The Robust Optimization of Low-Carbon Economic Dispatching for Regional Integrated Energy Systems Considering Wind and Solar Uncertainty

Abstract

In this paper, a two-stage robust optimization approach is employed to address the variability in renewable energy output by accounting for the uncertainties associated with wind and solar energy. The model aims to achieve a balanced system that is both low-carbon and economically efficient while also being resilient to uncertainties. Initially, a regional integrated energy system model is developed, integrating electricity, gas, and heat. The variability of wind and photovoltaic power outputs is represented using a modifiable uncertainty set. A resilient optimal scheduling model is formulated in two stages, with the objective of minimizing costs under worst-case scenarios. This model is solved iteratively through a column and constraint generation approach. Additionally, the scheduling model incorporates horizontal time shifts and vertical complementary substitutions for carbon trading costs and demand-side loads to avoid excessive conservatism and to manage carbon emissions and energy trading in the regional integrated energy system (RIES). Results show that the two-stage robust optimization approach significantly enhances the system’s resilience to risks and minimizes economic losses. The inclusion of carbon trading mechanisms and the demand response prevents the system from becoming overly robust, which could impede economic growth, while also reducing carbon emissions. The proposed method effectively achieves balanced optimal scheduling for a robust, economical, and low-carbon system.

1. Introduction

The escalating severity of global warming and the depletion of fossil resources highlight the urgent need for advanced solutions. Addressing the energy crisis and environmental pollution requires comprehensive advancements in renewable energy, which represents the primary direction for future energy development [1]. However, renewable sources like photovoltaics and wind power are notably unpredictable and irregular, leading to potential imbalances between the energy supply and demand. Additionally, the integration of wind and solar energy complicates the operation and dispatch of traditional power systems [2]. Consequently, developing a robust, efficient, and environmentally conscious energy infrastructure is crucial.

By integrating a diverse array of energy sources, the Regional Integrated Energy System (RIES) overcomes barriers between different energy types, facilitates the coordinated operation of various energy subsystems, achieves efficient energy conversion, and significantly enhances the utilization of renewable energy. However, the dispatching of resources, including renewable energy, within the RIES is marked by considerable uncertainty, posing significant challenges to the overall safety and operational efficiency of the system [3]. Therefore, it is essential to explore the impact of wind and solar energy uncertainty on RIES dispatch [4].

Currently, the predominant methods for addressing uncertainties can be classified as stochastic optimization or robust optimization [5]. Robust optimization provides a reliable and effective solution by constructing an indeterminate set based on the characteristics of uncertain variables without the need for precise modeling of their distribution functions [6]. This approach ensures that if the values of the variables fall within the defined set, the robust optimization model will deliver a feasible solution, thereby significantly simplifying the problem-solving process [7]. For power dispatching between the main grid and a microgrid incorporating solar panels and batteries, a robust algorithm has been proposed that ensures resilience to uncertainty while managing the variability in solar power generation and avoiding an excessively conservative dispatch through time-varying soft constraints [8]. Additionally, a two-stage robust multi-period economic dispatch problem, as discussed in [9], is addressed using a column and constraint generation method. This method employs robust optimization to handle uncertainties related to renewable energy generation and load consumption in active distribution networks.

To maximize the economic benefits of renewable energy systems, it is essential to consider demand. The demand response (DR) effectively harnesses the user-side potential, ensuring operational safety [10], enhancing the integration of renewable energy sources [11], and improving the economics and flexibility of the RIES operation [12]. Reference [13] proposed a two-stage stochastic p-robust optimal energy transaction management strategy that accounts for uncertainties in microgrid operations and integrates a hybrid demand response to improve robustness and to minimize operating costs. Reference [14] introduced a two-stage robust scheduling model for optimizing the operation of solar–wind–hydro–thermal pump (SWHTP) systems in a competitive environment, which enhances system flexibility by implementing demand response programs. This approach leverages adjustable and curtailable elastic load characteristics to improve the continuous energy supply reliability and to reduce overall energy costs in a deregulated market.

Nevertheless, the aforementioned research primarily focuses on model resilience and cost-effectiveness, often neglecting the goal of achieving low-carbon operation for the system. To ensure robust scheduling, it is necessary to anticipate and prepare for increased unit production and energy acquisition to enhance the resilience against risks [15], which, in turn, may lead to higher carbon emissions. Carbon trading allows for the exchange of predetermined CO₂ emissions allowances [16], facilitating a low-carbon operation by incorporating the cost of carbon trading into the overall cost, thereby leveraging economic incentives [17]. Reference [18] introduced a two-stage adjustment robust optimization model for multi-energy virtual power plants, accounting for various uncertainties and carbon trading. This model optimizes costs in the worst-case scenario, dynamically balancing economic and robust dispatching strategies while guiding energy conservation and emission reduction efforts. Reference [19] proposed a method for planning economic and environmental power distribution network assets, incorporating carbon emissions trading and the demand response. This approach mitigates carbon emissions at the distribution level and addresses the uncertainties of distributed generation and power demand through a two-stage robust optimization model.

In summary, existing studies often fall short in addressing the equilibrium between reduced carbon emissions, economic considerations, and system resilience. Moreover, there is a lack of research evaluating the balanced optimization of these three factors. Integrating carbon trading mechanisms and the demand response into the robust optimization model for Regional Integrated Energy Systems (RIESs) is crucial for achieving a well-balanced system that is low-carbon, cost-effective, and resilient [20]. This paper employs a robust optimization approach to tackle the uncertainties associated with wind power and photovoltaics. Additionally, it incorporates a carbon trading system and demand response to control carbon emissions, enhance energy flexibility, and optimize the balance for low-carbon, cost-effective, and resilient operations [21]. The proposed system model is validated with a concrete example, demonstrating its strong resilience and cost-effectiveness while effectively maintaining a low-carbon operation.

