IN SITU EFFICIENCY DETERMINATION OF INDUCTION MOTOR: A COMPARATIVE STUDY OF EVOLUTIONARY TECHNIQUES

Abstract

Induction motor is the most frequently used electric machine in industrial and commercial applications because of its well-known advantages, including robustness, low price, maintenance free, and line start. The exact knowledge of some of the induction motor parameters is very important to implement efficient control schemes and its in situ efficiency determination. These parameters can be obtained by no-load test that is not easily possible for the motors working in process industries where continuous operation is required. In this present study, particle swarm optimization, differential evolution, and some of their recent variants are used for in situ efficiency determination of induction motor (5 hp) without performing no-load test. Detailed information about the methods for efficiency measurement is given. Results are compared with genetic algorithm and a physical efficiency measurement method, called torque-gauge method. The performances in terms of objective function (error in the efficiency) and convergence time prove the effectiveness of optimization algorithms used in this article.

INTRODUCTION

Induction motors have become the most widely used machines in any industry, consuming around 70% of the electricity used. Process industries are found to be energy intensive (4% of energy cost in the total input cost in case of textile industry) compared with other industries, such as chemical, food, and computer manufacturing (Palanichamy et al. 2001). Therefore, a small increment in the efficiency of these motors can result in substantial savings over a long period. In addition, motors operating at light or partial load (no balance in between copper and iron losses) result in considerable reduction in the efficiency and power factor. Adjustment of motor excitation with respect to motor load and speed helps energy conservation in these motors. But adoption of energy conservation by adjusting motor excitation or replacing it by energy-efficient motors depends highly on the savings and payback periods. In this situation an accurate in situ efficiency determination of these motors is essential but requires the motor's electrical parameters.

Many nonlinear programming techniques like the Newton-Raphson technique, cyclic method, Hook, Jeeves, and Rosenbrock methods have been applied to parameter estimation and hence efficiency determination of induction motors. The optimum determined by the Newton-Raphson technique depends heavily on the initial guess of the parameter, with the possibility of a slightly different initial value, causing the algorithm to converge to an entirely different solution (Nangsue et al. 1999). Also, this algorithm needs derivative during the optimization process, which may be difficult to calculate. Bounekhla, Zaim, and Rezzoug (2005) proved the Rosenbrock method is better than scatter search and Hook and Jeeves methods in terms of fast and efficient search. Apart from conventional methods, some of the evolutionary techniques like genetic algorithm (GA) (Pillay et al. 1998), genetic programming (Nangsue, Pillay, and Conry 1999), particle swarm optimization (PSO) (Benaidja and Khenfer 2006), differential evolution (DE) (Ursem and Vadstrup 2003), evolution strategy (Ursem and Vadstrup 2004), and a PSO variant (Ursem and Vadstrup 2004) have been successfully applied to induction motor parameter (electrical and mechanical) estimation.

In continuation with the above studies, in the present study we propose the use of several variants of PSO, quantum PSO (QPSO), and DE and their variants (ATREPSO, GMPSO, SMPSO, LXDE, and DE-QI) to the in situ efficiency determination of induction motor. A wide comparison is performed on the results obtained from these algorithms along with GA (Pillay et al. 1998) and the torque-gauge method (Ontario Hydro Report 1990).

The article is organized as follows. Section 2 explains the standards for induction motor efficiency determination. Section 3 describes the optimization algorithms used in this study. In section 4 mathematical model of the in situ efficiency determination is given, and section 5 gives the experimental settings of all algorithms. In section 6 results and discussions are given, and section 7 is the conclusion.

STANDARDS FOR INDUCTION MOTOR EFFICIENCY DETERMINATION

The methods for efficiency measurements can roughly be divided into two categories: direct and indirect methods. In the direct method shaft torque is measured directly and efficiency is calculated by using the ratio of motor's output power to input power. The preferred method of determining efficiency in three-phase induction motor, however, is the summation-of-losses method.

The international standards IEEE 112 (United States; IEEE standard 2004), IEC 34-2 (European; IEC standard 1972), and IS 325: subrule IS 4029 (Indian standard 2002) represent the most important references for the three-phase induction motor efficiency measurements. These standards recommend different measurement procedures, in particular for the stray losses determination and the temperature corrections of the copper losses.

IEEE 112-2004 (previously known as IEEE 112-1996) (IEEE standard 2004) consists of five basic methods to determine the efficiency: A, B, C, E, and F. In method A the input and output powers are measured and the efficiency is directly obtained. This method is only used for very small machines. Method B uses a direct method to obtain the stray load losses but not a direct method for determining the motor efficiency. To reduce the influence of the measuring error, a linear regression is made of the stray load losses at different loads versus the torque squared.

