1

Layout and size optimization of sanitary sewer network using intelligent

ants

R. Moeinia, M. H. Afsharb

a (corresponding author) PhD Student, School of Civil Engineering, Iran University of

Science and Technology, P.O. 16765-163, Narmak, Tehran, Iran, E-mail:

rmoeini@iust.ac.ir, Tel: 0098-9196201603, Fax: 0098-21-77240398.

b Associate professor, School of Civil Engineering & Enviro-hydroinformatic Center of

Excellence, Iran University of Science and Technology, P.O. 16765-163, Narmak,

Tehran, Iran, E-mail : mhafshar@iust.ac.ir, Tel: 0098-21-77240097-, Fax: 0098-21-

77240398.

Abstract

The incremental solution building capability of ant algorithm is exploited in this paper for

the efficient layout and pipe size optimization of sanitary sewer network. Layout and pipe

size optimization of sewer networks requires that the pipe locations, pipe diameters and

pipe slopes are optimally determined. This problem is a highly constrained mixed-integer

nonlinear programming (MINLP) problem presenting a challenge even to the modern

heuristic search methods. In this paper, the Ant Colony Optimization Algorithm (ACOA)

is equipped with a Tree Growing Algorithm (TGA) to efficiently solve the sewer network

layout and size optimization problem and its performance is compared with the

conventional application of the ACOA. The method is based on the assumption that a

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base layout including all possible links of the network is available. The TGA is used to

construct feasible tree-like layouts out of the base layout defined for the sewer network,

while the ACOA is used to optimally determine the pipe diameters of the constructed

layout. An assumption of sewer flow at maximum allowable relative depth is made

allowing for the calculation of the optimal pipe slopes in the absence of any pump and

drop in the network. Two different formulations are, therefore, proposed and their

performances are tested against hypothetical problems. In the first formulation, ACOA is

used in a conventional manner for pipe size optimization while an ad-hoc engineering

concept for the layout determination. In the second formulation, however, ACOA

equipped with TGA is used to simultaneously determine both the layout and pipe sizes of

the network. Proposed formulations are used to solve three hypothetical test examples of

different scales and the results are presented and compared. The results indicate the

ability of the proposed method to optimally solve the problem of layout and size

determination of sewer networks.

Key works: Ant Colony Optimization Algorithm, Tree Growing Algorithm, sanitary

sewer network, layout, pipe size.

1. Introduction

Networks are an important part of any society. Physical networks such as gas pipelines,

sewer collection networks, water distribution systems, electricity and telecommunication

grids are only some examples of these networks. Construction of these networks is very

costly mainly due to cost of the large number of components they are composed of and

the operations required for their constructions. A relatively small change in the

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component and construction cost of these networks, therefore, leads to a huge saving.

Considerable researches have been focused on developing useful optimization techniques

for optimal design of such networks in recent years.

The network design optimization problem has appeared in many fields of applications

and solved with various optimization methods. A general network design optimization

problem includes two sub-problems of design of optimal layout (location) and design of

optimal size of network components. These sub-problems are strongly coupled and,

therefore, can not be solve separately if an optimal solution to the whole problem is

required. Due to the complex nature of the problem, however, most of the research in the

field is carried out on either layout determination or component sizing [25]. Some

researchers have addressed the joint layout and size optimization of networks with a

particular emphasis on the water distribution networks. For example, Martin [28]

developed a Dynamic Programming (DP) model for the joint layout and pipe size

optimization of branched water distribution networks. Morgan and Goulter [30]

developed a model that used two linked Linear Programming (LP) sub-models to find

least-cost solution for the problem of optimal layout and size of looped water distribution

systems. Jabbari and Afshar [21] applied Mixed Integer Linear Programming (MILP) for

layout and size optimization of a water supply line. Lejano [25] used MILP to

simultaneously solve the problem of optimal layout and pipe size design of a branching

water distribution system. Cembrowicz [12] combined Genetic Algorithm (GA) and LP

for water distribution network design optimization problem in which GA determined the

optimal layout and LP determined the least-cost pipe diameters. Hassanli and Dandy [19]

developed a GA model for layout and size optimization of a branched pipe networks.

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Afshar [1] used Max–Min Ant System (MMAS) for joint layout and pipe size

optimization of pipe networks. Afshar and Marino [3] applied ACOA for layout

optimization of tree pipe networks. Afshar [5] applied GA for simultaneous layout and

pipe size optimization of looped water distribution networks. More recently, Afshar [7]

used ACOAs for the optimal layout and size design of the branched pipe networks.

Sanitary sewer network as one of the physical networks is essential structure of any urban

area. Urban areas without an effective sanitary sewer networks may encounter many

problems, such as public health threats and environmental damages. Having a suitable

and cost-effective sewer network is normally interpreted as finding solution for sewer

network design optimization problem that minimizes infrastructural cost, without

violating operational requirements. The cost of construction a sewer system can be

significantly reduced if the system configuration (layout), pipe diameters and pipe slopes

can be effectively optimized.

Most of the works for the optimal design of sewer networks, however, are restricted to

the optimal design of the component size. For example, Walsh and Brown [34],

Templeman and Walters [33], Yen et al. [36], Kulkarni and Khanna [24], Botrous et al.

[11] and Diogo et al. [15] used DP to solve sewer design optimization problem. Desher

and Davis [13] developed a model named Sanitary Sewer Design (SSD) to find the least

cost solution of a sanitary sewer network. Elimam et al. [16] developed a model that

combined LP and a heuristic approach for optimal design of large-scale storm water

networks. GA was used to find the least-cost solution for storm network design

optimization problem by Heaney et al. [20]. Swamee [32] solved the sewer line design

optimization problem by iterative application of the Lagrange-multiplier method. Liang

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et al. [27] used GA and Tabu Search (TS) for optimal design of sewer networks. Guo [17]

proposed a sewer design model using Cellular Automata (CA) concept for the design of

storm sewers networks. Guo et al. [18] proposed a hybrid sewer design model by

combining GA and CA for sewer network design optimization problem. Afshar et al. [4]

applied GA for storm sewer design optimization problem. Afshar [6] proposed partially

constrained ACOA and applied it to storm sewer network design. Izquierdo et al. [23]

used Particle Swarm Optimization (PSO) algorithm for optimal design of wastewater

collection networks. Pan and Kao [31] developed a model that combined a GA with a

Quadratic Programming (QP) model to solve sewer system design optimization problem.

Afshar [2] used Continuous Ant Colony Optimization Algorithm (CACOA) for optimal

component sizing of storm sewer networks proposing constrained and unconstrained

approach.

Only a few researchers have addressed the problem of layout and in particular the joint

layout and size optimization of sewer networks. For example, Argaman et al. [10], Mays

and Wenzel [29] used DP for the optimal design of the gravity sewer network with a

single outlet using a simplifying assumption that for every pipe of the network the

direction of flow was fixed. Therefore, their methods are only suitable to very ideal cases

where natural topography is inclined in only one way to the outlet. Walters [35] also used

DP for layout and size optimization of the sewer networks which could be used to drain a

set of sources with fixed positions. Li and Matthew [26] proposed a nested approach for

the optimization of urban drainage systems in which a searching direction method used

for the layout determination and a Discrete Differential Dynamic Programming (DDDP)

for the optimal component sizing of any given layout, requiring very high computational

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effort. Jang [22] used DDDP to develop a model for design of optimal layout of urban

storm sewer network neglecting the component sizing part of the problem. Recently,

Diogo and Graveto [14] developed a comprehensive enumeration model and a Simulated

Annealing (SA) model for the layout and component size optimization of sewer networks.