2. The RIES Operational Architecture

The RIES is a complex structural system that integrates multiple energy sources. It optimizes the interplay between electricity, heat, and gas to improve energy efficiency and meet diverse customer needs while ensuring a reliable and consistent energy supply [22]. This study constructs an RIES framework that incorporates the demand response for electricity, gas, and heat. As shown in Figure 1, the framework is based on a carbon trading mechanism.

The energy supply side of the RIES includes an advanced power grid and an advanced gas grid, as well as wind power and solar panels [23]. There are four main types of energy conversion equipment: power-to-gas (P2G) systems, gas boilers (GBs), gas turbines (GTs), and waste heat boilers (WHBs) [24]. The system operates in a combined heat and power (CHP) coupling mode, allowing it to adapt to various system conditions. Additionally, there are three types of energy storage: electric, gas, and thermal. The energy demand-side-response resources within the system consist of the electricity load, gas load, and heat load [25].

2.1. Ladder Carbon Trading Mechanism

This article evaluates solar power, wind turbines, and natural gas as energy sources through a life cycle assessment perspective. It converts carbon emissions generated throughout the entire life cycle of energy and energy storage equipment—encompassing production, transportation, operation, and disposal—into carbon emission costs. These costs are integrated into optimization objectives to examine their impact on system optimization outcomes when considering full life-cycle carbon emission costs. For detailed measurement procedures of each component, please refer to reference [26].

The government or relevant agencies allocate carbon emission allowances to each carbon emission source within an RIES [27]. If a manufacturer’s actual CO₂ emissions are below the allocated quota, they can sell the surplus allowances for cash. Conversely, if their emissions exceed the quota, manufacturers must purchase additional allowances to cover the excess. The carbon trading mechanism model mainly consists of three components:

2.1.1. The Initial Quota Model

The primary sources of carbon emissions in the RIES include superior power purchases, gas turbines (GTs), gas boilers (GBs), and demand-side gas loads. The initial quota model is as follows:

$$\left\{\begin{array}{l}{E}_c=E_{c,buy}+E_{c,GT}+E_{c,GB}+E_{g,load}\\ E_{c,buy}={\beta }_e^{*}{\displaystyle \sum _{t=1}^T{P}_{e,buy}}(t)\Delta t\\ E_{c,GT}={\beta }_h^{*}\left({\displaystyle \sum _{t=1}^T{P}_{GT,h}}(t)\Delta t+{\phi }_{e,h}{\displaystyle \sum _{t=1}^T{P}_{GT,e}}(t)\Delta t\right)\\ E_{c,GB}={\beta }_h^{*}{\displaystyle \sum _{t=1}^T{P}_{GB,h}}(t)\Delta t\\ E_{g,load}={\phi }_{g,load}{\displaystyle \sum _{t=1}^T{P}_{g,load}}(t)\Delta t\end{array}\right.$$

(1)

where $E_c$, $E_{c,buy}$, $E_{c,GT}$, $E_{c,GB}$, and $E_{g,load}$ are the free carbon emission rights quotas of the RIES and the system purchase of electricity from superiors, gas turbines, gas boilers, and demand-side gas loads, respectively; ${\beta }_e^{*}$ and ${\beta }_h^{*}$ represent the allocated quotas for carbon emissions that are gained by generating electricity and heat, respectively; ${\phi }_{g,load}$ represents the allocated amount of carbon emissions that is permitted for each unit of gas load used; and ${\phi }_{e,h}$ is the conversion parameter of gas turbine power generation to a calorific value. During the t period, $P_{e,buy}(t)$ represents the power that the system purchases from the superior, $P_{GT,h}(t)$ represents the amount of electric power that the gas turbine supplies, $P_{GT,e}(t)$ is the amount of thermal power that the gas turbine supplies, $P_{GB,h}(t)$ is the amount of thermal power that the gas boiler supplies, $P_{g,load}(t)$ is the amount of gas load that is consumed during the t period, and T is the period that the system is scheduled to run.

2.1.2. The Actual Model

During the power-to-gas conversion process, some CO₂ will be utilized by the system, and significant CO₂ emissions will be generated by the demand-side gas load. Given these factors, the original model requires adjustments. The revised model can be summarized as follows:

$$\left\{\begin{array}{l}{E}_{c,a}=E_{e,buy,a}+E_{GBGT,a}+E_{g,load,a}-E_{P2G,a}\\ E_{e,buy,a}={\displaystyle \sum _{t=1}^T[a+b{P}_{e,buy}(t)+c{P}_{e,buy}^2(t)]}\Delta t\\ E_{GTGB,a}={\displaystyle \sum _{t=1}^T[d+e{P}_{GTGB}(t)+f{P}_{GTGB}^2(t)]}\Delta t\\ P_{GTGB}(t)=P_{GT,h}(t)+P_{GT,e}(t)+P_{GB,h}(t)\\ E_{g,load,a}=\phi {\displaystyle \sum _{t=1}^T{P}_{g,load}}(t)\Delta t\\ E_{P2G,a}=\theta {\displaystyle \sum _{t=1}^T{P}_{P2G,g}}(t)\Delta t\end{array}\right.$$

(2)

where the actual carbon emissions from the RIES are denoted by $E_{c,a}$, while those from the power the system bought from the superior are denoted by $E_{e,buy,a}$, and the real carbon emissions from gas turbines and boilers are denoted by $E_{GBGT,a}$. In the t period, $P_{GTGB}(t)$ stands for the equivalence of the gas turbine and gas boiler output power; in the case of the demand-side gas load, $E_{g,load,a}$ represents the real carbon emission, $E_{P2G,a}$ stands for the CO₂ absorbed by the P2G equipment during the electricity-to-gas conversion process, $P_{P2G,g}(t)$ represents the power output from natural gas produced by the P2G equipment in the given time period, and $\phi$ is the equivalent carbon emission parameter per unit gas load. For specific values of the $\theta$ parameters for absorbing CO₂ in this process, please refer to reference [27]. The carbon emission parameters for coal power units are represented by the variables a, b, and c, while the carbon emission characteristics for units consuming natural gas are represented by the variables d, e, and f.

