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SIMULATION OF THE TRANSIENT
RESPONSE OF SYNCHRONOUS MACHINES
Honours Thesis
by BOK ENG LAW
Supervisor: DR ALAN WALTON
THE UNIVERSITY OF QUEENSLAND
School of Information Technology and Electrical Engineering
Submitted for the degree of Bachelor of Engineering
in the division of Electrical and Electronics Engineering
October 2001
Bok Eng Law
1/157 Hawken Drive
St Lucia
Queensland 4067
19TH October 2001
The Dean
Faculty of Engineering
University of Queensland
St. Lucia QLD 4072
Dear Sir,
In accordance with the requirements of the degree of Bachelor of Engineering in the
division of Electrical and Electronics Engineering, I present the following thesis
entitled ” Simulation of the Transient Response of Synchronous Machines„ . This work
was performed under the supervision of Dr. Allan Walton.
I declare that the work submitted in this thesis is my own except as acknowledged in
the text and footnotes, and has not been previously submitted for a degree at the
University of Queensland or any other institution.
Yours sincerely,
Bok Eng Law
ACKNOWLEDGEMENTS
I
ACKNOWLEDGEMENTS
I would like to thank first and foremost my thesis supervisor Dr Allan Walton, for his
patience, care and guidance given to me throughout the duration of my thesis. Thank
you for your inspiration at times when I was feeling blue.
To my fellow peers and friends whom I have spent the year with ’ thanks for all the
ideas and support.
To my family for the support and encouragement given to strive for my goals.
To my beloved Grandparents, I have fulfilled your wish.
Special thanks to Mr. Ivan Lim Kian Tiong for lending me your laptop during the
seminar.
Last but not least, I would like to dedicate my success to my love - Livins Tay, for
your understanding during this period of time. The love you gave has been
tremendous and invaluable. I wouldnδt have succeeded without you as my other half.
TABLE OF CONTENT
II
CONTENTS
ACKNOWLEDGEMENTS I TABLE OF CONTENTS II LIST OF FIGURES V LIST OF TABLES IX ABSTRACT X CHAPTER 1 INTRODUCTION 1
1.1 Thesis Outline 2
1.2 Thesis Objective 3
1.3 Limitations of The Simulation Model 4
CHAPTER 2 LITERATURE REVIEWS 6
2.1 Determination of Machine Parameters Using Results from The Frequency
Response Tests 7
2.1.1 Operational Inductance 7
2.1.2 Time Constant Extraction 8
2.1.3 Equivalent Circuit Parameters 10
2.2 Nonlinear Excitation Control 12
2.2.1 Feedback Linearization 13
2.2.2 Nonlinear Controller Design 15
2.2.3 Nominal Response of Excitation System 16
2.3 Turbine - Governor Control 18
2.3.1 Relationship of Governor, Turbine And Generator 20
CHAPTER 3 THEORY 22
3.1 The Two-Axis Theorem 22
3.1.1 Direct Axis 24
3.1.2 Quadrature Axis 26
3.2 Inertia Constant and Swing Equation 28
3.3 Power ’ Load Angle 30
TABLE OF CONTENT
III
3.4 Speed Governor and Excitation System 33
3.4.1 Excitation System Model 33
3.4.2 Prime Mover and Governing System Models 34
CHAPTER 4 SIMULATION MODEL DESIGN 38
4.1 Selection of Simulation Software 38
4.1.1 Power Systems Simulator for Engineering (PSS/E) 38
4.1.2 Power Systems Computer-Aided Design (PSCAD) 39
4.1.3 MATLAB ’ Simulink 41
4.2 Concept of Modelling The Synchronous Machine in The Power System 43
4.2.1 Exciter Model 44
4.2.2 Generator Model 46
4.2.3 Sensor Model 47
4.2.4 Automatic Voltage Regulator (AVR) with PID Controller 48
4.2.5 Turbine Model 49
4.2.6 Governor Model 50
4.2.7 Automatic Generation Control (AGC) 51
4.2.8 Combining AGC and Excitation System 53
CHAPTER 5 SIMULATION PROCESS & EVALUATION 56
5.1 Simulation Inputs 56
5.2 Simulation Procedure And Results 59
5.2.1 PID Controller ’ Change in Kp Only 63
5.2.2 PID Controller ’ Change in Ki Only 64
5.2.3 PID Controller ’ Change in Kd Only 64
5.2.4 Change in KI of the AGC Only 65
5.2.5 Change in Excitation Gain (KE) 66
5.2.6 Lower Order Models 67
5.3 Evaluation 68
TABLE OF CONTENT
IV
CHAPTER 6 CONCLUSION & FUTURE IMPROVEMENTS 71
6.1 Future Improvements 71
6.2 Conclusion 72
BIBLIOGRAPHY 75
APPENDIX A ’ PHASOR DIAGRAM 78
APPENDIX B ’ EXCITATION CONTROL SYSTEM 79
APPENDIX C ’ SYSTEM MODEL WITH GOVERNOR & AVR 80
APPENDIX D ’ SIMULATIONS OF LOWER ORDER MODELS 81 D.1 First Order Model Simulation 82
D.2 Second Order Model Simulation 83
D.3 Third Order Model Simulation 84
LIST OF FIGURES
V
LIST OF FIGURES
CHAPTER 2
Figure 2.1 Equivalent circuit of a third order model 7
Figure 2.2 A synchronous generator connected to the infinite bus 12
Figure 2.3 Feedback linearization power system 15
Figure 2.4 System configuration under proposed nonlinear control 15
Figure 2.5 Nominal excitation system response 16
Figure 2.6 Typical arrangement of excitation components 17
Figure 2.7 Steady-state load-control band 19
Figure 2.8 Speed governor and turbine in relationship to generator 20
CHAPTER 3
Figure 3.1 Illustration of the position of d-q axis on a two-pole machine 23
Figure 3.2 Salient-pole rotor with damper windings 23
Figure 3.3 Diagram of windings in the direct axis 24
Figure 3.4 Direct axis equivalent circuit 25
Figure 3.5 Quadrature axis equivalent circuit 26
Figure 3.6 Generator and load block diagram 29
Figure 3.7 Block diagram of a load model derived from the swing equation 29
Figure 3.8 Equal area rule 31
Figure 3.9 Expected dynamic behaviour when α increases 31
Figure 3.10 Expected dynamic behaviour when α decreases 32
Figure 3.11 Governor characteristic 34
Figure 3.12 Speed governing system 35
Figure 3.13 Block diagram of governing system for a hydraulic turbine 36
LIST OF FIGURES
VI
CHAPTER 4
Figure 4.1 Simulation model for PSCAD 39
Figure 4.2 Output responses of the proposed PSCAD model 40
Figure 4.3 Blockset of a nonlinear control of hydraulic turbine and generator 41
Figure 4.4 Block diagram of a fourth order model synchronous machine 42
Figure 4.5 Schematic diagram of governor and AVR of the synchronous machine 43
Figure 4.6 Block diagram of governor and AVR of the synchronous machine 43
Figure 4.7 Block diagram of an exciter model 44
Figure 4.8 Exciter saturation curves 45
Figure 4.9 Block diagram of a simple generator model 46
Figure 4.10 Block diagram of a simple automatic voltage regulator (AVR) 47
Figure 4.11 Block diagram of the proposed AVR system with PID controller 48
Figure 4.12 Isolated power system load frequency control (LFC) block diagram 50
Figure 4.13 Block diagram of AGC in an isolated power system 51
Figure 4.14 Simulation model for the fourth order machine time constants 54
CHAPTER 5
Figure 5.1 Diagram of a fourth order synchronous machine model in MATLAB
Simulink 59
Figure 5.2 Initial PID controller values for fourth order model 60
Figure 5.3 Simulink parameter settings 61
Figure 5.4 Terminal voltage Vt of the fourth order model 62
Figure 5.5 Frequency deviation step response ∆ω of the fourth order model 62
Figure 5.6 ” Zoom in„ detail of Figure 5.4 62
Figure 5.7 Terminal voltage when Kp = 1 63
Figure 5.8 Terminal voltage when Ki = 0.2 64
Figure 5.9 Terminal voltage when Kd = 0.7 64
Figure 5.10 Frequency deviation step response ∆ω when Kd = 0.7 65
Figure 5.11 Terminal voltage when KI = 7 65
LIST OF FIGURES
VII
Figure 5.12 Response of ∆ω when KI = 7 65
Figure 5.13 Response of Vt when KE = 10 66
Figure 5.14 Response of ∆ω when KE = 10 66
Figure 5.15 Output voltage response when Kp is set too high 68
Figure 5.16 Output voltage response when Kp is set too low 68
Figure 5.17 ” Zoom in„ response of terminal voltage with the new setting 69
Figure 5.18 New possible feedback loop 69
LIST OF FIGURES
VIII
APPENDIX A
Figure A.1 Phasor diagram of a synchronous machine in steady state 78
APPENDIX B
Figure B.1 Synchronous excitation control system 79
APPENDIX C
Figure C.1 Power system block diagram with governor and voltage regulator 80
APPENDIX D
Figure D.1 Diagram of first order synchronous machine model in Simulink 82
Figure D.2 Frequency deviation step response ∆ω of the first order model 82
Figure D.3 ” Zoom in„ terminal voltage Vt of the first order model 82
Figure D.4 Diagram of second order synchronous machine model in Simulink 83
Figure D.5 Frequency deviation step response ∆ω of the second order model 83
Figure D.6 ” Zoom in„ terminal voltage Vt of the second order model 83
Figure D.7 Diagram of third order synchronous machine model in Simulink 84
Figure D.8 Frequency deviation step response ∆ω of the third order model 84
Figure D.9 ” Zoom in„ terminal voltage Vt of the third order model 84
LIST OF TABLES
IX
LIST OF TABLES
Table 4.1 Classification of steam turbine 49
Table 5.1 Optimum time constants 56
Table 5.2 Values of the constants required for turbine and governing system 57
Table 5.3 Values of the constants required for excitation control system 57
ABSTRACT
X
ABSTRACT
Modern power systems are highly complex and non-linear and their operating
conditions can vary over a wide range. The overall accuracy of the system is primarily
decided by how correctly the synchronous machines within the system are modelled.
