Research on a Low-Carbon Optimization Strategy for Regional Power Grids Considering a Dual Demand Response of Electricity and Carbon

Abstract

Considering the characteristics of the power system, where “the source moves with the load”, the load side is primarily responsible for the carbon emissions of the regional power grid. Consequently, users’ electricity consumption behavior has a significant impact on system carbon emissions. Therefore, this paper proposes a multi-objective bi-level optimization strategy for source-load coordination, considering dual demand responses for both electricity and carbon. The upper layer establishes a multi-objective low-carbon economic dispatch model for power grid operators, aiming to minimize the system’s total operating cost, the total direct carbon emissions of the power grid, and the disparity in regional carbon emissions. In the lower layer, a low-carbon economic dispatch model for load aggregators is established to minimize the total cost for load aggregators. To obtain the dynamic carbon emission factor signal, a complex power flow tracking method that considers the power supply path is proposed, and a carbon flow tracking model is established. NSGA-II is used to obtain the Pareto optimal frontier set for the upper model, and the ‘optimal’ scheme is determined based on the fuzzy satisfaction decision. The example analysis demonstrates that the interactive carbon reduction effect under the guidance of dual signals is the most effective. This approach fully exploits the carbon reduction potential of the flexible load, enhancing both the economic efficiency and low-carbon operation of the system.

1. Introduction

In the context of a clean and low-carbon power energy system, the traditional economic dispatch strategy of power systems needs to be adjusted accordingly. The scheduling objective should incorporate low-carbon targets alongside system economy and security [1,2]. Additionally, it is crucial to consider the spatial and temporal distribution characteristics of the scheduling subjects and propose corresponding low-carbon scheduling strategies for different times and regions. Moreover, various carbon emission reduction mechanisms, such as carbon market transactions and green certificate market transactions, should also be taken into account.

In Reference [3], a multi-objective low-carbon economic dispatch model incorporating wind power is established to minimize the total carbon emissions of the system. The results demonstrate a reduction in carbon emissions during the power generation process. In Reference [4], a carbon trading model is introduced into the scheduling of a power system with wind power, establishing an optimization model aimed at minimizing the sum of carbon trading costs and power generation costs. However, this method does not consider the impact of user-side electricity consumption behavior on system carbon emission reduction.

In Reference [5], the influence of the demand side on scheduling results is illustrated by establishing a two-stage model with load fluctuation and system comprehensive cost as the objectives. References [6,7] introduce a step-by-step carbon trading mechanism into the IES low-carbon economic dispatch problem, incorporating carbon transaction costs and demand response costs into the optimization goal. The results indicate that the model effectively reduces system operating costs and carbon emissions. In Reference [8], a low-carbon economic dispatch model with complementary sources and loads was established for the coordination of carbon capture power plants and wind power. In Reference [9], the influence of source-load uncertainty and user demand response on the results of low-carbon optimal scheduling is also considered.

Research on the cooperative interaction between sources and loads in low-carbon economic dispatch within the “power perspective” has become more extensive. However, users often lack a clear understanding of how their electricity consumption behavior affects carbon emissions across different periods, which reduces their motivation to adjust consumption behavior for carbon reduction. The carbon emission flow theory of power systems emphasizes that the system carbon emission factor is as crucial for low-carbon power dispatch research as system voltage is for power flow analysis [10]. Therefore, another crucial incentive signal that encourages users to actively reduce carbon emissions, the ‘electricity dynamic carbon emission factor’, must be considered. Building upon the theory of carbon emission flow in power systems, Reference [11] constructed a low-carbon optimization model for source-load coordination. They utilized carbon pricing as the nexus for interaction between sources and loads, aiming to stimulate active emission reduction on the demand side. In Reference [12], a two-layer low-carbon optimal scheduling model was established with carbon taxation as the signal. The results demonstrate that the model achieves low-carbon economic dispatch in the modern power system. Reference [13] proposed transmitting the dynamic carbon emission factor of electricity to users through incentive signals, enabling users to perceive real-time changes in carbon emission factors. Reference [14] established an IES low-carbon economic dispatch model incorporating multi-load carbon potential demand response, enhancing the low-carbon outcomes of IES dispatch. Building on node carbon potential, Reference [15] developed a low-carbon economic dispatch model for distributed resources in distribution networks. The example validates that the model can decrease line network losses, lower system operating costs, and reduce total carbon emissions.

The literature above does not explore the impact of source-load collaborative optimization on regional carbon emission management under the electricity–carbon dual demand response mechanism following the introduction of dynamic carbon emission factors. Additionally, it does not address how to meet regional demands for carbon emission balance in each region. Therefore, based on the carbon emission flow theory of power systems, this paper proposes a complex power flow tracking method that considers the power supply path. Subsequently, it establishes a carbon flow tracking model to derive the dynamic carbon emission factor of the entire system. Taking the dynamic carbon emission factor and time-of-use tariff as the incentive signals at the same time, a multi-objective bi-level optimization model of source-load coordination under the dual demand response of electricity and carbon is established, which shows the advantages of dual mechanism and source-load coordination of power system in carbon emission reduction. In addition to the system operation cost and the total carbon emission target, the regional carbon emission balance is also taken into account to prevent excessive carbon emissions in some regions. The New England 39-bus system is used for simulation verification, and the low-carbon optimal operation strategy of the regional power grid is given.

The rest of this article is organized as follows. In Section 2, a complex power flow tracking method considering the power supply path is proposed, and a carbon flow tracking model is established to obtain the solution of the dynamic carbon emission factor signal. In Section 3, the principle of the low-carbon demand response mechanism is expounded, the carbon emission accounting model of the flexible load is established, and the research framework of the multi-objective two-layer model is drawn. In Section 4, a multi-objective bi-level optimization model of source-load coordination is established, and the solution method is given. In Section 5, numerical experiments are given. Section 6 is the conclusion of this paper.

2. Obtaining Dynamic Carbon Emission Factors

2.1. Introduction of Power Flow Tracking

2.1.1. The Principle of Complex Proportional Sharing

Based on the converged power flow calculation results, the power flow tracking of the power system is carried out. The power flow tracking method characterizes the tracking of the actual complex power flow along the flow direction (from the power supply to the load) and the reverse flow direction (from the load to the power supply), so as to determine the utilization degree of the user side to the source side and the transmission network. The assumption is based on the complex proportional sharing principle. As shown in Figure 1, assuming that the total complex power of the inflow node n is ${\mathrm{S}}_{\mathrm{n}}$, the complex power of the i-th branch of the inflow node n is ${\mathrm{S}}_i^{\mathrm{in}}$, and the complex power of the j-th branch of the outflow node n is ${\mathrm{S}}_j^{\mathrm{out}}$, then the complex power of the j-th outflow branch from the i-th inflow branch is shown as follows.

$$S_{i,\text{in-}j,\mathrm{o}\mathrm{u}\mathrm{t}}={\displaystyle \frac{S_i^{\mathrm{i}\mathrm{n}}}{S_n}}{S}_j^{\mathrm{o}\mathrm{u}\mathrm{t}}$$

