th
5 International Advanced Technologies Symposium (IATS’09), May 13-15, 2009, Karabuk, Turkey
THREE-DIMENSIONAL SPACE-VECTOR MODULATION ALGORITHM
FOR ALL TYPES OF MULTILEVEL CONVERTERS USING ABC
COORDINATE
a, *
a
a
N. GHIASNEZHAD , M. Tavakoli BINA , M. A. GOLKAR
a
K. N. Toosi University of Technology, Tehran, Iran
E-mail: nima.ghiasnezhad@ee.kntu.ac.ir, tavakoli@kntu.ac.ir, golkar@kntu.ac.ir
are performed using algebraic operations, while the user of
the algorithms has no need to have an exact perspective
Among different constant frequency modulation techniques, on geometric three-dimensional analysis. For a given time,
space vector modulation, because of many advantages, is four normalized switching status are passed into a voltmore applicable for three phase converters, especially for- second equation in order to balance with that of the
wire types. The most important part of SVM method is reference vector. Every instantaneous phase voltage is
determination of proper switching states in each sampling located between two adjacent DC levels, providing eight
period. This step should also be quick to avoid undesirable possible switching choices for the three-phase voltages. To
delay. Conventional methods often use αβο transformation build the reference voltages, a comparative solution is
in both two-dimension (for three- wire systems without suggested in order to choose four out of eight possible
zero-sequence)
and three-dimension (for four-wire switching statuses. Then the duty ratios are calculated for
systems involve zero-sequence) which increases its the four chosen switching states [3]. To verify the
complexity with increment of converter’s level. In this paper suggested selection of switching states, a three-arm fulla simple and quick technique which uses abc coordination, bridge inverter (including the neutral-wire) (Fig.1) is
is applied. This method can be used for all converters with simulated with MATLAB. An unbalanced condition is
every type (cascade, diode clamp,), and with every level, considered in which the neutral wire carries zero sequence
current, and the three-dimensional SVM with the
without any change in the center part of modulation.
suggested switching selection is applied to the inverter
Keywords: Three-dimensional SVM, switching states, switches. Simulations confirm that the inverter output
vector track efficiently the reference voltage vector using
multilevel converters, duty ratios.
the suggested algorithm.
Abstract
1. Introduction
The space vector modulation (SVM) technique uses a fixed
frequency in synthesizing the reference vector, enabling
many power applications to lower their switching losses.
This clearly improves the efficiency, which is vital for high
power applications such as DC/AC converters that are
interconnected to power systems through transformers.
While the SVM can be practically considered as a voltagebased technique for the AC/DC converter output, one
should obtain the reference voltage vector based on the
required objectives of a certain application. Unlike the
voltage-based SVM techniques, the current-based
modulation techniques (e.g. hysteresis modulation) impose
high losses despite their easy approach towards providing
required reference current vector. Hence, the problem can
be narrowed down to two problems for the SVM technique;
first, provision of a three-phase reference voltage vector for
the AC/DC converter output, and, second, making up the
provided reference vector using the different switching
status produced by a three-dimensional SVM technique.
This latter can also be detailed in various sections.
Conventionally, the instantaneous phase quantities abc
are transferred to the so called αβο space to achieve a
better distinction of zero sequence component for four-wire
systems, in particular the active power filters. [1],
[2] .Nevertheless, this increases the needed computations
as well as the complexity of seeking the location of
instantaneous reference vector. This paper introduces a
complementary three dimensional SVM algorithm for an nlevel converter within the abc space to build up the
available reference voltage vector using the neighboring
switching status of the AC/DC converter. The propositions
© IATS’09, Karabük University, Karabük, Turkey
b
a
Sa1
Sb1
Sa2
Sb2
vd
vd
Sa3
Sa4
c
Sc1
Sc 2
Sc3
Sc4
vd
Sb3
Sb4
n
Figure 1. Four-wire Three-level cascade voltage source
inverter
2. The Proposed Method
In brief, consider the instantaneous three-phase voltage
quantities in the abc space in which the proposed
method is described. Also, a three-dimensional SVM
assumes that three instantaneous phase voltages are
independent because zero sequence components could
be generally available. Conventionally, the reference
voltages are made up using the four closest neighboring
switching states (V0, V1, V2, V3), followed by a set of voltsecond balance equations that eventually gives the
Ghiasnezhad N., Bina M. T. and Golkar M. A.
