Optimal Unit Scheduling Considering Multi-Scenario Source–Load Uncertainty and Frequency Security

Abstract

In response to the significant challenges posed by the volatility and low-inertia characteristics of new energy outputs to the safe operation of power systems, a day-ahead scheduling model for generating units that takes into account source–load uncertainty and frequency security is proposed. To address source–load uncertainty, kernel density estimation and copula theory are employed to model the correlation between wind and solar energy and generate a set of typical daily scenarios. Considering the low-inertia characteristic, frequency security constraints are incorporated into the scheduling model, and an optimization model for generating units that takes into account multi-scenario uncertainty is constructed. By solving the expected value of the objective function, economic and safe scheduling of the system under uncertain environments is achieved. An experimental analysis and case studies verify the advantages and feasibility of the proposed model through a comparison of scheduling costs and frequency security.

1. Introduction

In recent years, wind and solar energy, as representative forms of renewable energy, have witnessed extensive application and development due to their widespread distribution, abundant resources, and renewable characteristics. However, the large-scale integration of renewable energy for the transmission of clean electricity concurrently poses significant challenges to grid security [1,2]. In particular, the frequency security issues brought about by grid connections have become increasingly prominent. The intermittency and volatility of high proportions of renewable energy, as well as the low-inertia characteristics of power electronic interfaces, have significantly reduced the grid’s ability to cope with disturbances [3,4]. In recent years, incidents of grid safety accidents caused by new energy sources have occurred with increasing frequency. For instance, the bipolar blocking incident of the Jinping–Suzhou UHVDC transmission line in the East China Power Grid in 2015 [5]; the large-scale power outage in London, UK on August 9, 2019; and the extensive blackout event in Texas, USA in 2021 serve as notable examples. These incidents underscore the inadequacy of high-proportion renewable energy power systems in frequency regulation capabilities.

To address the aforementioned issues, the current research primarily focuses on integrating frequency security constraints into day-ahead scheduling strategies, thereby achieving the convergence of frequency security and optimal scheduling. Sun et al. established a grid-forming inverter model based on flywheel energy storage and proposed a grid frequency regulation strategy suitable for scenarios with high proportions of new energy access [6]. Musau et al. proposed methods to improve frequency stability by analyzing the frequency responses of new energy power plants after disturbances [7]. Park et al. examined the variability of renewable energy sources within the Korean power system, assessing the requisite capacity of adjustable-speed, pumped-storage generators to maintain frequency stability [8]. However, the majority of these studies predominantly focused on the impact of wind power integration on grid frequency, while overlooking the challenges posed by the intermittency and volatility of renewable energy generation to grid dispatch operations.

To address the challenges posed by the variability of renewable energy outputs, the existing research has proposed a variety of optimized dispatch strategies. Yan et al. proposed an optimization scheme for hybrid energy storage systems, incorporating the correlation between wind and photovoltaic power to mitigate power fluctuations. To address the challenges posed by the variability of renewable energy outputs, various optimization dispatch strategies have been proposed in the existing research [9]. Tong conducted a comprehensive study on the optimization techniques for large-scale energy storage capacity configuration in hybrid wind–solar power grids utilizing big data analytics, with the objective of reducing the load power deficiency rate and enhancing the operational stability of the power system [10]. Zhang proposed a day-ahead optimal scheduling model that accounts for the stochastic nature of wind and solar energy, as well as the application of energy storage systems [11]. Gu’s approach integrates robust optimization with solar energy scenarios synthesized through generative adversarial networks to enhance photovoltaic grid integration capabilities [12].

Furthermore, with the widespread application of stochastic optimization and artificial intelligence methods in power system dispatch, stochastic dual dynamic programming (SDDP) has been introduced to address multi-stage stochastic scheduling problems. By combining dynamic programming with dual theory, SDDP effectively resolves high-dimensional stochastic optimization challenges and has been applied to solve multi-stage economic dispatch problems, enabling optimal unit commitment and decentralized decision-making in smart grid operations. Lan et al. proposed the FSDDP method, which ensures finite convergence by solving a bilevel problem to generate deterministic updates [13]. On the other hand, reinforcement learning (RL)-based unit commitment methods have also attracted significant attention in recent years. RL agents learn optimal scheduling strategies through interaction with their environment, enabling effective handling of high-dimensional, nonlinear, and highly uncertain dispatch problems. These approaches show particular promise in real-time scheduling and frequency control applications. Qin et al. combined optimization methods with integrated deep reinforcement learning to solve unit commitment (UC) problems more efficiently and accurately [14]. While both stochastic dual dynamic programming (SDDP) and reinforcement learning (RL) offer distinct advantages in addressing uncertain scheduling problems, they exhibit limitations in simultaneously tackling the dual challenges of frequency security and economic efficiency. SDDP’s strong reliance on precise mathematical models makes it difficult to flexibly incorporate complex frequency dynamics constraints, whereas RL methods still lack mature mechanisms and theoretical guarantees for enforcing safety-critical constraints.

To more clearly demonstrate the focuses and limitations of existing research studies,

Table 1. Comparison of focuses and limitations among different dispatch methods.

