The Successive Differentiation Load Flow algorithm

A. Benítez Domínguez, F.M. Echavarren, L. Rouco

Summary:

Polynomial approximation methods to solve the power flow (PF), of which the Holomorphic Embedding Load-flow Method (HELM) is the best-known example, express the bus voltages as power series in an embedding parameter and obtain the series coefficients from a sequence of linear systems that share a single invariant matrix. HELM obtains them by substituting the series into the embedded PF equations and iterating over each polynomial degree. This paper presents the Successive Differentiation Load-Flow (SDLF) algorithm, which obtains the same coefficients by directly differentiating the embedded PF equations. In rectangular coordinates, the power injections are quadratic in the state variables, so the N-th differentiation of the homotopy yields a linear system whose matrix is the Jacobian at the starting point, and whose right-hand side is a weighted sum of Hessian bilinear forms of lower-order sensitivities. The weights are shown to be binomial coefficients, which closes the recursion in explicit form, and the first-order SDLF is shown to coincide with one Newton–Raphson iteration. SDLF and HELM have been shown to compute the same Taylor coefficients, as verified on the New England 39-bus system. Because the reactive power of PV buses is recovered by substitution rather than embedded as an additional series, the SDLF linear system is smaller than the HELM system by exactly the number of PV buses. On the IEEE 39- and 118-bus systems, four Spanish island systems, and two Spanish peninsular cases, this reduction lowers the factorization time of the invariant matrix by 32–69% and the direct solution time by up to 10%. Thus, the SDLF offers an interesting alternative among polynomial approximation methods.

Spanish layman's summary:

El Successive Differentiation Load-Flow (SDLF) es un método que expresa las tensiones en nudos como series de potencias cuyos coeficientes se obtienen diferenciando las ecuaciones aumentadas del flujo de cargas. Reduce el tamaño del sistema frente al HELM original, ahorrando tiempo de computación.

English layman's summary:

The Successive Differentiation Load-Flow (SDLF) is a method that expresses the bus voltages as power series whose coefficients are obtained by directly differentiating the embedded power flow equations. It reduces system size compared to the original HELM, saving computational time.

Keywords: Computational efficiency, HELM, polynomial approximation methods

Registration date: 23-sep-2026