Method B is the recommended method for testing of induction machines up to 180 kW. Method C is a back-to-back machine test. The total stray load losses are also obtained via a separation of losses for both motor and generator. The stray load losses are then divided between the motor and generator proportional to the rotor currents.

E and E₁ are indirect methods where the output power is not measured. In method E the stray load losses are directly measured using the reverse rotation test. In method E₁ the stray load losses are set to an assumed value. It has been shown experimentally that IS 325 is similar to IEEE 112-E (Thangaraj, Agarwal, and Srivastava 2007).

In methods F and F₁ the equivalent circuit of the machine is used and is more advantageous when tests under load are not feasible or not desirable (Pillay et al. 1998). The stray load losses are again directly measured, or in the case of F₁ an assumed value is used. The method developed in Otaduy (1996) requires only speed measurement and nameplate data to construct an equivalent circuit with electrical parameters.

OPTIMIZATION ALGORITHMS USED FOR THIS STUDY

Standard Particle Swarm Optimization

PSO was proposed by Kennedy and Eberhart (1995). The mechanism of PSO is inspired from the complex social behavior shown by the natural species. For a D-dimensional search space the position of the ith particle is represented as X _i  = (x _i1, x _i2, … , x _iD ). Each particle maintains a memory of its previous best position P _besti  = (p _i1, p _i2, … ,p _iD ). The best one among all the particles in the population is represented as P _gbest = (p _g1, p _g2, … , p _gD ). The velocity of each particle is represented as V _i  = (v _i1, v _i2, … ,v _iD ). In each iteration the P vector of the particle with best fitness in the local neighborhood, designated g, and the P vector of the current particle are combined to adjust the velocity along each dimension and a new position of the particle is determined using that velocity. The two basic equations that govern the working of standard PSO (SPSO) are that of velocity vector and position vector given by

The first part of Eq. (1) represents the inertia of the previous velocity, the second part is the cognition part that tells us about the personal experience of the particle, and the third part represents the cooperation among particles and is named as the social component. Acceleration constants c ₁, c ₂ and inertia weight w are the predefined by the user and r ₁, r ₂ are the uniformly generated random numbers in the range of [0, 1].

SMPSO Algorithm

The SMPSO algorithm (Pant et al. 2008) is an extension to the SPSO algorithm by including the component of mutation in it. The mutation operator defined in the SMPSO algorithm uses quasi-random Sobol sequence and is called a systematic mutation (SM) operator. In the SMPSO algorithm the worst particle of the swarm is mutated by the use of SM operator.

The SM operator is defined as

where R ₁ and R ₂ are random numbers in a Sobol sequence.

The idea behind applying the mutation to the worst particle is to push the swarm from the back. The quasi-random numbers used in the SM operator allows the worst particle to move forward systemically.

GMPSO Algorithm

In GMPSO, Gaussian mutation operator is developed for updating the position of the swarm particles (Pant, Thangaraj, and Singh 2008). The mutation operator is activated only when the diversity of the swarm becomes less than a certain threshold (say d _low).

The algorithm uses the general Eqs. (1) and (2) for updating the velocity and position vectors. At the same time a track of diversity is also kept that starts decreasing slowly and gradually after a few iterations because of the fast information flow between the swam particles, leading to clustering of particles. It is at this stage that the Gaussian mutation operator given as X _t+1[i] = X _t [i] + η ∗ Rand, where Rand is a random number generated by Gaussian distribution, is activated with the hope to increase the diversity of the swarm population. Here η is a scaling parameter.

ATREPSO Algorithm

The ATREPSO (Pant et al. 2007) is a simple extension of the attractive and repulsive PSO (ARPSO) proposed by Riget and Vesterstrom (2002), where a third phase called in-between phase or the phase of positive conflict is added. In ATREPSO the swarm particles switch alternately between the three phases of attraction, repulsion, and an in-between phase, which consists of a combination of attraction and repulsion. The three phases are defined as follows:

• Attraction phase (when the particles are attracted towards the global optimal)

• Repulsion phase (particles are repelled from the optimal position)

• In-between phase (neither total attraction nor total repulsion)

In the in-between phase the individual particle is attracted by its own previous best position p _id and is repelled by the best known particle position p _gd . In this way there is neither total attraction nor total repulsion but a balance between the two.

The swarm particles are guided by the following rule:

Quantum Particle Swarm Optimization

QPSO is also a variant of SPSO. In the quantum model of a PSO, the state of a particle is depicted by wave function,Ψ(x, t), instead of position and velocity. The dynamic behavior of the particle is widely divergent from that of the particle in traditional PSO systems. In this context the probability of the particle's appearing in position x _i from probability density function|Ψ(x, t)|², the form of which depends on the potential field the particle, lies in Liu, Sun, and Xu (2006).

