Upload
tinax
View
105
Download
5
Tags:
Embed Size (px)
Citation preview
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
2 EEH – Power Systems Laboratory
State space formulation of dynamical system
Autonomous dynamical linear system with initial condition:
Rate of change of each state is a linear combination of all states:
Transformation to diagonal form in order to derive solution easily:
Review: Closed-Loop Stability
0, ( 0)x Ax x t x= = =
1 11 12 1
2 21 22 2
1 11 1 12 2
2 21 1 22 2
x a a xx a a x
x a x a xx a x a x
=
= += +
1
1 1 1
1 1(0) t
z zz z eλ
λ=
= ⋅
3 EEH – Power Systems Laboratory
Review: Closed-Loop Stability State space formulation of dynamical system Our aim is to transform the equation to the “easy“ form:
1 1 1
2 2 2
00
z zz z
z zλ
λ
= ⋅ ⇔ = Λ ⋅
1 2
1 2
[ , ..... ]( , ....., )
( ) 0det( ) 0
n
n
i i i i i
i
diagA A I
A I
φ φ φλ λ λ
φ λ φ λ φλ
Φ =Λ =⋅ = ⋅ ⇒ − ⋅ =
− =
x zx z= Φ ⋅= Φ ⋅
.................
i
i
eigenvaluesright eigenvectors
λφ
Linear coordinate transformation:
1
z A zz A z−
Φ ⋅ = ⋅Φ ⋅
= Φ ⋅ ⋅Φ ⋅Λ
z z= Λ ⋅
This is equivalent to:
4 EEH – Power Systems Laboratory
Review: Closed-Loop Stability Eigenvalues, stability, oscillation frequency and damping ratio
Let be a real eigenvalue of matrix . Then holds: : The corresponding mode is stable (decaying exponential). : The corresponding mode is unstable (growing exponential). : The corresponding mode has integrating characteristics.
Let be a complex conjugate pair of eigenvalues of . Then: : The corresponding mode is stable (decaying oscillation). : The corresponding mode is unstable (growing oscillation). : The corresponding mode is critically stable (undamped osc.).
The following dynamic properties can be established:
Oscillation frequency:
Damping ratio:
1,2 jλ σ ω= ±
A1λ
A
1 0λ >
1 0λ =
1 0λ <
1,2Re 0λ <1,2Re 0λ >
1,2Re 0λ =
2f ω
π=
2 2
σζσ ω−
=+
5 EEH – Power Systems Laboratory
Voltage deviation in d- and q-axis:
with Linearized swing equation:
Third-Order Model of the Synchronous Machine
02 fsπδ ω∆ = ∆
1 ( )2 m e
D
T THs K
ω∆ = ∆ −∆+
6 EEH – Power Systems Laboratory
Heffron-Phillips Model
Electrical torque change
Purpose: Simplified representation of synchronous
machine, suitable for stability studies: “Small Signal Stability” linearized model Basis: Third-order Model of synchronous
machine Starting point for derivation: Single-Machine Infinite-Bus (SMIB) System Linearized generator swing equation:
02 fsπδ ω∆ = ∆
1 ( )2 m e
D
T THs K
ω∆ = ∆ −∆+
7 EEH – Power Systems Laboratory
Singel Machine Infinite Bus (SMIB)
8 EEH – Power Systems Laboratory
Generator terminals
Infinite bus
(Voltage magnitude and phase
constant)
Power line Generator
AVR tu
settu
Fe∆
Heffron-Phillips Model
Electrical torque change
Purpose: Simplified representation of synchronous
machine, suitable for stability studies: “Small Signal Stability” linearized model Basis: Third-order Model of synchronous
machine Starting point for derivation: Single-Machine Infinite-Bus (SMIB) System Linearized generator swing equation:
02 fsπδ ω∆ = ∆
1 ( )2 m e
D
T THs K
ω∆ = ∆ −∆+
9 EEH – Power Systems Laboratory
Heffron-Phillips Model
Electrical torque change
10 EEH – Power Systems Laboratory
Heffron-Phillips Model … including the composition of the electric torque:
Approximation of torque with power:
After linearization and some substitutions:
11 EEH – Power Systems Laboratory
Heffron-Phillips Model … including the effect of the field voltage equation:
Field voltage equation: After linearization and some substitutions: with:
12 EEH – Power Systems Laboratory
Influence of torque angle on internal voltage
Heffron-Phillips Model … including the model of the terminal voltage magnitude:
Terminal voltage: Linearization and substitution: with
13 EEH – Power Systems Laboratory
Influence of torque angle on internal voltage
Fe−∆
Fe−∆
4Fe K δ∆ + ∆
Heffron-Phillips Model Full model:
14 EEH – Power Systems Laboratory
Influence of torque angle on internal voltage
Heffron-Phillips Model Simulink implementation
15 EEH – Power Systems Laboratory
Dynamic Analysis of the Heffron-Phillips Model
δ∆
ω∆
δ ω∆ = ⋅∆ + ⋅∆e Sync DampT K K
DampK
SyncK
eT∆
Splitting between synchronizing and damping torque
Exercise 3!
