Design of an Anti-Corona Device for HVAC Substation Connectors

Ci a ion: La ea, A.M.; De La Hoz, M.; E xega ai, A.; Mazon, A.J.; A anzabal, I. Design o an An i-Co ona De ice o HVAC Subs a ion Connec o s. Ene gies 2022, 15, 5781. h ps://doi.o g/ 10.3390/en15165781 Academic Edi o s: Meng Huang, Yunxiao Zhang and Chenyuan Teng Recei ed: 18 July 2022 Accep ed: 4 Augus 2022 Published: 9 Augus 2022 Publishe ’s No e: MDPI s ays neu al wi h ega d o ju isdic ional claims in published maps and ins i u ional a il- ia ions. Copy igh : © 2022 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/). ene gies A icle Design o an An i-Co ona De ice o HVAC Subs a ion Connec o s Ane Mi en La ea , Manuel De La Hoz , Agu zane E xega ai , Angel Ja ie Mazon * and I xaso A anzabal Depa men o Elec ical Enginee ing, Facul y o Enginee ing o Bilbao, Uni e si y o he Basque Coun y UPV/EHU, 48013 Bilbao, Spain *Co espondence: ja ie [email p o ec ed] Abs ac : One o he aspec s o conside du ing high- ol age (HV) equipmen design is he educ ion in he p obabili y o co ona e ec onse . Indeed, he co ona e ec is ela ed o high elec ic ield alues beyond he equipmen ’s insula ion le els and insula ion s eng h, among o he ac o s. This issue can be add essed du ing he design s ep, ei he by modi ying he geome y o he elec ical de ice o by including addi ional elemen s in he equipmen s uc u e o smoo h ou he ol age g adien along c i ical egions, such as an i-co ona de ices. The s udy o an i-co ona de ices o HV insula o s is well documen ed, in con as o subs a ion connec o s. The e o e, he p esen s udy p oposed he design o a no el an i-co ona de ice o HV subs a ion connec o s, including a me hod o he selec ion o i s dimensions. This s udy shows ha he ela ionship be ween he dimensional design a iables and he c i ical elec ical ield on he connec o is desc ibed by linea and a ional unc ions. Thus, he design p ocess imes a e cu down due o a educ ion in he numbe o simula ions equi ed o un he assessmen o he an i-co ona de ice a angemen impac . Keywo ds: compu e -aided manu ac u ing; connec o ; co ona ing; elec ic ields; high ol age 1. In oduc ion The appea ance o he co ona e ec is common in high- ol age (HV) o e head lines. The phenomenon co esponds o a discha ge a he poin whe e he elec ic ield is g ea e han he dis up i e s eng h o he insula ion ha su ounds he su ace o he conduc o and has se e al physical aspec s, such as sound, luminance, and elec omagne ic pulse wa es ha p oduce ene gy losses on he ansmission sys em [ 1 ]. In he wo s case, a co ona e ec could de elop in o an a c ha causes he des uc ion o elec ic de ices ins alled in o e head lines, such as insula o s o connec o s [2,3]. The s udy o he co ona e ec in ol es he calcula ion o he maximum elec ic ield ha insula ion can wi hs and, which is a common s ep in he design o HV ins alla ions (e.g., subs a ions) [ 4 ] and HV equipmen (e.g., swi chgea , insula o s, connec o s) [ 5 ]. The ools used in powe sys em analysis s udies gene ally include app oxima ed modelling ega ding he geome ies o he ansmission lines and owe s [ 6 ]. Howe e , in some speci ic geome ies o insula o s and connec o s, a mo e de ailed analysis is equi ed [ 7 , 8 ]. In all cases, ini e elemen analysis (FEA) is a aluable ool o calcula ing he elec ic ield dis ibu ion. The elec ic ield ha is gene a ed a he su ace and a ound insula o s and line connec- o s depends di ec ly on he applied ol age and he geome y o he elec ode. Howe e , i is necessa y o conside he en i onmen al and insula ion condi ions o unde s and he co ona e ec comple ely [ 7 ]. Connec o s a e only subjec o he line ol age, whe eas in he case o insula o s, one end is connec ed o he line ol age and he o he end is g ounded. The la ge di e ence in he ol age suppo ed by he insula o s causes an elec ic ield g adien [ 9 ]. In con as , o connec o s, i is due o he smalle cu a u e in hei geome y [10] since he e is no ol age di e ence o e hem. The ins alla ion o an i-co ona de ices o insula o s is well documen ed in he li - e a u e. In he case o insula ing s ings, Ilhan and Ozdemi [ 11 ] conside ed he elec ic Ene gies 2022,15, 5781. h ps://doi.o g/10.3390/en15165781 h ps://www.mdpi.com/jou nal/ene gies Ene gies 2022,15, 5781 2 o 12 ield su ace o di e en poin s om he g ading ing and he insula o s ing. Then, based on a sensi i i y analysis, possible solu ions we e es ed by conside ing he adio in e e ence ol age p oduced by he con igu a ion. O he au ho s applied op imisa ion me hods [ 12 ]. Howe e , hese wo ks did no show an e iden solu ion o he loca ion and size o he