In most cases, the second order model of synchronous generator is used as it is
assumed to be sufficient to simulate the response of the machine. Yet this is
inadequate for transient study as units of microseconds or milliseconds are crucial to
the performance of the synchronous machine. Hence, there is a need to analyse
exclusively the model of synchronous machine in the power system.
A simulation model of a basic power system is set up to examine the response of the
synchronous machine during transient state. The power system simulation model is
designed to manage lower order (first and second order) machine time constants and
as well as handling higher order (third and fourth order) machine time constants. The
effects of using the PID controllers comprising a higher order model of a synchronous
machine in the power system are investigated and discussed. The other influencing
factors of using different types of turbines and various component parts within the
power system are briefly discussed.
This thesis demonstrates the simulation of the transient response of synchronous
machine connected to an infinite bus. Several improvements on the simulation model
are included. With proper modelling of the synchronous machine in the power system,
a better understanding of how the machine reacts under sudden large disturbances
during transient conditions can be achieved and hence a better controller of the
synchronous machine can be designed.
ABSTRACT
XI
CHAPTER 1
1
CHAPTER 1
1. INTRODUCTION
Modern power systems are highly complex and non-linear. Their operating conditions
can vary over a wide range. In stability studies, the overall accuracy is primarily
decided by how correctly the synchronous machines within the system are modelled.
Power system stability can be defined as the tendency of power system to react to
disturbances by developing restoring forces equal to or greater than the disturbing
forces to maintain the state of equilibrium (synchronism). Stability problems are
therefore concerned with the behaviour of the synchronous machine after they have
been perturbed. The increasing size of generating units, the loading of the
transmission lines and the operation of high-speed excitation systems nearer to their
operating limit are the main causes affecting small signal stability of power systems.
Generally, there are three main categories of stability analysis. They are namely
steady state stability, transient state stability and dynamics stability. Steady state
stability is defined as the capability of the power system to maintain synchronism
after a gradual change in power caused by small disturbances. Transient state stability
refers to as the capability of a power system to maintain synchronism when subjected
to a severe and sudden disturbance. This disturbance in the network connections is
brought about by faults and by sudden large increment of loads. The third category of
stability, which is the dynamic stability, is an extension of steady state stability. It is
concerned with the small disturbances lasting for a long period of time.
This thesis is focused on the transient response and stability of synchronous machine
in a typical power system using higher order models of synchronous machines.
CHAPTER 1
2
1.1 THESIS OUTLINE
This thesis provides a means of determining the transient response of any
synchronous machine in a power system by computerised simulation. Extensive
studies of various component parts are essential to closely simulate a working model.
Some of these studies include:
• Speed governor control
• Automatic voltage regulator (AVR) control
• Effects of using different types of prime mover (turbine)
• Proportional Integral Derivative (PID) control for excitation system
• Direct-quadrature axis theorem
Due to the wide scope of studies in power systems, this thesis will be limited to the
study of the synchronous machine only. The focus will be on designing the power
system that manages lower order (first and second order) machine time constants and
is capable of handling higher order (third and fourth order) machine time constants
accurately. The thesis will also discuss the effects that the PID controllers and
feedback control circuits have when comprising a higher order model of a
synchronous machine in the power system. The use of different AVRs, turbines and
governors will not be included in the scope of this thesis so as to probe into the effects
of the synchronous machine only.
With proper modelling of the synchronous machine in the power system, we can
better understand how the machine reacts under sudden large disturbances during
transient conditions and hence design a better controller of the synchronous machine.
This thesis will be the pioneering study of the simulation of the transient response
using higher order model of synchronous machines and will serve as a basis for
simulation of more comprehensive power systems in the future theses.
CHAPTER 1
3
1.2 THESIS OBJECTIVE
Generally, most studies of the power system assume that using second order model of
synchronous generator would be sufficient to simulate the response of the machine.
This, however, is inadequate for transient study as units of microseconds or
milliseconds are crucial to the performance of the synchronous machine. Hence, there
is a need to analyse exclusively the model of synchronous machine in the power
system. A simulation model of a basic power system will be set up to examine the
response of the synchronous machine during transient state.
The inclusion of a power system stabilizer in the power system may not be necessary
if the response of synchronous machine is correctly understood. The effects of using
higher order models of a synchronous machine will be investigated and its possible
responses and effects on the conventional elements in the power system will be
examined.
The aim of this thesis is therefore to produce a program that can closely simulate the
operation of the synchronous machine using a range of transfer functions in order to
determine the transient response for any synchronous machine.
CHAPTER 1
4
1.3 LIMITATIONS OF THE SIMULATION MODEL
There are some assumptions made prior to the design of the simulation model. They
are as follows:
q A single turbine is used and will produce a constant torque with a constant
speed maintained during steady state operation (at synchronous speed).
q The output terminals of the generator are connected to infinite busbar that has
constant load.
q Only basic and linear models of the power system components (i.e. turbines,
feedback sensors, exciter, governor etc) will be used except for the model of
synchronous generator.
q The time constants of the synchronous machine used in this thesis are assumed
to be the optimum time constants extracted based on the values given in
Walton [1].
q The investigations beyond fourth order model are outside the scope for this
thesis.
CHAPTER 1
5
CHAPTER 2
6
CHAPTER 2
2. LITERATURE REVIEW
Many comprehensive articles, journal and conference papers can be found describing
the investigation of the synchronous machine and its operational parameters. In spite
of this, none of these papers directly analyses the response of higher order models of
the synchronous machine during the transient state. Most authors have simply taken
the second order models as their reference to discuss their investigation. However,
this may be inadequate in some cases when the precision of machine response
matters. Nevertheless, some literatures that are indirectly related to this thesis are used
as a basis in this discussion.
Discussion of a wide range of related issues, generated from the study of these
publications and investigations is required as a foundation for this thesis. The
following sections in this chapter will discuss about some conference papers and
articles.
CHAPTER 2
7
2.1 DETERMINATION OF MACHINE PARAMETERS USING RESULTS
FROM THE FREQUENCY RESPONSE TESTS
The process for the extraction of the time constants of synchronous machines obtained
from the results of frequency response tests has been developed since the 1950s [2]. It
has become evident that frequency response methods are of major benefit in
determining machine parameters especially over the more traditional methods of
sudden short circuit and open circuit tests that may cause damage to the machines.
The advantage of doing so is that the conventional sudden short circuit tests can only
be used to determine parameters of second order models in direct axis while the
standstill frequency response (SSFR) tests are capable of achieving information on
both the direct and quadrature axis parameters. The paper by Walton [1] on the
method for determining parameters of synchronous machines from the results of
frequency response tests, describes three stages of the test which are:
q The conversion of impedance to an operational inductance,
q The extraction of the time constant of the machine from the operational
inductance
q The determination of parameters of the branches of the equivalent circuit for
the machine using these time constants and inductances.
2.1.1 Operational Inductance
Figure 2.1 Equivalent circuit of a third order model [3]
CHAPTER 2
8
The use of higher order models requires an equivalent circuit of the form shown in
Figure 2.1. The operational inductance Ld(s) can be obtained from the measured
impedance Zd(s) from the equation as follows:
Ld(s) = Zd(s) - Ra
s ----- (2.1)
where Ra is the armature resistance
The asymptotic value of Ld(s), regardless of the order of model used, will be Ld
(which is equal to La + Lm) as the frequency tends to zero. Eventually, the equation for
Ld(s) is just transfer function of the time constants. An example of the transfer
function of Ld(s) for a fourth order equivalent circuit would be:
Ld(s) = Ld (1 + sT1) (1 + sT3) (1 + sT5) (1 + sT7)
(1 + sT2) (1 + sT4) (1 + sT6) (1 + sT8) ----- (2.2)
where T1, T3, T5, T7 and T2, T4, T6, T8 are the time constants of the zeros and poles
respectively that are required to be found. Further discussion in obtaining these time
constants of the poles and zeros using direct axis and quadrature axis can be found in
the section of next chapter on the two-axis theorem.
2.1.2 Time Constant Extraction
Using an analytical approach and applying characteristic of lag circuits rather than
numerical curve-fitting techniques, it is possible to extract the time constants of the
synchronous machine to a better degree of accuracy. The extraction of these time
constants is based on the fact that the circuit must be represented by a series of poles
and zeros in the complex frequency domain along the negative real axis. They are
CHAPTER 2
9
produced by the individual R-L branches connected in parallel in the circuit shown in
Figure 2.1.
Two characteristics of a lag function are shown as:
(i) the maximum phase lag (φ) at the center frequency (λ m) of the pole-zero pair (Fc)
is determined from
sin φ = (π - 1)
(π + 1) ----- (2.3)
(ii) the overall gain change due to pole-zero pair is given by
∆Gain (dB) = -20 log π ----- (2.4)
The time constant values of the pole (Tp) and zero (Tz) can then be obtained from
Tp = √π
2ω Fc ----- (2.5)
Tz = Tp
π
----- (2.6)
The values of π and Fc can be obtained directly from the operational inductance data
since it is easy to identify the point in the frequency and the phase domain at which
the peak occurs. Hence, this simplifies the calculation of Tp and Tz from the given
equation (2.5) and (2.6). Measurement errors occur in both the phase and magnitude
which can be used in uniquely different ways to determine the best time constants.
The optimisation process involves varying Fc and π rather than Tp and Tz, and then re-
evaluating the time constants using the equations from (2.2) to (2.6). The process of
varying Fc is to adjust its value by 10% about the initial value. A similar approach is
used to find the optimum value of π, but the variation is much smaller as the change
in π has greater effects than the variation of Fc.
