Minimal-order observers for locating leaks in a pipeline with a branch ? Lizeth Torres *,** Cristina Verde * Jorge Rojas * * Instituto de Ingenier´ ıa, Universidad Nacional Aut´ onoma de M´ exico (e-mails: ftorreso@iingen.unam.mx, verde@unam.mx). ** C´ atedras CONACyT Abstract: This article introduces a novel method to locate leaks in trunk pipelines with a unique branch. The method is based on three state observers; each one with a specific assigned task. The aim of the first state observer (the orientation observer) is to indicate whether the leak is to the left (upstream) or to the right (downstream) of the branch. Depending on the indication of the orientation observer, one of the other two state observers (the locating observers) can be used to estimate the position of the leak. The three observers are based on second-order models that were built from the so-called rigid water column model of a pipeline and hydraulic gradient relations. Keywords: Leak detection, fault detection and isolation, state observers, nonlinear systems. 1. INTRODUCTION A pipeline is the most popular means for transporting large volumes of fluids. For this reason, governments and companies that operate pipelines devote a colossal effort to manage them in the most appropriate way. This effort may involve various tasks of control, maintenance, optimization and diagnosis. Un- fortunately, despite all these devoted tasks, undesirable events continue to occur causing accidents and irreversible damage to the society and the environment. The most common unwanted events are leaks, which are caused mainly by the aging of the pipelines (corrosion, erosion, tuberculation), failures in the installation (particularly in joints and valves), natural events (earthquakes, landslides, floods), environment conditions (hu- midity, salinity), illegal extractions and terrorist sabotages. For this reason, the concerned sectors invest generous resources for the development of new robust and reliable leak diagnosis sys- tems, which can be classified into two categories: external and internal systems. External systems use infrared radiometers, thermal cameras, vapor sensors, acoustic microphones or fiber optic cables to monitor the external physical variables of the pipeline. Internal systems use field instrumentation (e.g. flow rate, pressure or temperature sensors) to monitor the internal parameters of the pipeline. These latter are continuously en- hanced by the engineering community since they require sensor measurements with which a pipeline is normally instrumented. Internal methods can be classified into two categories: time- domain and frequency-domain methods. With regard to the location of leaks in branched pipelines, most of the approaches that have been proposed are based on a model formulated in the frequency domain that involves leak parameters such as its position and size (e.g. Mpesha et al. (2001); Kim (2014, 2016); Duan (2017, 2018); Capponi et al. (2017b)). The methodology of such approaches, which stand on the inverse transient analysis (ITA) of hydraulic variables, can ? Funded by CONACyT: Atenci´ on a Problemas Nacionales, Convocatoria 2017; and CONACyT-Secretar´ ıa de energ´ ıa, Proyecto 280170: Cero incidentes en la red de ductos de M´ exico. be summarized into the following steps: (1) the pipeline is set in transient state by closing a valve, usually a lateral downstream valve, (2) the pressure head is recorded and (3) an optimization algorithm, which minimize the error between the numerical solution of the frequency-domain model (a synthetic pressure head) and the pressure head recording, is launched to identify the leak parameters. The key feature of these methods is that the model must be formulated to involve the leak parameters such a position and size. There are three main approaches to do it: the transfer matrix method (Chaudhry, 1979), the impedance method (Wylie and Streeter, 1978), and the admittance matrix method (Capponi et al., 2017a; Capponi and Ferrante, 2017). The advantages of the above-mentioned approaches are really outstanding, for example: (a) they require few sensors for recording the transient and (b) they can be used for pipelines with complex configuration because complex pipelines can be easily modeled in the frequency domain. Their limitations are due to modeling assumptions; for example, the linearization of the quasi-steady friction term and the omission of the unsteady friction, but also due to the difficulty of ensuring convergence and uniqueness in the solution of the inverse problem by using optimization methods. In time domain, some approaches have also been proposed to detect leaks in branched pipelines, e.g. the methods proposed in Verde and Torres (2015); Verde et al. (2016), which are based on estimators of state variables. i.e. state observers, which in turn are based on reduced-order versions of the governing equations of the flow in a branched pipeline with a leak. The principal advantage of using state observers is the stability of the solution for the leak inverse problem, i.e, the estimation of the position and size of a leak from a set of observations (concretely, flow rate and pressure measurements). Moreover, the use of observer-based methods has additional benefits, for example: they can diagnose leaks in real-time, they can be programmed in small single-board computers (thanks to their mathematical simplicity), they use information that normally is measured in pipeline systems (pressure and flow rate) and they provide a diagnosis without alter the behavior of the flow for