Identification of customer profiles from electricity consumption data

Lappeen an a Uni e si y o Technology School o Enginee ing Science Compu a ional Enginee ing and Technical Physics In elligen Compu ing Mas e ’s Thesis Kali a Manjang IDENTIFICATION OF CUSTOMER PROFILES FROM ELECTRICITY CONSUMPTION DATA Examine s: P o . Lasse Lensu Assoc. P o . Samuli Honkapu o Supe iso s: Adjunc P o ., D . Xiao-Zhi Gao Associa e P o . A o Kaa na P o . Lasse Lensu 2 ABSTRACT Lappeen an a Uni e si y o Technology School o Enginee ing Science Compu a ional Enginee ing and Technical Physics In elligen Compu ing Kali a Manjang Iden i ica ion o cus ome p o iles om elec ici y consump ion da a Mas e ’s Thesis 2018 63 pages, 23 igu es, 16 ables. Examine s: P o . Lasse Lensu Assoc. P o . Samuli Honkapu o Keywo ds: K-means clus e ing, gene ic algo i hm, powe use p o iling, Da ies-Bouldin index, Silhoue e index, Calinski-Haba asz index. The elec ic powe supplie s a e in e es ed in iden i ying and ca ego ising hei con- sume s’ p o iles in o di e en ca ego ies acco ding o hei ene gy consump ion habi s. The p o iling o use s can help wi h unde s anding how he use s consume he ene gy and how he ene gy usage may a ec he elec ici y dis ibu ion g id. Howe e , he p i acy o he elec ici y use s is well p o ec ed by he cu en law. This s udy ocuses on da a mining me hods o ex ac he ele an knowledge based on anonymous da a ob ained om sma me e s. The K-means clus e ing algo i hm was used in g ouping he ene gy consump ion da a. To imp o e he quali y o he clus e s o med ia he K-means clus e ing and o ackle he common p oblem o local op imum, he gene ic algo i hm (GA) was adop ed in e ining he clus e s. The use o wo alidi y indices o compa e he me hods showed ha combining K-means and GA did indeed imp o e he clus e ing quali y. 3 PREFACE Bismillahi, Rahmani, Rahmeen all p aise be o Allah. I would like o hank my supe - iso s o hei undi ided a en ion, dedica ion and guidance p o ided o me du ing he cou se o his mas e ’s hesis. This wo k would no ha e o he wise been achie ed wi hou hei suppo . I hank my beau i ul wi e o he pa ience. To my pa en s, hank you o suppo ing my d eams o ge ing a highe educa ion and ins illing in me discipline, espec and ha d wo k. Finally, I would like o hank he LUT adminis a ion o he schola ship I was o e ed o pu sue a double deg ee p og am in his p es igious ins i u ion. I am g a e ul. Lappeen an a, Augus 31, 2018 Kali a Manjang 4 CONTENTS 1 INTRODUCTION 7 1.1 Backg ound................................. 7 1.2 Objec i es and delimi a ions . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 S uc u eo he hesis............................ 9 2 ELECTRICITY CONSUMER PROFILING 10 2.1 Powe use p o iling............................. 10 2.2 Techniques used in powe use p o iling . . . . . . . . . . . . . . . . . . 11 2.2.1 Neu al app oaches . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.2 Clus e ing algo i hms . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.3 S a is ical app oaches . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.4 Fuzzyapp oaches.......................... 12 2.2.5 Hyb idme hods........................... 13 2.3 Re iewo echniques............................ 13 3 PROPOSED APPROACH FOR ELECTRICITY POWER PROFILING 17 3.1 K-meansclus e ing ............................. 17 3.2 Gene icalgo i hm.............................. 19 4 EXPERIMENTS AND RESULTS 21 4.1 Desc ip iono da a ............................. 21 4.2 P e-p ocessing................................ 21 4.3 Dimensionali y educ ion . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.4 E alua ionc i e ia.............................. 23 4.4.1 Silhoue eindex .......................... 23 4.4.2 Da ies–Bouldin index . . . . . . . . . . . . . . . . . . . . . . . 24 4.4.3 Calinski-Ha abasz index . . . . . . . . . . . . . . . . . . . . . . 25 4.5 Implemen a ion o expe imen s . . . . . . . . . . . . . . . . . . . . . . . 27 4.6 Resul s.................................... 29 4.6.1 Selec ing he numbe o clus e s o he annual load p o iles . . . 29 4.6.2 Selec ing he numbe o clus e s o he daily load p o iles . . . . 31 4.6.3 The wi hin-clus e sum o squa es o annual load p o iles . . . . 33 4.6.4 The wi hin clus e sum o squa es o daily p o iles . . . . . . . . 34 4.7 The clus e ep esen a ion o he annual load p o iles . . . . . . . . . . . 35 4.8 The clus e ep esen a ion o he daily load p o iles . . . . . . . . . . . . 41 4.8.1 Simila i y measu e o annual p o iles . . . . . . . . . . . . . . . 44 4.8.2 Simila i y measu e o daily load p o iles . . . . . . . . . . . . . 47 5 4.8.3 Annual weekend load p o ile . . . . . . . . . . . . . . . . . . . . 48 4.8.4 Re ining annual load p o iles . . . . . . . . . . . . . . . . . . . . 50 4.8.5 Re ining daily load p o iles . . . . . . . . . . . . . . . . . . . . . 