The particles move according to the following iterative equations (Sun, Feng, and Xu 2004; Sun, Xu, and Feng 2004):

where

Mean best (mbest) of the population is defined as the mean of the best positions of all particles; u, k, c ₁, and c ₂ are uniformly distributed random numbers in the interval [0, 1]. The parameter β is called the contraction-expansion coefficient.

Differential Evolution

DE is an evolutionary algorithm proposed by Storn and Price (1995). DE is similar to other evolutionary algorithms, particularly GAs (Goldberg 1986), in the sense that it uses the same evolutionary operators like selection recombination and mutation like that of GA. However, the significant difference is that DE uses distance and direction information from the current population to guide the search process. The performance of DE depends on the manipulation of target vector and difference vector to obtain a trial vector. Mutation is the main operator in DE. A brief working may be described as follows: For a D-dimensional search space, each target vector x _i,g , a mutant vector is generated by

where r ₁, r ₂, r ₃ ∈ {1, 2, … ,NP} are randomly chosen integers, different from each other and also different from the running index i. F (>0) is a scaling factor that controls the amplification of the differential evolution . To increase the diversity of the perturbed parameter vectors, crossover is introduced (Goldberg 1986). The parent vector is mixed with the mutated vector to produce a trial vector u _ji,g+1,

where rand _j  ∈ [0, 1]; CR is the crossover constant takes values in the range [0, 1] and j _rand  ∈ (1, 2, … ,D) is the randomly chosen index.

The selection step chooses the vector between the target vector and the trial vector with the aim of creating an individual for the next generation.

LXDE Algorithm

The LXDE algorithm (Pant, Thangaraj, and Abraham 2009) is a modified version of classical DE algorithm. In the LXDE algorithm a new mutant vector is proposed by the use of Laplace distribution. The mutation operation in LXDE makes use of only two candidate vectors, in place of three candidate vectors like the usual mutation operation given in Eq. (10), to produce a mutant vector. Further, instead of using the fixed value of F as in basic, we use a random number based on Laplace distributed random numbers as scaling or amplitude factor.

Here, L is a random number following Laplace distribution and the remaining symbols have the usual meaning as defined in the previous section.

DE-QI Algorithm

The DE-QI algorithm (Pant, Thangaraj, and Singh 2009) is also a modified version of classical DE algorithm. In the DE-QI algorithm a mutation probability P _qi having a certain threshold value is fixed by the user. In every iteration, if the uniformly distributed random number U(0, 1) is less than P _qi , then the mutant vector is generated by using QI operator; otherwise, the mutant vector follows the classical DE method. The QI operator, based on quadratic interpolation (Mohan and Shankar 1994), is a nonlinear operator that produces a new candidate solution lying at the point of minima of the quadratic curve passing through the three selected candidates.

It uses a = X _min, and two other randomly selected particles {b, c} (a, b, and c are different particles) from the population to determine the coordinates of the new particle where

The nonlinear nature of the QI operator used in this work helps in finding a better solution in the search space. The interpolation approach increases the exploratory capabilities of the DE as it keeps on looking for points lying in between the three selected candidates. The selection of the best candidate (a = X _min) makes it behave like a greedy because it is always looking for a new candidate in the vicinity of the best solution obtained so far.

Genetic Algorithms

GAs are perhaps the most commonly used evolutionary algorithm for solving optimization problems. The natural phenomenon that forms the basis of GA is the concept of survival of the fittest. GAs were first suggested by John Holland and his colleagues (1975). The main operators of GA are selection, reproduction, and mutation. They work with a population of solutions called chromosomes. The fitness of each chromosome is determined by evaluating it against an objective function. The chromosomes then exchange information through their operators; crossover or mutation. More detail on the working of GAs may be obtained from Goldberg (1986).

IN SITU EFFICIENCY DETERMINATION

The method described here considers IEEE E₁ and F₁ methods of efficiency determination but no-load measurements are not performed. We followed the same procedure as that of Pillay et al. (1998) but instead of using GA we used PSO, DE, and their variants to solve algebraic equations. The general block diagram of in situ efficiency determination of induction motor using optimization algorithms is shown in Figure 1.

FIGURE 1 Block diagram of induction motor in situ efficiency determination.

First, the stator line resistance is measured after shutting down the motor. 5hp, four-pole induction motor considered as test motor. “Summation of losses” method for efficiency determination is used with the assumption of stray load losses. The winding arrangement of a star connected motor is shown in Figure 2 and the resistance per phase is calculated as in Eq. (13).

where r _1line is stator line resistance and r ₁ is stator phase resistance.

