Optimal design of Sewer network using Cellular Automata

Transcription

1 SCIREA Journa of Hydrauic Engineering November 17, 2016 Voume 1, Issue1, October 2016 Optima design of Sewer networ using Ceuar Automata M. ROHANI PhD Student, Schoo of Civi Engineering, Iran Univ. of Science and Tech., Tehran, Iran, Emai: M. H. AFSHAR Associate Professor, Schoo of Civi Engineering, Iran University of Science and Technoogy, P.O. Box: , Narma, Tehran, Iran. Emai: Abstract. In this paper, the Genera Ceuar Automata (GCA) method is proposed for the optima design of sewer networ probems with the fixed ayout. The networ nodes and upstream and downstream pipe cover depths are considered as CA ces and ce states, respectivey, and the ins around each ce are taen into account as neighborhoods. The updating rue is derived mathematicay from the origina objective function of the probem. The proposed method is a genera and fexibe method for optimization of sewer networs as it can be used to optimay design both gravity and pumped networ due to the use of pipe noda cover depths as decision variabes. The proposed method is tested against three gravitationa sewer networs with different sizes and the 1

2 comparison of resuts with other methods such as Ceuar Automata, and Ant Coony Optimization Agorithm show the efficiency and effectiveness of the proposed method. Keywords: Pumped Sewer Networ, Ceuar Automata, Optimization methods. Introduction The soution of sewer networ optimization probems requires determination of pipe diameters and average pipe cover depths, minimizing the tota cost of sewer networ subject to operationa constraints. Different numerica optimization approaches have been introduced and appied to the optima design of sewer networs such as Linear Programming (LP) (Dajani and Gemme 1971, Froise and Burges 1978), Non-Linear Programming (P) (Price 1978), Dynamic Programming (DP) (Yen et a. 1984, and Kuarni and Khanna 1985), and Evoutionary Agorithms. In the past decades, evoutionary agorithms have been widey appied for the optima design of sewer systems due to their simpicity and fexibiity. Pan and Kao (2009) used Quadratic Programming (QP) with Genetic Agorithm (GA) to sove the sewer networ optimization probem for gravity and pumped aternatives within an acceptabe computationa time. Afshar (2010) appied the Continuous Ant Coony Optimization Agorithm (CACOA) for the optima design of gravity sewer networs. Yeh et a. (2011) appied Tabu Search (TS) and Simuated Anneaing (SA) for the optimization of sewer networ probems. Haghighi and Bahshipour (2012) deveoped an adaptive GA for the optima design of gravity and pumped sewer networs. Paumbo et a. (2014) proposed a genera method based on a standard GA for the optima design of urban drainage networs. Recenty, Ceuar Automata (CA) has been introduced to optimization probem. CA has four basic components, ce, ce state, neighborhood, and transition (updating) rue. Each ce has a finite possibe vaue caed ce state. The new states of a ces are defined simutaneousy using an updating rue, which is a function of previous state of the ce itsef and its neighborhoods. In the eary appications in water resource probems, CA was used to produce good initia popuations for a GA eading to improved performance of the GA (Keedwe and hu 2005, and Guo et a. 2007b). Then CA was appied as a stand-aone optimizer with the transition rues 2

