Modeling of TCR and VSI Based FACTS
Controllers
Claudio A. Ca~nizares
University of Waterloo
Department of Electrical & Computer Engineering
Waterloo, ON, Canada N2L 3G1
c.canizares@ece.uwaterloo.ca
Internal Report for ENEL and POLIMI
September 9, 1999
Contents
1 Introduction
2 Modeling TCR-based Controllers
1
2
3 VSI-based Controllers
9
2.1 SVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 TCSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.1 STATCOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 SSSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 UPFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Conclusions
26
1
List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
Block diagram of a SVC with voltage control. . . . . . . . . .
Transient stability model of a SVC. . . . . . . . . . . . . . . .
Basic SVC controller model for voltage control. . . . . . . . .
Typical steady state V-I characteristics of a SVC. . . . . . . .
Handling of limits in the SVC steady state model. . . . . . . .
Block diagram of a TCSC operating in current control mode. .
Transient stability model of a TCSC. . . . . . . . . . . . . . .
Block diagram of a STATCOM with PWM voltage control. . .
Transient stability model of a STATCOM with PWM voltage
control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Basic STATCOM PWM voltage control. . . . . . . . . . . . .
Typical steady state V-I characteristics of a STATCOM. . . .
Handling of limits in the STATCOM steady state model. . . .
Block diagram of a SSSC with PWM current control. . . . . .
Transient stability model of a SSSC. . . . . . . . . . . . . . .
Block diagram of a UPFC. . . . . . . . . . . . . . . . . . . . .
Transient stability model of a UPFC. . . . . . . . . . . . . . .
Basic shunt branch control of UPFC. . . . . . . . . . . . . . .
Basic series branch dq control of UPFC. . . . . . . . . . . . .
2
3
4
4
5
6
7
7
10
11
13
15
15
17
18
21
22
23
24
Abstract
This report concentrates on presenting transient stability and power ow
models of Thyristor Controlled Reactor (TCR) and Voltage Sourced Inverter (VSI) based Flexible AC Transmission System (FACTS) Controllers.
The models discussed in detail are for the Static Var Compensator (SVC),
the Thyristor Controlled Series Compensator (TCSC), the Static Var Compensator (STATCOM), the Static Synchronous Source Series Compensator
(SSSC), and the Unied Power Flow Controller (UPFC), and are appropriate
for voltage and angle stability studies.
Chapter 1
Introduction
The large recent development and use of FACTS controllers in power transmission systems has led to many applications of these controllers to improve
the stability of power networks [1, 2]. Thus, many studies have been carried
out and reported in the literature on the use of these controllers in a variety
of voltage and angle stability applications, proposing diverse control schemes
and location techniques for voltage and angle oscillation control [2].
Several distinct models have been proposed to represent FATCS in static
and dynamic analysis [3]. This report concentrates on describing in detail
the most appropriate models available for these types of studies for systems
that include the following controllers: SVC, TCSC, STATCOM, SSSC and
UPFC. These models allow to accurately and reliably carry out power ow
and transient stability studies of systems and its controllers for voltage and
angle stability analyses.
Chapter 2 describes in detail the models for TCR-based controllers, concentrating specically on the SVC and TCSC, and Chapter 3 discusses in
detail the models for VSI-based controllers, namely, the STATCOM, the
SSSC and the UPFC. Finally, Chapter 4 summarizes the models presented
on this report and discusses the limitations of these models.
1
Chapter 2
Modeling TCR-based
Controllers
Basic models for SVCs and TCSCs built around a TCR structure are described in this section. These models are based on representing the controllers as variable impedances that change with the ring angle of the TCR,
which is used to control voltages, current and/or powers in the system.
