Two-Stage Optimization Research of Power System with Wind Power Considering Energy Storage Peak Regulation and Frequency Regulation Function

Abstract

Addressing the problems of wind power’s anti-peak regulation characteristics, increasing system peak regulation difficulty, and wind power uncertainty causing frequency deviation leading to power imbalance, this paper considers the peak shaving and valley filling function and frequency regulation characteristics of energy storage, establishing a day-ahead and intraday coordinated two-stage optimization scheduling model for research. Stage 1 establishes a deterministic wind power prediction model based on time series Autoregressive Integrated Moving Average (ARIMA), adopts dynamic peak-valley identification method to divide energy storage operation periods, designs energy storage peak regulation working interval and reserves frequency regulation capacity, and establishes a day-ahead 24 h optimization model with minimum cost as the objective to determine the basic output of each power source and the charging and discharging plan of energy storage participating in peak regulation. Stage 2 still takes the minimum cost as the objective, based on the output of each power source determined in Stage 1, adopts Monte Carlo scenario generation and improved scenario reduction technology to model wind power uncertainty. On one hand, it considers how energy storage improves wind power system inertia support to ensure the initial rate of change of frequency meets requirements. On the other hand, considering energy storage reserve capacity responding to frequency deviation, it introduces dynamic power flow theory, where wind, thermal, load, and storage resources share unbalanced power proportionally based on their frequency characteristic coefficients, establishing an intraday real-time scheduling scheme that satisfies the initial rate of change of frequency and steady-state frequency deviation constraints. The study employs improved chaotic mapping and an adaptive weight Particle Swarm Optimization (PSO) algorithm to solve the two-stage optimization model and finally takes the improved IEEE 14-node system as an example to verify the proposed scheme through simulation. Results demonstrate that the proposed method improves the system net load peak-valley difference by 35.9%, controls frequency deviation within ±0.2 Hz range, and reduces generation cost by 7.2%. The proposed optimization scheduling model has high engineering application value.

1. Introduction

Driven by peak carbon and carbon-neutral targets, the share of renewable energy in the power system, represented by wind power, continues to climb [1]. Wind generation capacity in China crossed the 400 GW threshold by late 2023, accounting for 15.8% of overall electricity generation capacity. Nevertheless, the intermittent, stochastic, and counter-peak characteristics of wind power output pose multiple challenges to the safe and economical operation of power systems [2].

Firstly, power imbalance caused by wind power uncertainty will cause system frequency deviation [3]. Secondly, in traditional power systems, thermal power units serve as the primary power sources responsible for maintaining system power balance, with their installed capacity typically dominating the system and undertaking the majority of load demand, making their operating costs constitute the major portion of total system costs. The anti-peak regulation characteristics of wind power exacerbate the peak-to-valley difference in the system’s net load, increasing the peak regulation burden [4,5]. Conventional thermal power units are forced to perform deep peak regulation or even frequent start-stop, which not only increases operating costs but also affects unit lifespan. In addition, wind curtailment problems caused by wind power accommodation difficulties seriously restrict the efficient utilization of clean energy. Energy storage with time-shifting characteristics can participate in peak regulation according to system requirements, effectively alleviating the operational pressure on thermal power units, and with its rapid response capability, energy storage can not only quickly respond to wind power prediction errors and provide virtual inertia support, but also effectively alleviate frequency deviation and ensure frequency quality requirements.

In response to the aforementioned issues, scholars both domestically and internationally have conducted extensive research and achieved significant progress. However, existing studies still exhibit notable limitations that urgently require further improvement.

The optimization scheduling problem of wind power integration is fundamentally a single-objective or multi-objective optimization problem that satisfies various constraint conditions after establishing generation plans across single or multiple time scales and predicting wind power through different methods. Wind power forecasting includes deterministic and uncertainty-based approaches. References [6,7,8] developed generation planning models with wind-storage systems, but they used deterministic wind power forecasting without considering prediction errors. Uncertainty-based wind power prediction provides comprehensive information by examining fluctuation ranges, probability distributions, and possible scenarios. Uncertainty prediction methods include interval prediction [9], probabilistic prediction [10], and scenario prediction. Common scenario prediction approaches include time series methods [11], machine learning approaches [12,13], Latin hypercube sampling [14], and Monte Carlo methods.

Zhang et al. [15] established a robust optimization method using budget uncertainty sets. This method utilizes the regulation advantages of conventional units and large-scale battery energy storage stations for addressing wind power variations. Li et al. [16] proposed a dynamic economic evaluation model for hundred-megawatt electrochemical energy storage systems. The model considers battery equivalent lifetime based on power-capacity relationships during peak-shaving and valley-filling processes. It analyzes the dynamic economic benefits of different storage technologies throughout power grid lifecycles. Crespo-Vazquez et al. [17] developed a stochastic planning framework for wind-storage stations participating in electricity markets. It uses machine learning for scenario generation to handle wind power and electricity price uncertainties. Wu et al. [18] proposed a risk-constrained day-ahead scheduling model for gravity energy storage systems and wind turbines. This model uses Information Gap Decision Theory (IGDT) to achieve profits against wind power and market price uncertainties. These studies provide technical foundations for energy storage systems addressing wind power uncertainty problems. However, wind power forecasting accuracy improves as time scales shorten. Single-stage scheduling cannot achieve corrections from subsequent stages to previous stages. This makes it difficult to consider unit allocation in real-time stages that account for actual wind power fluctuations.

Energy storage system applications focus on exploring system value from different functional perspectives. Yu et al. [19] utilized demand response resources and energy storage systems as novel wind power accommodation approaches. It coordinates thermal units, wind power, and storage to minimize system operating costs while improving wind utilization rates and reducing peak regulation pressure. Bian et al. [20] introduced an optimized dispatch method based on a two-stage stochastic power system model incorporating thermal power-energy storage peak regulation pricing. Zhang et al. [21] utilized deep regulation characteristics of thermal power units and energy storage time-shifting properties to reduce grid net load peak-valley differences. It established an optimized dispatch model for high-proportion renewable energy power systems. Xiao et al. [22] constructed a bilevel optimization dispatch model introducing net load fluctuation indicators in the upper model to optimize net load curves and satisfy system peak regulation requirements. Krommydas et al. [23] utilized energy storage systems to provide reserve services for reducing the uncertainty impacts of renewable energy generation on power systems. It established a two-stage stochastic unit commitment model optimizing reserve capacity allocation. Large-scale wind power integration increases net load peak-valley differences, intensifies peak regulation burdens, and increases potential operational risks. These studies introduced energy storage systems to alleviate system peak regulation pressure. However, energy storage primarily serves peak regulation or reserve services without fully considering the economic optimization of coordinated peak regulation and frequency regulation functions within scheduling frameworks.

