Mitigating Risk and Emissions in Power Systems: A Two-Stage Robust Dispatch Model with Carbon Trading

Abstract

The large-scale integration of renewable energy sources is crucial for reducing carbon emissions. Integrating carbon trading mechanisms into electricity markets can further maximize this potential. However, the inherent uncertainty in renewable power generation poses significant challenges to effective decarbonization, renewable energy accommodation, and the security and cost efficiency of power system operations. In response to these challenges, this paper proposes a two-stage robust power dispatch model that incorporates carbon trading. This model is designed to minimize system operating costs, risk costs, and carbon trading costs while fully accounting for uncertainties in renewable energy output and the effects of carbon trading mechanisms. This model is solved using the column-and-constraint generation algorithm. Validation of an improved IEEE 39-bus system demonstrates its effectiveness, ensuring that dispatch decisions are both robust and cost-efficient. Compared to traditional dispatch models, the proposed model significantly reduces system risk costs, enhances the utilization of renewable energy, and, through the introduction of a ladder carbon trading mechanism, achieves substantial reductions in carbon emissions during system operation.

1. Introduction

With the increasing proportion of renewable energy sources in the power system, wind and solar power poses challenges to the safe and economic operation of the system. The electricity sector is the primary source of carbon dioxide emissions in China, and promoting low-carbon operation of the power system is crucial for its sustainable development [1]. The dispatch strategies of the power system need to address the uncertainty of new energy sources while also promoting the development of low-carbon electricity, which is a key step in achieving the low-carbon transition of the power system.

In the context of a low-carbon economy, carbon trading mechanisms are an effective means of reducing carbon emissions. Numerous studies have explored carbon trading, carbon taxes, and their impact on low-carbon dispatch [2,3,4,5]. Ref. [6] introduces an approach that considers both direct and indirect carbon emissions, achieving a delicate balance between cost and environmental considerations. Ref. [7] delves into the equilibrium between economic viability and carbon emission policies by examining their effects on single-firm models. In terms of optimizing power system dispatch, some studies have considered the cost of carbon dioxide emissions and established optimization dispatch models for power systems with large-scale integration of new energy sources [8,9]. Refs. [10,11] introduce carbon trading models into wind power system models, prioritizing the dispatch of units with low carbon emissions to balance low-carbon benefits and economic benefits. Refs. [10,12] finds that the ladder carbon trading mechanism is more conducive to promoting the reduction in carbon emissions and the absorption of new energy by comparing different carbon trading mechanisms. Ref. [13] proposes constraints on carbon quota trading volumes and determines a reasonable range for carbon quota based on a two-stage robust optimization model to achieve the economic and environmental friendliness of dispatch.

Uncertainty optimization dispatch models mainly include stochastic optimization (SO) [14,15], chance-constrained programming [16,17], and robust optimization (RO) [18,19]. SO requires probability distribution parameters for uncertain factors, with computational accuracy and efficiency limited by the number of scenarios. Chance-constrained programming involves non-convex functions, making the solution process complex. RO does not rely on precise probability distributions and can ensure the robustness of solutions under uncertainty, making them more suitable for practical engineering applications in large-scale power systems. Ref. [20] established an adaptive robust dispatch model for systems with high wind power penetration, enhancing the applicability of dispatch decisions. Ref. [21] proposed a robust dispatch model for wind power with multiple scenarios, ensuring that conventional units can maintain robust operation trajectories under various wind power scenarios. Ref. [22] constructed a new type of uncertainty set based on the variance of system net load, and through the solution of robust dispatch models, it was found that this method is more accurate than traditional robust dispatch with uncertainty sets. Ref. [23] established a two-stage distributionally robust optimization model for power systems considering the uncertainty of wind power by clustering historical wind power data to obtain typical output scenarios.

To tackle the challenges of integrating renewable energy sources on a large scale into the power system and to foster low-carbon growth, this paper develops a two-stage robust low-carbon optimization model for day-ahead dispatch. This model considers both carbon trading mechanisms and the uncertainties associated with wind and solar power generation. First, we construct a day-ahead optimization dispatch model for the power system that integrates a carbon trading mechanism. Then, we address the uncertainties in wind and solar power output through a two-stage robust optimization approach, employing a column and constraint generation algorithm to solve this model. Finally, we conduct case study simulations to analyze the effects of carbon trading mechanisms and uncertainty parameters on low-carbon dispatch outcomes, thereby ensuring that the dispatch plan is both robust and economically feasible.

2. Optimal Dispatch Model of Power System with Carbon Trading

2.1. Carbon Trading Model

2.1.1. Carbon Source Model

Carbon emission trading, commonly referred to as “carbon trading”, encompasses the trading of greenhouse gas emission rights and related financial transactions conducted through market mechanisms to reduce greenhouse gas emissions. Under this system, governments initially allocate a certain amount of carbon emission quotas to enterprises. These enterprises then adjust their production activities based on the limits they receive and engage in trading on the carbon market. Incorporating the carbon trading mechanism into the power system and imposing restrictions on traditional power generation units with high carbon emissions can effectively control the carbon emissions within the system.

