Start-Up and Steady-State Regimes Automatic Separation in Induction Motors by Means of Short-Time Statistics

Abstract

Induction motors are widely used machines in a variety of applications as primary components for generating rotary motion. This is mainly due to their high efficiency, robustness, and ease of control. Despite their high robustness, these machines can experience failures throughout their lifespan due to various mechanical, electrical, and environmental factors. To prevent irreversible failures and all the implications and costs associated with breakdowns, various methodologies have been developed over the years. Many of these methodologies have focused on analyzing various physical quantities, either during start-up transients or during steady-state operations. This involves the use of specific techniques depending on the focus of the methodology (start-up transients or steady-state) to obtain optimal results. In this regard, it is of great importance to develop methods capable of separating and detecting the start-up transient of the motor from the steady state. This will enable the development of automatic diagnostic methodologies focused on the specific operating state of the motor. This paper proposes a methodology for the automatic detection of start-up transients in induction motors by using magnetic stray flux signals and processing by means of statistical indicators in time-sliding windows, the calculation of variances with a proposed method, and obtaining optimal values for the design parameters by using a Particle Swarm Optimization (PSO). The results obtained demonstrate the effectiveness of the proposed method for the start-up and steady-state regimes automatic separation, which is validated on a 0.746 kW induction motor supplied by a variable frequency drive (VFD).

1. Introduction

Electric motors are widely used across industries due to their robustness, efficiency, and ease of control. According to data from the ABB Group [1], approximately 300 million electric motors are installed worldwide, with an annual increase of 10%. These motors account for 28% of global electricity consumption. Moreover, 90% of these motors operate continuously at maximum speed. Keeping them in good working condition to ensure operational efficiency and to avoid costly downtime is an essential task. Although electric motors are considered robust machines, they are susceptible to failures due to various factors such as environmental conditions, operational stresses, and insufficient maintenance. Moreover, research by García et al. [2] indicates that the performance of these machines can be adversely affected by such malfunctions, leading to elevated energy consumption costs. If left unchecked, these failures can develop into irreversible damage, directly impacting the kinematic and production chains in which they operate. Therefore, it is necessary to develop techniques for monitoring and detecting faults in electric machines using different physical quantities inherent to the motors, such as current analysis, mechanical vibrations, acoustic signals, infrared imaging, and stray magnetic flux, among others [3]. In recent years, two types of methodologies have been developed for detecting faults in induction motors: on the one hand, there are those focused on the steady state, and on the other hand, there are those focused on the transient state. Within these methodologies, different techniques have been applied depending on the working regime of the analyzed machine (i.e., transient state or steady-state) [4]. Then, there arises the need to develop automatic diagnostic strategies focused on the specific operating state of the motor, as it is not feasible for a specialist technician constantly to monitor the machine for faults. In addition, this type of diagnostic practice allows for efficiency in spare parts, repair times, and monetary resources by obtaining a clear idea about the state of the motor in real-time and by knowing the evolution of any existing breakdown in it and predicting the level of severity [5].

Concerning the aforementioned, some methodologies have been developed in order to diagnose faults in electric machines. For example, the authors in [6] developed an online monitoring system for diagnosing faults, utilizing Motor Current Signature Analysis (MCSA) and the Extended Park’s Vector Approach (EPVA). By combining these two techniques, they successfully detected faults and diagnosed the condition of the stator and rotor in electric motors. Similarly, the authors in [7] developed a method to detect incipient mechanical faults in induction motors through vibration analysis using the Fast Fourier Transform (FFT) and artificial intelligence systems. This method includes a neural network algorithm that can identify common faults such as imbalance, misalignment, and bent shafts, and determine the severity of the problem based on the amplitude of vibration components. Likewise, the authors in [8] explored fault detection in 6 kV induction motors used for water pumping. Their analysis focused on motor current signals and magnetic stray flux, which proved to be a more sensitive alternative to many MCSA misdiagnoses. For instance, they highlighted the misinterpretation of faults in rotor air cooling ducts, where broken bars were detected within the motor. Additionally, in [9] the authors describe the use of MCSA and FFT to detect and analyze faults in induction motors caused by unsymmetrical supply voltages and broken rotor bars.

