Parameter Identification of PMSG-Based Wind Turbine Based on Sensitivity Analysis and Improved Gray Wolf Optimization

Abstract

With the large-scale integration of wind power, it is essential to establish an electromagnetic transient (EMT) model of a wind turbine system. Focusing on the problem of the difficulty in obtaining the parameters of the direct-driven permanent magnet synchronous generator (PMSG) model, this manuscript proposes a method based on trajectory sensitivity analysis and improved gray wolf optimization (IGWO) for identifying the parameters of the PMSG EMT model. First, a model of a PMSG wind turbine is established on an EMT simulation platform. Then, the key parameters of the model are determined based on the sensitivity analysis. Five control parameters are selected as the key parameters for their higher sensitivity indexes. Finally, the key parameters are accurately identified, using the proposed IGWO algorithm. The final case study demonstrates that the proposed IGWO algorithm has better optimization performance compared with the GWO algorithm and particle swarm optimization (PSO) algorithm. In addition, the simulation waveforms show that the identified parameters are accurate and applicable to other operating conditions.

1. Introduction

As the scale of wind power generation continues to grow, the operation of the wind power system will be inevitably influenced by access to large-scale wind power [1,2,3,4]. To accurately reproduce the operating performance of a wind power generation system, a grid-connected wind turbine simulation model with detailed and correct parameters is necessary. However, the accuracy of the wind turbine model cannot be consistently guaranteed, due to the inability of wind turbine manufactures to provide control parameters. In addition, the parameters of the turbines may be significantly different from the nameplate values after the long-term operation, which makes it difficult for the simulation model to reproduce the real characteristics of the power system linked with the wind turbine. Therefore, it is necessary to perform parameter identification on the simulation model of the wind turbine [5,6].

Parameter identification methods mainly comprise the least squares (LS) method, the Kalman filter (KF) method, and the model reference adaptive parameter identification method. In [7], a damped least squares method-based parameter identification method was proposed for the control system of the direct-driven wind turbine. In [8], the recursive least squares method was adopted for identifying the parameters of the generator. In [9], an estimation method that combined the extended Kalman filter (EKF) and LS was proposed, to identify the parameters of the permanent magnet synchronous motor (PMSM). An EKF-based method was proposed for the parameter identification of the doubly fed induction generator (DFIG) in [10], which could identify all the electrical parameters of the DFIG. In [11], a model reference adaptive parameter identification-based method was presented for identifying the parameters of the DFIG. It had a good identification effect on the mutual inductance, the leakage inductance, and the grid-connected stator.

The methods mentioned above all have good performance in the parameter identification of the wind turbine. However, these methods require the known input variables for their application and have high requirements for the size of the signal perturbations. In addition, a grid-connected wind turbine system is a highly nonlinear system, which makes it difficult to describe its output characteristic with accurate expressions. Then, parameter identification of the nonlinear system using traditional methods may lead to significant errors. Many population intelligence optimization algorithms have been applied to identifying the control parameters of wind turbine inverters [12]. In [13], an improved cuckoo search algorithm-based method was proposed for the parameter identification of the PMSM. In [14], a particle swarm optimization (PSO) algorithm-based parameter identification of a permanent magnet synchronous generator (PMSG) was proposed and implemented. The electrical parameters of the PMSG were identified using the proposed method based on the experimental tests. A genetic algorithm (GA)-based parameter identification of the PMSG model was proposed in [15]. In [16], a PSO gradient search optimization-based method was presented for identifying the parameters of the dynamic equivalent model of the DFIG. In [17], a modified GA method for the calibration of the parameters of a comprehensive wind farm model was introduced. In [18], an improved flower pollination algorithm was proposed, to identify the parameters of the PMSM. In [19], a chaotic mapping and Gaussian perturbation-based PSO algorithm for the parameter identification of the PMSG motor was introduced. However, these optimization methods have problems, such as slow convergence speed, easy premature convergence, and low identification accuracy. Focusing on these problems, [20] proposed a GWO algorithm with an information-sharing search strategy to solve the parameter identification problem of the DFIG. In [21], a chaotic adaptive search GWO algorithm was presented, to identify the parameters of the PMSM.

Based on the research results of the parameter identification, an improved gray wolf optimization (IGWO) algorithm-based parameter identification method is proposed for identifying the parameters of the PMSG wind turbine in this manuscript. The trajectory sensitivity-based method is performed first, for selecting the key parameters of the PMSG wind turbine. Then, the IGWO algorithm is proposed, for identifying the key parameters with the larger sensitivity indexes. Finally, the proposed method accurately yields the identification results of the PMSG model.