Wind power and solar power energy, as green energy sources, do not produce greenhouse gas emissions during the power generation process. Therefore, the carbon emissions of the system primarily originate from conventional thermal power units. The carbon emissions of a unit within a dispatch cycle can be represented as

$$Q_p=\eta {\displaystyle \sum _{t=1}^T{P}_t^G}$$

(1)

where $Q_p$ is the system carbon emission; $T$ is the dispatch cycle; $\eta$ is the carbon emission coefficient (t/(MW/h)); $P_t^G$ is the output of the thermal power unit in t period.

2.1.2. Carbon Emission Quota

The method of carbon quota allocation refers to the process of distributing carbon emission quotas to various participants or enterprises. Currently, there are two approaches to carbon quota allocation: auction allocation and free allocation. The specific method of allocation depends on the particular circumstances of different regions and industries. Among them, the methods for determining the free allocation of carbon emission quotas include the baseline method, historical intensity method, and historical emissions method. This article adopts the baseline method for the free allocation of carbon quotas to power generation companies. This method considers the level of power generation and historical emission levels and can be expressed as

$$Q_q=\sigma {\displaystyle \sum _{t=1}^T{P}_t^G}$$

(2)

where $Q_q$ is the system carbon quota; $\sigma$ is the free allocation ratio of carbon emission allowances.

2.1.3. Basic Carbon Price Model

With the principle of free carbon allowance allocation in place, carbon trading provides a platform for enterprises to make secondary allocations of carbon emission rights. The amount of carbon actually traded by a system participating in the carbon market is the difference between the carbon emissions generated by the operation of the system and the carbon allowances allocated to the system. When the carbon emissions of a power producer are higher than its allocated carbon allowances, it needs to purchase additional allowances to pay for the excess carbon emissions. When carbon emissions are lower than carbon allowances, the excess carbon allowances can be sold for a profit. The baseline carbon price cost model can be obtained as follows:

Under the principle of free carbon quota allocation, carbon trading provides enterprises with a platform for the secondary distribution of carbon emission rights. The actual volume of carbon trading in which the system participates in the carbon market is the difference between the carbon emissions generated by the system’s operations and the carbon quotas allocated to the system. When a company’s carbon emissions exceed its allocated carbon quotas, it needs to purchase additional quotas to compensate for the excess carbon emissions. If the carbon emissions are below the carbon quotas, the company can sell the surplus carbon quotas to gain revenue. From this, the benchmark carbon price cost model can be derived as follows:

$$C_{Carbon}^1=\gamma \left(Q_p-Q_q\right)$$

(3)

where $C_{Carbon}^1$ is the unit carbon trading price. When $C_{Carbon}^1$ is negative, it indicates that the actual carbon emissions are less than the carbon quota ($Q_p<Q_q$), meaning that the system has a surplus of carbon emission quotas. This surplus can be traded in the carbon emission market to obtain additional carbon revenue. Conversely, when $C_{Carbon}^1$ is positive, it means that the actual carbon emissions exceed the carbon emission quota ($Q_p\ge Q_q$), necessitating the purchase of additional carbon emission quotas, which incurs additional carbon trading costs.

2.1.4. Ladder Carbon Trading Model

Compared to the traditional carbon trading mechanism, the ladder carbon trading mechanism adopts the principle of ladder pricing, which segments the carbon trading volume according to certain standards and sets different transaction prices for each interval. This approach more effectively regulates corporate carbon emissions and limits the actual carbon emissions of the system. As the system’s carbon emissions $Q_p$ increase, the amount of carbon emission quotas $Q_q$ that need to be purchased also increases, and correspondingly, the carbon trading price becomes higher. The schematic diagram of the ladder carbon trading costs is shown in Figure 1. The calculation formula is shown in Equation (4).

$$C_C=\left\{\begin{array}{l}\gamma \left(Q_P-Q_q\right),Q_P-Q_q\le l\\ \gamma \left(1+\lambda \right)\left(Q_P-Q_q-l\right)+\gamma l,l<Q_P-Q_q\le 2l\\ \gamma \left(1+2\lambda \right)\left(Q_P-Q_q-2l\right)+\left(2+\lambda \right)\gamma l,2l<Q_P-Q_q\le 3l\\ \gamma \left(1+3\lambda \right)\left(Q_P-Q_q-3l\right)+\left(3+\lambda \right)\gamma l,3l<Q_P-Q_q\le 4l\\ \gamma \left(1+4\lambda \right)\left(Q_P-Q_q-4l\right)+\left(4+\lambda \right)\gamma l,4l<Q_P-Q_q\end{array}\right.$$

(4)

where $C_C$ is the ladder carbon trading cost; $\lambda$ is the carbon trading price of each step up the growth rate; this paper takes 25%; $l$ is the carbon trading interval length, considering the actual system unit power situation; the interval length is 200 t.

