Mathematical Modeling of Li-Ion Battery Using Genetic Algorithm Approach for V2G
2.5 Model Validation
2.5.1 Performance Characteristics of BM
Case (i) - EIG Battery: Fig. 2.11 shows the comparison of battery characteristics with 2.5V, 8Ah EIG battery at 1Cr/1Dr. The charging and discharging voltage with respect to S OCcrand DODcr are shown in Figs. 2.11 (a) and (b). The simulations are observed for different S OCcr and DODcr values from 0-100% and 100%-0%. The end of charge value (EOCV) is 4.3V and end of discharge value (EODV) is 1.3V. It is observed from Figs. 2.11 (a) and (b), the end of charge and discharge voltages are higher and lower with respect to increase in S OCcr and DODcr.
0 20 40 60 80 100
1.5 2 2.5 3 3.5
State of charge (%)
Charging voltage (V)
Calculated ( V c
C ) Measured ( V c M )
(a)
0 20 40 60 80 100
1 1.5 2 2.5 3
Depth of discharge (%)
Discharging voltage (V)
Calculated ( V d
C ) Measured ( V d M )
(b)
0 20 40 60 80 100
14 16 18 20 22 24
State of charge (%)
Charging power (W)
Calculated ( P c
C ) Measured ( P c M )
(c)
0 20 40 60 80 100
10 12 14 16 18 20 22
Depth of discharge (%)
Discharging power (W)
Calculated ( P d
C ) Measured ( P d M )
(d)
0 20 40 60 80 100
0 5 10 15 20
State of charge (%)
Stored energy (Wh)
Calculated ( E stor
C ) Measured ( E
stor M )
(e)
(e)
0 20 40 60 80 100
0 5 10 15 20
Depth of discharge (%)
Available energy (Wh)
Calculated ( E avail
C ) Measured ( E
avail M )
(f)
Figure 2.11: Comparison of proposed model result with EIG battery at 1Crand 1Dr.
It is observed from Figs. 2.11 (a) and (b), for large DODcr value the EODV is lower, while for large S OCcr value the EOCV is higher. The initial portion of the curve is linear in fashion and after
certain period it has varied. The comparison of charging (Pc) and discharging (Pd) power with respect to S OCcr and DODcr are shown in Figs. 2.11 (c) and (d). Figs. 2.11 (e) and (f) show the comparison of charging (Ec) and discharging (Ed) energy with respect to S OCcrand DODcr. It is clear from these plots that energy and power values are maximum at the end of S OCcrand DODcr value.
0 20 40 60 80 100 0 20 40 60 80
1.8 2 2.2 2.4 2.6 2.8
State of charge (%)
Charging voltage (V)
Measured ( V c
M ) Calculated ( V c C )
(a)
0 20 40 60 80 100 0 20 40 60
1.5 2 2.5
Depth of discharge (%)
Discharging voltage (V)
Measured ( V d
M ) Calculated ( V d C )
(b)
0 20 40 60 80 100 0 20 40 60 80
55 60 65 70 75 80 85 90
State of charge (%)
Charging power (W)
Measured ( P c
M ) Calculated ( P c C )
(c)
0 20 40 60 80 100 0 20 40 60
45 50 55 60 65 70 75 80
Depth of discharge (%)
Discharging power (W)
Measured ( P d
M ) Calculated ( P d C )
(d)
(d)
0 20 40 60 80 100 0 20 40 60 80
0 5 10 15 20
State of charge (%)
Stored energy (Wh)
Measured ( E stor
M ) Calculated ( E stor C )
(e)
0 20 40 60 80 100 0 20 40 60
5 10 15 20
Depth of discharge (%)
Available energy (Wh)
Measured ( E avail
M ) Calculated ( E avail C )
(f)
Figure 2.12: Comparison of proposed model result with EIG battery at 4Crand 4Dr.
Fig. 2.12 shows the comparison results of battery performance at 4Cr/4Dr. The EOCV and EODV for this case is 2.75V and 1.5V. As can be seen in Fig. 2.11 (a) and Fig. 2.12 (a), for constant current charging the charging voltage is equal to the battery terminal voltage. However, in discharging case, it has an additional voltage drop. The comparison of Pc, Pd, Estor, Eavail with respect to S OCcr and DODcr at 4Cr and 4Dr are shown in Figs. 2.12 (c) - (f). It can be observed that the energy extracted from the battery is inversely proportional to Drand directly proportional to battery voltage and current.
Notably for high Cr/Dr, the energy stored/extracted to/from the battery can be done in a short period of time, while for low Cr/Dr it takes long time. This affects the net energy stored/extracted from/to the battery. The time taken to charge or discharge the battery at 4Cror 4Drtakes only 0.333hr, while for 1Cr or 1Drit requires 1hr.
