Power System Stability, Training Course

1

Fundamentals of Power System Stability 1

Power System StabilityTraining Course

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 ofall 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=T is power provided to the system by each generating unit.• Assuming synchronism:

• Power shared according to generator inertia

nn

elmelm

PPPTTJ

j

i

j

i

ini

JJ

PP

PJ

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

PP

PRPR

PRPK

ffKP iii

ii 1

Primary Control

5


Turbine 1

Turbine 2

Turbine 3

Generator 1

Generator 2

Generator 3

Network

SecondaryControl

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 remainin 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

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Frequency Stability

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Frequency Stability - Analysis

Frequency stability improved by:

-Under-Frequency Load Shedding relaysadjusted according to system-wide criteria.

Automatic Loadshedding

-Tuning / replacing of governor controls.Improvement ofPrimary Control action

-Dispatching more generators-Interruptible loads-Power Frequency controllers of HVDC links

Increase of PrimaryReserve and SystemInertia


Frequency Stability

Typical methods to improve frequency stability:

- Increase of spinning reserve and system inertia (dispatching moregenerators)

- 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)


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 operatingpoint

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-areaoscillations, 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 typeof 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

DIgSILENTTransient Stability Subplot/Diagramm Date: 11/11/2004

Annex: 1 /3

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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

DIgSILENTTransient Stability Subplot/Diagramm Date: 11/11/2004

Annex: 1 /3

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Transient Stability


Transient Stability

Typical methods to improve transient stability:

- Reduction of transmission system impedance (additional lines, seriescapacitors, 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 tomaintain steady voltages at all buses in the system after beingsubjected 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


- Dynamic models (short-term),special importance on dynamicload modeling, stall effects etc.

Short-Term

- P-V-Curves (load flows)of the faulted state.- Long-term dynamic modelsincluding tap-changers, var-control, excitation limiters, etc.

- P-V-Curves (load flows)- dv/dQ-Sensitivities- Long-term dynamic modelsincluding tap-changers, var-control, excitation limiters, etc.

Long-Term

Large-Signal- System fault- Loss of generation

Small-Signal:- Small disturbance

Voltage Stability - Analysis


Long-Term vs. Short-Term Voltage Stability

Reactive power control:

High contributionHigh contributionSVC/TSC

High contributionNo contribution(switching times toohigh)

Switchable shunts

Limited byoverexcitation limitors

Large (thermal overloadcapabilities)

Q- contribution ofsynchronous gen.

Long-TermShort-Term

14


Voltage Stability

Outage of large generator

All generators in service


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.

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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 MvarSC: Voltage in p.u., P=1400MWSC: Voltage in p.u., P=1600MWSC: Voltage in p.u., P=1800MWSC: Voltage in p.u., P=2000MW

P=2000MW

P=1800MW

P=1600MW

P=1400MW

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const. P, variable Q


• Dynamic voltage stability problems are resulting from suddenincrease 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 ofreactive power after a heavy disturbance -> voltage recoveryproblems.

-> Consequences: Slow voltage recovery can lead to undervoltagetrips 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

x-Axis: GWT: Speed in p.u.GWT: Reactive Power in Mvar

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Dynamic Voltage Stability –Induction Generator (Motor)


1.041.031.021.011.00

3.00

2.00

1.00

0.00

-1.00

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

x-Axis: GWT: Speed in p.u.GWT: Reactive Power in Mvar

Constant X = 1.008 p.u.

-1.044 Mvar

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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

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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.

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Dynamic Voltage Collapse

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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.

