An Evaluation of a Metaheuristic Artificial Immune System for Household Energy Optimization

Resea ch A icle An E alua ion o a Me aheu is ic A i icial Immune Sys em o Household Ene gy Op imiza ion Ma ia Na a o-Cace es , 1 P amod He a h, 2 Gab iel Villa ubia , 1 F ancisco P ie o-Cas illo , 1,2 and G. Kuma Venyagamoo hy 3,4 1 Uni e si y o Salamanca, s/n. Salamanca, 37003 Espejo, Spain 2 Media Lab, Massachuse s Ins i u e o Technology, 20 Amhe s S , Camb idge, MA, USA 3 Holcombe Depa men o Elec ical and Compu e Enginee ing, Real-Time Powe and In elligen Sys ems Labo a o y and Clemson Uni e si y, Clemson, SC 29634, USA 4 School o Enginee ing, Uni e si y o KwaZulu-Na al, Du ban, Sou h A ica Co espondence should be add essed o Ma ia Na a o-Cace es; [email p o ec ed] Recei ed 25 Oc obe 2017; Accep ed 4 Ma ch 2018; Published 2 July 2018 Academic Edi o : Hugo Mo ais Copy igh © 2018 Ma ia Na a o-Cace es e al. This is an open access a icle dis ibu ed unde he C ea i e Commons A ibu ion License, which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed. De ices in a sma home should be connec ed in an op imal way; his helps sa e ene gy and money. Among nume ous op imiza ion models ha can be ound in he li e a u e, we would like o highligh a ificial immune sys ems, which use special bioinspi ed algo i hms o sol e op imiza ion p oblems effec i ely. The aim o his wo k is o p esen he applica ion o an a ificial immune sys em in he con ex o diffe en ene gy op imiza ion p oblems. Likewise, a case s udy is pe o med in which an a ificial immune sys em is inco po a ed in o de o sol e an ene gy managemen p oblem in a domes ic en i onmen . A ho ough analysis o he diffe en s a egies is ca ied ou o demons a e he abili y o an a ificial immune sys em o find a success ul op ima which sa isfies he p oblem cons ain s. 1. In oduc ion A Home Ene gy Managemen Sys em (HEMS) is a key ele- men in a domes ic en i onmen ha imp o es household economy h ough au oma ed echnologies. In he ecen yea s, diffe en domes ic buildings equipped wi h communica ion channels (sma houses) ha e ac i ely pa icipa ed in elec ical ne wo ks [1] as building blocks in sma g ids (SGs). The e o e, hey play an impo an ole in op imizing he scheduling o elec ic powe [1, 2]. The e a e a numbe o s a egies ha employ diffe en echniques o op imize he scheduling o ene gy use in he home. Among many o he s a egies, s a is ical models a e one example o hem. We can see how hey a e le e aged in he wo k o [3], which models con ollable loads using a Ma ko ian app oach. These loads depend on wea he condi- ions. In [4], a demand- esponse p og am is au oma ically applied om classical me hods o con ol he de ices connec ed o he ne wo k unde he unce ain y o he ou - side empe a u e and he p ice o elec ici y. In [5], h ee p oblems ela ed o HEMS ha e been sol ed by applying an obse able Ma ko ian decision p ocess. This wo k made i possible o educe domes ic ene gy cos s in he elec ici y p ice ma ke . Classical app oaches had some limi a ions [6]; hus, new pa adigms ha e been applied o sol e HEMS. One success- ully de eloped pa adigm is ha which uses bioinspi ed algo- i hms o sol e op imiza ion p oblems. These algo i hms y o mimic he beha io o some biological en i ies o find solu- ions which, applying classical compu a ion, would be oo cos ly o e en implausible in e ms o ime and esou ces. Some widely used and no ewo hy algo i hms a e a ificial neu al ne wo ks (ANN), gene ic algo i hms (GA), o swa m in elligence [7]. Some bioinspi ed algo i hms wo k in diffe - en con ex s, and hey ende good esul s. One o hese algo- i hms is he a ificial immune sys em (AIS) which ollows Hindawi Complexi y Volume 2018, A icle ID 9597158, 11 pages h ps://doi.o g/10.1155/2018/9597158 he p inciples o he e eb ae immune sys em o find solu- ions in an op imiza ion p oblem. The AIS algo i hm can be designed in a a ie y o ways. F om he diffe en a ian s, in his wo k, i is decided o use Op -aiNe [8], which has been used success ully o he op imiza ion o unc ions in diffe en con ex s [8]. Op -aiNe allows finding se e al solu ions in pa allel. By using ope a ions such as mu a ion, cloning, and supp ession, each solu ion co esponds o diffe en op ima (maxima o minima) poin s in he op imiza ion unc ion. In he a ea o in elligen ne wo k op imiza ion, se e al wo ks ha ollow he bioinspi ed pa adigm ha e been p o- posed. These include [9], who p opose an ene ge ic se ices modeling me hod based on he pa icle swa m op imiza ion (PSO) algo i hm. Soa es e al. [10] p opose a mul iobjec i e gene ic app oach o scheduling domes ic cha ge in an ene gy managemen sys em. Yuce e al. [11] p esen a neu al ne wo k wi h a gene ic algo i hm (ANN-GA) o op imize ene gy managemen in he domes ic sec o . Howe e , o he au ho s’knowledge, AIS has only been implica ed in some p elimina y achie emen s in powe managemen , such as sol ing powe supply p oblems, o elec ical econfigu a- ions. Fo example, in [12], an AIS is used o con ol he mal uni s in esiden ial buildings, and in [13], he au ho s op i- mize a wind ene gy-gene a ing sys em also wi h an AIS. In