An Interpretable Approach to Load Profile Forecasting in Power Grids using Galerkin-Approximated Koopman Pseudospectra

TL;DR

The results indicate that the Koopman-based method surpasses a separately optimized deep learning (LSTM) architecture in both accuracy and computational efficiency, while providing deeper insights into the underlying dynamics of the power grid.

Abstract

This paper presents an interpretable machine learning approach that characterizes load dynamics within an operator-theoretic framework for electricity load forecasting in power grids. We represent the dynamics of load data using the Koopman operator, which provides a linear, infinite-dimensional representation of the nonlinear dynamics, and approximate a finite version that remains robust against spectral pollutions due to truncation. By computing $ε$-approximate Koopman eigenfunctions using dynamics-adapted kernels in delay coordinates, we decompose the load dynamics into coherent spatiotemporal patterns that evolve quasi-independently. Our approach captures temporal coherent patterns due to seasonal changes and finer time scales, such as time of day and day of the week. This method allows for a more nuanced understanding of the complex interactions within power grids and their response to various exogenous factors. We assess our method using a large-scale dataset from a renewable power system in the continental European electricity system. The results indicate that our Koopman-based method surpasses a separately optimized deep learning (LSTM) architecture in both accuracy and computational efficiency, while providing deeper insights into the underlying dynamics of the power grid\footnote{The code is available at \href{https://github.com/Shakeri-Lab/Power-Grids}{github.com/Shakeri-Lab/Power-Grids}.

Figures (12)

• Figure 1: (a) Computing Koopman eigenfunctions. (b) Load forecasting using Koopman.
• Figure 2: Overview of dataset. (a) Distribution of generators. (b) Transmission grid and the 3-year average power of each generator in MWh/h. (c) Average load of all power plants. The sampling rate is 1 hour.
• Figure 3: Examples of spatiotemporal load reconstruction. The load data of 1494 stations during January 2014 are reconstructed using 100 Koopman eigenfunctions.
• Figure 4: Spatial and temporal patterns with multiple scales recovered by the load data during January 2014. The left, middle, and right columns display the Power Spectrum (PS), time evolution, and spatial pattern induced by the Koopman eigenfunction, respectively. The first row represents the trivial Koopman eigenfunction ($\omega=0$), followed by subsequent rows associated with frequencies scaled with a month ($3.86\times 10^{-7} \text{Hz}$), a week ($1.65\times 10^{-6} \text{Hz}$), a day ($1.15\times 10^{-5} \text{Hz}$), and a half day ($2.31\times 10^{-5} \text{Hz}$), arranged in ascending order.
• Figure 5: Example spatial patterns showing the frequency scaled with a day.