Battery Testing with the Calculated Discharge Curve Method-3D Mathematical Model

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1 Journal of Power and Energy Engneerng, 05, 3, 37-5 Publshed One January 05 n ScRes. hp:// hp://dx.do.org/0.436/jpee Baery Tesng wh he Calculaed Dscharge Curve Mehod-3D Mahemacal Model Aleksandar B. Djordjevc *, Dusan M. Karanovc Mrjevsk Bulevar 48, Belgrade, Serba Emal: * aldjordj@open.elekom.rs, * aldjordj@eune.rs Receved 8 December 04; acceped 0 January 05; publshed 5 January 05 Copyrgh 05 by auhors and Scenfc Research Publshng Inc. Ths work s lcensed under he Creave Commons Arbuon Inernaonal Lcense (CC BY). hp://creavecommons.org/lcenses/by/4.0/ Absrac The calculaed dscharge curve mehod s based on hermodynamcally reversble work: The produc of he open-crcu volage, nal curren, and me,.e., he sum of useful energy and energy losses. A calculaed dscharge curve s based on he consan sep change of he baery volage n correspondence wh a cardnal number se. The essenal soluon s he ransformaon of he dscharge daa volage vs. me no me vs. volage usng basc equaons (hree-pon operaors: power of nernal ressance and me), whch are vald for all baery elecrochemcal sysems, baery desgns and dscharge condons. The mono and mul-cell baery operang condons conss of he followng: ) The four dscharge modes by consan loads: ressor, curren, volage, and power; ) Two load regmes: Self-drvng and devce-drvng (galvanosa, poenosa) or baery connecon (seral, parallel, combne); and 3) Connual and nermen dscharge. The baery average cell and cell/baery average characerscs, regardng me and capaces, are nroduced as he new baery characerscs. Keywords N-Se, Volage, Tme Inerval, Tme Sub-Inervals, Capaces, Energes. Inroducon Baery R & D, manufacure, produc accepance, performance presenaon, and exploaon are suppored by esng and defnng of baery characerscs. Baery esng s performed n accordance wh eher nernaonal (IEC, IEE, DIN, ec.) or nernal (laboraory, producer, user, ec.) sandards. These sandards defne baery re- * Correspondng auhor. How o ce hs paper: Djordjevc, A.B. and Karanovc, D.M. (05) Baery Tesng wh he Calculaed Dscharge Curve Mehod-3D Mahemacal Model. Journal of Power and Energy Engneerng, 3, hp://dx.do.org/0.436/jpee

2 A. B. Djordjevc, D. M. Karanovc quremens, funconal characerscs, es mehods and procedures. The Calculaed Dscharge Curve Mehod (CDCM) was developed o asss baery engneers n baery characerzaon [] [] a any sage of baery lfe. The Calculaed Dscharge Curve Algorhm (CDCA) and 3D mahemacal modelg are mahemacal ools. In accordance wh baery desgn and parallel and/or seres connecon, baeres are eher mono or mul-cell blocks. CDCM defnes he average baery cell as he represenave cell of a mul-cell baery ha can be compared o oher ndvdual cells. A baery elecrochemcal sysem requres defnng an operang volage nerval,.e., open crcu, Uo V, sarng, V o V and fnal V n V volage. The sarng volage s an nsananeous volage drop and s excluded from hs arcle. CDCM s used o analyse baery performance: Profles and capaces of volage and curren, energy on load and energy losses, cycle lfe, baery comparson wh elecrochemcal sysems, desgn, and producon, ec. The CDCM approach o baery esng s based on he overall energy volume (OEV) as he sum of he energy on load and energy losses. The OEV s n he form of a rgh parallelepped. The base of he parallelepped s he produc of he nal volage and curren, UoIo/VA, referred o as he nal power plane (IPP). The pars of he IPP need o be defned wh a cell/baery (C/B) elecrochemcal model. In hs arcle, he pars of he IPP are dfferen ypes of power: (a) On load; (b) Inernal ressance; (c) Exernal ressance; and (d) CDCM, polarsaon ressance. Durng baery dschargng, power on load s decreasng, whle power losses are ncreasng. By mulplyng he volage-curren,.e., power surface wh me alude, he energy parallelepped s defned. CDCA defnes me aludes as follows: (a) Dscharge me nerval; and (b) Dscharge me sub-nervals, he sum of whch s he me nerval. The cardnal number se 0< n = N, named n-n-se, defnes he consan sep change of volage and corresponds, one o one, o he n-se of me sub-nervals. CDCA reas dscharge me as a dependen varable,.e., he duraon of a predeermned change of baery sae of charge. The dscharge me nerval, s, s he dscharge duraon of he open crcu volage unl he acual dscharge volage. A me sub-nerval, s, s he duraon of he volage change for he consan volage sep. In accordance wh he S shape of any dscharge curve, he longes me sub-nerval les beween he nal and fnal me sub-nerval, nal < < max > > fnal. Ths approach leads o he CDCA basc equaon,.e., me dependence on volage. Boh he dscharge me and me sub-nervals, defne he OEV parallelepped n he hree forms: (a) The nal power plane mulpled by he dscharge me,.e., hermodynamcally reversble energy; (b) The sum of he four rgh paralleleppeds wh power bases (on load and nernal, exernal and CDCM losses) ha are mulpled by he dscharge me; and (c) The sum of he horzonal layers ha have aludes ha are he me sub-nervals and comprse he above four bases, whch are changed sep by sep of dscharge, buldng up he hll of energy on load. Only he sum of he horzonal layers represens he real baery energy balance represened by rregular nsead of rgh paralleleppeds. The known mng of he IPP segmens,.e., dscharge me and sub-nerval, defne he (a) volage and curren capacy, (b) energy on load, and energy losses, whch nclude (c) CDCM, (d) nernal and (e) exernal. When averagng he baery parameers n regard o me, CDCA nroduces he followng: average volage, curren, and power, by whch an rregular form of he OEV surfaces and spaces are ransformed no he regular forms,.e., parallelograms and paralleleppeds [] []. The baery operang condon comprses ) four modes of consan load: ressor, curren, volage, and power; ) wo load regmes: self-drvng and devce-drvng (galvanosa, poenosa or baery connecon: seral, parallel, and combned); and 3) connual and nermen dscharge. The fve baeres wh dencal, as much as possble, sae of charge and sae of healh dsplay he fve real dscharges, self-drvng (ressor, curren, and volage) and devce-drvng (curren and volage) (Table ). Table. CDCM real and vrual baery dscharges. Parameer Ressor Curren Volage Power Baery Response Ressor * Real Vrual Vrual Vrual Volage/Curren vs. Tme Curren Vrual Real Vrual Vrual Volage vs. Tme Volage Vrual Vrual Real Vrual Curren vs. Tme Power Vrual Vrual Vrual Real ** Tme * Self-drvng only; ** Reversble dscharge. 38

