Effects of Voltage Interharmonics on Cage Induction Motors

energies

Article Effects of Voltage Interharmonics on Cage Induction Motors

Piotr Gnaci ´nski 1,* , Damian Hallmann 1 , Piotr Klimczak 2, Adam Muc 3 and Marcin Pepli ´nski 1

1 Department of Ship Electrical Power Engineering, Faculty of Marine Electrical Engineering, Gdynia Maritime University, Morska St. 83, 81-225 Gdynia, Poland; [email protected] (D.H.); [email protected] (M.P.) 2 Zakład Maszyn Elektrycznych “EMIT” Cantoni Group, Diagnostic and Service of Electrical Motors Department, Narutowicza St. 72, 99-320 Zychlin,˙ Poland; [email protected] 3 Department of Ship Automation, Faculty of Marine Electrical Engineering, Gdynia Maritime University, Morska St. 83, 81-225 Gdynia, Poland; [email protected] * Correspondence: [email protected]; Tel.: +48-58-5586-382

Abstract: In a power system, the voltage waveform usually contains harmonics and sometimes interharmonics, often deﬁned as components of frequency greater than the fundamental voltage component but not of its integer multiple. Previous studies have reported a minor effect of voltage interharmonics on a cage induction machine. This paper reveals their extraordinary harmfulness for induction motors. Namely, voltage interharmonics may cause high vibration, which can result in machine damage. In addition, interharmonics can lead to torque pulsations corresponding to the natural frequency of the ﬁrst elastic mode. Consequently, possible torsional resonance may cause destruction of a power train. In this study, the results of investigations on undesirable phenomena due to interharmonics are presented for seven motors with a rated power 3 kW–5.6 MW.

Keywords: induction motors; interharmonics; power quality; power system; torsional vibration; vibration; voltage fluctuations Citation: Gnaciński,P.; Hallmann, D.; Klimczak, P.; Muc, A.; Pepliński, M. Effects of Voltage Interharmonics on Cage Induction Motors. Energies 1. Introduction 2021, 14, 1218. https://doi.org/ The universal application of power electronic equipment and rapid development 10.3390/en14051218 of renewable energy industry result in the common occurrence of voltage waveform distortions. Usually, voltage waveform distortions are restricted to voltage harmonics, Academic Editor: Frede Blaabjerg although, in some power systems, voltage waveform contains interharmonics, understood

Received: 5 February 2021 in this study as components of frequencies that are greater than the fundamental harmonic Accepted: 19 February 2021 but not its integer multiple. The main reason for interharmonic contamination is the work Published: 24 February 2021 of non-linear and changeable loads, double-conversion systems and a renewable source of energy. Interharmonics are generated, among the other things, by power electronic

Publisher’s Note: MDPI stays neutral appliances connecting two AC systems of various frequencies through a DC link [1–4], with regard to jurisdictional claims in such as inverters [2–4] or HVDC [1,2]. Other sources of interharmonics are cycloconverters published maps and institutional afﬁl- and loads changing over time [2], which include AC motors driving compressors [5–7], iations. arc furnaces (especially AC types) [1,2,8,9] or laser printers [2]. Moreover, time-varying phenomena, like rapid changes of magnitude and phase, can generate interharmonics [10]. Sources of substantial interharmonics are photovoltaic plants [11–14] and wind power stations [1,2,15,16].

Copyright: © 2021 by the authors. Interharmonics of comparatively high values, namely mains signalling voltages, can be Licensee MDPI, Basel, Switzerland. superimposed on the fundamental voltage component for information transmission [1,17]. This article is an open access article For the frequency range of 110–500 Hz, the standard [17] tolerates mains signalling voltages distributed under the terms and of up to 9% of the rated network voltage. It is also worth mentioning that periodic voltage conditions of the Creative Commons ﬂuctuations can be regarded as a superposition of interharmonics and subharmonics [18] Attribution (CC BY) license (https:// (also called subsynchronous interharmonics—components having frequencies less than the creativecommons.org/licenses/by/ fundamental harmonic). 4.0/).

Energies 2021, 14, 1218. https://doi.org/10.3390/en14051218 https://www.mdpi.com/journal/energies Energies 2021, 14, 1218 2 of 13

Examples of interharmonic occurrences in power systems are provided in [1,3,7,13,19]. For instance, [3] reported various subharmonics and interharmonics components of value in the range 0.89–1.17%, occurring simultaneously. The voltage waveform distortions were caused by high-power variable speed drives, supplied from diesel-driven generators. Furthermore, in [19], high interharmonic contamination was reported in various industrial, urban and rural medium-voltage feeders. In some of them the average value was up to 0.9%. Interharmonics can disturb the work of power electronic equipment, measurement and control systems [20–22], induction motors [23], and light sources, including commonly applied LEDs [24]. At the same time, power quality standards generally do not provide limitations of interharmonics. For example, the standard IEEE-519: IEEE Recommended Practices and Requirements for Harmonic Control in Electric Power Systems [25] presents the limits for informative purposes only. Additionally, the following comment is provided: “It is important to recognize that the suggested voltage interharmonic limits are based on lamp flicker (...) correlate with a short-term flicker severity Pst value equal to 1.0 (...). The recommended limits (...) are not based on the effects of interharmonics on other equipment and systems such as generator mechanical systems, motors, transformers, signalling and communication systems, and filters. Due consideration should be given to these effects...”. Furthermore, in the standard EN 50160 Voltage Characteristics of Electricity Supplied by Public Distribution Systems [17], the following note occurs: “Levels are under consideration, pending more experience.” In summary, imposing limitations on interharmonics requires comprehensive investigations on their impact on various elements of a power system, including induction motors. Undesirable phenomena due to the interharmonics’ occurrence have been discussed in numerous works [20–24,26–35]. In [26–30], induction motors under voltage fluctuations were analysed. As periodic voltage fluctuations can be considered as the superposition of subharmonics and interharmonics, the results of the research cannot be used for imposing a limitation on interharmonics occurring as a single power quality disturbance. The influence of interharmonics appearing as a single power quality disturbance on an induction motor was investigated in [18,23,31,32]. References [18,23,31,32] show that voltage interharmonics caused the flow of rather moderate current subharmonics and interharmonics and resulted in a minor increase in power losses, windings temperature and vibration, slightly exceeding its admissible level for long-term continuous machine operation [33]. It should be noted that in [23] the vibration was investigated for the interharmonics of frequencies not exceeding 70 Hz because of the speed limit of the alternator used for the generation of interharmonics. It is also worth mentioning that some of the previous works were mostly focused on the adverse effects of subharmonics [23,32] or were limited solely to them [34,35]. To sum up, previous works [18,23,31,32] have reported minor maliciousness of interharmonics on induction motors. Additionally, experimental investigations were restricted to interharmonic frequencies of up to 70 Hz. Furthermore, torque pulsations due to interharmonics occurring as single power quality disturbances were not analysed. Based on the state of the art, the goals of the paper were formulated. The first goal is to point out that interharmonics can exert extraordinarily harmful effects on induction motors. The second goal is to initiate in-depth investigations on induction motors supplied with voltage containing interharmonics. The results of these investigations should contribute to the revision of power quality standards and rules.

