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Design of power system stabilizers using modern control theory

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Title:
Design of power system stabilizers using modern control theory
Creator:
Kang, Seok-Yoon
Publication Date:
Language:
English
Physical Description:
vii, 42 leaves : illustrations ; 29 cm

Thesis/Dissertation Information

Degree:
Master's ( Master of science)
Degree Grantor:
University of Colorado Denver
Degree Divisions:
Department of Electrical Engineering, CU Denver
Degree Disciplines:
Electrical engineering

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Subjects / Keywords:
Electric power systems -- Control ( lcsh )
Electric power systems -- Control ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Bibliography:
Includes bibliographical references.
General Note:
Submitted in partial fulfillment of the requirements for the degree, Master of Science, Electrical Engineering.
General Note:
Department of Electrical Engineering
Statement of Responsibility:
by Kang Seok-Yoon.

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University of Colorado Denver
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Auraria Library
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
28985718 ( OCLC )
ocm28985718
Classification:
LD1190.E54 1993m .K36 ( lcc )

Full Text
DESIGN OF POWER SYSTEM STABILIZERS
USING MODERN CONTROL THEORY
by
Kang Seok- Yoon
B.S., University of Colorado, 1989
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado at Denver in partial fulfillment
of the requirments for the degree of
Master of Science
Department of Electrical Engineering
1993
|


This thesis for the Master of Science
degree by
Kang Seok Yoon
has been approved for the
Department of
Electrical Engineering
by
William R. Roemish
Miloje S. Radenkovic
Date


Kang, Seok -Yoon (M.S., Electrical Engineering)
Design of Power System Stabilizers Using Modem Control Theory
Thesis directed by Professor Pankaj K. Sen
The thesis addresses the design of Power System Stabilizers(PSS)
using a linear optimal regulator model and a lead compensation method.
Transient responses following step changes in reference voltage by computer
simulation are obtained by means of two types of stabilizers: the conventional
PSS (Lead Compensator) and the optimal PSS (Linear Optimal Regulator).
The PSS was simulated by using MATLAB (a software for control
systems). For the conventional PSS, the system block diagram connection
method was' employed to get transfer function, step responses, and
eigenvalues. For the optimal PSS, a state-space model was used to get
response to a step input change in reference voltage and to get eigenvalues for
the closed loop state representation system.
Simulation results show that the linear optimal stabilizer yields better
system dynamic performance than the conventional stabilizer in the sense of
having greater damping in response to a step change.
The form and content of this abstract are approved. I recommend its
publication.
Signed
Panktfj K. Sen


CONTENTS
Figures ................................................. vi
Tables ................................................. vii
CHAPTER
1. INTRODUCTION ............................................ 1
2. SYSTEM MODELING ......................................... 3
Power System Stability ............................ 3
Excitation System ................................. 6
Swing Equation .................................... 9
Simplified Linear Synchronous Machine..............11
Single Machine Connected to Infinite Bus .......... 14
3. POWER SYSTEM STABILIZER .................................19
4. DESIGN OF PSS ........................................... 22
Conventional Lead PSS ............................. 22
Reduced Model ............................... 22
Inertial Transfer Function .................. 24
Procedure .................................. 24
Linear Optimal Regulator .......................... 26
Procedure ................................... 27
iv


5. SYSTEM SIMULATION
29
6. CONCLUSION ...................................... 37
APPENDIX
A. PROGRAM 7a.m Uncompensated System Response .... 38
B. PROGRAM 7b.m Lead Compensator Response ........ 40
C. PROGRAM 7c.m Optimum Regulator Design ......... 42
BIBLIOGRAPHY .......................................... 43
V


FIGURES
Figure
2.1. Typical Swing Curve .......................................... 4
2.2. Power System Stability Fundamentals .......................... 5
2.3. Principal Controls of a Generating Unit ...................... 6
2.4. IEEE Type 1 Excitation System ................................ 8
2.5. Block Diagram of the Swing Equation ..........................11
2.6. Simplified Linear Model of a Synchronous Machine ............ 12
2.7. One Machine Infinite Bus System ............................. 14
2.8. Block Diagram for the One Machine Infinite Bus System ....... 15
3.1. General Block Diagram of the System with PSS ................ 20
3.2. Block Diagram of the Complete System with PSS ............... 21
4.1. Reduced Model ............................................... 23
4.2. Generalized Block Diagram including PSS ..................... 27
5.1. Original System Response without PSS ........................ 32
5.2. System Response with Lead PSS ............................... 33
5.3. System Response with Optimal PSS ............................ 34
5.4(a). Comparison of Responses (t = 0 to 5 sec ) ................. 35
5.4(b). Comparison of Responses (t = 0 to 2 sec ).................. 36
VI


TABLES
Table
2.1. System Parameters ........................................... 18
5.1. System Eigenvalues Comparison ............................... 30
vii


