Phd thesis

Acknowledgements
I would like to express my sincerest gratitude to Dr.R.Jha (Project Guide, Head of Instrumentation and Control Department, National Institute of Technology Jalandhar) for rendering all possible assistances and facilities for the accomplishment of the project on State Space Modeling Of Power Systems. I am also heartily grateful to all my teachers and staff without whose expert guidance my project would not have been a successful one. It was a great learning experience to have undertaken this project under our respected Dr.R.Jha sir and I am fully confident that it will help me in building up a great future.

Supratim Ghosh

I

CONTENTS
Name of the topic………………………………………………Page no.
a) Acknowledgements………………………………………………………..………..I b) Contents……………………………………………………………………..……..II Synopsis……………………………………………………………………………….1 1. Introduction……………………….……………………………………………...2-6 1.1 An introduction to modeling…..……………………………………….……2-3 1.2 A brief introduction to state-space modeling…………………….………….4-6 2. Modeling of Power Systems ……………………………………………………7-21 2.1 Single generator connected to infinite-bus system…………………………..7-9 2.2 Power System using speed governor………………………………….…..10-11 2.3 Power System with an Excitation System and AVR…….………………..12-14 2.4 Multi-machine Power Systems……………………………………………15-17 2.5 Analysis of Power Systems……………………………………………….18-21 3. Transient Stability of Power Systems……………………………………….....22-31 3.1 Introduction…………………………………………………………….....22-23 3.2 Lyapunov’s Functions…………………………………………………….24-26 3.2.1 Power System without governor……………………………………26-29 3.2.2 Power System with governor…………………………………….....29-31 4.Power System Stabilizers ……………………………………………………...32-35 5. Conclusions…………………………………………………………………….…36 6. Future Scope of Work………………………………………………………….…37 8. References………………………………………………………………………..38

II

Synopsis
Any dynamical system requires us to undergo the following three broad stages for system studies: i) Mathematical Modeling ii) System Analysis and iii) System Synthesis Depending on purposes, classical methods for modeling have been found in use for power systems till recently. Modern control system theory, over the past many years has been gaining great importance for being potentially applied to power systems. State variable modeling and control of Power systems has successfully been carried out in variety of problems from the early 1960s. This project will include analysis of Power Systems and developing State Space model of the Power Systems starting from the swing equation and performing the stability domain analysis using: a) Single machine connected to an infinite bus which will include power systems using: i) Exciter ii) Governor b) Multi-machine systems. This project will also include study of the application of the Power System Stabilizers and their position in Power Systems. Furthermore we intend to analyze the models thus developed using unit values of the constants (per-unit analysis) and plot the response of output versus the input using CSMP.

1

1. INTRODUCTION
1.1 An introduction to modeling
Any dynamical system requires us to undergo the following three broad stages for system studies: 1) Mathematical Modelling 2) System Analysis and 3) System Synthesis Depending on purposes, classical methods for modeling have been found in use for power systems till recently. Modern control system theory, over the past many years has been gaining great importance for being potentially applied to power systems. State variable modeling and control of Power systems has successfully been carried out in variety of problems from the early 1960s.An attempt has been made to discuss state variable model as a core of mathematical techniques and its application to Power system modeling for control and analysis of systems. It is hoped that all concerned will appreciate the universality and convenience and advantages of the technique in system studies. System Modeling Modeling of a dynamic system is mathematical representation which determines quantitative features describing operation of the system. Quantitative does not always employ mathematical models. Since mathematical theories are always precise, dynamical systems are conveniently described mathematically. Modeling needs complete knowledge of the dynamical process. Mathematical model describing process dynamics truly is called optimal model. It also provides a clue for design of associated devices viz controllers for the systems

2

Model Classification Classification of system modeling depends on purposes for which it is developed Physical systems can be modeled in the following types ¾ Mathematical Models ¾ Physical Models ¾ Empirical Models ¾ Scaled Models ¾ Analog Models Mathematical Models (a) Classical Model (b) State Variable i) Stochastic Model ii) Deterministic Model Mathematical models are derived from fundamental laws. For example ¾ Mechanical System: Newton’s Laws of Motion ¾ Electrical Systems: K. Laws including Ohm’s law ¾ Thermal & Fluidic: Laws of Thermodynamics & Thermal Phenomenon Mathematical model is preferred as : 1) It is often difficult to experiment to observe how dynamics of process reacts to various inputs 2) Even if process equipment is available for experimentation, the modeling procedure is quite expensive.

3

1.2 A brief introduction to State Space modeling
Here three variables are considered i) ii) iii) Input or control variables, u(t) Output or controlled variables, y(t) Intermediary or state variables, x(t)

The following terms are now defined to introduce the concept: System States: The states of above system is a minimal set of variables such that the knowledge of the variables, x at t=t0 together with knowledge of inputs or control signals for t≥t0, completely determines the behavior of the system for all t, t>t0.

State Variables: The state variables of a dynamic system are the variables making up the smallest set of variables that determine the state of the dynamic system. If at least n variables x1, x2…..xn are needed to completely describe the behavior of a dynamic system (so that once the input is given for t ≥ t0 and the initial state at t=t0 is specified, the future state of the system is completely determined), then such n variables are a set of state variables.

State Vector: If n state variables are needed to completely describe the behavior of a given system, then these n state variables can be considered the n components of a vector x. Such a vector is called a state vector.

State Space: The n-dimensional space whose co-ordinates axes consist of x1-axis, x2-axis,….xn-axis is called a state space. Any state can be represented by a point in the state space.

4

State Trajectory: The locus of point P(x) in state space for time interval t0≤t≤tf is called state trajectory.

