In-situ efficiency and parameter estimation for induction motors using heuristic optimization

Abstract

Electric motors, especially three-phase induction motors, consume a significant portion of industrial electricity, making their efficiency critical for energy conservation. Traditional efficiency assessments require intrusive, labor-intensive laboratory tests, limiting in-situ evaluations. We propose a non-intrusive, heuristic optimization-based method compliant with IEEE Std-112 A/F1 to simultaneously estimate equivalent circuit parameters and in-situ efficiency using only field-measurable electrical quantities. Eight meta-heuristic algorithms were applied across six motors of varying power ratings and load conditions, with results validated against standard test data. The method achieved mean absolute errors below 0.7% at full load, demonstrating high accuracy and stability. This approach enables reliable, low-cost, and continuous efficiency monitoring under real operating conditions, offering practical benefits for industrial energy management and motor performance assessment.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-025-34932-1.

Introduction

In recent years, particularly since the energy crisis of the 1970s and the rapid industrialization that followed, concerns over energy consumption and efficient energy use have become increasingly prominent¹. Electric motors consume a significant proportion of the electricity produced in industry, accounting for two-thirds of the total. This increases both energy costs and greenhouse gas emissions. Consequently, it boosts energy demand while exacerbating environmental problems². Similar patterns are observed in developing countries such as South Africa, where motorized systems account for 60% of total electricity consumption. In contrast, developed countries like the US and Canada have responded to rising energy costs and sustainability concerns by implementing demand management programs aimed at improving energy efficiency³.

Given this global emphasis on energy conservation, the role of electric motors becomes even more critical. Due to their simple design, durability, and low maintenance costs, induction motors (IMs) are preferred and used in industry. Nearly all motors used in industry, 99.7% to be exact, are below 500 horsepower. The fact that these motors are operated at 60% of their rated load causes significant efficiency losses⁴. Consequently, these efficiency losses reduce energy efficiency and lead to economic losses. In response to rising energy expenses associated with inefficient motor operation, manufacturers have been motivated to develop more efficient electric motors⁵.

However, improving motor efficiency requires not only better motor design but also accurate assessment of the efficiency of motors already in operation. In industrial applications, low-efficiency machines draw significantly more power to deliver the same output, making energy efficiency a critical concern. Although manufacturers provide nameplate efficiency data, these values may not always be reliable¹. According to international standards (IEEE 112, IEC 60,034–2-1), accurate efficiency determination requires laboratory-based testing under controlled unloaded conditions. However, such procedures are labor-intensive, time-consuming, and impractical for many businesses⁶. As a result, there is a strong need for reliable, non-intrusive efficiency estimation methods that can be applied directly in the field while the motors remain in operation⁷.

Accurate determination of equivalent-circuit parameters is essential for the modeling, simulation, and control of induction motors. Although manufacturers typically provide these parameters, they are often unavailable or incomplete. Direct measurement of unknown parameters using specialized equipment is generally impractical for operational motors⁸. Parameter estimation methods are therefore widely used; however, the limited availability of measurement data and the fact that the parameters cannot be expressed in a closed-form mathematical solution make the estimation process challenging⁹. Because equivalent-circuit parameters directly affect the accuracy of both motor modeling and efficiency estimation, uncertainties in these parameters can significantly reduce the reliability of efficiency calculations. Conventional standards such as IEEE Std. 112-A/B (direct input–output tests) and IEEE Std. 112-F1 (segregation of losses) are considered the benchmarks for efficiency determination^10,11. Nonetheless, these methods require costly laboratory facilities, dynamometers, and torque transducers, making them unsuitable for in-situ applications^6,10. To address these limitations, researchers have developed non-intrusive efficiency estimation techniques based on identifying equivalent-circuit parameters from field measurements (voltage, current, power, and speed)^11–14. Heuristic and meta-heuristic optimization algorithms, including PSO, GA, WOA, GWO, and various hybrid approaches, have shown promise in reducing modeling errors between measured and simulated performance^15–18. Besides these efforts, GWO, WOA, and GA have been widely applied in control-oriented applications, including direct torque control and maximum power point tracking^19,20.

However, several important research gaps remain:

Most studies employ only one or two optimization algorithms, limiting the ability to evaluate robustness across a wider algorithmic set^11,14,21.

Existing research predominantly focuses on low-power motors (< 10 kW), while medium- and high-power machines—which account for the majority of industrial electricity consumption—are insufficiently investigated.

Parameter constraints are often chosen arbitrarily, which can result in slow convergence, premature stagnation, or unstable solutions.¹⁵.

In this study, a comprehensive approach is proposed for both equivalent-circuit parameter estimation and non-intrusive efficiency determination of three-phase induction motors. The approach employs eight state-of-the-art heuristic optimization algorithms (PSO, GA, GWO, HHO, TSA, DSA, WOA, and FOX) to estimate the per-phase equivalent-circuit parameters using only field-measurable quantities such as voltage, current, input power, and speed. Two estimation schemes (Method I and Method II) were developed to address uncertainty in nameplate output power and to evaluate the robustness of the algorithms under different constraint formulations. The proposed method was applied to six induction motors with different power ratings, enabling systematic comparison of algorithm performance across four load levels and validating the results against IEEE Std-112 A/F1 efficiency measurements.

The results show that the proposed methodology provides high accuracy—particularly at full load—without requiring intrusive laboratory tests, dynamometers, or torque sensors. By integrating physics-based parameter-range constraints with heuristic optimization, the approach improves convergence stability, reduces modeling errors, and enhances repeatability across different motor sizes. The contributions of this study are summarized as follows:

• A unified method is developed for the simultaneous estimation of induction motor efficiency and equivalent-circuit parameters, eliminating the need for invasive laboratory tests and torque sensors while remaining fully aligned with IEEE Std-112 A/F1 procedures.

• A comprehensive comparison of eight meta-heuristic algorithms (PSO, GA, GWO, HHO, TSA, DSA, WOA, FOX) is performed across six induction motors at four load levels (25–100%).

• A physical parameter-range determination strategy is introduced based on nameplate information and AEMOT manufacturer data, significantly improving optimization stability, convergence behavior, and repeatability.

Literature review

Today, induction motors are a fundamental component of industry and are widely used in applications ranging from compressors and conveyors to rail systems and electric vehicles. Their popularity stems from their low cost, simple design, robustness, and minimal maintenance requirements^22,23. The frequent use of induction motors and current energy efficiency concerns have made in-situ efficiency estimation a critical issue. In real industrial environments, stopping motors from performing traditional laboratory tests is impractical and can disrupt operations. Therefore, there is a growing need for methods that can estimate motor efficiency directly in the field, without interrupting operation or requiring intrusive procedures²⁴. This has attracted the attention of researchers.

In the equivalent-circuit approach, motor efficiency can be evaluated not only under the measured operating conditions but also across different load levels. Once the equivalent-circuit parameters are accurately identified, it becomes possible to compute performance at various loading stages without performing additional tests²⁵. Optimization algorithms, especially heuristic methods, are widely used for parameter estimation due to their flexibility and ability to effectively search for large solution spaces. Relevant studies in the literature are summarized below.

