# Parameter Estimation of Induction Machine Single-Cage and Double-Cage Models Using a Hybrid Simulated Annealing–Evaporation Rate Water Cycle Algorithm

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## Abstract

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## 1. Introduction

## 2. Induction Machine Equivalent Circuits

_{1}, R

_{2}, R

_{m}, X

_{1}, X

_{2}, and X

_{m}represent the stator resistance, rotor resistance in reference to stator side, core loss resistance, stator leakage reactance, rotor leakage reactance resistance in reference to stator side, and magnetizing reactance, respectively [4]. Therefore, in general, this circuit has six different parameters. However, in many papers dealing with induction machine parameter estimation, the value of the core loss resistance is ignored (for example in [18,26,51,65] and so on). The steady-state equivalent circuit of the double-cage IM, shown in Figure 1b, contains, in general, eight electrical parameters. In this circuit, parameters R

_{s}and X

_{sd}correspond to stator variables, while X

_{12}, X

_{1d}, X

_{2d}, R

_{11}, and R

_{22}correspond to rotor variables (one cage is represented by X

_{1d}and R

_{11}, while the second is defined with X

_{2d}and R

_{22}). The magnetizing part of the circuit is represented by X

_{m}. However, in some papers dealing with the double-cage IM, the value of the stator reactance X

_{sd}and/or the value of the mutual rotor reactance X

_{12}are ignored [5,13,20].

## 3. Parameter Estimation Methods Based on Steady-State Models: An Overview

#### 3.1. IEEE and IEC standards

#### 3.2. Methods Based on Catalog/Manufacturer/Nameplate Data

#### 3.3. Methods Based on Measured Data

#### 3.4. Methods Based on Optimization Algorithms

## 4. SA-ERWCA

_{pop}and N is the number of design variables (or dimension of the problem), a population is a matrix with dimensions N

_{pop}× N. In the original ERWCA, the population is initialized randomly between the upper bound (UB) and the lower bound (LB) of the design variables. In the hybrid SA-ERWCA proposed in this paper, the SA algorithm is used to initialize the population of the ERWCA, similarly to the relay-collaborative strategy presented in [81,88].

_{0}) given in Table 1.

_{k}and L

_{k}, are the temperature and number of transitions generated at some iteration k. They are calculated as explained in [91]. Also, rand represents a vector of random numbers between 0 and 1. After the initialization process, the obtained population must be sorted according to the value of the fitness function of each individual. Namely, the best individual, which has the minimum fitness function value, is chosen to be the sea. Besides the sea, the population consists of rivers and streams. The predefined parameter of the ERWCA is denoted as N

_{r}and represents the number of rivers. Thus, N

_{r}individuals of the initial population with the minimum fitness function value (except the sea) are chosen to be the rivers. Finally, the rest of the population is considered as streams: N

_{streams}= N

_{pop}− N

_{sr}, where N

_{sr}stands for the number of rivers plus the sea (N

_{sr}= N

_{r}+1). According to the water cycle process in nature, each stream flows directly or indirectly to the rivers or sea. The number of streams for each river and sea is calculated as follows:

_{n}represents the number of streams that flow to the nth river (or the sea of n is equal to 1). Since it was highlighted that streams continue their flow to either other rivers or directly to the sea, the next step in the ERWCA is to mathematically model the flow of streams. To that end, two update equations for the position of streams that flow to rivers and the sea are given (3) and (4), respectively:

_{1}, presented in Table 2, where t

_{max}stands for the maximum number of iterations.

_{2}, represented in Table 3.

_{3}, presented in Table 4.

_{max}is an adaptive parameter calculated as follows:

_{SA-ERWCA}) of the SA-ERWCA is presented in Table 5. Also, a flow chart of the SA-ERWCA is illustrated in Figure 2.

## 5. Simulation Results

#### 5.1. Simulation Results for Machine 1

#### 5.2. Simulation Results for Machine 2

#### 5.3. Simulation Results for Machine 3

#### 5.4. Simulation Results for Machine 4

#### 5.5. Simulation Results for Machine 5

## 6. Experimental Results

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ABC | Artificial bee colony |

