Cerman: Software for simulating streamer propagation in dielectric liquids based on the Townsend–Meek criterion

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Contents lists available atScienceDirect

Computer Physics Communications

www.elsevier.com/locate/cpc

Cerman ^✩ : Software for simulating streamer propagation in dielectric liquids based on the Townsend–Meek criterion ^✩✩

I. Madshaven

^a

, O.L. Hestad

^b

, P.-O. Åstrand

^a^,∗

aDepartmentofChemistry,NTNU–NorwegianUniversityofScienceandTechnology,7491Trondheim,Norway bSINTEFEnergyResearch,7465Trondheim,Norway

a rt i c l e i n f o a b s t r a c t

Articlehistory:

Received22July2020

Receivedinrevisedform8January2021 Accepted30March2021

Availableonline20April2021

Keywords:

Streamerbreakdown Dielectricliquid Simulationmodel Python

Computationalphysics

Wepresentasoftwaretosimulatethepropagationofpositivestreamersindielectricliquids.Suchliquids arecommonlyusedforelectricinsulationofhigh-powerequipment.Wesimulateelectricalbreakdown in aneedle–plane geometry, where the needle and the extremities of the streamer are modeledby hyperboloids, which are used to calculate the electric field in the liquid. If the field is sufficiently high,electronsreleased fromanionsinthe liquidcan turnintoelectronavalanches,and thestreamer propagates if an avalanche meets the Townsend–Meek criterion. The software is written entirely in Pythonand releasedunderanMIT license.Wealsopresentasetofmodel simulationsdemonstrating thecapabilityandversatilityofthesoftware.

©²⁰²¹^TheÂuthor(s).^Published^byÊlsevier^B.V.^Thisîsânôpenâccessârticleûnder^the^CC^BY^license (http://creativecommons.org/licenses/by/4.0/).

1. Introduction 1.1. Streamersinliquids

Dielectricliquids,speciﬁcallytransformeroils,areusedaselec- tricinsulationinhigh-powerequipmentsuchaspowertransform- ers [1]. Equipment failure is always a possibility, and ina world with ever-growing need for energy, there is a continuous effort tomakeequipmentbetter,cheaper,morecompact,andmoreenvi- ronmentallyfriendly.Topreventequipmentfailureduetoelectrical discharges,new insulatingliquids aswell asadditivesare tested, experimentsarecarriedout tobetterunderstandthephysicalna- tureofthephenomena,andsimulationsareperformedtotestthe validityofpredictivemodels[2,3].

Since electrical discharge events are rare at operating conditions, modelexperimentsaredesignedtoinducedischarge inthe liquid.Inonesuchmodelexperiment,aneedleelectrodeisplaced opposing a planar electrode, where the needle–plane gap is in- sulated by a liquid [2]. If highvoltage is applied, resulting in a suﬃcientlystrongelectricﬁeldclosetotheneedle,theliquidwill loseits insulatingpropertiesandbeginto conductelectricity,and subsequent (partial) discharges from the needle into the liquid canoccur. Thechargetransportedintotheliquidcanincrease the

✩ Availableat:github.com/madshaven/cerman.

✩✩ ThereviewofthispaperwasarrangedbyProf.DavidW.Walker.

*

Correspondingauthor.

E-mailaddress:[email protected](P.-O. Åstrand).

electric ﬁeld and lead to partial discharges in new regions in a self-inducedmanner.Shadowgraphicimagesrevealthat agaseous channel,a“streamer”,isformedandhowitbranchesasitpropa- gatesthroughtheliquid [4].Ifastreamerbridgesthegapbetween twoelectrodes, an electricdischarge can follow,possiblydestroy- ingtheaffectedequipment.

Streamersarecommonlyclassifiedbytheir polarityandspeed ofpropagationfromtheslow1stmodetothefast4thmode,rang- ing from below 0.1 km/s to well above 100 km/s [2]. Streamers withnegativepolaritytypicallyhavealowerinceptionvoltagethan streamerswithpositivepolarity(positivestreamers),however,positivestreamers typicallyleadtobreakdownsatlowervoltagethan negativestreamers,andassuch,researchismainlyconcernedwith positive streamers. The streamer phenomenon involves processes covering several length and time scales. Speed and branching is studied in gaps of different sizes (mm–m), while many of the interesting processes, such asfield ionization, high-field conduction, electro-hydrodynamic movement, bubble nucleation, cavitation, electron avalanches, photoionization, occur on a μm-scale [2,3].A streamerusually stops orleads toa breakdown ona μs- scale(km/s=^mm/μs),^whereas^processes^suchâsrecombinationof electronsandanions canoccurwithin picoseconds.Consequently, experimentationaswellassimulationischallenging.

1.2.Modelingandsimulations

While sophisticated equipment is required for experiments, simulationsofteninvestigatetheeffectofgivenprocessesthrough https://doi.org/10.1016/j.cpc.2021.107987

0010-4655/©²⁰²¹^TheÂuthor(s).^Published^byÊlsevier^B.V.^Thisîsânôpenâccessârticleûnder^the^CC^BY^license(http://creativecommons.org/licenses/by/4.0/).

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relativelysimplemodels.Thefractalnatureofthestreamerstruc- turecanbesimulatedbyconsideringalatticewhereeachpointis either partof an electrode, the liquid,or the streamer [5]. Here, the streamerexpands tonewlattice points when some criterion, such aselectricfieldstrength,isobtained. Throughkinetic Monte Carlomethods,thestochasticnatureandthephysicaltimecanalso bestudiedinsuchsimulations[6–8].Radialexpansionofstreamer channels andconductionwithin thestreamer have alsobeen investigated [9–12]. The latter mechanism, channel conduction, is alsoincludedin[13,14],wherethestreamerpropagationcriterion is a function of the square of the electric field. The charge and conduction ofa streamer can also be studiedby considering the streamerasanelectricalnetworkofresistorsandcapacitors,with- outnecessarilyconfiningthepointsofthenetworktoagrid [15].

Modelswherethestreamerconsistsofasetofdiscretepointsare simplisticbutalsoefficient.Conversely, withahigherdemandfor computationalpower,computationalfluiddynamic(CFD)methods canbeappliedtosolvetheequationsforgenerationandtransport ofchargedparticles(theflowofnaturalparticlesisoftenignored) during a streamer discharge [16,17], while the stochasticity and branching of streamers can be introduced by adding impurities [18].SuchCFD-calculationsoftenignorethephasechangefromliq- uidtogasaswellandareconfinedtoasmallregionbecauseofthe computational complexity.However, forsimplified,single-channel streamers,boththephasechangeandtheprocessesinthechannel canbesimulated[19,20].Codeusedforsimulationofstreamersin liquids is rarely published, in fact, we found just a single example [21].