In this paper, the Ant Colony Optimization Algorithm (ACOA) is equipped with a Tree

Growing Algorithm (TGA) to efficiently solve the sewer network layout and size

optimization problem and its performance is compared with the conventional application

of the ACOA. The method is based on the assumption that a base layout including all

possible links of the network is available. The TGA is used to construct feasible tree-like

layouts out of the base layout defined for the sewer network, while the ACOA is used to

optimally determine the pipe diameters of the constructed layout. An assumption of

sewer flow at maximum allowable relative depth is made allowing for the calculation of

the optimal pipe slopes in the absence of any pump and drop in the network. Two

different formulations are, therefore, proposed and their performances are tested against a

hypothetical problem. In the first formulation, ACOA is used in a conventional manner

for pipe size optimization while an ad-hoc engineering concept for the layout

determination. In the second formulation, however, ACOA equipped with TGA is used to

simultaneously determine both the layout and pipe sizes of the network. Proposed

formulations are used to solve three hypothetical test examples of different scales and the

results are presented and compared. The results indicate the ability of the proposed

method to optimally solve the problem of layout and size determination of sewer

networks.

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2. Sanitary Sewer System Design

Sanitary sewer system consists of sewer pipes, manholes, pumping stations, and other

appurtenances. The system is used to collect and transport wastewater by gravity from

house to wastewater treatment plants through pipes and manholes of the network. The

design of a sewer system consists of generating an adapted sanitary sewer network layout

that conforms to population served, street layout and local topography of the planning

area, and performing hydraulic design to find pipe sizes, excavation depths and other

hydraulic and design parameters of the specific layout. The problem of finding the best

layout and hydraulic design of sanitary sewer network is a complex task that cannot be

performed without using optimization techniques considering the vast number of

alternatives for the location of the pipes, their size and other components of the sewer

network. The design process of sewer system may be divided into two phases: (1)

Selection of optimal layout, (2) Optimal hydraulic design of the network components.

Determination of the location of pipes or layout of a sanitary sewer system is of great

importance because it serves as the foundation of the hydraulic design and, therefore,

influences the final cost of the network.

The problem of joint layout and component size determination of a sanitary sewer

network as a collection of pipes connected with manholes, here without any pump and

drop, can be mathematically defined as determining the pipe connections, diameter and

cover depth of each pipe such that the network construction cost is minimized:

M

m

mm

N

l

llpl hKEdKLCMinimize11

)(),( (1)

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Where, C= cost function of sanitary sewer network; N= total number of sewer pipes; M=

total number of manholes; lL the length of pipe l (l=1,…., N); pK = the unit cost of

sewer pipe provision and installation defined as a function of its diameter (d1) and

average cover depth (l

E ); and mK = the cost of manhole construction as a function of

manhole height ( mh ).

Subject to the topological and hydraulic constraints as follows:

1- Cover depth: It is necessary to have adequate cover depth for sewer pipes for three

reasons: (1) To provide protection against imposed loads, particularly vehicle loads, (2)

To allow an adequate fall on house connections, and (3) To reduce the possibility of

cross-contamination of water mains by making sure that, wherever possible, sewer pipes

are located below water mains. Normally, these conditions are met by requiring that the

average cover depth of a sewer pipe be within a range defined by maximum and

minimum cover depth defined as:

NlEEE l ,.....,1maxmin (2)

Where, Emin= minimum cover depth of sewer pipe; Emax = maximum cover depth of

sewer pipe; l

E = average cover depth of sewer pipe l; and N = total number of network

pipes.

2- Sewer flow velocity: When designing a sewer network, it is important to ensure that

sewer pipes will be capable of conveying sewer flow with a velocity which does not

exceed the maximum allowable velocity to prevent pipe erosion. Also, the sewer flow

velocity should be greater than the self-cleaning velocity at least once a day to prevent

solids form being deposited on the bottom of the pipe. These constraints may be

formulated as:

9

NlVV

VV

cleanl

l

,.....,1*

max

(3)

Where, lV = flow velocity of pipe l at the design flow; *

lV = the maximum flow velocity

of pipe l at the beginning of the operation; Vclean= self-cleaning flow velocity of sewer;

and Vmax = maximum allowable flow velocity of sewer.

3- Minimum sewer pipe slope: Minimum slope should be considered for sewer pipe to

avoid adverse slope caused by inaccurate construction or settlement. This constraint may

be formulated as:

NlSSl ,.....,1min (4)

Where, lS = slope of the sewer pipe l; and minS = minimum sewer pipe slope which is

usually considered as a very small positive slope.

4- Maximum and minimum relative flow depth: These constraints are used to avoid

pressurized flow and deposition of the solids in the sewer pipes defined as:

Nld

y

l

,.....,1maxmin

(5)

Where, ld diameter of sewer pipe l; ly = sewer flow depth in pipe l; max = maximum

allowable relative flow depth; andmin = minimum allowable relative flow depth.

5- Commercial pipe diameters: The sewer pipe should be selected form a set of

commercially available sewer pipe diameters defined as:

Nld l ,.....,1d (6)

Where, ld diameter of sewer pipe l; and d = discrete set of commercially available

sewer pipe diameters.

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6- partially-full pipe condition: Conduits that are used for conveying sewer flows are

designed to operate in partially-full pipe conditions to avoid pressurized flow condition.

Several equations are generally used to evaluate the hydraulic parameters of sewer

network. In this paper, Manning’s equation is used to find hydraulic parameters such as

velocity distribution defined as:

NlSRAn

Q llll ,.....,11

21

32

(7)

Where, lQ = discharge of sewer pipe l; lA = wetted cross section area of sewer pipe l;

lR = hydraulic radius of the sewer pipe l; n = Manning constant; and lS = slope of the

sewer pipe l.

7- Progressive pipe diameters: For each pipe, the leaving sewer pipe diameter can not

be smaller than anyone of the converging sewer pipes diameter defined as:

Nld ll ,.....,1 d (8)

Where, ld diameter of sewer pipe l; and dl = set of upstream pipe diameters of pipe l.

8- Branched layout: The sewer networks are designed such that the flow of the sewer in

the system is due to gravity requiring that the networks have a branched layout. For this,

information regarding the topography of the areas considered for design of the network

such as: layout of the streets, the wastewater treatment plants and existing subsystems are

collected and used to define the generic locations of the sewer network components.

Normally, definition of sewer network components leads to a connected undirected graph

often referred to as the base layout containing all possible looped and branched layout

configurations. The graph vertices (nodes) represent the fixed position manholes, outlet

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locations or treatment plants while graph edges represent the sewer pipes excited between

manholes or manholes and outlets or wastewater treatment plants [14].

In the form of arborescent structure, the sewer pipe configuration should be branched

defined by a set of root nodes and a number of non-root nodes and edges located between

nodes. The root nodes represent the outlets or the treatment plants, the non-root nodes

represent the system manholes, and the edges define the pipes of the sewer network. In an

arborescent structure, only one edge leaves from each node. This limitation governs the

layout determination of sewer networks which can be mathematically defined as:

KjiXX jiij ,......,1,1 (9)

KiXiN

j

ij ,.....,111

(10)

Where, ijX a binary variable with a value of 1 for each pipe l with a flow direction

from node i to node j and zero otherwise; iN the number of pipe connected to node i

and K= the total number of nodes.

This constraint has to be augmented with the continuity requirement at each node i of the

network defined as:

KiQXQXii N

j

lijl

N

j

ji ,.....,1011

(11)

Where, lQ the discharge of pipe l considered between node i and node j.

The sewer network design problem formulated above is clearly a Mixed Integer

Nonlinear Programming (MINLP) which can not be solved using conventional method.

The complexity of the problem is mostly due to the constraints 9 to 11 requiring a

systematic and efficient method to enforce them if an optimal solution is required.