ACA | Ant colony algorithm |

AFSA | Artificial fish swarm algorithm |

BFT | Bacterial foraging technique |

CSS | Charged system search |

DCIM | Double cage IM |

DEn | Dynamic encoding |

DEA | Differential evolution algorithm |

DGEA | Diversity-guided evolutionary algorithm |

ER | Evaporation rate |

ERWCA | Evaporation rate water cycle algorithm |

ES | Evolution strategy |

GA | Genetic algorithm |

GP | Genetic programming |

HGAPSO | Hybrid of genetic algorithm and particle swarm optimization |

IA | Immune algorithm |

IM | Induction machine |

LSA | Least-squares algorithm |

MFO | Moth-flame optimization |

MVO | Multi-verse optimizer |

MVMO | Mean-variance mapping optimization |

MSFLA | Modified shuffled frog-leaping algorithm |

OF | Objective function |

PC | Pseudo-code |

PSO | Particle swarm optimization |

SA | Simulated annealing |

SA-ERWCA | Simulated annealing–evaporation rate water cycle algorithm |

SCIM | Single cage IM |

SDLS | Steepest descent local search |

SEA | Simple evolutionary algorithm |

SFLA | Shuffled frog-leaping algorithm |

SGOA | Sparse grid optimization algorithm |

SRS | Simple random search |

SS | Scatter search |

VCM | Vector constructing method |

## Nomenclature

AVG | Average value |

C | Parameter used in the SA- ERWCA |

c_{k} | Temperature at the kth iteration |

$\left|e\right|$ | Absolute error |

d_{max} | Adaptive parameter |

f | Nominal frequency |

I | Phase current |

I_{st} | Starting current |

I_{fl} | Full load current |

LB | Lower bound of the design variables. |

L_{k} | Number of transitions |

MED | Median value |

n | Dimension |

n_{r} | Rated speed |

N | Number of design variables |

N_{pop} | Size of the population |

N_{r} | Number of rivers |

N_{streams} | Number of streams |

NS_{n} | Number of streams which flow to the nth river |

P_{n} | Nominal power |

p | Pole pairs number |

pf, pf_{fl} | Power factor and pf at full load |

R_{1} | Stator resistance |

R_{2} | Rotor resistance in reference to stator side |

R_{m} | Core loss resistance |

R_{11} | Resistance of first rotor cage (for double-cage machine) |

R_{22} | Reactance of second rotor cage (for double-cage machine) |

R_{th} | Thevenin equivalent resistance |

rand | Vector of random numbers between [0, 1] |

randn (1, N) | Vector of N standard Gaussian numbers |

STD | Standard deviation |

s_{fl} | Slip value at full load |

s_{max} | Slip value at maximal torque |

t | Current iteration |

t_{max} | Maximum number of iterations |

T | Torque |

T_{st} | Starting torque |

T_{fl} | Full load torque |

T_{max} | Maximal torque |

X_{1} | Stator leakage reactance |

X_{2} | Rotor leakage reactance resistance in reference to stator side |

X_{m} | Magnetizing reactance |

X_{sd} | Stator leakage reactance (for double-cage machine) |

X_{12} | Mutual rotor leakage reactance (for double-cage machine) |

X_{1d} | Reactance of first rotor cage (for double-cage machine) |

X_{2d} | Resistance of second cage (for double-cage machine) |

X_{th} | Thevenin equivalent reactance |

${\overrightarrow{X}}_{i},\forall i=1,2,\dots {N}_{pop}$ | Individual ranges from i to N_{pop} |

UB | Upper bound of the design variables |

V | Nominal voltage |

V_{ph} | Nominal phase voltage |

V_{th} | Thevenin equivalent voltage |

μ | Coefficient used in the SA- ERWCA |

## Appendix A

_{m}, are given as follows:

## Appendix B

## References

- Kazmierkowski, M.P. Electric Motor Drives: Modeling, Analysis and Control; Krishan, R., Ed.; Prentice-Hall: Upper Saddle River, NJ, USA, 2001. [Google Scholar]
- IEEE Standard 112. Test Procedure for Polyphase Induction Motors and Generators; IEEE: New York City, NJ, USA, 2004. [Google Scholar]
- IEC Standards 60034-28, Rotating Electrical Machines–Part 28: Test Methods for Determining Quantities of Equivalent Circuit Diagrams for Three-Phase Low-Voltage Cage Induction Motors; IEC: Geneva, Switzerland, 2012.
- Al-Badri, M.; Pillay, P.; Angers, P. A Novel In Situ Efficiency Estimation Algorithm for Three-Phase IM Using GA, IEEE Method F1 Calculations, and Pretested Motor Data. IEEE Trans. Energy Convers.
**2015**, 30, 1092–1102. [Google Scholar] [CrossRef] - Pedra, J.; Sainz, L. Parameter estimation of squirrel-cage induction motors without torque measurements. IEEE Proc. Electr. Power Appl.
**2006**, 153, 263–270. [Google Scholar] [CrossRef] - Toliyat, H.A.; Levi, E.; Raina, M. A review of RFO induction motor parameter estimation techniques. IEEE Trans. Energy Convers.
**2003**, 18, 271–283. [Google Scholar] [CrossRef] - Lindenmeyer, D.; Dommel, H.W.; Moshref, A.; Kundur, P. An induction motor parameter estimation method. Int. J. Electr. Power Energy Syst.
**2001**, 23, 251–262. [Google Scholar] [CrossRef] - Tang, J.; Yang, Y.; Blaabjerg, F.; Chen, J.; Diao, L.; Liu, Z. Parameter Identification of Inverter-Fed Induction Motors: A Review. Energies
**2018**, 11, 2194. [Google Scholar] [CrossRef] [Green Version] - Odhano, S.A.; Pescetto, P.; Awan, H.A.; Hinkkanen, M.; Pellegrino, G.; Bojoi, R. Parameter Identification and Self-Commissioning in AC Motor Drives: A Technology Status Review. IEEE Trans. Power Electron.
**2019**, 34, 3603–3614. [Google Scholar] [CrossRef] [Green Version] - Yamamoto, S.; Hirahara, H.; Tanaka, A.; Ara, T. A simple method to determine double-cage rotor equivalent circuit parameters of induction motors from no-load and locked-rotor tests. IEEE Trans. Ind. Appl.
**2019**, 55, 273–282. [Google Scholar] [CrossRef] - Natarajan, R.; Misra, V.K. Parameter estimation of induction motors using a spreadsheet program on a personal computer. Electr. Power Syst. Res.
**1989**, 16, 157–164. [Google Scholar] [CrossRef] - Akbaba, M.; Taleb, M.; Rumeli, A. Improved estimation of induction machine parameters. Electr. Power Syst. Res.
**1995**, 34, 65–73. [Google Scholar] [CrossRef] - Haque, M.H. Determination of NEMA Design Induction Motor Parameters from Manufacturer Data. IEEE Trans. Energy Convers.
**2008**, 23, 997–1004. [Google Scholar] [CrossRef] - Perez, I.; Gomez-Gonzalez, M.; Jurado, F. Estimation of induction motor parameters using shuffled frog-leaping algorithm. Electr. Eng.
**2013**, 95, 267–275. [Google Scholar] [CrossRef] - Nangsue, P.; Pillay, P.; Conry, S.E. Evolutionary algorithms for induction motor parameter determination. IEEE Trans. Energy Convers.
**1999**, 14, 447–453. [Google Scholar] [CrossRef] - Sakthivel, V.; Bhuvaneswari, R.; Subramanian, S. Multi-objective parameter estimation of induction motor using particle swarm optimization. Eng. Appl. Artif. Intell.
**2010**, 23, 302–312. [Google Scholar] [CrossRef] - Sakthivel, V.; Bhuvaneswari, R.; Subramanian, S. Artificial immune system for parameter estimation of induction motor. Expert Syst. Appl.
**2010**, 37, 6109–6115. [Google Scholar] [CrossRef] - Mohammadi, H.R.; Akhavan, A. Parameter Estimation of Three-Phase Induction Motor Using Hybrid of Genetic Algorithm and Particle Swarm Optimization. J. Eng.
**2014**, 2014, 148204. [Google Scholar] [CrossRef] [Green Version] - Nikranajbar, A.; Ebrahimi, M.K.; Wood, A.S. Parameter identification of a cage induction motor using particle swarm optimization. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng.
**2010**, 224, 479–491. [Google Scholar] [CrossRef] [Green Version] - Gomez-Gonzalez, M.; Jurado, F.; Pérez, I. Shuffled frog-leaping algorithm for parameter estimation of a double-cage asynchronous machine. IET Electr. Power Appl.
**2012**, 6, 484–490. [Google Scholar] [CrossRef] - Abro, A.G.; Mohamad-Saleh, J. Multiple-global-best guided artificial bee colony algorithm for induction motor parameter estimation. Turkish J. Electr. Eng. Comput. Sci.
**2014**, 22, 620–636. [Google Scholar] [CrossRef] - Çanakoğlu, A.İ.; Yetgin, A.G.; Temurtaş, H.; Turan, M. Induction motor parameter estimation using metaheuristic methods. Turkish J. Electr. Eng. Comput. Sci.
**2014**, 22, 1177–1192. [Google Scholar] [CrossRef] - Lv, J.Y. Improved Artificial Fish Swarm Algorithm Applied on the Static Model of the Induction Motor Parameter Identification. Appl. Mech. Mater.
**2012**, 220–223, 753–761. [Google Scholar] [CrossRef] - Huang, K.S.; Wu, Q.H.; Turner, D.R. Effective identification of induction motor parameters based on fewer measurements. IEEE Trans. Energy Convers.
**2002**, 17, 55–60. [Google Scholar] [CrossRef] - Huynh, D.C.; Dunnigan, M.W. Parameter estimation of an induction machine using advanced particle swarm optimisation algorithms. IET Electr. Power Appl.
**2010**, 4, 748–760. [Google Scholar] [CrossRef] - Sakthivel, V.P.; Bhuvaneswari, R.; Subramanian, S. Bacterial Foraging Technique Based Parameter Estimation of Induction Motor from Manufacturer Data. Electr. Power Compon. Syst.
**2010**, 38, 657–674. [Google Scholar] [CrossRef] - Ursem, R.K.; Vadstrup, P. Parameter identification of induction motors using stochastic optimization algorithms. Appl. Soft Comput.
**2004**, 4, 49–64. [Google Scholar] [CrossRef] - Benaïdja, N.; Khenfer, N. Identification of Asynchronous Machine Parameters by Evolutionary Techniques. Electr. Power Compon. Syst.
**2006**, 34, 1359–1376. [Google Scholar] [CrossRef] - Duan, F.; Zivanovic, R.; Al-Sarawi, S.; Mba, D. Induction Motor Parameter Estimation Using Sparse Grid Optimization Algorithm. IEEE Trans. Ind. Inform.
**2016**, 12, 1453–1461. [Google Scholar] [CrossRef] - Kim, J.-W.; Kim, S.W. Parameter Identification of Induction Motors Using Dynamic Encoding Algorithm for Searches (DEAS). IEEE Trans. Energy Convers.
**2005**, 20, 16–24. [Google Scholar] [CrossRef] - Guedes, J.J.; Castoldi, M.F.; Goedtel, A.; Agulhari, C.M.; Sanches, D.S. Parameters estimation of three-phase induction motors using differential evolution. Electr. Power Syst. Res.
**2018**, 154, 204–212. [Google Scholar] [CrossRef] - He, Y.; Wang, Y.; Feng, Y.; Wang, Z. Parameter Identification of an Induction Machine at Standstill Using the Vector Constructing Method. IEEE Trans. Power Electron.
