APROJECT REPORT

ON

VISUALIZATION OF EQUAL AREA CRITERIONUSING JAVA APPLET

Conducted atI.I.T. Mumbai

Submitted in partial fulfillment of the requirements for degree of

BACHELOR OF ELECTRICAL ENGINEERING(UNIVERSITY OF MUMBAI)

Prepared byPRAKASH NAIR

BHUPENDRA AMODEKARROHIT BHUTEANWESH DAS

Department of Electrical EngineeringSardar Patel College of Engineering

Munshi Nagar, Andheri (West),Mumbai - 400058

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CERTIFICATETHIS TO CERTIFY THAT

AMODEKAR BHUPENDRA VASANT SEAT NO: 2912BHUTE ROHIT V. SEAT NO: 2915

DAS ANWESH SEAT NO: 2917 NAIR PRAKASH SEAT NO: 2906

HAVE PRESENTED A PROJECT ENTITILED“VISUALIZATION OF EQUAL AREA CRITERION

USING JAVA APPLET”AND THE SAME IS APPROVED FOR THE DEGREE OF

BACHELOR OF ELECTRICAL ENGINEERINGUNIVERSITY OF MUMBAI

_______________ _______________

Internal Guide External Guide

_______________ _______________

Internal Examiner External Examiner

_______________

Head of Department

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ABSTRACT

This project presents a cross-platform application which provides a better comprehension of

power system stability using equal area criterion. A distinctive feature of this application is

interactive plotting of power angle curves with on-line parameter adjustment showing

accelerating/decelerating areas, operating and clearing angles, stability margins, etc. It

computes the critical clearing angle (CCA) and other stability parameters. It provides the user

with a choice to select the computational method and its step size for calculating the solution

to the swing equation. Some scenarios that are used to illustrate equal area criteria are sudden

change in mechanical input sudden short circuit on one of the parallel lines (3Φ-symmetrical

fault). This application concentrates on the second scenario, and handles lossless lines and

three phase symmetrical faults. Thus, the objective is to develop an interactive graphical applet

for the analysis of single-machine Infinite-Bus (SMIB) power system transient stability using

Equal Area Criteria (EAC).

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ACKNOWLEDGEMENT

While most project groups will consider themselves lucky to have one project guide, we were

extremely lucky to have two. We thank Prof. S. M. Pandit of Sardar Patel College of

Engineering who was our project guide and Prof. S. A. Soman of I.I.T. Mumbai who was our

external project supervisor. We are particularly indebted to our guide Prof. S. M. Pandit who

gave us complete freedom in choosing the project topic. We extend our thanks to I.I.T.

Mumbai and Prof. S. A. Soman, Prof. A. M. Kulkarni for giving us the chance to work on this

project . We are extremely grateful to Mr.Rajeev Gajbhiye (PhD Student , electrical

department , I.I.T. Mumbai) who helped us in this project. We wish to thank the staff of

S.P.C.E. for allowing us to use the resources of the college while working on the project. We

are also grateful to our classmate, Amey Kulkarni, who helped us test the software and locate

its bugs.

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Table of Contents

• Abstract 03

• Acknowledgement 04

• Table of Contents 05

• List of Figures 06

1. Introduction 08

2. Dynamics of Synchronous Machine and Swing Equation 10

3. Transient Stabiltiy 15

4. Numerical Techniques for Solution of Swing Equation 18

5. Equal Area Criterion 35

6. Application of EAC to SMIB 38

7. Factors affecting Power System Stability 45

8. Java and Applets 48

9. The PSS Applet 50

10. Conclusion and Scope for further development 62

11. Bibliography 64

12. Appendix 1 – PowerSystem class code 65

13. Appendix 2 – PowerSystem class documentation 75

14. Appendix 3 – General Public License 88

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List of Figures

Figure 2.1 : Single machine infinite bus system

Figure 2.2 : Flow of mechanical & electrical power in a synchronous machine

Figure 4.1 : Actual and assumed values of Pa , ω, d as a function of time

Figure 4.2 : Flowchart of point - by - point method

Figure 4.3 : Plot of x vs. t (Euler’s method)

Figure 4.4 : Flowchart for Euler’s method

Figure 4.5 : Plot of x vs. t (Modified Euler’s method)

Figure 4.6 : Flowchart for Modified Euler method

Figure 4.7 : Plot of x vs. y (RK 4th order method)

Figure 4.8 : Flowchart of RK4 method

Figure 5.1 : Plot of d vs t for stable & unstable systems

Figure 5.2 : Single machine connected to infinite bus through two parallel lines

Figure 5.3 : Illustration of equal area criterion

Figure 6.1 : Single machine connected to infinite bus through two parallel lines

Figure 6.2 : Single machine connected to infinite bus through one bus

Figure 6.3 : Illustration equal area criterion for the opening of one of the line

Figure 6.4 (a)&(b): Short circuit at one end of the line

Figure 6.5 : Equal area criterion illustrated

Figure 6.6 (a),(b)&(c) Simplification of circuit using delta-star & star-delta conversion

Figure 6.7 : Equal area criterion applied to fault on middle of one line of the system

Figure 6.8 : Fault on middle of one line of the system case of critical clearing angle

Figure 9.1 : MVC architecture

Figure 9.2 : Actual MVC architecture

Figure 9.3 : Main flowchart

Figure 9.4 : Help menu

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Figure 9.5 : Input Parameters

Figure 9.6 : Input parameters explanation

Figure 9.7 : Single machine tied to infinite bus as displayed in applet

Figure 9.8 : Model with flashing CBs

Figure 9.9 : Calculated Output

Figure 9.10 : Output Explained

Figure 9.11 : Plot of d vs. t for stable & unstable systems

Figure 9.12 : EAC Plot for stable system & unstable system

Figure 9.13 : Help menu which describes plots in detail

Figure 9.14 : Error mesages

Figure 9.15 : Snapshot before submission of data

Figure 9.16 : Snapshot on submission of valid data

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Introduction

Power system stability may be defined as that property of a system that enables the

synchronous machines in the system to respond to a disturbance and return to normal

operating conditions.

Depending upon the magnitude and nature of the disturbance , stability studies are often

classified into the three following categories ─ steady state , dynamic and transient stability

studies .

The difference between the steady state and dynamic stability studies is the degree of detail

used to model the machines . In dynamic stability studies , the excitation system and the

turbine governor systems are represented along with synchronous machine models . On the

other hand , steady state stability problems use a very simple generator model , which treats

the generator as a constant voltage source . However , both of the above mentioned stability

studies cover the case of system reaction to relatively small disturbances . This allows the

linearization of the non-linear differential and algebraic equations arising from a study of the

system .

Transient stability studies differ from the other two categories in the fact that it covers the case

of severe disturbances to the system , such as transmission system faults , sudden load changes

and loss of generating units or line switching . Transient stability studies are much more

commonly carried out thereby reflecting their greater importance in practice . Also , since the

disturbances are large , the aforementioned linearization of the differential equations cannot be

used . Thus , numerical methods are to be used for the solution of these systems . Transient

stability studies can be further subdivided into two categories ─ first swing and multi-swing

stability problems .

First swing stability is based on a simple generator model without representation of control

systems whereas multi-swing stability study , which extends over a longer period , must

consider effects of generator control systems .

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Thus we begin the analysis of Single Machine Infinite Bus (SMIB) system in the next chapter .

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2.Dynamics of Synchronous Machine and Swing Equation

Dynamics of synchronous machine :

Dynamics or kinetics is the study of the motion of a body under the action of one or several

forces . In this section , we analyse the motion of the rotor and derive a correlation between the

machine rating and it's basic property of inertia.

Figure 2.1 shows a single machine infinite bus system . The machine rating is G , and an input

power of Pm is fed to it by the prime mover .

Figure 2.1 : Single machine infinite bus system

The kinetic energy of the rotor at synchronous speed is

K.E. = (½)*J*ωsm2

where

J = rotor moment of inertia kg-m2

ωsm = synchronous speed in rad(mech)/sec.

But

ωs = ½*P*ωsm = rotor speed in rad(elect)/sec ........... (1)

where

P = number of machine poles

Thus,

K.E. = ½ * {J*(2/P)2 * ωs * 10-6} * ωs

10

= ½*M*ωs

where

M = J * (2/P)2 * ωs*10-6

= moment of inertia in MJ-s/elect rad.

Now we shall define the inertia constant H such that

GH = K.E. = (½)*M*ωs MJ [5]

where

G = machine rating (base) in MVA (3phase)

H = inertia constant in MJ/MVA.

