Identiﬁcation of Inter-Turn Short-Circuits in Induction Motor Stator Winding Using Simulated Annealing

: This paper presents a method of inter-turn short-circuit identiﬁcation in induction motors during load current variations based on a hybrid analytic approach that combines the genetic algorithm and simulated annealing. With this approach, the essence of the method relies on determining the reference matrices and calculating the distance between the reference matric values and the test matrix. As a whole, it is a novel approach to the process of identifying faults in induction motors. Moreover, applying a discrete optimization algorithm to search for alternative solutions makes it possible to obtain the true minimal values of the matrices in the identiﬁcation process. The effectiveness of the applied method in the monitoring and identiﬁcation processes of the inter-turn short-circuit in the early stage of its creation was conﬁrmed in tests carried out for several signiﬁcant state variables describing physical magnitudes of the selected induction motor model. The need for identiﬁcation of a particular fault is related to a gradual increase in its magnitude in the process of the induction motor’s exploitation. The occurrence of short-circuits complicates the dynamic properties of the measured diagnostic signals of the system to a great extent.


Introduction
The basic issue in the exploitation of different types of devices, machines, or technical systems is to provide continuous and failure-free operation. This reliability is the key requirement in the growing industry 4.0 era. Technical diagnostics is a tool to provide failure-free operation of technical objects. Diagnostics is a process in which the actual condition of an object is evaluated and, based on the result, one decides about its further exploitation or decides to subject it to a repair process. The process of technical condition assessment of electric machines in industry or in transport or other application fields includes the detection and identification unwanted operational states. The set of operational states is a series of deliberate actions performed by the machine in a specified period. It is required that the diagnostic system detects and identifies the occurrence of faults in real time or in early stages of their creation [1]. Nowadays, these requirements can only be satisfied for the systems built using high-tech technology, based on microprocessor techniques and software that implements effective methods of electric machine diagnostics. In complex electro-mechanical objects, the analysis and classification of diagnostic signals in the time-frequency domain is performed using transformation methods [2]. In recent years, it has been possible to observe an increase in neural network applications in technical diagnostics. For a wide range of neural network applications in technical diagnostics, it is worth pointing out their application in the modeling, identification, emergency state decision making and detection of faults occurring in complex processes and objects. The − The identification of inertia on the induction motor's shaft from the analysis of wavelet scalograms performed using a clustering method based on the k-mean harmonics technique [18] or other methods described in the literature [19][20][21][22][23][24][25][26][27].
The paper introduces a new method for fault diagnosis based on the results of the calculated differences between the values obtained from the identification algorithm and those of the reference matrix values determined for a selected group of tests. The appropriate values were obtained by means of the simulated annealing and genetic algorithm.
The applied genetic algorithm presents a new method of determining the length of the binary chain of individuals in the population by means of calculations performed for the obtained results representing the grouping appropriately normalized measurement data.
The developed method used in identifying interturn short-circuits makes it possible to use the obtained results of computer simulations in order to carry out an accurate analysis of the values of physical quantities recorded during the laboratory measurements of an induction motor.
The use of technology based on AI supported by a discrete optimization algorithm in the analyzes of the tested signals made a large contribution in terms of the obtaining of correct results in the process of identifying interturn short-circuits.
The remaining part of this paper is organized in the following way: the second part presents the development of the connection of the system's elements during the performance of simulation tests with various cases of inter-turn short circuits. The parameters of the examined induction motor model include the physical magnitudes investigated in the performed simulation tests and the variation of the factor of fault occurrence in the performed simulation tests. The third part contains the description of the developed fault identification algorithm, with a juxtaposition of the required parameters for the testing of the selected diagnostic method and the method of calculation of the factor required for fault assessment.
In the next section, laboratory tests and descriptions of the obtained results are presented. The conclusions are contained in the last part of this paper.

Materials and Theoretical Basis
Briefly, the detection of interturn-faults at an early stage is an important task in the maintenance of electrical machinery in the power industry. This problem has not been fully researched and resolved to this day. Therefore, it is still an interesting research issue.