co ona ings. P oposed op imisa ion algo i hms a e he ba lying algo i hm [ 13 ], pa icle swa m op imisa ion [ 14 , 15 ], and su ace esponse [ 16 ]. Gene ally, some minimum elec ical ield alue is conside ed as an objec i e a a pa icula ly c i ical poin (e.g., he insula ion iple poin o an insula ing s ing [ 17 ]) o along a cha ac e is ic line leng h (e.g., he pe iphe y line o he insula ing s ing [ 18 ]). Rega ding he co ona e ec onse a subs a ion connec o s, Gui e as e al. [ 19 ] sugges ed ins alling shielding pa s a ound he connec o s o o co e hem wi h a sphe ical cage unde he same ol age. This solu ion educes he elec ic ield, and he e o e, i can be used on sys ems ope a ing unde highe ol age. Ne e heless, o he speci ic p oblem condi ions, he au ho s concluded ha a edesign o he geome y o he connec o is p e e ed. This app oach is only alid o ce ain connec o s since i may no be easible o ce ain conduc o a angemen s. The edesign o he connec o s also implies ano he kind o s udy o he connec o ( he mal, mechanical, e c.), and he e o e, mo e indus ial es s o be ca ied ou . Th ee essen ial equi emen s a e desi able in he design, manu ac u ing, and ins al- la ion o co ona ings. Fi s , he ings mus be simple o build. Second, he ins alla ion o he co ona ings on si e mus be s aigh o wa d, in ol ing a minimal numbe o s eps. Consequen ly, in he p esen s udy, uncon en ional su aces we e no analysed, such as in Pa anadech e al. [ 20 ] and Liao e al. [ 21 ]. Howe e , i could be a alid ype o op imisa ion when applied o insula o s [ 22 ]. Thi d, he poin o a achmen o he o oid is also ele an . I s posi ion in space mus be suppo ed by some objec o which i can be mechanically linked using an adjus men o achie e ixa ion [ 23 ]. Fo connec o s, his means using hei ci il wo k suppo , i hey ha e one, o ixing he ings on he conduc o s ha en e he connec o . The las op ion was conside ed o his wo k. O he ime minimisa ion echniques in he es de elopmen a e implemen ed in he modelling o an i-co ona ings, such as he design o expe imen s (DOE) and analysis o a iance (ANOVA) [ 16 ], in his case, o insula o s. The limi a ion o his me hod lies in he ac ha he ela ionships be ween he mul iple ac o s conside ed mus be linea [24]. The p esen s udy p oposed he design o a no el an i-co ona de ice, i.e., an an i- co ona ing, o HV subs a ion connec o s, including a me hod o ing dimension selec ion. The co ona ing shape was mainly selec ed because i is easie o manu ac u e and main ain compa ed wi h sphe ical cages o o he geome ies. The applied me hod was ocused on conside ing whe he algo i hmic exp essions can be de i ed o he sizing o co ona ings o educe he analysis ime in he design s age, as well as applying i o usual cases on subs a ion connec o s. I educes he design p ocess calcula ions by conside ing he ela ionship be ween he geome ic dimensions o he co ona ings and he alue o he maximum su ace elec ic ield o he subs a ion connec o nea he an i-co ona ing. The objec i e was o educe he numbe o ials o es ablish he possible ela ionship be ween design pa ame e s, hus minimising he simula ion esou ces equi ed o an i e a i e op imisa ion p ocess. The o he ad an age comes om he simula ion o es ing condi ions using FEA. Al hough we conside ed he elec ical ield only a he connec o su ace, he connec o geome y is challenging o model and mus be done wi hou any dimensional simpli ica ion ( om a h ee-dimensional model o an axis-symme y one, o ins ance). Any educ ion in he nume ical me hods equi ed speeds up he design p ocess and inc eases he possibili y o educed es ing. This pape is o ganised as ollows. In he i s pa , geome ical de ails o he subs a ion connec o s, es ing se up, and he an i-co ona de ice a angemen a e in oduced. Thus, a nume ical FEA model ha was used o simula e s anda d HV labo a o y condi ions is p esen ed, and he loca ion o he maximum su ace elec ic ield is gi en. Based on p e ious esul s, in he second pa , he speci ic posi ions o he co ona ings nea he Ene gies 2022,15, 5781 3 o 12 subs a ion connec o s unde s udy ha educed he elec ic ield a e gi en. The esul s o se e al simula ions o de e mine he e ec o he p oposed an i-co ona ing’s geome ical ac o s on he elec ic ield a e p esen ed. The las pa o e s in o ma ion abou he algeb aic ela ionship be ween a iables ha diminish he design ime, educe he numbe o simula ions, and conside he su ace elec ic ield as an indica o . The me hod can also be used on si e o imp o e he loca ion o he co ona ings o s anda d condi ions. 2. Desc ip ion o he Connec o s and he Tes ing Se up 2.1. Geome y o he