It immediately follows that

M = (2*GH)/ωs = GH/(π∗f) MJ-s/elect rad

Taking G as base, inertia constant in p.u. is

M (pu) = H/(π∗f) s2/elect rad

= H/(180*f) s2/elect degree

Swing Equation :

Figure 2.1 : Flow of mechanical & electrical power in a synchronous machine[5]

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Figure 2.1 shows the torque , speed and flow of mechanical and electrical powers in a

synchronous machine . It is assumed that the windage , friction and iron loss torque is

negligible .

The equation governing the motion of rotor of a synchronous machine is based on the

elementary principle in dynamics which states that accelerating torque is a product of the

moment of inertia of rotor and its angular acceleration . This is basically Newton's second law

applied to rotational motion .

J(d2θm/dt2) = Tm – Te ........... (2)

where

J = total moment of inertia of rotor masses in kg-m2

θm = the angular displacement of rotor with respect to stationary axis in mechanical

radians

t = time in seconds

Tm = the mechanical or shaft torque supplied by the prime mover after deducting the

retarding torque due to rotational losses in Nm

Te = the net electrical torque Nm

While the dynamics of the machine rotor is governed by equation (2) , the actual change in

speed for very small amounts of time (of the order used for transient stability studies) , is not

significant . Thus , the speed for the period of the transient operation can be assumed to be the

synchronous speed , ωsm. Hence , equation (1) can be converted into the more convenient

power form by multiplying both sides by ωsm ─

ωsm * J(d2θm/dt2)*10-6 = Pm – Pe ........... (3)

where

Pm : mechanical power input in MW

Pe : electrical power output in MW(stator Cu loss is assumed negligible)

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Since

θm = (2/P)*θe

where

θe = angle in rad(elect)

Thus , we have ─

d2θm/dt2 = (2/P)* d2θe/dt2

Using the above result and equation (1), we can rewrite equation (3) as ─

{J*(2/P)2 * ωs*10-6}* (d2θe/dt2) = Pm – Pe

or

M * (d2θe/dt2) = Pm – Pe ........... (4)

Since synchronism is a key element in the operation of a synchronous machine , it is

convenient to measure the angular position of rotor with respect to synchronously rotating

frame of reference . Let

δ = θe – ωs *t ........... (5)

be rotor angular displacement from synchronously rotating reference frame (called torque

angle /power angle)

From above equation

d2θe/dt2 = d2δ/dt2

Hence equation (4) can be written in terms of δ as

M *(d2δ /dt2) = Pm – Pe MW

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With M defined earlier this can be further simplified as

(GH/πf)*(d2δ /dt2) = Pm – Pe MW

Dividing above expression by G we can write

(H/πf)*(d2δ/dt2) = Pm – Pe = Pa (pu)

where

H/πf = M (pu)

and

Pa = accelerating power , which is the power being fed to the rotor above what it is

supplying to the load. This power is used by the rotor to accelerate.

We have [5] ─

M*(d2δ/dt2) = Pm – Pe = Pa ........... (6)

The above 2nd order nonlinear differential equation defines dynamic behavior of rotor and is

known as the swing equation .

The swing equation, as derived above is a key to understanding rotor dynamics for any kind of

stability studies . Following chapter discusses how we shall apply the swing equation for

transient stability study .

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3. Transient stability

From equation (6) , as obtained in the last chapter , we see that the dynamics of a single

synchronous machine connected to infinite bus bar is governed by the equation ─

M(d2δ/dt2) = Pm – Pe

Pe can be represented as ─

Pe = Pmax* sin δ [5]

where Pmax is a function of the generator and infinite bus voltages and the effective

impedance between them. Thus , the swing equation becomes ─

M(d2δ/dt2) = Pm – Pmax*sin δ

A direct solution of this equation is very difficult to achieve . For small disturbances , as is

covered in steady state stability studies , it is possible to linearize the equation . This allows us

to obtain a condition , satisfaction of which ensures the stability of the system . Such a

simplistic approach is not applicable to the solution of swing equation for large disturbances as

in the case of transient stability . The method of solution is more involved . It requires use of

such tools such as numerical techniques to approximate the non-linear system to a discrete

form with arbitrarily close approximation . Thus , the general approach we use consists of

taking into account the disturbances in the system and solving the equation in the presence of

said disturbances .

First , in accounting for the disturbances in the system , we need to construct a mathematical

model of the system which will consist of the parameters affecting the dynamic behavior of

the machine . The complexity of the model depends upon the type of transient , system and the

accuracy required . The factors that should be kept in mind for constructing the model are as

follows ─

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1. The network before , during and after transient .

2. The loads and their characteristics .

3. The parameters of the synchronous machines .

4. The mechanical turbine and the speed governor .

5. Other supplementary controls, like tie-line controls are necessary for a mathematical

description of the system .

Thus , for the solution of the swing equation , we need information about the steady state of

the system prior to the occurrence of the disturbance and the mathematical model of the

system .

Generally , a factor affecting the dynamics of the system is represented by a differential

equation . For simplifying the analysis without significant loss of accuracy , we make the

following assumptions [5] ─

1. Transmission lines and synchronous machine resistance is ignored, thus inherent

damping has been neglected, giving pessimistic results .

2. Damping term contributed by synchronous machine damper winding has been ignored.

3. Rotor speed is assumed to be synchronous. In fact, it varies insignificantly during the

course of the transient .

4. Mechanical input is assumed to be constant during transient, which means the

regulating action of governor loop is ignored .

5. Voltage behind transient reactance is assumed to be constant , that is , action of voltage

regulating loop is ignored .

The basic outline of steps to be followed for stability studies is as ─

1. From prefault loading, determine the voltage behind transient reactance and the torque

angle δ0 of the machine with reference to the infinite bus .

2. For specified fault, determine the power transfer equation P(δ) during fault .

16

3. From the swing equation starting with δ0 as obtained in step 1 , calculate δ as a function

of time using a numerical technique for solving the non linear differential equation .

4. After clearance of the fault once again determine P(δ) and solve further for δ(t) .

5. After transmission line is switched on (reclosure) , again find Pe(δ) and continue to

calculate δ(t) .

6. If δ(t) goes through a maximum value and starts to reduce , the system is regarded as

stable. It is unstable if δ(t) continues to increase . Calculation is ceased after suitable

period of time .

In the following chapter , we discuss the numerical techniques that we have decided to use for

solution of the swing equation .

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4. Numerical Techniques for Solution of Swing Equation

It is not possible to obtain an analytical solution of coupled non-linear differential equations .

Hence , we have used the numerical techniques mentioned earlier. These allow us to obtain a

reasonably accurate solution of the equations . Some of the commonly used numerical

techniques for this purpose are ─

• Point by point method

• Euler's method

• Euler's modified method

• Runge-Kutta method

Quantities Pa and ωr , δ are continuous function of time t but here we discretise the system in

order to implement them on digital system, therefore we make following assumption

1. The accelerating power Pa at the beginning of an interval is constant from middle of

the preceding interval to the middle of interval considered.

2. The angular velocity is constant throughout any interval at the value computed for the

middle of the interval.

Point - by - point method:

Point by point solution is also known as step-by-step solution . The following two steps are

carried out alternately ─

1. First , compute the angular position δ , and angular speed dδ/dt at the end of the time

interval using the formal solution of the swing equation from the knowledge of the

assumed value of the accelerating power and the values of δ and dδ/dt at the beginning

of the interval .

2. Then compute the accelerating power of each machine from the knowledge of the

angular position at the end of the interval as computed in step 1 .

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Figure 4.1 : Actual and assumed values of Pa , ω, δ as a function of time[5]

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Point by point method applied to Swing curve solution:

In this method, the accelerating power during the interval is assumed constant at its value

calculated for the middle of the interval . The pertaining diagrams have been depicted in figure

4.1 .

The desired formula for computing the change in Δδ during the nth time interval is

Δδn =Δδn-1 + {(Δt)2 /M} Pan-1

where

Δδn = change in angle during the nth time interval

Δδn-1 = change in angle during the (n-1)th time interval

Δt = length of time interval

Pan-1 = accelerating power at the beginning of the nth time interval

Due to occurence or clearance of faults , or some switching operations , discontinuities occur

in the accelerating power curve . These are accounted for by taking the effective value of

accelerating power . If the discontinuity occurs at the beginning of an interval , the average

value of the accelerating power at the ends of the interval is taken . Thus , in computing the

increment of angle occurring during first interval after a fault is applied at t=0 , the above

equation becomes ─

Δδ1 = {(Δt)2/M} * (Pa0+)/2

where Pa0+ is the accelerating power immediately after the occurrence of the fault .

