Optimal Design of Sewer Networks using hybrid cellular automata and genetic algorithm
Yufeng Guo*, Godfrey Walters**, Soon-Thiam Khu***, Edward Keedwell**** * Centre for Water Systems, School of Engineering, Computer Science and Mathematics, University of Exeter, Exeter, EX4 4QF, UK, e-mail:
[email protected] ** Centre for Water Systems, School of Engineering, Computer Science and Mathematics, University of Exeter, Exeter, EX4 4QF, UK, e-mail:
[email protected] *** Centre for Water Systems, School of Engineering, Computer Science and Mathematics, University of Exeter, Exeter, EX4 4QF, UK, e-mail:
[email protected] ****Centre for Water Systems, School of Engineering, Computer Science and Mathematics, University of Exeter, Exeter, EX4 4QF, UK, e-mail:
[email protected] Abstract Optimal sewer design aims to minimize capital investment on infrastructure whilst ensuring a good system performance under specific design criteria. One of the state-of-the-art optimization techniques for this problem is the Genetic Algorithm (GA), which is commonly combined with a sewer hydraulic simulator during the optimization. However, this approach can be prohibitively time-consuming especially for designing large networks. Firstly, GAs normally take a large number of generations to achieve performance improvement. Secondly, many forms of GA rely on randomly generated initial populations which are often poor solutions. To overcome this intractable problem, this paper introduces a robust hybrid optimization method, named CA-GASiNO (Cellular Automata and Genetic Algorithm for Sewers in Network Optimization). It fulfils the design task at two stages. A local agent approach based on Cellular Automata (CA) principles is firstly applied to obtain a set of preliminary solutions, which are employed to seed a multi-objective Genetic Algorithm (MOGA) at the second stage for final polished designs. The CA based approach provides a good initial population at a remarkably small computational cost and hence saves computation for the following genetic algorithm runs. The GA targets the global optimal which is fundamentally troublesome to the localised CA approach. Two sewer networks, one small artificial network and one large real network, were used for case studies. All results indicate that the proposed method outperforms the standard multi-objective GA in terms of its optimization efficiency whilst achieving a better Pareto front. Keywords Cellular Automata, Genetic Algorithm, Sewer Design
1. Introduction An urban catchment without an effective sewer system may easily encounter inundation and other consequential problems, such as public health threats and environmental damage. Optimal sewer design aims to find a cost-effective solution, which minimizes capital investment whilst ensuring a good system performance under specific design criteria such as zero surface flooding under a 25-year frequency storm. Various optimization techniques have been developed and applied for this task such as Linear Programming (Dajani et al. 1972), Non-linear Programming (Price, 1978), Dynamic Programming (Walters & Templeman, © IWA Publishing 2006. Published by IWA Publishing, London, UK.
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1979), and recently Evolutionary Computation (Cembrowicz 1994; Diogo, et al. 2000). Genetic Algorithm (Holland, 1975), a specific version of Evolutionary Computation, is the most popular optimization technology currently explored in practice. However, GAs usually require a high computational cost to find sound solutions. Firstly, similar to the natural evolutionary process, GAs take a large number of generations to achieve performance improvement, with increased computational costs for complex systems. Secondly, many forms of GA rely on randomly generated initial populations which are often poor solutions. Therefore, the approach can be prohibitively time-consuming especially for designing large networks. Guo (2005) developed a highly efficient sewer design method, called CASiNO (Cellular Automata for Sewers in Network Optimization), which is loosely based on cellular automata principles. However, taking optimization actions only on the basis of local relationships and without any evaluation on overall system performance, CASiNO is principally categorized as a local-search strategy. Hence, no global optimum can be guaranteed. Some researches on GA (Neppalli et al., 1996; Harik and Goldberg, 2000; Hopper and Turton, 2001; cited by Keedwell and Khu, 2005) have found that if prior knowledge exists or can be generated at a low computational cost, seeding GAs with good initial estimates may generate better solutions with faster convergence. Keedwell and Khu (2005) developed a hybrid optimization approach using a cellular automata based method to seed a GA for water distribution network design problems. Basically their idea was to knit the strongpoint of both methods meanwhile counteracting each other s major deficiency. Two stages of this hybrid approach can be identified. Firstly, the cellular automata based strategy is applied to quickly obtain a set of preliminary solutions, which are then adopted as good seeds for the genetic algorithm implemented at the second stage. In this way, vast computation costs can be saved by providing the GA execution with a high-quality initial population. In light of their findings, the authors proposed a novel hybrid sewer design method, named as CA-GASiNO (Cellular Automata and Genetic Algorithm for Sewers in Network Optimization), by combining CASiNO and a multi-objective GA, specifically a constrained non-dominated sorting genetic algorithm (NSGAII) (Deb et al. 2000; Deb 2001). In this paper, brief introductions to the problem formulation and to the NSGAII and CASiNO algorithms are firstly given. The methodology of CA-GASiNO is then described in detail. In the case studies, the proposed approach is applied to two sewer networks, one small artificial and one large real sewer network, followed by discussions and analysis on the optimization results. A general conclusion is drawn in the end.
