Grid Converters for Photovoltaic and Wind Power Systems

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Appendix A
Space Vector Transformations
of Three-Phase Systems
A.1 Introduction
A generic three-phase electrical system consists of a set of three voltages and three currents
interacting with each other to deliver electrical power. A practical three-phase system cannot
be considered as the simple addition of three independent single-phase subsystems. Actually,
particular relations exist between the phase variables of a three-phase system, such as those
resulting from the Kirchhoff laws or regarding phase sequences, which invite the application
of certain space vector transformations to obtain a more elegant and meaningful representation
of its variables. Generally, the control system of a power converter connected to a three-phase
system is based on these transformed variables. This appendix reviews the most commonly
used space vector transformations and highlights their applications in control grid-connected
power converters.
A.2 Symmetrical Components in the Frequency Domain
In 1918, a young engineer from the Westinghouse Electric and Manufacturing Company in
Pittsburgh, C. L. Fortescue, proposed a method for analysing unbalanced polyphase networks,
which offered a new point of view in the analysis of three-phase systems and soon became
known as the method of symmetrical components [1]. In simple mathematical terms, this
method consists of a system of Lagrangian reference frames following the different sequences
existing in a generic polyphase system, which results in a coordinate system especially suited
to analyse all types of polyphase problems. The symmetrical components method allows
decomposition of the steady-state phasors of an unbalanced three-phase system into a set of
sequence components, namely the positive-, the negative- and the zero-sequence components.
This new approach not only offered an elegant and systematic method for analysing polyphase
systems under unbalanced sinusoidal conditions, but also allowed a rigorous explanation, with
mathematical and physical meaning, of the phenomena that take place under such operating
conditions.
Grid Converters for Photovoltaic and Wind Power Systems Remus Teodorescu, Marco Liserre and Pedro Rodríguez
© 2011 John Wiley & Sons, Ltd. ISBN: 978-0-470-05751-3

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356 Grid Converters for Photovoltaic and Wind Power Systems
0 10 20 30 40
-150
-100
-50
0
50
100
150
t [ms]
v
abc
b
v
a
v
c
v
b
V
a
V
c
V
(b)
(a)
148.7 3.6º
67.5 61.5º
112.8 165.5º
a
b
c
V
V
V
= ∠
= ∠ −
= ∠
Figure A.1 Unbalanced three-phase system: (a) instantaneous voltage waveforms and (b) phase voltage
phasors
The steady-state voltage waveforms of a three-phase unbalanced system together with their
phasor representation on a Gauss plane are shown in Figure A.1.
Applying the symmetrical components method, these three unbalanced phasors representing
the three-phase voltages can be transformed into a new set of three phasors representing the se-
quence components of one of the phases of the three-phase system. For example, the positive-,
negative- and zero-sequence phasors of phase a (
V +
a ,
V −
a and
V 0
a ) can be calculated by the
following transformation matrix:
V+−0(a) = [T+−0] Vabc (A.1)
with
Vabc =
⎡
⎢
⎣
Va
Vb
Vc
⎤
⎥
⎦ =
⎡
⎢
⎣
Va∠θa
Vb∠θb
Vc∠θc
⎤
⎥
⎦ , V+−0(a) =
⎡
⎢
⎣
V +
a
V −
a
V 0
a
⎤
⎥
⎦ =
⎡
⎢
⎣
V +
a ∠θ+
a
V −
a ∠θ−
a
V 0
a ∠θ0
a
⎤
⎥
⎦
[T+−0] = 1
3
⎡
⎢
⎣
1 α α2
1 α2
α
1 1 1
⎤
⎥
⎦ (A.2)
where α = ej2π/3
= 1∠120◦
is known as the Fortescue operator. The phasors representing the
sequence components for phases b and c are given by
V +
b = α2
V +
a ;
V −
b = α
V −
a
V +
c = α
V +
a ;
V −
c = α2
V −
a
(A.3)

