Control Strategy of PMSM for Variable Pitch Based on Improved Whale Optimization Algorithm

Abstract

A PI control approach grounded in an optimized improved whale algorithm is devised to tackle the characteristics of multivariable, nonlinear, strong coupling, and uncertain and fluctuating wind speeds in electric variable pitch systems. In the improved whale algorithm optimization algorithm, the reverse learning mechanism is utilized within the population initialization stage, the nonlinear inertial weight coefficient is introduced in the global and local search processes of whale predation, and the convergence factor is updated by the exponential function, which effectively addresses the issue of sluggish convergence speed and low convergence efficiency of the whale optimization algorithm. The position control of the electric variable pitch system is implemented with the application of the improved whale optimization algorithm. According to the performance index of the position ring, the appropriate objective function is established, and the adaptive control of the position ring is realized through the adaptive adjustment of PI parameters. The simulation outcomes demonstrate that the PI control, which is founded on an improved whale optimization algorithm, is superior to the PI control based on the whale optimization algorithm in dynamic and steady performance. When the load torque changes, using PI control based on the improved whale optimization algorithm, the pitch angle reaches the steady-state value in 0.06 s without overshoot, while using PI control based on the whale optimization algorithm, the pitch angle reaches the steady-state value in 0.09 s with a maximum overshoot of 2.4°. When the load torque is constant, PI control based on the improved whale optimization algorithm can achieve pitch angle tracking in 0.16 s, while PI control based on the whale optimization algorithm can achieve pitch angle tracking in 0.48 s.

1. Introduction

Actively developing non-petrochemical energy, enhancing the share of consumption of green energies such as wind and solar power, and establishing a new power system with new energy at its core are the chief means for the power system to attain the “carbon peak and carbon neutrality” target [1,2,3]. In this article, wind power has been vigorously developed, with wind power installed capacity growing rapidly every year and single unit capacity continuously increasing. Variable pitch wind turbines have become the mainstream of large-scale wind turbines. Pitch control supports grid stability indirectly by providing a stable power base and necessary reserve capacity through accurate power regulation, which enables the converter to perform its grid-supporting functions effectively. Therefore, it is necessary to conduct a detailed comparative analysis and research on the pitch control system of megawatt level wind turbines. The common pitch control methods are mainly electric pitch control and hydraulic pitch control. Among these, electric pitch control has been extensively utilized due to its compact and reliable transmission mechanism, without problems such as leakage and jamming. The selected schemes for pitch control motors include AC induction motors, brushless DC motors, PMSM, etc. However, considering the advantages of PMSM over other types of motors and the current development status of high-performance rare earth permanent magnet materials, and considering that the optimization of the magnetic pole shape of PMSM is more convenient, the waveform of air gap magnetic density and the waveform of back electromotive force can be closer to sine, and the variable pitch control system’s efficiency and stability can be enhanced. Thus, many wind turbines use PMSM as the actuator of the pitch control system.

In pitch control systems, PI controllers are widely used, but they have inherent limitations such as slow response speed, insufficient robustness, and difficulty in adjusting parameters online. Due to the strong coupling and nonlinearity of the pitch system itself, improper control parameters can significantly reduce the performance of the PI controller, leading to fluctuations in the output power of wind turbines and adverse effects on the stable operation of the power grid. The intelligent coordinated optimization control algorithm has become an effective way to improve pitch control due to its good adaptability, high robustness, and efficient solving ability [4,5,6].

In related research, various intelligent algorithms have been applied to optimize pitch control. Reference [7] proposes a PID pitch controller based on a fuzzy RBF neural network, which can dynamically adjust the pitch angle and change the aerodynamic torque when the actual wind speed deviates from the rated value, thereby stabilizing the output power near the rated value. Simulation results show that this control strategy has faster response speed, smaller wind energy utilization coefficient overshoot, and more stable power output compared to traditional PID and conventional RBF neural network PID control. Reference [8] adopted an improved grey wolf optimization algorithm to achieve autonomous tuning of the parameters of the self-disturbance rejection controller, which improved the response speed and disturbance rejection capability of the pitch control system. Reference [9] applied the artificial bee colony algorithm to PID parameter self-tuning, and the results showed that the controller can effectively suppress output power and wind turbine speed fluctuations, improve constant power control accuracy, and reduce transmission chain fatigue load and unit structural response, thereby extending unit life. Reference [10] proposed an independent variable pitch PID parameter tuning method based on the whale swarm algorithm, and its effectiveness and practicality were verified by comparing it with the particle swarm algorithm and the genetic algorithm.

