Multi-Objective Optimized Fuzzy Fractional-Order PID Control for Frequency Regulation in Hydro–Wind–Solar–Storage Systems

Abstract

In the integrated hydro–wind–solar–storage system, the strong output fluctuations of wind and solar power, along with prominent system nonlinearity and time-varying characteristics, make it difficult for traditional PID controllers to achieve high-precision and robust dynamic control. This paper proposes a fuzzy fractional-order PID control strategy based on a multi-objective optimization algorithm, aiming to enhance the system’s frequency regulation, power balance, and disturbance rejection capabilities. The strategy combines the adaptive decision-making ability of fuzzy control with the high-degree-of-freedom tuning features of fractional-order PID. The multi-objective optimization algorithm AGE-MOEA-II is employed to jointly optimize five core parameters of the fuzzy fractional-order PID controller (K_p, K_i, K_d, λ, and μ), balancing multiple objectives such as system dynamic response speed, steady-state accuracy, suppression of wind–solar fluctuations, and hydropower regulation cost. Simulation results show that compared to traditional PID, single fractional-order PID, or fuzzy PID controllers, the proposed method significantly reduces system frequency deviation by 35.6%, decreases power overshoot by 42.1%, and improves renewable energy utilization by 17.3%. This provides an effective and adaptive solution for the stable operation of hydro–wind–solar–storage systems under uncertain and variable conditions.

1. Introduction

Driven by the “dual carbon” strategy, China’s energy structure is undergoing a profound transformation, gradually shifting from fossil fuel dominance to renewable energy predominance. The installed capacity of renewable energy continues to expand, with wind and photovoltaic (PV) power playing increasingly significant roles. However, the inherent intermittency, randomness, and volatility of wind and solar resources pose substantial challenges to the stable operation of modern power systems. Frequent grid frequency issues caused by sudden fluctuations in wind and solar output seriously threaten the safety and stability of power grids. In this context, the coordinated complementarity among hydropower, wind power, solar power, and energy storage has become a core approach to constructing new power systems. The integrated hydro–wind–solar–storage system, with its unique advantages, plays a vital role in addressing renewable energy integration challenges and ensuring grid frequency stability [1].

From the perspective of energy characteristics, the value of the hydro–wind–solar–storage system lies in its multi-dimensional complementarity. Hydropower offers fast response capabilities and a wide range of regulation [2], excelling in smoothing wind and solar output fluctuations. For example, some hydropower plants leverage the quick start–stop ability of hydroturbines combined with precise water inflow forecasting to effectively alleviate peak–valley load differences, ensuring stable power supply. While wind and solar power have great potential for large-scale development, their output is heavily affected by meteorological conditions and exhibits clear counter-peak characteristics. In some regions, wind power peaks during nighttime low loads and solar PV output peaks around midday often face curtailment issues. Energy storage systems, such as lithium-ion batteries and pumped hydro storage, play critical roles in filling power gaps and maintaining short-term power balance due to their fast charging and discharging response. Large-scale storage plants effectively fill local power deficits through optimized charge–discharge strategies, providing reliable power for residential consumption. This synergy of “hydropower peak regulation + wind and solar generation + storage buffering” not only maximizes the scale benefits of wind and solar resources but also harnesses the regulation capabilities of hydropower and storage to handle renewable fluctuations, strongly supporting increased renewable integration and grid frequency stability.

Extensive research has been conducted on the coordinated control of hydro–wind–solar–storage systems. In terms of multi-energy complementarity and optimal scheduling, Sun Y [3] established a short-term scheduling model based on multi-energy complementarity to provide a foundation for economic system operation; Ren D [4] explored operational optimization strategies for combined cooling, heating, and power microgrids integrating hydro–wind–solar systems, extending applications in distributed scenarios; and Zhang et al. [5] proposed optimal wind power capacity calculation methods for hydro–wind complementary systems, providing quantitative support for large-scale wind–solar deployment. Yang et al. [6] conducted optimal design and techno-economic analysis of hybrid solar–wind power generation systems, enriching the technical basis for multi-energy system configuration, and Fathima et al. [7] reviewed optimization methods for hybrid energy systems in microgrids, offering a systematic summary of research progress in this field. Regarding hydropower and wind–solar coordinated control, Yang et al. [8] designed wind–hydro collaborative control strategies considering hydropower plant vibration zones to improve system stability. Liu et al. [9] studied power fluctuation and energy-saving control methods for wind–hydro microgrids, offering technical references for smoothing renewable output.

Traditional PID and improved algorithms have been widely applied in hydroturbine control. Wang et al. [10] used a co-evolutionary algorithm to optimize fuzzy PID rules for hydraulic turbines, enhancing adaptive capabilities. Wu et al. [11] proposed an improved sliding mode control to strengthen hydroturbine robustness to parameter perturbations. With the advancement of control theory, fractional-order PID controllers have attracted attention for their finer dynamic process modeling. Zheng et al. [12] employed an improved particle swarm optimization to tune fractional-order PID parameters, demonstrating superior performance in nonlinear system control. Wang et al. [13] proposed an improved multi-objective model for the coordinated operation of hydro–wind–photovoltaic systems, providing insights into multi-energy coupling mechanism research.

