476 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 30, NO. 2, JUNE 2015

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1 476 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 30, NO. 2, JUNE 2015 Comparison of Frequency and Time-Domain Iron and Magnet Loss Modeling Including PWM Harmonics in a PMSG for a Wind Energy Application Damian Kowal, Peter Sergeant, Luc Dupré, and Haran Karmaker, Life Fellow, IEEE Abstract This paper presents the calculation of the electromagnetic losses for a 2.1-MW permanent magnet synchronous generator for wind energy application. The focus is on recognizing the significance of including the analysis of higher harmonics in the electromagnetic loss calculation. The analyzed harmonics include the ones resulting from the use of a pulse width modulation (PWM) of the voltage of the generator. The magnet losses calculated for the PWM current are several times higher than the ones calculated for the sinusoidal current. In addition, frequency-domain and time-domain models for iron loss calculation are compared. The frequency-domain model that assumes a sinusoidal variation and considers only the fundamental component of the magnetic induction in the stator core material underestimates the iron losses in the machine. Especially, when the additional losses resulting from the higher harmonics, rotational fields, and minor loops are taken into account. Finally, it is shown how the composition and thickness of the electrical steel used in the stator core of the generator influences the total core losses. Index Terms Electrical steel, higher harmonics, iron losses, loss modeling, permanent magnet synchronous generator (PMSG), pulse width modulation (PWM). I. INTRODUCTION WHEN REVIEWING the current status of the wind energy market, it can be easily observed that the permanent magnet synchronous generators (PMSGs) have attracted a lot of attention among large scale wind turbine manufacturers. PMSGs for large scale wind turbines exist in two generator systems: 1) direct-drive; and 2) geared systems. When observing the market trends, it is difficult to conclude which of the systems is preferred. In this paper, the focus is on the high speed (1600 r/min) 2.1-MW rated PMSG with a gearbox. The challenge in designing a modern generator is to meet the needs of the energy efficiency and the market costs. Moreover, the market situations impose the maximal exploitation of new topologies and materials. Therefore, to meet the requirements Manuscript received February 23, 2014; revised November 8, 2014; accepted November 11, Date of publication December 12, 2014; date of current version May 15, Paper no. TEC D. Kowal and L. Dupré are with the Department Electrical Energy, Systems, and Automation, Ghent University, B-9000 Ghent, Belgium ( damian.a.kowal@gmail.com; luc.dupre@ugent.be). P. Sergeant is with the Department Electrical Energy, Systems, and Automation and the Department Industrial Technology and Construction, Ghent University, B-9000 Ghent, Belgium ( peter.sergeant@ugent.be). H. Karmaker is with the Global R&D Center, TECO-Westinghouse Motor Co., Round Rock, TX USA ( karmakeh@tecowestinghouse.com). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TEC of new designs, deep insight in the modeling of electromagnetic losses in machines is necessary. In recent years, a lot of attention is drawn to the modeling of losses particularly in the magnets of the PMSG. This is because an incorrect estimation of the losses may lead to an underestimation of losses, which in the end may cause overheating of the magnets leading to their permanent demagnetization. Since the magnet losses are caused by the harmonics in the magnetic induction in the air gap, a number of publications deal with a challenge of analytical modeling of the harmonics in the air gap [1], [2], studying the influence of the stator slotting [3], winding distribution in the stator slots [4], [5], and [6], as well as the influence of pulse width modulation (PWM) harmonics [7], [8]. In order to reduce the energy losses in the magnets and to prevent demagnetization, the segmentation of the magnets is extensively studied. In [9] [12], the effects of the axial and circumferential segmentation were studied and proved to be effective for magnet loss reduction. In this paper, the magnet losses are studied for the 2.1-MW PMSG for wind energy application. The finite-element method (FEM) was chosen for the investigation of the magnet losses in this machine under different load conditions. Different loads enable the study of the influence of different harmonic sources on the total magnet losses in the machine. The study of the influence of the higher harmonics resulting from the use of the PWM for the control of the voltage of the generator is based on the current waveforms recorded in the existing machine. In addition, a study of the iron losses in the stator core is performed. The statistical theory of loss separation [13], [14] is used. Two models are used and compared. The first is a simple frequency-domain model that assumes a sinusoidal flux variation of magnetic induction in the stator steel. The second is a time-domain model that calculates iron losses for arbitrary flux waveforms. Finally, the additional losses resulting from rotational field and minor loops are taken into account. The iron losses are studied for two load conditions, viz, the sinusoidal current as well as for the PWM current. The iron losses for the 2.1-MW generator are compared for different electrical steel grades. The influence of the permeability and the lamination thickness of steel on the total iron losses of the machine is studied. II. FEM MODEL OF THE GENERATOR The generator is a three-phase, high-speed (1600 r/min) radial flux, outer stator PMSG [15]. The generator was modeled with 2-D FEM using commercial software COMSOL. Due to IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See standards/publications/rights/index.html for more information.