CHAPTER 2
10
2.1.3 Equivalent Circuit Parameters
Ld(s) is the leakage reactance in series with the parallel combination of the
magnetising reactance and rotor impedance Zr(s). The expression for Zr(s) is therefore
Zr(s) = sLm[Ld(s) - La]
Lm + La - Ld(s) ----- (2.7)
and the rotor impedance can also be interpreted as
Zr = Rp (1 + sTf) (1 + sTj) (1 + sTk)
(1 + sTx) (1 + sTy) ----- (2.8)
where Rp is the parallel combination of the three rotor circuit resistances
suffixes f, j and k refer to the rotor branches in Figure 2.1.
Tx and Ty, are linearly related to the time constants of Ld(s) and the values of
Lm and La.
The relationship between the time constants and the unknown parameters can be
calculated using the matrix given below:
Through developing the variables using the same technique and deriving additional
time constants from the original response, this method can be used to determine for
higher order models of the synchronous machine.
1
Tx +Ty
Tx * Ty
----- (2.9)
CHAPTER 2
11
With this information, this thesis is able to probe more in depth by using these
optimum time constants to simulate the operation of the machine closely.
Because this thesis is concerned with the modeling of the generator, it is axiomatic
that the AVR will be most important and that the governor will be of secondary
importance.
CHAPTER 2
12
2.2 NONLINEAR EXCITATION CONTROL
In the paper by Kennedy [4] on a nonlinear geometric approach to power system
excitation and stabilization, a method of solving dynamic instability by adding a
power system stabilizer to the excitation controller is described. Also, the paper
proposed a nonlinear geometric control by using the input-output feedback
linearization to transform into the state space system model so that the terminal
voltage becomes a linear function of the control input. In addition to these, it can be
tuned to provide optimum damping of power angle oscillations at a particular
setpoint. Consequently, the controller is capable of tracking step changes in reference
voltage exactly without using a high gain as is normally required.
A Parkδs third order, nonlinear, time-invariant, state space model of a synchronous
generator is used in this paper [4]. As illustrated in Figure 2.2, the system is modeled
by a constant voltage source with constant magnitude and frequency that is also
known as infinite bus system.
Figure 2.2 A synchronous generator connected to the infinite bus
CHAPTER 2
13
2.2.1 Feedback Linearization
The selected state variables of the synchronous generator/infinite bus system model
are the power angle α, the power angle derivative λ , and the field flux linkage º f,
resulting in a system model described by
where the control input is the field voltage vf. The electric power Pe and the field
current if are nonlinear functions of the state shown as
A, B, C, D and B are constant matrices and their values depend on the physical
parameters of the system. The control output is the terminal voltage vt which is shown
as
Similarly, G and H are constant matrices and their values also depend on the system
parameters. The matrix G is nonzero which is the case in practice. The main purpose
----- 2.10
----- 2.11
----- 2.12
----- 2.13
----- 2.14
----- 2.15
CHAPTER 2
14
of these equations is to force vt to track a predetermined reference while ensuring the
power angle α is within the desired operating range. However, the nonlinear
characteristic of the system model hinders the process of achieving this purpose.
A more desirable way to overcome this problem is by using the technique of input-
output feedback linearization. The details of this technique will not be included in this
discussion as the main objective is to discover how nonlinearities in the system affect
the excitation control element. When this technique is being applied, it is capable of
holding on to a large part of the state space and no practical limitation on the
operating region is required.
As described in Kennedy [4], the following state space equations are obtained by
applying the input-output feedback linearization technique:
Defining v as new control signal, the control input vf is therefore
The system equations will then become
----- 2.16
----- 2.17
----- 2.18
----- 2.19
----- 2.20
----- 2.21
----- 2.22
CHAPTER 2
15
2.2.2 Nonlinear Controller Design
The power system can be divided into
(i) a linear electrical subsystem [equation (2.20)] which depends solely on
control input v, and
(ii) the remaining subsystem [equation (2.21) and (2.22)] which represents the
mechanics of motion driving the electrical subsystem.
The process of feedback linearization is illustrated in Figure 2.3.
Figure 2.3 Feedback linearization power system [4]
The proposed control is shown in the block diagram of Figure 2.4.
Figure 2.4 System configuration under proposed nonlinear control [4]
CHAPTER 2
16
The design of the controller should provide a means of asymptotic tracking of the
reference signal vref and damp the power angle oscillations. Under the proposed
control, the feedback voltage (vt - vref) will handle the tracking requirement and the
power angle derivative λ is fed back to damp the power angle oscillations. However,
the power angle α is not required to be fed back because it would interfere with the
tracking component as it has a nonzero value at steady state.
2.2.3 Nominal Response of Excitation System
In the IEEE standard definitions [5] for excitation system for synchronous machines,
the nominal excitation system response is defined as the rate of increase of the
excitation system output voltage determined from the excitation system response
curve, divided by the rated field voltage as illustrated in Figure 2.5. It should be
understood that the ideal excitation response is ac rather than ab. Therefore in most
cases, the output response of the excitation is assumed to be linear which is not the
case in practice as saturation occurs.
Figure 2.5 Nominal excitation system response
Even though the simulation model proposed in this thesis is hoped to be adequate
using linear excitation control without using the power system stabilizer, it is
CHAPTER 2
17
important to understand the characteristic of other forms of excitation control so as to
know the limitations of the simulation model in this thesis.
The excitation control is one of the important factors in the transient study of power
system analysis. Through understanding the paper by Kennedy [4] and the
ANSI/IEEE Standard [5], it is normally the requirement to have a high gain for the
excitation controller as an effective means of providing transient stability. In this way
when a disturbance occurs, the excitation controller can moderate the control signal
quickly and provide good damping of oscillations in the system. A typical relationship
between the excitation control system and the generator is illustrated in Figure 2.6.
Figure 2.6 Typical arrangement of excitation components
Generator
Auxiliary Control
Exciter Voltage Regulator
Input torque from
prime mover
Exciter power source
Output voltage and current
Desired voltage
CHAPTER 2
18
2.3 TURBINE - GOVERNOR CONTROL
There is a need to consider the speed/load control transient response of a power
system as specified in ANSI/IEEE Std 122-1985 [6]. An understanding of the
characteristics of typical turbine is essential in power system studies. The prime
mover plays a vital role in contributing to the stability of the whole system. Optimum
transient response of a closed loop control system to an external disturbance depends
not just on the transfer function of the excitation controller, generator and sensors but
also the speed/load controller as well.
Various types of steam turbines have been introduced in this standard and have been
classified according to their functions and characteristics. Several speed regulations
were briefly mentioned and definitions of various terms in the area of
turbine/governor were defined. Instructions towards setting the regulations were given
to handle specified models. This standard gives a good overview of how the detail of
steam turbine is being illustrated in block diagrams.
In terms of speed regulation, different types of turbine have different ways of
calculating the regulation. Taking automatic extraction and mixed pressure turbines
for example, the speed regulation will be
Rs = No ’ Nm
Nr
* Pr
Pm * 100% ----- (2.23)
where Rs = steady-state speed regulation
No = speed at zero power output
Nr = rated speed
Nm = speed at Pm
Pm = maximum power output at which zero extraction or induction conditions
are permitted
Pr = rated power output
CHAPTER 2
19
For all other types of turbine, the speed regulation can be expressed as follows:
Rs = No ’ N
Nr
* 100% ----- (2.24)
where Rs = steady-state speed regulation
No = speed at zero power output
Nr = rated speed
N = speed at rated power output
Careful consideration in selecting the turbine model is essential as from the above
examples, it is evident that there are different operating characteristics when using
various turbine models for simulation.
Stability of the turbine depends on the way the speed/load-control system positions
the control valves so that a sustained oscillation of the turbine speed or of the power
output as produced by the speed/load-control system does not exceed a specified
value during operation under steady-state load demand or following a change to a new
steady-state load demand. This steady-state load demand is being expressed in terms
of a range of values in a control band. This band is called steady-state load-control
band ∆Pb, which is shown in Figure 2.7
Figure 2.7 Steady-state load-control band [6]
∆Pb
Time
Power
CHAPTER 2
20
2.3.1 Relationship of Governor, Turbine And Generator
Figure 2.8 Speed governor and turbine in relationship to generator [7]
The turbine-governor models are designed to give representations of the effects of the
power plants in the power system stability. [7] A functional diagram of the
representation used and its relationship to the generator is exemplified in Figure 2.8.
Various kinds of turbine can be found for different environments. They are ranged
from the commonly used gas turbine, hydro turbines, to steam turbine. Some of the
characteristics of these turbines can be found in the PSS/E User Handbook [7].
Even though in this thesis, the main focus is on the synchronous machine rather than
the prime mover and its control element, it is useful in understanding how different
types of turbines contribute to the stability of the power system.
CHAPTER 2
21
CHAPTER 3
22
CHAPTER 3
3. THEORY
In power system studies, there are many elements affecting stability of the system.
These factors require to be addressed before proceeding to design any simulation
program for the power system.
For example, when stability analysis involves simulation times longer than about one
second, any effects due to machine controllers such as automatic voltage regulators
(AVR) and speed governor must be incorporated. The AVR has a substantial effect on
transient stability when varying the field voltage to try to maintain the terminal
voltage constant. On the other hand, we should not discard the stability factor
contributed by the turbine in the system as the variation of mechanical power may
occur from time to time.
Given these reasons, we are required to have necessary background knowledge in
order to understand the actual processes that take place in the power system in order
to design a power system simulation as closely as possible. The following sections are
essential in order to commence on the design of simulation model in this thesis.
3.1 THE TWO-AXIS THEOREM
The electrical characteristic equations describing a three-phase synchronous machine
are commonly defined by a two-dimensional reference frame. This involves in the use
of Parkδs transformations [8] to convert currents and flux linkages into two fictitious
windings located 90η apart. A typical synchronous machine consists of three stator
windings mounted on the stator and one field winding mounted on the rotor. These
axes are fixed with respect to the rotor (d-axis) and the other lies along the magnetic
neutral axis (q-axis), which model the short-circuited paths of the damper windings.