2.1 SVC
The basic structure of an SVC operating under typical bus voltage control is
depicted in the block diagram of Figure 2.1. Assuming balanced, fundamental
frequency operation, an adequate transient stability model can be developed
assuming sinusoidal voltages [4]. This model is depicted in Figure 2.2 and
can be represented by the following set of p.u. equations:
"
x_ c
_
#
= f (xc ; ; V; Vref )
2
Be , 2 , sin 2 ,XL(2 , XL =XC )
6
6
6
0 = 666 I , Vi Be
6
4
|
Q , Vi2Be
{z
g(; V; Vi ; I; Q; Be)
2
(2.1)
3
7
7
7
7
7
7
7
5
}
V
I
Filters
a:1
Vi
Zero
Crossing
Switching
Logic
C
L
Magnitude
Vref
α
Controller
Figure 2.1: Block diagram of a SVC with voltage control.
where most variables are clearly dened on Figure 2.2, and xc and f () stand
for the control system variables and equations, respectively. These equations
allow to represent limits not only on the ring angle , but also on the current
I , the control voltage V and the capacitor voltage Vi, as well as to model
other types of controllers such as reactive power Q control.
The dierential equations represented by f () in (2.1) vary with the type
of control system used. Figure 2.3 depicts the typical the voltage control
block diagram, which includes a droop to avoid continuos operation of the
controller and to allow for proper coordination with other voltage controllers
in the network. It is important to highlight the fact that an admittance
model is numerically more stable than the corresponding impedance model,
i.e., using Be on the model averts numerical problems when close to the
controller's resonant points [5]. The bias o for this controller is determined
by solving the equations resulting from forcing Be = 0 in (2.1), i.e., this
value corresponds to the resonant point of the SVC (I = 0) that results from
solving the nonlinear equation
2o , sin 2o , (2 , XL=XC ) = 0
The steady sate V-I characteristics for this controller are depicted in
Figure 2.4, and correspond to the well-known control characteristics of a
3
V
I
Filters
Q
a :1
Vi
Magnitude
Vref
Controller
α
Be(α)
Figure 2.2: Transient stability model of a SVC.
αmax − αo
V
KM
1+ S TM
K (1+ S T1 )
+
V
KD+ S T2
αmin− αo
ref
∆α
+
+
αo
Figure 2.3: Basic SVC controller model for voltage control.
4
α
V
XL
X SL
α max
Vref
(αo )
XC
α min
XC
I
Figure 2.4: Typical steady state V-I characteristics of a SVC.
typical SVC [2]. A SVC steady sate model can be obtained by replacing the
dierential equations in (2.1) with the equations representing these steady
state characteristics; thus, the \power ow" equations of the SVC in this
case are
2
3
V
,
V
ref + XSL I
7
0 = 64
(2.2)
5
g(; V; Vi; I; Q; Be)
which can be directly included in any power ow program, as discussed in [5].
However, for the model to be complete, all SVC controller limits should be
adequately represented. The proper handling of ring angle limits is depicted
o , until reaches
in Figure 2.5 [5], where Vref is kept xed, say at a value Vref
a limit, at which point Vref is allowed to changed while is kept at its limit
o .
value; voltage control is regained when Vref returns to its xed value Vref
2.2 TCSC
Figure 2.6 shows the block diagram for a TCSC controller operating under
current control. The model for balanced, fundamental frequency operation is
shown in Figure 2.7, and can be represented by the following set of equations,
5
α < αmin
α = αmin
o
Vref >Vref
αmin< α < αmax
o
Vref <Vref
o
Vref =Vref
α > αmax
α = αmax
o
o
Vref >Vref
0 < Vref < Vref
Figure 2.5: Handling of limits in the SVC steady state model.
