654
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 2, FEBRUARY 2014
Modulation Strategy Based on Mathematical
Construction for Matrix Converter Extending
the Input Reactive Power Range
Xing Li, Mei Su, Yao Sun, Member, IEEE, Hanbing Dan, and Wenjing Xiong
Abstract—In this paper, a modulation strategy based on mathematical construction is proposed to extend the input reactive power
range for the three-phase matrix converter, which offers clear physical meanings and less computational efforts. This strategy is developed based on the construction of the modulation matrix composed
by the sum of several matrices, one of which is used to generate the
required output voltage. The others are intended to provide more
degrees of freedom for control such that the matrix converter can
produce the input reactive power as much as possible. In the framework of mathematical construction method, an optimization problem for the maximum input reactive power is formulated, whose
analytical solution is difficult to obtain. Usually, optimization problem can be solved by using some numerical methods, but lots of time
will be consumed. Therefore, a suboptimal method is presented to
mitigate the computational burden. Besides, the proposed strategy
is compared with the optimum-amplitude and indirect SVM methods, in terms of the maximum input reactive power for different
operating conditions. It is shown that the proposed method can
obtain the maximum input reactive power over most situations.
Finally, the correctness of the proposed method is confirmed by
simulation and experimental results.
Index Terms—Input reactive power, matrix converter, modulation strategy, optimal.
I. INTRODUCTION
HE three-phase matrix converter is an ac–ac power converter, featured by the advantages such as sinusoidal input
and output currents, bidirectional energy flow, controllable input
power factor as well as a compact design, which has received
an increasing attention in recent years [1]–[4]. Due to the efforts of many researchers, it has found many applications in
adjustable-speed drives, power supply, wind energy conversion
system (WECS), flexible ac transmission systems (FACTS), and
so on. In most applications, the operation at unity input power
factor is preferred for a matrix converter. However, when matrix
converter is applied in WECS and FACTS [5], [6], the ability to generate and absorb input reactive power should be paid
T
Manuscript received July 18, 2012; revised November 27, 2012; accepted
March 30, 2013. Date of current version August 20, 2013. This work was supported by the National High-tech R&D Program of China (863 Program) under
Grants 2012AA051601 and 2012AA051603. Recommended for publication by
Associate Editor K.-B. Lee.
The authors are with the School of Information Science and Engineering, Central South University, Changsha 410083, China (e-mail: xingliaaa@
gmail.com; yaosuncsu@gmail.com; sumeicsu@csu.edu.cn; daniel698@
sina.cn; csu.xiong@163.com).
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/TPEL.2013.2256929
more attention. Because controllable reactive power generated
by matrix converter is helpful in reducing real power losses in
transmission lines and improving the voltage stability.
Usually, it is easy for the matrix converter to operate at unity
input power factor if the input filter influence can be neglected,
but it is difficult to realize with enough accuracy under the lightload condition. Usually, to compensate the reactive current, two
compensation methods were proposed in [7], where one is an
open-loop method based on the input filter and power supply
parameters; the other is a closed-loop method using a PI controller, which is robust to parameter variation. Besides, a control
scheme was proposed based on predictive control by selecting
a proper switch-combination to minimize one objective function [8], which is formed by adding the input reactive power
reference with other targets.
Unified power-flow controller (UPFC) is the combination of
a static synchronous compensator (STATCOM) and a static synchronous series compensator (SSSC), both of them can control
the reactive power freely. Compared with the UPFC based on
the back-to-back converter, the matrix converter-based UPFC
has smaller volume and better reliability. In the UPFC based
on matrix converter, the task of STATCOM will be completed
by the input stage of matrix converter; thus, the input reactive
power should be considered carefully. In addition, it is suitable
for the matrix converter to drive a doubly fed induction generator (DFIG) in WECS [9]. However, based on the grid codes,
the DFIG should be able to provide the appropriate reactive
power to support the grid voltage under abnormal network conditions [10]. For the reactive power requirement of the above
two types of system, it is necessary to investigate the input reactive power capability of matrix converter. However, due to the
nonlinear relations among the voltage transfer ratio, loads and
modulation methods, it is not easy to control the input reactive
power as required.