2 KOWAL et al.: COMPARISON OF FREQUENCY AND TIME-DOMAIN IRON AND MAGNET LOSS MODELING INCLUDING PWM HARMONICS 477 Fig. 1. Geometry of the 2-D electromagnetic FE model for the 2.1-MW generator. the symmetry in the generator, only one pole pair of it was modeled in order to limit the computational time. Fig. 1 shows the modeled geometry. Notice that four neighboring magnets have the same magnetization orientation, e.g., the magnetization of the permanent magnets enumerated 1 4 in Fig. 1 is oriented toward the air gap, while the other four magnets are oriented in the opposite way. According to data provided, the magnet remanence is around 1.26 T at 20 C. It is expected that the magnet remanence will drop with the increase of the magnet temperature during operation of the generator. To take this into account a 10% drop of the magnet remanence was chosen and the value of B r used in the simulations was set to 1.13 T for an average temperature of 110 C. For the finite-element modeling, the equivalent relative permeability value of the magnets was set to 1 and the electrical conductivity value was set to 0.6 MS/m. The 2.1-MW high-speed PMSG is a commercial generator for wind energy application. The voltage of the generator is enforced and controlled through PWM, as a result, the current of the generator contains PWM-related harmonics. The threephase PWM current of this machine was measured as shown in Fig. 2. Moreover, Fig. 2(b) presents the harmonic content of the PWM current without the base frequency. III. MAGNET LOSS MODELING According to [9] and [12], eddy currents are likely to occur in the permanent magnets and may cause a significant increase of the magnet temperature that can lead to a partial irreversible demagnetization. The currents occur due to the higher harmonics appearing in the air gap magnetic field and they origin from the reluctance effect [3], winding distribution [4], [6] and the fact that the stator current is not sinusoidal [7]. Due to the relatively large size of the permanent magnet per pole in the considered machine, large eddy currents are expected. To prevent overheating and demagnetization of the magnets, a segmentation of the magnets is considered. Fig. 1 Fig. 2. (a) Three-phase PWM current imposed in the 2-D FE model. The waveforms are reproduced based on measurement results. (b) Harmonic content without base frequency. presents how the circumferential segmentation is realized in this machine. However, to be able to take into account a finite axial length and to make the investigation of magnet segmentation in axial direction possible, the 2-D 3-D model is used [12] (see Section III-A). The magnet losses in the 2.1-MW generator are investigated for three load conditions of the machine, such as, no-load, fullload with sinusoidal stator current, and full-load with PWM stator current (see Fig. 2). A. Methodology The 2-D 3-D method for the magnet loss calculation consists of three steps. The first step is using a 2-D electromagnetic model, where only one pole pair of the machine is considered due to the symmetry in the machine (see Fig. 1). A series of static computations is performed using a nonlinear material model of the steel in the stator. The time step used for the consequent static computations is equal to 0.01 ms for a machine rotating at 1600 r/min. A total of 1250 steps are taken to cover one electrical period. The magnetic induction vector is recorded during each time step over the magnet domain. The recording is performed for points distributed over the surface of the magnet. During the second step of the numerical procedure, the values of B t are calculated locally in the permanent magnet, based on the recorded variations of the magnetic induction in time. This is done in order to be able to apply a computationally inexpensive method for calculating the magnet losses in the 3-D model in step 3, as presented in [12]. In the third and final step, a 3-D FEM model of one magnet segment is considered. In this method, it is assumed that eddy

3 478 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 30, NO. 2, JUNE 2015 currents in the magnet are resistance limited, and permanent magnet segments are electrically isolated from each other. Since eddy currents are resistance-limited, their influence on the inducing magnetic field distribution is negligible. Therefore, the equations that govern the resistance-limited eddy currents are equivalent to those of a linear magnetostatic field problem [12] H m = J m E = B (1) t B m = μh m J e = σe (2) where μ is the absolute permeability of the magnetic material, σ is the electrical conductivity of the permanent magnet material when H m E, J m B t, and μ σ. The left-hand side of (1) and (2) represents the equations governing the linear megnetostatic field problem. The source term B t, which is recorded from 2-D FEM in the xy-plane (the plane shown in Fig. 1), is imposed over the whole 3-D domain of the magnet. This means that the source term is not varying in z-direction because the source term is not a function of z. Nevertheless, the induced currents computed by the 3-D model have a 3-D path (including the z-direction): they circulate within the magnet. To summarize, based on the induction values in the magnet recorded in 2-D simulation during one electrical period, the value of B t in the PMs is calculated in the second step of the procedure. These values are then used as a source term in the 3-D static computation as the equivalent J e. The source term in time and space is obtained through an interpolation of the recorded values. Finally, a series of static 3-D computations is performed to calculate the eddy current losses at each time instant and for each segment of the magnet. B. No Load Magnet Losses