Day-Ahead Planning and Scheduling of Wind/Storage Systems Based on Multi-Scenario Generation and Conditional Value-at-Risk

Abstract

The volatility and uncertainty of wind power output pose significant challenges to the safe and stable operation of power systems. To enhance the economic efficiency and reliability of day-ahead scheduling in wind farms, this paper proposes a day-ahead planning and scheduling method for wind/storage systems based on multi-scenario generation and Conditional Value-at-Risk (CVaR). First, based on the statistical characteristics of historical wind power forecasting errors, a kernel density estimation method is used to fit the error distribution. A Copula-based correlation model is then constructed to generate multi-scenario wind power output sequences that account for spatial correlation, from which representative scenarios are selected via K-means clustering. An objective function is subsequently formulated, incorporating electricity sales revenue, energy storage operation and maintenance cost, initial state-of-charge (SOC) cost, peak–valley arbitrage income, and penalties for schedule deviations. The initial SOC of the storage system is introduced as a decision variable to enable flexible and efficient coordinated scheduling of the wind/storage system. The storage system is implemented using a 1500 kWh/700 kW lithium iron phosphate (LiFePO₄) battery to enhance operational flexibility and reliability. To mitigate severe profit fluctuations under extreme scenarios, the model incorporates a CVaR-based risk constraint, thereby enhancing the reliability of the day-ahead plan. Finally, simulation experiments under various initial SOC levels and confidence levels are conducted to validate the effectiveness of the proposed method in improving economic performance and risk management capability.

1. Introduction

In recent years, with the continued advancement of the “dual carbon” strategy, wind power—being a green and clean renewable energy source—has seen increasing penetration in power systems, becoming a key pillar in the development of a new-type power system [1]. However, as wind power output is highly dependent on meteorological conditions, it exhibits significant randomness and volatility. This results in forecasting errors in wind power output, leading to discrepancies between the reported schedule and actual generation, thereby increasing the operational risk and cost of system dispatching [2].

To address the uncertainty of wind power, early studies predominantly adopted deterministic optimization methods for day-ahead scheduling, in which the forecasted power was directly used as the submitted schedule [3]. Although this approach features a simple structure, it fails to account for the risk of deviation caused by forecasting errors, often resulting in large-scale wind curtailment or substantial penalty costs. To improve the feasibility of scheduling plans, some studies have introduced probabilistic constraint optimization frameworks to control the probability of wind power schedule deviations under a specified confidence level [4]. Reference [5] proposed a hydro–wind–thermal joint dispatching method based on the confidence interval of wind power output. By incorporating the confidence interval to model wind power uncertainty and leveraging the regulation capabilities of hydropower and thermal power, the method facilitates the accommodation of wind power uncertainty while minimizing system costs.

Meanwhile, robust optimization methods have been increasingly applied to wind power dispatch models to enhance the system’s resilience against adverse scenarios [6]. Reference [7] proposed a robust modeling approach for multiple wind farms based on high-dimensional non-parametric Copula functions, which effectively captures the correlation structure of wind power outputs. Reference [8] integrated an improved Wasserstein metric with a combined heat and power (CHP) coordination mechanism to enhance both the conservativeness and economic efficiency of system scheduling. Building on this, Reference [9] proposed a piecewise stochastic robust optimization approach, which divides the uncertainty set of wind power and introduces piecewise linear decision rules to coordinate AGC (Automatic Generation Control) and P2G (Power-to-Gas) resources, thereby improving real-time dispatch efficiency under wind power uncertainty. Although robust optimization performs well in handling extreme risks, its inherently conservative nature may limit the economic performance of dispatch strategies to some extent.

To strike a balance between risk control and profit maximization, multi-scenario optimization methods have been extensively studied and applied in wind power scheduling [10]. By constructing multiple representative wind power output scenarios, this approach incorporates the randomness of forecasting errors into the scheduling model, thereby enhancing the stability and flexibility of the dispatch strategy. Reference [11] addressed the uncertainty of load response by proposing a two-stage multi-scenario demand response strategy based on forecast deviation compensation. The prediction-adjustment dual-stage framework effectively mitigates the impact of forecasting errors on plan execution. Reference [12] considered the coordinated regulation capabilities of industrial loads and energy storage systems and developed a multi-timescale optimization model under high wind power penetration scenarios. This approach improves both system flexibility and the economic efficiency of wind power integration. Reference [13] further incorporated environmental wind conditions and proposed an efficient scenario generation method that integrates scenario screening with incremental risk control. This method enhances the robustness and computational efficiency of wind power scheduling models under chance-constrained frameworks. Reference [14] proposed a rolling intraday scheduling strategy for wind/storage systems based on multi-scenario synergy. By incorporating real-time adjustments of energy storage dispatch in response to wind power fluctuations, their method enhances operational flexibility and short-term economic benefits.

Accurate modeling of wind power forecast errors and their dependencies is essential for generating realistic multi-scenario datasets. Traditional parametric models—such as Beta, Levy-stable, or Gamma-like distributions—often fail to capture the skewness, heavy tails, and multimodal characteristics observed in actual data [15]. To address these limitations, this study adopts kernel density estimation (KDE) for flexible marginal distribution modeling without predefined assumptions. In addition, Copula-based methods are employed to characterize the spatio-temporal dependencies among forecast errors more accurately than traditional linear correlation models, providing a more realistic foundation for scenario generation [16].

