Power System Stability, Training Course

Power System Stability Training Course DIgSILENT GmbH

Fundamentals of Power System Stability

1

General Definitions


2

1

Power System Stabili Stability ty •

„Stability“ „Stab ility“ - genera generall defin definition: ition:

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

• •

Small distu disturbanc rbance e effec effects ts Large distur disturbanc bance e effects effects (nonline (nonlinear ar dynamic dynamics) s)

• • • •

Power System System Stability - definition according to CIGRE/IEEE CIGRE/IEEE:: Roto Ro torr an angl gle e stability (oscillatory, transient-stability) Vol olta tag ge stability (short-term, long-term, dynamic) Freq Fr eque uenc ncy y stability


3

Frequency Stability


4

2

Power System Stabili Stability ty •

„Stability“ „Stab ility“ - genera generall defin definition: ition:

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

• •

Small distu disturbanc rbance e effec effects ts Large distur disturbanc bance e effects effects (nonline (nonlinear ar dynamic dynamics) s)

• • • •

Power System System Stability - definition according to CIGRE/IEEE CIGRE/IEEE:: Roto Ro torr an angl gle e stability (oscillatory, transient-stability) Vol olta tag ge stability (short-term, long-term, dynamic) Freq Fr eque uenc ncy y stability


3

Frequency Stability


4

2

Frequency Stability

Ability of a power system to compensate for a power deficit: 1. Inert Inertial ial reserve reserve (network (network time time constant) constant)  Lost power is compensated by the energy stored in rotating masses masses of all generators -> Frequency decreasing 2. Primary Primary reser reserve: ve:  Lost power is compensated by an increase in production production of primary controlled units. -> Frequency drop partly compensated 3. Secondary Secondary reserv reserve: e:  Lost power is compensated by secondary controlled controlled units. Frequency and area exchange flows reestablished 4. Re-D Re-Dispatc ispatch h of Gener Generation ation


5

Frequency Stability

•

Frequency Freque ncy disturb disturbance ance follow following ing to an unbalance unbalance in active active power power Frequency Frequency Deviation according t o UCTE design criterion 0,1 t in s

0 -10

-0,1

0

10

20

30

40

50

60

70

80

90

-0,2 -0,3 -0,4 -0,5 -0,6 -0,7 -0,8 -0,9

dF in Hz

Rotor Inertia


Dynamic Governor Action

Steady State Deviation

6

3

Inertial Reserve

•

Mechanical Equation of each Generator:

  T m  T el  J  • •

P m  P el

 n



 P  n

P=T is power provided to the system by each generating unit. Assuming synchronism:

   P i J i n

 P i J i   P j J j •

Power shared according to generator inertia


7

Primary Control

•

Steady State Property of Speed Governors:

 P i  K i  f   f  •

Total frequency deviation:

1 K i

 P i  Ri  P

 P tot   K i  f   f 

 P tot

 K

i

•

Multiple Generators:

Ri  P i  R j  P j

 P i R j   P j Ri •

Power shared reciprocal to droop settings


8

4

Secondary Control

Set Value

Turbine 1

Set Value

Turbine 2

Set Value

Turbine 3

PT

Generator 1

PT

Generator 2

PT

Generator 3

PG

PG

Network

PG

f Contribution

P A

Secondary Control

•

Bringing Back Frequency

•

Re-establishing area exchange flows

•

Active power shared according to participation factors


9

Frequency Stability

Frequency drop depends on: •

Primary Reserve

•

Speed of primary control

•

System inertia

Additionally to consider: •

Frequency dependence of load


10

5

Frequency Stability - Analysis

•

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


11

Frequency Stability T N E L I S g I D

1.025 1.000 0.975 0.950 0.925 0.900 0.875 0.00

5.00

10.00

15.00

[s]

20.00

10.00

15.00

[s]

20.00

G 1: Turbine Power in p.u. G2: Turbine Power in p.u. G3: Turbine Power in p.u. 0.125

0.000

-0.125

-0.250

-0.375

-0.500

-0.625 0.00

DIgSILENT

5.00 Bus 7: Deviation of the El. Frequency in Hz

Nine-bus system Sudden Load Increase


Mechanical

Date: 11/10/2004 Annex: 3-cycle-f. /3

12

6

Frequency Stability - Analysis

Frequency stability improved by:

Increase of Primary Reserve and System Inertia

-Dispatching more generators

Improvement of Primary Control action

-Tuning / replacing of governor controls.

