Sensitivity Analysis of Topological Subgraphs within Water Distribution Systems

Abstract

Sensitivity analysis of the actual hydraulic state of water distribution systems is a valuable tool with number of applications in hydraulic systems analysis. Sensitivity matrices include the information of the response of the hydraulic state variables (flows, pressures) to changes in model parameters (e.g. demands, roughness, control parameters) for a specific hydraulic state of the system. For calculation, there exists in addition to finite difference approximations also exact solutions that include the inversion of the system matrix (the Schur Complement of the Jacobian of the hydraulic network equations). In combination with hydraulic network simulations, the factorization (for example Cholesky matrix decomposition) of the matrix that has already been done by the hydraulic solver in the computation process can be used for the efficient calculation of the inverse matrix. However, for large realworld networks the sensitivity calculation is a time and memory consuming process because the inverse of the system matrix of a connected network has no zero elements. In this paper a new method is presented that allows for the exact calculation of sensitivities of a particular subgraph of interest, the topological minor or supergraph within a water distribution system network graph. Supernodes are the most important nodes in terms of connectivity redundancy within the network graph. Superlinks replace all pipes (links) in series between two supernodes. It will be shown that the sensitivities that are calculated for the subgraph deliver exactly the same results as the inversion of the entire system matrix reduced to supernodes. This paper focuses on the derivation of the equations for the reduced system matrix inversion of the topological subgraph. In addition, the paper includes the proof of equivalence of the matrix inverses for the topological minor subgraph. This inverse represents the fundamental sensitivities of nodal heads and pipe flows with respect to nodal demands in demand driven analysis. The results presented can be extended to other sensitivities, since the matrix inverse in question is included in all other derived parameter sensitivities.