Parameter Identification of Multiphase Induction ... - IEEE Xplore

1056

IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 27, NO. 4, DECEMBER 2012

Parameter Identification of Multiphase Induction Machines With Distributed Windings—Part 1: Sinusoidal Excitation Methods Alejandro G. Yepes, Member, IEEE, Jose A. Riveros, Jesús Doval-Gandoy, Member, IEEE, ´ Federico Barrero, Senior Member, IEEE, Oscar López, Member, IEEE, Blas Bogado, Student Member, IEEE, Martin Jones, and Emil Levi, Fellow, IEEE

Abstract—Multiphase induction machines (IMs) are gaining increasing interest in industry due to their numerous advantages over the conventional three-phase ones. A lot of different parameter estimation methods have been developed for three-phase IMs, but the existing literature regarding specific identification techniques for multiphase IMs is almost nonexistent at this point. This paper proposes simple offline methods to estimate the stator resistance and stator leakage inductance of multiphase IMs with distributed windings, under different conditions, utilizing the machine’s degrees of freedom associated with the nonflux/torque producing current components. Once these parameters are identified, the rotor ones can be easily calculated by combination with the total values obtained from locked-rotor tests. The procedure enables segregation of the stator and rotor parameters in a simple manner, something that is very difficult to achieve in three-phase IMs where, usually, equality of leakage inductances and a constant stator resistance are assumed. In this manner, the magnetizing inductance can be then also more accurately assessed from no-load tests, because the error in its estimation that would be caused by assuming both leakage inductances to be equal is avoided. The proposed methods are experimentally tested on two different five-phase IMs. Index Terms—Experimental testing, multiphase induction machines (IMs), parameter estimation.

NOMENCLATURE Variables h

ir

Harmonic order. The sign of its value indicates positive- or negativesequence. Rotor current.

Manuscript received May 16, 2012; revised August 17, 2012; accepted September 14, 2012. Date of publication October 16, 2012; date of current version November 16, 2012. This work was supported by the Spanish Ministry of Science and Innovation and by the European Commission, European Regional Development Fund (ERDF) under Project DPI2009-07004 and Project DPI2012-31283. Paper no. TEC-00171-2012. ´ López are with the Department A. G. Yepes, J. Doval-Gandoy, and O. of Electronics Technology, University of Vigo, Vigo 36310, Spain (e-mail: [email protected]; [email protected]; [email protected]). J. A. Riveros and F. Barrero are with the Department of Electronic Engineering, University of Seville, Seville 41004, Spain (e-mail: [email protected]; [email protected]). B. Bogado is with the Department of Systems and Automatics, University of Seville, Seville 41004, Spain (e-mail: [email protected]). M. Jones and E. Levi are with the School of Engineering, Technology and Maritime Operations, Liverpool John Moores University, Liverpool, L3 3AF, U.K. (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TEC.2012.2220967

is k ∈ [1, ρ] Llr Lls Lm Lr = Llr + Lm Ls = Lls + Lm n nω = (ω s − ω r ) /ω s p ∈ [0, n − 1] Rr Rs ρ = n/2 − 2 s vs ωk ≥ 0 ωr ωs Other symbols x x x

Stator current. Identifier of x–y plane. Rotor leakage inductance. Stator leakage inductance. Mutual inductance. Rotor self-inductance. Stator self-inductance. Total number of phases. Slip (normalized). Phase identifier. Takes a value 0, 1, 2, . . . to denote phases a, b, c, . . ., respectively. Rotor resistance. Stator resistance. Total number of x–y planes. Generic signal that may be replaced by ir , is or v s . Stator voltage. Frequency in the kth x–y plane. A sign is added in front to indicate positive- or negative-sequence. Rotor angular speed (electrical). Synchronous frequency (electrical).

Estimated value of x. Complex conjugate of x. Maps x to the smallest following integer (ceiling function). x Maps x to the largest previous integer (floor function). |x| Absolute value of x. |x| Module of x. ∠x Phase of x. Inverse of [X]. [X]−1 Nonconjugated transpose of [X]. [X]T mth row of [X], with m ≥ 1. [Xm√] Imaginary unit. j = −1 Bold symbols denote complex values and nonbold symbols denote real values. Capital letters inside brackets denote arrays.

I. INTRODUCTION

M

ULTIPHASE induction machines (IMs) offer several advantages over their three-phase counterparts [1], [2].

0885-8969/$31.00 © 2012 IEEE

YEPES et al.: PARAMETER IDENTIFICATION OF MULTIPHASE INDUCTION MACHINES WITH DISTRIBUTED WINDINGS—PART 1

The rating of the inverter per leg is reduced as the number of phases increases, which is especially interesting in high-power scenarios. Furthermore, when designed with near-sinusoidal magneto-motive force (MMF) distribution these machines provide additional degrees of freedom, apart from the usual two in charge of controlling the torque and flux, that can be used for other purposes. Distributed winding arrangement, which yields near-sinusoidal MMF, provides some advantages over the alternative concentrated winding arrangement [1]. In particular, spatial harmonics of the MMF are reduced, fault-tolerant operation is easily obtainable by using the additional degrees of freedom, and it is also possible to independently control a multitude of multiphase machines in an opportune series connection while supplying the whole group from a single converter. Accurate knowledge of machine parameters is essential to achieve a high performance when using some of the most popular control strategies, such as field-oriented control (FOC). As a consequence, a huge effort has been put in the past into the development of both offline and online procedures for three-phase IM parameter identification. A comprehensive, but almost a decade old, survey is available in [3]. The topic however continues to attract continuous attention, as evidenced by some very recent publications in this area [4]–[7]. Although multiphase drive systems have been in the research focus for the best part of the last ten years, there is however little evidence that new parameter identification techniques have been developed. A methodology for parameter identification in an eleven-phase IM with concentrated windings has been described in [8]. Since the machine model is quite different when the windings are with near-sinusoidal MMF distribution, the method would not be suitable for this kind of machines. An online estimation technique for stator resistance and stator leakage inductance in a symmetrical six-phase IM with distributed windings has been developed in [9]. The method is based on injecting any of the nonflux/torque producing current components into the machine, but eventually the zero-sequence component is adopted because it allows for a lower number of sensors (when using the procedure exposed in [9]). The estimation process is further executed by using a least squares minimization algorithm. Practically, an identical estimation scheme, for the same parameters, has also been described in [10], but this time for a four-phase IM. Stator resistance and stator leakage inductance estimation methods of [9], [10] for multiphase IMs are in essence an extension of the principle developed in [11] for three-phase IMs using the zero-sequence model. In this paper, two offline methods are proposed to estimate the stator resistance and the stator leakage inductance of multiphase IMs with distributed windings, by the injection of nonflux/torque producing currents. However, in contrast to [9] and [10], in these techniques the current is injected into the nonflux/torque producing plane rather than into the zero-sequence axis, and they are essentially offline rather than online (so that only two sensors are required), aimed at parameter identification for subsequent use in drive control. An inherent advantage of this approach when the phase number is odd, over the one based on the zero-sequence model, is that the neutral point of the winding does not have to be accessible. The estimation takes

