01_-_Stability_Fundamental-libre (1).pdf

1

Fundamentals of Power System Stability 1

Power System Stability

Seminar

DIgSILENT GmbH


General Definitions

2


• „Stability“ - general definition:

Ability of a system to return to a steady state after a disturbance.

• Small disturbance effects

• Large disturbance effects (nonlinear dynamics)

• Power System Stability - definition according to CIGRE/IEEE:

• Rotor angle stability (oscillatory, transient-stability)

• Voltage stability (short-term, long-term, dynamic)

• Frequency stability

Power System Stability


Frequency Stability

3


Ability of a power system to compensate for a power deficit:

1. Inertial reserve (network time constant)

� Lost power is compensated by the energy stored in rotating masses of all generators -> Frequency decreasing

2. Primary reserve:

� Lost power is compensated by an increase in production of primary controlled units. -> Frequency drop partly compensated

3. Secondary reserve:

� Lost power is compensated by secondary controlled units. Frequency and area exchange flows reestablished

4. Re-Dispatch of Generation

Frequency Stability


• Frequency disturbance following to an unbalance in active power

Frequency Deviation according to UCTE design criterion

-0,9

-0,8

-0,7

-0,6

-0,5

-0,4

-0,3

-0,2

-0,1

0

0,1

-10 0 10 20 30 40 50 60 70 80 90

dF in Hz

t in s

Rotor Inertia Dynamic Governor Action Steady State Deviation

Frequency Stability

4


• Mechanical Equation of each Generator:

• ∆P=ω∆T is power provided to the system by each generating unit.

• Assuming synchronism:

• Power shared according to generator inertia

nn

elmelm

PPPTTJ

ωωω

∆=

−≈−=�

j

i

j

i

ini

J

J

P

P

PJ

=∆

∆

∆=ωω �

Inertial Reserve


• Steady State Property of Speed Governors:

• Total frequency deviation:

• Multiple Generators:

• Power shared reciprocal to droop settings

( )∑

∑∆

=∆⇒∆=∆i

totitot

K

PffKP

i

j

j

i

jjii

R

R

P

P

PRPR

=∆

∆

∆=∆

PRPK

ffKP ii

i

ii ∆=∆=∆⇒∆=∆1

Primary Control

5


Turbine 1

Turbine 2

Turbine 3

Generator 1

Generator 2

Generator 3

Network

Secondary Control

PT PG

PT PG

PT PG

f PA

Set Value

Set Value

Set Value

Contribution

• Bringing Back Frequency

• Re-establishing area exchange flows

• Active power shared according to participation factors

Secondary Control


Frequency drop depends on:

• Primary Reserve

• Speed of primary control

• System inertia

Additionally to consider:

• Frequency dependence of load

Frequency Stability

6


• Dynamic Simulations

• Steady state analysis sometimes possible (e.g. generators remain

in synchronism):

• Inertial/Primary controlled load flow calculation

- Frequency deviation

• Secondary controlled load flow calculation

- Generation redispatch

Frequency Stability - Analysis


20.0015.0010.005.000.00 [s]

1.025

1.000

0.975

0.950

0.925

0.900

0.875

G 1: Turbine Power in p.u.

G2: Turbine Power in p.u.

G3: Turbine Power in p.u.

20.0015.0010.005.000.00 [s]

0.125

0.000

-0.125

-0.250

-0.375

-0.500

-0.625

Bus 7: Deviation of the El. Frequency in Hz

DIgSILENT Nine-bus system Mechanical

Sudden Load Increase

Date: 11/10/2004

Annex: 3-cycle-f. /3

DIg

SIL

EN

T

Frequency Stability

7


Frequency Stability - Analysis

Frequency stability improved by:

Increase of Primary

Reserve and System

Inertia

-Dispatching more generators

-Interruptible loads

-Power Frequency controllers of HVDC links

Improvement of

Primary Control action

-Tuning / replacing of governor controls.

Automatic Load

shedding

-Under-Frequency Load Shedding relays

adjusted according to system-wide criteria.


