Improved Estimation Procedure of Cage-Induction-Motor-Equivalent Circuit Parameters Based on Two-Stage PSO Algorithm

Abstract

This paper analyzes errors in the estimation of induction-motor-equivalent circuit parameters using an improved combined two-stage Particle Swarm Optimization (PSO) method. The proposed method accounts for variations in rotor parameters based on both linear and square root speed approximations, as well as two different approaches for the stator and rotor leakage reactance ratios. The first approach assumes that the starting rotor leakage reactance is equal to the stator leakage reactance, while the second considers them as distinct. Improvement of the algorithm consists of increasing the accuracy of the approximations of parameter changes on the rotor. Thanks to more accurate determination of the initial rotor parameters, both approximations provide better results in parameter estimation. The analysis involved sixteen induction motors with four different power ratings and four different pole numbers. The analysis aimed to assess the impact of these approximations and assumptions on equivalent circuit parameter estimation errors. The estimated torque-speed characteristics closely matched the manufacturer’s reference data, including starting, maximum, and full-load torques. The deviation of the estimated torque-speed characteristics from the reference characteristics, within a defined speed range, is defined as the mean absolute percentage error. Based on the obtained results, the mean absolute percentage error is complex and depends on rotor parameter speed approximations, stator and rotor leakage reactance ratios, and the full power of the induction motor.

1. Introduction

Induction motors are a cornerstone of industrial and commercial applications due to their robustness, simplicity, and efficiency. They are widely used across various industries, from manufacturing and transportation to powering devices such as pumps, conveyors, and compressors. Their ability to handle varying loads and compatibility with modern control systems makes them essential components of modern automation. Since controller performance depends on the accuracy of motor parameters used by the control algorithm, it is crucial to determine the equivalent circuit parameters precisely [1,2]. Various methods were proposed for estimating the steady-state-equivalent circuit parameters of induction motors using different optimization techniques. The goal was to improve accuracy by minimizing the error between estimated parameters and manufacturer data. Different methods were applied, including the following: improved moth flame optimization (IMFO) [3], improved particle swarm optimization (IPSO) [4], a two-stage optimization approach based on the Engineering Method [5], an artificial bee colony algorithm (ABC) [6], an iterative method using nameplate data [7], an immune algorithm (IA) [8], a PSO-based approach [9], a charged system search algorithm (CSS), and differential evolution algorithm (DEA) [10]. These optimization techniques estimate parameters by solving nonlinear equations and validate the results by comparing them with manufacturer data, classical methods, or other optimization techniques. Each approach aimed to enhance computational efficiency and accuracy, leveraging optimization strategies such as swarm intelligence, evolutionary algorithms, and iterative refinement to achieve more reliable parameter estimates.

In addition to equivalent circuit parameter estimation, metaheuristic optimization methods are widely used in motor design. These methods are applied in areas such as the design optimization of permanent magnet synchronous motor [11] and the optimization of the geometric parameters of induction motors to achieve high efficiency [12].

To reduce the number of unknown parameters and computation time in estimating induction-motor-equivalent circuit parameters, single-cage models often assume that the stator and rotor leakage reactances are equal. Alternatively, a constant ratio based on NEMA motor design categories can be applied [13,14,15,16,17]. In double-cage models, parameter estimation is simplified by assuming that the stator leakage reactance equals the starting rotor leakage reactance [18,19,20,21].

The assumption of equal stator and rotor leakage reactances is valid for wound rotor induction motors due to the symmetry in the three-phase stator and rotor winding. Based on this symmetry, identifying the parameters of the series branch of the induction-motor-equivalent circuit through a short circuit test is analogous to the short circuit test of three-phase transformers. However, in squirrel cage induction motors, where the rotor has a fundamentally different design, such as in short circuit cages with deep bars or double-cage rotors, this symmetry is absent. Moreover, users typically lack data on the geometry of the rotor and stator slots, which significantly influence the leakage reactances, even when the motor’s catalog data are available.

In [22], it was noted that during transient processes, such as starting an induction motor by direct connection to the grid or during load changes, the ratio of stator-to-rotor leakage reactances is not constant. Therefore, using constant parameters in the machine model results in inaccurate dynamic performance predictions. As a solution, a direct method was proposed to estimate and separate the stator and rotor leakage reactance parameters under both normal operating conditions and when the core is deeply saturated. The method uses a two-dimensional time-stepping finite element method (FEM) with a coupled circuit.

In [23], a straightforward, non-iterative method was introduced to estimate the equivalent circuit parameters of an induction motor using manufacturer data. This method treats the starting rotor leakage reactance and the stator leakage reactance as distinct and was applied to data from nearly 3000 motors. The dataset covered IEC and NEMA standards, voltages from 220 to 575 V, frequencies of 50 and 60 Hz, 1 to 4 pole pairs, and power ratings from 0.12 to 675 kW.

The study in [23] also highlighted the skin effect in double-cage motors during direct grid start-up. At start-up, current primarily flows through the upper cage, which has a smaller cross-sectional area, increasing the starting rotor resistance and torque. The upper cage slots in the rotor are much smaller than the stator slots, resulting in a longer leakage flux path around them. Consequently, the stator and rotor leakage reactances differ significantly at start-up, leading to distinct values for the starting rotor leakage reactance and stator leakage reactance. The variation in rotor parameters due to the skin effect in deep bars or double squirrel-cage rotors was modeled and presented as a function of slip. The additional importance of considering the influence of rotor geometry on motor performance was highlighted in [24], where the effect of rotor bar designs on motor performance was studied using the FEM.

These considerations led the authors of this paper to assume that the starting rotor leakage reactance and the stator leakage reactance are different during the estimation of equivalent circuit parameters and to analyze the resulting estimation errors. The analyses include comparisons of starting, maximum, and full-load torque values, torque-speed characteristics, and mean absolute percentage errors. These comparisons are based on two approaches: one assuming equal starting rotor and stator leakage reactances and the other treating them as distinct.