his pape , an in-dep h e iew o he AIS concep and i s applica ion o diffe en elec ical p oblems is made. F om he esul s o his e iew, a case s udy on a p oblem o home ene gy managemen op imiza ion is desc ibed and sol ed using his algo i hm. We aim o demons a e ha AIS can be success ully applied o elec ical managemen p oblems in domes ic se ings. Pa o ou objec i e has been o adap he Op -aiNe algo i hm o include complex cons ain s on he op imiza ion p oblem and o wo k efficien ly wi h a la ge numbe o a iables. This pape p esen s a simple elec ic con ex wi h diffe en de ices, namely, a pho o ol aic panel (PV), a ba e y sys em, a space hea e o hea e , a wa e hea e , and mus - un se ices. All o hem a e connec ed in a sma home, wi hin an elec ical sys em. I is aimed a op imizing he scheduling o he nex 24 hou s so ha he elec ical benefi is maximized be ween he ene gy ha is sold and he ene gy ha is bough . Two s a egies a e designed in ou case s udy o ep e- sen wo diffe en elec ical si ua ions. In s a egy 1, he HEMS manages he elec ic powe wi h he elec ici y g id wi hou conside ing any in e nal es ic ion. In o he wo ds, we do no conside any a iable ela ed o he main enance o he domes ic elec ic cha ge h ough he elec ical ene gy p oduced by he PV sys em. The e o e, his s a egy only seeks o op imize ene gy benefi s. How- e e , s a egy 2 is aimed a supplying he elec ici y demand au onomously whene e possible. The e o e, he su plus gene a ed by he PV is s o ed in he ba e y. HEMS will sell elec ici y o he g id when he ba e y is ully cha ged. Also, he ba e y is discha ged when he elec ical demand is g ea e han he powe gene a ed by he PV. I he ba e y canno supply all he elec ical cha ge, hen he HEMS mus buy elec ici y om he powe g id. Based on hese wo s a egies, h ee diffe en expe imen s we e de eloped. Fi s ly, a compa ison o AIS wi h wo diffe en bioinspi ed algo i hms is made, namely, he classical gene ic algo i hm (GA) and he pa icle swa m op imiza ion (PSO). Secondly, bo h s a egies a e compa ed o analyze he influence o he ba e y in he home ne wo k. Finally, a deep analysis is ca ied ou wi h diffe en si ua ions o he ba e y cha ge in he home ne - wo k. The esul s ob ained in all si ua ions a e expec ed o alida e he AIS as an app op ia e algo i hm o he op i- miza ion o HEMS. This documen is s uc u ed as ollows. Sec ion 2 p o ides an o e iew o he design o AIS and a e iew abou i s in ol emen in elec ical p oblems. Sec ion 3 desc ibes he echnical de ails o he add essed elec ical p oblem. Sec ion 4 p esen s he configu a ion o AIS and i s applica ion o he elec ical p oblem. In Sec ion 5, he esul s ob ained in he h ee case s udies a e ou lined and discussed. Finally, Sec ion 6 p esen s he conclusions o ou esea ch and u u e wo k. 2. A i icial Immune Sys ems The o ganisms o many species ha e de eloped immune sys ems o p o ec hem om ex e nal agen s. Abo e all, e eb a e immune sys ems consis o diffe en molecules, cells, and o gans ha a e dis ibu ed h oughou he body and a e no con olled by any cen al en i y. F om an immunological poin o iew, any elemen p esen in he immune sys em is called an an igen. I his an igen belongs o he in e nal o ganism o p o ec he body, i is called sel -an igen o an ibody. O he wise, he an igens om he ex e nal en i onmen a e called non-sel -an igens and can p o oke diffe en diseases. The e o e, immune sys ems a e aimed a dis inguishing be ween sel -an igens and non-sel - an igens h ough a pa e n ecogni ion p ocess, a acking only hose ha a e ha m ul o he body [14]. D awing on he concep o he immune sys em, [15] de eloped he CLONALG algo i hm, a clonal selec ion p oce- du e ha allows mu a ing some an ibodies acco ding o hei affini y o an ex e nal an igen; he e o e, in o de o pe o m pa e n ecogni ion, i gene a es copies o he an ibodies acco ding o hei affini y wi h he an igen. The copies a e mu a ed ollowing a a e δin e sely p opo ional o hei affini y wi h he an igen (1). δ=e i β,1 whe e βis a cons an ob ained empi ically o no malize he effec o he fi ness alue i o each cell. These new indi iduals a e added o he gene al popula ion and ee alua ed o be ep oduced and mu a ed again. In o de o gi e a new solu ion o he op imiza ion o unc ions, [8] de eloped Op -aiNe , an a ificial immune sys- em (AIS) based on he CLONALG beha io . The in o ma- ion is encoded as an igens which should be ecognized by he an ibodies o ou immune sys em. Then, he fi ness alue o an an igen is defined as he affini y be ween he an igen and he an ibody and can be compa ed wi h a dis ance 2 Complexi y me ic. Hence o h, small dis ances be ween an an igen and an an ibody ep esen high affini y, whe eas longe dis ances ep esen lowe affini y. The Op -aiNe algo i hm ollows he gene al desc ip ion o an a ificial immune sys em. Fi s ly, an ibodies, which ep- esen he diffe en da a o op imize, a e andomly gene a ed. Then, hey a e p esen ed o he an igens, which encode he objec i e unc ion, in o de o calcula e he affini y be ween hem when he da a a e applied o he unc ion. I one an i- body ob ains a good a ing in he objec i e unc ion, ha means i has high