2.2. Optimal Dispatch Model Considering Carbon Trading Mechanism

2.2.1. Objective Function

The uncertainty of wind and solar power generation can lead to the occurrence of wind curtailment, solar curtailment, and load shedding in the system. Therefore, the costs of wind and solar curtailment and load shedding are used here to quantify the potential risks that the uncertainty of wind and solar forecasting brings to the system’s operation. Considering the carbon trading mechanism, the objective function of the optimized dispatch model includes the start-up and shutdown costs and fuel costs of thermal power units, tiered carbon trading costs, and risk costs. A day-ahead optimized dispatch model that minimizes the total operation cost of the system is constructed, with the objective function as follows:

$$\min C=C_G+C_C+C_R$$

(5)

where $C$ is the total cost of system operation; $C_G$ is the operating cost of thermal power units; $C_R$ is the cost of risk.

• Thermal power unit operating costs

The generation cost of conventional thermal power units is composed of two parts: the start-up and shutdown costs $C_{G1}$ and the fuel costs $C_{G2}$.

$$C_G=C_{G1}+C_{G2}$$

(6)

$$C_{G1}={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_G}{\kappa }_{Gi}}}{x}_{i,t}\left(1-x_{i,t-1}\right)$$

(7)

$$C_{G2}={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_G}{a}_i{P}_{i,t}^{G2}}}+b_i{P}_{i,t}^G+c_i$$

(8)

where $t$ is the dispatch time; $T$ is the dispatch cycle; $i$ is the index of thermal power units; $N_G$ is the total number of thermal power units in the system; $a_i$, $b_i$, and $c_i$ are the secondary coefficient, the primary coefficient, and the constant term of coal consumption of thermal power units; ${\kappa }_{G_i}$ is the start/stop cost of thermal power units; $u_{i,t}$ is the start/stop state of thermal power units at the time $t$;

2.

Carbon trade costs

Carbon trade cost calculations are detailed in the ladder carbon trade model;

3.

Risk costs

Risk costs encompass the costs of wind curtailment, solar curtailment, and load shedding, with the calculation formulas as follows:

$$C_R=C_{WT}+C_{PV}+C_{Load}$$

(9)

To enhance the accommodation of wind and solar energy, this paper incorporates the costs of wind and solar curtailment as penalty terms in the calculation, which is calculated as follows:

$$C_{WT}={\displaystyle \sum _{t=1}^T{\kappa }_{wt}\left(P_{WT,t}^{pre}-P_t^{WT}\right)}$$

(10)

$$C_{PV}={\displaystyle \sum _{t=1}^T{\kappa }_{pv}\left(P_{PV,t}^{pre}-P_t^{PV}\right)}$$

(11)

where ${\kappa }_{wt}$, ${\kappa }_{pv}$ are the penalty coefficients for wind and solar curtailment; $P_{WT,t}^{pre}$, $P_{PV,t}^{pre}$ are the predicted power of wind power and solar energy in time period t; $P_t^{WT}$, $P_t^{PV}$ are the actual output for wind and solar power energy in time period t.

When the system experiences a power deficit, load shedding is required to maintain the power balance, resulting in load shedding costs. The total load-shedding cost for the entire dispatch period is

$$C_{Load}={\displaystyle \sum _{t=1}^T{\kappa }_{load}{P}_{Load}^{loss}}$$

(12)

where ${\kappa }_{load}$ is the load shedding penalty factor; $P_{Load}^{loss}$ is the load shedding amount of the system.

2.2.2. Constraints

• Thermal Power Unit constraint

Thermal power units must meet constraints on output limits, ramping limits, minimum start–stop time, and maximum start–stop frequency during operation, as shown in Equations (13)–(17).

$$x_{i,t}{\underset{\_}{P}}_i^G\le P_{i,t}^G\le x_{i,t}{\overline{P}}_i^G$$

(13)

$$\left\{\begin{array}{l}{P}_{i,t}^G-P_{i,t-1}^G\le x_{i,t-1}\left(R_i^U-S_i^U\right)+S_i^U\\ P_{i,t-1}^G-P_{i,t}^G\le x_{i,t}\left(R_i^D-S_i^D\right)+S_i^D\end{array}\right.$$

(14)

$$\left\{\begin{array}{l}{\displaystyle \sum _{k=t}^{t+T_i^{off}-1}\left(1-x_{i,k}\right)\ge T_i^{off}}\left(x_{i,t-1}-x_{i,t}\right)\\ {\displaystyle \sum _{k=t}^{t+T_i^{on}-1}{x}_{i,k}\ge T_i^{on}}\left(x_{i,t}-x_{i,t-1}\right)\end{array}\right.$$