Connec o s and he Tes ing Se up The no el an i-co ona de ice p oposed in he p esen s udy was in es iga ed o i s applica ion in speci ic aluminium connec o s used in HV elec ical subs a ions. Howe e , he a angemen is alid o any subs a ion connec o p o ided ha a co ona ing is ins alled in each o he connec o s coupled h ough a subs a ion connec o ollowing he design me hod in oduced in Sec ion 4. The connec o s unde s udy a e shown in Figu e 1, whe e hei main dimensions a e de ined, including hei leng h, wid h, heigh , and minimum adius. The dimensions o each connec o a e shown in Table 1. Ene gies 2022, 15, x FOR PEER REVIEW 3 o 12 This pape is o ganised as ollows. In he i s pa , geome ical de ails o he subs a- ion connec o s, es ing se up, and he an i-co ona de ice a angemen a e in oduced. Thus, a nume ical FEA model ha was used o simula e s anda d HV labo a o y condi- ions is p esen ed, and he loca ion o he maximum su ace elec ic ield is gi en. Based on p e ious esul s, in he second pa , he speci ic posi ions o he co ona ings nea he subs a ion connec o s unde s udy ha educed he elec ic ield a e gi en. The esul s o se e al simula ions o de e mine he e ec o he p oposed an i-co ona ing’s geome ical ac o s on he elec ic ield a e p esen ed. The las pa o e s in o ma ion abou he alge- b aic ela ionship be ween a iables ha diminish he design ime, educe he numbe o simula ions, and conside he su ace elec ic ield as an indica o . The me hod can also be used on si e o imp o e he loca ion o he co ona ings o s anda d condi ions. 2. Desc ip ion o he Connec o s and he Tes ing Se up 2.1. Geome y o he Connec o s and he Tes ing Se up The no el an i-co ona de ice p oposed in he p esen s udy was in es iga ed o i s applica ion in speci ic aluminium connec o s used in HV elec ical subs a ions. Howe e , he a angemen is alid o any subs a ion connec o p o ided ha a co ona ing is in- s alled in each o he connec o s coupled h ough a subs a ion connec o ollowing he design me hod in oduced in Sec ion 4. The connec o s unde s udy a e shown in Figu e 1, whe e hei main dimensions a e de ined, including hei leng h, wid h, heigh , and minimum adius. The dimensions o each connec o a e shown in Table 1. Figu e 1. Geome y o connec o s. Table 1. Geome ical dimensions o connec o s. Connec o Leng h (mm) Wid h (mm) Heigh (mm) Radius Min. (mm) Connec o 1 470 470 100 6 Connec o 2 585 270 95 3 Connec o 3 240 120 230 4 In he p esen s udy, he in es iga ion o he elec ic ield a ound he connec o s was based on he ec ea ion o a labo a o y es ing se up, as shown in Figu e 2. In he es ing a angemen , se e al ubes allowed he connec o o be suspended in he model, e lec ing i s possible ins alla ion in he subs a ion, which would be he con en ional assembly o be es ed in he labo a o y. Thei diame e s a e he same as hose used unde s anda d in- s alla ion equi emen s. One o he simples and mos e ec i e solu ions o educing he elec ical s ess gen- e a ed a he ends o he conduc o s in he es bench is o use co ona-shielding sys ems (sphe es and ings) a hei ends [25] o p e en he elec ic ield om inc easing d ama - ically. Gene ally, hese me al pa s ha e a ing, sphe ical, o bowl shape [8]. No e ha Figu e 1. Geome y o connec o s. Table 1. Geome ical dimensions o connec o s. Connec o Leng h (mm) Wid h (mm) Heigh (mm) Radius Min. (mm) Connec o 1 470 470 100 6 Connec o 2 585 270 95 3 Connec o 3 240 120 230 4 In he p esen s udy, he in es iga ion o he elec ic ield a ound he connec o s was based on he ec ea ion o a labo a o y es ing se up, as shown in Figu e 2. In he es ing a angemen , se e al ubes allowed he connec o o be suspended in he model, e lec ing i s possible ins alla ion in he subs a ion, which would be he con en ional assembly o be es ed in he labo a o y. Thei diame e s a e he same as hose used unde s anda d ins alla ion equi emen s. One o he simples and mos e ec i e solu ions o educing he elec ical s ess gen- e a ed a he ends o he conduc o s in he es bench is o use co ona-shielding sys ems (sphe es and ings) a hei ends [ 25 ] o p e en he elec ic ield om inc easing d ama i- cally. Gene ally, hese me al pa s ha e a ing, sphe ical, o bowl shape [ 8 ]. No e ha hese elemen s a e independen o he co ona ings. The es ing enclosu e consis ed o a box wi h dimensions 15 m × 15 m × 10 m and he a angemen was loca ed a he cen e, as shown in Figu e 2. Ene gies 2022,15, 5781 4 o 12 Ene gies 2022, 15, x FOR PEER REVIEW 4 o 12 hese elemen s a e independen o he co ona ings. The es ing enclosu e consis ed o a box wi h dimensions 15 m × 15 m × 10 m and he a angemen was loca