At first, the magnet losses were investigated for the no-load condition. Fig. 3(a) presents the magnetic induction B changing within one electrical period of the machine. For this figure, B is recorded in the center of one of the magnet segments (see Fig. 1) on the surface facing the air gap and the stator. Fig. 3(b) shows the higher harmonic content of the magnetic induction waveform without the dc term. It can be observed that the harmonic with the highest amplitude is the 24th harmonic. This harmonic and a multiple of it are the result of the reluctance effect of the teeth of the stator (the machine has 24 teeth per pole pair). The amplitude of the dc term of the magnetic induction equals 0.76 T and is much higher than the amplitude of the harmonics. In Fig. 4(a) the instantaneous magnet losses are shown for one electrical period of the investigated generator. The magnet losses are computed for one segment of the magnet. In Fig. 4(b), the Fourier analysis results are shown. The most significant harmonics are a multiple of the 24th harmonic, related to the number of slots per pole pair. C. Magnet Losses in Case of Nominal Sinusoidal Current The second set of simulations was performed with a balanced three-phase sinusoidal current (1950-A rated current) Fig. 3. (a) Time-dependent magnetic induction in the magnet area facing the air gap. Curve obtained from no load simulation. (b) Harmonic content of the magnetic induction. Fig. 4. (a) Time-dependent magnet losses for one magnet segment as a result of 2-D 3-D FE simulation under no load conditions. (b) Harmonic content of the magnet losses per one magnet segment. implemented in the winding domains of a 2-D FE model (first step of magnet losses calculation). The winding scheme is shown in Fig. 1. In the simulations, the cos ϕ = 0.84, which corresponds with the testing condition of the machine. This power factor corresponds to the waveforms measured for the machine. The simulation was also performed for one electrical period of the machine. Fig. 5(a) presents the magnetic induction changing within one electrical period of the machine. Here, again, B is recorded in the center of the magnet segment on the surface facing the air gap and the stator. Fig. 5(b) presents the higher harmonic content

4 KOWAL et al.: COMPARISON OF FREQUENCY AND TIME-DOMAIN IRON AND MAGNET LOSS MODELING INCLUDING PWM HARMONICS 479 Fig. 5. (a) Time-dependent magnetic induction in the magnet area facing the air gap. Curve obtained from a series of static 2-D FE computations for sinusoidal current (1950-A rms). (b) Harmonic content of the magnetic induction. Fig. 6. (a) Time-dependent magnet losses for one magnet segment as a result of 3-D FE computation. Simulation with three-phase sinusoidal current (1950-A rms). (b) Harmonic content of the magnet losses per magnet. of the magnetic induction waveform without the dc term. It can be observed that the harmonic with the highest amplitude is again the 24th harmonic. On the other hand, the sixth harmonic and the multiples of it result from the nonsinusoidal distribution of the winding in the stator. Considering the different mechanisms inducing electric currents in the PMs, one may write for the local current density in the PMs J(ϕ r,t)= J i (ϕ r ) cos(iω s t + ϕ i ) (3) i where i refers to the harmonics appearing in the magnetic induction waveform in the magnet [1], [2]. ϕ r is the rotor angular variable and ω s is the stator current velocity. The following equation shows which harmonics can then be expected in the magnet losses: p m (ϕ r,t)= J 2 (ϕ r,t) σ = 1 J i J j cos [(i + j)ω s t + ϕ i + ϕ j ] 2σ i + 1 2σ j J i J j cos [(i j)ω s t + ϕ i ϕ j ] i j where p m (ϕ r,t) are the instantaneous magnet losses. Equation (4) shows that the harmonics expected in the magnet losses will have the order i, i + j, and i j with i and j harmonics appearing in the induced currents in the PMs. In Fig. 6(a), the instantaneous magnet losses can be observed for one electrical period of the investigated generator. The magnet losses are computed for one segment of the magnet. When comparing magnet losses calculated for the sinusoidal current (4) with the no load losses, it can be observed that the additional harmonics appear in case of the sinusoidal current. These space harmonics are caused by the nonsinusoidal distribution of the winding. The presence of the additional harmonics in the magnet loss waveform can be observed in Fig. 6(b). In the spectrum of the magnet losses, the combination of all the harmonics present in the magnetic induction waveform given in Fig. 5(b) can be observed. Notice, for example, that the harmonic 36 in the losses result from the 12th, 24th, and 48th harmonic in the magnetic induction. Indeed, according to (4), one has i j =48 12 = 36 and i + j =24+12=36. 1) Penetration Depth of Harmonics: Fig. 5(b) presents the harmonic content of the magnetic induction in the magnet in a point facing the air gap. For that case the amplitude of the 24th harmonic is high compared to the other harmonics. However, the depth of the penetration of a particular harmonic inside the magnet depends on the origin of harmonics. Fig. 7 presents how the amplitudes of different harmonics behave for increasing depth in the magnet domain. Notice that the depth of penetration of some of the harmonics is not related with the skin depth of these harmonics. For example, the depth of penetration of 24th harmonic is lower than is expected from calculation of skin depth. This is due to the fact that 24th harmonic originates from the reluctance effect of stator teeth. D. Magnet Losses for the PWM Current The final set of simulations of the magnet losses has the PWM current implemented in the winding domains of the 2-D FE model (see Fig. 2). In the machine, the PWM voltage is enforced and the current containing additional harmonics is a result of this enforcement. In the simulations, we enforce the current, but it produces the induced voltage same as the input PWM voltage. The simulation conditions are the same as for

