Modelling and Evaluation of Battery Packs with Different ... - MDPI

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Modelling and Evaluation of Battery Packs with Different Numbers of Paralleled Cells Fengqi Chang 1,2, * 1 2

*

ID

, Felix Roemer 1,2 , Michael Baumann 2 and Markus Lienkamp 1,2

TUM CREATE Ltd., 1 CREATE Way, #10-02 CREATE Tower, Singapore 138602, Singapore; [email protected] Institute of Automotive Technology, Technical University of Munich, Boltzmannstraße 15, Garching, 85748 Bavaria, Germany; [email protected] (M.B.); [email protected] (M.L.) Correspondence: [email protected]; Tel.: +65-9866-6779; Fax: +65-6601-4016

Received: 2 May 2018; Accepted: 4 June 2018; Published: 7 June 2018

Abstract: To better evaluate the configuration of battery packs in electric vehicles (EV) in the early design phase, this paper proposes a mathematic model for the simulation of battery packs based on the elementwise calculations of matrices. This model is compatible with the different battery models and has a fast simulation speed. An experimental platform is built for the verification. Based on the proposed model and the statistic features of battery cells, the influence of the number of paralleled cells in a battery pack is evaluated in Monte-Carlo experiments. The simulation results obtained from Monte-Carlo experiments show that the parallel number is able to influence the total energy loss inside the cells, the energy loss caused by the balancing of the battery management system (BMS) and the degradation of the battery pack. Keywords: battery model; prediction; simulation; efficiency; state of charge

1. Introduction The production technology of battery cells has greatly progressed, but the unevenness of the cell properties, e.g., the capacity, the inner resistance and the polarization time constants, between different cells is still inevitable [1,2]. The unevenness of cells will certainly result in extra charging and discharging processes when the cells are connected in a battery pack. For the paralleled cells in the pack, due to the difference of the polarization time constants and the inner resistance, a self-balancing process between cells can be observed after a charge or discharge [3–6]. For the cells in series, the cells with higher capacity will have a higher state of charge (SOC) than the average value during the operation of the battery pack. The extra SOC in those cells must be dissipated by the passive or active balancing of a battery management system (BMS) to prevent the over charge or over discharge of the weak cells [2,7]. As these additional cyclings of the batteries cannot be eliminated due to the unevenness of the cell properties, it is important to understand how the configuration of battery packs can influence those additional processes and how these additional processes can influence the electric and degradation behaviours of a battery pack. This knowledge can find a better configuration of a battery pack for different applications in early design phases. In this paper, the focus is the impact caused by paralleling more cells of the same type in a battery pack. Previous researchers also noticed the importance of the parallel number and the additional charging or discharging processes in the battery pack. The conclusions, however, are different. Reference [8] compared the degradation behaviours of a battery pack and a single cell in an experiment, in which the high quality and homogenous cells were used. No obvious difference of degradation between a single cell and a battery pack was identified. References [6,9] concluded that a significant accelerated degradation can be observed when the capacities or resistances of two paralleled cells are largely different from each World Electric Vehicle Journal 2018, 9, 8; doi:10.3390/wevj9010008

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degradation can be observed when the capacities or resistances of two paralleled cells are largely different from each In addition, the conclusion of [5] is of that when the difference of inner other. In addition, the other. conclusion of [5] is that when the difference inner resistances and capacities resistances capacities between cells is in large, connecting more in on parallel can reducecell the between cellsand is large, connecting more cells parallel can reduce thecells stress each individual stress on each individual cell and therefore decelerate the degradation of the whole battery pack. and therefore decelerate the degradation of the whole battery pack. Onepossible possibleexplanation explanation contradiction of those conclusions in different is all thatthe all One forfor thethe contradiction of those conclusions in different teststests is that the aforementioned depended on theobtained results from obtained fromnumber a limited number aof aforementioned studiesstudies largelylargely depended on the results a limited of samples, samples, a coincidence which could thus be taken as the conclusion for the whole population. coincidence which could thus be taken as the conclusion for the whole population. Secondly, all the Secondly, all the drawn conclusions drawn based onbetween the comparison battery pack conclusions were basedwere on the comparison only onebetween battery only packone with a specific with a specific configuration single cell.of The of the conclusions bedifference caused by configuration and a single cell.and Thea difference thedifference conclusions could be causedcould by the the difference of the battery pack configurations. Therefore, in order to obtain a more general view of the battery pack configurations. Therefore, in order to obtain a more general view of the influence theparallel influence of the aparallel a statistic approach be taken in athe research and ofofthe number, statisticnumber, approach should be taken inshould the research and large number ofa large number of different battery packs should be tested. different battery packs should be tested. However,testing testingaalarge largenumber numberofofbattery batterypacks packsininexperiments experimentsisistoo tootime-consuming time-consumingand and However, ineffectiveinin terms of cost. efficient method to simulate battery thus required. [3,4] ineffective terms of cost. An An efficient method to simulate battery packs packs is thusisrequired. Zhang and used Simulink to simulate the current distribution between paralleled cells, but expanding the Miyatake’s group [3,4] used Simulink to simulate the current distribution between paralleled cells, but model to thousands of interconnected cells is difficult, as every cell in the battery pack needs to be expanding the model to thousands of interconnected cells is difficult, as every cell in the battery pack set and connected manually in the software. built a model ofgroup a battery pack abased onofthe models needs to be set and connected manually in the[3] software. Zhang’s [3] built model a battery of the cells in the complex frequency domain. This model can be easily expanded to simulate large pack based on the models of the cells in the complex frequency domain. This model can be easily battery packs, but it contains a time-consuming iterationa and therefore is not suitable large scale expanded to simulate large battery packs, but it contains time-consuming iteration andfor therefore is simulations or degradation simulations. [10] built a battery cell model and realized an effective and not suitable for large scale simulations or degradation simulations. Wei’s group [10] built a battery accurate estimation. [11–14] developed pack models for theMany sameresearch purpose.[11–14] [15] further takes cell modelSOC and realized an effective and accurate SOC estimation. developed the differences of the cells in a battery packresearch into consideration SOC pack models for the same purpose. Zhong’s [15] further for takes theestimation. differencesHowever, of the cellsasinthe a main focus of these studies is SOC or SOH estimation, the models are not directly appropriate battery pack into consideration for SOC estimation. However, as the main focus of these studies for is largeorscale evaluate configurations. SOC SOHsimulation estimation,tothe modelsbattery are notpack directly appropriate for large scale simulation to evaluate Therefore, to enable the quick evaluation of battery packs while taking the statistic view, this battery pack configurations. paper firstly proposes a model of battery pack based on the elementwise of matrices, Therefore, to enable the quick evaluation of battery packs while taking thecalculation statistic view, this paperin whichproposes different cell models canpack be based implemented. Experimental results of arematrices, obtained for the firstly a model of battery on the elementwise calculation in which verification. Then the distributions followed by the properties of battery cells are obtained from the different cell models can be implemented. Experimental results are obtained for the verification. measurement results of 50 NCR18650PF cells. Based on the model and the statistic features Then the distributions followed by the properties of battery cells are obtained from the measurementof NCR18650PF cells, different battery packs aremodel generated andstatistic simulated in Monte-Carlo experiments results of 50 NCR18650PF cells. Based on the and the features of NCR18650PF cells, with the focus on how the number of cells in parallel can influence the efficiency and on the different battery packs are generated and simulated in Monte-Carlo experiments with the focus degradation of battery packs. how the number of cells in parallel can influence the efficiency and the degradation of battery packs. BatteryPack PackModel Model 2.2.Battery Tobuild buildthe the battery battery pack model, cell model [3] [3] in Figure 1. The To model, this thispaper paperuses usesa asecond-order second-order cell model in Figure 1. resistor is the Direct Current (DC) inner resistance. The open circuit The resistor R is the Direct Current (DC) inner resistance. The open circuit voltage (OCV) e is , 0j,k j,k, is generated according to the state of charge, . is the output current of the cell while the generated according to the state of charge, SOCj,k . ij,k, is the , output current of the cell while the terminal terminal voltage isbydenoted by , . voltage is denoted uj,k .

