APPLICATION OF HOLT-WINTERS METHOD IN WATER CONSUMPTION PREDICTION

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1 APPLICATION OF HOLT-WINTERS METHOD IN WATER CONSUMPTION PREDICTION Edward KOZŁOWSKI, Dariusz MAZURKIEWICZ, Beaa KOWALSKA, Dariusz KOWALSKI Summary: One of he mos sraegic managemen ass of waer supplying companies is waer consumpion predicion. This consumers demand forecas is done using several, specially designed models which generae daa necessary for planning forhcoming operaional aciviies. However, sill one can observe a lac of universal modeling mehod. For his reason, he auhors have proposed previously a soluion, wherein hourly waer consumpion is deermined by rend analysis and harmonic analysis and now coninue he research on predicion mehods enabling effecive forecass wih he applicaions of Hol- Winers mehod. Key words: waer supply sysem, waer demand, ime series, Hol-Winers mehod.. Inroducion Waer demand forecasing in ciy waer supply sysem is one of he ey ass of municipal companies. This is because hey have o ensure proper sandards of service wih reduced operaional coss, aing ino consideraion he mos imporan in economic calculaions - cos of energy required for pumping. Waer consumpion depends on a number of facors, including periodiciy, ime, emperaure and randomness. This is why waer demand predicion is done using waer several consumpion models specially developed for his purpose, wha was in deailed analyzed in []. Forecasing daa obained wih such models are among ohers used o manage, conrol and modernize he exising infrasrucure, as well as o design new elemens of he sysem. However, sill one can observe a lac of universal modeling mehod. For his reason, he auhors have proposed previously a soluion, wherein hourly waer consumpion is deermined by rend analysis and harmonic analysis [], presening wo mehods for modeling hourly waer consumpion in a waer supply sysem. The firs mehod involved he use of linear phase rends dependen on ambien emperaure. The oher was based on he Fourier ransform, wherein he harmonics are linearly dependen on ambien emperaure. Srucural parameers of he above models were esimaed based on he dae regarding wo lines, Puławy Eas and Puławy Wes. The applicaion of he wo waer demand forecasing models demonsraed ha he real par of heoreical values for he Fourier ransform and he values of he expeced phase rends are similar. Boh he maching of hese models o he empirical daa and he forecass obained wih hese models reflec relaively accuraely he variaions in daily waer consumpion. Some differences beween he heoreical and empirical daa on waer consumpion indicaed ha here exis some addiional facors affecing waer consumpion. This is why now we coninue he research on predicion mehods enabling effecive forecass wih he applicaions of Hol-Winers mehod, wha in deails is discussed in his paper. In [] i was found ha boh ambien emperaure and woring/free days affec he waer inae. Below, he Hol-Winers mehod will be presened o idenify 627

2 he rend and seasonaliy in he ime series. Invesigaions resuls show ha his is a good ool for shor-erm waer demand forecass. 2. Hol - Winers mehod in ime series analysis Le ( Ω, F, P) be a complee probabiliy space and he ime series { x n follows: s be defined as x = a + b + + ε () where he value a presens a level around which he elemens of he series { x evolve, he value b corresponds o he slope facor and series { s n describe he n seasonal effec in ime series{ x n, { ε n is a sequence of independen idenically disribued random variables (a sequence of exernal disurbances) wih normal disribuion 2 N ( 0, σ ). Le us assume, ha he series{ x n have a frequency p > (he number of seasonaliy componens per uni of ime). In he presened formula () he elemens of ime series { x n are presened as a sum of rend, seasonal effec and random disurbances. On ( Ω, F, P) we define a family of sub-σ -fields F = σ { xi : i =,2,.., for n. In order o predic he behavior of a ime series, firs we need o deermine he level a, he slope b and he seasonal effec { s. The as consiss in he consrucion of esimaors n s aˆ = E a F = φ X, of unnown parameers a, b and seasonal effec { as n ( ) ( ) bˆ = E( b F ) = ϕ( X ), sˆ = E( s F ) = ψ ( X ), where φ ( ), ϕ ( ), ( ) and he sample X { x x,..., ψ denoe he saisics =, 2 x. The predicion of behavior of series field F (observaion X ) is defined as: x ( x+ F ) = a + bˆ s+ p E + + = ˆ x + based on σ - ˆ, (2) where = means he remainder of he division by p. p The Hol-Winers mehod (see e.q. [2,3]) has been nown for a long ime. This mehod is based on exponenial smoohing and allows us o esimae he level, slope and seasonaliy componen in a simple way. The unnown parameers a, b,{ s deermined as follows: aˆ bˆ ( x sˆ ) + ( α )( aˆ bˆ ) = p + sˆ wih iniial values are n α, (3) ( aˆ ˆ ) ( ) ˆ a + β b ( x aˆ ) + ( γ ) s p β, (4) = = γ ˆ (5) a ˆ = x, b ˆ = 0 and s ˆ = 0 for p and 0 < α <, 0 < β < and 0 < γ < denoe he smoohing parameers. Someimes he observaion 628