Method	Focus	Limitations
Proposed Method	Unified Framework: Correlation Awareness + Frequency Security	Higher Computational Load Than Deterministic Methods
Deterministic UC	Economic dispatch with fixed reserves; simplicity	Ignores renewable uncertainty; prone to security risks
Two-Stage Stochastic	Day-ahead and real-time cost minimization	Assumes independence between renewables; overlooks frequency security
Robust Optimization	Worst-case scenario preparedness	Overly conservative; high operational costs
SDDP	Multi-stage decision-making; scalability	Struggles with nonlinear constraints (e.g., frequency dynamics)
RL-Based Methods	Adaptive control; handling high-dimensional problems	Black box nature; lacks safety guarantees and interpretability

provides a systematic comparison between representative studies and the proposed method in terms of their core emphases and methodological constraints.

Although significant progress has been made in mitigating the volatility of renewable energy outputs through these studies, few have concurrently addressed both the variability of renewable energy generation and frequency security constraints. Consequently, the development of an optimal dispatch strategy that simultaneously considers frequency security and renewable energy output variability represents a critical issue that urgently requires resolution.

The model’s characteristics are as follows: (1) A scenario-based method using the copula function to establish the joint probability density function of wind and solar power is employed to characterize the uncertainty of renewable energy output, generating typical scenarios and achieving a discrete description of this uncertainty. (2) An optimization framework is established that simultaneously considers expected costs and frequency security, with the objective function designed to solve for expected costs across multiple scenarios. This approach enhances the economic efficiency of unit scheduling optimization by employing probability weighting, thereby avoiding the conservatism inherent in traditional robust optimization methods that focus solely on worst-case scenarios. (3) A framework combining the Copula–KDE model and chance-constrained programming is proposed, which simultaneously addresses the correlation between wind and photovoltaic power generation outputs while strictly ensuring the safety of the system frequency in the UC problem.

2. Typical Day Scenario Generation for Wind and Solar Power

2.1. Overview of Typical Wind–Solar Power Scenarios

Wind and solar energy exhibit intermittency, volatility, and uncertainty, posing significant challenges to power system operational security. To ensure the safe and reliable operation of generation units, these inherent characteristics must be accounted for during both planning and operational stages. The general steps are as follows.

First, historical wind and solar power output data are fitted using a non-parametric kernel density estimation (KDE) method (Gaussian kernel) to obtain the probability density functions (PDFs) for each hour over a 24 h period. Compared to traditional theoretical distribution models, KDE does not require predefined distribution assumptions, enabling more flexible capture of real-world fluctuation patterns, thereby improving the modeling accuracy.

Next, the spatiotemporal correlation between wind and solar power is incorporated based on the univariate PDFs. Due to their negative complementary relationship (e.g., strong solar irradiation at noon coincides with weak wind, while nighttime has no solar but higher wind speeds), the Frank copula function is selected to construct the joint probability distribution model. This function effectively captures both positive and negative dependencies between variables. Through marginal distribution transformation and parameter optimization, an efficient joint distribution model for wind–solar power outputs is achieved.

Finally, large-scale sampling is performed based on the joint PDF, and inverse transform sampling yields time series curves of wind–solar power outputs. To reduce the computational complexity and extract typical features, the K-means clustering algorithm is applied to the sampled data, generating five representative typical daily scenarios.

2.2. Generation of Probability Density Function of Wind–Solar Output by Kernel Density Estimation

When analyzing wind speed and solar irradiance, two primary methods are typically employed: theoretical distribution models [15] and kernel density estimation (KDE) [16].

Theoretical distribution models rely on parametric estimation, requiring predefined assumptions about the underlying data distribution. For instance, wind speed is often assumed to follow a Weibull distribution [17], while solar irradiance is modeled using a beta distribution [18]. However, such approaches may overlook the intrinsic characteristics of wind and solar power curves, leading to significant deviations between fitted results and actual data [19].

Kernel density estimation, in contrast, is a nonparametric method that extracts distribution features directly from observed samples without imposing prior assumptions on data structure. This flexibility allows KDE to more accurately capture the true shape of the data distribution [20].

Given these advantages, this study adopts KDE with a Gaussian kernel function to generate 24 h probability density functions (PDFs) for wind and solar power outputs. The PDFs for the wind power output x and PV power output y at hour t are given by Equation (1):

$$\left\{\begin{array}{l}{\widehat{f}}_n^j(x^j)={\displaystyle \frac{1}{nh}}{\displaystyle \sum _{i=1}^{n}K}\left({\displaystyle \frac{x^j-X_i^j}{h}}\right)\\ {\widehat{f}}_n^j(y^j)={\displaystyle \frac{1}{nh}}{\displaystyle \sum _{i=1}^{n}K}\left({\displaystyle \frac{y^j-Y_i^j}{h}}\right)\end{array}\right.$$

(1)

where $n$ is the number of historical samples; $h$ represents bandwidth parameters; $K(\cdot )$ is the Gaussian kernel function; $X_i^j$ and $Y_i^j$ are the respective historical wind and PV outputs at hour $j$ of day $i$.

2.3. Establishment of the Probability Distribution Function of the Combination of Scenery and Wind Based on Copula Theory

Copula theory enables the characterization of dependence structures between random variables by linking their marginal distributions to the joint distribution, thereby independently capturing their interdependencies. Copulas are primarily categorized into elliptical copulas (e.g., Gaussian) and Archimedean copulas.

