Dynamic optimal power flow of active distribution network based on LSOCR and its application scenarios

Ci a ion: Meng, W.; Song, D.; Deng, X.; Dong, M.; Yang, J.; Rizk-Allah, R.M.; Snášel, V. Dynamic Op imal Powe Flow o Ac i e Dis ibu ion Ne wo k Based on LSOCR and I s Applica ion Scena ios. Elec onics 2023,12, 1530. h ps://doi.o g/ 10.3390/elec onics12071530 Academic Edi o : Ahmed Abu-Siada Recei ed: 9 Feb ua y 2023 Re ised: 16 Ma ch 2023 Accep ed: 21 Ma ch 2023 Published: 24 Ma ch 2023 Copy igh : © 2023 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h ps:// c ea i ecommons.o g/licenses/by/ 4.0/). elec onics A icle Dynamic Op imal Powe Flow o Ac i e Dis ibu ion Ne wo k Based on LSOCR and I s Applica ion Scena ios Weiqi Meng 1, Dong an Song 1, Xiao ei Deng 2,*, Mi Dong 1, Jian Yang 1, Rizk M. Rizk-Allah 3,4 and Václa Snášel 4 1School o Au oma ion, Cen al Sou h Uni e si y, Changsha 410083, China 2School o In o ma ion Technology and Managemen , Hunan Uni e si y o Finance and Economics, Changsha 410205, China 3Depa men o Basic Enginee ing Science, Facul y o Enginee ing, Menou ia Uni e si y, Shebin El-Kom 32511, Egyp 4Facul y o Elec ical Enginee ing and Compu e Science, VŠB-Technical Uni e si y o Os a a, 70800 Os a a, Czech Republic *Co espondence: [email p o ec ed]; Tel.: +86-139-7434-1334 Abs ac : Op imal powe low (OPF) is a c ucial aspec o dis ibu ion ne wo k planning and ope a- ion. Con en ional heu is ic algo i hms ail o mee he sys em equi emen s o speed and accu acy, while linea ized OPF app oaches a e inadequa e o dis ibu ion ne wo ks wi h high R/X a ios. To add ess hese issues and ca e o mul i-pe iod scena ios, his s udy p oposes a dynamic linea ized second-o de cone p og amming-based (SOCP) OPF model. The model is buil by i s es ablishing a dynamic OPF model based on linea ized second-o de conic elaxa ion (LSOCR-DOPF). The compo- nen s o he ac i e dis ibu ion ne wo k, such as enewable ene gy powe gene a ion uni s, ene gy s o age uni s, on-load- ap-change s, s a ic a compensa o s, and capaci o banks, a e hen sepa a ely modeled. The model is implemen ed in MATLAB and sol ed by YALMIP and GUROBI. Finally, h ee ep esen a i e scena ios a e used o e alua e he model accu acy and e ec i eness. The esul s show ha he p oposed LSOCR-DOPF model can ensu e calcula ion ime wi hin 3 min, ol age s abili y, and e o con ol wi hin 10 −6 o all h ee applica ions. This me hod has s ong p ac ical alue in he ields o ac i e dis ibu ion ne wo k day-ahead dispa ch, accu a e modeling o ZIP load, and eal- ime ope a ion. Keywo ds: op imal powe low (OPF); ac i e dis ibu ion ne wo k; linea ized second-o de conic elaxa ion (LSOCR); ne wo k econ igu a ion; ZIP load 1. In oduc ion Recen ly, esea che s ha e shown an inc eased in e es in he ac i e dis ibu ion ne wo k. The in eg a ion o a ious dis ibu ed gene a ions, ene gy s o age uni s, and ac i e managemen de ices has p esen ed new challenges o he planning and ope a ion o dis ibu ion ne wo ks [ 1 ], especially in he ield o ac i e managemen (AM) o dis- ibu ion ne wo ks [ 2 ]. I is pa icula ly u gen o de elop op imiza ion algo i hms and high-pe o mance compu ing ools applicable o a ious ields o ac i e dis ibu ion ne - wo ks. Re . [ 3 ] analyzed h ee kinds o op imiza ion p oblems o he sma g id: op imal powe low (OPF), uni commi men , and ope a ion planning. Thei essence is dis ibu ion ne wo k op imiza ion, while ha ing di e en op imiza ion scales. The OPF is o g ea sig- ni icance in he de elopmen p ocess o he dis ibu ion ne wo k, and is he mos common and undamen al op imiza ion p oblem in powe sys ems [ 4 ]. Resea ch on dis ibu ion ne wo k OPF has mainly ocused on he al e na ing cu en powe low (AC-OPF). Ex- plo ing a solu ion me hod o enhance he solu ion speed o dis ibu ion ne wo k AC-OPF while ensu ing i s op imal ope a ion and ul illing he equi emen s o ac i e dis ibu ion ne wo k planning and ope a ion has been a majo conce n in he ield o powe sys em Elec onics 2023,12, 1530. h ps://doi.o g/10.3390/elec onics12071530 h ps://www.mdpi.com/jou nal/elec onics Elec onics 2023,12, 1530 2 o 27 esea ch. As a non-con ex op imiza ion p oblem, he OPF is di icul o sol e. I is easy o all in o local op imum in he p ocess o sol ing and has been p o en o be an NP-ha d p oblem [ 5 , 6 ]. The powe low cons ain s a e cha ac e ized by nonlinea i y, and hence he essence o OPF lies in nonlinea p og amming. The me hods o sol ing OPF p oblems can be b oadly ca ego ized in o he ollowing h ee ca ego ies: (1) Pu sui o local op imal solu ions, including ea ly classical me hods [ 7 ] (such as Simpli ied G adien Me hod, New on Me hod, Sequen ial Quad a ic P og amming, In e io Poin Me hod) and ecen apidly-de eloping heu is ic algo i hms [3,8]; (2) App oxima ion o powe low equali y cons ain s. Fo example, he AC-OPF con- s ain s can be app oxima ely linea ized as di ec cu en powe low cons ain s, and he esul an di ec cu en op imal powe low (DC-OPF) p oblem can be acco dingly sol ed [9]; (3) Relaxa ion o he powe low equali y cons ain using con ex elaxa ion echniques [ 10 ]. Me hods o seeking local op imal solu ions, due o hei ad an ages such as simplici y and ease in simula ing complex cons ain s, ha e been widely applied in sol ing nonlinea p og amming models [ 11 ]. Howe e , due o he non-con ex na u e o he OPF p oblem, hese me hods canno gua an ee he quali y o he solu ion, and i is impossible o measu e he gap be ween he locally op imal solu ion and he globally op imal solu ion. Me hods ha app oxima e he powe low equa ion cons ain s, such as he DC-OPF, p esen se e al ob ious disad an ages. Fi s ly, hey can be challenging o apply o esea ch a eas ela ed o ol age and eac i e powe , as well as dis ibu ion ne wo ks wi h high R/X a ios. Secondly, he op imal solu ion o he DC-OPF p oblem may no be a easible solu ion o he o iginal OPF p oblem, leading o he need o cons an adjus men o he igh ness o DC-OPF cons ain s and he need o sol e again du ing he ac ual op imiza ion p ocess [ 12 ]. Wi h he nume ous issues associa ed wi h seeking local op ima and app oxima ing he cu en low equali y cons ain s, i equi es new me hods o deal wi h he cu en low equali y cons ain s. The con ex elaxa ion echnique has gained signi ican a en ion in ecen yea s due o i s ad an ages and po en ial in sol ing OPF p oblems in he ield o powe sys em op imiza ion. The use o con ex elaxa ion echniques, pa icula ly second-o de cone p og amming (SOCP) elaxa ion, has become inc easingly p e alen . The SOCP mainly con e s he o iginal model in o a con ex p og amming o m, hus ob aining he globally op imal solu ion and a good compu a ional speed. Re s. [ 13 , 14 ] sys ema ically es ablished a b anch low model (BFM) based on Dis low [ 15 ] o sol e he OPF model amewo k, and p esen ed wo elaxa ion s eps: (1) Elimina ion o ol age and cu en phase angles; (2) Second O de Conic Relaxa ion (SOCR). The au ho s also demons a ed he elaxa ion accu acy o SOCR. As a ypical ep esen a i e o SOCP, SOCR can be summa ized in he ma hema ical ield as he classic “dimensionali y elaxa ion- e u n mapping” p ocess: A new a iable is in oduced o ele a e he dimensionali y o he o iginal p oblem; hen, in he ele a ed p oblem, non-con ex cons ain s a e elaxed and a solu ion is ob ained; inally, he solu ion o he o iginal p oblem is eco e ed h ough a e u n mapping. The main challenge in using SOCR o sol e he op imal powe low lies in accu a ely sa is ying he condi ions o he elaxed model. Re s. [ 16 , 17 ] p o ided a comp ehensi e summa y o he su icien condi ions o he accu acy o SOCR unde adia ed ne wo k condi ions, which a e di ided in o h ee ca ego ies: powe injec ion cons ain s, ol age ampli ude cons ain s, and node ol age phase angle de ia ions, and co esponding explana ions a e gi en o he op imal powe low objec i e unc ion. A e he examina ion o Re . [ 17 ], u he esea ch on he exac elaxa ion su icien condi ion has been ex ensi ely conduc ed. Re . [ 18 ] expanded upon he su icien condi ions o he accu acy o SOCR in he p esence o he high pene a ion o dis ibu ed gene a ion. Re s. [ 19 , 20 ] highligh ed he sho comings o he second su icien condi ion in Re . [ 17 ], as i neglec ed he impac o he g ounding b anch powe low on line capaci y, and p oposed an imp o ed su icien condi ion. Addi ionally, al hough he OPF model o he ac i e dis ibu ion ne wo k based on MISOCP has achie ed a ela i ely high solu ion e iciency [ 21 ], s ic ly speaking, i is s ill Elec onics 2023,12, 1530 3 o 27 a