ed a he cen e, as shown in Figu e 2. Figu e 2. Modelled se up o he connec o s. The box ep esen s he es ing oom. 2.2. Con igu a ion o he Simula ion Tool The elec ic ield o he connec o was calcula ed wi h FEA. The assump ions consid- e ed we e as ollows. Fi s , he FEA model simula ion was ca ied ou unde HV labo a- o y condi ions, acco ding o no mal p essu e, empe a u e, and humidi y, as es ablished in s anda d IEC 61284 [26]. The FEA so wa e ool o p e e ence o his wo k was COM- SOL Mul iphysics. Second, when simula ing a piece ha is connec ed o conduc o s, i s ends a e ex- posed. Tha is, an in ini e conduc o canno be de ined. Howe e , his had no impac when e alua ing he piece, p o ided i s leng h is conside ably longe han he connec o unde analysis. E en so, a sphe e co e ed he ends o a oid he maximum elec ic ield o he whole assembly being ound a he ends o he conduc o , gi en i s sha p geome y. The es ing se up in Figu e 2 was modelled in he FEA so wa e. Then, he elec ic ield was only analysed a i s mos c i ical poin , i.e., he maximum alues a each poin o he connec o su ace we e iden i ied, omi ing he ac ha he ol age a ied o e ime ( he maximum alue is a he peak o he sinusoidal wa e). The simula ion enclosu e was g ounded o de ine an app op ia e en i onmen . The es o he pieces we e connec ed o a line ol age o 450 kV peak as he e e ence. The cha ac e is ics o he mesh a e a key ac o when ca ying ou an analysis by means o FEA. To ob ain p ecise esul s, he “ e y ine mesh” op ion in he meshing s age on COMSOL Mul iphysics was selec ed. O he impo an cha ac e is ics ela ed o he ype o mesh used a e desc ibed in Annex I. A e de ining meshing cha ac e is ics, a s a iona y s udy was ca ied ou . Once he elec ic ield on he connec o was calcula ed, he poin wi h he highes elec ic ield alue was ob ained. 3. Geome y and Design o he An i-Co ona De ice 3.1. Geome y o he An i-Co ona De ice The elec ic ield depends di ec ly on he geome y o a powe sys em elemen unde ol age. The e o e, i he geome y o he elemen is a ied, i has an impac on i s elec ic ield. To educe he elec ic ield o subs a ion connec o s, he p esen s udy p oposed o include a ing a ound each conduc o (Figu e 3). In his way, i was possible o al e he geome y o he whole assembly (connec o , ings, and conduc o s) and, in u n, he elec- ic ield. By a aching he ings o he conduc o s, a s able hold was allowed. Thus, a dis- ance was kep be ween he ing and he connec o , as de ined by he designe . The ing Figu e 2. Modelled se up o he connec o s. The box ep esen s he es ing oom. 2.2. Con igu a ion o he Simula ion Tool The elec ic ield o he connec o was calcula ed wi h FEA. The assump ions con- side ed we e as ollows. Fi s , he FEA model simula ion was ca ied ou unde HV labo a o y condi ions, acco ding o no mal p essu e, empe a u e, and humidi y, as es ab- lished in s anda d IEC 61284 [ 26 ]. The FEA so wa e ool o p e e ence o his wo k was COMSOL Mul iphysics. Second, when simula ing a piece ha is connec ed o conduc o s, i s ends a e exposed. Tha is, an in ini e conduc o canno be de ined. Howe e , his had no impac when e alua ing he piece, p o ided i s leng h is conside ably longe han he connec o unde analysis. E en so, a sphe e co e ed he ends o a oid he maximum elec ic ield o he whole assembly being ound a he ends o he conduc o , gi en i s sha p geome y. The es ing se up in Figu e 2was modelled in he FEA so wa e. Then, he elec ic ield was only analysed a i s mos c i ical poin , i.e., he maximum alues a each poin o he connec o su ace we e iden i ied, omi ing he ac ha he ol age a ied o e ime ( he maximum alue is a he peak o he sinusoidal wa e). The simula ion enclosu e was g ounded o de ine an app op ia e en i onmen . The es o he pieces we e connec ed o a line ol age o 450 kVpeak as he e e ence. The cha ac e is ics o he mesh a e a key ac o when ca ying ou an analysis by means o FEA. To ob ain p ecise esul s, he “ e y ine mesh” op ion in he meshing s age on COMSOL Mul iphysics was selec ed. O he impo an cha ac e is ics ela ed o he ype o mesh used a e desc ibed in Annex I. A e de ining meshing cha ac e is ics, a s a iona y s udy was ca ied ou . Once he elec ic ield on he connec o was calcula ed, he poin wi h he highes elec ic ield alue was ob ained. 