which includes the control system equations and assumes sinusoidal currents
in the controller [5, 6]:
"
x_ c
_
#
= f (xc; ; I; Iref )
(2.3)
2
P + Vk Vm Be sin(k , m)
6
6
6
6
,Vk2Be + Vk Vm Be cos(k , m) , Qk
6
6
6
6
6
,Vm2 Be + Vk Vm Be cos(k , m) , Qm
6
6
6
6
6
Be , Be()
6
6
4 q
P 2 + Q2k , I Vk
{z
|
g(; Vk ; Vm ; k ; m; I; P; Qk; Qm; Be)
0 =
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
}
where most variables are dened on Figure 2.7; xc stands for the internal
control system variables; and
Be ()
=
kx4 , 2kx2 + 1
h
cos
kx ( , )=
XC kx4 cos kx( , )
, cos kx( , ) , 2 kx4 cos kx( , )
2
4
+2 kx cos kx ( , ) , kx sin 2 cos kx( , )
2 sin 2 cos k ( , ) , 4 k3 cos2 sin k ( , )
+kx
x
x
xi
2
,4 kx cos sin cos kx( , )
s
kx
=
XC
XL
6
Vk δ k
Vm δ m
C
I
L
Switching
Logic
Zero
Crossing
Magnitude
I ref
α
Controller
Figure 2.6: Block diagram of a TCSC operating in current control mode.
Vk δ k
Be(α)
P +jQk
-P +jQm
Vm δ m
I
Magnitude
I ref
α
Controller
Figure 2.7: Transient stability model of a TCSC.
A simple PI controller with limits can be used to control the current
directly through the ring angle ; in this case, the dierential equations f ()
in (2.3) can be replaced by the equations of the corresponding control system.
Observe, however, that more sophisticated controls such as impedance or
power control can be readily implemented on this model.
A steady state model for this TCSC controller can be obtained by replacing the dierential equations on (2.3) with the corresponding steady state
control equations. For example, for an impedance control model with no
droop, which yields the simplest set of steady state equations from the nu-
7
merical point of view [5], the \power ow" equations for the TCSC are
0 =
2
6
4
3
7
5
Be , Beref
(2.4)
g(; Vk ; Vm ; k; m; I; P; Qk ; Qm; Be)
As previously indicated, it is important to adequately implement the controller limits on the steady state model to accurately represent its operation
[5].
8
Chapter 3
VSI-based Controllers
In this section, the basic models of the most common VSI-based FATCS
controllers, namely, STACTOM, SSSC and UPFC, are discussed. All the
models presented here are based on the power balance equation
Pac = Pdc + Ploss
which basically represents the balance between the controller's ac power Pac
and dc power Pdc under balanced operation at fundamental frequency. For
the models to be accurate, it is important to represent all losses of the controllers (Ploss ), especially those related to the inverters, as discussed below.
Although PWM control is currently not practical in typical high-voltage
applications of VSI-based controllers, given the high switching losses of GTOs,
there have been some new recent developments on power electronic switches
that will probably allow for the practical use of PWM control techniques on
these kinds of applications in the near future. Hence, on the models discussed in this paper, PWM control techniques are assumed, as these allow to
develop more general models that can readily be adapted to represent other
control techniques such as phase angle controls.
3.1 STATCOM
The basic structure of a STATCOM with PWM-based voltage controls is
depicted in Figure 3.1. Eliminating the dc voltage control loop on this gure
would yield the basic block diagram of a controller with typical phase angle
controls.
9
V δ
I
Filters
θ
a:1
Vi
Zero
Crossing
α
Switching
Logic
PLL
Magnitude
α
Vref
m
(PWM)
Controller
C
V
dc
PWM
Magnitude
Vdc
ref
Figure 3.1: Block diagram of a STATCOM with PWM voltage control.
10
V δ
Filters
θ
I
P+jQ
Magnitude
a:1
R+jX
Vref
k Vdc
α
Controller
α
k (PWM)
PWM
Vdc
C
RC
Vdc
ref
Magnitude
Figure 3.2: Transient stability model of a STATCOM with PWM voltage
control.