Several studies in the literature have focused on estimating induction motor efficiency and equivalent-circuit parameters using heuristic and meta-heuristic optimization techniques.

Pillay et al.¹ employed genetic algorithms for efficiency estimation and showed that using measurements from two loading points and incorporating additional constraints improves accuracy. They also demonstrated that incorporating rated power in the objective function reduces errors when output power remains close to its nominal value. Charette et al.²⁶ further advanced GA-based non-intrusive efficiency estimation by accounting for temperature effects at multiple load points, achieving errors below 1%. Similarly, Sakthivel et al.¹² proposed an immune-algorithm-based approach using a modified equivalent circuit that includes rotor leakage load loss, validating its accuracy against torque-sensor measurement methods.

Other studies have explored multi-algorithm comparisons and the limitations of traditional parameter estimation. Lu et al.²⁷ evaluated Newton–Raphson, Simulated Annealing, and PSO using input-impedance phasor data, but the method suffered from sensitivity to operating-point selection and simplified treatment of stray-load loss, yielding errors around 2–3%. Sakthivel et al.^13,28 also applied a bacterial foraging algorithm with a modified equivalent circuit that models PSLL through an added rotor-series resistance. While multi-load (25–100%) measurements improved accuracy, errors above 3% remained—particularly under low-load conditions—due to sensitivity to operating points and modeling assumptions. Thangaraj et al.¹⁴ compared GA, PSO, and Differential Evolution (DE) for in-situ efficiency estimation on a 5 hp motor, while Chandrakanth et al.²⁹ used PSO and validated results through physical measurements, achieving low error across multiple objective function formulations.

Several researchers have aimed to improve the robustness of the estimation process by modifying objective functions or broadening the measurement set. Al-Badri et al.¹¹ used a three-layer GA structure incorporating input power, current, phase angle, and output power. Although accuracy improved, reliance on output power is a disadvantage in practical conditions where this information is rarely available. Siraki and Pillay³⁰ proposed an evolutionary-based algorithm requiring only current and active power, which performs well under balanced and unbalanced conditions and avoids intrusive no-load tests.

Studies focusing on parameter-range formulation and search-space improvement have also emerged. Bijan et al.¹⁵ optimized parameter consraints using GA and validated the method on three motors. Diarra et al.³¹ introduced a hybrid QPSO–TRA technique that improves parameter determination, reduces computational burden, and enables real-time monitoring. Selek and Terzioğlu¹⁶ applied DEA and PSO for parameter identification, illustrating that nonlinear and time-varying parameters can be estimated with low error. Chayakulkheeree et al.¹⁷ proposed a PSO-based equivalent-circuit estimation method (PSOBECE), demonstrating significantly higher accuracy than slip and current-based techniques.

Additional contributions include Pillay et al.³² who showed that GA offers greater robustness and accuracy than Newton–Raphson for transient torque prediction using limited manufacturer data. Ramos and Lopez³³ assessed three evolutionary algorithms statistically for parameter estimation, while Sun et al.³⁴ estimated motor load and efficiency using PSO with hysteresis breakdown behavior. Guo et al.¹⁸ used GWO to identify optimal magnetic flux, showing efficiency improvements at the optimum point. Silva et al.³⁵ investigated temperature, magnetic saturation, and skin-effect influences on efficiency. Srited and Chayakulkheeree³⁶ obtained equivalent-circuit parameters by measuring voltage, current, real power, and power factor at multiple operating points, demonstrating high estimation accuracy. Bhowmick and Chowdhury³⁷ propose a PSO-based method to estimate the equivalent circuit parameters, losses, and rotor inertia of three-phase induction motors. Relying on transient current and voltage data, the proposed approach provides a high-accuracy and low-cost monitoring solution suitable for motor control and fault diagnosis applications. Geravandi and Moradi³⁸ present an air-gap torque (AGT)-based non-intrusive technique for estimating the efficiency of low-voltage induction motors using only nameplate information and line measurements. The method demonstrates significantly higher accuracy than conventional AGT approaches, even under unbalanced supply or harmonic distortion conditions. Manoharan et al.³⁹ propose an enhanced algorithm termed Adaptive Weighted Grey Wolf Optimization (AWGWO) for identifying the complex internal parameters of commercial induction motors. Experimental validation on eight different motor models confirms that the proposed algorithm achieves faster convergence and higher solution accuracy compared to existing methods. Paramo-Balsa et al.⁴⁰ introduce a novel non-intrusive approach for load and efficiency estimation without interrupting motor operation. By optimizing models based on input power and speed measurements, the method outperforms conventional techniques, achieving estimation errors below 3.5%.

Overall, the literature demonstrates that heuristic and meta-heuristic approaches offer a practical foundation for non-intrusive efficiency estimation. However, existing methods are often limited by a narrow range of algorithms, restricted validation scope, and insufficiently defined parameter constraints. Most prior studies concentrate on small motors or simplified models, resulting in limited understanding of medium- and high-power industrial machines. To overcome these gaps, this study proposes an integrated approach that applies eight meta-heuristic algorithms together with nameplate-based parameter coinsraints to estimate both the equivalent-circuit parameters and efficiency. The methods were tested on six motors at different load levels and validated using IEEE Std-112 A/F1 procedures. These results provide a broader and experimentally supported foundation for in-situ efficiency assessment and offer a practical tool for industrial monitoring and energy-efficiency applications.

Methodology

Proposed method

For efficiency estimation, heuristic algorithms can be employed to identify the equivalent-circuit parameters of induction motors, after which the efficiency is calculated in full compliance with IEEE Std-112 Form F2/Method F1 procedures⁴¹. In the literature, such approaches are generally referred to as optimization-based efficiency estimation methods. However, the effectiveness of these algorithms strongly depends on the accurate specification of the permissible search ranges for each parameter. Improperly defined bounds may lead to slow convergence or physically unrealistic solutions. To address this issue, the present study introduces a systematic technique that determines realistic parameter ranges using nameplate information and manufacturer data provided by AEMOT Electric Motors & Generators. This range-determination approach was then integrated into various optimization algorithms and incorporated into the overall efficiency estimation method.

The proposed approach was evaluated under two different scenarios to assess its performance. In the first scenario, the objective function was constructed using only the stator current, input power, and the phase angle between the stator voltage and current. In this case, the motor’s nominal power was intentionally excluded, and efficiency estimation relied solely on the parameter bounds determined by the proposed range-determination technique. In the second scenario, the nominal power value was additionally incorporated into the objective function, enabling the optimization algorithm to utilize both the range constraints and the motor’s rated operating characteristics. The results obtained from these two scenarios were compared with each other and further validated against experimentally measured data to evaluate their accuracy and robustness.

