Harmonic Analysis of a DFIG for a Wind Energy Conversion System-bPs

IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 25, NO. 1, MARCH 2010 181

Harmonic Analysis of a DFIG for a Wind EnergyConversion System

Lingling Fan, Senior Member, IEEE, Subbaraya Yuvarajan, Senior Member, IEEE,

and Rajesh Kavasseri, Senior Member, IEEE

Abstract—This paper develops a framework for analysis of har-monics in a doubly fed induction generator (DFIG) caused by non-sinusoidal conditions in rotor and unbalance in stator. Nonsinu-soidal rotor voltages are decomposed into harmonic componentsand their corresponding sequences are identified. Induced harmon-ics in stator are analyzed and computed, from which the torquesproduced by these interactions between stator and rotor harmoniccomponents can be found. During unbalanced stator conditions,symmetric component theory is applied to the stator voltage to getpositive-, negative-, and zero-sequence components of stator androtor currents. The steady-state negative-sequence equivalent cir-cuit for a DFIG is derived based on the reference frame theory.Harmonic currents in the rotor are computed based on the se-quence circuits. In both scenarios, the harmonic components of theelectromagnetic torque are calculated from the interactions of theharmonic components of the stator and rotor currents. Three casestudies are considered, namely: 1) nonsinusoidal rotor injection;2) an isolated unbalanced stator load scenario; and 3) unbalancedgrid-connected operation. The analysis is verified with results fromnumerical simulations in MATLAB/Simulink. For illustration, thesecond case is verified using experiments. The simulation resultsand experimental results agree well with the results from analysis.

Index Terms—Doubly fed induction generator (DFIG), harmon-ics, inverter, unbalance, wind generation.

NOMENCLATURE

iqs , ids q-Axis and d-axis stator currents.

i′qr , i′dr q-Axis and d-axis rotor currents referring to

the stator side.

Is , Ir Stator and rotor current vectors.

s, r Stator and rotor.

q, d Rotating reference frame.

ωe Nominal angular speed.

ωs , ωr , ωm Stator, rotor, and rotating angular speed.

+, − Positive and negative components.

I. INTRODUCTION

DOUBLY fed induction generators (DFIGs) are widely used

in wind generation. The possibility of getting a constant-

frequency ac output from a DFIG while driven by a variable-

speed prime mover improves the efficacy of energy harvest from

Manuscript received December 3, 2008; revised April 8, 2009. First pub-lished November 24, 2009; current version published February 17, 2010.Paper no. TEC-00470-2008.

L. Fan is with the University of South Florida, Tampa, FL 33617 USA(e-mail: [email protected]).

S. Yuvarajan and R. Kavasseri are with the Department of Elec-trical and Computer Engineering, North Dakota State University,Fargo, ND 58105 USA (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TEC.2009.2032594

wind [1]. Unlike a squirrel-cage induction generator, which has

its rotor short-circuited, a DFIG has its rotor terminals accessi-

ble. The rotor of a DFIG is fed with a variable-frequency (ωr )

and variable-magnitude three-phase voltage. This ac voltage in-

jected into the rotor circuit will generate a flux and a stator

voltage/current with a frequency ωr if the rotor is standing still.

When the rotor is rotating at a speed ωm , the net flux linkage and

the stator voltage/current will have a frequency ωs = ωr + ωm .

When the wind speed changes, the rotor speed ωm will change,

and in order to have the net flux linkage at a frequency 60 Hz,

the rotor injection frequency should also be adjusted. A key

requirement of a DFIG is to have its three-phase rotor circuit in-

jected with a voltage at a controllable frequency and controllable

magnitude.

The three-phase ac voltage can be synthesized using var-

ious switching techniques, including six-step switching [2],

pulsewidth modulation (PWM) [1], and space vector PWM [3].

To reduce the switching losses while having a simple control

circuit, a six-step switching technique is widely used in thyristor-

based inverters. The high-power capability of thyristor attracts

the implementation of thyristor-based converters in wind energy

systems and six-step switching has kindled a new interest in

wind energy systems [4], [5]. The six-step switching technique

generates quasi-sine ac voltages, which possess 6n ± 1 harmon-

ics. Under such conditions, the rotor currents contain harmonic

components, which, in turn, induce corresponding harmonics in

the stator. This leads to pulsating torques.

