Two-Stage Distributed Robust Optimal Allocation of Integrated Energy Systems under Carbon Trading Mechanism

Abstract

The development of renewable energy and the construction of a comprehensive energy system with multiple complementary energy sources have gradually become the main direction of China’s energy development. As the penetration rate of renewable energy increases, the intermittent and fluctuating output of wind and solar power has a more significant impact on the system. This article conducts research on the optimization configuration of integrated energy system (IES) considering photovoltaic output uncertainty under a ladder carbon trading mechanism. Firstly, a two-stage distributed robust optimization (DRO) configuration model for integrated energy system is established. In detail, a deterministic model aimed at minimizing investment costs is given in the first stage and an uncertainty model aimed at minimizing operating costs in the probability distribution of the worst scenario is built in the second stage. Then, a data-driven distributed robust optimization method is adopted to deal with the uncertainty of photovoltaic output using MATLAB software (R2020A). Finally, the column and constraint generation (C&CG) algorithm is used to solve the problem, and the optimal investment capacity and cost results of the integrated energy system considering demand response under a ladder carbon trading mechanism are obtained. Through analysis, the proposed method achieves a 5.54% reduction in carbon emission costs while maintaining nearly unchanged investment costs, thus balancing economic and environmental benefits. Additionally, the CCG algorithm can effectively improve computational efficiency and guarantee the optimality of the solution.

1. Introduction

Currently, the situation of fossil fuel depletion and global climate change is becoming increasingly severe. Fully developing renewable energy has become a trend in future development, which can alleviate the energy crisis and environmental pollution [1,2,3]. However, renewable energy sources such as wind and solar power have the characteristics of strong intermittency and high volatility. The intermittent output of wind and solar power can easily lead to source load imbalance in the energy system. On the other hand, the integration of wind and solar energy has increased the complexity of traditional power system operation and scheduling [4,5]. Therefore, it is urgent to build a clean, efficient, and highly absorptive energy system.

The integrated energy system (IES), as an important carrier for the realization of the energy Internet [6], integrates various links from production to utilization of multiple energy sources, and can conduct comprehensive management and economic scheduling of multiple energy sources, providing an important solution for the realization of comprehensive utilization of energy [7]. Currently, many scholars have conducted research on the optimization and scheduling of integrated energy systems. Reference [8] establishes a low-carbon economic operation model for an integrated energy system of electricity and gas containing carbon capture, demonstrating that carbon capture can effectively enhance the carbon reduction capacity of the integrated energy system. Reference [9] introduces carbon capture technology into the optimization scheduling model of thermal electric joint systems, effectively improving the peak shaving capacity of units and the economic benefits of the system, and promoting the consumption of new energy. However, the above research mainly builds models based on deterministic conditions, without considering the impact of uncertainty in wind and solar power output on system optimization scheduling problems. Reference [10] establishes a low-carbon economic dispatch model for carbon capture and storage (CCS)-electricity to gas (P2G) coordinated operation. In addition to the traditional low-carbon economic dispatch model, the carbon sink problem is also considered. Reference [11] introduces the P2G and CCS coupling model into a park level integrated energy system that includes combined heat and power (CHP) units. Reference [12] utilizes oxygen rich combustion capture technology to low-carbon retrofit traditional gas turbines, achieving coupling between natural gas and wind power hydrogen production systems, improving gas turbine efficiency, and reducing carbon capture energy consumption. Reference [13] provides a detailed modeling of hydrogen and thermal energy in the P2G reaction process, and establishes a two-stage coupled model of P2G, CCS, and gas hydrogen blending. Unfortunately, the above research mainly builds models based on deterministic conditions and does not consider the impact of uncertainty in wind and solar power output on system optimization scheduling problems.

Currently, the commonly used methods for dealing with uncertainty factors are: stochastic optimization [14] and robust optimization [15]. In detail, reference [16] constructed a virtual power plant consisting of photovoltaic and wind turbines, modeled the multiple uncertainties of the virtual power plant based on scenario decision-making, and generated probability density functions for each uncertain parameter. However, the accuracy of the prediction curve is often limited by the modeling of probability distribution models. The more precise the uncertain parameter probability distribution model, the higher the complexity of the model, which ultimately leads to difficulty in obtaining results and consequently affects the reliability of stochastic optimization [17,18]. In contrast, robust optimization solutions are reliable and efficient, and can construct corresponding uncertain sets for solving based on the characteristics of uncertain variables, without the need for accurate modeling of the distribution function of uncertain variables [19]. Therefore, as long as the values of variables are within the set, the robust optimization model can definitely obtain feasible solutions, greatly reducing the difficulty of model solving [20]. Reference [21] constructs a two-stage robust optimization scheduling model for microgrids. By formulating the worst-case daily unit output plan, it avoids serious economic losses caused by supply-demand imbalance on both sides of the source load during the day, and provides good robustness for the system. Reference [22] constructed a two-stage robust optimization scheduling model for a multi energy complementary system considering source load dual uncertainty, achieving a balance between system economy and robustness. Reference [23] proposes a two-stage robust stochastic optimization scheduling method and applies it to a multi energy virtual power plant to avoid overly conservative scheduling schemes and improve model economy. However, the above studies only constructed models from the perspectives of robustness and economy, ignoring the low-carbon operation goals of the system. Because robust optimization scheduling of systems often requires planning more unit output and purchasing electricity in advance to improve risk resistance, the operation of the system will lead to higher carbon emissions. Although carbon capture can absorb the CO₂ generated by the operation of system units [23], it cannot reduce the indirect carbon emissions that the system needs to bear due to trading electricity [24].

In terms of market mechanism, the construction of carbon trading mechanism also provides an effective way for the low-carbon of IES [25]. Reference [26] is based on a two-stage refined model of carbon capture power plants and electricity to gas conversion, introducing a green certificate carbon trading mechanism, which effectively improves the consumption level and system economy of renewable energy. Reference [27] proposed a carbon trading market incentive mechanism for P2G and used lifecycle assessment methods to analyze greenhouse gas emissions from different energy chains in the park, taking into account the environmental and economic aspects of IES. Reference [28] constructed a multi time scale integrated energy system optimization scheduling model considering a stepped carbon trading mechanism, which effectively reduces the system’s carbon emissions while achieving economic efficiency. The carbon trading mechanism trades CO₂ emission rights based on quotas, including carbon trading costs in the total cost, and achieves low-carbon operation of the system through the economic leverage effect. Reference [29] utilizes a two-stage robust optimization model to address source load uncertainty and utilizes a carbon trading mechanism to balance the economic and environmental aspects of the power grid. Reference [30] introduced a carbon trading mechanism into multi energy virtual power plants and analyzed the economic, robust, and low-carbon performance of the model. The above studies have all verified that the introduction of carbon trading mechanisms effectively reduces the indirect carbon emissions of robust optimization models such as microgrids and virtual power plants.