Since the matrix converter topology was first proposed in
1976, many modulation schemes have been presented, including
space vector modulation (SVM) [3], [11], optimum-amplitude
method [12], double input voltages synthesis [13], and some
others [14]–[16]. Different modulation schemes may reflect different input reactive power capability, so do different topologies
under the same modulation scheme. For instance, the input reactive power capability of the indirect matrix converter with the
SVM method is weaker than that of the conventional matrix converter, because the instantaneous dc-link voltage of the former
must be greater than zero. To extend the reactive power control range of the matrix converter, a hybrid modulation scheme
0885-8993 © 2013 IEEE
LI et al.: MODULATION STRATEGY BASED ON MATHEMATICAL CONSTRUCTION FOR MATRIX CONVERTER
expressed as
⎡
ua
⎤
655
⎡
cos(ωi t)
⎤
⎢
⎥
⎢ ⎥
ui = ⎣ ub ⎦ = Uim ⎣ cos(ωi t − 2π/3) ⎦
⎡
uc
ia
⎤
⎡
cos(ωi t + 2π/3)
cos(ωi t − ϕi )
(1)
⎤
⎢
⎥
⎢ ⎥
ii = ⎣ ib ⎦ = Iim ⎣ cos(ωi t − ϕi − 2π/3) ⎦
cos(ωi t − ϕi + 2π/3)
ic
Fig. 1.
Three-phase to three-phase matrix converter.
was proposed in [17], and the literature [18] also proposed a
three-vector-scheme for a FACTS device based on the matrix
converter.
In this paper, a novel modulation method based on mathematical construction is proposed for the matrix converter, which
can enlarge the control range of the input reactive power greatly.
In the framework of mathematical construction method, an optimization problem is presented for maximizing the input reactive current. However, it is difficult to determine the globaloptimum solution by simple analytical formulation. Generally,
some numerical methods can be employed, but much time will
be consumed to complete the calculation process. Therefore, a
suboptimal method is proposed to mitigate the computational
effort, which is easier to implement, but will lead to a relatively
smaller range. To illustrate its superiority and characteristics,
the proposed method is compared with the indirect SVM and
optimum-amplitude methods, under various given output voltages and load conditions. The results show that the input reactive
power capability of the proposed method is superior to that of
indirect SVM for all cases, and that of the optimum-amplitude
method for most cases. Simulation and experiment results verify
the correctness of the proposed method and related theoretical
analysis.
II. BASIC MODULATION METHOD
A. The Matrix Converter System
The three-phase to three-phase matrix converter studied in
this paper is shown in Fig. 1, which mainly consists of the
power grid, input filter, and matrix converter array formed by
nine bidirectional switches sij . Through these switches, each
output phase can connect to any one of the input phases. The
power grid is represented by three-phase symmetric voltage
sources. The second-order (LC) input filter is used to smoothen
the input current waveforms supplied to the grid, where the
damping resistors are in parallel with the inductors.
B. Description for the Basic Method
In this section, one modulation method based on mathematical construction, referred to as the basic method, will be introduced. For the sake of convenience, the input filter influence is
not emphasized here.
Assume the three-phase input phase-to-neutral voltages and
input line currents are symmetric and sinusoidal, which can be
(2)
where Uim and ωi are the amplitude and angular frequency of
the input voltages, respectively, Iim is the amplitude of the input
currents, and ϕi denotes the input displacement angle.
In addition, the desired output phase-to-neutral voltages referenced to input neutral are given by
⎤
⎡
⎤
⎤ ⎡
⎡
cos(ωo t − ϕo )
uA
ucom
⎥
⎢
⎥
⎥ ⎢
⎢
uo = ⎣ uB ⎦ = Uom ⎣ cos(ωo t − ϕo − 2π/3) ⎦ + ⎣ ucom ⎦
uC
cos(ωo t − ϕo + 2π/3)
ucom
(3)
where Uom and ωo are the amplitude and angular frequency of
the output voltages, respectively, ϕo is the initial phase angle,
and ucom is the zero sequence output voltage, which does not
contribute to the output currents.
For the symmetric and linear inductive loads, the output line
currents can be written as
⎡
⎤
⎡ ⎤
cos(ωo t − ϕo − ϕL )
iA
⎢
⎥
⎢ ⎥
io = ⎣ iB ⎦ = Iom ⎣ cos(ωo t − ϕo − ϕL − 2π/3) ⎦ (4)
iC
cos(ωo t − ϕo − ϕL + 2π/3)
where Iom is the amplitude of the output currents, and ϕL
is the load displacement angle. Assume the semiconductor switches of the matrix converter are ideal, each switch
sij (i = A, B, C; j = a, b, c) has two possible states: sij = 1
if the switch is closed and sij = 0 if the switch is open. The
states of these switches can be represented as an instantaneous
switching matrix
⎤
⎡
sA a sA b sA c
⎥
⎢
S = ⎣ sB a sB b sB c ⎦
(5)
sC a
sC b
sC c
with the constraint sia + sib + sic = 1, (i = A, B, C).
According to Fig. 1, the relation between the input and output
of matrix converter at any instant can be expressed as
uo = Sui
(6)
T
ii = S io
(7)
where T denotes the transpose. As the switching frequency
is much higher than the input and output frequencies, the
local-averaged value of switching states sij can be defined as
mij (i, j = 1, 2, 3). Therefore, a 3 × 3 modulation matrix M is
defined as
⎤
⎡
m11 m12 m13
⎥
⎢
M = ⎣ m21 m22 m23 ⎦ .
