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{\large An Abstract for a Poster at} \\
{\large \bf The Scalable Parallel Libraries Conference} \\
{\large \bf 6-8 October 1993}
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{\large \bf Parallel Differential Algebraic Equations (DAE) Solvers for\\
Electrical Power System Transient Stability Analysis} \\
by \\
{\large Kamala Anupindi, Anthony Skjellum, Paul Coddington, Geoffrey Fox}
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\bigskip

Transient Stability Analysis examines the dynamic behavior of a Power
System for as much as several seconds following a power-flow disturbance.
The reaction of a single generator to variations in its load can be
modeled using ODEs as follows:

\begin{center}

{\em $\dot{y}$  = f(y,z) }

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\noindent However, there are multiple generators supplying power to the network
and the generators are coupled through the power network. The power network
can be modeled by non-linear algebraic equations as follows:

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{\em 0 = g(y,z) }
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\noindent and this entire system of DAE must be solved simultaneously. Transient
Stability Analysis is computationally intensive because of the extremely
large number of equations to be solved. 
The dynamics of each generator are individually 
modeled with as few as two and with as many as forty Ordinary Differential 
Equations(ODEs) while the
generators are coupled via the algebraic equations that describe the
electrical network. This requires the solution of large systems of
simultaneous sparse non-linear equations.

\medskip
Real-time or faster-than-real-time power system transient stability 
simulations will have significant impact on the future design and
operations of both individual electrical utility companies and large
interconnected power systems. Several sequential DAE solvers exist, that
use various numerical techniques for solving the DAE. Prominent among these
are the following:

\begin{enumerate}

\item LIMEX - an extrapolation integration technique \cite{Brenan}.
\item RADAU5 - an implicit Runge-Kutta method of order 5 \cite{Hairer}.
\item RODAS - an embedded Rosenbrock technique of order(3)4 \cite{Hairer}.
\item SODEX - an extrapolation technique based on linearly implicit mid-point rule \cite{Hairer}.
\item SEULEX - an extrapolation technique based on linearly implicit Euler method \cite{Hairer}.
\item DASSL - a multistep, backward differentiation formulas technique
\cite{Brenan}.
\item LSODI - a fixed leading coefficient method using BDF technique
\cite{nmg-h83,nmg-Hi82,nmg-Hi85b}.

\end{enumerate}

But to date, there has been no explicit mention in the 
power system transient stability analysis literature of the application
of DAE solvers. These solvers from the numerical analysis
community often use higher order implicit Runge-Kutta integration
techniques or Backward Differentiation Formulas (BDF) 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.
\medskip

CDASSL (Concurrent Dassl) \cite{Skj90d}
is the only attempt that has been made at
parallelizing these DAE solvers, as far as we know. There is enormous potential for research in this field. 
Our research is examining the scalability of various DAE solution
techniques. Our research is concentrated on selecting the most suitable 
solver for the power system application, and creating a scalable library
for an application based on it. The parallel version of this DAE 
solver will then be developed for use by the power system community,
and in related applications {\em e.g.}\cite{Skj90d} .

\medskip
We will describe our research on the parallelization of the selected
DAE solver, with reference to the power system problem. We will then
present the results thus far for the parallel DAE solver using
Multicomputer Toolbox \cite{Skj93c} ; this software is currently
based on the portable message-passing system {\sl Zipcode\/} \cite{Skj93b};
 the Toolbox will port to MPI next year and hence our work will run on all
reasonable multicomputers.

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