Layout and size optimization of sanitary sewer network using intelligent. ants

Transcription

1 Layout and size optimization of sanitary sewer network using inteigent ants R. Moeini a, M. H. Afshar b a (corresponding author) PhD Student, Schoo of Civi Engineering, Iran University of Science and Technoogy, P.O , Narmak, Tehran, Iran, E-mai: rmoeini@iust.ac.ir, Te: , Fax: b Associate professor, Schoo of Civi Engineering & Enviro-hydroinformatic Center of Exceence, Iran University of Science and Technoogy, P.O , Narmak, Tehran, Iran, E-mai : mhafshar@iust.ac.ir, Te: , Fax: Abstract The incrementa soution buiding capabiity of ant agorithm is expoited in this paper for the efficient ayout and pipe size optimization of sanitary sewer network. Layout and pipe size optimization of sewer networks requires that the pipe ocations, pipe diameters and pipe sopes are optimay determined. This probem is a highy constrained mixed-integer noninear programming (MINLP) probem presenting a chaenge even to the modern heuristic search methods. In this paper, the Ant Coony Optimization Agorithm (ACOA) is equipped with a Tree Growing Agorithm (TGA) to efficienty sove the sewer network ayout and size optimization probem and its performance is compared with the conventiona appication of the ACOA. The method is based on the assumption that a 1

2 base ayout incuding a possibe inks of the network is avaiabe. The TGA is used to construct feasibe tree-ike ayouts out of the base ayout defined for the sewer network, whie the ACOA is used to optimay determine the pipe diameters of the constructed ayout. An assumption of sewer fow at maximum aowabe reative depth is made aowing for the cacuation of the optima pipe sopes in the absence of any pump and drop in the network. Two different formuations are, therefore, proposed and their performances are tested against hypothetica probems. In the first formuation, ACOA is used in a conventiona manner for pipe size optimization whie an ad-hoc engineering concept for the ayout determination. In the second formuation, however, ACOA equipped with TGA is used to simutaneousy determine both the ayout and pipe sizes of the network. Proposed formuations are used to sove three hypothetica test exampes of different scaes and the resuts are presented and compared. The resuts indicate the abiity of the proposed method to optimay sove the probem of ayout and size determination of sewer networks. Key works: Ant Coony Optimization Agorithm, Tree Growing Agorithm, sanitary sewer network, ayout, pipe size. 1. Introduction Networks are an important part of any society. Physica networks such as gas pipeines, sewer coection networks, water distribution systems, eectricity and teecommunication grids are ony some exampes of these networks. Construction of these networks is very costy mainy due to cost of the arge number of components they are composed of and the operations required for their constructions. A reativey sma change in the 2

3 component and construction cost of these networks, therefore, eads to a huge saving. Considerabe researches have been focused on deveoping usefu optimization techniques for optima design of such networks in recent years. The network design optimization probem has appeared in many fieds of appications and soved with various optimization methods. A genera network design optimization probem incudes two sub-probems of design of optima ayout (ocation) and design of optima size of network components. These sub-probems are strongy couped and, therefore, can not be sove separatey if an optima soution to the whoe probem is required. Due to the compex nature of the probem, however, most of the research in the fied is carried out on either ayout determination or component sizing [25]. Some researchers have addressed the joint ayout and size optimization of networks with a particuar emphasis on the water distribution networks. For exampe, Martin [28] deveoped a Dynamic Programming (DP) mode for the joint ayout and pipe size optimization of branched water distribution networks. Morgan and Gouter [30] deveoped a mode that used two inked Linear Programming (LP) sub-modes to find east-cost soution for the probem of optima ayout and size of ooped water distribution systems. Jabbari and Afshar [21] appied Mixed Integer Linear Programming (MILP) for ayout and size optimization of a water suppy ine. Lejano [25] used MILP to simutaneousy sove the probem of optima ayout and pipe size design of a branching water distribution system. Cembrowicz [12] combined Genetic Agorithm (GA) and LP for water distribution network design optimization probem in which GA determined the optima ayout and LP determined the east-cost pipe diameters. Hassani and Dandy [19] deveoped a GA mode for ayout and size optimization of a branched pipe networks. 3

4 Afshar [1] used Max Min Ant System (MMAS) for joint ayout and pipe size optimization of pipe networks. Afshar and Marino [3] appied ACOA for ayout optimization of tree pipe networks. Afshar [5] appied GA for simutaneous ayout and pipe size optimization of ooped water distribution networks. More recenty, Afshar [7] used ACOAs for the optima ayout and size design of the branched pipe networks. Sanitary sewer network as one of the physica networks is essentia structure of any urban area. Urban areas without an effective sanitary sewer networks may encounter many probems, such as pubic heath threats and environmenta damages. Having a suitabe and cost-effective sewer network is normay interpreted as finding soution for sewer network design optimization probem that minimizes infrastructura cost, without vioating operationa requirements. The cost of construction a sewer system can be significanty reduced if the system configuration (ayout), pipe diameters and pipe sopes can be effectivey optimized. Most of the works for the optima design of sewer networks, however, are restricted to the optima design of the component size. For exampe, Wash and Brown [34], Tempeman and Waters [33], Yen et a. [36], Kukarni and Khanna [24], Botrous et a. [11] and Diogo et a. [15] used DP to sove sewer design optimization probem. Desher and Davis [13] deveoped a mode named Sanitary Sewer Design (SSD) to find the east cost soution of a sanitary sewer network. Eimam et a. [16] deveoped a mode that combined LP and a heuristic approach for optima design of arge-scae storm water networks. GA was used to find the east-cost soution for storm network design optimization probem by Heaney et a. [20]. Swamee [32] soved the sewer ine design optimization probem by iterative appication of the Lagrange-mutipier method. Liang 4