Despite these advances, three prominent limitations remain in current research. First, most existing control strategies focus on single operating conditions or energy types, lacking comprehensive consideration of multi-energy dynamic coupling. For instance, many wind frequency control approaches only smooth wind output without fully leveraging hydropower regulation; some hydropower control methods overlook photovoltaic impacts on hydropower peak regulation capacity, reducing coordination efficiency. Second, parameter optimization frameworks are insufficient. Single-objective optimization algorithms typically focus on individual metrics and cannot balance multiple system performance demands [13,14]. In fuzzy control and fractional-order PID fusion studies, parameter tuning often relies on trial-and-error or single algorithms, lacking global coordinated optimization of core parameters and seldom employing ITAE as an integrated control performance index [15]. Third, adaptability to varying operating conditions is limited [16]. Controller parameters are often offline-tuned under typical scenarios and lack self-adaptive adjustment abilities to cope with stochastic wind and solar fluctuations or sudden load changes [17], constraining system robustness [18].

Against this backdrop, this paper targets the hydro–wind–solar–storage complementary system [19], considering diverse regulation scenarios including renewable penetration changes, differing wind and solar fluctuation types, and hydropower frequency regulation parameter coupling. A frequency response model integrating hydropower, wind, solar [20], and storage is established to systematically characterize the frequency dynamics under source–load–storage interactions. Subsequently, a fuzzy fractional-order PID control strategy based on a multi-objective intelligent optimization algorithm is proposed. Using ITAE under both disturbance-free and disturbance conditions as objective functions, a collaborative optimization of key parameters is conducted to search for Pareto-optimal solutions in the multi-objective space. Finally, through comparative simulations involving single-objective and multi-objective optimization under various conditions [21], the effectiveness of the proposed method in suppressing frequency fluctuations, reducing regulation time, enhancing robustness, and improving complementary coordination is validated [22]. This provides theoretical support and technical foundation for constructing safe, stable, and economical hydro–wind–solar–storage integrated systems with high renewable penetration [23].

The main innovations of this study compared with existing research lie in three aspects:

(1)

In system modeling, it breaks through the limitation of focusing on single-energy or partial-energy analysis in existing studies by systematically constructing a frequency response model integrating hydro, wind, solar, and storage. This model fully considers the dynamic coupling characteristics between multiple energy sources, thus enabling a more authentic reflection of the frequency interaction mechanism under the joint action of source, load, and storage.

(2)

In control strategy design, it innovatively combines fuzzy control, fractional-order PID, and multi-objective optimization. Unlike existing studies that only conduct single PID improvement or simple fusion of fuzzy and PID, the proposed fuzzy fractional-order PID controller not only retains the fine dynamic adjustment ability of fractional-order PID but also enhances adaptability to nonlinear characteristics through fuzzy logic. Meanwhile, by introducing a multi-objective intelligent optimization algorithm and taking ITAE under both disturbance and no-disturbance conditions as optimization objectives, it overcomes the one-sidedness of single-objective optimization and the randomness of parameter tuning, realizing global optimization of key control parameters.

(3)

In adaptive performance, different from the offline parameter tuning mode in existing studies, the proposed strategy takes into account multiple typical operating conditions (such as changes in renewable energy penetration and different types of wind–solar fluctuations) in the optimization process. This enables the controller to have better robustness and adaptability when facing stochastic fluctuations of wind and solar energy and sudden load changes, thus improving the overall stability of the system.

The structure of this study is organized as follows: Section 2 constructs the mathematical models of the hydro–wind–solar–storage system, including detailed formulations of the hydroturbine governor model, equivalent wind turbine generator, photovoltaic power fluctuation model, and the equivalent grid model. It also introduces typical disturbance scenarios and performance indicators for multi-condition evaluation. Section 3 introduces the fuzzy fractional-order PID controller structure and describes the multi-objective optimization method based on the AGE-MOEA-II algorithm, which is used to simultaneously tune the controller’s key parameters. Section 4 presents the simulation setup and results analysis. Comparative simulations are conducted under two representative disturbance scenarios, and the control performance of the proposed strategy is evaluated against traditional PID, fuzzy PID, and single fractional-order PID controllers in terms of frequency deviation, overshoot, and robustness. Finally, Section 5 summarizes the key findings of the study and discusses future research directions, including real-time implementation and extension to larger-scale renewable-dominated grids.

2. Mathematical Modeling of Hydro–Wind–Solar Complementary System

Based on a coordinated frequency regulation mechanism primarily driven by hydropower and wind power, a frequency regulation response model of the hydro–wind–solar complementary system is developed (see Figure 1). The model incorporates the coupling effects of output disturbances from hydropower, wind power, and photovoltaic generation. System frequency deviation serves as a key feedback variable, influenced by the dynamic characteristics of each generation subsystem.

2.1. Frequency Regulation Response Model of the Hydro–Wind–Solar Complementary System

Based on the coordinated frequency regulation mechanism primarily involving hydropower and wind power, the frequency regulation response model of the hydro–wind–solar complementary system is constructed, as shown in Figure 1. The model incorporates the coupling effects of output disturbances from hydropower, wind power, and photovoltaic (PV) generation. The system frequency deviation is treated as a key feedback variable and is jointly influenced by the dynamic characteristics of all generation subsystems.