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Appendix A: Space Vector Transformations of Three-Phase Systems 357
The inverse transformation to pass from the phasors representing symmetrical components of
phase a to the phasors representing the unbalanced phase voltages is given by
Vabc = [T+−0]−1
V+−0(a) (A.4)
with
[T+−0]−1
=
⎡
⎣
1 1 1
α2
α 1
α α2
1
⎤
⎦ (A.5)
As an example, the phasors representing the sequence components of the unbalanced voltages
of Figure A.1, together with their corresponding instantaneous waveforms, are shown in
Figure A.2.
A.3 Symmetrical Components in the Time Domain
Lyon extended the work of Fortescue and applied the method of the symmetrical components
in the time domain [2]. When the Fortescue transformation matrix of (A.2) is applied to the
following set of three-phase unbalanced sinusoidal waveforms:
vabc =
⎡
⎣
va
vb
vc
⎤
⎦ = v+
abc + v−
abc + v0
abc
= V +
⎡
⎢
⎣
cos(ωt)
cos(ωt − 2π
3
)
cos(ωt + 2π
3
)
⎤
⎥
⎦ + V −
⎡
⎢
⎣
cos(ωt)
cos(ωt + 2π
3
)
cos(ωt − 2π
3
)
⎤
⎥
⎦ + V 0
⎡
⎢
⎣
cos(ωt)
cos(ωt)
cos(ωt)
⎤
⎥
⎦ (A.6)
the resultant instantaneous variables are given by
v+−0 = [T+−0] vabc (A.7)
v+−0 =
⎡
⎢
⎣
v+
v−
v0
⎤
⎥
⎦ =
⎡
⎢
⎣
1
2
V +
ejωt
+ 1
2
V −
e− jωt
1
2
V +
e− jωt
+ 1
2
V −
ejωt
V 0
cos(ωt)
⎤
⎥
⎦ (A.8)
It is worth mentioning that the Lyon transformation is usually defined by the following
normalized matrix:
T
+−0
=
√
3 [T+−0] =
1
√
3
⎡
⎣
1 α α2
1 α2
α
1 1 1
⎤
⎦ (A.9)
where
T
+−0
−1
=
T
+−0
T
.

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358 Grid Converters for Photovoltaic and Wind Power Systems
(a)
(b)
(c)
0 10 20 30 40
-150
-100
-50
0
50
100
150
t [ms]
v
abc a
v+
b
v+
c
v+
a
V +
c
V +
b
V + 100 30º
100 90º
100 210º
a
b
c
V
V
V
+
+
+
= ∠
= ∠ −
= ∠ −
0 10 20 30 40
-150
-100
-50
0
50
100
150
t [ms]
v
abc
a
v−
c
v−
b
v− b
V −
a
V −
c
V −
50 40º
50 80º
50 160º
a
b
c
V
V
V
−
−
−
= ∠ −
= ∠
= ∠ −
0 10 20 30 40
-150
-100
-50
0
50
100
150
t [ms]
v
abc
0 0 0
a b c
v v v
= =
0 0 0
, ,
a b c
V V V
0
0
0
25 20º
25 20º
25 20º
a
b
c
V
V
V
= ∠ −
= ∠ −
= ∠ −
Figure A.2 Sequence components of the unbalanced three-phase system of Figure A.1: (a) positive-
sequence phasors, (b) negative-sequence phasors and (c) zero-sequence phasors