CHAPTER 3
23
Electrical quantities can then be expressed in terms of d and q-axis parameters. Figure
3.1 presents the diagram of d-q axis in the machine. Phasor diagram [9] of the
synchronous machine for steady state has been included in Appendix A.
Figure 3.1 Illustration of the positions of d-q axis on a two-pole machine [10]
There is the need for damper windings to reduce mechanical oscillations of the rotor
around the synchronous speed. The damper windings act in both the d-axis and q-axis,
however not equally. Illustrated in Figure 3.2 is the general construction of the
damper windings on the poles of the rotor.
Figure 3.2 Salient-pole rotor with damper windings
CHAPTER 3
24
There have been several methods used to determine the parameters of a synchronous
machine. All of these methods base their analysis on acquiring the operational
inductance - obtaining some time constants from the inductance data and then using
this to determine the parameters of the machine.
3.1.1 Direct Axis
When a synchronous machine is running at synchronous speed with no field current
flowing and with the field winding slip rings short-circuited, the total flux linkages ’ f
with the field windings are:
’ f = ( Lf + Md ) If ’ Md Id ≡ 0 ----- (3.1)
where Lf = leakage inductance of field winding
La = leakage inductance of armature winding
Md = mutual inductance between the field and d-axis winding
If = current in field winding
Id = current in d-axis winding
and Ra = Rf = 0
These windings are illustrated in Figure 3.3.
Figure 3.3 Diagram of windings in the direct axis
CHAPTER 3
25
With the aid of the diagram shown in Figure 3.4, the transfer function [11] for the
direct axis operational inductance can be expressed as follows:
Ld(s) = (1 + sTdδ)(1 + sTd„ )
(1 + sTd0δ)(1 + sTd0„ ) Ld ----- (3.2)
The direct axis reactance during transient is not the same as that in the steady state.
The value of Xd to be used during transients is called the direct axis transient
reactance Xdδ.
Xdδ = Xa + Xmd
Xf
Xmd + Xf ----- (3.3)
From this equation, it is obvious that the armature leakage reactance is in series with
the parallel combination of Xmd and Xf. Figure 3.4 shows the direct axis equivalent
circuit including the winding resistances.
Figure 3.4 Direct axis equivalent circuit
Since during transients the flux linkages with the field winding change, they will also
change with any closed circuit on the rotor.
CHAPTER 3
26
The leakage reactance of damper windings is negligible in the steady state but during
sub-transient and transient state, it will be significant as it affects the time constants in
those periods. The equation for direct axis sub-transient reactance is:
Xdδδ = Xa +
Xmd Xf Xkd
Xmd Xf + Xmd Xkd + Xf Xkd
----- (3.4)
3.1.2 Quadrature Axis
The quadrature axis equivalent circuit as shown in Figure 3.5 is similar to direct axis
equivalent circuit but it has no field winding [11].
Figure 3.5 Quadrature axis equivalent circuit
Xqδδ = Xa + Xmq
Xkq
Xmq + Xkq ----- (3.5)
From Figure 3.5, the quadrature axis sub-transient reactance can be determined as
shown in equation (3.5).
With the diagram shown in Figure 3.5, the transfer function [11] for the quadrature
axis operational inductance can be expressed as:
CHAPTER 3
27
Lq(s) = (1 + sTq„ )
(1 + sTq0„ ) Lq ----- (3.6)
Consequently, various time constants can be obtained as follows [11]:
Td0δ = 1
λ 0 Rf (Xmd + Xf) ----- (3.7)
Tdδ = 1
λ 0 Rf ( Xf +
Xmd Xa
Xmd + Xa ) ----- (3.8)
Td0δδ = 1
λ 0 Rkd ( Xkd +
Xmd Xf
Xmd + Xf ) ----- (3.9)
Tdδδ = 1
λ 0 Rkd ( Xkd +
Xmd Xa Xf
Xmd Xf + Xmd Xa + Xf Xa
) ----- (3.10)
Tq0δδ = 1
λ 0 Rkq (Xkq + Xmq) ----- (3.11)
Tqδδ = 1
λ 0 Rkq ( Xkq +
Xmq Xa
Xmq + Xa ) ----- (3.12)
Tkd = Xkd
λ 0 Rkd ----- (3.13)
CHAPTER 3
28
3.2 INERTIA CONSTANT AND SWING EQUATION
The stability of a synchronous machine depends on the inertia constant and the
angular momentum. The rotational inertia equations describe the effect of unbalance
between electromagnetic torque and mechanical torque of individual machines. By
having small perturbation and small deviation in speed, the swing equation [12]
becomes:
d∆λ
dt =
1
2H (∆Pm - ∆Pe) ----- (3.14)
where H = per unit inertia constant
∆Pm = change in per unit mechanical power
∆Pe = change in per unit electrical power
∆λ = change in speed
After Laplace transformation, equation (3.14) will then become
∆λ (s) = 1
2Hs [∆Pm(s)- ∆Pe(s)] ----- (3.15)
A more appropriate way to describe the swing equation is to include a damping factor
that is not accounted for in the calculation of electrical power Pe. Therefore a term
proportional to speed deviation should be included. The speed-load characteristic of a
composite load describing such issue is approximated by
∆Pe = ∆PL + KD∆λ ----- (3.16)
where KD is the damping factor or coefficient in per unit power divided by per unit
frequency. KD∆λ is the frequency-sensitive load change and ∆PL is the nonfrequency-
sensitive load change. Figure 3.7 presents a block diagram representation derived
from the swing equation using equation (3.16).
CHAPTER 3
29
∆Pm(s)
∆PL(s)
∆– (s)
∆Pm(s)
∆PL(s)
∆– (s)
Figure 3.6 Generator and load block diagram
Figure 3.7 Block diagram of a load model derived from the swing equation
∑ 1
2Hs + KD
∑ 1
2Hs
KD
CHAPTER 3
30
3.3 POWER न LOAD ANGLE
Generator control is needed to keep the operation of generator stable as soon as
possible after disturbances caused by some unexpected system faults. Two
performance indices are concerned. One is the system transfer capability. The more
power is transferred, the better it is. The other is the oscillating time, or system
damping. The faster, the better it is. To achieve this, consider a single machine
connected to an infinite bus system, the power output of generator can be expressed as
Pe = EgVt
Xs sinα ----- (3.17)
where Eg = generated EMF
Vt = constant terminal voltage of the infinite bus
Xs = constant synchronous reactance of the machine
If there is a fault occurred within the power system, the machine would operate along
Curve II during the fault period as shown in Figure 3.8. When the fault disappears, the
machine would operate along Curve I. Area A is the accelerating energy and Area B
is the decelerating energy. In order to damp system as soon as possible, Area A and B
must be minimized which can be achieved by either reducing the mechanical power
Pm input, or increasing the electrical power Pe output.
CHAPTER 3
31
Figure 3.8 Equal area rule of generator oscillation in first swing
The expected running curve is Curve IIδ during the fault period and Curve Iδ after
fault. Then the maximum internal angle is decreased from α2 to α2δ. This operation can
be achieved by increase the voltage and decrease Pm. The behavior after the first
swing will follow same argument: increasing the voltage and decreasing the
mechanical power when machine is in acceleration, decreasing the voltage and
increasing the mechanical power when machine is in deceleration. Figure 3.9 and 3.10
describe the processes as mentioned [9].
Figure 3.9 Expected dynamic behavior when α increases
CHAPTER 3
32
Figure 3.10 Expected dynamic behavior when α decreases
CHAPTER 3
33
3.4 SPEED GOVERNOR AND EXCITATION SYSTEM
The issue of power system stability is becoming more crucial. The excitation and
governing controls of the generator play an important role in improving the dynamic
and transient stability of the power system. Typically the excitation control and
governing control are designed independently. Changes in the values of these controls
affect the transient response of the machine. Different types of governors and AVRs
would then have different output characteristics that must be considered in this thesis
in order to simulate the response with a set of accurate time constants of the
synchronous machine.
3.4.1 Excitation System Model
Typically the excitation system is a fast response system where the time constant is
small. Its basic function is to provide a direct current to the field winding.
Furthermore, the excitation system performs control and protective functions essential
to secure operation of the system by controlling the field voltage. Hence the field
current is within acceptable levels under a range of different operating conditions.
The protective functions of the excitation system ensure that the limits of the
synchronous machine, excitation system and other controlling equipments are not
exceeded. Its control functions include the monitoring of voltage and reactive power
flow. These contribute as an important factor in power system stability. Appendix B
illustrates typical excitation systems within a control system.
The design of a simulation model based on a single machine connected to infinite bus
system is normally used as shown in Figure 2.2. The regular governing control is a
traditional PID control, which is similar to IEEE type 1 model. The excitation control
in this thesis will assume a linear optimal control.
CHAPTER 3
34
3.4.2 Prime Mover and Governing System Models
The prime mover governing system provides a means of controlling real power and
frequency. The relationship between the basic elements associated with power
generation and control is shown in Figure 2.7. A basic characteristic of a governor is
shown in Figure 3.11.
Figure 3.11 Governor characteristic
From Figure 3.11, there is a definite relationship between the turbine speed and the
load being carried by the turbine for a given setting. The increase in load will lead to a
decrease in speed. The example given Figure 3.11 shows that if the initial operating
point is at A and the load is dropped to 25%, the speed will increase. In order to
maintain the speed at A, the governor setting by changing the spring tension in the fly-
ball type of governor will be resorted to and the characteristic of the governor will be
indicated by the dotted line as shown in Figure 3.11.
Figure 3.11 illustrates the ideal characteristic of the governor whereas the actual
characteristic departs from the ideal one due to valve openings at different loadings
[14].
In contrast to the excitation system, the governing system is a relatively slow response
system because of the slow reaction of mechanic operation of turbine machine.
