1. Introduction
Due to their high efficiency, simple structure, low cost, wide speed range, and high environmental adaptability, Synchronous Reluctance Machines (SynRMs) are good alternatives to induction motors (IM) in fans, pumps, and compressors [
1] and powerful competitors of permanent magnet synchronous machines (PMSM) in traction applications [
2]. SynRMs are a highly nonlinear and strongly coupled system due to their self- and cross-saturation effects, which directly affect the torque output, power factor, excitation level, and stable operation area. To control SynRMs effectively, it is critical to evaluate the actual nonlinear behaviors of magnetic circuits based on accurate models [
3,
4,
5]. Therefore, a nonlinear magnetic model [
6] that satisfies the reciprocity conditions and inherent properties is introduced. However, the accuracy of the model is still limited by the rationality of the parameter identification algorithm because of the multimodal problems.
Many researchers have contributed to the parameter identification algorithms to obtain optimal solutions in recent decades [
7], such as genetic algorithm (GA) [
8], ant colony optimizer (ACO) [
9], particle swarm optimization (PSO) [
10], monarch butterfly optimization (MBO) [
11], and grey wolf optimization (GWO) [
12]. These metaheuristic algorithms have achieved satisfactory results in single objective optimization. However, the multimodality of parameter identification places more stringent requirements on traditional metaheuristic algorithms. To cope with the problems mentioned above of traditional metaheuristic algorithms in multimodality, various algorithms have been developed. In [
13], multistage ant colony optimization (MSACO) was raised for direct instantaneous torque control of switched reluctance motor drives. This algorithm considered the multi-dimension condition. However, it overemphasized the exploitation ability, similar to the original ant colony optimization, which induced the solution trapped into pause best solution. Paper [
14] combined Harris Hawks optimization with a Sine-Cosine algorithm and applied it to a hybrid renewable energy system. Paper [
15] mixed biogeography-based optimization with a Sine-Cosine algorithm to satisfy high-dimensional global optimization problems [
16]. Mohamed I. Abdelwanis mixed Particle Swarm Optimization (PSO) with Jaya optimization to fit the rapid response demand of induction motors [
17]. Rahimi, A. promoted a Bat-inspired algorithms by considering chaotic behavior of PMSM [
18]. Dalia Yousri promoted Grey Wolf Optimization (GWO) by introducing a chao map for applications to PMSM [
19].Moreover, researchers made many efforts to promote the stochastic optimization algorithms applied in the parameter identification for induction motors [
20,
21]. However, their mutation directions are uncertain, and their mutation strategies are based on gradient. Thus, their calculation speed is relatively slow, and they tend to converge prematurely when dealing with multimodal problems. Additionally, an uncomplicated but valid algorithm named the Rao algorithm is proposed [
22]. The distinguishing feature of Rao algorithms from other metaheuristic algorithms is the novel update mechanism of the optimal solution search path. The interaction of the algorithm among individuals was guided by fitness evaluation, which enhanced the exploitability. However, the interaction process ignored diversity’s value, which also induced premature convergence.
To solve the problem of early convergence, a self-adaptive synergistic optimization (SSO) was proposed to identify the parameters of the SynRM magnetic model [
23]. The mutation strategy of the algorithm is guided by introduced parameters based on probability, which could offer updated directions in the global scope and avoid falling into local optimums. However, when the current solutions are certainly not at the global optimum, the searchability is provided by algorithm Rao-3 [
24]. The searchability of Rao-3 is relatively weak, so it is hard to get rid of the current worst solutions. Recently, a simple yet effective metaheuristics method named the Sine-Cosine algorithm (SCA) has attracted much attention [
14] due to its strong exploration ability, simple structure and self-adaptive balance capability.
However, SCA tends to get stuck in adjacent optima and uneven exploitation [
25], resulting from its weak exploitability. In [
26], SCA was used in urban traffic light scheduling. In [
27], SCA was applied to constrained engineering optimization problems. However, when the complexity of the problem increased, SCA will over-consume calculation resources in exploration, also inducing premature convergence.
