Fractional Order PID Optimal Control Method of Regional Load Frequency Containing Pumped Storage Plants

Abstract

The pumped storage unit has good adjustment characteristics of a fast power response and convenient start and stop, which provides support for the safe and stable operation of the power system. To this end, this paper proposes a fractional order PID (FOPID) optimization control method for the regional load frequency of pumped-storage power plants. Specifically, based on IEEE standards, this paper established a single-region model of pumped storage. Then, a fractional order PID (FOPID) controller was designed, and the parameters of the controller were optimized via using the chaos particle swarm optimization (CPSO) algorithm. The effectiveness of the proposed method is verified by example simulation in the two-zone model of the pumped storage based on IEEE standards. The results of the example show that the proposed method exhibits stronger robustness and stability in the regional load frequency control.

1. Introduction

To cater the dual-carbon strategic goal of “carbon neutrality and carbon peaking”, renewable energy such as wind power will achieve great development in the next 30 years, and the power system will face great challenges in safe, economical and low-carbon operation [1]. As a mature energy storage technology, pumped storage has the advantages of large capacity, good economy, environmental protection and cleanness, which plays a key role in energy storage, peak shaving and frequency regulation in the power system and is critical for China ‘s 14th Five-Year Plan and the development of future power systems. The addition of pumped storage plants can improve the frequency regulation performance of the interconnected power system [2,3] and effectively solve the problem of grid frequency fluctuation caused by the large-scale grid connection of renewable energy, which is of great significance [4,5].

Efficient load frequency control (LFC) is the key to ensure the stable operation of power grid and user safety, power quality assessment and power monitoring. The design of the LFC controller is divided into controller design and parameter optimization. The purpose is to improve the dynamic performance of the system and achieve rapid stability of the grid frequency. PID controller is often used as LFC controller in practical engineering. However, due to the complexity of power system models and the enhancement of coupling between regions, the traditional PID controller is difficult to meet the regulation requirements of high-quality control of power systems.

To make sure supplying sufficient and reliable electric power with good quality, scholars have proposed a series of methods to tackle the issues for LFC. The history of control strategies and control methodologies for LFC were summarized in [6]. To deal with the random time–delay attack on a power system, a robust LFC scheme was presented in [7]. To address renewable energy uncertainties, a data-driven method for LFC based on deep reinforcement learning was proposed in [8]. A memory-based event-triggering H∞ load frequency control (LFC) method was presented in [9]. Reference [10] combined the advantages of sliding mode control and PI control and proposed a load frequency control method for multi-area interconnected power systems. At the same time, the advantages of proportional integral control and sliding mode control based on the control deviation of the new area were brought into play. On the basis of fuzzy control, a parameter self-tuning fuzzy PI controller to improve the robustness of the system in a variety of uncertain environments was proposed in [11]. Reference [12] established a two-zone control model for power generation conditions and pumping conditions of a pumped storage plant, designed a fuzzy logic controller based on a nonlinear load frequency control model and used a traditional PI controller and fuzzy logic controller to simulate the nonlinear load frequency control model.

By introducing additional parameters, the fractional order theory expands the dynamic adjustment range of the system and has better control ability. Reference [13] analyzed and compared the robustness of a fractional order PID controller and traditional PID controller in a two-region load frequency control model and proved that a fractional order PID (FOPID) controller had stronger robustness in the multi-region interconnection model. The FOPID controller was used in the simulation analysis of the two-region pumped storage power station in [14] and realized the approximation of fractional-order differential operators through the inter-proximity algorithm and compared it with the traditional PID controller, and the results showed that the FOPID controller had a better control effect. A type 2 fuzzy-based fractional order PID control scheme was proposed in [15] for the damping control of a microgrid penetrated power system. A FOPID controller design method for LFC with a communication delay was presented in [16]. However, the parameters of the controller were not optimized and solved, and the control potential of the controller could not be fully realized. The PID control and FOPID control mainly set parameters according to experience. On the one hand, it was impossible to objectively compare the control differences between the two controllers in the actual situation. On the other hand, the simulation results obtained by empirical parameters were difficult to objectively verify whether the difference in the simulation results was the advantage of the controller itself or a defect caused by insufficient parameter setting, which is easy to limit the control performance of the controller.

