73 CHAPTER 4 MODELLING OF UPFC AND STATCOM FORMULTIMACHINE SYSTEM STABILITY 4.1 INTRODUCTION Stability is a condition of equilibrium among opposing forces. The method by which interconnected synchronous machines keep on synchronism with another machine is through restoring forces, which acts when there are forces tending to increase or decrease its speed on one or more machines with respect to other machines. Instability in a power system is described by depending upon the system configuration and operating mode. Generally, the stability problem has been maintaining of synchronous operation. This aspect of stability is influenced by the dynamics of machine rotor angles. But, instability may also be encountered without loss of synchronism. For example, a system can go unstable because of the collapse of load voltage. Maintaining synchronism is not an issue in this instance; instead, the concern is stability and control of voltages. In this work, the discussion is restricted to voltage and rotor angle stability. Rotor angle stability is the ability of interconnected synchronous machines of a power system to remain in synchronism [2]. This stability problem is concerned with the behavior of a synchronous machine after it has been perturbed. Under steady state conditions, there is equilibrium between the input mechanical torque and the developed electrical torque of each machine. This equilibrium is upset during perturbation of the system. The torque unbalance is caused by a change in load, generation or any other network condition. In any case, for the system to be stable all the machines must remain operating in parallel and at the same speed. However, the statement declaring
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73
CHAPTER 4
MODELLING OF UPFC AND STATCOM
FORMULTIMACHINE SYSTEM STABILITY
4.1 INTRODUCTION
Stability is a condition of equilibrium among opposing forces. The
method by which interconnected synchronous machines keep on synchronism
with another machine is through restoring forces, which acts when there are
forces tending to increase or decrease its speed on one or more machines with
respect to other machines. Instability in a power system is described by
depending upon the system configuration and operating mode. Generally, the
stability problem has been maintaining of synchronous operation. This aspect
of stability is influenced by the dynamics of machine rotor angles. But,
instability may also be encountered without loss of synchronism. For
example, a system can go unstable because of the collapse of load voltage.
Maintaining synchronism is not an issue in this instance; instead, the concern
is stability and control of voltages. In this work, the discussion is restricted to
voltage and rotor angle stability.
Rotor angle stability is the ability of interconnected synchronous
machines of a power system to remain in synchronism [2]. This stability
problem is concerned with the behavior of a synchronous machine after it has
been perturbed. Under steady state conditions, there is equilibrium between
the input mechanical torque and the developed electrical torque of each
machine. This equilibrium is upset during perturbation of the system. The
torque unbalance is caused by a change in load, generation or any other
network condition.
In any case, for the system to be stable all the machines must remain
operating in parallel and at the same speed. However, the statement declaring
74
the power system to be stable is not meaningful unless the conditions under
which this stability has been examined are clearly stated. This includes the
operating conditions as well as the type of perturbation (which can be large or
small) given to the system.
Transient stability is the ability of the power system to maintain
synchronism when subjected to a large disturbance [2]. The resulting system
response involves large excursions of generator rotor angles and is influenced
by the nonlinear power-angle relationship. Stability depends on both the
initial operating state of the system and the severity of the disturbance. This
disturbance is usually so large that it alters the post disturbance equilibrium
conditions relative to those existing prior to the disturbance. The work
presented in this thesis is focused on the power system behavior when
subjected to large disturbances and the enhancement of this stability using
FACTS controller.
The most common form of instability between interconnected
generators is loss of synchronism, monotonically, in the first few seconds
following a fault due to lack of synchronizing torque and damping torque.
The first step in a stability study is to make a mathematical model of
the system. The elements included in the model are those affecting the
machine. The complexity of the model depends upon the type of stability
study. Generally, the components of the power system that influence the
electrical and mechanical torques of the machines are included in the model.
Such components are the loads and their characteristics, the network during
the disturbance and the parameters of synchronous machines (such as inertia
of the rotating mass). Thus, the basic requirements for these studies are initial
conditions of the power system prior to the start of the disturbance and the
mathematical description of the main components of the system that might
affect the behavior of synchronous machines.
75
Generally,differential equations are used to describe the various
components. The system equations for small signal stability analysis are
usually nonlinear. The behavior of any dynamic system, such as a power
system, is described by a set of n first order non-linear differential equations
of the form given by (4.1).
t~ ,u~ ,x~f=•~x (4.1)
In Equation (4.1), f is a vector of nonlinear functions. The column
vector~x is referred to as a state vector and
~u is the vector of inputs to the
system. The study of dynamic behavior of the system is based on the nature of
these differential equations.