Figure 1 illustrates flowchart of heuristic-based efficiency estimation method. The process begins with in-situ acquisition of electrical and mechanical measurements, including stator voltage, current, input power, and shaft speed. Based on these measurements, parameter search ranges are defined using a combination of nameplate specifications and manufacturer data. An initial population of candidate solutions is then generated for the optimization algorithm. During each optimization iteration, the objective function is evaluated, and candidate solutions are updated according to the specific heuristic technique employed (PSO, GA, GWO, etc.), while the best-performing candidates are retained. This iterative process continues until the convergence criterion is satisfied. Once convergence is reached, the set of equivalent-circuit parameters corresponding to the best candidate is selected. Using these parameters, motor efficiency is computed under different load conditions. Finally, the estimated efficiency values are compared with the reference IEEE Std-112 A/F1 results to validate the accuracy of the proposed method.

Fig. 1.

Flowchart of optimization-based efficiency estimation (Method I and Method II).

At each iteration, the objective function (ff) is computed as the weighted sum of absolute deviations between measured and predicted quantities, given by:

1

2

3

4

5

6

Here, denotes the phase angle between the input current () and the input voltage, P_çkş,etiket represents the rated output power, and P_çkş,tahmin denotes the estimated output power. These functions constitute the sub-functions of the objective function, and the minimization-based objective function is defined as follows. As illustrated in the unified flowchart in Fig. 1, the proposed framework accommodates two distinct estimation strategies Method I and Method II within a single optimization process. The selection between these methods is determined by the formulation of the objective function. For Method I (Nameplate Independent), the objective function is constructed by deliberately excluding the nameplate output power term . This configuration aims to estimate parameters relying solely on field measurements, minimizing the influence of potential inaccuracies in the manufacturer’s labeled data. Conversely, for Method II (Nameplate Dependent), the term is directly included in the objective function. In this mode, the algorithm utilizes the motor’s rated power as an active constraint, leveraging the manufacturer’s specifications to guide the convergence process.

Parameter range determination

Accurate determination of equivalent circuit parameters (ECPs) is critical for modeling, performance analysis, and efficiency estimation of induction motors (IMs). The per-phase equivalent circuit of an IM, shown in Fig. 2⁴², includes the stator resistance , stator leakage reactance , rotor resistance , rotor leakage reactance , magnetizing reactance , and core loss resistance . The mechanical losses, such as friction and windage losses , and stray-load losses are also considered¹⁶.

Fig. 2.

Induction machine per-phase equivalent circuit⁶.

Each parameter has a distinct physical meaning that influences the torque–speed characteristic, efficiency, and power factor of the motor. Stator resistance is closely related to copper losses and varies with temperature, typically accounting for 25–40% of total losses⁴³. Rotor resistance affects the shape of the torque–speed curve and determines the slip at which maximum torque occurs⁴⁴. The magnetizing reactance and core loss resistance represent the magnetization branch, modeling the energy conversion and magnetic losses in the stator core⁹.

The stator resistance affects both the shape of the torque–speed characteristic and the slip at which the breakdown torque occurs⁴⁴. The most common method for measuring is through a direct current (DC) resistance test. Loading the induction motor causes heating in the stator windings, which in turn increases the value of . Therefore, determining or knowing the motor temperature under different loading conditions is essential for accurately estimating . According to IEEE Std-112, the stator resistance at various load levels can be calculated as follows⁴¹.

7

where T_hot represents the operating temperature of the motor, and T_cold is the temperature measured when the motor is at rest, which may also be considered as the ambient temperature. denotes the stator resistance measured at T_cold. The constant K equals 234.5 for windings made of pure copper conductors and 225 for aluminum conductors.

The leakage reactances and are independent of temperature and are therefore assumed to remain constant under all load conditions. Based on the NEMA design classification, the ratio of stator to rotor leakage reactance (/) can be determined. The ratios given in⁴⁵ were used according to the design class of the motors.

According to IEEE Std-112, stray-load loss (PSLL) refers to the residual portion of total motor losses caused by complex electromagnetic effects—such as leakage flux harmonics and eddy currents—and is typically estimated empirically as about 0.9–1.8% of the rated input power.The stray-load loss value for the full-load condition was determined according to the rated power of the motors given in⁴¹.

Friction and windage losses (P_FW) arise from bearing friction and rotor airflow, typically accounting for 5–15% of total motor losses, varying with the square of speed, and are often approximated as a constant fraction of rated power. Al-Badri et al.⁴⁶ determined P_FW using a database of 182 motors. If the tested motor has no equivalent in the database, P_FW can be determined based on the number of poles as shown below.

8

9

10

Accurate parameter interval selection is essential for ensuring optimization convergence and preserving the physical consistency of the induction motor’s equivalent-circuit model. In this study, the limits of each electrical parameter are determined analytically using relationships derived from standard motor electrical tests.

Rotor Resistance (R₂) By using the following equation, the feasible range of is constrained to physically valid limits, enabling faster and more reliable convergence of the optimization algorithm⁴¹:

11

12

Before being used as a constraint, the minimum and maximum values of R₂ must be calibrated with respect to motor temperature⁴¹. Once the nominal temperature is determined, Eq. (12) can be employed to obtain the final interval of R₂.

Core-Loss Resistance () Core losses account for approximately 15% of the total losses in an induction motor and are considered independent of load⁴³. A suitable range for the core-loss resistance can be approximately defined by its lower and upper bounds as follows⁴⁷:

13

A study on 182 induction motors shows that maximum core loss at nominal operating conditions is about 6% of the input power⁴⁸. This criterion is used to narrow the allowable range of . First, the nominal input power is computed, and is taken as 6% of this value. The minimum permissible is then obtained using Eq. (13)⁴⁷.

Stator Leakage Reactance (X₁) The Stator Leakage Reactance (X₁) is a critical parameter in the design and performance analysis of induction motors. Neglecting the magnetizing branch components ( and ) in the equivalent circuit, the maximum torque expression can be rewritten as:

14

In this equation, , V₁, and R₁ are known, while and (X₁ + X₂) remain unknown. According to NEMA MG-1, the minimum allowable for each motor is expressed as a percentage of the rated torque and must not be lower than the specified value⁴⁵. Using this assumption, the range of the parameter ( X₁ ) is determined⁴⁷.

Magnetizing Reactance () The minimum magnetizing reactance () can be determined using the no-load current ratio the full-load current. For example, for motors rated between 1—7.5 HP, the expression can be written as:

15

The parameter ranges of the induction motors used in this study were determined according to the procedures described above, and the results are presented in Table 1. These ranges, customized based on the motor’s rated power, number of poles, and other characteristics, effectively narrow the search space, thereby improving both the convergence speed and the estimation accuracy of the optimization algorithms. This method provides highly precise parameter limits, especially for small-valued parameters like , whose range is extremely narrow compared to wider intervals of and .

Table 1.

Parameter ranges for different motor power ratings.

