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Parallel Di erential-Algebraic Equation Solvers for
Power System Transient Stability Analysis
Transient Stability Analysis (TSA) 1].

Real-time or faster-than-real-time power system tran-
Transient Stability Analysis is a compute-intensive
sient stability simulations will have signi cant im-
problem requiring the solution of nonlinear, discontin-
pact on the future design and operations of both in-
uous systems of DAEs for the purpose of simulating
dividual electrical utility companies and large inter-
such electrical power systems phenomena as network
connected power systems. The analysis involves so-
component and transmission line failures. Existing
lution of extremely large systems of di erential and
DAE solvers often use higher order implicit Runge-
algebraic equations. Di erential-Algebraic Equation
Kutta integration techniques or Backward Di eren-
(DAE) solvers have been used to solve problems simi-
tiation Formulas (BDF), techniques that yield more
lar in nature to the transient stability analysis (TSA)
accurate solutions or permit larger time-steps. DAE
problem. This paper discusses the possibility of the
solvers have been especially developed to handle dis-
use of the existing DAE solvers to solve the transient
continuities in functions, a condition encountered in
stability analysis application. We also discuss our re-
search in developing a scalable, parallel DAE solver
for use by the power system community and in related
This paper describes research on the use of existing
DAE solvers to solve the power system TSA problem.

In section 2, a detailed description of the power sys-
tem problem is given. In section 3, we discuss various
existing DAE solvers and their potential for applica-
tion to the TSA problem. Our ongoing research in the
In engineering and science, many problems give rise
benchmarking of various sequential solvers and the de-
to smooth and/or discontinuous systems of di eren-
velopment of parallel solvers is discussed in section 4.

tial equations coupled with nonlinear algebraic equa-
Lastly, in section 5, our conclusions on the application
tions. These coupled systems of di erential and al-
of DAE solvers for TSA problem are discussed.

gebraic equations are commonly known as di eren-
tial/algebraic equations (DAEs). Research in the eld
of DAEs started about a decade ago and is still in
its early stages. A number of engineering and sci-
ence problems have been solved with the help of DAE
solvers developed by 2, 3, 5, 10, 13]. But to this date,
Transient Stability Analysis (TSA) examines the dy-
there has been no investigation of the application of
namic behavior of a power system for as much as sev-
DAE solvers to power systems problems such as the
eral seconds following a power- ow disturbance. The
general nature of the transient stability problem is ex-
This work has been supported in part by Niagara Mohawk
Power Corporation,the New York State Scienceand Technology
plained in 11] as follows: Consider a simple mechani-
Foundation, the NSF under co-operative agreement No. CCR-
cal analog where a number of masses, representing the
9120008, and ARPA under contract No. DABT63-91-K-0005.

generators in an electric power system, are suspended

from a \network" consisting of elastic strings repre-
senting the electric transmission lines. Now one of
the strings is suddenly cut, corresponding to a sudden
loss of a transmission line. Thereafter the masses will
experience transient coupled motion, and the forces
j and 0q are the time constants involved in the
in the strings will uctuate. This sudden disturbance
may cause one of the following two e ects:
axis, -axis, -axis and -axis components of the ele-
1. The system will settle down to a new equilibrium
ments described. Detailed explanation on the notation
state, characterized by a new set of string forces
involved in the described equations can be found in 1].

corresponding to the power voltages in the elec-
There are multiple generators supplying power to
2. Because of the transient forces, one or more addi-
the network and the generators are coupled through an
tional strings will break, causing a weaker net-
electrical power network. The power network can be
work, resulting in a chain reaction of broken
modeled by nonlinear algebraic equations as follows:
strings and total collapse of the system. If the sys-
tem has the strength to survive the disturbance
and settle into a new steady state, it is referred
where is the network current, is the network volt-
to as \transient stable for the fault in question."
age and is the network admittance. Apart from
these two sets of equations, TSA involves equations
The transient stability analysis problem can be de-
for loads (which can be di erential or algebraic) and
scribed by a system of DAEs where the generators are
equations for the generator control (that are di eren-
represented by di erential equations and the power
tial). All these equations together form a system of
system network interconnecting the generators is rep-
DAEs that must be solved simultaneously.

resented by nonlinear algebraic equations. The reac-
tion of a single generator to variations in its load can
be modeled using the following Ordinary Di erential
bility analysis DAEs are non-symmetric in nature.

They are of bordered-block-diagonal form wherein
blocks of generator equations along the diagonal are
coupled with the power system distribution network.