If the fault is cleared at the beginning of the mth interval, then for this interval ,

Pa(m-1) = {Pa(m-1)- + Pa(m-1) +}/2

where Pa(m-1)- is the accelerating power before clearing and Pa(m-1)+ is that immediately after

clearing the fault . If the discontinuity occurs at the middle of the interval , no special

treatment is needed .

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This is the most widely used way of solving the swing equation , but there are several other

numerical methods available to compute swing curve .

The flowchart for the process involved in point-by-point solution of the swing equation is

shown in figure 4.2.

21

Figure 4.2 : Flowchart of point - by - point method

22

Euler’s method:

One of the simpler numerical integration techniques available , this is used for functions for

which the value of the gradient can be determined at any point .

This differs from the point-by-point method of solution in the fact that this belongs to the

Runge Kutta family of techniques . This uses the value of the slope of curve at a point to

approximate the next point . Hence , the discretization is not as explicit as point-by-point

method .

Consider a continuous and differentiable function x(t) , which has the function x'(t) defining

the gradient at t .

Figure 4.3 : Plot of x vs. t (Euler’s method)[2]

At x1 , the value of the slope can be obtained as x'(t1) . Thus the approximate value of the

ordinate at the end of the time interval is –

x2p = x1 + x'(t1)*h

[where h = Δt]

As can be seen from figure 4.3 , the non-linear graph has been approximated by a constant

slope. Thus, the approximation departs from the actual value . This introduces considerable

23

error in the approximation .

Euler’s method applied to Swing curve solution :

For the solution of the swing curve , we assume the following initial conditions –

ω0 = 0 δ0 = δinitial Pa = Pm – Pmax*sin δ0

For the nth iteration , the steps are –

ωn = ωn-1 + Δt*(1/M)*Pa(δn-1)

δn = δn-1 + Δt* ωn

Figure 4.4 shows the flowchart for the process of Euler's method applied to solution of swing

curve .

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Figure 4.4 : Flowchart for Euler’s method

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Modified Euler method:

This is an improvement over the Euler's method for solution of the swing equation . Where the

Euler method used one slope for an approximation , this uses two values of the slopes , at the

beginning and the end of the time interval , to obtain a better approximation of the final value .

Consider a continuous and differentiable function x(t) of which the derivative is known as the

function x'(t) . Let the point (t1, x1) be known . The successive point (t2, x2) is evaluated using

modified Euler’s method as follows –

The step size is taken as Δt . At (t1, x1) , the value of the slope of the curve is x'(t1) . This is line

L1 in figure 4.5 . Now , by a simple approximation , the predicted value of x(t) at t = t2 is –

xp = x1 + x'(t1)* Δt

Figure 4.5 : Plot of x vs. t (Modified Euler’s method)[7]

At t = t2, x' = x'(t2). This is line L3 . Thus , we get the slopes at the beginning and the end of

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the time interval . We take the corrected value of the slope in the time interval as an average of

the slope values at the ends .

Thus –

xc = x + ½*(x'(t1) + x'(t2))* Δt

The corrected slope lies somewhere between the values of the other two as is represented by

line L2 . As can be seen in figure 4.5 , the error in the value of x(t) is reduced by this method .

Modified Euler’s method applied to Swing curve solution:

Since we have to solve a coupled system of differential equations , it is necessary to break it

down to a system of two first order differential equations . We can do this by using the

property that the angular velocity is a derivative of the angular position .

Since , from equation (5) derived in chapter 2 ─

dθe/dt = dδ/dt

Here , we take the equations as –

dδ/dt = ω

dω/dt = Pa/M

To get the composite equation –

d2δ/dt2 = Pa/M

For this set of equations , at any point in the iterative solution , the following sequence of

operations take place [7] –

ωp = ω + (Pa(δ)/M)*Δt

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δp = δ + ωp*Δt

Thus , we get the predicted values of the functions after a time interval Δt . Now , we shall get

the values of the slope at the predicted values of the function –

dδ/dt = ωp

[Note : We can omit the above step, since we already know the value of ωp]

dω/dt = (Pa(δp)/M)

Thus we have the slopes at the end of the time interval . Now , in one go we shall evaluate the

corrected values of δ and ω by averaging their slopes –

δc = δ + (1/2)*(ω + ωp)*Δt

ωc = ω + (1/2M)*(Pa(δ) + Pa(δp))*Δt

These values are updated values for the next iteration .

Note : Pa(δ) is a designation that Pa is a function of the variable δ and that the value of δ has to

be substituted in the function to obtain the value of Pa . Similarly , Pa(δp) denotes that the

value of δp has to be substituted in the function to obtain its value .

Note : Pa has to be evaluated in each iteration twice using the appropriate expression .

The flowchart for the above process is shown in figure 4.6.

28

Figure 4.6 : Flowchart for Modified Euler method

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Runge Kutta 4th order method:

The Runge Kutta (RK) methods are of the family of methods which use the slope or its

suitably modified value to give the point on the curve at the end of the interval . The first order

RK method is the Euler method itself . Runge-Kutta methods achieve the accuracy of a Taylor

series without the need to calculate higher derivatives . All of the variations of this technique

can be cast in the form [2] –

yi+1 = yi + φ(xi, yi, h)h

where φ(xi, yi, h) is the increment function , which can be interpreted as a representative slope

over the interval . It can generally be expressed as [2] –

φ = a1k1 + a2k2 + …. + ankn

where the a’s are constants and the k’s are the values of the slope at the coordinates specified

by f(x,y) . This function is basically the function that gives the value of the gradient at a

specific coordinate . The k's are evaluated as [2] ─

k1 = f(xi,yi)

k2 = f(xi+p1h, yi + q11k1h)

.

.

.

kn = f(xi+pn-1h, yi+qn-1,1k1h+qn-1,2k2h+…..+qn-1,n-1kn-1h)

where p’s and q’s are constants .

The fourth order RK method has the iteration –

yi+1 = yi + (1/6)*(k1 + 2k2 + 2k3 + k4)*h

30

where ,

k1 = f(xi,yi)

k2 = f(xi+h/2, yi+k1h/2)

k3 = f(xi+h/2, yi+k2h/2)

k4 = f(xi+h, yi+k3h)

Figure 4.7 : Plot of x vs. y (RK 4th order method)[2]

We have used the fourth-order RK method in solution of swing equation . The process

involved in the 4th order RK has been depicted in figure 4.7.

Runge Kutta 4th order method method applied to Swing curve solution:-

For the iterative solution of the swing equation , we have the equations (as has been discussed

in section 4.3) –

dδ/dt = ω

dω/dt = (1/M)*Pa(δ)

The iteration steps are as follows –

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1. k1 = ω*Δt

q1 = (Δt/M)*(Pa(δ))

2. k2 = (ω+q1/2)* Δt

q2 = (Δt/M)*(Pa(δ +k1/2))

3. k3 = (ω +l2/2)* Δt

q3 = (Δt/M)*(Pa(δ +k2/2))

4. k4 = (ω +l3)* Δt

q4 = (Δt/M)*(Pa(δ +k3))

5. Δδ = (k1+2k2+2k3+k4)/6

Δω = (q1+2q2+2q3+q4)/6

6. δ(1) = δ + Δδ

ω(1) = ω + Δω

The flowchart for the above process is shown in figure 4.8.

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Figure 4.8 : Flowchart of RK4 method

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NOTE:

In all of the above methods we have ─

when t < tc –

Pa = Pm – Pmaxdf*sinδ

when t = tc –

Pa = (1/2)*((Pm-Pmaxdf*sinδ) + (Pm-Pmaxaf*sinδ))

and finally, when t > tc –

Pa = Pm-Pmaxaf*sinδ

Following table gives a comparison between the numerical techniques used for the solution of

the swing equation .

Comparison between numerical methods:

Property Euler methodmodified Euler

methodRK 4th order

method

No. of slope approximations 1 2 4

Calculative effort low higher highest

Speed of processing fast slower slowest

Accuracy not good good best

Convergence good better best

Although the swing equation provides a means to study the dynamics of the machine , it is a

tedious process involving lot of calculative effort . The Equal Area Criterion (EAC) is a tool

which gives results much faster and facilitates the stability analysis . We discuss this in the

following chapter .

34

5. Equal Area CriterionIn the system where one machine is connected to an infinite bus, it is possible to study

transient stability by means of a simple criterion , without resorting to the numerical solution

of swing equation .