Harmonics can also be introduced by unbalanced stator condi-

tions. Unbalanced stator voltages can be resolved into positive-,

negative-, and zero-sequence voltages. Negative-sequence com-

ponents in the stator cause high-frequency components in rotor

currents and torque pulsations, which lead to several undesir-

able conditions such as overheating. Mitigating the effects of

unbalanced stator conditions on a DFIG by appropriate control

has been the focus of the works in [6]–[16]. The focus in these

works is to develop control schemes to minimize overcurrents

and pulsating torques.

Harmonic analysis of induction motor drives has been well

documented in textbooks [2] and [17] . Slip-energy recovery in-

duction motor drives have a topology similar to that of DFIGs,

with a unidirectional power electronic interface. Harmonic anal-

ysis of these kinds of drives can be found in [18]–[20]. Harmonic

analysis of a DFIG has also been seen in [21]–[23]. The work

in [21] focuses on harmonic components from a machine design

perspective such as: 1) nonsinusoidal distribution of the stator

and rotor windings, referred to as MMF space harmonics, and

2) variations in reluctance due to slots, referred to as slot har-

monics. The work in [22] calculates the distortion components

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182 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 25, NO. 1, MARCH 2010

in the rotor and stator currents due to a six-pulse rotor-side con-

verter. An equivalent circuit is also presented to calculate stator

currents, but the resulting variations in torque are not calculated.

The work in [23] focuses on the effects of speed ripples due to

rotor-side harmonic injection, which is an important considera-

tion. However, this phenomenon is sensitive (see [23, Fig. 8]) to

the moment of inertia of the rotor. Their experiments are carried

out on a small test machine (1/3 hp) with a very small moment

of inertia (0.0035 kg·m2) where the effects of the ripple can be

quite pronounced.

In [7] and [9]–[13], expressions for torque under unbalanced

stator conditions have been derived in terms of the complex

power and speed. The expressions for complex power are de-

rived using space vector concept or instantaneous reactive power

theory.

Our study will focus on harmonic components resulting from

operating conditions due to: 1) nonsinusoidal rotor injection and

2) unbalanced stator conditions. These two conditions are often

encountered in wind energy systems, and are thus, important

to analyze. Overall, the aim of this paper is to present a gener-

alized platform for harmonic analysis in DFIG systems under

such conditions. The objective of the analysis is to estimate the

magnitudes of the frequency components in the stator and rotor

currents, and therefore, in the torque. The currents are computed

from the steady-state equivalent machine circuit and the torque

is calculated from the interactions of the harmonic components

of the stator and the rotor currents.

Some basic assumptions are made in this paper: 1) the speed

is assumed to be constant and ripple-free to simplify our analysis

and 2) the winding distribution is sinusoidal, and hence, there

are no MMF space harmonics and slot harmonics.

In the first scenario, nonsinusoidal rotor voltages are decom-

posed into harmonic components and their corresponding se-

quences are identified. Then, the induced harmonics in the rotor

and stator of a DFIG are analyzed and computed, from which

the torques produced by these interactions between stator and

rotor harmonic components are found.

In the second scenario, for unbalanced stator conditions, sym-

metric component theory is applied to the stator voltage to get

the positive-, negative-, and zero-sequence components of stator

and rotor currents. The steady-state negative-sequence equiva-

lent circuit for a DFIG is derived based on the reference frame

theory. Harmonic currents in the rotor are obtained based on

the positive- and negative-sequence circuits. Harmonic compo-

nents in the torque due to the interactions between stator and

rotor sequence components are found.

The rest of the paper is organized as follows. Section II

gives the steady-state equivalent circuit of the DFIG under

positive-sequence, negative-sequence, and harmonic scenarios.

Section III presents the harmonic analysis of a DFIG with non-

sinusoidal rotor circuit injection. Section IV presents the har-

monic analysis of a DFIG under unbalanced stator condition.

Three case studies are given in Section V, namely: 1) nonsi-

nusoidal rotor injection; 2) an isolated unbalanced load; and 3)

unbalance in grid-connected operation. The analysis, along with

experimental results and numerical simulations, in MATLAB/

Simulink is also given. Section VI concludes the paper.

Fig. 1. Steady-state induction machine circuit representation.

II. STEADY-STATE EQUIVALENT CIRCUIT OF A DFIG

For analysis, the per-phase steady-state equivalent circuit of

a DFIG based on [24] and [25] is shown in Fig. 1. Here, N ,

ωs , and slip are defined based on sequence and harmonic con-

ditions. For example, when N = 1, ωs = ωe , and slip = s, the

circuit corresponds to the well-known positive-sequence equiva-

lent circuit of an induction machine. For harmonic and negative-

sequence conditions, the parameters are modified, and details

are presented in Sections III and IV, respectively.

