EXPERIMENT 1

SMALL SIGNAL STABILITY ANALYSIS OF A SINGLE MACHINE INFINITE BUS SYSTEM

1.1 AIM:

To become familiar with the various aspects of small signal stability of Single Machine Infinite Bus System (SMIB).

1.2 OBJECTIVES:

To understand modeling and analysis of small signal stability of a SMIB system. To obtain linearized swing equation and to determine the roots of characteristic

equation damped frequency of oscillation and undamped natural frequency.

1.3 SOFTWARE REQUIRED:

MATLAB

1.4 THEORETICAL BACKGROUND:

Small signal stability is the ability of the power system to maintain synchronism when subjected to small disturbances. A disturbance is considered to be small if the equations that describe the resulting response of the system can be linearized for the purpose of analysis. Instability is of two forms

1). Steady increase in generator rotor angle due to synchronizing torque.

2) Rotor oscillations of increasing amplitude due to insufficient damping torque.

For small disturbance, the nonlinear equations characterizing the dynamics of the system may be linearized around an operating point for the purpose of analysis. Small signal stability using linear techniques provides valuable information about the dynamics of the system and assists in its design. The location of the roots of the characteristic equation of the linearized differential equation will reveal the stability status of the

system. We can also obtain the response of rotor angle and frequency of the rotor of the synchronous machines.

1.4.1 MODELLING FOR SMALL SIGNAL STABILITY:

Let us consider a system as shown in the Fig 1.1

Fig 1.1

For the purpose of analysis the system of Fig 1.1 a) may be reduced to the form of Fig 1.1 b) by using Thevenin’s equivalent of the transmission network external to the machine and the adjacent transmission. Because of the relative size of the system to which the machine is supplying power, dynamics associated with the machine will cause virtually no change in the voltage and frequency of Thevenin’s voltage VB. Such a voltage source of constant voltage and frequency is referred to as infinite bus.

1.4.2 GENERATOR REPRESENTED BY CLASSICAL MODEL:

With the generator represented as classical model and all resistances neglected, the system representation is shown in figure. 1.2.

Here E' is the voltage behind Xd'. Its magnitude is assumed to remain constant at the pre-disturbance value. Let δ be the angle by which E’ leads the infinite bus voltage EB .As the rotor oscillates during a disturbance δ changes.

With E’ as reference phasor,

Fig 1.2 Equivalent circuit of SMIB system

The complex power behind Xd' is given by

(1.1)

With stator resistance neglected, the air gap power (Pe) is equal to the terminal power (P). In per unit, the air gap power is equal to the air gap torque

(1.2)

Linearising (1.2) about an initial operating condition represented by δ=δo yields

(1.3)

The equation of motion in per unit is

(1.4)

Where ∆ωr is the per unit speed deviation, δ is the rotor angle in electrical radians

ωo is the base rotor electrical speed in radians per second and p is the differential

Operator d/dt with time t in seconds

Linearising(1.4) and substituting for ∆Te we get

(1.5)

Where Ks is the synchronizing torque coefficient given by

(1.6)

Writing the equations (1.5) and (1.4) in vector matrix form we get

(1.7)

This is of the form X’=AX +bu. The elements of the state matrix are seen to be dependent on the system parameters KD ,H, XT and the initial operating condition represented by the values of E’ and δo . The block diagram representation shown in the Fig 1.4 can be used to describe small signal performance.

From the block diagram (1.3) we have

Rearranging we get

(1.8)

Therefore the characteristic equation is given by

(1.9)

This is of the general form

Therefore the undamped natural frequency is

and the damping ratio is

As the synchronizing torque coefficient Ks increases, the natural frequency increases and the damping ratio decreases. An increase in damping torque coefficient KD increases the damping ratio, whereas an increase in inertia constant decreases both ωn and ζ.

Fig 1.3 block diagram of Single Machine Infinite Bus System with classical generator model

1.5 EXERCISE:

Fig 1.4 shows a system representation applicable to a thermal generating station consisting of four 555MVA, 24 KV,60 Hz units.

Fig 1.4

The network reactance shown in the fig. are in per unit on 2220 MVA, 24 KV base (referred to the LT side of the step up transformer). Resistances are assumed to be negligible.

The objective of this example is to analyze the small signal stability characteristics of the system about the steady state operating condition following the loss of circuit 2. The post fault condition in per unit on the 2220 MVA, 24 KV base is as follows.

The generators are to be modeled as a single equivalent generator by the classical model with the following parameters expressed in per unit on 2220 MVA, 24 KV base:

Xd'=.3 H=3.5MWs/MVA

a) Write the linearized state equations of the system. Determine the eigen values , damped frequency oscillation in Hz, damping ratio and undamped natural frequency for each of the following values of damping coefficient (in pu torque/pu speed)

i) KD=0 ii) KD=10 iii) KD=-10

b) For the case with KD=10, find the left and right eigen vectors and participation matrix. Determine the time response if at t=0, ∆δ=5º and ∆ω=0

PROGRAM:

clc;clear all;P=0.85;Q=0.52;H=3.07;Xd=0.253;f=50;Vt=1;Xe=0.65;w=(2*pi*f);I=(P-(sqrt(-1)*Q))Vinf=(Vt-(sqrt(-1)*I*Xe))E=(Vt+(sqrt(-1)*I*Xd))del1=angle(E);del2=angle(Vinf);Ks=abs((E*Vinf*cos(del1-del2))/(Xe+Xd))Kd=[0 10 -10];for i=1:3disp('KD=');disp(Kd(i));A=[(-Kd(i)/(2*H)) -(Ks/(2*H));w 0]B=[(1/(2*H)) 0]wn=5.958;zeta=(0.0137*Kd)fn=(wn/(2*pi))wd=wn*sqrt(1-zeta(i)^2);fd=(wd/(2*pi))if Kd(i)==10 e=eig(A)%---REV=Right Eigen Vector for the eigen values of A----% [REV,Dr]=eig(A); REV LEV=inv(REV) P=[REV(1,1)*LEV(1,1) REV(1,2)*LEV(2,1);REV(2,1)*LEV(1,2) REV(2,2)*LEV(2,2)]%--------Time Response--------% delw0=0; deldel0=0.0873; a=real(REV(1,1));

b=imag(REV(1,1)); c=real(LEV(2,2)); d=imag(LEV(2,2));delw=((b*c)-(a*d))*(-2*deldel0)

deldel=(2*deldel0*sqrt(c^2+d^2))alpha=real(Dr(1,1));beta=imag(Dr(1,1));endendt=0;for i=1:1000w(i)=delw*(exp(alpha*t))*sin(beta*t);dt(i)=deldel*(exp(alpha*t))*cos((beta*t)-(atan(d/c)));at(i)=t; t=t+0.02;endplotyy(at,w,at,dt);

EXPERIMENT 2

INDUCTION MOTOR STARTING ANALYSIS

2.1 AIM:

To perform induction motor starting analysis under various conditions and to find torque –slip characteristics.

