Distributed Parameter Identification Framework Based on Intelligent Algorithms for Permanent Magnet Synchronous Wind Generator

Abstract

Parameter identification of a permanent magnet synchronous wind generator (PMSWG) is of great significance for condition monitoring, fault diagnosis, and robust control. However, the conventional multi-parameter identification approach for a PMSWG is plagued by deficiencies, including its sluggish identification speed, subpar accuracy, and susceptibility to local optimization. In light of these challenges, this paper proposes a distributed parameter identification framework based on intelligent algorithms. The proposed approach involves the deployment of SSA, DBO, and PSO algorithms, leveraging golden sine ratio and Gaussian variation strategies for multi-parameter optimization and performance enhancement. Second, the optimal solutions of each intelligent algorithm are aggregated to achieve overall optimization performance enhancement. The efficacy of the proposed method is substantiated by a 6 MW PMSWG parameter identification practice simulation result, which demonstrates its superiority. The proposed method was shown to identify parameters more quickly and effectively than the underlying algorithms, which is of great significance for condition monitoring, fault diagnosis, and robust control of the PMSWG.

1. Introduction

In the context of the worldwide increase in demand for renewable energy sources, wind power technology has become a primary focus of research and implementation [1,2]. Among the diverse array of wind turbine technologies, permanent magnet synchronous wind generators (PMSWGs) have emerged as a preferred option within the industry owing to their exceptional energy conversion efficiency, unparalleled system reliability, and reduced maintenance costs [3,4]. However, this technological advancement is accompanied by a series of new challenges that require further exploration and resolution by industry.

In practical applications, PMSWGs encounter the repercussions of fluctuating operating environments and conditions including torque, angular velocity, and temperature [5]. These factors result in substantial nonlinear characteristics and pronounced coupling effects on the performance parameters of the generator. In dynamic working environments, the motor parameters of the PMSWG are subject to variation owing to external conditions, resulting in discrepancies between the motor control parameters and design parameters [6]. This not only affects the performance of the PMSWG system but also increases the complexity of the parameter identification work. Consequently, the effective identification of PMSWG parameters is crucial for technological development and application.

This study explores the two most common parameter identification techniques used for PMSWGs: offline and online identification methods. Offline identification relies on data-driven methods performed under the steady-state conditions of a system, such as Finite Element Analysis (FEA) and Static Frequency Response (SSFR) techniques [7]. However, offline identification methods are often deficient in terms of accuracy and robustness when the PMSWG is in a dynamic state. This results in difficulties in accurately reflecting the dynamic characteristics of the parameters during PMSWG operation. Online identification methods enable real-time or near real-time monitoring and adjustment of the system, facilitating more precise and responsive parameter estimation. Consequently, it enhances the stability and optimization of the system control [8,9]. The most common online identification methods currently include least squares, model reference adaptation (MRAS), the extended Kalman filter (EKF), and artificial intelligence algorithms. Wang et al. [10] proposed a parameter identification method based on the Coupled Recursive Total Least Squares (CRTLS) method. Li et al. [11] used the MRAS to identify stator resistance and d-q axis inductance, introducing a voltage drop compensation method based on Popov’s superstability theory without additional hardware. Vesely et al. [12] validated the parameter identification effect of the MARS method on the dSpace platform. Qi et al. [13] proposed an improved parameter identification method based on the MRAS, employing a variable bandwidth Linear Active Disturbance Rejection Control (ADRC) adaptive law, which effectively enhanced the interference resistance against high-frequency noise. Shi et al. [14] utilized the MRAS to estimate the rotor position and speed of an Interior Permanent Magnet Synchronous Motor (IPMSM) while employing the EKF to identify the permanent magnet flux. Yuan et al. [15] combined a Particle Filter (PF) and the EKF, proposing the EKPF algorithm. Udomsuk et al. [16] employed the EKF for online parameter estimation of three-phase induction motors.