State Space Equations: In state space analysis we are concerned with three types of variables that are involved in the modeling of dynamic systems: input variables, output variables and state variables. The state space representation for a given system is not unique, except that the no. of state variables is the same for any of the state space representations of the same system. An nth order system with three classes of variables is schematically shown:

u1 um

x1

Linear Plant

OBSERVER
(Output Element)

y1 yp

xn

The above system may be described by

x(n) = F[x, x(1), x(2)….x(n-1), u, u(1)…u(m-1),t]
where superscript (n) and (m) etc. indicate order of differentiation with respect to ‘t’ We define state variables as

x1 = x x2 = x(1)

xn = x(n-1)

5

Above defined S.V. and system equations can be combined to give

x(1) = x2 x(2) = x3 x(n-1) = xn x(n) = F(x1, x2… xn u1, u2,..um,t)
For Linear Lumped time invariant (LLTIV)-SISO system, the above eqn can be rewritten as o x ( t ) = Ax ( t ) + Bu ( t ) y (t ) = c T x (t ) + d T u (t )

Advantages of State Variable Modeling

1. The SV modeling is valid for nonzero initial conditions and is applicable to both MIMO & SISO, LTIV & LTV (L-systems) & NL systems and continuous and discrete time systems. 2. It incorporates the information regarding internal behavior of the system in addition to accepting input-output relationship. 3. It offers a general framework of modeling & system optimization. 4. It is powerful technique for state estimation and can be used for filtering, smoothing and forecasting problems. 5. S.V model enables us to split an nth order differential equation. into n-first order differential equations which is easily amenable to solution through digital computers.

In view of advantages & flexibility associated with S.V. modeling covering all necessary topics, it enables researchers to comprehend various research findings. Therefore, an attempt has been made to introduce the concept of S.V. modeling as a core of mathematical technique.

6

2. Modeling of Power Systems
2.1 Single-generator connected to ∞-bus system

We study dynamics of rotor of Generator under post-fault condition. Subject to simplifying assumptions, swing equation can be derived. The following assumptions are normally made to start with: 1. The voltage behind the transient reactance is assumed constant. 2. Damping power is assumed to be directly proportional to the slip velocity and this is primarily due to mechanical friction and asynchronous torque. 3. The Mechanical power input to the machine, Pm, is assumed constant during the transient period. 4. The effect of saturation is ignored. 5. Resistances of stator winding of machine and transmission lines are neglected. 6. The voltage regulator action is not fast enough to be effective during transient period. 7. The angular momentum of machine is taken to be constant. 8. For simplicity the effect of saliency is not considered and the nonsalient (round) rotor machine is considered.

Power balance relation gives

d ( KE ) + P = P − P m e d dt
7

(1)

W o d KE ( KE ) = 2 fo dt

d Δf dt

(2)

where W ° = KE of Generator at synchronous speed f 0 = frequency at synchronous KE speed.

d P = d δ (t ) d dt
where δ (t ) = θ − w t is rotor angle o = d (θ − ω ) = d (ω − ω ) = d (2ΠΔ f ) o o Pm = mechanical input to the machine Pe = electrical power output of the machine given by

(3)

o

(4)

Pe= EV sin δ (t ) ΣX
Substituting in equation (1) we have

(5)

2W o KE d EV ( Δ f ) + d ( 2ΠΔ f ) = Pm − sin δ ΣX fo dt
Dividing (6) by base MVA one gets o EV H oo δ (t ) + D δ (t ) = [sin δ o − sin δ ( t )] Π fo ΣX where
M = H (moment of inertia), D= damping factor. Π fo

(6)

(7)

Equation (7) is swing equation for the Generator under post fault condition. We define state variables as
⎧ ⎪ x = δ (t ) − δ 1 ⎪ ⎪ ⎨ ⎪ o ⎪x = (t ) δ ⎪ ⎩ 2

o

8

Using states defined in (8), equation (7) can be cast into o x ( t ) = Ax ( t ) + bf ( y ) : SE

(9) (10)

y (t ) = cT x (t )
⎡ with A = ⎢ 0 ⎢ ⎣0

: OE

⎡1 ⎤ ⎡ 0 ⎤ ⎤ 1 H ⎥, C = ⎢ ⎥, M = ⎥, B = ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ − D /M ⎦ Π fo ⎣0 ⎦ ⎣ − 1⎦

The n.l. function, f(y) is given by

f ( y) =

EV [sin(y + δ o) − sinδ o ] ΣXM

(11)

The linear sub-system of Lure’ problem (9-10) is given by

G(s) = −cT (sI − A)−1b =

1 s( s + D / M )

(12)

The plot of f(y) can be obtained as shown in Fig.1, with slope at y=0 given by

FIG. 1 Plot OF f(y) Vs y for δo given

EV cosδ ° , over − Π − 2 δ ° < y < Π − 2 δ ° . ΣXM

9

2.2 Power System Using Speed Governor
Governor Action Speed governors vary prime mover output(torque) automatically for changes in system speed (frequency). The speed sensing device is usually a flyball assembly for mechanical-hydraulic governors and a frequency transducer for electro-hydraulic governors. The output of the speed sensor passes through signal conditioning and amplification (provided by a combination of mechanical-hydraulic elements, electronic circuits, and/or software) and operates a control mechanism to adjust the prime mover output (torque) until the system frequency change is arrested. The governor action arrests the drop in frequency, but does not return the frequency to the pre-upset value (approximately 50 Hz) on large interconnected systems. Returning the frequency to 50 Hz is the job of the AGC (Automatic Generation Control) system. The rate and magnitude of the governor response to a speed change can be tuned for the characteristics of the generator that the governor controls and the power system to which it is connected.