**2012**, 27, 905–915. [Google Scholar] [CrossRef] - Pedra, J.; Candela, I.; Sainz, L. Modelling of squirrel-cage induction motors for electromagnetic transient programs. IET Electr. Power Appl.
**2009**, 3, 111–122. [Google Scholar] [CrossRef] - Farias, E.R.C.; Cari, E.T.; Erlich, I.; Shewarega, F. Online Parameter Estimation of a Transient Induction Generator Model Based on the Hybrid Method. IEEE Trans. Energy Convers.
**2018**, 33, 1529–1538. [Google Scholar] [CrossRef] - Guedes, J.J.; Castoldi, M.F.; Goedtel, A. Temperature influence analysis on parameter estimation of induction motors using differential evolution. IEEE Lat. Am. Trans.
**2016**, 14, 4097–4105. [Google Scholar] [CrossRef] - Klimenta, D.; Hannukainen, A.; Arkkio, A. Estimating the parameters of induction motors in different operating regimes from a set of data containing the rotor cage temperature. Electr. Eng.
**2018**, 100, 139–150. [Google Scholar] [CrossRef] - Ranta, M.; Hinkkanen, M. Online identification of parameters defining the saturation characteristics of induction machines. IEEE Trans. Ind. Appl.
**2013**, 49, 2136–2145. [Google Scholar] [CrossRef] [Green Version] - Córcoles, F.; Monjo, L.; Pedra, J. Parameter estimation of squirrel-cage motors with parasitic torques in the torque–slip curve. IET Electr. Power Appl.
**2015**, 9, 377–387. [Google Scholar] - Lindsay, J.; Barton, T. Parameter Identification for Squirrel Cage Induction Machines. IEEE Trans. Power Appar. Syst.
**1973**, PAS-92, 1287–1291. [Google Scholar] [CrossRef] - Jaramillo-Matta, A.; Guasch-Pesquer, L.; MartíNez-Salamero, L.; Barrado-Rodrigo, J.A. Operating points estimation of three-phase induction machines using a torque-speed tracking technique. IET Electr. Power Appl.
**2011**, 5, 307–316. [Google Scholar] [CrossRef] - Haque, M.H. Estimation of three-phase induction motor parameters. Electr. Power Syst. Res.
**1993**, 26, 87–193. [Google Scholar] [CrossRef] - Guimaraes, J.M.C.; Bernardes, J.V.; Hermeto, A.E.; Bortoni, E.D.C. Parameter Determination of Asynchronous Machines from Manufacturer Data Sheet. IEEE Trans. Energy Convers.
**2014**, 29, 689–697. [Google Scholar] [CrossRef] - Bhowmick, D.; Manna, M.; Chowdhury, S.K. Estimation of Equivalent Circuit Parameters of Transformer and Induction Motor from Load Data. IEEE Trans. Ind. Appl.
**2018**, 54, 2784–2791. [Google Scholar] [CrossRef] - Bechouche, A.; Sediki, H.; Ould Abdeslam, D.; Haddad, S. A Novel Method for Identifying Parameters of Induction Motors at Standstill Using ADALINE. IEEE Trans. Energy Convers.
**2012**, 27, 105–116. [Google Scholar] [CrossRef] - Kostov, I.; Spasov, V.; Rangelova, V. Application of genetic algorithms for determining the parameters of induction motors. Teh. Vjesn.
**2009**, 16, 49–53. [Google Scholar] - Abdelhadi, B.; Benoudjit, A.; Nait Said, N. Identification of Induction Machine Parameters Using a New Adaptive Genetic Algorithm. Electr. Power Compon. Syst.
**2004**, 32, 767–784. [Google Scholar] [CrossRef] - Boglietti, A.; Cavagnino, A.; Lazzari, M. Computational Algorithms for Induction-Motor Equivalent Circuit Parameter Determination—Part I: Resistances and Leakage Reactances. IEEE Trans. Ind. Electron.
**2011**, 58, 3723–3733. [Google Scholar] [CrossRef] - Boglietti, A.; Cavagnino, A.; Lazzari, M. Computational Algorithms for Induction Motor Equivalent Circuit Parameter Determination—Part II: Skin Effect and Magnetizing Characteristics. IEEE Trans. Ind. Electron.
**2011**, 58, 3734–3740. [Google Scholar] [CrossRef] - Ling, Z.; Zhou, L.; Guo, S.; Zhang, Y. Equivalent Circuit Parameters Calculation of Induction Motor by Finite Element Analysis. IEEE Trans. Magn.
**2014**, 50, 833–836. [Google Scholar] [CrossRef] - Bae, D.; Kim, D.; Jung, H.K.; Hahn, S.Y.; Koh, C.S. Determination of induction motor parameters by using neural network based on FEM results. IEEE Trans. Magn.
**1997**, 33, 1924–1927. [Google Scholar] - Monjo, L.; Kojooyan-Jafari, H.; Corcoles, F.; Pedra, J. Squirrel-Cage Induction Motor Parameter Estimation Using a Variable Frequency Test. IEEE Trans. Energy Convers.
**2015**, 30, 550–557. [Google Scholar] [CrossRef] [Green Version] - Seok, J.K.; Moon, S.I.; Sul, S.K. Induction machine parameter identification using PWM inverter at standstill. IEEE Trans. Energy Convers.
**1997**, 12, 127–132. [Google Scholar] [CrossRef] - Yamazaki, K.; Suzuki, A.; Ohto, M.; Takakura, T. Circuit Parameters Determination Involving Stray Load Loss and Harmonic Torques for High-Speed Induction Motors Fed by Inverters. IEEE Trans. Energy Convers.
**2013**, 28, 154–163. [Google Scholar] [CrossRef] - Sonnaillon, M.O.; Bisheimer, G.; Angelo, C.D.; García, G.O. Automatic induction machine parameters measurement using standstill frequency-domain tests. IET Electr. Power Appl.
**2007**, 1, 833–838. [Google Scholar] [CrossRef] - Repo, A.K.; Arkkio, A. Numerical impulse response test to identify parametric models for closed-slot deep-bar induction motors. IET Electr. Power Appl.
**2007**, 1, 307–315. [Google Scholar] [CrossRef] [Green Version] - Lalami, A.; Wamkeue, R.; Kamwa, I.; Saad, M.; Beaudoin, J.J. Unscented Kalman filter for non-linear estimation of induction machine parameters. IET Electr. Power Appl.
**2012**, 6, 611–620. [Google Scholar] [CrossRef] - Moonl, S.; Keyhani, A. Estimation of Induction Machine Parameters from Standstill Time-Domain Data. IEEE Trans. Ind. Appl.
**1994**, 30, 1609. [Google Scholar] [CrossRef] - Castaldi, P.; Geri, W.; Montanari, M.; Tilli, A. A new adaptive approach for on-line parameter and state estimation of induction motors. Control Eng. Pract.
**2005**, 13, 81–94. [Google Scholar] [CrossRef] - Chai, H.; Acarnley, D. Induction motor parameter estimation algorithm using spectral analysis. IEE Proc. B Electr. Power Appl.
**1992**, 139, 165–174. [Google Scholar] [CrossRef] - Holtz, J. Sensorless control of induction motor drives. Proc. IEEE
**2002**, 90, 1359–1394. [Google Scholar] [CrossRef] [Green Version] - Khang, H.V.; Arkkio, A. Parameter estimation for a deep-bar induction motor. IET Electr. Power Appl.
**2012**, 6, 133–142. [Google Scholar] [CrossRef] - Reed, D.M.; Hofmann, H.F.; Sun, J. Offline identification of induction machine parameters with core loss estimation using the stator current locus. IEEE Trans. Energy Convers.
**2016**, 31, 1549–1558. [Google Scholar] [CrossRef] - Babau, R.; Boldea, I.; Miller, T.J.E.; Muntean, N. Complete Parameter Identification of Large Induction Machines from No-Load Acceleration–Deceleration Tests. IEEE Trans. Ind. Electron.