Consider swing equation

d2δ/dt2 = (1/M)*(Pm – Pe ) = Pa/M

where M = H/πf in pu system

Pa = accelerating power

H = inertia constant

Figure 5.1 : Plot of δ vs t for stable & unstable systems[5]

For an unstable system , δ continues to increase indefinitely with time and the machine loses

synchronism. On the other hand , if the system is stable δ(t) performs oscillations whose

amplitude decreases in actual practice but as damping term is neglected in swing equation ,

therefore in this case oscillations are of constant amplitude . However , the response δ (t) falls

into two broad categories as shown in figure 5.1. It can easily be visualized now that for a

35

stable system , indication of stability will be given by first swing where δ goes to a maximum

and will start to reduce . This fact can be stated as a stability criterion , that the system is stable

if at some time instant , dδ/dt = 0 and unstable if dδ/dt > 0 for a sufficiently long time (more

than 1 second is generally accepted) .

Application of EAC:

The aforementioned stability criterion for power system can be converted into a simple and

easily applicable form for SMIB:

Consider a single machine infinite bus system . The swing equation of the system is given by

M*(d2δ/dt2) = Pm – Pe = Pa

Multiplying both sides of the equation by (2/M)*(dδ/dt) , we get

2*(dδ/dt)(d2δ/dt2) = (2Pa/M)(dδ/dt) ............ (7)

Integrating equation (7) , we get

(dδ/dt)2 = (2/M) ∫Pa dδ

When the machine comes to rest with respect to the infinite bus - a condition , which may be

taken to indicate stability - requiring that

∫Pa dδ=0 ............ (8)

The integral in equation (8) may be interpreted graphically as the area under a curve of Pa

plotted against δ between limits δ0 , the initial angle , and δ2 , the final angle . The condition

for stability is that there exists a δ for which the area under the curve is zero . We will briefly

illustrate the application of equal area criterion by applying it (to the following simple case)

for a synchronous generator connected to infinite bus through a double-circuit line as shown in

figure 5.2 .

36

Lets us assume that 3-Φ symmetrical fault has occurred on transmission line with reactance

X2 . Then , for the assumed lossless system , EAC plot can be drawn as shown in figure 5.3.

Figure 5.2 : Single machine connected to infinite bus through two parallel lines[5]

Figure 5.3 : Illustration of equal area criterion[5]

Area shaded in magenta and area shaded brown may be interpreted in terms of kinetic energy

gained and lost by the synchronous generator , respctively . Thus for stable system

accelerating area must be equal to decelerating area .

37

6. Application of EAC to SMIB

Problem statement:

Following system of single machine connected to infinite bus will be used in our application to

illustrate the EAC concept .

Model, as shown in figure 6.1 , is described as :

Pm-constant mechanical input power provided by prime mover

H-inertia constant

E-generated emf behind transient reactance

V-voltage of infinite bus (maintained at constant voltage and frequency irrespective of

any power flow)

Xt1,Xt2,X1,X2- transient reactances

Figure 6.1 : Single machine connected to infinite bus through two parallel lines[5]

Above system is considered under faulted condition at

1. Short circuit at one end of the parallel line

2. Short circuit away from line ends (Midline fault)

38

Sudden fault on one of the parallel lines

Case a: Fault at one end of the parallel line, near a bus

Let us now assume that the disturbance to be a short ciruit at the generator end of line 2 of a

double circuit line as shown in figure 6.2(a) . We shall assume the fault to be a three phase

one.

Figure 6.2 (a) : Short circuit at one end of the line[5]

Figure 6.2 (b) : Short circuit at one end of the line[5]

Before the occurrence of a fault the power angle curve is given by

│E'│ │V│

Pbfmax = , Pbf = Pbfmax sin δ

{Xt1+Xt2+(X1|| X2)}

39

Upon occurrence of a three phase fault at the generator end of the line 2 , the generator gets

isolated from the power system for the purpose of power flow as shown in figure 6.2(b). Thus

during the period the fault lasts –

Pdf=0

The rotor therefore accelerates and angle δ increases . Synchronism will be lost unless the fault

is cleared in time .

The circuit breakers at the two ends of the faulted line open at time tc - the clearing time

(corresponding to δc) , disconnecting the faulted line . The power flow is now restored via

healthy line (through higher line reactance X1 in place of X1|| X2) , with power angle curve

│E'│ │V│

Pafmax = ; Paf = Pafmax sin δ

( Xt1+Xt2+X1)

Obviously , Pafmax < Pbfmax. The rotor now starts to decelerate . The system will be stable if a

decelerating area A2 can be found equal to accelerating area A1 before δ reaches the

maximum allowable value δmax . As area A1 depends upon clearing time tc (corresponding to

clearing angle δc) , clearing time must be less than a certain value (critical clearing time) for

the system to be stable . It is to be observed that the equal area criterion helps to determine

critical clearing angle and not critical clearing time . Critical clearing time can be obtained by

numerical solution of the swing equation .

40

Figure 6.3 : Equal area criterion illustrated[5]

It also easily follows that larger initial angle (corresponding to magnitude of Pm) increases A1

for a given clearing angle (and time) and therefore quicker fault clearing would be needed to

maintain stable operation.All of these are shown in figure 6.3.

Case b: Fault away from line ends

When the fault occurs away from the line ends (say in the middle of a line) there is some

power flow during the fault , though considerably reduced , as different from case (a) where

Pdf = 0 . Circuit model of the system during fault is now shown in figure 6.4 (a). This circuit

reduces to that of figure 6.4(c) through one delta-star and one star-delta conversion . The

power angle curve during fault is therefore given by

│E'│ │V│

Pdfmax= , Pdf = Pdfmax sin δ

X II

41

Figure 6.4 (a)

Figure 6.4 (b)&(c) Simplification of circuit using delta-star & star-delta conversion[5]

Pbf and Paf as obtained previously and Pdf as obtained above are all plotted in figure 6.5.

Accelerating area A1 corresponding to given clearing angle δc , is less in this case than in case

(a) , giving a better chance for stable operation .

42

Figure 6.5 : Equal area criterion applied to fault on middle of one line of the system[5]

Stable system is shown in fig 6.5 , wherein it is possible to find an area A2 equal to A1 for δ2 =

δmax . As the clearing angle δc is increased , area A1 increases and to find A2=A1 , δ2 increases

till it has a value δmax , the maximum allowable for stability . This case of critical clearing angle

is shown in figure 6.6.

Figure 6.6 : Fault on middle of one line of the system case of critical clearing angle[5]

43

Applying equal area criterion to the case of critical clearing angle of figure , we can write ,

δcr δmax

∫(Pm – Paf sinδ)dδ = ∫(Paf sin δ – Pm)d δ

δo δcr

where

δmax = π - sin-1(Pm/Paf)

simplifying we get ,

Pm (δmax – δ0) – Pdf cos δ0 + Paf cos δmax

cos δcr =

Paf- Pdf

Critical clearing angles can be calculated from equation above . The angles in this equations

are in radians and can be converted to degrees by multiplying π/180.

We have simulated both case a and case b in this applet .

The exact nature of the profiles and curves shown above depend on various factors . These

factors have been discussed in the next chapter .

44

7. Factors Affecting Power System Stability

There are various parameters of the SMIB system that contribute to its behaviour under

disturbances . Variation in the values or nature of these factors thus affects the stability of

SMIB system . These factors are ─

1. Synchronous machine parameters : These include the generator parameters such as

the inertia constant H and the generator bus voltage E . The bus voltage decides the

profile of the power curve . An increase in the inertia constant H of the machine

reduces the angle through which it swings in a given interval of time , thereby

improving the stability . This method cannot be employed practically because of

economic reasons and also because it slows down the response of the speed governor

loop (which can become oscillatory) apart from excessive rotor weight .

2. Transmission line parameters : We observe from the power angle curves that an

increase in the maximum power limits of the curves causes the accelerating area to

decrease and the decelerating area to increase , for a given value of clearing angle .

One of the methods for achieving this is to decrease transfer reactance . Reducing the

system transfer reactance is also a practical method of increasing stability limits . The

reactance can be decreased by increasing spacing and by increasing the conductor size.

Spacing is controlled by other factors like lightning protection and minimum clearance

to prevent arcing from one phase to other . Diameter can be increased by using hollow

cores and also by using material of low conductivity . The use of bundled conductors

decreases series reactance . Compensation for line reactance by series capacitor is an

economical method of increasing stability . Series compensation is more effective and

economical if it is switched on to increase the degree of compensation upon the

occurrence of a disturbance likely to cause instability . It also reduces the load voltage

fluctuation . Switching shunt capacitor on also increases the stability limits . But this is

preferred only if control of voltage is implemented because MVA rating required for

this is three to six times rating of switched series reactance .

