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TRANSIENT STABILITY12Power system stability CIGRE IEEE Definition of Power system stability:It is the ability of an electric power system, with initial operating condition, to regain a state of operating equilibrium, when subjected to a physical disturbance, with most system variables bounded so that practically the entire system remains intact.Short termRotor angle stabilityFrequency stabilityVoltage stabilityTransient stabilitySmall disturbance Angle stabilityLong termShort termLarge disturbance voltage stabilitySmall disturbance voltage stabilityLong termShort termPower system stability3Transient Stability Studies44

Classical generator modelFig. 1: Single machine system connected to infinite busFig. 2: System representation with generator classical model5(radian)t secondsUnstableStable0.501Fig. 3: Variation of rotor angle with time abc6Assumptions for Transient stability studyTransmission line as well as synchronous machine resistance are ignored.Damping term contributed by synchronous machine damper windings are ignored.Rotor speed is assumed to remain constant.Mechanical input to machine is assumed to remain constant during the transient.Internal machine voltages are assumed constant.Loads are modeled as constant admittances.7Equal area criterion

8Response to a short circuit fault

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Stable case10

Response to a fault cleared in tc2 seconds - unstable caseUnstable case11(a)How heavily the generator is initially loaded.(b)The generator output during the fault. This depends on the fault location and type.(c)The fault clearing time.(d)The post-fault transmission system reactance.(e)The generator reactance. A lower reactance increases peak power and reduces initial rotor angle.(f)The generator inertia. The higher the inertia, the slower the rate of change angle. This reduces the kinetic energy gained during fault, i.e. area A1 is reduced.(g)The generator internal voltage magnitude (El). This depends on the field excitation.(h)The infinite bus voltage magnitude EB.Factors influencing Transient stability12Numerical Integration methodsEuler methodModified Euler methodRunge-Kutta method

Euler method

13

Modified Euler methodModified euler method consists of 2 steps:a) Predictor step: By using the derivative at the beginning of the step, the value at the end of the step is calculatedb) Corrector step14

Fourth order Runge Kutta methodThe general formula giving the value of x at (n+1)th step is:Where, 15Multi-machine transient stability The first step is to solve the initial load flow and to determine the initial bus voltage magnitudes and phase angles. The machine currents prior to disturbance are calculated from:All unknown values are determined from the initial power flow solution. The generator armature resistances are usually neglected and the voltages behind the transient reactances are then obtained:

All load are converted to equivalent admittances by using the relation:

(1)(2)(3)16Fig.5: Power system representation for transient stability analysisTo include voltages behind transient reactances, m buses are added to the n bus power system network. The equivalent network with all load converted to admittances is shown in Fig.1

17Nodes n+1, n+2, . . ., n+m are the internal machine buses, i.e., the buses behind the transient reactances. The node voltage equation with node 0 as reference for this network, is shown in (4).

(4)

Or,(5)Ibus - vector of the injected bus currentsVbus - vector of bus voltages measured from the reference node.18Off-diagonal elements are equal to the negative of the admittance between the nodes.The diagonal elements of the bus admittance matrix are the sum of admittances connected to it.Since no current enters or leaves the load buses, currents in the n rows are zero. The generator currents are denoted by the vector Im and the generator and load voltages are represented by the vector E m and Vn. Then, Equation (4), in terms of submatrices, becomes:To eliminate the load buses, the bus admittance matrix in (4) is partitioned such that the n buses to be removed are represented in the upper n rows.All nodes other than the generator internal nodes are eliminated using Kron reduction formula.

(6)19The reduced admittance matrix is:Now substituting into (8), we haveFrom (7),The voltage vector Vn may be eliminated by substitution as follows:

(7)(8)

(9)

(10)The reduced bus admittance matrix has the dimensions (m x m), where m is the number of generators.

(11)20Expressing voltages and admittances in polar form, i.e., E'i = |E'i | i and Yij = | Yij | ij , and substituting for Ii in (12), result inWherePrior to disturbance, there is equilibrium between the mechanical power input and the electrical power output, and we haveThe electrical power output of each machine can now be expressed in terms of the machines internal voltages:

or,

(12)

(13)

(14)

(15)21Yij - elements of the faulted reduced bus admittance matrixHi - inertia constant of machine i expressed on the common MVA base SB.The electrical power of the ith generator in terms of the new reduced bus admittance matrices are obtained from (14).The new bus admittance matrix is reduced by eliminating all nodes except the internal generator nodes. The generator excitation voltages during the fault and postfault modes are assumed to remain constant.The classical transient stability study is based on the application of a three-phase fault. A solid three-phase fault at bus k in the network results in Vk = 0. This is simulated by removing the kth row and column from the prefault bus admittance matrix.

The swing equation with damping neglected, for machine i becomes:

(16)22In transient stability analysis problem, we have two state equations for each generator. When the fault is cleared, which may involve the removal of the faulty line, the bus admittance matrix is recomputed to reflect the change in the networks. Showing the electrical power of the ith generator by Pef and transforming (16) into state variable mode yields:If HGi is the inertia constant of machine i expressed on the machine rated MVA SGi, then Hi is given by:

(17)

(18)(19)23Next, the postfault reduced bus admittance matrix is evaluated and the postfault electrical power of the ith generator shown by Ppfi is readily determined. Using the postfault power Ppfi, the simulation is continued to determine the system stability, until the plots reveal a definite trend as to stability or instability.

Usually the slack generator is selected as the reference machine, and phase angle difference of all other generators with respect to the reference machine are plotted. If the angle differences do not increase, the system is stable. If any of the angle differences increase indefinitely, the system is unstable.24

Phase angle difference (fault cleared at 0.4 sec)25

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