A mathematical investigation on the invariance problem of some hydraulic indices

• Published in: Applied mathematics and computation 2021-11, Vol.409, p.125726, Article 125726
• Main Authors: Caldarola, Fabio, Maiolo, Mario
• Format: Article
• Language: English
• Subjects: Hadamard and scalar product; Invariance problems; Linear algebra; Local performance indices; Mathematical models; Resilience indices; Water distribution networks
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2020.125726

•Through mathematics, the paper aims to bring attention to the general problems of invariance in hydraulic engineering.•Section 3 presents a mathematical-like formalization of a water network and provides the change formulas for the fundamental energy paprameters.•As an explicit case, considered two widely used resilience indices, they turn out to be non-invariant in a PDA approach.•In addition to numerical calculations and simulations, the mathematical reasons for their non-invariance are investigated.•Two new invariant resilience indices, which modify one of those considered, are presented as an example of mathematical solutions in a new specific framework. In recent decades many mathematical models, both theoretical and computational, have been applied to hydraulic networks with considerable success, and recently this trend appears to be growing exponentially. Yet there are important problems of a mathematical nature that have very often had little consideration in this field, such as that of the invariance of the models with respect to the reference adopted. In this paper, a mathematical framework new in the field is used and, starting from the discussion of the invariance problem of local indices, the behavior of some widespread global ones that evaluate the resilience of a network will be investigated from both a theoretical and computational point of view. The authors also give suitable changing formulas in the local and global case and describe the conditions that ensure invariance. Through a mathematical-like formalization of the hydraulic network concept, the new framework finally allows to find a series of mathematical solutions to problems of this kind, two of which will be provided in the text.