Analysis of Power System Power and Energy Balance Considering Demand-Side Carbon Emissions

Abstract

As substantial incorporation of variable renewable generation technologies, particularly wind and photovoltaic systems, becomes more common, the complexities of power supply and demand characteristics are increasing, making it essential to conduct a detailed power and energy balance analysis. Aiming at regional power systems with multi-source structures and internal transmission interface constraints, this paper proposes a power and energy balance analysis method that considers demand-side carbon emissions. First, a closed-loop mechanism of “carbon signal–load response–balance optimization” based on nodal carbon potential (NCP) is constructed. In this framework, NCP is utilized to generate carbon signals that guide the active response of flexible loads, which are subsequently integrated into the coordinated optimization of power and energy balance. Second, a power and energy balance optimization model adapted to multi-source structures is established, where transmission power limits between zones are directly embedded into the constraint system, overcoming the defects of traditional heuristic methods that require repeated iterations to correct interfaces. Finally, an improved hybrid solution strategy for large-scale balance analysis is designed, significantly reducing the variable scale through the aggregation of similar units within zones. Case studies show that this method can effectively guide the load to shift toward low-carbon periods and nodes, significantly reducing total system carbon emissions and improving renewable energy consumption while ensuring power and energy balance.

1. Introduction

In the grand backdrop of active global response to climate change, constructing a novel power system underpinned by renewable energy sources has become a crucial initiative for driving energy structure transformation [1,2,3]. Currently, as the share of volatile power sources like wind and solar energy capacities is steadily expanding across the electrical network, the equilibrium dynamics of supply and demand in the electrical grid have undergone a fundamental shift, posing severe challenges to the traditional “generation following load” mode [4,5,6]. Meanwhile, as the main battlefield for carbon emission reduction, the power system can no longer achieve the dual objectives of economic efficiency and sustainable operation by relying solely on clean substitution on the supply side; there is an urgent need to excavate the regulation potential of demand-side resources [7,8,9,10]. Deeply integrating carbon emission indicators into power and energy balance analysis, as well as utilizing carbon signals to guide active response on the load side, holds considerable theoretical importance as well as engineering value for achieving low-carbon operation while ensuring regional system adequacy and section security.

Revolving around power and energy balance analysis, both domestic and international researchers have conducted extensive investigations regarding balance method systems, low-carbon characteristic characterization, and model solution strategies.

Regarding calculation methods for power and energy balance, reference [11] focused on hydro-thermal systems, conducting static balancing based on annual maximum loads and representative hydrological years. To address the increasing complexity of modern grids, reference [12,13] reviewed various uncertainty modeling methods. However, most engineering practices remain deterministic. As pointed out in reference [14], static methods based on single or average scenarios struggle to capture the time-sequential operational characteristics under the strong stochasticity of new energy and extreme weather conditions, which often leads to significant deviations in balance results.

In the research of low-carbon characteristics and source–load interaction, references [15,16] explored pathways to enhance the low-carbon economy of systems by introducing carbon trading mechanisms. In particular, reference [17] utilized the carbon emission flow (CEF) theory to deeply couple carbon flow with power flow, enabling real-time tracking of emissions. Subsequent studies such as [18,19] further advanced carbon estimation methods and energy routing strategies, respectively. Nevertheless, these studies primarily focus on post-assessment or excavating the emission reduction potential of the source side. Although reference [20] attempted to guide active distribution systems using carbon signals, there is a scarcity of studies that directly embed the nodal carbon intensity into long-cycle power and energy balance models to establish a closed-loop feedback mechanism for diverse flexible loads.

Regarding the topological modeling and spatial synergy of regional multi-zone power grids, reference [21] criticized the traditional single-node “copper plate model” and emphasized the importance of capturing operational details. While reference [22] proposed incorporating network operation constraints to ensure engineering feasibility, reference [23] noted that refined network models often result in an exponential increase in variables, leading to a “curse of dimensionality” in long-cycle simulations. Furthermore, regarding inter-zonal coordination, reference [24] investigated coordinated transaction scheduling in multi-area markets. However, existing zonal balance models mostly adopt fixed transmission schedules and lack flexible mechanisms to dynamically optimize tie-line power based on the supply–demand status and carbon intensity signals of each zone, making it difficult to fully leverage the advantages of spatial resource mutual aid.

However, a deeper analysis of the aforementioned studies reveals significant research gaps in achieving profound “carbon–electricity” coordination. On the one hand, existing power and energy balance research has yet to achieve a deep coupling between nodal carbon intensity and the system’s long-cycle operational logic. Consequently, carbon emission factors are predominantly treated as independent external constraints, resulting in a lack of a closed-loop feedback architecture capable of utilizing real-time nodal carbon potential (NCP) signals to guide active demand-side adjustments. On the other hand, in complex scenarios involving multi-regional interconnections, current balance optimization models often simplify the treatment of inter-zonal transmission constraints, typically relying on preset, fixed power transfer schedules. This overlooks the potential for dynamic synergistic optimization between source-side low-carbon power and demand-side flexible response through the grid infrastructure, particularly under stringent cross-section security limits.

To address the practical challenges and pain points where conventional regional power systems struggle to balance “carbon signal guidance” with “computational efficiency for large-scale analysis” in long-cycle studies, this paper proposes a power and energy balance analysis method considering demand-side carbon emissions. The primary contributions are as follows: First, a closed-loop feedback mechanism of “carbon signal–load response–balance optimization” based on NCP is constructed. By conveying carbon responsibility to the demand side, this mechanism achieves deep coordination between decarbonization goals and power balance, reducing system carbon emissions by 22.34%. Second, a balance optimization model incorporating inter-regional transmission constraints is established to ensure precise verification of physical boundary limits, enhancing the renewable energy consumption rate from 82.5% to 100%. Finally, an improved hybrid solution strategy featuring resource aggregation is designed to effectively mitigate the “curse of dimensionality” in large-scale balance analysis [25], achieving convergence within an average of only five iterations.

The remainder of this manuscript is organized as follows: Section 2 introduces the framework for power and energy balance analysis. Section 3 presents the power and energy balance model based on carbon flow guidance. Section 4 introduces the solution process of this paper. Section 5 presents the basic data and result analysis. Finally, Section 6 concludes this paper.

2. Power and Energy Balance Analysis Framework

This paper constructs a regional power system model encompassing multi-source generation structures and internal transmission-section constraints. On this basis, a closed-loop mechanism of “carbon signal–load response–balance optimization” based on NCP is introduced. Consequently, a comprehensive power and energy balance analysis framework considering demand-side carbon emissions is established, as illustrated in Figure 1.

