Resource management with kernel-based approaches for grid-connected solar photovoltaic systems

Heliyon 7 (2021) e08609 Con en s lis s a ailable a ScienceDi ec Heliyon jou nal homepage: www.cell.com/heliyon Resea ch a icle Resou ce managemen wi h ke nel-based app oaches o g id-connec ed sola pho o ol aic sys ems V.S. Bha a h Ku uku ua, Ah eshamul Haquea, Mohammed Ali Khanb, F ede Blaabje gc,∗ aAd ance Powe Elec onics Resea ch Lab, Depa men o Elec ical Enginee ing, Jamia Millia Islamia, New Delhi, India bDepa men o Elec ical Powe Enginee ing, Facul y o Elec ical Enginee ing and Communica ion, B no Uni e si y o Technology, B no, Czech Republic cDepa men o Ene gy, Aalbo g Uni e si y, Denma k A R T I C L E I N F O A B S T R A C T Da ase link: h p://si es .ieee .o g /pes - es eede s /files /2017 /08 / eede 123 .zip Keywo ds: Pho o ol aic powe Sma in e e s Reac i e powe con ol Ke nels Powe loss Vol age egula ion The inc easing pene a ion o pho o ol aic (PV) powe gene a ion in o he dis ibu ion g ids has esul ed in equen e e se ac i e powe flows, apid fluc ua ions in ol age magni udes, and powe loss. To o e come hese challenges, his pape iden ifies he esou ce managemen o g id-connec ed PV sys ems wi h ac i e and eac i e powe injec ion capabili ies using sma in e e s. This app oach is aimed o minimize he ol age de ia ions and powe losses in he g id-connec ed sys ems o accommoda e he high pene a ion o PV sys ems. A ke nel- based app oach is p oposed o lea n policies and e alua e he eac i e powe injec ions wi h sma in e e s o imp o ing g id p ofile, minimizing powe losses, and main aining sa e ope a ing ol age limi s. The p oposed app oach pe o ms in e e coo dina ion h ough nonlinea con ol policies using an icipa ed scena ios o load and gene a ion. To assess he pe o mance o he p oposed app oach, nume ical simula ions a e pe o med wi h a single-phase g id-connec ed PV sys em connec ed o an IEEE bus sys em. The esul s show he effec i eness o he p oposed app oach in minimizing powe losses and achie ing a good ol age egula ion. 1. In oduc ion Pho o ol aics (PV) is conside ed as a logical solu ion o handle he d awbacks in con en ional gene a ion esou ces due o hei local a ail- abili y, alling p ices, and sus ainabili y. Ne e heless, he inc easing sha e o enewable ene gy sou ces in he ne wo k is causing se ious p oblems o he g id, such as e e se powe flow, ol age fluc ua ion, e c. [1]. Mo eo e , hese p oblems a e caused due o he emo e injec- ion o enewable ene gy ha e s ained he appa en powe capabili ies o subs a ion ans o me s [2]. Besides, he fluc ua ions obse ed a he esiden ial PV gene a ions b ing up he issue o unce ain y in gene - a ion depending mos ly on clima e and geog aphical loca ion o he sys em. This has esul ed in a highly dynamic and unp edic able eal powe gene a ion. Thus, o a oid hese fluc ua ions and ha e a s able g id ope a ion, ol age egula ion is equi ed. T adi ionally, ol age egula ion is ca ied ou using diffe en ech- niques like on-load ap changing (OLTC) in subs a ion ans o me s, swi ching o capaci o banks, and s ep ol age egula o s. In [3], [4], he issue o eac i e powe -sha ing is sol ed ia consensus-based dis- ibu ed ol age con ol. He e, he de eloped ol age con olle is com- *Co esponding au ho . E-mail add ess: [email p o ec ed] (F. Blaabje g). bined wi h a con en ional d oop-based con ol echnique o elimina - ing he line impedance misma ch. In [5], [6], a coo dina ed con ol s a egy is p oposed o eac i e powe injec ion wi h a g id in e- g a ed dis ibu ed gene a ion (DG) sys em. This app oach coo dina es he DGs and con ollable de ices by cons aining sys em a iables unde a p esc ibed ope a ing condi ion. I is iden ified ha hese echniques c i ically challenge he eac i e powe con ol due o he inc easing unce ain y in eal powe gene a ion. The e o e, o alle ia e hese p ob- lems, he use o sma in e e echnology wi h DG sys ems is widely adop ed. T adi ionally, PV sys ems a e in e aced wi h in e e s p ima ily o MPPT and DC-AC con e sion, and o achie ing g id in eg a ion o o m a DG sys em. In he p esen day scena io, hese in e e s a e upg aded by in e acing hem wi h ad anced communica ion, me e ing, and con- ol unc ionali ies [7]. These in e e s p o ide sma mul i-uni con ol by egula ing he eal powe limi , achie ing amp a e o eal powe limi , con olling eac i e powe ou pu o powe ac o (PF), ide- h ough capabili y o specific g id dis u bances, bi-di ec ional powe flow capabili y, and al e na i es o con en ional ans e ip schemes [7]. The use o sma in e e s o eac i e powe con ol p o ides a as h ps://doi.o g/10.1016/j.heliyon.2021.e08609 Recei ed 27 Ap il 2021; Recei ed in e ised o m 28 Oc obe 2021; Accep ed 13 Decembe 2021 2405-8440/©2021 The Au ho (s). Published by Else ie L d. This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/). V.S.B.Ku uku u,A.Haque,M.A.Khane al. Heliyon 7 (2021) e08609 esponding solu ion o a ious g id objec i es such as powe loss min- imiza ion and ol age egula ion [2]. In [8], a me hod o compensa e ol age imbalance is pe o med by injec ing ac i e and eac i e powe con ol h ough he powe condi ioning sys em o in e e s. Mo eo e , in [9, 10, 11, 12, 13], he coo dina ed con ol schemes a e p oposed o conse ing ol age educ ion in a sma in e e . The echniques coo dina ed he ope a ions o au oma ed ol -VAR con olle