Ricerca
Area Trasmissione e Dispacciamento
Modeling and Implementation of TCR
and VSI Based FACTS Controllers
C. Ca~
nizares
University of Waterloo, Waterloo, ON, Canada
S. Corsi, M. Pozzi
ENEL, Ricerca, Area Trasmissione e Dispacciamento
Dicembre 1999
AT-Unita Controllo e Regolazione n 99/595
Contents
1 Introduction
2 Modeling TCR-based Controllers
1
3
2.1 SVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 TCSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 VSI-based Controllers
10
4 Implementation and Results
27
5 Conclusions
39
3.1 STATCOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 SSSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 UPFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.1 Test System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 UWPFLOW Results . . . . . . . . . . . . . . . . . . . . . . . 31
4.3 EASY5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 32
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
4.1
4.2
4.3
4.4
4.5
4.6
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. . . . . . . . . . . . .
Test system. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
EMTP results for the test system. . . . . . . . . . . . . . . . .
EMTP generator AVR model for the test system. . . . . . . .
EMTP phase control for the STATCOM in the test system. . .
PV curves for the test system obtained with UWPFLOW for
the STATCOM operating under phase control. . . . . . . . . .
PV curves for the test system obtained with UWPFLOW for
the STATCOM operating under PWM control. . . . . . . . . .
2
5
5
6
6
7
8
8
11
12
14
16
17
18
19
22
23
24
25
28
29
30
30
32
33
4.7 STATCOM phase controller in EASY5. . . . . . . . . . . . . .
4.8 STATCOM PWM controller in EASY5. . . . . . . . . . . . . .
4.9 Fault simulation results for the test system obtained with
EASY5 for the STATCOM operating under phase control. . .
4.10 Fault simulation results for the test system obtained with
EASY5 for the STATCOM operating under PWM control. . .
3
35
36
37
38
List of Tables
4.1 Generator data in p.u. with respect to a 200 MVA and 13.8
kV base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Transmission system data in p.u. with respect to a 240 MVA
base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 SVC data in p.u. with respect to a 150 MVA base. . . . . . . . 31
4
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. Appropriate models for voltage and angle stability studies are discussed in detail 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). Finally, the implementation of the STACOM model
into steady sate and transient stability programs is discussed through the results obtained from stability studies of a simple test system that includes a
STATCOM controller.
Chapter 1
Introduction
The relatively 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 FACTS in static
and dynamic analyses [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.
Details of the implementation and use of the STATCOM model proposed
here in two programs that can be used for steady state and transient stability
analyses of power systems are discussed in this report. The programs where
the model has been implemented are UWPFLOW [4], a program designed
for voltage stability analysis of power systems, and EASY5 [5], a program
designed for modeling general linear and nonlinear control systems that has
been successfully used at ENEL for validation studies of models as well as
steady state and transient stability analyses of small power systems. The
results of using these programs to study the stability of a simple test system
are also presented here.
Chapter 2 describes in detail the models for TCR-based controllers, concentrating specically on the SVC and TCSC, and Chapter 3 discusses in
1
detail the models for VSI-based controllers, namely, the STATCOM, the
SSSC and the UPFC. In Chapter 4, the implementation of the STATCOM
model into two programs used for steady state stability and transient stability of power systems are discussed, and the results of using these programs to
study the stability of a simple power system are presented. Finally, Chapter 5
summarizes the implementation issues associated with the models explained
on this report and discusses their limitations.
2
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 [6]. 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)
3
(2.1)
3
7
7
7
7
7
7
7
5
}
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. It is important
to highlight the fact that the current variable I used in this model does not
only represent the current magnitude, as it is allowed to become negative
when the controller delivers reactive power (capacitive mode). This is done
to simplify the modeling, and is possible due to the fact that the model is
purely reactive (no active power ows in this model).
The dierential equations represented by f () in (2.1) vary with the type
of control system used. Figure 2.3 depicts a typical 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 mention 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 [7]. 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
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 [7].
It is important to point out that the Jacobean of equations (2.2) becomes
singular for = 180 ; hence, when solving these equations, the value of alpha
must be kept strictly below 180 , i.e., max < 180 .