In order to develop a more effective algorithm for the early detection of interturn short-circuits, experimental tests were carried out during which the measurement data were obtained for the development of an effective diagnostic algorithm.

Description of the Identification Process of the Inter-Turn Short-Circuit in the Induction Motor Model
The object of interest was the Sg-112M-4 cage induction motor, with the following parameters: P n = 4.0 kW, U n = 380 V, I n = 8.6 A and n n = 1435 rpm. The motor was mechanically coupled to the PZM5545 DC generator by the clutch. Figure 1 shows the DC generator with an autotransformer and a bridge rectifier used for the induction motor's load changing. The load of the DC generator comprised two resistive heaters with an overall power of 4 kW. Load torque was measured with the Data FLEX 22/50 torque meter. The torque meter also outputted a voltage signal that was proportional to the measured rotational speed [19]. The DC generator, which was used as a load for the tested motor, had the following parameters: Pn = 4.5 kW, Un = 230 V, In = 19.6 A, nn = 1450 rpm and If = 0.86 A. Figure 2 shows a wiring diagram of the turns in one stator coil of the tested motor. The windings were connected in wye configuration and supplied from a three-phase low-voltage network. The winding number from which the turn was wired to the terminal on the terminal board on the machine's housing is marked with a proper number in the diagram. Short circuiting of the selected turns occurred when the proper terminals were connected to a resistor and an ammeter, which was used for current control.  In order to decrease the current and provide long-term protection of the motor against overheating, an additional resistor, of resistance equal to 5 ÷ 6 Ω, was used. The additional resistor had no impact on the effects of faults that could be observed for different values of components of the measured diagnostic signals [27]. Simulation tests were carried out in five test groups with the following values of the load current Iload: 1 A, 2 A, 3 A, 4 A and 5 A. Each test group contained eight different cases of inter-turn short-circuiting. Test results for all physical magnitudes and for each case of short circuiting were The DC generator, which was used as a load for the tested motor, had the following parameters: P n = 4.5 kW, U n = 230 V, I n = 19.6 A, n n = 1450 rpm and I f = 0.86 A. Figure 2 shows a wiring diagram of the turns in one stator coil of the tested motor. The windings were connected in wye configuration and supplied from a three-phase low-voltage network. The winding number from which the turn was wired to the terminal on the terminal board on the machine's housing is marked with a proper number in the diagram. Short circuiting of the selected turns occurred when the proper terminals were connected to a resistor and an ammeter, which was used for current control. The DC generator, which was used as a load for the tested motor, had the following parameters: Pn = 4.5 kW, Un = 230 V, In = 19.6 A, nn = 1450 rpm and If = 0.86 A. Figure 2 shows a wiring diagram of the turns in one stator coil of the tested motor. The windings were connected in wye configuration and supplied from a three-phase low-voltage network. The winding number from which the turn was wired to the terminal on the terminal board on the machine's housing is marked with a proper number in the diagram. Short circuiting of the selected turns occurred when the proper terminals were connected to a resistor and an ammeter, which was used for current control.  In order to decrease the current and provide long-term protection of the motor against overheating, an additional resistor, of resistance equal to 5 ÷ 6 Ω, was used. The additional resistor had no impact on the effects of faults that could be observed for different values of components of the measured diagnostic signals [27]. Simulation tests were carried out in five test groups with the following values of the load current Iload: 1 A, 2 A, 3 A, 4 A and 5 A. Each test group contained eight different cases of inter-turn short-circuiting. Test results for all physical magnitudes and for each case of short circuiting were In order to decrease the current and provide long-term protection of the motor against overheating, an additional resistor, of resistance equal to 5 ÷ 6 Ω, was used. The additional resistor had no impact on the effects of faults that could be observed for different values of components of the measured diagnostic signals [27]. Simulation tests were carried out in five test groups with the following values of the load current I load : 1 A, 2 A, 3 A, 4 A and 5 A. Each test group contained eight different cases of inter-turn short-circuiting. Test results for all physical magnitudes and for each case of short circuiting were stored in matrix X 1[8,500,000] . The elements of matrix X 1 were defined for each value of load current I load .