In summary, current research still lacks emphasis on the triangular balance of low-carbon, economic, and robust systems, and most literature lacks analysis of the balance optimization problem among the three. However, introducing a carbon trading mechanism into the robust optimization model of integrated energy systems is of great significance for achieving a balanced optimization of low-carbon, economic, and robust systems. Based on this, this article constructs a comprehensive energy system that adopts a two-stage robust optimization method to deal with the uncertainty of photovoltaics, while introducing a carbon trading mechanism to constrain the carbon emissions of the system, achieving a balance optimization of low-carbon, economic, and robust. Finally, the proposed system model was validated through examples to not only have good robustness and economy, but also effectively maintain the low-carbon operation of the system.

2. Mathematical Model of Equipment in the Integrated Energy System

As shown in Figure 1, the comprehensive energy system constructed in this article includes photovoltaic power generation, combined heat and power (CHP) unit, power to gas (P2G) equipment, carbon capture system (CCS), gas boiler (GB), absorption chillers (AC), and energy storage equipment, thus constructing a comprehensive energy system for combined supply of electricity, heat, and cold. Currently, based on differences in capture principles, capture technologies can be divided into three types: pre-combustion capture, post-combustion capture, and oxygen-enriched combustion technology. Unlike the other two technologies, oxygen-enriched combustion technology first uses oxygen separation technology to separate oxygen from the air. Fossil fuels are then fully burned in the obtained pure oxygen, resulting in only high-concentration CO₂ and water vapor as products. Therefore, the separation effect of CO₂ is better, and capture can be achieved directly. However, the disadvantage of this technology is that the investment cost of the oxygen production process is excessively high, and the energy consumption during the oxygen production process is excessive. Based on the above introduction, it can be seen that these three types of technologies have their own advantages and disadvantages. Among them, post-combustion capture technology is the most mature and relatively economical, with a wider application range suitable for different types of fuel units. Therefore, this article adopts post-combustion capture technology.

(1)

Model of photovoltaic power generation

The output of photovoltaic power generation is mainly affected by two factors: light intensity and environmental temperature. The photovoltaic output can be calculated by Equation (1):

$$P_{PV}=f_{PV}{N}_{PV}{P}_{STC}\frac{G_t}{G_{STC}}\left[1+k\left(T_t-T_{STC}\right)\right]$$

(1)

Here, $P_{PV}$ represents the actual output power of the photovoltaic array; $f_{PV}$ represents the performance degradation parameters of components in photovoltaic power generation; $N_{PV}$ represents the number of photovoltaic modules; $P_{STC}$ represents the maximum output power of a single photovoltaic module under standard testing conditions. The adjustment coefficient $k$ between power and temperature is generally between −0.2% and −0.5%; $G_t$ represents the actual radiation intensity of light at time t; $T_t$ represents the working temperature of the surface of the solar panel at time t.

(2)

Model of P2G equipment

The process of electric to gas conversion is divided into two stages: hydrogen generation and methanation. In the IES, hydrogen can serve as a supplementary energy source, working in tandem with other energy forms to enhance the system’s flexibility and reliability. As a storable energy form, hydrogen can provide supplementation when renewable energy sources are insufficient, contributing to the diversification and stability of energy supply. In the first stage, the electrolytic cell (EL) electrolyzes hydrogen from aquatic products and input its hydrogen gas into the methane reactor (MR) in the second stage, which is supplied to the gas thermal power unit and gas network. In addition, considering the operational flexibility of the P2G process, hydrogen storage tanks and hydrogen fuel cells (HFCs) are installed to store and reuse hydrogen energy that cannot be absorbed by MR. Hydrogen fuel cells can directly convert hydrogen energy into electrical and thermal energy. The model of EL is shown in (2).

$$\{\begin{array}{c}{P}_{EL,t}={\eta }_{H2}{P}_{P2G,t}\\ P_{P2G}^{\min }\le P_{P2G,t}\le P_{P2G}^{\max }\\ \Delta P_{P2G}^{\min }\le P_{P2G,t+1}-P_{P2G,t}\le \Delta P_{P2G}^{\max }\end{array}$$

(2)

Here, $P_{EL,t}$ represents the hydrogen power output by EL during time period t; ${\eta }_{H2}$ represents the hydrogen conversion efficiency of EL; $P_{P2G,t}$ represents the electrical energy consumed by EL during time t; $P_{P2G,t}^{\max }$ and $P_{P2G}^{\min }$ are the upper and lower power limits of EL, respectively; $\Delta P_{P2G}^{\max }$ and $\Delta P_{P2G}^{\min }$ are the upper and lower power climbing limits for EL. Then, the model of MR can be described by (3).

$$\{\begin{array}{c}{P}_{MR,t}={\beta }_{MR}{P}_{MR,t}^{EL}\\ T_{MR,t}^{C{O}_2}={\displaystyle \sum _t^T\chi P_{MR,t}}\\ P_{MR,\min }^{EL}\le P_{MR,t}^{EL}\le P_{MR,\max }^{EL}\\ \Delta P_{MR,\min }^{EL}\le P_{MR,t+1}^{EL}-P_{MR,t}^{EL}\le \Delta P_{MR,\max }^{EL}\end{array}$$

(3)

Here, $P_{MR,t}$ is the gas power converted by MR during time period t; ${\beta }_{MR}$ is the conversion rate of MR; $P_{MR,t}^{EL}$ is the hydrogen power input from EL into MR during time period t; $T_{MR,t}^{C{O}_2}$ is the amount of carbon dioxide required during the methanation process in MR; χ represents the carbon dioxide conversion coefficient; $P_{MR,\max }^{EL}$ and $P_{MR,\min }^{EL}$ are the upper and lower limits of hydrogen power input to the MR, respectively; $\Delta P_{MR,\max }^{EL}$ and $\Delta P_{MR,\min }^{EL}$ are the upper and lower climbing limit of MR. Finally, the model of HFC can be described as (4).