The Frank copula, a member of the Archimedean family, is particularly suitable for modeling symmetric dependencies and can effectively represent negative correlations [21]. Given the negative complementary relationship between wind and solar power (e.g., high solar irradiance at noon coincides with low wind speeds, while nighttime exhibits no solar generation but higher wind output), this study adopts the Frank copula for dependency modeling. For hour $j$, the wind power ($w^j$) and PV power ($v^j$) outputs are first transformed using their respective marginal distribution functions $F_X(x)$ and $F_Y(y)$, as shown in Equation (2):

$$\left\{\begin{array}{l}{w}^j=F_{X^j}(x^j)\\ v^j=F_{Y^j}(y^j)\end{array}\right.$$

(2)

The Frank copula $C_{\theta }(w^j,v^j)$ is defined in Equation (3):

$$C_{\theta }(w^j,v^j)=-{\displaystyle \frac{1}{\theta }}ln(1+{\displaystyle \frac{((e^{(-\theta w^j)}-1)(e^{(-\theta v^j)}-1))}{(e^{(-\theta )}-1)}})$$

(3)

where $\theta$ is the copula parameter (estimated from historical data) that quantifies the dependence strength.

Finally, the joint distribution $F^j(x^j,y^j)$ for hour $j$ is given by Equation (4):

$$F^j(x^j,y^j)=C_{\theta }(F_{X^j}(x^j),F_{Y^j}(y^j))$$

(4)

2.4. Generate Typical Scenarios of Wind–Solar Power Generation

For each time period, samples are drawn from the joint probability distribution function to obtain a set of sample points $(w^j,v^j)$, Using the inverse transform of the joint distribution $x^j=F_{X^j}^{-1}(w^j)$ and $y^j=F_{Y^j}^{-1}(v^j)$, the corresponding wind and PV power output data are generated, forming representative daily power curves. Due to the large sample size, K-means clustering is applied to group similar daily curves into distinct clusters. This process yields five representative typical day scenarios, effectively capturing the key operational characteristics of renewable generation.

3. Optimal Scheduling Model

3.1. Object Function

This study proposes a stochastic optimization approach for unit scheduling that simultaneously considers source–load uncertainty and frequency regulation security. The method first generates multiple scenarios to capture renewable generation and load demand uncertainties using the techniques described in Section 2 (including kernel density estimation and copula theory). These scenarios are then incorporated into a stochastic optimization model that minimizes the expected total cost, including generation, startup and shutdown, renewable curtailment, frequency regulation reserves, and carbon emission trading costs. Unlike traditional deterministic approaches, this scenario-based method evaluates system performance through expected costs across multiple scenarios, avoiding overly conservative scheduling results while ensuring both economic efficiency and operational security under uncertainty. The comprehensive cost function is formulated as shown in Equation (5):

$$\min F={\mathbb{E}}_{\omega \in \Omega }{F}_1(\omega )$$

(5)

where $F$ is the total cost; $\omega$ is the uncertainty scenario; $\Omega$ is the uncertainty set.

The optimization objective $F_1(\omega )$ is the total cost in the uncertainty scenario, comprising the operational cost, upstream and downstream reserve costs, renewable curtailment penalties, frequency regulation reserve costs, and carbon emission trading expenses, as formulated in Equation (6):

$$F_1(\omega )={\displaystyle \sum _{t=1}^T(}{C}_{t,g}(\omega )+C_{t,cut}(\omega )+C_{t,ud}(\omega )+C_{t,PFR}(\omega )+C_{t,SFR}(\omega )+C_{carbon}(\omega ))$$

(6)

Generation operating costs refer to expenses from fuel consumption during power production, typically exhibiting a nonlinear relationship with the unit output, while startup and shutdown costs represent additional losses incurred during transitions between offline and online states (or vice versa). The combined commitment and operation costs are formulated in Equation (7), where the first term quantifies generation costs and the second term captures startup and shutdown expenses.

$$C_{t,g}(\omega )={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^N{a}_i}}{P}_{g,i,t}^2(\omega )+b_i{P}_{g,i,t}(\omega )+c_i+{\displaystyle \sum _{i=1}^T{\displaystyle \sum _{i=1}^N{S}_{i,t}}}{u}_{i,t}(\omega ){\left(1-u_{i,t-1}(\omega )\right)}_k$$

(7)

where $C_{t,g}(\omega )$ is the unit start–stop operation cost; $a_i,b_i,c_i$ are the coal consumption coefficient of thermal power units; $P_{g,i,t}(\omega )$ is the output value of thermal power unit $i$ during time period $t$ (MW); $S_{i,t}$ is the startup and shutdown cost of thermal power unit $i$ during time period $t$; $u_{i,t}(\omega )$ is the start–stop status of thermal power unit $i$ during time period $t$.

Accounting for uncertainties in the renewable generation and load, inflexible thermal power output, and economic operational requirements, dispatch plans may lead to either underutilized wind or solar power or insufficient total generation to meet demands, resulting in renewable energy curtailment or load shedding. These scenarios generate curtailment and shedding costs as formulated in Equation (8).

$$C_{t,cut}(\omega )={\mu }_{w,cut}{P}_{t,w,cut}(\omega )+{\mu }_{v,cut}{P}_{t,v,cut}(\omega )+{\mu }_{d,cut}{P}_{t,d,cut}(\omega )$$

(8)

where $C_{t,cut}(\omega )$ is the sum of the costs incurred by the unit due to curtailment of wind, solar, and load profiles, respectively; ${\mu }_{w,cut},{\mu }_{v,cut},{\mu }_{d,cut}$ are the cost coefficients of curtailed wind, solar, and load profiles, respectively; $P_{t,w,cut},P_{t,v,cut},P_{t,d,cut}$ are the abandoned wind, solar, and load power profiles, respectively.