1057

place at standstill in the first method, while in the second method the identification is conducted during the machine’s rotation. Once the stator resistance and leakage inductance have been estimated by any of these two strategies, the corresponding parameters of the rotor can be calculated by using the total resistance and leakage inductance values obtained from the lockedrotor test. This fact reveals an inherent advantage of multiphase IMs over their three-phase counterparts, in which it is difficult to segregate the rotor and stator parameters (unless neutral point is accessible and zero-sequence model can be used in conjunction with single-phase excitation). Actually, equality between the leakage inductances and a constant stator resistance are usually assumed [12], or complicated techniques are employed to separate rotor and stator values [6], [13]–[15]. The stator resistance is often considered to be greater than the dc value by an arbitrary factor between approximately 1.15 and 1.6 because of frequency-dependent phenomena such as hysteresis losses and eddy currents in the vicinity of stator slots due to the stator leakage flux and skin effect [16]; the error derived from this assumption is avoided in this paper by the assessment of the stator resistance actual variation with frequency. In addition, once the proposed procedure is applied, the magnetizing inductance can be simply identified by the subtraction of the resulting stator leakage inductance from the stator selfinductance obtained from the no-load test, without the error that would be caused by assuming the two leakage inductances to be equal [13]. All the tests, described in this paper, are performed using a two-level five-phase voltage source inverter. Experimental results obtained by the application of the developed testing procedure to two different five-phase IMs are presented and compared. To further ascertain the correctness of the results obtained by experiments, a completely different procedure for a five-phase IM parameter determination is reported in [17]. The comparison of the values obtained using the methods based on sinusoidal excitation and the method of [17] is included in [17]. This paper is organized as follows. Section II summarizes the dynamic and steady-state models of multiphase IMs with distributed windings. The parameter estimation methods based on the injection of nonflux/torque producing harmonic currents at standstill and with rotating machine are developed in Sections III and IV, respectively. The identification of the remaining parameters, based on no-load and locked rotor tests with inverter supply, is addressed in Section V. Section VI presents the experimental results, while Section VII concludes this study. II. MODEL OF A MULTIPHASE IM An n-phase IM with distributed windings is considered. Its model can be expressed in the 1 + n/2 orthogonal subspaces [1], [2], where the first plane (α–β) is where the coupling with the rotor takes place (flux/torque producing plane), the next ρ planes are nonflux/torque producing (x–y) planes, and finally there are one or two zero-sequence axes depending on whether n is odd or even, respectively. Each of the ρ + 1 planes may contain space vectors rotating in either the positive- or negative-sequence direction. This

1058


subspace decomposition is achieved through Clarke’s decoupling transformation matrix [C], which is shown in power invariant form in (A1), in the Appendix, where κ = 2π/n is the spatial displacement between two consecutive phases [2]. The last row in (A1), which corresponds to 0− , is only present in case of an even number of phases. This transformation converts a vector of per-phase instantaneous real values [S] = [ sa

sb

···

sn ]T

sβ

sx 1

sy 1

···

sx ρ

sy ρ

s0 +

s0 − ]T = [C] [S] . (2)

The dynamic model of the multiphase machines under consideration, expressed with respect to these n axes, is [2] d s d i + Lm irα (3) dt α dt d d vβs = Rs isβ + Ls isβ + Lm irβ (4) dt dt d d (5) 0 = Rr irα + Lr irα + Lm isα + ω r Lr irβ + Lm isβ dt dt d d 0 = Rr irβ + Lr irβ + Lm isβ − ω r (Lr irα + Lm isα ) (6) dt dt d vxs k = Rs isx k + Lls isx k k = 1, 2, . . . , ρ (7) dt d vys k = Rs isy k + Lls isy k k = 1, 2, . . . , ρ (8) dt d v0s + = Rs is0 + + Lls is0 + (9) dt d v0s − = Rs is0 − + Lls is0 − . (10) dt Equation (10) only exists when n is even, and is0 + , which obeys (9), can only flow if neutral connection is available. The identification procedures proposed in this paper, for the sake of generality, and in contrast to [9] and [10], are not based on any of these two equations; hence, they are discarded. vαs = Rs isα + Ls

B. Steady-State IM Model To operate with space vectors instead of real values, transformation (A2), shown in the Appendix, can be applied instead of (A1) to [S] [18]: sxy1

sxy2

···

sxyρ ]T = [D] [S]

(11)

where sαβ = sα + j sβ ;

subspace (if there is more than one, the superposition principle is applied, so that the one of interest is isolated). Under such conditions, the derivative operator is replaced by j ω, with ω being the frequency under consideration [8], [18]: s = (Rs + j ω s Ls ) isαβ + j ω s Lm irαβ vαβ

A. Dynamic IM Model

[ sαβ

Steady-state model of multiphase IM. (a) α–β plane. (b) xy k plane.

