1. Introduction
Induction devices are suitable for harsh operating settings due to their robust and uncomplicated design. Because of their adaptable, affordable, and dependable architecture, direct grid supply, and single or multi-phase manufacturing capacity, tiny or big powers are widely utilized in daily life and various industries [
1]. Among the most favored basic varieties of induction machines are squirrel-cage asynchronous machines. Compared to comparable permanent magnet machines, squirrel-cage induction machines are easier to assemble, less expensive, and require no driver [
2]. Understanding the comparable circuit model and its variables is crucial to operating squirrel-cage induction machines correctly and consistently. To the extent that one intends to use the machine in real-world scenarios, examine and test it in lab or simulation settings, and completely optimize it, these characteristics must be understood accurately or very near to reality [
3]. Maximum torque, current data, and no-load and locked-rotor tests or full-load tests are utilized to calculate the equivalent circuit variables of squirrel-cage machines [
4]. But running these tests is not always safe and feasible, especially when using large, powerful equipment. Machine manufacturers provide information about machine specs in their catalogs [
5].
However, the manufacturer’s data does not provide the machine’s corresponding circuit variables. It is possible to approximate the device’s equivalent circuit parameters by using information provided by the manufacturer. Recent studies on asynchronous machine parameter estimation have seen a sharp rise. The literature presents a few important and thorough investigations of methods based on calculating the machine parameters and performance of asynchronous machines [
6,
7,
8]. A novel approach to determining the double-cage method parameters of squirrel-cage induction motors was presented in a study. The machine impedance was measured at several places to approximate the motor characteristics [
9]. Through the use of machine labels and manufacturer catalog information, a new, simple, and non-repetitive methodology for deriving equivalent circuit characteristics of an asynchronous machine was described in another study. A slip function was used to simulate the variation in the rotor parameters [
10].
The motor variables’ accuracy utilized by the control algorithm determines how successful the controller is, which is why parameter determination of the induction motor is a significant topic in the literature on electric drives [
11]. Squirrel-cage induction motor models are often constructed using single- or double-cage versions. These methods’ variables can be acquired in two different models [
12]:
Accurate forecasting of the behavior and induction motors’ efficiency has challenged designers. Efficiency forecasting has become more accurate when electromagnetic field problems are solved numerically [
14]. The finite element method is one of the most effective and widely utilized machine analysis and design tools [
11,
15]. However, the numerical approach’s primary deficiency is the high processing time and resource requirement [
16]. The finite element method can be used during machine design to examine performance and optimize the motor design iteratively.
Moreover, the complete computing procedure may be repeated while creating a special machine that satisfies various constraints and requirements [
17]. Consequently, developing a model replicating finite element data and determining the necessity and adequate machine efficiency more quickly while retaining high precision would be an invaluable and crucial design instrument [
18]. Artificial neural networks (ANNs) fit such a method well [
19].
As previously mentioned, an induction motor’s reactance is a crucial factor influencing its efficiency and performance. It represents the resistance to the alternating current flow provided by the motor’s windings. In this way, the ANNs can be applied to several parts of the study of induction motors, such as the motor’s reactance prediction and modeling. ANNs are robust computational models modeled after the human brain’s composition and operation [
20]. They are made up of artificial neurons that are networked together and can learn from data to anticipate the future or carry out tasks. Recently, various techniques for determining the characteristics of induction motors have been developed by researchers [
12,
21,
22]. One of these techniques was used to identify the characteristics of induction motors operating in the electromechanical mode using the transient stator current.
Regarding the squirrel-cage model of induction motors presented in Ref. [
23], new parameters are determined by estimating the mechanical speed and instantaneous electrical power using a free acceleration test. Authors have recently proposed a two-step method to detect the characteristics of an induction motor using a nonlinear method that incorporates magnetic satiation [
24]. In Ref. [
25], the genetic algorithm (GA) presents a unique method for predicting the electrical parameters of an equipollent circuit for a three-stage induction motor. In [
26], the topic of variable determination for induction motors is covered. The proposed algorithm for an artificial bee colony (ABC) is compared with the most recent approaches. The shuffling frog-leaping algorithm—initially introduced in Ref. [
27]—and standard manufacturer data can be utilized to approximate the variables of the double-cage asynchronous device. Depending on the number of physical factors, the variables of the induction motor were obtained from a poor initial approximation using nonlinear regression approaches based on a least-squares two-step strategy.