SPEED ω
LOAD PM
99% 98%
25% 50%
A ∆ω
Slope = -R where R = speed regulation
CHAPTER 3
35
Figure 3.12 Speed governing system [14,18]
In Figure 3.12, the schematic diagram of a speed governing system that controls the
real power flow in the power system is shown. As shown, the speed governor is made
up of the following parts:
1. Speed Governor: As shown in Figure 3.12 is a fly-ball type of speed
governor. The mechanism provides upward and
downward vertical movements proportional to the
change in speed.
2. Linkage Mechanism: Provide a movement to the control valve in the
proportion to change in speed.
3. Hydraulic Amplifier: Low power level pilot valve movement is converted
into high power level piston valve movement which is
necessary to open or close the steam valve against high
pressure steam.
4. Speed Changer: Provides a steady-state power output setting for the
turbine.
Lower
Raise
Speed Governor
Hydraulic amplifier
To open
To close
To governor -controlled
valves
Speed Changer
CHAPTER 3
36
When selecting a prime mover to model in the simulation, special considerations are
required as different types of turbine required different operating conditions and
hence the effects on the power system stability will be different.
Using an example of hydraulic turbines, a large transient (temporary) droop with a
long resetting time is needed for stable control performance because of the ” water
hammer„ effect, a change in gate position generates an initial turbine power change
which is opposite that which is desired. The transient droop provides a transient gain
reduction compensation that limits the gate movement until the water flow and power
output have time to catch up. Figure 3.13 describes this process using block diagram.
Figure 3.13 Block diagram of governing system for a hydraulic turbine [9]
1
1 + sTP Ks
1
s
1
1 + sTG
Dead Band
Pilot Valve and servomotor μ max open
μ max close
max gate position=1
min gate position=0
RP
RT sTR
1 + sTP
Gate servomotor
Gate Position �
Permanent droop
Transient droop X 2
λ ref
λ r
CHAPTER 3
37
CHAPTER 4
38
CHAPTER 4
4. SIMULATION MODEL DESIGN
This chapter discusses the selection of simulation software, derivation of the
simulation model and the implementation of the proposed design model. The design
models must be able to meet the thesis objective specified despite having limitations
and assumptions listed under Chapter 1.
4.1 SELECTION OF SIMULATION SOFTWARE
Currently, there are many software programs available for analyzing comprehensive
power system simulation. There were three possible choice of simulation software
available for this thesis. They are listed in the following sections.
4.1.1 Power Systems Simulator for Engineering (PSS/E)
There are a wide variety of electromechanical equipment models in PSS/E library. Its
ability to interface with other data formats has gained recognition from the IEEE. The
advantage that PSS/E has is its software package provides comprehensive models of
power system components and details of such models are printed on its operation
manual. The disadvantage is that the ready-made models are tedious to modify.
Complex FORTRAN programming is required before the user can modify any
simulation models in its library. Therefore, it is less user-friendly as compared with
the selected software. Moreover, there are few articles and books that make use of the
new PSS/E as the simulation program. In addition to these, PSS/E is mainly used for
simulation of faults in the transmission lines, transformers or buses.
CHAPTER 4
39
4.1.2 Power Systems Computer-Aided Design (PSCAD)
PSCAD consists of a set of programs which enable the efficient simulation of a wide
variety of power system networks. With the integration of EMTDC (Electromagnetic
Transient and DC) functions, it is suitable for transient simulation. Figure 4.1 shows a
simulation model of synchronous machine.
Figure 4.1 Simulation model for PSCAD
Using this PSCAD model of synchronous machine, the output responses of several
variables are obtained and shown in Figure 4.2. The synchronous machine model
developed for PSCAD/EMTDC is based on Parkδs equations, with damping windings
and a solid-state exciter [15].
CHAPTER 4
40
Figure 4.2 Output responses of the proposed PSCAD model.
These output responses are for a second order model being generated by using the
built-in functions of PSCAD. An investigation in a later stage found that the built-in
functions in PSCAD are incapable of handling machine models that are higher than a
CHAPTER 4
41
second order model. The new creation of a higher order machine model is only
possible through extensive FORTRAN programming. Moreover during the research
of using PSCAD, it has shown signs of program ” bugs„ which sometimes disable the
execution of simulating the model. Under such circumstances, a more stable and
powerful software is required for this thesis.
4.1.3 MATLAB ” Simulink
The final option was to use MATLAB Simulink to design the model. However, there
are two ways in Simulink to design the machine model which are:
1. Using power system blockset [16] which is a set of ready-made machine
models in MATLAB Simulink.
2. Using blocks of transfer functions of the machine to manipulate the design
model.
Figure 4.3 illustrates a power system blockset model of nonlinear control of a
hydraulic turbine and a synchronous generator. The limitation of using blockset is
similar to PSS/E and PSCAD, as most of the ready-made models of the synchronous
machine cannot handle higher order time constant inputs.
Figure 4.3 Blockset of a nonlinear control of hydraulic turbine and generator
CHAPTER 4
42
VF(s) Vt(s)
First order time constants
Second order time constants
Third order time constants
Fourth order time constants
However, using blocks of the transfer function to represent the components in the
power system is capable of having higher order machine time constants as inputs.
This can be achieved by the illustration shown in Figure 4.4.
Figure 4.4 Block diagram representing a fourth order model synchronous machine.
where KG = Gain of the generator
Tz = Time constant of the zero
Tp = Time constant of the pole
VF = Field voltage of the synchronous generator
Vt = Terminal voltage of the synchronous generator
As a result, MATLAB Simulink was chosen for its flexibility in terms of designing
the simulation model and its powerful solvers for solving transfer function equations.
(1 + sTz2)
(1 + sTp2)
(1 + sTz3)
(1 + sTp3)
(1 + sTz1)
(1 + sTp1)
(1 + sTz4)
(1 + sTp4)
KG
CHAPTER 4
43
v
ω
Vref Vout
4.2 CONCEPT OF MODELLING THE SYNCHRONOUS MACHINE IN THE
POWER SYSTEM
In order to design the simulation program, a schematic diagram of the required
components for the simulation is shown in Figure 4.5.
Figure 4.5 Schematic diagram of governor and AVR of the synchronous machine
A simplified block diagram of this schematic is shown in Figure 4.6 below.
Figure 4.6 Block diagram of governor and AVR of the synchronous machine
In Figure 4.5 and 4.6, the diagrams give a general view of how the synchronous
machine should be modelled. However in order to incorporate the functions that can
accommodate higher order time constants, the block diagram in Figure 4.6 will need
to be explicitly redefined.
∑
Turbine
Generator ∑
Automatic Voltage Regulator (AVR)
Load Frequency Control (LFC)
CHAPTER 4
44
VR(s) VF(s)
4.2.1 Exciter Model
The most important component other than the synchronous machine in the power
system is the excitation system. The most basic form [12] of expressing the exciter
model can be represented by a gain KE and a single time constant TE as shown in
equation (4.1).
VF(s)
VR(s) =
KE
1 + sTE ----- (4.1)
where VR = the output voltage of the regulator (AVR)
VF = field voltage
Discussion and proof about this type of closed loop equation can be found in any
standard control text such as Phillips [17]. Therefore, in terms of expressing equation
(4.1) in the form of block diagram will be
Figure 4.7 Block diagram of an exciter model
There are many different types of excitation systems available. Some of which uses ac
power source through solid-state rectifiers such as SCR [14]. As a result, the output
voltage of the exciter becomes a nonlinear function of the field voltage due to the
saturation effects which occur in the magnetic circuit shown in Figure 4.8.
Consequently, there is no straightforward relationship between the field voltage and
the terminal voltage of the exciter. However, the modern exciter can be estimated as a
linearised model, taking account for major time constant and ignoring the saturation
and other nonlinearities. Therefore, the simplest form of representing a basic exciter is
expressed as equation (4.1) which will be used to represent the exciter model in the
simulation of this thesis.
KE
1 + sTE
CHAPTER 4
45
Air gap line No-load saturation
Constant resistance load saturation
Exciter Voltage, VF
Exciter Field Current, i B
A
SE = f(VF) = A - B
B =
A
B - 1
Figure 4.8 Exciter saturation curves
The excitation system amplifier may be a rotating amplifier, a magnetic amplifier or
modern electronic amplifier. In any case, a linearized characteristic of the amplifier is
assumed. The amplifier is represented similarly by a gain KA and a time constant TA.
The transfer function of the amplifier is
VR(s)
Ve(s) =
KA
1 + sTA ----- (4.2)
where Ve = reference voltage Vref - output voltage of the sensor VS
Typically, the time constant of the amplifier is very small and is often neglected.
Therefore it is very often the case, as well as in the thesis, to represent the amplifier
(neglecting the time constant) and the exciter as a single block model since the time
constant for the exciter is also very small.
CHAPTER 4
46
VF(s) Vt(s)
4.2.2 Generator Model
The basic generator model will be similar to the exciter model in terms of its transfer
function. A simple linearized model of generator can be expressed as shown in Figure
4.9 where Vt is the terminal voltage of the synchronous generator. KG and TG are the
generator gain and time constant of the generator. For this simple model, the typical
value of KG varies between 0.7 to 1, and TG between 1 second to 2 seconds from full-
load to no-load since these constants are load dependent. However, taking into the
consideration of higher order models of the synchronous machine, a more defined
model is required.
Figure 4.9 Block diagram of a simple generator model
Given the equation (2.2) in Chapter 2 and also the papers by Walton [3] and Keyhani
[20], a higher order generator model can be defined. As shown in Figure 4.4 a fourth
order model of the synchronous machine consists of a generator gain plus four pairs
of pole-zero time constants derived from the operational inductance equation (2.2). In
terms of expressing it as transfer function, it is shown in equation (4.3) below
Vt(s)
VF(s) =
KG (1 + sTz1) (1 + sTz2) (1 + sTz3) (1 + sTz4)
(1 + sTp1) (1 + sTp2) (1 + sTp3) (1 + sTp4) ----- (4.3)
The result of increasing from a fourth order model to a fifth order model is an
additional pole-zero pair time constants being added to equation (4.3) and so on as the
order goes higher. Similarly if a third order model is in place with the fourth one, a
pair of pole-zero time constant will be remove from equation (4.3).