Therefore, parameter optimization to improve the control performance of the controller has become a hot research topic. The Ant Colony Optimization (ACO) algorithm and Ziegler Nichols method were used to fine-tune the parameters of a FOPID controller in [17]. The particle swarm optimization (PSO) algorithm was used in [18] to obtain the PID and FOPID controller’s optimal values. Human learning optimization was adopted to design the controller for the water level control of steam generators [19] and temperature control in heat exchangers [20]. In [21], the adaptive particle swarm optimization (PSO) algorithm was used to tune the FOPID control parameters and applied it to typical nonlinear systems.

In the actual two-region load frequency simulation, the performance comparison of the controller, exists the problem of whether the empirical parameter setting is reasonable. In the bygone optimization examples, most of the single Integrated Time and Absolute Error (ITAE) index was a fitness function to optimize the controller parameters, resulting in excessive overshoot in the actual simulation. Therefore, it is particularly important to design the fitness function, optimize the solution and avoid the setting of empirical parameters, which is particularly important to improve the dynamic characteristics of the system regional frequency control.

To deal with the above-mentioned problems, this paper proposes a fractional order PID optimal control strategy for regional load frequency-containing pumped storage plants. Firstly, a single-region model is built based on IEEE standards. Secondly, the FOPID controller is designed, and the PSO and chaos particle swarm optimization (CPSO) algorithm are used to optimize and compare the parameters of the controller. In order to improve the control effect and avoid overshoot, the fitness function is improved based on the ITAE standard. Finally, the frequency regulation effect is verified by simulation under power generation and pumping conditions, and the experimental results verify the effectiveness of the proposed control method. This article is organized as follows: the Section 2 establishes the model, the Section 3 designs the FOPID, the Section 4 presents the solution methodology, the simulation analysis is carried out in the Section 5 and the Section 6 concludes this article.

2. Modeling of the Pumped Storage Plant Regulation System

2.1. Control Architecture

The main function of the load frequency control is to realize the normal exchange of power between systems and maintain the frequency stability of the system. Using the linear combination of regional power exchange and frequency deviation, the regional control error is used as the input signal of the controller, wherein the calculations of the regional control error ACE are shown in Equation (1). The controller sends the control signal to the units of power plants participating in the Load Frequency Control (LFC) adjustment; for the steam engine or hydraulic turbine, the active output of the prime mover is changed by changing the intake volume or the control valve of the water intake—that is, changing the ΔP_c to achieve the adjustment of the frequency [12].

$$ACE=\Delta P_{tie}+\beta \Delta f$$

(1)

where Δf is the system frequency deviation when a disturbance occurs, and ΔP_tie is the power exchange deviation of the tie lines.

2.2. Models of Units

The transfer function model of the turbine unit governor is:

$$Gov(s)=\frac{1}{T_{g}s+1}$$

(2)

where T_g is the time constant of the governor.

The transfer function model of the governor considering the speed regulation dead zone is described as Equation (3) [13]:

$$Gov(s)=\frac{N_2/{\omega }_{0}s+N_1}{T_{g}s+1}$$

(3)

where N₁ and N₂ are the linearization coefficients.

In the present system, the digital electrohydraulic speed regulation system is generally used to describe the operating conditions of the turbine governor, as shown in Equation (4); in the system containing a hydraulic turbine, Equation (4) can accurately reflect the real operation of the turbine [22].

$$G_d(s)=\frac{K_d^{\prime }{s}^2+K_p^{\prime }s+K_i^{\prime }}{K_d^{\prime }{s}^2+(K_p^{\prime }+f/R)s+K_i^{\prime }}$$

(4)

where R is the adjustment coefficient of the hydraulic turbine; $K_p^{\prime }$, $K_i^{\prime }$, and $K_d^{\prime }$ are the proportional, integral, and differential gain of the digital electrohydraulic speed control system, respectively, and f is the system frequency.

The reheat steam turbine model is expressed as Equation (5):

$$Gen(s)=\frac{1}{{\tau }_{t}s+1}\cdot \frac{1+K_{\tau }{T}_{\tau }s}{T_{\tau }s+1}$$

(5)

The hydraulic turbine model is expressed as Equation (6):

$$G_t(s)=\frac{1-T_{w}s}{1+0.5T_{w}s}$$

(6)

where K_τ is the proportion of the power produced by steam in the high-pressure cylinder section of the total power of the steam turbines, T_τ is the time constant of the reheater, τ_t is the time constant of the steam chamber and the main inlet volume, T_w is the water start time and s is the Laplace transform operator. This paper is based on the single-region model shown in Figure 1 for the simulation analysis.