CHAPTER 1
INTRODUCTION
In the mid 1960's while interconnecting the power systems of the
western United States to optimize the use of available generation, undamped
and low frequency oscillations appeared which caused tie lines to open
without a disturbance. It was shown by test that the oscillations disappeared if
the Automatic Voltage Regulator (AVR) was removed from service. Since it
is not practical to run generators without the AVR, it became necessary to
develop a supplemental signal which produced positive damping for low
frequency oscillations. The Power System Stabilizer (PSS) was developed to
provide the proper supplemental signal to solve the oscillation problem.
The purpose of the PSS is to compensate the phase lag resulting from the
voltage regulator/exciter and generator. The PSS generates a supplementary
damping torque which is in phase with rotor speed. And the supplementary
damping torque can be employed to enhance the dynamic stability of the
power system. The function of PSS is to change generator terminal voltage
varies in phase with system frequency swings or generator speed change.
Several papers illustrating the application of the PSS to synchronous
machine exitation have appeared over the past decades. The PSS adopted by
the utility industry is usually a lead-lag compensator using the shaft speed or


frequency deviation or a combination of accelerating power and shaft speed
as input. The purpose of this thesis is to design an improved PSS using
modem control theory. In the thesis two types of PSS were designed: the
conventional PSS which uses the lead compensation method, and the optimal
PSS which uses the linear optimal regulator design method.
A sample system for a single machine connected to the infinite bus
through a transmission line is given to illustrate the effectiveness of the
optimal stabilizer. The two PSS were simulated using "MATLAB", a
computer aided control system design software.
2


CHAPTER 2
SYSTEM MODELING
Power System Stability
Power system stability is that property of a power system which
insures that all synchronous machines connected to the power system will
remain in operating equilibrium through normal and abnormal conditions.
Given a steady state equilibrium condition (e.g. synchronous speed and a
given power angle) of the system and a finite disturbance, if the system and
the machines return to a new equilibrium point in finite time, the system is
called "stable". The nature of a power system or its component parts to
develop forces to maintain synchronoism and equilibrium is known as
stability.
Stability is usually classified into three types: "Steady State",
"Dynamic", and "Transient". These classifications are based on the type of
changes or disturbances to the system. Swing curve in Figure 2.1 shows
that disturbance takes place at time t0 and the initial power angle is &, the
final equilibrium point is S{.
Steady state stability is concerned with very small (infinitely small)
gradual changes. The steady state limit refers to the maximum power which
can be transmitted stabily. It is based not only on the transmission line
characteristics but also on the machine types and the machine control.


Dynamic stability considers the response of the system to small
changes in operating conditions or to badly set control parameters. The
dynamic stability limit is a measure of the damping or settling time.
Figure 2.1. Typical Swing Curve
Transient stability is concerned with response to a sudden large
disturbance such as a fault. These responses are of short duration (first
swing), fraction of seconds for multiswing studies. The transient stability
limit applies to a specific disturbance and is defined by the maximum
predisturbance level of power transmission that allows system recorvery.
Figure 2.2 summerizes the three categories of power system stability.
4


1. STEADY STATE STABILITY:
* Infinitely Small Impact, Small Angle Changes
* Time Invariant
* AEpo = 0 (No Change in Field Voltage)
* Manual Control of Voltage, No Automatic Voltage Regulator
2. DYNAMIC STABILITY:
* Smaller or Normal Random Impacts
* System Equations are Linear (or have been linearized about an
operating point)
* X = AX + BU( Linear State Model)
* Eigenvalues of A Matrix determine Stability, AEpp ^ 0 (Voltage
Regulator Operates)
* Multiple Swing (Decay of Oscillations)
3. TRANSIENT STABILITY:
* Nonlinear
* Caused by Large Impacts
* X = f ( X. U. t) (Nonlinear State Model)
* Time Solutions: Use Digital Computer
* First Swing and more for Multi-machine Problem
Figure 2.2. Power System Stability Fundamentals
Three principal control systems directly affect a synchronous
generator: the boiler control, governor, and exciter. The simplified diagram is
5


shown in Figure 2.3. This thesis deals exclusively with the excitation control
system including AYR as shown in the figure.
Power at
Steam at Pressure,? Torque at Voltage, V
Power Set-point flat, w
Hef. V
Figure 2.3. Principal Controls of a Generating Unit!1!
In power system, the dynamic performance of exciter, governor,
voltage regulator following a disturbance are represented by a nonlinear set of
differential equations. Since a nonlinear equation can be linearlized for small
disturbances (or dynamic stability problems), linear equations and linear
control theory can be used in analyzing the power system dynamic stability.
Excitation System
The excitation system controls the generator excitation voltage and
therefore controls not only the output voltage but the power factor and current
magnitude as well.
6