In modern power system operations, upward and downward reserves serve as critical tools for maintaining active power balance, particularly in grids with high renewable energy penetration. The costs associated with providing these flexibility reserves are formulated in Equation (9):

$$C_{t,ud}(\omega )={\sigma }_{g,up}{\displaystyle \sum _{i=1}^N{F}_{g,i,t,up}(\omega )}+{\sigma }_{g,down}{\displaystyle \sum _{i=1}^N{F}_{g,i,t,down}(\omega )}$$

(9)

where $C_{t,ud}(\omega )$ is the total cost of upregulation and downregulation reserves of the system during time period $t$; ${\sigma }_{g,up},{\sigma }_{g,down}$ are the adjust the cost coefficients of the crew up or down; $F_{g,i,t,up},F_{g,i,t,down}$ are the respective upward and downward adjustments of the reserve capacity of unit $i$ during time period $t$.

To address the low-inertia challenges arising from the integration of renewable energy units, primary and secondary frequency regulation reserves are allocated to ensure system frequency stability, with the associated costs formulated as shown in Equation (10):

$$\left\{\begin{array}{l}{C}_{t,PFR}(\omega )={\sigma }_{g,PFR}{\displaystyle \sum _{i=1}^N{F}_{g,i,t,PFR}(\omega )}\\ C_{t,SFR}(\omega )={\sigma }_{g,SFR}{\displaystyle {\displaystyle \sum }^N}{F}_{g,i,t,SFR}(\omega )\end{array}\right.$$

(10)

where $C_{t,PFR}(\omega )$ is the primary frequency regulation cost; $C_{t,SFR}(\omega )$ is the secondary frequency modulation cost; ${\sigma }_{g,PFR},{\sigma }_{g,SFR}$ are the respective primary and secondary frequency regulation cost coefficients of generating units; $F_{g,i,t,PFR},F_{g,i,t,SFR}$ are the respective primary and secondary frequency regulation reserve capacities of the generating unit during time period $t$.

To better align with energy conservation and emission reduction requirements, a tiered carbon trading mechanism is introduced to constrain carbon emissions, with the associated carbon trading cost formulated as shown in Equation (11):

$$C_{carbon}(\omega )=c\cdot {\displaystyle \sum _{k=0}^{⌊\frac{E_C(\omega )}{d}⌋-1}(1+k\alpha )}d+c\left(1+⌊{\displaystyle \frac{E_C(\omega )}{d}}⌋\alpha \right)\left(E_C(\omega )-d⌊{\displaystyle \frac{E_C(\omega )}{d}}⌋\right)$$

(11)

where $C_{carbon}(\omega )$ is the carbon trading costs; $⌊\cdot ⌋$ is the floor function; $\alpha$ is the carbon trading step penalty coefficient; $E_C$ is the difference between the actual carbon emissions and the initial carbon quota; $d$ is the length of the carbon emission range for each step; $c$ is the base price of carbon emissions for the day.

3.2. Constraints

For each scenario $\omega$, the following constraints should be considered based on the actual situation and safety standards.

3.2.1. Power Balance Constraint

To ensure the active power balance of the power system, the sum of the generating power of the units and the grid-connected power of new energy should be equal to the sum of the system load power, as shown in Equation (12):

$${\displaystyle \sum _{i=1}^N{P}_{g,i,t}}+P_{w,t}+P_{v,t}-P_{wQ,t}-P_{vQ,t}=P_{d,t}-P_{dQ,t}$$

(12)

where $P_{g,i,t}$ is the electricity generation of thermal power unit $i$ in period $t$; $P_{w,t}$ is the electricity generation of wind power in period $t$; $P_{v,t}$ is the power generation of photovoltaic power generation in period $t$; $P_{d,t}$ is the load volume in period $t$; $P_{wQ,t}$ is the curtailed wind power in period $t$; $P_{vQ,t}$ is the curtailed power of photovoltaic power generation in time period $t$; $P_{dQ,t}$ is the curtailed load in period $t$.

3.2.2. Thermal Unit Reserve Constraint

This paper decomposes the reserve constraints of thermal power units into system upward and downward flexibility constraints, primary frequency regulation reserve constraints, and secondary frequency regulation reserve constraints. The system flexibility constraints are shown in Equation (13).

$$\left\{\begin{array}{l}{F}_{\mathrm{g}.i.t.\mathrm{up}}=min\left(R_{\mathrm{g}.i.\max }^{+}\Delta t,{\mu }_{i,t}{P}_{\mathrm{g}.i,\max }-P_{\mathrm{g}.i,t}\right)\\ F_{\mathrm{g}.i.t.\mathrm{down}}=min\left(R_{\mathrm{g}.i.\max }^{-}\Delta t,P_{\mathrm{g}.i,t}-{\mu }_{i,t}{P}_{\mathrm{g}.i.\min }\right)\end{array}\right.$$

(13)

where $F_{g,i,t,up},F_{g,i,t,down}$ are the respective upward and downward adjustments of the reserve capacity of unit $i$ during time period $t$; $R_{\mathrm{g}.i.\max }^{+},R_{\mathrm{g}.i.\max }^{-}$ are the respective maximum upward and downward ramp rates of unit $i$; ${\mu }_{i,t}$ is the operating status of unit $i$ in period $t$, which is 1 when it starts and 0 when it stops; $P_{\mathrm{g}.i,\max },P_{\mathrm{g}.i,\min }$ are the respective maximum and minimum technical outputs of unit $i$.

