Definitions and reference values for battery systems in electrical power grids

Transcription

1 efiitios ad erece alues for battery systems i electrical power grids Hubert Rubebauer * ad Stefa Heiger Siemes AG, Freyeslebestraße, 9058 Erlage, Germay hair of Electrical Eergy Systems, Uiersity Erlage-Nuremberg, auerstraße 4, 9058 Erlage, Germay * orrespodig author: hubert.rubebauer@siemes.com Abstract: Sice more ad more large battery based eergy storage systems get itegrated i electrical power grids, it is ecessary to harmoize the wordig of the battery world ad of power system world, i order to reach a commo uderstadig. I this regard this article presets differet battery cotet alues ad their relatio to each other. tery operatios typically lead to a chage of the battery cotet. Based o a simplified battery model the basic alues ecessary to describe battery operatios are clarified. The the erece alues ad some acceptace criteria for batteries ad secodary cells are defied. Also alues describig limited usable eergy cotet caused by operatioal restrictios are proided. I order to be as close as possible to existig defiitios ad practical applicatios, care is take to be coform to curret stadards whereer possible i this article. The gie set of cosistet battery defiitios ca be used for a agreed desig of battery storage systems ad proides optios for battery performace criteria. Keywords: state of eergy, eergy storage capacity, usable eergy storage capacity, P-rate, costat power time, usable costat power time Highlights: - Harmoizatio of performace alues of battery systems for a better uderstadig betwee battery maufacturers ad power system itegrators - Presetatio of a suitable defiitio for battery eergy storage capacity ad desigatio of state of eergy (SOE) - efiitio of a appropriate erece (test) power alue ad explaatio of the term P-rate - Itroductio of the usable eergy storage capacity alue for descriptio of limited usable battery cotet caused by operatioal restrictios - larificatio of time alues regardig costat power battery chargig or dischargig

2 Abstract Sice more ad more large battery based eergy storage systems get itegrated i electrical power grids, it is ecessary to harmoize the wordig of the battery world ad of power system world, i order to reach a commo uderstadig. I this regard this article presets differet battery cotet alues ad their relatio to each other. tery operatios typically lead to a chage of the battery cotet. Based o a simplified battery model the basic alues ecessary to describe battery operatios are clarified. The the erece alues ad some acceptace criteria for batteries ad secodary cells are defied. Also alues describig limited usable eergy cotet caused by operatioal restrictios are proided. I order to be as close as possible to existig defiitios ad practical applicatios, care is take to be coform to curret stadards whereer possible i this article. The gie set of cosistet battery defiitios ca be used for a agreed desig of battery storage systems ad proides optios for battery performace criteria. Keywords: state of eergy, eergy storage capacity, usable eergy storage capacity, P-rate, costat power time, usable costat power time Itroductio Although batteries are a quite old ad pricipally well kow techology there is still ot always a commo uderstadig about characteristic ad erece alues of primary ad secodary cells, batteries ad battery systems. Especially sice huge battery systems get more ad more iterestig as statioary storage solutios for electrical power systems besides well kow alues like capacity i ampere-hours ad -rate also typical electrical alues like eergy ad power shall be proided by the system itegrator. Theore this article gies a oeriew about some characteristic ad erece alues of battery systems, primary ad secodary cells. tery model For all defiitios i this documet the simplified battery model with the circuit diagram i Figure is used ([], []). The battery ope-circuit oltage OV (t) describes the source oltage of the battery ( OV (t) > 0). (t) i Z eq, i ith battery curret ad battery oltage ( (t) 0) the battery power p (t) at the battery termials ca be deried p i. () I cosumer erece system the sig of the battery power also specifies, if the battery is charged (p (t) > 0) or discharged (p (t) < 0). Optioally the geerator erece system may be used (Figure, right side), i which the opposite sigs of battery curret ad battery power specify, if the battery is charged (i (t) < 0, p (t) < 0) or discharged (i (t) > 0, p (t) > 0). (t) i (t) Z eq, i ( ) OV t (t) i Figure : osumer ad geerator erece system Z eq, i (t) ( ) OV t (t) ( ) OV t After explaatio of differet battery cotet alues i the followig the basic alues which are importat, to describe the battery operatio are preseted. The battery erece alues ad possible acceptace criteria are proided. Fially alues for costricted battery operatig rages are characterized. Figure : Simplified battery model with ope circuit oltage The equialet, iteral impedace Z eq,i is typically specified by maufacturer ad summarizes all iteral resistaces of the battery. Besides ohmic parts also the diffusio resistace ad the charge-trasfer resistace hae a ifluece o the iteral resistace of a battery. The iteral resistace also depeds o battery s temperature ad state of charge. If for simplificatio a costat resistace with oly ohmic iflueces is take ito accout, the iteral impedace Z eq,i is equal to iteral ohmic battery resistace Z eq,i R i. The battery curret i (t) flows through the iteral battery resistace. I this article the idex sigalizes that the dedicated alue is preset at the battery termials. I o-load operatio (i (t)0) it follows (t) OV (t). As erece system of the battery curret i (t) the cosumer erece system (Figure, left side) is used i this article. Theore i (t) > 0 sigalizes battery chargig ad i (t) < 0 battery dischargig. The battery oltage (t) at the battery termials ca be calculated by ( OV eq, i t t) + Z i ( ). () tery cotet alues. Electric charge Oe mai characteristic of a battery is the ability to store electric charge. Theore electric charge t), which is usable for applicatios, is a importat alue. Stored electric charge t) The electric charge which a battery ca delier uder specified discharge coditios betwee its preset electric charge cotet ad its empty state is called electric charge. For electric charge which is expressed i ampere hours (Ah) the sig t) is used. Full state of a battery is the state of charge wherei the battery has bee completely charged i accordace with the maufacturer s recommeded chargig coditios (see also SO alue below). Accordigly empty state of a battery is usually defied by the battery supplier. ith q Start ad q Ed as electric charge at the begiig ad at the ed of a chargig or dischargig process the chage of electric charge ΔQ ca be calculated by:

3 qed Q dq (3) qstart I cosumer erece system a egatie sig of ΔQ sigalizes that at the ed of the time period the battery cotais less electric charge tha at the begiig ( discharged ); a positie sig of ΔQ sigalizes that at the ed of the time period the battery has got more electric charge tha at the begiig ( charged ). Furthermore battery curret i (t) (charge or discharge), the start time t Start ad the ed time t Ed of the curret flow ca be used to derie the chage of electric charge ΔQ ( oulomb outig ): ted Q i (4) I regard to oulomb outig also time alues ca be used to describe battery s electric charge cotet. See also costat curret discharge time ad costat curret charge time below. apacity The (actual) capacity of a battery is the electric charge which a fully charged cell or battery ca delier uder specified discharge coditios, betwee its full state ad its empty state. urig lifetime of a battery the capacity decreases i compariso to the capacity at begiig of life (BOL). Theore a idex ca be added to the capacity which specifies the battery agig. For example EOL expresses that the gie capacity is alid at the ed of life (EOL). As for electric charge the SI uit for capacity is coulomb (As) but i practice, capacity is usually expressed i ampere hours (Ah). tery ope-circuit oltage OV (q) or OV (t) The battery ope-circuit oltage OV (q) show i Figure is the termial oltage of a battery whe the battery curret is zero (accordig to [3]). Sice typically electric charge t) is a fuctio of time also battery ope-circuit oltage OV (t) ca be see as time depedet. For battery ope-circuit oltage, which geerally expresses the electrical potetial of the d.c. source i a battery circuit, also the terms source oltage ad electromotie force (EMF) ca be used. I Figure 3 battery ope-circuit oltage is show i depedece o the electric charge t) ad i depedece o SO respectiely. (The oltage alues V EO ad V EO i Figure 3 are explaied i sectio 3. ad rated capacity is gie i sectio 4.) tery ope-circuit oltage is usually defied at specified eirometal coditios (e.g. 5 ± 5 ). ue to agig the cure of ope-circuit oltage related to electric charge of the battery ca chage durig lifetime (see Figure 4). Figure 3: Typical battery ope-circuit oltage OV cure related to SO ad q at BOL State of charge SO State of charge SO of a battery is the amout of electric charge t) related to the actual capacity : electric charge t) SO SO (5) (actual)capacity ( 0 % SO 00% ) Full state (SO 00%) is the erece alue for electric charge t) ad meas t). Empty state (SO 0%) meas t) 0Ah. The other way roud electric charge of a battery ca be expressed by usig the SO alue: q ( SO) SO (6) Sice the alue of capacity chages durig lifetime due to battery agig, a idex of SO ca specify the capacity, which is the erece for SO alue. For example SO BOL meas that the capacity at begiig of life (BOL) is used as SO erece alue ( BOL ). SO EOL meas that the capacity at ed of life (EOL) is used as SO erece alue ( EOL ).. tery ope-circuit oltage Sice battery ope-circuit oltage OV (q) maily depeds o electric charge, the ope-circuit oltage of a battery is well suited to be used i defiitios of battery cotet. Howeer i reality slightly differet alues for OV (q) result at the same electric charge q depedig o whether the battery was charged or discharged before determiatio of OV (q). 3 Figure 4: Typical chage of battery ope-circuit oltage OV due to chage of capacity from BOL to EOL.3 Electrochemical eergy Stored (electrochemical) eergy E (q) or E (t) Stored eergy E (q) (of cells or batteries) is the electrochemical eergy which is curretly i the cell or battery erred to maufacturer s erece poit. Sice typically electric charge t) is a fuctio of time also eergy E (t) ca be see as time depedet. For specified coditios it ca be calculated by usig electric charge ad battery ope-circuit oltage OV (t) betwee its actual electric charge cotet ad its empty state (see Figure 5 ad Figure 6): SO) E ( q) ( q dq (7) SO 0%) OV )

4 Stored eergy is for high eergy applicatios ofte expressed i kilo watt hours (kh) or by usig SOE alue (see below). The adatage of this eergy defiitio deried from electric charge ad ope-circuit oltage oly is that the resultig eergy alue is idepedet of iteral resistace ad battery curret. For each electric charge alue a defiite eergy alue is gie - idepedet of preious or curret battery operatio. ith q Start ad q Ed as electric charge at the begiig ad at the ed of chargig or dischargig operatio the chage of eergy ΔE ca be calculated by qed E, OV ( q) dq qstart E,Ed de E,Start I the cosumer erece system a egatie sig of ΔE sigalizes that at the ed of the time period the battery has got less electrochemical eergy tha at the begiig (the battery was discharged); a positie sig of ΔE sigalizes that at the ed of the time period the battery has got more electrochemical eergy tha at the begiig (the battery was charged). (8) time t Ed of the curret flow ca be used to calculate the chage of eergy ΔE : ted E, OV () () t i t Usig the defiitio of eergy the eergy storage capacity ca be specified. Eergy (storage) capacity E Accordig to [4] the (actual) eergy storage capacity E is the amout of (electrochemical) eergy a cell or battery ca store ad delier, withi established desig limits ad maiteace iteral coditios. Eergy storage capacity of a cell or battery ca be calculated by usig (actual charge) capacity ad battery opecircuit oltage OV (t) betwee full ad empty state: SO 00%) SO 0%) ( q) E dq (0) OV Eergy storage capacity is usually expressed i kilo watt hours (kh). For eergy storage capacity also the terms eergy capacity, actual eergy capacity, actual maximum eergy cotet or (actual) electrochemical eergy capacity ca be used. The (actual) eergy storage capacity ca be lower tha the rated eergy storage capacity (see E i sectio 4) due to agig (e.g. see E at EOL i Figure 7 ad E at BOL i Figure ). Eergy storage capacity E, as well as eergy, caot be measured directly. It is a calculated alue. The adatage of upper defiitio of eergy storage capacity is that the resultig eergy alue is idepedet of battery curret ad iteral battery impedaces. Similar to the defiitio of SO a state of eergy (SOE) alue ca be calculated by usig electric eergy ad actual eergy storage capacity. (9) Figure 5: Stored eergy E (q) deried from ope-circuit oltage at BOL State of eergy SOE State of eergy SOE is accordig to state of charge SO the amout of eergy related to the actual eergy storage capacity: E SOE () E ( SO) E ( SO) SO) SO 0%) SOE () SO 00%) E SO 0%) OV OV ( q) dq ( q) dq Figure 6: Stored eergy E (t) deried from ope-circuit oltage at EOL Furthermore battery curret i (t) (charge or discharge), battery ope-circuit oltage OV (t), the start time t Start ad the ed 4 The state of eergy is expressed i percetage. Full state (SOE00% ad SO00%) is the erece alue for eergy E (t) ad meas E (t) E. So state of eergy is the degree to which a cell or battery has bee charged relatie to this erece poit. Empty state (SOE0% ad SO0%) meas E (t) 0 kh. Accordig to equatio (8) the chage of eergy ΔE ca also be calculated by usig the SOE start alue SOE Start ad the SOE ed alue SOE Ed of a chargig or dischargig process: E SOEEd ) E,Ed OV, ( q) dq de (3) SOEStart ) E,Sta rt