(15)

$$\left\{\begin{array}{l}{\displaystyle \sum _{t=1}^T\left(x_{i,t}-x_{i,t-1}\right)\le {\overline{T}}_i^{on}}\\ {\displaystyle \sum _{t=1}^T\left(x_{i,t-1}-x_{i,t}\right)\le {\overline{T}}_i^{off}}\end{array}\right.$$

(16)

$$S_i^U=S_i^D={\displaystyle \frac{1}{2}}\left({\underset{\_}{P}}_i^G+{\overline{P}}_i^G\right)$$

(17)

where ${\underset{\_}{P}}_i^G$, ${\overline{P}}_i^G$ are the minimum and maximum output of the thermal power unit $i$; $R_i^U$, $R_i^D$ are the upper and lower ramping rate limits for the thermal unit $i$; when the minimum output required for a unit to start is greater than the ramp-up rate, the constraint will prevent any stopped units from starting. In this paper, the maximum ramp-up rate for starting and the maximum ramp-down rate for shutting down are adopted as shown in Equation (17); $T_i^{off}$, $T_i^{on}$ are the minimum shutdown and start-up time for thermal power unit $i$; ${\overline{T}}_i^{on}$, ${\overline{T}}_i^{off}$ are the maximum number of shutdown and start-up for the thermal unit $i$;

2.

Renewable energy sources output constraint

$$\left\{\begin{array}{l}0\le P_t^{WT}\le P_{WT,t}^{pre}\\ 0\le P_t^{PV}\le P_{PV,t}^{pre}\end{array}\right.$$

(18)

3.

Wind and solar power curtailment constraint

$$\left\{\begin{array}{l}{P}_t^{shed,WT}\le {\xi }_{WT}{\overline{P}}_t^{WT}\\ P_t^{shed,PV}\le {\xi }_{PV}{\overline{P}}_t^{PV}\end{array}\right.$$

(19)

where ${\xi }_{WT}$, ${\xi }_{PV}$ are the maximum allowable curtailment rates for wind power and solar power units;

4.

The generation–load power balance constraint

$$P_{i,t}^G+P_{i,t}^{WT}-P_{i,t}^{shed,WT}+P_{i,t}^{PV}-P_{i,t}^{shed,WT}=P_{i,t}^{Load}-P_{i,t}^{shed,Load}$$

(20)

5.

Network security constraint

$${\underset{\_}{P}}^l\le {\displaystyle \sum _{i=1}^N{G}^{l-i}{P}_{i,t}^G}+{\displaystyle \sum _{m=1}^{N_{WT}}{G}^{l-m}{P}_t^{WT}}+{\displaystyle \sum _{n=1}^{N_{PV}}{G}^{l-n}{P}_t^{PV}}-{\displaystyle \sum _{j=1}^{N_L}{G}^{l-j}{P}_t^l}\le {\overline{P}}^l$$

(21)

where ${\underset{\_}{P}}^l$, ${\overline{P}}^l$ are the lower and upper limits of the line power transmission; $G^{l-i}$ is the power transfer distribution factor of unit $i$ to line $l$; $G^{l-i}$, $G^{l-m}$, $G^{l-n}$, and $G^{l-j}$ are also power transfer distribution factor; $N_{WT}$, $N_{PV}$ are the number of wind farms and solar power farms.

3. Two-Stage Robust Dispatch Model with Carbon Trading

3.1. Wind and Solar Output Uncertainty Model

With the continuous expansion of wind and solar power installations, the penetration rate of renewable energy is also steadily increasing. However, the inherent randomness and variability of these new energy sources introduce new risks and challenges to the secure dispatch of the power grid. To describe the uncertainty of wind and solar power output, this paper employs a boxed uncertainty set to characterize the fluctuations in their output. The boxed uncertainty set uses symmetric intervals to depict the range of fluctuations in uncertain factors, with wind and solar uncertainty parameters taking values at the boundary points. This approach ensures that the dispatch strategy can handle any fluctuations in wind and solar power output within the set, providing robustness against uncertainties.

$$U=\left\{\begin{array}{l}u={\left[u_{WT,t},u_{PV,t}\right]}^T\in {ℝ}^{T\times 2},t=1,2\cdots T\\ u_{WT,t}\in \left[{\widehat{u}}_{WT,t}-\Delta u_{WT,t}^{\max },{\widehat{u}}_{WT,t}+\Delta u_{WT,t}^{\max }\right]\\ u_{PV,t}\in \left[{\widehat{u}}_{PV,t}-\Delta u_{PV,t}^{\max },{\widehat{u}}_{PV,t}+\Delta u_{PV,t}^{\max }\right]\end{array}\right.$$

(22)

where $U$ is the boxed uncertainty set constructed for wind and solar power output. $u_{WT,t}$ and $u_{PV,t}$ are the uncertainty variables introduced for wind and solar power output after considering the uncertainty; $\Delta u_{WT,t}^{\max }$ and $\Delta u_{PV,t}^{\max }$ are the maximum allowable fluctuation deviations for wind and solar output, both of which are positive.