3. Fractional Order PID Controller

Fractional order controllers are the development and expanding of traditional integer order controllers, which the orders of differential and integral do not limit to the integer; instead, the complex number is evenly fractioned, rendering a more extensively applicable ability.

The fractional calculus operator is usually defined as the following piecewise function:

$${}_a{D}_t^{\alpha }=\left\{\begin{array}{l}\frac{d^{\alpha }}{d{t}^{\alpha }},\alpha >0\\ 1,\alpha =0\\ {\displaystyle {\int }_a^t{(d\tau )}^{-\alpha },\alpha <0}\end{array}\right.$$

(7)

where a and t are the upper and lower limits of the differential or integral, and α is the order of calculus [23].

The FOPID controller adds two parameters λ and μ to the traditional PID controller; hereby, its transfer function is as follows:

$$C(s)=K_p+K_i{s}^{-\lambda }+K_d{s}^{\mu }$$

(8)

where K_p is the proportional coefficient, K_i is the integral coefficient, K_d is the differential function, λ is the integral order and μ is the differential order.

When λ and μ are equal to 1, it is the traditional PID controller.

The key to designing the model of the FOPID controller is the approximation of the fractional order calculus operator. The Oustaloup approximation theory can achieve better approximation accuracy.

First, the fractional calculus operator s^α is placed in the set frequency band [ω_b, ω_h], which can be described using a fractional transfer function [24], as shown in Equation (9):

$$K(s)={\left(\frac{1+\frac{s}{d{\omega }_b/b}}{1+\frac{s}{b{\omega }_h/d}}\right)}^{\alpha }$$

(9)

where 0 < α < 1, α is the fractional differential order, s = jw and b > 0, d > 0 is an adjustable parameter; through the introduction of b, d, the two parameters can improve the approximation effect at both ends of the approximate frequency band [25].

The specific expression for the Oustaloup approximation is described by Equation(10):

$$G(s)=K{\displaystyle \prod _{k=1}^N\frac{1+s/{\omega }_k^{\prime }}{1+s/{\omega }_k}}$$

(10)

where k = 1, 2, …, N.

The expressions of pole ω_k, zero ${\omega }_k^{\prime }$ and gain K are as follows:

$$\left\{\begin{array}{l}{\omega }_k={\omega }_b{\left(\frac{{\omega }_h}{{\omega }_b}\right)}^{(2k-1+\alpha )/2N}\\ {\omega }_{{}_k}^{\prime }={\omega }_b{\left(\frac{{\omega }_h}{{\omega }_b}\right)}^{(2k-1-\alpha )/2N}\\ K={\omega }_h^{\alpha }\end{array}\right.$$

(11)

The Oustaloup filter is built according to Equation (10) and Equation (11). Figure 2 shows the controller structure. The frequency band selected in this paper is [0.001, 1000], and the filter order is 5; according to the determined fractional order, α, ${\omega }_k$, ${\omega }_k^{\prime }$ and K can be determined by Equation (11). The transfer function of the Oustaloup filter is constructed by Equation (10). The fractional order differentiator or integrator is directly approximated by the integer order transfer function module [26].

4. Algorithm Design

4.1. Basic Principle of CPSO Algorithm

The essence of the particle swarm optimization algorithm is to update the velocity of the next iteration based on the individual experience and social influence of particles. The velocity update and position update of the particles are shown in Equation (12).

$$\left\{\begin{array}{ll}{v}_i(t+1)\hfill & =\omega v_i(t)+c_{1}rand(x_{pbest}-x_i(t))\hfill \\ & +c_{2}rand(x_{gbest}-x_i(t))\hfill \\ x_i(t+1)\hfill & =x_i(t)+v_i(t+1)\hfill \end{array}\right.$$

(12)

where ω is the inertia factor, c₁ and c₂ are the acceleration factors, rand is the random number between [0, 1], x_pbest is the individual history optimal value and x_gbest is the global optimal value [27].

