Extending the battery model from a single cell to a

ECAI 2015 - International Conference – 7th Edition Electronics, Computers and Artificial Intelligence 25 June -27 June, 2015, Bucharest, ROMÂNIA

Extending the battery model from a single cell to a battery pack with BMS Bogdan-Adrian Enache, Magdalena Emilia Alexandru, Luminita-Mirela Constantinescu Department of Electronics, Computer and Electrical Engineering University of Pitesti Pitesti, Romania [email protected] Abstract – This paper presents the means for extending a battery model from a single cell to a battery pack with Battery Management System (BMS. Several modeling strategies are analyzed and a new one based on the cell with the minimum capacity is developed and implemented. Keywords- battery model, LFP battery, BMS, series connection, parallel connection

I. INTRODUCTION A multicellular battery is made up of a big number of cells connected in series and/or in parallel. A group of several cells in parallel form a battery unit and several units connected in series form the entire battery. The extension of the model from one cell to the whole battery cannot be made directly because there are a series of factors which need to be taken into consideration and which belong to the structure of the entire battery. Among them the most important ones are: intrinsec imbalances between individual cells, the connections between them, the cooling method of the battery, the management system (BMS), etc. In order to avoid these inconveniences there were developed a series of modelling strategies specific to multicellular batteries. The first strategy involves the modelling of each component cell and the making up the battery model based on the connections between them [1]. This method is of a great accuracy but at the same time it has a great complexity degree because it treats all nonuniformities between the cells in an individual way. Even though it is successfully applied for batteries with a reduced number of cells, it is inadequate for the batteries in the automotive industry which uses batteries with large number of cells. At the opposite pole it is the second modelling strategy which treats multicellular battery as a whole and determines its model as if it were a single cell [2]. This strategy is quickly to implement and naturally includes all the differences between the cells but does not offer any information about their state at a certain moment. The third strategy is based on the extension of the model from a single cell to the entire battery. In [3] it is suggested such a method based on a statistical

approach. It is started from the Thévenin model of one single cell and then there are analyzed the differences between the cells in order to extend the model. These differences are mainly of two types: caused by the non-uniformities of the cell capacities which are reflected in the battery’s overall capacity and differences of parameters which influence the battery’s behavior when discharging. In order to assess these differences the author introduces the concept of normalized polarization resistance according to the capacity. By performing the division between polarization resistance and normalized polarization resistance it is proved that there is a very narrow range of ratios for the 100 cells LiCoO2 of 300 mAh analyzed. This report is the basis for approximating the capacity, internal resistance and polarization resistance for each cell. In order to validate the method a model for a group of 3 cells connected in series is developed. The results obtained after the modelling are compared to the ones obtained after discharging a 3 cells battery pack at constant current from the range [In/3, In]. Errors reported by the suggested model are within the limits of 1.5 – 2 %. Another strategy to extend the model from a single cell to a multicellular battery is based on screening techniques to choose batteries within a group which have the same electrochemical properties [68], [69], [70]. The screening process takes place in two stages. In phase 1, batteries are classified according to their effective capacity and are kept only those which show differences less than ±5% of the capacity. After this process the number of batteries kept is reduced to almost a half [4], [5]. In phase two, batteries are classified according to internal resistance rate and are kept only those showing variations less than ±1%. After this stage, the number of batteries is again reduced by 20% [4], [6]. When the screening process is over [4] proves that the parameters of a cell and the parameters of a multicellular battery are: OCVseries  m  OCV Riseries  m  Ri

(1)

Rpseries  m  Rp

OCV parallel  OCV Ri parallel  Ri Rp parallel

n Rp  n

(2)

2

Bogdan-Adrian Enache, Magdalena Emilia Alexandru, Luminita-Mirela Constantinescu

where OCVseries, OCVparallel, Riseries and Rpseries are the parameters of the multicellular battery, OCV, Ri and Rp are the parameters of a single cell, and m and n represent the number of cells in series, respectively in parallel. The presented relations are experimentally validated by using an algorithm based on Extended Kalman Filter in order to determine the SoC of a LiIon battery with m cells in series and n cells in parallel, reporting an error of less than 5%. Similar screening techniques are also presented in [6] for a series of LiFePO4 (LFP) cells, where, because of the strongly non-linear feature of these batteries, (1) and (2) no longer present the same accuracy for modelling. In order to eliminate these problems, a new strategy based on the “Mean – plus- Difference” technique is suggested. Thus, for the same cells two models are built. A model which represents the mean behavior of the cells – the Thévenin model and a submodel of differences obtained based on spectroscopic analysis in low frequency of internal resistance. The model obtained is validated for a battery with 8 cells in series submitted to discharge after 7 different driving cycles obtaining approximately 3% error for the 70 – 40% State of Charge (SoC) range. All these techniques are based on the achievement of multicellular batteries formed of cells which are very close as electrochemical properties. This situation is very rarely found in practice because battery manufacturers also use cells which have capacity differences exceeding ±5% when batteries are made. A way to solve this issue is the quantitatively and qualitatively analysis of the differences which occur between a battery’s cells [7], [8]. Thus, in [7], it is started from the analytical model of a cell – (3), after which, by using a filter with particles there are approximated the parameters for the first cell reaching the maximum admissible voltage upon charging and the first cell reaching the minimum admissible voltage upon discharging. Based on these values it is built the model for a multicellular battery with different variants of BMS demanding to keep those rates. k1  k2  SoC (t )  SoC (t ) k3  ln( SoC (t ))  k4  ln(1  SoC (t ))  R  i (t )

