LIFE PREDICTION OF SATELLITE LITHIUM BATTERY BASED ON MULTI TIME-SCALE EXTENDED KALMAN FILTER (EKF) ALGORITHM

Transcription

1 U.P.B. Sci. Bu., Series C, Vo. 80, Iss. 4, 2018 ISSN LIFE PREDICION OF SAELLIE LIHIUM BAERY BASED ON MULI IME-SCALE EXENDED KALMAN FILER (EKF) ALGORIHM Qian-Qian LIU 1, Bing CHEN 2, Jingyuan ZHANG 3 State-of-Charge (SOC) and the maximum avaiabe caacity estimation are the most imortant arts of sateite ithium battery ife rediction. However, the arameters of the maximum avaiabe caacity change sowy comared with the fast time-varying SOC. his aer rooses a muti time-scae of Extended Kaman Fiter (EKF) agorithm on SOC and the maximum avaiabe caacity for estimation on different time scaes. he estimated vaue of SOC is used as an observation on the macro time-scae to udate the maximum avaiabe ower. he simuation resuts of exeriment on NCA/C sace borne ithium battery show that SOC and the maximum avaiabe caacity estimation from the roosed muti time-scae EKF agorithm has higher accuracy and comutationa efficiency comared with Dua EKF. Keywords: Sateite ithium battery SOC, the maximum avaiabe caacity, the venin circuit mode, Muti time-scae Extended Kaman Fiter (EKF) agorithm 1. Introduction Lithium batteries are critica to the sateite ower distribution systems and are graduay reacing traditiona batteries as the third-generation sateites with energy storage [1].Owing to charge, discharge management and erformance recession of ithium battery [2], its woring state monitoring, erformance degradation and residua ife rediction (RUL) have become the ey in the fied of sateite system faut rediction and heath management (PHM) research. he maximum avaiabe caacity of battery is often used as degradation characteristics of battery ife. Precise SOC estimation can not ony be used to assess the reiabiity of equiment but refect the residua service ife of the battery. In addition to the traditiona oen circuit votage method and the amere-hour integra method, iterature [3] [4] [5] anayzed Kaman fiter method aied to sateite ithium battery SOC estimation in detai. In Literature [6], the EKF agorithm was imroved. he Sigma oint set is constructed by using the state quantity and variance matrix. he Kaman fiter agorithm based on Sigma oint can achieve 1 Coege of Eectronic Engineering, Nava University of Engineering,WuHan,430033, China; e-mai: iuhe1984@163.com 2 Coege of Eectronic Engineering, Nava University of Engineering,WuHan,430033, China; 3 Coege of Weaonry Engineering, Nava University of Engineering,WuHan,430033, China;

2 68 Qian-Qian Liu, Bing Chen, Jingyuan Zhang better accuracy. In iterature [7], the Kaman fitering agorithm was combined with the amere-hour method. In iterature [8], unscented Kaman fiter was used for ithium battery SOC rediction. he dua EKF estimation agorithm ut forward in [9] can mae rea-time SOC and maximum avaiabe caacity with votage and current measurements with noise. he accuracy of maximum avaiabe caacity estimation obtained by using state and arameter estimation technoogy is oor, which has two reasons. One is that the votage of the battery is the ony measurement, but the connection between the maximum avaiabe caacity and the battery votage is very wea; additionay, due to the strong correation between SOC and the maximum avaiabe caacity, inaccurate maximum avaiabe caacity estimation woud further ead to inaccurate SOC estimation, and vice versa. In terms of comutationa efficiency, maximum avaiabe caacity is the sow-time variabe of indicator system of heath (SOH) state [12], [13]. If the maximum avaiabe caacity and fast-time variabe SOC are oerated on the same time scae, which woud ead to high comutationa comexity. In order to sove these difficuties, this aer rooses a muti timescae EKF agorithm to estimate the SOC and maximum avaiabe caacity resectivey. Contribution mainy incudes: (1) Muti time-scae SOC and the maximum avaiabe caacity of time scae searation estimation agorithm are roosed; (2) he estimated SOC is used to udate the rediction vaue of maximum avaiabe caacity dynamicay. As a technoogy of amere-hour method and EKF fitering, this agorithm reaized higher recision and efficiency than Dua EKF. 2. Sateite ithium battery system descrition 2.1hevenin battery mode he equivaent circuit mode of sateite ithium battery is Rint mode, RC mode; hevenin mode and PNGV mode [4] and so on. hevenin mode [9] considered the mutation and gradua change of battery votage under the excitation of current, the structure is shown in Fig.1.he mode arameters described as foows. Uoc is the oen circuit votage (OCV) of battery; Rt is used to describe the charge accumuation and dissiation of battery ohm resistance in doube eectric ayers; R describes the battery oarization resistance, C describes the battery oarization caacitance. he RC networ they constructed is used to simuate the dynamic characteristics of sateite battery showing in the rocess of generation and eimination in the oarization henomenon. is the oarization votage on the RC networ, U describes the termina votage of battery, I describes the oad current of battery (assuming discharge current is ositive, and charge current is negative). U

