1
Analytical modelling of a square-wave controlled
cascaded multilevel STATCOM
R. Sternberger, Student Member, IEEE, and D. Jovcic, Senior Member, IEEE
Abstract—The aim of this paper is to present an analytical, statespace model of an indirect, voltage-controlled cascaded-type
multilevel Static Synchronous Compensator (STATCOM) with
‘square wave control’. The multilevel converter model is
segmented into a dynamic and static part in order to accurately
represent all internal feedback connections. Each voltage
component is analyzed in detail and described mathematically by
an averaged expression with an equivalent capacitance. The
STATCOM model is linearised and linked with a DQ frame ac
system model and the controller model, and implemented in
MATLAB. The controller gains are selected by analyzing the root
locus of the analytical model, to give optimum responses. The
validity and accuracy of the proposed model is verified against
non-linear digital simulation PSCAD/EMTDC in the time and
frequency domain. The model is very accurate in the subsynchronous range, and it is adequate for most control design
applications and practical stability issues below 100Hz.
Furthermore, the developed model can be used for multilevel
cascaded converters which are exchanging real power.
Index Terms—Modelling, multilevel converter, STATCOM,
static VAR compensators, state-space methods.
I. INTRODUCTION
In recent years, the multilevel converters have become
increasingly popular in high power transmission/distribution
systems and in industry applications [1]-[5]. In contrast to a
conventional two-level voltage source converter, that works
with Pulse Width Modulation (PWM), such multilevel
converters use a number of (low voltage) series-connected
capacitors to generate a high ac voltage. This allows higher
power handling capability with reduced switching power
losses and harmonic distortion [3].
The main types of multilevel converters are: diode-clamped,
flying-capacitor and cascaded inverter [3]-[5]. By comparing
these different topologies, whilst considering harmonic level,
losses and component costs, the cascaded multilevel converter
with ‘square wave control’ is found to be the optimum
solution for STATCOM applications [3],[5]-[8]. The modular
structure of this converter, with a number of identical Hbridges, makes this converter very flexible in terms of power
handling capability. The use of ‘square wave control’ results
in a single switch-on and -off per cycle for each switch, which
brings benefits of low switching losses. In addition, since the
control angle at each capacitor can be used to cancel one
harmonic, a low harmonic distortion can be achieved [6],[9].
The dynamics of such complex converters are mostly
analyzed using EMTP type programs, like PSCAD/EMTDC
The authors are with the Engineering department, University of Aberdeen,
Aberdeen, AB24 3UE, UK: r.sternberger@abdn.ac.uk, d.jovcic@abdn.ac.uk
[10]. However, these simulators provide only trial and error
type studies in time domain. The demanding analysis and
design tasks become very time-consuming since a laboured
search has to be executed to find the best solution.
Alternatively, a suitable and accurate analytical system
model would provide faster design routes. Such a model
would provide the ability to use dynamic systems analysis
techniques (e.g. eigenvalue and frequency domain methods)
and modern control design theories, resulting in a shorter
design time and advanced controller configurations [8],[11].
Comparing to conventional voltage source converters, the
analytical modelling of multilevel converters is more
challenging. A conventional PWM-controlled voltage source
converter has fixed structure and the switching frequency is
high which implies that intervals between switchings are short
and conventional averaging approaches can represent
converter dynamics [12]. A multilevel converter with squarewave control, on the other hand, fundamentally changes its
structure as the capacitors are switched in and out of the
current path and therefore these changes may have significant
influence on the system dynamics. Further, each cell is
switched only twice per cycle and therefore variables may
undergo notable excursions within a single conduction
interval. One of the primary challenges in the dynamic
modelling of a square-wave converter is therefore the
adequate dynamic representation of cell variables between the
switching instants.
The multilevel converter model in [8] is in convenient statespace form but it adopts overly simplified equivalent
capacitance which is only valid for a particular number of
levels and does not consider firing angles on individual cells.
Similarly, the state space model in [9] is of little practical use,
since the equivalent capacitance has to be tuned for every
change in converter parameters, using identification methods.
The researchers in [13] developed a frequency domain model
of a square-wave controlled converter. Since it only considers
a single cell with uncontrollable, full 180deg conduction
periods, this model does not address the aforementioned
modelling challenge. The research presented in [14] represents
the structure of a multilevel converter at a wide range of
frequencies in order to provide a model for phasor studies and
harmonic analysis. However, this is essentially a static model,
and it ignores completely the control influence. Moreover,