1.3. Avalanchemodel

We havepreviouslydescribed ourstreamer modelforpositive streamerswherethepropagationisbasedonanelectronavalanche mechanism[22].Here,theavalancheprocesstakesplaceintheliq- uid,inthehigh-fieldregioninfrontofthestreamer,andcancause a direct transition to a streamer channel. We assume this is the main propagation mechanismof positive second-modestreamers innon-polarliquids.However,asmentionedabove,thereareother possiblepropagationmechanisms.Thefocusofthemodelisonthe streamerextremitiesandthehigh-fieldregionintheliquidinfront ofthem,whereasthestreamerchannelitself,andthemechanisms therein,aregivenasimplifiedrepresentation.Themodelhasbeen extendedtoaccountforconductanceinthestreamerchanneland capacitance between the streamer andthe planar electrode [23], aswellasphotoionization infrontofthestreamer[24],thelatter asamechanismforthetransitionfromslowtofastpropagation.

Streamerpropagationissimulatedinasetupresemblingmodel experiments, a needle–plane gap ﬁlled with a model liquid, see Fig. 1 fordetails. The needle andstreamer give rise to an electric ﬁeld, affecting charged particles in the liquid. A number of anions, “seeds” forelectron avalanches,is modeled within a vol- ume surroundingthe streamer, a “region of interest” (ROI). Elec- tronsreleasedfromtheanionscancreateelectronavalanches,and thestreamerpropagateswhenanavalanchemeetstheTownsend–

Meekcriterion,i.e.exceedsacriticalnumberofelectrons[22].The needle andthe streamer heads (the extremities ofthe streamer) aremodeledashyperboloids,whichsimplifiescalculatingtheelec- trical field since theLaplacian is analytic inprolate spheroid coordinates [25]. The electric field and potential are calculated by consideringelectrostaticshielding[22],aswellastheconductance in the channel andthe capacitance towards the planar electrode [23]. The streamer undergoes atransitioninto a fastpropagation modewhenradiationfromthestreamerheadcanionizemolecules directly infront ofthe streamer[24].More details onthe model aregiven insection3.

Fig. 1.Illustrationofthemaincomponentsinthesimulationmodel.Theneedleelec- trodeandthestreamerheadsarehyperboloids,eachwithapotentialVi.Aregionof interest(ROI)isusedtolimitthecomputationalefforttoaregionsurroundingthe activepartofstreamer.TheROIcontrolsthepositionofthe“seeds”,whichareclas- sifiedasanions,electrons,oravalanches,dependingontheelectricfieldstrengthat theirposition.The“shadowgraphic”imageofthestreameriscreatedbyplottingall formerpositionsofstreamerheads.(Forinterpretationofthecolorsinthefigure(s), thereaderisreferredtothewebversionofthisarticle.)

The main output of the simulations includes the propagation speed,the streamer shape (branching),andpropagation distance.

Inaddition,propertiessuchastheinitiationtime,thepotentialof individualstreamerheads,electricbreakdownwithinthestreamer channel, andavalanche growth,can also be investigated.Simula- tions show how various parameters affectthe results,where the gapsize, applied voltage, andtype ofliquid are important para- metersforasimulation.Furthermore,otherparameterssuchasthe sizeofastreamer head,theconductivityofthestreamerchannel, propertiesof additives, and avalanche growth parameters can be variedtovalidatewhethertheunderlyingphysicalmodelsarerea- sonable.

1.4.Scope

The present work describes the use, functionality and imple- mentationof Cerman[26], asoftware todo simulationswithour model [22–24], with the purpose to make the software publicly available. Section 2 demonstrates how to set up, simulate, and evaluateresultsofarelevantproblem.Furtherdetailsonthemodel anditsimplementation aregivenin section 3,whereas section4 outlines thecurrentfunctionality andsome prospectsforthe fu- ture.Asummaryisthengiveninsection5.Furthermore,detailson thealgorithm areincluded in AppendixA, simulationparameters inAppendixB,andsimulationexampleinputﬁlesinAppendixC.

2. Simulation–usingthesoftware 2.1. Softwareoverview

ThesoftwarenameCermanisanabbreviationofceraunomancy, which means tocontrol lightningor to use lightningto gain in- formation.TheimplementationisdoneinPython,anopen-source, interpreted, high-level, dynamic programminglanguage [27], and the software is available on GitHub [26] under an MIT license.

Thesoftwareisscript-based,andcontrolledthroughthecommand cerman,which isusedforcreationofinputﬁles, runningsimu- lations,and evaluating the results.When a simulation is started,

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Listing1:ExampleofJSON-inputﬁle, cmsim.json,deﬁningasimulationserieswithseveralvaluesfortheneedlevoltage V₀ andthe threshold forbreakdownin thestreamer channel E_bd,both withandwithout photoionization enabled. Furthermore,each permutation is tobe carriedout 10timeswithdifferent initialseed positions.Note,by setting alphakind to A1991,

α

iscalculated by(6).See AppendixBforadescriptionoftheparameters.

the simulationparameters are loadedfroman input ﬁle, andthe classesforthevariousfunctionsareinitiated.SeeAppendixBfora summaryofsimulationparameters.Thesimulationitselfisessen- tiallyaloopwhereseedsintheliquidaremovedandthestreamer structure isupdateduntilthe streamerstopsorleadsto abreak- down.ThealgorithmisdetailedinAppendixA.

2.2. Gettingstarted

DownloadCermanfromGitHub [26] andinstallitbyrunning pip install .

from the downloaded folder. This installs the python package and the script cerman. Python 3.6 or above is required, as well as the packages numpy [28], scipy [29], matplotlib, simplejson,and statsmodels. The dependencies are automatically installed by pip.The softwarehas been developed in OSXandhasbeentestedonLinuxaswell.

2.3. Createsimulationinput

EachsimulationrequiresaJSON-formattedinputfilewherethe parameters aregiven.Such filescanbecreatedfromamasterin- putfile,specifyingtheparametersforasimulationseries.Amaster input file can be created froma regular input file by changinga parametervalueintoalistofvalues.Morethanoneparametercan containlists,andallpossiblecombinationsofvaluesarefound to createtheinputfilesforthesimulationseries.Todemonstrate,we use cmsim.json inListing1,whichspecifiesasimulationseries exploring the influenceof various parameters.The ten valuesfor theappliedvoltagearespecifiedthrougha linspace-command, while thevalues forthe thresholdforbreakdown in thechannel and photoionization (fast mode) enabled are given in list form.

Furthermore,simulation_runs speciﬁesthenumberofsimilar simulations, onlydiffering by random_seed.If random_seed is null,theneachinputﬁleiscreatedwitharandomnumberas

random_seed,andwhenanumberisspecified,arangeofnum- berisgenerated,inthiscase,thenumbers1through10.Notethat random_seed referstotheseednumberforinitializingtheran- domnumbergenerator,nottotheseedswithintheROI.However, agiven random_seed doescorresponds to agiven initialposi- tionsoftheseedanions.Definingfixeda random_seed foreach simulationseriesmakes it easiertoseehow achange ina given valueaffectsthesimulation,butforanalyzingalargerassembleof simulations,itisusuallypreferablethatthesimulationsareuncor- related,i.e.haverandominitialanionplacement.