(1)

into n real values, which correspond to the components in the n axes that form those 1 + n/2 orthogonal subspaces: [ sα

Fig. 1.

sxyk = sx k + j sy k

k = 1, 2, . . . , ρ (12) so that each pair of equations in (3)–(8) can be reduced to a single complex equation. The machine model can be further simplified by assuming that the steady state is reached and that only one frequency of voltage and current is considered in each

(13)

0 = [Rr + j (ω s − ω r )Lr ] irαβ + j (ω s − ω r )Lm isαβ (14) s vxy = Rs ± j ωk Lls isxyk k = 1, 2, . . . , ρ. (15) k Note that ωk ≥ 0 is considered and the sign associated with ωk is explicitly indicated so that the positive- and negative-sequence cases (+ωk and −ωk , respectively) can be separately studied in what follows. Fig. 1 represents the steady-state model that corresponds to (13)–(15). III. ESTIMATION AT STANDSTILL USING x–y PLANE SINUSOIDAL EXCITATION The current in an x–y plane does not contribute to torque or rotor flux, since no coupling with the rotor is present in (15) [2]. Taking advantage of this property, an alternating voltage may be applied to the x–y axes without causing rotation. Equation (A3), given in the Appendix, shows the symmetrical components transform [E] (also known as Fortescue transform) in power invariant form, where a = ej κ [19]. Note that an element in any√row f ∈ [1, n] and column g ∈ [1, n] is given by a(f −1)(g −1) / n. Let us define a vector of n per-phase voltages or currents in complex form [S] = [ sa such that [S] =

···

sb

sn ]T

(16)

1 [S] + [S] . 2

(17)

Transform [E] converts [S] to 2 (n − 1)/2 complex vectors: [E][S] = s0 + sαβ+ sxy+ . . . 1

...

sxyρ+

s0 −

sxyρ− . . .

T sxy1−

sαβ−

. (18)

As opposed to transform [D], which uses a vector of real values [S] as input [see (11)], the fact that in this case the per-phase variables are complex permits [E] to separate the positive- and negative-sequences into different and independent subspaces. In this manner, the αβ and xyk planes are now decomposed into a pair of planes each: αβ + , αβ − , xyk+ , and xyk− , respectively. Thus, the number of planes under study is doubled: from ρ + 1


to 2ρ + 2. It should be noted that, as in [C], the 0− row should be removed from (A3) in case of an odd number of phases. Taking into account that sxyk = sxy+ + sxy−

(19)

k

k

which is demonstrated in the Appendix, combination of (15) and (18) yields s vxy k

vs

+

xyk is + xyk

=

s + vxy − k

=

[Ek+ 2 ][V s ] + [En−k−1 ][V s ]

. [Ek+ 2 ][I s ] + [En−k−1 ][I s ] (20) When only a frequency ωk is considered, either the conjugated or nonconjugated terms in (20) are zero, depending on whether the sequence is positive or negative, respectively. In this manner, for the positive-sequence, the combination of (20) and (15), in which the positive sign is chosen for ωk , yields isxyk

vs

+

isxy− k

+

xyk s i + xyk

=

[Ek+ 2 ][V s ] = Rs + j ωk Lls . [Ek+ 2 ][I s ]

(21)

For the negative-sequence, the combination of (20) and (15), in which the negative sign is selected for ωk , results in s vxy − k

isxy−

[En−k−1 ][V s ] = Rs + j ωk Lls . = [En−k−1 ][I s ]

(22)

k

The symmetrical components transform satisfies [E]−1 = [E]T = [E]. Therefore, the left multiplication of both sides in (18) by [E]−1 yields [S] = [E] s0 + sαβ+ sxy+ . . . 1

...

sxyρ+

s0 −

T

sxyρ− . . .

sxy1−

sαβ−

. (23)

Hence, if the only voltage supplied to the machine is a signal of frequency ±ωk in the plane xyk± , the stator voltage and current in phase p ∈ [0, n − 1] are vps

=

s ∓p(k +1) vxy ; ±a k

isp

=

isxy± a∓p(k +1) . k

vps isp

s

ls

= R + j ωk L .

same reasoning also applies to the expressions developed for parameter estimation in the following sections. The magnitude and phase of vps and isp can be calculated by, for instance, a discrete Fourier transform (DFT) algorithm. Given the asymmetries that the machine and the converter present in practice, as well as the low-order harmonics produced by the nonlinearities (e.g., dead-time), the machine may start to rotate from a certain (relatively high) current in the x–y axes. Hence, if it is desired to perform the test also with large current (e.g., to observe a heavier saturation of Lls ), additional measures may be needed in order to prevent the shaft from moving. One possibility is to externally lock the rotor. Alternatively, FOC can be implemented in the α–β plane with zero speed reference and certain magnetizing current, at the expense of increasing the number of current sensors to 2 (n − 1)/2 and of requiring an approximate knowledge of the parameters in order to tune the controller. Nevertheless, as will be proven in Section VI, to control only the x–y axes, without locking the rotor or using FOC, usually suffices. IV. ESTIMATION WITH ROTATING MACHINE USING x–y PLANE SINUSOIDAL EXCITATION In order to analyze any additional effects on Lls and Rs that may be present under different operating conditions, these parameters can be also estimated while the machine is rotating. The rotor can be brought into rotation by a controller in the α–β plane, either an open-loop or a closed-loop one. The latter would require 2 (n − 1)/2 current sensors and knowledge of approximated values of the parameters. A harmonic component of order h may be imposed on an x–y plane [i.e., ±ωk = hω s in (15)] during the operation of the machine without affecting its mechanical behavior. From (23): s −p vps = vαβ + v s ± a∓p(k +1) + a xyk s vp

isp

(24)

From (21), (22), and (24), division of phase p voltage by its corresponding current results in

=

−p

a

is p

1

h

+ isxy± a∓p(k +1) k

is p

(28)

.