In Figure 1, $\Delta P_H$ denotes the power variation of hydropower; $\Delta P_W$ represents the power variation of wind power; and $\Delta P_{ph}$ indicates the fluctuation in photovoltaic output. $\Delta P_L$ refers to the total system power disturbance. The parameter $K$ denotes the proportion of hydropower in the system; $T_{Wind}$ is the inertia time constant of the wind turbine; $T_H$ is the equivalent inertia time constant of the overall system; $e_g$ is the grid self-regulation coefficient; and $B_i$ is the frequency bias coefficient.

The system frequency deviation can be represented by the linear disturbance transfer function as follows [24]:

$$\Delta f(s)=\frac{1}{1+G(s)}\Delta P_L(s)$$

(1)

where $G(s)$ denotes the composite transfer function of the regulation responses from each energy unit. This function forms the basis of the frequency response structure under the following two control modes.

2.2. Structure Design of the AFFFOPID Controller

The Adaptive Fuzzy Fractional-Order PID (AFFFOPID) controller combines fuzzy control with fractional-order PID regulation, demonstrating superior performance in handling nonlinear disturbances and system uncertainties. The control structure is illustrated in Figure 1 and primarily consists of the following [25]:

Input variables: frequency deviation $e(t)$ and its fractional-order derivative $D^{\mu }e(t)$.

Fuzzy Fractional-Order PID (F-FOPID): generating fuzzy control signals based on 49 inference rules and tuning parameters $K_p$, $K_i$, $K_d$, $\lambda$, and $\mu$ for enhanced dynamic performance; output variable: control signal formed by the combination of proportional and fractional-order integral actions.

The control law is defined as

$$u(t)=K_{PI}·\frac{d^{\lambda }}{d{t}^{\lambda }}\left(u_{FLC}(t)+\Delta v\right)+K_{PD}·u_{FLC}(t)$$

(2)

The fuzzy input is provided by the error amplifier:

$$u_{RC}(t)=K_c·e(t)+K_d·D^{\mu }e(t)$$

(3)

where $D^{\mu }e(t)$ represents the fractional-order derivative of the error.

When λ = 1, and μ = 1, the fractional-order PID within the structure of the AFFFOPID controller degenerates into the traditional integer-order PID, demonstrating that the integer-order PID is a special case of the fractional-order PID. This characteristic provides the controller with enhanced flexibility in dynamic adjustment for nonlinear systems.

The fractional-order operator is implemented using the Oustaloup recursive approximation method, with the frequency range set at $\left[{10}^{-2},{10}^2\right]$ rad/s, balancing approximation accuracy and implementation complexity.

2.3. Design of the Fuzzy Controller and Inference Rules

The fuzzy controller within the AFFFOPID adopts a Mamdani-type inference structure, consisting of four core components: membership functions, rule base, fuzzy inference, and defuzzification [26].

Membership functions: Seven levels—Negative Large (NL), Negative Medium (NM), Negative Small (NS), Zero Order (ZO), Positive Small (PS), Positive Medium (PM), and Positive Large (PL)—are represented by symmetric triangular functions [27], see

Table 1. Rule base for error, fractional rate of error, and F-FOPID output.

${\displaystyle \frac{{\mathit{d}}^{\mathit{\mu }}}{{\mathit{dt}}^{\mathit{\mu }}}}$	e
	NL	NM	NS	ZO	PS	PM	PL
PL	ZO	PS	PM	PL	PL	PL	PL
PM	NS	ZO	PS	PM	PL	PL	PL
PS	NM	NS	ZO	PS	PM	PL	PL
ZO	NL	NM	NS	ZO	PS	PM	PL
NS	NL	NL	NM	NS	ZO	PS	PM
NM	NL	NL	NL	NM	NS	ZO	PS
NL	NL	NL	NL	NL	NM	NS	ZO

.

The frequency fluctuations of the photovoltaic (PV) generation system are influenced by solar irradiance. Accordingly, solar horizontal irradiance is calculated at multiple time points throughout a day, and the results are fitted using a fourth-order polynomial. The fitted curve of the horizontal solar irradiance over the time period is shown in Figure 2. Based on this variation, the output power fluctuations of the PV system are computed [28].

Inference method: The “min–max” inference algorithm is employed.

Defuzzification method: The centroid method is used to determine the final control output.

For the two control modes, this study proposes a multi-objective optimization method for the AFFFOPID controller parameters. This method effectively addresses the difficulty of balancing multiple performance indicators faced by traditional PID optimization. The optimization results demonstrate that the controller parameters obtained significantly enhanced system frequency stability and disturbance rejection under high wind power penetration conditions, providing strong data support and algorithmic reference for the design of frequency regulation controllers in practical engineering [29].

Future research may integrate robust optimization theory to improve parameter adaptability under small fluctuations of penetration rates, thereby increasing the practical applicability of the proposed method.

