Influence of inter and subharmonics on the chaotic behavior of the stator current waveform in an induction motor

1. Introduction

A characteristic feature of power systems are disturbances in voltage waveform, which are most often caused by the operation of power electronic devices. Usually it is related to the presence of voltage harmonics, but in some systems [1–3] there are also voltage components with frequencies lower than the fundamental frequency, known as subharmonics (subsynchronous interharmonics – components having frequencies lesser than the fundamental harmonic) and interharmonics, whose frequencies are not equal to an integer multiple of the fundamental harmonic. The main reasons for the occurrence of voltage subharmonics and interharmonics include the operation of non-linear receivers, as well as energy production by the ship’s shaft generators and renewable energy sources [1]. It should be emphasized that the problem of waveform distortion is more visible in the case of the so-called soft grids, which include the ship system.

It was shown in [2,4,5] that even a small content of interharmonics and subharmonics causes shortening of the life of induction motors. Taking into account the motor, which is a non-linear device and additionally supplied with a distorted voltage, a chaotic or close to chaotic response (stator currents) is possible. This causes additional vibrations or an increase in power losses and winding temperature [5,6].

In classical literature related to the theory of chaos [7–9], one can find many different indicators of chaotic behavior, such as: signal time course; signal phase trajectory; autocorrelation function; power spectrum; Poincare map; Lyapunov exponents. The above indicators can be divided into qualitative and quantitative. Most of the proposed quantitative indicators require additional parameters such as phase space dimension, embedding dimension or time delay. It should be emphasized that, according to many authors [7–9], there is no universal method that determines these parameters for each data set.

Therefore, an attempt was made to investigate the influence of such a distorted voltage on two selected indicators of chaotic behavior, i.e. phase trajectories and Poincare maps. In addition, the definition of a new quantitative index based on standard deviation, which can indicate the degree of irregular signal distortion, is presented. The field-circuit model of the squirrel cage induction motor implemented in the ANSYS Electronics Desktop environment was used for the research. The percentage share of these interharmonics and subharmonics in relation to the fundamental harmonic was assumed in the range of 1–2%. This range reflects the observations of harmonic analysis presented in the papers [2,3].

2. Research methodology and short description of the field model

The engine model used for the research was numerically and experimentally verified in the papers of the authors [5,6,10,11]. The field model built allowed a large number of simulation studies to be carried out for different inter and sub harmonic contents even for unquantifiable frequencies, which would be experimentally inconvenient and difficult to carry out due to obtaining the exact value of the sub or inter harmonic frequencies. The squirrel cage induction motor model was made on the basis of a real object, the rated parameters of which are: 3 kW, 380 V, 6.9 A, and rated load 20.25 Nm. The discussed model has a field-circuit character and is based on the solution of the system of electromagnetic field equations and the equation of motion. At the same time, the field part was coupled with a part of the electric circuit, which constitutes a three-phase supply (Fig. 1a). Figure 1b shows an example of the flux density distribution in a 3D model of the engine. The authors are in the possession of 2D and 3D models of the engine. However, due to the need for a long simulation time needed to compute the Poincare map, a 2D model was used in the article. The division grid of the model used contains approximately 22,000 elements.

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Fig. 1.

(a) Motor stator power supply circuit in ANSYS Electronics Desktop environment. (b) Flux density distribution in a 3D motor model.

The simulation tests carried out assumed the existence of an additional subharmonic component Uha, interharmonic component Uha2 or both in the supply voltage. The percentage share of these interharmonics and subharmonics in relation to the fundamental harmonic was assumed in the range of 1–2%. As an example of simulations obtained from the model, Fig. 2 shows the current responses in Fig. 2b and Fig. 2c, respectively, for an ideal voltage (Fig. 2a) and a voltage containing an interharmonic component with a frequency of 95 Hz and an amplitude of 2%U _N .

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Fig. 2.

(a) Motor voltage supply waveform. (b) Motor current for ideal voltage. (c) Motor current for voltage with interharmonic.

In addition to the quality indices of chaotic behaviour, selected power quality indices as per IEC 61000-4-7 [14] were determined. Of these, the TWD index and the THDS index were chosen. At the same time, for the THDS, no courses were presented in the paper and only the maximum value was given.

From among the qualitative indicators (of chaotic behavior) mentioned in the introduction, Poincare maps and phase trajectories were used in the study. In addition, propositions of quantitative indicators were defined, which will allow for easier determination of the degree of irregularities discussed in the following parts of the article [12,13]. The first such indicator is the standard deviation from the ideal point (2). The ideal point is a Poincare map for an ideal sinusoidal signal whose coordinates depend on the initial phase angle (1). I d e a l P o i n t = [ I m sin ⁡ ( S t a r t P h a s e ) ; I m cos ⁡ ( S t a r t P h a s e ) ](1) where: I _m is the amplitude of the fundamental and StartPhase is the starting phase of the signal. σ I D E A L = ∑ k = 0 n [ ( m a p k , 0 − I d e a l P o i n t 0 ) 2 + ( m a p k , 1 − I d e a l P o i n t 1 ) 2 ] n + 1(2) where: map_k,0, map_k,1 are the coordinates of the Poincare map points, n is the number of samples.