The remainder of the manuscript is presented in five sections. Section 2 gives the structure and the control strategy of the PMSG. A trajectory sensitivity-based method to obtain the key parameters is introduced in Section 3. In Section 4, the traditional GWO algorithm and the IGWO algorithm for the parameter identification of the PMSG are discussed. In Section 5, a simulation case is conducted. Section 6 concludes the manuscript.

2. Model of Direct-Driven Permanent Magnet Synchronous Generator

The schematic diagram of the PMSG model used in this manuscript is depicted in Figure 1. It mainly includes the wind turbine, the PMSG, the converters, and the control system [22]. The control system consists of the maximum power point tracking (MPPT) control, the pitch angle control, the machine-side converter control (MSC), and the grid-side converter control (GSC). In the PMSG grid-connected system, the turbine drives the PMSG to rotate, which converts the captured wind energy into alternating electrical energy with varying amplitude and frequency. Then, the PMSG is connected with the grid through the converter, and the electrical energy with the fixed amplitude and frequency is finally obtained and input into the grid. To better identify the parameters of the PMSG model, the structures and the control methods of these two converters are introduced below.

2.1. The Structure and the Control Strategy of the MSC

In this manuscript, the MSC mainly consists of the uncontrolled bridge rectifier and the boost circuit [23]. In this structure, the current of the DC side can be controlled by adjusting the duty cycle of the boost circuit. Then, the electromagnetic torque and the speed of the generator can be changed and the MPPT can be achieved. Specifically, the change of the current of the DC side can operate the wind turbine at the optimal tip speed ratio, which can maximize the power captured by the wind turbine. In the MPPT control, the reference rotor speed is given by [22]:

$${\omega }_{\mathrm{ref}}={\displaystyle \frac{{\lambda }_{\mathrm{opt}}{V}_{\mathrm{wind}}}{R}}$$

(1)

where ${\omega }_{\mathrm{ref}}$ represents the reference wind turbine rotor speed; ${\lambda }_{\mathrm{opt}}$ represents the optimal tip speed ratio; $V_{\mathrm{wind}}$ represents the wind speed; R indicates the radius of the wind turbine. The output of the MPPT control is input into the boost circuit as the switching signal of the insulated gate bipolar transistor (IGBT).

2.2. The Structure and the Control Strategy of the GSC

The GSC used in this paper is composed of a controllable inverter bridge made up of IGBTs. Its primary functions are to ensure the stability of the DC-side capacitor and to deliver the specified reactive power according to the reactive power reference value. The GSC uses a grid voltage-oriented control to align the grid voltage vector along the d-axis [22]:

$$\left\{\begin{array}{c}{u}_{\mathrm{d}}=e_{\mathrm{g}}\hfill \\ u_{\mathrm{q}}=0\hfill \end{array}\right.$$

(2)

where $u_{\mathrm{d}}$ and $u_{\mathrm{q}}$ represent the d-axis and q-axis voltages of the grid, respectively; $e_{\mathrm{g}}$ indicates the magnitude of the grid voltage space vector. Then, the active and reactive power delivered to the grid by the GSC are calculated by [22]

$$\left\{\begin{array}{c}P={\displaystyle \frac{3}{2}}{e}_{\mathrm{g}}{i}_{\mathrm{gd}}\hfill \\ Q=-{\displaystyle \frac{3}{2}}{e}_{\mathrm{g}}{i}_{\mathrm{gq}}\hfill \end{array}\right.$$

(3)

where P and Q represent the active and reactive power output by the GSC, respectively; $i_{\mathrm{gd}}$ and $i_{\mathrm{gq}}$ represent the d-axis and q-axis currents of the grid, respectively. It can be seen through (3) that the active and reactive power are controlled separately by adjusting the values of $i_{\mathrm{gd}}$ and $i_{\mathrm{gq}}$.

When GSC is in steady state, the GSC model under the d–q rotating coordinate system can be given by [22]

$$\left\{\begin{array}{c}{u}_{\mathrm{gd}}=e_{\mathrm{g}}-R_{\mathrm{g}}{i}_{\mathrm{gd}}-L_{\mathrm{g}}{\displaystyle \frac{\mathrm{d}{i}_{\mathrm{gd}}}{\mathrm{d}t}}+{\omega }_{\mathrm{g}}{L}_{\mathrm{g}}{i}_{\mathrm{gq}}\hfill \\ u_{\mathrm{gq}}=-R_{\mathrm{g}}{i}_{\mathrm{gq}}-{\omega }_{\mathrm{g}}{L}_{\mathrm{g}}{i}_{\mathrm{gd}}-L_{\mathrm{g}}{\displaystyle \frac{\mathrm{d}{i}_{\mathrm{gq}}}{\mathrm{d}t}}\hfill \end{array}\right.$$