Remarks:

1) Positive-sequence circuit:

N = 1ωs = ωe

slip = s.

2) Negative-sequence circuit:

N = −1ωs = −ωe

slip = 2 − s.

3) 6n − 1 harmonic rotor injection:

N = −1ωs = (6n − 1)ωr − ωm

slip =(6n − 1)ωr

(6n − 1)ωr − ωm.

4) 6n + 1 harmonic rotor injection:

N = 1ωs = (6n + 1)ωr + ωm

slip =(6n + 1)ωr

(6n − 1)ωr + ωm.

III. HARMONIC ANALYSIS WITH QUASI-SINE

ROTOR VOLTAGE INJECTION

In this section, harmonic analysis in the stator circuits with

quasi-sine rotor voltage injection will be investigated. The ac

voltage injected to the rotor usually comes from a dc/ac bridge

converter shown in Fig. 2. While PWM technique is widely

used for rotor injection [1], a six-step switching technique is

another possibility to simplify the control circuit and reduce the

switching losses. Six-step switching introduces 6n ± 1 harmon-

ics in the voltages and the resultant output is called a quasi-sine

waveform. Unlike PWM, this does not need sine and triangu-

lar waves. It is easy to adjust the rotor injection frequency by

simply varying a control voltage. The output line voltage of the


FAN et al.: HARMONIC ANALYSIS OF A DFIG FOR A WIND ENERGY CONVERSION SYSTEM 183

Fig. 2. Power circuit of a three-phase bridge inverter.

Fig. 3. Quasi-sine waveforms of line and phase voltages applied to the rotor.

inverter is a quasi-sine wave with levels 0, VB , and −VB , and

one of the three line voltages and a phase voltage are shown in

Fig. 3.

A. Harmonic Components in Stator and Rotor Currents

For the quasi-sine waveforms of Fig. 3, triple-n harmonics

(3, 6, 9, 15, . . .) are absent. The voltage waveform in phase A

can be expressed in the mathematical form as [2]

vra(t) = VS

∞∑

k

1

ksin(kωt), k = 1, 5, 7, . . . (1)

where VS = (2/π)VB .

The 120 phase displacements among the three phase voltages

can be conveniently represented as

vra(t) = Im

(

VS

∞∑

k

1

kejkωt

)

(2)

vrb(t) = Im

(

VS

∞∑

k

1

kejkω (t−τ )

)

(3)

vrc(t) = Im

(

VS

∞∑

k

1

kejkω (t−2τ )

)

(4)

where k = 1, 5, 7, 11, . . ., and ωτ = 2π/3.

For the 5th and any (6n − 1)th (n > 0) harmonic, the wave-

forms of the rotor voltages are given by

vra(t) = Im

(

VS1

6n − 1ej (6n−1)ωt

)

vrb(t) = Im

(

VS1

6n − 1ej (6n−1)ωtej (2π/3)

)

vrc(t) = Im

(

VS1

6n − 1ej (6n−1)ωte−j (2π/3)

)

.

Thus, the set of (6n − 1) harmonics can be considered as a

negative-sequence set. The frequency of the waveform is equiv-

alent to −(6n − 1)ω. Throughout the paper, a negative sign for

the frequency implies a negative sequence for the corresponding

quantity.

Using the same analogy, for the 7th or any (6n + 1)th har-

monic, the three-phase set is considered as a positive-sequence

set and the frequency of the waveform is (6n + 1)ω.

Therefore, the following harmonic components can be ob-

served in the stator current: ωr + ωm , −5ωr + ωm , 7ωr + ωm ,

−11ωr + ωm , 13ωr + ωm , etc. The lowest order harmonic

(LOH) observed has a frequency of |5ωr − ωm |.The magnitudes of the harmonics in the stator current can

be computed based on the steady-state equivalent circuit in

Fig. 1.

B. Harmonic Components and Magnitudes of Electromagnetic

Torque

The harmonic analysis of a DFIG is similar to the harmonic

analysis of an induction machine presented in [2]. For an in-

duction motor with a quasi-sine ac power supply, the harmon-

ics present in stator currents and electromagnetic torque are at

n × 60 Hz, while in a DFIG with quasi-sine rotor injection, the

harmonics in stator currents and electromagnetic torque are de-

pendent on the injected frequencies, which will be discussed in

the following paragraphs.

The constant or steady torques are developed by the reac-

tion of harmonic air gap fluxes with harmonic rotor MMFs, or

currents, of the same order. Since the 6n − 1 harmonics are

negative-sequence harmonics, the induced torques oppose the

torques produced by the fundamental MMFs and 6n + 1 har-

monics [2].