Meanwhile, the primary frequency regulation capacity of the unit should account for 6% to 10% of the rated capacity of the unit, for which the primary frequency regulation reserve constraint is as shown in Equation (14).

$$F_{\mathrm{g}.i.t,PFR}\le 0.08\times P_{\mathrm{g}.i,\max }$$

(14)

In addition, when the unit is performing primary frequency regulation, its output power cannot exceed the maximum output power of the unit. Considering that the unit reserves a certain amount of flexibility reserve, the flexibility reserve and the unit’s frequency regulation constraints can be expressed as Equation (15).

$$\left\{\begin{array}{l}{F}_{g,i,t,PFR}+F_{g,i,t,SFR}+F_{g,i,t,up}\le {\mu }_{i,t}{P}_{g,i,max}-P_{g,i,t}\\ F_{g,i,t,down}\le P_{g,i,t}-{\mu }_{i,t}{P}_{g,i,min}\end{array}\right.$$

(15)

3.2.3. Unit Output Constraint

The output constraints of the generating units include the upper and lower limits of the output, the ramping constraints of the output, and the startup and shutdown constraints of the generating units, which are respectively shown in Equations (16)–(18).

$$P_i^{min}{u}_{i,t}\le P_{g,i,t}{u}_{i,t}\le P_i^{max}{u}_{i,t}$$

(16)

$$\left\{\begin{array}{l}{P}_{g,i,t}-P_{g,i,t-1}\le \Delta P_i^U{u}_{i,t}+P_i^{min}\left(u_{i,t}-u_{i,t-1}\right)+P_i^{max}\left(1-u_{i,t}\right)\\ P_{g,i,t-1}-P_{g,i,t}\le \Delta P_i^D{u}_{i,t}+P_i^{min}\left(u_{i,t}-u_{i,t-1}\right)+P_i^{max}\left(1-u_{i,t-1}\right)\end{array}\right.$$

(17)

$$\left\{\begin{array}{l}\left[T_{on,i,t-1}-M_{on,i}\right]\left[u_{i,t-1}-u_{i,t}\right]\ge 0\\ \left[T_{off,i,t-1}-M_{off,i}\right]\left[u_{i,t}-u_{i,t-1}\right]\ge 0\end{array}\right.$$

(18)

where $\Delta P_i^U,\Delta P_i^D$ are the respective upward and downward ramping rates of unit $i$; $T_{on,i,t-1},T_{off,i,t-1}$ are the respective continuous operation time and continuous shutdown time of the thermal power unit $i$ in the $(t-1)th$ period; $M_{on,i},M_{off,i}$ are respective the minimum continuous operation time and the minimum continuous shutdown time of the thermal power generating unit $i$.

3.2.4. Renewable Curtailment Constraint

In the operation of power systems, new energy generation is intermittent and unstable, and its output is greatly affected by natural conditions, making it difficult to precisely control. To ensure the safe, stable, and economically efficient operation of the system, curtailment constraints are set to reasonably limit its output and avoid excessive impact on the power grid. The curtailment constraints for new energy include wind curtailment, photovoltaic curtailment, and load curtailment, as shown in Equation (19).

$$\left\{\begin{array}{l}0\le P_{wQ,t}\le P_{w,t}^f\\ 0\le P_{vQ,t}\le P_{v,t}^f\\ 0\le P_{dQ,t}\le P_{d,t}^f\end{array}\right.$$

(19)

where $P_{wQ,t},P_{vQ,t},P_{dQ,t}$ are the curtailed amounts of wind, solar and load energy, respectively; $P_{w,t}^f,P_{v,t}^f,P_{d,t}^f$ are the wind, solar, and load forecast values, respectively.

3.2.5. Frequency Security Constraint

To ensure that the system can return to its previous frequency after experiencing a power shock, the total frequency modulation reserve of the generating units should be no less than the power shock, as shown in Equation (20):

$${\displaystyle \sum _{i=1}^N{F}_{g,i,t,\mathrm{PFR}}}+{\displaystyle \sum _{i=1}^N{F}_{g,i,t,\mathrm{SFR}}}\ge \Delta P_{0,t}$$

(20)

where $\Delta P_{0,t}$ is the power impact quantity.

To ensure the safety of the system after being impacted and to limit the frequency drop of the system, the minimum frequency constraint of the system is added, as shown in Equation (21):

$${\displaystyle \sum _{i=1}^N{F}_{g,i,t,\mathrm{PFR}}}+k_D{P}_{d,t}{\displaystyle \frac{\Delta f_m}{f_0}}\ge \Delta P_{0,t}$$

(21)

where $k_D=5$ is the load regulation coefficient; $\Delta f_m=0.5\mathrm{Hz}$ is the maximum system change frequency; $f_0=50\mathrm{Hz}$ is the initial reference frequency of the system.