Single generator connected to infinite bus with governor dynamics included
Including Speed Governor enables us to relax one of the assumptions above i.e. “The Mechanical power input to the machine, Pm, is assumed constant during the transient period.” Here mechanical power is varying during the transient period. Here speed governor is considered to have time constant (Te) comparable with transient period. System description is given by
oo o EV sin δ (t ) M δ (t ) + D δ (t ) = Pm (t ) − ΣX

(13) (14)

o δ (t ) T e P m ( t ) + Pm ( t ) = P o m − G dt
where G and Te are gain constant and time constant of the governor, P0m= initial value of Pm

10

Defining system sates
Δ x (t ) = δ (t ) − δ ° 1

Δ o x (t ) = x = o (t ) δ 2 1 Δ o x ( t ) = Pm − Pm ( t ) 3
Using states defined by (15) in system equations (13-14), S.V. model takes the form o x ( t ) = Ax ( t ) + bf ( y ) : SE

(15)

y (t ) = cT x (t )
with A=
⎡0 ⎢ ⎢ ⎢0 ⎢ ⎢0 ⎣

:OE

(16)
⎡ 0⎤ ⎢ ⎥ ⎢ ⎥ ⎢−1 ⎥ ⎢ ⎥ ⎢ 0⎥ ⎣ ⎦
⎡1 ⎤ ⎢⎥ ⎢⎥ ⎢0 ⎥ ⎢⎥ ⎢0 ⎥ ⎣⎦

1 − D/ M G/Te

⎤ ⎥ −1/ M ⎥ ⎥ ⎥ 1/Te ⎥ ⎦

0

B=

C=

The linear sub system is described by T.F.

G ( s ) = − cT ( sI − A) − 1b =

s+h D 1 s[ s 2 + ( h + )s + ( hd + g )] M M

(17)

with g=G/Te and h=1/Te, being parameters of speed governor.

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2.3 Power System with an Excitation System and Automatic Voltage Regulator
An Excitation System is required to provide d.c. power to the synchronous machine field of the power system. A simplified block diagram of an Excitation System is given below:

Vref +

∑

Amplifier And Saturation

Exciter

Generator

Vt

Vdc Rectifier Block
Parts 1. Rectifier Block- The rectifier block consists of a potential transformer and rectifier which converts the three phase ac voltage (transmission voltage Vt) to a corresponding d.c. voltage Vdc. The transfer fuction of the block is given by Vdc =

K RVt where τR is 1+τ Rs

the time constant due to filtering or first order smoothing in the transformer-rectifier assembly and Vt is the sum of the rms values of the three phase voltages. Since τR is very small of the order of 0.06s, it can be neglected and the transfer function can be simplied to Vdc= KRVt where KR is the gain of the rectifier block.] 2. Voltage regulator and reference(comparator)- This block compares the voltage Vdc against a fixed reference and supplies an output voltage Ve called the error voltage which is proportional to the difference i.e. Ve = k (V REF − Vdc ) where VREF is the reference voltage applied. This procedure can be accomplished using an electronic difference amplifier.

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3. The amplifier- The amplifier portion of the excitation system may be a rotating amplifier, magnetic amplifier, or conceivably an electronic amplifier. In any case we will assume linear voltage amplification KA with time constant τA.

VR =

K AVe 1+τ As

As with any amplifier a saturation value must be specified such as Vmin< VR< Vmax. 4. The exciter- The output of the boost-buck exciter can be given as

Vex =
where Vex= Exciter output voltage

VR − Vex S E KE +τ Es

KE, τE= Exciter gain and time constant respectively. SE= Saturation function of the exciter We can neglect the effect of saturation to get a simplified model and a first order system results. 5. The generator- It produces an output three phase voltage in response to the excitation system response. It can also be approximated as a first order system as

Vt =

K GVex 1+τGs

where KG and τG are the gain and time constant of the generator respectively.

Block diagram representation showing all the trasfer functions

VREF +

∑

Vdc

KA 1 +τ As

VR

1 KE +τ Es

Vex

KG 1+τGs

Vt

KR
13

Overall transfer function of the system is given as

Vt K AK G = VREF (1 + τ A s )(1 + τ G s )(K E + τ E s ) + K R Vt K AKG = VREF ( K E + K R ) + (K Eτ A + K Eτ G + τ E )s + (K Eτ Aτ G + τ Aτ E + τ Eτ G )s 2 + (τ Aτ Eτ G )s 3
Inverting Laplace transforms, putting all intial conditions equal to zero and making the differential equation model we get,

τ Aτ Eτ G V t + (K Eτ Aτ G + τ Aτ E + τ Eτ G )Vt + (K Eτ A + K Eτ G + τ E )Vt + (K E + K R )Vt = K A K GV REF
.
..

...

..

.

Let the state variables be x1 = Vt , x 2 = V t and x3 = V t such that x1 = x 2 and x 2 = x3 and output y=Vt Now,

.

.

x3 =

.

K A KG

τ Aτ Eτ G

VREF −

K Eτ Aτ G + τ Aτ E + τ Eτ G

τ Aτ Eτ G

x3 −

K Eτ A + K Eτ G + τ E

τ Aτ Eτ G

x2 −

KE + KR

τ Aτ Eτ G

x1

Giving us the state equation
⎡ . ⎤ ⎡ 0 ⎢ x.1 ⎥ ⎢ ⎢ ⎢x ⎥ = 0 ⎢ .2 ⎥ ⎢ K + K E R ⎢ x .3 ⎥ ⎢ ⎢ ⎦ ⎥ ⎢ ⎣ ⎣ τ Aτ Eτ G 1 0 K Eτ A + K E τ G + τ E ⎤ ⎡ ⎤ ⎥ ⎡ x1 ⎤ ⎢ 0 ⎥ ⎥⎢x ⎥ + ⎢ 0 ⎥ V REF 2 K Eτ Aτ G + τ Aτ E + τ Eτ G ⎥ ⎢ ⎥ ⎢ K A K G ⎥ ⎥⎢ ⎥ ⎣ x3 ⎥ ⎦ ⎢ τ Aτ Eτ G τ τ τ ⎥ ⎢ ⎦ ⎣ A E G⎥ ⎦ 0 1

[

]

τ Aτ Eτ G

and the output equation

[ y ] = [1

0

⎡ x1 ⎤ ⎥ 0 ]⎢ ⎢ x2 ⎥ ⎢ ⎦ ⎣ x3 ⎥

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2.4 Multi-machine Power Systems ¾ The use of single-machine connected to ∞-bus involves a gross simplification of
actual system dynamics.
¾ In fact, most power systems consist of a number of synchronous generators interconnected together through the transmission networks ¾ We now present n-machine system using Willem’s approach (1970)

