Small-Signal Stability and
Power System Stabilizer
Dynamics and Control of Electric Power Systems
Contents




Review: Closed-Loop Stability
Third-Order Model of the Synchronous Machine
Heffron-Phillips Model
Dynamic Analysis of the Heffron-Phillips Model




Split between damping and synchronizing torque
SMIB with classical generator model
SMIB including field circuit dynamics
SMIB including excitation system
 Power System Stabilizer
 Block diagram
 Effect on system dynamics
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Review: Closed-Loop Stability
State space formulation of dynamical system
 Autonomous dynamical linear system with initial condition:
=
x Ax, x(=
t 0)= x0
 Rate of change of each state is a linear combination of all states:
 x1   a11 a12   x1 
 x  =  a
 x 
a
 2   21 22   2 
=
x1 a11 x1 + a12 x2
=
x2 a21 x1 + a22 x2
 Transformation to diagonal form in order to derive solution easily:
z1 = λ1 z1
=
z1 z1 (0) ⋅ eλ1t
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Review: Closed-Loop Stability
State space formulation of dynamical system
 Our aim is to transform the equation to the “easy“ form:

 z1  λ1 0   z1 
 z  =  0 λ  ⋅  z  ⇔ z = Λ ⋅ z
 2 
2  2
Linear coordinate transformation:
x = Φ⋅z
x = Φ ⋅ z
 This is equivalent to:
Φ ⋅ z= A ⋅ Φ ⋅ z
z = Λ ⋅ z
−1
⋅ A ⋅
Φ⋅z
z = Φ