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3. Ant Colony Optimization Algorithm (ACOA)

In the ACOA a colony of artificial ants cooperate to find good solutions to discrete

optimization problems. Application of the ACOA to any combinatorial optimization

problem requires that the problem can be projected on a graph [8]. Consider a graph G =

(D,L,C) in which D= n21 d,....,d,d is the set of decision points at which some decisions

are to be made, L= ijl is the set of options j=1, 2,…,J at each of the decision points

i=1,2,…,n and finally C= ijc is the set of costs associated with options L=

ijl . The

components of sets D and L may be constrained if required. A solution is found by

constructing a path on the problem graph.

The basic steps of the ACOA may be defined as follows [8]:

1- The total number of ants, m, in the colony is chosen and the amount of pheromone trail

on all options L= ijl are initialized to some proper value.

2- Starting from an arbitrary or pre-selected decision point i, ant k is required to build a

solution by selecting from available options using the following transition rule:

J

j

ijij

ijij

ij

t

ttkp

1

][)]([

][)]([),(

(12)

Where, ),( tkpij = the probability that the ant k selects option j of the ith decision point,

ijl , at iteration t; )t(ij = the concentration of pheromone on option ijl at iteration t;

)(1

ijij c is the heuristic value representing the local cost of choosing option j at point i,

and and are two parameters that control the relative weight of pheromone trail and

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heuristic value referred to as pheromone and heuristic sensitivity parameters, respectively.

3- Once the option at the current decision point is selected, the ant k moves to the next

decision point and a solution is incrementally created by the ant k as it moves from one

decision point to the next one. This procedure is repeated until all decision points of the

problem are covered and a complete trial solution,k)( , is constructed by the ant k.

4- The cost, k)(f , of the trial solution is calculated.

5- Steps 2 to 4 are repeated for all ants leading to the generation of m trial solutions and

calculation of the corresponding costs referred to as an iteration.

6- At the end of each iteration, the amount of pheromone on each and every of the

options are updated using the solutions created at the current iteration. In this work, only

the ant which has produced the best solution of the iteration is allowed to contribute to

pheromone change as follows [8]:

)t()1t( ijij best

ij (13)

Where, )1t(ij = the amount of pheromone trail on option ijl at iteration t+1; )t(ij =

the concentration of pheromone on option ijl at iteration t; )10( = the coefficient

representing the pheromone evaporation and best

ij = the change in pheromone

concentration associated with option ijl defined as:

otherwise 0

ant best theby chosen isj option if )(f

Rbestbest

ij (14)

Where, best)(f = the cost of the solution produced by the best ant and R = a quantity

related to the pheromone trail called pheromone reward factor.

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7- The process defined by steps 2 to 6 is continued until the iteration counter reaches its

maximum value defined by the user or some other convergence criterion is met.

Premature convergence to sub-optimal solutions is an issue that can be experienced by all

ACOAs, especially those that use an 'elitist' strategy of pheromone updating as defined in

Eq. 13. To overcome this problem the Max-Min ant system (MMAS) is used here [8] in

which the pheromone trail intensities are bounded within a maximum and minimum

values as follows [8]:

)(1

1max bestf

R

(15)

).(

))(1.(/1

/1

max

min n

bestavg

n

best

pJ

p

(16)

Where, min , max represent the lower and upper limit on the pheromone trail strength,

respectively; bestp = the probability that the best solution is constructed again; Javg = the

average number of options available at decision points of the problem; and n = the

number of decision points as defined earlier.

4. Layout and size optimization of sanitary sewer network by ACOA

Formulation of the sanitary sewer network design as an optimization problem by ACOA

requires that the problem is defined as a graph. The graph used for the application of

ACOAs consists of a set of nodes referred to as decision points and edges referred to as

options available at each decision points. The problem graph is very much dependant on

the decision variables selected for the problem. In the absence of pumps and drops,

different sets of decision variables such as pipe slopes, pipe diameters or nodal elevations

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of the sewer can be used to define a network with known layout. Here the pipe diameters

of the base layout are considered as the decision variables of the component sizing

problem leading to an easy definition of the problem graph. An assumption of sewer flow

at maximum allowable relative depth is also made allowing for the calculation of the

optimal pipe slopes in the absence of any pump and drop in the network.

Having addressed the pipe sizing side of the problem, a question remains as how to

determine the optimal layout of the network. Here an ACOA equipped with a TGA,

referred to as ACOA-TGA, is used in which the TGA guides the ants to create the

required tree structure of the network when deciding on the pipe sizing aspect of the

problem. Furthermore, a conventional application of the ACOA is also suggested for

comparison purpose in which the layout of the network is decided upon after pipe sizing

side of the problem is fully addressed.

4.1. Conventional Application of ACOA

In this formulation, the pipes of base layout are taken as decision points of the graph. The

options available at each decision point are, therefore, represented by finite number of

commercially available sewer pipe diameters. In this formulation each ant is required to

select one option (diameter) from the set of available options (commercial diameters) for

each pipe. The graph representation of the problem for the application of ACOA is shown

in Figure 1, where bold small lines represent the components of options (pipe diameters,

j=1,…..J) at each decision point i (i=1,…..n), dashed lines represent the potential

components of the solution at each decision point i, and finally the bold lines represent a

trial solution on the graph constructed by an arbitrary ant.

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The layout of the network is decided upon once each ant constructed a trial solution. An

ad-hoc method based on engineering judgment is used to define the layout of the network

with pre-defined pipe diameters as follows:

1. Attribute an integer number to each node of the network, referred to as the rank of

the node, defined by the minimum number of pipes between the current node and

the root note. Starting from the root node with a zero rank, all nodes connected to

the root node are ranked one. The process is continued by attributing a rank to each

node defined as the minimum of the rank of the nodes connected to the current node

plus one, until all nodes are covered.

2. The flow direction at each pipe of the network is defined from the node of higher

rank to the node of lower rank. For pipes with the nodes of the same rank, the flow

direction is decided upon randomly.

3. Starting from the root node and moving towards the nodes of highest rank, the

nodes are checked for the number of leaving pipes. If more than one pipe leaves

from each node, the pipe with largest diameter is considered as the one leaving from

node and connection of other pipes to this node is cut to recover the tree structure of

the layout. A dummy node is then considered at adjacency of the node to indicate

the cut. Finally, by application this formulation the directed tree-like layout is

constructed from the base layout of sewer network with kwon pipe diameters and

other parameters can be calculated for this directed tree-like layout.

Figure 2 schematically shows the process of layout determination out of a typical base

layout in which the circles represent the network nodes, the bold circle represents the

root node, the lines represent the pipes, the numbers in brackets represent the nodal

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rankings and the numbers in the parenthesis represent the pipe diameters determined by

ants.

Having determined the pipe diameters and the layout of a trial sewer network, the

remaining task is to determine the average cover depth of each sewer pipe. For this, an

assumption of sewer flow at maximum allowable relative depth, as indicated in Eq. 5

bymax , is made allowing for the calculation of nodal cover depths of the network

considering the fixed elevation of the root node. Calculation of the average cover depth

of the cut pipes, however, requires that the cover depth of the dummy nodes introduced

upstream of the cut pipes are also calculated. Two different methods are used here for the

calculation of dummy nodes cover depth. In the first method, referred to hereafter as

ACOA1, the cover depths of dummy nodes are computed by the assumption of sewer

flow at maximum allowable relative depth in the cut pipes while in the second method,

referred to as ACOA2, the cover depths of the dummy nodes are taken equal to the

minimum allowable cover depth.

4.2. Proposed intelligent ACOA-TGA method

In the second formulation named ACOA-TGA, incremental solution building capability

of ACOA is used to simultaneously construct tree-like feasible layouts out of the base

layout while determining the pipe sizes. In this approach, the TGA is responsible to keep

the ants’ options limited to forming tree layouts while the ACOA determines the pipe

sizes. This concept has already been used by Afshar and Marino [3] for layout

optimization of tree pipe networks. In this formulation, the nodes of the base layout are

taken as decision points of the graph and the options available at each decision point is

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defined by the aggregation of the commercially available diameters for all the pipes

which can contribute to a tree-like layout which is defined by TGA at each decision

point. In this formulation, therefore, the number of options at each decision point is

determined by the decisions already taken up to the decision point under consideration.