2.2 OBJECTIVE:

To integrate whether the starting motor can be successfully started under various operating conditions.

2.3 SOFTWARE REQUIRED:

MATLAB

2.4 THEOREY:

2.4.1 MOTOR STARTING ANALYSIS:

During the motor starting period, the starting motor appears to the system as small impedance connected to a line. It draws a large amount of current from the system, about six times the motor rated current, which therefore results in voltage drops in the system and imposes disturbances to the normal operation of other systems loads. Since the accuracy torque is dependent on motor not be able to reach its rated speed to extremely low terminal voltage, in some cases the starting motor may not be able to reach its rated speed to extremely low terminal voltage. This makes it necessary to perform a motor starting analysis.

The purpose of performing motor study is to investigate whether the starting motor can be successfully started under the operating conditions and to see it starting the motor will seriously impede the normal operation of the other loads in the system.

The motor starting analysis chapter is divided in to the following sections. The motor starting tool bar section explains how to start a motor, starting calculation to open and view an output report of to select display options.

The motor starting study case section explains how you can create a new study case what parameters are required to specify a steady state.

The display option sector explains available option for displaying some key system parameters of output result of the one line diagram.

Calculation section describes calculation method using the module.

2.4.2 DYNAMIC MOTOR ACCELERATION:

In the acceleration, a dynamic model throughout the whole simulation models the accelerating model in the accelerating motor. For this study you can also need to specify the a load performance torque model for the load that the motor is driving.in the model page you can specify motor dynamic model from the 5 different types

Single 1- equivalent (Thevenin’s) circuit model with constant rotor resistance and reactance.

Single2-circuit model with deep bar effect rotor resistance and reactance

Dbl1- double cage circuit model with independent motor cages.

Dbl2- double cage circuit model with independent motor cages.

TSC- torque – slip characteristics curve model.

All the three models are locked on an electrical circuit representation of the motor. The TSC model allows modeling a starting motor directly from manufacture performance curve.

2.4.3 STATIC MOTOR STARTING:

In this model , it is assumed that the starting motor can always be started as u specified from the motor – editor acceleration time at 0% and of load and the module interpolates the acceleration time for the motor based on three two values.

2.4.4 STARTING A DEVICE:

A starting motor (a) have one of the 13 types of starting devices modeled in the motor starting studies

2.4.5 FEATURES OF MOTOR STARTING STUDY

A state load can be switched on and off at any time during a simulation with user defined category.

The motor can be switched on and off repeatly at any time during simulation.

The motor can be specified by an individual load or by loses at starting category.

2.4.6 MOTOR STARTING ANALYSIS REQUIRED DATA

Bus data Branch data Capacitor data Induction motor data Static load data Lumped load data additional

2.4.7 BUS DATA

Bus IN

Nominal KV Load diversity factor

2.4.8 BRANCH DATA:

Branch I∞

Cable and transmission line length and unit.

2.4.9 INDUCTION MOTOR DATA

Induction motor I∞

Rated KW/HP & KV Loading category Id & % loading Equipment cable data

2.4.10 CAPACITOR DATA:

Capacitor I∞

Rated KV, KVAR/bus & no. Of buses

2.4.11 LUMPED LOAD DATA:

Rated KV, MVA, power factor Loading category ID & % loading

2.4.12 ADDITIONAL DATA:

Motor acceleration at – no load & full load Starting device –data when needed No load & full load acceleration – time

2.5 PROGRAM:

clc;

clear all;freq=60;j=63.87;Ns=1800;KW=1678.5;KV=2.3;FLA=526.675;BaseKV=((KW*1000)/(FLA*1.732));n=4;V1=1.026;V2=1.026;V3=1.026;S3ph=125;Ssys=10;Zline=Ssys/S3ph;P13=0;Q13=0;Qc3=0;Zt=0.001+0.0001i;Zs=0;r1=0.029;r2=0.022;x1=0.226;x2=0.226;Te=0;g=0;t=0;t1=0;%speed irpfTIdata=[0 6 9 010 5.9 18 020 5.8 20 030 5.7 36 040 5.6 50 050 5.5 60 060 5.5 71 070 5.4 81 080 5.2 86 090 4 88 0 96 3.2 89 098.5 1.7 91 0]for i=1:12n(i)=(((data(i,3)/100)))+(sqrt(-1)*sin(acos(data(i,3)/100)));Zm(i)=(n(i))/data(i,2);I(i)=abs((V1)/(Zm(i)+Zt+Zline));V3(i)=abs(I(i)*Zm(i)*BaseKV);h=(data(i,1)*(1800/100));s(i)=(Ns-h)/Ns;sI(i)=(Ns-g)/Ns;d(i)=((r1+(r2/s(i)))*(r1+(r2/s(i))))+((x1+x2)*(x1+x2));Id2(i)=(V1/sqrt(d(i)));omega=(2*pi*Ns)/60;Tm(i)=(3*V3(i)*V3(i)*(r2/s(i)))/(omega*d(i));Tacc(i)=(Tm(i)-(data(i,4)/100));dels(i)=(s(i)-sI(i));delt(i)=abs(((j*1000)*dels(i))/Tacc(i));g=h;

N(i)=Ns*(1-s(i));t=t+delt(i);at(i)=t;endZm%s%Tm%N%atplot(s,Tm)%plot(N,Tm)%plot(at,Tm)%plot(at,I)xlabel('Slip');ylabel('Torque');

EXPERIMENT 3

HARMONIC ANALYSIS OF A RECTIFIER FEEDING RL LOAD

3.1 AIM:

To perform the harmonic analysis of a full bridge rectifier feeding RL load and to note down the harmonic level.

3.2 OBJECTIVES:

To compute the harmonic analysis of the output voltage and output current. To find the harmonic order by conducting FFT analysis.

3.3 SOFTWARE REQUIRED:

MATLAB SIMULINK

3.4 THEORETICAL BACKGROUND:

HARMONICS:

Harmonics are sinusoidal components of a periodic wave or quantity having a frequency that is an integral multiple of the fundamental frequency. The fundamental together with its frequency multiples is referred as harmonic series. The frequency multiples of 3f, 5f, 7f etc.. are odd harmonics while, the frequency multiples of 2f, 4f, 6f etc.. are even harmonics. The FFT analysis is done to find the harmonic order.