Frequency Stability

Typical methods to improve frequency stability:

- Increase of spinning reserve and system inertia (dispatching more generators)

- Power-Frequency controllers on HVDC links

- Tuning / Replacing governor systems

- Under-Frequency load shedding relays adjusted according to system-wide criteria

- Interruptible loads

8


Rotor Angle Stability


Two distinctive types of rotor angle stability:

- Small signal rotor angle stability (Oscillatory stability)

- Large signal rotor angle stability (Transient stability)

Rotor Angle Stability

9


Small signal rotor angle stability (Oscillatory stability)

Ability of a power system to maintain synchronism under small disturbances

– Damping torque

– Synchronizing torque

Especially the following oscillatory phenomena are a concern:

– Local modes

– Inter-area modes

– Control modes

– (Torsional modes)

Oscillatory Stability


Small signal rotor angle stability is a system property

Small disturbance -> analysis using linearization around operating

point

Analysis using eigenvalues and eigenvectors


10



Typical methods to improve oscillatory stability:

- Power System Stabilizers

- Supplementary control of Static Var Compensators

- Supplementary control of HVDC links

- Reduction of transmission system impedance ( for inter-area oscillations, by addition of lines, series capacitors, etc.)


Large signal rotor angle stability (Transient stability)

Ability of a power system to maintain synchronism during severe

disturbances

– Critical fault clearing time

Large signal stability depends on system properties and the type

of disturbance (not only a system property)

– Analysis using time domain simulations

Transient Stability

11


3.2342.5871.9401.2940.650.00 [s]

200.00

100.00

0.00

-100.00

-200.00

G1: Rotor angle with reference to reference machine angle in deg

DIgSILENT Transient Stability Subplot/Diagramm

Date: 11/11/2004

Annex: 1 /3

DIg

SIL

EN

T

4.9903.9922.9941.9961.000.00 [s]

25.00

12.50

0.00

-12.50

-25.00

-37.50

G1: Rotor angle with reference to reference machine angle in deg

DIgSILENT Transient Stability Subplot/Diagramm

Date: 11/11/2004

Annex: 1 /3

DIg

SIL

EN

T

Transient Stability


Transient Stability

Typical methods to improve transient stability:

- Reduction of transmission system impedance (additional lines, series capacitors, etc.).

- High speed fault clearing.

- Single-pole breaker action.

- Voltage control ( SVS, reactor switching, etc.).

- Improved excitation systems ( high speed systems, transient excitation boosters, etc.).

- Remote generator and load tripping.

- Controls on HVDC transmission links.

12


Voltage Stability


Voltage stability refers to the ability of a power system to

maintain steady voltages at all buses in the system after being

subjected to a disturbance.

• Small disturbance voltage stability (Steady state stability)

– Ability to maintain steady voltages when subjected to small

disturbances

• Large disturbance voltage stability (Dynamic voltage stability)

– Ability to maintain steady voltages after following large disturbances

Voltage Stability

13


Small-Signal:

- Small disturbance

Large-Signal

- System fault

- Loss of generation

Long-Term - P-V-Curves (load flows)

- dv/dQ-Sensitivities

- Long-term dynamic models

including tap-changers, var-

control, excitation limiters, etc.

- P-V-Curves (load flows)

of the faulted state.

- Long-term dynamic models

including tap-changers, var-

control, excitation limiters, etc.

Short-Term - Dynamic models (short-term),

special importance on dynamic

load modeling, stall effects etc.

Voltage Stability - Analysis


Long-Term vs. Short-Term Voltage Stability

Reactive power control:

Short-Term Long-Term

Q- contribution of

synchronous gen.

Large (thermal overload

capabilities)

Limited by

overexcitation limitors

Switchable shunts No contribution

(switching times too

high)

High contribution

SVC/TSC High contribution High contribution

14


Voltage Stability

Outage of large generator

��


20.0015.0010.005.000.00 [s]

1.25

1.00

0.75

0.50

0.25

0.00

-0.25

APPLE_20: Voltage, Magnitude in p.u.

SUMMERTON_20: Voltage, Magnitude in p.u.

LILLI_20: Voltage, Magnitude in p.u.

BUFF_330: Voltage, Magnitude in p.u.

DIg

SIL

EN

T

Fault with loss of transmission line

Large-Signal Long-TermVoltage Instability

15


Voltage Stability – Q-V-Curves

1762.641462.641162.64862.64562.64262.64

1.40

1.20

1.00

0.80

0.60

0.40

x-Achse: SC: Blindleistung in Mvar

SC: Voltage in p.u., P=1400MW




P=2000MW

P=1800MW

P=1600MW

P=1400MW

DIg

SIL

EN

T

const. P, variable Q


• Dynamic voltage stability problems are resulting from sudden increase in reactive power demand of induction machine loads.