Although techniques based on steady-state signal analysis, mainly using the FFT algorithm, have proven highly effective in detecting diverse faults appearing in induction motors, the sensitivity of this technique poses challenges in identifying other types of faults [10]. It can lead to erroneous diagnostics when analyzing the machine’s steady regime, since the FFT-based diagnosis focuses solely on specific amplitudes and peaks, which can be masked by unrelated phenomena such as axial air ducts and oscillating loads [11]. In recent years, innovative techniques have been developed for analyzing various types of signals in various motor operating regimes, including the start-up transient. This is of great importance as several faults have been reported in the literature that cannot be detected in the steady-state or that tend to generate similar fault frequencies between one or more faults, which makes them difficult to distinguish. However, if the signals are analyzed during the start-up transient, there is a clear difference between the time-frequency pattern, thus allowing false diagnoses to be ruled out and showing a more robust behavior than in the steady state. Thus arises the need to develop an automatic method that allows for the separation of the start-up transient state—acting as the first step in an automatic methodology based on the diagnosis of transient state faults [12]. In this regard, Daviu [13] provides an overview of monitoring electric motors under transient conditions, noting that transient signals analysis has provided valuable diagnostic information, helping to eliminate the false interpretations that may arise with traditional methods. This research concludes that this new method measures and analyzes the corresponding variables using time-frequency transforms. The results of this analysis provide reliable patterns of failure, which are highly unlikely to be caused by other phenomena unrelated to the anomaly. Various investigations, such as [14], have been conducted for fault detection in induction motors using the analysis of different physical quantities during transient start-up. Among the analyzed techniques are Gabor distribution (Time-Frequency Distribution of Gabor), Morlet time-frequency scalogram (Time-Frequency Morlet Scalogram), multiple-signal classification (MUSIC), and Fast Fourier Transform [15]. Consequently, it can be observed that the diagnosis and interest in the analysis of faults in electric motors during transient states have increased in recent years, aiming to address false fault diagnoses and to detect one or more faults in the study of a single magnitude. An example of this kind of methodology is the paper presented in [16] where a study of several works is carried out. The authors base their research on the analysis of motor transient current signatures (MTCSA), demonstrating and highlighting its effectiveness in the diagnosis of faults in this regime. Similarly, in [17] an array of convolutional neural networks (CNN) and the analysis of the current signature during the transient state are used to achieve the automatic detection of broken rotor bars considering different severity levels and highlighting the importance of analyzing the transient regime for applications where the operating regime varies continuously, or in situations where it is required to diagnose the equipment before experiencing a prolonged uptime. Pointing out the diagnosis of faults in transient regimes is the work of Fernandez [18], who mentions how the detection of faults in induction motors operating in non-stationary regimes has become a necessity in the current industry due to the large number of motors fed by inverters. Then, to carry out this diagnosis successfully, it is necessary to use appropriate techniques. These mathematical techniques belong to the time-frequency decomposition (TFD) tools field and must be applied only in the transient state to obtain optimal results. In conclusion, as observed in the works above, it is necessary to apply specialized techniques depending on whether the machine is operating in a steady state or in a start-up transient state in order to have an automatic methodology. Therefore, it is necessary to have a methodology/technique that allows for isolating the signal obtained during the start-up transient from that obtained in the steady state in order to be able to apply the appropriate techniques.

