Time-Domain Derived Frequency-Domain Voltage
Source Converter Model for Harmonic Analysis
P. A. Gray, Student Member, IEEE, and P. W. Lehn, Senior Member, IEEE
Setup network with VSCs modelled as
multi-frequency current sources­
amplitudes initialized to zero
Abstract-It is well recognized that for voltage source con­
verters (VSCs) there exists strong coupling between AC-side
and DC-side harmonics; this harmonic coupling can be rep­
resented through a Frequency Coupling Matrix (FCM). This
paper develops a frequency-domain steady-state harmonic model
for loads/generators interfaced to the grid through VSCs. The
purpose of developing this model is to allow for improved
harmonic analysis of distribution grids and to derive an intuitive
FCM in which the relationships between the various state­
variables are readily available to the user. A space-vector time­
domain derived approach is used to derive the FCM. The FCM
is validated through PSCAD simulations. The derived FCM
converges faster than previous time-domain derived models and
results from interfacing the model with the OpenDSS harmonic
analysis software are shown.
In dex Terms-VSC, harmonics, converter, FCM, harmonic
coupling, non-linear loads, modelling.
M
mp X
I.
Imp
INTRODUCTION
RID Harmonics tend to increase system losses, can
excite resonant frequencies, and cause equipment mal­
function and ageing; therefore, as the proportion of non-linear
loads and generation on the grid continues to increase, the
ability to predict the harmonics generated by these elements
becomes increasingly important. Loads and generation inter­
faced to the grid through VSCs make up an important subset
of the non-linear elements typically seen on a distribution grid.
VSCs mostly appear in the front end of three-phase drives, as
well as solar photovoltaic and wind generation.
A harmonic model of a VSC can be derived in either
in the time or frequency domain. The frequency domain is
the method used in this paper since the time-requirement of
time-domain studies is infeasible for any reasonably sized
distribution system. There are many methods that have been
developed for modelling harmonic sources in the frequency­
domain, some of the most notable include transfer function
and frequency coupling matrices [1]-[16]. The latter is the
approach taken in this paper.
This work derives the VSC model similarly to [17] and [18]
and approaches the calculation of the FCM similar to [16],
but uses a space-vector derived approach along with all real
valued components. This has the effect of greatly reducing the
number of states being solved for, leading to a much faster
convergence. The approach to solving the FCM proposed
in this paper, requires the positive and negative sequence
components to be decoupled, which leads to a more intuitive
>-____ NO
for all VSCs
< tolerance
G
Fig. l.
Model Integration into OpenDSS
FCM matrix. The model incorporates dc-link voltage and
reactive power control to ensure the VSC operating point and
its corresponding harmonic spectrum are accurately identified.
An algorithm is also developed to incorporate the model into
an available harmonic analysis software.
In this paper, the algorithm used to interface the derived
model to a harmonic analysis software is first discussed. The
derivation of the model is then explored - this includes the
iterative outer control loop and the derivation of the FCM.
Finally, the accuracy of the model is assessed and the results
of interfacing the model with the OpenDSS harmonic analysis
software are shown.
The authors are with the Department of Electrical and Computer Engi­
neering, University of Toronto, Toronto, ON, M5S 3G4 Canada (emails:
p.gray@mail.utoronto.ca, lehn@ecf.utoronto.ca).
978-1-4673-1943-0/12/$31.00 ©2012 IEEE
100%
512
II.
INTERFACING MODEL WITH
OPENDSS
A goal of this paper is to develop an accurate and compu­
tationally efficient frequency-domain model of VSCs for har­
monic analysis studies. The model is interfaced with OpenDSS
to perform these harmonic analysis studies. The algorithm is
outlined in Fig. 1.
From Fig. 1, it is shown that the bus voltages of the system
are first initialized by solving the zero load power flow (power
2
Fig. 4.
Fig. 2.
Calculation of Switching Times
Assumed V SC Topology
Vdc, Qac,
v'aP, I d,
I
,-
the modulation index (used in the PWM switching scheme)
and the phase angle offset between the positive sequence line­
frequency component of the VSC terminal voltages
and
the PCC voltages
The converter has 8 possible configurations depending on
the states of the switches. For one period of steady-state
operation, if the times of all switching transitions are known,
the converter can be fully specified through conventional time­
domain circuit analysis. This is the approach used in the FCM
derivation.