Despite the success of these methods in relevant parameter identification practices, their inherent limitations are evident in practical applications. For instance, in the context of the least squares method, the convergence of the algorithmic iteration is found to be relatively slow, the utilization of the data is found to be limited, and the capacity to effectively address the nonlinear characteristics of PMSWGs during operation is found to be inadequate. Model reference adaptation is constrained by computational costs, and this method may exhibit substantial steady-state errors. The Kalman filter method has been shown to have suboptimal dynamic tracking performance during the identification process because of significant computational demands.

To overcome the shortcomings of these methods, researchers are exploring the advantages of integrating multiple methods or developing new algorithms and technologies. Avdeev et al. [17] utilized the GA algorithm to identify PMSM parameters; Bui et al. [18] addressed the issues of inverter nonlinearity and rotor position measurement errors in traditional parameter estimation through the use of deep neural networks. Zhai et al. [19] proposed a parameter identification method for the PMSG based on trajectory sensitivity analysis and an improved gray wolf optimization (IGWO) algorithm. Jiang et al. [20] proposed a multi-parameter identification method based on the IGWO algorithm. Parameter sampling is conducted by injecting negative sequence d-axis current. Gao et al. [21] proposed a parameter identification method for the PMSG based on the PSO algorithm. Zhou et al. [22] proposed a modified fuzzy particle swarm optimization algorithm (MDFPSO), which effectively enhanced the convergence speed and identification accuracy of the algorithm. Su et al. [23] introduced an adaptive simulated annealing algorithm and proposed a multi-parameter identification method for the PMSM based on the MRAS and simulated annealing particle swarm optimization (MRAS-SAPSO).

Nevertheless, the aforementioned PMSWG parameter identification methods based on artificial intelligence algorithms continue to exhibit inherent limitations. The parameter identification method based on a neural network algorithm requires a substantial amount of training data, is computationally complex, and is difficult to deploy in practical applications. The parameter identification method based on the traditional swarm intelligence optimization algorithm exhibits a slow convergence speed and is susceptible to becoming stuck in local optima, resulting in identification results that are still subject to errors. To address the aforementioned gaps, this study proposes a distributed parameter identification framework based on intelligent algorithms for the precise and rapid identification of PMSWG parameters. The primary innovations of this study are as follows:

(1) A distributed parameter identification framework based on intelligent algorithms was proposed. This framework integrates the superior performance of different intelligent algorithms to achieve accurate and fast identification of the PMSWG parameters with simple deployment.

(2) Two novel approaches were introduced to enhance the performance of various intelligent algorithms. These approaches achieve this enhancement by incorporating the golden ratio improvement strategy and the Gaussian mutation strategy. These strategies facilitate a balance between convergence performance and global optimality.

(3) The domain search strategy was implemented in the distributed framework aggregation process to enhance the overall optimization performance of the system.

(4) The distributed computing framework under consideration has been demonstrated to possess scalability and the capacity to combine the underlying algorithms in a manner that facilitates the integration of performance. This integration process enables the accurate identification of the PMSWG parameters, thereby demonstrating its efficacy in complex computing environments.

The remainder of this paper is organized as follows. Section 2 introduces the PMSWG model. In Section 3, several warm intelligence optimization algorithms are introduced. In Section 4, we introduce the distributed parameter identification framework and the improved algorithms. The proposed method is validated through simulation experiments in Section 5. Finally, the conclusions and discussions are presented in Section 6.

2. Mathematical Model

2.1. PMSWG Model

APMSWG is a multivariate, strongly coupled nonlinear system, and the following assumptions are established for ease of study:

(1) The stator winding is symmetrical;

(2) Ignore the influence of magnetic saturation and harmonics;

(3) Excluding eddy current loss and hysteresis loss;

(4) There is no damping winding on the rotor, and there is no damping effect on the permanent magnet.

According to the above assumptions, the model of the PMSWG d-q rotating coordinate system is established as shown in Figure 1.

