#### Abstract

With the increasing penetration of the permanent-magnet direct-drive wind power system, the maximum wind-energy capture and the generation speed control are more and more important. In the literature, the dynamic performance of the generator speed is well documented by the inverse system method. However, conventional inverse system methods have parameter dependency that is not sufficient to meet the dynamic requirements for permanent magnet synchronous generator (PMSG) speed tracking. Therefore, this paper introduces a support vector regression machine (SVR) method, especially for the inverse system model, which could solve the inaccurate parameters problems. As the SVR has the nonlinear approximation ability to identify and adjust the parameters online, thus, the system robustness could be improved. Finally, the dynamic performance of generator speed is evaluated by using the SVR method. Proposed theoretical developments are verified by the Simulink Test and experimental test.

#### 1. Introduction

In the past decades, wind power is always the one of the fastest increasing renewable energy. The small wind turbine is widely used in industry due to its advantages characteristics such as low inertia, low design cost, and low requirements for construction sites [1]. Compared with traditional fixed speed WT systems, variable speed WT systems not only have the advantages of low production cost but also have advantages in increasing energy capture [2, 3].

The existing maximum wind-energy capture method, namely maximum power point tracking (MPPT) strategies for variable speed WT, can be classified into three methods: the optimal tip speed ratio control (TSR), the optimal torque control, and the perturbation observation method. Among the three methods, the TSR is the most widely used due to its quick response to the changing wind speed [4].

For the traditional TSR method, the linear controller is usually used to realize the decoupling control of PMSG. The design principle and structure of the linear controller are simple, but the linear regulation method is limited due to the controller linear nature [5, 6]. As PMSG is a nonlinear and strong coupling system, direct closed-loop control with a linear controller will lead to poor dynamic performance [7–9]. Under randomness and uncertainty of the wind speed condition, a faster and stable tracking of the rotor speed is necessary to ensure the maximum wind-energy capture efficiency [10, 11]. Considering that linear controllers such as proportional integral (PI), proportional derivative (PD), and proportional integral derivative (PID), they are more widely accepted in the industry [12, 13]. PMSG can be linearly compensated to make it more suitable for linear control law.

The inverse system method has been used for linear decoupling control such as applied for buck-converter control, which potentially improves the anti-interference performance with inverse system theory [14–16]. By constructing the inverse system and cascading the original system, the nonlinear original system has pseudolinear properties and is suitable for the linear control law [17]. Literature [18] establishes the mathematical model of the PMSG inverse system, and the pseudolinear system formed by cascading with the original system improves the dynamic performance of the maximum wind-energy capture control. The mathematical model of PMSG is usually established in an ideal state. In actual working conditions, the inverse system model established by mathematical analysis is very sensitive to the parameters. When parameter perturbation occurs, the model needs to be reconstructed [19]. Recently, the machine learning method has been applied in various research fields, and it is used to imitate the learning ability of the human brain [20]. Literature [21] combined the machine learning method with the MPPT control of photovoltaic systems, which significantly improved the tracking speed and efficiency. Literature [22] applied the machine learning method to the wind power generation system to improve the system robustness in the case of wind speed fluctuations. Using machine learning methods, an inverse model can be obtained through a lot of training, which could improve wind-energy capture efficiency and avoid parameter perturbation problems.

The inverse model, based on machine learning and the neural network method, was used to the decoupling control of motor [23]. However, neural networks have black-box properties, and neural networks based on empirical risk minimization are easy to fall into local minimums and excessive reliance on experience, which reduces the use effect. Compared with neural networks, support vector machines (SVM) have the characteristics of small samples and high generalization ability. Literature [24] uses SVM instead of the neural network to establish an inverse system model when the accurate model of the motor is unknown, which effectively avoids problems such as overlearning and local minimization. Literature [25] uses the least square method to optimize the SVM algorithm and proposes an inverse system decoupling method based on LS-SVM. Using LS-SVM’s ability to approximate and fit nonlinear functions, the model of the inverse system is obtained through training samples.