3 derived by engineering judgment (Keedwe and Khu 2006a, 2006b; Guo et a. 2007a). Afshar et a. (2011) proposed a singe stage CA for the optima design of sewer networs in which the networ nodes were considered as the CA ces and the corresponding noda excavation depths as ce states. Afshar and Rohani (2012) extended the singe stage CA method of Afshar et a. (2011) into a two-stage CA method, caed Hybrid CA (HCA), with the pipe diameters and networ noda cover depths as decision variabes. However, these methods were unabe to design pumped sewer networs due to the restive choice of noda excavation depths as decision variabes. In this paper, Genera Ceuar Automata (GCA) method is proposed for design of pumped or gravity sewer networ with fixed ayout in which, pipe noda cover depths are considered as decision variabes. The nodes considered as the CA ces and upstream and downstream pipe cover depths as ce states and the ins around each ce are taen into account as neighborhoods. The CA updating rue is derived by requiring that the networ cost is minimized in the neighborhood of each ce. The proposed method is a genera and fexibe method for optimization of sewer networs as it can be used to optimay design both gravity and pumped networ due to the use of pipe noda cover depths as decision variabes. The GCA method is used to design three benchmar exampes and comparison the resuts with the existing ones shows the efficiency and effectiveness of the method to sove the sewer design optimization probems. Sewer networ Size Optimization Optima sewer networ design with a fixed ayout aims to find a cost-effective soution by determining the pipes diameters and sopes which minimizes the capita investment whist ensuring a good system performance under specific design criteria. The probem of sewer networ design for a fixed networ ayout can be formuated as: MinC networ Subject to: Cp NN Cm NN NN Cd NN Kdhd KppQ, hp Km hm NN Cpump 1 1 NN 1 L Kp D, H, H V V V 1,..., (2) min max min max 1,..., (3) 3 i j (1)

4 H min i j H, H H max 1,..., (4) S S S 1,..., (5) min max D D 1,..., (6) D D ' 1,..., (7) Where, C networ is the tota cost of the networ, Cp is the instaation cost of th pipe, Cm is the cost of th manhoe, Cd is the cost of drop instaed at the th node if required, Cpump represents the instaation and/or operation cost of the pump at the th node, is the number of pipes in the networ, NN is the number of nodes in the networ, L is the ength of th pipe, Kp is the unit cost of th pipe defined as a function of its diameter ( D ) and upstream and downstream noda cover depths of th i pipe ( H, H j ), Km is the cost of manhoe construction as a function of manhoe depth ( hm ), Kd is the cost of drop construction as a function of drop height ( hd ), and Kpp is the coefficient of pumping instaation and/or operation cost as a function of pumping discharge ( Q ) and pumping height ( hp ) Eqs. (2) to (7) represents the constraints of veocity, water-depth ratio, pipe noda cover depth, pipe sope, commerciay avaiabe pipe diameter, and progressive diameter for the sewer networ probem, respectivey, where, V is the veocity of th y pipe, D, y is the fow depth of th pipe, S is the sope of th pipe, D is the set of commerciay avaiabe pipe diameters, ' refers to the set of pipe ocated downstream of pipe, and and maximum parameters, respectivey. min, max are the aowabe minimum Genera Ceuar Automata (GCA) As it is mentioned before, CA has four basic components, ce, ce state, neighborhood, and updating rue, and appication of CA to any probem requires that these parameters are propery defined. In optimization probems, generay, ce states are decision variabes and depend on the defined ces. 4

5 In Genera Ceuar Automata (GCA) method, each node of the sewer networ is regarded as a ce and pipe noda cover depths are considered as the ce states (decision variabes). Choosing the pipe noda cover depths as decision variabes resuts in considering pumping station in the nodes that noda cover depth of connected pipes are not coincident. The set of the pipes connected to each node of the networ is considered as the ce neighborhood. The convergence criterion is met when the soution is fixed or repeated in the successive iteration. Decision variabes are updated through the updating rue. This rue is achieved by considering objective function on the ce and its neighborhoods. The objective function (Eq. 1), in the absence of drop cost and appying a penaty method for the satisfaction of veocity and pipe sope constraints defined by Eqs. (2) and (5), can be written as: MinC Cp NN 1 1 Cm CSVv CSVs 1 NN Cpump (8) Where, is the penaty parameters with arge enough positive vaue, and 1 CSVv, CSVs represent the vioation from the constraints of veocity and sope for each pipe, respectivey, CSVv 1 1 V min Vmax V V S, S CSVs 1 1 S min S max Cm Km, Cpump KppQ, hp i hm i Subject to: H min., i j L KpD,H, H Cp, i j H, H H max 1,..., (9) The pipe noda cover depths are cacuated in a manner that the objective function over the ce neighborhood ( ) is minimized. Since the constraint of water-depth ratio is considered in cacuation of pipe diameters when the optimization process is finished, this constraint is automaticay satisfied. Pipe sopes are obtained using the optima vaues of upstream and downstream pipe noda cover depths cacuated with CA and then for getting the pipe diameters, starting from the smaest diameter, and increased the pipe diameter unti the water-depth ratio attains its maximum vaue, which does not vioate the constraint of maximum water-depth ratio. The resuting diameter is then considered as the optima diameter for the corresponding pipe. Therefore, a the constraints of optimization probem are automaticay satisfied. 5