In the d-q rotating coordinate system, the circuit model of the PMSWG is as follows:

$$\left\{\begin{array}{l}{u}_d=R{i}_d+L_d{\displaystyle \frac{d{i}_d}{dt}}-L_q\omega i_q\hfill \\ u_q=R{i}_q+L_q{\displaystyle \frac{d{i}_q}{dt}}+L_d\omega i_d+\omega {\psi }_f\hfill \end{array}\right.$$

(1)

where $u_d$ and $u_q$ are the voltages of the d-axis and q-axis, respectively; $R$ is the stator resistance; $i_d$ and $i_q$ are the currents of the d-axis and q-axis, respectively; $L_d$ and $L_q$ are the inductances of the d-axis and q-axis, respectively; $\omega$ is the rotor electrical angular velocity; and ${\psi }_f$ is the magnetic flux.

2.2. Parameter Identification Model

According to model (1), in theory, the presence of current signals with different characteristics should result in the corresponding voltage signals. Consequently, the corresponding parameters can be accurately calculated by employing the established voltage–current mapping relationships. However, owing to practical limitations, in PMSWG parameter identification practice, orthogonal current signals or step excitation signals are usually injected into the d-axis to obtain electrical signals in different states. Consequently, by injecting the excitation signals into the d-axis, the voltage vector equation of the PMSWG under different signals is obtained as follows:

$$\left\{\begin{array}{l}{u}_d^{\prime }=R{i}_d^{\prime }+L_d{\displaystyle \frac{d{i}_d^{\prime }}{dt}}-L_q\omega i_q\hfill \\ u_q^{\prime }=R{i}_q+L_q{\displaystyle \frac{d{i}_q}{dt}}+L_d\omega i_d^{\prime }+\omega {\psi }_f\hfill \\ u_d^0=-L_q\omega i_q\hfill \\ u_q^0=R{i}_q+L_q{\displaystyle \frac{d{i}_q}{dt}}+\omega {\psi }_f\hfill \end{array}\right.$$

(2)

In the formula, $u_d^{\prime }$ and $u_q^{\prime }$ represent the voltages on the d-axis and q-axis under the excitation signal, respectively. Where $i_d^{\prime }\ne 0$ is the excitation signal, which can be an orthogonal signal or a step signal; $u_d^0$ and $u_q^0$ represent the case in which the d-axis current is 0.

As demonstrated by the aforementioned model, the voltage vector equation of the PMSWG is full rank when the excitation signals are introduced. Consequently, by solving Equation (2) under a specified current, the parameters $〈R,L_d,L_q,{\psi }_f〉$ of the PMSWG can be determined. In practical applications, the following discretization of the aforementioned equation is required to facilitate the solution process:

$$\left\{\begin{array}{l}{u}_d^{\prime }\left(k\right)=R{i}_d^{\prime }\left(k\right)-L_q\omega i_q\left(k\right)\hfill \\ u_q^{\prime }\left(k\right)=R{i}_q\left(k\right)+L_d\omega i_d^{\prime }\left(k\right)+\omega {\psi }_f\hfill \\ u_d^0\left(k\right)=-L_q\omega i_q\left(k\right)\hfill \\ u_q^0\left(k\right)=R{i}_q\left(k\right)+\omega {\psi }_f\hfill \end{array}\right.$$

(3)

As indicated by the aforementioned equation, the objective of the PMSWG parameter identification model is to minimize the voltage error. The identified parameters function as decision variables and are defined as follows:

$$\begin{array}{cc}Min& {\displaystyle \sum _{k=1}^K\left[{\left(u_d\left(k\right)-{\widehat{u}}_d\left(k\right)\right)}^2+{\left(u_q\left(k\right)-{\widehat{u}}_q\left(k\right)\right)}^2\right]}\\ s.t.& R,L_d,L_q,{\psi }_f\in R^{+}\end{array}$$

(4)

where $u_d\left(k\right)$ and $u_q\left(k\right)$ represent the actual voltages measured on the d-axis and q-axis, respectively, under different current signals; ${\widehat{u}}_d\left(k\right)$ and ${\widehat{u}}_q\left(k\right)$ represent the theoretical voltages calculated based on Equation (3) under the given parameters, respectively.

Therefore, by injecting excitation current signals with different characteristics into the d-axis, the actual voltage signal characteristics at different moments are obtained. Theoretically, the theoretical voltage value under the corresponding current signal is calculated according to Equation (3) under the corresponding motor parameters without considering loss and with sufficiently high data accuracy. Therefore, minimizing the objective function (4) corresponds to the optimal parameter values.