V  t   k0 

and for a parallel connection of m batteries, the bigger the number of cells is, the less significant the influence of the different cell is. The main contributions of the paper are represented by the novel strategy of extending the battery model from the minimum capacity cell to the entire battery pack (Section 6). The starting point is represented by the third order RC network model of the cell (Section 2) which is proven to be the best model for EV applications in ours latest paper [9]. After which the influence of the BMS (Section 3) and the connection between cells – series (Section 4) and parallel (Section 5) are analyzed. II. THE COMPONENCE OF THE BATTERY PACK In order to study the influence of the minimum capacity cell over a battery pack 10 fresh 3.3 V and 1400 mAh LFP cells were purchased. They were numbered from LFP1 to LFP10 and for each of them it was determined the effective capacity according to the USABC procedure [10] – Fig.1, and the parameters necessary for the third order RC network model – Fig. 2. The parameters were determined in concordance with the HPPCT procedure [11], but only for discharge. After each discharge the batteries were charged in concordance with the manufacturer’s instructions and then they were left to rest at least one hour.

Figure 1. Cells efective capacity

R1

R2

R3

C1

C2

C3

R0

(3)

where ki are the model’s coefficients, SoC – state of charge, i –discharging current; R – internal resistance of the battery. In [8] it is presented the modelling of a multicellular battery of a special construction which includes n identical cells and a cell different from the point of view of capacity ad internal resistance. There are analyzed two distinct situations: one in which the different cell has a capacity larger and an internal resistance smaller than the rest of the cells and another in which the capacity of the cell is smaller and the internal resistance is larger. By using these cells there are achieved different battery topologies proving that for a series connection of n batteries the group’s capacity is given by the cell with a minimum capacity

E0

RL

Figure 2. Third order RC network battery model

The parameters involved in the third order RC network model are: E0 – voltage source equal to the cell Open Circuit Voltage (OCV), R0 – cell internal resistance, R1, R2 and R3 – polarization resistances and C1, C2 and C3 – the capacitors from the RC branches. The time constants of the three RC branches are τ1 = 10 s, τ2 = 3 s and τ3 = 5 s chosen according to the evolution of the battery during discharging. The evolution of parameters C1, C2 and C3 will not presented because it is similar to the evolution of parameters R1, R2 and R3 as they are calculated based on the branch time constant.

3

Extending the battery model from a single cell to a battery pack with BMS 3.55

3.4

0.05

0.04 R3 [ohm]

3.45

E0 [V]

0.06

LFP1 LFP2 LFP3 LFP4 LFP5 LFP6 LFP7 LFP8 LFP9 LFP10

3.5

3.35

3.3

0.03

0.02 3.25

3.2

0.1

0.01

0.2

0.3

0.4

0.5 0.6 SoC [%]

0.7

0.8

0.9

1

0 0.1

0.2

0.3

0.4

0.5 0.6 SoC [%]

0.7

0.8

0.9

1

Figure 3. The evolution of the parameter E0 for different SoC Figure 7. The evolution of the parameter R3 for different SoC 0.26

0.24

0.22

R0 [ohm]

The battery balancing and management functions were achieved with 2 BMS 3S (for 3 cells in series) and 1 BMS 4S (for 4 cells in series). The BMS used are D-type (dissipative) and achieve the balance by dissipation of the cell energy when the voltage of a cell is greater than the average voltage.


0.2

0.18

III.

0.16

0.14

0.12 0.1

0.2

0.3

0.4

0.5 0.6 SoC [%]

0.7

0.8

0.9

1

Figure 4. The evolution of the parameter R0 for different SoC

BATTERY PACKS WITH SERIES CONNECTIONS

After an extensive experimental analysis it was determined that the cell with the minimum capacity is the LFP6 cell – Fig. 1. So in order to analyze its influence over a series connection three multicellular batteries were assembled: S1: LFP1 + LFP5 + LFP6, S2: LFP1 + LFP3 + LFP4 + LFP6 and S3: LFP1 + LFP2 + LFP3 + LFP4 + LFP5 + LFP6, provided with BMS (3S, 4S and 3S+3S) and without it.