2. Theoretical Basis

In the previous paper by the authors [25], a combined method for estimating the equivalent circuit parameters of cage induction motors using the two-stage PSO was presented. In the first stage of optimization, the electrical parameters of the motor for nominal operating mode were estimated (R_s-stator resistance, R_r-rotor resistance, X_s-stator leakage reactance, X_r-rotor leakage reactance, R_Fe-core loss resistance and X_m-magnetization reactance), while in the second stage, the rotor parameters at motor start-up were estimated (R_{r_st}-starting rotor resistance and X_{r_st}-starting rotor leakage reactance). To simplify the estimation procedure, it was assumed that the starting rotor leakage reactance is equal to the stator leakage reactance (X_{r_st} = X_s).

The first objective function has the following form [25]:

$$O{F}_1=F_1^2+F_2^2+F_3^2+F_4^2,$$

(1)

where

$$F_1={\displaystyle \frac{\frac{3p}{{\omega }_s}{\left|I_r(n_{fl})\right|}^2\frac{R_r}{s_{fl}}–T_{fl\_mf}}{T_{fl\_mf}}},$$

(2)

$$F_2={\displaystyle \frac{\frac{3p}{{\omega }_s}{\left|I_r(n_{\max })\right|}^2\frac{R_r}{s_{\max }}–T_{\max \_mf}}{T_{\max \_mf}}},$$

(3)

$$F_3={\displaystyle \frac{\frac{P_n}{P_{fl}}–{\eta }_{fl\_mf}}{{\eta }_{fl\_mf}}},$$

(4)

$$F_4={\displaystyle \frac{\cos \left[\arctan \left(\frac{Q_{fl}}{P_{fl}}\right)\right]–p{f}_{fl\_mf}}{p{f}_{fl\_mf}}}.$$

(5)

More details about the first objective function (OF₁), including expressions and constraints, can be found in [25].

Modification of the Algorithm for Cage-Induction-Motor-Equivalent Circuit Parameter Estimation

To evaluate the error resulting from the assumption (X_{r_st} = X_s), the electrical parameters of induction motors with different rated powers and pole numbers were estimated using the two previously mentioned approaches. For the first approach (X_{r_st} = X_s), the two-stage PSO algorithm from [25] was employed. To implement the second approach (X_{r_st} ≠ X_s), the algorithm described in [25] was modified to account for the increased number of unknown electrical parameters. Since the assumption (X_{r_st} ≠ X_s) in the second approach does not affect the estimation procedure of electrical parameters for nominal operating mode, their estimation procedure remains the same as in [25] (i.e., no changes will be made to the objective function OF₁).

However, introducing a new unknown electrical parameter in the second stage of optimization (the starting rotor leakage reactance X_{r_st}) requires an extension of the objective function OF₂ from [25].

In the objective function OF₂, the starting rotor resistance R_{r_st} and starting rotor leakage reactance X_{r_st} are obtained by minimizing the deviation between the manufacturer and calculated values for starting torque and starting power factor. The objective function OF₂ is expressed as follows:

$$O{F}_2=F_5^2+F_6^2,$$

(6)

where

$$F_5={\displaystyle \frac{\frac{3p}{{\omega }_s}{\left|I_r(n_{st})\right|}^2\frac{R_{r\_st}}{s_{st}}–T_{st\_mf}}{T_{st\_mf}}},$$

(7)

$$F_6={\displaystyle \frac{\cos \left[\arctan \left(\frac{Q_{st}}{P_{st}}\right)\right]–p{f}_{st\_mf}}{p{f}_{st\_mf}}}.$$

(8)

The starting power factor can be calculated using the following expression:

$$p{f}_{st}={\displaystyle \frac{P_{st}}{\sqrt{{P_{st}}^2+{Q_{st}}^2}}},$$

(9)

Meanwhile the apparent, active, and reactive powers at motor start-up are obtained using the following expressions:

$$\underset{¯}{S}(n_{st})=3\cdot {\underset{¯}{U}}_{ph}\cdot {\left[{\underset{¯}{I}}_s(n_{st})\right]}^{\ast },$$

(10)

where ${\underset{¯}{U}}_{ph}$ is the stator phase voltage and $\underset{¯}{I}{}_s(n)$ is the stator current as a function of speed.

$$P_{st}=\mathrm{Re}\left\{\underset{¯}{S}(n_{st})\right\},Q_{st}=\mathrm{Im}\left\{\underset{¯}{S}(n_{st})\right\}.$$

(11)

The starting stator and rotor currents were obtained based on the following expressions:

$$\underset{¯}{I}{}_s(n_{st})={\displaystyle \frac{{\underset{¯}{U}}_{ph}}{R_s+j{X}_s+{\underset{¯}{Z}}_{cr}(n_{st})}},$$

(12)

$${\underset{¯}{I}}_r(n_{st})={\displaystyle \frac{{\underset{¯}{Z}}_{cr}(n_{st})\cdot {\underset{¯}{I}}_s(n_{st})}{\frac{R_{r\_st}}{s_{st}}+j{X}_{r\_st}}},$$

(13)

where the starting equivalent impedance ${\underset{¯}{Z}}_{cr}(n_{st})$ is represented by the following equation that can be written based on the induction motor equivalent circuit:

$${\underset{¯}{Z}}_{cr}(n_{st})={\displaystyle \frac{1}{\frac{1}{\frac{R_{Fe}\cdot j{X}_m}{R_{Fe}+j{X}_m}}+\frac{1}{\frac{R_{r\_st}}{s_{st}}+j{X}_{r\_st}}}}.$$

(14)

The following constraints are applied in the second stage of the optimization:

$$R_{r\_st}>R_r,X_{r\_st}<X_r.$$

(15)