(4)

where $u_{\mathrm{gd}}$ and $u_{\mathrm{gq}}$ represent the d-axis and q-axis output voltages of the GSC, respectively; $R_{\mathrm{g}}$ is the resistance at the AC side; $L_{\mathrm{g}}$ indicates the filter inductor; ${\omega }_{\mathrm{g}}$ represents the grid synchronous electrical angular velocity. To avoid the influence of the coupling voltage terms and grid voltage, the feedforward decoupling control is introduced and the output voltages of the GSC are given by [22]

$$\left\{\begin{array}{c}{u}_{\mathrm{g}\mathrm{d}}=-\left(K_{\mathrm{p}1}+{\displaystyle \frac{K_{\mathrm{i}1}}{s}}\right)\left(i_{\mathrm{dref}}-i_{\mathrm{g}\mathrm{d}}\right)+\Delta u_{\mathrm{d}}\hfill \\ u_{\mathrm{g}\mathrm{q}}=-\left(K_{\mathrm{p}2}+{\displaystyle \frac{K_{\mathrm{i}2}}{s}}\right)\left(i_{\mathrm{qref}}-i_{\mathrm{g}\mathrm{q}}\right)+\Delta u_{\mathrm{q}}\hfill \end{array}\right.$$

(5)

with [22]

$$\left\{\begin{array}{c}\Delta u_{\mathrm{d}}=e_{\mathrm{g}}-R_{\mathrm{g}}{i}_{\mathrm{gd}}+{\omega }_{\mathrm{g}}{L}_{\mathrm{g}}{i}_{\mathrm{gq}}\hfill \\ \Delta u_{\mathrm{q}}=-R_{\mathrm{g}}{i}_{\mathrm{gq}}-{\omega }_{\mathrm{g}}{L}_{\mathrm{g}}{i}_{\mathrm{gd}}\hfill \end{array}\right.$$

(6)

where $K_{\mathrm{p}1}$, $K_{\mathrm{i}1}$, $K_{\mathrm{p}2}$, and $K_{\mathrm{i}2}$ represent the PI parameters of the inner-loop current control; $i_{\mathrm{dref}}$ and $i_{\mathrm{qref}}$ represent the reference values of the d-axis and q-axis currents of the GSC, respectively.

3. A Trajectory Sensitivity-Based Method to Obtain the Key Parameters for Identification

In the above PMSG model, the parameters are numerous and have varying impacts on the system’s transient responses. If all the parameters are identified, the computational burden will greatly increase and the identification results will show great dispersion [24]. Therefore, identification should only be performed on the key parameters. In the transient response of the PMSG model, the pitch angle control generally does not work and its parameters do not need identification. In addition, the values of the static components, such as the inductors, capacitors, and resistors generally do not change, which can be fixed as the typical values [25]. Therefore, the control parameters of the PMSG model are determined as the parameters for the sensitivity calculation.

Actually, the sensitivity analysis plays a crucial role in guiding the parameter identification process by determining which parameters are most influential and should be prioritized for identification. Specifically, it can quantify the sensitivity of the system output to the input, and it can rank the influence of each input parameter on the output of the system. After the sensitivity analysis, the parameters with low sensitivity among the input parameters are generally difficult to obtain accurately, which can be considered as the rated values. Then, the parameters that have high sensitivity indexes are determined for the identification. This approach significantly improves the convergence rate of the identification program and ensures more accurate and less dispersive results.

To better select the key parameters of the PMSG model, a trajectory sensitivity-based method was adopted in this paper, which is given below [26]:

$$S\left(x\right)={\displaystyle \frac{1}{K}}\sum _{i=1}^K\left|{\displaystyle \frac{y_1^i\left(x_1\right)-y_2^i\left(x_2\right)}{2\left(\Delta x_0/x_0\right)}}\right|$$

(7)

with

$$\left\{\begin{array}{c}{x}_1=x_0+\Delta x_0\hfill \\ x_2=x_0-\Delta x_0\hfill \end{array}\right.$$