since the model neglects the transfer of active power, it cannot
be used for dynamic modelling.
This paper aims to develop a dynamic analytical model of a
multilevel cascaded STATCOM converter that is convenient
for system stability studies and for analysis of interactions
with ac systems. We attempt to establish generic modelling
principles that are applicable for a range of multilevel
2
II. MULTILEVEL TEST STATCOM CONFIGURATION
The selected test STATCOM, which contains a cascaded
(chain-circuit) multilevel converter and its connection to the
ac system, is shown in Figure 1. The multilevel STATCOM
system includes a 3-phase transformer and three strings of
single-cells that are connected in series. A single-cell is here a
full-bridge, single-phase inverter with a capacitor as dc source
[15]. Note that only chain circuit A with its IGBTs, diodes and
capacitors is shown, but similar groups of single-cells are also
connected in branches B and C.
The STATCOM parameters that determine the main circuit
are: number of cells per phase X, dc capacitances CDC1 to
CDCX, buffer inductance LDC, transformer leakage inductance
LTR and transformer ratio nTR. The relationship between the
number of single-cells per string X, and the number of levels N
for a cascaded multilevel converter is given by X=(N-1)/2.
In further studies it is assumed that all capacitances are equal
CDC=CDC1=…=CDCX and that the steady-state voltages across
these capacitors are also identical VDC0=VDC01=…=VDC0X (with
the help of a suitable controller). To ensure low converter
losses and dc voltage ripple, the STATCOM system
parameters are optimized by following the algorithm proposed
in [16] and their values can be found for the selected test
systems in Table I in the Appendix.
At the Point of Common Coupling (PCC), the STATCOM is
connected to ac circuit represented by resistance RAC and
inductance LAC. The ratio between XAC (ω*LAC) and RAC is
assumed to be constant (XAC/RAC=10).
The converter switching losses are reduced by the use of
‘square wave control’ where each device has only one turn-on
and turn-off per cycle. The harmonic distortion is reduced by
calculating the firing angle at each cell to cancel one
harmonic. In view of the fact that triplen harmonics are
cancelled in the converter delta connection, the firing angles
Y/∆
RAC
VAC 0
230kV
50Hz
abc
Phase
Locked
Loop
magnitude
VAC +
Vs ϕm
o
CDC1
VGrid
CDCX
LAC .
VDC0X
Transformer
STR, LTR, nTR
PI
Controller
+
-
VDC01
PCC
LDC
3-phase ∆ -connected cascaded
converter with X cells
IS
ϕm
Firing pulses
Gate Pattern Logic
αa=constant
VRefAC
Controller
Fig. 1: Multilevel STATCOM test system with indirect control.
Mm=1
for the individual cells are calculated for cancellation of the
lowest odd, non-triple harmonics [9], and their values are
shown for selected test systems in Table II in Appendix. Note
that these calculated firing angles can cancel only steady-state
harmonics but as loading changes, harmonics increase [14].
The ideal stepped output voltage of a 13-level converter is
shown in Figure 2.
The STATCOM uses indirect control [15], as shown in
Figure 1. This control has as benefit a constant modulation
ratio Mm=1, resulting in constant control angles αι and
therefore in low harmonics even at low converter voltages.
The dc voltage is affected by controlling the angle ϕm (i.e. the
active power transfer) [15].
VS , VAC [pu]
cascaded converters, even those which exchange real power.
A modular modelling approach is adopted to represent the
complexity of the system with the benefit that each individual
subsystem can be analysed independently. This analytical
model employs all parameters and variables with physical
meaning and therefore, it can study variations in ac and dc
system structures.
6
ϕm
4 α
2
2 α1
α
0
−π/2
α4
α
α
vAC(t)
6
vS(t)
5
3
−π/4
0
π/4
π/2
ω*t [rad]
Fig. 2: Ideal stepped output voltage of 13-level converter (ω=2*π*f)
III. CASCADE MULTILEVEL CONVERTER MODEL
A. General Voltage and Current Representations
The phasor diagram of the STATCOM system is shown in
Figure 3. It is assumed that the system is balanced and all
variables are represented with fundamental frequency phasor.
Note that all ac voltages are shown as line-neutral magnitudes.
With reference to Figure 3, the instantaneous ac voltage
vAC(t) and the ac current iS(t) can be presented as:
v AC (t ) = VAC ⋅ cos(ωt )
(1)
iS (t ) = I S ⋅ cos(ωt + ϕ I ) .
(2)
The equation (2) can also be represented using dqcomponents in the rotating dq frame, where d and q indices
appropriately label axes:
iS (t ) = I Sd ⋅ cos(ωt ) − I Sq ⋅ sin(ωt )
(3)
where: I Sd = I S ⋅ cos(ϕ I )
(4)
I Sq = I S ⋅ sin(ϕ I )
(5)
q
IS
ISq
ϕm
ϕI
ISd
VAC
VS
d
VAC - AC voltage at PCC ϕI - Current phase angle
VS - STATCOM voltage ϕm - Control output (STATCOM
voltage phase angle)
IS - STATCOM current
Fig. 3: STATCOM phasor diagram where the general reference frame is
linked with VAC.
B. Instantaneous Non-linear Dynamic Model of a Single-cell
In the first part, we investigate a single-cell of a cascaded
multilevel converter, as shown in Figure 4. The single-cell
S
v DC
(t )
S
− v DC
(t )
;− π2 + α − ϕ m ≤ ωt ≤
; π2 + α − ϕ m ≤ ωt ≤
π
2
3π
2
− α − ϕm
− α − ϕm
(6)
; elsewhere
0
It is also seen that the converter is supplied by a rectified ac
current (positive: S1,S4 ON; negative: S2,S3 ON) and that the
current iSDC(t), which flows through the capacitor, has the
following form:


S
i DC (t ) = 


i S (t )
;− π2 + α − ϕ m ≤ ωt ≤
− i S (t )
; 2 + α − ϕ m ≤ ωt ≤
π
0
π
2
3π
2
− α − ϕm
− α − ϕm
(7)
; elsewhere
The single-cell variables in Figure 5 are shown under steadystate conditions for ideal converter. Under dynamic transients
and assuming some internal losses, the instantaneous variables
will have different waveforms. Figure 6 shows the single-cell
variables assuming non-zero control angle ϕm, which results in
an active power transfer. The transient curves in Figure 6 have
different traces comparing with Figure 5 since:
• The current will have some non-zero average value, ISDCa.
• The dc voltage vSDC(t) is not constant, but it is increasing
along a sine curve in the conducting interval (from initial
value VDC0).
The capacitor dc voltage vSDC(t) can be represented as:
V DC 0
;0 ≤ ωt ≤ − π2 + α − ϕ m

(8)

S
S
v DC
(t ) = V DC 0 + v DCdyn
(t )
;− π2 + α − ϕ m ≤ ωt ≤ π2 − α − ϕ m

S
; π2 − α − ϕ m ≤ ωt ≤ π2
V DC 0 + V DCdyn (π2 − α − ϕ m )
where vSDCdyn(t) stands for the dynamic voltage component
which depends on the current and the control angle, and has
therefore a direct influence on the converter dc and ac voltage:
S1
iS(t)
vSS(t)
S3
D1 CDC
vSDC(t) D3
S2
iSDC(t) S4
D2
Fig. 4: Single-cell of a cascaded converter
D4
VS
[pu] IS
[pu]
DC
DC
VS
[pu]
S
1
v
AC
vS
(t)
S
(t)
0
iS(t)
−1
S1,S4
S2,S4
1
0
−1
1
S2,S3
S1,S3
S2,S4
iS
(t)
DC
V
0
1
V
0
vS (t)
DC0
DC
(t)
vS
S
DC0
ϕm
−1 α
−π/2 −π/4
α ϕ
m
0
π/4
π/2 3∗π/4
π
ω *t [rad]
Fig. 5: Currents and voltages of a STATCOM system in steady-state
5∗π/4
 1
S
(t )dt ;− π2 + α − ϕ m ≤ ωt ≤ π2 − α − ϕ m (9)
⋅ i DC

S
v DCdyn
(t ) =  C DC ∫

0
; elsewhere

VDC0 is the average steady-state value of dc voltage of each
single-cell. It is shown in [16] that VDC0 is constant and can be
found using static (power flow) equations:


1
V DC 0 = VS −
⋅ IS 


ω
C
⋅
eq1


with Ceq1 =
4 X

 ⋅ ∑ [cos(α i )] ⋅ M m 
π
i =1


π
X
X
i =1
i =1
X ⋅ π − 2∑ [α i ] − ∑ [sin( 2α i )]
⋅ C DC
(10)
(11)
where Mm defines the converter modulation ratio (Mm=1 if
STATCOM is using indirect control).
We observe that VDC0 expression does not involve any
dynamics, but it will have a direct static influence on the
magnitude of the converter ac voltage. This component
depends on the current magnitude IS, the firing angles α and
capacitance CDC, but not on the control angle ϕm.
S1,S4
S1,S3
IS
[pu]
DC