s and ag- g ega ed he eac i e powe con ol. Fu he , wi h he inc easing pen- e a ion le el o PV powe in o he g id mo e sophis ica ed ules o in e connec ion a e eme ging oo. In elligen solu ions o he p oblems p esen in he g id ha nessing he in e e con ol capabili ies will be he key o success ul implemen a ion o la ge-scale PV gene a ion in he dis ibu ion g id [14]. In [15], a hie a chical coo dina ed ol -VAR op imiza ion me hodology is p oposed. The issue ela ed o mul iple ob- jec i es has been add essed by implemen ing a uzzy decision-making me hod and an 𝜀-cons ain me hod. These in elligen solu ions inco - po a e a la ge ange o con ol unc ions in o newe PV in e e designs, which enhances he ope a ion o he dis ibu ion g id [7]. Mo eo e , acco ding o he amended IEEE 1547 s anda d [16], in- e e s a e allowed o ope a e a non-uni PF, gi ing hem he eedom o imp o e he g id ol age p ofile [17]. Besides, wi h he inc easing numbe o in e e s in he g id, i mus be no ed ha he coo dina- ion o each in e e needs o be conside ed o achie e g id s abili y. Gene ally, he PV gene a ion and ins an aneous loads om any node in a ypical dis ibu ion g id se up a e communica ed o a cen al u ili y con olle [18]. This con olle compu es he eac i e powe injec ion se -poin s and communica es hem o he in e e s a diffe en nodes o minimize he ohmic losses subjec o ol age egula ion cons ain s. A his ins ance, he u ili y con olle has he ask o iden i ying he op i- mal se -poin s o achie ing eac i e powe injec ion o he in e e s. This can be defined as an op imal powe flow ask, which is gene - ally non-con ex. In adial ne wo ks, his ope a ion can be eased in o a second-o de cone p og am h ough pola coo dina es [19], whe e he p oblem o powe loss and ol age de ia ion minimiza ion is sol ed. To alle ia e he complexi y o he in ol ed op imiza ion p oblems, app ox- ima e g id models ha e been employed in [20, 21, 22]. The eac i e powe con ol p oblem can be sol ed using cen alized, decen alized, o local echniques [23, 24]. The cen alized app oaches need a good communica ion se up as global in o ma ion is needed o con ol ac ions [25, 26], whe eas, he decen alized me hods equi e local o neighbo - ing inpu s o e alua ing he con ol se ings o single and unbalanced mul iphase g ids [27, 28]. The pu ely localized schemes p o ide eac- i e powe suppo using only local measu emen s [29, 30]. Mo eo e , i is iden ified ha he cen alized schemes incu high compu a ional complexi y due o he communica ion o la ge da ase s be ween he con olle and each in e e [31]. Besides, he decen alized schemes exchange mul iple communica ions among in e e s [32, 33], and he local schemes ha e no gua an eed pe o mance as he con ol se poin s depend only on he local inpu s. This makes he sys em e y un eliable o dis u bances om o he nodes. [34]. In [35, 36], he combina ion o cen al and local ac i e/ eac i e powe con ol was adap ed o ol age egula ion in DG sys ems. As seen om he p io wo ks, mos o he ex- is ing app oaches ei he sol e p oblems locally o cen ally o h ough a combina ion o bo h while conside ing a linea decision ule on he inpu pa ame e s. Besides, he con ol o sma in e e s in he li e a- u e did no lea n he inpu /ou pu pai s o he in e e independen ly o achie ing op imal powe flow. Ins ead, hey combined as a mul i- unc ion lea ning ask by linea ly ela ing o he op imal powe flow p oblem. This o mula ion ailed o yield a spa se con ol because o he ol age de ia ion a he in e e ou pu s. The significance o spa se con ol is o join ly lea n he in e e ules by posing he op imal powe flow p oblem as a mul i- unc ion lea ning ask. This is conside ed as a esou ce ul ep esen a ion o in e e con ol de elopmen as i sa es he equi emen o communica ion elemen s. In ligh o hese issues, his pape p oposes a mapping o eac i e powe con ol app oaches as linea o nonlinea policies conce ning hei inpu ea u es. I is iden ified ha he linea policies a e es ic ed o cap u ing linea ela ions be ween he ea u es and dependen a i- ables, and e y o en can only cap u e second-o de s a is ical ela ions. Such limi a ions call o ex ensions o nonlinea and highe -o de al- go i hms. This is achie ed by adap ing ke nel-based lea ning o mod- elling he eac i e powe con ol policies. The majo con ibu ions o his pape a e: ∙A decen alized app oach is de eloped o e alua ing he eac i e powe con ol policies o a ol age egula ion cons ained p ob- lem. ∙The in e e coo dina ion is pe o med h ough nonlinea con ol policies designed on a slowe imescale using an icipa ed scena ios o load and gene a ion. ∙A ke nel-based lea ning algo i hm is u ilized o e alua e he con ol policies on he basis o inpu scena io da a. ∙The ke nel-based policies a e modeled as a nonlinea unc ion o inpu ea u e ec o making i p ac ically easible o achie ing he pe o mance and complexi y ade-off. The emaining sec ions o he pape a e o ganized as ollows: Sec- ion 2discusses he g id modeling o e alua e he eac i e powe dis- pa ch in a adial ne wo k. Sec ion 3discusses a ious p oblems wi h he exis ing con ol models and iden ifies he sho comings o diffe en me hods. The p oposed ke nel-based policies o eac i e powe con ol a e discussed in sec ion 4and he nume ical simula ions a e de eloped in sec ion 5. The esea ch is finally concluded in sec ion 6. 