3. Methodology

In this paper, three commonly used and better-performing swarm intelligence optimization algorithms are implemented in the underlying framework. These algorithms include the sparrow search algorithm (SSA), the particle swarm optimization algorithm (PSO), and the dung beetle optimization algorithm (DBO). The following is a methodological description of these algorithms.

3.1. Sparrow Search Algorithm

SSA is a swarm intelligence optimization algorithm inspired by the foraging behavior of sparrows and the natural behavior of evading foragers proposed by Xue et al. [24]. The algorithm simulates the foraging and anti-foraging behaviors of sparrows. In the avian population under study, individual sparrows are distinguished as explorers, followers, and scouts. In this instance, the formula for updating the explorer’s position is presented as follows:

$$x_i^{t+1}=\left\{\begin{array}{l}{x}_i^t\ast \exp \left({\displaystyle \frac{-i}{a\ast Maxiteration}}\right),R_2<ST\hfill \\ x_i^t+Q\ast L,R_2\ge ST\hfill \end{array}\right.$$

(5)

where $x_i^t$ represents the i-th individual in the t-th iteration; $a\in \left[0,1\right]$ is a random number; $R_2\in \left[0,1\right]$ and $ST\in \left[0.5,1\right]$ denote warning and safety values, respectively; Q is a random number obeying a normal distribution; and L denotes a row vector where each element is 1.

For followers, they need to enforce the following rules:

$$x_i^{t+1}=\left\{\begin{array}{l}Q\ast \exp \left({\displaystyle \frac{x_{worst}^t-x_i^t}{i^2}}\right),i>n/2\hfill \\ x_p^{t+1}+\left|x_i^t-x_p^{t+1}\right|\ast A^{+}\ast L,i\le n/2\hfill \end{array}\right.$$

(6)

where $x_{worst}^t$ represents the worst individual in the t-th iteration; $x_p^{t+1}$ represents the best individual in the t + 1-th iteration; and $A^{+}$ denotes a row vector where each element is randomly assigned the value 1 or −1.

The formula for updating the scouts’ position is presented as follows:

$$x_i^{t+1}=\left\{\begin{array}{l}{x}_{best}^t+\beta \ast \left|x_i^t-x_{best}^t\right|,f_i>f_g\hfill \\ x_i^t+K\ast \left({\displaystyle \frac{\left|x_i^t-x_{worst}^t\right|}{\left(f_i-f_w\right)+\epsilon }}\right),f_i=f_g\hfill \end{array}\right.$$

(7)

where $\beta$ is the step control parameter, which is defined as a normally distributed random number with a mean of 0 and a variance of 1; $x_{best}^t$ represents the global best individual; $K\in \left[-1,1\right]$ is a random number; and $f_i$, $f_b$, and $f_w$ are the fitness values for the i-th individual, the global best individual and the worst individual, respectively.

3.2. Particle Swarm Optimization Algorithm

The PSO algorithm is derived from the behavioral study of bird flocks’ predation. The fundamental premise of the algorithm is predicated on the notion of identifying optimal solutions through the collaborative efforts and information exchange among individuals within the group. PSO has gained significant traction in various fields, including function optimization, neural network training, and fuzzy system control, among others [25]. In this algorithm, the position and velocity of the particles are updated by the following equation:

$$v_i^{t+1}=w\ast v_i^t+c_1\ast r_1\ast \left(p_{best}-P_i^t\right)+c_2\ast r_2\ast \left(g_{best}-P_i^t\right)$$

(8)

$$P_i^{t+1}=P_i^t+v_i^{t+1}$$

(9)

where $v_i^t$ represents the velocity of the i-th individual in the t-th iteration; $w$ is the inertia weight, which is used to control the effect of the particle’s current velocity; $c_1$ and $c_2$ are the learning factors, which are used to control the degree to which particles move closer to the individual best position and the global best position; $p_{best}$ is the individual best position in the t-th iteration; $g_{best}$ is the global best position; and $P_i^t$ is the i-th individual in the t-th iteration.