(1)

2.1.2. Power Supply Path Analysis

Through the analysis of the above complex proportional sharing principle, the complex power relationship between adjacent branches of the system can be obtained. However, for the whole system, it is necessary to clarify the complex power transmission process; that is, through which branches the complex power is transmitted from the power supply side to the load side. This requires an in-depth analysis of the system power supply path, and its core is directed graph theory. For a directed graph, $\Gamma$, with N nodes, the adjacency relationship between nodes is described by an adjacency matrix, D. The rows and columns of D correspond to nodes, where i is the upstream node of j, and each element is defined as follows:

$$d_{\mathit{ij}}=\left\{\begin{array}{cc}1& i \mathrm{i}\mathrm{s} \mathrm{a}\mathrm{d}\mathrm{j}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t} \mathrm{t}\mathrm{o} j\\ 0& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{s}\end{array}\right.$$

(2)

In this paper, the breadth-first search algorithm is employed to explore the power supply path between the source and the load. The fundamental approach involves expanding the search outward from the starting node in a radial manner. Initially, all nodes adjacent to the starting node are accessed. To prevent revisiting nodes, these nodes are marked as visited and then explored layer by layer until the target node is reached. When the queue is empty, the search result is output.

2.1.3. Equivalent Treatment of Line Loss

As depicted in Figure 2a, this method represents a conventional approach for handling line losses, but it leads to an increase in the number of system nodes, the order of network matrices, and computational complexity. Therefore, this paper adopts an equivalent approach by reallocating the network loss load to the upstream node of the line, as illustrated in Figure 2b. Here, the power flow and direction of the line remain consistent with its original end. It is crucial to note that the network loss load cannot be transferred to the downstream node of the line, as doing so would require the incoming power at the downstream node to also account for the network loss, leading to a logical inconsistency.

2.2. Power Flow Tracking Method Based on Complex Power

Countercurrent tracking characterizes the contribution of upstream generator power and line inflow power of node load to load power. The generator contribution factor can be defined as the proportion of a certain generator’s power to the total injected power of its downstream nodes, as follows:

$$G_{c,k}={\displaystyle \frac{S_{G,k}}{S_k^{{ }^{{\displaystyle \sum +}}}}}\Rightarrow S_{G,k}=G_{c,k}·S_k^{{ }^{{\displaystyle \sum +}}}$$

(3)

where $S_{G,k}$ is the generator complex power of node k, $S_k^{{ }^{{\displaystyle \sum +}}}$ is the total injection complex power of node k.

The branch contribution factor is defined as the proportion of a branch’s power i to the total injected power of its downstream nodes, as follows:

$$B_{c,i}={\displaystyle \frac{S_{L,i}}{S_{N,j}}}$$

(4)

where $S_{L,i}$ is the power of upstream branch i and $S_{N,j}$ is the total injection power of downstream node j corresponding to upstream branch i.

Considering the existence of multiple branch chains between the power source G_k and node f where the load D is located, let L be the corresponding branch chain set, where any branch chain b contains a branch sequence of b₁, b₂, …, b_m. Therefore, based on the directed path search method based on the breadth-first search idea mentioned above and combined with the branch contribution factor B_c,i, the power contribution ratio of the branch chain set L to node f where load D is located can be obtained as follows:

$$B_{c,L}^{\mathrm{c}\mathrm{h}\mathrm{a}}={\displaystyle \sum _{b\in L}\left({\displaystyle \prod _{x=1}^m{B}_{c,b_x}}\right)}$$

(5)

Combining the generator contribution factor and the branch chain contribution factor in the above formula, the contribution factor of the power supply G_k to node f where load D is located can be obtained as follows:

$${\Gamma }_{G_k,f}=G_{c,k}\times B_{c,L}^{\mathrm{c}\mathrm{h}\mathrm{a}}={\displaystyle \frac{S_{G,k}}{S_k^{{ }^{{\displaystyle \sum +}}}}}·{\displaystyle \sum _{b\in L}\left({\displaystyle \prod _{x=1}^m{B}_{c,b_x}}\right)}$$

(6)

The combined Formula (6) can obtain the matrix calculation method of the contribution factor of any power source to any node of the system, as follows.

$$\begin{array}{c}\mathbf{\Gamma }={\mathit{G}}_{\mathit{c}}·{\mathit{B}}_{\mathit{c}}={\mathit{S}}_{\mathit{G}}·{\mathit{S}}_{\mathit{N}}^{-1}·{\mathit{B}}_{\mathit{c}}\\ {\mathit{S}}_{\mathit{G}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(S_{G,1},S_{G,2},\cdots ,S_{G,n}\right)\\ {\mathit{S}}_{\mathit{N}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(S_{N,1},S_{N,2},\cdots ,S_{N,n}\right)\end{array}$$

(7)

Thus, the actual load and equivalent load of all nodes of the system can be obtained by the power provided by the system generator, as follows:

$$\begin{array}{c}{\mathit{S}}_{\mathit{d},\mathit{G}}=\mathbf{\Gamma }·{\mathit{S}}_{\mathit{d}}\\ {\mathit{S}}_{\mathit{d},\mathit{G}}^{\mathit{l}\mathit{o}\mathit{s}\mathit{s}}=\mathbf{\Gamma }·{\mathit{S}}_{\mathit{d}}^{\mathit{l}\mathit{o}\mathit{s}\mathit{s}}\\ {\mathit{S}}_{\mathit{d}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(S_{d,1},S_{d,2},\cdots ,S_{d,i},\cdots ,S_{d,n}\right)\\ {\mathit{S}}_{\mathit{d}}^{\mathit{l}\mathit{o}\mathit{s}\mathit{s}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(S_{d,1}^{loss},S_{d,2}^{loss},\cdots ,S_{d,i}^{loss},\cdots ,S_{d,n}^{loss}\right)\\ S_{d,i}^{loss}={\displaystyle \sum _{j\in {\mathrm{\Omega }}_i}{S}_{ij,loss}}\end{array}$$

(8)

where $S_{d,i}^{loss}$ is the combined value of the downstream branch line loss of the adjacent node i, and ${\mathrm{\Omega }}_i$ is the downstream node set of node i.

2.3. Carbon Flow Tracking Model

Based on the above complex power flow tracking method, a carbon flow tracking model is established. The node carbon emission factor characterizes the weighted average of the carbon emissions provided by each generator to each node. Therefore, according to the carbon emission intensity of each generator, combined with the power provided by the generator to each node in the above countercurrent tracking method, the node carbon potential of node f can be obtained, as follows:

$$E_f={\displaystyle \frac{1}{\mathrm{Re}\left(S_f\right)}}·{\displaystyle \sum _{i=1}^k\left(\mathrm{Re}\left(S_{f,i}\right)·E_{G,i}\right)}$$

(9)

where $S_f$ is the total injection power of node f, $S_{f,i}$ is the power provided by the i-th generator to node f, $E_{G,i}$ is the carbon emission intensity of the i-th generator.

