Optimal operations of large scale water distribution systems.

Improving water distribution system operations can be achieved through the use of optimally generated pump schedules by minimizing the overall pumping cost while satisfying the water pressure and flow requirements and subject to all physical constraints. This study is devoted to developing a general methodology to determine the optimal operations of large scale water distribution systems. Rather than solving the original mixed integer nonlinear programming problem directly, a two-level hierarchical optimization scheme is proposed. The first level is a NLP problem where the decision variables are each pump station's discharge and added head, pump speeds, valve control settings, nodal pressure heads and tank water elevations. At the second level a simple DP or a direct conversion method is applied to find the best pump combinations based on the optimal solutions obtained from the first level NLP optimization. Prior to solving the two level optimization problem, pre-optimization work is performed which produces a lumped energy function for each pump station to approximate the relationship between consumed energy and pump station's added head and discharge. To make the first level NLP problem solvable, a reduction technique is proposed which uses the network simulation model to reduce the number of constraints and decision variables. This reduction, however, results in a NLP problem with implicit decision variables which are not directly controlled by the decision variables. One strategy proposed is to consider the constraints of the implicit decision variables in a penalty term appended to the objective function. The problem is then structured using an augmented Lagrangian algorithm and solved with a NLP code. The second strategy is to use an active set method. The entire NLP problem is solved using successive quadratic programming where only the active constraint set is considered during the solution process of each quadratic programming subproblem. Two case studies were performed to determine the optimal schedules and compare the two NLP approaches. The Lagrangian seemed to perform better when many constraints were initially violated, while the active set method seemed appropriate to solve systems with few initial active constraints