0.1 LFP1 LFP2 LFP3 LFP4 LFP5 LFP6 LFP7 LFP8 LFP9 LFP10

0.09 0.08 0.07

R1 [ohm]

0.06

The characteristic of the series connection is that it has the same discharge current through each cell. In this case, if the cells were identical, the relation between the group parameters and component cells would be in accordance with (1). However, in reality, imbalances between them make the application of (1) not possible.

0.05 0.04

In order to assess the total influence of the LFP6 cell over the batteries two different categories of influences were analyzed: influences over the battery capacity and influences over the model parameters.

0.03 0.02 0.01 0 0.1

0.2

0.3

0.4

0.5 0.6 SoC [%]

0.7

0.8

0.9

1

Figure 5. The evolution of the parameter R1 for different SoC 0.045


0.04 0.035

R2 [ohm]

0.03 0.025 0.02

A. Influence of the minimum capacity cell over the capacity of a series connection A battery’s capacity is the quantity of electric charge which can be stored within the battery. Theoretically, in the case of a series multicellular battery, the battery’s capacity is given by the sum between the minimum capacity which can be charged and the minimum capacity which can be discharged [71]: Cseries  min  SoCi  Ci   min   SoCi  1  Ci 

0.015

1i  m

1i  m

(4)

0.01 0.005 0 0.1

0.2

0.3

0.4

0.5 0.6 SoC [%]

0.7

0.8

0.9

1

Figure 6. The evolution of the parameter R2 for different SoC

where Cseries – capacity of the series connection, Ci – capacity of the i cell, SoCi – state of charge of the i cell, m – number of cells connected in series. Considering the case of the 3 batteries S1, S2 and S3 without BMS, their theoretical capacity when all the cells are charged is 1.4438 Ah, given by the cell with the minimum capacity LFP 6. After the

4


experimental measurement of battery capacity (according to the USABC procedure), it was noticed that in all the cases the capacity obtained exceeded the theoretical capacity – Table I.

In order to prevent this over-discharging the batteries were equipped with BMS, which besides monitoring the battery parameters it also records the behavior of each individual cell and the experimental test were repeated. The new results are presented in Table II and from them it can be seen that the measured capacities are much closer to the theoretical capacity because, when the minimum capacity cell reaches the minimum admissible voltage, even if the voltage at the battery terminals is higher than the 7.5 V threshold, the BMS cuts the load and the discharge process is stopped.

13.3 13.2

E0[V]

13.1 13 12.9 12.8 12.7 12.6 12.5 0.1

0.2

0.3

0.4

WITHOUT BMS

Battery

Theoretical capacity [Ah]

S1

1.455056

1.4438

S2

1.444722

1.4438

S3

1.460538

1.4438

TABLE II.

0.7

0.8

0.9

1

0.55 S1 S1BMS 0.5

0.45

0.4

0.35

0.2

0.3

0.4

THE CAPACITY OF THE BATTERIES S1, S2 AND S3 Measured capacity [Ah]

0.5 0.6 SoC[%]

Figure 8. Comparisson analysys of the influence of the BMS over the E0 parameter for S2 battery

0.1

TABLE I.

S2 S2BMS

13.4

R0[ohm]

After a closer examination upon the phenomena which occurred within the battery S1 it was noticed that when reaching the minimum admissible voltage during discharge 7.5 V, the LFP6 cell exceeded the threshold of its own minimum admissible voltage of 2.5 V and its own capacity. The same over-discharging phenomenon of the minimum capacity cell was also found with the rest of the batteries and it is responsible for exceeding theoretical capacity.

13.5

0.5 0.6 SoC[%]

0.7

0.8

0.9

1

Figure 9. Comparisson analysys of the influence of the BMS over the R0 parameter for S1 battery

0.12 S1 S1BMS 0.1

THE CAPACITY OF THE BATTERIES S1, S2 AND S3

0.08

Battery

Measured capacity [Ah]


S1

1.440964

1.4438

S2

1.449214

1.4438

S3

1.447421

1.4438

R1[ohm]

WITH BMS 0.06

0.04

0.02

0 0.1

B. Influence of the BMS over the parameters of a series connection In the case of a multicellular battery which includes a series connection and is equipped with BMS, the battery parameters are influenced both by the component cells and by the management system. In these conditions, to evaluate only the influence of the minimum capacity cell it is required to initially determine the BMS influence over the battery parameters. For that, there were obtained the parameters of batteries S1, S2 and S3 in both cases with BMS and without.