Besides the steady torques, there are also pulsating torque

components, which are produced by the reaction of harmonic

rotor MMFs with harmonic rotating fluxes of different order.

The reaction between the fundamental rotor MMF and the fifth

harmonic component in the stator current will cause a pulsat-

ing torque with frequency of ωr1 + ωm − ωs5 = 6ωr . Simi-

larly, the reaction between the fifth harmonic rotor MMF and

the fundamental component in the stator current will produce a

pulsating torque with frequency of ωr5 + ωm − ωs1 = −6ωr .

The reactions between the fundamental rotor MMF and the sev-

enth harmonic component in the stator current, and the seventh

harmonic rotor MMF and the fundamental component in the

stator current will also produce pulsating torques of the same

frequency 6ωr .

Similarly, the interactions between the fundamental rotor

MMF and the 11th (13th) harmonic component in the stator

current will produce a pulsating torque with frequency of 12ωr .

In general, the pulsating torque contains harmonics of frequency

6nωr (n > 0).

The torque can be expressed in terms of stator and rotor

currents in the same reference frame [25] or current space vector

[2]. The torque in terms of the q–d currents and the current space



vector can be written as

Te =

(

3

2

) (

P

2

)

M(iqsi′dr − idsi

′qr ) (5)

= 3

(

P

2

)

M Im[Is I∗r ] (6)

where Is = (iqs − jids)/√

2 and Ir = (i′qr − ji′dr )/√

2.

The steady torque can be expressed in current space vector as

Te0 = 3

(

P

2

)

M Im∑

k

[Isk I∗rk ] (7)

where k = 1, 5, 7, 9, 11, . . ., Ik = (iqk − jidk )/√

2. The q–dvariables are referred to the rotating reference frame with the

same speed as the frequency of the stator harmonic component.

The pulsating torque of frequency 6ωr can be expressed as

Te6 = 3

(

P

2

)

M Im[Is1 I∗r5e

j6ω r ]

+ 3

(

P

2

)

M Im[Is1 I∗r7e

−j6ω r ]

+ 3

(

P

2

)

M Im[Is5 I∗r1e

−j6ω r ]

+ 3

(

P

2

)

M Im[Is7 I∗r1e

j6ω r ].

The currents iqs1 and ids1 are in the reference frame with a

rotating speed of ωs1 , iqr5 and idr5 are in the reference frame

with a rotating speed of ωs5 , and iqr7 and idr7 are in the reference

frame with a rotating speed of ωs7 .

The pulsating torque of Te6 can be expressed in real variables

as

Te6 = Te cos 6 cos(6ωr t) + Te sin 6 sin(6ωr t) (8)

where Te cos 6 and Te sin 6 can be expressed by currents in[

Te6 cos

Te6 sin

]

=3

2

P

2M

×[

−(ids5+ids7) (iqs5 + iqs7) −ids1 iqs1 −ids1 iqs1

−iqs5+iqs7 (ids5+ids7) iqs1 ids1 −iqs1 −ids1

]

×

iqr1

idr1

iqr5

idr5

iqr7

idr7

. (9)

IV. HARMONIC ANALYSIS FOR THE UNBALANCED

STATOR CONDITION

The purpose of the analysis is to investigate the DFIG opera-

tion at unbalanced stator conditions and study the waveforms of

the rotor currents and the electromagnetic torque. It is assumed

that sinusoidal voltages are injected into the rotor and that the

rotor injection voltage magnitude is constant during the system

disturbance.

Fig. 4. Two reference frames: synchronous and negative synchronous.

There are two steps in the analysis. The first step is to identify

the harmonic components in the rotor currents and the electro-

magnetic torque, and the second step is to estimate the magni-

tude of each harmonic component.

A. Harmonic Components in Stator and Rotor Currents

The stator frequency is assumed to be 60 Hz. During stator

unbalance, the magnitudes of the three phase voltages will not

be the same. Also, the phase angle displacements of the three

voltages will not be 120. Using symmetric component theory, a

set of three-phase voltages can be decomposed into a positive-,

a negative-, and a zero-sequence component. The stator cur-

rents will, in turn, have positive-, negative-, and zero-sequence

components.

For an induction machine, the sum of the rotor injection fre-

quency and the rotor frequency is equal to the stator frequency or

ωr + ωm = ωs . For the positive-sequence voltage set with fre-

quency ωs applied to the stator side, the resulting rotor currents

or flux linkage have a frequency ωr = ωs − ωm = sωs .

