The main purpose of the research project is simulate an electrical power system from the view of multimachine transient stability. Simulink is advanced software ...
Republic of Iraq Ministry of Higher Education and Scientific Research Middle Technical University College of Electrical and Electronics Techniques
Simulation of Multi-Machine Transient Stability
A RESEARCH PROJECT SUBMITTED TO THE COLLEGE COUNCIL OF ELECTRICAL AND ELECTRONIC TECHNIQUES AS PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF HIGHER DIPLOMA OF TECHNOLOGY IN ELECTRICAL POWER ENGINEERING TECHNIQUES.
By Saif Aldeen Saad Obayes Al-Kadhim Engineer BSc. Electrical Engineering
Supervisor Dr. Majli Nema Hawas Assist. Professor
November 2010
Thoul Hijjah 1431
Acknowledgement
I am heartily thankful to my supervisor, Dr. Majli N. Hawas whose encouragement, guidance and support and his whole-hearted interest throughout the preparation of this project. Deep appreciation is also expressed to all of those who supported me in any respect during the completion of the project.
I
ABSTRACT:
The
power
sudden
system
disturbance
transient
to
which
stability
the
depend
system
is
on
the
state
of
such
as
simulate
an
subjected,
fault state on the system. The electrical
main
purpose
of
power
system
from
the
research
the
view
project of
is
multimachine
transient
stability. Simulink
is
advanced
software
by
Mathworks,
which
is
increasingly being used as a basic building block in many areas of project. As such, it also holds great potential in the area of power system simulation. In
this
example
to
project,
we
demonstrate
have the
taken
a
features
multimachine and
scope
power
of
a
system
Simulink-
based model for transient stability analysis. A
self
sufficient
model
has
been
given
with
full
details,
which can work as a basic structure for an advanced and detailed study. In and
this
yet
an
performance tool.
research
Hope
efficient of that
a
project
approach
practical this
has
been to
power
attempt
demonstrated
study system,
will
add
information in this important and unexhausted domain.
II
the with some
a
simplified
transient Simulink more
stability as
a
practical
List of Contents Subject
Page
Acknowledgments
I
Abstract
II
List of Contents
III
Main Symbols and Abbreviations
V
Chapter One – Introduction and Literature Survey 1.1 Introduction
1
1.2 Literature survey
3
1.3 Project Importance
5
1.4 Project Layout
5
Chapter Two – Theoretical background 2.1 Introduction
6
2.2 Steady State Stability
10
2.3 Transient stability
11
2.4 Numerical Solution Of Swing Equation
12
2.5 Multimachine Transient Stability Studies
16
2.6 Digital Computer Solution of Swing Equation
19
2.7 State Variable Formulation of Swing Equations
21
2.8 Computational Algorithm for Obtaining Swing Curves Using
21
Modified Euler's Method Chapter Three – Project Procedure 3.1 Introduction
24
3.2 Numerical technique for solution of swing equation
25
3.3 Illustrative system example
29
3.4 System Modeling
32
3.5 Mathematical modeling
34
3.6 Simulink models
36
3.7 Modeling of power system components
38
III
Subject
Page
Chapter Four – Simulation & Discussion Results 4.1 Introduction
45
4.2 The Case Study of the system
45
4.3 Simulation Results
46
Chapter Five – Conclusions and Recommendations 5.1 Conclusion
51
5.2 Prospects of future work
52
References
53
Appendix-I
56
Appendix –II
57
Appendix –III
58
IV
List of Symbols
Rotor position with respect to a rotating frame of reference, rad .
o
Prefault steady state rotor angle, rad .
cr
Rotor angle at the instant of critical fault clearance, rad .
Angular position of rotor, rad .
Angular acceleration, rad/Sec2.
H
Inertia constant, MW.
J
Moment of inertia, kg. M2.
M
Angular moment, Joul. Sec/rad.
S
Complex power (PjQ).
P
Active power (watt).
Q
Reactive power (VAR).
Pa
Acceleration power, watt.
Pe
Electric power output, watt.
Pm
Mechanical power input, watt.
Pmax
Maximum power.
tc
Time clearing, sec
tcc
Critical clearing time, sec.
tcc+Δt
Critical clearing time with increment step, sec.
Vt
Infinite bus voltage, volte.
ω
Angular velocity (degrees or radians).
X
Transmission line reactance, p.u.
Xd
Direct axis synchronous reactance (p.u).
Xd
Direct axis transient reactance (p.u).
XL
Transmission Line reactance (p.u ).
XT
Transformer reactance (p.u ).
V
Eg
Generator Internal generated voltage (volt).
E'g
Transient Generator Internal generated voltage (volt).
f
Frequency.
F
Fault.
x1k
Angle devotion of .
x1k
The rotor derived the point of view for The Times ∂δ/ ∂t.
x2 k
Deviation in the speed of the rotor ω.
x1(Kr )
Angle devotion of for generator rotor No. (k) at iteration (r).
x1(Kr 1)
Angle devotion of for generator rotor No. k at iteration (r+1).
x2( rK)
Deviation in the speed generator rotor ω No. (k) at iteration (r).
x2( rK1)
Deviation in the speed generator rotor ω No. (k) at iteration (r+1).
Ek ( o )
Voltage primary internal generator No. (k) before fault.
Ek(r )
Voltage internal generator No. (k) at iteration (r).
Ek( r 1)
x1(Kr )
Voltage internal generator No. (k) at iteration (r+1). Angle devotion of derived for generator rotor No. (k) at iteration (r).
x1(Kr 1)
Angle of derived for generator rotor No. (k) at iteration (r+1).
x1(Kr ) ave
Average value for angle of derived for generator rotor No. (k) at
x 2( rK) ave
Average value for speed devotion of derived for generator rotor No.
iteration (r).
(k) at iteration (r).
VI
List of Abbreviations MATLAB EMTP ATP EMTDC SPICE
Matrix laboratory Electro Magnetic Transients Program. Alternative Transient Program. Electro Magnetic Transients for D.C. Simulation Program with Integrated Circuit Emphasis.
BJTs
Bipolar junction transistors.
JFETs
Junction field effect transistors.
PSPICE A/D
Commercial version of SPICE by MicroSim
ABM
Analog Behavioral Modeling
PSB
Power System Block.
PMUs
Phasor Measurement Units.
RTS
Real Time Station.
AGC
Automatic Generation Control.
PSS
Power System Stabilizer.
SVC
Static VAR Compensator.
TCSC FCT WSCC
Thyristor Control Series Capacitor. Fault Clearing Time. Western System Coordinated Council.
MHC
Mechanical Hydraulic Control.
ANN
Artificial Neural Network.
VII
Chapter One
Introduction and Literature Survey 1
Chapter One Introduction and Literature Survey 1.1 Introduction The stability of power systems has been and continues to be of major concern in system operation. Modern electrical power systems have grown to a large complexity due to increasing interconnections, installation of large generating units and extra-high voltage tie-lines etc. Transient stability is the ability of the power system to maintain synchronism when subjected to a severe transient disturbance, such as a fault on transmission facilities, sudden loss of generation, or loss of a large load. The system response to such disturbances involves large excursions of generator rotor angles, power flows, bus voltages, and other system variables. It is important that, while steadystate stability is a function only of operating conditions, transient stability is a function of both the operating conditions and the disturbance(s)[1]. This complicates the analysis of transient stability considerably. Repeated analysis is required for different disturbances that are to be considered. In the transient stability studies, frequently considered disturbances are the short circuits of different types. Out of these, normally the three-phase short circuit at the generator bus is the most severe type, as it causes maximum acceleration of the connected machine[2]. Historically, simulation of transient phenomena related to power systems has been carried on using the Electro Magnetic Transients Program (EMTP) or one of its variants, such as the Alternative Transient Program (ATP) or Electro Magnetic Transients for D.C. (EMTDC), which are all based on the trapezoidal integration rule and the nodal approach[3]. SPICE (Simulation Program with Integrated Circuit Emphasis) is a general purpose circuit simulation program, which was developed at the University of California, Berkeley [4].