0.2

0.3

0.4

0.5 0.6 SoC[%]

0.7

0.8

0.9

1

Figure 10. Comparisson analysys of the influence of the BMS over the R1 parameter for S1 battery

By analyzing Fig. 8 it can be noticed that the evolution of the voltage source E0 is very similar in the two cases with BMS and without. The maximum relative error registered is below 0.5%, less than the error introduced by the equipment used for discharge, thus the influence of the BMS is insignificant. On the other hand, over the internal resistance R0 Fig. 9, the BMS has a strong influence mainly because the way of its connection – in series with the battery. Due to the semiconductor elements used by the BMS it influences not only the value of R0, but also its evolution. In order to quantify the effects produced by the BMS it was determined the difference in value for the two situations and for each SoC and then it was determined the average rate of these differences. This value was approximated to be the BMS internal

5

Extending the battery model from a single cell to a battery pack with BMS resistance and it was found to be 0.091 Ω for the 3S BMS, respectively 0.158 Ω for the BMS 4S. In the case of battery S3 – 6S which includes two distinct 3S BMS, its internal resistance is 0.186 Ω, approximately two times higher than for only one 3S BMS.

By submitting the results obtained for batteries S1, S2 and S3, in both cases with BMS and without, to a global analysis, it can be concluded that the BMS: 

Does not influence the evolution of the voltage source E0;



It behaves as a 0.091 Ω (BMS 3S) resistance, respectively of 0.158 Ω (BMS 4S) connected in series with the internal resistance of the battery;



Does not significantly influence polarization resistance value.

the

After deciding the impact of the BMS over batteries S1, S2 and S3, the influence of the minimum capacity cell is determined in the situation without BMS. IV.

EXTENDING THE MODEL FROM A SINGLE CELL TO A SERIES CONNECTION BATTERY PACK

The extension of the model from a single cell to the entire battery needs to take into consideration both thermodynamic effects (voltage source and internal resistance) and the polarization effects (the resistor and capacitor from the model branches) [67]. These effects are also present within the model of the cell used, in our case, the minimum capacity cell, requiring now only extension to the entire battery.

3

2.5

2

1.5

1

3

3.5

4

4.5 No of cells

5

5.5

6

Figure 12. The influence of the minim capacity cell over the internal resitance R0

In order to establish the influence of the minimum capacity cell over the 3 batteries the ratio between the voltage source of the batteries S1, S2 and S3 and the voltage source of the LFP6 cell, as well as the ratio between internal resistance and the internal resistance rate of the minimum capacity cell for each SoC of the battery were calculated. The results shown in Fig.11 and Fig. 12, prove that the variation of the parameters with the number of cells is linear. So to determine the equations which best model this variation it should be taken into consideration that: 

The voltage source value is the smallest for the minimum capacity cell – Fig. 3;



Internal resistance rate is the greatest for the minimum capacity cell – Fig. 4;



Current going through each cell is the same (series connection) and so the value of the voltage source and the internal resistance for the battery pack can be approximated by the sum of values of the components.

In this hypothesis the relations between the value of the voltage source and internal resistance value can be described as: E0 series  E0min   E0min  c1    E0min  c2   ... E0 series  m  E0min  c1  c2  ...  cm 1 

6

E0 series

5.5 E0 serie/E0 min

3.5

  E0min  cm 1  

6.5

E0min

5

4 3.5 3

3

3.5

4 4.5 5 No of cells conected in series

5.5

(5)

 m  CSE

R0 series  R0min   R0min  c1    R0min  c2   ...

SoC100 SoC90 SoC80 SoC70 SoC60 SoC50 SoC40 SoC30 SoC20 SoC10

4.5

2.5


4

R0 serie/R0 min

In the case of the polarization resistances the influence of BMS over parameters R1, R2 and R3 is much harder to be evaluated because of their very low rates – Fig. 10. For SoC of up to 50% it can be noticed that polarization resistance has generally a higher value for the battery with BMS, while for SoC below 50% its value is lower. The average rate of the difference between the two cases is 0.0083 Ω for R1, while for R2 and R3 this difference is reduced even more reaching 10-7 Ω.

4.5

  R0min  cm 1   R0 series  m  R0min  c1  c2  ...  cm 1  R0 series 6

Figure 11. The influence of the minim capacity cell over the voltage source E0

R0min

(6)

 m  CRS

where E0series – battery voltage source, E0min – minimum capacity cell voltage source, R0series – battery internal resistance, R0min – minimum capacity cell internal resistance, c1, c2, …cm-1 – correction coefficients, m – number of cells connected in series, CES –global coefficient for the value of the voltage

6


source, CRS –global coefficient for the value of the internal resistance. The parameters CES and CRS incorporate all the imbalances between the cells and they were determined through curve fitting techniques of their average values (due to their little variance at different SoC). E0 series E0min

 m  0.01287

R0 serie R0min

 m  1.884

(7)

(8)

