In this chapter the aspects of sewer network design relevant to the optimization problem were introduced. It was stated that in this investigation the network is to be optimized in terms of capital investment cost. The function used to calculate the investment cost as well as the unit cost functions associated with it were introduced. The use of a penalty function to avoid infeasible solutions was motivated.
The two parts of the layout design problem, spatial and directional layout design, were introduced. Spatial design of the layout is to be completed a-prior to the optimization procedures which are concerned with the directional design of the network. The concept of a base layout, which includes all required manholes and pipes of the nal solution, was described. The design constraints relevant to the layout creation algorithms, namely that no cycles may be present in the nal layout, was reviewed. The adjacency node implementation used by Moeini and Afshar (2012) to avoid cycles in the layout was described.
Constrains on the hydraulic design parameters relevant to the optimization was reviewed and their inclusion motivated. It was stated that the suitability of a given set of hydraulic parameters for a design is dependent on the network layout. The a-priori design variables, namely ground elevations at manholes and in ow rates due to service connections, were introduced and the required data format described. The calculation of the downstream cumulative ow rates using the contributor hydrograph model was determined to be su ciently accurate for use in the design techniques described here. The method used to attenuate ow rates
with time delay and a conservative linear interpolation procedure to determine intermediate ow values of the hydrographs was introduced.
The tness warping phenomenon of algorithms which select both layout and hydraulic parameters simultaneously was introduced. To avoid this phenomenon, a computationally e cient hydraulic optimization model (cf. Chapter 3) which can be used to optimize the hydraulic parameters of each layout produced by a metaheuristic layout optimization algorithm (cf. Chapter 4) is proposed by this investigation, thus avoiding tness warping.
This chapter did not go into any detail on how the optimization algorithms incorporate the design constraints or how the layout creation algorithms use the concept of adjacency nodes. Rather, the focus was on formulating the problem of sewer network design as an optimization problem.
The process by which an optimized design can be obtained comprises layout optimization and hydraulic optimization for each candidate layout. In chapter 3 the hydraulic optimization procedure is described, followed by the layout optimiza- tion procedure in chapter 4. It should be noted that the hydraulic optimization procedure is useful for any given layout, i.e. it can be used even if no attempt is made to optimize the network layout.
Chapter 3
Hydraulic Optimization
Hydraulic optimization is the component of the sewer network optimization prob- lem in which element sizes, installation depths and slopes are determined for a given layout. Due to the highly constrained nature of hydraulic optimization and complexity of simultaneous solution algorithms this part has seen considerably more work than the layout optimization problem (Lejano, 2006). In this chap- ter an overview of the hydraulic optimization problem is presented. The state of the art solutions to the problem are reviewed and their shortcomings, relevant to this investigation, identi ed. A computationally e cient heuristic optimization algorithm which relies on required slope information to systematically solve for all hydraulic parameters, is developed and applied to two case studies from the literature. It is shown to obtain near optimal solutions while requiring very little computation time.
3.1 Problem Statement
The hydraulic optimization problem can be seen as an element size selection, or pipe diameter selection, problem. While other approaches may be viable, this is certainly the best option as diameters can only be selected from a discrete set of commercially available diameters. The hydraulic optimization problem is catego- rized as a mixed integer linear programming optimization problem.
Referring to Equation 2.10, Manning's equation for open channel ow, it is clear that ow rate, Q, pipe slope, S and diameter, d are dependent variables. The ow area, A, and wetted perimeter, P , are dependent on the diameter as well as the ow depth, y, within a pipe. Full ow conditions are often assumed when performing hydraulic calculations for sewer networks as this greatly simpli es the required calculations. All three variables, namely the ow rate, slope and diameter,
have to be determined, either by some mechanism of the optimization algorithm, or by calculation. Most commonly diameters are selected by a mechanism of the optimization algorithm from the discrete set of available diameters. Cumulative ow rates for pipes are often prede ned in optimization benchmarking problems, so no hydraulic analysis to determine ow rates is required. This only leaves the slope to be determined. As Manning's equation requires the ow area and wetted perimeter the slope can, at this stage, still not be calculated as the ow depth needs to be determined. However, as stated it is common practice to assume full ow conditions and calculate ow area and wetted perimeter based on this as- sumption. This allows calculation of the required slope at full ow conditions. If the assumption of full ow conditions is not made the ow depth within a pipe has to be determined. Calculating the ow depth requires a highly implicit equation in terms of the ow depth to be solved using some form of numerical analysis.
The problem is further complicated if ow rates are not prede ned and need to be obtained using hydraulic analysis. In this case values for the slopes are often estimated and adjusted if need be after hydraulic analysis. However, it is possible in the case of a branched network layout to perform the hydraulic analysis without the need to estimate slopes as at the upper ends of the network the in ow into a pipe is known at the start of the analysis. The hydraulic analysis procedure can thus start at the upper pipes in the network and proceed downstream.
If the hydraulic optimization procedure is to be successful it must overcome all of these challenges simultaneously, with or without assumptions. In this inves- tigation as few as possible assumptions are made during the analysis procedure. Existing hydraulic models are modi ed or new procedures developed to allow all variables to be calculated accurately from engineering theory and principles rather than to rely on estimations.