Figure 1. Equivalent circuit of a battery cell in the parallel row j and the column k. Figure 1. Equivalent circuit of a battery cell in the parallel row j and the column k.

Two RC circuits are used to demonstrate the polarization during charge or discharge. The four parameters, , , , , , , , , can be identified by measuring the transient voltage when the

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Two RC circuits used to demonstrate World Electric Vehicle Journalare 2018, 9, x FOR PEER REVIEW the

polarization during charge or discharge. The 3 offour 14 parameters, R1j,k , C1j,k , R2j,k , C2j,k , can be identified by measuring the transient voltage when the cell cell is discharged byconstant a constant current. are denoted to have a smaller constant is discharged by a current. R1j,k and C1j,k are, denoted to have a smaller timetime constant than , and than of and, Rand , so that the polarization in the short term (several seconds) and the long that that of R2j,k , so that the polarization in the short term (several seconds) and the long term , 2j,k term (in minutes level) candemonstrated be demonstrated (in minutes level) can be by Rby uc1j,k and uc2j,k are , , and , Rand , respectively. , and 1j,k , C1j,k 2j,k , R2j,k, ,respectively. the, voltage C1j,k and voltage of thevoltage cell. Inof this resistance are theofvoltage ofC2j,k,. uand . , is the terminal themodel, cell. Inthe this model, and the j,k is the, terminal capacitance arecapacitance assumed toare stay constant. The dependency the SOC is neglected. resistance and assumed to stay constant. Theon dependency on the SOC is neglected. Foraabattery batterypack packcomposed composedof ofnncells cellsin inparallel paralleland andmmrows rowsof of paralleled paralleledcells cellsconnected connectedin in For series (represented (represented by this paper), thethe quantity in Figure 1 with j,k means series by npms npmsthereafter thereafterinin this paper), quantity in Figure 1 subscript with subscript j,k the properties of the cell thecell jth parallel rowparallel and therow kth column the column pack, as of demonstrated in means the properties of in the in the jth and theofkth the pack, as Figure 2. Qj,k isinthe capacity corresponding output current theoutput pack is current represented by demonstrated Figure 2. of, the is the capacity ofcell. the The corresponding cell.ofThe of the ipack . is The output of the is the current the voltage each cell in the the voltage battery pack, when pack represented by model . The output of and the model is theofcurrent and of each cell the in load of thepack, battery pack, , is of given. the battery when theipack load the battery pack, , is given.

Figure 2. Cell numbering inside a battery pack. Figure 2. Cell numbering inside a battery pack.

To calculate the output variables with a given input, paralleled cells in one row are analysed the output variables with a given input, paralleled cells in one row are analysed first. To In calculate a given row, e.g., the jth row, since all the battery cells are connected in parallel, the first. In avoltage given row, e.g.,cells the jth since all the battery are connected in parallel, the terminal terminal of the is row, the same, denoted by cells . Using Kirchhoff’s law of current, the voltage ofgroup the cells the same,can denoted by uj .inUsing Kirchhoff’s law of current, the following group of following of is equations be derived Equation (1): equations can be derived in Equation (1):

i

=i +i

+ + i

j ,1 j ,2 j ,n (  pack (1) iu = i + i + . . . + = e j , kj,1− uC1j,2 − uC 2 j , k i−j,nR0 j , k i j , k ( k = 1, 2, , n) j j ,k pack (1) u j = e j,k − uC1j,k − uC2j,k − R0j,k i j,k (k = 1, 2, · · · , n) In Equation (1), by transforming the cell terminal voltage equations into the cell current equations and then all thethe cell current values in the pack current equation, Equation (2) In Equation (1),substituting by transforming cell terminal voltage equations into the cell current equations can obtained. The ujall andthe correspondingly the current of each cell can be calculated by (2) Equation andbe then substituting cell current values in the pack current equation, Equation can be (2) if the polarization and OCVthe are current known.of each cell can be calculated by Equation (2) if the obtained. The uj and voltage correspondingly polarization voltage and OCV are known. n n





e

−u

−u



1

C1 j ,k C 2 j ,k  ipack = n j , k − un j e j,k −uC1j,k −uC2j,k 1R  i R 1 k =∑ = − u 0 , j k j ∑ kR=10j,k 0 j , k pack R0j,k k =1  i = (ek=1− u − uCC2j,k − u j )j )//R R0 0j,k = 1,1,2, (( kk = 2, · · ,· n,)n) 1 j ,k − u 2 j ,k − u j , k − uC j ,k i j,kj , k = (e j,k C1j,k

(2) (2)

However, the OCV and polarization voltage are state variables in this model, which cannot be calculated directly with only the information at a given point of time. Equations of state should thus be built to continue the modelling of the battery pack. Using the integer l to represent the lth step of