The motor’s torque-speed characteristics were obtained using the following expression [16,25]:

$$T(n)={\displaystyle \frac{3p}{{\omega }_s}}{\left|{\displaystyle \frac{{\underset{¯}{Z}}_{cr}(n)\cdot {\underset{¯}{I}}_s(n)}{\frac{R_r(n)}{s}+j{X}_r(n)}}\right|}^2{\displaystyle \frac{R_r(n)}{s}},$$

(16)

where ${\underset{¯}{Z}}_{cr}(n)$ is the equivalent impedance representing the parallel connection of the magnetizing branch and the rotor branch in the function of speed, given by the following expressions [25]:

$$\underset{¯}{I}{}_s(n)={\displaystyle \frac{{\underset{¯}{U}}_{ph}}{R_s+j{X}_s+{\underset{¯}{Z}}_{cr}(n)}},$$

(17)

$${\underset{¯}{Z}}_{cr}(n)={\displaystyle \frac{1}{\frac{1}{\frac{R_{Fe}\cdot j{X}_m}{R_{Fe}+j{X}_m}}+\frac{1}{\frac{R_r(n)}{s}+j{X}_r(n)}}}.$$

(18)

The variables R_r(n) and X_r(n) represent approximations of the changes in rotor resistance and rotor leakage reactance as functions of speed. These approximations are used to obtain part of the motor’s torque-speed characteristic, ranging from the starting torque point to the maximum torque point. Depending on the motor power [25], they can be modeled as either a square root function or a linear function of speed. Approximations of the change in rotor parameters as a function of the square root of the speed are as follows [25,26]:

$$R_r(n)=R_{r\_st}-{\displaystyle \frac{R_{r\_st}-R_r}{\sqrt{n_n}}}\sqrt{n},$$

(19)

$$X_r(n)=X_{r\_st}+{\displaystyle \frac{X_r-X_{r\_st}}{\sqrt{n_n}}}\sqrt{n}.$$

(20)

Approximations of the change in rotor parameters as a linear function of speed are as follows [25,27]:

$$R_r(n)=R_{r\_st}-{\displaystyle \frac{R_{r\_st}-R_r}{n_n}}n,$$

(21)

$$X_r(n)=X_{r\_st}+{\displaystyle \frac{X_r-X_{r\_st}}{n_n}}n,$$

(22)

According to [25], square root approximation is recommended for motors with a power of up to 15 kW, while linear approximation is preferred for motors exceeding this power.

3. Reference Curves and Standards

Sixteen ABB induction motors were considered, and their manufacturer’s data are provided in

Table 1. Motors’ manufacturers data [28].

Manufacturer’s Code	Pn [kW]	Pole Numb.	Uph [V]	fn [Hz]	nn [r/min]	Ifl [A]	Ist/Ifl	Tfl [Nm]	Tst/Tfl	Tmax/Tfl	pffl	pfst	ηfl
3GBP 091 530-ASK	2.2	2	230	50	2900	7.00	8.3	7.2	2.9	3.5	0.89	0.48	0.859
3GBP 102 810-ASK		4	230	50	1454	7.45	8.9	14.4	3.1	4.1	0.83	0.45	0.867
3GBP 113 390-ASK		6	230	50	967	9.00	6.5	21.7	2.4	3.5	0.69	0.49	0.843
3GBP 134 210-ASK		8	230	50	725	10.05	5.2	29	2.0	3.0	0.64	0.48	0.819
3GBP 131 260-ADK	5.5	2	230	50	2901	16.80	7.9	18.1	2.3	3.4	0.91	0.39	0.892
3GAA132 300-ADJ		4	230	50	1460	19.05	6.6	36	2.2	3.3	0.82	0.39	0.896
3GBP 133 280-ADK		6	230	50	966	20.96	5.0	54	1.83	2.7	0.73	0.31	0.880
3GBP 164 420-ADK		8	230	50	732	22.52	5.0	72	2.0	2.4	0.69	0.50	0.862
3GBP 251 210-ADK	55	2	400	50	2963	94.57	5.9	177	2.1	2.5	0.89	0.35	0.943
3GBP 252 210-ADK		4	400	50	1485	98.21	7.9	354	3.0	3.3	0.85	0.42	0.946
3GBP 283 230-ADK		6	400	50	990	99.59	6.8	531	2.4	2.6	0.85	0.37	0.941
3GBP 314 210-ADK		8	400	50	742	106.69	7.1	708	1.6	2.7	0.80	0.27	0.925
3GBP 281 230-ADL	90	2	400	50	2976	155.02	7.4	289	2.1	2.9	0.89	0.32	0.950
3GBP 282 230-ADL		4	400	50	1485	159.35	7.0	579	2.5	2.9	0.86	0.37	0.952
3GBP 313 240-ADK		6	400	50	994	169.74	7.2	865	2.4	2.9	0.81	0.33	0.949
3GBP 314 230-ADK		8	400	50	741	171.13	7.4	1160	1.8	2.7	0.82	0.27	0.934

. Motors with different power ratings (2.2 kW, 5.5 kW, 55 kW, and 90 kW) were selected, and for each power rating, four motors with different pole numbers (2, 4, 6, and 8) were chosen. The first criterion for motor selection was to avoid motors with the same power ratings as those used in the authors’ previous work [25]. The second criterion was to include both low-power motors, for which the square root approximation of rotor parameter variations is used, and medium-power motors, for which the linear approximation is applied. The third criterion was to include motors with 2, 4, 6, and 8 poles to cover the full range of motors commonly used in the industry. The induction motor data from Table 1 were used as input parameters in the PSO algorithm for the identification of equivalent circuit parameters.