In the proposed diagnostic method, simulation tests were run for the following physical magnitudes:

Description of Diagnostic Algorithm Used in Physical Magnitude Signal Processing Using the Genetic Algorithm of Simulated Annealing
In the identification tests performed for each case of inter-turn short-circuiting, the matrix X 3 values obtained from normalization of matrix X 1 values were used, followed by sorting of the elements of matrix X 2 into the right order.
Simulations were run with 50 kHz sampling frequency, and the simulated period was 10 s.
Elements of matrix X 3 , obtained from sorting the elements of matrix X 2 in descending order for all examined physical magnitudes, were used in the diagnostic procedure.
This means that in the simulation tests, the values of matrix X 3 were defined as follows: Moreover, two types of identification tests using normalization of the elements of matrix X 1 were applied: − Tests performed for the assumed first four cases of inter-turn short-circuiting using the parameters defined for acoustic pressure p s ; − Tests performed for the assumed last four cases of inter-turn short-circuiting using the parameters defined for axial flux φ 1 .
The reference matrix elements values and the values of examined matrix calculated by the genetic algorithm of simulated annealing were used in identification tests in the diagnostic procedure.
The values of reference matrix were obtained after performing calculations in the genetic algorithm using the values of matrix X 2 defined in test group for load current The values of matrix X 2 elements were obtained by limiting the values of matrix X 1 to the range [a 3 , a 4 ] for all investigated physical magnitudes according to the following formula: where: X 1 -values of the matrix elements; a 1 -minimum value of the matrix X 1 elements defined in the test; a 2 -maximum value of the matrix X 1 elements defined in the test; a 3 -initial value of the range containing normalized values of matrix X 1 calculated in tests for acoustic pressure p s and axial flux φ 1 ; a 4 -end value of the range containing normalized values of matrix X 1 calculated in tests for acoustic pressure p s and axial flux φ 1 ; i-the number of the investigated cases of inter-turn short-circuits. The values of variables a 1 , a 2, a 3 and a 4 were calculated using the following formulas: The variables a 5 and a 6 were used in the calculation of matrix X 2 . The values of these variables were arithmetic means and standard deviations calculated in tests for acoustic pressure p s and axial flux φ 1 using the following formulas: The values of tested matrix were calculated by applying the genetic algorithm, which used the matrix X 2 values obtained in the test for a given load current I load .
Calculation of the variables a 5 and a 6 was performed after the following conditions for the selected physical magnitudes, such as acoustic pressure p s and axial flux φ 1 for the load current I load = 3 A, were met: a 7 > a 8 ; f or i ∈ 1, 4 (9) where: i-the number of the investigated inter-turn short-circuit case. The variables a 7 , a 8 , a 9 and a 10 were calculated using the following formulas: The values of matrix X 4 were defined according to the formula given below: The genetic algorithm with simulated annealing was used for the proposed diagnostic method.
The simulated annealing algorithm is an iterative method that searches the space of alternative solutions to a problem in order to find an optimal solution. The solution depends on the obtained difference in values between the old and the new solution and the current temperature value. The temperature value is constantly lowered and it is possible to adjust the selection of successive approximations of the solutions to the problem. The probability of choosing the best solution increases with the decreasing of the temperature value in subsequent iterations of the simulated annealing algorithm. Over time, the results stabilize and the subsequent changes are not accepted.
In the applied genetic algorithm, normalization of individuals was performed using the values of the matrix X 6 elements.
The values of the matrix X 6 elements were calculated by summing the consecutive elements of matrix X 5 .
The values of matrix X 3 were used in calculations of the sequence values stored in matrix X 5 .
The values of matrix X 6 were the results of calculations of five elements according to the formula: where: X 5 -the values of the matrix elements calculated in the test. Matrix X 6 contained the successively summed arithmetic means of the respective values of the X 3 matrix. This operation increased the range of changes for the values of the matrix X 3 elements for different cases of inter-turn short-circuiting. The values of matrix X 5 can be effectively used in identification process of investigated faults.