3 Life rediction of sateite Li battery based on muti time-scae extended Kaman fiter (EKF) 69 U oc R C U R t I Fig.1 hevenin equivaent circuit mode SOC is defined as the ratio of avaiabe caacity to nomina caacity. When the environment temerature is certain, the reationshi between OCV and SOC is not ony a refection of static characteristics of ithium battery [11], but aso discernibe. Set z = SOC, where f (z) is used to describe the determined reationshi between OCV and SOC. It is imortant to oint out that the connection of battery SOC and its termina votage has been enhanced through the f (z), which is very imortant for the imrovement of SOC rediction accuracy. It can be nown from Fig. 1 that the mathematica reationshi of hevenin mode arameters can be exressed as foows: U = U U IR = f ( z) U IR (1) Amere-hour method is the simest SOC estimation method which is currenty used more frequenty [14], [15]. It maes integra cacuation with the current fowing through the battery in run-time to cacuate the fowing in or out of battery. If the initia battery SOC vaue is avaiabe, it can be used for obtaining the battery remaining ower. Eq. (3) is the foundation of SOC state equation: where i is the current, Q is the maximum avaiabe ower consumtion, t is the time and is Couomb effective factor, which is defined as the ratio required by charging and discharging energy restoring to the origina ower. is ess than or equa to 1. For exame, in the discharge mode, when the minimum discharge votage is reached, it is thought that the battery has been cometey discharged, and SOC is 0. hrough the combination of hevenin equivaent circuit mode and Amere-hour method, the battery arameters, SOC and its termina votage are associated. aing SOC, oarization votage as state variabes, state Eq. is obtained as foows: oc t t U = U / ( R C ) + I / C SOC = 1 ( i dt) / Q U (2) (3)

4 70 Qian-Qian Liu, Bing Chen, Jingyuan Zhang SOC = 1 ( i dt) / Q (4) U = U / ( RC ) + I / C aing termina votage U as measurement vaue, the observation Eq. is: 2.2 Discretization mode of muti time-scae he aication object of EKF is noninear discrete systems, therefore, the continuous mode in the above section needs discretization. For systems with very different arameter variations, we can set two-time scaes: the macroscoic timescae and the microscoic time-scae. he system voume on the macro time-scae changes sowy with time but on the micro time-scae, it changes raidy with time. For convenience, and is used as time index of macro time-scae and micro time-scae. Any moment can be exressed as t, and there are reationshis:, t, (, ), is a fixed time interva, = t,0 + t,0 = t 1, L = 1,2,...,L = 1,2,..., between two adjacent oints. It shoud be noticed that L reresents the eve of time-scae searation. According to the change of system arameters, simuation time stes of the macro time-scae can be determined; between time ste and +1, according to the change of system state variabe, the saming eriod can aso be determined, i.e. the micro simuation time stes is determined. Considering that the mode arameters is sow time-varying,we assume that the battery is the time-invariant system, and oad current is constant at each saming interva.hen we can get the anaytic soution Eq. (2): Battery mode shown in Fig. 1 taes State of Charge z and oarization votage U as state variabes, oad current I as the inut, termina votage U as outut, it obtains after discretization in muti time-scae: Set z = SOC, Eq. (8) is obtained from Eq. (3),, U = f () z U IRt t I( ) U (( + 1) ) = ex( ) U ( ) + ex( ) dt R C R C C 0 U = ex( ) U + (1 ex( )) R I,, 1, RC RC (5) (6) (7) I, 1 z, = z, 1 Q Set = R C, Eq. (9) is obtained from Eq. (4) to (8), (8)