For the steady state model to be complete, all SVC controller limits should
be adequately represented. The proper handling of ring angle limits is
4
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.
V
I
Filters
Q
a :1
Vi
Magnitude
Vref
Controller
α
Be(α)
Figure 2.2: Transient stability model of a SVC.
5
αmax − αo
KM
V
1+ S TM
∆α
K (1+ S T1 )
-
KD+ S T2
+
V
α
+
αmin− αo
ref
+
αo
Figure 2.3: Basic SVC controller model for voltage control.
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.
o , until
depicted in Figure 2.5 [7], where Vref is kept xed, say at a value Vref
reaches a limit, at which point Vref is allowed to changed while is kept
at its limit value; voltage control is regained when Vref returns to the xed
o .
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,
6
α < αmin
α = αmin
o
Vref >Vref
αmin< α < αmax
o
Vref <Vref
o
Vref =Vref
α > αmax
o
Vref >Vref
α = αmax
o
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 [7, 8]:
"
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
4q
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 and f () stand for the
internal control system variables and equations, respectively; and
Be ()
=
kx4 , 2kx2 + 1
h
cos
kx ( , )=
XC kx4 cos kx( , )
, cos kx( , ) , 2 kx4 cos kx( , )
2 cos kx ( , ) , k4 sin 2 cos kx( , )
+2 kx
x
2
3 2
+kx sin 2 cos kx ( , ) , 4 kx cos sin kx ( , )
i
,4 kx2 cos sin cos kx( , )
s
kx
=
XC
XL
7
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-
8
merical point of view [7], 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 realistically represent its operation
[7].
9
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 [9]. 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.
10
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.
11
V δ
Filters
θ
I
P+jQ
Magnitude
a:1
R+jX
Vref
k Vdc
α
Controller
α
k (PWM)
Vdc
PWM
C
RC
Vdc
ref
Magnitude
Figure 3.2: Transient stability model of a STATCOM with PWM voltage
control.
12
Assuming balanced, fundamental frequency voltages, the controller can
be accurately represented in transient stability studies using the basic model
shown in Figure 3.2 [10, 11, 12]. 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_
2
V_dc = CV VI cos( , ) , GCC Vdc , CR VI
dc
dc
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; GC = 1=RC ; G +
jB = (R + jX ),1, which is used to represent the transformer impedance, any
ac series lters, and the \switching
inertia" of the inverter due to its high
q
frequency switching; k = 3=8 m, and hence is directly proportional to the
modulation index m; and xc and f () stand for the internal control system
variables and equations, respectively.
A simple PWM voltage controller is shown in Figure 3.3 [13, 14], 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. Typically, the modulation index control would be \faster" than the phase angle control, as there
is a signicant charging and discharging \inertia" of the capacitor due to its
relative large value, whereas the modulation index has an immediate eect
13
I max
Vref
K ( 1 + S T1 )
KD+ S T 2
+
-
+
+
I min
KM
ac
m
mo
1 + S TM
ac
V
Vdc
Vdcref
+
KP +
Vdc
KM
dc
min
KI
S
max
+
α
+
δ
1 + S TM
dc
Vdc
Figure 3.3: Basic STATCOM PWM voltage control.
on the output voltage of the controller.
The controller limits are directly dened in terms of the GTO current
limits, which are the basic limiting factor in VSI-based controllers, for the
modulation index, and the dc voltage for the phase angle. This direct implementation of limits allows to closely duplicate the steady state V-I characteristics of the controller shown in Figure 3.4, as well as adequately representing
the way these limits are actually implemented in a real controller. In simulations, the integrator blocks are \stopped" whenever the converter current
I or dc voltage Vdc reach a limit. A simpler way to simulate these limits is
to determine the values of the modulation index m and phase angle corresponding to the current and dc voltage limits, respectively, by solving the
steady state equations of the converter, as discussed in [15].
14
Note that the controllers have a bias, which corresponds to the steady
state value of the modulation index mo for the voltage magnitude controller,
and to the phase angle of the output voltage of the STATCOM for the
dc voltage controller (notice that this value changes as the system variables
change during the simulation). Although the latter complicates the simulation, it is needed to guarantee a direct control of the charging and discharging
of the capacitor, which basically depends on the power ow between the VSI
and the STATCOM bus, i.e., it depends on ( , ). This can be simplied by
setting the bias of the dc voltage control to the constant value o = o, where
o stands for the bus voltage phase-shift when the STATCOM is disconnected
from the system, as suggested in [15].