FIGURE 2 Winding arrangement of star connected motor.

Some measurements on the motor are required before to run the optimization algorithm to estimate the motor parameters: stator line to line voltage V ₁, stator current I ₁, input power P _inp, and rpm at difference load points. Then, power factor can be calculated as in Eq. (14). The measured and calculated values of the test motor for a wide range of loads and the equivalent circuits considered here are taken from Pillay et al. (1998) and are shown in Table 1 and Figures 2 and 3, respectively.

TABLE 1 Measured Data of the Test Motor

Percent load	Voltage V, V	Current I, A	Input power P inp , W	Power factor, pf	Speed, rpm
25	460	2.7	381	0.177	1794
50	460	4	2047	0.642	1764
75	460	5.3	3272	0.775	1741
100	460	6.7	4326	0.81	1719

FIGURE 3 Equivalent circuit of induction motor with stray loss resistance.

Two variations were performed in these circuits from the conventional exact equivalent circuit for comparison: (1) inclusion of stray load loss resistance r _st , shown in Figure 3 and (2) parallel connection of X _m , r _m is transformed into series connection as , shown in Figure 4, and its calculation can be performed as in Eq. (15).

where X _m is mutual inductance and r _m is core loss component.

FIGURE 4 Equivalent circuit of induction motor (modified).

The next step is to find the stray load losses at any load point from its assumed value at full load as per the recommendation in IEEE standards section 5.7.4 (IEEE standard 2004). The recommended value of stray load losses for different capacity motors is shown in Table 2. In the present study we considered this loss at full load is 1.8% and its calculation at different load point is shown in Eq. (16). The remaining Eqs. (17) through (32) of induction motor that are involved in the present study are as follows (Pillay et al. 1998):

TABLE 2 Stray Load Losses for the Different Capacity Motors

Motor rated power	Stray load losses relative to the output power (%)
0.75–90 kW	1.8
91–375 kW	1.5
376–1800 kW	1.2
1800 kW and higher	0.9

where P _st , P _stfl are stray load losses at any point and full load, respectively, and I ₂, I _2fl are rotor currents at these load points.

The stray load loss resistance r _st is

where s _fl slip of the motor under full load.

Temperatures of stator and rotor windings are assumed to be the same and calculated as in Eq. (18) with the IEEE recommended reference temperature.

where I ₁, I _fl are the measured and nameplate stator currents, I ₀ is the stator current under no-load DC test, T _r  = 75°C is the reference temperature for the insulation system of class A (IEEE standard 2004), and T _s  = 25°C is the ambient temperature.

Equations (19) and (20) are used to correct the stator and rotor resistances to the test temperature:

where r ₁ is the stator resistance measured during DC test and r ₂ is the assumed rotor resistance.

The complex admittances of the branches of the equivalent circuits of Figures 3 and 4 are given below (Pillay et al. 1998).

For the equivalent circuit in Figure 3:

For the equivalent circuit in Figure 4:

The stator current can be estimated as

where .

The power factor and rotor current can be estimated as

The current through r _m for the circuit of Figure 3:

The current through r _m for the circuit of Figure 4:

The input power of the circuit of Figure 3 can be estimated as

The input power of the circuit of Figure 4 can be estimated as

The output power can be estimated as

The efficiency can be estimated as

The goal of optimization algorithms is to minimize the errors between the measured and calculated parameters. For this purpose, four methods as given below are considered in this study.

In method 1, the objective function is

where f ₁ = (I _1est  − I ₁)100/I ₁ and f ₂ = (P _infest  − P _inf)100/P _inf.

In method 2, the objective function includes stator current, input power, and power factor, which is

where f ₁, f ₂ are as same as in objection function ff ₁ and f ₃ = (pf _est  − pf)100/pf.

In method 3, the objective function includes stator current, input power, and output power, which is

where f ₁, f ₂ are as same as in objection function ff ₁ and f ₄ = (P _outest  − P _out )100/P _out .

In method 4, the objective function includes four input parameters: stator current, input power, power factor, and output power. The objective function is

The decision variables of the above objective functions are x ₁, r ₂, x _m (or x _m ′) and r _m (or r _m ′). The optimization algorithms are used to determine the above said unknown variables. The assigned parameters of the given motors are K _a  = 225, K _c  = 234.5, r ₁ = 1.635 ohm.

EXPERIMENTAL SETTINGS

For all the PSO algorithms the inertial weight w is taken to be linearly decreasing from 0.9 to 0.5 and the acceleration constants c ₁ and c ₂ are taken as 2.0 each. For QPSO algorithm, the parameter β is linearly decreasing from 1.0 to 0.5.