KG
1 + sTG
CHAPTER 4
47
Vref(s) Vt(s)
Sensor
Amplifier Exciter Generator
Ve(s) VR(s) VF(s)
VS(s)
4.2.3 Sensor Model
The terminal voltage of the synchronous generator is being fed back by using a
potential transformer that is connected to the bridge rectifiers. The sensor is also being
modelled, likewise as the exciter, by a first order transfer function
VS(s)
Vt(s) =
KR
1 + sTR ----- (4.4)
where VS = output voltage of the sensor, i.e. the output of the bridge rectifiers.
By combining the various models from Section 4.2.1 to 4.2.3, a simple automatic
voltage regulator (AVR) is created with the combination of a first order model of
synchronous generator.
Figure 4.10 Block diagram of a simple automatic voltage regulator (AVR) [12,19] Therefore the closed-loop transfer function relating the generator terminal voltage
Vt(s) to the reference voltage Vref(s) is
Vt(s)
Vref(s)
= KA KE KG KR (1 + sTR)
(1 + sTA) (1 + sTE) (1 + sTG1) (1 + sTR) + KA KE KG KR
----- (4.5)
KA
1 + sTA
KE
1 + sTE
KR
1 + sTR
KG
1 + sTG
CHAPTER 4
48
Vref(s) Vt(s)
Sensor
PID Controller
Exciter Generator
Ve(s) VF(s)
VS(s)
4.2.4 Automatic Voltage Regulator (AVR) with PID Controller
A three term controllers of proportional-integral-derivative action called the PID
controller, is introduced to the excitation system. It improves the dynamic response
and also reduces or eliminates the steady state error.
However, the use of a high derivative gain will result in excessive oscillation and
instability when the generators are strongly connected to an interconnected system.
Therefore an appropriate control of derivative gain is required. The proportional and
integral gains can be chosen to result in the desired temporary droop and reset time.
The transfer function of a PID controller is
GC(s) = Kp + Ki
s + Kds ----- (4.6)
Therefore, the proposed AVR system block diagram for simulating a fourth order
model of synchronous generator with the rest of the appropriate excitation system
components is shown in Figure 4.11. Note that the amplifier block shown in Figure
4.10 has merged with the exciter block in Figure 4.11.
Figure 4.11 Block diagram of the proposed AVR system with PID controller
KE
1 + sTE
KG (1 + sTz1) (1 + sTz2) (1 + sTz3) (1 + sTz4)
(1 + sTp1) (1 + sTp2) (1 + sTp3) (1 + sTp4)
KR
1 + sTR
PID
CHAPTER 4
49
4.2.5 Turbine Model
As mentioned in the earlier chapters different types of turbines have different
characteristics. This source of mechanical power can be a hydraulic turbine, steam
turbine and others. Six types of steam turbine models are discussed in an IEEE
transaction report [21]. There are listed in the following table:
STEAM TURBINE TYPE DESCRIPTION
A Non-reheat
B Tandem compound, single reheat
C Tandem compound, double reheat
D Cross compound, single reheat
E Same as D but with different shaft arrangement
F Cross compound, double reheat
Table 4.1 Classification of steam turbine
The simplest form of model for a non-reheat steam turbine can be approximated by
using a single time constant TT. The model for turbine associates the changes in
mechanical power ∆Pm with the changes in steam valve position ∆PV. Hence the
transfer function is
GT(s) = ∆Pm(s)
∆PV(s) =
1
1 + sTT ----- (4.7)
CHAPTER 4
50
∆Pref(s)
Droop
Governor & Turbine Rotating Mass & load
∆PL(s)
∆– (s) ∆Pg(s) ∆Pm(s)
4.2.6 Governor Model
The speed governor mechanism works as a comparator to determine the difference
between the reference set power ∆Pref and the power (1/R)∆ω shown in Figure 3.11.
The speed governor output ∆Pg is therefore
∆Pg(s) = ∆Pref(s) - 1
R ∆ω(s) ----- (4.8)
where R represents the speed regulation [19]
From the speed governing system illustrated in Figure 3.12, speed governor output
∆Pg is being converted to steam valve position ∆PV through the hydraulic amplifier
[18]. Assuming a linearized model with a single time constant Tg:
∆PV(s) = 1
1 + sTg ∆Pg(s) ----- (4.9)
Consequently from section 4.2.5, 4.2.6 and the load model in Figure 3.7, the proposed
load frequency control (LFC) for this thesis is created which is illustrated in below
Figure 4.12 Isolated power system load frequency control (LFC) block diagram
Note that the governor and turbine have merged to form a single block for simplicity.
1
(1 + sTg) (1 + sTT)
1
R
1
2Hs + KD
CHAPTER 4
51
∆Pref(s)
Droop
Governor & Turbine Machine Dynamics
∆– (s) ∆Pg(s) ∆Pm(s)
∆PL(s)
Integrator
4.2.7 Automatic Generation Control (AGC)
The generic functions [18] of AGC include the following aspects:
1. Load frequency control (LFC)
2. Economic dispatch
In this thesis, only the first aspect is discussed since it is involved in the transient
response of the machine. In the case of a steam turbine, if the load on the system is
increased, the turbine speed decelerates before the speed governor can detect and
adjust the input of the steam to cater for the new load. As the change in the value of
the speed diminishes, the error signal becomes smaller and the position of the
governor flyball moves closer to the point needed to maintain constant speed. Yet this
constant speed is not the set point and an offset occurs.
Figure 4.13 Block diagram of AGC in an isolated power system
By adding an integrator, it can restore the speed or frequency to its apparent value by
monitoring the average error over a period of time to correct the offset. Considering
the AGC in a single area system and in an interconnected system with the primary
LFC loop, any change in the load of the system will cause a steady state frequency
1
(1 + sTg) (1 + sTT)
1
R
1
2Hs + KD
KI
s
CHAPTER 4
52
deviation depending on the governor speed regulation. A reset control is needed to
reduce this frequency deviation to zero by introducing a secondary loop, which
consists of an integral unit, shown in Figure 4.13.
The integral controller gain KI is fine tuned to obtain the optimum transient response
of the system. In order to restore the system to its set point, the integrator is added on
to the load reference setting to change the speed set point. This forces the final
frequency deviation to zero.
CHAPTER 4
53
4.2.8 Combining AGC and Excitation System
Due to the weak coupling relationship between the AVR and AGC, the voltage and
frequency are regulated separately. The study of coupling effects of the linearized
AVR and AGC can be found in Kundur [12] and Anderson [13].
In [12] and [13], they have mentioned that a small change in the electrical power ∆Pe
is the product of the synchronizing power coefficient PS and the change in the power
angle ∆α. Taking account of the voltage proportional to the main field winding flux
Eδ, the following linearized equation is obtained:
∆Pe = K2∆α + K1Eδ ----- (4.10)
where K1 is the change in electrical power for a change in the direct axis flux linkages
with constant rotor angle and K2 = PS.
By modifying the generator field transfer function (one time constant lag model) and
taking into account the effect of rotor angle α, the equation for stator EMF can be
expressed as
Eδ = KG
1 + sTG (Vf ’ K3∆α ) ----- (4.11)
where K3 = the demagnetizing effect of a change in the rotor angle (at steady state)
The small effect of this rotor angle α upon the generator terminal voltage can be
expressed as
∆Vt = K4∆α + K5Eδ ----- (4.12)
where K4 = change in terminal voltage with the change in rotor angle for constant Eδ
K5 = change in terminal voltage with the change in Eδ for constant rotor angle
More detail discussion on equation (4.10) to (4.12) can be found in [12], [13] and
[14]. A representation of these constants is shown in Appendix C.
CHAPTER 4
54
Therefore, the simulation model for a fourth order machine time constants is
generated in Figure 4.14. The actual model may vary slightly in presentations due to
some limitations in the MATLAB graphics interface. Nonetheless, the simulation
model in MATLAB follows closely as shown in Figure 4.14.
Figure 4.14 Simulation model for the fourth order machine time constants.
Note that ∆VL and ∆PL are included to simulate the load change in voltage and power
respectively, which are effectively the change in reactive and real power.
CHAPTER 4
55
CHAPTER 5
56
CHAPTER 5
5. SIMULATION PROCESS AND EVALUATION
This chapter provides the required simulation inputs and produces the simulation
results so as to make evaluations on the models used.
5.1 SIMULATION INPUTS
Before commencing the simulation, some data are required to input to the model.
They are the gains of the various controllers, the coupling coefficients, the speed
regulation and a set of synchronous machine time constants.
As mentioned in Section 1.3, the optimum time constants extracted from the result of
Walton [1] are used. These time constant values are shown in Table 5.1 below
Rotor Time Constants
Circuit Poles Zeros
F 3.9517 0.9087
J 0.1481 0.1257
K 0.00838 0.00688
L 0.000937 0.000775
Table 5.1 Optimum time constants
The suffixes of f, j, k and l refer to the rotor branches corresponding to Figure 2.1 with
the additional rotor branch of l. With all these rotor branches being considered, they
represent a fourth order model of the synchronous generator. Therefore, using only
rotor branch of f represents first order model; using f and j represent second order
model; using f, j and k represent third order model and so on.
CHAPTER 5
57
The gain values of various components as well as other constants required in this
simulation (corresponding to the simulation diagram Figure 4.14) can be found in
Table 5.2 and Table 5.3 below. The values chosen are typically values gathered from
papers, articles and books [1, 5, 6, 12, 14].
∆Pref KD R H Tg TT ∆PL 0 0.8 0.05 10 0.2 0.5 0.2
Table 5.2 Values of the constants required for turbine and governing system (all
values are in per unit).