(8)

where $x_0$ is the initial value of parameter x; $\Delta x_0$ is a small variation of $x_0$; $y_1^i\left(x_1\right)$ and $y_2^i\left(x_2\right)$ are the dynamic responses of the system obtained according to parameter $x_1$ and $x_2$, respectively; K is the number of i in the transient response trajectory; $S\left(x\right)$ is the calculated sensitivity index. After finishing the sensitivity calculation according to (7) and (8), the parameters are sorted based on the values of their corresponding $S\left(x\right)$. The parameters corresponding to the larger $S\left(x\right)$ are determined as the key parameters. In the sensitivity calculation of the PMSG model, the x represents the selected parameters to be identified, mainly the PI parameters of the PMSG control system; y indicates the transient power response of the PMSG model when a three-phase short-circuit fault occurs, including the active power and reactive power. This method can greatly improve the calibration process by effectively eliminating parameters that exhibit zero or minimal sensitivity to model performance. Actually, the ranking of the parameters sensitivity indexes are not the same under different operating conditions and transient faults. Therefore, the trajectory sensitivity calculation for the parameters is necessary for selecting the key parameters accurately when dealing with different faults.

4. Improved Gray Wolf Optimization Algorithm

4.1. Gray Wolf Optimization Algorithm

The GWO algorithm is a novel intelligent optimization algorithm, which mimics the predatory behavior of gray wolf groups [27]. In the algorithm, a gray wolf can be considered as a candidate solution for the problem. The gray wolves are categorized into four ranks, including $\alpha$ wolf, $\beta$ wolf, $\delta$ wolf, and $\omega$ wolves. The schematic diagram of the social dominance hierarchy of gray wolves is illustrated in Figure 2. In the application of the GWO algorithm, it is considered that $\alpha$ wolf represents the optimal solution; $\beta$ wolf and $\delta$ wolf represent the second- and third-best solutions, respectively. The other solutions are considered as the $\omega$ wolves, whose hunting behavior is dominated by $\alpha$ wolf, $\beta$ wolf, and $\delta$ wolf.

In the GWO algorithm, the predatory behavior of a gray wolf population is composed of three steps: encircling, hunting, and attacking.

The equations simulating the encircling stage of the wolves are expressed as follows:

$$\left\{\begin{array}{c}\mathit{D}=|{\mathit{CZ}}_{\mathit{p}}\left(t\right)-\mathit{Z}\left(t\right)|\hfill \\ \mathit{Z}(t+1)={\mathit{Z}}_{\mathit{p}}\left(t\right)-\mathit{BD}\hfill \end{array}\right.$$

(9)

where t represents the $t\mathrm{th}$ iteration; $\mathit{D}$ indicates the distance between the wolf and the prey; ${\mathit{Z}}_{\mathit{p}}$ and $\mathit{Z}$ represent the position of the wolf and the prey, respectively; $\mathit{B}$ and $\mathit{C}$ represent the vector coefficients, which are given by [28]:

$$\left\{\begin{array}{c}\mathit{B}=2\mathit{b}·{\mathit{r}}_{\mathbf{1}}-\mathit{b}\hfill \\ \mathit{C}=2·{\mathit{r}}_{\mathbf{2}}\hfill \end{array}\right.$$

(10)

where ${\mathit{r}}_{\mathbf{1}}$ and ${\mathit{r}}_{\mathbf{2}}$ are both random vectors between $[0,1]$; $\mathit{b}$ indicates the convergence factor vector. The elements of $\mathit{b}$ reduce from 2 to 0 during the iteration.

The hunting stage is mainly dominated by $\alpha$ wolf, $\beta$ wolf, and $\delta$ wolf. According to the positions of $\alpha$ wolf, $\beta$ wolf, and $\delta$ wolf, the other wolves are updated, using the following formulas [28]:

$$\left\{\begin{array}{c}{\mathit{D}}_{\alpha }=|{\mathit{C}}_{\mathbf{1}}·{\mathit{Z}}_{\alpha }-\mathit{Z}|\hfill \\ {\mathit{D}}_{\beta }=|{\mathit{C}}_{\mathbf{2}}·{\mathit{Z}}_{\beta }-\mathit{Z}|\hfill \\ {\mathit{D}}_{\delta }=|{\mathit{C}}_{\mathbf{3}}·{\mathit{Z}}_{\delta }-\mathit{Z}|\hfill \end{array}\right.$$

(11)

$$\left\{\begin{array}{c}{\mathit{Z}}_{\mathbf{1}}={\mathit{Z}}_{\alpha }-{\mathit{B}}_{\mathbf{1}}·{\mathit{D}}_{\alpha }\hfill \\ {\mathit{Z}}_{\mathbf{2}}={\mathit{Z}}_{\beta }-{\mathit{B}}_{\mathbf{2}}·{\mathit{D}}_{\beta }\hfill \\ {\mathit{Z}}_{\mathbf{3}}={\mathit{Z}}_{\delta }-{\mathit{B}}_{\mathbf{3}}·{\mathit{D}}_{\delta }\hfill \end{array}\right.$$

