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OSCILLATORY VOLTAGEINSTABILITY:
A FUNDAMENTAL STUDY Luigi VanfrettiECSE 6963 Nonlinear Phenomena in Engineering and Biology
0.88
0.90.92
0.94 11.5
22.5
0.012
0.014
0.016
0.018
E f (pu)
Trayectories Sorrounding the Limit Cycle
Eq (pu)
s ( p u
)
Unstable TrajectorySEP
StableTrajectory
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SEP
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2
Outline
Background and Case StudiedModeling and Implementation
Modeling of the network and apparatusComputer Implementation
Analysis: Nonlinear Systems PerspectiveBifurcation Diagram and Eigenvalue LocusBifurcation ClassificationOscillatory voltage instability:
simulation of stable and unstable pointsAnalysis of Hopf Bifurcation andClassificationController Tuning to Avoid VoltageInstabilityEffect of Controller Limits on Stability
Conclusions
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Background and Case Studied
Background:Short-term voltage instability arises when dynamic loads attempt to restore consumedpower in a very short time frame (one - few seconds) [1, 2]. Oscillatory voltage instability[3] is a type of short-term voltage instability that originates due to the interaction of twoor more load restoration processes acting in the same short-term time scale.
Case Studied
The system consists of a synchronous generator feeding an isolated three phase induction
motor.The mechanical load of the motor is considered constant.
The generator has a first-order excitation system with a proportional AVR. AVR limits willbe considered to evaluate their impact on stability.
Assumption: frequency transients have no impact on the response of the system neglected.
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G
M
1
t V
2
2V jX
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Modeling of the SystemThe system is represented by:
The algebraic variables arefunction of the network andstate variables.
Network Modeling CurrentInjection:
Generator Modeling: field flux decay eqn.
Active power is constant as the frequency is heldconstant by the generator.
The d-axis current can be obtained from:
Where the generator injected currents are
' ' ' '0 ( )Generator:
AVR: ( )2 ( , )Motor:
d q f q d d d
f o t f
m e
T E E E x x i TE K V V E
Hs T T V s
1 12 1 12 1 2
1 12 1 12 1 2
2 21 2 21 2 1
2 12 2 21 2 1
1 1
2 2
( ) 0( ) 0( ) 0( ) 0
and : Components of the currentinjected by the generator.
and : Components of the
x S y y y
y S x x x
x S y y y
y S x x x
x y
x y
i B v B v v
i B v B v v
i B v B v v
i B v B v v
i i
i i
2 21 1
2 22 2
currentinjected by the load.
: Gen. terminal voltage
: Motor Voltage
t x y
x y
V v v
V v v
1 1 1 1 1 1 1 1 1Re ( )( )x y x y x x y y P v jv i ji v i v i
1 1 1
1
1
1 1
sin coscos sinsin cos
x y
x d
y q
d x y
i i ji
i i
i i
i i i
2 2' '
1 1 1' '
2 2' '
1 1 1' '
sin2 1 1 cos sin( cos ) ( sin )2
cos sin sin2 1 1( cos ) ( sin )
2
x x q y q q q d d
y x q y q
q q d d
i v E v E x x x x
i v E v E
x x x x
' ' ' '0'
'0
( ): behind transient reactance: filed voltage: open-circuit transient time constant
d q f q d d d
q
f
d
T E E E x x i E emf E T
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Modeling ContinuedAVR Model
Induction
Motor
Load Model
Two components: induction motor, and shuntcompensation.
Induction motor currents are function of the equivalentseries resistance and slip. (omitted here)
Shunt Compensation:
Summary:Third order model.
Algebraic variables are function of the network andstate variables but not simple tractable functions.
Torque expression complicates analytical derivations.
Not a straightforward formulation to treat itanalytically.
Approach: use a Power System Simulation Software todetermine fixed points, bifurcation diagrams,
linearization analysis, and to do nonlinear systemsimulations.
