Small Signal Stability • The Synchronizing and Damping Torques • small rotor oscillations are governed by an approximate second order differential equation given by
Small Signal Stability
The Synchronizing and Damping Torques
small rotor oscillations are governed by an approximate second order differential equation given by
Small Signal Stability
The Synchronizing and Damping Torques
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
Small Signal Stability
The Synchronizing and Damping Torques The rotor angle deviation can be written as
Small Signal Stability
The Synchronizing and Damping Torques
The synchronizing and damping components are
Ksync and/or Kdamp 0 Unstable Ksync 0 Motonic instability. Kdamp 0 Growing oscillations.
Small Signal Stability
The Synchronizing and Damping Torques
Case 1 : Without AVR
Small Signal Stability
The Synchronizing and Damping Torques
Case 1 : Without AVR
Small Signal Stability
The Synchronizing and Damping Torques
Case 1 : Without AVR
Note:
Small Signal Stability
The Synchronizing and Damping Torques
Case 2 : With Fast Excitation System (TE 0 )
Small Signal Stability
The Synchronizing and Damping Torques
Case 2 : With Fast Excitation System (TE 0 )
Small Signal Stability
The Synchronizing and Damping Torques
Case 2 : With Fast Excitation System (TE 0 )
Small Signal Stability
The Synchronizing and Damping Torques
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.
Small Signal Stability
The Synchronizing and Damping Torques
Solution
Small Signal Stability
The Synchronizing and Damping Torques
Small Signal Stability
The Synchronizing and Damping Torques
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.
Small Signal Stability
The Synchronizing and Damping Torques
Example
Small Signal Stability
The Synchronizing and Damping Torques
Example
Small Signal Stability
The Synchronizing and Damping Torques
Example
Small Signal Stability
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.
Small Signal Stability
Modal Analysis Swing mode oscillations can be further grouped into:
Small Signal Stability
Modal Analysis
Small Signal Stability
Modal Analysis
Small Signal Stability
Modal Analysis
(orthogonality; if i j)
Small Signal Stability
Modal Analysis
Small Signal Stability
Modal Analysis Find eigenvalues eigenvectors
Solution
Small Signal Stability
Modal Analysis Find V, W,
Solution
v1
v2
v3
W1
W2
W3
Small Signal Stability
Modal Analysis
These products are not yet I and because the eigenvectors need to be appropriately normalized.
WT=
Small Signal Stability
Modal Analysis
is is called mode of system
Small Signal Stability
Modal Analysis
Small Signal Stability
Modal Analysis
Small Signal Stability
Modal Analysis
State Transition Matrix
,
State Transition Matrix
Small Signal Stability
Modal Analysis
Theorem:
To evaluate , take entry-by-entry inverse transform of
Small Signal Stability
Modal Analysis
Example
Find the state transition matrix of A
Solution
By taking the inverse Laplace transform
Small Signal Stability
Modal Analysis
Small Signal Stability
Modal Analysis
Small Signal Stability
Modal Analysis
Small Signal Stability
Modal Analysis
Small Signal Stability
Modal Analysis
Small Signal Stability
Modal Analysis
Small Signal Stability
Modal Analysis
Small Signal Stability
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.
Small Signal Stability
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.
Small Signal Stability
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
Small Signal Stability
Modal Analysis
Possible combinations of eigenvalue pairs
Small Signal Stability
Modal Analysis
Possible combinations of eigenvalue pairs
Small Signal Stability
Modal Analysis
Possible combinations of eigenvalue pairs
Small Signal Stability
Example: Modal Analysis of Spring-Mass System
A multi-spring system with the parameters below is shown. Find the state matrix equations.
Small Signal Stability
Example: Modal Analysis of Spring-Mass System
Solution
The state equations are:
Small Signal Stability
Example: Modal Analysis of Spring-Mass System
In the matrix format:
Small Signal Stability
Example: Modal Analysis of Spring-Mass System
Or,
Small Signal Stability
Example: Modal Analysis of Spring-Mass System
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.
Small Signal Stability
Example: Modal Analysis of Spring-Mass System
Small Signal Stability
Example: Modal Analysis of Spring-Mass System
Small Signal Stability
Example: Modal Analysis of Spring-Mass System
Small Signal Stability
Example: Modal Analysis of Spring-Mass System
Note:
Small Signal Stability
Example: Modal Analysis of Spring-Mass System
Small Signal Stability
Example: Modal Analysis of Spring-Mass System
Small Signal Stability
Example: Modal Analysis of Spring-Mass System
Small Signal Stability
Example: Modal Analysis of Spring-Mass System
Small Signal Stability
Example: Modal Analysis of Spring-Mass System
Small Signal Stability
Example: Modal Analysis of Spring-Mass System
Note:
Small Signal Stability
Example: Modal Analysis of Spring-Mass System
Observations
Small Signal Stability
Example: Modal Analysis of Spring-Mass System
Small Signal Stability
Example: Modal Analysis of Spring-Mass System
Small Signal Stability
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.
Small Signal Stability
Example: Spring-Mass System with Damping Coefficient
Solution
(*)
Small Signal Stability
Example: Spring-Mass System with Damping Coefficient
Small Signal Stability
Example: Spring-Mass System with Damping Coefficient
Small Signal Stability
Example: Spring-Mass System with Damping Coefficient
Small Signal Stability
Example: Spring-Mass System with Damping Coefficient
Small Signal Stability
Example: Spring-Mass System with Damping Coefficient
By numerical solution of (*)
Small Signal Stability
Example: Spring-Mass System with Damping Coefficient
Small Signal Stability
Example: Spring-Mass System with Damping Coefficient
Small Signal Stability
Example: Spring-Mass System with Damping Coefficient
Small Signal Stability
Example: Spring-Mass System with Damping Coefficient
Small Signal Stability
Example: Spring-Mass System with Damping Coefficient
Small Signal Stability
Example: Spring-Mass System with Damping Coefficient
Observations
Small Signal Stability
Example: Spring-Mass System with Damping Coefficient