While improving the executability of wind power scheduling, energy storage systems have also been integrated into the dispatch framework as flexible regulation resources. Reference [17] proposed a frequency control strategy based on state-of-charge (SOC) regulation to enable coordinated operation between wind and storage systems. Reference [18] developed a fluctuation characteristic analysis and typical day clustering model based on actual wind and solar output data, and proposed an energy storage capacity optimization strategy aimed at smoothing power fluctuations. However, most existing studies regard energy storage primarily as a regulation resource, without fully exploring its arbitrage potential under time-of-use pricing mechanisms. Moreover, current scheduling models often fail to incorporate the initial SOC of storage systems into the joint optimization framework.

On the other hand, although multi-scenario approaches can improve the average profit of dispatch plans, they still face the risk of revenue decline under extreme scenarios. Therefore, Conditional Value-at-Risk (CVaR), as a risk measurement tool, has been introduced into power dispatch optimization models to quantify tail risk [19]. Reference [20] incorporated CVaR into a wind power investment equilibrium model under multiple market trading mechanisms to characterize the risk preferences on the power production side. Reference [21] introduced CVaR into the scheduling model of an integrated energy system with uncertain wind power, enabling the quantification and management of risks caused by wind power volatility. However, existing studies that incorporate CVaR into dispatch models mainly focus on the formulation and constraint of the risk metric itself, with insufficient attention to the participation mechanism of energy storage systems in risk mitigation. In particular, the lack of joint modeling of initial state-of-charge and time-of-use price response strategies limits the ability of wind/storage systems to achieve coordinated optimization between profitability and robustness.

While existing studies have explored wind/storage scheduling through multi-scenario optimization, demand-side coordination, or rolling intraday control, many of them assume a fixed SOC and do not explicitly incorporate risk constraints. In contrast, this study focuses on day-ahead scheduling and introduces a unified framework that integrates non-parametric modeling of forecast errors, Copula-based scenario generation, initial SOC optimization, and CVaR-based risk control. This approach captures both the economic and risk dimensions of decision-making under uncertainty, distinguishing it from prior works that either overlook initial SOC flexibility or treat uncertainty in a simplified manner. The main contributions of this study are as follows:

(1)

Based on historical wind power forecasting errors, kernel density estimation is employed to fit the marginal distributions, while a Gaussian Copula function is used to capture the temporal correlation among forecasting errors across different time periods. This enables the generation of wind power output scenarios with sequential dependence. Representative scenarios are then extracted using K-means clustering to enhance scenario representativeness and modeling accuracy;

(2)

The initial SOC of the energy storage system is introduced as a decision variable in the scheduling framework. By jointly considering electricity sales revenue, operation and maintenance costs, penalties for schedule deviations, and time-of-use pricing, the model enables economic arbitrage through charging and discharging during peak and off-peak price periods, thereby improving overall system profit and operational flexibility;

(3)

A CVaR-based risk control mechanism is incorporated into the multi-scenario optimization framework to effectively constrain profit losses under extreme scenarios, thereby enhancing the robustness and risk controllability of the scheduling strategy under high uncertainty.

The remainder of this paper is organized as follows: Section 2 introduces the modeling of wind power forecasting errors and the generation of representative scenarios. Section 3 develops the day-ahead optimization scheduling model for the wind/storage system, with a detailed description of the objective function, constraints, and the CVaR-based risk control mechanism. Section 4 presents case studies based on multi-scenario wind power outputs, in which simulation experiments are conducted under different optimization strategies, initial SOC settings, and CVaR confidence levels. Section 5 concludes the paper and outlines directions for future research.

2. Multi-Scenario Generation Based on Wind Power Forecasting Errors

The data used in this study are obtained from the SCADA system of a wind farm in Jiangsu Province, covering a typical month (30 days). The scheduling horizon is 24 h with a 10 min sampling interval, totaling 4320 time points. The 10 min interval complies with industry standards, offering sufficient resolution to capture power fluctuations while ensuring computational efficiency. Figure 1 presents a comparison between the forecasted and actual wind power outputs for the entire month. The forecasting errors derived from the forecasted and actual outputs over the first 29 days are used for statistical analysis and multi-scenario modeling. The forecasted power on the 30th day serves as the input for scheduling optimization.

2.1. Statistical Analysis of Wind Power Forecasting Errors

To characterize the forecasting error characteristics of wind power output, statistical analysis is first conducted based on historical data from 144 time intervals per day over a one-month period. Let the forecasted wind power at time interval t of a given day be denoted as $\widehat{p}(t)$, and the actual wind power be denoted as $p(t)$. The wind power forecasting error is then defined as follows:

$$e(t)=p(t)-\widehat{p}(t), t=1,2,\dots ,144$$

(1)

Based on the historical data from the first 29 days, the forecasting errors are grouped by time interval to form an error sample set for each time period, denoted as $e_i(t)$, where $i=1,2,\dots ,29$.