An excitation system consists of two components: the exciter itself
which supplies the direct current to the generator rotor and the automatic
voltage regulator which controls the exciter. A change in the terminal
conditions of the generator causes the voltage regulator to change the signals
to the exciter.
There are five types of exciter models available in the industry!1]:
1. Type 1 System Continuously acting regulator and
exciter
2. Type Is System Controlled rectifier system with
terminal potential supply only
3. Type 2 System Rotating rectifier system
4. Type 3 System Static with terminal potential and
current supplies
5. Type 4 System Noncontinuous acting
In the thesis, Type 1 exciter was employed for the study because Type
1 is widely used in industry.
The excitation control system with a continuously acting voltage
regulator and exciter for a simpified machine referred as IEEE Type 1 is
shown in Figure 2.4. Provision can be made for first-order smoothing or
filtering of the terminal voltage with a filter time constant. Usually filter time
constant is very small and is often approximated as zero.
The quantities for Figure 2.4 are:
Ero = Exciter output voltage
Ka= Regulator amplifier gain
Ke = Exciter constant related to self excited field
7


Kf = Regulator stabilizing circuit gain
SE = Exciter saturation function
Ta = Regulator amplifier time constant
Te = Exciter time constant
Tf = Regulator stabilizing circuit time constant
V3 = Regulator stabilizing circuit output
^Rmax = Limits maximum regulator output
^Rmin = Limits minimum regulator output
VE = Error voltage
Tr = Regulator input filter time constant
Vj = Regulator input filter output
VreF ~ Regulator reference voltage setting
Vs = Stabilizing input signal
Figure 2.4. IEEE Type 1 Excitation System[i]
8


The equations which describe the dynamic behavior of the excitation
system are:
V, = (KR / TR ) Vt ( 1 / TR )V,
V3 = (Kf/Tf)E5D-(1/Tf)V3
vR = (KA/ ta) VE ( 1 / Ta ) vr , < vr <
Epd = (1 /Te)Vr [(S'E + Ks) / TE]Ero
VE - + Vs-V3-V, (2.1)
The state-space description of the excitation system described in
Figure 2.4 and equation 2.1 is given by equation 2.2.
#
V,
V.
v*
T*
0
l.
0
0
Tc
Ka
0
0 0
K Kf(Sc_ -k
TfTs tfts
1 A
ta u
1 (Ss+K B)
Tc Tc
V, ' ^S-V.
V3 Ls, 0
V,
La
0
(2.2)
Swing Equation
The rotor dynamics can be expressed^ by the linearized swing
equation 2.3. The swing equation governs the motion of the machine rotor
relating the inertia torque to the resultant of the mechanical and electrical
torque on the rotor.
9


(2.3)
T = T
Aa Am
2H
2H
Te = 8=w pu
Wr
Wr
The quantities for equation 2.3 are:
Ta = Acceleration torque in per unit
Te = Electrical torque in per unit
Tm = mechanical turbine torque in per unit
H = Inertia constant in seconds
w = The angular velocity of the revolving magnetic field in rad/sec
wr = Base system radian frequency (377 r/s for 60 Hz)
8 = Generator rotor angle in radians
Kj = Positive constant change in electrical torque
D = Damping
The block diagram representation of the swing equation defined by
equation 2.3 is shown in Figure 2.5. This block diagram generates the rotor
angle 8 .
Figure 2.5 also shows a damping component due to rotor bearing
friction, windage, magnetic losses and other drag torques that oppose rotation.
The damping component is a function of the rotor shaft velocity and is
shown by the dotted line. Braking torques are also developed in phase with
the machine rotor angle, 8 and in phase with the machine rotor speed, co, at
system power swings.
10


Figure 2.5. Block Diagram of the Swing Equation
Braking torquefK^J) in phase with 5 are referred to as "synchronizing
torque", while braking torques in phase with oo are reffered to as damping
torques as shown in Figure 2.5 t5l.
Simplified Linear Synchronous Machine
A simplified d-axis linear model[5] for a synchronous machine
connected to an infinite bus including the swing equation is shown in Figure
2.6.
The following assumptions are made for the linear model:
1. Amortisseur winding effects are neglected
2. Stator winding resistance is neglected
3. Balanced conditions are assumed
4. Saturation effects of the synchronous generator are neglected
ll


Figure 2.6 Simplified Linear Model of a Machine
The quantities for Figure 2.6 are:
Kj = Positive constant change in electrical torque for a change in
rotor angle with constant flux linkages in the d-axis
Tdo = Direct axis open circuit time constant in sec
K9 = Positive constant representing the change in electrical power
for a change in the d-axis flux linkages with the constant rotor
angle
K3 = Positive constant impedance factor
K4 = Positive constant representing the demagnetizing effect of a
change in rotor angle
E'q = Voltage proportional to the main field winding flux in per unit
Vt = Machine terminal voltage in per unit
K5 = Change in terminal voltage Vt for a small change in rotor angle
12