(8)
m31
m32
m33
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 2, FEBRUARY 2014
The equations (6) and (7) are still held in the local-averaged
manner, by replacing S with M . In (8), mij has to meet the
restriction as follows:
0 ≤ mij ≤ 1.
(9)
Consider that the input voltages should never be shortcircuited and the output currents should never be left open
due to the inductive loads, mij also has to meet the following
constraints
⎧
⎪
⎨ m11 + m12 + m13 = 1
⎪
⎩
m21 + m22 + m23 = 1
(10)
m31 + m32 + m33 = 1.
Usually, it is not easy to obtain the modulation matrix M
directly. Therefore, a simple method based on mathematical
construction is used. As is well known that the modulation of
matrix converter can be divided into two processes in conception: the virtual rectifier and virtual inverter ones, which can be
expressed as follows:
udc = R(ωi )T ui
(11)
uo = I(ωo )udc
(12)
where udc is the virtual dc-link voltage, and the rectifier and
inverter modulation vectors are expressed as
⎤
⎡
cos(ωi t − ϕi )
⎥
⎢
R(ωi ) = ⎣ cos(ωi t − ϕi − 2π/3) ⎦
(13)
cos(ωi t − ϕi + 2π/3)
⎤
⎡
cos(ωo t − ϕo )
⎥
⎢
I(ωo ) = m ⎣ cos(ωo t − ϕo − 2π/3) ⎦ .
(14)
cos(ωo t − ϕo + 2π/3)
Therefore, based on (11) and (12), the modulation matrix can
be represented as M = I(ωo )R(ωi )T . In this way, ϕi and m are
the adjustable parameters that can be used to achieve different
control targets, such as desired output voltage and input reactive
power. If substituting (13), (14) into (11), (12), after some simple
calculations, it can lead to
⎤
⎡
cos(ωo t − ϕo )
3
⎥
⎢
uo = mUim cos (ϕi ) ⎣ cos(ωo t − ϕo − 2π/3) ⎦ . (15)
2
cos(ωo t − ϕo + 2π/3)
To synthesize the desired output voltage, Uom =
(3/2) mUim cos (ϕi ) must be fulfilled. If q is the voltage transfer ratio, defined as q = Uom /Uim , we can get
3
m cos (ϕi ) .
(16)
2
However, until now, the modulation matrix M does not meet
the constraints (9) and (10). For this reason, it cannot be utilized
directly. In order to obtain a practical modulation matrix, a
modified modulation matrix M is presented as
⎤
⎡
x1 x2 x3
⎥
⎢
M = M + M0 = I(ωo )R(ωi )T + ⎣ x1 x2 x3 ⎦ . (17)
q=
x1
x2
x3
Note that such a modification only changes ucom , but the
output line-to-line voltages remain the same. To make M meet
(9) and (10), the offset values x1 , x2 , and x3 should be chosen
from
⎧
⎪
⎨ x1 ≥ − min(m11 , m21 , m31 )
⎪
⎩
x2 ≥ − min(m12 , m22 , m32 )
(18)
x3 ≥ − min(m13 , m23 , m33 )
x1 + x2 + x3 = 1.
(19)
By taking appropriate values of x1 , x2 , and x3 , M will be a
feasible modulation matrix for realization. Now, the key problem
is whether there exists such a set of offset values, and how to
choose them. After analyzing the modulation matrix M , it can
be found that at any instant, two of the minimum values in each
column of M always appear in the same row, and the third one
belongs to the other row. Therefore, let x1 , x2 , and x3 take their
minimum values in (18) first. Without loss of generality, assume
at one instant, m11 , m12 , and m23 are the minimum values in
each column of M , which means the values of x1 , x2 , and x3
are
⎧
⎪
⎨ x1 = − min(m11 , m21 , m31 ) = −m11
⎪
⎩
x2 = − min(m12 , m22 , m32 ) = −m12
(20)
x3 = − min(m13 , m23 , m33 ) = −m23 .
Then, we could get
x1 + x2 + x3 =
√
3mρ ≤
√
3m
(21)
2π/3).
Acwhere ρ = cos(ωo t − ϕo + π/6) cos(ωi t − ϕi + √
cording to (21), it can be found that if m ≤ 3 3, then
x1 + x2 + x3 ≤ 1. In that way, if x1 + x2 + x3 < 1, the additional offset 1 − (x1 + x2 + x3 ) needs to be distributed to
x1 , x2 , and x3 in some way to meet the constraint
√ (19). Therefore, the feasible offset values exist if m ≤ 3 3, which cor√ responds to the voltage transfer ratio limit q ≤ qm ax = 3 2
exactly.