2. Methodology The results of investigations are presented for seven cage induction motors with a rated power of 3 kW–5.6 MW, denoted (depending on rated power in kW and number of poles) as motor 3-p4, motor 4-p4, motor 200-p2, motor 200-p4, motor 200-p6, motor 200-p8 and motor 5M-p8. Their basic parameters are listed in Table1. More details are provided in [34,35]. Energies 2021, 14, 1218 3 of 13

Table 1. Basic parameters of the investigated motors.

Rated Rated Rated Power Number Rated Speed Motor Type Voltage Current (kW) of Poles (Rpm) (V) (A) motor 3-p4 TSg100L-4B 3 4 1420 380 ∆ 6.9 motor 4-p4 1LE1003-1BB22-2AA4 4 4 1460 400 Y 7.9 motor 200-p2 SLgm 315 ML2B 200 2 2982 400 ∆ 335 motor 200-p4 SEE 315 L4 200 4 1487 400 ∆ 346 motor 200-p6 3SIE 355 ML6A 200 6 989 400 ∆ 350 motor 200-p8 SEE 355 ML8B 200 8 740 400 ∆ 384 motor 5M-p8 Sfw 900 HV8D 5 600 8 745 10500 Y 375

For the purpose of the paper, analysis with the finite element method (FEM) was employed. The 2D numerical computations were carried out using the MAXWELL-ANSYS environment for motor 3-p4 and the ANSYS Electronics Desktop environment (ANSYS Electromagnetics Suite 18.0.0) for the other motors. The applied Tau meshes [34] contained from 4100 to 53,596 elements. The parameters of the models were identified on the basis of constructional and experimental data. It is also worth mentioning that the motors (except motor 3-p4 and 4-p4) are produced by the company of one of the co-authors— Zakład Maszyn Elektrycznych “EMIT” S.A. Cantoni Group. Thanks to the courtesy of its management he had free access to constructional documentations and the test results of the motors. A detailed description of the used field models is presented in [34,35]. In addition to the FEM computations, experimental methods were applied. Because of the high power of most of the investigated machines, the empirical research (including model verification) was restricted to motor 3-p4 and motor 4-p4. The applied measurement setup was composed of an AC programmable power source, a power quality analyser, a vibration measurement system and the investigated motors. The AC programmable power source type Chroma 61512+A615103 consisted of two master/slave modules working in parallel, with a total rated power of 36 kVA. The power source enables the generation of programmable voltage subharmonics and interharmonics within the frequency range of 0.01–2400 Hz. For the measurement of current subharmonics and interharmonics, a PC-based power quality analyser was used. The vibration was measured with a Bruel & Kjear (B&K) system, consisting of a standalone four-channel data acquisition module (B&K type: 3676-B-040), a three-axis acelerometer (B&K type: 4529-B), an accelerometer calibrator (B&K type: 4294) and a computer with the BK Connect software. The accelerometer was fixed to an additional steel stand, screwed into the aluminium motor casing (Figure1). The accelerometer was calibrated before starting the measurements. The investigations on vibration were performed for two newly commissioned motors 4-p4, of which one was coupled with a DC generator, and the other was uncoupled but mounted on a rigid frame (Figure1). A simplified diagram of the measurement stand is shown in Figure2. In the next diagrams (Figures3 and4) the comparison of the results of measurements and computations is presented for motor 3-p4, supplied with the voltage containing interharmonics. The appropriate numerical and empirical experiments were performed for the uncoupled motor and the fundamental voltage component determined to achieve the nom- inal flux under no-load. Figure3 shows current interharmonics vs. the frequency of voltage interharmonics, and Figure4—current subharmonics accompanying current interharmonics. The summary of the results of measurements and FEM analysis (Figures3 and4 ) shows that the accuracy of the field calculations is acceptable. Energies 2021, 14, 1218 4 of 13 Energies 2021, 14, 1218 4 of 13

Energies 2021, 14, 1218 4 of 13 Energies 2021, 14, 1218 4 of 13

FigureFigure 1. 1.Motors Motors 4-p4 4-p4 and and the the accelerometer accelerometer (bottom (bottom left). left). Figure 1. Motors 4-p4 and the accelerometer (bottom left). Figure 1. Motors 4-p4 and the accelerometer (bottom left).

PC- based power quality PC- based PCpower- analyser based quality Programmable AC power source power quality type Chroma 61512+A615103 analyser Programmable AC power source analyser Programmabletype Chroma AC61512+A615103 power source type Chroma 61512+A615103

supply voltage supply voltage input supply voltage inputmodule accelerometer moduleinput accelerometer module accelerometer

induction induction DC PC inductionmotor inductionmotor generatorDC computerPC inductionmotor inductionmotor generatorDC computerPC motor motor generator computer Figure 2. Simplified diagram of the measurement stand. Figure 2. SimplifiedSimpliﬁed diagram of the measurement stand. Figure 2.12.0 Simplified diagram of the measurement stand. 12.0 computations 12.0 computationsmeasurements 10.0 computations 10.0 measurements measurements 10.08.0 8.0 8.06.0

Iih Iih [%] 6.0 6.0 Iih Iih [%] 4.0 Iih Iih [%] 4.0 4.02.0 2.0 2.00.0 0.0 55 60 65 70 75 80 85 90 95 100 0.0 55 60 65 70 75 80 85 90 95 100 55 60 65 70 75fih [Hz]80 85 90 95 100 fih [Hz] Figure 3. Measured and computedf ihcurrent[Hz] interharmonics (related to the rated current) vs. the Figurefrequency 3. Measured of voltage and interharmonics computed current for motor interharmonics 3-p4 and voltage (related containing to the rated interharmonics current) vs. the of FigureFigure 3. 3. MeasuredMeasured and and computed computed current current interharmonics interharmonics (related (related to the to rated the rated current) current) vs. the vs. the frequencyvalue of voltage interharmonics for motor 3-p4 and voltage containing interharmonics of frequencyfrequency of of voltage voltage interharmonics interharmonics for for motor motor 3 3-p4-p4 and and voltage voltage containing containing interharmonics interharmonics of of value valueUih = 1 % U1. value Uihih == 11%%U U1.1 . Uih = 1% U1.