To describe the distribution characteristics of wind power forecasting errors at different time intervals, statistical analysis is further conducted on the error samples of each interval to compute the sample mean $\mu (t)$ and sample standard deviation $\sigma (t)$. The corresponding formulas are as follows:

$$\mu (t)={\displaystyle \frac{1}{29}}{\displaystyle \sum _{i=1}^{29}{e}_i(t)}$$

(2)

$$\sigma (t)=\sqrt{{\displaystyle \frac{1}{29}}{\displaystyle \sum _{i=1}^{29}{(e_i(t)-\mu (t))}^2}}$$

(3)

2.2. Kernel Density Estimation for Error Distribution Fitting

To characterize the distribution of wind power forecasting errors across different time intervals, this study employs the kernel density estimation (KDE) method to fit the error samples derived from the historical data of the first 29 days [22]. Compared with traditional parametric distribution fitting methods, KDE—as a non-parametric technique—can adaptively capture the true underlying distribution of the data without assuming any specific distribution form in advance.

$${\widehat{f}}_t(e)={\displaystyle \frac{1}{d\cdot h}}{\displaystyle \sum _{i=1}^{d}K(\frac{e-e_i(t)}{h})}$$

(4)

where d is the number of historical sample days, and $K(\cdot )$ is the kernel function. In this study, the Gaussian function is selected, which is defined as follows:

$$K(u)={\displaystyle \frac{1}{\sqrt{2\pi }}}\exp (-{\displaystyle \frac{u^2}{2}})$$

(5)

In Equation (4), h denotes the bandwidth parameter, which directly affects the smoothness of the kernel density estimation. To determine an appropriate bandwidth, this study adopts Silverman’s rule of thumb [23], which defines the bandwidth as follows:

$$h=1.06\times \sigma \times d^{-1/5}$$

(6)

where $\sigma$ is the standard deviation of the forecasting error sample set, and d is the number of historical samples used for error distribution estimation. If the bandwidth is set too small, the KDE curve becomes overly steep and sensitive to noise; conversely, if the bandwidth is too large, the estimated distribution may become overly smooth, potentially resulting in the loss of important detail. Therefore, Silverman’s rule of thumb is adopted in this study to determine the bandwidth parameter.

In addition, to further validate the fitting performance of the kernel density estimation method, two typical parametric approaches—namely, the normal distribution and the t-distribution—are selected for comparative analysis. The normal distribution fitting method derives the probability density function using the sample mean and standard deviation of the historical forecasting errors, estimated via the maximum likelihood method. The t-distribution fitting method accounts for the heavy-tailed characteristics of the data, making it more suitable for modeling extreme forecasting errors [24]. Figure 2 presents a comparison between the histogram of forecasting errors from the first 29 days and the fitted curves obtained from the three methods.

As shown in Figure 2, the normal distribution fitting curve performs well near zero forecasting errors, but its accuracy deteriorates in regions with larger errors. Although the t-distribution curve improves the fitting performance to some extent compared to the normal distribution, its unimodal nature prevents it from accurately capturing multimodal patterns or local density fluctuations present in the actual data. In contrast, kernel density estimation, as a non-parametric method, is not constrained by any specific distributional assumptions and can flexibly adapt the fitting curve according to the actual data characteristics. It effectively captures the skewness, multimodality, and heavy-tailed features of forecasting error data. The KDE fitting curve exhibits a significantly better match with the error histogram than both the normal and t-distribution fitting methods.

2.3. Error Correlation Modeling Based on Copula Functions

In practical wind power scenarios, forecasting errors not only exhibit random fluctuations within individual time intervals but also demonstrate significant temporal correlation. Traditional single-period probabilistic modeling methods fail to capture the interdependence among forecasting errors across different time intervals, which in turn compromises the accuracy of subsequent scenario generation and system scheduling. To address this limitation, this study introduces the Copula function to construct a joint probabilistic model that captures the dependency structure among wind power forecasting errors across multiple time intervals [25].

A Copula function describes the dependency structure among multivariate random variables. It allows for the separation of marginal distributions from the joint distribution, thus avoiding rigid assumptions about the form of the data distribution and offering broad applicability. According to Sklar’s theorem, for any set of random variables, the joint distribution can be expressed as a combination of the marginal distributions and a Copula function:

$$F(x_1,x_2,\dots ,x_n)=C\left[F_1(x_1),F_2(x_2),\dots ,F_n(x_n)\right]$$

(7)

where $F(x_1,x_2,\dots ,x_n)$ is the joint distribution function of the n-dimensional random variables, $F_i(x_i)$ is the marginal distribution function of the i-th variable, and $C(\cdot )$ denotes the Copula function.

Among various Copula functions, the Gaussian Copula is widely used to characterize the dependence structure among complex random variables due to its advantages of simple parameter estimation, clear functional form, and strong adaptability [26]. Therefore, the Gaussian Copula is selected to model the temporal correlation of wind power forecasting errors. Its mathematical formulation is given as follows:

$$C(u_1,u_2,\dots ,u_3\left|\Sigma \right.)={\mathrm{\Phi }}_{\Sigma }({\mathrm{\Phi }}^{-1}(u_1),{\mathrm{\Phi }}^{-1}(u_2),\dots ,{\mathrm{\Phi }}^{-1}(u_n))$$

(8)

where ${\mathrm{\Phi }}_{\Sigma }$ is the joint cumulative distribution function (CDF) of an n-dimensional standard normal distribution with covariance matrix $\Sigma$; ${\mathrm{\Phi }}^{-1}(\cdot )$ is the inverse of the standard normal CDF; and $u_i$ denotes the marginal cumulative probability of the forecasting error at time interval i.

To accurately capture the temporal correlation characteristics of wind power forecasting errors, a joint probability distribution model is constructed using the Gaussian Copula function. The specific implementation steps are as follows:

• Marginal Distribution Fitting

First, kernel density estimation is applied to the historical forecasting error data from the first 29 days for each time interval, yielding the empirical distribution function of forecasting errors at each time step. Based on this, the original forecasting error data are transformed into a unified [0,1] probability space using their respective empirical cumulative distribution functions, thereby forming a set of standardized random variables with consistent marginal distributions. This transformation eliminates the influence of scale differences among error data on the modeling of correlation structures;

2.