v SS (t ) = 


VDC [pu]
consists of one capacitor and four switching devices (each
contains one IGBT and an antiparallel diode).
The instantaneous ac voltage of such a single-cell vSS(t) is
shown in Figure 5 together with the ac voltage at the PCC
vAC(t), the capacitor voltage VDC(t), and the converter ac
current iS(t). The variables iS(t) and vSS(t) (all variables in top
graph of Figure 5) are at system-level, for a multilevel
converter. These variables are assumed to be of ideal sine
shape because of averaging action of multiple levels and
filtering action of passive components.
Since a single-cell is switched into the electrical circuit
within the interval -π/2+α-ϕm≤ωt≤π/2-α-ϕm, (S1,S4 ON or
S2,S3 ON), the converter ac voltage vSS(t) in time-domain can
be represented as:
VAC, I S [pu]
3
1
0
α -ϕm
−1
S2,S4
iS
(t)
DCa
α +ϕm
iS
(t)
DC
vS
(t)
DCa
vS
(t)
DC
1
vS
(t)
DCdyna
VDC0
0
−π/2
−π/4
0
π/4
π/2
ω *t [rad]
Fig. 6: Dynamic dc current and dc voltage of a single-cell
C. Dynamic DC Voltage Component VDCdyn
The dynamic component of the STATCOM ac voltage in (9)
plays an important role in the dynamics of the overall system.
Since the non-linear current waveform in (7), which is driving
the dynamic component VSDCdyn, cannot be represented directly
4
in a continuous model, an average dc current iSDCa(t) will be
derived firstly. This average current, also illustrated in Figure
6, will result in average value of dynamic dc voltage
component VSDCdyna that shows the same cumulative dc voltage
change over one half-cycle as the instantaneous dynamic
component VSDCdyn, i.e. vSDCdyna(π)=vSDCdyn(π). Since system
dynamics are assumed to be slower than structural changes
within one-half cycle, there will be no loss in accuracy if all
variables are represented as constant/linear curves, which give
accurate values at the end of each half-cycle. The average dc
current ISDCa of the non-linear current waveform iSDC(t) over
one half-cycle can be found averaging (7):
VSdS =
VSqS =
S
I DCa
=
1
π
⋅
−
∫i
π
S
DC
(t ) ⋅ d (ωt ) =
2
1
π
⋅
π 2 −α −ϕ m
∫i
Placing (3) in (12), we can derive expression for the
averaged current. At this stage, we assume that ISd and ISq in
(3) are constant over the integration period in (12). This can be
justified since the current dynamics are slower than the
voltage dynamics and can be neglected over the short segment
of half fundamental cycle. The dc current equation becomes:
[
]
(13)
The voltage expression in (9) can not be directly averaged
since the essential dynamic properties of the converter would
be lost. Firstly, the equation (9) is differentiated, which gives a
non-linear but algebraic expression on the right side. The
resultant expression on the right side of (9) can be averaged as
in (12). Following these steps we obtain dynamic equation for
the average dynamic component of STATCOM voltage in sdomain (LaPlace transformation):
S
S
s ⋅ V DCdyna
= 1 C DC ⋅ I DCa
(14)
where s denotes coefficient of Laplace transformation. The
average dynamic voltage VDCdyna in (14) uses linear
approximation to represent a sine-shaped dynamic voltage
component, as seen in Figure 6. Replacing (13) in (14), the
formula for dynamic dc voltage component is obtained as:
S
s ⋅ V DCdyna
=
2
π ⋅ C DC
[
⋅ cos(α ) ⋅ I Sd ⋅ cos(ϕ m ) + I Sq ⋅ sin(ϕ m )
S
DC
−π 2 + − m
π 2 −α −ϕ m
2
∫v
S
DC
−π 2 +α −ϕ m
π
(t ) ⋅ cos(ωt ) d (ωt )
(16)
(t ) ⋅ cos(ωt + π 2)d (ωt )
(17)
S
VSdS = VSdynad
+ VSS0 d
=V
S
Sq
S
Sdynaq
+V
(18)
(19)
S
S 0q
(12)
(t ) ⋅ d (ωt )
S
−π 2 +α −ϕ m
S
I DCa
= 2 π ⋅ cos(α ) ⋅ I Sd ⋅ cos(ϕ m ) + I Sq ⋅ sin(ϕ m )
∫αv ϕ
π
The Fourier coefficients in (16)-(17) are chosen considering
the coordinate frame alignment in Figure 3. Replacing (8) in
(16)-(17), the ac voltage will consist of two components:
V
π 2
π 2 −α −ϕ m
2
]
(15)
Assuming a typical STATCOM application, we will have
ϕm≈0 and it can be seen in (15) that the dynamic ac voltage
will depend mainly on the active current ISd and the control
angle α. Therefore, in a typical STATCOM application, the
component ISq in (15) can be neglected with a limited loss in
accuracy. However, the converters with active power flow
should include the complete model.
D. Converter dynamic Model at Fundamental Frequency
The fundamental ac voltage of the switched dc voltage
components can be found if the Fourier Transformation at
frequency ω, (Fω) is used. We calculate first coefficients of the
Fourier series to obtain d and q axis components of ac voltage:
1) Dynamic Voltage Component VSSdyna:
In order to preserve the dynamic nature of the converter
voltage in (15), we need to apply Fourier transformation to
differential of converter voltage. For a given function y, the
following property of Fourier transformation is employed:
sFω(y)= Fω(sy)
(20)
Placing (15) in (16)-(17), using (20) and assuming a constant
current magnitude IS across the integration interval, the
dynamic equations for the ac voltage components are:
[
⋅ [I
]
)]sin(ϕ
S
s ⋅ VSdynad
= 1 CeqS 2 ⋅ I Sd ⋅ cos(ϕ m ) + I Sq ⋅ sin(ϕ m ) cos(ϕ m )
s ⋅V
S
Sdynaq
=1 C
with C eqS 2 =
S
eq 2
Sd
⋅ cos(ϕ m ) + I Sq ⋅ sin(ϕ m
π2
8 ⋅ cos 2 (α )
⋅ C DC
m
)
(21)
(22)
(23)
2) Static Voltage Component VSS0:
Placing (10) in (16)-(17), the corresponding static ac voltage
components become:
VSS0 d = k S ⋅ VDC 0 ⋅ cos(ϕ m )
(24)
= k ⋅ VDC 0 ⋅ sin(ϕ m )
(25)
V
S
S 0q
S
with k S = 4 ⋅ cos(α ) π
(26)
The expressions for equivalent capacitance in (11) and (23)
represent some imaginary capacitance which enables us to
represent multilevel converter dynamics using simple
equations for a single-capacitor system. It is observed that
these capacitances depend only on the capacitance CDC and the
control angle α, which are constant for a given STATCOM
topology. Therefore, the equivalent capacitance will not
change as the operating conditions are changing.
It is noted that the q components in (22) and (25) will be
small in a typical STATCOM.
E. Non-linear Dynamic Model for a Multilevel Converter
The above modelling approach can be adopted for each cell
of a multilevel converter. The fundamental component of ac
voltage for a cascaded STATCOM with X cells is obtained as:
x
V Sd = ∑ V SdS _ i
i =1
(27)
5
x
V Sq = ∑ V SqS _ i
(28)
i =1
VGrid
VSd, VSq
DC Electrical
System
AC System
ISd, ISq
The capacitor voltage balancing is normally used with
multilevel converters and such control function is very fast
compared with main control responses [17]. Consequently, the
DC voltages on individual cells can be assumed identical. This
enables us to use same equations for each cell (21)-(26), and
only the firing angles will differ. Replacing (21)-(26) in (27)
and (28), it can be shown that the fundamental ac voltage
components of a multilevel converter are similar to (21)-(22)
and (24)-(25). The equivalent capacitance Ceq2 and the
parameter k for a multilevel converter are different, however,
and result in:
[
]
 x

