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Date: Thu, 16 Sep 93 09:41:20 CDT
From: kamala@CS.MsState.Edu
Message-Id: <9309161441.AA03453@dale.CS.MsState.Edu>
To: paulc@npac.syr.edu
Subject: LaTex file sum.tex (for DAE solvers)
Status: R

\documentstyle[11pt]{article}
\nonstopmode
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\begin{document}
\begin{center}
\large{
{\bf Study of DAE solvers for Transient Stability Analysis}}\\
\end{center}
Transient Stability Analysis examines the dynamic behaviour 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

\vspace{-0.15in}
\begin{center}
{\em $\dot{y}$ = f(y,z) }\\
\end{center}
In our case, the exact equations are 
\begin{center}
 I = VY 
\end{center}
\vspace{-0.15in}

\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

\vspace{-0.15in}
\begin{center}
{\em 0 = g(y,z) }
\end{center}
The exact equations are: 
\begin{center}
{\em
E +$x_{d}I_{d} = E^{'}_{q} + x^{'}_{d}$\\
\medskip
$E_d + x_{q}I_{q} = E^{'}_d + x^{''}_{q}I_{q}$\\
\medskip
${\tau}^{'}_{q0}{\dot{E}}^{'}_{d} = -E^{'}_{d} - (x_{q} - x^{'}_{q} )I_{q}$ \\
\medskip
$\dot{E}^{'}_{q} = {\frac{1}{T^{'}_{d0}}}(E_{FD} - E)$ \\
\medskip
$T_{e} = E^{'}_{d}I_{d} + E^{'}_{q}I_{q} - (L^{'}_{q} - L^{'}_{d})I_{d}I_{q}$\\
\medskip
${\tau}_{j}{\dot\omega} = T_{m} - D{\omega}-[E^{'}_{d}I_{d}+E^{'}_{q}I_{q} - (L^{'}_{q} - L^{'}_{d})I_{d}I_{q}]$\\
\medskip
$\dot{\delta} = \omega - 1 $\\
\medskip

}
\end{center}
\vspace{-0.15in}

\noindent and this entire system of DAEs must be solved simultaneously.


{\em \bf NOTE: For the terminology description of the above equations, please
refer to: Power System Stability and Control by Anderson.}\\

Depending on the load equations and control equations that need
to be added to this system, our system will become {\em Index-1} or {\em
Index-2} system. The algebraic system as described above presents a very
interesting scenario, in that it is a system of complex variables. We
then are faced with the choice of solving it as a system on N complex
equations or 2-N real equations. But we may lose the sparsity structure 
in the process. We are investigating this issue now.

The generator equations by themselves do not pose any problem except
when the load equations and control equations are added. But they still
will be able to satisfy the standard input format of most DAE solvers.
Hence one of the reasons to pick a particular DAE solver will be the
form of equations it requires as input. Keeping this in mind, the 
following solvers were analysed and here are the findings:\\

\begin{enumerate}

\item LIMEX - an extrapolation integration technique. \cite{Brenan}.
      This can be used to solve semi-explicit index-1 DAEs and uses the
      semi-implicit Euler method. This is a strong contender because of
      its ability to handle frequent discontinuities, which is a contidtion
      that frequently occurs in Power Systems TSA problem.

\item RADAU5 - an implicit Runge-Kutta method of order 5. \cite{Hairer}.

\item RODAS - an embedded Rosenbrock technique of order(3)4. \cite{Hairer}.
             This is similar to R-K except that at each step the Jacobian
         has to be evaluated and the linear systems have to be solved again. 
Also, this can handle only Index-1 DAEs.

\item SODEX - an extrapolation technique based on linearly implicit mid-point rule. \cite{Hairer}.
     One disadvantage of this method is that it takes large step sizes, which
may cause exponential overflow when evaluating the right hand side of ODEs.

\item SEULEX - an extrapolation technique based on linearly implicit Euler method. \cite{Hairer}.
\item DASSL - a multistep, backward differentiation formulas technique.\cite{Brenan}. 

      This is designed for DAEs of Index one or zero. CDASSL (concurrent
      DASSL) has been developed already.

\item LSODI - a fixed leading coefficient method using BDF technique.\cite{nmg-h83,nmg-Hi82,nmg-Hi85b}.
      In contrast to DASSL (which is a fixed leading coefficient method),
LSODI is a 'fixed coefficient' method. The fixed coefficient methods can be
implemented very efficiently for smooth problems, but suffer from 
inefficiency, or possible instability, for problems that require frequent
stepsize adjustments, which is not the case with power systems problem.
Hence LSODI promises to be our best choice currently.
     
\end{enumerate}

{\em The backward differentiation formulas (BDFs) methods achieve the
same order of convergence for this class of DAEs as they do for ODEs.
Implies, even if our system is of Index-2, we can achieve convergence.
Also, BDF methods can handle discontinuities, which is a condition
frequently encountered in our problem.
Thus there is some theory providing the foundation for the software
based on these methods. However, there are other issues that need to
be considered too while selecting a solver for the problem in question.}

For now, our target is LSODI and LIMEX , with possible diversion to a implicit
version of VODE called VODI. Solvers like LSODI and LIMEX have been used
extensively and we have data that proves their efficiency. Solvers like
RADAU5, RODAS, SEULEX and SODEX might be efficient and probably more
suitable but we have absolutely no data or opinions from any users. Hence
the choice of LSODI (since a parallel version of DASSL is underway...)
seems apt.

\vspace{-0.15in}
\noindent
\bibliographystyle{plain}
\bibliography{sum}
\end{document}
\endinput

\end{thebibliography}
\end{document}
\endinput