Gaussian Copula Parameter Estimation

Next, to determine the parameters reflecting the temporal correlation of errors, the standardized data in the [0,1] space are further transformed into the standard normal space using the inverse of the standard normal distribution function. Then, the covariance matrix parameter of the Gaussian Copula function is estimated using the maximum likelihood estimation method. This allows the model to fully capture the intrinsic temporal dependence among forecasting errors across different time intervals;

3.

Forecasting Error Scenario Generation

Finally, based on the parameters of the Gaussian Copula model, stochastic scenarios of wind power forecasting errors are generated. The generation process consists of two stages. First, using the established Gaussian Copula parameters, random samples are drawn from the joint distribution in the standard normal space. Then, the sampled data are mapped back to the original error space by applying the inverse of the fitted empirical cumulative distribution function (CDF) for each time interval. In this way, a set of candidate wind power output scenarios is constructed.

2.4. Multi-Scenario Generation of Wind Power Output

Let the forecasted wind power output for each time interval on Day 30 be denoted as ${\widehat{p}}_{30}(t), t=1,2,\dots ,144$, with a data sampling interval of 10 min. To generate candidate scenarios, S samples are first randomly drawn from the joint distribution constructed using the wind power data from the first 29 days, resulting in a standardized forecasting error matrix. The mathematical expression is given by the following:

$$U_s=\left[\begin{array}{cccc}{u}^1(1)& u^1(2)& \dots & u^1(t)\\ u^2(1)& u^2(2)& \dots & u^2(t)\\ ⋮& ⋮& \ddots & ⋮\\ u^s(1)& u^s(2)& \dots & u^s(t)\end{array}\right]$$

(9)

where $u^s(t)\in [0,1]$ represents the standardized forecasting error at time interval t in the s-th candidate scenario. By applying the inverse of the empirical cumulative distribution function ${\widehat{F}}_t(e)$ for each time interval, the standardized sampled values can be transformed into actual forecasting errors:

$$e^s(t)={\widehat{F}}_t^{-1}(u^s(t))$$

(10)

By adding the inverse-transformed actual forecasting errors to the forecasted wind power values of each time interval on Day 30, the wind power output scenarios for Day 30 can be constructed as follows:

$$p_{30}^s(t)={\widehat{p}}_{30}(t)+e^s(t)$$

(11)

Directly using all candidate scenarios may lead to severe data redundancy, making it difficult for the scheduling model to accurately capture the characteristics of wind power uncertainty, and substantially increasing the computational complexity of the optimization problem. Therefore, k-means clustering is employed to reduce the dimensionality of the S candidate scenarios by grouping them into K clusters, from which representative wind power output scenarios are extracted [27]. This reduction not only preserves the statistical structure of the original scenario set but also significantly improves the computational efficiency of the subsequent optimization process. Let $n_k$ denote the number of candidate scenarios in the k-th cluster. The weight assigned to the representative scenario of this cluster is defined as follows:

$$w_k={\displaystyle \frac{n_k}{k}}$$

(12)

Figure 3 illustrates the overall process of the proposed approach, from data acquisition and error modeling to scenario generation, reduction, and scheduling optimization. It outlines the key steps involved in constructing the wind/storage day-ahead scheduling model.

3. Day-Ahead Planning and Scheduling Model for Wind/Storage Systems Based on Multi-Scenario Generation and CVaR

3.1. Objective Function

For a single wind power output scenario, the objective function comprehensively considers the electricity sales revenue from the reported wind power schedule, the operation and maintenance cost of the energy storage system, the cost of the initial SOC, the penalty cost for scheduling deviations, and the arbitrage revenue from peak–valley electricity prices. This objective function is used to guide the wind/storage scheduling system in balancing revenue and cost, thereby achieving optimal decision-making for wind power reporting, energy storage charge/discharge strategies, and management of scheduling deviations. The mathematical expression of the objective function for the k-th scenario is given as follows:

$$C_k={\displaystyle \sum _{t=1}^T\left[C^{sell}(t)-C_k^{cf}(t)-C^b(t)-C^{soc}(t)+C^{fg}(t)\right]}$$

(13)

$$\left\{\begin{array}{l}{C}^{sell}(t)=p_{sell}(t)\cdot p_j(t)\cdot \Delta t\\ C_k^{cf}(t)={\lambda }_k(t)\cdot {\Delta }_k(t)\\ C^b(t)=C_b\cdot (p^c(t)+p^d(t))\cdot \Delta t\\ C^{soc}(t)=C_{soc}\cdot E_b\cdot SO{C}_0\\ C^{fg}(t)=\gamma \cdot ({\alpha }^f(t)\cdot p^d(t)\cdot p_{sell}(t)-{\alpha }^g(t)\cdot p^c(t)\cdot p_{buy}(t))\end{array}\right.$$

(14)

where $C_k$ is the total revenue of the wind farm under the k-th scenario. $C^{sell}(t)$ denotes the electricity sales revenue. $C_k^{cf}(t)$ is the penalty cost for scheduling deviation. $C^b(t)$ is the operation and maintenance cost of the energy storage system. $C^{soc}(t)$ is the cost associated with the initial SOC. $C^{fg}(t)$ represents the arbitrage revenue from peak–valley electricity price differences. $p_{sell}(t)$ is the electricity selling price. $p_j(t)$ is the scheduled wind power, and $\Delta t$ is the time interval. ${\Delta }_k(t)$ is the scheduled deviation energy under scenario k. ${\lambda }_k(t)$ is the unit penalty cost for scheduling deviation, and $C_b$ is the cost coefficient of energy storage. $p^c(t)$ and $p^d(t)$ are the charging and discharging power of the energy storage system. $C_{soc}$ is the cost coefficient of the initial SOC. $E_b$ is the rated capacity of the energy storage system, and $SO{C}_0$ is the initial SOC. $\gamma$ is the arbitrage incentive coefficient. ${\alpha }^f(t)$ and ${\alpha }^g(t)$ are binary variables indicating peak and valley price periods, respectively (0–1). $p_{buy}(t)$ is the electricity purchase price from the grid.