Based on the above factors, a hybrid optimizer for the parameter identification of SynRM is proposed in this paper, with superior exploration and exploitation. The main contributions of this paper can be summarized as follows:
- 1)
An improved position updating mechanism with hybridized SCA, weighted fitness values, and personal historical best is designed in SSO search;
- 2)
A collaboration-based hybrid SSO-SCA optimizer (SCSSO) is developed.
The rest of the paper is organized as follows.
Section 2 presents the magnetic model of SynRMs.
Section 3 and
Section 4 briefly introduced the SSO algorithm and SCA, respectively. The working principle of the proposed SCSSO algorithm has been illustrated in
Section 5.
Section 6 revealed the statistical results of SCSSO and other compared algorithms. Finally, the conclusions are drawn in
Section 7.
4. Sine-Cosine Algorithm
Seyedali Mirjalili proposed the Sine-Cosine algorithm in 2016. The algorithm introduced a trigonometric function to stochastic population-based optimization [
14] and realized a wide range search and gradually approaching solutions in the exploitation process. The working principle of SCA is introduced.
Its mutation strategy is as follow:
where
,
,
are the same as ones in (21) to (23), a, t, T represents constant, the current number of iteration and the maximum number of iteration, respectively. The range of
and search detail of SCA are as
Figure 3 shows.
As
Figure 3 shows, the value of
decided the search direction of the SCA algorithm. Additionally, four parameters decide the search path of SCA.
defines the direction.
dictates how far the movement should be towards or outwards the destination.
provides a random weight for the destination in order to stochastically emphasize (
) or deemphasize (
) the effect of destination in defining the distance. Finally, the parameter
equally switches between the sine and cosine components. As mentioned above, SCA holds much randomness. Algorithm 3 shows the pseudo code of SCA.
Algorithm 3 Pseudo code of Sine-Cosine algorithm |
1: Initialize a set of serach agents (solutions)(X); |
2: Do |
3: Evaluate each of the search agents by the objective function; |
4: Update the best solution obtained so far (P = X*); |
5: Update r1, r2, r3, and r4; |
6: Update the position of search agents using (24); |
7: While (t < maximum number of iterations) |
8: Return the best solution obtained so far as the global optimum. |
In Algorithm 3, X* represents the best individual in current searching process. The Sine-Cosine algorithm’s search process is adaptive. There is not another parameter that needs to be trained. Moreover, it has a wider search range.
5. Sine-Cosine Self-Adaptive Synergistic Optimization
As stated above, when the current candidate is absolutely not the best solution, and the value of
is small, SSO uses Equation (24) to update the direction of exploration. The searching process is shown as
Figure 4.
The next position region is restricted by two reasons. One is the worst solutions. Another is the last candidates. Resulting from the two factors, the next position region is the Rhombic region.
As stated above, the search region of SCA is relatively narrow. However, the search range of SCA is a circle around current candidate, as shown in
Figure 3. Moreover, SCA and SSO both automatically balance exploration and exploitation, and SCA has a much wider search range. Thus, if we combine SSO with SCA, the new SC-SSO remains self-adaptive. In addition, SCA replaced Rao-3, offering stronger searching ability.
Figure 5 shows the framework of SC-SSO.
As shown in
Figure 5, Rao-3 was replaced after the random selection process. When two individuals are randomly selected, a random number decides the equal to modify the solution. If Ps > rand, it proves that current solutions are still valuable and thus continue to exploit around the solution. However, Ps < rand proves that current solutions are worse than surrounding candidates. Then, we use a method to explore.
The standard SSO uses the Rao-3 algorithm to explore, while Rao-3’s search ability is relatively weak. Therefore, SCA is a better alternative.
As shown in
Figure 5, the structure of the self-organization mechanism is simple and clear and has no extra parameters. During the operation of the algorithm, the direction of evolution is automatically balanced, which means the convergence strength is self-adaptively managed.
In SCSSO,
Ps remains the index of evaluating the capability of current candidates. When the
xk is the best individual, the next searching direction is decided by Equation (21), as
Figure 6 shows.
