###### Abstract  
Bayesian Optimization (BO) with Gaussian process regression is a popular framework for the optimization of time-consuming cost functions. However, the joint exploitation of BO and parallel processing capabilities remains challenging, despite intense research efforts over the last decade. In particular, the choice of a suitable batch-acquisition process, responsible for selecting promising candidate solutions for batch-parallel evaluation, is crucial. Even though some general recommendations can be found in the literature, many of its hyperparameters remain problem-specific. Moreover, the limitations of existing approaches in terms of scalability, especially for moderately expensive objective functions, are barely discussed. This work investigates five parallel BO algorithms based on different batch-acquisition processes, applied to the optimal scheduling of Underground Pumped Hydro-Energy Storage stations and classical benchmark functions. Efficient management of such energy-storage units requires parallel BO algorithms able to find solutions in a very restricted time to comply with the responsive energy markets. Our experimental results show that for the considered methods, a batch of four candidates is a good trade-off between execution speed and relevance of the candidates. Analysis of each method's strengths and weaknesses indicates possible future research directions. Bayesian Optimization; parallel computing; Gaussian Processes; efficient global optimization; electrical engineering; hydro-energy storage +
Footnote †: This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons. The complexity of physical phenomena in UPHES devices usually calls for model-based approximations [3]. Using these approximations, evaluation of the expected profit is fast and classical optimization methods (e.g., dynamic programming, genetic algorithms) can be applied to optimize the UPHES scheduling within a reasonable time budget [4; 5]. However, approximations may lead to unreliable simulations, motivating the recent proposal of a more robust model-based UPHES simulator in [PERSON] et al.[3]. This comes at a much higher simulation cost compared to conventional model-based approximations. With this increased simulation cost, existing optimization approaches become impractical, as operational constraints require scheduling optimization to be completed within approximately 30 min. This motivates us to investigate the use of surrogate models to (partially) replace the time-consuming simulator, in conjunction with parallel evaluations of the expected profit. To the best of our knowledge, surrogate-based optimization has never been applied to the UPHES scheduling problem. In our previous work [2], we demonstrated that Bayesian Optimization (BO) can be handy in management problems in electrical engineering. More precisely, Parallel BO (PBO) is emerging as a powerful framework for such problems. This surrogate-guided optimization approach relies on a surrogate model--often Gaussian Process (GP) regression. Based on that model, an Acquisition Function (AF) is defined to assess the value of any candidate point. Optimizing the AF yields the best candidate point according to this specific AF. In this way, only valuable (still in the sense of the AF) points are evaluated in parallel using the time-consuming simulator. Consequently, the overall optimization time can be considerably reduced. This is essential due to the time constraint arising from both the organization of energy markets and the complex modeling of physical and economic constraints of pumped-hydro systems. More generally speaking, we refer to the process responsible for selecting the next candidate points as the Acquisition Process (AP). Clearly, the choice of the AP is of crucial importance to balance the exploration and exploitation during the optimization process.