Mathematical model

Differential model of ith interconnected machine is oo o M i δ i ( t ) + D i δ ( t ) = Pmi − Pei ( t ) i= 1, 2, 3…n Electrical output power Pei is given by
Pei = E i2 G ii + n j = 1, i ≠ j
∑

E E Y cos( θ − δ + δ ) i j j ij i j i = 1, 2 ,... n

Since B

c Π where c= 1, 5… , as ij = Y ij sin θ ij B ij ≅ Y ij θ ij ≅ 2 Pei leads to give
P ei = E i2 G ii + i = 1 , 2 ,..... n n E i E j B ij sin( δ i − δ j ) Σ j = 1, i ≠ j

State variable model can be represented as

Let x = ⎡ x M x ⎤T be a 2n-column state vector. ⎢ 2⎥ ⎣ 1 ⎦ T o ⎞ ⎛ o o ⎟ ⎜ x = ⎜δ δ ...........δ n ⎟ 1 ⎜ 1 2 ⎟ ⎠ ⎝

T o,δ −δ o,...... o⎞ ⎜ ⎟ x =⎛ δ − δ δ − δ n n⎟ 2 ⎜ 2 ⎝ 1 1 2 ⎠

15

δ i°
i.e.

can be computed from power equilibrium relation

Pei = Pmi
n i=1
∑

P

mi

=

n i=1
∑

E 2G i ii

Output variables

Output is an m-vector,
y m ×1 =C x 2m × n 2

m=

n(n − 1) given by 2

n ×1

Considering interaction free case, nonlinearity is

T ⎤ f ( y) = ⎡ ⎢ f ( y ) f ( y )....... f m ( y m )⎥ 2 ⎣ 1 1 2 ⎦
where ith component of f(y) is written as o o⎤ f i ( yi ) = E p E q B pq ⎡ ⎢sin( y + y ) − sin y ⎥ i i i ⎣ ⎦ with p and q being internal nodes of generator on which ith non-linear element depends. Now defining
⎡A 1 A= ⎢ ⎢ ⎢A ⎣ 2

where A1= M D: n A ⎤ 3⎥ ⎥ A ⎥ 4⎦ M =diag (Mi) C2=an mxn mat.

-1

× n matrix, A2=A4=Onxn, A4=Inxn

with D = diag {-di}, i = 1,2,……..n. , i = 1,2,…….. n C = [C1:C2], C1 = Omxn, Such that
⎤ B = ⎡ ⎢B MB ⎥ 2⎦ ⎣ 1

T

, B = M − 1C T , B = O n× m 2 1 2

16

and structure of B1 is
⎡ 1 ⎢ ⎢M ⎢ 1 ⎢ ⎢ −1 ⎢ ⎢M 2 ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ . ⎢ ⎢ . ⎢ ⎢ 0 ⎢ ⎢ ⎣

1 M 1 0 −1 M 3 . . 0

. .

. .

B = M − 1C T = 1 2

0 . . . . . 0 .

1 M 1 1 M 2 1 −M 3 . . 1 −Mn

0 1 M 2 0 . . 0

0 . . 0 . . 0 . . . . . . . . 0 . .

0. 0 0 . . 1 −Mn

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

With above definitions, the n-m/c P.System can be expressed by o x ( t ) = Ax ( t ) − Bf ( y )

y(t ) = Cx(t )
Schematic representation of above system is as shown

r=0

+

∑

-u

Y

G(s)

u=f(y)

f(.)

Linear s/s is described by TF
G ( s ) = − C ( sI − A ) − 1 B = C [ s 2 M − sD ]− 1C T

2

2

NL functions f(y) is taken to satisfy sector conditions (i) f ( o ) = 0 i (ii) 0 ≤ y f ( y ) ≤ k y 2 , i = 1 , 2 ,......... i i i i i in finite intervals
o − ( Π + 2δ o pq ) ≤ y i ≤ ( Π − 2 δ pq )

.n

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2.5 Analysis of Power Systems
a) Per-unit analysis of power system having a single generator with a governor connected to an infinite bus

A power system having governor is represented by the equations:
oo o EV sin δ (t ) M δ (t ) + D δ (t ) = Pm (t ) − ΣX

o δ (t ) T e P m ( t ) + Pm ( t ) = P o m − G dt
where M=moment of inertia of the generator, D= damping constant, Pm= mechanical power input, δ= rotor angle Ge, Te= gain and time constant of the governor respectively
Assumptions for the above system ¾ We consider the constants D= Te= Ge= 1 (for per unit analysis) ¾ We also consider the voltages behing the trasient reactance of both the motor and

source to be 1 and the impedence of the load be 1.
¾ We consider that the synchronous frequency of the rotor is 100Hz. ¾ We consider that the rotor angle variation is small so sinδ=δ thereby linearizing

the model at the start of the generator.
¾ We have M= H/πf0=0.0031. ¾ We consider Pm to be a ramp input of magnitude 0.1 /sec. ¾ We neglect the initial value of Pm i.e. Pm0=0. ¾ Simulation time interval is taken to be 0.2sec and the system is sampled for a

period of 10 secs. With these assumptions we construct the CSMP program as follows. CONST D=1, Te=1, Ge=1, E=1, V=1, X=1, M=0.0031, INPUT= 0.1*t D2DOT= -(D/M) D1DOT – (E*V/XM) D + (1/M) INPUT INPUT1DOT= -G*D1DOT – (1/Te) INPUT INPUT= INTGRL (0.0, INPUT)

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D1DOT= INTGRL (0.0, D2DOT) D= INTGRL (0.0, D1DOT) Pe= (E*V/X) SinD TIMER DEL=0.05, OUTDEL=0.2, FINTIM=10.0 PRTPLT Pe END LABEL (“ Variation of generator power wrt time”) STOP
Output: Watts

8.0 6.0 4.0 2.0 0.0 -2.0 -4.0 -6.0 -8.0 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6

Time (in secs) Variation of generator power wrt time

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b) Power Systems having single generator connected to infinite bus having speed exciter:

A power system having an exciter is given by the equation:

τ Aτ Eτ G V t + (K Eτ Aτ G + τ Aτ E + τ Eτ G )Vt + (K Eτ A + K Eτ G + τ E )Vt + (K E + K R )Vt = K A K GVREF
In per-unit analysis of this system e make the following assumptions:
¾ We assume all the constants to be unit i.e. τA= τE= τG= KA= KE= KG= KR= 1. ¾ We assume our reference voltage to be a dc of 5 V i.e. a step input of magnitude

...