Chapter One
Introduction and Literature Survey 2
It contains models for basic circuit elements (R, L, C, independent and controlled sources, transformer, transmission line), switches and most common semiconductor devices: diodes, bipolar junction transistors (BJTs), junction field effect transistors (JFETs), MESFETs and MOSFETs. SPICE is mainly applied to simulate electronic and electrical circuits for different analyses, including d.c., a.c., transient, zero pole, distortion, sensitivity, and noise. SPICE uses the nodal approach with a variable time step integration algorithm, so that it can correctly simulate switching power electronic circuits. The simulation of control systems in PSPICE A/D (a commercial version of SPICE by MicroSim) is facilitated by using the Analog Behavioral Modeling (ABM) blocks. However, there are no specific models for power systems and drives, such as electrical machines, circuit breakers, surge arresters, thyristors, etc. To simulate a power system, the user has to build the needed models using SPICE primitives and basic elements, so the simulation setup can be highly time consuming[4]. Simulink is an interactive environment for modeling, analyzing, and simulating a wide variety of dynamic systems. Simulink provides a graphical user interface for constructing block diagram models using ‘drag and drop’ operations[5]. A system is configured in terms of block diagram representation from a library of standard components. A system in block diagram representation is built easily and the simulation results are displayed quickly. Simulation algorithms and parameters can be changed in the middle of a simulation with intuitive results, thus providing the user with a ready access learning tool for simulating many of the operational problems found in the real world. Simulink is particularly useful for studying the effects of nonlinearity on the behavior of the system, and as such, is also an ideal research tool. The key features of Simulink are [6]: a) interactive simulations with live display;
Chapter One
Introduction and Literature Survey 3
b) a comprehensive block library for creating linear, nonlinear, discrete or hybrid multi-input/output systems; c) unlimited hierarchical model structure; d) scalar and vector connections; e) mask facility for creating custom blocks and block libraries; The user can also derive many features and in-built components from the Power System Block set (PSB) [7]. PSB by itself gives the detailed three-phase representation of machine models and other components. Considering the overall complexity and data requirements (which might not be available in many cases) of a complete three-phase representation as required with PSB[8]. We have considered its parent software package Simulink as a main tool in our present study. Excitation systems, turbine and governor blocks from PSB can be readily used with Simulink blocks as and when required. The user also has access to numerous design and analysis tools provided in MATLAB and its toolboxes. Use of Simulink is rapidly growing in many areas of research work and so also in the field of power systems[9,10]. 1.2 Literature survey 1.2.1 Antoine Vidalinc,1997 [11] This thesis investigates and develops a direct method for transient stability analysis using the energy approach and the Phasor Measurement Units (PMUs). The originality of this new method results from a combination of a prediction of the post-fault trajectory based on the PMUs and the Transient Energy Function of a multimachine system 1.2.2 Chan, K.H. Parle, J.A. Acha, E.,2004 [12] This paper details implementation of the mathematical model of a synchronous generator in the direct time phase domain suitable for transient
Chapter One
Introduction and Literature Survey 4
analysis in a real-time simulation environment. The network model, which is a full multimachine model in the phase domain, has been implemented on a multipurpose real time station (RTS). 1.2.3 HAMEED,FADHIL HAMEED,2006 [13] Transient stability is the main object of this research, which includes the electrical power system behavior when it exposed to large disturbances. MATLAB program is used in this research to study and analysis transient stability of multi machine power system. MATLAB program has simplicity and accuracy in output and diagrams which represents the different relations , through these relations, it is decided whether the system is stable or not. 1.2.4 Rasheed ,Anwar Hameed,2006 [14] The main purpose of the research is to study and developed computer program to study and analysis an electrical power system from the view of stability in both its type state of the system after any change has taken in its normal state. This program is build to solve the mathematical model used "Euler's method" of single machine connected to infinite bus. The program is used to study the problem of transient stability and determine the critical clearing time for different location of faults in the network. 1.2.5 Mahmood ,Pshtiwan Kamal,2007 [15] The objective of this project is to investigate and understand the effect of fault location on power system stability, with the main focus or stability theories and power system modeling, through simulation the effect fault location on power system stability by the following methods: 1- Equal Area Criteria. 2- Step by Step Method. By using the technique (MATLAB.7, MATLAB.SIMULINK AND
Chapter One
Introduction and Literature Survey 5
MATLAB.PROGRAMMING) to analysis and simulation of the transient stability of two test systems at different fault location.
1.3 Project Importance: To simulate a multimachine system and study the critical clearing time & critical clearing angle whenever occurs fault in more place of the network for stability of the power system by using MATLAB Simulink. 1.4 Project Layout: This Project is structured as follows: In addition to this introduction (chapter one), the project consists of four other chapters. Chapter two presents a background on review of the main concepts on power system stability used here. It also describes the models and stability definitions, formulation of the swing equation and methods of solution, which include indirect methods like step-by-step, numerical analysis digital computer solution steps for state variable formulation of swing equations and modified Euler's method. Chapter Three presents theory solution of the illustrative system example, system modeling , mathematical modeling and Simulink models . Chapter four presents practical tests by using MATLAB/Simulink and discussed the simulation results. Chapter five presents conclusion and suggestions for future work.
Chapter Two
Theoretical Background
6
Chapter Two Theoretical Background 2.1 Introduction The mechanical electrical transient of a power system that has experienced a large disturbance can evolve into two different situations. In the first situation, the relative rotor angles among generators exhibit swing (or oscillatory) behavior, but the magnitude of oscillation decays asymptotically; the relative motions among generators eventually disappear, thus the system migrates into a new stable state, and generators remain in synchronous operation. The power system is said to be transiently stable. In another situation, the relative motions of some generator rotors continue to grow during the mechanical electrical transient, and the relative rotor angles increase, resulting in the loss of synchronism of these generators. The system is said to be transiently unstable. When a generator loses synchronization with the remaining generators in the system, its rotor speed will be above or below what is required to produce a voltage at system frequency, and the slip motion between the rotating stator magnetic field (relative to system frequency) and rotor magnetic field causes generator power output, current and voltage to oscillate with very high magnitudes, making some generators and loads trip and, in the worst case, causing the system to split or collapse[16]. A necessary condition that a power system maintains normal operation is the synchronous operation of all generators. Therefore, analyzing the stability of a power system after a large disturbance is equivalent to analyzing the ability of generators to maintain synchronous operation after the system experiences a large disturbance, this is called power system transient stability analysis[16].