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However, the OCV and polarization voltage are state variables in this model, which cannot be calculated directly with only the information at a given point of time. Equations of state should thus be built to continue the modelling of the battery pack. Using the integer l to represent the lth step of the simulation, assuming that the initial SOC values of all the cells are already known, and that the polarization voltage of all the cells is zero initially, the equations of state in Equation (3) can be obtained. ∆t is the time step of the simulation. In each step, four state variables, the SOC, the OCV and the polarization voltage values are updated first. Then the current distribution within the paralleled cells in the jth row and the cell voltage are recalculated, which will be used to update the state variables in the next simulation step. For the battery cells in other rows, the results could be obtained by sweeping the subscript j from 1 to m. In this way, the current and voltage of all the cells in the npms battery pack can be obtained. In addition, it should be noted that the assumptions in this model are common in reality, because most battery tests should start at a known SOC value and after a long time of relaxation eliminate the polarization voltage. Therefore, the model in Equation (3) can be reliably used to simulate a battery pack.  i j,k (l −1)∆t   SOCj,k (l ) = SOCj,k (l − 1) − Q j,k    i j,k (l −1) R1j,k ∆t+uC1j,k (l −1) R1j,k C1j,k   uC1j,k (l ) =   ∆t+ R1j,k C1    i j,k (l −1) R2j,k ∆t+uC2j,k (l −1) R2j,k C2j,k   uC2j,k (l ) =  ∆t+ R2j,k C2j,k e ( l ) = f ( SOC ( l )) j,k j,k   n e j,k (l )−uC1j,k (l )−uC2j,k (l )   −ipack (l )  ∑ R0j,k   k =1  u ( l ) =  n j   ∑ R1  (3)  k =1 0j,k   i j,k (l ) = [e j,k (l ) − u j (l ) − uC1j,k (l ) − uC2j,k (l )]/R0j,k k = 1, 2, · · · , n j = 1, 2, · · · , m Initial conditions : SOCj,k (0)known uC1j,k (0) = uC1j,k (0) = 0 i j,k (0) = 0 As the model has no nonlinearly coupled variables, the model can be rewritten as the elementwise calculations of matrices as in Equation (4), in which all the operations of matrices are elementwise operations. Compared to calculating each variable in loops, the transformation (also known as vectorisation) of the model can significantly improve the calculation speed if the Intel Math Kernel Library (MKL) is implemented in the simulation environment [16]. An example of such a simulation environment is MATLAB. On an Intel i7 6600U CPU, using this model on MATLAB, a simulation, in which a 72p108s battery pack (the configuration of the battery pack in Tesla Model S) discharged by a

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SOC(l ) = SOC(l − 1) − I (l − 1) Δt

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 I (l − 1) R 1 Δt + U C1 (l − 1) R 1 C1  U C1 (l ) = t + then R 1 Cleft  s (time step 0.1 s) Δ 1 fluctuating load for 1800 and stand for relaxation for another 800 s, is finished ( l − 1) Δ t + I  R U C2 (l − 1) R 2 C 2  2 in 51.87 s.  U (l ) = C2  SOC  (l ) = SOC(l Δ−t +1)R−2 IC(l2−Q1)∆t      I(l −(1l)· ∆t+UC1 (l −1)R1 ·C1 EU(l ) =(l )f =  (SOC ))R1 ∆t  C1 + R1 · C1    n  R2u∆t+U(C2 eI(j ,lk−(1l ))·− l ) (−l −u1C2)Rj , k2 (Cl2)  C1j , k  U ( l ) =  ∆t+R2 ·C2 − ipack (l )  C2  (4)   E(l ) = kf=1(SOC(l )) R 0 j ,k u (l ) = ( j = 1, 2,  , m) n  j 1   n e ( l )− u ( l )− u ( l )  j,k C1j,k  R C2j,k −ipack (l)  ∑  R0j,k  k =1  0 j ,k k =1 (4)   u j (l ) = ( j = 1, 2, · · · , m) n  1 − U (l )] / R   l − u l − U l = I l E ) ( ) ( ) ( ) [ (  ∑ j C1 C2 0  R   k=1 0j,k

  I(l ) c= [E( l ) − s: u j (l ) − UC1 (l ) − UC2 (l )]/R0 ondition Initial Initial conditions SOC(0)kn own : SOC (0)known U C1 (0) = U C2 (0) = 0 UIC1 (0()0=) 0= UC2 (0) = 0 I(0) = 0 It should also be noted that the second-order cell model used in this paper can be replaced by shouldadvanced also be noted that the second-order modelof used in this paper candependency be replaced by otherIt more cell models for different cell purposes research, e.g., the of other more advanced cell models for different purposes of research, e.g., the dependency of parameters parameters on the SOC can be further considered. The derivation procedure of the battery pack on the SOC can be further considered. derivation procedure the battery packpack model in this model in this paper still applies. As theThe main contribution of the of paper, the battery model is paper still applies. As the main contribution of the paper, the battery pack model is mainly proposed mainly proposed as a calculation framework. Hence the model can be easily used by other as a calculation Henceon the model canbehaviours. be easily used by other researchers with different researchers withframework. different focuses cell or pack focuses on cell or pack behaviours.

3. Experimental Verification of the Proposed Model 3. Experimental Verification of the Proposed Model In order to verify the model, an experiment platform was built (Figure 3). This platform is used In order to verify the model, an experiment platform was built (Figure 3). This platform is used to to measure and record the current and voltage of four paralleled battery cells when they are measure and record the current and voltage of four paralleled battery cells when they are discharged discharged or charged. The cells are Panasonic NCR18650PF cells rated at 2.9 Ah. The identification or charged. The cells are Panasonic NCR18650PF cells rated at 2.9 Ah. The identification is done at is done at 80% of SOC. However, as stated in Section 2, the parameters of cells in this paper are 80% of SOC. However, as stated in Section 2, the parameters of cells in this paper are assumed to be assumed to be SOC-independent. Errors can thus occur when the influence of SOC on parameters SOC-independent. Errors can thus occur when the influence of SOC on parameters are significant. are significant. The line resistance from each cell to the load is also measured accurately by a Kelvin The line resistance from each cell to the load is also measured accurately by a Kelvin bridge before the bridge before the experiment, as the line resistance is comparable with the DC resistance of cells experiment, as the line resistance is comparable with the DC resistance of cells and has a large impact and has a large impact on the current distribution. The cell parameters and the line resistance Rline on the current distribution. The cell parameters and the line resistance Rline are listed in Table 1. are listed in Table 1.

Figure 3. Experimental Platform to test paralleled battery cells. Figure 3. Experimental Platform to test paralleled battery cells.


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Table 1. Parameters of the four cells in the experiment. Table 1. Parameters of the four cells in the experiment. Cell No. Cell No. Rline Rline R0j,k R0j,k Cell1 81.3 mΩ81.3 mΩ 37 mΩ Cell1 37 mΩ Cell2 41.9 mΩ 36 mΩ Cell2 41.9 mΩ 36 mΩ Cell3 15.8 mΩ 36 mΩ Cell3 15.8 mΩ 36 mΩ Cell4 22.6 mΩ 32 mΩ

Cell4

22.6 mΩ

32 mΩ

R1j,k R1j,k 8.6 8.6mΩ mΩ 17.7 mΩ 17.7 mΩ 14.1 mΩ 14.1 mΩ 9.3 mΩ

9.3 mΩ

C1j,k C1j,k 204 204 FF 446 446 F F 436 F 436 F 322 F 322 F

C2j,k R2j,k R2j,k C2j,k 54.7 mΩ 54.7 mΩ 791 F 791 F 64.3 mΩ 64.3 mΩ 1252 F 1252 F 72.3 mΩ 1170 F 72.3 mΩ 1170 F 1033 F 61.2 mΩ 61.2 mΩ 1033 F

The four byby a 1aC1constant current load,load, starting fromfrom 80% SOC The four paralleled paralleled cells cellsare arefirstly firstlytested tested C constant current starting 80% until 20% SOC. The current is recorded for another 2500 s after the removal of the load. The measured SOC until 20% SOC. The current is recorded for another 2500 s after the removal of the load. The and simulated current waveforms of each cellofare compared in Figure simulated measured and simulated current waveforms each cell are below compared below4.in The Figure 4. The waveforms follow the measured waveforms accurately. The maximum error of the simulation is simulated waveforms follow the measured waveforms accurately. The maximum errorresult of the 0.06 A, stillresult withinis0.1 A,A, and it within is found the simulation result of simulation cell 1. simulation 0.06 still 0.1inA, and it is found in the result of cell 1.