6 The objective function over the ce neighborhood is defined as: C Cp CSVv CSVs Cm Cpump (10) Minimization of the oca objective function of Eq. (10) with respect to pipe noda cover depths i j ( H, H 1,..., ) eads to the noninear system of equations to be soved with the Newton- Raphson method which resuts in the updated pipe noda cover depths: K 1 X F X H1, H2, H3,... H X X X (11) Where, K is the stiffness matrix, F is the right-hand side vector, is the noninear iteration index, F F H, and H functions of the is the change in the vaue of the ce state. F and K are both impicit which can be cacuated using the chain rue of differentiation and H Manning equation. This procedure is repeated for the ce under consideration unti the convergence is met and the process of updating is repeated for a ces of the networ. Test exampes In this section, the performance of GCA method is investigated by appying the mode to three hypothetica design probems with different sizes previousy proposed and used by Moeini and Afshar (2012 a,b) for the simutaneous ayout and size optimization of sewer networ using ACOA based methods. The optima ayouts obtained by Moeini and Afshar (2012 b), shown in Figure 1, is used here to assess the efficiency and effectiveness of GCA method. More detais and constraints of the sewer networs can be found in the wor of Moeini and Afshar (2012 a,b) and Rohani and Afshar (2014). The terms of pipe and manhoe construction costs are defined as (Moeini and Afshar, 2012 a,b): K K p h 10.93e 41.46h 3.43D m 0.012X X 1.47 D (12) Where, D is the pipe diameter (m), X is the buried depth (m), and h m is the depth of manhoe (m). These test probems are here soved using GCA method and the resuts are presented in Tabe 1 and compared with CA (Rohani and Afshar, 2014), HCA (Rohani and Afshar, 2014), and those 6

7 of ACOA-TGA (Moeini and Afshar, 2012 a) and CACOA-TGA (Moeini and Afshar, 2012 b) methods using a 2 MHz Pentium 4. It can be seen that the cost of the GCA method is near the optima soution obtained by other methods, whie requiring comparabe computationa effort. Since GCA method requires an initia guess for decision variabes of the probem, pipe noda cover depths, to start off the soution procedure, a sensitivity anaysis is carried out here to assess the sensitivity of the fina soution to the initia guess. Tabe 2 represents the maximum, minimum and average soution costs over 10 runs using different initia designs aong with the scaed standard deviation of the soutions defined as the ratio of the standard deviation to the average soution. This tabe emphasizes on the insensitivity of the CA methods to the initia popuation. Detais of the optima soution obtained by the GCA method for three sewer networs are aso shown in Tabe 3, 4, and 5, respectivey. Concuding remars In this paper, Genera Ceuar Automata approach was used for the optima soution of sewer networ design probems. Considering the pipe noda cover depths as decision variabes in the proposed method resuted in fexibiity of the method for optimization of both gravity and pumped sewer networs. The networ nodes and upstream and downstream pipe cover depths were considered as CA ces and ce states, respectivey, and the ins around each ce were taen into account as neighborhoods. The GCA method was used to sove three benchmar exampes in the iterature and the comparison of the resuts with other CA methods and two versions of Ant Coony Optimization Agorithm indicated the abiity and efficiency of the GCA method to produce near optima resuts. 7