3 of ime series { x n is divided ino wo ses { x mp and { x mp n. The firs of + hem conains m full seasons (periods) of hisorical daa and is used o esimae he iniial values of level, slope and seasonal effec based on { x mp. The iniial values are deermined as follows: a = mp ˆ, mp x i mp i= xmp x = mp ˆ b sˆ m = x m i= 0 mp and ( m ) p+ + im mp (6) aˆ Nex, for he imes mp < n he esimaors of level, slope and seasonal effec are calculaed by usage he formulas (3) (5). From (3) (5) we see, ha he a he momen he esimaors a ˆ = aˆ ( α, β, γ ), b ˆ = ˆ ( α, β, γ ), ˆ sˆ ( α, β, γ ) b α β, γ s = depend on he values of smoohing parameers,. In accordance wih above o deermine he unnown parameers (level, slope and seasonal effec) we mus solve he as: min J 0 < α, β, γ < ( α, β, γ ) where he obecive funcion is defined as a sum of squared errors of predicion: for J n (, β, γ ) = ( x+ x ) = p+ 2 + α (8) p. The value J ( α, β, γ ) denoes a sum of squared -sep-ahead predicion errors (SS PE). 3. Waer demand analysis as ime series daa model The daa of waer consumpion for region "Puławy Eas" from /07/205 o 3/08/205 were used o analyze and forecas he waer demand. The waer consumpion is differen during he day, whereas if we loo a he waer demand during he wee or monh, i is no difficul o noice ha he waer demand has a seasonal componens. For example, he Figure a shows he waer demand from 08/07/205-00:00 o 6/07/205-00:00 in region "Puławy Eas". (7) 629

4 Fig.. Waer consumpion for Puławy Eas The Figure b represens a paern of waer demand, which is based on hourly average waer consumpion (he doed line). To deermine he rend componen and seasonal effec in equaion () he Hol-Winers mehod was applied. The iniial values of level, slope and seasonal effec were esimaed based on he waer consumpion during he firs five days of July, and nex he esimaors of level, slope and seasonal effec were calculaed wih he use of he formulas (3)-(5). Taing ino consideraion he as (7) for performance crierion J ( α, β, γ ), which is defined by formula (8), i was deermined he se of smoohing parameers. The opimal values of hese parameers are: α = 0.54, β = 0. 64, γ = This way he waer demand predicion wih he use of Hol-Winers mehod was achieved. The final resul was presened on Figure 2a and Figure 2b. The blac solid line (Fig. 2a) illusraes he real values of waer consumpion during he ime period from 3/07/205-00:00 o 9/07/205-00:00 (as elemens of he ime series { n ) whereas he red doed line presens he expeced values for -sep-ahead predicion (as elemens of x ime series { + n ). x 630

5 Fig. 2. Real and expeced values of waer consumpion for Puławy Eas Applying he Hol-Winers mehod we have successfully calculaed he prediced values of waer demand for he firs hree days of Augus. In Figure 2b he blac solid line illusrae he real waer consumpion for he ime period from 26/07/205-00:00 o 04/08/205-00:00. The green doed line show he expeced values of waer consumpion (as elemens of ime series { + n x ) for he ime period from 26/07/205-00:00 o 3/07/205-00:00, while he red dashed curve presens he forecas for he nex hree days x. (72 hours) as elemens of he series { + 72 Analyzing he Figure 2b we can noice, ha he shor-ime predicion wih he use of Hol-Winers mehod is quie good, whereas he long-ime predicion does no give expeced effecs. Analysis of esimaed componens sequences in he equaion () gives a promising answer o he quesion why he long-erm forecass are less accurae han he shor-erm predicion? Figure 3 shows he realizaions of sequences of values of level componen{ a ˆ, slope componen { mp+ n b ˆ, seasonal componens { s ˆ and mp+ n mp+ n ε. On he Figure 3a and Figure 3 we can observe he realizaions of residuals { mp+ n sequences { a mp+ n ˆ and { b mp+ n ˆ (hey describe he behaviors of inercep and slope esimaors respecively) which have a flucuaions. Nex he ess were carried ou o analyze he significance and sabiliy of he inercep esimaors and slope esimaors. 63