Power (kW)	1.1	2	22	30	90	132
(Ω)	2.02	0.98	0.78	0.7	0.24	0.17
(Ω)	12.12	5.89	4.69	4.19	1.47	1
(Ω)	89.35	46	29.2	20	7.84	5.12
(Ω)	1007	2138	1233	904	305	210
(Ω)	698	1481.2	854	626	211	145
(Ω)	2617	5554	3204	2350	792	546
(Ω)	2.93	1.52	0.33	0.13	0.06	0.02
(Ω)	7.15	3.7	0.88	0.34	0.17	0.05

Optimization algorithms

Inspired by intelligent behaviors observed in nature, these algorithms iteratively search for optimal or near-optimal solutions within predefined bounds and constraints. Although they do not guarantee the global optimum, they typically achieve solutions very close to it within reasonable computational time, making them highly effective for complex and nonlinear engineering problems. In this section, the meta-heuristic algorithms employed in the study are introduced.

Particle swarm optimization algorithm

In the PSO algorithm, each particle represents a d-dimensional potential solution. Particles initially receive random values, and their positions are updated iteratively to reach the global optimal solution. The positions of the particles are determined in each generation based on their local best positions (p_best) and the best position of the global swarm (g_best). Each particle is evaluated based on its proximity to the desired minimum or maximum numerical value according to the specified objective function. In the algorithm, the velocity values of the particles and their positions, which depend on these velocity values, are updated in each iteration until the termination criterion is reached. These equations are given below (Eqs. 16, 17)⁴⁹.

16

17

where, is the coefficient of inertia and is [0, 1]. and are the acceleration coefficients and are [0, 2]. and are random numbers between 0 and 1. represents the speed of the ith particle at the kth iteration, represents the best value of the particle so far, and represents the value of the global best.

Genetic algorithm

In GA, as in other metaheuristic algorithms, the process is initiated with random individuals. Each individual is evaluated using an objective function, and its fitness value and probability of selection (Eq. 18) are determined based on this value⁵⁰.

18

According to the selection operator, individuals are transferred to the next generation. Here, represents the fitness value of the ith individual, and represents the probability of selection. In genetic algorithms, the crossover operator is used to ensure genetic enrichment and gene transfer. With this operator, individuals share values within a certain size range, similar to gene sharing, as shown below.

19

20

where, and are the crossover points, are the individuals, and and are the genes that constitute the individual. The mutation operator is used to make small changes to the individual to prevent the algorithm from getting stuck in local optima. This operator performs the reverse () on randomly selected individuals.

Harris hawks optimization

In the Harris Hawk Optimization (HHO), while determining the perching strategy () in the exploration phase, attack strategies are used according to the prey’s escape energy () and escape chance () in the exploitation phase. If the value is less than 0.5, it uses the location of other family members for perching; otherwise, a random location is selected (Eq. 21)⁵¹.

21

where, represents the current position of the hawk, represents the randomly selected hawk position, represents the best hawk position, represents the average position of the hawk population, and represent random numbers in the range [0,1]. The transition phase is governed by the prey’s escape energy. The initial energy of the prey is denoted as , and this value is updated in the range [-1,1] throughout the iteration (max iteration number , current iteration ). The escape energy is calculated using Eq. 22.

22

In the soft besiege case of and , Eq. (23) is used, in the hard besiege case of and , Eq. (24) is used. is random numbers in the range [0,1].

23

24

The and values representing the new generation are calculated. Here, the value is calculated differently depending on the prey’s energy and escape probability. In the calculation of , in the soft besiege with progressive rapid dives case of and , Eq. 25 is used, in the hard besiege with progressive rapid dives case of and , Eq. 26 is used.

25

26

The following equations (Eqs. 27, 28, 29) are used to calculate , which can be used in the new generation selection.

27

28

29

where, represents the problem size, is a random vector of size , is the levy flight function, and are random numbers in the range [0,1], and represents the flight step and is taken as a constant 1.5.

Grey wolf optimizer

Grey Wolf Optimizer (GWO) is a metaheuristic algorithm that mathematically models the discovery, encirclement, and attack phases of prey, with a leadership hierarchy of alpha (), beta (), delta (), and omega () wolves⁵². Alpha represents the best solution, while beta, delta, and omega represent the best solutions after alpha, depending on the hierarchy within the wolf pack. The movement of wolves around the prey is modeled as specified in the following equation.

30

where, is location of prey (optimal solution-prey), is location of wolves. and are random vectors in the range [0,1], and is a number that decreases linearly from two to zero. To mathematically model the hunting behavior of gray wolves, we assume that alpha has the best information about the potential location of prey, while beta and delta have better information than other wolves. Therefore, the top three best solutions obtained so far are stored, and other search agents (including omegas) are required to update their positions based on the positions of the best search agents. The following Eq. (31) and Eq. (32) are used for hunting behavior.

31

32

where, represent the positions of alpha, beta, and delta wolves, respectively. and are vectors controlling the balance of exploration and exploitation.

FOX optimization algorithm

The FOX optimization algorithm is based on models of the foraging behaviors of red foxes while hunting in nature⁵³. Foxes must first locate their targets during the search phase. To do this, they must detect the sound waves produced by their prey. Based on the intensity of these sound waves, they determine the distance between the fox and the prey. Equation (33) is used to determine this distance. However, Eq. (34) is used to calculate the final distance (taking into account the travel of the sound).

33

34

where, is the speed of sound, is the vector representing the best solution obtained so far, and is a vector of randomly generated numbers between [0,1]. The actual distance vector is . After detecting its prey, how the fox will approach, and attack is calculated using the following equation.

35

where, 9.81 is the gravitational constant, and is the average sound transfer time. The parameter in the equation above takes either 0.18 or 0.82. Which of these values is used depends on the value. If the value is greater than 0.18, 0.82 is used; otherwise, 0.18 is used.

Whale optimization algorithm

Algorithm (WOA) is a successful algorithm that balances exploration and exploitation phases⁵⁴. When searching for prey, whales begin with random solutions. They determine the distance between themselves and the prey and then follow the best solution (Eq. 36). Once prey is detected, the encircling process is initiated by making circular movements around them.

36

37

where, the value represents the distance vector between the whale with the best solution up to that iteration and the whale whose position will be updated. is the vector representing the current whale’s position, and values are multiplying vectors that take random values in the range [0,2].

38

Whales use a bubble net attack strategy when attacking their prey. In this strategy, whales use both a spiral updating position (Eq. 38) for hunting and a shrinking encircling approach. The choice of strategy depends on the probability . If the value of is greater than 0.5, they use the spiral update position; otherwise, they use the shrinking encircling approach. is a random number in the range [− 1,1], and is a coefficient.

Differential search algorithm

Many species in nature, including birds, butterflies, fireflies, honeybees, and whales, exhibit migratory behaviors to locate optimal environmental conditions, a process mathematically modeled by Civicioglu through the Differential Search Algorithm (DSA)⁵⁵. In the DSA algorithm, superorganisms first attempt to find their own donor locations (Eq. 39). Different approaches (elitist1, elitist2, bijective, surjective) can be used in the donor location finding process.

39

where, is the position vector of the donor and represents the superorganism.