In addition, some studies have further improved control performance through hybrid strategies. Reference [11] constructs a novel optimal control law by introducing inverse hyperbolic sine functions, trigonometric functions, and quadratic functions, and improves the whale optimization algorithm (WOA) based on the refraction principle to optimize the self-disturbance rejection control parameters, effectively reducing speed and torque oscillations. Reference [12] proposes a chaotic whale swordfish composite algorithm, which uses the whale algorithm for local search and the swordfish algorithm for global search and combines segmented chaotic mapping to initialize the population, significantly enhancing the system’s anti-interference ability and convergence speed.

The whale optimization algorithm, put forward by Mirjalili and Lewis in 2016, is a novel swarm intelligence optimization algorithm. It mimics the hunting behavior and spiral motion of humpback whales as described in [13]. It features simple implementation, a small number of adjustable parameters, and a powerful capacity to escape from local optima. When dealing with complex nonlinear problems, it exhibits superior optimization speed and convergence performance in comparison with other swarm intelligence optimization algorithms. The advantage of metaheuristic algorithms such as the whale optimization algorithm (WOA) is that they can independently search for robust and adaptable controller parameters by directly optimizing system performance indicators without relying on precise system models. This makes it particularly suitable for dealing with such uncertain systems. This algorithm has rarely been applied in wind turbine pitch control research, so this paper chooses to study its application in wind turbine pitch control, which has certain reference significance.

Pitch position control constitutes a significant aspect of electric pitch control systems. Improving the system’s ability to resist load disturbances and suppress position fluctuations are the performance requirements for electric pitch system position control. The PI control optimized based on the improved whale optimization algorithm will be applied in the position loop control of the pitch servo system. By adaptively adjusting the PI parameters, the position loop can be adaptively controlled to enhance the control performance of the pitch system in both dynamic and steady-state aspects.

2. Working Principle and Mathematical Model of Variable Pitch System

In the electric variable pitch control system, the PMSM drives a small gear to rotate, which meshes with the reduction gearbox. The output shaft of the reduction gearbox is connected to the blade through a rotating support device. By changing the speed and position of the PMSM, the pitch angle of the fan blade can be changed; thus, the torque of the blade ultimately affects the PMSM, as shown in Figure 1.

2.1. Mathematical Model of PMSM

The PMSM represents a nonlinear system, characterized by multiple variables and strong coupling, as noted in [14]. In accordance with the established convention, the positive direction of each physical quantity is defined. Additionally, the following assumptions are posited for the mathematical model of the PMSM: firstly, the rotor damping winding is disregarded. Secondly, during motor operation, the saturation of the magnetic circuit is overlooked, and the magnetic circuit is presumed to be linear with unaltered inductance parameters. Thirdly, the eddy current and hysteresis losses within the motor core are neglected. Fourthly, it is assumed that the induced electromotive force generated by the three phase stator armature winding of the motor exhibits a sinusoidal waveform, without higher harmonics. Under these ideal conditions:

2.1.1. The Voltage Equation in a Two-Phase Rotating Coordinate System

$$\left\{\begin{array}{l}{u}_{\mathrm{d}}=R_{\mathrm{s}}{i}_{\mathrm{d}}+{\displaystyle \frac{d{\psi }_{\mathrm{d}}}{dt}}-{\omega }_{\mathrm{e}}{\psi }_{\mathrm{q}}\\ u_{\mathrm{q}}=R_{\mathrm{s}}{i}_{\mathrm{q}}+{\displaystyle \frac{d{\psi }_{\mathrm{q}}}{dt}}+{\omega }_{\mathrm{e}}{\psi }_{\mathrm{d}}\end{array}\right.$$

(1)

Among them, $u_{\mathrm{d}}$ and $u_{\mathrm{q}}$ are the dq axis voltages; $i_{\mathrm{d}}$ and $i_{\mathrm{q}}$ are the dq axis currents; ${\psi }_{\mathrm{d}}$ and ${\psi }_{\mathrm{q}}$ are the total magnetic flux of the dq axis; $R_{\mathrm{s}}$ is the armature resistance; ${\omega }_{\mathrm{e}}$ is the electrical angular velocity.

2.1.2. Magnetic Flux Equation in Two-Phase Rotating Coordinate System

$$\left\{\begin{array}{l}{\psi }_{\mathrm{d}}=L_{\mathrm{d}}{i}_{\mathrm{d}}+{\psi }_{\mathrm{f}}\\ {\psi }_{\mathrm{q}}=L_{\mathrm{q}}{i}_{\mathrm{q}}\end{array}\right.$$

(2)

Among them, $L_{\mathrm{d}}$ and $L_{\mathrm{q}}$ are dq axis inductors; ${\psi }_{\mathrm{f}}$ is the magnetic flux of the permanent magnet.