Using the ergodicity of chaotic variables can effectively prevent falling into the local optimal solution and solve the problem of premature convergence caused by particle stagnation. Mapping x_gbest to the domain of the Logistic Equation by Equation (13):

$$y_i^k=\frac{x_{gbest}^k-R_{i\_\min }}{R_{i\_\max }-R_{i\_\min }}$$

(13)

$$y_i^k(j+1)=\mu y_i^k(j)(1-y_i^k(j))$$

(14)

$$x_i^k(j)=R_{\min }^k+(R_{\max }^k-R_{\min }^k)y_i^k(j)$$

(15)

The algorithm steps of CPSO are as follows:

Step 1: Initialize the particle swarm and set the particle swarm size, maximum number of iterations and related parameters;

Step 2: Assign values to K_p, K_i and K_d or K_p, K_i, K_d, μ and λ according to the particle position;

Step 3: Run the two-region load control model and return the fitness value f_gbest;

Step 4: Chaotically optimize the group optimal value x_gbest of the particle swarm:

(1)

By Equation (13), x_gbest is mapped to the domain of the logistic equation [0, 1];

(2)

Use Equation (14) to iterate $y_i^k$ for M times and gain chaotic sequences $y_i^k(1)$, $y_i^k(2)$, …, $y_i^k(M)$;

(3)

According to Equation (15), the chaotic sequence is mapped back to the original solution space and attains the feasible solution sequences $x_i^k(1)$, $x_i^k(2)$, …, $x_i^k(M)$;

(4)

The optimal solution $f_{gbest}^{\prime }$ after chaos optimization is obtained by running the two-region load control model, where $f_{gbest}^{\prime }=\min f\left(x_i^k(j)\right)$;

(5)

Randomly substitute the optimal particle swarm in the chaotic sequence for any particle swarm in the current particle swarm;

(6)

Update the fitness value. If $f_{gbest}^{\prime }<f_{gbest}$, then $f_{gbest}=f_{gbest}^{\prime }$, $x_i(t)=x_i^k(j)$;

Step 5: Update the particle velocity and position based on the individual best fitness and global best fitness, return to Step 2 and proceed with the optimization [28,29];

Step 6: Output K_p, K_i and K_d or K_p, K_i, K_d, μ and λ.

4.2. Design of Fitness Function

The fitness function determines the goal and direction of the parameter optimization; hence, the selection of the fitness function is very important when optimizing and setting the parameters of the controller. The PID controller generally judges whether the parameter setting is appropriate through the steady-state indicators such as maximum overshoot, rise time and adjustment time based on experience. In the process of optimization, it is necessary to establish a comprehensive performance index to accurately describe the control effect of the controller.

Usually, based on the instantaneous error e(t) of the system, the control effect of the controller is described by the following performance evaluation index [30].

Time multiplied by the absolute error integral index:

$$J_{ITAE}={\displaystyle {\int }_0^{\infty }t\left|e(t)\right|dt}$$

(16)

Since the ITAE index has better practicability and selectivity in practical problems [31], this paper selects the regional control deviation signal of region 1 in the interconnected power grid to establish the fitness function of the system, as shown in Equation (17).

$$f_{fitness}={\displaystyle {\int }_0^{40}t\left|AC{E}_1(t)\right|dt}$$

(17)

Since Equation (17) contains only two variables, the region control error and the time, it is difficult to determine whether the error in the system is a positive error or a negative error, and at the same time, it will sacrifice the overshoot and reduce the rising time of the system. In addition, Equation (17) does not directly reflect the overshoot, rise time and adjustment time index of the system.

Therefore, the fitness function in Equation (17) is modified by adding a penalty link to avoid excessive overshoot [32]. The modified fitness function is shown as Equation (18).

$$f_{fitness}^{\prime }(K_p,K_i,K_d,\lambda ,\mu )={\displaystyle {\int }_0^{40}(t\left|AC{E}_1(t)\right|+{\xi }_1\left|AC{E}_1(t)\right|)}dt+{\xi }_2{t}_r$$

(18)

where t_r is the rise time, and ξ₁ and ξ₂ are the weight coefficients.

The penalty function is used to realize dynamic adjustment; when ξ₁≫ξ₂ is in Equation (18), the rise time of the system is sacrificed to reduce or eliminate the overshoot of the system; when ξ₁≪ξ₂ is in Equation (18), the overshoot of the system increases, but the rise time of the system decreases.

