ECE 422 Power System Operations & Planning
6 ‐ Small Signal Stability
Spring 2015
Instructor: Kai Sun
1
References
•Saadat’s Chapter 11.4
•EPRI Tutorial’s Chapter 8 – Power Oscillations
•Kundur’s Chapter 12
2
Power Oscillations
• The power system naturally enters periods of oscillation as it continually adjusts to new operating conditions or experiences other disturbances. • Typically the amplitude of the oscillations is small and their lifetime is short. • When the amplitude of the oscillations becomes large or the oscillations are sustained, a response may be required. • A system operator may have the opportunity to respond and eliminate harmful oscillations or, less desirably, protective relays may activate to trip system elements.
3
Small Signal Stability
•Small signal stability (also referred to as small‐disturbance stability or steady‐state stability) is the ability of a power system to maintain synchronism when subjected to small disturbances
– In this context, a disturbance is considered to be small if the equations that describe the resulting response of the system may be linearized for the purpose of analysis
– It is convenient to assume that the disturbances causing the changes disappear (the details of the disturbance is not important)
– The system is stable if it returns to its original state, i.e. a stable equilibrium point.
– Such a behavior can be determined in the linearized model of the power system
4
Consider the Classical Model
(Te)
(Tm)
P+jQ’
Pt+jQt
PB+jQB
Linearize swing equations at =0 :
(Pe=Te)
=Tm-Te
Complex power behind X’d:
Define synchronizing torque coefficient
With resistance (RT) neglected:
KS =
E ¢E B
cos d0 = Pmax cos d0
XT
Te= Pe= P= Pt= PB=
Pmax= E’EB/XT
5
w0DTm
d 2Dd K D d Dd K s w0
+
+
D
d
=
dt 2
2 H dt
2H
2H
•
Apply Laplace Transform:
s 2Dd +
•
w DT
Kw
KD
sDd + s 0 Dd = 0 m
2H
2H
2H
Characteristic equation:
Note: PS =KS and D0=KD in Saadat’s book
w0DTm
d 2Dd Dw0 d Dd Ps w0
+
+
D
d
=
dt 2
2 H dt
2H
2H
s2 +
Dw0
Pw
s+ s 0 =0
2H
2H
6
（K S =
E ¢E B
cos d0 = Pmax cos d0）
XT
• Compared to the general form of a 2nd order system (0<<<1 for a generator):
s 2 + 2zwn s + wn 2 = 0
Two conjugate complex roots:
 ‐ Damping ratio
n – Nature frequency
s1 , s2 = -s  jwd = -zwn  jwn 1- z 2
Zero‐input response is damped sinusoidal oscillation:
Dd (t ) = A0 e-st sin(wd t + q )
• Natural frequency:
Damping ratio:
wn = K S
1
z=
2
s = zwn
w0
pf
= PS 0 rad/s
2H
H
KD
D
=
2
K S 2 H w0
p f0
HPS
Note: transmission resistance
is ignored here, so the actual
damping is higher than 
• When H  or KS, oscillation frequency (n or d) 
How does n change when XT or ଴ increases?
7
System Response after a Small Disturbance
Y( s )  X ( s )  ( sI  A ) 1  x(0)＋BU ( s )
Zero-input Zero-state
x1 = Dd
·
x2 = Dwr = D d / w0
é  ù
ê x1 ú é
0
w0 ù
ê  ú=ê
ú
ê ú ê-w 2 w - 2zw ú
0
nû
ê x2 ú ë n
êë úû
DTm
Du =
2H
é x1 ù é 0 ù D Tm
ê ú+ê ú
ê x 2 ú ê1 ú 2 H
ë û ë û