C. Input Reactive Power Analysis
Assume the expected output voltage has been determined, as
seen from (16), the parameters m and ϕi are the two degrees
of freedom that can be utilized to maximize the input reactive
power. According to the definition, the input reactive power can
be expressed as Qi = Po tan (ϕi ), where Po is the output active
power, and Po = 1.5Uom Iom cos (ϕL ). Under certain output
active power, the larger is ϕi , the more is Qi . In this case,
m should take its maxima, then, the corresponding maximum
ϕi m ax equals to cos−1 (q/qm ax ). Therefore, it can be concluded
that the larger is q, the smaller is |ϕi m ax |. In specific, if q =
qm ax , ϕi m ax should be zero.
According to the aforementioned analysis, the maximum input reactive power can be written as
Qi m ax = Po tan (ϕi m ax ) = ±Po (qm ax /q)2 − 1. (22)
From (22), the sign of ϕi m ax determines the reactive power
type. If ϕi m ax < 0, Qi m ax is negative and the reactive power
shows capacitive; while ϕi m ax > 0, the inductive reactive
LI et al.: MODULATION STRATEGY BASED ON MATHEMATICAL CONSTRUCTION FOR MATRIX CONVERTER
power (Qi m ax > 0) is produced. Furthermore, the maximum
reactive power that can be supplied to the grid is affected by the
output active power, related to the load properties. If the load
is purely inductive, the input side will not produce any power,
neither active nor reactive power. This basic method requires
strictly reactive power to facilitate active power transmission.
III. NOVEL METHOD FOR MAXIMIZING
INPUT REACTIVE POWER
To increase the input reactive power capability, some modifications are made on the basic modulation method. Analogously,
the modulation matrix is still denoted as M = M + M0 , where
M is formulated as follows:
M = M1 + M 2
(24)
M2 = mq Iq (ωo )Rq (ωi )T
(25)
and
⎡
cos(ωi t)
⎤
⎥
⎢
Rp (ωi ) = ⎣ cos(ωi t − 2π/3) ⎦
⎡
cos(ωi t + 2π/3)
cos(ωo t − ϕo )
cos(ωo t − ϕo + 2π/3)
j
mij = 1.
cos(ωo t − ϕo + φ)
(30)
Obviously, it is difficult to derive the accurate analytical solution of the global optimum. Alternatively, a suboptimal method
is presented in the following section.
First, divide M2 into two parts, expressed as
⎡
cos(ωo t − ϕo + φ )
(31)
⎤
cos(ωo t − ϕo + φ + 2π/3)
⎤
cos(ωi t + 2π/3 ± π/2)
If the first term in (31) is merged with M1 , we will get
M1 + mq 1 Ip (ωo )Rq (ωi )T = (m2p + m2q 1 )Ip (ωo )R (ωi )T
(32)
where
⎤
⎡
cos(ωi t ± ϑ)
R (ωi ) = ⎣ cos(ωi t ± ϑ − 2π/3) ⎦
cos(ωi t ± ϑ + 2π/3)
⎤
cos(ωo t − ϕo + φ + 2π/3)
.
In (23), M1 is used to generate the expected output voltages,
and M2 is dedicated to produce input reactive current. It is
worth noting that the sign of ±π/2 in Rq (ωi ) determines the
input reactive power type (inductive or capacitive). If it is set
to −π/2, the reactive power is inductive; while it is +π/2, the
reactive power shows capacitive. In this modulation, the free
parameters that can be adjusted are mp , mq , and φ, as well
as M0 . To obtain the desired output voltages, mp should be
determined first, which is
2
(26)
mp = q.
3
Then, the input currents of the matrix converter can be expressed as
3
Iom mp cos(ϕL ) [Rp (ωi )]
2
id
3
+ Iom mq cos(ϕL + φ) [Rq (ωi )] .
2
iq
Subject to 0 ≤ mij ≤ 1,
(29)
⎥
⎢
Iq (ωo ) = ⎣ cos(ωo t − ϕo + φ − 2π/3) ⎦ .
⎥
⎢
Iq (ωo ) = ⎣ cos(ωo t − ϕo + φ − 2π/3) ⎦
ii =
Maximize Iq (mq , φ)
where
⎤
cos(ωi t ± π/2)
⎥
⎢
Rq (ωi ) = ⎣ cos(ωi t − 2π/3 ± π/2) ⎦
⎡
3
Iom mq cos(ϕL + φ).
(28)
2
For given output voltage and load condition, we can choose
appropriate mq and φ rigorously to increase the reactive power
control range under the physical restriction of feasible modulation implementation. Thereby, it seems opportune to define
a constrained optimization problem for maximizing the input
reactive current, which can be formulated as follows:
Iq =
M2 = [mq 1 Ip (ωo ) + mq 2 Iq (ωo )]Rq (ωi )T
⎥
⎢
Ip (ωo ) = ⎣ cos(ωo t − ϕo − 2π/3) ⎦
⎡
As seen, mp controls the active components of the input
currents; while mq and φ determine the reactive ones. If mq = 0
or ϕL + φ = π/2, unity input power factor is realized. Thus, the
reactive current amplitude can be written as
(23)
M1 = mp Ip (ωo )Rp (ωi )T
657
(27)
m
and ϑ = arctan( mqp1 ).
