Outer approximation methods for the solution of co-design optimisation problems in water distribution networks

In the present manuscript, we {\color{red} investigate and demonstrate the use of outer approximation methods} for simultaneously optimising the placement and operation of control valves in water distribution networks. The problem definition results in a mixed-integer nonlinear program with nonconvex constraints. We simplify the formulation, compared to previous literature, in order to reduce the degree of nonlinearity in the constraints and decrease the total problem size. We then {\color{red} formulate the application of} outer approximation based methods for the generation of good quality local optimal solutions for {\color{red} the considered co-design problem}. Finally, we present the results of applying the developed techniques to two case studies, and also comparing the performances of the outer approximation approaches with those of other local mixed integer nonlinear programming solution methods.