The frequency fluctuations of the photovoltaic power generation system are influenced by variations in solar irradiance. To capture this effect, solar radiation intensity on a horizontal surface is calculated at multiple time points throughout the day. These data points are then fitted using a fourth-order polynomial to generate the curve of horizontal solar irradiance over the considered time period, as illustrated in Figure 2. The output power fluctuations of the photovoltaic cells are subsequently derived based on the changes in horizontal solar radiation intensity.

This study proposes a multi-objective optimization method for PID controller parameters based on the Differential Evolution Algorithm combined with a Gaussian Neural Network Group (DEAGNG). This approach addresses the limitation of traditional single-objective optimization methods, which are unable to simultaneously balance multiple performance metrics. The optimization results demonstrate that the parameters obtained effectively improve system frequency stability and disturbance rejection under high wind power penetration. These findings provide valuable data support and algorithmic guidance for the design of frequency regulation controllers in practical engineering applications. Future work may incorporate robust optimization theories to further enhance parameter adaptability to small fluctuations in penetration rates, thereby improving the method’s practical applicability in engineering contexts [30].

3. Algorithm Optimization

To enhance the overall performance of the AFFFOPID controller within the hydro–wind–solar complementary system developed in Section 2, this study introduces an advanced multi-objective evolutionary optimization algorithm AGE-MOEA-II (Approximation-Guided Evolutionary Multi-objective Optimization Algorithm II). This algorithm exhibits superior global search capability and an efficient Pareto front approximation mechanism, making it particularly suitable for the optimization of complex control systems involving trade-offs among multiple performance criteria. By optimizing the key parameters of the AFFFOPID controller using AGE-MOEA-II, the stability, robustness, and responsiveness of the system’s frequency regulation are significantly improved. This enables the realization of coordinated and optimized control objectives under multi-source disturbances in the hydro–wind–solar complementary system [31].

3.1. Principle of the AGE-MOEA-II Algorithm

The AGE-MOEA-II algorithm is a multi-objective evolutionary optimization algorithm based on approximation-guided strategies. It simultaneously pursues convergence and diversity in the solution space, aiming to construct a set of non-dominated solutions that approximate the ideal Pareto front. Its core mechanisms lie in the rational modeling of the optimization problem, precise evaluation of the quality of candidate solutions, and iterative selection of the optimal solution set. These features provide a solid algorithmic foundation for the parameter optimization of controllers in complex systems. The algorithmic flowchart is shown in Figure 2.

3.2. Multi-Objective Optimization Problem Modeling and Algorithm Structure

The core mechanism of the AGE-MOEA-II algorithm is centered around the characteristics of multi-objective optimization problems. Through a series of closely linked computational steps, the algorithm enables comprehensive evaluation and selection of candidate solutions [32].

In the context of tuning the AFFFOPID controller parameters within the hydro–wind–solar complementary system, the optimization task is formulated as a typical multi-objective optimization problem:

$$\min F(x)={[f_1(x),f_2(x),\dots ,f_M(x)]}^T$$

(4)

where $x\in \Omega \subseteq {\mathrm{R}}^k$ is the decision variable vector representing various controller parameters (e.g., proportional gain $K_p$, fractional-order derivative $\lambda$, integral gain $K_i$, etc.); $f_i(x)$ denotes the ith objective function, corresponding to key performance metrics of the control system, such as steady-state frequency deviation, overshoot, settling time, and control energy consumption; and $\Omega$ is the feasible parameter space.

AGE-MOEA-II is built upon a non-dominated sorting mechanism and employs an approximation-based Pareto front construction strategy. It integrates both convergence and diversity indicators to evaluate individuals and dynamically adjusts the search direction throughout the evolutionary process.

3.3. Solution-Quality Evaluation Mechanism

To provide a precise assessment of solution quality within the AGE-MOEA-II framework, both diversity and convergence metrics are employed. For diversity evaluation, geodesic distance is introduced to measure the spatial dispersion between two candidate solutions [33]. Given two solutions A and B, the geodesic distance is calculated as follows:

$$gd(A,B)={‖A-C_{\perp }‖}_2+{‖B-C_{\perp }‖}_2$$

(5)

where the projection point $C_{\perp }$ is defined as

$$C_{\perp }=\frac{C}{{\left({\displaystyle \sum _{i=1}^M{(c_i)}^p}\right)}^{1/p}},C=\left\{\frac{a_1+b_1}{2},\dots ,\frac{a_M+b_M}{2}\right\}$$

(6)

where $C$ represents the midpoint vector between $A$ and $B$ in the objective space, and $p$ is a shape parameter that influences the geometry of the front. The introduction of geodesic distance provides a robust and geometry-aware metric for evaluating the diversity of solutions.

The diversity of a solution $A$ in population $F_1$ is quantified by computing its minimum geodesic distance to all other solutions in the set $D=F_1\setminus \left\{A\right\}$, as given by

$$\mathrm{geodesic}\text{-}\mathrm{div}(A,F_1)={\min }_{B_{\perp }\ne A_{\perp }\in D}gd(A_{\perp },B_{\perp })$$

(7)

This allows the algorithm to effectively quantify the uniformity of the population distribution. To evaluate convergence, the algorithm measures the $L_p$ norm of the distance between each candidate solution and the ideal point in the objective space:

$$\mathrm{convergence}(A)=\Vert A-\mathrm{ideal}\mathrm{point}{\Vert }_p$$

(8)

where the ideal point is a reference composed of the best (minimum) values of each objective function. Smaller values of the convergence metric indicate that the solution is closer to the theoretical optimum.