2. G id modelling The g id-connec ed sys em is modeled by conside ing a adial single- phase g id wi h 𝑁+1 buses (indexed by 𝑛 =1, … , 𝑁+1), and 𝑀 b anches. Gene ally, o a adial sys em wi h se e al b anches 𝑀=𝑁, e e y bus 𝑛 =1, … , 𝑁is connec ed o a unique pa en bus 𝜋𝑛 ia dis i- bu ion line shown in Fig. 1. He e, an app oxima ed linea ized dis ibu- ion flow (LDF) model is used o e alua e he eac i e powe dispa ch o he in e e s. The g id is modeled by he b anch flow equa ions gi en as [37] 𝑠𝑛=∑ 𝑘∈𝐶𝑛 𝑆𝑘−𝑆𝑛+𝑙𝑛(𝑟𝑛+𝑗𝑥𝑛)(1) 𝑣𝑛=𝑣𝜋𝑛−2Re[(𝑟𝑛−𝑗𝑥𝑛)𝑆𝑛]+𝑙𝑛(𝑟2 𝑛+𝑥2 𝑛)(2) |||𝑆2 𝑛|||=𝑣𝜋𝑛𝑙𝑛(3) whe e o e e y line 𝑛 he line impedance is 𝑧𝑛=𝑟𝑛+𝑗𝑥𝑛, 𝑙𝑛is he squa e o cu en magni ude in line 𝑛, 𝑆𝑛=𝑃𝑛+𝑗𝑄𝑛is he complex powe flow om he sending bus 𝜋𝑛 o bus 𝑛, 𝑠𝑛=𝑝𝑛+𝑗𝑞𝑛is he complex powe injec ion a bus 𝑛, 𝑣𝑛is he squa ed ol age magni ude a bus 𝑛, 𝐶𝑛is he se o child en buses o 𝑛, and he ini ial condi ion 𝑠0=∑𝑘∈𝐶0𝑆𝑘. Fo all nodes 𝑛 =1, … , 𝑁 he eal powe injec ion is collec ed as a ec o in p ∶= [𝑝1,…,𝑝 𝑁]T, and he eac i e powe injec ion in q ∶= [𝑞1,…,𝑞 𝑁]T, whe e hese injec ions can be w i en as p=p 𝑔−p 𝑐(4) q=q 𝑔−q 𝑐(5) whe e p𝑔is he ac i e powe gene a ion and he DG side, p𝑐is he inelas ic load powe , and q𝑔and q𝑐a e he eac i e powe injec ions a in e e , and load, espec i ely. Mo eo e , he complex nodel injec ions a e gi en by 𝑠 =𝑝 +𝑗𝑞, and he squa ed ol age magni udes a e s acked as 𝑣 ∶= [𝑣1,…,𝑣 𝑁]T. Be- sides, all lines ha e esis ance, eac ance collec ed oge he as ∶= [𝑟1,…,𝑟 𝑁]T, and 𝑥 ∶= [𝑥1,…,𝑥 𝑁]T, espec i ely. The eal and eac i e line flows a e defined as P ∶= [𝑃1,…,𝑃 𝑁]T, and Q ∶= [𝑄1,…,𝑄 𝑁]T, e- spec i ely, and he complex powe flows a e gi en as S =P +𝑗Q. F om (3) i is known ha he e exis s a nonlinea i y ha complica es he 2 V.S.B.Ku uku u,A.Haque,M.A.Khane al. Heliyon 7 (2021) e08609 Fig. 1. Dis ibu ion line 𝑙𝑛 om pa en bus 𝜋𝑛 o bus 𝑛. powe flow equa ions o he dis ibu ion line. Hence, o o e come his, he dis ibu ion g id is o en emodeled as a linea model using he lin- ea d i ing o ce (LDF) model [37]. As he line esis ance, and eac ance a e small and hei mul iplica ion wi h squa ed cu en magni ude is less o e alua ing he powe flow equa ions a a fla ol age p ofile, he las summands in (1) and (2) can be d opped o o mula e hem as a linea ized model. The g id connec i i y is cap u ed in he b anch-bus injec ion ma ix  𝐴.  𝐴∈{0,±1}𝑀×(𝑁+1) and can be pa i ioned as  𝐴=[𝑎0𝐴]. This e- duces he b anch-bus injec ion ma ix o 𝐴. 𝐴, which is an in e ible squa e ma ix 𝐹∶= 𝐴−1. He e, A ollows 𝑎0+A 1=0.(6) Using his connec i i y ma ix, he LDF can be ew i en as 𝑠=A TS(7) A = 2 Re [𝑑𝑔(𝑟−𝑗𝑋)𝑆]−𝑎0𝑣0(8) whe e 𝑣0is he squa ed ol age magni ude a he subs a ion. Using (6) 𝑆can be elimina ed om (7) and (8) gi ing he squa ed bus ol age magni ude o all buses 𝑛 =1, … , 𝑁as [38] ≃ 2Rp + 2Xq + 𝑣01𝑁(9) whe e R∶=F Tdg( )F (10) X∶=F Tdg(x)F (11) Since F ≥0, he ma ices R, Xa e also R ≥0, X ≥0. Mo eo e , by he p ope ies o he ma ices, i can be easily seen ha Rand Xa e sym- me ic posi i e defini e wi h posi i e en ies. Hence, bus ol ages o all he buses in he g id a e seen o inc ease i eal o eac i e powe in- jec ions inc ease in he g id. Since losses ha e been igno ed in (3), he squa ed ol age magni udes a e an o e es ima e conce ning i s o igi- nal squa ed ol age magni udes (9) wi h he bias depending on 𝑙′ 𝑛𝑠. Bu s ill, acco ding o he nume ical es s, he app oxima ion e o s in ol - age magni udes a e seen o be less han 0.001 𝑝.𝑢. 3. P oblem o mula ion Gene ally, he ac i e and eac i e powe injec ions 𝑝, 𝑞can be de- composed in o gene a ion and inelas ic load componen s as shown in (4) and (5). Fo known PV gene a ion 𝑝𝑔 𝑛and o comply wi h i s ap- pa en powe limi 𝑠𝑔 𝑛, he eac i e powe injec ed by in e e 𝑛is cons ained h ough linea inequali ies as ||𝑞𝑔 𝑛||≤𝑞𝑔 𝑛∶= √(𝑠𝑔 𝑛)2−(𝑝𝑔 𝑛)2.(12) Mo eo e , o ca e o he ol age egula ion in IEEE 1547 [16], a linea se o inequali ies can be added. ≤ ≤ (13) whe e , a e se acco ding o he egula ion guidelines and a e usually aken as ±(3% −5%)abou he nominal alue [16]. To e alua e ol age de ia ions a each bus in he g id, le he sum o squa ed ol age mag- ni ude de ia ions ∑𝑁 𝑛=1 (𝑣𝑛−𝑣0)2. Using he app oxima ion in (9), and by igno ing he inconsequen ial scaling ac o , he squa ed ol age de- ia ions a e Δ𝑠(q𝑔)∶= ‖‖‖Rp+X q‖‖‖ 2 2.(14) Besides ol age de ia ion, he ohmic powe losses a e ano he c i ical quan i y in he dis ibu ion g id ope a ion. The ac i e powe losses can be exp essed as 𝐿 =∑𝑁 𝑛=1 𝑟𝑛𝑙𝑛o ∑𝑁 𝑛=1 𝑟𝑛 𝑃2 𝑛+𝑄2 𝑛 𝑣𝜋𝑛 . Fo small ol age de i- a ions, as ad oca ed in [2], he powe losses (𝐿)can be app oxima ed as 𝐿=𝑣−1 0[PTdg ( )P+Q Tdg( )Q] (15) Using (10) and igno ing he inconsequen ial scaling by 𝑣−1 0≃1, he powe losses can be exp essed as 𝐿=p TRp + qTRq (16) Since pTRp is a cons an o a gi en se o da a, he con ol a iable qin he second summand in (16) is he unc ion o in e es . The powe loss unc ion can be exp essed as 3 V.S.B.Ku uku u,A.Haque,M.A.Khane al. Heliyon 7 (2021) e08609 𝐿(q𝑔)∶= qTRq.