54 4.9 Me hodcompa ison............................. 56 5 DISCUSSION 58 6 CONCLUSION 60 REFERENCES 61 6 LIST OF ABBREVIATIONS BCSS Be ween-Clus e Sum o Squa es CDI Clus e ing Dispe sion Indica o CFSFDP Fas Sea ch and Find o Densi y Peaks DB Da ies–Bouldin EA E olu iona y Algo i hms FCM Fuzzy Clus e ing Means GA Gene ic Algo i hm SAX Symbolic Agg ession App oxima ion SVM Suppo Vec o Machine SOM Sel O ganizing Maps WCSS Wi hin-clus e Sum o Squa es 7 1 INTRODUCTION 1.1 Backg ound Wi h he eme gence o sma me e s, mo e in o ma ion abou a use ’s elec ici y consump- ion can be collec ed easily. P io knowledge abou he g oup a pa icula use belongs o is known by he ene gy company o some ex en . This is achie ed h ough knowledge abou he ype o appliances in use o ype o hea ing sys em used in he buildings. This in o ma ion is s o ed and used in cus ome g ouping. One sho coming o his me hod is ha he eco ded in o ma ion is seldom upda ed. Wi h ime, he ene gy consump ion o he use , o example, he ype o hea ing o elec ical appliance usage migh change e- ma kably om he known beha iou s. In his ega d, his single-sho me hod o consume ca ego iza ion is ine icien . The adi ional ene gy use g ouping is pe o med using h ee use ca ego ies: indus ial, esiden ial and comme cial use s. An example o he indus ial use s a e ac o ies, com- me cial use s a e he shops, es au an s and supe ma ke s, and esiden ial use s e e o homes and apa men buildings. The consump ion pa e n o he ene gy use s is much mo e complex han hese men ioned g oups [1]. The ene gy use s should be ca ego ized based on he pa e n o elec ici y beha iou hey exhibi . The gene al applica ion o elec ici y use p o iling is ha he knowledge o how he cus ome s use he elec ici y can help he ene gy companies o c ea e impo an policies anging om ne wo k planning, demand esponse, and load o ecas ing [1]. A mo e de ailed applica ion o load p o iling is desc ibed as ollows [2]: Dis ibu ion ne wo k ope a ion •Real- ime ecogni ion o ne wo k loadings and ol ages. •I ensu es ha he ne wo k is kep wi hin i s ope a ing limi s. •I can manage pos - aul supply es o a ion. Sho - e m ope a ion planning •Ve y use ul in conges ion o ecas ing. 8 •Load p o iles ha e immense applica ion in ne wo k econ igu a ion, o ins ance, o minimize ne wo k los . •Planned ou ages p epa a ion. Dis ibu ion ne wo k planning •New base p o ile o p obabilis ic ne wo k planning. •Ensu ing he ne wo k can cos -e icien ly hos all o eseeable loads and gene a o s. O he applica ions o load p o iling a e he design o a i s, a ge sales based on cus- ome s load p o iles, he load p o ile can also be used in he adjus men o he elec ici y e ail o ecas s when new cus ome s a e con ac ed o old ones los [2]. By i ue o his impo an demand, inding and unde s anding clus e s using da a min- ing echniques (scien i ic me hods) is wo hwhile [3]. Since he use iden i y is p o ec ed unde he Eu opean Union p i acy laws [4], he speci ic loca ions and iden i ies o he pa icipan s will emain anonymous. This esea ch applies da a mining me hods (clus e - ing) o he p o ided da a se o show use s wi h simila ene gy consump ion pa e ns and g oup hese use s oge he . 1.2 Objec i es and delimi a ions This mas e ’s hesis aims a achie ing he ollowing objec i es: •To s udy and use he K-means clus e ing algo i hm o he pu pose o load p o iling. •To choose and e i y he app op ia e numbe o clus e s o use in he ca ego iza ion o he ene gy use s. •To use he GA o imp o e he quali y o he clus e s o med. This s udy is limi ed in ha he esul s ha will be ob ained canno be e i ied because o he anonymi y o he pa icipan s. 9 1.3 S uc u e o he hesis The ou line o his mas e ’s hesis is as ollows. Chap e 2 desc ibes ela ed wo k on powe use p o iling, Chap e 3 con ains he p oposed me hods o powe use p o iling and he algo i hms o hese me hods a e p esen ed. In Chap e 4, he applica ion o hese speci ic me hods o elec ici y load da a and he esul s de i ed om he expe imen a e analysed. Chap e 5 con ains he discussions and he challenges aced du ing he s udy. Finally, he concluding ema ks a e p esen ed in Chap e 6. 16 Table 2. Summa y o he K-means algo i hm Ad an ages Disad an ages Re s. K-means ·Simple ·Sensi i e o selec ion o ini ial cen- oids. [22–24] ·Scalable and E icien ·Numbe o clus e has o be de ined. ·Can handle big da a ·Sensi i e o noise and ou lie s. ·Linea complexi y ·Local op imum solu ion. The K-means algo i hm e mina es p ima ily i he da a poin s in he espec i e clus e s a e no eassigned o a di e en clus e o i he maximum numbe o allowed i e a ions is a ained. Fo his s udy, he la e is used. 17 3 PROPOSED APPROACH FOR ELECTRICITY POWER PROFILING 3.1 K-means clus e ing The K-means algo i hm is a pa i ion clus e ing algo i hm. I was in oduced by J.B. MacQueen in 1967 [25]. The algo i hm is based on unsupe ised lea ning used wi h unlabeled mul idimensional da a. The goal o he algo i hm is o g oup he unlabeled mul idimensional da a in o K clus e s (K is ixed a p io i). The K a iable ep esen s he numbe o g oups o he pa i ion. I wo ks by i e a i ely assigning da a poin s o one o he K g oups based on he p o ided ea u es. Each da a poin is assigned o one unique g oup. The algo i hm