From a physical topology perspective, the regional power grid is divided into multiple supply zones by transmission cross-sections, and power exchanges among zones are strictly constrained by cross-section transmission capacities. On the supply side, the system integrates diversified resources including wind power, photovoltaic generation, energy storage systems, and thermal power units. Regarding the demand side, flexible assets like electric vehicles (EVs) [26], curtailable loads, and shiftable loads are aggregated [27].

From an operational mechanism perspective, an interactive iterative framework between the grid operator and load aggregators is established. The grid operator formulates generation schedules by reducing generation expenses and costs associated with carbon trading [28] based on carbon allowance coefficients from the carbon trading platform and computes NCP using CEF tracing techniques [29]. The NCP signals, together with time-of-use electricity prices, are then transmitted to load aggregators. Based on these signals, aggregators design incentive prices to guide flexible load responses and determine optimal demand adjustments by comprehensively considering electricity procurement and carbon trading costs [30,31,32]. The grid operator subsequently updates generation schedules and NCP in response to load variations. Through iterative information exchange and adjustment, coordinated optimization and balance between supply and demand are ultimately achieved.

3. Power and Energy Balance Model Based on Carbon Flow Guidance

3.1. Carbon-Flow-Tracking-Driven Demand Response Model

3.1.1. Carbon Flow Tracking Model Grounded in the Principle of Proportional Sharing

To meet the requirement for refined analysis of demand-side carbon emissions, this model adopts a carbon flow tracking approach grounded in the proportional sharing principle. This principle has been widely proven to satisfy both nodal power conservation and carbon mass conservation, making it the method most consistent with physical laws currently available for mapping power–carbon flows. By tracking carbon emissions originating from the generation sector along the DC power flow, NCPs are calculated, and carbon emission responsibilities are accurately allocated to each load node, effectively revealing the distribution of carbon elements within power flows.

In this paper, let $\Omega$ denote the set of zones ${\Omega }^{\mathrm{g}},{\Omega }^{\mathrm{w}},{\Omega }^{\mathrm{p}\mathrm{v}}$, and ${\Omega }^{\mathrm{e}\mathrm{s}s}$, which denote the set of regions connected with thermal power, wind power, photovoltaic power, and energy storage devices, respectively. It is assumed that each region i contains M nodes, among which I nodes are connected to generation units and K nodes have loads. The NCP of node m is determined by the carbon emissions generated by connected generators and the carbon emissions flowing in from other nodes:

$$e_m^{\mathrm{R}}={\displaystyle \frac{{\displaystyle \sum _{s\in S^{+}}}{P}_s^{\mathrm{B}}{\rho }_s+P_i^{\mathrm{g}}{e}_i^{\mathrm{g}}}{{\displaystyle \sum _{s\in S^{+}}}{P}_s^{\mathrm{B}}+P_i^{\mathrm{g}}}}{P}_i^{\mathrm{g}}$$

(1)

where $S^{+}$ denotes the collection of branches delivering power to node m; ${\rho }_s$ denotes the carbon flow density along branch s; $P_s^{\mathrm{B}}$ denotes the active power of branch s; $e_i^{\mathrm{g}}$ denotes the carbon emission intensity for generator i; $P_i^{\mathrm{g}}$ denotes the active power generated by unit i.

The carbon emissions of node m during the dispatching period are expressed as

$$D_m^{\mathrm{L}}={\displaystyle \sum _{t\in T}{P}_{m,t}^{\mathrm{b}\mathrm{u}\mathrm{y}}{e}_{m,t}^{\mathrm{R}}\Delta t}$$

(2)

where T denotes the scheduling period; $P_{m,t}^{\mathrm{b}\mathrm{u}\mathrm{y}}$ denotes the load at node m during time t; $\Delta t$ denotes the time step; $e_{m,t}^{\mathrm{R}}$ denotes the NCP of node m at time t.

3.1.2. Flexible Load Response Model Based on NCP

Building upon the acquisition of NCP data via carbon flow tracing technology, this paper introduces an incentive mechanism to guide the load side in adjusting electricity consumption behavior guided by carbon potential signals. In order to quantify the carbon emission levels subsequent to demand response, a flexible load response framework driven by NCP is constructed. This model specifically encompasses three types of flexible resources: EVs, interruptible loads, and transferable loads. The associated carbon emission calculation formulas are presented as follows.

1.

Carbon Emission Model of EVs

The carbon output of EVs $D_t^{\mathrm{E}\mathrm{V}}$ is determined by taking the emissions generated from charging and subtracting the reductions in emissions that occur during discharging at time t:

$$D_t^{\mathrm{E}\mathrm{V}}=e_{m,t}^{\mathrm{R}}{P}_t^{\mathrm{E}\mathrm{V}}\Delta t$$

(3)

$$P_t^{\mathrm{E}\mathrm{V}}={\displaystyle \sum _{n\in N^{ev}}(P_{n,t}^{\mathrm{e}\mathrm{v}\mathrm{c}}-P_{n,t}^{\mathrm{e}\mathrm{v}\mathrm{d}})}$$

(4)

where $P_t^{\mathrm{E}\mathrm{V}}$ denotes the aggregate charge/discharge power of all EVs connected to the grid at time t; $N^{ev}$ denotes the number of EVs; $P_{n,t}^{\mathrm{e}\mathrm{v}\mathrm{c}}$ and $P_{n,t}^{\mathrm{e}\mathrm{v}\mathrm{d}}$ denote the power drawn or injected by the n-th EV during time interval t.

2.

Carbon Emission Model of Curtailable/Transferable Loads

The carbon emissions from curtailable and transferable loads following demand response are represented as

$$D_t^{\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}=e_{m,t}^{\mathrm{R}}(P_t^{\mathrm{t}\mathrm{r}\mathrm{a},\mathrm{i}\mathrm{n}}-P_t^{\mathrm{t}\mathrm{r}\mathrm{a},\mathrm{o}\mathrm{u}\mathrm{t}}-P_t^{\mathrm{c}\mathrm{u}\mathrm{t}})\Delta t$$

(5)

where $P_t^{\mathrm{t}\mathrm{r}\mathrm{a},\mathrm{i}\mathrm{n}}$ and $P_t^{\mathrm{t}\mathrm{r}\mathrm{a},\mathrm{o}\mathrm{u}\mathrm{t}}$ denote the transferred-in and transferred-out load of transferable loads at time t; $P_t^{\mathrm{c}\mathrm{u}\mathrm{t}}$ denotes the load amount shed at time t.