-> Consequences: Undervoltage trip of one or several machines, dynamic voltage collapse

• Small synchronous generators consume increased amount of reactive power after a heavy disturbance -> voltage recovery problems.

-> Consequences: Slow voltage recovery can lead to undervoltage trips of own supply -> loss of generation

Dynamic Voltage Stability

16


1.201.161.121.081.041.00

3.00

2.00

1.00

0.00

-1.00

x-Axis: GWT: Speed in p.u.

GWT: Electrical Torque in p.u.

1.201.161.121.081.041.00

0.00

-2.00

-4.00

-6.00

-8.00


GWT: Reactive Power in Mvar

DIg

SIL

EN

T

Dynamic Voltage Stability –Induction Generator (Motor)


1.041.031.021.011.00

3.00

2.00

1.00

0.00

-1.00


GWT: Electrical Torque in p.u.

Constant Y = 1.000 p.u. 1.008 p.u.

1.041.031.021.011.00

0.00

-1.00

-2.00

-3.00

-4.00

-5.00

-6.00


GWT: Reactive Power in Mvar

Constant X = 1.008 p.u.

-1.044 Mvar

DIg

SIL

EN

T


17


2.001.501.000.500.00 [s]

1.20

1.00

0.80

0.60

0.40

0.20

0.00

G\HV: Voltage, Magnitude in p.u.

MV: Voltage, Magnitude in p.u.

2.001.501.000.500.00 [s]

80.00

40.00

0.00

-40.00

-80.00

-120.00

Cub_0.1\PQ PCC: Active Power in p.u.

Cub_0.1\PQ PCC: Reactive Power in p.u.

2.001.501.000.500.00 [s]

1.06

1.04

1.02

1.00

0.98

GWT: Speed

DIg

SIL

EN

T



3.002.001.000.00 [s]

60.00

40.00

20.00

0.00

-20.00

-40.00

Cub_0.1\PQ RedSunset: Active Power in p.u.

Cub_0.1\PQ RedSunset: Reactive Power in p.u.

3.002.001.000.00 [s]

60.00

40.00

20.00

0.00

-20.00

-40.00

Cub_0.2\PQ BlueMountain: Active Power in p.u.

Cub_0.2\PQ BlueMountain: Reactive Power in p.u.

3.002.001.000.00 [s]

60.00

40.00

20.00

0.00

-20.00

-40.00

-60.00

Cub_1.1\PQ GreenField: Active Power in p.u.

Cub_1.1\PQ GreenField: Reactive Power in p.u.

3.002.001.000.00 [s]

1.125

1.000

0.875

0.750

0.625

0.500

0.375

GLE\1: Voltage, Magnitude in p.u.

GLZ\2: Voltage, Magnitude in p.u.

WDH\1: Voltage, Magnitude in p.u.

DIg

SIL

EN

T

Dynamic Voltage Collapse

18


3.002.001.000.00 [s]

1.20

1.00

0.80

0.60

0.40

0.20

0.00

HV: Voltage, Magnitude in p.u.

MV: Voltage, Magnitude in p.u.

3.002.001.000.00 [s]

120.00

80.00

40.00

0.00

-40.00

-80.00

-120.00

Cub_1\PCC PQ: Active Power in p.u.

Cub_1\PCC PQ: Reactive Power in p.u.

DIg

SIL

EN

T

Dynamic Voltage Stability –Voltage Recovery (Synchronous Generators)


Time-domain Analysis

19


Fast Transients/Network Transients:

Time frame: 10 mys…..500ms

� Lightening

� Switching Overvoltages

� Transformer Inrush/Ferro Resonance

� Decaying DC-Components of short circuit currents

Transients in Power Systems


Medium Term Transients / Electromechanical Transients

Time frame: 400ms….10s

� Transient Stability

� Critical Fault Clearing Time

� AVR and PSS

� Turbine and governor

� Motor starting

� Load Shedding


20


Long Term Transients / Dynamic Phenomena

Time Frame: 10s….several min

� Dynamic Stability

� Turbine and governor

� Power-Frequency Control

� Secondary Voltage Control

� Long Term Behavior of Power Stations



Stability/EMT

Different Network Models used:

Stability:

EMT:

ILjV ω= VCjI ω=

dt

diLv =

dt

dvCi =

21


Short Circuit Current EMT

0.50 0.38 0.25 0.12 0.00 [s]

