## 2 Material and Methods
### Experimental Protocol  
All algorithms are implemented within the Python-based BOTorch framework [12]. In all three approaches, Message Passing Interface for Python (MPI4 Py [46]) is used for parallelization. The UPHES simulator itself is implemented in Matlab and the domain-specific RAO language. As the black-box UPHES simulator executable requires a software license, experiments are performed on a university cluster, preventing us from conducting the analysis on large batch sizes. The dedicated node possesses 2 Intel(R) Xeon(R) CPU E5-2630 v3 2.40 GHz, with 8 cores each.  
Due to the responsive energy market operational constraints, the optimization must be completed within tens of minutes. Hence, to remain consistent with the time constraints, the global budget of each optimization run is chosen as a time budget of 20 min, without initial sampling. According to usual recommendations in BO, a fraction of the budget is allocated to the initial sampling. The initial sampling budget is set to \(16\times n_{batch}\). The limiting factor being the time, the budget in terms of simulations increases proportionally to the batch size (and degree of parallelism) \(n_{batch}\). However, in order to avoid too much case-specific conclusions, the choice is made to consider a fixed time of 10 s for a simulation. Indeed, one execution of a simulation lasts for 9 to 10 s and a non-negligible overhead results from parallel calls to the black-box simulator. This overhead is only case-specific since the simulator resorts to RAO language, under the form of an executable program which necessitates interfaces between programs not suited for parallel computing. Therefore, the parallel overhead is independent of the parallel framework and is only caused by the software-dependent simulator. Experiments are performed for \(n_{batch}=1,2,4,8,16\). For any  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name** & **Expression** & \(\mathbf{\Omega}\) & \(f_{\text{min}}\) \\ \hline Rosenbrock & \(\sum\limits_{i=1}^{11}100(x_{i}^{2}-x_{i+1})^{2}+(x_{i}-1)^{2}\) & \([-5;10]^{12}\) & 0 \\ [PERSON] & \(-20 exp(-0.2\sqrt{\frac{1}{\Omega}\sum\limits_{i=1}^{12}x_{i}^{2})-exp(\frac{1 }{\Omega}\sum\limits_{i=1}^{12}cos(2\pi x_{i}))+20+e}\) & \([-5;10]^{12}\) & 0 \\ [PERSON] & \(418.9828872724338*12-\sum\limits_{i=1}^{12}x_{i}\sin\left(\sqrt{|x_{i}|}\right)\) & \([-500;500]^{12}\) & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Definition of the benchmark functions.  
Figure 1: Topology of the modeled PSH unit on Maizeret site [21].  
batch size and any algorithm, the duration of the optimization is around 25 min (initial sampling included), and the budget allocation is summarized in Table 2.  
The following hyperparameters concern all algorithms involved in this study. The surrogate model is a GP model with a homoskedastic noise level, constant trend and Materm-\(\frac{5}{2}\) kernel with automatic relevance discovery. Single criterion approaches are executed with the EI criterion (or \(q\)EI for multi-points AF) and UCB is used as supplementary AF for mic-\(q\)-EGO. The optimization of the AF is realized with the _optimize_acq_ function of the BOTorch package using L-BFGS-B algorithm. The experimental setup is summarized in Table 3.