Therefore, the actual carbon emissions of loads in each region can be obtained as

$$D^{\mathrm{L}\mathrm{A}}={\displaystyle \sum _{t\in T}{e}_{m,t}^{\mathrm{R}}{P}_t^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\Delta t+D_t^{\mathrm{E}\mathrm{V}}+D_t^{\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}}$$

(6)

where $P_t^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}$ denotes the initial load at time t.

3.

Stepwise Carbon Trading Model Based on NCP

In an effort to impose stricter regulation on carbon emissions and enhance the motivation of demand-side energy conservation and emission reduction, a stepwise carbon trading model based on NCP is established. The demand-side carbon emissions are calculated based on NCP. To determine the responsibility strictly, the baseline method is adopted where the initial carbon emission allowance $E^{\mathrm{q}\mathrm{u}\mathrm{o}\mathrm{t}\mathrm{e}}$ is allocated based on the actual electricity consumption multiplied by the unit carbon quota coefficient $\delta$. This mechanism ensures that users are incentivized to consume electricity when the real-time NCP is lower than $\delta$. Should the actual carbon emissions surpass the assigned quota, additional carbon emission permits must be purchased; otherwise, surplus allowances can be sold for profit. The actual carbon emission trading volume of the demand side is expressed as

$$D^{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}=D^{\mathrm{L}\mathrm{A}}-D^{\mathrm{Q}}$$

(7)

$$D^{\mathrm{Q}}={\displaystyle \sum _{t\in T}{E}^{\mathrm{q}\mathrm{u}\mathrm{o}\mathrm{t}\mathrm{e}}(P_t^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}+P_t^{\mathrm{E}\mathrm{V}}-P_t^{\mathrm{c}\mathrm{u}\mathrm{t}}+P_t^{\mathrm{t}\mathrm{r}\mathrm{a},\mathrm{i}\mathrm{n}}-P_t^{\mathrm{t}\mathrm{r}\mathrm{a},\mathrm{o}\mathrm{u}\mathrm{t}})\Delta t}$$

(8)

where $D^{\mathrm{Q}}$ denotes the baseline carbon quota allocated to the load; $E^{\mathrm{q}\mathrm{u}\mathrm{o}\mathrm{t}\mathrm{e}}$ denotes the carbon allowance coefficient per unit of power consumption.

When calculating carbon trading costs, the demand-side carbon emissions are segmented into multiple intervals with predefined interval lengths. As the purchased carbon emission quotas increase, the corresponding carbon trading price for each interval increases accordingly. The stepwise carbon trading cost $C_{\mathrm{L}\mathrm{A}}^{{\mathrm{c}\mathrm{o}}_2}$ is expressed as

$$C_{\mathrm{L}\mathrm{A}}^{{\mathrm{c}\mathrm{o}}_2}=\left\{\begin{array}{ll}\lambda D^{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}& \hfill D^{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}⩽d\hfill \\ \lambda (1+\alpha )(D^{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}-d)+\lambda d& \hfill d<D^{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}⩽2d\hfill \\ \lambda (1+2\alpha )(D^{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}-2d)+\lambda (2+\alpha )d& \hfill D^{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}>2d\hfill \end{array}\right.$$

(9)

where $\lambda$ denotes the fundamental price for carbon trading; $d$ denotes the duration of the carbon emission interval; $\alpha$ denotes the price escalation coefficient.

In particular, the interval length d is determined based on the standard deviation of the nodal carbon potential distribution to cover normal operational fluctuations, while $\alpha$ is calibrated via sensitivity analysis to ensure sufficient demand response incentives.

3.2. Regional Power and Energy Balance Optimization Model with Source–Load Coordination

3.2.1. Economic Dispatch Model for the System Operator with Multiple Energy Sources

1.

Objective Function

The primary goal of the grid operator is to reduce overall expenses, which encompass the expenses related to coal usage in thermal generation units $C^{\mathrm{g}}$, wind–solar–storage costs $C^{\mathrm{w},\mathrm{p}\mathrm{v}}$, energy storage device costs (including depreciation and operation and maintenance costs) $C^{\mathrm{e}\mathrm{s}}$, carbon trading cost of the grid operator $C^{\mathrm{c}{\mathrm{o}}_2}$, and penalty terms for transmission-section power violations $C^{\mathrm{u}\mathrm{b}\mathrm{P}}$. The optimization objective function for the grid operator is formulated as

$$z_1=\min (C^{\mathrm{g}}+C^{\mathrm{w},\mathrm{p}\mathrm{v},\mathrm{c}}+C^{{\mathrm{c}\mathrm{o}}_2}+C^{\mathrm{u}\mathrm{b}\mathrm{P}})$$

(10)

$$C^{\mathrm{g}}={\displaystyle \sum _{t\in T}{\displaystyle \sum _{i\in \Omega }[a_i{(P_{i,t}^{\mathrm{g}})}^2+b_i{P}_{i,t}^{\mathrm{g}}+c_i]}}\Delta t$$

(11)

$$C^{\mathrm{w},\mathrm{p}\mathrm{v}}={\displaystyle \sum _{t\in T}{\displaystyle \sum _{i\in \Omega }(a_i^{\mathrm{w}}{P}_{i,t}^{\mathrm{w}}+a_i^{\mathrm{p}\mathrm{v}}{P}_{i,t}^{\mathrm{p}\mathrm{v}})\Delta t}}$$

(12)

$$C^{\mathrm{e}\mathrm{s}}={\displaystyle \sum _{t\in T}{\displaystyle \sum _{i\in \Omega }(a_i^d+a_i^o)P_{i,t}^{\mathrm{e}\mathrm{s}}\Delta t}}$$

(13)

$$C^{{\mathrm{c}\mathrm{o}}_2}={\displaystyle \sum _{t\in T}{\displaystyle \sum _{i\in \Omega }(e_i^{\mathrm{g}}-e^{\mathrm{q}\mathrm{u}\mathrm{o}\mathrm{t}\mathrm{e}})P_{i,t}^{\mathrm{g}}\Delta t}}$$

(14)

$$C^{\mathrm{u}\mathrm{b}\mathrm{P}}={\displaystyle \sum _{t\in T}{\displaystyle \sum _{l\in L}{M}^{\mathrm{p}+}{P}_{t,l}^{\mathrm{p}+}+M^{\mathrm{p}-}{P}_{t,l}^{\mathrm{p}-}}}$$