Based on the above analysis, addressing the challenges of increased system peak regulation difficulty caused by the anti-peak regulation characteristics of wind power and frequency deviations resulting from power imbalances due to wind power uncertainty, this paper proposes a two-stage optimization scheduling method for wind power integrated power systems considering the coordinated peak and frequency regulation functions of energy storage. The main contributions of this paper are as follows:

(1)

Addressing the limitation that existing wind power scheduling research predominantly employs single deterministic forecasting methods without enabling inter-stage corrections, this paper adopts ARIMA time series method for deterministic modeling in the day-ahead stage and employs Monte Carlo scenario generation combined with improved K-means clustering and simultaneous backward reduction algorithms for uncertainty modeling in the intra-day stage, fully utilizing the characteristic that wind power forecasting accuracy improves as time scales shorten to achieve precise corrections of subsequent stages on preceding stages.

(2)

Different from traditional single-function energy storage applications, this paper establishes a dynamic peak-valley identification method based on net load characteristics and a State of Charge (SOC) zoning management mechanism, enabling energy storage to perform peak shaving and valley filling tasks during peak regulation periods and participate in frequency regulation using reserved capacity during non-peak regulation periods, significantly improving energy storage system utilization rates and comprehensive benefits while resolving the single-function limitation of energy storage in existing research.

(3)

Establishing a multi-unit coordinated frequency regulation mechanism based on dynamic power flow. This approach overcomes the limitation in conventional power flow calculations where unbalanced power is solely borne by idle buses. It establishes a distribution mechanism whereby wind, thermal, load, and storage resources coordinate to share unbalanced power according to their respective frequency characteristic coefficients. Concurrently, it accounts for the impact of virtual inertia support from energy storage on the initial rate of frequency change. By integrating constraints on both the initial rate of frequency change and steady-state frequency deviation, the dispatch scheme ensures compliance with system frequency quality requirements.

This paper is divided into the following sections.

Section 2 establishes the two-stage wind power forecasting framework. The day-ahead stage uses the ARIMA time series method for deterministic forecasting. The intra-day stage employs Monte Carlo scenario generation. This method combines improved K-means clustering and SBR scenario reduction to handle wind power uncertainty. Section 3 presents the coordinated peak and frequency regulation strategy for energy storage. This section establishes SOC-based working region division methods. It also proposes dynamic peak-valley identification methods based on net load characteristics. Section 4 constructs the two-stage optimization scheduling model considering energy storage functions. The model includes day-ahead optimization scheduling and intra-day real-time scheduling based on dynamic power flow. This section also introduces optimization algorithm solution methods. Section 5 presents the results, followed by discussion, conclusion, future work, acknowledgements, conflicts of interest, and references.

2. Wind Power Forecasting Under Two-Stage Optimal Scheduling

Given the uncertainty of wind power, predicting wind farm output power has become a key condition in the optimization and dispatch decision-making of power systems that include wind power. Since wind power output and corresponding decisions exhibit temporal correlation, with forecasting accuracy improving as time scales become shorter [24], this scheduling adopts different forecasting methods at different stages based on comprehensive consideration of simplicity and accuracy. The first stage uses a longer time scale and focuses on establishing an efficient foundation for second-stage optimization. It employs the time series method with substantial historical data to forecast wind speed. However, discrepancies inevitably arise between first-stage wind speed predictions and second-stage realized values, and such errors lead to more pronounced system power imbalance. Therefore, the second stage’s real-time optimization employs the Monte Carlo method for comprehensive wind power forecasting as the time scale shortens. This approach offers greater precision, flexibility, and universal applicability. The forecasting results are then refined through scenario reduction using combined K-means clustering and simultaneous backward reduction (SBR) algorithms, yielding more intuitive results. Conducting two-stage optimal scheduling containing wind power based on wind farm output power forecasting aligns more closely with practical conditions.

2.1. Stage 1 Deterministic Wind Power Forecasting Based on ARIMA Time Series

Due to the temporal and discrete nature of wind speed data, traditional linear regression models struggle to effectively handle these complex characteristics. Consequently, the ARIMA methodology is implemented for day-ahead wind velocity forecasting within this research framework. The ARIMA model can capture both the stationary components of a time series and the non-stationary components through differencing, making it more adaptable to data with pronounced time-varying characteristics, such as wind speed. The mathematical structure underlying ARIMA wind speed forecasting is outlined below:

(1)

Let the original wind speed data be {x₁, x₂, x₃, …, x_t}. First, determine whether this data series is stationary. If it is non-stationary, introduce the ordered difference operator ∇ = 1 − K, and perform d-order differencing transformation on the above series until it becomes a stationary series {y₁, y₂, y₃, …, y_t}.

(2)

The number of terms in the wind speed prediction model is determined by calculating the autocorrelation function and partial autocorrelation function of the series {y₁, y₂, y₃, …, y_t}.

(3)

Parameter estimation using the least squares method involves minimizing the sum of the squares of the residuals to determine the unknown parameters of the model.

(4)

The model residuals are tested to determine whether the white noise is stationary. If so, the model is appropriate; otherwise, the model order requires modification, and parameters must be re-estimated.

(5)

The appropriate model performs wind velocity prediction to obtain the predicted wind speed value series for the next 24 h.

After obtaining the wind speed forecast values, as they are directly correlated with wind turbine output power, substituting the predicted wind speed values into Equation (1) directly yields the 24 h active power output of wind turbines. The piecewise linear power curve model in Equation (1) is based on two fundamental assumptions widely adopted in power system scheduling studies: wind speed remains constant within each dispatch time interval, and the power coefficient remains fixed within each piecewise segment. This simplified treatment effectively transforms complex nonlinear relationships into linear constraints suitable for optimization scheduling [25], providing data for the first stage optimization.

$$P_w=\left\{\begin{array}{ll}0& v<v_{cr} \mathrm{or} v>v_{co}\\ {\displaystyle \frac{P_r}{v_r-v_{cr}}}\cdot v-{\displaystyle \frac{P_r}{v_r-v_{cr}}}\cdot v_{cr}& v_{cr} <v<v_r\\ P_r& v_r<v<v_{co}\end{array}\right.$$

(1)

2.2. Stage 2 Wind Power Uncertainty Forecasting Based on Multi-Scenario Probability Method

Given the high unpredictability of intra-day wind generation output, single deterministic forecasting cannot meet the accuracy requirements of real-time scheduling. Multi-scenario probability-based methods generate multiple possible wind power realization paths. These methods comprehensively characterize uncertainty distribution and provide a robust decision-making basis for scheduling optimization. This research employs Monte Carlo simulation combined with K-means clustering and simultaneous backward reduction (SBR) algorithms for multi-scenario probability modeling during the intra-day phase. This integrated approach accurately captures the stochastic characteristics of wind power uncertainty.