This is in contrast to the conventional application of ACOA in which the pipes of the

base layout was taken as the decision point with fixed number of options equal to the

number of the commercially available diameters.

While in the conventional ACOA, each ant can start the solution construction, pipe size

determination, from an arbitrary decision point, in ACOA-TGA each ant starts from the

root node and makes a decision before moving to the next decision point. The options

available to the ants at each decision point are determined by the TGA, and the decision

made by the ants determines which decision point to move to. The role of the TGA is to

define the edges of the base layout which can contribute to a tree layout. A tabu list is

then created for the ant which includes all potential diameters of the edges already

defined by the TGA. Any decision of the ant by choosing one diameter from the tabu list

will lead to simultaneous definition of the pipe to be included in the tree layout and its

diameter. This process is continued until all decision points of the problem, base layout

nodes, are covered.

Figure 3(b) shows the graphical representation of the proposed ACOA-TGA method for

the typical problem of Figure 3(a), in which the circles represent the network nodes, the

bold circles represent the decision points of the ACOA, the numbers in the circles

represent the node numbers, the numbers in the parenthesis indicate the pipe numbering,

Dl represent the set of available commercial diameters to be used for pipe l, and the

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brackets represent the options available to the ant at each decision point to form a tree

layout, and the bold Dl indicates that a diameter from Dl is chosen by ant leading to the

inclusion of pipe l in the tree under construction. This would automatically define the

other node of pipe l as the next decision point. Any path on the problem graph shown in

Figure 3(b) represents a tree layout out of the base layout shown in Figure 3(a)

represented by the links of bold Dls on the path. For example the path (1-2-3-5-4) on the

graph represents a tree layout composed of links (1,3,4,5) while the path (1-2-4-3-5)

denotes another tree layout composed of links (1,5,2,4). It should be note, however, there

might be some different paths on the graph Figure 3(b) which correspond to the same tree

layout but different solution due to probably different sizes selected for the pipes.

The ACOA-TGA is easily implemented using four vectors B, A, AA, and T in which B=

the set of nodes contained within the growing tree; A= the set of edges, pipes, within the

growing tree; AA= the set of edges adjacent to the growing tree; and T= the tabu list

containing the options available to the ants at each decision point. Starting from the root

node Root-Node.

1- Initialize B= [Root-Node], A= [ ], and AA= [edges in the base graph connected to

root node].

2- Form the tabu list T as the aggregation of all available diameters for the components

of AA.

3- Let the ant chose a diameter, d, at random from T.

4- Determine to which edge, a, of AA the chosen diameter, d, belong and set A=A+ [a].

5- Identify the other node of edge a as the newly connected node i and set B=B+ [i].

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6- Identify edges connected to node i in the base graph and update AA by removing

edge a and any newly infeasible edges, and adding any of the edges connected to node

i that are feasible candidates to form a tree layout. AA now contains all feasible

choices for the next edge of the tree. The feasibility of the edges is determined by the

condition that each node of the network is visited once and only once.

7- Repeat from (2) until AA is empty.

It should be noted that the vector AA acts as the tabu list for the ants which is updated at

each decision point of the problem. This process is repeated for all ants in the colony at

each iteration of the optimization algorithm.

The process defined above leads to the construction of a spanning tree network with

known diameters. The resulting network, however, does not contain some of the pipes in

the looped base layout. To complete the construction of a tree-like network containing all

pipes of the base layout, the connection of the edges which are absent in the constructed

layout are cut at the node of higher rank and included in the constructed spanning tree to

form the final network layout. It should be remarked that rank of the network nodes is

defined here in a manner different from the one used in the conventional application of

ACOA. In ACOA-TGA, the rank of each node is defined as the minimum number of

pipes between the root node and the current node in the constructed spanning tree layout

rather than the base layout which is easily calculated during the solution construction.

The resulting network will obviously be a tree-like network containing all pipes of the

base layout.

The next step in the construction of a trial solution is to define the nodal cover depths of

the resulting network. For this, an assumption of sewer flow at maximum relative depth

21

is made allowing for the calculation of pipe slopes of the spanning tree using the fixed

elevation of the root node. The calculated pipe slopes are in turn used to calculate the

cover depth for all nodes of the network except for the dummy nodes introduced at the

cut positions since the cut pipe diameters are not known yet. A minimum cover depth is

considered for the dummy nodes upstream of the cut pipes from which the diameters of

the cut pipes are calculated. Different methods can be employed to calculate the cut pipe

diameters using the known slope of these pipes. Here the cut pipe diameters are

calculated such that all the constraints defined by Eq. (3), (5) and (6) are fully satisfied, if

possible.

The proposed formulations generate a directed tree-like layout out of the base layout as

defined earlier. Calculation of the nodal cover depths for each of the algorithms, as

defined before, requires that the pipe slopes are known. Calculation of the pipe slopes

using the assumption of maximum relative depth requires that the design discharge of

each pipe is first determined. For every directed tree-like layout constructed, the design

discharge of the pipes in the resulting network can be calculated using continuity

equation, direction of the flow in each pipe, and the local discharge of the pipes. The

local discharge is calculated using the service population of each pipe at the end of the

design period, the average water consumption per person per day, and the return factor.

It should be noted that the trial solutions obtained by the application of the proposed

formulations may violate some of the constraints of problem. To encourage the ants to

make decision leading to feasible solutions, a higher cost is associated to the solutions

that violate the problem constraints defined by Eqs. (2) to (11). This may be done via the

22

use of a penalty method in which the total cost of the problem is considered as the sum of

the problem cost and a penalty cost as:

G

g

gpp CSVFF1

(17)

Where, Fp = penalized objective function; F = original objective function defined by

Eq.1; gCSV = a measure of the violation of constraint g; G= total number of constraints

and p = the penalty constant assumed large enough so that any infeasible solution has a

total cost greater than that of any feasible solution. The penalized term in Eq. (17) will be

zero for feasible solutions.

5. Results and Discussions

Performance of the proposed algorithms is now tested against three hypothetical test

examples of sanitary sewer network design. The networks required for three quadrangle

zones with the size of 200 (meter) * 200 (meter), 400 (meter) * 400 (meter) and 800

(meter) * 800 (meter) are used to test the efficiency of the proposed methods. The

population of the area is assumed to be uniformly distributed over the area with the value

of 2500 and 4000 person per hectare at the beginning and the end of the design period,

respectively. The average water consumption per person per day at the beginning and the

end of the design period is taken as 250)*( capday

lit . The coefficients of the maximum and

minimum sewer flow rate are assumed to be constant and equal to 2.8 and 0.6,

respectively. Note that the simplifying assumption of constant coefficients of the

maximum and minimum sewer flow rate does not affect the generality of the proposed

algorithm since it only affect the design flow rate for each pipe of a given layout. The

23

return factor used for calculation the sewer discharge of area is assumed to be 0.8. The

geometry of the area along with the ground elevation at the benchmark point is shown in

Figure 4 in which a uniform decrease of the ground elevation from upstream to the

location of the wastewater treatment plants and from the center to the right and left of the

zone is assumed. Three different base layouts are considered with increasing scales to

assess the ability of the proposed methods to solve small, medium and large scale sewer

network design problems. The first test example is considered to have 9 nodes and 12

edges while the second test example has 25 nodes and 40 edges and finally the third test

example has 81 nodes and 144 edges as shown in Figure 5(a) to 5(c). All the networks are

supposed to deliver the collected sewer to two treatment plant of fixed elevation located

at the bottom corner of the area. The set of diameters ranging from 100 mm up to 1500

mm with an interval of 50 mm from 100 mm to 1000 mm and an interval of 100 mm

from 1000 mm to 1500 mm is used as the set of commercially available pipe diameters

for all the pipes. The pipes lengths of three networks are assumed to be constant and

equal to 100 meter. Since the test examples are hypothetical, the numerical values of the

other parameters used to solve the problems are taken from other similar works such as

those by Afshar et al. [9] and by Diogo and Graveto [14] as follows: Manning

coefficient, n = 0.015; Maximum cover depth, Emax = 10 (meter); Minimum cover depth,

Emin = 2.5 (meter); Maximum relative flow depth, max = 0.83; Minimum relative flow

depth,min =0.1; Self-cleaning sewer flow velocity, Vclean = 0.75(

sm ); and Maximum

sewer flow velocity, Vmax = 6 (s

m ).