is a ou ed in many applica ion a eas such as compu e ision, image p ocessing, business analy ics e c. I s popula i y is due o he simplici y and linea complexi y, de ined as O(I∗n∗K∗D), whe e I ep esen s he numbe o i e a ion, n is he numbe o inpu ea u es, Kis he clus e numbe and Dis he dimension o he ea u es [26]. The K-means algo i hm includes wo s eps: 1.Clus e Assignmen s ep 2. Mo e cen oid s ep. In he clus e assignmen s ep, he idea is o de ine K cen oids o he clus e s, one o each clus e . The K-means esul is sensi i e o he ini ial cen oids, di e en ini ial cen oid yield di e en esul s. The e o e, a good choice is o place hem a he away om each o he . The nex s ep in ol es examining each da a poin and assign he da a poin o he closes cen oid. In he mo e cen oid s ep, he algo i hm calcula es he a e age o all he da a poin s in each clus e and he cen oid is mo ed o ha loca ion. This con inues un il no changes in he clus e s occu o un il some s opping c i e ion is me . The algo i hm aims a minimizing an objec i e unc ion, which in his case is he squa ed e o unc ion: J= k X j=1 n X i=1 kx(j) i−cjk2 whe e k=numbe o clus e s, n=numbe o da a poin s, and (1) kx(j) j−cjk2is he dis ance me ic used. Tha is he dis ance be ween he load p o ile x(j) j and he clus e cen e cj. The dis ance me ic used in his case is he Euclidean dis ance. The Euclidean dis ance o mula is gi en in Equa ion 2. 18 d(x, c) = u u x X k=1 (xk−ck)2.(2) Assuming Xis he se o load p o iles wi h X=x1, x2, ..., xnand V= 1, 2, ..., kis he se o clus e cen oids, he K-means algo i hm p oceeds by he ollowing s eps: 1. cclus e cen oids a e andomly selec ed. 2. The dis ances be ween each load p o ile and he clus e cen oids a e calcula ed. 3. Assign he load p o ile o a clus e cen oid wi h he minimum dis ance. 4. Recalcula e he new clus e cen oids as ollows: Vi=1 |ci| ci X j=1 xi(3) ci ep esen s he numbe o da a poin s in he i h clus e . 5. Recalcula e he dis ance be ween each load p o ile and he new clus e cen oids. 6. I no single load p o ile is eassigned o a clus e cen oid , he algo i hm s ops else p oceed o s ep 3. 19 3.2 Gene ic algo i hm Gene ic algo i hms a e biologically-inspi ed heu is ic sea ch op imiza ion algo i hm. They a e inspi ed by Cha les Da win’s heo y o e olu ion i.e he su i al o he i es . The al- go i hm exhibi he p ocess o na u al selec ion in which he i es indi iduals a e chosen o ep oduce he o sp ing o he nex gene a ion. The gene ic algo i hm essen ially epli- ca e he way in which li e uses e olu ion o ind solu ions o eal wo ld p oblems [27]. The e a e i e phases conside ed in a gene ic algo i hm [27]: 1. Ini ial popula ion 2. Fi ness unc ion 3. Selec ion 4. C osso e 5. Mu a ion A b ie desc ip ion o hese phases is gi en below: Ini ial popula ion Popula ion o andomly gene a ed solu ions o he p oblem. Clea ly, andomly gene a ed solu ions o he p oblem migh no be oo ideal. Fi ness The i ness quan i a i ely e alua es how i a gi en solu ion is o how i indi iduals can be p oduced om he gi en solu ions i.e., he i ness abili y o an indi idual o compe e wi h o he s. A i ness sco e is assigned o each indi idual, he selec ion o an indi idual o ep oduc ion depends on i s i ness sco e [27]. 20 Selec ion In he s i e o achie e con e gence, he bes o sp ings a e selec ed as pa en s in he new pa en al popula ion. The selec ion o he o sp ings a e based on hei i ness alues [27]. C osso e Du ing c osso e he gene ic ma e ial o he pa en s a e combined. This can be hough o as mimicking he ma ing p ocess in eal li e. By combining ce ain ai s om wo o mo e indi iduals, he hope is ha a ’ i e ’ o sp ing will e ol e wi h he bes ai s inhe i ed om he pa en s [27]. Mu a ion Mu a ion-ope a o s p o ides andom changes o he popula ion by dis u bing hem. Mu- a ion ypically allow e y small changes a andom o he indi idual genomes [27]. Mu- a ion main ains di e si y wi hin he popula ion and help p e en as con e gence. Te mina ion When con e gence is a ain he algo i hm e mina es. A his poin i can be said ha he algo i hm has p o ided a solu ion o he p oblem. The GA cycle is gi en in Figu e 3. Figu e 3. Gene ic Algo i hm cycle [27]. 21 4 EXPERIMENTS AND RESULTS The p oposed algo i hms a e implemen ed on Ma lab R2017a e sion, on a Windows 10 machine wi h 8GB o RAM. The Ma lab inbuil unc ion ’Kmeans’ and ’ga’ we e used o he implemen a ion. The so wa e p o ides lexibili y in eading and displaying s o ed iles. 4.1 Desc ip ion o da a The da a se used in he expe imen a e ime se ies hou ly elec ici y consump ion da a o 13601 households in Sou he n Finland. The da a a e based on hou ly loads eco ded o a span o one yea . The ows in he aw da a se ep esen he ime-s amps and he columns he espec i e cus ome s. The dimension o he load da a is 8760 ×13601. 