800.0

600.0

400.0

200.0

0.00

-200.0

4x555 MVA: Phase Current B in kA

Short Circuit Current with complete model (EMT-model) Plots

Date: 4/25/2001

Annex: 1 /1

DIg

SIL

EN

T


Short Circuit Current RMS

0.50 0.38 0.25 0.12 0.00 [s]

300.0

250.0

200.0

150.0

100.0

50.00

0.00

4x555 MVA: Current, Magnitude in kA

Short Circuit Current with reduced model (Stability model) Plots

Date: 4/25/2001

Annex: 1 /1

DIg

SIL

EN

T

22


Phenomena RMS-Simulation EMT-Simulation

Critical fault clearing time X (X)

Dynamic motor startup

Peak shaft-torque

X

0

(X)

X

Torsional oscillations

Subsynchronous resonance

X

0

X

X

Dynamic voltage stability

Self excitation of ASM

X

0

(X)

X

Oscillatory stability X ((X))

AVR and PSS dynamics X (X)

Transformer/Motor inrush 0 X

HVDC dynamics (X) X

Switching Over Voltages 0 X

RMS-EMT-Simulation


Frequency-domain analysis

23


Small signal stability analysis

• Small signal stability is the ability of the power system to maintain

synchronism when subjected to small disturbances.

• Disturbance is considered to be small when equation describing the response

can be linearized.

• Instability may result as: steady increase in rotor angle (lack of synchronizing

torque) or rotor oscillations of increasing amplitude (lack of damping torque)



• Linear model generated numerically by Power Factory.

• Calculation of eigenvalues, eigenvectors and participation factors

• Calculation of all modes using QR-algorithm -> limited to systems up to

500..1000 state variables

• Calculation of selected modes using implicitly restarted Arnoldi method ->

application to large systems

24



• Linear System Representation:

• Transformation:

• Transformed System

• Diagonal System

bAxx +=�

xTx ~=

TbxTATx += − ~~ 1�

TbxDx += ~~�



• State Space Representation:

• State of a system is the minimum information at any instant necessary

to determine its future behaviour. The linearly independent variables

describing the state of the system are called state variables x.

• Output variables:

• Initial Equilibrium :

• Perturbation:

),...,,;,...,,( 2121 rnii uuuxxxfx =�

),...,,;,...,( 2121 rnii uuuxxxgy =

iii

iii

iii

xxx

uuu

xxx

�� ∆+=

∆+=

∆+=

0

0

0

0),...,,;,...,,( 02010020100== rnii uuuxxxfx�

25



• As perturbations are small, the nonlinear functions f and g can

be expanded using the Taylor series:

• Using Vector-Matrix notation:

r

r

jj

n

n

jj

rnjj

r

r

iin

n

iirnii

uu

gu

u

gx

x

gx

x

guuuxxxgy

uu

fu

u

fx

x

fx

x

fuuuxxxfx

∆∂

∂++∆

∂

∂+∆

∂

∂++∆

∂

∂+=

∆∂

∂++∆

∂

∂+∆

∂

∂++∆

∂

∂+=

......),...,;,...,,(

......),...,,;,...,,(

1

1

1

1

0201002010

1

1

1

1

0201002010�

]][[]][[][

]][[]][[][

uDxCy

uBxAx

∆+∆=∆

∆+∆=∆�



• Taking the Laplace transform of the previous equations:

• Block Diagram of the state-space representation:

)](][[)](][[)]([

)](][[)](][[)]0([)]([

suDsxCsy

suBsxAxsxs

∆+∆=∆

∆+∆=∆−∆

26



• Poles of [∆x(s)] and [∆y(s)] are the root of the characteristic equation of matrix

[A]:

• Values of s which satisfy above equation are the eigenvalues of [A]

• Real eigenvalues correspond to non oscillatory modes. Negative real

eigenvalues represent decaying modes.

• Complex eigenvalues occur in conjugate pairs. Each pair correspond to an

oscillatory mode.

0])[][det( =− AIs



• An oscillatory system mode is given by a pair of eigenvalues

• The real component σ gives the damping. A negative real part represents a

damped (decreasing) oscillation.

• The imaginary component ω gives the frequency of the oscillation in rad/s.

