1
Power System StabilityTraining Course
DIgSILENT GmbH
Fundamentals on Power System Stability 2
General Definitions
2
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
3
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
4
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
5
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
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Frequency Stability
6
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
7
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
8
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
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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
9
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
10
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
11
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
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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
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Dynamic Voltage Stability –Induction Generator (Motor)
12
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
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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
13
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.
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Dynamic Voltage Stability –Voltage Recovery (Synchronous Generators)
Fundamentals on Power System Stability 26
Time Domain Simulation
14
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
15
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 =
16
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
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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
18
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
19
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
20
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
21
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:
22
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 =&&
23
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
24
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
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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
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const. Power factor, variable P
Voltage Stability – P-V-Curves
25
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)
26
Fundamentals on Power System Stability 51
Small signal analysis
• Linear System Representation:
• Transformation:
• Transformed System
• Diagonal System
bAxx +=&
xTx ~=
TbxTATx += − ~~ 1&
TbxDx += ~~&