3 A. B. Djordjevc, D. M. Karanovc CDCA generaes hree vrual dscharges based on he parameers of he real dscharge. The magned consan power dscharge, as he self-drvng dscharge, s acceped as he reversble dscharge. Any cell n seral or parallel connecon dscharges wh seral curren or parallel volage and may dsplay dfferen characerscs n comparson wh he dscharge across he ressor,.e., here s a dfference beween self and devce drvng. Any dscharge mode may be eher connual or nermen. Afer he end of connual dscharge as well a he end of dscharge perods, relaxaon curves may be recorded for he new open crcu volage,.e., sae of charge []. Baery dscharge parameers, ncludng volage, curren, capaces, energy on load and losses, hea generaon, AC mpedance, DC ressance, and conducvy, may be monored durng dscharge n accordance wh he baery es mehod and daa acquson echnque, n addon o beng analyzed by CDCM.. CDCM 3D Mahemacal Model A 3D mahemacal model can be used o represen baery elecrochemcal pars wh he CDCA parameers (curves, surfaces and volumes) expressed by algebrac calculaons, placng all of hem nsde he OEV parallelepped [] []. CDCA s able o deermne he OEV wh relevan C/B exploaon characerscs of he same elecrochemcal sysem and varous C/B desgns, szes, dscharge modes, nenses, ec. Insead of volagecurren-me, he N-n-me space (where N V = Uo ) may be appled o show parameers of he varous elecrochemcal sysems... Power Plane CDCA dvdes he baery dscharge volage nerval wh consan volage seps, V = Uo N, o defne he volage sequence: U > V > V > > V > V > V > > V > V > V () o + n n n whch s n one-o-one correspondence wh he N-n-se: N 0< < < < ( ) < < ( + ) < < ( n ) < ( n ) < n = () as well as wh he over volage sequence, η = Uo V: < η < η < < η < η < η < < η < η < η (3) 0 + n n n The sequences of curren, Io > I > In, and curren losses, 0 < ν < νn, are calculaed wh he known load ressor, Rl/Ohm. In hs arcle, he four elecrochemcal power segmens are as follows: ( )( ) ( ) ( ) U I = V + η I + ν = VI + Vν + η I + ην o 0 = VI. + V Io I + Uo V I+ ην = Pload, + Pex, + Pn, + PCDCM, on load exernal nernal CDCM and are shown n Fgure. The nex equaons represen he sep changes of CDCA powers: (4) Fgure. CDCM baery power plane, LR0VARTA, 0 Ohm,. Fnal power on load, (VI) n/va;. Acual power on load, (VI)/VA; 3. Acual average power on load, (VI) avg, /VA; 4. Monocell baery nal power, UoIo/VA. 39