16 EEH – Power Systems Laboratory
SMIB with classical generator model (mechanical damping torque KD = 0)
Real Imaginary Damping Ratio f [Hz] 0 6.385 - 1.016 1,2λ
s Ksync Kdamp
0.757 0 1,2λ
Eigenvalues
Synchronizing and damping torque coefficients
Dynamic Analysis of the Heffron-Phillips Model
Eigenvalues on imaginary axis system is critically stable
±
17 EEH – Power Systems Laboratory
Dynamic Analysis of the Heffron-Phillips Model SMIB including field circuit dynamics
Real Imaginary Damping Ratio f [Hz] – 0.109 6.411 0.0170 1.020 – 0.204 0 1.0
1,2λ
s Ksync Kdamp
– 0.0008 1.5333 – 0.7651 0
1,2λ
Eigenvalues
3λ
Synchronizing and damping torque coefficients due to field circuit
3λ
Eigenvalues moved to the left because field circuit adds damping torque
±
18 EEH – Power Systems Laboratory
Dynamic Analysis of the Heffron-Phillips Model SMIB including excitation system
Real Imaginary Damping Ratio f [Hz]
0.8837 10.7864 – 0.0816 1.7167 – 33.8342 0 1.0 0 –18.4567 0 1.0 0
1,2λ
s Ksync Kdamp
0.2731 -10.6038 – 19.8103 0 – 7.0126 0
Eigenvalues
3λ
Synchronizing and damping torque coefficients due to exciter
±
4λ
1,2λ
4λ3λ
19 EEH – Power Systems Laboratory
Dynamic Analysis of the Heffron-Phillips Model SMIB including excitation system
Eigenvalues moved to the right by the excitation system System is unstable!
Generator tripping
might eventually result in Blackout!
20 EEH – Power Systems Laboratory
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:
Washout filter: Suppress effect of low-frequency speed changes
Gain: Tuning parameter
for damping torque increase
Phase compensation: Provide phase-lead characteristic to compensate for lag between
exciter input and el. torque
21 EEH – Power Systems Laboratory
Power System Stabilizer Block diagram
22 EEH – Power Systems Laboratory
Power System Stabilizer Effect on the system dynamics
23 EEH – Power Systems Laboratory
Power System Stabilizer Effect on the system dynamics
24 EEH – Power Systems Laboratory
Real Imaginary Damping Ratio f [Hz] – 1.0052 6.6071 0.1504 1.0516 – 19.7970 12.8213 0.8394 2.0406 – 39.0969 0 - - – 0.7388 0 - -
1,2λ
s Ksync Kdamp
– 0.145 22.761 10.838 290.163
– 30.306 0 –1.072 0
Eigenvalues
3,4λ
Synchronizing and damping torque coefficients due to PSS
±
5λ±
6λ
1,2λ3,4λ
5λ
6λ
s Ksync Kdamp
0.21 – 8.69 – 1.27 – 13.00 1.16 0 0.30 0
Synchronizing and damping torque coefficients due to exciter
1,2λ3,4λ
5λ
6λ
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.
25 EEH – Power Systems Laboratory