Based on the physical process of the indirect modulation
method, at least, the parameters mp , mq 1 , and mq 2 should satisfy the following constraint:
√
3
2
2
.
(33)
mp + m q 1 + m q 2 ≤
3
In this case, rather than (28), the reactive current is rewritten
as
3
Iq = Iom [mq 1 cos(ϕL ) + mq 2 cos(ϕL + φ )].
(34)
2
To maximize the input reactive current, let φ = −ϕL and
mq 2 takes the value as
√
3 2
− (mp + m2q 1 ).
(35)
mq 2 =
3
Then, substitute φ = −ϕL and (35) into (34), it leads to
√
3
3
2
2
Iq = Iom
+ mq 1 cos(ϕL ) − (mp + mq 1 ) . (36)
2
3
658
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 2, FEBRUARY 2014
Since mp is determined by q, the last unknown parameter
is mq 1 . Thus, take the derivative of the reactive current with
respect to mq 1 , yields
⎡
⎤
∂Iq
3
m
q1
⎦.
= Iom ⎣cos(ϕL ) − (37)
∂mq 1
2
2
2
(m + m )
p
q1
Let ∂Iq /∂mq 1 = 0; then
mq 1 =
mp
.
tan(ϕL )
(38)
Considering the constraint (33), after some manipulations,
the optimal mq 1 can be summarized as
⎧
2q
⎪
⎪
⎨ 3 tan(ϕL ) , when q ≤ qm ax sin(ϕL )
mq 1 = ⎪
⎪
⎩ 1 − 4 q 2 , when qm ax sin(ϕL ) < q ≤ qm ax .
3 9
(39)
With the parameters mp , mq 1 , and mq 2 in (26), (35), and (39),
not only the expected output voltage, but also the maximum
input reactive power generated by the matrix converter with the
proposed method will be obtained.
IV. INPUT REACTIVE POWER CAPABILITY
To verify the merit of the proposed method, it is necessary to
compare it with some other methods. For this purpose, the input
reactive power limits, obtained with regard to different q and
ϕL , are computed and investigated for the optimum-amplitude
method, indirect SVM, and the proposed method, respectively.
Without loss of generality, the comparisons are made under
the following two types of load: Type I: current source loads,
whose output currents are independent on the output voltage.
Type II: inductive loads with the impedance of ZL (ωo ) = R +
jωo L = |ZL (ωo )| ∠ϕL .
A. The Optimum-Amplitude Method
The optimum-amplitude method was presented in [12], where
the maximum
√ voltage transfer ratio can reach the intrinsic
qm ax = 3 2. Due to its computational complexity, the detailed derivation is not stated here. For the purpose of reactive
power analysis, the related constraints are reviewed briefly. According to [12], the operation constraints can be represented
as
|p| + a ≤ 1
(40)
√
where a = 2 |θ| q, p = (2q − a) 3, q is the voltage transfer
ratio, and θ = tan (ϕi )/tan (ϕL ). After some manipulations
with (40), it leads to
√
q(|1 − |θ|| + 3 |θ|) ≤ qm ax .
(41)
From (41), if ϕi = 0, unity input power factor is obtained.
However, with the increase of |ϕi |, the voltage output ability
will change accordingly. After some simplifications, (41) can
be stated as follows:
⎧
tan(ϕL ) qm ax
⎪
√
+
1
,
)|
≤
|tan(ϕ
⎪
i
⎨
q
3+1
⎪
qm ax
⎪
L)
⎩ |tan(ϕi )| ≤ tan(ϕ
√
−1 ,
q
3−1
if q ≤ 0.5
if q > 0.5.
(42)
By inspection of (42), if q ≤ 0.5, |ϕi m ax | is more than ϕL ;
while q > 0.5, it becomes −ϕL ≤ ϕim ax ≤ ϕL . Note that if ϕL
is zero, ϕi must also be zero.
For the first type of load, the load currents are represented as
(4). If the power loss caused by the semiconductor switches is
neglected, the maximum reactive power can be formulated as
|Qi m ax | = 1.5Uim Iom sin (ϕL )
(qm ax − qsgn(b))
√
3 − sgn(b)
where b = q − 0.5, and sgn(b) can be expressed as
1,
b≥0
sgn(b) =
−1, b < 0.
