COMPARISON OF METHODS OF PUMP SCHEDULING IN WATER SUPPLY SYSTEMS. G Waterworth, K Darbyshire

Transcription

1 COMPARISON OF MEHODS OF PUMP SCHEDULING IN WAER SUPPLY SYSEMS G Waerworh, K Darbyshire School of Engineering, Leeds Meropolian Universiy, Leeds, LS 3HE England g.waerworh@lmu.ac.uk, k.darbyshire@lmu.ac.uk. Absrac: In he domesic waer supply indusry, he reducion of pumping coss is a coninuing objecive. Wih he efficien scheduling of pumping operaions, i is considered ha % of he annual expendiure on energy and relaed coss may be saved. A ypical cos funcion will include all of he expendiure caused by he pumping process and also consider he elecrical cos of pumping aking ino accoun he various elecrical ariffs, as well as peak demand and pump swiching coss. Using only fixed speed pumps, i is possible o use an efficien dynamic programming based mehod, provided ha he sorage reservoir levels are known. Oher echniques ha are showing fruiful resuls in opimisaion are geneic programming and simulaed annealing. his paper compares hese mehods and discusses which is more appropriae in his ype of pump scheduling problem. INRODUCION A ypical problem in he waer supply indusry is he ransfer of waer beween inerconneced reservoir sysems using a leas cos operaion. he auhors are currenly engaged on a projec in collaboraion wih he waer indusry o invesigae cos-effecive conrol of waer ransfer using fixed speed pumps, which may be eiher on, or off. Making cerain assumpions regarding decoupling and simplificaion, allows us o consider he sysem as a source reservoir supplying a conrolled reservoir via a pumping saion and an equivalen pipe-line wih a waer demand from he conrolled reservoir which is assumed o be known in advance. he supply sysem considered in his paper is a secion of waer supply nework using four fixed speed pumps o supply a reservoir from a large capaciy source reservoir. he pumps may be swiched a hourly inervals, and he schedule is opimised over a 24-hour period. Sysem consrains are considered, which consis of upper and lower reservoir level consrains, and sar and finish reservoir level expecaions. he ariffs for day and nigh elecriciy consumpion are considered, bu peak ariff and swiching coss are ignored for he purpose of his sudy. he proposal considers he pump characerisics in erms of heir pumping capaciies and he amoun of elecriciy used. he scheme is used as a means of invesigaing a variey of echniques for he opimisaion of pumping o give minimum cos. 2 PROBLEM FORMULAION he pump-scheduling problem proposed by Mackle e al [] is used as a saring poin for seing up a model which can hen be addressed for opimizing. he same problem is discussed by Savic e al [2]. he problem a his sage considers he pumping capaciies of he fixed speed pumps and he amoun of elecriciy used per hour as shown in able. he sysem considered consiss of one waer able Pumping capaciies of he fixed speed pumps Pump Amoun of waer pumped in one hour [cu m] Amoun of elecriciy used in one hour [kwh] Pump 5 2 Pump2 5 3 Pump Pump4 5 8 disribuion reservoir, which is, supplied by four fixed speed pumps hrough a single waer main. he opimizaion period is se o one day, as hisoric paerns of he waer demand of an average day are