5 480 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 30, NO. 2, JUNE 2015 Fig. 7. Penetration of the higher harmonics in the magnet: the figure shows the amplitude of the radial component of the magnetic induction as a function of depth in the magnet relative to its height. Here, zero corresponds with the middle point on the surface of the magnet facing the air gap. Fig. 9. (a) Time-dependent magnet losses for one magnet segment as a result of the 3-D FE computation. Simulation with three-phase PWM current (1950-A rms). (b) Harmonic content of the magnet losses per magnet. Fig. 8. (a) Time-dependent magnetic induction in the magnet area facing the air gap. Curve obtained from a series of static 2-D FE computations for the PWM current (1950-A rms). (b) Harmonic content of the magnetic induction. the previous simulation. The harmonic content of the magnetic induction waveform is presented in Fig. 8(b). The most dominating harmonic in the spectrum of the magnetic induction waveform in the magnet is the 24th harmonic originating from the reluctance effect. Fig. 9 presents the time waveform of magnet losses for the PWM current, together with the analysis of the harmonics content. The total magnet losses per machine for three load conditions are equal 57, 95, and 751 W for no load, full-load sinusoidal, and PWM current, respectively. The magnet losses per machine for no load are 40% lower than in the case of the load conditions with pure sinusoidal current. When the sinusoidal current was replaced by the PWM current originating from the measurements, the magnet losses increased by 4.5 times compared to the magnet losses for the sinusoidal current. It is important to notice that none of the amplitudes of the higher harmonics in the Fig. 10. Effect of the axial segmentation of the magnets on the total magnet losses of the generator for two load conditions: sinusoidal and PWM stator current. PWM current is exceeding 2% of the amplitude of the current corresponding with the fundamental frequency. E. Effect of Axial Segmentation of the Magnets Until now the computations were performed for peripherally segmented magnets. With the use of the FE model, the influence of the axial segmentation on the total value of magnet losses in the machine is investigated. Here, we consider an axial segmentation by varying from 1 to 70 segments. The simulations were performed with a balanced three-phase sinusoidal current (1950-A rms), where cosϕ equals 0.84 as well as with the PWM current. The magnet losses presented in Fig. 10 are the total magnet losses for the machine. It can be observed that the axial segmentation has a significant influence on the total magnet losses of the machine. Increasing the number of axial segments of the magnets from 1 to 70, decreases the total magnet losses per machine by nearly a factor of 9 when the sinusoidal current

6 KOWAL et al.: COMPARISON OF FREQUENCY AND TIME-DOMAIN IRON AND MAGNET LOSS MODELING INCLUDING PWM HARMONICS 481 is considered. For the case of the PWM current, the studied segmentation range is between 50 and 70 axial segments. For larger segments, the magnet losses are not resistance limited and the FE model will overestimate the magnet losses. In [12], by using 20 axial segments, the magnet losses are reduced 2 to 12 times compared to a solid magnet. The scale of reduction depends also on the number of circumferential segments. In [9], the magnet losses are reduced more than four times by using 20 axial segments instead of one. IV. IRON LOSS MODELING The iron losses for different steel grades were computed using the same 2-D electromagnetic FE model as used for magnet losses computation. The investigation is performed on different steel grades to understand the phenomena causing the losses. Actual steel grade used in the manufactured machine is not analyzed, since the required properties were not available from the vendor. Two statistical methods for the iron loss computation are used. The statistical loss theory of [13] and [14], that relies on the principle of loss separation into hysteresis, classical, and excess losses is used. The first method is a frequency-domain method, which assumes sinusoidal induction waveforms and requires the computation of a peak magnetic induction value in each elementary volume, where the iron losses are computed. The second method is the time-domain method that requires the time waveform of the magnetic induction in each elementary volume in which losses are computed. To acquire necessary data for both methods in the FE model of one pole pair of the machine, the domain of the iron core of one pole of the stator is divided into elementary volumes in which the induction values (B x and B y ) are computed. In 2-D, the volume of each of the 12 teeth is divided into 40 elementary volumes and the part of the stator yoke covering one pole is divided into 300 elementary volumes. The time variation of the local magnetic induction patterns is obtained by a proper sequence of static 2-D FE simulations. Each of the static computations corresponds with a different value of the rotor angle with respect to the stator. The considered angles are within the range of one electrical period of the machine. One step of the rotor angle corresponds with 0.01 ms for the generator rotating with 1600 r/min (1250 points are recorded per electrical period). For both methods, used for the iron losses computation, the corresponding, material-related coefficients