5 et a. [27] used GA and Tabu Search (TS) for optima design of sewer networks. Guo [17] proposed a sewer design mode using Ceuar Automata (CA) concept for the design of storm sewers networks. Guo et a. [18] proposed a hybrid sewer design mode by combining GA and CA for sewer network design optimization probem. Afshar et a. [4] appied GA for storm sewer design optimization probem. Afshar [6] proposed partiay constrained ACOA and appied it to storm sewer network design. Izquierdo et a. [23] used Partice Swarm Optimization (PSO) agorithm for optima design of wastewater coection networks. Pan and Kao [31] deveoped a mode that combined a GA with a Quadratic Programming (QP) mode to sove sewer system design optimization probem. Afshar [2] used Continuous Ant Coony Optimization Agorithm (CACOA) for optima component sizing of storm sewer networks proposing constrained and unconstrained approach. Ony a few researchers have addressed the probem of ayout and in particuar the joint ayout and size optimization of sewer networks. For exampe, Argaman et a. [10], Mays and Wenze [29] used DP for the optima design of the gravity sewer network with a singe outet using a simpifying assumption that for every pipe of the network the direction of fow was fixed. Therefore, their methods are ony suitabe to very idea cases where natura topography is incined in ony one way to the outet. Waters [35] aso used DP for ayout and size optimization of the sewer networks which coud 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 ayout determination and a Discrete Differentia Dynamic Programming (DDDP) for the optima component sizing of any given ayout, requiring very high computationa 5

6 effort. Jang [22] used DDDP to deveop a mode for design of optima ayout of urban storm sewer network negecting the component sizing part of the probem. Recenty, Diogo and Graveto [14] deveoped a comprehensive enumeration mode and a Simuated Anneaing (SA) mode for the ayout and component size optimization of sewer networks. In this paper, the Ant Coony Optimization Agorithm (ACOA) is equipped with a Tree Growing Agorithm (TGA) to efficienty sove the sewer network ayout and size optimization probem and its performance is compared with the conventiona appication of the ACOA. The method is based on the assumption that a base ayout incuding a possibe inks of the network is avaiabe. The TGA is used to construct feasibe tree-ike ayouts out of the base ayout defined for the sewer network, whie the ACOA is used to optimay determine the pipe diameters of the constructed ayout. An assumption of sewer fow at maximum aowabe reative depth is made aowing for the cacuation of the optima pipe sopes in the absence of any pump and drop in the network. Two different formuations are, therefore, proposed and their performances are tested against a hypothetica probem. In the first formuation, ACOA is used in a conventiona manner for pipe size optimization whie an ad-hoc engineering concept for the ayout determination. In the second formuation, however, ACOA equipped with TGA is used to simutaneousy determine both the ayout and pipe sizes of the network. Proposed formuations are used to sove three hypothetica test exampes of different scaes and the resuts are presented and compared. The resuts indicate the abiity of the proposed method to optimay sove the probem of ayout and size determination of sewer networks. 6

7 2. Sanitary Sewer System Design Sanitary sewer system consists of sewer pipes, manhoes, pumping stations, and other appurtenances. The system is used to coect and transport wastewater by gravity from house to wastewater treatment pants through pipes and manhoes of the network. The design of a sewer system consists of generating an adapted sanitary sewer network ayout that conforms to popuation served, street ayout and oca topography of the panning area, and performing hydrauic design to find pipe sizes, excavation depths and other hydrauic and design parameters of the specific ayout. The probem of finding the best ayout and hydrauic design of sanitary sewer network is a compex task that cannot be performed without using optimization techniques considering the vast number of aternatives for the ocation 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) Seection of optima ayout, (2) Optima hydrauic design of the network components. Determination of the ocation of pipes or ayout of a sanitary sewer system is of great importance because it serves as the foundation of the hydrauic design and, therefore, infuences the fina cost of the network. The probem of joint ayout and component size determination of a sanitary sewer network as a coection of pipes connected with manhoes, here without any pump and drop, can be mathematicay defined as determining the pipe connections, diameter and cover depth of each pipe such that the network construction cost is minimized: Minimize C N 1 L K p M ( d, E ) K ( h ) (1) m 1 m m 7

8 Where, C= cost function of sanitary sewer network; N= tota number of sewer pipes; M= tota number of manhoes; L the ength of pipe (=1,., N); K p = the unit cost of sewer pipe provision and instaation defined as a function of its diameter (d1) and average cover depth ( E ); and K m = the cost of manhoe construction as a function of manhoe height ( h m ). Subject to the topoogica and hydrauic constraints as foows: 1- Cover depth: It is necessary to have adequate cover depth for sewer pipes for three reasons: (1) To provide protection against imposed oads, particuary vehice oads, (2) To aow an adequate fa on house connections, and (3) To reduce the possibiity of cross-contamination of water mains by making sure that, wherever possibe, sewer pipes are ocated beow water mains. Normay, 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: E E E 1,..., N (2) min max Where, Emin= minimum cover depth of sewer pipe; Emax = maximum cover depth of sewer pipe; E = average cover depth of sewer pipe ; and N = tota number of network pipes. 2- Sewer fow veocity: When designing a sewer network, it is important to ensure that sewer pipes wi be capabe of conveying sewer fow with a veocity which does not exceed the maximum aowabe veocity to prevent pipe erosion. Aso, the sewer fow veocity shoud be greater than the sef-ceaning veocity at east once a day to prevent soids form being deposited on the bottom of the pipe. These constraints may be formuated as: 8