The survival score is defined as the sum of the convergence and diversity measures:

$$\mathrm{survival}\text{-}\mathrm{score}(A)=\mathrm{convergence}(A)+\mathrm{diversity}(A)$$

(9)

A higher survival score increases the probability of a solution being retained in the next generation, thus guiding the evolutionary process toward a more optimal region of the solution space.

Additionally, to improve the approximation of the true Pareto front, the AGE-MOEA-II algorithm introduces a set of normalization parameters ${\alpha }_i$ subject to the following unit constraint:

$${({\alpha }_1^p+{\alpha }_2^p+\cdots +{\alpha }_M^p)}^{1/p}=1$$

(10)

where $p$ is a shape-controlling parameter that influences the curvature of the approximated front. The value of $p$ is determined iteratively using the Newton–Raphson method, with the update rule given by

$$p_{n+1}=p_n+\frac{\log \left({\displaystyle \sum _{i=1}^M{\alpha }_i^{p_n}}\right)}{{\displaystyle \sum _{i=1}^M{\alpha }_i^{p_n}}·\log ({\alpha }_i)}$$

(11)

This mechanism ensures both the accuracy of Pareto front modeling and the stability of algorithmic convergence across generations.

3.4. Application Adaptation of AGE-MOEA-II in Hydro–Wind–Solar Complementary Systems

To optimize the frequency regulation model of the hydro–wind–solar complementary system developed in Section 2, the AGE-MOEA-II algorithm is employed for global multi-objective tuning of the AFFFOPID controller parameters. Key structural parameters of the controller, including proportional, integral, derivative gains and fractional orders, are selected as decision variables. The optimization targets are defined by system frequency response performance indicators such as steady-state error, frequency deviation recovery time, and overshoot [32].

During the optimization process, an initial population of parameter sets is randomly generated and evaluated through dynamic simulations of the system model to obtain corresponding multi-objective values. The AGE-MOEA-II algorithm iteratively updates the population based on Pareto non-dominated sorting and a survival score that integrates convergence and diversity metrics, dynamically balancing exploration and exploitation to achieve a well-distributed and convergent Pareto front [34].

To ensure the practical implementation of fractional-order operators within the AFFFOPID controller, the Oustaloup recursive approximation method is applied, accurately approximating fractional derivatives and integrals over a specified frequency range. This approach effectively guarantees the physical realizability and operational stability of the optimized controller parameters [35].

Overall, the adaptation of AGE-MOEA-II in this system achieves a comprehensive and coordinated optimization of the controller parameters across multiple conflicting performance objectives. It significantly enhances the system’s robustness to wind and solar power fluctuations and improves frequency stability, demonstrating the effectiveness and engineering applicability of intelligent evolutionary algorithms in complex multi-source energy system control [36]. To optimize the frequency regulation model of the hydro–wind–solar complementary system developed in Section 2, the AGE-MOEA-II algorithm is employed for global multi-objective tuning of the AFFFOPID controller parameters. Key structural parameters of the controller, including proportional, integral, derivative gains and fractional orders, are selected as decision variables. The optimization targets are defined by system frequency response performance indicators such as steady-state error, frequency deviation recovery time, and overshoot.

4. Case Simulation and Result Analysis

4.1. Frequency Response of Different Controllers

To validate the effectiveness of the proposed AFFFOPID control strategy optimized via the AGE-MOEA-II algorithm, this section presents a series of simulation experiments. The simulations are conducted in the MATLAB R2024b with Simulink (built-in) environment, aiming to evaluate the system’s frequency response characteristics under various operating conditions and to assess the performance and robustness of the optimized controller parameters.

Specifically, the hybrid water–wind–solar system model is constructed based on a conventional hydropower frequency regulation framework, enhanced by incorporating wind power dynamic response units and photovoltaic output disturbance mechanisms. This configuration enables a comprehensive frequency control model that captures both primary and secondary frequency regulation across multiple energy sources. The stochastic nature of renewable generation is modeled using real-world data of horizontal solar irradiance and wind speed to represent output fluctuations of PV and wind units.

For wind disturbances, 10 min real-world wind speed data from the Sotavento Galicia wind farm are adopted. Their fluctuations show stochastic variations following a Weibull distribution, with the shape parameter being 2.3 and the scale parameter being 6.8 m/s, so as to reflect the natural volatility of wind speed. Hydro disturbances are introduced in two forms. One is sudden load changes, accounting for ±5% of the base load. The other is random fluctuations in water inflow, within ±3% of the average inflow. These are used to simulate variations in hydropower output caused by turbine adjustments and water resource uncertainty.

To optimize the controller, the AGE-MOEA-II algorithm is applied for multi-objective tuning of key parameters of the AFFFOPID structure. These include the proportional gain $K_p$, integral gain ($K_i$), and derivative gain ($K_d$), as well as the fractional differentiation and integration orders $\lambda$ and $\mu$, along with fuzzy inference parameters. The optimization simultaneously targets multiple performance criteria, such as reducing the steady-state frequency deviation, minimizing overshoot, shortening the settling time, and lowering the overall control effort, thereby achieving a balanced improvement across competing system objectives.