The negative-sequence voltage set can be seen as a three-

phase balanced set with a negative frequency −ωs . Thus, the

induced flux linkage in rotor circuit and the rotor currents have

a frequency of −ωs − ωm = −(2 − s)ωs .

Observed from the synchronous reference frame qd+ with a

rotating speed ωe = ωs , the first component (positive sequence)

has a frequency of sωs − (ωe − ωm ) = 0, or a dc component,

and the second component (negative sequence) has a frequency

of −(2 − s)ωs − (ωe − ωr ) = −2ωe , i.e., 120 Hz. Observed

from the negative synchronous reference frame qd−, which ro-

tates clockwise with the synchronous speed ωe , the positive-

sequence component has a frequency of sωs − (−ωe − ωm ) =2ωe , and the negative-sequence component has a frequency of

−(2 − s)ωs − (−ωe − ωm ) = 0. The two reference frames are

shown in Fig. 4, and Table I shows the components of the rotor

currents in abc and the two reference frames.

The rotor currents in both reference frames will have a dc

component and a high-frequency component. To extract the



TABLE IROTOR CURRENT COMPONENTS OBSERVED IN VARIOUS REFERENCE FRAMES

Fig. 5. Scheme for extracting dc components.

harmonic components in the rotor currents, both a synchronous

reference frame qd+ and a negative synchronous rotating refer-

ence frame qd− (see Fig. 4) will be used. A low-pass filter with

a suitable cutoff frequency can be used to extract the dc compo-

nents, which correspond to the magnitudes of the two harmonic

components. The scheme for extracting the dc components is

shown in Fig. 5.

B. Magnitudes of the Harmonic Components in Stator

and Rotor Currents

In this section, the derivation of the steady-state circuit for the

negative-sequence components is given. The equivalent circuit

is derived by establishing the relationship of the voltages and

currents in qd variables and further in phasors.

For the negative sequence, the q-axis, d-axis, and 0-axis vari-

ables become dc variables at steady state when the reference

frame rotates at a frequency of −ωe . The derivatives of the flux

linkages are zero at steady state. Hence, the voltage and current

relationship is expressed in qd0 as

v−eqs = rsi

−eqs − ωeλ

−eds (10)

v−eds = rsi

−eds + ωeλ

−eqs (11)

v′−eqr = r′r i

′−eqr + (−ωe − ωr )λ

′−edr (12)

v′−edr = r′r i

′−edr − (−ωe − ωr )λ

′−eqr . (13)

The relationship between a phasor Fas at a given frequency

and the corresponding qd variables in the reference frame rotat-

ing at the same frequency can be expressed as

√2Fa = Fq − jFd (14)

where F can be voltages, currents, or flux linkages in the stator

or rotor circuits.

Therefore, the stator and rotor voltage, current, and flux link-

age relationship can be expressed in phasor form as

V −eas = rs I

−eas − jωe λ

−eas (15)

V ′−ear = r′r I

′−e

ar − j(ωe + ωr )λ−ear . (16)

The rotor relationship can be further expressed as

V ′−ear

2 − s=

r′r2 − s

I ′−e

ar − jωe λ−ear . (17)

The equivalent circuit in Fig. 1 now has N = −1, ωs = ωe ,

and slip = 2 − s. If the rotor voltage injection is assumed to be a

balanced sinusoidal three-phase set, then the negative-sequence

component V ′−ear = 0.

Thus, the stator and rotor currents are induced by both the

positive-sequence voltages and negative-sequence voltages. The

rotor currents have two components: one at the low frequency

sωe having an rms magnitude of Ias+ and the other at the high

frequency (2 − s)ωe with a magnitude of Ias−.

C. Harmonic Components and Magnitudes of Electromagnetic

Torque

The zero-sequence stator currents will not induce a torque

[25]. Meanwhile, the zero-sequence stator circuit and the zero-

sequence rotor circuit are completely decoupled. Hence, the

zero-sequence variables presented at the stator side will not

induce any voltage at the rotor side. In machines where wye

connection is used, even the stator side does not have zero-

sequence currents.

Under the unbalanced stator condition, the stator cur-

rent has two components: positive-sequence components Is+

and negative-sequence components Is−. The rotor current

also has two components: positive-sequence components Ir+

and negative-sequence components Ir−. The electromagnetic

torque is produced by the interactions between the stator

and rotor currents. The torque can be decomposed into four

components

Te = Te1 + Te2 + Te3 + Te4 (18)

where Te1 is due to the interaction of Is+ and Ir+ , Te2 is due to

the interaction of Is− and Ir−, Te3 is due to the interaction of

Is+ and Ir−, and Te4 is due to the interaction of Is− and Ir+ .

