Robust Energy System Design via Semi-infinite Programming

TL;DR

This paper tackles the challenge of robustly designing energy systems under time-series uncertainty from renewable sources by introducing Robust Energy System Design (RESD), a semi-infinite programming framework that identifies worst-case uncertainty realizations adaptively. It combines clustering-based representative scenarios with PCA-driven dimensionality reduction and an ESIP relaxation (with a lifting option for convex lower-level problems) to guarantee feasibility across all plausible future conditions. The authors demonstrate RESD on a La Palma island case, achieving a high renewable share (≈92%) and a total annualized cost of €28.0, while highlighting the trade-off between dimensionality reduction and solution accuracy. Despite its computational intensity, RESD provides a rigorous approach to robust design beyond traditional feasibility-time-step heuristics, offering a pathway to reliable, low-risk energy-system configurations in settings with nonconvex operational behavior.

Abstract

Time-series information needs to be incorporated into energy system optimization to account for the uncertainty of renewable energy sources. Typically, time-series aggregation methods are used to reduce historical data to a few representative scenarios but they may neglect extreme scenarios, which disproportionally drive the costs in energy system design. We propose the robust energy system design (RESD) approach based on semi-infinite programming and use an adaptive discretization-based algorithm to identify worst-case scenarios during optimization. The RESD approach can guarantee robust designs for problems with nonconvex operational behavior, which current methods cannot achieve. The RESD approach is demonstrated by designing an energy supply system for the island of La Palma. To improve computational performance, principal component analysis is used to reduce the dimensionality of the uncertainty space. The robustness and costs of the approximated problem with significantly reduced dimensionality approximate the full-dimensional solution closely. Even with strong dimensionality reduction, the RESD approach is computationally intense and thus limited to small problems.

Figures (6)

• Figure 1: The processing of historical time-series data to obtain the representative scenarios and uncertainty bounds for the RESD problem: Time-series data with the length of the representative period are collected for different quantities, normalized, and concatenated such that a single vector is obtained for each representative period in the historical data. Representative scenarios are obtained by clustering the concatenated and normalized data according to the framework by teichgraeberClusteringMethodsFind2019. Uncertainty bounds are obtained by first reducing the dimensionality of the historical data and then determining bounds in the lower-dimensional space.
• Figure 2: Mean and standard deviation of the daily historical electricity demand on the island of La Palma in 2013-2019. Demand data were obtained from the Spanish electricity distribution system operator redelectricadeespanaPalmaElectricityDemand2024. Furthermore, the 15 representative scenarios used to compute the operational costs are depicted as dashed lines. There is significant variation in demand over the course of a day, with a wide peak during the morning and a sharp peak in the evening.
• Figure 3: Installed component capacities for the robust design (blue) identified by the RESD approach using 16 time steps and 9 principal components. For comparison, the currently installed capacities (orange) gobiernodecanariasAnuarioEnergeticoCanarias2023 are shown. The current design has much higher conventional generation capacity. However, plans to switch to higher renewable electricity generation have already been announced gobiernodecanariasConsejeriaTransicionEcologica2023. Note that we did not consider possible component failures in the RESD approach, and hence do not have backup capacities, which are considered in the current design.
• Figure 4: Average solution times in logarithmic scale for the ESIP and the lifting approach. The average solution time in seconds is plotted against the number of principal components (PCs) for varying numbers of time steps. The solution time increases both with an increasing number of PCs and time steps. The lifting approach scales better with an increasing number of PCs than the ESIP approach.
• Figure 5: Optimal total annualized cost (TAC) of energy system designs obtained using the lifting approach against the number of principal components (PCs) for varying number of time steps (top). Dashed lines show results obtained with the feasibility time-step heuristic bahlTimeseriesAggregationSynthesis2016teichgraeberExtremeEventsTime2020 discussed in Section \ref{['special']}. For a low number of PCs, a significant approximation error can be seen. If the number of PCs is chosen sufficiently large, i.e., greater than or equal to $5$, the TAC of full-dimensional problems is approximated closely by the RESD approach. Explained variance ratio plotted against the number of PCs for a varying number of time steps (bottom). A small number of PCs can explain a majority of the variance in the historical data.