3. Geome y and Design o he An i-Co ona De ice 3.1. Geome y o he An i-Co ona De ice The elec ic ield depends di ec ly on he geome y o a powe sys em elemen unde ol age. The e o e, i he geome y o he elemen is a ied, i has an impac on i s elec ic ield. To educe he elec ic ield o subs a ion connec o s, he p esen s udy p oposed o include a ing a ound each conduc o (Figu e 3). In his way, i was possible o al e he geome y o he whole assembly (connec o , ings, and conduc o s) and, in u n, he elec ic ield. By a aching he ings o he conduc o s, a s able hold was allowed. Thus, a dis ance was kep be ween he ing and he connec o , as de ined by he designe . The ing a angemen p esen ed in his wo k allowed o a s aigh o wa d eplacemen o he ings i necessa y and a pe ec as ening a a dis ance om he piece as de e mined by he designe . In his way, he same connec o could be used unde di e en ope a ion condi ions by changing he geome y o dis ance o he ings ins ead o changing o Ene gies 2022,15, 5781 5 o 12 edesigning he connec o . In addi ion, compa ed wi h sphe ical cages, ings a e easie o manu ac u e and main ain. Ene gies 2022, 15, x FOR PEER REVIEW 5 o 12 a angemen p esen ed in his wo k allowed o a s aigh o wa d eplacemen o he ings i necessa y and a pe ec as ening a a dis ance om he piece as de e mined by he designe . In his way, he same connec o could be used unde di e en ope a ion condi- ions by changing he geome y o dis ance o he ings ins ead o changing o edesigning he connec o . In addi ion, compa ed wi h sphe ical cages, ings a e easie o manu ac u e and main ain. The design o he an i-co ona de ice was classi ied acco ding o he ollowing pa am- e e s: hickness o he ings (A), inne diame e o he ings (Din ), and dis ance be ween he ing and he connec o (dis ), as iden i ied in Figu e 4. Figu e 3. Connec o wi h ings ins alled a ound he conduc o s. Figu e 4. Design o indi idual ings ins alled a ound he conduc o s. 3.2. Modelling and Simula ion o he Se up wi h he An i-Co ona De ice The elec ic ield o he connec o was calcula ed wi h FEA again, bu now o se e al cases ha conside ed di e en pa ame e alues. Fi s , he es ing se up in Figu e 2 wi h he an i-co ona de ice was modelled in he FEA so wa e again. Then, he maximum elec- ic ield in he posi ion ound in Sec ion 2.2 has been eco ded. Two o he pa ame e s we e always cons an and he las one was modi ied in o de o ind ou he dependency wi h he maximum elec ic ield on he connec o su ace. The simula ion enclosu e was g ounded o de ine an app op ia e en i onmen . The es o he pieces we e connec ed o a line ol age o 450 kV peak as a e e ence, including he an i-co ona de ice, since i was ixed o he cables o he se up. 4. Simula ion Resul s 4.1. Simula ion Resul s wi hou he An i-Co ona De ice The elec ic ield a he su ace o he connec o s unde s udy wi hou any an i-co ona de ice ins alled is shown in Figu e 5. I can be obse ed ha he maximum elec ic ield alue mos ly occu ed in he zone wi h he highes cu a u e. Howe e , he no mal ec o om hese c i ical su aces was no always in he same di ec ion as he axis o he ubes, as shown o connec o 3 in Figu e 5. Figu e 3. Connec o wi h ings ins alled a ound he conduc o s. The design o he an i-co ona de ice was classi ied acco ding o he ollowing pa ame- e s: hickness o he ings (A), inne diame e o he ings (Din ), and dis ance be ween he ing and he connec o (dis ), as iden i ied in Figu e 4. Ene gies 2022, 15, x FOR PEER REVIEW 5 o 12 a angemen p esen ed in his wo k allowed o a s aigh o wa d eplacemen o he ings i necessa y and a pe ec as ening a a dis ance om he piece as de e mined by he designe . In his way, he same connec o could be used unde di e en ope a ion condi- ions by changing he geome y o dis ance o he ings ins ead o changing o edesigning he connec o . In addi ion, compa ed wi h sphe ical cages, ings a e easie o manu ac u e and main ain. The design o he an i-co ona de ice was classi ied acco ding o he ollowing pa am- e e s: hickness o he ings (A), inne diame e o he ings (Din ), and dis ance be ween he ing and he connec o (dis ), as iden i ied in Figu e 4. Figu e 3. Connec o wi h ings ins alled a ound he conduc o s. Figu e 4. Design o indi idual ings ins alled a ound he conduc o s. 3.2. Modelling and Simula ion o he Se up wi h he An i-Co ona De ice The elec ic ield o he connec o was calcula ed wi h FEA again, bu now o se e al cases ha conside ed di e en pa ame e alues. Fi s , he es ing se up in Figu e 2 wi h he an i-co ona de ice was modelled in he FEA so wa e again. Then, he maximum elec- ic ield in he posi ion ound in Sec ion 2.2 has been eco ded. Two o he pa ame e s we e always cons an and he las one was modi ied in o de o ind ou he dependency wi h he maximum elec ic ield on he connec o su ace. The simula ion enclosu e was g ounded o de ine an app op ia e en i onmen . The es o he pieces we e connec ed o a line ol age o 450 kV peak as a e e ence, including