To ensure that the system can return to its normal operating level after being impacted, the quasi-steady-state frequency deviation constraint of the system is added, as shown in Equation (22):

$$f_{ss}\le {\displaystyle \frac{f_0\Delta P_{0,t}-f_0{\displaystyle \sum _{i=1}^N{F}_{g,i,t,PFR}}}{k_D{P}_{d,t}}}\le f_{ss,max}$$

(22)

where $f_{ss}$ is the system’s quasi-steady-state frequency deviation; $f_{ss,max}=0.1\mathrm{Hz}$ is the maximum system change frequency; $f_0=50\mathrm{Hz}$ is the initial reference frequency of the system.

3.2.6. Carbon Trading Constraint

Carbon trading constraints are divided into total carbon emission constraints and carbon emission constraints at different time periods, as shown in Equations (23) and (24).

$${\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^N{e}_i}}{P}_{g,i,t}\le Q_{total}$$

(23)

$${\displaystyle \sum _{i=1}^N{e}_i}{P}_{g,i,t}\le Q_t,\forall t$$

(24)

where $e_i$ is the carbon emission coefficient per unit power; $Q_{total}$ represents the total carbon emissions; $Q_t$ represents the total carbon emissions per unit time period.

4. Case Studies Analysis

In order to verify the effectiveness and feasibility of the proposed scheduling model, this study employs the modified IEEE 39 bus system for case construction. The case includes 7 conventional thermal power units, 1 photovoltaic power station, 1 wind farm, and 1 external DC transmission line. Among them, the configuration of the DC interconnection line provides a reserved interface for the follow-up research of multi-region power grid coordinated dispatching and mutual aid, which reflects the scalability of the model. The structure is illustrated in Figure 1.

The proposed optimization model was tested on a computer equipped with an AMD Ryzen 5 5600U with Radeon Graphincs from AMD and 16 GB of RAM, utilizing the CPLEX 12.10 solver from IBM for computation. The total solution time was 20.646 s, demonstrating the computational efficiency and practical feasibility of the approach for power system dispatch problems.

4.1. Example Setting

This study focused on scenario generation and uncertainty interval planning for the annual photovoltaic (PV) and wind power generation within a specific regional power grid. Utilizing the scenario generation method detailed in Section 1, five representative scenarios were constructed. These scenarios not only effectively covered the full range of forecasted fluctuations in renewable energy outputs but also clearly demonstrated the expected negative correlation between wind power and the PV output, a key characteristic of the regional energy system.

In terms of wind power performance across scenarios, the output consistently peaked during early morning and nighttime hours, with a notable decline observed at midday. Among the five scenarios, scenario 1 reflects a high-load scenario, with consistently elevated electricity demand over the day and pronounced morning and evening peaks. Scenario 2 corresponds to a low-wind–high-solar scenario, featuring a relatively stable wind power output and a prominent peak in solar generation during midday (12:00–14:00). Scenario 3 serves as the baseline scenario, with wind, solar, and load profiles closely following forecasted values and exhibiting relatively smooth variations, reflecting typical operating conditions. Scenario 4 represents a low-load scenario, where the total system electricity demand is significantly reduced, while scenario 5 represents a high-wind–low-solar scenario, characterized by peak wind power outputs in the early morning and nighttime with lower values at noon, alongside significant overall fluctuations. The solar PV generation remains at low levels throughout the day. The complete wind–PV–load scenarios and uncertainty intervals are presented in Figure 2.

The study solves the unit scheduling problem considering the uncertainty intervals of renewable generation and load factors, while evaluating and comparing system performance metrics using scenario data.

As shown in

Table 2. Dispatch mode setting.

Dispatch Mode	Considerations	Comparison Metrics
Proposed Method	Frequency security and uncertainty	N/A (baseline)
Case Study 1	Ignores uncertainty	Adjustment costs
Case Study 2	Excludes frequency security constraints	Frequency security status

, three distinct grid operation modes are compared and analyzed.

4.2. Decision-Making Plan Analysis

The day-ahead dispatch plan of the units based on the model presented in this paper is shown in Figure 3. It fully demonstrates the operational characteristics of power systems with a high proportion of renewable energy, presenting the following notable advantages.

Firstly, the optimal allocation of base load and peaking units ensures the stability of the power system. Unit 1 maintains a stable output range of 594–660 MW, providing continuous and reliable power supply, thereby ensuring the stability of system frequency and voltage parameters. Units 2 and 3 dynamically adjust their outputs in response to load variations, reducing operational power during the peak photovoltaic generation period (10:00–14:00) while increasing the output during the evening peak hours (18:00–22:00). This strategic operation effectively mitigates load fluctuations, minimizes frequent adjustments to base load units, and extends their service life.

Secondly, the prioritization of renewable energy in dispatch operations has significantly enhanced the utilization rate of clean energy resources. During the peak photovoltaic generation period (10:00–14:00), units 6 and 7 are temporarily deactivated to avoid competition with solar power generation, thereby maximizing the utilization of zero-marginal-cost clean energy and reducing fossil fuel consumption and carbon emissions. During periods of high wind power penetration at night, unit 2 reduced its output to 180 MW, thereby creating capacity for wind power integration and enhancing the proportion of wind energy utilization, which aligns with the development trend of low-carbon power systems.

