Optimal design of storm sewer networks
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This document provides a review of past, present and future approaches to optimal design of storm sewer networks. It discusses how optimization techniques have been used since the 1960s to minimize construction costs while ensuring system performance, moving from linear programming and non-linear programming to more advanced techniques like dynamic programming and discrete differential dynamic programming. The document also outlines key advantages of optimal design over traditional design methods.
Optimal design of storm sewer networks
- 1. Optimal design of storm sewer networks: Past, Present and Future Yufeng Guo1, Godfrey Walters2* and Dragan Savic2 1 EDAW, 1019 Nanhai Boulevard, Nanshan District, 518067, Shenzhen, China 2 Centre for Water Systems, School of Engineering, Computing and Mathematics, University of Exeter, North Park Road, Exeter, EX4 4QF, England, UK *Corresponding author, e-mail G.A.Walters@exeter.ac.uk ABSTRACT Optimal design of storm sewer networks aims to minimize construction costs whilst ensuring good system performance under specified design criteria. It has proved to be a complex NP (Non-deterministic Polynomial time) optimization problem, encompassing multimodal, discontinuous (or mixed discrete-continuous), non-convex features. This problem has been heavily studied since its concept was proposed in the late 1960s. However, there has been limited success in real-world practice due to the intrinsic complexity of the problem. This paper is aimed at providing a systematic and up-to-date review of achievements in this field and discussing problems and key issues today in the context of future research needs in sewer optimization. KEYWORDS Separate System; Storm Sewer; Network Design; Optimization INTRODUCTION Storm sewers play an important role in wet weather management. Without efficient drainage, storm water may cause urban flooding with severe consequential problems, such as public inconvenience, economic and environmental damage, infectious disease, and even threat to public safety. Therefore, it is vital to maintain reliable performance of storm sewer systems. However, in the face of tight budgets and more stringent regulation, sewer engineers are confronted with a significant challenge and urged to pursue cost-effective strategies for design, operation and management of storm sewers. This study focuses on the first of these issues: the design of storm sewer networks.
- 2. In conventional sewer design, the underlining principle is that all sewer conduits should be designed to deliver a free-surface flow, thus an un-pressurized or just full condition could be ensured. The relationship between pipe size and its capacity is approximated based on the hydraulic resistance equations, typically the Manning equation and Colebrook-White equation. Given a design flow (velocity or discharge) and pipe roughness, these equations can be used to determine the size of a sewer. To simplify calculations, a steady flow approximation is mostly used. In designing a network, each pipe is considered as a separate entity and in a sequence from upstream to downstream. Without any concept of optimization, the design is simply based on the idea of keeping pipe slopes as flat as possible, giving a unique but over-expensive solution. 11th International Conference on Urban Drainage, Edinburgh, Scotland, UK, 2008 2 Optimal design of storm sewer networks: Past, Present and Future Optimal storm sewer design aims to minimize the network construction cost whilst ensuring a good system performance. Depending on the problem formulation, the problem can be considered as a single-objective or multiple-objective optimization problem whilst satisfying a set of constraints. Its general formulation can be defined as follows: T f ( x ) min f ( x ), f ( x ), , fn ( x ) r K r r r r 1 2 (1) Subject to g ( x ) 0 i rfor i = 1, 2,…, k h ( x ) 0 j rfor j =1,2,…, l Where: T m x x ,x ,L,x r 1 2 decision variable vector, T problem dimension g x rinequality constraints h x requality constraints
- 3. Often a multiple objective problem can be transformed into a single objective problem by setting up a linear combination of the objectives with different assigned weighting factors, or keeping one key objective and transforming others into design constraints. Complete enumeration of all possible solutions can truthfully find the globally optimal solution for the problem. However, due to the problem complexity in nature, an exhaustive search using complete enumeration is usually computationally unaffordable and inapplicable for continuous variables. Optimization techniques turn out to be essential in addressing the problem effectively and locating optimal solutions efficiently. During the design, the optimization technique is usually integrated with a sewer network hydraulic simulator, which evaluates the hydraulic performance of each potential solution. Compared to the traditional design method, optimal sewer design demonstrates several distinct advantages: a. It renders a potentially valuable and practical solution for rigorously incorporating local economic considerations into the hydraulic design process (Dajani, et al. 1972). b. It targets obtaining the cheapest design solution whilst providing more reliable serviceability. In this way, it can reduce the use of oversized pipes, which may lead to reduced flow velocities and increase the risk of sediment deposition and subsequent blockage in pipes (DoE/NWC, 1981). c. It enables sewer engineers to investigate a great number of scenarios and deliver more design alternatives, implementing potential tradeoffs among various design objectives. d. It significantly eases the whole design procedure by automatic computer- based design and releases designers from tedious manual calculations associated with the design. e. It can work closely with sophisticated simulation models, providing the possibility of