The sinusoidal voltage or current which is dependent on time is represented by the following expression

v(t) = Vsin(wt)

i(t) = Isin(wt+ф)

Fig 3.1 Sinusoidal Voltage and Current Waveform

Non sinusoidal voltage wave is represented in fourier series as follows.

v(t) =V0+ V1 sin(wt) + V2 sin(2wt) + V3 sin(3wt) ……..

Fig 3.2 Non-Sinusoidal Voltage Waveform

Fig 3.3 Fundamental Voltage Waveform

3.5 SOURCES OF HARMONICS:

Any electrical load that presents a constant impedance to the power source throughout the cycle is a linear load. Linear loads draw sinusoidal current from the source and do not cause harmonics.

A non-linear load can be defined as one which draws current discontinuously or its impedance varies with each cycle of the ac waveform. It is the source of harmonics. Some of the non-linear loads which cause harmonics are given below:

Diode and thyristor rectifiers. Switched mode power supplies. Variable speed drives. Electronic power supplies. DC motor drives. Battery charges. Electronic ballasts. Arc furnaces.

3.6 EFFECTS OF HARMONICS:

Harmonics causes increase in system line losses.

Interaction with power system equipment and causes additional line losses, overheating, over loading.

Leads to voltage distortion.

Leads to low power factor.

Interference with telecommunication lines.

Errors in metering device.

Low efficiency.

Low power factor.

Harmful disturbances to neighboring appliances.

3.7 SYSTEM MODEL:

Fig 3.4 Single-phase bridge rectifier with RL load

3.8 SYSTEM PARAMETERS:

Frequency=60Hz.Load resistance=0.5Ω.Load inductance=6.5mH.

EXPERIMENT 4

AVAILABLE TRANSFER CAPABILITY

4.1 AIM:

To estimate the available transfer capability between the sending area and receiving area through the tie-lines connecting the two areas.

4.2 OBJECTIVES:

To determine the ATC for maintaining system reliability while serving a wide range of transmission transactions.

To study the methods of determining the ATC and the importance of it.

4.3 SOFTWARE USED:

MATLAB

4.4 THEORETICAL BACKGROUND:

4.4.1 DEFINITION:

Available Transfer Capability (ATC) is a measure of the transfer capability remaining in the physical transmission network for further commercial activity over and above the already committed uses.

ATC can be expressed as

ATC=TTC-TRM-ETC+CBM

Where, TTC (Total Transfer Capability) is defined as the amount of electric power that can be transferred over the interconnected transmission network or particular path or

interface in a reliable manner while meeting all of a specific set of defined pre and post contingency conditions.

TRM (Transfer Reliability Margin) is defined as the amount of transmission transfer capability necessary to ensure that the interconnected network is secure under a reasonable range of uncertainties in the system conditions.

CBM (Capacity Benefit Margin) is the amount of transmission transfer capability reserved by the load serving entities to ensure access to generation from interconnected systems to meet generation reliability requirements. It also helps to reduce the installed capacity of the plant.

ETC (Existing Transfer Capability) refers to the power transfer capability that must be reserved for already committed transactions.

Existing transaction is the power flow over the transmission paths at the desired time at which ATC should be calculated. This is the already committed used power on the transmission path. Utilities would have to determine adequately their ATC’s to ensure that system reliability is maintained while serving a wide range of transmission transactions. ATC between and within areas of the interconnected power system and ATC for critical transmission paths between areas would be continuously updated and posted changes in scheduled power transfers between areas.

TRM

CBM

ATC

TTC ETC

Fig 3.1 Basic Definition of ATC

4.4.2 GOVERNING PRINCIPLES:

The following ATC principles govern the development of the definition and determination method of ATC

The ATC value produced by the calculation must give a reasonable and dependable indication of transfer capabilities available to the electric power market.

ATC calculation must recognize the dependency of ATC on the points of electric power injection, the direction of transfer across the interconnected transmission network and the points of power extraction.

Regional or wide-area co-ordination is necessary to develop the post information that reasonably reflects the ATC of the interconnected transmission network.

ATC calculations must confirm to regional, sub-regional, power pool and individual system reliability planning and operating policies, criteria or guides.

The determination of ATC can accommodate reasonable uncertainties in system conditions and provide operating flexibility to ensure the secure operation of the interconnected network.

4.4.3 SIGNIFICANCE OF ATC

The information of ATC as an important indicator of the system performance, is useful in restructured energy market in the following ways

It provides the knowledge of power system capability about the present system condition.

Running the system under ATC limits also ensures system security and reliability to some extent, since the calculation of ATC is based on the security constraints with the consideration of critical contingencies that can lead the system normal state to alert state.

The ATC is required in making decisions for the transactions between market participants. The market participants check for the power contract themselves.

The ATC is also useful in enhancing the system stability. With the knowledge of the limiting condition for the ATC, the system operator can take some operating or planning decision to avoid this limiting condition and thus enhance system capability.

The ATC can also serve as an indicator of power congestion through transmission lines.

The ATC is useful in transmission costing function. The ISO can put more transmission cost for the transaction through transmission path having low value of ATC.

4.4.4 METHODS OF STATIC ATC DETERMINATION:

Method based on linear sensitivity factors

Method based on Continuation power flow

Method based on Optimization power flow

4.4.4.1 METHOD BASED ON LINEAR SENSITIVITY FACTORS:

It offers a great potential for real time calculation of ATC. Use of these factors offers an approximate but extremely fast model for static ATC determination. In this we are having DC power transfer distribution factor, AC power transfer distribution factors, and line outage distribution factors for ATC determination.

4.4.4.1.1 PTDF-DEFINITION:

For ATC determination the MW flows must be allocated to each line or group of line in proportion to the MWs being transmitted by each transaction. This is accomplished through the use of the linear Power Transfer Distribution Factors (PTDF). From the power flow point of view, a transaction is a specific amount of power that is injected into the system at one bus by a generator and removed at another bus by a load.

The coefficient of linear relationship between amount of transaction and flow on a line is called PTDF. It is also called as sensitivity because it relates the amount of one change transaction amount to another change line power flow. The PTDF is the fraction of the amount of transaction from one bus to another that flows over a given transmission

line. PTDFij,mn is the fraction of transmission from bus m to n that flows over a transmission line connecting bus i to j.

Linear sensitivity factors method use DC power flow. The DC power flow model assumes that only the angles of the complex bus voltage vary, and the variation in small voltage magnitudes is assumed to be constant. Transmission lines are assumed to have no resistance and hence no losses.

