Optimal Low-Carbon Economic Dispatch Strategy for Active Distribution Networks with Participation of Multi-Flexible Loads

Abstract

Optimization dispatch with flexible load participation in new power systems significantly enhances renewable energy accommodation, though the potential of flexible loads remains underexploited. To improve renewable utilization efficiency, promote wind/PV consumption and reduce carbon emissions, this paper establishes a low-carbon economic optimization dispatch model for active distribution networks incorporating flexible loads and tiered carbon trading. First, a hybrid SSA (Sparrow Search Algorithm)–CNN-LSTM model is adopted for accurate renewable generation forecasting. Meanwhile, multi-type flexible loads are categorized into shiftable, transferable and reducible loads based on response characteristics, with tiered carbon trading mechanism introduced to achieve low-carbon operation through price incentives that guide load-side participation while avoiding privacy leakage from direct control. Considering the non-convex nonlinear characteristics of the dispatch model, an improved Beluga Whale Optimization (BWO) algorithm is developed. To address the diminished solution diversity and precision in conventional BWO evolution, Tent chaotic mapping is introduced to resolve initial parameter sensitivity. Finally, modified IEEE-33 bus system simulations demonstrate the method’s validity and feasibility.

1. Introduction

With the continuous increase in electricity demand and grid peak loads [1], to effectively alleviate grid operational pressure, enhance the grid’s emergency regulation capacity, and guide users in reasonably optimizing their electricity loads, flexible loads, as an important component of demand-side response, are gradually being introduced into the power system to achieve a balance between electricity supply and demand and improve users’ economic benefits [2]. Therefore, making full use of the advantages of resources such as renewable energy, energy storage, flexible loads, and conventional units, and fully exploring their regulation potential, is an effective way to improve grid operational flexibility and achieve rational optimization of resource allocation. By aggregating various forms of energy, such as renewable energy, energy storage, flexible loads, and conventional units, through advanced communication means [3], and summarizing and analyzing multiple types of information at a control center to participate in grid ancillary services and dispatching [4], the safety and economy of grid operation can be significantly improved.

There is a large number of active response loads in distribution networks, commonly referred to as flexible loads. Upon receiving information from power companies, flexible loads proactively alter their traditional electricity consumption patterns to achieve load reduction, shifting, or translation [5], thereby reducing the peak-to-valley difference in loads [6], which plays a crucial role in the coordinated operation of transmission and distribution systems. Currently, numerous scholars have focused on such flexible resources to enhance the economic efficiency and security of distribution network operations. Reference [7] considers electric vehicles participating in optimized grid dispatching through charging and discharging, establishing a multi-agent two-layer game model to balance the interests of multiple agents. Reference [8] takes into account flexible electrical loads, gas loads, and thermal loads, proposing an optimized operation strategy that incorporates a stepped carbon trading mechanism and dual flexibility responses of supply and demand to reduce the operational costs and carbon emissions of integrated energy systems. Considering the significant uncertainty on the demand side in integrated communities, Reference [9] establishes a flexible load uncertainty model based on price incentives and analyzes the impact of incentive schemes on the operation of integrated community energy systems under different operational scenarios.

From the level of passive dispatching, controllable loads can serve as demand response resources, providing effective dispatching resources and regulatory means for the power grid. Modeling based on user responses to the grid can reflect the characteristics of user responses to incentives or electricity prices, facilitating the acquisition of the temporal characteristics of load response behaviors. Reference [10] constructs a demand-side electricity price response model, incorporating time-of-use pricing mechanisms and energy storage technologies into a wind power accommodation model to improve wind power consumption levels by altering system load distribution. Reference [11] analyzes the revenue model for controllable loads participating in demand response, suggesting that the charging and discharging characteristics of grid-connected electric vehicles and the flexible electricity consumption characteristics of industrial users enable them to respond to both electricity prices and incentives. In contrast, commercial and residential users, with fixed electricity consumption times, can only respond to incentives by signing policies in advance. Reference [12] establishes a user response model to electricity prices based on demand theory, where incentive-based users can increase or decrease their loads under constraint conditions to obtain electricity compensation, thereby constructing a scheduling cost model for incentive-based demand response. The geographical dispersion and diverse types of controllable loads make it difficult for the grid dispatching center to directly obtain dispatchable power, affecting the grid’s ability to tap into the response capabilities of controllable loads. Modeling controllable loads from an aggregation perspective typically involves using load aggregators as intermediaries to achieve an overall response from large-scale and dispersed controllable loads to the grid. Reference [13] estimates the aggregated power of air conditioning loads based on an aggregated power model and the law of large numbers, establishing an assessment model for the aggregated response potential of air conditioning loads to deeply explore their response potential. Reference [14] employs the K-means clustering algorithm to cluster three types of controllable loads and introduces electricity consumption satisfaction to quantitatively analyze the electricity consumption behaviors of various users. The constructed optimized scheduling model can enhance electricity consumption satisfaction for users participating in demand response while ensuring that the revenues of aggregators remain largely unaffected. Reference [15] aggregates electric vehicles and renewable energy sources through virtual power plant technology, constructing an optimized scheduling model for virtual power plants participating in day-ahead and reserve markets while considering the uncertainty of electric vehicle behavior. This model maximizes the revenue of virtual power plants while improving the overall revenue of electric vehicle users.

From the level of active dispatching, modeling from the perspective of user electricity consumption characteristics can account for the influence of various factors on user response behaviors, such as consumer psychology and user satisfaction, facilitating the acquisition of temporal characteristics of response behaviors. Reference [16] addresses the issue of neglecting user-side willingness in current load resource scheduling modeling by constructing a two-stage scheduling model incorporating user satisfaction. The results demonstrate that this model can adjust user electricity consumption behaviors and optimize the system’s load curve. Reference [17] tackles the problem of peak-to-valley differences in power systems caused by large-scale wind power and air conditioning loads by designing a novel user-side electricity consumption model that considers users’ voluntary declarations of their electricity consumption intentions. This model enables users’ declarations to fully reflect their actual production needs, providing new insights for user-side participation in grid interactions. Reference [18] addresses the limitation of traditional fixed models for electric vehicles, which fail to reflect the impact of user behavior on electric vehicle models, by establishing a charging and discharging stage model for electric vehicles, enhancing the accuracy of electric vehicle scheduling models. Reference [19] addresses the overly simplistic nature of time-of-use electricity price models established by statistically analyzing user responses to electricity prices by developing multiple time-of-use electricity price models that consider loads with different electricity consumption characteristics, enabling the final model to effectively reduce peak-to-valley differences. Currently, commonly used user response modeling methods rely on users’ historical statistical data, reflecting macroscopic manifestations of user responses to electricity prices/incentives, but lack modeling of physical characteristics, which, to a certain extent, affects the accuracy of the models in dynamic applications.

However, current research exhibits the following shortcomings in characterizing the uncertainty of renewable energy output and optimizing load dispatch: First, renewable energy output is significantly influenced by meteorological conditions, and its uncertainty possesses complex characteristics such as nonlinearity and spatio-temporal coupling. Existing methods often describe it through probability distributions, but frequently assume adherence to specific distributions, making it difficult to balance portrayal accuracy and computational efficiency. Excessive simplification overlooks actual fluctuation characteristics and increases decision-making conservatism, while complex models, due to excessive parameters and high computational costs, are challenging to apply in practical systems. Meanwhile, after incorporating the load dispatch model, it is necessary to simultaneously consider the interactions of multiple variables such as renewable energy output, user demand response, and grid constraints, leading to a dramatic increase in the dimensionality of the optimization problem. Traditional meta-heuristic algorithms are prone to getting trapped in local optima during solution-finding, and their convergence speed significantly decreases as the problem scale expands, limiting the economic efficiency and reliability of dispatch strategies. Particularly for large-scale power systems with a high proportion of renewable energy, the contradiction between accuracy and efficiency in existing methods becomes more prominent.