5 charge (Idex ) or discharge (Idex ). I cosumer erece system it holds i (t) > 0A ad i (t) < 0A. Figure 7: Eergy storage capacity E deried from ope-circuit oltage at EOL Similar to electric charge ad SO i (6) also eergy is usually expressed by usig SOE alue which relates eergy E to (actual) eergy storage capacity E: ( SOE) SOE E E (4) 3 tery operatig alues After characterizig charge ad eergy cotet alues the importat alues durig battery operatio, i.e. durig chargig or dischargig are preseted. At first battery curret ad its related characteristic alues expressig the chage of electric charge of a battery are explaied. 3. tery termial curret alues tery curret i (t) The battery curret (see Figure ) is electric curret deliered or cosumed by a battery at the battery termials durig its discharge or charge (accordig to [3]). The battery curret i (t) ca be expressed i amperes (A) or as fractios or multiples of a erece curret I. -rate -rate is the rate at which a battery is charged or discharged (see discharge rate i [3]). The -rate specifies the battery curret related to the battery erece curret I (see sectio 4, I > 0). i - Rate : (5) I Sice I is directly related to (see sectio 4) i the term -rate idicates that charge or discharge curret is gie i relatio to the rated capacity. For example i cosumer erece system charge with -rate of 0.5 meas that the battery curret is i (t) +0.5I with I > 0. A discharge with -rate of 0. meas that the battery curret i (t) is equal i (t) -0.I. Theore i (t) +I meas chargig with a -rate of. Otherwise a battery curret i (t) -I meas dischargig with a -rate of accordigly. Importat battery termial curret alues are: tery charge curret i (t) ad discharge curret i (t) The battery charge ad discharge curret are the electric currets cosumed or deliered by a battery at its termials durig its 5 ostat curret discharge mode Accordig to defiitio of costat power discharge i [5] costat curret discharge mode is the battery operatio i which the battery discharge curret is held costat ad where the power ad oltage freely adjust. For costat curret discharge mode also the expressio dischargig is used. I regard to oulomb outig the followig time alue for dischargig with costat curret is iterestig. ostat curret discharge times t The time how log a battery is discharged (Idex ) with costat curret (Idex ) is called costat curret discharge time t. ith t t Ed - t Start ad costat discharge curret i (t) -I cost the chage of electric charge ΔQ ca be calculated i accordace to equatio (4): ted Q i -I t (6) cost The costat discharge curret rate ca be added i the idex. For example t x meas that battery is discharged (Idex ) with a costat curret with a -rate of x (i.e. i (t) -xi ). To specify from which state of charge the dischargig process was started, the SO alue ca also be gie as a idex, e.g. t,so65%. As eeded, also the coditios durig dischargig (e.g. ambiet temperature, battery age or SO start or ed alue) ca be expressed i the idex of t. ostat curret charge mode Accordig to defiitio of costat power charge i [5] costat curret charge mode is the battery operatio i which the battery charge curret is held costat ad where the power ad oltage freely adjust. For costat curret charge mode also the expressio chargig is used. I this regard the followig time alue is iterestig. ostat curret charge times t The time duratio how log a battery is charged (Idex ) with costat curret (Idex ) is called costat curret charge time t. ith t t Ed - t Start ad with costat charge curret i (t) I cost the chage of electric charge ΔQ ca be calculated: ted Q i I t (7) cost The costat charge curret ca be added i the idex. For example t y meas that battery is charged (Idex ) with a costat curret (Idex ) with a -rate of y (i.e. i (t) yi ). To specify from which state of charge the chargig process starts, the SO alue ca also be gie as a idex, e.g. t,so40%. As eeded also the coditios durig chargig (e.g. ambiet temperature, battery age or SO start ad ed alue) ca be expressed i the idex of t. Maximum battery charge curret I,max ad maximum discharge curret I,max Maximum battery charge or discharge currets of the battery are the maximum charge or discharge currets, which are allowed oly for a short period of time (e.g. some secods) at the battery termials because of heatig reasos. Usually the maufacturer

6 specifies maximum battery charge or discharge currets for certai coditios ad time duratios. Maximum cotiuous battery charge curret I cot,,max ad discharge curret I cot,,max Maximum cotiuous battery charge ad discharge currets are the maximum allowed charge ad discharge currets of the battery, which the battery ca cosume ad delier cotiuously at certai coditios specified by maufacturer. If maximum cotiuous battery charge curret is applied cotiuously to the battery uder the specified ambiet coditios, the battery is typically kept i thermal balace. Theore for these alues the thermal coordiatio of the battery is decisie. Fiishig charge rate I,fiish The fiishig charge rate I,fiish (accordig to [3]) is the electric curret at which a battery is charged towards the ed of charge. This meas that battery s full state (SO00%) is reached, if (t) V EO (see sectio 3.) ad discharge curret is I,fiish (i.e. i (t) I,fiish ). Fiishig discharge rate I,fiish The fiishig discharge rate I,fiish (accordig to fiishig charge rate) is the electric curret at which a battery is discharged towards the ed of discharge. This meas that battery s empty state (SO0%) is reached, if (t) V EO (see sectio 3.) ad discharge curret is I,fiish (i.e. i (t) -I,fiish ). 3. tery termial oltage alues tery (termial) oltage (t) tery oltage (t) or battery termial oltage respectiely is the oltage which is preset betwee the battery termials. The battery termial oltage ( (t) 0) depeds o operatioal coditios of the battery (e.g. o-load or discharge). As depicted i Figure ad show i formula () the battery termial oltage (t) ca be calculated by usig ope-circuit oltage, battery curret i (t) ad iteral impedace Z eq,i. Importat battery termial oltage alues are: ostat oltage charge mode ostat oltage charge is the battery charge operatio i which the battery oltage is held costat ad where the power ad curret freely adjust. ( V chargig ) ostat oltage discharge mode ostat oltage discharge is the battery discharge operatio i which the battery oltage output is held costat ad where the power ad curret freely adjust. ( V dischargig ) tery charge oltage (t) ad battery discharge oltage (t) tery charge ad discharge oltages (accordig to [3]) are the oltages ( (t) > 0 ad (t) > 0) which are preset betwee the battery termials durig battery chargig (Idex ) ad dischargig (Idex ). ue to equatio () ad the differet curret flow directios battery charge oltage is typically higher ad battery discharge oltage is typically lower tha battery opecircuit oltage. tery ed-of-discharge oltage V EO tery ed-of-discharge oltage (accordig to [5]) is the specified battery termial oltage at which the battery discharge is termiated. For battery ed-of-discharge oltage (Idex EO ) the symbol V EO is used (V EO > 0). For ed-of-discharge oltage also the terms fial oltage, cutoff oltage ad ed-poit-oltage are applied. tery ed-ofdischarge oltage is assumed to be equal oer whole battery lifetime. The ed-of-discharge oltage V EO is typically declared by the maufacturer. The ed-of-discharge oltage may be used to iitiate the termiatio of the discharge process or to start with costat oltage discharge mode ( V dischargig ). I this mode the battery termial oltage (t) is held costat at V EO by reducig battery discharge curret i (t). The ed of dischargig is reached, whe battery discharge curret becomes lower tha fiishig discharge rate I,fiish. ostat curret ed-of-discharge times t mi,eo, ad t EO, Startig from a certai SO alue the miimum time how log the battery ca be discharged with costat curret till reachig ed-of-discharge oltage V EO is called miimum costat curret ed-of-discharge time t mi,eo,. Like for costat curret discharge time t the costat discharge curret ca be added i the idex. For example t mi,eo,x meas that battery is discharged with a costat curret with a -rate of x. Also the startig SO alue ca be gie as a idex, e.g. t mi,eo,so85%. I compariso to the miimum costat curret ed-of-discharge time, which is a importat alue regardig rated capacity (see sectio 4), the really measured time how log a battery is discharged with costat curret till ed-of-discharge oltage is reached, is called costat curret ed-of-discharge time t EO,. If a battery is discharged to ed-of-discharge oltage t mi,eo, t EO, is alid. tery ed-of-charge oltage V EO tery ed-of-charge oltage (accordig to [3]) is the specified oltage attaied at the ed of a chargig process (see also fiishig charge rate ). For battery ed-of-charge oltage the symbol V EO is used (V EO > 0). tery ed-of-charge oltage is assumed to be equal oer whole battery lifetime. Usually battery termial oltage is i the rage V EO (t) V EO. The ed-of-charge oltage V EO is typically declared by the maufacturer. The ed-of-charge oltage is used to iitiate the termiatio of the charge process or to start with costat oltage charge mode ( V chargig ). I this mode the battery termial oltage (t) is held costat at V EO by reducig battery charge curret i (t). The ed of chargig is reached, whe battery charge curret becomes lower tha fiishig charge rate I,fiish. ostat curret ed-of-charge times t mi,eo, ad t EO, Startig from a certai SO alue the miimum time duratio how log the battery ca be charged with costat curret till reachig ed-of-charge oltage V EO is called miimum costat curret ed-of-charge time t mi,eo,. Like for costat curret charge time t the costat charge curret ca be added i the idex. For example t mi,eo,y meas that battery is charged with a costat curret with a -rate of y. Also the 6

