Numerical Efficiency Analysis of the Holomorphic Embedding Method

While the holomorphic embedding method (HEM) theoretically guarantees convergence superior to iterative power flow methods, its numerical performance limits practical applications. This paper presents a comprehensive numerical efficiency analysis of HEM through computational profiling of its core algorithmic components: embedding initialization, power series recursion, Padé approximant construction and solution recovery. Tests are conducted on 14-bus to 2383-bus systems under varying convergence criteria, loading levels, and impedance ratios. Simulation results reveal that power series recursion and Padé approximant construction dominate the runtime ($80-99 \%$ of the total execution time). The primary bottleneck shifts between the two depending on system scale and operating conditions: power series recursion becomes increasingly dominant in large-scale systems, while Padé approximant construction exhibits significant computational complexity under strict convergence tolerances or heavily loaded scenarios. Efficiency deteriorates sharply near system’s loadability limit and in high R/X ratio networks. Those findings could provide valuable guidance for performance optimization in HEM implementations.