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Power System StabilityTraining Course
DIgSILENT GmbH
Fundamentals on Power System Stability 2
General Definitions
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Fundamentals on Power System Stability 3
• „Stability“ - general definition:
Ability of a system to return to a steady state after a disturbance.
• Small signal effects• Large signal 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 on Power System Stability 4
Ability of a power system to compensate for a power deficit:1. Inertial reserve (network time constant)
Lost power is compensated by the energy stored in rotating masses of all generators -> Frequency decreasing
2. Primary reserve:Lost power is compensated by an increase in production of primary controlled units. -> Frequency drop partly compensated
3. Secondary reserve:Lost power is compensated by secondary controlled units. Frequency and area exchange flows reestablished
4. Re-Dispatch of Generation
Frequency Stability
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Fundamentals on Power System Stability 5
• 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
Fundamentals on Power System Stability 6
• Mechanical Equation of each Generator:
• ∆P=ω∆T is power provided to the system be each generating unit.• Assuming synchronism:
• Power shared according to generator inertia
nn
elmelm
PPPTTJωω
ω ∆=
−≈−=&
j
i
j
i
ini
JJ
PP
PJ
=∆∆
∆=ωω &
Inertial Reserve
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Fundamentals on Power System Stability 7
• 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
RR
PP
PRPR
=∆∆
∆=∆
PRPK
ffKP iii
ii ∆=∆=∆⇒∆=∆1
Primary Control
Fundamentals on Power System Stability 8
Turbine 1
Turbine 2
Turbine 3
Generator 1
Generator 2
Generator 3
Network
Secondary Control
PT PG
PT PG
PT PG
f PA
Set Value
Set Value
Set Value
Contribution
• Bringing Back Frequency• Re-establishing area exchange flows• Active power shared according to participation factors
Secondary Control
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Fundamentals on Power System Stability 9
Frequency drop depends on:• Primary Reserve• Speed of primary control• System inertia
Additionally to consider:• Frequency dependence of load
In case of too severe frequency drops:• Load shedding
Frequency Stability
Fundamentals on Power System Stability 10
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 MechanicalSudden Load Increase
Date: 11/10/2004
Annex: 3-cycle-f. /3
DIg
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Frequency Stability
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Fundamentals on Power System Stability 11
• Dynamic Simulations
• Sometimes possible: Inertial/Primary controlled or secondary controlled load flows
Frequency Stability - Analysis
Fundamentals on Power System Stability 12
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
Rotor Angle Stability
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Fundamentals on Power System Stability 13
Small signal rotor angle stability (Oscillatory stability) is a system property
Small disturbance -> analysis using linearization around operating point
Analysis using eigenvalues and eigenvectors
Rotor Angle Stability
Fundamentals on Power System Stability 14
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
Rotor Angle Stability
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Fundamentals on Power System Stability 15
3.2342.5871.9401.2940.650.00 [s]
200.00
100.00
0.00
-100.00
-200.00
G1: Rotor angle with reference to reference machine angle in deg
DIgSILENT Transient Stability Subplot/Diagramm
Date: 11/11/2004
Annex: 1 /3
DIg
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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
DIgSILENT Transient Stability Subplot/Diagramm
Date: 11/11/2004
Annex: 1 /3
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Transient Stability
Fundamentals on Power System Stability 16
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 signal voltage stability (Dynamic voltage stability)
– Ability to maintain steady voltages after following large disturbances
Voltage Stability
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Fundamentals on Power System Stability 17
- Dynamic models (short-term), special importance on dynamic load modeling, stall effects etc.
Short-Term
- P-V-Curves (load flows)of the faulted state.- Long-term dynamic models including tap-changers, var-control, excitation limiters, etc.
- P-V-Curves (load flows)- dv/dQ-Sensitivities- Long-term dynamic models including tap-changers, var-control, excitation limiters, etc.
Long-Term
Large-Signal- System fault- Loss of generation
Small-Signal:- Small disturbance
Voltage Stability - Analysis
Fundamentals on Power System Stability 18
151.30138.80126.30113.80101.3088.80
1.10
1.00
0.90
0.80
0.70
0.60
0.50
x-Axis: U_P-Curve: Total Load of selected loads in MWAMBOWS51: Voltage, Magnitude in p.u.ANGONS51: Voltage, Magnitude in p.u.BELLES51: Voltage, Magnitude in p.u.BISSES51: Voltage, Magnitude in p.u.BISSES61: Voltage, Magnitude in p.u.