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
limits. Hence, the steady state equations for the PWM controller are
0 =
2
V , Vref XSL I
6
6
6
6
Vdc , Vdcref
6
6
6
6
6
P , GC Vdc2 , R I 2
6
6
4
g(; k; V; Vdc ; ; I; ; P; Q)
3
7
7
7
7
7
7
7
7
7
7
7
5
(3.2)
where on the rst equation, the positive sign is used when the device is
operating on the capacitive mode (Q < 0) and negative for the inductive
mode (Q > 0). 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. This type of controller is used in
the example discussed in Chapter 4.
These equations can be directly used to compute the control steady state
values and biases as well as its limits, as previously mentioned. Thus, the
modulation index bias and steady state is determined by setting I = 0,
15
V
X SL
Vref
(mo ,α o )
I (Q<0)
I
min
Capacitive
Inductive
Imax
I (Q>0)
Figure 3.4: Typical steady state V-I characteristics of a STATCOM.
yielding
mo =
s
8 Vref
3 Vdcref
| {z }
ko
Modulation index and the phase-shift control limits can be computed by
solving equations (3.2) [15].
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 needed to properly model FACTS controllers in steady state studies
[7]. 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, as shown in Figure 3.5.
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
16
I > I max
Q> 0
I = I max
o
Vref >Vref
o
Vref <Vref
I < I max &Q > 0
I < I min &Q< 0
o
Vref =Vref
I > I min
Q< 0
I = I min
o
o
Vref >Vref
0 < Vref < Vref
Figure 3.5: Handling of limits in the STATCOM steady state model.
is shown in Figure 3.7 [11], and can be represented by the following p.u.
equations:
2
x_ c 3
6 _ 7
4 5
m_
= f (xc; ; m; I; Vdc; Iref ; Vdcref )
2
V_dc = CV VI cos( , ) , GCC Vdc , CR VI
dc
dc
2
Pk , Vk I cos(k , )
6
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)
17
(3.3)
3
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
7
5
}
Vk δ k
I
Vm δ m
δ
V
θ
a:1
Vi
Zero
Crossing
Switching
Logic
PLL
Magnitude
I ref
β
β
C
m
(PWM)
Controller
V
dc
PWM
Magnitude
Vdc
ref
Figure 3.6: Block diagram of a SSSC with PWM current control.
q
where most variables are dened on Figure 3.7, k = 3=8 m, and xc and f ()
stand for the internal 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
phase angle . The PWM controller represented on the SSSC block diagrams 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,
18
Pk +jQk
Vk δk
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.
are then
0 =
2
I , Iref
6
6
6
6
Vdc , Vdcref
6
6
6
6
6
P , GC Vdc2 , R I 2
6
6
6
6
4 g (; k; Vdc ; Vk ; Vm ; V; k ; m; ;
I; ; Pk ; Pm ; P; Qk ; Qm; Q)
3
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
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; the corresponding transient stability model
19
directly reects this fact, as shown in Figure 3.9. Thus, the model equations
can be dened as follows [16]:
2
x_ c
6
6
_
6
6
_
6
6m
4 _ sh
m_ se
3
7
7
7
7
7
7
5
; msh; mse; Vk ; Vl; Vdc ;
= f(x;c; ;
;
P
k l lref ; Qlref ; Vkref ; Vdcref )
V_dc = VCk VIsh cos(k , sh ) + CVmVIl cos(m , l)
dc
dc
2
R
R
G
sh Ish
se Il2
C
, C Vdc , C V , C V
dc
dc
2
Psh , Vk Ish cos(k , sh)
0 =
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
|
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)
20
(3.5)
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
}
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
6
6
6
4
Pk , Psh , Vk Il cos(k , l)
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 )
2
3
I
cos(
)
,
I
cos(
)
,
I
cos(
)
k
k
sh
sh
l
l
6
7
0 =
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
P
,
V
I
cos(
,
)
k
k k
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 )
3
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
7
5
}
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 physical interconnection 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 basi21
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.