: Regulator Gain: Regulator Time Constant: Reference Voltage
: Generator Terminal Voltage
o
t
K T V
V
max
min
00
( )
0 if and ( )0 if and ( )
o t f
f f f o t f
f f o t f
K V V E
T
E E E K V V E E E K V V E
2
2 ( ):Motor Slip:Mechanical Torque
( , ) : Electrical TroqueDeveloped by the Motor
m e
m
e
Hs T T s
s
T
T V s
2
2 2
2 2 21 1
1 1
22
( , )
( ) ( )
, : Equivalent series resistance and reactance, : Stator resistance and reactance, : Rotor resistance and rea
r m
e
r r s s m
s s
r r
RX
T V s R
R X X R X
s
s
R
R
X
V
X
X
X
R
ctance: Magnetizing Reactancem X
3
3
x xM xC
y yM yC
i i i i i i
2 2
2 2
2 2
C S xC yC
xC S y
yC S x
i jB V i ji
i B v
i B v
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Implementation with PSAT
The system was implemented using the Power System Analysis Toolbox [4] viathe PSAT-Simulink Library.
Command line usage and scripting:For the construction bifurcation diagram and eigenvalue location plotsCommand Line Usage of the software had to be learnt. Routines for
automatic variation of Tm where written. Needs careful analysis of thesoftware data types and structures.
Implementation of perturbations:For the analysis of Hopf bifurcation, AVR tuning and Controller Limit AnalysisPerturbation Files where written to introduce a torque pulse as a small
perturbation required skillful programming.
Bus2Bus1
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Bifurcation Diagram: Torque Voltage Curve
The fixed points of the system arecomputed for many Tms variedmonotonically to obtain the bifurcationdiagram.
A Saddle-Node Bifurcation point islocated at point SN.
Point HB indicates a Hopf bifurcationpoint, which determines the actualstability limit.
The effect of Capacitive Compensationat Bus 2 is shown by the red curve.
Observe that even tough the SN* pointoccurs at a higher Tm than SN, the HB*point moves to the left with respect toHB, thereby reducing the stabilitymargin.
0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
1.2
Mechanical Torque, T m (pu)
V o l t a g e ,
V 2
( p u
)
Bifurcation Diagram
SS HB
SN
HB*
SN*
BS2
=0
BS2
=0.6
Note: because the bifurcation diagram was obtainedusing a Power System Simulation program, it is onlypossible to obtain the section of the bifurcationdiagram which contains fixed points which containsacceptable voltage solutions.
Other fixed points will yield low voltage solutions,this type of solutions are not computed by the powerflow routine.
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Eigenvalue Locations
The variation of the eigenvalues of the state matrix as the fixed pointmoves from point SS to point HB on the torque voltage curve for B = 0 andB = 0.6 was obtained. The results for B = 0 are shown here.
The eigenvalue locations are plotte for for Tm:{0.4,1.11}, they are difficult
to interpret at the first glance. Therefore the eigenvalue locations areploted for different ranges of the variation of Tm.
These results are used to construct a more insightful eigenvalue locus.
-20 -10 0 10 20-4
-2
0
2
4
Real
I m a g
i n a r y
Eigenvalue Locations for 0.71 T m 1.11
-15 -10 -5 0 5
x 1063
-4
-2
0
2
4
Real
I m a g
i n a r y
Eigenvalue Locations for 0.4 T m 0.71
-1 -0.8 -0.6 -0.4 -0.2 0 0.2
-2
0
2
Real
I m a g
i n a r y
Eigenvalue Locations for 0.4 T m 0.71
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Eigenvalue Locus
From the eigenvalue plots the eigenvaluelocus is constructed.
At the HB point the pair of complexeigenvalues crosses the jw-axis: thesystem will experiment a HopfBirfurcation.For point HB and SS the third eigenvalueis real and far to the left (not shown in theplots)
At point SNB one eigenvalue becomes
zero, one negative, and one positive. Thisis a Saddle-Node Bifurcation point.
At the SNB point the system is unstableboth before and after it due to thepositive eigenvalue.