To address this, this paper proposes a low-carbon economic optimization dispatch model for active distribution networks incorporating flexible loads and tiered carbon trading. The innovations and main contributions of this paper are summarized as follows:

(1)

A SSA (Sparrow Search Algorithm)–CNN-LSTM hybrid model is employed to achieve high-precision prediction of renewable energy generation power. By optimizing model parameters with SSA, combining convolutional neural networks to extract spatial features and long short-term memory networks to handle temporal dependencies, it effectively resolves the conservative prediction issues caused by simplified assumptions in traditional methods.

(2)

In terms of solution algorithms, this paper has made dual improvements to the BWO algorithm: it introduces the Tent chaotic map to initialize the population, addressing the sensitivity of the traditional BWO algorithm to initial parameters and enhancing global search capabilities. During the algorithm development phase, the Levy flight strategy is embedded to improve local search efficiency and the ability to escape local optima. Additionally, the chaotic map is used to adjust the whale fall step size to balance the exploration and exploitation processes, ensuring both solution speed and accuracy.

2. New Energy Output Prediction Based on SSA-CNN-LSTM Hybrid Model

New energy output data exhibits significant local correlations in time series. Through its convolutional kernels and pooling operations, CNN can efficiently and automatically extract local features and patterns from these time series, which traditional time series models and fully connected networks lack. Moreover, new energy output is influenced by weather systems, circadian rhythms, etc., demonstrating notable long-term dependencies. The gating mechanism of LSTM can effectively capture long-term dependencies in time series and avoid the vanishing gradient problem, making it highly suitable for time series forecasting. The SSA module automates hyperparameter tuning, avoiding local optima. Although the CNN-LSTM model is powerful, its performance highly depends on hyperparameter settings. Traditional manual tuning or grid search methods entail extremely high computational costs and are prone to getting stuck in local optima. The Sparrow Search Algorithm (SSA) is a novel meta-heuristic optimization algorithm characterized by fast convergence and strong optimization capabilities. This paper utilizes SSA to automatically optimize the hyperparameters of CNN-LSTM, aiming to find globally optimal or near-globally optimal hyperparameter combinations, thereby fully unleashing the forecasting potential of the CNN-LSTM model and avoiding performance degradation caused by improper parameter settings.

For the source-load sides with temporal characteristics and diversified features, the SSA-CNN-LSTM model is employed for data prediction to enhance the accuracy and stability of forecasts on both the source and load sides. The LSTM model selectively discards minor information by adding gate mechanisms, endowing it with excellent predictive capabilities for multi-input multi-output time series data. The CNN model excels in feature extraction, compensating for the deficiencies of the LSTM model. The CNN-LSTM model consists of six parts: first, the input layer takes in raw data; then, the convolutional layer extracts data features, the pooling layer reduces data dimensionality, and the fully connected layer classifies the processed data; subsequently, the data is fed into the LSTM layer for training, and the trained data is input into the output layer to obtain the output values. The structural diagram of the CNN-LSTM model is shown in Figure 1.

The key parameters of the CNN-LSTM model are primarily selected by researchers based on experience. To prevent the CNN-LSTM model from falling into local optima, the SSA is employed to optimize its key parameters. The implementation steps of SSA are as follows.

First, the population consists of n sparrows, and the dimensionality of the problem variables is denoted by m, with the population represented by Equation (1).

$$X=\left[\begin{array}{cccc}{x}_{1,1}& x_{1,2}& \dots & x_{1,m}\\ x_{2,1}& x_{2,2}& \dots & x_{2,m}\\ ⋮& ⋮& ⋮& ⋮\\ x_{n,1}& x_{n,2}& \dots & x_{n,m}\end{array}\right]$$

(1)

Here, X represents the sparrow population; $x_{n,m}$ denotes the position of sparrow n in the m-dimensional space in the matrix X. In the algorithm, discoverers with high fitness have priority in obtaining food, attract other individuals to join, and possess a larger search range. During the iteration process, discoverers continuously update their positions. Fitness is represented by Equation (2).

$$F_X=\left[\begin{array}{c}f([x_{1,1},x_{1,2},\dots ,x_{1,m}])\\ f([x_{2,1},x_{2,2},\dots ,x_{2,m}])\\ ⋮\\ f([x_{n,1},x_{n,2},\dots ,x_{n,m}])\end{array}\right]$$

(2)

Here, F_X represents the fitness value of the population; f(·) denotes the fitness function. The position update of discoverers is represented by Equation (3).

$$X_{i,j}^{d+1}=\left\{\begin{array}{c}{X}_{i,j}^d·\exp (-{\displaystyle \frac{i}{a\times i_{\max }}}),R_2<S\\ X_{i,j}^d+Q·L,R_2\ge S\end{array}\right.$$

(3)

Here, $X_{i,j}^d$ represents the position at the d-th iteration, i.e., the current position of the discoverer; a is a random number; i_max is the maximum number of iterations; Q is a random number following a normal distribution; L is a 1 × m matrix; R₂ is the warning value; S is the safety value. When discoverers find better food, other sparrows will join in to compete for it. If successful, they are termed joiners, and the position update of joiners is represented by Equation (4).

$$X_{i,j}^{d+1}=\left\{\begin{array}{c}Q·\exp ({\displaystyle \frac{X_{worst}^d-X_{i,j}^d}{i^2}}),i>n/2\\ X_p^{d+1}+\left|X_{i,j}^d-X_p^{d+1}\right|·A^{+}·L,others\end{array}\right.$$

(4)

Here, $X_{worst}^d$ represents the global worst position in the d-th iteration; $X_p^{d+1}$ is the optimal position found by the discoverer during the iterations; $A^{+}$ is a 1 × d matrix where each element is randomly assigned a value of 1 or −1. When i > n/2, the i-th joiner, having relatively low fitness, needs to fly to other locations to forage for energy. Within the sparrow flock, there exists a group of vigilant sparrows that perceive surrounding threats and randomly take on alerting roles to protect their population. This behavior is represented by Equation (5).

$$X_{i,j}^{d+1}=\left\{\begin{array}{c}{X}_{best}^d+\beta ·\left|X_{i,j}^d-X_{best}^d\right|,f_i>f_g\\ X_{i,j}^d+K·({\displaystyle \frac{\left|X_{i,j}^d-X_{worst}^d\right|}{(f_i-f_w)+\epsilon }}),f_i=f_g\end{array}\right.$$

(5)

Here, $X_{best}^d$ represents the global optimal position in the d-th iteration; β is the step size control parameter; f_i and f_g are the fitness values of the current sparrow’s position and the global optimal position, respectively; f_w is the global worst fitness value; K is a random number within the range [−1,1]; ε is a number close to zero.

3. Optimal Scheduling Model Considering Flexible Loads and Carbon Trading Mechanisms

3.1. Introduction to the Tiered Carbon Trading Approach

The regulation of flexible loads can compensate for the power deficit caused by the output fluctuations of wind and photovoltaic power, thereby enhancing the integration level of renewable energy and reducing the system’s carbon emissions. When establishing the scheduling model, the introduction of a tiered carbon trading mechanism and the setting of an incentive coefficient are considered to further limit carbon emissions and reduce system scheduling costs.