DE parameters crossover rate C _r and scaling factor F are taken as 0.2 and 0.5, respectively, for all DE algorithms. Besides these settings a total of 30 runs for each experimental setting were conducted and the average efficiency was recorded. Results for GA are noted from the literature (Pillay et al. 1998).

RESULTS AND DISCUSSION

The comparison results of all algorithms corresponding to Figure 3 are given in Tables 3 through 10. Tables 11 through 14 show the comparison results of given algorithms corresponding to Figure 4. Performance curves of algorithms are described in Figures 5 through 11. Figures 12 through 15illustrate the comparison among the optimization algorithms at 25% load for Figure 4.

FIGURE 5 Performance curves of algorithms using objective function ff ₁ of Figure 3.

FIGURE 6 Performance curves of algorithms using objective function ff ₂ of Figure 3.

FIGURE 7 Performance curves of algorithms using objective function ff ₃ of Figure 3.

FIGURE 8 Performance curves of algorithms using objective function ff ₁ of Figure 4.

FIGURE 9 Performance curves of algorithms using objective function ff ₂ of Figure 4.

FIGURE 10 Performance curves of algorithms using objective function ff ₃ of Figure 4.

FIGURE 11 Performance curves of algorithms using objective function ff ₄ of Figure 4.

FIGURE 12 Comparison of algorithms for objective function ff ₁ at 25% load corresponding to Figure 4.

FIGURE 13 Comparison of algorithms for objective function ff ₂ at 25% load corresponding to Figure 4.

FIGURE 14 Comparison of algorithms for objective function ff ₃ at 25% load corresponding to Figure 4.

FIGURE 15 Comparison of algorithms for objective function ff ₄ at 25% load corresponding to Figure 4.

TABLE 3 Comparison Results of Algorithms in Terms of Efficiency and Error: Objective Function ff ₁ of Figure 3

	25%		50%		75%		100%
Algorithm	Efficiency	Error	Efficiency	Error	Efficiency	Error	Efficiency	Error
SPSO	61.5622	13.7622	89.374	5.074	87.7029	2.9029	85.5525	2.0525
SMPSO	48.2914	0.4914	89.2255	4.9255	87.7029	2.9029	85.522	2.022
GMPSO	47.818	0.018	89.1885	4.8885	87.7027	2.9027	85.5286	2.0286
ATREPSO	57.4311	9.6311	89.1885	4.8885	87.7023	2.9023	85.5266	2.0266
QPSO	55.8678	8.0678	89.0964	4.7964	87.681	2.881	85.5479	2.0479
DE	56.9297	9.1297	89.0775	4.7775	87.7051	2.9051	85.5337	2.0337
LXDE	54.9057	7.1057	89.0764	4.7764	87.6568	2.8568	85.4922	1.9922
DE-QI	55.4705	7.6705	89.0741	4.7741	87.7023	2.9023	85.5241	2.0241
GA	68.99	21.19	89.44	5.14	86.68	1.88	85.86	2.36
TG	47.8		84.3		84.8		83.5

TABLE 4 Comparison Results of Algorithms in Terms of Efficiency and Error: Objective Function ff ₂ of Figure 3

	25%		50%		75%		100%
Algorithm	Efficiency	Error	Efficiency	Error	Efficiency	Efficiency	Error	Efficiency
SPSO	60.0291	12.2291	89.393	5.093	87.7468	2.9468	85.5198	2.0198
SMPSO	57.6565	9.8565	89.3888	5.0888	87.7125	2.9125	85.5181	2.0181
GMPSO	58.3305	10.5305	89.3791	5.0791	87.7062	2.9062	85.5174	2.0174
ATREPSO	59.9916	12.1916	89.3659	5.0659	87.7108	2.9108	85.5166	2.0166
QPSO	60.5208	12.7208	89.263	4.963	87.7503	2.9503	85.5122	2.0122
DE	58.4146	10.6146	89.2604	4.9604	87.7106	2.9106	85.5169	2.0169
LXDE	52.2689	4.4689	89.1821	4.8821	87.687	2.887	85.4965	1.9965
DE-QI	55.0697	7.2697	89.2566	4.9566	87.7103	2.9103	85.5117	2.0117
GA	65.47	17.67	87.04	2.74	85.87	1.07	84.77	1.27
TG	47.8		84.3		84.8		83.5

TABLE 5 Comparison Results of Algorithms in Terms of Efficiency and Error: Objective Function ff ₃ of Figure 3