C eq 2 = π 2  8 ⋅ ∑ cos 2 (α i )  ⋅ C DC
 i=1

x


k =  4 ⋅ ∑ [cos(α i )] π
 i =1

(29)
(30)
where αi are the control angles on individual cells.
The final d-component VSd of the multilevel converter model
is therefore obtained using (21) and (24):
[
]
s ⋅ VSdynad = 1 Ceq 2 ⋅ I Sd ⋅ cos(ϕ m ) + I Sq ⋅ sin(ϕ m ) cos(ϕ m )
(31)
VS 0 d = k ⋅ VDC 0 ⋅ cos(ϕ m )
(32)
and using (22) and (25), the q-component VSq is:
[
]
s ⋅ VSdynaq = 1 Ceq 2 ⋅ I Sd ⋅ cos(ϕ m ) + I Sq ⋅ sin(ϕ m ) sin(ϕ m )
(33)
VS 0 q = k ⋅ VDC 0 ⋅ sin(ϕ m )
(34)
IV. ANALYTICAL STATCOM SYSTEM MODEL
A. Model Structure
The STATCOM system consists of the multilevel converter,
a transformer, a control system with Phase Locked Loop
(PLL) and it is connected with an ac system.
The STATCOM model schematic diagram is shown in
Figure 7 with all linking variables between the sub-systems. It
can be seen that the STATCOM model is segmented in two
main sub-systems: an ac system and a dc system, which is a
conventional approach with converter modelling [11]. Each of
these subsystems is represented as a stand-alone state-space
model, and the dc system includes the converter model, the
controller model and the PLL model.
The final state-space model can be written in the following
matrix form: [9], [11]
s ⋅ x S = AS ⋅ x S + BS ⋅ u Ref
(35)
y S = C S ⋅ x S + DS ⋅ u Ref
(36)
where:
BDCAC C AC 
 ADC
AS = 
AAC 
 B ACDC C DC
T
u Ref = [VRefAC VGrid ]
y S = [V AC ]
(37)
ϕAC
ϕPLL
PLL System
ϕm
VAC
PI Controller
Feedback Filter
ω²f
s²+2ζωfs+ω²f
KP
-
+
+
KI 1/s
- +
ϕRefAC
Delay Filter
1
Tf*s+1
DC Model
VRefAC
ϕRefAC - Calculated firing angle
ϕPLL - PLL reference angle
ϕm
- Actual firing angle
VRefAC
- Reference ac Magnitude
VAC, ϕAC - Magn., Phase angle of νAC
νAC=VAC cos(ωt+ϕAC)
Fig. 7: Structural diagram of STATCOM system model
In (35) and (36), xS is the state-variable vector, uRef
represents the input variable vector (including reference
voltage VRefAC and disturbance voltage VGrid) and yS stands for
the output variable vector. The sub-matrices in the system
matrix AS are associated with the individual subsystems: dc for
dc subsystem and ac for ac subsystem. The indices with input
matrix B and output matrices C indicate the actual model (first
index) and the linking subsystem (second index).
B. AC Model
Assuming a symmetrical and balanced ac system, the ac grid
can be modelled as a single-phase dynamic system as shown
in Figure 8. The instantaneous circuit variables are used as the
states. Since the ac grid voltage VAC cannot be measured
directly in a state-space model, an artificial capacitance CART,
which has a very small value (no dynamic effects to the
system), is used to accommodate state-space modelling.
It is shown in [16] that conduction losses are dominant in a
converter with square-wave control and therefore, the
converter losses are represented in the model as a seriesconnected resistance Rloss.
The final model is subsequently converted to the dq rotating
frame using Park’s transformation as described in [11].
IAC
IS
RAC
VGrid
LTR+LDC*nTR² Rloss*nTR²
LAC
CART
VAC
VS
Fig. 8: Equivalent ac circuit
C. Converter Model
The final STATCOM model is presented in the dc submodel and the ac voltage is the sum of dynamic and static
voltage components (31)-(34). Since these expressions are
strongly non-linear, they are linearised in a usual manner.
The linearised STATCOM model is linked with the PLL and
the controller model, as shown below in the next section, to
create the final dc model.
D. Controller and PLL
The STATCOM system uses indirect control, which consists
of a proportional plus integral (PI) controller, to regulate the
ac voltage as seen in Figure 7. The ‘PI’-controller produces
6
Imag
Gain increase
0
Selected Eigenvalue
-50
-100
-60
-30
-20
-10
Real
Fig. 9: Dominant branch of the root locus for test system 1.
-50
-40
0
B. Reference Step Input
1) 13-level converter
A reference step response for test system 1 (13-level
converter) is shown in Figure 10. It can be seen that the
developed model shows good accuracy for all considered
variables, i.e. the voltage magnitude, the STATCOM current
and the control angle ϕm.
Figure 11 shows the model verification for test system 2 (13level converter). It is concluded that the analytical model
shows very good accuracy even for different system
topologies.
2) 9-level converter
Figure 12 shows the system response when test system 3 is
