Dynamics and Stability of Matrix-Converter Based Permanent Magnet Wind Turbine Generator

Bingsen Wang Department of Electrical and Computer Engineering

Michigan State University

East Lansing, MI, USA

Email: bingsen@egr.msu.edu

Abstract-This paper proposes a boost configuration of wind turbine generation system based on matrix converters. The proposed boost configuration features i) the limitation of the voltage transfer ratio of 0.866 will not limit the terminal voltage at the stator winding under normal operating mode; and ii) low voltage ride through capability is enhanced when grid disturbances occur. The main attention of this paper is focused on the dynamics and stability issue of the matrix-converter based wind turbine generation system, which very much distinguishes itself from the systems that consist of voltage source inverters or current source inverters, where the dynamics of input and output state variables are naturally decoupled by the large­enough passive components in the dc link. Stability criterion that will guide proper design of the system is proposed and validated through numerical simulation.

1. INTRODUCTION

Matrix converters offer solid-state intensive solutions to

various applications that require sinusoidal input and output

voltages and currents [1], [2]. The inherent bidirectional

power flow capability and potential very compact realization

due to miniaturized passive components have attracted well

attended research interests since the high-frequency synthesis

was first proposed by Alesina [3]. Significant research effort

has been directed to the modulation of the matrix converters.

An improved modulation strategy with a voltage transfer ratio

of 0.866 was published in 1989 [4]. The understanding of

the modulation process has been advanced by the indirect

modulation methods based on a fictitious dc link concept and

the further-developed space vector modulation scheme [5],

[6]. Among the various modulation schemes, it is worth

noting that unified understanding has been achieved by the

general approaches proposed in [7]-[10]. Furthermore, various

modulation schemes have been explored to achieve reduced

hannonic distortion, lower common mode, and higher effi­

ciency among other perfonnance metrics [11]-[18].

In addition to the study of modulation methods used, topo­

logical development of the indirect matrix converter (IMC) has

resulted in simplified implementations under certain operating

conditions [19], [20]. As an alternative to the nine-switch con­

ventional matrix converter (CMC), the IMC features a simple

and robust commutation. As the matrix converter technologies

approach increased maturity, application oriented research

effort has been reported in literature. Beyond the induction-

Giri Venkataramanan Department of Electrical and Computer Engineering

University of Wisconsin-Madison

Madison, WI, USA

Email: giri@engr.wisc.edu

machine based electric drive applications [21], expanded range

of applications of matrix converter have been found in wind

energy generation [22], [23] and energy storage [24]. In order

to achieve the desired dynamic performance of the overall

system in the specific application context, suitable modeling

tool is needed. A real- and reactive-power decoupling control

scheme has been proposed for matrix-converter based grid

interface of distributed generation [25]. However, the devel­

opment of the suitable models that are compatible with the

commonly adopted dq-model of electric machines has been

rarely found in literature. In particular, the stability issue

associated with the bidirectional power flow has not been

thoroughly explored.

This paper is focused on the development of the dynamic

model of the matrix-converter based wind turbine generation

system that uses a pennanent magnet ac (PMAC) generator.

In particular, low voltage ride-through (LVRT) requirements

for wind turbine systems dictate that wind turbines be able

to maintain grid connection during sag conditions all the

way down to zero volts [26]. This requires that the matrix

converter system be configured such that the PMAC generator

maintained at a high enough voltage beyond the cut-in speed of

the turbine. This leads the configuration of the matrix converter

drive-train in a boost configuration as opposed to the more

common buck configuration.

In addition, the stability issue that has been observed under

boost operating mode is investigated both analytically and nu­

merically. The stability criterion has been proposed to ensure

successful functioning of the overall system. The proposed

scheme is experimentally verified on an electric drive system

that includes a PMAC machine in a boost configuration.

The rest of the paper is organized as the following. Sec­

tion II presents the modulation process of both indirect matrix

converter and conventional matrix converters. It has been

shown that any modulation scheme developed for IMC can

be mapped to CMC. The averaged models in abc- and dq­

reference frames of the matrix converter are developed in

Section m. A boost configuration of the matrix-converter

interfaced penn anent-magnet wind turbine generator system

is proposed in Section IV and its dynamic model is presented.