7 startig SO alue ca be gie as a idex, e.g. t mi,eo,so0%. The really measured time how log a battery is charged with costat curret till ed-of-charge oltage is reached, is called costat curret ed-of-charge time t EO,. If a battery is charged to ed-of-charge oltage t mi,eo, t EO, is alid. Full state, iitial discharge oltage V full,iitial, Full state, iitial discharge oltage (accordig to [3]) is the discharge oltage of a fully charged cell or battery at the begiig of the discharge with a certai discharge curret or power immediately after ay trasiets hae subsided (see Figure 8). For full state, iitial discharge oltage the symbol V full,iitial, is used (V full,iitial, > 0). Agai the discharge curret rate ca be added i the idex. So V full,iitial,x meas that battery is iitially discharged with a curret with a -rate of x (i.e. i (t) -xi ). Figure 8: Full state, iitial discharge oltage Empty state, iitial charge oltage V empty,iitial, Empty state, iitial charge oltage is the charge oltage of a empty (SO0%) cell or battery at the begiig of the charge with a certai charge curret or power immediately after ay trasiets hae subsided (see Figure 9). For empty state, iitial charge oltage the symbol V empty,iitial, is used (V empty,iitial, > 0). The charge curret rate ca also be added i the idex. Theore oltage V empty,iitial,y meas that battery is iitially charged with a curret with a -rate of y (i.e. i (t) yi ). 3.3 tery termial power alues Usig battery oltage (t) ad battery curret i (t) the battery power p (t) at battery termials ca be calculated as gie i equatio (). Especially i power system applicatios electrical power is a importat alue. tery (termial) power p (t) Similar to battery curret the battery power is electric power deliered or cosumed by a battery durig its discharge or charge at the battery termials. The sig of the battery power specifies if the battery power is a battery charge power or a battery discharge power. I cosumer erece system a egatie sig of battery power p (t) specifies battery discharge; a positie sig specifies battery charge. tery power is a actie power ad is expressed i watt or kilowatt (k). Furthermore battery power p (t) ca be expressed as fractios or multiples of a erece power P (see also P-rate). If iteral battery resistace is completely ohmic (Z eq,i R i ) the battery termial oltage (t) is, + R i. (8) OV i So battery power p (t) ca be calculated by p ( i, OV + Ri i ) i (9) P-rate P-rate is the power rate at which a battery is charged or discharged. The P-rate is the battery power at the battery termials related to battery erece power (see sectio 4, P > 0): p P - Rate : (0) P Sice P is directly related to (see sectio 4) P i the term P-rate idicates that charge or discharge power is gie i relatio to the rated capacity. For example i cosumer erece system battery chargig with a P-rate of 0.6P meas that the battery power p (t) is equal to p (t) +0.6 P. A discharge with P-rate of 0.3P meas that the battery power p (t) is equal to p (t) -0.3 P. Furthermore p (t) +P meas chargig with a P-rate of P. Accordigly battery power of p (t) -P meas dischargig with a P-rate of P. Importat battery power alues are: ostat battery power charge Accordig to [5] costat battery power charge is the battery operatio i which the charge power iput, i.e. the product of charge curret ad charge oltage, is held costat ad where battery curret ad battery oltage freely adjust. ( P chargig ) ostat battery power discharge Accordig to [5] costat battery power discharge is the battery operatio i which the discharge power output, i.e. the product of discharge curret ad discharge oltage, is held costat ad where battery curret ad battery oltage freely adjust. ( P dischargig ) Figure 9: Empty state, iitial charge oltage 7 tery charge power p (t) ad discharge power p (t) Similar to charge ad discharge curret the battery charge ad discharge power are the electric powers cosumed or deliered by a battery at its termials durig its charge (Idex ) or discharge (Idex ). I cosumer erece system it holds p (t) > 0 ad p (t) < 0.