he an i-co ona de ice, since i was ixed o he cables o he se up. 4. Simula ion Resul s 4.1. Simula ion Resul s wi hou he An i-Co ona De ice The elec ic ield a he su ace o he connec o s unde s udy wi hou any an i-co ona de ice ins alled is shown in Figu e 5. I can be obse ed ha he maximum elec ic ield alue mos ly occu ed in he zone wi h he highes cu a u e. Howe e , he no mal ec o om hese c i ical su aces was no always in he same di ec ion as he axis o he ubes, as shown o connec o 3 in Figu e 5. Figu e 4. Design o indi idual ings ins alled a ound he conduc o s. 3.2. Modelling and Simula ion o he Se up wi h he An i-Co ona De ice The elec ic ield o he connec o was calcula ed wi h FEA again, bu now o se e al cases ha conside ed di e en pa ame e alues. Fi s , he es ing se up in Figu e 2wi h he an i-co ona de ice was modelled in he FEA so wa e again. Then, he maximum elec ic ield in he posi ion ound in Sec ion 2.2 has been eco ded. Two o he pa ame e s we e always cons an and he las one was modi ied in o de o ind ou he dependency wi h he maximum elec ic ield on he connec o su ace. The simula ion enclosu e was g ounded o de ine an app op ia e en i onmen . The es o he pieces we e connec ed o a line ol age o 450 kV peak as a e e ence, including he an i-co ona de ice, since i was ixed o he cables o he se up. 4. Simula ion Resul s 4.1. Simula ion Resul s wi hou he An i-Co ona De ice The elec ic ield a he su ace o he connec o s unde s udy wi hou any an i-co ona de ice ins alled is shown in Figu e 5. I can be obse ed ha he maximum elec ic ield alue mos ly occu ed in he zone wi h he highes cu a u e. Howe e , he no mal ec o om hese c i ical su aces was no always in he same di ec ion as he axis o he ubes, as shown o connec o 3 in Figu e 5. Ene gies 2022,15, 5781 6 o 12 Ene gies 2022, 15, x FOR PEER REVIEW 6 o 12 Figu e 5. Elec ic ield (kV/cm) a he connec o su ace. The maximum ield alues a he su ace o he connec o s wi hou any ing we e 45.2 kV/cm (connec o 1), 46.5 kV/cm (connec o 2), and 65.3 kV/cm (connec o 3). The poin o e alua ion o he elec ic ield o he simula ions in Sec ion 4.2 is indi- ca ed in Figu e 5 by E bc . I coincided wi h he loca ion o he maximum elec ic ield E max , excep in he s udy o connec o 2. The p oposed solu ion educed he elec ic ield mag- ni ude in he closes poin s o he ings and he elec ic ield ec o mus be almos pa allel o he ing axis. The e o e, o connec o 2, he poin selec ed o analyse E bc was a poin nea he plane o he an i-co ona de ice ins alla ion wi h an elec ic ield alue o 20.0 kV/cm. 4.2. Simula ion Resul s wi h an An i-Co ona De ice Fo he di e en con igu a ions in Figu e 3, se e al simula ions we e ca ied ou by changing he ing pa ame e s: hickness (A), inne diame e (Din ), and dis ance be ween he ing and he connec o (dis ). Figu e 6 desc ibes he maximum elec ic ield alue ound a he connec o su ace unde each speci ic pa ame e while keeping he o he s cons an . I can be no ed ha o any con igu a ion wi h an i-co ona de ices, he elec ic ield was weake han wi hou he an i-co ona de ices, as indica ed in Sec ion 4.1, and he co ona incep ion ol age was also educed. (a) 5 10 15 20 25 30 35 40 45 15 35 55 75 95 E [kV/cm] A [mm] RING THICKNESS (A) Connec o 1 Connec o 2 Connec o 3 Din 1= 100 mm dis 1= 200 mm Din 2= 100 mm dis 2= 250 mm Din 3=80mm dis 3= 250 mm Figu e 5. Elec ic ield (kV/cm) a he connec o su ace. The maximum ield alues a he su ace o he connec o s wi hou any ing we e 45.2 kV/cm (connec o 1), 46.5 kV/cm (connec o 2), and 65.3 kV/cm (connec o 3). The poin o e alua ion o he elec ic ield o he simula ions in Sec ion 4.2 is indica ed in Figu e 5by E bc . I coincided wi h he loca ion o he maximum elec ic ield E max , excep in he s udy o connec o 2. The p oposed solu ion educed he elec ic ield magni ude in he closes poin s o he ings and he elec ic ield ec o mus be almos pa allel o he ing axis. The e o e, o connec o 2, he poin selec ed o analyse E bc was a poin nea he plane o he an i-co ona de ice ins alla ion wi h an elec ic ield alue o 20.0 kV/cm. 4.2. Simula ion Resul s wi h an An i-Co ona De ice Fo he di e en con igu a ions in Figu e 3, se e al simula ions we e ca ied ou by changing he ing pa ame e s: hickness (A), inne diame e (Din ), and dis ance be ween he ing and he connec o (dis ). Figu e 6desc ibes he maximum elec ic ield alue ound a he connec o su ace unde each speci ic pa ame e while keeping he o he s cons an . I can be no ed ha o any con igu a ion wi h an i-co ona de ices, he elec ic ield was weake han wi hou he an i-co ona de ices, as indica ed in Sec ion 4.1, and