The torque-speed characteristics presented in Figure 1 and Figure 2 were not obtained using the equations for the induction motor’s torque-speed characteristics. These curves were derived graphically by plotting numerical data on motor torque and speed at 42 points, extracted from the MotSize 7.1.1 software [28]. The data for one motor were retrieved in the form of a table with two columns and 42 rows. The first column represents the motor speed, while the second column contains the motor torque. The curves in Figure 1 and Figure 2 were plotted based on the original numerical data, over which the authors had no influence in determining the torque values. In this study, the authors considered these curves as reference characteristics, experimentally determined by the motor manufacturer.

A graphical comparison was performed for motors with the same rated power but different pole numbers. The results of this comparison are presented in Figure 1 and Figure 2, showing that the torque-speed characteristics of motors with power ratings of 2.2 kW and 5.5 kW are similar. Similarly, the 55 kW and 90 kW induction motors also exhibit comparable torque-speed characteristics. According to the NEMA MG-1-2009 standard [29,30,31], there are four classes of torque-speed characteristics depending on motor construction (rotor design), as shown in Figure 3. The same standard defines the stator and rotor leakage reactance ratios for different motor designs, as given in

Table 2. Ratio of stator to rotor leakage reactance in the steady state based on the NEMA class of torque-speed characteristics [29].

Xs/Xr	Class of Torque-Speed Characteristics
1	A and D
0.76	B
0.43	C

.

By comparing the shapes of the characteristics in Figure 1 and Figure 2 with those in Figure 3, it can be observed that the characteristics of the 2.2 kW and 5.5 kW motors correspond to Classes A and B. In contrast, the characteristic shapes of the 55 kW and 90 kW motors align with Class C. As noted in references [32,33], motors with characteristics corresponding to class A are typically designed with a single cage rotor, while class B motors are designed with deep-bar or double-cage rotors. Class C motors are usually built with a double-cage rotor.

4. Results

The combined two-stage PSO method was developed in Matlab R2017b and run on a PC with an AMD Ryzen 7 3700U processor and 12 GB of RAM. The control parameters of the algorithm were set as follows: cognitive and social acceleration coefficients c₁ and c₂ were both set to 2; the maximum and minimum inertia weights w_max and w_min were set to 0.9 and 0.4, respectively. The population size and the maximum number of iterations were set to 100 [25]. This method combines Particle Swarm Optimization (PSO) with rotor parameter approximations as a function of speed, considering the skin effect. While these approximations are not directly used in the parameter estimation algorithm, they help determine the motor’s torque-speed characteristics. To achieve this, a two-stage optimization process was used: in the first stage, the equivalent circuit parameters were estimated at the nominal operating mode, while in the second stage, rotor parameters at motor start-up were determined. These parameters were obtained by minimizing the error between the calculated values and the manufacturer data.

In the subsequent research process, the equivalent circuit parameters for sixteen induction motors were first estimated under the assumption X_{r_st} = X_s (approach 1). The parameters were then re-estimated for the same sixteen motors using the assumption X_{r_st} ≠ X_s (approach 2). This resulted in new values for the electrical parameters of the rotor at motor start-up, while the equivalent circuit parameters for the nominal operating mode remained unchanged. Based on these equivalent circuit parameters, the torque-speed characteristics of the motors, as shown in Figure 4, Figure 5, Figure 6 and Figure 7, were obtained.

Figure 4 compares the torque-speed characteristics of induction motors with a rated power of 2.2 kW (p = 1, 2, 3, 4) obtained using the first and second approaches with the square root and linear approximations against the reference torque-speed characteristic.

The dashed black line represents the characteristics obtained using the first approach with the square root approximation. The solid blue line represents the characteristics obtained using the second approach with the square root approximation. The dashed pink line represents the characteristics obtained using the first approach with the linear approximation. The solid green line represents the characteristics obtained using the second approach with the linear approximation. The reference characteristic provided by the manufacturer is shown using circle symbols.

In Figure 4, Figure 5, Figure 6 and Figure 7, ‘SRA’ denotes square root approximation, while ‘LA’ denotes linear approximation.

From Figure 4, it can be seen that, in all cases, the torque-speed characteristics obtained using the first approach with the square root approximation closely align with those obtained using the second approach with the square root approximation. They also deviate from the reference characteristic in the same manner. Similarly, characteristics obtained using the first approach with the linear approximation closely aligned with those obtained using the second approach with the linear approximation. However, the characteristics obtained using the linear approximation deviate more from the reference characteristic compared to those obtained using the square root approximation.

Figure 5 compares the torque-speed characteristics of induction motors with a rated power of 5.5 kW (p = 1, 2, 3, 4) obtained using the first and second approaches with the square root and linear approximations against the reference torque-speed characteristic. Similarly to the 2.2 kW motors, the torque-speed characteristics obtained using the first approach with the square root approximation align closely with those obtained using the second approach with the square root approximation. Additionally, the characteristics obtained using the first approach with the linear approximation align closely with those obtained using the second approach with the linear approximation. As with the 2.2 kW motor, the characteristics obtained using the linear approximation deviate more from the reference characteristic than those obtained using the square root approximation.

Figure 6 compares the torque-speed characteristics of induction motors with a rated power of 55 kW (p = 1, 2, 3, 4) obtained using the first and second approaches with the square root and linear approximations against the reference characteristic. The results show that the torque-speed characteristics obtained using the first approach with the square root approximation deviate from those obtained using the second approach with the square root approximation. Similarly, the characteristics obtained using the first approach with the linear approximation deviate from those obtained using the second approach with the linear approximation. Among these, the characteristics obtained using the second approach with the linear approximation show the best alignment with the reference characteristic.

Figure 7 compares the torque-speed characteristics of induction motors with a rated power of 90 kW (p = 1, 2, 3, 4) obtained using the first and second approaches with the square root and linear approximations against the reference torque-speed characteristic. Similarly to the 55 kW motor, the characteristics obtained using the first approach deviate from those obtained using the second approach. The closest alignment with the reference characteristic is achieved using the second approach with the linear approximation.

