“…There are indeed quite a few existing methods for finding one or many load flow solutions [1, 5, 8-10, 16, 18-20, 23, 30, 31, 38, 41, 43-45, 47, 49, 52, 54, 57-60, 62, 63] (see [48] for a recent review). Out of the few methods that guarantee to find all load flow solutions, i.e., the interval based approach [57], Gröbner bases technique [16,54,58,59] and the numerical polynomial homotopy continuation (NPHC) method [8,9,41,47,49,60], the NPHC method appears most promising in scalability with increasing system sizes in that it has already found all load flow solutions of up to IEEE 14 bus systems [49] (and 18 oscillators case for the Kuramoto model [47]) and is inherently parallel: formulating load flow equations as system of polynomial equations, the NPHC method, whose roots are in complex algebraic geometry, finds all isolated complex systems which obviously include all the isolated real solutions. In all these computational methods, the knowledge of the number of solutions play a crucially important role.…”