5 Life rediction of sateite Li battery based on muti time-scae extended Kaman fiter (EKF) 71 U, ex( ) 0 U, 1 (1 ex( )) R = + I, (9) z, z 0 1, 1 / Q he discretization state transition and measurement Eq. on muti time-scae are, ex( ) 0 (1 ex( )) R x x = + u 0 1 / Q y, = f ( z, ) U, Rt u, t, + 1,, where x, = [ z, U, ], = Q, u,, and is the measurement, = I, y, = U, U, vaue of the battery termina votage at t., In order to mae the foowing discussion be more genera, Eq. (10) is changed to the foowing noninear state sace mode, ransition: x = F( x, u, ) + w, = + r where, x is the system state vector at time, is the, t, = t,0 +, = 1,2,..., L fixed time interva between the neighboring measurement oint. It shoud be noticed that L reresents the eve of time searation, and x. is the,0 = x 1, L vector of system mode arameters at t,0 ; u, is the inut of externa observation source; y is the vector of system observation vaue or measurement vaue., w, and are rocess noise vector of state and mode arameters. is the vector of measurement voice, F( x,, u,, )and G( x,, u,, )are the functions of state transition and state measurement, resectivey. 3. Muti time-scae EKF agorithm, + 1,,, + 1 Measurement: y = G( x, u, ) + v,,,, r v, As for the discrete mode of the system on the muti time-scae, muti timescae EKF agorithm is used to redict the SOC and maximum avaiabe caacity. Muti time-scae EKF agorithm rediction rocess is divided into six stes, macro EKF and micro EKF execute together in the form of a nested oo. Within each macro time ste, macro EKF executes time udate ste, state rediction ste and measurement udate ste. Within each micro time ste, micro EKF executes time udate ste and measurement udate ste. When = 1 cyces to = L, the macro time ste ends and enters into the next macro time ste +1. Regardess of macro or micro EKF, they both needs to get the exerience vaue based on rior information to mode arameters and state x for initiaization before conduction. Covariance matrix and for estimation error mae initiaization according x (10) (11)

6 72 Qian-Qian Liu, Bing Chen, Jingyuan Zhang to riori information too. Basic stes are summarized in abe 1. Stes Ste 1 Ste 2 Ste 3 Ste 4 Ste 5 Ste 6 abe 1. Muti time-scae EKF agorithm stes Contents ˆ ˆ ˆ ˆ 0 = E[ 0], = E[( 0 0)( 0 0) ], initiaization (12) x ˆ = E[ x ], = E[( x x ˆ )( x x ˆ ) ] On macro-scae {1,..., }, time udate of macro EKF ˆ = ˆ, = + (13) State rediction of macro EKF x = F ( xˆ, u, ˆ ) (14) Measurement udate of macro EKF On micro-scae {1,...,L}, Measurement udate of micro EKF 3.1. Macro EKF (15) (16) (17) (18) (19) F( x, u ˆ ˆ, 1, 1 ) G( x, u,, 1) Where, x A (20) K, 1 =, C, = x x Within each macro time ste, arameter estimation vaue ˆ and at the revious ste are cacuated according to Eq. (13). And then we use micro EKF for redictive state according to Eq. (14), F0 ( L x, u 1,0-1,L-1, )is the iteration form of state transfer function F( x,, u,, )in Eq. (11). Comare with the time and measurement udate,the comutationa comexity needed by 0 L 1,0-1,L-1 0,0 0,0 x 0,0 0,0 0,0 0,0, F ( x, u, )cacuation on micro time ste L. On the measurement udate rocedures, redicted state cacuated by macro EKF and micro EKF estimation state vaue are different, and difference is used to obtain the osteriori 1 1 r 1 1, L 0 L 1,0 1, L1 1, 1, 1 = ( ) [ ( ) + ] n K C C C ˆ ˆ = + K [ xˆ x ], = (1 K C ) L L ime udate of micro EKF xˆ F x u ˆ A A ˆ, = (, 1,, 1, 1 ), = x, 1, 1, x +, 1 w, 1 x x x x 1, = (, ) [, (, ) + ] x, x, v, K C C C xˆ xˆ K y G xˆ u ˆ I K C x x x, =, +, [, (,,,, 1 )], = (,, ) x, x, C x= xˆ, 1 x= xˆ, df ˆ 0 L ( x 1,0, u 1,0: L1, ) = d ˆ = (21)