2.2.1. Scenario Generation

The Monte Carlo stochastic simulation method is a numerical calculation method based on random sampling, capable of effectively characterizing the probability distribution characteristics of wind power. Within the present research, Monte Carlo algorithms produce extensive wind power scenarios via these implementation stages:

(1)

For the random variable wind power P, the sampling time is divided into 24 points, namely a sample size of 24, expressed as {P₁, P₂, …, P₂₄};

(2)

For a specific time point of wind power P_m, after obtaining the parameters of its probability distribution function from historical wind power data, simulation methods are utilized to generate random wind power, thereby obtaining a set of randomly generated samples;

(3)

Take a sample size M, continue sampling in the sample capacity, and finally obtain the sample group {P₁, P₂, …, P₂₄}. Take M = 1000, generate 1000 wind power scenarios based on the Monte Carlo simulation method, and obtain a 1000 × 24 array containing the wind power of all time periods in the sample capacity.

2.2.2. Scenario Reduction

Monte Carlo simulation techniques can comprehensively capture the stochastic characteristics of wind power, providing rich information for uncertainty analysis. However, due to the vast number of generated scenarios, directly incorporating all scenarios into the optimization model would lead to excessive computational complexity. Consequently, the application of scenario reduction methodologies is essential, with the objective of extracting a highly representative streamlined subset from the initial scenario set, ensuring computational efficiency while preserving key probabilistic features. The scenario reduction process in this paper is implemented in two steps. The first step classifies the initial scenarios through an improved K-means clustering algorithm, grouping similar scenarios together; the second step applies a simultaneous backward reduction algorithm based on Kantorovich distance within each group to select the most representative scenario, thereby obtaining a significantly reduced set that still effectively characterizes the initial scenario collection [26]. The formal expression for the Kantorovich distance is presented in the following equation:

$$\begin{array}{l}{D}_l\left(S_c,S_c^{\prime }\right)=\min \left\{{\displaystyle \sum _{sc\in S_c,s{c}^{\prime }\in S_c^{\prime }}d\left(sc,sc’\right)\eta \left(sc,s{c}^{\prime }\right)}\right.\\ |\eta \left(sc,s{c}^{\prime }\right)\ge 0,{\displaystyle \sum _{s{c}^{\prime }\in S_c^{\prime }}\eta \left(sc,s{c}^{\prime }\right)=p_{sc}^{\prime }}\\ \left.{\displaystyle \sum _{sc\in S_c}\eta \left(sc,s{c}^{\prime }\right)=p_{sc}^{\prime },sc\in S_c,s{c}^{\prime }\in S_c^{\prime }}\right\}\end{array}$$

(2)

During implementation, a critical technical issue must be addressed: the K-means clustering algorithm operates solely on numerical similarity without incorporating probability information, while the subsequent SBR algorithm requires probability-based selection criteria. This creates a methodological gap in probability handling. To bridge this gap, scenarios within each cluster are treated as equiprobable, with normalized probabilities summing to unity. Furthermore, recognizing that Monte Carlo-generated scenarios inherently possess equal probability distributions and that post-reduction each cluster retains only a single representative scenario, it becomes necessary to reassign probability weights to these representative scenarios. The probability update mechanism follows the following expression:

$$p_c=N_c/N$$

(3)

Through the implementation of the above scenario reduction techniques, 1000 initial wind power scenarios can be efficiently reduced to four representative core scenarios while retaining the statistical characteristics of the initial scenario set. This method not only ensures that the selected scenarios fully reflect the uncertainty characteristics of wind power output but also significantly improves the computational efficiency of subsequent optimization solutions.

3. Coordinated Optimization Operation Strategy for Energy Storage System Peak and Frequency Regulation

When energy storage is used solely for single peak regulation scenarios, it only performs charging and discharging actions during peak load periods or valley load periods. During other periods, it remains idle, significantly reducing energy storage utilization rates. Therefore, this paper adopts a coordinated operation strategy for energy storage peak regulation and frequency support functions. Through rational capacity allocation and temporal scheduling, energy storage systems execute peak regulation tasks while utilizing reserved capacity to rapidly respond to system power imbalances. Specifically, during peak and valley load periods, energy storage performs peak shaving and valley filling tasks while simultaneously reserving partial capacity for power balance response. During non-peak periods, energy storage utilizes more available capacity to participate in compensating for system power imbalances, reducing frequency deviations through power adjustments, and providing virtual inertia support to reduce the initial rate of change of frequency (RoCoF).

3.1. SOC-Based Working Region Division Method

Subject to energy storage capacity limitations, when energy storage is applied to coordinated peak and frequency regulation control, the impact of SOC must be considered. When energy storage operates in the valley-filling region, the energy storage is in a charging state, with its SOC rising from 0.1 to 0.9. When operating in the peak-shaving region, it is in a discharging state, with SOC decreasing from 0.9 to 0.1. When the energy storage system completes peak regulation tasks and switches to the frequency regulation region, the initial SOC value may be at 0.1 or 0.9. However, frequency deviations possess bidirectional possibilities. Therefore, when SOC is 0.1, energy storage cannot improve frequency drop issues through discharging; when SOC is 0.9, energy storage cannot improve frequency rise issues through charging. To avoid placing SOC at critical values, which would limit frequency regulation capability, this paper sets the SOC peak regulation working range as 0.15–0.85, reserving a capacity margin of 0.05 for frequency regulation functions. Figure 1 shows the schematic diagram of the SOC working region division under coordinated peak and frequency regulation control of energy storage.

3.2. Dynamic Identification of Peak and Valley Periods with Net Load Characteristics

In traditional power systems, the load curve directly reflects the system’s regulation requirements. Nevertheless, following substantial wind generation integration, the actual regulation pressure faced by the system is no longer determined solely by load, but rather by the combined effect of load and wind energy output. Accordingly, the net load curve must be derived from the load profile combined with the 24 h wind power forecast generated during Stage 2 of Section 2.1, and employ a dynamic identification method utilizing net load characteristics for temporal segmentation into peak and valley intervals. The fundamental principle of this method is to optimize energy utilization during peaking and valley filling through adjusting positions of the peaking line and valley filling line under energy storage capacity constraints. The specific execution process is outlined below:

(1)

Import the net load P_L, obtain the net load maximum value P_max and the minimum value P_min, which serve as the initial boundaries for the peaking line and the filling line.