The only variable term in this part of the control system is KI which is adjusted
accordingly in order to satisfy the transient response of the machine. The speed
regulation is set as 5% which eventually becomes a gain of 20 and the load change in
real power is set at 20%. For a 0.8% in load change, there is a 1% change in
frequency which corresponds to KD = 0.8. The value of the generator inertia constant
is assumed to be 10 seconds and the time constants of the governor and turbine are 0.2
second and 0.5 second respectively
Vref KE KG KR K1 K2 K3 K4 K5 TE TR ∆VL 1 200 1 1 0.2 1.5 1.4 -0.1 0.5 0.05 0.05 0.05
Table 5.3 Values of the constants required for excitation control system (all values are
in per unit).
The variable terms in the excitation control system are the values of the PID controller
(namely Kp for proportional gain, Ki for integral gain and Kd for derivative gain) that
are varied to obtain the optimum output responses of the machine, which are the
terminal voltage Vt and the frequency deviation step response ∆ω. The exciter model
in Figure 4.14 includes an amplifier unit. Due to the small time constants of the
amplifier, it has been ” merged„ with the exciter unit to form a single block. Therefore
the constant values shown for the exciter includes the values of the amplifier. The
typical exciter gain is high as mentioned in Section 2.2.3. In this case, the gain KE is
CHAPTER 5
58
200 and the time constant TE is 0.05 sec. The gain of the feedback sensor and the
generator is 1 and the step input reference voltage is set to 1.
In simulating a reactive load change, a voltage change in load is assumed to be 5%.
For a stable system, K1, K2 (which is equal to Ps), K3 and K5 are positive and K4 may
be negative.
As a summary, the variables in this simulation are the feedback integrator gain KI of
the AGC, the PID controller values of proportional gain Kp, integral gain Ki and
derivative gain Kp. The rest of the controller gains and their time constants are fixed
unless otherwise stated.
CHAPTER 5
59
5.2 SIMULATION PROCEDURE AND RESULTS
With all the values of the constants given, a fourth order synchronous generator model
corresponds to Figure 4.14 has been set up using MATLAB Simulink as shown in
Figure 5.1.
Figure 5.1 Diagram of a fourth order synchronous machine model in MATLAB
Simulink
CHAPTER 5
60
The values of the PID controller are set as shown below
Figure 5.2 Initial PID controller values for fourth order model
These values for the PID controller are typical values recommended in Kundur [12]. It
is found that the derivative gain, which was being recommended as 0.5, has to be
lower to obtain a better response. The initial value of KI in the governing control loop
can be seen to be 5.2 in Figure 5.1.
In Simulink, the selection of solvers is important as it affects the accuracy and
efficiency of the model. In this simulation, a standardized setting throughout the
whole investigation is set as shown in Figure 5.3.
The solver selected is ode23(Bogacki-Shampine) because it was found to produce a
more efficient response when compared to the other solvers available in Simulink.
Further details of the selection of solver can be found in Appendix A of Saadat [14].
In addition to this, a detail explanation of the usage of these solvers can be found in
the help features within the MATLAB program.
The period of simulation is set as 30 seconds so as to verify that there are no further
oscillations.
CHAPTER 5
61
Figure 5.3 Simulink parameter settings
The corresponding outputs in terms of the terminal voltage Vt and the frequency
deviation step response ∆ω are generated in Figure 5.4 and Figure 5.5 respectively.
Note that due to the limitations in MATLAB of labeling axis on the generated graphs,
all the plots generated using Simulink will have
1. x axis as time scale in seconds, and
2. y axis as per unit scale
From the results shown in Figure 5.4 to 5.6, the primary objective of this thesis is the
correct simulation of the output response of this fourth order model. The PID
controller was adjusted to the values shown in Figure 5.2 to achieve this set of
satisfactory responses.
CHAPTER 5
62
Figure 5.4 Terminal voltage Vt of the fourth order model
Figure 5.5 Frequency deviation step response ∆ω of the fourth order model
In Figure 5.5, the response for ∆ω oscillates for a period of 12.5 seconds before
settling down to zero deviation. There is an overshoot error occurring at 3.5 seconds.
This overshoot error has to be minimized by adjusting the values of the PID
controllers and KI. The ideal response is to keep the deviation (oscillation) as close to
zero as possible at the minimum period of time.
Figure 5.6 ” Zoom in„ detail of Figure 5.4
CHAPTER 5
63
Due to the fact that a high excitation gain is used, the compensating effect is fast and
the response of the terminal voltage happens in a split second. Figure 5.4 may not
display this resultant effect vividly. Therefore the ” zoom in„ detail of Figure 5.4 is
shown in Figure 5.6. The model, reacting to the resistive and reactive load changes at
the same time, is able to restore the terminal voltage back to the nominal step input
value of 1 at about 0.2 seconds. After restoring the voltage back to nominal value, the
model remains stable as shown in Figure 5.4.
5.2.1 PID Controller ” Change in Kp Only
Setting the proportional gain Kp to 1 and keeping the rest of the variables at their
initial values, the response of ∆ω is similar to the one in Figure 5.5. The detailed
response of the terminal voltage is then
Figure 5.7 Terminal voltage when Kp = 1
It can found that with a set of values for other variables, the improper setting of Kp
leads to the additional increase or decrease in excitation controlled by the voltage
regulator. In this case the previous value of Kp gave an acceptable response. Now by
decreasing the value of Kp, there is an overshoot from 0.5 second and settling down to
1V after 6 seconds. Even though, it is usually the practice to generate a very small
overshoot in the terminal voltage that gradually settles at the desired set value which
in this case is 1, the response that is theoretically preferred should be the one shown in
Figure 5.4 rather than Figure 5.7. Nevertheless, the setting of Kp = 1, with respect to
the rest of the variable values, does not produce a good overall response.
CHAPTER 5
64
5.2.2 PID Controller - Change in Ki Only
By decreasing Ki to 0.2 and keeping the rest of the variables the same as their initial
values, the frequency deviation step response ∆ω has similar response to the one
shown in Figure 5.5. The ” zoom in„ detail of the response for terminal voltage is
illustrated in Figure 5.8.
Figure 5.8 Terminal voltage when Ki = 0.2
After Ki has decreased, the response of the machine is ” slower„ as compared to Figure
5.5 when Ki is at 0.7. The time taken for the terminal voltage to reach the value of 1 is
now 0.5 seconds.
5.2.3 PID Controller - Change in Kd Only
Figure 5.9 Terminal voltage when Kd = 0.7
Similarly, the values of all other variables remain the same and Kd is increased from
0.2 to 0.7, the terminal voltage is shown in Figure 5.9. The frequency deviation step
CHAPTER 5
65
response ∆ω is also shown in Figure 5.10. From these results, it can be seen that the
time for terminal voltage to reach 1V has increased to 15 seconds. Moreover, there are
small but slow oscillation in ∆ω after 20 seconds. Generally, except for the slight
oscillation, the initial response of ∆ω is the same as the previous few cases. It can be
deduced that the value of Kd was set too high and the optimal value should be smaller.
Figure 5.10 Frequency deviation step response ∆ω when Kd = 0.7
5.2.4 Change in KI of the AGC Only
By changing the gain of the integrator KI from 5.2 to 7 in the AGC feedback loop, the
results are as follows:
Figure 5.11 Terminal voltage when KI = 7 Figure 5.12 Response of ∆ω when KI = 7
The response of the terminal voltage in Figure 5.11 is similar to the one in Figure 5.6.
In contrast, the responses of ∆ω are different in these two cases. Comparing Figure
CHAPTER 5
66
5.5 and 5.12, the first positive peak is higher for the latter and also small constant
oscillation is identified in Figure 5.12. From these results, the value of KI is obviously
too high and causes the system response to oscillate and not enable to achieve a final
state when ∆ω = 0 is constant.
5.2.5 Change in Excitation Gain (KE)
There is another factor which can be considered for simulation purpose. That is the
excitation gain. It was mentioned in the earlier chapter that it is usual in practice to
have a high excitation gain to correct and adjust the system to the desired level of
output quickly. The initial value that was used in the fourth order model was 200
which is considered high. Assuming that the excitation gain is now 10 and in this
case, since the amplifier unit has formed a single function block with the exciter unit
in this thesis, KE is now decreased to 10. The gain values of the PID controller and KI
remain at their initial values (as in Figure 5.2). The results are as follows:
Figure 5.13 Response of Vt when KE = 10 Figure 5.14 Response of ∆ω when KE = 10
A good response required the voltage regulator to sense the changes in the output
voltage (and current) and correct the difference in voltage as quickly as possible.
Regardless speed of the response of the exciter, it will not alter its response until the
voltage regulator has ” instructed„ it on how much excitation to produce in order to
correct the error occurred.
CHAPTER 5
67
In this setting, because the gain is low, the excitation control response to changes will
be slow. This means that the time taken for the voltage regulator to restore the output
voltage back to the desired level will be longer. The terminal voltage response in
Figure 5.13 as compared to Figure 5.4 is slower. The latter has a sharp rise time to
attain the final desired output level of 1V. The setting in this model required about 4
seconds to achieve and maintain a constant of 1V in the output voltage. Comparing
this to the 0.2 seconds in Figure 5.6, the delay is significant.
The response of ∆ω for KE = 10 is slightly better than in Figure 5.5. Most of the points
along the two plots (Figure 5.5 and Figure 5.14) are the same other than the first
positive peak. The response in Figure 5.14 is slightly better because its maximum
point at the first positive peak is closer to the one in Figure 5.5. As an overall view,
due to the poor output voltage response, this setting of low gain (KE = 10) is not better
than the initial setting (KE = 200).
5.2.6 Lower Order Models
The output responses of the lower order models are shown in Appendix D. These
results are generated using the same initial setting shown in Figure 5.2. The resultant
plots are almost exactly similar to the fourth order model. Theoretically, if the power
system simulation model is able to handle higher order machine time constants, there
should be no problem handling lower ones except that the accuracy of the machine
response tends to be slightly poorer as lower order models are used.
