  It is well known that closed-form analytical solutions for AC power flow
equations do not exist in general. This paper proposes a multi-dimensional
holomorphic embedding method (MDHEM) to obtain an explicit approximate
analytical AC power-flow solution by finding a physical germ solution and
arbitrarily embedding each power, each load or groups of loads with respective
scales. Based on the MDHEM, the complete approximate analytical solutions to
the power flow equations in the high-dimensional space become achievable, since
the voltage vector of each bus can be explicitly expressed by a convergent
multivariate power series of all the loads. Unlike the traditional iterative
methods for power flow calculation and inaccurate sensitivity analysis method
for voltage control, the algebraic variables of a power system in all operating
conditions can be prepared offline and evaluated online by only plugging in the
values of any operating conditions into the scales of the non-linear
multivariate power series. Case studies implemented on the 4-bus test system
and the IEEE 14-bus standard system confirm the effectiveness of the proposed
method.