Chapter Two
Theoretical Background
7
The aforementioned power system transient stability analysis typically involves the short term (within some 10 s) dynamic behavior of a system, nevertheless, sometimes we have to study system midterm (10 s to several minutes) and long term (several minutes to tens of minutes) dynamic behavior, this would be termed power system midterm and long-term stability analysis. Midterm and long-term stability mainly concerns the dynamic response of a power system that experiences a severe disturbance. A severe disturbance can cause system voltages, frequency, and power flows to undergo drastic changes; therefore, it is meaningful to look into certain slow dynamics, control, and protection performance that are not addressed in a short-term transient stability analysis. The response time of devices that affect voltage and frequency can be from a few seconds (such as the response time of generator control and protection devices) to several minutes (such as the response time of a prime mover system and on-load tap changing regulators, etc) [17]. A long-term stability analysis focuses on the slow phenomenon with long duration that occur after a large disturbance has happened, and the significant mismatch between active/reactive power generation and consumption. The phenomena of concern include: boiler dynamics, water gate and water-pipe dynamics of hydraulic turbines, Automatic Generation Control (AGC), control and protection of power plant and transmission system, transformer saturation, abnormal frequency effects of load and network, and so forth. When performing long-term stability analysis, one is often concerned about the responses of a system under extremely severe disturbances that are not taken into consideration in system design. After the occurrence of an extremely severe disturbance, a power system can undergo cascading faults and can be split into several isolated parts. The question a stability study has to answer is whether or not each isolated part can reach acceptable stable operation after any load-shedding occurs[17].
Chapter Two
Theoretical Background
8
Midterm response refers to response whose timeframe is between that of short term response and long-term response. Midterm stability study investigates the synchronous power oscillations among generators, including some slow phenomena and possibly large voltage and frequency deviations[18]. Large disturbances are severe threatens to power system operation, but in reality they cannot be avoided. The consequence of losing stability after a power system experiences a disturbance is in general very serious, it can even be a disaster. In fact the various large disturbances, such as short circuit, tripping or committing of large capacity generator, load, or important transmission facility, appear as probabilistic events, therefore when designing and scheduling a power system, one always ensures that the system can maintain stable operation under a set of reasonably specified credible contingencies, rather than requiring that the system can sustain the impact of any disturbance. Because every country has their own stability requirements, the selection of credible contingencies can be based on different standards. To check if a power system can maintain stable operation under credible contingencies, one needs to perform transient stability analysis. When the system under study is not stable, efficient measures that can improve system stability need to be sought. When a system experiences extremely severe stability problems, fault analysis is required to find the weak points in the system, and develop corresponding strategies. In power system stability analysis, the mathematical models of system components not only directly relate to the analysis results, but also have a significant effect on the complexity of the analysis. Therefore, if appropriate mathematical models for each system component are developed, stability analysis can be made simple and accurate. This is a crucial step in stability analysis[18].
Chapter Two
Theoretical Background
9
PSS Eqs
Primer-mover & Governor Eqs
Excitation System Eqs
Rotor motion Eqs
Rotor Circuit Eqs
Stator Voltage Eqs
Coordinate Transformation Eqs
Network Eqs
Other generator, Loads, DC system, Other dynamic devices such as SVC,TCSC,etc
Fig.(2-1) Conceptual framework of mathematical models for stability studies[18] Fig.(2.1) conceptualizes the mathematical model of all system components for power system stability studies. From the figure one can see that the mathematical model consists of the models of synchronous machines and the associated excitation systems, prime mover and speed-governing system, electrical load, and other dynamic devices and electrical network. Apparently, all the dynamic components of the system are independent; it is the electrical network that connects them with each other[18].
Chapter Two
Theoretical Background
10
2.2 Steady State Stability The stability of an electric power system is a property of the system motion around an equilibrium set, i.e., the initial operating condition. Steady state analysis consists of assessing the existence of the steady state operating points of a power system. At steady state, time derivatives of state variables are assumed to be zero. Consequently, the overall differential algebra equations describing the system are reduced to pure algebraic equations . No solution for the algebraic equations means that the system cannot operate under specific conditions. One solution means that a unique operating point exists. Multiple solutions means that further investigation is required to study the characteristics of each solution and find the stable solution. Conventionally, power flow equations are applied to conduct the steady state analysis. In the past, voltage stability was studied by steady state analysis methods [19]. Steady state stability limit, as signified earlier, is the greatest power that can be transfer via a particular transmission system under definite operation condition in the steady state, without breaking from synchronism. It is therefore expected that, “under the same operating condition” the steady state stability limit of specified circuit be greater than the transient stability limit. One can, therefore conclude that the system can be operate normally above its transient stability limit, but not above its steady state stability limit[20]. The steady state stability limit of a given system fixes a “ceiling” which cannot be exceeded through improvements in relays and circuit breakers. This power limit is primarily established by the impedance of transmission network, generators and other power system components[20].
Chapter Two
Theoretical Background
11
A well designed protection with fast clearing of faults enables the transient limit to come very close to the steady state stability limits. Improved methods of system operation and “out of step” relaying considerably lessen the severity of perturb due to the loss of synchronism [20]. It is also worth noting that improved reliability of the power system component, such as transmission lines and associated equipment can result in considerable decrease in the frequency of fault occurrence, thereby reducing the risk of instability consequence upon a fault[20]. 2.3 Transient stability When the disturbing forces are large and sudden such as large change in loads, loss of generations, transmission facilities, switching operations and fault etc, it become a case of transient stability. These disturbing forces case a large mismatching between the input and output power and as a result the rotor of the generating units are subjected to accelerating or decelerating torques. However, due to the fact that the inertia of different machines are not the same and the amount of accelerating power or decelerating powers available to the different units are equal, the rotor at different location of the system undergo different change in their angular position and velocity. This can be determined with the help of a set of differential equations, commonly known as swing equations the change in the angular displacement of the generator rotor subsequently produces a variation of the power flow through the system. The study of transient stability, consists of solving alternatively the algebraic network equation and differential equations of motion of the rotor in order to assess the dynamic behavior of the system for few seconds after the disturbance[21]. In order to determine the angular displacement between the generating unit of a power system following a major disturbance, it is
Chapter Two
Theoretical Background
12
necessary to solve a set of differential equations describing the motion of the rotor, which is called “swing Equation”[21].
2.4 Numerical Solution of Swing Equation In most practical systems, after machine lumping has been done, there are still more than two machines to be considered from the point of view of system stability. Therefore, there is no choice but to solve the swing equation of each machine by a numerical technique on the digital computer. Even in the case of a single machine tied to infinite bus bar, the critical clearing time cannot be obtained from equal area criterion and we have to make this calculation numerically through swing equation. There are several sophisticated methods now available for the solution of the swing equation including the powerful Runge-Kutta' method. Here shall treat the point-bypoint method of solution which is a conventional, approximate method like all numerical methods but a well tried and proven one. shall illustrate the pointby-point method for one machine tied to infinite bus bar. The procedure is, however, general and can be applied to every machine of a multimachine system. Consider the swing equation: δ
M
P π
P
sin δ
P
or in p. u system M
where : M = angular momentum. Pm = Input mechanical power. Pmax = Maximum power. Pa = Acceleration power. H = is inertia constant. f = frequency.
M ……………………………………(2.1) π
..……………………………………..(2.2)
Chapter Two
Theoretical Background
13
The solution δ(t) is obtained at discrete intervals of time with interval spread of ∆t uniform throughout. Accelerating power and change in speed which are continuous functions of time are discredited as below[2]: 1. The accelerating power Pa computed at the beginning of an intervals assumed to remain constant from the middle of the preceding interval to the middle of the interval being considered as shown in Fig.(2-2).