(15)

where $a_i$, $b_i$, and $c_i$ denote the coal consumption cost coefficient of thermal units in region i; $P_{i,t}^{\mathrm{g}}$ denotes the output of thermal units in region i at time t; $a_i^{\mathrm{w}}$ and $a_i^{\mathrm{p}\mathrm{v}}$ denote the generation cost factor of wind and photovoltaic power in region i; $a_i^d$ and $a_i^o$ denote the depreciation cost factor and operation and maintenance cost factor of energy storage device in region i; $a_i^{\mathrm{c}}$ denotes the cost factor of energy storage within region i; $P_{i,t}^{\mathrm{w}}$ and $P_{i,t}^{\mathrm{p}\mathrm{v}}$ denote the actual output generated by wind and PV units within region i at time t; $P_{i,t}^{\mathrm{e}\mathrm{s}}$ denotes the charge/discharge power rates of energy storage devices in region i at time t; $e_i^{\mathrm{g}}$ and $e^{\mathrm{q}\mathrm{u}\mathrm{o}\mathrm{t}\mathrm{e}}$ denote the carbon emission intensity per unit of electricity produced alongside the initial carbon quota coefficient; $P_{t,l}^{\mathrm{p}+}$ and $P_{t,l}^{\mathrm{p}-}$ denote the power violation amounts in forward and reverse directions for section l at time t; $M^{\mathrm{p}+}$ and $M^{\mathrm{p}-}$ denote the penalty coefficients for forward and reverse transmission-section violations.

2.

Constraints

The thermal unit output constraints are as follows:

$$P_i^{\mathrm{g},\mathrm{m}\mathrm{i}\mathrm{n}}⩽P_{i,t}^{\mathrm{g}}⩽P_i^{\mathrm{g},\mathrm{m}\mathrm{a}\mathrm{x}},\forall i\in {\Omega }^g,\forall t\in T$$

(16)

where $P_i^{\mathrm{g},\mathrm{m}\mathrm{i}\mathrm{n}}$ and $P_i^{\mathrm{g},\mathrm{m}\mathrm{a}\mathrm{x}}$ denote the minimum and maximum generation bounds of thermal power units in region i.

The thermal unit ramping constraints are as follows:

$$|P_{i,t}^{\mathrm{g}}-P_{i,t-1}^{\mathrm{g}}|⩽P_{i,r}^{\mathrm{g}},\forall i\in {\Omega }^g,\forall t\in T$$

(17)

where $P_{i,r}^{\mathrm{g}}$ denotes the ramping power limit of thermal units in region i.

The wind power output constraints are as follows:

$$0⩽P_{i,t}^{\mathrm{w}}⩽P_i^{\mathrm{w},\mathrm{m}\mathrm{a}\mathrm{x}},\forall i\in {\Omega }^{\mathrm{w}},\forall t\in T$$

(18)

where $P_i^{\mathrm{w},\mathrm{m}\mathrm{a}\mathrm{x}}$ denotes the maximum output cap for wind power units in region i.

The photovoltaic power output constraints are as follows [33]:

$$0⩽P_{i,t}^{\mathrm{p}\mathrm{v}}⩽P_i^{\mathrm{p}\mathrm{v},\mathrm{m}\mathrm{a}\mathrm{x}},\forall i\in {\Omega }^{\mathrm{p}\mathrm{v}},\forall t\in T$$

(19)

where $P_i^{\mathrm{p}\mathrm{v},\mathrm{m}\mathrm{a}\mathrm{x}}$ denotes the maximum output of photovoltaic units in region i.

The energy storage power constraints are as follows:

$$-P_i^{\mathrm{e}\mathrm{s}\mathrm{m}}⩽P_{i,t}^{\mathrm{e}\mathrm{s}}⩽P_i^{\mathrm{e}\mathrm{s}\mathrm{m}},\forall i\in {\Omega }^{\mathrm{e}\mathrm{s}},\forall t\in T$$

(20)

where $P_i^{\mathrm{e}\mathrm{s}\mathrm{m}}$ denotes the rated power of energy storage devices in region i.

The inter-regional transmission-section power boundary constraints are as follows:

$$P_{l,t}^{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}}={\displaystyle \sum _{i\in \Omega }{S}_{i,l}^{\mathrm{S}}(P_{i,t}^{\mathrm{S}}+P_{i,t}^{\mathrm{G}})}-{\displaystyle \sum _{i\in \Omega }{S}_{i,l}^{\mathrm{l}\mathrm{d}}{P}_{i,t}^{\mathrm{l}\mathrm{d}}}$$

(21)

$$P_{i,t}^S=P_{i,t}^{\mathrm{g}}+P_{i,t}^{\mathrm{w}}+P_{i,t}^{\mathrm{p}\mathrm{v}}+P_{i,t}^{\mathrm{c}},\forall i\in \Omega ,\forall t\in T$$

(22)

where $P_{l,t}^{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}}$ denotes the transmission power flowing on section l at time t; $P_{i,t}^S$ and $P_{i,t}^{\mathrm{l}\mathrm{d}}$ denote the total generation and load of region I at time t; $S_{i,l}^S$ and $S_{i,l}^{\mathrm{l}\mathrm{d}}$ denote the power transfer factors of generation and load in region i with respect to section l; $P_{i,t}^G$ denotes the power supplied by the upper-level grid to region i.

To ensure the solvability of the optimization model, relaxed transmission-section boundary constraints are introduced:

$$P_l^{\min }-{\alpha }_{l,t}^{-}\le P_{l,t}\le P_l^{\max }+{\alpha }_{l,t}^{+},\forall l\in {\Omega }^l,\forall t\in T$$

(23)

$${\alpha }_{l,t}^{+}\ge 0,{\alpha }_{l,t}^{-}\ge 0,\forall l\in {\Omega }^l,\forall t\in T$$

(24)

where $P_l^{\max }$ and $P_l^{\min }$ denote the maximum transmission capacity limits in the forward and reverse directions for section l; ${\alpha }_{l,t}^{+}$ and ${\alpha }_{l,t}^{-}$ denote the relaxation variables.