Therefore, based on the K clustered wind power output scenarios, the objective function for the wind farm’s multi-scenario expected revenue is defined as follows:

$$\max C={\displaystyle \sum _{k=1}^K{w}_k}\cdot C_k$$

(15)

where $w_k$ is the weight of the k-th wind power output scenario, $C_k$ is the total revenue of the wind farm under this scenario, and K is the total number of clustered scenarios.

3.2. Conditional Value-at-Risk

Although the weighted summation of the K clustered scenarios can enhance the economic returns of the wind farm to a certain extent, it remains difficult to effectively mitigate significant profit drops caused by extreme scenarios, particularly under conditions of large forecasting errors or distributional deviations. To further enhance the robustness of the submitted schedule across different scenarios, a Conditional Value-at-Risk (CVaR) constraint is incorporated into the profit maximization objective function.

CVaR is a risk management tool used to measure the expected loss beyond a specified threshold under the worst-case scenarios. Compared with traditional risk metrics, CVaR captures tail risks more effectively and better reflects potential losses under extreme scenarios, thus offering enhanced robustness when dealing with wind power forecasting errors and other uncertainties. A confidence level $\alpha$ is specified, and two auxiliary variables, $\zeta$ and ${\tau }_k$, are introduced into the model. CVaR is typically used to assess losses under extreme scenarios, whereas the objective function in this study focuses on economic gains in wind power systems. Therefore, to apply CVaR in the context of revenue optimization, the profit is negated and transformed into a loss function. Through this transformation, maximizing CVaR becomes equivalent to minimizing losses in worst-case scenarios, thereby enhancing the system’s performance under extreme conditions. The specific expression is given as follows:

$$CVa{R}_{\alpha }(C)=\xi -{\displaystyle \frac{1}{1-\alpha }}{\displaystyle \sum _{k=1}^K{w}_k{\tau }_k}$$

(16)

where $\zeta$ is the baseline value representing the quantile. ${\tau }_k$ denotes the deviation between the revenue in scenario k and the quantile baseline. When the revenue in scenario k falls below $\zeta$, ${\tau }_k$ increases accordingly.

3.3. Constraints

(1)

Energy Storage Constraints

$$SOC(t+1)=SOC(t)+(p^c(t)\times {\eta }_c-{\displaystyle \frac{p^d(t)}{{\eta }_d}})\times \Delta t$$

(17)

$$0\le p^c(t)\le P_{rated}$$

(18)

$$0\le p^d(t)\le P_{rated}$$

(19)

$$SO{C}_{\min }\le SOC(t)\le SO{C}_{\max }$$

(20)

(2)

CVaR Risk Constraints

To incorporate risk control into the objective function and prevent revenue degradation under extreme scenarios, a CVaR constraint is introduced.

$${\tau }_k\ge \xi -C_k$$

(21)

$${\tau }_k\ge 0$$

(22)

where $\zeta$ is the baseline value representing the revenue quantile threshold. ${\tau }_k$ is the deviation between the revenue in scenario k and the quantile baseline. $C_k$ is the total revenue of the wind farm in scenario k.

(3)

Linear Constraints

In each scenario k, the actual wind power output at time t, denoted as $p_{30}^k(t)$, is an uncertain quantity determined by the scenario generation process. Ideally, the scheduled wind power $p_j(t)$ combined with the net energy storage dispatch $p^c-p^d$ should match this actual output, leading to a power balance constraint of the following form:

$$p_j(t)+p^c-p^d=p_{30}^k(t)$$

(23)

However, since $p_{30}^k(t)$ is not a decision variable but a scenario-dependent parameter, the above equality cannot be directly used in the MILP formulation. To ensure tractability and retain linearity, a non-negative deviation variable $z_k(t)$ is introduced to quantify the scheduling shortfall under each scenario. The equality is then relaxed to a linear inequality:

$$p_j(t)+p^c-p^d-z_k(t)\le p_{30}^k(t)$$

(24)

This constraint ensures that the total scheduled output and storage dispatch do not exceed the available wind power in each scenario. The deviation variable $z_k(t)$ is penalized in the objective function, encouraging solutions that approximate the ideal balance while maintaining tractability in the MILP framework.

3.4. Solution Method

In this study, a mixed-integer linear programming (MILP) approach is adopted to solve the wind/storage power planning and scheduling model under multi-scenario wind power uncertainty. The YALMIP 20230622 toolbox in MATLAB R2021b is used as the modeling interface, and CPLEX 12.10 is selected as the solver. Although CPLEX provides robust and efficient solutions for MILP models, its computational efficiency may decline for very large-scale instances with extensive scenario sets. Therefore, the proposed method is most suitable for day-ahead scheduling problems with a moderate number of representative scenarios. The decision variables in the optimization model include the following: (1) wind power scheduling plan $p_j(t)$; (2) energy storage charging and discharging power $p^c(t)$ and $p^d(t)$; and (3) initial state of charge of the energy storage system SOC₀.

4. Discussion

4.1. Case Parameters

This study selects a wind/storage system from an industrial park in Jiangsu Province, China, as the research object. The scheduling horizon is set to 24 h with a time interval of 10 min, resulting in 144 sampling points per day. The data are obtained from the actual operation records of the wind farm in the park, covering a typical month (30 days) of wind power forecasts and corresponding actual output data. The forecasting error data from the first 29 days are used to construct the probabilistic model and scenario set of wind power output, while the forecasted data for the 30th day serve as the input for optimization and scheduling.

During the generation of wind power output scenarios for Day 30, a combination of the Copula method and KDE is employed to generate 100 candidate output scenarios via random sampling. Subsequently, the k-means clustering method is applied to extract 10 representative scenarios along with their corresponding probability weights, in order to reduce computational complexity while preserving the representativeness of the scenario set. Figure 4 illustrates the curve distribution of the 10 representative wind power output scenarios selected via k-means clustering.

The confidence level for the CVaR constraint is set to 0.90. The parameters of the LiFePO₄ battery energy storage system are listed in

Table 1. Energy storage system parameters.

Parameter	Value	Parameter	Value
Rated Capacity	1500 kWh	Minimum SOC	0.1
Rated Power	700 kW	Maximum SOC	0.9
Charging Efficiency	0.95	Terminal SOC	0.5
Discharging Efficiency	0.95	Maintenance Cost	0.02 CNY/kWh

. In particular, the initial SOC of the storage system is treated as a decision variable for optimization, while the terminal SOC is fixed at 0.5 to ensure that the system retains a certain level of dispatchability at the end of the scheduling horizon. The cost coefficient for the initial SOC is set to 0.45 CNY/kWh. Moreover, to account for the impact of electricity price fluctuations on the charge/discharge strategy of the storage system, a time-of-use (TOU) pricing mechanism is introduced, as shown in

Table 2. Time-of-use electricity pricing.

Period Type	Time Range	Purchase Price (CNY/kWh)	Selling Price (CNY/kWh)
Peak	10:00–15:00 18:00–21:00	0.83	0.65
Flat	7:00–10:00 15:00–18:00 21:00–23:00	0.49	0.38
Valley	23:00–7:00	0.17	0.13

. The electricity purchase price refers to the cost of buying electricity from the grid, while the selling price corresponds to the feed-in tariff for wind power. On this basis, to further incentivize arbitrage scheduling by leveraging the peak–valley electricity price spread, a peak–valley arbitrage incentive coefficient $\gamma$ is introduced, with a value of 0.5.

To ensure the practicality and reproducibility of the proposed method, the wind power system is modeled using a representative commercial wind turbine. Specifically, a 4 MW horizontal-axis wind turbine is considered, with a cut-in wind speed of 3 m/s, a rated wind speed of 12 m/s, and a cut-out wind speed of 25 m/s.

To further reduce the economic losses caused by wind power scheduling deviations across scenarios, the objective function incorporates a deviation penalty cost, which is jointly determined by the electricity purchase price and a dynamic penalty coefficient. The penalty cost for each time period is listed in

Table 3. Penalty cost for scheduling deviations in different time periods.

Period Type	Purchase Price (CNY/kWh)	Dynamic Penalty Coefficient	Unit Penalty Cost for Scheduling Deviation (CNY/kWh)
Peak	0.83	2.0	1.66
Flat	0.49	1.5	0.735
Valley	0.17	1.5	0.255

. Specifically, a higher penalty coefficient (2.0) is assigned to peak periods to reflect their elevated economic impact. For flat and valley periods, a unified coefficient of 1.5 is adopted to balance their differing electricity prices with the system’s overall sensitivity to deviations during non-peak hours.

4.2. Case Simulation and Results Analysis

Figure 5 presents a comparison between the original forecasted wind power and the optimized scheduled power. Figure 6 shows the SOC curve of the energy storage system along with the corresponding time-of-use electricity price curve at each time interval. Since the SOC curve includes the initial SOC, a total of 145 sampling points are displayed.

Table 4. Comparison of day-ahead economic benefits and risk indicators of the wind/storage system.

Indicator	Value
Initial SOC	0.1
Expected Revenue (CNY)	16,233.78
Minimum Scenario Revenue (CNY)	15,532.48
Maximum Scenario Revenue (CNY)	16,263.00
CVaR (CNY)	15,903.29

further summarizes the comparison of day-ahead economic benefits and risk indicators of the wind/storage system under the optimized scheduling results.

According to the results in Figure 5 and Figure 6 and Table 4, the optimized initial SOC is 0.1. This decision is mainly influenced by the time-of-use electricity price structure and the cost associated with energy storage. During valley periods, the electricity purchase price is only 0.17 CNY/kWh, which is significantly lower than the unit cost of the initial SOC (0.45 CNY/kWh). If the initial SOC is set too high, the storage system will have little room for charging during low-price hours, making it impossible to take advantage of cheaper electricity and missing the opportunity to reduce operational costs. At the same time, the feed-in tariff during valley periods is also low, making discharging less profitable and reducing the cost-effectiveness of selling stored energy. In contrast, during peak periods (10:00–15:00 and 18:00–21:00), the high electricity price encourages the wind/storage system to discharge the storage unit in coordination with wind power to increase the scheduled output and achieve peak-period arbitrage gains. The final optimization results show that the expected revenue across multiple scenarios is CNY 16,233.78, with a CVaR of CNY 15,903.29. The minimum and maximum scenario revenues are CNY 15,532.48 and CNY 16,263.00, respectively.