22
V δl
Pl +jQl l
Vk δ k
Vm δ m
Pk +jQk
V
Il θl
Ik θk
I sh
+
δ
-
R l +jX l
ase: 1
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.
23
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.
cally the same one described for the STATCOM above, but without droop.
The series controller, originally proposed in [17], is a PQ controller based
on a dq-axis decomposition to decouple the active and reactive powers of
the inverter [13, 14, 16]; this PQ controller performs better than other PQ
controls proposed in the literature [14]. However, a current control strategy
for the SSSC could also be 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
24
11
00
00
11
Il
d _
Pl
ref
2
/
+
Ild
x1
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.
25
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 , GC Vdc , 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.
26
Chapter 4
Implementation and Results
The STATCOM model described here was programmed into two software
packages that can be used for the stability analysis of power systems, namely,
UWPFLOW and EASY5. The results obtained with these programs were
compared to the Electromagnetic Transient Program (EMTP) results presented in [11] for the same test system to validate the implementation of the
model. Both of these programs were then used to study the stability of the
test system. The details of the STATCOM model implementation on these
programs and the results obtained with them are discussed in this chapter.
4.1 Test System
The test system depicted in Figure 4.1 is used here to validate implementation
of the STATCOM model into the programs UWPFLOW and EASY5. This
is the same system used in [11] to validate the STATCOM model presented
in this report. Thus, in [11], the EMTP [18] is used to compare the results
obtained with a detailed switching model of the STATCOM versus the results
obtained for the model described here for the controller operating under phase
control. The results of this comparison are depicted on Figure 4.2; observe
how close the results are for the detailed and reduced models for a 3-phase
fault at Bus 6 applied at 4.5s and cleared by opening the breaker 3-5 at 4.65s.
All the data and controls required for typical stability studies of the given
test system were extracted from the detailed 3-phase EMTP data of the
system, and are depicted in Figures 4.3 and 4.4, and Tables 4.1, 4.2 and 4.3.
27
Bus 3
0110
10
Bus 5
Bus 4
245.5 MW
Generator
13.8 kV
Bus 7
Bus 6
15 mi
Transf.
238 MW
16 Mvar
3 Phase Fault
Bus 1
Bus 2
Bus 8
90 mi
125 mi
Bus 9
Bus 10
15 mi
Transf.
Infinite bus
Z Thevenin
238-321-384 MW
16-43-64 MVar
Filter
230 kV
STATCOM
Figure 4.1: Test system.
Variable Value [p.u./s] Variable Value [p.u./s]
Poles
2
H
2.7113
Ra
0.001096
D
0
Xl
0.15
Td0o
6.19488
0
Xd
1.7
Tqo
0
00
Xq
1.64
Tdo
0.028716
Xd00
0.238324
Tq00o
0.07496
00
00
Xd
0.18469
Xq
0.185151
Table 4.1: Generator data in p.u. with respect to a 200 MVA and 13.8 kV
base.
28
Generator Phase Angle
degrees
50
0
−50
4
4.2
4.4
4.6
4.8
5
5.2
5.4
Generator Terminal Voltage
5.6
5.8
6
4.2
4.4
4.6
4.8
5
5.2
Load Voltage
5.4
5.6
5.8
6
4.2
4.4
4.6
4.8
5
5.2
Statcom DC Voltage
5.4
5.6
5.8
6
4.2
4.4
4.6
4.8
5
Alpha
5.2
5.4
5.6
5.8
6
4.2
4.4
4.6
4.8
5
5.2
5.4
5.6
5.8
6
1.3
p.u.
1
0.5
0
4
p.u.
1.5
1
0.5
0
4
kV
15
10
5
0
4
degrees
10
0
−10
4
Figure 4.2: EMTP results for the test system with the STATCOM operating
under phase control [11]. The continuous lines correspond to the detailed
EMTP model, whereas the dashed lines correspond to the simplied transient
stability model.
29
V4
1
1 + S 0.001
1
+
750
+
(0.11+S 0.01)(0.17+S 0.95)
Vf
+
-
1
S 0.04
1+S
Figure 4.3: EMTP generator AVR model for the test system.