The equation for PTDF is

PTDFij,mn= (Xim- Xjm– Xin+ Xjn) / xij

Where ,

xijis the reactance of transmission line connecting bus i and bus j

Ximis entry in the ith row and mth column of the bus reactance matrix X

and the maximum amount of power transfer is given by

Pmn,ijmax = (Pij

max- Pijo)/ PTDFij,mn

Where

Pijmax is the line’s flow limit

Pijo is the base case flow on the line

ATC is the minimum of the maximum allowable transactions over all the lines

ATCmn = minij Pmn,ijmax

4.4.2 METHOD BASED ON CONTINUATION POWER FLOW:

This method runs AC load flow for each increment of transaction between an interface and checks whether any of the operating conditions such as line flow limit or bus voltage limit is violated. For the increment of transactions, continuation power flow is also used to find out maximum load ability or voltage instability point.

4.4.3 METHOD BASED ON OPTIMIZATION OF POWER FLOW:

ATC determination can be formulated as an optimization problem which finds out maximum value of the transaction between given interface while satisfying the network power balance and security constraints such as line flow, voltage limits and voltage instability points.

4.5 ALGORITHM:

STEP 1. Start

STEP 2. Read the number of buses and their type.

STEP 3. Read the power generation (Pg) and power demand (Pd) at all the buses and calculate the net power injection (P).

STEP 4. Read the voltage at each buses or if not available initialize to flat start.

STEP 5. Read the voltage limits Vmin and Vmax.

STEP 6. Read the bus load angles or if not available initialize to flat start.

STEP 7. Read the Transmission Reliability Margin (TRM) & Capacity Benefit Margin (CBM) values for all lines.

STEP 8. Read or compute the Y-bus matrix.

STEP 9. Set a suitable convergence tolerance and run Fast Decoupled Load Flow (FDLF).

STEP 10. From FDLF find the new power flows at all the buses.

STEP 11. Calculate the power flows through all lines Pij=Pi-Pj.

STEP 12. Existing Transfer Commitments (ETC) = Flow through the lines.

STEP 13. Increase the power demand at all the load buses by a small set value.

STEP 14. Run steps 8,9,10.

STEP 15. Check for voltage limit violation, if not violated go to 12 else 15.

STEP 16. Calculate the power flows through all lines Pij=Pi-Pj.

STEP 17. Total Transfer Commitments (TTC) = Flow through the lines.

STEP 18. Calculate Available Transfer Capability ATC=TTC-TRM-ETC+CBM

4.6 EXERCISE:

To determine the ATC of the given sample bus system.

1 .02+.04i 2

.01+.03i .0125+.025i 256.6+110.2i

3

138.6+45.2i

Fig 4.2 Single line diagram of three bus system

4.6.1 DATA OF THREE BUS SYSTEM:

TABLE A.1 LINE DATA OF THREE BUS SYSTEM

BUS RESISTANCE (R) REACTANCE (X) SUSCEPTANCE (B)FROM TO

1 2 .02 .04 .051 3 .01 .03 .032 3 .0125 .025 .06

TABLE A.2 BUS DATA OF THREE BUS SYSTEM

BUS NO BUS CODE V (pu) ANGLE LOAD GENERATORMW MVAR MW MVAR

1 1 1 0 0 0 0 02 0 1 0 256.6 110.2 0 03 0 1 0 138.6 45.2 0 0

PROGRAM:

clc;clear all;n=3;Y=[20.0000-50.0000i -10.0000+20.0000i -10.0000+30.0000i; -10.0000+20.0000i 26.0000-52.0000i -16.0000+32.0000i;-10.0000+30.0000i -16.0000+32.0000i 26.0000-62.0000i];bi=imag(Y);br=real(Y);th=angle(Y);mY=abs(Y);pd=[0 4 0];pg=[0 0 2];ps=pg-pd;pd1=pd(2);qd=[0 2.5 0];qg=[0 0 0];qs=qg-qd;v=[1.05 1 1.04];dl=[0 0 0];dv=[0 0 0];for iter=1:100p=[0 0 0 ];q=[0 0 0];dv=[0 0 0]; y=1;if v(2)<.95 ATCbreak;elsefor iteration=1:100000if y<0.000001break;elsefor i=2:np(i)=0;q(i)=0;

for j=1:n p(i)=(v(i)*v(j)*mY(i,j)*cos(th(i,j)-dl(i)+dl(j)))+p(i); q(i)=-(v(i)*v(j)*mY(i,j)*sin(th(i,j)-dl(i)+dl(j)))+q(i);endenddelp(1:2)=ps(2:3)-p(2:3);delq(1)=qs(2)-q(2);delpq=[delp delq];for i=2:nfor j=1:nif i~=j j1(i,j)=-(v(i)*v(j)*mY(i,j)*sin(th(i,j)-dl(i)+dl(j))); j2(i,j)=(v(i)*v(j)*mY(i,j)*cos(th(i,j)-dl(i)+dl(j))); j3(i,j)=-(v(i)*v(j)*mY(i,j)*cos(th(i,j)-dl(i)+dl(j))); j4(i,j)=-(v(i)*v(j)*mY(i,j)*sin(th(i,j)-dl(i)+dl(j)));endendendfor i=2:nj1(i,i)=-q(i)-(bi(i,i)*v(i)^2); j2(i,i)=p(i)+(br(i,i)*v(i)^2);j3(i,i)=p(i)-(br(i,i)*v(i)^2);j4(i,i)=q(i)-(bi(i,i)*v(i)^2);endJ11(1:2,1:2)=j1(2:3,2:3);J12(1:2,1)=j2(2:3,2);J21(1,1:2)=j3(2,2:3);J22(1,1)=j4(2,2);J=[J11 J12;J21 J22];J;del=(inv(J)*transpose(delpq));ddl(2:3)=del(1:2);dl=dl+ddl;dv(2)=del(3);v=v+dv;y=(abs(dv(2)));end

end ATC=pd(2)-pd1;pd(2)=pd(2)*1.01;qd(2)=qd(2)*1.01;pg(3)=pg(3)*1.01;qg(3)=qg(3)*1.01;ps=pg-pd;qs=qg-qd;endend

EXPERIMENT 5

LOAD FLOW ANALYSIS WITH STATCOM

5.1 AIM:

To understand the modeling of STATCOM (static synchronous compensator) for power flow studies using Newton Raphson method.

5.2 OBJECTIVES:

To investigate the operating principle and power injection modeling of statcom.

To investigate the control and operating limits of STATCOM

5.3 SOFTWARE REQUIRED:

C/C++ or MATLAB or ETAP

5.4 THEORETICAL BACKGROUND:

5.4.1 POWER INJECTION MODEL OF STATCOM:

Accurate power flow calculation should consider the steady state losses of STATCOM. The converter losses include mainly three parts: the power losses in the dc capacitor, the switching losses, and conduction losses. The percentage of each loss component relates to the control mode of STATCOM and the steady-state operating point.