2 Demand cu m/hr commonly used for pump scheduling [3]. he ime inerval over which he elecriciy ariff srucure is repeaed is also modelled o be on 24 hours. As is common in many elecriciy supply sysems, he model incorporaes a cheap nigh and more expensive day ariff. For he problem used in his paper he day-ime ariff cos is se o be wice ha charged during he nigh. he period for which he higher day-ariff applies is, for convenience, aken as being from 8 h hrough o 2 h. Waer Demand Unlimied waer supply Reservoir x(k) pumps u(k) d(k) ime of Day Figure Waer Supply Sysem Schemaic Figure 2 Waer demand from reservoir he opimizaion period is divided ino inervals of one hour, i.e. he pumps can be eiher swiched on or off during any hour of he day. As each of he four pumps can be run during any ime inerval here are 6 possible combinaions of he pumps during each hour of he day. A schemaic of he sysem is shown in Figure where: u(k) is he oal quaniy of pumped waer (cu m/hr) d(k) is he waer demand in cu m/hr x(k) is he reservoir level x() is iniial reservoir level he consrains are xmin < x(k) < xmax - min and max allowable levels. he hourly demand is aken from Coulbeck [4] and is as shown in Figure 2. Elecriciy ariffs are 2.86p for peak ariff 8-2 hrough he day (8 hours) and.2p for off peak ariff 2-8 hrough he nigh (6 hours). oal reservoir capaciy is 25 cu m 3 OPIMISAION ECHNIQUES CONSIDERED he fundamenal aim of opimisaion is o esablish a cos funcion, which is a measure of he performance of some aspec of he process under consideraion. Classical mehods of opimisaion include linear programming, dynamic programming, hill climbing, saisical mehods such as simulaed annealing, and evoluionary programming mehods such as geneic algorihms. Linear and dynamic programming mehods make a sequence of decisions, which ogeher consiue an opimal policy. his approach lends iself well in principle o he pump-scheduling problem. Hill climbing and evoluionary mehods generae a deerminisic sequence of rial soluions based on he gradien of he cos funcion. 3. Dynamic Programming Dynamic programming was developed by Bellman o solve problems in which a sequence of decisions is required o be made. he mehod lends iself ideally, in heory, o he problem of pump scheduling. Bellmans saemen of he Principle of Opimaliy [5] is ha an opimal policy has he propery ha whaever he iniial sae and iniial decisions are, he remaining decisions mus consiue an opimal policy wih regard o he sae resuling from he firs decision. Le x be he sae a any ime. Le U be he conrol acion a ime. In his case conrol of he pumps is by pumps being eiher on or off. Le V ( x ) be he cos of reaching x a ime. Le g ( x, y ) be he cos of ransiion from sae x a ime (>). hen he minimum principle can be expressed as:

3 Reservoir Level min( V ( x )) min[ g U ( x, x ) min( V ( x ))] i.e. he minimum of (he cos of ransiion from x o x + minimum of geing o he poin x ) he minimum principle holds for any form of cos funcion and whaever he effec of he conrols on he sysem. Dynamic programming is, in effec, he repeaed applicaion of he minimum principle a a sequence of inervals. In he pump-scheduling problem considered, he sae of he sysem is aken as he volume of waer in he reservoir. In order o calculae he minimum cos, he reservoir is divided ino discree levels. he sages are aken as he hourly decisions made over he 24-hour period. [6]. In order o give sufficien accuracy in he calculaion, i is necessary o divide he reservoir ino abou 5 levels. Using four pumps wih herefore 6 combinaions, he number of possible ransiions a each sage is 6x5. For hourly decisions over he 24-hour period, he cos mus herefore be sored a 6x5x24. Clearly as more reservoirs and more pumps are considered, he ime and sorage requiremens of dynamic programming become very large. 3.. Opimisaion of Pumping using Dynamic Programming he classical approach for opimal scheduling of waer pumping is: Minimise{pumping cos+reamen cos} Subjec o: Waer nework equaions, pressure consrains a criical nodes, flow consrains in criical pipes, reservoir level consrains [7 ]. he resuling conrolled reservoir level over he 24-hour period is shown in Figure 3 wih he pump schedule in Figure Conrolled Reservoir Level Variaion - Dynamic Programming ime of Day Figure 3 Reservoir level wih opimal scheduling Figure 4 Pumps ON or OFF during 24 of pumping using dynamic programming hour period P, P2, P3 and P4 3.2 Simulaed Annealing he hear of he mehod of simulaed annealing is an analogy wih hermodynamics called annealing. his is he process by which liquids freeze and crysallise, or meals cool and anneal. A high emperaures, he molecules of a liquid move freely wih respec o one anoher. As he liquid is slowly cooled, hermal mobiliy reduces. he aoms line hemselves up and form a pure crysal which is compleely ordered. he crysal is he sae of minimum energy of he sysem. A each emperaure during he annealing process, slow cooling enables he sysem o achieve equilibrium. If he emperaure is lowered oo quickly, he sysem does no have sufficien ime o achieve equilibrium, and he resuling configuraion migh have many defecs in he form of high-energy, measable, locally opimal srucures. Naures minimisaion algorihm is based on he Bolzmann probabiliy disribuion, prob(e) ~ exp(-e/k)