4.2 P e-p ocessing The gi en da a se was p e-p ocessed o emo e missing alues. In checking he load da a o missing in o ma ion he Ma lab inbuil unc ions ’isnan’ was used. A single use ’s da a was ound wi h missing in o ma ion. Only ha pa icula use was excluded om he inal da a se used o he expe imen s. The K-means algo i hm, in his case, uses he Euclidean dis ance me ic. The Euclidean dis ance is known o be biased due o he scale o he measu emen s, o his ega d he aw elec ici y consump ion da a was s anda dized, so ha i has ze o mean and uni a iance. The ollowing o mula was used o he s anda diza ion: ¯ X=(Xi−µi) σi .(4) whe e Xis he load p o ile da a, µis he mean and σ ep esen he a iance. Fo each espec i e use load, µiand σi ep esen he mean and a iance espec i ely o he en i e da a o ha pa icula use . 22 4.3 Dimensionali y educ ion As he numbe o dimensions inc eases, he dis ance be ween any wo poin s in he same da a se s con e ges ( he maximum dis ance and he minimum dis ance be ween any wo poin s will be iden ical) [28]. This ends o be an issue wi h he Euclidean dis ance me ic. Reducing he dimensionali y p io o he K-means clus e ing can alle ia e his p oblem and conside ably help wi h he compu a ion. The dimensionali y educ ion echnique used was adop ed om [29]. To educe he load p o ile da a om ndimensions o Ndimensions, he da a was di ided in o zequally-sized ames. The mean o all he da a wi hin his ame was compu ed and a ec o No all he mean alues de i ed becomes he new ep esen a ion o he o iginal da a. This dimension educ ion was needed only o de i ing he annual p o iles. The whole da a se was conside ed in building he annual p o iles hence, he need o comp ess he size o he da a. Fo his s udy, he da a was di ided in o 24 equally-sized ames (8760 ows in o 24 equal- sized ames), 24 because each use p o ides 24 da a poin s a day. In simple e ms, he a e age o he load da a p o ided in a day ep esen s he elec ici y consump ion on ha pa icula day. Fo he annual load p o iles, he dimension o he da a is educed om 8760×13600 o 365×13600. Figu e 4 shows he ull load p o iles and he co esponding dimensional educed p o iles. The dimensionali y educed p o ile was ob ained by he me hod desc ibe abo e. I can be seen ha he shape o he wo p o iles has no changed. (a) (b) Figu e 4. Load p o ile o esiden ial la s: (a) A ull load p o ile. (b) Dimensionali y educed load p o ile. 23 4.4 E alua ion c i e ia Va ious me hods can be used o quan i y he pe o mance o a clus e ing algo i hm as well as o p o ide a echnique o he selec ion o he app op ia e numbe o clus e s. The e alua ion c i e ia can be ca ego ized as simila i y-o ien ed and classi ica ion-o ien ed [30]. In de e mining he op imal numbe o clus e s, h ee alidi y indices Silhoue e, Da id- Bouldin and Calinski-Ha abasz index we e used: 4.4.1 Silhoue e index In silhoue e analysis, he sepa a ion dis ance be ween he clus e s is s udied. I gi es a measu e o closeness be ween he poin s in one clus e o he poin s in he neighbou ing clus e s. The o mal de ini ion o his quali y index was adop ed om [31]. Le X=x1, ..., xNbe he load p o ile da a se and le C=c1, ..., ckbe i s clus e ing in some kclus e s. Le us deno e d(xk, xi) o be he dis ance be ween xkand xi. Le cj=xj i, ..., xj mjbe he j h clus e whe e j= 1, ..., k and mj=|cj|.aj ideno es he a e age dis ance be ween he i h ec o in he clus e cjand he ec o s in he same clus e . The a e age dis ance aj iis hence gi en by : aj i=1 mj−1 mj X k=1,k6=i d(xj i, xj k), i = 1, ...., mj.(5) The minimum a e age dis ance be ween he i h ec o cjand all he ec o s clus e ed in clus e ck, whe e k= 1, ..., K and k6=jis gi en as ollows : bj i= min n=1,...,k,n6=j1 mn mn X k=1 d(xj i, xn k), i = 1, ..., mj.(6) The i h ec o silhoue e wid h in clus e cjis gi en below: sj i=bj i−aj i max(bj i, aj i).(7) 24 The silhoue e wid h is in he ange [−1,1]. The silhoue e o a clus e cjgi en as: sj=1 mj mj X i=1 sj i.(8) The algo i hm o de e mining he op imal numbe o clus e s using he Silhoue e index is gi en as ollows: 1. Pe o m K-means clus e ing o he ange o alues o K. 2. Fo each alue in he ange, an a e age Silhoue e was calcula ed o he obse a- ion. 3. A plo o he cu e acco ding o he a e age silhoue e was gene a ed. 4. The loca ion o he maximum is he op imal numbe o clus e s. The Ma lab inbuil unc ion ’e alclus e s’ was used o achie e his. 4.4.2 Da ies–Bouldin index The Da ies–Bouldin (DB) index was in oduced in 1979 by Da id L. Da ies and Donald W. Bouldin. I is he a io be ween he wi hin-clus e dis ances and he be ween-clus e dis ances and compu ing he a e age o e all clus e s [31]. The o mal de ini ion o he Da ies-Bouldin index was adop ed om [32]. Le δkdeno e he mean dis ance o he poin in he clus e ck o hei cen oids Gk: δk=1 nk nk X i=1 kMk i−Gkk.(9) whe e Mk iis he n-dimensional ea u e ec o assigned o clus e ck, and nkis he size o he clus e . Le us deno e also, ∆kk0=d(Gk, Gk0) = ||Gk−Gk 0 ||.(10) 25 ha is, he dis ance be ween he cen oid Gkand Gk 0 o clus e s ckand ck0. Fo all indices k06=k, he Da ies-Bouldin index is as ollows: C=1 K K X i=1 maxk06=kδk+δ0 k ∆kk0,whe e K=numbe o clus e s.(11) The algo i hm o ind he op imal alue using he Da ies-Bouldin index is simila o he Silhoue e me hod. The algo i hm is gi en below as: 1. Pe o m K-means clus e ing o he ange o alues o K 2. Fo he alues in his ange, calcula e he Da ies-Bouldin index. 