Analytical Optimization of Vertical Closed-Loop Ground Source Heat Pump Systems

In this paper, we study the optimization of the operation of closed-loop ground source heat pump systems with any layout and any number n of vertical boreholes. Given the total required heat load, q T , the goal is to maximize the rate of thermal gains from the ground or, equivalently, to minimize the disturbance of the ground temperature at the location of the boreholes. This is achieved by optimizing the distribution of q T to the individual boreholes. We prove analytically that, at any time, the weighted temperature disturbance is minimal when the following condition holds: the temperature change is the same at the locations of all boreholes. Our proof is based on the analogy between heat transfer due to conduction and water flow through aquifers, and we make use of the results obtained for pumping cost minimization from systems of wells under transient groundwater flow conditions in infinite confined aquifers. Finally, we present a procedure to calculate the optimal distribution of the total heat load to the individual boreholes at any given time. The procedure entails the solution of a linear system of n equations and n unknowns, which is explained by means of two theoretical application examples. Accuracy of the results is also discussed.