It is crucial to have a precise understanding of the electrical properties of induction machines due to their wide applications in industry [
28]. The electrical equivalent circuit characteristics of the induction motor are necessary to compute the motor’s efficiency [
29], load variations’ response, control driver parameters [
30,
31], and estimate breakdown behavior [
32]. Consequently, determining electrical and mechanical parameters requires accurate parameter estimations and induction motor modeling [
33]. However, their nonlinear models make obtaining these motors’ mathematical methods somewhat difficult [
34]. This study examined how training affected the parameter determination of squirrel-cage IMs.
This research improves the modeling of IMs utilizing analogous circuits, neural networks, and numerical simulations. ANNs have been used to solve numerous engineering issues [
35,
36]. Before being expanded to accommodate more polyphase rotating induction motors [
37], a neural network was first utilized to produce a single-sided linear IM [
38]. In these early research articles, the neural network transferred the input machine’s geometrical design variables to the output machine’s efficiency. The closely linked and interdependent input–output variables in this model are one of its drawbacks; this makes it challenging and ineffective to create data patterns and train the network. The same circuit approach can considerably reduce the coupling effect. It links machine performance through circuit variables to geometric design elements. The circuit parameters of an induction motor can all be connected to particular machine parts [
39]. The neural network will be trained more effectively as a result. It is possible to predict and measure the impacts of the airgap, stator, and rotor independently on the behavior and performance of a machine. Using this method will enable you to identify the necessary efficiency attributes swiftly. Because the circuit variables are estimated using the finite element approach, which also considers the complex geometry and excitations, as well as the non-linear magnetic characteristics, this model retains a high level of accuracy.
Using a novel approach proposed in this publication, variables of the double and single-cage patterns of induction motors are estimated. The proposed method uses six network (ANN) techniques. To do this, the manufacturer data [
4] use twenty induction motors with synchronous speeds varying from two to eight poles and a 400 V line voltage.
2. Structure and Equivalent Circuit Models of Squirrel-Cage Induction Motors
Induction motors are preferred in the industry since they require less maintenance than many other motors, in addition to their simple, robust, and inexpensive structures. The structure of squirrel-cage induction motors, one of the most preferred, is also straightforward, robust, and valuable. In addition to these features of squirrel-cage induction motors, their load-dependent speed regulation is also quite good. Squirrel-cage rotors can be manufactured as single or double-cage rotors. The squirrel cage machine’s rotor is manufactured by cutting laminated steel sheets into the appropriate geometry and pressing them into blocks. The cage structure is formed by placing aluminum or copper rods in the channels opened following the surface of the rotor, which has been turned into a block, using casting or other techniques [
40]. The rotor formed in this way is called a squirrel-cage rotor. Squirrel-cage bars may have different combinations and geometries. The stator of the induction motor of the squirrel-cage approach and the double and single-cage rotor structure are illustrated in
Figure 1.
Obtaining equivalent circuit parameters with a high degree of accuracy from the manufacturer’s data sheet of a squirrel-cage machine allows electrical and mechanical analysis of the machine without purchasing or using it directly in a system—in other words, without risking it. This gives us significant advantages in terms of cost, time, and practicality in many applications.
