Multi-objective optimization of water supply network rehabilitation

Transcription

1 Proceedngs of the 2009 IEEE Internatonal Conference on Systems, Man, and Cybernetcs San Antono, TX, USA - October 2009 Mult-obectve optmzaton of water supply network rehabltaton Wenyan Wu Faculty of Computng, Engneerng and Technology Staffordshre Unversty, Beaconsde Stafford ST 18 0DF, UK Emal: w.wu@staffs.ac.uk Abstract Water network rehabltaton s a complex problem, and many facets should be concerned n the solvng process. It s a dscrete varables, non-lnear, mult-obectve optmal problem. An optmzaton approach s dscussed n ths paper by transformng the hydraulc constrants nto obectve functons of optmzaton model of water supply network rehabltaton problem. The non-domnated sortng Genetc Algorthm-II (NSGA-II) was adopted to solve the altered multobectve optmal problem. The ntroducton of NSGA-II for water supply network optmal rehabltaton problem results n solvng the conflct between one ftness value of standard genetc algorthm (SGA) and mult-obectves of rehabltaton problem. Moreover, t benefts to control the uncertantes brought by usng weghtng coeffcents or punsh functons n conventonal methods. In order to accelerate the convergence speed of populaton, ths paper ntroduces the artfcal nducement mutaton (AIM). It not only mproves the convergence speed, but also mproves the ratonalty and feasblty of solutons. Keywords water supply system; water supply network; optmal rehabltaton; mult-obectve; non-domnated sortng genetc algorthm (NSGA) I. INTRODUCTION Optmal rehabltaton of water supply network has been a man research subect n water supply area for a long tme. Mantenance and rehabltaton are long-term tasks after water utltes have been bult. The most mportant and hardest part of mantenance and rehabltaton obs n water supply system s water supply network rehabltaton. Therefore how to mantenance and rehabltate water supply network feasbly, economcally s essental ssue to managers of water supply companes. Water network rehabltaton s a complex problem, and many facets should be concerned n the solvng process. It s dscrete varables, non-lnear, mult-obectve optmal problem [1]. The concepts of buldng decson models for ths problem are vared, the man methods are: general rehabltaton gudes, prortzaton models and optmzaton models [2]. The methods of rehabltaton gudes and prortzaton models have been appled n the early perod. Lmted by theoretcal bass and algorthm level, they cannot evaluate rehabltaton schemes n a scale of whole network. Several obectves were used as the crtera for mans rehabltaton, such as net present value, crtcal break rate [3] and cost X Jn Department of Muncpal Engneerng, Wuhan Unversty of Technology, Wuhan HuBe Chna Jnlang Gao School of Muncpal and Envronment Engneerng, Harbn Insttute of Technology, Po box 2624,Harbn, HeLongJang, Chna savngs n pumpng and energy [4]. These methods analyze the ppes n solaton from water supply network. It s dffcult to measure the mprovement of rehabltaton. Wth the development of optmzaton theory and modelng technology n water supply system, usng a more comprehensve and detal optmal model to solve rehabltaton problem becomes doable.. Optmzaton technques consder the nteracton of each man wth the system as a whole, whch enable both the performance and the cost of the rehabltated system to play a role n the formulaton of the rehabltaton program [5][6]. It allows for the trade-off between system performance and cost of rehabltaton. However, such technques requre large numbers of tral evaluatons to obtan near-global optmal solutons. By usng models of water network system and new optmzaton methods (SGA, partcle swarm optmzaton (PSO), NSGA, NSGA-II), solvng these optmzaton model of rehabltaton becomes possble. II. REHABILITATION FORMATION A. Optomalzaton model for rehabltaton Currently the popular optmzaton model for rehabltaton of water supply network s usng mnmzaton of rehabltaton cost and energy cost per year n the nvestment perod as obectve and hydraulc performance of network as constrants. Wth ths concept, the general form of rehabltaton optmzaton model can be expressed by equatons below [7]. Obectve m α Mn. W = ( P + ) ( a + bd ) L 100 N 3 H, Q, ( r ET ) η = 1 N, s T I c (1 + I c ) T (1 + I c ) 1 P = (2) Where where a, b, c are coeffcents n formula of ppe constructon cost; Dk s dameter (mm) of ppe k; E s electrcty prce n perod (Yuan/(kW h)); H s pressure of pump n perod ; I s norm yeld rate (%); Lk s length of ppe k; m s lft to save a rate of captal repars fund (%); N s set of ppes whch need be rehabltated; ns s set of pump statons; P s equvalent coeffcent; Q s flow of pump n (1) /09/$ IEEE 3634