When the
xk is not the best solution, a method of comparing
Ps with random numbers has been carried out. If
Ps is greater than a random number, we could use Equation (23) to search the best solution around current solutions, but if
Ps is smaller than a random number, a much larger search region should be carried out. The search region of SCA is a circle around the destination, as shown in
Figure 3, so it could help to get rid of the current worst solution.
6. Experimental Results
In the field of optimization using meta-heuristics and evolutionary algorithms, several tests and experiments should be carried out in order to test the efficiency, accuracy, and robustness of proposed algorithms. In these experiments, FEA data are selected as a benchmark, while the error between the calculation results of object functions and reference data are inferred to RMSE, so that the accuracy and robustness could be evaluated. The analysis software used in this paper is MATLAB. Algorithms are compared by the best RMSE values in 30 populations. Through the best (B), and worst (W) values, we can evaluate the accuracy of them, and the standard deviation (SD) and mean (M) represent the stability of them.
6.1. Reference Data
The reference data come from the finite element analysis of the designed 2 kW prototype SynRMs. The design parameters of the prototype are summarized in
Table 1.
The result of FEA illustrated the nonlinearity of SynRMs flux linkage. As
is increases, the nonlinearity of
ψs becomes increasingly obvious. As shown in
Figure 7a, when
id reaches 10A, the curve of
ψd rising becomes slower. The same phenomena appeared in
ψq. This is so-called self-saturation. Additionally, when
id is fixed, the value of
ψd decreases as
iq increases. In
Figure 7b, the
ψq decreases more rapidly than
ψd in (a). This
id,
iq influencing mutual flux linkage phenomena is called cross-saturation.
To prove the performance metrics of the enhanced SC-SSO, comparative experiments are carried out with other well-performed algorithms, including original Rao algorithms, Ant Lion Optimization (ALO) algorithm [
33], differential evolution with biogeography-based optimization (DE/BBO) [
34] and JAYA algorithm [
35]. For the sake of fairness, different algorithms retain the same parameter configurations, as
Table 2 shows.
6.2. Results on q-Axis Flux Linkage Model
For the q-axis flux linkage model, 11 algorithms are independently implemented 30 times to obtain their self-saturation and cross-saturation parameters. RMSEq is taken as the index of accuracy evaluation. The best parameters and best RMSE are shown to compare in
Table 3.
In
Table 4, B M and W mean the best mean and worst RMSE values in 30 populations. SD represents standard deviation of 30 RMSE values for each algorithm. As
Table 4 shows, the SCSSO offered the best RMSEq value (2.2339 × 10
−4); the second-best value (2.2443 × 10
−4) was offered by RAO-1; and the worst value (0.0022) was offered by CLPSO, which was ten times that of SCSSO. The best RMSEq of SSO and SCA was 1.56 times and 10 times that. of SCSSO. Compared with other well-known algorithms, SCSSO is highly performed. To further illustrate the excellent results of the parameter identification, the
q-axis flux linkage map is reconstructed using the best results of the SCSSO algorithm.
Table 4 explains the statistical results, and SCSSO ranks first in the comparison. Moreover, the RAO-1 was the second best RMSE, but its mean value was 0.0017, nearly 20 times that of SCSSO. The second-best mean value (4.8678 × 10
−4) was provided by BLPSO, 1.76 times that of the proposed algorithms. Additionally, SCSSO is superior to SCA in statistical results. Although the standard deviation of SCSSO is inferior to SSO, the best RMSEq, worst RMSEq and mean value are superior.
Figure 8 is the plots of the best RMSEq of the proposed algorithm and other compared ones. As indicated above, the standard deviation of the new algorithm was the smallest in the 11 algorithms, and the best value was the smallest in this comparison.
It can be seen in
Table 3 and
Table 4 that the best value, mean value and worst value of traditional simple algorithms are one hundred times larger than other improved algorithms. The standard deviation is at least ten times larger than the other 13 algorithms. This proved that the traditional algorithms are not suitable for multidimensional problems.
Figure 9a takes FEA data as a surface and uses the SCSSO algorithm data as the fitting ball to illustrate the accuracy of the SCSSO algorithm.
Figure 9b illustrates the mismatch between estimated data and FEA data.