To further optimize the carbon trading mechanism, this paper constructs a tiered carbon trading cost calculation model. It sets a total carbon emissions value, divides it evenly into several intervals, and linearizes the actual carbon emissions. When carbon emissions are below the free allocation of carbon emission allowances, energy supply enterprises can sell surplus carbon emission quotas on the carbon trading market, with higher carbon trading prices corresponding to intervals with lower carbon emissions. If the system’s carbon dioxide emissions exceed the free allocation quota, energy suppliers must purchase allowances on the carbon trading market, with higher carbon trading prices corresponding to higher levels of carbon emissions. When solving the model, this paper employs the tiered carbon trading mechanism as shown in (6):

$$C_t^{CO2}=\left\{\begin{array}{l}-cv(2+\lambda )-c(1+2\lambda )(-2v-\Delta E),\hfill \\ -2v<\Delta E\le -v;\hfill \\ -cv-c(1+\lambda )(-v-\Delta E),-v<\Delta E\le 0;\hfill \\ c\Delta E,0<\Delta E\le v;\hfill \\ cv+c(1+a)(\Delta E-v),v<\Delta E\le 2v;\hfill \\ cv(2+a)+c(1+2a)(\Delta E-2v),\hfill \\ 2v<\Delta E\le 3v;\hfill \\ cv(3+3a)+c(1+3a)(\Delta E-3v),3v<\Delta E.\hfill \end{array}\right.$$

(6)

Here, ΔE represents the system’s carbon emissions trading volume, where ΔE = E_p−E_c; E_p is the total carbon emissions from power generation, and E_c is the total allocated carbon emissions allowance for power generation. Additionally, $C_t^{CO2}$ denotes the total carbon trading cost; λ is the incentive coefficient; c is the carbon trading price in the market; α is the incremental increase in carbon trading price for each tier; and v is the length of the carbon emissions interval. λ represents the intensity of carbon trading incentives, with its value determined through historical data fitting and sensitivity analysis to balance carbon reduction effectiveness with system economic efficiency. When renewable energy penetration is high, appropriately increasing λ strengthens low-carbon dispatch guidance. v evenly divides the total carbon emission allowance E_c into intervals, with each v set based on the system’s historical carbon emission fluctuation range and renewable energy forecast errors.

3.2. Analysis of Flexible Load Characteristics

Flexible loads refer to loads that can be flexibly adjusted within a certain time period and are classified into multiple types based on their distinct regulatory characteristics. As a controllable load resource, flexible loads in the power grid have an increasingly pronounced impact on the power system. By guiding users to actively participate, flexible loads can utilize their adjustable properties to flexibly respond to electricity demand in the power system, enriching grid scheduling operations and regulatory methods. For the system, this achieves economic benefits and overall resource allocation, thereby earning certain compensation. The privacy protection mentioned in this paper is not achieved through the traditional direct encryption or anonymization of user data, but rather indirectly through the innovative design of load modeling methods. Specifically, during the modeling process, only macro power characteristics of electrical loads, such as response speed and power fluctuation range, are extracted for load classification, rather than collecting or analyzing detailed user electricity consumption data. This macro modeling approach fundamentally avoids the exposure of sensitive information on the user side. For example, for energy-intensive enterprises such as electrolytic aluminum plants, their core business information, like actual production output and production processes, is often strongly correlated with detailed electricity consumption data. By avoiding the collection and modeling of such data, effective protection of user privacy is inherently achieved. This paper categorizes controllable loads into three types: transferable loads, interruptible loads, and reducible loads. This classification method fundamentally differs from the traditional ZIP load modeling approach, which includes constant impedance, constant current, and constant power models. The ZIP model focuses on describing load voltage/frequency characteristics through static parameters, representing an equivalent circuit modeling approach at the purely physical level. In contrast, the classification proposed in this paper originates from the scheduling potential of demand-side resources, emphasizing the time-dimensional adjustability of loads, making it more suitable for dynamic optimization scenarios.

Based on how flexible loads participate in scheduling, they are generally categorized into three types:

(a)

Shiftable loads: The load profile is shifted in its entirety across different time periods.

(b)

Transferable loads: Within a scheduling cycle, the total load remains constant, but the load amounts in different time periods can be flexibly regulated.

(c)

Reducible loads: Electricity consumption is reduced according to demand.

(1)

Shiftable Loads

According to the above definition, shiftable loads can be treated as a whole and shifted from one continuous time interval within the scheduling cycle to another. Within the scheduling cycle, if the interval for shiftable loads is denoted as T, the shifted time period can be expressed as:

$$T=[t_{sh}^{-},t_{sh}^{+}-t^{*}+1]$$

(7)

Here, $t_{sh}^{-}$ and $t_{sh}^{+}$ represent the start and end times of load shifting; t^∗ is the duration of the shift for the shiftable load. The power of the shiftable load after participating in scheduling can be expressed as:

$$P_{sh}^{after}={\displaystyle \sum _{i=1}^t{u}_i^{sh}}·P_{sh}^{ahead}$$

(8)

Here, $u_i^{sh}$ is the state variable of the shiftable load scheduled for the i-th time period, where $u_i^{sh}$ = 1 indicates that power shifting occurs during this period, and $u_i^{sh}$ = 0 indicates no shifting occurs; $P_{sh}^{ahead}$ represents the power of the shiftable load before participating in scheduling.

(2)

Transferable Loads

Transferable loads can be partially or entirely shifted to other time periods within the scheduling cycle, while maintaining a constant total load. Let the time interval for transferable loads be denoted as $[t_{trs}^{-},t_{trs}^{+}]$, and the power of transferable loads is subject to the following constraints:

$$v_i^{trs}·P_{trs,\min }\le P_{trs,i}\le v_i^{trs}·P_{trs,\max }$$

(9)

Here, $P_{trs,\min }$ and $P_{trs,\max }$ represent the minimum and maximum output power of transferable loads, respectively; $v_i^{trs}$ is the state variable for the transferable load scheduled in the i-th time period, where $v_i^{trs}$ = 1 indicates power transfer occurs during this period, and $v_i^{trs}$ = 0 indicates no transfer occurs; $P_{trs,i}$ is the power of the transferable load participating in scheduling during the i-th time period. The magnitude of transferable load power is jointly determined by the load’s inherent physical constraints and system dispatch requirements: at the physical level, it is restricted by equipment capacity, power fluctuation ranges, and other characteristics; at the system level, specific transfer quantities are dynamically determined based on renewable energy output forecasts, grid balancing needs, and other factors to achieve the minimum overall dispatch cost.

Meanwhile, the total power remains unchanged before and after transfer, subject to the following constraint (10):

$${\displaystyle \sum _{i=t_{trs}^{-}}^{t_{trs}^{+}}{P}_{trs,i}^{after}}={\displaystyle \sum _{i=1}^t{P}_{trs,i}^{ahead}}$$

(10)

where $P_{trs,i}^{ahead}$ and $P_{trs,i}^{after}$ represent the power of the transferable load before and after transfer, respectively, during the i-th time period.