8 Maximum battery charge power P,max ad maximum discharge power P,max Maximum battery charge or discharge powers of the battery are the maximum charge or discharge power alues, which are allowed oly for a short period of time (e.g. some secods) at the battery termials because of heatig reasos. Usually the maufacturer specifies maximum battery charge or discharge powers for certai coditios ad time duratios. More iterestig are the power alues which are related to more or less cotiuously battery operatio. For these power alues (depicted for example i Figure 8) both cotiuous battery currets ad battery oltage limits are decisie: Maximum cotiuous battery charge power at empty state P cot,,max,empty Maximum cotiuous battery charge power at empty state is the maximum charge power of the battery, which is cotiuously applicable at the battery termials startig from empty state. Typically maximum cotiuous battery charge power at empty state is gie by P cot,,max, empty I cot,,max V empty, iitial, () wherei I cot,,max is the maximum cotiuous battery charge curret ad V empty,iitial, is the empty state, iitial charge oltage erred to maximum cotiuous battery charge curret for a empty battery. Sice battery agig ad eirometal coditios hae got a decisie ifluece o maximum cotiuous battery charge power of a cell or a battery, the maufacturer typically proides alues for P cot,,max,empty at least for begiig of life (BOL) ad ed of life (EOL) at a ambiet temperature of 5. Maximum cotiuous battery charge power at full state P cot,,max,full Maximum charge power, with which the battery ca be cotiuously charged to full state (SO00%), is called maximum cotiuous battery charge power at full state. ith battery ed-of-charge oltage V EO ad fiishig charge curret rate I,fiish it ca be calculated by P, cot,,max, full I,fiish, VEO () P cot,,max,full ca also be see as fiishig charge power rate P,fiish. Maximum cotiuous battery charge power P cot,,max Maximum battery charge power, which ca be cotiuously applied at the battery termials, is the maximum cotiuous battery charge power. Typically it is gie by P, cot,,max I, cot,,max VEO (3) Thereby I cot,,max is the maximum cotiuous battery charge curret ad V EO is the battery ed-of-charge oltage. So P cot,,max,empty P cot,,max ad P cot,,max,full P cot,,max are alid. If maximum cotiuous battery charge power is applied cotiuously to the battery uder specified ambiet coditios, the battery temperature always stays withi allowed operatioal rage. But due to the oltage limit V EO charge power P cot,,max ca usually ot be applied cotiuously but oly for a short time. Maximum cotiuous battery discharge power at empty state P cot,,max,empty Maximum cotiuous battery discharge power at empty state is the maximum discharge power of the battery, which is cotiuously applicable at the battery termials till reachig empty state. Typically maximum cotiuous battery charge power at empty state is gie by P, cot,,maxempty, I,fiish, VEO (4) wherei I,fiish is the fiishig discharge curret ad V EO is the battery ed-of-discharge oltage of the cell or battery as declared by the maufacturer (V EO > 0). P cot,,max,empty ca also be see as fiishig discharge power rate P,fiish. Maximum cotiuous battery discharge power at full state P cot,,max,full Maximum battery discharge power at full state is the maximum allowed discharge power of the battery startig at SO00%. Typically maximum battery discharge power at full state is gie by P cot,,max, full I cot,, max Vfull,iitial, (5) I this regard I cot,,max is the maximum cotiuous battery discharge curret ad V full,iitial, is the iitial discharge oltage erred to maximum cotiuous battery discharge curret for a fully charged battery. tery agig ad eirometal coditios hae got a decisie ifluece o maximum battery discharge power of a cell or a battery. Theore the maufacturer shall proide alues for maximum battery discharge power at least for begiig of life (BOL) ad ed of life (EOL) at a ambiet temperature of 5. Maximum cotiuous battery discharge power P cot,,max Maximum cotiuous battery discharge power is the maximum discharge power of the battery, which ca be cotiuously applied at the battery termials. Typically maximum cotiuous battery discharge power P cot,,max is equal to maximum battery discharge power at full state P cot,max Pcot,max,, full, (6) If maximum cotiuous battery discharge power is applied cotiuously to the battery uder specified ambiet coditios, the battery temperature always stays withi allowed operatioal rage. But due to a decreasig ope-circuit oltage at battery dischargig the discharge power P cot,,max ca usually ot be applied cotiuously but oly for a short time. Usig battery termial power the battery termial eergy ca be deried. 3.4 tery termial eergy alues tery (termial) eergy or termial The battery termial eergy ca be calculated usig battery power p (t) or battery curret i (t) ad battery oltage (t) at the battery termials: p t) i ( t E ( (7) ) The sig of the battery termial eergy specifies, if the battery deliered or cosumed eergy at the battery termials i total oer the cosidered time period. I cosumer erece system a egatie sig of battery termial eergy specifies eergy deliery i total oer the cosidered time period; a positie sig characterizes eergy cosumptio i total. I case of costat battery power charge ( P chargig ) or discharge ( P dischargig ) also time alues ca be used to specify battery termial eergy. These time alues ca also iclude iformatio about operatig ad agig coditios. 8

9 ostat power discharge time t P, The costat power discharge time t P, specifies the time, how log the battery at certai coditios is discharged with costat battery power at the battery termials. I accordace to equatio (7) with costat discharge power p (t) -P cost ad with t P, the battery termial eergy ca be calculated: ted E p -P t (8) cost P, The costat discharge power ca be added i the idex. For example t P,x meas that battery is discharged (Idex ) with a costat power (Idex P ) with a P-rate of xp (i.e. p (t) -xp ). Also the specified coditios for t P, like SOE start ad SOE ed alue, ambiet temperature, battery age ad actual battery capacity ca be proided as idices. For example t P,0.7,SOE90%,EOL,T5 specifies the time how log the battery is discharged at ed of life (EOL) with a costat discharge power of 0.7P startig from a SOE alue of 90% at a ambiet temperature of 5. ostat power charge time t P, The costat power charge time t P, specifies the time, how log the battery at certai coditios is charged with costat battery power at the battery termials. I accordace to equatio (7) with costat charge power p (t) P cost ad with t P, the battery termial eergy ca be calculated: ted E p P t (9) cost P, The costat charge power ca be added i the idex. For example t P,z meas that battery is charged (Idex ) with costat power (Idex P ) with a P-rate of zp (i.e. p (t) zp ). Also the specified coditios like SOE start ad SOE ed alue, ambiet temperature, battery age ad actual battery capacity ca be proided as idices. For example t P,.,SOE0%,BOL,T30 specifies the time durig which the battery is charged at begiig of life (BOL) with a costat charge power of.p startig from a SOE alue of 0% at a ambiet temperature of 30. ostat power ed-of-discharge times t mi,eo,p ad t EO,P The ed of discharge depeds o V EO. The time alue t mi,eo,p is called miimum costat power ed-of-discharge time ad is the miimum time duratio how log the battery ca be discharged with costat power till reachig ed-of-discharge oltage V EO startig from a certai SOE alue. To specify from which state of eergy the chargig process is started, the SOE alue ca be gie as a idex, e.g. t EO,P,SOE70%. Also the P-rate of costat discharge power ca be added to the idex. The really measured time how log a battery is discharged with costat power till ed-of-discharge oltage is reached, is called costat power ed-of-discharge time t EO,P. If a battery is discharged to ed-of-discharge oltage t mi,eo,p t EO,P is alid. ostat power ed-of-charge times t mi,eo,p ad t EO,P The ed of charge depeds o V EO. The time alue t EO,P is called costat power ed-of-charge time ad is the miimum time duratio how log the battery ca be charged with costat power till reachig ed-of-charge oltage V EO startig from a certai SOE alue. To specify from which state of eergy the chargig process is started, the SOE alue ca be gie as a idex, e.g. t EO,P,SOE30%. Also the P-rate of costat charge power ca be added to the idex. The really measured time how log a battery is charged with costat power till ed-of-charge oltage is reached, is called costat power ed-of-charge time t EO,P. If a battery is charged to ed-of-charge oltage t mi,eo,p t EO,P is alid. By measurig battery termial eergy ad with usage of eergy alues differet efficiecy alues ca be deried. 3.5 tery termial efficiecy alues tery discharge efficiecy η tery discharge efficiecy (Idex ) at the battery termials ca be calculated by h E E - E ( 0 h ) loss, E - loss, (30) urig battery dischargig i cosumer erece system it holds < 0, ΔE < 0 ad ΔE. ith ad ted ted p, t), i ted ( () (), OV t i t E, discharge efficiecy ca be deried by h E ted ted OV i () t i () t (3) (3) (33) tery charge efficiecy η tery charge efficiecy (Idex ) at the battery termials ca be calculated by E E h (34) E + loss, ( 0 h ) urig battery chargig i cosumer erece system it holds > 0, ΔE > 0 ad ΔE ith ad ted ted p, t) ; ( i (35) ted () (), OV t i t E, (36) battery charge efficiecy ca be deried by 9