40

41

When determining where superorganisms will stopover, donor locations and the scale factor are important during position updates. The scale factor represents the rate of position change. This rate of change is achieved by generating gamma () and uniform random numbers (, , ). The stopover location vector is denoted here as .

Tree-seed algorithm

The scattering of numerous seeds around the tree will result in the formation of new trees. However, high-quality seeds have a higher probability of becoming trees. This natural phenomenon has been modelled, and the Tree-Seed Algorithm (TSA) has been proposed⁵⁶.

42

43

TSA utilizes Search Propensity () to scan the search space. is randomly generated in the range [-1,1]. If the probability value is less than , it performs a local search using Eq. (42); otherwise, it performs a global search using Eq. (43). While the exploration process is conducted with local search, the exploitation process is conducted with global search. Here, represents the seed vector, represents the tree vector of the best solution, represents the tree position value, and represents the position vector of the randomly selected tree. is the scaling factor vector and takes a random value in the range [-1,1].

Experimental results and discussions

In this section, the performance of the proposed efficiency estimation approach is experimentally evaluated and discussed. Eight meta-heuristic algorithms—GA, PSO, DS, TSA, GWO, WOA, FOX, and HHO—were applied to the in-situ operating data of induction motors to estimate the equivalent-circuit parameters and the corresponding efficiencies. Using measurements obtained at or near nominal load, the estimated results were compared with experimentally validated reference values to assess the accuracy and robustness of each algorithm. The parameter settings of all optimization algorithms are summarized in Table 2.

Table 2.

Parameter settings of the algorithms.

PSO (particle swarm optimization) Population size: 200 Number of iterations: 200 c₁ = 2, c₂ = 2 wmax = 0.9 wmin = 0.4	HHO (Harris Hawks optimization) Population size: 200 Number of iterations: 200 Escape Energy (E): 1 Lévy Flight Parameter: 1.5
FOX (FOX optimization) Population size: 200 Number of iterations: 200 c₁ = 0.18 c₂ = 1	WOA (whale optimization algorithm) Population size: 200 Number of iterations: 200 Spiral Update Coefficient (b): 1
GA (genetic algorithm) Population size: 200 Number of iterations: 200 Mutation rate: 0.1 Crossover rate: 0.8	GWO (grey wolf optimization) Population size: 200 Number of iterations: 200
DS (differential search) Population size: 200 Number of iterations: 200	TSA (tree-seed algorithm) Population size: 200 Number of iterations: 200

The technical specifications of the induction motors used in this study are presented in Table 3. The experimental reference data were primarily obtained from the AEMOT Electric Motors & Generators company, whose fully automated, standards-compliant test infrastructure—capable of testing motors across various power ratings—is shown in Fig. 3. Test data for the 22 kW, 30 kW, 90 kW, and 132 kW motors were supplied directly by AEMOT. For the 1.1 kW motor, the necessary test setup was established by the authors, and all measurements were conducted accordingly. Additionally, the 2 kW motor data were taken from the study by Arslan et al.⁵⁷. All tests were carried out based on the IEEE Std 112 F1 procedure, and both no-load and locked-rotor tests were performed to determine the equivalent circuit parameters and efficiency.

Table 3.

Technical specifications of induction motors.

Power (kW)	1.1	2	22	30	90	132
Vline (V)	380	380	500	400	400	500
Iline (A)	2.7	5.3	34	50	157	192
F (Hz)	50	50	50	50	50	50
Connection	Y	Y	Δ	Δ	Δ	Δ
Speed (d/d)	1415	1424	987	992	1489	1493
NEMA	A	A	B	A	A	A
Pole	4	4	6	6	4	4

Fig. 3.

AEMOT induction motor test setup (with permission from AEMOT).

According to the IEEE Std-112 standard, the efficiency values measured and calculated using Methods A and F1 are presented in Table 4. Method A determines efficiency directly through the measurement of input and output powers, whereas Method F1 computes efficiency using the equivalent circuit parameters obtained from the no-load and locked-rotor tests, based on the F2 formula defined in IEEE Std-112.

Table 4.

Efficiencies determined according to IEEE Std-112 methods A and F1.

kW	1.1		2		22		30		90		132
Load (%)	A	F1	A	F1	A	F1	A	F1	A	F1	A	F1
100	80.69	80.64	80.91	79.80	93.77	93.48	93.70	92.15	95.00	94.31	95.91	94.03
75	80.67	79.61	81.44	80.53	93.92	93.29	93.52	91.59	95.34	94.41	95.50	93.77
50	78.59	75.68	81.25	78.49	93.21	92.14	92.23	89.78	95.16	93.82	94.25	92.62
25	66.32	63.54	70.04	68.70	89.72	87.68	87.34	83.31	92.86	90.73	90.12	88.26

Certain differences were observed between the efficiency results obtained from the two methods. These discrepancies mainly arise from uncertainties in the calculation of equivalent circuit parameters. In particular, the reliance on experimental data for stray load losses, the approximate estimation of motor temperature based on load conditions, and the way core losses are determined are significant sources of error. Although core losses increase with motor load in practice, they are represented by a constant resistance in the equivalent circuit model and assumed to be fixed across all load levels. This assumption may lead to deviations in efficiency estimation, especially at higher loads. As the load increases, the voltage drops across and increase, whereas the voltage across decreases, resulting in reduced core losses. Therefore, the primary reason why the efficiency values obtained from the equivalent circuit method appear lower than those measured directly using Method A at low loads is the assumption of constant core losses. For this reason, the efficiency values directly measured using Method A were taken as the reference in the subsequent analyses, and the efficiency results estimated by the optimization algorithms were compared against these reference values.

Method I and Method II were compared with the efficiency values calculated according to the IEEE 112-A standard. The performance of the methods was evaluated based on the directly measured efficiency values, the estimated efficiencies, and the absolute error rates between them. The tests were conducted at four different load levels: 100%, 75%, 50%, and 25% of the rated load.

In-situ efficiency estimation for induction motor

The performances of the eight optimization algorithms and the two proposed methods in estimating the efficiency of the GMK_1-1 kW induction motor—based on the IEEE 112-A reference values—were comparatively evaluated, and the results are summarized in Table 5.

Table 5.

Error rates (%) between the measured and estimated efficiencies of the GMK_1-1 kW motor.