2 perod I (L/s); r s coeffcent of pumpng energy n perod ; T s repayment perod of nvestment (year); t s tme n perod (h); s effcency of pump n perod (%) 1 = 2 3 Peak perod of electrcty consumpton Low perod of electrcty consumpton Normal perod of electrcty consumpton Constrants Contnuous equaton Q = 0 ( = 1,2,..., n ) (3) q V Energy balance equaton ( h ) l = 0 (4) where Q s nodal demand (L/s) of node ; q s ppe flow (L/s) from node to node ; V s adacent nodes set of node I; h s head loss of ppe whch from node to node ; l s loop number Node pressure constrant H mn H H max J (5) Ppe velocty constrant v mn v v max Ps (6) Ppe dameter standard constrant d Ds = { D1, D2,, D z} (7) where H s pressure of node ; Hmn s the mnmum servce pressure ; Hmax s the maxmum servce pressure ; J s node set of network. v s velocty of ppe (m/s); v max s upper boundary of velocty of ppe (m/s); v mn s lower boundary of velocty of ppe (m/s); PS s ppe set of network. D s dameter (mm) of ppe ; DS s avalable standard dameter set Ths optmzaton model s a sngle-obectve model wth mnmzaton of constructon cost and energy cost as obectve, and network performances and avalable dameters as constrants. In conventonal solvng methods, constrants wll transform as parts of obectve functon by weghtng method or -constrant method [8]. These transformatons enable the tradtonal algorthms to be used n solvng optmzaton models of water supply network rehabltaton. Unfortunately, the soluton obtaned by ths process largely depends on the values assgned to the weghtng factors used or the desgn of -constrant functon. Ths approach does not provde a dense spread of the Pareto ponts. So the best way to solve rehabltaton problem s to abstract the problem as a multobectve functons and solve these functons wth a true multobectve orented algorthm. B. Alternatve optomalzaton model for rehabltaton The concept of alteraton s regardng the constrants that are transformed as parts of obectve functon wth weghtng method or -constrant functon as solate obectve functons. So the constrants whch represent performance of water supply network should be expressed as obectve functon. Obectve functon of ppe load s calculated as the sum of the velocty shortfalls or excesses at ppes (), weghted by the ppe lengths (L ) and dameters (D, here use unt of meter): Mn. W2 = α ( Δv L D ) + β ( Δv L D ) (8) pm px Where pm s set of ppes wth velocty below the lower boundary of velocty; px s set of ppes wth velocty above the upper boundary of velocty; and, are weghts to allow dfferent emphass on velocty shortfalls or excess. Obectve functon of node pressure s calculated as the sum of the pressure shortfalls or excesses h at consumer nodes(), weghted by the nodal demands(q ) : m Mn. W3 = ρ ( Δh Q ) + σ ( Δh Q ) (9) x Where m s set of nodes wth pressure below the lower boundary pressure; x s set of nodes wth pressure above the upper boundary pressure; and are weghts to allow dfferent emphass on pressure shortfalls or excess. So the new mult-obectve optmzaton model can be expressed as functons below: Obectve functons m c Mn. W 1 = ( P + ) ( a + bd ) L N ( r ET ) H, Q η, = 1 N, s Mn. W2 = ( Δv L D ) + β( Δv L D ) pm px (10) α (11) m Mn. W3 = ρ ( Δh Q ) + σ ( Δh Q ) (12) x where ρ, σ are weght coeffcents; Δh s node pressure shortfalls or excesses wth mnmum servce pressure ; Q s nodal demand (L/s); Jm s set of nodes wth pressure below the mnmum servce pressure; Jx s set of nodes wth pressure above mnmum servce pressure. constrants V Q = 0 ( = 1,2,..., n ) (13) q ( h ) l = 0 (14) d D = D, D,, D } (15) { 1 2 z III. SOLUTION PROCEDUARE In case of mult-obectve optmzaton, nstead of obtanng a unque optmal soluton, a set of equally good (non-domnatng) optmal solutons s usually obtaned (Pareto sets). Wthn a Pareto set, one obectve functon mproves whle the other deterorates. In absence of any other hgh level addtonal nformaton, a decson maker normally cannot choose any one of these non-domnant optmal solutons snce 3635