The values of the arithmetic means m 1 , m 2 , m 3 , m 4 and m 5 were calculated as follows: Calculation of reference matrix elements values used in identification of inter-turn short-circuits was a crucial step in the presented diagnostic method. The calculations in the presented diagnostic procedure were performed according to the genetic algorithm with simulated annealing [14]. A block diagram containing the order of the calculations run in the genetic algorithm is presented in Figure 3.  For the applied genetic algorithm, it was crucial to specify an objective function. The choice of objective function was made using a series of simulation tests performed for various objective functions.
The correct results of the proposed diagnostic procedure were observed for F6 Schaffer's function with two variables [14].
This function was chosen as an objective function for the applied genetic algorithm. In this way, a representation of an individual in the population, for which each of the two variables were encoded as a binary string of specified length, needed to satisfy the requirements of accuracy [14].
The operation of the genetic algorithm with simulated annealing started with the initialization of the population individuals.
The initial values of the individuals were obtained by randomizing binary strings containing values of 0 or 1.
Binary strings created in this way for the first variable used in objective function calculation were stored in matrix X 7 .
In the case of the second variable used in objective function calculation, the binary strings were stored in matrix X 8 .
The overall length of binary string for a population individual was obtained from summing the number of bits of each variable binary string used in objective function calculation.
Based on the results obtained for a series of simulation tests, the population size used in the identification tests performed using the genetic algorithm with simulated annealing was set to 25 or 50.
When the stop criterion of the genetic algorithm was achieved, the processing of population individuals in the next generations was stopped.
In each iteration (generation), a random selection of a binary string of one k-th individual and a new binary string l from the vicinity of the k-th individual out of all population individuals was performed. For a randomly selected binary string of the l-th individual from the vicinity of the k-th individual, one random bit was changed [14]. A block diagram of the proposed GA-SA algorithm is presented in Figure 3.
In this proposed algorithm, the value of the objective function of the l-th individual new binary string is greater than the value of objective function of the k-th individual, then the binary string of the k-th individual is changed accordingly. Otherwise, a function returning a random number in a range of [0,1] is called and one checks if the obtained value is lower than the probability distribution of the new binary string of the l-th individual.
If the random number satisfies this condition, an appropriate exchange of k-th individual binary string occurs [14].
The end condition that is used in simulated annealing checks if the probability distribution of the newly selected string for the l-th individual is similar to the Boltzmann distribution.
In this way, the so-called thermal equilibrium was defined. Temperature T is decreased in each iteration of the genetic algorithm [14].
In order to lower the temperature, logarithmic cooling scheme is used [4,20]. For varying values of the adapted parameter, this scheme improves the quality of the results of inter-turn short-circuit identification.
In order to realize the above-described steps of the simulated annealing procedure, the operations described below were performed.
For a new l-th individual, singular bits in binary strings used for objective function calculation were randomly selected and created in the following way: 1 − X 7(l,j) ; i ∈ 1, 8 ; j = random < 1, s 1(i) >; k = l; l = random 1, n ; n ∈< 25, 50 > X 8(l,j) = 1 − X 8(l,j) ; i ∈ 1, 8 ; j = random < 1, s 2(i) >; k = l; l = random 1, n ; n ∈< 25, 50 > where: s 1 -the number of bits s 1 calculated in the test; s 2 -the number of bits s 2 calculated in the test; n-the number of population individuals. The numbers of bits s 1 and s 2 used in the procedure were defined by two variables, a 17 and a 18 . The values of these variables were obtained by increasing, with a certain accuracy, the variables a 11 and a 12 .
The values of variables a 11 and a 12 create a range of variation that is used in the calculation of the length of bits for variables used in the objective function calculation.
The values of these variables were obtained by calculating the differences between the maximum and minimum values of appropriate elements of the X 6 matrix. In the calculations, the following formulas were applied: where: a 13 , a 15 -the minimum values of the matrix X 6 elements; a 14 , a 16 -the maximum values of the matrix X 6 elements. The values of variables a 13 , a 14 , a 15 and a 16 were defined in the following way: a 14(i) = max X 6(i)(1) , X 6(i)(2) ; i ∈ 1, 8 (28) a 15(i) = min X 6(i)(4) , X 6(i)(5) ; i ∈ 1, 8 (29) a 16(i) = max X 6(i)(4) , X 6(i)(5) ; i ∈ 1, 8 The calculated values of variables a 11 and a 12 defined a range of variation of variables used in the calculation of the objective function performed in the following generations.