4 A. B. Djordjevc, D. M. Karanovc Overvolage, ( ) Vν = V ν + ηi + η ν = N η ν + η ν + η ν (5) ( V ) V ( V )( ) V ( N ) ν = ν + η ν ν = ν ην + η ν = + η ν (6) ( I) I ( )( I ) I ( N ) η = η η η + ν = η η ν + η ν = + η ν (7) ( ) ( )( ) ( ) ην = ην η η ν ν = η ν + ην η ν = η ν (8) η V may be he sum of he componens defned by he elecrochemcal model... OEV Geomerc Forms and Parameers CDCM acceps he elecrochemcal defnon of dscharge curve ([3], p. 59): Dscharge curves show eher he open crcu volage of a cell or half-cell as a funcon of he fracon of dscharge compleed, or, much more commonly, he cell volage durng a deep dscharge, usually under a fxed load or a consan curren. The abscssa may be calbraed n erms of he quany of elecrcy passed (Ah) or as a percenage of he heorecal capacy. A CDCA dscharge curve of any elecrochemcal sysem, baery desgn, dscharge mode and regme, and load nensy les nsde an OEV parallelepped whose base s UI o o= N η ν CDCA descrbes he OEV geomercally n accordance wh an elecrochemcal model. The parallelogram η ν s he smalles parallelogram of IPP. In hs arcle, N η ν parallelograms are shared wh he four elecrochemcal parallelo- grams. The dsrbuon of N parallelograms beween paral parallelograms changes by me, sep by sep, as he dscharge me expres. In he case of consan ressve load, Rl/Ohm, he frs dscharge me sub-nerval,, defnes he nal horzonal layer: UI o o,.e., N V I base of alude, whch s energy on load only because of he nonexsence of energy losses. Durng he sub-nerval, he power on load s reduced o he VI parallelogram,.e., ( N ) V I parallelograms, and he ( N ) parallelograms appear as power losses of he sub-nerval,.e., he second horzonal layer ha comprses energes on load and losses. The nex layers are n accordance wh Equaons (5)-(8), and hey form he horzonal layers of he decreasng power on load and ncreasng power of energy losses all he way o he las power layer. The hll of he energy on load (see Fgure 9) s rsng o he n-h horzonal layer, VI n n. Boh hll sldes, n an S shape, sar a an nal volage and curren and are placed unl he fnal layer, whch covers one-fourh of he nal power plane. If he n successve me aludes are placed a he e parallel eher wh he volage or curren axes, hey consue a D dscharge curve. If he n successve me aludes are placed a any of he power ( V I, η ν) es, hey consue a 3D dscharge curve (see Fgure 9). The possble baery model, defned by dscharge modes and regmes, dvdes he OEV no he dfferen pars, bu all are placed nsde he same OEV s volume, as he IPP, dscharge me and me sub-nervals reman consan. CDCM baery analyss s based on reversble and rreversble and real and vrual dscharges of varous elecrochemcal sysems, dscharge modes, regmes, and nenses.... Reversble and Irreversble Dscharges If he ressve load ends o nfny, hermodynamcally reversble baery dscharge occurs. An energy balance of rreversble baery dscharge emanaes from he reversble dscharge. A baery and s ressve load may be placed n an adabac calormeer [4]. Durng he dscharge, hea wll evolve on he load ressor and he baery. Ther sum forms he calormeer readng, and CDCM consders hs energy as baery reversble work. Ths work s presened as an OEV parallelepped, nsde whch CDCA places he energy on load and energy losses. If he baery and ressve load are placed n he wo separae calormeers, he work on he load ressor and he baery hea wll be measured, separaely, and her sum s equal o he reversble work. CDCA defnes energy losses n accordance wh he baery model OEV comprses he n = N horzonal energy layers, nsde whch are he pars of energy on load and energy losses. The OEV parallelepped base,.e., he IPP remans consan, whereas he bases of he parcular energes,.e., horzonal layers as well as sub-nerval s aludes, are changed.... Elecrochemcal and CDCA Dscharge Curve Elecrochemcally, a dscharge me curve s expressed by pars: volage vs. me. The regsered pars are as 40

5 A. B. Djordjevc, D. M. Karanovc follows 0 < < < < < < + < < n < n < n = fnal (9a) U > V > V > > V > V > V > > V > V > V = V (9b) o ( ) ( + ) n ( ) n ( ) n fnal The me s he ndependen varable, and volage s he dependen varable. The CDCM approach s reversed: volage s he ndependen varable, whereas dscharge me s he dependen varable. The CDCA ask s o ransform volage vs. me o me vs. volage. The dscharge me nerval, s, s equal o he sum of he me subnervals, s. In hs manner, an ordered rple s formed: < < < < < < < < < = (0) 0 + n n n fnal 0< < < < < > > > > > 0 () max max max + n n 0 < y < y < < ymax < ymax > ymax + > > yn > yn > yfnal () where y may be a baery parameers. By usng hese ses, ncludng IPP parameers, he elecrochemcal and CDCA baery characerscs can be calculaed [] [] [5]. The CDCA parameers are n one o one correspondence wh he N-n-se. Tme aludes may be placed a he IPP pon, e or power surface n accordance wh he baery elecrochemcal parameers...3. CDCA Inpu Daa The npu daa for baery esng, volage vs me, may be obaned eher as a low-frequency recordng/readng (manually regsered or graph readng) or as a hgh-frequency recordng/readng. The ses of volage vs me pars are as follows:. Auomacally, programmed = f( V,, I, ) 0 < < < < j < j < j < < k < k < k = fnal = programmed (3a) < < < < < < < < < < = (3b) 0 η η ηj ηj ηj+ ηk ηk ηk ηfnal. Publshed (graphcally presened or abulaed), eher graph = consan or V = consan 0 < < < < < < < < < < < = (4a) j j j j+ j+ fnal fnal graph 0 < η < η < < ηj < ηj < ηj+ < < ηfnal < < ηfnal 3. Manually readng: begnnng beween fnal (4b) 0 < < < < j < j < j+ < < fnal < fnal (5a) 0 < η < η < < ηj < ηj < ηj+ < < ηfnal < ηfnal (5b) Insead of dscharge me, all of he me-dependen characerscs may be CDCA npu daa. n..4. Inpu Daa n Ths Arcle The daa of he 6 dscharges are he npu daa n hs arcle: a CDCA dscharge curve from publshed daa [] on he mono-cell baery LR0VARTA, connual dscharge, load 0 Ohm, HP 3054 acquson un; b CDCA dscharge daa from a publshed graph [] concernng Mallory alkae manganese baeres (T. R. Crompon, Small Baeres, (98) 5), 0 baeres, baery load, Rl/Ohm, wh a baery average cell volage (.6 V) as follows;. 9K6, wo cells, U o = 3. V, V fnal =.6 V, baery load/ohm: 50, 00, 50;. PX4, wo cells, U o = 3. V, V fnal =.6 V, baery load/ohm: 00, 66; 3. PX, hree cells, U o = 4.8 V, V fnal =.4 V, baery load/ohm: 50, 50; 4. 7K67, four cells, U o = 6.4 V, V fnal = 3. V, baery load/ohm: 00, 50,

