Applying the Sine-Cosine Optimization Algorithm to the Parametric Estimation Problem in Three-Phase Induction Motors

The steady-state analysis of electrical machines requires a detailed characterization of their equivalent electrical circuit, which adequately represents the transformation and interaction between electrical and mechanical energy. This research aims to characterize the equivalent circuit of three-phase induction motors by minimizing the mean square error between the measured and calculated torque variables. These torques are obtained from data provided by the manufacturer, including starting, peak


X th
Thevenin inductance (Ω).r 3 Random number between 0 and 2π with a uniform distribution.

Parameters
r 4 Random number between 0 and 1 with a uniform distribution.

Iterative counter.
t max Maximum number of iterations.

V ph
Single-phase voltage applied to the induction motor (V).
X max 1 Maximum value for the stator reactance (Ω).

X min 1
Minimum value for the stator reactance (Ω).

X max 2
Maximum value for the rotor reactance (Ω).

X min 2
Minimum value for the rotor reactance (Ω).
X max

M
Maximum value for the magnetizing reactance (Ω).

M
Minimum value for the magnetizing reactance (Ω).

Sets and indices
i Position of the solution in the matrix of potential solutions.

Variables R 1
Resistive effect assigned to the windings of the stator in the induction motor (V).
R 2 Resistive effect assigned to the windings of the rotor in the induction motor (Ω).
Potential solution for the next iteration (i.e., t+1), using the sine rule.
Potential solution for the next iteration (i.e., t+1), using the cosine rule.
Vector containing the i th potential solution at iteration t.

General context
The processes related to the final use of electrical energy have gained significant relevance in terms of both operation and costs (Chauhan, Chauhan, & Badar, 2022).Therefore, energy efficiency plays a fundamental role in new industrial developments (Bouakkaz, Mena, Haddad, & Ferrari, 2021).This has been demonstrated multiple times through the increasing rate of vehicles that require less fuel and the growing efficiency of appliances (lower electricity consumption), e.g., lamps that consume a quarter of the energy compared to classic lighting with incandescent bulbs (Nota, Nota, Peluso, & Lazo, 2020;Abo-Khalil et al., 2022).
At the industrial level, one of the most common elements corresponds to induction motors, which are present in all economic sectors and are considered to be the cornerstone of the modern industry (Sengamalai et al., 2022).It is estimated that engines use 65% of the electricity generated in the world, so they can contribute greatly to reducing energy consumption and CO 2 emissions (Pay án, Fernandez, Ortega, & Santos, 2019).On the other hand, environmental policies based on energy efficiency, together with those related to the energy transition, comprise a large number of actions aimed at reducing global warming; the less energy is used, the fewer pollutants associated with the energy sector will be produced (Friederici, 2021).

Motivation
In order to contribute to the study of induction motors, large-scale industries with hundreds of engines must be analyzed.Here, due to intensive use, multiple internal parameters related to efficiency calculations can change over time (Trisha, Gupta, & Kumar, 2021).To update the internal parameters of induction motors (i.e., series and parallel reactances), the specialized literature has proposed multiple optimization approaches that avoid physical interventions and favor classical laboratory tests (V éliz-Tejo, Travieso-Torres, Peters, Mora, & Leiva-Silva, 2022).These optimization algorithms focus on torque measurements at the terminals of the induction motor for different load conditions.With these measurements, a nonlinear non-convex optimization model has been proposed which minimizes the expected error between the calculated and the measured torques (Mohammadi & Akhavan, 2014).This research proposes the application of the sine-cosine algorithm (SCA) to obtain the internal parameters of induction motors by solving the equivalent optimization model, since metaheuristic optimizers have demonstrated to be efficient and robust in solving parametric estimation problems associated with electrical devices, i.e., induction machines, distribution transformers, and photovoltaic (PV) modules, among others.