• The damping ratio ζ determine the rate of decay of the amplitude of the

oscillation and is given by:

ωσλ j±=

22 ωσ

σζ

+

−=

27


-0.8000-1.6000-2.4000-3.2000-4.0000 Neg. Damping [1/s]

3.5000

2.9000

2.3000

1.7000

1.1000

0.5000

Damped Frequen

Stable EigenvaluesUnstable Eigenvalues

Y = 1.500 Hz

Y = 2.000 Hz

Y = 3.000 Hz

-0.8000-1.6000-2.4000-3.2000-4.0000 Neg. Damping [1/s]

3.5000

2.9000

2.3000

1.7000

1.1000

0.5000

Damped Frequen

Stable EigenvaluesUnstable Eigenvalues

Y = 0.800 Hz

DIg

SIL

EN

T

Eigenvalue Analysis without and with PSS

Without PSS

With PSS


Voltage Stability

Fundamental Concepts

28


0E

eQX

'

GE

( )

( )( )GGG

e

GG

e

EEX

EQ

X

EEP

ϕ

ϕ

cos

sin

0

''

'

0

−=

=

Voltage Stability


Voltage stability: basic concepts

( ) ( )2 2

s

LN LD LN LD

EI

Z cos Z cos Z sin Z sinθ φ θ φ=

+ + +

1 s

LN

EI

ZF= ( )

2

1 2LD LD

LN LN

Z ZF cos

Z Zθ φ

= + + ⋅ ⋅ −

2

R LD

sLDR R

LN

V Z I

EZP V I cos cos

F Zφ φ

= ⋅

= =

con

29


Voltage stability: basic concepts

Voltage collapse depends on the load characteristics


Study case: Tap changer

30


1762.641462.641162.64862.64562.64262.64

1.40

1.20

1.00

0.80

0.60

0.40

x-Achse: SC: Blindleistung in Mvar





P=2000MW

P=1800MW

P=1600MW

P=1400MW

DIg

SIL

EN

T

const. P, variable Q

Voltage Stability – Q-V-Curves


1350.001100.00850.00600.00350.00100.00

1.00

0.90

0.80

0.70

0.60

0.50

x-Achse: U_P-Curve: Total Load of selected loads in MW

Klemmleiste(1): Voltage in p.u., pf=1

Klemmleiste(1): Voltage in p.u., pf=0.95

Klemmleiste(1): Voltage in p.u., pf=0.9

pf=1

pf=0.95

pf=0.9

DIg

SIL

EN

T

const. Power factor, variable P

Voltage Stability – P-V-Curves

31

Fundam

enta

ls of P

ow

er S

yste

m S

tability

61

Ro

tor A

ng

le S

tab

ility

Fundam

enta

ls of P

ow

er S

yste

m S

tability

62

On

e M

ach

ine S

yste

m

DIg

SIL

EN

T

Pow

erF

acto

ry 12

.1.1

78

Exa

mp

le

Pow

er S

yste

m S

tab

ility an

d C

on

trol

On

e M

ach

ine P

rob

lem

Pro

ject: T

rain

ing

Gra

ph

ic: G

rid

Date

: 4/1

9/2

00

2

An

nex: 1

G~

G1Gen 2220MVA/24kV(1)

1998.000 MW967.920 Mvar

53.408 kA1.163 p.u.-0.000 p.u.

Trf

50

0kV

/24

kV

/22

20

MV

A

-1998.00 MW-634.89 Mvar

2.56 kA

1998.00 MW967.92 Mvar

53.41 kA

CC

T 2

Type

CC

T1

86

.00

km

-698.60 MW30.44 Mvar

0.90 kA

698.60 MW221.99 Mvar

0.90 kA

CC

T1

Type

CC

T1

00

.00

km

-1299.40 MW56.62 Mvar

1.67 kA

1299.40 MW412.90 Mvar

1.67 kA

V~

Infinite Source

-1998.00 MW87.07 Mvar

2.56 kA

Infinite Bus500.00 kV 450.41 kV

0.90 p.u.0.00 deg

HT500.00 kV 472.15 kV

0.94 p.u.20.12 deg

LT24.00 kV 24.00 kV

1.00 p.u.28.34 deg

DIgSILENT

32


One Machine System

0E

ePX

'

GE

Equivalent circuit, transferred power:


One Machine System

• Power transmission over reactance:

• Mechanical Equations:

0

0

ωωϕ

ωωω

−=

−≈

−=

G

emem PPPPJ

�

�

( )

( )( )GG

Ge

GG

e

EEX

EQ

X

EEP

ϕ

ϕ

cos

sin

0

''

'

0

−=

=

33


One Machine System

• Differential Equation of a one-machine infinite bus bar system:

• Eigenvalues (Characteristic Frequency):

• Stable Equilibrium points (SEP) exist for:

GGG

m

G

m

G

PPPPPJ ϕϕ

ωϕ

ωωϕ

ωωϕ ∆

−−≈−=

0

0

max

0

0

max

00

max

0

cossinsin��

0

0

max2/1 cos G

J

Pϕ

ωλ −±=

0cos 0 >Gϕ


One-machine System

180.0144.0108.072.0036.00 0.00

4000.

