs for each subsystem of the operational model. (Ten thousand CNY).

Models	PGS	HES	CHP and SE	PE	SE	PH	PG	SH	Total Cost
1	5.51	0.22	0.10	1.43	3.56	2.27	2.76	1.55	7.19
2	5.51	0.23	0.10	1.45	3.41	0	2.18	0.99	5.07
3	5.48	0.22	0.10	1.43	2.83	2.55	2.78	1.48	8.26
4	5.49	0.24	0.10	1.45	2.81	0.01	2.02	0.47	6.03

, it can be observed that from the perspective of considering HESHR, the total cost decreases by 29% in the deterministic model and 27% in the uncertain model. This is because the system reduces the hydrogen sales volume for hydrogen fuel cells and methane conversion equipment, thereby reducing the costs of purchasing heat and gas for the system and improving the operational economy of the system.

5.5. Feasibility Validation

To validate the effectiveness of the HESHR-EHG-IES robust operation method, this study will compare the feasibility of solutions from the deterministic model under source and load uncertainty conditions, further demonstrating the advantages of the decision solutions.

Uncertain parameter $\mathsf{\Gamma }=18$, robust level $\mathsf{\Delta }u=0.15$, and ARO solve the HESHR-EHG-IES operational model. Latin hypercube sampling is used to uniformly sample 2000 sets of data from the uncertain set of source loads. By fixing the first-stage decision variable $x$, the solutions of the robust model and the deterministic model are iteratively input into the deterministic HESHR-EHG-IES operational model for solving, obtaining the economic cost and feasibility of the plans as shown in

Table 5. Feasibility verification results.

—	Deterministic Model (CNY)	ARO Model (CNY)
Average cost of feasible solutions	59,085	68,523
Percentage of feasible solutions	61.59%	100%

.

According to Table 5, it is not difficult for us to understand:

(1) From an economic perspective, the operating cost of the ARO operational model is higher than that of the deterministic model. This is because ARO makes decisions based on the worst-case scenarios, aligning with the optimization principle of ARO. The second stage represents a compensatory decision made under the worst-case scenario in the first stage.

(2) In terms of the feasibility of the solutions, there exists a certain gap between the two approaches. Firstly, the deterministic model encounters some infeasible scenarios when solving for all possible scenarios. These are characterized by an inability to maintain system source-load balance, requiring an urgent shedding of some loads to sustain the energy balance of the HESHR-EHG-IES. This occurs because the deterministic model does not consider the uncertainties associated with the source-load in the first stage. If there is a significant deviation between real-time and predicted values, the deterministic model lacks the ability to make further compensations, leading to the emergence of infeasible solutions. On the other hand, the ARO, being a two-stage adjustable robust optimization, takes into account the uncertainties on both sides of the source load in the HESHR-EHG-IES. The decisions made under the worst-case scenario are feasible for all sample scenarios, thus demonstrating the effectiveness of the ARO approach in dealing with real-time uncertainties.

5.6. Comparison of Different Methods

To analyze the advantages and disadvantages of ARO compared to traditional uncertainty handling methods, this section will compare the optimization results of HESHR-EHG-IES under source-load uncertainty using SO and RO approaches. We consider uncertain parameter $\mathsf{\Gamma }=18$ and robustness level $\mathsf{\Delta }u=0.2$, employing the ARO method to solve the HESHR-EHG-IES model. Multiple random approximate sample values were generated using Latin hypercube sampling; however, the excessive data volume imposes a significant computational burden. To address this, Gaussian mixture clustering was used to reduce the scenarios to five representative typical scenarios, as illustrated in Figure 11.

Using the Latin hypercube sampling method, we uniformly sampled 2000 sets of data from the source-load uncertainty set $\left(\mathsf{\Delta }u=0.2\right)$ and applied them to the first-stage decision variable from the SO, RO, and ARO schemes for real-time rolling optimization. The feasibility levels and average economic costs are presented in

Table 6. Feasibility verification results.

SO		ARO		RO
Economic Cost	Feasibility Rate	Economic Cost	Feasibility Rate	Economic Cost	Feasibility Rate
60,594	76.51%	74,573	100%	77,542	100%

.

Analyzing Table 6, it is evident that the operational scheme obtained through SO has a certain economic cost advantage compared to ARO and RO. However, in terms of robustness, both RO and ARO outperform SO. This indicates that, compared to SO, ARO and RO exhibit stronger resilience in the face of source-load fluctuations, thereby enhancing the robustness of the HESHR-EHG-IES. At the same time, compared to RO, ARO demonstrates better economic performance, avoiding overly conservative decision-making solutions. Therefore, in dealing with system uncertainties, ARO demonstrates significant advantages in enhancing its robustness and flexibility compared to traditional uncertainty handling methods.

6. Conclusions

The present study focuses on the utilization of the HESHR model and, based on this, constructs a HESHR-EHG-IES model that incorporates wind and PV power output. It fully considers the uncertainty of various load demands and wind and PV power output in the system. A two-stage ARO model is developed, and the C&CG algorithm is employed for solving it. The simulation analysis primarily aims to verify the superiority of the proposed model and its comprehensiveness in generating day-ahead energy trading plans under the most adverse scenarios of uncertainty.

(1) The HESHR model possesses the ability to flexibly adjust electric heating and heat storage. Based on the hydrogen and heat load requirements of the system, the model can flexibly control the storage and regulation of heat energy, thereby improving the operational flexibility of the hydrogen energy system.

(2) The model fully considers the uncertainty of various load demands and wind and PV power output in the system. By setting uncertain parameters and maximum allowable deviation parameters, the conservativeness of the day-ahead energy scheduling obtained from the ARO model can be flexibly adjusted. This helps decision-makers at higher levels to make reasonable choices between cost and operational risk.

(3) By comparing the operational results of the deterministic model and the ARO model in various typical scenarios, it is demonstrated that using the C&CG algorithm to solve the two-stage ARO model can provide a day-ahead scheduling plan with the minimum system operating cost for the comprehensive energy system in the “worst-case” scenario.

(4) Compared with the deterministic optimization model, the scheduling plan obtained by the ARO model under the uncertainty of supply and demand has the ability to resist the real-time fluctuation risk on both sides of the supply and demand; that is, it has stronger robustness.