7 Life rediction of sateite Li battery based on muti time-scae extended Kaman fiter (EKF) 73 arameter estimation, seen Eq. (16). Macro EKF has two distinct features: since it udate S on the macro scae ( L ), the comutationa comexity is greaty reduced; macro EKF maes state estimation from micro EKF firsty and then udates the measurement. According to the Eq. (14) for state rediction achievement, the arameter estimated vaue generated is decouing through state rediction. 3.2 EKF Micro EKF As for state transition, micro EKF uses the estimated vaue of maximum avaiabe caacity achieved in the above macro time ste rocedure (seen Eq. (17)). It is worth mentioning that in the initiation of each macro time stes, i.e., t, 1,0 micro EKF sends a state estimation vaue to macro EKF, and then redicts the state according to the state of Eq. (14) at the macro time ste. After cometing the state rediction at each macro ste, i.e., t, micro EKF sends another state estimated 1, L 1 vaue to macro EKF to comare with the redicted estimation and use its difference for correcting the arameter estimation vaue in the measurement udate rocedure of Eq. (16). 3.3 Numerica imementation: Recursive differentia cacuation On the Muti time-scae EKF agorithm, C cacuation in macro EKF invoves in the tota derivative of the state rediction function concerning the arameters: df ˆ 0 L ( x 1,0, u 1,0, ) C = (22) d State amount x is aso the function concerning system arameter, therefore, the tota derivative is needed to sit into artia derivative for circuated cacuation. he foowing Eq. can be obtained: C can be obtained according to the recursive cacuation of Eq.s (23)-(26). he artia derivative of state rediction arameters on state x and arameter can be cacuated easiy according to the confirmed function form. = ˆ df ( xˆ, u, ) F ( xˆ, u, ) F ( xˆ, u, ) dxˆ = + d xˆ d 0 L 1,0 1,0 0 L 1,0 1,0: L1 0 L 1,0 1,0: L1 1,0 1,0 dxˆ dxˆ dg ( xˆ, u, ) -1,0-1,0 x 0 L -1,0-1,0: L1 = K, 1 d d d dg ˆ ˆ ˆ ˆ 0 L ( x,0, u,0 : L1, ) G0 L ( x,0, u,0 : L1, ) G0 L ( x,0, u,0 : L1, ) dx,0 = + d xˆ d dxˆ F ( xˆ, u, ) G ( xˆ, u, ) dxˆ,0 0 L 1,0 1,0: L1 0 L 1,0 1,0: L1 1,0 = + ˆ 1,0 d x d,0 (23) (24) (25) (26)