(3)

Reducible Load

Certain reducible loads can be curtailed to a certain extent as needed. The power of the load after reduction can be expressed as Equation (11):

$$P_{cut,i}^{after}=(1-w_i^{cut}·\epsilon )P_{cut,i}^{before}$$

(11)

Here, $P_{cut,i}^{before}$ represents the load power of the reducible load in the i-th time period before reduction; $P_{cut,i}^{after}$ represents the load power of the reducible load in the i-th time period after reduction; $w_i^{cut}$ is the state variable of the reducible load in the i-th time period, where $w_i^{cut}$ = 1 indicates that power reduction occurs in the i-th time period, and $w_i^{cut}$ = 0 indicates no reduction occurs; ε is the reduction coefficient of the reducible load, with ε ∈ [0, 1]. It should be noted that there is no need for an in-depth analysis of user comfort or satisfaction metrics in this paper, primarily due to the following two reasons: First, users on the demand side voluntarily participate in load regulation through price incentives—time-of-use pricing guides industrial users to adjust production shifts and encourages electric vehicles to charge during periods of high renewable energy generation. This market-based participation mechanism implicitly assumes that users have accepted the adjustment conditions, thereby avoiding potential satisfaction losses at the source. Second, the study focuses primarily on industrial loads, manufacturing loads, and electric vehicles, rather than air conditioning loads. The regulation of these loads is mostly achieved through adjustments to production plans rather than directly impacting end-user experiences, eliminating the need for additional comfort quantification metrics.

3.3. Mathematical Model Description of Optimal Scheduling

The diversification of coordinated scheduling strategies for multiple flexible resources will lead to significant differences in the amount of renewable energy consumed and the operational costs of the distribution network. To reasonably leverage the flexible and interactive characteristics of multiple flexible resources, this paper further incorporates the coordinated scheduling costs of these resources, in addition to considering the costs of curtailed power, purchased power, and network losses. The objective is to minimize the total operational cost of the distribution network, as expressed in the objective function (12).

$$obj=\min {\displaystyle \sum _{t=1}^T(C_t^{Curt}+C_t^{Buy}+C_t^{Loss}+C_t^{Flex}+C_t^{CO2})}$$

(12)

Here, $C_t^{Curt}$, $C_t^{Buy}$, and $C_t^{Loss}$ represent the curtailed power cost, purchased power cost, and network loss cost of the distribution network, respectively; $C_t^{Flex}$ denotes the coordinated scheduling cost of multiple flexible resources in the distribution network. The specific calculation formulas are given as (13)–(15).

$$C_t^{Curt}={\displaystyle \sum {\lambda }_{curt}^{DG}{P}_{i,t}^{DG,curt}}$$

(13)

$$C_t^{Buy}={\displaystyle \sum ({\lambda }_t^{DG}{P}_{i,t}^{DG}+{\lambda }_t^{buy}{P}_{i,t}^{buy}-{\lambda }_t^{sell}{P}_{i,t}^{sell})}$$

(14)

$$C_t^{Loss}={\lambda }^{Loss}{\displaystyle \sum I_{ij,t}^2}{R}_{ij}$$

(15)

The coordinated scheduling cost of multiple flexible resources in the distribution network mainly includes the flexible regulation cost of controllable loads, which can be calculated according to (16).

$$C_t^{Flex}={\displaystyle \sum ({\lambda }^{DR}\left|P_{i,t}^{load,O}-P_{i,t}^{load}\right|)}$$

(16)

$P_{i,t}^{load}$ is obtained by summing up the formulas mentioned above (8)–(11). $P_{i,t}^{load,O}$ represents the original power of the controllable load at node i, and ${\lambda }^{DR}$ is the unit scheduling cost for the controllable load.

The constraints related to this model include:

(1)

Renewable energy output constraints

$$\left\{\begin{array}{c}{P}_{i,t}^{DG}=P_{i,t}^{DG,av}-P_{i,t}^{DG,curt}\\ -P_{i,t}^{DG}\tan (\arccos P{F}_{i,down}^{DG})\le Q_{i,t}^{DG}\le P_{i,t}^{DG}\tan (\arccos P{F}_{i,up}^{DG})\end{array}\right.$$

(17)

$${(P_{i,t}^{DG})}^2+{(Q_{i,t}^{DG})}^2\le {(S_{i,t}^{DG})}^2$$

(18)

where $P_{i,t}^{DG,av}$ represents the active power output of new energy at node i; $P{F}_{i,down}^{DG}$ and $P{F}_{i,up}^{DG}$ are the lower and upper power factor limits of new energy at node i, respectively; $S_{i,t}^{DG}$ is the capacity of the new energy inverter at node i.

(2)

Constraints on the operating state of energy storage equipment

To protect the service life of energy storage and prevent overcharging and over-discharging, the energy storage system must consider its state of charge (SOC) constraints and balance constraints (19) as well as charging and discharging power constraints (20) when undergoing flexible regulation of charging and discharging.

$$\left\{\begin{array}{c}SO{C}_{i,t+1}=SO{C}_{i,t}+{\displaystyle \frac{{\eta }_{i,cha}^{ESS}{P}_{i,t,cha}^{ESS}}{S_{i,ESS}^{total}}}-{\displaystyle \frac{P_{i,t,dis}^{ESS}}{{\eta }_{i,dis}^{ESS}{S}_{i,ESS}^{total}}}\\ SO{C}_{t,\min }\le SO{C}_{i,t}\le SO{C}_{t,\max }\end{array}\right.$$

(19)

$$\left\{\begin{array}{c}0\le P_{i,t,cha}^{ESS}\le P_{i,cha}^{ESS,\max }\\ 0\le P_{i,t,dis}^{ESS}\le P_{i,dis}^{ESS,\max }\\ P_{i,t,cha}^{ESS}{P}_{i,t,dis}^{ESS}=0\end{array}\right.$$

(20)

In addition, leveraging its surplus capacity and in conjunction with the operating range of the power factor, the energy storage inverter can perform continuous dynamic reactive power compensation within the four-quadrant operating range. The inverter constraints and power factor constraints for energy storage reactive power compensation are given by (21) and (22), respectively.

$$\left\{\begin{array}{c}{(P_{i,t,cha}^{ESS})}^2+{(P_{i,t}^{ESS})}^2\le {(S_{i,rate}^{ESS})}^2\\ {(P_{i,t,dis}^{ESS})}^2+{(P_{i,t}^{ESS})}^2\le {(S_{i,rate}^{ESS})}^2\end{array}\right.$$

(21)

$$-(P_{i,t,cha}^{ESS}+P_{i,t,dis}^{ESS})\tan (\arccos P{F}_{i,down}^{ESS})\le Q_{i,t}^{ESS}\le (P_{i,t,cha}^{ESS}+P_{i,t,dis}^{ESS})\tan (\arccos P{F}_{i,up}^{ESS})$$

(22)

Here, $SO{C}_{i,t}$, $SO{C}_{t,\min }$ and $SO{C}_{t,\max }$ represent the SOC and its lower and upper limits, respectively, for the energy storage system at node i; $P_{i,cha}^{ESS,\max }$ and $P_{i,dis}^{ESS,\max }$ are the maximum allowable charging and discharging powers, respectively, for the energy storage system at node i; $S_{i,ESS}^{total}$ and $S_{i,rate}^{ESS}$ denote the rated capacity of the energy storage and the capacity of the energy storage inverter, respectively, at node i; $P{F}_{i,up}^{ESS}$ and $P{F}_{i,down}^{ESS}$ are the upper and lower power factor limits, respectively, for the energy storage system at node i.

