Analytical Solution of Jacobian Matrices of WDS Models

The Jacobian matrices of water distribution system (WDS) models contain the gradient information of input parameters to observations, which are the key to construct gradient vectors for model calibration via gradient-based methods. In existing studies, the perturbation method is most often applied to approximate Jacobian matrices, however it is computation expensive for larger WDSs. Besides, although other methods have also been presented to compute the analytical solution of Jacobian matrices directly, they fail to provide the explicit expressions of Jacobian matrices leading to the difficulty in application. This paper presents a novel matrix analysis method to deduce the analytical solution of Jacobian matrices of WDS models, including the nodal demand, pipe roughness and pipe diameter to nodal pressure and pipe flow, respectively. All solutions are expressed in a concise matrix form with advantages of being easy to understand. A numerical case is used to detail the construction of the involved matrices.