have to be identified based on the loss measurements for the applied steel grades. The material characteristics implemented in the FEM for the stator core are nonlinear for all simulations. A. Frequency Domain Iron Losses Computation In the frequency domain, the equation for iron losses for the case of sinusoidal unidirectional field can be expressed as follows [13]: P t = P h + P c + P e = W h.f σπ2 d 2 B 2 p f 2 +2B p f ( n 2 0 V π2 σgsv 0 B p f n 0 V 0 ). (5) TABLE I STATOR IRON LOSSES FOR THE 2.1-MW GENERATOR FOR DIFFERENT STEEL GRADES Steel grade M330 M330P M400 M400XP M600 M A 50 A 50 A 50 A 50 A 100 A Iron losses per machine (kw) The iron losses were computed using the frequency-domain method. The simulation was done at nominal sinusoidal current and rated speed (1600 r/min). where P h, P c, and P e are hysteresis, classical, and excess losses, respectively, f is the frequency of the flux variation, W h is a hysteresis stored energy per cycle (J/m 3 ), σ is the electrical conductivity of the steel, d is the lamination thickness of the steel sheet, B p is the peak value of the magnetic induction in the steel, and n 0 and V 0 are material-dependent microstructural parameters (dependent on the B p ). P t is the total value of the iron losses expressed in W/m 3. Equation (5) can be simplified and calculated per elementary volume as ) P FeV = V Fe (a ˆB p α f + b ˆB p 2 f 2 + c ˆB p f( 1+e ˆB p f 1) (6) where V Fe is the elementary volume of the iron and B p is the peak induction value in the elementary volume, where iron losses are calculated. The a, b, c, e and α are the steel-gradedependent parameters that can be calculated and fitted based on the material properties and the measurements of iron losses. P FeV is the total iron losses per defined mass of the steel expressedinw. Parameter b can be easily defined based on the electrical conductivity value and the thickness of the lamination d of the steel grade investigated. To define parameters a and α, the iron loss measurements for the electrical frequency close to 0 Hz are necessary. The parameters c and e can be fitted using the least-square method with the iron loss measurements performed for a range of frequencies and magnetic induction peak values. It is important to notice that the parameters c and e fitted in this way will be independent from peak induction value B p [compare (5)]. Based on the FE simulations and obtained induction waveforms for each of the considered elementary volumes in the stator geometry, the peak value B p is calculated. Then, the computed B p values are used to calculate the losses per assigned elementary volume [see (6)]. Finally, summation of the iron losses for all elementary volumes covering the whole stator iron gives the total iron losses in the stator. The iron losses were computed and compared for three pairs of steel grades. Two pairs with the same specific loss value and thickness of the lamination but different permeability value (M330-50A with M330P-50A, and M400-50A with M400XP- 50A). One pair of steel grades with the same specific loss value but different thickness of the lamination (M600-50A with M A). The total iron loss value for the 2.1-MW machine was calculated for all steel grades (see Table I). The values are

7 482 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 30, NO. 2, JUNE 2015 Fig. 11. Comparison of iron loss values between grade M330-50A and M330P-50A for different induction levels and three frequencies (10, 50, and 100 Hz). Fig. 13. Comparison of iron loss values between two steel grades for induction level 1.7 T and different frequency values. Fig. 12. Comparison of iron loss values between grade M400-50A and M400XP-50A for different induction levels and three frequencies (10, 50, and 100 Hz). calculated at the nominal speed of the machine (1600 r/min). It can be observed, that the grades with lowest specific loss value have the lowest iron losses. Table I presents the difference in total losses per machine for steel grades with the same specific loss value and lamination thickness but different permeability. Compare grade M330-50A with M330P-50A and M400-50A with M400XP-50A. Grades denoted with P have a higher induction in the useful range of H. Figs. 11 and 12 present the measured iron losses as a function of B p for 10, 50, and 100 Hz. Notice that the value of losses is the same for both pairs of grades for the B p equal to 1.5 T and 50-Hz frequency but differ significantly for higher induction levels and higher frequency. The difference in composition of P grades results in higher electrical conductivity when compared with low permeability grades. This makes the losses for higher frequencies (above 50 Hz) higher than losses in case of low permeability grades. Since the nominal frequency, the considered generator is 80 Hz and the operational magnetic induction higher than 1.5-T peak induction value, both grades with higher permeability (M330P-50A and M400XP-50A) are expected to give higher total iron losses for this machine than the low permeability grades (M330-50A and M400-50A). Moreover, two grades with the same specific loss value and different thickness of the lamination are compared with respect to the total iron loss value for the considered machine. The generator using a thinner grade M600-50A (0.5-mm lamination thickness) has a lower value of total losses than the thicker grade M A (1 mm). The difference comes from the fact that the specific loss value is defined for 1.5 T and 50 Hz, while the rated frequency of the generator is 80 Hz and the induction level is typically higher than 1.5 T. Fig. 13 shows the iron loss measurements for the grades M600-50A and M A for the induction level of 1.7 T and different frequencies. The calculation