2.1.3. The Calculation Model

The carbon emissions trading share in the carbon trading market can be determined by considering the following two components:

$$E_{RIES,c}=E_{\mathrm{c},a}-E_c$$

(3)

where $E_{RIES,c}$ represents the system’s share in the carbon trading market for carbon emissions.

The carbon trading cost calculation model uses a hierarchical approach to classify CO₂ emissions into various ranges. As CO₂ emissions rise, both transaction prices and costs increase. A mathematical model representing carbon trading costs is formulated as a piecewise linear function with multiple intervals for different emission levels. Surplus carbon can be sold at a specified price at the corresponding trading center, generating a subsidy. The carbon trading cost can be expressed using a piecewise function as follows:

$$F_{c{o}_2}=\left\{\begin{array}{c}c{E}_{RIES,c},E_{RIES,c}<d\\ c(1+\lambda )(E_{RIES,c}-d)+cd,d\le E_{RIES,c}<d\\ c(1+2\lambda )(E_{RIES,c}-2d)+c(2+\lambda )d,2d\le E_{RIES,c}<3d\\ c(1+3\lambda )(E_{RIES,c}-3d)+c(3+3\lambda )d,3d\le E_{RIES,c}<4d\\ c(1+4\lambda )(E_{RIES,c}-4d)+c(4+6\lambda )d,4d\le E_{RIES,c}<5d\\ c(1+5\lambda )(E_{RIES,c}-5d)+c(5+10\lambda )d,5d\le E_{RIES,c}\end{array}\right.$$

(4)

where $F_{c{o}_2}$ represents the stepped carbon trading cost, $\lambda$ indicates the price increase rate, c is the carbon trading benchmark price, and d is the interval length for carbon emissions.

2.2. IDR

This paper introduces a demand response approach utilizing a horizontal time shift and vertical complementary alternative strategy for managing electricity, gas, and heat [28]. Operators can develop various scheduling plans in response to energy price fluctuations to accommodate different energy demands and to ensure the economical, flexible, and efficient operation of the RIES. In line with traditional power demand categorization, the demand response loads in the RIES—electricity, gas, and heat—are classified into three groups based on their interactive response characteristics: fixed load, transferable load, and replaceable load.

The transferable load versus alternative load models are shown below:

$$\left\{\begin{array}{c}{P}_{\mathrm{k},load}^{n*}=P_{\mathrm{k},load}^n(t)+\Delta P_{\mathrm{k},load}^n(t)\\ \Delta P_{\mathrm{k},load}^n(t)=v_{k,in}^n{P}_{\mathrm{k},in}^n(t)-v_{k,out}^n{P}_{\mathrm{k},out}^n(t)\\ v_{k,in}^n+v_{k,out}^n=1\\ {\displaystyle \sum _{t=1}^T\Delta P_{\mathrm{k},load}^n(t)}=0\\ P_{\mathrm{k}}^{n,\min }\le \Delta P_{\mathrm{k},load}^n(t)\le P_{\mathrm{k}}^{n,\max }\end{array}\right.$$

(5)

where n represents the different types of loads (n = p, c); when n = p, $P_{\mathrm{k},load}^{\mathrm{p}*}$ denotes the magnitude of the transferable load’s power during the k-th load t time following its involvement in the demand response; when n = c, $P_{\mathrm{k},load}^{\mathrm{c}*}$ represents the power after the substitutable load in the k-th load t period participates in the demand response. $v_{k,in}^n$ and $v_{k,out}^n$ are the binary parameters that reflect the transfer-in and transfer-out for the k-th load in the t period, respectively; $P_{\mathrm{k},in}^n(t)$ and $P_{\mathrm{k},out}^n(t)$ denote, correspondingly, that power is transported to and from the k-th load in t periods. For the k-th load taking part in the demand response, $P_{\mathrm{k}}^{n,\min }$ and $P_{\mathrm{k}}^{n,\max }$ represent the minimum and maximum values, respectively.

In summary, the DR model can adjust the energy demand through various types of demand responses, as detailed below:

$$\left\{\begin{array}{l}{P}_{\mathrm{k},load}^{*}=P_{\mathrm{k},load}(t)+\Delta P_{\mathrm{k},load}(t)\\ =P_{\mathrm{k},load}(t)+\Delta P_{\mathrm{k},load}^p(t)+\Delta P_{\mathrm{k},load}^c(t)\end{array}\right.$$

(6)

where $P_{\mathrm{k},load}^{*}$ denotes the power level following the involvement of the k-th load in the DR during the time t, while $\Delta P_{\mathrm{k},load}(t)$ illustrates the strength of the k-th load taking part in the DR within the given time period t.

3. Robust Optimization

This research addresses the unpredictability in wind power and solar output using a two-stage robust optimization approach. The model’s min–max–min structure aims to minimize the worst-case day-ahead dispatching cost. The first phase involves an outer minimization problem, optimizing variables such as energy storage equipment and the energy purchase state to ensure that the system can handle any scenario within the uncertain set [29]. The second phase involves an internal maximum–minimum problem, where, after determining the first-stage optimization variables, the goal is to identify a set of worst-case scenarios that maximize the day-ahead scheduling cost. This stage includes finding the worst-case scenario and deriving a deterministic optimization problem from it [30]. The objective is to maximize the output of each unit under the worst-case scenario while minimizing the potential day-ahead scheduling cost. Figure 2 illustrates the detailed framework of the optimization model.