	25%		50%		75%		100%
Algorithm	Efficiency	Error	Efficiency	Error	Efficiency	Efficiency	Error	Efficiency
SPSO	64.6357	16.8357	88.9229	4.6229	87.7134	2.9134	85.9546	2.4546
SMPSO	54.3489	6.5489	88.8516	4.5516	87.4601	2.6601	85.9497	2.4497
GMPSO	61.2841	13.4841	88.7275	4.4275	87.4898	2.6898	85.947	2.447
ATREPSO	61.772	13.972	88.6545	4.3545	87.626	2.826	85.9731	2.4731
QPSO	63.5507	15.7507	89.0644	4.7644	87.4738	2.6738	85.9489	2.4489
DE	66.1704	18.3704	89.121	4.821	87.4895	2.6895	85.9515	2.4515
LXDE	64.2549	16.4549	88.7976	4.4976	87.4413	2.6413	85.9426	2.4426
DE-QI	60.2366	12.4366	88.9832	4.6832	87.3902	2.5902	85.942	2.442
GA	71.24	23.44	88.88	4.58	88.04	3.24	86.36	2.86
TG	47.8		84.3		84.8		83.5

TABLE 6 Comparison Results of Algorithms in Terms of Efficiency and Error: Objective Function ff ₄ of Figure 3

	25%		50%		75%		100%
Algorithm	Efficiency	Error	Efficiency	Error	Efficiency	Efficiency	Error	Efficiency
SPSO	55.4919	7.6919	88.4656	4.1656	87.6683	2.8683	85.997	2.497
SMPSO	55.4919	7.6919	88.1818	3.8818	87.4651	2.6651	85.9256	2.4256
GMPSO	57.2915	9.4915	88.2892	3.9892	87.6405	2.8405	85.9753	2.4753
ATREPSO	57.2962	9.4962	88.2231	3.9231	87.5096	2.7096	85.9435	2.4435
QPSO	55.7984	7.9984	88.9104	4.6104	87.4671	2.6671	85.9165	2.4165
DE	55.3947	7.5947	88.8762	4.5762	87.5096	2.7096	85.9461	2.4461
LXDE	53.7438	5.9438	88.725	4.425	87.4778	2.6778	85.9091	2.4091
DE-QI	57.0967	9.2967	88.0563	3.7563	87.4972	2.6972	85.9447	2.4447
GA	71.66	23.86	88.85	4.55	87.91	3.11	86.15	2.65
TG	47.8		84.3		84.8		83.5

TABLE 7 Comparison Results of Algorithms in Terms of NFE and Time (sec): Objective Function ff ₁ of Figure 3

	25%		50%		75%		100%
Algorithm	NFE	Time	NFE	Time	NFE	Time	NFE	Time
SPSO	15030	1.7	8007	0.9	5985	0.7	4521	0.6
SMPSO	15030	1.8	7161	0.8	5823	0.7	4491	0.5
GMPSO	15030	1.8	6357	0.8	4752	0.6	4062	0.5
ATREPSO	15030	2.0	6357	0.9	4644	0.6	4140	0.5
QPSO	15030	1.8	5704	0.8	4329	0.5	4161	0.4
DE	15030	1.7	4104	0.4	2679	0.2	2751	0.2
LXDE	15030	1.8	2949	0.3	2370	0.2	2346	0.2
DE-QI	15030	1.9	2808	0.4	2085	0.2	2172	0.2
GA	NA	NA	NA	NA	NA	NA	NA	NA

TABLE 8 Comparison Results of Algorithms in Terms of NFE and Time (sec): Objective Function ff ₂ of Figure 3

	25%		50%		75%		100%
Algorithm	NFE	Time	NFE	Time	NFE	Time	NFE	Time
SPSO	93	0.01	6921	0.8	5958	0.7	5010	0.6
SMPSO	93	0.01	6921	0.7	6246	0.6	4890	0.6
GMPSO	74	0.01	5463	0.7	4842	0.7	4209	0.6
ATREPSO	80	0.01	5484	0.7	4902	0.7	4368	0.6
QPSO	288	0.1	5431	0.6	4392	0.5	4167	0.5
DE	324	0.1	4248	0.5	2973	0.3	3024	0.3
LXDE	117	0.01	2835	0.3	2337	0.3	2661	0.4
DE-QI	303	0.1	2817	0.3	2079	0.3	2319	0.3
GA	NA	NA	NA	NA	NA	NA	NA	NA

TABLE 9 Comparison Results of Algorithms in Terms of NFE and Time (sec): Objective Function ff ₃ of Figure 3

	25%		50%		75%		100%
Algorithm	NFE	Time	NFE	Time	NFE	Time	NFE	Time
SPSO	87	0.01	858	0.2	1938	0.3	3660	0.4
SMPSO	87	0.01	750	0.1	1886	0.3	3510	0.4
GMPSO	87	0.01	789	0.1	1746	0.2	3246	0.4
ATREPSO	87	0.01	825	0.2	1731	0.2	3414	0.5
QPSO	93	0.01	792	0.1	1448	0.2	2059	0.4
DE	165	0.01	1194	0.2	1710	0.3	2403	0.3
LXDE	126	0.01	927	0.2	1557	0.2	2130	0.3
DE-QI	135	0.01	915	0.1	1353	0.1	1947	0.2
GA	NA	NA	NA	NA	NA	NA	NA	NA