$$\{\begin{array}{c}{P}_{H,t}^e={\eta }_{HFC}^e{P}_{HFC,t}^{EL}\\ P_{H,t}^h={\eta }_{HFC}^h{P}_{HFC,t}^{EL}\\ k_{HFC}^{\min }\le P_{H,t}^h/P_{H,t}^e\le k_{HFC}^{\max }\\ \begin{array}{c}{P}_{HFC,\min }^{EL}\le P_{HFC,t}^{EL}\le P_{HFC,\max }^{EL}\\ \Delta P_{HFC,\min }^{EL}\le P_{HFC,t+1}^{EL}-P_{HFC,t}^{EL}\le \Delta P_{HFC,\max }^{EL}\end{array}\end{array}$$

(4)

Here, $P_{H,t}^e$ and $P_{H,t}^h$ are the electrical and thermal power of HFC during period t; ${\eta }_{HFC}^e$ and ${\eta }_{HFC}^h$ are the output efficiency of HFC’s electrical and thermal power; $P_{HFC,t}^{EL}$ is the hydrogen power input from EL into HFC during time period t; $k_{HFC}^{\max }$ and $k_{HFC}^{\min }$ are the upper and lower limits of the thermoelectric ratio of HFC; $P_{HFC,\max }^{EL}$ and $P_{HFC,\min }^{EL}$ are the upper and lower limits of hydrogen power input to HFC, respectively; $\Delta P_{HFC,\max }^{EL}$ and $\Delta P_{HFC,\min }^{EL}$ represent the upper and lower climbing limit of HFC.

(3)

Model of CHP-CCS-P2G joint operation

The carbon flow process in the CHP-CCS-P2G system can be expressed as (5):

$$\{\begin{array}{c}{T}_{CHP,t}+T_{MT,t}=T_t^{out}+T_{cc,t}\\ T_{cc,t}=T_{cc,t}^c+T_{P2G,t}\\ T_{cc,t}^c\le T_{cc,\max }^c\end{array}$$

(5)

Here, $T_{CHP,t}$ and $T_{MT,t}$ respectively represent the carbon dioxide generated by cogeneration units and micro gas turbines; $T_t^{out}$ represents the amount of carbon dioxide directly emitted into the atmosphere; $T_{cc,t}$ represents the amount of carbon dioxide captured by CCS; $T_{cc,t}^c$ is the amount of carbon dioxide stored; $T_{cc,\max }^c$ is the maximum storage capacity; $T_{P2G,t}$ is the carbon dioxide consumed by P2G.

The electricity consumption of the CHP-CCS-P2G system during carbon capture and utilization is as follows, with a portion of photovoltaic power used for input into the grid, and the remaining portion used for energy consumption of carbon capture equipment and electric to gas conversion equipment, as shown in (6).

$$\{\begin{array}{c}{P}_{CCP,t}=P_{CC,t}+P_{P2G,t}+P_{CHP,t}\\ P_{CC,t}+P_{P2G,t}=P_{PV,t}^{cp}+P_{G,t}^{cp}\\ P_{PV,t}=P_{PV,t}^{cp}+P_{PV,t}^g\end{array}$$

(6)

Here, $P_{CCP,t}$ is the energy consumption of the CHP-CCS-P2G system during time period t; $P_{CC,t}$ is the electricity consumption of carbon capture during time period t; $P_{P2G,t}$ is the electricity consumption of the P2G process during time period t; $P_{CHP,t}$ is the electricity consumption of the cogeneration unit during time period t; $P_{G,t}^{cp}$ is the grid purchasing power of P2G and CCS during time period t when the photovoltaic power is insufficient. $P_{PV,t}^{cp}$ represents the energy consumption provided by photovoltaic power for CCS-P2G equipment during time period t; $P_{PV,t}^g$ is the grid connected power of photovoltaic power during time t.

The expression for the CHP unit model is (7):

$$\{\begin{array}{c}{P}_{CHP,t}=P_{CHP,t}^e+P_{CHP,t}^h\\ P_{CHP,t}^e=V_{CHP,t}{H}^{C{H}_4}{\eta }_{CHP}^e\\ P_{CHP,t}^h=V_{CHP,t}{H}^{C{H}_4}{\eta }_{CHP}^h\\ \begin{array}{c}{P}_{CHP,\min }^e\le P_{CHP,t}^e\le P_{CHP,\max }^e\\ P_{CHP,\min }^h\le P_{CHP,t}^h\le P_{CHP,\max }^h\end{array}\end{array}$$

(7)

Here, $P_{CHP,t}$ is the total power output of CHP unit during time period t; $P_{CHP,t}^e$ and $P_{CHP,t}^h$ are the electrical and thermal power of the CHP unit during time t; $V_{CHP,t}$ is the natural gas consumption of the CHP unit during time t; ${\eta }_{CHP}^e$ and ${\eta }_{CHP}^h$ are the electrical efficiency and thermal efficiency of the CHP unit, respectively; $P_{CHP,\min }^e$ and $P_{CHP,\max }^e$ are the lower and upper electrical power limits of CHP unit, respectively; $P_{CHP,\min }^h$ and $P_{CHP,\max }^h$ are the lower and upper thermal power limits of CHP, respectively.

(4)

Model of gas boiler

The mathematical expression of gas boiler is shown in (8):

$$\{\begin{array}{c}{H}_t^{\mathrm{GB}}={\eta }^{\mathrm{GB}}{G}_t^{\mathrm{GB}}\\ 0\le H_t^{\mathrm{GB}}\le H_{\max }^{\mathrm{GB}}\end{array}$$

(8)

Here, $H_t^{\mathrm{GB}}$ represents the thermal power output of the gas boiler at time t; $G_t^{\mathrm{GB}}$ represents the amount of natural gas consumed by the gas boiler at time t; ${\eta }^{\mathrm{GB}}$ represents heating efficiency; $H_{\max }^{\mathrm{GB}}$ represents the upper limit value of the thermal power output of the gas boiler.