affini y and he e o e is selec ed. These chosen an ibodies a e ep oduced and mu a ed based on hei fi ness alue acco ding o he CLONALG algo i hm and he βpa ame e . In o de o p ese e di e si y, an ibod- ies whose affini y is lowe han a gi en h eshold s a e emo ed om he popula ion. A ificial immune sys ems, in pa icula Op -aiNe , a e able o find se e al op ima o he objec i e unc ion in pa al- lel. This means ha AIS can find a se o good candida es o he solu ion o op imiza ion p oblems ha a e diffe en om one ano he . Addi ionally, AIS can p ese e hose indi iduals ha a e good enough o be ep oduced and mu a ed in con- secu i e i e a ions. 2.1. AIS Applica ions in Ene gy Con ex s. The concep o a nex -gene a ion powe sys em such as sma g id, efficien ene gy managemen , and be e powe sys em planning canno be achie ed wi hou elec ical load o ecas ing [16]. Consequen ly, mul iple ime ho izons which a e asso- cia ed wi h he egula ion, dispa ching, scheduling, and uni commi men o he powe g id a e analyzed and sol ed using diffe en me hods. A ificial in elligence (AI) is widely applied o a a ie y o applica ions, as i can han- dle he complexi y de i ed om such elec ical p oblems. In pa icula , bioinspi ed algo i hms, such as a ificial neu- al ne wo ks o swa m in elligence, a e especially effec i e in sol ing his kind o p oblems. In his sec ion, a b ie bu comp ehensi e li e a u e e iew o a special bioin- spi ed algo i hm, he a ificial immune sys ems, is p o- ided. AIS was applied in diffe en con ex s wi h posi i e esul s: when sol ing combina o ial p oblems [17, 18], o de ec in usions in wi eless senso ne wo ks [19], o e en o gene a e cho d p og essions [20]. The majo goal o his sec ion is o e iew, iden i y, e alua e, and analyze he pe - o mance o AIS in powe sys ems and model esea ch. Rega ding he elec ical con ex , he e a e plen y o p o- posals ocusing on di e se fields. One o hem is ela ed o he con ol o a iables and configu a ion o an elec ical sys em. de Mello Hono io e al. [21] model an op imal powe flow (OPF), which is a nonlinea , noncon ex, and la ge-scale p oblem wi h bo h con inuous and disc e e con- ol a iables, using a modified a ificial immune sys em (AIS). The AIS makes use o hype mu a ion, which is esponsible o local sea ch, and ecep o edi ion, which explo es diffe en a eas in he solu ion space. The p oposed AIS is combined wi h a g adien ec o o imp o e he final esul s. This combina ion is also aimed a collec ing aluable in o ma ion du ing he hype mu a ion p ocess, dec easing he numbe o gene a ions and clones, and, consequen ly, speeding up he con e gence p ocess while educing he compu a ional ime. Belkacemi and Feliachi [22] use a mul- iagen sys em (MAS) which ollows he human immune sys em beha io o p opose a new echnique o powe sys- em econfigu a ion and es o a ion, applied o a model o Sou he n Cali o nia Edison’s Ci cui o he Fu u e. Each ele- men o he MAS ep esen s a na u al immunological ele- men ha in e ac s wi h he o he elemen s o heal he body. Simila ly, he MAS is able o de ec and isola e aul s and es o e powe o he affec ed loads aking in o conside - a ion line capaci y, ol age p ofile, and powe losses [22]. de Oli ei a e al. [23] p esen a me hodology o he econfigu- a ion o adial elec ical dis ibu ion sys ems o minimize ene gy losses making use o he bioinspi ed me aheu is ic a ificial immune sys em. The AISs ha e o plan he sys em ope a ion conside ing bo h adiali y and connec i i y con- s ain s and diffe en load le els. Consequen ly, he AIS algo i hm is adap ed o accommoda e he ea u es o he p oblem be e and o imp o e he sea ch p ocess. The algo- i hm de eloped is es ed in well-known dis ibu ion sys- ems, wi h e y success ul esul s. Souza e al. [24] sol e he econfigu a ion p oblem o elec ical dis ibu ion sys- ems (EDSs) wi h a iable demand, using he a ificial immune algo i hm. As he econfigu a ion p oblem wi h a iable demand is a complex p oblem o a combina o ial na u e, Cop -aiNe (a ificial immune ne wo k o combina- o ial op imiza ion), which is a combina o ial e sion o he algo i hm Op -aiNe , is applied o iden i y he bes adial opology o an EDS in o de o minimize he cos o ene gy losses in a gi en ope a ion pe iod. A specialized sweep load flow o adial sys ems was used o e alua e he easibili y o he opology wi h espec o he ope a ional cons ain s o he EDS and o calcula e he ac i e powe losses o each demand le el. The ob ained esul s we e compa ed wi h hose in he li e a u e in o de o alida e and p o e he effi- ciency o he p oposed algo i hm. Souza e al. [25] also aim o sol e he econfigu a ion p oblem o EDS by compa ing he esul s o he Cop -aiNe (a ificial immune ne wo k o combina o ial op imiza ion) and he Op -aiNe (a ificial immune ne wo k o op imiza ion) algo i hms. A specialized o wa d/backwa d adial powe flow was used o e alua e each o he p oposed solu ions in o de o de e mine i s powe losses and i s easibili y ega ding he ope a ional cons ain s o he EDS. To alida e he use o an AIS, he final esul s we e compa ed wi h o he solu ions ob ained wi h o he algo i hms in he li e a u e. O he impo an