Battery Model

Mathematical Modeling of Li-Ion Battery Using Genetic Algorithm Approach for V2G

2.2 Battery Model

the proposed model result with manufactures’ data and summary of the present work is given in Section 2.6.

R₁ = (a₁+a₂x+a₃x²)e^−a4y+(a₅+a₆x+a₇x²) (2.1) R2 =(a8+a9x+a10x²)e^−a11y+(a12+a13x+a14x²) (2.2) C =−(a₁₅+a₁₆x+a₁₇x²)e^−a18y+(a₁₉+a₂₀x+a₂₁x²) (2.3) V₀ =(a₂₂+a₂₃x+a₂₄x²)e^−a25y+(a₂₆+a₂₇y+a₂₈y²+a₂₉y³)−a₃₀x+a₃₁x² (2.4) In the above set of equations, R₁, R₂, C and V₀ are represented in terms of polynomial equations and there are 31 coefficients (from a1 to a31) in total. The detailed extraction method to find these polynomial coefficients has been explained in Section 2.3. The battery parameters for charging pro- cess can be derived by replacing x and y with Cr and S OCcr, while for discharging process with Dr

and (1−DOD_cr). Here, (1−DOD_cr) is chosen as V₀decreases with increase in DOD_crfor discharging voltage. These equations are used for calculating battery terminal voltage for charging or discharging at different C_r and D_r. The terminal charge or discharge voltage of the battery (V_c^C_i or V_d^C

j) changes with capacity of the battery, S OC_cr/DOD_cr levels and C_r/D_r. The parameters of non-linear relation of V_c^C_i/V_d^C

j can also be represented in terms of polynomial equations, where i and j denotes i^th and j^th calculated value of the charging and discharging voltage. Therefore, under constant current the bat- tery terminal voltage for charging and discharging scenario with respect to time is given in Eq. (2.5) and Eq. (2.6). In Appendix A, the calculated battery terminal voltage for charging and discharging scenarios has been mentioned.

V_c^C_i = Q_r

C +IcR2

exp − t_c R2C

!!

+V0−(Ic(R1+R2)) (2.5)

V^C_d

j =

Qr

C +IdR2

exp − td

R₂C

!!

+V0−(Id(R1+R2)) (2.6)

where Q_r is the remaining capacity of the battery, t_c,I_c,t_dand I_d are charge time, charge current, discharge time and discharge current, respectively. Thus, Eq. (2.5) and Eq. (2.6) can accurately represent the behavior of any battery types, if the parameters are well defined. This equations capture the non-linear behavior of the battery which depends on the actual battery charge/discharge voltage.

Fig. 2.3 shows the representation of electric circuit based battery model with its non-linear equations.

switch

Source Controlled voltage Signal

electric circuit (EC) V^C_d =_Qr

C +I_dR₂

×exp

−_R^t₂^d_C

+V₀−(I_d(R₁+R₂))

R I^∗<0

I^∗>0

Icor Id

I^∗reference current V^C_c =_Qr

C +I_cR₂

×exp

−R^t2^cC

+V₀−(I_c(R₁+R₂))

R₁

V0

V_d^C

j

S OCcr,DODcr,

V_c^C_ior

Calculate: Qr, I^∗,t_c, t_d

V_c^C_i or V_d^C

j

R2

C

Figure 2.3: Non-linear battery model.

2.2.1 Charge/Discharge Rate and SOC Calculations

The charge or discharge rate algorithm is used to determine the amount of energy stored or ex- tracted from the EV battery. The C_r and S OC_cr of the battery varies depending on the present condi- tion of the battery.

Fig. 2.4 explains the calculation of C_r and S OC_cr for charging scenario. The control algorithm developed inside the battery checks the battery status and then calculates the current charge rate (C^crt_r ) of the EV battery. It has also taken into account of user defined C_rlimit (C^lmt_r ) and initial battery SOC (S OCini). The C_r^crtand D^crt_r of the battery can be expressed as given in Eq. (2.7) - Eq. (2.8).

C_r =C_r^crt= Ic

Q_r (2.7)

Dr=D^crt_r = Id

Qr

(2.8)

This is calculated based on current status of the battery, which is the ratio of current and remaining capacity of the battery. The algorithm chooses the minimum of charge rate based on the C_r^lmt and C^crt_r to regulate the charge current of the battery. Similar type of control algorithm is used for discharging scenario.

if

if Yes if

Yes No

No

Yes Yes

Stop

if Calculation for

Calculation for

No No

To BM and CFM

≥S OCmin

≤S OCmax

S OCiniis Cr^crt≤C^lmtr

tc

tc tc

tc=tc+(Ts−Ts^pre) tc=0 Ic

C_r^crt

Ts=0 C_r^crt=C^lt_r

sign of Ic

I^∗×I^∗pre>1 S OCini

Ts S OCmax C^lmt_r S OCmin Qr

∆tc=(tc−delay(tc)) S OCini=S OCcr

tc=0

Ts^pre= delay(Ts); I^∗_pre=delay(I^∗) Ic

S OCcr=S OCini+_Q3600^Ic×∆tc S OCmax,S OCmin,S OCini,Ts,Ic,Qr,I^∗,tc

Ic

I^∗ tc

Figure 2.4: Functional flow chart for Crand S OCcr.

The S OC_cr and DOD_cr can be calculated from Eq. (2.9) - Eq. (2.10). The S OC_cr and t_c is estimated from the C_r control algorithm which is given in Fig. 2.4. Similarly, the DOD_cr and t_d is calculated for discharging scenario.

S OCcr=S OCini+ Ic∆tc

Qr3600

!

(2.9)

DODcr=DODini+ Id∆td

Q_r3600

!

(2.10)

Here, S OC_iniis the initial SOC of the battery. The S OC_max and DOD_max are the maximum user defined S OC and DOD limits. If S OC_crand DOD_crof the battery reaches S OC_maxand DOD_max, then the control algorithm should not allow to charge or discharge the battery to prevent over charging or discharging. The control algorithm used to charge/discharge the EVs’ batteries from/to grid have been explained in Chapter 3. The sample calculation for S OC_cr, C_r, etc. has been given in Appendix A.

2.2.2 Battery Power and Processed Energy

The battery power for charging (P_c) and discharging (P_d) scenario is given in Eq. (2.11) - Eq.

(2.12).

P_c =V_c^C_iI_c (2.11)

P_d = V_d^C

jI_d (2.12)

The amount of stored energy (E_stor) during charging process depends on increase in V_c^C

i and S OC_cr, which is given in Eq. (2.13).

Estor =V_c^C_iQr∆S OCcr (2.13)

where,∆S OC_cris the change in current S OC_cr. The processed energy (PE_c) for charging scenario is given in Eq. (2.14).

PEc =X

Estor (2.14)

The available energy (E_avail) in the battery during discharging process decreases V_d^C

i with increase in DOD_cr, which can be calculated using Eq. (2.15).

Eavail=V_d^C_jQr∆DODcr (2.15)

where, ∆DOD_cr is the change in current DOD_cr. The processed energy (PE_d) for discharging scenario is given in Eq. (2.16).

PE_d = X

E_avail (2.16)

The total processed energy (E_total) of the battery in a cycle is calculated using Eq. (2.17).

E_total =X

(PE_c+PE_d) (2.17)

Eq. (2.11) to Eq. (2.17) represents the real-time performance of the battery during charging and discharging process. Simulations are done based on these equations for the developed BM and has been validated with the manufacturers’ catalogue which is discussed in Section 2.5. In Appendix A, battery power and processed energy calculation details has been mentioned.