The first step is to do a load flow study as discussed [2], to obtain the
initial steady state conditions. After establishing initial conditions, a
mathematical model of the power system is formed as discussed in [4]. In this
model the effects of AVR and PSS are taken into account except the effects of
governor. The mathematical model obtained is set of non-linear differential
equations. Solving these equations by using Runge-Kutta method the state
variables are determined. After giving a large disturbance for a particular
period the system is again restored and the behavior of the system is studied at
the same time the effect of change in load and change in mechanical input
power of the system is also studied.
4.2 SYSTEM MODELLING
This section presents the mathematical models used for the power
system components such as generator, exciter and PSS. Mathematical
modeling of STATCOM and UPFC is also discussed in this section. Feed
Back Linearizing Controller (FBLC) modeling and implementation to
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STATCOM and UPFC is also described in this chapter. Further tuning of
FBLC is done using two intelligence techniques Bacterial Foraging Algorithm
and Differential Evolution is discussed.
4.2.1 Synchronous Machine Model
Synchronous machine is represented by means of the single-axis model
[4]. The state variables are rotor angle i , rotor angular velocity, and the
voltage proportional to main field flux linkage, 'qE . The sub transient
reactance, saturation and turbine governor dynamics are neglected.
In developing equations for a mathematical model of a multi-machine
power system, the following assumptions were made
1. Mechanical power input Pm is constant.
2. The mechanical rotor angle of a machine coincides with the angleofthe
voltage behind the transient reactance.
3. Loads are represented by passive impedance.
The resulting differential – algebraic equations for the ‘m’ machine,
‘n’ bus system with exciter model is given below as the state equations in p.u.
dididiqifdiqi ixxEE
dtdE
)'(''
T'doi (4.2)
midt
di
i ,.....,11 (4.3)
miiixxiETdt
djqididiqiqiqmi
i ,...,1/})''('{ (4.4)
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Where j - fHs
id - direct axis current
iq - quadrature axis current
xd - direct axis synchronous reactance
xd’ - direct axis transient reactance
xq - quadrature axis synchronous reactance
xq’ - quadrature axis transient reactance
Eq’ - Voltage proportional to main field flux linkage
T’do - direct axis open circuit time constant
Efd - Equivalent stator emf corresponding to field voltage
Hs - inertia constant of synchronous machine
Tm - Mechanical torqueof synchronous machine
Equation (4.4) has dimensions of torque in per-unit.When the stator
transients were neglected, the electrical torque became equal to the per-unit
power associated with the internal voltage source. The dynamic performances
of STATCOM and UPFC have been analyzed with different types of
disturbance thus damping has not been included in equation 4.4, because
STACOM and UPFC can improve the damping of the system. The system
data have been given in Appendix 3.
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4.2.2 Algebraic Equation The stator algebraic equation can be written in the form of
m1,...,i
ejEix-xE-
)eji)(ijx(ReV0
)2
-j'qiqi
'di
'qi
'di
)2
-j
qidi'disii
i
ii
(4.5)
Where 'diE = 0, since one axis model has been considered.
4.2.3 Excitation System and PSS Model
IEEE Type 1S [4] excitation system model is considered in the sample
system. The block diagram of the excitation system with PSS is shown in
Figure 4.1. The state equations are given below.
1V•
= (Vt-V1)/TR (4.6)
Efd = (KAVe- Efd)/TA (4.7)
3
•
V = {[KF(KAVe-Efd)/TA]-V3}/TF (4.8)
Where Ve - VREF+VF-V1-V3+ upss (4.9)
Vt - terminal voltage of synchronous machine
V1 - output signal of filter
V3 - output signal of stabilizing circuit
VR - regulator output signal
VF - supplementary stabilizing signal
VREF - regulator reference voltage
Ve - error voltage
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KA - gain of amplifier excitation system
KF - gain of stabilizing feedback
TF - time constant of stabilizing feedback
TA - time constant of amplifier of excitation system
TR - regulator time constant
upss - PSS output signal.
The PSS is represented by a washout filter and a cascade of lead lag
controllers of the following form
2
1
11
1 sTsT
sTwsTwKU stabpss (4.10)
Where, the rotor speed deviation is taken as the input to the PSS. Kstab, T1
and T2 are stabilizer gain and time constants respectively.
The parameters Kstab, T1 and T2 are to be determined to enhance the
system damping for the electromechanical mode.Practically, the washout
block has little phase compensation effect and its time constant Tw is fixed to
ten seconds in advance. The Exciter, PSS data for 3 machine, 9-bus system
are given in Table A 3.5 and Table A 3.6 of Appendix 3.
