For a (N+1)-bus power system, possibly 2N solutions exists. One of these solutions
is known as the high-voltage (HV) solution or operable solution. The rest of the solutions
are the low-voltage (LV), or large-angle, solutions.
In this report, a recently developed non-iterative algorithm for solving the power-
flow (PF) problem using the holomorphic embedding (HE) method is shown as
being capable of finding the HV solution, while avoiding converging to LV solutions
nearby which is a drawback to all other iterative solutions. The HE method provides a
novel non-iterative procedure to solve the PF problems by eliminating the
non-convergence and initial-estimate dependency issues appeared in the traditional
iterative methods. The detailed implementation of the HE method is discussed in the
While published work focuses mainly on finding the HV PF solution, modified
holomorphically embedded formulations are proposed in this report to find the
LV/large-angle solutions of the PF problem. It is theoretically proven that the proposed
method is guaranteed to find a total number of 2N solutions to the PF problem
and if no solution exists, the algorithm is guaranteed to indicate such by the oscillations
in the maximal analytic continuation of the coefficients of the voltage power series
After presenting the derivation of the LV/large-angle formulations for both PQ
and PV buses, numerical tests on the five-, seven- and 14-bus systems are conducted
to find all the solutions of the system of nonlinear PF equations for those systems using
the proposed HE method.
After completing the derivation to find all the PF solutions using the HE method, it
is shown that the proposed HE method can be used to find only the of interest PF solutions
(i.e. type-1 PF solutions with one positive real-part eigenvalue in the Jacobian
matrix), with a proper algorithm developed. The closet unstable equilibrium point
(UEP), one of the type-1 UEP’s, can be obtained by the proposed HE method with
limited dynamic models included.
The numerical performance as well as the robustness of the proposed HE method is
investigated and presented by implementing the algorithm on the problematic cases and
large-scale power system.