The regional total power balance constraint is as follows:

$${\displaystyle \sum _{i\in I}{P}_{i,t}^S}={\displaystyle \sum _{i\in I}{P}_{i,t}^{\mathrm{l}\mathrm{d}}}$$

(25)

The regional capacity balance constraint is as follows:

$$R_t^{\mathrm{p}\mathrm{e}\mathrm{n}}+{\displaystyle \sum _{i\in \Omega }(R_{i,t}^{\mathrm{S}}+R_{i,t}^{\mathrm{G}})}\ge (1+r^{\mathrm{l}\mathrm{d}}){\displaystyle \sum _{i\in \Omega }{P}_{i,t}^{\mathrm{l}\mathrm{d}}}+R_t^{\mathrm{e}\mathrm{m}}$$

(26)

$$R_{i,t}^{\mathrm{S}}=R_{i,t}^{\mathrm{g}}+R_{i,t}^{\mathrm{w}}+R_{i,t}^{\mathrm{p}\mathrm{v}}+R_{i,t}^{\mathrm{c}}$$

(27)

where $R_t^{\mathrm{p}\mathrm{e}\mathrm{n}}$ denotes insufficient adjustable capacity in the entire area; $R_{i,t}^{\mathrm{S}}$ denotes the adjustable capacity provided by all types of power sources in region i at time t, including thermal power $R_{i,t}^{\mathrm{g}}$, wind power $R_{i,t}^{\mathrm{w}}$, photovoltaic power $R_{i,t}^{\mathrm{p}\mathrm{v}}$, and energy storage $R_{i,t}^{\mathrm{c}}$; $R_{i,t}^{\mathrm{G}}$ denotes the adjustable capacity provided by the upper-level grid; $R_t^{\mathrm{e}\mathrm{m}}$ denotes the contingency reserve capacity; $r^{\mathrm{l}\mathrm{d}}$ denotes the load reserve margin. The value of this coefficient is determined based on the distribution of forecast errors in historical load data. Typically, to accommodate short-term load fluctuations, the coefficient ranges from 2% to 5%.

3.2.2. Demand-Side Optimal Dispatch Model Responding to Carbon Signals

The objective of the demand-side optimal dispatch model is to minimize total costs by responding to NCP signals and optimizing flexible load consumption strategies to lower carbon emissions.

1.

Objective Function

By utilizing price incentives, the demand-side aggregator motivates users to take part in low-carbon demand response, providing them with economic benefits. Specifically, the cooperative interaction between carbon signals and electricity prices is achieved through cost superposition in the objective function, where the equivalent incentive price perceived by users is dynamically determined by the sum of the time-of-use tariff and the marginal carbon cost derived from NCP. The total cost $z_2$ is expressed as

$$z_2=\min (C^{\mathrm{b}\mathrm{u}\mathrm{y}}+C^{{\mathrm{c}\mathrm{o}}_2}+C^{\mathrm{c}\mathrm{t}}+C^{\mathrm{e}\mathrm{v}\mathrm{d}})$$

(28)

$$C^{\mathrm{b}\mathrm{u}\mathrm{y}}={\displaystyle \sum _{t\in T}{\displaystyle \sum _{i\in \Omega }{q}_{i,t}{P}_{i,t}^{\mathrm{b}\mathrm{u}\mathrm{y}}\Delta t}}$$

(29)

$$C^{\mathrm{c}\mathrm{t}}={\displaystyle \sum _{t\in T}{\displaystyle \sum _{i\in \Omega }(q^{\mathrm{c}\mathrm{u}\mathrm{t}}{P}_{i,t}^{\mathrm{c}\mathrm{u}\mathrm{t}}+q^{\mathrm{t}\mathrm{r}\mathrm{a},\mathrm{i}\mathrm{n}}{P}_{i,t}^{\mathrm{t}\mathrm{r}\mathrm{a},\mathrm{i}\mathrm{n}}+q^{\mathrm{t}\mathrm{r}\mathrm{a},\mathrm{o}\mathrm{u}\mathrm{t}}{P}_{i,t}^{\mathrm{t}\mathrm{r}\mathrm{a},\mathrm{o}\mathrm{u}\mathrm{t}})\Delta t}}$$

(30)

$$C^{\mathrm{e}\mathrm{v}\mathrm{d}}={\displaystyle \sum _{t\in T}{\displaystyle \sum _{n\in N^{\mathrm{ev}}}\varphi P_{n,t}^{\mathrm{e}\mathrm{v}\mathrm{d}}\Delta t}}$$

(31)

where $C^{\mathrm{b}\mathrm{u}\mathrm{y}}$ denotes the cost of electricity purchased from the grid on the demand side; $C^{\mathrm{c}\mathrm{t}}$ denotes compensation for users participating in low-carbon initiatives; $C^{\mathrm{e}\mathrm{v}\mathrm{d}}$ denotes economic compensation for EV discharge; $P_{i,t}^{\mathrm{b}\mathrm{u}\mathrm{y}}$ denotes the electricity purchased by region i procured from the grid operator at instant t; $q_{i,t}$ represents the tariff rate within the time-of-use scheme; $q^{\mathrm{c}\mathrm{u}\mathrm{t}}$ denotes the compensation coefficient per unit of curtailed load; $q^{\mathrm{t}\mathrm{r}\mathrm{a},\mathrm{i}\mathrm{n}}$ and $q^{\mathrm{t}\mathrm{r}\mathrm{a},\mathrm{o}\mathrm{u}\mathrm{t}}$ denote the unit compensation factors of transferred-in and transferred-out load; $\varphi$ denotes the compensation coefficient for EV discharging.

2.

Constraints

The power balance constraint is as follows [34]:

$${\displaystyle \sum _{i\in \Omega }{P}_{i,t}^{\mathrm{b}\mathrm{u}\mathrm{y}}}+{\displaystyle \sum _{n\in N^{\mathrm{ev}}}{P}_{n,t}^{\mathrm{e}\mathrm{v}\mathrm{d}}}={\displaystyle \sum _{i\in \Omega }{P}_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}}+{\displaystyle \sum _{n\in N^{\mathrm{ev}}}{P}_{n,t}^{\mathrm{e}\mathrm{v}\mathrm{c}}}-{\displaystyle \sum _{i\in \Omega }{P}_{i,t}^{\mathrm{c}\mathrm{u}\mathrm{t}}}+{\displaystyle \sum _{i\in \Omega }{P}_{i,t}^{\mathrm{t}\mathrm{r}\mathrm{a},\mathrm{i}\mathrm{n}}}-{\displaystyle \sum _{i\in \Omega }{P}_{i,t}^{\mathrm{t}\mathrm{r}\mathrm{a},\mathrm{o}\mathrm{u}\mathrm{t}}}$$

(32)

The EV constraints governing charge and discharge operations are as follows:

$$0\le y_{n,t}^{\mathrm{d}}{P}_{n,t}^{\mathrm{e}\mathrm{v}\mathrm{d}}\le P^{\mathrm{e}\mathrm{v}\mathrm{d},\mathrm{m}\mathrm{a}\mathrm{x}},\forall n\in N^{\mathrm{e}\mathrm{v}},\forall t\in T$$