Literature review
Although there are many methods to estimate the parameters of induction motors, iterative methods based on least squares are the most widespread, given their simplicity and reasonable convergence times (Lindenmeyer, Dommel, Moshref, & Kundur, 2001;Toliyat, Levi, & Raina, 2003;Pedra & Corcoles, 2004;Gupta, Wadhwani, & Kapoor, 2011).In addition, multiple metaheuristic optimization algorithms have been applied to solve the nonlinear non-convex optimization problem regarding parametric estimation in electrical machines, as they are easily programmable and require low computational efforts.These algorithms include bee colony optimization (Aminu, 2019), particle swarm optimization (Huynh & Dunnigan, 2010), the gravitational search algorithm (Avalos, Cuevas, & G álvez, 2016), and the water cycle algorithm ( Ćalasan, Micev, Ali, Zobaa, & Aleem, 2020), among others.Some of the most recent applications belonging to the family of metaheuristic optimization algorithms for parametric estimation in induction motors are discussed below.
Mohammadi2014 combined the classical genetic algorithm and the particle swarm optimizer to obtain a hybrid optimization approach aimed at determining the electrical parameters of three-phase induction machines.
The optimization problem was formulated as a nonlinear programming model, where the mean square error between manufacturer data and the calculated starting, maximum, and full-load torques was considered as the objective function.Numerical results confirmed that this hybrid approach provides better numerical results regarding the objective function value when compared to the genetic algorithm and particle swarm optimization.
The work by Wu2018 employed the polynomial regression approach to determine the electrical parameters of induction machines.Numerical results provided accurate estimations when compared to the experimental motor curve and the procedure established by the IEEE Standard 112 Test.
In Guedes2018, the authors applied the differential evolution algorithm to estimate the electrical parameters of three-phase induction motors via dynamic simulation.Theoretical results with the proposed optimization algorithm and experimental validations confirmed the accuracy of this approach.However, the authors provided no comparative analysis with additional optimization algorithms.
The study by Rezk2019 presented an effective optimization method to estimate parameters in three-phase induction motors.
The particle swarm optimization algorithm was used in combination with experimental validations, with the purpose of reducing the error between the theoretical model and the experimental setup.Numerical results confirmed that the particle swarm optimizer yields acceptable results.However, comparative analyses with other metaheuristic optimizers were not presented to demonstrate the effectiveness of the proposed approach.Joodaki2020 applied the cuckoo search algorithm to estimate parameters in three-phase induction motors while considering manufacturer torque data.Numerical comparisons in two machines with a genetic algorithm, the water cycle algorithm, and the bacterial proliferation approach, among others, confirmed the effectiveness of the cuckoo search algorithm in minimizing the error between the measured and the calculated torques.
The main characteristics of the approaches mentioned above are the following: (i) the manufacturer data regarding the starting, maximum, and full-load torques are typically used to formulate a minimization problem aimed at finding the electrical parameters of the motor, which makes the theoretical and the calculated torques equal; and (ii), given the complexity of the optimization model, most studies focus on the application of metaheuristic optimization algorithms to obtain an efficient solution.

Contributions and scope
In light of the above, this research presents the application of the sine-cosine algorithm (SCA) as an efficient optimization technique to estimate the parameters of the electrical circuits in induction motors.The SCA is implemented to solve the exact nonlinear non-convex optimization model for minimizing the error between measured and calculated torques under different load conditions (i.e., starting, full, and maximum load torques).This algorithm was identified as a potential solution methodology because it has already demonstrated its effectiveness and robustness in solving similar problems, as is the case of parametric estimation in single-phase transformers and PV modules (Bocanegra, Montoya, & Molina, 2021;Montoya, Gil-Gonz ález, & Grisales-Nore ña, 2020).
Regarding the scope of this research, note that all measurements of the induction machine have been taken with specialized torque measurement systems, which have been reviewed and conditioned (filtered) prior to their evaluation in the proposed SCA.These data have also been provided by the manufacturer of the induction machine.In addition, a comparison between different metaheuristic optimizers (the genetic algorithm, the particle swarm optimizer, and a combination of the two) has been performed to validate our proposal.

