Tensor Power Flow Formulations for Multidimensional Analyses in Distribution Systems

TL;DR

The paper addresses the need for rapid, large-scale power-flow analyses in distribution systems by developing two multidimensional fixed-point iteration (FPI) formulations cast as tensor power flows. It introduces dense and sparse tensor formulations (TensorPowerFlow) that exploit CPU/GPU parallelism, proves convergence to a unique high-impedance operating point under a contraction condition, and demonstrates robust performance advantages over Newton-Raphson and backward/forward sweep methods. Key contributions include a geometric interpretation of solution existence, a practical software tool, and extensive comparative results showing substantial speedups—up to 164× for yearly 1-minute-resolution studies—across grid sizes and problem dimensionalities. The work enables scalable, high-resolution probabilistic analyses and hosting-capacity studies in distribution networks by leveraging multidimensional PF, fixed-point convergence, and mixed computing resources ($k<1$, $|z_s|$ versus $|z_l|$).

Abstract

In this paper, we present two multidimensional power flow formulations based on a fixed-point iteration (FPI) algorithm to efficiently solve hundreds of thousands of power flows in distribution systems. The presented algorithms are the base for a new TensorPowerFlow (TPF) tool and shine for their simplicity, benefiting from multicore \gls{cpu} and \gls{gpu} parallelization. We also focus on the mathematical convergence properties of the algorithm, showing that its unique solution is at the practical operational point, which is the solution of high-voltage and low-current. The proof is validated using numerical simulations showing the robustness of the FPI algorithm compared to the classical \gls{nr} approach. In the case study, a benchmark with different PF solution methods is performed, showing that for applications requiring a yearly simulation at 1-minute resolution the computation time is decreased by a factor of 164, compared to the NR in its sparse formulation.

Figures (5)

• Figure 1: Geometry of the solutions of the power flow problem for a two-bus system with parameters $z_s=1.0 + \boldsymbol{j}0.5$ and $\|v_o\|=1.0$. (a) and (b) are the contour graphs for the circular formulation \ref{['eq:circle_pl']} and \ref{['eq:circle_ql']}, for $p_l$ and $q_l$, respectively; black circles highlight a feasible combination of active and reactive power. (c) The red circles highlight an infeasible value of the load power (no crossing between the circles), while the feasible values have two points of solution that form a line that passes through the origin. (d) All the solutions for the critical load power, which has only one point of contact between the circles, form a circle with radius $\|z_s\|$.
• Figure 2: Example of feasibility regions depicted by \ref{['eq:conic_section']} and \ref{['eq:region']}. (a) Each point of the maximum power transfer circle $\|z_s\|$ in the impedance plane $(r_s, x_s)$ parametrize the parabola in \ref{['eq:conic_section']} that defines the feasible load power values. (b) The vertexes of the parametrized parabolas are rotating around the circle defined by \ref{['eq:region']} in the load power plane $(p_l,q_l)$. Only points of the first quadrant are shown, highlighting three example points ($z_{s,1}$, $z_{s,2}$, $z_{s,3}$) with their respective parabolas. Feasibility region for the case of $z_{s,2}$ is emphasised with red and yellow. The green region is the union of all possible feasible regions for all parabolas.
• Figure 3: Convergence analysis of NR and FPI algorithms. (a) and (c) corresponds to the regions of convergence for different initial values of voltages $v_0$ for NR and FPI algorithms, respectively. The green region corresponds to the high voltage (high impedance) and the red region to the low voltage (low impedance) solutions. (b) and (d) correspond to the number of iterations required for convergence, for the NR and FPI algorithms, respectively.
• Figure 4: Example of the tensor power flow formulations for a simulation with a four-dimensional power tensor $\boldsymbol{S} \in \mathbb{C}^{ p \times r \times t \times b\phi}$, for $p=r=2$ and $t=b\phi=3$. (a) Visualization of the tensor dense formulation of \ref{['eq:dense_tensor']}. (b) In the case of constant power, the reshaped tensors use resources efficiently as the operations highlighted in red are performed concurrently. (c) Visualization of the tensor sparse formulation in \ref{['eq:sparse_tensor']}, where tensors $\boldsymbol{\mathcal{M}}$ and $\boldsymbol{\mathcal{H}}$ are sparse. (d) Reshaped sparse formulation \ref{['eq:sparse_reshape']}, which is solved iteratively using a direct sparse solver.
• Figure 5: Comparison of the performance of the algorithms. Computational wall-times ($t_c$) increase the bus-phases size, $b\phi$, for (a) 1 power flow, and (b) 500 power flows. (c) Test for increased dimensional tensor elements $\tau$, for a grid size of 500 bus-phases ($b\phi$). (d) Computational complexity fit with asymptotic complexity model $t_c = c \cdot n^k$ (solid lines are $c$, dotted $k$). Tensor dense, sparse, and GPU are the proposed algorithms in this paper.