Vt = Machine terminal voltage in per unit
K5 = Change in terminal voltage Vt for a small change in rotor angle
K<5 = Positive constant. Change in terminal voltage Vt for a small
change in the d-axis linkage
The swing equations which describes the dynamics of the synchronous
machine are :
E'q = -(1/K3 T'd0)E'q (K4/ Td0)^ + (1+ T'd0)EFD
w = -( K2/2H)E'q (K^/ 2H)8 + ( 1/ 2H)Tm
8 = w (2.4)
By designating the state variables as E'q, w ,8 and the input signals as
Epo and Tm, the equation 2.5 is in the desired state-space form.
X = AX + BU (2.5)
where
XT = [E'q w S]
U7 = (Efd Tm]
[A] =
K,TJ0
-i- 0
2H
0 1
K.l 1 0
Td0 T' -1 dO
K, [B] = 0
2H 2H
0 0 0
13


Single Machine Connected to Infinite Bus
Single machine connected to infinite bus is utilized in this thesis. The
machine model is linearized at the equilibrium state. The synchronous
generator is equipped with an IEEE Type-1 Exciter with a continuously
acting voltage regulator. The system is shown in Figure 2.7.
[rmriTTP
Bus
Figure 2.7. Single Machine Connected to Infinite Bus
A complete block diagram, for the infinite system is shown in Figure
2.8M.
This system is described by the following state-space representation:
X = AX + BU
Y = CX
where
XT = [ AS Aw AEq AEro AVR AV3 AVT ]
Y1 = [ AS Aw ] (2.6)
14


Figure 2.8. Block Diagram for the One Machine Infinite Bus System
T oi


The quantities for equation 2.6 are:
A = Denotes deviation from operating point
8 = Torque angle
w = Speed
E'q = Voltage proportional to d-axis flux linkages
EFd = Generator field voltage
VR = Regulator output voltage
V3 = Regulator stabilizing circuit output
V] = Feedback voltage from the infinite bus
U = Supplementary control voltage
The driving functions are VreF and Tm assuming that Vs in equation
2.2 is zero. The A subscripts are dropped for convenience.
s ' s ' 0
w w 0
K Eq 0
E J'FD = [A] Ero + [B] 0
VR vR v v REF
V3 v3 0
. V,. . v,. 0
Y = [8 w]T = [C] [XT] (2.7)
The matrices A, B, and C for the equation 2.7 are given in equation
16


[A]
[B]
0 w 0 0 0 0 0
D 0 0 0 .0
2H 2H 2H
k4 0 1 1 o o o
l H a o K3T'd0 T'do
0 0 0 (S's +ke) 1 0 0
Te Te
1 Ka Ka
0 0 0 0
TA Ta Ta
0 0 0 KF(S-E+KE) kf 1 0
tfte tfte Te
K5Kr 0 K5Kr 0 0 0 1
tr tr T 1R J
T
K.
0 0 0 0 1 o o r|n<
1 0 0 0 0 0 0
0 1 0 0 0 0 0
(2.8)
Assuming TR = 0 and KR = 1, the system is unit feedback from
terminal voltage. New state-space form of the system is in the equation 2.9.
X = AX + BU
Y = CX
where
X7 = [ A5 Aw AE'q AEjtd AVr AVe ]
YT = [ AS Aw ] (2.9)
17


Eauauon 2.3 can be rewmren on the eqnanon 2.10.
[A] =
0 w 0 0 0 0
JEl ~2H D *2H "2H' 0 0 0
K, 0 1 l r\ A
o f- U
0 0 0 (S, A) Tc 1 Tc 0
n KA n 1 KA
T* TA ' \ ta
0 0 0 K,(S, *K,) K, T,T, 1 \
[B] =
0 0 0 0
KA
-it
0 0
[C] =
10 0 0 0 0 0
0 1 0 0 0 0 0
(2.10)
Data for this thesis are taken from [1] and they are given in Table 2.1
Table 2.1. System Parameters
Kj = 1.4479 Ka = 400
K2 = 1.3174 Tdo = 5.9
K3 = 0.3072 KF = 0.04
= 1.805 Ke = -0.17
K5 = 0.0294 2H = 4.74
= 0.5257 D = 2.0
Ta = 0.05 TF = 1.0
TE = 0.95 = T* Kr = 1.0
K s = 400 0 11
18


CHAPTER 3
POWER SYSTEM STABILIZER
The excitation system as discussed in the previous chapter introduces a
large phase lag at low system frequencies just above the natural frequency of
the excitation system. Thus it can be assumed that the voltage regulator
introduces negative damping. To offset this effect and to improve the system
damping in general, artificial means of producing torques in phase with the
speed are introduced. These are called supplementary stabilizing signals and
the networks used to generate these signals are known as "power system
stabilizer" networks!1!.
Stabilizing signals are introduced in excitation systems at the summing
junction where the reference voltage and the signal produced from the
terminal voltage are added to obtain the error signal fed to the regulator-
excitation system. Generalized block diagram of one machine infinite bus
model with PSS is shown in Figure 3.1. In the Figure 3.1, the stabilizing
signal is indicated as the signal Vs. The signal usually obtained from
shafr(rotor) speed co, the torque angle 5, or in some cases accelerating power
into the AVR.
The complete system model including exciter with voltage
regulator(AVR) and PSS is shown in Figure 3.2.