(29)

h

The combination of (21), (22), (28), and (29) leads to

(25)

It should be remarked that, if no neutral connection is available, the phase-to-neutral signals vps and isp can be replaced by vps − vps and isp − isp , and these equations will still hold true. The

s vp

1

isαβ+

vps h

Thus, an estimation of Lls and Rs can be finally obtained as

s

1 vp

ls

sin ∠vps − ∠isp (26) L = ωk

is

p

s

v

s = p cos ∠v s − ∠is . R (27) p p

s

ip

1059

isph

= Rs + j |h| ω s Lls .

(30)

Hence, Lls and Rs can be estimated by ls = L

s = R

s 1 |vph | s s sin ∠v − ∠i ph ph |h| ω s |isph |

|vps h | |isph |

cos ∠vps h − ∠isph .

(31)

(32)

Special care is needed to filter out the fundamental component of (28) and (29), so that the hth harmonic components are properly extracted. A DFT algorithm tuned at ωk = hω s provides effective cancellation of signals of frequency ω s , so the DFT would be also suitable for this method.

1060


A. Ls and Lm Estimation by No-Load Test According to (23), during no-load tests the stator voltage and current of phase p satisfy s −p vps = vαβ ; + a

isp = isαβ+ a−p

respectively. From (13) and (33), and assuming vps isp

irαβ

(33) ≈ 0:

= Rs + j ω s Ls .

(34)

Thus, Ls can be estimated as |vps | s = 1 L sin(∠vps − ∠isp ). s ω |isp |

(35)

In fact, the usual procedure to calculate Ls consists in applying expressions equivalent to (35) [12]. Regarding Lm identification, it is usually estimated as m =L ls s − L L

Fig. 2. Control schemes for the estimation method proposed in Section IV. (a) Open-loop control in α–β plane. (b) Closed-loop control in α–β plane.

Another issue that should be taken into account is described in the following. Due to asymmetries and nonideal behavior of the inverter (e.g., dead-time), xy components appear in the stator voltage, even if a controller is implemented only in the α–β subspace. It has been proved in [20] that this can lead to large undesired currents, since the impedance in these subspaces is very low [see (15)]. Therefore, a closed-loop current controller is required in the x–y planes to actively eliminate these current components. Thus, the hth harmonic component in (28) should not be added in an open-loop manner, because the current control would try to reject it as a perturbation. Instead, isph should be imposed by the current controller itself. Fig. 2, in which θr is the estimated rotor flux angle and * denotes reference signals, depicts the control schemes proposed for this approach, depending on whether open-loop V /f [see Fig. 2(a)] or closed-loop FOC [see Fig. 2(b)] is implemented in the α–β plane. In both structures, the proportional + integral controller in the xyk plane ∗ imposes the current harmonic (of amplitude isxyk−dq ) aimed at parameter estimation, while reducing the current components caused by system nonlinearities. Additional synchronous reference frames (not shown in Fig. 2) may be added in parallel, tuned at other harmonics or x–y planes, to further improve the cancellation of undesired currents [20].

V. IDENTIFICATION OF THE REMAINING PARAMETERS Once the stator resistance and the stator leakage inductance have been estimated by the proposed methods, the resulting values can be used to identify some of the remaining parameters, more accurately than without their knowledge.

(36)

ls obtained through a locked-rotor test and with the aswith L lr ≈ L ls [12]. Since this hypothesis is not really sumption that L true in practice, a certain error is introduced not only in the stator leakage inductance, but also in the magnetizing inductance [13]. Alternatively, it is suggested to avoid this error by simply using in (36) the Lls value estimated by means of the methods developed in Sections III and IV.

B. Llr and Rr Estimation by Locked-Rotor Test During locked-rotor tests, (33) is also satisfied for any phase p. The combination of (33) with (13), (14) and ω r = 0 results in vps isp

=

(ω s )2 Lr (Rs Lr + Rr Ls ) + Rs (Rr )2 (Rr )2 + (ω s Lr )2 (ω s )2 Rr Lls Lr + Lm Llr − (Rr )2 + (ω s Lr )2 +j

+j

ω s Rr (Rs Lr + Rr Ls ) − ω s Lr Rs Rr (Rr )2 + (ω s Lr )2 (ω s )3 Lr Lls Lr + Lm Llr (Rr )2 + (ω s Lr )2

(37)

from which Rr and Llr can be calculated, because all the other variables in (37) are known at this point and (37) can be seen as two independent equations, depending on whether the real or the imaginary parts of both sides are considered. For the sake of simplicity, the Lm parallel branch is often ignored during these tests, due to its large impedance. In this manner, the following is obtained: vps isp

= Rs + Rr + j ω s (Lls + Llr ).

(38)

which also coincides with what results from assuming that Lm tends to infinite in (37). Then, the segregated stator and rotor values are normally calculated by assuming both leakage


TABLE I PARAMETERS OF THE IMS USED IN THE EXPERIMENTS

1061

As the first step, the stator resistance was measured with dc supply. The resulting values, for machines A and B, are 2 Ω and 12.9 Ω, respectively. A. Lls and Rs Estimation at Standstill Using x–y Plane Excitation Only

Fig. 3.