From (7) and (8) the global coefficients for a series connection are calculated as CES = 0.01287, respectively CRS = 1.884. Their value is strongly depend on the minimum capacity cell and, in order for them to be used by another cell it is also necessary that their value should be expressed according to the cell’s minimum capacity rate. Statistically speaking, most of the cells from a battery have close electrochemical properties. In [3] there are analyzed 100 LiCoO2 cells and 83% of them show a variation of the capacity within ±1.5%. A series of 16 cells LiFePO4 with the nominal capacity of 7.2 Ah are analyzed in [5], and 50% of them fit into a ±1% marge around the 7.6 Ah capacity (average capacity). Generally, LFP cells show effective capacities with rates higher than nominal capacity which means that if the minimum capacity cell has a rate higher than the nominal capacity then it is very close to the average capacity of the cells. Based on this Q hypothesis it is introduced the ratio nom which Qmin minimizes the effects of global parameters for the series connection as the cell’s minimum capacity gets closer to the battery average capacity. In this terms the new values for the global coefficients are expressed as:

V.

BATTERY PACKS WITH PARALLEL CONNECTIONS

In order to study the effects produced by the minimum capacity cell over a parallel connection there multicellular battery were constructed: P1 – LFP1||LFP6, P2 – LFP1||LFP5||LFP6 and P3 – LFP1||LFP3||LFP4||LFP6, where LFP6 is the minimum capacity cell. Unlike the series connection, in the case of a parallel connection the voltage of each cell is the same. In this case, if cells were identical, the relation between the parameters of the connection and the component cells would be in accordance with (2). However, in reality, non-uniformities between them lead to the occurrence of unequal discharge currents through component cells and thus o the impossibility of using (2). As with the series connection, the overall influence of LFP6 cell over the entire batteries was highlighted over the battery capacity and over the model parameters. A. Influence of the minimum capacity cell over the capacity of a parallel connection Theoretically, in the case of a parallel multicellular battery the battery’s capacity is given by the amount of the individual capacities of component cells [7]: n

C parallel   Ci

where Cparallel – capacity of the parallel connection, Ci – the capacity of the i cell, n – number of cells connected in parallel. After the experimental determinations of the capacities for the batteries P1, P2 and P3 (according to the USABC procedure), it was noticed that in all cases the capacity obtained is greater that the theoretical capacity – Table III. TABLE III.

CES  CRS 

Qnom 1.4438 Q   0.01287  nom  0.01327 (9) Qmin 1.40 Qmin Qnom 1.4438 Q  1.884  nom  1.9429 Qmin 1.40 Qmin

where Qnom = 1.40 Ah and Qmin = 1.4438 Ah.

1 p  3

1 p  3

THE CAPACITY OF THE BATTERIES P1, P2 AND P3

Battery

Measured capacity [Ah]


P1

3.041111111

2.990944444

P2

4.62525

4.510722222

P3

6.143666667

6.030305556

(10) 1.6

1.4

Teoretic LFP6 LFP1

1.2

Current [A]

A similar approach to the one used in the quantification of thermodynamic effects was also tested to determine the polarization effects. However, now, because the polarization resistances of the minimum capacity cell does not have the greatest values (compared to the rest of the cells) – Fig.5, Fig.6 and Fig. 7, this approach was inconclusive. In this situation, taking into account that most of the polarization resistance rates are very low (less than 0.05 Ω) the evolution of polarization resistances for batteries S1, S2 and S3 can be approximated according to (1): Rpseries  m  Rp

(11)

i 1

1

0.8

0.6

0.4

0.2 0.1

0.2

0.3

0.4

0.5 0.6 SoC [%]

0.7

0.8

0.9

1

Figure 13. The evolution of the discharge current through the cells of battery P1

7

Extending the battery model from a single cell to a battery pack with BMS

The inequalities of the discharging currents are kept for most of the discharging process, but around the rate of SoC = 40% electrochemical properties of the cells are getting balanced and the discharging currents are closer to the theoretical discharging current (cell self-balancing). Cell discharging at different currents influences their theoretical capacity, in concordance with Peukert’s Law and so the differences in capacities are explained. In order to determine the minimum capacity cell effects over the capacities of the three batteries it was tried an approach similar to the one used in determining the effects on the voltage source rate for a series connection. With this in mind the ratios between capacities of the batteries (Qparallel) and the capacity of the minimum cell (Qmin) were calculated. As expected Q parallel the evolution of the ratio is linear and Qmin considering (11) the evolution of the ratio can be expressed as:

Q parallel  Qmin  Qmin  c1   Qmin  c2   ... 