(a)

(b)

(c)

(d)

Figure 4. The simulated and measured current waveforms of four paralleled cells when discharged Figure 4. The simulated and measured current waveforms of four paralleled cells when discharged by by the 1 C constant current load. (a) Output current waveforms of cell 1; (b) Output current the 1 C constant current load. (a) Output current waveforms of cell 1; (b) Output current waveforms of waveforms of cell 2; (c) Output current waveforms of cell 3; (d) Output current waveforms cell 2; (c) Output current waveforms of cell 3; (d) Output current waveforms of cell 4.

of cell 4.

To evaluate of the model in theinautomotive use cases, four cells are also discharged To evaluatethe theaccuracy accuracy of the model the automotive usethe cases, the four cells are also by the C-rate curve in Figure 5. From 0 to 1800 s, the curve is the output current (in C-rate) of a discharged by the C-rate curve in Figure 5. From 0 to 1800 s, the curve is the output currentbattery (in Cpack of in an EV, when theinEV simulated by the C3 driving cycle. TheC3 following s isThe the rate) a battery pack anisEV, when the EV WLTP is simulated by the WLTP driving 800 cycle. relaxation period of the battery pack, during which the self-balancing current between paralleled cells following 800 s is the relaxation period of the battery pack, during which the self-balancing current can be observed. Therefore, load C-rate curve is able to manifest the possible between paralleled cells can this be observed. Therefore, this load C-rate curve is ablefluctuations to manifest of thea practical load in an EV. The current waveforms in the simulation and the experiment are compared in possible fluctuations of a practical load in an EV. The current waveforms in the simulation and the Figure 6. The simulation measured results can match accurately whole period of the experiment are comparedand in Figure 6. The simulation and measuredduring resultsthe can match accurately test. The maximum error, 0.39 A, appears in the simulation result of the cell 1 in the 1716th second, during the whole period of the test. The maximum error, 0.39 A, appears in the simulation result of while of all the four while cells inthe the remaining within 0.1remaining A. the cellthe 1 inerror the 1716th second, error of all thetime fouriscells in the time is within 0.1 A. Therefore, the two tests proved that the proposed model is able to accurately calculate the the Therefore, the two tests proved that the proposed model is able to accurately calculate current distribution within paralleled cells. The further results based on the models can thus also be current distribution within paralleled cells. The further results based on the models can thus also be reliable. The errors observed in the simulation results could be caused by the errors in the parameter reliable. The errors observed in the simulation results could be caused by the errors in the identification and also the assumption that parameters are SOC-independent. parameter identification and also the assumption that parameters are SOC-independent.


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Figure 5. C-rate curve of the load current obtained in a simulation with the WLTP C3 driving cycle. Figure 5. C-rate curve of the load current obtained in a simulation with the WLTP C3 driving cycle. Figure 5. C-rate curve of the load current obtained in a simulation with the WLTP C3 driving cycle.

(a) (a)

(b) (b)

(c) (c)

(d) (d)

Figure 6. The simulated and measured current waveforms of four paralleled cells when discharged Figure 6. The simulated and measured current waveforms of four paralleled cells when discharged current waveforms cellwhen 1; (b) Output by the 6. fluctuating driving load. current (a) Output Figure The simulated andcycle measured waveforms of four paralleledofcells discharged Output current waveforms of cell 1 ; (b) Output by the fluctuating driving cycle load. (a) current waveforms of cell 2 ; (c) Output current waveforms of cell 3 ; (d) Output by the fluctuating driving cycle load. (a) Output current waveforms of cell 1; (b) Outputcurrent current current waveforms of cell 2 ; (c) Output current waveforms of cell 3 ; (d) Output waveforms cell waveforms ofofcell 2; 4. (c) Output current waveforms of cell 3; (d) Output current waveforms of current cell 4.

waveforms of cell 4. 4. 4. Evaluation Evaluation of of Battery Battery Packs Packs Based Based on on Monte-Carlo Monte-Carlo Experiments Experiments 4. Evaluation of Battery Packs Based on Monte-Carlo Experiments In thischapter, chapter,thethe proposed battery model is in used in Monte-Carlo experiments to In this proposed battery packpack model is used Monte-Carlo experiments to evaluate In this chapter, thewith proposed battery packofmodel is used in Monte-Carlo experiments to evaluate battery packs different numbers parallel. For each configuration, hundreds of battery packs with different numbers of parallel. For each configuration, hundreds of battery packs evaluate battery packs with different numbers of parallel. For each configuration, hundreds of battery packs are simulated and thebased conclusion based on these willbe thus not be influenced by are simulated and the conclusion on these results willresults thus not influenced by possible battery packs are simulated and the conclusion based on these results will thus not be influenced by possible coincidences of a limited number of samples. coincidences of a limited number of samples. possible coincidences of a limited number samples. of paralleling more NCR18650PF cells instead This paper only only intends intends tofind find outthe theof influence This paper to out influence of paralleling more NCR18650PF cells instead of This paper only intends to find out the influence of paralleling more NCR18650PF cells battery instead of trying to find optimal the corresponding parallel number a given trying to find the the optimal cell cell sizesize and and the corresponding parallel number for a for given battery pack of trying to find the optimal cell size and the corresponding parallel number for a given battery pack capacity. Therefore, it should be noted that all the cells generated in this paper have the same capacity. Therefore, it should be noted that all the cells generated in this paper have the same nominal pack capacity. Therefore, should be noted that all different the cells generated in this paper have the same nominal capacity andbattery thatit the battery with parallel thus have different capacity and that the packs with packs different parallel numbers thusnumbers have different capacities. nominal capacity and that the battery packs with different parallel numbers thus have different capacities. capacities. 4.1. Statistic Features of Battery Cells 4.1. Statistic Features of Battery Cells In orderFeatures to generate the properties of the battery cells, i.e., the inner resistance (R0j,k ), capacity 4.1. Statistic of Battery Cells order to generate the(Rproperties battery cells, i.e., the (C inner 0j,k), capacity (Qj,k ),Inpolarization resistors , R ) and polarization capacitors C2j,k ) in the(RMonte-Carlo 1j,k 2j,k of the 1j,k , resistance order to generate the(Rproperties of the battery cells, i.e., the (C inner resistance 0j,k), capacity (Qj,k),Inpolarization resistors 1j,k, R2j,k) and polarization capacitors 1j,k, C 2j,k) in the(R Monte-Carlo (Q j,k ), polarization resistors (R 1j,k , R 2j,k ) and polarization capacitors (C 1j,k , C 2j,k ) in the Monte-Carlo experiments, the statistic features and the distributions of the parameters are analysed based on the experiments, features and the distributions of the parameters are analysed based on the measured datathe of statistic 50 Panasonic NCR18650PF cells. measured data of 50 Panasonic NCR18650PF cells.