Document structure
The remainder of this document is structured as follows.Section presents the general mathematical formulation associated with parametric estimation in three-phase induction motors while considering torque measurements.Section describes the general characteristics of the SCA and its application to the analyzed problem.Section outlines the main characteristics of the two induction machines examined with regard to their manufacturer data and the upper and lower bounds imposed on the decision variables.Section shows the numerical validations and a comparison between the SCA, the genetic algorithm (GA), the particle swarm optimizer (PSO), and a GA-PSO hybrid in the first Ingeniería e Investigaci ón 44(2) 3 test system.In addition, the multimodal nature of the optimization model is shown via three solutions provided by the SCA in the second test system.Finally, Section lists the main concluding remarks of this research and possible future studies using metaheuristic optimization methods.

Mathematical formulation
The parametric estimation problem in three-phase induction motors is formulated as a nonlinear non-convex optimization problem using the Thevenin equivalent of these machines under steady-state conditions.
Figure 1 presents the equivalent circuit representation of the induction motor.
Single-phase equivalent of the induction motor In Figure 1, R 1 and R 2 denote the resistive effects on the stator and rotor elements, X 1 and X 2 correspond to the reactance equivalent values of the stator and rotor windings, X M is the equivalent magnetization reactance, V ph means the single-phase voltage applied to the induction motor, and s is the sliding under normal operating conditions.Note that j is the imaginary unit.
To obtain the equivalent Thevenin representation of the induction motor in Figure 1, consider that the motor load (i.e., R 2 /s) is removed.Under this condition, the equivalent V th , R th , and X th are obtained.
Remark 1.Note that, in order to obtain (1), it was assumed that X M >> R 1 , which implies that the effect of R 1 can be neglected in the Thevenin voltage calculation and equivalent impedance (Mohammadi & Akhavan, 2014).In Equation 4, the general induced torque is defined.
where ω sinc is the synchronous speed.
In addition, the starting torque (τ st ) occurs when s = 1, which implies that it can be defined from (4) as follows: The maximum torque (τ max ) is reached at maximum converted power, which occurs when R2 s = R 2 th + (X th + X 2 ) 2 .This maximum torque is defined in (6).
Finally, the full-load torque (τ f l ) is reached when s = s f l in (4), which yields Considering the starting, maximum, and full-load torques defined from ( 5) to ( 7), the optimization model that includes the calculated and measured torques of an induction motor is formulated, using the sum of the errors as the objective function, which is defined in (8).
where E f is the objective function, and τ m st , τ m max , and τ m f l represent the measured values associated with the starting, maximum, and full-load torques, respectively.
Note that, in order to ensure that the solution to the objective function in ( 8) is subject to the torque equality constraints ( 5)-( 7), some typical bounds are imposed.These are associated with the stator and rotor resistances and reactances, including the magnetization reactance.These box-type constraints are listed below.
where Y min and Y max represent the lower and upper bounds assigned to the decision variables.
Remark 2. The problem regarding parametric estimation in three-phase induction motors corresponds to a nonlinear programming model aimed at minimizing the average square error defined in (8) and constrained by equalities (5)-( 7), which are related to the starting, maximum, and full-load torques (calculated), as well as by the box-type constraints (9)-( 13), which define the solution space where the decision variables exist (i.e., reactances and resistances).

Solution methodology
This study applies the SCA to solve the optimization problem expressed in ( 5)-( 13).The SCA is a metaheuristic optimization algorithm that belongs to the family of mathematics-inspired metaheuristics.It works from an initial (feasible) population that evolves during the iteration process while following trigonometric rules based on the sine and cosine functions (Attia, Sehiemy, & Hasanien, 2018).The main characteristics of the SCA approach are presented below.