Individualinputﬁlesarecreatedbyrunningthe cerman with theargument ci (createinput) andspecifying which ﬁle to ex- pandwith-f:

cerman ci -f <filename>

Thiscommand createsa number of newfiles by permutation of all lists in the given input file. The permutation of 10 random seeds,10appliedvoltages,5 breakdownthresholds,and2modes for photoionization in Listing 1 results in 1000 files. By default, therandomseedisexpandedfirst,followedbytheneedlevoltage, which is useful to consider when designing a simulation series.

Choosing an appropriate number of values for these two parameters makes it easier to search for simulation ﬁles withgiven properties.WhenexpandingtheexampleinListing1,theleastsig- niﬁcantdigit ??Xindicatesrandomseednumber,theseconddigit

?X? indicates theneedlevoltage,whilethemostsigniﬁcantdigit X?? indicatesthethresholdforbreakdowninthestreamerchan- nelandwhetherphotoionizationisenabledornot.

The action pp (plot parameters) creates a matrix representa- tion ofthe parameter variation inset of input ﬁles, andis used like

cerman pp -g <pattern>

The argument -g speciﬁes the pattern to search (or “glob”) for, e.g. cmsim_?00.json.Theﬁlesareplottedatthex-axisandthe

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Fig. 2.Visualization of the difference in parameter values between a selection of input ﬁles.

variedparametersonthe y-axis,seeFig.2foranexampleoutput.

Thenameoftheoutputfileisbasedonthefirstfileinthepattern.

After ensuring that theinput parametervalues areasdesired, simulationsarerunusing

cerman sims -g <pattern> -m <no>

whichcreatesaqueueofallfilesmatchingthepattern,andsimu- lates a given number in parallel. For instance, cerman sims -g "cmsim_?5?.json" -m 15,simulatesall inputfileswiththesamevoltage,creatingaqueuewhereupto15 separatesubprocesseseachrunonesimulation.Thesepythonpro- cessesaresinglethreaded,andworkbest if numpy islimitedtoa singlethreadaswell.Eachsimulationdumpstheinputparameters andprogressinformationtoalogfile.Foreachsimulationwethen have aparameter file, a logfile, andone ormore savefiles. The filesarenamedbyextendingthenameofthemasterfile,e.g.

cmsim.json # master file

cmsim_290.json # input parameters

cmsim_290.log # log file

cmsim_290_gp5.pkl # save file cmsim_290_stat.pkl # save file

2.4. Evaluateresults

The resultsareevaluated by parsingthedata storedfromone or more simulations. The input file, Listing 1, defines two “save specifications” as true, i.e. enabled, each defining various simulation data to be dumped to disk at given intervals or occa- sions.The save_spec calledstat savesiterationnumber,CPU time, simulationtime,leading head z position,numberofcritical avalanches,andthepositionofeachnewstreamerhead,forevery iteration.Thisenablesevaluationoftheshapeofthestreamerand itspropagationspeed.The save_speccalled gp5 savesmostof thedataavailable,includingthepositionofalltheseeds(anions/- electrons/avalanches),forevery5percentofstreamerpropagation, i.e.atotalof20timesforabreakdownstreamer.Thedataissaved using pickle andcan beloaded to analyzea giveniteration,a wholesimulation,orbycombiningdatafromseveralsimulations.

Evaluateiterations.Iterationdatacanbeusedtoanalyzethede- tailsofasimulation.Thisisparticularlyusefulwhenevaluatingthe validityofthesimulationparameters.Useforinstance

cerman pi seedheads -r <start stop> -g <pattern>

wherepi means“plotiteration”andseedheadsisascatterplot ofavalanchesandthestreamerheadconﬁguration.Theoption -r

controlsthe rangeofiterationstoplot.Figure 28in[22] showsa numberofsuchplots.

Evaluatesimulations. Use ps for simulation plots. These are mainlyplottedwiththez-positioninthegaponthe y-axis. Plot- tingthe x- or y-position ofstreamer heads on the x-axis, using shadow,givesa“shadowgraphic”plotofthestreamer:

cerman ps shadow -g <pattern> -o <options>

Similarly, plottingthe propagation time on the x-axisis done in a streak plot (seeFig. 3).Optionscan be addedtocontrol the limits/extentsoftheplot,thefiguresize,the behavioroftheleg- end, redefining the axislabels, starting each plot withan offset, savingtheplotteddatatoaJSON-file,andmuchmore.Use help toshowavailablecommandsandoptions,forinstance:

cerman help # for the main script cerman ps help # for plot simulation cerman ps shadow -o help # for shadow plot

Singlesimulationdatamaybeofinterest,butitisoftenbetter tocompareseveralsimulationsinthesameplotto visualizehow theinput parameters affect theresults. The gp5 save requiresa lotofdiskspace,butcanbeveryusefulinanalyzingthedata.For plottingthepotentialofeachhead,use

cerman ps headsestr -g <pattern> -o <options>

Thecurrent(active) headsof thestreamer areselected, theirpo- tential is scaled (electrostatic shielding, using a nnls-approach, cf. [22]),andthen,theelectricﬁeldattheirpositioniscalculated.

However,theoptionscanbeusedtospecifythemethodforscaling andwhichpositions tocalculatethe ﬁeld for,e.g. attheposition ofeachappended(new)streamerhead.Theelectricﬁeldstrength andelectricpotentialarepresentedasscatterplotsagainstthez- positionof each givenposition,aswell asdashed lineindicating theaveragevalue,seeFig.4.

Evaluateaseriesofsimulations.The difference in thesimulated parameterscanbe visualizedusing pp andglobing forthe pkl- filesor log-files.Anexisting log-fileindicatesthatasimulation wasinitiated.Thecommand

cerman psr -g <pattern>

parses log ﬁles andplots the reasons why the simulations were terminated.AnexampleofsuchaplotisshowninFig.5.Thisisa goodwaytoverifythatsimulationshavecompletedsuccessfully.

Whenallsimulations arecompletedandveriﬁed,parseallthe saveﬁlesandbuildacombineddatabaseoftheresults:

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Fig. 3.(left)Streakplotswhere“options”havebeen usedtolimitthex-axisandtoshow V, Ebd,andphotoionizationonthelegend.(right)Shadowplotswhereeach streamerisplottedwithanoffsetandthelegendishidden.Thelegendisthesameforbothplots.

Fig. 4.Theelectricstrength(left)andtheelectricpotential(right)atthetipofeachnewstreamerhead.Thestreamerheadsaresampledforevery5%ofpropagation.The

“options”areusedtocontrolwhichstreamerheadstouseforthecalculationandwhichpositionstocalculatefor.

Fig. 5.Example, parsing the log ﬁles to visualize how simulations have terminated. “Unknown” implies that the simulation is not complete (ongoing or aborted).