Motor	IEEE method-A measured efficiencies (%)				ALG	Estimated efficiencies (%)				Error (%)				STD deviation
	100%	75%	50%	25%		100%	75%	50%	25%	100%	75%	50%	25%
Method I
GMK_ 1-1KW	80.7	80.7	78.6	66.3	PSO	80.1	79.5	76.2	65.5	0.6	1.2	2.4	0.8	1.04e−02
					HHO	80.8	80.6	77.7	67.7	− 0.1	0.1	0.9	− 1.4	1.20e−02
					FOX	80.2	79.7	76.5	65.9	0.5	1.0	2.1	0.5	1.17e−01
					WOA	80.5	80.3	77.3	67.3	0.2	0.4	1.3	− 0.9	1.22e−02
					GA	80.9	80.4	77.3	66.9	− 0.2	0.3	1.3	− 0.6	1.80e−01
					GWO	80.1	79.6	76.3	65.7	0.6	1.1	2.3	0.6	1.17e−03
					DS	80.1	79.5	76.2	65.6	0.6	1.2	2.4	0.8	2.85e−04
					TSA	80.5	79.9	76.6	66.0	0.2	0.8	2.0	0.3	3.39e−03
Method II
GMK_ 1-1KW	80.7	80.7	78.6	66.3	PSO	80.1	79.6	76.4	65.9	0.6	1.0	2.2	0.4	2.13e−11
					HHO	79.6	79.1	75.7	64.8	1.1	1.6	2.9	1.5	1.43e−02
					FOX	80.5	80.1	77.1	66.7	0.2	0.5	1.5	− 0.4	1.23e−01
					WOA	80.1	79.6	76.3	65.6	0.6	1.1	2.3	0.7	1.59e−02
					GA	79.2	78.3	74.5	63.0	1.5	2.4	4.1	3.3	1.72e−01
					GWO	80.1	79.5	76.2	65.5	0.6	1.1	2.4	0.8	3.06e−04
					DS	80.1	79.5	76.2	65.5	0.6	1.2	2.4	0.8	3.28e−04
					TSA	80.3	79.7	76.3	65.5	0.4	1.0	2.3	0.9	4.97e−03

The best values are indicated in bold.

When Table 5 is examined, it is observed that the HHO algorithm provides the highest accuracy under Method I, whereas the FOX algorithm yields the best performance under Method II. Considering full-load operating conditions within the estimation process, these two algorithms achieve the closest agreement with the IEEE 112-A reference values and produce the most accurate efficiency predictions. The evaluation of efficiency estimates obtained using eight different optimization algorithms for GMK_1-1KW induction motor, based on the IEEE 112-A standard and assessed under Method I and Method II, is illustrated in Fig. 4.

Fig. 4.

Error rates (%) between the measured and estimated efficiencies of the GMK_1-1 kW motor.

For the 1.1 kW motor, the results show that Method II generally provides lower error rates for most algorithms. Under 100% and 75% load conditions, PSO, GWO, FOX, DS, and TSA achieve high accuracy with low error levels in both methods, while FOX demonstrates the best overall performance—particularly under Method II. In contrast, the GA algorithm produces the weakest results, exhibiting high error deviations across both estimation methods. Although Method II performs better for several algorithms, a broader comparison indicates that Method I yields slightly lower average error.

Table 6 presents the analysis results for the AE_280M4A_90 kW induction motor evaluated using Method I and Method II. In both methods, all algorithms produced highly accurate results under full-load conditions, with error rates remaining within ± 0.2%. However, as the load decreased—particularly at 50% and 25%—an increase in estimation error was observed. At these low load levels, the TSA algorithm achieved the best performance in both methods, yielding the smallest deviation with an approximate error of 2.2% at 25% load. In contrast, the FOX algorithm showed the largest errors at 50% and 25% load, with deviations of about 2.5–2.7%, making it the least accurate under low-load conditions.

Table 6.

Measured and estimated efficiencies of the AE_280M4A_90 kW induction motor.

Motor	IEEE method-A measured efficiencies (%)				ALG	Estimated efficiencies (%)				Error (%)				STD deviation
	100%	75%	50%	25%		100%	75%	50%	25%	100%	75%	50%	25%
Method I
AE_ 280M4A_ 90 kW	95.0	95.3	95.2	92.9	PSO	95.0	94.7	93.8	90.4	0.0	0.6	1.4	2.4	3.96e−04
					HHO	95.0	94.7	93.8	90.5	0.0	0.6	1.4	2.3	6.88e−03
					FOX	95.0	94.7	93.7	90.4	0.0	0.7	1.4	2.5	4.30e−01
					WOA	95.0	94.8	93.8	90.6	0.0	0.6	1.3	2.3	9.06e−03
					GA	95.0	94.8	93.8	90.6	0.0	0.6	1.3	2.2	4.33e−02
					GWO	95.0	94.7	93.8	90.5	0.0	0.6	1.4	2.4	4.66e−03
					DS	95.0	94.7	93.8	90.4	0.0	0.6	1.4	2.4	1.33e−02
					TSA	95.2	94.9	94.0	90.7	− 0.2	0.5	1.2	2.2	6.45e−03
Method II
AE_ 280M4A_ 90 kW	95,0	95,3	95,2	92,9	PSO	95.0	94.7	93.8	90.5	0.0	0.6	1.4	2.4	3.58e−04
					HHO	95.0	94.8	93.8	90.6	0.0	0.6	1.3	2.3	8.49e−03
					FOX	94.9	94.6	93.6	90.2	0.1	0.7	1.6	2.7	4.57e−01
					WOA	95.0	94.7	93.8	90.5	0.0	0.6	1.4	2.4	7.55e−03
					GA	95.1	94.8	93.9	90.5	− 0.1	0.5	1.3	2.3	3.98e−02
					GWO	95.0	94.7	93.8	90.5	0.0	0.6	1.4	2.3	4.50e−03
					DS	95.0	94.7	93.8	90.4	0.0	0.6	1.4	2.4	1.23e−02
					TSA	95.1	94.8	93.9	90.6	− 0.1	0.5	1.3	2.2	3.79e−03

The best values are indicated in bold.

An examination of Fig. 5 shows the error rates obtained from both methods across four loading levels. The figure clearly indicates that the errors increase for all algorithms at 25% load. Among the algorithms, TSA, GA, and WOA exhibit the smallest rise in error, whereas FOX shows the highest deviations under reduced-load conditions.

Fig. 5.

Error rates (%) between measured and estimated values for the AE_280M4A_90kW induction motor.

The relatively higher estimation error observed at 25% load can be primarily attributed to the assumptions adopted for core-loss resistance and stray-load losses in non-intrusive efficiency estimation methods. At light-load operation, the magnetizing current constitutes a larger portion of the stator current, increasing the relative contribution of iron losses. Since core losses are assumed to be load-independent in IEEE Std. 112-based formulations, this simplification leads to higher uncertainty under partial-load conditions^41,58. Moreover, stray-load losses are commonly modeled using fixed ratios recommended in IEC 60,034–2-1, although previous studies indicate that these losses vary nonlinearly with load^11,48. Consequently, the increased error at 25% load reflects inherent modeling assumptions rather than limitations of the proposed optimization algorithms.

Tables 5 and 6 report the standard deviation values obtained from repeated independent runs, which directly reflect the robustness and repeatability of the optimization algorithms for both Method I and Method II.