Figure 3.1. General Block Diagram of the System with PSS
20


Figure 3.2. Block Diagram of the Complete System with PSS


CHAPTER 4
DESIGN OF PSS
Optimal power system stabilizer as well as a conventional lead type
power system stabilizer is designed in the thesis.
Conventional Lead PSS
The lead compensator which improves the speed of response, and also
reduces the amount of overshoot is applicable as a PSS to compensate the
phase lag resulting from the voltage regulator, exciter and generator.
(a) Reduced Model
An approximate model for the excitation system is valid for low
frequencies because dynamic system performance is dominated by two pairs
of complex roots that are particularly significant at low frequencies. These
roots occur near the natural frequency.
We assume the folowing for the approximation:
1. The effect of the rate feedback Gf (S) in the Figure 3.2 can
be neglected at low frequencies, s = jw - 0
2. K5 is usually very small and is omitted in the approximate
model
3. The feedback path K4 provides a small positive damping
component that is usually considered negligible


4. Neglect the effect of saturation S'g
5. KR = 1 TR = 0
The resulting reduced order system is composed of two sub systems:
one representing the exciter-field effects and the other representing the
inertial effects. The reduced system is shown in Figure 4.1 where the exciter
and the generator have been approximated by simple first-order lags.
The exciter-regulator time constant Ts and gain Ks is to be used when
the excitation system is represented by one time constant In Figure 3.2, block
diagram between Ep^ and V2 is replaced by the block diagram between Epp
and V2 in Figure 4.1.
v3
G*(S)

Figure 4.1. Reduced Model
The function Gx (s) is given in the equation 4.1.
T, + K,TdD
l + K3KsKff
K,rd0 Tt 2
i + k3k5k/
23


(4.1)
K2Ke/
/Tl T
1 dO is
k2k£/
'T'nn X
dO
S2 +2CWxS + Wx
d(s)
Where wx is the undamped natural frequency and is the damping
ratio:
Wx =
K6K / c = (T, + K3rd0 )/
/T'do T£ ^ /2 wx
K3T'd0 T£
(b) Inertial Transfer Function
The inertial transfer function from Figure 2.8 can be obtained by
inspection. For the case when damping is present,
T.F. =
- 8
wx
w.
'2H
y2K
Te.
'2 2
^ S
+ + K>w* s2 + 2£nwns + wr
(4.2)
2H 2H
Where wn is the natural frequency of the rotating mass and is the
damping factor:
w
KiWr
2H
" 4Hwn 2^/2HK,wr
(c) Procedure
The excitation system phase lag is large because wosc > wx, and C,K is
small, and phase compensation is likely to be required.
24


For good dynamic performance, i.e., for good damping characteristics,
a resonable value of <£n is 1/V2 = 0.7071. For typical values of the gains and
time constant, we usually have T'd0 Te Kg 1.
From equations 4.1 and 4.2 with parameters given in Table 2.1, the
results are:
wx = 6.1251 r/s wn = wosc = 10.73 r / s
Cx = 0.131 Cn = 0.096
From equation 4.1, the characteristic equation of Gx (s) is obtained by
substituting s = jw in the denominator of the Gx (s):
d(s) = d (jw) = w2 + j 1.6048 w +37.516
The dominant frequency of oscillation is approximately near the
natural frequency wn = 10.73 r / s. At the frequency of interest (wosc) we
have
d (jwosc) = 77.616 +j 17.2195
lag (osc) = tan*1 (17.2195/-77.616) = 167.5
Finally, it needs to compute that the parameters of the PSS required to
exactly compensate for the excitation control system lag of 167.5.
The PSS shown on Figure 3.2 has a feedback element from the shaft
speed and is given by the equation 4.3.
Gs (s)
K0T0s (1 + aTs)
(1 + T0s)L (1 + Ts)
when n is the number of lead stages.
The parameter used in the equation 4.3 is given as
(4.3)
25


a = ( 1 + sin <|>m) / ( 1 sin m}
(4.4)
where m = desired phase lead which occur at the median frequency
wm
wm= 1/ (T-Ja ) =
T = 1 / ( wm 4a ) (4.5)
where wm occurs at the geometric near of the comer frequency.
We assume three lead stage, then the phase lead per stage is as
follows:
((>m = 167/3 = 55.8
a = 10.588
wm = wosc = 10.73 r/s
T= 0.02864
aT = 0.3032
/. Gs(s)
' K0T0 s ' 1 +0.3032s '
.1 + To s_ .1 +0.02864s.
We choose the specifications as follows:
T0 = 10 sec
Ko = 1
Then we have the PSS, Gs ( S ) as in equation 4.3:
G = 0.279 s4 + 2,759s3 +9.096 s2 + 10s
s^S 0.000235s4 +0.0240235s3 +0.8614s2 +10.086s + l
Linear Optimal Regulator
The closed loop state-space equation for the system is given by
26