(43)
(44)
For the second type of load, the maximum reactive power is
|Qi m ax | =
1.5
(qm ax − qsgn(b))
qU 2 sin (ϕL ) √
.
|ZL | im
3 − sgn(b)
(45)
B. The Indirect SVM Method
As is well known, the SVM method can be classified into
two categories: the indirect SVM and direct SVM. The indirect
SVM scheme characterized by its simple calculations and clear
physical meaning is widely used for the matrix converter. In
essence, the indirect SVM is explicitly equivalent to the basic
method as previously mentioned, from the input reactive power
point of view.
Similarly, for the two types of load, the maximum input reactive power can be expressed as
|Qi m ax | = 1.5Uim Iom cos (ϕL ) (qm ax )2 − q 2
(46)
1.5
|Qi m ax | =
qU 2 cos (ϕL )
|ZL | im
(qm ax )2 − q 2 .
(47)
As seen, in this method, the input reactive power is dependent
on the active power transfer. In particular, the matrix converter
does not generate any input reactive power under purely inductive load condition.
C. The Proposed Method
According to (26), (35), and (39), when matrix converter uses
the proposed method, for the two types of load, the maximum
input reactive power that can be generated will be as follows:
⎧
1.5Uim Iom [qm ax − q sin(ϕL )],
⎪
⎪
⎨
if q ≤ qm ax sin(ϕL )
(48)
|Qi m ax | =
2
2
⎪
⎪
⎩ 1.5Uim Iom cos(ϕL ) (qm ax − q ),
if qm ax sin(ϕL ) < q < qm ax
LI et al.: MODULATION STRATEGY BASED ON MATHEMATICAL CONSTRUCTION FOR MATRIX CONVERTER
|Qi m ax | =
659
⎧
1.5
2
⎪
⎪
qUim
[qm ax − q sin(ϕL )],
⎪
⎪
|Z
|
L
⎪
⎪
⎨
if q ≤ qm ax sin(ϕL )
(49)
1.5
⎪
⎪
2
2
⎪
qUim
cos(ϕL ) (qm
− q 2 ),
⎪
ax
⎪
⎪
⎩ |ZL |
if qm ax sin(ϕL ) < q < qm ax .
D. Comparisons
In this section, per unit value is used to assess the input reactive power capability for these methods. For the first type
of load, the input reactive power base is defined
as Qb1 =
1.5qm ax Uim Iom , and Qb2 = 1.5(qm ax Uim )2 |ZL | is for the
second type of load. From the results in the previous sections, Qi
can be regulated within the range of −Qi m ax ≤ Qi ≤ Qi m ax ,
and Qi m ax varies with ϕL and q.
According to (43)–(49), the maximum reactive power curves
with regard to q for the load type I and II are illustrated under
ϕL =0, π/6, π/3, and π/2, as shown in Figs. 2 and 3, respectively.
From Fig. 2(a), it can be found that if ϕL is near zero, the
optimum-amplitude method almost produce no reactive power
for the load type I; while the indirect SVM and the proposed
methods generate the same reactive power. Fig. 2(b) shows
the maximum reactive power with ϕL = π/6. Compared with
those in Fig. 2(a), the maximum reactive power of the optimumamplitude method has been enlarged with the increase of ϕL ;
it first increases linearly with q until q = 0.5, then decreases
to zero at q = qm ax . Moreover, the proposed method can obtain more input reactive power than those of the other two
methods for all q(q ≤ qm ax ). Fig. 2(c) shows the corresponding curves with ϕL = π/3, and the maximum reactive power
boundary of the optimum-amplitude method intersects with that
of the proposed method. It can be concluded that when ϕL is
greater than π/3, there exists an area, where the maximum reactive power of the optimum-amplitude method is greater than
that of the proposed method. The boundary curves of the input reactive power at ϕL = π/2 are illustrated in Fig. 2(d). As
seen, the indirect SVM method cannot generate the input reactive power any longer. On the contrary, the reactive power
capability of the optimum-amplitude method can reach its peak
(ϕL = π/2, q = 0.5).
Similarly, for the load of type II shown in Fig. 3, all the
curves increase first and then decrease with the increase of q. In
the optimum-amplitude method, the input reactive power limit
still appears at q = 0.5 and it gets higher as ϕL increases from
0 to π/2. In the indirect SVM scheme, the input reactive power
reaches its maxima at
qm ax
(50)
q= √
2
and it is inversely proportional to ϕL . However, the proposed
method offers a variable maxima at
qm ax
(51)
q=
2 sin (ϕL )
under ϕL ≤ π/4, while it becomes (50) under ϕL > π/4, and
displays the same trend related to ϕL as the indirect SVM.
Fig. 2. Maximum input reactive power curves for load type I under different
load displacement angles. (a) ϕ L =0. (b) ϕ L = π/6. (c) ϕ L = π/3. (d) ϕ L =
π/2.
Conclusively, in the optimum-amplitude method, the ability
to generate reactive power is weak at small ϕL , even zero at
ϕL = 0; but with ϕL becoming larger, it gets better gradually.