Input:
I
L
-
Initialize
o,ma
-
1,
----'---
Calculate
Switching Times
J
l
�i<-------'
,
�I
L�
I­
--,
-- I
Vtabe
Vsabe.
Calculate
FCM
update 0 and nta
Calculate
��alc , (fa�lc
IV. C ALCULATING
Ll.Vd" Ll.Qa,
It is assumed that the switches employ an exact sinusoidal
PWM switching scheme. The switching scheme is synchro­
nized with the positive-sequence line-frequency component of
the PCC voltages - this is conceptually illustrated in Fig. 4. In
this visualization, for each half-period of the carrier waveform,
three switching times are calculated - one for each of the
phases.
No
< tolerance
Yes
Output:
Fig. 3.
lsafJ,�
------"
Solution Algorithm
V. FCM
flow with all loads and generators disconnected). The loads
and generators are then re-connected - with the VSCs current
injections initialized to 0 - and the system voltages for
frequency multiples 1 through to h of the fundamental line
frequency are solved for, The injected current harmonics for
all VSCs are then updated, using the solved system voltages
as input The system voltages are re-solved, This iterative
procedure continues until the injected current harmonics be­
tween successive iterations remains within a pre-determined
tolerance value for each vse The model is developed in
MATLAB, and a COM interface is used to interface the model
to OpenDSS,
III.
MODEL DERIVATION
The derivation assumes the VSC topology of Fig, 2 for all
VSCs, The algorithm used to derive the VSC model is outlined
in Fig. 3.
The model assumes the following as inputs: PCC bus volt­
ages
dc-link reference voltage
reactive power
and the dc-side injection currents
The PCC bus voltages
are directly obtained from the previous OpenDSS solution and
the reactive power
is the reactive power drawn by the
VSC at the pce The
and 15 referred to in Fig. 3 are
Vsabe,
V2e'
Ide.
Qae,
SWITCHING TIMES
Qae,
ma
513
DERIVATION
The FCM is used to map the input vector contmmng
the PCC voltage and the dc-side current harmonics, into an
output vector containing the ac-side current and dc-link voltage
harmonics; this relationship is shown in (1)
�= FCM'1!:
(1)
where,
1!:=
Vs--a h
Vsf] h
I-sah
rsf]h
V+sa h
Vs+h
f]
Re{Ide}
Im{I�J
Is+ha
I+sf]h
Re{VdOJ
Im{VdOe}
, and �=
Re{I,tm}
Im{I,tm}
Re{Vd�m}
Im{Vd�m}
The vector 1!: contains the PCC positive and negative se­
quence harmonic voltages. For example, the negatively rotat­
ing space vector at frequency multiple h is expressed in (2),
V-saf]h = ( v-sa h J-Vs-f] h) e -h)wt
+
j(
(2)
k [-h,+h]
1 j' (Zsa
. + ].Zs(3
. ) e-jkwtdt
Isak + ]'Is(3k (7)
-T
t- T
An expression for : (I:a + j1:(3 ) can be solved - indirectly
- by looking at (8), as is done in [16]
dz(t
)
(8)
----;It = m z(t) + hq(t)
where, z(t) and q(t) are time-varying complex variables, and
m and h are constants
The solution of (8) for an arbitrary time t is
t
z(t)=emtz(t-T)+ J em(t-T)hqdt (9)
t- T
where, Vs�� represents the negative sequence voltage at har­
monic h.
Vector Jl also contains the dc-side injection current harmon­
ics; these are represented in the conventional phasor domain.
vector current harmonic at frequency k, where
Itm = I Itm l LeI+=; therefore, Re{Itm} =
I Itm l coseI+= and Im{Itm} = I Itm l sineI+=.
For example,
__
de
In order tcfCierive the FCM, the input Jl and 'the output J!.. are
represented as state variables of an extended dynamic system.
It is convenient to combine the states into a single vector this yields the autonomous system shown in (3),
(3)
where,
isa
is(3
Ji. = [Vde]
, and
M(t) is a linear time-variant matrix.
Since, we are solving for steady-state quantities, matrix
E [0, ] , where
is the
period.
Due to the properties of the state-variables,
has the
following structure
M(t) must be specified for all t
T
T
Comparing (9) and (7), the following substitutions can be
made to equate the two equations
= -j k
= �TejwkT
q = isa + jis(3
m
M(t)
[A(t)
M(t) = �
Using the assumed VSC topology, the differential equations
for the variables of vector x can be derived in (5a) - (5c).