SVR, a learning machine based on the principle of statistical learning and structural risk minimization, introduces insensitivity coefficients on the basis of SVM, obtains a regression support vector machine, and achieves good performance [26, 27]. Compared with the neural network and least squares method and other empirical risk minimization methods, SVR can guarantee the global optimal solution and avoid the problem of getting into the local minimums [28]. Unfortunately, in previous studies, the inverse system and SVR methods were not used in PMSG to improve the efficiency of wind-energy capture.

In this paper, a maximum wind-energy capture control method based on the SVR inverse system (SIS) is proposed for PMSG WT. Two inverse models based on accurate analysis are designed. The SVR inverse model is used to replace the analytical model, and the two PID controllers are combined to achieve closed-loop decoupling control. The main contribution of this paper is to use the SVR method to train the analytical inverse system and obtain a model that does not depend on motor parameters so that the controller is robust to parameter perturbation and solve the model inaccuracy problem under various conditions. The Simulink and experiments results are provided to verify the conclusion of the theoretical study.

The remainder of this paper is organized as follows. The mathematical model of PMSG and problem analysis is presented in Section 2. The design process and control law, as well as the detailed analysis of the analytic inverse system and the SVR inverse system, are discussed in Section 3. The simulation results and experimental results are presented in Section 4. Section 5 concludes this paper.

#### 2. Mathematical Modeling and Problem Analysis

The research object of this paper is a small wind power generation system. As shown in Figure 1, the wind turbine (WT) and PMSG are directly coupled and connected to DC-Link through a three-phase converter.

##### 2.1. WT Model

The power obtained by wind turbine blades from wind energy can be expressed as follows:

where is the air density, *S* is the rotor swept area, *V* is the wind speed, *R* is the radius of the WT blade, and is the power coefficient, which indicates the efficiency of WT to capture wind energy and estimated bywhere is the pitch angle of the WT and is tip speed ratio, which is the ratio between the WT blade tip speed and the wind speed. TSR is given as follows:where (rad/s) is the rotor speed and (r/min) is the rotational speed of the WT.

##### 2.2. PMSG Model

In order to facilitate the analysis of the reversibility of the PMSG system, the mathematical model of PMSG’s two-phase rotating coordinate is used as follows:where is the stator resistance; and are the stator voltages of the *d*-axis and *q*-axis, respectively; and are the stator currents of the *d*-axis and *q*-axis, respectively; and are the inductances of the *d*-axis and *q*-axis, respectively ( = = *L*); is the rotor electrical speed; is the permanent-magnet flux; is the number of pole pairs; *J* is the moment of inertia of the rotor; is the electromagnetic torque; and TL is the load torque.

Therefore, the differential equation of PMSG in the two-phase rotating coordinate system can be expressed as follows:

##### 2.3. Maximum Wind-Energy Capture for Wind Turbines

The maximum wind-energy capture control occurs in the variable power stage of the wind power generation system when the wind speed is lower than the rated wind speed. The power coefficient can be obtained by (2) and (3), as shown in Figure 2.

**(a)**

**(b)**

Figure 2 shows the relationship between the , , and . When fixed at an optimal value, there is an optimal tip speed ratio to keep the at the maximum value, which provides a theoretical basis for the maximum wind-energy capture of the wind power generation system. The control goal of this paper is to maximize the of the WT by controlling the speed of PMSG. This method sets the optimal tip speed ratio value as the reference speed. Moreover, the traditional TSR usually uses vector decoupling control based on PI, which will lead to poor dynamic performance.