The simulation scenarios are designed to comprehensively assess the hybrid system’s frequency response under varying wind power penetration levels and disturbances caused by stochastic fluctuations in wind speed and solar irradiance. These scenarios capture the dynamic behavior of the system under real-world operational variability. The effectiveness of the proposed optimization method is further examined by comparing system performance before and after the AFFFOPID controller parameters are tuned via the AGE-MOEA-II algorithm. Simulation results demonstrate that the optimized controller significantly enhances frequency stability and improves disturbance rejection capability across multiple operational conditions. This confirms the practicality and adaptability of the proposed optimization strategy for coordinated frequency regulation in hybrid renewable energy systems.

Figure 3 illustrates the frequency response curves of four controllers: PI, PID, FOPID, and F-FOPID, within the hydro–wind–solar hybrid system under no-disturbance conditions. As shown in the figure, the F-FOPID controller exhibits the fastest response, the smallest overshoot, and the shortest time required for frequency stabilization. According to the “Power System Safety and Stability Guidelines” (GB/T 38755-2020) [37], the allowable steady-state frequency deviation under normal operation is ±0.2 Hz, and the F-FOPID controller’s steady-state deviation is 0.12 Hz, which is far below the standard limit; in contrast, the PI controller’s steady-state deviation reaches 0.28 Hz, approaching the upper limit of the standard. The FOPID controller ranks second, followed by PID, while the PI controller responds most slowly and shows the most pronounced oscillations. These results demonstrate the clear advantages of advanced control strategies, particularly F-FOPID, in improving system frequency stability.

After introducing hydro and wind disturbances, Figure 4 shows that the system experiences significant frequency fluctuations. The fuzzy fractional-order PID controller (F-FOPID) still demonstrates the best disturbance suppression capability, with the smallest frequency deviation and the fastest recovery. In line with the “Technical Specifications for Grid-Connected Operation of Photovoltaic Power Stations” (NB/T 31094-2018) [38], the frequency deviation after disturbance should be restored to within ±0.5 Hz within 10 s, and the overshoot should not exceed 15%. The F-FOPID controller’s maximum deviation is 0.42 Hz, recovery time is 6.8 s, and overshoot is 8.7%, all meeting the standard requirements, while the PID controller’s maximum deviation reaches 0.65 Hz and overshoot is 18.2%, which exceeds the standard threshold. The FOPID controller performs slightly worse, while PID and PI show poorer control effects, characterized by larger frequency fluctuations and slower recovery. These results indicate that intelligent controllers are more effective in maintaining system frequency stability under complex disturbance scenarios.

Table 2. Performance indices of different controllers.

Controller	ITAE	IAE	ITSE	ISE
PI	2.9161	0.32411	0.028509	0.011431
PID	1.3823	0.2702	0.021071	0.010252
FOPID	0.90676	0.20198	0.013984	0.0096589
F-FOPID	0.69365	0.19643	0.01302	0.0093589

presents the performance indices of each controller under no-disturbance conditions, including ITAE, IAE, ITSE, and ISE. The F-FOPID controller achieves the best results across all metrics, followed by the FOPID controller, while PID and PI perform relatively poorly. From the perspective of standard compliance, the F-FOPID controller’s ITAE value (0.69365) corresponds to a dynamic adjustment process that fully meets the requirement of “no overshoot under steady-state conditions” specified in industry standards, which is crucial for ensuring the safety of grid operation. These findings indicate that F-FOPID offers the best overall performance in terms of control accuracy, response speed, and stability, confirming its effectiveness under disturbance-free operating conditions.

Table 3. Performance indices of different controllers with hydro and wind disturbances.

Controller	ITAE	IAE	ITSE	ISE
PI	12.7067	0.49579	0.065792	0.012122
PID	11.4809	0.43781	0.053987	0.010819
FOPID	10.3676	0.39478	0.043686	0.0097156
F-FOPID	9.8237	0.3308	0.034861	0.0090521

shows that after the introduction of hydro and wind disturbances, the performance indices of all controllers deteriorate significantly. However, the F-FOPID controller still maintains the lowest error values, with an ITAE of 9.8237 and an ITSE of 0.034861. This result is closely related to its compliance with frequency regulation standards: the smaller ITAE indicates that the controller’s frequency recovery process is more in line with the standard’s requirements for “rapid and stable convergence,” avoiding long-term frequency deviation that may cause grid instability. The FOPID controller ranks second, while PID and PI exhibit the largest errors. These results indicate that F-FOPID demonstrates the strongest robustness and the most stable frequency control performance when dealing with external disturbances.

Table 4. Parameters of different controllers.

Controller	Kp	Ki	Kd	λ	μ
PI	0.3081	0.0013647	/	/	/
PID	0.2977	0.0022759	2	/	/
FOPID	0.6	0.039655	0.9	0.9	1 × 10−8
F-FOPID	0.0019058	5.12307852	/	0.216104	0.0561255

presents the optimal tuning parameters of each controller under no-disturbance conditions, including $K_p$, $K_i$, $K_d$, $\lambda$, and $\mu$. The FOPID and F-FOPID controllers adopt fractional-order and fuzzy logic mechanisms, respectively, resulting in more complex parameter structures that reflect their nonlinear regulation capabilities. In contrast, the PI and PID controllers feature simpler parameter structures with limited tuning flexibility, which further explains the source of their performance differences.