Here the following explicit relation is used for the pipe installation and manhole costs

which are a modified form of the one used in Afshar et al. [9]:

24

mm

lll

d

p

hK

dEEeK l

46.41

437.0012.093.1047.153.143.3

(18)

Three test examples are solved using proposed formulations using MMAS and the result

are presented and compared. A set of preliminary runs are first conducted to find the

proper values of MMAS parameters for each test example as shown in Table 1. The

results are obtained using 500, 1000 and 2000 iterations and colony size of 50, 100, 200

amounting to 25000, 100000 and 400,000 function evaluations for test example I, II and

III, respectively.

Table 2 shows the results of 10 runs carried out using different randomly generated initial

guess for the test examples along with the scaled standard deviation, standard deviation

of the final solution costs produced in ten runs scaled by the average solution cost, and

the number of feasible final solutions obtained on a 3 GHZ Pentium PC and CPU. It is

clearly seen from Table 2 that all measures of the quality of the final solutions such as the

minimum cost, maximum coat, average cost, and the scaled standard deviation

representing the sensitivity of the method to initial guesses are improved when using

ACOA-TGA compared to other formulations in which the layout is created in an ad-hoc

manner. It is particularly seen that the number of final feasible solution created by the

ACOA-TGA is always greater than those of ACOAs in which an ad-hoc engineering

concept is used for layout determination. It is worth noting that ACOA2 has been able to

outperform the ACOA1 regarding the quality of the solution and the number of final

feasible solutions due to the fact that solution constructed by the ACOA2 will never be

infeasible regarding the minimum nodal cover depth leading to smaller search space for

the method. In fact, ACOA2 has been able to produce near-optimal solutions for all three

25

cases while being outperformed by the ACOA-TGA regarding the number of final

feasible solutions.

It is worth noting that based on the allowable considered cover depths, [1, 10], the

manhole heights of the final design obtained by ACOA-TGA vary from 3.4 (meter) to 7

(meter), from 2.6 (meter) to 7 (meter) and from 2.6 (meter) to 9.2 (meter) for test

example I, II and III, respectively. The reason for such large manhole depths is twofold:

First, the operational cost of the network including the maintenance of the manholes is

not considered in this work and the optimization is carried our only based on the

construction cost. Second, the manhole cost defined in Eq. (18) apparently constitutes a

small part of the total cost as evident from the final solutions. For example, the manhole

costs of the final design obtained by ACOA-TGA are about 1871.3, 4935.3 and 16361.8

for test example I, II and III, respectively, which are about 7%, 5% and 4% of the total

cost values.

Figures 6, 7 and 8 show the optimal tree-like layout for the test examples I, II and III,

respectively, obtained using ACOA-TGA. Tables 3, 4 and 5 show the optimal

characteristics of the networks obtained for test example I, II and III, respectively, using

ACOA-TGA. Figure 9 shows convergence curves of the minimum solution costs

obtained in ten runs using proposed methods for test example II indicating superior

performance of the ACOA-TGA compared to other algorithms. First, the solution cost of

ACOA-TGA method remains lower than those of other method during the evolution

process leading to lower cost final solution. Second, the ACOA-TGA solutions become

feasible earlier than those of other algorithms indicated by the fact that the ACOA-TGA

convergence curve appear to the left of the other curves.

26

Finally, it is worth noting that although the test examples considered are symmetric,

however, no use of symmetry has been made to solve these problems. The asymmetrical

cases can, therefore, be easily considered and solved in the same manner without any

special arrangements.

6. Concluding Remarks

In this paper, two different ACOA was proposed for the efficient layout and pipe size

determination of sanitary sewer network out of a base layout including all possible links

of the network is available. In the first formulation, ACOA was used in a conventional

manner for pipe size optimization while an ad-hoc engineering concept was used for the

layout determination. In the second formulation, ACOA equipped with TGA was used to

simultaneously determine both the layout and pipe sizes of the network. The TGA was

used to construct feasible tree-like layouts out of the base layout defined for the sewer

network, while the ACOA was used to optimally determine the pipe diameters of the

constructed layout. Proposed formulations were used to solve a hypothetical test example

considered of different scales and the results were presented and compared. The results

indicated the ability of the proposed methods and in particular the ACOA-TGA approach

to optimally solve the problem of layout and size determination of sewer networks. While

all proposed algorithms showed good performance in solving the problems under

consideration, the ACOA-TGA algorithm was shown to produce better results and to be

less sensitive to the randomly generated initial guess required to start the solution process

represented by the scaled standard deviation of the solutions produced in ten different

runs. Furthermore, the proposed ACOA-TGA method was shown to enjoy higher success

27

rate indicated by the number of final feasible solution in particular for the problems of

larger scale.

References

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Texas at Austin, 1984.

31

Figure1. Problem graph used for ACOA1 or ACOA2.

Figure2. Different steps of construction tree-like layout for typical base layout using

ACOA1 or ACOA2.

(a) Typical Base layout (b) Diameter determination

(c) Nodal ranking (d) Cut determination

(400)

(300)

(250)

(200)

(250)

[0]

[1] [1]

[2] [2]

(350)

(400)

(300)

(250)

(200)

(250)

(350) (400)

(250)

(200)

(250)

(350)

(300)

d1 d… j=1

2

J-1

di dn

J

32

Figure 3. Graph representation of the typical problem for ACOA-TGA.

2 3

1

(3)

(1) (2)

(4) (5)

(6)

(a) Typical Base layout

4

1

2

3

3

4

2

5

d1

4 5

4

3

5

5

4

2

4

5

5

3

4

5

4

2

d2 d3 d4 d5

[D1, D2]

[D1, D2]

[D1, D3, D4]

[D1, D3, D4]

[D2, D3, D5]

[D2, D3, D5]

[D2, D3, D6]

[D2, D3, D6]

[D1, D3, D6]

[D1, D3, D6]

[D1, D3, D5]

[D2, D3, D4]

[D5, D6]

[D5, D6]

[D4, D5]

[D4, D5]

[D4, D6]

[D4, D6]

[D4, D6] [D4, D5]

[D4, D5] [D5, D6]

(b) Graph representation

5

33

Figure 4. Geometry of the areas used for three test examples.

W.T.P W.T.P

1000

S=2% S=2%

S=2%

S=2%

W.T.P W.T.P

W.T.P W.T.P

34

Figure 5. Base layouts of three proposed test examples.

