1 Introduction

1.1 Motivation

Polyphase Induction Motors (PPIMs) are the largest, commonly used electrical machines in the industry [1]. These motors account for about 60% of the electrical-to-mechanical energy conversion [2]. For PPIMs to achieve the desired control performance, modeling quality is a critical factor. The large nonlinearity of the PPIM models is a critical issue [3]. Finding the unknown parameters of these machines is thus a difficult challenge. The problem of the PPIMs parameter estimation is described as a non-smooth and nonlinear problem of optimization [3]. The target of this problem is to minimize the deviation of estimated parameters from the actual ones. In this regard, the minimization of the difference between the actual parameters and the estimated ones represents the objective function, keeping these parameters within the permissible boundaries of operation. Many optimization algorithms have been proposed to ensure accurate parameter identification to follow the PPIM models requirements. This paper proposes the IISA as a new variant to find the optimal unknown PPIM parameters in this context.

1.2 Literature review

Modeling of induction motors in ac drive systems is seen to be a significant task. Especially, for PPIMs, the model’s parameters estimation target is to get the unknown variables with accuracy because of operation aspects and control drives [1, 4]. With high degree of accuracy, it is necessary to optimize the induction motors parameters costs effectively. According to IEEE Std 112-1991 IEEE Std 112-2004 [5] the locked rotor and no-load tests are used in to get the parameters of equivalent circuit the traditional methods.

The accuracy of the experimentally based methods estimated parameters is dependent on the accuracy degree of the monitoring, hardware cost and implementation procedures. As a result of these restrictions, various optimization technique providers offer a variety of innovative ways to achieve the goal of estimating parameters to an acceptable level [6]. So, several methods of optimization have been developed to estimate optimally the PPIMs equivalent circuits' parameters [7,8,9].

Several methods for determining the unknown parameters of PPIMs have been used in the literature. Various methods developed to estimate unknown PPIMs parameters have been reviewed in references [6, 10]. Previous initiatives are summarized in this section as follows:

  • Reference [11] proposed the artificial immune system to extract the PPIM parameters for optimization of the IMs parameters from manufacturer data and experimental tests.

  • Reference [12] addressed the Neural-network has been presented in form of training mechanism to find the parameter estimation problem solution.

  • Multi-objective algorithm has been developed in Ref. [13] to minimize the estimated results deviation from manufacturer data.

  • Reference [14] developed the shuffled frog-leaping has been presented to extract the PPIMs equivalent circuit parameters using manufacturer data.

  • Reference [15] estimated the six-phase induction motor parameters using modified standard tests. An improved equivalent circuit has been used to precede a zero-sequence test for improving the estimated parameters accuracy.

  • The particle swarm optimization technique was presented in Ref. [16].

  • A simplified model for PPIMs parameter estimation was proposed in Ref. [17].

  • Reference [18] introduced the shuffled frog-leaping algorithm for extracting the parameters estimation of induction motors using standard manufacturer data.

  • A differential evolution aims at finding the PPIMs parameter estimation has been developed in Ref. [19].

Also, the application of numerous optimization approaches to deal with the parameter estimation issue is demonstrated in the previous review; the field of optimization is both renewable and intriguing. For many real engineering problems, many optimization algorithms have been developed such as: particle swarm optimization (PSO) [20, 21], coyote optimization algorithm [22, 23], moth-flame optimization algorithm [24, 25], wind driven optimization algorithm [26], water cycle optimization algorithm [27], sunflower optimization algorithm [28, 29], fruit fly optimization [30, 31], forensic optimization algorithm for PV parameter estimation [32] and cat swarm optimizer [33]. For synchronous motor design, an intelligent diagnosis approach is proposed [34], for the PID controller parameters adjustment optimization method has been developed in Ref. [35], to assess the improved model of solar cells and generation production cost has been developed in Ref. [36], to solve the optimal power flow problem [37].

The interior search algorithm (ISA) [38] represents a novel inspired optimization method based on the decoration and interior design for global optimization. It introduces based on coupling of different search mechanisms, mirror search and composition optimization. The ISA contains only one tunable parameter with two different strategies are used to adjust it. Several constrained and unconstrained benchmark problems have been solved using this algorithm [38,39,40].

