Pumps running as turbines are suitable hydraulic machines for micro hydropower applications. The selection of the proper pumps to install in a given site still remains a major challenge, as pump manufacturers do not provide the characteristic curves data in the turbine mode. Also, the accurate prediction and modelling of the pumps running as turbines characteristic curves still remain a major difficulty as existing methodologies still lack accuracy, especially in the part load and full load operating regions. This paper proposes a new two-step methodology based on the Hermite polynomial chaos expansion for predicting the characteristic curves of pumps running as turbines and modelling their variable speed operation, aiming at improving the prediction accuracy. Firstly, bivariate continuous surrogate functions are established for predicting the turbine mode and the extended operation mode characteristic curves inside a closed interval of unit specific speed values. These surrogate functions are developed by calibrating empirical coefficients based on collected experimental data. Secondly, a hill chart model is determined for describing the variable speed operation of a given pump running as a turbine. This hill chart model allows identifying the discharge and the rotational speed set points for maximising efficiency for a given operating condition. The proposed prediction surrogate functions and the variable speed hill chart model are useful engineering tools for improving the design of pump as turbine hydropower plants and for optimising the pump running as turbine control settings to maximise the produced energy.