The equivalent circuit of STATCOM is shown in Fig 5.1. In this circuit, the inductance, xs represents the leakage inductance of the transformer, the series resistance, rs represents the sum of the switching and conduction losses, and the shunt resistance, rp

represents the power losses in the dc capacitor. From Fig.5.1, it can be seen that the VSC acts as an ac voltage source behind equivalent impedance, where both magnitude and Phase angle of the source are controllable. The voltage vector is given as

(5.1)

Where

Vdc: Dc voltage across the capacitor

kc : Converter dc to ac gain

kt : Ratio of the coupling transformer.

m : Modulation ratio of PWM control

α : Control angle of VSC

Fig5.1Equivalent circuit of STATCOM

The current injected by STATCOM into the ac system is given as

(5.2)

Where is the voltage of ac system bus s, and

(5.3)

The active and reactive power injections of STATCOM into the ac system are given as,

(5.4)

(5.5)

Then the power balance equations at the ac system bus, sare expressed as,

(5.6)

(4.7)

Where = PS – PD (5.8)

= QS - QD (5.9)

and are calculated real and reactive powers respectively at bus s.

and are fixed for load buses. is fixed for generation buses. As the

STATCOM being connected to the ac system at bus s, and are not fixed.

Since real and reactive power exchange between the ac system and STATCOM is varying.

The active power output at the ac side of VSC is given as,

(5.10)

The real power of dc side of VSC is given as,

(4.11)

The power balance between the ac and dc side of the STATCOM is given as,

(5.12)

The active power exchange equation is given as,

(5.13)

By substituting Equations (5.10) and (5.11) in Eq.(5.13) and setting , the active

power exchange equation is reduced to,

(5.14)

5.4.2 CONTROL STRATEGIES OF STATCOM

With the developments on power electronic switches, PWM control is becoming a more practical technology for power system applications. In PWM control, switching losses associated with the relatively fast switching of the electronic devices have an important effect on the power flow calculation, as these have a direct effect on the charging and discharging of the dc capacitor, and hence should be considered in the modeling. The proposed model can readily be adapted to represent PWM and phase control of STATCOM. In Fig.5.1, eliminating the dc voltage control loop will yield the basic diagram of a controller with a typical phase control strategy.

The output voltage magnitude of the VSC relates to the dc side voltage and the control strategy of itself. If the VSC applies the phase control, the magnitude is a function of the dc side voltage and the phase angle of the VSC. If the PWM mode is applied, the output voltage magnitude is a function of the control phase angle and the modulation ratio of the PWM.

5.4.3 OPERATING LIMITS OF STATCOM:

It is very important to adequately address the system limits when studying voltage stability. Voltage collapse often occurs as a consequence of some devices hitting their limits in a heavily stressed power system. For a STATCOM, the limits on the ac current, Is modulation ratio, m phase angle, α and dc voltage, Vdc. can be directly introduced into this model. Because the current limit is the main limiting factor in VSC-based devices,

this thesis will implement it in NR power flow. Either the maximum limit, or

minimum limit, is reached, depending on whether the controller is operating in the

inductive or capacitive domain.

5.4.4 NR (NEWTON RAPHSON) EQUATIONS WITH CONTROL AND OPERATING LIMITS OF STATCOM:

In order to implement the STATCOM model in NR power flow, two augmented bus types are introduced, which are used to representing the specific control strategy and the operating limits of STATCOM.

One is an augmented PV bus, namely APV. This is a controlled bus where the active and reactive power injections and the voltage magnitude are specified, while the voltage phase angle of ac system bus, control angle and/or modulation ratio and/or dc voltage of STATCOM are treated as the state variables.

Fig 5.2 Bus types conversion and enforcement of the operating limit

The other is an augmented PQ bus, namely APQ. This is an uncontrolled bus where the active and reactive power injections are specified, while the voltage magnitude and angle of ac system bus and the dc voltage of STATCOM are treated as

the state variables in its place. When the ac current Islies between and the bus

is treated as APV type, otherwise it is APQ type. Fig.5.2 shows the conversion of bus types and the enforcement of the current limit.

In order to implement the model in NR power flow, the STATCOM control strategy must be properly expressed. As mentioned above, there are two control strategies for STATCOM and different variables must be selected depending on whether a PWM or phase control is used in the controller. For PWM control, the control variables are modulation ratio and control phase angle α, and the control objective is to maintain the dc voltage Vdc and ac voltage Vsat their reference value. For phase control, the control variable is phase angle α, and control objective is to maintain the ac voltage Vsat the reference value.

According to the operating principles, control strategies and operating limit of STATCOM, the augmented NR power flow equations are described as follows:

(I) PHASE CONTROL STRATEGY:

In unconstraint operation and phase control strategy of STATCOM, the modulation ratio m is given a certain value, and ac bus s is set as APV type, Jacobian elements in detail are illustrated in Appendix B and the augmented NR power flow equations are given by,

(5.15)

(II) PWM CONTROL STRATEGY

In unconstraint operation and PWM control strategy of STATCOM, the dc voltage Vdc is assigned a certain value, and bus s is set as APV type, Jacobian elements in detail are illustrated in Appendix B and the augmented NR power flow equations are given by,

(5.16)

(III) ENFORCEMENT OF THE CURRENT LIMIT:

Whether STATCOM is operating with PWM control or phase control, if the ac current

limit ( or )is violated, it will act as an unregulated voltage compensator whose

production or absorption reactive power capabilities will be a function of the nodal voltage at the STATCOM point of connection. In this situation, the phase angle α and/or

modulation ratio m of the VSC are fixed at the values corresponding with or , and

the dc voltage, Vdc is uncontrollable. Then the ac system bus s is converted to APQ type, and the augmented NR power flow equations are given as,

(5.17)

5.5 EXERCISE:

To conduct the load flow analysis for three bus system shown in figure 3 with and without STATCOM (placed at bus three) and demonstrate the voltage control by STATCOM for various loading conditions

1 .02+.04i 2

.01+.03i .0125+.025i 256.6+110.2i

138.6+45.2i 3

Fig 5.3 Single line diagram of three bus system with statcom

STATCOM PARAMETERS:

rs =.01

Xs=.1

rp=200

K=Kc*Kt=.9

DATA OF THREE BUS SYSTEM:

Table A.1: Line Data of three bus system

BUS RESISTANCE (R) REACTANCE (X) SUSCEPTANCE (B)FROM TO

1 2 .02 .04 .051 3 .01 .03 .032 3 .0125 .025 .06

Table A.2 Bus data of three bus system

BUS NO BUS CODE V (pu) ANGLE LOAD GENERATORMW MVAR MW MVAR

1 1 1 0 0 0 0 02 0 1 0 256.6 110.2 0 03 0 1 0 138.6 45.2 0 0

EXPERIMENT 6

TRANSIENT STABILITY ANALYSIS WITH STATCOM

6.1 AIM:

To become familiar with the various aspects of transient stability studies of 3-bus STATCOM connected system.