Section V explores the stability issues associated with the

system under different operating conditions and the stability

978-1-4673-2421-2/12/$31.00 ©2012 IEEE 6069

'--v--' Currem Source Bridge

'----------y--­Voltage Source Bridge

Fig. I. Schematic of the indirect matrix converter that is composed of single­pole-double-throw and single-pole-triple-throw switches.

criterion is proposed. The proposed stability criterion is veri­

fied by numerical simulation in Section VI. A summary and

further discussions in Section VII conclude this paper.

II. MODULATION OF MATRIX CONVERTERS

The modulation of IMC is based on the ideaJ converter

as shown in Fig. 1. The schematic in Fig. 1 illustrates the

IMC that consists of two bridges, namely, a current source

bridge (CSB) and a voltage source bridge (VSB). The CSB

is composed of two single-pole-triple-throw (SPTT) switches,

Sl and S2. The VSB is further composed of three single-pole­

double-throw (SPOT) switches S3, S4, and S5. The ideal SPTTs and SPOTs can be realized with semi­

conductor devices such as insulated gate bipolar transistors

(IGBTs) and diodes. As an illustrative example, the SPTT

switch Sl and the SPOT switch S3 can be reaJized by the

four-quadrant throws as shown in Fig. 2(a) and the two­

quadrant throws as shown in Fig. 2(b), respectively. Although a

reduced-switch-count realization of the SPTT is possible if the

displacement factor on the output side (VSB side) is limited

to a certain operating range, the discussion of aJternative

reaJization of the IMC is beyond the scope of this paper.

Without loss of generality, the input terminals in Fig. 1 are

connected to a set of voltage-stiff sources ViI, Vi2, and Vi3

while the output terminals are connected to a set of current­

stiff sources iol, io2, and io3. Due to the modulation of the

switches, the synthesized input currents iiI, ii2, and ii3 and

output voltages Vol, vo2, and Vo3 are discontinuous in time.

Fig. 3 illustrates the schematic of a conventionaJ matrix

converter that consists of three SPTT switches Sl, S2, and

S3. Each of the three SPTTs can be realized by the four­

quadrant throws as shown in Fig. 2(a). It is worth noting

that the semiconductor realization of conventional matrix

converter does not depend on the operating condition, which

distinguishes the CMC from IMC. Again, the designation of

input and output terminal of the CMC only carries notationaJ

significance.

The description of modulation process of the matrix convert­

ers can be mathematically facilitated by switching functions.

The switching function hxy of throw Txy with x, Y E {I, 2, 3}

(a) (b)

Fig. 2. Semiconductor realization the ideal switches in Fig. 1: (a) realization of the SPTT switch 81; (b) realization of the SPDT switch 83.

Sl/--<T1 lJ-!.----==----+-

',-'-...:-"'- :.:.T-i.!.J1�/+-_,.,.'-" /��>\�2

, , Ti2 \ Vo2 lJ-!.----==---�--+--+--T,Q �OT,----��

, '

u-c----==---+----:::±:-+--'

''''"'''

' T 2�//

Fig. 3. Schematic of the direct matrix converter that is composed of three single-pole-triple-throw switches.

denotes the switching state of the particular throw and discon­

tinuously varies in time.

throw Txy is on and conducts current

throw Txy is off and blocks voltage (1)

The synthesized input currents and output voltages of the

IMC are related to the voltage-stiff sources and current-stiff

sources by the reciprocal relationships in (2).

Vo = HVSBHcSBVj (2)

ij = H�SBH�SBio

where the superscript ,T, denotes the transpose of a matrix; the

vectors vo, ij, Vj, io and matrices HVSB and HCSB are defined

by

[VOl] [i'l] [V'l] [iOl]

Vo = Vo2 ; i, = i,2 ; v, = V,2 ; io = �02 Vo3 2,3 V,3 Zo3

HVSB = [hll ; HCSB =

h2l

(3)

(4)

In a similar manner, the modulation process of the CMC

can be described by the reciprocal relationships as

Vo = HCMCvj

ij = H�MCio (5)

(6)

6070

A comparison of (2) and (5) suggests that the any modulation

functions developed for the IMC can be extended to CMC by

the following equivalency.