**2007**, 54, 1962–1972. [Google Scholar] [CrossRef] - Kojooyan-Jafari, H.; Monjo, L.; Corcoles, F.; Pedra, J. Using the Instantaneous Power of a Free Acceleration Test for Squirrel-Cage Motor Parameters Estimation. IEEE Trans. Energy Convers.
**2015**, 30, 974–982. [Google Scholar] [CrossRef] [Green Version] - Benzaquen, J.; Rengifo, J.; Albanez, E.; Aller, J.M. Parameter Estimation for Deep-Bar Induction Machines Using Instantaneous Stator Measurements from a Direct Startup. IEEE Trans. Energy Convers.
**2017**, 32, 516–524. [Google Scholar] [CrossRef] - Grantham, C.; McKinnon, D.J. Rapid parameter determination for induction motor analysis and control. IEEE Trans. Ind. Appl.
**2003**, 39, 1014–1020. [Google Scholar] [CrossRef] - Lin, W.M.; Su, T.J.; Wu, R.C. Parameter Identification of Induction Machine with a Starting No-Load Low-Voltage Test. IEEE Trans. Ind. Electron.
**2012**, 59, 352–360. [Google Scholar] [CrossRef] - Kojooyan-Jafari, H.; Monjo, L.; Córcoles, F.; Pedra, J. Parameter Estimation of Wound-Rotor Induction Motors from Transient Measurements. IEEE Trans. Energy Convers.
**2014**, 29, 300–308. [Google Scholar] - Lee, S.H.; Yoo, A.; Lee, H.J.; Yoon, Y.D.; Han, B.M. Identification of Induction Motor Parameters at Standstill Based on Integral Calculation. IEEE Trans. Ind. Appl.
**2017**, 53, 2130–2139. [Google Scholar] [CrossRef] - Khemici, L.; Bounekhla, M.; Boudissa, E. Alienor method applied to induction machine parameters identification. Int. Journal El. and Comp. Eng.
**2020**, 10, 223–232. [Google Scholar] [CrossRef] - Pereira, L.A.; Perin, M.; Pereira, L.F.; Ruthes, J.R.; de Sousa, F.L.; de Oliveira, E.C. Performance estimation of three-phase induction motors from no-load startup test without speed acquisition. ISA Trans.
**2020**, 96, 376–389. [Google Scholar] [CrossRef] - Debbabi, F.; Nemmour, A.L.; Khezzar, A.; Chelli, S.E. An approved superiority of real-time induction machine parameter estimation operating in self-excited generating mode versus motoring mode using the linear RLS algorithm: Ideas & applications. Int. J. Electr. Power Energy Syst.
**2020**, 118105725. [Google Scholar] - Souza Ribeiro, L.A.; Jacobina, C.B.; Lima, A.M.N.; Oliveira, A.C. Real-time estimation of the electric parameters of an induction machine using sinusoidal PWM voltage waveforms. IEEE Trans. Ind. Appl.
**2000**, 36, 743–754. [Google Scholar] [CrossRef] - Stephan, J.; Bodson, M.; Chiasson, J. Real-time estimation of the parameters and fluxes of induction motors. IEEE Trans. Ind. Appl.
**1994**, 30, 746–759. [Google Scholar] [CrossRef] - Xiao, J.; Wang, S.; Dinavahi, V. Multi-rate real-time model-based parameter estimation and state identification for induction motors. IET Electr. Power Appl.
**2013**, 7, 77–86. [Google Scholar] - Jabbour, N.; Mademlis, C. Online Parameters Estimation and Autotuning of a Discrete-Time Model Predictive Speed Controller for Induction Motor Drives. IEEE Trans. Power Electron.
**2019**, 34, 1548–1559. [Google Scholar] [CrossRef] - Haroon, S.S.; Malik, T.N. Evaporation Rate-Based Water Cycle Algorithm for Short-Term Hydrothermal Scheduling. Arab. J. Sci. Eng.
**2017**, 42, 2615–2630. [Google Scholar] [CrossRef] - Haroon, S.S.; Malik, T.N. Evaporation rate based water cycle algorithm for the environmental economic scheduling of hydrothermal energy systems. J. Renew. Sustain. Energy
**2016**, 8, 44501. [Google Scholar] [CrossRef] - Kler, D.; Sharma, P.; Banerjee, A.; Rana, K.P.S.; Kumar, V. PV cell and module efficient parameters estimation using Evaporation Rate based Water Cycle Algorithm. Swarm Evol. Comput.
**2017**, 35, 93–110. [Google Scholar] [CrossRef] - Ćalasan, M.; Abdel Aleem, S.H.E.; Zobaa, A.F. On the root mean square error (RMSE) calculation for parameter estimation of photovoltaic models: A novel exact analytical solution based on Lambert W function. Energy Convers. Manag.
**2020**, 210, 112716. [Google Scholar] [CrossRef] - Rodriguez, F.J.; Garcia-Martinez, C.; Lozano, M. Hybrid Metaheuristics Based on Evolutionary Algorithms and Simulated Annealing: Taxonomy, Comparison, and Synergy Test. IEEE Trans. Evol. Comput.
**2012**, 16, 787–800. [Google Scholar] [CrossRef] - Herrera, F.; Lozano, M. Gradual distributed real-coded genetic algorithms. IEEE Trans. Evol. Comput.
**2020**, 4, 43–63. [Google Scholar] [CrossRef] [Green Version] - Alba, E. Parallel Metaheuristics; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2005. [Google Scholar]
- Thompson, D.R.; Bilbro, G.L. Sample-Sort Simulated Annealing. IEEE Trans. Syst. Man Cybern. Part B
**2005**, 35, 625–632. [Google Scholar] [CrossRef] [PubMed] - Xavier-de-Souza, S.; Suykens, J.; Vandewalle, J.; Bollé, D. Cooperative behavior in coupled simulated annealing processes with variance control. In Proceedings of the 2006 International Symposium on Nonlinear Theory and its Applications, Bologna, Italy, 11–14 September 2006; pp. 114–119. [Google Scholar]
- Chen, D.J.; Lee, C.Y.; Park, C.H.; Mendes, P. Parallelizing simulated annealing algorithms based on high-performance computer. J. Glob. Optim.
**2007**, 39, 261–289. [Google Scholar] [CrossRef] - Mühlenbein, H. Parallel genetic algorithms, population genetics and combinatorial optimization. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Springer Nature: Berlin, Germany, 1991; Volume 1, pp. 398–406. [Google Scholar]
- Feng-Tse, L.; Cheng-Yan, K.; Ching-Chi, H. Applying the genetic approach to simulated annealing in solving some NP-hard problems. IEEE Trans. Syst. Man. Cybern.
**1993**, 23, 1752–1767. [Google Scholar] [CrossRef] - Talbi, E.G. A Taxonomy of Hybrid Metaheuristics. J. Heuristics.
**2002**, 8, 541–564. [Google Scholar] [CrossRef] - Krasnogor, N.; Smith, J. A Tutorial for Competent Memetic Algorithms: Model, Taxonomy, and Design Issues. IEEE Trans. Evol. Comput.
**2005**, 9, 474–488. [Google Scholar] [CrossRef] [Green Version] - Delahaye, D.; Chaimatanan, S.; Mongeau, M. Simulated Annealing: From Basics to Applications BT—Handbook of Metaheuristics; Springer International Publishing: Cham, Switzerland, 2019; pp. 1–35. [Google Scholar]
- Li, S.; Chen, H.; Wang, M.; Heidari, A.A.; Mirjalili, S. Slime mould algorithm: A new method for stochastic optimization. Future Gener. Comput. Syst.
**2020**, 111, 300–323. [Google Scholar] [CrossRef] - Zobaa, A.F.; Aleem, S.H.E.A.; Abdelaziz, A.Y. Classical and Recent Aspects of Power System Optimization; Academic Press/Elsevier: Cambridge, MA, USA, 2018. [Google Scholar]