5. Discussion

The influence of approach 1 (X_{r_st} = X_s) and approach 2 (X_{r_st} ≠ X_s) on the estimation of equivalent circuit parameters is analyzed by comparing the errors in the estimated motor torque values at three key points: starting torque, maximum torque, and full-load torque. These errors represent the relative deviation between the obtained torque values, based on the estimated equivalent circuit parameters from

Table 3. Estimated values of the induction-motor-equivalent circuit parameters.

Pn [kW]	Pole Numb.	Motor Parameters
		Rs [Ω]	Rr [Ω]	Xs [Ω]	Xr [Ω]	Xm [Ω]	RFe [Ω]	(Xr_st = Xs)		(Xr_st ≠ Xs)
								Rr_st [Ω]	Xr_st [Ω]	Rr_st [Ω]	Xr_st [Ω]
2.2	2	1.1825	2.1240	3.5027	4.9274	147.446	550.5262	2.7396	3.5027	2.7111	3.4711
	4	1.0414	1.9234	2.8904	4.2004	105.305	913.7245	1.9235	2.8904	2.0812	3.1052
	6	1.7062	1.8940	3.4572	3.8029	59.0461	994.7469	2.3437	3.4572	2.1179	3.1433
	8	1.9746	1.8239	3.8181	4.5281	49.7985	1397.929	2.4024	3.8181	2.1540	3.4407
5.5	2	0.3680	0.8415	1.6164	1.9441	74.6671	404.2879	1.0525	1.6164	0.9177	1.4225
	4	0.4520	0.6610	1.5315	2.0061	42.6501	426.9546	0.9380	1.5315	0.8344	1.3721
	6	0.1553	0.8063	1.8501	2.7356	31.7261	291.7224	1.0513	1.8501	1.2412	2.1317
	8	0.7529	0.5215	2.0497	2.3513	25.0096	383.9699	1.8239	2.0497	2.0589	2.2455
55	2	0.1209	0.0960	0.4445	1.1041	24.4347	260.3490	0.2258	0.4445	0.2816	0.5412
	4	0.1246	0.0793	0.3228	0.7911	16.1477	325.2691	0.1809	0.3228	0.2212	0.3863
	6	0.0862	0.0769	0.4670	1.0227	16.6089	218.5872	0.2906	0.4670	0.3570	0.5560
	8	0.0417	0.0807	0.6142	0.8197	13.5210	116.0817	0.3188	0.6142	0.3965	0.7411
90	2	0.0388	0.0390	0.2692	0.5612	12.9662	127.8566	0.1291	0.2692	0.1913	0.3762
	4	0.0520	0.0483	0.2670	0.5602	11.4274	128.8865	0.1604	0.2670	0.1861	0.3039
	6	0.0553	0.0285	0.2555	0.5509	8.5306	119.8540	0.1399	0.2555	0.1723	0.3066
	8	0.0067	0.0565	0.3700	0.5321	10.1234	96.5582	0.2116	0.3700	0.2633	0.4455

and the catalog values from Table 1. Table 3 presents the estimated equivalent circuit parameters for the motor’s nominal operating mode and the rotor electrical parameters at motor start-up for the two approaches considered.

Table 4. Estimated values for starting, maximum, and full-load torques, along with their errors.

Pn [kW]	Pole Numb.	Xr_s = Xs						Xr_s ≠ Xs		e1−e4
		Tst		Tmax		Tfl		Tst
		Value [Nm]	Error (e1) [%]	Value [Nm]	Error (e2) [%]	Value [Nm]	Error (e3) [%]	Value [Nm]	Error (e4) [%]
2.2	2	20.87954	0.00222	25.06878	0.52073	7.15011	0.69289	20.87986	0.00069	0.00153
	4	44.70970	0.15614	58.97634	0.10782	14.57668	1.22694	44.63458	0.01214	0.14400
	6	52.08053	0.00102	75.44506	0.66483	21.95273	1.16465	52.07961	0.00075	0.00027
	8	57.99982	0.00031	86.45417	0.62739	29.23668	0.81613	58.00008	0.00014	0.00017
5.5	2	41.62890	0.00265	61.67958	0.22681	18.08793	0.06670	41.62971	0.00070	0.00195
	4	80.64353	0.00438	119.00864	0.17562	35.94635	0.14904	80.63921	0.00098	0.00340
	6	98.81734	0.00269	145.83268	0.02242	54.37018	0.68551	98.81978	0.00023	0.00246
	8	144.00199	0.00138	172.92663	0.07328	71.96895	0.04313	143.99859	0.00098	0.00040
55	2	371.77452	0.02005	442.79902	0.06758	176.86406	0.07680	371.71209	0.00325	0.01680
	4	1061.64468	0.03346	1169.37983	0.10100	352.45933	0.43522	1062.03128	0.00294	0.03052
	6	1274.51085	0.00870	1369.14998	0.82935	532.34476	0.25325	1274.38432	0.00123	0.00747
	8	1132.57379	0.01997	1912.87389	0.06664	706.48690	0.21371	1132.79282	0.00063	0.01934
90	2	606.77281	0.02096	830.79009	0.87220	287.89329	0.38295	606.92321	0.00382	0.01714
	4	1447.65759	0.01089	1667.84742	0.67016	575.00282	0.69036	1447.47606	0.00165	0.00924
	6	2077.14989	0.05539	2505.61489	0.11501	865.08823	0.01020	2075.55873	0.02126	0.03413
	8	2087.77875	0.01060	3170.43832	1.22728	1162.74373	0.23653	2087.89505	0.00503	0.00557

presents the estimated values of starting torque, maximum torque, and full-load torque, along with their relative errors. The relative errors in Table 4 are defined as follows: e₁ represents the relative error in the starting torque, e₂ represents the relative error in the maximum torque, and e₃ represents the relative error in the full-load torque.

