lecture20-2

Small Signal Stability

The Synchronizing and Damping Torques

small rotor oscillations are governed by an approximate second order differential equation given by



Kdamp is the damping of the entire synchronous machine (including field winding, excitation, and PSS if connected).

It shall not be confused with the mechanical damping constant KD of the swing equation.

For calculating Ksync and Kdamp for a given eigenvalue of the system, we calculate the transfer function K(s) between the rotor angle deviation and the electrical torque deviation

For a given eigenvalue , K is a complex number


The Synchronizing and Damping Torques The rotor angle deviation can be written as



The synchronizing and damping components are

Ksync and/or Kdamp 0 Unstable Ksync 0 Motonic instability. Kdamp 0 Growing oscillations.



Case 1 : Without AVR



Case 1 : Without AVR

Note:



Case 2 : With Fast Excitation System (TE 0 )



Example A synchronous generator is connected to an infinite bus through an external reactance Xe = 0.4 pu. Compute the Heffron-Phillips constants, K1 to K6 at the operating point.



Solution



Example For the system considered in the previous example, compute the eigenvalues for the two operating conditions and (i) without AVR (ii) with AVR of TE = 0.05 sec, KE = 200.



Example


Classification of Power System Oscillations

1) Swing mode (electromechanical) oscillations. For an n generator system, there are (n-1) swing (oscillatory) modes associated with the generator rotors. A swing mode oscillation is characterized by a high association of the generator rotor in that mode.

2) Control mode oscillations. Control modes are associated with generating units and other controls. Poorly tuned exciters, speed governors, static var compensators, etc are the usual causes of instability of these modes.

3) Torsional mode oscillations. These oscillations involve relative angular motion between the rotating elements (synchronous machine, turbine, and exciter) of a unit, with frequencies ranging from 4Hz and above. Dc lines, static converters, series-capacitor-compensated lines can excite torsional oscillations such as and other devices.


Modal Analysis Swing mode oscillations can be further grouped into:


Modal Analysis


Modal Analysis

(orthogonality; if i j)


Modal Analysis


Modal Analysis Find eigenvalues eigenvectors

Solution


Modal Analysis Find V, W,

Solution

v1

v2

v3

W1

W2

W3


Modal Analysis

These products are not yet I and because the eigenvectors need to be appropriately normalized.

WT=


Modal Analysis

is is called mode of system


Modal Analysis


Modal Analysis

State Transition Matrix

,

State Transition Matrix


Modal Analysis

Theorem:

To evaluate , take entry-by-entry inverse transform of


Modal Analysis

Example

Find the state transition matrix of A

Solution

By taking the inverse Laplace transform


Modal Analysis


Modal Analysis

Pjk gives the sensitivity of j to the diagonal element akk of A:

The participation factor (or residue)-based analysis is valid only if the eigenvalues are distinct.


Time-Domain Solution

Stability can be verified by numerical solution of non-linear deferential-algebraic system of equations, for small and large disturbances. Disadvantages The choice of disturbance and selection of variables to be observed in

time response are critical. The input, if not chosen properly, may not provide substantial excitation of the important modes.

For a large power system it is not possible to identify any desired mode and study their characteristics.


Modal Analysis

The damping ratio determines the rate of decay of the amplitude of the oscillation. The time constant of amplitude decay is 1/||. Namely, the

amplitude reduces to 1/e or 37% of the initial amplitude in 1/|| seconds.

Damping ratio and frequency of oscillation


Modal Analysis

Possible combinations of eigenvalue pairs


Example: Modal Analysis of Spring-Mass System

A multi-spring system with the parameters below is shown. Find the state matrix equations.



Solution

The state equations are:



In the matrix format:



Or,



k23 >> k12 means M2 and M3 coupled rigidly than M2, M3 with M1.

M1 >> M2, M3. Thus, low frequency oscillations are due to M1 and high frequency oscillation are due to M2, M3.



Note:



Observations


Example: Spring-Mass System with Damping Coefficient

The system is modified to include the effect of damping coefficient, assuming B1 = 1 N/m/s, B2 = B3 = 0. Write system differential equations.



Solution

(*)



By numerical solution of (*)



Observations

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lecture20-2

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