Computational Complexity Analysis of the Holomorphic Embedding Method and Parallel Optimization Strategies for Improving Efficiency

The holomorphic embedding (HE) method has gained significant attention in power flow analysis. However, there is no consensus regarding its computational efficiency compared to the traditional Newton-Raphson (NR) method. This paper presents a comprehensive analysis of the computational complexity of the HE method focusing on key steps such as matrix factorization, higher-order power series calculation, and Padé approximation. A theoretical investigation of these steps is conducted, followed by the implementation of multi-core parallel computing using Numba to enhance performance. Case studies on test systems ranging from 300 to 25,000 buses are used to evaluate the parallel strategies for accelerating computation. The results show a significant improvement in the computational efficiency of the HEM, demonstrating its scalability and practical applicability for large-scale power flow calculations. This study provides both theoretical and practical foundations for the broader adoption of the HE method in power system analysis.