#### 3. Design of the SVR Inverse System Model

##### 3.1. Analytic Inverse System of the PMSG

PMSG original system (hereafter, it is referred to as the original system) is considered a nonlinear system with two inputs and two outputs. The next step is to analyze the reversibility of the original system. , , and were set as the input, state, and output variables, respectively, as presented by

The following equation represents the original system state equation of PMSG:

The output equation of the original system is given by

Adopting the theory of inverse system and interactor, the output *y* is differentiated until *u* occurs in the result, as shown in

To achieve and , the Jacobin matrix can be obtained as shown in

The full rank of the two Jacobian matrices in (10) is given in

The relative order of the vector can be obtained according to (11): and , where *n* is the order of the original system of PMSG. This proves the reversibility of the original PMSG system. was selected as the input variable of the inverse system; , which is the output variable of the inverse system; and *T*_{L} is the load torque, and its value is obtained from the PMSG original system observation. When these are combined with (9), the inverse system can be expressed by

Equation (12) can be rewritten as follows:

Equation (13) is the inverse-system analytical expression based on the PMSG mathematical model. The inverse system consists of two inputs (, ) and two outputs (, ).

The PMSG is dynamically decoupled into a first-order linear current subsystem and a second-order linear speed subsystem through the inverse system method. The pseudolinear system constructed by cascading the two systems is shown in Figure 3.

##### 3.2. Inverse System Modeling of SVR

The training sample of the inverse model is set to , where is the input sample containing six features , and the two features and are contained in the output sample . The function model determined by the SVR method can be regarded as a hyperplane in the kernel space, as follows:where is the weight vector of hyperplane , is a bias term, and means to map data to a high-dimensional feature space .

The optimization problems of the SVR are expressed as follows:where is the minimized structural risk and the is the minimized sample error. Two nonnegative variables and are used to weaken the constraints of some sample points. The positive number is a compromise between structural risk and sample error.

The constraint conditions are expressed as follows:where is the introduced insensitive coefficient to avoid overfitting. In order to solve (15), Lagrange multipliers and are introduced. Reconstruct the above optimization problem as a Lagrangian function, as follows:

Introduce a kernel function that satisfies the Mercer condition to replace the inner product of the high-dimensional kernel space in the (17); here, the Gaussian kernel is used. The kernel function can be written as follows:

The optimized problem can be expressed as follows:

and the constraint conditions are expressed as follows:

In the solution of (22), part of the coefficient is not zero. The sample corresponding to nonzero is the support vector. The optimized (14) is expressed as follows:

Therefore, the inverse model based on SVR can be expressed as follows (Figure 4):

As shown in Figure 4, the SVR model is used instead of the analytical model. The SVR inverse system model gets rid of the dependence on the precise mathematical model of the control object.

##### 3.3. A Control Strategy Based on SVR Inverse System Decoupling for Maximum Wind-Energy Capturing

To maximize the wind-energy capture, a vector control strategy of *id* = 0 is adopted in combination with the PMSG speed control based on the PI vector control and inverse system (Figure 4). The real-time wind speed of the WT is measured, and the reference speed is calculated according to (3) to maintain the WT running at the optimal blade-TSR. The reference speed is used as the input of the controller speed outer loop. The inner loop is the current loop and is set to 0. Figure 5 shows the equivalent control block diagram of the MWECIS.

Figure 6 shows the PMSG control block diagram based on the SVR inverse system decoupling. In Figure 6, the inverse system is merged with the original system to form a pseudolinear system. Therefore, the system will adapt to the linear control law. A PID control can achieve the good dynamic decoupling effect so that the speed of PMSG can promptly follow the change in the wind speed and keep the WT always running at the optimal blade-TSR.

#### 4. Decoupling Control of PMSG Based on SVR Inverse System

##### 4.1. Simulation Results

In order to verify the effectiveness of the proposed SIS method, the SIS is compared with the analytical inverse system method (AIS) and the traditional TSR method. The AIS method will be designed as two controllers. For the AIS-I type controller, the parameters in the mathematical model are completely matched with the actual parameters of the motor. For the AIS-II controller, there is a 30% error between the parameters in the mathematical model and the actual parameters of the motor.

The parameters of the SIS, TSR, AIS-I, and AIS-II were well tuned through trial and error. The parameters of PMSG and WT are shown in Table 1.

###### 4.1.1. Simulation of AIS-I, TSR, and SIS Controllers

In order to reflect the improvement of the dynamic performance of the control system by the inverse system method, the selected wind speed is shown in Figure 7.