(12)

$$\mathit{Z}(t+1)={\displaystyle \frac{{\mathit{Z}}_{\mathbf{1}}+{\mathit{Z}}_{\mathbf{2}}+{\mathit{Z}}_{\mathbf{3}}}{3}}$$

(13)

where ${\mathit{C}}_{\mathbf{1}}$, ${\mathit{C}}_{\mathbf{2}}$, ${\mathit{C}}_{\mathbf{3}}$, ${\mathit{B}}_{\mathbf{1}}$, ${\mathit{B}}_{\mathbf{2}}$, and ${\mathit{B}}_{\mathbf{3}}$ can be calculated according to (10); $\mathit{Z}(t+1)$ is the new position of the wolf.

In the attacking stage, the value of b in the vector $\mathit{b}$ decreases to 0 as the iteration proceeds. In this process, the gray wolves gradually converge towards the position of the prey. After reaching the upper limit of iterations, the optimal solution to the problem can be obtained.

In conclusion, GWO initializes the positions of the population within the solution space and assesses the positions of the wolves according to the value of its fitness function. Throughout the optimization process, the aforementioned three stages, including encircling, hunting, and attacking the prey, are repeatedly executed until the upper limit of iterations is reached. After that, the corresponding position of $\alpha$ is considered as the optimal solution found using GWO.

4.2. Improvements on GWO

Although GWO has the advantages of strong optimization capability and simple operation, the local optimum problem is prone to occurring when dealing with a complex and high-dimensional optimization problem such as the parameter identification of the PMSG. To suppress the premature convergence problem of GWO, this manuscript proposes an IGWO method to identify the parameters of the PMSG model.

4.2.1. Cubic Mapping

The initialization of the population in GWO adopts a random generation strategy, which may cause the local optimum. In comparison, chaotic sequences have better regularity and ergodicity [29]. Therefore, cubic mapping is employed for the initialization of the population in IGWO, to increase the diversity of the population. Compared with other mapping methods, cubic mapping has the advantages of easy implementation and high computational efficiency. Cubic mapping is given by [30]

$$s_{n+1}=\rho s_n(1-s_n^2)$$

(14)

where $s_n\in [0,1]$ and $\rho$ is the control parameter. The chaos of the cubic mapping is greatly influenced by the value of $\rho$. The value of $\rho$ is set as $(1.5,3)$ and the corresponding bifurcation diagram is shown in Figure 3a. It can be found that the value of $s_{n+1}$ is distributed between 0 and 1 when $\rho =2.596$. Furthermore, the distribution of $s_{n+1}$ is illustrated in Figure 3b after 3000 iterations when $s_0=0.3,\rho =2.596$. It is obvious that cubic mapping has a good chaotic traversal property for the interval $[0,1]$ in this setting. However, the selected parameters in the PMSG model have different value ranges. Therefore, after generating a series of data using cubic mapping, the corresponding values of the parameters can be calculated as follows [30]:

$$x_{pi}=x_i^{\min }+s_i·(x_i^{\max }-x_i^{\min })$$

(15)

where $s_i$ is the ith number of the cubic mapping sequence; $x_{pi}$ indicates the ith parameter of the positions of the wolves; $x_i^{\min }$ and $x_i^{\max }$ represent the lower and upper value of the parameter, respectively.

4.2.2. Nonlinear Convergence Factor

The value of $\mathit{B}$ plays a crucial role in balancing the global optimization and local search capabilities of the GWO algorithm. When $\left|\mathit{B}\right|<1$, the wolves will gradually approach the target and the local search capability of GWO will be enhanced. When $\left|\mathit{B}\right|>1$, the wolves are compelled to move away from the prey and the global optimization capability of GWO is enhanced. In each iteration, the value of the convergence factor $\mathit{b}$ directly influences the value of $\left|\mathit{B}\right|$. In traditional GWO, the value of $\mathit{b}$ is updated using a linear strategy, which may increase the risk of the algorithm getting stuck in the local optimum. In this manuscript, the tangent function-based nonlinear update strategy is introduced, to update $\mathit{b}$, which can be calculated as follows [31]:

$$b=b_0-\left(b_0-b_1\right)\times tan\left({\displaystyle \frac{1}{\lambda }}·{\displaystyle \frac{g}{g_{max}}}\pi \right)$$

(16)

where $b_0$ and $b_1$ are the initial and final values of the convergence factor, respectively; $\lambda$ indicates the adjustment parameter and is set as 4; g and $g_{\max }$ indicate the current and maximum number of iterations, respectively.

