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Differential-Algebraic Equation Solvers for Power System Transient Stability Simulations

A transient stability simulation is composed of three major software components:

  1. the user interface
  2. the DAE solver, and
  3. 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:
  1. iterated timing analysis
  2. time-domain parallelism techniques
  3. waveform Newton techniques
  4. waveform relaxation techniques, and
  5. network analysis software similar to that used in"SPICE".
Each of these techniques requires additional numerical analysis techniques such as:
  1. numerical integration techniques using the trapezoidal rule
  2. solutions of non-linear equations by Newton's methods
  3. solutions of non-linear equations by relaxation methods
  4. solutions of non-linear equations by Picard iteration
  5. solutions of linear equations by direct methods, and
  6. 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:
  1. 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.
  2. 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:

  1. multistep backward differentiation formulas (BDF) techniques
  2. extrapolation techniques
  3. Runga-Kutta techniques, and
  4. 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:

  1. DASSL --- a multistep, backward differentiation formulas (BDF) technique [9]
  2. LIMEX --- an extrapolation technique [9]
  3. LSODI --- a BDF technique [9]
  4. RADAU5 --- a multistep, implicit Runga-Kutta (IRK) based technique [21]
  5. RODAS --- a Rosenbrock technique [21], and
  6. 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 up previous
Next: Relevant Transient Stability Up: Power Systems Transient Stability Previous: Techniques to Speedup



David P. Koester
Sun Oct 22 16:35:54 EDT 1995