3.3. Dung Beetle Optimization Algorithm

DBO was developed by Xue et al. in 2022. Inspired by the biological behavioral process of dung beetles, DBO exhibits characteristics such as strong optimization ability and rapid convergence speed [26]. The DBO algorithm comprises four phases: Dung Beetle Rolling, Egg Laying, Dung Beetle Feeding, and Dung Beetle Stealing. During the Dung Beetle Rolling process, the dung beetle’s navigation is primarily dependent on sunlight, with the position of the individual beetle updated by the following equation:

$$D{P}_i^{t+1}=D{P}_i^t+\alpha \ast k\ast D{P}_i^{t-1}+b\ast \Delta P$$

(10)

where $D{P}_i^t$ is the i-th individual in the t-th iteration; $\alpha$ takes the value −1 or 1; $k\in \left(0,0.2\right]$ is a random number; $b\in \left(0,1\right)$ is a random number; $\Delta P=\left|P_i^t-P_{worst}\right|$, and $P_{worst}$ is the worst individual.

When a dung beetle is impeded in its forward progress and is unable to continue, the position is consequently updated by Equation (11) with a probability of occurrence of 0.1.

$$D{P}_i^{t+1}=D{P}_i^t+\tan (\theta )\ast \left|D{P}_i^t-D{P}_i^{t-1}\right|$$

(11)

where $\theta \in \left[0,\pi \right]$ denotes the angle of deflection.

In the spawning stage, the search boundary is in dynamic form. The lower and upper boundaries of the optimization problem are denoted by Lb and Ub, respectively. The boundary of the spawning area is defined as follows:

$$\left\{\begin{array}{l}L{b}^{\ast }=\max \left\{\left(1-R\right)\ast P^{\ast },Lb\right\}\hfill \\ U{b}^{\ast }=\max \left\{\left(1+R\right)\ast P^{\ast },Ub\right\}\hfill \end{array}\right.$$

(12)

where $R=1-t/Maxiteration$; $P^{\ast }$ is the current optimal solution; and $L{b}^{\ast }$ and $U{b}^{\ast }$ represent the lower and upper boundaries of the dynamic spawning area, respectively.

Based on this, individuals are updated as follows:

$$D{P}_i^{t+1}=P^{\ast }+b_1\ast \left(D{P}_i^t-L{b}^{\ast }\right)+b_2\ast \left(D{P}_i^t-U{b}^{\ast }\right)$$

(13)

where $b_1$ and $b_2$ are random vectors.

When the ovoid hatches into a baby dung beetle, the dung beetle establishes an optimal foraging area to guide the baby dung beetle to be fed on, as follows:

$$D{P}_i^{t+1}=P^{\ast }+C_1\ast \left(D{P}_i^t-L{b}^b\right)+C_2\ast \left(D{P}_i^t-U{b}^b\right)$$

(14)

where $C_1$ and $C_2$ are random vectors; $L{b}^b$ and $U{b}^b$ represent the lower and upper boundaries of the dynamic spawning area based on the best individual, respectively.

During the stealing phase, the position of dung beetle individuals was updated by Equation (15).

$$D{P}_i^{t+1}=P^b+S\ast g\ast \left(\left|D{P}_i^t-P^{\ast }\right|+\left|D{P}_i^t-P^b\right|\right)$$

(15)

where S is a constant; g is a vector; and $P^b$ is the global optimal solution.

4. Distributed Parameter Identification Framework

4.1. Distributed Framework

Distributed systems represent a paradigmatic manifestation of collective intelligence, comprising individuals who are capable of collecting information, making decisions, and generating information. These individuals are interconnected through a distributed communication network, thereby forming a unified entity capable of accomplishing system-level tasks through collaborative means. In comparison with existing parameter identification methods, distributed parameter identification methods exhibit several notable advantages. Compared with the existing parameter identification methods, the distributed parameter identification method avoids the requirements of centralized ultra-large-scale computing and demonstrates significant advantages in data privacy protection and security, structural flexibility, functional robustness, convergence, and global optimality. Therefore, to address the limitations of traditional methods in PMSWG parameter identification practice, this paper proposes a distributed parameter identification framework based on intelligent algorithms. The framework structure and process are shown in Figure 2.