11
Assuming balanced, fundamental frequency voltages, the controller can
be accurately represented in transient stability studies using the basic model
shown in Figure 3.2 [7, 8, 9]. The p.u. dierential-algebraic equations (DAE)
corresponding to this model are
2
3
x_ c
6 _ 7 = f (x ; ; m; V; V ; V ; V
(3.1)
4
5
c
dc ref dcref )
m_
I2
V_dc = CV VI cos( , ) , R 1 C Vdc , R
C Vdc
dc
C
2
3
P , V I cos( , )
0 =
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
Q , V I sin( , )
P , V 2 G + k Vdc V G cos( , )
+k Vdc V B sin( , )
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
Q + V 2 B , k Vdc V B cos( , )
+k Vdc V G sin( , )
|
{z
}
g(; k; V; Vdc ; ; I; ; P; Q)
where most of the variables are explained on Figure 3.2; G+jB = (R+jX ),1,
which is used to represent the transformer impedance, any ac series lters,
and the
q \switching inertia" of the inverter due to its high frequency switching;
k = 3=8 m, and hence is directly proportional to the modulation index m;
and xc stands for the internal control system variables.
A simple PWM voltage controller is shown in Figure 3.3 [10, 11], which
basically denes the dierential equations represented by f () in (3.1). Observe that the ac bus voltage magnitude is controlled through the modulation
index m, as this has a direct eect on the VSI voltage magnitude, whereas
the phase angle , which basically determines the active power P owing
into the controller and hence the charging and discharging on the capacitor,
is used to directly control the dc voltage magnitude. The controller limits are
dened in terms of the controller current limits, which are directly related to
the GTO current limits, as these are the basic limiting factor in VSI-based
controllers. In simulations, these limits can be directly dened in terms of
the maximum and minimum converter currents Imax and Imin, respectively,
12
mmax(Imax ) - mo
Vref
K ( 1 + S T1 )
KD+ S T 2
+
-
+
m
+
mmin (Imin ) - mo
KM
ac
mo
1 + S TM
ac
V
Vdcref
αmax (Imax )− αo
+
KP +
-
KI
S
αmin (Imin )− αo
KM
dc
+
α
+
αo
1 + S TM
dc
Vdc
Figure 3.3: Basic STATCOM PWM voltage control.
i.e., the integrator blocks are \stopped" whenever the converter current I
reaches a limit, which would allow to closely duplicate the steady state V-I
characteristics of the controller shown in Figure 3.4. Another option is to
compute these limits by solving the steady state equations of the converter,
as discussed below; these equations are also used to compute the biases mo
and o.
The steady state model can be readily obtained from (3.1) by replacing
the dierential equations with the steady state equations of the dc voltage
and the voltage control characteristics of the STATCOM (see Figure 3.4
[2]). Notice that the controller droop is directly represented on the V-I
characteristic curve, with the controller limits being dened by its ac current
13
limits. Hence, the steady state equations for the PWM controller are
2
3
V
,
V
+
X
I
ref
SL
6
7
0 =
6
6
6
Vdc , Vdcref
6
6
6
6
6
P , Vdc2 =RC , R I 2
6
6
4
g(; k; V; Vdc ; ; I; ; P; Q)
7
7
7
7
7
7
7
7
7
7
5
(3.2)
A phase control technique can be readily modeled by simply replacing the dc
voltage control equation in (3.2) with an equation for k, i.e., for a 12-pulse
VSI, replace 0 = Vdc , Vdcref with 0 = k , 0:9. In this case the dc voltage
changes as changes, thus charging and discharging the capacitor to control
the inverter voltage magnitude.
These equations can be directly used to compute the biases and limits of
the PWM controller. The biases are determined by setting I = 0, yielding
mo =
s
8 Vref
3 |Vdcref
{z }
ko
o = o
where o stands for the bus angle phase shift when the STATCOM is disconnected from the system. The modulations index limits mmax and mmin , and
the phase-shift limits max and min can be computed by solving equations
(3.2) for Imax and Imin, respectively. Thus, the modulation index limits can
be shown to be equal to
s
mmax = 83 Vref ,V XSL Imax
dcref
|
{z
}
k
max
s
mmin = 83 Vref ,V XSL Imin
dcref
|
{z
}
kmin
The phase-shift limits, on the other hand, do not have a simple close form
solution and must be obtained numerically.
