Jochen Deuerlein

Reformulated Co-Tree Flows Method Competitive with the Global Gradient Algorithm for Solving Water Distribution System Equations

AbstractMany different methods have been devised to solve the nonlinear systems of equations that model water distribution networks. Probably the most popular is Todini and Pilati’s global gradient algorithm (GGA). Given the GGA’s success, alternative methods have not aroused much interest. One example is the co-tree method, which requires some cumbersome steps in its implementation. In this paper, a reformulated co-trees method (RCTM) is presented that simplifies the procedure by manipulating the incidence matrix into trapezoidal form: a lower triangular block at the top representing a spanning tree and rectangular block below it representing the corresponding co-tree. This reordering leads to significant efficiencies that make the RCTM competitive with the GGA in certain settings. The new method has some similarities to the loop flows corrections formulation, and it is shown, by application to a set of eight case study networks with between 932 and 19,647 pipes and between 848 and 17,971 nodes, to be be...

The Never Ending Story of Modeling Control-Devices in Hydraulic Systems Analysis

Difficulties of simulation in existing hydraulic models arising from combinations of pressure and flow controlling devices in water distribution systems have been discussed in a number of previous papers. For instance, examples for non-convergence or wrong results of the hydraulic solver EPANET (version 2.00.10) were first published by Simpson in 1999. It may be shown that the problems were caused by a singularity of the equation system that appears if in an iteration two interacting control devices are active at the same time. In terms of graph theory the part of the network between the two active valves in this case is disconnected from the rest of the system leading to the singularity. In the new EPANET version 2.00.12 that has been released recently this problem is tackled by adding a virtual coefficient to all matrix columns and rows corresponding to nodes of active flow control valves. Mathematically this method is equivalent to adding a very small diameter pipe to the actual network in parallel to the FCV resulting in a nonsingular system. The examples of networks published by Simpson (1999) where EPANET 2.00.10 failed to converge or converged to wrong results now can be solved successfully. Nevertheless the latest release of EPANET still has difficulties in modeling of combinations of control devices. Whereas the former version of EPANET (version 2.00.10) often failed to calculate the correct valve states (active, closed, open) the problems of the new version consist of numerical inexactness that is caused by the addition of the virtual matrix terms for FCVs. In addition examples can be found where version 2.00.12 of EPANET still fails to converge.

Modeling the Behavior of Flow Regulating Devices in Water Distribution Systems Using Constrained Nonlinear Programming

Currently the modeling of check valves and flow control valves in water distribution systems is based on heuristics intermixed with solving the set of nonlinear equations governing flow in the network. At the beginning of a simulation, the operating status of these valves is not known and must be assumed. The system is then solved. The status of the check valves and flow control valves are then changed to try to determine their correct operating status, at times leading to incorrect solutions even for simple systems. This paper proposes an entirely different approach. Content and co-content theory is used to define conditions that guarantee the existence and uniqueness of the solution. The work here focuses solely on flow control devices with a defined head discharge versus head loss relationship. A new modeling approach for water distribution systems based on subdifferential analysis that deals with the nondifferentiable flow versus head relationships is proposed in this paper. The water distribution equations are solved as a constrained nonlinear programming problem based on the content model where the Lagrangian multipliers have important physical meanings. This new method gives correct solutions by dealing appropriately with inequality and equality constraints imposed by the presence of the flow regulating devices (check valves, flow control valves, and temporarily closed isolating valves). An example network is used to illustrate the concepts.

A Robust, Rapidly Convergent Method That Solves the Water Distribution Equations for Pressure-Dependent Models

AbstractIn the past, pressure-dependent models (PDMs) have suffered from convergence difficulties. In this paper conditions are established for the existence and uniqueness of solutions to the PDM problem posed as two optimization problems, one based on weighted least squares (WLS) and the other based on the co-content function. A damping scheme based on Goldstein’s algorithm is used and has been found to be both reliable and robust. A critical contribution of this paper is that the Goldstein theorem conditions guarantee convergence of the new method. The new methods have been applied to a set of eight challenging case study networks, the largest of which has nearly 20,000 pipes and 18,000 nodes, and are shown to have convergence behavior that mirrors that of the global gradient algorithm on demand-dependent model problems. A line search scheme based on the WLS optimization problem is proposed as the preferred option because of its smaller computational cost. Additionally, various consumption functions, i...

Calibration of Water Demand Multipliers in Water Distribution Systems Using Genetic Algorithms

AbstractHydraulic models have been widely used for design, analysis, and operation of water distribution systems. As with all hydraulic models, water demands are one of the main parameters that cause the most uncertainty to the model outputs. However, the calibration of the water demands is usually not feasible attributable to the limited quantity of available measurements in most real water networks. This paper presents an approach to calibration of the demand multiplier factors under an ill-posed condition where the number of measurements is less than the number of parameter variables. The problem is solved using a genetic algorithm (GA). The results show that not only is the GA able to match the calibrated values at measured locations, but by using multiple runs of the GA model, the flow rates and nodal heads at nonmeasured locations can be estimated. Three case studies are presented as an illustration of the problem. The first case study is a small network that demonstrates the calibration model. The ...