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Appendix A: Space Vector Transformations of Three-Phase Systems 359
in
v
3
1
2
in
v
α
( )
2
2
( ) in
in
LPF s
s
ω
ω
=
+
LPF
Figure A.3 Simple implementation of the a operator in the time domain
From (A.8) it can concluded that, independently of the scale factor used in the Lyon
transformation, the resulting vector consists of two complex elements,
v+
and
v−
, plus a
real element v0
. The complex elements
v+
and
v−
can be represented by instantaneous space
vectors, having the same amplitude and rotating in opposite directions. Therefore,
v+
and
v−
should not be mistaken for the positive- and negative-sequence voltage vectors v+
abc and
v−
abc. The real element v0
is directly related to the zero-sequence component of the original
three-phase voltage vector.
To calculate the positive- and negative-sequence voltage vectors, v+
abc and v−
abc, from the
unbalanced input vector vabc, it is necessary to translate the Fortescue operator, α, from
the frequency domain to the time domain. This translation can be performed by using a
simple time-shifting operator, provided the frequency of the sinusoidal input is a well-known
magnitude. In such a case, a 2
/3T time-shifted sinusoidal signal, with T the signal period, can
be understood as a 120◦
leaded version of the original sinusoidal signal. This operator in the
time domain is named a in this appendix. Since α = −1 2 + j
√
3 2, the a operator can be
implemented by using a proper filter to generate the 90◦
phase-shifting associated to the j
operator [3]. As an example, Figure A.3 shows a simple implementation of the a operator
based on a second-order low-pass filter (LPF) tuned at the input frequency with a damping
factor ξ = 1. The a2
operator can be implemented by multiplying the output signal of the LPF
by −1.
Therefore, the following expressions can be used to calculate the instantaneous positive-
and negative-sequence components of vabc:
v+
abc = [T+] vabc;
⎡
⎢
⎣
v+
a
v+
b
v+
c
⎤
⎥
⎦ =
1
3
⎡
⎢
⎣
1 a a2
a2
1 a
a a2
1
⎤
⎥
⎦
⎡
⎢
⎣
va
vb
vc
⎤
⎥
⎦ (A.10)
v−
abc = [T−] vabc;
⎡
⎢
⎣
v−
a
v−
b
v−
c
⎤
⎥
⎦ =
1
3
⎡
⎢
⎣
1 a2
a
a 1 a2
a2
a 1
⎤
⎥
⎦
⎡
⎢
⎣
va
vb
vc
⎤
⎥
⎦ (A.11)
A.4 Components αβ0 on the Stationary Reference Frame
Since the complex elements
v+
and
v−
in (A.8) are not independent from each other, only
three independent real components can be found among the elements resulting from the
transformation of (A.7). Therefore, a possible set of independent elements can be defined

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360 Grid Converters for Photovoltaic and Wind Power Systems
from (A.8) as
v+
,
v+
, v0
, although other combinations are also possible. At this
point, the following real transformation matrix can be written:
⎡
⎢
⎣
(
v+
)
(
v+
)
v0
⎤
⎥
⎦ =
1
3
⎡
⎢
⎣
1 (α) (α2
)
0 (α) (α2
)
1 1 1
⎤
⎥
⎦
⎡
⎢
⎣
va
vb
vc
⎤
⎥
⎦ (A.12)
A similar reasoning caused Clarke to reformulate the Lyon transformation and to propose the
following transformation matrix [4]:
vαβ0 =
Tαβ0
vabc (A.13)
⎡
⎢
⎣
vα
vβ
v0
⎤
⎥
⎦ =
2
3
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 −
1
2
−
1
2
0
√
3
2
−
√
3
2
1
√
2
1
√
2
1
√
2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎣
va
vb
vc
⎤
⎥
⎦ (A.14)
where
Tαβ0
−1
=
Tαβ0
T
.
It is worth remarking here that the input and output vectors have the same norm when the
normalized transformation of (A.14) is applied, i.e.
v2
α + v2
β + v2
0 = v2
a + v2
b + v2
c (A.15)
As a consequence, when the normalized transformation of (A.14) is applied to the voltages
and currents of a three-phase system, power calculations will give rise to equivalent results for
both the abc and the αβ0 reference frames, i.e.
p = vαβ0 · iαβ0 = vabc · iabc (A.16)
The transformation
Tαβ0
can be rescaled as shown in the following equation when the
amplitude of the sinusoidal signals on the abc and the αβ0 reference frames should be equal,
i.e. when V̂α = V̂a:
⎡
⎢
⎣
vα
vβ
v0
⎤
⎥
⎦ =
2
3
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 −
1
2
−
1
2
0
√
3
2
−
√
3
2
1
√
2
1
√
2
1
√
2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎣
va
vb
vc
⎤
⎥
⎦ (A.17)
The αβ0 reference frame is graphically depicted in Figure A.4. In this figure, the αβ plane holds
all the symmetrical vectors, i.e. those vαβ vectors with no zero sequence (va + vb + vc = 0),
whereas the 0 axis is aligned with the space diagonal of the abc reference frame.