(2)

Set the nominal power and nominal capacity of the energy storage device to P_m and E_m, with the current availability of the energy storage system being E_n. The initial value of the peak shaving line is P_max, and it moves downward with step size ∆P, yielding the real-time peak shaving line P_f = P_max − k × ∆P and its intersection points t₁, t₂ with the load curve. The electricity released by energy storage participating in peak shaving at this time is as follows:

$$S_f={\displaystyle {\int }_{t_1}^{t_2}\left(P_{\max }-P_f\right)}dt$$

(4)

If S_f < E_n, then the iteration count k = k + 1, updating the peak shaving line P_f(t) at time t, until S_f = E_n, at which point the iteration terminates, yielding the optimal peak shaving line.

(3)

Take the initial valley filling line P_min and move it upward with step size ∆P. The valley filling line, P_g = P_min + k × ∆P, intersects with the load curve at points t₃ and t₄. The electricity absorbed by energy storage participating in valley filling at this time is as follows:

$$S_g={\displaystyle {\int }_{t_3}^{t_4}\left(P_g-P_{\min }\right)}dt$$

(5)

If S_g < E_n, then the iteration count k = k + 1, updating the valley filling line P_g(t) at time t, until S_g = E_n, at which point the iteration terminates, yielding the optimal valley filling line.

3.3. Coordinated Optimization Operation Strategy for Peak and Frequency Regulation

According to Section 3.2, the planned peak shaving line P_f(t) and planned valley filling line P_g(t) can be obtained, whilst the frequency regulation output of energy storage caused by system disturbances during non-peak regulation periods is P_PRF(t). Following the working region division and output rules for the coordinated peak and frequency regulation scenario of energy storage, the energy storage output P_bess(t) at each moment can be obtained. The specific implementation process is as follows.

(1)

Valley-filling region: During this period, the load is at valley values with low electricity demand, in a state of supply exceeding demand. Energy storage is in a charging state, absorbing surplus electrical energy. At this time, the energy storage output P_bess(t) = P_g(t) < 0.

(2)

Frequency regulation region: During this period, there is no peak regulation demand. If the frequency deviation exceeds the set upper limit, energy storage is in a charging state, with P_bess(t) = P_PRF(t) < 0; if the frequency deviation falls below the set lower limit, energy storage is in a discharging state, with P_bess(t) = P_PRF(t) > 0; if the frequency deviation is within the dead zone, energy storage output is zero, with P_bess(t) = 0.

(3)

Peak shaving region: During this period, the load is at peak values with high electricity demand, in a state of demand exceeding supply. Energy storage is in a discharging state, with energy storage output at this time being P_bess(t) = P_f(t) > 0.

4. Coordinated Optimization of Peak Load and Frequency Regulation Strategies for Energy Storage Systems

This study establishes a two-stage optimization scheduling framework that accounts for wind power stochasticity and the peak-shaving and frequency regulation functions of energy storage. The first stage operates on a 1 h time scale with planning executed 24 h in advance. It employs the deterministic wind power forecasting method from Section 2.1. Energy storage implements the coordinated peak and frequency regulation strategy from Section 2.1 to minimize total cost as the optimization objective. This stage determines each power source’s output as the base for the second stage. The second stage addresses wind power uncertainty using a 15-min time scale. It employs the method in Section 2.2 to reforecast wind power while incorporating the initial rate of change of frequency constraints and steady-state frequency error constraints. The optimization maintains the objective of minimizing total cost while correcting power source outputs to achieve optimal dispatch.

4.1. Stage 1 Day-Ahead Optimization Scheduling Model

4.1.1. Objective Function

The day-ahead stage takes minimum generation cost as the objective function, where f_G represents the operating cost of thermal power units; f_E represents the charging and discharging cost of energy storage.

$$\min F_1=f_G+f_E$$

(6)

$$f_G={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_G}{a}_i\left(P_{Gi,t}^2\right)+b_i{P}_{Gi,t}+c_i}}$$

(7)

$$f_E={\displaystyle \sum _{t=1}^T{c}_{sc}}\left(P_{i,t}^{dis}+P_{i,t}^{ch}\right)$$

(8)

4.1.2. Constraints Conditions

(1)

Upper and lower limit constraints for thermal power unit output:

$$P_{Gi,\min }\le P_{Gi,t}\le P_{Gi,\max }$$

(9)

$$Q_{Gi,\min }\le Q_{Gi,t}\le Q_{Gi,\max }$$

(10)

(2)

Thermal power unit power regulation constraints:

$$-d_{Gi}\le P_{Gi,t}-P_{Gi,t-1}\le f_{Gi}$$

(11)

(3)

Conventional power flow equation equality constraints:

$$\Delta P_{i,t}=P_{Gi,t}+P_{Wi,t}+P_{i,t}^{dis}-P_{i,t}^{ch}-P_{L,t}-U_{i,t}{\displaystyle \sum _{j=1}^{N_i}{U}_{j,t}\left(G_{ij}\cos {\theta }_{ij,t}+B_{ij}\sin {\theta }_{ij,t}\right)=0}$$

(12)

$$\Delta Q_{i,t}=Q_{Gi,t}+Q_{Wi,t}-Q_{L,t}-U_{i,t}{\displaystyle \sum _{j=1}^{N_i}{U}_{j,t}\left(G_{ij}\sin {\theta }_{ij,t}-B_{ij}\cos {\theta }_{ij,t}\right)=0}$$

(13)

(4)

Voltage phase angle constraints:

$$U_{i,\min }\le U_{i,t}\le U_{i,\max }$$

(14)

$${\theta }_{i,\min }\le {\theta }_{i,t}\le {\theta }_{i,\max }$$

(15)

(5)

Transmission line power flow constraints:

$$P_{ij}^{\min }\le P_{\mathrm{ij},\mathrm{t}}\le P_{ij}^{\max }$$

(16)

(6)

SOC constraints:

$$\left\{\begin{array}{l}SO{C}_{i,t}=SO{C}_{i,t-1}+{\displaystyle \frac{P_{i,t}^{ch}{\eta }_c\Delta t}{E_n}}-{\displaystyle \frac{P_{i,t}^{dis}\Delta t}{{\eta }_{\mathrm{d}}{E}_n}}\\ SO{C}_{i,p\min }\le SO{C}_{i,t}\le SO{C}_{i,p\max }\end{array}\right.$$