Automatic Load shedding

-Under-Frequency Load Shedding relays adjusted according to system-wide criteria.

-Interruptible loads -Power Frequency controllers of HVDC links


13

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 systemwide criteria

-

Interruptible loads


14

7

Rotor Angle Stability


15


Two distinctive types of rotor angle stability: -

Small signal rotor angle stability (Oscillatory stability)

-

Large signal rotor angle stability (Transient stability)


16

8

Oscillatory Stability

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)


17


Small signal rotor angle stability is a system property Small disturbance -> analysis using linearization around operating point Analysis using eigenvalues and eigenvectors


18

9


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.)


19

Transient Stability

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


20

10

Transient Stability T N E L I S g I D

25.00

12.50

0.00

T N E L I S g I D

200.00

-12.50

100.00 -25.00

0.00

-37.50 0.00

1.00 1.996 G1: Rotor angle with reference to reference machine angle in deg

T ra ns ie nt S ta bi il ty

DIgSILENT

2.994

3.992

S ub pl ot /D ia gr am m

[s]

4.990

Date: 11/11/2004 Annex: 1 /3

-100.00

-200.00 0.00

DIgSILENT

0.65 1.294 G1: Rotor angle with reference to reference machine angle in deg

1.940

T ra ns e i nt S ta bi il ty

2.587

S ub pl ot /D ia gr amm

[s]

3.234

Date: 11/11/2004 Annex: 1 /3


21

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.


22

11

Voltage Stability


23

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


24

12

Voltage Stability - Analysis

Small-Signal:

Large-Signal

- Small disturbance

- System fault - Loss of generation

Long-Term - P-V-Curves (load flows) - dv/dQ-Sensitivities - Long-term dynamic models including tap-changers, varcontrol, excitation limiters, etc.

Short-Term

- P-V-Curves (load flows) of the faulted state. - Long-term dynamic models including tap-changers, varcontrol, excitation limiters, etc.

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


25

Long-Term vs. Short-Term Voltage Stability

Reactive power control: Short-Term

Long-Term

Q- contribution of synchronous gen.

Large (thermal overload Limited by capabilities) overexcitation limitors

Switchable shunts

No contribution (switching times too high)

High contribution

SVC/TSC

High contribution

High contribution


26

13

Voltage Stability

All generators in service

Outage of large generator


27

Large-Signal Long-Term Voltage Instability

T N E L I S g I D

1.25

Fault with loss of transmission line 1.00

0.75

0.50

0.25

0.00

-0.25 0.00

5.00 APPLE_20: Voltage, Magnitude in p.u.

10.00

15.00

[s]

20.00

SUMMERTON_20: Voltage, Magnitude in p.u. LILLI_20: Voltage, Magnitude in p.u. BUFF_330: Voltage, Magnitude in p.u.


28

14

Voltage Stability – Q-V-Curves T N E L I S g I D

1.40

1.20

1.00

P=2000MW P=1800MW P=1600MW

0.80

P=1400MW

0.60

const. P, variable Q 0.40 262.64 x-Achse:

562.64

862.64

1162.64

1462.64

1762.64

SC: Blindleistungin Mvar SC: Voltage in p.u., P=1400MW SC: Voltage in p.u., P=1600MW SC: Voltage in p.u., P=1800MW SC: Voltage in p.u., P=2000MW


29

Dynamic Voltage Stability

•

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


30

15

Dynamic Voltage Stability – Induction Generator (Motor)

T N E L I S g I D

3.00

2.00

1.00

0.00

-1.00 1.00 x-Axis:

1.04 GWT: Speed in p.u. GWT: Electrical Torque in p.u.

1.08

1.12

1.16

1.20

1.04 GWT: Speed in p.u. GWT: Reactive Power in Mvar

1.08

1.12

1.16

1.20

0.00

-2.00

-4.00

-6.00

-8.00 1.00 x-Axis:


31


T N E L I S g I D

3.00

2.00

1.00

Constant Y= 1.000 p.u.

1.008 p.u.

0.00

-1.00 1.00 x-Axis:

0.00

-1.00

1.01 GWT: Speed in p.u. GWT: Electrical Torque in p.u.

1.02

1.03

1.04

1.02

1.03

1.04

Constant X = 1.008 p.u.