-15 -10 -5 0 5 10 15-4
-2
0
2
4
Real
I m a g
i n a r y
Eigenvalue Locations for 0.4 T m 1.11
-15 -10 -5 0 5 10 15-4
-2
0
2
4
Real
I m a g
i n a r y
Eigenvalue Locus
SS
SS
SNSN
HB
HB
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Oscillatory Voltage Instability:Nonlinear Simulation of a Stable Point
Post-disturbance Stable PointChosen before the HB point.
State variables settle at a stable fixed point.
3D plot shows the stable fixed point
System is stable.
0.8972
0.8972
0.8972 1.19361.1936
1.19361.1936
9.4523
9.4524
9.4524
x 10-3
Ef
(pu)
Stable Point Simulation
Eq (pu)
s
( p u
)0 10 20 30 40 500.8972
0.8972
0.8972
E q
Stable Point Simulation
0 10 20 30 40 501.1935
1.1936
E f
0 10 20 30 40 500.9905
0.9905
0.9905
1 - s
0 10 20 30 40 500.9425
0.9425
0.9425
V 2
Time (sec)
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0 2 4 6 8 10 120
0.2
0.4
0.6
0.8
1
1 - s
Time (sec)
Unstable Point Simulation
0 2 4 6 8 10 120.3
0.4
0.5
0.6
0.7
0.8
V 2
Time (sec)
Unstable Point Simulation
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Nonlinear Simulation of an Unstable Point
11.1
1.21.3
2
3
4
0
0.1
0.2
Ef
(pu)
Unstable Point Simulation
Eq (pu)
s ( % )
Post-disturbance Unstable PointThe post-disturbance condition is chosen in an unstableoperating fixed point after the HB point.
Ef and Eq settle at fixed points while the slip(s)
growsunboundedly (partially shown in center plot).
Unstable voltage oscillations force the motor to stall when theoscillation amplitude grows beyond an unstable fixed point.
3D plot shows the trajectory, the red dot indicates the lastpoint of the simulation, not a stable fixed point.
This is an example of Oscillatory Voltage Instability
MotorStalling
Unstable FixedPoint0 2 4 6 8 10 12
1
1.2 E
q
Unstable Point Simulation
0 2 4 6 8 10 12
2
3
4
E f
Time (sec)
f f l
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Hopf Bifurcation
In a previous slides it was noted that aHopf bifurcation takes place when acomplex pair of eigenvalues from thestate matrix cross the jw-axis as themechanical torque is varied.
What type of Hopf bifurcation is it?
A transient torque pulse is imposed ona stable fixed point before the HBpoint.
By adjusting the duration of the pulsea critical trajectory can be found.
The trajectory is trapped in the stablemanifold of the unstable limit cycle.
12
0.90.95
1
0
1
2
0
0.01
0.02
Eq (pu)
Unstable Trajectory Moving Away from the Limit Cycle
E f (pu)
s ( p u )
0 10 20 30 40 50 600.8
0.9
1
E q
Hopf Bifurcation Analysis
0 10 20 30 40 50 600
2
4
E f
0 10 20 30 40 50 600.95
1
1 - s
0 10 20 30 40 50 600.5
1
1.5
V 2
Time (sec)
1t
0t 1t t
0T
T
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0.880.9
0.920.94 1
1.52
2.5
0.012
0.014
0.016
0.018
E f (pu)
Trayectories Sorrounding the Limit Cycle
Eq (pu)
s ( p u
)
Analysis of Hopf Bifurcation andClassification
From the simulation of theunstable post-disturbancepoint we observe thatthere are no attracting limitcycles after the HB.
This is evidence that is likelya subcritical HB.
Figure shows a limit cyclesurrounded by unstable
trajectories moving awayfrom the limit cycle, andstable trajectories movingtowards the SEP.
Therefore, an unstable
limit cycle exists April 24, 2008
13
Unstable Trajectory SEP
StableTrajectory
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0.89 0.9 0.91 0.92 0.93 0.94 11.5
20.013
0.014
0.015
0.016
0.017
E f (pu)E
q(pu)
Unstable Limit Cycle Shrinking
SEP s ( p u
)
Shrinking Unstable Limit Cycle!