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experiments, the statistic features and the distributions of the parameters are analysed based on the measured data of 50 Panasonic NCR18650PF cells. The distribution of the cell capacity is studied first. The frequency histogram of the capacity is in Figure 7a. The Kolmogorov-Smirnov test (KS-test) shows the probability of the null hypothesis that the capacity data complies with a normal distribution is 66%. As this probability cannot directly decide if the null hypothesis should be accepted, the skewness of the data is also calculated. The skewness 0.1941 and the histogram show that the data has a clear skew on the right side. Therefore, this paper proposes to use the skew normal distribution instead of the normal distribution to generate the capacity of cells [1]. The choice can be reasonable, because the cell capacity must not be lower than the nominal value in the datasheet, which could result in a tendency in the production to make the distribution of the cell capacity skew to the right side.

that the capacity data complies with a normal distribution is 66%. As this probability cannot directly decide if the null hypothesis should be accepted, the skewness of the data is also calculated. The skewness 0.1941 and the histogram show that the data has a clear skew on the right side. Therefore, this paper proposes to use the skew normal distribution instead of the normal distribution to generate the capacity of cells [1]. The choice can be reasonable, because the9 ofcell World Electric Vehicle Journal 2018, 9, 8 15 capacity must not be lower than the nominal value in the datasheet, which could result in a tendency in the production to make the distribution of the cell capacity skew to the right side. Secondly, the The histogram histogram is is in in Figure 7b, the distribution distribution of the the inner inner resistance resistance is is discussed. discussed. The which resembles the normal distribution better than that of the cell capacity. The skewness of the capacity. inner resistance data is 0.11. Therefore, although the probability of the null hypothesis is 51% in the Kolmogorov-Smirnov test, the normal distribution is still adopted for the inner resistance due to the lower skewness and the histogram. The four , ,CC1j,k four polarization polarization parameters parameters in inthe theequivalent equivalentcircuit, circuit,RR 1j,k 1j,k,, R R2j,k 2j,k and C C2j,k 2j,k,, are more 1j,k difficult to measure, as there there is is no no equipment equipment specially specially designed designed for the automatic automatic measurement of those parameters. parameters. Therefore, Therefore,the thepolarization polarizationparameters parametersofofonly only cells identified manually 1010 cells areare identified manually to to obtain their statistic features. 10 results cannot show an obvious pattern in histograms, the histograms, obtain their statistic features. As As 10 results cannot show an obvious pattern in the the the statistic indices of the four polarization parameters arecalculated calculatedtotoidentify identifytheir their distributions. distributions. statistic indices of the four polarization parameters are Results are listed in Table 2, in in which which the p-value p-value of of KS-test KS-test is is the the probability probability that that the the tested tested dataset complies with a normal distribution. From the p-values of the KS-test and the skewness values, it is seen that that the the CC2j,k 2j,k quite possibly possibly follows follows the thenormal normaldistribution distributionwhile whileRR 1j,k,,R R2j,k 2j,k and and CC1j,k 1j,k should be 1j,k generated by skew normal distributions. 0.4 0.35

Probability

Probability

0.3 0.25 0.2 0.15 0.1 0.05 0 2.87 2.88 2.89 2.9 2.91 2.92 2.93 2.94 Capacity of the cells in Ah

0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 28

30 32 34 36 38 Inner resistance of the cells in m

(a)

40

(b)

Figure 7.7.The Thefrequency frequency histograms of capacity the capacity and the resistance inner resistance the(a)50Frequency cells (a) Figure histograms of the and the inner of the 50of cells Frequency of histogram of the (b) histogram Frequencyof histogram of the DC inner resistance. histogram the capacity; (b)capacity; Frequency the DC inner resistance. Table 2. Statistic Statistic indices indices of of polarization polarization indices. indices. Table 2.

Indices R1j,k C1j,k R1j,k C1j,k Mean 0.0122 Ω 326.6 F Mean Deviation0.0122 Ω 0.003 Ω 326.6 F83.9 F Standard Standard Deviation 0.003 Ω Skewness 1.0831 83.9 F 0.372 Skewness 1.0831 0.372 p-Value of KS-test 0.441 0.441 0.628 0.628 p-Value of KS-test Indices

R2j,k 0.06 Ω 0.06 Ω Ω 0.012 0.012 Ω −0.9416 −0.9416 0.539 0.539 R2j,k

C2j,k 1020.4 F 1020.4 145.5 FF 145.5 F 0.034 0.034 0.902 0.902 C2j,k

Additionally, the interdependence between the six parameters should also be identified. The Additionally, the interdependence between the six parameters should also be identified. correlation between every two parameters is tested with the null hypothesis that the tested two The correlation between every two parameters is tested with the null hypothesis that the tested parameters are linearly independent. Similar to a covariance matrix, the p-value (possibilities of the two parameters are linearly independent. Similar to a covariance matrix, the p-value (possibilities null hypothesis) of each test is listed in the symmetric matrix in Table 3 and located by the column of the null hypothesis) of each test is listed in the symmetric matrix in Table 3 and located by the and the row labelled by the two corresponding parameters. The values on the diagonal are not column and the row labelled by the two corresponding parameters. The values on the diagonal are not important, because they correspond to the self-dependence and must be 1. As all the p-values are lower than 50%, the null hypotheses cannot be declined. The distributions of the parameters can be considered as linearly independent of each other.


