*4.3. Battery Model*

A Li-Ion battery has a small AC voltage ripple for large AC current ripple due to small internal impedance. The adaptive Randle equivalent circuit model (AR-ECM) of a Li-Ion battery can explain the internal impedance. AR-ECM is shown in Figure 7b. It consists of an open circuit voltage source, *Vbat* , battery inductance, *Lbat*, Ohmic resistance, *R*Ω, charge transfer resistance, *RCT*, double layer capacitance, *CDL*, Warburg impedance, *ZW*, solid electrolytic interface resistance, *RSEI*, and capacitance, *CSEI*. The overall internal battery impedance in frequency domain, *Z*(*s*) is expressed by (29) [19].

$$Z(s) = sL\_{\text{flat}} + R\_{\Omega} + \frac{1}{\frac{1}{R\_{SEI}} + sC\_{SEI}} + \frac{1}{\frac{1}{R\_{CT} + \sigma\sqrt{\frac{2}{s}}} + sC\_{DL}} \tag{29}$$

where, *σ* is Warburg coefficient, and Warburg impedance *ZW*=*σ* <sup>√</sup>2/*s*. *<sup>Z</sup>*(*s*) is taken into account for versatile current controller design.

For a Valence U-12XP 40 Ah 13.8 V Li-Ion battery, the model components are *<sup>L</sup>* = 0.34 <sup>μ</sup>H, *<sup>R</sup>*<sup>Ω</sup> = 5.65 mΩ, *CDL* = 4.29 F, *RCT* = 1.23 m<sup>Ω</sup> and *<sup>σ</sup>* = 2.05 × <sup>10</sup>−<sup>3</sup> at 25% state of charge [9]. The bode plot of the battery impedance is shown in Figure 8. The magnitude is calculated by 20*log*10(|*Z*|/*Rbase*), where *Rbase* is 1 Ω. The magnitude of impedance is ≈ −40 dBΩ i.e., a small voltage perturbation will cause large current perturbation. The phase plot indicates that at low frequency perturbation the battery behaves in capacitive manner. However, at high frequency the inductive part becomes dominant.

**Figure 8.** Bode-plot of internal impedance: (**a**) magnitude, (**b**) phase [9].

#### *4.4. Open Loop Transfer Function*

The output current is controlled by duty perturbation. Therefore, output current to duty transfer function is defined by *Gid* as in (30).

$$G\_{id} = \frac{I\_{out}(s)}{d(s)}\tag{30}$$

*Gid* can be determined by circuit analysis from Figure 7a. The overall impedance at the midpoint can be expressed by (31).

$$Z\_{\rm mid}(s) = sL + \frac{\frac{1}{sC}Z(s)}{\frac{1}{sC} + Z(s)}\tag{31}$$

The impedance at the output node can be expressed by (32).

$$Z\_{out}(s) = \frac{\frac{1}{s\mathbb{C}}Z(s)}{\frac{1}{s\mathbb{C}} + Z(s)}\tag{32}$$

The inductance current, *IL*(*s*), can be expressed by capacitor current, *IC*(*s*), and output current, *Iout*(*s*), based on Kirchhoff's current law as in (33).

$$I\_L(\mathbf{s}) = I\_{\mathbb{C}}(\mathbf{s}) + I\_{out}(\mathbf{s}) \tag{33}$$

The equation for output voltage to midpoint voltage can be derived by (34) and (35).

$$H\_v(\mathbf{s}) = \frac{V\_{\rm out}(\mathbf{s})}{V\_{\rm mid}(\mathbf{s})} = \frac{I\_L(\mathbf{s})Z\_{\rm out}(\mathbf{s})}{I\_L(\mathbf{s})Z\_{\rm mid}(\mathbf{s})} \tag{34}$$

$$H\_v(s) = \frac{1}{1 + \frac{sL}{Z(s)} + s^2LC} \tag{35}$$

The battery ripple voltage transfer function, *Gvd*, can be expressed by (36)

$$G\_{vd} = \frac{V\_{out}(s)}{d(s)} = \frac{1}{1 + \frac{sL}{Z(s)} + s^2LC} V\_{in} \tag{36}$$

The *Vout*(*s*) can be expressed by (37).

*Vout*(*s*) = *Iout*(*s*)*Z*(*s*) (37)

Using (34) and (37), we can write (38).

$$H\_v(\mathbf{s}) = \frac{I\_{out}(\mathbf{s})Z(\mathbf{s})}{V\_{mid}(\mathbf{s})} \tag{38}$$

For average mode control, (38) can be expressed by (39)

$$H\_v(s) = \frac{I\_{out}(s)Z(s)}{V\_{in}d(s)} = \frac{Z(s)}{V\_{in}}G\_{id} \tag{39}$$

Using (35) and (38), *Gid* can be determined by (40)

$$G\_{\rm id} = \frac{1}{Z(s) + sL + s^2 L \mathbb{C}Z(s)} V\_{\rm in} \tag{40}$$

The bode plot of *Gid* is shown in Figure 9 for the proposed system. The bode plot is based on selected parameters of Table 2 and *Z*(*s*) from Figure 8. The *LC* resonance peak is not visible in bode plot due to small internal impedance. The internal parameters of a battery changes with state of charge and aging. Therefore, the nominal value of internal resistance can be used as an alternative to *Z*(*s*). At lower frequency, *Gid* has a very high gain (≈70 dB). This means a very small duty perturbation causes a very high current perturbation which leads to instability i.e., 1% duty perturbation at 5 Hz would cause 31 A current perturbation whereas recommended current is only 20 A. This instability happens because of very low magnitude of internal impedance (≈−40 dBΩ). The instability is removed by designing a proper feedback and feedforward controller.