From Equation (8), the contribution power of all generators to the nodes of the system can be obtained. Combined with the carbon emission intensity of the generator, the system node load and network loss carbon emissions can be calculated as follows:

$$\begin{array}{c}{\mathit{E}}_{\mathit{d}}=\mathrm{Re}\left({\mathit{S}}_{\mathit{d},\mathit{G}}\right)·{\mathit{E}}_{\mathit{G}}\\ {\mathit{E}}_{\mathit{l}\mathit{o}\mathit{s}\mathit{s}}=\mathrm{Re}\left({\mathit{S}}_{\mathit{d},\mathit{G}}^{\mathit{l}\mathit{o}\mathit{s}\mathit{s}}\right)·{\mathit{E}}_{\mathit{G}}\end{array}$$

(10)

where ${\mathit{E}}_{\mathit{G}}$ represents the column vector of unit carbon emission intensity.

Then the branch carbon flow density and branch carbon flow rate can be determined, as follows:

$$\begin{array}{c}\mathit{\rho }=\mathit{M}·{\mathit{E}}_{\mathit{N}}\\ {\mathit{M}}_{ij}=\left\{\begin{array}{cc}1 \mathrm{b}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{h} i \mathrm{i}\mathrm{s} \mathrm{a}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{c}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} \mathrm{b}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{h} j,& \mathrm{R}\mathrm{e}\left(S_{j+}\right)\ge 0\\ 0& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{s}\end{array}\right.\\ {\mathit{R}}_{\mathit{B}}=\mathrm{R}\mathrm{e}\left({\mathit{S}}_{\mathit{l}}\right)·\mathit{\rho }\\ {\mathit{S}}_{\mathit{l}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(S_{l,1},S_{l,2},\cdots ,S_{l,m},\cdots ,S_{l,n}\right)\end{array}$$

(11)

where E_N is the node carbon potential column vector, M is the carbon flow adjacency matrix, Re(S_j+) is the positive power from the node to the other node of the branch i, and S_l is the line power.

Combining the carbon flow tracking model established in this paper with the unit combination method proposed in Reference [16], the real-time carbon flow of the whole system can be determined.

3. Source-Load Cooperative Multi-Objective Bi-Level Model Framework

3.1. Low-Carbon Demand Response Mechanism

Currently, most studies on low-carbon power systems focus on two aspects: reducing direct carbon emissions from the power supply side and considering the impact of the user side on indirect carbon emissions. The latter involves introducing a power demand response mechanism that uses price signals (price DR) to encourage users to modify their electricity consumption behavior. However, both of these studies stay in the perspective of power and do not consider the user-side guidance signal of more detailed time granularity under the carbon emission flow theory of power systems. Therefore, this paper proposes a low-carbon demand response (incentive DR) based on power demand response, aiming to guide the user-side carbon reduction through the double-demand response mechanism of electric carbon. Low-carbon demand response means that the energy dispatching department subsidizes and incentivizes users to participate in dispatching by developing contracts.

The process of obtaining dynamic carbon emission factors is central to low-carbon demand response, as elaborated in the carbon flow tracking model. Figure 3 illustrates the carbon emission reduction principle of the low-carbon demand response mechanism based on dynamic carbon emission factors.

3.2. Carbon Emission Accounting of Flexible Loads Based on the Dynamic Carbon Emission Factor

In this section, considering the weakening and transferable characteristics of flexible loads, the regional carbon emissions in step 4 above are evaluated, and the carbon emission reduction accounting model of the flexible load after the low-carbon demand response is established. The accounting formula for the carbon emission reduction of flexible loads is as follows:

$$\begin{array}{c}{E}_t^{\mathrm{D}}={\epsilon }_{n,t}(P_t^{\mathrm{T}\mathrm{L},1}-P_t^{\mathrm{T}\mathrm{L},2}-P_t^{\mathrm{C}\mathrm{L}})\Delta t\\ \begin{array}{c}0⩽{\mu }_t^{\mathrm{CL}}{P}_t^{\mathrm{CL}}⩽P_{\max }^{\mathrm{CL}}\hspace{1em}t\in [t_{\mathrm{st}}^{\mathrm{CL}},t_{\mathrm{end}}^{\mathrm{CL}}]\\ \begin{array}{c}{\displaystyle \sum _{t=1}^{N_t}{\mu }_t^{\mathrm{CL}}}=t_{\max }^{\mathrm{CL}}\hspace{1em}t\in [t_{\mathrm{st}}^{\mathrm{CL}},t_{\mathrm{end}}^{\mathrm{CL}}]\\ 0⩽{\mu }_t^{\mathrm{TL},1}{P}_t^{\mathrm{TL},1}⩽P_{\max }^{\mathrm{TL},1}\hspace{1em}t\in [t_{\mathrm{st}}^{\mathrm{TL},1},t_{\mathrm{end}}^{\mathrm{TL},1}]\\ 0⩽{\mu }_t^{\mathrm{TL},2}{P}_t^{\mathrm{TL},2}⩽P_{\max }^{\mathrm{TL},2}\hspace{1em}t\in [t_{\mathrm{st}}^{\mathrm{TL},2},t_{\mathrm{end}}^{\mathrm{TL},2}]\\ 0⩽{\mu }_t^{\mathrm{TL},1}+{\mu }_t^{\mathrm{TL},2}⩽1\\ {\displaystyle \sum _{t=1}^{N_t}{\mu }_t^{\mathrm{TL},1}{P}_t^{\mathrm{TL},1}\Delta t={\displaystyle \sum _{t=1}^{N_t}{\mu }_t^{\mathrm{TL},2}{P}_t^{\mathrm{TL},2}\Delta t}}\end{array}\end{array}\end{array}$$

(12)

where ${\epsilon }_{n,t}$ represents the carbon emission factor of node n at time t, $P_t^{\mathrm{C}\mathrm{L}}$, $P_t^{\mathrm{T}\mathrm{L},1}$, $P_t^{\mathrm{T}\mathrm{L},2}$ represent the reduction power, transfer power, and transfer power of the flexible load under demand response, respectively, ${\mu }_t^{\mathrm{CL}}$, ${\mu }_t^{\mathrm{TL},1}$, ${\mu }_t^{\mathrm{TL},2}$ represent the (0,1) state variables of the reducible load and transferable load at time t, respectively, $P_{\max }^{\mathrm{CL}}$, $P_{\max }^{\mathrm{TL},1}$, $P_{\max }^{\mathrm{TL},2}$ represent the maximum adjustment of reduction power, transfer power, and transfer power of the flexible load under demand response, respectively, $t_{\mathrm{st}}^{\mathrm{CL}}$, $t_{\mathrm{end}}^{\mathrm{CL}}$, $t_{\mathrm{st}}^{\mathrm{TL},1}$, $t_{\mathrm{end}}^{\mathrm{TL},1}$, $t_{\mathrm{st}}^{\mathrm{TL},2}$, $t_{\mathrm{end}}^{\mathrm{TL},2}$ represent the start and end times of demand response of the reducible load and transferable load, respectively.

