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Transient stability analysis is the study of a power system for several
seconds following a disturbance. It involves the study of loads, electrical
network and electro-mechanical equations of motion. The dynamics of each
generator in the system are described by several differential equations.
The coupling of the generators with the electrical network is decsribed
by algebraic equations. \\

The variation in the load of a single generator can be modeled using 
ordinary differential equations (ODEs):

\begin{equation}
 \dot{y} = f(y,z) 
\end{equation}

The power system network itself can be modeled by non-linear algebraic
equations:

\begin{equation}
 0= g(y,z) 
\end{equation}

The above two equations together lead to a system of differential-algebraic
equations ( DAEs) that must be solved simultaneously. Also, since the large
system of DAEs needs to be solved at small time increments to minimize the
errors in numerical integration of potentially computationally-stiff ODEs,
the transient stability analysis becomes a compute intensive problem.\\

The transient stability simulation can be described by three software 
components:

\begin{enumerate}
\item user interface
\item DAE solver
\item analytical software to classify network
\end{enumerate}

The number of equations required to solve the transient stability DAEs are
extremely large and sparse. For the NiMo system, we have a total of 1766
buses with (292) generator buses and (1473) load buses. If we have twenty
differential equations to describe each generator and two equations to
describe the complex voltage/current at each network bus, we then have
8788 non-linear equations that must be solved simultaneously. This leads 
to the vast majority of machine
cycles of the stability simulation being taken up by a DAE solver.\\

Various numerical techniques have been used to calculate the solution of
DAEs of the transient stability problem. But there has been no
mention in the power system transient stability analysis literature of
applications using public domain DAE solvers. DAE solvers exist that
handle the discontinuities in functions which is the conditon encountered
in transient stability calculations.\\

Traditional approaches to solving the transient stability DAEs have used:

\begin{enumerate}
\item $partitioned~ approach$ where the differential equations and non-
linear algebraic equations are decoupled and all generator equations are
updated for a time step. The network equations are then solved to get an
improved update for an estimate of the voltage at taht sample time.
 
\item $linearization~ approach$ where
the generator equations are linearized at a recent point on the solution
trajectory by taking numerical differences and the non-linear algebraic
equations are then solved.

\end{enumerate}

But the main problem with the above techniques is that they use algorithms
that are inherently sequential i.e. they break up the problem (matrix) into
sub problems (sub-matrices) and solve them sequentially. This approach
destroys the natural physical structure and sparsity of the full order
system. Therefore, it is desirable to solve the system in their original
DAE form. This is conducive to parallelization too as by preserving the 
original form of the matrix, it helps in re-ordering the matrix to a
bordered block diagonal form. This structure in turn helps in
 $load~ balancing$.\\

Numerous computing techniques have been reported to solve DAEs that have
been used as the basis for various DAE solvers. Some are:

\begin{enumerate}
\item DASSL -- a multistep, backward diefferentiation formulas (BDF)
technique
\item GEARS -- a multistep, modified Newton's iteration technique
\item LIMEX -- an extrapolation technique
\item RADAU5 -- a multistep, implicit Runge-Kutta based technique
\item RODAS -- a Rosenbrock technique
\item SEULEX -- an extrapolation technique
\end{enumerate}

We believe that using Rosenbrock techniques might be
advantageous as they completely avoid non-linear systems of equations while
providing the advantages of accurate solutions with stiff differential
equations. This requires that the DAEs be of $index~one$.
$Index~ One$
DAEs are those whose algebraic equations when differntiated $once$, can be
represented as a system of explicit differential equations which is true
of power system transient stability DAEs.
RADAS is based on this technique. A parallel version of DASSL
might be suitable too for this problem as it is based on a multistep BDF
technique and also
has the capabilities of locating discontinuities which is a condition 
encountered in transient stability analysis. All software discussed above 
is available through internet.\\

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