ECE 422/522
Power System Operations &
Planning/Power Systems Analysis II :
6 - Small Signal Stability
Spring 2014
Instructor: Kai Sun
1
References
• Kundur’s Chapter 12
• Saadat’s Chapter 11.4
• EPRI Tutorial’s Chapter 8 – Power Oscillations
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
Small Signal Stability of
a Single-Machine-Infinite-Bus System
• Because of the relative size of
the large system to which the
machine is supplying power,
dynamics associated with the
machine will cause virtually no
change in the voltage and
frequency of Thevenin’s EB
• The large system is often
referred to as an Infinite Bus
(constant voltage and constant
frequency)
• The voltage phasor at the
infinite bus can be the reference
of the rotor angle, i.e. EB∠0
5
(Te)
Consider the Classical Model
P+jQ’
Pt+jQt
PB+jQB
(Tm)
(Pe=Te)
Linearize swing equations at δ=δ0 :
Complex power behind X’d:
=∆Tm-∆Te
Define synchronizing torque coefficient
With resistance (RT) neglected:
KS 
Te= Pe= P= Pt= PB=
Pmax= E’EB/XT
6
E EB
cos 0  Pmax cos 0
XT
0Tm
d 2 K D d  K s 0





2 H dt
2H
2H
dt 2
•
Apply Laplace Transform:
s 2 
•
K
 T
KD
s  s 0   0 m
2H
2H
2H
Characteristic equation:
Note: KS =PS and KD=Dω0 in Saadat’s book
0Tm
d 2 D0 d  Ps 0





dt 2
2 H dt
2H
2H
s2 
D0
P
s s 0 0
2H
2H
7
（K S 
E EB
cos 0  Pmax cos 0）
XT
• Compared to the general form of a 2nd order system (0<ζ<<1 for a generator):
s 2  2n s  n 2  0
Undamped natural frequency:
Note: resistance RT is ignored
here, whose effect is to result in
more damping
Damping ratio:
Two conjugate complex roots: s1 , s2  n  jn 1  2  n  jd
Damped oscillation frequency:
• When H ↑ or KS↓, oscillation frequency (ωn or ωd) ↓
(e.g. when XT ↑ or δ0 ↑)
8
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
9
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 →
Loss of 24% load
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 (ωn or ωd) ↓ when H ↑ or KS↓ (e.g. when XT ↑ or δ0 ↑)
10
700
System 800
islanding
and
blackouts
System Response after a Small Disturbance
Y( s ) =
X( s) =
( sI − A ) −1 [ x(0)＋B∆U ( s )]
Zero-input Zero-state
x1  

x2  r    / 0
  
 x1  
0
0 
  

   2  2 
0
n
 x2   n
 
Tm
u 
2H
 x1  0 Tm
  
 x2  1  2H
   

x=
(t ) Ax(t ) + B∆u (t )
1 0  x1 
  
y (t )  x(t )  
 0 1  x2 
 
sX ( s ) − x=
(0) AX ( s ) + B∆U ( s )
U ( s ) 
u
s
 s  2n 0 


n 2 0 s 
  x(0)＋BU ( s ) 
X ( s )  2
s  2n s  n 2
 s  2n 0 


2
 0 s 
  ( s ) 


  2 n
r ( s ) s  2n s  n 2




  (0)   0 

)
(
r (0)  u 

  s 


Zero-input Zero-state
Usually ∆ωr(0)≈0 following a disturbance
11
 s  2n 0 


2

  ( s ) 
n 0 s 



r ( s ) s 2  2n s  n 2





  (0)   0 

)
(
r (0)  u 

  s 


Note:
u 
Zero-input response
•
•
E.g. when there is a small increase in
mechanical torque ∆Tm (= ∆Pm in pu)
ω0 ∆u
∆δ ( s ) =
s ( s 2 + 2ζωn s + ωn2 )
( s  2n ) (0)
 ( s )  2
s  2n s  n 2
 
  (0) / 0
s 2  2n s  n 2
2
n
 (0)
n t
e
Taking inverse
Laplace
transforms
=
∆δ
sin(d t   )
1  2
  (0) nt
e
r   n
sin d t
2
0 1  
θ = cos −1 ζ
τ
=
1
4H
=
ζωn K D
Tm
2H
Zero-state response
E.g. when the rotor is suddenly perturbed by
a small angle ∆δ(0)≠0 and assume ∆ωr (0)=0
r ( s )  
r    / 0  (r  0 ) / 0
∆u
∆ωr ( s ) =
s 2 + 2ζωn s + ωn2

ω0 ∆u 
1
−ζω t
1
sin
ω
θ
e
t
−
+

( d
)
2
ωn2 
1− ζ

n
∆ωr =
∆u
ωn 1 − ζ 2
(Response time constant)
12
e −ζωnt sin ωd t
 s  2n 1


2


  ( s ) 
n
s

  2
2
 ( s ) s  2n s  n
Note: Saadat’s book defines


  (0)   0 

)
(
 (0)  u 
 s 

       0
Zero-input response
•
•
E.g. when there is a small increase in
mechanical torque ∆Tm (= ∆Pm in pu)
∆u
∆δ ( s ) =
s ( s 2 + 2ζωn s + ωn2 )
( s  2n ) (0)
 ( s )  2
s  2n s  n 2
 
