# Optimization of Sewerage System Using Simulated Annealing

DOI : 10.17577/IJERTCONV6IS11005

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#### Optimization of Sewerage System Using Simulated Annealing

Santosh Kumar

Reaserch Scholer, Dept. of Civil Engg. Malaviya National Institute of Technology, Jaipur, India

Praveen Kumar Navin

Assistant Professor, Dept. of Civil Engg. Vivekananda Institute of Technology, Jaipur, India

Yogesh Prakash Mathur

Professor, Dept. of Civil Engg.

Malaviya National Institute of Technology, Jaipur, India

Abstract- Sewer networks are an important part of the infrastructure of any society. Since, the investment needed for construction and maintenance of these large scale networks is so huge and, thus any saving in the cost of these networks may result in considerable reduction of total construction cost. This study focuses on the issues of the design of sewer networks.In this paper, a new and powerful stochastic method, called Simulated Annealing (SA) is adopted for solving the sewer network optimization problem. Simulated Annealing (SA) is a probabilistic method proposed for finding the global minimum of a cost function that may possess several local minima. A sewer network is considered to show the Simulated Annealing algorithm performance, and the results are presented. The results show the capability of the proposed technique for optimally solving the problems of sewer networks.

Keywords – Sewer network, Simulated Annealing, Optimal sewer design

1. INTRODUCTION

Sewerage or the wastewater system is the system of pipes used to collect and carry rain, wastewater and trade waste away for treatment and disposal. Sewage collection and disposal systems transport sewage through cities and other inhabited areas to sewage treatment plants to protect public health and prevent disease. The design of a sewerage system in general involves selection of a suitable combination of pipe sizes and slopes so as to ensure adequate capacity for peak flows and adequate self cleansing velocities at minimum flow. In a conventional design procedure, efforts are made to analyze several alternative systems (each meeting the physical and hydraulic requirements) and the least cost system is

such as genetic algorithms [8, 9], ant colony optimization algorithms [10, 11], cellular automata [12] and particle swarm optimization algorithms [13], have received significant consideration in sewer network design problems. Recently, Ostadrahimi et al. [14] used multi- swarm particle swarm optimization (MSPSO) approach to present a set of operation rules for a multi-reservoir system. Haghighi and Bakhshipour [15] developed an adaptive genetic algorithm. Therefore, every chromosome, consisting of sewer slopes, diameters, and pump indicators, is a feasible design. The adaptive decoding scheme is set up based on the sewer design criteria and open channel hydraulics. Using the adaptive GA, all the sewer systems constraints are systematically satisfied, and there is no need to discard or repair infeasible chromosomes or even apply penalty factors to the cost function. Moeini and Afshar [16] used tree growing algorithm (TGA) for efficiently solving the sewer network layouts out of the base network while the ACOA is used for optimally determining the cover depths of the constructed layout. Karovic and Mays [17] used simulated annealing within Microsoft Excel to sewer Ssystem design optimization.

In this paper, simulated annealing algorithm is applied to get optimal sewer network component sizes of a predetermined layout.

2. SEWER NETWORK DESIGN PROBLEM

1. Sewer Hydraulics

In circular sewer steady-state flow is described by the continuity principle (Q= VA) and Mannings equation which is

selected. Obviously, the outcome of such a procedure depends to a large extent on the designer experience and efforts. It is practically almost impossible to incorporate all

v 1 R1/ 3S1/ 2

n

(1)

feasible design alternatives, and an optimal solution is not necessarily reached. Only a resources to computer oriented optimal designing may be a solution.

Many optimization techniques have been applied and developed for the optimal design of sewer networks, such as linear programming [1, 2], nonlinear programming [3, 4] anddynamic programming [57]. Evolutionary strategies,

where Q = sewage flow rate, V = velocity of sewage flow, A = cross-sectional flow area, R = hydraulic mean depth, n

= Mannings coefficient and S = slope of the sewer. Common, partially full specifications for circular sewer sections are also determined from the following equations:

D 2 2

d 1 1 cos

r D sin

(2)

(3)

temperature in physical annealing), then find the initial value of the objective function. While these choices of

starting solution and temperature are unique to each application, SA is normally fairly insensitive to the starting

4

a

D2 sin 8

(4)

conditions. In the application to structural optimization, this step establishes the initial physical characteristics of the structural components, ensures that all constraints are met, and determines the initial weight of the structure.

D = sewer diameter, = the central angle in radian and

(d/D) = proportional water depth, a = flow area while running partially full,r = hydraulic mean radius.