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Table 3. The matrix of p-Values obtained from correlation tests of the six parameters. p-Values of Correlation Tests

Qj,k

R0j,k

R1j,k

C1j,k

R2j,k

C2j,k

Qj,k R0j,k R1j,k C1j,k R2j,k C2j,k

1.00 0.04 0.21 0.22 0.33 0.10

0.04 1.00 0.08 0.18 0.40 0.35

0.21 0.08 1.00 0.00 0.18 0.00

0.22 0.18 0.00 1.00 0.23 0.00

0.33 0.40 0.18 0.23 1.00 0.31

0.10 0.35 0.00 0.00 0.31 1.00

Based on the discussions above and the statistic indices, the properties of the cells in Monte-Carlo experiments could be generated by the distributions in Equation (5). r0j,k and C2j,k comply with normal distributions, represented by N in Equation (5), while the other four properties are generated according to the skew normal distributions, represented by SKEWN in Equation (5). In Equation (5), µ1 , µ2 and σ1 , σ2 are respectively the mean values and standard deviations of the two normal distributions. ω 1 –ω 4 , α1 –α4 and ε1 –ε4 are the parameters of the skew normal distributions. In the possibility density function (PDF) of the skew normal distribution, Φ and φ are respectively the cumulative distribution function (CDF) and the PDF of a standard normal distribution N(0,1). The values of these parameters are selected to ensure that the distributions have the same mean values, standard deviations and skewness values as those of the measured data. The detailed values and calculation processes are not given in this paper due to the length limit.  Q j,k ∼ SKEW N (ω1 , α1 , ε 1 )      R0j,k ∼ N (µ1 , σ1 2 )    R 1 j,k ∼ SKEW N ( ω2 , α2 , ε 2 )  C1 j,k ∼ SKEW N (ω3 , α3 , ε 3 )     R2 j,k ∼ SKEW N (ω4 , α4 , ε 4 )    C2 j,k ∼ N (µ2 , σ2 2 ) in which −ε )) SKEW N (ω, α, ε) ∼ PDF : ω2 φ( x )Φ(α( xω

(5)

4.2. Simulative Evaluations with Monte-Carlo Experiments Using the distributions in Equation (5), battery packs composed of randomly generated NCR18650PF cells are simulated to evaluate the influences of the parallel number in terms of efficiency and degradation. Firstly, battery packs with 36p108s, 18p108s, 9p108s, 4p108s, 2p108s and 1p108s configurations are generated. The parameter of every cell in all tested packs is randomly generated according to the distribution derived in Section 4.1. In addition, 500 battery packs of each configuration (3000 packs in total) are generated to form enough samples in the Monte-Carlo experiments. The series connection numbers of all the pack configurations are kept constantly 108 to reach the required nominal voltage 400 V, while the parallel number starts at 36, half of the parallel number of the Tesla Model S battery pack, then declines following an equal ratio progression until 1 to make the difference brought by different parallel numbers more noticeable. The parallel number 9 is followed by 4 instead of 4.5, because the parallel number should be integer. Secondly, to evaluate the efficiency of different configurations, a discharge simulation from 100% to 20% of SOC, repetitive using the C-rate curve in Figure 5 for 10 times, is performed on every battery of the 3000. The total energy dissipated by the BMS balancing and the total energy loss inside the cells of every battery pack are recorded. To obtain the proportions, the two losses count in the total output, they are then divided by the net output energy of the battery packs. In that way, the losses of different battery packs can be compared. The medians, means, minimums and maximums of the proportions of the two losses over the parallel numbers are plotted in Figure 8a,b.

to 20% of SOC, repetitive using the C-rate curve in Figure 5 for 10 times, is performed on every battery of the 3000. The total energy dissipated by the BMS balancing and the total energy loss inside the cells of every battery pack are recorded. To obtain the proportions, the two losses count in the total output, they are then divided by the net output energy of the battery packs. In that way, the losses of World Electricbattery Vehicle Journal 15 different packs2018, can9, 8be compared. The medians, means, minimums and maximums 11ofofthe proportions of the two losses over the parallel numbers are plotted in Figure 8a,b.

(a)

(b)

Figure 8. The influence of the parallel number on the losses of battery packs; (a) The influence on the Figure 8. The influence of the parallel number on the losses of battery packs; (a) The influence on the Battery Management System (BMS) balancing loss; (b) The influence on the total in-cell energy loss. Battery Management System (BMS) balancing loss; (b) The influence on the total in-cell energy loss.

In the results, it is observed that as the parallel number increases, the energy consumed by the In the of results, is observed as the the energy consumed by balancing BMS itdecreases. Thethat reason is parallel that thenumber sum ofincreases, several independent and identical the balancing of BMS decreases. The reason is that the sum of several independent and identical distributions always has a lower relative standard deviation (standard deviation divided by mean distributions always lower relative standard deviation mean value) compared to has thea original distribution as deviation deduced (standard in Equation (6). SDdivided means by standard value) compared distributionfunction. as deduced Equation (6). SD means deviation while to thethe E original is the expectation Theinfirst sub equation showsstandard that thedeviation standard while the E is the expectation function. The first sub equation shows that the standard deviation of the deviation of the sum value counts for√a portion of σ/μ√ in the mean value, while in the second sum value counts for a portion σ/µ variable N in the meanfor value, whileof inσ/μ the in second sub the sub equation the deviation of a of single counts a portion the mean value, which √ equation deviation of a single variable counts for a portion of σ/µ in the mean value, which is N times higher is √ times higher than that of the summed value. In Equation (6) X1, X2 …, XN are not physical than that ofinthe Equation X1 , Xof physicaldistributed variables invariables the battery 2 . .independent . , XN are not normal variables thesummed battery value. model,Inbut just a (6) group to model, group of independent variables prove that theof deviation of prove but thatjust thea deviation of the sum normal value isdistributed much smaller thantothe deviation individual the sum value is much smaller thaninthe deviation individual variables. Connecting moredifferent cells in variables. Connecting more cells parallel willoflower the capacity difference between parallel will lower the capacity difference between different rows of paralleled cells, and thus reduces rows of paralleled cells, and thus reduces the proportion of the energy consumed by the balancing the proportion of the BMS. of the energy consumed by the balancing of the BMS.  N NN   SD(N∑   Xi))/E (( ∑ XXi ) )== √σσ  SD ( X / E µ N i i 





i =1 i =1 μ N (6) i =1σ =1  SD(i X  i ) /E ( X ) = µ   σ (6)  SD  X , X , ..., X ∼ M (µ, σ2 )  1( X i2) / E ( NX ) = μ  2 Additionally, an increasing number paralleled cells to result in a higher loss in the XN ~ M ( μ ,isσexpected )  X 1 , Xof2 ,...,

cells, because as the parallel number increases, a cell with the properties significantly different from an increasing of paralleled cells is expected to result in a higher loss in thoseAdditionally, of the other cells in parallel number is more likely to appear. This cell intensifies the energy exchanges the cells,the because as the increases, a cell properties significantly different between cells and thusparallel makes number the energy loss inside thewith cells the higher. fromThe those of the other cells inand parallel is more likely to appear. This cell intensifies the energy curves of the maximum minimum values manifest that the distribution of the total in-cell exchanges between the cells and thus makes the energy loss inside the cells higher. loss tends to converge as the parallel number increases in the Monte-Carlo experiments. Therefore, The the curves the maximum minimum values manifest that the distribution total inalthough totalofin-cell loss tendsand to grow together with the parallel number, this lossofisthe also more cell losstotends to far converge the parallel number increases in the Monte-Carlo experiments. unlikely diverge from theasexpectation value. Therefore, although the total loss tends to grow with parallel number, this loss is However, the influence ofin-cell the parallel number on thetogether efficiency of athe battery pack is not significant, also unlikely to diverge from the 0.1% expectation value. as themore changes of losses count far lower than in the net output energy when the parallel number However, the influence of the parallel number on efficiency a battery pack is not varies. Nonetheless, if the small influences accumulate over athe longer time, itof is possible that the aging significant,ofasthe thebattery changes of are losses count lower than 0.1% in the net output energy when the behaviours packs different. Therefore, as the third step of the Monte Carlo experiment, one battery pack with respectively 72p1s, 36p1s, 18p1s, 9p1s, 4p8s, 2p1s and 1p1s configuration is generated and simulated for 500 discharging cycles (the cycle life of Panasonic NCR18650PF cells) to observe the long term influence of the parallel number. In each cycle of simulation, the load curve in Figure 5 is repeated for 10 times to discharge the battery pack from 100% SOC to 20% SOC. The degradation model in [17] is used.