The induction motor is one of the most commonly utilized motor types in high-efficiency drive applications, which requires full knowledge of some, if not all of the induction motor variables of the control schemes of drivers [
41]. It is necessary to establish the equivalent circuit method of the machine and know the parameters of this model correctly in terms of detecting critical operating points of the machine, performance analysis, control, protection, malfunction, and operation [
7]. Depending on the single or double-cage rotor structure, squirrel-cage induction motors can be modeled as a single or double-cage equivalent circuit. The most commonly utilized equivalent circuit model of the squirrel-cage induction motor is the constant parameter equivalent circuit model with a single-cage rotor structure [
11]. Contrary to popular belief, obtaining equivalent circuit parameters is complex, and it is even more challenging to find the typical amounts of the double-cage induction motor variables [
42,
43]. Since the single-cage equivalent circuit method does not adequately present induction motors, the double-cage equivalent circuit method should be utilized. The equivalent circuit models of the induction motor with a single and double-cage rotor structure are shown in
Figure 2 and
Figure 3. Under steady-state conditions, the equivalent circuit model consists of five electrical variables, including R
r, R
s, X
sd, X
m, and X
rd, for a single-cage structure, and seven electrical parameters, including R
s, R
2, R
1, X
sd, X
m, X
2d, and X
1d, for a double-cage structure. Among these parameters, X
1d and R
1 demonstrate the inner cage, and the variables of X
2d and R
2 demonstrate the outer cage [
4,
44].
Equations (1)–(10) illustrated the mathematical induction motor model [
45]. The mathematical modeling of an induction motor involves expressing its electrical and mechanical characteristics through a set of equations. An induction motor is an alternating current (AC) motor widely used in various industrial applications. The basic mathematical model consists of electrical equations representing the stator and rotor windings, and mechanical equations describing the motor’s motion. A proper predictive model can avoid complicated mathematical processes. The proposed predictive network obtained from this study can be implemented as a practical solution by producing design charts.
For balanced systems, and are zero; – are the stator phase currents and voltages; and are the d-q axis stator voltages; are the d-q axis stator currents; are the d-q axis rotor fluxes; is the rotor winding resistance with referred to stator; is the equivalent resistance; is the leakage factor; is the stator phase winding resistance; and are the angular frequency of stator and rotor currents; is the angular speed of the rotor; is the electromagnetic moment; is the load moment; is the damping coefficient; is the moment of inertia; is the stator inductance; is the rotor inductance with referred to stator; is the magnetization inductance; and is the number of double poles.
3. Established Dataset
The data utilized in this study are taken from Ref [
4]. The producer’s evaluations, displayed in
Table 1, are used to calculate the experimental data using a numerical method presented by Monjo, Kojooyan-Jafari [
46], Pedra and Corcoles [
47]. The models’ input parameters have rated power
P(
KW), full load power factor cos(
FL), maximum torque to total load torque
/
, the proportion of initial torque to total load torque
/
, the proportion of initial current to total load current
/
, angular velocity
(rpm), and full load efficiency
. The induction motor’s output variable is
. The empirical data are split into two categories to train and test the networks. Eighty percent of the empirical findings are employed for network training, and the remaining twenty percent are employed to assess the efficiency of the trained methods. To find the best ANN structures, various ANN structures (networks with various numbers of concealed neurons and layers in each concealed layer) were tested and optimized in this research.
The experimental data are randomly chosen to form the data training (the more significant data collection) and data assessment. The various ANN structures are initially developed using the training data (the training procedure). The precision of the developed (trained) network is then evaluated using the data set evaluation, which is unidentified to the network. A computer program was created using MATLAB software R2020 to train the ANN methods.
Table 2 displays the optimal architectures of the ANN method.