Initial population
The SCA is a population-based optimizer that explores and exploits the solution space from an initial population generated upon the basis of the upper and lower bounds of the decision variables in order to make it 100% feasible.The structure of the solution individual i at iteration t (with t = 0) that conforms to the initial population is presented below.
where each vector component is obtained as a random value between the lower and upper bounds defined in ( 9)-( 13), following a uniform distribution.

Evolution of the population
The SCA applies evolution rules based on the trigonometric sine and cosine functions in order to explore the solution space.To determine whether an individual s t+1 i will be part of the population, the following evolution rule is applied: where s t+1 i,1 and s t+1 i,2 are two potential candidate solutions derived from the sine and cosine rules in ( 14) and ( 15), r 3 is a random number with a uniform distribution between 0 and 2π, r 4 is a random number between 0 and 1 with a uniform distribution, and N s is the number of potential individuals that make up the population.Note that s t best represents the best current solution in the population.In addition, r 1 controls the exploration and exploitation stages of the optimization algorithm by following a linear rule defined below.
To determine whether one of the individuals s t+1 i,1 or s t+1 i,2 will be part of the next population, the following criteria are applied. i.
iii.Otherwise, s t+1 i = s t i .
Remark 3. To ensure that each potential solution (s t+1 i,1 and s t+1 i,2 ) is feasible, each of its components is reviewed/corrected, with the purpose of maintaining their values between the lower and upper bounds (see box-type constraints (9)-( 13)).

Stopping criteria
One of these stopping criteria must be met to determine whether the SCA has finished exploring and exploiting the solution space.
i.If the number of iterations t max has been reached, or ii. if, during k max iterations, the value of the objective function (i.e., E f s t best ) has not improved.
Note that k max is set as 20% of the total number of iterations.

Summary of the SCA
The application of the SCA to the parametric estimation problem in three-phase induction motors while considering torque measurements is summarized in Algorithm 1.

Test systems
Two induction machines were considered to validate the proposed SCA with regard to the studied problem.The first system corresponds to a three-phase induction motor with 5 hp, 460 V, and 60 Hz.Its main characteristics are reported in Table 1, which have been adapted from Mohammadi2014.
The second machine is a three-phase induction motor with 25 hp, 460 V, and 60 Hz, whose main characteristics are reported in Table 2.
Data: Read data of the three-phase induction motor under analysis Define the lower and upper bounds of the decision variables; Define N s , t max , and k max ; Generate the initial population while observing the lower and upper bounds in ( 9)-( 13); Evaluate the starting, maximum, and full-load torques in ( 5), (6), and ( 7); Evaluate the objective function in (8); Find the best current solution s t best ; Make k = 0; for t = 1 : t max do Obtain the value of r 1 using ( 16); for i = 1 : N s do Obtain the random numbers for r 3 and r 4 ; Generate the potential solutions s t+1 i,1 and s t+1 i,2 using Equations ( 14) and ( 15); Check the feasibility of s t+1 i,1 and s t+1 i,2 and correct if necessary; Evaluate the starting, maximum, and full-load torques for s t+1 i,1 and s t+1 i,2 in ( 5), ( 6), and ( 7); Evaluate the objective functions E f s t+1 i,1 and The lower and upper limits of the decision variables for both test systems are listed in Table 3.

Numerical results and discussion
For the computational implementation of the proposed SCA, the MATLAB software (version 2021b) was employed on a PC with an AMD Ryzen 7 3700 2.3 GHz processor and 16.0 GB RAM, running a 64-bit version of Microsoft Windows 10 Single Language.To implement the SCA, a population size (i.e., N s ) of 100, 1000 iterations, and 100 repetitions were employed.

