# ee4031 5 transient stability

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The HK Polytechnic University Transient Stability Analysis

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1 Power System Stability A large power system consists of a number of synchronous machines operating

in synchronism synchronous generators located at great and small distancesapart and of varied ratings and characteristics have to continuously operate atthe same frequency (electrical speed = pole pairs mechanical speed).

It is necessary that they should maintain synchronism despite the ever presentdisturbances induced by nature and by man load changes, switching actions,faults and so on.

When the system is subject to some form of disturbance, there is a tendency forthe system to develop forces to bring it to a normal or stable condition. Theability of a system to reach a normal or stable condition after being disturbed iscall stability.

Synchronous stability may be divided into two main categories depending uponthe magnitude of the disturbance steady-state, dynamic and transient stability.

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The steady-state stability is the ability of a system to bring it to a stablecondition after a small disturbance. The study of steady-state stability isbasically concerned with the effect of gradual infinitesimal power changes andthe dynamics of rotating machines will be excluded from the studies.

However in practice the dynamics of rotating machines will be effected even theincrease in load is gradual. Dynamic stability is an extension of steady-statestability where the dynamic effects of synchronous machines and automaticcontrol devices such as governors and voltage regulators are included. Thedynamic stability is concerned with small disturbances lasting for a long timeand in dynamic stability studies non-linearities are neglected.

The transient stability is the ability of a system to bring it to a stable conditionafter a large disturbance. Transient stability is concerned with sudden and largechanges in the network conditions. The large disturbances can occur due tosudden changes in application or removal of loads, line switching operations,line faults, or loss of excitation.

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The HK Polytechnic University Transient Stability Analysis

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A mechanical analog of power system transient stability is given in the figurebelow, in which a number of masses representing synchronous machines areinterconnected by a network of elastic strings representing transmission lines.

When one of the strings is cut, representing the loss of a transmission line, themasses undergo transient oscillations and the forces on the strings fluctuate.The system will then either settle down to a new steady-state operating point oradditional strings will break, resulting in an even weaker network and eventualsystem collapse.

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2 Stability Limits The stability limit is the maximum power that can be transferred in a network

between sources and loads without loss of synchronism.

The steady-state stability limit is the maximum power that can be transferredwithout the system becoming unstable under steady-state conditions.

The dynamic stability limit is the maximum power that can be transferred withoutthe system becoming unstable under small disturbances lasting for a long time.

The transient stability limit is the maximum power that can be transferred withoutthe system becoming unstable when a sudden or large disturbance occurs.

In general, the transient stability limit is the lowest whilst the steady-state limit isthe largest.

i.e. transient stability limit < dynamic stability limit < steady-state stability limit

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The HK Polytechnic University Transient Stability Analysis

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3 Synchronism and Steady-State Stability3.1 Two Finite Machines

First consider case of two identical unloaded machines that are subsequentlydisturbed.

E1

X X

E2

E2 E2E1

E1

Ic

E - E1 2

I =c 2jXE - E1 2

Ic

Say some disturbance occurs that causes generator 1 to accelerate withrespect to generator 2. Observed that power comes out of the faster machineterminals and passes into the slower machine terminals. The disturbance iscorrected, opposed and (hopefully) overcome.

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Suppose the machines were initially loaded and equally sharing a load. Apartfrom losses, both machines would have a mechanical input equal to theelectrical output.

E1

X X

E2

E2 E2E1

E1

Ic

E - E1 2

IcI1 I2

I = I1 2

I1

I2

Once again say the same disturbance causing generator 1 to accelerate withrespect to generator 2 were to occur. The effects of this are shown in the abovefigure. As can be seen, the main effect is to increase the electrical burden onthe faster machine and to reduce it on the slower machine. The resulting powerunbalance with respect to mechanical input tends to correct the disturbance.The power flow between the generators is called the synchronous power flow.

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The HK Polytechnic University Transient Stability Analysis

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3.2 Infinite Bus

In a power system, normally more than two generators operate in parallel. Themachines may be located at different places. A group of machines located atone place may be treated as a single large machine. Also, the machines notconnected to the same bus but separated by lines of low reactance, may begrouped into one large machine.

The operation of one machine connected in parallel with such a large systemcomprising many other machines is of great interest. The capacity of the systemis so large that its voltage and frequency may be taken constant. Theconnection or disconnection of a single small machine on such a system wouldnot affect the magnitude and phase of the voltage and frequency.

Such a system of constant voltage and constant frequency regardless of theload is called infinite busbar system or simply infinite bus. Physically it is notpossible to have a perfect infinite bus.

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3.3 Single Machine on Infinite BusThe stability problem can be studied with an analysis of the behaviour of asynchronous generator through a line to an infinite busbar as shown below.

jXV 0oE

VEInfinite Bus

where V = phase voltage of the infinite busE = excitation phase voltage of the generatorX= total synchronous reactance from the source to infinite bus = load angle, i.e. phase angle between V and E

and the power transfer P from the generator to the infinite bus is given by

P =EV

Xsin

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The HK Polytechnic University Transient Stability Analysis

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Maximum power Pmax will be transferred when = 90o, i.e.

Pmax =EV

X

and the power transfer can be rewritten as

P = Pmax sin (3.3.1)

Equation (3.3.1) represents the steady-state stability limit of the power system. It isclean from the equation that steady-state stability limit Pmax can be increased by1. Increasing system voltages E or V : by increasing the excitation2. Decreasing system reactance X by : use of double-circuit line

use of bundled conductors series compensation of the reactance use of machines of low impedances

In fact, increasing steady-state stability limit by decreasing reactance is economicaland also effective.

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The graphical representation of power received P and the load angle is called thepower-angle diagram as shown below.

1. Magnitude of P (output) depends onmagnitude of .

2. There is a maximum limit to the powerthat can be extracted from a generator forgiven E, V and X .

3. The sign of the output power depends onthe sign of (Generating/Motoring).

4. There is a stable range of operation from = -90o to = 90o and stable operationis not possible outside this range.

Pmax

-Pmax

90 180-90-180

Power angle curve

Generator

Motor

P

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3.4 Synchronizing Power (and Torque) CoefficientThe system is stable if and only if for an increase in rotor angle (load) thetransmitted power also increases, i.e. the dP

dshould be positive.

The rate dPd

is called the synchronizing power coefficient and is taken as themeasure of the stability of a system, i.e.

synchronizing power coefficient, ps =dP

d= Pmax cos

Hence, the steady-state synchronous stability criterion for a simple system isps > 0, i.e. the synchronizing power coefficient is positive. The steady-statestability limit is reached when ps = 0 and if ps < 0, then system is unstable.

If 0 is the synchronous speed, then

synchronizing torque coefficient, ts =ps0

=Pmax0

cos

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4 Synchronous Machine Models4.1 The Voltage Behind Synchronous Reactance Model The previous model in which X = Xs the synchronous reactance is true only

for slowly changing conditions. Since resistance is neglected, there should beno significant transmission line length prior to connection to a large busbarsystem. Since saliency is neglected, the machine should be of cylindrical poleconstruction.

Most of these assumptions are rarely true. The most unsatisfactory one isregarding slow changes. This assumption is not acceptable for transient anddynamic stability studies.

4.2 Connect

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