- 4. detailed investigation of the dynamic drainage process, and leading to optimization solutions that are more hydraulically correct and reliable. HISTORICAL REVIEW OF SEWER OPTIMIZATION The topic of optimal sewer design has been heavily studied. Its concept was first proposed in the mid 1960s (Deininger, 1966; Holland, 1966) when advances in the computer power shined light on engineering research. Comprehensive cost-effective designs incorporating early simulation models and optimization technologies became computationally tractable and flourished in 1970s and 1980s. Various early optimization techniques were developed, including Linear Programming (LP) (Deininger, 1966; Dajani and Hasit, 1974), Non-linear Programming (NLP) (Holland, 1966; Price, 1978), and Dynamic Programming (DP) (Mays and Yen, 1975; Walters and Templeman, 1979). 11th International Conference on Urban Drainage, Edinburgh, Scotland, UK, 2008 Guo et al. 3 LP is a special form of mathematical programming. It can easily handle a large number of decision variables and implement the optimization in an efficient, reliable and deterministic manner. Nevertheless, the approach poses several strict requirements for its implementation: (i) its implementation basically requires all objective functions and constraints to be linear. However, highly dynamic hydraulic conditions are hardly possible to have a linear relationship with decision variables, such as pipe diameters and slopes; (ii) LP requires individual segments of the problem to operate as well independently as together. This inevitably requires each pipe to be designed separately, and implies that every pipe flow is independent of flows in adjacent pipes, which, even for a tree-like network, is only true under a steady state condition (Walters, 1992); (iii) All decision variables are treated as continuous variables. Its solutions often encompass continuous diameters, which have to be adjusted by
- 5. rounding each continuous diameter up to its nearest commercial size. NLP techniques can generally deal with non-linear objective functions and constraints, but entail much increased computational difficulty due to the discontinuous and nondifferentiable objective function. Moreover, most of them could not deal with discrete diameters (Price, 1978; Gidley, 1986). Because of various difficulties encountered with their application, mathematical programming techniques, like LP and NLP, had limited success and soon fell out of favour with researchers when more advanced optimization techniques emerged. DP and its modified version Discrete Differential Dynamic Programming (DDDP) (Mays and Wenzel, 1975) were popularly applied techniques, and DDDP is still well- liked in some current studies. DP became popular and favoured mainly because many sewer design requirements can be met by its key features: a. DP needs the optimization problem to be represented in a sequential form and be divided into stages, this corresponding well to the design of tree-like drainage networks. Argaman, et al. (1973) proposed “Drainage Lines”, which pass through all manholes being the same number of pipes away from the system outfall, to divide a sewer network into multiple levels. The Drainage lines were later renamed as “Isonodal Lines” by Mays and Wenzel (1976). b. At each stage, the implementation of DP is based on sets of states and decisions. An input state is transformed by a decision into an output state, incurring a cost (stage return). For each output state, the cumulatively optimal decision can be identified over the decision variables, and consequently incur an optimal cost up to this stage. The process of making stage decisions continues until all stages have been traversed, usually from upstream nodes to the system outfall, at which point, the overall optimal set of decisions can be traced back through the system.
- 6. c. DP is flexible with regard to the form of the objective function. It can deal with nondifferentiable and discontinuous functions. Moreover design constraints generally do not pose a problem for its implementation (Gidley, 1986). d. Dynamic programming tackles the decision in a discrete manner at each stage. It directly produces solutions with discrete diameters. Hence it is free of the problem experienced by mathematical programming that the final solution may not be optimal after the post-processing to round up continuous pipe sizes. Since finer discretization of the state variables, such as manhole depth, induces greater computation, Mays and Yen (1975) developed Discrete Differential Dynamic Programming (DDDP) to address this problem. DDDP defined the discrete depth in a more useful way by