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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 TransientsTime frame: 400ms….10s

Transient Stability Critical Fault Clearing Time AVR and PSS Turbine and governor Motor starting Load Shedding


20


Long Term Transients / Dynamic PhenomenaTime 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 VCjI

dtdi

Lv dtdv

Ci

21


Short Circuit Current EMT

0.500.380.250.120.00 [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

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Short Circuit Current RMS

0.500.380.250.120.00 [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

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(X)X

X0

Dynamic voltage stabilitySelf excitation of ASM

X(X)HVDC dynamics

X0Switching Over Voltages

X0Transformer/Motor inrush

(X)XAVR and PSS dynamics

((X))XOscillatory stability

XX

X0

Torsional oscillationsSubsynchronous resonance

(X)X

X0

Dynamic motor startupPeak shaft-torque

(X)XCritical fault clearing time

EMT-SimulationRMS-SimulationPhenomena

RMS-EMT-Simulation


Frequency-domain analysis

23


Small signal stability analysis

• Small signal stability is the ability of the power system to maintainsynchronism when subjected to small disturbances.

• Disturbance is considered to be small when equation describing the responsecan be linearized.

• Instability may result as: steady increase in rotor angle (lack of synchronizingtorque) 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 to500..1000 state variables

• Calculation of selected modes using implicitly restarted Arnoldi method ->application to large systems

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• 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 necessaryto determine its future behaviour. The linearly independent variablesdescribing the state of the system are called state variables x.

• Output variables:

• Initial Equilibrium :

• Perturbation:

),...,,;,...,,( 2121 rnii uuuxxxfx

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

iii

iii

iii

xxxuuuxxx

0

0

0

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

25



• As perturbations are small, the nonlinear functions f and g canbe expanded using the Taylor series:

• Using Vector-Matrix notation:

rr

jjn

n

jjrnjj

rr

iin

n

iirnii

uu

gu

u

gx

x

gx

x

guuuxxxgy

uuf

uuf

xxf

xxf

uuuxxxfx

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

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

11

11

0201002010

11

11

0201002010

]][[]][[][]][[]][[][

uDxCyuBxAx



• Taking the Laplace transform of the previous equations:

• Block Diagram of the state-space representation:

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

suDsxCsysuBsxAxsxs

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 realeigenvalues represent decaying modes.

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

0])[][det( AIs



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

• The real component gives the damping. A negative real part represents adamped (decreasing) oscillation.

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

• The damping ratio determine the rate of decay of the amplitude of theoscillation and is given by:

j

22

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-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 Frequency [Hz]

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 Frequency [Hz]

Stable EigenvaluesUnstable Eigenvalues

Y = 0.800 Hz

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Eigenvalue Analysis without and with PSS

Without PSS

With PSS


Voltage Stability

Fundamental Concepts

28


0E

eQX

'GE

GGG

e

GG

e

EEXE

Q

XEE

P

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 sin

1 s

LN

EI

ZF

2

1 2LD LD

LN LN

Z ZF cos

Z Z

2

R LD

sLDR R

LN

V Z I

EZP V I cos cos

F Z

con

29


Voltage stability: basic concepts

Voltage collapse depends on the load characteristics


Study case: Tap changer

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1762.641462.641162.64862.64562.64262.64

1.40

1.20

1.00

0.80

0.60

0.40

x-Achse: SC: Blindleistung in MvarSC: Voltage in p.u., P=1400MWSC: Voltage in p.u., P=1600MWSC: Voltage in p.u., P=1800MWSC: Voltage in p.u., P=2000MW

P=2000MW

P=1800MW

P=1600MW

P=1400MW

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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 MWKlemmleiste(1): Voltage in p.u., pf=1Klemmleiste(1): Voltage in p.u., pf=0.95Klemmleiste(1): Voltage in p.u., pf=0.9

pf=1

pf=0.95

pf=0.9

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const. Power factor, variable P

Voltage Stability – P-V-Curves

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One Machine System

DIgSILENT

PowerFactory 12.1.178

Example

Power System Stability and ControlOne Machine Problem

Project: Training

Graphic: GridDate: 4/19/2002

Annex: 1

G ~ G1

Gen

2220

MV

A/2

4kV

(1)

1998

.000

MW

967.

920

Mva

r53

.408

kA1.

163

p.u.

-0.0

00p.

u.