X = AX + BU
= AX + B(-GX) + BNU0
= (A-BG)X + BNU0
= A.X + Bc U0
Y = C X + DU
= CX-DGX + DNUq
= (C-DG)X + DNU0
= Cc X + Dc U0 (4.5)
Where, U = NU0 -GX
Aj = A>BG : Closed Loop A Matrix
Bc = BN : Closed Loop B Matrix
Cc = C D G : Closed Loop C Matrix
The block diagram representation for the state-space form of the entire
system including the PSS is shown in Figure 4.2.
Figure 4.2. Generalized Block Diagram including PSS
27


(a) Procedure
Given a linearized system state equation
X = AX + BU (4.6)
an optimal control can be found from the minimization of a chosen
quadratic performance function (J) as defined by:
J = J" ( XT Q X + UT RU ) dt (4.7)
where Q = State weighting matrix
R = Control weighting matrix
subject to the system dynamics constraint.
The optimal control is obtained when,
U = -GX = -R-i BT K X (4.8)
and the Riccati matrix K is solved from the nonlinear matrix algebraic
equation
0 = K A +AT K K B R-1 BT K + Q (4.9)
The closed loop system equations become
X = AcX (4.10)
where,
A,, = A BG = A-BR-'BTK (4.11)
thus the eigenvalues of the closed loop system Ac depend upon the selection
of Q and R for equation J in equation 4.7. The choice of the state weighting
matrix Q depends on what the system designer is trying to achieve. There are
many other methods are also suggested in the literature!8].
With software that is now widely available, it is a simple matter to
solve for Gs(s) of PSS given A, B, Q, and R.
28


CHAPTER 5
SYSTEM SIMULATION
Computer simulation performed on a PC-MATLAB are described in
this section. The system under consideration is a synchronous machine which
is equipped with an IEEE Typel exciter tied to an infinite bus.
Performances of PSS designed in chapter 4 are analyzed through the
eigenvalues of the closed loop system and the transient responses of the
system following a step change in the reference voltage.
The Use of Computer Software. MATLAB for PC was used to
simulate the system performances. For the original system (not including the
PSS), "Connection Method" was applied to get state-space models required in
order to bring system analysis and design into use. The "Connect" command
is a function that can help form the matrices (A, B, C, D) of state-space
models from system block diagrams.
The Original System. Starting with the system block diagram in Figure
2.8, the "Connect" was to get a state-space model, the eigenvalues of this
system is shown in Table 5.1, and the transient response in Figure 5.1.
The Lead Type PSS. The state-space model brought in the original
systeman anlysis is also applied to the Lead Type PSS. The transient response
for the Lead Type PSS is shown in Figure 5.2 and the eigenvalues are shown
in Table 5.1.
The Optimal PSS. The state-space model given by equation 2.10 was
used to get an Optimal PSS in MATLAB.


In this thesis shaft speed w as PSS input was adopted to simulate the
system with optimum PSS. In the program Thesis7c.m(Appendix c), matrix C
is 1 x 6. This reflects only one input of PSS has been considered.
The eigenvalues are given in Table5.1, whereas the system response to
a step input is depicted in Figure 5.3.
Figure 5.4 shows all three response plots (using the same scale) for
comparison original system without PSS Lead PSS, and Optimal PSS
responses.
The codes for all these calculations are added in the Appendix for
reference only.
Eigenvalues analysis. The eigenvalues of the system using different
control schemes are given in Table 5.1 for comparison.
Table 5.1. System Eigenvalues Comparison
OPEN LOOP without PSS LEAD COMPENSATOR OPTIMAL PSS
-9.2381 + j 14.904 -46.4794 +j 21.135 -12.9876 +j 16.555
-9.2381 j 14.904 -46.4794-j 21.135 -12.9876-j 16.555
-0.4284 + j 10.744 -6.1484 +j 21.441 -6.1804 +j 15.881
-0.4284 +-j 10.744 -6.1484-j 21.441 -6.1804-j 15.881
-1.2309 +j 0.8883 -1.1353 +j 9.0772 -11.6278
-1.2309-j 0.8883 -1.1353 j 9.0772 -1.2283 +j 0.8908 -1.2283 j 0.8908 -0.1 -13.9394 -1.0017
30


Transient Response Analysis. Transient responses are obtained when
the system is disturbed in reference voltage. If 5 % step change is applied to
the system, the result of the rotor speed w following the step change is shown
in Figure 5.3. This plot shows that the optimal PSS is very effective.
31