Experimental setups. (a) Machine A. (b) Machine B.

inductances to be equal, as well as a constant Rs , as already commented. Instead, it is possible to avoid the error introduced by those approximations in the rotor parameters by simply introducing, in either (37) or (38), the Lls and Rs values identified through the procedures of Sections III and IV. VI. EXPERIMENTAL RESULTS The experiments are carried out with two different five-phase IMs (machines A and B), which have been obtained from commercial three-phase IMs. Machine A has new stator laminations with 40 slots, while stator core of the machine B is the original one. A five-phase winding was rewound in both machines. Available information regarding the machines is given in Table I. Corresponding experimental setups are shown in Fig. 3. The acronym DSP in this figure stands for digital signal processor. It is important to note that the primary purpose of the testing here is to obtain, using the developed methods, the behavior of the machine parameters when subjected to different operating conditions. On the other hand, the procedure introduced in [17] performs parameter identification for a single operating point only. As shown in [17], values obtained for the specific operating point fit reasonably well with the values reported here for the same operating conditions.

The results of Lls estimation, obtained by the method proposed in Section III [i.e., by means of (26)], for IMs A and B, are shown in Fig. 4(a) and (b), respectively (rated frequencies of IMs A and B are 60 and 50 Hz, respectively). The x–y current isxyk can be increased up to about 0.7 Arms without causing rotation in IM A, and to about 1 Arms in IM B (the rotation is eventually caused by the parasitic spatial MMF harmonics, since the MMF distribution can never be truly sinusoidal). To test the behavior of Lls for higher values of current, FOC with zero speed reference is implemented in the α–β plane. From these figures, the stator leakage inductance is practically independent from frequency in IM A, while it shows some frequency dependence in IM B. On the other hand, results in Fig. 4(a) indicate substantial parameter dependence on current for IM A (saturation effect), whereas there is practically no change of the leakage inductance with current for IM B [see Fig. 4(b)]. This difference can be attributed to the fact that the saturation of Lls is greatly affected by constructive aspects, depending on the extent to which the leakage-flux paths cross air-gap or iron area, as well as on the magnetic characteristics of the iron in these paths [21]. Fig. 4(c) and (d) depicts the Rs estimations calculated using (27) for machines A and B, respectively. From these plots, the estimated resistance changes significantly with frequency and is practically current-independent. It is important to note that conventional iron losses are negligibly small in this test due to the very low values of the applied voltage (very small impedance of the x–y plane). To avoid temperature related variations, special care was taken during measurements to supply currents during brief periods of time. It can be also observed that the dispersion of the estimated Rs values at low currents in Fig. 4(c) is much higher than that of the leakage inductance in Fig. 4(a), due to the predominantly inductive behavior, but it reduces as the current magnitude increases. Finally, it can be remarked that matching results were obtained by using sinusoidal 50-Hz single-phase excitation of the zero-sequence model in (9) (which requires accessible neutral connection). B. Lls and Rs Estimation While the Machine Rotates, Using x–y Plane Excitation To experimentally verify the Lls estimation method of Section IV, a negative-sequence third-order harmonic (i.e., h = −3) is injected in the x–y plane. FOC is implemented in the α–β plane to produce the shaft rotation, in accordance with the scheme shown in Fig. 2(b). Next, Lls is estimated according to (31). The results for machines A and B are presented in Fig. 5(a) and (b), respectively. It can be observed that, in machine A, the saturation effect that was present in the tests at standstill [see ls hardly deFig. 4(a)] does not appear in this case; in fact, L pends on current in Fig. 5(a). It is also worth noticing that the

1062

Fig. 4.


l s in Estimates obtained by the standstill method of Section III. (a) L

l s in IM B. (c) R s in IM A. (d) R s in IM B. IM A. (b) L

Fig. 5.

ls Estimates obtained by the Section IV method, with h = −3. (a) L

l s in IM B. (c) R s in IM A. (d) R s in IM B. in IM A. (b) L


1063

average value of inductance in Fig. 5(a) is quite similar to what would have been obtained at zero current in Fig. 4(a) (by extrapolation of the results). Hence, this strategy is an interesting option if the desire is to estimate the Lls value that results when saturation is not present. s values obtained by means Fig. 5(c) and (d) shows the R of (27) in machines A and B, respectively. In comparison to the curves obtained at standstill [see Fig. 4(c) and (d)], the frequency in the x–y plane is now about three times higher. s values shown in Fig. 5(c) and This fact explains the larger R (d). For instance, the curve for 60 Hz in Fig. 4(c) is practically the same as the one obtained at 600 r/min (i.e., ω s /2π = 20 Hz and ωk /2π = 60 Hz) in Fig. 5(c). This means that no significant ls ) are to be expected when the s (as opposed to L changes in R machine is running instead of being at standstill (excluding the temperature related changes, of course). C. Ls and Lm Estimation by No-Load Tests Standard no-load tests were performed at several frequencies and for various magnetizing current values. Fig. 6(a) and (b) shows the estimated Ls [according to (35)] obtained from these tests in machines A and B, respectively. The most noticeable variation of this parameter is its decrease as the current rises, due to the magnetizing flux saturation. ls curves obtained through the standstill technique Next, the L are subtracted from the curves in Fig. 6(a) and (b) to calculate the magnetizing inductance [according to (36)], and the resulting plots are shown in Fig. 6(c) and (d). To reduce the inaccuracies introduced by systematic and random errors, it may be interesting for practical applications to average the curves obtained at different frequencies, by assuming the magnetizing inductance to be practically frequency independent. D. Llr and Rr Estimation by Locked-Rotor Tests Finally, locked rotor tests were performed to obtain the rotor parameters. The testing frequencies are lower than those used in the previous experiments, due to the fact that in this case ω r = ω s but during normal operation ω r ω s . Both Llr and Rr are calculated by means of (37). The values of Lm , Lls , and Rs introduced in (37) have been obtained by means of the standstill method, which includes the saturation behavior, under the corresponding conditions. Using the inductance values estimated with rotating machine would produce significant error in the estimation of the rotor parameters, because of the fact that they do not reflect the dependence on current. Fig. 7(a) and (b) shows the Llr estimates for machines A and B, respectively. These figures reveal that, in both IMs, Llr is practically independent of frequency, and that the leakage flux saturation with increasing current is heavier than that for the stator leakage inductance [see Fig. 4(a) and (b)]. On the other hand, Fig. 7(c) and (d) presents the estimated Rr in machines A and B, respectively. From Fig. 7(c), Rr of the first machine changes mainly with frequency, and its dependence on it is also more considerable, in relative terms, than that of the stator resistance [see Fig. 4(c)]. To the contrary, there is very little