The first number of bits for a binary string of a population individual calculated for each generation corresponded to a variable from the range [a 13(i) , a 14(i) ]. The second number of bits for a binary string of a population individual calculated for each generation corresponded to a variable from the range [a 15(i) , a 16(i) ].
The obtained variation ranges a 11 and a 12 were used with a specified accuracy equal to 10 5 in calculations of the variables a 17 and a 18 that are required when defining the number of bits of applied binary strings according to the following formulas: Using the calculated values of variables a 17 and a 18 , one can define the number of bits s 1 and s 2 using the following formulas: The values of binary strings of one k-th individual and one new l-th individual were changed to values in the decimal system using the formulas given below: An important step in the diagnostic procedure was the defining of the most advantageous variations of values of the k-th and l-th individuals in the population, calculated using the variables a 13 , a 14 , a 15 and a 16 using the formulas given below: a 25(i) = a 13(i) + a 21(i) · a 14(i) − a 13(i) The obtained values of the variables a 23 , a 24 , a 25 and a 26 were used in the calculations of objective function values.
The values of the objective function for the k-th and l-th individuals were calculated using the formulas below: a 28(i) = 0.5 + sin 2 a 25(i) 2 + a 26(i) 2 − 0.5 where: a 27 -the objective function value of the k-th individual; a 28 -the objective function value of the l-th individual.
The binary string of the k-th individual changed when one of the two conditions presented below was satisfied: where: b 1 -a random value from the range [0,1] defined in the test; T-the temperature value for the current iteration of the genetic algorithm; m-the number of the current iteration of the genetic algorithm. The exchange of binary strings of one k-th individual was realized in the following way: . . s 1(i) ; k, l = random 1, n ; k = l; n ∈< 25, 50 > (47) . . s 2(i) ; k, l = random 1, n ; k = l; n ∈< 25, 50 > The stop criterion of the genetic algorithm was defined as shown below: where: a 29 -the value of the variable required for the stopping of the genetic algorithm (experimentally set in the test).
The value of temperature T in the following iterations m of simulated annealing decreased according to the logarithmic cooling scheme, as shown in the formula below [4,20]: where: λ-the value of variable adapted for the logarithmic cooling scheme calculated in the test.
In the following iterations m of simulated annealing, the value of the adapted variable λ was calculated using the formula given below [4,20]: where: δ-the parameter representing the accuracy of the equilibrium state for the logarithmic cooling scheme defined in the range (0.1,10); σ-the value of the standard deviation of the objective function for all individuals generated at a given temperature and calculated in each iteration of simulated annealing m.
The standard deviation σ of objective function in each iteration m of simulated annealing was calculated in the following manner: where: F 1 -the value of the objective function for a specified population individual, calculated in the test for each iteration m of the genetic algorithm; m 6 -the arithmetic mean of the objective function of all individuals calculated for each iteration m of the genetic algorithm.
The values of objective function F 1 of all population individuals were calculated using the formula presented below: where: Calculations of the elements' values of matrices F 2 and F 3 were performed using the following formulas: where: a 13 , a 15 -the minimum values of the matrix X 6 elements defined by formulas (27) and (29); a 14 , a 16 -the maximum values of the matrix X 6 elements defined by formulas (28) and (30); X 9 -the values of the matrix containing binary strings of population individuals in a decimal system; X 10 -the values of the matrix containing binary strings of population individuals in a decimal system.
The values of the X 9 and X 10 matrices were calculated using the formulas shown below: Identification of different cases of inter-turn short-circuiting was performed using the genetic algorithm according to the order defined in matrix K 1 .
For all examined physical magnitudes in the applied diagnostic procedure, reference matrix calculation was performed for eight assumed cases of inter-turn short-circuiting.