  (0)
s 2  2n s  n 2
2
n
 (0)
n t
e
θ = cos −1 ζ
τ
=

∆u 
1
−ζωn t
−
+
1
e
sin
t
ω
θ

( d
)
2
ωn2 
1− ζ

∆=
δ
sin(d t   )
1
4H
=
ζωn K D
∆u
∆ω ( s ) =
s 2 + 2ζωn s + ωn2
Taking inverse
Laplace
transforms
1 
  (0) nt
e
   n
sin d t
2
1 
2
0Tm
2H
Zero-state response
E.g. when the rotor is suddenly perturbed by
a small angle ∆δ(0)≠0 and assume ∆ωr (0)=0
 ( s )  
u 
∆ω =
∆u
ωn 1 − ζ 2
(Response time constant)
13
e −ζωnt sin ωd t
Saadat’s Example 11.2 and Example 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) ∆δ(0)=10o=0.1745 rad
(2) ∆P=0.2pu
δ(0)=16.79+10o=26.79o
14
Zero-input response: ∆δ(0)=10o
Zero-state response: ∆P=0.2pu
δ(0)=16.79+10=26.79o
δ(∞)=16.79+5.76=22.55o
15
Consider the Excitation System
(see Kundur’s 12.4 for aij and b1)
Te | fd 
K 2 K 3 [ K 4 (1  sTR )  K 5Gex ( s )]

s 2T3TR  s (T3  TR )1  K 3 K 6Gex ( s )
16
• The effect of the AVR on damping
and synchronizing torque
components is primarily influenced
by Gex(s) and K5
• With K5<0, the AVR may introduce
a positive synchronizing torque
Te | fd 
K 2 K 3 [ K 4 (1  sTR )  K 5Gex ( s )]

2
s T3TR  s (T3  TR )1  K 3 K 6Gex ( s )
KR and KI are respectively the real and
imaginary parts of the coefficient of ∆δ
• For a given oscillation frequency s=jω:
Te | fd  K R   K I j  K R  
K I 0
r

=K S (  fd )  K D (  fd )r
j  j /   s / 
 r 0 / 
Synchronizing and damping torque coefficients due to ∆ψfd
17
Example on effects of different AVR settings
• Steady-state synchronizing torque coefficient:
The effect of the AVR is to
increase the synchronizing torque
component at steady state
• Damping and synchronizing torque components at rotor oscillation frequency
10 rad/s (s=jω=j10)
18
∆Te= ∆TS+∆TD= KS ∆δ+KD∆ω
• KS = KS(∆ψfd)+KS(gen & network)
KD = KD(∆ψfd)+KD(gen & network)
Usually, KS (gen & network)>0
KD(gen & network)>0
• Constant field voltage (KA=0):
– KD>0
– Perhaps,
KS=KS(gen & network) + KS(∆ψfd)<0
• With excitation control (large KA)
– KS>0
– Perhaps,
KD =KD(gen & network) + KD(∆ψfd)<0
19
Power System Stabilizer
• The basic function of a power system stabilizer (PSS) is to add damping to the
generator rotor oscillations by controlling its excitation using non-voltage
auxiliary stabilizing signal(s)
– If the transfer function from PSS’s output to ∆Te was pure gain, a direct
feedback of ∆ωr would create a positive damping torque component.
20
• However, actual generators and exciters exhibit
frequency dependent gain and phase-lag
characteristics
G( s) 
1
1 s
• Therefore, GPSS(s) should provide phase-lead
compensation to create a torque in phase with ∆ωr
G(s)
1/τ1
GPSS ( s ) 
21
1/τ2
1  1s
(1   2 )
1 2 s
PSS Model
• Stabilizer gain KSTAB
– determines the amount of
damping introduced by PSS
• Signal washout block:
– High-pass filter with TW long enough (typically 1~20s) to allow signals
associated with oscillations in ωr to pass unchanged. However, if it is too long,
steady changes in speed would cause generator voltage excursions
• Phase compensation block:
– Provides phase-lead compensation over the frequency range of interest
(typically, f=0.1~2.0 Hz, i.e. ω=0.6~12.6 rad/s)
– Two or more first-order blocks, or even second-order blocks may be used.
– Generally, some under-compensation is desirable so that the PSS results in a
slight increase of the synchronizing torque as well
22
Kundur’s Example 12.4
23
PSS in NERC Interconnections
• The WECC and the MRO Regions have operating requirements that
mandate the use of PSS. All units with fast excitation systems must
be equipped with well tuned PSS in these Regions. PSS are typically
installed in the majority of generating units in the problem area.
• Several units in the PJM system are equipped with PSS to address
local oscillatory stability concerns
24
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
25
Homework
• Problems 11.10~11.13 in Saadat’s book (3rd ed), due by
April 15 (Tuesday) in class or by email
26