For the GMK_1–1 kW induction motor, Method I (Table 5) shows that DS and GWO achieve the lowest standard deviation values, indicating very stable convergence behavior. TSA also demonstrates good repeatability, while PSO, HHO, and WOA present moderate but still acceptable variability for practical in-situ efficiency estimation. In contrast, GA and FOX exhibit noticeably higher dispersion, revealing a stronger sensitivity to random initialization and reduced robustness. Under Method II, the same motor shows a clearer separation among algorithms. PSO exhibits almost negligible variability, indicating near-deterministic convergence. GWO and DS also maintain very low dispersion, while TSA remains stable within the 10⁻³ range. HHO and WOA show increased variability, whereas GA and FOX again present the highest dispersion levels, confirming their limited repeatability.

Similar trends are observed for the AE_280M4A_90 kW induction motor in Table 6. For Method I, PSO, GWO, and TSA provide low standard deviation values, while HHO, WOA, and DS remain within acceptable limits. FOX, however, shows significantly higher dispersion, indicating robustness issues that persist regardless of motor power. Method II yields comparable results for the high-power motor. PSO maintains very low variability, followed by GWO and TSA with strong repeatability. GA and HHO exhibit moderate dispersion, while FOX again demonstrates the highest variability.

Overall, the standard deviation analysis clearly indicates that PSO, GWO, DS, and TSA consistently deliver stable and repeatable performance across different motors and estimation methods. In contrast, GA and particularly FOX show high dispersion, pointing to reduced numerical stability.

Figure 6 presents a comparative analysis of the mean absolute error (MAE, %) values for six induction motors with different power ratings, evaluated using eight optimization algorithms under Method I and Method II. The MAE values were calculated using the Eq. 38.

44

Fig. 6.

Comparison of average full-load efficiency estimation errors for six induction motors using Method I and Method II.

The standard deviation values reported in Tables 5 and 6 quantify the variability observed across multiple independent runs and provide a direct measure of algorithmic robustness and repeatability for both Method I and Method II.

According to Fig. 6, the mean absolute error values of the algorithms range between 0.3% and 0.7%. Method II yields slightly lower average errors for several algorithms and delivers more consistent performance overall. Among the algorithms, WOA, PSO, and DS stand out with the lowest error values under both methods, while GA and FOX illustrate higher deviations, highlighting the importance of selecting an appropriate algorithm–method combination. In general, the average full-load MAE remained below 0.7% in both methods, demonstrating the reliability of the proposed approach. The similarity between the two methods is largely since the motors tested in this study had not undergone repair and their nameplate data were accurate. For repaired motors or those with incorrect nameplate information, the discrepancy between the methods would be more pronounced, in which case Method I—where the output power term is excluded from the objective function—would be more suitable.

Parameter estimation for induction motor

The performance of eight optimization algorithms (PSO, HHO, FOX, WOA, GA, GWO, DS, TSA) was evaluated for estimating the fundamental equivalent-circuit parameters of the induction motor (, , , and ) using two different approaches: Method I and Method II. The estimated parameters were compared with the experimentally obtained reference values determined in accordance with the IEEE 112-F standard.

Table 7 presents the parameter estimation performance of the eight heuristic algorithms for the GMK_1-1 kW induction motor. The results indicate that each algorithm demonstrates varying levels of performance for each parameter. GA shows particularly strong performance for , the GWO algorithm for and , while the DS and FOX algorithms perform well for and . However, when all four parameters are evaluated together, PSO achieves the lowest overall mean error in Method I. Under Method II, the WOA and GWO algorithms also provide generally successful estimations. FOX (Method I) further yields accurate results particularly for the magnetizing reactance and core-loss resistance.

Table 7.

Parameter values estimated by the algorithms and the IEEE 112-F reference values for the GMK_1-1 kW motor.

Motor		ALG	Parameters
GMK_ 1-1KW	Method I	PSO	8.47	128.7	1377	4.80
		HHO	9.99	134.2	1538	4.61
		FOX	8.96	130.3	1392	4.74
		WOA	8.78	135.3	1516	5.06
		GA	6.74	121.4	1682	4.90
		GWO	9.05	131.4	1363	4.73
		DS	8.73	129.5	1370	4.77
		TSA	9.09	138.4	1356	4.57
	Method II	PSO	10.07	134.2	1339	4.65
		HHO	10.12	134.1	1239	4.67
		FOX	9.04	130.7	1482	4.72
		WOA	8.71	128.9	1376	4.74
		GA	6.83	121.4	1242	4.94
		GWO	8.61	129.0	1375	4.78
		DS	8.63	129.2	1373	4.78
		TSA	9.31	131.1	1334	4.39
	IEEE 112- F		5.73	125.2	1384	4.87

Reference values are indicated in bold.

Figure 7 illustrates the convergence behavior of the PSO algorithm for estimating the equivalent-circuit parameters of the GMK_1-1 kW induction motor in two independent runs (Run-1 and Run-2), along with the reference values determined according to the IEEE 112-F standard. The four subplots illustrate the evolution of , , , and as a function of iteration count. The results indicate that PSO exhibits stable convergence across all parameters. A strong agreement with the reference values is observed particularly for and , while reaches its steady state value within only a few iterations. For , however, a noticeable deviation from the reference value occurs, indicating a weaker estimation performance for this parameter. Overall, the convergence profiles confirm that PSO is capable of reliably estimating the key equivalent-circuit parameters.

Fig. 7.

Convergence graphs of PSO algorithm parameters during the first two iterations.

Table 8 presents the fundamental parameters of the AE_ 280M4A_ 90 kW motor estimated using eight different algorithms and two methods, and the results are compared with the reference values determined according to the IEEE 112-F standard. High accuracy was achieved for the estimated , and parameters, with the algorithms producing values that are close to the reference. The parameter, however, exhibited larger deviations. Nevertheless, PSO, HHO, FOX, and DS algorithms illustrated reference-aligned outputs in both methods. In particular, the DS algorithm statistically stood out by providing low-error and stable results across all parameters. Figure 8 illustrates the convergence behavior of the equivalent-circuit parameter estimates for the AE_280M4A_90 kW induction motor obtained from two independent runs of the FOX algorithm, along with the reference values. The FOX algorithm provided high accuracy and stable convergence for the , and parameters. The resulting values 0.33–0.39 Ω for ,, approximately 18 Ω for , and 0.08 Ω for , are very close to their respective reference values. In contrast, the parameter was estimated in the range of 350–411 Ω, which is lower than the 618 Ω reference value. Overall, the algorithm demonstrated its best performance particularly for the and parameters.

Table 8.

Parameter values estimated by the algorithms and the IEEE 112-F reference values for the AE_280M4A_90 kW motor.

Motor		ALG	Parameters
AE_ 280M4A_ 90 kW	Method I	PSO	0.33	18.1	721	0.08
		HHO	0.33	18.0	738	0.08
		FOX	0.33	18.1	712	0.08
		WOA	0.38	18.6	740	0.08
		GA	0.40	18.8	740	0.08
		GWO	0.33	18.1	729	0.08
		DS	0.34	18.2	716	0.08
		TSA	0.34	17.9	792	0.07
	Method II	PSO	0.34	18.1	720	0.08
		HHO	0.34	18.2	748	0.08
		FOX	0.33	18.0	669	0.08
		WOA	0.38	18.6	712	0.08
		GA	0.40	18.9	754	0.08
		GWO	0.34	18.5	728	0.08
		DS	0.34	18.1	716	0.08
		TSA	0.31	17.8	766	0.07
	IEEE 112- F		0.30	17.3	618	0.08

Reference values are indicated in bold.