(17)

(7)

Energy storage charging and discharging power constraints:

$$\left\{\begin{array}{l}{P}_{i,\min }^{dis}\le P_{i,t}^{dis}\le P_{i,\max }^{dis}\\ P_{i,\min }^{ch}\le P_{i,t}^{ch}\le P_{i,\max }^{ch}\end{array}\right.$$

(18)

Furthermore, as the energy storage system cannot charge and discharge simultaneously during the same period, the energy storage system must also satisfy the following conditions.

$$P_{i,t}^{ch}{P}_{i,t}^{dis}=0$$

(19)

4.2. Stage 2 Intra-Day Optimization Scheduling Model

After completing the day-ahead output allocation for each power source, discrepancies arise between forecast and actual wind speeds during practical grid operation. These errors create more pronounced system power imbalances, exacerbating frequency issues. Therefore, this study establishes intra-day real-time scheduling with a 15-min time scale to accurately simulate actual conditions. The scheduling employs the initial rate of change of frequency and steady-state frequency deviation as constraints. It introduces dynamic power flow to enable wind, thermal, load, and storage resources to coordinately share unbalanced power through joint frequency regulation. This approach ensures the scheduling scheme meets system frequency quality requirements. The second stage employs the same optimization objective as previously stated; hence, the specific formulation is omitted here.

4.2.1. Initial Rate of Change of Frequency Constraints

The initial rate of frequency change reflects the speed at which frequency drops or climbs when the system experiences disturbances. At the onset of a power deficit, the rate of frequency change reaches its maximum value, while the frequency deviation remains minimal, approximating zero. To effectively handle the complex frequency response characteristics of multiple power sources within wind-storage hybrid power systems, this paper employs a segmented linearization approach. By analyzing the temporal power variation trends of various power sources during frequency regulation, it approximates the frequency response processes of wind turbines, thermal power units, energy storage systems, and other sources during the inertia response phase. This simplifies the computational complexity arising from the system’s diverse power source types. During the inertial response phase, to prevent excessive frequency drop rates within the system, the primary consideration must be whether the system inertia is sufficient. Specifically, the initial rate of change of frequency (RoCoF) must satisfy the constraint:

$$RoCoF={\displaystyle \frac{\Delta P_t}{2H_{sys}}}{f}_N\le RoCo{F}_{\max }$$

(20)

It should be noted that while inertial support typically operates during the initial seconds following a disturbance, affording time for other control mechanisms to activate, the present study focuses on optimizing system operational scheduling rather than real-time frequency dynamic control. This is achieved by optimizing power supply capacity to meet frequency constraints. The grid equivalent inertia parameter equals the capacity-weighted sum of inertial contributions from all operating generation equipment. In traditional power systems, this value is primarily governed by the rotational inertia inherent in synchronous generation units. However, in power systems containing wind and storage, wind power and energy storage systems can provide virtual inertia support through virtual synchronous generator control technology. The expression for the system total inertia is as follows:

$$H_{sys}={\displaystyle \frac{{\displaystyle \sum H_{Gi}{S}_i}+{\displaystyle \sum H_W{S}_w}+{\displaystyle \sum H_E{S}_e}}{S_{BASE}}}$$

(21)

This study employs an adaptive virtual inertia regulation strategy for energy storage systems [27]. The approach divides power disturbance magnitudes into distinct intervals, with each interval assigned a specific virtual inertia value. In practical power system operations, power imbalance detection and quantification are achieved through real-time monitoring by the Energy Management System (EMS) at dispatch centers. The system continuously monitors power output from all generation sources and load demand to calculate real-time power balance deviations. When the detected system power imbalance falls within different intervals, the energy storage virtual inertia switches to the corresponding preset value for that interval. The system switches between predefined discrete inertia values based on the magnitude of power disturbances. This enables real-time adjustment according to system operating conditions and requirements.

4.2.2. Power Flow Equation Equality Constraints Based on Dynamic Power Flow Imbalanced Power Sharing

Dynamic power flow considers that all power sources and loads with frequency regulation capability in the system share the imbalanced power according to their frequency static characteristic coefficients [28]. The calculation results are more realistic than those from conventional power flow calculations, where imbalanced power is shared only by the slack bus, and can quantify the steady-state frequency deviation value of the system.

Power flow equation equality constraints:

$$\Delta P_{i,t}^{\prime }=P_{Gi,t}^{\prime }+P_{Wi,t}^{\prime }+P_{i,t}^{dis’}-P_{i,t}^{ch’}-P_{L,t}^{\prime }-U_{i,t}^{\prime }{\displaystyle \sum _{j=1}^{N_i}{U}_{j,t}^{\prime }\left(G_{ij}\cos {\theta }_{ij,t}^{\prime }-B_{ij}\sin {\theta }_{ij,t}^{\prime }\right)=0}$$

(22)

$$\Delta Q_{i,t}^{\prime }=Q_{Gi,t}^{\prime }+Q_{Wi,t}^{\prime }-Q_{L,t}^{\prime }-U_{i,t}^{\prime }{\displaystyle \sum _{j=1}^{N_i}{U}_{j,t}^{\prime }\left(G_{ij}\sin {\theta }_{ij,t}^{\prime }-B_{ij}\cos {\theta }_{ij,t}^{\prime }\right)=0}$$

(23)

$$\begin{array}{cc}{P}_{acc}& ={\displaystyle \sum P_{Gi,t}^{\prime }}+{\displaystyle \sum P_{Wi,t}^{\prime }}+{\displaystyle \sum P_{i,t}^{dis’}}-{\displaystyle \sum P_{i,t}^{ch’}}-{\displaystyle \sum P_{loss,t}}\hfill \\ & =\left({\displaystyle \sum K_{Gi}+{\displaystyle \sum K_L+{\displaystyle \sum K_{Wi}}+{\displaystyle \sum K_{Ei}}}}\right)\Delta f\hfill \\ & =K_{\sum }\Delta f\hfill \end{array}$$