As illustrated in Figure 2, various population intelligence algorithms were implemented at the foundation of the distributed parameter identification framework. These algorithms include the SSA, PSO, and DBO. These three algorithms are among the most frequently employed intelligent optimization algorithms for the target, owing to their superior optimization capabilities and operational efficiency. It is noteworthy that in the underlying algorithms, operators have the flexibility to replace algorithms according to their preferences and even to freely add and deploy algorithms such as genetic algorithms and gray wolf optimization algorithms. Therefore, this distributed framework is scalable.

4.2. Improvement Strategy

As illustrated in Figure 2, the performance of the distributed parameter identification method was enhanced through the implementation of three improvement strategies: the golden sine ratio, Gaussian mutation, and domain search strategies. Two initial improvement strategies were employed to enhance the performance of the underlying algorithm, with the objective of accelerating the convergence rate and improving the optimality of the underlying algorithm. The third strategy, the domain search strategy, was employed during the aggregation stage to identify the optimal PMSWG parameters. The improvement strategies employed at various stages achieved a balance between convergence performance and global optimality.

The golden sine algorithm is a novel metaheuristic algorithm that was proposed by Tanyildizi et al. [27]. According to the relationship between the sine function and unit circle, the golden sine algorithm can traverse all points on the sine function and scan the entire unit circle, similar to searching for space in optimization problems. The incorporation of the golden sine ratio strategy into the underlying algorithm was demonstrated to enhance the convergence speed while maintaining a balanced global and local search capability, thereby improving the accuracy of the solution. The iterative method of the algorithm based on the golden sine ratio strategy is as follows.

$$X_{i,j,d}^{t+1}=X_{i,j,d}^t\left|\sin \left(r_1\right)\right|-r_2\sin \left(r_1\right)\ast \left|k_1{V}_{i,j,d}^{t+1}-k_2{X}_{i,j,d}^t\right|$$

(16)

Among them, $X_{i,j,d}^t$ represents the d-th parameter value of the j-th individual of the i-th algorithm in the t-th iteration; $r_1$ is a random number, $r_1\in \left[0,2\pi \right]$; $r_2$ is a random number, $r_2\in \left[0,\pi \right]$; and $k_1$ and $k_2$ use the golden ratio coefficient, which is calculated as follows:

$$k_1=a\left(1-\tau \right)+b\tau$$

(17)

$$k_2=a\tau +b\left(1-\tau \right)$$

(18)

where $\tau$ is the golden ratio, and its value is $\left(1-\sqrt{5}\right)/2$; $a$ and $b$ are constants, $a=-\pi$ and $b=\pi$, respectively.

Gaussian mutation is a frequently employed mutation operation strategy in optimization algorithms [28]. The implementation of this strategy enables the introduction of randomness and exploration, thereby allowing individual solution vectors to navigate the solution space of the problem. It has been demonstrated that an increase in the standard deviation leads to an enhancement in variability, thereby promoting the probability of a global search. Conversely, a decrease in the standard deviation is more conducive to the local search and convergence of the algorithm. The optimal solution update method based on the Gaussian mutation algorithm is as follows:

$$X_i^{\ast }=X_i^{\ast }\ast \left[1+k\times Gaussian\left(0,1\right)\right]$$

(19)

where $X_i^{\ast }$ denotes the optimal solution obtained by the i-th algorithm, $k$ is a random number, and $k\in \left[0,1\right]$; $Gaussian\left(0,1\right)$ represents a Gaussian-distributed random vector with a mean of 0 and a variance of 1.

In the aggregation stage, the optimal parameters are obtained by aggregating the optimal solutions of the various underlying algorithms. To enhance the global optimality of the aggregation stage, a domain search strategy is employed for parameter updates, with the update method outlined as follows:

$$X_{best}=X_{best}+\delta \left(X_{best}\right)$$

(20)

where $X_{best}$ is the optimal parameter aggregated by each algorithm and $\delta \left(X_{best}\right)$ is the domain of the optimal parameter $X_{best}$.