14
V
X SL
Vref
(mo ,α o )
I
Imax
min
I
Figure 3.4: Typical steady state V-I characteristics of a STATCOM.
I = I max
o
Vref >Vref
I > I max
o
Vref <Vref
I min < I < I max
o
Vref =Vref
I < I min
I = I min
o
o
Vref >Vref
0 < Vref < Vref
Figure 3.5: Handling of limits in the STATCOM steady state model.
The limits on the current I , as well as any other limits on the steady state
model variables, such as the modulation ratio represented by k or the voltage
phase angle , can be directly introduced in this model. It is important to
properly represent the control mode switching when these limits are reached,
as this is a signicant factor for properly modeling FACTS controllers in
steady state studies [5]. Thus, the mode switching logic depicted in Figure
2.5 for the SVC can be readily modied to represent the steady state control
mode switching for the STATCOM, by simply replacing the ring angle limits
with current limits, as shown in Figure 3.5.
15
3.2 SSSC
For the SSSC, the basic controller structure operating on current control
mode is depicted in Figure 3.6. The corresponding transient stability model
is shown in Figure 3.7 [8], and can be represented by the following p.u.
equations:
2
x_ c 3
6 _ 7 = f (x ; ; m; I; V ; I ; V
(3.3)
4 5
c
dc ref dcref )
m_
2
V_dc = CV VI cos( , ) , R 1 C Vdc , CR VI
dc
C
dc
2
3
P
k , Vk I cos(k , )
6
7
0 =
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
|
Qk , Vk I sin(k , )
Pm + Vm I cos(m , )
Qm + Vm I sin(m , )
P , Pk + Pm
Q , Qk + Qm
P , V 2 G + k Vdc V G cos( , )
+k Vdc V B sin( , )
Q + V 2 B , k Vdc V B cos( , )
+k Vdc V G sin( , )
{z
g(; k; Vdc; Vk ; Vm; V; k ; m; ;
I; ; Pk ; Pm; P; Qk ; Qm; Q)
q
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
}
where most variables are dened on Figure 3.7, k = 3=8 m, and xc and
f () stand for the dynamic variables and equations of the control system,
respectively.
Dierent kinds of controls can be implemented for various controller variables. The simplest is a PI current controller that directly operates on the
16
Vk δ k
I
Vm δ m
δ
V
θ
a:1
Vi
Zero
Crossing
Switching
Logic
PLL
Magnitude
β
I ref
β
m
(PWM)
Controller
C
V
dc
PWM
Magnitude
Vdc
ref
Figure 3.6: Block diagram of a SSSC with PWM current control.
phase angle . The PWM controller represented on the SSSC gures in this
report, indirectly controls the current I by operating on the phase angle and the capacitor voltage Vdc, i.e., the current is controlled by direct control
of the series voltage V 6 . A more sophisticated dq controller to control the
active and reactive powers on the line is discussed on the next section for the
series branch of a UPFC, which is basically a SSSC.
The steady state model equations, for a PWM controller with no droops,
are then
2
3
I , Iref
0 =
6
6
6
6
Vdc , Vdcref
6
6
6
6
6
P , Vdc2 =RC , R I 2
6
6
6
6
4 g (; k; Vdc ; Vk ; Vm ; V; k ; m; ;
I; ; Pk ; Pm ; P; Qk ; Qm; Q)
7
7
7
7
7
7
7
7
7
7
7
7
7
5
(3.4)
For a phase controller, the dc voltage equation is replace by an equation
17
Vk δk
Pk +jQk
I θ
Magnitude
Pm+jQm
V δ
Vm δm
a:1
aI θ
P+jQ
R+jX
I ref
k Vdc β
Controller
β
k (PWM)
PWM
Vdc
C
RC
Vdc
ref
Magnitude
Figure 3.7: Transient stability model of a SSSC.
dening the variable k. Once again, it is important to properly model the
controller limits in order to have an adequate steady state model of the SSSC.