nonlinea model, and he e iciency o sol ing nonlinea second-o de cone cons ain s will dec ease wi h he inc ease in he numbe o dis ibu ion ne wo k nodes. In o de o achie e e icien sol ing o he OPF p oblem in u u e la ge-scale dis ibu ion ne wo ks, i is necessa y o simpli y he second-o de cone p og amming cons ain s while ensu ing he accu acy and e iciency o dis ibu ion ne wo k sol ing. Re . [ 22 ] has p o en ha he second-o de cone in dimension N, can be ou e -app oxima ed o an a bi a y accu acy by a polyhed al cone in an ex ended space. Polyhed al app oxima ions a e e y powe ul o sol ing MISOCP as he bene i s o wa m-s a ing o LPs can be u ilized h oughou he b anch-and-bound algo i hm [ 23 ]. This is pa icula ly impo an o p oblems ha can be o mula ed as MISOCP, such as he OPF p oblem in his pape . Meanwhile, Re s. [24,25] ex- ended he cons an powe load model o a ZIP load model unde a ec angula coo dina e sys em and applied i o he solu ion o he OPF. Meanwhile, mos o he exis ing dis i- bu ion ne wo k models only include powe gene a ion uni s and ene gy s o age de ices, and a ely conside eac i e powe compensa ion de ices and o he ac i e managemen de ices a he same ime [ 26 ]. In addi ion, in he ac i e dis ibu ion ne wo k planning and ope a ion op imiza ion model, disc e e a iables will ine i ably appea wi h he ac i e managemen de ices conside ed, which u ns he o iginal p oblem in o a mixed in ege linea p og amming (MILP) [ 27 ]. Wi h con inuous de elopmen and ma u i y, comme cial op imiza ion so wa e (GUROBI, CPLEX, MOSEK, e c.) has been widely used in dis ibu- ion ne wo k econs uc ion [ 28 ], eac i e powe op imiza ion [ 29 ], dis ibu ion ne wo k planning [11], e c. I is no ewo hy ha he abo e s udies we e mos ly limi ed o he adi ional single- pe iod s a ic OPF ca ego y, while he ac ual op imiza ion equi es he o e all coo dina ion o mul i-pe iods, which is ac ually a dynamic op imal powe low (DOPF). Addi ionally, compa ed o he ec angula coo dina e sys em, he cu en o m in pola coo dina es is mo e common. The e o e, in o de o mee he ope a ional equi emen s o mul i-pe iod scena ios, a ious complex cons ain s, and as la ge-scale solu ions o ac i e dis ibu ion ne wo ks, his pape p oposes a linea ized second-o de cone elaxa ion dynamic op imal powe low (LSOCR-DOPF) model o he dis ibu ion ne wo k and explo es he linea modeling me hod o key cons ain s o ac i e dis ibu ion ne wo k pa icipa ing elemen s (such as on-load- ap-change (OLTC), s a ic a compensa o (SVC), capaci o banks (CB), he ene gy s o age sys em (ESS), e c.). The nonlinea OPF p oblem o he dis ibu ion ne wo k is ans o med in o a compu a ionally e icien solu ion u ilizing MILP, wi h a comp ehensi e explana ion o he ne wo k’s cons ain s on adia ion and connec i i y. On his basis, his pape u he p esen s an inno a i e app oxima ion o he ZIP load model unde pola coo dina es and alida es he e ec i eness o he OPF amewo k h ough h ee scena ios: powe coo dina ion op imiza ion, ne wo k econ igu a ion, and ZIP load applica ion. The main con ibu ions o his s udy can be summa ized as ollows: • P esen he LSOCR-DOPF model, which is based on b anch powe low analysis, o he ac i e dis ibu ion ne wo k. • Explo e he linea modeling me hod o he cons ain s o a ious ac i e managemen uni s, including OLTC, SVC, CB, ESS, e c. • Valida e he LSOCR-DOPF model h ough simula ion expe imen s in h ee ypical scena ios: powe coo dina ion op imiza ion, ne wo k econ igu a ion, and ZIP load applica ion. The a icle is o ganized as ollows: The Sec ion 2p esen s he LSOCR-DOPF model o ac i e dis ibu ion ne wo ks and he design o a ious ac i e managemen uni s based on b anch powe low analysis. The Sec ion 3discusses he esul s o h ee simula ion expe imen s and p o ides a comp ehensi e analysis and discussion o he indings. Finally, he Sec ion 4p esen s he conclusion and u u e pe spec i es. Elec onics 2023,12, 1530 4 o 27 2. Me hodology 2.1. LSOCR-DOPF Model o he Dis ibu ion Ne wo k 2.1.1. Basic S uc u e o Dis ibu ion Ne wo ks In mos dis ibu ion ne wo ks, he s eady-s a e powe ope a ion mode is adial and i s s uc u e is depic ed in Figu e 1. Fo a adial opology ne wo k, he node di ec ed g aph can be used o equi alen analysis. Fu he mo e, Sij and Si ep esen complex powe , Sij =Pij +Qiji and Si=pi+qii . B anch complex impedance Zij = ij +xiji . Se B ep esen s he se o all nodes in he ne wo k. In he adi ional OPF, he ol age emains cons an . I OLTC is ins alled, he ol age will change wi h he OLTC ans o ma ion a io; E ep esen s he collec ion o all b anches in he ne wo k. The e a e Nsub subs a ions, Nbus nodes, and NLine line b anches in he ne wo k. Elec onics 2023, 12, x FOR PEER REVIEW 4 o 27 based on b anch powe low analysis. The hi d sec ion discusses he esul s o h ee sim- ula ion expe imen s and p o ides a comp ehensi e analysis and discussion o he ind- ings. Finally, he ou h sec ion p esen s he conclusion and u u e pe spec i es. 2. Me hodology 2.1. LSOCR-DOPF Model o he Dis ibu ion Ne wo k 2.1.1. Basic S uc u e o Dis ibu ion Ne wo ks In mos dis ibu ion ne wo ks, he s eady-s a e powe ope a ion mode is adial and i s s uc u e is depic ed in Figu e 1. Fo a adial opology ne wo k, he node di ec ed g aph can be used o equi alen analysis. Fu he mo e,  and  ep esen complex powe ,  = +i and =+i. B anch complex impedance  = +i. Se  ep- esen s he se o all nodes in he ne wo k. In he adi ional OPF, he ol age emains cons an . I OLTC is ins alled, he ol age will change wi h he OLTC ans o ma ion a io; E ep esen s he collec ion o all b anches in he ne wo k. The e a e sub subs a ions, bus nodes, and Line line b anches in he ne wo k. i S j S k S k V j V i V ij S ij I jk S jk I 0 1 2 3456 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Figu e 1. S uc u e o adial dis ibu ion ne wo k. 2.1.2. Basic OPF Model Based on BFM Gene ally, he basic model o op imal powe low based on b anch powe low (BFM- OPF, Figu e 1) is exp essed as ollows [14]:   min , , , , , p q P Q V I (1) s. .             2 2 2 2 , , , , j jk ij ij ij j j k j i j j jk ij ij ij j j k j i j p P P I g V j B ij E q Q Q I x b V j B ij E                                  (2)     2 2 2 2 2 2 , , , j i i j i j i j i j i j i j i j V V P Q x I x i j B i j E         (3) 2 2 2 2, , ij ij ij i P Q I j B ij E V       (4) , ij ij ij I I I ij E       (5) , j j j V V V j B       (6) , p j j q j j p R j B q R          (7) Figu e 1. S uc u e o adial dis ibu ion ne wo k. 2.1.2. Basic OPF Model Based on BFM Gene ally, he basic model o op imal powe low based on b anch powe low (BFM- OPF, Figu e 1) is exp essed as ollows [14]: min (p,q,P,Q,V,I)(1) s. .    pj=∑k∈δ(j)Pjk −∑i∈π(j)Pij −I2 ij ij+gjV2 j,∀j∈B,∀ij ∈E qj=∑k∈δ(j)Qjk −∑i∈π(j)Qij −I2 ijxij+bjV2 j,∀j∈B,∀ij ∈E(2) V2 j=V2 i−2Pij ij +Qijxij+I2 ij 2 ij +x2 ij,∀i,j∈B,∀ij ∈E(3) I2 ij =P2 ij +Q2 ij V2 i ,∀j∈B,∀ij ∈E(4) I −ij ≤Iij ≤− Iij,∀ij ∈E(5) V −j ≤Vj≤− Vj,∀j∈B(6) (pj∈Rp j qj∈Rq j ,∀j∈B(7) whe e, pj and qj ep esen he ac i e and eac i e powe injec ions, espec i ely, a each node; he b anch ij ep esen s he posi i e di ec ion o low di ec ion om node i o node j ; δ(j) is he collec ion o b anch end nodes wi h j as he head node, and π(j) is he collec ion o b anch end nodes wi h j as he end node; Pjk and Qjk deno e he ac i e and eac i e powe a he head node o he b anch ij , espec i ely; Pij and Qij deno e he ac i e and eac i e cu en low in each b anch; ij and xij co espond o he indi idual esis ance and eac ance o each b anch; gjand bja e he sepa a e g ound conduc ance and g ound suscep ance o Elec onics 2023,12, 1530 5 o 27 node j ; Vj and − Vj a e he uppe and lowe limi s o node ol age, espec i ely; Iij and − Iij a e he uppe and lowe limi s o b anch ij cu en , espec i ely. F om Equa ions (1)–(7), i can be in e ed ha : (1) The op imiza ion a iables o he OPF consis o node injec ion powe (p,q) , b anch powe low (P,Q) , node ol age (V) , and b anch cu en (I) , wi h he subs a ion node ol age no conside ed as an op imiza ion a iable. (2) Equa ion (1) ep esen s he objec i e unc ion, which can be he minimiza ion o ne wo k losses and subs a ion node powe pu chases. (3) The well-known b anch low equa ion [ 30 ] is exp essed by Equa ions (2) and (3), while Equa ion (4) ep esen s he powe calcula ion equa ion. Equa ions (5) and (6) deno e he sa e y cons ain equa ions o b anch cu en s and nodal ol ages, espec i ely. Equa ion (7) in oduces node-dependen cons ain s ha a e subjec o change based on he employed model. 