8 Figures and Tabes 9 [12] 8 [11] [10] [9] 6 [7] 5 [6] 7 [8] 4 25 [36] 20 [40] 24 [39] [35] [31] 19 [30] [38] [37] [34] [33] 18 [29] 17 [28] [32] [5] 3 [4] [2] [1] [3] 1 [27] 15 [26] [22] 14 [21] [25] [24] 13 [20] 12 [19] [23] 11 W.T.P 2 (a) W.T.P [18] 10 [17] [13] 9 [12] [16] 8 [15] [11] 7 [10] [14] 6 [9] [8] [7] [6] [5] 5 [4] 4 [3] 3 [2] 2 [1] 1 W.T.P (b) W.T.P [136] 81 [144] [143] 80 [142] [135] [134] [133] [141] 77 [132] [140] 76 [131] [139] 75 [130] [138] 74 [129] [137] 73 [128] [127] [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 (c) Figure 1: Networ ayouts for three exampes: a) sma scae sewer networ, b) medium scae sewer networ, c) arge scae sewer networ. 8

9 Tabe 1: Optima networ cost obtained by different methods. Test Exampe Mode Cost Time (mii second) ACOA-TGA (Moeini and Afshar, 2012 a) CACOA-TGA (Moeini and Afshar, 2012 b) Sma Scae Networ CA (Rohani and Afshar 2014) HCA-Discrete (Rohani and Afshar 2014) HCA-Continuous (Rohani and Afshar 2014) GCA ACOA-TGA (Moeini and Afshar, 2012 a) CACOA-TGA (Moeini and Afshar, 2012 b) Medium Scae Networ CA (Rohani and Afshar 2014) HCA-Discrete (Rohani and Afshar 2014) HCA- Continuous (Rohani and Afshar 2014) GCA ACOA-TGA (Moeini and Afshar, 2012 a) CACOA-TGA (Moeini and Afshar, 2012 b) Large Scae Networ CA (Rohani and Afshar 2014) HCA-Discrete (Rohani and Afshar 2014) HCA- Continuous (Rohani and Afshar 2014) GCA

10 Tabe 2: Maximum, Minimum and Average soution costs over 10 runs. Cost Test Exampe Mode Minimu m Maximu m Averag e Scaed Standard Deviation ACOA-TGA (Moeini and Afshar, 2012 a) CACOA-TGA (Moeini and Afshar, 2012 b) Sma Scae Networ CA (Rohani and Afshar 2014) HCA-Discrete (Rohani and Afshar 2014) HCA-Continuous (Rohani and Afshar 2014) GCA ACOA-TGA (Moeini and Afshar, 2012 a) Medium Scae Networ CACOA-TGA (Moeini and Afshar, 2012 b) CA (Rohani and Afshar 2014) HCA-Discrete (Rohani and Afshar 2014) HCA- Continuous (Rohani and Afshar 2014) GCA ACOA-TGA (Moeini and Afshar, 2012 a) CACOA-TGA (Moeini and Afshar, 2012 b) Large Scae Networ CA (Rohani and Afshar 2014) HCA-Discrete (Rohani and Afshar 2014) HCA- Continuous (Rohani and Afshar 2014) GCA

11 Tabe 3: Resuts obtained from GCA method for the sma scae sewer networ. Cover Depth (m) Pipe D (mm) Upstream Downstream V (m/s) y/d

12 Tabe 4: Resuts obtained from GCA method for the medium scae sewer networ. Cover Depth (m) Cover Depth (m) Pipe D(mm) Upstream Downstream V (m/s) y/d Pipe D(mm) Upstream Downstream V (m/s) y/d

13 Tabe 5: Resuts obtained from GCA method for the arge scae sewer networ. Cover Depth (m) Cover Depth (m) Pipe D(mm) Upstream Downstream V (m/s) y/d Pipe D(mm) Upstream Downstream V (m/s) y/d