**Figure 6.**Curves of Machine 2: (

**a**) Torque versus slip; (

**b**) power factor versus slip; and (

**c**) phase current versus slip.

**Figure 7.**Curves of Machine 3: (

**a**) Torque versus slip; and (

**b**) phase current versus slip obtained from the acceleration test [64].

**Figure 11.**Experimental results: (

**a**) Phase current versus slip; (

**b**) power factor versus slip; (

**c**) active power versus slip; and (

**d**) convergence characteristics of considered machine when applying the proposed algorithm.

Pseudo-Code of the SA Algorithm (PC_{0}) |
---|

For each individual ${\overrightarrow{X}}_{i},\forall i=1,2,\dots {N}_{pop}$ |

Enter the input data: k = 0, c_{k} = c_{0}, L_{k} = L_{0} |

${\overrightarrow{X}}_{i}=rand\times (UB-LB)+LB$ |

Repeat |

For l = 0 to L_{k} |

Generate a solution ${\overrightarrow{X}}_{j}$ from the neighborhood of the current solution ${\overrightarrow{X}}_{i}$ |

If $OF({\overrightarrow{X}}_{j})<OF({\overrightarrow{X}}_{i})$ then ${\overrightarrow{X}}_{j}$ becomes the current solution $({\overrightarrow{X}}_{i}={\overrightarrow{X}}_{j})$ |

Else ${\overrightarrow{X}}_{j}$ becomes the current solution with the probability ${e}^{\left(\frac{f({\overrightarrow{X}}_{i})-f({\overrightarrow{X}}_{j})}{{c}_{k}}\right)}$ |

k = k + 1 |

Compute L_{k} and c_{k} |

Until c_{k} ≅ 0 |

Pseudo-Code (PC_{1}) |
---|

for i = 1: N_{sr} − 1 |

If (exp(−t/t_{max}) < rand) & (NS_{i} < ER) |

Perform rain process represented by (7) |

End for |

Pseudo-Code (PC_{2}) |
---|

If $|{\overrightarrow{X}}_{sea}-{\overrightarrow{X}}_{river}^{i}|<{d}_{max}$ or $rand<0.1,\text{}i=1,2,\dots {N}_{sr}-1$ |

Perform rain process represented by (7). |

Pseudo-Code (PC_{3}) |
---|

If $|{\overrightarrow{X}}_{sea}-{\overrightarrow{X}}_{stream}^{i}|<{d}_{max},\text{}i=1,2,\dots N{S}_{1}$ |

Perform rain process represented by (8). |

Pseudo-Code (PC_{SA-ERWCA}) |
---|

Enter the parameters: N_{sr}, d_{max}, N_{pop}, and t_{max} |

Initialize the population using the SA algorithm |

t = 1 |

while (t < t_{max}) |

Calculate the intensity of flow for rivers and sea using (1) and (2) |

Calculate the positions of the streams using (3) and (4) |

If a certain stream finds a better solution than the rivers/sea then exchange the positions |

Calculate the positions of the rivers according to (5) |

If a river obtains a better solution than the sea; then, exchange the positions |

Calculate the evaporation rate ER as given by (6) |

Check the evaporation condition among rivers and streams and calculate the new positions using PC_{1} |

Similarly, to the previous step, check the evaporation conditions between sea and streams/rivers and calculate new positions PC_{2} and PC_{3} |

Update the value of d_{max} using (9) |

t = t + 1 |

End while |

**Table 6.**Four benchmark test functions [92].

Function | Dimension | Range | f_{min} |
---|---|---|---|

${f}_{1}={\displaystyle {\displaystyle \sum}_{i=1}^{n}}{x}_{i}^{2}$ | n | [−100, 100] | 0 |

${f}_{2}={\displaystyle {\displaystyle \sum}_{i=1}^{n}}\left|{x}_{i}\right|+{\displaystyle {\displaystyle \prod}_{i=1}^{n}}|{x}_{i}|$ | n | [−10, 10] | 0 |

${f}_{3}={\displaystyle {\displaystyle \sum}_{i=1}^{n}}{\left({\displaystyle {\displaystyle \sum}_{j-1}^{i}}{x}_{j}\right)}^{2}$ | n | [−100, 100] | 0 |

${f}_{4}={\displaystyle {\displaystyle \sum}_{i=1}^{n-1}}[100({x}_{i+1}-{x}_{i}^{2}\text{})+{({x}_{i}-1)}^{2}]$ | n | [−30, 30] | 0 |

Functions | f_{1} | f_{2} | ||||

Algorithms | AVG | STD | MED | AVG | STD | MED |

SA-ERWCA | 1.12 × 10^{−10} | 7.80 × 10^{−11} | 9.01 × 10^{−11} | 3.88 × 10^{−4} | 2.9 × 10^{−4} | 2.6810^{−4} |

MFO | 1670 | 3790 | 1667 | 35.3 | 24.5 | 35.3 |

MVO | 3.11 × 10^{−3} | 7.04 × 10^{−4} | 596 | 3.84 × 10^{−2} | 1.3 × 10^{−2} | 11.13 |

PSO | 101 | 14.3 | 111.3 | 46.9 | 3.54 | 51.56 |

DE | 2.38 × 10^{−2} | 2.48 × 10^{−2} | 5.56 × 10^{−2} | 1.18 × 10^{−2} | 3.99 × 10^{−3} | 1.7 × 10^{−2} |

Functions | f_{3} | f_{4} | ||||

Algorithms | AVG | STD | MED | AVG | STD | MED |

SA-ERWCA | 0.3437 | 0.2297 | 0.3265 | 21.915 | 2.00 | 21.82 |

MFO | 15,800 | 10,800 | 15,785 | 2.69 × 10^{6} | 1.46 × 10^{7} | 2.68 × 10^{6} |

MVO | 0.37 | 0.31 | 1613 | 66.8 | 94.5 | 3.59 × 10^{4} |

PSO | 185 | 27.6 | 220.5 | 8.98 × 10^{4} | 1.83 × 10^{4} | 1.08 × 10^{5} |

DE | 1390 | 773 | 6275 | 30.8 | 18.1 | 32.59 |

**Table 8.**Data of Machine 1 [14].

Parameter | Value | Parameter | Value | Design Variables |
---|---|---|---|---|

P_{n} | 40 HP | T_{fl} | 190 | $0.1\le {R}_{1}\le 0.6$ |

V | 400 V | T_{max} | 370 | $0.2\le {R}_{2}\le 0.6$ |

f | 50 Hz | pf_{fl} | 0.8 | $0.1\le {X}_{1}\le 0.5$ |

p | 2 | s_{fl} | 0.09 | $0.3\le {X}_{2}\le 1.0$ |

T_{st} | 260 | $4\le {X}_{m}\le 11$ |

Parameter (Ω) | DE [14] | GA [14] | PSO [14] | SFLA [14] | MSFLA [14] | SA-ERWCA |
---|---|---|---|---|---|---|

R_{1} | 0.4993 | 0.4875 | 0.3555 | 0.3437 | 0.270719 | 0.27821 |

X_{1} | 0.3264 | 0.3264 | 0.3455 | 0.3360 | 0.357274 | 0.20111 |

R_{2} | 0.3510 | 0.3556 | 0.4353 | 0.4345 | 0.477311 | 0.38795 |

X_{2} | 0.3510 | 0.3556 | 0.4353 | 0.4345 | 0.477311 | 0.80380 |

X_{m} | 5.6967 | 6.6072 | 6.4223 | 6.2629 | 7.543194 | 7.87820 |

Manufacturer Data | DE | GA | PSO | |||

Value | $\left|\mathit{e}\right|$ | Value | $\left|\mathit{e}\right|$ | Value | $\left|\mathit{e}\right|$ | |