2.1.3. Mul i-Pe iod LSOCR-OPF Model In o de o con e powe low cons ain s in o quad a ic cone cons ain s, addi ional a iables o cone op imiza ion need o be se :      − Iij =I2 ij,∀ij ∈E − Vj=V2 j,∀j∈B (8) Subs i u e Equa ion (8) in o Equa ions (2)–(6) o change he powe low cons ain o he second-o de cone cons ain [31] as ollows: min (p,q,P,Q,V,I)(9) s. .        pj=∑k∈δ(j)Pjk −∑i∈π(j)Pij −− Iij ij+gj − Vj,∀j∈B,∀ij ∈E qj=∑k∈δ(j)Qjk −∑i∈π(j)Qij −− Iijxij+bj − Vj,∀j∈B,∀ij ∈E (10) − Vj=− Vi−2Pij ij +Qijxij+− Iij 2 ij +x2 ij,∀j∈B,∀ij ∈E(11)      2Pij 2Qij − Iij −− Vj    2 ≤− Iij +− Vj,∀j∈B,∀ij ∈E(12) I2 −ij ≤− Iij ≤−2 Iij,∀ij ∈E(13) V2 −j≤− Vj≤−2 Vj,∀j∈B(14) whe e, || ||2 ep esen s he L2 no m. Wi h his, he SOCR-OPF (a ypical Mixed-In ege Second O de Cone P og amming, MISOCP) model is ully modeled. Among hem, he second-o de cone p og amming ep esen ed by Equa ion (12) can be expanded and simpli ied as: qP2 ij +Q2 ij ≤− Vj − Iij,∀j∈B,∀ij ∈E(15) Equa ion (15) can be uni o mly desc ibed as: qx2 1+x2 2≤x3;x1=Pij,x2=Qij,x3= − Vj − Iij,∀j∈B,∀ij ∈E(16) Ben-Tal and Nemi o ski [ 22 ] showed ha Equa ion (16) could be app oxima ed by a sys em o linea homogeneous equali ies and inequali ies in e ms o x1 , x2 , x3 , and 2 (k+1) Elec onics 2023,12, 1530 6 o 27 a iables ak , bk o k= 0, . . . , K . K is a pa ame e o he polyhed al- elaxed app oxima ion. The app oxima e exp ession o he polyhed on o he h ee-dimensional SOCR cons ain (Equa ion (16)) is:        a0⩾|x1|;b0⩾|x2|;ak=ak−1cos π 2k+1+bk−1sin π 2k+1 bk⩾−ak−1sin π 2k+1+bk−1cos π 2k+1 aK⩽x3;bK⩽aK an π 2k+1 ,k=1, 2, · · · ,K(17) whe e, K ep esen s he numbe o ace s in a polyhed on. a0 and bk a e auxilia y a iables. The polyhed al app oxima ion gi en by (17) can be educed by using he linea equal- i y cons ain s ak=ak−1cos (π/2k+1+bk−1sin (π/2k+1 o sol e o ak(k=0, . . . , K) in e ms o a0 and ak(k=0, . . . , K) and hen subs i u e ak ou o Equa ion (17). The esul ing sys em will only ha e linea inequali y cons ain s in e ms o x1 , x2 , x3,a0 , and he (k+1) a iables bk o k= 0, . . . , K . The e o ( ε(K) ) [ 22 ] o he polyhed al app oxima ions o he h ee-dimensional second-o de cone cons ain (Equa ion (16)) is: ε(K) = 1 cosπ 2K+1−1 (18) Acco ding o Equa ion (18), when K= 11, he e o is abou 3×10−7 . The e o e, he MISOCP model o ac i e dis ibu ion ne wo k econ igu a ion is app oxima ely equi alen o he MILP model. A his poin , he imp o ed e sion o SOCR-OPF, namely he LSOCR- OPF model, is ully modeled. So a , he powe low cons ain has been ans o med om a nonlinea p og amming model o a MISOCP model composed o Equa ions (9)–(14). Then, h ough he polyhed on app oxima ion me hod, which is used o linea ize Equa ion (16), he MISOCP model can be app oxima ely con e ed in o a MILP model o solu ion. The p e iously desc ibed model ep esen s he con en ional single-pe iod OPF model. Howe e , since mos p ac ical applica ions in ol e mul i-pe iod op imiza ion, his s udy con e s he single-pe iod (s a ic) powe low model in o a mul i-pe iod (dynamic) OPF model. Fo cla i y o exp ession, he ollowing dynamic OPF model, based on LSOCR, can be exp essed in ec o o m:                          min ∑ ∈T (x ) s. . x ∈X ,∀ Aij, x ≤bT ij, x ,∀ ,∀ij ∈E ∑ ∈T B x ≤c ∑ ∈T C x =d (19) whe e, is he pe iod iden i ica ion, T is he o al numbe o ime pe iods. x ∈X , ∀ ep esen s he cons ain ela ionship in he adi ional single-pe iod OPF model, such as uppe and lowe limi cons ain s, powe low equa ion cons ain s, e c. ∑ ∈TB x ≤c is he second-o de cone cons ain ela ionship unde each b anch a each ime. Equa ion (19) adds he linea coupling ela ionship be ween mul i-pe iod pe iods o he objec i e unc ion and cons ain condi ions. Some elemen s will be desc ibed in de ail in he nex sec ion, such as OLTC, CB, ESS, e c. Hence, he u he imp o ed e sion o LSOCR-OPF, namely he LSOCR-DOPF model, is ully modeled. In o de o acili a e unde s anding, addi ional explana ion is equi ed o he en i e model ans o ma ion p ocess ( om AC-OPF o LSOCR-DOPF): Equa ions (8) and (16) espec i ely embody he phase angle elaxa ion and second-o de cone elaxa ion o LSOCR- Elec onics 2023,12, 1530 7 o 27 DOPF, and Figu e 2depic s he schema ic diag am o he wo-s ep elaxa ion p ocess. The non-con ex easible egion CAC−OPF o he o iginal AC-OPF p oblem will be elaxed in o a con ex second-o de cone easible egion COPF−c a e phase angle elaxa ion and second- o de cone elaxa ion. Then, he con ex easible egion o he second-o de cone is u he linea ized in o he con ex easible egion COPF−linea o he in ege p og amming by he polyhed al app oxima ions. A his ime, he op imal powe low p oblem in he o iginal o mula ion has al eady been ans o med in o a con ex op imiza ion p oblem. Nume ous s udies, as demons a ed in Re s. [ 13 , 14 , 16 , 17 ], ha e subs an ia ed he s ic accu acy o he second-o de cone elaxa ion (SOCR) app oach o mos dis ibu ion ne wo k s uc u es, when he objec i e unc ion is bo h a con ex and s ic ly inc easing unc ion. Elec onics 2023, 12, x FOR PEER REVIEW 7 o 27 () T ,, s. . , ,, T ij ij T T min x xX A xbx ijE Bx c Cx d ∈ ∈ ∈   ∈∀  ≤∀∀∈  ≤  =      (19) whe e, 𝑡 is he pe iod iden i ica ion, 𝑇 is he o al numbe o ime pe iods. 𝑥∈𝑋,∀𝑡 ep esen s he cons ain ela ionship in he adi ional single-pe iod OPF model, such as uppe and lowe limi cons ain s, powe low equa ion cons ain s, e c. ∑∈ 𝐵𝑥≤𝑐 is he second-o de cone cons ain ela ionship unde each b anch a each ime. Equa ion (19) adds he linea coupling ela ionship be ween mul i-pe iod pe iods o he objec i e unc ion and cons ain condi ions. Some elemen s will be desc ibed in de ail in he nex sec ion, such as OLTC, CB, ESS, e c. Hence, he u he imp o ed e sion o LSOCR-OPF, namely he LSOCR-DOPF model, is ully modeled. In o de o acili a e unde s anding, addi ional explana ion is equi ed o he en i e model ans o ma ion p ocess ( om AC-OPF o LSOCR-DOPF): Equa ions (8) and (16) espec i ely embody he phase angle elaxa ion and second-o de cone elaxa ion o LSOCR-DOPF, and Figu e 2 depic s he schema ic diag am o he wo-s ep elaxa ion p o- cess. The non-con ex easible egion 𝐶 o he o iginal AC-OPF p oblem will be e- laxed in o a con ex second-o de cone easible egion 𝐶 a e phase angle elaxa- ion and second-o de cone elaxa ion. Then, he con ex easible egion o he second- o de cone is u he linea ized in o he con ex easible egion 𝐶 o he in ege p og amming by he polyhed al app oxima ions. A his ime, he op imal powe low p oblem in he o iginal o mula ion has al eady been ans o med in o a con ex op imi- za ion p oblem. Nume ous s udies, as demons a ed in Re s. [13,14,16,17], ha e subs an- ia ed he s ic accu acy o he second-o de cone elaxa ion (SOCR) app oach o mos dis ibu ion ne wo k s uc u es, when he objec i e unc ion is bo h a con ex and s ic ly inc easing unc ion. C OPF-c C OPF-a Angle elaxa ion Eq (8) Conic elaxa ion Eq (12)/(16) C AC-OPF SOCR-OPF: MISOCP A R-OPF: MINLP OPF: MINLP polyhed al app oxima ions Eq (17) C OPF-linea Z LSOCR-DOPF: MILP block diag am Feasible Region： Objec i e： Equa ion(1) 【Cons ain s】： (2)~(7) Objec i e： Equa ion(9) 【Cons ain s】： (10)~(11)+(13)~(14)+(4)+(8) Objec i e： Equa ion(9) 【Cons ain s】： (10)~(14)+(8) Objec i e：Equa ion(9) 【Cons ain s】： (10)~(11)+(13)~(14)+(17)+(8) 010 2 1 1 11 11 11 31 ;; cos sin 22 sin cos 1, 2, , 22 an , ;2 kk k kk kk k kk KKK k axbxaa b ba b k K axba ππ ππ π −− ++ −− ++ +         =+ −+ =      { 24T+= Equa ion (19) Equa ion (8) Equa ion (12) Equa ion (17) 2 2 2,, ij ij ij ij j j P QjBijEIV IV −− −− ∀+∈≤∀∈ −           2 2 , , ij ij jj I IijE VV jB − −     ∀∈ =∀∈  = Figu e 2. The p oposed modeling p ocess o LSOCR o sol ing AC-OPF. 2.2. Ac i e Dis ibu ion Ne wo k Modeling In his sec ion, he ac i e managemen componen s o he ac i e dis ibu ion ne wo k a e conside ed, including: ① eac i e compensa ion de ice (SVC and CB); ② ac i e Figu e 2. The p oposed modeling p ocess o LSOCR o sol ing AC-OPF. 2.2. Ac i e Dis ibu ion Ne wo k Modeling In his sec ion, he ac i e managemen componen s o he ac i e dis ibu ion ne wo k a e conside ed, including: 1  eac i e compensa ion de ice (SVC and CB); 2  ac i e powe egula ion de ice (ene gy s o age sys em and elec ic ehicle mobile ene gy s o age sys em); 3  on-load ap change (OLTC); 4  dis ibu ed gene a ion powe egula ion. Gi en ha he objec i e unc ion in he OPF is ei he linea o quad a ic, he quad a ic o m can be e ec- i ely add essed by means o piecewise linea iza ion. This pape linea izes he cons ain s ela ed o he ac i e managemen equipmen . Gi en ha he ne wo k econ igu a ion is o g ea signi icance o ac i e dis ibu ion ne wo k planning and op imal ope a ion [ 32 ], his pape discusses he cons ain s ela ed o g id econ igu a ion, such as adial cons ain s. In addi ion, in o de o ensu e he applicabili y o he model, his pape u he conside s he ZIP load. 2.2.1. Ac i e Dis ibu ion Ne wo k Uni s Modeling 1. Ac i e powe egula ion de ice (1) Modeling o disc e e eac i e powe compensa ion (CB).    