The maximum reactive power always increases first and reduces
afterward with regard to q, and reaches the best point at q = 0.5.
In the indirect SVM method, when both ϕL and q are small,
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 2, FEBRUARY 2014
Fig. 4. 3-D diagrams of the maximum reactive power for different q and
ϕ L : (a) optimum-amplitude method. (b) Indirect SVM method. (c) Proposed
method.
Fig. 3. Maximum input reactive power curves for load type II under different
load displacement angles. (a) ϕ L =0. (b) ϕ L = π/6. (c) ϕ L = π/3. (d) ϕ L =
π/2.
the maximum reactive power is relatively high, then decreases
with the increase of ϕL , until drops to zero at ϕL = π/2. In the
proposed method, when ϕL is less than π/3, the matrix converter
can always produce the highest input reactive power at any q.
Even though ϕL is at π/2, the ability to generate reactive power
is still the best of the three methods under q ≤ 0.4, and a little
worse than that of the optimum-amplitude method at higher q.
In addition, Fig. 4(a), (b), and (c) show the corresponding
three-dimension diagrams of the optimum-amplitude, indirect
SVM, and proposed methods, respectively, which is obtained for
different q and ϕL . The three-dimension diagrams can reflect
more information about the input reactive power ranges for the
three methods.
LI et al.: MODULATION STRATEGY BASED ON MATHEMATICAL CONSTRUCTION FOR MATRIX CONVERTER
Fig. 5.
Control schematic diagram of the matrix converter system.
TABLE I
PARAMETERS USED IN THE SIMULATIONS
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Fig. 6. Simulation results for capacitive input reactive power. (a) Basic
method. (b) Proposed method.
V. SIMULATION RESULTS
Numerical simulations are carried out on the matrix converter system to verify the correctness of the comparison results
mentioned previously. Fig. 5 shows the control diagram for the
proposed method, which includes the power supply, input filter,
matrix converter, load, modulation, and measurement. The measurement of the input voltages and output currents are necessary
for the proposed method, but the input current measurement is
used to calculate the input reactive power. Both the basic and
proposed methods are utilized to generate the maximum input
reactive power under the same given conditions. In the simulations, the input voltage and the parameters for the input filter
are listed in Table I. The inductive loads are |ZL | = 4.3 Ω and
ϕL = π/3. The output frequency is set to 60 Hz. To reflect
the reality, the dead time for commutation and forward voltage drops due to the IGBTs and diodes are considered in the
simulations.
First, the voltage transfer ratio is arranged as: q is set to 0.2
before t = 0.05 s, and after t = 0.05 s, q equals to 0.3. When
the capacitive reactive power is considered, Fig. 6(a) shows the
input voltage ua , input current ia , and the relevant per-phase
input reactive power Qi for the basic method, and Fig. 6(b)
shows the same waveforms for the proposed method. Moreover,
the corresponding simulation results for the inductive reactive
power are shown in Fig. 7. As seen from Figs. 6 and 7, with the
increase of q, all the input currents and the absolute values of
Qi increase. Under the given condition, by using the proposed
method, the input capacitive or inductive reactive power of the
matrix converter is more than those by the basic method.
To further verify the correctness of the theoretical results, load
type I (Iom = 100 A) and load type II (|ZL | = 10 Ω) are used
for the three methods mentioned above with different q and ϕL .
Note that the input capacitive reactive powers are considered.
Fig. 7. Simulation results for inductive input reactive power: (a) Basic method.
(b) Proposed method.
Fig. 8.
Profile of the voltage transfer ratio.
In this case, the voltage transfer ratio is arranged as that shown
in Fig. 8.
For the load type I, the measured input reactive powers with
ϕL being π/6 are shown in Fig. 9(a). When ϕL is set to be π/3,
the measured results are shown in Fig. 9(b). For the load type
II, with different ϕL as π/6 and π/3. Fig. 9(c) and (d) shows
the corresponding input reactive power results, respectively.
As seen, the simulation results agree well with the theoretical
analysis in the previous sections. For instance, the trends of
the maximum reactive power with the increase of q decrease
gradually, in the proposed method for the load type I. The matrix
converter system with the proposed method can always obtain
more input reactive power than that of the indirect SVM method.
Besides, from Figs. 9(b) and (d), the reactive power curves of the
optimum-amplitude and proposed methods almost coincide with
each other at certain voltage transfer ratio. However, considering
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 2, FEBRUARY 2014
Fig. 10.
Prototype of the converter.
TABLE II
COMPARISON RESULTS OF INPUT REACTIVE POWER
VI. EXPERIMENTAL SETUP
Fig. 9. Input reactive power curves under different load displacement angles:
(a) ϕ L = π/6 for the load type I. (b) ϕ L = π/3 for the load type I. (c) ϕ L =
π/6 for the load type II. (d) ϕ L = π/3 for the load type II.
the large calculation efforts of the optimum-amplitude method,
the proposed method is easier to realize and exhibits superior
performance comprehensively.