6.2 OBJECTIVES:

To understand modeling of STATCOM for transient stability analysis using explicit numerical integration method.

6.3 SOFTWARE REQUIRED:

MATLAB / PSAT/ETAP

6.4 THEORETICAL BACKGROUND:

Voltage collapse of the transmission system has caused more than 90% of the blackouts. Voltage magnitude regulation in the network can be achieved by controlling the production, absorption, and flow of reactive power throughout the system. The industry needs fast-acting dynamic sources of reactive power to counter fast voltage-collapse events. Sources or sinks are the Flexible AC Transmission System (FACTS) sometimes permanently connected and others are automatically adjusted to maintain fixed voltage magnitude at connection points.

The static synchronous compensator is a main member of the FACTS family of voltage source converter (VSC) based devices. It is being to improve voltage regulation as well as to increase load ability margin. The STATCOM is a shunt connected reactive power compensation device that is capable of generating and/or absorbing reactive power and its output can be varied to control the specific parameters of an electric power system. It is a solid state switching converter capable of generating or absorbing independently controllable real and reactive power at its output terminals when it is fed from an energy source at its input terminals.

6.5 MODELLING OF POWER SYSTEM COMPONENTS

+

The modeling of various power system components are presented below.

6.5.1 Synchronous Machine model

The synchronous generators are modelled as classical machines with rotor angle ( ) and speed ( ) as state variables.

(6.1)

Fig.6.1 Synchronous Machine model

The equivalent circuit for classical model of the generator is given in Fig.6.2.a. This equivalent circuit is the Thevenin’s form. The Norton form of the equivalent circuit is shown in Fig.2.b.

E' = V + (Ra+jXd' ) I INorton= YNortonV + I

E'

Ra + jXd' I

V

INorton

V

YNorton

I

Fig 6.2. Modelling of Generators and Representation for Network Solution

INorton is the Norton current source. The expressions for the current source and the admittance is given by

INorton = E' /(Ra+jXd' ) (6.3)

YNorton = 1 / (Ra+jXd' ) (6.4)

6.5.2 Transmission line model

Transmission line is modelled as lumped П- model. The transmission lines are modeled as π-circuits using positive sequence parameters: series resistance, series reactance and half line charging.6.5.3 Load Model

The loads are modelled as admittances. The admittances are computed from the initial load flow solution as shown below.

YL = ( PL – j QL) / |VL|2 (6.5)

where, PL and QL are the active and reactive powers of the load

|VL| is the magnitude of the voltage of the load bus computed by load flow analysis

YL is the load admittance and it gets added to the diagonal of bus admittance matrix Y corresponding to the node where the load is connected.

6.5.4 Modelling OF STATCOM

(b) Norton equivalent

The STATCOM consists of a voltage source converter which produces a set of 3 phase ac output voltages each in phase with and coupled to the corresponding ac system voltage through a leakage inductance of coupling transformer. The dc input voltage is provided by an energy storage capacitor. The functional model of STATCOM is shown in Fig7.3 where a VSI is connected to a utility bus through magnetic coupling. The exchange of reactive power between the converter and the ac system can be controlled by varying the amplitude of 3-phase voltage of the VSI. If the amplitude of the output voltage of VSI is greater than the amplitude of utility bus voltage, then a current flows from VSI to ac system. If the amplitude of the output voltage of VSI is lesser than the amplitude of utility bus voltage, then a current flows from ac system to converter. Adjusting the phase shift between converter output voltage and ac system voltage can control real power exchange between converter and ac system voltage.

Fig 6.3 Functional Model of STATCOM

6.5.4.1 Transient stability model

Assuming balanced, fundamental frequency voltages, STATCOM with PWM voltage control can be accurately represented in transient stability studies using the basic model shown in Fig.7.4.

Pac = P dc + P loss. (6.6)

Equation (2.6) basically represents the balance between the controller’s ac power Pac and dc power Pdc under balanced operation at fundamental frequency

AC System

Coupling

Transformer

KVdc

VdcIdc

C

Voltage

Source

Inverter

+ -

+

1mac

mac

K

sT

1

2

(1 )p

D

K sT

K sT

0m

mrefV

maxI

minI

+

Vi

_

+

+

Σ Σ

Fig.7.5 Voltage magnitude controller

m1V dpac

V pac

+

6.5.4.2 Voltage magnitude controller

This controller controls the AC output voltage magnitude of the voltage source inverter by controlling the modulation index, m of the PWM controller. The controller has a bias

corresponding to the steady state value of the modulation index, .

The differential equations are derived from the controller block shown in Fig 6.5

Magnitude

Magnitude

Controller

PWMK (PWM)

+

+

-

-

+

-

P + j Q

Rt + j Xt

+-

Σ

Σ

Σ

Fig 7.4. Transient stability model of STATCOM

1mdc

mdc

K

sT

KidcK pdc

S

+

_

+

+

refdcV

dcV

Σ Σ

maxdcV

mindcV

Fig.7.6 DC voltage controller

α1

V dcp

V dcdp

(6.7)

(6.8)

6.5.4.3 DC VOLTAGE CONTROLLER

The DC voltage controller controls the phase angle of the output voltage of the voltage source inverter. The phase angle determines the exchange of active power between the controller with the ac system and is used to directly control the DC voltage magnitude. The controller has a bias corresponding to the phase angle, .

The differential equations are derived from the controller blocks given in Fig 6.6 as,

(6.9)

(6.10)

From power balance condition,

Pac = P dc + P loss (6.11)

From Fig 4 Pac= (6.12)

Pdc = Vdc I dc (6.13)

The capacitor dynamic equation is

(6.14)

Discretized form of these equations and interfacing STATCOM as a current with network are given in Appendix

6.6 ALGORITHM TO ADVANCE SIMULATION BY ONE TIME STEP

The stepwise computations to be performed to advance the simulation by one step from t-∆t to t are as follows. The time step width (∆t) used in the algorithm is 0.0001 sec.

Assumptions:

The machines are considered to be classical(no controllers) Damping ignored Loads are assumed as constant admittances

Preparation:

The initial conditions for δ, ω and voltages are obtained from load Note:

i. The loads are converted into the constant admittances and these are pushed in to diagonal elements of the corresponding load buses.