6 A. B. Djordjevc, D. M. Karanovc c Lead-acd baery, A5/4.0 Type No , publshed by Sonnenschen, he fve consan curren loads dscharge daa were pcked up by publshed graph. The seven pars: volage vs me mark ou he S shape of he dscharge curve (Table ). The fve dscharge me ses, n = 30, for he consan curren loads were generaed by he same CDCA procedure as for he prmary baery [] and are shown n Fgure. One par, volage vs me, needs o be chosen from used daa as he frs and parcular par for CDCA me generaon and needs o sasfy he nex crera: ( ) ( ) ( ) η = Uo V a, = η η b, η = U N c, 0 < η < η ( d) (6) 0 n The expermenal overvolage value: η V, needs o be placed as close as possble o he predeermned CDCA se value, η0 < η < ηn. The complee CDCA npu daa are he followng ordered rple: 0< < < < j < j < j < j+ < j+ < < j = N (7a) < η < η < < η < η < η < < η < η = U V (7b) 0 j j j+ n n 0 n 0 < < < < j < j < j+ < < n < n < n = fnal (7c) The S profle s marked wh he npu se of he dscharge curve, whereas he generaed se s he fnal and complee dscharge curve. Any baery elecrochemcal parameer may be used nsead of he overvolage n Equaon (7b). In hs arcle, he 0 generaed dscharge curves represen he followng: he wo elecrochemcal sysems, fve prmary and one lead-acd baery desgns, and consan ressve and curren loads. The am of usng such npu daa s as follows: To (a) verfy he CDCA basc equaon wh four CDCA operaors; (b) subm a repor abou he unque mehodology for baery characerzaon; and (c) esablsh he parameers vs he common abscssa: volage and/or N-se. n Fgure. Dscharge mes vs baery volage, SONNEN-SCHEIN, lead-acd baery, dryfa5/4.0 Symbols: npu me daa (7 pons), /s, Lnes: generaed me daa, /s, Consan curren/a and fnal mes/s:.,.4, 4.8, 7.,.0, 7,000, 33,30, 4,400, 8780, Table. Lead-acd baery dscharge daa. LOAD/A V V 889 0, V 43,846 8, V 55,6 4,009 0, V 63,000 7,34, V 70,05 30,980 3, V 7,000 33,30 4, V 7,000 33,30 4,

7 A. B. Djordjevc, D. M. Karanovc 3. CDCA Dscharge Tme Generaon 3.. CDCA Basc Equaon Invesgaon of he possble algebrac relaon me = f(volage, curren) s founded on he fac ha he IPP s defned by volage and curren nervals as elecrochemcal sysem characerscs, as well as on he concluson ha consan volage seps dvde he dscharge curve of he S-shape no he me nervals, whch sar as very shor and hen pass hrough he maxmum values wh decreases oward he end. Ths s n accordance wh he caon ([6], p. 8): xy, daa The fundamenal curve-fng problem n wo dmensons s o predc from a dscree se of ( ) pars ( x, y ) hrough (, ) xn y n he value of y when he value of x s specfed. The leas-square approxmaon of Chaper 4 s one form of curve fng n whch we seek he parameers of a gven model ha bes f he daa. In hs chaper, we look a oher mehods, whch do no accoun for random errors n he x and y values. We nsead, rea he daa as f hey were accurae and deermne a curve ha passes hrough he daa pons exacly. CDCA uses an emprcal equaon o ransform volage vs me no me vs volage o represen CDCM baery esng []. In hs arcle, boh he (a) expermenal pars volage vs. me and (b) he CDCA generaed se, n = 30, LR0VARTA, 0 Ohm me vs volage [], are he npu daa used o derve he CDCA basc equaon. The npu daa,.e., pars, have a role n defnng he S shape of he dscharge me, whch needs o be sasfed by he CDCA generaed me. Beween he npu pars, only one par, volage vs. me, needs o be chosen as he nal par for he generaon procedure. Ths chosen par needs o be placed n correspondence wh he CDCA volage as well as wh he n-se n he role of he sole common par of he npu and generaed se. All oher npu values have a conrol role. The generaed me curve needs o cover he expermenal markers, sasfyng he S shape. The dscharge me generaon wh he CDCA basc equaon, sarng from he chosen par, s performed oward he end as well as a he very begnnng of dscharge. The hree-pon me values CDCA operaors member s placed a he poson as well as he hrd, he greaer me value. CDCA dervaon of he basc equaon s sared by defnng he power of he nernal ressance se: 0 < Pn, I Pn, n. The profle of he Pn se s changeable by changng he value of he open crcu volage, Uo V. The rend e of he npu me values (Lne 3), and he rend e of generaed me se ( 30 ) (Lne 4), are shown n Fgure 3. = 64553P R = (8) npu generaed n, = 63653P R = (9) n, Fgure 3. Inpu me and generaed me vs. power of nernal ressance, Pn/VA, LR0VARTA, 0 Ohm,. /s npu daa, pons-symbols;. /s generaed dscharge me, 30 pons-e; 3. Trend e for npu daa: npu, I = 4553(Pn, )^5.4; 4. Trend e for generaed dscharge me: generaed, = 6365(Pn, )^