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12.0 computations 10.0 measurements

8.0

6.0

Ish[%] 4.0

2.0

0.0 55 60 65 70 75 80 85 90 95 100

fih [Hz] FigureFigure 4. 4. MeasuredMeasured and and computed computed current current subharmonics subharmonics (related (related to the to rated the rated current) current) vs. the vs. fre- the quencyfrequency of voltage of voltage interharmonics interharmonics for formotor motor 3-p4 3-p4 and and voltage voltage containing containing interharmonics interharmonics of value of value Uih = 1% U1. Uih = 1%U1. 3.3. Results Results 3.1.3.1. Preliminary Preliminary Remarks Remarks InIn the the following subsections,subsections, thethe results results of of investigations investigations on on currents, currents, torque torque pulsations pulsationsand vibrationand vibration are presented. are presented. Because Because the powerthe power quality quality disturbance disturbance under under consideration consider- ationcauses causes fluctuations fluctuations of the of the rotational rotational speed speed [18 [18],], the the impact impact ofof voltage interharmonics interharmonics onon induction induction motors motors significantly significantly depends depends on on the the moment moment of of load load inertia inertia [31]. [31]. For For this this reason,reason, the the FEM FEM computa computationstions are are shown shown for for the the two two extreme extreme cases: cases: a a load load of of a a negligible negligible momentmoment of of inertia inertia (NMI) (NMI) and and the the work work with with a a constant constant rotational rotational speed speed (CRS) (CRS) due due to to a a highhigh moment moment of of load load inertia. inertia. For For the the computations, computations, the the torque torque load load and and the the fundamental fundamental voltagevoltage components components are are assumed assumed to to be be equal equal to to its its rated rated values. values. All All the the presented presented results results of numerical and experimental investigations were performed for the fundamental voltage of numerical and experimental investigations were performed for the fundamental volt- component of its rated value and voltage interharmonics equal to 1%, as similar levels are age component of its rated value and voltage interharmonics equal to 1%, as similar levels reported in [3,19]. are reported in [3,19]. 3.2. Effect of Interharmonics on Current 3.2. Effect of Interharmonics on Current Voltage interharmonics cause the flow of current interharmonics and subharmonics (CIS)Voltage through interharmonics motor windings. cause The the interaction flow of current between interharmonics these current components and subharmonics and the (CIS)rotational through electromagnetic motor windings. field The [18 interaction] results in between tangential these electromagnetic current components forces [ and36] and,the rotationalconsequently, electromagnetic vibration and field torsional [18] results vibration. in tangential electromagnetic forces [36] and, consequently,Below, Figures vibration5–9 andpresent torsional the computed vibration. CIS vs. the frequency of voltage interharmonics,Below, for Figures motor 3-p4,5–9 present 200-p2, the motor computed 200-p4, CIS motor vs. the 200-p6, frequency motor of 200-p8 voltage and interhar- motor monics,5M-p8. for For motor CRS 3 (Figures-p4, 2005- p2,and motor6), the 200 current-p4, motor interharmonics 200-p6, motor did 200 not-p8 exceed and motor roughly 5M- p8.5% For of theCRS rated (Figure currents 5 and (Irat 6),), andthe current subharmonicsinterharmonics were did not below exceed 1% ofroughlyIrat. Notably, 5% of rat rat themaxima rated forcurrent motor (I 200-p8), andand current a frequency subharmonicsfih ≈ 60 were Hz resulted below 1% from of theI . interferenceNotably, maxima of CIS forcaused motor by 200 voltage-p8 and interharmonics a frequency fih and ≈ 60 CIS Hz causedresulted by from slots the and interference teeth (CIS alsoof CIS occurs caused for bythe voltage purely interharmonicssinusoidal supply and voltage). CIS caused Then, by for slots NMI and (Figures teeth 7 (CIS and 8 also), the occur highests for CIS the purelyoccurred sinusoidal for motor supply 200-p2 voltage). and the frequencyThen, for NMIfih = 57–58(Figure Hz.s 7 Bothand current8), the highest interharmonics CIS oc- curredand subharmonics for motor 200 reached-p2 and approximatelythe frequency fih 9.5% = 57– of58Irat Hz.. A Both current current spectrum interharmonics for this case and is subharmonicsgiven in Figure reached9. In turn, approximately the lowest 9.5% CIS appeared of Irat. A current for motor spectrum 200-p8 for and this motor case is 5M-p8, given inmaximising Figure 9. In at turn, 5.5% the and lowest 4.8% CIS of Irat appeared, respectively. for motor 200-p8 and motor 5M-p8, maximis- ing at 5.5% and 4.8% of Irat, respectively. The shape of the characteristics presented in Figures 7 and 8 results from the occurrence of rigid-body resonance, similar to that observed for voltage subharmonics [34,35]. Namely, current interharmonics cause torque pulsations and fluctuations of the rotational speed, whose frequency is defined as the following (based on [18]):

Energies 2021, 14, 1218 6 of 13 EnergiesEnergies 2021 2021, 14, ,14 1218, 1218 6 of6 13of 13

푓푝 = 푓𝑖ℎ − 푓1, (1) 푓푝 푓=푝 =푓𝑖ℎ푓𝑖−ℎ −푓1,푓 1, (1)( 1) where fih and f1 are the frequencies of current interharmonics and the fundamental voltage Energies 2021, 14, 1218 6 of 13 wherewherecomponent, fih fandih and f1 arerespectively.f1 are the the frequencies frequencies Speed of fluctuations ofcurrent current interharmonics interharmonics lead to the flow and and ofthe thecurrent fundamental fundamental subharmonics voltage voltage of component,component, respectively. respectively. Speed Speed fluctuations fluctuations lead lead to tothe the flow flow of ofcurrent current subharmonics subharmonics of of frequency fsh (based on [5,6,18]): frequencyfrequency fsh f(basedsh (based on on [5,6,18]): [5,6,18]):