of losses mentioned in Table I was performed for the nominal speed of the machine, which corresponds with the nominal electrical frequency of 80 Hz. Taking that into account and considering Fig. 13, it is clear why the grade M600-50A has lower iron losses than the grade M A. The grade M600-50A has slightly higher losses than M A for frequencies lower than 35 Hz and lower losses for higher frequencies. B. Time-Domain Iron Losses Computation In this method of iron loss calculation, the computed B x and B y values for each of the considered elementary volumes of the stator pole pair are used to determine the time waveform of the induction. It can be observed that the magnetic induction vector is changing direction in some parts of the geometry. Therefore, for each of the volumes, the dominant direction is recognized and the induction vector for each time point is then projected on the dominant (major) direction (see Fig. 14). Consequently, iron losses are first computed as losses under unidirectional magnetic induction. The time-domain iron loss model used for unidirectional arbitrary flux waveforms neglects the additional hysteresis losses when the minor loops occur. The calculation of losses for minor loops is then added as a next step, similar to the estimation of losses from elliptical or rotational effects. 1) Hysteresis Loss Component: The hysteresis energy losses are calculated as follows. First, the total energy losses are measured for 2 Hz. Then, the classical energy losses are calculated

8 KOWAL et al.: COMPARISON OF FREQUENCY AND TIME-DOMAIN IRON AND MAGNET LOSS MODELING INCLUDING PWM HARMONICS 483 Fig. 14. Magnetic induction vector locus, where recording point was set in the crossing point of the mean diameter of the yoke and the symmetry axis of the tooth. The simulation was performed for the full-load condition with sinusoidal current injected. for 2 Hz using (5). Finally, the hysteresis energy losses are calculated by subtracting the classical from total energy loss. 2) Classical Loss Component: The energy loss per cycle of the lamination with thickness d and conductivity σ depends on the time derivative of B(t) W cl = 1 T ( ) 2 db 12 σd2 dt. (7) 0 dt 3) Excess Loss Component: An expression for the instantaneous excess loss is given in [14] p exc = n 0V 0 2 ( 1+ 4σGS n 2 0 V 0 db dt ) 1 db dt. (8) Here, n 0 and V 0 are functions of B p and should be fitted from measured loops at several frequencies and amplitudes; n 0 is the number of simultaneously active magnetic objects for frequency f 0 and V 0 defines the statistics of the magnetic objects. The dimensionless coefficient G = and the lamination cross section S are known constants. The fitting of n 0 and V 0 is done separately for many B p values. From [13, pp ], it is shown that n depends on the excess field H exc, expecting, as a first approximation, a linear dependence n(h exc )=n 0 + H exc. (9) V 0 Here, n 0 and V 0 can be fitted starting from the loss of a measured loop P meas that is obtained for several B p and frequencies. Given P meas, the hysteresis loss and the classical loss computed as explained previously, H exc can be found from H exc = P meas P h P cl (10) 4B p f wherein P h = fw h is the hysteresis loss in W/m 3, and the classical loss is obtained by multiplying (7) by f. With this H exc, the number of active magnetic objects is n(h exc )= 4σGSB pf. (11) H exc Fig. 15. Magnetic induction vector locus recorded in the center of a tooth. The simulation was performed for the full-load condition with sinusoidal current injected. With (10) and (11), the curves (9) for every B p can be fitted. The parameter S was calculated based on the dimensions of steel samples. The function n(h exc ) is approximated by a straight line. 4) Rotational Losses: Figs. 14 and 15 present the variation of B x and B y over one electrical period for a stator tooth and stator yoke, respectively. In the stator tooth, the magnetic induction is computed in the center of the tooth. In the stator yoke, the saving point was set in the intersection of the symmetry axis of the tooth and the circle defined by the mean diameter of the yoke. The simulation was performed for the full-load condition with sinusoidal current injected. Analyzing Figs. 14 and 15, it can be assumed that the additional losses originating from the rotational field might influence the value of total losses. A simple model was used to take into account the rotational losses. The model presented in [16] assumes recognizing two directions in the loci (see Fig. 15) the major (dominant) and minor direction. Then, the projection of the computed induction vector on both directions is performed. The iron losses are calculated for both directions using two iron loss models, frequency- and time-domain model. Finally, for each of the elementary volumes, the iron loss value for major and minor direction is added giving the total iron loss value (see Table III). 5) Sinusoidal Current: At first, the series of static computations were performed for sinusoidal three-phase currents injected in the copper areas of the 2-D FE model. The simulation parameters are the same as for the frequency-domain model. Results were obtained for two steel grades: M330-50A and M600-50A. The simulations are performed for the rated speed of the machine (1600 r/min ), that means that the electrical frequency is equal to 80 Hz. Comparing iron losses per machine for the same steel grades between Tables I and II, it can be observed that losses computed by the time-domain loss model are higher than the losses computed by the frequency-domain loss model. This is due to the assumption taken for frequency-domain calculations, i.e., having a sinusoidal flux waveform. In reality, the time variation of the local magnetic induction for several parts of the stator core is different from a sinusoidal time variation and this results in additional iron losses calculated