0 20 40 60 80 100
1.5 2 2.5 3 3.5 4 4.5 5
State of charge (%)
Charging voltage (V)
Calculated ( V c
C ) Measured ( V c M )
(a)
0 20 40 60 80 100
2.5 3 3.5 4 4.5
Depth of discharge (%)
Discharging voltage (V)
Calculated ( V d
C ) Measured ( V d M )
(b)
0 20 40 60 80 100
2.5 3 3.5 4 4.5 5 5.5 6 6.5 7
State of charge (%)
Charging power (W)
Calculated ( P c
C ) Measured ( P c M )
(c)
0 20 40 60 80 100
3.5 4 4.5 5 5.5 6 6.5
Depth of discharge (%)
Discharging power (W)
Calculated ( P d
C ) Measured ( P d M )
(d)
0 20 40 60 80 100
0 1 2 3 4 5 6
State of charge (%)
Stored energy (Wh)
Calculated ( E stor
C ) Measured ( E
stor M )
(e)
0 20 40 60 80 100
0 1 2 3 4 5 6
Depth of discharge (%)
Available energy (Wh)
Calculated ( E avail
C ) Measured ( E
avail M )
(f)
Figure 2.13: Comparison of proposed model result with Sony US18650 battery at 1Crand 1Dr. Case (ii) - Sony US18650 Battery: Fig. 2.13 shows the comparison of battery characteristics with 3.7V, 1.4Ah Sony US18650 battery. Figs. 2.13 (a) and (b) show the simulated characteristics of terminal voltage plotted with S OCcrand DODcr. It can be seen from these figures, EOCV and EODV for Sony battery are 4.1V and 2.7V. The results of battery power and energy shown in Figs. 2.13 (c) - (f) for charging and discharging cases are in good agrement. It is observed from Figs. 2.12 (e) the maximum stored energy is 17.46Wh, however in Figs. 2.12 (f) the initial extracted energy is only
17.32Wh due to capacity loss in the battery.
Fig. 2.14 shows the comparison of performance characteristics of Sony US18650 battery with 2Cr and 2Dr at 2.8A. The terminal voltage curves for charging and discharging scenarios are shown in Figs. 2.14 (a) and (b). The EOCV and EODV observed in this case are 4.42V and 2.5V. The comparison of battery power (Pc, Pd) and energy (Estor, Eavail) with the manufacturers’ data are shown in Figs. 2.14 (c) - (f). As it can be seen from Figs. 2.14 (e) - (f), the amount of energy stored into the battery and extracted (available) from the battery has reduced with respect to S OCcr and DODcr.
0 20 40 60 80 100 0 20 40 60 80
2 2.5 3 3.5 4 4.5 5
State of charge (%)
Charging volatge (V)
Measured ( V c
M ) Calculated ( V c C )
(a)
0 20 40 60 80 100 0 20 40 60 80
2.5 3 3.5 4 4.5
Depth of discharge (%)
Discharging voltage (V)
Measured ( V d
M ) Calculated ( V d C )
(b)
0 20 40 60 80 100 0 20 40 60 80
6 7 8 9 10 11 12 13 14 15
State of charge (%)
Charging power (W)
Measured ( P c
M ) Calculated ( P c C )
(c)
0 20 40 60 80 100 0 20 40 60 80
7 8 9 10 11 12
Depth of discharge (%)
Discharging power (W)
Measured ( P d
M ) Calculated ( P d C )
(d)
0 20 40 60 80 100 0 20 40 60 80
0 1 2 3 4 5 6 7
State of charge (%)
Stored energy (Wh)
Measured ( E stor
M ) Calculated ( E stor C )
(e)
0 20 40 60 80 100 0 20 40 60 80
0 1 2 3 4 5 6
Depth of discharge (%)
Available energy (Wh)
Measured ( E avail
M ) Calculated ( E avail C )
(f)
Figure 2.14: Comparison of proposed model result with Sony US18650 battery at 2Crand 2Dr. Case (iii) - Panasonic Battery: Fig. 2.15 shows the comparison results with 4.5V, 0.83Ah Pana- sonic battery at 1Crand 1Dr. The terminal voltage characteristics with respect to S OCcrand DODcr at 1Crand 1Drare shown in Figs. 2.15 (a) and (b). The EOCV and the EODV of this case are 4.3V
and 3V. Figs. 2.15 (c) - (f) show the performance characteristics of battery power and energy. In all the cases, the measured (catalogue value) data agrees with the proposed data.
0 20 40 60 80 100
3.4 3.6 3.8 4 4.2 4.4
State of charge (%)
Charge voltage (V)
Calculated ( V c
C ) Measured ( V c M )
(a)
0 20 40 60 80 100
3 3.5 4 4.5
Depth of discharge (%)
Discharging voltage (V)
Calculated ( V d
C ) Measured ( V d M )
(b)
0 20 40 60 80 100
1.9 2 2.1 2.2 2.3 2.4 2.5
State of charge (%)
Charging power (W)
Calculated ( P c
C ) Measured ( P c M )
(c)
0 20 40 60 80 100
2 2.5 3 3.5 4
Depth of discharge (%)
Discharging power (W)
Calculated ( P d
C ) Measured ( P d M )
(d)
0 20 40 60 80 100
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
State of charge (%)
Stored energy (Wh)
Calculated ( EstorC ) Measured ( EstorM )
(e)
0 20 40 60 80 100
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Depth of discharge (%)
Available energy (Wh)
Calculated ( E avail
C ) Measured ( E
avail M )
(f)
Figure 2.15: Comparison of proposed model result with Panasonic battery at 1Crand 1Dr.