4 Reservoir Level his describes he process by which a sysem in hermal equilibrium a a emperaure has is energy probabilisically disribued among all differen energy saes E. Even a low emperaures, here is a small chance of a sysem being in a high energy sae. his means ha here is a corresponding chance for he sysem o ge ou of a local minimum energy sae in favor of finding a beer, more global, minimum. he quaniy k is Bolzmann s consan and is a consan of naure, which relaes emperaure o energy. he original incorporaion of hese principles ino numerical calculaions was firs carried ou by Meropolis [8]. Consider a sysem which is assumed o change is configuraion from energy E o energy E2 wih probabiliy p = exp[-(e-e2)/k]. If E2 < E hen he sysem is arbirarily assigned a probabiliy p=, and he sysem will always ake his opion (i is a reducion in energy). If however E2 > E hen he scheme may ake his opion wih a probabiliy p. he scheme is one, which goes in he general direcion of reducing energy, bu someimes allowing an energy increase. he requiremens of a Meropolis algorihm are: A descripion of he possible sysem configuraions 2 A means of generaing random changes in he configuraion. hese are he opions presened o he sysem 3 An objecive funcion E, which is an analogue of he energy. he goal of he procedure is he minimisaion of his funcion 4 A conrol parameer which is he analogue of emperaure. he emperaure is lowered a each successive pass hrough he algorihm using an annealing schedule Opimisaion of Pumping using Simulaed Annealing As a problem in simulaed annealing, he pump opimisaion problem is handled as follows: Configuraion. he four pumps may be eiher on or off. his may be represened as a or a. Since we are considering he sae of hese four pumps on an hourly basis over 24 hours we have 4X24 = 96 possibiliies. 2 Random Rearrangemens. wo useful rearrangemens suggesed by Lin [9] are (a) o remove a secion of he possibles and replace hem in he opposie order and (b) o remove a secion and replace in a differen posiion. 3 Objecive Funcion. he cos funcion in he case considered is a combinaion of he elecriciy cos ogeher wih he penalies for no meeing he end reservoir level requiremen and for exceeding he upper and lower level limis. E = Sum of Elec coss a various ariffs + sum of exceed limis + final error 4 Annealing Schedule. his is arrived a by experimenaion. I may be appropriae o choose some random rearrangemens and use hem o deermine he range of values of delae ha will be obained. his hen allows he choice of wih saring value considerably larger han delae, and hen gradually reducing in muliplicaive seps unil eiher he limiing number of emperaure seps has been reached, or he number of successful reconfiguraions has reduced o zero [][]. he resuls of he scheduling opimisaion using simulaed annealing are shown in Figure 5 25 Conrolled Reservoir Level Variaion 2 max level 9% % 72% min level 5% ime of Day Figure 5 Reservoir level - resuls of he scheduling opimisaion using simulaed annealing Figure 6 Minimum esimaion problem similar o pump opimisaion scheme