Throughout this section we discussed several methodologies that have proposed using different processing tools in this field. Currently, a way to adapt these technologies to problems related to electric machines and to obtain optimal results has been sought, such as bio-inspired optimization algorithms. See, for example, the work reported in [19] where the authors propose a method for motor fault detection using the wavelet transform (WT), combining a back propagation neural network with a particle swarm optimization (PSO) algorithm analyzing motor current signals with bearing damage, winding damage, and bar breakage—obtaining results close to 97% of correct classification. Another work is shown in [20], using a model reference adaptive system and simulated annealing particle swarm optimization (MRAS-SAPSO) to identify the electrical parameters of the motor, such as stator winding resistance, transverse inductance, and magnetic linkage—obtaining simulation values very close to the real ones. Similarly, in [21], the authors employ an improved particle swarm optimization method (IPM) to facilitate the parameterization of active disturbance rejection controllers in a 2-motor control system, thus improving the response time and robustness of the system. As it was possible to observe in these works, this type of bio-inspired algorithm has been of great help in the diagnosis of faults in different electrical machines, allowing an increase in the percentages of effectiveness of diverse methodologies in development. Due to the interest in studying the transient state of an induction motor, discriminating between steady-state and transient start-up has become necessary. While the literature includes studies on the analysis of start-up transient state, this regime is often manual or relies on known motor start-up conditions due to the employed technologies. In this regard, Valtierra et al. [22] propose an expert system for early detection of broken bars in induction motors, including half-broken bars and one and two broken bars, by analyzing current signals and considering transient start-up and steady-state to enhance result reliability. In the study, different methodologies focused on diagnosing faults applied only in the transient state have been observed, solving a latent need for automatic diagnosis. This work applies an automatic detection and separation of transient start-up and steady-state, using the Hilbert transform and particle swarm optimization method to separate both current regimes. However, there is not yet a technique that allows for the automatic discrimination of the transient state to which to apply these methodologies, which would be the first step toward achieving a fully automated methodology. Unfortunately, while isolating the transient state of an electric motor is crucial for analysis, few works in the literature have been conducted to detect this regime automatically, and most need to specify the technology used for motor start-up. Since the transient regime fluctuates and is unique to each specific motor, developing a novel methodology for automatically detecting start-up transient would be convenient. The main contribution of this paper is a novel methodology for automatically detecting start-up transients in induction motors using magnetic stray flux signals and their processing through the calculation of statistical indicators in sliding time windows, the calculation of variances with a proposed method, and obtaining optimal values for the design parameters by Particle Swarm Optimization (PSO).

2. Materials and Methods

2.1. Magnetic Stray Flux

The researchers in [23] describe stray flux as the magnetic flux emitted outside a machine’s boundaries, produced by the currents in both the stator and the rotor. These currents, and thus the stray flux, change when there is a fault in the electric motor. As noted in [24], the stray flux can be dissected into two magnetic elements: axial and radial. The axial stray flux originates from the currents in either the end windings of the stator or in the cage of the end ring, occurring along the axis of the machine. Conversely, the radial stray flux exists in a plane orthogonal to the axis. According to the findings in [25], it is feasible to detect stray flux components—individually or combined—by positioning sensors appropriately on the motor’s frame. Illustrated in Figure 1, the presumed pathways of the axial and radial field magnetic lines are shown along with the sensor positions for their detection. The sensor at position 1 is for the axial field magnetic, at position 3 for the radial field magnetic, and at position 2 for capturing both types of stray flux.

2.2. Statistical Indicators

The primary objective of employing statistical indicators is to extract relevant information on the behavior of signals in the time domain. When a system operates under various conditions, its signals manifest through diverse statistical parameters. It is anticipated that a state of failure will be evident in these characteristics or indicators [26]. A primary goal of using statistical indicators is to identify failure patterns by measuring parameters of any physical magnitude related to the system, such as vibrations, electric current and voltage, magnetic stray flux, etc. [27]. Statistical indices in the time domain have been used to analyze the steady-state regime of motors, including the mean or average, RMS value, standard deviation, and variance.

Table 1. Statistical time-domain features.

