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Step by Step Eigenvalue Analysis with EMTP Discrete
Time Solutions
PhD University Oral Exam, September 29th 2006 J. A. Hollman
SupervisorDr. J. R. Martí
The University of British ColumbiaDepartment of Electrical & Computer Engineering
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Presentation Outline
• Introduction• Framework• New methodology• Test cases • Conclusions
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Motivation• Evolution of power systems
– Generation capacity 171GW (‘60), 1049 GW (‘06)• De-regulated environment (only US)
– Capacity margin 25% (‘05) 18.8% (‘06) 15.5% (‘08)
– Transmission lines High Voltage 237,009 km (‘93)255,250 km (‘02)
– Revenue all sectors198.2b $ (‘93) 270.4b $ (‘04)
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Motivation (cont.)
• Previous operational paradigm– Reliability oriented
• New operational paradigm– Revenue oriented
• Power systems stability analysis– Off-line vs. real-time simulation
• Research opportunity– Dynamic location of limits – Support allocation of investment– Locally coordinated operation
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UBC power systems group simulation projects: Integration and evolution
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Framework
• The problem– Limitation of current Transient Stability analysis tools
• Fast time-domain (FTD)• Prony spectral analysis• Transient rotor angle analysis
• EMTP extended capabilities– Step by step trajectory analysis– Non-linearities modelling– Modelling accuracy
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Framework (cont.)
(TSAT) Trapezoidal, 10 ms FTD
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New Methodology
Eigenvalue analysis from EMTP solution
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State-space formulation continuous time
and for the single input/output discrete system
Multiple input/multiple output
Output of the system
Dynamics of the system
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Discrete state-space formulation
Classical form – Forward Euler
Output of the system
Dynamics of the system
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EMTP solution
Trapezoidal rule
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Output equation
Dynamic part
Discrete state-space from EMTP
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branch voltage and current
updating formula
Discrete time state-space equation of basic elements - Inductor
thus, the discrete state-space equation of a self inductor
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Discrete time state-space equation of basic elements – Series RL and RC
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Treatment of series branches
branch history terms
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Where
and the relationship between branch and node voltages is given by the incidence matrix
the Transition matrix [A] is then given by
Treatment of series branches (cont.)
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Treatment of parallel branches
We can keep the identity of each component or treat them as a new equivalent aggregatedparallel RLC.
The general formulas are maintained and we define [A] as for the single branch case
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Treatment of Non-linear elements
• Non-linear elements can be made up of piecewise linear segments.
• A change of piecewise segment corresponds to a new set of eigenvalue.
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Hybrid real-time/soft real-time simulator layout
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Discrete to continuous time mapping
For Trapezoidal
The continuous time eigenvalues can be reconstructed from the discrete ones by
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Automatic EMTP time step selection scheme
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Discretization time considerations
• Linearization of differential equations: Nyquist frequency.
• Non-linear elements: small time step for accurate representation of region change.
• As long as eigenvalue frequency is below the Nyquist freq. reconstructed cont. time eigenvalues are “exact”
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Test cases
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Comparison of state-space formulation between continuous and discrete time domains
continuous time state-space system equations
selecting VR(t) and VC(t) as statesvin(t) as input and VL(t) as output
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Discrete time state-space system equation
the branch histories
the discrete transition matrix [A] for the RLC series computed from the nodal eq.
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Continuous time eigenvalues
Discrete time eigenvalues
Reconstructed Continuous time eigenvalues
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Eigenvalue trajectory of a RLC series with a non lineal L
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Eigenvalue trajectory of a RLC series with a non lineal L (cont.)
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Identification of segmentation areas - Latency application
Continues time domain
Continues time eigenvalues
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Identification of segmentation areas - Latency application (cont.)Discrete time domain
Reconstructed Continues time eigenvaluesDiscrete time eigenvalues
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Identification of segmentation areas - Latency application (cont.)
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Voltage collapse of a radial system
Continuous time eigenvalues Load increment profile
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Voltage collapse of a radial system (cont.)
Discrete time eigenvalues Voltage collapse
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Voltage collapse of a radial system (cont.)
Reconstructed Continuous time eigenvalues Voltage collapse
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Voltage collapse of a radial system (cont.)
Anticipation of voltage drop from eigenvalue trajectory
250ms (A1) ; 300ms (B) ; 200ms (C) ; 450ms (D)
40-60 ms 500kV interrupter operation / 120-200 ms DAG 1000 km (optic/microwave)
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Conclusions
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Research Contributions
The description and implementation of a new and original power system stability assessment
methodology that identifies the system’s eigenvalues trajectories in a real-time EMTP
solution incorporating the effect of switching and non-linear behaviour.
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Research Contributions (cont.)Advantages of Discrete state-space formulation from EMTP
• Trajectory tracking of non-linear elements eigenvalues moment by moment.
• In the context of OVNI, the capability of identifying suitable network partitioning schemes for application of multi-step integration solution in a hybrid power system simulator environment.
• Visualization of eigenvalues trajectories in discrete time domain for the purpose of assessing power system’s dynamic behaviour.
• Automatic selection of discretization step from discrete time eigenvalue information
• Extension of EMTP capabilities to perform transient and voltage stability studies
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Possible application extensions & future work• Distributed intelligent control solutions based on embedded OVNI and
eigenvalue trajectories.
• Integration of discrete state space eigenvalue methodology with Latency.
• Discrete state space eigenvalue methodology within UBC’s OVNI-NET simulator for stability analysis and determination of segmentation schemes.
• Discrete state space eigenvalue methodology within UBC - JIIRP’s I2Sim simulator for identification of trajectories of critical interdependencies among Critical Infrastructures.
• Development of new Visualization tools to provide simplified information about stability system trajectory to control center operators.
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Thank you