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Appendix A: Space Vector Transformations of Three-Phase Systems 361
α
β
0
0
v
α
v
β
v
plane
α β
−
0 abc
αβ
= =
v v v
αβ
v
a
b
c
Figure A.4 Graphical representation of the αβ0 reference frame
A.5 Components dq0 on the Synchronous Reference Frame
Any voltage vector rotating on the αβ plane can be expressed on a synchronous reference frame
by using the Park transformation [5]. As depicted in Figure A.5, the synchronous reference
frame, also known as the dq reference frame, is based on two orthogonal dq axes, rotating at
frequency ω, which are placed at the θ = ω t angular position on the αβ plane. Thanks to its
rotating character, this transformation has been extensively used in the analysis of electrical
machines.
The transformation matrix to translate a voltage vector from the αβ0 stationary reference
frame to the dq0 synchronous reference frame is given by
vdq0 =
Tdq0
vαβ0 (A.18)
d
v
q
v
α
β
q
d
ω
α
v
β
v
θ
plane
α β
−
0
0
v
0 0
dq abc
αβ
= = =
v v v v
αβ
v
Figure A.5 Graphical representation of the dq0 reference frame

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362 Grid Converters for Photovoltaic and Wind Power Systems
⎡
⎣
vd
vq
v0
⎤
⎦ =
⎡
⎣
cos(θ) sin(θ) 0
− sin(θ) cos(θ) 0
0 0 1
⎤
⎦
⎡
⎣
vα
vβ
v0
⎤
⎦ (A.19)
where
Tdq0
−1
=
Tdq0
T
. Therefore, the transformation matrix to translate a voltage vector
from the abc stationary reference frame to the dq0 synchronous reference frame is given by
vdq0 = [Tθ ] vabc (A.20)
⎡
⎣
vd
vq
v0
⎤
⎦ =
2
3
⎡
⎢
⎢
⎢
⎢
⎣
cos(θ) cos(θ − 2π
3
) cos(θ + 2π
3
)
− sin(θ) − sin(θ − 2π
3
) − sin(θ + 2π
3
)
1
√
2
1
√
2
1
√
2
⎤
⎥
⎥
⎥
⎥
⎦
⎡
⎣
va
vb
vc
⎤
⎦ (A.21)
where [Tθ ]−1
= [Tθ ]T
.
The normalized transformations shown in (A.14) and (A.21) allow the norm of the voltage
vector to be conserved in all the reference frames; thus
v2
d + v2
q + v2
0 = v2
α + v2
β + v2
0 = v2
a + v2
b + v2
c (A.22)
Expressing voltage and currents using space vectors allows instantaneous phenomena in three-
phase systems to be studied using an efficient and elegant formulation. This formulation is
particularly useful to control active and reactive power components in three-phase systems.
References
[1] Fortescue, C. L. ‘Method of Symmetrical Coordinates Applied to the Solution of Polyphase Networks’. Transac-
tions of the AIEE, Part II, 37, 1918, 1027–1140.
[2] Lyon, W. V., Application of the Method of Symmetrical Components, New York: McGraw-Hill, 1937.
[3] Iravani, M. R. and Karimi-Ghartemani, M., ‘Online Estimation of Steady State and Instantaneous Symmetrical
Components’. Proceedings of the IEE on Generation, Transmission and Distribution, 150(5), September 2003,
616–622.
[4] Clarke, E., Circuit Analysis of AC Power Systems, Vol. 1, New York: John Wiley Sons, Inc., 1950.
[5] Park, R. H., ‘Two Reaction Theory of Synchronous Machines. Generalized Method of Analysis – Part I’. In
Proceedings of the Winter Convention of the AIEE, 1929, pp. 716–730.