T_{fl} | 190.902 | 0.902 | 192.788 | 2.788 | 190.453 | 0.453 |

T_{st} | 265.669 | 5.669 | 268.016 | 8.016 | 263.337 | 3.337 |

T_{max} | 349.842 | 20.158 | 354.092 | 15.908 | 363.730 | 6.27 |

pf_{fl} | 0.8065 | 0.0065 | 0.817 | 0.017 | 0.7883 | 0.0117 |

OF | 3.5 × 10^{−3} | 3.5 × 10^{−3} | 6.6 × 10^{−4} | |||

Manufacturer Data | SFLA | MSFLA | SA-ERWCA | |||

Value | $\left|\mathit{e}\right|$ | Value | $\left|\mathit{e}\right|$ | Value | $\left|\mathit{e}\right|$ | |

T_{fl} | 195.106 | 5.106 | 192.197 | 2.197 | 190.001 | 0.001 |

T_{st} | 262.467 | 2.467 | 261.687 | 1.687 | 260.002 | 0.002 |

T_{max} | 368.036 | 1.964 | 373.852 | 3.852 | 370.000 | 0.0004 |

pf_{fl} | 0.7860 | 0.014 | 0.7995 | 0.0005 | 0.8 | 0.0000 |

OF | 1.1 × 10^{−3} | 2.8 × 10^{−4} | 1.6 × 10^{−10} |

**Table 11.**Data of Machine 2 [20].

Parameter | Value | Parameter | Value | Design Variables |
---|---|---|---|---|

P_{n} | 148 HP | T_{max} | 1094.3 N | $0.02\le {R}_{s}\le 0.06$ |

V | 400 V | pf_{fl} | 0.9 | $0.03\le {X}_{sd}\le 0.09$ |

f | 50 Hz | s_{fl} | 0.0077 | $2\le {X}_{m}\le 5$ |

p | 2 | I_{st} | 1527.2 A | $0.005\le {R}_{11}\le 0.030$ |

T_{st} | 847.2 N | I_{fl} | 184 A | $0.05\le {R}_{22}\le 0.2$ |

T_{fl} | 353 N | $0.1\le {X}_{1d}\le 0.2$ $0.04\le {X}_{2d}\le 0.20$ |

Parameter (Ω) | PAMP | MSFLA | SA-ERWCA |
---|---|---|---|

R_{s} | 0.0375 | 0.0377 | 0.037614 |

X_{sd} | 0.0692 | 0.0691 | 0.050454 |

X_{m} | 3.7385 | 3.7475 | 3.767293 |

R_{11} | 0.0109 | 0.0109 | 0.010833 |

R_{22} | 0.1031 | 0.1032 | 0.135273 |

X_{1d} | 0.1424 | 0.1422 | 0.159068 |

X_{2d} | 0.0692 | 0.0691 | 0.112364 |

Manufacturer Data | MSFLA | PAMP | SA-ERWCA | |||
---|---|---|---|---|---|---|

Value | $\left|\mathit{e}\right|$ | Value | $\left|\mathit{e}\right|$ | Value | $\left|\mathit{e}\right|$ | |

T_{fl} | 355.306 | 2.306 | 355.373 | 2.373 | 353.007 | 0.007 |

T_{st} | 847.169 | 0.031 | 846.924 | 0.276 | 847.199 | 0.001 |

T_{max} | 1094.230 | 0.77 | 1094.112 | 0.288 | 094.315 | 0.015 |

pf_{fl} | 0.9005 | 0.0005 | 0.9001 | 0.0001 | 0.8999 | 0.0001 |

I_{fl} | 185.216 | 1.216 | 185.130 | 1.13 | 183.99 | 0.01 |

I_{st} | 1527.262 | 0.062 | 1527.225 | 0.025 | 1527.196 | 0.004 |

OF | 8.07 × 10^{−5} | 8.90 × 10^{−5} | 4.73 × 10^{−9} |

**Table 14.**Data of Machine 3 [64].

Parameter | Value | Parameter | Value |
---|---|---|---|

P_{n} | 75 HP | n_{r} | 1480 rpm |

V | 400 V | T_{max}/T_{fl} | 4.7 |

f | 50 Hz | T_{st}/T_{fl} | 3.8 |

p | 2 | I_{st}/I_{fl} | 5.9 |

Parameter (Ω) | Acceleration Test | SA-ERWCA |
---|---|---|

R_{s} | 0.11691 | 0.10001 |

X_{sd} | 0.10688 | 0.15559 |

X_{m} | 3.2023 | 5.1957 |

R_{11} | 0.03904 | 0.04089 |

R_{22} | 0.38144 | 0.29293 |

X_{1d} | 0.23872 | 0.21717 |

X_{2d} | 0.10688 | 0.01001 |

**Table 16.**Comparison of acceleration test and SA-ERWCA results with manufacturer data for Machine 3.

Marked Data | Value | Acceleration Test | SA-ERWCA | ||
---|---|---|---|---|---|

Value | $\left|\mathit{e}\right|$ | Value | $\left|\mathit{e}\right|$ | ||

T_{max}/T_{fl} | 4.7 | 4.556 | 0.144 | 4.644 | 0.056 |

T_{st}/T_{fl} | 3.8 | 3.529 | 0.271 | 3.788 | 0.012 |

I_{st}/I_{fl} | 5.9 | 6.175 | 0.275 | 5.86 | 0.04 |

OF | 0.1698 | 0.00488 |

**Table 17.**Data of Machine 4 and measured data: three-phase induction motor [45].

Parameter | Value | Measured Data | ||
---|---|---|---|---|

P_{n} | 0.75 kW | Slip | Stator current (A) | Power factor |

V | 380 V | 0.06 | 1.86 | 0.62 |

f | 50 Hz | 0.10 | 2.39 | 0.74 |

p | 1 | 0.15 | 3.07 | 0.78 |

Parameter (Ω) | GA [45] | SA-ERWCA |
---|---|---|

R_{1} | 10.28 | 10.094 |

X_{1} | 8.19 | 9.506 |

R_{2} | 10.48 | 10.238 |

X_{2} | 19.21 | 17.315 |

X_{m} | 143.17 | 141.961 |

Slip | Measured Data | GA [45] | SA-ERWCA | ||

Stator Current(A) | Value | $\left|\mathit{e}\right|$ | Value | $\left|\mathit{e}\right|$ | |