(17) Fo a posi i e defini e ma ix R, he posi i e con ex quad a ic unc ion o 𝐿(q𝑔)is gua an eed. The objec i es o ol age de ia ions Δ𝑠(q𝑔)and powe loss 𝐿(q𝑔)a e con adic ing in gene al. Thus, a mul i-objec i e p oblem can be sol ed o ca e o hese con adic ions. A con ex com- bina ion o he objec i es can be posed o o mula e he eac i e con ol op imiza ion p oblem gi en as min q𝑔𝜆Δ𝑠(q𝑔)+(1−𝜆)𝐿(q𝑔)s. .q∈(18) whe e he se  ⊆ℝ𝑁cap u es he linea cons ain s in (12) o all 𝑛 ∈ . The ol age de ia ions and he ohmic losses a e minimized by his o mula ion o diffe en alues o he pa ame e 𝜆 ∈[0,1], ela ed o he appa en powe limi . I should be no ed ha in he abo e exp ession no ol age egula ion limi s apply, bu he ol age de ia ion is minimized as a unc ion o cos . To apply he ol age egula ion limi , he p oblem can also be o mula ed as min q𝑔𝐿(q𝑔)s. o q ∈ , ∈(19) whe e he se ⊆ℝ𝑁cap u es he linea cons ain s in (13) o all 𝑛 ∈ . The e o e, in his o mula, he ohmic losses a e kep o a minimum wi h espec o he appa en powe limi and he ol age egula ion limi . 4. Me hodology 4.1. Con ol policies To minimize ne powe loss and ol age d op o he ne wo k, he model should es ima e he eac i e powe injec ed in o each node. This injec ion depends on he size, layou , ne wo k opology and configu a- ion o he in e e . In his s udy, he injec ion o eac i e powe 𝑞𝑔 𝑛by in e e 𝑛can be modeled as, 𝑞𝑔 𝑛(𝓏𝒾𝑛)=𝑓𝑛(𝓏𝑛)+𝑏𝑛(20) whe e he inpu s 𝓏𝒾𝑛, 𝑓𝑛, and 𝑏𝑛co espond o he con olle inpu ec o , con olle unc ion, and in e cep , espec i ely. Con olle inpu s: Vec o 𝓏𝒾𝑛∈𝐼𝑛⊆ℝ𝑀𝑛is gi en as an inpu o he in e e o e alua e he eac i e powe injec ion a node 𝑛. This pu ely depends on i s local alues (𝓏𝒾𝑛∶= [𝑝𝑔 𝑛−𝑝𝑐 𝑛𝑞𝑔 𝑛𝑞𝑐 𝑛]Twhe e 𝑞𝑔 𝑛∶= √(𝑠𝑔 𝑛)2−(𝑝𝑔 𝑛)2) o ha e ew nonlocal o neighbo ing inpu s like eal powe flow, squa ed magni ude. The en y o he local inpu 𝓏𝒾𝑛is a anged in he ollowing o de : he ac i e powe inpu a node 𝑛, he maximum possible inpu o eac i e powe a node 𝑛, and he eac i e load a his node. In addi ion, 𝓏𝒾𝑛may be changed o comply wi h he managemen policy. This can be done by adding some impo an global inpu s o he 𝓏𝒾𝑛 ec o . I has been ound in he li e a u e ha adding he squa e o he ol age alue 𝑣𝑛 o 𝓏𝒾𝑛will make i difficul o analyze he s abili y o he esul ing closed-loop con ol, e en i 𝑓𝑛is linea [38, 39]. Since hese inpu s a e a ailable locally, he e is a minimum bu den on communica ion channels and he e alua ion is quick. Ideally, i he e a e abundan communica ion esou ces, he unce ain quan i ies om all buses {𝑞𝑔 𝑛,𝑝 𝑐 𝑛−𝑝𝑔 𝑛,𝑞𝑐 𝑛}𝑛∈could be o wa ded o all in e e s. So, in his case he con ol inpu o 𝓏𝒾𝑛is equal o and g ea e han all in e e s in he g id. Also, he inpu a iables a e e y flexible and can a y depending on he a ailable ne wo k bandwid h. Non-local and common con ol inpu s can be added o he inpu ec o 𝓏𝒾𝑛 o s udy he esul s o he common local and global inpu s o he in e e . This gi es 𝓏𝒾𝑛∶= [𝑝𝑔 𝑛−𝑝𝑐 𝑛𝑞𝑔 𝑛𝑞𝑐 𝑛𝑃𝑖𝑃𝑗𝑃𝑘]T, whe e 𝑖, 𝑗, and 𝑘 ep esen he line numbe s, and 𝑃𝑖is he eal powe flow on line 𝑖. These lines a e selec ed on he bases o he opology o each ne wo k and hese inpu s will be iden ical o each in e e . Con olle unc ion: The nex s ep includes he con ol unc ion policy 𝑓𝑛. Con ol unc ions can be e alua ed as linea o non-linea policies as defined below. Using ke nel-based lea ning heo y, he eac i e powe con ol o he in e e 𝑛is assumed o be in he Rep oducing ke nel Hilbe space (RKHS) [40]. 𝜅𝑛∶= {𝑓𝑛(𝓏𝒾𝑛)= ∞ ∑ 𝑡=1 𝐾𝑛(𝓏𝒾𝑛,𝓏𝒾𝑛,𝑡)𝑎𝑛,𝑡,𝑎 𝑛,𝑡∈ℝ}(21) ha is uniquely de e mined by he ke nel unc ion 𝐾𝑛∶𝐼𝑛×𝐼𝑛→ℝ. The linea policies can be implemen ed by e alua ing a linea ke nel 𝐾𝑛(𝓏𝒾𝑛,𝑡,𝓏𝒾𝑛,𝑡′)=𝓏𝒾T 𝑛,𝑡𝓏𝒾𝑛,𝑡′and he nonlinea policies can be designed by selec ing a polynomial ke nel 𝐾𝑛(𝓏𝒾𝑛,𝑡,𝓏𝒾𝑛,𝑡′)=(𝓏𝒾T 𝑛,𝑡𝓏𝒾𝑛,𝑡′+𝛾)𝛽, o a Gaussian ke nel 𝐾𝑛(𝓏𝒾𝑛,𝑡,𝓏𝒾𝑛,𝑡′)=exp(−‖‖𝓏𝒾𝑛,𝑡 −𝓏𝒾𝑛,𝑡′‖‖2 2∕𝛾)wi h design pa ame e s 𝛽and 𝛾>0, o by a linea combina ion o linea , polynomial, and gaussian ke nels. In e cep : The con ol unc ion also needs o e alua e an in e cep alue 𝑏𝑛∈ℝin (20). Al hough i could be inco po a ed in o 𝑓𝑛by augmen ing 𝓏𝒾𝑛wi h a cons an en y o 1, i is usually kep sepa a e o a oid i s penaliza ion h ough ‖𝑓‖𝜅𝑛. 4.2. Lea ning policies om scena ios The p oposed app oach deals wi h mul iple gene a ion uni s wi h mul iple in e e s ha communica e wi h each o he and communi- ca e wi h he ope a o . In gene al, he p ocess o da a communica ion be ween diffe en in e e s es ablishes 𝑁in e e u ili y communica- ion links and equi es ano he 𝑁u ili y in e e communica ion links o communica e wi h he ope a o . This leads o affic bo h be ween he ope a o and he in e e . To o e come his, ope a o s can use a scena io sample app oach. Ins ead o assessing he p oblem o e a long pe iod o ime, he ope a o can decide o ou pu he se ings less equen ly, o example e e y 10 minu es. A e he con ol unc- ion and inpu ec o a e comple ed, he eac i e powe con ol policy (20) should be e alua ed be ween he inpu da a se ings. He e, he 𝑛𝑡ℎ en y o q𝑔 o any gi en scena io 𝑡can be eplaced wi h he pol- icy 𝑞𝑔 𝑛(𝓏𝒾𝑛,𝑡 )=𝑓𝑛(𝓏𝒾𝑛,𝑡 )+𝑏𝑛 om (20). Thus, he algo i hm e alua es he op imal unc ion, and he in e cep pai { 𝑓𝑛, 𝑏𝑛}𝑁 𝑛=1, which can be ound ia he unc ional minimiza ion as min 𝑇 ∑ 𝑡=1 𝐶(y𝑡,{𝑓𝑛(𝓏𝒾𝑛,𝑡)}𝑛,b)+𝜇𝑃 ({‖‖𝑓𝑛‖‖𝜅𝑛}) (22) o e {𝑓𝑛∈𝜅𝑛}𝑁 𝑛=1 ,b(23) s. o |||𝑓𝑛(𝓏𝒾𝑛,𝑡)+𝑏𝑛|||≤𝑞𝑔 𝑛,𝑡,∀𝑛, 𝑡 (24) 𝑣𝑛,𝑡 ≤𝑟𝑛(𝑝𝑔 𝑛,𝑡 −𝑝𝑐 𝑛,𝑡)+𝑥𝑛(𝑓𝑛(𝓏𝒾𝑛,𝑡)+𝑏𝑛−𝑞𝑐 𝑛,𝑡)≤𝑣𝑛,𝑡,∀𝑛, 𝑡 (25) whe e b ∶= [𝑏1,…,𝑏 𝑁]T, cons ain (24) ep esen s he appa en powe cons ain , and (25) ep esen s he ol age egula ion cons ain . The egula ize 𝑃({‖‖𝑓𝑛‖‖𝜅𝑛})has been added in (22) o a oid o e fi ing o con ol policies o scena io da a. The usual machine lea ning eg ession se ings analyze he depen- dencies be ween ea u e da a and a ge da a and e alua e he closes fi . In his o mula ion, he ne wo k a iables supplied