It will be convenient to use both the synchronous reference

frame and the negative synchronous reference frame to compute

Te . For example, Te1 can be identified as a dc variable in the

synchronous reference frame. Te2 can be identified as a dc vari-

able in the negative synchronous reference frame. Te3 and Te4

are pulsating torques with a frequency 2ωe . The expressions for

the torque components are as follows:

Te1 = 3

(

P

2

)

M Im[Ies+ I∗er+ ] (19)

Te2 = 3

(

P

2

)

M Im[I−es− I∗−e

r− ] (20)

Te3 = 3

(

P

2

)

M Im[Ies+ I∗er−] (21)

= 3

(

P

2

)

M Im[Ies+ I∗−e

r− ej2ωe t ] (22)



Fig. 6. Stand-alone wind energy system configuration.

Te4 = 3

(

P

2

)

M Im[Ies−I∗er+ ] (23)

= 3

(

P

2

)

M Im[I−es− I∗er+e−j2ωe t ] (24)

where Is = (iqs − jids)/√

2 and I ′r = (i′qr − ji′dr )/√

2, and

F e+ denotes the qd variables of the positive-sequence compo-

nent in the synchronous reference frame, F e− denotes the qd vari-

ables of the negative-sequence component in the synchronous

reference frame, F−e+ denotes the qd variables of the positive-

component in the negative synchronous reference frame, and

F−e− denotes the qd variables of the positive-sequence compo-

nent in the negative synchronous reference frame.

The torque expression under unbalanced stator condition

is Te = Te0 + Te sin 2 sin(2ωst) + Te cos 2 cos(2ωst), where the

expressions for Te0 , Te sin 2 , and Te cos 2 can be found in

Te0

Te sin 2

Te cos 2

=3PM

4

ieqs+ −ieds+ i−eqs− −i−e

ds−i−eds− i−e

qs− −ieds+ −ieqs+

i−eqs− −i−e

ds− ieqs+ −ieds+

i′edr+

i′eqr+

i′−edr−

i′−eqr−

(25)

The harmonic components in the torque can be computed

from positive- and negative-sequence stator/rotor currents. In

the following sections, case studies will be performed.

V. CASE STUDIES—ANALYSIS AND SIMULATION

A. Case Study 1—-Nonsinusoidal Rotor Injection

A four-pole 5 hp DFIG with parameters in Table VII (see

the Appendix) is considered. The stator is connected to a wye-

connected resistive load with 22 Ω in each phase. The configura-

tion of the system is shown in Fig. 6. The injected rotor voltages

are quasi-sine.

The fundamental frequency of the rotor injection is 24 Hz.

The rotating speed is 1080 r/min for the four-pole 5 hp DFIG

and the corresponding electrical frequency is 36 Hz. The har-

monic orders of the rotor and stator currents are computed and

listed in Table II. The simulation (MATLAB/Simulink) results

are shown in Fig. 7. The fifth harmonic in the injected rotor

voltage causes a low harmonic in stator current at 84 Hz of neg-

ative sequence (24 × 5 − 36). Hence, distortions are observed

in the stator current waveforms. The torque is shown to have a

TABLE IIANALYZED FREQUENCY COMPONENTS IN THE ROTOR AND STATOR CURRENTS

DUE TO NONSINUSOIDAL ROTOR INJECTION

Fig. 7. DFIG with quasi-sine rotor injection. (a) Phase A rotor voltage.(b) Phase-to-phase rotor voltage. (c) Phase A rotor current. (d) Phase A sta-tor current. (e) Electromagnetic torque.

steady-state dc value and a pulsating component of 144 (6 × 24)

Hz. The discrete Fourier transform (DFT) of the simulated elec-

tromagnetic torque, and stator and rotor current waveforms are

shown in Fig. 8. The DFT results show that the torque has a 144

Hz harmonic, the stator current has a fundamental component

at 60 Hz and a harmonic at about 84 Hz, and the rotor current

has a fundamental component at 24 Hz and harmonics at 120

(5 × 24) and 168 (7 × 24) Hz. The DFT results agree well with

the analytical results in Table II.