0.06 | 1.86 | 1.8554 | 0.0046 | 1.8591 | 0.0009 |

0.10 | 2.39 | 2.3840 | 0.006 | 2.3921 | 0.0021 |

0.15 | 3.07 | 3.0542 | 0.0158 | 3.0685 | 0.0015 |

Slip | Measured Data | GA [45] | SA-ERWCA | ||

Power Factor | Value | $\left|\mathit{e}\right|$ | Value | $\left|\mathit{e}\right|$ | |

0.06 | 0.62 | 0.6193 | 0.0007 | 0.6203 | 0.0003 |

0.10 | 0.74 | 0.7366 | 0.0034 | 0.7375 | 0.0015 |

0.15 | 0.78 | 0.7812 | −0.0012 | 0.7819 | 0.0019 |

OF | 2.18 × 10^{−4} | 2.31 × 10^{−5} |

**Table 20.**Data of Machine 5 [46].

Parameter | Value |
---|---|

P_{n} | 4 kW |

V | Δ/Y 220/380 |

I_{fl} | Δ/Y 14.2/8.2 A |

pf_{fl} | 0.88 |

n_{r} | 2870 |

**Table 21.**Measured results from [46].

Speed (rpm) | Stator Current (A) | Angle (degree) | Power Factor |
---|---|---|---|

0 | 45.70 | 57.0 | 0.5446 |

2842 | 10.00 | 25.0 | 0.9063 |

2878 | 8.20 | 27.0 | 0.8910 |

2902 | 7.00 | 28.5 | 0.8788 |

2931 | 5.90 | 31.0 | 0.8572 |

2950 | 4.55 | 40.0 | 0.7660 |

2952 | 4.25 | 43.0 | 0.7314 |

2960 | 3.80 | 52.0 | 0.6157 |

2968 | 3.55 | 58.0 | 0.5299 |

2994 | 3.05 | 76.0 | 0.2419 |

Parameter (Ω) | AGA [46] | SA-ERWCA |
---|---|---|

R_{1} | 1.4460 | 1.6794 |

X_{1} | 2.0735 | 1.1164 |

R_{2} | 1.1994 | 1.0372 |

X_{2} | 2.0735 | 3.0241 |

X_{m} | 72.728 | 78.723 |

Speed (rpm) | Stator Current (A) | AGA [46] | SA-ERWCA | |||

Value | $\left|\mathit{e}\right|$ | Value | $\left|\mathit{e}\right|$ | |||

0 | 45.7 | 45.3571 | 0.3429 | 45.5731 | 0.1269 | |

2842 | 10 | 9.3893 | 0.6107 | 10.6616 | 0.6616 | |

2878 | 8.2 | 7.5945 | 0.6055 | 8.5931 | 0.3931 | |

2902 | 7.0 | 6.3968 | 0.6032 | 7.1816 | 0.1816 | |

2931 | 5.9 | 4.9958 | 0.9042 | 5.4826 | 0.4174 | |

2950 | 4.55 | 4.1558 | 0.3942 | 4.4225 | 0.1275 | |

2952 | 4.25 | 4.0738 | 0.1762 | 4.3165 | 0.0665 | |

2960 | 3.80 | 3.7634 | 0.0366 | 3.9089 | 0.1089 | |

2968 | 3.55 | 3.4871 | 0.0629 | 3.5366 | 0.0134 | |

2994 | 3.05 | 2.9566 | 0.0934 | 2.7801 | 0.2699 | |

Speed (rpm) | Angle (degree) | Power Factor | AGA [46] | SA-ERWCA | ||

Value | $\left|\mathit{e}\right|$ | Value | $\left|\mathit{e}\right|$ | |||

0 | 57 | 0.5446 | 0.5318 | 0.0128 | 0.5471 | 0.0025 |

2842 | 25 | 0.9063 | 0.9026 | 0.0037 | 0.9179 | 0.0116 |

2878 | 27 | 0.8910 | 0.8829 | 0.0081 | 0.9075 | 0.0165 |

2902 | 28.5 | 0.8788 | 0.8553 | 0.0235 | 0.8900 | 0.0112 |

2931 | 31 | 0.8572 | 0.7861 | 0.0711 | 0.8409 | 0.0163 |

2950 | 40 | 0.7660 | 0.6947 | 0.0713 | 0.7683 | 0.0023 |

2952 | 43 | 0.7314 | 0.6815 | 0.0499 | 0.7571 | 0.0257 |

2960 | 52 | 0.6157 | 0.6194 | 0.0037 | 0.7025 | 0.0868 |

2968 | 58 | 0.5299 | 0.5398 | 0.0099 | 0.6273 | 0.0974 |

2994 | 76 | 0.2419 | 0.1361 | 0.1058 | 0.1686 | 0.0733 |

OF | 0.4667 | 0.2582 |

Parameter (Ω) | Single Cage | Parameter (Ω) | Double Cage |
---|---|---|---|

R_{1} | 3.2342 | R_{s} | 3.3486 |

R_{2} | 3.5253 | X_{sd} | 0.1004 |

X_{1} | 5.7459 | X_{m} | 195.486 |

X_{2} | 9.2029 | R_{11} | 4.5219 |

X_{m} | 192.2646 | R_{22} | 21.9186 |

X_{1d} | 20.2025 | ||

X_{2d} | 59.8289 | ||

OF | Single cage | 0.0744 | |

Double cage | 0.0178 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ćalasan, M.; Micev, M.; Ali, Z.M.; Zobaa, A.F.; Abdel Aleem, S.H.E.
Parameter Estimation of Induction Machine Single-Cage and Double-Cage Models Using a Hybrid Simulated Annealing–Evaporation Rate Water Cycle Algorithm. *Mathematics* **2020**, *8*, 1024.
https://doi.org/10.3390/math8061024

**AMA Style**

Ćalasan M, Micev M, Ali ZM, Zobaa AF, Abdel Aleem SHE.
Parameter Estimation of Induction Machine Single-Cage and Double-Cage Models Using a Hybrid Simulated Annealing–Evaporation Rate Water Cycle Algorithm. *Mathematics*. 2020; 8(6):1024.
https://doi.org/10.3390/math8061024

**Chicago/Turabian Style**

Ćalasan, Martin, Mihailo Micev, Ziad M. Ali, Ahmed F. Zobaa, and Shady H. E. Abdel Aleem.
2020. "Parameter Estimation of Induction Machine Single-Cage and Double-Cage Models Using a Hybrid Simulated Annealing–Evaporation Rate Water Cycle Algorithm" *Mathematics* 8, no. 6: 1024.
https://doi.org/10.3390/math8061024