Thirdly, the generating units possess the capability to flexibly adapt to net load variations, thereby optimizing the economic efficiency of system operations. Anti-peak units (such as unit 4) rapidly ramp up (from 132 MW to 330 MW) in the evening when the photovoltaic output drops sharply (16:00–18:00), making up for the power gap caused by the decline in photovoltaic output, maintaining the balance between supply and demand, and avoiding the high-cost activation of backup units.

4.3. Economic Analysis

The operating costs of different operation modes are shown in

Table 3. An economic analysis.

Dispatch Mode	Operational Cost/$	Flexibility Reserve Cost/$	Renewable Curtailment Penalties/$	Frequency Regulation Reserve Costs/$	Carbon Emission Trading Expenses/$	Total Cost/$
Proposed Method	689,450	45,654	13,299 (Mean value)	28,478	21,351	798,232
Case 1	627,150	37,959	3451	26,923	19,422	714,905
Case 2	615,500	44,436	12,708 (Mean value)	0	19,061	691,705

. To ensure frequency security and adapt to the uncertainty of wind and solar outputs, the total cost of the method proposed in this paper is the highest, which is 11.33% and 13.74% higher than for case 1 and case 2, respectively. The increase is mainly due to the cost of the unit output and flexibility reserve. Both the proposed method and case 2 reserve more upward and downward reserve capacity to cope with the uncertainty of wind and solar outputs. However, case 1 does not consider uncertainty modeling and configures reserve capacity based on predicted values, underestimating actual demand. Therefore, the flexibility reserve cost of the method proposed in this paper and case 2 is 16.85% and 14.57% higher than for case 1, respectively. Considering the grid frequency fluctuations caused by load uncertainty, the frequency regulation reserve cost of the method proposed in this paper is 5.46% higher than that of case 1. The method proposed in this paper has a conservative nature in robust dispatch, requiring comprehensive consideration of multi-scenario wind and solar output fluctuations, leading to early curtailment of the wind or solar output or load shedding to avoid exceeding limits in the worst-case scenario, resulting in a higher average cost of source–load curtailment. Meanwhile, case 1 does not consider uncertainty and maximizes the use of renewable energy, thereby resulting in lower source–load curtailment costs. The method proposed in this paper relies on thermal power unit reserves, leading to increased carbon emissions and higher carbon trading costs, but in exchange, it achieves frequency stability and adjustment flexibility.

Here, we verify the advantages of the scheduling plan proposed in this paper in multiple scenarios and compare it with the scheduling plan given in example 1 without considering uncertainty. The adjustment costs for each scenario are shown in

Table 4. Adjustment cost comparison.

Dispatch Mode	Scenario 1	Scenario 2	Scenario 3	Scenario 4	Scenario 5
Proposed Method’s Adjustment Cost/$	106.64	39,101.67	16,620.68	15,144.65	24,905.74
Case Study 1’s Adjustment Cost/$	34,954.63	64,942.69	16,466.61	43,417.98	12,116.52

. As the unit’s daily output plan proposed in this paper considers the worst-case scenario, the scheduling plan covers most of the fluctuation situations. Therefore, when facing different scenarios, only adjustments need to be made, and the costs are significantly reduced compared to deterministic scheduling. Among them, when facing scenario 3 with the highest probability of occurrence, the adjustment costs of the two are almost the same, indicating that their performances are similar in general situations. However, when facing scenario 1 with the lowest probability of occurrence, this paper’s method only needs to make adjustments, greatly reducing the adjustment costs when there are changes in wind and solar load outputs. The cost results in fewer extreme operations. In the high penetration rate of new energy in the power grid, the unit scheduling scheme obtained by using the method in this article has good reliability, can reduce real-time balancing pressure, and ensures the safety of the power grid.

4.4. Frequency Security Analysis

A frequency security analysis is a crucial component of a power system stability assessment, primarily encompassing a nadir frequency analysis and quasi-steady-state frequency analysis. The nadir frequency serves as the most direct indicator of system frequency stability, reflecting the system’s viability under extreme disturbances. If the nadir frequency falls below the protection action threshold, it may trigger underfrequency load shedding (UFLS) or generator protection actions, potentially leading to cascading failures. The quasi-steady-state frequency, on the other hand, reflects the system’s medium- and long-term regulation capabilities, serving to verify the system’s reliability.

4.4.1. Nadir Frequency Analysis

Taking scenario 3 with the highest probability of occurrence as an example, the comparison between the primary frequency regulation reserve and load disturbance is shown in Figure 4. Since scenario 2 does not consider the reserved frequency regulation reserve, when subjected to load shocks, the remaining thermal reserve of the unit is used for frequency regulation, and its reserve amount is far lower than the load disturbance amount. The worst situation occurs in period 12, with a power deficit of up to 85.38 MW, indicating a severe reserve shortage. The analysis shows that this is due to the rapid increase in load caused by the concentrated startup of industrial and commercial enterprises, which amplifies the reserve shortage issue, consistent with the vulnerability characteristics of the power grid during the morning peak. The scheduling plan obtained using the method proposed in this paper reserves the frequency regulation reserve, which is close to or exceeds the load disturbance amount in each period. There is no power deficit in period 12, and the maximum deficit occurs in period 18, with a power difference of 20.96 MW. This may be due to a sudden increase in power caused by residential life. It can be seen that the reserve amount provided by the method in this paper can cover most power shock situations.