used, i.e. with a 9-level converter. It is seen that excellent
model matching occurs for all compared variables.
C. Reference Disturbance Input
1) 13-level converter
The perturbation on the remote ac source tests the model
ability to represent perturbations on remote ac source and the
coupling between the ac and dc sub-models. The model
verification for disturbance inputs is shown in Figure 13 and
14 for test system 1 and test system 2, respectively. It can be
seen that a good response matching exists for both systems,
and similar matching was confirmed for all the other variables
that are not shown.
230.67
230.66
Reference
Average model
PSCAD
230.65
0
0.05
ωt [sec]
0.1
0.15
a) System voltage
0.176
0.174
0.172
0.17
Average Model
PSCAD
0
0.05
ωt [sec]
0.1
0.15
b) AC Current IS
Control angle ϕm [deg]
A. Selection of Controller Gains
The controller gains are selected in this study by analysing
the dominant eigenvalues of the system and they are tuned to
meet the following requirements:
- Rise time less than 3 cycles of fundamental frequency,
- Overshoot below 10%,
- Settling time less than 10 cycles of fundamental frequency.
Figure 9 shows the root locus of test system 1 as the open
loop gain changes with the nominal system configuration. The
PI controller zero is selected for this particular test system at
z1=–KI /KP=-2.8, which is close to the slowest system pole, in
order to improve speed of response and robustness.
It can be seen that for small open-loop gains the dominant
eigenvalues are only real poles resulting in slow rise times and
large damping. It can also be seen that large controller gains
will result in significant overshoots, faster rise times and a
reduced damping. Best controller gains are then selected for
each individual test system from its root locus to give
optimum responses according with the above requirements,
and final controller gains are given in Table I in the Appendix.
50
AC voltage VAC [kV]
V. MODEL VERIFICATION
The derived analytical model is implemented in Matlab and
tested in the time domain against the detailed, non-linear
simulation software PSCAD/EMTDC.
Three, largely different, test systems are simulated in order
to verify the accuracy and flexibility of the analytical model,
and all parameters are given in Table I in the Appendix. The
test system 1 is a 30kV, 70Mvar 13-level (X=6) STATCOM
system, whereas test system 2 is operating at 15kV and has a
rated reactive output power of 25Mvar. To confirm the model
accuracy with different ac system parameters, the grid strength
in test system 2 is reduced by the factor of 3, compared with
test system 1. A third test system is introduced to study the
model robustness for a STATCOM system with different
number of levels. This system is similar to test system 1, but it
involves a 9-level converter (X=4). The operating point of all
systems is set close to their rated reactive power QST. The
MATLAB model enables insightful study of changes in
parameters or structure since all parameters are directly
reflected in the symbolic form in the final model.
The perturbation signal used in step responses is the
reference voltage VRefAC, as seen in Figures 1 and 6. In
disturbance tests, a step on the remote source voltage
magnitude, given by VGrid in Figures 1 and 7, is introduced.
100
AC current IAC [kA]
the phase angle ϕm, which is common for all cells. The
reference signal VRefAC is provided by an external or a system
control [15].
The delay filter in Figure 7 is introduced to represent firing
control circuit delay and does not represent actual system
dynamics. This filter accounts for the delay in the signal
transfer in a discrete system [18], and the delay time constant
is determined as Tf=1/(50*3*2*X), where 50 represents
fundamental frequency, ‘3’ is the number of branches and ‘2’
considers two switchings per cycle per each branch.
The main role of PLL is to synchronise the converter firing
signals with the ac system, and the latest D-Q-Z type [18],
which has the advantage of accurate phase information even
with a distorted ac system voltage waveform, is used here.
Average model
PSCAD
0.23
0.21
0.19
0.17
0
0.05
ωt [sec]
0.1
0.15
c) Control angle ϕm
Fig. 10: Test system 1 response following a 0.02kV voltage reference step
Reference
Average Model
PSCAD
230.75
230.7
0
0.05
ωt [sec]
0.1
0.15
0.07
230.65
0.066
Average Model
PSCAD
0.062
0.05
ωt [sec]
0.1
0.15
Average Model
PSCAD
0.5
0.4
0.3
0.2