The costs of RIES scheduling and the benefits of low-carbon emissions are predominantly influenced by the worst-case scenario simulated in the second stage of the robust optimization. When the actual wind power and photovoltaic outputs deviate from expected levels, the RIES employs robust optimization strategies to address the risks associated with uncertainty, especially in worst-case scenarios [31]. Generally, using robust optimization results in a more cautious approach, leading to higher anticipated unit outputs and increased energy purchases. This approach raises dispatching costs and reduces the benefits of low-carbon energy [32].

3.1. Uncertainty Ensemble Modeling

Due to factors such as illumination, wind speed, and geographical location, renewable energy production is highly variable. Therefore, this study employs a robust optimization strategy to model the uncertainty in wind and solar outputs, typically using an uncertainty set to characterize the nature of this uncertainty. The optimization outcomes are significantly influenced by the choice of uncertainty sets. Accurate modeling of the uncertainty set ensures that the optimization results closely reflect the actual scenarios, although it can make the model more complex and difficult to solve. Conversely, if the uncertainty set is too loose, the optimization approach becomes overly conservative. To effectively manage the level of conservatism in the scheduling plan, a flexible uncertainty set is used to describe the variability in energy output, as indicated in Equation (7).

$$P^{RE}=\left\{\begin{array}{l}{P}_{ele,t}^{WT}=\overline{P_{ele,t}^{WT}}-u_{\mathrm{wt},\mathrm{t}}{d}_{wt,t}\\ {\displaystyle \sum _t{u}_{\mathrm{wt},\mathrm{t}}\le {\Gamma }^{WT}}\\ P_{ele,t}^{PV}=\overline{P_{ele,t}^{PV}}-u_{\mathrm{pv},\mathrm{t}}{d}_{pv,t}\\ {\displaystyle \sum _t{u}_{\mathrm{pv},\mathrm{t}}\le {\Gamma }^{PV}}\\ \forall t\in T\end{array}\right.$$

(7)

where $\overline{P_{ele,t}^{WT}}$ and $\overline{P_{ele,t}^{PV}}$ are the predicted values of wind and solar outputs in period t, and the wind power and solar output fluctuations have upper and lower limitations, $\Delta P_{+}^{RE}$ and $\Delta P_{-}^{RE}$, respectively. In the t period, $d_{wt,t}$ and $d_{pv,t}$ stand for the wind power and photovoltaic fluctuation values, respectively, and $u_{\mathrm{wt},\mathrm{t}}$ and $u_{\mathrm{pv},\mathrm{t}}$, which are 0–1 variables, indicate the wind power and photovoltaic fluctuation states, respectively. The worst-scenario output is denoted by $P_{ele,t}^{WT}$ and $P_{ele,t}^{PV}$; ${\Gamma }^{WT}$ and ${\Gamma }^{PV}$ are the unknowns, which stand for the total amount of time periods when the cycle’s wind and solar output changes. The larger the uncertainty value, the more periods during which the wind and solar output fluctuates, and the more conservative the scheduling scheme is.

3.2. Objective Function

The overall operational cost F is taken into account as the objective function of the RIES in this study and includes the energy purchase cost $F_{buy}$, carbon trading cost $F_{C{O}_2}$, demand response compensation cost $F_{DR}$, and equipment operation and maintenance cost $F_{MC}$, which are specifically expressed as follows:

$$F_{\min }=F_{buy}+F_{C{O}_2}+F_{MC}+F_{DR}$$

(8)

• Energy purchase cost $F_{buy}$.
  $$F_{buy}={\displaystyle \sum _{t=1}^T{\alpha }_t}{P}_{e,buy}(t)+{\displaystyle \sum _{t=1}^T{\beta }_t}{P}_{g,buy}(t)$$

  (9)

  where $P_{g,buy}(t)$ represents the gas purchase volume in period t, and ${\alpha }_{\mathrm{t}}$ and ${\beta }_t$ reflect the prices of electricity and gas, respectively, during the time period t.
• $F_{C{O}_2}$ is shown in Equation (4).
• Demand response compensation cost $F_{DR}$.
  $$F_{DR}={\displaystyle \sum _{k=1}^3{\displaystyle \sum _{t=1}^T({\mu }_p|\Delta P_{k,load}^p(t)|+{\mu }_c|\Delta P_{k,load}^c(t)|)}}$$

  (10)

  where ${\mu }_p$ and ${\mu }_c$, respectively, represent the unit compensation coefficients of transferable loads and replaceable loads participating in the demand response.
• Equipment operation and maintenance cost $F_{MC}$.
  $$F_{MC}={\displaystyle \sum _{t=1}^T(}{\displaystyle \sum _{i=1}^n{P}_i}(t)\cdot C_i+{\displaystyle \sum _{j=1}^m{P}_j}(t)\cdot C_j+P_{p2g}(t)\cdot C_{p2g}+|P_{es}(t)|\cdot C_{es})$$

  (11)

  where $C_i$ stands for the cost of operating and maintaining the i-th coupling equipment per unit of power, and $C_j$ is the cost of operating and maintaining new energy equipment per unit of power. The operational power of the i-th coupling equipment is represented by $P_i(t)$, whereas the operating power of the new energy equipment is represented by $P_j(t)$; $C_{p2g}$ shows how much the power-to-gas technology costs to run and maintain per unit of power; $P_{es}(t)$ represents the energy storage device’s operational power for a given time period; and $C_{es}$ shows how much money it costs the energy storage systems to run and maintain each unit.