(33)

$$0\le y_{n,t}^{\mathrm{c}}{P}_{n,t}^{\mathrm{e}\mathrm{v}\mathrm{c}}\le P^{\mathrm{e}\mathrm{v}\mathrm{c},\mathrm{m}\mathrm{a}\mathrm{x}},\forall n\in N^{\mathrm{e}\mathrm{v}},\forall t\in T$$

(34)

$$0\le y_{n,t}^{\mathrm{d}}+y_{n,t}^{\mathrm{c}}\le 1,\forall n\in N^{\mathrm{e}\mathrm{v}},\forall t\in T$$

(35)

where $y_{n,t}^{\mathrm{d}}$ and $y_{n,t}^{\mathrm{c}}$ denote the charging/discharging status of the n-th EV at time t; $y_{n,t}^{\mathrm{d}}$ = 1 and $y_{n,t}^{\mathrm{c}}$ = 1 take the value of 1 to indicate discharging and charging behaviors; $P^{\mathrm{e}\mathrm{v}\mathrm{c},\mathrm{m}\mathrm{a}\mathrm{x}}$ and $P^{\mathrm{e}\mathrm{v}\mathrm{d},\mathrm{m}\mathrm{a}\mathrm{x}}$ denote the power limits for EV charging or discharging.

The EV state-of-charge (SOC) continuity constraint is as follows:

$$E_{n,t+1}=E_{n,t}+({\eta }^{\mathrm{c}}{P}_{n,t}^{\mathrm{e}\mathrm{v}\mathrm{c}}-{\eta }^{\mathrm{d}}{P}_{n,t}^{\mathrm{e}\mathrm{v}\mathrm{d}})\Delta t,\forall t\in [t_n^{\mathrm{a}\mathrm{r}\mathrm{r}},t_n^{\mathrm{d}\mathrm{e}\mathrm{p}}]$$

(36)

where $E_{n,t}$ denotes the energy stored in the n-th EV at time t; ${\eta }^{\mathrm{c}}$ and ${\eta }^{\mathrm{d}}$ denote the charging and discharging efficiencies; $t_n^{\mathrm{a}\mathrm{r}\mathrm{r}}$ and $t_n^{\mathrm{d}\mathrm{e}\mathrm{p}}$ denote the arrival and departure times of the EV.

The SOC capacity constraint is as follows:

$$S^{\min }\le E_{n,t}/E^{\mathrm{c}\mathrm{a}\mathrm{p}}\le S^{\max }$$

(37)

where $E^{\mathrm{c}\mathrm{a}\mathrm{p}}$ denotes the rated battery capacity; $S^{\min }$ and $S^{\max }$ denote the minimum and maximum allowable states of charge.

The departure energy requirement is as follows:

$$E_{n,t^{\mathrm{dep}}}\ge E_n^{target}$$

(38)

where $E_n^{target}$ denotes the expected energy required for the next trip [35].

4. Power and Energy Balance Model Based on Carbon Flow Guidance

For annual-scale power balance analysis problems, directly constructing an optimization model on an annual cycle would inevitably lead to a massive number of variables and computational intractability. To address this problem, this paper designs a daily rolling solution scheme based on a daily cycle. The detailed procedural steps of the proposed algorithm are described below, as illustrated in the model solution flowchart in Figure 2.

Figure 2. Model solution flowchart.

Figure 2. Model solution flowchart.

Step 1: Initialize generation unit parameters, transmission line parameters, and load parameters. With sections as boundaries, similar types of power sources within each sub-region are aggregated into an equivalent unit based on similarity evaluation indicators including regulation rate, capacity, and carbon emission intensity. Pre-assessment indicates that the total system error relative to the detailed model is less than 0.1%, satisfying accuracy requirements.

Step 2: Execute the grid operator’s economic dispatch optimization to determine the power output for each unit type within every sub-region, and then calculate NCP using the CEF model.

Step 3: Guide demand response (DR) using NCP and electricity price as signals. An optimal dispatch model based on NCP and DR is then solved to obtain the load demand for each sub-region after incentive-based DR.

Step 4: Once satisfied for region i between iteration $\phi$ and $\phi -1$ (with $\epsilon$ set to 0.03 MW), the solution procedure terminates, and the optimal dispatch results for that day are output. Otherwise, Steps 2 and 3 are repeated.

Step 5: Building upon the preceding steps, day-by-day rolling calculations are performed. During this process, the terminal energy state of storage devices and the remaining carbon quota from the current day are inherited as the initial boundary conditions for the next day, thereby yielding annual generation profiles, utilization hours for various power sources, and other annual indicators pertinent to power balance.

Notably, in Step 4, to prevent oscillations, a bisection method is employed to constrain the line flows. This is achieved by providing a feasible interval for the load demand, gradually narrowing the search interval by adjusting the respective lower or upper limits at each step until the interval size is less than or equal to the convergence tolerance. Furthermore, the Gurobi optimizer is utilized to solve the models in Step 2 and Step 3.

5. Case Studies

To systematically evaluate the practical efficacy of the suggested framework in incentivizing demand-side low-carbon behavior, improving renewable energy utilization, and maintaining power supply–demand balance, this study employs a modified IEEE 30-bus test system (partitioned into three independent power supply zones) for simulation analysis. The simulation programs are developed based on the MATLAB platform, and the commercial solver Gurobi is invoked to solve the optimization model.

5.1. Basic Data

In this study, a typical-day operation dataset from a regional power grid is selected as the test background [36]. The modified IEEE 30-bus system is partitioned into three supply sub-regions according to geographical location and transmission corridor constraints, as shown in

Table 1. Regional grid partition and power supply configuration.

Zone	Buses	Thermal Installed Capacity (MW)	Wind Installed Capacity (MW)	PV Installed Capacity (MW)	Storage Configuration (MW/MWh)
1	1–8	600	300	100	30/60
2	9–20	200	100	200	40/80
3	21–30	200	100	300	30/60

. Furthermore, the operational parameters of thermal units are listed in

Table 2. Operational parameters of thermal units.

Zone	Rated Capacity (MW)	Minimum Output (MW)	Ramp Rate (MW/h)	Fuel Cost Coefficient a (CNY/MW2)	Fuel Cost Coefficient b (CNY/MW)	Fuel Cost Coefficient c (CNY)
1	600	180	100	0.0022	120	500
2	200	60	100	0.0025	180	400
3	200	60	100	0.0025	210	400

, and the demand-side resources and carbon economic parameters are presented in

Table 3. Demand-side resources and carbon economic parameters.