B. Influence of the minimum capacity cell over the parameters of a parallel connection In the same manner as the influence of the minimum capacity cell was evaluated over a series connection the issue was also addressed in the case of parallel connection. In order to determine the thermodynamic effects the ratios between the voltage source rate of the batteries P1, P2 and P3 and the voltage source rate of the LFP6 cell, as well as the ratios between the internal resistance rate and the internal resistance rate of the minimum capacity cell, for each SoC of the battery were calculated. The evolution of the voltage ratios – Fig. 14, shows very small variations regardless of the number of cells and SoC and it is between 0.9983 and 1.0416 having an average rate of 1.0067. In these conditions its value is approximated to 1, which gives to the minimum capacity cell a dominant character over the voltage of a parallel connection. 1.045 1.04


1.035 1.03

E0paralel/E0min

This phenomenon occurs because of to the inequality between the discharging currents. Thus, through the minimum capacity cell, because of the high internal resistance, the discharging current is initially lower while through the rest of the cells the discharging current has a higher rate so as to compensate the current from the minimum cell – Fig. 13.

1.025 1.02 1.015 1.01 1.005

 Qmin  cn 1  

1

Q parallel  n  Qmin  c1  c2  ...  cn 1 

(12)

Q parallel

 n  CPQ Qmin where Qparallel – capacity of the parallel connection, Qmin – minimum capacity cell, c1, c2, …cn-1 – correction coefficients, n – number of cells connected in parallel, CPQ – global coefficient for a parallel connection.

0.995

2

2.5

3 3.5 No of cells connected in parallel

4

Figure 14. The influence of the minim capacity cell over the voltage source E0

1

0.9

Q parallel Qmin

 n  0.1883

(13)

Qnom 1.4438 Q   0.1883  nom  0.19419 Qmin 1.40 Qmin

0.7

0.6

0.5

The correction parameter rate depends on the minimum capacity rate and it should be reduced as the capacity of this cell comes close of the average capacity rate. As such, the expression of the new global parameter for capacity becomes: CPQ 

R0parallel/R0min

0.8

The global parameter is deduced through curve fitting techniques and is found to be:

(14)

0.4

2

2.5

3 3.5 No of cells connected in parallel

4

Figure 15. The influence of the minim capacity cell over the internal resistance R0

8


Considering the evolution of the ratio for the internal resistance – Fig. 15 and the fact that the minimum capacity cell has the greatest internal resistance, the relation between the internal resistance of the parallel connection and the internal resistance of LFP6 cell can be described as: 1 1 1 1 1     ...  R0 parallel R0 min R0 min  c1 R0 min  c2 R0 min  cn 1 1 R0 parallel 



(R 0 min  c1 )( R0 min  c2 )...( R0 min  cn 1 )  ...  R0 min ( R0 min  c1 )...( R0 min  cn 1 )

R0 min ( R0 min  c1 )...( R0 min  cn  2  R0 min ( R0 min  c1 )...( R0 min  cn 1 ) n 1 0 min n 1 0 min 0 min

n 2 1 0 min n 2 1 0 min

Figure 16. Experimental setup

(15)

1 n( R  C R  ...  C R  Cn 1 )   R0 parallel R ( R  K R  ...  K R  K n 1 ) R0 parallel 

1 n  2 0 min 1 n  2 0 min

R R0 min 1  K RP  0 parallel   CRP n R0 min n

where R0parallel –internal resistance of the battery, R0min – internal resistance of the minimum capacity cell, c1, c2, …cn-1, C1, C2, …Cn-1, K1, K2, …Kn-1 – correction coefficients, n – number of cells connected in parallel, KRP, CRP – global parameters for the internal resistance. As in the previous cases establishing the value of the global coefficient CRP is done through curve fitting followed by expressing its value in concordance with the minimum capacity of the cell which has a big influence over it. R0 parallel R0 min

CRP 



1  ( 0.2075) n

Qnom 1.4438 Q   0.2075   nom  0.2139 Qmin 1.40 Qmin

1 p  3

Validating the models was done for three different SoC = 100%, SoC = 70% and SoC = 40% by direct comparison of the results obtained from the models with the data obtained from discharging the two batteries in concordance with the Extra Urban Driving Cycle (EUDC) and Urban Dynamometer Driving Schedule (UDDS) [12]. Based on (9), (10), (17) and Fig. 3 – 7 the parameters for the two batteries were calculated. For a better understanding of the proposed model performances the results obtained were also compared with the ones from a model developed from an average capacity cell (LFP4, QLFP4 = 1.5155 Ah), which is the closest to the average of the entire cells lot (Qaverage = 1.5065 Ah).

(16) TABLE IV.

(17)

The evolution of the polarization resistances is hard to approximate because of the non-linear feature of LFP batteries, but their very low rates leads to their approximation with an expression of the type (2): Rp parallel 

The developed batteries contain as the minimum capacity cells LFP6 which was used in the previous experiments and LFP9 which is a fresh cell.

Rp parallel

SoC

100%

70%

40%

E0 [V]

13.31796

13.23618

13.1544

R0 [Ω]

0.441496

0.44664

0.40515

R1 [Ω]

0.029118

0.048531

0.038825

R2 [Ω]

0.009706

0.038826

0.009706

R3 [Ω]

0.029119

0.029119

0.009707

TABLE V.