Figure 4.1Block diagram of IEEE Type-1s Excitation model with PSS
80
VjQ-P
=y 2i
LiLiLi
4.2.4 Internal Node Model
This is a widely used reduced-order multi-machine model in first-
swing transient stability analysis. In this model, the loads are assumed to be
constant impedances and converted to admittances as [41]
n1,....,i (4.11)
Where Liy load admittance at ith bus, Vi – Voltage at ith bus
LiP - real power load at ith bus and LiQ - reactive power load at ith bus.
There is a negative sign for Liy , since loads are assumed as injected
quantities.
(4.12)
Where,
'
1
dijXDiagy i = 1,2,….., m (4.13)
m -no of machines
and 000
1yYY NN (4.14)
y y 0
y1NY
0
augY
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If transmission line resistances are neglected, then the network
admittance matrix is
ijN BjY (4.15)
Where Bij=suspectance betweenith and jth bus.
Adding Liy to the diagonal elements of the N1Y matrix and makes it )(12 LiNN yDiagYY .
The modified augmented Y matrix becomes
(4.16)
Figure 4.2 Augmented Y matrix with constant impedance
y y 0
y 2NY
0
augnewY
82
The passive portion of the network is shown in Figure 4.2. The
network equations for the new augmented network can be written as
B
A
DC
BAA
VE
YYYY
0I (4.17)
where yYA ,
0YB y ,
0y-YC and 2DY NY
The n network buses can be eliminated, since there is no current
injection at these buses. Thus
ACD1
BAA EYYYYI (4.18)
AintA EI Y
where ,the elements of AI and AE are respectively,
QiDi2
j
qidii jIIejIIIi
and iii EE . (4.19)
Where i = 1,…,m.
intY = CD1
BA YYYY (4.20)
The elements of intY are ijijij jBGY . Since the network buses have
been eliminated, the internal nodes are such as 1,.., m, for ease of notation.
j
1i EI
m
jijY
83
i= 1, ……, m (4.21)
Real electrical power out of the internal node ‘i’ from Figure 4.2 is given by
i*
iei IEReP
m
1j
*j
*ijiei EYeEReP i
m
1j
-jijijiei
ji eEBGeEReP j
m
1jjiijijei sincosEEBGReP jiji jj
(4.22)
4.3 INTERFACING OF STATCOM AND UPFC
For 3 machines, 9-bus system the STATCOM has been connected at
8th bus, which is a load bus and UPFC has been connected between line 7-8
which is shown in Figure 4.3 and 4.4
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Figure 4.3 Line diagrams of 3 machines, 9-bus system with STATCOM
Figure 4.4 Line diagrams of 3 machines, 9-bus system with UPFC
4.3.1 Modelling of UPFC for Multi Machine System
The mathematical model of the UPFC is derived here in the d-q
(synchronously rotating at the system angular frequency ) frame of
reference. This is followed by a detailed description of the conventional PI
control strategy employed for active and reactive power control using UPFC.
The equivalent circuit model of a power system equipped with a UPFC
is shown in figure A3.2 inAppendix 3. The series and shunt VSIs are
represented by controllable voltage sources Vc and Vp, respectively.Rp and Lp
represent the resistance and leakage reactance of the shunt transformer
respectively. Leakage reactance and resistance of series transformer have
been neglected.
The mathematical model of UPFC is derived by performing standard
d-q transformation [4] of the current through the shunt transformer and series
85
transformer. They are as given below ( is the angular frequency of the
voltages and currents).
4.3.2 Modeling of Shunt Converter
The dynamic equations governing the instantaneous values of the
three-phase voltages across the two sides of STATCOM and the current
flowing into it are given by [40]
R + L i = V V (4.23)
= R + V V (4.24)
Where:i = i i i , V = [V V V ] , V = V V V
= 0 0
0 00 0
and = 0 0
0 00 0
Under the assumption that the system has no zero sequence
components, all currents and voltages can be uniquely represented by
equivalent space phasors and then transformed into the synchronous d-q-o
frame by applying the following transformation (q is the angle between the d-
axis and reference phase axis):
=
cos cos +
sin sin sin + (4.25)
Thus, the transformed dynamic equations are given by,
= R + V V + i (4.26a)
86
= R + V V i (4.26b)
Where, is the angular frequency of the AC bus voltage.