2. Method 2.1 Optimization formulation Sewer design is commonly interpreted as an optimal pipe sizing problem, set up with two optimization objectives: minimization of flooding within the sewer system and minimization of capital cost of the network. The objective functions can be formulated into the following equations: n
Ftotal
Fmax,i
(1)
Cm Li
(2)
i 1 n
Ctotal i 1
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Where: Ftotal total flooding volume in the system throughout the simulated period (m3) Fmax,i maximum flooding volume occurring at the ith manhole (m3) Ctotal total capital cost of the designed network (£) Cm integrated cost per unit length for the pipe with mth pipe diameter (£/m) Li length of the ith pipe (m) In published literature, optimal sewer design applications are mainly implemented in a single-objective optimization manner by aggregating two objectives together with assigned weighting factors. However, it is always a troublesome issue to assign a proper weighting value to each objective. In this study the design task is treated as a multi-objective optimization problem. The multi-objective optimization considers all objectives in parallel and eventually generates a set of trade-off solutions, called Pareto-Optimal solutions, which can construct a trade-off surface called the Pareto Front. Since all solutions on the Pareto Front have the same level of optimization performance, multi-objective optimization can offer decision makers a variety of options. 2.2 NSGAII There are a large number of multi-objective optimization algorithms developed so far, and many review or comparison works on different algorithms can be found in the literature (Zitzler et al., 2000). However, no concrete conclusion has been drawn as to which is the best algorithm. As one of the most popular multi-objective optimization techniques, the NonDominated Sorting Genetic Algorithm II (NSGAII) (Deb et. al, 2000) is employed in this study. NSGAII is an elitist approach, which ensures that the best solutions are transferred from one generation to the next generation. Mainly benefiting from a fast non-dominated sorting 2
scheme to rank solutions, it only has the overall complexity of O (mN ) . A crowded comparison operator is used to keep the diversity in each generation and uniformly spread out the Pareto Front. Because our major interest is to design a system with little or no flooding instead of solutions on the whole Pareto front, a constrained NSGAII is applied to set an upper limit on the total flooding volume of each solution. The flood constraints are generally small values compared to the maximum flooding that could be encountered. During the optimization, a high rank and survival priority is assigned to individuals fitting this constraint. In this way the algorithm easily narrows down the search space into the area of interest, and can find preferred solutions more efficiently than an unconstrained version. 2.3 Cellular Automata and CASiNO Cellular automata are spatially and temporally discrete dynamic systems characterized by local interaction, self-production and universal computation. Its concept was firstly conceived by von Neumann in the late 1940s (Chopard and Droz 1998). CA were mostly utilized as a simulation environment: only in the last decade, has the use of CA been extended to optimization problems. A standard cellular automaton consists of a regular lattice of cells, each of which has a finite number of states. At every modelling step, the states of all cells are updated synchronously by an identical set of transition rules. For each cell, the updating only involves the previous state of the cell and of its predefined neighbours. Despite the simple principles of cellular automata, very complex behaviours can be generated.