QCB j, =yCB j, QCB,s ep j yCB j, ≤YCB,m j ,∀ ,∀j∈BCB (20) whe e, BCB is he se o CB nodes; yCB j, is he numbe o g oups pu in o ope a ion and he disc e e a iable alue; YCB,max j is he uppe limi o he Elec onics 2023,12, 1530 8 o 27 numbe o CB g oups connec ed by node j ; QCB, s ep j is he compensa ion powe o each g oup o CB . Conside ing ac o s such as equipmen li e o economy, disc e e eac i e compensa ion is mos ly limi ed by he numbe o adjus men s, so i gene ally includes he o al numbe o ope a ions in mul iple pe iods; NCB,max jis he uppe limi o ope a ion imes: ∑ ∈TyCB j, −yCB j, −1≤NCB,m j,∀ ,∀j∈BCB (21) In addi ion, o he absolu e alue cons ain in he abo e equa ion, add an auxilia y a iable δCB j, =yCB j, −yCB j, −1 ha ep esen s he change in CB com- pensa ion capaci y be ween adjacen pe iods, co esponding o:    ∑ ∈T δCB j, ≤NCB,m j −δCB j, YCB,m j≤yCB j, ≤δCB j, YCB,m j ,∀ ,∀j∈BCB (22) (2) Modeling o con inuous eac i e powe egula ion de ice (SVC). QSVC,min j≤QSVC j, ≤QSVC,m j,∀ ,∀j∈BSVC (23) whe e, BSVC is he node se con aining SVC ; QSVC, min j and QSVC,max j a e he lowe limi and uppe limi o SVC compensa ion powe , espec i ely. Con- side ing ha , in he p ocess o ac i e dis ibu ion ne wo k ope a ion, wi h he inc easing pene a ion o dis ibu ed gene a ion (DG) such as pho o ol aic powe gene a ion, he sys em powe low may be e e sed and o e ol age p oblems may occu , he lowe limi Q o SVC compensa ion in his pape is: QSVC,min j<0. 2. OLTC model The OLTC is used o adjus he ol age alue a he low- ol age side o he bus node. The e o e, he subs a ion bus node V0is u he con e ed o he adjus able a iable:      V2 j≤VBase j, 2 j, ≤V2 j min j≤ j, ≤ max j ,∀ ,∀j∈BOLTC (24) whe e, BOLTC e e s o he node se o subs a ion con aining OLTC; VBase j, is he ol age alue a he high ol age side o he ans o me , which is a cons an alue; max jand min j a e he squa e o he uppe and lowe limi o he OLTC adjus able ans o ma ion a io; j, is he squa e o he OLTC ans o ma ion a io, de ined as he a io o he seconda y side o he p ima y side, which is ac ually a disc e e alue a iable, and can be u he ea ed as he ollowing ela ionship including 0–1 a iables: j, = min j+∑ s j,sσOLTC j,s, ,∀ ,∀j∈BOLTC (25) whe e, j,s ep esen s he di e ence be ween OLTC gea s and he squa e o gea s− 1, which is he adjacen adjus men inc emen . σOLTC j,s, is a 0–1 iden i ica ion a iable. I Elec onics 2023,12, 1530 9 o 27 i is conside ed o be cons ained by he limi o adjus men imes in p ac ice, i can be u he cons ained as: OLTC                      σOLTC j,1, ≥σOLTC j,2, ≥OLTC j,SRj, δOLTC,IN j, +δOLTC,DE j, ≤1 ∑sσOLTC j,s, −∑sσOLTC j,s, −1≥δOLTC,IN j, −δOLTC,DE j, SRj ∑sσOLTC j,s, −∑sσoLTC j,s, −1≤δOLTC,IN j, SRj−δOLTC,DE j, ∑ ∈TδOLTC,N j, +δOLTC,DE j, ≤NOLTC,max j ,∀ ,∀j∈BOLTC (26) whe e, δOLTC,N j, and δOLTC,DE j, ep esen he OLTC gea adjus men change sign, which is 0–1 a iable; i δOLTC,IN j, = 1, hen he gea alue o OLTC a ime − 1 is g ea e han he gea alue a ime , δOLTC,DE j, is simila ; SROLTC j is he maximum ange o gea change; NOLTC, max j is he maximum allowable adjus men imes o he OLTC gea a ime T. 3. ESS model Du ing his pa , he modeling o he ene gy s o age sys em akes in o accoun mul iple-pe iod cons ain s, including es ic ions on i s cha ging and discha ging s a us, cha ging and discha ging powe , as well as capaci y limi a ions. (1) Powe limi .    udischa ge j, Pdischa ge,min j≤Pdischa ge j, ≤udischa ge j, Pdischa ge,max j ucha ge j, Pcha ge,min j≤Pcha ge j, ≤ucha ge j, Pcha ge,max j ,∀ ,∀j∈BESS (27) (2) Cha ge and discha ge s a us limi . udischa ge j, +ucha ge j, ≤1, ∀j∈BESS,∀ (28) (3) Capaci y cons ain s. (EESS j, +1=EESS j, +αcha ge jPcha ge j, −αdischa ge jPdischa ge j, EESS,min j≤EESS j, ≤EESS,max j ,∀j∈BESS,∀ (29) whe e, BESS is he node se con aining ESS ; Equa ion (23) indica es ha he ESS canno be simul aneously cha ged and discha ged a he same ime, ucha ge j, and udischa ge j, is Pcha ge, min j and Pdischa ge,min j a e he uppe and lowe limi s o ESS cha ging and discha ging powe , espec i ely; EESS j, is he powe o he pe iod o ESS , EESS,max j and EESS, min j a e he uppe and lowe limi alues conside ing ac o s such as ESS li e; αcha ge j and αdischa ge j a e he cha ge and discha ge e iciency coe icien , espec i ely, gene ally 0 <αcha ge j< 1, αdischa ge j> 1. As a no el ac i e managemen echnique, elec ic ehicles can be conside ed as mobile ac i e powe ene gy s o age sys ems [ 33 ]. Thei basic model is la gely simila o ha o Ene gy S o age Sys ems (ESS). The equi alen injec ion powe a each bus node in he dis ibu ion ne wo k can be ep esen ed as he clus e ing ou comes o indi idual elec ic ehicles. 4. Dis ibu ed gene a ion model Respec i ely modeling DG wi h o wi hou eac i e powe : (1) DG modeling wi hou conside ing eac i e powe . Elec onics 2023,12, 1530 16 o 27 Elec onics 2023, 12, x FOR PEER REVIEW 16 o 27 (a)SVC (b)CB Figu e 10. 24-h eac i e compensa ion powe (SVC + CB) diag am. I can be seen om Figu es 5–10 ha : (1) Du ing he peak load pe iods (08:00–15:00, 19:00–21:00), he ac i e load demand o he sys em can be me while he Ene gy S o age Sys em (ESS) is unable o abso b he su plus clean ene gy due o i s own cha ging limi a ions, esul ing in powe abandon- men . The ESS discha ges du ing bo h he peak load pe iod and he low peak pe iod o enewable ene gy gene a ion, e ec i ely educing he peak- alley di e ence o he equi - alen load; (2) Du ing pe iods when he p opo ion o enewable ene gy ou pu o load is ela- i ely high (6:00–15:00, 18:00–22:00), eac i e powe compensa ion de ices (S a ic Va Compensa o s (SVC) and Capaci o Banks (CB)) abso b he excess eac i e powe o he sys em, a oiding o e ol age. Hence, he LSOCR-DOPF model p oposed in his pape demons a es signi ican e - ec s on he op imiza ion o bo h ac i e and eac i e powe . 3.1.3. Model Validi y Analysis The LSOCR-DOPF is analyzed unde h ee si ua ions: “ elaxa ion accu acy, calcula- ion e iciency, and compa ison o di e en op imiza ion cases”. (1) Relaxa ion accu acy The elaxa ion accu acy o he objec i e unc ion, such as ne wo k loss, has been p o ed and e i ied in he pu e load ne wo k en i onmen . Howe e , conside ing he u - he analysis o he model accu acy a e inc easing he main ne wo k ou pu , powe aban- donmen , and load loss penal y cos s, i is necessa y o de ine he e o index: Δ, diff = , +, −, ,. Figu e 11 shows he e o sca e diag am o each b anch in one day. Ob iously, he de ia ion a e elaxa ion mee s he equi emen s o accu a e ope a ion, which is 10–. 6 Figu e 11. Sca e cha o e o unde each b anch in each pe iod. (2) Calcula ion imeliness Figu e 11. Sca e cha o e o unde each b anch in each pe iod. (2) Calcula ion imeliness Table 1p esen s he solu ion speed and espec i e simula ion esul s unde a ious ob- jec i e unc ions. Speci ically, he objec i e unc ions o each simula ed ope a ing condi ions a e as ollows: Case 1: minimized ne wo k loss; Case 2: minimized powe pu chase cos o he main g id; Case 3: minimum powe loss; Case 4: minimized ne wo k loss and main g id powe pu chase cos ; Case 5: minimized ne wo k loss, main g id powe pu chase cos , and powe abandonmen . Table 1shows he solu ion speed and co esponding simula ion esul s unde di e en objec i e unc ions. Among hem, he objec i e unc ion Case 1 o each simula ed ope a ion condi ion is he minimum ne wo k loss; Case 2: he minimum powe pu chase cos o he main ne wo k; Case 3: minimal powe loss; Case 4: ne wo k loss and main ne wo k powe pu chase cos a e he leas ; Case 5: he ne wo k loss, main ne wo k powe pu chase cos , and powe abandonmen a e he minimum. Table 1. Op imiza ion esul s unde ou ypes o objec i e unc ions. Case Time (s) Ta ge (103$) Ne wo k Loss Powe Pu chase Cos Powe Abandonmen 1 20.394 0.244 10.937 22.939 2 18.581 3.274 1.890 10.728 3 8.555 10.013 4.757 6.752 4 125.044 0.337 1.890 13.673 5 200.482 4.796 1.890 9.087 Based on Table 1, wi h he inc ease in he objec i e unc ion, he calcula ion ime con inues o inc ease, bu he calcula ion speed is s ill accep able, he maximum is 200 s, which mee s he ime equi emen s o day-ahead dispa ching and