As shown in Figure 5, in case 2 (orange broken line), without considering the frequency modulation reserve, the frequency fluctuates violently after being impacted. Only four time periods remain above the safety threshold of 49.5 Hz, while the remaining time periods operate below this limit for a long time, which greatly increases the likelihood of triggering underfrequency load shedding or unit disconnection, leading to power outages. The worst condition occurs in the 12th time period, where the frequency plummets to 49.2 Hz, far below the safety threshold.

On the other hand, the method proposed in this paper (blue broken line) takes into account the primary frequency modulation reserve, resulting in the lowest frequency valley occurring in the 18th time period, which only drops to 49.8 Hz. The frequency remains stable above the safety threshold of 49.5 Hz throughout the entire time period, effectively avoiding the risk of triggering safety control measures due to excessive frequency drops and ensuring the stability of system operation.

4.4.2. Transient Stability Frequency Analysis

As shown in Figure 6, load disturbances are introduced into the system, and the frequency recovery capability of the proposed method is compared with that of example 2 after such disturbances. The dispatch plan derived from the proposed method maintains the quasi-steady-state frequency at approximately 50 Hz. Although the quasi-steady-state frequency reaches its minimum in the 18th time period, dropping from 50 Hz to 49.97 Hz, it recovers to 50 Hz by the 20th time period. This indicates that the dispatch plan obtained via the proposed method can satisfy the system’s secondary frequency regulation requirements and facilitate the restoration of normal system operation. In contrast, example 2 operates in a low-frequency state for an extended period, with the lowest steady-state frequency of 49.6 Hz occurring in the 12th time period. Meanwhile, the dispatch plan generated by the proposed method remains at 50 Hz.

4.5. Method Practicability Verification

This study conducted extensive validation of the proposed frequency-constrained stochastic unit commitment model based on actual data from a provincial-level power grid in China. This system comprises 54 conventional thermal units, 8 wind farms, and 4 photovoltaic power stations. With a solution time of 329.69 s, the model demonstrates excellent computational efficiency and engineering practicality even at the provincial grid scale.

The unit commitment results, shown in Figure 7, clearly illustrate how thermal units significantly reduce their output during periods of high solar generation (midday) to accommodate renewable energy integration, and rapidly ramp up in the evening as solar generation declines to effectively meet the net load peak.

This study successfully applies the proposed method to a large-scale provincial grid in China. The solution time of 329.69 s confirms that the method maintains superior computational performance while handling high-dimensional uncertainties, complex unit commitment, and frequency security constraints. The results fully validate the feasibility and practical value of the proposed framework for actual provincial grids with high renewable energy penetration, providing system operators with a decision-making tool that balances security and economic efficiency.

5. Conclusions and Outlook

5.1. Conclusions

This study proposes a method for probabilistic modeling and a scenario analysis of a combined wind and solar power output based on the copula function. By establishing a joint probability density function for wind and solar power and utilizing scenario generation technology, the continuous uncertainty of the new energy output is discretized into five representative typical scenarios. These scenarios fully cover various operating states of wind and solar power generation, including high-output, low-output, and transitional conditions with strong volatility, and can accurately reflect the spatiotemporal correlation characteristics of new energy output.

In terms of modeling, this study adopts a multi-scenario optimization scheduling framework, which simultaneously considers the multiple uncertainties of wind power, photovoltaic power, and load, as well as system frequency security constraints. Through a comparative analysis with traditional deterministic scheduling models and optimization models that ignore frequency security constraints, the superiority of the proposed method in handling high-proportion renewable energy integration into power systems is verified. This provides a new solution for the optimal scheduling of power systems with high-proportion renewable energy, which is especially suitable for handling practical engineering scenarios where multiple uncertainties coexist.

5.2. Outlook

The current research has established a joint probability model for wind and solar power based on the copula function. In the future, further consideration can be given to the impacts of extreme weather events (such as typhoons and sandstorms) on new energy outputs by introducing extreme scenario generation techniques to enhance the robustness of the model under abnormal operating conditions. It is possible to explore the integration of deep learning techniques, such as generative adversarial networks (GANs) or variational autoencoders (VAEs), to generate more representative new energy output scenarios, capturing their spatiotemporal coupling characteristics.

With the development of new flexible resources such as hydrogen energy storage, electric vehicle V2G, and virtual power plants, it is necessary to study their collaborative scheduling mechanisms under frequency security constraints, explore the market-oriented mechanism for distributed resource aggregation to participate in system frequency regulation, and design corresponding incentive policies and settlement methods.

To enhance the practicality and scalability of the model, our future work will integrate N-1 contingency security constraints, uncertainty in reserve capacity requirements, and electricity price volatility in market environments into the optimization framework. This holistic approach will significantly improve the applicability and robustness of dispatch strategies in real-world power system operations, ensuring both economic efficiency and operational reliability under complex scenarios.

From a computational perspective, we will explore strategies that combine multi-stage stochastic programming with distributed optimization algorithms to enhance the feasibility and efficiency of large-scale power system computations. The proposed method will be evaluated for its applicability in provincial and regional power grids using real-world operational data. Through comparative analyses with traditional deterministic optimization and stochastic optimization methods, we will quantify the benefits of the proposed framework in improving both economic efficiency and system security.