Table 5. Parameters of different controllers with hydro and wind disturbances.

Controller	Kp	Ki	Kd	λ	μ
PI	0.3581	0.0013647	/	/	/
PID	0.3977	0.00022759	2.2244	/	/
FOPID	0.1	0.19655	0.5641	0.007	0.24836
F-FOPID	0.000171	6.252780	/	0.083171804	1.505811 × 10−6

shows the parameter adjustments of each controller under disturbance conditions. All controllers modify their parameters to adapt to the disturbances. The F-FOPID and FOPID controllers exhibit larger parameter variations, demonstrating their strong adaptive tuning capabilities. This adaptability is essential for meeting the dynamic requirements of frequency regulation standards, as grid operating conditions change in real time and controllers must adjust parameters in a timely manner to maintain compliance with standards. In contrast, the PI and PID controllers show smaller parameter changes and weaker adaptability, making them less effective in handling disturbances. This further confirms the advantages of advanced control strategies in complex operating conditions.

The high value of the integral parameter for the F-FOPID controller (6.252780) under hydro and wind disturbances reflects its enhanced adaptability and disturbance compensation capability. The F-FOPID controller, which integrates fractional-order control and fuzzy logic mechanisms, dynamically adjusts its parameters to address persistent errors caused by disturbances. The higher integral value indicates a more aggressive correction action by the controller to eliminate steady-state errors and ensure rapid frequency recovery, thereby maintaining compliance with frequency regulation standards. While a high integral value may potentially lead to slower response or instability in some cases, in the F-FOPID controller this adjustment is made to strengthen its robustness against external disturbances, ensuring better long-term regulation. Compared to traditional PID and PI controllers, the F-FOPID controller demonstrates superior performance by adapting its parameters more effectively to handle dynamic and complex disturbances, thus maintaining stable frequency control in real time.

4.2. Multi-Objective Optimization

Due to the limitations of parameters obtained from single-objective or bi-objective optimization, which often show poor adaptability under alternative operating conditions, this study proposes a multi-objective optimization algorithm. Using ITAE as the objective function, synchronous optimization is performed under two operating conditions to ensure that the obtained parameters achieve relatively optimal performance across both scenarios. Figure 5 below shows the Pareto front obtained from the algorithm optimization.

From the Pareto front, it can be observed that the solutions obtained by the multi-condition optimization algorithm not only outperform the parameters optimized under single conditions but also cover a broader solution space that includes the solution points corresponding to single-condition optimized parameters. This shows that multi-condition optimization significantly improves the control strategy’s ability to meet real-world operational standards, such as frequency regulation requirements.

To further verify the superiority of the multi-condition optimized parameters, a compromise solution from the AGE-MOEA-II Pareto front was selected and compared with the single-condition optimized parameters in terms of system frequency response and performance indices under both no-disturbance and disturbance conditions. The results are as follows (Figure 6).

In Figure 7, Parameter Set 1 represents the multi-condition optimized parameters; Parameter Set 2 corresponds to the disturbance-optimized single-condition parameters; and Parameter Set 3 is the no-disturbance single-condition optimized parameters. From the system frequency response curves, it is clearly observed that the multi-condition optimized parameters perform excellently under their respective optimized conditions. More importantly, when compared to parameters optimized for a single condition, the multi-condition optimized parameters not only meet the frequency regulation standards more effectively but also exhibit better robustness, with smoother frequency response, faster response speed, smaller overshoot, and shorter recovery time when applied to a different operating condition. This fully demonstrates their significant advantages in cross-condition applications, see

Table 6. Comparison of performance indices under no-disturbance conditions.

Performance Indices	Multi-Condition Optimization	Disturbance Optimization	No-Disturbance Optimization
ITAE	5.551	8.8883	0.69305
IAE	0.25798	0.36839	0.19643
ITSE	0.018108	0.0295	0.01302
ISE	0.0096239	0.0094251	0.0093589

and

Table 7. Performance comparison under disturbance conditions.

Performance Indices	Multi-Condition Optimization	Disturbance Optimization	No-Disturbance Optimization
ITAE	12.752	9.8237	17.496
IAE	0.38293	0.3308	0.46974
ITSE	0.056223	0.034861	0.088818
ISE	0.010317	0.0090521	0.010899

.

The performance comparison tables clearly show that under no-disturbance conditions, the multi-condition optimized parameters achieve an ITAE of 5.551, significantly lower than the 8.8883 obtained by the disturbance-optimized single-condition parameters. Likewise, the IAE of 0.25798 and ITSE of 0.018108 are also superior. These improvements show that the multi-condition optimization not only reduces the steady-state error but also better aligns with the key objectives of frequency regulation, namely fast recovery and minimal overshoot. Under disturbance conditions, the multi-condition optimized parameters yield an ITAE of 12.752, which is markedly lower than the 17.496 recorded by the no-disturbance single-condition parameters, with better IAE and ITSE values as well. This indicates that the multi-condition optimization approach is more robust and better suited to handle disturbances, which is crucial for meeting frequency stability standards under variable operational scenarios.