10 h E ; ted ted OV () t i () t ; i, (37) ad is the time base i hours for which the rated capacity is declared. If for example the rated capacity declared by maufacturer is 5 5Ah, the erece test curret is I I t /h 5 /h 5Ah /h 5A. tery roud trip efficiecy η RT For determiatio of battery roud trip efficiecy (Idex RT ) at the battery termials it is importat that after discharge ad charge operatio (or charge ad discharge operatio) the ed alue of SOE (or E, SO or t) respectiely) is the same as the alue at begiig of battery chargig or dischargig operatio. The the battery roud trip efficiecy ca be calculated by h RT h h E E E ( 0 h ) RT E - + loss, loss, (38) Sice losses mostly deped o curret flow battery roud trip efficiecy η RT depeds o dischargig ad chargig curret. Theore the charge ad discharge currets (or charge ad discharge powers respectiely) should be proided together with battery roud trip efficiecy η RT. Furthermore also the chage of electric charge t), eergy E, SOE or SO ( depth of discharge (O)) is decisie for battery roud trip efficiecy η RT. Theore the maximum ad miimum alue of SO, SOE, t) or E should also be gie with battery roud trip efficiecy η RT. 4 tery erece alues Rated apacity As erece alue for capacity the rated capacity is used [6]. For rated capacity also the terms omial capacity or istalled capacity ca be foud. The rated capacity is the quatity of electricity i Ah (ampere-hours) declared by the maufacturer which a sigle cell or battery ca delier durig a hour period whe chargig, storig ad dischargig with costat battery curret / I uder specified coditios (see below rated capacity erificatio test ). For example for lithium-io batteries typically 5 (hours) is used which idicates a rated capacity 5. Ofte rated capacity is related to begiig of life (BOL) of a battery. This meas that at BOL ad SO00% (fully charged) the capacity is equal to the rated capacity. Referece (test) curret I or I t Accordig to [7] the erece (test) curret is specified as I t I. (39) h Therei I or I t respectiely is the erece (test) curret (I > 0 ad I t > 0) i amperes, is the rated capacity of the cell or battery as declared by the maufacturer ( > 0) i ampere-hours Rated costat curret ed-of-discharge time t mi,eo,/ or The rated costat curret ed-of-discharge time t mi,eo,/ is equal to the dimesioless alue (see [6]), which is the time base i hours (h) for which the rated capacity is declared. The rated costat curret ed-of-discharge time t mi,eo,/ or respectiely h specifies the miimum time, how log a battery, which is fully charged accordig to a gie charge procedure, ca be discharged with costat battery curret / I at room temperature (5 ± 5 ) ad other coditios specified by maufacturer (see below rated capacity erificatio test ). For differet discharge rate type cells or batteries differet rated time bases exist for declaratio of rated capacity (see [6]). For really resultig costat curret ed-of-discharge time t EO,/ it holds t, teo, ³ tmi, EO, / h (40) Rated capacity erificatio test The rated capacity erificatio test is used to proof that uder specified coditios the battery or cell has got at least the rated electric charge cotet. Accordig to [6] the rated capacity erificatio test shall be performed as follows:. Prior to chargig, the cell or battery shall be discharged at 5 ± 5 at a costat curret of / I, dow to a specified battery ed-of-discharge oltage V EO.. The empty cells or batteries shall be fully charged i a ambiet temperature of 5 ± 5, usig the followig method: a. hargig at costat curret of / I up to a specified battery ed-of-charge oltage V EO b. hargig at costat ed-of-charge oltage V EO dow to a fiishig charge rate I,fiish. 3. The cell or battery shall be i a ambiet temperature of 5 ± 5, for ot less tha h ad ot more tha 4 h. 4. I the same ambiet temperature the cell or battery shall the be discharged at a costat curret of / I to battery edof-discharge oltage V EO (see Figure 0). The electric charge ΔQ, deliered durig dischargig, shall ot be less tha the rated capacity, i.e. for the costat curret discharge time t / till ed-ofdischarge oltage is reached it is alid t / h ad t / t mi,eo,/. Table : Rated capacity erificatio test hargig procedure for test purposes Idle time ischarge performace with costat curret I this regard determies the fractio or multiple of I. For example, if 5, the the discharge curret used to erify the rated capacity shall be -0. I. If, the discharge curret used to erify the rated capacity shall be.0 I. osiderig I > 0 the followig formula ca be used for calculatio of the chage of electric charge ΔQ durig 0

11 discharge i rated capacity erificatio test. ith costat curret discharge time t till ed-of-discharge oltage is reached it results Q t t full 0 I æ I ö ç - è ø t ³ I ( t ³ t h (- I ), mi, EO, / ) h I t, t full 0 h (4) If rated capacity is ideally determied, it is t t mi,eo,/ h. That is Q q 0 SO 0%) q dq dq SO 00%) (4) ΔE t t full 0 ³ OV, BOL () t (- I ) OV BOL () t,, t t (44) t full 0 mi, EO, / (- I ) OV, BOL () t E tfull 0 ( t ³ t h, mi, EO, / ) æ I ç - è ö ø If rated eergy storage capacity is ideally determied, it is t t mi,eo,/ h ad ΔE E. Figure 0: Verificatio of rated capacity specified for begiig of life (BOL) Rated eergy storage capacity E The erece alue rated eergy storage capacity E is a calculated alue. It caot be measured directly. The rated eergy storage capacity of a cell or battery betwee full ad empty state is deried by usig rated (charge) capacity ad battery opecircuit oltage OV (t) at coditios specified by the maufacturer: E SO 00%) SO 0%) q ( q) dq ( q), dq (43) OV q 0 OV Rated eergy storage capacity is a eergy alue ad usually expressed i kilo watt hours. For rated eergy storage capacity also the terms rated eergy capacity, rated maximum eergy cotet, rated electrochemical eergy capacity, omial eergy capacity or istalled eergy capacity ca be foud. Similar to rated capacity the rated eergy storage capacity is usually related to begiig of life (BOL) of a battery. This meas that at BOL ad SOE00% (fully charged) the eergy storage capacity E is equal to the rated eergy storage capacity (see Figure ). For rated capacity erificatio test the followig equatio ca be used for calculatio of the chage of eergy ΔE at begiig of life (BOL). ith costat curret discharge time t till ed-of-discharge oltage is reached it results Figure : Rated eergy storage capacity E deried from opecircuit oltage at BOL Nomial battery oltage V omial Nomial oltage (accordig to [3]) is a suitable approximate alue of the oltage used to desigate or idetify a cell, a battery or a electrochemical system. The omial oltage is typically specified by the maufacturer. The followig formula ca be used to determie a reasoable alue for omial battery oltage E V omial (45) Referece (test) power alues P ad P, The erece power (P > 0) shall be specified as P V, I (46) EO with I as erece (test) curret (I > 0) ad V EO as edof-discharge oltage as declared by the maufacturer (V EO > 0). So if for example the maufacturer declares ed-of-discharge oltage as V EO 650V ad the erece curret as I 60A, the erece power is P 39k. The erece power P is itetioally defied by usig the edof-discharge oltage V EO ad ot by usig omial battery oltage V omial. By this defiitio a kid of erificatio test discharge power P,, with which SO0% ca be reached, ca be defied as P P, V EO I (47)