In a test group for the load current I load = 3 A, the values of the reference matrix were calculated and stored in matrices N 1 and N 2 of size 8 × 3. Tested values were stored in matrices N 3 and N 4 of size 1 × 3.
Reference and tested matrices were derived from matrix X 12 after the genetic algorithm had finished its operation. This matrix contained the values obtained by sorting the elements of matrix X 11 in descending order.
The values of matrix X 11 were obtained by changing the entire binary string of a population individual that was transforming the binary strings X 7 and X 8 to the decimal system. The calculations were performed according to the formula: The values of X 12 were defined as: (2) . . . . ≥ X 11(i)(n) ; a = 1, 2 . . . n; i ∈ 1, 8 ; n ∈ 25, 50 The values of the matrix X 12 elements were stored in matrices containing the reference values N 1 and N 2 and in matrices containing the test values N 3 and N 4 after the genetic algorithm finished its operation. The values of matrices N 1 , N 2 , N 3 and N 4 were calculated using the formulas below: The results obtained from identification tests: − Were stored in the reference matrix N 1 and in the tested matrix N 3 for the normalization of matrix X 1 , which was performed using the parameters calculated for axial flux. − Were stored in the reference matrix N 2 and in the tested matrix N 4 for the normalization of matrix X 1 , which was performed using the parameters calculated for acoustic pressure p s .
The calculation of the values of matrices H 1 and H 2 made it possible to perform the correct identification of cases of inter-turn short circuiting for the calculated diagnostic signals of the examined physical magnitudes at the specified load current I load .
The calculations of the values of matrices H 1 and H 2 were performed using the Manhattan metric as shown below: In general, the calculations of matrix H 1 are performed directly after the genetic algorithm stops. After calculating the index nr 3 , it may be necessary to calculate the values of matrix X 2 . The aim is to obtain the index nr 3 ∈ < 1, 4 >; the calculated values of matrix H 1 might point to an incorrect number of inter-turn short-circuit case. Otherwise, the index nr 3 (for nr 3 ∈ < 5, 8 >) defined in matrix H 1 makes it possible to calculate the column number in matrix K 1 containing the right case of inter-turn short-circuiting for the examined induction motor model.
In this study, the index nr 3 was defined using the minimum value of the matrix H 1 as shown below: This means that column number i in matrix K 1 corresponded to calculated index nr 3 (i = nr 3 ).
Generally, when obtaining index nr 3 ∈ < 1, 4 >, the calculation of index nr 4 in matrix H 2 is performed. This allows for the examined induction motor model to define the correct case of inter-turn short circuiting out of four first assumed cases. The calculated index nr 4 (i = nr 4 ) corresponds to the column number of matrix K 1 .
Here, by defining the minimum value of matrix H 2 , index nr 4 was defined.

Results of Research on Diagnostic Algorithm Application in the Identification of Inter-Turn Short-Circuiting in Induction Motor Models
In Tables 1-4, several cases of the inter-turn short-circuits that were investigated in the identification process are contained in the column labelled Test parameters and also the correct results of the calculation of the matrices H 1 and H 2 obtained during the identification process of inter-turn short-circuiting are marked in bold. In the performed identification tests, the initial temperature of the genetic algorithm with simulated annealing was equal to T = 100. Based on the simulation results in Tables 1-4, one can see that for all investigated physical magnitudes, it is possible to obtain correct results in the identification of shortcircuit cases when using appropriate values of the variable a 29 and the accuracy parameter δ.
Using the variations of test parameters in inter-turn short-circuit identification process allows for the obtaining of improvements in the calculation results for matrices H 1 and H 2 .
In addition, it can be concluded on the basis of the results presented in Tables 1-4 that increasing the population size and the precision parameter δ increases the scope of both the search for solutions for the variable neighborhood of the individual k and the random selection of binary chains of individuals from the neighborhood of the individual k. Thus, the occurrence of the stochastic process during the operation of the simulated annealing algorithm increases intensively, thus enabling the reduction in the number of iterations as well as the calculated minimum value in the matrices H 1 and H 2 , respectively.