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Inspired by the work of Keedwell and Khu (2006) for designing water distribution systems, the authors developed a fast sewer network design method loosely based on Cellular Automata, namely CASiNO (Guo, 2005). The basic mechanism of CASiNO follows the definition of cellular automata and sewer design theory. 1) Each manhole and its immediate downstream conduit are regarded as one cell. Its spatial structure fully follows the layout of the designed sewer network. Variable conduit lengths cause variable cell sizes. As a consequence, a two-dimension cellular automaton is constructed with different dimension widths. 2) Two variables are set to represent the states of each cell. One is the diameter of the conduit, and the other reflects the hydraulic situation at the manhole based on the maximum flooding volume occurring at the manhole during the simulation. 3) Corresponding to the layout of the network and the hydraulic process, the neighbourhood scheme involves only the direct upstream and downstream cells. 4) The transition rules are defined in a heuristic way which reflects an expert s decisions when facing the same sewer hydraulic situations. Benefiting from the inherent CA features of homogeneity and parallelism, CASiNO can achieve very high optimization efficiency. However, driven by the locality of neighbourhood interactions, CASiNO principally lacks the ability to target the global optima. 2.3 CA-GASiNO As noted before, an optimal design method based on either Genetic Algorithms or Cellular Automata has particular advantages but also its own weaknesses. Hence, the basic motivation of the proposed approach is to combine the benefits of both algorithms while counteracting each other s disadvantages. Two stages of this approach can be identified. CASiNO is firstly applied to obtain a set of preliminary solutions, which are adopted to seed NSGAII implementation in the second stage. In this way, CASiNO execution greatly saves on the computational cost, which would be required for NSGAII to reach a similar level of preliminary solutions. The following GA execution then aims the optimization at globaloptimal solutions. EPA-SWMM (Rossman, 2005) is adopted as the sewer hydraulic simulator to evaluate the hydraulic performance of solutions. Before the optimization, a preliminary sewer network model is set up in SWMM according to catchments characteristics, network configurations and the designed storm event. All steps to be taken for executing CA-GASiNO are described below in detail and are also presented in the flowchart of Figure 1. Stage One: CASiNO execution 1) Generate pipe-sizing scenarios of the network as initial conditions for the CASiNO simulation. The first two initial scenarios are specially set: one with the maximum pipe diameter for all pipes, and the other with the minimum pipe diameter for all pipes. These two represent extreme scenarios with the highest and the lowest possible capital costs for the network. All initial conditions afterwards are generated via a random process. 2) Simulate the model configured with the newly-updated pipe-sizing scenario in SWMM. 3) Evaluate the solution in respect of the optimization objectives
Y. Guo, G. Walters, S.T. Khu and E. Keedwell
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Figure 1 The processes of CASiNO-GA execution
4) Check Termination Criteria 1: (a) a predefined maximum number of CASiNO steps is reached; (b) a homogeneous or periodic behaviour is detected. This determines whether or not to stop the current CASiNO simulation, which starts from a certain initial condition. 5) Apply the CASiNO rules to revise the pipe-sizing scenario of the network. 6) Steps 2-5 are repeated until one of Termination Criterion 1 has been satisfied, and then the results of the current CASiNO simulation are recorded. 7) Check the Termination Criteria 2: CASiNO simulations from all initial conditions have been executed. If yes, go for the next step, otherwise Steps 1-6 are repeated. 8) Find non-dominated solutions from the results of all CASiNO simulations. Stage Two: NSGAII execution 1) Generate the initial population based on the preliminary solutions from Stage One. Several practical issues are related to this step: (a) if the number of preliminary solutions is larger than the required population size, the surplus solutions are discarded by randomly selecting
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from all candidate solutions except 5 solutions with least flooding; (b) if the number of the preliminary solutions is less than the population size, a quasi-mutation process is applied whereby the missing individuals are generated by randomly selecting the required number from the preliminary solutions and randomly altering a few variables in each solution with a predefined probability. 2) Evaluate every individual according to the optimization objectives. 3) Via a selection procedure, all individuals of the current generation are paired to create a mating pool. 4) Each pair of parents produces two children through the crossover and mutation processes. 5) Newly generated children are simulated and evaluated as in Step 2. Then, parents and children are grouped and then ranked using constraint checking first and non-dominated sorting second. Solutions with higher rank are then given priority to fit into the population of the next generation. 6. Check Termination Criterion 3: maximum number of GA generations is reached, which also implies that the maximum number of system simulations in SWMM has been achieved. 7. Create a Pareto Front in the end by discarding duplicate and dominated solutions from the last population.