4.3.3 Cascade Control Strategy for Shunt Converter
The conventional control strategy for this inverter concerns with the
control of ac-bus and dc-link voltage. The dual control objectives are met by
generating appropriate current reference (for d and q axis) and, then, by
regulating those currents. PI controllers are conventionally employed for both
the tasks while attempting to decouple the d and q axis current
regulators.In this study, the strategy adopted in [40] [41] for shunt current
control has been taken.The inverter current ( pi ) is split into real (in phase with
ac-bus voltage) and reactive components.The reference value for the real
current is decided so that the capacitor voltage is regulated by power
balance.The reference for reactive component is determined by ac-bus voltage
regulator.As per the strategy, the original currents in d-q frame )i ,i( pqpd are
now transformed into another frame, qd frame, where d axis coincides
with the ac-bus voltage (Vs), as shown in Figure 4.5.
Figure 4.5 Phasor diagram showing d-q and d’-q’ frame
Thus, in qd frame, the currents dpi and qpi represent the real and
reactive currents and they are given by:
87
= cos + sin (4.27)
= cos sin (4.28)
Now, for current control, the same procedure as outlined in [57] has
been adopted by re-expressing the above differential equations as:
= R + V V + i (4.29)
= R + V i (4.30)
Where
= cos + sin (4.31)
= cos sin (4.32)
= +
The VSI controlled voltages are as follows:
V = L i + L u (4.33)
V = L i + V L u (4.34)
By putting the above expressions for dpV and qpV in equations (4.29)
and (4.30) the following set of decoupled equations are obtained.
= i + u (4.35)
= i + u (4.36)
88
Conventionally, the control signals du and qu are determined by
linear PI controllers. The complete cascade control architecture is shown
below in Figure 4.6, where dpqiqpicpcitpt K,K,K,K,K,K,K and diK are the
respective gains of the PI controllers.
In this study, the above design has been used for demonstration of
STATCOM control.This approach leads to good control as illustrated by the
simulation results.
The final and important stage of the design of PI based STATCOM
involves tuning of parameters of STATCOM, which is posed as an
optimization problem. In this problem the optimal output gain K0 are
determined by maximizing the damping out of transient voltage oscillations of
the load bus voltage and dc capacitor voltage being controlled. This is in
effect carried out by minimizing Sum Squared Deviation (SSD) of the load
bus voltage and dc capacitor voltage being controlled from the desired value
through non-linear simulation of power system under typical operating
condition and disturbance. The non-linear simulation is carried out using a
Transient Stability Algorithm [4] employing a Runge-Kutta fourth order
method. To get the original currents it is again transformed to d -q
Figure 4.6 PI-Control Structure of STATCOM
4.3.4 Modeling of Series Converter
= + ( sin ) (4.37)
89
= + cos (4.39)
For fast voltage control, the net input power should instantaneously
meet the charging rate of the capacitor energy. Thus, by power balance,
= + + + ( + )
= Vdcidc= +
= + + + ( ) + (4.40)
An appropriate series voltage (both magnitude and phase) should be
injected for obtaining the commanded active and reactive power flow in the
transmission line, i.e., uu Q ,P in this control. The current references are
computed from the desired power references and are given by,
= (4.41)
= (4.42)
The power flow control is then realized by using appropriately
designed controllers to force the line currents to track their respective
reference values. Conventionally, two separate PI controllers are used for this
purpose. These controllers output give the amount of series injected voltages
)V ,V( cqcd . The corresponding control system diagram is shown in Figure 4.7.
P I cqi
Equations (4.39) & (4.40)
sK
K idpd
sK
K iqpq
P I
refP
refQ
refcqi
refcdi
cdi
+
_
_
+
cdV
cqV
90
Figure 4.7 PI Control of Series Converter
4.4 IMPLEMENTATION OF FBLC FOR UPFC IN SMIB
ANDMULTI MACHINE
4.4.1 Implementation of FBLC for UPFC in SMIB
In this section, the design steps for the feedback linearizing control of
UPFC have been presented followed by simulation results under various
transient disturbances. A brief review of nonlinear control using feedback
linearization is presented in the Appendix 2.
4.4.2 FBLC design
In UPFC control, there are four objectives. They are (i) active power
control, (ii) reactive power control, (iii) ac-bus voltage (Vs) control and (iv)
dc link voltage (Vdc) control. Tracking of active and reactive power are
indirectly translated to tracking of line currents to their respective reference
values computed from Pref and Qref.The differential equations of the line
currents (ibd, ibq) and Vdcare already derived. Thus, for control of Vs, its
differential equation need to be derived, i.e, Vsis taken as an additional state
in this control design.
Now, for the control design, the complete state space model is
expressed in the form of Equations. (A.1) and (A.2) in Appendix 1 as follows: