# A Reversible Hydropump-Turbine System
###### Abstract  
Water-pumped storage systems have become an ideal alternative to regulate the intermittent power delivered by renewable energy sources. For small-scale operations, a type of centrifugal pump coupled to asynchronous machines represents an adequate solution due to their techno-economic feasibility in addition to their ability to operate as reversible systems. This work provides a novel port-Hamiltonian modelling approach to an integrated reversible hydropump-turbine system, that can be switched from motor pump to turbine-generator by employing a conventional hydraulic switch. Our modelling strategy provides a clear physical interpretation of the energy flow from the mechanical to electrical domains. Then, the model was built with multi-domain storing and dissipating elements and the interconnection of well-defined input-output port pairs. The system's internal energy, i.e., Hamiltonian function, can be exploited for energy-shaping control strategies. The performance of our modelling approach is validated via numerical simulations. port-Hamiltonian framework; hydropump-turbine systems; nonlinear systems; multi-domain modelling +
Footnote †: journal: Applied Sciences  
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Footnote 0: Copyright** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ([[https://creativecommons.org/licenses/py/4.0/](https://creativecommons.org/licenses/py/4.0/)]([https://creativecommons.org/licenses/py/4.0/](https://creativecommons.org/licenses/py/4.0/))). L.S.E.S. different domains (electrical, mechanical, and hydraulic) to be easily coupled to each other, in addition to facilitating the design of efficient control strategies. Several models of centrifugal pumps with their advantages and disadvantages are recapitulated by [4]. For instance, in [5], the general model of a solar-powered centrifugal pump connected to an induction motor was introduced with a type of equivalent circuit optimization technique. The work of [6] suggests a modelling approach to a centrifugal pump system where its input pressure depends directly on a quadratic form of its rotational speed. The description of pumped storage systems found in [7] presents second-order differential equations that consider the pump-turbine's nonlinearities. Furthermore, in [8], a model of variable speed doubly-fed induction-generator pumped storage power is presented, which is described as a nonlinear eighth-order state equation with eight state variables. Moreover, a five-order nonlinear dynamic system of the variable-speed pumped storage power station was developed in [9], while in [10], a variable-speed pumped storage model is shown via algebraic differential equations. In addition to the above mentioned examples, a nonlinear model of a water distribution network based on a port-Hamiltonian description is presented in [11]. Moreover, a Hamiltonian model of a hydro-power system is presented in [12], which provides a theoretical basis for its safe and stable operation during transient processes. Such a model is used later on in [13] where a passivity-based control is designed. Their control strategy guarantees stability and efficient performance under different operating conditions. In [14], an energy-based modelling approach is developed where a solar-powered hydro pump system is coupled to a direct current motor. A model based on a port-Hamiltonian approach to an asynchronous motor is introduced by [15; 16], and later on extended to the modelling of an asynchronous machine configured as a generator by [17]. A trajectory tracking control strategy for the system in [17] is presented in [18]. Finally, in [11], a controller for a water-distribution system is designed, using a port-Hamiltonian approach to model the pump, allowing them to analyze stability, including nonlinear elements.