8 A. B. Djordjevc, D. M. Karanovc These values are very close o one anoher. By changng he coeffcen and exponen, Equaon (9a), he generaed me profle changes bu may no overlapped he dscharge npu daa profle (as well as he prevously generaed) me values,.e., he CDCA basc equaon s no obaned. In he nex sep, on he bass of he emprcal equaon ([], Equaon (7)), CDCA nroduces he rao of he adjacen pons of (a) he nal power of nernal ressance, Pn/VA (Lne ), (b) he rend e of npu me, /s (Lne ), and (c) he generaed me values, /s (Lne 3) and pus hese n relaon vs. he rao of Pn, /Pn, n logarhm form (Fgure 4): P n, P n, Power of Inernal Ressance : = R = Pn, Pn, P Inpu me : n, 6 = R = Pn, P Generaed me : n, = R = Pn, Thus, a ear relaon beween he npu and generaed me raos s obaned. However, he S profle of he dscharge npu me values s no acheved by changng he coeffcens n Equaon (0b). Because of hs fac, CDCA acceps, as a useful soluon, he raos of he adjacen raos (cross-rao) of boh power and me, wh he background ha he adjusable power operaor se generaes a me operaor se,.e., me profles ha sasfy he npu daa may be acheved. The operaor members, power and me, comprse he hree successve values [] [] found a he, and posons. Boh operaors are found a he poson. As he frs sep n me generaon, CDCA defnes he nal me operaor comprsng he followng: (a) Expermenal me from he solo common par (a ); (b) The me akes o se up (a ); and (c) The me value ha s calculaed (a ). The nex me operaor member conans wo known me values from he prevous (forward or backward drecon) operaor member, and he hrd me value s calculaed usng he known power operaor member. Ths algebrac operaon, sarng from he real solo common par s performed o he end as well as from he very begnnng of dscharge. The dscharge curve generaed by he nal power operaor se s based on he npu daa. Adjusmen proceeds by changng he coeffcens of he power operaor se (Equaon () unl he nex creron s sasfed: npu, generaed,, and (0a) (0b) (0c) = b < b < < R () Based on hs, he generaed me se may be acceped as he CDCA dscharge curve me vs volage nsead of he expermenal volage vs me. In accordance wh such a resul, CDCA acceps he four nex forms of he CDCA basc equaon: Fgure 4. Lnear dependence he raos of he adjacen members,.e.. The power of nernal ressance, (Pn, /Pn, ) = (Pn, /Pn, );. The rend e of he npu daa, n Fgure 3, (/ ) = 5.560[(Pn, /Pn, )]; 3. Trend e for generaed dscharge me: (/ ) = 7.089[(Pn, /Pn, )] 0.04 on he raos of he power of he nernal ressance adjacen members. 44

9 A. B. Djordjevc, D. M. Karanovc P n, 3 P n, a + b + c + d = ( a) = ( b) 3 n, A B C D P P n, n, n, n, P P = P P n, ( c) 3 a + b + c + d = 3 A + B + C + D ( d) In addon o he power operaor, CDCA nroduces he polynomal rao, Equaons (b), (d). CDCA also defnes generalzed equaons o nerrelae (dfference, sum, rao, ec.) he CDCA baery characerscs. P y P y y y = ( a) = ( b) y P y P y y n, n, P n, y P n, y n, n, y P y n, P n, P P = P P where y may be me and me-dependen parameers. Usng Equaon (a), he nex me calculaon equaons are obaned: Pn, Pn, P n, = Pn, Pn, Pn, P n, = Pn, Sarng from he nal expermenally known me, he me se may be generaed n he forward and backward drecons. In he forward drecon, he known values are and, whereas he hrd,, s calculaed. In he backward drecon, he known values are and, whereas he hrd,, would be calculaed. The backward generaon leads o he very begnnng of dscharge. If a backward generaon s performed from he fnal me, s he verfcaon of he forward generaed curve. The precse adjusng of he me operaor se (lef-hand sdes of Equaon ()) by changng he power operaor se (rgh-hand sdes of Equaon ()) leads o sasfyng he eary of npu daa o he generaed dscharge curve (Equaon ()). CDCM baery esng s exended and mproved regardng he common parameers for elecrochemcal sysems, dscharge modes, and regmes. 3.. Operaors n Ths Arcle In hs arcle, usng he npu daa, CDCA generaes 0 operaors (Fgure 5), 0 dscharge me curves (Fgure 6), and, n addon, 0 ses of me sub-nervals (Fgure 7). These resuls cover he followng: a Mono-cell baery LR0VARTA, 0 Ohm [], 5 operaor pars: Equaons (a)-(d), curves: - 4 forward and Equaon (a), -backward drecon; b Mul-cell MALLORY baeres, 4 baery desgns [], 0 dscharges, average cell, curves: 6 5 (Equaon (a)); c Mul-cell lead-acd baery, 6 cells, 5 consan curren loads, average cell (Equaon (a)), curves: 6-0, and Fgure. The dscharge curve can be shown eher as a se of pons,.e., curve, or as a se of columns placed on he power plane and along he volage, curren and power es. n, n, n, n, () c () (3) (4) (5) 45