..

.

5.0.
¾ We assume the rectifier assembly to behave linearly overa period of time. ¾ We assume that the amplifier is not driven into saturation. ¾ We assume all initial conditions to be equal to zero.

With these assumptions we proceed to construct the model in CSMP as follows: CONST KE=1, KA=1, KG=1, KR=1 TOUA=1, TOUG=1, TOUE=1, INPUT= 5.0*STEP(0.0) VT3DOT=(1/(TOUA*TOUG*TOUE))*(KA*KG)*INPUT - (KE*TOUA8TOUG+TOUA*TOUE+TOUE*TOUG) VT2DOT - (KE*TOUA+ KE*TOUG + TOUE) VT1DOT - ((KE+KR)/ (TOUA*TOUG*TOUE)) VT VT2DOT = INTGRL (0.0, VT3DOT) VT1DOT = INTGRL (0.0, VT2DOT) VT = INTGRL (0.0, VT) TIMER DEL= 0.05, OUTDEL =0.2, FINTIME = 10.0 PRTPLT VT END LABEL (“ Variation of Generator output with time”) STOP

20

Output: Volts 2.5

2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8

Time (in secs) Variation of Generator output with time

21

3. Transient Stability of Power Systems
3.1 Introduction:
With the advent of large power systems came a renewed interest in the stability properties of such systems. Indeed, the tendency of a system to lose synchronism and the possibility of the existence of oscillations in the power transfer between interconnected systems appear to be much more prevalent for large systems than for relatively isolated groups. Most stability studies are based on direct simulation: the post fault system behavior is simulated and the stability properties of the solutions are considered for the various values of the switching time; that is, when normal operating conditions are restored. For low value of this switching time the system regains synchronism. The largest acceptable value of the switching time, i.e. the largest value, for which stability prevails, is generally called the critical switching time. For a system consisting of a single machine connected to an infinite bus, a direct method called the equal-area or the energy- integral criterion has been known for a long time. This system however has no obvious analog for larger systems a direct method for estimating the domain of attraction of a given equilibrium point (i.e. the set of initial conditions for which the resulting motion approaches this equilibrium) is given by the direct method of Lyapunov. This method, which can be applied to any dynamic system, has been used in studies concerning the single-machine and multi-machine power systems. The difficulty in the application of Lyapunov’s direct method is that in general there is no obvious choice for a function suitable for use as a Lyapunov’s function. In most systems describing a physical system, the energy stored in the system appears to be a natural candidate.

22

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3.2 Lyapunov’s Functions:
Consider the dynamic system described by the differential equation: d(x)/dt = Ax(t) – Bƒ[Cx(t)] ------- (1) where A, B, and C are, respectively, (2n × 2n), (2n × m), and (m × 2n) matrices, and ƒ(σ) maps the m-dimensional vector δ = col (σ1, σ2, . . . . . σm) into the m-dimensional vector ƒ(δ) = col [ƒ1(σ1), ƒ2(σ2), . . . . ƒ(σm)]. Note that the nonlinearity ƒ(δ) is time invariant, memory less, and of the diagonal type. It is furthermore assumed that ƒ(0) = 0 and that all the Eigen values of the matrix A have non-positive real parts. As shown above, the motion of the state of a multi-machine power system around an equilibrium state is described by an equation such as (1). The stability of such kind of systems can be described by Popov’s stability theorem to systems with multiple nonlinearities. It concerns the asymptotic stability in the large of the solution x(t) ≡ 0 of (1) and this ensures that for any initial condition x(t = 0) = x0, the ensuring solution x(t) satisfies lim x(t) = 0. t →α
Theorem 1:

The null-solution of differential equation (1) is asymptotically stable in the large if: 1) 0 < σi ƒi(σi) for all i = 1, 2, . . . . m and σi ≠ 0, and 2) there exist a diagonal (m × n) matrix Q = diag (q1); qi ≥ 0, such that Z(s) = (Im + Qs) C(sI2n – A)-1 B is a positive real matrix; i.e. Z(jω) + Z τ(-jω) is a nonnegative definite Hermitian matrix for all real ω ≥ 0. The question of whether or not, for particular matrices, A, B and C exist; can in general be resolved quite readily using graphical techniques or directly using analytical means. Above theorem can be proved by constructing Lyapunov function that involves solving certain algebraic matrix equations involving the matrices A, B, C, and Q. The nonlinearities appearing in the mathematical description of the multimachine power system do not satisfy the conditions 1) of the above theorem since they are of the type ƒi(σi) = sin (σi + σio ) – sin σio , and hence the inequality σi ƒi(σi) ≥ 0 is only satisfied for a

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range of values σ. Hence asymptotic stability in the large can not be concluded. However with the aid of the function V(x) which is to be found as explained above, one can obtain an estimate (through consideration of V(x)) of domain of attraction. This estimate will be large if the Lyapunov function V(x) “fits” well in the system. For the three machine system it is hence required to find a diagonal matrix Q = diag (qi), qi ≥ 0, such that Z(s) = (I + Qs)G(s) is a positive real matrix where