eal- ime op imiza ion o he ac i e ope a ion in he ac i e dis ibu ion ne wo k. Fu he mo e, based on Case 1, his pa compa es he sol ing e iciency o he MINLP (In e io Poin -DOPF, Mixed-In ege Nonlinea P og amming) model, he MISOCP (SOCR- DOPF) model, and he MILP (LSOCR-DOPF) model o he DOPF p oblem, and he esul s a e summa ized in Table 2. I can be seen om Table 2 ha he accu acy and e iciency o LSOCR and SOCR a e highe han hose o he adi ional in e io poin me hod. The dis i- bu ion ne wo k loss op imiza ion esul o he MILP model linea ized by SOCR is equal o he esul o he MISOCP model, bu he solu ion speed has been imp o ed o a ce ain ex en . This ully shows ha he op imiza ion e iciency can be imp o ed by he LSOCR, which can each 25~30% (compa ed wi h SOCR). I is shown ha he LSOCR me hod is sligh ly be e han he SOCR me hod in e ms o comp ehensi e solu ion e iciency, bu i s solu ion imeliness is en i ely be e han he adi ional in e io poin me hod. Elec onics 2023,12, 1530 17 o 27 Table 2. Compa ison o op imiza ion esul s o DOPF. Ne wo k Ta ge (103$) Time (s) MINLP MISOCP MILP MINLP MISOCP MILP IEEE33 0.352 0.244 0.244 >5 h 20.394 14.564 IEEE69 1.556 1.356 1.356 >5 h 27.396 20.509 (3) Compa ison o di e en op imiza ion cases Acco ding o Tables 1and 2, i can be seen ha : 1  In Case 1, only he op imiza ion o ne wo k loss is conside ed, leading o subs an ial was e o clean ene gy powe . On he o he hand, Case 2 p io i izes he minimiza ion o main g id powe pu chase cos s, which esul s in inc eased consump ion o clean ene gy, educing he main g id powe pu chase and, in u n, educing powe was e. Con e sely, Case 3 ocuses on minimizing powe loss. In compa ison o Case 1, i equi es mo e u iliza ion o clean ene gy o educe main g id powe pu chases, bu esul s in an inc ease in ne wo k loss; 2  In Case 4, he objec i e is o simul aneously minimize bo h he ne wo k loss and he main ne wo k’s powe pu chase. The esul s show ha while he main ne wo k’s powe pu chase emains unchanged, he ne wo k loss dec eases, bu he amoun o powe abandonmen inc eases. The pape assumes ha he cos o ne wo k loss and he penal y cos o powe abandonmen a e bo h equal o 5000 $/MWh, which sugges s ha excessi e access o clean ene gy may inc ease he ne wo k loss o he sys em; 3  Case 5 aims o minimize he ne wo k loss, main ne wo k powe pu chase, and powe abandonmen . The main ne wo k powe pu chase emains he same, while he powe abandonmen co espondingly dec eases, and he ne wo k loss inc eases, which means ha he ne wo k loss caused by inc easing he access o clean ene gy is less han he penal y o clean ene gy powe abandonmen , so he powe abandonmen is u he educed. 4  To assess he impac o he ac i e managemen equipmen (SVC, CB, ESS) in- oduced in his pape on he pe o mance o he dis ibu ion ne wo k, h ee es s we e designed and included: (a) No addi ion o ene gy s o age and eac i e powe compensa ion equipmen ; (b) Addi ion o an ene gy s o age de ice in place o a eac i e compensa ion de ice; (c) Simul aneous addi ion o bo h an ene gy s o age de ice and a eac i e compen- sa ion de ice. I can be seen om Figu e 12 ha adding he ac i e managemen uni can imp o e he uni o m dis ibu ion o ol age in he dis ibu ion ne wo k o some ex en , which shows some unc ions o he ac i e managemen uni . I is wo h men ioning ha he ela i ely ele a ed node ol age depic ed in Figu e 12 can be a ibu ed o he ne wo k loss speci ied by he objec i e unc ion. Elec onics 2023, 12, x FOR PEER REVIEW 18 o 27 mo e u iliza ion o clean ene gy o educe main g id powe pu chases, bu esul s in an inc ease in ne wo k loss; ② In Case 4, he objec i e is o simul aneously minimize bo h he ne wo k loss and he main ne wo k’s powe pu chase. The esul s show ha while he main ne wo k’s powe pu chase emains unchanged, he ne wo k loss dec eases, bu he amoun o powe abandonmen inc eases. The pape assumes ha he cos o ne wo k loss and he penal y cos o powe abandonmen a e bo h equal o 5000 $/MWh, which sugges s ha excessi e access o clean ene gy may inc ease he ne wo k loss o he sys em; ③ Case 5 aims o minimize he ne wo k loss, main ne wo k powe pu chase, and powe abandonmen . The main ne wo k powe pu chase emains he same, while he powe abandonmen co espondingly dec eases, and he ne wo k loss inc eases, which means ha he ne wo k loss caused by inc easing he access o clean ene gy is less han he penal y o clean ene gy powe abandonmen , so he powe abandonmen is u he educed. ④ To assess he impac o he ac i e managemen equipmen (SVC, CB, ESS) in o- duced in his pape on he pe o mance o he dis ibu ion ne wo k, h ee es s we e de- signed and included: (a) No addi ion o ene gy s o age and eac i e powe compensa ion equipmen ; (b) Addi ion o an ene gy s o age de ice in place o a eac i e compensa ion de ice; (c) Simul aneous addi ion o bo h an ene gy s o age de ice and a eac i e com- pensa ion de ice. I can be seen om Figu e 12 ha adding he ac i e managemen uni can imp o e he uni o m dis ibu ion o ol age in he dis ibu ion ne wo k o some ex- en , which shows some unc ions o he ac i e managemen uni . I is wo h men ioning ha he ela i ely ele a ed node ol age depic ed in Figu e 12 can be a ibu ed o he ne wo k loss speci ied by he objec i e unc ion. (a)Wi hou ESS,Reac i ePowe Compensa o (b)ESS,Wi hou Reac i ePowe Compensa o (c)ESS,Reac i ePowe Compensa o Figu e 12. Vol age o di e en nodes (compa a i e e i ica ion). Based on he simula ion esul s o he scena io in his sec ion, an analysis o h ee impo an pa ame e s, calcula ion ime, solu ion e o , and node ol age dis ibu ion, is conduc ed o e alua e he e ec i eness o he p oposed me hod in his pape . Fi s ly, in e ms o powe g id s abili y, as shown in Figu e 12c, he ol age dis ibu- ion o each node in he IEEE33 node ne wo k can mee he se ange equi emen o [0.94, 1.06], and he dis ibu ion is ela i ely uni o m. Secondly, in e ms o powe g id scheduling equency, as shown in Tables 1 and 2, he six se s o compa ison cases (IEEE33 + IEEE69) se in his scena io can un wi hin 3 min (minimum o 8.555 s), which can mee he eal- ime powe g id scheduling equency equi emen s. Thi dly, in e ms o sol ing accu acy, as shown in Figu e 11, he magni ude o he e o in he simula ion esul s o his scena io is s ic ly con olled wi hin 10-6, which can ensu e he equi emen o powe g id sol ing accu acy. 3.2. Ne wo k Recon igu a ion Figu e 12. Vol age o di e en nodes (compa a i e e i ica ion). Elec onics 2023,12, 1530 18 o 27 Based on he simula ion esul s o he scena io in his sec ion, an analysis o h ee impo an pa ame e s, calcula ion ime, solu ion e o , and node ol age dis ibu ion, is conduc ed o e alua e he e ec i eness o he p oposed me hod in his pape . Fi s ly, in e ms o powe g id s abili y, as shown in Figu e 12c, he ol age dis ibu ion o each node in he IEEE33 node ne wo k can mee he se ange equi emen o [0.94, 1.06], and he dis ibu ion is ela i ely uni o m. Secondly, in e ms o powe g id scheduling equency, as shown in Tables 1and 2, he six se s o compa ison cases (IEEE33 + IEEE69) se in his scena io can un wi hin 3 min (minimum o 8.555 s), which can mee he eal- ime powe g id scheduling equency equi emen s. Thi dly, in e ms o sol ing accu acy, as shown in Figu e 11, he magni ude o he e o in he simula ion esul s o his scena io is s ic ly con olled wi hin 10-6, which can ensu e he equi emen o powe g id sol ing accu acy. 3.2. Ne wo k Recon igu a ion The dis ibu ion ne wo k econ igu a ion scena io is analyzed based on he s anda d IEEE 33 sys ems (Appendix A), and he load alue is he o iginal sys em da a. In o de o e i y he e ec i eness o second-o de cone elaxa ion in dis ibu ion ne wo k econs uc- ion, his example uses s a ic single-pe iod econs uc ion o analysis. In he wo examples, node 1 is a subs a ion node wi h a ol age ampli ude o 1.06 p.u.. I is assumed ha each b anch is equipped wi h a sec ion swi ch, and he op imiza ion calcula ion is ca ied ou wi h he goal o minimizing he ne wo k loss. The esul s a e shown in Table 3. This also sol es he e o sca e diag am o each b anch, as shown in Figu e 13. Table 3. Recon igu a ion scheme and op imiza ion esul s. IEEE33 Time (s) Ne wo k Loss (MW) O iginal Ne wo k Loss (MW) Disconnec ed Swi ch Single-pe iod 1.355014 0.0256 0.0368 7 (6–7), 9 (8–9), 14 (13–14), 32 (31–32), 37 (24–28) Mul i-pe iod 153.404963 1.7080 2.4964 Elec onics 2023, 12, x FOR PEER REVIEW 19 o 27 The dis ibu ion ne wo k econ igu a ion scena io is analyzed based on he s anda d IEEE 33 sys ems (Appendix A), and he load alue is