45

Increasing number of lines in parallel between transmission points not only reduces the

transfer reactance but also increases the reliability of transmission line . However , this

is not economical .

3. Type of fault : Transient stability is greatly affected by the type of fault and location

of fault . In our case we have selected a 3-phase fault which is more severe from point

of view of power transfer .

4. Circuit breaker and relay characteristics :As majority of fault are transient in

nature , rapid switching and isolation of unhealthy lines followed by rapid reclosing

improves the stability margins . The modern circuit breaker technology has now made

it possible to clear a line in as fast as two cycles . Majority of faults are line to ground

in nature . It is natural that methods have been developed for selective single pole

opening and reclosing which further aid the stability limits . If L-G fault occurs at the

generator bus the power flow is not zero as in the case of three phase symmetrical fault

so breaker pole corresponding to faulty line is opened and other two lines remain in the

circuit and power continues to flow unlike in case of three pole switching where

power flow is reduced to zero . Thus single pole switching and reclosing aids the

stability problem and is widely accepted as it prevents large swing and consequently

voltage dips . Single pole switching is not economical in term of relaying and also

introduces the problem of over voltages caused due to single pole opening , owing to

line capacitances .

5. Excitation system and governor characteristics : When a fault takes place , the

voltage at all buses is reduced . At the generator terminal it is sensed by the Automatic

Voltage Regulator (AVR) which helps restore the voltage by acting within the

excitation system .

46

Using the analysis done up till now , we have designed an applet for simulating the SMIB

system behaviour under fault conditions, as a part of the project .

The effect of the parameters , discussed in this chapter , on system stability can be studied in

the software by:

1. Varying the inertia constant H

2. Varying various line impedances

3. Varying clearing time , tc

4. Varying generator emf , E or bus voltage , V; to vary the voltage and power profile

while keeping the other parameters constant .

47

8. Java and Applets

A software based analysis of the Equal Area Criterion is advantageous for various reasons .

Chief among them is the rapidity with which the effect of a particular combination of

parameters on system stability can be studied . Since the applet is made available on the

internet , it can be freely used for transient stability study . Also , the applet presents an

interface which allows the user to specify a wide range of inputs . This allows the study of the

effect of each parameter on the study of the system . The software also presents the

opportunity to study the effect of various numerical integration methods and step sizes on the

solution of the swing equation . Thus it helps to get a better understanding of the nuances of

stability study procedures .

Traditionally , programming languages have been split into two types – compiler and

interpreter based .

Languages like C/C++ are compiler based languages . Their source code is targeted at a

particular combination of hardware and software . The compiler gives a compact executable

file which is run on the target . While the final executable is fast , it is not portable . To run on

another combination of hardware/software , you have to recompile the code , possibly with

appropriate modifications .

Languages like Perl , on the other hand , are interpreted languages . The source code is

interpreted by the Perl interpreter . Thus , you only need to install the appropriate interpreter in

order to run the program . However platform-specific issues (like special libraries or

resources) can still hamper executing the code . Also , interpreter based programs are slow

because the code is checked and compiled each time it is called up for execution . There is also

no guarantee that the program may execute in the same way on different platforms – because

the interpreters are designed to account for and work around the differences in various

hardware/software combinations .

Java combines the features of both the categories of programming languages . The Java

48

compiler generates bytecode like a compiler language , which can then be run on any

compatible Java Virtual Machine (JVM) like an interpreted language . Thus , the twin goals of

final executable speed and wide range of targets for running the program is achieved . Thus ,

Java is a good choice for developing cross-platform applications . This is because of its

“compile once run anywhere” nature of development as has been mentioned .

Java is also more conducive to object oriented development as its syntax resembles native

Object Oriented Development (OOD) and Unified Modeling Language (UML) terminology .

Object oriented development simplifies the process of modeling and mapping physical

systems into any software .

An applet is an independent Java bytecode which can be downloaded from networks and run

in a browser . The only difference between a Java application and an applet is that an applet is

executed in a far more restricted environment . It usually involves behind-the-scenes

interaction with the browser in which the applet is being run . That being said , applets are

incredibly easy to design and use . Also , Java's Swing library gives an unsurpassed look-and-

feel that is maintained across platforms – not only does your program run the same way on all

platforms , but it also looks the same way .

All factors considered , Java (and applets) is a very good choice for rapid development of GUI

based applications for engineering purposes – both educational and critical . Hence , we have

used Java as the programming language for developing the software .

In the following chapter we present the basic functioning and the appearance of the applet .

49

9. The PSS Applet

We have used the classical Model-View-Controller architecture to design the applet . In this

architecture , the physical system being modeled (in our case , the power system) is

encapsulated in the Model class . It is separated from the user interface (UI) (or the View

class) which is all the the end-user of the software sees . The Controller class acts as the

connection between the two .

Figure 9.1 : MVC architecture

However , for all practical purposes , the Controller is a sub-set of the View class . The

actually realized architecture is shown below . This is what we have used in the applet .

Figure 9.2 : Actual MVC architecture

The PowerSystem class which has been documented in the appendix (see Appendix 2) is the

Model shown in the above figure . The pssApplet class is the View . Functions that control

interaction between the pssApplet and PowerSytem classes will be included in the Controller ,

but as a rule , they usually are already a part of the View component , at least , for our applet .

The analysis done so far has been implemented in the PowerSystem class . The GUI features

50

that have been implemented in the pssApplet class are shown ahead .

The software consists of modules for input , plotting , data output , power system simulation ,

error handling , etc. The functioning of these modules is shown in the main flowchart diagram

in figure 9.3

51

Figure 9.3 : Main flowchart

52

Figure 9.3 : Main flowchart (contd.)

53

The interface designed for the user to interact with the applet is presented below through

snapshots of the various components .

Help menu:

Any application is incomplete without a help menu . We have kept a help menu at top most

left corner .

Figure 9.4 : Help menu

Every thing in help menu except About is explained in following sections . About contains

information about copyright , authors and the GPL license (see Appendix 3 for the license) .

Inputs:

The user gives the values of various parameters , either from text-boxes or drop-down menus .

When ever applet is loaded all the boxes are filled with default numerical values . These

values can be deleted and restored by using Reset and Defaults buttons . Input block and

corresponding help menu views as follows ─

54

Figure 9.5 : Input Parameters

Figure 9.6 : Input Parameters Explanation

55

Model:

To the top left corner of window we can see model with labels for which we are carrying out

analysis ,

Before fault it looks as follows ─

Figure 9.7 : Single machine tied to infinite bus as displayed in applet

On submission of valid input data the model changes and depicts the fault location , and CB

which are in operation flashes red , which indicates that the transmission line is isolated .

Figure 9.8 : model with flashing CBs

56

Output:

Critical parameters are computed and displayed in the output widgets .

Figure 9.9 : Calculated Output

The outputs are explained in the help menu as shown.

Figure 9.10 : Output Explained

57

The comprehension is enhanced when the simulation results are displayed graphically rather

than as data files . A single click displays the results of the simulation on the graphics

terminal . Getting results from one program and plotting them on another is a time consuming

process . Thus our output also includes the plotting of EAC (P-δ ) as well as swing curve (δ-t) .

These plots can be seen as follows ─

Figure 9.11 : Plot of δ vs. t for stable & unstable systems

:

Figure 9.12 : EAC Plot for stable system & unstable system

58

The corresponding help menus of the plots describes legend and other specifications .

Figure 9.13 : help menu which describes plots in detail

Appropriate error messages are also displayed .

Error messages sre displayed in the following cases ─

Values of either of maechanical power input , bus voltages or reactances is zero or

negative or the value of clearing time is negative .

Improper selection of fault location or method of integration .

Selection of step size is out of range .

An accelerating area cannot be obtained .

A decelerating area cannot be obtained .

Sample error messages have been shown below ─

Figure 9.14 : Error mesage

59

SNAPSHOTS:

Figure 9.15 : Snapshot before submission of data

60

Figure 9.16 : Snapshot on submission of valid data

61

10. Conclusions and scope for further development

The significance of education in any evolving society is paramount . System modelling in a

graphical environment on a PC can greatly improve the students' ability to grasp difficult

concepts .

This Java applet will be helpful in creating an interactive environment of "learning by doing" .

Beyond its ability to better convey certain concepts , the applet can increase motivation and

instill greater interest among students , and encourage them to be more actively involved in the

learning process . Consequently , their understanding of the course content can further

improve . Such an endeavor can also be useful to a teacher during instruction and otherwise in

their course work .