Time-Domain Features	Mathematical Equations
Arithmetic mean	$T1={\displaystyle \frac{1}{N}}\times {\sum }_{i=1}^N\left|X_i\right|$	(1)
Maximum value	$T_2=\max \left(x\right)$	(2)
RMS	$T_3=\sqrt{{\displaystyle \frac{1}{N}}\times {\sum }_{i=1}^N{\left(x_i\right)}^2}$	(3)
SRM	$T_4={\left({\displaystyle \frac{1}{N}}\times {\sum }_{i=1}^N\sqrt{{|x}_i|}\right)}^2$	(4)
Standard Deviation	$T_5=\sqrt{{\displaystyle \frac{1}{N}}\times {\sum }_{i=1}^N{(x_i-T_1)}^2}$	(5)
Variance	$T_6={\displaystyle \frac{1}{N}}\times {\sum }_{i=1}^N{(x_i-T_1)}^2$	(6)
Skewness	$T_7={\displaystyle \frac{\sum \left[{\left(x_{i-}{T}_1\right)}^3\right]}{T_5^3}}$	(7)
5th Moment	$T_8={\displaystyle \frac{\sum \left[{\left(x_{i-}{T}_1\right)}^5\right]}{T_5^5}}$	(8)
6th Moment	$T_9={\displaystyle \frac{\sum \left[{\left(x_{i-}{T}_1\right)}^6\right]}{T_5^6}}$	(9)

presents various statistical indicators and their mathematical equations. These indicators are recommended for their ability to provide insights into the general trend of system behavior. It is worth noting that these indicators can be easily implemented in signal processing devices due to their low computational demand and minimal memory resources required to compute them. In the equations presented in Table 1, $x_k$ represents the data series in the time domain, and N denotes the number of data points per series. These indicators are straightforward to compute, enhancing their applicability in real-time systems where computational efficiency is critical.

2.3. Particle Swarm Optimization (PSO)

The PSO algorithm is one of the variants of evolutionary optimization algorithms, which is inspired by the behavior of swarms of animals, such as flocks or shoal, which exhibit intelligent collective behavior. This algorithm has several advantages in that it is able to deal with complex, high-dimensional problems. The general operation of the PSO is shown in Figure 2, by means of a flowchart, which consists of a group of particles (known as a swarm) navigating a multi-dimensional search space (called design space), where the position of each particle (PBest) depends on the design variables and each one represents a potential solution to the problem. During each iteration, the particles adjust their velocity and position by considering both their current and previous positions (GBest). Finally, the position of the optimal GBest is obtained by performing this process until the convergence condition given by a maximum number of iterations is met, carrying out a comparison between the maximum number of iterations (T) and the current iteration (t). This process allows each particle to be influenced by the entire set of particles in the swarm, with the goal of finding the best global position that optimizes the objective function, seeking both its maximum and minimum [28].

2.4. Short-Time Statistics and Short-Time Variance

Short-time statistics are used to calculate the indicators shown in Table 1. In addition, Figure 3 shows an example of how to obtain the indicators proposed here. As shown there, it is a signal captured in the time domain, over which square sliding windows are generated with or without overlapping occurring between each one of them. The size of the windows is selected by the proposed optimization algorithm (PSO) in order to avoid as much as possible the processing of disturbances in the captured signal. Then, for each window created, the proposed indicators are calculated using the expressions (1) to (9). Finally, a new raw signal corresponding to the statistical analysis in a short time is obtained. This new raw signal will correspond to the statistical amplitude along the acquired signal in the full-time domain, including the start-up transient and the steady state. Likewise, for the calculation of the short-time variance, the same process is carried out concerning the segmentation of the windows and obtaining the number of windows of the particle optimization algorithm (PSO), with the difference that once the signal has been windowed only the variance in each of them is calculated. The input signal at this stage is the signal resulting from the process above (Shor-time statistics) and results in a new signal based on the variance.