PV-curves U_P-Curve
Date: 11/11/2004
Annex: 1 /1
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Small-Signal Voltage Stability –PV-Curves
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Fundamentals on Power System Stability 19
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
Fundamentals on Power System Stability 20
• 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 undervoltagetrips of own supply -> loss of generation
Dynamic Voltage Stability
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Fundamentals on Power System Stability 21
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
DIg
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Dynamic Voltage Stability –Induction Generator (Motor)
Fundamentals on Power System Stability 22
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
DIg
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Dynamic Voltage Stability –Induction Generator (Motor)
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Fundamentals on Power System Stability 23
2.001.501.000.500.00 [s]
1.20
1.00
0.80
0.60
0.40
0.20
0.00
G\HV: Voltage, Magnitude in p.u.MV: Voltage, Magnitude in p.u.
2.001.501.000.500.00 [s]
80.00
40.00
0.00
-40.00
-80.00
-120.00
Cub_0.1\PQ PCC: Active Power in p.u.Cub_0.1\PQ PCC: Reactive Power in p.u.
2.001.501.000.500.00 [s]
1.06
1.04
1.02
1.00
0.98
GWT: Speed
DIg
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Dynamic Voltage Stability –Induction Generator (Motor)
Fundamentals on Power System Stability 24
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 on Power System Stability 25
3.002.001.000.00 [s]
1.20
1.00
0.80
0.60
0.40
0.20
0.00
HV: Voltage, Magnitude in p.u.MV: Voltage, Magnitude in p.u.
3.002.001.000.00 [s]
120.00
80.00
40.00
0.00
-40.00
-80.00
-120.00
Cub_1\PCC PQ: Active Power in p.u.Cub_1\PCC PQ: Reactive Power in p.u.
DIg
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Dynamic Voltage Stability –Voltage Recovery (Synchronous Generators)
Fundamentals on Power System Stability 26
Time Domain Simulation
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Fundamentals on Power System Stability 27
Fast Transients/Network Transients:Time frame: 10 mys…..500ms
LighteningSwitching OvervoltagesTransformer Inrush/Ferro ResonanceDecaying DC-Components of short circuit currents
Transients in Power Systems
Fundamentals on Power System Stability 28
Medium Term Transients / Electromechanical TransientsTime frame: 400ms….10s
Transient StabilityCritical Fault Clearing TimeAVR and PSSTurbine and governorMotor startingLoad Shedding
Transients in Power Systems
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Fundamentals on Power System Stability 29
Long Term Transients / Dynamic PhenomenaTime Frame: 10s….several min
Dynamic StabilityTurbine and governorPower-Frequency ControlSecondary Voltage ControlLong Term Behavior of Power Stations
Transients in Power Systems
Fundamentals on Power System Stability 30
Stability/EMT
Different Network Models used:
Stability:
EMT:
ILjV ω= VCjI ω=
dtdiLv =
dtdvCi =
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Fundamentals on Power System Stability 31
Short Circuit Current EMT
0.50 0.38 0.25 0.12 0.00 [s]
800.0
600.0
400.0
200.0
0.00
-200.0
4x555 MVA: Phase Current B in kA
Short Circuit Current with complete model (EMT-model) Plots
Date: 4/25/2001
Annex: 1 /1
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Fundamentals on Power System Stability 32
Short Circuit Current RMS
0.50 0.38 0.25 0.12 0.00 [s]
300.0
250.0
200.0
150.0
100.0
50.00
0.00
4x555 MVA: Current, Magnitude in kA
Short Circuit Current with reduced model (Stability model) Plots
Date: 4/25/2001
Annex: 1 /1
DIg
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Fundamentals on Power System Stability 33
(X)X
X0
Dynamic voltage stabilitySelf excitation of ASM
X(X)HVDC dynamicsX0Switching 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 on Power System Stability 34
Rotor Angle Stability
Fundamental Concepts
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Fundamentals on Power System Stability 35
One Machine Problem
DIgSILENT
PowerFactory 12.1.178
Example
Power System Stability and Control One Machine Problem
Project: Training Graphic: Grid Date: 4/19/2002 Annex: 1
G ~ G1
Gen
222
0MV
A/2
4kV
(1)
1998
.000
MW
967.
920
Mva
r53
.408
kA
1.16
3 p.
u.-0
.000
p.u
.
Trf500kV/24kV/2220MVA
-199
8.00
MW
-634
.89
Mva
r2.
56 k
A
1998
.00
MW
967.
92 M
var
53.4
1 kA
CCT 2Type CCT186.00 km
-698
.60
MW
30.4
4 M
var
0.90
kA
698.