2.1.3. Electromagnetic Torque Equation

$$T_{\mathrm{e}}={\displaystyle \frac{3}{2}}p({\psi }_f{i}_{\mathrm{q}}+(L_{\mathrm{d}}-L_{\mathrm{q}})i_{\mathrm{d}}{i}_{\mathrm{q}})$$

(3)

$$T_{\mathrm{e}}={\displaystyle \frac{30P_{\mathrm{e}}}{\mathsf{\pi }{n}_{\mathrm{m}}}}$$

(4)

Among them, $p$ is the number of pole pairs; $n_{\mathrm{m}}$ is the motor speed; $P_{\mathrm{e}}$ is the electromagnetic power of the PMSM.

2.1.4. Mechanical Motion Equation

$$T_{\mathrm{e}}-T_{\mathrm{L}}-B{\omega }_{\mathrm{m}}=J{\displaystyle \frac{d{\omega }_{\mathrm{m}}}{dt}}$$

(5)

Among them, ${\omega }_{\mathrm{m}}$ is the mechanical angular speed of the motor; $T_{\mathrm{L}}$ is the motor load torque; $J$ is the motor moment of inertia; and $B$ is the motor damping coefficient.

2.2. Mathematical Model of Transmission System

According to the connection relationship, the connection among speed, power, and torque can be deduced as (6).

$$\left\{\begin{array}{l}{n}_1=i_1{n}_{\mathrm{m}}\\ P_1=P_{\mathrm{e}}{\eta }_1\\ T_1=T_{\mathrm{e}}{\eta }_1/i_1\end{array}\right.$$

(6)

Among them, $i_1$ is the speed increase ratio; ${\eta }_1$ is the transmission efficiency of the main transmission gearbox, with ${\eta }_1=0.975$; $n_1$ is the output speed; $P_1$ represents output power; $T_1$ is the output torque.

2.3. Electric Pitch Control System

The electric pitch control system in the article uses a permanent magnet synchronous motor as the actuator motor. A control scheme for an electric pitch servo system with a position loop, speed loop, and current loop—three closed loop controls—was designed using $i_{\mathrm{d}}=0$ vector control method and space vector pulse width modulation technology. The control framework of the pitch servo system is depicted in Figure 2. ACR in the figure represents the automatic current loop regulator, ASR represents the automatic speed loop regulator, and APR represents the automatic position loop regulator.

After a series of simplification procedures, the control object within the current loop can be considered as a dual inertia link that possesses two time constants, and the sizes of these two time constants differ greatly. Adopting a PI-type current controller, its transfer function is as shown in (7).

$$W_{\mathrm{ACR}}={\displaystyle \frac{K_{\mathrm{i}}({\tau }_{\mathrm{I}}s+1)}{{\tau }_{\mathrm{I}}s}}$$

(7)

Among them, $K_{\mathrm{i}}$ represents the proportional coefficient within the current loop, while ${\tau }_{\mathrm{I}}$ stands for the time constant of the current loop.

The open-loop transmission function of the current loop is shown in (8).

$$W_{\mathrm{opi}}={\displaystyle \frac{K_{\mathrm{i}}({\tau }_{\mathrm{I}}s+1)}{{\tau }_{\mathrm{I}}s}}{\displaystyle \frac{{\alpha }_{\mathrm{i}}{K}_{\mathrm{S}}/R}{(T_{1}s+1)(T_{2}s+1)}}$$

(8)

The current loop mainly considers its tracking performance and can be verified as a typical Type I system. $T_2$ is much smaller than $T_1$. Choose ${\tau }_{\mathrm{I}}=T_1$ and correct the current loop to a typical Type I system. Formula (8) can be rewritten as (9).

$$W_{\mathrm{opi}}={\displaystyle \frac{K_{\mathrm{I}}}{s(T_{2}s+1)}}$$

(9)

Among them, $K_{\mathrm{I}}=K_{\mathrm{i}}{\alpha }_{\mathrm{i}}{K}_{\mathrm{S}}/(R{\tau }_{\mathrm{I}})$ is the open-loop gain of the current loop. Design $K_{\mathrm{I}}$ based on $K_{\mathrm{I}}{T}_2=0.5$ with a damping ratio of 0.707 in the project.

The speed loop adopts a PI-type speed controller, and its transfer function is shown in (10).

$$W_{\mathrm{ASR}}={\displaystyle \frac{K_{\mathrm{s}}({\tau }_{\mathrm{s}}s+1)}{{\tau }_{\mathrm{s}}s}}$$

(10)

Among them, $K_{\mathrm{s}}$ is the proportional coefficient of the velocity loop, and ${\tau }_{\mathrm{s}}$ is the time constant of the velocity loop.