d0
d1
d2
Va (t) = U an (t) / Vdc
d3
(2)
Vb (t) = U bn (t) / Vdc
Ts
x2
Sx
x1
y2
Sy
y1
z2
Sz
z1
V0
V1
V2
V3
Figure 2. Four-wire Three-level cascade voltage source
inverter
r
⋅T s = V 1 ⋅ t 1 + V 2 ⋅ t 2 + V 3 ⋅ t 3 + V
T s = t 0 + t1 + t 2 + t 3
2) Consider each instantaneous normalized reference
voltage separately; the closest switching states are
related to the two DC levels right above and below the
reference voltage. Thus, a vector is formed for
magnitude of each phase using the upper (U) and
lower (L) switching states of an n-level converter,
totally three vectors as below:
L 
L 
L 
a =  a  b= b  c=  c 
U a 
U b 
U c 
Where U a , La , U b , Lb , U c , Lc
needed duty ratios for the switches as below:
V


Vc (t) = U cn (t) / Vdc
0
⋅t0
(1)
Where Ts is the switching period, Vr is the reference
vector, and t0, t1, t2 and t3 are on-time durations related to
the four switching states, respectively.
The proposed method follows this conventional algorithm,
but avoids seeking the exact location of the reference
voltage vector within the three-dimensional space. This not
only saves the modulating times of the switches, also
lowers significantly the complexity of the conventional
methods. The suggested algorithm can be applied without
any changes to all voltage-sourced converter (VSC)
topologies. Also, there is no need to look up tables, can be
used practically on-line.
The simplest way is chosen for the arrangement of
switching pulses in order to reach a compromise between
the switching losses and the harmonic performance of the
VSC. This is done in a way that only one phase performs
one switching changeover when transiting from one vector
to another state.
 Lk < Vk < U k

 U k − Lk = 1
(3)
∈ Ζ , and

 k = a, b, c

(4)
3) It is now possible to introduce eight different switching
states for each sampling period based on the six switching
states expressed by the three vectors in (3). These are [La,
t
t
t
t
t
Lb, Lc] , [Ua, Lb, Lc] , [La, Ub, Lc] , [La, Lb, Uc] , [Ua, Ub, Lc] ,
t
t
t
[Ua, Lb, Uc] , [La, Ub, Uc] and [Ua, Ub, Uc] .These eight
vectors introduce neighbors of the normalized reference
vector. Two methods are presented to choose four out of
eight possible vectors and their related duty ratios. Two
vectors V0 = [La, Lb, Lc]t (the beginning of the switching
t
period) and V3 = [Ua, Ub, Uc] (end of the switching period)
are similarly chosen for both methods. Note that one can
change the order of switching vectors based on the need
for optimization of number of switching or the symmetry of
the waveforms. Selection of the two middle vectors V1 and
V2 determines which phase should reach earlier from the
bottom to the top.
4) First Method: The first method is based on measuring
the distance of the each phase voltage from the lower-level
state (∆a, ∆b and ∆c)as below:
2.1. Suggested Algorithm
The following steps describe the whole algorithm in finding
the four neighboring switching states around the reference
vector as well as the related duty ratios for the four vectors:
1) Assume each capacitor has a DC voltage Vdc,, and then
three reference voltages (Uan(t), Ubn(t) and Ucn(t)) are
normalized by
Vdc
like (2), giving a new reference vector
Vr = [Va(t), Vb (t), Vc (t)]t . It is usual to choose Vdc based on
the maximum AC voltage needed at the n-level converter
output. Theoretically, the converter output AC voltages
vary within [-n Vdc, n Vdc], and the normalized reference
within [-n, n].
Figure 3. The simulated circuit including a cascaded threelevel inverter that supplies a four-wire load.
Ghiasnezhad N., Bina M. T. and Golkar M. A.
Figure 4. The reference three-phase voltages that are to be generated by the cascaded converter.