x(t )  Ax(t )  Bu (t )
é1 0ù é x1 ù
ú ê ú
y (t ) = x(t ) = ê
êë 0 1úû ê x2 ú
ë û
sX ( s )  x(0)  AX ( s )  BU ( s )
DU ( s ) =
Du
s
é s + 2zwn w0 ù
ê
ú
ê-wn 2 w0 s ú
û [ x (0)＋BDU ( s ) ]
X ( s ) = ë2
s + 2zwn s + wn 2
é s + 2zwn w0 ù
ê
ú
2
ê-w w0 s ú
é Dd ( s ) ù
û
ê
ú = ë2 n
êDwr ( s )ú s + 2zwn s + wn 2
ë
û
é
ù
é Dd (0) ù ê 0 ú
ú+ê
ú)
(ê
êDwr (0)ú ê Du ú
ë
û ê s ú
ë
û
Zero-input Zero-state
Usually r(0)0 following a disturbance
8
é s + 2zwn w0 ù
ê
ú
2
ê
ú
é Dd ( s ) ù
-w / w0 s û
ê
ú = ë2 n
êDwr ( s )ú s + 2zwn s + wn 2
ë
û
é
ù
é Dd (0) ù ê 0 ú
ú+ê
ú)
(ê
êDwr (0)ú ê Du ú
ë
û ê s ú
ë
û
·
Dwr =Dd / w0 = (wr -w0 ) / w0 in pu
Du =
DTm
pu
2H
Zero‐input response
•
Zero‐state response
E.g. when the rotor is suddenly perturbed by
a small angle (0)0 and assume r (0)=0
•
E.g. when there is a small increase in
mechanical torque Tm (= Pm in pu)
( s + 2zwn )Dd (0)
Dd ( s ) = 2
s + 2zwn s + wn 2
Dwr (s) = -
Dd in rad =
 (s) 
w Dd (0) / w0
s 2 + 2zwn s + wn 2
2
n
Dd (0)
-zwn t
e
Taking inverse Laplace transforms
sin(wd t + q )
1- z
w Dd (0) -zwnt
e
Dwr in rad/s = - n
sin wd t
2
1- z
  cos 1 
2

1
n

4H
KD
0u
s ( s 2  2  n s   n2 )
 r ( s ) 
 T
  in rad  0 m2
2H  n
  r in rad /s 
u
s 2  2 n s   n2

1
1 
1

2
e
 0  Tm
2 H n 1  
2
  n t

sin  d t    

e   n t sin  d t
(Response time constant)
9
Saadat’s Examples 11.2 and 11.3
• H=9.94 MJ/MVA, D=0.138, P=0.6 pu with 0.8 power factor. Obtain the zero‐input and zero‐state responses for the rotor angle and the generator frequency:
(1) (0)=10o=0.1745 rad
(2) P=0.2pu
(0)=16.79+10o=26.79o
10
Zero‐input response: (0)=10o
(0)=16.79+10=26.79o
Zero‐state response: P=0.2pu
()=16.79+5.76=22.55o
11
Characteristics of Small‐Signal Stability Problems
• Local or machine‐system modes (0.7~2Hz): oscillations involve a small part of the system
– Local plant modes: associated with rotor angle oscillations of a single generator or a single plant against the rest of the system; similar to the single‐machine‐
infinite bus system
– Inter‐machine or interplant modes: associated with oscillations between the rotors of a few generators close to each other
• Inter‐ or intra‐area modes (0.1~0.7Hz): machines in one part of the system swing against machines in other parts
– Inter‐area model (0.1~0.3Hz): involving all the generators in the system; the system is essentially split into two parts, with generators in one part swinging against machines in the other parts.
– Intra‐area mode (0.4~0.7Hz): involving subgroups of generators swinging against each other.
• Control or torsional modes: – Due to inadequate tuning of the control systems, e.g. generator excitation systems, HVDC converters and SVCs, or torsional interaction with power system control
12
High vs. Low Frequency Oscillations in Realistic Systems
• When power flows, I2R losses occur. These energy losses help to reduce the amplitude of the oscillation. The higher the frequency of the oscillation, the faster it is damped. High frequency (>1.0 HZ) oscillations are damped more rapidly than low frequency (<1.0 HZ) oscillations.
• Usually, in realistic systems:
• Power system operators do not want any oscillations. However, it is better to have high frequency oscillations than low frequency. The power system can naturally dampen high frequency oscillations. Low frequency oscillations are more damaging to the power system, which may exist for a long time, become sustained (undamped) oscillations, and even trigger protective relays to trip elements
13
Blackout Event on August 10, 1996
970 MW loss
1. Initial event (15:42:03):
Short circuit due to tree contact  Outages of 6 transformers and lines
2. Vulnerable conditions (minutes)
Low‐damped inter‐area oscillations 
Outages of generators and tie‐lines
2,100 MW loss
11,600 MW loss
15,820MW loss
3. Blackouts (seconds)
Unintentional separation 
load
Loss of 24% Malin-Round Mountain #1 MW
1500
15:42:03
15:47:36
15:48:51
1400
0.276 Hz
oscillations
Damping>7%
1300
1200
1100
200
300
0.264 Hz
oscillations
3.46% Damping
400
500
0.252 Hz
oscillations
Damping 1%
Time in Seconds
600
Oscillation frequency (n or d)  when H  or KS (e.g. when XT  or ଴ )
700
System 800
islanding
and
14
blackouts