6.0 6.06.0 5.0 5.05.0 4.0 4.04.0 3.0

3.03.0

Iih Iih [%] Iih Iih [%] Iih Iih [%] 2.0 2.02.0 3-p4 200-p2 1.0 3-p43-p4200-p4 200-p2200-p2200-p6 1.01.0 200-p4200-p4200-p8 200-p6200-p65M-p8 0.0 200-p8200-p8 5M-p85M-p8 0.00.0 55 60 65 70 75 80 85 90 95 100 55 55 60 60 65 65 70 70 75 75 80 80 85 85 90 90 95 95 100100 fih [Hz] f [Hz] fih [Hz]ih FigureFigure 5. 5. ComputedComputed current current interharmonics interharmonics vs. vs. the the frequency frequency of voltage of voltage interharmonics interharmonics for the for the FigureFigure 5. Computed5. Computed current current interharmonics interharmonics vs. vs. the the frequency frequency of ofvoltage voltage interharmonics interharmonics for for the the variousvarious motors motors and and the the case case of of constant constant rotational speed ( (CRS).CRS). variousvarious motors motors and and the the case case of ofconstant constant rotational rotational speed speed (CRS (CRS). ). 1.0 1.01.0 3-p4 200-p2 0.9 3-p43-p4200-p4 200-p2200-p2200-p6 0.90.9 0.8 200-p4200-p4200-p8 200-p6200-p65M-p8 0.80.8 200-p8200-p8 5M-p85M-p8 0.7 0.70.7 0.6 0.60.6 0.5 0.50.5 0.4

Ish[%] 0.40.4 Ish[%] Ish[%] 0.3 0.30.3 0.2 0.20.2 0.1 0.10.1 0.0 0.00.0 55 60 65 70 75 80 85 90 95 100 55 55 60 60 65 65 70 70 75 75 80 80 85 85 90 90 95 95 100100 fih [Hz] f [Hz] fih [Hz]ih Figure 6. Computed current subharmonic vs. the frequency of voltage interharmonics for the vari- FigureFigureousFigure motors6. Computed6. 6. ComputedComputed and the current casecurrent current of subharmonic CRS. subharmonic subharmonic vs. vs. the the frequency frequency of ofofvoltage voltagevoltage interharmonics interharmonicsinterharmonics for for for the the the vari- various vari- ousousmotors motors motors and and and the the the case case case of of CRS. ofCRS. CRS. 10 10 10 3-p4 200-p2 3-p43-p4200-p4 200-p2200-p2200-p6 8 200-p4200-p4 200-p6200-p6 8 8 200-p8 5M-p8 200-p8200-p8 5M-p85M-p8 6 6 6

Iih Iih [%] 4 Iih Iih [%] Iih Iih [%] 4 4

2 2 2

0 0 0 50 55 60 65 70 75 80 85 90 95 100 50 55 60 65 70 75 80 85 90 95 100 50 55 60 65 70 75fih [Hz]80 85 90 95 100 f [Hz] fih [Hz]ih Figure 7. Computed current interharmonics vs. the frequency of voltage interharmonics for the FigureFigurevariousFigure 7. Computed7. 7.motors ComputedComputed and current thecurrent current case interharmonics interharmonicsofinterharmonics negligible momentvs. vs. the vs. the fr equency theoffr equencyinertia frequency of (NMI ofvoltage voltage of). voltage interharmonics interharmonics interharmonics for for the the for the variousvariousvarious motors motors motors and and and the the thecase case case of ofnegligible of negligible negligible moment moment moment of ofinertia of inertia inertia (NMI ( (NMI).NMI). ).

EnergiesEnergiesEnergies 2021 20212021, ,14 14, 14, ,1218 1218, 1218 77 of 7of of13 13 13

1010 3-p43-p4 200-p2200-p2 88 200-p4200-p4

66 200-p6200-p6

200-p8200-p8 Ish[%] Ish[%] 44 5M-p85M-p8

00 5050 5555 6060 6565 7070 7575 8080 8585 9090 9595 100100 ff [Hz][Hz] shsh FigureFigureFigure 8. 8. Computed 8.ComputedComputed current current current subharmonics subharmonics subharmonics vs. vs. the the vs. frequency frequency the frequency of of voltage voltage of voltage interharmonics interharmonics interharmonics for for the the for the variousvariousvarious motors motors motors and and and the the the case case case of of ofNMI. NMI. NMI.

2020 1818 1616 1414 1212 1010

88 Ish, Iih I1, [%] Ish, Iih I1, [%] 66 44 22 00 00 1010 2020 3030 4040 5050 6060 7070 8080 9090 100100 f [Hz] fihih[Hz] FigureFigureFigure 9. 9. Compu 9.CompuComputedtedted current current current spectrum spectrum spectrum of of motor ofmotor motor 200 200 200-p2,--p2,p2, supplied supplied supplied with with with voltage voltage voltage containing containing containing interhar- interhar- interhar- monicsmonics of of frequency frequency f ihfih = = 58 58 Hz Hz for for the the case case of of NMI. NMI. monics of frequency fih = 58 Hz for the case of NMI.