1 p  3

n

VI. VALIDATING THE MODEL In order to validate the proposed method for extending the model from the minimum capacity cell to the entire battery two multicellular batteries were constructed: B1(2P4S) – LFP1 || LFP6 + LFP3 || LFP4 + LFP2 || LFP5 + LFP7 || LFP8 and B2(3P3S) – LFP7 || LFP8 || LFP9 + LFP3 || LFP4 || LFP5 + LFP1 || LFP2 || LFP10 equipped with a 4S BMS and a 3S BMS.

THIRD ORDER RC NETWORK PARAMETERS FOR BATTERY B1 DEVELOPED FROM LFP6


SoC

100%

70%

40%

E0 [V]

10.00769

9.976989

9.915581

R0 [Ω]

0.180195

0.189115

0.198034

R1 [Ω]

0.033993

0.033993

0.053417

R2 [Ω]

0.02428

0.014568

0.02428

R3 [Ω]

0.004856

0.004856

0.02428

TABLE VI.


SoC

100%

70%

40%

E0 [V]

13.27723

13.19564

13.15485

R0 [Ω]

0.46879

0.439654

0.420229

R1 [Ω]

0.029136

0.009712

0.048561

R2 [Ω]

0.019425

0.029138

0.019424

R3 [Ω]

0.04856

0.019424

0.029137

9

Extending the battery model from a single cell to a battery pack with BMS TABLE VII.


10 9.8

100%

70%

40%

9.957921

9.896733

9.866139

R0 [Ω]

0.246395

0.231827

0.222114

R1 [Ω]

0.014568

0.004856

0.02428

R2 [Ω]

0.009712

0.014569

0.009712

R3 [Ω]

0.02428

0.009712

0.014569

9.6 9.4

Voltage[V]

SoC

E0 [V]

9.2 9 8.8 8.6 8.4

Ubatt measured U model LFP6 U model LFP4

13

8

0

200

400

600 800 Time [s]

1000

1200

1400

Figure 17. UDDS driving cycle for battery B2 at SoC=40%

12.5

Voltage [V]

U batt masurat U model LFP9 U model LFP4

8.2

13.5

12

TABLE VIII.

THE REPORTED ERRORS FOR EUDC DRIVING CYCLE FOR BATTERY B1

11.5

SoC

Model

Max rel error [%]

Avr rel Error [%]

LFP6 model

5.697289

3.592793

LFP4 model

8.040771

5.140201

LFP6 model

7.777310

4.867517

LFP4 model

5.847456

3.664613

LFP6 model

8.919012

3.335474

LFP4 model

10.75958

4.567450

11

100% 10.5

0

50

100

150

200 Time [s]

250

300

350

400

Figure 18. EUDC driving cycle for battery B1 at SoC=100%

70% 40%

13.5

TABLE IX.

13

Voltage[V]

12.5

SoC

12

100%

THE REPORTED ERRORS FOR UDDS DRIVING CYCLE FOR BATTERY B1 Model

Max rel error [%]

Avr rel Error [%]

LFP6 model

8.79988

2.462573

LFP4 model

11.05996

3.272907

LFP6 model

10.40136

2.993744

LFP4 model

8.521090

2.513357

LFP6 model

6.216460

1.751663

LFP4 model

14.21397

4.180929

11.5

70% 11

10.5

0

200

400

600 800 Time [s]

1000

U batt masurat U model LFP6 U model LFP4 1200 1400

Figure 19. UDDS driving cycle for battery B1 at SoC=70%

40%

TABLE X.

SoC

10.2 U batt masurat U model LFP9 U model LFP4

10

100%

9.8

Voltage[V]

9.6

70%

9.4 9.2

40%

9

THE REPORTED ERRORS FOR EUDC DRIVING CYCLE FOR BATTERY B2 Model

Max rel error [%]

Avr rel Error [%]

LFP9 model

6.854890

4.490042

LFP4 model

10.98189

7.30916

LFP9 model

6.519650

4.080071

LFP4 model

8.765851

5.68338

LFP9 model

10.0972

6.298075

LFP4 model

8.738791

5.43896

8.8 8.6

TABLE XI.