3. A plo is gene a ed o each alue o K. 4. The loca ion o he minimum is conside ed o be he op imal numbe o clus e s. 4.4.3 Calinski-Ha abasz index In he Calinski-Ha abasz index, he compa ison o he be ween-clus e s a iance o he wi hin-clus e a iance is made. The index was i s in oduced in 1974. The o mal de ini ion is de i ed om [32] and i is gi en as: C=BGSS WGSS ×N−K K−1(12) whe e K ep esen he numbe o clus e s, Nis he o al numbe o load p o iles. The o e all wi hin-clus e a iance is deno ed W GSS and he o e all be ween-clus e a i- ance as BGSS [32]. BGSS is calcula ed as he o al sum o squa es sub ac ed om WGSS. The o al sum o squa es is he squa ed dis ance o all he load p o iles om he cen oids. 32 Figu e 7. Silhou e, Da ies-Bouldin and Calinski-Ha abasz index. In Figu e 8 he wo daily weekday load p o iles a e ep esen ed. Figu e 8. Daily weekday load p o iles, 6910 and 6690 load p o iles in each espec i e clus e . 33 Figu e 6 and 8 p esen he new clus e s o he annual weekday p o iles and he daily weekday p o iles espec i ely. F om he Figu es, he clus e s seem o show only wo g oups o use s i.e. use s wi h a la elec ici y consump ion pa e n and hose use s whose elec ici y consump ion a ies ac oss he yea o day. An obse a ion o he o med clus- e s showed ha a lo o a e aging occu ed and some po en ial unique ai s o he espec- i e use s a e no exhibi ed. Also, conside ing he numbe o use s in he da a, 2clus e s is oo small o ep esen he elec ici y consump ion beha iou s o hese use s. P e ious wo k conside ed in his s udy had he op imum numbe o clus e s highe ha 2as seen in Table 1. The e o e, i is sa e o a gue ha 2is no app op ia e o he ca ego iza ion o he load p o iles. O he ways o choosing he app op ia e numbe o clus e s we e examined. A di e en ange o alues needs o be conside ed o he app op ia e numbe o clus- e s. The Da ies-Bouldin index is chosen because he o he wo alidi y indices, in his case, do no seem o be he app op ia e me hod o selec ing he numbe o clus e s. The Da ies-Bouldin index alues ha seem o be he po en ial solu ions a e looked a , hese alues a e 4,8,12 and 15 o he annual weekday load p o iles and 6and 14 o he daily weekday p o iles. Be o e he po en ial Da ies-Bouldin alues a e analysed in de ail ,we i s s udy he he wi hin-clus e sum o squa es o bo h he daily and annual load p o iles 4.6.3 The wi hin-clus e sum o squa es o annual load p o iles To u he analyze he app op ia e numbe o clus e s, he wi hin-clus e sum o squa es was applied. The deg ee o a iabili y o he load p o iles in each clus e is gi en by he wi hin-clus e sum o squa es (WCSS). The WCSS dec eases as he numbe o clus e s inc ease. The app op ia e numbe o clus e s can be selec ed his way, he hin is o choose he numbe o clus e s om which he WCSS d op is no e y la ge. In Figu e 9, he WCSS d op is no la ge a ound 8 he e o e, he annual load p o iles can be ca ego ized in o 8clus e s. The WCSS is supposed o dec ease and s ay low as he numbe o clus e s inc ease. The si ua ion is di e en when he numbe o clus e s is 12 and 15, he WCSS inc eased in- s ead o s aying low. The pe cen age o a iance as a unc ion o he numbe o clus e s is looked a . A numbe o clus e s should be chosen so ha an addi ion o ano he clus e does no gi e a be e modelling o he da a. In his iew, e en hough he WCSS d op did no s ay low a 12 and 15, he wo alues do no gi e a much be e esul han when he numbe o clus e s was 8. 34 Figu e 9. Wi hin-clus e sum o squa es o annual load p o iles. 4.6.4 The wi hin clus e sum o squa es o daily p o iles The wi hin-clus e sum o squa es is also u ilized o s udy he app op ia e numbe o clus e s o ca ego izing he daily load p o iles. Figu e 10 p o ides he WCSS plo . Figu e 10. Wi hin-clus e sum o squa es o daily load p o ile. 35 F om he plo in Figu e 10 i is seen ha he WCSS d op is no subs an ial a ound clus e 6and 7. These alues can, he e o e, sugges he app op ia e numbe o clus e s. The WCSS did no s ay low o all he alues analyzed. A 12 he WCSS inc eased ins ead o d opping. The same a gumen used wi h he annual load p o iles also applies he e. The alue a 12 is s ill lowe han he alue a he app op ia e numbe o clus e s. 4.7 The clus e ep esen a ion o he annual load p o iles Fo each o he cases conside ed, i.e., Case 1, Case 2, Case 3and Case 4(Co esponding o 4,8,12 and 15 clus e s espec i ely), he numbe o p o iles in each case is gi en in Table 5. Table 5. The numbe o consume s in each o he cases conside ed abo e. Clus e Case 1 (4 p o iles) Case 2 (8 p o iles) Case 3 (12 p o iles) Case 4 (15 p o iles). 