4.2.3. Improved Position Update Equation

In traditional GWO, the change of the positions of the wolves is mainly determined by the positions of $\alpha$, $\beta$, and $\delta$ wolves. This way of updating may reduce the optimization effect of the algorithm. To strengthen the communication between the individual and the population, the position update strategy of PSO is brought into IGWO. The specific position-update equations of the IGWO algorithm are as follows [28]:

$$\left\{\begin{array}{c}m=g/g_{\max }\hfill \\ {\mathit{Z}}_{g\mathrm{G}}=({\mathit{Z}}_{\mathbf{1}}+{\mathit{Z}}_{\mathbf{2}}+{\mathit{Z}}_{\mathbf{3}})/3\hfill \end{array}\right.$$

(17)

$$\begin{array}{c}\hfill {\mathit{Z}}_{g\mathrm{P}}=m\times r_3\times {\mathit{Z}}_{g\mathrm{G}}+{\displaystyle \frac{(1-m)}{2}}\times \left[r_4\times \left({\mathit{Z}}_{\alpha }-{\mathit{Z}}_g\right)+r_5\times \left({\mathit{Z}}_{g\mathrm{G}}-{\mathit{Z}}_g\right)\right]\end{array}$$

(18)

$${\mathit{Z}}_{g\mathrm{IG}}=\left\{\begin{array}{cc}{\mathit{Z}}_{g\mathrm{G}},\hfill & \mathrm{fitness}\left({\mathit{Z}}_{g\mathrm{G}}\right)<\mathrm{fitness}\left({\mathit{Z}}_{g\mathrm{P}}\right)\hfill \\ {\mathit{Z}}_{g\mathrm{P}},\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(19)

where m is a linear learning parameter; $r_3$, $r_4$, and $r_5$ are all random numbers between 0 and 1; ${\mathit{X}}_g$ is the current position of the wolf; ${\mathit{X}}_{g\mathrm{G}}$ is the position updated using the GWO algorithm; ${\mathit{X}}_{g\mathrm{P}}$ is the position updated using the improved position update equation.

The aforementioned equations apply the idea of the PSO algorithm to improving the GWO position update strategy, which enhances the global optimization capability of GWO. In each iteration, the final position of the wolf is determined by comparing the fitness values. It is obvious that the introduction of the PSO algorithm can enhance the connection between the individual and the population, which is beneficial to improving the optimization efficiency of GWO.

To evaluate the performance of the proposed IGWO algorithm, six standard fitness test functions are optimized using GWO and the IGWO algorithm. The used functions are listed in

Table 1. The standard fitness test functions.

Function	Dim	Range	${\mathit{f}}_{\mathit{min}}$
$f_1\left(x\right)={\displaystyle \sum _{i=1}^n}{x}_i^2$	30	$[-100,100]$	0
$f_2\left(x\right)=10n+{\displaystyle \sum _{i=1}^n}\left(x_i^2-10cos\left(2\pi x_i\right)\right)$	30	$[-5.12,5.12]$	0
$f_3\left(x\right)=-20exp\left(-0.2\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{x}_i^2}\right)-exp\left({\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}cos\left(2\pi x_i\right)\right)+20+e$	30	$[-32.7,32.7]$	0
$f_4\left(x\right)=1+{\displaystyle \frac{1}{4000}}{\displaystyle \sum _{i=1}^n}{x}_i^2-{\displaystyle \prod _{i=1}^n}cos\left({\displaystyle \frac{x_i}{\sqrt{i}}}\right)$	30	$[-600,600]$	0
$f_5\left(x,y\right)=2x^2-1.05x^4+{\displaystyle \frac{x^6}{6}}+xy+y^2$	2	$[-5,5]$	0
$f_6\left(x\right)={\sum }_{i=1}^n{x}_i^2+{\left({\sum }_{i=1}^{n}0.5i{x}_i\right)}^2+{\left({\sum }_{i=1}^{n}0.5i{x}_i\right)}^4$	30	$[-5,10]$	0

. The value of dim represents the number of variables, which is equal to n; $f_{\min }$ indicates the minimum value of the function. During the optimization, the size of the gray wolf population is set to 30 and the maximum number of iterations is set to 500. The obtained convergence curves of these two algorithms are illustrated in Figure 4. It can be found that as the iteration proceeds, the IGWO algorithm converges significantly faster than the GWO algorithm, which well demonstrates the effectiveness of the proposed IGWO algorithm.