field o applica ion is he o ecas ing o elec ical a iables (loads and powe gene a ion o consump- ion). Abdul Hamid and Abdul Rahman [26] p opose an a i- ficial neu al ne (ANN) ained ollowing he beha io o an a ificial immune sys em (AIS) o gene a e a sho - e m load o ecas ing model. Two se s o elec ical ene gy demand da a we e used o es he capabili y o he p oposed algo i hm. The esul s p esen ed in he manusc ip show ha he p o- posed AIS lea ning algo i hm is capable o p o iding a o e- cas compa able o ha o an a ificial neu al ne wo k wi h an in eg a ed back p opaga ion (BP) algo i hm. Conse- quen ly, he AIS is an al e na i e lea ning algo i hm o an a ificial neu al ne wo k. The wo k p oposed by [27] is one 3Complexi y o he fi s s udies using an in eg a ed AIS simula ion o imp o ed o ecas ing o elec ici y consump ion wi h an- dom a ia ions. They de elop a new sys em wi h diffe en algo i hms, namely, AIS, gene ic algo i hm (GA), and pa i- cle swa m op imiza ion (PSO), o simula e annual elec ici y consump ions in selec ed coun ies. The mean absolu e pe - cen age e o (MAPE) is applied o e alua e he esul s and selec he bes o ecas ing model. A case s udy wi h da a o he annual elec ici y consump ions o 16 coun ies om 1980 o 2006 is analyzed. Fo he selec ed coun ies, he AIS me hod wi h he clonal selec ion algo i hm (CLONALG) shows sa is ac o y esul s when applied wi h simula ed da a and has been selec ed as he p e e ed me hod. He nandez e al. [28] model a hyb id a ificial immune sys em (AIS) combining he back p opaga ion me hod wi h he a ificial immune sys em, o achie e highe accu acy, lesse inpu load da a equi emen , and as e con e gence. The hyb id app oach is implemen ed, and i s esul s a e compa ed wi h a GA and a PSO. This analysis e eals ha AIS sol es he p oblem in a mo e efficien way han do GA and PSO. Dudek [29] p oposes a sho - e m load o ecas model based on an AIS o p edic he hou ly load demand o a week. In his p oposed echnique, each an igen o AIS, which con- ains he ime se ies load sequences (some pa is a o ecas sequence), is compa ed wi h his o ical load pa e ns. MAPE is also used o e alua e he pe o mance o he p oposed o e- cas model. The sys em achie es a minimum MAPE o 1.77%, which means he AIS ob ains e y success ul esul s. AISs a e also applied o economic op imiza ion in an elec i- cal en i onmen . Dynamic economic dispa ch de e mines he op imal scheduling o online gene a o ou pu s wi h p e- dic ed load demands o e a ce ain pe iod o ime aking in o conside a ion he amp a e limi s o he gene a o s [30]. Basu [31] p esen s an a ificial immune sys em algo i hm ha sol es a hea and powe economic op imiza ion p ob- lem. The AIS is adap ed o his p oblem, adding new ope a- ions such as hype mu a ion and ou namen , and is hen used in a p elimina y es sys em. Basu [30] implemen s adap i e cloning, hype mu a ion, aging ope a ion, and ou namen selec ion. In o de o al- ida e he new AIS, nume ical esul s o a en-uni sys em wi h uel cos unc ion ha e been de eloped. The esul s ob ained om he p oposed algo i hm a e compa ed wi h hose ob ained om pa icle swa m op imiza ion and e o- lu iona y p og amming. F om nume ical esul s, i is shown ha he p oposed AIS p o ides a mo e efficien solu ion han do pa icle swa m op imiza ion and e olu- iona y p og amming in e ms o minimum cos and com- pu a ion ime. A agón e al. [32] p esen an AIS-inspi ed algo i hm, called IA EDP, which ies o sol e an eco- nomic dispa ch p oblem. I makes use o wo e sions o a edis ibu ion powe ope a o which ies o keep he solu ions ha i finds. The p oposal is applied o eigh p oblems aken om he li e a u e. The esul s a e com- pa ed wi h hose de i ed om se e al o he app oaches o de e mine he ad an ages o he IA EDP agains classi- cal e olu iona y compu ing. This b ie backg ound leads us o he conclusion ha AIS can be applied o a a ie y o elec ical con ex s wi h e y success ul esul s. This ac encou ages us o wo k wi h a specific AIS, called Op -aiNe , also le e aged in di - e en pape s [24, 25] and o adjus i specifically o ou case s udy. The classical Op -aiNe usually wo ks wi h a low numbe o a iables (each indi idual con ains abou 6 a iables a mos ) and wi hou cons ain s encoded as ma hema ical unc ions. In he p esen wo k, his algo- i hm is adjus ed o admi up o 336 a iables and 25 lin- ea cons ain s (inequali ies and equa ions). 