Mathematical Modeling of Li-Ion Battery Using Genetic Algorithm Approach for V2G
2.5 Model Validation
2.5.2 Capacity Fade Analysis
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%.
0 20 40 60 80 100 0 20 40 60 80 2.5
3 3.5 4 4.5
State of charge (%)
Charging volatge (V)
Measured ( VcM ) Calculated ( VcC )
(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
7 8 9 10 11 12 13
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 13
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
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 7
Depth of discharge (%)
Available energy (Wh)
Measured ( E avail
M ) Calculated ( E avail C )
(f)
Figure 2.18: Comparison of proposed model result with Sanyo battery at 2Cr/2Dr.
2.22 reports the results obtained from capacity fade study. The parameters of capacity fade analysis have been taken from the capacity loss data provided by the battery manufacturers’ catalogue. Due to limited data, the capacity loss comparison only done for specified charge and discharge rate. Due to unavailability of the battery manufacturers’ capacity loss data at different Cr and Dr, only specified Cr and Dr is discussed in this work. The results obtained after n number of cycle for each types of batteries are summarized from Table 2.4. The variation of capacity loss is shown with respect to cycle number and total processed energy. The remaining capacity of the battery has been given in each case to show the net processed energy.
Case (i) - EIG Battery: Fig. 2.19 shows the comparison of capacity fade characteristics of EIG battery. Figs. 2.19 (a) shows the capacity loss as a function of cycle number for EIG battery.
0 500 1000 1500 2000 2500 3000 0
0.5 1 1.5 2
Cycle
Capacity loss (Ah)
Calculated ( QlC ) Measured ( QlM )
(a)
0 25 50 75 100
0 0.5 1 1.5 2
Total processed energy (kWh)
Capacity loss (Ah)
Measured ( Q l
M ) Calculated (Q l C )
(b)
(b)
0 500 1000 1500 2000 2500 3000
14 15 16 17 18 19
Cycle
Processed energy for charging (Wh)
Calculated ( PEcC ) Measured ( PEcM )
(c)
0 500 1000 1500 2000 2500 3000
14 15 16 17 18 19 20
Cycle
Processed energy for discharging (Wh)
Calculated ( PEdC ) Measured ( PEdM )
(d)
0 500 1000 1500 2000 2500 3000
6 6.5 7 7.5 8 8.5 9
Cycle
Remaining capacity (Ah)
Calculated ( Q r
C ) Measured ( Q r M )
(e)
0 25 50 75 100
6.5 7 7.5 8 8.5
Total processed energy (kWh)
Remaining capacity (Ah)
Calculated ( Q r
C ) Measured ( Q r M )
(f) Figure 2.19: Capacity loss characteristics of EIG battery.
In Fig. 2.19 (b), the capacity loss is observed as a function of total processed energy. As it can be seen in Figs. 2.19 (a) and (b), the capacity loss as a function of cycle number and processed energy has a rising behavior. Figs. 2.19 (c) and (d) show the required and available energy plots with respect to cycle number. The observation is done till 3000 cycles to check the increase and degradation of charge and discharge energy. This results indicate, the charge and discharge rate plays a strong role in determining the capacity fade of batteries. Figs. 2.19 (e) and (f) show the remaining capacity curve as a function of cycle number and total processed energy.
Case (ii) - Sony US18650 Battery: The measured (catalogue value) and calculated capacity fade characteristics of Sony US 18650 is shown in Fig. 2.20. In this case, the capacity fade study has been observed for 300 cycles. The capacity fade as a function of cycle number and total processed energy
are shown in Figs. 2.20 (a) and (b).
0 50 100 150 200 250 300
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Cycle
Capacity loss (Ah)
Calculated ( QlC ) Measured ( QlM )
(a)
0 500 1000 1500 2000 2500 3000
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Total processed energy (Wh)
Capacity loss (Ah)
Calculated ( Q l
C ) Measured ( Q l M )
(b)
0 50 100 150 200 250 300
4.7 4.8 4.9 5 5.1 5.2 5.3 5.4
Cycle
Processed energy for charging (Wh)
Calculated ( PE c
C ) Measured ( PE c M )
(c)
0 50 100 150 200 250 300
4.6 4.7 4.8 4.9 5 5.1 5.2 5.3
Cycle
Processed energy for discharging (Wh)
Calculated ( PE d
C ) Measured ( PE d M )
(d)
0 50 100 150 200 250 300
1.26 1.28 1.3 1.32 1.34 1.36 1.38 1.4 1.42 1.44
Cycle
Remaining capacity (Ah)
Calculated ( Q r
C ) Measured ( Q r M )
(e)
0 500 1000 1500 2000 2500 3000
1.26 1.28 1.3 1.32 1.34 1.36 1.38 1.4 1.42 1.44
Total processed energy (Wh)
Remaining capacity (Ah)
Claculated ( Q r
C ) Measured ( Q r M )
(f)
Figure 2.20: Capacity loss characteristics of Sony US18650 battery.
Figs. 2.20 (c) and (d) show the characteristics of charge and discharge energy as a function of cycle number, while the characteristics of remaining capacity and total processed energy are shown in Figs. 2.20 (e) and (f). It is noticed from the plots, the energy and remaining capacity of battery with increase in cycle has a falling behavior.
