Dynamics and Stability of Matrix-Converter Based
Permanent Magnet Wind Turbine Generator
Bingsen Wang
Giri Venkataramanan
Department of Electrical and Computer Engineering
Michigan State University
East Lansing, MI, USA
Email: bingsen@egr.msu.edu
Department of Electrical and Computer Engineering
University of Wisconsin-Madison
Madison, WI, USA
Email: giri@engr.wisc.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.
I. I NTRODUCTION
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
harmonic distortion, lower common mode, and higher efficiency among other performance metrics [11]–[18].
In addition to the study of modulation methods used, topological 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 conventional 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-
978-1-4673-2420-5/12/$31.00 ©2012 IEEE
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 development 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 permanent 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 numerically. 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. Section 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 dqreference frames of the matrix converter are developed in
Section III. A boost configuration of the matrix-converter
interfaced permanent-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
6073
ip1
S1
-
vi1
vi2
vi3
T11
+
+
+
ii1
+
T31
T12
T32 T41
ii2
v12
ii3
S2
T21
T23
S3
T42 T51
S4
T22
T31
S1
T13
vo1
io1
vo2
io2
vo3
io3
T11
Current Source Bridge
-
T32
T12
S1
T11
(a)
T52
S5
ip2
T31
T13
T12
T32
T13
(b)
Fig. 2. Semiconductor realization the ideal switches in Fig. 1: (a) realization
of the SPTT switch S1 ; (b) realization of the SPDT switch S3 .
Voltage Source Bridge
S1
Fig. 1. 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 verified by numerical simulation in Section VI. A summary and
further discussions in Section VII conclude this paper.
-
vi2
vi3
T11
+
+
+
ii1
T13
T12
ii2
T21
S2
T22
ii3
T23
vo1
io1
vo2
io2
vo3
io3
T32
II. M ODULATION OF M ATRIX C ONVERTERS
The modulation of IMC is based on the ideal 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,
S1 and S2 . The VSB is further composed of three single-poledouble-throw (SPDT) switches S3 , S4 , and S5 .
The ideal SPTTs and SPDTs can be realized with semiconductor devices such as insulated gate bipolar transistors
(IGBTs) and diodes. As an illustrative example, the SPTT
switch S1 and the SPDT switch S3 can be realized by the
four-quadrant throws as shown in Fig. 2(a) and the twoquadrant 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 alternative
realization 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 vi1 , vi2 , and vi3
while the output terminals are connected to a set of currentstiff sources io1 , io2 , and io3 . Due to the modulation of the
switches, the synthesized input currents ii1 , ii2 , and ii3 and
output voltages vo1 , vo2 , and vo3 are discontinuous in time.
Fig. 3 illustrates the schematic of a conventional matrix
converter that consists of three SPTT switches S1 , S2 , and
S3 . Each of the three SPTTs can be realized by the fourquadrant 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 notational
significance.
The description of modulation process of the matrix converters can be mathematically facilitated by switching functions.
The switching function hxy of throw Txy with x, y ∈ {1, 2, 3}
vi1
S3
T31
T33
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 discontinuously varies in time.
1 throw Txy is on and conducts current
(1)
hxy =
0 throw Txy is off and blocks voltage
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 = HVSB HCSB vi
ii = HTCSB HTVSB io
(2)
where the superscript ‘T ’ denotes the transpose of a matrix; the
vectors vo , ii , vi , io and matrices HVSB and HCSB are defined
by
⎡ ⎤
⎡ ⎤
⎡ ⎤
⎡ ⎤
vo1
ii1
vi1
io1
vo = ⎣vo2 ⎦ ; ii = ⎣ii2 ⎦ ; vi = ⎣vi2 ⎦ ; io = ⎣io2 ⎦ (3)
vo3
ii3
vi3
io3
⎤
⎡
h31 h32
h11 h12 h13
⎦
⎣
HVSB = h41 h42 ; HCSB =
(4)
h21 h22 h23
h51 h52
In a similar manner, the modulation process of the CMC
can be described by the reciprocal relationships as
vo = HCMC vi
ii = HTCMC io
where the matrix HCMC is defined by
⎤
⎡ h11 h12 h13
HCMC = ⎣h21 h22 h23 ⎦
h31 h32 h33
6074
(5)
(6)
(7)
-
vi2
vi3
+
^ii1
+
^ii2
+
^ii3
^
+ v12
-
Voltage Source Bridge
⎤
h31 h13 + h32 h23
h41 h13 + h42 h23 ⎦
h51 h13 + h52 h23
(8)
Hence, the subsequent analysis based on the IMC is equally
applicable to the CMC as well.