Step Response--Mo PSS ( See Appendix
^4 L.0 IT, LO A. k M
S s

IS i 1 S (T*l cV i
ftapn'^-T I
Figure 5.1. Original System Response without PSS
32


xl.0 3 S tep Response---UitR PSS ( See Appendix
Figure 5.2. System Response with Lead PSS
33


x10-4 Step Response -Opt ima 1 Regulator^ See Appendix
o
LO
1 A

f!-apniaj dwy
LB
S
CNj
Figure 5.3. System Response with Optimal PSS
34
T iwe-Sec


Step Response
Figure 5.4(a). Comparison of Responses (t = 0 to 5 sec)
35
T i me-Sec 7rt : 7R


.Step Response-
Figure 5.4(b). Comparison of Responses (t = 0 to 2 sec)
36
TimeSeo 7A: 7B:+ 7C:


CHAPTER 6
CONCLUSION
A study of the design of Power System Stabilizers( PSS ) is presented
in this thesis. The results obtained in chapter 5 demonstrate that the optimal
PSS is more effective than the lead compensator design.
Comparing three systems, the optimal stabilizer yields results in better
system dynamic performance than the conventional stabilizer in the sense of
having greater damping in response to a step change in reference voltage.
The programs that are used in this thesis will enable the use of
different exciter models by simple manipulation in block diagram. If the
parameters are available for different exciters and generators, the
program(technique adopted in this thesis) performs to get the state-space
model, transfer function, responses, and eigenvalues. Furthermore, the
program realizes not only the simple simulation steps but also actual design
and study of the PSS.
It is recommended that the adaptive control theory be also used to
enable to design self-tunning PSS including governor system for total system
stability study and also apply to multi machine system.


APPENDICES
Appendix A
Uncompensated System Response
7. THE5IS7A.M.
7. SYSTEM IS FROM THESIS COMPLETE BLOCK DIAGRAM
7. This file is "o get STEP RESPONSE, TRANSFER FUNCTION and ROOTS
7. from comolete olock diagram
/. Using MATLABCONNECTION COMMAND
7,
7. with out-pur on w
7.
7. *** NO PSS ***
7.
innum=inDut('INPUT: number of input block ~ (1) );
7.
outnum=inout( INPUT: number of output block = (7) );
7.
^Constants
7.
Ka=400;Ke=-.17; : K5=.0294; Wr=377;
KBlock descriptions
nl=CO 0 Ka]; dl=CTa*Te
n2=C .04 0] ; d2='Il 1] i
n3=C0 K31; d3=C:<3*Tdo
n 4= C K 4 3; d4=C1];
nS=C K2I; d5=Cl];
nA=CKH; d6=C 1 ] ;
n7=C0 i ] ; d7=.C2KH 0] ;
nQ=C0 Wr]; dS=C1 0];
n9=CD]; d9=C1];
n10=C K5 j ; dlO=Cij;
n11 = C K6]; dll=f:];
V.n I2=C Kr ] ; al2=C13;
ni2=Cl];dl2=Cl];
; Kl = L.4479;K2=1- 3174; K3=.3072; K4=1.S052;
.05; Te=.95; Tr=O.0; H=2.37;- D=2.0 ; Ttio=S. 9 ;
Ke#Ta-t-Te Ke];
7. Kp = 0.04 is from thesis data
1]5 7. TF = |.o
7. : 7. open loop
7.
nblocks= 12: 7. MAKE SURE THE NUM
blkbui1d
d=C i -2 -12 0
2 10 0
3 1-ao
4 3 0 0
3 3 0 0
6 3 0 0
7 -3 -6 -9
3 7 0 0
9 7 0 0
10 3 O 0
11 3 0 0
7. 12 14 0 0
12 10 11 0
7.
9 is the 9th 31ock, which is connected from no.7
7. 3 14 CONNECTED WITH
38


iy=Coutnum]; 7. iu means input 91ock, iy means, output 91ock
Cac,Pc,cc,dc]=connect(a,b,c,d,q,iu,iy);
C a cm, bcm, ccm, dc.m ]=minrea l (ac,bc,cc,dc) ;
[nura,den ]=ss2tf ( acm, bcm, ccm,dan,1) ; '/. 1 means first thing
/.
[aa,bb,cc,dd]=tf2ss(num,den)
[numer,denom]=ss2tf ( aa,bb, cc,dd, JL)
rootrium=roots (numer)
rootden=>roots (denom)
/.
t-0:.05:5;
yl-step(aa,bb,cr,dd,l,t);
plot (t, y 1), titi e (' Step ResponseNo PSS') ;
xlabel ( TimeSec' ) ,y label C AmplitudeW' ) ;
grid
39