3 all of them are equally compettve and none of them can domnate each other. Several methods: goal attanment method, -constrant method, versons of NSGA and NSGA-II [9] avalable to solve mult-obectve optmzaton problems, NSGA-II[10][11] s used here to obtan the Pareto set. Use of penalty functon s a very popular way of handlng constrants. But tunng of the penalty parameter appearng n the penalty functon s very tme consumng and normally performed on the bass of tral and error. Unless tuned properly, one may get msdrected totally n the search space. NSGA-II based constrant-handlng technque, allows one to get rd of the above stated problem of penalty functon. A. Codng The code strng was made wth all rehabltated ppes dameter code shown n Table1 and Table2. Suppose that the IDs of rehabltated ppe are 1,2,3,4,5. Table 1 Code of standard ppe dameters Table 1 Code of standard ppe dameters Dameters code Dameters code Table2 code of chromosome of ndvduals of ntal generaton ndex Ppes dameter scheme( Chromosome order s ppe1-ppe5) Code 1 800,1800,1400,800,100 7,14,11,7, ,2000,500,300,400 15,15,4,2, ,1600,1000,700,1400 3,13,9,6, ,2000,200,200,1800 2,15,1,1, ,2200,400,1500,100 11,16,3,12, ,1800,900,800,900 15,14,8,7, ,2000,100,300,1400 6,15,0,2, ,200,1200,1000,800 15,1,10,9, ,1800,2000,1000,100 2,14,15,9, ,2200,900,400,100 10,16,8,3,0 B. Selcton operaton Dsposal of obectve values The three obectve functons of rehabltaton model are all mnmzaton functons, n non-domnated sortng, the ndvduals that have the larger obectve values wll domnate the one wth smaller obectve values. So some dsposal should be done to obectve values and make the better ndvdual n the preferental rank. The dsposal of obectve values can be expressed wth the functon below: W = Const /( W + Const ), (16) where W s orgnal obectve value; Const, Const are constants varyng wth dfferent obectves. Indvdual sortng The NSGA-II non-domnated sortng process s used a fast non-domnated sortng approach. Indvduals wll be sorted accordng to two parameters: non-domnated rank and crowdng dstance. After obtan the non-domnated sortng rank and crowdng-dstance of ndvduals, ndvduals can be sorted by Crowded Comparson Operator (CCO). The CCO gudes the selecton process at the varous stages of the algorthm toward a unformly spread-out Pareto optmal front. Selecton operator In ths paper roulette wheel selecton was selected as the selecton method, whch t benefts for selectng potentally useful solutons for recombnaton. In the selecton process, the ftness value of each ndvdual was calculated frstly, and then transfers these ftness values to selecton probabltes, whose transfer functon s as below Pop _ sze p = f f. (17) k k = 1 The ftness value of ndvdual f s transferred to selecton probablty p. The cumulatve probablty (p k ) of kth ndvdual can be obtaned by addng the ndvdual probabltes startng from top of the lst tll the kth member. The kth ndvdual s represented by the cumulatve probablty value between p k1 and p k. C. Crossover operaton Crossover operator s responsble for searchng of new ndvduals, whch could possbly have better ftness. There are also several crossover methods: Here we use the one pont crossover method as the crossover operator D. Mutaton operaton Snce mutaton operator has the random attrbute, although t can acheve the goal of dversty preservaton, t also tampers wth the constrngency of genetc algorthm. Sometme the bad gene that mutaton operator brought n wll be gotten rd of by many generatons. In order to brngng beneft gene n the mutaton operaton, a new mutaton method: artfcal nducement mutaton (AIM) was ntroduced n the NSGA-II n ths paper. Ths operator can steer the populaton convergence to the feld of feasble solutons acceleratng, and then use normal mutaton operator searchng for the best soluton n the feasble solutons. For the optmzaton model of rehabltaton, the goal of AIM s to make the selected dameters of rehabltated ppes follow the drecton of meetng the constrant of ppe velocty, untl the solutons converge to the feasble feld. 3636