1 1538 942 850 794 2 2166 1068 497 466 3 4610 3809 525 2142 4 5286 1244 569 410 5•2304 1477 460 6•850 289 242 7•742 487 437 8•2641 1811 683 9• • 569 1619 10 • • 2659 1799 11 • • 1384 1235 12 • • 2483 1481 13 ••• 385 14 ••• 1044 15 ••• 403 36 Case 1: 4Annual weekday p o iles In Clus e 1 ound in Figu e 11, a peak appea ed in he i s mon h o he yea . The high elec ici y consump ion a e declined as he yea p oceeds, his end con inued un il mid- yea . Gene ally, he wea he in Finland is iendlie a ound his ime o he yea hence elec ic hea ing is no a necessi y, e iden in he ela i ely s able elec ici y consump- ion showed. A ound he end o he yea , he consump ion a es a e shown o be high once again, his ise in he pa e n o consump ion is a ibu ed o he d op in empe a u e which is expe ience a ound he beginning o he win e season. Ano he peak in elec- ici y consump ion was obse ed a his ime. This p o ile ep esen esiden ial homes whe e dis ic hea ing is no p o ided so esiden ha e o esul o p o iding hea ing o hemsel es du ing he win e pe iod. Some con idence can be gi en o his claim due o he majo peaks eco ded a ound he ime when he empe a u es a e e y low. Figu e 11. Case 1: 4Annual weekday p o iles. In Clus e 1 ound in Figu e 11, a peak appea ed in he i s mon h o he yea . The high elec ici y consump ion a e declined as he yea p oceeds, his end con inued un il mid- yea . Gene ally, he wea he in Finland is iendlie a ound his ime o he yea hence elec ic hea ing is no a necessi y, e iden in he ela i ely s able elec ici y consump- ion showed. A ound he end o he yea , he consump ion a es a e shown o be high once again, his ise in he pa e n o consump ion is a ibu ed o he d op in empe a u e 37 which is expe ience a ound he beginning o he win e season. Ano he peak in elec- ici y consump ion was obse ed a his ime. This p o ile ep esen esiden ial homes whe e dis ic hea ing is no p o ided so esiden ha e o esul o p o iding hea ing o hemsel es du ing he win e pe iod. Some con idence can be gi en o his claim due o he majo peaks eco ded a ound he ime when he empe a u es a e e y low. In Clus e 2o Figu e 11, he elec ici y consump ion du ing he beginning o he yea showed a s able beha iou , bu a peak in he consump ion a es was eco ded du ing he middle o he i s mon h ( oughly a ound he 15 h). This ise in consump ion co esponds o he beginning o he yea when empe a u es a e a hei lowes . The consump ion a e became s able o he es o he yea . The consump ion a e akes up again a ound he end o he yea when li le peaks a e seen o appea . These a e mos p obably houses wi h elec ical hea ing. The di e ence be ween his p o ile and he i s p o ile in Figu e 11 is ha he consump ion a e showed a mo e s able beha io . In Clus e 3o Figu e 11, a low bu no iceable peak appea ed a ound Janua y a e which he beha iou o he p o iles eco ded a s able bu downwa ds end, his pa e n s ays consis en . A ound June/July, he consump ion a e declined e en u he . A his ime, he days a e no mally wa me compa ed o he es o he yea . A s able inc ease is eco ded igh a e wa ds. The cus ome s in his p o ile can be a ibu ed o esiden ial homes we e a o m o ai condi ioning o cooling sys em is absen . The absen o a cooling sys em is no iceable in he downwa d end in consump ion a es exhibi ed o wa me days. The p o iles in Clus e 4o Figu e 11 eco ded a s able consump ion pa e n in elec ici y usage in he beginning o he yea . A ise in he elec ici y consump ion begins sho ly a e wa ds, e iden in he peaks seen a oughly a ound June. This p o ile is a ibu ed o esiden ial homes in which some o m o cooling sys em is p esen . This should explain he peaks a ound ha ime o he yea when empe a u es a e no mally no low. The majo peaks declined a a ound July/Augus . The pa e n s ays almos he way h ough he es o he yea . As he yea elapsed he consump ion a es began o inc ease again. Case 2: 8Annual weekday p o iles Clus e 1,2,4and 5o Figu e 12 show simila consump ion beha iou o p o iles in Figu e 11 i.e, Clus e 1,2,3and 4 espec i ely. Clus e 3o Figu e 12 showed he same pa e n o ene gy consump ion h oughou he 38 Figu e 12. Case 2: 8Annual weekday p o iles. yea . An inc ease in he elec ici y consump ion is seen in he o m o peaks du ing he ea ly mon hs o he yea , bu his phenomenon did no las long as he cons an pa e n con inued. Besides ew indi idual peaks, a majo peak also occu ed a he end o he yea . In Clus e 6o Figu e 12, a ise in he ene gy consump ion is eco ded a e June. In his p o ile, as he end o he yea app oach, he a e o elec ici y consump ion is no iceably seen o be on he ise. Also, some majo peaks o med a he la e end o he yea . In Clus e 7o Figu e 12, he consump ion pa e n showed almos he same beha iou as Clus e 3o he same igu e. The only no iceable di e ence is ha as he yea p og essed a decline in he ene gy consump ion is seen in Clus e 7. This end s ays consis en o he whole yea . Besides ew indi idual peaks, no majo peaks can be seen. These p o iles can be a ibu ed o load cu es p oduced by indus ial o comme cials cus ome s. The consump ion o elec ici y by use s in Clus e 8o Figu e 12 began e y s able. De- spi e ew indi idual peaks shown by di e en use s. The ene gy consump ion beha iou emained e y much he same un il a ound May. A s eady ise in ene gy usage is no- iced a ound June. This ended un il he end o he yea . The consump ion a e sligh ly 39 declined a some poin , bu quickly ook up again as he yea ends. Case 3: 12 Annual