TRANSIENT STABILITY ANALYSIS

TRANSIENT STABILITY ANALYSIS

ByProf. C. Radhakrishna

ByProf. C. Radhakrishna

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CONTENTS

TRANSIENT STABILITY ANALYSIS

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TRANSIENT STABILITY ANALYSIS

The synchronous generators are represented in varyinglevels of detail depending on the phenomena being studied.

For transient stability studies dealing with theelectromechanical behavior of the power system, thenetwork is assumed to be in a quasi steady state.

The network is represented by a system of algebraicequations based on the fundamental principles of thepower flow analysis.

Load modeling is another important feature of transientstability studies

Coupled set of differential and algebraic equations. These are solved numerically to obtain the condition of the

system at each instant in time. Various important quantities like relative rotor angles,

voltages at key buses, and power flows on criticaltransmission lines are plotted and observed.

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The behavior of these variables is then used to judgestability or instability.

The starting point for any transient stability simulation isthe predisturbance power flow solution.

The system is always assumed to be in a steady state whenthe initiating disturbance occurs.

Using the predisturbance power flow solution, all the initialvalues of the state variables that govern the differentialequations are calculated.

After these are obtained, the disturbance is simulated. Different disturbances can occur on the system. The simulation of the disturbance causes a mismatch

between the mechanical power input to the generators andthe electrical power output of the generators.

The equilibrium is upset, and the state variables governedby the various differential equations coupled with algebraicequations change their values.

The change in behavior is tracked by numericallyintegrating the coupled set of differential and algebraicequations.

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Two broad categories of numerical techniques Implicit integration techniques and explicit integration

techniques. The time evolution of the state variables and other system

variables is observed to determine the behavior of the system. The classical model is used to study the transient stability

of a power system for a period of time during which thedynamic behavior of the system is dependent largely on thestored energy in the rotating inertias.

This is the simplest model used in stability studies andrequires a minimum amount of data.

Assumptions made1) The mechanical power input to each synchronous machine

is constant.2) Damping or asynchronous power is negligible.3) The synchronous machines are represented electrically, by

constant-voltage-behind-transient-reactance models.4) The motion of each synchronous machine rotor (relative to

a synchronously rotating reference frame) is at a fixed anglerelative to the angle of the voltage behind the transientreactance.

5) Loads are represented by constant impedances.

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This type of analysis is usually called first swing analysis-Assumption 2 can be relaxed by assuming a linear dampingcharacteristic. A damping torque D is included in the swing equation.

Figure 1: Representation of a synchronous machine by constant voltage behind transient reactance.

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Figure 2. Multimachine system representation (classical model)

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To prepare the system data for a stability study, the followingpreliminary calculations are made:

1. The system data are converted to a common system base; asystem base of 100 MVA is conventionally chosen.

2. The load data from the prefault power flow are converted toequivalent impedances or admittances.The necessary information for this step is obtained from the resultof the power flow

where yLi = gLi + jbLi is the equivalent shunt load admittance.

3. The internal voltages of the generators are calculated fromthe power flow data using the predisturbance terminal voltages

*

2 2Li Li Li Li

LiLi Li Li

I S P jQyV V V

0i iE

ai iV '

i i ai di iE V jx I

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Since the actual terminal voltage angle is i, we obtain theinitial generator angle by adding the predisturbance voltageangle i to or

4. The Ybus matrices for the prefault, faulted, and postfaultnetwork conditions are calculated. In obtaining thesematrices, the following steps are involved:

(a)The equivalent load admittances calculated in step 2 areconnected between the load buses and the reference node.Additional nodes are provided for the internal generatornodes (nodes 1, 2, . . . , n in Figure 2) and the appropriatevalues of admittances corresponding to are connectedbetween these nodes and the generator terminal nodes.

0i

'i

0 'i i i

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(b) In order to obtain the Ybus corresponding to the faulted system, weusually only consider three-phase to ground faults.

The faulted Ybus is then obtained by setting the row and columncorresponding to the faulted node to zero.

(c) The postfault Ybus is obtained by removing the line that would havebeen switched following the protective relay operation.

Figure 3: Generator representation for computing initial angle

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The power injected into the network at node i, which is the electricalpower output of machine i, is given by Pgi = Re (Ei Ii* ) . Theexpression for the injected current at each generator bus Ii in termsof the reduced admittance matrix parameters is given above .

0nn nr nn

rn rr r

Y Y VIY Y V

1( )n nn nr rr rn nI Y Y Y Y V

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5. In the final step we eliminate all the nodes except theinternal generator nodes using Kron reduction.

We get

2

1

sin( ) cos(n

Gi i ii i j ij i j ij i jjj i

P E G E E B G

i = 1,2, . . . ,n

for a multimachine systemi = 1,2, . . . ,n0

i i M i GiM P P 20

1

sin( ) cos(n

i i Mi i ii i j ij i j ij i jjj i

M P E G E E B G

i = 1,2, . . . ,n

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The value of the mechanical power for each machine isdetermined from the prefault conditions.

The mechanical power is set equal to the (active) electricalpower output of each generator at the prefault conditions.

This provides the equilibrium conditions and the initial anglesfor each generator as given by 0i

20

1

sin( ) cos(n

i i Mi i ii i j ij i j ij i jjj i

M P E G E E B G

i = 1,2, . . . ,ni i

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REFERENCES : [ 1 ] Arthur R. Bergen & Vijay Vittal : Power System Analysis , 2ndedition, Prentice Hall, Inc., 2000.

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CONCLUSIONS

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THANK YOU