As shown in
Figure 9b, the calculation results of the proposed algorithm have only a 6% mismatch with the reference data in the edge of the model, which is attributed to the modeling method and the calculation error of FEA at the model boundary. Under many circumstances, the error of mismatch does not exceed 1%, which means the model with the identified parameters can be applied in actual engineering.
6.3. Results on d-Axis Flux Linkage Model
As the second stage of SynRMs magnetic model parameter identification, the extraction of the d-axis model parameters is similar to the process mentioned in the previous subsection. The best cross-saturation coefficients are obtained in the previous process. Therefore, only four self-saturation parameters need to be identified.
It can be seen from
Figure 10 that the estimated data suit the reference perfectly. Moreover, the errors of d-axis flux linkage are below 1%.
Table 5 lists the best estimated parameters and the best RMSEd (4.0816 × 10
−4) of different algorithms. The SCSSO keeps the best RMSEd value (4.0816 × 10
−4).
Table 6 illustrates that the best RMSE values of the 11 algorithms are resultant, but the mean value, the worst RMSE and standard deviation of the 11 compared algorithms are orders of magnitude larger than the proposed algorithm. Especially with DE\BBO, its mean value was almost 10 times that of SCSSO. The worst RMSE value of GWO was nearly 20 times that of the proposed algorithms. Additionally, the standard deviation of GWO was nearly 60 times that of SCSSO. The SCSSO has superior performance to SCA and SSO in any aspect.
Figure 11 shows the plots of the best RMSEd of the proposed algorithm and other compared algorithms. As shown above, the best and worst RMSE, mean value and standard deviation were the smallest of all.
An experiment in an SynRM prototype was carried out based on the 2 kW SynRM prototype and MicroLabBox dSPACE experimental platform, as shown in
Figure 12. Its parameters are shown in
Table 1. The hysteresis brake is used as the brake load. The torque sensor is installed on the shaft between the SynRM prototype and the hysteresis brake to measure the torque generated by the SynRM; the customized 55 kW three-phase inverter is powered by the 9 kW DC power supply to drive the motor; a MicroLabBox dSPACE ds1202 with a 2 GHz dual-core real-time processor and programmable FPGA is used as the system controller for implementing the control algorithm, sending pulses to the inverter, and sampling and analyzing information of the measured currents, position and torque of the motor. In the experiment tests, the sampling time is chosen as 100 μs.
To verify the applicability of the proposed algorithm, the results of the proposed algorithm obtained were compared with the traditional look-up table method.
The experimental results are shown in
Figure 13.
It can be seen from
Figure 13 that the flux linkages and incremental inductances can highly fit the traditional method. Additionally, because of the differentiability of objective function, the incremental inductances are continuous.
6.4. Results Discussion
The comparison results demonstrate that the proposed SCSSO algorithm has much better accuracy and effectiveness for solving the parameters identification problem of the SynRMs magnetic model, and its performance is competitive in contrast with all compared algorithms. The difference between the best RMSEd and the best RMSEq provided by the proposed SCSSO algorithm can be clearly observed. On one hand, the accuracy of the experimental results in the
q-axis model is considerable, and the algorithm error is inevitably introduced into the
d-axis model with the application of cross-saturation parameters. On the other hand, the energy loss of the rotor region corresponding to the
d- and
q-axis is different. Those losses are not reflected in the cross-saturation parameters under conservative field assumptions. This makes the influence of the loss contained in the standard datasets equivalent to a kind of noise, which is accumulated and introduced into the identification result of the
d-axis self-saturation parameters. Fortunately, the values of best RMSEd and RMSEq are in the same order of magnitude. Thus, the superior accuracy and robustness of the proposed SCSSO algorithm can be explained in one aspect, while its stability was contrasted by itself under a different NP. Thus, an experiment was carried out to test the robustness and stability of SCSSO. In this experiment, the SCSSO algorithm with NP = 10, 20, 30, 40 and 50 was tested and compared, maintaining the Max_FES at 30,000. The statistical results are presented in
Table 7.
It can be clearly seen that due to the different dimensions of the problem, the optimal population size for different problems is also different. Hence, the most appropriate population size for the parameter identification of d- and q-axis flux models is set to 30.