Efficient method to compute all the type-1 low-voltage power flow solutions in distribution systems

The power flow equations usually have multiple solutions including a high-voltage solution and many low-voltage solutions, among which only the type-1 solutions (where the power flow Jacobian matrix has only one positive real-part eigenvalue) are closely related to the voltage instability phenomenon. This study proposes an efficient method to compute all the type-1 low-voltage power flow solutions in distribution systems. First, the geometric properties of the power flow solution space of distribution systems have been studied. Second, the propositions which can guarantee to locate all the type-1 power flow solutions have been suggested and proved. Finally, the conventional implicit Z-bus method is modified to compute all the type-1 germ solutions, based on the suggested propositions; and then the Newton–Raphson method is utilised to trace all the type-1 low-voltage power flow solution branches which originate from the known type-1 germ solutions. The 7-node, 33-node, and 69-node systems are used to validate and demonstrate the proposed method.