It can be seen from Figure 7 that the wind speed fluctuates randomly between 2 m/s and 8 m/s. Figure 7 is used as the input wind speed, and the speed response of the generator is shown in Figure 8.

**(a)**

**(b)**

**(c)**

As shown in Figure 8(a), the PMSG rotor speed based on the AIS-I controller can quickly and stably follow the reference speed. As shown in Figure 8(b), the PMSG rotor speed based on the TSR controller produces a large overshoot when the reference speed is abruptly changed. For example, at 1.5 s, the speed response of PMSG is 150 r/min, while the reference speed is 200 r/min. As shown in Figure 8(c), the controller based on the SIS method almost perfectly approximates the AIS-I controller, and the PMSG rotor speed can quickly and stably follow the reference speed. The error between the speed response and the reference speed of the PMSG is shown in Figure 9.

It can be seen from Figure 9 that the controller based on the SIS method has the same rotational speed error as the controller based on the AIS-I method. At 3.5 s–3.6 s, the PMSG rotational speed error of the TSR method is twice that of the SIS method and the AIS-I method. The tip speed ratio of the three control methods is shown in Figure 10.

As shown in Figure 10, the optimum tip speed ratio for the wind turbine is 8.1. In 1.2 s–1.4 s, the of the TSR method fluctuated around 8.1, while the of the SIS and AIS-I methods remained at 8.1. The power coefficient of the WT is shown in Figure 11.

**(a)**

**(b)**

**(c)**

It can be seen from Figure 11 that the power coefficient of WT under different methods and the maximum of is 4.8. As shown in Figure 11(a), the AIS-I method can keep the WT in the state of maximum wind-energy capture. When the wind speed changes suddenly, the drops briefly and then returns to 4.8. As shown in Figure 11(b), compared with ASI-I, the average of the TSR method is lower. When the wind speed changes, the adjustment time based on the TSR method is longer. As shown in Figure 11(c), the based on the SIS method can well approximate the of the AIS-I method. The output power (Pm) response based on different methods is presented in Figure 12.

As shown in Figure 12, it is the output power of the wind turbine. For the same wind speed input, the wind turbine output power of AIS-I and SIS methods is higher than that of the TSR method.

Compared with the traditional TSR method, both AIS-I and SIS methods can make PMSG obtain better dynamic performance. The faster and more stable control of PMSG rotor speed enables wind turbines to maintain high efficiency and capture more wind energy. The proposed SIS method can well approximate the AIS-I method and achieve the same control effect.

###### 4.1.2. Simulation of AIS-I and AIS-II Controllers

The analytical inverse system method is designed as two controllers to study the influence of inaccurate PMSG parameters on the control effect. The input wind speed is shown in Figure 7, and the speed responses of the two controllers are shown in Figure 13.

**(a)**

**(b)**

As can be seen from Figure 13, it is the rotational speed response of the PMSG. As shown in Figure 13(a), the speed response based on the AIS-I method can follow the reference speed quickly and stably, and the overshoot and adjustment time are very small. As shown in Figure 13(b), compared with AIS-1, the speed response based on the AIS-2 method will have a larger overshoot when the wind speed is abruptly changed. The error between the speed response and the reference speed of the PMSG is shown in Figure 14.

Figure 14 shows the rotational speed error of two different methods. Between 2.4 s and 3.2 s, the rotational speed error of the AIS-2 method is twice that of the AIS-1 method. The power coefficient of the WT is shown in Figure 15.

**(a)**

**(b)**

It can be seen from Figure 15 that the power coefficient of WT under different methods and the maximum of is 4.8. As shown in Figure 11(a), the AIS-1 method can keep the WT in the state of maximum wind-energy capture. When the wind speed changes suddenly, the drops briefly and then returns to 4.8. As shown in Figure 11(b), compared with AIS-1, the adjustment time based on the AIS-2 method is longer when the wind speed changes. The output power (Pm) response based on different methods is presented in Figure 16.