3. Home Ene gy Managemen P oblem In he designed case s udy, we conside a home elec ical sys em ha has some household appliance connec ed o i (Figu e 1). The con ex can be hough as a domes ic g id wi h a gene a ion pa and a consump ion pa , connec ed o he powe g id. As shown in Figu e 1, he gene a ion sys em o he PV sys em includes he PV gene a o and he ba - e y. The consump ion pa s a e he elec ic loads which con ain he ollowing appliances: a space hea e , a s o age wa e hea e , and mus - un se ices. To balance he p ofi o ene gy se ices be ween he PV sys em and he loads, he schedule is connec ed o he g id. The schedule aims a maximizing he p ofi o ene gy se ices p o ided in a Space hea e Powe g id Ba e y PV panel Schedule Wa e s o age hea e Mus - un se ices Figu e 1: Schema ic image o domes ic elec ical sys em. 4 Complexi y domes ic ene gy managemen sys em h ough (2) OF, which is he objec i e unc ion o op imize. OF = 〠 λsoldPsold −λbough Pbough −〠 j∈ELs VOLLjLj shed −Vs p 2 OF is a linea combina ion o ou elec ical ac o s. λsold,λbough , VOLLjand Vp a e cons an s ha p o ide he p ices pe uni o ene gy load and a e gi en by he ma ke . The fi s e m λsoldPsold ep esen s he income om he sale o ene gy p oduced by he PV panel o he elec ici y g id. The second ac o is he o al cos o elec ical ene gy ha is bough om he ne wo k, λbough Pbough . The alue o elec ical ene gy is no se ed, meaning he lo is encoded in he hi d pa ,∑j∈ELsVOLLjLj shed. Finally, he spillage cos s o PV panels, Vs p Sp , a e ep esen ed in he las e m o he equa ion. We need o balance he loads be ween he ene gy gene - a ed ( he PV sys em Pp , he ene gy p o ided by he ba - e y Pb,ou , and he ene gy bough om he powe g id Pbough ) and he ene gy consumed, meaning he elec ical loads o he diffe en se ices Lj −Lj shed (hea e , s o age wa e hea e , and mus ung se ices) and he ba e y cha ge Pb,in (3). Addi ionally, he powe flow limi a ion h ough he dis ibu ion line is s a ed in (4), whe e max is a cons an se o 6 acco ding o [33]. Pbough +Pp +Pb,ou =〠 j∈ELs Lj −Lshed j +Pb,in , 3 − max ≤Pbough −Psold ≤ max 4 The specificdefini ions o all domes ic appliances a e desc ibed in he ollowing subsec ions. 3.1. PV Sys em. The PV sys em can gene a e he powe ou - pu Pp o he g id, which can be modelled h ough Equa ion 5. Pp =Pp ,p −Sp , 5 whe e S e e s o he spillage cos s o he PV sys em and P p ,p ( ) is he po en ial powe gene a ion o he PV sys em. P p ,p ( ) is limi ed o maximum and minimum bands due o he p edic ion o he PV powe gene a ion, ollowing (6). σdown p and σ p up a e down and up p edic ion a iances o he PV sys em, espec i ely, and a e calcula ed ollowing [34]. Pp ed p is he p edic ed powe gene a ed by he PV sys- em. This amoun is posi i e o equal o ze o and is limi ed o he ac ual powe gene a ion o he PV, P p ,p ( ), as ep esen ed in (7). In o he wo ds, he PV sys em can po en ially gene a e his powe bu HEMS canno ope a e i because o economic and echnical cons ain s. Pp ed p −σdown p ≤Pp ,p ≤Pp ed p −σup p ,6 0≤Sp ≤Pp ,p 7 3.2. Elec ical Loads. Elec ical loads include loads ha can be con ollable and/o shi able. In his case s udy, h ee ypes o loads a e modelled: space hea e , L sh ( ), which is a con ollable load, s o age wa e hea e , L swh ( ), which is a shi able load, and mus - un se ices, L m s ( ), which a e noncon ollable-shi able loads. Equa ions 8 and 9 define he o al elec ical load and o al load shedding o ou domes ic g id, espec i ely. These loads a e desc ibed in he ollowing subsec ions. 〠 j∈ELs Lj =Lsh +Lswh +Lm s ,8 〠 j∈ELs Lshed j =Lshed sh +Lshed swh +Lshed m s 9 3.2.1. Space Hea e . The space hea e p o ides he desi ed indoo empe a u e. Equa ion 10 ep esen s he pe o mance o he space hea e based on he ela ionship be ween he indoo empe a u e and i s elec ical load. In (10), θ 0 is he ini ial indoo empe a u e in ime = 1, which is assumed o be equal o he desi ed empe a u e. Ris he he mal esis- ance, and Cis he he mal capaci y o θin +1 =θin e−1/RC +Lsh R 1−e−1/RC +θp ed ou 1−e−1/RC , ≥2θin =θ0=θdes,  =1 10 Equa ion 11 ep esen s he limi a ion o he indoo empe a u e. In ou case s udy, his limi a ion is se o 1 ° C mo e o less han he desi ed empe a u e. Finally, due o physical ac o s, he loads and he load sheddings a e bo h limi ed by hei maximum and minimum con- s ain s (12) and (13). −1≤θin −θdes ≤1, 11 Lmin sh ≤Lsh ≤Lmax sh ,12 0≤Lsshed sh ≤Lsh 13 3.2.2. S o age Wa e Hea e . The s o age wa e hea e is esponsible o p ese ing he hea in he wa e anks. The maximum and minimum limi a ions o he s o age wa e hea e ’s load a e s a ed in (14). The maximum ene gy con- sump ion o he s o age wa e hea e should be less han he maximum capaci y o he ank U swh , acco ding o (15). Finally, he maximum o he load shedding o he s o age wa e hea e is always less han he ene gy consump ion o he appliance (16). Lmin swh ≤Lswh ≤Lmax swh ,14 〠 N =1 Lswh =Uswh,15 5Complexi y 0≤Lshed swh ≤Lswh 16 3.2.3. Mus -Run Se ices. Mus - un se ices consis o loads ha should be p o ided quickly, and he e o e, i is no easy o p edic hem, o example, ligh ing and en e ain- men . Fo he pu poses o his pape , i is assumed ha he e is no unce ain y in p edic ing he elec ical loads o mus - un se ices (17). As in he s o age wa e hea e , he maximum o he load shedding Lshed m s mus always be less han he ene gy consumed L m s ( ) (18). Also, he load shedding cons ain is s a ed in (18). Lm s =Lp ed m s ,17 o≤Lshed m s ≤Lm s 18 3.3. Ba e y Sys em. The ba e y sys em can be used o apply he cha ge and discha ge s a egies in he HEMS. A flowcha (Figu e 2) is designed o ope a e wi h he ba e y in he domes ic en i onmen . The sys em aims a p o iding he equi ed elec ical demand, maximizing i s benefi s. When he e is a su plus P b,in ( ) o he ene gy gene a ed (i.e., he PV panel gene a es mo e ene gy P p ( ) han he o al load TL( ) demands), i is s o