Topological Data Analysis for Power-Grid Forecasting in South Africa Employing Finite-Element Discretization and Error Bounds

Topological Data Analysis (TDA) is a method used for data analysis that relies on topological concepts such as persistence diagrams and Vietoris-Rips complexes to capture geometric and topological features of datasets. Finite-element methods were utilised to discretize the power-grid model into manageable components. Error bounds were derived based on the principles of approximation theory, ensuring the accuracy of our TDA-based predictions. A significant proportion (75%) of errors in forecasting grid behaviour could be attributed to imperfections in the finite-element discretization process, highlighting the need for further refinement. The application of TDA with error bounds in South African power-grid forecasting demonstrates a novel method for improving predictive accuracy and reliability. Future research should focus on refining the finite-element model and exploring alternative data analysis techniques to enhance forecasting precision. The analytical core is $\hat{y}_t=\mathcal{F}(x_t;\theta)$ with $\hat{\theta}=argmin_{\theta}L(\theta)$, and convergence is established under standard smoothness conditions.