Trf500kV/24kV/2220MVA

-199

8.00

MW

-634

.89

Mva

r2.

56kA

1998

.00

MW

967.

92M

var

53.4

1kA

CCT 2Type CCT186.00 km

-698

.60

MW

30.4

4M

var

0.90

kA

698.

60M

W22

1.99

Mva

r0.

90kA

CCT1Type CCT100.00 km

-129

9.40

MW

56.6

2M

var

1.67

kA

1299

.40

MW

412.

90M

var

1.67

kA

V ~

Infin

iteS

ourc

e

-199

8.00

MW

87.0

7M

var

2.56

kA

Infin

iteB

us50

0.00

kV45

0.41

kV0.

90p.

u.0.

00de

g

HT

500.

00kV

472.

15kV

0.94

p.u.

20.1

2de

g

LT24

.00

kV24

.00

kV1.

00p.

u.28

.34

deg

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One Machine System

0E

ePX

'GE

Equivalent circuit, transferred power:


One Machine System

• Power transmission over reactance:

• Mechanical Equations:

0

0

G

emem PPPPJ

GGG

e

GG

e

EEXE

Q

XEE

P

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:

GGGm

Gm

G

PPPPPJ

0

0

max0

0

max

00

max

0

cossinsin

00

max2/1 cos GJ

P

0cos 0 G


One-machine System

180.0144.0108.072.0036.000.00

4000.

3000.

2000.

1000.

0.00

-1000...

x-Axis: Plot Power Curve: Generator Angle in degPlot Power Curve: Power 1 in MWPlot Power Curve: Power 2 in MW

Pini y=1998.000 MW

DIgSILENTSingle Machine Problem P-phi Date: 4/19/2002

Annex: 1 /4

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SEP UEP

stable unstable

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Large disturbances (Transient Stability)

• Energy Function:

• At Maximum Angle:

0)(

21

0

2

potkinem

G EEdPP

JG

0max G

0)(max

0

dPP

EG

empot

0kinE


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 degPlot Power Curve: Power 1 in MWPlot Power Curve: Power 2 in MW

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E1

E2

0 c

max

SEP UEP

critPm

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Large disturbances: Equal Area Criterion

21 EE

c

dPE m

0

11

max

)sin(1

max2

c

dPPE m

Stable operation if:


Large disturbances: Equal Area Criterion

)(1

01

cmPE

)cos(cos)( maxmax

max2 ccm PP

E

000 cossin)2(cos c

Setting and equating E1 and -E2:0 crit

36


Large-disturbances: Critical Fault Clearing Time

• During Short Circuit:

• Differential Equation:

• Critical Fault Clearing Time:

02

02

c

mc t

JP

0eP

0 m

GP

J


Small disturbances (Oscillatory Stability)

G~G

ener

ator X

V ~In

finite

bus

Assumptions:1. Constant excitation2. Constant damping from synchronous machine, Ke3. Simplified generator model, Pe = Te (in per unit)4. Constant mechanical torque

'gE oE

37


Small disturbances

oo

ee

gee

PT

T

X

EETP

cos

sin

max

'0

Equation of electrical circuit…

Equation of motion…

0)(2

)(2

)(2

)(

2

2

2

eem

emem

emem

emem

TKKsHs

KKsHsTT

KKsHsTT

KKJTT

Combined… 0cos22

max2

oem

HP

HKK

ss

HP o

n 2cosmax


Small disturbances:Structure of linearised generator model

*K0

eT

*K

• Damping torque: a torque in phase with• Synchronising torque: a torque in phase with

Exciter Generator Shafts1

mT

eT

tu

0 refu Exciter Generator Shafts1

mT

tu

0 refu

38


Linear model of generator + AVR + PSS

PSSu


tu

PSSu


0 mT

eT

tu

PSS

oPSSeTangleWant 0)(

Phase lag

Phase lead compensation

Power System Stability, Training Course

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Transcript of Power System Stability, Training Course