3.3. System Operation Constraints

• Constraints on electricity generation from wind and solar systems.
  $$\left\{\begin{array}{l}0\le P_{WT}(t)\le P_{WT}^{\max }\\ 0\le P_{PV}(t)\le P_{PV}^{\max }\end{array}\right.$$

  (12)

  where $P_{WT}^{\max }$ is the wind power’s maximum output power, and $P_{PV}^{\max }$ is the highest power that can be produced by photovoltaics.
• CHP operation constraints.
  $$\left\{\begin{array}{l}{P}_{CHP,e}(t)={\epsilon }_{CHP}^e{P}_{g,CHP}(t)\\ P_{CHP,h}(t)={\epsilon }_{CHP}^h{P}_{g,CHP}(t)\\ P_{g,CHP}^{\min }\le P_{g,CHP}(t)\le P_{g,CHP}^{\max }\\ \Delta P_{g,CHP}^{\min }\le P_{g,CHP}(t+1)-P_{g,CHP}(t)\le \Delta P_{g,CHP}^{\max }\\ {\omega }_{CHP}^{\min }\le P_{CHP,h}(t)/P_{CHP,e}(t)\le {\omega }_{CHP}^{\max }\end{array}\right.$$

  (13)

  where $P_{g,CHP}(t)$ represents the quantity of natural gas consumed by the CHP over the given time period t. $P_{CHP,e}(t)$ and $P_{CHP,\mathrm{h}}(t)$ are the thermal and electrical power generated by the CHP in the time period t, while ${\epsilon }_{CHP}^e$ and ${\epsilon }_{CHP}^h$ demonstrate the relative effectiveness of transforming the CHP into thermal energy and electricity; the two extremes of the power input to CHP from natural gas are denoted as $P_{g,CHP}^{\min }$ and $P_{g,CHP}^{\max }$, respectively. As for the CHP, its lower ascending limit is $\Delta P_{g,CHP}^{\min }$, and its maximum climbing limit is $\Delta P_{g,CHP}^{\max }$; the lowest and highest values for the thermoelectric ratio of the CHP are given by ${\omega }_{CHP}^{\min }$ and ${\omega }_{CHP}^{\max }$, respectively.
• GB running constraints.
  $$\left\{\begin{array}{l}{P}_{GB,h}(t)={\phi }_{GB}{P}_{GB,g}(t)\\ P_{GB,g}^{\min }\le P_{GB,g}(t)\le P_{GB,g}^{\max }\\ \Delta P_{GB,g}^{\min }\le P_{GB,g}(t+1)-P_{GB,g}(t)\le \Delta P_{GB,g}^{\max }\end{array}\right.$$

  (14)

  where ${\phi }_{GB}$ indicates how well the GB converts energy, and $P_{GB,g}(t)$ shows the GB input power from natural gas in the time period t; the two extremes of the power input to GB from natural gas are denoted as $P_{GB,g}^{\min }$ and $P_{g,CHP}^{\max }$, respectively. As for the GB, its lower ascending limit is $\Delta P_{GB,g}^{\min }$, and its maximum climbing limit is $\Delta P_{g,CHP}^{\max }$.
• Electric power balance constraints.
  $$\left\{\begin{array}{l}{P}_{e.\mathrm{buy}}(t)=P_{e.load}(t)+P_{e.P2G}(t)+P_{ES,cha}^e(t)\\ -P_{PV}(t)-P_{ES,dis}^e(t)-P_{DG}(t)-P_{GT,e}(t)\\ 0\le P_{e.buy}(t)\le P_{e,buy}^{\max }\end{array}\right.$$

  (15)

  where $P_{e.load}(t)$ stands for the electric load in period t, $P_{e.P2G}(t)$ stands for the electric power of input P2G in period t, $P_{ES,cha}^e(t)$ stands for the power input to the electric energy storage system in period t, $P_{ES,dis}^e(t)$ stands for the power output from the electric energy storage system in period t, and $P_{e,buy}^{\max }$ stands for the maximum purchased power from the superior power grid.
• Gas power balance constraints.
  $$\left\{\begin{array}{l}{P}_{g.\mathrm{buy}}(t)=P_{g.load}(t)+P_{GT,g}(t)+P_{ES,cha}^g(t)\\ +P_{GB,g}(t)-P_{e.P2G}(t)-P_{ES,dis}^g(t)\\ 0\le P_{g.buy}(t)\le P_{g,buy}^{\max }\end{array}\right.$$

  (16)

  where in the given time period, $P_{ES,cha}^g(t)$ represents the power input to the natural gas energy storage system, and $P_{ES,dis}^g(t)$ denotes the power output from the same system. $P_{g,buy}^{\max }$ is the highest possible power of the gas purchase from the superior natural gas network, and $P_{GT,g}(t)$ is the gas power rate of the input gas turbine in time t.
• Heat power balance constraints.
  $$P_{CHP,h}(t)+P_{GB,h}(t)=P_{h.load}(t)+P_{ES,cha}^h(t)-P_{ES,dis}^h(t)$$

  (17)

  where in each given time period, $P_{h.load}(t)$ stands for the heat load, $P_{ES,cha}^h(t)$ stands for the power input to the thermal energy storage system, and $P_{ES,dis}^h(t)$ stands for the power output from the system.
• Energy storage operation constraints.
  $$\left\{\begin{array}{l}{P}_{es,cha}^{\min }(t)\le P_{es,cha}(t)\le P_{es,cha}^{\max }(t)\\ P_{es,dis}^{\min }(t)\le P_{es,dis}(t)\le P_{es,dis}^{\max }(t)\\ S_{es,\min }(t)\le S_{es}(t)\le S_{es,\max }(t)\\ S_{es}(0)=S_{es}(24)\end{array}\right.$$

  (18)

  where $P_{es,cha}^{\max }(t)$ and $P_{es,cha}^{\min }(t)$ stand for the maximum and minimum values of the power storage capacity for electricity, heat, and gas during the time period t; $P_{es,dis}^{\max }(t)$ and $P_{es,dis}^{\min }(t)$ represent the maximum and minimum values of the power supply for these energy storage mediums during the time period t; $S_{es,\max }(t)$ and $S_{es,\min }(t)$ stand for the maximum and minimum percentages of the energy condition of gas, heat, and electricity storage throughout the specified time period; and $S_{es}(0)$ and $S_{es}(24)$ stand for the percentages representing the energy storage device’s initial and final states during the scheduling cycle.