We have designed and created an user-friendly applet to illustrate the EAC method of power

system transient stability analysis for the Single Machine Infinite Bus Power System (SMIB-

PS).

Although the method is practically useful for single-machine systems only , and can only

provide the critical clearing angle (CCA) for a specific fault (solid 3Φ symmetrical fault) as of

now , it gives a much clearer understanding of the various nuances of power system stability .

This program is very useful for students to determine stability of system with variation of

system parameters such as E , tc , H , Pm etc . This will basically involve running the

application for the same set of data , varying only the test parameter .

Thus , it would be particularly useful for in the following respects –

1. Observing stability results over a great range of input parameters, rapidly .

2. Fast and intuitive interpretation of results from the graphical outputs .

For students , it provides an insight into the transient stability concepts . For instructors , it can

help in designing drill and exam problems .

62

There are other salient features in the program .

1. Any kind of illegal data is handled and notified to user .

2. Applet is started with set of default values which can be erased using the Reset button

and retrieved using the Default button .

3. It also enables the user to select proper step size so that more accurate results can be

obtained and effect of step size on the accuracy of the swing curve can be studied .

The scope of this application can be further increased by updating existing features such as

allowing user to use more methods of integration . The application's model can be extended

to analyze different types of faults such as L-G , L-L-G , L-L , etc .

We have done analysis for loss less system but modifications can be made in the swing

equation and the model can be refined with regards to the assumptions that have been made for

design . For example , the damper resistance has been neglected in the swing equation . This

could be included in the swing equation to give more accurate results . Though the inaccuracy

is quite negligible by the standards set by the purpose of this work (educational purpose) , it

may be intolerable for more critical applications . Similarly , governor action may also be

considered to the same effect . Thus same system can be modified into lossy and practical

systems .

63

11.BIBLIOGRAPHY:

1. Presi, W. et al , "Numerical Recipies in C" , Cambridge University Press , 2nd edition,

1999

2. Chapra, S.C. and Canale, R.P.,"Numerical Methods for Engineers", TMH Publication, 4th

edition, 2004

3. Stevenson, W.D., "Elements of Power System Analysis", TMH Publication, 4th edition,

2000

4. Schildt, H. and Naughton, P., "The Complete Reference – Java2 ", TMH Publication, 5th

edition, 2001

5. Nagrath, I.J. and Kothari, D.P., "Modern Power System Analysis", TMH Publication, 3rd

edition, 2003

6. Khan, A.Z. and Shahzad, F., "A PC based software package for the equal area criterion

of power system transient stability" IEEE Trans. Power Systems, Vol. 13, No. 1,

pp.21 26, Feb. 1998

7. Hadi, S., "Power System Analysis", TMH Publication, 1st edition, 1998

8. Gross, C., "Power System Analysis", John Wiley & Sons Inc, 2nd edition, 1998

64

Appendix 1

This is the source code for the PowerSystem class whose documentation is provided in Appendix 2.

001 /*002 Power System Stability - a JAVA applet to study power system stability using003 Equal Area Criterion004 005 Copyright (c) 2006 authors006 007 Prof. S. A. Soman, Electrical Engineering Dept., I.I.T., Mumbai008 Prof. Shubha Pandit, Electrical Engineering Dept., S.P.C.E., Mumbai009 010 Bhupendra Amodekar, B.E. (Electrical Engineering), S.P.C.E., Mumbai011 Rohit Bhute, B.E. (Electrical Engineering), S.P.C.E., Mumbai012 Anwesh Das, B.E. (Electrical Engineering), S.P.C.E., Mumbai013 Prakash Nair, B.E. (Electrical Engineering), S.P.C.E., Mumbai014 015 This program is free software; you can redistribute it and/or modify016 it under the terms of the GNU General Public License as published by017 the Free Software Foundation; either version 2 of the License, or018 (at your option) any later version.019 020 This program is distributed in the hope that it will be useful,021 but WITHOUT ANY WARRANTY; without even the implied warranty of022 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the023 GNU General Public License for more details.024 025 You should have received a copy of the GNU General Public License026 along with this program; if not, write to the Free Software027 Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA028 You can also see the license at <http://www.gnu.org/copyleft/gpl.html>029 */030 031 import static java.lang.Math.*;032 033 /**034 *This class contains the basic SMIB model. It takes in the parameter values,035 *validates them and determines system stability using EAC. Various I/O 036 *functions are provided to utilize the model parameters; inlcuding those 037 *specific for graphic routines. The class can be extended to increase its038 *functionality. Variables are explained in the code listing.039 */040 041

65

042 public class PowerSystem {043 // miscellaneous variables044 protected int FLAGS, STABLE;045 protected double eNum, freq = 50, rem, stability_margin;046 047 // system model parameters obtained from input048 protected double049 Pm, // mechanical power input050 H, // inertia constant051 E, // voltage behind transient reactance052 Xt1, // reactance before generator and parallel link053 X1, // healthy transmission line054 X2, // soon-to-be faulty transmission line055 Xt2, // reactance betn parallel link & infinite bus056 V, // infinite bus assumed at angle 0057 tc, // fault clearing time058 ts; // step time for swing curve calculation059 060 protected int061 loc, // location of fault - near a bus or midline062 met; // method of integration for swing curve063 064 // system parameters in three stages065 protected double066 // before fault X, Pmax and delta zero067 Xbf, Pmaxbf, delta_0, 068 069 //during fault X, Pmax070 Xdf, Pmaxdf,071 072 // after fault X, Pmax073 Xaf, Pmaxaf,074 delta_max, // max delta for stable system 075 delta_cr; // critical clearing angle076 077 // swing curve variables078 protected double079 M, // inertia constant in elect. degrees080 delta_c, // clearing delta081 Paminus, Paplus;082 protected double[] ang; // data array for delta (in deg)083 protected int fault_cleared_index;084 085 086 // EAC variables087 protected double delta2, a_area, d_area, rem_d_area; 088 // acc. area, dec. area089 090 // methods091 092 /**093 *This method sets the model parameters to a default value.094 */095 public void SetDefaults() {096 Pm = 1; H = 2.52; E = 1.2; 097 Xt1 = 0.25; X1 = 0.5; X2 = 0.4;

66

098 Xt2 = 0.05; V = 1; tc = 0.12; 099 ts = 0.0001; loc = 1; met = 1;100 }101 102 /**103 *This method sets the model parameters to zero. Not recommended for use in104 *model initialization. Use SetDefaults() instead.105 */106 public void SetZero() {107 Pm = 0; H = 0; E = 0; 108 Xt1 = 0; X1 = 0; X2 = 0; 109 Xt2 = 0; V = 0; tc = 0; 110 ts = 0; loc = 0; met = 0;111 }112 113 // constructor114 PowerSystem() {115 SetDefaults();116 }117 118 /**119 *This method assigns input values to model parameters.120 *The following values are transferred through a vector: Pm, H, E, Xt1, 121 *X1, X2, Xt2, V, tc, ts122 *@param arg_list vector of above values123 *@param location location of fault124 *@param method method of integration for swing curve solution125 */126 public void Initialize(double[] arg_list, int location, int method) {127 Pm = arg_list[0];128 H = arg_list[1];129 E = arg_list[2];130 Xt1 = arg_list[3];131 X1 = arg_list[4];132 X2 = arg_list[5];133 Xt2 = arg_list[6];134 V = arg_list[7];135 tc = arg_list[8];136 ts = arg_list[9];137 loc = location;138 met = method;139 140 // call ApplyEAC141 ApplyEAC();142 }143 144 /**145 *This method calculates the model in the three stages - before fault,146 *during fault and after fault. It also validates the parameter values.147 *@return model status - non-zero for errors 148 */149 protected int CalculateModel() {

