Eyal Price

Iterative Linearization Scheme for Convex Nonlinear Equations: Application to Optimal Operation of Water Distribution Systems

Convex equations exist in different fields of research. As an example are the Hazen-Williams or Darcy-Weisbach head-loss formulas and chlorine decay in water supply systems. Pure linear programming (LP) cannot be directly applied to these equations and heuristic techniques must be used. This study presents a methodology for linearization of increasing or decreasing convex nonlinear equations and their incorporation into LP optimization models. The algorithm is demonstrated on the Hazen-Williams head-loss equation combined with a LP optimal operation water supply model. The Hazen-Williams equation is linearized between two points along the nonlinear flow curve. The first point is a fixed point optimally located in the expected flow domain according to maximum flow rate expected in the pipe (estimated through maximum flow velocities and pipe diameter). The second point is the calculated flow rate in the pipe resulting from the previous iteration step solution. In each iteration step, the linear coefficients are altered according to the previous steps flow rate result and the fixed point. The solution gradually converges closer to the nonlinear head-loss equation results. The iterative process stops once both an optimal solution is attained and a satisfactory approximation is received. The methodology is demonstrated using simple and complex example applications. DOI: 10.1061/(ASCE)WR.1943-5452.0000275.

Discrete Pump Scheduling and Leakage Control Using Linear Programming for Optimal Operation of Water Distribution Systems

AbstractPump station scheduling is a major issue in optimal water system operation. Pump operation may be of an on/off form or of a fluctuating form using a variable-frequency drive (VFD). This research proposes an iterative linear discrete pump-scheduling algorithm using linear programming (LP). The examined problem includes nonlinear convex headloss, leakage, and varying total-head pump energy consumption constraints. A discrete pump operation index is proposed to select time steps on which to enforce a discrete pump operation constraint. After each iteration step, the index is recalculated based on the previous iteration steps’ results and the discrete operation constraint is added or removed from the time steps accordingly. The iterative process stops when all time steps have been discretely evaluated. The algorithm is first demonstrated on a small illustrative example application and compared to the global minimal results found by enumeration. Next, the algorithm is demonstrated on two complex exampl...

Graph Theory Modeling Approach for Optimal Operation of Water Distribution Systems

AbstractA graph theory-based algorithm is demonstrated for optimal pump scheduling of two example application water networks. The hydraulic part of the problem is solved using a dedicated and efficient hydraulic solver. The pump scheduling part of the problem is solved using a skeletonized operational graph, representing only the basic logic operational relations existing in the network required for pump selection: the pumping units (with nominal operating costs), water tanks and clustered demand nodes. The hydraulic solver advances one time step at a time. After each time step advance, the nodes of the model are checked to see if satisfy minimum service pressure and minimum water tank level. For nodes not satisfying the service constraints, the Dijkstra’s shortest path algorithm is applied to the skeletonized graph to determine the optimal pumping unit to be activated and then updating the pumps operation pattern in the model. The hydraulic solver is then reinitialized to resolve and recheck the time ste...

Optimal Pump Scheduling in Water Distribution Systems Using Graph Theory under Hydraulic and Chlorine Constraints

AbstractFinding the optimal pump operation in water distribution systems, taking into account hydraulic and water quality constraints, is a complex problem due to the nonlinear relationship between dynamic head loss and flow rate and between chlorine decay and water age, and due to the size of the problem. The proposed algorithm, for minimum cost pump scheduling, utilizes the operational graph algorithm applied to hydraulic and quality constraints. The proposed algorithm utilizes a graph algorithm that considers hydraulic and water quality constraints to find the pump scheduling that minimizes pump operational costs. The algorithm has short solution times and therefore is suitable for real-time water system control or, if used offline, for giving a recommendation on the pump operation taking into account hydraulic and water quality constraints. The algorithm results were compared with the best results found by enumeration to show that the operational graph algorithm returns a global minimal solution and, ...

Successive Linear Programming Approach Applied to BBLAWN

AbstractThe battle of background leakage assessment for water networks (BBLAWN) challenge was approached using successive linear programming. A linear representation was solved successively for the nonlinear constraints of headloss, leakage, pump energy consumption, and pipe sizing. The optimization model returned minimal cost pump scheduling and pipe sizing while minimizing leakage and maintaining minimum service pressures to the consumers. Pressure reducing valves, pump, and water tank sizing were performed manually and their effect was examined using the optimization model. Parallel pipes were added along the main supply pipes from the pumping stations to the water tanks, to allow for minimum service pressures. Pressure reducing valves were added to pipes branching from the main supply pipes to lower excess pressures to the secondary supply pipes.

Optimal Water System Operation Using Successive Shortest Path Graph Algorithm

1 PhD Student, Faculty of Civil and Environmental Engineering, Technion Israel Institute of Technology, Haifa 32000, Israel; PH: +972-3-6924603; FAX: 972-36924550; E-mail: price-e@tahal.com 2 Associate Professor, Faculty of Civil and Environmental Engineering, Technion Israel Institute of Technology, Haifa 32000, Israel; PH: +972-4-8292782; FAX: 9724-8228898; E-mail: ostfeld@tx.technion.ac.il

A Successive Linear Programming Scheme for Optimal Operation of Water Distribution Networks

Optimal operation of water distribution systems is a well explored problem defined as finding the scheduling of pumping units over time which minimize cost while maintaining flow, pressure, and tank water levels constraints. One of its major complexities is the inherent non linearity and non-smoothness relationship of the headloss equation (e.g., the Hazen Williams or Darcy Weisbach formulas). This study suggests a method for the linearization of the Hazen-Williams headloss equation which enables the non-linear hydraulic problem to be addressed and solved as a linear programming scheme. The methodology is demonstrated on a small illustrative example.