Comparison of models for optimal operation of multiquality water supply networks

The solution of models for the optimal operation of multiquality networks is difficult because both the objective function and the constraints are nonlinear. Methods of nonlinear optimization are sensitive to scaling, values of gain factors in the penalty terms, distance of the initial solution point from the optimal point, and the size of the network. The original nonlinear problem has been simplified by selecting proper variables as unknowns. An equivalent optimization problem with nonlinear objective functions and linear constraints is presented. A comparison is made between two methods: the suggested one, decomposed projected gradient (DPG), and sequential quadratic programming (SQP), on several case studies. It was found that SQP obtains slightly better solutions for small networks but is sensitive to the gain factor, and to the choice of initial point. For networks containing 20–50 pipes and nodes, SQP did not reach a feasible optimal solution. To overcome the scaling problem, two projected gradient approaches – full mixing step (FMS) and partial mixing step (PMS) – are suggested and tested against the SQP and a unit adjustment (PMS-O) optimization methods. Both methods, PMS and FMS, result in good steady solutions even for a complicated case of a regional network with multiple water quality factors and treatment plants. The SQP and PMS-O methods failed due to scaling, size of network, and distance of initial points from the optimal solution.