Fig. 6. Estimates obtained by no-load tests. (a) Ls in IM A. (b) Ls in IM B. m in IM A. (d) L m in IM B. (c) L

variation with frequency in the second machine, as illustrated in Fig. 7(d). With regard to the influence of current magnitude, the resistance of machine A shows an initial slight decrease as the current increases at 30 Hz but is practically current-independent at 10 Hz. The rotor resistance of machine B exhibits a differ-

1064


ent behavior; there is a slight Rr increase with current [see Fig. 7(d)], which could be due to an increase in temperature during the locked-rotor tests. As a final remark, it is worth noticing that in the machine B the rotor leakage inductance differs considerably from the stator leakage inductance [see Fig. 7(b) versus Figs. 4(b) and 5(b)], so the error that would be caused by using the hypothesis Llr = Lls would have been considerable. The difference in values of stator and rotor leakage inductance is much smaller for machine A [see Fig. 7(a) versus Figs. 4(a) and 5(a)]. VII. CONCLUSION Two offline techniques have been developed for the estimation of the stator leakage inductance and stator resistance in multiphase IMs with distributed windings. They are based on the injection of currents into the nonflux/torque producing planes. One of the methods is suitable for identification at standstill, which is interesting in applications where the machine is not allowed to rotate during the commissioning process. On the other hand, the other approach consists in injecting a harmonic current, which is also mapped into a nonflux/torque producing plane, while the machine is rotating, so that the parameters can be identified at different operating conditions. The obtained stator parameter values make it possible to calculate the rotor resistance and rotor leakage inductance, by utilizing the standard locked-rotor test; this fact reveals an advantage over three-phase IMs, in which it is difficult to segregate the rotor and stator parameters, unless neutral point of the winding can be accessed and the zero-sequence model utilized for stator leakage inductance identification. Moreover, a more accurate assessment of the magnetizing inductance is also possible, without the error that would be introduced by assuming that the stator and rotor leakage inductances are equal. No-load test is additionally required for this purpose. The proposed methods are experimentally tested on two different five-phase IMs. As a general conclusion, it can be said that the parameter values depend on the operating conditions and that use of a single value for each parameter, as the case would be in FOC, requires parameter identification in very well defined operating points. Such a procedure is described next in [17], and the values obtained serve a purpose of further verifying the results reported here. APPENDIX The transformation matrices [C], [D], and [E] are shown in (A1), (A2), and (A3), respectively, at the top of the next page. In the following, the identity in (19) is developed. Left multiplication by [Dk+ 1 ] is applied to both sides of (17):

[Dk+ 1 ] [S] = l r in IM A. (b) L l r in Fig. 7. Estimates obtained by locked-rotor test. (a) L r r IM B. (c) R in IM A. (d) R in IM B.

=

[Dk+ 1 ][S] + [Dk+ 1 ][S] 2 [Dk+ 1 ][S] + [Dk+ 1 ][S] . 2

(A4)


α

⎡

⎢ 0 β ⎢ ⎢ ⎢ 1 x1 ⎢ ⎢ y1 ⎢ ⎢ 0 ⎢ 1 x2 ⎢ ⎢ 2 y ⎢ 0 · 2 ⎢ [C] = n . ⎢ ⎢ . .. ⎢ .. ⎢ ⎢ 1 xρ ⎢ ⎢ ⎢ 0 yρ ⎢ ⎢ √ 0+ ⎢ ⎣ 1/√2 0− 1/ 2

⎡ ⎢ ⎢ ⎢ [D] = ⎢ ⎢ ⎣

[C2ρ+1 ] + j

cos (2κ)

cos (3κ)

···

sin (κ)

sin (2κ)

sin (3κ)

···

cos (2κ)

cos (4κ)

cos (6κ)

···

sin (2κ)

sin (4κ)

sin (6κ)

···

cos (3κ)

cos (6κ)

cos (9κ)

···

sin (3κ) .. .

sin (6κ) ...

sin (9κ) .. .

···

cos [(ρ + 1) κ]

cos [2 (ρ + 1) κ]

cos [3 (ρ + 1) κ]

···

sin [(ρ + 1) κ] √ 1/ 2 √ −1/ 2

sin [2 (ρ + 1) κ] √ 1/ 2 √ 1/ 2

sin [3 (ρ + 1) κ] √ 1/ 2 √ −1/ 2

···

⎡

1 a ⎢ 1 a2 xy1 ⎢ ⎢ ⎥ [C4 ] ⎢ ⎥ 2 xy2 ⎢ 1 ⎥ a3 · ⎢ ⎥= ⎥ n .. ⎢ .. .. ⎦ . ⎢ . ⎢. ⎣ [C2ρ+2 ] ρ+1 xyρ 1 a

[C1 ] + j [C2 ] [C3 ] + j .. .

cos (κ)

1

⎤

αβ

0+

⎡

⎢ αβ + ⎢ ⎢1 ⎡ ⎤ + ⎢ xy1 ⎢ 1 [Cn −1 ] ⎢ ⎢ [D ] ⎥ xy2+ ⎢ ⎢1 ⎢ ⎥ 1 ⎢ ⎢ ⎥ ⎢ ⎥ . . .. ⎢ .. ⎢ .. ⎢ ⎥ ⎢. ⎢ ⎥ ⎢ ⎥ + ⎢ ⎢ ⎥ xy [D ] ρ ⎢ 1 1 ⎢ ρ+ 1 ⎥ ⎢1 [E] = √ ⎢ ⎥ = √ · 0− ⎢ ] [C ⎢1 ⎢ ⎥ n n 2⎢ ⎢ ⎥ − ⎢ ⎢ [Dρ+ 1 ] ⎥ xyρ ⎢ 1 ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ .. .. ⎢ . ⎢ ⎥ . . ⎢ ⎢ .. ⎣ ⎦ ⎢ − [D1 ] xy2 ⎢ 1 ⎢ xy1− ⎢ ⎢1 ⎣ − αβ 1

···

n −1

···

a2(n −1)

a6 .. .

a9 .. .