(3)

Branch power flow constraints

$$\left\{\begin{array}{c}{P}_{ij,t}=g_{ij}{U}_{i,t}^2-g_{ij}{U}_{i,t}{U}_{j,t}\cos {\theta }_{ij,t}-b_{ij}{U}_{i,t}{U}_{j,t}\sin {\theta }_{ij,t}\\ Q_{ij,t}=-b_{ij}{U}_{i,t}^2+b_{ij}{U}_{i,t}{U}_{j,t}\cos {\theta }_{ij,t}-g_{ij}{U}_{i,t}{U}_{j,t}\sin {\theta }_{ij,t}\end{array}\right.$$

(23)

Here, $P_{ij,t}$ and $Q_{ij,t}$ represent the active and reactive power of branch ij, respectively; $g_{ij}$ and $b_{ij}$ are the conductance and susceptance of branch ij, respectively; ${\theta }_{ij,t}$ is the voltage phase difference between nodes i and j; $U_{i,t}$ and $U_{j,t}$ are the voltage magnitudes at nodes i and j, respectively. The power flow formulation remains applicable to distribution networks with a high proportion of renewable energy, primarily because it can accurately describe the radial topology and branch power flow characteristics of distribution networks. By recursively calculating branch power losses and node voltages, this model effectively handles bidirectional power flows caused by the volatility of renewable energy generation.

(4)

Node power balance constraints

$$\left\{\begin{array}{c}{P}_{i,t}^{buy}-P_{i,t}^{sell}+P_{i,t}^{DG}+P_{i,t,dis}^{ESS}-P_{i,t,cha}^{ESS}-P_{i,t}^{Load}={\displaystyle \sum P_{ij,t}}\\ Q_{i,t}^{DG}+Q_{i,t}^{ESS}-Q_{i,t}^{Load}={\displaystyle \sum Q_{ij,t}}\end{array}\right.$$

(24)

(5)

Node voltage and line current constraints

$$\left\{\begin{array}{c}{U}_{i,\min }\le U_{i,t}\le U_{i,\max }\\ \left|I_{ij,t}\right|\le I_{ij,\max }\end{array}\right.$$

(25)

where $U_{i,\max }$ and $U_{i,\min }$ are the upper and lower limits of the voltage magnitude at node i, respectively. $I_{ij,\max }$ is the maximum allowable current-carrying capacity of branch ij.

(6)

Power purchase and sale constraints

$$\left\{\begin{array}{c}0\le P_{i,t}^{buy}\le P^{tie}\\ 0\le P_{i,t}^{sell}\le P^{tie}\end{array}\right.$$

(26)

Here, $P^{tie}$ represents the maximum allowable power transmission capacity of the tie line.

4. Solution Method Based on an Improved Beluga Whale Optimization Algorithm

When constructing a low-carbon economic dispatch model for active distribution networks, the optimization problem is high-dimensional, nonlinear, and non-convex. Traditional optimization algorithms such as genetic algorithms (GA) and particle swarm optimization (PSO) algorithms are prone to issues like premature convergence, getting trapped in local optima, and slow convergence when solving such problems. Therefore, this paper selects the improved BWO algorithm as the solution tool, primarily based on strong global search capability, adaptive balancing mechanism, introduction of Tent chaotic mapping and applicability to high-dimensional problems.

The traditional BWO is an intelligent heuristic algorithm derived by simulating the swimming, foraging, and whale fall behaviors of beluga whales. In the BWO algorithm, adaptive variables—the balancing factor $B_f$ is utilized to facilitate the transition between the exploitation process and the exploration process. The calculation formulas are as follows:

$$B_f=B_0(1-{\displaystyle \frac{Z}{2Z_{\max }}})$$

(27)

Here, Z represents the current iteration number; Z_max denotes the maximum iteration number; B₀ signifies a random number within the interval (0, 1). When B_f > 0.5, the algorithm is in the exploration phase; when B_f ≤ 0.5, the algorithm is in the exploitation phase.

(1)

Exploration Phase

When the algorithm is in the exploration phase, the position update is determined by the paired beluga whale swimming behavior. A position update model is established based on this swimming behavior, as follows:

$$\left\{\begin{array}{c}{X}_{i,j}^{Z+1}=(X_{r,P_1}^Z-X_{i,P_j}^Z)(1+r_1)\sin (2\pi r_2)+X_{i,P_j}^Z,j=even\\ X_{i,j}^{Z+1}=(X_{r,P_1}^Z-X_{i,P_j}^Z)(1+r_1)\cos (2\pi r_2)+X_{i,P_j}^Z,j=odd\end{array}\right.$$

(28)

Here, $X_{i,j}^{Z+1}$ represents the new position of the updated individual of the i-th beluga whale in the j-th dimension; P_j represents a randomly selected dimension from the d-dimensional space; $X_{i,P_j}^Z$ represents the new position of the updated individual of the i-th beluga whale in the randomly selected dimension P_j; $X_{r,P_1}^Z$ represents the current position of the i-th beluga whale and the r-th beluga whale in the j-th dimension; r₁ and r₂ represent random values within the interval (0, 1); 2sin(2πr₂) and 2cos(2πr₂) represent the fin orientations of the mirrored beluga whale towards the water surface under the conditions of even and odd selections, respectively.

(2)

Exploitation Phase

The predatory behavior is designed as the exploitation phase of the BWO algorithm. To enhance the algorithm’s convergence, the Levy flight strategy (29) is introduced.

$$X_i^{Z+1}=r_3{X}_{best}^Z-r_4{X}_i^Z+C_1\cdot L_F(X_r^Z-X_i^Z)$$

(29)

In the formula, $X_i^Z$ represents the i-th beluga whale; $X_r^Z$ represents a randomly selected beluga whale; $X_{best}^Z$ represents the most outstanding beluga whale individual in the population; $X_i^{Z+1}$ represents the new position of the i-th beluga whale; r₃ and r₄ are random values within the interval (0, 1); $L_F$ represents the Levy flight function; $C_1$ represents the random jump intensity, which is used to measure the strength of the Levy flight, and its formula is given by (30).

$$C_1=2r_4(1-Z/Z_{\max })$$

(30)

(3)

Whale Fall Phase

When beluga whales encounter external threats, they update their positions to ensure the survival and maintenance of the population size. The mathematical model for this process is as follows:

$$X_i^Z=r_5{X}_i^Z-r_6{X}_i^Z+r_7{X}_{step}$$

(31)

Here, r₅, r_6, and r₇ represent random values within the interval (0, 1); $X_{step}$ represents the step length of the whale fall, with its calculation formula given by (32):

$$X_{step}=(u_b-l_b)\exp [-(S·Z)/Z_{\max }]$$

(32)

In this formula, S represents the step factor related to the whale fall probability W_f and the population size, and its calculation formula is provided by (33) and (34).

$$S=2·W_f·n$$

(33)

$$W_f=0.1-0.05·(Z/Z_{\max })$$

(34)