((a))
((b))
Chapter Two
Theoretical Background
14
((c)) Fig.(2-2) Point-by-point solution of swing equation[2]. 2. The angular rotor velocity ω
δ
(over and above synchronous
velocity ω ) is assumed constant throughout any interval, at the value computed for the middle of the interval as shown in Fig.(2-2). In Fig.(2-2), the numbering on
axis pertains to the end of intervals. At the
∆
end of the (n-1) the interval, the acceleration power is P
P
P
sin δ
has been previously calculated. The change in velocity ω
where δ
ω
/
δ
,
,assumed constant over ∆t from ( - 3/2) to ( - 1/2) is
caused by the P ω
…………………………..………………..(2.3)
∆
/
…………………………………….…….(2.4)
The change in δ during the (n - 1)th interval is ∆δ
δ
∆tω
δ
/
……………………………….……..(2.5)
and during the nth interval ∆δ
δ
∆tω
δ
/
…………………………………….……..(2.6)
Subtracting Eq. (2.5) from Eq. (2.6) and using Eq. (2.4), we get ∆δ
∆
∆δ
P
…….……….………………………..……..(2.7)
Using this, we can write δ
δ
∆δ
…………………………….…………..…………...…(2.8)
Chapter Two
The obtain P
Theoretical Background
process , ∆δ
and δ
of
computation
is
now
repeated
15
to
.The time solution in discrete form is thus carried
out over the desired length of time, normally 0.5 s. Continuous form of solution is obtained by drawing a smooth curve through discrete values as shown in Fig.(2-2). Greater accuracy of solution can be achieved by reducing the time duration of intervals. The occurrence or removal of a fault or initiation of any switching event causes a discontinuity in accelerating power Pa . If such a discontinuity occurs at the beginning of an interval, then the average of the values of Pa before and after the discontinuity must be used. Thus, in computing the increment of angle occurring during the first interval after a fault is applied at t = 0, Eq. (2.8) becomes[2]:
∆δ
∆
where P
…………………………………………………………..(2.9)
is the accelerating power immediately after occurrence of fault.
Immediately before the fault the system is in steady state, so that P
=0 and
δ is a known value. If the fault is cleared at the beginning of the nth interval, in calculation for this interval one should use for P
,where P
clearing and P
P
is the accelerating power immediately before is that immediately after clearing the fault. If the
discontinuity occurs at the middle of an interval, no special procedure is needed. The increment of angle during such an interval is calculated, as usual, from the value of Pa at the beginning of the interval [2].
Chapter Two
Theoretical Background
16
2.5 Multimachine Transient Stability Studies The first step in the transient stability analysis is solving the initial load flow and determine the initial bus voltage magnitudes and phase angles. The machine current prior to disturbance is calculated from,
S I V
i
(P
i
i
V
jQ ) i
,
i=1, 2….m
.................................(2.10)
i
where, m is the number of generators
vi is the terminal voltage of the ith generator Pi and Qi are the generator active and reactive powers of the ith generator All unknown values are determined from the initial power flow solution. The generator armature resistance was usually neglected and the voltage behind the transient reactance was obtained .[6]
EEii = vVii +jX jXdd . IIi i
………..........(2.11)
All loads are represented as constant impedance to ground ,to include voltage behind transient reactance, m buses are added to the n bus power system network. The equivalent network with all load converted is shown figure (2-2).
Fig(2-3) Power system representation for transient stability analysis[22] Nodes n+1, n+2….n+m are the internal machine buses, i.e, the buses behind the transient reactance. The voltage equation with node zero as reference for this networks is
Chapter Two
11
1 . .
n n 1
.
. .
. .
. . . .
n1 (n 1)1
. .
. .
.
Theoretical Background
n m (n m)1
. . . . . . . .
1n . .
1(n 1) . .
nn n(n 1) (n 1)n (n 1)(n 1) . .
. .
(n m)n (n 1)(n m)
. . . . . . . . . . . . . . . .
1(n m)
17
v1
. .
. .
n( n m) v n (n 1)(n m) n 1 . .
…….(2.12)
. .
(n m)(n m) n m
Or
I bus Y bus vbus
…………..…........…….(2.13)
where Ibus is the vector of the injected bus currents
v
bus
is the vector of bus
voltages measured from the reference node 0 The diagonal elements of bus admittance matrix are the sum of admittance connected to it, and the offdiagonal elements are equal to the negative of the admittance between the nodes. The reference is that additional node added to include the machine voltages behind transient reactance. Also, diagonal elements are modified to include the load admittances. To simplify the analysis, all nodes other than the generator internal nodes are eliminated using (Kron reduction formula). To eliminate the load buses, the admittance matrix in eq. (2.12) is partitioned such that the n buses to be removed are represented in the upper n rows. Since no current enters or leaves the load buses, currents in the n rows are zero. The generators current are denoted by the vector Im and the generator and load voltages are represented by the vector Em and Vn, respectively. Then, eq. (2.12), in terms of sub matrices, becomes 0 Y nn Y nm vn t I m Y nmY mm E m
………….…..………….(2.14)
The voltage vector Vn may be eliminated by substitution as follows.
vn Y nm E m
………………….… … (2.15)
Y nm vn Y mm E m
............................................(2.16)
0 Y
I
nn t
m
From eq. (2.15)
Chapter Two
v
Theoretical Background
1
n
18
……………..……………..(2.17)
Y nn Y nm E m
Now substituting into eq. (2.16), it yield
I
m
Y
1
t
mm
Y nm Y nm Y nm
E
red
m
……………………………(2.18)
Y bus E m
The reduced admittance matrix is 1
red
..........................................(2.19)
Y bus Y mm Y nmY nnY mn
The reduced bus admittance matrix has the dimensions (m×m), where m is the number of generators. The electrical power output of each machine can now be expressed in terms of the machine's internal voltages.
…………………………….(2.20)
* S eleciS*eiE= i IE i i Ii
Or
* (E Pei =Re Re(E Peleci ii IIii) )
.…………………………….(2.21)
Where
Ii
m
m
Ii =Ej Y Ejij Yij j 1
…………………………..….(2.22)
j=1
Expressing
voltage
and
admittances
in
polar
form,
Ei=|Ei|∟δi
and,Yij=|Yij|∟θij and substituting for Ii in eq. (2.11)resulting m
m
j 1
j=1
( ij i j ) PeleciPei E i iE j Y ij cos = E .E (ij -i+j) ij.cos j.Y
……...….… (2.23)
When taking into connect the losses and neglecting the transfer conductance from equation, this equation becomes .
Pe
m
E 'i Gii
Pei = E
2
2
i
E ' E ' B sin ( m) i
j
ij
i
j
Gii + E i . E j . Bij . sin (i -j) j 1 i j
……………………..(2.24)
j=1 ij
A three-phase fault at bus k in the network result in
vk=0. This is simulated
by removing the kth row and column from the pre fault bus admittance
Chapter Two
Theoretical Background
19
matrix. The new bus admittance matrix is reduced by eliminating all nodes except the internal generator nodes. The generator excitation voltages during the fault and post fault modes are assumed to remain constant. The electrical power of the ith generator in terms of the new reduced bus admittance matrices are obtained from eq. (2.24) The swing equation with damping neglected, therefore machine i becomes [2,6,22,]:
H d f dt 2
i
2
i
Pmechi Peleci
………….……………….. (2.45)
o
2.6 Digital Computer Solution of Swing Equation
The above swing equations (during fault followed by post fault) can be solved by the point-by-point method presented earlier or by the Euler's method presented in the later part of this chapter. The plots of δ and δ are given in Fig.(2-4) for a clearing time of 0.275 s and in Fig. 2.5 for a clearing time of 0.08 s. For the case (i), the machine 2 is unstable, while the machine 3 is stable but it oscillates wherein the oscillations are expected to decay if effect of damper winding is considered. For the case (ii), both machines are stable but the machine 2 has large angular swings[2]. If the fault is a transient one and the line is reclosed, power angle and swing equations are needed for the period after reclosure. These can be computed from the reduced Ybus matrix after line reclosure[2].