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Only one pack of each configuration is simulated, because of the long simulation time of the degradation test. In addition, the series connection number of battery packs is reduced to one, because only the cells in parallel can influence each other in terms of degradation and one parallel row is thus already enough to demonstrate the aging behaviours. The simulation time is also in this way further reduced. The simulation contains no temperature model and the temperature of the cells is set to be constantly 25 ◦ C. Therefore, this simulation implies the assumption that the battery packs have a good cooling system to maintain the constant temperature. The statistic indices of the capacity and the inner resistance of all cells in each battery pack before and after the aging test are listed in Table 4. The aging curves of the average capacity and the average inner resistance after every 100 cycles are plotted in Figure 9. Table 4. The capacity and the inner resistance of cells in each battery pack before and after the degradation test. Configuration of Battery Packs

New

After 500 cycles test

1p1s

2p1s

4p1s

9p1s

18p1s

36p1s

72p1s

E(Qj,k ) SD(Qj,k )

2.910 Ah -

2.915 Ah 0.0133

2.910 Ah 0.0058

2.918 Ah 0.0062

2.913 Ah 0.0073

2.913 Ah 0.0076

2.914 Ah 0.0082

E(r0j,k ) SD(r0j,k )

38.16 mΩ -

34.51 mΩ 0.0033

34.91 mΩ 0.00451

36.42 mΩ 0.00263

35.64 mΩ 0.00242

33.94 mΩ 0.00269

34.26 mΩ 0.00284

E(Qj,k ) E(Qj,k ) change SD(Qj,k )

2.372 Ah −18.48% -

2.361 Ah −19.02% 0.0255

2.362 Ah −18.84% 0.0301

2.367 Ah −18.89% 0.0177

2.365 Ah −18.81% 0.0113

2.365 Ah −18.82% 0.0160

2.365 Ah −18.84% 0.0185

E(r0j,k ) E(r0j,k ) change SD(r0j,k )

53.33 mΩ 39.77% -

48.90 mΩ 41.69% 0.00512

49.00 mΩ 40.36% 0.00513

51.33 mΩ 40.93% 0.00357

50.10 mΩ 40.58% 0.00308

47.75 mΩ 40.70% 0.00343

48.22 mΩ 40.75% 0.00352


(a)

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(b)

Figure 9. The aging process of the average capacity and the average inner resistance of each battery Figure 9. The aging process of the average capacity and the average inner resistance of each battery pack; (a) Aging process of the average capacity; (b) Aging process of the average inner resistance. pack; (a) Aging process of the average capacity; (b) Aging process of the average inner resistance.

Another noteworthy phenomenon is that the degradation of the 1p1s battery pack (single cell) It is observed theother influence of the parallel number is more in terms of degradation. is slower than thatthat of the battery packs as seen in Figure 9.conspicuous To verify if this is a coincidence of In Table 4 the highest increase of the mean inner resistance is 41.69% in the 2p1s battery pack while the a single sample, another 15 single cells, i.e., 1p1s battery packs, are simulated. The maximum inner lowest increase is 39.77% in the reduction 1p1s battery (single cell). Theand difference is which aroundare 2%.still lower resistance growth and capacity arepack respectively 40.31% −19.76%, Moreover, as the parallel number increases, instead of a monotonic trend, Figure 9a,b shows this that than the corresponding changes of the other battery packs tested in this paper. Therefore, the changes of the average inner resistance and the average capacity tend to converge (to respectively phenomenon can be confirmed as a general situation. An explanation is that a single cell is not around 40.7% and −18.8% in this after 500 cycles and of degradation test. The convergence is also involved in any extra charging or paper) discharging processes thus has a lower cycle aging. supported by the standard deviations: When the parallel number is higher than nine, the post-test Based on the results collected in Monte-Carlo experiments, it can be concluded that the parallel standard of the capacity and the inner resistance of cellsof area significantly lower than those number isdeviations able to influence the efficiency and the degradation battery pack. If the parallel in the 2p1s and 4p1s packs, although the initial deviations of the 2p1s and 4p1s packs can be lower. number increases, the BMS balancing loss tends to decrease while the energy loss inside the cells This means cell properties do not tend to diverge to follow consistent pattern the tends to rise.the However, when evaluating battery packsbut composed ofahomogeneous cells when as in this paper, the influence on the efficiency is only marginal. The influence on the degradation is in comparison more conspicuous. If more cells are paralleled in a battery pack, the degradation behaviours of the battery pack tend to be more predictable. 4.3. Discussion