Results obtained for the first test system
The first test system was taken from Mohammadi2014, where three metaheuristic optimizers were applied: the PSO (Gulbahçe & Karaaslan, 2021), the classical GA (Fortes, Ferreira, & Coelho, 2013), and a hybrid between the two (HGAPSO) (Mohammadi & Akhavan, 2014).The numerical results obtained with each comparison method were contrasted with those of the proposed SCA.
Table 4 presents a comparative analysis between the measured torques (provided by the manufacturer) and the values calculated via the optimal solution reported by the methods used for comparison (Rengifo-Santana, Benzaquen-Su ñe, Aller-Castro, Bueno-Montilla, & Restrepo-Zambrano, 2015).Note that the errors in Table 4 correspond to the absolute error, which was calculated as presented below: where z m corresponds to the measured data and z c to the calculated value for each variable.
In comparison with those reported in the literature (Mohammadi & Akhavan, 2014), the results in Table 4 show that: i.The SCA provides the best estimates for the starting, maximum, and full-load torques.Regarding the starting torque, the estimation error was about 8.38 × 10 −4 %, while the HGAPSO approach reported about 3 × 10 −2 %, i.e., about 35 times higher than the SCA.
As for the maximum torque, the SCA reached an estimation error of about 4.58 × 10 −4 %, which is 65 times lower than the result reported for the HGAPSO approach.In the case of the full-load torque, the SCA reported an estimation error of about 1.90 × 10 −3 , followed by the HGAPSO with a value of 0.58%, i.e., the precision of the SCA is about 305 times better than that of the HGAPSO for this torque estimation.
ii.The GA and the PSO, when implemented independently, exhibit higher estimation errors in comparison with the HGAPSO approach, which is evident in the case of the starting and maximum torques.However, in the case of the full-load torque, these three methods have a similar behavior, with values between 0.58 and 0.76%.
iii.As for the objective function value, it is worth mentioning that the GA showed a value of about 2.75 × 10 −4 , the PSO reported about 3, 48 × 10 −5 , the HGAPSO found about 3.41 × 10 −5 , and the SCA reached about 4.63×10 −10 .These results confirm that the proposed SCA is the best optimization method to estimate parameters in three-phase induction machines while considering manufacturer data, as the objective function value is more than 70 000 times lower than that of the HGAPSO as reported by Mohammadi2014.
Table 5 compares the estimated parameters with respect to the manufacturer data and the results obtained with the metaheuristic optimizers.
In comparison with the specialized literature (Mohammadi & Akhavan, 2014), these numerical results show the following: i. Regarding the resistive and reactance parameters in the stator and rotor (i.e., R 1 , R 2 , X 1 , and X 2 ), the SCA exhibited the best numerical estimations, with errors lower than 0.30%.Meanwhile, the HGAPSO approach reported errors higher than 0.70% for the same parameters, which confirms the effectiveness of the SCA in dealing with the analyzed optimization problem.
ii.The magnetization reactance, as estimated by the SCA, was 36.5475Ω, with an error of about 4.82% compared to the manufacturer data.The HGAPSO approach reported 36.8888Ω, with an estimation error of about 3.94%.However, even though the SCA showed a higher estimation error in this parameter, given its magnitude in comparison with the stator and rotor resistances and reactances, this has no significant effects on the final objective function value, which is much better for the SCA when compared to the HGAPSO approach.Furthermore, these results confirm that the problem under study is in fact a multimodal optimization problem, i.e., it involves multiple combinations of variables to minimize the objective function (Bocanegra et al., 2021).