···

a3(n −1) .. .

a2(ρ+1)

a3(ρ+1)

···

..

1

···

a

a2

a3

···

a2

a4

a6

···

3

6

9

a .. .

a .. .

a .. .

···

aρ+1

a2(ρ+1)

a3(ρ+1)

···

−1

1

−1

···

aρ+3 .. .

a2(ρ+3) .. .

a3(ρ+3) .. .

···

an −3

a2(n −3)

a3(n −3)

···

n −2

2(n −2)

3(n −2)

··· ···

a

a

a

an −1

a2(n −1)

a3(n −1)

(A5)

Combination of (A4) and (A5) results in [Dk+ 1 ] [S] = [Ek+ 2 ][S] + [En−k−1 ][S].

···

a6

a

1

[Dk+ 1 ] = 2[En−k−1 ].

.

a4

On the other hand, it can be derived from (A3) that [Dk+ 1 ] = 2[Ek+ 2 ];

⎥ ⎥ ⎥ cos [2 (n − 1) κ] ⎥ ⎥ ⎥ sin [2 (n − 1) κ] ⎥ ⎥ ⎥ cos [3 (n − 1) κ] ⎥ ⎥ sin [3 (n − 1) κ] ⎥ ⎥ ∈ Mn ×n . ⎥ .. ⎥ ⎥ . ⎥ ⎥ cos [(ρ + 1) (n − 1)κ] ⎥ ⎥ sin [(ρ + 1) (n − 1)κ] ⎥ ⎥ √ ⎥ ⎥ 1/ 2 ⎦ √ −1/ 2

···

a

(A6)

Finally, (19) is obtained from (11), (18), and (A6). REFERENCES [1] E. Levi, “Multiphase electric machines for variable-speed applications,” IEEE Trans. Ind. Electron., vol. 55, no. 5, pp. 1893–1909, May 2008.

..

..

.

.

a

.

⎤

sin [(n − 1) κ]

3

2

1

1

..

cos [(n − 1) κ]

1065

a(n −1)(ρ+1)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ∈ M(ρ+1)×n . ⎥ ⎥ ⎥ ⎦

(A1)

(A2)

⎤

1

⎥ ⎥ ⎥ ⎥ 2(n −1) ⎥ a ⎥ ⎥ a3(n −1) ⎥ ⎥ ⎥ .. ⎥ . ⎥ ⎥ (ρ+1)(n −1) ⎥ a ⎥ ⎥ ∈ Mn ×n . ⎥ −1 ⎥ ⎥ a(ρ+3)(n −1) ⎥ ⎥ ⎥ .. ⎥ . ⎥ ⎥ (n −3)(n −1) ⎥ a ⎥ ⎥ (n −2)(n −1) ⎥ a ⎦ an −1

a(n −1)

(A3)

2

[2] E. Levi, R. Bojoi, F. Profumo, H. Toliyat, and S. Williamson, “Multiphase induction motor drives—A technology status review,” IET Elect. Power Appl., vol. 1, no. 4, pp. 489–516, Jul. 2007. [3] H. Toliyat, E. Levi, and M. Raina, “A review of RFO induction motor parameter estimation techniques,” IEEE Trans. Energy Convers., vol. 18, no. 2, pp. 271–283, Jun. 2003. [4] F. Salmasi and T. Najafabadi, “An adaptive observer with online rotor and stator resistance estimation for induction motors with one phase current sensor,” IEEE Trans. Energy Convers., vol. 26, no. 3, pp. 959–966, Sep. 2011. [5] Y. He, Y. Wang, Y. Feng, and Z. Wang, “Parameter identification of an induction machine at standstill using the vector constructing method,” IEEE Trans. Power Electron., vol. 27, no. 2, pp. 905–915, Feb. 2012. [6] W.-M. Lin, T.-J. Su, and R.-C. Wu, “Parameter identification of induction machine with a starting no-load low-voltage test,” IEEE Trans. Ind. Electron., vol. 59, no. 1, pp. 352–360, Jan. 2012.