Therefore, the formula for the total carbon emissions of flexible loads is as follows:

$$E^{\mathrm{D}}={\displaystyle \sum _{t=1}^{N_t}({\epsilon }_{n,t}}{P}_t^{\mathrm{s}\mathrm{t}}\Delta t+E_t^{\mathrm{D}})$$

(13)

where $P_t^{\mathrm{s}\mathrm{t}}$ denotes the initial load of the flexible load at time t.

3.3. Source-Load Cooperative Multi-Objective Bi-Level Model Framework

Based on the above analysis, this paper comprehensively considers dual demand response for electricity and carbon, source-load interaction, and other factors. It establishes a multi-objective bi-level optimization model for source-load coordination, as depicted in Figure 4. At the level of the power grid operator, a multi-objective function is formulated to minimize the operator’s costs, total direct carbon emissions from the power grid, and regional disparities in carbon emissions. The Pareto optimal frontier set is obtained by the second-generation non-dominated sorting genetic algorithm (NSGA-II). The fuzzy satisfaction decision-making based on the logistic membership function transforms multi-objective optimization into a single-objective problem. The power grid operator guides the power management center to adjust the unit output plan based on the multi-objective optimization results, and based on the carbon flow tracking model, the dynamic carbon emission factor of the system demand side is obtained. The upper grid operator passes the dynamic carbon emission factor and time-of-use tariff to the lower load aggregator. At the level of the load aggregator, a single objective function is established for minimizing aggregator costs, which is solved using Gurobi 10.0.3. The aggregator formulates demand response incentive contracts based on signals of dynamic carbon emission factors. Flexible loads under the aggregator’s jurisdiction consider incentive contracts and time-of-use tariffs comprehensively, optimizing their electricity consumption behaviors accordingly. The load aggregator communicates the energy consumption of demand response loads back to the upper power management center. The operator then re-optimizes the multi-objective low-carbon economy using the updated load information.

4. Source-Load Collaborative Multi-Objective Bi-Level Optimization Model

4.1. Upper Model: Multi-Objective Low-Carbon Economic Dispatch Model of Power Grid Operator

4.1.1. Objective Function

Building upon traditional power system economic dispatch, the upper model incorporates three objective functions focusing on low-carbon economic dispatch: minimizing the total cost of power grid operators, reducing total direct carbon emissions from the power system, and achieving carbon emission balance across regional power grids. The objective function is formulated as follows:

The low-carbon economic dispatch target with the minimum total cost of the power grid operator is as follows:

$$F_1=\mathrm{m}\mathrm{i}\mathrm{n}(F_{\mathrm{G}}+F_{\mathrm{W}}+F_{\mathrm{c}\mathrm{e}\mathrm{t}}+F_{\mathrm{r}\mathrm{p}\mathrm{s}})$$

(14)

where $F_{\mathrm{G}}$ represents the power generation cost of traditional thermal power units, $F_{\mathrm{W}}$ denotes the composition cost of wind turbines, $F_{\mathrm{c}\mathrm{e}\mathrm{t}}$ represents the carbon transaction cost, and $F_{\mathrm{r}\mathrm{p}\mathrm{s}}$ signifies the green certificate transaction cost.

Among them, the power generation cost of traditional thermal power units is generally a quadratic function, and its expression is as follows:

$$F_G={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_{\mathrm{P}\mathrm{G}}}(a_i{P}_{\mathrm{G}i,t}^2}}+b_i{P}_{\mathrm{G}i,t}+c_i)$$

(15)

where $a_i$, $b_i$, $c_i$ represent the cost coefficient power generation values of the i-th thermal power unit, and $P_{\mathrm{G}i,t}$ denotes the power generation of the i-th thermal power unit within unit time, t.

For the cost composition of wind turbines, this paper considers the full life cycle operational costs, which approximately scale linearly with power generation. In addition, the penalty cost of wind curtailment is introduced to limit the wind curtailment behavior. The specific expression is as follows:

$$F_{\mathrm{W}}={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_{\mathrm{P}\mathrm{W}}}({\mathsf{\zeta }}_1{P}_{\mathrm{W}i,t}}}+{\mathsf{\zeta }}_2(P_{\mathrm{S}i,t}-P_{\mathrm{W}i,t}))$$

(16)

where ${\mathsf{\zeta }}_1$ and ${\mathsf{\zeta }}_2$ represent the power generation cost coefficient values of wind turbines and the penalty cost coefficient of wind abandonment, respectively, $P_{\mathrm{W}i,t}$ denotes the power generation of the i-th wind turbine in unit time t, and $P_{\mathrm{S}i,t}$ signifies the predicted power generation of the i-th wind turbine in unit time t.

For carbon transaction costs, this paper calculates the difference between the direct carbon emissions of the system units and the carbon quotas set by the government, as follows:

$$F_{\mathrm{c}\mathrm{e}\mathrm{t}}={\mu }_{\mathrm{c}\mathrm{e}\mathrm{t}}(E_m-E_q)$$

(17)

where ${\mu }_{\mathrm{c}\mathrm{e}\mathrm{t}}$ is the carbon price, $E_m$ is the direct carbon emission of the unit, and $E_q$ is the free carbon quota set up by the energy management department.

For the direct carbon emission $E_m$ of the unit, the carbon emission intensity of thermal power and wind power can be obtained based on the carbon emission accounting model of the whole life cycle of the unit. The specific expression is as follows:

$$E_m={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_{\mathrm{P}\mathrm{G}}}\epsilon P_{\mathrm{G}i,t}}}$$

(18)

where $\epsilon$ is the carbon emission factor of unit fuel.

The free carbon quota, E_q, is proportional to the total power generation of the system unit, and the specific expression is as follows:

$$E_q={\displaystyle \sum _{t=1}^T\alpha ·({\displaystyle \sum _{i=1}^{N_{\mathrm{P}\mathrm{G}}}{P}_{\mathrm{G}i,t}+{\displaystyle \sum _{j=1}^{N_{\mathrm{P}\mathrm{W}}}{P}_{\mathrm{W}j,t}}}})$$

(19)

where $\alpha$ is the carbon quota per unit of electricity.