(24)

$$\left\{\begin{array}{l}{P}_{Gi,t}^{\prime }=P_{Gi,t}+{\displaystyle \frac{K_{Gi}}{K_{\Sigma }}}{P}_{acc}\\ P_{Wi,t}^{\prime }=P_{Wi,t}+{\displaystyle \frac{K_{Wi}}{K_{\Sigma }}}{P}_{acc}\\ P_{i,t}^{dis’}=P_{i,t}^{dis}+{\displaystyle \frac{K_{Ei}}{K_{\Sigma }}}{P}_{acc}\\ P_{i,t}^{ch’}=P_{i,t}^{ch}+{\displaystyle \frac{K_{Ei}}{K_{\Sigma }}}{P}_{acc}\\ P_{L,t}^{\prime }=P_{L,t}+{\displaystyle \frac{K_L}{K_{\Sigma }}}{P}_{acc}\end{array}\right.$$

(25)

Steady-state frequency deviation constraints:

$$\Delta f\le \Delta f_{\max }$$

(26)

As the SOC under energy storage peak regulation conditions and frequency regulation conditions operates in different intervals, the SOC constraints in the second stage are as follows:

$$SO{C}_{i,f\min }\le SO{C}_{i,t}\le SO{C}_{i,f\max }$$

(27)

Except for replacing the power flow equation equality constraints in the first stage with power flow equation equality constraints based on dynamic power flow imbalanced power sharing, and replacing the SOC constraints with constraints under frequency regulation conditions, all constraints from the first stage remain applicable in the second stage, with only the corresponding parameters changed to those of the second stage. These will not be listed individually here.

4.3. Model Solution Method

This paper implements single-objective optimization in the first stage, where, after inputting initial parameters, the output of each power source is obtained through iterative solution within the feasible region using conventional power flow and an improved particle swarm optimization algorithm with chaotic mapping and adaptive weighting [29,30], serving as the base output for the second stage. The second stage introduces dynamic power flow to consider RoCoF and frequency deviation constraints, and accounts for the frequency regulation capability of all power sources and loads to jointly share the system’s imbalanced power. The final power source output scheme results from implementing the enhanced particle swarm optimization method.

Based on the standard particle swarm optimization method, a tent chaotic mapping is applied to initialize the population particles. This approach exploits the ergodic and non-repetitive properties of chaos to generate initial solutions closer to the global optimum. These enhanced initial conditions accelerate convergence and strengthen the algorithm’s global optimization capability. The algorithm also implements an adaptive weighting strategy to balance global and local search capabilities. The inertia weight starts with larger values to promote global exploration, then gradually decreases to facilitate local refinement. This dynamic adjustment enables effective exploration in early iterations while ensuring precise convergence in later stages.

The basic concept of the algorithm is to randomly generate several groups of solutions, namely, power allocation schemes for each power source. The objective function resulting from each existing allocation scheme is calculated, and the allocation scheme with the lowest objective function is selected as the current optimal scheme. The remaining power allocation schemes continuously approach the current optimal scheme through transformation calculations and iterations, with the scheme having the minimum objective function after each iteration selected as the current local optimal scheme. Through updating particle positions and velocities in each iteration, after completing the iterations, the allocation scheme with the minimum scheduling objective function is the global optimal power allocation scheme. The objective function is the fitness function; thus, when substituting the allocation scheme to derive the fitness function, the smaller its value, the better its fitness. The particle’s current position is the local optimal value, and its velocity is the iterative convergence speed. Velocity updates are governed by the particle’s current status, historical personal optimum, and collective best solution. Each particle advances to its next location using the velocity computed in the current iteration. As iterations continue to advance, the entire particle swarm gradually completes the search for the optimal allocation scheme in the decision space. The two-stage optimization scheduling framework is shown in Figure 2.

5. Case Studies

This paper employs an improved IEEE 14-bus system for case simulation analysis. A wind farm comprising 60 units of 2 MW Doubly Fed Induction Generator (DFIG) is connected to bus 13, and an energy storage system using lithium iron phosphate batteries is connected to bus 9. The locations of buses containing each power source are shown in Figure 3, and the specific parameters of the energy storage system are shown in

Table 1. Energy storage system parameters.

Parameter Type	Parameter Values
Rated capacity/MWh	300
Rated power/MW	50
Charge/discharge efficiency	0.95/0.95
Initial value of SOC	0.5
Maximum/minimum value of SOC	0.9/0.1

. The system contains 5 thermal power units, with relevant parameters shown in

Table 2. Some operating parameters of thermal power units.

Thermal Power Units	Power/MW		Price/(USD/MW2·h)
	Pmax	Pmin	a	b	c
G1	332	50	0.044	20	0
G2	140	15	0.250	20	0
G3	100	15	0.010	30	0
G4	100	15	0.010	30	0
G5	100	15	0.010	30	0

. The wind turbine operational parameters include: start-up velocity of 3.5 m/s, nominal speed of 6 m/s, and shutdown threshold of 22 m/s; the wind turbine inertia time constant is set to 7.5 s, and the battery energy storage inertia time constant adopts the adaptive adjustment strategy for energy storage virtual inertia; the system rated frequency is 50 Hz, with the rate of change of frequency limit set to 0.5 Hz/s [31,32], the quasi-steady-state frequency deviation limit set to 0.2 Hz, the energy storage frequency regulation dead zone set to ±0.033 Hz, and 10% capacity reserved for frequency regulation.

5.1. Wind Power Forecasting and Scenario Analysis

Actual wind speed data from a day with significant wind speed fluctuations at a certain location is selected as the historical data for wind speed forecasting. The first stage day-ahead wind speed forecasting results are shown in Figure 4. The power forecasting curves for wind power and load are shown in Figure 5.

Considering the variable and stochastic nature of wind generation, this paper selects the period of 10:00–13:00 with the most severe wind speed variations for intra-day real-time scheduling research. This period coincides with the system’s peak load period, providing an ideal test scenario for verifying the suggested technique’s capability in challenging operational environments. Therefore, after the intra-day wind power forecast is processed by the multi-scenario probability method, only the period with significant wind speed fluctuations from 10:00 to 13:00 is selected for analysis. The results are shown in Figure 6. After reduction using the scenario reduction method combining improved K-means clustering with the SBR algorithm, the intra-day forecast is reduced to 4 scenarios: S1, S2, S3, and S4. The probabilities of the 4 scenarios are 0.288, 0.328, 0.150, and 0.234, respectively. It can thus be seen that scenario S2 has a relatively higher probability of occurrence. Figure 7 shows the comparison of two-stage wind power forecasting results, reflecting the wind power forecasting errors between the two stages. To validate the effectiveness of the scenario reduction method, Figure 6 presents the actual wind power curve during this period. The comparison results demonstrate that the four representative scenarios can effectively envelope the variation range of actual wind power with good trend consistency, confirming that the proposed scenario reduction method accurately preserves the key statistical characteristics of the original wind power data.