HCMC = HvsBHcsB

h32h12 + h32h22 h42h12 + h42h22 h52h12 + h52h22

(7)

h31h13 + h32h23] h41h13 + h42h23 h51 h13 + h52h23

(8)

Hence, the subsequent analysis based on the IMC is equally

applicable to the CMC as well.

III. AVERAGED MODEL OF MATRIX CONVERTERS

To model the matrix converter in control perspective, an

averaged model compatible with the space vector modeling

practice is developed in this section. The fundamental frequen­

cies on each side of the matrix converter are different under

typical operating conditions. Consequently, it is appropriate

to transform the quantities on the input side to the dq refer­

ence frame synchronously rotating at input frequency, while

the quantities on output side are transformed to tile output

frequency. If an electric machine is connected on either side of

the converter, the transformation can be conducted in stator or

rotor reference frame. From control point of view, it is useful

to represent the matrix converter in an averaged model that

neglects the high frequency switching actions. Accordingly,

the system can be treated as continuous system and various

controller design tools from the very rich library of control

theory can be applied.

The averaged model is based on the modulation functions rather than the switching functions that are used in Section

II. With reference to Fig. 1, the modulation function mxy for

throw Txy is related to its switching function hxy by

mxy(t) = � it hxy(T)dT

s t-Ts (9)

where Ts is the switching period. With the further assumption

of the stiff input voltages and the output currents, the variation

of the stiff quantities in over each switching period Ts is

negligible. The averaged dc link current and voltage are related

to the output currents and input voltages, respectively, as in

(10).

ip1 = m31 io1 + m41 i02 + m51 i03 V12 = (mn - m21)vi1 + (m12 - m22)vi2 + (m13 - m23)vi3

(10)

where ,

A

, denotes the averaged quantity. For instance, the

averaged dc link current ip1 is related to its instantaneous

quantity ip1 by

(11)

�----"'v----' Voltage Source Bridge

'--------y----­Current Source Bridge

Fig. 4. Averaged model of an indirect matrix converter in abc reference frame. The averaged quantities are determined by equations (15) and (16).

For the VSB, tile modulation functions m31, m41, m51 for

each throw are related to the phase-leg modulation functions

mol, m02, m03 as

mol + 1 m31 = 2 m03 + 1 m51 = 2 (12)

where the phase-leg modulation functions mol, m02, m03 are

further determined by

mol = Mo cos (wot) m02 = Mo cos (Wot - 27r /3) m03 = Mo cos (wot + 27r /3)

(13)

It is worth noting the modulation index Mo can be time

varying such that the averaged dc link current ip1 is time

varying as well. Thus the switching frequency of the CSB can

be reduced without generation of low-order harmonics in the

synthesized input currents if tile variation of ip1 matches seg­

ments of the desired input current waveforms. More detailed

discussion of the modulation can be referenced to [27].

The phase-leg modulation functions for the CSB are related

to the modulation functions of the throws in CSB by

mil = mn - m21 mi2 = m12 - m22 (14)

The averaged dc link current ip1 and V12 in (10) can be

rewritten with reference to (12) and (14) as the following.

c mol io1 + m02i02 + m03i03 ·p1 = 2 (15)

V12 = mil Vi1 + mi2vi2 + mi3vi3 Furthermore, the averaged input currents and output voltages

can be obtained. A A

�i1 = mi1�p1 �i2 = mi2�p1 ii3 = mi3ip1

Vol = mol V12/2 V02 = m02v12/2 V03 = m03v12/2

(16)

Hence, an equivalent circuit of the IMC in abc reference frame

can be derived as shown in Fig. 4 .