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Fig. 6.Example,combinationplots:(left)propagationspeedusingbothmarkersandcolors,and(right)propagationlengthforonlyagivenvalueofoneparameterwiththe otherparametercolored.

cerman ca -g <pattern> -o mode=reload

Theoption reload forcesﬁlestobeparsed,eveniftheyalready are in the database. When parsing the save ﬁles, a number of propertiesareextractedorcalculated.Resultssuchaspropagation length (ls), averagepropagation speed (psa), inception time (ita), average jump distance (jda), simulation time (st), and computationaltime (ct)are savedinthedatabase. Theresults areplotted byusing

cerman pr <parameter> -f <file> -o <options>

The parameter gives the x-axis of the plot,for instance v (needle voltage). The results to plot is given through the option, e.g. -o ykeys=ls_psa. The software inspects the database of parsed save ﬁles for varied parameters and automatically create plotsofallpossiblepermutationsusingcolorsandmarkers.Fig.6 shows a selection of such plots, where needle voltage is on the x-axisandtheotherparametershavebeenaddedautomatically.

Asillustrated,thesoftwarehasbeendesignedtofacilitatesimu- latingandanalyzingalargesimulationsseriesinasemi-automatic fashion. Theindividual simulationshave theirparameters defined indedicatedfiles,whilethebehaviorofthe cerman-scriptisde- finedbycommandlinearguments.

3. Modelimplementation

Thissectiondescribestheimplementationofourmodel[22–24]

inmoredetail.Anoverviewofthemodelisalreadygiveninsec- tion1.3,andFig.1isusefulforunderstandingtheprincipalsetup, a needle–plane gap where electron avalanches grow from single electronseeds.

Theelectricfield. The needle electrode andthe streamer heads are modeled as hyperboloids. The Laplacian electric potential Vi andelectricfield Ei froma streamer headi iscalculated analyt- ically in prolate spheroid coordinates [22]. For a position r, the electricpotential V(r)andfield E(r)isgivenbythesuperposition principle,

V

(

r

) =

i

k_iV_i

(

r

)

and E

(

r

) =

i

k_iE_i

(

r

) ,

(1)

wherethecoeﬃcientskiareintroducedtoaccountforelectrostatic shielding. Anoptimizationis performedsuchthat the unshielded potential atthetip ofanystreamerhead j equals thesumofall theshieldedpotentialsatthat position,Vj(rj)=

kiVi(rj) [22].

The needleand theextremities ofthe streamer, i.e.each electrical hyperboloid, probably interacts to a greater extent than the method of superimposing and scaling the hyperboloids accounts for. To include such interactions or to calculate the full Poisson ﬁeld requires a much greater computational effort. However, the appliedmethodscalesthepotentialattheverytipofeachhyper- boloid,tobetterestimatetheLaplacianﬁeldinitsvicinity.

Regionofinterest(ROI).TheROI isa cylindricalvolume usedto control the position of seeds, see Fig. 1. Note, the seeds in this respectincludeallanions, electrons,andevenall avalanches.The numberofseeds inasimulationisgivenbythespeciﬁed density ofseeds andthevolumeoftheROI.Initially, theseedsareplaced atrandompositionswithintheROI.Whenaseedfallsbehindthe ROI, collideswith thestreamer, orcreates a criticalavalanche, it isremovedandreplaced by anew seed.The newseed isplaced witha z-positionone ROI-heightclosertotheplane, atarandom xy-position (at a radius less than the ROI radius). The ROI vol- umeisdeﬁnedbya distancefromthe z-axis,anda givenlength aboveandbelowtheleading streamer head,whichisthepartof thestreamerclosesttotheplanarelectrode.Whenanewstreamer headiscreatedcloserto theplane, thestreamer propagates, and theROImovesaswell.

Anions,electrons,andelectronavalanches.Eachseed j isclassiﬁed accordingtotheelectricﬁeld E_jatitslocation:

Ej

≥

Ec

=⇒

avalanche

,

Ej

≥

E_d

=⇒

electron

,

or

Ej

<

Ed

=⇒

anion

,

(2)

where the avalanche threshold Ec and detachment threshold Ed aregivenasinputparameters.Seedsmoveintheelectricﬁeldwith aspeed dependenton theirmobility

μ

,whichgivesthedistance theseed moves, s=^E

μ

t,for atime step t. Thenumber of electrons Neinan electronavalanche,startingfromasingleelec- tron,canbecalculatedby [22]

Ne

=

exp

i

si

α

i

=

exp

i

Ei

μ

e

α

me⁻^Eα^/^Eⁱ

ti

,

(3)

where

α

istheavalanchegrowth,

μ

eistheelectronmobility,tis thetimestep,andi isaniteration,whereastheavalanchegrowth parameters

α

mandEα havebeenobtainedfromexperiments[22].

Inpractice,however,we onlycalculateandstoretheexponentin (3), Qe=^ln^Ne,

Q_e

=

i

Q_i

=

i

s_i

α

_i

,

(4)

(7)

foreachelectron avalanche.Whena low-IPadditiveispresent,

α

ismodiﬁedbyaddingafactor [22]

α

i,add

= α

i

(

1

−

xadd

+

xadde^kα⁽Î^b⁻Îâ⁾

)

(5) whichisdependentonthemolefractionoftheadditivex_add,and thedifferenceinIPbetweenthebaseliquidIbandtheadditive Ia modiﬁed by afactorkα asprescribed by [30].Thisis thedefault settingforthesoftware.Anothermodelfor

α

givenin [31] isalso implemented:

α

_i_,_mod

₌

^{3e E}

2α I_bE_ie⁻

Eα

Ei

.

(6)

ThismethodisappliedinListing1.

Expanding thestreamerstructure. According to the Townsend–

Meek criterion [32], streamer breakdown occurs when an ava- lancheexceedsacriticalnumberofelectronsN_c=^exp(Q_c).When an avalanche obtains Qe>Qc, weplace anewstreamer headat its position [22]. The initial potential of a new streamer head is calculatedbyconsideringthecapacitanceandpotentialoftheclos- est existing streamer heads[23]. If addingthe newheadimplies removing another head (see the paragraph below), the potential changesslightly,mimickingtransferofcharge.However,ifboththe new andthe present head stays, they share the “charge”, which givesamoderatereductioninthepotentialofbothheads.

Optimizingthestreamerstructure.Therearethreecriteriaforre- movingheads.Astreamerheadi isremovedif

ν

j

(

r_i

) < ν

j

(

r_j

)

or k_i

<

k_c or

(|

^ri

−

^rj

| <

d_m

)

and

(

z_i

>

z_j

)

,

(7)

are satisﬁed forany other streamer head j [22]. The ﬁrstcondi- tioncheckswhetherthetipofonehyperbole(r_i)isinsideanother hyperbole, a collision (the

ν

-coordinate describes a hyperboloid, speciﬁcally the asymptotic angle). The second condition removes heads whose potential are to a high degree shielded by other heads (ifthe coeﬃcientk_i in (1) lower thana thresholdk_c). The thirdconditioncheckswhethertwostreamerheadsarecloserthan dm andshouldbe mergedtoasingle head,wheretheone atthe highestz-coordinateisremoved.