In Figure 7, the intra-day stage forecast adopts scenario S2 with the highest probability. The data points on the day-ahead forecast curve are expanded from the original 24 points at 1 h intervals to 96 points at 15-min intervals (with the intra-day forecast maintaining the same value for all 4 points within each hour, equal to the day-ahead forecast value for that hour). The 10:00–13:00 time period is extracted for analysis, comparing the wind power forecasting errors between the two stages during 10:00–13:00. Using the first stage power source output as the base output for each intra-day hour, the output is adjusted in real-time every 15 min according to the actual load variations and wind power forecasting errors.

5.2. Energy Storage Peak and Off-Peak Period Division

The effectiveness of energy storage peak regulation largely depends on the accurate identification of peak and valley periods. Traditional fixed period division methods are based on empirical settings and struggle to adapt to the actual variation characteristics of the load. This paper adopts a dynamic division method based on equal capacity with variable power, achieving maximized utilization of energy storage capacity through iterative optimization. The load peak and valley period division is shown in Figure 8.

The final determined dynamic period division results are: valley period from 3:00–8:00, totalling 5 h; peak period from 9:00–14:00, totalling 5 h; with the remaining 14 h as normal periods. This dynamic division method possesses significant advantages compared to traditional fixed division, as the dynamic division method accurately identifies the periods that truly require peak shaving and valley filling, fully utilizing the regulation potential of energy storage.

5.3. Day-Ahead Scheduling Results Analysis

5.3.1. Day-Ahead Scheduling Results Overview

During phase one, following wind generation prediction using historical velocity data, and with known parameters for each thermal power unit, conventional power flow calculations are performed, and the enhanced particle swarm optimization algorithm is used for optimization with the objective of minimizing generation costs. The day-ahead scheduling unit outputs are shown in Figure 9, with these unit outputs serving as the base output plan for the intra-day real-time stage.

Figure 10 shows the net load profile variations with and without storage system engagement in peak management, whilst Figure 11 shows the storage power dynamics and state-of-charge evolution throughout the operational cycle. The energy storage system demonstrated excellent load profile smoothing capabilities in day-ahead scheduling. To optimize load profile smoothing performance, the energy storage system begins charging during the low load period from 3:00 to 8:00, raising the minimum points in the demand profile, and begins discharging during the noon peak period, lowering the peak values in the demand profile, with its state of charge ratio consistently satisfying energy storage operation constraints. Through analysis of Figure 10 and Figure 11, it can be seen that when energy storage does not participate in peak regulation, the net load exhibits obvious peak and valley characteristics, with the highest net load reaching 548.7 MW, the lowest net load at 349.3 MW, and a peak-valley difference of 199.4 MW. After energy storage participates in peak regulation, the system’s net load characteristics are significantly improved, with the highest net load reduced to 510.4 MW, the lowest net load increased to 382.5 MW, and the net load peak-valley difference reduced to 127.9 MW. Particularly during the peak load period from 11:00 to 12:00, continuous energy storage discharge support effectively alleviates the peak regulation pressure on thermal power units.

5.3.2. Typical Period Operation Analysis

Assuming wind power costs are not considered, taking the 11:00–12:00 period as an example, the output of each power source and generation cost values with and without energy storage participation in peak regulation are shown in

Table 3. Day-ahead optimization results with energy storage participation in peak regulation.

Time	Power Source Output (MW)							Cost
	G1	G2	G3	G4	G5	W	E
11	225.7	85.4	67.2	64.6	67.5	71.7	38.0	16,698.3

and

Table 4. Day-ahead optimization results without energy storage participation in peak regulation.

Time	Power Source Output (MW)						Cost
	G1	G2	G3	G4	G5	W
11	298.4	69.4	59.0	60.6	60.9	71.7	18,001.1

, respectively.

By comparing the optimization scheduling results with and without energy storage participation in peak regulation, the economic improvement during the 11:00–12:00 period is analyzed in detail. The G1 unit output is significantly higher than that of other units, serving as the system’s balancing unit, with thermal power unit output values inversely proportional to their comprehensive cost coefficients. With energy storage participation, the G1 unit output decreases from 298.4 MW to 225.7 MW, effectively avoiding its operation in high coal consumption intervals, whilst reducing the average generation cost from 18,001.1 USD to 16,698.3 USD, improving the system’s economic efficiency. Through the day-ahead stage optimization scheduling, the output scheme for each power source is obtained, laying the foundation for the intra-day stage.

5.4. Intra-Day Scheduling Results Analysis

The second stage employs more accurate real-time wind speed forecasting, not only with shorter time scales but also incorporating analysis of wind power uncertainty. Building upon these findings and accounting for frequency deviations caused by power mismatches, the intra-day real-time scheduling achieves further cost reduction through optimizing thermal power unit output allocation and energy storage charging and discharging strategies.

In Figure 7, the two-stage wind power forecasting error is most pronounced during the 11:30–11:45 period. Taking this period as an example, to better illustrate the effectiveness of considering wind and storage frequency regulation based on dynamic power flow in the optimization process, a comparative analysis is conducted between the optimization effects of applying dynamic power flow calculations considering wind, thermal, and load simultaneously sharing imbalanced power, and the optimization effects based on dynamic power flow calculations considering wind, thermal, load, and storage simultaneously sharing imbalanced power.

(1)

Intra-day optimization applying dynamic power flow calculations considering wind, thermal, and load simultaneously sharing imbalanced power. If traditional power flow methods were adopted, all imbalanced power would be borne by the balancing unit G1, potentially causing its power to exceed limits. Applying dynamic power flow calculations allows wind, thermal, and load to jointly share imbalanced power. The corrected power source outputs, frequency deviations, and RoCoF values for each scenario during the 11:30–11:45 period are shown in

Table 5. Power source output, frequency indicators, and cost under different scenarios with wind, thermal, and load jointly sharing imbalanced power.

Scenario	G1	G2	G3	G4	G5	PW	∆f	ROCOF	Cost
S1	183.9	57.9	87.8	85.2	88.1	94.3	0.039	0.163	15,218.9
S2	194.3	59.3	89.2	86.7	89.5	85.2	0.111	0.464	15,811.7
S3	188.1	58.5	88.4	85.8	88.7	90.6	0.068	0.286	15,460.6
S4	186.9	58.3	88.2	85.6	88.5	91.7	0.059	0.249	15,387.8

. The wind power unit outputs in Table 5 correspond to the power source output values for each scenario at 11:30 in Figure 6. Compared to the first stage wind turbine output in Table 3, there is an increase, with the resulting imbalanced power shared jointly by wind, thermal, and load.

