(710g) A Semi-Infinite Programming Approach to Adress Renewable Energy Uncertainty in Energy Systems Design

Increasing shares of variable renewable energy sources (VRES) introduce higher volatility and uncertainty in energy systems [1]. Using mathematical optimization, this uncertainty can be accounted for in systems design by incorporating historical time series data [2]. However, excessive use of historical data leads to large and thus intractable optimization problems [3]. Therefore, time series aggregation methods are applied to distill few representative periods from the historical data, but they may neglect extreme scenarios which are known to disproportionally drive system costs [4, 6].

Extreme periods, e.g., times with lowest solar irradiation, may be identified a-priori by statistical measures [6,7]. In many cases, however, the identification is difficult. For instance, the combined influence of multiple time series, e.g., wind speeds, solar irradiation, and electricity demand, may depend on the system design itself. Recently, approaches to iteratively select extreme periods have been suggested where the design optimization and the underlying operational optimization are performed separately [5, 7]. Such a strategy keeps the size of the optimization problem manageable and guarantees the feasibility of the solution on the historical data set, but it is limited to finite sets of uncertainty realizations.

We propose an approach for energy systems design based on semi-infinite programming with existence constraints (ESIP) [8, 9] to identify worst-case scenarios during the optimization. Our ESIP approach guarantees that the system can operate feasibly for all possible VRES and electricity demand realizations within predefined uncertainty ranges, i.e., box constraints. Representative days based on historical time series data [4] are only used for operational cost estimation. To identify the bounds for the worst-case scenario search and enhance computational tractability, we consider a lower-dimensional representation of the historical data using principal component analysis [10] and the convex hull, leveraging the observation that energy time series data typically lie on lower dimensional manifolds [11]. We restrict ourselves to energy systems design problems where determining the optimal operational strategy is a convex problem which allows us to reformulate the ESIP problem into a bi-level SIP problem, by using strong duality of the third level, and solve it using our solver library libDIPS [12]. We demonstrate our approach on the illustrative case of designing a robust renewable energy supply system for the island of La Palma, Canary Islands. We validate the obtained robust design by examining operational feasibility on historical time series data and investigate the trade-off between solution accuracy and CPU time that results from different degrees of dimensionality reduction.

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[12] Zingler, A., Mitsos, A., Jungen, D., & Djelassi, H. (2023). libDIPS — Discretization-Based Semi-Infinite and Bilevel Programming Solvers (24914). Optimization Online. https://optimization-online.org/?p=24914