W.T.P W.T.P

[2] [1] [4] [3]

[5] [6] [7] [8] [9]

[11] [10] [13] [12]

[20] [19] [22] [21]

[29] [28] [31] [30]

[38] [37] [40] [39]

[14] [15] [16] [17] [18]

[23] [24] [25] [26] [27]

[32] [33] [34] [35] [36]

1 3 2 4 5

6 8 7 9 10

11 13 12 14 15

16 18 17 19 20

21 23 22 24 25

5(b): Test example (II)

W.T.P W.T.P

1 3 2 4 5 7 6 8 9

10 12 11 13 14 16 15 17 18

19 21 20 22 23 25 24 26 27

28 30 29 31

32 34 33

35 36

37 39 38 40 41 43 42 44 45

46 48 47 49 50 52 51 53 54

55

0 57 56 58 59 61 60 62 63

64 66 65 67 68 70 69 71 72

73 75 74 76 77 79 78 80 81

[2] [1] [4] [3] [6] [5] [8] [7]

[9] [10] [11] [12] [13] [14] [15] [16] [17]

[19] [18] [21] [20] [23] [22] [25] [24]

[26] [27] [28] [29] [30] [31] [32] [33] [34]

[36] [35] [38] [37] [40] [39] [42] [41]

[43] [44] [45] [46] [47] [48] [49] [50] [51]

[53] [52] [55] [54] [57] [56] [59] [58]

[60] [61] [62] [63] [64] [65] [66] [67] [68]

[70] [69] [72] [71] [74] [73] [76] [75]

[77] [78] [79] [80] [81] [82] [83] [84] [85]

[87] [86] [89] [88] [91] [90] [93] [92]

[94] [95] [96] [97] [98] [99] [100] [101] [102]

[104] [103] [106] [105] [108] [107] [110] [109]

[138] [137] [140] [139] [142] [141] [144] [143]

[111] [112] [113] [114] [115] [116] [117] [118] [119]

[121] [120] [123] [122] [125] [124] [127] [126]

[128] [129] [130] [131] [132] [133] [134] [135] [136]

5(c): Test example (III)

W.T.P W.T.P

[2] [1]

[3] [4] [5]

[7] [6]

[12] [11]

[8] [9] [10]

1

2

3

4 6 5

7 9 8

5(a): Test example (I)

35

Figure 6. Optimal tree-like layout of test example I obtained using ACOA-TGA.

Figure 7. Optimal tree-like layout of test example II obtained using ACOA-TGA.

W.T.P W.T.P

[2] [1] [4] [3]

[5] [6] [7] [8] [9]

[11] [10] [13] [12]

[20] [19] [22] [21]

[29] [28] [31] [30]

[38] [37] [40] [39]

[14] [15] [16] [17] [18]

[23] [24] [25] [26] [27]

[32] [33] [34] [35] [36]

1 3 2 4 5

6 8 7 9 10

11 13 12 14 15

16 18 19 20

21 23 22 24 25

17

W.T.P W.T.P

[2] [1]

[3] [4] [5]

[7] [6]

[12] [11]

[8] [9] [10]

1

2

3

4 6 5

7 9 8

36

Figure 8. Optimal tree-like layout of test example III obtained using ACOA-TGA.

85000

90000

95000

100000

105000

110000

115000

120000

125000

0 20000 40000 60000 80000 100000

Function evaluations

cost

valu

e

ACOA1ACOA2ACOA -TGA

Figure 9. Variation of minimum solution cost values of test example II using ACOA1,

ACOA2 and ACOA-TGA.

W.T.P W.T.P

1 3 2 4 5 7 6 8 9

10 12 11 13 14 16 15 17 18

19 21 20 22 23 25 24 26 27

28 30 29 31

32 34 33

35 36

37 39 38 40 41 43 42 44 45

46 48 47 49 50 52 51 53 54

55

0 57 56 58 59 61 60 62 63

66 65 67 68 70 69 71 72

73 75 74 76 77

79 78

80 81

[2] [1] [4] [3] [6] [5] [8] [7]

[9] [10] [11] [12] [13] [14] [15] [16] [17]

[19] [18] [21] [20] [23] [22] [25] [24]

[26] [27] [28] [29] [30] [31] [32] [33] [34]

[36] [35] [38] [37] [40] [39] [42] [41]

[43] [44] [45] [46] [47] [48] [49] [50] [51]

[53] [52] [55] [54] [57] [56] [59] [58]

[60] [61] [62] [63] [64] [65] [66] [67] [68]

[70] [69] [72] [71] [74] [73] [76] [75]

[77] [78] [79] [80] [81] [82] [83] [84] [85]

[87] [86] [89] [88] [91] [90] [93] [92]

[94] [95] [96] [97] [98] [99] [100] [101] [102]

[104] [103] [106] [105] [108] [107] [110] [109]

[138] [137] [140] [139] [142] [141] [144] [143]

[111] [112] [113] [114] [115] [116] [117] [118] [119]

[121] [120] [123] [122] [125] [124] [127] [126]

[128] [129] [130] [131] [132] [133] [134] [135]

64

[136]

37

Table 1: values of MMAS parameters. test

example iteration ant

bestp

I 500 50 1 0.2 0.95 0.3 II 1000 100 1 0.2 0.95 0.3 III 2000 200 1 0.2 0.95 0.3

Table 2: Maximum, minimum and average solution cost values over 10 runs obtained

using proposed methods.

test

example Formulation

Cost value Scaled

Standard

deviation

No. of runs with

final feasible

solution Minimum Maximum Average

I

ACOA1 25053.4 25869.5 25298.2 0.0156 10

ACOA2 24693.2 29770.2 25680.6 0.0650 10

ACOA-TGA 24514.9 25290.5 24722.4 0.0120 10

II ACOA1 93166.7 155536 100080 0.1948 9

ACOA2 90602.2 95358.7 93141.3 0.0180 10

ACOA-TGA 89568.3 94425.8 91951.1 0.0173 10

III ACOA1 420544 9360800 3184060 1.3341 6

ACOA2 402695 6433280 1300630 1.5208 7

ACOA-TGA 397563 830097 461305 0.2831 9

Table 3: Characteristics of the optimal network obtained by ACOA-TGA (Example I).

pipe No.

Node No. Cover Depth (m) Design flow (m3/s)

Diameter (mm)

Slope y/d V (m/s) Up Down Up Down

1 2* 1 2.500 4.500 0.00648 100 0.040 0.574 1.389

2 2 3 5.545 4.500 0.0324 200 0.010 0.830 1.162

3 4 1 3.666 4.500 0.02592 150 0.028 0.830 1.653

4 5 2 4.712 5.545 0.02592 150 0.028 0.830 1.653

5 6 3 5.125 4.500 0.03888 200 0.014 0.830 1.395

6 5* 4 2.500 3.666 0.01296 150 0.032 0.487 1.516

7 5* 6 2.500 5.125 0.01296 150 0.046 0.437 1.745

8 7* 4 2.500 3.666 0.00648 100 0.032 0.619 1.268

9 8* 5 2.500 4.712 0.01296 150 0.042 0.449 1.686

10 9 6 5.531 5.125 0.01944 150 0.016 0.830 1.240

11 7 8 3.283 6.823 0.00648 100 0.015 0.830 0.930

12 8 9 6.823 5.531 0.01296 150 0.007 0.830 0.827

Note: *k is the dummy node considered at the adjacency of node k representing the upstream node of cut

pipe l.

38

Table 4: Characteristics of the optimal network obtained by ACOA-TGA (Example II).

pipe No.

