## POWER SYSTEM MODELS FORMULATED AS EIGENVALUE PROBLEMS AND PROPERTIES OF THEIR SOLUTIONS

##### Abstract

Eigenvalue problems incorporate multiplicative nonlinearities in otherwise linear equations.
Two power systems models with multiplicative nonlinearities are reformulated as eigenvalue
problems. Reformulation allows for the application of eigenvalue theory and solution
techniques to these models.
The first model involves determination of the initial conditions for induction machine internal
variables in a non-linear dynamic power system analysis. The internal variables of stator
and rotor currents, rotor speed, and mechanical torque must be determined from the given values
of input real power, stator voltage magnitude, and stator voltage angle. This problem is
posed in an eigenvalue formulation that can be solved using standard linear algebra techniques.
The eigenvalue formulation allows for determination of all possible solutions for the internal
variables rather than the single solution obtained from traditional iterative methods. Additionally,
the absence of non-zero real eigenvalues indicates that the problem has no solution. Both
single-cage and double-cage induction machines are considered, and numeric examples are
presented.
The second model involves reformulation of the power flow equations as a multiparameter
eigenvalue problem. In this formulation, both the eigenvalues and the eigenvectors are composed
of the d and q orthogonal components of the bus voltages. The two parameter formulation
of the power flow equations for two bus systems can be solved directly by decomposing
the problem into two generalized eigenvalue problems that must be simultaneously satisfied.
Since n bus systems require 2 (n ? 1) parameter eigenvalue problems, which do not yet have
a general solution method for n > 2, systems with more than two buses are not yet directly
solvable from the multiparameter eigenvalue formulation.
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In addition to direct solution, two other research avenues for the multiparameter eigenvalue
formulation of the power flow equations are pursued. First, motivated by eigenvalue problem
structure, a reformulation of the multiparameter eigenvalue form of the power flow equations
is presented. Although it is without known practical applications, this reformulation and the
intermediate results in its derivation are interesting from a theoretical standpoint. Second, an
eigenvalue sensitivity analysis is performed on the multiparameter eigenvalue formulation of
the power flow equations. The linearization obtained from the eigenvalue sensitivity analysis
is equivalent to the linearization obtained from the power flow Jacobian.
Future developments in multiparameter eigenvalue theory may provide additional insights
into solutions of the power flow equations. General solution techniques for multiparameter
eigenvalue problems with more than two parameters may enable direct solution of the power
flow equations. A method for determining the number of solutions to multiparameter eigenvalue
problems would be useful as a stopping condition for continuation power flows. Conditions
for the existence of any solutions to multiparameter eigenvalue problems would be useful
for finding the point of voltage collapse and for analyzing power systems in heavily loaded
conditions.