14 Cover Depth (m) Cover Depth (m) Pipe D(mm) Upstream Downstream V (m/s) y/d Pipe D(mm) Upstream Downstream V (m/s) y/d

15 Cover Depth (m) Cover Depth (m) Pipe D(mm) Upstream Downstream V (m/s) y/d Pipe D(mm) Upstream Downstream V (m/s) y/d

16 References: [1] M.H., Afshar, 2010, A parameter free Continuous Ant Coony Optimization Agorithm for the optima design of storm sewer networs: Constrained and unconstrained approach, Advances in Engineering Software, 41(2), pp [2] M.H., Afshar, M., Shahidi, M., Rohani, M., Sargozaei, 2011, Appication of ceuar automata to sewer networ optimization probems, Scientia Iranica A, 18 (3), pp [3] M.H., Afshar, M., Rohani, 2012, Optima design of sewer networs using ceuar automata-based hybrid methods: Discrete and continuous approaches, Engineering Optimization, 44(1), pp [4] J.S., Dajani, R.S., Gemme, 1971, Economics of wastewater coection networs, Civi Eng. Res. Rep. 43, Water Resour. Center, Univ. of I. At Urbana-Champaign, Urbana. [5] S., Froise, S.J., Burges, 1978, Least-cost design of urban drainage networs, J. Water Resources Panning and Management Division, ASCE, 104(1), pp [6] Y., Guo, G.A., Waters, S.T., Khu, E., Keedwe, 2007a, A nove ceuar automata based approach to storm sewer design, Engineering Optimization, 39 (3), pp [7] Y., Guo, E.C., Keedwe, G.A., Waters, S.T., Khu, 2007b, Hybridizing Ceuar Automata Principes and NSGAII for Muti-objective Design of Urban Water Networs, Evoutionary Muti-Criterion Optimization, pp [8] A., Haghighi, A.E., Bahshipour, 2012, Optimization of sewer networs using an adaptive genetic agorithm, Water Resources Management, 26(12), pp [9] E., Keedwe, S.T., Khu, 2005, Using ceuar automata to seed genetic agorithms for water distribution networ design probems, Engineering Appications of Artificia Inteigence, 18(4), pp [10] E., Keedwe, S.T., Khu, 2006a, Nove ceuar automata approach to optima water distribution networ design, Journa of Computing in Civi Engineering, 20(1), pp [11] E., Keedwe, S.T., Khu, 2006b, A nove evoutionary meta-heuristic for the muti-objective optimization of rea-word water distribution networs Engineering Optimization, 38(3), pp

17 [12] V.S., Kuarni, P., Khanna, 1985, Pumped wastewater coection systems optimization, J Environ Eng, ASCE, 111(5), pp [13] R., Moeini, M.H., Afshar, 2012a, Layout and size optimization of sanitary sewer networ using inteigent ants, Advances in Engineering Software, 51, pp [14] R., Moeini, M.H., Afshar, 2012b, Constrained Ant Coony Optimisation Agorithm for the ayout and size optimisation of sanitary sewer networs, Urban Water Journa, pp [15] T.C., Pan, J.J., Kao, 2009, GA-QP mode to optimize sewer system design, Journa of Environmenta Engineering, 135 (1), pp [16] A., Paumbo, L., Cimorei, C., Covei, L., Cozzoino, C., Mucherino, D., Pianese, 2014, Optima design of urban drainage networs, Civi Engineering and Environmenta Systems, 31(1), pp [17] R.K., Price, 1978, Design of storm water sewers for minimum construction cost, In Proc. 1 st Internationa Conference on Urban Strom Drainage, Southampton, United Kingdom, pp [18] M., Rohani, M.H., Afshar, 2014, Sewer Networs Optimization Using Ceuar Automata, Studies in Engineering and Technoogy, 1 (1), pp [19] B.C., Yen, S.T., Cheng, B.H., Jun, M.L., Voohees, H.G., Wenze, 1984, Iinois east cost sewer system design mode, User s guide, Department of Civi Engineering, University of Texas at Austin. [20] S.F., Yeh, C.W., Chu, Y.J., Chang, M.D., Lin, 2011, Appying tabu search and simuated anneaing to the optima design of sewer networs, Engineering Optimization, 43 (2), pp