3.2. Two-Stage Robust Optimization Model

This paper constructs a robust two-stage optimization model aimed at finding the dispatch plan that minimizes the total system operating cost under the “worst-case” scenario, ensuring that the solution obtained has the ability to withstand uncertainties or risks. Based on the objective function and constraints of the model presented in this paper, the mathematical model for the two-stage robust problem is as follows:

$$\begin{array}{l}\min c^{T}x+\underset{u\in U}{\max }\underset{y\in \Omega \left(x,u\right)}{\min }{b}^{T}y\\ s.t.\left\{\begin{array}{l}Ax\ge d\\ Bx+Cy\ge e-Du\\ Ex=f\\ Fy+Gu=g\\ Hu\ge h\end{array}\right.\end{array}$$

(23)

where x and y are the decision variables in the first and second stages, as expressed by

$$s.t.\left\{\begin{array}{l}x={\left[x_{i,t}\right]}^T,x\in \left\{0,1\right\}\\ y={\left[P_{i,t}^G,P_{i,t}^{WT},P_{i,t}^{PV},P_{i,t}^{Load},P_{i,t}^{shed,WT},P_{i,t}^{shed,WT},P_{i,t}^{shed,Load}\right]}^T,t=\left(1,2\dots T\right)\end{array}\right.$$

(24)

where $c$ and $b$ are the parameter vectors associated with the decision variables of the first and second stages, respectively, which correspond to the coefficients in the objective function; $u$ is the uncertainty parameter, which is the parameter associated with the decision variables in the second stage, and $u\in U$; $A, B, C, D, E, F, G, H$ are the coefficient matrices of the variables under the corresponding constraints; $d, e, f, g, h$ are the vectors of constant columns.

3.3. Solution Algorithm

Robust optimization methods are dedicated to finding the optimal solution under the worst-case scenario. In this paper, a column-and-constraint generation (C&CG) algorithm is employed to iteratively solve the interaction between the master and subproblems. By using strong duality theory, the two-stage robust optimization problem is decomposed, and the master problem (MP) is obtained as follows:

$$\begin{array}{l}MP:\min c^{T}x+\eta \\ s.t.\left\{\begin{array}{l}\eta \ge b^T{y}_l\\ Ax\ge d\\ Ex=f\\ Bx+C{y}_l\ge e-D{u}_l^{*}\\ F{y}_l+G{u}_l^{*}=g\end{array}\right.\end{array}$$

(25)

The master problem includes the first-stage decision variables $x$ and constraints that depend only on $x$, as well as cuts returned by the subproblem. In this equation, $\eta$ is an auxiliary variable representing the value of the subproblem’s objective function; $y_l$ is the decision variable related to the subproblem that is updated after the $l$-th iteration; $u_l^{*}$ is the worst-case scenario obtained after the $l$-th iteration.

The two-stage robust optimization problem is decomposed using strong dyadic theory to obtain the subproblem problem as

$$S{P}_1:\underset{u\in U}{\max }\underset{y\in \Omega \left(x,u\right)}{\min }{b}^{T}y$$

(26)

The two-stage problem is a bi-level optimization problem with a maximization problem at the outer level and a minimization problem at the inner level. Solving this under normal circumstances is challenging; hence, the strong dyadic theory condition is used to transform the bilevel model of the subproblem into a single-level mixed-integer programming problem for solution. The transformation process, based on strong duality theory, converts the inner minimization structure of the subproblem into a maximization structure, which is then merged with the outer maximization structure. This results in the following bilinear programming problem:

$$\begin{array}{l}S{P}_2:\underset{u\in U}{\max }{\left(e-Bx-Du\right)}^T\lambda +{\left(g-Gu\right)}^T\mu \\ s.t.\left\{\begin{array}{l}{C}^T\lambda +F^T\mu \le b\\ \lambda \ge 0\end{array}\right.\end{array}$$

(27)

where “$\lambda$” and “$\mu$” are the dual variables corresponding to each constraint. The objective function contains two bilinear terms, denoted as “$u^T\lambda$” and “$u^T\mu$”. An Alternating Optimization Procedure (AOP) is employed to linearize these terms, thereby decomposing the original nonlinear problem into a series of linearized problems that are solved iteratively for each unit.

To prevent the model from converging to a local optimum, multiple initial values can be set for an iterative solution. The two-stage robust solution algorithm that combines C&CG and AOP is illustrated in Figure 2.