The admittance matrix involved is extremely large,
complex and sparse. Research is being conducted 9]
to reorder this matrix into block-diagonal-bordered
form in order to exploit the structure for parallelism.

tions involved in the transient stability analysis solu-
tion is extremely large. Consider an example of 20 dif-
ferential equations to describe each generator and two
equations to describe the complex voltage/current at
each network bus. Then an interconnected power sys-
The equations described above roughly describe the
tem with 2000 buses and 300 generators could generate
\structure" of the di erential equations involved in
a sparse, unsymmetric system of 10,000 nonlinear al-
the transient stability analysis DAEs. The notation
gebraic equations that must be solved simultaneously.

used in the above equations is as follows:
For regional power systems, the number could be ve
is the magnitude of source voltage of the gen-
of a system is the minimal number of analytical di er-
entiations needed such that the di erential equations
e and m are the electrical and mechanical
of the DAE can be transformed by algebraic manip-
ulations into an explicit ODE system. The transient
stability problem leads to an index-one or higher-index
Euler method and can handle sti ODEs. This
DAE system with the possibility of the ODEs involved
code has been successfully used in the solution of
problems arising in combustion modeling where
there are frequent discontinuities in time. Be-
cause of its ability to handle sti di erential equa-
tions of the TSA are complex in nature (the voltages
tions and frequent discontinuities, LIMEX is an
and currents involved are both complex). These equa-
excellent choice for the power system TSA solu-
tions are also nonlinear. Research in developing a
solver that can be used directly to solve a system of
complex, nonlinear algebraic equations has been lim-
ited and leaves the eld wide open for further research.

lation technique for the solution of DAEs.

(IRK) method of order ve. Research on solution
consist of many individual power systems, it is highly
of sti ODEs has focused a great deal on Implicit
desirable that the DAE solver used for the transient
Runge-Kutta methods. RADAU5 is designed to
stability analysis be scalable. Also, the power net-
solve index one, two, and three systems. IRK
works can vary in size depending on the region under
methods have a de nite advantage over multistep
methods such as BDFs methods when the DAEs
exhibit frequent discontinuities. Because of their
one-step nature, IRK methods can be started at
a higher order after every discontinuity, whereas
multistep methods must be restarted, usually at
a low order, after every discontinuity. This fact
The power system community has been solving the
helps the IRK methods to be more e cient. Also,
transient stability problem approximately as a decou-
with IRK methods, it is possible to construct high
pled system of di erential and algebraic equations. As
order A-stable or nearly A-stable IRK formulas,
explained in 2], this method of reduction of large
which is important while solving sti ODEs with
DAE system to a system of explicit ODEs not only
eigenvalues lying close to the imaginary axis. The
destroys the sparsity of the system but also prevents
generator ODEs in the transient stability analy-
the exploitation of the system structure (symmetry of
sis may be of this type. Since the TSA DAEs can
the Jacobian). Several sequential DAE solvers exist
be of index higher than one, RADAU5, with its
that use various numerical techniques for their solu-
ability to handle higher index systems makes an
tion. Prominent methods among the existing solvers
excellent choice for the transient stability analysis
tiation technique. It uses a variable step-size,
variable-order xed-leading-coe cient implemen-
tation of BDFs formulas to advance solution from
Backward Di erentiation Formulas (BDF) meth-
one time step to the next. DASSL can solve
su er no order reduction for index one systems.

Also, BDFs methods achieve the same order of
Our research is concentrated on analyzing the follow-
convergence for this class of DAEs as they do for
ODEs. DASSL has a robust order selection strat-
egy for DAEs with eigenvalues close to the imag-
inary axis in the complex plane. DASSL can also
discussed in 2], one-step methods such as ex-
be used to solve semi-explicit index-two systems
trapolation techniques have an inherent advan-
of DAEs. DASSL by itself cannot handle discon-
tage over BDFs methods when applied to DAEs
tinuities, but a variation of DASSL, DASRT, has
with frequent discontinuities. LIMEX is a code
root nding capabilities to locate discontinuities
developed for semi-explicit index one DAEs. It
when they are su ciently large that DASSL can-
implements an extrapolation of the semi-implicit
not integrate through without intervention. Re-
search is being conducted at the Northeast Paral-
best suited for the TSA problem. We wil also present
lel Architectures Center (NPAC) 10] to solve the
a comparative review of their speed and performance.

The test data being used is the standard IEEE 118-
bus data but we will also benchmark larger systems at
coe cient implementation of BDFs formulas to
solve linearly implicit system of DAEs. The
xed-coe cient methods can be implemented e -
ciently for smooth problems, but su er from inef-
ciency and possible instability for problems re-
CDASSL (Concurrent DASSL) 13] is the rst attempt
quiring frequent step-size adjustments. LSODI
that has been made at parallelizing these DAE solvers.