To validate the effectiveness of the proposed day-ahead planning and scheduling model for wind/storage systems based on multi-scenario generation and CVaR, three comparison schemes are designed using the same set of wind power output scenarios. The details of the three schemes are shown in

Table 5. Comparison of different scheduling schemes.

Scheme	CVaR Considered	Peak–Valley Arbitrage Considered
1	Yes	Yes
2	No	Yes
3	Yes	No

.

Table 6. Comparison of day-ahead economic benefits and risk indicators under three scheduling schemes.

Indicator	Scheme 1	Scheme 2	Scheme 3
Initial SOC	0.1	0.1	0.1
Expected Revenue (CNY)	16,233.78	16,903.17	14,894.26
Minimum Scenario Revenue (CNY)	15,532.48	14,445.90	14,192.96
Maximum Scenario Revenue (CNY)	16,263.00	17,721.76	14,923.48
CVaR (CNY)	15,903.29	-	14,563.77

presents the comparison of day-ahead economic benefits and risk indicators of the wind/storage system under the three scheduling schemes.

As shown in Table 6, Scheme 2 achieves the highest expected revenue of CNY 16,903.17, which is CNY 669.39 higher than Scheme 1 (CNY 16,233.78). Its maximum scenario revenue also reaches CNY 17,721.76, exceeding that of Scheme 1 (CNY 16,263.00) by CNY 1437.27. However, Scheme 2 does not incorporate CVaR and thus fails to effectively manage risk under extreme scenarios. As a result, its minimum scenario revenue drops to CNY 14,445.90, which is CNY 1086.58 lower than that of Scheme 1 (CNY 15,532.48). In Scheme 3, which excludes peak–valley arbitrage based on Scheme 1, the expected revenue decreases to CNY 14,894.26—lower than both Scheme 1 and Scheme 2. Its minimum scenario revenue is also the lowest, at CNY 14,192.96. Therefore, incorporating both CVaR and peak–valley arbitrage is crucial for mitigating risk and enhancing the economic performance of the wind/storage system.

4.3. Analysis of Results Under Different Initial State-of-Charge Levels

To verify the effectiveness of optimizing the initial SOC in improving the economic performance of the wind farm, this study analyzes comparison cases with initial SOC levels set at 0.5, 0.9, and the optimized value of 0.1. Figure 7 and Figure 8 show the SOC variation curves when the initial SOC is set to 0.5 and 0.9, respectively.

Table 7. Comparison of expected revenues under different initial state-of-charge levels.

Initial SOC	0.1	0.5	0.9
Expected Revenue (CNY)	16,233.78	16,044.70	15,923.12

compares the expected revenues of the wind farm under different initial SOC levels.

As shown in Table 7, when the initial SOC is set to 0.1, the wind farm achieves the highest expected revenue of CNY 16,233.78. In contrast, the expected revenues at initial SOC levels of 0.5 and 0.9 are CNY 16,044.70 and CNY 15,923.12, respectively, both lower than that of the 0.1 case. Further insights from Figure 7 and Figure 8 show that during valley periods, a lower initial SOC facilitates the use of surplus wind power for energy storage charging, thereby reducing the cost associated with the initial SOC. During peak periods, the storage system discharges to take advantage of higher electricity prices, thus improving overall revenue through arbitrage. Therefore, optimizing the initial state of charge is of great significance for enhancing the overall economic performance of the wind farm.

4.4. Analysis of Results Under Different Confidence Levels

Different confidence levels represent the wind/storage system’s tolerance for extreme scenarios. A higher confidence level requires the system to maintain relatively high revenue even in worst-case conditions, while a lower confidence level allows for higher expected returns at the cost of increased risk exposure. By comparing scheduling results under different confidence levels, the model’s performance in scenario-based risk management can be comprehensively evaluated.

As shown in

Table 8. Comparison of day-ahead economic benefits and risk indicators of the wind/storage system under different confidence levels.

Confidence Level	0.8	0.85	0.9	0.95
Initial SOC	0.1	0.1	0.1	0.1
Expected Revenue (CNY)	16,447.05	16,453.56	16,233.78	15,772.14
Minimum Scenario Revenue (CNY)	15,163.38	15,136.98	15,532.48	15,736.71
Maximum Scenario Revenue (CNY)	16,500.54	16,508.42	16,263.00	15,773.62
CVaR (CNY)	16,076.46	16,166.63	15,903.29	15,676.59

, when the confidence level is relatively low (0.8 and 0.85), the expected revenues are higher—CNY 16,453.56 and CNY 16,447.05, respectively. However, the corresponding minimum scenario revenues are relatively low, at CNY 15,136.98 and CNY 15,163.38. When the confidence level increases to 0.9 and 0.95, the expected revenues decrease to CNY 16,233.78 and CNY 15,772.14, respectively, but the minimum scenario revenues rise significantly to CNY 15,532.48 and CNY 15,736.71. Meanwhile, the CVaR values also decline as the confidence level increases, dropping from CNY 16,166.63 at 0.8 to CNY 15,676.59 at 0.95. These results indicate that higher confidence levels lead to more conservative scheduling strategies, which can effectively mitigate significant revenue drops under extreme scenarios. Therefore, selecting an appropriate confidence level is essential for achieving a balance between risk control and revenue maximization.

Based on the above analysis, the confidence level should be determined according to the operator’s risk preference and scheduling objectives. A higher confidence level, such as 0.9 or 0.95, is appropriate when the focus is on reducing the impact of extreme scenarios and ensuring more stable returns. Conversely, when improving expected revenue is prioritized and the system can tolerate certain deviations, a lower level, such as 0.8 or 0.85, may be considered. This allows for flexible adjustment of the scheduling strategy in line with different operational requirements.