4 gve results generate ntal generaton calculate obectve value of ndvduals W W1 W2 YES meet termnate condton NO non-domnated sortng and calculatng crowdeddstance sortngwth Crowded-Comparson Operator andcalculatngftness valueofndvduals roulette wheel selecton one pont crossover Table 4 Data of nodes wth low pressure Node ID elevaton demand (L/S) pressure HGL The postons of chock ppes and lower pressure nodes are demonstrated n fgure 2. YES count(esppes)>1 NO AIM operaton one-pont random mutaton operaton AIM operator Fgure 1 Flow chart of NSGA-II wth AIM IV. CASE STUDY A. Introducton The example network [12] wth the ncrease of water consumpton, the phenomenon of over fast velocty, hgh hydraulc slope and lower node pressure has appeared. The chock ppes and lower pressure nodes are shown n table 3 and table 4. Table 3 Data of chock ppes n case network Ppe ID Da (mm) length hydraulc slope (m/km) velocty (m/s) Fgure 2 Poston of chock ppes and low pressure nodes n case network The optmzaton rehabltaton model of case network was solved by NSGA-II wthout AIM operator and wth AIM operator respectvely. Snce AIM s always beneft to populaton, so n the algorthm of NSGA-II wth AIM operator, ts probablty s 1, means use AIM uncondtonal. B. Results and dscusson 3D non-domnated Pareto optmal front n ntal, 100 and 200 generatons of NSGA-II wthout AIM are shown n Fg 3.(a)(b)(c). The detal of ndvduals of non-domnated Pareto optmal front n the 200th generaton of NSGA-II wthout AIM s shown n Table 5. It can be seen from Fg. 3(a) n ntal generaton only 4 non-domnated Pareto optmal solutons were obtaned. Wth successve generatons the domnated solutons were elmnated and replaced by better solutons so that the number of non-domnated Pareto optmal solutons ncreased to 37 (there are cases that some ndvduals are superposton) after 200 generatons. It s also seen that wth the ncrease n the number of generatons, better solutons are obtaned, for example each evaluaton value of ndvduals extends follow the postve drecton. And n generaton 100 node pressure evaluaton value acheves ts the best, whch means all nodes n the network meet the pressure constrant 3637

5 From Table 5 t can be seen that the rehabltaton schemes was mproved n some extent. But t s found that although the node pressure constrant was met, there are stll some ppes, whch do not satsfed the velocty constrant. So all these solutons are not the feasble solutons, although the cost was cheap, they can not be adopted. Ths s manly because that although new dameters can be brought n by mutaton operaton, the mutaton probablty s low and mutaton operaton has a random attrbute, so the searchng scope was lmted, and the constrngency of algorthm was not good. If we want to search n a larger soluton scope, a larger populaton sze and generatons should be assgned, and also the qualty of ndvduals of ntal generaton s very mportant. But all these operatons wll result n the very tme consumng stuaton. Ths shortcomng can be solved by the ntroducton of AIM. 3D non-domnated Pareto optmal front n ntal, 10, 50 generatons of NSGA-II wth AIM are shown n Fg 4 (a)(b)(c) respectvely. The detal of ndvduals of non-domnated Pareto optmal front n the 50th generaton wth AIM s shown n Table 6. Fgure 3(a)(b)(c) 3D Pareto optmal front of ntal, 100 and 200 generaton of NSGA-II wthout AIM Table 5 Indvduals of Pareto optmal front n the 200th generaton of NAGA-II wthout AIM Number Number Index of nodes of ppes Rehabltaton Energy cost that do that not cost per year per year not meet meet (Yuan) Yuan pressure velocty constrant constrant Fgure.4 (a)(b)(c) 3D Pareto optmal front of ntal,10 th and 50th generatons of NSGA-II wth AIM It can be seen from Fg.4(c) that ndvduals whch meet both constrants of node pressure and ppe velocty were generated. Ths ndcates that artfcal mutaton does gude the 3638