4. Simulation Verification

4.1. Simulation Description

To validate the rationality of the model in this paper, an improved IEEE-39 bus system is used for verification and analysis, with the system connection diagram shown in Figure 3. This system consists of 10 thermal power units, one wind farm, and one solar power station. The carbon emission quota per unit of electricity is determined by the “Grid Emission Factor” published by the Ministry of Ecology and Environment, which is 0.581 t/(MWh). The carbon emission factor for each unit is 0.3 t/(MWh). The computational environment for the model is a computer with Windows 10 as the operating system, 64 bit system type, 16 GB of RAM, and Inter Core i7-10700 processor at 2.90 GHz. The software used is Python 3.9.7 and the Gurobi solver was invoked to perform the solution.

The uncertainty fluctuation coefficient for wind and solar power generation is set at 15% based on historical experience. The forecast power curves for wind and solar energy are shown in Figure 4 and Figure 5.

4.2. Analysis of Arithmetic Results

To compare and analyze the strengths and weaknesses of robust scheduling models, two scheduling models were established. Model I is a traditional dispatch model that considers wind and solar power based on forecasted output; Model II is a robust dispatch model that accounts for the uncertainty of wind and solar power output, with a fluctuation allowance set at 15%. Concurrently, to analyze the impact of the uncertainty of wind and solar power output and carbon trading mechanisms on the operation of the power system, two models were constructed to simulate and solve three scenarios. Scenario A is a traditional dispatch model without considering carbon trading, with the objective function intending to minimize the system’s operational costs.; Scenario B incorporates a baseline carbon price model, with the objective function aiming to minimize the sum of system operational costs and carbon trading costs. Scenario C considers a ladder carbon trading model, with the objective function seeking to minimize the sum of system operational costs and ladder carbon trading costs. The dispatch solutions for different scenarios under both models are presented in

Table 1. Dispatch results.

Dispatch Scenario		Model I			Model II
		Scenario A	Scenario B	Scenario C	Scenario A	Scenario B	Scenario C
Operation costs (104 CNY)	Unit operating costs	149.61	180.98	183.25	176.63	221.81	214.80
	Carbon trading costs	-	2.37	1.88	-	2.69	2.15
	Total costs	149.61	183.35	175.13	176.63	224.50	217.45
Risk costs (104 CNY)	Wind/solar power abandoned costs	25.11	12.23	9.11	23.11	10.82	10.06
	Load cutting costs	13.14	11.75	10.14	10.05	8.30	6.24
	Total costs	38.25	23.98	19.25	33.16	19.12	16.30
Dispatch total costs (104 CNY)		187.86	207.33	194.38	218.79	243.62	233.25
System carbon emissions (tCO2)		7138	5991	5479	8583	6420	5321

.

To analyze the impact of the carbon trading mechanism on power system dispatch, this paper conducts a comparative analysis of the dispatch results from various scenarios, considering multiple perspectives such as operational costs, risk costs, comprehensive costs, and system carbon emissions. The cost comparison for each scenario under Models I and II is shown in Figure 6.

From the perspective of operational costs, Scenario A, which does not consider the carbon trading model, results in higher carbon emissions compared to other scenarios. When carbon trading models are taken into account, the total costs increase due to the addition of carbon trading costs; hence, the operation costs for Scenarios B and C are higher than those for Scenario A. The overall operation costs for Model II are higher than those for Model I. This is because Model II is a robust optimization model that accounts for the uncertainty of wind and solar power generation. To ensure the safe and stable operation of the system under the worst-case scenario, the system increases the output of thermal power units to accommodate fluctuations in wind and solar power, resulting in higher operational costs for Model II.

From the perspective of risk costs, Model I does not consider the uncertainty of wind and solar power fluctuations when formulating the dispatch plan. When there is a significant deviation between the actual output and the forecasted output of wind and solar power, its dispatch plan struggles to cope with the output fluctuations, leading to risk behaviors such as wind and solar curtailment and load shedding to maintain the safe and stable operation of the system. This results in higher amounts of wind and solar curtailment and load shedding and, thus, higher risk costs for Model I. The dispatch results obtained from Model II consider the uncertainty of wind and solar power output and can handle a certain degree of deviation from wind and solar power forecasts. This increases the absorption of wind and solar power, reduces the amount of wind and solar curtailment and load shedding, and effectively reduces the risk level.

From the perspective of view of dispatch total costs, the total costs of scenarios A, B, and C in Model II are, respectively, 20.93 × 10⁴, 36.29 × 10⁴, and 38.87 × 10⁴ CNY higher than those in Model I. This increase is attributed to the Model II approach to expand the regulatory space in the dispatch plan to accommodate fluctuations in wind and solar power, which, in turn, raises operation costs. While this approach enhances the system’s ability to cope with risks to a certain extent, it results in higher total costs and more conservative dispatch outcomes.