Λ
Φ =[φ1 , φ2 .....φn ]
Λ =diag (λ1 , λ2 ....., λn )
φi ⋅ λi = A ⋅ φi ⇒ ( A − λi I ) ⋅ φi = 0
det( A − λi I ) =
0
λi ........eigenvalues
φi .........right eigenvectors
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Review: Closed-Loop Stability
Eigenvalues, stability, oscillation frequency and damping ratio
 Let λ1 be a real eigenvalue of matrix A . Then holds:
λ1 < 0 : The corresponding mode is stable (decaying exponential).
λ1 > 0 : The corresponding mode is unstable (growing exponential).
λ1 = 0 : The corresponding mode has integrating characteristics.
 Let λ1,2= σ ± jω be a complex conjugate pair of eigenvalues of A . Then:
Re λ1,2 < 0 : The corresponding mode is stable (decaying oscillation).
Re λ1,2 > 0 : The corresponding mode is unstable (growing oscillation).
Re λ1,2 = 0 : The corresponding mode is critically stable (undamped osc.).
The following dynamic properties can be established:
 Oscillation frequency: f =
 Damping ratio:
ζ =
ω
2π
−σ
σ 2 + ω2
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Third-Order Model of the Synchronous Machine
 Voltage deviation in d- and q-axis:
with
 Linearized swing equation:
=
∆ω
1
(∆Tm − ∆Te )
2 Hs + K D
2π f 0
∆=
∆ω
δ
s
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Heffron-Phillips Model
Purpose:
 Simplified representation of synchronous
machine, suitable for stability studies:
“Small Signal Stability”  linearized model
Basis:
Electrical
torque change
 Third-order Model of synchronous
machine
Starting point for derivation:
 Single-Machine Infinite-Bus (SMIB) System
 Linearized generator swing equation:
1
(∆Tm − ∆Te )
=
∆ω
2 Hs + K D
2π f 0
∆=
∆ω
δ
s
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Singel Machine Infinite Bus (SMIB)
Generator terminals
Power line
Generator
∆eF
AVR
ut
Infinite bus
(Voltage magnitude and phase
constant)
set
t
u
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Heffron-Phillips Model
Purpose:
 Simplified representation of synchronous
machine, suitable for stability studies:
“Small Signal Stability”  linearized model
Basis:
Electrical
torque change
 Third-order Model of synchronous
machine
Starting point for derivation:
 Single-Machine Infinite-Bus (SMIB) System
 Linearized generator swing equation:
1
(∆Tm − ∆Te )
=
∆ω
2 Hs + K D
2π f 0
∆=
∆ω
δ
s
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Heffron-Phillips Model
Electrical
torque change
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Heffron-Phillips Model
… including the composition of the electric torque:
Approximation of torque with power:
After linearization and some substitutions:
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Heffron-Phillips Model
… including the effect of the field voltage equation:
Influence of torque angle on internal voltage
Field voltage equation:
After linearization and some substitutions:
with:
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Heffron-Phillips Model
… including the model of the terminal voltage magnitude:
∆eF + K 4 ∆δ
Influence of torque angle on internal voltage
−∆eF
−∆eF
Terminal voltage:
Linearization and substitution:
with
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Heffron-Phillips Model
Full model:
Influence of torque angle on internal voltage
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Heffron-Phillips Model
Simulink implementation
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Dynamic Analysis of the Heffron-Phillips Model
Splitting between synchronizing and damping torque
∆ω
K Damp
∆Te
K Sync
Exercise 3!
∆δ
∆=
Te K Sync ⋅ ∆δ + K Damp ⋅ ∆ω
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Dynamic Analysis of the Heffron-Phillips Model
SMIB with classical generator model (mechanical damping torque KD = 0)
Eigenvalues on imaginary axis
 system is critically stable
Eigenvalues
λ1,2
Real
0
Imaginary
± 6.385
Damping Ratio
f [Hz]
-
1.016
Synchronizing and damping torque coefficients
s
λ1,2
Ksync
Kdamp
0.757
0
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Dynamic Analysis of the Heffron-Phillips Model
SMIB including field circuit dynamics
Eigenvalues moved to the
left because field circuit
adds damping torque
Eigenvalues
λ1,2
λ3
Real
– 0.109
Imaginary
± 6.411
– 0.204
0
Synchronizing and damping torque
coefficients due to field circuit
Damping Ratio
f [Hz]
0.0170
1.020
1.0
s
λ1,2
λ3
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Ksync
Kdamp
– 0.0008
1.5333
– 0.7651
0
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Dynamic Analysis of the Heffron-Phillips Model
SMIB including excitation system
Eigenvalues
λ1,2
λ3
λ4
Real
Imaginary
Damping Ratio
f [Hz]
– 0.0816
1.7167
± 10.7864
0.8837
– 33.8342
0
1.0
0
–18.4567
0
1.0
0
Synchronizing and damping torque
coefficients due to exciter
s
λ1,2
λ3
λ4
Ksync
Kdamp
0.2731
-10.6038
– 19.8103
0
– 7.0126
0
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Dynamic Analysis of the Heffron-Phillips Model
SMIB including excitation system
 Generator tripping
 might eventually result in Blackout!
Eigenvalues moved to the right
by the excitation system
 System is unstable!
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Power System Stabilizer
 Purpose:
provide additional damping torque component in order to prevent the
system from becoming unstable
 Approach:
insert feedback between angular frequency and voltage setpoint
 Block diagram:
Gain:
Washout filter:
Phase compensation:
Tuning parameter
Suppress effect
Provide phase-lead characteristic
for damping torque
of low-frequency
to compensate for lag between
increase
speed changes
exciter input and el. torque
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Power System Stabilizer
Block diagram
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Power System Stabilizer
Effect on the system dynamics
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Power System Stabilizer
Effect on the system dynamics
Eigenvalues
λ1,2
λ3,4
λ5
λ6
Real
– 1.0052
– 19.7970
Imaginary
Damping Ratio
f [Hz]
0.1504
1.0516
0.8394
2.0406
± 6.6071
±12.8213
– 39.0969
0
-
-
– 0.7388
0
-
-
Synchronizing and damping torque
Synchronizing and damping torque
coefficients due to exciter
coefficients due to PSS
s
λ1,2
λ3,4
λ5
λ6
Ksync
Kdamp
0.21
– 8.69
– 1.27
– 13.00
1.16
0
0.30
0
s
λ1,2
λ3,4
λ5
λ6
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Ksync
Kdamp
– 0.145
22.761
10.838
290.163
– 30.306
0
–1.072
0
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Coming up …
Exercise 3: Power System Stabilizer
 Contents:
Stability analysis of Heffron-Phillips Model, PSS design and testing
 Date and time:
Tuesday, 29 May 2012
 Handouts will be sent around one week in advance.
Please prepare the exercise at home, timing is tight!
 Attendance is compulsory for the “Testat“.
Please notify us in case you cannot attend  substitute task.
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