0
0.05
ωt [sec]
0.1
0.15
AC voltage VAC [kV]
b) Control angle ϕm
Fig. 11: Test system 2 response following a 0.1kV voltage reference step
230.75
Reference
Average model
PSCAD
230.7
230.65
0
0.05 ωt [sec]
0.1
0.15
AC current IAC [kA]
a) System voltage
0.2
0.19
0.18
Average Model
PSCAD
0.17
0
0.05
ωt [sec]
0.1
0.15
Control angle ϕm [deg]
b) AC Current IS
Average model
PSCAD
0.8
0.4
0.2
0.05
ωt [sec]
0.1
AC voltage VAC [kV]
230.65
230.63
Reference
Average model
PSCAD
0
0.05
ωt [sec] 0.1
0.15
Fig. 13: Test system 1 response following a 0.02kV disturbance step change
0.15
230.71
0
-10
-20
-30
-40
Average model
PSCAD
-200
-300
-400
10
Average model
PSCAD
30
100
Frequency [rad]
300
600
Fig. 16: Test system 1 frequency response of the linear model compared with
PSCAD/EMTDC frequency response
0
-10
-20
-30
-40
Average model
PSCAD
-100
V
AC voltage VAC [kV]
ωt [sec] 0.1
-100
0.15
c) Control angle ϕm
Fig. 12: Test system 3 response following a 0.1kV voltage reference step
230.64
0.05
D. Frequency Response of AC Grid with STATCOM
Since a time domain study cannot reveal the frequency
bandwidth for the model accuracy, the test systems 1 and 3 are
also tested against PSCAD in the frequency domain. As
PSCAD does not possess the frequency response ability, the
individual frequency components are injected (1Hz step) with
measurement of the gain and phase of the system voltage VAC
at the PCC. The experimental frequency response is performed
in the frequency range 1.6Hz<f<100Hz, and using VRefAC as
the injection point. The readings at individual frequencies are
then linked to create gain and phase curves as shown in
Figures 16 and 17 for test system 1 and 3 respectively.
Comparing with the PSCAD model, it can be observed that
very good accuracy can be achieved for both test systems in
V
0.6
0
0
Fig. 15: Test system 3 response following a 0.1kV disturbance step change
Magnitude VAC [dB]
Control angle ϕm [deg]
b) AC Current IS
Reference
Average model
PSCAD
230.55
Magnitude VAC [dB]
0
230.6
Phase φ [deg]
AC current IAC [kA]
a) System voltage
2) 9-level converter
Figure 15 shows the ac voltage at PCC, in response to a
disturbance step change in VGrid, if test system 3 is used. It is
concluded that the linear model readily accommodates
structural changes (number of cells) in the multilevel
converter.
AC voltage VAC [kV]
230.8
Phase φ [deg]
AC voltage VAC [kV]
7
230.7
230.69
Reference
Average Model
PSCAD
0
0.05
ωt [sec] 0.1
0.15
Fig. 14: Test system 2 response following a 0.02kV disturbance step change
-200
-300
10
Average model
PSCAD
30
100
Frequency [rad]
300
600
Fig. 17: Test system 3 frequency response of the linear model compared with
PSCAD/EMTDC frequency response
8
the frequency range f<40Hz, and reasonably good accuracy at
frequencies below 100Hz. It is important that the dominant
complex poles, which are located just above 50Hz, are well
represented by the linear model. PSCAD however points to
the existence of additional complex poles and zeros close to
the dominant poles. The slope of the bode curves is well
matched by the analytical model indicating that the order of
the analytical model is accurate.
VI. CONCLUSIONS
A suitable and accurate analytical model of an indirectly
controlled cascaded multilevel STATCOM with square-wave
control is presented in this paper.
The converter voltage components are analyzed in detail for
a single-cell and the results are then generalised for a
multilevel cascaded converter. The converter ac voltage
waveform is of a non-linear, discrete and dynamic nature,
which is described mathematically by appropriate averaged
expressions.
The dynamic, analytical state-space model is built of subsystems to enable model application to a wide range of system
configurations and various dynamic studies.
The developed STATCOM model is linearised and
implemented in MATLAB. Eigenvalue studies are conducted
for each particular test system in order to select optimum
open-loop controller gains.
The validity and accuracy of the proposed model is verified
against non-linear PSCAD simulations, and good matching is
observed in the time domain for a range of outputs and for
three different test systems. The model is also tested in
frequency domain and it is concluded that the presented model
can be used for dynamic studies below 100Hz.
VII. APPENDIX
TABLE I - SYSTEM PARAMETER VALUES
Description
Notation System 1 System 2 System 3
AC system data:
STATCOM rms output voltage
VS1(rms)
30kV
15kV
30kV
STATCOM rated output power
QST
±70Mvar ±25Mvar ±70Mvar
STATCOM dc capacitance
CDC
3.0mF
4.2mF
5.0mF
STATCOM dc buffer inductance
LDC
0.95mH 0.675mH 2.5mH
STATCOM series resistance
Rloss
71mΩ
80.7mΩ 59.3mΩ
Transformer rating
STR
70MVA 70MVA 70MVA
Transformer inductance
LTR
0.1pu
0.1pu
0.1pu
AC circuit resistance
RAC
0.22Ω
0.66Ω
0.22Ω
AC circuit inductance
LAC
7.01mH 21.05mH 7.01mH
ON-state resistance IGBT/diode
RON
3mΩ
3mΩ
3mΩ
ON-state voltage IGBT/diode
VON
2V
2V
2V
Controller data:
Proportional gain
Kp
6.5
4.5
15
Integral gain
Ki
0.045 1/s 0.04 1/s 0.05 1/s
Open loop gain
Kc
0.5
0.7
0.5
Firing time delay constant
Tf
1/3900 s 1/3900 s 1/2700 s
Damping factor feedback filter
1
1
1
ζf
Cut-off frequency feedback filter
188.5rad/s 188.5rad/s 188.5rad/s
ωf
PLL proportional gain
PLL kp
75
75
75
PLL integral gain
PLL ki
500 1/s 500 1/s 500 1/s
TABLE II - SWITCHING ANGLES FOR A 9- AND 13-LEVEL CONVERTER
Capacitor
1
2
3
4
5
6
13-level
1.56°
11.2° 28.26°
38.76° 45.44° 77.66°
9-level
13.98° 29.93°
51°
64.21°
-
VIII. REFERENCES
[1] G.Joos, X.Huang, B.-T.Ooi, “Direct-Coupled Multilevel Cascaded Series
Var Compensators”, IEEE Trans. On Industrial Applications., Vol.34,
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[2] R.Betz, B.Cook, T.Summers, R.Palmer, A.Bastiani, S.Shao, K. Willis,
“Design and Development of an 11kV H-Bridge Multilevel STATCOM”,
AUPEC ‘07, Perth, W. Australian, Conf. Proceedings, Dec. 2007.
[3] C. Hochgraf, R. Lasseter, D. Divan, T.A. Lipo, “Comparison of
Multilevel Inverters for Static Var Compensation“, Conf. Rec. – IAS
Annual Meeting Denver, CO, USA, Vol.2, pp.921-928, 1994
[4] J.S.Lai, F.Z.Peng, “Multilevel Converter - A new Breed of Power
Converter”, IEEE Transactions On Industrial Applications, Vol.32, No.3,
pp.509-517, May 1996
[5] F.Z.Peng, J.W.McKeever, D.J.Adams “A power line conditioner using
cascade multilevel Inverters for Distribution Systems“, IEEE Trans. On
Industry Applications, Vol.34, No.6, pp.1293-1298, November 1998
[6] S.Sirisukprasert, J.S.Lai, T.H.Liu, “Optimum harmonic reduction with a
wide range of modulation indexes for multilevel converters“, IEEE Trans.
On Ind. Electr., Vol.49, No.4, pp.875-881, Aug. 2002
[7] J.D. Ainsworth, M. Davies, P.J. Fitz, K.E. Owen, D.R. Trainer, “Static Var
compensator (STATCOM) based on single-phase circuit converters“, IEE
Proc.-Gener. Transm., Vol. 145, No. 4, pp. 381-386, July 1998
[8] D.Soto, R.Pena, “Nonlinear Control Strategies for Cascaded Multilevel
STATCOMs“, IEEE Transactions On Power Delivery, Vol.19, No.4,
pp.1919-1927, Oct. 2004
[9] R.Sternberger, D.Jovcic, “Small Signal Multi-level STATCOM Model”,
2006 Power Eng. Soc. Gen. Meet. Conf. Proc., June 2006
[10] Manitoba HVDC Research Center, “PSCAD/EMTDC 4.1 Users manual”,
Tutorial manual, 2004
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analytical model” IEEE Trans. on Power Delivery, Vol 4, pp 1455-1461,
Oct. 2003
[12]A.Yazdani, R.Iravani, “An Accurate Model for the DC-Side Voltage
Control of the Neutral Point Diode Clamped Converter”, IEEE Trans. On
Power Delivery, vol. 21, no. 1, pp. 185-193, Jan. 2006.
[13] M.Mohaddes, A.M.Gole and S.Elez “Steady-state frequency response of
STATCOM”, IEEE Transactions on Power Delivery, Vol 16, no 1, pp1823, Jan. 2001
[14] D.Jovcic, R.Sternberger, “Frequency Domain Analytical Model
for a Cascaded Multi-level STATCOM”, IEEE Trans. On Power
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IX. BIOGRAPHIES
Ronny Sternberger (S’06) obtained a Dipl.-Ing.(FH) degree in Electrical
Energy Engineering at the FHTW Berlin, Berlin, Germany in 2004. He is
currently studying towards his Ph.D. degree at the University of Aberdeen,
Aberdeen, Scotland.
His current research interests are in FACTS devices, multilevel converter
and control systems.
Dragan Jovcic (SM’06, M’00, S’97) obtained a Diploma Engineer degree
in Control Engineering from the University of Belgrade, Yugoslavia in 1993
and a Ph.D. degree in Electrical Engineering from the University of Auckland,
Auckland, New Zealand in 1999.
He is currently a lecturer with the University of Aberdeen, Aberdeen,
Scotland where he has been since 2004. He also worked as a lecturer with
University of Ulster, in period 2000-2004 and as a design Engineer in the New
Zealand power industry in period 1999-2000. His research interests lie in the
areas of FACTS, HVDC and control systems.