60 M
W22
1.99
Mva
r0.
90 k
A
CCT1Type CCT100.00 km
-129
9.40
MW
56.6
2 M
var
1.67
kA
1299
.40
MW
412.
90 M
var
1.67
kA
V ~
Infin
ite S
ourc
e
-199
8.00
MW
87.0
7 M
var
2.56
kA
Infin
ite B
us50
0.00
kV
450.
41 k
V0.
90 p
.u.
0.00
deg
HT
500.
00 k
V47
2.15
kV
0.94
p.u
.20
.12
deg
LT24
.00
kV24
.00
kV1.
00 p
.u.
28.3
4 de
g
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Fundamentals on Power System Stability 36
One Machine Problem
0E
ePX
'GE
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Fundamentals on Power System Stability 37
One Machine Problem
• Power transmission over reactance:
• Mechanical Equations:
0
0
ωωϕωω
ω
−=
−≈
−=
G
emem PPPPJ
&
&
( )
( )( )GGG
e
GG
e
EEXEQ
XEEP
ϕ
ϕ
cos
sin
0'
'
'0
−=
=
Fundamentals on Power System Stability 38
One Machine Problem
• Differential Equation of a one-machine infinite bus bar system:
• Eigenvalues (Characteristic Frequency):
• Stable Equilibrium points (SEP) exist for:
GGGm
Gm
G
PPPPPJ ϕϕ
ωϕ
ωωϕ
ωωϕ ∆⎟⎟
⎠
⎞⎜⎜⎝
⎛−−≈−= 0
0
max0
0
max
00
max
0
cossinsin&&
00
max2/1 cos GJ
P ϕω
λ −±=
0cos 0 >Gϕ
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Fundamentals on Power System Stability 39
Small Signal Stability
180.0144.0108.072.0036.00 0.00
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
DIgSILENT Single Machine Problem P-phi
Date: 4/19/2002
Annex: 1 /4
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SEP UEP
Fundamentals on Power System Stability 40
Transient Stability
• Energy Function:
• At Maximum Angle:
( ) 0)(21
0
2 =+=−
+ ∫ potkinem
G EEdPPJG
ϕω
ϕϕ
ϕ
&
0max =Gϕ&
0)(max
0
=−
= ∫ ϕω
ϕ
ϕ
dPPEG
empot
( )0=kinE
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Fundamentals on Power System Stability 41
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
DIgSILENT Single Machine Problem P-phi Date: 4/19/2002
Annex: 1 /4
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E1
E2
0ϕ cϕ
maxϕ
SEP UEP
critϕPm
Fundamentals on Power System Stability 42
Equal Area Criterion
21 EE −=
∫=c
dPE m
ϕ
ϕ
ϕω
0
11
( )∫ −=max
)sin(1max2
ϕ
ϕ
ϕϕω
c
dPPE m
Stable operation if:
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Fundamentals on Power System Stability 43
Equal Area Criterion
)(101 ϕϕ
ω−= cmPE
)cos(cos)( maxmax
max2 ccm PPE ϕϕ
ωϕϕ
ω−+−=
000 cossin)2(cos ϕϕϕπϕ −−=c
Setting and equating E1 and -E2:0ϕπϕ −=crit
Fundamentals on Power System Stability 44
Critical Fault Clearing Time
• During Short Circuit:
• Differential Equation:
• Critical Fault Clearing Time:
02
02ϕ
ωϕ += c
mc t
JP
0=eP
0ωϕ m
GPJ =&&
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Fundamentals on Power System Stability 45
Voltage Stability
Fundamental Concepts
Fundamentals on Power System Stability 46
0E
eQX
'GE
( )
( )( )GGG
e
GG
e
EEXEQ
XEEP
ϕ
ϕ
cos
sin
0'
'
'0
−=
=
Voltage Stability
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Fundamentals on Power System Stability 47
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
SILE
NT
const. P, variable Q
Voltage Stability – Q-V-Curves
Fundamentals on Power System Stability 48
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
SILE
NT
const. Power factor, variable P
Voltage Stability – P-V-Curves
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Fundamentals on Power System Stability 49
Dynamic Stability / Eigenvalue Analysis
Fundamental Concepts
Fundamentals on Power System Stability 50
Small signal analysis
• Linear model automatically generated by linearizing the stability model.
• 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 (released in Summer 2001)
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Fundamentals on Power System Stability 51
Small signal analysis
• Linear System Representation:
• Transformation:
• Transformed System
• Diagonal System
bAxx +=&
xTx ~=
TbxTATx += − ~~ 1&
TbxDx += ~~&