4.5. Sensitivity Analysis on the Number of Clusters

To investigate the impact of the number of representative scenarios on scheduling performance, a sensitivity analysis was conducted by varying the cluster count K used in the scenario reduction step. Specifically, the optimization model was solved for K = 5, K = 10, and K = 15, and the corresponding results are presented in

Table 9. Sensitivity of scheduling performance to the number of clusters in scenario reduction.

Number of Clusters	5	10	15
Initial SOC	0.1	0.1	0.1
Expected Revenue (CNY)	16,180.98	16,233.78	16,379.67
Minimum Scenario Revenue (CNY)	15,476.75	15,532.48	15,269.88
Maximum Scenario Revenue (CNY)	16,233.98	16,263.00	16,439.82
CVaR (CNY)	15,636.42	15,903.29	15,770.80

.

From Table 9, it can be observed that increasing the number of clusters from 5 to 15 leads to a general improvement in expected revenue, with the highest value (CNY 16,379.67) achieved when K = 15. However, this improvement is accompanied by a slight reduction in CVaR, decreasing from CNY 15,903.29 at K = 10 to CNY 15,770.80 at K = 15, as well as a decline in the minimum scenario revenue. This indicates that while a larger number of clusters provides a more comprehensive representation of wind power uncertainty, it may also introduce more extreme scenarios that negatively affect risk indicators.

4.6. Grid-Friendliness Evaluation Based on Power Ramp Rate

To further evaluate the grid-interactive characteristics of the proposed wind/storage dispatch model, this section investigates the smoothness of the system’s net power output by analyzing its power ramp rate. The power ramp rate, defined as the change in net power output to the grid over consecutive time intervals, serves as a critical indicator of grid-friendliness. Excessive fluctuations in output power can lead to frequency instability, increase the burden on grid regulation resources, and hinder the reliable integration of renewable energy. In this study, the net power output is defined as the sum of the scheduled wind power and the difference between discharging and charging power of the battery system. The power ramp rate is calculated as the variation in net output between two consecutive 10 min intervals. The variation of net power ramp rate over 10 min intervals is shown in Figure 9.

As shown in Figure 9, the net power ramp rate remains relatively stable over the scheduling horizon, with limited short-term fluctuations. This reflects that the optimized dispatch strategy avoids large power swings between consecutive intervals. Combined with the scenario-based modeling and the inclusion of time-of-use pricing and CVaR constraints, the dispatch results demonstrate a smooth power output trajectory that is compatible with grid operational requirements.

To quantitatively assess the output smoothness of the wind/storage system,

Table 10. Ramp rate metrics of net power over 10 min intervals.

Parameter	Value (kW/10 min)
Maximum Ramp Rate	1083.26
Average Ramp Rate	179.36

presents the maximum and average power ramp rates calculated over consecutive 10 min intervals.

As shown in Table 10, the maximum and average ramp rates are 1083.26 kW/10 min and 179.36 kW/10 min, corresponding to 27% and 4.5% of the turbine’s rated power, respectively. This falls within the acceptable range based on domestic and international grid standards, confirming the proposed strategy’s capability to deliver grid-friendly power fluctuations.

5. Conclusions

This paper proposes a day-ahead planning and scheduling model for wind/storage systems that integrates multi-scenario generation with CVaR constraints. Wind power output scenarios are generated using non-parametric modeling combined with Copula functions. The model also incorporates initial SOC optimization and a peak–valley arbitrage mechanism, effectively addressing the scheduling challenges posed by wind power uncertainty.

Simulation results show that the proposed model significantly enhances both economic performance and risk control. Compared with the benchmark without CVaR and arbitrage design, the expected revenue increases by CNY 340.67, the CVaR improves by CNY 420.74, and the minimum scenario revenue rises by CNY 563.95. When the initial SOC is optimized to 0.1, the expected revenue is CNY 189.08 and CNY 310.66 higher than those with SOC values of 0.5 and 0.9, respectively. Under different confidence levels, raising the level from 0.8 to 0.95 leads to a CVaR increase of CNY 739.96 and a minimum revenue gain of CNY 1599.73, at the cost of CNY 681.42 in expected revenue, reflecting a clear trade-off between risk and return. In addition, increasing the number of clusters from 5 to 10 improves both CVaR and expected revenue, while further increasing to 15 results in only a slight revenue gain (CNY 145.89) but a decline in CVaR by CNY 132.49, indicating diminishing returns and potential overfitting.

Although the proposed model improves the economic efficiency and robustness of the wind/storage system, it primarily focuses on day-ahead scheduling and does not fully capture the dynamic response capabilities required during real-time operation. This limitation may affect its adaptability under highly volatile conditions. Future work will aim to incorporate real-time dispatch mechanisms to enhance the responsiveness and practical applicability of the proposed approach.

6. Patent

Chinese invention patents result from the work as follows:

(1)

A Double Energy Storage Battery Management Method for Wind Power Grid Integration Plan Deviation Compensation, Patent No.: ZL 202410373465.5, authorized on 24 December 2024;

(2)

A Wind-Storage Power Planning Method with Coordinated Dual Energy Storage Scheduling, Patent No.: ZL 202311324413.0, authorized on 2 August 2024;

(3)

A Microgrid Economic Scheduling Method Based on Dynamic Pricing and Battery Degradation, Application No.: CN 202411929009.0, Application Date: 25 December 2024.