Manhole No. Cover Depths (m) Design flow (m3/s)

Diameter (mm)

Slope y/d V(m/s) Up Down Up Down

1 2 1 5.375 4.500 0.10368 300 0.011 0.830 1.653

2 3* 2 2.500 5.375 0.00648 100 0.049 0.540 1.498

3 3 4 5.101 4.056 0.0324 200 0.010 0.830 1.162

4 4 5 4.056 4.500 0.05184 200 0.024 0.830 1.860

5 6 1 3.526 4.500 0.10368 250 0.030 0.830 2.381

6 7 2 5.099 5.375 0.09072 250 0.023 0.830 2.083

7 8 3 4.268 5.101 0.02592 150 0.028 0.830 1.653

8 9* 4 2.500 4.056 0.01296 150 0.036 0.471 1.584

9 10 5 5.388 4.500 0.15552 350 0.011 0.830 1.822

10 7* 6 2.500 3.526 0.01296 150 0.030 0.494 1.491

11 8* 7 2.500 5.099 0.01296 150 0.046 0.438 1.742

12 8* 9 2.500 3.569 0.01296 150 0.031 0.492 1.499

13 9 10 3.569 5.388 0.0648 200 0.038 0.830 2.325

14 11 6 3.563 3.526 0.08424 250 0.020 0.830 1.934

15 12 7 3.280 5.099 0.0648 200 0.038 0.830 2.325

16 13* 8 2.500 4.268 0.01296 150 0.038 0.463 1.618

17 14 9 4.195 3.569 0.03888 200 0.014 0.830 1.395

18 15 10 5.425 5.388 0.08424 250 0.020 0.830 1.934

19 12* 11 2.500 3.563 0.01296 150 0.031 0.492 1.498

20 13 12 3.905 3.280 0.03888 200 0.014 0.830 1.395

21 13* 14 2.500 4.195 0.01296 150 0.037 0.466 1.606

22 14* 15 2.500 5.425 0.01296 150 0.049 0.430 1.786

23 16 11 4.402 3.563 0.0648 250 0.012 0.830 1.488

24 17* 12 2.500 3.280 0.01296 150 0.028 0.506 1.444

25 18 13 3.072 3.905 0.02592 150 0.028 0.830 1.653

26 19* 14 2.500 4.195 0.01296 150 0.037 0.466 1.606

27 20 15 6.263 5.425 0.0648 250 0.012 0.830 1.488

28 17 16 4.530 4.402 0.04536 200 0.019 0.830 1.627

29 18* 17 2.500 4.530 0.01296 150 0.040 0.455 1.659

30 18* 19 2.500 6.889 0.01296 100 0.064 0.813 1.895

31 19 20 6.889 6.263 0.03888 200 0.014 0.830 1.395

32 21 16 5.693 4.402 0.01296 150 0.007 0.830 0.827

33 22 17 4.936 4.530 0.01944 150 0.016 0.830 1.240

34 23* 18 2.500 3.072 0.01296 150 0.026 0.518 1.403

35 24* 19 2.500 6.889 0.01296 100 0.064 0.813 1.895

36 25 20 6.669 6.263 0.01944 150 0.016 0.830 1.240

37 22* 21 2.500 5.693 0.00648 100 0.052 0.530 1.535

38 23* 22 2.500 4.936 0.00648 100 0.044 0.556 1.445

39 23 24 2.971 2.511 0.00648 100 0.015 0.830 0.930

40 24 25 2.511 6.669 0.01296 100 0.062 0.830 1.860

Note: *k is the dummy node considered at the adjacency of node k representing the upstream node of cut

pipe l.

39

Table 5: Characteristics of the optimal network obtained by ACOA-TGA (Example III).

pipe No.

Node No. Cover Depths (m) Design flow (m3/s)

Diameter (mm)

Slope y/d V(m/s) Up Down Up Down

1 2 1 4.827 4.500 0.07776 250 0.017 0.830 1.785

2 3 2 3.734 4.827 0.05832 200 0.031 0.830 2.092

3 4 3 4.359 3.734 0.03888 200 0.014 0.830 1.395

4 5 4 4.765 4.359 0.01944 150 0.016 0.830 1.240

5 5* 6 2.500 3.629 0.00648 100 0.031 0.622 1.262

6 6 7 3.629 4.463 0.02592 150 0.028 0.830 1.653

7 7 8 4.463 5.077 0.0972 250 0.026 0.830 2.232

8 8 9 5.077 4.500 0.11664 300 0.014 0.830 1.860

9 10 1 4.087 4.500 0.76464 550 0.024 0.830 3.627

10 11* 2 2.500 4.827 0.01296 150 0.043 0.445 1.703

11 12* 3 2.500 3.734 0.01296 150 0.032 0.484 1.529

12 13* 4 2.500 4.359 0.01296 150 0.039 0.460 1.632

13 14* 5 2.500 4.765 0.01296 150 0.043 0.447 1.694

14 15* 6 2.500 3.629 0.01296 150 0.031 0.489 1.510

15 16 7 2.644 4.463 0.0648 200 0.038 0.830 2.325

16 17* 8 2.500 5.077 0.01296 150 0.046 0.439 1.739

17 18 9 3.139 4.500 0.69984 600 0.034 0.571 4.196

18 11 10 4.330 4.087 0.1296 300 0.018 0.830 2.066

19 12 11 3.356 4.330 0.10368 250 0.030 0.830 2.381

20 13 12 3.683 3.356 0.07776 250 0.017 0.830 1.785

21 14 13 2.850 3.683 0.02592 150 0.028 0.830 1.653

22 14* 15 2.500 3.270 0.01296 150 0.028 0.507 1.442

23 15 16 3.270 2.644 0.03888 200 0.014 0.830 1.395

24 16* 17 2.500 3.765 0.01296 150 0.033 0.483 1.534

25 17 18 3.765 3.139 0.03888 200 0.014 0.830 1.395

26 19 10 4.456 4.087 0.62856 550 0.016 0.830 2.982

27 20* 11 2.500 4.330 0.01296 150 0.038 0.461 1.628

28 21* 12 2.500 3.356 0.01296 150 0.029 0.502 1.459

29 22 13 4.309 3.683 0.03888 200 0.014 0.830 1.395

30 23* 14 2.500 2.850 0.01296 150 0.023 0.532 1.356

31 24* 15 2.500 3.270 0.01296 150 0.028 0.507 1.442

32 25* 16 2.500 2.644 0.01296 150 0.021 0.547 1.309

33 26* 17 2.500 3.765 0.01296 150 0.033 0.483 1.534

34 27 18 4.028 3.139 0.65448 600 0.011 0.830 2.609

35 20 19 5.033 4.456 0.11664 300 0.014 0.830 1.860

36 21 20 6.172 5.033 0.09072 300 0.009 0.830 1.447

37 22* 21 2.500 6.172 0.01296 150 0.057 0.413 1.881

38 23* 22 2.500 4.309 0.01296 150 0.038 0.462 1.624

39 23 24 4.226 4.670 0.05184 200 0.024 0.830 1.860

40 24 25 4.670 4.342 0.07776 250 0.017 0.830 1.785

41 25 26 4.342 3.467 0.10368 300 0.011 0.830 1.653

42 26 27 3.467 4.028 0.33696 500 0.026 0.534 3.161

40

Table 5: (continued.) pipe No.

Node No. Cover Depths (m) Design flow (m3/s)

Diameter (mm)