(5)

Model of absorption chiller

The working principle of an absorption chiller is to directly use the waste heat of the CHP unit and gas boiler as the driving source to achieve refrigeration, thereby realizing the transformation of thermal energy to cold energy and energy cascade utilization, and reducing waste heat emissions. The specific mathematical model is represented as (9):

$$\{\begin{array}{c}{C}_t^{\mathrm{AC}}={\eta }^{\mathrm{AC}}{H}_t^{\mathrm{AC}}\\ 0\le C_t^{\mathrm{AC}}\le C_{\max }^{\mathrm{AC}}\end{array}$$

(9)

Here, $C_t^{\mathrm{AC}}$ represents the cooling power of an absorption chiller; ${\eta }^{\mathrm{AC}}$ represents the refrigeration efficiency of an absorption chiller; $H_t^{\mathrm{AC}}$ represents the input thermal power of an absorption chiller; $C_{\max }^{\mathrm{AC}}$ represents the upper limit cooling power of the absorption chiller.

(6)

Model of energy storage equipment

Battery are the most widely used form of electrical storage equipment today, due to their mature technology, ease of maintenance, flexibility in configuration, and small footprint. Therefore, the electrical storage equipment utilized in this IES is battery. The battery stores and releases electrical energy through the mutual conversion between chemical energy and electrical energy. The expression for the charging and discharging states is shown in (10):

$$\{\begin{array}{c}{S}_t^{\mathrm{bat}}=S_{t-1}^{\mathrm{bat}}·(1-\mu )+(P_t^{\mathrm{bat},\mathrm{cha}}{\eta }_{\mathrm{cha}}-\frac{P_t^{\mathrm{bat},\mathrm{dis}}}{{\eta }_{\mathrm{dis}}})·\Delta t\\ S_0^{\mathrm{bat}}=S_T^{\mathrm{bat}}\end{array}$$

(10)

Here, $S_t^{\mathrm{bat}}$ represents the real-time electricity amount stored in the battery; ${\eta }_{\mathrm{cha}}$ and ${\eta }_{\mathrm{dis}}$ represent the charging and discharging efficiency of the battery separately $P_t^{\mathrm{bat},\mathrm{cha}}$ and $P_t^{\mathrm{bat},\mathrm{dis}}$ represent the charging and discharging power of the battery at time t respectively; $\mu$ represents the self-discharge efficiency of the battery; $\Delta t$ represents the time interval. At the end of a scheduling cycle T, due to the continuity of the battery state, the initial state of the energy storage device at the next moment is equal to the final state of the previous moment.

(7)

Model of demand response

Demand response refers to users actively changing demand side loads based on price signals or incentive measures, ensuring stable operation of the power grid, and improving the consumption capacity of renewable energy. In this article, the demand response resources mainly considered are transferable loads and reducible loads, and the specific mathematical expressions are shown in Equations (11) and (12).

$$\{\begin{array}{c}\begin{array}{l}0\le P_{t,k,in}^{\mathrm{SL}}\le {\mu }_{t,k,in}\left(P_{t-1,k,in,\max }^{\mathrm{SL}}-P_{t-1,k,in}^{\mathrm{SL}}\right)\\ 0\le P_{t,k,out}^{\mathrm{SL}}\le {\mu }_{t,k,out}\left(P_{t-1,k,out,\max }^{\mathrm{SL}}-P_{t-1,k,out}^{\mathrm{SL}}\right)\\ {\mu }_{t,k,in}{\mu }_{t,k,out}=0\end{array}\\ \begin{array}{l}{\displaystyle \sum _t\left({\mu }_{t,k,in}+{\mu }_{t,k,out}\right)}=0,t\notin \left[t_{k,sat}^{\mathrm{SL}},t_{k,end}^{\mathrm{SL}}\right]\\ {\displaystyle \sum _{t=1}^T\left({\mu }_{t,k,in}{P}_{t,k,in}^{\mathrm{SL}}-{\mu }_{t,k,out}{P}_{t,k,out}^{\mathrm{SL}}\right)=0}\end{array}\end{array}$$

(11)

$$\{\begin{array}{l}0\le P_{t,k}^{\mathrm{CL}}\le {\mu }_{t,k}^{\mathrm{CL}}{P}_{t,k,\max }^{\mathrm{CL}}\\ {\displaystyle \sum _{t=1}^T\left({\mu }_{t,k}^{\mathrm{CL}}-{\mu }_{t-1,k}^{\mathrm{CL}}\right){\mu }_{t,k}^{\mathrm{CL}}=n_{t,k}^{\mathrm{CL}}}\\ {\displaystyle \sum _t{\mu }_{t,k}^{\mathrm{CL}}}=0,t\notin \left[t_{k,sat}^{\mathrm{SL}},t_{k,end}^{\mathrm{SL}}\right]\\ 0\le n_{t,k}^{\mathrm{CL}}\le n_{t,k,\max }^{\mathrm{CL}}\\ T_{k,\min }^{\mathrm{SL}}\le T_k^{\mathrm{SL}}\le T_{k,\max }^{\mathrm{SL}}\end{array}$$

(12)

Here, $P_{t,k,in}^{\mathrm{SL}}$ and $P_{t,k,out}^{\mathrm{SL}}$ respectively represent the equivalent charging and discharging power of the k-th transferable load at time t; ${\mu }_{t,k,in}$ and ${\mu }_{t,k,out}$ are 0–1 variables and represent the equivalent charging and discharging operating states of the k-th transferable load at time t, respectively; $P_{t-1,k,in,\max }^{\mathrm{SL}}$ and $P_{t-1,k,out,\max }^{\mathrm{SL}}$ represent the upper limits of equivalent charging and discharging power for the k-th transferable load at time t − 1, respectively; $t_{k,sat}^{\mathrm{SL}}$ and $t_{k,end}^{\mathrm{SL}}$ represent the start and end scheduling moments for the transferable load, respectively. $P_{t,k}^{\mathrm{CL}}$ represents the reducible power of the k-th load at time t; ${\mu }_{t,k}^{\mathrm{CL}}$ is 0–1 variable and represents whether to call for load reduction at time t. $P_{t,k,\max }^{\mathrm{CL}}$ represents the upper reducible power limit of the k-th load at time t. $t_{k,sat}^{\mathrm{SL}}$ and $t_{k,end}^{\mathrm{SL}}$ represent the start and end scheduling time to carry out load reduction, respectively. $n_{t,k}^{\mathrm{CL}}$ and $n_{t,k,\max }^{\mathrm{CL}}$ represent the number of demand response that can reduce load power during the scheduling cycle and the maximum number of demand response. $T_{k,\max }^{\mathrm{SL}}$ and $T_{k,\min }^{\mathrm{SL}}$ represent the upper and lower duration limits of the single demand response for the k-th type of load reduction, respectively.