⎤ ⎡ 1 1 1 1 + − ⎥ ⎢ M1s + a1 M2s + a2 0 0 ⎤ ⎥⎡E1E2B12 ⎢M1s + a1 M2s + a2 1⎢ 1 1 1 1 ⎢ ⎥ G(s) = 0 E1E3B13 0 ⎥ + ⎢ ⎥ ⎥ ⎢ s M1s + a1 M1s + a1 M3s + a3 M3s + a3 ⎢ ⎥ 0 0 E E B ⎥ ⎢ 2 3 23⎦ 1 1 1 ⎥⎣ ⎢ − 1 + M2s + a2 M3s + a3 M2s + a2 M3s + a3 ⎦ ⎥ ⎢ ⎣
If all the damping coefficients ai is zero, then qi → α is the only possible choice. If a1, a2, a3 >0, then Z(s) can be made positive real not only with qi → α, but also for finite values of the constant qi, and many possible functions V(x) can thus be constructed.
Theorem 2:

Let δΓ denote the boundary of Γ and let V1 denote the minimum of V(x) over all x in δΓ. The equation V(x) = V1 defines a bounded surface inside Γ and contains 0. The region R1 enclosed by this surface belongs to the domain of attraction of 0. The above theorem follows rather easily from the fact that V(x) ≤ 0, for all x in Γ and the usual estimates of the domain of attraction based on Lyapunov functions. It is possible to obtain a larger domain of attraction by taking into consideration the particular structure of the nonlinear differential equation containing a single nonlinear element. Let Si be the part of δΓ whether either cix = σim or cix = σiM, and let Li be the intersection of Si and the set of all x for which ciΤ[Ax – Bƒ(Cx)] = 0.
Theorem 3: Let V2 denote the minimum of V(x) over all x in Li, i = 1, 2, . . . . m. (clearly V2 ≥ V1 and

equality holds exceptionally). The equation V(x) = V2 defines a bounded surface inside Γ

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and contains 0. The region R2 enclosed by this surface belongs to the domain of attraction of 0. Clearly this theorem gives the larger domain of attraction than what is predicted by Theorem 2. The application of Theorem 3 generally leads to the considerable enlargement of the domain of attraction compared to what can be obtained from Theorem 2.

3.2.1 Lyapunov’s Function for Power System without Governor:
The normalized equations of a synchronous generator – infinite bus system is the post fault state can be written as

δ + D δ = Pi − sin δ

..

.

---------- (2)

Where δ is power angle, Pi the mechanical power input, and D the constant damping coefficient. The equilibrium state δ0 = sin-1Pi, δ 0 = 0 can be transferred to the origin by defining a new variable x = δ - δ0
.
.

------------- (3)

So that (2) becomes
..

x = − D x + Pi − sin ( x + δ 0 ) = − D x + Pi (1 − cos x) − 1 − Pi 2 sin x By defining x1 = x x2 = x ƒ(x1) = − Pi (1 − cos xi ) + 1 − Pi 2 sin xi ------------- (5)
.

.

------- (4)

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We obtain the system equations in state variable form, as ⎡ .. ⎢ x..1 ⎢x ⎣ 2 ⎤ ⎡0 ⎥ = ⎢ ⎥ ⎣0 ⎦ 1 ⎤ ⎡ x1 ⎤ ⎡ 0 ⎤ + ⎢ ⎢ ⎥ ⎥ ƒ(σ) ⎥ − D ⎦⎣x2 ⎦ ⎣ − 1⎦

⎡ x1 ⎤ σ = [1 0]⎢ ⎥ ⎣ x2 ⎦

------------ (6)

r=0 + -

σ

N. L

ƒ(σ)

1 S+D

-x2

1 S

-x1

Block diagram representation of single machine system without governor

The transfer function of the linear part is G ( s) = 1 s ( s + 1) -------- (7)

So that the system can be represented by block diagram as given above. The nonlinear function defined in equ. (5) Satisfies the “sector condition” for – (π + 2δ0) < σ < (π - 2δ0). Therefore, if q is determined from the application of Popov’s method, the value of M can be determined as M = min {M 1, M 2 } = M 2

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Because the system characteristic equation has a zero eigen value, we can reduce equation (6) to the form below:x = − Dx − ƒ(σ )
.

ξ = −ƒ(σ )
σ =−
1 1 x+ ξ D d

.

-------- (8)

An easy way to arrive at this form is to use the partial-fraction expansion of G(s) of equation (7) for this system, ρ = 1/D r = q(ρ + hTg) = 0 ------- (9) Application of the Popov condition to this system results in Re (1 + qjω) Or 1 jω ( jω + D ) ≥0

qD − 1 ≥ 0 for all real ω > 0 D2 + ω 2

n where n ≥ 1. D For this system, ψ(s) = (s + D), so that the expression is,

This is satisfied if qD ≥ 1. let us choose q =

W(ω) = (qD – 1) = ( n – 1)
∴θ ( z ) = − n − 1
v ( z ) = n − 1 as r = 0.

The vector u reduces to a scalar u for this system, and u = Then
V ( x, ξ , σ ) =

n − 1 . The Lyapunov matrix

equation reduces to the scalar equation (-D)b + b(-D) = - (n – 1) so that b = (n – 1)/2D.
(n − 1) 2 1 2 n σ x + ξ + ∫ ƒ(σ )dσ -------- (10) 2D 2D D 0

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The variables x and ξ are related to the variables x1 and x2 by the relations x = x2 ξ = ( Dx1 + x2 )
~

So that
V ( x1 , x 2 , σ ) = D 2 n 2 n σ x1 + x 2 + x1 x 2 + ∫ ƒ(σ )dσ 2 2D D 0

------- (11)

Different choices of n will lead to different Lyapunov’s functions. .to illustrates the computational procedure, a choice of n = 1 + D3 is made, which gives D 2 (1 + D 3 ) 2 (1 + D 3 ) σ V ( x1 , x 2 , σ ) = x1 + x 2 + x1 x 2 + ƒ(σ )dσ 2 2D D ∫0

------ (12)