2. Design Constraints

For a given network, the optimal sewer design is defined as a set of pipe diameters, slopes and excavation depths which satisfies all the constraints. Typical constraints of sewer network design are:

1. Each pipe flow velocity should be greater than the minimum permissible velocityfor self cleaning capability and less than the maximum permissible velocity for preventing from scouring.

2. Flow depth ratio: wastewater depth ratio of the pipe should be less than 0.8.

3. Choosing pipe diameters from the commercial list.

4. Maintaining the minimum cover depth to avoid damage to thesewer line and adequate fall for house connections. The minimum cover depthof 0.9 m and maximum cover depth of 5.0 m has been adopted.

5. For each manhole, assigning the outlet pipe diameter equal to or greater than the upstream inlet pipes.

The optimal design of a sewer system for a given layout is to determine the sewer diameters, cover depths and sewer slopes of the network in order to minimize the total cost of the sewer system. The objective function can be stated as

n

The term temperature is a holdover from the physical process of annealing, where it refers to the actual heat content of a casting. In simulated annealing, the temperature is a parameter that controls the probability of accepting a new solution that is "worse" than the old one. The higher the temperature, the greater the chance of accepting a "worse" solution. This probability of accepting a worse solution is the feature that allows SA to leave a local minimum and continue to search for the global minimum.

1. The second step in the algorithm is to randomly perturb the system. In explaining combinatorial optimization, Kirkpatrick, et.al. [18]described a random search method that accepts only lower values of the objective function at each iteration. It usually gets stuck in the local minimum closest to the starting point. This algorithm is often called the Greedy Algorithm because, in its "greed" to find any optimum, it will likely miss the global optimum and accept a local instead (McLaughlin, 1989:25). In 1985, Cerny [19] presented a Monte Carlo algorithm to fid approximate solutions to the traveling salesman problem. "The algorithm generates randomly the permutations of the stations of the traveling salesman trip, with a probability depending on the length of the corresponding route. This offers one method for generating random perturbations to a

Minimize C (TCi PCi )

i1

(5)

system. In structural optimization, this step corresponds to a random change in the physical dimension of one or more

Where i = 1,, n (total number of sewers), TCOSTi (total cost) = (Cost of seweri + Cost of manholei + Cost of earth worki) and PCi = penalty cost (it is assigned if the design constraint is not satisfied).

3. SIMULATED ANNEALING (SA) Simulated Annealing (SA) is a fairly new process for

numerical optimization of many classes of problems. It is modeled after the centuries-old annealing process for metal and glass castings. Manufacturers anneal castings to make them tougher, by reducing their internal energy (McLaughlin, 1989) between Simulated Annealing and the physical process of annealing. In each case, a system of many variables is minimized. SA uses many steps in a random search to find the optimum of the system. Other random search algorithms are prone to selecting the first local optimum encountered. However, SA has a feature that helps it find the global optimum rather than a local optimum. The many steps required in SA are possible with modern computers, and the more capable computers become, the more useful SA will be.

1. Procedure of Simulated annealing algorithm

1. The first step in the algorithm is to choose a starting configuration and control parameter (analogous to

components.

1. The third step is to evaluate the new solution. The specific mechanics of this evaluation depend on the application. For structural optimization, this step determines the total weight of the structure with the new dimensions.

2. In the fourth step, accept or reject the new solution. If the new solution gives a lower value for the objective function, accept it. However, if the new solution gives a higher value, consider accepting it. This possibility of accepting the "worse" solution gives the SA algorithm the ability to leave a local optimum, and continue to search for the global optimum. This is the key feature that sets SA apart from other random search algorithms. From statistical mechanics, Kirkpatrick, et.al. [18] described the Metropolis procedure to overcome the Greedy Algorithm's problem of stalling at a local optimum. The Metropolis procedure from statistical mechanics provides a generalization of iterative improvement in which controlled uphill steps can also be incorporated in the search for a better solution [18]. This makes it possible for the algorithm to climb out of a local minimum and find a better local minimum, or the global minimum. Control for the uphill steps is given by the Boltzmann distribution:

Pr (E)

1

Z(T)

E

K T

exp B

(6)

function reaches a stable value for a certain number of iterations [20].

• If there is a certain target value of the function (a

Where, () is the probability of accepting the uphill step,

() is a normalizing factor depending on the assigned temperature(), is the average energy level, and is the Boltzmann constant. The value of is a natural constant, determined by experimentation, which adjusts the shape of the Boltzmann distribution to model the physical annealing process. It normally would not represent a valid constant in the SA process, but a different constant may be appropriate. For a given change in temperature, when the temperature is high, the probability of accepting an uphill step is high. As the temperature is reduced, the probability of accepting the uphill step is reduced.