3. A Two-Stage Optimization Configuration Model for Integrated Energy Systems

3.1. Introduction of Ladder Carbon Trading Mechanism

The carbon trading cost of IES is equal to the difference between the system’s carbon emissions and the initial carbon emission quota, and its value can be positive or negative. It mainly represents the cost expenditure or benefit generated when there is insufficient or excess carbon emission quota during system operation, and it is necessary to purchase or sell carbon emission quotas. Therefore, the carbon trading cost model is shown in Equation (13):

$$C_{{\mathrm{CO}}_2}={\displaystyle \sum _{t=1}^T\rho }\left(E_t-E_c\right)$$

(13)

Here, $C_{{\mathrm{CO}}_2}$ represents the cost of carbon trading; $\rho$ represents the market unit carbon trading price; $E_c$ represents initial quota for system carbon trading; $E_t$ represents actual system carbon emissions. To further reduce the total carbon emissions, a ladder carbon trading model is established to divide the difference between carbon emissions and carbon emission quotas into different intervals corresponding to different carbon trading prices. The diagram of ladder carbon trading model is shown in Figure 2 and corresponding mathematical model is shown in (14).

$$C_{{\mathrm{CO}}_2}=\{\begin{array}{c}-\rho \left(1+\zeta \right)\left(E_t-E_c-\delta \right),E_t-E_c<-\delta \\ -\rho \left(1+2\zeta \right)\delta -\rho \left(1+\varsigma \right)\left(E_t-E_c\right),-\delta <E_t-E_c<0\\ \rho \left(E_t-E_c\right),0<E_t-E_c<+\delta \\ \rho \delta +\rho \left(1+\sigma \right)\left(E_t-E_c-\delta \right),\delta <E_t-E_c<2\delta \\ \rho \left(2+\sigma \right)\delta +\rho \left(1+2\sigma \right)\left(E_t-E_c-2\delta \right),2\delta <E_t-E_c<3\delta \\ ⋮\\ \rho \left[\left(n-1\right)+\frac{\left(n-1\right)\left(n-2\right)}{2}\sigma \right]\delta +\rho \left[1+\left(n-1\right)\sigma \right]\left[E_t-E_c-\left(n-1\right)\delta \right]\\ ,\left(n-1\right)\delta <E_t-E_c<n\delta \end{array}$$

(14)

Here, $\zeta$ represents the reward coefficient; $\delta$ represents the step size of carbon emission interval; $\sigma$ represents price growth ratio.

3.2. Objective Function and Constraints in the First Stage

The investment cost of various equipment within the integrated energy system is the center of attention in the first stage, which optimizes the capacity configuration of energy conversion equipment and energy storage equipment within the system, and then solves for the minimum investment cost. The objective function for this stage is to minimize the total daily construction cost of CHP-CCS-P2G combined operation cogeneration units, gas boiler (GB), absorption chillers (AC), and energy storage equipment. The expression is shown in Equation (15):

$$\min C_{s1}=\frac{1}{365}{\displaystyle \sum _{i\in {\Omega }_i}\frac{{\left(1+a\right)}^{t_i}{M}_i{c}_i^{\mathrm{inv}}}{\left[{\left(1+a\right)}^{t_i}-1\right]}}$$

(15)

Here, $C_{s1}$ is the objective function of the first stage and represents the daily average investment cost of the equipment; $t_i$ represents the planned service life of device i; $c_i^{\mathrm{inv}}$ represents the cost of constructing unit capacity equipment; $M_i$ represents the construction capacity of each device.

The relevant constraint is shown in (16):

$$0\le M_i\le M_{i,\max }$$

(16)

Here, $M_{i,\max }$ represents the upper limit value of the investment capacity of each device.

3.3. Objective Function and Constraints in the Second Stage

The second stage model takes the minimum daily operating cost of the system as the objective function, and optimizes the operating cost of the system in the worst scenario based on the optimal configuration capacity of each device obtained in the first stage. The daily operating cost includes the cost of purchasing energy from external sources, demand response cost, depreciation cost of energy storage, carbon trading cost, and total operating cost of CHP-CCS-P2G joint operation. The expression is shown in Equation (17):

$$\begin{array}{l}\min C_{s2}=C^{\mathrm{buy}}+C^{\mathrm{dr}}+C^{\mathrm{bat}}+C^{{\mathrm{CO}}_2}+C^{\mathrm{CHP}}\\ \{\begin{array}{c}{C}^{\mathrm{buy}}={\displaystyle \sum _{t=1}^T\left(u_t^{\mathrm{buy},\mathrm{gd}}{P}_t^{\mathrm{buy},\mathrm{gd}}+u^{\mathrm{gas}}{G}_t^{\mathrm{gas}}\right)}\\ C^{\mathrm{dr}}={\displaystyle \sum _{t=1}^T\left(c_e^{\mathrm{SL}}{P}_{t,e}^{SL}+c_h^{\mathrm{SL}}{P}_{t,h}^{SL}+c_c^{\mathrm{SL}}{P}_{t,c}^{SL}+c_e^{\mathrm{CL}}{P}_{t,e}^{CL}\right)}\\ C^{\mathrm{bat}}={\displaystyle \sum _{t=1}^T\left({\lambda }_{\mathrm{cha}}{P}_t^{\mathrm{bat},\mathrm{cha}}+{\lambda }_{\mathrm{dis}}{P}_t^{\mathrm{bat},\mathrm{dis}}\right)}\\ C^{\mathrm{CHP}}={\displaystyle \sum _{t=1}^T\left[a_1\left(P_t^{\mathrm{CHP}}+P_t^{\mathrm{CCS}}+P_t^{\mathrm{P}2\mathrm{G}}\right)+b_1{P}_t^{\mathrm{CCS}}+c_1{P}_t^{\mathrm{P}2\mathrm{G}}\right]}\end{array}\end{array}$$

(17)