67

150 // first validate inputs151 if (Pm <= 0)152 return 1;153 if (H <= 0)154 return 2;155 if (E <= 0)156 return 3;157 if (Xt1 < 0)158 return 4;159 if (X1 <= 0)160 return 5;161 if (X2 <= 0)162 return 6;163 if (Xt2 < 0)164 return 7;165 if (V <= 0)166 return 8;167 if (tc < 0)168 return 9;169 if (ts > 0.1 || ts < 0.000001)170 return 10;171 if (loc != 1 && loc != 2)172 return 11;173 if (met != 1 && met != 2 && met != 3 && met !=4)174 return 12;175 176 // before fault model177 Xbf = Xt1+X1*X2/(X1+X2)+Xt2;178 Pmaxbf = E*V/Xbf;179 delta_0 = asin(Pm/Pmaxbf);180 181 // during fault model182 if (loc == 1) // 1 - near bus fault183 Pmaxdf = 0; // Xdf is not needed, hence ignored184 185 if (loc == 2) // 2 - midline fault186 {187 double188 Za = X2/2, // midline, Za and Zb can be in any ratio189 Zb = X2/2,190 Z10 = Xt1+X1*Za/(X1+X2), // delta - star conversion191 Z20 = Xt2+X1*Zb/(X1+X2), // we lump the reactances in the limbs192 Z30 = Za*Zb/(X1+X2),193 Xdf = Z10+Z20+Z10*Z20/Z30; // star - delta, for Xdf only194 Pmaxdf = E*V/Xdf;195 if (Pm <= Pmaxdf*sin(delta_0))196 return 13; // we won't get an accelerating area197 }198 199 // after fault model200 Xaf = Xt1+X1+Xt2;201 Pmaxaf = E*V/Xaf;202

68

203 if (Pmaxaf <= Pm) // we won't get a decelerating area204 return 14; 205 206 delta_max = PI-asin(Pm/Pmaxaf);207 // we will use eNum in the expression for delta_cr208 eNum = Pm*(delta_max-delta_0)-Pmaxdf*cos(delta_0)+Pmaxaf*cos(delta_max);209 delta_cr = acos(eNum/(Pmaxaf-Pmaxdf)); 210 211 // if we get this far, the user's model has no errors212 return 0;213 } 214 215 /**216 *This method calculates the accelerating power, Pa, at a given instant.217 *@param delta angle for which Pa has to be caluclated218 *@param i value which determines instant of Pa - pre,post or during fault219 *@return Pa220 */221 protected double accPower(double delta, int i) {222 double P = 0;223 if (i < fault_cleared_index)224 P = Pm-Pmaxdf*sin(delta);225 226 if (i == fault_cleared_index)227 {228 Paminus = Pm-Pmaxdf*sin(delta);229 Paplus = Pm-Pmaxaf*sin(delta);230 P = (Paminus+Paplus)/2;231 }232 233 if (i > fault_cleared_index)234 P = Pm-Pmaxaf*sin(delta);235 236 return P;237 }238 239 240 /**241 *The main method in this class. Look here when things act funny.242 *This method does the following: calculate swing curve using either of 243 *the 4 available methods and then determine system stability using EAC.244 */245 protected void ApplyEAC() {246 FLAGS = CalculateModel();247 248 if (FLAGS == 0)249 {250 // Swing Curve251 double 252 t = 0, tf = 2.0, // plot till tf in steps of ts 253 delta = delta_0, d_delta = 0, // delta and

69

incremental delta 254 Pa = 0; // accelerating power255 256 if (tc > 0.25) // plot for atleast 2 sec.257 tf = 8*tc;258 259 int size = (int)(tf/ts); //size of data arrays260 ang = new double[size];261 262 int i;263 M = H/(PI*freq);264 fault_cleared_index = (int)(tc/ts);265 266 // step by step solution of swing equation267 if (met == 1)268 {269 for (i = 0; i < size; i++)270 {271 if (i == 0)272 {273 ang[i] = toDegrees(delta_0);274 Paminus = 0;275 Paplus = Pm-Pmaxdf*sin(delta);276 Pa = (Paminus+Paplus)/2;277 continue;278 }279 280 d_delta = d_delta+(ts*ts*Pa/M);281 delta = delta+d_delta;282 ang[i] = toDegrees(delta);283 284 // calculate Pa for next iteration 285 Pa = accPower(delta, i);286 }287 288 }289 290 // euler291 if (met == 2)292 {293 double w = 0;294 for (i = 0; i < size; i++)295 {296 if (i == 0)297 {298 ang[i] = toDegrees(delta_0);299 w = 0;300 Paminus = 0;301 Paplus = Pm-Pmaxdf*sin(delta);302 Pa = (Paminus+Paplus)/2;303 continue;304 }305 306 w = w+(ts*Pa/M);307 delta = delta+ts*w;

70

308 ang[i] = toDegrees(delta);309 310 // calculate Pa for next iteration311 Pa = accPower(delta, i);312 }313 }314 315 // modified euler316 if (met ==3)317 {318 double w = 0, wp = 0, dp = 0, k = 0, q = 0;319 for (i = 0; i < size; i++)320 {321 if (i == 0)322 {323 ang[i] = toDegrees(delta_0);324 w = 0;325 Paminus = 0;326 Paplus = Pm-Pmaxdf*sin(delta);327 Pa = (Paminus+Paplus)/2;328 continue;329 }330 331 wp = w+Pa*ts/M;332 dp = delta+wp*ts;333 334 k = accPower(delta, i)/M;335 q = accPower(dp, i)/M;336 337 delta = delta+ts*(w+wp)/2;338 w = w+ts*(k+q)/2;339 ang[i]=toDegrees(delta);340 341 // Pa for next iteration342 Pa=accPower(delta, i);343 }344 }345 346 // RK-4347 if (met == 4)348 {349 double 350 w = 0, 351 k1 = 0, k2 = 0, k3 = 0, k4 = 0,352 q1 = 0, q2 = 0, q3 = 0, q4 = 0;353 354 for (i = 0; i < size; i++)355 {356 if (i == 0)357 {358 ang[i] = toDegrees(delta_0);359 w = 0;360 Paminus = 0;361 Paplus = Pm-Pmaxdf*sin(delta);362 Pa = (Paminus+Paplus)/2;

71

363 continue;364 }365 366 k1 = w;367 q1 = accPower(delta, i)/M;368 369 k2 = w+ts*q1/2;370 q2 = accPower((delta+ts*k1/2), i)/M;371 372 k3 = w+ts*q2/2;373 q3 = accPower((delta+ts*k2/2), i)/M;374 375 k4 = w+ts*q3;376 q4 = accPower((delta+ts*k1), i)/M;377 378 w = w+ts*(q1+2*q2+2*q3+q4)/6;379 delta = delta+ts*(k1+2*k2+2*k3+k4)/6;380 ang[i]=toDegrees(delta);381 }382 }383 384 385 // to find delta_c 386 delta_c = toRadians(ang[fault_cleared_index]);387 388 // Equal Area Criterion389 double e; // e = a_area-d_area, should be zero390 391 for (delta2 = delta_c; delta2 <= delta_max+0.5; delta2 = delta2+0.0001)392 {393 a_area = Pm*(delta_c-delta_0)+Pmaxdf*(cos(delta_c)-cos(delta_0));394 d_area = Pmaxaf*(cos(delta_c)-cos(delta2))-Pm*(delta2-delta_c);395 e = a_area-d_area;396 if (e < 0) /* practically will check for zero */397 break;398 }399 400 // system stability check401 if (delta2 <= delta_max)402 {403 STABLE = 1; 404 rem_d_area = Pmaxaf*(cos(delta2)-cos(delta_max))-Pm*(delta_max-delta2);405 stability_margin = rem_d_area/(rem_d_area+d_area); 406 }407 408 else409 {410 STABLE = 0;411 stability_margin = -1;412 }413

72

414 415 416 } // end FLAGS check417 } 418 419 /**420 * This method fetches the model status for use outside the class.421 */422 public int GetStatus() {423 return FLAGS;424 }425 426 /**427 *This method is used to extract various power angles for use outside 428 *the class. The vector contains: initial angle, critical clearing angle,429 *clearing angle, maximum angle for system to be stable, angle which the430 *system swings up to (according to EAC), stability status and stability431 *margin.432 *@return vector 433 */434 public double[] GetParameters() {435 double[] param_list = new double[7];436 param_list[0] = rint(toDegrees(delta_0)*10000)/10000;437 param_list[1] = rint(toDegrees(delta_cr)*10000)/10000;438 param_list[2] = rint(toDegrees(delta_c)*10000)/10000;439 param_list[3] = rint(toDegrees(delta_max)*10000)/10000;440 param_list[4] = rint(toDegrees(delta2)*10000)/10000;441 param_list[5] = (double)STABLE;442 param_list[6] = rint(stability_margin*1000000)/10000;443 return param_list;444 }445 446 /**447 *This method is used to get data related to Swing Curve448 *@return vector449 */450 public int[] GetSwingCurveParameters() {451 int[] param_list = new int[2];452 param_list[0] = fault_cleared_index;453 param_list[1] = ang.length;454 return param_list;455 }456 457 /**458 *This method retrieves the data actually used to plot a swing curve459 *@return vector 460 */461 public double[] GetSwingCurveData() {462 return ang;463 }464

73

465 /**466 *This method extracts the four power parameters from the model -467 *mechanical input power - Pm; and power during the three phases - pre-, 468 *during- and post-fault.469 *@return vector470 */471 public double[] GetEACPowerValues() {472 double[] param_list = new double[4];473 param_list[0] = Pm;474 param_list[1] = Pmaxbf;475 param_list[2] = Pmaxdf;476 param_list[3] = Pmaxaf;477 return param_list;478 }479 480 481 /**482 *This method gives the values of delta angles necessary for plotting 483 *EAC curves - initial angle, clearing angle and maximum swing angle.484 *@return vector485 */486 public double[] GetEACDeltaValues() {487 double[] param_list = new double[3];488 param_list[0] = delta_0;489 param_list[1] = delta_c;490 param_list[2] = delta2;491 return param_list;492 }493 }

74

Appendix 2Class Documentation

The main class in this software is the PowerSystem class whose documentation has been provided here. This documentation has been generated using the javadoc utility from comments embedded in the source code.