9 484 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 30, NO. 2, JUNE 2015 TABLE II IRON LOSSES FOR GRADE M330-50A, M600-50A Type of iron losses per machine M330-50A M600-50A Sinusoidal stator current Classical losses [kw] Hysteresis losses [kw] Excess losses [kw] total losses [kw] PWM stator current, no minor loops Classical losses [kw] Hysteresis losses [kw] Excess losses [kw] total losses [kw] PWM stator current, including minor loops Classical losses [kw] Hysteresis losses [kw] Excess losses [kw] total losses [kw] The iron losses were computed using the time-domain method. The simulation was done at nominal current and rated speed (1600 r/min). Only unidirectional flux was considered. TABLE III IRON LOSSES FOR GRADE M330-50A, M600-50A M330-50A M600-50A Type of iron Freq Time Freq Time loss model domain domain domain domain Iron losses in major direction B [kw] Iron losses in minor direction B [kw] Total iron losses including rotational fields [kw] The simulation was done at nominal sinusoidal current and rated speed (1600 r/min). Unidirectional and rotational flux was considered. by the time-domain iron loss model (compare with Figs. 16 and 17). Moreover, in the frequency-domain model, the parameters c and e, which represent the microstructural parameters n 0 and V 0 in time domain, are considered independent from B p.this assumption simplifies the frequency-domain model making it less accurate. 6) PWM Current: Second, the current given as an input to the FE model was changed from a sinusoidal one to the measured PWM current (see Fig. 2). As can be observed in Table II, the total iron losses per machine have increased when compared to the case of sinusoidal current. The increase of the iron losses can be observed for both steel grades. Note that, the applied iron loss model does not take losses due to minor loops into account. By analyzing the magnetic induction time waveforms computed for the stator geometry, the minor loops are identified. Based on (6), the additional hysteresis loops were estimated for both materials. The total iron losses for the PWM current with estimated additional losses due to the minor loops are also presented in Table II. Table II presents the separation of losses into three components for both machines using different materials. It can be Fig. 16. Magnetic induction time waveforms computed in the center of the stator tooth. Comparison of results for the simulation with sinusoidal stator current and with PWM stator current. (a) Magnetic induction computed during one electrical period. (b) Magnetic induction computed during 1.75 ms. observed that the largest loss component for both steel grades is the hysteresis loss component. The least significant is the excess loss component. The machine with the low loss grade M330-50A has significantly lower total iron losses when compared to the high loss grade M600-50A. Figs. 16 and 17 present the magnetic induction waveforms for the center of the stator teeth and the center of the stator yoke, above the stator slot. The presented figures indicate the difference between the real-time waveform of the induction in the stator and the sinusoidal waveform assumed in the frequencydomain iron loss computation method. By analyzing Figs. 16 and 17, it can be observed how the use of PWM influences the magnetic induction time waveforms, when compared to a pure sinusoidal stator current. V. CONCLUSION In this paper, the electromagnetic losses in the 2.1-MW PMSG are calculated, taking into account the influence of higher harmonics. The magnet losses were computed using 2-D 3-D models and static computations for acquiring the magnet losses. It is shown that, in the considered generator for wind energy application, the higher harmonics in the current, resulting from the use of PWM, can increase the magnet losses significantly. Due to that, the higher harmonic analysis should be considered at the design stage of the machine to prevent damaging magnets with excessive heating source. Segmentation

10 KOWAL et al.: COMPARISON OF FREQUENCY AND TIME-DOMAIN IRON AND MAGNET LOSS MODELING INCLUDING PWM HARMONICS 485 Fig. 17. Magnetic induction time waveforms computed in the center of the stator yoke. Comparison of results for the simulation with sinusoidal stator current and with PWM stator current. (a) Magnetic induction computed during one electrical period. (b) Magnetic induction computed during 1.75 ms. of magnets proved to be an effective method of limitation of eddy currents in the magnet. In addition, it is shown that a simplified iron loss model that assumes a sinusoidal variation and considers only the fundamental component of the magnetic induction in the stator core material, will underestimate the iron losses in the machine. Especially, when the additional losses resulting from higher harmonics, rotational fields, and minor loops are taken into account. Finally, it is shown how the composition and thickness of the electrical steel used in the stator core of the particular machine influences the total core losses. REFERENCES [1] S. R. Holm, Modelling and optimization of a permanent magnet machine in a flywheel, Delft Univ. Technol., Delft, The Netherlands, [2] Z. Q. Zhu and D. Howe, Instantaneous magnetic field distribution in brushless permanent magnet dc motors Part III: Effect of stator slotting, IEEE Trans. Magn., vol. 29, no. 1, pp , Jan [3] H. Xuan, D. Lahaye, H. Polinder, and J. Ferreira, Influence of stator slotting on the performance of permanent-magnet machines with concentrated windings, IEEE Trans. Magn., vol. 49, no. 2, pp , Feb [4] J. Li, D. Choi, D. Son, and Y. Cho, Effects of MMF harmonics on rotor eddy-current losses for inner-rotor fractional slot axial flux permanent magnet synchronous machines, IEEE Trans. Magn., vol. 48, no. 2, pp , Feb [5] A. M. El-Refaie and T. M. Jahns, Impact of winding layer number and magnet type on synchronous surface PM machines designed for wide constant-power speed range operation, IEEE Trans. Energy Convers., vol. 23, no. 1, pp , Mar [6] D. Ishak, Z. Zhu, and D. Howe, Eddy-current loss in the rotor magnets of permanent-magnet brushless machines having a fractional number of slots per pole, IEEE Trans. Magn., vol. 41, no. 9, pp , Sep [7] A. Soualmi, F. Dubas, D. Depernet, A. Randria, and C. Espanet, Study of copper losses in the stator windings and pm eddy-current losses for pm synchronous machines taking into account influence of PWM harmonics, presented at the 15th International Conference on Electrical Machines Systems (ICEMS), Sapporo, Japan, [8] T. L. Mthombeni and P. Pillay, Lamination core losses in motors with nonsinusoidal excitation with particular reference to PWM and SRM excitation waveforms, IEEE Trans. Energy Convers., vol. 20, no. 4, pp , Dec [9] M. Mirzaei, A. Binder, and C. Deak, 3D analysis of circumferential and axial segmentation effect on magnet eddy current losses in permanent magnet synchronous machines with concentrated windings, presented at the 19th International Conference on Electrical Machines (ICEM), Rome, Italy, Sep. 6 8, [10] W. Huang, A. Bettayeb, R. Kaczmarek, and J. Vannier, Optimization of magnet segmentation for reduction of eddy-current losses in permanent magnet synchronous machine, IEEE Trans. Energy Convers., vol. 25, no. 2, pp , Jun [11] J. Klotzl, M. Pyc, and D. Gerling, Permanent magnet loss reduction in pm-machines using analytical and FEM calculation, presented at the International Symposium on Power Electronics Electrical Drives Automation and Motion (SPEEDAM), Pisa, Italy, [12] J. D. Ede, K. Atallah, G. W. Jewell, J. B. Wang, and D. Howe, Effect of axial segmentation of permanent magnets on rotor loss in modular permanent-magnet brushless machines, IEEE Trans. Ind. Appl., vol. 43, no. 5, pp , Sep/Oct [13] G. Bertotti, Hysteresis in Magnetism, for Physicists, Material Scientists, and Engineers. San Diego, CA, USA: Academic, [14] E. Barbisio, F. Fiorillo, and C. Ragusa, Predicting loss in magnetic steels under arbitrary induction waveform and with minor hysteresis loops, IEEE Trans. Magn., vol. 40, no. 4, pp , Jul [15] H. Karmaker, G. A. Knierim, H. Mantak, and B. Palle, Methodologies for testing a 2 MW permanent magnet wind turbine generator, presented at the IEEE Power Energy Society General Meeting, San Diego, CA, USA, [16] C. A. Hernandez-Aramburo, T. C. Green, and A. C. Smith, Estimating rotational losses in an induction machine, IEEE Trans. Magn., vol. 39, no. 6, pp , Nov Damian Kowal received the M.S. degree in electrical engineering from the University of Science and Technology, Cracow, Poland, in 2008, and the Ph.D. degree in electromechanical engineering from Ghent University, Ghent, Belgium, in He became a Postdoctoral Researcher at Ghent University in His current research interests include electrical machine modeling and design, particularly for wind energy application. Peter Sergeant received the M.Sc. degree in electromechanical engineering in 2001 and the Ph.D. degree in engineering sciences in 2006 from Ghent University, Ghent, Belgium. In 2001, he became a Researcher at the Electrical Energy Laboratory, Ghent University. He became a Postdoctoral Researcher at Ghent University in 2006 (Postdoctoral Fellow of the Research Foundation Flanders) and at Ghent University College, Ghent, in Since 2012, he has been an Associate Professor at Ghent University. His current research interests include numerical methods in combination with optimization techniques to design nonlinear electromagnetic systems, in particular, electrical machines for sustainable energy applications.

11 486 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 30, NO. 2, JUNE 2015 Luc Dupré was born in He received the Graduate degree in electrical and mechanical engineering in 1989 and the Doctorate degree in applied sciences in 1995 from the University of Ghent, Ghent, Belgium. He is currently a Full Professor with the Faculty of Engineering and Architecture, Ghent University, Ghent. His research interests mainly include numerical methods for electromagnetics, modeling, and characterization of soft magnetic materials, micromagnetism, inverse problems, and optimization in (bio)electromagnetism. Haran Karmaker received the Ph.D. degree in electrical engineering from the University of Toronto, Toronto, ON, Canada, in He served as an Electromagnetics Team Leader of the R&D team, General Electric, Canada, for 30 years. He joined TECO-Westinghouse Global R&D in 2004, where he is currently a Principal Engineer. He has published 58 technical papers in international journals including ICEM. He is a coauthor of the Design Handbook on Electric Motors (Marcel & Dekker, 2004). He is a coauthor of the patent WO 2011/ on power converter for use with wind generator. He has contributed technically to the development of many international standards. Dr. Karmaker received the Best Paper Award by the IEEE Power and Energy Society for his paper on investigations on skewed stator synchronous machines published in the IEEE TRANSACTIONS ON ENERGY CONVERSION in He was Chair of the working group on the 2009 edition of the IEEE standard 115 Synchronous Machines Test Procedures. He is currently a Chair of a new IEEE working group on the development of a standard on Guide for Testing Permanent Magnet Machines. He received the Outstanding Engineer Award of IEEE Canada in 2004.