The averaged model of the matrix converter can be de­

veloped in the dq reference frame by transforming the abc

variables to space vectors. First, the three-phase variables on

the VSB side are transformed to the dq reference frame as

6071

, 3 ( ) v =-m m·v _0 4-0 _I _i

'-v--' VSB & CSB

Fig. 5. Averaged model of an indirect matrix converter in space-vector notation.

follows. The space vectors of the modulation functions and

the currents on the VSB side (output side) are defined as

_ 2 ( + + 2 ) -jWot mo - 3 mol am02 a m03 e

. _2 ( . . 2')-jwot 10 - 3 Zol + aZ02 + a Z03 e

(17)

where Wo is the output fundamental frequency on the output

side and a = ej27r /3. With the assumption of io1 + i02 + i03 = 0, the dot product

of these two complex vectors is determined to be

1 mo'10 = 2 (mo 1� + m� 10)

2 = 3 (mo1io1 + m02i02 + m03i03) (18)

In a similar manner, the space vectors of the modulation

functions and the voltages on the CSB side (output side) are

defined as

2 ( 2 ) -J·w·t m· = - m'l + am'2 + a m'3 e ' -t 3 ' , , 2 ( 2 ) -J·w·t v· = - V'l + av'2 + a V'3 e ' -t 3 t , ,

(19)

where Wi is the input fundamental frequency on the input side.

With the assumption of Vi1 + Vi2 + Vi3 = 0, the dot product

of mi and 12.i is determined to be

2 mi . 12.i = 3 (mil Vii + mi2Vi2 + mi3vi3) (20)

Based on (18) and (19), the averaged current and voltage in

(15) can be alternatively expressed as functions of the output

current and input voltage space vectors, respectively.

c 3( . ) A

3( ) Zp1 = =1 mo . 10 ; V12 = 2 mi . 12.i (21)

The synthesized input current space vector and output

voltage space vector are

c 3 ( . ) Ii = =1mi mo . 10 ; (22)

Due to the absence of passive components in the matrix

converter, the input and output are related by the aJgebraic

equations (22). This is in great contrast to the ac/dc/ac

converter case where dynamic equations would be involved

between the input and output variables. The relationship de­

scribed by (22) can be pictoriaJly represented by the equivalent

circuit in Fig. 5.

.d 10

Fig. 6. Equivalent circuit of the averaged model of the [MC in synchronous dq reference frame.

'-----v-------' Turbine&Generator

+

TI3

1

T3I 12 T32 T41

VI2 T42 T"

Tn j T52 T23

'-----.r------' '------v------­Current Source Bridge Voltage Source Bridge

Fig. 7. Schematic of the PMAC-based wind generation that is interfaced to a grid through an [MC and works in boost mode.

In order for the averaged matrix converter model to be

interfaced with machine models that are commonly presented

in Cartesian coordinate system, (22) is transformed to syn­

chronous dq reference frame and is written in rectangular

form.

Y..� = img(m;v; + mfvf) iJd = :J.md(mqvq + mdvd) -0 4 0 1. 'l. 1. 1-

(23)

The corresponding equivalent circuit is shown in Fig. 6. It

is apparent that the equivalent circuit in dq reference frame is

compatible with the commonly adopted dq model of electric

machines.

IV. BOOST CONFIGURATION OF PMAC GENERATOR

The configuration of an electrical drive is shown in Fig. 7

where a permanent magnet ac (PMAC) generator is fed by

an IMC. Unlike the commonly adopted buck configuration of

electric drives, the boost configuration has been chosen due to

i) the limitation of the voltage transfer ratio of 0.866 will not

limit the available terminal voltage at the stator winding; and

ii) lower grid voltage enables LVRT compatibility.

Based on the well-documented dq model of the machine

in [28], in conjunction with the model of the matrix converter

in (23), the dynamic model of the overall plant can be

6072

developed in space vector notation as (24).

Lg � 1g = - (Rg + jwgLghg + �mg(ms . Qs) - Qg

C d . C . 3 ( . ) s dtQs = -JWr sQs + ls - 4ms mg 'lg (24)

Ls :t1s = - (Rs + jwrLshs + Wrflpm - Qs

where flpm is the space vector of field flux linkage produced

by permanent magnets. is is the space vector of the stator

currents. Qs is the space vector of the stator voltage. 19 is the

space vector of the output currents that are fed to grid. Wr is

electricaJ rotor speed of the PMAC generator.

The manipulated inputs of the system are the modula­

tion function vectors ms and mg for the CSB and VSB,

respectively. They are coupled to the states that affect the

output voltage. The modulation inputs ms and mg may be

algebraically decoupled in favor of an alternative control input

Q; and e;, which are reference vector of the synthesized

voltage (both amplitude and phase angle) on the grid side of

the IMC and the phase angle of the synthesized current on the

machine side, respectively, using feedback of the state Qs as

follows. v* . '"

m -g. m = eJBs -g = 3

. ejB;' -s 4Qs

(25)

where e; is the phase angle of the synthesized current space

vector on the stator side of the IMC.