5. Simulation Analysis

The proposed method is simulated and verified in the two-region frequency control model based on IEEE standards, and the PID controller and FOPID controller are utilized to test the dynamic performance of the system. The simulation model is established in MATLAB/Simulink 2018b, and the controller parameters are optimized via using the PSO algorithm and the CPSO algorithm.

Table 1. Parameters of PSO and CPSO.

Parameter	c1	c2	ω	Size	Maximum Iterations
Value	2	2	0.6	40	40

shows the basic parameters of PSO and CPSO. The value range of the PID controller parameters K_p, K_i and K_d are [0, 200], [0, 100] and [0, 100], respectively. The value range of the FOPID controller parameters K_p, K_i and K_d are [0, 200], [0, 100] and [0, 100], respectively. The range of the integral order μ and differential order λ is [0, 2] [33]. The main parameters of the energy transfer and power regulation of the unit are given in

Table 2. Basic parameters of the interconnected grid model.

Parameter	Value	Parameter	Value	Parameter	Value
Tij	0.545 p.u.	τt	10 s	Tri	20 s
αij	−1 p.u.	Kτ	5 s	$K_d^{\prime }$	4
βi	0.425 p.u. MW/Hz	TW	1 s	$K_p^{\prime }$	1
Ri	2.4 Hz/p.u.	Tg	0.08 s	$K_i^{\prime }$	5
Tτ	0.3 s	Kpi	120 s	PL_nom	1000 MW

.

5.1. Comparative Analysis of Optimization Algorithms

According to the two-region LFC model established in Figure 1, a step disturbance signal of 0.01 p.u. is added to region 1, and the controller parameters are optimized, with $f_{fitness}^{\prime }$ as the fitness value. Since the PSO algorithm is a population-based stochastic optimization algorithm, it is extremely important to analyze its convergence through multiple experiments.

Table 3. Comparison of the parameters’ optimization results of two controllers by using PSO and CPSO.

Algorithm	Controller	${\mathit{f}}_{\mathit{f}\mathit{i}\mathit{t}\mathit{n}\mathit{e}\mathit{s}\mathit{s}}^{\prime } \mathbf{Initial} \mathbf{Fitness} \mathbf{Value}$			${\mathit{f}}_{\mathit{f}\mathit{i}\mathit{t}\mathit{n}\mathit{e}\mathit{s}\mathit{s}}^{\prime } \mathbf{Final} \mathbf{Fitness} \mathbf{Value}$
		MIN	MAX	AVG	MIN	MAX	AVG
PSO	PID	4.118	5.561	4.625	4.045	4.062	4.058
	FOPID	2.122	5.004	3.237	1.447	1.906	1.488
CPSO	PID	4.118	5.561	4.618	4.036	4.062	4.058
	FOPID	1.885	3.609	2.655	1.433	1.499	1.463

shows the optimization results of different controller parameters after 30 online optimizations.

According to Table 3, since the PID controller only needs to optimize three parameters, the optimization results of the two algorithms are basically the same, and the minimum fitness value optimized by the CPSO algorithm is slightly smaller. When optimizing the five parameters of the FOPID controller, the CPSO algorithm shows certain advantages. Among the 30 online optimization results, the average value of the initial value of the fitness function is 2.655 and the minimum fitness value is 1.433 when using the CPSO algorithm to optimize, while the average value of the initial value of the fitness function is 3.237 and the minimum fitness value is 1.447 when using the PSO algorithm to optimize, which demonstrates that the CPSO algorithm has better solution accuracy and a stronger search ability. In order to better compare the quality of the optimal solution, Figure 3 shows the iteration process to converge. When the PSO algorithm is used, it converges around the 15th generation, while the CPSO algorithm converges around the 9th generation. It can therefore be observed that the introduction of chaos theory improves the global search ability of the PSO algorithm and effectively avoids the premature problem caused by particle stagnation.

5.2. Comparison of the Controller’s Performance

Table 4. Optimal parameters of the controller.

Algorithm	Controller	Parameters
		Kp	Ki	Kd	μ	λ
PSO	PID	67.8069	8.8716	14.4256	-	-
	FOPID	49.0901	14.3847	13.5162	1.7108	0.9092
CPSO	PID	49.5711	6.5176	10.5171	-	-
	FOPID	198.98	50.1701	25.2283	1.6806	0.9156

shows the optimal parameters of the controller. To compare the controller’s performance, based on the parameters in Table 4, the grid frequency fluctuation under the step disturbance of 0.01 p.u. is shown in Figure 4. What can be observed is that the two control algorithms can find the optimal solution with higher quality after 30 times of optimization.