12 Thereby is the time base for which the rated capacity is declared. Typically it holds P, P cot,,max,empty ad I,fiish / I (see Figure 8). Two other characteristic performace alues are the followig chargig or dischargig times kow from uiterruptible power supplies (UPS). Stored eergy time t E, Stored eergy time (accordig to [8]) is the miimum time durig which a battery, uder specified serice coditios, esures cotiuity of load power. So t E, is the miimum time how log a battery with a certai eergy alue ca be discharged with costat power at the battery termials. Typically it holds t E, t mi,eo,p. Thereby the battery is assumed sufficietly charged accordig to re eergy time (see below). Accordig to [8] the erece temperature of the battery is usually 5. Additioally the temperature of the battery ca be measured i order to derie a ecessary adjustmet to the expected eergy time. urig eergy time test ed-of-discharge oltage shall ot fall below the specified alue before eergy time t E, has elapsed. Stored eergy times ca be proided with a idex which may iclude the costat battery termial power as well as iformatio about dischargig ad battery agig. For example the time alue t E,,x,BOL specifies that at begiig of life (BOL) the discharge with a costat discharge power p (t) -xp betwee full charge ad reachig V EO lasts at least t E,,x,BOL. Re eergy time t E,re Re eergy time t E,re (accordig to [8]) is the maximum time required to, uder ormal mode of operatio ad with the chargig capacity istalled, recharge the battery so that eergy time ca agai be achieed. I this regard also oe costat power chargig methods may be applied, to reach the ecessary charge alue or eergy alue respectiely, e.g. V chargig. So usually it holds t E,re t mi,eo,p. Rated battery discharge efficiecy η, Rated battery discharge efficiecy is the discharge efficiecy for dischargig the battery with i (t) -/ I for rated costat curret ed-of-discharge time t mi,eo,/ h startig from SO 00% h, E t full 0 t full 0 tmi, EO, / tmi, EO, / (- I ) (- I ) OV, BOL tmi, EO, / t full 0 t full 0 tmi, EO, / OV, BOL (48) Typically rated battery discharge efficiecy η, is determied at begiig of life (BOL) ad for certai coditios specified by battery maufacturer. So rated battery discharge efficiecy ca be determied durig rated capacity erificatio test ad may be used as battery acceptace criterio. If rated capacity is properly defied by the maufacturer, at begiig of life the real costat curret ed-of-discharge time t EO, is equal or higher tha t mi,eo,/. The ed alue for itegrals i equatio (48) is always t mi,eo,/. If rated eergy storage capacity is ideally determied, it is t EO, t mi,eo,/ h. Tha it holds ΔE E ad tmi, EO, p t full 0 h, (49) E Assumig that the battery termial oltage (t) is oly depedig o ope-circuit oltage OV (t) ad o the oltage drop oer the iteral battery resistace R i, it is, + R i (50) OV i Thereby p (t) ca be calculated by p i + R i ( ). (5) OV i t ith i (t) -/ I it results tmi, EO, / tmi, EO, / t full 0 æ I ö ç - è ø t full 0 E - By that it is alid h, E OV tmi, EO, / t full 0 OV R I i i t + R i i é ù ê æ I ö æ I ö ú ê OV ç Ri ç ú ê ø è ø ú ê4444 è ú ë < 0 ³ 0 û E Ri I - t E æ I - R ç i è - mi, EO, / R I mi, EO, / i E t ö ø tmi, EO, / t full 0 mi, EO, / (53) (5) If rated eergy storage capacity is ideally determied, it holds at begiig of life ad at specified coditios h Ri, BOL I h, - (54) E 5 haracteristic alues for costricted operatig rages The height of battery curret ad battery power is decisie for the SO alue to which a battery ca be charged or discharged. Theore operatioal alues limit the SO rage i which a certai costat curret or power ca be applied o the battery. The electric charge which is aailable betwee these SO limits is the usable capacity use at certai operatioal coditios. There are differet alues which describe the costricted operatig rages of batteries.

13 Usable capacity regardig costat battery curret use, The usable capacity use, is the capacity of a cell or a battery which ca be used at certai operatioal coditios with certai charge or discharge currets. Like capacity the usable capacity use, is specified at certai operatioal coditios. The (actual) capacity of a battery is always equal or higher tha the usable capacity. ue to possibly higher battery discharge currets tha i (t) -/ I the ed-of-discharge oltage ca be reached at a higher SO alue tha 0%. This is depicted i Figure, i which the three blue cures show the termial oltage (t), if the battery is discharged with three differet battery currets i (t) startig from a fully charged battery. The graphs are alid for begiig of life ad it holds /<x3<x<x. If oly the discharge rate is gie i the idex of usable capacity use,, e.g. for discharge ad x for the -rate, the dischargig to determie the usable capacity use, x always starts from SO00% (see Figure ). Besides discharge rate also further specific battery discharge coditios like battery agig alue ca be added i the idex of usable capacity use,. Similarly also at a higher charge rate the ed-of-charge oltage is reached at lower SO alues tha at lower charge currets. I Figure 3 the three gree cures show the termial oltage (t), if the battery is charged with three differet battery currets i (t) startig from a empty battery. The graphs are alid for begiig of life ad it holds /<y3<y<y. If oly the charge rate is gie i the idex of usable capacity use,, e.g. for charge ad y for the -rate, the chargig to determie the usable capacity use,y always starts from SO0% (see Figure 3). Besides charge rate also further specific battery charge coditios like ambiet temperature ca be added i the idex of usable capacity use,. Figure ad Figure 3 ca be combied to a set of cures as depicted i Figure 4, from which the usable capacity use, ca be deried for differet costat charge ad discharge currets. For example i Figure 4 both, a certai discharge rate (e.g. x ) as well as a certai charge rate (e.g. y3 ), are used to specify the usable capacity use,y3,x. The resultig usable SO rage is typically smaller tha 00%. Figure 3: Reductio of usable capacity use by higher charge currets at BOL Figure 4: Set of cures for determiatio of usable capacity use for differet costat charge ad discharge currets at begiig of life (BOL) Figure : Reductio of usable capacity use by higher discharge currets at BOL Sets of cures as depicted i Figure 4 shall be proided by the maufacturer for dischargig currets i the rage I, max i (t) -/ I, ad for chargig currets i the rage / I, i (t) I, max. Figure 5: Set of cures for determiatio of usable capacity use for differet costat charge ad discharge currets at ed of life (EOL) 3

