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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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=

= ⋅

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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:

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:

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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−

=+

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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∆ = ∆ −∆+

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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∆ = ∆ −∆+

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Singel Machine Infinite Bus (SMIB)

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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∆ = ∆ −∆+

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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:

Field voltage equation: After linearization and some substitutions: with:

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Influence of torque angle on internal voltage

Heffron-Phillips Model … including the model of the terminal voltage magnitude:

Terminal voltage: Linearization and substitution: with

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Influence of torque angle on internal voltage

Fe−∆

Fe−∆

4Fe K δ∆ + ∆

Heffron-Phillips Model Full model:

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Influence of torque angle on internal voltage

Heffron-Phillips Model Simulink implementation

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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!

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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

±

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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

±

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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

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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!

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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:

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

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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

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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.

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