(lOa)
w
n
(4)
Setting
(11)
z(t -
T) = 0,
(lOb)
(lOc)
and substituting (10) into (9), results in
d (Ik + 'Ik ) -'k (Ik + 'Ik)+ 1 jkwT(t'sa+ .ts(3
.
] )
dt sa ] s(3 -] sa ] s(3 T e
W
Converting (11) into vector format,
0
T0 [
-kW] [I:ka] + � [1 ]
Is(3
1
o
where, Te is the Clarke transform, and
[]
o
c=
0
'Vr
E
(6)
+mw
o
�_((TT))] =
y(T) i=l
2:= !:::.ti = T
i=l
n
[Ji.(0)]
II exp (M(i)!:::.ti) Jl(O)
Jl
J!..(O)
The Fourier series decomposition is used to derive the dif­
ferential equations for the variables of vector J!... The following
analysis derives the differential equations for the ac-side space514
(14)
n
(which is assigned to variable <1»
0
(13)
The periodicity property of steady-state solutions can be
exploited in conjunction with (3) to give an expression of the
state-variables in terms of the initial conditions
<1>
n
n
(M(i)!:::.ti)
i
=l
has the following structure
Due to the structure of matrix M(t),
-mw
0
'Vk
[O,+m].
where,
o
(12)
The sub-matrix entries D and E can be solved by extrapo­
lating the results from (12) and (13) for
E [-h,+h] and
0
+hw
0
_
1 or 0 when the corresponding switch is on or off, respectively.
The sub-matrix entries A(t) and B can be extrapolated from
(5a) - (5c).
Since, the variables of vector Jl are sinusoidally time­
varying inputs, the sub-matrix C can be simply derived as
shown in (6) for the ac-side voltage harmonics ranging from
to +h and dc-side current harmonics ranging from
to
-hw
]
r
d [Re{VIJ] - [ -rw] [Re{VIJ] + 1 [1] Vde
T
Im{Vle}
dt Im{Vle} rw
Sa, Sb, and Se are binary variables - each having t e value of
+m:
isa
zs(3
(11)
The derivation of the dc-link voltage time-derivatives is sim­
ilar. The results for dc-link voltage harmonic r are shown in
(13), where E [0,+m]
Sa
Sabe(t) = Sb
Se
-h
E
t
[
= �T �; � ]
DT FT ET
exp
(15)
4
If the periodicity constraints are invoked (namely, J:( O ) =
J:(T)), the combination of (14) and (15) can be rearranged to
yield the following expression for y( T ) in terms of -y(O) and
TABLE I
INPUT PARAMETERS - WHERE
�(O)
1
U(T) = (DT [I - ATr BT + FT) �(O) + ETU(O)
Input
R
L
(16)
where, I is a 3 by 3 Identity matrix
Rdc
Cdc
Q
Vdoc
�c
Alternatively, if U(O) is initialized to 0, (16) equates to the
expression in (1) - giving the expression for the FCM in (17)
FCM=
1
(DT [Ir - ATr BT + FT)
(17)
I
V+1
sa
+3
Vsa
+3
Vs(3
V-5
sa
V-5
sa
+7
Vsa
+7
Vs(3
v.l-l
base
31>
Pbase
This FCM when multiplied with the input vector �(O) yields
the fundamental and harmonic currents injected by the VSC
as well as the dc-link voltages.
The Newton-Raphson iterative technique is used to solve for
the unique combination of the PWM phase angle offset 0 and
modulation index
that yields a dc-link average voltage and
reactive power that matches the
values specified
and
by the user. The Jacobian expression used is shown in (18)
ma
where,
J
E
[- OVdoc
(18)
to
Q��lc,
Vi Ii
s(3 a
_
sa (3
(20)
Qac
P"c
P",c
I�c
I�c
. .
.
.
2) Only the power contnbuted by the posItIve-sequence lIne
frequency component is considered
3) The voltage at the terminals of the VSC is approximated
with the following expression
�
�
+1
ta
v:
�
�
+
1
jV+
t(3 =
ma VdOc
__ cosO
2
+
�c (I
ma Vdc .lll (5:))
----wLs
I
ma VdOc sino
J. -_
2
(21)
(22)
Expanding (22), subbing in (21), and extracting the real and
and
imaginary components yields expressions relating
Vdoc
Qac
515
ma Vdc
U
(23)
�2 I Vs�11 2 (wLI V+sa(31I
- R (wL)
ma Vd
---c wLcos(0))
_
+
ma Vdc .