10 A. B. Djordjevc, D. M. Karanovc Fgure 5. CDCA, me and power of nernal ressance operaors vs. n-se, mulpled by 0.375, LR0VARTA, 0 Ohm, Equaon (), e:. forward (a);. backward (a); 3. forward (b); 4. forward (c); 5. forward (d). MALLORY: consan ressve loads, Equaon (a), 9K6: Ohm; Ohm; Ohm, PX4: Ohm; Ohm, PX:. 50 Ohm;. 50 Ohm, 7K67:3. 00 Ohm; Ohm; Ohm. SONNENSCHEIN lead-acd baery, consan currens, Equaon (a): 6.. A, [F() = G(P)]x0.475, 7..4 A, [F() = G(P)]x0.45, A, F() = G(P)]x0.45, A [F() = G(P)]x0.4, 0.,0 A, [F() = G(P)]x0.375 Fgure 6. CDCA, dscharge mes vs. n-se, generaed wh operaors n Fgure 5, (Ohm/seconds): LR0- VARTA, 0 Ohm, Equaon (), e:-forward (a)/308,000s, -backward (a)/308,000s, 3-forward (b)/ 308,000s, 4-forward (c)/308,000s, 5-forward (d)/308,000s. MALLORY: consan ressve loads, Equaon (a), 9K6: Ohm/35,00s; Ohm/7,00s; Ohm/79,00s. PX4: Ohm/ 7,000s, Ohm/54,80s. PX:. 50 Ohm/7,000s;. 50 Ohm/54,80s. 7K67: Ohm/ 35,00s, Ohm/7,00s; Ohm/79,00s. SONNENSCHEIN, consan currens, Equaon (a): 6.. A/7,000s; 7..4 A/33,30s; A/4,400s; 9. 7.A/8780s; 0..0A/4600s. Fgure 7. Dscharge me sub-nervals vs. n-se, LR0VARTA 0 Ohm, curves:,, 3, 4, 5, MALLORY: 9K6, curve/ohm: 6/50, 7/00, 8/50, PX4, curve/ohm: 9/00, 0/60, PX, curve/ohm: /50, /50, 7K67, curve/ Ohm: 3/00, 4/00, 5/500, SONNESCHEIN, curve/curren: 6/.A, 7/.4A, 8/4.8A, 9/7.A, 0/A. 46

11 A. B. Djordjevc, D. M. Karanovc 4. CDCA Baery Characersc Currenly, many elecrochemcal sysems exs, and mos can be used relably n a wde range of applcaons. If he average cell represens a baery of one of he elecrochemcal sysems, he closed dscharge volage nerval of he cell s he vald abscssa for any baery desgn. In addon o volage, Uo > V Vfnal, and he overvolage se, 0 < η ηfnal, CDCM nroduces he N-n-se, n = Uo ( V) = N, as he common abscssa for every elecrochemcal sysem. The dscharge me and baery characerscs depend on he baery sae of charge, sae of healh, and dscharge condons. The CDCA algebrac equaons of baery characerscs, he (a) volage and curren capaces and (b) energy volumes, are based on he volage nerval and baery me response. The me response ncludes he dscharge me and me sub-nervals. Numercal summaons are performed across he nerval 0 < n. 4.. Tme Volage and Curren Surfaces-Capaces The overall volage capacy, S V, as volage losses: η, s Uo Vs, s dvded no wo pars: volage on load, Sv, U = V + η = V + V + η + η = V + η 0 V, and overvolage, (6) (7) S = V = V + V = U η V, I 0 (8) Sη = η = η+ η = U V, I 0 The overall curren capacy and pars are obaned by mulplyng Equaons (6)-(8) wh he recprocal of he consan ressve load, Rl/Ohm: 4.. OEV Energy Volumes (9) I = I + ν = I + I + ν + ν = I + ν 0 (30) Q = I = I + I = I ν I, 0 (3) Q = ν = ν + ν = I I I, 0 CDCA places he dscharge mes, s, as he parallelepped s aludes on he power plane o defne he energy s pars n he form of paralleleppeds: U I = V + η I + ν = P + P + P + P (3) ( )( ) 0 0 load, ex, n, CDCM, Overall on load curren volage CDCM losses. To defne he OEV pars, CDCA places dscharge me, 0 < s < n s.e., he sum of me sub-nervals, 0 < Σ s < Σ n s, along power es: on load, exernal, nernal and CDCM losses n Equaon (3) and calculaes volumes by summaon by pars: where U I = P + P + P + P + P + P 0 0 load, load, ex, ex, n, n, + P + P. CDCM CDCM, P = P + P ++ P (34) load, n, ex, CDCM, s (33) 47

12 A. B. Djordjevc, D. M. Karanovc 0 0 (35) U I = VI + η I + Vν + ην Usng Equaon (35), he OEV parallelepped s defned as he sum of he n horzonal layers ha conan he four elecrochemcal pars, and each of he pars has he same me sub-nerval: (36) E = E = VI + S I + V Q load, load, V, I, (37) n, = n, = η = E E I V Q E = E = Vν + S ν (38) ex, ex, V, (39) CDCM, = CDCM, = η ν E E The appled elecrochemcal model and me sub-nervals confgure he horzonal layers Dscharge Tme and Tme Sub-Inervals vs n-n-se Treang he dscharge me, Equaon (0), and me sub-nervals, Equaon (), as he range vs. N-n-se as he doman, CDCA defnes he followng relaons: = + ( a) = + b ( ) ( ) = + = + ( ) ( ) ( ) c d The elecrochemcal meanngs of hese equaons are obaned by mulplyng Equaon (40) by consan volage/curren seps ( V, η, I, ν), as well as by summaon, o defne capacy surfaces and energy volumes. Furher analyses of hese relaons are no addressed n hs arcle. The dscharge me and he me sub-nerval curves are shown n Fgure 6 and Fgure Baery Average Characerscs CDCA nroduces he baery average characerscs [] [] ha ransform he rregular surfaces and volumes of he real dscharge no he regular quadrlaeral form of he vrual dscharges, wh her exens unchanged. By mulplyng Equaon (6) wh ( ) =, he overall volage capacy s dvded no he average volage and overvolage parallelograms of he vrual dscharge: (4) U 0 = + V + η = Vavg, + ηavg, A any sae of dscharge, he average volage, Vavg, V, s he sragh e of he changng dscharge curve of S shape. Ths means ha he vrual dscharge wh he consan average volage produces he same volage capacy as he real dscharge by he dscharge curve of S shape. The average volages of dscharge n hs arcle are shown n Fgure 8. Mulplyng Equaon (4) wh he recprocal of load ressance, (Rl/Ohm), as well as mulplyng Equaon (9) wh ( ) =, he average dscharge curren and curren losses are obaned: I0 = I + ν = Iavg, + νavg, (40) (4) The acual power on load, VI VA and baery average power, ν avg, I, are shown n Fgure. The known avg, average volage and curren defne he average overvolage and average curren losses: η = U V ( a) ν = I I ( b) (43) avg, 0 avg, avg, 0 avg, 48