The open-loop transmission function of the speed loop is shown in (11).

$$W_{\mathrm{ops}}={\displaystyle \frac{K_{\mathrm{s}}({\tau }_{\mathrm{s}}s+1)}{{\tau }_{\mathrm{s}}s}}{\displaystyle \frac{{\alpha }_{\mathrm{s}}/{\alpha }_{\mathrm{i}}}{T_{3}s+1}}{\displaystyle \frac{1}{T_{4}s}}={\displaystyle \frac{K_{\mathrm{s}}{\alpha }_{\mathrm{s}}({\tau }_{\mathrm{s}}s+1)}{{\tau }_{\mathrm{s}}{\alpha }_{\mathrm{i}}{T}_4{s}^2(T_{3}s+1)}}$$

(11)

The speed loop mainly considers its anti-interference performance and can be verified as a typical II system. $T_4$ is much smaller than $T_3$. Choose ${\tau }_{\mathrm{s}}=T_3$ and correct the speed loop to a typical Type II system. Formula (11) can be rewritten as (12).

$$W_{\mathrm{ops}}={\displaystyle \frac{K_{\mathrm{S}}({\tau }_{\mathrm{s}}s+1)}{s^2(T_{3}s+1)}}$$

(12)

Among them, $K_{\mathrm{S}}=K_{\mathrm{s}}{\alpha }_{\mathrm{s}}/({\tau }_{\mathrm{s}}{\alpha }_{\mathrm{i}}{T}_4)$ is the open-loop gain of the velocity loop. In engineering, calculate ${\tau }_{\mathrm{s}}$ by taking the intermediate frequency width $h=5$, where ${\tau }_{\mathrm{s}}=h{T}_3$, and then calculate $K_s$ based on $K_{\mathrm{s}}=(h+1)/(2h^2{T}_3^2)$.

3. Improve Whale Optimization Algorithm

3.1. Whale Optimization Algorithm

The whale optimization algorithm represents a novel group intelligence optimization algorithm that mimics the hunting behavior of whales in nature. In addition, the hunting process of whales includes surrounding prey, bubble net attacks, and searching for prey [15]. The specific modeling process is as follows.

3.1.1. Surrounding Prey

Assuming that in the d-dimensional space, the current position of the optimal whale individual $X^{*}$ is $(X_1^{*},X_2^{*},\dots ,X_d^{*})$, and the position of whale individual $X^j$ is $(X_1^j,X_2^j,\dots ,X_d^j)$. The calculation formula for the next position $(X_1^{j+1},X_2^{j+1},\dots ,X_d^{j+1})$ of whale individual $X^j$ under the influence of the optimal whale individual is as follows:

$$\begin{array}{l}{X}_k^{j+1}=X_k^{*}-A_1•D_k\\ D_k=\left|C_1•X_k^{*}-X_k^j\right|\\ C_1=2r_2\\ A_1=2a{r}_1-a\end{array}$$

(13)

Among them, $X_k^{j+1}$ represents the k-th component of the spatial coordinate $X^{j+1}$. The $\mid \mid$ in the $D_k$ calculation formula represents absolute value. $A_1$ and $C_1$ are matrix coefficients. $a$ serves as the convergence factor, and it decreases in a linear manner from 2 to 0 along with the growth of the number of iterations; it is linearly related to its iteration coefficient $t$. The relevant formula is calculated as $a=2-2t/t_{\max }$, where $t_{\max }$ represents the maximum quantity of iterations throughout the entire predation process. Both $r_1$ and $r_2$ are random numbers that lie in the interval from 0 to 1.

3.1.2. Bubble Network Attack (Local Search)

Bubble net attack represents a distinctive exhalation bubble feeding behavior exhibited by humpback whales; two mathematical models have been devised to represent the feeding behavior. Suppose that within the d-dimensional space, the position of the currently optimal whale individual is $(X_1^{*},X_2^{*},\dots ,X_d^{*})$ and the position of the whale individual is $(X_1^j,X_2^j,\dots ,X_d^j)$.

1. Shrink-wrapping. This hunting behavior is highly similar to the mathematical model of the surrounding prey behavior described earlier, with the difference being the range of values for $A_1$. As the significance of shrink-wrapping is to make the whale individual at the present position nearer to the whale individual at the current optimal position, the value range of $A_1$ is adjusted from $\left[-a,a\right]$ to $\left[-1,1\right]$, while other formulas stay unaltered. When the value of $A_1$ falls within the range of $\left[-1,1\right]$, the subsequent position of the whale is capable of being any position lying between its current position and the position of its prey. When $A_1<1$, the whale attacks its prey.

