Bifurcation and Manifold Based Approach for Voltage and Oscillatory Stability Assessment and Control

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Bifurcation and Manifold Based Approach for
Voltage and Oscillatory Stability Assessment and
Control

• V. Ajjarapu(vajjarap_at_iastate.edu)
• Iowa State University
• Presentation at NSF workshop on applied math for
  power , Nov. 4, 2003

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Outline

• Need For Nonlinear Techniques
• Bifurcation Frame Work
• Motivation
• A single tool for cost based fast monitoring and
  control for voltage and oscillatory stability
  (including timing decisions )
• Formulation
• Research Needs
• Impact of this work

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Jackson
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Motivation Formulate power flow , optimal
power flow or time domain simulation as solving a
set of nonlinear equations in the form
H(x,y,t,..) 0 Apply Continuation and
Manifold based techniques
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Margin Boundary Tracing (MBT)
V
P
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Security Margin Tracing
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Power System DAE Model and Parameterization
Load
Generation
Control
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Manifold Based Method to Identify Voltage Hopf
Points
For Voltage Singularity of Atotal Let the
maximum eigenvalue of is ?1, then ?1 0 is
the necessary condition for Hopf
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Identification of Hopf Cont..
Only Need to Detect The Singularity
of (Atotal AtotalT) Similar to
Continuation Power Flow
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Identification Cont
If ?H1 or ?H2 Changes sign
Hopf bifurcation

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Tracing Oscillatory Instability Boundary
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Tracing Cont..
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Tracing Cont..
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The approach
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Hopf Detection
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Example For Voltage Stability Boundary Reactive
Power Variation
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1400
Voltage
Oscillatory
1300
1200
1100
maximum total system load margin(MW)
1000
900
800
700
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
single contingency line 6-31 outage
Voltage and Oscillatory stability margins with
line outage
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Damping Margin
Damping Ratio
Case A
Case B
100MW
100MW
D0
Parameter a
0
a1 a
a2
a4
a3
Two cases with different damping ratio margins
and the same oscillatory stability margin
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Eigenvalue Movement
Im
Re
0
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Extensions Eigenvalue Tracing
The eigenvalue ? of an any parameterized matrix
M(?) can be obtained at any ? by
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Extension to Power System DAE Model
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Real Part of Dominant Eigenvalue
Real part of dominant eigenvalue
0
Hopf
Tangent slope is
Conventional method uses secant to predict.
The tangent information can be used for
predicting the next parameter value
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Numerical Results Eigenvalue Tracing
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Derivative information
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Ranking Critical Eigenvalues
Represents Influence of parameter change on an
eigenvalue crossing the imaginary axis.
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Top 3 Eigenvalues Tracing for New
England System
Hopf Point Eigenvalue Position
Ranked No. 3 Eigenvalue
Ranked No. 2 Eigenvalue
Base Case Eigenvalue Position
Ranked No. 1 Eigenvalue
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Damping Ratio Margin

• At each iteration, the damping ratio is given by
  
• The derivative of damping ratio can be
• Similar method with predictor and corrector can
  be applied to calculate the damping margin.

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Damping Ratio Margin
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Damping for the Top Three Critical Eigenvalues
Ranked No. 3 Eigenvalues Damping
Ranked No. 2 Eigenvalues Damping
Ranked No. 1 Eigenvalues Damping
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Optimal Margin Boundary Tracing
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Basic Formulation
For a sequence of
specified margin levels  
Minimize a cost function f(U)  
Subject to Equality
Conditions F, G
Inequality constraints I

Optimality conditions
Margin boundary
conditions Cobt   Results in a set of
nonlinear equations of the form  

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Formulation (Cont.)

• The formulation could automatically generate a
  whole set of cost based optimal control
  configurations with each optimal control
  configuration corresponding to specific margin
  level that can be realized.
• 
  

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Sample Result for OMBT
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Continuation Based Time Domain Simulation
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Key Points

• No divergence due to singularity of
• Time step size is adjusted according to the
  variation of dS/dt
• Small time steps for fast dynamics and large time
  steps for slow dynamics

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Trajectory Before and After a Line Outage (with
load variation)
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QSS Simulation
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Overall Strategy
Credible Contingency Set CjC1,C2, ,Cm
j j 1
Yes
Check for (AAT) singularity
No
Trace Eigenvalue for Hopf and Damping Margin
Yes
Yes
No
No
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Research Needs

• On line monitoring Fast calculations
• Accuracy vs Speed
• Timing Information Fast vs Slow Collapse
• Integration of Cost of Voltage and Oscillatory
  Stability Control in Overall Security Assessment

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IMPACT

• Cost Based Preventive and Corrective strategies
  that are very effective for stressed power
  systems
• When, Where and How Much control to Maintain
  Required Margins
• A single tool can be adapted for steady state,
  optimization as well as time domain simulation.
• By Products
• Various Sensitivities
• What if Analysis