For the transaction cost of green certificates, if the output of renewable energy is substantial, the operator holds more green certificates than the minimum required by energy authorities for renewable energy consumption. In this case, they can sell surplus green certificates to generate profits. Conversely, if the renewable energy output is insufficient, the operator must purchase green certificates from the market. Therefore, the calculation model for green certificate transaction costs is as follows:

$$\begin{array}{c}{Q}_{\mathrm{r}\mathrm{p}\mathrm{s}}=\kappa (\varphi (P_t^{st}+P_t^{\mathrm{T}\mathrm{L},1}-P_t^{\mathrm{T}\mathrm{L},2}-P_t^{\mathrm{C}\mathrm{L}})-{\displaystyle \sum _{i=1}^{N_{\mathrm{P}\mathrm{W}}}{P}_{\mathrm{W}i,t}})\\ F_{\mathrm{r}\mathrm{p}\mathrm{s}}={\mu }_{\mathrm{r}\mathrm{p}\mathrm{s}}{Q}_{\mathrm{r}\mathrm{p}\mathrm{s}}\end{array}$$

(20)

where $\kappa$ represents the conversion coefficient of the actual power generation of renewable energy into the number of green certificates; ${\mu }_{\mathrm{r}\mathrm{p}\mathrm{s}}$ and $\varphi$ denote the green certificate price and the minimum consumption coefficient of renewable energy, respectively; $Q_{\mathrm{r}\mathrm{p}\mathrm{s}}$ indicates the additional amount of green certificates held by operators or the amount of green certificates to be purchased.

Minimize the total direct carbon emissions of the power grid as follows:

$$F_2=\mathrm{m}\mathrm{i}\mathrm{n}{\displaystyle \sum _{t=1}^T({\displaystyle \sum _{i=1}^{N_{\mathrm{P}\mathrm{G}}}{E}_{\mathrm{G},i}}}+{\displaystyle \sum _{j=1}^{N_{\mathrm{P}\mathrm{W}}}{E}_{\mathrm{W},j}})$$

(21)

where $E_{\mathrm{G},i}$ represents the carbon emissions of the conventional thermal power unit i of the system, and $E_{\mathrm{W},j}$ represents the carbon emissions of the wind turbine j of the system.

The difference in direct carbon emissions between different regions is as follows:

$$F_3=\mathrm{m}\mathrm{i}\mathrm{n}{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_{\mathrm{\Omega }}-1}{\displaystyle \sum _{j=i+1}^{N_{\mathrm{\Omega }}}\left|E_{\mathrm{G}}^{{\mathrm{\Omega }}_i}-E_{\mathrm{G}}^{{\mathrm{\Omega }}_j}\right|}}}$$

(22)

where $E_{\mathrm{G}}^{{\mathrm{\Omega }}_i}$ and $E_{\mathrm{G}}^{{\mathrm{\Omega }}_j}$ represent the total carbon emissions of regions ${\mathrm{\Omega }}_i$ and ${\mathrm{\Omega }}_j$, respectively. The smaller the difference in carbon emissions between different regions, the more balanced the regional carbon emissions.

4.1.2. Constraint Condition

The system power balance constraints are as follows:

$${\displaystyle \sum _{n\in {\mathrm{\Omega }}_{\mathrm{G}i}}{P}_{\mathrm{G}i,t}^n}+{\displaystyle \sum _{m\in {\mathrm{\Omega }}_{\mathrm{W}i}}{P}_{\mathrm{W}i,t}^m}={\displaystyle \sum _{k\in {\mathrm{\Omega }}_{\mathrm{B}i}}{P}_{\mathrm{B}i,t}^k}+P_{\mathrm{L}i,t}$$

(23)

where $P_{\mathrm{G}i,t}^n$ and $P_{\mathrm{W}i,t}^m$ are the injection power of the thermal power unit and the wind turbine at node i at time t, respectively, $P_{\mathrm{B}i,t}^k$ and $P_{\mathrm{L}i,t}$ are the branch power and load power connected to node i at time t, respectively.

The system node voltage security constraints are as follows:

$$V_{i,t}^{\mathrm{m}\mathrm{i}\mathrm{n}}⩽V_{i,t}⩽V_{i,t}^{\mathrm{m}\mathrm{a}\mathrm{x}}$$

(24)

where $V_{i,t}$ is the voltage of node i at time t, $V_{i,t}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $V_{i,t}^{\mathrm{m}\mathrm{i}\mathrm{n}}$ are the upper and lower limits of the voltage of node i at time t.

The unit operation constraints are as follows:

$$\begin{array}{c}{R}_{Gi}^{\mathrm{m}\mathrm{i}\mathrm{n}}⩽\left|P_{\mathrm{G}i,t}-P_{\mathrm{G}i,t-1}\right|⩽R_{\mathrm{G}i}^{\mathrm{m}\mathrm{a}\mathrm{x}}\\ P_{\mathrm{G}i}^{\mathrm{m}\mathrm{i}\mathrm{n}}⩽P_{\mathrm{G}i,t}⩽P_{\mathrm{G}i}^{\mathrm{m}\mathrm{a}\mathrm{x}}\\ P_{\mathrm{W}j}^{\mathrm{m}\mathrm{i}\mathrm{n}}⩽P_{\mathrm{W}j,t}⩽P_{\mathrm{W}j}^{\mathrm{m}\mathrm{a}\mathrm{x}}\end{array}$$

(25)

where $R_{\mathrm{G}i}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $R_{\mathrm{G}i}^{\mathrm{m}\mathrm{i}\mathrm{n}}$ represent the uphill and downhill climbing rates of the i-th thermal power unit, $P_{\mathrm{G}i}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $P_{\mathrm{G}i}^{\mathrm{m}\mathrm{i}\mathrm{n}}$ represent the upper and lower limits of the active power of the i-th thermal power unit, $P_{\mathrm{W}j}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $P_{\mathrm{W}j}^{\mathrm{m}\mathrm{i}\mathrm{n}}$ represent the upper and lower limits of the active power of the j-th wind turbine.

The branch flow constraints are as follows:

$$\left|P_{\mathrm{B}i}\right|⩽P_{\mathrm{B}i}^{\mathrm{m}\mathrm{a}\mathrm{x}}$$

(26)

where $P_{\mathrm{B}i}$ represents the active power of branch i, and $P_{\mathrm{B}i}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ represents the upper limit of active power of branch i.

4.2. Lower Level Model: Economic Dispatch Model of Load Aggregator under Electricity–Carbon Dual Demand Response

Building on the analysis of demand response in the previous section, the lower model establishes an electricity–carbon dual demand response framework from the perspective of the load aggregator, to minimize the total cost of the load aggregator, as follows:

$$F=\mathrm{m}\mathrm{i}\mathrm{n}(F_{\mathrm{B}}+F_{\mathrm{c}\mathrm{e}\mathrm{t},\mathrm{D}}+F_{\mathrm{C}\mathrm{L},\mathrm{T}\mathrm{L}})$$

(27)

where $F_{\mathrm{B}}$ represents the power purchase cost of the load aggregator to the superior, $F_{\mathrm{c}\mathrm{e}\mathrm{t},\mathrm{D}}$ denotes the carbon transaction cost, and $F_{\mathrm{C}\mathrm{L},\mathrm{T}\mathrm{L}}$ represents the subsidy cost of the flexible load demand response managed by the aggregator.