--Rs lll (0)
2
(24)
2
Taking the derivatives of (23) and (24) yields the Jacobian
entries in (19).
+
Once the tolerance requirement:
< E,
has been met, the ac-side space-vector currents and dc-link
voltages in vector U(T) are solved. The ac-side space-vector
currents are transformed back into the abc reference frame, and
the current spectrum of the corresponding VSC in OpenDSS
is updated.
J� Vlc
VII.
Using these assumptions, the complex power drawn at the PCC
is
I
)
Qaccalc = 1m (scalc
31>
+
Vi Ii
To solve for the Jacobian entries in (19), a number of approxi­
mations are made since an analytical expression relating 0 and
to
is not available:
and
ma Vic
1) VOdc
100 kW
0.001
1
"2
R V+
sa(3 - -2-Rcos(0)
R2 + (wL)2
2
To obtain
the imaginary component of the complex
power drawn by the VSC at the PCC is calculated
i=-h
480 V
ma and 0 shown in (23) and (24)
vltalc = Re (S�¢lc ) / �c
=
(19)
oQac
'"'
�
.
2 2
-0.4
1.0
0. 0 2
0. 0 2
0. 0 2
0. 0 2
0.01
0. 0 2
I
Using the FCM in conjunction with the input vector y(O),
.
· d.
can Immed·lateIy be obtame
Qaccalc =
m = 18
Value
(p.u.)
0. 0 2
0. 2
108
0.45
0.3
31Vs�11
00
+h
AND
Qac
Vdoc
00
Vodd calc
m f = 15, h = 18,
�Q�c
RESULTS AND DISCUSSION
The derived model is implemented in MATLAB. A simula­
tion of the model is performed using the parameters listed in
Table I as input.
An equivalent model is implemented in the time-domain
simulation software PSCAD to evaluate the accuracy of the
derived model. The most significant outputs of the model
TABLE II
SIMULATION RESULTS
Output
ma
0
11a{3-17
1-5a{3
r1a{3
1a{3+1
1a{3+3
1a{3+7
1a{3+11
1a{3+13
Vd+
e2
TABLE III
COMPARING DERIVED MODEL AND [16] - TIME REQUIRED TO SOLVE
FCM
Model
(p.u.)
0.8666
-6.944°
PSCAD
(p.u.)
0.8668
-6.952°
Error
(%)
0.02
0.12
0.08140
0.08144
0.05
0.02728
0.02729
0.04
0.3939
0.3996
1.43
0.6279
0.6277
0.03
0.1207
0.1208
0.08
0.01647
0.01646
0.06
0.01043
0.01058
1.42
0.1082
0.1082
0.00
0.3648
0.3691
1.16
h
7
18
27
45
27
:
' '
40
FCM for Test Case
are listed in Table II, for both the MATLAB and PSCAD
simulations. The PSCAD simulation uses a time-step of 0.2
/LS.
The error column in Table II contains the percent discrep­
ancy between the two simulation results. As can be observed,
the data is well correlated between the two simulation methods
with a maximum error of less than 1.5%.
The FCM derived as a by-product of generating the results
in Table II is shown in Fig. 5. The FCM can be seen as having
the four distinct regions outlined in (25).
FCM
=
81a{3
8Va{3
8Vde
8Va{3
(25)
The cross-coupling between the harmonic orders is directly
observable from the FCM. The majority of large magnitude
terms in the FCM are along the main diagonal going into
1
' ,4_ ------'- -6-7-1 _�.;2
i
652
50
r
Derived Model
0.35s
2.1s
6.5s
29.2s
2.6s
Model in [16]
2.2s
31.4s
79.6s
300s
25.0s
646
Fig. 6.
Fig. 5.
m
7
18
27
45
0
6-.Z5
___
680
IEEE 13-Bus Benchmark System
the page. Harmonic cross-coupling is significant for the low­
order harmonics, as is evidenced by the off-diagonal peaks
surrounding the fundamental frequency largest-diagonal peak.
