## 1 Introduction
### Optimization of MHPP  
The problem of optimizing MHPP is addressed in the literature from a wide variety of perspectives. For example, some works propose solutions to reduce the environmental impact of the plants [2,3], to quantify the potential of these installations [4], to improve the MHPP viability [5], or to operate the plants under water scarcity conditions [6,7]. Although the seasonal character of water availability induces variations in the energy production of MHPP, these plants remain one of the best solutions for isolated, rural locations with no access to the power grid, and therefore a large effort has been made in the literature to improve their design and viability. In line with this, some studies have focused on the overview and characterization of the main elements and variables of these installations [8], the design of the turbines (see [9] for a nice survey about this topic), and the analysis of the most important economic indexes [10]. A very relevant line of work is that related to the optimal components selection and/or operation of MHPP in situations in which the plant layout, that is, the location of its main elements (namely water intake, penstock profile, and powerhouse), has been fixed. For example, Ref. [11] presents a numerical tool to maximize power production or economic profit by selecting the adequate penstock diameter and type of turbines. The work takes into account the dependence of the turbine efficiency on variables such as water flow, suction head, and rotational speed. Moreover, ref. [12] presents the most important features of the turbine model from a control point of view, and, in [13], the use of model predictive control to deal with prediction errors in the available water in the reservoir and is proposed. In general, the number of variables and constraints and its nonlinear character involve a degree of complexity that makes the use of traditional gradient-based optimization strategies difficult. For this reason, the utilization of computational intelligence methods has gained great attention to deal with the previous optimization problems. Some examples of this include [14], where the authors present a stochastic evolutionary algorithm to find the blade of a Turgo water turbine that maximizes the hydraulic efficiency. Similarly, in [15], the bucket shape of Pelton turbines in various operation conditions is optimized using a Lagrangian formulation and solving the problem with evolutionary algorithms. In addition, ref. [16] proposes a complex simulation algorithm to analyze plant operation yearly and compute energy production and economic indices. The authors' findings show that the optimal sizing in terms of some economic indicators is not the same as the one maximizing the exploitation of the hydraulic potential. Moreover, ref. [17] investigates the potential of using a proposed metaheuristic method to provide optimal operations for multireservoir systems, with the aim of optimally improving hydro-power generation. Figure 1: Basic scheme of an MHPP. This work addresses the problem of optimizing the layout of a MHPP. The problem can be roughly stated as follows: given the profile of a river and its surrounding terrain, find the location of the water intake, the turbine, and the penstock layout in such a way that the mechanical stress in the pipe is below a certain level, \(\sigma_{max}\); the obtained power is above a desired value, \(P_{min}\); and the cost of the installation is minimized. Typically, this problem has been tackled, making some simplifications to use traditional analysis or optimization tools. For example, in [18], the authors propose a theoretical analysis to find the optimal penstock layout and diameter for low head plants. The optimization problem considers a 2D formulation of the MHPP layout, which, in addition, is considered composed of a unique straight segment.