## 2 Material and Methods  
### Underground Pumped Hydro-Energy Storage  
Due to their ability to quickly and cost-effectively mitigate energy imbalances, Pumped Hydro-Energy Storage (PHES) stations offer an appropriate storage solution. PHES plants are composed of (at least) a lower and an upper reservoir from which water is exchanged to either produce or store energy. In off-peak periods, production might exceed consumption such that energy is saved by pumping water from the lower basin into the upper one. It then provides a substantial reserve of energy that can be later released when needed, e.g., to maintain the transmission grid stability. Recent progress in power electronics has enabled PHES units to operate with a reliable variable-speed feature in both pump and turbine modes. The flexibility offered by these facilities is highly valuable. Indeed, it improves the economic efficiency of existing resources such as wind farms or thermal power plants [15; 16], and provides ancillary services to ensure grid stability (such as frequency control or congestion management). The inherent potential of PHES units leads to the development of new technological solutions such as Underground PHES (UPHES) for which the lower basin is located underground. A significant advantage of UPHES is the limitation of expenses from civil engineering works thanks to the recycling of end-of-life mines or quarries. These stations have a very limited impact on the landscape, vegetation and wildlife, and are not limited by topography so that more sites can be exploited [17]. In the current competitive framework governing the electricity sector, UPHES units are exploited with the objective of return on investment. Consequently, the profit of such stations must be maximized. This task is challenging since the UPHES operation is governed by two main nonlinear effects that cannot be easily modeled with traditional analytical models [3]. Firstly, groundwater exchanges between the reservoirs and their hydro-geological porous surroundings may occur. This situation typically arises for UPHES when the waterproofing work is not feasible or uneconomical [18]. Secondly, UPHES units are generally subject to important variations of the net hydraulic head (i.e., the height difference between water levels in the reservoirs). These variations are referred to as the head effects [19], and are typically quantified through laboratory measurements on a scaled model of the hydraulic machines [20]. This characterization of head effects is important since the head value defines both the safe UPHES operating range as well as the efficiency of both pump and turbine processes. In this way, the safe operating limits in pump and turbine modes continuously vary over time with regard to head variations. In general, the performance curves of UPHES stations are difficult to model since they present a non-convex and non-concave behavior. Directly integrating these effects into model-based optimization (which maximizes the UPHES profit in the different market floors) implies a high computational burden or strong assumptions that may jeopardize the feasibility of the obtained solution. To address these issues, a simulation-based BO strategy has been developed in this work. The simulator,which is a black box from the user perspective (developed independently without access to the source code), returns the daily UPHES profit accounting for all techno-economic constraints. The full description of physical and economical constraints can be found in [PERSON] et al. (2018). This simulator is denoted as \(f\) in the following and, according to a decision vector \(\mathbf{x}\in\mathbb{R}^{12}\), it returns the expected profit \(y=f(\mathbf{x})\in\mathbb{R}\). The 12-dimensional decision vector includes 8 decision variables to participate in the different time slots of the energy market, and 4 to the reserve market (i.e., provision of ancillary services). The number of decision variables is set in accordance with standard recommendations of electrical engineering.