Results obtained for the second test system
The second test system is a three-phase induction motor that has not previously been reported in the specialized literature on parameter estimation via metaheuristic optimization.Therefore, considering the effective, efficient, and robust performance of the proposed SCA in the first test system, its best three results for this system are presented.Table 6 presents a comparison between the manufacturer torque data and the calculated values for each of these solutions (Pedra & Corcoles, 2004).In contrast, Table 7 presents the estimated resistances and reactances, as well as their comparison against manufacturer data.
These results confirm that the best three solutions reached with the SCA report objective function values lower than 2.17 × 10 −8 , with excellent performance regarding maximum torque, and slight deviations for the starting and full-load torques.
The numerical results in Table 7 show that: i.The parameter with the lowest estimation error regarding the manufacturer data is the stator resistance (R 1 ).
ii. Solution 1 is the best one regarding the objective function value and the estimation errors in each parameter, with values lower than 2% in the case of stator and rotor resistances and reactances and about 15.43% in the case of the magnetization reactance.
iii The parameter with the highest estimation error is the magnetization reactance, which varies from 15 to 47% in all solutions.However, these variations did not affect the torque calculations reported in Table 6.
To demonstrate that the best solutions reached with the SCA (Tables 6 and 7) adequately model the torque curve in the analyzed three-phase induction machine, the torque behavior for all solutions, including the benchmark case, is depicted in Figure 3.Note that, regardless of the estimation difference exhibited by this particular parameter in Table 7 with respect to the manufacturer data, the best three solutions reached with the SCA allow adequately reproducing the induced torque defined in Equation ( 4), which confirms the results regarding the starting, maximum, and full-load torques in Table 6.These results confirm the multimodal nature of the parametric estimation problem in three-phase induction motors.
It is worth mentioning that an optimization problem is considered to be multimodal when multiple combinations of variables provide the same numerical objective function value, i.e., the solution is not unique.However, in the case of parametric estimation in electrical machines, this behavior is typical, given the nonlinearities between parameters and electrical variables.Notwithstanding, if one solution is selected for numerical simulations and physical implementations, the motor's expected dynamic and static behavior will exhibit slight variations with respect to other potential solutions.

Conclusions and future work
This paper proposed an efficient solution methodology to determine electrical parameters in three-phase induction machines while considering measurement data provided by the manufacturer regarding starting, maximum, and fullload torques.This problem was formulated as a nonlinear programming model, with the aim of minimizing the sum of the errors between the measured and the calculated torques.
According to a comparative analysis with three metaheuristic optimization algorithms (i.e., GA, PSO, and HGAPSO) in the first test system analyzed, the proposed SCA was the most effective optimization algorithm for the studied objective function, with a final value of about 4.63 × 10 −10 , while the best literature report (the HGAPSO) found an objective function value of 3.41×10 −5 .In addition, regarding each particular parameter (i.e., stator and rotor resistances and reactances), the SCA exhibited the best numerical performance, with the lowest estimation errors.However, in the case of the magnetization reactance, the HGAPSO approach reported a better estimation, which could be attributed to the multimodal nature of the optimization problem under study.
Numerical results in the second test system confirmed that there are multiple combinations of resistances and reactances that allow for an adequate reproduction of the induced torque in the entirety of the operating range.In addition, the parameter with the highest deviation with respect to the manufacturer data was the magnetization reactance, which can also be attributed to the multimodal nature of the optimization model and the lack of information regarding the active power behavior of the induction motor.
In this study on parametric estimation, the highest errors were reported for the magnetizing reactance (i.e., X M ).However, these errors can be explained by the fact that this is the largest parameter in the induction motor, with a parallel connection to the equivalent circuit motor.According to circuit theory, the sum of two or more parallel elements is always lower than the small parameter, which confirms that, for variations in the largest parameter, this effect is minimal or negligible in the final solution.
As future work, the following studies could be conducted: (i) the application of new metaheuristic optimization algorithms to solve the nonlinear programming model that represents the studied problem; (ii) the use of the exact model of the induction machine without simplifications

Figure 2 Figure 2 .
Figure 2 depicts the equivalent Thevenin circuit representing the induction motor per phase shown in Figure 1.

Figure 3 .
Figure 3. Torque behavior in the second test system for the best three solutions reached with the SCA

Table 3 .
Upper and lower bounds of the decision variables in both test systems

Table 4 .
Comparative torque analysis for the first test system

Table 5 .
Comparative analysis for the first test system

Table 6 .
Comparative torque analysis for the second test system