To prove that thebifurcation is asubcritical Hopf wefurther require to showthat the limit cycle
shrinks as the HB pointis approached
3D plot was obtainedby imposing differentperturbations.
Observe that thebifurcation emergesfrom the shrinking ofthe unstable limit cyclethat exists before the
Hopf Bifurcation.April 24, 2008
14
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Controller Tuning to Avoid VoltageInstability
An operating condition immediately before the HB is chosen.
A transient torque pulse is imposed at this fixed point.
The response of the system will be oscillatory with the current AVR settings.
Appropriate AVR tuning helps to avoid oscillatory voltage instability.
AVR Model:
Two cases analyzed:CASE 1: Oscillatory condition with the transient torque pulse used in theHopf bifurcation section (unstable behavior onset).Case 2: Oscillatory condition from an increased torque pulse.
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1K
Ts f E
o V
t V
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Controller TuningCASE 1
Torque pulse used in HB illustration
Designed Controller: K = 30 (Nominal), T =0.105 (was 0.5) sec
Slow Response Time constant canbe made smaller: equipment limits
Increase gain with caution.
CASE 2Increased torque pulse
Designed Controller: K = 175, T = 0.105sec
Fast Response at cost ofrequirement higher rating for thefield current: equipment limits needto be assessed
16
0 10 20 30 40 50
1
1.5
2
2.5
3
3.5
Time (sec)
E f
( p u
)
CASE 2: Field Voltage
0 10 20 30 40 50
1
1.5
2
2.5
Time (sec)
E f
( p u
)
CASE 1: Field Voltage
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0 10 20 30 40 500.9
0.95
1
E q
CASE 2
0 10 20 30 40 501
2
3 E
f
0 10 20 30 40 50
0.98
0.985
0.99
1 - s
0 10 20 30 40 50
0.8
0.85
0.9
V 2
Time (sec)
Controller Tuning Continued17
0 10 20 30 40 500.9
0.95 E q
CASE 1
0 10 20 30 40 50
11.5
22.5
E f
0 10 20 30 40 50
0.98
0.99
1 - s
0 10 20 30 40 50
0.750.8
0.850.9
V 2
Time (sec)
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Effect of Controller Limits on Stability
Assumes that the controller cannot be made faster: equipment limitations.
Same cases as in controller tuning are studied.
Controller limits are set to evaluate their effect on stability. Depending on the maximumlimit the AVR may be able (or not) to stabilize the system.
AVR Model:
Four cases discussed:CASE 1a: conditions from CASE 1 tuning, system is stabilized by the AVR.
CASE 1b: conditions from CASE 1 tuning, AVR is not able to stabilize the system.
CASE 2a: conditions from CASE 2 tuning, system is stabilized by the AVR.
CASE 2b: conditions from CASE 2 tuning, AVR is not able to stabilize the system.
18
1K
Ts f E
o V
t V min0 f E
max f E
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Controller Limits Cases 1a and 1b
CASE 1aSystem is stabilized by the AVR.
Controller Limits:Efmax = 2.625
Limit only affects first swing.
CASE 1bAVR is not able to stabilize the system.
Controller Limits:Efmax = 2.622
Minimal decrease in controller limit yieldsinstability
19
0 10 20 30 40 500
0.5
1
1.5
2
2.5
3
Time (sec)
E f ( p u )
CASE 1a: Field Voltage
0 10 20 30 40 500
0.5
1
1.5
2
2.5
3
Time (sec)
E f ( p u )
CASE 1b: Field Voltage
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Controller Limits Continued20
0 10 20 30 40 500.5
1
1.5
E q
CASE 1a
0 10 20 30 40 500
2
4
E f
0 10 20 30 40 500.95
1
1 - s
0 10 20 30 40 50
0.7
0.8
0.9
V 2
Time (sec)
0 10 20 30 40 500.5
1
1.5
E q
CASE 1b
0 10 20 30 40 500
2
4
E f
0 10 20 30 40 50-20
0
20
1 - s
0 10 20 30 40 50
0
0.5
1
V 2
Time (sec)
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Controller Limits Cases 2a and 2b
CASE 2aSystem is stabilized by the AVR.