9 V V * V V max cean 1,..., N (3) Where, V = fow veocity of pipe at the design fow; * V = the maximum fow veocity of pipe at the beginning of the operation; Vcean= sef-ceaning fow veocity of sewer; and Vmax = maximum aowabe fow veocity of sewer. 3- Minimum sewer pipe sope: Minimum sope shoud be considered for sewer pipe to avoid adverse sope caused by inaccurate construction or settement. This constraint may be formuated as: S Smin 1,..., N (4) Where, S = sope of the sewer pipe ; and S min = minimum sewer pipe sope which is usuay considered as a very sma positive sope. 4- Maximum and minimum reative fow depth: These constraints are used to avoid pressurized fow and deposition of the soids in the sewer pipes defined as: y min max 1,..., N (5) d Where, d diameter of sewer pipe ; y = sewer fow depth in pipe ; max = maximum aowabe reative fow depth; and min = minimum aowabe reative fow depth. 5- Commercia pipe diameters: The sewer pipe shoud be seected form a set of commerciay avaiabe sewer pipe diameters defined as: d d 1,..., N (6) Where, d diameter of sewer pipe ; and d = discrete set of commerciay avaiabe sewer pipe diameters. 9

10 6- partiay-fu pipe condition: Conduits that are used for conveying sewer fows are designed to operate in partiay-fu pipe conditions to avoid pressurized fow condition. Severa equations are generay used to evauate the hydrauic parameters of sewer network. In this paper, Manning s equation is used to find hydrauic parameters such as veocity distribution defined as: Q A R S 1,..., N (7) n Where, Q = discharge of sewer pipe ; A = wetted cross section area of sewer pipe ; R = hydrauic radius of the sewer pipe ; n = Manning constant; and S = sope of the sewer pipe. 7- Progressive pipe diameters: For each pipe, the eaving sewer pipe diameter can not be smaer than anyone of the converging sewer pipes diameter defined as: d d 1,..., N (8) Where, d diameter of sewer pipe ; and d = set of upstream pipe diameters of pipe. 8- Branched ayout: The sewer networks are designed such that the fow of the sewer in the system is due to gravity requiring that the networks have a branched ayout. For this, information regarding the topography of the areas considered for design of the network such as: ayout of the streets, the wastewater treatment pants and existing subsystems are coected and used to define the generic ocations of the sewer network components. Normay, definition of sewer network components eads to a connected undirected graph often referred to as the base ayout containing a possibe ooped and branched ayout configurations. The graph vertices (nodes) represent the fixed position manhoes, outet 10

11 ocations or treatment pants whie graph edges represent the sewer pipes excited between manhoes or manhoes and outets or wastewater treatment pants [14]. In the form of arborescent structure, the sewer pipe configuration shoud be branched defined by a set of root nodes and a number of non-root nodes and edges ocated between nodes. The root nodes represent the outets or the treatment pants, the non-root nodes represent the system manhoes, and the edges define the pipes of the sewer network. In an arborescent structure, ony one edge eaves from each node. This imitation governs the ayout determination of sewer networks which can be mathematicay defined as: X ij X ji 1 i, j 1,..., K (9) N i j 1 X ij 1 i 1,..., K (10) Where, X a binary variabe with a vaue of 1 for each pipe with a fow direction ij from node i to node j and zero otherwise; N the number of pipe connected to node i i and K= the tota number of nodes. This constraint has to be augmented with the continuity requirement at each node i of the network defined as: Ni N i j 1 X ji Q X ijq 0 i 1,..., K j 1 (11) Where, Q the discharge of pipe considered between node i and node j. The sewer network design probem formuated above is ceary a Mixed Integer Noninear Programming (MINLP) which can not be soved using conventiona method. The compexity of the probem is mosty due to the constraints 9 to 11 requiring a systematic and efficient method to enforce them if an optima soution is required. 11

12 3. Ant Coony Optimization Agorithm (ACOA) In the ACOA a coony of artificia ants cooperate to find good soutions to discrete optimization probems. Appication of the ACOA to any combinatoria optimization probem requires that the probem can be projected on a graph [8]. Consider a graph G = (D,L,C) in which D= d,d,..., is the set of decision points at which some decisions 1 2 d n are to be made, L= ij is the set of options j=1, 2,,J at each of the decision points ij i=1,2,,n and finay C= c is the set of costs associated with options L=. The components of sets D and L may be constrained if required. A soution is found by constructing a path on the probem graph. The basic steps of the ACOA may be defined as foows [8]: 1- The tota number of ants, m, in the coony is chosen and the amount of pheromone trai on a options L= ij are initiaized to some proper vaue. 2- Starting from an arbitrary or pre-seected decision point i, ant k is required to buid a soution by seecting from avaiabe options using the foowing transition rue: ij p ( k, t) ij J [ ( t)] j 1 ij [ ( t)] ij [ ] ij [ ] ij (12) Where, p ij ( k, t) = the probabiity that the ant k seects option j of the ith decision point, ij, at iteration t; ij( t ) = the concentration of pheromone on option ij at iteration t; 1 ij is the heuristic vaue representing the oca cost of choosing option j at point i, ( c ) ij and and are two parameters that contro the reative weight of pheromone trai and 12

