Assessment of the gradient of an objective function by analytical derivation for optimization-based design of ground-coupled heat pump systems

Optimization-based design of ground heat exchangers requires derivation of the objective function with respect to the design parameters, which is usually done through finite-differentiation of the cost or utility function. The approach is however prone to approximation errors and can result in convergence issues or long optimization time. By deriving analytically the ground heat exchanger transfer function, it is possible to obtain an exact representation of the objective function gradient and avoid numerical instabilities. To illustrate the advantages of using analytical expressions, a common design task is expressed as an optimization problem. It is shown that by using an analytical derivation of the gradient in conjunction with strong Wolfe conditions during a line search may reduce significantly computation time by comparison to a finite-differentiation of the gradient. INTRODUCTION Two approaches are commonly used to design a ground-coupled heat pump system. The first one relies on sizing equations, like the one suggested by ASHRAE (Kavanaugh and Rafferty, 2014; Philippe et al., 2010; Bernier et al., 2008), while the second approach consists of iteratively using a simulation method to optimize a cost or utility function. The optimization process can be achieved by trial and error, which can be cumbersome, or automated through nonlinear optimization algorithms (Retkowski and Thöming, 2014; Huang et al., 2015; Hénault et al., 2016). The latter however requires the computation of the derivative of the objective function with respect to the n design parameters in order to find a suitable descent direction. The objective function gradient is usually computed by finite difference through 1  n simulations of the ground-coupled heat pump system. Although being easy to implement, computation of objective function gradient through finite differentiation often leads to inaccurate estimations of the descent direction, which may significantly increase solution time. The objective of this paper is to present two efficient computational approaches to derive the gradient of an objective function corresponding to a common design task encountered by designers.