3.3 UPFC
As shown in Figure 3.8, the UPFC can be viewed as a STATCOM and a
SSSC with a shared dc branch, and the corresponding transient stability
model reects this fact, as shown in Figure 3.9. Thus, the model equations
then can be dened as follows [12]:
2
3
x
_
c
6
6
_ 777
6
; msh; mse; Vk ; Vl; Vdc ;
6
(3.5)
_ 77 = f(kx;c;l;
6
;
P
l
6m
7
ref ; Qlref ; Vkref ; Vdcref )
4 _ sh 5
m_ se
V_dc = VCk VIsh cos(k , sh ) + CVmVIl cos(m , l)
dc
dc
2
1
R
sh Ish Rse Il2
, R C Vdc , C V , C V
C
dc
dc
18
0 =
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
Psh , Vk Ish cos(k , sh)
Qsh , Vk Ish sin(k , sh )
Psh , Vk2 Gsh + ksh Vdc Vk Gsh cos(k , )
+ksh Vdc Vk Bsh sin(k , )
Qsh + Vk2 Bsh , ksh Vdc Vk Bsh cos(k , )
+ksh Vdc Vk Gsh sin(k , )
{z
|
gsh (; ksh ; Vk ; Vdc ; k; Ish; sh ; Psh ; Qsh)
2
3
P
k , Psh , Vk Il cos(k , l)
6
7
0 =
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
|
Qk , Qsh , Vk Il sin(k , l)
Pl , Vm Il cos(m , l)
Ql , Vm Il sin(m , l)
Pk , Pl , Psh , Pse
Qk , Ql , Qsh , Qse
Pse , V 2 Gse + kse Vdc V Gse cos( , )
+kse Vdc V Bse sin( , )
Qse + V 2 Bse , kse Vdc V Bse cos( , )
+kse Vdc V Gse sin( , )
{z
gse (; kse ; Vdc; Vk ; Vl; V; k ; l; ; Il; l;
Pk ; Pl; Psh ; Pse ; Qk ; Ql; Qsh; Qse )
19
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
}
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
}
0 =
2
3
I
cos(
)
,
I
cos(
)
,
I
cos(
)
k
k
sh
sh
l
l
6
7
6
7
6
7
6
I
k sin(k ) , Ish sin(sh ) , Il sin(l ) 7
6
7
6
7
6
7
6
7
6
7
Pk , Vk Ik cos(k , k )
6
7
6
7
4
5
Qk , Vk Ik sin(k , k )
|
{z
}
gcon (Vk ; k ; Ik; Ish; Il; k ; sh; l; Pk ; Qk )
Most of these variables are dened in Figure 3.9. Observe that these equations are basically a combination of the STATCOM and SSSC equations (3.1)
and (3.3). The main dierence is in the corresponding control system equations and variables, represented here by f () and xc, and in the additional set
of algebraic constraints gcon () = 0, which stand for the connection between
the shunt (STATCOM) and series (SSSC) branches of the UPFC.
A control system diagram for the UPFC's shunt and series branches are
depicted in Figures 3.10 and 3.11, respectively. The shunt controller is basically the same one described for the STATCOM above, but without droop.
The series controller, originally proposed in [13], is a PQ controller based
on a dq-axis decomposition to decouple the active and reactive powers of
the inverter [12, 10, 11]; this PQ controller performs better than other PQ
controls proposed in the literature [11]. However, a current control strategy
for the SSSC could be also used in this case.