ed in he ba e y (C b ( )). I he ba - e y is ully cha ged Cb >Cmax b, hen i is sold o he g id P sold ( ). On he con a y, he sys em makes use o he ene gy s o ed in he ba e y P b,ou ( ) when he elec ical demand is highe han he powe gene a ion o he PV panel. Addi ion- ally, i he ba e y canno p o ide he ene gy needed (i.e., is una ailable o comple ely discha ged, Cb >Cmin b), he sys- em will buy he elec ici y P bough ( ) om he powe g id. 4. Expe imen al Se ing To assess he pe o mance o he p oposed HEMS, some pa ame e s ha e been se o op imize he sys em. The maxi- mum powe p oduced by he PV sys em is 2kW. The ba e y can s o e be ween Cb min 48 and Cb max 2.4 kWh. The maximum hea ing powe o he space hea e (SH) Lmax sh equals 2 kW o main ain he empe a u e o he house wi hin ±1 o he desi ed empe a u e (θ des 23 ° C). The he mal esis ance, R, o he building shell is equal o 18 ° C/kW, and he capaci y Cequals 0.525 kWh/ ° C. The ene gy capaci y o he s o age wa e hea e (U swh ) is 10.46 kWh (180 L) which has 2 kW as maximum o hea ing elemen s Lmax swh . Table 1 displays he p e- dic ed da a ha has been used in [33]. Table 2 gi es he p ice da a o he sys em. VOLL and spillage cos s o PV powe gene a ion a e shown in Table 3. These da a a e used in ou HEMS o op imize he unc ion (2). The objec i e unc ion is in eg a ed in o Op -aiNe o ge an op imized schedule o 24 hou s in a domes ic en i onmen . Following he Op -aiNe p ocedu e, he ini ial popula ion an- domly gene a ed, whe e each indi idual is a se o elec ical alues o 24 hou s, mus comply wi h he cons ain s modelled in he elec ical managemen p oblem. Fo his pu pose, each da um o he indi idual is conside ed as an elec ical pa ame e o op imize o 24 hou s. Ini ially, he pa ame e s which only depend on some fixed bounda ies (6), (7), (12), (14), and (17) a e gene a ed. Then, hese pa ame e s a e used as new bounda ies o hose elec ical pa ame e s ha depend on hem ( he es o heequa ionsa egi enin hemodelo Sec ion3). This me hodology is ecu si ely applied un il all he pa ame e s needed o each indi idual a e gene a ed. Finally, o ob ain he pa ame e s ela ed o he ba e y load, he pa ame e s p e i- ously calcula ed and he flowcha o Figu e 2 a e applied o gene a e he new ones. Wi h he indi iduals gene a ed, Op -aiNe wo k ollows hese s eps: (1) Ini ia e Npopula ion ollowing he me hod abo e, o espec he linea cons ain s and he flowcha i i is he case. (2) E alua e each indi idual acco ding o he op imiza- ion unc ion gi en in (2). (3) C ea e N c clones o each indi idual. The elemen s o each clone should be sligh ly changed acco ding o he mu a ion equa ion (1). (4) Fo each cell o an ibody, selec he bes clone wi h he highes objec i e alue. (5) I he mean objec i e o he las i e a ion and he p esen one a e below a limi , hen simila indi iduals a e supp essed acco ding o he simila i y h eshold s ha measu es dis ances be ween wo an ibodies. (6) I some indi iduals a e supp essed, hen i is needed o add a new andom popula ion. These new solu- ions a e gene a ed ollowing he me hod gi en abo e o espec he cons ain s and he flowcha i ha is he case. (7) This wo k flow is epea ed un il he con e gence c i- e ion (maximum numbe o i e a ions gen). The esul is one o mo e indi iduals wi h an op imum objec i e alue. As we can see, AIS con ains fi e pa ame e s, namely, numbe o indi iduals N, numbe o clones N c , simila i y h eshold s , maximum numbe o gene a ions gen, and he mu a ion pa ame e β. Depending on he p oblem, hey can ake se e al alues and a e essen ial o a co ec ope a ion. AIS needs some pa ame e s o be se be o e- hand in o de o op imize a p oblem co ec ly. These pa ame e s a e ela ed o he cloning and mu a ion p o- cess, he supp ession algo i hm, and he con e gence c i e ion. Fo each i e a ion, a numbe o clones N c is gen- e a ed pe cell. This numbe N c is se empi ically and can influence he final esul s. Gene ally, i N c is se wi h a e y low alue, he con e gence c i e ion can be delayed, as we a e no able o find enough di e si y o selec be e indi- iduals o each cell. O he wise, i oo many clones a e gene a ed, he ime upon con e gence migh be longe han expec ed. We empi ically se he AIS pa ame e s acco ding o Table 4, which ga e he op imal pe o mance in e ms o fi ness and ime. The nex sec ion will desc ibe he simula ions and esul s wi h Op -aiNe configu ed o he p esen ed HEMS. 6 Complexi y 5. Simula ion Resul s The e alua ion is wo old. Fi s ly, i is expec ed ha Op - aiNe ob ains posi i e esul s in sol ing op imiza ion p oblems in he c ea ed se ing and compa ing hem wi h o he classical bioinspi ed app oaches. Consequen ly, a compa a i e analysis was ca ied ou be ween a classical gene ic algo i hm (GA) and a pa icle swa m op imiza- ion (PSO). Addi ionally, we analyzed he impac o he flowcha (Figu e 2) on ou sys em, when he ba e y was in ol ed. The e o e, wo s a egies ha ep esen wo elec ical si ua ions a e designed. In he fi s s a egy (s a egy 1), he domes ic en i onmen does no conside any a iable ela ed o he main enance o he domes ic elec ic cha ge h ough he elec ical ene gy p oduced by he PV sys em. This