o each con- olle se e as cha ac e is ic da a and he eac i e injec ion se es as he se alue. Ideally, he designed unc ion should wo k well wi h unc- ional a ge pai s no ound du ing he aining o adap a ion p ocess. In a di ec analogy, he con ol concep o an in e e is p esen ed as a gene al ask o adap ing unc ions based on scena io da a. Once you ha e designed ea u es (policies), you can apply hem o hidden ea u e da a. The appa en powe limi (24) applies o he aining da a, bu he guidelines ob ained by (22) (25) impose an appa en powe limi o 𝓏𝒾𝑛,𝑡′’s wi h 𝑡′∉{1,…,𝑇 }because he policy was ained only on he 4 V.S.B.Ku uku u,A.Haque,M.A.Khane al. Heliyon 7 (2021) e08609 da a up o scena io 𝑇whe e he limi is ac i e. This limi a ion o ke nel- based lea ning also occu s in scena io-based and andom designs [41]. To o e come his limi a ion he eac i e powe e alua ed a 𝑡′scena io o node 𝑛can be heu is ically p ojec ed wi hin [−𝑞−𝑔 𝑛,𝑡′,+𝑞−𝑔 𝑛,𝑡′]as [𝑞𝑔 𝑛(𝓏𝒾𝑛,𝑡′)]𝑞𝑔 𝑛,𝑡′ ∶= max {min {𝑞𝑔 𝑛(𝓏𝒾𝑛,𝑡′),𝑞𝑔 𝑛,𝑡′},−𝑞𝑔 𝑛,𝑡′}(26) He e, he op imal policies ( unc ions) a e e alua ed indi idually o each in e e 𝑛. The e o e, he in e e policies a e linked ia he loss pa ame e 𝐿, since ol age de ia ion and powe loss a e affec ed by each eede supplying eac i e powe . Simila mul i unc ional se ings can be ound in collabo a i e fil e ing o mul i asking lea ning [42, 43]. In his s udy, a egula ize ha can be di ided in o all in e e s as shown in (27) [44] is adop ed. 𝑃({‖‖𝑓𝑛‖‖𝜅𝑛}𝑁 𝑛=1)= 𝑁 ∑ 𝑛=1 ‖‖𝑓𝑛‖‖2𝜅𝑛(27) The well-known Rep esen e Theo em [45] can be applied successi ely o e 𝑛in (22)–(25). This ensu es ha  𝑓𝑛(𝓏𝒾𝑛)= T ∑ 𝑡=1 𝐾𝑛(𝓏𝒾𝑛,𝓏𝒾𝑛,𝑡)𝑎𝑛,𝑡 (28) s ill holds o all 𝑛. Thus, a e he op imal policies a e e alua ed, he coefficien s {𝑎𝑛,𝑡}𝑛,𝑡, he con ol policies { 𝑓𝑛}can be e alua ed a any o he poin . As seen ea lie , he in e e policy  𝑓𝑛o e he es da a {𝓏𝒾𝑛,𝑡}𝑇 𝑡=1 can be examined as  𝑛=K 𝑛 a𝑛,∀𝑛(29) whe e [K𝑛]𝑡,𝑡′=𝐾𝑛(𝓏𝒾𝑛,𝑡, 𝓏𝒾𝑛,𝑡′) o 𝑡, 𝑡′∈{1, … , 𝑇}, and  a𝑛∶= [ 𝑎𝑛,1, … , 𝑎𝑛,𝑇 ]T. Mo eo e , he egula ize e m, he RKHS no ms can be ex- p essed as ‖‖𝑓𝑛‖‖2𝜅𝑛= aT 𝑛K𝑛 a𝑛,∀𝑛(30) 4.3. Op imal policy Vol age D op and Powe Loss Minimiza ion: A e e alua ing he con- ol policy, i is necessa y o e alua e he op imal abili y o minimize ol age d op and powe loss in he ne wo k. Thus, he p oblem o mini- mizing he ohmic loss in (27) and he egula ize (22) can be exp essed as a linea bounded quad a ic p og am. Lemma 1. I he da a-fi ing e m in (22) is selec ed as 𝐶(y𝑡,{𝑓𝑛(𝓏𝒾𝑛,𝑡)}𝑛,b)=‖‖Cq𝑔 𝑡+y 𝑡‖‖2 2,𝑡=1,…,𝑇 (31) and he egula izing e m as 𝑃({‖‖𝑓𝑛‖‖𝜅𝑛})= 𝑁 ∑ 𝑛=1 ‖‖𝑓𝑛‖‖2𝜅𝑛(32) he unc ional op imiza ion in (22)–(25) can be ans o med in o he ec o minimiza ion min 1 𝑇(‖CQ + Y‖2 𝐹+𝜇 𝑁 ∑ 𝑛=1 aT 𝑛K𝑛a𝑛)(33) o e Q ∈ ℝ𝑁×𝑇,{a𝑛∈ℝ𝑇}𝑁 𝑛=1 ,b∈ℝ𝑁(34) 𝑠.𝑡𝑜 QT=[K1a1+𝑏11,…,K𝑁a𝑁+𝑏𝑁1](35) −q𝑔 𝑛≤K𝑛a𝑛+𝑏𝑛1≤q𝑔 𝑛,∀𝑛(36) whe e Y ∶= [y1, … , yT]and he en ies o he ec o q𝑔 𝑛∶= [𝑞𝑔 𝑛,1, … , 𝑞𝑔 𝑛,𝑇 ]T ha e been defined in (12). P oo o Lemma 1.Based on (20), and (29), he eac i e powe in- jec ion o in e e 𝑛 o scena ios 𝑡 =1, … , 𝑇can be exp essed by he ec o K𝑁a𝑁+𝑏𝑁1. Then, he appa en powe cons ain o in e e 𝑛and ac oss all scena ios can be exp essed as gi en in (36). Conside - ing he fi s summand in (33) and based on he equali y in (35), he 𝑡𝑡ℎ column o Qdeno ed by q𝑔 𝑡con ains he eac i e injec ions om all in e e s o scena io 𝑡. The squa ed F obenius no m o a ma ix equals he sum o he squa ed 𝑙2-no ms o i s columns, which ollows ∑𝑇 𝑡=1 ‖‖Cq𝑔 𝑡+y 𝑡‖‖2 2=‖CQ + Y‖2 𝐹. Mo eo e , he second summand in (33) ollows di ec ly om (30). □ Powe Loss Minimiza ion unde Vol age Cons ain s: The ne wo k imple- men s a model o minimize powe loss, aking in o accoun he lim- i a ions o o e all powe and ol age egula ion. This model ocuses on keeping he ol age wi hin specified limi s. The e o e, a e e alua - ing he con ol policy, he op imal unc ion o minimizing he powe loss associa ed wi h he cons ain as desc ibed abo e is es ima ed. This p oblem can also be exp essed in linea bounded quad a ic p og am- ming. Lemma 2. I he da a-fi ing e m in (22) is selec ed o minimize losses gi en as 𝐶(y𝑡,{𝑓𝑛(𝓏𝒾𝑛,𝑡)}𝑛,b)=‖‖‖‖R1 2(q𝑔 𝑡−q 𝑐 𝑡)‖‖‖‖ 2 2 ,𝑡=1,…,𝑇 (37) and he egula izing e m as 𝑃({‖‖𝑓𝑛‖‖𝜅𝑛})= 𝑁 ∑ 𝑛=1 ‖‖𝑓𝑛‖‖2𝜅𝑛(38) he unc ional op imiza ion in (22) can be ans o med in o he ec o mini- miza ion min 1 𝑇(‖‖‖‖R1 2(Q−Q 𝑐)‖‖‖‖ 2 𝐹 +𝜇 𝑁 ∑ 𝑛=1 aT 𝑛K𝑛a𝑛)(39) o e Q ∈ ℝ𝑁×𝑇,{a𝑛∈ℝ𝑇}𝑁 𝑛=1 ,b∈ℝ𝑁(40) 𝑠.𝑡𝑜 QT=[K1a1+𝑏11,…,K𝑁a𝑁+𝑏𝑁1](41) −q𝑔 𝑛≤K𝑛a𝑛+𝑏𝑛1≤q𝑔 𝑛,∀𝑛(42) V=R(P𝑔−P 𝑐)+X(Q−Q 𝑐)+𝑣01𝑁×𝑇(43) 𝑡≤ 𝑡≤ 𝑡,∀𝑡(44) whe e eal and eac i e powe consump ion ec o s a e s acked as columns o he 𝑁×𝑇ma ix P𝑐∶= [p𝑐 1,…,p𝑐 𝑇], and Q𝑐∶= [q𝑐 1,…,q𝑐 𝑇], espec- i ely. The ol age a each bus a e s acked as columns o he 𝑁×𝑇ma ix 𝑉∶= [ 1,…, 𝑇]. Simila ly, eal powe gene a ion is P𝑔∶= [p𝑔 1,…,p𝑔 𝑇]and he en ies o he ec o 𝑡∶= [ 1,…, 𝑁]T, 𝑡∶= [ 1,…, 𝑁]T, q𝑔 𝑛∶= [q𝑔 𝑛,1,…,q𝑔 𝑛,𝑇 ]Twhe e he limi s o ol age egula ion a e defined in (13), he eac i e powe has been defined in (12). P oo o Lemma 2.Based on (20) and (29), he eac i e powe injec- ion o in e e 𝑛 o scena ios 𝑡 =1, … , 𝑇can be exp essed by he ec o K𝑛a𝑛+𝑏𝑛1. Then he limi o he appa en powe o in e e 𝑛and all scena ios can be exp essed as (42). In he LDF equa ion o (9), he ol - age limi is exp essed as (43). A linea limi is added o all 𝑁buses in all scena ios gi en in (44) o keep he ol age wi hin ce ain limi s. Conside ing he fi s e m in (39) and based