The power purchase cost of the load aggregator from the superior is as follows:

$$F_{\mathrm{B}}={\displaystyle \sum _{t=1}^T{\mathsf{\rho }}_t{P}_t^{\mathrm{B}}}\Delta t$$

(28)

where $P_t^{\mathrm{B}}$ indicates that the load operator purchases power from the superior at time t, and ${\mathsf{\rho }}_t$ indicates time-of-use tariff.

For the carbon trading costs incurred by the load aggregator to participate in the carbon market, the carbon emissions of the load are first calculated using the dynamic carbon emission factor. The difference between this calculated emission and the initial carbon quota represents the carbon quota trading volume of the flexible load in the carbon market, as shown in Equation (29). This paper employs a step-by-step carbon transaction cost approach, with the specific expression detailed in Equation (30), as follows:

$$\begin{array}{c}{E}^{tr}=E^D-E^Q\\ E^Q={\displaystyle \sum _{t=1}^{N_t}{E}^{\mathrm{q}\mathrm{u}}}(P_t^{st}+P_t^{\mathrm{T}\mathrm{L},1}-P_t^{\mathrm{T}\mathrm{L},2}-P_t^{\mathrm{C}\mathrm{L}})\Delta t\end{array}$$

(29)

$$F_{\mathrm{c}\mathrm{e}\mathrm{t},\mathrm{D}}=\left\{\begin{array}{cc}\begin{array}{l}\sigma E^{tr}\\ \sigma (1+\omega )(E^{tr}-d)\\ \sigma (1+2\omega )(E^{tr}-2d)+\sigma (2+\omega )d\\ \sigma (1+3\omega )(E^{tr}-3d)+\sigma (3+3\omega )d\\ \sigma (1+4\omega )(E^{tr}-4d)+\sigma (4+6\omega )d\end{array}& \begin{array}{l}{E}^{tr}⩽d\\ d⩽E^{tr}⩽2d\\ 2d⩽E^{tr}⩽3d\\ 3d⩽E^{tr}⩽4d\\ E^{tr}>4d\end{array}\end{array}\right.$$

(30)

where $E^Q$ represents the initial carbon quota of the flexible load, $E^{\mathrm{q}\mathrm{u}}$ denotes the carbon quota coefficient corresponding to the unit purchase quantity, $\sigma$ represents the benchmark price of carbon trading, $\omega$ denotes the price growth coefficient, and d denotes the price growth interval.

The aggregator subsidizes the cost of demand response for the managed flexible load, as follows:

$$F_{\mathrm{C}\mathrm{L},\mathrm{T}\mathrm{L}}={\displaystyle \sum _{t=1}^T({\delta }^{\mathrm{C}\mathrm{L}}{P}_t^{\mathrm{C}\mathrm{L}}+{\delta }_{{ }^1}^{\mathrm{T}\mathrm{L}}{P}_t^{\mathrm{T}\mathrm{L},1}}+{\delta }_{{ }^2}^{\mathrm{T}L}{P}_t^{\mathrm{T}\mathrm{L},2})\Delta t$$

(31)

where ${\delta }^{\mathrm{C}\mathrm{L}}$, ${\delta }_{{ }^1}^{\mathrm{T}\mathrm{L}}$ and ${\delta }_{{ }^2}^{\mathrm{T}\mathrm{L}}$, respectively, represent the unit load reduction subsidy coefficient, the unit transferable load transfer-in and transfer-out subsidy coefficient under the demand response stipulated in the incentive contract, $P_t^{\mathrm{C}\mathrm{L}}$, $P_t^{\mathrm{T}\mathrm{L},1}$ and $P_t^{\mathrm{T}\mathrm{L},2}$, respectively represent the power reduction, transfer-in power, and transfer-out power under the demand response.

The constraints include the total power balance constraints of the system, and the specific expression is as follows:

$$P_t^{\mathrm{B}}=P_t^{st}+P_t^{\mathrm{T}\mathrm{L},1}-P_t^{\mathrm{T}\mathrm{L},2}-P_t^{\mathrm{C}\mathrm{L}}$$

(32)

4.3. Model Solving

4.3.1. Non-Dominated Sorting Genetic Algorithm-II

The low-carbon economic dispatch conducted by the upper grid operator is a multi-objective optimization problem. This paper employs the second-generation non-dominated sorting genetic algorithm (NSGA-II) to solve it. NSGA-II is known for its speed and effectiveness in elite multi-objective optimization. Compared to other algorithms, NSGA-II excels in achieving a well-distributed set of solutions and superior convergence near the Pareto optimal frontier.

NSGA-II builds upon NSGA with three key improvements: It introduces a fast non-dominated sorting method, incorporates crowding distance and crowding comparison operators, and implements an elite strategy selection. These enhancements significantly enhance NSGA-II’s computational efficiency and optimization performance.

4.3.2. Fuzzy Satisfaction Decision-Making Based on the Logistic Membership Function

In the last section, the NSGA-II algorithm can obtain the Pareto optimal frontier set of the upper-level multi-objective model. In multi-objective optimization problems, since the Pareto solution set is not unique, the key point is the degree of preference for the Pareto solution. This paper adopts the interactive fuzzy satisfaction decision based on the logistic membership function proposed in the literature [17] to obtain the “optimal” scheme of the Pareto set, and select the Sigmoid membership function (monotonically decreasing in [0, 1]) to fuzzy the original objective function, as shown in Equation (33), as follows:

$$L_{F_i}(X)={\displaystyle \frac{1}{1+e^{{\tau }_i(N[F_i(X)]-{\lambda }_i)}}},i=1,2,3;X\in \Psi$$

(33)

where ${\tau }_i$ and ${\lambda }_i$ represent the parameters of the Sigmoid membership function, which are used to determine the fuzzy value and the intermediate point of $F_i$, respectively, where ${\tau }_i$ = 10, ${\lambda }_i$ = 0.5; $N(·)$ represents the objective function after linear normalization; $\Psi$ represents the Pareto optimal solution set.

Reference [17] introduces the concept of reference membership to deal with multi-objective problems. First, the decision-maker should set the reference membership value for each objective, and then convert the multi-objective problem into a min–max problem by finding the difference between the reference membership value and the membership function target value, as shown in Equation (34):

$$\mathrm{m}\mathrm{i}\mathrm{n}\left.\left\{\mathrm{m}\mathrm{a}\mathrm{x}(\left|u_{rp}-L_{F_i}(X)\right|\right.)\right\},i=1,2,3;X\in \mathrm{\Psi };u_{rp}\in [0,1]$$

(34)

where u_rp represents the reference membership value given by the decision maker, which can characterize the importance of its corresponding target. The larger u_rp is, the more important F_i(X) is.

4.3.3. Model Solving Process

Figure 5 illustrates the solution process of the two-layer model. The optimal solution of the upper multi-objective model is obtained using NSGA-II and fuzzy satisfaction decision-making. The lower model utilizes the Gurobi solver, interacting iteratively with the upper layer. The dynamic carbon emission factor signal updated by the upper layer serves as a parameter for the lower layer. Based on this parameter, the load aggregator formulates incentive contracts and time-of-use tariffs. Through price incentives, flexible loads adjust their electricity consumption behaviors. The aggregator then transmits optimized load data back to the upper layer for further iterative solutions. When $\left|P_{t,\sigma }^{\mathrm{B}}-P_{t,\sigma -1}^{\mathrm{B}}\right|<\mathsf{\varsigma },\mathsf{\varsigma }=0.05 \mathrm{M}\mathrm{W}$, the solution process ends and the optimal result is output.