Conductionandbreakdowninthestreamerchannel.Conductionin thestreamerchannelincreasesthepotentialofeachstreamerhead i eachiteration,

V_i

=

^V0

−

^Vi

→

^Vi

=

^V0

−

V_ie⁻^t^/^τⁱ

.

(8) Thetimeconstant

τ

i=^RC

τ

0(foragivenheadi)iscalculatedfrom theresistanceRinthechannelandthecapacitanceC towardsthe plane[23]

R

∝ ,

and C

∝

ln4z

+

2rs

r_s

₋1

,

(9)

where is the channel length (distance to theneedle), rs is the tip curvatureradiusofthestreamerhead,andz isthe z-position of the streamer head. As such, each streamer head is treatedas an individual RC-circuit, e.g. the three streamer heads in Fig. 1 wouldeach havean individual resistance (channel)aswell asan individual capacitance towards the planar electrode. The capacitance in (9) isbased on a single hyperboloid above a grounded plane [23]. As such, superimposing each streamer branch would overestimate the capacitance of a branched streamer, and there- fore this calculation is not performed within our simulations. In fact,wedonotevencalculatetheabsolutecapacitanceofindivid- ualbranches,thecapacitanceofastreamerheadisalwayskeptas ameasurerelativetoeithertheneedleelectrodeoranotherpartof

thestreamer.Inadditiontothecalculationoft_i in(8), thecapac- itanceisusedtodeterminethechangeinstreamerheadpotential whenthestreamerpropagatestonewpositions [23].Thelinearre- sistancein(9) isa simplemodelincomparisonto modelswhich alsoestimates expansionand relaxationwithin thechannels, e.g.

[11,14].However,wealsomodelbreakdownswithinthestreamer channel,forwhichtheresistanceisgreatlyreduced.Iftheconduc- tionislow,thepotentialdifferenceVi increasesasthestreamer propagates, and the electric ﬁeld within the streamer channel Es=Vi/i may increase aswell. If Es exceeds a threshold E_b, abreakdownoccurs,equalizingthepotentialofthestreamerhead andtheneedle,whichisachievedby setting

τ

i=^{0 in}^{(8) for}^the giveniteration.

Photoionization.Photoionizationisapossiblemechanismforfast streamer propagation [33]. We have proposed a mechanism in which the propagation speed of a streamer increases if the liq- uidcannotabsorbradiationenergytoexcitedstates,asaresultof astrongelectricfield reducingtheionizationpotential [24].Since thefullmodel,consideringfluorescentradiationfromthestreamer head,andafield-dependentphotoionization absorptioncrosssec- tion,iscomputationallyexpensive, asimplermodelisusedinthe simulations.InsteadwecalculatethefieldstrengthE_w requiredto reduce the ionization potential below the energy ofthe fluores- centradiation.Ineachiteration,iftheelectricfield E atthetipof astreamerheadiexceedstheparameter E_w,theheadismoveda distancesitowardstheplanarelectrode,

s_i

= −

^vw

t

(

E

(

r_i

) −

^Ew

)

^z

ˆ ,

(10) where vw is the photoionization speed, and is the Heaviside stepfunction.Formoredetailsontheentiremodel,seeourprevi- ouswork[22–24].

4. Currentfunctionalityandfutureprospects

The main function of the model and the software is to sim- ulatestreamers ina point–plane gap,using the Townsend–Meek criterion for propagation. The propagation criterion is metwhen electronavalanchesobtaina givensize.Thismodelandthealgo- rithm are ﬁxed, but there are several parameters which can be adjusted.Changingexperimentfeatures suchasneedletipradius, gap size, voltage, liquid properties, or the parameters of the al- gorithms, is straightforward. Proposals to extendthe software to encompassnewfunctionalityisgiveninthissection.

In [22] we explored the fundamental features of the model, i.e. a streamer consisting of charged hyperbolic streamer heads, and streamer growth by electron avalanches initiating from anions.Themodel predictsseveralaspects ofstreamer propagation, andshows how they are linked towards the values of givenin- putparameters. The predictedpropagationspeed andthe degree ofbranching were both lower than expected. We foundhow the speedwasdependentmainlyonthenumberofelectronavalanches andtheir growth,whilethebranchingwasmainlyrelatedtohow thestreamerheadswereconﬁguredandmanaged,whichismainly controlledbytheparameterskc anddm in(7).

Theelectronavalanche modelwas calibratedtohavestreamer inceptionat30 kV[22] according thepropagationcriterion given in [30]. However, the simulated streamers do not propagate far at this voltage, and potentials above 60 kV is required for a breakdown [22]. For small gaps, 3 mm to 10 mm, the inception ofpropagationstreamersoccursbetween14 kV to 36 kV,depend- ing on how “propagation” is deﬁned [30,34,35]. The higher end of these values can typically also give breakdown, whereas fast streamers canoccur atvoltages above60 kV [35]. Inlargergaps, 50 mm to 77 mm,wellabove100 kVisrequiredforabreakdown, and the acceleration voltage is just 10 kV to 30 kV above that

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[34,36]. The focus of ourwork has been on usingcorrect values forourparameters,ratherthan,forinstance,argueforanincrease innseed or

μ

etoincreasethespeedofthesimulatedstreamers.

When new streamer heads were added, their potential was set assuming a constant electric ﬁeld within the channel, result- inginamoderatevoltagedropbetweentheneedleelectrodeand the streamer head [22]. To better representthe dynamics of the streamer channel, an RC-model was developed [23]. In the RC- model, the potential ofnew streamer heads is dependent of the potentialoftheclosestexistingstreamerheads.Iftheconductance ofthestreamerchannelishigh,thenthepotentialofthestreamer headiskeptclosetothatoftheneedle,givingresultscomparable tothosewithouttheRC-model.Conversely,havinglowconduction regulatesthespeed ofthestreamer,increasingthelikelihoodthat more branchesareable to propagate.Furthermore,the RC-model also allows for simulation of a breakdown within the streamer channelitself,whichislikelywhatoccursduringare-illumination.

Thisbreakdownoccurswhentheelectricﬁeldwithinthechannel exceedsagiventhreshold.

The importance of photoionization during a streamer breakdown is unknown.We explored different aspects ofphotoioniza- tionin [24], andimplementedamodelforchangetoafastprop- agating mode. Molecules excited by energetic processes, such as electronavalanches,canrelaxtoalowerenergystateby emitting radiation.We arguedthat fluorescent radiationcanbe important, andmodeledhowthisradiationcancauseionizationinhigh-field areas,sincethehigh-fieldreducestheionization potential.Forin- stance,forcyclohexane,weassumethatradiationfromthelowest electronicallyexcitedstate(about7 eV)cancauseionizationwhen the localelectric field isin the range1.4 GV/m to 3.1 GV/m [24].

Molecular dissociation within the streamer and radiation from other molecules than the base molecules are interesting aspects of thestreamer phenomenon, butare not considered inthe cur- rentmodel.Moreover,theenergyavailable informofradiationis unknownanddiﬃculttoquantify.