As demonstrated above, the balancing unit G1 output in each scenario in Table 5 is reduced compared to Table 3. Units G2–G5 make corresponding adjustments according to their respective cost characteristics, with the G2 unit output relatively reduced due to its higher cost characteristic coefficient, whilst units G3–G5 output relatively increases, and wind power unit output relatively increases. The four scenarios describe four different possibilities, representing wind power uncertainty. In Table 5, the greater the fluctuation of wind power output compared to that in Table 3, the larger the frequency deviation and the greater the initial rate of change of frequency.

(2)

Applying dynamic power flow calculations considering wind, thermal, load, and storage jointly sharing imbalanced power. The optimized and corrected outputs, frequency indicators, and costs for each component when applying dynamic power flow calculations are shown in

Table 6. Power source output, frequency indicators, and cost under different scenarios with wind, thermal, load, and storage jointly sharing imbalanced power.

Scenario	G1	G2	G3	G4	G5	Pdis	∆f	ROCOF	Cost
S1	183.7	57.8	87.6	85.1	87.9	41.9	0.031	0.162	15,192.9
S2	193.8	58.9	88.8	86.3	89.1	44.5	0.090	0.449	15,737.1
S3	187.8	58.2	88.1	85.6	88.5	43.5	0.055	0.281	15,414.8
S4	186.5	58.1	88.0	85.4	88.3	43.3	0.048	0.244	15,347.3

.

Taking scenario S2 with the highest probability in Table 6 as an example, the outputs of thermal power units G1–G5 and energy storage are 193.8 MW, 58.9 MW, 88.8 MW, 86.3 MW, 89.1 MW, and 44.5 MW, respectively. Compared to the outputs of thermal power units G1–G5 and energy storage in Table 3 of 225.7 MW, 85.4 MW, 67.2 MW, 64.6 MW, 67.5 MW, and 38.0 MW, the imbalanced power is jointly shared by wind, thermal and storage power sources, resulting in relatively reduced outputs of thermal power units G3–G5 after first stage correction, whilst wind power unit output increases and energy storage power source output increases, effectively compensating for the power deficit caused by wind power forecasting errors. Meanwhile, the balancing unit output is relatively reduced, avoiding the adverse issues of slack bus selection on conventional power flow calculations, thereby making the power distribution in the network more reasonable. Through this coordinated optimization, the generation cost for this period is reduced to 15,737.1 USD, further achieving economic operation of the system.

The energy storage system demonstrated excellent frequency regulation capability in intra-day scheduling. The RoCoF values for the four scenarios in Table 6 are reduced compared to those in Table 5, indicating that energy storage provides certain inertia support, alleviating the initial frequency change. Additionally, the frequency deviation values for the four scenarios in Table 6 are also reduced to some extent compared to those in Table 5, indicating that energy storage participation in sharing imbalanced power can reduce the steady-state frequency deviation value compared to scheduling schemes without energy storage participation. Due to the greater flexibility of energy storage, which can rapidly charge and discharge to achieve power adjustment, systems without energy storage are likely to trigger protective device actions when encountering large disturbances, resulting in load shedding. This scheme reduces the risk of system load shedding and effectively improves the system’s frequency stability.

Through the above analysis of energy storage peak and frequency regulation effects and system operation economics, the optimization scheduling model proposed in this paper can alleviate the peak regulation pressure on thermal power units whilst improving system frequency quality, significantly enhancing operational economics under the premise of ensuring safe and stable system operation.

6. Conclusions

This paper constructs a two-stage optimization scheduling model for wind power integrated power systems, considering the peak and frequency regulation effects of energy storage. With coordination between day-ahead long-timescale and intraday short-timescale scheduling, the power system realizes secure, high-quality, and economical operation. The main research outcomes and contributions are as follows:

(1)

Wind power forecasting adapted to different optimization stages was established. In the day-ahead optimization stage requiring long time scales, the ARIMA time series method is employed for deterministic wind power modeling, simplifying computational complexity whilst ensuring forecasting accuracy, providing forecast data for day-ahead 24 h scheduling plan formulation. In the intra-day optimization stage requiring short time scales, Monte Carlo scenario generation technology is employed, combined with a method integrating improved K-means clustering and SBR scenario reduction for wind power uncertainty modeling, enabling more accurate capture of short-term wind power fluctuation characteristics. This differentiated forecasting approach fully considers the characteristic that wind power forecasting accuracy improves as the time scale shortens, laying a solid data foundation for two-stage optimization scheduling.

(2)

A day-ahead and intra-day two-stage coordinated optimization scheduling framework was constructed, achieving synergy between energy storage peak and frequency regulation functions. In the day-ahead stage with a 1 h time scale, through energy storage peak regulation periods determined by the dynamic peak-valley identification method, charging during load valleys and discharging during peaks can be achieved, effectively reducing the system net load peak-valley difference and alleviating thermal power unit peak regulation pressure. The intra-day stage is shortened to a 15-min time scale, utilizing the reserved capacity of energy storage to rapidly respond to power imbalances caused by wind power forecasting errors, reducing frequency deviations. Through reasonable design of energy storage SOC working regions and operation mode switching mechanisms, efficient utilization of energy storage resources across different time scales is achieved, avoiding resource idleness during non-peak regulation periods.

(3)

The mechanism established based on dynamic power flow for wind, thermal, load, and storage to coordinately share imbalanced power according to frequency characteristic coefficients breaks through the limitation of traditional power flow calculations, where imbalanced power is borne solely by the balancing unit, resulting in reduced output from units with higher cost characteristic coefficients. By considering the initial rate of change of frequency constraints and frequency deviation constraints, the two-stage optimization scheduling scheme ensures economic optimality whilst meeting system frequency quality requirements. Simulation results demonstrate that the energy storage system can not only provide inertia support to reduce initial frequency changes, but its participation in the imbalanced power sharing mechanism can also reduce steady-state frequency deviation. Meanwhile, the output of each power source is further optimized, reducing generation costs and improving the quality and economics of system operation.

Future research directions include extending the proposed framework to larger power systems, such as IEEE 39-bus and IEEE 118-bus networks, to validate scalability. Investigation of varying energy storage capacities will provide insights into the relationship between storage sizing and optimization performance. Additionally, the application of advanced methods, including deep reinforcement learning and other state-of-the-art optimization techniques, could enhance computational efficiency and solution quality. These extensions will enable a more comprehensive assessment of system scheduling performance at deeper operational levels.