Substituting (25) into (24) leads to the reformulated state

equations in terms of the new control inputs.

Lg :t1g = - (Rg + jWgLghg + Q; - Qg

d q*'q + d*'d C . C . Vg Zg Vg Zg jB* s-V =-JWr sQs + 1s e s

deS v� cos( e;) - v� sin( e;)

Ls :t1s = - (Rs + jwrLshs + Wrflpm - Qs (26)

With the dynamic model of the system shown in (26), the

stability anaJysis is readily to be conducted.

V. STABILITY ANALYSIS

Clearly, the system described in (26) features various non­

linearities. In order to examine its behavior in terms of stability

and design suitable controllers, it may be linearized around

a desired steady state operating point. The linearized system

may be described in the scalar form of

d -x = Ax+Bu dt (27)

where x is the vector of state variables given by x = [iZ i� vg v� i1 i�lT. The A matrix is determined by computing

the Jacobin of (26) at the operating point of interest. The A matrix is block-triangular as given by.

A = [All 0 ] A21 A22 (28)

where the sub-matrices All, A21, and A22 are further deter­

mined by

All =

A2l =

A22 =

[-�; Wg

CS(VS4 cos e; - V/ sin e;) - vgq* sin e;

Cs(Vsq cos e; - V/ sin e;) o o

w s -Wr

CS(VS4 cos e; - V/ sin e;) v.qd* sin e:

Cs(Vsq cos e; - V/ sin e;) o o

1 Cs

Wr r -K ,,,,'e'

-Kwsin2 e; 0 C, ? 1 1 0 Rs L, -r:; -Wr 0 1 -Wr R,

L, -r;

_ vgq* Ii+Vgd* It

where Kw - C (V" 8* _ V d' 8*)2' s 8 COS s t sin s Since the A matrix is a Dlock-triangular matrix, its deter-

minant may be factored into the following form:

det(A) = det(All) det(A22) (29)

where det(·) denotes the determinant of a matrix. Due to this

factorial decomposition, the eigenvalues of A are the union

of the eigenvaJues of All and the eigenvalues of A22. If the

synthesized current is controlled to be in phase with the stator

terminaJ voltage, i.e. e; = 0, the generator will operate under

unity power factor condition, which will minimize the stator

losses for a given power converted from wind turbine shaft

to the stator terminal. Under such condition, sub-matrix A22 becomes (30). A preliminary examination of the A22 matrix in

(30) indicates the presence of a large positive diagonaJ element

at the (1,1) position when Viq* Ii + Vid* If > O. Detailed

numericaJ studies over a range of parameters indicate the large

positive element at (1,1) potentially leads to the eigenvalues

in the left half plane.

v;' * I; + vgd* It -Wr 1

X 1 C,(V,")2 C,

A22 = Wr 0 0 (30) 1 0 Rs -Wr L, - L,

0 1 -Wr _& Ls L,

The conditions for stability under typicaJ range of design

parameters have been studied empirically and are presented in

the form of an impedance matching criterion described further.

Let the incremental admittance looking forward into the

matrix converter, as illustrated in Fig. 8 be defined as

Vq* Iq + Vd* Id y;

g ' g ' m -

2(Vn2 (31)

The output impedance looking into the second order L-C filter

network consisting of the machine stator inductance Ls with

damping provided by stator winding resistance Rs and filter

capacitor, as illustrated in Fig. 8, is defined by

1 + ..£.. Z(s) = Rs Wz

2 (32) 1 + Q n

sWn

+ �;

6073

Fig. 8. lllustration of the machine/filter capacitor input impedance and converter output admittance.