The shape of the characteristics presented in Figures7 and8 results from the occurrence 푓푓 ==푓푓 −−푓푓, , ((22)) of rigid-body resonance, similar to that observed푠푠ℎℎ 11 for푝푝 voltage subharmonics [34,35]. Namely, currentForFor the the interharmonics frequency frequency f pfp close close cause to to torquethe the natural natural pulsations frequency frequency and of of fluctuations the the rigid rigid--bodybody of the mode mode rotational f NrfNr-b- b[5], [5], speed, CIS CIS imposedimposedwhose frequencytorque torque pulsations pulsations is defined and, and, as consequently, theconsequently, following (basedspeed speed fluctuations. onfluctuations. [18]): The The fluctuations fluctuations are are il- il- lustratedlustrated inin FigureFigure 10.10. ForFor motormotor 33--p4p4 thethe speedspeed fluctuationsfluctuations areare asas highhigh asas aboutabout 1.9%,1.9%, andand for for t thehe other other motors motors up up to to about about 0.7%. 0.7%.fp =The Thefih magnified magnified− f1, speed speed fluctuations fluctuations additionally additionally (1) increasedincreased CISCIS (based(based onon [5,6]). [5,6]). Consequently,Consequently, inin FigureFiguress 77 andand 8,8, currentcurrent peakspeaks occurredoccurred where f and f are the frequencies of current interharmonics and the fundamental voltage aroundaround the theih frequency frequency1 corresponding corresponding to to the the rigid rigid body body resonance. resonance. Contrastingly, Contrastingly, for for a a hi highgh component, respectively. Speed fluctuations lead to the flow of current subharmonics of momentmoment of of load load inertia, inertia, the the speed speed fluctuations fluctuations were were negligible negligible and, and, as as a a result, result, CIS CIS were were frequency fsh (based on [5,6,18]): comparativelycomparatively low low (Figure (Figuress 5 5 and and 6). 6). The The natural natural frequency frequency f NrfNr-b- bdepends depends on on machine machine size; size; fsh = f1 − fp, (2) thethe frequenciesfrequencies fNrfNr-b- b areare tenstens ofof HzHz forfor smallsmall machinesmachines andand aa fewfew HzHz forfor largelarge machinesmachines [5,6,18][5,6,18]For. . ForFor the thethe frequency reason,reason, forforfp close motormotor to 33 the--p4p4 natural thethe characteristicscharacteristics frequency of inin the FigureFigure rigid-bodyss 77 andand mode 88 reachreachfNr-b thethe[5 ], maximummaximumCIS imposed for for the the torque frequency frequency pulsations f ihfih closed closed and, consequently,toto 80 80 Hz, Hz, while while speed for for the the fluctuations. other other motors motors The itit fluctuations is is about about 54 54– are– 6363illustrated Hz. Hz. in Figure 10. For motor 3-p4 the speed fluctuations are as high as about 1.9%, andTheThe for flow flow the otherof of CIS CIS motors caused caused up a a minor tominor about increase increase 0.7%. Theinin power power magnified losses losses speed [31] [31] as as fluctuations well well as as more more additionally harmful harmful phenomenaphenomenaincreased— CIS—torquetorque (based pulsationspulsations on [5,6]). and Consequently,and vibrations,vibrations, in whichwhich Figures areare7 analyandanaly8,ssed currented inin thethe peaks nextnext occurred subsec-subsections.tions.around the frequency corresponding to the rigid body resonance. Contrastingly, for a high moment of load inertia, the speed fluctuations were negligible and, as a result, CIS were comparatively low (Figures5 and6). The natural frequency fNr-b depends on machine size; the frequencies fNr-b are tens of Hz for small machines and a few Hz for large machines [5,6,18]. For the reason, for motor 3-p4 the characteristics in Figures7 and8 reach

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2.00 3-p4 1.75 200-p2 Energies 2021, 14, 1218 200-p4 8 of 13 1.50 200-p6 1.25 200-p8 5M-p8 Energies 2021, 14, 1218 1.00 8 of 13

the[%]n maximum for the frequency fih closed to 80 Hz, while for the other motors it is about 54–63D 0.75 Hz. 0.50 2.00 0.25 3-p4 1.75 200-p2 0.00 200-p4 1.50 50 60 70 80 90 200-p6 100 200-p8 1.25 fih [Hz] 5M-p8

Figure1.00 10. Computed amplitude of speed fluctuation (related to rated speed) vs. the frequency of n [%]n

voltageD interharmonics for the various motors and the case of NMI. 0.75 3.3. Effect0.50 of Interharmonics on Torque Pulsations The0.25 results of research on electromagnetic torque pulsations are presented in Figures 11–13. In Figure 11, exemplary torque waveforms are shown for motor 200-p2 and for two 0.00 supply cases.50 The first60 case concerns70 the purely80 sinusoidal90 voltage100 supply, and the second case is for the supply with voltage fcontaining[Hz] interharmonics of a frequency correspond- ih ing to the rigid-body resonance. For both cases, the torque waveform contains pulsating Figure 10. Computed amplitude of speed fluctuation (related to rated speed) vs. the frequency of componentsFigure 10. Computed due to the amplitude presence of speedof slots fluctuation and teeth. (related Additionally, to rated speed) for the vs. latter the frequency case, the of voltage interharmonics for the various motors and the case of NMI. mainvoltage alternating interharmonics component for the variousis caused motors by the and interharmonic the case of NMI. injection, with an amplitude of approximately 20% of the rated torque Trat and frequency fp = 7 Hz—see (1). 3.3. EffectThe of flow Interharmonics of CIS caused on Torque a minor Pulsations increase in power losses [31] as well as more harmful For further consideration, the amplitude of the torque-pulsating component of the fre- phenomena—torqueThe results of research pulsations on electromagnetic and vibrations, torque which pulsations are analysed are inpresented the next in subsections. Figures quency described with (1) is denoted as ΔTih. Its characteristics vs. the frequency of voltage 11–13. In Figure 11, exemplary torque waveforms are shown for motor 200-p2 and for two interharmonics3.3. Effect of Interharmonics are given in onFigure Torques 12 Pulsations and 13. For CRS (Figure 12), the amplitude ΔTih is supplynearly cases.6.6%. TheFor first motor case 200 concerns-p8, the the characteristic purely sinusoidal curve showsvoltage a supply, maximum and thefor second the fre- case is Thefor the results supply of research with voltage on electromagnetic containing torqueinterharmonic pulsationss of are a presentedfrequency in correspond-Figures 11–13. quency fih of about 60 Hz, which was caused by the interaction of torque pulsations due ingIn to Figure the rigid 11, exemplary-body resonance. torque waveformsFor both cases, are shown the torque for motor wavef 200-p2orm contains and for twopulsating supply tocases. voltage The interhamonics first case concerns and torque the purely pulsations sinusoidal from voltage slots and supply, teeth and (even the for second the purely case is componentssinusoidal supply,due to the the presence torque waveform of slots and contains teeth. Additionally, pulsating components for the latter of frequencycase, the mainfor thealternating supply with component voltage containingis caused by interharmonics the interharmonic of a frequency injection, correspondingwith an amplitude to the closerigid-body to 10 Hz). resonance. In turn, For for both NMI cases, (Figure the 13), torque the waveformmaximal value contains of the pulsating amplitude components ΔTih is of approximately 20% of the rated torque Trat and frequency fp = 7 Hz—see (1). roughly 18% and 20% of Trat for motor 3-p4, motor 200-p2, respectively, and the frequen- Fordue further to the consideration, presence of slots the and amplitude teeth. Additionally, of the torque for- thepulsating latter case, component the main of alternating the fre- ciescomponent fih = 57 Hz is and caused fih = 80 by Hz. the The interharmonic lowest amplitudes injection, appeared with an for amplitude motor 5M of- approximatelyp8 and motor quency described with (1) is denoted as ΔTih. Its characteristics vs. the frequency of voltage rat 20020%-p8 of— theup ratedto 9.3% torque and 12.5%Trat and of frequencyT , respectivelyfp = 7. Hz—see (1). interharmonics are given in Figures 12 and 13. For CRS (Figure 12), the amplitude ΔTih is nearly 6.6%. For motor 200-p8, the characteristic curve shows a maximum for the fre- 900 quency fih of about 60 Hz, which was caused by the interactiona of torque pulsations due to voltage850 interhamonics and torque pulsations from slotsb and teeth (even for the purely sinusoidal800 supply, the torque waveform contains pulsating components of frequency close to 10 Hz). In turn, for NMI (Figure 13), the maximal value of the amplitude ΔTih is 750 roughly 18% and 20% of Trat for motor 3-p4, motor 200-p2, respectively, and the frequen- ih 700 ih

