A Fast Holomorphic Embedding Power Flow Approach for Meshed Distribution Networks

Abstract

This paper proposes a novel holomorphic embedding approach for solving the nonlinear power flow equation for meshed electric distribution networks with ZIP load model. In the proposed approach, bus voltages are modelled as holomorphic functions in the constant power injections and then expanded using Maclurin series. The Z-bus matrix is implicitly used to calculate Maclurin series coefficients for the expanded voltage functions in a recursive manner. The necessary and sufficient conditions for the convergence of expanded voltage functions are found. Performance evaluations show that the proposed approach solves the nonlinear power flow equations faster than the existing approaches when applied to 18-, 33-, 69-, 141-, 3239-, 5701-, and 6921-bus distribution network test cases.

A Review on Applications of Holomorphic Embedding Methods

“…Various HEM-based power flow models introduced before mainly focusing on AC transmission networks. There are papers extending the HEM to other types of power systems such as DC systems [33] , AC/DC systems [55][56][57][58][59] , three-phase distribution networks [60][61][62][63][64] , integrated systems [35,42,65,66] and isolated systems [58,67] . It is important to note that unlike the AC power flow, in which one has to take particular attention to deal with the complex conjugation operation by considering additional independent variables , all voltages variables involved in DC systems [33] are real instead of complex, so is required and satisfied, ensuring that physical solutions are real when is real.…”

“…For distribution systems, there are several key features, first the distribution system is characterized by radial or weakly meshed structures [62] , second there are many unbalanced three-phase operations [61] . Also, the accurate modeling of ZIP loads is necessary which will introduce more nonlinearity to the system [63] . Furthermore, the high R/X ratio in distribution systems can lead to slow or non-convergence of the NR method because it may encounter difficulties in finding the appropriate initial guesses for voltage magnitudes and angles.…”

“…However, it is highlighted in Ref. [63] that the holomorphy assumption for solution functions might not be met for that derived embedded system in Ref. [60].…”

“…It is crucial to note that the existence, uniqueness, and holomorphism of solutions in Eq. ( 2) are not straightforward, particularly when dealing with non-polynomial systems involving elements like ZIP loads in distribution networks [63] , where the unknown complex functions in the embedded system are not algebraic in general. One must carefully design the embedded system among infinite choices.…”