Graph Decomposition in Risk Analysis and Sensor Placement for Water Distribution Network Security

Over the last decade considerable research in water distribution network modeling has focused on the security of water supply against terrorist attacks. In this paper, specific issues of urban and regional distribution systems as to their vulnerability against terrorist attacks with CBRN (chemical, biological, radiological, nuclear) substances, detection methods and emergency plans are investigated. As a first step, a risk analysis is carried out based on topological properties of different parts of the network, classification of building developments and customers. The decomposition of the network graph enables the differentiation of network components (into treed components, blocks and bridges). Following this subdivision, the impacts and detection of attacks on various parts can be assessed. For example, looped parts (blocks) of the network are at a higher risk than branched subsystems (trees) since toxic matter can be distributed using alternate paths that depend on the particular water demand loading case whereas in a branched network only the system downstream the intrusion point is affected. The results of the risk analysis will be used for the creation of risk maps, efficient placement of sensors and the development of emergency plans. The specific differences of urban (mainly looped) and regional (mainly branched) supply networks will be demonstrated. The observation of water distribution systems with water quality sensors has been studied by researchers in detail over recent years. The application of the decomposition of the network graph to the sensor allocation problem will be demonstrated using the example network 2 of the Battle of the Water Sensor Networks (BWSN). The network is subdivided into tree, bridge and looped block components. It will turn out that tree structures should be excluded from application of sensor allocation algorithms using mathematical optimization since they are mostly responsible for non-detections. Decrease in calculation time can be further reached if separated blocks are identified in a preliminary analysis and the information is used for the solving the allocation problem. The crucial point of all sensor networks is that a full coverage of the system won’t be reachable and there will be a more or less long time to detection. An alarm is not actually generated until the toxic substance has passed a sensor. In this study another approach is proposed. The case of intrusion of the toxic substance by pumping against the pressure of the network will be considered. This event is considered as a “positive” leak. Leak detection methods that are normally used for the observation of pipelines are applied to the investigation of the water hammer event caused by the intrusion. Finally, conditions for the practical applicability of the method will be put up for discussion.

Alternative approaches for solving the sensor placement problem in large networks

ABSTRACT Positioning sensors in a water supply network is a NP–hard task. We propose three algorithms – one based on integer linear programming (ILP) and the other two based on the Greedy paradigm. We apply these algorithms to real case networks and com-pare the results of these algorithms with the results of an algorithm based on NSGA II, a genetic algorithm. We come to the conclusion that our algorithms outperform NSGA II in every single case. The algorithm based on linear integer programming may be applied as a competitor to the algorithm implemented in TEVA –SPOT (Ber-ry, 2009), while the first Greedy algorithm may replace the ILP algorithm in large networks due to its faster running time. The second Greedy algorithm approaches the question on finding those nodes which are the most sensitive to variations in pressure and are thereby ideal places to monitor the hydraulic state of a water distribution network. KEYWORDS Graph Theory, Sensor location layout, Greedy Algorithm, Genetic Algorithm, Integ-er Linear Programming, Sensitivity

Fast Graph Matrix Partitioning Algorithm for Solving the Water Distribution System Equations

AbstractIn this paper a method that determines the steady-state hydraulics of a water distribution system, the graph matrix partitioning algorithm (GMPA), is presented. This method extends the technique of separating the linear and nonlinear parts of the problem and using the more time-consuming nonlinear solver only on the nonlinear parts of the problem and faster linear techniques on the linear parts of the problem. The previously developed forest-core partitioning algorithm (FCPA) used this approach to separate the network graph’s external forest from its looped core but did not address the fact that within the core of a network graph there may be many internal trees—nodes in series—for which a more economical linear process can be used. This extension of the separation process can significantly reduce the dimension of the nonlinear problem that must be solved: GMPA applied to eight case study networks with between 900 and 20,000 pipes show reductions to between 5 and 55% of the core dimension (after F...

Local Sensitivity of Pressure-Driven Modeling and Demand-Driven Modeling Steady-State Solutions to Variations in Parameters

AbstractThe first-order sensitivity matrices (matrices of sensitivity or influence coefficients) have application in many areas of water distribution system analysis. Finite-difference approximations, automatic differentiation, sensitivity equations, and the adjoint method have been used in the past to estimate sensitivity. In this paper new, explicit formulas for the first-order sensitivities of water distribution system (WDS) steady-state heads and flows to changes in demands, resistance factors, roughnesses, relative roughnesses, and diameters are presented. The formulas cover both pressure-dependent modeling (PDM) and demand-dependent modeling (DDM) problems in which either the Hazen-Williams or the Darcy-Weisbach head-loss models are used. Two important applications of sensitivity matrices, namely calibration and sensor placement, are discussed and illustrative examples of the use of sensitivity matrices in those applications are given. The use of sensitivity matrices in first-order confidence estima...

Reliability analysis of water distribution systems using graph decomposition

Water supply utilities are increasingly being confronted with growing maintenance requirements in order to provide a reliable and cost efficient operation of the water distribution system in order to deliver high standard drinking water. One specific challenge is to find a satisfactory compromise between the contradictory objectives of saving investment cost and meeting the increasing expectations of customers. A valuable tool for the planner is reliability analysis of the system that for instance provides information about the importance of each particular pipe in determining the total reliability of the system. This information can be used to decide which pipes should be replaced next or potentially duplicated to reduce the risk of system isolation. Since structural (topological) reliability only considers the connectivity of the network a new approach for the calculation of structural hydraulic reliability is presented in this paper. The algorithmic implementation is based on a decomposition method of the network graph that greatly enhances both the calculation of pure structural reliability as well as structural hydraulic reliability.