Slope y/d V(m/s) Up Down Up Down

43 28 19 4.703 4.456 0.50544 500 0.018 0.830 2.901

44 29* 20 2.500 5.033 0.01296 150 0.045 0.440 1.732

45 30 21 7.010 6.172 0.0648 250 0.012 0.830 1.488

46 31* 22 2.500 4.309 0.01296 150 0.038 0.462 1.624

47 32 23 4.851 4.226 0.03888 200 0.014 0.830 1.395

48 33* 24 2.500 4.670 0.01296 150 0.042 0.450 1.680

49 34* 25 2.500 4.342 0.01296 150 0.038 0.461 1.630

50 35 26 5.134 3.467 0.22032 500 0.003 0.830 1.265

51 36 27 3.846 4.028 0.31104 450 0.022 0.639 2.897

52 29 28 5.039 4.703 0.49248 500 0.017 0.830 2.827

53 30* 29 2.500 5.039 0.01296 150 0.045 0.440 1.733

54 31 30 2.634 7.010 0.03888 150 0.064 0.830 2.480

55 32* 31 2.500 2.634 0.01296 150 0.021 0.548 1.307

56 32* 33 2.500 8.050 0.01296 100 0.075 0.751 2.049

57 33 34 8.050 6.675 0.11664 350 0.006 0.830 1.366

58 34 35 6.675 5.134 0.14256 400 0.005 0.830 1.279

59 35* 36 2.500 3.846 0.01296 150 0.033 0.479 1.548

60 37* 28 2.500 4.703 0.00648 100 0.042 0.565 1.416

61 38 29 5.546 5.039 0.46656 500 0.015 0.830 2.678

62 39* 30 2.500 7.010 0.01296 100 0.065 0.805 1.913

63 40* 31 2.500 2.634 0.01296 150 0.021 0.548 1.307

64 41 32 4.017 4.851 0.02592 150 0.028 0.830 1.653

65 42 33 2.565 8.050 0.09072 250 0.075 0.534 3.404

66 43* 34 2.500 6.675 0.01296 100 0.062 0.829 1.863

67 44 35 3.315 5.134 0.0648 200 0.038 0.830 2.325

68 45 36 3.928 3.846 0.2916 450 0.019 0.639 2.716

69 37 38 2.826 5.546 0.324 500 0.007 0.830 1.860

70 39 38 6.122 5.546 0.11664 300 0.014 0.830 1.860

71 40 39 5.846 6.122 0.09072 250 0.023 0.830 2.083

72 41* 40 2.500 5.846 0.01296 150 0.053 0.420 1.841

73 41* 42 2.500 2.565 0.01296 150 0.021 0.554 1.291

74 42* 43 2.500 3.940 0.01296 150 0.034 0.476 1.564

75 43 44 3.940 3.315 0.03888 200 0.014 0.830 1.395

76 44* 45 2.500 3.928 0.01296 150 0.034 0.476 1.562

77 46 37 4.162 2.826 0.31104 500 0.007 0.830 1.785

78 47* 38 2.500 5.546 0.01296 150 0.050 0.427 1.802

79 48* 39 2.500 6.122 0.01296 150 0.056 0.414 1.875

80 49 40 6.684 5.846 0.0648 250 0.012 0.830 1.488

81 50* 41 2.500 4.017 0.01296 150 0.035 0.473 1.577

82 51 42 3.404 2.565 0.0648 250 0.012 0.830 1.488

83 52* 43 2.500 3.940 0.01296 150 0.034 0.476 1.564

84 53* 44 2.500 3.315 0.01296 150 0.028 0.504 1.451

85 54 45 5.036 3.928 0.27216 450 0.009 0.830 1.929

86 47 46 5.227 4.162 0.27864 450 0.009 0.830 1.975

41

Table 5: (continued.) pipe No.

Node No. Cover Depths (m) Design flow (m3/s)

Diameter (mm)

Slope y/d V(m/s) Up Down Up Down

87 48 47 6.810 5.227 0.13608 400 0.004 0.830 1.221

88 48* 49 2.500 6.684 0.01296 150 0.022 0.544 1.318

89 50 49 5.850 6.684 0.02592 150 0.028 0.830 1.653

90 50* 51 2.500 3.404 0.01296 150 0.029 0.500 1.468

91 51* 52 2.500 4.118 0.01296 150 0.036 0.469 1.594

92 52 53 4.118 2.815 0.12312 350 0.007 0.830 1.442

93 53 54 2.815 5.036 0.20088 350 0.042 0.599 3.339

94 55 46 3.328 4.162 0.02592 150 0.028 0.830 1.653

95 56 47 5.470 5.227 0.1296 300 0.018 0.830 2.066

96 57 48 8.113 6.810 0.12312 350 0.007 0.830 1.442

97 58* 49 2.500 6.684 0.01296 100 0.062 0.828 1.864

98 59* 50 2.500 5.850 0.01296 150 0.054 0.420 1.841

99 60 51 4.029 3.404 0.03888 200 0.014 0.830 1.395

100 61 52 5.129 4.118 0.0972 300 0.010 0.830 1.550

101 62 53 3.653 2.815 0.0648 250 0.012 0.830 1.488

102 63 54 3.218 5.036 0.0648 200 0.038 0.830 2.325

103 56* 55 2.500 3.328 0.01296 150 0.028 0.504 1.454

104 57* 56 2.500 5.470 0.01296 150 0.050 0.429 1.792

105 58 57 7.499 8.113 0.0972 300 0.026 0.571 2.331

106 59 58 8.967 7.499 0.07128 300 0.005 0.830 1.137

107 59* 60 2.500 4.029 0.01296 150 0.035 0.472 1.579

108 60* 61 2.500 5.129 0.01296 150 0.046 0.437 1.746

109 61* 62 2.500 3.653 0.01296 150 0.032 0.488 1.514

110 62* 63 2.500 3.218 0.01296 150 0.027 0.510 1.432

111 64* 55 2.500 3.328 0.00648 100 0.028 0.644 1.213

112 65 56 6.346 5.470 0.10368 300 0.011 0.830 1.653

113 66* 57 2.500 8.113 0.01296 100 0.076 0.748 2.056

114 67* 58 2.500 7.499 0.01296 100 0.070 0.777 1.980

115 68 59 7.874 8.967 0.05832 200 0.031 0.830 2.092

116 69* 60 2.500 4.029 0.01296 150 0.035 0.472 1.579

117 70 61 2.509 5.129 0.07128 200 0.046 0.830 2.557

118 71 62 4.278 3.653 0.03888 200 0.014 0.830 1.395

119 72 63 3.347 3.218 0.04536 200 0.019 0.830 1.627

120 64 65 2.752 6.346 0.01944 150 0.016 0.830 1.240

121 66 65 6.971 6.346 0.03888 200 0.014 0.830 1.395

122 67* 66 2.500 6.971 0.01296 100 0.065 0.808 1.907

123 67* 68 2.500 7.874 0.01296 150 0.034 0.478 1.553

124 68* 69 2.500 2.638 0.01296 150 0.021 0.548 1.308

125 69 70 2.638 2.509 0.04536 200 0.019 0.830 1.627

126 70* 71 2.500 4.278 0.01296 150 0.038 0.463 1.620

127 71* 72 2.500 3.347 0.01296 150 0.028 0.503 1.457

128 73 64 3.212 2.752 0.00648 100 0.015 0.830 0.930

129 74 65 7.391 6.346 0.0324 200 0.010 0.830 1.162

130 75* 66 2.500 6.971 0.01296 100 0.065 0.808 1.907

42

Table 5: (continued.) pipe No.

Node No. Cover Depths (m) Design flow (m3/s)

Diameter (mm)

Slope y/d V(m/s) Up Down Up Down

131 67 76 2.617 5.326 0.01296 150 0.007 0.830 0.827

132 77 68 8.920 7.874 0.0324 200 0.010 0.830 1.162

133 78 69 3.044 2.638 0.01944 150 0.016 0.830 1.240

134 79* 70 2.500 2.509 0.01296 150 0.020 0.559 1.277

135 80* 71 2.500 4.278 0.01296 150 0.038 0.463 1.620

136 81 72 2.513 3.347 0.02592 150 0.028 0.830 1.653

137 73* 74 2.500 7.391 0.00648 100 0.029 0.639 1.223

138 75 74 3.233 7.391 0.01296 100 0.062 0.830 1.860

139 76* 75 2.500 3.233 0.00648 100 0.027 0.652 1.196

140 76 77 5.326 8.920 0.01944 150 0.016 0.830 1.240

141 77* 78 2.500 3.044 0.00648 100 0.025 0.668 1.162

142 78* 79 2.500 4.210 0.00648 100 0.037 0.588 1.349

143 79 80 4.210 2.919 0.01296 150 0.007 0.830 0.827

144 80 81 2.919 2.513 0.01944 150 0.016 0.830 1.240

Note: *k is the dummy node considered at the adjacency of node k representing the upstream node of cut

pipe l.