Category	Parameter	Value
Load Characteristics	System Peak Load	1200 MW
	Load Spatial Distribution	Z1: 20%; Z2: 50%; Z3: 30%
Flexible Resources	EV Fleet Size	5000 units
	Maximum EV Charging/Discharging Power	7 kW
	Curtailable Load Ratio	5%
	Transferable Load Ratio	10%
Carbon Emission Parameters	Thermal Unit Carbon Emission Intensity	0.85–0.95 kg/kWh
	Carbon Emission Intensity of Purchased Electricity	0.58 kg/kWh
	Renewable Energy (Wind/PV) Carbon Emission	0 kg/kWh
Carbon Trading Parameters	Base Carbon Trading Price	50 CNY/t
	Unit Quota Coefficient	0.7 kg/kWh
	Price Escalation Coefficient	1.2

.

Zone 1: A wind-rich area comprising Buses 1–8, where the installed capacity is dominated by wind power and thermal generation.

Zone 2: The main load center containing Buses 9–20, characterized by a high photovoltaic penetration level and significant demand.

Zone 3: The receiving-end system including Buses 21–30, which mainly relies on power imported through tie lines.

5.2. Case Study Scenario Settings

To examine the performance of the model, this paper constructs the following three scenarios for comparison:

Scenario 1: Unequal carbon emission constraints are not considered. Only thermal unit operating costs are minimized to achieve the system power–energy balance. The load does not participate in regulation and is treated as an inelastic demand.

Scenario 2: A carbon trading mechanism is introduced on the generation side, in which thermal units bear both fuel costs and carbon taxes. However, demand-side response is not incorporated, and loads remain inelastic.

Scenario 3: Both generation-side and demand-side regulation are considered. A locational marginal carbon price is calculated based on nodal carbon marginal costs. Through the combined effect of price signals and incentives, flexible loads are guided to participate in carbon-responsive scheduling, thereby achieving coordinated supply–demand optimization.

5.3. Case Study Results and Analysis

5.3.1. Power and Energy Balance Analysis

To explore the influence of varying scheduling modes on the supply–demand characteristics of regional power grids—particularly to confirm the efficacy of the suggested framework regarding leveraging demand-side carbon-responsive interactions to enhance absorption of clean energy and alleviate supply-side strain—this study examines the power-energy balancing outcomes under the three scenarios. By comparing the typical operating profiles of conventional economic dispatch, carbon trading on the generation side, and coordinated low-carbon optimization of both supply and demand sides, the variations in thermal unit operation, renewable utilization, and residual load characteristics are analyzed. Figure 3 illustrates the typical daily power–energy balancing results for Zone 1 under the three scenarios.

Under Scenario 1, representing the traditional dispatch mode, the absence of demand-side response and the omission of carbon costs result in significant wind curtailment during nighttime. The wind output is constrained by the local load valley and the transmission export limits, leading to substantial unused wind energy.

Under Scenario 2, after incorporating carbon costs on the generation side, thermal units are driven to operate near their minimum technical output, which slightly alleviates wind curtailment. However, because the load profile remains inelastic, the fundamental bottleneck in renewable energy absorption persists.

Under Scenario 3, enabled by the proposed source–load interactive mechanism, nighttime low-carbon signals incentivize large-scale participation of EVs and shiftable loads. This additional flexible demand successfully accommodates the wind energy that would otherwise be curtailed. Consequently, the system achieves zero wind curtailment, while thermal units remain at their minimum output, resulting in an optimal power–energy balance.

5.3.2. Analysis of Nodal Carbon Marginal Signals and Flexible Load Response Characteristics

To validate the effectiveness of the proposed carbon-flow-driven guidance mechanism, this section examines the load center nodes under Scenario 3 and illustrates the relationship between nodal carbon marginal signals and the response rate of flexible loads.

As shown in Figure 4, the NCP, acting as a price-based signal, accurately reflects the cleanliness of the primary energy mix in the power system. A clear inverse coupling relationship is observed between the carbon potential and the level of flexible load response:

Low-carbon response: During 02:00–06:00, when wind generation is abundant, the NCP drops to its lowest level. At this time, flexible loads—represented by the blue bars—respond rapidly by significantly increasing electricity consumption (mainly due to concentrated EV charging). This not only reduces user energy costs but also provides additional space for wind energy accommodation.

High-carbon avoidance: During the high-emission period of 18:00–20:00, the NCP rises sharply. Flexible loads actively reduce consumption or suspend charging to avoid high-carbon periods, thereby achieving a “load-following-carbon” response behavior.

It is worth noting that the load response shown in Figure 4 is determined by the global optimization of costs and physical constraints, rather than fixed elasticity coefficients. This mechanism captures the dynamic sensitivity of flexible loads to NCP signals while ensuring operational feasibility.

5.3.3. Impact of Transmission Corridor Constraints on Power–Energy Balance

In response to the transmission capacity limitations between the regional zones discussed in this paper, this section investigates the impact of corridor constraints on system balancing performance. The analysis focuses on the key transmission interface connecting Zone 1 and Zone 2.

As shown in Figure 5, in the unconstrained transmission case—where the thermal stability limits of the interface are ignored—excess wind generation from Zone 1 is transmitted in large quantities to Zone 2. Although this results in the lowest overall system carbon emissions, power flow calculations indicate that the interface exceeds its limit by 25% during Period 14, posing a significant security risk. After incorporating the transmission corridor constraint, the proposed model automatically adjusts its operational strategy.

Export limitation: During Period 14, the transmitted power from Zone 1 to Zone 2 is restricted within the secure operating boundary.

Source-side adjustment: A small amount of wind curtailment occurs in Zone 1 due to the export limitation.

Load-side response: The low carbon potential strongly stimulates flexible loads within Zone 1, leading to a substantial increase in local consumption. This shifts the system behavior from “source following load” to “load following source,” enabling maximum in-region utilization of clean energy while ensuring interface security.

5.3.4. Comparative Analysis and Statistical Evaluation of Integrated Indicators

To quantitatively assess the comprehensive benefits of the proposed method in long-term system operation, the system performance under the three scenarios is evaluated.

Table 4. Comparison of integrated operational indicators under the three scenarios.