5. Case Analysis

5.1. Parameter Settings

In this paper, the improved New England 39-node system is used for simulation analysis. The system topology and area division are shown in Figure 6. The operating parameters of the system are shown in

Table 1. System operating parameter.

Parameter	Numerical Value
The cost of wind power generation (CNY/(MW·h))	200
Penalty cost for wind power curtailment (CNY/(MW·h))	150
Carbon quota per unit of electricity (t/(MW·h))	0.639
Carbon prices (CNY/t)	45
Green Certificate Price (CNY/copy)	200
Minimum consumption weight for renewable energy	5%
Conversion coefficient of renewable energy and green certificate quantity	1
Compensation coefficient of unit weakened energy (CNY/(MW·h))	34.25
Unit transfer, transfer capacity compensation coefficient (CNY/(MW·h))	34.25

, and the flexible load subsidy cost reference [14].

5.2. Upper Model Optimization Results Analysis

To validate the rationality and effectiveness of the multi-objective model proposed by the upper layer and considering the impact of wind turbine placement on unit scheduling results and regional carbon emission management, three scenarios are configured without optimizing the scheduling of load demand response: mode 1 is the wind turbine 1 access area 1; mode 2 is the wind turbine 1 access area 2; mode 3 is the wind turbine 1 access area 3.

5.2.1. Pareto Optimal Set Analysis

Figure 7a,b and Figure 8 depict the Pareto optimal frontier sets of multi-objective functions for the three models. It is evident that the Pareto optimal frontier sets exhibit considerable variation across the three models, providing decision-makers with numerous unit scheduling options to choose from. All three models demonstrate that reducing the total direct carbon emissions (F₂) of the power grid leads to a notable increase in the total cost (F₁) for the power grid operator. Similarly, when aiming to enhance the balance of regional carbon emissions while keeping the total direct carbon emissions unchanged, the total cost for the power grid operator also increases significantly. Moreover, the choice of wind turbine access region 1 exerts a significant influence on the fuzzy optimization outcomes.

5.2.2. Influence of Wind Turbine Access Location on Dispatching Results

As shown in

Table 2. The fuzzy optimization results in each mode.

Mode Type	Objective Function Value			Regional Carbon Emissions/Ten Thousand Tons
	F1/CNY 10,000	F2/Ten Thousand Tons	F3/Ten Thousand Tons	Area 1	Area 2	Area 3
Mode 1	631.0868	9.2586	8.4524	5.3613	2.7622	1.1351
Mode 2	630.0857	9.3034	8.5521	5.4021	2.7751	1.1262
Mode 3	632.1659	9.3029	8.4820	5.3834	2.7771	1.1424

, the fuzzy optimization results under each mode show that the three objective function values under mode 1 are small, and the fuzzy optimization effect is good. In addition, the direct carbon emissions of area 1 under mode 1 are the lowest. This is because the carbon emission intensity of each unit in region 1 is large and belongs to the heavy load area. After wind turbine 1 is connected to area 1, the output of units with large conventional carbon emission intensity will be limited. The overall carbon emissions in the region decrease and the difference between carbon emissions in other regions is also greatly reduced. Therefore, integrating wind turbines into area 1 can improve the economy and carbon efficiency of regional dispatching.

5.3. Multi-Objective Bi-Level Model Optimization Results

To validate the electricity–carbon dual demand response mechanism proposed in this paper and assess the system’s low-carbon and economic aspects following carbon trading and green certificate trading, the following comparative examples are established:

In scenarios where only low-carbon economic dispatch is considered at the upper grid operator level, the following cases are set up:

Scene 1: only considering the power demand response mechanism;

Scene 2: only consider the low-carbon demand response mechanism;

Scene 3: considering electricity–carbon dual demand response mechanism, without considering carbon trading;

Scene 4: consider the electricity–carbon dual demand response mechanism, without considering the green certificate transaction;

Scene 5: consider the electricity–carbon dual demand response mechanism, taking into account both carbon trading and green certificate trading;

At the upper grid operator level, which considers both low-carbon economic dispatch and regional carbon emission balance, the following scenario is established:

Scene 6: consider the electricity–carbon dual demand response mechanism.

5.3.1. Multi-Objective Bi-Level Model Scheduling Results

As shown in

Table 3. Comparison of results in different operating scenarios.

Name of System Operating Parameters	Scene 1	Scene 2	Scene 3	Scene 4	Scene 5
Total system operating cost/CNY 10,000	510.3041	513.4438	486.7663	538.0776	502.1581
Direct carbon emissions from the grid/Ten thousand tons	8.2407	8.2672	8.1481	8.1376	8.1376
Operating costs of thermal power units/CNY 10,000	370.6182	373.1072	364.7025	364.8986	364.8975
Wind power operating cost/CNY 10,000	168.2840	168.2840	168.2840	157.9826	168.2840
Operators’ carbon trading costs/CNY 10,000	16.4421	16.8028	-	15.1964	15.1968
Green certificate transaction costs/CNY 10,000	−45.0402	−44.7502	−46.2202	-	−46.2202

, the economic and low-carbon impact of scenario 1 is slightly better than that of scenario 2. The total operating cost of the system is reduced by 0.611% compared to scenario 2, and the total direct carbon emissions of the power grid decrease by 0.321% compared to scenario 2. Scenario 5 demonstrates superior economic and low-carbon dispatching results compared to scenarios 1 and 2. The total operating cost of the system decreases by 1.596% and 2.198%, respectively, and the total direct carbon emissions of the power grid decrease by 1.251% and 1.568%, respectively, compared to scenarios 1 and 2. These results indicate that the electricity–carbon dual demand response mechanism proposed in this paper can significantly reduce the direct carbon emissions of the power grid while controlling the operating costs of the system, thereby achieving a balance between economic efficiency and carbon reduction goals.

Comparing scenarios 3, 4, and 5, scenario 5 demonstrates superior system economics compared to scenario 4, and better low-carbon performance compared to scenario 3. The total operating cost of the system is reduced by 8.429%, and the total carbon emissions are reduced by 0.129% compared to scenario 3. Figure 9 illustrates that the collaboration between carbon trading and green certificate trading reduces reliance on units with high carbon emissions intensity and promotes the development and utilization of renewable energy to a certain extent.

5.3.2. Optimal Scheduling Results Considering Regional Carbon Emission Balance

As shown in

Table 4. Bi-level optimization results considering regional carbon emission balance.