3.4. Model Solution

The RIES two-stage resilient optimal scheduling model is formulated using constraints and objective functions. For ease of solution, the model is presented in a streamlined format, as shown in Equation (19).

$$\left\{\begin{array}{l}\underset{x}{\min }(\underset{u\in P_{RE}}{\max }\underset{y\in \Omega (x,u)}{\min }{c}^{T}y)\\ s.t.\\ Dy\ge d\\ Ey=0\\ Fx+Gy\ge h\\ I_{u}y=u\end{array}\right.$$

(19)

$$\left\{\begin{array}{l}x=[u_{ess,t},u_{tss,t},v_{ele,t},v_{ther,t}]\\ y=[P_{e,t}^w,P_{dis,t}^i,P_{ch,t}^i,P_{e,t}^{buy},P_{e,t}^{sell},P_{e,t}^l]\\ u=[P_{ele,t}^{PV},P_{ele,t}^{WT}]\end{array}\right.$$

(20)

where c is the symbol for the objective function’s corresponding coefficient matrix; x and y represent the vectors of system output variables and state variables, respectively; the coefficient matrices D, E, F, and G correspond to the restrictions; and the system operational parameters d and h are shown.

The initial problem (Equation (19)) is decomposed into the primary problem (outer minimization problem) as shown in Equation (21), and the secondary problem (inner max–min problem) is depicted in Equation (22). The column and constraint generation (C&CG) algorithm is used to solve it iteratively. This approach accelerates convergence by systematically integrating variables and constraints related to the sub-problems while solving the main problem. Initially, a set of numerical values is designated as the uncertainty parameters, which are then input into the sub-problem. The solution of the sub-problem identifies the factors that lead to the worst-case scenario. These parameters are iteratively reintroduced into the main problem for further resolution. The sub-issue is reformulated as a single-layer maximization problem to determine the worst-case scenario’s uncertainty parameters. The procedure is detailed in Figure A1 (Appendix B), while the equipment specifications and energy prices are listed in

Table A1. Main parameters of the RIES equipment.

Equipment	Total Capacity/kW	Conversion Efficiency/%	Climbing Constraints/%
P2G	500	60	20
GT	1000	22 (e→g), 72 (g→h)	20
WHB	600	80	20
GB	800	82	20
Electrical storage	450	——	20
Thermal storage	500	——	20
Gas storage	300	——	20

and

Table A2. Time-sharing price of the grid network power.

Peak and Valley Periods	Peak Period	Ordinary Period	Valley Period
Time period	08:00–11:00 18:00–21:00	06:00–08:00婉11:00–18:00 21:00–22:00	22:00–06:00
Electricity purchase price [CNY(kW·h)−1]	1.167	0.718	0.338

. The anticipated values of wind and photovoltaic power, representing the initial uncertainty parameters, are shown in Figure A3 in Appendix B.

$$MP\left\{\begin{array}{l}\underset{x}{\min }\theta \\ s.t.\\ \theta \ge c^T{y}_k\\ D{y}_k\ge d\\ Fx+G{y}_k\ge h\\ I_u{y}_k=u_k^{*}\end{array}\right.$$

(21)

where the sub-problem solution after the k-th iteration is denoted as $y_k$, and $u_k^{*}$ is the value of the wind power and solar outputs in the most unfavorable scenario computed after the k-th iteration.

$$SP\left\{\begin{array}{l}\underset{u\in P_{RE},\gamma ,\lambda ,\nu ,\pi }{\max }{d}^{T}y+{(h-Fx)}^T\nu +u^T\pi \\ s.t.\\ D^{T}y+K^T\lambda +G^T\nu +I_u^T\pi \le c\\ \gamma \ge 0,\nu \ge 0,\pi \ge 0\end{array}\right.$$

(22)

Because the maximization problem obtained by dual transformation of the max–min problem of the sub-problem has the bilinear term $u^T\pi$, it is linearized using the large M approach. In Equation (23), we can see the sub-problem in its ultimate linearized form.

$$\left\{\begin{array}{l}\underset{u\in P_{RE},\gamma ,\lambda ,\nu ,\pi }{\max }{d}^{T}y+{(h-Fx)}^T\nu +\overline{u^T\pi }+\Delta u^T{B}^{\prime }\\ s.t.\\ D^{T}y+K^T\lambda +G^T\nu +I_u^T\pi \le c\\ 0\le B^{\prime }\le M\cdot B\\ \pi -M\cdot (1-B)\le B^{\prime }\le \pi \\ \gamma \ge 0,\nu \ge 0,\pi \ge 0\end{array}\right.$$

(23)

where the dual variable has the upper bound of a big-enough positive real number M, and the auxiliary variables $\Delta u=[d^{pv,t},d^{wt,t}]$ and $B^{\prime }=[B_{pv,t}^{\prime },B_{wt,t}^{\prime }]$ are also included.