Indicator Category	Specific Metric	Scenario 1	Scenario 2	Scenario 3
Economic Efficiency	Total system operating cost (CNY 10,000)	425.6	423.1	410.8
	Fuel cost of thermal units (CNY 10,000)	280.4	265.4	248.3
	Wind curtailment penalty cost (CNY 10,000)	28.5	0	0
Low-Carbon Performance	Total carbon emissions (t)	6850	6340	5870
	Average carbon marginal signal (kg/kWh)	0.68	0.62	0.51
Renewable Utilization Level	Renewable energy utilization rate (%)	82.5%	88.1%	100.0%
Operational Flexibility	Total flexible resource utilization (MWh)	-	-	480
Co-benefits	Carbon intensity reduction rate (%)	-	8.8%	25%
	Unit carbon emission reduction cost (CNY/t)	-	−49.02	−151.02

presents a comparative analysis of economic efficiency, carbon reduction performance, renewable energy utilization, and operational flexibility across the three scenarios. Specifically, the total system operating cost is composed of four components: generation-side physical costs, net carbon trading costs, penalties for load shedding or wind curtailment, and demand-side response compensation costs.

The results can be summarized as follows:

Breakthrough in renewable energy utilization: By incorporating energy storage and DR, Scenario 3 increases wind power utilization from 82.5% in Scenario 1 to 100%, fully eliminating wind curtailment in Zone 1.

Economic–environmental co-benefits: Although Scenario 3 introduces additional storage losses and compensation costs for flexible loads, it completely avoids expensive wind-curtailment penalties and substantially reduces thermal fuel consumption during peak periods. As a result, the total system operating cost becomes the lowest among all scenarios, while total carbon emissions decrease by 22.34% compared with Scenario 1.

Value of flexibility resources: A comparison between Scenarios 2 and 3 shows that relying solely on supply-side adjustments has inherent limits. Only by integrating energy storage and flexible loads—thereby breaking the rigid supply–demand structure—can the system achieve optimal operation under high renewable penetration.

Overall, the proposed method effectively guides load shifting toward low-carbon periods and low-carbon nodes. As verified by the comparative results in Table 4, the proposed mechanism successfully breaks the rigid supply–demand constraints of traditional modes (Scenarios 1 and 2), which not only mitigates local transmission congestion and reduces wind curtailment but also significantly enhances the system’s low-carbon economic performance.

To intuitively reflect the impact of the proposed mechanism on the transmission interface, the transmission power of the key cross-section (Zone 1–2) under the three scenarios is compared in

Table 5. Comparison of transmission power of Zone 1–2 under three scenarios (unit: MW).

Time (h)	1	2	3	4	5	6	7	8	9	10	11	12
Scenario 1	60.00	60.00	60.00	60.00	60.00	60.00	58.42	52.10	45.30	38.60	32.50	30.20
Scenario 2	60.00	60.00	60.00	60.00	60.00	60.00	58.15	51.80	44.90	38.20	32.10	29.80
Scenario 3	60.00	60.00	60.00	60.00	60.00	60.00	56.30	48.50	42.10	35.40	28.90	26.50
Time (h)	13	14	15	16	17	18	19	20	21	22	23	24
Scenario 1	35.40	48.90	55.60	59.80	60.00	60.00	60.00	60.00	58.20	54.30	58.90	60.00
Scenario 2	35.10	48.50	55.20	59.50	60.00	60.00	60.00	60.00	57.90	54.10	58.60	60.00
Scenario 3	31.20	45.30	52.80	58.10	59.50	58.80	57.20	58.40	55.60	52.30	58.10	60.00

. The thermal stability limit of the section is set to 60 MW. In Scenario 1 and Scenario 2, due to the lack of demand-side flexibility, the transmission channel remains saturated (hitting the upper limit) for extended periods during the night (01:00–06:00), leading to wind curtailment as excess generation cannot be exported or consumed locally. In contrast, Scenario 3, while also maintaining high transmission levels to maximize renewable export, dynamically adjusts the flow during peak load periods (e.g., 19:00–21:00) and coordinates with local flexible loads to absorb excess wind power locally, thereby strictly adhering to security constraints while optimizing the overall energy balance.

5.4. Convergence Analysis and Model Effectiveness Evaluation

To verify the solvability of the proposed master–slave interactive mechanism, representative operational data are selected to test the convergence performance of the iterative interaction process between the system operator and load aggregators. The convergence tolerance is set to 0.03 MW, meaning that when the maximum deviation of load adjustments between two consecutive iterations falls below this threshold, the system is considered to have reached equilibrium.

Figure 6 illustrates the iterative variation in the maximum deviation of load adjustments and the corresponding convergence behavior. As shown, the proposed algorithm exhibits good convergence characteristics. When the tolerance is set to 0.025 MW, the deviation rapidly decreases and stabilizes within a small range, clearly demonstrating that the iterative interaction between the operator and load aggregators converges effectively without oscillation.

For the selected test case, the algorithm reaches convergence after 12 iterations. Furthermore, the convergence trends of other key indicators were monitored during the iteration process. Statistical data shows that the NCP, total system carbon emissions, and renewable energy consumption rate exhibit synchronous convergence characteristics with the load deviation. Specifically, after the 12th iteration, the variation rate of total system carbon emissions drops below 0.01%, and the renewable energy consumption rate stabilizes at 100%. This verifies that the proposed algorithm can achieve coordinated convergence across the multi-dimensional objectives of economy, low-carbon performance, and energy efficiency. Compared with a full spatiotemporal coupled equilibrium model, the proposed method significantly reduces computational complexity while maintaining convergence reliability. This satisfies the engineering requirements for power–carbon coupled dispatch and multi-regional operation analysis.

6. Conclusions

Addressing the complex supply–demand challenges within regional grids characterized by a high share of renewables, this paper proposes a power and energy balance analysis method considering demand-side carbon emissions, verified on an improved IEEE 30-node system. The primary findings are summarized below:

(1) Carbon flow tracing is used to guide active response from loads such as EVs, achieving “load following carbon.” Case studies demonstrate that this mechanism reduces system carbon emissions by 22.34%, effectively balancing low-carbon objectives with economic efficiency.

(2) The model overcomes the limitations of traditional methods that ignore transmission limits by directly incorporating key sections into the constraints. By guiding load response to resolve wind curtailment caused by congestion, the model increases renewable energy utilization from 82.5% to 100% while ensuring operational security.

(3) To address the heavy computational burden, a strategy of aggregating similar units within zones is adopted. Tests indicate that the source–load interaction process requires an average of only five iterations to reach equilibrium, with single-day optimization taking approximately 12 s. This significantly outperforms full-time domain coupling models and meets the timeliness requirements for annual analysis.

Future research will focus on constructing a carbon-constrained multi-time-scale balance framework and deepening flexible resource coordination. Emphasis will be placed on refined analysis for low-carbon, secure, and economic operation under market and source–load uncertainties. Furthermore, advanced intelligent optimization algorithms will be adopted to address complex renewable output characteristics, thereby enhancing source-side modeling accuracy [37].