TABLE 10 Comparison Results of Algorithms in Terms of NFE and Time (sec): Objective Function ff ₄ of Figure 3

	25%		50%		75%		100%
Algorithm	NFE	Time	NFE	Time	NFE	Time	NFE	Time
SPSO	60	0.01	798	0.1	2082	0.3	3702	0.4
SMPSO	60	0.01	600	0.1	1800	0.3	3420	0.4
GMPSO	60	0.01	741	0.1	1719	0.2	3507	0.4
ATREPSO	60	0.01	720	0.1	1800	0.3	3570	0.5
QPSO	60	0.01	898	0.1	1765	0.3	2954	0.4
DE	60	0.01	1077	0.1	1749	0.3	2739	0.3
LXDE	60	0.01	849	0.2	1692	0.3	2115	0.3
DE-QI	60	0.01	939	0.1	1461	0.2	2010	0.3
GA	NA	NA	NA	NA	NA	NA	NA	NA

TABLE 11 Comparison Results of Algorithms in Terms of Efficiency and Error: Objective Function ff ₁ of Figure 4

	25%		50%		75%		100%
Algorithm	Efficiency	Error	Efficiency	Error	Efficiency	Error	Efficiency	Error
SPSO	52.8327	5.0327	87.3836	3.0836	87.2989	2.4989	85.6585	2.1585
SMPSO	50.3145	2.5145	87.3403	3.0403	86.6205	1.8205	85.3841	1.8841
GMPSO	47.866	0.066	86.4765	2.1765	86.9479	2.1479	85.5267	2.0267
ATREPSO	57.8555	10.0555	85.9917	1.6917	86.468	1.668	85.5712	2.0712
QPSO	64.3479	16.5479	86.5772	2.2772	86.2004	1.4004	84.8355	1.3355
DE	64.7172	16.9172	83.4023	−0.8977	86.3019	1.5019	84.7322	1.2322
LXDE	59.011	11.211	83.8205	−0.4795	84.9311	0.1311	83.5694	0.0694
DE-QI	56.8356	9.0356	84.2825	−0.0175	86.2282	1.4282	84.2291	0.7291
GA	56.38	8.58	83.51	−0.79	85.59	0.79	83.18	−0.32
TG	47.8	0	84.3	0	84.8	0	83.5	0

TABLE 12 Comparison Results of Algorithms in Terms of Efficiency and Error: Objective Function ff ₂ of Figure 4

	25%		50%		75%		100%
Algorithm	Efficiency	Error	Efficiency	Error	Efficiency	Error	Efficiency	Error
SPSO	58.0905	10.2905	85.3179	1.0179	84.8403	0.0403	83.8705	0.3705
SMPSO	48.6996	0.8996	85.7073	1.4073	84.2272	−0.5728	82.9782	−0.5218
GMPSO	53.8224	6.0224	84.8432	0.5432	85.9331	1.1331	83.2451	−0.2549
ATREPSO	50.5116	2.7116	84.8611	0.5611	85.9443	1.1443	83.8515	0.3515
QPSO	52.2799	4.4799	84.8585	0.5585	85.1353	0.3353	82.7034	−0.7966
DE	56.5057	8.7057	85.5052	1.2052	86.6821	1.8821	84.3514	0.8514
LXDE	54.7454	6.9454	84.1863	−0.1137	86.6216	1.8216	84.7473	1.2473
DE-QI	54.4074	6.6074	85.6781	1.3781	85.95	1.15	84.7072	1.2072
GA	58.77	10.97	86.93	2.63	85.74	0.94	82.91	−0.59
TG	47.8	0	84.3	0	84.8	0	83.5	0

TABLE 13 Comparison Results of Algorithms in Terms of Efficiency and Error: Objective Function ff ₃ of Figure 4