The 144 Hz pulsating torque component is produced by the

reaction of harmonic rotor MMFs with harmonic rotating stator

MMFs of a different order. In this case study, the fifth har-

monic component produces an 84 Hz component with negative

sequence in the stator current. The reaction between the funda-

mental rotor MMF (24 Hz) and the fifth harmonic component

in the stator current (−84 Hz) will cause a pulsating torque at

fr1 + fm − fs5 = 144 Hz. Similarly, the reaction between the

fifth rotor MMF (120 Hz negative sequence) and the funda-

mental component in the stator current (60 Hz) will produce a

pulsating torque at fr5 + fm − fs1 = −144 Hz. The reactions

between the fundamental rotor MMF and the seventh harmonic

component in the stator current, and the seventh rotor MMF

and the fundamental component in the stator current will also

produce pulsating torques of ∓144 Hz. The dc component and

the pulsating component of the torque can be computed from

(7) and (9) and are shown in Table III. The analytical results

agree with the simulation results in Fig. 7 and the DFT results

in Fig. 8.



Fig. 8. DFT of the electromagnetic torque, and stator and rotor current wave-forms in Fig. 7.

TABLE IIIHARMONIC COMPONENTS IN THE ELECTROMAGNETIC TORQUE FROM

ANALYSIS

Experimental results of the stator and rotor currents can be

found in [26].

B. Case Study 2—A Stand-Alone DFIG System

In the second case study, the setup used is same as in Fig. 6

except that the resistance in phase A is varied to create an

unbalanced stator condition. The rotor injection is assumed to

be balanced sinusoidal. Two scenarios will be considered. In

scenario 1, the frequency of the injected rotor voltages is 20

Hz. In scenario 2, the frequency of the injector rotor voltages is

15 Hz.

In the experiment, the wind turbine is replaced by a dc motor

and a 5 hp wound rotor induction generator is used as a DFIG.

A sine-wave power source provides the injection voltage at any

desired frequency. The speed of the dc motor is adjusted to

have a 60 Hz stator voltage. The experimental setup is shown in

Fig. 9.

Figs. 10 and 11 show the waveforms of phase A and phase B

stator voltages, and phase A rotor current and stator current. The

experiment was conducted for two injection frequencies: 20 and

15 Hz. In both cases, the rotor current contains high-frequency

harmonics. The results of DFT analysis of the waveforms are

given in Fig. 12 and Table IV.

The unbalanced load condition is created by short-circuiting

a portion of the load resistance in phase A, and hence, the

equivalent circuit under single phase to ground fault can be

used to obtain the stator current and rotor current phasors of

positive sequence and negative sequence. Since the load is wye-

Fig. 9. Experimental setup for case study 2.

Fig. 10. Stator voltages, rotor current, and stator current under unbalancedload condition. Rotor injection frequency = 20 Hz.

Fig. 11. Stator voltages, rotor current, and stator current under unbalancedload condition. Rotor injection frequency = 15 Hz.

connected, there is no zero-sequence current. The equivalent

circuit is shown in Fig. 13.

An analysis of the circuit in Fig. 13 gives the harmonic com-

ponents in the rotor currents. The phasors and the corresponding

harmonic components are shown in Table IV. The experimental



Fig. 12. DFT of the rotor currents. (a) Rotor injection frequency = 20 Hz.(b) Rotor injection frequency = 15 Hz.

TABLE IVCOMPONENTS OF ROTOR CURRENTS FROM EXPERIMENTS AND ANALYSIS OF

FIG. 13 DURING UNBALANCED STATOR CONDITION

Fig. 13. Equivalent circuit for the DFIG under unbalanced stator load condi-tion.

waveforms and the DFT analysis results agree well with the

analytical results.

C. Case Study 3—-A DFIG Connected to Grid

A 3 hp DFIG is used for analysis and simulation. The machine

parameters are shown in the Appendix. The initial condition of

the machine is the stalling state. A balanced three-phase voltage

to the stator and a mechanical torque 10 N·m are applied at

t = 0 s. The rms value of the rotor voltage is kept at 10 V. The

Fig. 14. Grid-interconnected DFIG system configuration.

Fig. 15. Dynamic responses of rotor speed, electromagnetic torque, phase Astator current, and phase A rotor current.

Fig. 16. Dynamic response of ieq r , iedr , i−eq r , and i−e

dr.

system configuration is shown in Fig. 14. At t = 1 s, the voltage

of phase A drops to zero. The fault is cleared at t = 1.5 s. The

simulation is performed in MATLAB/Simulink, and the results

are shown in Figs. 15–17.

Fig. 15 shows the dynamic responses of the rotor speed, elec-

tromagnetic torque, and stator and rotor currents in phase A.