5. Results and Discussion
The motor reactance (
Xm) was predicted using several hybrid prediction models combined with multilayer perceptrons such as MVO, COA, HBO, LCA, OOA, and STOA. MATLAB was used to develop the algorithms. The correlation between the dependent and independent parameters was then determined using the networks fed by these datasets. A trial and error procedure was employed to determine the optimal intricacy of the predictive models. The step to execute an MLP is determining the testing and training phases. The training and testing datasets, which accounted for 80% and 20% of the dataset, respectively, were selected in line with earlier reports from other researchers. The designed models are dependable and straightforward, and agree with the original experimental results. This demonstrates that the developed technique is a versatile and dependable instrument. To determine the suitable population size, this study used parametric research. Different population sizes, such as 500, 450, 400, 350, 300, 250, 200, 150, 100, and 50, were used for several MLP analyses. The effectiveness of the error mentioned above and the trial procedure was approximated using the MSE reduction process. In light of the error procedures depicted in
Figure 7, the optimal efficiency was demonstrated in the smaller amount of the means square error (MSE = 429.2693) reached by the multi-verse optimization with a population size of 400. This structure was introduced as the ideal MVO-MLP architecture to facilitate future assessments of motor reactance. The other five methods, COA-MLP, HBO-MLP, LCA-MLP, OOA-MLP, and STOA-MLP, also showed an appropriate result in terms of MSE with values of 665.0085, 2080.2834, 795.005, 3298.092, and 642.6149, and relative population sizes of 350, 400, 450, 500, and 500, respectively.
To evaluate the proposed ANN approaches’ performance, the attained results and the known results are compared with each other.
Figure 8,
Figure 9,
Figure 10,
Figure 11,
Figure 12 and
Figure 13 compare the outcomes of the suggested ANN approaches with the empirical data for testing and training.
Figure 8,
Figure 9,
Figure 10,
Figure 11,
Figure 12 and
Figure 13 show a regression analysis of empirical and forecasted amounts to provide more detailed research into the suggested ANN models. The correlation factor (CF) can be used to validate the appropriateness of the suggested ANN models. The CF is determined as follows:
where
N represents the data frequency, and ‘
X(
Exp)’ and ‘
X(
pred)’ indicates the empirical and anticipated (ANN) amounts, respectively.
Figure 8,
Figure 9,
Figure 10,
Figure 11,
Figure 12 and
Figure 13 demonstrate a positive correlation between the recommended MVO-MLP, COA-MLP, HBO-MLP, LCA-MLP, OOA-MLP, and STOA-MLP methods, respectively, and the empirical amounts for the motors’ reactance.
Figure 8,
Figure 9,
Figure 10,
Figure 11,
Figure 12 and
Figure 13 make it understandable that the suggested ANN models’ anticipated outputs are reasonably close to the empirical findings, demonstrating the appropriateness of ANN as a precise and dependable method for the simulation of the induction motors for Squirrel-cage kind. It is evident that the coefficient of determination was 0.9962, 0.9937, 0.9858, 0.9929, 0.9691, and 0.9946 in the testing phase, and 0.99598, 0.99358, 0.98561, 0.99252, 0.97008, and 0.99395 in the training phase for the MVO-MLP, COA-MLP, HBO-MLP, LCA-MLP, OOA-MLP, and STOA-MLP techniques, respectively. Additionally, based on the enquired findings, it can be observed that the recommended MVO-MLP approach is more precise than the other approaches regarding the R
2 values.
To confirm the consistency of the learning system, the training mechanism was repeated several times for each structure. Six models introduced in
Table 3,
Table 4,
Table 5,
Table 6,
Table 7 and
Table 8 all underwent the same procedure. It is worth mentioning that a greater frequency of neurons results in more complex networks with elevated accuracy. The researchers opted for seven concealed layers as the fittest structure based on the precision of testing results and a modest increase in testing R
2 value (and a negligible reduction in the RMSE value). Consequently, an MLP architecture with a network structure of 8 × 7 × 1 was selected as the best possible structure for an overall better hybridization process (for example, MVO-MLP, COA-MLP, HBO-MLP, LCA-MLP, OOA-MLP, and STOA-MLP). For the testing and training datasets, the proposed MVO-MLP model yielded R
2 of 0.9962 and 0.99598 and RMSE of 20.80626 and 20.31492, respectively. The COA-MLP, HBO-MLP, LCA-MLP, and OOA-MLP model, nevertheless, had R
2 of 0.9937, 0.9858, 0.9929, and 0.9691 in training, and 0.99358, 0.98561, 0.99252, and 0.97008 in testing; and RMSE amounts of 26.6653, 39.97074, 28.3406, and 58.72181 in training, and 25.64152, 38.31342, 27.67516, and 55.03676, in testing. Additionally, for the training and testing stages of the STOA-MLP, the R
2 are 0.9946 and 0.99395, and the RMSE are 24.80362 and 24.89665, respectively. According to
Table 9, which summarizes the findings of all six tables, the formed hybrid MVO-MLP model precisely anticipates the motor’s reactance. In light of this, the proposed hybrid MVO-MLP model can be used or recommended as a cutting-edge, accurate model for predicting the motor’s reactance.