he o iginal sys em da a. In o de o e i y he e ec i eness o second-o de cone elaxa ion in dis ibu ion ne wo k econ- s uc ion, his example uses s a ic single-pe iod econs uc ion o analysis. In he wo ex- amples, node 1 is a subs a ion node wi h a ol age ampli ude o 1.06 p.u.. I is assumed ha each b anch is equipped wi h a sec ion swi ch, and he op imiza ion calcula ion is ca ied ou wi h he goal o minimizing he ne wo k loss. The esul s a e shown in Table 3. This also sol es he e o sca e diag am o each b anch, as shown in Figu e 13. Table 3. Recon igu a ion scheme and op imiza ion esul s. IEEE33 Time (s) Ne wo k Loss (MW) O iginal Ne wo k Loss (MW) Disconnec ed Swi ch Single-pe iod 1.355014 0.0256 0.0368 7 (6–7), 9 (8–9), 14 (13–14), 32 (31–32), 37 (24–28) Mul i-pe iod 153.404963 1.7080 2.4964 33 1 2 3456 7 8 910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1 2 3456 7 8 910 11 12 13 14 15 16 17 26 27 28 29 30 31 32 23 24 22 19 20 21 33 35 34 36 37 Figu e 13. Op imized and econs uc ed IEEE33 ne wo k (node se ial numbe is gi en by black numbe s, b anch se ial numbe is shown by ed numbe s, and ed c oss and ed do ed line indica e disconnec ed b anch). As shown in Figu e 14, i is no di icul o ind ha he e o is 10– o de s o mag- ni ude, mee ing he econs uc ion equi emen s. I is wo h no ing ha , in his example, no ac i e managemen equipmen is added. Fo IEEE 33 nodes, all nodes ha e load alues, so only Equa ion (33) can be added o he adia ion cons ain . Fo some nodes in he IEEE69 sys em wi hou load, Equa ion (34) mus be added o ensu e ha all nodes a e connec ed and ope a e wi hou islands and adia ion. Based on he simula ion esul s o his sec ion, he h ee impo an pa ame e s o cal- cula ion ime, solu ion e o , and node ol age dis ibu ion a e e alua ed. Fi s ly, in e ms o powe g id s abili y, as shown in Figu e 15, he ol age dis ibu ion o each node in he IEEE33 node ne wo k can mee he se ange equi emen o [0.94, 1.06], and he ol age dis ibu ion in mul i-pe iod is ela i ely uni o m. Secondly, ega ding powe g id sched- uling equency, as shown in Table 3, he wo se s o compa a i e cases (single-pe iod and mul i-pe iod) ensu e ha he ope a ing ime is wi hin 3 min ( he sho es being 1.355 s), mee ing he equi emen s o eal- ime powe g id scheduling equency. Thi dly, in e ms o sol ing accu acy, as shown in Figu e 14, he e o le el o he simula ion esul s in his scena io is s ic ly con olled wi hin 10−9, which can well gua an ee he accu acy equi e- men s o powe g id sol ing. Figu e 13. Op imized and econs uc ed IEEE33 ne wo k (node se ial numbe is gi en by black numbe s, b anch se ial numbe is shown by ed numbe s, and ed c oss and ed do ed line indica e disconnec ed b anch). As shown in Figu e 14, i is no di icul o ind ha he e o is 10 –9 o de s o magni- ude, mee ing he econs uc ion equi emen s. I is wo h no ing ha , in his example, no ac i e managemen equipmen is added. Fo IEEE 33 nodes, all nodes ha e load alues, so only Equa ion (33) can be added o he adia ion cons ain . Fo some nodes in he IEEE69 sys em wi hou load, Equa ion (34) mus be added o ensu e ha all nodes a e connec ed and ope a e wi hou islands and adia ion. Elec onics 2023,12, 1530 19 o 27 Elec onics 2023, 12, x FOR PEER REVIEW 20 o 27 9 Figu e 14. Sca e diag am o dis ibu ion ne wo k econ igu a ion e o . (a)Single-pe iod(24hou s) (b)Mul i-pe iod Figu e 15. Vol age o di e en nodes. 3.3. ZIP Load Applica ion This scena io adop s he PG69 node sys em es (Figu e 16). In o de es he s a ic ol age cha ac e is ics o he load, his pa does no conside he ac i e managemen uni , and only analyzes he calcula ion esul s a e ci ing he s a ic ol age cha ac e is ics o he load. Th ee ypes o loads, namely cons an powe , cons an cu en , and cons an im- pedance, a e added o each node. The ol age dis ibu ion o each node a each ime is calcula ed as shown in Figu e 17. Figu e 18 shows he compa ison o load ac i e demand conside ing ol age s a ic cha ac e is ics o no . Figu e 14. Sca e diag am o dis ibu ion ne wo k econ igu a ion e o . Based on he simula ion esul s o his sec ion, he h ee impo an pa ame e s o calcula ion ime, solu ion e o , and node ol age dis ibu ion a e e alua ed. Fi s ly, in e ms o powe g id s abili y, as shown in Figu e 15, he ol age dis ibu ion o each node in he IEEE33 node ne wo k can mee he se ange equi emen o [0.94, 1.06], and he ol age dis ibu ion in mul i-pe iod is ela i ely uni o m. Secondly, ega ding powe g id scheduling equency, as shown in Table 3, he wo se s o compa a i e cases (single-pe iod and mul i-pe iod) ensu e ha he ope a ing ime is wi hin 3 min ( he sho es being 1.355 s), mee ing he equi emen s o eal- ime powe g id scheduling equency. Thi dly, in e ms o sol ing accu acy, as shown in Figu e 14, he e o le el o he simula ion esul s in his scena io is s ic ly con olled wi hin 10 −9 , which can well gua an ee he accu acy equi emen s o powe g id sol ing. Elec onics 2023, 12, x FOR PEER REVIEW 20 o 27 9 Figu e 14. Sca e diag am o dis ibu ion ne wo k econ igu a ion e o . (a)Single-pe iod(24hou s) (b)Mul i-pe iod Figu e 15. Vol age o di e en nodes. 3.3. ZIP Load Applica ion This scena io adop s he PG69 node sys em es (Figu e 16). In o de es he s a ic ol age cha ac e is ics o he load, his pa does no conside he ac i e managemen uni , and only analyzes he calcula ion esul s a e ci ing he s a ic ol age cha ac e is ics o he load. Th ee ypes o loads, namely cons an powe , cons an cu en , and cons an im- pedance, a e added o each node. The ol age dis ibu ion o each node a each ime is calcula ed as shown in Figu e 17. Figu e 18 shows he compa ison o load ac i e demand conside ing ol age s a ic cha ac e is ics o no . Figu e 15. Vol age o di e en nodes. 3.3. ZIP Load Applica ion This scena io adop s he PG69 node sys em es (Figu e 16). In o de es he s a ic ol age cha ac e is ics o he load, his pa does no conside he ac i e managemen uni , and only analyzes he calcula ion esul s a e ci ing he s a ic ol age cha ac e is ics o he load. Th ee ypes o loads, namely cons an powe , cons an cu en , and cons an impedance, a e added o each node. The ol age dis ibu ion o each node a each ime is calcula ed as shown in Figu e 17. Figu e 18 shows he compa ison o load ac i e demand conside ing ol age s a ic cha ac e is ics o no . Elec onics 2023,12, 1530 20 o 27 Elec onics 2023, 12, x FOR PEER REVIEW 21 o 27 1 2 3456 7 8 910 11 12 13 14 15 16 17 28 29 30 31 47 48 49 68 69 18 19 20 21 22 23 24 25 26 27 36 37 38 39 32 33 34 35 40 41 42 43 44 45 46 53 54 55 56 57 58 59 60 61 62 64 65 63 50 66 67 51 52 Cons an powe load Cons an cu en load Cons an impedanceload Figu e 16. IEEE69 es ac i e dis ibu ion ne wo k. Figu e 17. Vol age dis ibu ion diag am o each node. Figu e 18. Compa ison o load ac i e demand conside ing ol age s a ic cha ac e is ics o no . As can be seen om Figu es 17 and 18, since mos o he node ol age is lowe han 1.0 p.u., he load demand will be educed a e conside ing he s a ic ol age cha ac e is- ics o he load. I is impo an o conside he s a ic ol age cha ac e is ics o ac i e dis i- bu ion ne wo k ine simula ion. In addi ion, he second-o de cone elaxa ion e o o each pe iod is shown in Figu e 19. Ob iously, he elaxa ion e ec is also highly sa is ac o y. Figu e 16. IEEE69 es ac i e dis ibu ion ne wo k. Elec onics 2023, 12, x FOR PEER REVIEW 21 o 27 1 2 3456 7 8 910 11 12 13 14 15 16 17 28 29 30 31 47 48 49 68 69 18 19 20 21 22 23 24 25 26 27 36 37 38 39 32 33 34 35 40 41 42 43 44 45 46 53 54 55 56 57 58 59 60 61 62 64 65 63 50 66 67 51 52 Cons an powe load Cons an cu en load Cons an impedanceload Figu e 16. IEEE69 es ac i e dis ibu ion ne wo k. Figu e 17. Vol age dis ibu ion diag am o each node. Figu e 18. Compa ison o load ac i e demand conside ing ol age s a ic cha ac e is ics o no . As can be seen om Figu es 17 and 18, since mos o he node ol age is lowe han 1.0 p.u., he load demand will be educed a e conside ing he s a ic ol age cha ac e is- ics o he load. I is impo an o conside he s a ic ol age cha ac e is ics o ac i e dis i- bu ion ne wo k ine simula ion. In addi ion, he second-o de cone elaxa ion e o o each pe iod is shown in Figu e 19. Ob iously, he elaxa ion e ec is also highly sa is ac o y. Figu e 17. Vol age dis ibu ion diag am o each node. Elec onics 2023, 12, x FOR PEER REVIEW 21 o 27 1 2 3456 7 8 910 11 12 13 14 15 16 17 28 29 30 31 47 48 49 68 69 18 19 20 21 22 23 24 25 26 27 36 37 38 39 32 33 34 35 40 41 42 43 44 45 46 53 54 55 56 57 58 59 60 61 62 64 65 63 50 