13 heuristic vaue referred to as pheromone and heuristic sensitivity parameters, respectivey. 3- Once the option at the current decision point is seected, the ant k moves to the next decision point and a soution is incrementay created by the ant k as it moves from one decision point to the next one. This procedure is repeated unti a decision points of the k probem are covered and a compete tria soution, ( ), is constructed by the ant k. 4- The cost, k f ( ), of the tria soution is cacuated. 5- Steps 2 to 4 are repeated for a ants eading to the generation of m tria soutions and cacuation 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 soutions created at the current iteration. In this work, ony the ant which has produced the best soution of the iteration is aowed to contribute to pheromone change as foows [8]: ij( ij t 1) ( t ) best ij (13) Where, ij( t 1) = the amount of pheromone trai on option ij at iteration t+1; ij( t ) = the concentration of pheromone on option ij at iteration t; ( 0 1) = the coefficient representing the pheromone evaporation and best ij = the change in pheromone concentration associated with option ij defined as: R if option j is chosenby the best ant best best ij f ( ) (14) 0 otherwise Where, best f ( ) = the cost of the soution produced by the best ant and R = a quantity reated to the pheromone trai caed pheromone reward factor. 13

14 7- The process defined by steps 2 to 6 is continued unti the iteration counter reaches its maximum vaue defined by the user or some other convergence criterion is met. Premature convergence to sub-optima soutions is an issue that can be experienced by a ACOAs, especiay those that use an 'eitist' strategy of pheromone updating as defined in Eq. 13. To overcome this probem the Max-Min ant system (MMAS) is used here [8] in which the pheromone trai intensities are bounded within a maximum and minimum vaues as foows [8]: max 1 R best 1 f ( ) (15) 1/ n max.(1 ( pbest) ) min (16) 1/ n J.( p ) avg best Where, min, max represent the ower and upper imit on the pheromone trai strength, respectivey; p best= the probabiity that the best soution is constructed again; Javg = the average number of options avaiabe at decision points of the probem; and n = the number of decision points as defined earier. 4. Layout and size optimization of sanitary sewer network by ACOA Formuation of the sanitary sewer network design as an optimization probem by ACOA requires that the probem is defined as a graph. The graph used for the appication of ACOAs consists of a set of nodes referred to as decision points and edges referred to as options avaiabe at each decision points. The probem graph is very much dependant on the decision variabes seected for the probem. In the absence of pumps and drops, different sets of decision variabes such as pipe sopes, pipe diameters or noda eevations 14

15 of the sewer can be used to define a network with known ayout. Here the pipe diameters of the base ayout are considered as the decision variabes of the component sizing probem eading to an easy definition of the probem graph. An assumption of sewer fow at maximum aowabe reative depth is aso made aowing for the cacuation of the optima pipe sopes in the absence of any pump and drop in the network. Having addressed the pipe sizing side of the probem, a question remains as how to determine the optima ayout 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 probem. Furthermore, a conventiona appication of the ACOA is aso suggested for comparison purpose in which the ayout of the network is decided upon after pipe sizing side of the probem is fuy addressed Conventiona Appication of ACOA In this formuation, the pipes of base ayout are taken as decision points of the graph. The options avaiabe at each decision point are, therefore, represented by finite number of commerciay avaiabe sewer pipe diameters. In this formuation each ant is required to seect one option (diameter) from the set of avaiabe options (commercia diameters) for each pipe. The graph representation of the probem for the appication of ACOA is shown in Figure 1, where bod sma ines represent the components of options (pipe diameters, j=1,..j) at each decision point i (i=1,..n), dashed ines represent the potentia components of the soution at each decision point i, and finay the bod ines represent a tria soution on the graph constructed by an arbitrary ant. 15

16 The ayout of the network is decided upon once each ant constructed a tria soution. An ad-hoc method based on engineering judgment is used to define the ayout of the network with pre-defined pipe diameters as foows: 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, a 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 pus one, unti a nodes are covered. 2. The fow direction at each pipe of the network is defined from the node of higher rank to the node of ower rank. For pipes with the nodes of the same rank, the fow direction is decided upon randomy. 3. Starting from the root node and moving towards the nodes of highest rank, the nodes are checked for the number of eaving pipes. If more than one pipe eaves from each node, the pipe with argest diameter is considered as the one eaving from node and connection of other pipes to this node is cut to recover the tree structure of the ayout. A dummy node is then considered at adjacency of the node to indicate the cut. Finay, by appication this formuation the directed tree-ike ayout is constructed from the base ayout of sewer network with kwon pipe diameters and other parameters can be cacuated for this directed tree-ike ayout. Figure 2 schematicay shows the process of ayout determination out of a typica base ayout in which the circes represent the network nodes, the bod circe represents the root node, the ines represent the pipes, the numbers in brackets represent the noda 16

17 rankings and the numbers in the parenthesis represent the pipe diameters determined by ants. Having determined the pipe diameters and the ayout of a tria sewer network, the remaining task is to determine the average cover depth of each sewer pipe. For this, an assumption of sewer fow at maximum aowabe reative depth, as indicated in Eq. 5 by max, is made aowing for the cacuation of noda cover depths of the network considering the fixed eevation of the root node. Cacuation 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 aso cacuated. Two different methods are used here for the cacuation 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 fow at maximum aowabe reative depth in the cut pipes whie in the second method, referred to as ACOA2, the cover depths of the dummy nodes are taken equa to the minimum aowabe cover depth Proposed inteigent ACOA-TGA method In the second formuation named ACOA-TGA, incrementa soution buiding capabiity of ACOA is used to simutaneousy construct tree-ike feasibe ayouts out of the base ayout whie determining the pipe sizes. In this approach, the TGA is responsibe to keep the ants options imited to forming tree ayouts whie the ACOA determines the pipe sizes. This concept has aready been used by Afshar and Marino [3] for ayout optimization of tree pipe networks. In this formuation, the nodes of the base ayout are taken as decision points of the graph and the options avaiabe at each decision point is 17

