ORACLE: A rigorous metric and method to explore all near-optimal designs for energy systems

TL;DR

This work addresses the inadequacy of single-solution optimization in energy systems by targeting the near-optimal design space. It introduces ORACLE, a convexity-based algorithm that builds inner and outer approximations of the near-optimal region $\mathcal{Z}_\epsilon$ and uses a convergence metric $d_{\mathcal{OI}}$ to guarantee complete exploration within a prescribed tolerance. The method reformulates a max-min distance problem into a MILP to efficiently guide exploration and, once converged, enables rapid sampling of near-optimal designs with controlled coverage. Empirical evaluation on a large-scale sector-coupled energy model of Switzerland shows ORACLE outperforms existing MGA methods in convergence and uncovers near-optimal designs that heuristics miss, underscoring the practical value of rigorous near-optimal analysis for energy-transition planning.

Abstract

Optimization models are fundamental tools for providing quantitative insights to decision-makers. However, models, objectives, and constraints do not capture all real-world factors accurately. Thus, instead of the single optimal solution, real-world stakeholders are often interested in the near-optimal space -- solutions that lie within a specified margin of the optimal objective value. Solutions in the near-optimal space can then be assessed regarding desirable non-modeled or qualitative aspects. The near-optimal space is usually explored by so-called Modelling to Generate Alternatives (MGA) methods. However, current MGA approaches mainly employ heuristics, which do not measure or guarantee convergence. We propose a method called ORACLE, which guarantees generation and exploration on the \emph{entire near-optimal} space by exploiting convexity. ORACLE iteratively approximates the near-optimal space by introducing a metric that both measures convergence and suggests exploration directions. Once the approximations are refined to a desired tolerance, any near-optimal designs can be generated with negligible computational effort. We compare our approach with existing methods on a sector-coupled energy system model of Switzerland. ORACLE is the only method able to guarantee convergence within a desired tolerance. Additionally, we show that heuristic MGA methods miss large areas of the near-optimal space, potentially skewing decision-making by leaving viable options for the energy transition off the table.

Figures (7)

• Figure 1: The key steps of the ORACLE algorithm. ORACLE is initialized by constructing an initial outer and inner approximation ($\mathcal{O}$ and $\mathcal{I}$) using known near-optimal points and system information. In Step 2, the main loop of ORACLE begins by finding the point in $\mathcal{O}$ furthest from $\mathcal{I}$. If the distance is below a desired tolerance, the algorithm terminates, and the resulting polytope(s) can be used to generate near-optimal designs at negligible computational effort. Otherwise, ORACLE locates the nearest feasible point to the outer point (Step 3), and uses this solution to grow the inner approximation, and (if possible) cut the outer approximation (Step 4), before going back to Step 2.
• Figure 2: Distance between inner and outer approximations of the near-optimal space for state-of-the-art MGA methods and the ORACLE algorithm. ORACLE's non-monotonic convergence is a numerical artifact, as problem \ref{['eq: maxmin_dist_prob']} is solved with relative and absolute termination criteria of 10% and 0.05 respectively.
• Figure 3: Volumes of the inner (solid lines) and outer (dashed lines) approximations of state-of-the-art MGA methods and of the ORACLE algorithm. ORACLE terminates after 144 iterations, having converged to within 0.1 GW (see Figure \ref{['fig:distance-oracle-small-tech-list']}).
• Figure 4: Identification of strategies not found by the literature MGA methods. Each radial plot identifies a design found by ORACLE and the closest design found by the respective MGA method. The installed capacity of each technology is normalised against its maximum potential capacity.
• Figure 5: Ratio of the volume of the inner and outer approximations of state-of-the-art MGA methods and ORACLE. ORACLE terminates after 144 iterations, having converged to within 0.1 GW, which corresponds to a volume ratio of 0.98.