3.2.2 Lyapunov’s Function for Power System with Velocity Governor:
The introduction of a velocity governor modifies the block diagram. The additional state variable x3 corresponds to the governor power, while g and a are the parameters of the governor, which is approximately by one time constant. The system equations in state variable form are
⎡ . ⎤ ⎢ x.1 ⎥ ⎡ 0 ⎢ x ⎥ = ⎢0 ⎢ .2 ⎥ ⎢ ⎢ x3 ⎥ ⎢ ⎣0 ⎢ ⎥ ⎣ ⎦ 1 −D g 0 ⎤ ⎡ x1 ⎤ ⎡ 0 ⎤ ⎢ ⎥ ⎢ ⎥ − 1⎥ ⎥ ⎢ x 2 ⎥ + ⎢ − 1⎥ ƒ( σ ) − a⎥ ⎦⎢ ⎣ x3 ⎥ ⎦ ⎢ ⎣0 ⎥ ⎦

σ = [1

0

⎡ x1 ⎤ ⎥ 0 ]⎢ ⎢ x 2 ⎥ ------------ (13) ⎢ ⎦ ⎣ x3 ⎥

The transfer function of the linear part is
G(s) = ( s + a) ------ (14) s[ s + (a + D ) s + (aD + g )
2

29

Reducing the above equation, we obtain ⎡ . ⎤ ⎡ 0 ⎢ x.1 ⎥ = ⎢ ⎢ x ⎥ ⎣ − ( aD + g ) ⎣ 2⎦ 1 ⎤ ⎡ x1 ⎤ ⎡ 0 ⎤ ⎢ ⎥ − ⎢ ⎥ ƒ( σ ) − (a + D )⎥ ⎦ ⎣ x 2 ⎦ ⎣1 ⎦

ξ = −ƒ(σ )

.

σ = ⎢− ⎣ aD + g
Notice here that

⎡ (a 2 − g )

− a ⎤ ⎡ x1 ⎤ a ξ ------ (15) ⎥⎢x ⎥ + aD + g ⎦ ⎣ 2 ⎦ aD + g

ρ=

a aD + g r = q ( ρ + h T g ) = 0 ---------- (16)

Application of the Popov’s condition to the transfer function given in equation (13) results in
Re(1 + qjω ) jω + a ≥0 jω[(a + D ) jω + (aD + g − ω 2 )]

Or (qD − 1)ω 2 + qa(aD + g ) + g − a 2 ≥0 (aD + g − ω 2 ) 2 + (a + D) 2 ω 2

For all real ω > 0. For this to hold, it is necessary and sufficient that qD – 1 ≥ 0 And qa(aD + g) + g – a2 ≥ 0 If we choose q = 1/D, the second of the above inequalities reduces to 30

g (a + D) ≥0 D

Which holds always, since g, a, and D are positive numbers. For the system under consideration

ψ ( s ) = [ s 2 + (a + D) s + (aD + g )]

So that
W (ω ) = g (a + D ) D

And

θ ( z ) = g ( a + D) / D
And

v( z ) = + g (a + D / D , r = 0.
The vector u is given by

⎡ ⎤ ⎡ u1 ⎤ ⎢ g (a + D) ⎥ u=⎢ ⎥= D ⎥ ⎣u 2 ⎦ ⎢ 0 ⎣ ⎦ The next step is to solve the Lyapunov matrix equation which in this case becomes
⎡0 − ( aD + g ) ⎤ ⎡b11 ⎢ 1 − ( a + D ) ⎥ ⎢b ⎣ ⎦ ⎣ 12 b12 ⎤ ⎡b11 +⎢ b22 ⎥ ⎦ ⎣b12 ⎡ g (a + D ) b12 ⎤ ⎡ 0 1 ⎤ D ⎢ − ( aD + g ) − ( a + D ) ⎥ = − ⎢ b22 ⎥ ⎢ ⎦ ⎦⎣ 0 ⎣ ⎤ 0⎥ 0⎥ ⎦

From this, one can obtain 2 × 2 matrix B, after which equality can be deduced.

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4. Power System Stabilizers
Introduction:

The power system stabilizer (PSS) is an optional control that is part of the excitation system for generator control. The PSS acts to modulate the generator field voltage to damp electrical power-speed oscillations. It is desirable to ensure that the generator controls are equipped to support transmission system reliability by enhancing the excitation controls. The PSS provides supplementary control that improves dynamic stability by increasing damping of power swing oscillations. Excitation system with high gain and fast response times greatly aid transient stability (synchronizing torque), but at the same time tend to reduce small signal stability (damping torque). The main objective of the PSS is to increase damping of generator rotor angle swings, which can occur in a broad range of frequencies in the power system. PSS typically utilizes phase compensation and adjusting phase compensation is the main task in tuning PSS parameters. Phase compensation is accomplished by adjusting the PSS to compensate for phase lags through the generator excitation system, and power system such that PSS provides torque changes in phase with speed changes. Tuning should be performed when system configurations and operating conditions result in the least damping.
Functioning of PSS:

The purpose of a PSS is to extend the angular stability limits of a power system by providing supplemental damping to the oscillation of synchronous machine rotors through generator excitation. This damping is provided by electric torque applied to the rotor that is in phase with the speed variation. Once the oscillations are damped, the stability limits improve. This supplementary control is very beneficial during the line outages and large power transfers. The basic function of PSS is to add damping to the generator rotor oscillations by controlling its excitation by using auxiliary stabilizing signal. To provide damping, the stabilizer must produce a component of electrical torque in phase with the rotor speed deviations.

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PSS are designed with various types of inputs. They include speed, frequency, power, accelerating power and integral of accelerating power. The PSS may derive these quantities from generator terminal voltage and current measurements.
Block diagram of a Power system having PSS

+

Σ

δ

..

Pm

K 1 + sT

δ

.