Case (iii) - Panasonic Battery: Fig. 2.21 shows the measured (catalogue value) and calculated characteristics of capacity fade for Panasonic battery. Here the study has been done till 500 cycles.
Figs. 2.21 (a) and (b) show the capacity loss as a function of cycle number and energy. The capacity loss as a function of total processed energy is shown in Fig. 2.21 (b).
The characteristics of required and available energy as a function of cycle number are shown
in Figs. 2.21 (c) and (d). It can be seen, the energy stored or extracted from or to the battery has decreased with increased in cycle number due to capacity loss. Figs. 2.21 (e) and (f) show the remaining capacity curve as a function of cycle number and total processed energy. As like previous cases, the energy and capacity loss characteristics have falling behavior with increase in cycle number and processed energy.
0 50 100 150 200 250 300 350 400 450 500
0 20 40 60 80 100 120
Cycle
Capacity loss (mAh)
Calculated ( Q l
C ) Measured ( Q l M )
(a)
0 500 1000 1500 2000 2500 3000
0 20 40 60 80 100 120 140
Total processed energy (Wh)
Capacity loss (mAh)
Calculated ( Q l
C ) Measured ( Q l M )
(b)
0 50 100 150 200 250 300 350 400 450 500
2.8 2.9 3 3.1 3.2 3.3 3.4 3.5
Cycle
Processed energy for charging (Wh)
Calculated ( PE c
C ) Measured ( PE c M )
(c)
0 50 100 150 200 250 300 350 400 450 500
2.6 2.7 2.8 2.9 3 3.1 3.2 3.3
Cycle
Processed energy for discharging (Wh)
Calculated ( PE d
C ) Measured ( PE d M )
(d)
0 50 100 150 200 250 300 350 400 450 500
700 720 740 760 780 800 820 840
Cycle
Remaining capacity (mAh)
Calculated ( QrC ) Measured ( QrM )
(e)
0 500 1000 1500 2000 2500 3000
700 720 740 760 780 800 820 840
Total processed energy (Wh)
Remaining capacity (mAh)
Calculated ( Q r
C ) Measured ( Q r M )
(f) Figure 2.21: Capacity loss characteristics of Panasonic battery.
Case (iv) - Sanyo Battery: Fig. 2.22 shows the comparison of capacity fade profiles of Sanyo battery. The capacity loss as a function of cycle number and total processed energy are shown in Figs.
2.22 (a) and (b). As can be seen from these plots, the characteristics have rising behavior with respect to cycle and total processed energy.
Figs. 2.22 (c) and (d) show the characteristics of required and available energy as a function of
0 50 100 150 200 250 300 350 400 450 500 0
0.25 0.5
Cycle
Capacity loss (Ah)
Calculated ( QlC ) Measured ( QlM )
(a)
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0
0.25 0.5
Total processed energy (Wh)
Capacity loss (Ah)
Calculated ( QlC ) Measured ( QlM )
(b)
0 50 100 150 200 250 300 350 400 450 500
4 4.5 5 5.5 6
Cycle
Processed energy for charging (Wh)
Calculated ( PE c
C ) Measured ( PE c M )
(c)
0 50 100 150 200 250 300 350 400 450 500
4 4.5 5 5.5 6
Cycle
Processed energy for discharging (Wh)
Calculated ( PE d
C ) Measured ( PE d M )
(d)
0 100 200 300 400 500
1.1 1.2 1.3 1.4 1.5 1.6
Cycle
Remaining capacity (Ah)
Calculated (Q r
C) Measured (Q r M)
(e)
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0
0.1 0.2 0.3 0.4 0.5
Total processed energy (Wh)
Capacity loss (Ah)
Calculated ( Q r
C ) Measured ( Q r M )
(f) Figure 2.22: Capacity loss characteristics of Sanyo battery.
cycle number. The values are observed for 500 cycles to check its performance. The comparison of remaining capacity curve as a function of cycle number and total processed energy are shown in Figs.
2.22 (e) and (f).
Table 2.4: Results obtained from battery and capacity fade model.
Manufacturer→ EIG (2.5V, 8Ah) Sony (3.7V, 1.4Ah) Panasonic (4.5V, 0.83Ah) Sanyo (3.7V, 1.5Ah) Parameter↓ Measured Calculated Measured Calculated Measured Calculated Measured Calculated
Qr(Ah) 6.61 6.55 1.27 1.28 0.73 0.72 1.14 1.22
Ql(Ah) 1.39 1.45 0.13 0.15 0.12 0.10 0.41 0.15
Etotal(kWh) 94.98 95.12 2.87 2.78 3.04 2.81 4.67 4.83
Estor(kWh) 14.87 14.65 4.76 4.79 2.88 2.9 4.18 4.49
Eavail(kWh) 14.58 14.27 4.65 4.69 2.69 2.72 4.08 4.32
1stcycle Estor(kWh) 17.99 17.89 5.33 5.36 3.19 3.19 5.55 5.81
1stcycle Eavail(kWh) 17.53 17.43 5.15 5.16 3.01 2.95 5.67 5.74
It can be observed from Fig. 2.19 to Fig. 2.22, the capacity fade model characteristics exactly
fits with the different battery manufacturers’ curves. The typical simulation and measured (catalogue value) results of different battery types such as for EIG (till 3000 cycle), Sony (till 300 cycles), Panasonic (till 500 cycles) and Sanyo (till 500 cycles) are shown in Table 2.4.