From the perspective of system carbon emissions, Scenario A, which does not account for carbon trading constraints, results in a higher level of carbon emissions. Scenarios B and C, which incorporate carbon trading models, have, to some extent, curbed the system’s carbon emissions, leading to their reduction compared to Scenario A. Notably, Scenario C employs a ladder carbon trading model, where the cost of carbon trading increases in a stepped manner as carbon emissions rise. This model exerts greater cost pressure on units with higher carbon emissions, thereby achieving the lowest carbon emissions. Compared to Scenarios A and B, the reduction in system carbon emissions for Scenario C is 23.24% and 8.55% and 38.01% and 17.12%, respectively. It is evident that the ladder carbon trading model is highly effective in reducing carbon emissions.

4.3. Comparison and Analysis of Dispatch Results

Taking Model I Scenario A as the basic scenario, this scenario employs a deterministic model, which assumes that wind and solar forecasts are certain values and does not consider carbon trading. Figure 7 illustrates the output results of thermal units and renewable energy units under the deterministic model. The total system costs are 187.86 × 10⁴ CNY.

From the dispatch results of the deterministic model, it is evident that the wind and solar power output in the system is comparable to the total output of thermal power units. During the daytime, when the load demand is high, wind, solar, and thermal power units share the output plan. Economically favorable units operate for longer periods, while those with poorer economics operate for fewer hours. At night, when the load demand is lower, only wind and thermal power units contribute to the output. The two peak load periods occur at noon and night. At these times, the load level is high, and the combined output of wind, solar, and thermal power units cannot fully meet the load requirements, necessitating load-shedding measures, which result in higher operation costs. This model does not consider the uncertainty of wind and solar power output, so the dispatch results may not adapt to the actual fluctuations in renewable energy output, failing to fully reflect the impact of uncertainty on the scheduling outcomes. Below is an analysis of the scheduling results for each scenario of both models, as shown in Figure 8.

Overall, the output of thermal power units in Model II has increased significantly compared to Model I because Model II takes into account the variability of wind and solar power generation when formulating the dispatch plan. To meet the load balance and renewable energy accommodation requirements under the worst-case scenario, Model II introduces thermal power units G5 and G7 into the dispatch plan, with both units maintaining a high level of output. Among them, thermal power unit G2 remains shut down due to its high fuel costs. Although G1 has fuel costs comparable to G2, it maintains a certain level of output throughout the dispatch period due to its lower carbon emission factor. This approach helps to avoid the start-up costs and generation costs associated with frequent activation of other units and also contributes to reducing the system’s carbon emissions. Since Scenarios B and C consider both economic efficiency and low carbon emissions, these scenarios prioritize the dispatch of low-carbon thermal power units and wind and solar power to control carbon trading costs and the system’s carbon emissions.

4.4. Impact of Changes in the Parameters of the Carbon Trading Mechanism

Carbon trading costs are a crucial component of the objective function in robust low-carbon dispatch models and are closely related to the parameter settings of the carbon trading mechanism. To investigate the impact of variations in carbon trading prices and price growth rates on the system’s dispatch outcomes, this section analyzes Scenario C of Model II. The curves of different carbon trading prices and prices at growth rates versus system dispatch costs and carbon emissions are shown in Figure 9.

As carbon trading prices and price growth rate increase, the dispatch costs of the system increase, while the carbon emissions decrease. It is evident that the system’s carbon emissions exhibit a gradual decline with the escalation of carbon prices. When the carbon trading price fluctuates between 0 and 30 CNY/tCO₂, the reduction in system dispatch costs is relatively slow. This is because, at this stage, the carbon trading cost constitutes a smaller proportion of the total costs, and the variation in carbon trading prices has a minimal impact on carbon emissions. When the carbon trading price changes between 30 and 50 CNY/tCO₂, the decrease in scheduling costs becomes more significant. Once the carbon trading price exceeds 50 CNY/tCO₂, the trend of system carbon emissions slows down. The dispatch costs show a steady increase with the rise in carbon trading prices. When carbon prices are low, the carbon trading costs are relatively small, so the system prioritizes calling upon low-cost units to reduce operational costs. As carbon prices rise, the carbon trading costs generated by continuing to operate low-cost units become relatively high. Consequently, the system shuts down some high-emission units and increases the output of low-emission units to reduce carbon trading costs and system carbon emissions, achieving optimal operation costs.

With the increase in price growth rates, the system’s carbon emissions generally follow a trend of rapid decline followed by a gradual reduction. When the price growth rate varies within the range of 0 to 0.25, the proportion of carbon trading costs in the system increases, leading to a corresponding increase in system dispatch costs. When the price growth rate exceeds 0.25, the system becomes less sensitive to changes in the price growth rate; the downward trend in system carbon emissions slows, and the growth rate of system dispatch costs decreases.

4.5. Impact of Uncertainty of New Energy Output

Considering the impact of uncertainties in actual power grid operations, the fluctuation levels of wind and solar power output in Scenario C of Model II are set to vary within the range of 0.05–0.25. This analysis examines the influence of uncertain factors on the dispatch plan. The dispatch results under different levels of wind and solar power fluctuations are illustrated in Figure 10.

