Thermoeconomic optimization of vertical ground-source heat pump systems through nonlinear integer programming

Vertical ground-source heat pump systems (GSHPSs) use the ground's undisturbed relative constant temperature as a source for space heating of residential and commercial buildings. The design of GSHPSs is focused in finding the optimal depth and amount of boreholes and also the connected power requirement like the amount and size of heat pumps. In this paper a mixed-integer nonlinear programming (MINLP) approach to solve the design problem of a vertical GSHPS is presented. The resulting mathematical model includes the calculation of the total annual costs (TAC) and the coefficient of performance to obtain estimates of both economic and ecological relevance to design an optimal equipment set-up. For desired constraints the numerically optimal values of the design parameters (borehole depth, mass flow rate, number of boreholes, type and number of the heat pumps) were calculated. Two numerical solution alternatives are investigated, namely Generalized Reduced Gradient (GRG2) and evolutionary algorithm. The GRG2 approach provides a more stable and faster optimal solution. Calculated results are presented through a validation example. The evaluation of the proposed objectives and studied sensitivity effects present the applicability of the model. This method was able to improve the TAC about more than 10%.