18 defined by the aggregation of the commerciay avaiabe diameters for a the pipes which can contribute to a tree-ike ayout which is defined by TGA at each decision point. In this formuation, therefore, the number of options at each decision point is determined by the decisions aready taken up to the decision point under consideration. This is in contrast to the conventiona appication of ACOA in which the pipes of the base ayout was taken as the decision point with fixed number of options equa to the number of the commerciay avaiabe diameters. Whie in the conventiona ACOA, each ant can start the soution 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 avaiabe 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 roe of the TGA is to define the edges of the base ayout which can contribute to a tree ayout. A tabu ist is then created for the ant which incudes a potentia diameters of the edges aready defined by the TGA. Any decision of the ant by choosing one diameter from the tabu ist wi ead to simutaneous definition of the pipe to be incuded in the tree ayout and its diameter. This process is continued unti a decision points of the probem, base ayout nodes, are covered. Figure 3(b) shows the graphica representation of the proposed ACOA-TGA method for the typica probem of Figure 3(a), in which the circes represent the network nodes, the bod circes represent the decision points of the ACOA, the numbers in the circes represent the node numbers, the numbers in the parenthesis indicate the pipe numbering, D represent the set of avaiabe commercia diameters to be used for pipe, and the 18

19 brackets represent the options avaiabe to the ant at each decision point to form a tree ayout, and the bod D indicates that a diameter from D is chosen by ant eading to the incusion of pipe in the tree under construction. This woud automaticay define the other node of pipe as the next decision point. Any path on the probem graph shown in Figure 3(b) represents a tree ayout out of the base ayout shown in Figure 3(a) represented by the inks of bod Ds on the path. For exampe the path ( ) on the graph represents a tree ayout composed of inks (1,3,4,5) whie the path ( ) denotes another tree ayout composed of inks (1,5,2,4). It shoud be note, however, there might be some different paths on the graph Figure 3(b) which correspond to the same tree ayout but different soution due to probaby different sizes seected for the pipes. The ACOA-TGA is easiy impemented 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 ist containing the options avaiabe to the ants at each decision point. Starting from the root node Root-Node. 1- Initiaize B= [Root-Node], A= [ ], and AA= [edges in the base graph connected to root node]. 2- Form the tabu ist T as the aggregation of a avaiabe 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, beong and set A=A+ [a]. 5- Identify the other node of edge a as the newy connected node i and set B=B+ [i]. 19

20 6- Identify edges connected to node i in the base graph and update AA by removing edge a and any newy infeasibe edges, and adding any of the edges connected to node i that are feasibe candidates to form a tree ayout. AA now contains a feasibe choices for the next edge of the tree. The feasibiity of the edges is determined by the condition that each node of the network is visited once and ony once. 7- Repeat from (2) unti AA is empty. It shoud be noted that the vector AA acts as the tabu ist for the ants which is updated at each decision point of the probem. This process is repeated for a ants in the coony at each iteration of the optimization agorithm. The process defined above eads to the construction of a spanning tree network with known diameters. The resuting network, however, does not contain some of the pipes in the ooped base ayout. To compete the construction of a tree-ike network containing a pipes of the base ayout, the connection of the edges which are absent in the constructed ayout are cut at the node of higher rank and incuded in the constructed spanning tree to form the fina network ayout. It shoud be remarked that rank of the network nodes is defined here in a manner different from the one used in the conventiona appication 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 ayout rather than the base ayout which is easiy cacuated during the soution construction. The resuting network wi obviousy be a tree-ike network containing a pipes of the base ayout. The next step in the construction of a tria soution is to define the noda cover depths of the resuting network. For this, an assumption of sewer fow at maximum reative depth 20

21 is made aowing for the cacuation of pipe sopes of the spanning tree using the fixed eevation of the root node. The cacuated pipe sopes are in turn used to cacuate the cover depth for a 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 cacuated. Different methods can be empoyed to cacuate the cut pipe diameters using the known sope of these pipes. Here the cut pipe diameters are cacuated such that a the constraints defined by Eq. (3), (5) and (6) are fuy satisfied, if possibe. The proposed formuations generate a directed tree-ike ayout out of the base ayout as defined earier. Cacuation of the noda cover depths for each of the agorithms, as defined before, requires that the pipe sopes are known. Cacuation of the pipe sopes using the assumption of maximum reative depth requires that the design discharge of each pipe is first determined. For every directed tree-ike ayout constructed, the design discharge of the pipes in the resuting network can be cacuated using continuity equation, direction of the fow in each pipe, and the oca discharge of the pipes. The oca discharge is cacuated using the service popuation of each pipe at the end of the design period, the average water consumption per person per day, and the return factor. It shoud be noted that the tria soutions obtained by the appication of the proposed formuations may vioate some of the constraints of probem. To encourage the ants to make decision eading to feasibe soutions, a higher cost is associated to the soutions that vioate the probem constraints defined by Eqs. (2) to (11). This may be done via the 21

22 use of a penaty method in which the tota cost of the probem is considered as the sum of the probem cost and a penaty cost as: F p G F CSV (17) p g 1 g Where, Fp = penaized objective function; F = origina objective function defined by Eq.1; CSV g = a measure of the vioation of constraint g; G= tota number of constraints and p = the penaty constant assumed arge enough so that any infeasibe soution has a tota cost greater than that of any feasibe soution. The penaized term in Eq. (17) wi be zero for feasibe soutions. 5. Resuts and Discussions Performance of the proposed agorithms is now tested against three hypothetica test exampes of sanitary sewer network design. The networks required for three quadrange 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 popuation of the area is assumed to be uniformy distributed over the area with the vaue of 2500 and 4000 person per hectare at the beginning and the end of the design period, respectivey. The average water consumption per person per day at the beginning and the end of the design period is taken as 250 it ( day * cap). The coefficients of the maximum and minimum sewer fow rate are assumed to be constant and equa to 2.8 and 0.6, respectivey. Note that the simpifying assumption of constant coefficients of the maximum and minimum sewer fow rate does not affect the generaity of the proposed agorithm since it ony affect the design fow rate for each pipe of a given ayout. The 22

