Power System Stability, Training Course
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Transcript of Power System Stability, Training Course
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Fundamentals of Power System Stability 1
Power System StabilityTraining Course
DIgSILENT GmbH
Fundamentals of Power System Stability 2
General Definitions
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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
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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
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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
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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
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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
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Frequency Stability
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
DIg
SIL
EN
T
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
DIg
SIL
EN
T
const. Power factor, variable P
Voltage Stability – P-V-Curves
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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
DIg
SIL
EN
T
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32
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
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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
DIg
SIL
EN
T
SEP UEP
stable unstable
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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
DIg
SIL
EN
T
E1
E2
0 c
max
SEP UEP
critPm
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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
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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
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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
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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