solves DAEs of index zero but if the coe cient
More recently, attempts have also been made to de-
matrix is singular, then it solves the resulting in-
velop a data-parallel and message-passing versions of
dex one DAE system. It can solve systems of sti
DASPK 12] on the CM5. Our research is examin-
and non-sti ODEs as well. LSODI has been suc-
ing the scalability of the various DAE solution tech-
cessfully used to solve systems arising from PDE
niques. We are concentrating on selecting the most
problems. Another version of LSODI, LSODA,
suitable solver for the power system application, and
switches automatically between sti (BDF) and
creating a scalable library for an application based on
non-sti (Adams) methods. Because of its abil-
it. The parallel version of this DAE solver is being de-
ity to handle sti ODEs and frequent discontinu-
veloped for the use by the power system community,
ities, LSODI presents a viable alternative to other
and in related applications such as the chemical plant
solvers discussed for the power system TSA prob-
simulations (e.g., 13]) and the electrical circuit sim-
ulations 2]. We are using the Multicomputer Toolbox
15] for the development of the parallel solver. This
software is currently based on the portable message-
nique of order four. Rosenbrock methods have
a big advantage over IRK methods in that they
MPI 4] next year and hence our programs will run
completely avoid non-linear systems of equations,
on all reasonable message-passing multicomputers.

at the same time providing the advantages of ac-
curate solutions with sti di erential equations.

RODAS can solve DAEs that are expressible in
semi-explicit form and that are of index one. Be-
cause of the lack of data about the application of
We have discussed the power system transient stability
RODAS to any signi cant problem, and because
analysis problem (TSA). We also presented a review of
of the varied advantages of other solvers discussed
some of the existing DAE solvers that we consider best
above, our research will not involve RODAS, even
suited for this problem. Benchmarking of sequential
though it has the qualities of being a good con-
DAE solvers and the development of a parallel, scal-
able DAE solver are in progress. This research is being
closely coordinated with parallel sparse matrix solver
9] research being performed at the Northeast Parallel
Architectures Center (NPAC) at Syracuse University.

Currently at NPAC, we are involved in the following
two issues: Transient Stability Analysis Benchmarks,
We wish to thank David Koester and Alvin Leung, of
NPAC, for their help with this research.

We are setting up transient stability analysis bench-
marks with interfaces to di erent sequential solvers
1] P. M. Anderson and A. A. Fouad. Power Sys-
like LIMEX, LSODI, RADAU5 and DASSL. This is
tem Control and Stability. Iowa State University
to test our empirical conclusion that these solvers are
2] K. E. Brenan, S. L. Campbell, and L. R. Petzold.

14] Anthony Skjellum. The Design and Evolution of
The Numerical Solution of Initial Value Problems
Zipcode. Parallel Computing, 1993. (Invited Pa-
in Di erential-Algebraic Equations. Elsevier Sci-
15] Anthony Skjellum, Alvin P. Leung, Steven G.

3] P. Deu ard, E. Hairer, and J. Zugck. One Step
Smith, Robert D. Falgout, Charles H. Still, and
and Extrapolation Methods for Di erential- Alge-
Chuck H. Baldwin. The Multicomputer Toolbox{
braic Systems. Numer. Math., 51, pages 501{516,
First-generation Scalable Libraries. Technical Re-
port 930805, Mississippi State University, Com-
4] Message Passing Interface Forum. Document for
puter Science Dept., June 18 1993. Submitted
a standard message-passing interface. Technical
to HICSS{27: Minitrack on Tools and Languages
Report Technical Report No. CS-93-214, Univer-
for Transportable Parallel Applications.

5] E. Hairer and G. Wanner. Solving Ordinary Dif-
ferential Equatios II | Sti and Di erential-
Algebraic Problems. Springer-Verlag, 1991.

6] A. C. Hindmarsh. Large ordinary di erential
equation systems and software. IEEE Control
7] A. C. Hindmarsh. ODEPACK, a systematized
collection of ODE solvers. In R. S. Stepleman
et al., editors, Scienti c Computing, pages 55{64.

8] A. C. Hindmarsh. The ODEPACK solvers. In
R. C. Aiken, editor, Sti Computation, pages
9] David Koester, Sanjay Ranka, and Geo rey Fox.

Parallel Block-Diagonal-Bordered Sparse Linear
Solvers for Electrical Power System Applications.

Scalable Parallel Libraries Conference, 1993.

10] Alvin Leung and Anthony Skjellum. Concurrent
Dassl: A Second Generation DAE Solver. Scal-
able Parallel Libraries Conference, 1993.

11] Iris Mack. Block Implicit One-Step Methods for
Solving Smooth and Discontinuous Systems of
Di erential/Algebraic Equations. PhD thesis,
Applied Mathematics, Harvard University, May
12] Linda Petzold. Solving Large-Scale Di erential-
Algebraic Systems via DASPK on the CM5. Scal-
able Parallel Libraries Conference, 1993. (Invited
13] Anthony Skjellum. Concurrent Dynamic Simu-
lation: Multicomputer Algorithms Research Ap-
plied to Ordinary Di erential-Algebraic Process
Systems in Chemical Engineering. PhD the-
sis, Chemical Engineering, California Institute of

Source: http://www.new-npac.org/users/fox/pdftotal/sccs-0563.pdf

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