Chapter Two
Theoretical Background
20
Fig.(2-4) Swing curves for machines 2 and 3 for case(i)[2].
Fig.(2-5) Swing curves for machines 2 and 3 for case(ii)[2].
Chapter Two
Theoretical Background
21
2.7 State Variable Formulation of Swing Equations The swing equation for the kth generator is P
P
;k
1,2, … , m
………………….….(2.10)
For the multimachine case, it is more convenient to organize Eqs. (2.10) in state variable form define ∠
′
Then ,
1,2, … ,
………………..……………(2.11)
Initial state vector (upon occurrence of fault) is ∠
;
0
…………………………..……(2.12)
The state form of swing equations Eq.(2.11) can be solved by the many available integration algorithms (modified Euler's method is a convenient choice)[2].
2.8 Computational Algorithm for Obtaining Swing Curves Using Modified Euler's Method [2] 1. Carry out a load flow study (prior to disturbance) using specified voltages and powers. 2. Compute voltage behind transient reactance's of generators ( E
V
) using
jX I .This fixes generator e.m.f magnitudes and initial
rotor angle (reference slack bus voltage
).
3. Compute Ybus (during fault, post fault, line reclosed). 4. Set time count r = 0.
Chapter Two
Theoretical Background
22
5. Compute generator power outputs using appropriate YBUS with the help
of
the
general
E ′ E′ |Y | cos θ
form
of
δ .This gives P
δ
P
Eq.
E′
G
for t =t(r).
Note: After the occurrence of the fault, the period is divided into uniform discrete time intervals (∆t) so that time is counted as t(0),t(1),A typical value of ∆t is 0.05s. ,
6. Compute from
,
1,2, … ,
,Eqs. (2.11).
7. Compute the first state estimates for t =t(r+l) as ∆ ∆
,
1,2, … ,
8. Compute the first estimates of cos
sin
9. Compute P
, (appropriate Ybus and Eq. (2.8)). ,
10. Compute
,
1,2, … ,
, from Eqs. (2.11).
11. Compute the average values of state derivatives , ,
,
1 2
1,2, … ,
12. Compute the final state estimates for t=t(r+l). ,
∆
,
∆
,
13. Compute the final estimate for Ek at t=t(r+l) using cos 14. print
,
,
1,2, … ,
sin .
1,2, … ,
Chapter Two
Theoretical Background
23
15. Test for time limit (time for which swing curve is to be plotted), i.e., check if r > rfinal .If not, r= r + 1 and repeat from step 5 above otherwise print results and stop. The swing curves of all the machines are plotted. If the rotor angle of a machine (or a group of machines) with respect to other machines increases without bound, such a machine (or group of machines) is unstable and eventually falls out of step. The computational algorithm given above can be easily modified to include simulation of voltage regulator, field excitation response, saturation of flux paths and governor action.
Chapter Three
Project Procedure
24
Chapter Three Project Procedure 3.1 Introduction: Complex
systems
usually
comprise
a
number
of
components interconnected together to form a unit which performs a specific task. In order to study the dynamic characteristics of such systems, a model needs to be developed. It is advantageous that the model can incorporate fault conditions in any part of the system, so that both normal and abnormal operating conditions may be simulated. Such a model is often described by a set of differential and algebraic equations. Detailed models may require a very large number of such equations, some of which may be non-linear. The full set of these equations forms the basis for investigation into the system’s dynamics via detailed simulation. The main features that need to be considered are discussed next[23]. Modularity:
As
large-scale
interconnected
systems
are
physically comprised of a number of subsystems, it is natural that each of these subsystems be modeled separately and, through input-output signals, be interconnected to others. In this way a number of advantages can be realized, amongst which are flexibility, simplicity, facilitation of configuration, and ease of implementation. As a result, a highly modular based model of the interconnected system can be obtained, one that will permit the user to analyze the behavior of the system as a whole just as easily as analyzing any particular subsystem[23]. Numerical Considerations: Through simulation, the dynamic behavior and main characteristics of the system can by studied and analyzed. Digital simulation of complex dynamical systems
Chapter Three
Project Procedure
25
involves using numerical methods to arrive at approximate solutions to differential equations. This numerical approximation problem is in itself as challenging as the actual problem of modeling of the original system. This is especially true for highly nonlinear systems which are ill-conditioned, and have dynamics over a wide range of time-scales . This complicates the solution process considerably. In this specific case, numerical solution procedures applicable for “stiff” systems are often better suited than ordinary solvers for the problem. Therefore it is important that any modeling should be done in conjunction with simulation[24]. Debugging: Another important issue that needs to be addressed when modeling and simulating complex systems is considered is debugging. For complex nonlinear systems debugging can be very messy and time consuming. Therefore a unified approach to modeling, debugging and simulation should be taken when dealing with large complex nonlinear systems. we present a block-diagram graphical modular approach to the modeling, simulation and debugging of complex non-linear dynamic
systems.
The
approach
utilizes
the
graphical
capabilities of the Matlab and Simulink software packages. The approach is generic and can apply to any complex interconnected systems. However, for the sake of illustration, we take the specific case of modeling of power systems, and illustrate the main features of the approach through two case studies[25]. 3.2 Numerical technique for solution of swing equation The transient stability analysis requires the solution of a system of coupled non linear differential equations. In general, no analytical solution of these equations exists. However,
Chapter Three
Project Procedure
26
techniques are available to obtain approximate solution of such differential equations by numerical methods and one must therefore resort to numerical computation techniques commonly known as digital simulation. Some of the commonly used numerical techniques for the solution of the swing equation are[26]: i. ii. iii.
Point by point method. Euler's modified method. Runge-Kutta method.
3.2.1 Point by point method Point by point solution, also known as step by step solution is the most widely used way of solving the swing equation. The following two steps are carried out alternately. 1. First, compute the angular position δ , and angular speed ω= δ / t at the end of the time interval using the formal solution of the swing equation from the knowledge of the assumed value of the accelerating power and the values of δ and
δ / t at the beginning of the interval.
2. Then compute the accelerating power of each machine from the knowledge of the angular position at the end of the interval as computed in step 1. In this method the accelerating power during the interval is assumed constant at its value calculated for the middle of the interval. The desired formula for computing the change in δ during the nth time interval is ∆
∆
∆
P
………...……………….(3.1)
where, ∆δ = change in angle during the nth time interval.
Chapter Three
∆δ
Project Procedure
27
= change in angle during the (n-1)th time interval.
∆ = length of time interval. P
= accelerating power at the beginning of the nth time
interval. Due attention is given to the effects of discontinuities in the accelerating power Pa which occur, for example, when a fault is applied or removed or when any switching operation takes place. If such a discontinuity occurs at the beginning of an interval, then the average of the values of Pa before and after the discontinuity must be considered. Thus, in computing the increment of angle occurring during first interval after a fault is applied at t=0, the above equation becomes: ∆
∆δ
…………..………………………(3.2)
where, P
is the accelerating power immediately after the occurrence of
the fault. If the fault is cleared at the beginning of the mth interval, then for this interval, Pa(m-1) = 0.5 [Pa(m-1)- + Pa(m-1) +]
…………………….(3.3)
where, Pa(m-1)- is the accelerating power before clearing and Pa(m-1) + is that immediately after clearing the fault.. If the discontinuity occurs at the middle of the interval, no special treatment is needed[26]. 3.2.2 Modified Euler’s Method[27] The modified Euler’s method gives greater improvement in accuracy over the original Euler’s method. Here the core idea is that we use a line through (x0, y0) whose slope is the average of the slopes at (x0, y0) and (x1, y1(1)) where y1(1) = y0 +hf(x0, y0).
Chapter Three
This
line
Project Procedure
approximates
the
curve
in
the
interval
28
(x0,
x1).Geometrically, if L1 is the tangent at (x0, y0), L2 is a line through (x1, y1(1)) of slope f(x1, y1(1)) and L is the line through (x1, y1(1)) but with a slope equal to the average of f(x0, y0) and f(x1, y1(1)) then the line L through (x0, y0) and parallel to
is used
to approximate the curve in the interval (x0, x1). Thus the ordinate of the point B will give the value of y1. Now, the eqn. of the line AL is given by
Fig.(3-1) Geometrical explanation of the modified Euler’s method. ,
,
,
,
……………...……..(3.4)
A generalized form of Euler’s modified formula is ;
0,1,2, …(3.5)
where y1(n) is the nth approximation to y1. The above iteration formula can be started by choosing y1(1) from Euler’s formula ,
……………………………….(3.6)
Chapter Three
Project Procedure
29
Since this formula attempts to correct the values of yn+1 using the predicted value of yn+1 (by Euler’s method), it is classified as a one-step predictor-corrector method. 3.2.2.1 Algorithm of Modified Euler’s Method
1. Function F(x)=(x–y)/(x+y) 2. Input x(1),y(1),h,xn 3. yp=y(1)+h*F(x(1),y(1)) 4. itr=(xn–x(1))/h 5. Print x(1),y(1) 6. For i=1,itr 7. x(i+1)=x(i)+h 8. For n=1,50 9. yc(n+1)=y(i)+(h/2*(F(x(i),y(i))+F(x(i+1),yp)) 10. Print n,yc(n+1) 11. p=yc (n+1)-yp 12. If abs(p) p and where ai are constant and dependent on the method. For some implicit methods the characteristic root is equivalent to a Pade approximant to ehʎ.The Pade approximant of a function f ( t ) is given by ∑ ∑
………….……….(3.11)
Chapter Three
Project Procedure
31
and if then _
∑
∑
∑ ∑
.(3.12)
If the approximant is to have accuracy of order M + N and if f ( 0 )= PMN( f ( 0 ) ) then ∑
∑
∑
∑
It has been demonstrated that for approximations of
.. (3.13) where
M = N, M = N + 1 and M = N + 2, the modulus is less than unity and thus a method with a characteristic root equivalent to these approximants is A-stable as well as having an order of accuracy of M + N [1] . 3.2.3.1 Algorithm of Runge-Kutta Method
1. Function F(x)=(x-y)/(x+y) 2. Input x0,y0,h,xn 3. n=(xn-x0)/h 4. x=x0 5. y=y0 6. For i=0, n 7. k1=h*F(x,y) 8. k2=h*F(x+h/2,y+k1/2) 9. k3=h*F(x+h/2,y+k2/2) 10. k4=h*F(x+h,y+k3) 11. k=(k1+(k2+k3)2+k4)/6 12. Print x,y 13. x=x+h 14. y=y+k 15. Next i
Chapter Three
Project Procedure
32
16. Stop Notations used in the Algorithm: x0 is the initial value of x. y0 is the initial value of y. h is the spacing value of x. xn is the last value of x at which value of y is required. 3.3 Illustrative system example Consider to the popular Western System Coordinated Council (WSCC) 3machine, 9-bus system shown in Fig.(3-2) This is also the system appearing in [1] and [2]. The base MVA is 100 and 230 KV base, and system frequency is 60Hz. The system data are given in Appendix I [28]. The system has been simulated with a classical model for the generators. The disturbance initiating the transient is a threephase fault occurring near bus 7 at the end of line 5–7. The fault is cleared by opening line 5–7. The system, while small, is large enough to be nontrivial and thus permits the illustration of a number of stability concepts and results[28].
Chapter Three
Project Procedure
33
Fig.(3-2) WSCC 3-machine, 9-bus system; all impedances are in p.u on a 100MVA & 230KV base[28].
3.4 System Modeling The complete system has been represented in terms of Simulink blocks in a single integral model. It is self-explanatory with the mathematical model given below. One of the most important features of a model in Simulink is its tremendous interactive capacity. It makes the display of a signal at any point readily available; all one has to do is to add a Scope block or, alternatively, an output port. Giving a feedback signal is also as easy as drawing a line. A parameter within any block can be controlled from a MATLAB command line or through an m-file program. This is particularly useful for a transient stability study as the power system configurations differ before, during and after fault. Loading conditions and control measures can also be
Chapter Three
Project Procedure
34
implemented accordingly. 3.5 Mathematical modeling[25] Once the Y matrix for each network condition (pre-fault, during and after fault) is calculated, we can eliminate all the nodes except for the internal generator nodes and obtain the Y matrix for the reduced network. The reduction can be achieved by matrix operation with the fact in mind that all the nodes have zero injection currents except for the internal generator nodes. In a power system with n generators, the nodal equation can be written as:
0
……………………………………………. (3.14)
Where the is subscript n used to denote generator nodes and the subscript r is used for the remaining nodes. Expanding eqn. (3.14),
=
+
,0=
+
From which we eliminate
=
–(
)
to find
……………………………………… (3.15)
Thus the desired reduced matrix can be written as follows:
= It has dimensions (n
……………………………………………(3.16) n) where n is the number of generators.
Note that the network reduction illustrated by Eqs (3.15),(3.16) is a convenient analytical technique that can be used only when the loads are treated as constant impedances. For the power system under study, the reduced matrices are calculated.
Chapter Three
Project Procedure
35
Appendix II gives the resultant matrices before, during and after fault. The power into the network at node i, which is the electrical power output of machine i, is given by:
1,2,3, … . . , … 3.17
Where, ∠ = negative of the transfer admittance between nodes i and j ∠ = driving point admittance of node i The equations of motion are then given by
∑
…. (3.18)
1,2,3, … . , ……………………...…………. (2.19)
It should be noted that prior to the disturbance (t = 0) Pmi0 = Pei0; Thereby ∑ The subscript
0
……………. (2.20)
is used to indicate the pre-transient conditions.
As the network changes due to switching during the fault, the corresponding values will be used in above equations.
Chapter Three
Project Procedure
36
Bus data: , ,Pload,Qload,Pgen,Qgen,… Line data: Rij , Xij , Bij ,….. Generator data:Xd ,Hi,……
Determine the bus admittance matrix
Power Flow Analysis
Enter Fualted bus number
Enter the line number of the removed faulty line
Simulate the transient process of power system based on swing equation
Evaluate the transient stability determine the critical clearing time
Fig(3-3) Flow chart of transient stability analysis for a multimachine power system
3.6 Simulink models The complete 3-generator system, given in Fig.(3-2), has been simulated as a single integral model in Simulink. The mathematical model given above gives the transfer function of the different blocks. Fig.(3-4) shows the complete block diagram of a classical system representation for transient stability study. The subsystems 1,2,3,4,5 and 6 in Fig.(3-4) are meant to
Chapter Three
Project Procedure
37
calculate the value of electrical power outputs (Pe) for different generators; The model also facilitates the choice of simulation parameters, such as start and stop times, type of solver, step sizes, tolerance and output options etc. The model can be run either directly or from the MATLAB command line or from an m-file program. In the present study, the fault clearing time, the initial values of parameters as well as the changes in network due to fault, are controlled through an m-file program in MATLAB. For the simulation for multimachine stability the three summation boxes Add, Add 2 & Add 3 give the net accelerating powers Pa1, Pa2 and Pa3 .The gains of the gain blocks Gain2, Gain1,Gain4,Gain3,Gain5 and Gain5 are set equal to pi*f/H2, damp2, pi*f/H1, damp1, pi*f/H3 and damp3, respectively. The accelerating power Pa is then integrated twice for each machine to give the rotor angles δ , δ , δ .The initial conditions for
integrator
blocks
integrator1,integrator2,integrator3
integrator4,integrator5 and integrator6 are set to 0, D1/rtd, 0, D2/rtd ,0 and D3/rtd respectively. The gain blocks Gain7, Gain8 and Gain9 convert the angles δ , δ , δ
into degrees and hence
their gains are set to rtd. The electrical power Pe1 is calculated by using two subsystems 1 and 2.The detailed diagram for subsystem 1 is shown in Fig.(3-5). Subsystem 1 gives two outputs
Complex voltage E ∠δ Current
of
.
generator
I1
which
is
equal
to
E ∠δ *Yaf(1,1)+ E ∠δ *Yaf(1,2)*E ∠δ *Yaf(1,3) . The switches are used to switch between during fault and post fault conditions for each machine and their threshold values
Chapter Three
Project Procedure
38
are adjusted to FCT.
3.7 Modeling of power system components The classical system model represented above can be supplemented with other power system components for a detailed
study
or
for
implementation
of
the
stability
improvement measures.[1] and [2] give the simplified and generic models for many such components and transient stability improvement schemes. The block diagram models can be simulated within the Simulink environment almost in the same form. However, the representation of the transfer functions in the form of an integrator and gain with unity feedback is more convenient, when initial conditions have to be specified. Figs (3.6) and Figs (3.7) give the Simulink models of a Mechanical Hydraulic Control (MHC) governing system and that of a single reheat
tandem-compound
steam
turbine,
respectively.
The
typical parameter values are given in [1]. These values can be either defined in an m-file program or can be directly supplied to the Simulink models as shown in Appendix III.
Chapter Three
Project Procedure
Fig.(3-4) Complete classical system model for transient stability study.
39
Chapter Three
Project Procedure
Fig.(3-4) (continued) Complete classical system model for transient stability study.
Fig.(3-4) (continued) Complete classical system model for transient stability study.
40
Chapter Three
Project Procedure
Fig.(3-4) (continued) Complete classical system model for transient stability study.
Fig.(3-5) Computation of electrical power output of gen.(1) by Subsystem 1.
41
Chapter Three
Project Procedure
Fig.(3-6) Computation of electrical power output of gen.(2) by Subsystem 3.
42
Chapter Three
Project Procedure
43
Fig.(3-6) Simulink model of MHC governor.
Fig.(3-7) Simulink model of single reheat tandem-compound steam turbine.
Chapter Three
Project Procedure
Fig.(3-8) Simulink Blocks that used in the model.
44
Chapter Four
Simulation Results & Discussion 45
Chapter Four Simulation & Discussion Results 4.1 Introduction To simulate, essentially means to 'mimic' or to try to duplicate a real world process or system over time. Whether done by hand or on a computer, simulation involves the generation of an artificial history of a system. and the observation of that artificial history to draw inferences concurring the operating characteristics of the real system. This means that the more we know about the system, better will be the simulation and more reliable the results from the simulation.. The behavior of a system as it evolves over time is studied by developing a simulation model. This model is based on a set of assumptions concerning the operation of the system. Just as modeling is an art in getting a re-presentation of a phenomenon or system, simulation is an art of depicting the facts understood from the observations such that more experimentation can yield better understanding of the phenomenon or the system being studied. In contrast to optimization models, simulation models are "run" rather than solved, even though there may be a few exceptions. Given a set of inputs and model characteristics, the model is run for a set of inputs and the simulated behavior observed. The process of changing inputs and model characteristics results in a set of scenarios that are evaluated. A good solution, either in the analysis of the existing system or in the design of a new system is then recommended for implementation. 4.2 The Case Study of the system The system chosen for the study is a Three Machine, Nine node and Six line system. The disturbance initiating the transient is a three phase fault occurring near bus 7 at the end of line 5–7. The fault is cleared by opening line 5–7. The system, while small, is large enough to be nontrivial and thus permits the illustration of a number of stability concepts and results.
Chapter Four
Simulation Results & Discussion 46
4.3 Simulation Results The Simulink model had been run to analyze transient stability for power system as shown in Fig.(3-2) and the results had been recorded as figures and tables. System responses are given for different values of fault clearing time (FCT). Figs.(4-1a) show all generator rotor angles and (4-1b) show the individual generator rotor angles and the difference angles for the system with FCT=0.125sec, where as Figs.( 4-1c) and (4-1d) show the rotor angular speed deviations and accelerating powers for the same case. The results show that the power system is stable in this case. We can see in the complete model of Fig.(3-4) that output ports 7, 8 and 9 give the individual generator angles of the respective machines. Ports 10 and 11 (or alternatively Scopes 4 and 5) give the relative angular positions of generators 2 and 3 respectively, with generator 1 as reference. Similarly, ports 4, 5 and 6 give the angular velocities of the machines, whereas Scopes 1–3 (or the corresponding ports) display the accelerating powers. The system becomes unstable for FCT = 0.126sec, as the system responses in Figs.(4-2a), (4-2b), (4-2c) and (4-2d) indicate. Deducing that the system will be stable at 0.125(sec) and will be instable at 0.126(sec).
Chapter Four
Simulation Results & Discussion 47
Fig.(4-1a) Rotor angle for tcc =0.125(sec) and represent FCT.
Fig.(4-1b) Rotor angle at tcc = 0.125sec and represent FCT.
Chapter Four
Simulation Results & Discussion 48
Fig.(4-1 c) Deviation in speed at tc=tcc=0.125 sec.
Fig.(4-1 d) Accelerating powers at tc=0.125 sec.
Chapter Four
Simulation Results & Discussion 49
(stable) (stable)
(unstable)
Fig.(4-2a) Rotor angle for tcc+Δt = 0.126(sec) and represent FCT.
Fig.(4-2b) Rotor angle at tcc+Δt = 0.126(sec) and represent FCT.
Chapter Four
Simulation Results & Discussion 50
Fig.(4-2c) Deviation in speed at tcc+Δt = 0.126(sec).
Fig.(4-2d) Accelerating powers at tcc+Δt = 0.126(sec).