Appendix B
Lead Compensator Response
THE5IS7B.iv1
7. System is from Thesis Slack Diagram
7.
7. *xx with PSS ( Yoon' Lead comoensatDr ) #xx
7. xxx outout on w inout On Vref XXX
7.
7. xxx programdeterminations for the << Step Response, roots >>
7. with PSS innum=inout('INPUT: number of input block = (1) ');
7.
outnum3input( INPUT: number or output block = (7) ' ) ;
7.
7. Constants (Tp= 1.0 ; Kf = 0,04 )
7.
Ka=400; <5=.0294; <6=.5257;Ta=.05; Te=.95; Tr=O.0; H=2.37; D=2.0 ;Tdo=5.9
Wr=377;
ZBlock descriptions
nl = C0 0 n2=C0.04 0]; d2=Cl 1]; - 7. n3=C0 <33; d3=C<3*Tdo 1];
n4=C <4]; d4=C1];
n5=C<2]; d5=Cl];
n6=C <1]; d6=C1];
n7=T0 1]; d7=C2XH 0];
n3=C0 Wr ] ; d3=CI 0];
n9=CD]; d9=Cl];
nl0=C<5]; dl0=C1];
nll = r <6]; d11 = C1]j
7,nl2=C n12=C1];dl2=C1];
nl3=C 279 2.759 9.096 10.0 0.0]; d!3=C- 000235 .0240235
7. XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
7. xxxxxxxxxxxxxxxxxxxx
.3614 10.036 1.0];
7. xxxxxxxxxxxxxxxxxxxx
nblocks=13;
blkbul1d
q=Ci -2 -12 13
2 10 0
3 1-^0
4 8 0 0
5 3 0 0
6 3 0 0
7 -5 -6 -9
3 7 0 0
9 7 0 0 7. 9 is the 9th Slock, which is connected from no. 7
10 3 0 0
11 3 0 0
7. 12 14 0 0 7. "4 14 CONNECTED WITH
7. 12 10 11 0 i ;
12 10 11 0
40


/.
/.
13 7 O 0]:; 7. *## 13 is the PSS 3lock from *7-w to 41-Oref-
iu=C innum];
iy=Coutnum] ; 7. iu means inout Block, iy means output Slock
Cac,bc,cc,dc]=connect(a,b,c,d,q,iu,iy);
Cacm, dcm, ccm, acm]a Cnum,den]=ss2tf(acm,bcm,ccm,dcm,1); '/. 1 means first thing
7.
Cal,bl,cl,dl]=tf2ss(num,den)
Cnumer,denom]=ss2tf(al,bl,cl,dl,1)
rootnumaroots(numer)
rootdenaraots(denom)
7.
C z, p, k ]=ss2zp ( al bl, cl, dl, 1.)
7.
t=0:-05:5;
y2=step(al,bl,cl,dl, 1, t) ;
plot(t,y2),title( Step Response With PSS );
xlabelC' Time-Sec '),ylabel( Amplitude ');
grid
7.
41


Appendix C .......................Optnnum Reguiaior Design
'/. THES IS7C ."1
7.-------------:----------
7.xxx actinium regulator design -* Step response with Closedloop
7. State Space *x*
7. xx State Equation X = C delta, W, Eg Efd, Vr, oe ]' xx
7.
7.
a=C0.0 377 0.0 0 0 0 ;
-0.3055 -0.4219 -O.2779 0 0 0 ;
-O.3060 0 -0.5517 0.1695 0 0 ;
0 0 0 0.1709 1.0526 0 ;
-235.2 0 -4205.6 0 -20 -6000 ;
0 0 0 , 0.00716 0.04211 -1 ];
7. xxx 0=2, Kf=O.04 as thesis data xxx
7,
5=C0 0 0 o aooo 0]- ;
C=C 0100001;
7.
d=C0];
7.
f=C0.0000001] ;
ff=C 1.00000] ;
fff=C 10000000] ;
7.
q=Cfff 00000;
000000;
O O f 0 O 0;
0 0 0 f 0 0;
0 0 0 0 f 0;
0 0 0 0 0 f f ] ;
r=C7T];
7.
q- 1 ar ( a, b q ,) 7. xxx full state feed back gain xxx
7.
acl=aoXg; 7, Xxx Closed loop C A ] *xx
cci=c-dXg; 7. %xx Closed loop [ C ] xxx
n=Cl 1 1 1 1 1]; 7.xxx n is reference metrix CN] xxx
bn=oxn 7. xxx Closed looo C 9 ] xxx
dn=dXn 7, xxx Closed loop C 0 ] xxx.
7.
C z,p,k]=ss2zd(aci,bn,ccl,dn,l)
[numTden]=5s2tf(a,o,c,d,l)
rootsnum=roots(num)
rootsden=roots(cen )
7.
Cnumcl,dencl]=ss2tf(acl,bn,ccl,dn,1)
rootnuci=roots(numcl)
rootdecl=roDts(dencl)
7.
t=0:.05:5;
y=steo(acl,bn,ccl,dn,l,t);
plot(t,y),title('Step Response-Optimal Regulator');
x1 aoe1( 'Time-Sec' ) ,y1abel( 'Amoletude-W' );grid
7.
42


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