This paper aims at estimating the electrical and mechanical parameters of the PPIMs based on current and speed measurements through developing an improved ISA variant (IISA) through two improvements that are inserted into the original ISA. These two improvements are:

  • The composition group based-crossover strategy with the aim to provide enriched diversity of the solutions.

  • Employing a chaotic concept as a local search around the best solution to refine its quality during the searching process.

The performance of IISA is assessed compared with different modern optimization algorithms, namely genetic algorithm (GA), differential evolution (DE), Jaya algorithm (Jaya), gray wolf optimizer (GWO), sine cosine algorithm (SCA) [41], basic ISA and some state-of-the-art advanced optimization algorithms including chaotic antlion optimization (CALO), chaotic gravitational search algorithm (CGSA), chaotic electromagnetic field optimization algorithm (CEFOA), crossover-based sine cosine algorithm (CBSCA), crossover-based gravitational search (CBGSA), and crossover-based monarch butterfly optimization (CBMBO) for both six- and three-phase induction motors.

1.3 Contribution

The following are the primary characteristics of this paper:

  • The PPIM steady-state model is derived while the IISA is investigated to obtain the PPIM optimal unknown parameters from name plate data.

  • The deviation between the experimental data for the tested PPIMs from the estimated parameters is very small. Then, the capability of the proposed IISA is proved.

  • The performance of IISA is assessed and compared with basic state-of-the-art algorithms, GA, DE, Jaya, GWO, SCA, and ISA and some advanced state-of-the-art algorithms, CALO, CGSA, CEFOA, CBSCA, CBGSA, and CBMBO, for both PPIMs.

  • The assessment study clarifies the fast response of IISA compared with other competitive algorithms for the accurate identified parameters. The results also confirmed the potential robustness of the proposed IISA in comparison to others.

  • In terms of the quality indices, it can be revealed that the IISA is more promising algorithm as the identified parameters by the IISA can more accurately describe the actual characteristics of the induction motor models.

1.4 Organization

The sections that follow are arranged as: In Sect. 2, the mathematical representation of the PPIM steady-state characteristics is presented. In Sect. 3, the problem is designed as an optimization problem that specifies the constraints and objective. The proposed design procedure of IISA is presented in Sect. 4. In Sect. 5, the experimental setup and algorithms results, and discussion are presented. In Sect. 6, the findings are concluded.

2 The polyphase induction motor steady-state model

The steady-state equivalent circuit of the polyphase induction motor (PPIM) is shown in Fig. 1. The per-phase steady-state equivalent circuit of PPIM without mutual inductance separation of the stator winding is presented in [16]. The Thevenin equivalent circuit of PPIM is shown in Fig. 2. The voltage equation of the Thevenin equivalent circuit is computed as:

$$V_{th} = \frac{{jX_{m} }}{{R_{s} + jX_{s} + jX_{m} }} V_{ph}$$
(1)
Fig. 1
figure 1

PPIM equivalent circuit

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Fig. 2
figure 2

Thevenin equivalent circuit

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The Thevenin impedance of PPIM is computed as:

$$\begin{aligned} Z_{th} & = R_{th} + jX_{th} = \frac{{jX_{m} (R_{s} + jX_{s} )}}{{R_{s} + jX_{s} + jX_{m} }} \\ & = \frac{{R_{s} X_{m} }}{{X_{s} + X_{m} }} + j\frac{{X_{m} X_{s} }}{{X_{s} + X_{m} }} \\ \end{aligned}$$
(2)

The rotor current of PPIM can be determined as:

$$I_{2} = \frac{{V_{th} }}{{Z_{th} + Z_{2} }} = \frac{{V_{th} }}{{\left( {R_{th} + \frac{{R_{2} }}{s}} \right) + j\left( {X_{th} + X_{2} } \right)}}$$
(3)

The electromechanical torque can be calculated by Eq. (4) as:

$$\begin{aligned} T_{d} & = \frac{m}{{\omega_{s} }}I_{2}^{2} \frac{{R_{2} }}{s} \\ & = \frac{m}{{\omega_{s} }}\frac{{V_{th}^{2} }}{{\left( {R_{th} + \frac{{R_{2} }}{s}} \right)^{2} + \left( {X_{th} + X_{2} } \right)^{2} }}\frac{{R_{2} }}{s} \\ \end{aligned}$$
(4)

The slip \(s_{mT}\) of PPIM at maximum torque can be determined from

$$s_{mT} = \frac{{R_{2} }}{{\sqrt {\left( {R_{th} } \right)^{2} + \left( {X_{th} + X_{2} } \right)^{2} } }}$$
(5)

the maximum torque at maximum slip can be calculated from:

$$T_{\max } = \frac{m}{{2\omega_{s} }}\frac{{V_{th}^{2} }}{{\left[ {R_{th} + \sqrt {\left( {R_{th} } \right)^{2} + \left( {X_{th} + X_{2} } \right)^{2} } } \right]}}$$
(6)

The torque at starting can be obtained from

$$T_{st} = \frac{m}{{\omega_{s} }}\frac{{V_{th}^{2} }}{{\left( {R_{th} + R_{2} } \right)^{2} + \left( {X_{th} + X_{2} } \right)^{2} }}R_{2}$$
(7)

The power factor of input current can be calculated from

$$pf = \cos \left( {\tan^{ - 1} \left( {\frac{{X_{th} + X_{2} }}{{R_{th} + \frac{{R_{2} }}{s}}}} \right)} \right)$$
(8)

3 Problem formulation: objective function statement

The problem of parameter estimation is developed as a nonlinear optimization model. The aim of the objective function is to find the smallest deviation between the calculated and the manufacturer's nameplate data, such as the torques (starting, rated, and maximum), and the power factor at full load. The proposed method is constructed to estimate the motor parameters (leakage reactance of rotor and stator parts and resistance, and the magnetizing reactance of the motor's steady-state equivalent circuit). The objective function is calculated as:

$$\Delta F = \Delta T_{d}^{2} + \Delta T_{\max }^{2} + \Delta T_{st}^{2} + \Delta pf^{2}$$
(9)

The four terms in Eq. (9) are obtained through:

$$\Delta pf = \frac{epf - mpf}{{mpf}}$$
(10)
$$\Delta T_{d} = \frac{{eT_{d} - mT_{d} }}{{mT_{d} }}$$
(11)
$$\Delta T_{\max } = \frac{{eT_{\max } - mT_{\max } }}{{mT_{\max } }}$$
(12)
$$\Delta T_{st} = \frac{{eT_{st} - mT_{st} }}{{mT_{st} }}$$
(13)

where \(\Delta F\) denoted the required objective function is represented by square error for the power factor of rated input current, starting, full load, and maximum torque, which should be minimized.

4 Optimization methodologies

In the following subsections, the basics of the ISA and the proposed IISA method are described in detail.

4.1 The basics of interior search algorithm (ISA)

ISA presents one of the stochastic optimization algorithms to solve optimization tasks [38]. It was developed based on the decoration and interior design architectures, where two main stages are involved. The first one presents the composition phase, where the composition of elements (i.e., solutions from the optimization viewpoint) is performed to achieve more fantastic and beautiful environment (i.e., best fitness from optimization viewpoint). The second stage represents the mirror search which aims for exploring best sights among the elements and the better one. The main features of the ISA can be described as:

  1. (1)

    Create a random population of elements within the problem space bounds, upper limit (\(\theta_{UL}\)) and lower limit (\(\theta_{LL}\)), and also report the fitness values regarding all elements.

  2. (2)

    (2) Record the best element that has been achieved minimum value for the objective function (i.e., for minimization problem). The best element is defined by \(\theta_{gb}^{k}\) for iteration \(k\), where \(gb\) represents the global best.

  3. (3)

    Randomly divide the population into two groups, namely the mirror group and composition group, through employing a switching parameter (\(\lambda\)). In this context, for the element \(A\), if \(r_{A} \le \lambda\), carryout the mirror or else the composition group is performed. Here \(r_{i}\) defines a random number within 0 and 1, \(i=1,2,\ldots ,PS\).\(PS\) denotes the population size.

  4. (4)

    For composition group, update their elements by the following rule as follows.

    $$\theta_{i}^{k} = \theta_{LL}^{k} + r_{2} \cdot \left( {\theta_{UL}^{k} - \theta_{LL}^{k} } \right),\quad i = 1,2, \ldots ,PS$$
    (14)

    where \(\theta_{i}^{k}\) denotes the \(i{\text{th}}\) elements at the \({k}^{th}\) iteration and \(r_{2}\) defines a random number lies within interval [0, 1].

  5. (5)

    For the mirror group, the \(i{\text{th}}\) element at \(k{\text{th}}\) iteration of the mirror location (\(\theta_{m,i}^{k}\)) can be updated s follows.

    $$\theta_{m,i}^{k} = r_{3} .\,\theta_{i}^{k - 1} + (1 - r_{3} ).\,\theta_{gb}^{k}$$
    (15)

    where \(r_{3}\) denotes a random number within the interval [0, 1]. In this regard, the virtual location for the candidate element depends on the mirror position, where this scenario can be implemented as follows.

    $$\theta_{i}^{k} = 2\theta_{m,i}^{k} - \theta_{i}^{k - 1}$$
    (16)
  6. (6)

    To enhance and improve the position quality of the global best solution, a random walk phase can be carried out as a local search way to potentially improve the global best position. This phase can be formulated as:

    $$\theta_{gb}^{k} = \theta_{gb}^{k - 1} + r_{n} \times \gamma$$
    (17)

    where \(r_{n}\) defines a random numbers vector that are distributed normally and \(\gamma\) represents a user defined scaling factor which is taken as \(0.01\left( {\theta_{UL}^{k} - \theta_{LL}^{k} } \right)\).

  7. (7)

    Calculate the fitness for each element or position and perform the updating respect, if it is revival, where this step be considered as

    $$\theta_{i}^{k} = \left\{ {\begin{array}{*{20}c} {\theta_{i}^{k} } & {f(\theta_{i}^{k} ) < f(\theta_{i}^{k - 1} )} \\ {\theta_{i}^{k - 1} } & {else} \\ \end{array} } \right.$$
    (18)
  8. (8)

    Stop the procedures, if judgment criteria are met, otherwise repeat the procedures with starting from step 2. The pseudocode of the traditional version of ISA is illustrated in Fig. 3.

Fig. 3
figure 3

The pseudocode of the traditional ISA

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4.2 Proposed IISA algorithm

The improved ISA (IISA) is introduced based on two improvements, namely the composition group-based crossover strategy with the aim to provide enriched diversity of the solutions, while the second improvement employs a chaotic concept during the searching process as a local search around the best solution to refine its quality. The details behind the IISA are elaborated in the following subsections.

4.2.1 Composition phase using the crossover strategy

Within the composition group the traditional updating strategy of the element or the individual was developed based on random manner along with boundaries of the problem space which may deteriorate the performance behavior and diversity of the algorithm. To overcome these weaknesses, a crossover strategy is developed based on exploiting the experience of elements and thus to improve the diversity and the algorithm performance. Consider two individuals \(\theta_{l}^{k}\) and \(\theta_{r}^{k}\) are selected at random from the population. Consequently, the updating of the current element in terms of the crossover strategy is expressed as follows.

$$\theta_{i}^{k} = \left\{ {\begin{array}{*{20}c} {\theta_{i}^{k} + r_{2} .\left( {\theta_{r}^{k} - \theta_{l}^{k} } \right)} & {if\,f(\theta_{r}^{k} ) < f(\theta_{l}^{k} )} \\ {\theta_{i}^{k} + r_{2} .\left( {\theta_{l}^{k} - \theta_{r}^{k} } \right)} & {\,Otherwise\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\ \end{array} } \right.$$
(19)

4.2.2 Local search-based chaos concept

The main feature of chaos distribution is contained in the ergodicity and mixing properties that can perform higher speeds in carrying out iterative step search than the standard stochastic search. The local search-based chaos (LSC) concept is introduced to effectively improve the performance and enrich the superiority of the algorithm. The LSC works with different initial points that are coupled with chaos maps to potentially achieve more enhanced ones. The LSC procedure can be established as:

Step 1. Chaos quantities In this step, a set of chaotic values is generated from a set of chaotic maps, where these values are formulated as matrix (\(M_{k}\)) form with size of \(N \times D\) as follows:

$$M_{k} = \left[ {\begin{array}{*{20}c} {q_{11}^{k} } & {q_{12}^{k} } & \cdots & {q_{1D}^{k} } \\ {q_{21}^{k} } & {q_{22}^{k} } & \cdots & {q_{2D}^{k} } \\ \vdots & \vdots & \cdots & \vdots \\ {q_{N1}^{k} } & {q_{N2}^{k} } & \cdots & {q_{ND}^{k} } \\ \end{array} } \right]_{N \times D}$$
(20)

where \(N\) denotes the number of map functions,\(D\) defines the number of the decision (control) variables, \(q_{jd}^{k}\) defines the generated chaotic quantity inside the interval of (0, 1) for the chaotic map j on the dimension d at the iteration k.

Step 2. Mapping process In this step, the candidate solution, \({{\varvec{\uptheta}}} = \left( {\theta_{1} ,\theta_{2} ,...,\theta_{D} } \right)\), is mapped to another one according to the chaotic values of the matrix \(M_{k}\) as follows.

$$X_{k}^{{{\text{candidate}}}} = \left[ {\begin{array}{*{20}c} {{{\varvec{\uptheta}}}_{k} } \\ {{{\varvec{\uptheta}}}_{k} } \\ \vdots \\ {{{\varvec{\uptheta}}}_{k} } \\ \end{array} } \right]_{m \times N}$$
(21)
$$X_{k}^{FIC} = \theta_{LL} + M_{k} (\theta_{UL} - \theta_{LL} )$$
(22)
$$Z_{k}^{{{\text{chaotic}}}} = \alpha .X_{k}^{{{\text{candidate}}}} + \left( {1 - \alpha } \right)X_{k}^{FIC}$$
(23)

where \(Z_{k}^{{{\text{chaotic}}}}\) defines the matrix of mapped individual \({{\varvec{\uptheta}}}_{k}\) that is repeated \(N\) times, \(\lambda = k/T\) defines a weighting parameter and \(X_{k}^{FIC}\) denotes the matrix of new individuals that are generated chaotically. In this context, \(X_{k}^{{{\text{candidate}}}}\) is expressed as the best solution obtained from the ISA stage.

Step 3. Updating process Evaluate the matrix \(Z_{k}^{{{\text{chaotic}}}}\) and obtain the best one (\(Z_{k,best}^{{{\text{chaotic}}}}\)), where if \(f(Z_{k,best}^{{{\text{chaotic}}}} ) < f(\theta_{gb}^{k} )\) then replace \(\theta_{gb}^{k}\) with \(Z_{k,best}^{{{\text{chaotic}}}}\), otherwise maintains \(\theta_{gb}^{k}\). Stop the searching process if the maximum iteration is satisfied.

Based on above-mentioned modifications, the scenario of the proposed IISA is presented through the pseudocode of Algorithm 2 as appeared in Fig. 4, where the proposed IISA begins with a population of elements or solutions which are initialized randomly. These solutions go to the ISA phase to explore the search space and potentially attain the best solution by using the composition phase based on crossover strategy. Then, the best solution will enter the local search phase to refine its quality and to avoid predicament in the local solution rather than the global solution. In this regard, the better solution will be fed into the next iteration. This scenario is iteratively continued until the stopping condition is satisfied.

Fig. 4
figure 4

The procedure of the proposed IISA

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5 Applications

5.1 Experimental setup

DC and short- and open-circuit tests are three experimental tests that have been carried out on a 3 HP six-phase induction motors and 1/3 HP three-phase to get the equivalent circuit parameters. Figure 5 shows a photograph of the experimental set up at Faculty of Engineering; Kafrelsheikh University. According to the IEEE Std 112-2004 specifications for tests procedure for PPIM [5], the tests have been performed. Tables 1 and 2 represent, respectively, the recorded measurements for short-circuit and no-load tests for six- and three-phase induction motors.

Fig. 5
figure 5

Experimental installation at the Kafrelsheikh University Faculty of Engineering

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Table 1 Experimental tests of 1/3 HP three-phase induction motor [2]
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Table 2 Experimental tests of 3 HP six-phase induction motor [2]
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5.2 Parameter settings for competitive algorithms

In this section, the proposed IISA is evaluated on parameters identification of the polyphase induction machines, i.e., 1/3 HP three-phase induction motor, and 3 HP six-phase induction motor. In this regard, the superior performance of the IISA is validate by carrying out the comparisons with other well-established optimizers including the genetic algorithm (GA) [42], differential evolution (DE) [43], Jaya algorithm (Jaya) [44], gray wolf optimizer (GWO) [45], sine cosine algorithm (SCA) [41], and basic ISA. For fair comparison, all of the implemented algorithms are carried out with the same maximum number of iterations (100) and population size with 60 individuals. It is notable that the maximum number of iterations and population size of IISA and ISA are set after running a few trials. Besides, each of the implemented algorithms is tested 20 times independently for every problem to reduce statistical errors. The parameter settings for all compared optimizers are based on the acclaimed suggestions in the corresponding literature and recorded in Table 3.

Table 3 The parameters’ settings of the for the implemented algorithms
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Firstly, the comparisons are claimed on the best outcomes represented by the square error values (i.e., Eq. (9)) to show the accuracy of each algorithm. And then, the statistical indices, convergence rates, stability analysis and statistical pairwise test are illustrated to evaluate and assess the robustness and convergence rate of each algorithm.

5.3 Results on the 1/3 HP 3-phase induction motor

The 1/3 HP, 3-phase induction motor parameters are estimated by merging the competitive optimization techniques that are GA, DE, Jaya, GWO, SCA, HPJOA [2], ISA and the proposed IISA. Table 4 explains the best equivalent circuit estimated parameters of the 1/3 HP 3-phase induction motor by the proposed IISA and implemented methods. It is note that the estimated parameters are recorded based on the best value of the objective function (least value of the square error).In the context of operational indices, a comparison of experimental and estimated values of the full load power factor and starting, maximum, full load torques has been made. In this sense, the IISA has lower fitness function (zero value for the square error) compared to the Jaya method which equals 1.68 × 10–5, GWO method which equals 1.73 × 10–5, GA method which equals 7.51 × 10–7, DE method which equals 1.68 × 10–5, SCA method which equals 4.90 × 10–5, and classical ISA method which equals 7.77 × 10–7. Although the fitness values for the presented algorithms seem to be are very close to that of the proposed IISA, the tiny reduction in the fitness value is significant as it is a prerequisite for accurate modeling.

Table 4 The obtained parameters of the 3-phase induction motor
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To further confirm that the obtained results are not reached by chance, some statistical measures of the presented algorithms are carried out over 20 independent runs, where these measures are shown in Table 5. The mean value for fitness function quantifies the average accuracy, and standard deviation (St. dev.) of fitness function indicates the parameters estimation reliability.

Table 5 The statistical measures regarding the fitness function of the 3-phase induction motor
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From these results, the proposed IISA leads to the best competitive performance compared with other algorithms at acceptable statistical indices.

5.4 Results on the 3 HP, 6-phase, induction motor

For the six-phase induction motor, the best parameters of equivalent circuit for the six-phase induction motor obtained by presented algorithms that include the GA, DE, Jaya, GWO, SCA, HPJOA, ISA and the proposed IISA are recorded in Table 6. In this sense, the estimated parameters are obtained based on the best value of the fitness function (i.e., minimum value of the square error). The estimated motor parameters reveal that the proposed IISA achieves the lower fitness function (zero value for the square error as in Table 7) compared to the Jaya, GWO, GA, DE, SCA, HPJOA and ISA that have reached the fitness values with1.91 × 10–3, 6.02 × 10–4, 7.86 × 10–7, 1.91 × 10–3, 1.34 × 10–3, and 7.37 × 10–4, respectively. To further confirm that the obtained results are not reached by chance, some statistical measures of the presented algorithms are carried out over 20 independent runs, where these measures are shown in Table 7. The mean value for fitness function quantifies the average accuracy, and standard deviation (St. dev.) of fitness function indicates the parameters estimation reliability. The comparison between estimated and experimental parameters reveals that the proposed IISA can accurately predict the corresponding motor parameters with good level of precision.

Table 6 The obtained parameters of the6-phase induction motor
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Table 7 The statistical measures regarding the fitness function of the 6-phase induction motor
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5.5 Operating characteristic performance by estimated parameters for induction motor models

Figure 6 explains the torque-speed characteristic of 3-phase induction motor using the experimental test recorded parameters and the parameters estimated from presented algorithms including, Jaya, GWO, GA, DE, SCA, ISA and the proposed IISA. The torque-speed characteristic estimated by IISA parameters is very close to actual characteristic. The characteristics of rotor and stator current against speed using estimated parameters by the Jaya, GWO, GA, DE, SCA, ISA and the proposed IISA, and the presented algorithms are shown in Fig. 7a, b. Estimated current of stator and rotor using Jaya, GWO, GA, DE, SCA, ISA and the proposed IISA, the calculated parameters using IISA is very close to the actual characteristics.

