Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ensuring Solution Uniqueness in Fixed-Point-Based Harmonic Power Flow Analysis with Converter-Interfaced Resources: Ex-post Conditions

Antonio Di Pasquale, Johanna Kristin Maria Becker, Andreas Martin Kettner, Mario Paolone

Abstract

Recently, the authors of this paper proposed a method for the Harmonic Power-Flow (HPF) calculus in polyphase grids with widespread deployment of Converter-Interfaced Distributed Energy Resources (CIDERs). The HPF problem was formulated by integrating the hybrid nodal equations of the grid with a detailed representation of the CIDERs hardware, sensing, and controls as Linear Time-Periodic (LTP) systems, and solving the resulting mismatch equations using the Newton-Raphson (NR) method. This work introduces a novel problem formulation based on the fixed-point algorithm that, combined with the contraction property of the HPF problem, provides insights into the uniqueness of its solution. Notably, the effectiveness of the fixed-point formulation and the uniqueness of the solution are evaluated through numerical analyses conducted on a modified version of the CIGRE low-voltage benchmark microgrid.

Ensuring Solution Uniqueness in Fixed-Point-Based Harmonic Power Flow Analysis with Converter-Interfaced Resources: Ex-post Conditions

Abstract

Recently, the authors of this paper proposed a method for the Harmonic Power-Flow (HPF) calculus in polyphase grids with widespread deployment of Converter-Interfaced Distributed Energy Resources (CIDERs). The HPF problem was formulated by integrating the hybrid nodal equations of the grid with a detailed representation of the CIDERs hardware, sensing, and controls as Linear Time-Periodic (LTP) systems, and solving the resulting mismatch equations using the Newton-Raphson (NR) method. This work introduces a novel problem formulation based on the fixed-point algorithm that, combined with the contraction property of the HPF problem, provides insights into the uniqueness of its solution. Notably, the effectiveness of the fixed-point formulation and the uniqueness of the solution are evaluated through numerical analyses conducted on a modified version of the CIGRE low-voltage benchmark microgrid.
Paper Structure (12 sections, 1 theorem, 56 equations, 5 figures, 4 tables)

This paper contains 12 sections, 1 theorem, 56 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Harmonic Power-Flow mondel in the Presence of Converter-Interfaced Resources
  4. Model of the Grid
  5. Generic Model of the CIDERs
  6. Mathematical Formulation of the HPF Problem
  7. Fixed-Point Formulation and Uniqueness of the Solution
  8. Numerical Validation
  9. Validation of the Fixed-Point Algorithm
  10. Assessment of the Uniqueness of Solution
  11. Conclusion
  12. Modelling of Constant Impedance Loads

Key Result

Theorem 1

Assume that $\boldsymbol{\Phi}$ is continuously differentiable in a neighbourhood of a fixed point $\Tilde{\mathbf{W}}^*_{\rho}$ of $\boldsymbol{\Phi}$ and $\|\nabla\boldsymbol{\Phi}(\Tilde{\mathbf{W}}^*_{\rho})\|_{\infty}~<~1$. Then there exists a closed neighborhood $\Omega$ of $\Tilde{\mathbf{W}}

Figures (5)

  • Figure 1: Block diagram of the proposed generic state-space model of CIDERs. Note the modularity: power hardware $\pi$, control software $\kappa$, and grid $\gamma$ are represented by separate blocks, which are interfaced via coordinate transformations. The reference calculation $\mathbf{r}(\cdot)$ may be either linear (i.e., for $\mathit{Vf}$ control) or nonlinear (i.e., for $\mathit{PQ}$ control). The other blocks of the model are exactly linear (i.e., LTP systems and LTP transforms). Adapted from kettner2018properties.
  • Figure 2: Schematic diagram of the test system, which is based on the CIGRÉ low-voltage benchmark microgrid Rep:2014:CIGRE (in black) and extended by unbalanced impedance loads (in grey). For the cable parameters see Table \ref{['tab:grid:parameters']}. The resources are composed of constant impedance loads (Z) and constant power loads (P/Q), parameters given in Table \ref{['tab:resources:references']}.
  • Figure 3: Plot of indicators $\delta_f^{(k)}$ (in black) and $\delta_x^{(k)}$ (in grey) across iterations.
  • Figure 4: Results of fixed-point algorithm validation on the benchmark system. The plots show maximum absolute errors over all nodes and phases between the time domain simulation and the fixed point algorithm, for voltages (left column) and currents (right column) in magnitude (top row) and phase (bottom row).
  • Figure 5: Norm of the Jacobian of the $\boldsymbol{\Phi}$ over fixed-point iterations, illustrating the impact of increasing power in P/Q nodes.

Theorems & Definitions (5)

  • Remark 1
  • Definition 1
  • Definition 2
  • Remark 2
  • Theorem 1