4.3. Parameter Identification Process

According to the aforementioned framework explanation (as shown in Figure 2) and improvement strategy explanation, the parameter identification process based on the distributed PMSWG is as follows:

(1) First, the operator deploys relevant intelligent algorithms at the bottom layer and sets various algorithm parameters such as population size and iteration times.

(2) In accordance with the parameter identification principle of the PMSWG, as demonstrated in Equation (3), the operator acquires the corresponding voltage signal data by injecting current signals with distinct characteristics into the d-axis. Subsequently, the experimental data were integrated into the underlying algorithm, and model (4) was designated as the optimization objective.

(3) Each algorithm is subjected to iterative updates based on its underlying principles, thereby yielding the optimal solution. On the other hand, updates to the population of algorithms are derived from the golden sine and Gaussian mutation strategies.

(4) In the aggregation stage, the optimal solutions of each algorithm are integrated, resulting in an aggregated optimal solution.

(5) Obtain the optimal generator parameters by aggregating the optimal solutions for domain search.

5. Simulation

In this section, the effectiveness of the proposed method is illustrated through the implementation of the distributed parameter identification algorithm programmatically based on Matlab 2024b software. The software platform for the motor model is ANSYS Electronics software. All experimental environments utilize the Windows 11 system, 16 GB of running memory, and a 2.30 GHz central processing unit. The programming language employed is Matlab 2024b.

In order to verify the performance of the proposed method in optimization, a comparison was made between the proposed improved algorithm and the basic algorithms, differential evolution algorithm, and electric eel foraging algorithm used for CEC test functions. The detailed results of this comparison are shown in Appendix A.

5.1. Parameter Settings

In order to verify the effectiveness and superiority of the proposed distributed parameter identification method in the PMSWG, the method is applied to simulation practice in this section. The parameter settings of the PMSWG are shown in

Table 1. Parameter settings of PMSWG.

Parameter	Range
Rated power/kW	6000
Stator resistance/Ω	0.3
D-axis inductance/H	0.0085
Q-axis inductance/H	0.0085
Flux/Wb	8
Number of pole pairs	60
Rated line voltage/V	720

. The parameters to be identified are stator resistance $R$, D-axis inductance $L_d$, Q-axis inductance $L_q$, and magnetic flux ${\psi }_f$.

In this distributed framework, SSA, PSO, and DBO are deployed at the bottom layer. The parameter settings for each algorithm were as follows: the maximum number of iterations was set to 100 and the population size was 50.

5.2. Parameter Identification of PMSWG

In this section, the implementation method of the distributed parameter identification framework mentioned above is applied to the identification of PMSWG parameters. Several improved underlying algorithms were used for simulation comparison to demonstrate the effectiveness and superiority of the proposed method. The total identification error convergence curves of these algorithms and the distributed algorithm (DA) are compared in Figure 3. As illustrated in Figure 3, the proposed distributed parameter identification method demonstrates the most expeditious convergence performance compared with the underlying algorithm. The proposed distributed parameter identification method can converge to the optimal solution within a finite number of iterations, indicating that the method achieves optimal convergence. Furthermore, given that disparate optimization algorithms function on the basis of distinct threads, it is imperative to consider the temporal dynamics of these processes. Consequently, from a temporal perspective, while the duration of each algorithm varies, the proposed DA method is consistent with the algorithm that requires the least time, thereby ensuring its efficacy.

In the context of optimality, a comparative analysis was conducted on the parameter identification results and errors of the various algorithms and DA methods, as illustrated in

Table 2. Comparison of parameter identification results and errors between different algorithms and DA.