Table 9 demonstrates that the optimum population amounts for the MVO-MLP, COA-MLP, HBO-MLP, LCA-MLP, OOA-MLP, and STOA-MLP were estimated to be 400, 350, 400, 450, 500, and 500, respectively. The mistakes occurrence and the minimum mistake number in the MVO-MLP, COA-MLP, HBO-MLP, LCA-MLP, OOA-MLP, and STOA-MLP best-fitted structures are demonstrated in
Figure 14,
Figure 15,
Figure 16,
Figure 17,
Figure 18 and
Figure 19, respectively. The outcomes obtained from the testing and training database show a great agreement among the observed and estimated amounts of motor reactance. During the training stages, the MAE amounts of 13.6075, 19.9943, 33.1182, 21.0433, 43.0551, and 17.8025 were reached for the MVO-MLP, COA-MLP, HBO-MLP, LCA-MLP, OOA-MLP, and STOA-MLP, respectively. Also, the MSE amounts related to the MVO-MLP, COA-MLP, HBO-MLP, LCA-MLP, OOA-MLP, and STOA-MLP models are correspondingly equal to 433.1356, 712.2449, 1961.1447, 813.8206, 3683.0843, and 615.406. According to the error values, it is evident that MVO-MLP is a more reliable predictive network than the other proposed algorithms for approximating real-world induction motor reactance.
Taylor Diagram
In addition to the statistical parameters (RMSE, MAE, and R
2), the Taylor diagram [
61] was utilized to evaluate the mentioned methods’ accuracy. This graph accurately maps the predicted and observed data [
62]. Taylor represented a solo projection for representing various parameters of evaluation. Considerably, these are qualified for focus on the methods’ accuracy utilizing several plot points. The Taylor graph shows the standard deviation, RMSE amount, and correlation coefficient between observed and predicted amounts for better recognition of variations [
63].
Figure 20 demonstrates Taylor graphs for different best-fit methods. The radial length from the observed value is the RMSE quantity [
61]. As a result, the more accurate method is recognized by the point with the highest R
2 amount (R
2 = 1) and the RMSE with the minimum value. It was evident from
Figure 20 that all six methods show high accuracy in predicting the induction motor reactance, but MVO-MLP produced the best prediction.
6. Practical Implementation
As stated earlier, ANNs (e.g., including all hybridized techniques) can be trained on the historical data of induction motors, including their design specifications, geometrical parameters, operational conditions, and the corresponding reactance values. By learning from this data, the ANN can establish patterns and relationships between the motor characteristics and its reactance. Once trained, the ANN can predict a new motor’s reactance based on its parameters. Their ability to learn from large datasets and generalize from learned patterns makes them valuable tools in motor analysis, optimization, and fault detection. It is noteworthy that ANNs have advantages such as flexibility, adaptability, accuracy, generalization, real-time and fast predictions, data-based optimization, and integration with automation systems. These advantages make ANNs useful and valuable for industrial applications’ motor monitoring, control, and optimization. Estimating equivalent circuit variables of an induction motor using ANN can be applied in various ways in industry. In this way, the engine can optimize operations, increase reliability, reduce maintenance costs, and improve overall system efficiency. In addition, ANNs can help detect and diagnose engine faults early, prevent unexpected failures, and minimize downtime by monitoring equivalent circuit parameters. By analyzing the trends and patterns in the predicted parameters, it can be decided how to intervene according to the signs of deterioration or inefficiency in the engine’s operation.