23 return factor used for cacuation the sewer discharge of area is assumed to be 0.8. The geometry of the area aong with the ground eevation at the benchmark point is shown in Figure 4 in which a uniform decrease of the ground eevation from upstream to the ocation of the wastewater treatment pants and from the center to the right and eft of the zone is assumed. Three different base ayouts are considered with increasing scaes to assess the abiity of the proposed methods to sove sma, medium and arge scae sewer network design probems. The first test exampe is considered to have 9 nodes and 12 edges whie the second test exampe has 25 nodes and 40 edges and finay the third test exampe has 81 nodes and 144 edges as shown in Figure 5(a) to 5(c). A the networks are supposed to deiver the coected sewer to two treatment pant of fixed eevation ocated at the bottom corner of the area. The set of diameters ranging from 100 mm up to 1500 mm with an interva of 50 mm from 100 mm to 1000 mm and an interva of 100 mm from 1000 mm to 1500 mm is used as the set of commerciay avaiabe pipe diameters for a the pipes. The pipes engths of three networks are assumed to be constant and equa to 100 meter. Since the test exampes are hypothetica, the numerica vaues of the other parameters used to sove the probems are taken from other simiar works such as those by Afshar et a. [9] and by Diogo and Graveto [14] as foows: Manning coefficient, n = 0.015; Maximum cover depth, Emax = 10 (meter); Minimum cover depth, Emin = 2.5 (meter); Maximum reative fow depth, = 0.83; Minimum reative fow max depth, min =0.1; Sef-ceaning sewer fow veocity, Vcean = 0.75( m s ); and Maximum sewer fow veocity, Vmax = 6 ( m ). s Here the foowing expicit reation is used for the pipe instaation and manhoe costs which are a modified form of the one used in Afshar et a. [9]: 23

24 K K p m 10.93e 41.46h 3.43d m E E 1.47 d (18) Three test exampes are soved using proposed formuations using MMAS and the resut are presented and compared. A set of preiminary runs are first conducted to find the proper vaues of MMAS parameters for each test exampe as shown in Tabe 1. The resuts are obtained using 500, 1000 and 2000 iterations and coony size of 50, 100, 200 amounting to 25000, and 400,000 function evauations for test exampe I, II and III, respectivey. Tabe 2 shows the resuts of 10 runs carried out using different randomy generated initia guess for the test exampes aong with the scaed standard deviation, standard deviation of the fina soution costs produced in ten runs scaed by the average soution cost, and the number of feasibe fina soutions obtained on a 3 GHZ Pentium PC and CPU. It is ceary seen from Tabe 2 that a measures of the quaity of the fina soutions such as the minimum cost, maximum coat, average cost, and the scaed standard deviation representing the sensitivity of the method to initia guesses are improved when using ACOA-TGA compared to other formuations in which the ayout is created in an ad-hoc manner. It is particuary seen that the number of fina feasibe soution created by the ACOA-TGA is aways greater than those of ACOAs in which an ad-hoc engineering concept is used for ayout determination. It is worth noting that ACOA2 has been abe to outperform the ACOA1 regarding the quaity of the soution and the number of fina feasibe soutions due to the fact that soution constructed by the ACOA2 wi never be infeasibe regarding the minimum noda cover depth eading to smaer search space for the method. In fact, ACOA2 has been abe to produce near-optima soutions for a three 24

25 cases whie being outperformed by the ACOA-TGA regarding the number of fina feasibe soutions. It is worth noting that based on the aowabe considered cover depths, [1, 10], the manhoe heights of the fina 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 exampe I, II and III, respectivey. The reason for such arge manhoe depths is twofod: First, the operationa cost of the network incuding the maintenance of the manhoes is not considered in this work and the optimization is carried our ony based on the construction cost. Second, the manhoe cost defined in Eq. (18) apparenty constitutes a sma part of the tota cost as evident from the fina soutions. For exampe, the manhoe costs of the fina design obtained by ACOA-TGA are about , and for test exampe I, II and III, respectivey, which are about 7%, 5% and 4% of the tota cost vaues. Figures 6, 7 and 8 show the optima tree-ike ayout for the test exampes I, II and III, respectivey, obtained using ACOA-TGA. Tabes 3, 4 and 5 show the optima characteristics of the networks obtained for test exampe I, II and III, respectivey, using ACOA-TGA. Figure 9 shows convergence curves of the minimum soution costs obtained in ten runs using proposed methods for test exampe II indicating superior performance of the ACOA-TGA compared to other agorithms. First, the soution cost of ACOA-TGA method remains ower than those of other method during the evoution process eading to ower cost fina soution. Second, the ACOA-TGA soutions become feasibe earier than those of other agorithms indicated by the fact that the ACOA-TGA convergence curve appear to the eft of the other curves. 25

