Next: Relevant Transient Stability
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A transient stability simulation is composed of three major
software components:
- the user interface
- the DAE solver, and
- analytical software to classify network stability.
with the vast majority of machine cycles being utilized by the DAE
solver. Numerous general numerical analysis techniques have been
addressed in the power systems literature that can be applied to the
solution of the DAEs [13]. This list of techniques include:
- iterated timing analysis
- time-domain parallelism techniques
- waveform Newton techniques
- waveform relaxation techniques, and
- network analysis software similar to that used in"SPICE".
Each of these techniques requires additional numerical analysis
techniques such as:
- numerical integration techniques using the trapezoidal rule
- solutions of non-linear equations by Newton's methods
- solutions of non-linear equations by relaxation methods
- solutions of non-linear equations by Picard iteration
- solutions of linear equations by direct methods, and
- solutions of linear equations by iterative methods.
There have been several competing recurring general techniques used to
solve the DAEs.
[3] describes two techniques for solving the transient stability DAEs
that are indicative of the aforementioned list:
- the partitioned approach --- where at each time step, the generator differential
equations are solved independently to obtain estimates of generator
and load currents, then the non-linear algebraic equations are solved
to get an improved update for an estimate of the generator voltages at
the sample time.
- the linearization approach --- where the generator equations and network
equations are linearized, according to Newton's method, at a recent
point on the solution trajectory by taking numerical differences and
the linear algebraic equations are then solved by appropriate
techniques.
The iterated timing analysis is an application of the partitioned
approach, while waveform Newton techniques are an instance of a
linearization approach.
These numerical DAE solution techniques have been applied to
the transient stability problem, although, there has been no explicit
mention in the power systems transient stability analysis literature
of applications using public domain DAE solvers. These numerical
analysis techniques are based on the following concept: to avoid
numerical differentiations, instead perform analytical
differentiations of the given equations until they can be represented
as a system of explicit differential equations [21]. The
reason for avoiding numerical differentiation is to reduce errors when
taking numerical differences, and the reduction of errors in the
numerical techniques is a critical factor that yields compound
dividends. It is possible that the reduction in computational errors
may permit trade-offs by permitting a reduction in the number of
required state variables or generator equations, and trade-offs may be
possible that attempt to keep numerical error constant while
permitting an increase in time-step interval.
DAE solvers from the numerical analysis community often use
higher order implicit Runga-Kutta integration techniques that yield
more accurate solutions or permit larger time-steps. DAE solvers
exist that have been explicitly developed to handle discontinuities in
functions --- a condition encountered in transient stability
calculations. Much research is possible to determine the
applicability of various DAE solution techniques to both algorithmic
speedup and parallel processing speedup.
Only recently has the numerical analysis community provided
various public domain software packages that utilize specialized
techniques for the numerical solution of initial-value DAE problems
[9,21,49]. There is the distinct possibility that public
domain DAE solvers could significantly speedup the concurrent solution
of the transient stability differential-algebraic equations in a
manner similar to a petrochemical engineering application [37].
There are numerous competing techniques reported in the literature to
solve DAEs:
- multistep backward differentiation formulas (BDF) techniques
- extrapolation techniques
- Runga-Kutta techniques, and
- Rosenbrock techniques.
This list is not intended to be all inclusive, nevertheless, it does
illustrate a rich, untapped source of numerical analysis techniques
that could be utilized to improve the performance of transient
stability simulations on parallel architectures.
Various DAE solvers are available from the numerical analysis
community, for example:
- DASSL --- a multistep, backward differentiation formulas (BDF) technique [9]
- LIMEX --- an extrapolation technique [9]
- LSODI --- a BDF technique [9]
- RADAU5 --- a multistep, implicit Runga-Kutta (IRK) based technique [21]
- RODAS --- a Rosenbrock technique [21], and
- SEULEX --- an extrapolation technique [21].
These DAE solvers offer the promise of many benefits that can be
utilized by parallel implementations of transient stability analysis.
There are numerous versions of DASSL that have been designed for
specific purposes, including a concurrent version, CDASSL [37],
which utilizes a concurrent, direct, non-symmetric sparse matrix
solver. There is also a variant of DASSL, DASRT, that has root
finding capabilities to locate discontinuities when they are
sufficiently large that DASSL cannot integrate through without
intervention [24]. Such a capability could be important for
transient stability simulations because some of the physical phenomena
involved may be discontinuous [3]. IRK techniques may offer
potential advantages to the transient stability analysis of power
systems, because the generator ODEs may be stiff equations, with
eigenvalues lying close to the imaginary axis [41]. This
phenomena requires high order A-stable or nearly A-stable integration
formulas and may benefit from codes such as RADAU5 [21].
Rosenbrock methods have advantages over IRK-based methods
because they completely avoid non-linear systems of equations while
providing the advantages of accurate solutions with stiff differential
equations. However, to use Rosenbrock methods, the DAEs must be of
index one and expressible in semi-explicit form [21]. This
technique is implemented in the software RODAS. Extrapolation
techniques utilize linearly implicit Euler methods and can be
excellent choices when strict tolerances are required for implicit
index one DAEs. LIMEX and SEULEX are implementations of this
technique [9,21]. All software discussed above is
available through the Internet [21,49].
Next: Relevant Transient Stability
Up: Power Systems Transient Stability
Previous: Techniques to Speedup
David P. Koester
Sun Oct 22 16:35:54 EDT 1995