Here, $C_{s2}$ represents the objective function of the second stage, which is the daily operating cost of the system; $C^{\mathrm{buy}}$ represents the cost of purchasing energy from the power grid; $C^{\mathrm{dr}}$ represents the cost of demand response; $C^{\mathrm{bat}}$ represents the depreciation cost of energy storage equipment; $C^{{\mathrm{CO}}_2}$ represents the carbon trading cost; $C^{\mathrm{CHP}}$ represents the total operating cost of CHP-CCS-P2G. $u_t^{\mathrm{buy},\mathrm{gd}}$ and $u^{\mathrm{gas}}$ respectively represent the real-time prices for purchasing electricity and gas in the system. $P_t^{\mathrm{buy},\mathrm{gd}}$ and $G_t^{\mathrm{gas}}$ represent the amount of electricity and gas purchased by the system from the external network, respectively; T represents the scheduling period. $c_e^{\mathrm{SL}}$, $c_h^{\mathrm{SL}}$, $c_c^{\mathrm{SL}}$, $c_e^{\mathrm{CL}}$ respectively represent the compensation prices for transferable electricity, heating, cooling loads, and reducible electricity loads; $P_{t,e}^{\mathrm{SL}}$, $P_{t,h}^{\mathrm{SL}}$, $P_{t,c}^{\mathrm{SL}}$ and $P_{t,e}^{\mathrm{CL}}$ represent the power that can transfer electricity, heat, and cooling loads and reduce electricity loads at time t; ${\lambda }_{\mathrm{cha}}$ and ${\lambda }_{\mathrm{dis}}$ respectively represent the depreciation coefficients for charging and discharging; $a_1$, $b_1$ and $c_1$ represent the operating cost coefficients for CHP units, CCS, and P2G equipment, respectively.

In the second stage, in addition to the operational constraints (1)–(12) mentioned earlier, other conventional constraints need to be met, such as power balance constraints for electricity, heat, and cold, and interaction constraints between the system and external networks, which will not be discussed in detail here.

4. Solving Algorithm of the Two Stage DRO Model

Firstly, this paper adopts the data-driven optimization approach to generate typical uncertainty scenarios, which is based on actual historical data, thus establishing uncertainty sets under the constraints of 1-norm and ∞-norm, respectively. Compared with traditional methods, this approach can effectively utilize specific historical data without the need for dualization, resulting in simpler solutions and greater advantages in addressing the uncertainty of renewable energy generation. Photovoltaic power generation is influenced by factors such as light intensity and ambient temperature, exhibiting certain spatial and temporal distribution characteristics that are difficult to describe using models. To address the uncertainty of photovoltaic power output, we adopt a DRO approach that combines the probabilistic thinking of stochastic optimization with the worst-case scenario optimization strategy of robust optimization. This method combines the advantages of both and performs better in terms of solution time, economy, and robustness. Compared to stochastic optimization and robust optimization, this approach significantly reduces the requirement for probabilistic distribution information on uncertainty factors and significantly improves conservativeness. In this chapter, based on the initial probability distribution values of photovoltaic power output scenarios, we obtain an uncertainty set for photovoltaic power output using the comprehensive norm as a constraint. Utilizing the K-means clustering algorithm, we select s discrete scenario values from historical sample data to represent possible values of photovoltaic power output, solving the problem of difficulty in obtaining uncertainty sets.

The optimization variables mentioned in the previous text are divided into two stages of variables. The decision variable x in the first stage includes the construction plan of CHP units, CCS equipment, P2G equipment, gas boilers, electric refrigeration units, absorption refrigeration units, and energy storage equipment. The decision variable y in the second stage includes the actual output, carbon emissions, and purchased electricity of the equipment. Considering the uncertainty of photovoltaics, a generalized two-stage distributed robust optimization configuration model is constructed as follows:

$$\begin{array}{l}\underset{x\in X}{\min } {\mathit{a}}^{\mathrm{T}}\mathit{x}+\underset{P\left(\xi \right)\in \Omega }{\max }{E}_P\left({\mathit{b}}^{\mathrm{T}}\mathit{y}+{\mathit{c}}^{\mathrm{T}}\mathit{\xi }\right)\\ s.t.\{\begin{array}{c}\mathit{A}\mathit{x}\le \mathit{d}\\ \mathit{B}\mathit{y}\le \mathit{C}\mathit{\xi }\\ \mathit{D}\mathit{y}\le \mathit{d}\\ \begin{array}{c}\mathit{E}\mathit{y}=\mathit{e}\\ \mathit{G}\mathit{x}+\mathit{H}\mathit{y}\le \mathit{g}\\ \mathit{J}\mathit{x}+\mathit{K}y=\mathit{h}\end{array}\end{array}\end{array}$$

(18)

Here, $\mathit{\xi }$ represents the predicted photovoltaic output vector; $P\left(\mathit{\xi }\right)$ represents the probability distribution function of photovoltaic output; $E_P$ represents the expected value of the probability distribution function; ${\mathit{a}}^{\mathrm{T}}\mathit{x}$ represents the investment cost of equipment in the IES; ${\mathit{b}}^{\mathrm{T}}\mathit{y}+{\mathit{c}}^{\mathrm{T}}\mathit{\xi }$ represents the operating cost of the second stage. Furthermore, with the help of the uncertain set construction method mentioned in reference [23], the objective function in (18) can be rewritten as (19):

$$\underset{x\in X}{\min } {\mathit{a}}^{\mathrm{T}}\mathit{x}+\underset{\left\{p_s\right\}\in {\Omega }_s}{\max }{\displaystyle \sum _{s=1}^S{p}_s}\underset{y_s\in Y\left(x,{\xi }_s\right)}{\min }\left({\mathit{b}}^{\mathrm{T}}{\mathit{y}}_s+{\mathit{c}}^{\mathrm{T}}{\mathit{\xi }}_s\right)$$

(19)

A two-stage DRO configuration model in the form of min-max-min is established for (19). The C&CG algorithm is used to divide the model into a master problem (MP) and a sub problem (SP), and iterative solutions are performed. The MP is to obtain the optimal solution considering the probability distribution of the worst-case scenario, and transmit it to the SP. The optimization result is taken as the lower bound of (19):

$$\begin{array}{l}\left(\mathrm{MP}\right)\underset{x\in X,y_s\in Y\left(x,{\xi }_s\right),R}{\min }{\mathit{a}}^{\mathrm{T}}\mathit{x}+R\\ R\ge {\displaystyle \sum _{s=1}^S{p}_s^k\left({\mathit{b}}^{\mathrm{T}}{\mathit{y}}_s^k+{\mathit{c}}^{\mathrm{T}}{\mathit{\xi }}_s\right)},\forall k=1,2,\cdots ,n\end{array}$$

(20)