The steady state model can be obtained from the transient stability model
of equations (3.5) and the corresponding controls, resulting in the following
20
Vk δ k
Vm δ m
Il θl
Ik θk
V
+
δ
Line
ase: 1
I sh θ sh
ash: 1
Vi
sh
Vise β
α
+
C
Vdc
-
Switching
Logic
Switching
Logic
α msh
β mse
UPFC CONTROLLER
Pl
ref
Ql
ref
Vdcref
Vkref Vk δ k Vl δ l Vdc
Figure 3.8: Block diagram of a UPFC.
21
V δl
Pl +jQl l
Vk δ k
Vm δ m
Pk +jQk
V
Il θl
Ik θk
+
δ
-
R l +jX l
ase: 1
I sh θ sh
ase I l θ l
P se+jQ se
Psh+jQ sh
V δl
Pl +jQl l
ash: 1
R sh+jX sh
R se+jX se
+
ksh Vsh α
Pdc
C
Vdc
+
+
α ksh
RC
-
kse Vse
β
kse
β
UPFC CONTROLLER
Pl
ref
Ql
ref
Vdc
ref
Vkref Vk δ k Vl δ l Vdc
Figure 3.9: Transient stability model of a UPFC.
22
msh - m sho
max
V k ref
+
KPac +
KI ac
msh
KM
ac
S
+
+
- m sh
min
msh
msh
o
o
1+ S T M
ac
Vk
Vdc
ref
αmax- αo
+
KPdc +
αmin- αo
KM
dc
KI dc
S
+
α
+
αo
1 + S TM
dc
Vdc
Figure 3.10: Basic shunt branch control of UPFC.
23
Il
d _
Pl
ref
2
/
+
Ild
x1
11
00
00
11
Converter Model
+
KP+ KI /S
ref
_
ωB
01
1
S+K
+
x1
+
11
00
I ld
ωB
Vld
Vl d
ωB
ωB
Ql
Ilq
ref
2
/
+
ref
+
KP+ KI /S
+
_
0110
_
x2
1
S+K
+
01
Ilq
K
RT
XT
Vld
Vkd
Vkq
=
=
=
=
=
=
Vised
=
Viseq
=
Vise
=
mse
=
I lq
x2
RT !B
XT
Rl + Rse
X
pl + Xse
p2 Vl
p2 Vk cos(l , k )
2 Vk sin(l , k )
T
Vkd , Vld , X
!B x1
T
Vkq , X
!B x2
q
p1 Vi2sed + Vi2seq
2
r
=
11
00
00
11
8
3
Vise
Vdc
V
l , tan,1 Viseq
ised
!
Figure 3.11: Basic series branch dq control of UPFC with respect to the bus
voltage Vl 6 l. All variables are in p.u., and !B stands for the fundamental
frequency of the system in rad/s.
24
set of equations:
0 =
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
Vdc , Vdcref
7
7
7
7
7
Pse , Pseref
7
7
7
7
Qse , Qseref
7
7
7
7
2
2
2
Psh , Pse , Vdc =RC , Rsh Ish , Rse Il 777
7
7
gsh (; ksh ; Vk ; Vdc ; k ; Ish; sh; Psh ; Qsh) 77
7
7
gse (; kse ; Vdc; Vk ; Vl; V; k ; l; ; Il; l; 777
Pk ; Pl; Psh ; Pse ; Qk ; Ql; Qsh; Qse) 777
5
gcon (Vk ; k ; Ik ; Ish; Il; k ; sh; l; Pk ; Qk )
Vk , Vkref
(3.6)
As previously mentioned, it is important to properly model the controller
limits to obtain reliable results in steady state studies.
25
Chapter 4
Conclusions
The transient stability and power ow models presented here are based on
models that have been proposed on the current literature, and can be considered as the most adequate and simple models available for voltage and angle
stability studies of networks with these kinds of FACTS controllers.