cies [Nm] f = 57 Hz and f = 80 Hz. The lowest amplitudes appeared for motor 5M-p8 and motor T 200-p8650—up to 9.3% and 12.5% of Trat, respectively.

600 900 550 a 850 b 500 800 1.8 1.85 1.9 1.95 2 750 t [s] 700 FigureFigure[Nm] 11. 11. ComputedComputed torque torque waveform waveform of of motor motor 200 200-p2,-p2, supplied supplied with (a) purely sinusoidal voltagevolt- ageT and (b) voltage containing interharmonic of frequency fih = 57 Hz and the case of NMI. and650 (b) voltage containing interharmonic of frequency fih = 57 Hz and the case of NMI.

600

550

500 1.8 1.85 1.9 1.95 2 t [s] Figure 11. Computed torque waveform of motor 200-p2, supplied with (a) purely sinusoidal voltage and (b) voltage containing interharmonic of frequency fih = 57 Hz and the case of NMI.

Energies 2021, 14, 1218 9 of 13 EnergiesEnergies 2021 2021, 14, 14, 1218, 1218 9 9of of 13 13

7.07.0

6.06.0

5.05.0

4.04.0

3.0 3-p4

Tih Tih [%] 3.0 3-p4

Tih Tih [%] D D 200-p2200-p2 2.0 2.0 200-p4200-p4 200-p6 1.0 200-p6 1.0 200-p8200-p8 5M-p85M-p8 0.00.0 5050 6060 7070 8080 9090 100100 fih [Hz] fih [Hz] ih FigureFigureFigure 12. 12. 12. Computed ComputedComputed amplitude amplitude of ofof torque torquetorque pulsations pulsations pulsations ∆Δ ΔTTihih vs. vs. the the frequency frequency ofof of voltagevoltage voltage interharmonics interhar- interhar- monicsmonicsfor the for variousfor the the various various motors motors motors and the and and case the the of case CRS.case of of CRS. CRS.

2020 3-p43-p4 1818 200-p2200-p2 1616 200-p4200-p4 14 14 200-p6200-p6 12 12 200-p8200-p8 10 10 5M-p85M-p8 8

Tih Tih [%] 8

Tih Tih [%] D D 66 44 22 00 5050 6060 7070 8080 9090 100100 fih [%] fih [%] Figure 13. Computed amplitude of torque pulsations due to interharmonics ΔTih vs. interharmonic FigureFigure 13. 13. ComputedComputed amplitude amplitude of of torque torque pulsations pulsations due due to to interharmonics interharmonics Δ∆TTih ihvs.vs. interharmonic interharmonic frequency for the various motors and the case of NMI. frequencyfrequency for for the the various various motors motors and and the the case case of of NMI. NMI.

ItItFor should should further be be stresse consideration,stressedd that that the the the frequency frequency amplitude of of oftorque torque the torque-pulsatingpulsations pulsations due due to to component interharmonics interharmonics of the fp—see (1)—may correspond to the natural frequency of the first elastic-mode fN1e. Accord- fp—frequencysee (1)—may described correspond with (1)to the is denoted natural frequency as ∆Tih. Its of characteristics the first elastic vs.-mode the f frequencyN1e. Accord- of ingingvoltage to to [37], [37], interharmonics the the natural natural frequency frequency are given f inN1efN1e Figuresis is typically typically 12 and below below 13. Forthe the CRSnetwork network (Figure frequency frequency 12), the amplitudefor for large large and medium two-pole or four-pole motors driving turbomachinery. For example, for in- and∆T ihmediumis nearly two 6.6%.-pole For or motorfour-pole 200-p8, motors the driving characteristic turbomachinery. curve shows For a maximumexample, for for in- the ductionductionfrequency motors motorsfih of with with about rated rated 60 powers Hz,powers which of of 6.1 6.1 was MW MW caused and and 500 500 by hp, thehp, driving interactiondriving a a compressor compressor of torque and pulsationsand a a fan, fan, thethedue natural natural to voltage frequencies frequencies interhamonics f N1efN1e were were andreported reported torque as as pulsations 17, 17, 24 24 and and from28.5 28.5 Hz slotsHz (depending (depending and teeth on (evenon coupling coupling for the stiffness)stiffness)purely sinusoidal [37,38]. [37,38]. In In supply, the the authors’ authors’ the torque previous previous waveform study study contains [34], [34], similar similar pulsating frequencies frequencies components f N1efN1e were ofwere frequency deter- deter- minedminedclose with to with 10 an Hz).an analytical analytical In turn, method formethod NMI for for (Figure the the high high 13),-power- thepower maximal motors motors valueunder under of consideration. consideration. the amplitude Appro- Appro-∆Tih is priatepriateroughly calculations calculations 18% and 20%were were of carried Tcarriedrat for motorout out for for 3-p4, the the motorexemplary exemplary 200-p2, moment moment respectively, of of load load and inertia inertia the frequencies equal equal to to 500500fih%=% of 57 of the Hzthe andmotor motorfih moment= moment 80 Hz. and Theand lowestclutches clutches amplitudes of of a a sample sample appeared manufacturer. manufacturer. for motor The The 5M-p8 frequencies frequencies and motor f N1efN1e areare200-p8—up shown shown in in Table to Table 9.3% 2 2 (based and (based 12.5% on on [34]). of[34]).Trat , respectively. TheTheIt should amplitudes amplitudes be stressed Δ ΔTTih ihpresented presented that the frequency in in Figures Figures of 11 11 torque––1313 are are pulsations rather rather moderate. moderate. due to interharmonics However, However, under underfp— elasticelasticsee (1)—may mode mode resonance resonance correspond [37,38], [37,38], to the torque torque natural pulsations pulsations frequency ca ca ofnn be the be amplified ﬁrstamplified elastic-mode 50 50 times times f[38].N1e [38].. AccordingNotably Notably, , excessiveexcessiveto [37], thetorsional torsional natural vibration vibration frequency can can fcause N1ecauseis mechanical typicallymechanical below failures, failures, the such such network as as a a crack crack frequency in in the the shaft for shaft large or or couplingcouplingand medium [37,38]. [37,38]. two-pole For For this this reason, or reason, four-pole voltage voltage motors interharmonics interharmonics driving turbomachinery. should should be be regarded regarded For a example,a considera- considera- for blyblyinduction more more harmful harmful motors power power with ratedquality quality powers distu disturbance ofrbance 6.1 MWthan than andthe the previous 500previous hp, drivingworks works [18,23,31,32] a[18,23,31,32] compressor have have and indicated.indicated.a fan, the Another Another natural detrimental frequenciesdetrimental effect feffectN1e were of of torque torque reported pulsations pulsations as 17, due 24dueand to to interharmonics interharmonics 28.5 Hz (depending is is vibra- vibra- on tion.tion.coupling stiffness) [37,38]. In the authors’ previous study [34], similar frequencies fN1e were determined with an analytical method for the high-power motors under consideration. Appropriate calculations were carried out for the exemplary moment of load inertia equal