As the level of wind and solar power fluctuation increases, the system’s operation costs gradually rise, while the risk costs show a decreasing trend, and the total system dispatch costs increase accordingly. This indicates that with the increase in wind and solar power fluctuations, the number of periods where wind and solar power output reaches the boundary of the interval increases, taking more into account the uncertainties faced by the system. To cope with the fluctuations in wind and solar power output and ensure that the unit output adapts to all possible scenarios of wind and solar power output, the robust scheduling becomes more conservative in the arrangement of unit combinations. Dispatch decisions may involve starting more units or adjusting the output of existing units to respond to changes in uncertain factors. As a result, the costs of wind and solar power curtailment and load shedding for the system decrease, while the system’s operation costs increase, leading to a subsequent increase in total dispatch costs. This demonstrates that enhancing the robustness of the dispatch plan comes at a certain economic cost.

4.6. Comparing Different Methods of Uncertainty

To demonstrate the superiority of the two-stage robust optimization strategy in reducing system operation costs and carbon emissions, this paper compares the optimization problems of the same system using the certain model (CM), SO model, and RO model. The specific optimization objectives and constraints are consistent with Scenario C mentioned earlier. The operational results after solving are shown in

Table 2. Dispatch results.

Method	IEEE 6				IEEE 39				IEEE 118
	Objective	Risk Cost	CO2	Time	Objective	Risk Cost	CO2	Time	Objective	Risk Cost	CO2	Time
CM	28.93	12.08	3911	0.10	194.38	19.25	5479	2.80	1471.88	51.88	14,108	24.29
SO	24.03	2.81	1193	1.85	178.65	12.98	4812	44.14	1476.22	41.86	10,143	537.49
RO	40.39	8.41	2313	1.67	233.25	16.30	5321	27.65	1633.17	44.32	10,942	216.04

(10⁴ CNY).

.

In terms of new energy accommodation, the deterministic model relies on forecast curves. When there are significant fluctuations in wind and solar power output, the system still makes optimization decisions based on the predicted wind and solar power values, leading to more wind and solar curtailment and the lower accommodation of wind and solar energy. As a result, the deterministic model has the highest risk cost. At the same time, due to the lower operating costs of some thermal power units, it also has the highest carbon emissions. The SO model only considers the predicted output of wind and solar power to be distributed according to a certain probability, randomly generating wind and solar output within the fluctuation range. Although the total cost is lower, it does not sufficiently mine the probability information of historical data and cannot accurately simulate the uncertainty of wind and solar power output.

The RO model uses an uncertainty set to represent the output of wind and solar power, allowing the units within the system to fully cope with power fluctuations, thereby increasing the accommodation rate of wind and solar energy. Since traditional deterministic models and SO models are not the worst-case scenarios, their operating costs are both better than those of the RO model. RO is more conservative but provides better guarantees for the safety and stability of the system.

In terms of computational time, as the system size increases, the solution time for all models increases. The deterministic model has the best computational time, while the SO model has a longer computation time due to the large number of scenarios. The solution time for the RO model is between the two, effectively ensuring the safety and stability of system operation without sacrificing computational efficiency.

5. Conclusions

For the large-scale new energy grid-connected system, this paper considers the impacts of system power generation costs, carbon trading costs, risk costs, and uncertainty factors and establishes a carbon trading mechanism model that considers both the baseline carbon price and the laddered carbon price. The analysis of the arithmetic example results in the following conclusions:

• By comparing CM, SO, and RO models, it is evident that while RO models may increase some operation costs of the system, they significantly reduce the risks of wind and solar power curtailment, as well as load shedding. These models enhance the system’s ability to accommodate wind and solar energy, thereby improving the system’s robustness at the expense of some economic efficiency. Overall, they strike a balance between economic viability and low-carbon objectives;
• By comparing the solution results of the traditional low-carbon dispatch model and the low-carbon robust dispatch model, it can be concluded that the introduction of the carbon trading mechanism model can control the output level of thermal power units and the carbon emissions of the system. It helps to improve the low-carbon performance and environmental benefits of the power system;
• The carbon trading price and price growth rate of laddered carbon trading will affect the system’s carbon emissions and dispatch costs. In the actual dispatch operation, a reasonable carbon trading price and price growth rate should be set in consideration of the actual situation.

Due to the regional variations in global carbon markets, there are differences in carbon trading mechanisms, price levels, and the allocation of carbon quotas, which, together, pose a series of practical challenges for model dispatch decisions. Taking the EU, the US, and China as examples, these regions each have their own characteristics in their carbon markets. In particular, the carbon price in the EU is much higher than in China, which means that companies operating in the EU face a heavier burden of carbon costs, complicating the management of carbon emissions across different markets for multinational companies. To meet this challenge, multinational companies need to balance costs across different markets and develop differentiated carbon emission strategies to adapt to price fluctuations while ensuring compliance and cost-effectiveness in their global operations.