In addition, a similar index was tested by calculating the standard deviation from the mean point (3), where the coordinates of the mean center were determined on the basis of the all points of Poincare map. σ M E A N = ∑ k = 0 n [ ( m a p k , 0 − M e a n P o i n t 0 ) 2 + ( m a p k , 1 − M e a n P o i n t 1 ) 2 ] n + 1 .(3) This article presents the simulation results made for a sinusoidal voltage containing one disturbance (subharmonic f _sh or interharmonic f _ih ) with two amplitudes 1% and 2% and frequencies which are presented in Table 1.

The first test that was performed was a simulation for the rated conditions of the motor with an ideal sinusoidal voltage. The obtained results constituted a certain benchmark for subsequent tests with deformed voltage, presented later in the article. The Fig. 3 presents basic quantities such as voltage spectrum (Fig. 3a) and current (Fig. 3d), as well as the phase trajectory and Poincare map for voltage (Fig. 3b, 3c) and current (Fig. 3e, 3f).

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Fig. 3.

Spectra of voltage (a) and stator current (d); phase trajectories of voltage (b) and stator current (e); Poincare maps for voltage (c) and stator current (f) when the motor is supplied with sinusoidal voltage.

According to the theory, the determined phase trajectory for a sinusoidal voltage while maintaining the scale of both axes is a circle (Fig. 3b), and the Poincare map is a point (Fig. 3c) of this circle depending on the phase initial (selected moment). On the other hand, the phase trajectory of the current (Fig. 3e) is characterized by a visible uniqueness, and the Poincare map this time is not a point but moves along a certain closed curve (Fig. 3f). Such a response, despite the ideal voltage, is the result of a non-linear load.

For the case in question, research on quantitative indicators was also carried out. The rms current I _rms that was obtained for this calculation was 6.81 A. The values of the THDS (Total Harmonic Distortion Subgroups) and TWD (Total Wide Distortion) coefficients for the stator current were 3.6% and 6.7%, respectively. For indices related to chaotic behavior, the values of standard deviation in relation to the ideal and mean point for current were respectively 13.99% and 5.83%.

In the next step, the simulation was carried out when the motor was supplied with a voltage containing one subharmonic (in addition to fundamental component) the frequency of which was changed from 5 Hz to 45 Hz, with a step of 5 Hz. The results presented in Fig. 4 concern an additional component with a frequency of 10 Hz and the content of 2%U _N in the supply voltage (spectrum from the Fig. 4a). The voltage phase trajectory (Fig. 4b) is a spiral curve consisting of five circles, which results from the ratio of the fundamental harmonic frequency (50 Hz) and the subharmonic frequency (10 Hz). In the current spectrum shown in the Fig. 4d, it can be seen that the values of the subharmonics generated in the current are significant (15-fold increase to the voltage of subharmonics) while the magnitude of interharmonic that also occurs there is small. As a result of the presence of a subharmonic in the voltage, one can see the difference in the phase trajectory of the current (Fig. 4e) compared to the case for an ideal supply (Fig. 3e). There is also an evident influence of the subharmonic in voltage on the Poincare map of the current, in which three separate cycles can be seen (Fig. 4f). However, by watching the map-making animation, you could see that it actually consists of 5 overlapping cycles, which is related to the five points of Poincare’s voltage map.

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Fig. 4.

Spectra of voltage (a) and stator current (d); phase trajectories of voltage (b) and stator current (e); Poincare maps for voltage (c) and stator current (f) when the motor is supplied with a voltage containing a subharmonic.

It should be emphasized that the Poincare map in the literature on chaos [9] is defined as a set of points obtained from a phase trajectory for selected moments of time separated from each other by a fixed period for the entire recorded course. The period used to determine Poincare’s maps is the period of the basic sinusoid (20 ms), although the waveforms resulting from the sum of this sinusoid and the subharmonic change this period. The number of circles and points on the Poincare voltage map can be calculated by dividing the fundamental frequency by the value of the greatest common divisor of the fundamental frequency and the subharmonic frequency occurring in the supply voltage.

The Fig. 5 shows the values of the TWD (Fig. 5a), as well as the standard deviation with respect to the mean center (Fig. 5b) and to the ideal center (Fig. 5c).

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Fig. 5.