Figure 5. (a) the reference voltage of phase a at the converter output, and (b) the reference voltage on resistance of phase
a.
algorithms can be found in literatures. The four duty ratios
d0, d 1, d2 and d 3 (see (1) by dividing both sides by Ts) can
be calculated as follows (see Fig. 2):
∆ a = V a − La
∆b = V b − Lb
(5)
∆c = V c − Lc
The longest distance (let say ∆k) among the three
distances shows that the kth phase moves faster towards
the top level. Hence, V1 is obtained by changing Lk to Uk.
Then, the second longest distance is recognized. The
second selected phase is treated similarly by changing the
related element to get V2. Further, duty ratios can be
evaluated using (5). Three distances are sorted for three
phases as ∆x, ∆y and ∆z like (x, y and z are the sorted
phase indexes)
∆x > ∆ y > ∆z
(6)
Then, three switching states (Sx, Sy and Sz) and their
related duty ratios (dx, dy and dz) are shown in Fig. 2 based
on the sorted phase distances in (6). Note that Fig. 2 only
demonstrates a typical switching status and their
corresponding duty ratios. Different switching sequence
d x = V x − x 1 , d y = V y − y 1 , d z = V z − z 1

d 0 = 1 − d x , d 1 = d x − d y , d 2 = d y − d z , d 3 = d z
(7)
4) Second Method: The second method concentrates on
the distance of the reference vector from the six states
other than the switching states of (3) in order to choose the
middle vectors. Thus, two groups of vectors VB and VS are
defined as below:
VS = {[ La ; Lb ;U c ] , [ La ;U b ; Lc ] , [U a ; Lb ; Lc ]}

VB = {[ La ;U b ;U c ] , [U a ; Lb ;U c ] , [U a ;U b ; Lc ]}
(8)
The distances of the six vectors in (8) are calculated with
respect to the normalized reference vector Vr. Then, the
lowest distance in group Vs is regarded as V1, and the
Ghiasnezhad N., Bina M. T. and Golkar M. A.
Figure 6. Unbalanced reference voltages (different magnitudes) that are to be generated by the cascaded converter.
lowest distance in group VB as V2. Finally, the obtained
vectors can be put into (1) to calculate duty ratios.
3. Simulation Results
To verify the suggested algorithms, a cascaded topology is
simulated with MATLAB in which the neutral wire carries
the zero sequence current. Figure 3 shows the simulated
circuit in which the DC-link voltage of the inverter is 70V,
and the load includes a 50Ω along with a 1mH smoothing
inductor. Also, the topology of the cascaded inverter is
shown by Fig. 1 that includes the fourth-wire as the
neutral-point.
Two cases are simulated in order to show the capability of
the suggested simple methods. The first case concentrates
on a low frequency distorted waveform, and the second
one focuses on an unbalanced condition. These cases are
studied by the following subsections.
3.1. Synthesizing Distorted Inverter Output
Figure 4 introduces the three-phase reference voltages to
be synthesized by the cascaded converter at the AC
output using a fixed 5 kHz switching frequency. Three
distorted waveforms include a 40V (50 Hz) together with
120% third harmonic (150 Hz). Figure 5(a) also depicts the
reference voltage for phase-a.
Applying the two suggested methods in modulating the
reference voltages of Fig. 4 will result in similar outcomes.
In other words, the same vectors will be chosen in the two
suggested methods. Thus, here only one of the pictures is
demonstrated in Fig. 5(b), which shows the modulated
voltage seen on the 50Ω resistor of phase-a. Comparing
the simulations of Fig. 5(b) with the reference of Fig. 5(a)
clearly indicates that both waveforms are quite the same.
In fact, both simple methods are following the reference
voltage suitably.
However, one can argue on some parts of Fig. 5(b), which
somehow differs from those of Fig. 5(a). These parts are
produced exactly when the third harmonic waveform
reaches its maximum. The bigger the smoothing
inductance, the lower would be the small difference in the
compared pictures. These spike-like differences are also
available in other literatures that proposing threedimensional SVM techniques (e.g. [4]) that implementing
more sophisticated algorithms compared to the suggested
simple proposals.
3.1. Synthesizing Unbalanced Inverter Output
Different applications of cascaded multilevel converters
can be considered for reactive power compensation, active
filtering, and etc. All applications are potentially subjected
to the three-phase unbalance operation of power systems.