8 74 Qian-Qian Liu, Bing Chen, Jingyuan Zhang 3.4 Muti time-scae EKF agorithm to redict the sateite ithium battery SOC cacuation rocess Next, the EKF rediction agorithm execution rocess of sateite ithium battery system with muti time-scae is introduced in this aer. Fow chart is shown in Fig. 2. Agorithm consists of two arae extended Kaman fiters; the uer art (micro EKF) modifies SOC on the micro time-scae, the ower art (macro EKF) modifies the ower avaiabiity estimation vaue on macro time scae. Micro EKF sends SOC estimation vaue to macro EKF and receives maximum avaiabe caacity estimation vaue from macro EKF. x, -1 i, 1 micro time udate x, micro measurement udate y, i, x, micro time scae = L? x YES, NO x, Micro EKF (SOC) C 1 macro time udate C macro state rediction x, macro measurement udate C macro EKF(Maximum ower avaiabiity) Fig.2 Muti time-scae estimation fow chart of sateite ithium battery SOC based on EKF 4. Simuation and Exeriment Resut 4.1 Exerimenta rocess he NCA/C saceborne ithium battery we used in this exeriment is 1.6Ah. est system incudes comrehensive test equiment of sateite ithium battery, the temerature sensor modue, NCA/C saceborne ithium batteries, and the secia insta fixture. est equiment can suort 8 channes for exeriment at the same time. According to the oca test resuts, we use the data of battery No.37 for identification and simuation. he NCA/C saceborne ithium battery used in the exeriment and the exeriment equiment are shown in Fig Reationshi identification of oen circuit votage Uoc - SOC he rocess of mode arameter identification and sateite ithium battery SOC estimation are both invoved in the reationshi identification between EMF (Eectromotive Force) and SOC. Because EMF cannot be obtained by circuit exeriment directy, the resent researches mosty use the baanced votage when battery oens, which is OCV to reace EMF; OCV and EMF are thought equa aroximatey. In section 2.1, the battery oen circuit votage (OCV) is reaced with f (z), identified resuts are shown in Fig. 4.

9 Life rediction of sateite Li battery based on muti time-scae extended Kaman fiter (EKF) 75 (a) NCA/C saceborne ithium battery (b) comrehensive test equiment of sateite ithium battery Fig.3 sateite ithium battery and exeriment equiment. Fig.4 Oen votage and SOC reationshi Fitting It can be seen from Fig.4 that with the rising of SOC, oen circuit votage is amost a inear reationshi at 0.3<SOC<0.9; the oen circuit votage changes dramaticay when SOC < 0.2. herefore, the reationshi fitting between oen circuit votage at 0.3<SOC<0.9 and SOC is: Uoc = SOC + SOC (27) Mode arameter identification hevenin equivaent circuit mode contains arameter R, R and C, which are needed to be identified before using the mode. Exeriments are conducted under 25 C and the resuts of each arameter identification as shown in Fig.5: We found that in the rocess of discharge, athough the change ranges of R and C is arger, but the change trend is reativey stabe. It can be seen that the change trends of them are on the contrary, which maes =R C change itte; Rt fuctuates in the range of to 0.033, and there is no certain inear rue. t

10 76 Qian-Qian Liu, Bing Chen, Jingyuan Zhang (a) is the reationshi curve of discharge direction (b) is the reationshi curve of discharge direction R C and SOC; and SOC; (c) is the reationshi curve of discharge direction R t and SOC. Fig. 5 he reationshi between the arameters (he red ine is the fitted curve; discrete oints reresent the true vaue of identification exeriment) Exerimenta verification under the custom woring condition After mode arameter identification comementation, the rediction mode of sateite ithium battery SOC and maximum avaiabe caacity on muti time-scae is estabished. he discharge exeriment under a set of custom woring condition is used to test the estimation erformance of the agorithm ut forward on SOC. he custom condition test scheme used is as foows: firsty, et the battery fuy rest, then discharges 500 seconds with the constant current 0.5 C=0.8A, rest 50 seconds. And then et battery charge 50 seconds with the constant current 0.5 C=0.8A. Reeat the above stes, 12 times in a. he exeriment current is shown in Fig. 6. he exerimenta current is oaded to test equiment of sateite ithium battery according to the custom woring condition. he current and votage data of the battery is extracted after the exeriment. he battery arameters obtained in the revious exeriment corresonded to SOC from 0.3 to 0.9, in order to match the arameters, SOC in this exeriment a changes from 0.3 to 0.9. his section uses Dua EKF and muti time-scae EKF agorithm resectivey in the MALAB