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parallel number is higher. The battery pack properties will thus also distribute in narrower intervals after the degradation test. Although only one pack of each configuration is simulated in the aging test, the trend of convergence can still be confirmed by the degradation of the battery pack properties and the cell properties. Therefore, it can be concluded that the degradation behaviour of battery packs with a higher parallel number is more consistent or, in other words, more predictable. A possible explanation is that the energy exchange caused by the few particularly weak or strong cells is shared by more cells when the parallel number increases. The influence of these few cells on the aging of the battery pack degradation is thus limited. Another noteworthy phenomenon is that the degradation of the 1p1s battery pack (single cell) is slower than that of the other battery packs as seen in Figure 9. To verify if this is a coincidence of a single sample, another 15 single cells, i.e., 1p1s battery packs, are simulated. The maximum inner resistance growth and capacity reduction are respectively 40.31% and −19.76%, which are still lower than the corresponding changes of the other battery packs tested in this paper. Therefore, this phenomenon can be confirmed as a general situation. An explanation is that a single cell is not involved in any extra charging or discharging processes and thus has a lower cycle aging. Based on the results collected in Monte-Carlo experiments, it can be concluded that the parallel number is able to influence the efficiency and the degradation of a battery pack. If the parallel number increases, the BMS balancing loss tends to decrease while the energy loss inside the cells tends to rise. However, when evaluating battery packs composed of homogeneous cells as in this paper, the influence on the efficiency is only marginal. The influence on the degradation is in comparison more conspicuous. If more cells are paralleled in a battery pack, the degradation behaviours of the battery pack tend to be more predictable. 4.3. Discussion As proved by the Monte-Carlo experiments, the parallel number cannot significantly influence the efficiency of a battery pack. The reason is that the significance is limited by the highly consistent cell properties, as all the distributions of the properties are obtained from the measurement results of high quality cells. If cells with higher variance are used to design a battery pack, the influence on efficiency should be considered. Secondly, although the degradation simulation shows that the single cell has a slower aging process than that of the paralleled cells, it does not mean reducing the parallel number to one is an optimal choice. On the one hand there are more factors, e.g., the cost and reliability, that need to be considered. On the other hand, the degradation simulation largely depends on the degradation model and the temperature model. The conclusion could be changed if different models are selected to adapt to different scenarios of application. Nonetheless, although the simulation in this paper does not have a temperature model and the universality of the conclusions could thus be limited, the proposed approach can still be used to further evaluate the configuration of battery packs in other scenarios by implementing more detailed or more specific models, because of the compatibility of the battery pack model. 5. Conclusions The paper proposes a model for the simulation of battery packs, the accuracy of which is verified by experimental results. Using the proposed model and the statistic features of the battery cell properties, different battery packs are simulated in Monte-Carlo experiments to evaluate the potential influence of paralleling different numbers of the cells of the same type in a battery pack. When the parallel number changes, the influence on the degradation and the efficiency of the battery packs can be observed. The influence is not significant in terms of efficiency because the cell properties have a high homogeneity. The influence on the degradation is in comparison more conspicuous. A higher parallel number will make the degradation behaviours of the battery packs more predictable.


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The major contribution of this paper is the proposal of the battery pack model and the statistic approach to evaluate battery packs. In a practical design, if the statistical features of different battery cells are known, the proposed model and the approach can also be used to select the cells and correspondingly the configuration of the battery pack to achieve the best performance in terms of efficiency and degradation. Author Contributions: F.C. is the major author of the paper. He proposed the battery pack model and the approach to evaluate battery packs randomly generated cells in Monte-Carlo experiments. F.R. designed the experimental platforms and conducted the verification experiments. M.B. provided the capacity and inner resistance of the Panasonic NCR18650PF cells. M.L. made an essential contribution to the conception of the research project. He revised the paper critically for important intellectual content. M.L. gave final approval of the version to be published and agrees to all aspects of the work. As a guarantor, he accepts responsibility for the overall integrity of the paper. Acknowledgments: This work was financially supported by the Singapore National Research Foundation under its Campus for Research Excellence and Technological Enterprise (CREATE) programme. Conflicts of Interest: The authors declare no conflicts of interest

References 1. 2. 3. 4. 5.

6. 7. 8. 9. 10.

11.

12. 13.

14.

Brand, M.J.; Hofmann, M.H.; Steinhardt, M.; Schuster, S.F.; Jossen, A. Current distribution within parallel-connected battery cells. J. Power Sources 2016, 334, 202–212. [CrossRef] Cui, X.; Shen, W.; Zhang, Y.; Hu, C.; Zheng, J. Novel active LiFePO4 battery balancing method based on chargeable and dischargeable capacity. Comput. Chem. Eng. 2017, 97, 27–35. [CrossRef] Zhang, J.; Ci, S.; Sharif, H.; Alahmad, M. Modeling discharge behavior of multicell battery. IEEE Trans. Energy Convers. 2010, 25, 1133–1141. [CrossRef] Miyatake, S.; Susuki, Y.; Hikihara, T.; Itoh, S.; Tanaka, K. Discharge characteristics of multicell lithium-ion battery with nonuniform cells. J. Power Sources 2013, 241, 736–743. [CrossRef] Gong, X.; Xiong, R.; Mi, C.C. Study of the Characteristics of Battery Packs in Electric Vehicles with Parallel-Connected Lithium-Ion Battery Cells. In Proceedings of the 2014 Twenty-Ninth Annual IEEE Applied Power Electronics Conference and Exposition (APEC), Fort Worth, TX, USA, 16–20 March 2014; pp. 3218–3224. Gogoana, R.; Pinson, M.B.; Bazant, M.Z.; Sarma, S.E. Internal resistance matching for parallel-connected lithium-ion cells and impacts on battery pack cycle life. J. Power Sources 2014, 252, 8–13. [CrossRef] Bouchhima, N.; Schnierle, M.; Schulte, S.; Birke, K.P. Erratum: Corrigendum to ‘Active model-based balancing strategy for self-reconfigurable batteries’. J. Power Sources 2016, 326. [CrossRef] Campestrini, C.; Keil, P.; Schuster, S.F.; Jossen, A. Ageing of lithium-ion battery modules with dissipative balancing compared with single-cell ageing. J. Energy Storage 2016, 6, 142–152. [CrossRef] Shi, W.; Hu, X.; Jin, C.; Jiang, J.; Zhang, Y.; Yip, T. Effects of imbalanced currents on large-format LiFePO4/graphite batteries systems connected in parallel. J. Power Sources 2016, 313, 198–204. [CrossRef] Wei, J.; Dong, G.; Chen, Z. On-board adaptive model for state of charge estimation of lithium-ion batteries based on Kalman filter with proportional integral-based error adjustment. J. Power Sources 2017, 365, 308–319. [CrossRef] Hua, Y.; Cordoba-Arenas, A.; Warner, N.; Rizzoni, G. A multi time-scale state-of-charge and state-of-health estimation framework using nonlinear predictive filter for lithium-ion battery pack with passive balance control. J. Power Sources 2015, 280, 293–312. [CrossRef] Sun, F.; Xiong, R. A novel dual-scale cell state-of-charge estimation approach for series-connected battery pack used in electric vehicles. J. Power Sources 2015, 274, 582–594. [CrossRef] Zheng, Y.; Ouyang, M.; Lu, L.; Li, J.; Han, X.; Xu, L.; Ma, H.; Dollmeyer, T.A.; Freyermuth, V. Cell state-of-charge inconsistency estimation for LiFePO4battery pack in hybrid electric vehicles using mean-difference model. Appl. Energy 2013, 111, 571–580. [CrossRef] Plett, G.L. High-performance battery-pack power estimation using a dynamic cell model. IEEE Trans. Veh. Technol. 2004, 53, 1586–1593. [CrossRef]


15. 16. 17.

15 of 15

Zhong, L.; Zhang, C.; He, Y.; Chen, Z. A method for the estimation of the battery pack state of charge based on in-pack cells uniformity analysis. Appl. Energy 2014, 113, 558–564. [CrossRef] Intel. Intel Math Kernel Library. Reference Manual; Intel: Santa Clara, CA, USA, 2003. Schmalstieg, J.; Käbitz, S.; Ecker, M.; Sauer, D.U. A holistic aging model for Li(NiMnCo)O2 based 18650 lithium-ion batteries. J. Power Sources 2014, 257, 325–334. [CrossRef] © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).