The probability of a beluga whale falling gradually decreases from an initial value of 0.10 to 0.05, indicating that as the optimization process progresses and the whales get closer to the food source, they are less likely to die (i.e., encounter unfavorable conditions or suboptimal solutions). When dealing with dynamic problems like coordinated optimization, the standard BWO algorithm often suffers from low convergence accuracy and sensitivity to initial parameters, resulting in different optimization outcomes under varying parameter settings. This makes the BWO algorithm prone to getting stuck in local optima. In the field of optimization, replacing pseudorandom number generators with chaotic maps to generate chaotic numbers between 0 and 1 and initializing the population using chaotic sequences can effectively address the issue of sensitivity to initial parameters, thereby improving the algorithm’s optimization accuracy and convergence speed. Among various chaotic maps, the Tent map exhibits a more uniform distribution function and generates chaotic sequences with better global ergodicity. To address the aforementioned shortcomings, this paper introduces the Tent chaotic map to optimize and improve the BWO algorithm. The Tent chaotic mapping function is defined as follows:

$$x_{k+1}=\left\{\begin{array}{c}{\displaystyle \frac{x_k}{0.5}},0<x_k\le 0.5\\ {\displaystyle \frac{x_k(1-x_k)}{0.5}},0.5<x_k\le 1\end{array}\right.$$

(35)

The Tent-enhanced BWO algorithm employs a Tent chaotic parameter adjustment strategy during the exploitation phase of the BWO algorithm, enhancing its convergence accuracy and global search capability. The improved WBO algorithm is suitable for solving coordinated optimization problems that are sensitive to initial parameters. To gain a better understanding, a simplified flowchart is given in Figure 2.

5. Case Study

5.1. The Introduction of the Testing System

To verify the effectiveness and feasibility of the proposed method in this paper, an improved IEEE-33 node system is employed in this section. The topological diagram of the test system is shown in Figure 3 below. In detail, two wind farms are connected to the system through nodes 9 and 25, respectively, while a photovoltaic (PV) power generation unit is connected through node 30. Capacitor banks and an energy storage device are installed at nodes 20 and 15, respectively. Detailed information on the new energy output is illustrated in Figure 4 below. All computational programs were developed and executed using MATLAB R2024b. The simulations were performed on a personal computer equipped with 12 GB of RAM and a processor operating at a frequency of 3.6 GHz for an Intel Core i7 processor (Intel, Santa Clara, CA, USA).

5.2. Analysis of Equipment and Load Response Conditions

To validate the effectiveness of the new energy forecasting model proposed in this paper, this section compares it with two currently popular advanced forecasting models. The CNN-GRU model replaces LSTM with Gated Recurrent Units (GRU). The EMD-LSTM model first employs Empirical Mode Decomposition (EMD) to decompose the original unstable new energy sequence into multiple relatively stable Intrinsic Mode Functions (IMFs), then uses LSTM to forecast each IMF component separately, and finally sums the results to obtain the final forecast value. Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Coefficient of Determination (R²) are used as evaluation metrics, with lower RMSE and MAE values being preferable and a higher R² value being better. When tested on the same dataset, the results are shown in

Table 1. Performance Comparison Results of Different Forecasting Models.

Model	RMSE (kW)	MAE (kW)	R2	Total Operation Cost/$
SSA-CNN-LSTM	105.3	78.6	0.974	18,176.1
CNN-GRU	137.5	102.4	0.941	18,952.4
EMD-LSTM	152.8	115.7	0.926	19,159.3
CNN-LSTM	124.6	92.1	0.957	19,325.4

below:

Compared with the CNN-GRU model, the proposed model in this paper reduces RMSE by 23.4%, MAE by 23.2%, and significantly improves R². Although GRU has a simpler structure, when dealing with the complex temporal dependencies in new energy output, the sophisticated gating mechanism of LSTM can generally capture longer-term and more complex patterns than GRU, demonstrating stronger modeling capabilities. The optimization by SSA further ensures the superiority of LSTM parameters. In comparison with the EMD-LSTM model, the proposed model in this paper reduces RMSE by 31.1%, MAE by 32.1%, and shows a marked improvement in R². The EMD-LSTM model is an excellent one, but its performance is constrained by the mode-mixing issue in EMD. EMD may not be able to fully separate fluctuations of different scales, resulting in impure IMF components and affecting the subsequent forecasting accuracy of LSTM. Errors are generated in the forecasting of each IMF component, and these errors accumulate and amplify during final reconstruction. EMD has high computational complexity and requires training multiple LSTM models, resulting in significantly higher computational costs than a single hybrid model. The proposed model in this paper learns features directly from raw data in an end-to-end manner, avoiding the error accumulation problem caused by decomposition-reconstruction, and SSA optimization enhances its accuracy.

Compared with the unoptimized CNN-LSTM model, the proposed model in this paper reduces RMSE by 15.5% and MAE by 14.7%. This directly proves the effectiveness of the SSA optimization algorithm. The performance of the unoptimized CNN-LSTM model is limited by manually set hyperparameters and may be stuck in local optima. SSA finds a superior combination of hyperparameters through global search, thereby significantly improving the performance of the basic CNN-LSTM model. The higher the accuracy of new energy output forecasting, the more precisely the power system dispatch can predict its power generation in advance, allowing for the optimization of output arrangements for conventional energy units ahead of time. This reduces the need for reserve capacity and emergency start-stop operations of conventional units caused by fluctuations in new energy output, lowers fuel consumption and operation and maintenance costs for conventional units, avoids losses incurred from purchasing electricity at high prices or curtailing wind and solar power when new energy output is insufficient, and ultimately achieves a reduction in total economic dispatch costs.

During periods of high new energy output, the On-Load Tap Changer (OLTC) and capacitor banks (CBs) require frequent adjustments to ensure the safe and stable operation of the power system, as shown in Figure 5.

During periods of substantial new energy output, the power grid often experiences a high penetration of renewable energy. The fluctuating output from wind and photovoltaic power generation can lead to rapid changes in voltage and power flow distribution. OLTCs require frequent adjustments to maintain voltage stability, as the intermittency and anti-peak shaving characteristics of new energy sources may cause local voltage violations. By dynamically adjusting the transformer tap ratio, OLTCs balance system voltage to prevent overvoltage or undervoltage issues. Meanwhile, frequent switching of capacitor banks provides reactive power compensation. Although new energy generation equipment inherently possesses some reactive power regulation capability, capacitor banks must still respond quickly to supplement reactive power during sudden output changes or when grid impedance is high, improving the power factor, reducing line losses, and supporting voltage levels. The coordinated regulation of these two types of equipment addresses the complex operating conditions of the grid under high new energy penetration, ensuring voltage quality and system stability.

When new energy output is significant, the direction of power flow in the grid may reverse, such as when distributed photovoltaic systems feed power back to the main grid, rendering traditional voltage regulation methods ineffective or slow to respond. Frequent operation of OLTCs can accommodate these bidirectional power flow changes by adjusting tap positions to optimize voltage distribution, while capacitor bank switching dynamically compensates for reactive power deficits or surpluses. Particularly during periods of light load but high new energy output, capacitive reactive power may become excessive, necessitating the disconnection of some capacitor banks to prevent overvoltage. The essence of this frequent regulation lies in the randomness and volatility of new energy sources, which disrupt the steady-state operation mode of traditional grids, requiring OLTCs and capacitor banks to shift from passive to active adaptation to cope with rapidly changing grid conditions and maintain safe and economical system operation.

The response of flexible adjustable loads is illustrated in Figure 6 below. In scenarios with significant fluctuations in new energy output, dispatchable transferable loads, interruptible loads, and reducible loads can flexibly adjust the power demand curve to align with the peaks and troughs of new energy generation. For instance, during periods of high wind or photovoltaic output, incentivizing users to shift transferable loads to these times can increase local consumption of new energy power, reducing curtailment of wind and solar resources. Conversely, when new energy output drops suddenly, the rapid response of interruptible and reducible loads can lower net grid demand, alleviate supply–demand imbalances, and reduce reliance on traditional fossil fuel backup units, thereby enhancing new energy penetration and lowering carbon emissions.