8.4 8.2

0

50

100

150

200 Time [s]

250

300

350

400

Figure 20. EUDC driving cycle for battery B2 at SoC=70%

SoC

100% 70% 40%

THE REPORTED ERRORS FOR UDDS DRIVING CYCLE FOR BATTERY B2 Model

Max rel error [%]

Avr rel Error [%]

LFP9 model

7.018686

2.175156

LFP4 model

11.19646

3.742571

LFP9 model

6.390760

1.752152

LFP4 model

8.612478

2.692797

LFP9 model

7.352972

2.380943

LFP4 model

5.887710

2.170337

10


Analyzing the obtained data Table VIII – IX it can be noticed that the average relative errors for the EUDC driving cycle are between 2.46% and 6.29%, while for the UDDS driving cycle they are significantly lower between 1.75% and 2.99%. The maximum relative errors even though they are higher picking at a 10.40% for UDDS and 10.09% for UDDS are only reached locally in the points where the effects of low voltage source a high internal resistance are corroborated (SoC = 70% for LFP6 and SoC = 40% for LFP9). VII. DETERMINING THE MINIMUM CAPACITY CELL FROM A LOT

A key component in developing the model from a single cell to the entire battery is choosing the right cell, in our case the minimum capacity cell. One way for picking tis cell might be using the USABC procedure for establishing the effective capacity for the entire lot. This method while is very accurate took for 10 LFP cells all most 90 hours of experimental determinations using a six channels electronic load, so it’s very clear that this method can’t be used for high number battery packs. In [3] another approach is proposed based on the fact that the differences between capacities are related to physical factors. In this way all the 100 LiCoO2 cells used in the experiment are weighed. Even though this method looks promising by determining the correct number of low capacities the determined ones are not the correct ones, so method is inconclusive. Enchanted by the ease of use we tried this method for our 10 LFP cells but we had the same inconclusive results. Looking at the problem form a different angle we tried to correlate the capacity of the cell with their response to a high current step impulse. The idea for this correlation started from the very high values of the internal resistance of the minimum capacity cell. For this we used an 18 second high current pulse from the HPPC test and we recorded the end voltage. Based on this voltages we obtained the following distribution of the cells:

Figure 21. End voltage for the 10 LFP cells submited to the 18 s high current pulse discharge at SoC = 100%

Even due the distribution of the cells is not identical to the one regarding capacities – Fig. 1, they

are very similar and the minimum capacity cells are correct selected. CONCLUSIONS In a battery pack with BMS the minimum capacity cell has a strong influence over the battery function. The influence is not limited to the battery capacity (the battery capacity is equal to the cell capacity), but also to the battery parameters and especially to the voltage source value. In this terms extending the cell model to the entire battery is only a curve fitting problem for which the global coefficients incorporate all the imbalances between the cells. The proposed method is validated using real data from a battery submitted to discharge but also in comparison with a model developed from the average capacity cell. In general the proposed method errors are around the 5% threshold and compared to the average capacity cell brings an improvement of 22% in the terms of maximum relative error. REFERENCES C. Sen, N-C. Kar, “Battery Pack Modeling for the Analysis of Battery Management System of a Hybrid Electric Vehicle,” Proceedings of Vehicle Power and Propulsion Conference (VPPC) 2009, pp. 207-212. [2] J. Li, M. Mazzola, “Accurate battery pack modeling for automotive applications,” Journal of Power Sources, 237, (2013), pp. 215-228. [3] M. Dubarry, N. Vuillaume, B. Liaw, “From single cell model to battery pack simulation for Li-ion batteries,” Journal of Power Sources, 186, (2009), pp. 500–507. [4] J. Kim, B.H. Cho, „Screening process-based modeling of the multi-cell battery string in series and parallel connections for high accuracy state-of-charge estimation,” Energy, 57, (2013), pp. 581-599. [5] Y. Zheng, M. Ouyang and co, “Cell state-of-charge inconsistency estimation for LiFePO4 battery pack in hybrid electric vehicles using mean-difference model,” Applied Energy, 111, (2013), pp.571–580. [6] R. Xiong, F. Sun, X. Gong, H. He, “Adaptive state of charge estimator for lithium-ion cells series battery pack in electric vehicles,” Journal of Power Sources, 242, (2013), pp.699-713. [7] L. Zhong, C. Zhang, co, “A method for the estimation of the battery pack state of charge based on in-pack cells uniformity analysis,” Applied Energy, 113, (2014), pp. 558–564. [8] S. Miyatake, Y. Susuki, T. Hikihara, S. Itoh, K. Tanaka, “Discharge characteristics of multicell lithium-ion battery with nonuniform cells,” Journal of Power Sources, 241, (2013), pp. 736-743. [9] B-A Enache, M. E. Alexandru, L-M Constantinescu, A LiFePO4 Battery Discharge Simulator for EV applications– Part 1: Determining the Optimal Circuit Based Battery Model, in Proceedings of the 9th ADVANCED TOPICS IN ELECTRICAL ENGINEERING, ATEE 2015. [10] ***, “USABC Electric Vehicle Battery Test Procedures Manual,” Revision 2, 1996. [11] ***, “PNGV Battery Test Manual,” Revision 3, 2001. [12] T.J. Barlow, S. Latham, I.S. MCCrae, P.G. Boulter, “A refernce book of driving cycles for use in the measurement of road vehicle emissions,” published project report PPR354, version 3, TRL Limited, 2009. [1]