Y TRANBUSii = Y ii+ Y Li Here i is for all load busesii. The diagonal elements of Y bus corresponding to all synchronous generators are modified

as:

Y TRANBUSii = Y ii+ 1/ (R 1i+ j X d’)iii. The diagonal elements of Y bus corresponding to STATCOM are modified as:

Y TRANBUSii = Y ii+ 1/(R t+ j X t)

Explicit Method - Runge-Kutta Fourth Order Method. The general formula giving the value of x for the (n+1)st step is Xn+1 = Xn + 1/6*(k1 + 2 k2 + 2 k3 + k4).Where,

k1 = f(xn , tn)∆t,

k2 = f(xn+ k1/2 , tn+∆t/2)∆t

k3 = f(xn+ k2/2 , tn+∆t/2)∆t

k4 = f(xn+k3 , tn+∆t)∆t

Explicit algorithm. The stepwise computations to be performed to advance the simulation by one step from t-∆t to t are as follows. The time step width (∆t) used in the algorithm is 0.0001 sec.

Step 1. Find the Norton current injections at all the generator buses;

For synchronous machine :

For STATCOM connected bus :

Step2. Solve the network equations, which yields approximate bus voltage vector Vapprox. [Y]

TRANBUS [V] = [I] NOR

Step3. Application of Fourth Order Runge-Kutta (RK) method involves the following sequence of steps.

For synchronous generator:

First estimate. k1 δ)*∆t , l1 ω)*∆t

Second estimate. k2 δ+l1/2)*∆t , l2 ω+k1/2)*∆t

Third estimate. k3 δ+l2/2)*∆t , l3 ω+k2/2)*∆t

Fourth estimate. k4 δ+l3)*∆t, l4 ω+k3)*∆t

∆ δ= 1/6*(k1+2k2 2k3 k4), ∆ ω= 1/6*(l1+2l2 2l3 l4)

δ = δ0 + ∆ δ ,

ω = ω0 + ∆ ω.

Similarly include the STATCOM state variables also.

Step5. Repeat steps with updated values of state variables.

Step6. Introduce the fault and repeat step 6 and later the fault will be cleared and repeat the procedure.

Handling network discontinuities.

For the normal time advance and the first solution at the discontinuity, the induction generator is represented by the Norton equivalent. The second solution is iterative as far as terminal voltage is concerned since slip, being a state variable, cannot change across a discontinuity. The second solution is computed using the model described by Norton equivalent.

The pre-disturbance solution at a discontinuity is called the first solution and the post disturbance solution is called the second solution.

.

Fig6.7 Handling network discontinuities; dots denote the points computed by the algorithm

Solution (td-)

V

t

SecondSolution (td

+)

td td+Δt

First

td - Δt

6.7 EXERCISE:

To conduct the load flow analysis for three bus system shown in figure 3 with and without STATCOM (placed at bus three) and demonstrate the voltage control by STATCOM for various loading conditions

STATCOM PARAMETERS:

rs =.01, Xs=.1, rp=200,K=Kc*Kt=.9

1 .02+.04i 2

256.6+110.2i

.01+.03i .0125+.025i

3

138.6+45.2i

Fig 6.8 Single line diagram of three bus system with statcom

DATA OF THREE BUS SYSTEM:

Table A.1: Line Data of three bus system

BUS RESISTANCE (R) REACTANCE (X) SUSCEPTANCE (B)

FROM TO

1 2 .02 .04 .05

1 3 .01 .03 .03

2 3 .0125 .025 .06

Table A.2 Bus data of three bus system

BUS NO BUS CODE V (pu) ANGLE LOAD GENERATOR

MW MVAR MW MVAR

1 1 1 0 0 0 0 0

2 0 1 0 256.6 110.2 0 0

3 0 1 0 138.6 45.2 0 0

EXPERIMENT 7

RELAY COORDINATION OF RADIAL TRANSMISSION/DISTRIBUTION SYSTEM

7.1 AIM:

To perform relay coordination of a radial distribution system.

7.2 OBJECTIVES:

To understand the significance of relay coordination in power system protection.

To manually perform relay coordination of a radial distribution system, to determine the T.M.S, PSM settings and validate the results using ETAP/ MATLAB/MIPOWER.

7.3 SOFTWARE USED:

MATLAB / ETAP / MIPOWER

7.4 THEORETICAL BACKGROUND:

7.4.1 RELAY:

A protective relay is a complex electromechanical apparatus, often with more than one coil, designed to calculate operating conditions on an electrical circuit and trip circuit breakers when a fault is detected. Unlike switching type relays with fixed and usually ill-defined operating voltage thresholds and operating times, protective relays have well-established, selectable, time/current (or other operating parameter) curves. Such relays may be elaborate, using arrays of induction disks, shaded-pole magnets, operating and restraint coils, solenoid-type operators, telephone-relay style contacts, and phase-shifting networks. Protection relays respond to such conditions as over-current, over-voltage, reverse power flow, over- and under- frequency,

and even distance relays that would trip for faults up to a certain distance away from a substation but not beyond that point. An important transmission line or generator unit will have cubicles dedicated to protection, with a score of individual electromechanical devices. The various protective functions available on a given relay are denoted by standard ANSI device numbers.

Table 7.1 Abbreviated list of commonly used relay device numbers

Relay device function number Protective function21 Distance25 Synchronizing27 Undervoltage32 Directional power40 Loss of excitation(field)46 Phase balance (current balance , negative

sequence balance)47 Phase sequence voltage 49 Instantaneous Overcurrent50 Time Overcurrent51 Time - Overcurrent59 Over voltage60 Voltage balance67 Dirctional Overcurrent81 Frequency (Under and Over frequency)86 Lockout87 Differential

The different types of overcurrent relays used are

Instantaneous over current relay -Operation is instantaneous in the sense that the relay has no intentional time delay. Time setting is absent.

Definite time over current relay - A definite time delay can be set after which the relay operates. It has both time setting and plug setting.

Inverse definite time overcurrent relay- The operating time of the relay is inversely proportional to the magnitude of fault current.

7.4.2 NEED FOR RELAY COORDINATION:

Relay coordination is the process of determining the settings for the relays that will provide an orderly shutdown and determine the sequence of relay operation with sufficient margin. It is by means of relay coordination that we ensure that if a fault occurs in a network only the breaker nearest to the fault operates leaving the healthy sections in service. Thus can be achieved by time grading or discrimination by time in which the relay farthest from the source is set to operate first and others follows it.

If relay coordination is not performed it may result in

a) Loss of selectivity resulting in nuisance tripping.

b) Loss of reliability of power supply to the healthy parts of the power system.