4. Example Analysis

This article presents a case study utilizing the MATLAB environment and the CPLEX solver from the YALMIP toolkit to address the issue. The study develops the unit output, energy storage charging and discharging, and energy acquisition strategies for each time period using a two-stage robust optimization approach. The objective is to ensure that the RIES achieves the lowest cost even under adverse wind and solar output conditions. The model’s capacity to balance robustness, economics, and low-carbon benefits is validated by the operation scheduling results. The example data are configured as follows: The characteristics of the energy conversion and energy storage units are selected from the referenced source [33]; specifically, Table A1 in Appendix A. Real-time energy prices from the energy trading market are listed in Table A2 in Appendix A. Load prediction values, along with photovoltaic and wind power output figures, are depicted in Figure A2 and Figure A3 in Appendix B. References for actual carbon emission model parameters are given in [34]; the carbon emission right quota ${\beta }_e^{*}$ per unit of electricity generated is 0.798 kg/(kW · h), and the carbon emission right quota ${\beta }_h^{*}$ per unit of heat generated is 0.385 kg/(kW · h); the carbon emission quota ${\phi }_{g,load}$ = 0.180 kg/(kW · h) per unit gas load consumption; the interval length of the carbon emission d = 2000 kg, the price increase range $\lambda$ = 0.25, and the base price of carbon trading c = 0.251 CNY/kg. The transferable load on the demand side accounts for 10% of the total load, and the substitutable load accounts for 5% of the total load. The robust optimization model considers the extreme values of wind and solar production as the most unfavorable scenarios. However, the actual output may vary, falling within or outside this range. Consequently, during the robust optimization process, there can be a discrepancy between the wind and solar production predicted by the uncertainty set and the actual output of wind and solar sources.

4.1. Comparison and Analysis of Scene Optimization Results

To demonstrate the efficacy of the RIES robust optimization model in conjunction with the carbon trading mechanism and the demand response, this study constructs four optimization scenarios:

Scenario 1: This scenario uses deterministic optimization based on a stepped carbon trading mechanism without incorporating the demand response and considering the costs of carbon trading.

Scenario 2: This scenario focuses on carbon trading costs within the stepped carbon trading system using deterministic optimization while excluding the demand response.

Scenario 3: Deterministic optimization is employed in this scenario, which accounts for demand-side loads and carbon emission costs related to horizontal time shifts and vertical complementary substitutions.

Scenario 4: Robust optimization is applied in this scenario with the stepped carbon trading mechanism, considering both horizontal time shifts and vertical complementary substitutions of carbon emission costs and demand-side loads.

This section provides a comparison and analysis of the most efficient scheduling outcomes across the four scenarios. Power balance diagrams for all the scenarios are detailed in Appendix C. Scenario 1, which uses deterministic optimization and excludes the demand response and carbon trading prices, results in the highest carbon trading costs and actual carbon emissions. Scenario 2, by focusing solely on carbon trading costs and using deterministic optimization, achieves a 12.93% reduction in actual carbon emissions and a 29.37% decrease in carbon trading costs compared with Scenario 1. Additionally, reduced energy purchases in Scenario 2 further enhance the low-carbon benefits.

Scenario 3 incorporates both carbon emission costs and the demand response, resulting in a 12.49% reduction in energy procurement expenses and a 7.01% decrease in carbon trading costs compared with Scenario 2. This scenario also achieves a 6.17% reduction in system carbon emissions. The use of multiple energy sources in a complementary manner enhances the system’s economic and low-carbon performance.

In Scenario 4, robust optimization is applied to Scenario 3, leading to a 6.11% increase in energy purchase costs and a 3.42% rise in real carbon emissions. However, this robust model improves the system’s resilience by increasing energy purchases to mitigate the risks associated with fluctuations in wind and solar outputs. It effectively balances resilience and low-carbon objectives while maintaining a robust system operation.

4.2. Analyzing the Effects of the Base Price and Range Length in Carbon Trading

Since carbon trading is a key component of the objective function, fluctuations in its cost will influence the overall operating cost of the system. Figure 3 explores the relationships among variables such as the carbon trading base price (c) and interval length (d) and the system’s actual carbon emissions and trading costs. By analyzing these relationships, one can assess how changes in the carbon trading base price and interval length impact both the real carbon emissions and the costs associated with carbon trading.

The proportion of carbon trading costs increases directly with the value of c. As the carbon price rises, the system quickly adjusts, with the aim of minimizing operational expenses by significantly reducing carbon emissions. However, when the price exceeds the actual cost, carbon trading costs will continue to rise steadily with the increase in c. When c reaches or exceeds 0.35 CNY/kg, the operation of system components stabilizes, and carbon emissions drop to their lowest and remain steady. Beyond this point, further increases in c do not affect the system’s behavior but lead to higher overall running costs due to increased carbon trading expenses. In the two-stage robust optimization, the treatment of carbon trading costs needs to take into account the uncertainty of wind and solar resources. In the first stage, policymakers may need to set aside some flexibility to cope with possible future fluctuations in carbon trading costs. In the second stage, when faced with the actual wind and solar resource situation, the operation strategy needs to be adjusted to minimize the overall cost, including the carbon trading cost. The dispatch of wind and solar power generation should be optimized while meeting the needs and stability requirements of the system. Through these integrations, the model is able to optimize the overall performance of the system in the face of uncertainty, while considering the impact of carbon trading costs, thereby achieving the dual goals of economic and environmental benefits.

When the weight of d falls between 2000 kg and 4000 kg, the system needs to purchase additional carbon emission quotas at higher pricing tiers, leading to a spike in carbon trading costs. Despite the system’s prompt adjustment to minimize carbon emissions and reduce operational expenses, this higher pricing impacts overall costs. For weights between 4000 kg and 8000 kg, the system acquires extra quotas at lower prices, resulting in increased carbon emissions but significantly reduced trading costs. When the weight exceeds 8000 kg, the system continues to purchase quotas at the lower end of the carbon price scale, with minimal price escalation and stable carbon emission levels.