The flowchart of the introduced parameter identification method is illustrated in Figure 5. First, the trajectory sensitivity-based sensitivity calculation is performed on the parameters of the PMSG model. According to the calculation results, the parameters with higher sensitivity indexes are determined as the key parameters. Then, the IGWO algorithm is adopted for identifying the parameters of the PMSG. After the initialization of the values of the parameters, using the cubic mapping, the positions of the wolves are updated according to (11)–(13) and (17)–(19) in each iteration. When the iteration process ends, the corresponding position of the $\alpha$ wolf is considered as the final identification result of the PMSG model.

5. Case Study

To verify the correctness of the introduced method in identifying the parameters of the PMSG model, a PMSG model was constructed on the simulation platform CloudPSS, whose topology is illustrated in Figure 1. All the simulations were carried out on a desktop PC with 32G-RAM and 2.10 GHz Intel Core i7-13700. The rated capacity of the PMSG was set as 1.5 MW. The information of the PMSG model is listed in

Table 2. The parameters of the PMSG.

Parameter	Value
rated power	1.5 MVA
rated rotor speed	1500 rpm
rated wind speed	11.17 m/s
turbine radius	36.5 m
moment of inertia	30,000 $\mathrm{kg}·{\mathrm{m}}^2$
rotor flux linkage	4.727 Wb
number of pole pairs	60
stator resistance	0.008 $\mathsf{\Omega }$
stator inductance	0.68 mH

. For this system, a three-phase short-circuit fault that lasted 0.2 s occurred outside the GSC at $t=3.5$ s. In addition, the wind turbine started at $t=2 \mathrm{s}$ during the simulation.

During the parameter identification process of the PMSG, the values of the key parameters are represented by the positions of the wolves. The corresponding fitness value can be calculated by [6]

$$V_{\mathrm{F}}=\sqrt{{\displaystyle \frac{1}{N}}\sum {\left(P^{*}-P\right)}^2}+\sqrt{{\displaystyle \frac{1}{N}}\sum {\left(Q^{*}-Q\right)}^2}$$

(20)

where N is the number of sampling points; $P^{*}$ and $Q^{*}$ indicate the reference values of the active and reactive power between $t=3.5$ s and $t=4$ s, respectively; P and Q represent the simulation results of the active and reactive power, respectively.

5.1. Parameter Identification Using the Proposed Method

First, sensitivity calculations were performed on the parameters of the PMSG, to select the key parameters for identification. The identification ranges for the parameters are listed in

Table 3. The identification ranges of the parameters to be identified in the PMSG model.

Parameter	Quantity	Identification Range
$K_{\mathrm{p}1}$	voltage outer-loop control proportional gain in GSC	0–10
$K_{\mathrm{p}2}$	voltage inner-loop control proportional gain in GSC	0–1
$K_{\mathrm{p}3}$	voltage inner-loop control proportional gain in GSC	0–1
$K_{\mathrm{p}4}$	proportional gain in chopper control	0-10
$K_{\mathrm{p}5}$	proportional gain in MPPT control	0–200
$K_{\mathrm{i}1}$	voltage outer-loop control integral time constant in GSC	0–0.1
$K_{\mathrm{i}2}$	voltage inner-loop control integral time constant in GSC	0–1
$K_{\mathrm{i}3}$	voltage inner-loop control integral time constant in GSC	0–1
$K_{\mathrm{i}4}$	integral time constant in chopper control	100–300
$K_{\mathrm{i}5}$	integral time constant in MPPT control	0–0.001

. According to (7) and (8), the sensitivity indexes of the parameters were calculated and listed in

Table 4. The sensitivity indexes of the parameters to be identified.

Parameter	Real Power Sensitivity ($\times {10}^{-3}$)	Reactive Power Sensitivity ($\times {10}^{-3}$)
$K_{\mathrm{p}1}$	0.526	0.017
$K_{\mathrm{i}1}$	0.184	0.007
$K_{\mathrm{p}2}$	3.218	0.297
$K_{\mathrm{i}2}$	2.378	0.118
$K_{\mathrm{p}3}$	3.106	13.38
$K_{\mathrm{i}3}$	0.0402	2.221
$K_{\mathrm{p}4}$	0.139	0.008
$K_{\mathrm{i}4}$	0.021	0.001
$K_{\mathrm{p}5}$	0.56	0.017
$K_{\mathrm{i}5}$	259.45	2.501

. Considering the sensitivity of the PMSG transient power to the parameters, the table shows that five of the parameters had higher sensitivity than the other parameters. Therefore, $K_{\mathrm{p}2}$, $K_{\mathrm{i}2}$, $K_{\mathrm{p}3}$, $K_{\mathrm{i}3}$, and $K_{\mathrm{i}5}$ were determined as the key parameters.