on he equa ion in (41), he 𝑡𝑡ℎ column o Qas q𝑔 𝑡con ains he eac i e injec ion om all in e - e s o scena io 𝑡. The squa e o he F obenius no m o a ma ix is equal o he sum o he squa es o he 𝑙2-no m o he column which ollows ∑𝑇 𝑡‖‖‖‖R1 2(q𝑔 𝑡−q 𝑐 𝑡)‖‖‖‖ 2 2 =‖‖‖‖R1 2(Q−Q 𝑐)‖‖‖‖ 2 𝐹 . Mo eo e , he second summand in (39) ollows di ec ly om (30). The o al cos is no malized by T. □ 5 V.S.B.Ku uku u,A.Haque,M.A.Khane al. Heliyon 7 (2021) e08609 Fig. 2. The p oposed eac i e powe con ol me hodology o a g id sys em wi h PVs. 4.4. Implemen ing eac i e con ol policies A e se ing up he ype o policy and egula ize o he p oblem, he eac i e powe con ol policies ollow ou s eps. In he fi s s ep, he scena io da a {𝓍𝑡}𝑇 𝑡=1 is c ea ed by he ope a o o all he scena ios om 𝑡 =1, … , 𝑇. In he second s ep, he ope a o sol es (33). In he hi d s ep, each in e e 𝑛 ecei es he op imal policy coefficien s ( a𝑛, 𝑏𝑛) and aining da a {𝓏𝒾𝑛,𝑡}𝑇 𝑡=1 om he ope a o . Fo he final s ep, each in e e 𝑛collec s he new z𝑛,𝑡′and applies i s p ojec ed con ol policy o e he nex 𝜏mins. [𝑞𝑔 𝑛(𝓏𝒾𝑛,𝑡′)]𝑞𝑔 𝑛,𝑡′ =[𝑇 ∑ 𝑡=1 𝐾𝑛(𝓏𝒾𝑛,𝑡′,𝓏𝒾𝑛,𝑡)𝑎𝑛,𝑡 + 𝑏𝑛]𝑞𝑔 𝑛,𝑡′ (45) Fig. 2shows he p oposed me hodology di ided in o da a collec ion, managemen ule de elopmen , and wo k s eps. While de eloping con- ol ules, ope a o s collec inpu da a om o ecas s, eede dis o ions and his o ical da a. This da a is collec ed and so ed as {𝓍𝑡}𝑇 𝑡=1. Acco d- ing o (33), he ope a o sol es he p oblem o minimizing a quad a ic plan as a linea cons ain o policy s udies. In he nex s ep, he lea ned pa ame e s ( a𝑛, 𝑏𝑛)and i s aining da a {𝓏𝒾𝑛,𝑡}𝑇 𝑡=1 a e sen o each in- e e 𝑛. This s ep is epea ed e e y 𝜏minu es depending on he needs o he powe sys em and he a ailabili y o communica ion bandwid h. I is wo h men ioning ha i 𝓏𝒾𝑛,𝑡 ∈ℝ𝑀𝑛, he ope a o needs o send (𝑀𝑛+1 )𝑇+1 da a o in e e 𝑛. Besides, he numbe o scena ios 𝑇 affec s he amoun o da a being communica ed o each in e e . As 𝑇inc eases, he bandwid h mus be la ge o send da a quickly. To do his, he in e e applies he acqui ed con ol policy o he pa ame e s o he second s age. This ecalls ha all he lea ned pa ame e s ( a𝑛, 𝑏𝑛) and {𝓏𝒾𝑛,𝑡}𝑇 𝑡=1 a e al eady eadily a ailable o each in e e 𝑛, and he ou h s ep can be i e a ed as compa ed o 𝜏minu es o collec ing da a. Also, o pu e local con ol inpu 𝓏𝒾𝑛, con ol policy can be applied wi hou addi ional in o ma ion. In con as , i he con ol inpu s a e no common o local, hen a eal- ime eco d o inpu s 𝓏𝒾𝑛,𝑡 mus be sen by he ope a o o o igin be o e each in e e n. Also, b oadcas p o ocols wi h b oadband da a can educe communica ion o e head when inpu da a is sha ed by in e e s. 5. Nume ical simula ion 5.1. Simula ion de elopmen The ules de eloped o he eac i e con ol o in e e s we e es ed using he ecommended IEEE 123 bus es eede s [46, 47] shown in Fig. 3, con e ed o a single-phase ne wo k using he p ocedu e de- sc ibed in [48]. The node es eede eposi o y can be ob ained om [49]. A 12.35 kV base and a 100 kVA powe base was used. Residen- ial load (ac ual powe consump ion) and PV module p oduc ion da a we e gene a ed based on a Gaussian mix u e model o gi en mean and a iance. The a e age alues o ac ual powe gene a ion and powe consump ion we e a e aged 𝑝𝑚𝑒𝑎𝑛 𝑔=2.5 kW, and 𝑝𝑚𝑒𝑎𝑛 𝑐=10.25 kW, e- spec i ely. The a iance 𝜎was a ied om (0 − 20)% in s eps o 10%. The eac i e load ( he load wi h eac i e powe ) was aken a a con- s an lagging powe ac o o 0.97. Analyzes we e pe o med o 20%, 50%, and 100% PV pene a ion a es. Pene a ion a e is he a io o so- la powe ed buses o he o al numbe o buses consuming ene gy. To compensa e o eac i e powe , e en in he p esence o peak insola ion, i was assumed ha he in e e was made 10% la ge o gi e a max- imum powe 𝑠𝑔 𝑛=1.1𝑝𝑔 𝑛 o all 𝑛. The nume ical analysis es in ol ed fi e ci cui s. i) single powe ac o op ion whe e he in e e does no p o ide eac i e powe suppo ; ii) he fixed Wa -VAR managemen ules de ailed in [2]; iii) op imal eac i e powe se ing [26, 50]; i ) he ke nel-based app oach o (33) and (39) o he linea ke nel; and ) Gaussian ke nel app oaches o (33) and (39). Ke nel-based ules we e lea ned using he load and gene a ion da a obse ed in he mos ecen 𝑇=10 scena io, and he pa ame e s 𝜇 and 𝛾we e de e mined ia 5- old c oss- alida ion [51]. Con olle 𝑛 was es ed o 𝑇′=20 diffe en scena ios wi h only he local inpu s (LI) 𝓏𝒾𝑛=[𝑞𝑔 𝑛𝑝𝑐 𝑛−𝑝𝑔 𝑛𝑞𝑐 𝑛]T o each 𝑛, wi h global inpu s (GI) o powe flows 𝓏𝒾𝑛=[𝑞𝑔 𝑛𝑝𝑐 𝑛−𝑝𝑔 𝑛𝑞𝑐 𝑛𝑃15 𝑃16 𝑃17 ]T, whe e lines 15, 16 and 17 we e chosen as impo an lines di iding he g id in o h ee sepa a e b anches a he fi s le el. iii) A quad a ic so wa e ope a o pa i ion sol e was used o sol e he ) me hod [52]. We ini ially es ima ed he cos 𝜆Δ𝑠(q𝑔)+(1−𝜆)𝐿(q𝑔) o 𝜆 =1 o minimize he ol age de i- a ion in he p oblem o mula in (33). The esul s a e hen e alua ed o minimize powe loss and egula e he bus ol age. The cos 𝐿(q𝑔)is es ima ed acco ding o he desc ip ion o p oblem (39). 