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

Abstract

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 step’s 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.