2.5. Proposed Variance Method

We have proposed the variance method to detect the point at which the steady state of magnetic stray flux signals begins. This algorithm performs a first-degree polynomial fit between several signal points (i.e., window size). It must be ensured that the number of samples used for the fit is manageable in order to avoid a significant reduction of the total sample size, which could lead to a loss of information. Another critical aspect of this methodology is the proposal of a threshold, which corresponds to the average value of the variances of the last 10% of the signal (this portion is considered to be in a steady state), as shown at the bottom of the flowchart in Figure 4 as the “threshold of last samples”. This percentage can differ for different IMs depending on different aspects, such as the connected load and power supply technology. To this end, the methodology employs a Particle Swarm Optimization (PSO) algorithm, which allows the optimal estimate of the window size to perform the calculation of statistical and non-statistical indicators (Nwins) in the first stage of the algorithm, the window size for the variance calculation (NwinsVariance), and a second threshold to discard a part of the signal noise that could result in false positives as shown in the second block of the flowchart in Figure 4. This approach allows for an optimal threshold that adjusts to the values of each analyzed signal, as seen in the last block of the flowchart in Figure 4 labeled “Optimized threshold”. Once the stationary value of the signal has been determined, each of the selected variance values is compared with the first threshold calculated in the previous step. The transition from the transient state to the steady state is identified by looking for the most significant difference between two consecutive indices that has coincided with the value obtained at the threshold mentioned above, which will correspond to the time that the steady state ends. This process is illustrated graphically in Figure 4. As an additional step, the relative error of each sample was calculated using Equation (10). The real value corresponds to the time value of the start-up ramp programmed in the frequency inverter to carry out the tests, while the approximate value is the value obtained by the proposed methodology.

$$Relative error=\left|{\displaystyle \frac{Real value-Approximate value}{Real value}}\right| \ast 100$$

(10)

This relative error value is subsequently used to calculate the root mean square error (RMSE) of the total of the case studies, which is calculated by means of Equation (11). The RMSE value allows for the percentage of assertiveness of the proposed method.

$$RMSE=\sqrt{\sum _{i=1}^n{\displaystyle \frac{{\left(Real value-Approximate value\right)}^2}{Real value}}}$$

(11)

The signal received by the proposed variance method as input is the signal directly acquired by the magnetic flux sensor, and upon completion of the above process it delivers a new signal based on the variance as well as the optimized threshold.

3. Proposed Methodology

This part describes the proposed methodology, which is shown by a flow chart in Figure 5. In the first block, a normalization process is carried to the signals generated and stored by the data acquisition system (DAS): axial stray flux (${\varnothing }_1$), radial stray flux (${\varnothing }_2$), and axial+radial stray flux (${\varnothing }_3$). This normalization is performed by scaling the amplitude of the input signal to a range [−1, 1]. Then, a Savitzky-Golay type filtering is applied to this signal with a data window size of 49 points and order 5 for the filter’s polynomial approximation. These parameters are chosen due to the nature of the filter, which requires an odd window size and which is larger than the order to be used for the approximation. Once this process has been carried out, a pre-processed signal is obtained SG (${\varnothing }_i$), this signal will be the input of the next block, which is the proposed variance method (described in detail in the previous chapter). As explained above, the output of this block consists of a signal and an optimized threshold, which are used in the last block of automatic start-up detection explained below. The fourth block of the methodology corresponds to the detection of the transient state. For this purpose, a threshold (stable threshold) is established, which is obtained with the absolute value of the average of the last 10% of the total signal, since it is considered that at this point the signal is already in a stable state and will be used as a reference. In addition, the optimized threshold obtained in the previous block is used to overcome the noise peaks that may appear in the stable zone, as they can produce false positives and affect the start-up time detected by the methodology. Once the stable threshold has been obtained, this threshold is compared with each of the amplitudes of the signal resulting from the variance method. For those indices with a value lower than the threshold, a dashed line is drawn as a graphical reference only. Subsequently, once the entire signal has been compared, the most significant difference between two consecutive indices that has fulfilled the condition is sought, which will correspond to the time that the transient state lasts—allowing to find the point where the transient state ends and the steady state begins.