This coordinated regulation of demand-side resources can enhance the renewable energy consumption rate by 3.2% and reduce the peak-to-valley difference by up to 11.4%. By guiding user electricity consumption behavior through price signals or demand response mechanisms, the grid can more efficiently utilize intermittent renewable energy without additional storage or thermal power peaking capacity, reducing overall system operating costs. Simultaneously, flexible load management can delay the need for grid upgrades, improve the utilization of existing equipment, and provide critical support for the safe and stable operation of future high-renewable-penetration power systems.

5.3. Analysis of Economic and Environmental Performance

Introducing a tiered carbon trading mechanism into power system dispatch decision-making enables a more refined reflection of the differences in carbon emission costs among various power generation entities, thereby incentivizing the prioritized consumption of low-carbon energy. The carbon emissions for each time period are illustrated in Figure 7 below. Traditional carbon trading employs a fixed carbon price, imposing limited constraints on high-emission units. In contrast, the tiered carbon trading mechanism establishes different price ranges based on carbon emission intensity, with units emitting higher levels experiencing a more pronounced increase in carbon costs. This further diminishes the economic viability of traditional high-carbon power sources, such as coal-fired generation, in dispatch decisions while enhancing the competitiveness of zero-carbon energy sources like wind and photovoltaic power. Additionally, this mechanism encourages thermal power units to proactively undertake flexibility retrofits or participate in deep peak shaving, as moderately reducing output may place them in lower carbon price tiers, thereby reducing total carbon costs. Ultimately, it facilitates the power system in achieving optimal low-carbon operation while maintaining supply–demand balance.

Table 2. Quantitative evaluation of the metrics among different methods.

Method	Economic Cost ($)	New Energy Accommodation Rate	CO2 Emissions (kg)
The proposed method	18,176.1	97.4%	2225.8
Traditional demand response strategy	18,972.4	94.5%	2746.3
Uncoordinated dispatch	19,746.5	88.2%	3125.7

further provides a quantitative evaluation of the metrics among different methods. Here, traditional demand response strategies refer to guiding users to adjust their electricity consumption behaviors based on price changes through dynamically adjusting electricity prices; uncoordinated dispatch means that users only consider their own electricity demands without making power adjustments or responses.

Upon analyzing the data in the table above, it becomes evident that the tiered carbon trading mechanism possesses another notable benefit: its ability to dynamically adjust, aligning well with the inherent variability of new energy generation. In instances where a significant increase in new energy output results in a reduction in the overall system’s carbon emission intensity, the carbon price tiers have the potential to adjust automatically. This adjustment serves to avoid the risk of diminished emission reduction incentives caused by overly low carbon costs. In contrast, when new energy generation falls short and the share of thermal power increases, high-emission units will confront a more sharply inclined carbon cost curve, which in turn limits over-dependence on fossil fuels. Such a flexible pricing framework not only boosts the carbon market’s adaptability to the real-time operational status of the power system but also unlocks more value potential for flexibility resources, including energy storage and demand response. By stabilizing new energy fluctuations, these resources can indirectly contribute to lowering the system’s tiered carbon price levels, promoting a beneficial cycle of emission reduction and economic efficiency, and facilitating a faster transition of the power system towards low-carbon operations.

Figure 8 presents a comparison of convergence curves before and after algorithmic improvements. The convergence criterion in this paper is that the relative change in the objective function value is less than a preset threshold ε = 0.0001. The improved BWO Algorithm converges at the 32nd iteration, whereas the traditional algorithm converges only at the 56th iteration. Compared to the traditional BWO Algorithm, the improved method introduces Tent chaotic mapping to optimize the initial population distribution, enabling the algorithm to cover a broader solution space in the early iterations. This effectively avoids the issue of traditional methods getting trapped in local optima due to sensitivity to initial parameters, thereby enhancing search efficiency. Meanwhile, the randomness and ergodicity of chaotic mapping enhance population diversity, maintaining exploration capabilities throughout the evolutionary process and reducing the risk of premature convergence. Consequently, the algorithm can more rapidly approximate the global optimal solution when solving non-convex, nonlinear dispatch models, resulting in faster convergence and higher solution accuracy.

To validate the effectiveness of the improved BWO, this section compares it with the improved Particle Swarm Optimization (PSO) used in [20], improved Genetic Algorithm (GA) used in [21], and traditional BWO through simulations on test systems of different scales, including the IEEE-69 and IEEE-118 test systems, whose topological structures are shown in Figure 9 below. The relevant computational results are presented in

Table 3. Comparison of Computational Results for Test Systems of Different Scales.

Test System	Method	Convergence Iteration	Total Computation Time/s	Dispatch Cost/$	Renewable Energy Accommodation Rate/%
IEEE-33 node test system	Improved BWO	32	285	18,176.1	97.4
	Traditional BWO	56	420	18,492.3	95.8
	PSO	78	510	18,765.8	93.2
	GA	85	542	18,652.4	94.6
IEEE-69 node test system	Improved BWO	48	635	39,245.7	96.8
	Traditional BWO	82	985	39,872.4	94.1
	PSO	115	1320	40,568.9	91.5
	GA	129	1409	39,854.2	92.1
IEEE-118 node test system	Improved BWO	65	1240	75,832.3	96.1
	Traditional BWO	110	2050	77,125.6	93.5
	PSO	156	2880	78,459.4	90.2
	GA	171	2975	78,024.5	91.9

below.

By observing the above table, it can be found that the improved algorithm in this paper demonstrates a fast convergence speed (32 iterations). This is attributed to the Tent chaotic mapping, which enhances the quality of the initial population, increases population diversity, and avoids premature convergence, thereby enabling the algorithm to find superior solutions with fewer iterations. Additionally, it achieves the lowest dispatch cost due to its strong global search capability, allowing it to better approximate the optimal solution. The computation time is moderate because the algorithm’s efficient structure avoids redundant iterations. However, the algorithm’s structure is slightly complex, making its implementation more challenging than PSO and GA. Traditional BWO exhibits strong exploration capabilities but is sensitive to initial parameters, prone to getting trapped in local optima, and has a slow convergence speed. The lack of chaotic initialization leads to insufficient population diversity, resulting in a higher number of convergence iterations and increased costs. PSO has the advantages of simple implementation and relatively fast convergence, but it is prone to premature convergence and lacks sufficient global search capability. PSO tends to stagnate in local optimal regions, leading to higher dispatch costs and lower renewable energy accommodation rates. GA boasts strong robustness but suffers from low efficiency in crossover and mutation operations, requiring multiple iterations to converge and resulting in the longest computation time.

When applied to larger-scale test systems, including the IEEE-69-node and IEEE-118-node test systems, the increase in system nodes significantly raises the dimensionality of decision variables. The exponential expansion of the search space necessitates more iterations for any algorithm to explore and converge. Although the number of iterations and computation time increase for all algorithms, the improved BWO algorithm in this paper exhibits the smallest growth rate. In the 118-node system, the improved BWO converges in just 65 iterations, far fewer than the 110 iterations of traditional BWO and 156 iterations of PSO. This is attributed to the high-quality initial population generated by the Tent chaotic mapping, which improves search efficiency and avoids numerous ineffective iterations. Larger-scale systems encompass more controllable resources and more complex operational constraints, theoretically increasing both optimization potential and difficulty. The improved BWO maintains a leading position in various economic and environmental indicators. Its powerful global exploration capability ensures that high-quality solutions can still be found in high-dimensional solution spaces, resulting in lower dispatch costs and higher renewable energy accommodation rates. This demonstrates that the algorithm proposed in this paper has good scalability and can meet the optimization needs of future large-scale active distribution networks.