Controller Limits:Efmax = 3.10
Limit only affects first swing.
CASE 2bAVR is not able to stabilize the system.
Controller Limits:Efmax = 3.09
Minimal decrease in controller limit
yields instability
21
April 24, 2008
0 10 20 30 40 500.5
1
1.5
2
2.5
3
3.5
Time (sec)
E f ( p u
)
CASE 2a: Field Voltage
0 10 20 30 40 500.5
1
1.5
2
2.5
3
3.5
Time (sec)
E f ( p u
)
CASE 2b: Field Voltage
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Controller Limits Continued22
0 10 20 30 40 500.5
1
1.5
E q
CASE 2a
0 10 20 30 40 500
2
4
E f
0 10 20 30 40 500.95
1
1 - s
0 10 20 30 40 50
0.7
0.8
0.9
V 2
Time (sec)
0 10 20 30 40 500.5
1
1.5
E q
CASE 2b
0 10 20 30 40 500
2
4
E f
0 10 20 30 40 50-20
0
20
1 - s
0 10 20 30 40 50
0
0.5
1
V 2
Time (sec)
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Controller Limits Effect on Trajectories
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0.80.911.1
1.21.3
0
2
4
0
2
4
6
E f (pu)
Effect of Limits on Trajectories - Case 2b
Eq (pu)
s ( p u
)
Non-smoothTrajectories as a resultof controller limitsSystem Unstable
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ConclusionsBifurcation analysis showed the existence of a SNB and a HB.
Capacitive compensation reduces stability margin by moving the HB to the left.Linearization analysis provided evidence of the existence of a HB.
Stable operating points are found before the HB. Unstable operating points are found after the HB.Hopf Bifurcation:
Complex pair of eigenvalues cross the jw-axis.Simulation of post-disturbance point shows that there are no attracting limit cycles after the HB.A limit cycle is surrounded by unstable trajectories moving away from the limit cycle, and stable trajectories moving towards the SEPThe limit cycle is unstable .Showing that the limit cycle shrinks as the HB point is approached proves that the bifurcation is a subcritical Hopf bifurcation .Bifurcation emerges from the shrinking of the unstable limit cycle that exists before the HB.
AVR Tunning:Faster controller avoids oscillatory voltage instability.The burden on the field voltage increases as the perturbation grows important design constraint.
Effect of Controller Limits:Minimal decrease in controller limits may disable the ability of the AVR to stabilize the system.For engineering design: worst case scenario must be considered.
Remarks:The oscillatory behavior of the system emerges from the interaction of two load restoration processes acting in thesame short-term time scale.The load restoration due to the adjustment of the slip (which is product of the admittance increase) races againstthe voltage regulation provided by the AVR (which restores the load indirectly by restoring the voltage).
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References
1. Thierry Van Cutsem, and Costas Vournas, Voltage Stability of Electric PowerSystems. Boston: Kluwer Academic Publishers, 1998.
2. Carson Taylor, Power System Voltage Stability . New York: McGraw-Hill, 1994.
3. F. P. de Mello, and J. W. Feltes , Voltage Oscillatory Instability Caused byInduction Motor Loads, IEEE Transactions on Power Systems
, vol. 11, no. 3, pp.1279 1285.
4. F. Milano, An Open Source Power System Analysis Toolbox, IEEE Transactionson Power Systems , Vol. 20, No. 3, pp. 1199-1206, August 2005.
5. Steven H. Strogatz, Nonlinear Dynamics and Chaos. Reading, MA: Perseus
Books, 1994.6. Pal, M.K., "Voltage stability: analysis needs, modelling requirement, and
modelling adequacy," IEE Proceedings-Generation, Transmission and Distribution , vol.140, no.4, pp.279-286, Jul 1993
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