3000.

2000.

1000.

0.00

-1000...

x-Axis: Plot Power Curve: Generator Angle in deg

Plot Power Curve: Power 1 in MW


Pini y=1998.000 MW

DIgSILENT Single Machine Problem P-phi

Date: 4/19/2002

Annex: 1 /4

DIg

SIL

EN

T

SEP UEP

stable unstable

34


Large disturbances (Transient Stability)

• Energy Function:

• At Maximum Angle:

( ) 0)(

2

1

0

2=+=

−+ ∫ potkin

emG EEd

PPJ

G

ϕω

ϕϕ

ϕ

�

0max

=G

ϕ�

0)(max

0

=−

= ∫ ϕω

ϕ

ϕ

dPP

EG

em

pot

( )0=kin

E


Large disturbances : Equal Area Criterion

180.0144.0108.072.0036.000.00

4000.

3000.

2000.

1000.

0.00

-1000...

x-Axis: Plot Power Curve: Generator Angle in deg



DIgSILENT Single Machine Problem P-phi Date: 4/19/2002

Annex: 1 /4

DIg

SIL

EN

T

E1

E2

0ϕ cϕ

maxϕ

SEP UEP

critϕ

Pm

35


Large disturbances: Equal Area Criterion

21 EE −=

∫=c

dPE m

ϕ

ϕ

ϕω

0

11

( )∫ −=max

)sin(1

max2

ϕ

ϕ

ϕϕω

c

dPPE m

Stable operation if:


Large disturbances: Equal Area Criterion

)(1

01 ϕϕω

−= cmPE

)cos(cos)( maxmax

max2 ccm PP

E ϕϕω

ϕϕω

−+−=

000 cossin)2(cos ϕϕϕπϕ −−=c

Setting and equating E1 and -E2:0ϕπϕ −=crit

36


Large-disturbances: Critical Fault Clearing Time

• During Short Circuit:

• Differential Equation:

• Critical Fault Clearing Time:

0

2

02ϕ

ωϕ += c

mc t

J

P

0=eP

0ωϕ m

G

PJ =��


Small disturbances (Oscillatory Stability)

G~

Ge

ne

rato

r X

V ~

Infin

ite

bu

s

Assumptions:

1. Constant excitation

2. Constant damping from synchronous machine, Ke

3. Simplified generator model, Pe = Te (in per unit)

4. Constant mechanical torque

'

gE oE

37


Small disturbances

( )

( ) δδδδ

δ

∆⋅=∆∂

∂=∆

==

o

o

ee

g

ee

PT

T

X

EETP

cos

sin

max

'

0Equation of electrical circuit…

Equation of motion…

0)(2

)(2

)(2

)(

2

2

2

=∆+∆++∆

∆++∆=∆−∆

++=−

++=−

eem

emem

emem

emem

TKKsHs

KKsHsTT

KKsHsTT

KKJTT

δδ

δδ

δδ

δδ ��

Combined… 0cos22

max2 =+

++ o

em

H

P

H

KKss δ

H

P on

2

cosmax δω =


Small disturbances:Structure of linearised generator model

δ∆*K0

eT∆

ω∆*K

• Damping torque: a torque in phase with

• Synchronising torque: a torque in phase with

ω∆

δ∆

Exciter Generator Shafts

1

mT∆

−∆ eT ω∆ δ∆

tu∆

0=∆refu Exciter Generator Shaft

s

1

mT∆

ω∆ δ∆

tu∆

0=∆refu

38


Linear model of generator + AVR + PSS

PSSu∆


1

ω∆

δ∆

tu∆

PSSu∆


1

0=∆ mT

−∆ eT

ω∆

δ∆

tu∆

PSS

oPSSeTangleWant 0

)(=

∆

∆

ω

Phase lag

Phase lead compensation

01_-_Stability_Fundamental-libre (1).pdf

Documents