As shown in Figure 16, it is the output power of the wind turbine. For the same wind speed input, the wind turbine output power of AIS-1 is higher than that of the AIS-2 method.

From the simulation results, the method of AIS has better dynamic performance than the TSR method, which enables the WT to capture more wind energy. However, when the parameters of the PMSG are imprecise, the control effect of the AIS method becomes poor. The proposed SIS method can approximate the AIS method, avoid the problem of inaccurate parameters, and improve the dynamic performance and robustness of the system.

##### 4.2. Experiment Results

In order to ensure that the experimental data are closer to the actual working conditions, we obtained the 2020 annual wind speed data from NASA Power data [29] in Hunan Province, China (where the laboratory is located, Figure 17), as shown in Figure 18.

As shown in Figure 8, the wind speed in Hunan in 2020 varies from 3 m/s to 22.5 m/s, and the average wind speed is 7.9 m/s. Using Python, combined with Rayleigh law [30] to analyze wind speed data, Rayleigh law mainly depends on the average value of wind speed.

The probability density function (PDF) is expressed as follows:

The cumulative distribution function (CDF) is expressed as follows:

Figure 19 shows the output PDF and CDF after Rayleigh distribution processing on wind speed data.

**(a)**

**(b)**

As shown in Figure 19, the Rayleigh probability density function PDF is basically consistent with the measured wind speed data. Most wind speeds throughout the year are concentrated between 2.5 and 12.5 m/s, and the cumulative distribution function CDF also grows fastest in this interval. Therefore, we can determine the validity of the Rayleigh distribution and classify it as a normal distribution.

The analysis of wind speed data is applied to simulation and experiment, simulate similar curve distribution, meet 2.5–12.5 m/s variable wind speed, and provide corresponding aerodynamic torque to drive PMSG with an interval of 20 s.

As shown in Figure 20, it is a 20 s wind speed graph, and the change law conforms to the above-mentioned wind speed data analysis. The wind speed in Figure 20 will be used in experiments.

The experimental platform is shown in Figure 21, which includes the asynchronous motor and PMSG. The wind turbine is simulated by an asynchronous motor, which drives by a 7.5 kW ABB inverter. The experiment programming is completed by the YXSPACE rapid prototyping controller.

The key parameters of PMSG and WT in the experiment are listed in Table 2.

The input wind speed is shown in Figure 20; the experiment time is 30 s; and the result is shown in Figure 22.

**(a)**

**(b)**

Figure 22(a) shows the reference speed and response speed with the TSR method. It could be seen that the overshoot in the start-up phase is 10%, and the adjustment time is 3 s. When the wind speed is changed, the rotor speed differs from the reference speed by 5 r/min. Figure 22(b) shows the reference speed and response speed with the SIS method. It could be seen that a small speed overshoot in the start-up phase. During the wind speed change phase, the two curves basically coincide. When using the SIS method, the rotor speed can quickly and stably follow the reference speed.

#### 5. Conclusions

In this study, the proposed SIS method can successfully approach the analytical inverse system. The SVR model could successfully transfer the nonlinear PMSG speed control system into a pseudolinear system with linear properties. Using the maximum wind-energy capture control, SIS can avoid the parameter perturbation problem faced by the analytical inverse system method and has more stable performance under actual working conditions. Simulation and experimental results show that the proposed SIS control strategy provides stable and fast control effects for the rotor speed of PMSG and can achieve greater wind-energy capture under simulated real wind speed. Finally, the research in this paper focuses on using the SIS method to control WT to capture more wind energy, and the work of transferring energy into the grid is done by grid-side inverters. Therefore, it is strongly recommended to apply the SIS method to the grid-side inverter control in future work. The PMSG system will better control the transmission of active power and reactive power to the grid.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by the Natural Science Foundation of Hunan Province of China, under Grant nos. 2017JJ2293 and 2019JJ50119, major special projects of Changsha Science and Technology Plan, under Grant no. kq2105001, Hunan Education Department Outstanding Youth Program, under Grant no. 21B0780, and National Natural Science Foundation of China under Grant no. 51907061.