2. Spiral position update. The mathematical model of the hunting behavior, during which the current whale individual approaches the present optimal whale individual in a spiral pattern, is as follows:

$$\begin{array}{l}{X}_k^{j+1}=X_k^{*}+D_p•e^{bl}•\cos (2\pi l)\\ D_p=\left|X_k^{*}-X_k^j\right|\end{array}$$

(14)

Among them, $b$ is the logarithmic spiral shape constant, and $l$ is a random number between −1 and 1.

During the process of preying, humpback whales will not only shrink their enclosing formation but also approach their prey in a spiral manner. Consequently, each whale has a 50% likelihood of opting to either shrink the encirclement or advance toward the prey in a spiral way. The corresponding mathematical model is presented as follows:

$$X_k^{j+1}=\left\{\begin{array}{l}{X}_k^{*}-A_1•D_k p<0.5\\ X_k^{*}+D_p•e^{bl}•\cos (2\pi l) p\ge 0.5\end{array}\right.$$

(15)

3.1.3. Search for Predation (Global Search)

In the mathematical model of the shrinking surrounding predation behavior, the value of $A_1$ is limited to $\left[-1,1\right]$. However, when the value of $A_1$ is not in $\left[-1,1\right]$, the current whale specimen may not move towards the top performing whale specimen. Instead, it randomly picks whales from the existing whale population to approach. This is the principle of search predation. The pursuit of prey might cause the current individual whale to deviate from its goal, but it will improve the overall search proficiency of the whale group. Assume that within a d-dimensional space, the location of a randomly chosen whale individual $X^{rand}$ among the current top performing whale population is $(X_1^{rand},X_2^{rand},\dots ,X_d^{rand})$, and the position of a whale individual $X^j$ is $(X_1^j,X_2^j,\dots ,X_d^j)$. The mathematical model for searching for predation behavior is as follows:

$$\begin{array}{l}{X}_k^{j+1}=X_k^{rand}-A_1•D_k\\ D_k=\left|C_1•X_k^{rand}-X_k^j\right|\\ C_1=2r_2\\ A_1=2a{r}_1-a\end{array}$$

(16)

The decision for a whale to move towards the optimal individual or towards a random individual is determined by the value of $A_1$. When $\left|A_1<1\right|$, the whale makes a choice to approach the optimal individual. When $\left|A_1\ge 1\right|$, the whale opts to advance towards a randomly selected individual. Evidently, during the prey encirclement process, the search approach of the whale algorithm involves searching in the vicinity of the optimal individual or at a distance from the random individual.

3.2. Problems of Whale Optimization Algorithm

The whale optimization algorithm has a high dependence on the initial population. It uses a random method to generate the initial whale population, which has a high degree of randomness and leads to a significant difference in the superiority of the initial whale population, directly affecting the convergence speed and causing a large deviation between the convergence speeds. Therefore, improvements to the initial group identification process need to be considered.

The whale optimization algorithm has fixed inertia weight coefficients during the global and local search processes of predation, which exert a certain influence on the convergence speed and search efficiency.

In the standard whale algorithm, coefficient A constitutes a critical factor that governs the balance between the global exploration capacity and the local exploration capacity. When coefficient $\left|A\right|\ge 1$, the population enlarges the search scope in order to discover more favorable candidate solutions, and the algorithm performs global exploration; when the coefficient $\left|A\right|<1$, the population will reduce the search scope and conduct a detailed search within the local area, that is, the algorithm performs local exploration. However, from Equation (16), the value of coefficient A is set by the linearly decreasing convergence factor α, and this is unable to precisely depict the complex nonlinear search process.

3.3. Improved Whale Optimization Algorithm

Regarding the issues present in the aforementioned whale optimization algorithm, the corresponding improvement methods are proposed as follows:

3.3.1. Employing the Reverse Learning Mechanism in the Initialization Stage of the Population

Employing the reverse learning mechanism in the initialization stage of the population, since it takes into account both the population P and the reverse population P* concurrently. Therefore, compared to simple random initialization, it has a higher probability of reaching the optimal objective of the problem [14]. The initialization process of reverse learning is as follows:

Randomly initialize individuals of population P:

$$x_{ij}(0)=rnd(0,1)(U_j-L_j)+L_{j}i=1,2\dots N,j=1,2\dots D$$

(17)

Among them, $N$ is the population size; $D$ is the dimension of the solution space; $x_{ij}(0)$ is the j-dimensional variable value of the i-th individual of population P at the beginning; $U_j$ and $L_j$ symbolizes the upper and lower bounds of the j-dimensional variable within the solution space.

Compute the individuals within the reverse population P* $o{x}_{ij}(0)=L_j+U_j-x_{ij}(0)$.

Pick N top performing individuals from population P and reverse population P* to form the initial population.