3. The fifth step in the algorithm is to iterate at a given temperature and, when the system is at a stable average configuration for that temperature, reduces the temperature according to the annealing schedule. This schedule for reducing the temperature is critical to the success of either real or simulated annealing. According to Cerny experiments are done by careful annealing, first melting the substance, then lowering the temperature slowly, and spending a long time at temperatures in the vicinity of the freezing point. If this is not done, and the substance is allowed to get out of equilibrium, the resulting crystal will have many defects [19]. Quenching is the process of deliberately reducing the temperature quickly, without allowing the substance to reach equilibrium. This degenerates the SA algorithm to an ordinary random search like the Greedy Algorithm. In annealing, this process creates a brittle casting, but it is much quicker, and in some cases may be preferred to the slow annealing process. Quenching is not normally used in SA. To get the lowest possible cost with SA, the annealing schedule must allow the system to reach steady-state at each temperature. On the other hand, spending too much time at a given temperature wastes computer resources. So, the annealing schedule must allow the system to stabilize before changing temperature, and then change promptly.

The cooling schedule is often found by trial and error Brooks and Verdini [20]. However, Basu and Fraser [21] suggest that it may be cost effective to spend up to 80 percent of the total CPU time to establish the best cooling schedule. Collins et.al. [22] listed five different schemes for controlling the temperature, T:

• A constant value of T; T(t) = C

• An arithmetic function of T; T(t) = T(t – 1) C

• A geometric function; T(t) = a(t)T(t – 1)

• An inverse; T(t) = C/(1 + ta)

• A logarithmic function; T(t) = C/In(1 + t)

4. The last step in the SA algorithm is to iterate until the stopping criteria is met. Several classes of stopping criteria can be used [22].

• In the simplest criteria, a fixed amount of CPU time is allocated, and the process stops when the time runs out [20].

• Another approach is to compare the value of the objective function at each iteration with the value at previous iterations. Under this criteria, stop when the

known or estimated minimum), stop when the configuration meets the target [20].

• When the algorithm is near the optimum the ratio of accepted configurations to total configurations will become very small. The algorithm can stop when this ratio reaches a predetermined value [23].

If none of the other criteria are met, stop when the temperature reaches a value near zero [22]. At this point the algorithm degenerates to a random search, and the cost of further annealing should be compared to the benefit that might be gained. When the correct stopping criteria are met, the algorithm will have a solution closer to the global optimum.

According to the above-mentioned steps, a possible structure of the Simulated Annealing algorithm is shown in fig. 1.

Fig. 1. Flow chart of Simulated Annealing Algorithm

4. OPTIMIZATION OF SEWER NETWORK The sewer network example (Banjaran sewer network,

Laxmangarh, Rajasthan, India) is considered to check the

above-proposed approach. The Banjaran sewer network as shown in Fig. 2 consists of 105 manholes, 104 pipes and STP is located at Node Number 0.

The following steps were used to optimize the component sizing of sewer system using the Simulated Annealing algorithm:

• Calculate values of Hydraulic Mean Depth, Velocity, Depth of flow, and Discharge in partial flow condition.

• Calculate invert levels of upstream and downstream node of a particular link

• Calculate no of manholes, depth of excavation and earthwork.

• Calculate cost of sewer, cost of manholes and cost of earthwork.

• Calculate the total cost of the sewer network (TCOST)

• Add the respective penalty cost (PC) in TCOST where constraints are violated.

• Calculate feasible solution using SA

• Check solutions obtained are feasible or not.

• If feasible solution is not obtained repeat the process.

• If feasible solution is obtained, then take output.

• End.

The cost of pipe (RCC NP4 class), manhole and earth work was taken from theIntegrated schedule of Rates, RUIDP [24].

Fig. 2.Banjaran sewer network

5. RESULTS

The performance of the proposed Simulated Annealing procedure for optimization of the sewer system is now tested against Banjaran sewer network. The result exhibit a final total cost ofRs.8.505 Ã— 106. 100000 evaluations were done for a system having 100 iterations for each evaluation. Then after accepting the higher as well as lower

values of the function the global best solutions were achieved. The pipe diameter and slopes have been shown for the best solution. Accordingly the total cost of the sewerage system has been shown in the results. Table 1 shows the solution obtained by Simulated Annealing approach.