to 500% of the motor moment and clutches of a sample manufacturer. The frequencies fN1e are shown in Table2 (based on [34]).

Energies 2021, 14, 1218 10 of 13 Energies 2021, 14, 1218 10 of 13

TableTable 2. 2. NaturalNatural frequencies frequencies of of the the first ﬁrst elastic elastic mode. mode. Motor Motor Motor Motor Motor Motor motor motor motor motor motor Motor 200-p2 200-p4 200-p6 200-p8 5M-p8 200-p2 200-p4 200-p6 200-p8 5M-p8 NaturalNatural frequency frequency (Hz) (Hz) 30 3032.5 32.5 31 3030 14

The amplitudes ∆T presented in Figures 11–13 are rather moderate. However, under 3.4. Effect of Interharmonicsih on Vibration elastic mode resonance [37,38], torque pulsations can be ampliﬁed 50 times [38]. Notably, excessiveThe highest torsional vibration vibration due canto subharmonics cause mechanical and interharmonics failures, such as is aexpected crack in under the shaft no loador coupling (based on [37 [23]).,38]. ForTherefore, this reason, for the voltage demonstration interharmonics of extraordinary should be harmfulness regarded a con- of voltagesiderably interharmonics, more harmful the power case qualityof idle running disturbance was chosen. than the It previous should be works noted[18 that,23 idling,31,32] forhave most indicated. of the operational Another detrimentalperiod might effect correspond of torque to the pulsations standard due work to interharmonicsS6 15% [39]. is vibration.Recommendations concerning acceptable vibration levels are provided in the standard ISO 10816-1 Mechanical Vibration—Evaluation of Machine Vibration by Measure- ments3.4. Effect on Non of Interharmonics-rotating Parts on— VibrationPart 1: General Guidelines [33] and its updated version [40]. TheThe standard highest vibration [40] does due not tounivocally subharmonics specify and the interharmonics acceptable levels; is expected so, in this under study, no theload measured (based on vibrations [23]). Therefore, are referred for theto the demonstration recommendations of extraordinary included in harmfulness[33], similarly of asvoltage in [23]. interharmonics, the case of idle running was chosen. It should be noted that idling for mostThe broadof the- operationalband vibration period velocity might correspondcomponent tovs. the interharmonic standard work frequency S6 15% [39 fih]. is given Recommendationsin Figure 14 for motor concerning 4-p4 coupled acceptable with vibration the unloaded levels areDC provided generator in (see the standardSection 2).ISO For 10816-1 comparison, Mechanical Figure Vibration—Evaluation 15 presents the measured of MachineCIS. The highest Vibration vibration by Measurements velocity— 5.59on Non-rotatingmm/s—occurred Parts—Part in the horizontal 1: General directio Guidelinesn for the [ 33frequency] and its fih updated = 85 Hz version(Figure 14), [40]. whichThe standard is the approximate [40] does not frequency univocally corresponding specify the acceptable to the greatest levels; CIS so, in in Figure this study, 15. For the themeasured uncoupled vibrations motor (see are referredSection 2), to the recommendationsvibration level was included much less in than [33], for similarly the pre- as sentedin [23]. case. Equally of note, for supply voltage without interharmonic injection, the measured vibrationThe broad-band velocity vibration was roughly velocity 0.87 component mm/s and vs.0.34 interharmonic mm/s for the coupled frequency andfih uncou-is given pledin Figure motor, 14 respectively. for motor 4-p4 coupled with the unloaded DC generator (see Section2). For comparison,The highest Figure vibration 15 presents level in the Figure measured 14 (5.59 CIS. mm/s) The corresponds highest vibration to Zone velocity—5.59 D specified inmm/s—occurred the standard [33] in (vibration the horizontal velocity direction greater than for the 4.5 frequencymm/s in thefih case= 85 of Hz small (Figure electric 14), motors).which is According the approximate to [33], frequency “vibration corresponding values within to this the zone greatest are CISnormally in Figure considered 15. For theto beuncoupled of sufficient motor severity (see Sectionto cause2), damage the vibration to the machine level was” much[33]. For less some than other for the frequencies, presented thecase. vibrations Equally are of note, also excessive. for supply For voltage the frequencies without interharmonic fih around 51, injection, 60 and 110 the Hz, measured the vi- brationvibration levels velocity exceed was the roughly boundaries 0.87 mm/s of Zone and C— 0.341.8 mm/s mm/s for[33]. the For coupled this zone, and “ uncoupledvibration valuesmotor, are respectively. normally considered unsatisfactory for long-term continuous operation” [33].

6.0 H 5.0 V L 4.0

3.0

2.0

Vibration velocity [mm/s] 1.0

0.0 50 75 100 125 150 175 200

fih [Hz] FigureFigure 14. 14. MeasuredMeasured broad broad-band-band vibration velocityvelocity inin thethe horizontal horizontal (H), (H), vertical vertical (V) (V) and and longitudinal longitudinal(L) direction (L) direction vs. interharmonic vs. interharmonic frequency frequency for motor for motor 4-p4 coupled4-p4 coupled with with the DC the generator. DC generator.