Then, the key parameters were identified, using the proposed IGWO algorithm. The identification ranges of these five parameters were the same as those shown in Table 3. The size of the gray wolf was set to 30, and the upper limit of the iterations was set to 40. To test the performance of IGWO in parameter identification, IGWO, GWO, and PSO were all adopted, to identify the selected parameters of PMSG model. The identification results of these three algorithms are listed in

Table 5. The parameter identification results of the three optimization algorithms.

Parameter	PSO (Error)	GWO (Error)	IGWO (Error)	True Value
$K_{\mathrm{p}2}$	0.0902 (9.8%)	0.1 (0)	0.1003 (0.3%)	0.1
$K_{\mathrm{i}2}$	0.493 (1.4%)	0.486 (2.8%)	0.505 (1%)	0.5
$K_{\mathrm{p}3}$	0.118 (18%)	0.102 (2%)	0.1 (0)	0.1
$K_{\mathrm{i}3}$	0.206 (58.8%)	0.516 (9.8%)	0.5 (0)	0.5
$K_{\mathrm{i}5}$	0.00013 (30%)	0.0001 (0)	0.00011(10%)	0.0001

, and the convergence curves are illustrated in Figure 6. It is clear that the error of the IGWO parameter identification was the smallest, which shows that the IGWO algorithm has better optimization capability in identifying the parameters of the PMSG model compared with the PSO and GWO algorithms. Furthermore, it can be found that the fitness value of the IGWO algorithm decreased very rapidly in the initial few generations. This indicates that the introduction of cubic mapping can improve the initial values of the GWO algorithm and accelerate the optimization process. In addition, the IGWO algorithm can obtain the optimal solution at the 15th iteration, which demonstrates that the proposed IGWO algorithm has a faster convergence rate compared with the other two methods.

To validate the accuracy of the identified parameters in Table 5, simulations with the real parameters and the identified parameters were conducted and the simulation results are illustrated in Figure 7 and Figure 8. Since the wind turbine started at $t=2 \mathrm{s}$, the values of active power and reactive power were both zero between $t=0 \mathrm{s}$ and $t=2 \mathrm{s}$. Therefore, the waveforms of active power and reactive power both started at $t=2 \mathrm{s}$. As can be observed, the final results obtained with the identified parameters are generally consistent with the reference results. This indicates that the IGWO algorithm is effective for identifying the parameters of the PMSG model. In addition, the 2-norm errors for the active power and reactive power were 0.01 and 0.012, respectively, which were both tiny.

5.2. Model Validation under Different Operating Conditions

To validate the adaptability of the identification results under different operating conditions, simulations under different constant wind speeds were also conducted, and the simulation waveforms are depicted in Figure 9. It shows that the simulation results from the identified parameters are coincident with the reference results for three different wind speeds, which well demonstrates the accuracy of the identified parameters. Furthermore, the 2-norm errors for the simulation results under different wind speeds were calculated and are listed in

Table 6. The 2-norm errors for the simulation results for different wind speeds.

	${\mathit{V}}_{\mathit{wind}}=5\mathbf{m}/\mathbf{s}$	${\mathit{V}}_{\mathit{wind}}=7\mathbf{m}/\mathbf{s}$	${\mathit{V}}_{\mathit{wind}}=10\mathbf{m}/\mathbf{s}$
P	0.0067	0.0097	0.015
Q	0.0005	0.001	0.0021

. It can be found that the errors were all very small.

In addition to the constant wind speed, a simulation was also conducted with the identified parameters in which there was a step change in wind speed. The wind speed step time was set to $t=2.3 \mathrm{s}$ and the three-phase short-circuit fault was removed. The simulation results are illustrated in Figure 10. It can be seen that the simulation results from the identified parameters align closely with the reference results for the step wind speed, which demonstrates that the obtained parameters using the proposed method were accurate.

6. Conclusions

A trajectory sensitivity analysis and IGWO algorithm-based method was proposed for identifying the parameters of the PMSG model in this manuscript. Our theoretical analysis and simulation results show that the introduced method can obtain accurate results when dealing with the parameter identification of the PMSG model. The final results demonstrate that the proposed method has a better optimization effect and a faster convergence speed compared with other optimization algorithms. In addition, the simulation results show that the obtained results are also accurate under different operation conditions. The proposed method can identify the parameters of the PMSG-based wind turbine model, which can ensure that the simulation results can accurately characterize the physical objects. This is beneficial to enhancing the stability of wind power grid-connected system operation. In future research, the proposed method could be further developed in regard to optimization algorithm improvement and simulation parallel computing. In addition to this, it would also be valuable to further study the stability advance online analysis method based on parameter identification results in a scenario of random wind speed fluctuation.