13 A. B. Djordjevc, D. M. Karanovc Fgure 8. Average volages, Vavg, V : LR0VARTA 0 Ohm, curves:,, 3, 4, 5, MALLORY:9K6, curve/ Ohm: 6/50, 7/00, 8/50, PX4, curve/ohm: 9/00, 0/60, PX, curve/ohm: /50, /50, 7K67, curve/ Ohm: 3/00, 4/00, 5/500, Sonneschen: curve/curren: 6/.A, 7/.4A, 8/4.8A, 9/7., 0/A Average cell volages, Vcell/V: e prmary baeres, e lead acd baery. In parallel wh he elecrochemcal parameers, he parameers n Equaons (4)-(43) dvde he baery power plane (Fgure ) no he four analogue average power values: on load, exernal, nernal and CDCM,.e. Mulplyng Equaon (35) wh ( ) P = V I ( a) P = η I ( b) avg,load, avg, avg, avg,n, avg, avg, P = V ν ( c) P = η ν ( d) avg,ex, avg, avg, avg,cdcm, avg, avg, =, he nex equaons are obaned: U I P P P P (45) 0 0 = load, + ex, + n, + CDCM, 0 0 avg,load, avg,ex, avg,n, avg,cdcm, (44) U I = P + P + P + P (46) A baery OEV s represened by he hree fgures: (a) he four elecrochemcal rgh paralleleppeds, Equaon (3); (b) he sum of < n horzonal me sub-nerval layers, Equaon (35); and (c) he sum of he four CDCA averaged paralleleppeds, Equaon (44). All of hese paralleleppeds change smulaneously and successvely due o he power values and he me sub-nervals changng. Combnng hese hree fgure usng algebrac soluons, CDCA enables recalculaon of he real dscharge no vrual dscharges n accordance wh Table n hs arcle CDCA 3D Curve, Surface and Space In addon, CDCA can represen he baery characerscs n a 3D char where he capacy surfaces and energy volumes are vsble. In hs arcle, he LR0VARTA, Mallory prmary baeres, and lead acd baery are shown n Fgures 9-, respecvely. In he creaed 3D chars, every enh alude of n = 30 s drawn. The LR0VARTA dscharge s presened by he fve me ses. The frs wo are ploed a he IPP a 0.00 A and 0.08 A consan curren es. The wo nex ses are ploed a he 0.0 and 0.8 V consan volage es. The ses beween boh pars, curren and volage, are no ploed due o vsbly of he char, bu hey are dscharge surfaces n an S shape. The ffh se s ploed a he power-me (overvolage-curren losses,.e., volage-curren) e. The fnal me aludes, n/s, cover one-quarer of he IPP, UI o o4. The energy on load volume s n he form of a runcaed pyramd wh he wo curved surfaces n he S shape. The dfferences beween he OEV and energy on load are energy losses (whch conss of he hree pars) and are posoned n he empy upper half of he parallelepped. These real dscharge volumes may be recalculaed for oher dscharge modes and regmes. 3D OEVs for he 0 MALLORY dscharges are placed a he fnal average currens as volage-me surfaces (Fgure 0). Dscharge aludes are sared a Uo V and are posoned sep by sep, changng overvolage and me, oward he 0.8 V and fnal me. Fnally, OEVs are formed as prsms wh bases ha are me-overvolage surfaces, and he hrd dmenson s he consan fnal average curren value. 49

14 A. B. Djordjevc, D. M. Karanovc Fgure 9. 3D fgures of dscharge me ses (every enh alude), Equaon (a), LR0VARTA, 0 Ohm, = s, repeaed along: (a) consan volage: 0.8 V and 0 V, (b) consan curren: 0.08 A and 0 A; fnal (c) dscharge power: UI > VI UI 4. o o o o Fgure 0. 3D fgures of dscharge me ses (every enh alude) placed a fnal average currens, Iavg, n/a:, Equaon (a), MALLORY: /34,74s, /7,0s, /79,00s. PX4: / 7,000s, /54,80s. PX: /7,000s, 0.098/54,000s, 7K67: /35,94s / 7,00s, /79,00s. Fgure. 3D fgures of dscharge mese, /s (every enh alude), Equaon (a), lead acd VRLAS onnenschen Dryf A 5/4.0, placed a consan dscharge currens: I/A:.,.4, 4.8, 7. and.0. 3D OEVs for fve lead-acd VRLA dscharges (Fgure ) are placed as wh he MALLORY dscharges,.e., he volage capacy surfaces are placed a he real consan dscharge currens,.e., he hrd dmensons of he formed prsms. For he 3D baery model, he necessary se of equaons s descrbed. The four followng equaons are recommended for creave and more dealed analyss of an acual OEV: 50