s a egy only aims a op imizing i s ene gy benefi s. In he second s a egy (s a egy 2), he home en i onmen aims a supplying he elec ici y demand au onomously. The e o e, he su plus gene a ed by he PV is s o ed in he ba e y. In his s a egy, he elec ici y could be sold o he g id i he ba e y is comple ely cha ged. On he con a y, i he ba e y can- no supply all he elec ical cha ge, hen he HEMS mus buy elec ici y om he powe g id. S a Ini Pp ( ) > TL( ) Pp ( ) = TL ( ) Pb,in ( ) = Pp ( )−TL( ) Pbough ( ) = 0 Cb ( ) = Cb ( −1) + Pb,in ( )−Pb,ou ( ) Cb ( ) = Cb ( −1) + Pb,in ( )−Pb,ou ( ) Cb ( ) = Cbmin Cb ( ) = Cbmax Pb,in ( ) = Cbmax−Cb ( −1) +Pb,ou ( ) Psold ( ) = Pp ( )−TL( )−Pb,ou ( ) Pb,ou ( ) = TL( ) -Pp ( ) Pb,ou ( ) = Cb ( −1)− Cbmin +Pb,in ( ) Pbough ( ) = TL ( )− Pp ( )−Pb,ou ( ) Cb ( ) > Cbmax Cb ( ) > Cbmin Psold ( ) = 0 Psold ( ) = 0 Pb,in ( ) = 0 Psold ( ) = 0 Pbough ( ) = 0 Pb,ou ( ) = 0 Pb,in ( ) = 0 Pb,ou ( ) = 0 Yes No Yes Fin Yes Yes No No End End No Pbough ( ) = 0 Figu e 2: Flowcha modelling he ba e y pa ame e s. 7Complexi y Based on hese wo main goals, h ee diffe en expe i- men s a e conside ed: (i) Expe imen I: compa a i e s udy be ween GA, PSO, and AIS (ii) Expe imen II: compa ison be ween s a egy 1 and s a egy 2 (iii) Expe imen III: analysis o s a egy 2 when he ba - e y is disconnec ed o connec ed Expe imen I makes a compa ison be ween GA, PSO, and AIS. Expe imen II op imizes he pa ame e s ela ed o he PV sys em, he space hea e , he wa e hea e , and he mus - un se ices; he e o e, he pa ame e s ela ed o he ba e y cha ge a e no conside ed (s a egy I). The esul s ob ained om he op imiza ion p ocess a e compa ed wi h he esul s ha a e e ie ed when he pa ame e s o he ba - e y a e included in he sys em, al hough all a e se o 0. Tha means he ba e y is disconnec ed om he home en i on- men bu he AIS ollows he flowcha o op imize he sys em. Finally, expe imen III pe o med wo diffe en analyses: when he ba e y was disconnec ed and when he ba e y was connec ed, o s udy he impac o his de ice on he sys em. 5.1. Compa ison be ween GA, PSO, and AIS. In his sec ion, we aim o compa e he esul s ob ained wi h h ee diffe en bioinspi ed algo i hms: gene ic algo i hm, pa icle swa m op imiza ion, and a ificial immune sys em. The gene ic algo i hm (GA) is widely used in op imiza ion p oblems wi h many configu a ions. In his pape , a classical app oach o GA is applied so ha he mu a ion and c osso e as well as he numbe o gene a ions a e empi ically se o 0.3, 0.8, and 2000, espec i ely. Addi ionally, he selec ion unc ion chosen was he oule e algo i hm. The pa icle swa m op imiza ion (PSO) was fi s in o- duced by Ebe ha and Kennedy [35] and consis s o a popula ion-based op imiza ion algo i hm which is deemed o be a na u e-inspi ed op imiza ion me hodology. PSO employs a se o pa icles which would mo e h ough he sea ch space a e e y i e a ion and would calcula e he fi ness alue a each poin . A e a e mina ion condi- ion is me , he bes op imized alue is selec ed by choos- ing he bes alue ound in he his o y o pa icles. The success o PSO lies in i s “ eloci y equa ion”; his equa ion decides on he nex poin in space ha each pa icle would mo e o. The eloci y equa ion can be shown as Vid,k+1 =wVid,k+c1× and1×Xpbes id,k−Xid,k +c2× and2×Xgbes d,k−Xid,k 19 He e, V id,k+1 is he eloci y o he d h dimension o he i h pa icle in he nex i e a ion (k + 1s ), wis he ine ia o he pa icle, V id,k is he eloci y o he d h dimension o he i h pa icle in he cu en i e a ion (k h i e a ion), c 1 is he cogni i e accele a ion cons an , c 2 is he social accele a ion cons an , X pbes id,k is he posi ion o he pa i- cle in he d h dimension a which he bes solu ion was Table 4: Op ima alues se o N,N c ,gen, s , and βin bo h s a egies. NN c gen s β S a egy I 250 12 250 10 100 S a egy II 250 18 300 3 10 Table 1: P edic ed da a o unce ain a iables. P p p ed( )σ p up σ p down θou p ed( )Lp ed m s ( ) 1 0 0.03 0.01 5.5 0.3 2 0 0.03 0.01 5.5 0.3 3 0 0.03 0.01 5.2 0.3 4 0 0.03 0.01 5.2 0.3 5 0 0.03 0.01 4.8 0.4 6 0 0.03 0.01 5.5 0.6 7 0.25 0.03 0.01 6.5 0.8 8 0.75 0.03 0.01 7.5 0.8 9 1.25 0.03 0.01 9.8 0.7 10 1.75 0.03 0.01 10.1 0.55 11 1.9 0.03 0.01 11.5 0.5 12 1.9 0.03 0.01 12 0.5 13 1.9 0.03 0.01 12.5 0.5 14 1.75 0.03 0.01 12 0.5 15 1.25 0.03 0.01 11.5 0.6 16 0.75 0.03 0.01 10 0.8 17 0.25 0.03 0.01 9 1.5 18 0 0.03 0.01 8.5 1.8 19 0 0.03 0.01 8 1.7 20 0 0.03 0.01 7.5 1.1 21 0 0.03 0.01 7 0.9 22 0 0.03 0.01 6.5 0.7 23 0 0.03 0.01 6.2 0.6 24 0 0.03 0.01 6 0.4 Table 2: P ice da a o he sys em. P ice ($/MW) Time (hou ) λ i λne 23–7 2.2 0.0814 8–14 2.2 0.1408 15–20 2.2 0.3564 21-22 2.2 0.1408 Table 3: VOLL and spillage cos s. VOLL ($/MW) Spillage cos ($/MW) Time (hou ) SH SWH MRS PV 22–7 1 1 2.2 4 8–21 1 1 2.2 4 8 Complexi y ound so a by he i h pa icle, X gbes d,k is he alue o he d h dimension a which he bes solu ion so a was ound by he whole sys em, X id,k is he cu en posi ion o he i h pa icle in he d h dimension, and c 1 and c 2 a e andom numbe s. Using he eloci y calcula ed, he nex posi ion o be e alua ed is calcula ed as Xid,k+1 =Xid,k+Vid,k20 The abo e eloci y