5.2. Tes s o ol age d op minimiza ion To es ablish he p oblem, as shown in sec ion 4.3, he ol age d op was minimized o he appa en powe cons ain as (33). Each 𝑇=10 es scena io was op imized by e alua ing he op imal policy and es ed agains 𝑇=20 scena ios. E alua ed con ol policies we e compa ed o 5 egimens. Ci cui compa ison and lack o esponse con ol a e pe - o med o Mon e Ca lo simula ions. In his case, we ex ac he inpu ec o 𝓏𝒾𝑛 om he p e iously desc ibed se o Gaussian dis ibu ions and hen es ima e he eac i e powe inpu o each ci cui . A each 6 V.S.B.Ku uku u,A.Haque,M.A.Khane al. Heliyon 7 (2021) e08609 Fig. 3. IEEE 123- bus benchma k eede [47]. Table 1. Reac i e powe con ol o ol age d op minimiza ion. Ne wo k Local (%) Op imal (%) Linea Gaussian Pene a ion (%) Va iance (%) Local Inpu s (%) Global Inpu s (%) Local Inpu s (%) Global Inpu s (%) 20 10 4.47 99.51 95.93 98.895.99 99.29 20 20 74.25 99.98 96.799.31 97.77 99.42 50 20 88.46 99.99 97.74 99.66 97.95 99.62 50 20 90.92 99.99 92.12 99.593.199.56 100 20 69.65 99.99 94.67 99.16 94.86 99.25 100 20 96.35 99.99 96.12 99.197.04 99.21 s a , he pe cen age imp o emen o he lack o eac i e powe sup- po is es ima ed (q𝑔=0). imp o emen in cos (i)=1 𝑇 𝑇 ∑ 𝑡=1 𝐶|q𝑔=0 −𝐶|scheme(𝑖) 𝐶|q𝑔=0 × 100% (46) whe e 𝐶|scheme(𝑖)=(‖‖‖Rp+X q‖‖‖2|scheme(𝑖))is he cos o he op imiza ion, 𝑖 ∈{2,3,4,5} o each scheme as lis ed abo e. I should be no ed ha local con ol can pe o m much wo se han a ci cui wi hou eac i e suppo [2], which only e alua es eac i e powe con ol in e ms o sola cell gene a ion and local load. The e o e, o unde s and his con- di ion, he eac i e powe con ol o minimize he ol age d op Δ𝑠(𝑞𝑔)𝜆 is se o 1, and he imp o emen a e o he insufficien eac i e powe suppo 𝑞𝑔is ega ded as 0. These esul s a e lis ed in Table 1. Table 1shows ha he op imal con ol echniques ou pe o m he o he ou me hods. I e alua es he eac i e powe con ol o an op- imal alue wi h all inpu s on all buses. This is ue because op imal con ol wo ks well as i sol es he p oblem as a whole. The downside o his me hod is ha he policy is slow and insecu e wi h he commu- nica ion o e head ha equi es high bandwid h o ansmi da a om e e y node o he subs a ion a e e y momen . On he o he hand, since local egula ions es ima e eac i e powe supply based on local inpu and ake in o accoun a cons an 𝑥∕𝑟 a io o all lines, local egula ions canno con ol eac i e con ol e y accu a ely, esul ing in e y li le imp o emen . Fu he , he p oposed ke nel policy p o ides a high pe - cen age o imp o emen o bo h linea and non-linea policies. Also, he pe o mance is e y close o he op imal ule. As shown in Table 1, adding a global inpu ( eal powe flow h ough lines 15, 16, and 17 in Fig. 3.) o he policy esul s in an es ima ed be e eac i e powe de- li e y and a highe pe cen age imp o emen han using he local inpu alone. The ol age d op minimiza ion es esul s a e shown in Fig. 4, and 5. In he Fig. 4shows he alues o he log scale ‖‖‖Rp+X q‖‖‖2 o each scena io. In his figu e, i can be seen ha local con ol beha es simila o o wo se han he no eac i e con ol scheme, while he linea and nonlinea policies p oposed in his s udy beha e e y closely o he op- imal con ol me hod. When compa ing he esul s in Fig. 4, and Fig. 5 i is iden ified ha wi h he addi ion o he global inpu da a o he policy, he p oposed me hod wo ks much be e and app oaches op i- mal con ol. Adding hese global inpu s can inc ease da a ans e and cybe o e head, bu he inc ease will be small i he numbe o added inpu s emains small. Only h ee ea u es we e added o his analy- sis, and a significan imp o emen in policy effec i eness was obse ed. The e o e, adding mul iple global inpu s can significan ly imp o e he effec i eness o ke nel policies. I is wo h no ing ha he beha io o linea and nonlinea policies is e y simila . These esul s encou aged o e alua e he powe loss minimiza ion using ol age egula ion con- s ain s, whe e ne wo ks a e expec ed o beha e diffe en ly o linea and non-linea policies. 7 V.S.B.Ku uku u,A.Haque,M.A.Khane al. Heliyon 7 (2021) e08609 Fig. 4. Reac i e powe con ol wi hou global inpu s o each scheme a 20% pene a ion, and 10% a iance. Fig. 5. Reac i e powe con ol wi h global inpu s o each scheme a 20% pen- e a ion, and 10% a iance. 5.3. Tes s o powe loss minimiza ion unde ol age cons ain s The powe loss minimiza ion esul s wi h espec o ol age and ap- pa en powe cons ain s (39) a e desc ibed below. Vol age iola ions be ween 3% o he base ol age alue, i.e., =0.97 p.u. and =1.03 p.u. a e allowed in his implemen a ion. Each o he 𝑇=10 es scena io was op imized by e alua ing he op imal policy and es ed agains 𝑇=20 scena ios. The e alua ed con ol policies we e compa ed agains he fi e app oaches p e iously desc ibed. Fo each scena io, he a e age ol age d op ac oss he eede was es ima ed as Δ𝑣=‖‖‖Rp+X q‖‖‖1 𝑁(47) The a e age ol age d op was compa ed o he eac i e powe loss. Fig. 6shows he a e age ol age d op o each scheme wi h espec o 100% PV pene a ion and powe loss, and 10% de ia ion in eal powe consump ion. The local ule 𝜆 =0, minimizes he powe loss o he ol age d op, ega dless o he ule alone. Howe e , i u ns ou ha he ol age d op is e y high, al hough he powe loss is kep o a minimum. The e o e, local egula ions canno keep he ol age wi hin he specified limi s. Linea and nonlinea policies s ic ly ollow he op imal con ol scheme. The model can iola e he ol age cons ain s du ing es ing because he ol age cons ain s a e ac i e only du ing he aining s ep. As i is seen in he figu e hese iola ions a e no e y high on a e age. F om Fig. 6i can be seen ha o he global inpu , he a e age ol age-d op dec eases and he model beha es close o op imal. In his case, he non-linea policy p o ides be e egula ed ol age and lowe powe dissipa ion compa ed o he linea policy. F