According to Figure 4, for the PID controller, the parameters obtained by the two algorithms are adopted, and the control effect is not significantly different. For the FOPID controller, compared with the results optimized by the PSO algorithm, the results optimized by the CPSO algorithm have smaller frequency difference, and the curve is more stable.

The fitness value in Table 2 shows that the average fitness value of the FOPID controller is 1.463 with the parameters optimized by the CPSO algorithm, while the average fitness value of the PID controller is 4.058. Comparing the frequency variation curves of different controllers, for the PID controller, the maximum frequency difference of the system is −0.0794 p.u. and the adjustment time is 15 s; the maximum frequency difference of the FOPID controller is −0.06, and the adjustment time is about 20 s. The maximum frequency difference of the system adapting the FOPID controller is decreased by 32.33%, and the adjustment time is reduced by about 33.33%, which indicates that the regional frequency difference quickly returned to zero and remained stable, with a better dynamic performance. From the perspective of the performance indicators, the FOPID controller has a better control performance.

Figure 5 shows the tie line power exchange, the power fluctuation range of the FOPID controller is significantly shortened and the curve is fast and stable. From the perspectives of frequency fluctuation and tie line power fluctuation, the FOPID controller has a better dynamic response ability when confronted with a step disturbance.

In order to further verify the control performance of the FOPID controller in the actual scene, according to the method proposed in [34], the random wind power sequence is generated, and the wind farm power fluctuation per unit value (reference power is 10 MVA) curve is shown in Figure 6. The random fluctuation of the wind power, as shown in Figure 6, is introduced to region 1. The frequency deviation curve of region 1 is shown in Figure 7. It can be observed that the traditional PID controller is seriously affected by the random disturbance of the wind power. The frequency deviation of the region fluctuates deeply, with a large deviation value of 0.00142 Hz; the system oscillates near the stable point, and stability is difficult to maintain. On the contrary, when the FOPID controller is adopted, the frequency deviation of the system is obviously improved, and the maximum frequency difference is 0.0092 Hz. The curve is more stable when it is close to the steady point. The system can effectively track the random power disturbance of the wind power and has a stronger anti-disturbance ability and dynamic performance.

5.3. Pumped Storage Power Station Participation

In order to verify the effectiveness of the proposed control strategy, the pumped storage unit is added to the two-region model. The step signal of 0.01 p.u. is added at 5 s, and the frequency variation curve of the system is shown in Figure 8. The figure indicates that the addition of pumped storage units effectively decreases the frequency difference and the adjustment time, and the maximum frequency difference of the system is decreased to about 0.056 p.u. With the addition of the pumped storage unit, the adjustment time is shortened by about 2 s, which makes the system recover quickly and remain stable, and the reliability of the FOPID controller that, when the system structure changes, the system can still maintain a good dynamic performance without changing the parameters of the FOPID controller is further verified. The frequency variation curve after adding a pumped storage unit in region 1 under a random wind disturbance is shown in Figure 9. It can be found that, compared with the operation mode without pumped storage units, after adding the pumped storage unit, the maximum frequency deviation of the system is 0.00755 Hz and 0.00749 Hz, which is reduced by 21.85% and 22.83%, respectively.

6. Conclusions

This paper studies the controller design and controller parameter configurations in a regional power grid frequency control and realizes the in-depth exploration of the potential of traditional unit frequency regulation. The main conclusions are summarized in the following three sections:

(1)

Based on the theory of fractional calculus, a FOPID controller is designed. The performance of the controller in the region model is compared and analyzed under different disturbances. The results show that the FOPID controller has better dynamic performance and can fully exert the frequency regulation potential of traditional units.

(2)

The CPSO algorithm is introduced to optimize the controller parameters, and the fitness function containing multiple performance indices is used as the optimization objective, which fully exerts the control performance of the controller and enhances the robustness of the system.

(3)

The simulation results under the generator operation mode and pumping operation mode indicates that the FOPID controller optimized by CPSO can effectively stabilize the grid frequency fluctuation caused by different disturbances, which has a certain guiding significance for the study of the grid frequency control.

Future research interests will further focus on designing the FOPID controller, which parameters are optimized by other commonly used algorithms in the control system, such as human learning optimization.