III. AVERAGED M ODEL OF M ATRIX C ONVERTERS
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 frequencies 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 reference frame synchronously rotating at input frequency, while
the quantities on output side are transformed to the 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
t
1
mxy (t) =
hxy (τ )dτ
(9)
Ts t−Ts
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).
io1
^ip1
v^o2
io2
v^o3
io3
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, the modulation functions m31 , m41 , m51 for
each throw are related to the phase-leg modulation functions
mo1 , mo2 , mo3 as
mo1 + 1
mo2 + 1
mo3 + 1
; m41 =
; m51 =
(12)
2
2
2
where the phase-leg modulation functions mo1 , mo2 , mo3 are
further determined by
m31 =
mo1 = Mo cos (ωo t)
mo2 = Mo cos (ωo t − 2π/3)
(13)
mo3 = Mo cos (ωo t + 2π/3)
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 the variation of ip1 matches segments 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
mi1 = m11 − m21
mi2 = m12 − m22
(14)
mi3 = m13 − m23
The averaged dc link current îp1 and v̂12 in (10) can be
rewritten with reference to (12) and (14) as the following.
mo1 io1 + mo2 io2 + mo3 io3
2
= mi1 vi1 + mi2 vi2 + mi3 vi3
îp1 =
v̂12
(15)
Furthermore, the averaged input currents and output voltages
can be obtained.
îi1 = mi1 îp1 v̂o1 = mo1 v12 /2
îi2 = mi2 îp1 ; v̂o2 = mo2 v12 /2
îi3 = mi3 îp1 v̂o3 = mo3 v12 /2
îp1 = m31 io1 + m41 io2 + m51 io3
v̂12 = (m11 − m21 )vi1 + (m12 − m22 )vi2 + (m13 − m23 )vi3
(10)
where ‘ˆ’ denotes the averaged quantity. For instance, the
averaged dc link current îp1 is related to its instantaneous
quantity ip1 by
t
1
ip1 (τ )dτ
(11)
îp1 =
Ts t−Ts
v^o1
+
-
OR
⎡ ⎤
h11 h12 h13
⎣h21 h22 h23 ⎦ =
h
h32 h33
⎡ 31
h31 h11 + h32 h21 h32 h12 + h32 h22
⎣h41 h11 + h42 h21 h42 h12 + h42 h22
h51 h11 + h52 h21 h52 h12 + h52 h22
-
vi1
+
-
HCMC = HVSB HCSB
-
+
-
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.
(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 developed 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
6075
+
vi
3
iˆi mi mo ˜io 4
-
+
-
vˆo
3
mo mi ˜ vi 4
+
vi
q
io
3 q qq
mi mo io
4
3 q dd
mi mo io
4
++-
3 q q q
mo mi vi
4
3 q d d
mo mi vi
4
io
3 d qq
mi mo io
4
3 d dd
mi mo io
4
++-
3 d q q
mo mi vi
4
3 d d d
mo mi vi
4
io
-
q
VSB & CSB
+
Fig. 5. Averaged model of an indirect matrix converter in space-vector
notation.
vi
d
-
follows. The space vectors of the modulation functions and
the currents on the VSB side (output side) are defined as
2
mo =
mo1 + αmo2 + α2 mo3 e−jωo t
3
(17)
2
io1 + αio2 + α2 io3 e−jωo t
io =
3
where ωo is the output fundamental frequency on the output
side and α = ej2π/3 .