om all he cases shown in Fig. 6, he non-linea policy has sligh ly mo e powe Fig. 6. A e age ol age-d op in a eede o a ying powe loss unde ol age cons ain s o diffe en schemes a 100% pene a ion and 10% a iance wi h local and global inpu s. dissipa ion, bu he ol age egula ion is be e compa ed o he lin- ea policy. A simila end is shown in Fig. 7, which shows he a e age ol age d op o each ci cui o powe loss, sola de ice gene a ion, and ac ual ene gy consump ion de ia ion o 10% a 50% and 20% pene a- ion o PV modules. The figu e shows ha he nonlinea policy wo ks be e han he linea one. Also, he a e age ol age o he bus has a lowe ol age de ia ion om he a ed ol age o he non-linea policy compa ed o he linea policy. F om he expe imen s and esul s, i is clea ha local and op imal app oaches sol e p oblems locally o cen ally, aking in o accoun he linea ules o decision making on inpu pa ame e s. I has also been ound ha using local inpu s esul ed in subop imal esul s, whe eas global inpu s equi ed complex compu a ions. The co e me hod de el- oped om he dis ibu ed app oach effec i ely e alua es eac i e powe con ol policies along wi h he p oblem o ol age egula ion limi ing. This makes he de eloped app oach sui able o eal-wo ld implemen- a ion and allows you o find he igh app oach in e ms o balance be ween pe o mance and complexi y. A compa a i e analysis o he exis ing flexible and de eloped app oaches o he egula ion o eac i e powe in dis ibu ed gene a ion sys ems is p esen ed in Table 2. 6. Conclusion This pape de eloped a ke nel-based eac i e powe con ol ap- p oach o achie e esou ce managemen and mi iga ing he impac s o a ying loads and high PV pene a ions in he dis ibu ion g id. In he de eloped app oach, he policies ha e been designed o e alua e he con ol se -poin s o diffe en scena ios and es ima ion has been done o he eac i e powe con ol in eal- ime using inpu s and ou - pu s indi idually. Besides, eac i e powe con ol policies a e modeled by c ea i ely c oss-pollina ing ideas om machine lea ning and using he powe ul ool o ke nel-based lea ning, which is p ac ically easible. Tes s ha e been ca ied ou o minimiza ion o powe losses and ol - age egula ion on an IEEE 123 bus sys em modeled as a single-phase g id. Compa ed o he echniques in he li e a u e, he esea ch esul s depic ed a e flexible and o adjus able na u e. Fu he , his me hod can be ex ended o mul is age o mula ions, a ying con olle inpu s, and o e alua ing he combina ion o ke nels. Decla a ions Au ho con ibu ion s a emen V S Bha a h Ku uku u: Concei ed and designed he expe imen s; Pe o med he expe imen s; W o e he pape . Ah eshamul Haque: Pe - o med he expe imen s; Analyzed and in e p e ed he da a; W o e he 8 V.S.B.Ku uku u,A.Haque,M.A.Khane al. Heliyon 7 (2021) e08609 Fig. 7. A e age ol age-d op in a eede o a ying powe loss unde ol age cons ain s o diffe en schemes. Table 2. Compa a i e analysis o flexible eac i e powe con ol app oaches. Me hod Ad an ages Limi a ions/D aw- backs Rema ks Objec i e unc ions Pa icle swa m op imiza ion [14] Less con e gence ime, Reduced p oblem solu ion space using dep h fi s sea ch, and p io i ized loads du ing load shedding. Reac i e powe con ol is no conside ed wi h he in e e s in dis ibu ion gene a ion sys em. The op imiza ion p oblem is o mula ed conside ing ee Knapsack p oblem. The load p io i y emo es he leas load in he 1s s age. Leas a e age p inciple o es ima e he ol age de ia ion o selec ed nodes An sea ch algo i hm [24] Inc eased s a ic s abili y in s andalone mode ope a ion. Models eac i e powe con ol as a linea p og amming op imiza ion p oblem, and wo ks only wi h DC load Flow. The algo i hm con e gence speed is inc eased using DC load flow app oach. Load shedding is minimized du ing s andalone ope a ion. Adap i e op imiza ion app oach unde equency load shedding [18] The load shedding p oblem is o mula ed as a mixed in ege linea p og amming p oblem. Complex powe flow o mula ions ha a e no ideal o adial ne wo ks An app oxima ion o ini ial g oup AC ope a ional limi a ion is conside ed wi h he op imiza ion model du ing he islanding condi ion. Based on loca ion o load cu ailmen s. Ke nel-based app oach [P oposed] In e e coo dina ion h ough nonlinea con ol policies using an icipa ed scena ios o load and gene a ion. Minimizes powe losses and achie es ol age egula ion. Vol age con ol inpu s a e no conside ed o a ying con olle inpu s wi h he de eloped app oach. The p oposed app oach achie es desi able ade-off be ween eac i e con ol pe o mance and compu a ional equi emen s Linea ly-cons ained quad a ic p og am pape . Mohammed Ali Khan & F ede Blaabje g: Analyzed and in e - p e ed he da a; Con ibu ed eagen s, ma e ials, analysis ools o da a; W o e he pape . Funding s a emen This esea ch did no ecei e any specific g an om unding agen- cies in he public, comme cial, o no - o -p ofi sec o s. Da a a ailabili y s a emen Da a associa ed wi h his s udy has been deposi ed a he IEEE Powe & Ene gy Socie y unde he URL: h p://si es .ieee .o g /pes - es eede s / files /2017 /08 / eede 123 .zip. Decla a ion o in e es s s a emen The au ho s decla e no conflic o in e es . Addi ional in o ma ion No addi ional in o ma ion is a ailable o his pape . Re e ences [1] A. Sajadi, L. S ezoski, V. S ezoski, M. P ica, K.A. 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