Analysis and Comparison of Chebyshev–Halley Multipoint Methods for Power Flow Calculation in Monopolar Direct-Current Networks

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Original abstract

The increasing penetration of direct-current (DC) technologies in power transmission and distribution systems necessitates efficient and robust tools for steady-state analysis. This paper presents a comparative evaluation of the Chebyshev–Halley (CH) family of multipoint iterative methods against the classical Newton–Raphson (NR) method for power flow calculation in monopolar DC networks. Both methods were implemented in MATLAB and tested on four radial test systems of increasing complexity (10, 21, 33, and 69 nodes) under three distinct initialization scenarios: optimal (flat start), adverse (V(0)=0.5 p.u.), and random (V(0)∼U[0.8,1.2] p.u.). Performance was assessed using key metrics including iteration count, CPU time, solution accuracy, and convergence failure rate. The results demonstrate that the cubic convergence of CH consistently reduces the number of iterations by one when compared to NR across all systems. However, this reduction does not translate into computational savings, as CH exhibits median CPU times 1.36 to 2.44 times higher than those of NR, given its higher cost per iteration, which involves solving two additional linear systems. Under adverse starting conditions, both methods converge for the 10-, 21-, and 33-node systems, but CH fails on the 69-node network due to severe Jacobian ill-conditioning, from which NR recovers through an implicit regularization mechanism. Under random initializations, both methods show high failure rates, reaching 100% in the 69-node network. It is concluded that, while CH offers superior convergence order and final accuracy, NR remains more computationally efficient for small- to medium-scale networks under flat-start conditions. The CH family is best justified in high-precision applications or larger networks where the iteration reduction may offset its per-step overhead. Future work should focus on extending CH to meshed and multi-source DC networks, developing quasi-Newton variants to reduce its computational cost, and designing hybrid NR-CH strategies that combine global robustness with local cubic convergence.

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