These models are all based on the assumption that voltages and currents
are sinusoidal, balanced, and operate near fundamental frequency, which are
the typical assumptions in transient stability and power ow studies. Hence,
they have several limitations, especially when studying large system changes
occurring close to these FACTS controllers:
1. These models cannot be reliably used to represent unbalanced system
conditions, as they are all based on balanced voltage and current conditions.
2. Large disturbances that yield voltage and/or currents with high harmonic content, which is usually the case when large faults occur near
power electronics-based controllers, cannot be accurately studied with
these models, as they are all based on the assumptions of having sinusoidal signals.
3. The above also applies for cases where voltage and current signals undergo large frequency deviations.
4. Internal faults as well as some of the internal variables of the controller
cannot be reliably represented with these models.
26
For these cases, detailed EMTP types of studies are required to obtain reliable
results. Observe that these limitations also apply to most models typically
used to represent other devices in transient stability and power ow studies.
27
Bibliography
[1] N. G. Hingorani, \Flexible AC Transmission Systems," IEEE Spectrum,
April 1993, pp. 40{45.
[2] \FACTS Applications," technical report 96TP116-0, IEEE PES, 1996.
[3] Convener Terond, \Modeling of Power Electronics Equipment (FACTS)
in Load Flow and Stability Programs: A Representation Guide for Power
System Planning and Analysis," technical report TF 38-01-08, CIGRE,
September 1998.
[4] N. Christl, R. Heiden, R. Johnson, P. Krause, and A. Montoya, \Power
System Studies and Modeling for the Kayenta 230 KV Substation Advanced Series Compensation," AC and DC Power Transmission IEEE
Conference Publication 5: International Conference on AC and DC
Power Transmission, September 1991, pp. 33{37.
[5] C. A. Ca~nizares and Z. T. Faur, \Analysis of SVC and TCSC Controllers in Voltage Collapse," IEEE Trans. Power Systems, vol. 14, no.
1, February 199, pp. 158{165.
[6] S. G. Jalali, R. A. Hedin, M. Pereira, and K. Sadek, \A Stability Model
for the Advanced Series Compensator (ASC)," IEEE Trans. Power Delivery, vol. 11, no. 2, April 1996, pp. 1128{1137.
[7] E. Uzunovic, C. A. Ca~nizares, and J. Reeve, \Fundamental Frequency
Model of Static Synchronous Compensator," Proc. NAPS, Laramie,
Wyoming, October 1997, pp. 49{54.
[8] C. A. Ca~nizares, E. Uzunovic, J. Reeve, and B. K. Johnson, \Transient Stability Models of Shunt and Series Static Synchronous Compen28
[9]
[10]
[11]
[12]
[13]
sators," submitted for publication in IEEE Trans. Power Delivery and
available upon request, December 1998.
D. N. Koseterev, \Modeling Synchronous Voltage Source Converters in
Transmission System Planning Studies," IEEE Trans. Power Delivery,
vol. 12, no. 2, April 1997, pp. 947{952.
E. Uzunovic, C. A. Ca~nizares, and J. Reeve, \EMTP Studies of UPFC
Power Oscillation Damping," Proc. NAPS, San Luis Obispo, California,
October 1999.
E. Uzunovic, C. A. Ca~nizares, and J. Reeve, \Transient Stability Model
of Unied Power Flow Controllers and Control Comparisons," submitted for publication in IEEE Trans. Power Delivery and available upon
request, November 1999.
E. Uzunovic, C. A. Ca~nizares, and J. Reeve, \Fundamental Frequency
Model of Unied Power Flow Controller," Proc. NAPS, Cleveland, Ohio,
October 1998, pp. 294{299.
I. Papic, P. Zunko, and D. Povh, \Basic Control of Unied Power Flow
Controller," IEEE Trans. Power Systems, vol. 12, no. 4, November 1997,
pp. 1734{1739.
29