15 A. B. Djordjevc, D. M. Karanovc EVI, QII, = Sv, (a) Eη, QI, = Sη, E Q = S (d) E Q = S Vν, ν, ν, ην, ν, η, The presened CDCM approach o baery esng and 3D baery models may be furher mproved wh regard o he elecrochemcal model, operaor meanngs, and possble baery dscharge predcon. The mahemacal calculaons were performed usng Mcrosof Excel sofware. 5. Conclusons The calculaed dscharge curve mehod (CDCM) acceps ha a mono- and mul-cell baery (C/B) s a physcochemcal, hermodynamc, and elecrochemcal sysem n whch he energy delvered s he sum of he useful work and energy losses. C/B esng s conduced n defned condons, and he process s recorded as volage changes over me. CDCM defnes a baery average cell o represen a mul-cell baery and compares he average wh he mono-cell or sngle cell separaed from a seral and/or parallel connecon. The baery operang condons are classfed usng he load modes (ressor, curren, volage, power), regmes (self and devce drvng), and mng (connual and perodcal). The open-crcu volage and nal curren (defned by he ressve load) are he prncpal characerscs of a baery elecrochemcal sysem, and her nervals are dvded no N consan seps. Boh he volage and curren ses defne he baery nal power plane (IPP), whch comprses N smalles power parallelograms, whch are he bass of he 3D baery model,.e., he overall energy volume (OEV). The IPP comprses he power on load and powers of nernal, exernal and CDCM ressances ha appear a he dscharge sars. Durng baery dschargng, he power on load decreases, whereas he loss power ncreases,.e., dsrbued, sep by sep, beween he four pars of he IPP. The baery dscharge s recorded n he form volage vs. me, where me s he ndependen varable. CDCM was evolved from he Calculaed Dscharge Curve Algorhm (CDCA) as a mahemacal ool. CDCA defnes he basc equaon by whch dscharge daa from volage vs. me ransforms no me vs. volage. The resul s he ordered rple cardnal number-volage-me and s vald for all baery desgns and elecrochemcal sysems. Ths ordered rple defnes a 3D mahemacal model of OEV,.e., he baery reversble energy volume. All parameers ha defne and are necessary o analyze OEV reman n correspondence wh he nal ordered rple. The basc equaon s based on he hree-pon operaor se ha comprses he hree successve values of he power of nernal ressance, me, and me-dependen parameers. The operaor se member s placed as he las of he hree power and me values, and hey are n correspondence, one o one, wh all CDCA ses. The wo earler values of he operaor exs n he wo prevous power and me operaors. The basc equaon generaes he dscharge me se, whch overlaps he expermenal me values and s acceped by CDCM for characerzng he baery durng s exploaon. The dscharge me s dscharge duraon and comprses he se of me subnervals, whch are he duraon of he volage sep change. In accordance wh he S shape of any dscharge curve, he longes me sub-nerval les beween he shores nal and fnal me sub-nerval. Usng he megeneraon procedure, he 3D baery model s quanfed, and CDCA can calculae he baery sandard and new CDCM characerscs. For averagng capaces,.e., volage/curren-me surface o me and energy capaces, CDCA nroduces average volage, overvolage, curren and curren losses. The baery average parameers ransform he rregular forms of he OEV (capaces and volumes) no regular forms,.e., parallelograms and paralleleppeds. All baery ypes of he same elecrochemcal sysem may be presened on he common volage abscssa. The common abscssa for he baeres of all of he elecrochemcal sysems s a cardnal number se. The hree-pon operaors, ncludng me and me-dependen parameers, are he new baery characerscs, and hey are no examned n hs arcle. Examnaon of he nerrelaons of hese baery operaor nerrelaons may lead o new baery esng parameers. References (b) (d) (47) N parallelograms are [] Djordjevc, A.B. and Karanovc, D.M. (999) Cell Tesng by Calculaed Dscharge Curve Mehod. Journal of Power Sources, 83, hp://dx.do.org/0.06/s (99) [] Djordjevc, A.B. and Karanovc, D.M. (006) Baery Tesng by Calculaed Dscharge-Curve Mehod Consan Re- 5

16 A. B. Djordjevc, D. M. Karanovc ssve Load Algorhm. Journal of Power Sources, 6, hp://dx.do.org/0.06/j.jpowsour [3] Vnce, C.A. and Scrosa, B. (997) Modern Baeres. Arnold, London. [4] Bard, A.J. and Faulkner, R. (980) Elecrochemcal Mehods. John Wley & Sons, Inc., New York. [5] Akay, T.J. (980) Appled Numercal Mehods for Engneers. John Wley & Sons, Inc., New York. [6] Wu, M.-S., Ln, C.-Y., Wang, Y.-Y., Wan, C.-C. and Yang, C.R. (006) Elecrochmca Aca, 5, Nomenclaure Ls of symbols E: energy (VAs) Io: nal curren (A) I: curren (A) ΔI: consan curren sep (V) N: cardnal number se, 0 < < n < N Q: curren capacy (As) Rl: consan ressance (Ohm) S: volage capacy, Vs T: me (s) U : open crcu volage (V) o V: volage (V) ΔV: consan volage sep (V) x: common symbol y: common symbol η: overvolage (V) ν: curren losses (A) Subscrps avg: averagng wh me ex: exernal energy losses : dscharge progress seps n: fnal sep, < n< N n: nernal losses o: refer o nal sae 5

17