66 67 51 52 Cons an powe load Cons an cu en load Cons an impedanceload Figu e 16. IEEE69 es ac i e dis ibu ion ne wo k. Figu e 17. Vol age dis ibu ion diag am o each node. Figu e 18. Compa ison o load ac i e demand conside ing ol age s a ic cha ac e is ics o no . As can be seen om Figu es 17 and 18, since mos o he node ol age is lowe han 1.0 p.u., he load demand will be educed a e conside ing he s a ic ol age cha ac e is- ics o he load. I is impo an o conside he s a ic ol age cha ac e is ics o ac i e dis i- bu ion ne wo k ine simula ion. In addi ion, he second-o de cone elaxa ion e o o each pe iod is shown in Figu e 19. Ob iously, he elaxa ion e ec is also highly sa is ac o y. Figu e 18. Compa ison o load ac i e demand conside ing ol age s a ic cha ac e is ics o no . As can be seen om Figu es 17 and 18, since mos o he node ol age is lowe han 1.0 p.u., he load demand will be educed a e conside ing he s a ic ol age cha ac e is ics o he load. I is impo an o conside he s a ic ol age cha ac e is ics o ac i e dis ibu ion Elec onics 2023,12, 1530 21 o 27 ne wo k ine simula ion. In addi ion, he second-o de cone elaxa ion e o o each pe iod is shown in Figu e 19. Ob iously, he elaxa ion e ec is also highly sa is ac o y. Elec onics 2023, 12, x FOR PEER REVIEW 22 o 27 8 Figu e 19. E o sca e cha o ZIP load model. Based on he simula ion esul s o his scena io, he calcula ion ime, solu ion e o , and node ol age dis ibu ion a e e alua ed. Fi s ly, in e ms o powe g id s abili y, as shown in Figu e 17, he ol age dis ibu ion o each node in he IEEE33 node ne wo k can mee he equi emen s o he [0.94, 1.06] ange. Secondly, in e ms o powe g id dispa ch equency, he unning ime o his case is 5.394 s, which can mee he equi emen s o eal- ime powe g id dispa ch equency. Thi dly, in e ms o solu ion accu acy, as shown in Figu e 19, he e o le el o he simula ion esul in his scena io is s ic ly con olled wi hin 10–, which can gua an ee he equi emen o powe g id solu ion accu acy. Fi- nally, ega ding he accu a e modeling o dis ibu ion ne wo ks, as shown in Figu e 18, he in oduc ion o ZIP loads can change he load demand, demons a ing he impo ance o he ac i e and de ailed modeling o dis ibu ion ne wo ks. 4. Conclusions In his pape , a polyhed al linea app oxima ion me hod o he second-o de cone elaxa ion (LSOCR-DOPF) o he dynamic op imal powe low p oblem based on he b anch powe low model o he dis ibu ion ne wo k is p oposed and alida ed. De ailed linea cons ain modeling o ac i e managemen uni s is pe o med, and he e ec i eness o he p oposed solu ion me hod is e i ied h ough compa ison wi h o he solu ions in h ee majo applica ion scena ios. The esul s show ha he s a egy p oposed in his pa- pe can mee he enginee ing s anda ds o ypical applica ion scena ios o dis ibu ion ne wo ks: on one hand, he calcula ion ime is con olled wi hin 3 min, which sa is ies he equi emen o he gene al powe g id scheduling e esh a e and ensu es as solu ion o OPF p oblems in u u e la ge-scale ac i e dis ibu ion ne wo ks; on he o he hand, he calcula ion esul e o is con olled wi hin 10−6, e ec i ely a oiding he ca as ophic con- sequences caused by calcula ion e o s in p e ious wo k; mo eo e , he ol age dis ibu- ion o each node in he dis ibu ion ne wo k mee s he condi ion cons ain s [0.94, 1.06] se by au ho s, which can achie e he op imiza ion goal o educing ne wo k loss. I is no ewo hy ha he LSOCR-DOPF model sa is ies he equi emen s o calcula ion ime and accu acy in he ields o daily scheduling, eal- ime ope a ion, and he con ol o ac- i e dis ibu ion ne wo ks, demons a ing s ong p ac ical applica ion alue, and he case s udy sugges s ha conside ing he ZIP model o he ine simula ion o ac i e dis ibu ion ne wo ks is also o g ea signi icance. Cu en ly, he inclusion o disc e e a iables, which a e non-con ex sou ces, has an impac on he accu acy o he elaxa ion model, and he uni e sali y condi ion o he second-o de cone elaxa ion model equi es u he heo e - ical in es iga ion in u u e wo k. Fu he mo e, i is di icul o ensu e he accu acy o con ex elaxa ion using a ela- i ely uni e sal me hod due o he a ious bounda y condi ions ha need o be se in he model o p ac ical applica ions. The echnical challenges o applying con ex elaxa ion echniques in enginee ing can be summa ized as ollows: u he explo ing he in luence o he objec i e unc ion and easible egion o he OPF p oblem on con ex elaxa ion and Figu e 19. E o sca e cha o ZIP load model. Based on he simula ion esul s o his scena io, he calcula ion ime, solu ion e o , and node ol age dis ibu ion a e e alua ed. Fi s ly, in e ms o powe g id s abili y, as shown in Figu e 17, he ol age dis ibu ion o each node in he IEEE33 node ne wo k can mee he equi emen s o he [0.94, 1.06] ange. Secondly, in e ms o powe g id dispa ch equency, he unning ime o his case is 5.394 s, which can mee he equi emen s o eal- ime powe g id dispa ch equency. Thi dly, in e ms o solu ion accu acy, as shown in Figu e 19, he e o le el o he simula ion esul in his scena io is s ic ly con olled wi hin 10 –8 , which can gua an ee he equi emen o powe g id solu ion accu acy. Finally, ega ding he accu a e modeling o dis ibu ion ne wo ks, as shown in Figu e 18, he in oduc ion o ZIP loads can change he load demand, demons a ing he impo ance o he ac i e and de ailed modeling o dis ibu ion ne wo ks. 4. Conclusions In his pape , a polyhed al linea app oxima ion me hod o he second-o de cone elaxa ion (LSOCR-DOPF) o he dynamic op imal powe low p oblem based on he b anch powe low model o he dis ibu ion ne wo k is p oposed and alida ed. De ailed linea cons ain modeling o ac i e managemen uni s is pe o med, and he e ec i eness o he p oposed solu ion me hod is e i ied h ough compa ison wi h o he solu ions in h ee majo applica ion scena ios. The esul s show ha he s a egy p oposed in his pape can mee he enginee ing s anda ds o ypical applica ion scena ios o dis ibu ion ne wo ks: on one hand, he calcula ion ime is con olled wi hin 3 min, which sa is ies he equi emen o he gene al powe g id scheduling e esh a e and ensu es as solu ion o OPF p oblems in u u e la ge-scale ac i e dis ibu ion ne wo ks; on he o he hand, he calcula ion esul e o is con olled wi hin 10 −6 , e ec i ely a oiding he ca as ophic con- sequences caused by calcula ion e o s in p e ious wo k; mo eo e , he ol age dis ibu ion o each node in he dis ibu ion ne wo k mee s he condi ion cons ain s [0.94, 1.06] se by au ho s, which can achie e he op imiza ion goal o educing ne wo k loss. I is no ewo hy ha he LSOCR-DOPF model sa is ies he equi emen s o calcula ion ime and accu acy in he ields o daily scheduling, eal- ime ope a ion, and he con ol o ac i e dis ibu ion ne wo ks, demons a ing s ong p ac ical applica ion alue, and he case s udy sugges s ha conside ing he ZIP model o he ine simula ion o ac i e dis ibu ion ne wo ks is also o g ea signi icance. Cu en ly, he inclusion o disc e e a iables, which a e non-con ex sou ces, has an impac on he accu acy o he elaxa ion model, and he uni e sali y condi- Elec onics 2023,12, 1530 22 o 27 ion o he second-o de cone elaxa ion model equi es u he heo e ical in es iga ion in u u e wo k. Fu he mo e, i is di icul o ensu e he accu acy o con ex elaxa ion using a ela i ely uni e sal me hod due o he a ious bounda y condi ions ha need o be se in he model o p ac ical applica ions. The echnical challenges o applying con ex elaxa ion echniques in enginee ing can be summa ized as ollows: u he explo ing he in luence o he objec i e unc ion and easible egion o he OPF p oblem on con ex elaxa ion and seeking su icien condi ions o gua an ee accu a e con ex elaxa ion in heo y; cons uc ing igh e and mo e p ecise elaxa ions based on SOCP elaxa ion echniques and explo ing he possibili y o combining con ex elaxa ion echniques wi h o he OPF solu ion me hods. Au ho Con ibu ions: Concep ualiza ion, W.M. and X.D.; me hodology, W.M.; so wa e, W.M.; alida ion, W.M. and J.Y.; o mal analysis, W.M. and R.M.R.