Assuming the variables ${\mathit{x}}^{*}$ in the first stage are given, the SP obtains the worst-case probability distribution, which is then returned to the MP for iterative solution. The SP provides an upper bound for (19), and the expression for the SP is obtained as shown in (21):

$$\left(\mathrm{SP}\right)R\left(x^{*}\right)=\underset{\left\{p_s\right\}\in {\Omega }_s}{\max }{\displaystyle \sum _{s=1}^S{p}_s}\underset{y_s\in Y\left(x,{\xi }_s\right)}{\min }\left({\mathit{b}}^{\mathrm{T}}{\mathit{y}}_s+{\mathit{c}}^{\mathrm{T}}{\mathit{\xi }}_s\right)$$

(21)

Due to the independence of each scenario in the SP, the probabilities in each scenario $p_s$ are independent of the decision variables in the inner layer. Parallel solving methods can be used simultaneously to save solving time. Therefore, original model (19) can be rewritten as (22) and (23):

$$R_{s1}=\underset{y_s\in Y\left(x,{\xi }_s\right)}{\min }\left({\mathit{b}}^{\mathrm{T}}{\mathit{y}}_s+{\mathit{c}}^{\mathrm{T}}{\mathit{\xi }}_s\right)$$

(22)

$$R_{s2}=\underset{\left\{p_s\right\}\in {\Omega }_s}{\max }{\displaystyle \sum _{s=1}^S{p}_s}{R}_{s1}$$

(23)

By iteratively solving the main problem and sub problems, the iteration is stopped until it converges to the given accuracy value. Overall, employing the CCG algorithm to solve DRO problems is an effective approach. As outlined in the previous steps, the CCG algorithm is an iterative method that iteratively adds new columns (i.e., decision variables) and constraints to approximate the optimal solution. This approach is particularly suitable for tackling DRO problems with a large number of potential uncertainty scenarios. In the context of DRO, the core idea of the CCG algorithm is to iteratively incorporate uncertainty scenarios into the optimization problem to gradually approximate the true optimal solution. This typically involves two main steps: column generation and constraint generation. The column generation step expands the solution space of the problem by adding new decision variables to explore potentially better solutions. The constraint generation step, on the other hand, restricts the solution space by introducing new constraints to ensure the feasibility of the solution. Through multiple iterations, the CCG algorithm gradually improves the quality of the solution until it meets the predefined convergence criteria or reaches the maximum number of iterations. This method effectively balances the accuracy and computational efficiency of the solution when dealing with complex DRO problems.

5. Case Study

This paper uses the YALMIP toolbox in the MATLAB platform to establish the mathematical model of the IES, and then adopts the commercial solver CPLEX to solve the two-stage DRO model in MATLAB (R2020A). This section takes a typical day of IES in northern China as an case, with a planning period of 24 h and a time interval of 1 h. The time of use electricity price and natural gas price are shown in

Table 1. The time of use electricity price and natural gas price.

Type	Period of Time/h	Price/(Yuan/kWh)
Electricity price	9:00–12:00, 14:00–18:00, 20:00–23:00	1.25
	12:00–14:00, 18:00–19:00, 23:00–1:00	0.75
	1:00–9:00	0.42
Natural gas price	0:00–24:00	3.5

. The relevant parameters of various equipment are shown in

Table 2. Related parameters of equipment.

Equipment	Construction Unit Price/(Yuan/kW)	Maintenance Price/(Yuan/kW)	Life Cycle/(Year)
CHP	6500	0.04	20
P2G	2700	0.015	30
CCS	2300	0.03	35
GB	800	0.02	35
EC	2600	0.03	25
AC	700	0.05	20
Photovoltaic power generation	8000	0.05	25

, and the technical parameters are shown in

Table 3. The specific technical parameters of each equipment.

Name	Parameter Name	Value	Upper Limit of Installed Capacity/(kW)
CHP	Power generation efficiency	0.37	5250
	Gas-to-heat conversion efficiency	0.29
P2G	Electricity-to-natural gas coefficient	0.41	650
CCS	Carbon capture coefficient	0.3	650
GB	Gas-to-heat conversion efficiency	0.9	580
EC	Electricity-to-cooling conversion coefficient	3.85	3450
AC	Cooling coefficient	1.05	860

. The power curves of different energy demands is given in Figure 3. It should be noted that only the GB produces thermal power, while the remaining equipment generates electrical power. The corresponding cooling/heating power can be calculated based on the conversion efficiency given in Table 3.

5.1. Analysis of Optimized Configuration Results

The capacity configuration results of each equipment are shown in

Table 4. Capacity configuration results of equipment.

Equipment	Capacity/(kW)
CHP	1658.4
P2G	203.2
CCS	97.3
GB	0
EC	840.2
AC	0
Photovoltaic power generation	1800

. It can be seen that the configuration capacity of gas boilers and AC equipment is 0. This is because in the ladder carbon trading market environment, the operating cost of gas boilers is higher than that of CHP units, and the weighted cost of carbon emissions is also higher than that of CHP units. The configured CHP units can already meet the system’s heat load demand. In addition, the use of gas boilers not only increases operating costs but also emits excessive carbon dioxide, resulting in an increase in carbon trading costs. The configured EC equipment can fully meet the cooling load requirements of the system, without the need to configure AC equipment.

Table 5. Results of various costs and carbon emissions.

Type	Value
Investment cost/(yuan)	45,893.6
Operating cost/(yuan)	46,845.2
Energy purchase cost/(yuan)	39,744.6
Demand response compensation/(yuan)	841.5
Energy storage depreciation cost/(yuan)	72.6
Total operating cost of CHP/(yuan)	710.8
Carbon trading cost/(yuan)	1476.3
Carbon emissions/(t)	11,456.1

shows the results of various costs and carbon emissions of the integrated energy system. The total cost of the integrated energy system in the worst scenario is 147,040.7 yuan. The carbon emissions of the IES are mainly generated by the CHP unit, gas boiler, and electricity purchased from the upper-level grid during the joint operation of CHP-CCS-P2G. Under this planning scheme, the total carbon emissions of the energy system amount to 11,456.1 tons. Specifically, the CHP unit generates 6542.7 tons of CO₂, the gas boiler produces 3456.8 tons of CO₂, and electricity purchased from the upper-level grid contributes to 1456.6 tons of CO₂. In the case study, due to the consideration of the multi-energy flow power balance constraint, the energy quantities provided by each equipment are equal to the demand quantities shown in Figure 3. This is because the solution obtained through the CCG algorithm is globally optimal, enabling the satisfaction of the power balance constraints.