4. Experimental Setup

The experimental setup involves a WEG 00136APE48T induction motor (0.74 kW, 2.9 A, 220 V AC at 60 Hz) connected to a standard alternator, loading it at 25% of its nominal capacity. The IM is supplied via a variable frequency drive (VFD) that manipulates the start-up profiles through various ramps and target frequencies. Magnetic stray flux components: radial, axial, and radial + axial are captured using a triaxial Hall-effect sensor, with data captured at a rate of 1 kHz by a proprietary Data Acquisition system (DAS). This sensor is placed at a distance of 10 cm to avoid saturation and to obtain an accurate measurement of the magnetic flux. The sensor operates with a direct current power supply of 1.7 V to 3.6 V, has a measurement range of 1200 µT, and a sensitivity of 0.042 µT. The setup, illustrated in Figure 6, aims to validate the accuracy of the slope method (described in Section 2.3) for predicting start-up times, comparing these with controlled measurements according to the testbench matrix shown in

Table 2. Test Matrix for Database Generation.

Number of Repetitions		Time of Start-Up Transient (s)
		4	6	8	10
Power supply frequency	45 Hz	8	8	8	8
	50 Hz	8	8	8	8
	60 Hz	8	8	8	8

. Each configuration is tested under seven conditions to ensure robust statistical analysis and reliability of the results. The matrix includes tests conducted at three different frequencies: 45 Hz, 50 Hz, and 60 Hz, with four different start-up times: 4 s, 6 s, 8 s, and 10 s. In addition to the existing case studies, a further analysis involving a Direct-On-Line (DOL) induction motor was conducted. This allowed for a broader evaluation of the methodology, as the motor’s kinematic chain differed from those used in the other cases.

5. Results and Discussion

This section presents the results obtained by applying the proposed methodology to the samples collected, as detailed in the test matrix in Table 2. The matrix includes tests conducted at three different frequencies: 45 Hz, 50 Hz, and 60 Hz, with four different start-up times: 4 s, 6 s, 8 s, and 10 s, with a ramp-up profile. After acquiring the samples of the start-up profiles, the three components of each test are processed to obtain the start-up times for each of the statistics listed in Table 1.

The signal to be shown as an example in the following section corresponds to a magnetic flux component in the radial direction for a power supply frequency at 50 Hz. Furthermore, the following graphs represent the results obtained from the step-by-step methodology proposed in Section 3, showing the acquired signal without any processing, the normalization and filtering stage, the calculation of statistical and non-statistical indicators, the proposed variance method, the automatic detection block for the transient state, and the percentage error results of each case study.

In this regard, Figure 7 corresponds to the radial magnetic flux signal mentioned above, which does not have any type of processing other than the segmentation of the 30 s with which we worked throughout the proposed methodology.

Following the steps of the proposed methodology, a normalization stage is conducted, allowing scaled signals to the same range of values, which was between −1 to 1. In addition to implementing a Savitzky-Golay filter, this filter allows the numerical data to be smoothed and eliminates noise while preserving the main characteristics of the data. This graph is shown in Figure 8.

The next step corresponds to the results obtained from the calculation of the statistical and non-statistical short-time indicators mentioned in Table 1. As an example, Figure 9 shows the results obtained for the calculation of the arithmetic mean of the normalized and filtered signal, corresponding to the radial stray flux with a power supply frequency of 50 Hz, as shown above. As seen in Figure 9, in this step it is already possible to distinguish when the IM is operating in a steady state, however there is still a certain amount of noise, which could limit the accuracy of the proposed methodology. For this, the proposed variance method is applied.

Next, Figure 10 shows the graph obtained at the end of the signal processing stage, where the short-time variance is applied to the signal previously shown from the calculation of the statistical and non-statistical indicators. This signal clearly shows the peaks of change that exist in the transient state. This is due to the variance calculation because as the amplitude of the variance tends to zero, it can be inferred that this corresponds to the steady state of the signal, while the segment with a more significant variance belongs to the transient state. Once these limits are defined, several vertical lines (shown in red in Figure 11) are drawn for observation and clarification purposes, indicating the points on the graph where there is no considerable change in variance and, therefore, it is considered a steady state. This allows the detection algorithm to focus on finding the part with a more negligible variance difference, which it selects as the definitive point at which the steady-state begins.