Table 1 Results of the Banjaran sewer network obtained by Simulated Annealing

 Pipe no. Node no. Length (m) Design flow (m/s) Diameter (mm) Slope (1 in) vp (m/s) d/D Cover depths (m) Up Down Up Down 24 23 22 30 0.0001 200 250 0.17 0.05 1.12 1.422 39 37 36 28 0.0002 200 250 0.19 0.06 1.426 1.12 41 38 39 20 0.0001 200 80 0.2 0.03 1.14 1.12 42 39 40 24 0.0001 200 250 0.18 0.05 1.434 1.12 44 40 42 28 0.0003 200 250 0.24 0.08 1.12 6.487 45 41 28 29 0.0001 200 250 0.16 0.04 1.12 1.338 46 42 35 28 0.0004 200 250 0.26 0.09 6.487 2.182 47 43 44 30 0.0001 200 60 0.26 0.03 1.12 1.184 48 44 27 38 0.0002 200 250 0.21 0.07 1.184 1.538 52 49 48 35 0.0001 200 250 0.17 0.05 1.12 1.489 54 50 51 35 0.0001 200 250 0.17 0.05 1.125 1.12 55 51 52 34 0.0002 200 250 0.21 0.07 1.12 1.343 56 52 53 30 0.0621 300 200 1.09 0.73 1.343 1.781 57 53 54 35 0.0622 300 200 1.09 0.73 1.781 1.969 69 64 63 30 0.0001 200 250 0.17 0.05 1.12 1.541 83 69 68 30 0.0001 200 200 0.18 0.05 1.12 1.126 80 70 67 30 0.0001 200 250 0.17 0.05 1.12 1.259 77 71 66 30 0.0001 200 250 0.17 0.05 1.12 1.373 74 72 65 30 0.0001 200 250 0.17 0.05 1.12 1.164 107 87 88 30 0.0001 200 250 0.16 0.05 1.12 1.415
 102 88 83 33 0.0002 200 250 0.2 0.07 1.415 2.981 117 97 96 16 0.0002 200 250 0.21 0.07 1.12 1.593 120 98 99 30 0.0001 200 250 0.16 0.05 1.12 1.333 127 99 104 34 0.0002 200 250 0.21 0.07 1.333 1.297 122 100 101 30 0.0001 200 250 0.16 0.05 1.12 1.3 123 101 102 26 0.0002 200 250 0.2 0.06 1.3 1.362 126 104 103 30 0.0003 200 250 0.24 0.08 1.297 1.312 23 22 21 30 0.0002 200 250 0.21 0.07 1.422 1.701 36 36 35 27 0.0002 200 250 0.21 0.07 1.12 1.388 51 48 47 35 0.0002 200 250 0.21 0.07 1.489 1.832 71 63 79 30 0.0004 200 250 0.26 0.09 1.541 1.371 75 65 84 30 0.0004 200 250 0.26 0.09 1.164 1.498 78 66 89 30 0.0004 200 250 0.26 0.1 1.373 1.57 97 79 80 30 0.0005 200 250 0.27 0.1 1.519 1.12 98 80 81 17 0.0005 200 250 0.28 0.11 1.12 1.266 99 81 82 35 0.0007 200 250 0.31 0.13 1.266 1.493 101 82 83 30 0.0008 200 250 0.32 0.14 1.493 2.667 95 83 77 35 0.0011 200 250 0.36 0.16 2.981 2.849 103 84 85 30 0.0005 200 250 0.27 0.11 1.846 1.12 104 85 86 17 0.0005 200 250 0.28 0.11 1.12 1.238 106 86 91 35 0.0007 200 80 0.46 0.1 1.266 1.12 109 89 90 30 0.0005 200 80 0.4 0.08 1.57 1.3 110 90 91 18 0.0005 200 250 0.28 0.11 1.303 1.12 111 91 92 35 0.0015 200 70 0.6 0.13 1.12 1.624 113 92 93 30 0.0016 200 70 0.62 0.14 1.624 2.214 114 93 94 30 0.0017 200 70 0.63 0.14 2.214 2.919 115 94 21 29 0.0018 200 80 0.61 0.15 2.919 3.257 116 96 95 30 0.0003 200 250 0.22 0.08 1.593 1.637 124 103 102 33 0.0004 200 250 0.26 0.09 1.312 1.374 22 21 20 12 0.002 200 80 0.64 0.16 3.257 3.451 37 35 34 30 0.0007 200 250 0.31 0.13 2.182 2.433 50 47 46 27 0.0003 200 250 0.24 0.09 1.832 2.042 81 95 67 30 0.0003 200 250 0.25 0.09 1.637 1.855 125 102 57 29 0.0007 200 250 0.31 0.13 1.374 1.407 4 20 4 30 0.0021 200 80 0.64 0.16 3.451 3.853 38 34 30 18 0.0008 200 250 0.32 0.13 2.433 2.288 49 46 45 10 0.0011 200 250 0.36 