Real-time management of a sewage system: verification of the optimality and applicability of graphical linear programming compared to mixed linear programming
The first stage of real-time management of wastewater systems could, for example, consist of making sure that the use of controls such as valves and pumps can indeed minimise the discharge into the natural environment. This management step, referred to as reference management, is used to determine the control strategies over the entire duration of a rainfall event known in advance. The calculation is therefore performed at the end of the rainfall event and is used to determine what could have been done with the regulation components (e.g. in terms of minimising the volumes discharged). The calculation can also show whether or not it is necessary to control the valves and pumps during the rainfall occurrence (dynamic management) rather than fixing the flow rates in advance (static management) if the receiving body of water is to be protected from discharges. In the area of operational research, management controls can be determined with the help of linear programming. Here the aim is to minimise the linear function (f), generally called the cost function or objective function, under different linear constraints (g). There are several variants of linear programming. The first one is mixed linear programming, where some variables are required to be integers or even binary. Conventional calculation techniques such as branch-and-bound method provide a global minimum solution, but the calculation is very lengthy. Another variant of linear programming is graph programming. This optimisation technique consists of modelling the hydraulic behaviour of most of the constructions that can be found in wastewater systems (main drains, storm overflows, storm water basins, etc.). It has been applied to the wastewater system of Saverne in order to minimise the volumes discharged into the natural environment. In order to ensure that the constructions are modelled correctly and to optimise the functioning of the controls, saturation constraints had to be added to the choice of the arcs of the graph. The primal-dual algorithm no longer provides a global minimum solution when these constraints are added. In contrast, the calculation time is much shorter than that for mixed linear programming. This article is aimed at comparing the results in terms of the minimisation of the discharged volumes by means of linear programming with graphs and mixed linear programming, as part of the reference management applied to the wastewater system of Saverne. The final goal was to be able to select a compromise between the relevance or the accuracy of the results and the means to achieve them. We have shown that wastewater constructions such as main drains, storm basins and overflows can be modelled simply with the two techniques above. However, it is necessary to add binary variables in the case of mixed linear programming and a degree of arc saturation if the graph approach is used. The branch-and-bound algorithm used for mixed linear programming can be used to obtain a global minimum solution, with a very long calculation time. In contrast, even though the convergence time is very short for linear programming with graphs, the global minimum cannot be ensured because the algorithm used imposes independence with respect to the choice of the arcs to be saturated. Given the benefits and drawbacks of each approach, we have attempted to use the example of the wastewater system of Saverne to quantify the calculation time and the differences in terms of the discharged volume. The results have shown that mixed linear programming requires calculation times that can last over 24 h. In contrast, with the graph approach, the calculation takes approximately five minutes on average, with a global minimum in terms of volume that does not exceed 5% as compared to the solution obtained by mixed linear programming. We have shown that a solution requiring a much shorter calculation time is available and offers a compromise between exact determination and an optimised associated calculation time.