The input impedance has a dc value of Rs, followed by

a real zero at Wz given by Rs/ Ls, and a complex quadratic

pole at the resonant frequency Wn = 1/ v'LsCs, with a quality

factor Qn as defined below: %

Qn = Zn with Zc = JLs/Cs (33)

Rs The peak value of the impedance of the second order damped

L-C network is well known to be

For typical designs, Rs is much smaller than Zc, which results

in a large quality factor Qn. Under these conditions, (34) may

be approximated by

(35)

The introduced error will be less than 0.5% for typical values

of Qn > 10. With these definitions, for operating points when the PMAC

machine is under generation mode, i.e. �q* It + �d* If < 0, A22 has been found to have all its eigenvalues on the left half

plane if Y mZnpk < 1, which is alternatively expressed by the

impedance matching criterion.

1 Ym

> ZcQn (36)

It is worth noting that when the PMAC machine is under

motoring mode, �q* It + �d* If > 0, the (l,l) element of A22 is negative and all the eigenvalues of A22 are on the left half

plane regardless of the impedance matching criterion given by

(36). This would correspond to grid-powered start-up of the

wi nd-turbi ne.

VI. NUMERICAL VERIFICATION

A numerical eigenvalues analysis has been conducted to

verify the stability criterion in (36). The system parameters are

listed in Table 1. The numerical verification of the eigenvalues

of A22 suggests that when condition (36) is violated under

generation mode, the eigenvalues of the system becomes

unstable. This unstable condition has been observed when the

generator speed Wr or the value of the filter capacitance Cs

decreases as shown in Fig. 9.

A time-domain simulation is conducted to verify the op­

eration of the stability criteria, including an appropriate grid

TABLE I LIST OF THE SYSTEM PARAMETERS.

Os Filter capacitor at machine terminal 3 /-LF Rg Resistance of grid-side inductor 0.1 [1 Lg Inductance of grid-side inductor 1.0 mH Npp Number of pole pairs of the PMAC 24 PN Rated power to kW WmR Rated mechanical speed of PMAC rotor L50 rpm Rs Stator winding resistance 0.265 [1 Ls Stator winding inductance L.4 mH

6.---�----�-----.----�----�----�---,

�

� �

4

2

N a S

0fJ -2 .....

� c; -4 � '-<-' � -6

S -8

-10

-12L---�----�-----L----�----�----L---� 0.7 0.8 0.9 1.1 1.2 1.3

Fig. 9. The eigenvalues of the system will move to the right half plane as the speed decreases or the capacitance at the stator terminal decreases.

current regulator that controls the wind power generation. It

may be observed from the first equation of (26) that the grid

current dynamics with respect to the modified control input

v; is identical to that of a classical three-phase voltage source

converter, and hence may adopt well-established approaches

based on complex vector decoupling to realize excellent per­

formance, and is not discussed further herein [24]. The system

response to the wind power change is shown in Fig. 10. It

is evident the stability is preserved during steady state and

transients with appropriate choices of power circuit elements.

VII. CONCLUSIONS

This paper presented detailed modeling of a matrix con­

verter driven PMAC wind-turbine generation system in a

boost configuration. An average model of the indirect matrix

converter is developed in both abc reference frame and dq synchronous reference frame. The formulation of the dq model

of the matrix converter is readily compatible with the widely

adopted dq model of the electric machines. It is worth noting

that the developed model for the IMC equally applies to the

CMC with minimal reformulation. The stability issue that

originates from the algebraic coupling of the input and output

has been identified for the boost operation of the wind power

6074

200 mTITIUITm���ITm�mTI���XKKF��� 2: .,00 0

-200 2.9 2.95 3 3.05 3.1

�.-:;� 2.9 2.95 3 3.05 3.1

�·2� 2.9 2.95 3 3.05 3.1

�rn JIffl_lJIfJJJfflllfllJJJ 2.9 2.95 3 3.05 3.1

t (s)

Fig. 10. Simulated results: from top to bottom the waveforms are: back emf, stator terminal voltages, stator currents, generated torque, and grid currents.

generation system and a stability criterion is proposed to

ensure proper design of the system components and the tuning

of controller parameters. The analysis presented in this paper

has been validated by numerical simulation.

ACKNOWLEDGEMENT

A part of the work presented in this paper was funded by

the USA Department of Energy's 20% By 2030 Award Num­

ber, DE-EE0000544/001, titled "Integration of Wind Energy

Systems into Power Engineering Education Programs at UW­

Madison".

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