5 f(x) 3.3 Geneic Algorihms Geneic algorihms (GAs) use a sochasic global search mehod, which mimics naural geneic evoluion. hey operae on a populaion of poenial soluions by applying he principle of survival of he fies o produce increasingly beer approximaions o he soluion. A each generaion, a new se of approximaions is generaed by he process of selecing individuals according o heir level of finess of he cos funcion in he problem in which hey are being used. his leads o he creaion of a se of individuals beer suied o he environmen in which hey exis han he individuals from which hey were creaed. o progress from one populaion o he nex, members of he curren populaion may be modified using reproducion, crossover, muaion and inversion. A sring of numbers is ofen used o represen he decision variables and he possible choices. In he case of he pumping saion considered in his invesigaion i is convenien o represen each of he four pumps by a or a as o wheher i is on or off. his leads o a convenien represenaion of he populaion over a 24-hour period by a binary sring word lengh of 4 x 24 (i.e. 96). he overall cos funcion used wih he pumping schedule includes he oal elecriciy cos of pumping over he 24 hour period ogeher wih penaly coss associaed wih violaing consrains such as exceeding reservoir levels and no meeing final level requiremens Opimisaion of Pumping using Geneic Algorihms he GA search firs generaes a family of iniial populaion using random seeding. A finess funcion is hen used o assess he performance of each of he individual members of he populaion. A proporion of hese are hen chosen according o heir relaive finess and recombined o produce he nex generaion. Geneic operaors such as recombinaion and muaion are used o manipulae he chromosomes on he assumpion ha cerain individual s genes produce, on average, fier individuals. Finally he objecive funcion is evaluaed, a finess value assigned o each individual, and he individuals seleced again for maing according o heir finess. he process coninues hrough subsequen generaions unil eiher a cerain number of generaions have been compleed or a mean deviaion in he populaion has been achieved [2][3][4][5]. A ypical cos funcion graph for a range of chromosomes is shown in Figure 6 wih successive esimaes of minimum shown as circles. his figure shows he seeming randomness of he problem wih a problem such as pump scheduling. he corresponding minimisaion is shown in Figure 7. he reservoir level wih GA pump opimisaion is shown in Figure generaion Figure 7 Cos funcion minimisaion Figure 8 Reservoir level using geneic algorihm pump opimisaion 4 CONCLUSIONS In comparing he hree differen approaches o opimisaion, dynamic programming is generally faser han he ohers, bu has he problem ha as he complexiy of he sysem increases, he size of he

6 sorage requiremens becomes excessive. I is for example no appropriae for pump opimisaion sysems wih more han wo reservoirs. Simulaed annealing provides a sub-opimal resul aking abou minues o run wih he pump opimisaion problem. he developmen of an appropriae cos funcion is worh some consideraion, and may lead o a beer opimal resul. Geneic algorihms are easy o use, bu again i is he cos funcion, which presens he problem. he algorihm ook approximaely 2 minues o run in his case. 5 REFERENCES Mackle G e al Applicaion of Geneic Algorihms o Pump Scheduling of Waer Supply In Conf. On Geneic Algorihms in Engineering Sysems pp Savic D A e al Muliobjecive Geneic Algorihms for Pump Scheduling in Waer Supply Evoluionary Compuing AISB In Wkshp Seleced Papers Springer-Varlag Coulbeck and Orr Developmen of an Ineracive Pump Scheduling Program for Opimised Conrol of Bulk Waer Supply Proc In Conf on Sysems Eng pp Coulbeck B and Orr C H [984] Opimized pumping in waer supply sysems IFAC 9 h riennial World Congress, Budapes, Hungary 5 Bellman R. E. Dynamic Programming Princeon Universiy Press echnical Repor R232 Waer Research Cenre Swindon Brdys & Ulanicki Operaional Conrol of Waer Sysems Prenice Hall 8 Meropolis N, Rosenbluh A, Rosenbluh M, eller A and eller E Equaion of Sae Calculaion by Fas Compuing Machines J Chem Phys., Vol 2 p Lin S Bell Sysem echnical Journal Vol 44 p Press W H, Flannery B P, eukolsky S A, Veerling W Numerical Recipes he Ar of Scienific Compuing CUP 99 Sousa J e al On he Qualiy of a Simulaed Annealing Algorihm for Waer Nework Opimisaion Problems Waer Indusries Sysems Modelling and Opimisaion Applicaions, Ed Savic D A and 2 Savic, D.A., Waiers, G.A. and Schwab, M. (997). Muliobjecive geneic algorihms for pump scheduling in waer supply, Evoluionary Compuing workshop, AISB '97, Mancheser, 7-8 April. 3 Simpson, A.R., Dandy, G.C. and Murphy, L.J. (994). Geneic algorihms compared o oher echniques for pipe opimisaion, Journal of Waer Resources Planning and Managemen, ASCE, 2 (4), July/Augus, Dandy, G.C., Simpson, A.R. and Murphy, L.J. (996). An improved geneic algorihm for pipe nework opimisaion. Waer Resources Research, Vol. 32, No. 2, Feb., Mackle, G., Savic, D.A. and Waiers, G.A. (995). Applicaion of Geneic Algorihms o Pump Scheduling for Waer Supply, Conference on Geneic Algorihms in Engineering Sysems: Innovaions and Applicaions, GALESIA `95, IEE Conference Publicaion No. 44, Sheffield, UK, pp