Class PowerSystemjava.lang.Object PowerSystem

public class PowerSystemextends java.lang.Object

This class contains the basic SMIB model. It takes in the parameter values, validates them and determines system stability using EAC. Various I/O functions are provided to utilize the model parameters; inlcuding those specific for graphic routines. The class can be extended to increase its functionality. Variables are explained in the code listing.

Field Summaryprotecte

d double

a_area

protected

double[]

ang

protected

doubled_area

protected

doubledelta_0

protected

doubledelta_c

protected

doubledelta_cr

protected

doubledelta_max

75

protected

doubledelta2

protected

doubleE

protected

doubleeNum

protected int fault_cleared_index

protected int FLAGS

protecte

d double

freq

protected

doubleH

protected int loc

protecte

d double

M

protected int met

protecte

d double

Paminus

protected

doublePaplus

protected

doublePm

protected

doublePmaxaf

protected

doublePmaxbf

protected

doublePmaxdf

protected

doublerem

76

protected

doublerem_d_area

protected

doublestability_margin

protected int STABLE

protecte

d double

tc

protected

doublets

protected

doubleV

protected

doubleX1

protected

doubleX2

protected

doubleXaf

protected

doubleXbf

protected

doubleXdf

protected

doubleXt1

protected

doubleXt2

Constructor SummaryPowerSystem()

Method Summary

77

protected

doubleaccPower(double delta, int i) This method calculates the accelerating power, Pa, at a given instant.

protected void ApplyEAC()

The main method in this class.protected int CalculateModel()

This method calculates the model in the three stages - before fault, during fault and after fault.

double[]GetEACDeltaValues() This method gives the values of delta angles necessary for plotting EAC curves - initial angle, clearing angle and maximum swing angle.

double[]GetEACPowerValues() This method extracts the four power parameters from the model - mechanical input power - Pm; and power during the three phases - pre-, during- and post-fault.

double[]GetParameters() This method is used to extract various power angles for use outside the class.

int GetStatus() This method fetches the model status for use outside the class.

double[]GetSwingCurveData() This method retrieves the data actually used to plot a swing curve

int[] GetSwingCurveParameters() This method is used to get data related to Swing Curve

void Initialize(double[] arg_list, int location, int method) This method assigns input values to model parameters.

void SetDefaults() This method sets the model parameters to a default value.

void SetZero() This method sets the model parameters to zero.

Methods inherited from class java.lang.Objectclone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait Field Detail

78

FLAGS

protected int FLAGS

STABLE

protected int STABLE

eNum

protected double eNum

freq

protected double freq

rem

protected double rem

stability_margin

protected double stability_margin

Pm

protected double Pm

79

H

protected double H

E

protected double E

Xt1

protected double Xt1

X1

protected double X1

X2

protected double X2

Xt2

protected double Xt2

V

protected double V

80

tc

protected double tc

ts

protected double ts

loc

protected int loc

met

protected int met

Xbf

protected double Xbf

Pmaxbf

protected double Pmaxbf

delta_0

protected double delta_0

81

Xdf

protected double Xdf

Pmaxdf

protected double Pmaxdf

Xaf

protected double Xaf

Pmaxaf

protected double Pmaxaf

delta_max

protected double delta_max

delta_cr

protected double delta_cr

M

protected double M

82

delta_c

protected double delta_c

Paminus

protected double Paminus

Paplus

protected double Paplus

ang

protected double[] ang

fault_cleared_index

protected int fault_cleared_index

delta2

protected double delta2

a_area

protected double a_area

83

d_area

protected double d_area

rem_d_area

protected double rem_d_area

Constructor Detail

PowerSystem

PowerSystem()

Method Detail

SetDefaults

public void SetDefaults()

This method sets the model parameters to a default value.

SetZero

public void SetZero()

This method sets the model parameters to zero. Not recommended for use in model initialization. Use SetDefaults() instead.

Initialize

public void Initialize(double[] arg_list, int location, int method)

This method assigns input values to model parameters. The following values are transferred through a vector: Pm, H, E, Xt1, X1, X2, Xt2, V, tc, ts

84

Parameters:arg_list - vector of above valueslocation - location of faultmethod - method of integration for swing curve solution

CalculateModel

protected int CalculateModel()

This method calculates the model in the three stages - before fault, during fault and after fault. It also validates the parameter values.

Returns:model status - non-zero for errors

accPower

protected double accPower(double delta, int i)

This method calculates the accelerating power, Pa, at a given instant.

Parameters:delta - angle for which Pa has to be caluclatedi - value which determines instant of Pa - pre,post or during fault

Returns:Pa

ApplyEAC

protected void ApplyEAC()

The main method in this class. Look here when things act funny. This method does the following: calculate swing curve using either of the 4 available methods and then determine system stability using EAC.

85

GetStatus

public int GetStatus()

This method fetches the model status for use outside the class.

GetParameters

public double[] GetParameters()

This method is used to extract various power angles for use outside the class. The vector contains: initial angle, critical clearing angle, clearing angle, maximum angle for system to be stable, angle which the system swings up to (according to EAC), stability status and stability margin.

Returns:vector

GetSwingCurveParameters

public int[] GetSwingCurveParameters()

This method is used to get data related to Swing Curve

Returns:vector

GetSwingCurveData

public double[] GetSwingCurveData()

This method retrieves the data actually used to plot a swing curve

Returns:vector

86

GetEACPowerValues

public double[] GetEACPowerValues()

This method extracts the four power parameters from the model - mechanical input power - Pm; and power during the three phases - pre-, during- and post-fault.

Returns:vector

GetEACDeltaValues

public double[] GetEACDeltaValues()

This method gives the values of delta angles necessary for plotting EAC curves - initial angle, clearing angle and maximum swing angle.

Returns:vector

87

Appendix 3

GNU GENERAL PUBLIC LICENSE

Version 2, June 1991

Copyright (C) 1989, 1991 Free Software Foundation, Inc.

51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA

Everyone is permitted to copy and distribute verbatim copies

of this license document, but changing it is not allowed.

Preamble

The licenses for most software are designed to take away your

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License is intended to guarantee your freedom to share and change free

software--to make sure the software is free for all its users. This

General Public License applies to most of the Free Software

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When we speak of free software, we are referring to freedom, not

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To protect your rights, we need to make restrictions that forbid

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anyone to deny you these rights or to ask you to surrender the rights.

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For example, if you distribute copies of such a program, whether

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We protect your rights with two steps: (1) copyright the software, and

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98

If the program is interactive, make it output a short notice like this

when it starts in an interactive mode:

Gnomovision version 69, Copyright (C) year name of author

Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type `show w'.

This is free software, and you are welcome to redistribute it

under certain conditions; type `show c' for details.

The hypothetical commands `show w' and `show c' should show the appropriate

parts of the General Public License. Of course, the commands you use may

be called something other than `show w' and `show c'; they could even be

mouse-clicks or menu items--whatever suits your program.

You should also get your employer (if you work as a programmer) or your

school, if any, to sign a "copyright disclaimer" for the program, if

necessary. Here is a sample; alter the names:

Yoyodyne, Inc., hereby disclaims all copyright interest in the program

`Gnomovision' (which makes passes at compilers) written by James Hacker.

<signature of Ty Coon>, 1 April 1989

Ty Coon, President of Vice

This General Public License does not permit incorporating your program into

proprietary programs. If your program is a subroutine library, you may

consider it more useful to permit linking proprietary applications with the

library. If this is what you want to do, use the GNU Library General

Public License instead of this License.

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