The comparison of Panasonic battery with proposed model is shown in Fig. 2.16. Figs. 2.16 (a) and (b) show the comparison of terminal voltage characteristics with 0.9398Crat 0.78A and 1.8795Dr at 1.56A. The charge and discharge voltage characteristics are shown in Figs. 2.16 (a) and (b). The EOCV and EODV observed in this case are 4.3V and 3V. Figs. 2.16 (c) and (d) show the performance of battery power. The amount of stored and available energy are shown in Figs. 2.16 (e) and (f).
Case (iv) - Sanyo Battery: The battery used in this case for comparison is Sanyo which has a nominal capacity 1.5Ah and a voltage of 3.7V. Here the battery is charged and discharged with a constant current of 1.5A. Figs. 2.17 (a) and (b) show the terminal voltage curves of charge and
0 20 40 60 80 100 0 20 40 60 80 3.2
3.4 3.6 3.8 4 4.2 4.4 4.6
State of charge (%)
Charging voltage (V)
Measured ( V c
M ) Calculated ( V c C )
(a)
0 20 40 60 80 100 0 20 40 60 80
2.8 3 3.2 3.4 3.6 3.8 4
Depth of discharge (%)
Discharging voltage (V)
Calculated ( VdC ) Measured ( VdM )
(b)
0 20 40 60 80 100 0 20 40 60 80
2.6 2.8 3 3.2 3.4 3.6
State of charge (%)
Charging power (W)
Measured ( P c
M ) Calculated ( P c C )
(c)
0 20 40 60 80 100 0 20 40 60 80
3.8 4 4.2 4.4 4.6 4.8 5 5.2
Depth of discharge (%)
Discharging power (W)
Calculated ( P d
C ) Measured ( P d M )
(d)
0 20 40 60 80 100 0 20 40 60 80
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
State of charge (%)
Stored energy (Wh)
Measured ( E stor
M ) Calculated ( E stor C )
(e)
0 20 40 60 80 100 0 20 40 60 80
0 5 10 15 20 25 30
Depth of discharge (%)
Available energy (Wh)
Calculated ( E avail
C ) Measured ( E
avail M )
(f)
Figure 2.16: Comparison of proposed model result with Panasonic battery at 0.9398Crand 1.8795Dr. discharge characteristics. As can be seen from the plots, the EOCV and EODV in this case are 4.3V and 2.5V. In this figures, the variation of S OCcr and DODcr value has reflected in the charging and discharging voltage. The performance of power and energy characteristics are shown in Figs. 2.17 (c) - (f). The stored and extracted energy is maximum at very first cycle which is given in Fig. 2.17 (e) and Fig. 2.17 (f). The amount energy exchange is reduced after nth cycle due to capacity fade which is given in Fig. 2.22 (a).
Fig. 2.18 shows the comparison of measured (catalogue value) and calculated characteristics of Sanyo battery of 2Cr/2Dr at 3A. The EOCV and EODV observed in this case are 4.02V and 2.5V.
Figs. 2.18 (c) and (f) show the characteristics of charging and discharging battery power and energy (Pc, Pd, Eavail and Estor) at 2Cr and 2Dr. The tc required to store the energy in the EV battery is
0 20 40 60 80 100 0 20 40 60 80 2.5
3 3.5 4 4.5
State of charge (%)
Charging voltage (V)
Measured ( V c
M ) Calculated ( V c C )
(a)
0 20 40 60 80 100 0 20 40 60 80
2.5 3 3.5 4 4.5
Depth of discharge (%)
Discharging voltage (V)
Measured ( VdM ) Calculated ( VdC )
(b)
0 20 40 60 80 100 0 20 40 60 80
3.5 4 4.5 5 5.5 6 6.5 7 7.5
State of charge (%)
Charging power (W)
Measured ( P c
M ) Calculated ( P c C )
(c)
0 20 40 60 80 100 0 20 40 60 80
3.5 4 4.5 5 5.5 6 6.5 7 7.5
Depth of discharge (%)
Discharging power (W)
Calculated ( P d
C ) Measured ( P d M )
(d)
0 20 40 60 80 100 0 20 40 60 80
0 1 2 3 4 5 6 7
State of charge (%)
Stored energy (Wh)
Calculated ( E stor
C ) Measured ( E
stor M )
(e)
0 20 40 60 80 100 0 20 40 60 80
0 1 2 3 4 5 6 7
Depth of discharge (%)
Available energy (Wh)
Calculated ( E avail
C ) Measured ( E
avail M )
(f)
Figure 2.17: Comparison of proposed model result with Sanyo battery at 1Cr/1Dr.
reduced due to high Cr which is shown in Fig 2.18 (e). It is observed from Fig. 2.18 (f) the energy extracted from the battery is reduced due to the high Dr.
Thus, it is observed from Fig. 2.11 to Fig. 2.18, the proposed battery model characteristics exactly fits the manufacturers’ characteristics and in some cases it has an error less than 5%.