11 Life rediction of sateite Li battery based on muti time-scae extended Kaman fiter (EKF) 77 software for offine verification on NCA/C ithium battery SOC and maximum avaiabe caacity estimation. Exeriment is carried out fewer than 25 C. Fig. 6 Exeriment current of the custom woring condition 4.2 Exerimenta resuts and anaysis hevenin equivaent circuit mode is estabished in the MALAB software, the inut signa for simuation exeriment is current of the custom woring condition. SOC obtained using Dua EKF [11] and the curve of maximum avaiabe caacity estimation is shown in Fig. 7. (a) is the reationshi of termina votage rediction vaue and measurement vaue; (b) is the reationshi of SOC estimation vaue and truth-vaue; (c) is the reationshi of system arameter Q estimation vaue and truth-vaue. Fig.7 Simuation resuts of Dua EKF.

12 78 Qian-Qian Liu, Bing Chen, Jingyuan Zhang he curve of NCA/C saceborne ithium battery SOC and maximum avaiabe caacity rediction obtained using muti time-scae EKF agorithm is shown Fig. 8. (a) is the reationshi of termina votage rediction vaue and measurement vaue; (b) is the reationshi of SOC estimation vaue and truth-vaue; (c) is the reationshi of system arameter Q estimation vaue and truth-vaue. Fig.8 Estimation resuts of muti time-scae EKF agorithm. In terms of maximum avaiabe caacity estimation, the initia vaues we set are both ess than the true vaues in the two methods, as shown in Fig. 7 (c) and Fig. 8(c). From Fig. 7 (c), it can be nown that estimation of the maximum avaiabe caacity faied to cosey trac the rea maximum avaiabe caacity; eventuay, the estimated vaue converged in about 4.3% error range and incudes arger noise. In Fig. 8(c), the estimated vaue is converged to the true maximum avaiabe caacity graduay as the simuation stes increases. he accuracy of Dua EKF is ow due to measurement in SOC and maximum avaiabe caacity estimation and time scae couing. As shown in Eq. (28), for modifying the rediction vaue of maximum avaiabe caacity, Dua EKF uses termina votage of the battery as the measurement vaue to modify it. ˆ = ˆ + K (y G( xˆ, u )),,,,,, he difference is that muti time-scae EKF agorithm maes estimation on maximum avaiabe caacity according to Eq. (29) and (30) on the macro scae. It avoids the termina votage of the battery is the ony measurement for the arae estimation rocess of SOC and the maximum avaiabe caacity. (28)

13 Life rediction of sateite Li battery based on muti time-scae extended Kaman fiter (EKF) 79 x = F ( xˆ, u, ˆ ) 1, L 0 L 1,0 1, L1 (29) Moreover, there is no direct reationshi between the observations Eq. G( xˆ,, u, ) and the maximum avaiabe caacity. When we use observed quantity to udate the maximum avaiabe caacity, it just reies on the measurement of White Gaussian Noise on the Kaman fiter rocessing, with imited effect. Meanwhie, the state variabe x, = [ z, U, ] and the maximum avaiabe caacity have a direct connection. Using the state variabe for arameters udating can roduce more reiabe maximum avaiabe caacity estimation vaue. Comare Fig. 7(a) and Fig. 8 (a), it is easy to see that simuation vaues of the termina votage with muti time-scae are more cosey foowing the measured vaue change and contains ess noise. he error of SOC estimation vaue in Fig. 8 (a) is smaer than Fig. 7 (a). he deendency of SOC estimation on the maximum avaiabe caacity is arger; the deficiency of the maximum avaiabe caacity estimation accuracy woud reduce the veracity of SOC estimation. Muti time-scae EKF agorithm rovides a more accurate maximum avaiabe caacity estimation vaue and SOC estimation is more accurate. Muti time-scae EKF agorithm imroves the rediction erformance than Dua EKF to a certain extent. In addition, the two methods are aso comared in comutationa efficiency, shown in tabe 2. In order to minimize the infuence of randomness, the two methods are carried out 10 times, and then tae average for comarison. he time of the average cacuation is summarized in abe 2. abe 2. Comutationa efficiency contrast Agorithm ye comuting time efficiency Imrovement Dua EKF [11] Muti time-scae EKF agorithm % It can be observed that muti time-scae EKF agorithm consumes ess comutation time than Dua EKF. his hes to reduce the burden of comutation and the hardware in sateite faut rediction and heath management (PHM) system, aso imrove the PHM system aication fexibiity and efficiency. 5. Concusion ˆ ˆ [ ˆ = + K x 1, L x 1, L ] he muti time-scae EKF agorithm is an effective and accurate state and arameter estimation method for engineering systems with time-scae searation. In this aer, the muti time-scae EKF agorithm is used to redict the SOC of the sateite ithium battery on the microscoic scae, and the SOC estimation is used as the observation to forecast the maximum avaiabe eectricity. he effect is better (30)

