Pipe roughness calibration in water distribution systems using grey numbers

This paper presents a procedure based on the use of grey numbers for the calibration (with uncertainty) of pipe roughness in water distribution systems. The pipe roughness uncertainty is represented through the grey number amplitude (or interval). The procedure is of a wholly general nature and can be applied for the calibration (with uncertainty) of other parameters or quantities, such as nodal demands. In this paper, for the purpose of roughness calibration, a certain number of nodal head measurements made under different demand conditions is assumed to be available at different locations (nodes); all other topological and geometric characteristics of the system are considered to be known exactly. The general approach to pipe roughness calibration (taking account of uncertainty) focuses on identifying the grey roughness values which produce grey head values at the measuring nodes such as to encompass the observed values grouped on the basis of the different demand scenarios and, at the same time, have as small an 'amplitude' as possible. The proposed procedure was applied to two synthetic case studies and to one real network. The tests on the synthetic case studies show that the proposed procedure is able to correctly solve the inverse problem, i.e. it can identify the known grey roughness numbers even when they overlap; the same applies when the known grey roughness numbers collapse into known white roughness numbers. The test on the real case offers the possibility of highlighting the potentials of the procedure when applied within a context where measurement errors and other uncertainties are present. The procedure entails computing times that may become lengthy. However, it is possible to reduce these computing times considerably by replacing the hydraulic-simulator to which a number of calls must be made during the calibration procedure (for objective function evaluation)-with an approximation based on a first-order Taylor series expansion. This approach introduces acceptable approximations within the context of the problem considered.