3.3.2. Introduction of Nonlinear Inertial Weight Coefficient

The inertia weight coefficient has a certain impact on the convergence speed and search efficiency during the global and local search processes of hunting. To make the algorithm more accurate and efficient, in the early stage of the whale optimization algorithm, a smaller inertia weight coefficient is required in local search and a larger inertia weight coefficient is required in global search to improve its global search ability and obtain more solutions. In the later stage, a smaller inertia weight coefficient is required in global search, and a larger inertia weight coefficient is needed in local search to improve convergence speed. On the basis of the above principles, two nonlinear inertia weight coefficients ${\lambda }_1$ and ${\lambda }_2$ are introduced [16,17,18].

The relevant formula is as follows:

$${\lambda }_1={\lambda }_{\max }\times {\mathrm{e}}^{{\displaystyle \frac{-3t({\lambda }_{\max }-{\lambda }_{\min })}{T_{\max }}}}$$

(18)

$${\lambda }_2=1-{\lambda }_{\max }\times {\mathrm{e}}^{{\displaystyle \frac{-3t({\lambda }_{\max }-{\lambda }_{\min })}{{\mathrm{T}}_{\max }}}}$$

(19)

Among them, $T_{\max }$ is the maximum number of iterations; $t$ is the current generation; ${\lambda }_{\max }$ and ${\lambda }_{\min }$ are the maximum and minimum values of the inertia weight coefficients.

Due to ${\lambda }_1$ decreasing at a slower rate in the early iterations and more rapidly in later iterations, adding it to the step size of the iteration at any position of the optimized whale can improve the global search efficiency. The optimized formula is given by (20). The value ${\lambda }_2$ decreases slowly in the early stage of iteration and faster in the later stage; it is used to change the step size in the local search process, which has the ability to heighten the efficiency of local search. The optimized formula is demonstrated in (21).

$$X_k^{j+1}=X_k^{rand}-{\lambda }_1{A}_1•D_k$$

(20)

$$X_k^{j+1}=X_k^{*}-{\lambda }_2{A}_1•D_k$$

(21)

3.3.3. Nonlinear Convergence Factor

Within this article, an exponential function is incorporated to refresh the convergence factor. The convergence factor is initially set as the upper and lower limits within which it varies during the optimization process. Moreover, with the utilization of the information on evolutionary algebra and population, the convergence factor is adaptively modified, thereby enhancing the capacity to seek the optimal solution [19,20].

The relevant formula is as follows:

$$a=a_{\max }\times {\mathrm{e}}^{{\displaystyle \frac{-3t(a_{\max }-a_{\min })}{T_{\max }}}}$$

(22)

Compared to the linearly decreasing convergence factor α, the exponential function has excellent characteristics that make the nonlinear convergence factor approach linear descent in the early stages, which is beneficial for the algorithm to perform a global search; in the later stage of algorithm iteration, α begins to decrease exponentially, which is beneficial for the algorithm to perform a local search. Nonlinear convergence factors are more effective in balancing the global and local search capabilities of the algorithm, further improving its optimization performance. The flowchart of the improved whale optimization algorithm is shown in Figure 3.

3.4. Selection of Performance Indicators

In order to stabilize the output power of wind turbines and reduce system errors, the objective function for PI parameter tuning is selected based on the time-domain integrated time absolute error (ITAE), and the objective function is used as the fitness function [21]. The objective function is the integral of the product of the absolute value of the motor position deviation determined by the PI parameters and time. The formula is:

$$J=ITAE={\displaystyle {\int }_0^{\infty }t\left|e(t)\right|}\mathrm{d}t$$

(23)

In order to facilitate the application of MATLAB software for calculations, a sufficiently large simulation time can be selected, and the continuous time can be discretized into m equal extremely short time $\Delta t$. The objective function can be approximated as formula (24):

$$J\approx {\displaystyle \sum _{i=1}^{m}t(i)\left|e(i)\right|}\Delta t$$

(24)

4. Simulation Verification

4.1. Simulation Analysis on Test Functions

For the purpose of validating the capabilities of the improved whale optimization algorithm, the traditional ZDT1, ZDT2, and DTLZ2 were chosen as the test functions. The assessment of the test outcomes is carried out from two perspectives, namely convergence and distribution, by means of generational distance and distribution evaluation, as described in [15].

Experimental parameter settings: In the whale optimization algorithm, the maximum number of iterations is $T_{\max }=500$ and the population size is 100; the inertia weight coefficient is $\lambda =0.5$; the nonlinear convergence factor is $a=0.5$. The maximum number of iterations in the improved whale optimization algorithm is $T_{\max }=500$, and the population size is 100. The maximum value of the inertia weight coefficient is ${\lambda }_{\max }=0.9$ and the minimum value is ${\lambda }_{\min }=0.1$. The maximum value of the nonlinear convergence factor is $a_{\max }=0.7$ and the minimum value is $a_{\max }=0.3$.