With the assumption of io1 + io2 + io3 = 0, the dot product
of these two complex vectors is determined to be
1
(mo i∗o + m∗o io )
2
(18)
2
= (mo1 io1 + mo2 io2 + mo3 io3 )
3
In a similar manner, the space vectors of the modulation
functions and the voltages on the CSB side (output side) are
defined as
2
mi =
mi1 + αmi2 + α2 mi3 e−jωi t
3
(19)
2
vi1 + αvi2 + α2 vi3 e−jωi t
vi =
3
where ωi is the input fundamental frequency on the input side.
With the assumption of vi1 + vi2 + vi3 = 0, the dot product
of mi and v i is determined to be
Fig. 6. Equivalent circuit of the averaged model of the IMC in synchronous
dq reference frame.
+
T11
is1
2
(mi1 vi1 + mi2 vi2 + mi3 vi3 )
(20)
3
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.
3
3
(m · i ); v̂12 = (mi · v i )
(21)
4 o o
2
The synthesized input current space vector and output
voltage space vector are
Turbine& Generator
T31
T32
vs2
is3
vs1
T13
T12
is2
mo · io =
mi · v i =
d
v12
vs3
T22
+
T21
-
Cs
T23
Lg
T41
ig1
T42 T51
ig2
T52
ig3
+
+
+
vg1
vg2
vg3
-
Grid
-
Current Source Bridge
Voltage Source Bridge
Fig. 7. Schematic of the PMAC-based wind generation that is interfaced to
a grid through an IMC 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 synchronous dq reference frame and is written in rectangular
form.
q
îi = 34 mqi (mqo iqo + mdo ido );
d
îi
= 34 mdi (mqo iqo + mdo ido );
v̂ qo = 34 mqo (mqi viq + mdi vid )
v̂ do = 34 mdo (mqi viq + mdi vid )
(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.
îp1 =
3
3
(22)
îi = mi (mo · io ); v̂ o = mo (mi · v i )
4
4
Due to the absence of passive components in the matrix
converter, the input and output are related by the algebraic
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 described by (22) can be pictorially represented by the equivalent
circuit in Fig. 5.
IV. B OOST C ONFIGURATION O F PMAC G ENERATOR
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
6076
developed in space vector notation as (24).
d
3
i = −(Rg + jωg Lg )ig + mg (ms · v s ) − v g
dt g
4
d
3
Cs v s = −jωr Cs v s + is − ms (mg · ig )
dt
4
d
Ls is = −(Rs + jωr Ls )is + ωr λpm − v s
dt
Lg
(24)
where the sub-matrices A11 , A21 , and A22 are further determined by
R
− Lgg −ωg
A11 =
R
ωg
− Lgg
⎡
⎤
q∗
∗
d∗
∗
q
where λpm is the space vector of field flux linkage produced
by permanent magnets. is is the space vector of the stator
currents. v s is the space vector of the stator voltage. ig is the
space vector of the output currents that are fed to grid. ωr is
electrical rotor speed of the PMAC generator.
The manipulated inputs of the system are the modulation 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
v ∗g and θs∗ , 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 v s as
follows.
v ∗g
jθs∗
(25)
mg = 3
∗ ; ms = e
jθ
s
4 vs · e
where θs∗ 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.
d
i = −(Rg + jωg Lg )ig + v ∗g − v g
dt g
vgq∗ iqg + vgd∗ idg
∗
d
ejθs
Cs v s = −jωr Cs v s + is − q
dt
vs cos(θs∗ ) − vsd sin(θs∗ )
d
Ls is = −(Rs + jωr Ls )is + ωr λpm − v s
dt
(26)
Lg
With the dynamic model of the system shown in (26), the
stability analysis is readily to be conducted.
V. S TABILITY A NALYSIS
Clearly, the system described in (26) features various nonlinearities. 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
(27)
dt
where x is the vector of state variables given by x =
[iqg idg vsq vsd iqs ids ]T . 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.