Table 4 indicates that the obtained values for the motor’s starting, maximum, and full-load torques closely align with the manufacturer values. The last bold column presents the difference between the starting torque error obtained using the first approach and that obtained using the second approach (e₁−e₄). This difference is positive for all motors, indicating that the error e₄ is consistently smaller than the error e₁. This demonstrates that the second approach provides more accurate starting torque values compared to the first approach.

Since the assumption X_{r_st} = X_s does not affect the estimation of the motor’s electrical parameters for nominal operating mode in the first stage of optimization, the values of the maximum and full-load torque remain unchanged. As a result, the errors obtained by applying the second approach will be the same as those from the first approach.

The errors resulting from different approximations of rotor parameter variations are evaluated by comparing the motor’s torque-speed characteristics derived from both the SRA and LA, as well as the equivalent circuit parameters obtained using approaches 1 and 2. These errors, expressed as the mean absolute percentage errors, include both the errors in estimating electrical parameters using approaches 1 and 2 and the errors arising from approximating rotor parameter variations with speed.

According to [25], the mean absolute percentage error is determined based on the torque values obtained at different speeds. The number of operating points in the motor’s torque-speed characteristics, as provided by the induction motor manufacturer, is 41 [28]. The mean absolute percentage error is calculated using the following expression [25]:

$$e={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N\frac{\left|x_i-y_i\right|}{x_i}\cdot 100},$$

(23)

where N is the number of working points (41 points, for which the induction motor manufacturer gives torque values at the corresponding speeds [28]); x_i is the exact torque value at the i-th point, provided by the induction motor manufacturer [28]; and y_i is the obtained torque value at the i-th point.

The obtained values of the mean absolute percentage errors for the 16 motors considered are presented in

Table 5. Obtained values of the mean absolute percentage errors.

Pn [kW]	Pole Numb.	Errors
		(Xr_st = Xs)		(Xr_st ≠ Xs)
		SRA	LA	SRA	LA
		e5 [%]	e6 [%]	e7 [%]	e8 [%]
2.2	2	1.78	8.26	1.76	8.27
	4	4.85	10.24	4.75	9.86
	6	2.95	5.89	2.89	6.07
	8	3.68	8.04	3.51	8.22
5.5	2	3.65	8.36	3.54	8.72
	4	4.79	8.97	4.66	8.93
	6	4.43	4.85	3.67	5.22
	8	4.23	8.03	3.47	8.62
55	2	23.93	4.90	21.04	2.82
	4	21.64	3.41	18.98	2.16
	6	24.55	4.85	21.07	3.25
	8	19.25	5.50	15.84	4.39
90	2	24.66	5.87	17.96	3.82
	4	22.75	4.92	20.44	3.65
	6	27.19	4.39	23.04	3.08
	8	20.97	5.52	17.50	3.74

. The errors e₅ to e₈ have the following meanings:

e₅—the mean absolute percentage error of the estimated characteristic obtained by applying the first approach (X_{r_st} = X_s), and the square root approximation of the change in rotor parameters as a function of speed;

e₆—the mean absolute percentage error of the estimated characteristic obtained by applying the first approach (X_{r_st} = X_s), and the linear approximation of the change in rotor parameters as a function of speed;

e₇—the mean absolute percentage error of the estimated characteristic obtained by applying the second approach (X_{r_st} ≠ X_s), and the square root approximation of the change in rotor parameters as a function of speed;

e₈—the mean absolute percentage error of the estimated characteristic obtained by applying the second approach (X_{r_st} ≠ X_s) and the linear approximation of the change in rotor parameters as a function of speed.

Based on the results, for the 2.2 kW and 5.5 kW motors, the error e₇ is the smallest, indicating that the torque-speed characteristics derived using the second approach and the SRA best match the reference characteristics. For the 55 kW and 90 kW motors, the error e₈ is the smallest, showing that the second approach with the LA yields the best match to the reference torque-speed characteristics.

When approach 1 was used, the average execution time of the algorithm was about 0.285 s. However, with approach 2, the average computation time increased to about 1.39 s. This increase is due to the introduction of an additional unknown variable (X_{r_st}), which requires expanding the objective function to include an equation related to the motor’s starting power factor. Consequently, the optimization process becomes more complex, slowing the algorithm’s convergence and increasing the time needed to find the optimal solution.

To better demonstrate the influence of approaches 1 and 2 on estimating equivalent circuit parameters for motors with different rated powers, mean absolute percentage errors e₅ to e₈ are presented graphically in Figure 8 and Figure 9.

For 2.2 kW motors, the differences between mean absolute percentage errors e₅ and e₇, as well as e₆ and e₈, are small. For 5.5 kW motors, these differences are also minor but show an increasing trend as the number of poles increases. Notably, the errors e₆ and e₈ obtained using LA are significantly larger than the errors e₅ and e₇ obtained using the SRA. For motors with powers of 55 kW and 90 kW, the deviations between errors e₅ and e₇, as well as e₆ and e₈, are greater compared to those for 2.2 kW and 5.5 kW motors. Additionally, for 55 kW and 90 kW motors, errors e₅ and e₇ calculated using the SRA are significantly higher than errors e₆ and e₈ obtained using the LA.

This indicates that for motors with power ratings of 2.2 kW and 5.5 kW, the torque-speed characteristics obtained using the SRA align more closely with the reference torque-speed characteristics. Additionally, the second approach provides a slight improvement in the obtained torque-speed characteristics compared to the first approach. For motors with power ratings of 55 kW and 90 kW, the torque-speed characteristics derived from the LA show better alignment with the reference torque-speed characteristics. Similarly, the second approach achieves an improvement in the torque-speed characteristics compared to the first approach.

The error differences (e₅−e₇) and (e₆−e₈) shown in Figure 10 and Figure 11 represent the algebraic differences between the mean absolute percentage errors e₅ and e₇, as well as e₆ and e₈. The results indicate that the sign of the error difference (e₅−e₇) is always positive (e₅−e₇ > 0), meaning that error e₅ is consistently greater than error e₇ (e₅ > e₇).