V8
1
-
+
∆α max= 10o
3.9652 (1.5 + S 0.0115)
+
(1+S 0.001)(11+S 0.03)
∆α max=-10o
α
+
δ8
Figure 4.4: EMTP STATCOM phase control for the test system.
Element R
X
B=2
1-2
0.0003 0.0684 0
2-3
0.0159 0.2275 0.0754
3-8
0.022 0.316 0.1047
4-3
0
0.08 0
5-6
0.0026 0.0379 0.0126
6-7
0
0.12 0
8-9
0.0026 0.0379 0.0126
9-10
0
0.12 0
Filter 0.0087 -4.3
0
Table 4.2: Transmission system data in p.u. with respect to a 240 MVA base.
30
Variable
Value
R
0
X
0.145
GC
0.0017
C
0.0432
k (Phase)
0.9
Vdcref (PWM) 1
Imax
1
Imin
1
Table 4.3: SVC data in p.u. with respect to a 150 MVA base.
4.2 UWPFLOW Results
The program UWPFLOW, as described in [4], is a tool that can be used
to determine the steady state of power systems at various system conditions, and hence can be used to partially study the steady state stability
of these systems, especially their voltage stability with respect to various
system changes, in particular with respect to load changes. UWPFLOW is
basically a continuation power ow program with fairly detailed steady state
models of generators and HVDC links, including some of their controllers
and corresponding limits. It also contains the SVC and TCSC controller
models described in this report to represent the most popular TCR-based
FACTS controllers in power systems. This program was used in [7] to study
a variety of voltage stability issues in realistic systems with SVC and TCSC
controllers.
The model corresponding to equations (3.2) was programmed into UWPFLOW. To validate the model, the results obtained were successfully compared to those obtained with the EMTP in [11] and those obtained with
EASY5, as discussed below, for the test system of Figure 4.1. Once the
model was validated, this program was used to obtain the PV curves and
maximum loading conditions for this system for phase and PWM controls of
the STATCOM as shown in Figures 4.5 and 4.6. As expected, the loading
margin and voltage proles of the system are signicantly improved by the
introduction of the STATCOM, especially for the phase control mode, as the
dc voltage is free to change while the current is within its limits, whereas in
PWM control mode the dc voltage is kept constant at a value that could be
31
Profiles
250
200
kVBUS 3
kVBUS 7
kVBUS 8
kV
150
230.
230.
230.
BUS 10 230.
100
50
0
0
0.1
0.2
0.3
0.4
0.5
L.F. [p.u.]
0.6
0.7
0.8
0.9
Figure 4.5: PV curves for the test system obtained with UWPFLOW for the
STATCOM operating under phase control.
low for the system, as it is the case here. In both cases, the current limits
are reached before the maximum loading point.
4.3 EASY5 Results
EASY5 is a program by BOEING used for the simulation of linear and nonlinear control systems [5]. This program allows to graphically represent any
linear and nonlinear control system through the denition of its equations
and the icons associated with these equations. Linear matrix analysis tools
and nonlinear integration tools allow for the analysis of the steady state as
well as the transient stability of any control system dened by the user. Libraries can be readily developed, so that new systems and elements can be
easily dened and integrated into other simulations. The dierent element
32
Profiles
250
200
kVBUS 3 230.
kVBUS 7 230.
kVBUS 8 230.
kVBUS 10 230.
150
100
50
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
L.F. [p.u.]
Figure 4.6: PV curves for the test system obtained with UWPFLOW for the
STATCOM operating under PWM control.
33
models and the connections between these elements in a given system must
be dened together with the numerical analysis techniques required for its
analysis. Thus, as with SIMULINK-MATLAB, the program can be used to
graphically model power systems, which are basically complex control systems. This program has been successfully used at ENEL to model and test
a variety of power system devices [19].
The model corresponding to equations (3.1) for a phase and PWM controllers were basically implemented in EASY5. The STATCOM phase and
PWM controllers are depicted in Figures 4.7 and 4.8, respectively. The generator AVR and STATCOM phase controllers implemented in this program
are not exactly the same as the ones used in the EMTP for the test system,
particularly for the STATCOM phase and PWM controllers, which are signicantly dierent from the ones discussed in Chapter 3 in the way the limits
are implemented. The main reason for these dierences, particularly in the
implementation of the STATCOM limits, is to improve the dynamic voltage
control characteristics of the system controllers. Observe that a transient
overload of the STATCOM is allowed, thus improving its dynamic response;
however, this does not aect the steady state behavior of the device.