0.16 2.042 2.131 79 67 68 34 0.0007 200 250 0.31 0.13 1.855 1.561 82 68 54 24 0.0011 200 250 0.35 0.16 1.561 1.819 68 45 62 36 0.0063 200 200 0.64 0.37 2.131 2.888 58 54 55 30 0.0634 300 200 1.1 0.75 1.969 1.858 59 55 56 30 0.0635 300 200 1.1 0.75 1.858 1.971 60 56 57 15 0.0635 300 200 1.1 0.75 1.971 1.913 61 57 58 30 0.0643 300 200 1.1 0.76 1.913 1.792 62 58 59 30 0.0644 300 200 1.1 0.76 1.792 2.257 63 59 60 30 0.0645 300 200 1.1 0.76 2.257 2.555 64 60 24 34 0.0646 300 200 1.1 0.76 2.555 2.77 65 62 61 30 0.0064 200 200 0.64 0.37 2.888 2.9 26 24 25 30 0.0647 300 200 1.1 0.76 2.77 2.93 27 25 26 30 0.0648 300 200 1.1 0.76 2.93 2.866 28 26 27 32 0.0649 300 200 1.1 0.76 2.866 2.366 29 27 28 32 0.0652 300 200 1.1 0.76 2.366 1.939 30 28 29 30 0.0654 300 200 1.1 0.77 1.939 2.19 31 29 30 25 0.0655 300 200 1.1 0.77 2.19 3.038 32 30 31 30 0.0663 300 200 1.1 0.78 3.038 3.197 33 31 32 30 0.0664 350 250 1.04 0.63 3.197 2.744 34 32 33 30 0.0665 350 250 1.04 0.63 2.744 2.02 35 33 17 20 0.0666 350 250 1.04 0.63 2.02 1.888 67 61 73 30 0.0067 200 200 0.65 0.38 2.9 2.97 89 73 74 30 0.0068 200 200 0.65 0.38 2.97 3.05 90 74 75 17 0.0068 200 250 0.6 0.41 3.05 2.379 91 75 76 35 0.0077 200 250 0.62 0.43 2.379 2.819 93 76 77 30 0.0078 200 250 0.62 0.43 2.819 3.817 94 77 78 30 0.0091 200 250 0.65 0.47 3.817 4.304
 96 78 2 28 0.0092 200 250 0.65 0.47 4.304 3.154 1 2 3 30 0.0153 200 250 0.72 0.63 3.154 3.336 2 3 4 30 0.0154 200 250 0.72 0.64 3.336 2.991 3 4 5 30 0.0176 200 250 0.74 0.7 3.853 3.58 5 5 6 30 0.0177 200 250 0.74 0.7 3.58 3.593 6 6 7 30 0.0178 200 250 0.74 0.7 3.593 3.044 7 7 105 30 0.0179 200 250 0.74 0.71 3.044 3.132 128 105 8 7 0.0179 200 250 0.74 0.71 3.132 3.159 8 8 9 28 0.0192 200 250 0.75 0.74 3.159 3.595 10 9 10 28 0.0193 200 250 0.75 0.75 3.595 2.943 11 10 11 30 0.0195 200 250 0.75 0.75 2.943 3.105 13 11 12 22 0.0195 200 250 0.75 0.76 3.105 2.384 14 12 13 30 0.0266 250 250 0.83 0.62 2.384 2.078 15 13 14 21 0.0266 250 250 0.83 0.62 2.078 1.917 16 14 15 30 0.0267 250 250 0.83 0.62 1.917 1.737 17 15 16 30 0.0268 250 250 0.83 0.62 1.737 2.039 18 16 17 28 0.0269 250 250 0.83 0.62 2.039 2.601 19 17 18 30 0.0936 400 350 0.99 0.69 2.601 3.22 20 18 19 30 0.0936 400 350 0.99 0.69 3.22 3.531 21 19 1 26 0.0937 400 350 0.99 0.69 3.531 3.734
6. CONCLUSION

The optimization technique adopted in this work proved to be successful in optimal designing of the sewerage network. In this study, the Simulated Annealing (SA) method of optimization a stochastic approach was applied to the problem of finding optimal pipe diameters and slopes for the conjunctive least-cost design and operation of a sewerage system network. Using the SA approach, the total cost of the sewer system was Rs. 8.505

Ã— 106. The results indicated that the proposed approach is very promising and reliable, that must be taken as the key alternative to solve the problem of optimal design of the sewer network.

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