26 Finay, it is worth noting that athough the test exampes considered are symmetric, however, no use of symmetry has been made to sove these probems. The asymmetrica cases can, therefore, be easiy considered and soved in the same manner without any specia arrangements. 6. Concuding Remarks In this paper, two different ACOA was proposed for the efficient ayout and pipe size determination of sanitary sewer network out of a base ayout incuding a possibe inks of the network is avaiabe. In the first formuation, ACOA was used in a conventiona manner for pipe size optimization whie an ad-hoc engineering concept was used for the ayout determination. In the second formuation, ACOA equipped with TGA was used to simutaneousy determine both the ayout and pipe sizes of the network. The TGA was used to construct feasibe tree-ike ayouts out of the base ayout defined for the sewer network, whie the ACOA was used to optimay determine the pipe diameters of the constructed ayout. Proposed formuations were used to sove a hypothetica test exampe considered of different scaes and the resuts were presented and compared. The resuts indicated the abiity of the proposed methods and in particuar the ACOA-TGA approach to optimay sove the probem of ayout and size determination of sewer networks. Whie a proposed agorithms showed good performance in soving the probems under consideration, the ACOA-TGA agorithm was shown to produce better resuts and to be ess sensitive to the randomy generated initia guess required to start the soution process represented by the scaed standard deviation of the soutions produced in ten different runs. Furthermore, the proposed ACOA-TGA method was shown to enjoy higher success 26

27 rate indicated by the number of fina feasibe soution in particuar for the probems of arger scae. References [1] Afshar MH. Appication of a max min ant system to joint ayout and size optimization of pipe networks. Engineering Optimization 2006;38(3): [2] Afshar MH. A parameter free Continuous Ant Coony Optimization Agorithm for the optima design of storm sewer networks: Constrained and unconstrained approach. Advances in Engineering Software 2010;41: [3] Afshar MH, Marino MA. Appication of an ant agorithm for ayout optimization of tree networks. Engineering Optimization 2006;38(3): [4] Afshar MH, Afshar A, Marino MA, Darbandi AAS. Hydrograph-based storm sewer design optimization by genetic agorithm. Canadian Journa civi Engineering 2006;33(3): [5] Afshar MH. Evauation of Seection Agorithms for Simutaneous Layout and Pipe Size Optimization of Water Distribution Networks. Scientia Iranica 2007;14(1): [6] Afshar MH. Partiay constrained ant coony optimization agorithm for the soution of constrained optimization probems: Appication to storm water network design. Advances in Water Resources 2007;30(4): [7] Afshar MH. Layout and size optimization of tree-ike pipe networks by incrementa soution buiding ants. Canadian Journa civi Engineering 2008;35:

28 [8] Afshar MH, Moeini R. Partiay and Fuy Constrained Ant Agorithms for the Optima Soution of Large Scae Reservoir Operation probems. J. Water Resource Management 2008; 22(1): [9] Afshar MH, Shahidi M, Rohania M, Sargozaei M. Appication of ceuar automata to sewer network optimization probems. Scientia Iranica. Transactions A: Civi Engineering 2011; 18 (3): [10] Argaman Y, Shamir U, Spivak E. Design of Optima Sewerage Systems. Journa of the Environmenta Engineering Division 1973;99(5): [11] Botrous A, E-Hattab I, Dahab M. Design of wastewater coection networks using dynamic programming optimization technique. In: ASCE Nat. Conf. on Environmenta and pipeine Engineering, Kansas City, MO, United States; p [12] Cembrowicz RG. Evoution strategies and genetic agorithms in water suppy and waste water systems design. In: Water Resources and Distribution, edited by Bain W.R. et a., Southampton, United Kingdom; p [13] Desher DP, Davis PK. Designing sanitary sewers with microcomputers. Journa of Environmenta Engineering 1986;112(6): [14] Diogo AF, Graveto VM. Optima Layout of Sewer Systems: A Deterministic versus a Stochastic Mode, Journa of Hydrauic Engineering 2006;132(9): [15] Diogo AF, Waters GA, de Sousa ER, Graveto VM. Three-dimensiona optimization of urban drainage systems. Computer-Aided Civi and Infrastructure Engineering 2000;15(6): [16] Eimam AA, Charaambous C, Ghobria FH. Optimum design of arge sewer networks. J Environ Engineering 1989;115(6):

29 [17] Guo Y. Sewer Network Optima Design Based on Ceuar Automata Principes. In: 2005 XXXI IAHR Congress, Seou, Korea; p [18] Guo Y, Waters GA, Khu ST, Keedwe E. Optima Design of Sewer Networks using hybrid ceuar automata and genetic agorithm. In: IWA Word Water Congress, Beijing, China; [19] Hassani AM, Dandy GC. Optima ayout and hydrauic design of branched networks using genetic agorithms. Appied Engineering in Agricuture 2005;21(1): [20] Heaney JP, Wright LT, Sampe D, Fied R, Fan CY. Innovative methods for the optimization of gravity storm sewer design. In: 8th internationa conference on urban storm drainage, Sydney, Austraia; p [21] Jabbari I, Afshar A. Optimum ayout and design of a water suppy ine. Hydrauic Information Management 2002;52: [22] Jang SH. Urban Storm Sewer Optima Layout Design Mode by DDDP Technique, In: 2006 Asia Oceania Geosciences Society, AOGS 2006, Singapore; [23] Izquierdo J, Montavo I, Perez R, Fuertes VS. Design optimization of wastewater coection networks by PSO. Computers and Mathematics with Appications 2008;56(3): [24] Kukarni VS, Khanna P. Pumped wastewater coection systems optimization. J Environ Engineering 1985;111(5): [25] Lejano RP. Optimizing the ayout and design of branched pipeine water distribution systems. Irrigation and Drainage Systems 2006;20: [26] Li G, Matthew RGS. New approach for optimization of urban drainage systems. ASCE Journa of Environmenta Engineering 1990;116(5):