7.4.3 TO PERFORM RELAY COORDINATION:

The mathematical expression to arrive at the required characteristics of a relay has been specified by IEEE ad IEC

Where

I – Fault current level in CT secondary

Ip– Pick up current selected

t – Tripping time

Tp – TMS (Time Multiplier Setting)

The values of alpha, beta, L for the different types of over current relays have been specified by IEC as follows

Table 7.2 Relay types and the constant values

Curve description Standard Alpha beta L

Moderately Inverse IEEE .02 .0515 .114

Very Inverse IEEE 2.0 19.61 .491

Extremely Inverse IEEE 2 28.2 .1217

Standard Inverse IEC .02 .14 0

Very Inverse IEC 1 13.5 0

Extremely Inverse IEC 2 80 0

Long time inverse IEC 1 120 0

7.4.4 STEP BY STEP PROCEDURE:

Draw the single line diagram of the radial distribution system with the ratings of all the concerned equipment noted down.

Compute the nominal current that would flow through the system.

Compute the short circuit current (IEC standard or IEEE standard red book or buff book )

CT ratio is selected depending on the magnitude of the load current.

The relay to be used and its placement is then determined

Compute the pickup setting and time setting.

Plot the time Vs current characteristics of all relay and verify their coordination. (The plot should include the thermal characteristics of the relays also).

If characteristics intersect the procedure must be repeated for another suitable value of TMS.

7.5 EXERCISE:

To perform relay coordination of the radial distribution system given below i.e to suitably place the relays and determine the T.M.S (Time Multiplier Setting) and P.S.M (Plug Setting Multiplier)

Fig 7.1 Single line diagram of radial distribution system

EXPERIMENT 8

SMALL SIGNAL STABILITY ANALYSIS OF A MULTIMACHINE SYSTEM

8.1 AIM:

To become familiar with the various aspects of small signal stability of Multi Machine Power Systems.

8.2 OBJECTIVES:

To understand the modeling and analysis of small signal stability of a Multi Machine Power System.

To obtain linearized swing equation and to determine the roots of characteristic equation damped frequency of oscillation and undamped natural frequency.

8.3 SOFTWARE REQUIRED:

MATLAB

8.4 THEORETICAL BACKGROUND:

Small signal stability is the ability of the power system to maintain synchronism when subjected to small disturbances. A disturbance is considered to be small if the equations that describe the resulting response of the system can be linearized for the purpose of analysis. Instability is of two forms

1). Steady increase in generator rotor angle due to synchronizing torque.

2) Rotor oscillations of increasing amplitude due to insufficient damping torque.

For small disturbance, the nonlinear equations characterizing the dynamics of the system may be linearized around an operating point for the purpose of analysis. Small signal stability using linear techniques provides valuable information about the dynamics of the system and assists in its design. The location of the roots of the characteristic equation of the linearized differential equation will reveal the stability status of the system. We can also obtain the response of rotor angle and frequency of the rotor of the synchronous machines.

8.4.1 MODELLING FOR SMALL SIGNAL STABILITY:

For system stability studies it is appropriate to neglect the transmission network and machine stator transients. The dynamics of machine stator transients, machine rotor circuits, excitation systems, prime mover and other devices are represented by ODE. The result is that the complete system model consists of a large number of ordinary differential and algebraic equations.

Fig.8.1 Structure of a complete power system.

8.4.2 FORMULATION STATE EQUATION:

The linearized model of each dynamic device is expressed in the following form

Modeling of Synchronous machines:

Classical model of synchronous machines is considered with the q axis leading the d axis.

The dynamic equations are

(8.1)

Where,

Where, i - 1… m number of machines δ - Rotor angle in electrical radians ωs - Synchronous speed in electrical rad/s ω - Rotor angular velocity in rad/s Tm - Mechanical torque of synchronous generator H - Inertia constant D - Damping constant

The Equations (8.1) are linearized around a steady-state operating point in order to see the effect of small disturbances on the system. The steady-state operating point is obtained from the load flow results.

The linearized state Equations are:

(8.2)

The Equations (8.2) are functions of network variables. To express the network variables in terms of the state variables coordinated transformation is carried out.

8.4.3 COORDINATE TRANSFORMATION

Network equations for the 14 bus system with reduced Y-bus are given as:

Let T be the transformation matrix that transforms the d and q quantities of all machines to system frame, which a common reference frame is moving at synchronous speed. T is given as:

Network equations in individual machine d-q coordinates are:

(8.3)

Where M is given as

Linearizing Equation (8.3) we get

(8.4)

Where

From Equation (8.4) the real and imaginary parts can be separated out and the q and d

components of ∆I can be obtained. They are then substituted in the state equations to eliminate

the algebraic variables.

Expressing the state Equations in the standard canonical form

Where

A is the state space matrix

8.5 STEP BY STEP PROCEDURE:

1) Write down the algebraic equations in network coordinates in the nodal form. While doing this the machine internal nodes are explicitly included. The loads are represented as constant admittance matrices.

2) Eliminate all passive nodes. This results in a reduced set of algebraic equations.

3) Transform the reduced set of equations obtained in step 2 to individual machine qd coordinates.

4) Linearize the transformed equations obtained at the end of step 3.

5) Linearize the differential equations for the machine. The resulting equations will contain incremental changes in algebraic variables.

6) Eliminate the incremental changes in algebraic variables from the linearized the differential equations obtained in step 5 using the expressions obtained for incremental changes in step 4. The resulting equation will be in the state variable canonical form.

7) Apply eigenvalue technique for stability assessment.

8.6 EXERCISE:

To determine the small signal stability of the power system shown in Fig 8.2

Fig 8.2 Single line diagram of sample bus system

TABLE 8.1 ANDERSON AND FOUAD 3 MACHINE 9 BUS SYSTEM DATA

Generator 1 2 3

Rated MVA 247.5 192 128

Kv 16.5 18 13.8

Power factor 1.0 0.85 0.85

Type Hydro steam Steam

Speed 180 r/min 3600 r/min 3600 r/min

Xd(p.u) 0.1460 0.8958 1.3125

Xd’(p.u) 0.0608 0.1198 0.1813

Xq(p.u) 0.0969 0.8645 1.2578

Xq’(p.u) 0.0969 0.1969 0.25

Xl (p.u) 0.0336 0.0521 0.0742

Td0’ (s) 8.96 6 5.89

Tq0’ (s) 0 0.535 0.6

Stored energy at rated speed (MW-s)

2364 640 301

TABLE 8.2 BUS DATA

Bus No Pg (p.u) PD (p.u) QD (p.u) V (p.u) Qmax

(p.u)

Qmin

(p.u)

1 - - - 1.04 1 -1

2 1.63 - - 1.025 1 -1

3 0.85 - - 1.025 1 -1

4 - 0 0 - - -

5 - 1.25 0.5 - - -

6 - 0.9 0.3 - - -

7 - 0 0 - - -

8 - 1.0 0 - - -

9 - 0 0 - - -

Type 1-Slack Bus; Type 2-PV Bus; Type 3-PQ Bus