The following graph, shown in Figure 11, corresponds to the result of the detection stage for which the signal calculated by the variance is taken, and two horizontal lines are drawn whose values are given by the threshold calculated by the bio-inspired algorithm. This algorithm used ten initial particles and a maximum of five iterations. For each iteration, these particles are moved along the search space, looking for the best position obtained as well as the best global position, which would be defined by the error obtained in each case of study and modifying the three variables to optimize, namely: the window size for the calculation of indicators, the window size for the calculation of variance, and the size of the threshold to avoid noise in the signal. The window sizes are bounded between 1 and 100 with integers, and from 0 to 2 with floats for the threshold, looking for optimal values according to the PSO algorithm.

The following section presents an extra case to those proposed in the test matrix, which consists of the induction motor used in the previous tests but connected to a different kinematic chain which represents 20% load. In this case, the motor was fed at direct voltage. First, in Figure 12 the flux signal acquired can be observed, already normalized and filtered, and then the result obtained by applying the method proposed in this article is shown. It can be seen that the optimized threshold discards the noisy signal that is already in steady state and positions the end of the transient state in an approximate time of 2.39 s when the real steady state time corresponds to 2.5 s. It is important to note that these results were obtained by applying the parameters obtained by the methodology proposed for the previous cases, without the need to perform a new training and allowing for the corroboration of the efficiency of the method.

As final results, Figure 13 shows a box and whiskers plot, resulting from the calculation of the relative error obtained by completing the proposed methodology, calculating the error for each repetition of the four cases of study (4 s, 6 s, 8 s, and 10 s of start-up time), and for the power frequency of 45 Hz. This graph shows the analysis of variance (ANOVA) resulting from the eight repetitions for each start-up ramp used in the experimentation. It is noticeable the relatively low error percentages obtained, having an average of 2.7% error for the case of 4 s, 4.8% error for 6 s, 6.5% error for the case of 8 s, and 6.7% relative error for the 10 s start = up transient.

The next power supply frequency tested was 50 Hz, for which the same four motor starting times and the same number of repetitions are used. Similar to the previous figure, Figure 14 shows the graph obtained for the radial magnetic stray flux with the error percentages for these study cases, with a 1.8% error for the 4 s case, 0.5% error for the 6 s, 2.8% error for 8 s, and finally 2.7% error for the 10 s start ramp.

To conclude this section, Figure 15 shows the error percentages of the radial magnetic stray flux for a power frequency of 60 Hz for the cases of the study mentioned above. The results show a 9% error for the 4 s start-up ramp, being the highest error of all the cases, 5.5% error for the 6 s ramp, 1.25% error for the 8 s start-up, and 1.20% error for the 10 s start-up.

6. Conclusions

As could be observed throughout this work, the proposed methodology is capable of automatically detecting the onset of the steady state for an induction motor, with a high level of assertiveness considering the large number of case studies available, since by applying this methodology for all starting times (4 s, 6 s, 8 s, and 10 s) and for the three power supply frequencies (45 Hz, 50 Hz, and 60 Hz) a total mean square error of 11. 95% is obtained. These results are largely due to the application of the particle optimization algorithm that allowed the calculation of the optimum values for the three variables to be controlled (statistical and non-statistical window size, variance window size, and threshold value)—obtaining an optimal value of 26 samples for the statistical and non-statistical window size, 15 samples for the variance window size, and 1.970302843426175 for the threshold value. This represents a contribution in the field of automatic fault detection, as the calculations necessary for the development of this methodology do not represent large computational resources such as memory, so they can be easily implemented in a microcontroller or in a field-programmable gate array (FPGA). This characteristic allows the use of this methodology to be adapted to automatic diagnostic methods that require applying appropriate techniques depending on whether the IM is in a transient state or in the steady state. It should be noted that this methodology promises to be useful to achieve the automatic separation of the transient state, highlighted for its high percentage of assertiveness in general and the low percentages of error obtained in most of the particular cases: for the frequency of 45 Hz an average error of 5.175% was obtained, for the frequency of 50 Hz 1.95% was obtained, and finally for 60 Hz 4.23% was obtained. These percentages were obtained by averaging the results obtained for the four proposed start-up times.