For motors with power ratings of 2.2 kW and 5.5 kW, the sign of the error difference (e₆−e₈) is positive only for motors with 4 poles, while for motors with 2, 6, and 8 poles, this sign is negative. For motors with power ratings of 55 kW and 90 kW, the sign of the error difference (e₆−e₈) is consistently positive, indicating that error e₆ is always greater than error e₈.

Applying the first approach with the SRA to 2.2 kW motors (p = 2, 4, 6, 8) increases the difference in mean absolute percentage errors e₅−e₇, ranging from 0.02 to 0.17. When using the first approach and LA for a 2.2 kW motor with four poles, the difference in mean absolute percentage errors e₆−e₈ increases by 0.38. However, for 2.2 kW motors with 2, 6, and 8 poles, the first approach with the LA reduces the difference in mean absolute percentage errors e₆−e₈, ranging from −0.01 to −0.18.

For 5.5 kW motors, applying the first approach with the SRA increases the difference in mean absolute percentage errors e₅−e₇, ranging from 0.11 to 0.76. Using the first approach with LA for a 5.5 kW motor with 4 poles increases the difference in mean absolute percentage errors e₆−e₈ by 0.04. In contrast, for 5.5 kW motors with 2, 6, and 8 poles, this approach reduces the difference in mean absolute percentage errors e₆−e₈, ranging from −0.36 to −0.59.

Applying the first approach with the SRA to 55 kW and 90 kW motors increases the difference in mean percentage absolute errors e₅−e₇, ranging from 2.66 to 3.48 for the 55 kW motor and from 2.31 to 6.7 for the 90 kW motor. Similarly, applying the first approach with the LA increases the difference in mean absolute percentage errors e₆−e₈, ranging from 1.11 to 2.08 for the 55 kW motor and from 1.27 to 2.05 for the 90 kW motor.

6. Case Study for Low Power VSD

The two-stage PSO algorithm, utilizing the second approach for estimating the equivalent circuit parameters of squirrel cage induction motors, was validated on a 2.2 kW Siemens induction motor. This experiment was conducted in the Laboratory for Electrical Drives at the Faculty of Electronic Engineering, University of Niš, as shown in Figure 12a. Motor nameplate data were used to estimate its equivalent circuit parameters using the two-stage PSO algorithm and the second approach. Additionally, the same motor was applied for parameter identification using a Danfoss FC302 frequency converter through an Automatic Motor Adaptation (AMA) test. Nameplate of Danfoss FC302 frequency converter is given in Figure 12b. The Automatic Motor Adaptation (AMA) algorithm is designed to automatically determine the electrical parameters of a motor while it remains at a standstill, offering a practical and efficient solution for parameter estimation [34,35]. The motor’s manufacturer data are provided in

Table 6. Catalog data for the Siemens 2.2 kW motor.

Manufacturer’s Code	Pn [kW]	Pole Numb.	Uph [V]	fn [Hz]	nn [r/min]	Ifl [A]	Ist/Ifl	Tfl [Nm]	Tst/Tfl	Tmax/Tfl	pffl	ηfl
1LE10011AB422AA4-Z	2.2	4	230	50	1455	4.65	6.96	14.44	2.18	3.3	0.81	0.843

.

Using only the first approach and the square root approximation, the authors of this paper verified the two-stage PSO algorithm on the identical laboratory setting. The findings of their validation are shown in [36].

The obtained values of the equivalent circuit parameters are given in

Table 7. The obtained equivalent circuit parameters for the 2.2 kW Siemens induction motor.

Method	Motor Parameters
	Rs [Ω]	Rr [Ω]	Xr [Ω]	Xs [Ω]	Xm [Ω]	RFe [Ω]	Rr_st [Ω]	Xr_st [Ω]
AMA test	2.785	1.885	3.349	3.349	98.5796	1803.90	/	/
Two-stage PSO (approach 2)	2.69	1.77	3.22	3.68	94.28	970.53	1.84	2.83
Error e9 [%]	3.42	6.12	3.87	9.86	4.36	46.19	/	/

. Error e₉ represents the percentage deviation of the electrical parameters obtained using the two-stage PSO algorithm and the second approach from those identified by the AMA test. As presented in Table 7, the stator resistance, rotor leakage reactance, and magnetizing reactance values obtained using the two-stage PSO algorithm and the second approach show slight deviations from the AMA test results, all below 5%. The deviations for rotor resistance and stator leakage reactance are slightly higher, at 6.12% and 9.86%, respectively. The largest deviation is observed in the core loss resistance, reaching up to 46.19%.

Figure 13 shows the motor’s torque-speed characteristics obtained using electrical parameters identified by the two-stage PSO algorithm (approach 2) and the Danfoss converter through the AMA test. The solid blue line represents the characteristic based on parameters from the two-stage PSO algorithm (approach 2) and the square root approximation. The dashed black line represents the characteristic based on parameters from the AMA test. Both characteristics were calculated using Expression 16. To obtain the torque-speed characteristic based on AMA test parameters, the rotor parameters were assumed to be fixed over the entire speed range. This assumption was made because the AMA test does not provide rotor parameter values at motor start and during the acceleration.

In the speed range from 500 r/min to synchronous speed (1500 r/min), the characteristic based on the two-stage PSO algorithm (approach 2) parameters closely matches the characteristic from the AMA test parameters. However, within the speed range of 0 to 500 r/min, slight differences are observed between these two characteristics. A possible reason for these differences lies in the methods used to identify the equivalent circuit parameters of the induction motor. The AMA test, which falls under the category of offline methods for parameter identification, employs a specific algorithm to test the motor under a blocked rotor. The AMA test does not consider any changes in the rotor’s electrical parameters during motor start-up. The two-stage PSO algorithm, on the other hand, is a metaheuristic method that uses linear and square root approximations to take into account changes in the rotor’s electrical parameters as the motor starts up.