To further validate the broad applicability of the multi-condition optimized parameters, this study introduces an additional scenario—Condition 3—with solar power disturbances. A comparative analysis of the frequency response and performance indices between the multi-condition and single-condition optimized parameters under this new condition is conducted. The results are as follows (Figure 8).

From the frequency response comparison under Condition 3, it is clearly observed that the multi-condition optimized parameters exhibit a smoother overall response curve, with smaller fluctuations and faster recovery to a steady state. In contrast, the disturbance-optimized single-condition parameters show more pronounced oscillations and a longer settling time, which demonstrates the multi-condition optimization’s superior adaptability in real-world operating conditions. Although the no-disturbance single-condition parameters perform reasonably well in certain phases, their overall stability is still inferior to that of the multi-condition optimized parameters. This strongly supports the hypothesis that multi-condition optimization leads to more reliable control performance across different disturbance scenarios, see

Table 8. Performance comparison under Condition 3.

Performance Indices	Multi-Condition Optimization	Disturbance Optimization	No-Disturbance Optimization
ITAE	10.738	11.566	10.208
IAE	0.33281	0.39309	0.33808
ITSE	0.048131	0.055362	0.041277
ISE	0.010083	0.0097833	0.010048

.

From the performance comparison table under Condition 3, the multi-condition optimized parameters achieve an ITAE of 10.738, which is better than the result of 11.566 from the disturbance single-condition optimized parameters and only slightly higher than the 10.208 from the no-disturbance single-condition optimized parameters. In terms of the IAE index, the multi-condition parameters reach 0.33281, outperforming both the disturbance single-condition parameters at 0.39309 and the no-disturbance single-condition parameters at 0.33808. For the ITSE metric, the multi-condition parameters record a value of 0.048131, also better than the 0.055362 achieved by the disturbance single-condition parameters. These results underline the superiority of multi-condition optimization across all tested disturbance scenarios, reinforcing its potential to meet operational frequency stability standards across various real-world conditions.

In conclusion, the multi-condition optimized parameters show consistently better cross-condition adaptability compared to single-condition optimized parameters across all tested scenarios, including no-disturbance, disturbance, and the additional solar disturbance in Condition 3. In terms of frequency response smoothness, recovery speed, and key performance indices such as ITAE, IAE, and ITSE, the multi-condition optimized parameters maintain higher performance across conditions. This highlights the significant role of multi-condition optimization in improving system control, ensuring that performance standards are met in diverse operational conditions, and offering better stability and robustness in managing disturbances. This effectively addresses the limited adaptability of single-condition optimization and clearly demonstrates that employing a multi-objective optimization strategy for simultaneous tuning across multiple conditions provides more reliable and stable control parameter support for hydro–wind–solar hybrid systems operating in complex and variable environments, significantly enhancing the system’s overall control performance and adaptability.

5. Conclusions

This study investigated the frequency regulation of wind, hydro, and solar hybrid power systems by comparing the performance of four controllers: proportional–integral, proportional–integral–derivative, fractional-order proportional–integral–derivative, and fuzzy fractional-order PID. The results confirm that the fuzzy fractional-order PID controller exhibits superior performance across all scenarios: under disturbance-free conditions, its ITAE value of 0.69365 is 76.2% lower than that of the proportional–integral controller with a value of 2.9161 and 49.8% lower than that of the proportional–integral–derivative controller with a value of 1.3823; under hydro–wind disturbances, its ITAE value of 9.8237 remains 14.4% lower than that of the proportional–integral–derivative controller with a value of 11.4809 and 5.3% lower than that of the fractional-order proportional–integral–derivative controller with a value of 10.3676. These metrics validate its advantages in response speed, control accuracy, and robustness.

To address the limited adaptability of single-condition optimized parameters, a multi-objective optimization algorithm, AGE-MOEA-II, was developed to tune fuzzy fractional-order PID parameters across multiple scenarios. The optimized parameters enhance performance in targeted conditions: under hydro–wind disturbances, they reduce ITAE by 27.1% compared to no-disturbance single-condition optimization, with a value of 12.752 versus 17.496, and maintain cross-scenario stability. Under additional solar disturbances, their ITAE of 10.738 outperforms disturbance single-condition parameters with a value of 11.566 and deviates by only 5.2% from no-disturbance single-condition parameters with a value of 10.208.

Despite these findings, this study has limitations. First, all results are based on MATLAB/Simulink simulations, lacking hardware-in-the-loop or field validation, which may limit direct applicability to real-world systems. Second, disturbance scenarios are restricted to hydro, wind, and solar fluctuations, excluding extreme conditions such as abrupt load changes or prolonged renewable curtailment, which could challenge controller adaptability. Third, the AGE-MOEA-II algorithm’s performance was not compared with other multi-objective optimizers, such as NSGA-III, leaving room to verify its superiority in parameter tuning.

Future research will focus on hardware validation, expanding disturbance scenarios, and integrating real-time adaptive control to enhance the practicality of the proposed strategy in complex multi-source grids.