14 Sice battery agig has a decisie ifluece o the usable capacity (see Figure 4 ad Figure 5) the set of cures for usable capacity determiatio are typically proided at least for begiig of life (BOL) ad for ed of life (EOL). The adatage of this defiitio of usable capacity is, that a kid of usable capacity erificatio test ca be performed similar to the capacity erificatio test. Also time alues with costat charge or discharge curret ca be used to specify the (miimum) usable capacity of a battery. Usable costat charge ad discharge curret times Additioal to costat curret ed-of-discharge times t mi,eo,, t EO, ad costat curret ed-of-charge times t mi,eo,, t EO,, which oly cosider the lower or the upper SO limit, further usable costat curret times ca be gie for a certai usable capacity. For example regardig the usable capacity use,y3,x,bol show i Figure 4 the usable costat curret times t mi,use,y3,bol ad t mi,use,x,bol ca be specified. The usable costat curret time t mi,use,y3,bol defies the miimum time for chargig from miimum SO to maximum SO of usable capacity use,y3,x,bol with a costat charge curret of i (t) y3i at begiig of battery life. Accordigly the usable costat curret time t mi,use,x,bol meas the miimum time for dischargig from maximum SO to miimum SO of usable capacity use,y3,x,bol with a costat discharge curret of i (t) -xi at begiig of battery life. Usable costat curret times deped o battery age ad are defied for certai eirometal coditios specified by the maufacturer. Similar to the defiitio of usable capacity regardig costat battery curret the usable capacity regardig costat battery power ca be defied. Usable capacity regardig costat battery power use,p The usable capacity is the capacity of a cell or a battery which ca be used uder certai operatioal coditios with costat charge or discharge power. ue to possibly higher battery discharge powers tha p (t) -/ P the ed-of-discharge oltage ca be reached at a higher SO alue tha 0% (or at higher SOE alue tha 0% respectiely). Similarly at a higher charge rate the edof-charge oltage is reached at lower SO alues (or SOE alues respectiely) tha at lower charge powers. Furthermore for high charge rates the miimum SO alue, at which this high charge power is possible, ca be higher tha SO0% (see z4p i Figure 8). By usig the sets of cures as depicted i Figure 4 or Figure 5 eelopes for charge ad discharge operatios ca be deried as depicted i Figure 6 ad Figure 7 for BOL ad EOL. As show i Figure 8 ad Figure 9 these eelopes ca be used to determie the usable capacity use regardig costat battery charge ad discharge powers. For example the usable charge capacity use,p,z4,x specifies the usable capacity with costat discharge power xp ad costat charge power z4p. So x specifies dischargig with a costat power with a P-rate of x (p (t) -x P ) ad z4 specifies chargig with costat power with a P-rate of z4 (p (t) z4 P ). Agai the set of cures for usable capacity are typically proided at least for begiig of life (BOL) ad for ed of life (EOL) (see Figure 8 ad Figure 9). Figure 6: Eelope of allowed battery charge ad discharge powers at BOL I cot,,max I / yi y3i harge q [Ah] 00% EOL SO EOL -P / -I / -x3p -xp -xp p [] -x3i -xi -I cot,,max -xi yi ischarge Figure 7: Eelope of allowed battery charge ad discharge powers at EOL Figure 8: etermiatio of usable capacity regardig costat powers at BOL 4

15 Figure 9: etermiatio of usable capacity regardig costat powers at EOL Usable costat charge ad discharge power times For a costat power discharge startig from full state the miimum discharge times ( eergy times) ca be added i Figure 6 ad Figure 7 as depicted i Figure 0 ad Figure. Furthermore for SO rages at which a certai costat power charge is possible, the miimum charge times ( re eergy times) are put i Figure 0 ad Figure. It should be oted that it is ot possible to directly draw coclusios from the depicted SO rages to the actual costat charge ad discharge power times. These time alues are typically proided by the battery maufacturer regardig a certai SO rage, a specified battery age ad gie ambiet coditios. Usable costat power times ca be specified for usable capacities additioally. For example for the usable capacity use,p,z4,x,bol show i Figure 8 the usable costat power times t mi,use,p,z4,bol ad t mi,use,p,x,bol ca be specified (see Figure ). The usable costat power time t mi,use,p,x,bol meas the miimum time for dischargig from maximum SO to miimum SO of usable capacity use,p,z4,x,bol with a costat discharge power of p (t) -xp at begiig of battery life. The time duratio t mi,use,p,x,bol ca also be see as usable eergy time t E,,use,x,BOL. Accordigly the usable costat charge power time t mi,use,p,z3,bol defies the miimum time for chargig from miimum SO to maximum SO of usable capacity use,p,z4,x,bol with a costat charge power of p (t) z4p at begiig of life. Sice the battery is always charged with costat power, to restore the maximum SO alue of the usable capacity use,p,z4,x,bol the time duratio t mi,use,p,z4,bol ca also be see as usable re eergy time t E,re,use,z4,BOL. ith reductio of the usable capacity due to agig also the usable costat power times are reduced (see use,p,z4,x,eol i Figure 3). The time alues t mi,use,p,x,bol ad t mi,use,p,z4,bol shall be proided by battery maufacturer. I regard to UPS applicatios usable costat power times ad usable (re) eergy times ca be specified by the maufacturer for a certai battery age ad for certai eirometal coditios. Figure 0: Stored ad re eergy times for differet costat powers at BOL Figure : Stored ad re eergy times for differet costat powers at EOL Figure : Stored ad re eergy times for a certai usable capacity at BOL 5

16 Figure 3: Stored ad re eergy times for a certai usable capacity at EOL Usable eergy storage capacity E use The usable eergy storage capacity (or usable eergy capacity ) is the eergy storage capacity of a cell or a battery which ca be used uder certai operatioal coditios. For usable eergy storage capacity the sig E use shall be used. Equatio SOmax ) E use OV ( q) dq (55) SOmi ) ca be used to calculate usable eergy storage capacity E use (see Figure 4). Figure 5: Eelope of allowed battery charge ad discharge powers at BOL erred to SOE with usable eergy storage capacity As with usable capacity, also usable eergy storage capacity ca be specified regardig costat battery curret E use, ad regardig costat battery power E use,p. Similarly the usable eergy storage capacity at certai operatioal coditios ca be specified by the related miimum ad maximum SOE limits. The (actual) eergy storage capacity is always equal or higher tha the usable eergy storage capacity. Besides operatioal coditios also battery agig ad eirometal coditios hae got a decisie ifluece o usable eergy storage capacity of a cell or a battery. V EO V EO [V] 0 SO mi,use,bol E use,p,z4,x,bol use,p,z4,x,bol z4p OV,BOL -xp q [Ah] SO BOL SO max,use,bol Figure 4: etermiatio of usable eergy storage capacity regardig costat powers at BOL Additioally with formula SOE ( SO) E E SO) Q( SO 0%) Q( SO 00%) OV SO 0%) OV ( q) dq ( q) dq (56) ad from usable capacity use the usable eergy storage capacity ad its depedecy o SOE ca be deried (e.g. see Figure 8 ad Figure 5). 6 oclusios A cosistet set of characteristic battery alues is preseted i this article. Ee if there are some simplificatios ad some ideal assumptios the defiitios ca built a good base to achiee a commo uderstadig regardig battery performace alues for battery maufacturers ad itegrators of battery storage solutios i electrical power systems. Especially the preseted power ad eergy alues as well as the clarificatios regardig usable eergy storage capacity ad usable costat power times support the comprehesio of eergy proiders regardig performace capability of batteries. I geeral the alues characterized i this article are well suited to be used for the desig of battery storage systems ad as criteria to examie the performace of batteries ad secodary cells. I additio to the preseted set of battery alues further distictie battery alues applicable for determiatio of battery performace ca be foud i literature ad for practical applicatios. Regardig battery agig the terms state of health (SOH) ad equialet cycle are importat alues ot treated i this article (see [9]). Also the measuremet of iteral a.c. ad d.c. resistace as described i [6] ca be used as acceptace criterio of batteries. REFERENES [] Shig-Lih u, Hug-heg he, Shuo-Rog hou, Fast Estimatio of State of harge for Lithium-Io teries, Eergies Joural, 04, 7,