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{\large \bf Transient Stability Analysis for Power Systems}

An important aspect of the operation of an electrical power network
is the security model, which simulates the response of the network to 
disturbances such as component failure or routine maintenance.
These simulations allow power utility control-room operators to
identify possible sources of cascading failures which could cause 
a disruption of power generation or transmission, and to take pro-active
measures to prevent their occurrence. In order to maintain fault
tolerance in the network, extra generation and transmission capacity
must be available. There is a certain margin of error factored in to 
this extra capacity due to the limited accuracy of the simulations.
Faster and more accurate simulations would allow improved system security
and reduced operating margins, which would save energy and money, and 
reduce environmental impact.

Typically the security model is run every few minutes using data on the
current state of the network which is updated every few seconds.
However these simulations are based on component stability limits which
are determined very infrequently, using worst-case scenarios as input data.
Improved stability limits based on real-time data would greatly improve
the accuracy of security model simulations.

These limits are based on transient stability analysis, which examines
the dynamic behavior of a power system following a power flow disturbance.
It uses information concerning the electrical distribution network,
electrical loads, and the electro-mechanical equations of motion of the 
interconnected generators to set tolerance limits for network components
which allow stable operation of the system. Due to the computational
complexity of transient stability analysis, it currently cannot by used
for real-time simulation.

Techniques are examined to
solve differential-algebraic equations --- the computational heart of
the transient stability problem. 


Transient stability analysis involves the study of the electrical network
and generator dynamics for several seconds following a disturbance. 
The dynamics of each generator in the system is described by multiple 
differential equations, while the coupling of the generators to the 
electrical network is described by algebraic equations. 
Together these lead to a system of differential-algebraic equations (DAEs) 
that must be solved simultaneously. Since a large system of DAEs needs to 
be solved at small time increments to minimize the errors in numerical 
integration of the differential equations, the transient stability analysis
is a highly compute intensive problem.

Traditional approaches to solving transient stability DAEs, such as
partitioning and linearization, have used simple trapezoidal integration
techniques that require very small step sizes during the integration steps.
Meanwhile, there has been significant research in the numerical analysis
community on fast, accurate, and efficient DAE solvers. 
However, very little work has been done to date on using these more recent
public domain DAE solvers for the transient stability problem. 



    Sparse matrix calculations are some of the most challenging problems 
for parallel machines, but they are key to many power system applications,
including load flow and transient stability analysis,
for which they represent the majority of the computation.
Various organizations have efforts in progress to develop general concurrent
sparse matrix solvers.  There has already been significant progress in
this area, especially by Michael Heath at Oak Ridge National Laboratory, 
Ian Duff in Europe and groups at Sandia National Laboratory.

However, specialized parallel sparse matrix solvers which can capitalize on 
any regular structure in the sparse matrices for a particular problem will
generally give better performance than general solvers.
In-house research at Syracuse University is currently focussing on the
development of specialized sparse matrix solvers that exploit hierarchical
structure in the underlying power systems problems.  

There are distinct portions of the matrix that represent the 
generator equations and the power distribution network. 
Diakoptics theory and the actual matrix sparsity structure (the power 
distribution network structure) will be used to develop algorithms to 
reorder the sparse matrices resulting from power system applications 
into block-bordered diagonal form. 
The result of this reordering is that many of the operations that would be
sequential in a general solver are now highly parallel.  
This will simplify the development of scalable, portable algorithms.
Reordering can also be used to improve load balancing for both direct and
iterative techniques for matrix solvers.  

One of our graduate students is presently working on parallel sparse matrix
techniques especially suited for power systems applications. 
Another graduate student in our team has worked with a group at Lawrence 
Livermore National Laboratory on a related petrochemical simulation using
parallel DAE and sparse matrix solvers.  

Parallel sparse matrix solvers are critically dependent on a proper match 
between the particular problem and the machine architecture and ``general 
approaches'' are often less than optimal.
It is essential to have matrices of representative sizes to 
successfully research this real-life problem. 
We will work with NMPC to obtain the actual structure of the 
transmission network for three interconnected power grids:
 