In the current implementation, a square wave voltage is applied tothe needleat thebeginning ofthe simulation.It iseasy tochangethebehaviortoavoltageramp,fromzerotomaxovera giventime.Thiscanbethebasisforastudyonstreamerinception whereotherparameterssuchasneedlesizeandtheelectronprop- ertiesoftheliquidareinvestigatedaswell.Simulatingadynamic voltage,suchasalightningimpulse,requiressomemorework,but isalsoachievable.

We havefocused on cyclohexanesince many ofits properties arewell-known,butothernon-polarinsulatingliquidscanbestud- iedbychangingrelevantparameters.Theseeddensityn_ionisbased on the low-ﬁeld conductivity

σ

and electron mobility

μ

e of the liquid, andthe propagationspeed scales linearly withboth seed densityn_ionandelectronmobility

μ

e.In[22],wecalculatedn_ionin therange6×¹⁰¹¹^m⁻³ ^to⁶×¹⁰¹²^m⁻³^,^the^lower^value^from^the equilibrium generated by cosmic radiation, andthe higher value fromtheionmobilityandtheliquidconductivity.Forsimulations thereafter,n_ion=²×¹⁰¹² ^m⁻³ ^have ^beenûsed.^The êlectronâv- alanche growth parameters are also liquid-dependent, and Eα in particular has a big impact on the results [22]. Streamer parameters,suchasconductivityofthestreamerchannelandstreamer headradius,needto bereevaluated aswell forotherliquids.The propertiesofthestreamerchannelare alsoimportanttosimulate theeffectsofexternalpressure,whichmainlyaffectsprocessesin thegaseousphase[37].

The effect of additives with a low ionization potential (IP) is modeled as causing an increase in electron avalanche growth [22,30]. Other additives can easily be used as long as the IP of both the base liquid and the additive are known. Low-IP additives are known to facilitate the propagation of slow streamers andtoincreasetheaccelerationvoltage,possiblyasaresultofin-

creasedbranching[34],howeverthemechanismsinvolvedarenot known.Itispossiblethatlow-IPadditivesaresourcesofelectrons that can initiate avalanches, produced for instance through pho- toionizationorﬂuctuationsintheelectricﬁeld.Suchmechanisms canbe addedto themodelandsimulated,butwill requiresome work.Furthermore,themechanismsofaddedelectron scavengers canalsobeinterestingforfurtherinvestigation,andparticularlyif negativestreamersaretobesimulated[30].

Ourprimary concern has been with positive streamers, since theseare morelikelytoleadto acompletebreakdownthanneg- ativestreamers.The modelreliesonelectronsdetachingfroman- ions,moving towards regions wheretheﬁeld is higher,andthen forming electron avalanches. The polarity of our model can eas- ilybereversed,however,theelectronswouldthen driftawayand be unable to form avalanches. As such, a model for creation of newelectrons isneededto simulatenegative streamers withthe software.Charge injectionfromthe needleand thestreamer can beonesuchmechanism [38].Anotheroptionistomodelelectron generation(chargeseparation)inthehigh-ﬁeldregionsurrounding theneedleandthestreamer.Suchmechanismsareinteresting for simulatingpositivestreamersaswell.

The hyperbole approximation simpliﬁes the calculations of the electric ﬁeld, both from the needle electrode and from the streamer heads. Other experimental geometries, such as plane–

needle–plane, or even more realistic real-life geometries can be implemented.Thechallengeistosetthecorrectshieldingorscal- ing of the streamer heads according to the inﬂuence of the geometry.Simplergeometricrestrictionsareeasiertoimplement,for instanceby manipulatingtheROI. The streamercan berestricted toa tube by settinga low value forthemaximum radius ofthe ROI. Anothermethod is settingthe “merge threshold” very high, suchthatthestreamerisrestrictedtoasinglechannelwithasin- gle head, which can be representative for a streamer in a tube [39].

There aremany mechanismsthat can be added toinvestigate differentmethodsofstreamer propagation, forinstanceeffectsof Joule heating or electro-hydrodynamic cavitation [40]. There are alsoseveralpartsoftheexistingmodelthatcanbeimproved.Bet- tercalculationandbalancingofchargesandenergywouldgreatly improvethemodel.Forinstanceanelectricnetworkmodelwhere thestreamerisconsistingofseveralinterconnectedparts,incon- trasttothecurrentimplementationwhereall thestreamerheads areindividually “connected”to theneedle.Such an approachcan give a better understanding of the charge flow in the different parts and branches of the streamer, as well as a better repre- sentation of the electricfield. Developmenttowards a modelfor a space-charge limited field [41] can further improve the elec- tricfieldrepresentation,however,possiblyatahighcomputational cost.

5. Summary

Wepresentasoftwareforsimulatingthepropagationofposi- tivestreamers ina needle–planegapinsulatedby adielectricliq- uid.ThemodelisbasedontheTownsend–Meekcriterioninwhich anelectron avalanche hasto obtainagivensize forthestreamer topropagate.Thesoftwarewasdevelopedandusedforsimulating ourmodels forelectron avalanche growth [22], conductance and capacitance in the streamer channels [23], as well as photoion- izationinfront ofthe streamer [24].From the examples onhow to set up, run, andevaluate simulations, others can recreateour previous results or create their own set of simulations. Further- more,theoverviewoftheimplementationandalgorithmservesas agoodstarting pointforotherstochangeorextendthefunction- alityofthesoftware.

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Fig. 7.Thesimulationalgorithmconsistsofinitialization,aloopwheretheseedsandthestreamerstructureisupdated,andthenafinalization.Inthe loop,firsttheseeds (anions,electronsavalanches)areaffectedbytheelectricfield,thenthestreamerstructureismodifiedandthischangestheelectricfield,finallytheregionofinterest(ROI) isupdatedandthedatafromtheiterationisevaluated.Theloopconcludeswhenone(ofseveral)criterionismet,typicallylowpropagationspeedorreachingtheopposing electrode.Detailsoneachsteparefound inAppendixA.

Declarationofcompetinginterest

Theauthorsdeclarethattheyhavenoknowncompetingﬁnan- cialinterestsorpersonalrelationshipsthatcouldhaveappearedto inﬂuencetheworkreportedinthispaper.

Acknowledgement

ThisworkhasbeensupportedbyTheResearchCouncilofNor- way(RCN),ABBandStatnett,undertheRCNcontract228850.

Appendix A. Thealgorithm

This section describes the algorithm used to implement the model in more detail, essentially each part of Fig. 7, while ref- erencingrelevantpartsfromsection3.

Initializesimulation. The simulation input parameters are read andusedtoinitializeclassesforcodeprofiling,simulationlogging, calculationofavalanche growth,theneedle,thestreamer, there- gionofinterest(ROI),theseeds(anions,electron,avalanches),how tosave data,andhow to evaluate simulationdata. Theinitiation ofthelogfile includesdumping theinput parametersto thefile.