The experimental outcome represents the mean value obtained from ten calculations. The statistical details of the ultimate operation are illustrated in

Table 1. Convergence and diversity on test functions.

Test Functions	Algorithm	Generation Distance	Distribution Evaluation
ZDT1	the whale optimization algorithm	0.081	0.780
	the improved whale optimization algorithm	0.054	0.667
ZDT2	the whale optimization algorithm	0.116	0.398
	the improved whale optimization algorithm	0.065	0.237
DTLZ2	the whale optimization algorithm	0.066	0.674
	the improved whale optimization algorithm	0.023	0.126

. By observing the data within the table, it becomes evident that, in terms of convergence and diversity across the three test functions, the improved whale optimization algorithm surpasses the original whale optimization algorithm. Consequently, the enhancement strategies for the whale optimization algorithm proposed in this article prove to be both effective and practicable.

4.2. Simulation Analysis of Variable Pitch System

In order to confirm the efficacy of the devised control approach, simulation trials were conducted within the MATLAB/Simulink software environment. All simulations in this study were conducted in the MATLAB/Simulink R2021a environment. The simulations employed a variable-step solver, specifically the ode45 (Dormand-Prince) algorithm. The relative tolerance and absolute tolerance were set to 1e-3 and 1e-6, respectively. The inverter was modeled using an average-value model, which ignores the switching details of the power devices and focuses on the dynamic performance analysis of the control system. The motor was modeled using a detailed nonlinear model based on the d-q reference frame, incorporating nonlinearities such as magnetic saturation and cogging effect. The specific parameter configurations for the Permanent Magnet Synchronous Motor (PMSM) and the transmission system are presented in

Table 2. Parameter table of PMSM and transmission system.

Parameter	Set Value
number of poles	6
rated power/kW	11
rated voltage/V	400
voltage constant/mV * min	150
rated speed/1/min	1500
max speed/1/min	2600
stall torque/Nm	94
rated torque/Nm	70
torque constant/Nm/A	2.4
stall current/A	39.2
rated current/A	29.2
stator resistance/Ω	0.15
stator inductance/mH	4.4
moment of inertia/kg * cm2	190
gear ratio	2300

.

PI parameter setting: $k_{\mathrm{P}}=0.3$, $k_{\mathrm{I}}=20$ in the position loop; $k_{\mathrm{P}}=0.3$, $k_{\mathrm{I}}=20$ in the speed loop; $k_{\mathrm{P}}=0.3$, $k_{\mathrm{I}}=50$ in the current loop. At this point, the pitch angle oscillation is relatively small, and the effect is the best. In the whale optimization algorithm, different iterations and population numbers were selected, and through multiple tests and data analysis, the maximum iteration number $T_{\max }=500$ and population size of 100 were selected in the improved whale optimization algorithm; the maximum and minimum inertia weight coefficients are ${\lambda }_{\max }=0.9$ and ${\lambda }_{\min }=0.1$; the upper and lower limit values of the nonlinear convergence factor are $a_{\max }=0.7$ and $a_{\max }=0.3$.

4.2.1. Simulation of Load Torque Variation

During simulation, the pitch angle of the blade that is preset presents as a step signal, which changes from 10° to 20° at 0.75 s, and the load on the motor varies from 6 Nm → 11 Nm → 8 Nm → 5 Nm. Figure 4 shows the pitch angle position curve, Figure 5 shows the torque curve, Figure 6 shows the position loop output curve, and Figure 7 shows the motor current curve. As shown in Figure 4, when implementing PI control relying on the improved whale optimization algorithm, the pitch angle reaches the steady-state value in 0.06 s without overshoot. When using PI control, which is founded on the whale optimization algorithm, the pitch angle reaches the steady-state value in 0.09 s, and the maximum overshoot is 2.4°. As shown in Figure 5, the torque of the motor fluctuates when the pitch angle changes from 10° to 20° in 0.75 s. The overshoot and adjustment time of PI control based on the improved whale optimization algorithm and PI control based on the whale optimization algorithm are basically the same. As shown in Figure 6, when the pitch angle changes from 10° to 20° in 0.75 s, there will be fluctuations in the output curve of the position loop. The PI control based on the improved whale optimization algorithm has smaller fluctuations than the PI control based on the whale optimization algorithm. From Figure 7, it can be seen that there will be fluctuations in the motor current when the pitch angle changes from 10° to 20° in 0.75 s. The overshoot of PI control based on the whale optimization algorithm is significantly greater than that of the PI control based on the improved whale optimization algorithm.