1066


[7] A. Bechouche, H. Sediki, D. Abdeslam, and S. Haddad, “A novel method for identifying parameters of induction motors at standstill using ADALINE,” IEEE Trans. Energy Convers., vol. 27, no. 1, pp. 105–116, Mar. 2012. [8] A. Abdelkhalik, M. Masoud, and W. Barry, “Eleven-phase induction machine: Steady-state analysis and performance evaluation with harmonic injection,” IET Elect. Power Appl., vol. 4, no. 8, pp. 670–685, Sep. 2010. [9] C. Jacobina, C. de Azevedo, C. da Silva, A. Lima, and E. da Silva, “Online estimation of the stator resistance of a six-phase induction machine,” in Proc. IEEE Ind. Appl. Conf., Oct. 2002, vol. 2, pp. 746–75. [10] C. Jacobina, C. de Azevedo, A. Lima, and L. de Souza Ribeiro, “Online estimation of the stator resistance and leakage inductance of a four-phase induction machine drive,” IEEE Trans. Power Electron., vol. 19, no. 1, pp. 10–15, Jan. 2004. [11] C. Jacobina, J. Filho, and A. Lima, “On-line estimation of the stator resistance of induction machines based on zero-sequence model,” IEEE Trans. Power Electron., vol. 15, no. 2, pp. 346–353, Mar. 2000. [12] Standard Test Procedure for Polyphase Induction Motors and Generators, New York: Institute of Electrical and Electronics Engineers, IEEE Std. 112-2004, 2004 [13] A. Stankovic, E. Benedict, V. John, and T. Lipo, “A novel method for measuring induction machine magnetizing inductance,” IEEE Trans. Ind. Appl., vol. 39, no. 5, pp. 1257–1263, Sep./Oct. 2003. [14] J.-F. Brudny, J.-P. Lecointe, F. Morganti, F. Zidat, and R. Romary, “Use of the external magnetic field for induction machine leakage inductance distinction,” IEEE Trans. Magn., vol. 46, no. 6, pp. 2205–2208, Jun. 2010. [15] J.-W. Kim, T. Kim, Y. Park, and S. W. Kim, “On-load motor parameter identification using univariate dynamic encoding algorithm for searches,” IEEE Trans. Energy Convers., vol. 23, no. 3, pp. 804–813, Sep. 2008. [16] S. Herman, Alternating Current Fundamentals, 8th ed. Delmar Cengage Learning, 2011 [17] J. Riveros, A. G. Yepes, F. Barrero, J. Doval-Gandoy, B. Bogado, O. Lopez, M. Jones, and E. Levi, “Parameter identification of multiphase induction machines with distributed windings—Part 2: Time domain techniques,” IEEE Trans. Energy Convers., vol. 27, no. 4, pp. 1067–1077, 2012. [18] E. Levi, M. Jones, S. Vukosavic, and H. Toliyat, “Steady-state modeling of series-connected five-phase and six-phase two-motor drives,” IEEE Trans. Ind. Appl., vol. 44, no. 5, pp. 1559–1568, Sep./Oct. 2008. [19] J. Figueroa, J. Cros, and P. Viarouge, “Generalized transformations for polyphase phase-modulation motors,” IEEE Trans. Energy Convers., vol. 21, no. 2, pp. 332–341, Jun. 2006. [20] M. Jones, S. Vukosavic, D. Dujic, and E. Levi, “A synchronous current control scheme for multiphase induction motor drives,” IEEE Trans. Energy Convers., vol. 24, no. 4, pp. 860–868, Dec. 2009. [21] B. J. Chalmers and R. Dodgson, “Saturated leakage reactances of cage induction motors,” Proc. IEE, vol. 116, no. 8, pp. 1395–1404, Aug. 1969.

Alejandro G. Yepes (S’10–M’12) received the M.Sc. and Ph.D. degrees from the University of Vigo, Vigo, Spain in 2009 and 2011, respectively. Since 2008, he has been with the Department of Electronics Technology of the University of Vigo. His research interests include the control of switching power converters, ac drives, and power quality problems.

Jose A. Riveros received the B.Eng. degree in electronic engineering from the Universidad Nacional de Asunción, San Lorenzo, Paraguay, in 2005, and the M.Sc. degree from the University of Seville, Seville, Spain, in 2009, where he is currently working toward the Ph.D. degree in the Department of Electronic Engineering. He is a recipient of Scholarship from Itaipu Binacional/ Parque Tecnológico Itaipu-Py for the Ph.D. studies.

´ Doval-Gandoy (M’99) received the M.Sc. deJesus gree from the Polytechnic University of Madrid, Madrid, Spain, in 1991, and the Ph.D. degree from the University of Vigo, Vigo, Spain, in 1999. From 1991 to 1994, he was with industry. He is currently an Associate Professor at the University of Vigo. His research interests include the areas of ac power conversion.

Federico Barrero (M’04–SM’05) received the M.Sc. and Ph.D. degrees in electrical and electronic engineering from the University of Seville, Seville, Spain, in 1992 and 1998, respectively. In 1992, he joined the Department of Electronic Engineering, University of Seville, where he is currently an Associate Professor. Dr. Barrero received the Best Paper Award from the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS for 2009. ´ Oscar López (M’05) received the M.Sc. and Ph.D. degrees from the University of Vigo, Vigo, Spain, in 2001 and 2009, respectively. Since 2004, he has been an Assistant Professor with the Department of Electronics Technology, University of Vigo. His research interests include the areas of ac power switching converters technology.

Blas Bogado (S’10) was born in Paraguay in 1982. He received the M.Sc. degree from the University of Seville, Seville, Spain, in 2011, where he is currently working toward the Ph.D. degree in the Department of Systems and Automatics. His research interests include the area of control of multiphase motors. Mr. Bogado is a recipient of the training program for university lecturers from the Ministry of Education of Spain for the Ph.D. studies.

Martin Jones received the B.Eng. degree (First Class Hons.) in electrical engineering from the Liverpool John Moores University, Liverpool, U.K. in 2001. He was a research student at the Liverpool John Moores University from September 2001 till Spring 2005, when he received the Ph.D. degree. He was a recipient of the IEE Robinson Research Scholarship for the Ph.D. studies and is currently with Liverpool John Moores University as a Reader. His research interest includes the area of high performance ac drives. Emil Levi (S’89–M’92–SM’99–F’09) received the M.Sc. and Ph.D. degrees in electrical engineering from the University of Belgrade, Belgrade, Yugoslavia, in 1986 and 1990, respectively. From 1982 to 1992, he was with the Department of Electrical Engineering, University of Novi Sad. He joined Liverpool John Moores University, Liverpool, U.K., in May 1992, and is since September 2000 a Professor of Electric Machines and Drives. Dr. Levi serves as an Editor of the IEEE TRANSACTION ON ENERGY CONVERSION, a Co-Editor-in-Chief of the IEEE TRANSACTION ON INDUSTRIAL ELECTRONICS, and as the Editor-inChief of the IET Electric Power Applications. He is the recipient of the Cyril Veinott Award of the IEEE Power and Energy Society for 2009 and the Best Paper Award of the IEEE TRANSACTION ON INDUSTRIAL ELECTRONICS for 2008.