GiventheROI volumeandtheseeddensity,anumberofseedsis createdandplacedwithin theROIatrandompositions.Then,the saveﬁlesareinitializedbydumpingtheinitial data,mainlyinfor- mationconcerningtheneedleandtheseeds.

Updateseeds.Theelectricfield Eiscalculatedforeachseed(applying (1)) and the seeds are classified as avalanches, electrons, andanions.Alltheavalanchesmoveintheelectricfieldandgrow in size (see (3)). The procedure is repeated until an avalanche collideswithastreamerhead,an avalanchemeetstheTownsend–

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Meekcriterion,oratotalofNMSN repetitionshasbeenperformed.

The“time”spentinthisinnerloopsetsthetimestepforthecur- rentiterationofthemainsimulationloop.Finally,alltheelectrons and anions are moved. The inner loop over just the avalanches saves signiﬁcant computationaltime since the calculation of the electric ﬁeld is the most expensivepart of the computation and theavalanchesareusually asmallfractionofalltheseeds.

Photoionization. The electric field at the tip of each streamer headiscalculatedandcomparedwiththethresholdforphotoion- ization.Eachstreamer headwhere E>E_w is“moved”a distance vwt towardsz=^0.^Movingîmplies^creatingâ^new^head,^setting thepotentialby“transferringcharge”,andremovingtheoldhead.

Managenewheads. For each critical avalanche a new head is created.Ifthesimulationtimestepissetsufficientlylow,thereis usuallyzeroorjustonenewhead.Thenewheadisdiscardedifit satisfiesanyofthecriteriain(7),however,ifaddingitwillcause anothertoberemoved(later),thenewheadisclassifiedas“merging”.Ifnoneofthecriteriain(7) ismetbyaddingthenewhead, i.e.allheadsarekept,thenitisa“branching”head,sinceaddingit ispotentiallythestartofanewbranch.Thepotentialof“merging”

issetby“chargetransfer”fromtheclosesthead,while“branching”

headshavetheirpotentialsetby“sharingcharge”withtheclosest head,where thelatter methodalso modiﬁesthepotential of the existinghead[23].

Managestreamer.Thepotentialofeachstreamerheadisrelaxed towards thepotential oftheneedleby applying(8). Thisstepin- creasesthepotential ofthestreamer headtothepotential ofthe needle when a breakdown in the channel occurs. As mentioned above,thecalculationoftheelectricﬁeldfortheseedsiscompu- tationallyexpensive,anditactuallyscaleswithboththenumberof seedsandthenumberofstreamerheads.Itisthereforepreferable to keep the number of heads to a minimum. Superﬂuous heads aretrimmedaccordingtothecriteriain(7).Then,theelectrostatic shieldingissetforthetrimmedstructure.

UpdateROI. Seeds that have moved behind the ROI, collided with the streamer, or lead to a critical avalanche are removed and replaced by a newseed. When a seed is replaced, the new seedisplacedadistance,equaltotheheightoftheROI,closerto the plane, at a random xy-position within the ROI radius. If the streamer has moved closer to the plane, the ROI movesas well, andseedsbehindthenewpositionarereplaced.Ifastreamerhead isclosetotheedgeoftheROI,theROIexpandstowardsthemaxi- mumradius.Newseedsarecreatedatrandompositionwithinthe expandedregion.

Evaluateiteration. Iteration data is extracted from the various classes to be saved for later use. The data is used to evaluate whether any stop condition is fulﬁlled, and stored to dedicated classes. Which data to store and how often to sample the data iscontrolledbytheuserinput.Thesaveddataisdumpedtoaﬁle atregularintervalstokeepmemoryrequirementsoftheprogram low.Informationtomonitortheprogressisprintedtothescreen, atregularintervals.Finally,anumberoftemporaryvariables,rele- vantonlyto theiterationiscleared,andtheprogramisprepared foranewiteration.

Terminate? Ifnoneoftheconditions forstoppinga simulation aremet,thenextiterationisperformed. Theseconditionsinclude low streamerspeed,streamerheadclosetotheplane(breakdown), simulationtime exceeded,computation timeexceeded,andmore.

Whenacriterionismet,anyunsaveddataisdumpedtodisk,and aﬁnalloggingtoﬁleandscreenisperformed,beforetheprogram terminates.

Appendix B. Parameters

The parameters for a simulation are supplied by the user in a JSON-formatted ﬁle(see Listing1).A listofallimportant para-

metersisshown inTable1.Mostofthe defaultparametervalues were motivated, calculated, and/or tested for sensitivity in [22], whereas[23] and[24] introducedasomenewparametersandjus- tiﬁedtheirdefaultvalues.

Experimentalconditions. The potential, position andsize ofthe needle are important parameters in an experiment. These para- metersgivetheoriginandthemaximumpotentialofthestreamer inthesimulations.Wehavemostlydealtwithstreamersinsmall gaps dg=^{3 mm to} ^{10 mm} ^at^potentials ^Vn=^{30 kV to} ^{150 kV} [22–24],however,single-channel streamers ingapsup to 50 mm havealsobeensimulated [42].The potential isperhapsthe most important parameter of a simulation, and as such, great care should be takenwhen selecting its value. Higher potentials can, forinstance,requirealargerROIandsmallertimesteps.Morede- tailsfollowbelow.

Seedsandavalanches.Thecreationandmovement ofseeds(anions, electrons, and avalanches), as well as the growth of the avalanches, are controlled by the parameters of the liquid. We havebasedthesimulationsoncyclohexaneasamodelliquidsince mostofits propertiesarewell-known [22]. Thenumber ofseeds inasimulation isgivenbythe ROIvolume andtheseeddensity.

The latter property is calculated from the low-ﬁeld conductivity and the ion mobility, unless explicitly set by the user [22]. The remainder ofthe liquid parameters relatesmainly to the thresh- oldforelectrondetachment,electronavalanche growth,andcriti- cal avalanchesize, see(2) to (6).Further informationonelectron avalanchesandtheirparametersisfoundine.g.[22,30,31,43,44].

Streamerstructure.Theparametersforthestreamercanbesplit in two groups. The first group controls creation of the streamer headsandhowtheyaretreatedinrelationtoeachother,whereas thesecondgroupisrelatedtotheRC-model,controllingthepoten- tialofthestreamerheadsandtheelectricfieldwithinthestreamer channel. There is also the option to choose whether the electric field from the streamer, acting on the seeds, is calculated using 32- or 64-bit precision. The latterrequires about twice the time tocompute.Thetip radiusofthestreamer headsischosenbased on the inception of second-mode streamers [22], and this para- meterisessentialtocalculatetheelectricfield.Inthisrespect,dm andk_c,see(7),arealsoimportant.Thefirstonesetstheminimum distancebetweentwoheads,andthelatter,theminimumscaling ofahead.Alowdm allowsforafine-branched streamer,whereas highervaluescansuppressbranchingaltogether.Forhigherpoten- tials,it makes sense to reduce kc, allowing morestreamer heads to be kept during the simulation.The minimum potential differ- encebetweentheneedleandeachstreamer headisgivenbythe fieldwithinthechannel E_s[22].Thedifferencecanbelargerifthe conductanceislow,however,itcan belimitedtoa maximumby thevalueof Ebd[23].Itisalsopossibletochangethewaythere- sistanceandthecapacitance iscalculated,forinstanceasparallel plane orspherecapacitor, butthishasinsofar onlybeenused to exploredifferencesinoutcomes.Thetimeconstant

τ

0 (in(8))and E_bd are theactual main contributorsof theRC-model andareof importanceto thestreamer breakdownsimulations [23].Further- more,the parameters forphotoionization are includedin (10) to increasethespeedwhenagiventhresholdﬁeldisexceeded [24].