he co ona incep ion ol age was also educed. Figu e 6. Con . Ene gies 2022,15, 5781 7 o 12 Figu e 6. Elec ic ield on he connec o ’s su ace as a unc ion o he ing dimensions: ( a ) ing hickness, (b) in e nal diame e , and (c) ing–connec o dis ance. E en i he sweep ange was modi ied o he di e en s udy cases, he endency was simila o each pa ame e s udy. Depending on he elec ic ield p esen a he ins alla ion poin o he an i-co ona de ice, he esul s o Figu e 6allowed o deciding he dimensional cha ac e is ics o he an i-co ona de ice as a i s app oxima ion. As obse ed in Figu e 6a, he change in hickness o he ing (keeping i s in e nal diame e cons an ) ollowed a downwa d linea end. Analogously o he hickness, he inc ease in he in e nal diame e o he ing (Figu e 6b) ollowed he same end. Thus, he la ge he in e nal diame e o he ings (keeping he es o he pa ame e s cons an ), he smalle he elec ic ield. Finally, he a ia ion o he dis ance be ween he ing and he connec o in Figu e 6c did no ollow a linea end, bu a mo e complex end. I was e i ied ha he a he he ings we e om he connec o , he less in luence hey had on he elec ic ield. Ene gies 2022,15, 5781 8 o 12 Fo a mo e accu a e design, h ee i ing equa ions we e ex apola ed om Figu e 6 based on he pa ame e s analysed and he maximum elec ic ield alue. Equa ions (1)–(3) a e in ended o he selec ion o he co ona ing dimensions. Equa ions (1) and (2) co e- spond o he ela ionships be ween he elec ic ield E (kV/cm) and ing hickness A (mm) and be ween he elec ic ield E (kV/cm) and he in e nal diame e o he ing Din (mm) . Equa ion (3) ela es he elec ic ield E (kV/cm) and he dis ance be ween ings and connec- o s dis (mm). The alue o R 2 o all he i ings was abo e 0.92. The cons an alues C1 o C6 acco ding o he connec o and equa ion a e shown in Table 2. Emax =C1×A+C2 (1) Emax =C3×Din +C4 (2) Emax =C5 √dis +C6 (3) Table 2. Cons an alues o Equa ions (1) o (3) o each connec o . Connec o C1 C2 C3 C4 C5 C6 Connec o 1 −0.161 32.1 −0.103 37.3 −212 41.3 Connec o 2 −0.138 16.8 −0.082 20.8 −376 36.9 Connec o 3 −0.103 42.1 −0.074 47.3 −106 46.78 In Equa ions (1)–(3), he o he pa ame e s a e kep cons an , as speci ied in Table 2. A single i ed Equa ion (4) was deduced o he mos gene al case, whe e all he pa ame e s a e desc ibed oge he . The cons an alues L1 o L4 o each connec o a e shown in Table 3. Emax =L1 +L2 ×A+L3 ×Din +L4 √dis (4) Table 3. Cons an alues o Equa ion (4) o each connec o . Connec o L1 L2 L3 L4 Connec o 1 56.8 −0.161 −0.103 −212 Connec o 2 46.8 −0.138 −0.082 −376 Connec o 3 56.2 −0.103 −0.074 −106 Equa ion (4) is use ul o de e mining he elec ic ield in a combina ion o he pa- ame e s ha a e no included in he s udied cases, as long as he case conside ed is an in e pola ion. The dis ance be ween he ings and he connec o (dis ) is a key ac o when s udying he adequacy o an an i-co ona de ice. I physical es ic ions a e dis ega ded, he close he ings a e o he piece, he smalle he elec ic ield. In case he elec ic ield alue is no su icien ly low, he ings would ha e o be eplaced by o he ings o g ea e hickness o g ea e in e nal diame e . 5. Discussion Based on he con igu a ion o a speci ic connec o , i is possible o a oid he appea ance o he co ona e ec by dec easing he alue o he elec ic ield a i s su ace. Fo his pu pose, he p esen s udy p oposed ins alling co ona ings a ound each conduc o linked by he connec o . Unde hese condi ions, he incep ion ol age inc eased, allowing o use o his equipmen in applica ions wi h a highe elec ical ol age. The geome y o he ings was de ined by h ee pa ame e s: hickness (A), in e nal diame e (Din ), and dis ance (dis ), which can be calcula ed sepa a ely using Equa ions (1)–(3) o he connec o s in es iga ed in his s udy (Figu e 1). The equa ions we e e i ied o he connec o s unde s udy, which we e subjec ed o a line ol age o 450 kV peak AC. The p ocedu e can be gene alised, and Ene gies 2022,15, 5781 9 o 12 hus, i can be applied o o he connec o designs a any ol age le el. The gene alised design me hodology is in oduced in he p esen pape . The pape p esen s o he con ibu ions, oo. Fi s , he maximum elec ic ield unde some condi ions (e.g., connec o 2) was no loca ed in a a ou able posi ion o he an i- co ona ing o educe i s magni ude by a conside able pe cen age. The elec ic ield educ ion was s ongly in luenced by he sepa a ion be ween he an i-co ona ing and he connec o poin unde s udy. Hence, o e y long connec o s, he solu ion in oduced in he p esen wo k could be imp ac ical. Al e na i ely, he an i-co ona ing can ci cumsc ibe he poin whe e he elec ic ield