Quantitative indicators of the stator current as a function of the subharmonic frequency contained in the voltage: (a) TWD, (b) standard deviation from the mean point and (c) from the ideal point for a voltage containing a subharmonic.

From the graphs in Fig. 5a–c you can read the factor TWD (31.7%) obtained for the subharmonic frequency 15 Hz and its content in the supply voltage equal to 2%U _N . For the THDS factor (not presented in this paper) its maximum value occurs at 10 Hz and amounts to 4.24%. The graphs of the standard deviation with respect to the mean center (Fig. 5b) and the ideal center (Fig. 5c) shows that the highest value of the standard deviation that was obtained occurred at the same subharmonic frequency equal to 20 Hz in both cases. The values of these coefficients were respectively: 6.86% and 14.57%.

Analogous tests were carried out for a sinusoidal voltage containing one interharmonic, the content of which was 1%U _N or 2%U _N at the frequency varying from 55 Hz to 95 Hz with a step of 5 Hz. The results of the simulation of phase trajectories and Poincare maps for the interharmonic frequency of 85 Hz (2%U _N ) are shown in the Fig. 6. On the basis of the current spectrum, it can be seen that under the influence of the interharmonic occurring in the supply voltage (Fig. 6a), there is a several times greater interharmonic in the current (7% I _N ), as well as a subharmonic of a much smaller value (0.4% I _N ; Fig. 6d). The voltage phase trajectory has a similar shape and properties as for the case when there was a subharmonic in the sinusoidal supply voltage. From the Poincare map for current it is no longer possible to read separately groups of points as for an example with a voltage containing a subharmonic. Thus, the Poincare map tends to fill a certain area, which according to Schuster [13] may be an indicator of near-chaotic behavior.

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Fig. 6.

Spectra of voltage (a) and stator current (d); phase trajectories of voltage (b) and stator current (e); Poincare maps for voltage (c) and stator current (f) when the motor is supplied with a voltage containing a interharmonic

Figure 7 shows the TWD factor and the standard deviation studies with respect to the mean center and ideal center determined for the Poincare maps. For the frequency 75 Hz, there was a maximum of TWD (9.75%). For THDS (not presented in this paper) the maximum (3.61%) reached for the highest tested frequency (95 Hz). In the case of examining the standard deviation for the mean center (Fig. 7b) and the ideal center (Fig. 7c), it can be seen that at the frequency of 75 Hz there is actually a decrease in the value in which there is the local minimum for cases with the content of this disorder amounting to 2%U _N . For this frequency, the subharmonic of the current is close to the so-called resonant frequency of the tested engine [10]. From the graphs, in Fig. 7b,c the maximum values were read for the interharmonic frequency of 95 Hz (for the standard deviation with respect to the mean center (Fig. 7b) and 85 Hz (for the standard deviation with respect to the ideal center Fig. 7c), which were 5.92% and 13.99%, respectively.

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Fig. 7.

Quantitative indicators of the stator current as a function of the interharmonic frequency contained in the voltage: (a) TWD, (b) standard deviation from the mean point and (c) from the ideal point for a voltage containing a interharmonic.

In practice, the supply voltage may contain a wide spectrum of various disturbances. The studies presented in this article were aimed at examining the effect of only subharmonics or interharmonics in the supply voltage on the indicators determining the degree of current distortion and uniqueness (degree of chaotic behavior). Additionally, in order to emphasize this influence in relation to the deviation from the frequency of the fundamental component, the article presents cases only for rational frequencies. If the irrational frequency was adopted, the obtained phase trajectories and Poincare maps (not presented in this article) indicated a more chaotic behavior, which is the result of a finite representation of the irrational number.

In addition to studies involving either the subharmonic or the interharmonic separately, the authors also studied cases involving both components simultaneously according to Table 1. Observing chaotic behaviour indices and especially the Poincare maps of currents for voltage containing sub- and/or interharmonics in comparison with the current map obtained for the ideal supply voltage, a clear difference was found, which shows a significant influence of sub-harmonics and interharmonics on these qualitative indicators. At the same time, when examining this effect using quantitative indicators, it was found that subharmonics have a dominant influence on the indicators tested, except for THDS, where interharmonics dominate.

The electricity quality indicators used so far, such as TWD and THDS, do not include chaotic behaviors that may occur in various types of non-linear systems. It should be considered that the indicators of chaotic behavior such as the phase trajectory and the Poincare map can be a qualitative indicator of both the deformation of the waveform from the sinusoid and the chaotic behavior. In addition to the above qualitative indicators, it seems that in the future, the indicators of standard deviation from the mean center or ideal center can also be used to control on the one hand the degree of deformation and on the other hand the irregularity of the course (chaotic behavior). The nature of both of these quantitative indicators turned out to be very similar, which means that one of them can be selected for control.