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Fundamentals of Power System Stability 1
Power System StabilityTraining Course
DIgSILENT GmbH
Fundamentals of Power System Stability 2
General Definitions
2
Fundamentals of Power System Stability 3
• „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
Fundamentals of Power System Stability 4
Frequency Stability
3
Fundamentals of Power System Stability 5
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
Fundamentals of Power System Stability 6
• 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
Fundamentals of Power System Stability 7
• 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
Fundamentals of Power System Stability 8
• 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
Fundamentals of Power System Stability 9
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
Fundamentals of Power System Stability 10
Frequency drop depends on:• Primary Reserve• Speed of primary control• System inertia
Additionally to consider:• Frequency dependence of load
Frequency Stability
6
Fundamentals of Power System Stability 11
• 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
Fundamentals of Power System Stability 12
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
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Frequency Stability
7
Fundamentals of Power System Stability 13
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
Fundamentals of Power System Stability 14
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
Fundamentals of Power System Stability 15
Rotor Angle Stability
Fundamentals of Power System Stability 16
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
Fundamentals of Power System Stability 17
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
Fundamentals of Power System Stability 18
Small signal rotor angle stability is a system property
Small disturbance -> analysis using linearization around operatingpoint
Analysis using eigenvalues and eigenvectors
Oscillatory Stability
10
Fundamentals of Power System Stability 19
Oscillatory Stability
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.)
Fundamentals of Power System Stability 20
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
Fundamentals of Power System Stability 21
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
Fundamentals of Power System Stability 22
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
Fundamentals of Power System Stability 23
Voltage Stability
Fundamentals of Power System Stability 24
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
Fundamentals of Power System Stability 25
- 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
Fundamentals of Power System Stability 26
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
Fundamentals of Power System Stability 27
Voltage Stability
Outage of large generator
All generators in service
Fundamentals of Power System Stability 28
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
Fundamentals of Power System Stability 29
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
Fundamentals of Power System Stability 30
• 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
Fundamentals of Power System Stability 31
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)
Fundamentals of Power System Stability 32
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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Dynamic Voltage Stability –Induction Generator (Motor)
17
Fundamentals of Power System Stability 33
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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Dynamic Voltage Stability –Induction Generator (Motor)
Fundamentals of Power System Stability 34
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
18
Fundamentals of Power System Stability 35
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)
Fundamentals of Power System Stability 36
Time-domain Analysis
19
Fundamentals of Power System Stability 37
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
Fundamentals of Power System Stability 38
Medium Term Transients / Electromechanical TransientsTime frame: 400ms….10s
Transient Stability Critical Fault Clearing Time AVR and PSS Turbine and governor Motor starting Load Shedding
Transients in Power Systems
20
Fundamentals of Power System Stability 39
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
Transients in Power Systems
Fundamentals of Power System Stability 40
Stability/EMT
Different Network Models used:
Stability:
EMT:
ILjV VCjI
dtdi
Lv dtdv
Ci
21
Fundamentals of Power System Stability 41
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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Fundamentals of Power System Stability 42
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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Fundamentals of Power System Stability 43
(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
Fundamentals of Power System Stability 44
Frequency-domain analysis
23
Fundamentals of Power System Stability 45
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)
Fundamentals of Power System Stability 46
Small signal stability analysis
• 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
24
Fundamentals of Power System Stability 47
Small signal stability analysis
• Linear System Representation:
• Transformation:
• Transformed System
• Diagonal System
bAxx
xTx ~
TbxTATx ~~ 1
TbxDx ~~
Fundamentals of Power System Stability 48
Small signal stability analysis
• 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
Fundamentals of Power System Stability 49
Small signal stability analysis
• 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
Fundamentals of Power System Stability 50
Small signal stability analysis
• Taking the Laplace transform of the previous equations:
• Block Diagram of the state-space representation:
)](][[)](][[)]([)](][[)](][[)]0([)]([
suDsxCsysuBsxAxsxs
26
Fundamentals of Power System Stability 51
Small signal stability analysis
• 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
Fundamentals of Power System Stability 52
Small signal stability analysis
• 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
27
Fundamentals of Power System Stability 53
-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
Fundamentals of Power System Stability 54
Voltage Stability
Fundamental Concepts
28
Fundamentals of Power System Stability 55
0E
eQX
'GE
GGG
e
GG
e
EEXE
Q
XEE
P
cos
sin
0'
'
'0
Voltage Stability
Fundamentals of Power System Stability 56
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
Fundamentals of Power System Stability 57
Voltage stability: basic concepts
Voltage collapse depends on the load characteristics
Fundamentals of Power System Stability 58
Study case: Tap changer
30
Fundamentals of Power System Stability 59
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
Fundamentals of Power System Stability 60
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
31
Fundamentals of Power System Stability 61
Rotor Angle Stability
Fundamentals of Power System Stability 62
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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Fundamentals of Power System Stability 63
One Machine System
0E
ePX
'GE
Equivalent circuit, transferred power:
Fundamentals of Power System Stability 64
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
Fundamentals of Power System Stability 65
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
Fundamentals of Power System Stability 66
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
34
Fundamentals of Power System Stability 67
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
Fundamentals of Power System Stability 68
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
DIgSILENTSingle Machine Problem P-phi Date: 4/19/2002
Annex: 1 /4
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E1
E2
0 c
max
SEP UEP
critPm
35
Fundamentals of Power System Stability 69
Large disturbances: Equal Area Criterion
21 EE
c
dPE m
0
11
max
)sin(1
max2
c
dPPE m
Stable operation if:
Fundamentals of Power System Stability 70
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
Fundamentals of Power System Stability 71
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
Fundamentals of Power System Stability 72
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
Fundamentals of Power System Stability 73
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
Fundamentals of Power System Stability 74
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
Fundamentals of Power System Stability 75
Linear model of generator + AVR + PSS
PSSu
Exciter Generator Shafts1
tu
PSSu
Exciter Generator Shafts1
0 mT
eT
tu
PSS
oPSSeTangleWant 0)(
Phase lag
Phase lead compensation