weekday p o iles In Figu e 13, some o he clus e s o med ha e al eady been seen and add essed p e i- ously, i.e., in Case 1and 2. Some new p o iles di e en om hose ound ha e also been c ea ed. The p o iles show some simila i y isually, o ins ance, Clus e 8,9and 11. The mos no iceable cha ac e is ic exhibi ed by hese clus e s is he decline in he ene gy consump ion a es as yea ad ances. Figu e 13. Case 3: 12 Annual weekday p o iles. A mino ise in he elec ici y consump ion is obse ed in Clus e 7o Figu e 13. Al- hough hese peaks a e e y much no iceable, hey a e no majo peaks. The peaks s ayed cons an un il Feb ua y, bu p omp ly declined a e wa ds. A oughly a ound June, he consump ion pa e n begins o ise gen ly. This s eady ise is no ed h oughou he yea . This p o ile can be associa ed wi h homes whe e dis ic hea ing is absen as e iden in he 40 ise and decline o elec ici y consump ion, i.e., ene gy is demanded mo e in he colde mon hs and less in he wa me seasons. Clus e 2and Clus e 7ha e a simila pa e n o elec ici y consump ion, Howe e , he peaks in he beginning o he yea o Clus e 2a e s onge and mo e p ominen . Case 4: 15 Annual weekday p o iles Like Figu e 13, some o he p o iles in Figu e 14 ha e al eady been me . Again, a e Case 2, he esul ing clus e s o he o he cases a e no e y dis inc in many ways. In Figu e 14. Case 4: 15 Annual weekday p o iles. Figu e 14, Clus e 8,9,10,11 and 15 a e e y ela ed and pe haps hese p o iles should no o m indi idual clus e s bu a he be oge he . The same implies o clus e 5and 13. 41 4.8 The clus e ep esen a ion o he daily load p o iles Fo each o he cases conside ed o he daily load p o iles i.e. Case 1and Case 2(Co - esponding o 6and 14 clus e s espec i ely), he numbe o p o iles in each case is gi en in Table 6. Table 6. The numbe o use s in each o he cases conside ed o he daily load p o iles. Clus e Case 1 (6 p o iles) Case 2 (14 p o iles). 1 2333 881 2 4150 1990 3 1499 377 4 2390 1142 5 1059 374 6 2169 1373 7•1308 8•431 9•1005 10 •893 11 •1660 12 •1340 13 •396 14 •430 Case 1: 6daily weekday p o iles In Clus e 1o Figu e 15, du ing he ea ly hou s o he day, he need o elec ici y emains mode a ely low. A abou 6 : 00 he demand o ene gy inc eased. This end con inued un il 14 : 00 when inally he demand subsided. This dec ease in elec ici y consump ion con inued h oughou he day. In his p o ile, he ene gy equi emen s we e high in he mo nings and a g ea e pa o he a e noon when he demand s a ed o decline. This can be a ibu ed o homes we e esiden s spend a good pa o he mo nings and a e noons a home. This p o ile migh indica e households we e esiden s a e away om hei homes a mid-day, pe haps hey wo k in he e enings and nigh s. I can also indica e p o iles o es au an s and small ca e e ias. A close look a he p o ile suppo s his claim. The pa e n o ene gy consump ion is somewha highe a 1 : 00 in Clus e 2o Figu e 15 compa ed o he o he hou s (in he same p o ile). A educ ion in he elec ici y usage is e iden om a ound 5 : 00, his s eady dec ease ended o a couple o hou s un il 9 : 00 48 Table 12. CASE 2: Simila i y measu e o 14 daily load p o iles. Clus e 1234567891011121314 1 011111010 0 0 1 1 0 2 101111011 1 1 1 1 1 3 110111111 1 1 1 1 1 4 111001000 0 0 0 0 0 5 111001100 1 1 1 0 1 6 111110111 1 1 1 1 1 7 001011010 0 0 1 1 1 8 111001100 0 1 1 0 1 9 011001000 0 0 1 0 1 10 0 1 1 0 1 1 0 0 0 0 0 0 0 0 11 0 1 1 0 1 1 0 1 0 0 0 1 1 1 12 1 1 1 0 1 1 1 1 1 0 1 0 1 0 13 1 1 1 0 0 1 1 0 0 0 1 1 0 1 14 0 1 1 0 1 1 1 1 1 0 1 0 1 0 In case 1, i.e, Table 11, he cen oids o Clus e 4and 5a e simila acco ding o he Kolmogo o -Smi no es . The emaining use p o iles a e di e en acco ding o he same es . In he second case o he daily load p o iles, i.e, Table 12, he simila i y be ween hese load p o iles a e highe . An inc ease in he numbe o p o iles esul s in use s wi h simila elec ici y consump ion pa e ns ending up in di e en clus e s. This explains he high simila i y in Table 12. Clus e 3and 6a e he only dis inc p o iles in he able, he es o he clus e s ha e one o mo e use s p o iles hey a e simila o. Due o he high simila i y in Table 12 he app op ia e numbe o clus e s o he daily load p o iles canno be 14 clus e s. This lea es only one choice o he app op ia e numbe o clus e s, i.e, he clus e s in Table 11 . The simila i y measu e is in ui i e in he case o he daily load p o iles. Clus e 4and 5in Table 11 a e no only simila acco ding o he Kolmogo o - Smi no 2 sample es bu also ha e a simila pa e n o consump ion. These load p o iles can be me ged. Hence, he app op ia e numbe o clus e s o he daily load p o iles is, he e o e, 5clus e s. 4.8.3 Annual weekend load p o ile The annual weekend load p o iles we e c ea ed using 8clus e s ( he app op ia e num- be o clus e s o he annual load p o iles is 8). The elec ici y consump ion du ing he weekdays o he load p o iles ha a e conside ed o indica e esiden ial households is p esumed o a y conside ably om he weekend’s consump ion pa e ns (wo king class 49 esiden spend mo e ime a home du ing he weekends). Fo