Fig. 6
figure 6

Three-phase induction motor electromagnetic torque—slip characteristics (GWO, Jaya, GA, DE, SCA, ISA and IISA)

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Fig. 7
figure 7

3-phase induction motor performance characteristics (GWO, Jaya, GA, DE, SCA, ISA and IISA)

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The characteristics of power factor and magnetizing current versus speed of 3-phase induction motor using Jaya, GWO, GA, DE, SCA, ISA and IISA calculated parameters are shown in Fig. 7c, d. Estimated characteristics of power factor and magnetizing current using IISA calculated parameters has very small difference from the actual and give the better characteristics.

Figure 8 shows the torque-speed characteristics of 6-phase induction motor estimated from the measurements, Jaya, GWO, GA, DE, SCA, ISA parameters and IISA parameters. Estimated torque-speed characteristic using ISA calculated parameters has a little deviation from the measured torque-speed characteristic, while that using IISA is closer to real characteristic.

Fig. 8
figure 8

6-phase induction motor torque—slip characteristics (GWO, Jaya, GA, DE, SCA, ISA and IISA)

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The stator and rotor current-speed characteristics of 6-phase induction motor using Jaya, GWO, GA, DE, SCA, ISA and IISA calculated parameters are declared in Fig. 9a, b. It is clear that the calculated rotor and stator current characteristics using ISA calculated parameters has deviation from the characteristic calculated by measured values, while estimated characteristics using IISA calculated parameters is very close to the actual characteristics. The power factor and magnetizing current characteristics versus speed of 6-phase induction motor using IISA calculated parameters are declared in Fig. 9c, d. Estimated power factor and magnetizing current characteristics using IISA calculated parameters has very small deviation from the measured characteristics.

Fig. 9
figure 9

6-phase induction motor performance characteristics (GWO, Jaya, GA, DE, SCA, ISA and IISA)

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5.6 Comparison with state-of-the-art optimization methods

In this section, proposed IISA was further investigated by comparing its performance with some advanced state-of-the-art optimization algorithms including chaotic ant lion optimization (CALO) [46], chaotic gravitational search algorithm (CGSA) [47], chaotic electromagnetic field optimization algorithm (CEFOA) [48], crossover-based sine cosine algorithm (CBSCA) [49], crossover-based gravitational search (CBGSA) [50], and crossover-based monarch butterfly optimization (CBMBO) [51]. The parameters’ values for the above competitors used for comparison were suggested as recommended in the original articles.

Table 8 records the results of the fitness function achieved by the IISA and advanced optimization competitors. It can be revealed from the table that IISA provides superior performance compared to the other state-of-the-art competitors in terms of solution accuracy on the 3-phase and 6-phase models. Furthermore, the optimal parameters for 3-phase and 6-phase models obtained by the optimization competitors are recorded in Table 9. For the 3-phase model, the IISA achieved to the global optimal solution with zero value for the objective function (i.e., minimization of the square error), other counterparts were unable to find a good solution as the achieving of the optimal solution for unknown parameters is prerequisite for accurate modeling.

Table 8 Statistical results obtained by IISA and state-of-the-art optimization competitors for studied models
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Table 9 Estimated optimal parameters using the IISA and state-of-the-art optimization competitors for studied models
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The IISA won first place on 6-phase model, which reveals its very competitive results. Furthermore, it can be seen that the IISA presents a more stable performance in different runs as it possesses very small standard deviation close to zero value. The calculation time is also a significant metric for assessing an algorithm's performance. In order to further confirm the algorithm's efficiency, the average CPU calculation time of all competing optimizers on the two models regarding the induction machine is recorded in Table 10. From Fig. 10, it can be seen that the time taken by each algorithm on different models is different. No matter which model, CEFOA possesses the largest average CPU time, and ISA has the shortest average CPU time. IISA is not the fastest; it is relatively very close to other competing optimizers. Therefore, it can be affirmed that the IISA is a promising optimizer.