Fig. 8.

Convergence graphs of FOX algorithm parameters during the first two iterations.

Discussion

In Appendix A (see supplementary file), the convergence graphs of the parameters (, , and ) obtained from the first two runs of the algorithms (GA, PSO, DS, TSA, GWO, WOA, FOX, and HHO) for the AE_280M4A_90 kW motor are presented. The analyses indicate that all algorithms exhibit very high estimation accuracy for the parameter. For nearly all motors, the algorithms successfully estimated the value with close agreement to the IEEE 112-F reference value. Among these, the DS, PSO, and TSA algorithms demonstrated the highest consistency.

For the parameter, the estimated values generally exhibited low deviation levels. In most cases, the estimates approached the IEEE 112-F reference values within ± 0.03 Ω. The GA algorithm was observed to produce more inaccurate results for in some motors, whereas the PSO, HHO, DS, and FOX algorithms provided more accurate estimations for this parameter.

Regarding the parameter, the performance of the algorithms was generally consistent. The estimated values typically remained within ± 1.0–2.0 Ω of the reference value, indicating that the algorithms achieved a high level of accuracy for . In particular, the TSA algorithm produced estimates that matched the reference value exactly for certain motors and demonstrated stability, especially alongside the DS and PSO algorithms.

The parameter emerged as the parameter with the highest deviation across all motors. The estimated values often exceeded the IEEE 112-F reference values by 20% to 100%. This outcome is attributed to the limited set of input measurements available for induction motor parameter estimation. However, is the parameter that has the least influence on the efficiency characteristics of an induction motor. Therefore, the deviations observed in have minimal impact on the overall motor efficiency analysis.

“In-situ efficiency estimation for induction motor” section revealed important findings into the in-situ efficiency estimation performance. For all six motors, the full-load estimation errors remained below 0.7% across all algorithms, demonstrating that the proposed heuristic-based approach can reliably reproduce IEEE 112-A reference efficiencies under real operating conditions. At low load levels, particularly at 25%, all algorithms exhibited higher errors, reflecting the reduced observability of motor characteristics in lightly loaded operation. PSO, WOA, and DS consistently delivered the lowest error values across multiple load levels, while GA and FOX tended to show higher deviations, especially at low loads. Furthermore, Method II provided slightly lower average errors for several algorithms due to its use of rated power, whereas Method I gives a good result in situations where nameplate data may be uncertain.

Additionally, Figs. 9 and 10 present the convergence, efficiency, and torque curves obtained using the parameters estimated by the WOA and DS algorithms for the AE_280M4A_90 kW motor under Method I. The torque and efficiency curves generated using these estimated parameters exhibit a high degree of similarity and strong agreement with those obtained from the IEEE Std 112-F method.

Fig. 9.

Convergence, efficiency and torque characteristics obtained using the WOA-estimated parameters under Method II for the AE_280M4A_90kW induction motor.

Fig. 10.

Convergence, efficiency, and torque characteristics obtained using DS-estimated parameters under Method II for the AE_280M4A_90 kW induction motor.

Compared to conventional in-situ efficiency estimation methods reported in the literature—which typically rely on a single optimization algorithm, limited motor power ranges, or intrusive laboratory measurements—the proposed approach provides a broader and more robust framework. Classical IEEE Std-112 Method A requires direct torque measurement and laboratory infrastructure, while Method F1 depends on no-load and locked-rotor tests. In contrast, the proposed method achieves comparable accuracy using only field-measurable quantities, without interrupting motor operation. Furthermore, while most previous studies report efficiency errors in the range of 1–3% and focus on low-power motors, the proposed approach maintains full-load errors below 0.7% across six motors ranging from 1.1 kW to 132 kW. The inclusion of eight meta-heuristic algorithms and statistical robustness analysis further distinguishes this study from earlier works.

The proposed method is particularly suitable for industrial environments where induction motors operate continuously and cannot be removed from service for laboratory-based testing. Since the approach relies solely on field-measurable electrical quantities—such as voltage, current, input power, and speed—it can be applied during normal operation without production downtime. This makes the method practical for routine energy audits, condition-based maintenance, and efficiency monitoring in industrial facilities.

However, several limitations should be noted. Estimation accuracy decreases under light-load conditions (25%), which is attributed to reduced parameter observability and inherent assumptions of the equivalent-circuit model. In addition, Method II relies on accurate nameplate data; therefore, Method I is preferable for motors with uncertain or altered nameplate information (e.g., rewound motors). Finally, the use of meta-heuristic algorithms introduces higher computational effort compared to analytical methods, limiting real-time embedded implementation.

Conclusions

This study introduces a practical and non-intrusive approach for estimating the efficiency and equivalent-circuit parameters of three-phase induction motors under real operating conditions. By combining field-measurable electrical quantities with heuristic optimization, the method provides a flexible and low-cost alternative to traditional laboratory-based testing. Eight meta-heuristic algorithms—PSO, GA, GWO, HHO, TSA, DSA, WOA, and FOX—were evaluated across six motors with different power ratings and load levels using two estimation schemes (Method I and Method II) to assess their accuracy and robustness. The results demonstrate that both methods achieve high estimation accuracy, with the mean absolute error between measured and estimated efficiency values remaining below 0.7% at full load for all algorithms. Method II yielded lower overall error levels and more stable convergence, particularly under medium and high load conditions. Among the algorithms, PSO, WOA, and DSA produced the most accurate and repeatable efficiency estimates, whereas GA and FOX exhibited relatively higher deviations, especially at low-load operating points.

Parameter estimation results confirm that the proposed method can closely reproduce IEEE 112-F reference values for parameters such as , , and . The strong agreement observed between the simulated and experimentally obtained torque–efficiency curves reinforce the validity of the method and demonstrates that the estimated parameters remain physically meaningful across different motors and operating points.

Overall, the proposed method offers a practical solution for industrial energy-efficiency assessments and motor performance monitoring applications. By enabling continuous and reliable tracking of motor efficiency under real operating conditions, it contributes to energy management tasks.

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

M.G.: Investigation, Conceptualization, Methodology, Software, Data curation, Writing, Visualization, Validation, Review & editing. M.A.Ş.: Methodology, Software, Writing—original draft, Validation, Review & editing. M.Ç.: Conceptualization, Methodology, Writing, Visualization, Validation, Review & editing, Supervision.

Funding

There is no funding received for this research work.

Data availability

No datasets were generated or used in this study. All analyses were performed using experimentally obtained measurements, which are presented within the article.

Declarations

The authors declare that generative AI was used only for language editing and not for scientific content generation.

Competing interests

The authors declare no competing interests.