CHAPTER 5
68
5.3 EVALUATION
From the results of varying KI of the AGC, the PID controller values of proportional
gain Kp, integral gain Ki and derivative gain Kp, each of the variables has their role in
contributing to a stable response of a power system.
In the first case of changing Kp only, if excessive proportional gain is applied, it will
result in a ” spike„ response in the output voltage during the transient state shown
below where Kp is set to 30.
Figure 5.15 Output voltage response when Kp is set too high
This causes the machine to be driven to an extremely high voltage during transient
state. In contrast, if Kp is set too low (Kp = 0.3), the resultant response may oscillate
and become unstable. Both effects are undesirable.
Figure 5.16 Output voltage response when Kp is set too low
Therefore by varying Kp, the rest of the variables have to be adjusted to fit the overall
output response. For example, Kp can be set to 20 while Ki, Kd, KI and KE are having
values of 0.7, 1, 5.2 and 10 respectively. This will also give a satisfactory response
CHAPTER 5
69
though it is not as good as the initial setting. The result of its output voltage response
is shown in Figure 5.17. The response of ∆ω in this case is similar to Figure 5.5.
Figure 5.17 ” Zoom in„ response of terminal voltage with the new setting
Hence, if any of the variables are varied, the rest of the variables have to be adjusted
so as to satisfy the transient and steady state responses of the power system.
With the simulation of the first, second and third order models, they produce identical
responses as those in the fourth order model. This is due to the fact that since this
simulation model is being assumed to connect to an infinite bus, any changes may
have been too fast to be reflected by the Simulink program (since transient responses
are dealing in terms of milliseconds and microseconds). This suggests that a continual
investigation in the future thesis on the transient response of the synchronous machine
taking an additional feedback signal from the actual load (virtually). This chain of
connections is illustrated in Figure 5.18. This can be done by using a current feedback
at the output terminal of the machine, in addition to the existing voltage feedback.
Figure 5.18 New possible feedback loop
With this new suggested feedback, hopefully the simulation model is able to simulate
the machine response of a practical case.
G AVR + LFC
setpoint
Existing Feedback Loop, Vt
Bus Bus
Transmission lines
load
New Feedback Loop (virtual)
transformer
CHAPTER 5
70
CHAPTER 6
71
CHAPTER 6
6. CONCLUSION AND FUTURE IMPROVEMENTS
This chapter describes possible future developments on the existing simulation model
of the thesis. This is followed by some concluding remarks on the value of this thesis.
6.1 FUTURE IMPROVEMENTS
This thesis serves as a basis for simulation of more comprehensive power systems.
There are some areas in the simulation model that can be improved further on. The
ultimate goal of any power system simulation is to simulate, as closely as possible, the
actual behaviours of the controllers and machines within the system. The responses of
the machine in this thesis have been limited by the assumptions stated in Chapter 1. In
order to improve this model, several areas can be considered:
1. Using a more defined (higher order) turbine model instead of using first order
turbine model
2. Implementing nonlinear excitation control system since there are nonlinear
characteristics between the input and output of the excitation control in
practice.
3. Connecting the output terminal of the machine model to an actual load
(virtually) instead of using infinite bus for simulation. The actual load includes
the transformer, the actual bus, the transmission line and the end user loads.
This process can be achieved through monitoring the changes of output
current at the terminal instead of the hard-wired connection to the actual load.
Hopefully in the future theses, the above points being mentioned can be implemented
for a more complete and accurate power system simulation.
CHAPTER 6
72
6.2 CONCLUSION
The simulation of the transient response of synchronous machine has been successful.
The simulation model in this thesis is able to generate the responses of the first,
second, third and fourth order synchronous machine correctly.
The restriction that the model has when connecting to the infinite bus is that the small
changes in response are not reflected clearly due to the strong grid between the
machine and the infinite bus. The infinite bus forces the machine to run at the
synchronous speed and thus, changes in the transient response are almost impossible
to be detected with this model. The proposed solution to this problem is to connect the
output terminal to an actual load virtually by a mean of connecting an additional
feedback of current at the terminal output. In this way the model is able to reflect the
changes in the transient response of the machine.
Nevertheless, the objective of this thesis has been met as the simulation model is able
to simulate the response of various orders of the machines correctly in the specified
setting.
The knowledge that I have gained through the research and implementation of this
thesis has been tremendously valuable. The process involved in the investigation of
this thesis has widened my scope towards power system control. There are many
perspectives in the investigation of the machine responses in the power system. The
work presented in this thesis is considered a small part in power system control. More
developments can be made to the simulation model in order to achieve a higher
accuracy and to have a more complete control.
CHAPTER 6
73
CHAPTER 6
74
BIBLIOGRAPHY
75
BIBLIOGRAPHY
1. A. Walton, ” A Systematic Analytical Method for the Determination of
Parameters of Synchronous Machines from the Results of Frequency
Response Tests„ , Journal of Electrical Engineering-Australia, Vol. 20 No. 1,
2000, pp. 35-42.
2. S.K. Sen and B. Adkins ” The Application of the Frequency-Response Method
to Electrical Machines.„ Proc. IEE vol.103 part C 1956 pp378-391.
3. A. Walton, ” Characteristics of Equivalent Circuits of Synchronous Machines„ ,
IEE Proceedings. Electric Power Applications, Vol. 143 No.1 January 1996,
pp. 31-40.
4. D. Kennedy et al, ” A Nonlinear Approach to Power System Excitation Control
and Stabilization„ , International Journal of Electrical Power & Energy
Systems, Vol. 20 No. 8, 1998, pp. 501-515.
5. ANSI/IEEE Std 421.1-1986, IEEE Standard Definitions for Excitation
Systems for Synchronous Machines, American National Standards Institute
and The Institute of Electrical and Electronics Engineers, USA, 1986.
6. ANSI/IEEE Std 122-1985, IEEE Recommended Practice for Functional and
Performance Characteristics of Control Systems for Steam Turbine-Generator
Units, The Institute of Electrical and Electronics Engineers, USA, 1985.
7. PSS/E User Handbook, ” Speed Governor System Modeling„ , PSS/E-26
Program Application Guide, Vol. 2.
8. Y.N. Yu, Electric Power System Dynamics, Academic Press Inc., London,
1983.
BIBLIOGRAPHY
76
9. J.A. Momoh and M.E. El-Hawary, Electric Systems, Dynamics and Stability
with Artificial Intelligence Applications, Marcel Dekker Inc., New York, 2000.
10. J. Faiz et al, ” Closed-loop Control Stability for Permanent Magnet
Synchronous Motor„ , International Journal of Electrical Power & Energy
Systems, Vol. 19, Issue 5, June 1997, pp331-337
11. P.C. Krause et al, Analysis of Electric Machinery, IEEE Press, 1995.
12. P. Kundur, Power System Stability and Control, McGraw-Hill Inc., 1994.
13. P.M. Anderson and A.A. Fouad, Power System Control and Stability, IEEE
Press, Revised Printing, 1994.
14. H. Saadat, Power System Analysis, McGraw-Hill Inc., 1999.
15. J. Arrillaga and B. Smith, AC-DC Power System Analysis, IEE, London, 1998.
16. MATLAB User Guide Version 1, Power System Blockset for use with
Simulink, electronic copy from http://www.mathworks.com, visited on 27 May
2001.
17. C. L. Phillips and R. D. Harbor, Feedback Control Systems, 3rd ed. Prentice
Hall Inc, Englewood Cliffs, New Jersey, 1996.
18. A.R. Bergen and V. Vittal, Power Systems Analysis, 2nd ed. Prentice Hall Inc,
Upper Saddle River, New Jersey, 2000.
19. S.A. Nascar, Schaum�s Outline of Theory And Problems of Electric Power
Systems, McGraw-Hill Inc., 1990.
BIBLIOGRAPHY
77
20. A. Keyhani and S. Hao, ” The Effects of Noise on Frequency-Domain
Parameter Estimation of Synchronous Machine Models„ , IEEE Transaction
on Energy Conversion, Vol. 4, No. 4, December 1989.
21. IEEE Committee, ” Dynamics Models for Steam and Hydro Turbines in Power
System Studies„ , IEEE Trans. Power Appar. Syst., pp1904-1915, December
1973.
APPENDIX A
78
APPENDIX A
PHASOR DIAGRAM
Figure A.1 Phasor diagram of a synchronous machine in steady state. [9]
APPENDIX B
79
APPENDIX B
EXCITATION CONTROL SYSTEM
Figure B.1 Synchronous excitation control system [13]
This diagram, which is originated from IEEE Trans., vol. PAS-88, Aug 1969,
illustrates a typical excitation control system for a synchronous generator. It clearly
defines the elements of the various subsystems.
APPENDIX C
80
APPENDIX C
SYSTEM MODEL WITH GOVERNOR AND AVR
Figure C.1 Power system block diagram with governor and voltage regulator
APPENDIX D
81
APPENDIX D
SIMULATIONS OF LOWER ORDER MODELS
Using the values of PID controller as stated in Figure 5.2 which are Kp = 3, Ki = 0.7
and Kd = 0.2, the output responses of various lower order models, for example the first
to third order models, are being generated. The following diagrams from Figure D.1
to Figure D.9 are based on these settings with the feedback integrator KI = 5.2.
APPENDIX D
82
D.1 First Order Model Simulation
Figure D.1 Diagram of first order synchronous machine model in Simulink
Figure D.2 Frequency deviation step Figure D.3 ” Zoom in„ terminal voltage
response ∆ω of the first order model Vt of the first order model
APPENDIX D
83
D.2 Second Order Model Simulation
Figure D.4 Diagram of second order synchronous machine model in Simulink
Figure D.5 Frequency deviation step Figure D.6 ” Zoom in„ terminal voltage
response ∆ω of the second order model Vt of the second order model
APPENDIX D
84
D.3 Third Order Model Simulation
Figure D.7 Diagram of third order synchronous machine model in Simulink
Figure D.8 Frequency deviation step Figure D.9 ” Zoom in„ terminal voltage
response ∆ω of the third order model Vt of the third order model
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