Table 10 Average CPU time obtained by each algorithm on different models
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Fig. 10
figure 10

Average CPU time achieved by each algorithm on different models

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5.7 Statistical pairwise analysis: the Wilcox on signed-rank test (WSRT)

As in [52], the assessment of several optimization algorithms no free lunch theorems for optimization In this subsection, The WSRT is statistical pair wise test that was employed to determine the significant difference of obtained results by different methods [53]. This statistical evaluation metric rejects or accepts the hypothesis if the median of the achieved results by a pair of algorithms is different or equal. In this test, the statistical significance level (\(\alpha )\) is usually set to 0.05, and \(p\)-value examines that the results achieved by two different methods are significantly different or not. For a pair-side comparison on two individual samples, Table 11 shows the \(p\)-values computed using the nonparametric WSRT. The test was performed at a meaning level of 0.05, with 20 independent runs. Based on the reported results, it can be revealed that all \(p\)-values are less than 0.05, which confirms that the performance of IISA is significantly better than these state-of-the-art competing methods.

Table 11 Pairwise statistical comparison between the proposed IISA and all competitors using WSRT (\(\alpha =0.05\))
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5.8 Convergence analysis

Convergence speed is an essential assessment to evaluate the behavior of performance of any algorithm during the course of iterations. To justify the convergence behavior of the proposed IISA, Fig. 11 represents the convergence curves for the IISA and the compared basic state-of-the-art algorithms, GA, DE, Jaya, GWO, SCA, and ISA on 100 iterations. Also, Fig. 11 shows the convergence curves of the proposed IISA compared with advanced state-of-the-art algorithms, namely CALO, CGSA, CEFOA, CBSCA, CBGSA, and CBMBO, on 100 iterations for the 3-phase and 6-phase models. The figures portray convergence behavior with taking the x-axis for the growth of the iterations and the iteration's best fitness as y-axis. It can be revealed from in Fig. 11 that the newly developed IISA methodology converges faster than other compared peers for the studied models of induction motor, affirms the superior performance ability of the IISA in describing the actual characteristics of induction motor models.

Fig. 11
figure 11

Convergence curves by the competitive basics and advanced optimizers

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5.9 Stability analysis using the box plot behavior

Box plot behavior analysis is an important metric to evaluate the stability behavior of any optimization algorithm over the course of different independent runs. In this regard, the box plot diagram uses the inter quartile range to depict the box aspect, and the median value to represent the bar in the box.

The lowest width of the inter quartile range presents the most stable algorithm. As depicted in Fig. 12, it can be observed that the proposed IISA provides super narrow width for the inter quartile range than the compared basic and advanced state-of-the-art algorithms while dealing with the 3-phase and 6-phase models. Then, the depicted box plots advocate the supremacy of the IISA over the other peers.

Fig. 12
figure 12

The box plot analysis by the competitive basics and advanced optimizers

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6 Conclusions

This study has extracted the electrical equivalent circuit parameters of PPIM. The proposed IISA has been used to characterize the PPIM electrical model estimated parameters. Significant benefits of the proposed IISA compared with basic state-of-the-art algorithms like GA, DE, Jaya, GWO, SCA, and ISA and some advanced state-of-the-art algorithms like CALO, CGSA, CEFOA, CBSCA, CBGSA, and CBMBO are observed. Added to the estimated parameters obtained by using the competitive algorithms, the performance of the PPIMs is evaluated. High closeness between the experimental records and the estimated tests based on the extracted parameters. The results show that the proposed IISA methodology for extracting the optimal parameters of three- and six-phase induction devices is valid and reliable. The use of IISA also improves the operating performance of the PPIMs as it can accurately describe the actual characteristics of induction motor models. The suggested IISA's robustness is confirmed against other competing methods using some statistical indicators, including the Wilcoxon rank test, stability analysis, and convergence analysis. The record results and analyses have validated the developed IISA's superiority over the compared peers.

Although competitive performances and analysis have illustrated the efficiency of the proposed algorithms in estimating the parameters of the induction motor models over the other competitors, the incorporation of crossover strategy and chaotic concepts can take a slight time compared to their conventional counterpart. Moreover, there are still some limited issues that can be explored in future work. Firstly, the proposed IISA can be extended to deal with other complicated models of induction motor to exert its capability thoroughly. Secondly, the feasibility of the IISA can be further enhanced through the use of other optimization strategies and concepts. The authors would like to point out that IISA cannot be considered as a ubiquitous algorithm because no such algorithm exists that can deal with all optimization tasks as per the statement of the NFL theorem.