3. Results and discussion 3.1 Example Networks To illustrate the use of CA-GASiNO in practice, it is here applied to the design of two urban sewer systems. In Case I, a small artificial network is selected; and in Case II, a large real network is adopted (see Figure 2). Detailed information about network characteristics and simulation modelling is given in Table 1. Optimization performance is empirically investigated by comparing solutions with those of optimization solely using a constrained NSGAII. In order to make an impartial comparison, the optimization experiments are implemented in exactly the same fashion for both CAGASiNO and the constrained NSGAII: i.e the same random seeds and parameter settings are specified for the GA execution of both algorithms. Some attempts have also been made to tune the parameters related to each method in order to present their best performance. For all NSGAII executions, the probability of crossover is 0.9, the mutation rate is 0.01, and the population size is 100. Table 1 Network and simulation information of the sewer networks
Subject Network Pipe Manhole Outfall Pipe roughness Pipe gradient Pipe diameter Simulation period Hydraulic routing Routing time step Storm frequency Flood constraint
Case 1 An artificial network 29 circular conduits 29 standard manholes 1 outfall Uniform Manning n: 0.013 0.005, uniform 10 options, 0.15-1.20 m 6 hours Dynamic wave 10 seconds 10 years 100 m3
Case 2 A real network (Miljakovac, Belgrade) 112 circular conduits 112 standard manholes 1 outfall Uniform Manning's n: 0.013 Average: 0.059, range: 0.003-0.346 12 options, 0.20-1.80 m 8 hours Kinematic wave 30 seconds 25 years 100m3
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Outfall
Outfall
(a) The sewer network of Case 1
(b) The sewer network of Case 2
Figure 2 Outlines of the sewer networks optimized in the modelling experiments
From the practical viewpoint of sewer engineers, solutions with least cost and zero flooding are favoured. Hence this type of solution is considered as one performance indicator. In addition, the S-Metric (Zitzler 1999), a widely accepted numerical indicator for evaluating Pareto Fronts, is applied to evaluate the optimization performance of the different approaches. A detailed comparison is given in Table 2. The figures clearly indicate that CAGASiNO shows huge savings in computational costs whilst obtaining better solutions. It outperforms the GA on both optimization efficiency and performance by using 37.7% of the computational cost of the constrained NSGA II in Case 1 and only 25% of the GA computations in Case 2. A further graphical comparison between the Pareto Fronts obtained by the two algorithms is given in Figure 3. Table 2 Optimization results of CA-GASiNO and NSGAII models
CAGASiNO
CASiNO NSGAII
NSGAII
Best Solution (Cost, Flooding) Unit: (£, m3) Case 1 Case 2 (129287, 0) (1694075, 0) (125878, 0) (1679847, 0)
Case 1 0.735 0.747
Case 2 0.906 0.910
Number of SWMM Evaluations Case 1 Case 2 128 39 30000 30000
(127032, 0)
0.742
0.905
80000
(1740802, 0)
100
120000
100
Flooding volume
CA-GASiNO NSGAII
80 Flooding volume
S-Metrics
60 40 20
CA-GASiNO NSGAII
80 60 40 20
0
0
120000
122000
124000
126000
128000
1600000
1650000
Pipe cost
(a)
Case 1
1700000 Pipe cost
(b)
Case 2
Figure 3 Comparison of the Pareto Fronts obtained by CA-GASiNO and NSGAII
1750000
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4. Conclusion The CA-GASiNO approach combines the best of CASiNO and NSGAII in that the CA based approach uses a remarkably small number of computational steps to provide a good initial population for the following genetic algorithm runs. This method significantly outperforms the non-heuristic based GA in terms of its optimization efficiency and performance.
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