To validate the proposed method experimentally, a low-power
three-phase matrix converter prototype has been built in the
laboratory, as shown in Fig. 10. The controller board is mainly
composed of a floating-point DSP (TMS320F28335) and a field
programmable gate array (EP2C8T144C8 N) where the fourstep commutation strategy based on voltage information [19] is
implemented. The parameters of the input voltages, input filter,
and loads are the same as those in the first simulation above. The
output voltage frequency is set to 60 Hz. The sampling frequency
is 5 kHz, and the double-sided switching pattern is used.
For comparisons, both the proposed and basic methods are
tested. When the capacitive reactive power is considered, at
q = 0.2, the experimental waveforms of the input voltage ua ,
input current ia , output voltage uA C , and output current iB with
the basic and proposed methods are shown in Fig. 11(a) and (b),
respectively. And Fig. 11(c) and (d) illustrates the corresponding waveforms with the two methods at q = 0.3, respectively.
As seen in Fig. 11, in every case, the input current is before
the input voltage of a phase angle, which means the input capacitive reactive power is generated. Under the same q, the two
methods have almost the same output currents, but obviously
different input currents, because different input reactive currents
are obtained. In addition, Fig. 12 shows the corresponding results under the same operating conditions as those in Fig. 11,
when the inductive reactive power is generated by the matrix
converter. As seen, the input currents are different between the
basic and proposed methods, and are behind the input voltage
of certain phase angle.
With the experimental results aforementioned, Table II lists
the results of the relevant per-phase reactive power, obtained
by using the power analysis module of Tektronix oscilloscope.
From Table II, it can be found that the input reactive power of
LI et al.: MODULATION STRATEGY BASED ON MATHEMATICAL CONSTRUCTION FOR MATRIX CONVERTER
Fig. 11. Experimental results for capacitive input reactive power: (a) basic
method, q = 0.2. (b)Proposed method, q = 0.2. (c) Basic method, q = 0.3.
(d) Proposed method, q = 0.3.
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Fig. 12. Experimental results for inductive input reactive power. (a) Basic
method, q = 0.2. (b) Proposed method, q = 0.2. (c)Basic method, q = 0.3.
(d) Proposed method, q = 0.3.
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matrix converter in the proposed method is much more than that
in the basic method, either capacitive or inductive. Compared
with the results shown in Figs. 6 and 7, it can be found that the
experimental results are basically in agreement with those in the
simulations.
VII. CONCLUSION
In this paper, a novel modulation strategy based on mathematical construction method for matrix converter has been proposed, which can be used to enhance the control range of input
reactive power with low computational efforts. The comparisons
about the reactive power capability for three methods are made,
which include the optimum-amplitude method, indirect SVM
method, and proposed method. It is verified that the maximum
input reactive power with the proposed method is higher than
that of other schemes mentioned previously in most cases. In
particular, when the voltage transfer ratio is relatively low and
the load displacement angle is relatively large, the superiority
is more obvious. In addition, it is convenient to regulate the
free parameters of the proposed method to achieve the control
purpose efficiently. According to the first optimization problem
as stated in this paper, the maximum input reactive power can
be extended more in theory, which will be the focus of further
investigation.
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Xing Li was born in Hunan, China, in 1988. She received the B.S. degrees from the School of Information Science and Engineering, Central South University, Changsha, China, in 2009, where she is currently
working toward the PhD. degree.
Her research interests include matrix converter
and wind energy conversion system.
Mei Su was born in Hunan, China, in 1967. She
received the B.S., M.S., and Ph.D. degrees from
the School of Information Science and Engineering,
Central South University, Changsha, China, in 1989,
1992, and 2005, respectively. Since 2006, she has
been a Professor with the School of Information Science and Engineering, Central South University.
Her research interests include matrix converter,
adjustable speed drives, and wind energy conversion
system.
Yao Sun (M’13) was born in Hunan, China, in 1981.
He received the B.S., M.S., and Ph.D. degrees from
the School of Information Science and Engineering,
Central South University, Changsha, China, in 2004,
2007, and 2010, respectively. He is currently a Lecturer with the School of Information Science and Engineering, Central South University, China.
His research interests include matrix converter,
micro-grid, and wind energy conversion system.
Hanbing Dan was born in Hubei, China, in 1991. He
received the B.S. degree in automation from Central
South University, Changsha, China, in 2012, where
he is currently working toward the M.S. degree in
electrical engineering.
His research interests include matrix converter,
dc/dc converters, and V2G applications.
Wenjing Xiong was born in Hunan, China, in 1991.
She received the B.S. degree in automation from
Central South University, Changsha, China, in 2012,
where she is now working toward the M.S. degree in
electrical engineering.
Her research interests include power electronics
and power transmission.