Transient stability analysis is a detailed simulation of the power
system that models the dynamic behavior of the electrical
distribution networks, electrical loads, and the electro-mechanical
equations of motion of the interconnected generators [4].
Transient stability analysis can be used to perform selective detailed
analyses of generator commitment stability, and to support crisis
decision-making during network recovery. The transient stability
problem is modeled by differential algebraic equations (DAEs) with
differential equations representing the generators and non-linear
algebraic equations representing the power system network that
interconnects the generators. The DAEs are in natural non-symmetric
block-diagonal-bordered form, with diagonal blocks of generator
equations coupled by the power system distribution network. In this
representation, there are as many coupling equations as the entire
sparse admittance matrix. It is also possible to order the
admittance matrix to block-diagonal-bordered form in order to increase
available parallelism. This is illustrated in
figure . The size of the sparse matrices representing
the DAEs have as many as 10,000 complex equations for an individual
power system, while regional power authorities could have as many as
50,000 sparse complex equations in the matrix formed from the DAEs.
Figure: Ordering the Admittance Sub-Matrix in Transient Stability Differential-Algebraic Equations
It is also possible to solve the above equations by decoupling the
generator equations from the network equations. For decoupled
transient stability analysis, the transient stability
differential-algebraic equation matrix is partitioned into four
sub-matrices. In figure , the generator equations are
in the block-diagonal matrix labeled
. The generator
equations are solved independently of the network equations, then the
sparse admittance matrix is modified by the matrix coefficients in the
sparse borders, labeled B and C. The admittance matrix is
labeled
in this figure. Instead of the common practice
of decoupling the generator and network calculations in a transient
stability simulation, we hope to continue this research and eventually
examine the use of more powerful differential-algebraic equation
solvers for transient stability analysis that do not decouple the
generator and network equations. The fully-coupled
differential-algebraic equations will offer more potential for good
parallel performance because