-A.; in es iga ion, M.D.; esou ces, W.M.; da a cu a ion, W.M. and X.D.; w i ing—o iginal d a p epa a ion, X.D. and W.M.; w i ing— e iew and edi ing, V.S.; isualiza ion, W.M.; supe ision, D.S.; p ojec adminis a ion, X.D.; unding acquisi ion, X.D. All au ho s ha e ead and ag eed o he published e sion o he manusc ip . Funding: This esea ch was suppo ed by Na ional Na u al Science Founda ion o China (NSFC) unde G an 52177204; he Na u al Science Founda ion o Hunan P o ince (No. 2020JJ4744); he Inno a ion-D i en P ojec o Cen al Sou h Uni e si y (No.2020CX031); he in e nal g an p ojec o VSB-Technical Uni e si y o Os a a (SGS p ojec , g an numbe SP2022/77). Da a A ailabili y S a emen : No applicable. Con lic s o In e es : The au ho s decla e no con lic o in e es . Appendix A Elec onics 2023, 12, x FOR PEER REVIEW 23 o 27 seeking su icien condi ions o gua an ee accu a e con ex elaxa ion in heo y; cons uc - ing igh e and mo e p ecise elaxa ions based on SOCP elaxa ion echniques and explo - ing he possibili y o combining con ex elaxa ion echniques wi h o he OPF solu ion me hods. Au ho Con ibu ions: Concep ualiza ion, W.M. and X.D.; me hodology, W.M.; so wa e, W.M.; al- ida ion, W.M. and J.Y.; o mal analysis, W.M. and R.M.R.-A.; in es iga ion, M.D.; esou ces, W.M.; da a cu a ion, W.M. and X.D.; w i ing—o iginal d a p epa a ion, X.D. and W.M.; w i ing— e iew and edi ing, V.S.; isualiza ion, W.M.; supe ision, D.S.; p ojec adminis a ion, X.D.; unding acqui- si ion, X.D. All au ho s ha e ead and ag eed o he published e sion o he manusc ip . Funding: This esea ch was suppo ed by Na ional Na u al Science Founda ion o China (NSFC) unde G an 52177204; he Na u al Science Founda ion o Hunan P o ince (No. 2020JJ4744); he In- no a ion-D i en P ojec o Cen al Sou h Uni e si y (No.2020CX031); he in e nal g an p ojec o VSB-Technical Uni e si y o Os a a (SGS p ojec , g an numbe SP2022/77). Da a A ailabili y S a emen : No applicable. Con lic s o In e es : The au ho s decla e no con lic o in e es . Appendix A IEEE33 33 1 2 3456 7 8 910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Figu e A1. IEEE33 Radial Dis ibu ion Ne wo k. IEEE69 1 2 3456 7 8 910 11 12 13 14 15 16 17 28 29 30 31 47 48 49 68 69 18 19 20 21 22 23 24 25 26 27 36 37 38 39 32 33 34 35 40 41 42 43 44 45 46 53 54 55 56 57 58 59 60 61 62 64 65 63 50 66 67 51 52 Figu e A2. IEEE69 Radial Dis ibu ion Ne wo k. Table A1. Sys em da a o 69-bus adial dis ibu ion ne wo k (‘*’ deno es a ie-line). B anch Num- be Sending Bus Recei ing Bus Resis ance Ω Reac ance Ω Nominal Load a Re- cei ing Bus Maximum Line Capac- i y (kVA) P (kW) Q(kVA) 1 1 2 0.0005 0.0012 0.0 0.0 10,761 Figu e A1. IEEE33 Radial Dis ibu ion Ne wo k. Elec onics 2023, 12, x FOR PEER REVIEW 23 o 27 seeking su icien condi ions o gua an ee accu a e con ex elaxa ion in heo y; cons uc - ing igh e and mo e p ecise elaxa ions based on SOCP elaxa ion echniques and explo - ing he possibili y o combining con ex elaxa ion echniques wi h o he OPF solu ion me hods. Au ho Con ibu ions: Concep ualiza ion, W.M. and X.D.; me hodology, W.M.; so wa e, W.M.; al- ida ion, W.M. and J.Y.; o mal analysis, W.M. and R.M.R.-A.; in es iga ion, M.D.; esou ces, W.M.; da a cu a ion, W.M. and X.D.; w i ing—o iginal d a p epa a ion, X.D. and W.M.; w i ing— e iew and edi ing, V.S.; isualiza ion, W.M.; supe ision, D.S.; p ojec adminis a ion, X.D.; unding acqui- si ion, X.D. All au ho s ha e ead and ag eed o he published e sion o he manusc ip . Funding: This esea ch was suppo ed by Na ional Na u al Science Founda ion o China (NSFC) unde G an 52177204; he Na u al Science Founda ion o Hunan P o ince (No. 2020JJ4744); he In- no a ion-D i en P ojec o Cen al Sou h Uni e si y (No.2020CX031); he in e nal g an p ojec o VSB-Technical Uni e si y o Os a a (SGS p ojec , g an numbe SP2022/77). Da a A ailabili y S a emen : No applicable. Con lic s o In e es : The au ho s decla e no con lic o in e es . Appendix A IEEE33 33 1 2 3456 7 8 910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Figu e A1. IEEE33 Radial Dis ibu ion Ne wo k. IEEE69 1 2 3456 7 8 910 11 12 13 14 15 16 17 28 29 30 31 47 48 49 68 69 18 19 20 21 22 23 24 25 26 27 36 37 38 39 32 33 34 35 40 41 42 43 44 45 46 53 54 55 56 57 58 59 60 61 62 64 65 63 50 66 67 51 52 Figu e A2. IEEE69 Radial Dis ibu ion Ne wo k. Table A1. Sys em da a o 69-bus adial dis ibu ion ne wo k (‘*’ deno es a ie-line). B anch Num- be Sending Bus Recei ing Bus Resis ance Ω Reac ance Ω Nominal Load a Re- cei ing Bus Maximum Line Capac- i y (kVA) P (kW) Q(kVA) 1 1 2 0.0005 0.0012 0.0 0.0 10,761 Figu e A2. IEEE69 Radial Dis ibu ion Ne wo k. Elec onics 2023,12, 1530 23 o 27 Table A1. Sys em da a o 69-bus adial dis ibu ion ne wo k (‘*’ deno es a ie-line). B anch Numbe Sending Bus Recei ing Bus Resis ance Ω Reac ance Ω Nominal Load a Recei ing Bus Maximum Line Capaci y (kVA) P (kW) Q(kVA) 1 1 2 0.0005 0.0012 0.0 0.0 10,761 2 2 3 0.0005 0.0012 0.0 0.0 10,761 3 3 4 0.0015 0.0036 0.0 0.0 10,761 4 4 5 0.0251 0.0294 0.0 0.0 5823 5 5 6 0.3660 0.1864 2.60 2.20 1899 6 6 7 0.3811 0.1941 40.40 30.00 1899 7 7 8 0.0922 0.0470 75.00 54.00 1899 8 8 9 0.0493 0.0251 30.00 22.00 1899 9 9 10 0.8190 0.2707 28.00 19.00 1455 10 10 11 0.1872 0.0619 145.00 104.00 1455 11 11 12 0.7114 0.2351 145.00 104.00 1455 12 12 13 1.0300 0.3400 8.00 5.00 1455 13 13 14 1.0440 0.3450 8.00 5.50 1455 14 14 15 1.0580 0.3496 0.0 0.0 1455 15 15 16 0.1966 0.0650 45.50 30.00 1455 16 16 17 0.3744 0.1238 60.00 35.00 1455 17 17 18 0.0047 0.0016 60.00 35.00 2200 18 18 19 0.3276 0.1083 0.0 0.0 1455 19 19 20 0.2106 0.0690 1.00 0.60 1455 20 20 21 0.3416 0.1129 114.00 81.00 1455 21 21 22 0.0140 0.0046 5.00 3.50 1455 22 22 23 0.1591 0.0526 0.0 0.0 1455 23 23 24 0.3463 0.1145 28.00 20.0 1455 24 24 25 0.7488 0.2475 0.0 0.0 1455 25 25 26 0.3089 0.1021 14.0 10.0 1455 26 26 27 0.1732 0.0572 14.0 10.0 1455 27 3 28 0.0044 0.0108 26.0 18.6 10,761 28 28 29 0.0640 0.1565 26.0 18.6 10,761 29 29 30 0.3978 0.1315 0.0 0.0 1455 30 30 31 0.0702 0.0232 0.0 0.0 1455 31 31 32 0.3510 0.1160 0.0 0.0 1455 32 32 33 0.8390 0.2816 14.0 10.0 2200 33 33 34 1.7080 0.5646 9.50 14.00 1455 34 34 35 1.4740 0.4873 6.00 4.00 1455 35 3 36 0.0044 0.0108 26.0 18.55 10,761 36 36 37 0.0640 0.1565 26.0 18.55 10,761 37 37 38 0.1053 0.1230 0.0 0.0 5823 38 38 39 0.0304 0.0355 24.0 17.00 5823 39 39 40 0.0018 0.0021 24.0 17.00 5823 40 40 41 0.7283 0.8509 1.20 1.0 5823 Elec onics 2023,12, 1530 24 o 27 Table A1. Con . B anch Numbe Sending Bus Recei ing Bus Resis ance Ω Reac ance Ω Nominal Load a Recei ing Bus Maximum Line Capaci y (kVA) P (kW) Q(kVA) 41 41 42 0.3100 0.3623 0.0 0.0 5823 42 42 43 0.0410 0.0478 6.0 4.30 5823 43 43 44 0.0092 0.0116 0.0 0.0 5823 44 44 45 0.1089 0.1373 39.22 26.30 5823 45 45 46 0.0009 0.0012 39.22 26.30 6709 46 4 47 0.0034 0.0084 0.00 0.0 10,761 47 47 48 0.0851 0.2083 79.00 56.40 10,761 48 48 49 0.2898 0.7091 384.70 274.50 10,761 49 49 50 0.0822 0.2011 384.70 274.50 10,761 50 8 51 0.0928 0.0473 40.50 28.30 1899 51 51 52 0.3319 0.1114 3.60 2.70 2200 52 52 53 0.1740 0.0886 4.35 3.50 1899 53 53 54 0.2030 0.1034 26.40 19.00 1899 54 54 55 0.2842 0.1447 24.00 17.20 1899 55 55 56 0.2813 0.1433 0.0 0.0 1899 56 56 57 1.5900 0.5337 0.0 0.0 2200 57 57 58 0.7837 0.2630 0.0 0.0 2200 58 58 59 0.3042 0.1006 100.0 72.0 1455 59 59 60 0.3861 0.1172 0.0 0.0 1455 60 60 61 0.5075 0.2585 1244.0 888.00 1899 61 61 62 0.0974 0.0496 32.0 23.00 1899 62 62 63 0.1450 0.0738 0.0 0.0 1899 63 63 64 0.7105 0.3619 227.0 162.00 1899 64 64 65 1.0410 0.5302 59.0 42.0 1899 65 11 66 0.2012 0.0611 18.0 13.0 1455 66 66 67 0.0047 0.0014 18.0 13.0 1455 67 12 68 0.7394 0.2444 28.0 20.0 1455 68 68 69 0.0047 0.0016 28.0 20.0 1455 69 * 11 43 0.5000 0.5000 566 70 * 13 21 0.5 0.5 566 71 * 15 46 1.0 1.0 400 72 * 50 59 2.0 2.0 283 73 * 27 65 1.0 1.0 400 Elec onics 2023,12, 1530 25 o 27 Table A2. Sys em da a o 33-bus adial dis ibu ion ne wo k. B anch Numbe Sending Bus Recei ing Bus Resis ance Ω Reac ance Ω Nominal Load a Recei ing Bus P (kW) Q (kVA) 1 1 2 0.0922 0.047 100 60 2 2 3 0.493 0.2511 90 40 3 3 4 0.366 0.1864 120 80 4 4 5 0.3811 0.1941 60 30 5 5 6 0.819 0.707 60 20 6 6 7 0.1872 0.6188 200 100 7 7 8 0.7114 0.2351 200 100 8 8 9 1.03 0.74 60 20 9 9 10 1.044 0.74 60 20 10 10 11 0.1966 0.065 45 30 11 11 12 0.3744 0.1298 60 35 12 12 13 1.468 1.155 60 35 13 13 14 0.5416 0.7129 120 80 14 14 15 0.591 0.526 60 10 15 15 16 0.7463 0.545 60 20 16 16 17 1.289 1.721 60 20 17 17 18 0.732 0.574 90 40 18 2 19 0.164 0.1565 90 40 19 19 20 1.5042 1.3554 90 40 20 20 21 0.4095 0.4784 90 40 21 21 22 0.7089 0.9373 90 40 22 3 23 0.4512 0.3083 90 50 23 23 24 0.898 0.7091 420 200 24 24 25 0.896 0.7011 420 200 25 6 26 0.203 0.1034 60 25 26 26 27 0.2842 0.1447 60 25 27 27 28 1.059 0.9337 60 20 28 28 29 0.8042 0.7006 120 70 29 29 30 0.5075 0.2585 200 600 30 30 31 0.9744 0.963 150 70 31 31 32 0.3105 0.3619 210 100 32 32 33 0.341 0.5302 60 40 33 20 7 2.0000 2.0000 - - 34 8 14 2.0000 2.0000 - - 35 11 21 2.0000 2.0000 - - 36 17 32 0.5000 0.5000 - - 37 24 28 0.5000 0.5000 - -