As expected from the general ANNs, the hybrid methods used in this article are expected to contribute significantly to these applications. The preceding sections have established that the MVO algorithm exhibits a marginally lower convergence curve than other algorithms, as indicated in
Table 6. This proposes that, compared to alternative optimization strategies, the algorithm has achieved lower error rates when altering ANN parameters. Thus, the algorithm’s outcomes are provided here to create a predictive approach. Regarding ANN optimization computations, the last neuron involves eight variables and outputs. Seven prior neural layers, each with nine variables, supply this neuron. Metaheuristic techniques are used to optimize these sixty-three variables. Equation (45) uses the MVO method to determine reactance based on seven parameters that describe neural responses in the buried layer. These parameters are
O1,
O2,
O3,
O4,
O5,
O6, and
O7.
Reactance MVO-MLP = 0.2437 × O1 + 0.0850 × O2 + 0.5791 × O3 + 0.0937 × O4 − 0.3849 × O5 − 0.4100 × O6 + 0.4853 × O7 + 0.1542, where considering the labels presented in
Table 10,
In which,
Wi1,
Wi2, …,
Wi11, and
bi are in
Table 11.
Indeed, the output of this study can be employed to optimize the design of induction motors by adjusting various parameters to achieve desired reactance values. By training the ANN on a dataset that includes different motor designs and their corresponding reactance values, the network can learn to identify the optimal combination of parameters that yield the desired reactance. This can aid in improving motor efficiency, reducing losses, and enhancing performance.
7. Conclusions
This research improves the modeling of induction motors using neural networks, analogous circuits, and numerical simulations. The closely linked and interdependent input-output variables in this model are one of its drawbacks; this makes it challenging and ineffective to create data patterns and train the network. In this study, the COA, LCA, HBO, MVO, OOA, and STOA were applied to estimate the variables of the reactance of the motor. Comparing the earlier investigation and the ANN estimates demonstrates that the suggested MVO-MLP method with R2 of (0.9962 and 0.99598) and RMSE of (20.80626 and 20.31492) performed better than COA-MLP (R2 = 00.9937 and 0.99358 and RMSE = 26.6653 and 25.64152), HBO-MLP (R2 = 0.9858 and 0.98561 and RMSE = 39.97074 and 38.31342), LCA-MLP (R2 = 0.9929 and 0.99252 and RMSE = 28.3406 and 27.67516), OOA-MLP (R2 = 0.9691 and 0.97008 and RMSE = 58.72181 and 55.03676), and STOA-MLP (R2 = 0.9946 and 0.99395 and RMSE = 24.80362 and 24.89665). Not only has the suggested MVO-MLP generated a better outcome than others, but it is also very close to actual results. The suggested ANNs have yielded excellent results for the projected model.
According to the findings, metaheuristic solutions can offer valuable perspectives for optimizing induction motor reactance. Metaheuristic algorithms used in this study are well-suited for handling complex problems. These algorithms can address multimodal optimization problems by searching for multiple solutions simultaneously or adapting their search strategies to explore different regions of the search space. This flexibility allows for comprehensive design space exploration, leading to a better understanding of the trade-offs involved. Optimizing an induction motor’s reactance can enhance its overall efficiency. Metaheuristic algorithms can assist in identifying reactance values that minimize losses, reduce energy consumption, and improve the motor’s overall performance. This perspective aligns with the growing emphasis on energy efficiency and sustainability in various industries. Also, metaheuristic algorithms offer a highly efficient alternative by leveraging stochastic search strategies, reducing the computational burden, and providing reasonably good solutions within a reasonable timeframe. By incorporating domain knowledge and problem-specific constraints, these algorithms can be tailored to address the unique challenges and objectives in motor design and control. Further research and experimentation in this area can help to refine and improve the application of metaheuristic solutions in induction motor reactance optimization, leading to more efficient and optimized motor designs.