he indus ial and comme - cial cus ome s, he p oduc ion a e is lesse du ing he weekends (businesses open and close p oduc ion di e en ly du ing he weekends). The elec ici y consump ion pa e n is also impac ed by his. As a consequence, he annual weekend load p o iles we e ob ained and compa ed o he annual weekday load p o iles al eady add essed in Figu e 12. Figu e 17. Annual weekend load p o iles. Igno ing he posi ion o he clus e s in he espec i e igu es, he weekend load p o iles and he weekday p o iles a e compa able. Howe e , some di e ences can be no ed. In Clus e 2o he weekend load cu es, he elec ici y consump ion is below 5kw whe eas in he weekday load p o iles many o he p o iles ha e consump ion a es abo e 5kw. Also, all he use s in he weekend p o ile ha e consump ion below 10 kw excep o ew use s in Clus e 3and Clus e 8. In he weekday p o iles many o he p o iles ha e use s whose powe consump ion a es a e abo e 10 kw. The consump ion o ene gy is highe du ing he weekday in compa ison o he weekends. 50 4.8.4 Re ining annual load p o iles As seen p e iously, he numbe 4,8,12 and 15 we e iden i ied as he po en ial alue o he app op ia e numbe o clus e s in he case o he annual load p o iles. The clus e ep esen a ion o each o hese cases ha e been p esen ed al eady and he cha ac e is ics o he espec i e use p o iles elabo a ed. In his subsec ion, he clus e ep esen a ion o hese numbe s is eplica ed using a di e en me hod. The clus e s a e o med by p o iding he ini ial cen oids o he K-means clus e ing algo i hm a he han elying on he andom ini ial cen oids selec ion as was he case p e iously. To ensu e ha he global op imum is a ained, hese ini ial cen oids we e op imised using he GA. The clus e s o he a ious cases o he annual load p o iles a e gi en in Figu e 18, 19, 20, 21. Figu e 18. 4Re ined annual load p o iles. 51 Figu e 19. 8Re ined annual load p o iles. 52 Figu e 20. 12 Re ined annual load p o iles. 53 Figu e 21. 15 Re ined annual load p o iles. As expec ed he eplica ion o he load p o iles using a di e en me hod yielded simila p o iles as be o e. In he i s ins ance, o example, he ou clus e s p oduced a e simila o he clus e s ound in Figu e 11. The same implies o Figu e 12, 13 and 14. The posi ion o he load p o iles migh be di e en in each espec i e igu e, bu hese p o iles look alike. In Figu e 21, Clus e 10 is emp y ( he clus e con ains only he ini ial cen oids alue). Figu e 14 has he same numbe o clus e s as Figu e 21 howe e , no emp y clus e s a e p oduced in he o me . This is one no ed di e en . 54 4.8.5 Re ining daily load p o iles The daily load p o iles we e also eplica ed using he same me hod desc ibed o he annual load p o iles. In he annual load p o iles eplica ion, i has been seen ha he shapes o he load p o iles o pa e n o elec ici y consump ion did no change despi e he me hod used. The same cha ac e is ics we e also exhibi ed du ing he daily load p o iles e ining. Figu e 22 and 23 con ain he e ine daily p o iles. The load p o iles we e gene a ed using he po en ial app op ia e numbe o clus e s discussed ea lie (6and 14 clus e s). Figu e 22. Case 1: 6Re ined daily load p o iles. 55 Figu e 23. Case 1: 14 Re ined daily load p o iles. 56 4.9 Me hod compa ison The wo me hods we e compa ed in o de o de e mine i he op imiza ion p ocess will, in ac , imp o e he clus e ing esul . Since he clus e s p oduced ia he di e en me hods a e simila isually, a way is needed o compa e he pe o mance o he me hods. The alidi y index alues (Silhoue e and Da ies-Bouldin) o he wo me hods we e compu ed and compa ed. Each o he cases o he annual load p o iles we e compa ed o hei co esponding e ined clus e s. Table 13, 14, 15 and 16 gi e he alidi y index o he wo me hods. Table 13. Case 1 wi h 4 annual load p o iles: Resul s o he wo alidi y indices. Bold numbe s show he bes index alue. Me hods Silhoue e Da ies–Bouldin Kmeans wi h GA 0.1136 3.0778 Kmeans 0.1136 3.0778 Table 14. Case 2 wi h 8 annual load p o iles: Resul s o he wo alidi y indices. Bold numbe s show he bes index alue. Me hods Silhoue e Da ies–Bouldin Kmeans wi h GA 0.0741 3.2759 Kmeans 0.0741 3.2816 Table 15. Case 3 wi h 12 annual load p o iles: Resul s o he wo alidi y indices. Bold numbe s show he bes index alue. Me hods Silhoue e Da ies–Bouldin Kmeans wi h GA 0.0590 3.3956 Kmeans 0.0580 3.4046 Table 16. Case 4 wi h 15 annual load p o iles: Resul s o he wo alidi y indices. Bold numbe s show he bes numbe index alue. Me hods Silhoue e Da ies–Bouldin Kmeans wi h GA 0.0568 3.2626 Kmeans 0.0508 3.6130 57 F om Table 13, he alidi y indices o he wo me hods a e he same. This means e ining he clus e s did no imp o e o dec ease he quali y o he clus e ing esul s in ha case. As he numbe o clus e s inc eased i is no iced ha e ining he clus e s sligh ly imp o ed he quali y o he clus e ing esul . This can be obse ed in Table 14, 15 and 16. The wo alidi y indices es did indeed suppo ou assump ion ha he op imisa ion o he ini ial cen oids alues using GA will imp o e he quali y o he load clus e ing esul s.