6 populaton convergence to feasble solutons feld, acceleratng and searchng best solutons n feasble feld wth rest of evoluton process Table 6 Indvduals of Pareto optmal front of the 50th generaton of NSGA-II wth AIM ndex Rehabltaton cost per year (Yuan) Energy cost per year Yuan) Number of nodes that do not meet pressure constrant Number of ppes that not meet velocty constrant The hydraulc nformaton of rehabltated ppes n one of best soluton of NSGA-II wth AIM s shown n table 7, and the pressure of lower pressure nodes n the rehabltated network are shown n Table 8. Table 7 Data of chock ppes after rehabltaton ID Dameter Length hydraulc velocty (mm) slope (m/km) (m/s) Table 8 Data of lower pressure nodes after rehabltaton Node ID Pressure Node ID Pressure From Table 7 and 8, t can be seen that the chock ppes have been elmnated and pressure of the lower pressure nodes has been mproved to an acceptable level. In ths rehabltaton scheme, the dameters of some chock ppes have not been modfed, ths s because after optmzaton, the dameters of other ppes has been magnfed, these ppes hydraulc condton also mproved, so these ppes do not need to be modfed. V. CONCLUSION A mult-obectve optmal model of water supply network rehabltaton was dscussed and solved wth NSGA-II and AIM n ths paper. By ntroducton of NSGA-II, the problem of mult-obectve of optmal rehabltaton model wth one ftness value of conventonal GA s solved. The shortcomngs that brought n by weghtng method or -constrant method have been elmnated. By ntroducton of artfcal nducement mutaton (AIM), the populaton s drected to feasble solutons feld rapdly, and searchng the best soluton n the feasble feld. So the convergence of algorthm has been mproved and can gve more feasble and better solutons. By comparng the results of two NSGA-IIs wth and wthout artfcal nducement mutaton n the case study, the advantage and feasblty of artfcal nducement mutaton are shown and evaluated. In fact there are several dfferent concepts n buldng mult-obectve optmal model for water supply network rehabltaton. NSGA- II and AIM can stll work on solvng other mult-obectve optmal models. REFERENCES [1] [Pu Yhu, Zhao Hong-bn, and Zhou Jan-hua (2003) Solve optmzaton rehabltaton model of water supply network wth genetc algorthm. WATER&WASTE WATER29(12),89-92 [2] Engelhardt, M.O., P.J. Skpworth, D.A. Savc, A.J. Saul, G.A. Walters (2000) Rehabltaton strateges for water dstrbuton networks: a lterature revew wth a UK perspectve. Urban Water [3] [3] Walsk, T. M.(1982b) Economc analyss for rehabltaton of water mans. Journal of Water Resources Plannng and Management Dvson ASCE,108(WR3), [4] [4] Walsk, T. M. (1987) Water supply system rehabltaton, New York: Task Commttee on Water Supply System Rehabltaton, ASCE [5] [5] Engelhardt, M. O.(1999) Development of a strategy for the optmum replacement of water mans. Ph.D., Department of Cvl and Envronmental Engneerng, Unversty of Adelade, [6] [6] Halhal, D., Walters, G. A., Ouzar, D., and Savc, D. A. (1997), Water network rehabltaton wth a structured messy genetc algorthm. Journal of Water Resources Plannng and Management, 123(3), [7] [7] Zhao Hong-bn (2003) Water network system theores and analyss, Chna Archtecture & Buldng Press [8] [8] L Duan, JB YangM. P. Bswal (1999) Quanttatve parametrc connectons between methods for generatng nonnferor solutons n mult-obectve optmzaton, European Journal of Operatonal Research,117(1), [9] [9] Khu, S.T., Keedwell, E., Introducng more choces (flexblty) n the upgradng of water dstrbuton networks: the New York cty tunnel network example. Engneerng Optmzaton, 37(3): [do:10 [10] [10] Deb, K., (2001a) Mult-obectve Optmzaton Usng Evolutonary Algorthms. Wley, Chchester, UK [11] [11] Deb, K., Pratap, A., Agarwal, S., Meyarvan, T., (2002) Fast and Eltst Multobectve Genetc Algorthms: NSGA-II. IEEE Transactons on Evolutonary Computaton 6 (2), [12] [12] Rossman, L. A. (1993) EPANET users manual, U.S. Envronment Protecton Agency, Cncnnat, Oho 3639