Thus, here a three-phase unbalance situation is generated
for the inverter AC output to evaluate the performance of
the two suggested methods.
Assume the cascaded three-level inverter (Fig. 1) is used
to develop the three-phase reference voltages of Fig. 6.
These waveforms are unbalanced in terms of their
magnitudes, providing three sinusoidal 50 Hz voltages with
three peaks of 65 V, 50 V and 45 V respectively for phasea, phase-b and phase-c. Once again, the two methods are
applied to the three-dimensional SVM in order to produce
the waveforms of Fig. 6. Since both proposals give similar
converter output, only one set of results are shown and
described. Note that both resistor and inductor parameters
remain the same as those of Fig. 3.
Figure 7(a) shows the 65 V reference voltage and Fig. 7(b)
illustrates the same waveform which is simulated based on
the proposed methods(both in phase-a). It can be seen
from Fig. 7(b) that it is significantly the same as the 65 V
reference voltage. However, some sort of distortion is
available on the modulated waveform in Fig, 7(b),
especially when it passes through zero. This is also an
expected concern because of unbalance conditions. In
fact, a second harmonic can be modulated through the
converter when the three-phase system is interacting with
the unbalance-operating inverter.
Moreover, Figs. 8(a) and 9(a) introduce the reference
voltages related to the 50 V and 45 V references,
respectively. Simulations corresponding to these
references are shown by Figs. 8(b) and 9(b). Comparing
Ghiasnezhad N., Bina M. T. and Golkar M. A.
Figure 7. phase-a: The voltage drop on a 50 Ω resistor for the
unbalance inverter output (reference: 65 V peak), (a) the
reference voltage, and (b) the modulated voltage using the
proposed method.
Figure 9. phase-c: The voltage drop on a 50 Ω resistor for the
unbalance inverter output (reference: 45 V peak), (a) the
reference voltage, and (b) the modulated voltage using the
proposed method.
transformation is omitted, it is also expected to achieve
faster modulating response in comparison with other
methods.
4. Conclusion
Figure 8. phase-b: The voltage drop on a 50 Ω resistor for the
unbalance inverter output (reference: 50 V peak), (a) the
reference voltage, and (b) the modulated voltage using the
proposed method.
the modulated outcomes with the related references
illustrates that the suggested algorithms are wellperforming under unbalanced situation. Again, distortion of
the modulated waveforms can be observed, which are
bolder than the first waveform (65 V peak). Note that the
level of distortion depends on the unbalance condition,
including the available negative and zero sequence
components.
In brief, the suggested methods can simply achieve the
results that other methods gain through more complicated
algorithms. Further, the two proposals can be done in a
simple algebraic environment away from three-dimensional
mathematical analysis concerned with positive, negative
and zero sequence components. Hence, because αβo-
This paper presents two suggestions in choosing the
required switching vectors as well as deriving the
related duty ratios for a three-dimensional SVM. The
methods are simple to understand in an uncomplicated
algebraic formulation that avoids geometrical threedimensional complexity. The come up methods can be
applied to all voltage source converters, including
multilevel topologies, as a central control core of SVM
under a fixed switching frequency modulation. All
implementing steps are taken place on-line, where no
look-up table is needed. The methods are operating
straightforward on phase quantities, no transformation
is needed either. Thus, unlike conventional methods
require less implementing steps, resulting in faster
modulating process. To confirm the raised algorithms,
the methods are simulated and applied to a threephase cascaded inverter. Simulations are obtained
under two different conditions, which confirm the
capability of the suggested methods in comparison with
conventional algorithms.
References
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Lee, “Three-dimensional space vector modulation
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May 2002.
[2] D. Shen and P.W. Lehn," Fixed-frequency spacevector-modulation control for three-phase four-leg
active power filters"IEE Proc-Elec. Power
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[3] Manuel A. Perales, M. M. Prats, Ramón Portillo,
José L. Mora, José I. León, and Leopoldo G.
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Ghiasnezhad N., Bina M. T. and Golkar M. A.
Modulation in abc Coordinates for Four-Leg
Voltage Source Converters" IEEE Power
Electronics Letters,Vol.1,No.4, December 2003.
[4] Leopoldo Garcia Franquelo, Ma. Ángeles Martín
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