needs o be a enua ed. Howe e , he ixa ion o he ing should be u he s udied. Mo eo e , he design o he co ona ings sugges s he possibili y o op imisa ion. Equa ion (4) gi es a en a i e objec i e unc ion, whe e bo h economic and echnical aspec s should be conside ed. In any case, a sweep o e each o he s udy cases is use ul o analyse he beha iou o all he a iables oge he , as shown in Figu e 7. In each axis o Figu e 7, one o he pa ame e s in ol ed in Equa ion (4) is ep esen ed. The elec ic ield o each squa e has a colou ha ep esen s i s alue and co esponds o he case wi h he midpoin combina ion alues. The hype su ace does no allow o in e ing he beha iou o he design pa ame e s analy ically, bu i does o e a global idea o he co ona ing designs ha should be conside ed by he designe . Fo ins ance, we could quickly es ablish a eas wi h a limi alue (e.g., 30 kV/cm) and disca d hose designs on he hype su ace. Ene gies 2022, 15, x FOR PEER REVIEW 9 o 12 linked by he connec o . Unde hese condi ions, he incep ion ol age inc eased, allowing o use o his equipmen in applica ions wi h a highe elec ical ol age. The geome y o he ings was de ined by h ee pa ame e s: hickness (A), in e nal diame e (Din ), and dis ance (dis ), which can be calcula ed sepa a ely using Equa ions (1)–(3) o he connec - o s in es iga ed in his s udy (Figu e 1). The equa ions we e e i ied o he connec o s unde s udy, which we e subjec ed o a line ol age o 450 kVpeak AC. The p ocedu e can be gene alised, and hus, i can be applied o o he connec o designs a any ol age le el. The gene alised design me hodology is in oduced in he p esen pape . The pape p esen s o he con ibu ions, oo. Fi s , he maximum elec ic ield unde some condi ions (e.g., connec o 2) was no loca ed in a a ou able posi ion o he an i- co ona ing o educe i s magni ude by a conside able pe cen age. The elec ic ield educ- ion was s ongly in luenced by he sepa a ion be ween he an i-co ona ing and he con- nec o poin unde s udy. Hence, o e y long connec o s, he solu ion in oduced in he p esen wo k could be imp ac ical. Al e na i ely, he an i-co ona ing can ci cumsc ibe he poin whe e he elec ic ield needs o be a enua ed. Howe e , he ixa ion o he ing should be u he s udied. Mo eo e , he design o he co ona ings sugges s he possibili y o op imisa ion. Equa ion (4) gi es a en a i e objec i e unc ion, whe e bo h economic and echnical as- pec s should be conside ed. In any case, a sweep o e each o he s udy cases is use ul o analyse he beha iou o all he a iables oge he , as shown in Figu e 7. In each axis o Figu e 7, one o he pa ame e s in ol ed in Equa ion (4) is ep esen ed. The elec ic ield o each squa e has a colou ha ep esen s i s alue and co esponds o he case wi h he midpoin combina ion alues. The hype su ace does no allow o in e ing he beha - iou o he design pa ame e s analy ically, bu i does o e a global idea o he co ona ing designs ha should be conside ed by he designe . Fo ins ance, we could quickly es ab- lish a eas wi h a limi alue (e.g., 30 kV/cm) and disca d hose designs on he hype su - ace. Figu e 7. Elec ic ield on connec o 1 as a unc ion o he ing dimensions: hickness (A), in e nal diame e (Din ), and ing–connec o dis ance (dis ), ep esen ed simul aneously. 6. Conclusions The ins alla ion o co ona ings a ound a de ice o p o ec i , e.g., HV subs a ion con- nec o s, is a simple and e ec i e way o a oid he appea ance o he co ona e ec . The ings dec ease he elec ic ield a he c i ical poin s, and consequen ly, he p obabili y o Figu e 7. Elec ic ield on connec o 1 as a unc ion o he ing dimensions: hickness (A), in e nal diame e (Din ), and ing–connec o dis ance (dis ), ep esen ed simul aneously. 6. Conclusions The ins alla ion o co ona ings a ound a de ice o p o ec i , e.g., HV subs a ion connec o s, is a simple and e ec i e way o a oid he appea ance o he co ona e ec . The ings dec ease he elec ic ield a he c i ical poin s, and consequen ly, he p obabili y o he co ona e ec occu ing. The p esen s udy p oposed a no el an i-co ona de ice a angemen consis ing o he ins alla ion o a co ona ing pe conduc o coupled h ough subs a ion connec o s. This an i-co ona de ice gua an eed a co ec ixa ion and he co ona e ec onse was educed. The design was s udied o speci ic connec o s used o he connec ion o ou conduc o s unde a line ol age o 450 kVpeak AC. The design o co ona de ices is a key ac o when e alua ing he elec ic ield a a subs a ion connec o o in he case o subs a ion epowe ing. The e o e, se e al simula ions we e ca ied ou in he p esen wo k in o de o de e mine he mos adequa e an i-co ona