The FCM for the derived model converges faster than that
derived in [16]. This results from a reduction in the order
of states, as enabled by the space vector formulation. The
proposed formulation employs only 4h + 2m + 4 states in its
derivation as compared to 12h + 2m + 6 states as employed
in [16]. When computing the matrix exponentials, the reduced
number of states yields an order of magnitude reduction in
FCM computation time. A comparison of the time required to
solve for the FCM, for the two models, is shown in Table III. In
both cases MATLAB scripting language was employed. Since
the number of harmonic states being solved greatly impacts
computation time, h and m were varied in the comparison.
For ac system studies it is worth noting that computation
of dc link harmonics may be excluded, unlike in [15] for
example, as these harmonic equations are only employed for
output purposes; their exclusion has no impact the on the
underlying steady-state solution. The last row of data in Table
III shows the FCM computation time when 27 ac harmonics
are computed, while the amplitude of the DC harmonics are
not.
The practicality of using the derived model for harmonic
analysis studies is evaluated by interfacing the model with the
OpenDSS harmonic power flow software. The network used
to conduct the harmonic analysis study is the IEEE 13-bus
benchmark system, shown in Fig. 6. One VSC is added to the
network at bus 634. The input parameters for the VSC are
specified in Table IV.
The injection currents are solved using the algorithm out­
lined in Fig. 3. The most significant injected current harmonics
are shown in Fig. 7.
The IEEE 13-bus test system is by definition an unbalanced
516
6
TABLE IV
INPUT PARAMETERS - WHERE
Input
R
L
Rdc
Cdc
Q
Vdoc
I�c
V;l-l
base
31>
Pbase
E
m f = 15, h = 18,
AND
injections. Accuracy in the order of 1.5% was achieved when
compared to detailed time domain simulations.
VSC space vector harmonics are conveniently computed and
visualized for both negative and positive frequencies, which
correspond to negative and positive sequence quantities. This
allows for rapid identification of uncharacteristic harmonics
without the need for any post-processing of data. Results from
the 13-bus study system show the existence of many significant
uncharacteristic harmonic current injections from the VSC that
would not be predicted by commercially available harmonic
analysis software. These harmonics include low frequency in­
jections such as the positive sequence third and fifth harmonic.
m = 18
Value (p.u.)
0.02
0.2
108
0.45
-0.1
2.2
-0.4
480 V
100 kW
0.001
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0.6
0.5
:::-!
0.4
0.3
0.2
0.1
Harmonic Multiple
Fig. 7.
OpenDSS Harmonic Analysis Study with Derived Model
system. As such, harmonics are present at the bus connected
to the VSC, bus 634. From Fig. 7, it is shown that significant
amplitude uncharacteristic harmonics are injected into the sys­
tem by the VSe. The models currently being used to represent
VSCs in commercially available harmonic analysis software,
would not be able to pick up these low-order harmonics. This
could lead to measurable inaccuracies in harmonic analysis
studies; particularly if the system is weak, contains a lot of
harmonics, or a high proportion of the power is provided by
VSCs.
VIII.
CONCLUSION
This paper presents a time-domain derived model of a VSC
for harmonic analysis studies. The formulation employs a
space vector representation of ac variables, allowing acceler­
ated computation of the frequency coupling matrix of the VSe.
Computation is accelerated by over an order of magnitude
compared to previous time-domain derived models.
An iterative algorithm for interfacing the model to the
OpenDSS software is provided and a modified IEEE 13-bus
benchmark system is examined as a test case. The algorithm
is shown to enforce a pre-specified de voltage (thus de power)
constraint, as well as a pre-specified reactive power flow
constraint at the PCe. Enforcing these constraints ensures
accuracy of the operating point and the resulting harmonic
Philippe A. Gray (S'11) received the B.A.Sc. (Hons.) degree in engineering
science at the University of Toronto, ON, Canada in 2010. He is currently
pursuing his M.A.Sc. degree in electrical engineering at the University of
Toronto.
Peter W. Lehn (SM'05) received the B.Sc. and M.Sc. degrees in
electrical engineering from the University of Manitoba, Winnipeg, Canada,
in 1990 and 1992, respectively, and the P h.D. degree from the University of
Toronto, ON, Canada, in 1999. From 1992 to 1994, he was with the Network
P lanning Group of Siemens AG, Erlangen, Germany. Currently, he is a full
P rofessor at the University of Toronto.
517