Chapter Five
Conclusion and Suggestions 51
Chapter Five Conclusion and Suggestions for Future Works 5.1 Conclusions: A complete model for transient stability study of a multimachine power system was developed using Matlab/Simulink. It is basically a transfer function and block diagram representation of the system equations. A variety of component blocks are readily available in various Simulink libraries and also in other compatible toolboxes such as Power System Block set, Controls Toolbox, and Neural Networks Block set etc. Thus a Simulink model is not only best suited for an analytical study of a typical power system network, but it can also incorporate the state of the art tools for a detailed study and parameter optimization. A Simulink model is very user friendly, with tremendous interactive capacity and unlimited hierarchical model structure. Typically, for a transient stability study the model facilitates fast and
precise
solution
of
nonlinear
differential
equations
visualization the swing equation. The user can easily select or modify the solver type, step sizes, tolerance, simulation period, output options etc. with the help of an appropriate menu from within Simulink. Any parameter within any block or subsystem of the model can be easily modified through simple MATLAB commands to suit the changes in the original power system network due to fault or a corrective action.
Chapter Five
Conclusion and Suggestions 52
5.2 Prospects of future work: It is clear from the above study that Simulink offers a wide perspective for simulation and analysis of various power system networks. The features of a Simulink model are exhaustive and at the same time it is very easy to understand and implement. In the present study, a simple classical model of a multimachine system was considered. However, it explains very well the principles and the scope of the tool, typically for the study of transient stability in a power system. As indicated in the discussions in previous sections, the other factors such as effects of excitation, turbine, speed governor or any control measure, can be easily realized in a Simulink model, especially with the help of readily available and perfectly compatible tools like Power System Block set. It should also be noted that a Simulink model can generate an equivalent C code for embedded applications and for rapid prototyping of control systems. Furthermore, the optimization and application of advanced tools such as ANN and fuzzy logic, is also much easier as there are corresponding toolboxes available within MATLAB.
References 1.
53
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power system, HIGHER.D.PROJECT, The College of Electrics and Electronics Technical - Baghdad-Iraq, 2006. 15.Mahmood ,Pshtiwan Kamal , Fault Location Effect on Power System Stability, HIGHER.D.PROJECT, The College of Electrics and Electronics Technical - Baghdad-Iraq, 2007. 16.P. W. Sauer and M. A. Pai, Power System Dynamics and Stability ,Prentice Hall, Upper Saddle River, New Jersey, 1998. 17.J. Arrillaga, C. P. Arnold, Computer analysis of power systems,Wiley,1990. 18.Xi-Fan Wang, Yong-Hua Song,Malcolm Irving, Modern Power Systems Analysis, Springer Science Business Media, 2008. 19.Al Azzawi, F. J., Al Baldawi, F. F., C. L. Narayana , Fast methods for direct assessment of power system transient stability, IEEE proceedings,Vol.135 ,Pt.C,No.6, November 1988 . 20.B.M weedy, Electrical power system ,John Wiley & sons, 1998. 21.Boldea I. Nasar S.A. , Electric machine dynamics ,printed in U.S.A Machillan Publishing company New York,1986. 22.Abdul-Kareem, A. Amer, Comparative Study of power system transient stability parameter calculation ,M.Sc.thesis, the College of electrical and electronics techniques ,2008 . 23.D. William and Stevenson, Elements of power system analysis, McGrawHill, fourth Edition, USA, 1982 . 24.Lawrence F. Shampine, Numerical Solution of Ordinary Differential Equations, Chapman & Hall, 1994. 25.P. M. Anderson and A. A. Fouad, Power System Control and Stability, Iowa State University Press, Ames, IA, 1977. 26.Grainger, John; Stevenson Jr., William D ,Power Systems Analysis ,McGraw-Hill Education ,1994. 27.M. GOYAL ,Computer-Based Numerical & Statistical Techniques
References (Mathematics) , Firewall Media,2007. 28. J. Duncan Glover, Mulukucla S. Sarma, and Thomas J. Overbye, power
system analysis and design , Thomson Learning ,2008.
55
I
Appendix
Appendix I (generator data) [28]. Generator no. 1 Rated MVA 247.5 kV 16.5 H(s) 23.64 Power factor 1.0 Type Hydro Speed 180r/min 0.1460 0.0608 0.0969 0.0969 0.0336 (leakage) 8.96 0 Stored energy at rated speed 2364MWs
2 192.0 18.0 6.4 0.85 Steam 3600r/min 0.8958 0.1198 0.8645 0.1969 0.0521 6.00 0.535 640MWs
56
3 128.0 13.8 3.01 0.85 Steam 3600r/min 1.3125 0.1813 1.2578 0.25 0.0742 5.89 0.600 301MWs
Note: Reactance values are in p.u on a 100MVA base. All time constants are in seconds.
II
Appendix
Appendix II (Redused Y matrices) [28].
Prefault network: 0.8455 0.2871 0.2096
2.9883 1.5129 1.2256
0.2871 0.4200 0.2133
1.5129 2.7239 1.0879
0.2096 1.2256 0.2133 1.0879 0.277 2.3681
During fault: 0.6568
3.8160 0
0.0701
0 0.6306
0 5.4855 0
0.0701
0.6306 0
0.1740
2.7959
After fault network: 1.1386 0.1290 0.1824
2.2966 0.7063 1.0637
0.1290 0.3745 0.1921
0.7063 2.0151 1.2067
0.1824 0.1921 0.2691
1.0637 1.2067 2.3516
57
III
Appendix
Appendix III %Multimachine transient stability for power system analysis. %Thes code given below should be run prior to simulation shown in %fig.(3-4). %Prameters to operate the case study. clc clear all global n r y yr global Pm f H E Y ngg global rtd dtr %conversion factor rad/degree global Ybf Ydf Yaf f=60; ngg=3; r=9; nbus=r; rtd=180/pi; dtr=pi/180; % Gen. Ra gendata=[1 0 2 0 3 0
%no. of generator %no. of Buses
Xd' 0.0608 0.1198 0.1813
Ybf=[0.8455-2.9883i 0.2871+1.5129i 0.2096+1.2256i
H 23.64 6.4 3.01]; 0.2871+1.5129i 0.2096+1.2256i 0.4200-2.7239i 0.2133+1.0879i 0.2133+1.0879i 0.277-2.3681i];
Ydf=[0.8455-3.8160j 0 0 -5.4855j 0.0701+0.6306j 0
Yaf=[1.1386-2.2966j 0.1290+0.7063j 0.1824+1.0637j
0.1740-2.7959j];
0.1290+0.7063j 0.3745-2.0151j 0.1921+1.2067
fct=input('fault clearing time fct= '); %Damping factors damp1=0.0;
0.0701+0.6306j 0
0.1824+1.0637j 0.1921+1.2067 0.2691-2.3516j];
58
Appendix
damp2=0.0; damp3=0.0; %Initial generator Angels D1=8.9568; D2=3.0658; D3=-1.8534; %Initial Powers Pm1=2.9012; Pm2=1.6300; Pm3=0.8500; %Generator Inrternal Voltages E1=1; E2=1.0515; E3=1.0181; %Machine Ienertia Constants H1=gendata(1,4); H2=gendata(2,4); H3=gendata(3,4); %Machine Xd' xdd1=gendata(1,3); xdd2=gendata(2,3); xdd3=gendata(3,3);
III
59