Scene Type	Objective Function Value			Regional Carbon Emissions/Ten Thousand Tons
	F1/CNY 10,000	F2/Ten Thousand Tons	F3/Ten Thousand Tons	Region 1	Region 2	Region 3
Mode 3	632.1659	9.3029	8.4820	5.3834	2.7771	1.1424
Scene 6	546.2539	8.5878	8.0038	4.8694	2.8509	0.8675

, it can be observed that in scene 6, the three objective function values are significantly reduced by 13.590%, 7.687%, and 5.638% compared to mode 3. Scene 6 expands upon considerations of economic efficiency and carbon efficiency to further incorporate regional carbon emission balance, effectively preventing excessive carbon emissions in specific regions. Furthermore, comparing scene 6 with scene 5 in Table 3 reveals that taking the regional carbon emission balance into account results in a slight increase in both the operational costs and total carbon emissions of the power grid. This variation is influenced by the preferences of decision-makers for target values, where different reference membership degrees will impact target selection.

5.3.3. The Influence of Different Reference Membership on Fuzzy Optimization Results

As depicted in

Table 5. The optimization results under different reference membership degrees in scene 6.

Reference Membership urp			Objective Function Value
F1	F2	F3	F1/CNY 10,000	F2/Ten Thousand Tons	F3/Ten Thousand Tons
1	1	1	546.2539	8.5878	8.0038
1	1	0.8	545.4726	8.5714	8.2040
1	1	0.5	544.1835	8.5539	8.5221
1	1	0.1	545.7719	8.5461	8.9894
1	0.5	1	532.6980	8.6877	6.6390
0.5	1	1	575.6563	8.2790	8.2976

, when decision-makers assign equal importance to objective functions F₁ and F₂, with a decrease in emphasis on F₃, the target value of F₃ increases significantly, while the target values of F₁ and F₂ decrease slightly. Conversely, if the emphasis on F₂ is appropriately reduced and the emphasis on F₃ is increased, the target values of F₃ and F₁ decrease significantly. Therefore, decision-makers should select an appropriate solution by adjusting the emphasis on each objective, allowing for improvements in F₁ and F₃ while slightly increasing F₂.

5.3.4. Comparative Analysis of the Situation before and after Load Scheduling

Figure 10a, Figure 10b and Figure 10c show the comparison before and after load scheduling under scenarios 1, 2, and 5, respectively. The load adjustment rates are 12.381%, 8.194%, and 13.447%, respectively. In scene 5, the increase of the user’s electricity consumption in the off-peak period of 2:00–7:00 and the decrease of the user’s electricity consumption in the peak period of 18:00–23:00 are both better than those in scene 1 and scene 2. It embodies the advantages of the power-carbon demand response optimization model proposed in this paper.

5.3.5. Dynamic Carbon Emission Factor Distribution Results

Figure 11 depicts the distribution of the system’s dynamic carbon emission factors in scenarios 1 and 5 after optimization. It is evident that in scenario 5, both the overall temporal and spatial distribution of carbon emission factors is lower compared to scenario 1. The adjustment rate of carbon emission factors is 1.81%.

5.3.6. Comparative Analysis of Regional Carbon Emission Factors on the Power Generation Side and Demand Side

As shown in Figure 12, the change rate of carbon emission factor in the region adjacent to the power generation side and the demand side is compared and analyzed. It is evident that the change in the electric carbon emission factor on the demand side is significantly greater than that on the power generation side on the time scale. Compared with the power generation side, the increase of carbon emission factor from 6:00 to 7:00 on the demand side is obviously greater than the reduction of carbon emission factor from 7:00 to 8:00. The aggregator can opt to shift electricity loads from 7:00 to 8:00 to the period from 6:00 to 7:00, providing strong guidance for the commercial electricity behavior of the aggregator. This strategy facilitates electricity consumption during periods of low-carbon emissions.

6. Conclusions

In the context of low-carbon initiatives, user behavior significantly influences the carbon emissions of the system. To fully harness the potential for reducing carbon emissions on the demand side, this paper primarily explores the multi-objective low-carbon optimal scheduling strategy of source-load coordination under demand response. Firstly, considering the spatial and temporal differences of indirect carbon emissions in the power system, a complex power flow tracking method considering the power supply path is innovatively proposed, and a carbon flow tracking model is established to obtain the dynamic electricity carbon emission factor. Secondly, in the low-carbon economic dispatch of electric power, the electricity–carbon dual demand response mechanism is innovatively proposed. The dynamic carbon emission factor, time-of-use tariff, and demand response load energy consumption are simultaneously used as the coupling point of source-load interaction, and the regional carbon emission balance is taken into consideration. A multi-objective bi-level optimization model of source-load coordination under the dual demand response of electricity and carbon is established. Different from the traditional optimization model, this paper uses the NSGA-II multi-objective optimization algorithm to obtain the Pareto optimal frontier solution of the upper model and finds the ‘optimal’ solution based on the fuzzy satisfaction decision of the logistic membership function, and then iterates with the lower layer. Finally, the validity of the model and the solution method is verified by an example. The main conclusions are as follows:

(i) Wind power should be connected to areas with high carbon emission intensity and heavy load to effectively balance the disparity in carbon emissions between regions and reduce the overall carbon emissions of the system.

(ii) Compared with the single-signal demand response, the source-load interaction carbon reduction effect under the dual-signal guidance of dynamic carbon emission factor and time-of-use electricity price is better. The total operating cost of the dual mechanism is reduced by 1.596% and 2.198% compared to using only the power demand response mechanism and the low-carbon demand response mechanism, respectively. Additionally, the total direct carbon emissions of the grid under the dual mechanism are reduced by 1.251% and 1.568% compared to the power demand response mechanism and the low-carbon demand response mechanism, respectively. Implementing a dual demand response mechanism can fully tap into the carbon reduction potential of flexible loads, enhancing both the economic efficiency and the low-carbon performance of the system.

(iii) If regional carbon emission balance is considered alongside dual demand response, the operational costs and total carbon emissions of the power grid may increase slightly, depending on the decision maker’s preferences for target values. It is crucial to select an appropriate reference membership value to ensure that the model functions effectively as a carbon emission management tool in the hands of power grid operators.

In the future, more in-depth research on low-carbon electricity can be carried out from the following two aspects:

(i) The carbon flow tracking model established in this paper can realize the tracking of carbon information in the whole process of source–network load, but it does not study the real-time measurement technology of carbon information. Therefore, in the future, the power system carbon meter device can be studied to realize the ‘minute-level’ carbon measurement by monitoring the real-time carbon information of the system source–network load.

(ii) In this paper, renewable energy is connected to the grid under the premise of ensuring the safe and stable operation of the power system, and the low-carbon optimal operation of the regional power grid is realized through the friendly interaction between the source and the load. The low-carbon optimal operation strategy considering the security and stability constraints under the high proportion of renewable energy has not been studied, and the problem of network-load interaction has not been involved. Therefore, in the future, research can be carried out on the safe and stable operation of the system after the high proportion of renewable energy is connected to the grid, the network-load interaction technology, and other issues.