Fig. 17. Harmonic components after extracting strategy from Fig. 5.(a) ieq r+ —dc component of iq r observed in the synchronously rotating ref-

erence frame. (b) iedr+ —dc component of idr observed in the synchronously

rotating reference frame. (c) i−eq r−—dc component of iq r observed in the nega-

tively synchronously rotating reference frame. (d) i−edr−—dc component of idr

observed in the synchronously rotating reference frame.

TABLE VCALCULATED SEQUENCE COMPONENTS IN STATOR VOLTAGES, STATOR

CURRENTS, AND ROTOR CURRENTS ASSUMING SLIP = 4.5/60

TABLE VIHARMONIC COMPONENTS IN THE ROTOR CURRENTS AND THE

ELECTROMAGNETIC TORQUE FROM SIMULATION AND ANALYSIS DURING

UNBALANCED CONDITION (SLIP = 4.5/60)

Fig. 16 shows the dynamic responses of the rotor currents in

the synchronous reference frame and the negative synchronous

reference frame. At the steady-state balanced stator condition,

the rotor speed is equivalent to 57.2 Hz, while the slip frequency

of the rotor currents is 2.8 Hz. The two add up to 60 Hz. During

unbalance, it is found that the torque has a 120 Hz pulsating com-

ponent. It is seen from the simulation plots that the rotor current

consists of two components: a low-frequency component and a

high-frequency component. According to the analysis, the two

frequencies are the slip frequency and a frequency close to 120

Hz [(2 − s)ωe ]. The rotor currents observed in the synchronous

reference frame have a dc component and a 120 Hz component.

The magnitudes of the harmonic components can be com-

puted from the equivalent phasor circuits in Fig. 1.

The sequence components of the stator voltage, the stator cur-

rents, and the rotor currents are listed in Table V. A comparison

of the analysis results from the circuit in Fig. 1 and simulation

results shows that the rotor current and torque components from

the analysis agree with the simulation results in Table VI.

VI. CONCLUSION

This paper has presented a systematic method to analyze the

harmonics caused by nonsinusoidal rotor injection and unbal-

anced stator conditions in a DFIG. The key contributions of

the paper are: 1) a generalized steady-state equivalent circuit

for DFIGs suitable for analysis under harmonic and unbalanced

conditions; 2) a systematic method to calculate electromagnetic

torque by computing the interactions of harmonic stator and

rotor currents, derived from the equivalent circuit (see Fig. 1);

and Fig. 3) the development of positive- and negative-sequence

equivalent circuits, which enables one to analyze unbalanced

conditions on the stator side by a suitable interconnection of the

sequence circuits. The three case studies and experimental ver-

ifications demonstrate the effectiveness of the proposed method

in analyzing the harmonics and unbalanced operation of DFIGs.

APPENDIX

The 3 hp induction machine parameters, taken from [25], are

listed in Table VII. The 5 hp wound rotor induction machine

parameters are measured from experiments and are also listed

in the table.

TABLE VIIMACHINE PARAMETERS FOR 3 HP AND 5 HP DFIGS

ACKNOWLEDGMENT

The authors wish to thank the reviewers for their constructive

comments and suggestions, which have helped in improving the

quality of the manuscript.

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Lingling Fan (S’98–M’02–SM’08) received the B.S. and M.S. degrees fromSoutheast University, Nanjing, China, in 1994 and 1997, respectively, and thePh.D. degree from West Virginia University, Morgantown, in 2001, all in elec-trical engineering.

She was with Midwest ISO, St. Paul, MN (2001–2007) and North DakotaState University, Fargo, ND (2007–2009). She is currently an Assistant Profes-sor at the University of South Florida, Tampa, FL. Her research interests includemodeling and control of renewable energy systems, power system reliability,and economics.

Subbaraya Yuvarajan (SM’84) received the B.E. (Hons.) degree from theUniversity of Madras, Chennai, India, in 1966, and the M.Tech. degree and thePh.D. degree in electrical engineering from the Indian Institute of Technology,Chennai, in 1969 and 1981, respectively.

Since 1995, he has been a Professor in the Department of Electrical andComputer Engineering, North Dakota State University, Fargo. His current re-search interests include electronics, power electronics, and electrical machines.

Rajesh Kavasseri (SM’07) received the Ph.D. degree in electrical engineeringfrom Washington State University, Pullman, in 2002.

He is currently an Associate Professor in the Department of Electrical andComputer Engineering, North Dakota State University, Fargo. His researchinterests include power system dynamics and control, nonlinear systems, andalgebraic geometry application in power system analysis.


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