30 [27] Liang LY, Thompson RG, Young DM. Optimising the design of sewer networks using genetic agorithms and tabu search. Engineering Construction Architectura Management 2004;11(2): [28] Martin W. Optima design of water conveyance systems. J. of the Hydrauics Div. 1980;106(9): [29] Mays LW, Wenze HG. Optima Design of Mutieve Branching Sewer Systems. Water Resource Research 1976;12(5): [30] Morgan DR, Gouter IC. Least cost ayout and design of ooped water distribution systems. In: Ninth Int. Symposium on Urban Hydroogy, Hydrauics and Sediment Contro, University of Kentucky, Lexington, KY, USA; p [31] Pan TC, Kao JJ. GA-QP Mode to Optimize Sewer System Design. Journa of environmenta engineering 2009;135(1): [32] Swamee PK. Design of Sewer Line. Journa of Environmenta Engineering 2001;127(9): [33] Tempeman AB, Waters GA. Optima design of storm water drainage networks for roads. In: Inst. of Civi Engineers, London, 1979;67: [34] Wash S, Brown LC. Least cost method for sewer design. J. Environmenta Engineering Division 1973;99(3): [35] Waters GA. The design of the optima ayout for a sewer network. Engineering Optimization 1985;9(1): [36] Yen BC, Cheng ST, Jun BH, Voohees ML, Wenze HG. Iinois east cost sewer system design mode. User s guide, Department of Civi Engineering, University of Texas at Austin,

31 J J-1 2 j=1 d1 di d dn Figure1. Probem graph used for ACOA1 or ACOA2. (200) (250) (250) (300) (350) (400) (a) Typica Base ayout (b) Diameter determination [2] (200) [2] (200) (250) (250) (250) (250) [1] (300) [1] (300) (350) (400) (350) (400) [0] (c) Noda ranking (d) Cut determination Figure2. Different steps of construction tree-ike ayout for typica base ayout using ACOA1 or ACOA2. 31

32 4 (5) (6) (4) 5 (3) 2 3 (1) (2) 1 (a) Typica Base ayout [D 4, D 5] 5 [D 5, D 6] 4 3 [D 4, D 5] 4 [D 4, D 6] 5 [D 2, D 3, D 5] 2 [D 2, D 3, D 5] [D 2, D 3, D 6] 3 [D 4, D 6] 5 1 [D 1, D 2] [D 1, D 2] 4 [D 2, D 3, D 6] [D 4, D 5] 2 [D 4, D 5] [D 2, D 3, D 4] [D 5, D 6] [D 4, D 6] [D 1, D 3, D 4] [D 1, D 3, D 4] [D 1, D 3, D 6] 2 [D 5, D 6] 4 5 [D 1, D 3, D 6] 4 [D 1, D 3, D 5] 2 d1 d2 d3 d4 d5 (b) Graph representation Figure 3. Graph representation of the typica probem for ACOA-TGA. 32

33 1000 S=2% S=2% S=2% S=2% W.T.P W.T.P W.T.P W.T.P W.T.P Figure 4. Geometry of the areas used for three test exampes. W.T.P 33

34 [36] 25 [40] 24 [39] [35] 23 [38] 22 [37] 21 [34] [33] [32] 20 [31] 19 [30] 18 [29] 17 [28] 16 [27] [26] [25] [24] [23] 9 [12] 8 [11] 7 15 [22] 14 [21] 13 [20] 12 [19] 11 [10] [9] [8] [18] [17] [16] [15] [14] 6 [7] 5 [6] 4 10 [13] 9 [12] 8 [11] 7 [10] 6 [5] [4] [3] 3 [2] [1] 1 W.T.P 2 W.T.P 5(a): Test exampe (I) [9] [8] [7] [6] [5] 5 [4] 4 [3] 3 [2] 2 [1] 1 W.T.P 5(b): Test exampe (II) W.T.P [136] 81 [144] 80 [143] 79 [142] 78 [141] 77 [135] [134] [133] [132] [140] 76 [139] 75 [138] 74 [137] 73 [131] [130] [129] [128] 72 [127] 71 [126] 70 [125] 69 [124] 68 [123] 67 [122] 66 [121] 65 [120] 64 [119] [118] [117] [116] [115] [114] [113] [112] [111] 63 [110] 62 [109] 61 [108] 60 [107] 59 [106] 58 [105] 57 [104] 56 [103] 55 0 [102] [101] [100] [99] [98] [97] [96] [95] [94] 54 [93] 53 [92] 52 [91] 51 [90] 50 [89] 49 [88] 48 [87] 47 [86] 46 [85] [84] [83] [82] [81] [80] [79] [78] [77] 45 [76] 44 [75] 43 [74] 42 [73] 41 [72] 40 [71] 39 [70] 38 [69] 37 [68] [67] [66] [65] [64] [63] [62] [61] [60] 36 [59] 35 [58] 34 [57] 33 [56] 32 [55] 31 [54] 30 [53] 29 [52] 28 [51] [50] [49] [48] [47] [46] [45] [44] [43] 27 [42] 26 [41] 25 [40] 24 [39] 23 [38] 22 [37] 21 [36] 20 [35] 19 [34] [33] [32] [31] [30] [29] [28] [27] [26] 18 [25] 17 [24] 16 [23] 15 [22] 14 [21] 13 [20] 12 [19] 11 [18] 10 [17] [16] [15] [14] [13] [12] [11] [10] [9] 9 [8] 8 [7] 7 [6] 6 [5] 5 [4] 4 [3] 3 [2] 2 [1] 1 W.T.P W.T.P 5(c): Test exampe (III) Figure 5. Base ayouts of three proposed test exampes. 34