Properly adjusting the interval duration and carbon trading benchmark pricing can optimize economic benefits while simultaneously reducing carbon emissions.

4.3. Impact Analysis of the Demand Response Strategy Based on Horizontal Time Shifts and Vertical Complementary Substitutions

The data presented in

Table 1. Optimized scheduling outcomes for each scenario.

Scenario	Total Cost/CNY	${\mathit{F}}_{\mathit{b}\mathit{u}\mathit{y}}$/CNY	${\mathit{F}}_{\mathit{C}{\mathit{O}}_2}$/CNY	${\mathit{F}}_{\mathit{M}\mathit{C}}$/CNY	${\mathit{F}}_{\mathit{D}\mathit{R}}$/CNY	Actual Carbon Emissions/kg
1	19,418	5396	12,904	1118	0	23,807
2	15,658	5429	9114	1116	0	20,728
3	14,327	4751	8475	1019	82	19,450
4	15,051	5060	8819	1055	117	20,138

indicate that Scenario 3 results in a reduction of 1278 kg in carbon emissions compared with Scenario 2, along with a further savings of 1331 CNY in the system’s overall running cost. The demand response outcomes for Scenario 3 are illustrated in Figure 4. For detailed balance diagrams of electric, heat, and gas power in both scenarios, refer to Figure A5 and Figure A6 in Appendix C.

To reduce wind and solar curtailment, demand-side measures such as horizontal time shifts and vertical complementary alternatives are employed. Since electricity is more expensive during the day compared with gas, gas loads can replace some of the electricity and heat loads. Conversely, at night, electric loads can substitute for part of the gas and heat loads. Horizontal time shift demand responses achieve “peak shaving and valley filling” by redistributing load power from peak to lower demand periods. Vertical complementary substitutions allow for flexible consumption options across multiple energy loads. This demand response strategy adjusts the load curve, eases energy supply pressures, and fully leverages the optimization potential in the RIES operation.

4.4. Research on How New Energy Output Uncertainty Adjustment Parameters Affect Scheduling Results

Figure 5 illustrates the deviations between the photovoltaic power generation, wind power generation, and predicted values (${\Gamma }^{PV}$ = ${\Gamma }^{WT}$ = 0) at ${\Gamma }^{PV}$ values of 6 and 12 and ${\Gamma }^{WT}$ values of 12 and 24 under the scenario of 150% photovoltaic power.

Figure 5 illustrates that the most unfavorable scenario occurs at the minimum expected scenario output. When wind and solar outputs hit their lowest values within the specified interval, the RIES shows its highest operational cost, reflecting the “worst” scenario described in the study.

For comparison, three sets of new energy output uncertainty adjustment parameters have been selected. The values of these adjustment parameters and their solution results are detailed in

Table 2. The influence of different uncertainty adjustment parameters on various costs.

Optimization Method		Day-Ahead Cost/CNY	Intraday Cost/CNY	Total Cost/CNY
Deterministic optimization		14,327	4504	18,831
Two-stage robust	${\Gamma }^{WT}=0,{\Gamma }^{PV}=0$	15,051	3439	18,490
optimization	${\Gamma }^{WT}=12,{\Gamma }^{PV}=6$	15,937	2221	18,158
	${\Gamma }^{WT}=24,{\Gamma }^{PV}=12$	16,819	1168	17,987

.

Table 2 shows that as the uncertainty adjustment parameters increase, the system scheduling scheme becomes more robust and conservative, which is reflected in the rising operating costs and economic losses. The model generates scheduling schemes based on the risk preference of the schedulers to ensure the system is sufficiently robust to handle various worst-case scenarios. The day-ahead scheduling cost for the resilient optimization approach with the highest level of uncertainty is 14.82% higher compared with the deterministic optimization scheme. However, during the intraday operation, the system’s consideration of the worst-case scenario results in significant reductions in operational expenses. Consequently, the resilient optimization approach has a lower overall cost compared with the deterministic optimization approach.

The provided image demonstrates that the proposed model effectively reduces the overall system cost and improves economic performance by incorporating real wind and solar output scenarios and selecting appropriate uncertainty measures. This approach ensures the system’s robustness while optimizing cost efficiency.

5. Conclusions

Taking carbon trading and the demand response into account, this work develops a scheduling model aimed at optimizing a low-carbon economy within the context of a RIES. To address the unpredictability of wind and solar production, a two-stage robust optimization strategy is employed. Analysis of the scheduling results reveals the following:

• To enhance economic cooperation with reduced carbon emissions, a stepped carbon trading mechanism is utilized. This involves establishing a rational base price and range for carbon trading. A small trade-off in system resilience is made to improve low-carbon performance, thereby optimizing the balance between system resilience and low-carbon benefits.
• Incorporating the demand response into the stepped carbon trading mechanism improves the RIES’s efficiency in achieving energy savings and emission reduction goals. This approach also alleviates strain on the energy supply to some extent and effectively aligns the economic and low-carbon aspects of the system’s operation.
• By selecting an appropriate level of uncertainty for dispatching, considering the unique characteristics of wind and solar output, the system’s resilience to uncertainty-related risks is enhanced. Dynamically adjusting the level of caution in the scheduling scheme allows for a rational balance between cost-effectiveness and resilience.

This study concentrates exclusively on the uncertainties related to new energy sources and does not address the uncertainties associated with energy demand. Future research will aim to enhance the joint optimization of both the supply and demand sides, facilitating more efficient and stable system operations while promoting low-carbon and cost-effective practices.