Table 8. Obtained values for starting, maximum, and full-load torque for the 2.2 kW Siemens induction motor.

Method	Tst	Tmax	Tfl
	[Nm]	[Nm]	[Nm]
AMA test	29.39196	46.63259	13.67479
Two-stage PSO	31.32862	46.95063	14.46165
Error e10 [%]	6.57	0.68	5.75

shows the starting, maximum, and full-load torques of the motor obtained through the AMA test and the two-stage PSO algorithm utilizing the second approach. The error e₁₀ represents the difference between the values obtained using the two methods. The torque values obtained with the two-stage PSO algorithm (approach 2) are close to those from the AMA test. The full-load torque values nearly match, with a small deviation of 0.68%. The deviations for the starting and maximum torques are slightly larger, at 6.57% and 5.75%, respectively.

This experimental verification of the proposed parameter estimation algorithm has confirmed its effective application to the 2.2 kW, 4-pole motor.

Experimental verification results of the PSO algorithm obtained through the AMA test on a 2.2 kW, 4-pole motor could be conditionally extended to a 5.5 kW, 4-pole motor. This extrapolation assumes comparable performance can be achieved under the following conditions:

• Application of the same algorithm;
• Use of the same rotor parameter approximation as a function of speed;
• Potential utilization of the same frequency converter;
• Both motors (2.2 kW and 5.5 kW) belong to the low-power motor category;
• The motors have the same pole number and similar construction characteristics.

Therefore, the experimental validation results for the 5.5 kW, 4-pole motor are expected to be similar to those of the 2.2 kW, 4-pole motor.

The experimental verification results of the PSO algorithm obtained through the AMA test on a 2.2 kW, 4-pole motor could not be extended to a 55 kW and 90 kW motors with different pole numbers (2, 4, 6, 8) due to failure to meet the following conditions:

• Use the different rotor parameter approximation as a function of speed compared to low power motor;
• Utilization of the frequency converter with different power rating compared to the low power frequency converters rating;
• Both motors (50 kW and 90 kW) belong to the medium-power motor category opposite to the low power motor;
• The motors have different pole numbers and construction characteristics in relation to the low power motor.

The 55 kW and 90 kW motors have significantly higher power and a different rotor construction, so a linear approximation of rotor parameter variation was used. Due to these differences, the experimental validation performed on the low-power 2.2 kW motor cannot be applied to these motors.

As explained in Section 5, the number of poles in a motor plays a significant role in the accuracy of parameter estimation. Motors of the same rated power but with different numbers of poles have distinct design characteristics, which also results in different values of equivalent circuit parameters. This difference becomes even more pronounced in motors with different rated powers and pole numbers. Any change in the number of motor poles is expected to have a direct impact on the accuracy of the derived motor characteristics. Therefore, separate experimental verification of the results is necessary for each motor with a different number of poles.

7. Conclusions

This study presents the results of equivalent circuit parameter estimation for 16 induction motors with different rated powers and pole numbers, using two different approaches. The first approach assumes that the starting rotor leakage reactance and stator leakage reactance are equal, while the second approach assumes these parameters are different. The study also describes the results of equivalent circuit parameter estimation for a laboratory 2.2 kW induction motor using the two-stage PSO algorithm utilizing the second approach and the Danfoss FC302 frequency converter via an AMA test. Based on the results presented, the following conclusions can be drawn:

• Regardless of the pole numbers, the torque-speed characteristics obtained using the first approach are identical to those obtained using the second approach for motors with a rated power of 2.2 kW and 5.5 kW. In addition, the characteristics produced using the square root approximations align more precisely with the reference characteristics. This alignment is more accurate compared to the characteristics obtained through the linear approximation. The fact that this is the case suggests that the method of applying the square root approximation to the first approach is appropriate for these motors.
• For 55 kW and 90 kW motors, regardless of the pole numbers, the torque-speed characteristics obtained using the first approach differ from those obtained using the second approach. For these motors, the linear approximation yields characteristics that align more closely with the reference characteristics compared to the square root approximation. Graphical and numerical results indicate that applying the first approach with the square root approximation results in worse torque-speed characteristics for 55 kW and 90 kW motors. Conversely, the second approach with the linear approximation yields torque-speed characteristics that better match the reference characteristics.
• The implementation of the two-stage PSO algorithm results in estimated starting, maximum, and full-load torque values that closely align with those specified by the manufacturer. The second approach in the parameter estimation procedure improves the precision of the starting torque, while the maximum and full-load torques remain unchanged since this approach does not affect them.
• The algorithm’s average execution time for the first approach was 0.285 s. However, due to the introduction of an additional unknown variable (X_{r_st}), which expands the objective function, the execution time for the second approach increased to approximately 1.39 s.
• Experimental verification of the proposed parameter estimation PSO algorithm showed its effectiveness when applied to a 2.2 kW 4-pole motor. The results obtained through the PSO algorithm and the AMA test suggest that, under certain conditions, these findings can be conditionally extended to a 5.5 kW 4-pole motor. This extrapolation is valid provided that (1) the same algorithm is applied, (2) the same rotor approximation is used, (3) the same frequency converter is used, and (4) both motors have the same pole number and similar construction characteristics.
• However, the experimental validation results cannot be applied to medium-power motors (55 kW and 90 kW) due to significantly different motor power and rotor construction, and the use of different approximations for rotor parameter variation.
• In addition, the number of poles significantly influences the accuracy of parameter estimation. Motors with the same rated power but different pole numbers have distinct design characteristics, which also results in different values of equivalent circuit parameters. These differences become even more noticeable in motors with varying power ratings and pole numbers. Therefore, separate experimental verification of the results is necessary for each motor with a different number of poles.