The results of using this program and the corresponding voltage controls to model the test system fault depicted in Figure 4.1 for both types of
STATCOM controllers are shown in Figures 4.9 and Figures 4.10. As with
the EMTP, a 3-phase fault is applied at 4.5s and cleared at 4.65s by opening
the breaker 3-5. Observe that the results are qualitatively the same as those
obtained with the EMTP, which basically validates the models. There are
some dierences, particularly in the magnitude values, as the system models
are not identical and that the generator loading conditions are not the same.
Other FACTS controller models presented here are currently being implemented into EASY5.
34
Vdc =1.2
+
max
0.25
-
I max=-1
Vdc
+
M
I
N
0.2
-
1 + S 0.5
S 0.5
Q/V
Vdc =0.8
+
-Imin=-1
Vdc
+
0.2
-
M
A
X
Q/V
Vref= 1
+
α
+
δ8o
0.25
min
-
0.25
V8
Figure 4.7: STATCOM phase controller in EASY5.
35
Vdc =1.2
-
0.25
max
+
Imax
=-1
Vdc
+
M
I
N
0.2
-
1 + S 0.01
S 0.01
+
m
+
Q/V
Vdc =0.8
-
mo
0.25
min
+
-Imin=-1
Vdc
+
M
A
X
0.2
Q/V
Vref= 1
+
0.25
V8
Vdc = 1
+
1 + S 0.5
0.25
ref
S 0.5
Vdc
-
α
+
δ8o
Figure 4.8: STATCOM PWM controller in EASY5.
36
0.425
Generator Phase Angle
0.4
radians
0.375
0.35
0.325
0.3
0.275
4
1.02
4.5
5
s
5.5
6
5
s
5.5
6
4.5
5
s
5.5
6
4.5
5
s
5.5
6
4.5
5
s
5.5
6
4.5
5
s
5.5
6
Generator Terminal Voltage
1
p.u.
0.98
0.96
0.94
0.92
4
4.5
Generator Active Power
1
0.95
p.u.
0.9
0.85
0.8
0.75
0.7
4
1.11
Load Voltage
1.08
p.u.
1.05
1.02
0.99
0.96
0.93
0.9
4
9.3
Statcom DC Voltage
9
kV
8.7
8.4
8.1
7.8
7.5
4
0.02
Alpha
radians
0.01
0
-0.01
-0.02
4
Figure 4.9: Fault simulation results for the test system obtained with EASY5
for the STATCOM operating under phase control.
37
0.425
Generator Phase Angle
0.4
radians
0.375
0.35
0.325
0.3
0.275
4
1.02
4.5
5
s
5.5
6
5
s
5.5
6
4.5
5
s
5.5
6
4.5
5
s
5.5
6
4.5
5
s
5.5
6
4.5
5
s
5.5
6
Generator Terminal Voltage
1
p.u.
0.98
0.96
0.94
0.92
4
4.5
Generator Active Power
1
0.95
p.u.
0.9
0.85
0.8
0.75
0.7
4
1.11
Load Voltage
1.08
p.u.
1.05
1.02
0.99
0.96
0.93
0.9
4
8.6
Statcom DC Voltage
8.4
kV
8.2
8
7.8
7.6
7.4
4
0.02
Alpha
radians
0.01
0
-0.01
-0.02
4
Figure 4.10: Fault simulation results for the test system obtained with
EASY5 for the STATCOM operating under PWM control.
38
Chapter 5
Conclusions
The transient stability and power ow models presented in this report 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. The implementation of the STATCOM model into a couple of
stability analysis tools and the results obtained here for a simple test system
show how these models can be easily and reliably used for stability studies
in power systems.
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,
the models 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 to cases where voltage and current signals undergo large frequency deviations.
39
4. Internal faults as well as some of the internal variables of the controller
cannot be reliably represented with these models.
For all of 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 a variety of devices in transient stability and
power ow studies.
40
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