Parameter	ISSA		IDBO		IPSO		DA
	Value	Error (%)	Value	Error (%)	Value	Error (%)	Value	Error (%)
Stator resistance/Ω	0.3003	0.1	0.3003	0.1	0.3056	1.8667	0.3001	0.0333
D-axis inductance/H	0.0087	2.3529	0.0086	1.1765	0.0089	4.7059	0.00851	0.1176
Q-axis inductance/H	0.00851	0.1176	0.0080	5.8824	0.0082	3.5294	0.00851	0.1176
Flux/Wb	8.005	0.0625	8.001	0.0125	8.002	0.025	8.001	0.0125

. According to the findings, the proposed DA exhibited superior optimization performance in comparison with the underlying algorithm. With regard to stator resistance identification, the DA method demonstrated the lowest identification error of 0.0333%. In comparison to the ISSA, IDBO, and IPSO algorithms, the DA method reduced errors by 0.0667%, 0.0667%, and 1.8334%, respectively. Furthermore, with regard to the identification of d-axis inductance and q-axis inductance, the recognition accuracy of each method is ranked as DA $\succ$ IDBO $\succ$ ISSA $\succ$ IPSO and DA = ISSA $\succ$ IPSO $\succ$ IDBO. In terms of magnetic flux recognition, the recognition accuracy of each method was ranked as DA = IDBO $\succ$ IPSO $\succ$ ISSA. From the comparison results, it can be seen that the underlying algorithms have specific advantages in identifying different parameters; therefore, the recognition performance of each algorithm was aggregated through distributed aggregation, achieving accurate identification of PMSWG parameters. The lowest identification error result also indicates the effectiveness and superiority of the proposed DA method in PMSWG parameter identification.

In order to compare the performance of different algorithms, 10 times simulation tests were repeated. Based on the SPSS analysis tool, Games–Howell was used to perform multiple tests, and the results showed that the DA method was superior to the other three methods. The detailed results of the comparison are provided in Appendix A.

Furthermore, a comparative analysis was conducted on the convergence details of various methods across different parameters. Figure 4 presents the comparison results of the convergence curves of various methods in stator resistance identification, Figure 5 presents the comparison results of the convergence curves of various methods in d-axis inductance identification, Figure 6 presents the comparison results of the convergence curves of various algorithms in q-axis inductance identification, and Figure 7 presents the comparison of the convergence curves of various methods in magnetic flux identification. According to the comparison of the convergence curves of various methods, the proposed DA-based parameter identification methods can converge faster and more stably to the target parameter values during the parameter identification process. A comparison of the proposed DA-based parameter identification method with the underlying algorithm revealed that the former yields more stable and reliable identification results. It is noteworthy that during the recognition process of different parameters, each algorithm experiences oscillations, particularly the IPSO algorithm. This instability can be attributed to the optimization objectives of the underlying algorithms, which were designed to address the overall recognition error. Consequently, this instability can lead to variations in the recognition accuracy of the individual parameters. In contrast, the proposed DA method not only considers the overall recognition accuracy but also ensures the recognition accuracy and stability of individual parameters. The findings of this study demonstrate the efficacy and reliability of the proposed DA parameter identification method for multi-parameter identification of a PMSWG.

6. Conclusions

Aiming at the multi-parameter identification problem of the PMSWG, this paper proposes a distributed parameter identification framework and method based on intelligent algorithms. At the fundamental level, the SSA, DBO, and PSO algorithms were implemented, leveraging golden sine and Gaussian variational strategies to optimize the search for distributed parameters. In the aggregation phase, the optimal solutions of each intelligent algorithm were aggregated with optimality. Concurrently, a domain search strategy is employed to enhance the optimal solutions during the aggregation phase. The effectiveness and superiority of the proposed distributed parameter identification framework were verified in the 6 MW PMSWG parameter identification practice. The simulation results demonstrate that the proposed method identifies the parameters of the PMSWG at the fastest iteration and achieves optimal results when compared with the underlying algorithm. This finding is of great significance for the state monitoring, fault diagnosis, and robust control of the PMSWG.

In light of the potential ramifications of this research, such as encompass hardware-based deployment implementation and the potential repercussions of error signaling, our subsequent focus will be directed towards these aspects of research and analysis. Consequently, the proposed distributed framework is deployed with hardware systems to verify its practicability. Additionally, we will deliberate on more pragmatic scenarios to further enhance the proposed methodology, thereby ensuring the reliability of the proposed approach.