4.2.2. Simulation When the Load Torque Is Constant

During the simulation experiment, the assigned pitch angle of the blade varies according to the slope signal, with an initial value of 90° and a slope of 7.5°/s. At 1.5 s, the pitch angle remains constant at 78.75°. Figure 8, Figure 9, Figure 10 and Figure 11 show the pitch angle position curve, torque variation curve, position loop output curve, and motor current curve, respectively. As shown in Figure 8, employing PI control grounded on the improved whale optimization algorithm can achieve tracking of pitch angle in 0.16 s. When using PI control based on the whale optimization algorithm, the tracking of pitch angle can be achieved in 0.48 s. As shown in Figure 9 and Figure 11, when the pitch angle setpoint changes, there will be fluctuations in the torque and position loop output of the executing motor. The adjustment time of PI control based on the improved whale optimization algorithm is much smaller than that based on the whale optimization algorithm. As shown in Figure 11, when the pitch angle setpoint changes, there will be fluctuations in the motor current during execution. The overshoot of PI control based on the whale optimization algorithm is significantly greater than that based on the improved whale optimization algorithm.

5. Experimental Verification

To evaluate the effectiveness of the developed pitch controller, a pitch test platform was built for testing. Taking a 1 MW wind turbine as the research target, the load of the pitch system was analyzed and calculated, and the parameters and selection of the main components of the electric pitch mechanism were determined. The electric pitch system was designed. The variable pitch actuator motor adopts a customized three phase AC permanent magnet synchronous servo motor developed by Feiye Power according to the special application requirements of the wind power variable pitch system for the motor. The motor model is 5FSNA85-150R01KD0.13-1(Manufactured by Shanghai Feiye Power Technology Co., Ltd., Shanghai, China). The model of the brake that is compatible with the motor is BFK470 (Manufactured by KENDRION, Eisenach, Germany). Please refer to the data manual of the motor for specific parameters. Based on the design of the pitch mechanism and the functional requirements of the pitch system, an electric pitch controller was designed. The controller employs a dual-processor system consisting of a DSP (Digital Signal Processor) and an FPGA (Field-Programmable Gate Array). DSP primarily handles the implementation of vector control and communication functions of the pitch motor; FPGA is mainly responsible for the acquisition, output, and pitch logic control functions of switch values. This fully leverages the advantages of DSP motor control and FPGA parallel processing, making the designed hardware platform fully functional and easy to expand. Corresponding upper computer software has been developed to test the performance of the pitch control system. The platform is presented in Figure 12, and the test results are observed through the upper computer and oscilloscope. The trial results are presented in Figure 13.

Figure 13a shows the test results of the pitch angle given by the upper computer in the order of 45° → 90° → 0° in automatic mode. The findings indicate that the position control precision is excellent and there is no overshoot. The waveform in Figure 13b shows the process of adjusting the blade pitch angle in manual mode. The test results show that the designed pitch controller has complete functions, fast response speed, and high control accuracy, meeting the functional requirements and performance indicators set at the beginning of the design.

6. Conclusions

This article focuses on the problems of slow response speed, poor robustness, and inability to adjust parameters online in PI control. Based on the improved whale optimization algorithm, PI control is applied to the position loop control of electric pitch permanent magnet synchronous motors. Adopting a reverse learning mechanism to generate an initial population in improving the whale optimization algorithm. In the local and global exploration phases of the whale optimization algorithm, the inertia weights and convergence factors are dynamically adjusted based on evolutionary algebra and population characteristics to enhance the efficiency of locating optimal solutions. Through simulation experiments and physical testing, it has been proven that the controller has good adaptability, stronger real-time performance, and better control accuracy. By utilizing an improved whale optimization algorithm, the PI control parameters of the position loop can be adjusted in real-time online, resulting in better performance of vector control in pitch systems using permanent magnet synchronous motors as actuators. It can effectively solve the control problem of nonlinear and strongly coupled systems such as variable pitch systems. The pitch control system is the core component of large-scale wind turbines, so the research and design of this system have certain reference significance for the development of the domestic wind power industry. Meanwhile, analyzing and exploring the functions of the variable pitch system can help improve the safety and power generation efficiency of existing wind turbines. This article’s main focus lies in investigating the control mechanisms of permanent magnet synchronous motors within electric pitch systems. Through the research efforts, certain accomplishments have been made. Nonetheless, a number of aspects remain that have not been fully fathomed or intensively explored. The following outstanding issues necessitate further scrutiny and refinement: (1) a comprehensive analysis and precise simulation of permanent magnet synchronous motors; (2) research on the termination conditions of whale optimization algorithms in order to improve the efficiency of whale optimization algorithms.