equa ion wo ks well when he sys em is unde no cons ain s. Howe e , when he e a e cons ain s, some o he pa icles migh all off he easible egion. In his case, he pa icles should no upda e hei pe sonal bes when a pa icle is ou side he easible egion. Nei he should he global bes be upda ed in case o a pa icle being ou side he easible bounda y, ha ing a be e alue han he cu en global bes . In case a pa icle has no ound a pe sonal bes ha alls in he easible egion, he pa icle should ely only on global bes o guide i s mo emen . In his case, he calcu- la ion o eloci y would change o Vid,k+1 =wVid,k+c1+c2× and × Xgbes d,k−Xid,k21 Howe e , equali y cons ain sa is ac ion is mo e difficul han he inequali y cons ain sa is ac ion. To sol e his p ob- lem, a mending p ocedu e is ca ied ou a e e y i e a ion o make su e all pa icles sa is y he equali y cons ain . The mending p ocedu e calcula es he o e shoo o each o he pa icle and adds a co ec ion alue o co ec he e o . The co ec ion is equally added o each o he in e als o he pa icle. In o de o demons a e he efficiency o he a ificial immune sys em in he ene gy managemen op imiza ion p oblem, we pe o med a compa a i e es wi h a e sion o cons ained PSO adop ed om [36] and an app oach o he gene ic algo i hm. The goal was o p edic he op imum alues o each a iable du ing 24 hou s, ollowing s a egy 1 and s a egy 2. The linea cons ain s p oposed in he elec- ical model a e applied, and subsequen ly, he objec i e unc ion o he op imized a iables is measu ed. Table 5 shows he esul s ob ained when he objec i e unc ion and he ene gy a e bough and consumed when AIS, GA, and PSO a e applied wi h an op imal se ing. As Table 5 shows, AIS ob ains sligh ly be e esul s in he objec i e unc ion o bo h s a egies when bo h se ings a e op imal o sol e his p oblem. Tha means AIS is a be e configu a ion o he elec ical pa ame e s o op imize he p oblem s a ed. Likewise, AIS is able o find lowe alues o he amoun o ene gy bough , which means he algo i hm allows sa ing ene gy and money o he cus ome . The esul s o he AIS o he ene gy sold a e lesse han in PSO and in GA. Tha could be explained because in PSO and in GA, he mean amoun s o ene gy ha a e managed a e g ea e han in AIS. Howe e , PSO also ob ained posi i e esul s (a leas , be e han wi h he classical app oach o GA); he e o e, we p opose o make a deepe compa ison wi h mo e complex p oblems o alida e bo h algo i hms and s udy he limi a ions o each one. 5.2. Analysis o he Ene gy Managemen S a egy. In his sec- ion, he wo s a egies desc ibed below a e deeply analyzed and compa ed. The fi s s a egy looks o maximizing he domes ic ene gy p ofi . Howe e , he second s a egy aims a maximizing ene gy p ofi and ac ing as an au onomous ene gy sys em. I is expec ed ha s a egy II ob ains be e esul s, as his s a egy pu sues he au onomous managemen o ene gy, sa ing mo e money han in s a egy I. Op -aiNe wo ked wi h indi iduals o ec o o 264 elemen s wi h equali y and inequali y cons ain s, as each indi idual con ains all he a iables o 24 hou s. Each alue co esponds o he diffe en elec ic loads and pow- e s desc ibed in Sec ion 3 o he PV panel, he s o age wa e hea e , he space hea e , and he mus - un se ices. We un he algo i hm wi h wo diffe en configu a ions, one wi h s a egy 1, excluding all he cons ain s ela ed o ba e y managemen , and one wi h s a egy 2, including all he cons ain s ela ed o ba e y managemen , bu wi h he ba e y a iables se o 0. Table 6 shows he esul s acco ding o he fi ness alue o he objec i e unc ion OF and he pa ame e s o sold and bough ene gy. As we can see, he alue o he objec i e unc ion in s a - egy 1 is highe han ha o s a egy 2. Howe e , he ans- ac ed ene gy be ween home and powe g id is lesse in s a egy 1, which means ha S a egy 2 allows o he au on- omous managemen o ene gy a home. 5.3. Impac o he Ba e y. In his analysis, s a egy 2 is applied o s udy he influence o he ba e y in he domes ic en i onmen . In his case pa icula ly, indi iduals in he AIS ha e 336 elemen s because he pa ame e s co espond- ing o he ba e y cha ge and load a e se as elemen s o op i- miza ion. Each indi idual is cons uc ed ollowing he linea cons ain s and he flowcha be o e being inse ed in o he popula ion. Two diffe en execu ions a e made, fi s ly, wi h all he a iables ela ed o he ba e y managemen se o 0. In he second execu ion, he a iables o he ba e y could change acco ding o he co esponding equa ions and flowcha (Figu e 2). The esul s ob ained a e shown in Table 7. F om he da a in Table 7, we can see ha he ba e y sys- em can imp o e he alue o he objec i e unc ion. Table 7 also conside s a si ua ion in which he ba e y inc eases he amoun o elec ical ene gy sold om he sma home o he g id, and i dec eases he amoun o elec ical ene gy ha a home buys om he ne wo k. Table 5: Resul s o he objec i e unc ion and he ene gy bough and consumed when AIS and PSO a e applied wi h an op imal se ing. Objec i e alue Ene gy bough Ene gy sold S a egy 1 PSO 23.52 46.61 14.12 GA 22.48 46.94 14.09 AIS 23.86 45.66 14.22 S a egy 2 PSO 4.86 37.60 5.40 GA 4.92 32.66 5.22 AIS 5.11 26.53 4.64 9Complexi y