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interharmonics 10 subharmonics

6 Iih,Ish [%] 4

0 50 75 100 125 150 175 200 f [Hz] ih FigureFigure 15. 15. MeasuredMeasured current in interharmonicsterharmonics and subharmonics (related(related toto thethe ratedrated current)current) vs.vs. the thefrequency frequency of voltage of voltage interharmonics interharmonics for motorfor motor 4-p4 4- coupledp4 coupled with with the DCthe DC generator. generator.

InThe summary, highest vibration voltage interharmonics level in Figure 14reported (5.59 mm/s) in real corresponds power systems to Zone can cause D specified unac- ceptablein the standard vibration. [33 Comprehensive] (vibration velocity investigations greater than of 4.5the mm/svibration in theof induction case of small motors electric un- dermotors). interharmonics According will to [33 be], presented “vibration in values a separate within paper. this zone The arestudy normally includes considered a compari- to sonbe of of sufficient motors of severity various to rated cause power damages and to number the machine” of poles [33 as]. well For some as the other admissible frequencies, level ofthe voltage vibrations interharmonics, are also excessive. determined For theaccording frequencies to thefih criterionaround 51,of induction 60 and 110 motor Hz, thevi- bration.vibration levels exceed the boundaries of Zone C—1.8 mm/s [33]. For this zone, “vibration values are normally considered unsatisfactory for long-term continuous operation” [33]. 4. DiscussionIn summary, voltage interharmonics reported in real power systems can cause unacceptable vibration. Comprehensive investigations of the vibration of induction motors As mentioned in Section 1, previous studies have indicated a minor impact of voltage under interharmonics will be presented in a separate paper. The study includes a compari- interharmonics on an induction cage machine (the term interharmonics in this work in- son of motors of various rated powers and number of poles as well as the admissible level of clude components of frequency greater than the fundamental voltage component but not voltage interharmonics, determined according to the criterion of induction motor vibration. of an integer multiple). The reported impact [18,23,31,32] was generally limited to speed fluct4. Discussionuations, small increases in power losses and winding temperature, and moderate vi- brationsAs— mentionedslightly exceeding in Section 1 the, previous lower boundariesstudies have of indicated Zone C. a Consequently, minor impact of from voltage the pointinterharmonics of view of oninduction an induction motors, cage interharmonics machine (the term could interharmonics be regarded as in a this slightly work noxious include pocomponentswer quality of disturbance. frequency greater than the fundamental voltage component but not of an integerThe results multiple). of the Thecurrent reported research impact reveal [18 that,23,31 voltage,32] was interharmonics generally limited can exert to speed ex- traordinarilyfluctuations, smallharmful increases effect on in inductionpower losses motors. and winding Voltage interharmonics temperature, and of moderate levels re- portedvibrations—slightly in real power exceedingsystems may the cause lower vibration boundaries “of sufficient of Zone C. severity Consequently, to cause fromdamage the topoint the ofmachine view of” [33]. induction To compound motors, interharmonics the problem, interharmonics could be regarded may asresult a slightly in torque noxious pul- sationspower qualitycorresponding disturbance. to the natural frequency of the first elastic mode of power trains with mediumThe results- and of high the current-power induction research reveal motors. that For voltage the investigated interharmonics high-power can exert motors ex- andtraordinarily the negligible harmful moment effect onof inductionload inertia, motors. the observed Voltage interharmonicsamplitude of torque of levels pulsations reported wasin real up powerto 20% systems of Trat. This may value cause apparently vibration “of does sufficient not seem severity especially to cause detrimental. damage How- to the ever,machine” under [33 the]. To elastic compound mode resonance, the problem, torque interharmonics pulsations can may be result amplified in torque dozens pulsa- of timestions corresponding[38]. Consequently, to the the natural potential frequency occurrence of the of first the elastic torsional mode resonance of power may trains lead with to themedium- destruction and high-power of a clutch or induction a motor motors. shaft. For the investigated high-power motors and the negligibleAt the same moment time, ofpower load inertia,quality thestandards observed and amplitude rules should of torque provide pulsations effective was pro- up tectionto 20% offorT anyrat. Thiselectrical value equipment apparently against does not potential seem especially malfunction detrimental. due to excessive However, power under qualitythe elastic disturbances. mode resonance, Thus, torquelimitations pulsations on voltage can be interharmonics amplified dozens are ofurgently times [ 38needed.]. Conse- In practice,quently, thethis potential task requires occurrence the determination of the torsional of admissible resonance may levels lead for to various the destruction components of a ofclutch a power or a system, motor shaft. including induction motors. Therefore, in-depth research on induction motorsAt under the same considered time, power power quality quality standards disturbances and is rules required. should provide effective pro- tection for any electrical equipment against potential malfunction due to excessive power quality disturbances. Thus, limitations on voltage interharmonics are urgently needed. In practice, this task requires the determination of admissible levels for various components

Energies 2021, 14, 1218 12 of 13

of a power system, including induction motors. Therefore, in-depth research on induction motors under considered power quality disturbances is required.

5. Conclusions The presented results of investigations have revealed extraordinarily harmful effect of voltage interharmonics on induction motors. It was found out that they can cause unacceptable vibration. Additionally, voltage interharmonics lead to pulsations of the rotational torque, whose frequency may correspond to the natural frequency of the ﬁrst elastic mode. Under resonance the torque pulsations may be magniﬁed dozens of times, which could result in the destruction of the power train. Consequently, voltage interharmonics should be regarded as much more detrimental power quality disturbance than the previous works [18,23,31,32] have indicated. The results of the research also prove there is a necessity to impose limitations on voltage interharmonics. From the point of view of induction motors, the admissible levels of interharmonics (except subsynchronous ones, which are not covered by this study) should be determined according to the criterion of the allowable longitudinal and torsional vibration. Additionally, separate limits should be determined according to the criterion of the correct work of other electric equipment, like for example power electronic appliances, measurement and control systems or modern light sources. The lowest of the above limits should be included in the power quality standards as the admissible levels of voltage interharmonics.

Author Contributions: Conceptualisation, P.G.; methodology, D.H., P.K.; formal analysis, D.H, P.K., A.M. and M.P.; investigation D.H, P.K., A.M. and M.P.; writing—original draft preparation, P.G.; supervision, P.G. All authors have read and agreed to the published version of the manuscript. Funding: This project is financially supported under the framework of a program of the Ministry of Science and Higher Education (Poland) as “Regional Excellence Initiative” in the years 2019–2022, project number 006/RID/2018/19, amount of funding 11 870 000 PLN. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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