-1.044 Mvar

-2.00

-3.00

-4.00

-5.00

-6.00 1.00 x-Axis:

1.01 GWT: Speed in p.u. GWT: Reactive Power in Mvar


32

16


T N E L I S g I D

1.20 1.00 0.80 0.60 0.40 0.20 0.00 0.00

0.50

1.00

1.50

[s]

2.00

0.50

1.00

1.50

[s]

2.00

0.50

1.00

1.50

[s]

2.00

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

80.00

40.00

0.00

-40.00

-80.00

-120.00 0.00 Cub_0.1\PQ PCC: Active Power in p.u. Cub_0.1\PQ PCC: Reactive Power in p.u.

1.06

1.04

1.02

1.00

0.98 0.00 GWT: Speed


33

Dynamic Voltage Collapse 60.00

T N E L I S g I D

60.00

40.00

40.00

20.00 20.00 0.00 0.00 -20.00 -20.00

-40.00

-40.00

0.00

1.00 2.00 [s] Cub_0.2\PQ BlueMountain: Active Power in p.u. Cub_0.2\PQ BlueMountain: Reactive Power in p.u.

3.00

1.125

-60.00

0.00

1.00 2.00 Cub_1.1\PQ GreenField: Active Power in p.u. Cub_1.1\PQ GreenField: Reactive Power in p.u.

[s]

3.00

0.00

1.00 2.00 Cub_0.1\PQ RedSunset: Active Power in p.u. Cub_0.1\PQ RedSunset: Reactive Power in p.u.

[s]

3.00

60.00

1.000

40.00

0.875 20.00 0.750 0.00 0.625 -20.00

0.500

0.375

-40.00 0.00

1.00 GLE\1: Voltage, Magnitude in p.u. GLZ\2: Voltage, Magnitude in p.u.

2.00

[s]

3.00

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


34

17

Dynamic Voltage Stability – Voltage Recovery (S ynchronous Generators)

T N E L I S g I D

1.20

1.00

0.80

0.60

0.40

0.20

0.00 0.00

1.00

2.00

[s]

3.00

1.00

2.00

[s]

3.00

HV: Voltage, Magnitude in p.u. MV: Voltage, Magnitude in p.u. 120.00

80.00

40.00

0.00

-40.00

-80.00

-120.00

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


35

Time-domain Analysis


36

18

Transients in Power Systems

Fast Transients/Network Transients: Time frame: 10 mys…..500ms 

Lightening



Switching Overvoltages



Transformer Inrush/Ferro Resonance



Decaying DC-Components of short circuit currents


37


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


38

19


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


39

Stability/EMT

Different Network Models used:

Stability:

V  j L I

EMT:

v  L


di dt

I  j C V

i  C

dv dt

40

20

Short Circuit Current EMT T N E L I S g I D

800.0

600.0

400.0

200.0

0.00

-200.0

0.00 4x555 MVA: Phase Current B in kA

0.12

0.25

0.38

S ho rt C ir c ui t C ur re nt w it h c om pl et e m od el ( EM T- mo de l)

[s]

0.50

P lo ts Date: 4/25/2001 Annex: 1 /1


41

Short Circuit Current RMS T N E L I S g I D

300.0

250.0

200.0

150.0

100.0

50.00

0.00 0.00 4x555 MVA: Current, Magnitude in kA

0.12

0.25

S ho rt C ir cu it C ur re nt w it h r ed uc ed m od el ( St ab il it y m od el )

0.38

[s]

0.50

P lo ts Date: 4/25/2001 Annex: 1 /1


42

21

RMS-EMT-Simulation

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

(X)

X

0

X

HVDC dynamics Switching Over Voltages Fundamentals of Power System Stability

43

Frequency-domain analysis


44

22

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)


45


•

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


46

23


•

Linear System Representation:

x  Ax  b

•

Transformation:

~ x  T x

•

Transformed System

~  TAT 1 ~ x x  Tb

•

Diagonal System

~  D x ~  Tb x


47


xi

 f i ( x1, x2 ,..., xn ; u1 , u2 ,..., ur )

•

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:

yi

 g i ( x1 , x2 ,... xn ; u1 , u2 ,..., ur )

•

Initial Equilibrium :

xi 0

 f i ( x10 , x20 ,..., xn 0 ; u10 , u 20 ,..., u r 0 )  0

•

Perturbation:


 xi 0   xi ui  ui 0  ui xi  xi 0   xi xi

48

24


•

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

 f i  f  f  f  x1 ... i  xn  i u1 ... i ur  x1  xn u1 ur  g j  g  g  g y j  g j ( x10, x20,..., xn0;u10,u20,...ur 0 )   x1 ... j  xn  j u1 ... j ur  x1  xn u1 ur

xi  f i ( x10, x20,..., xn0;u10,u20,...,ur 0) 

•

Using Vector-Matrix notation:

[ x ]  [ A][ x]  [ B ][u ] [ y ]  [C ][ x ]  [ D ][u ]


49


•

Taking the Laplace transform of the previous equations:

s[ x( s )]  [ x (0)]  [ A][ x( s )]  [ B ][u ( s )] [  y( s )]  [C ][ x( s)]  [ D][u ( s )] •

Block Diagram of the state-space representation:


50

25


•

Poles of [ x(s)] and [ y(s)] are the root of the characteristic equation of matrix [ A]:

det( s[ I ]  [ A])  0

•

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.