1 s

δ

Governor & Turbine

Exciter & AVR

Eq Eex

Vref

+

Σ

PSS

Δω

Tr. Lines

Location of PSS

For oscillations in the range of 0.4 to 0.8 Hz, which are called as inter area oscillations, multi machine modeling of power systems is used. Generally the PSS are installed in those machines having higher MVA ratings but as the purpose of PSS is to provide a supplementary signal to the excitation system, the excitations systems also play an important role from excitation system point of view, PSSs are placed on exciters having high gain and fast acting excitation systems (voltage regulator time constant TA is very small). In general PSS location in a multi machine system is decided using two popular methods 1. participation factors 2. mode controllability

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Generator

Design of PSS

Once the selection of input signal is made, the next step is to decide the parameters of the transfer function of PSS so as to improve the damping of the system. Several methods are available in the literature like 1. Frequency domain method 2. Eigen value assignment method 3. Using optimal control theory and based on minimization of a quadratic performance index. 4. H-infinity control Out of several methods available first two methods are widely used and are easy to apply.
PSS design using frequency domain technique

By measuring the transfer function between the terminal voltage and stabilizer output it is possible to experimentally determine the phase characteristics of the plant. The determination of plant transfer function GEP[s] can be done analytically or experimentally from field test. The frequency at which the transfer function GEP[s] suddenly changes in phase is the oscillation frequency, for which the PSS is to be designed. Once the GEP[s] is determined the PSS time constant are adjusted by trial and error. To set the gain of the PSS root locus analysis is performed
PSS Design using Eigen value Assignment Technique

As the function of the PSS is to provide a damping torque, this design technique is centered on placing the rotor mode eigen value at desired location. The Eigen value location is decided based on the damping required and the undamped natural frequency of the open loop system. When the damping is negligible, the natural frequency of the system is related to synchronizing torque. Hence while improving the damping torque, the synchronizing torque should not be reduced,. Also, from the mathematical point of view once the location of one Eigen value is relocated to shift to the left of the imaginary axis, some other eigenvalue automatically moves to the right of the imaginary axis. This happens because the trace of the closed loop matrix remains the same.

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The eigenvalue placement method nearly concentrates on the relocation of the rotor mode. This method does not have any control, over the unassigned modes. The unassigned modes may move towards right of imaginary axis and may make the system unstable. Hence for this purpose all Eigen values of closed loop system are to be computed. With the changes in network as well as generation conditions, based on previous tuning experiences, time, effort and cost involved, PSSs are to be tuned periodically, at some regular intervals or as and when some operational problems are attributed due to PSS. The point to be emphasized is that, once the PSS parameters are tuned and PSS is made functional, the PSS parameters need not be fixed constant throughout the lifetime of exciter and machine.
A simplified model of PSS

Single Input Power System Stabilizer

The figure shows the generalized form of a power system stabilizer with a single input. Some common stabilizer input signals (VSI) are speed, frequency, and power. T6 may be used to represent a transducer time constant. Stabilizer gain is set by the term KS, and signal washout is set by the time constant, T5. In the next block, A1 and A2 allow some of the low-frequency effects of high-frequency torsional filters (used in some stabilizers) to be accounted for. When not used for this purpose, the block can be used to assist in shaping the gain and phase characteristics of the stabilizer, if required. The next two blocks allow two stages of lead-lag compensation, as set by constants T1 to T4.

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5. Conclusions
The importance of modeling with its various types was introduced. The various dvantages of state variable. modeling of physical systems viz. Power System were pointed out. The concept of system states and state variable modeling was indicated. An attempt to discuss state variable model as a core of mathematical technique and its applications to various cases of single-machine and multi-machine power systems for control and system analysis was made and presented. Because of universality and convenience of this modeling and its applications to power system problems, the technique may be appreciated well. The fascinating aspect of state variable modeling is that it finds applications to variety of disciplines as it follows a generalized system approach. S.V. modeling technique is reported to have gained maturity and popularity in engineering disciplines and has been successfully applied to variety of engineering problems both for off-line and on-line analysis, control and design. We after modeling the Power Systems arrived at the conclusion that they are potentially unstable due to the various disturbances and oscillations going on in the system. The governor stabilizes the system against the oscillations in the mechanical input power and the exciter stabilizes it against the variations in the terminal voltage of the generator. To improve the stability further PSS (Power System Stabilizers) should be used. The Power Systems are inherently non-linear and hence cannot be modeled using classical techniques. So the State-space model gives a near-perfect model for them after certain assumptions. For more detailed models Adaptive Cotrol and Fuzzy Logic should be used.

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6. Future Scope of Work
In future this model can be generalised to multi-machine power systems having:
¾ A common governor ¾ A common exciter ¾ Individual Governors ¾ Individual Exciters

There after the various stability criteria of such systems can be studied. The operation, design and model of multi-machine PSS can also be studied. The same systems can be modeled using various advances in control system like Fuzzy Logic, Adative Control, Robust Controllers etc. Then the model will be more acurate and hence the stability analysis can be done at ease. A compensator for the Excitation System can be designed and studied using any of the above control design techniques. In this manner this project has still a lot to offer for future work and study.

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7. References
1. Ogata K.: Modern Control Engineering, Third Edition, Prentice Hall of India, pp 710-785, 2000. 2. Chakrabarti A., Soni M.L., Gupta P.V. and Bhatnagar U.S.: A text book on Power System Engineering, Dhanpat Rai & Co., pp 346-417. 3. Pai M.A., Mohan M.A. and Rao J.G: Power System Transinet Stability Regions Using Popov’s Method, IEEE transactions on Power Apparatus and Systems, pp 788-794, May-June 1970. 4. Willems J.L. and Willems J.C.: The application of Lyapunov methods to the computation of transient stability regions for multi-machine power systems, IEEE transactions on Power Apparatus and Systems, pp 795-803, May-June 1970. 5. Bala Subraniyam: Power System Stabilizers in Power Systems, The Journal of CPRI, pp 73-84, March 2005. 6. Pai M.A. and Sawyer Peter W., Power Systems Dynamics and Stability, Second Edition, Pearson Education, 2003. 7. Anderson P.M. and Fouad A.A, Power System Control and Stability, Second Edition, John Wiley and Co., IEEE Press, 2004. 8. Kundur Prabha, Power System Stability and Control, Electric Power Research Institute, 1999.

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