14 80 Qian-Qian Liu, Bing Chen, Jingyuan Zhang than the dua EKF estimation, and the cacuation is imroved effectiveness. he next ste wi be to estabish a sateite ithium battery equivaent mode and OCV and SOC reationshi mode, taing into account the imact of temerature on the SOC, the imact of eectricity on the OCV, etc., to imrove the accuracy of the mode to obtain more accurate estimates. R E F E R E N C E S [1] Yuing Wu, Churong Wan, Changying Jiang, Secondary Battery of Lithium Ion, Chemica Industry Press, 20012:1-10. [2] Goebe, Kai, et a. "Prognostics in battery heath management." IEEE instrumentation &measurement magazine 11.4 (2008): [3] Pett, Gregory L. "Extended Kaman fitering for battery management systems of LiPB-based HEV battery acs: Part 1. Bacground." Journa of Power Sources 161.2(2006): [4] Pett, Gregory L. "Extended Kaman fitering for battery management systems of LiPB-based HEV battery acs: Part 2. Modeing and identification." Journa of ower sources (2004): [5] Arman Amini Badr, Navid aghizadegan Kaantari. " Reiabiity evauation of radia distribution systems incuding distributed generation based on an imroved cassification agorithm", U.P.B. Sci. Bu., Series C, Vo. 79, Iss. 4, 2017.: [6] Pett, Gregory L. "Sigma-oint Kaman fitering for battery management systems of LiPB-based HEV battery acs: Part 1: Introduction and state estimation." Journa of Power Sources (2006): [7] Huaizhong Chen, Aication of imroved Partice Swarm Otimization agorithm in Maximum Power Point racing for PV system. U.P.B. Sci. Bu., Series C, Vo. 79, Iss. 4, 2017.: [8] He, Wei, et a. "State of charge estimation for Li-ion batteries using neura networ modeing and unscented Kaman fiter-based error canceation." Internationa Journa of Eectrica Power & Energy Systems 62 (2014): [9] Lee, Seongjun, et a. "State-of-charge and caacity estimation of ithium-ion battery using a new oen-circuit votage versus state-of-charge." Journa of ower sources (2008): [10] Rahimi-Eichi, H., F. Baronti, and M-Y. Chow. "Modeing and onine arameter identification of Li- Poymer battery ces for SOC estimation." Industria Eectronics (ISIE), 2012 IEEE Internationa Symosium on. IEEE, [11] Simon D. Otima state estimation: Kaman, H-infinity, and noninear aroaches. John Wiey & Sons, [12] Goebe, Kai, et a. "Prognostics in battery heath management." IEEE instrumentation & measurement magazine 11.4 (2008): [13] Santhanagoaan S, Zhang Q, Kumaresan K, et a. Parameter estimation and ife modeing of ithium-ion ces. Journa of the Eectrochemica Society, 2008, 155(4): A345-A353. [14] Mi C, Li B, Buc D, et a. Advanced eectro-therma modeing of ithium-ion battery system for hybrid eectric vehice aications. IEEE Vehice Power and Prousion Conference. IEEE, 2007: [15] Fuer F, Doye M, Newman J. Simuation and otimization of the dua ithium ion insertion ce. Journa of the Eectrochemica Society, 1994, 141(1): 1-10.