5.2. Comparative Analysis of Different Scenarios

In order to further investigate the superiority of the two-stage distributed robust optimization model considering demand response under a stepped carbon trading mechanism in this chapter, the following five different scenarios are set up in this section:

Scenario 1: A deterministic optimization model considering demand response and a ladder carbon trading mechanism;

Scenario 2: A two-stage distributed robust optimization model considering demand response and traditional carbon trading mechanisms;

Scenario 3: A two-stage distributed robust optimization model considering demand response, but without considering a ladder carbon trading mechanism;

Scenario 4: A two-stage distributed robust optimization model considering a ladder carbon trading mechanism, but without considering demand response;

Scenario 5: A two-stage distributed robust optimization model considering demand response and a ladder carbon trading mechanism.

As shown in Figure 4 and Figure 5, compared with the traditional carbon trading mechanism in scenario 2, scenario 5 adopts a ladder carbon trading mechanism, which increases the configuration capacity of CHP units, increases the daily total cost by 1.9%, reduces carbon emissions by 5.54%, and increases carbon trading costs. The reason is that the actual carbon emissions are usually greater than the carbon emission quota. Scenario 2 adopts a traditional carbon trading mechanism with a constant carbon trading price, which has limited ability to limit carbon emissions under high carbon emissions, after considering the ladder carbon trading mechanism, the IES will further reduce carbon emissions and improve energy-saving and emission reduction technologies under the enormous economic pressure of carbon trading costs.

Compared with scenario 5, scenario 3 shows an increase in both investment and operating costs after considering the tiered carbon trading mechanism. The total cost of Scenario 5 has increased by 6.46%, while carbon emissions have decreased by 15.7%. This is because, without considering the carbon trading mechanism, the system is no longer equipped with P2G and CCS equipment due to economic considerations. Although the total cost of the system has increased after considering the tiered carbon trading mechanism, carbon emissions have significantly decreased. The system chooses to allocate clean equipment with lower carbon emissions under the economic pressure of carbon trading costs to achieve energy conservation and emission reduction as much as possible, which is in line with the direction of energy development.

Comparing scenario 4 and scenario 5, it can be seen that considering demand response, scenario 5 has fewer device capacities compared to scenario 4, indicating that demand response has limited response to extreme time periods. Scenario 5 considers that the various costs and carbon emissions after demand response have been reduced compared to scenario 4, indicating that considering demand response can save the cost of the comprehensive energy system while ensuring the low-carbon nature of the system.

On the other hand, the distributed robust optimization model proposed in this article can reduce the conservatism of traditional robust optimization. In order to demonstrate the advantages of the proposed model more intuitively, this article uses distributed robust optimization, stochastic optimization, and traditional robust optimization methods to optimize the capacity allocation of IES. The relevant calculation results are shown in

Table 6. Analysis and Comparison of IES with Different Optimization Methods.

Method	Total Cost/(Yuan)	Investment Cost/(Yuan)	Operating Cost/(Yuan)
Stochastic optimization	91,677.2	45,102.6	46,587.9
Robust optimization	92,456.8	46,756.3	44,345.6
The proposed method	90,413.2	45,158.1	44,457.7

.

From Table 6, it can be seen that the investment cost of the DRO method is between stochastic optimization and robust optimization, and its results are better than robust optimization, but slightly worse than stochastic optimization. This is because robust optimization considers harsh photovoltaic scenario information, resulting in more conservative results, leading to a significant increase in investment costs. Relatively speaking, DRO has lower costs and better economic efficiency. Compared to stochastic optimization, DRO considers the probability distribution of the worst-case scenario and requires more units to mitigate the uncertainty of photovoltaic output. Therefore, the investment cost is higher than stochastic optimization, but it has stronger robustness. Comparing the operating costs under different optimization methods, it can be found that robust optimization has the lowest operating cost, followed by distributed robust optimization, because robust optimization has already considered the worst-case scenario in the first stage. By comparing the total cost, it can be seen that DRO has significant advantages, with the lowest total cost. The primary objective of capacity planning for the integrated energy system in this article is to maximize energy efficiency and reduce carbon emissions, thus maintenance costs are not the main focus. Under this context, planners are more concerned with optimizing the production, distribution, and consumption of energy, rather than the maintenance and inspection of equipment. In long-term capacity planning, the initial investment costs and operating costs of equipment are usually considered as more significant factors, as these costs have a notable impact over a prolonged period. In contrast, the daily cost of maintenance is relatively small, therefore, this article neglects the maintenance costs. In summary, the DRO method fully combines the advantages of stochastic optimization and robust optimization, while also possessing the characteristics of economy and robustness.

6. Conclusions

The extensive integration of renewable energy sources has led to increased uncertainty in the system, significantly increasing the difficulty of system operation and planning decisions. This article optimizes the capacity allocation of integrated energy system considering demand response under a ladder carbon trading mechanism. A data-driven distributed robust optimization method is adopted to deal with the uncertainty of photovoltaic output. A two-stage distributed robust optimization configuration model is constructed, which includes deterministic optimization and probability distribution optimization considering the worst scenario. The C&CG algorithm is used to decompose the model into the master problem and sub problems for solution. Through the analysis of the case study, the following conclusions can be drawn:

(1) Through a comparative analysis of five different scenarios, the proposed method increases the configuration capacity of CHP units, increases the daily total cost by 1.9%, reduces carbon emissions by 5.54%, and increases carbon trading costs. Compared with the traditional carbon trading mechanism. Although the total cost of the system increases after considering the stepped carbon trading mechanism, the carbon emissions are significantly reduced. Under the economic pressure of carbon trading costs, the system opts to configure clean equipment with lower carbon emissions to achieve energy conservation and emission reduction as much as possible, which is in line with the direction of energy development.

(2) Compared with traditional robust and stochastic optimization methods, the proposed DRO method fully combines the advantages of stochastic optimization and robust optimization, while also possessing the characteristics of economy and robustness. Meanwhile, the CCG algorithm effectively ensures the optimality and convergence of the solution.

On the other hand, DRO algorithms usually need to deal with a large number of uncertainty scenarios, which can lead to a significantly increased scale of the optimization problem and consequently increased computational complexity. This may result in excessively long runtimes when solving practical problems, and it may even be impossible to find the optimal solution within a given time frame. In the future, designing more efficient solution algorithms and scenario screening algorithms are potential research directions.