**Open Access**-
**Total Downloads**: 23 -
**Authors :**Santosh Kumar, Praveen Kumar Navin, Yogesh Prakash Mathur -
**Paper ID :**IJERTCONV6IS11005 -
**Volume & Issue :**RTCEC – 2018 (Volume 6 – Issue 11) -
**Published (First Online):**31-05-2018 -
**ISSN (Online) :**2278-0181 -
**Publisher Name :**IJERT -
**License:**This work is licensed under a Creative Commons Attribution 4.0 International License

#### 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

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.

SEWER NETWORK DESIGN PROBLEM

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.

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:

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.

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

Choosing pipe diameters from the commercial list.

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.

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.

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).

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.

Procedure of Simulated annealing algorithm

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

components.

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.

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.

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)

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

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:

Start with the first link (I=1) of the first iteration(ITN=1)

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

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

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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Integrated Schedule of Rates: Rajasthan urban infrastructure development project (RUIDP),Government of Rajasthan (2013)