In addition to this and based on the results in Tables 1-4, it can be stated that increasing the population size and the accuracy parameter value leads to an augmenting of the solution search range within a varied neighborhood as well as a random selection of binary chains of individuals from within the neighborhood of the individual k. As a result, the probability of a stochastic process occurring increases dramatically, thus decreasing the number of iterations and the computed minimal values in the matrices H 1 i H 2 , respectively.
Next, Figure 4 shows the impact of changes in the population size and the accuracy parameter δ on the convergence of the proposed genetic algorithm. Next, Figure 4 shows the impact of changes in the population size and the accuracy parameter δ on the convergence of the proposed genetic algorithm.   Tables 1 and 2) and the calculated minimum values of the H2 matrix (see Tables 3 and 4).
Note that there were significant decreases in the minimum values calculated in the matrices H1 and H2, respectively, for the same investigated cases of the interturn short circuits, as well as in the obtained number of iterations occurring when there were large increases in the population size and the accuracy parameter value δ.

Conclusions
Based on the results of this research, one can see that in the proposed diagnostic method, the extraction of information from obtained time series for the investigated physical magnitudes of induction motors leads to increased fault detection and identification capabilities.
All variables used in the genetic algorithm provide information about fault occurrence symptoms. The assumed order of simulated annealing operations for each examined induction motor model makes it possible to efficiently solve the problem of inter-   Tables 1 and 2) and the calculated minimum values of the H 2 matrix (see Tables 3 and 4).
Note that there were significant decreases in the minimum values calculated in the matrices H 1 and H 2 , respectively, for the same investigated cases of the interturn short circuits, as well as in the obtained number of iterations occurring when there were large increases in the population size and the accuracy parameter value δ.

Conclusions
Based on the results of this research, one can see that in the proposed diagnostic method, the extraction of information from obtained time series for the investigated physi-cal magnitudes of induction motors leads to increased fault detection and identification capabilities.
All variables used in the genetic algorithm provide information about fault occurrence symptoms. The assumed order of simulated annealing operations for each examined induction motor model makes it possible to efficiently solve the problem of inter-turn short-circuit identification in the system using binary strings of population individuals. The correct calculation of these values is achieved through the use of the genetic algorithm with simulated annealing.
Proper functioning of diagnostic systems with effective selectivity of dynamic state identification requires the use of reference matrices derived from repetitive analyses of the examined induction motor models.
In the presented diagnostic algorithm, an effective identification of the dynamic states of the examined induction motor model is possible when using the created reference matrices.
In addition, one may state that an effective solution by means of the proposed diagnostic method involves the determination of the length of the binary chain/sequence of individuals in the population using math operations carried out based on the values of the arithmetic means calculated for the elements of the appropriate matrices and obtained as a result of the applied grouping and normalization of the values of the tested physical quantities recorded during the performed measurements.
In comparison to the GA-based methods used in the early-stage diagnostics of shortcircuit faults in the stators of induction motors [5,9], the method that uses a hybrid GA-PSO algorithm to determine the parameters of an induction motor (HGAPSO) [21], the DE methods used for the estimation of the electrical and mechanical parameters of three-phase induction motors [1] or the hybrid method based on the simulated annealing algorithm and the evaporation rate of the water cycle (SA-ERWCA) used to estimate the parameters of the reduced-order circuit of an induction machine [10], the presented algorithm, designed for the identification of interturn short-circuits, makes it possible to: − Obtain correct results when searching for a local optimal solution by calculating the length of the binary sequence of individuals with the parameters obtained by means of clustering in a fixed order and normalization of the data recorded via laboratory measurements and the adopted Schaffer F6 objective function; − Ensure the optimization of performance in terms of the quality of the results obtained via the identification process upon applying changes within the appropriate range of parameters of the genetic algorithm (population size, initial temperature value, accuracy parameter and convergence criterion); − Solve continuous and discrete optimization problems by ensuring a stochastic process during the GA operation. Funding: This research: which was carried out under the theme Institute E-2, and was funded by the subsidies for scientific research granted by the Polish Ministry of Science and Higher Education.
Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.

Conflicts of Interest:
The authors declare no conflict of interest.