Simulation algorithm. The parameters in the last section are mainlyrelatedtothesimulationalgorithmitself.Thetimestepand the numberof steps per loop, limitingthe maximum movement ofseeds, are essential foragood results.The random seed,used toinitializetherandomnumberengine,controlstheinitialplace- mentofseedanions.Choosingthesamerandomnumberenables the study of, for instance, changing voltage with the same ini- tialseedconﬁguration.Conversely,notsettingtherandomnumber makesthe simulationsuncorrelated, whichis betterforstatistics.

Thesizeofthe ROIdecides howmanyseedsthat areincludedin asimulation.Forinstance,anROIof10 mm³ incombinationwith

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Table 1

Main parametersfor simulation program. The experimentalconditions specify an overvoltage applied to mediumsizeneedle–planegap.Thevaluesofthephysicalparametersinseedsandavalanchesandstreamer structurearejustiﬁedinourpreviouswork[22–24].Parametervaluesrelatedtothesimulationalgorithm,or themodelingeneral,havealsobeendiscussedinpreviouswork.SeefurtherdescriptioninAppendixB.

Property Keyword Symbol Default

Experimental conditions

Distance from needle to plane gap_size dg 10 mm

Voltage applied to needle needle_voltage Vn 100 kV

Needle tip radius needle_radius rn 6.0 μm

Seeds and avalanches

Seed number density seeds_cn nseeds 2×¹⁰¹²/m³

Anion mobility liquid_mu_ion μion 0.30 mm²/Vs

Electron mobility liquid_mu_e μe 45 mm²/Vs

Liquid low-ﬁeld conductivity liquid_sigma σion 0.20 pS/m

Electron detachment threshold liquid_Ed_ion Ed 1.0 MV/m

Growth calculation method alphakind – I2009

Critical avalanche threshold Q_crit Qc 23

Electron multiplication threshold liquid_Ec_ava Ec 0.2 GV/m

Electron scattering constant liquid_Ealpha Eα 1.9 GV/m

Max avalanche growth liquid_alphamax αm 130 μm⁻¹

Additive IP diff. factor liquid_k1 kα 2.8 eV⁻¹

Base liquid IP liquid_IP Ib 10.2 eV

Additive IP additive_IP Ia 7.1 eV

Additive mole fraction additive_cn xadd 0.00

Streamer structure

Streamer head tip radius streamer_head_radius rs 6.0 μm

Minimum ﬁeld in streamer channel streamer_U_grad Es 2.0 kV/mm

Streamer head merge distance streamer_d_merge dm 25 μm

Electrostatic shielding threshold streamer_scale_tvl kc 0.20 Photoionization threshold ﬁeld streamer_photo_efield Ew 3.1 GV/m

Photoionization added speed streamer_photo_speed vw 20 km/s

Data type for ﬁeld calculation efield_dtype – float32

RC-model time constant rc_tau0 τ0 1 μs

RC resistance model rc_resistance – linear

RC capacitance model rc_capacitance – hyperbole

RC breakdown ﬁeld rc_breakdown Ebd 6 kV/mm

Simulation algorithm

Time step of avalanche movement time_step t 1.0 ps

Max avalanche steps per iteration micro_step_no NMSN 100

Seed for random number generator random_seed – None

Number of similar simulations simulation_runs – 10

ROI – behind leading head roi_dz_above z⁺ROI 1.0 mm

ROI – in front of leading head roi_dz_below z⁻ROI 1.0 mm

ROI – radius from center roi_r_max rROI 3.0 mm

Stop – low streamer speed stop_speed_avg vmin 100 m/s

Stop – streamer close to plane stop_z_min zmin 50 μm

Stop – avalanche time stop_time_since_avalanche t^ava_max 100 ns

Sequential start number seq_start_no – 0

Enable proﬁling of code profiler_enabled – False

Interval – dump save data to ﬁle file_dump_interv – 500

Interval – display data on screen display_interv – 500

Level of logging to ﬁle log_level_file – 20

Level of logging to console log_level_console – 20

aseeddensityof2×¹⁰¹²/m³ resultsin20 000seedsgenerated, a sizewhichiseasily treatedbymostcomputers. Thesize ofthe ROI should dependon themagnitudeof theelectricﬁeld [22]. A streamer branchlaggingbehind the ROIwill die,andtheROI ra- diusgivesthemaximumlateralmovementofthestreamer.Finally, severalparameters canbe usedto controlhow longa simulation will run, to prevent wasting computational time on uninterest- ing simulations,andtostopa simulationifthestreamerstopsor reachestheotherelectrode.Additionalparameterscontrolhowof- teninformationislogged,andhowdetailedtheloggingshouldbe.

Appendix C. Exampleﬁles

Thisappendixcontains anumberofexamplesofpossiblesim- ulations. TheﬁlesinListings 2to5are all includedinthefolder

examplesonGitHub [26].

Listing 2: The example ﬁle small_gap.json speciﬁes a smallgapandarangeofvoltagesalongwithmanyparameterval-

ues equal to the defaults (cf. Table 1). Although all values used in a simulation are stored in the log,it is nice to be explicit in theinput as well. Byspecifying 10 simulation_runs and10 voltages,a totalof100simulations iscreatedfromthisﬁle.Each simulationinitiatedwiththeseedsatuncorrelatedpositionssince random_seed is none.

Listing3:Theexampleﬁle small_gap_mod.json buildson Listing2, but a numberof parameters are modiﬁed, notably the seed density and the electron avalanche parameters, as well as severalparameters forthe streamer. All data is storedevery 5th percent of propagationby gp5 and every 100th nanosecond by ta07. Storing the properties of tens of thousands of seeds en- ables plotting of the development of seeds, but also requires a lot of disk space. Specifying streamer saves all the streamer heads every 0.1 percent of propagation and is used to evaluate thedevelopment of thestreamer. Since random_seed is1 and simulation_runs is 10, a range of random_seed from 1

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Listing 2: The example ﬁle small_gap.json speciﬁes simulations similar to the baseline studies in section 3.1 in [22].

Listing3: The exampleﬁle small_gap_mod.json speciﬁes simulations similarto theattempts tofacilitate branchingin section 3.5 in [22].

Listing 4: The example ﬁle rc.jsonspeciﬁes simulations similar to those performed in section 4 in [23].