0
A11
(28)
A=
A21 A22
A21
A22
Vg
cos Θs
−Vg
cos Θs
d
∗
d
∗
Θ∗
Cs (Vs cos Θ∗
s −Vs sin Θs )
s −Vs sin Θs ) ⎥
⎢ Cs (Vs cos q∗
⎢
⎥
−Vg sin Θ∗
Vgd∗ sin Θ∗
s
s
⎥
q
q
=⎢
⎢ Cs (Vs cos Θ∗s −Vsd sin Θ∗s ) Cs (Vs cos Θ∗s −Vsd sin Θ∗s ) ⎥
⎣
⎦
0
0
0
0
⎡
⎤
1
−Kω cos2 Θ∗s
−ωr
0
Cs
1 ⎥
⎢
ωr
−Kω sin2 Θ∗s
0
Cs ⎥
=⎢
Rs
1
⎣
0
− Ls −ωr ⎦
Ls
1
s
0
−ωr − R
Ls
Ls
q
V q∗ I q +V d∗ I d
g
g
i
i
where Kω = Cs (Vsq cos
d
∗ 2.
Θ∗
s −Vs sin Θs )
Since the A matrix is a block-triangular matrix, its determinant may be factored into the following form:
det(A) = det(A11 ) 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 eigenvalues of A11 and the eigenvalues of A22 . If the
synthesized current is controlled to be in phase with the stator
terminal voltage, i.e. Θ∗s = 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 diagonal element
at the (1, 1) position when Viq∗ Iiq + Vid∗ Iid > 0. Detailed
numerical 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 q∗ I q +V d∗ I d
⎤
1
− g Csi(Vsqg)2 i −ωr
0
Cs
⎢
1 ⎥
⎢
⎥
0
0
ωr
Cs ⎥
A22 = ⎢
(30)
R
1
s
⎣
0
−
−ω
r⎦
Ls
Ls
1
s
0
−ωr − R
Ls
Ls
The conditions for stability under typical 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
Ym =
Vgq∗ Iiq + Vgd∗ Iid
2(Vsq )2
(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
6077
Z(s) = Rs
s
ωz
s
s2
2
Qn ωn + ωn
1+
1+
(32)
+
es
Lg
Ls
+
-
Cs
-
PMAC Generator
TABLE I
L IST OF THE SYSTEM PARAMETERS .
+
vg
Cs
Rg
Lg
Np p
PN
ωmR
Rs
Ls
-
Z(s) Ym
VSB & CSB
Fig. 8.
Illustration of the machine/filter capacitor input impedance and
converter output admittance.
The peak value of the impedance of the second order damped
L-C network is well known to be
Zc
(34)
Znpk = 2 1 + Q22 − 2 + Q12
n
3
0.1
1.0
24
10
150
0.265
1.4
μF
Ω
mH
kW
rpm
Ω
mH
Unstable
max{real[eig(A22)]}
The input impedance has a dc value of Rs , followed by
a real zero at ωz given by Rs /Ls , and√a complex quadratic
pole at the resonant frequency ωn = 1/ Ls Cs , with a quality
factor Qn as defined below: %
Zn
Qn =
with Zc = Ls /Cs
(33)
Rs
Filter capacitor at machine terminal
Resistance of grid-side inductor
Inductance of grid-side inductor
Number of pole pairs of the PMAC
Rated power
Rated mechanical speed of PMAC rotor
Stator winding resistance
Stator winding inductance
Stable
Stable
Unstable
n
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
Znpk ≈ Zc Qn
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. Viq∗ Iiq + Vid∗ Iid < 0,
A22 has been found to have all its eigenvalues on the left half
plane if Ym Znpk < 1, which is alternatively expressed by the
impedance matching criterion.
1
> Zc Q n
Ym
|YmZnpk| as
(35)
(36)
It is worth noting that when the PMAC machine is under
motoring mode, Viq∗ Iiq +Vid∗ Iid > 0, the (1,1) 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
wind-turbine.
VI. N UMERICAL V ERIFICATION
A numerical eigenvalues analysis has been conducted to
verify the stability criterion in (36). The system parameters are
listed in Table I. 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 ωr or the value of the filter capacitance Cs
decreases as shown in Fig. 9.
A time-domain simulation is conducted to verify the operation of the stability criteria, including an appropriate grid
r
or Cs decreases
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
vg∗ 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 performance, 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. C ONCLUSIONS
This paper presented detailed modeling of a matrix converter 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
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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 Number, DE-EE0000544/001, titled “Integration of Wind Energy
Systems into Power Engineering Education Programs at UWMadison”.
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