*State of the Art*

In the literature, there are different approaches in the study of PAT that can be grouped as follows:


providing a methodology that allows the choice of the most suitable turbomachine to obtain electricity in those areas that do not have access to it, exploiting small hydroelectric resources.

All the proposed models can be improved because it is very difficult to foresee the performance curves of any PAT, given the very wide range of machines present on the market. For some PATs, these models provided acceptable results; for others, they did not. The PATs show phenomena of instability in the associated fluid, generating S-curves or cavitation phenomena and making the performance lower than a traditional Kaplan, Francis, or Pelton turbine. These instabilities are linked to the geometric configuration of the machine and to the deviation of the parameters calculated from the design values or measured in the laboratory. Given their use and their high energy consumption, it is therefore essential that their performance be optimized, and different models or design methodologies have been developed in this regard. Recent literature shows that in-depth studies have been carried out concerning the impact that the geometric parameters have on the performance of PAT, studies of a theoretical, numerical, and experimental nature [31]. A model has recently been developed those acts on the shape of the impellers of centrifugal compressors, given its influence on the overall performance of the machine [32]. It references genetic algorithms (GA) and a 3D simulation, which act on certain parameters such as the angle of the blades at the leading and trailing edges and the point where the splitter blades are connected. A new and performing design has therefore been obtained, which contributes to the research and development of compressors, without altering the technical characteristics of the fluid, to be able to replace low-consumption engines with ecological and economical fuel. A further solution, to optimize the PAT performances, refers to a numerical model which can determine the most advantageous geometric structure of the water cut [33]. Its finite thickness interferes with the flow at the entrance to the duct, generating swirling phenomena and deviations of the flow lines. Different stretching and cutting water thickness values at variable inclination are then analyzed using CFD simulations to identify the geometric features that have the greatest impact on machine performance. In recent literature, there is also a discussion that refers to PATs with low specific speed values in pico-hydropower plants [34]. This research was carried out by referring to regenerative pump models, given the characteristics that mark them in terms of stability and constructive simplicity. The approach used is both theoretical, referring to the momentum exchange theory, and through a 3D numerical simulation, to study the behaviour and performance of the machine in turbine operation. It is evident that PATs are the object of study for many researchers, as this technology is still under development and improvement. There are numerous contributions and optimizations that have been made in recent years, with different approaches and innovative ideas, both with a purely theoretical treatment and through a subsequent experimental verification.