51


•

An oscillatory system mode is given by a pair of eigenvalues

    j •

The real component  gives the damping . A negative real part represents a damped (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 the oscillation and is given by:

 


   2   2

52

26

Eigenvalue Analysis without and with PSS Without PSS

3.5000

T N E L I S g I D

Damped Freque 2.9000

2.3000

1.7000

1.1000 Y = 0.800 Hz

-4.0000

-3.2000

-2.4000

-1.6000

-0.8000

0.5000 Neg. Damping [1/s]

Stable Eigenvalues Unstable Eigenvalues

With PSS

3.5000 Y = 3.000 Hz

Damped Freque 2.9000

2.3000 Y = 2.000 Hz Y = 1.500 Hz

1.7000

1.1000

-4.0000

-3.2000

-2.4000

-1.6000

-0.8000

0.5000 Neg. Damping [1/s]

Stable Eigenvalues Unstable Eigenvalues


53

Voltage Stability

Fundamental Concepts


54

27

Voltage Stability

X

Qe

E G'

E 0

P e  Qe

E 0 E G' X



E G' X

sin  G 

 E

' G

 E 0 cos G 


55

Voltage stability: basic concepts

E s

I 

2

 Z LN cos  ZLD cos    ZLN

sin

 ZLD sin 

2

2

I  V R P R

1 E s F Z LN

con

 Z   Z  F  1   LD   2   LD   cos      Z LN   Z LN 

 Z LD  I  VR I cos  


Z LD F

 E s     Z LN 

2

cos 

56

28

Voltage stability: basic concepts

Voltage collapse depends on the load characteristics


57

Study case: Tap changer


58

29

Voltage Stability – Q-V-Curves T N E L I S g I D

1.40

1.20

1.00

P=2000MW P=1800MW P=1600MW

0.80

P=1400MW

0.60

const. P, variable Q 0.40 262.64 x-Achse:

562.64

862.64

1162.64

1462.64

1762.64

SC: Blindleistungin Mvar SC: Voltage in p.u., P=1400MW SC: Voltage in p.u., P=1600MW SC: Voltage in p.u., P=1800MW SC: Voltage in p.u., P=2000MW


59

Voltage Stability – P-V-Curves T N E L I S g I

1.00

D

0.90

pf=1 0.80

pf=0.95

0.70

pf=0.9

0.60

const. Power factor, variable P 0.50 100.00 x-Achse:

350.00

600.00

850.00

1100.00

1350.00

U_P-Curve: Total Load of selected o l ads 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


60

30



61

One Machine System T N E L I S g I D

e c r u

o S e t i n i f n I

~ V

r W a M v A k 0 M 0 6 . 4 9 . 8 4 . 0 9 0 6 - 3

r W M a v A k 0 M 0 . 6 8 7 . . 5 9 0 7 2 9 8 1 -

r W M a v A k 0 M 4 . 2 7 9 6 . . 6 9 6 2 5 1 1 V . g k u . e 1 p d 4 . 0 0 0 9 . . 0 5 0 4 0

r

a W v A M M k 0 9 0 6 . 9 . 9 . 8 1 0 9 2 6 2

CCT 2 Type CCT 186.00 km

r W a M v A k 0 M 4 . 0 7 . 6 9 9 . 9 2 1 1 2 1 4

CCT1 Type CCT 100.00 km

r a W v M A M k 0 9 0 . . 6 5 8 8 . 9 4 3 2 9 6 1 - -

Trf 500kV/24kV/2220MVA V . g k u . e 5 p d 2 1 . 4 1 . 2 9 7 . 0 4 0 2

T V H k 0 0 . 0 0 5

s V u k B 0 e 0 t . i 0 n 0 i f n 5 I

DIgSILENT

r W a . . u M v A k u . . 0 M p p 0 0 8 0 3 0 0 0 . 2 4 6 0 . . 1 8 9 . 9 7 3 . 0 9 6 5 1 1 9

r a W M v A k M 0 2 1 0 . . 4 . 8 9 3 9 7 6 5 9 9 1

g V . e k u . d 0 p 4 0 . 0 3 . 4 0 . 8 2 1 2

T L V k 0 0 . 4 2

Example Power System Stability and Control One Machine Problem

PowerFactory 12.1.178


~ G

) 1 ( V k 4 2 / A 1 V M G 0 2 2 2 n e G

Project: Training Graphic: Grid Date: Annex:

4/19/2 002 1

62

31

One Machine System

Equivalent circuit, transferred power:

X

P e

E G'

E 0


63

One Machine System

•

Power transmission over reactance:

P e Qe •

 

E 0 E G' X E G' X

sin  G 

 E

' G

 E 0 cos G 

Mechanical Equations:

 J  G 


P m  P e 



P m  P e  0

    0

64

32

One Machine System

•

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

G J  •



P m  0



P max  0

sin  G

P m  0



P max  0

sin  G 0

 P    max cos  G 0  G   0 

Eigenvalues (Characteristic Frequency):

 1/ 2 •



 

P max J  0

cos  G 0

Stable Equilibrium points (SEP) exist for:

cos  G 0  0


65

One-machine System T N E L I S g I D

4000.

stable

unstable

3000.

SEP

UEP Pini y=1998.000 MW

2000.

1000.

0.00

-1000... 0.00 x-Axis:

36.00 Plot Power Curve: Generator Angle in deg

72.00

108.0

144.0

180.0

Plot Power Curve: Power 1 in MW Plot Power Curve: Power 2 in MW

DIgSILENT Fundamentals of Power System Stability

Single Machine Problem

P-phi Date: 4/19/2002 Annex: 1 /4

66

33

Large disturbances (Transient Stability)

•

Energy Function:

1 2 •

 G

G  J 

2





( P m  P e )

 0



d   E kin

 E pot  0

At Maximum Angle:

 G max  0   G max

E pot





 0

( P m  P e ) 

 E  0 kin

d   0


67

Large disturbances : Equal Area Criterion T N E L I S g I D

4000.

3000.

SEP

UEP

E2

Pm

2000.

 crit

 max E1

1000.

0.00

 0  c -1000... 0.00 x-Axis :

36.00 Plot Power Curve: Generator Angle in deg Plot Power Curve: Power 1 in MW

72.00

108.0

144.0

180.0

Plot Power Curve: Power 2 in MW

DIgSILENT Fundamentals of Power System Stability

Single Machine Problem

P-phi Date: 4/19/2002 Annex: 1 / 4

68

34

Large disturbances: Equal Area Criterion

E 1



1

 c

P d    m

 0

E 2



1

 max

 P  P   m

max

sin( ) d 

 c

Stable operation if:

E 1

  E 2


69

Large disturbances: Equal Area Criterion

1

E 1



P m ( c 

E 2



P m

Setting



  0 )

( max

  c ) 

P max 

(cos  max

 cos  c )

 crit     0 and equating E1 and -E2:

cos  c

 (  2 0 ) sin  0  cos  0


70

35

Large-disturbances: Critical Fault Clearing Time

•

During Short Circuit:

P e  0 •

Differential Equation:

G J  •



P m  0

Critical Fault Clearing Time:

 c



P m 2 J  0

t c

2

  0


71

Small disturbances (Oscillatory Stability)

r o t a r e ~ n G e G

E g '

X

~ V

s u b e t i n i f n I

E o

Assumptions: 1. Constant excitation 2. Constant damping from synchronous machine, K e 3. Simplified generator model, Pe = T e (in per unit) 4. Constant mechanical torque


72

36

Small disturbances Equation of electrical circuit…

P e  T e

T e 



E 0 E g ' X

sin  

T e   P max cos o     o

 T e  J    ( K m  K e )  2 T m  T e  2 Hs   s ( K m  K e ) T m  T e  2 Hs 2   s ( K m  K e ) 2 2 Hs   s( K m  K e )  T e  0

Equation of motion…

T m

Combined…

s

2

K  K e  P max cos  o  0  s  m   2 H  2 H

 n



P max cos  o 2 H


73

Small disturbances: Structure of linearised generator model

T m

uref  0

Exciter

Generator

T e 

Shaft



1



s

ut

K * 

T e



•

Damping torque: a torque in phase with

•

Synchronising torque: a torque in phase with



0


K * 

74

37

Power System Stability, Training Course

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