6 Fig. 3. The values of esimaors of level, slope, seasonal and he ime series of residuals As a resul i was considered he influence of he ime facor on he values of esimaors ~ â and bˆ. For he sequences of esimaors { a b, where a ~ = aˆ + mp, ~ = ˆ b b + mp ~ and { d d and d = n mp we have calculaed and analyzed he dependences: a a a a ~ = λ 0 + λ + ζ, (9) ~ b b b b λ + λ + ζ, (0) a b where { ζ and { d d = 0 ζ represen he exernal disurbances in linear formula (9) and (0) respecively. To es he significance and sabiliy of he inercep esimaors and slope we analyze and explain he exisence of linear dependences in models (9) and (0) (see e.g. [4-7]). The able presens he resuls of hese invesigaions. Tab.. Analysis of linear regression of values of inerceps and slope esimaors = a = b i λi S λ i λ i P T > λ i > <2* * * We can see, ha for he formula (9) he hypohesis ha he muliple correlaion coefficien is equal zero (or no significanly differs from zero) should be reeced, herefore he value of he esimaor of inercep depends on he ime facor. For he formula (0) we have no basis o reec he hypohesis, ha he muliple correlaion coefficien is equal zero (or no significanly differs from zero), herefore he value of he esimaor of inercep does no depend on he ime facor. From he oher side we have no basis o reec he R 2 632

7 b hypohesis, ha he parameer λ 0 is equal zero. Remar. Thus, using he Hol-Winers filer (3) - (5) for model () we have noiced ha he esimaor of parameer a is no saionary (he values of esimaor depend on he ime) and he esimaors of parameer b no significanly differs from zero. Thus, he long ime predicion of waer demand based on he Hol-Winers mehod does no give a good effec. 4. Final conclusions I is highly required o coninue research on waer demand predicion mehods, a paricular challenge for mahemaical forecasing models being here long-erm forecass based on correcly seleced ses of inpu daa. This sems from he fac ha aside from he random naure of waer consumpion, hese mehods should also ae accoun of many addiional facors. The resuls presened in his aricle confirm ha he Hol-Winers mehod can be effecively used for waer demand forecasing and designing conrollers for waer supply pumps in order o opimize he coss of mainaining waer surface a he desired level. Curren analysis resuls show ha furher invesigaion are necessary o improve o predicion performance in long-ime period. Soluion is expeced o be found when modifying he performance crierion (8) o deermine more sable esimaors of inercep and slope in formula (). Lieraura. Kozłowsi E., Kowalsa B., Kowalsi D., Mazuriewicz D.: Waer demand forecasing by rend and harmonic analysis, Archives of Civil and Mechanical Engineering, 8, 208, pp Hol C.C.: Forecasing rends and seasonals by exponenially weighed moving averages, ONR Research Memorandum, Carnegie Insiue of Technology, 52, Winers P.R.: Forecasing sales by exponenially weighed moving averages, Managemen Science, 6, 960, pp Kozłowsi E.: Analiza i idenyfiaca szeregów czasowych. Wyd. Poliechnia Lubelsa, Lublin Zeliaś A., Pawełe B., Wana S.: Prognozowanie eonomiczne. Teoria, przyłady, zadania. PWN, Warszawa, Chow G.C.: Eonomeria. PWN, Warszawa, Hamilon J.D.: Time Series Analysis. Princeon Universiy Press, Princeon Baer M., Vreeburg J.H.G., Palmen L.J., Sperber V., Baer G., Rieveld L.C.: Beer waer qualiy and higher energy efficiency by using model predicive flow conrol a waer supply sysem. Journal of Waer Supply; Research and Technology AQUA, 62, 203, pp.: Losa A., Moczulsi W., Wyczółowsi R., Dąbrowsi A.: Inegraed sysem of conrol and managemen of exploiaion of waer supply sysem. Diagnosya, 7(), 206, pp.: Losa, A.: Scenario modeling exploiaion decision-maing process in echnical newor sysems. Esploaaca i Niezawodnosc Mainenance and Reliabiliy 9 (2), 207, pp.: , hp://dx.doi.org/0.753/ein

8 Dr Edward KOZŁOWSKI Wydział Zarządzania/ Kaedra Meod Ilościowych w Zarządzaniu Poliechnia Lubelsa Lublin, ul. Nadbysrzyca 38 Tel.: (0-8) e.ozlovsi@pollub.pl dr hab. inż. Dariusz MAZURKIEWICZ, prof. PL Wydział Mechaniczny, Kaedra Podsaw Inżynierii Produci Poliechnia Lubelsa Lublin, ul. Nadbysrzyca 36 Tel.: (0-8) d.mazuriewicz@pollub.pl dr hab. inż. Beaa KOWALSKA, prof. PL dr hab. inż. Dariusz KOWALSKI, prof. PL Wydział Inżynierii Środowisa/Kaedra Zaoparzenia w Wodę i Usuwania Ścieów Poliechnia Lubelsa Lublin, ul. Nadbysrzyca 40B Tel.: (0-8) b.owalsa@pollub.pl, d.owalsi@pollub.pl 634