	25%		50%		75%		100%
Algorithm	Efficiency	Error	Efficiency	Error	Efficiency	Error	Efficiency	Error
SPSO	54.3575	6.5575	90.2422	5.9422	89.3893	4.5893	86.1882	2.6882
SMPSO	50.8532	3.0532	88.7778	4.4778	87.2944	2.4944	86.1882	2.6882
GMPSO	51.6536	3.8536	88.7165	4.4165	87.2927	2.4927	86.1852	2.6852
ATREPSO	57.9641	10.1641	88.2677	3.9677	87.3953	2.5953	86.1882	2.6882
QPSO	57.9816	10.1816	90.5271	6.2271	89.4017	4.6017	86.1898	2.6898
DE	62.6156	14.8156	90.6982	6.3982	89.3959	4.5959	86.1923	2.6923
LXDE	53.3793	5.5793	89.5805	5.2805	87.3838	2.5838	85.9544	2.4544
DE-QI	58.0465	10.2465	88.5345	4.2345	87.3911	2.5911	86.1886	2.6886
GA	72.37	24.57	88.87	4.57	87.95	3.15	86.13	2.63
TG	47.8	0	84.3	0	84.8	0	83.5	0

TABLE 14 Comparison Results of Algorithms in Terms of Efficiency and Error: Objective Function ff ₄ of Figure 4

	25%		50%		75%		100%
Algorithm	Efficiency	Error	Efficiency	Error	Efficiency	Error	Efficiency	Error
SPSO	50.7735	2.9735	88.3947	4.0947	87.3534	2.5534	85.2512	1.7512
SMPSO	50.1228	2.3228	88.1714	3.8714	87.1219	2.3219	84.4474	0.9474
GMPSO	51.4569	3.6569	88.2749	3.9749	84.7294	−0.0706	83.9022	0.4022
ATREPSO	51.4188	3.6188	88.1275	3.8275	84.8454	0.0454	82.8624	−0.6376
QPSO	59.0119	11.2119	88.7229	4.4229	85.7053	0.9053	83.6247	0.1247
DE	59.3771	11.5771	86.4578	2.1578	86.6514	1.8514	84.2524	0.7524
LXDE	59.2987	11.4987	86.2706	1.9706	85.6621	0.8621	84.1691	0.6691
DE-QI	55.763	7.963	86.4489	2.1489	86.3441	1.5441	84.061	0.561
GA	72.92	25.12	89.02	4.72	88.04	3.24	86.19	2.69
TG	47.8	0	84.3	0	84.8	0	83.5	0

Comparison among the four methods in terms of error in the efficiency estimation indicate that the methods corresponding to Figure 4, which are considered the equivalent circuit with a series connection of , gave better results than one with parallel connection of . The addition nameplate output power (method 3) to the inputs of the algorithm helps to achieve minimum convergence time and number of function evaluations (NFE) at all the loads. However, no positive effect of this variable is in the error minimization. The standard deviation of method 3 in each run is 6.08e⁻⁷, which proves its superiority, whereas the standard deviation value in method 1 is 4.65e⁻⁵. Method 4, which considered all four inputs to the algorithm, produced poor results compared with method 3 in terms of convergence time and NFE.

The results from methods 3 and 4 indicate that the effect of third (power factor) input parameter in the performance of the algorithms is almost negligible. But its contribution in method 2 is significant because it gave good performance compared with method 1. Hence, the parameters, namely current, input power, speed, and output power, are sufficient to determine the motor parameters quickly. If the knowledge of output power is not available, power factor should be considered in addition with the first two inputs to get better performance.

In comparison with the optimization algorithms in terms of error, the results indicate that LXDE outperformed the other algorithms at 100% load. ATRE-PSO and DE-QI algorithms produced better results at 75% and 50% loads, respectively. At 25% load, variants of PSO (SMPSO and ATRE-PSO) outperformed the other algorithms.

In comparison with the algorithms in term of convergence speed, the results showed that DE-QI offered better performance than all other algorithms at 100% and 75% loads; SMPSO at 50% load and almost all algorithms at 25% load performed well.

Overall, we can say that method 2, which is solved by the variants of optimization algorithms based on diversity and mutation, gave better results in most cases in terms of accuracy in efficiency estimation. Method 3 solved by the variants of algorithms based on mutation offered very good results in terms of convergence time at all the load points.

CONCLUSION

In this article we compared PSO, DE, and their variants (six improved versions) with GA and torque-gauge methods for in situ efficiency determination of an induction motor through parameter identification. This problem was framed by four different methods. The differences in methods were based on the number of input parameters used in the optimization algorithms and modifications in the equivalent circuit of the motor. All algorithms proved their numerical stability and their robustness toward error minimization in a short time.

In summary, diversity- and mutation-based variants of PSO and DE outperformed the other algorithms in most cases in terms of efficiency evaluation. In the case of convergence speed, mutation-based variants of PSO and DE were winners in all cases. The influence of output power as an input parameter of the algorithm is significant. Finally, induction motor in situ efficiency can be accurately calculated by using input power, current, speed, and output power as the input parameters of optimization algorithms. Modification in the equivalent circuit of the induction motor helped to estimate the efficiency with high accuracy.