5.4. Sensitivity Analysis Results of Key Parameters and Discussion on Practical Applications

To thoroughly investigate the response characteristics of the dispatch model constructed in this paper to different key parameters, this section conducts sensitivity analyses on the benchmark price in the carbon trading market, the cost coefficient for load-side flexibility adjustment, and the renewable energy penetration rate of the system. By varying these parameters, the impacts on the system’s total cost, carbon emissions, and new energy accommodation rate are quantitatively analyzed, providing a reference basis for policy formulation and system operation. The core of the stepped carbon trading mechanism is the carbon price. By setting different benchmark carbon prices while keeping other parameters constant, changes in the system dispatch results are analyzed, as shown in Figure 10.

Observing Figure 10 above, it can be seen that as the carbon price increases, the total system cost initially drops rapidly and then stabilizes, carbon emissions continuously decline, and the new energy accommodation rate steadily improves. When the carbon price is low, the cost of carbon emissions accounts for a relatively small proportion of the total cost, and the cost advantage of traditional coal-fired units still exists. Consequently, the dispatch strategy leans more towards economic efficiency, resulting in lower total costs but higher carbon emissions. As the carbon price rises, the cost of carbon emissions becomes a non-negligible factor. The stepped carbon pricing mechanism causes the marginal cost of excess emissions to increase sharply. To minimize total costs, the optimization model prioritizes the dispatch of zero-carbon wind and solar energy sources and incentivizes flexible loads to respond in order to match new energy output, thereby significantly reducing carbon emissions and improving the accommodation rate. Once the carbon price reaches a certain high level, the system is almost entirely dominated by new energy and flexible loads, leaving limited room for further carbon emission reductions. At this point, the total cost is primarily composed of the operation and maintenance costs of wind and solar power, compensation costs for flexible loads, and network losses, and it is no longer sensitive to carbon prices. This reflects the law of diminishing marginal effects of carbon pricing policies.

Load flexibility is key to accommodating new energy. Users’ willingness to participate in demand response is significantly influenced by compensation prices. By setting different adjustment cost coefficients, the analysis results are shown in Figure 11.

Observing Figure 11 above, it can be seen that as the cost of mobilizing flexible loads increases, the total system cost rises, carbon emissions increase, and the new energy accommodation rate decreases. When compensation prices are low, users have a strong willingness to participate in demand response, and flexible load resources are abundant and inexpensive. The model can extensively utilize these resources to track new energy output, resulting in the highest accommodation rate, the lowest carbon emissions, and the lowest total cost due to savings on electricity purchase and carbon costs. As compensation prices rise, the economic efficiency of mobilizing flexible loads diminishes. The optimization model re-evaluates the trade-off between the cost of calling upon flexible loads and the cost of wind/solar curtailment or using traditional energy sources. It reduces the utilization of flexible loads and instead accepts more wind/solar curtailment or increases the output of traditional generating units, leading to a decline in the new energy accommodation rate and an increase in carbon emissions. Consequently, the total cost also rises significantly. This indicates that formulating reasonable demand response subsidies or electricity pricing policies is crucial for unlocking the potential on the load side.

Renewable energy penetration rate is a core indicator of energy transition. Different penetration rate scenarios are simulated by varying the rated installed capacity of wind and solar power in the system, with the results shown in Figure 12.

Observing Figure 12 above, it can be seen that as the renewable energy penetration rate increases, the total system cost initially decreases and then rises, carbon emissions continue to decline significantly, and the new energy accommodation rate gradually decreases. With the increase in wind and solar installed capacity, clean energy replaces a substantial amount of fossil energy, making the continuous decline in carbon emissions an inevitable trend. However, the higher the penetration rate, the greater the challenge posed by the volatility and intermittency of new energy output to system balance. In high-penetration scenarios, even with full utilization of all flexibility resources in the model proposed in this paper, wind and solar curtailment may still occur in extreme cases, leading to a decrease in the accommodation rate. This highlights the necessity of introducing more flexible resources. When the penetration rate increases from 30% to 50%, the cost rise is not significant, as the saved fuel and carbon costs still outweigh the additional costs of wind and solar investments and flexible dispatch. However, when the penetration rate exceeds 70%, the balancing cost of the system increases sharply: to accommodate fluctuating energy sources, more frequent use of higher-cost flexibility resources is required, and the cost of resource waste due to wind and solar curtailment also rises, causing the total cost to increase instead of decrease. This indicates that achieving a high proportion of renewable energy systems requires support from lower-cost flexibility technologies and more advanced dispatch strategies.

Overall, carbon pricing is an effective policy tool to drive the system towards low-carbon operation, with higher carbon prices effectively incentivizing new energy accommodation and carbon reduction. The cost of load flexibility is a key economic factor influencing the effectiveness of demand response implementation, and a reasonable compensation mechanism is crucial for unlocking the potential on the load side. The renewable energy penetration rate is not necessarily higher the better; its development needs to be coordinated with system flexibility construction, otherwise it may lead to an increase in total system costs and resource waste. The joint optimization model proposed in this paper can effectively integrate various flexibility resources and achieve favorable economic and environmental benefits under different penetration rate scenarios, demonstrating its robustness and applicability.

6. Conclusions

This study presents a comprehensive low-carbon economic optimization dispatch framework for active distribution networks, integrating flexible load participation and tiered carbon trading mechanisms to maximize renewable energy utilization and minimize carbon emissions. By leveraging a hybrid SSA-CNN-LSTM model for high-precision renewable generation forecasting and systematically classifying flexible loads into shiftable, transferable, and reducible categories, the proposed approach effectively harnesses demand-side flexibility through price-based incentives rather than direct control. The introduction of a tiered carbon trading mechanism further aligns economic objectives with carbon reduction goals, creating a market-driven pathway toward sustainable operation. To address the inherent non-convex nonlinearities of the dispatch model, an enhanced BWO algorithm is developed, incorporating Tent chaotic mapping to overcome limitations in solution diversity and parameter sensitivity observed in conventional BWO methods. Simulation results on a modified IEEE-33 bus system validate the effectiveness of the proposed framework, demonstrating significant improvements in renewable consumption, carbon emission reductions, and computational robustness compared to traditional approaches. The limitations of the current research are primarily reflected in the aspects of model validation and adaptability to real-world scenarios: On one hand, the algorithm’s performance has only been validated through a modified IEEE-33 bus simulation system, lacking testing with complex operational data from real distribution networks, which may result in insufficient robustness under actual operating conditions such as noise interference and data missing. On the other hand, the study has not fully considered practical constraints such as instruction execution delays caused by communication latency, scheduling deviations resulting from renewable energy forecasting errors, and users’ refusal of load regulation due to production continuity requirements. These factors may undermine the actual effectiveness of flexible loads participating in grid interaction. Future research could construct a testing platform based on historical operational data from real distribution networks, introduce communication reliability constraint models and prediction error correction mechanisms, and combine user behavior analysis to quantify the impact of uncertain factors on scheduling results, thereby developing a multi-objective optimization framework that balances economic efficiency and engineering feasibility.