**3. Adaptive Control Scheme for Estimating OCV and IRs**

Online estimation of parameters open-circuit voltage (OCV) and internal resistances (IRs) of new battery is illustrated in [16]. These two parameters are relative to SOC and SOH [23,24]. The adaptive control algorithm guarantees estimating error converges to zero, by applying Lyapunov stability criteria. As a result, OCV and IRs can be precisely estimated from input parameters, i.e., working voltage and current. The detailed mathematical modeling in Simulink/MATLAB, according to an adaptive control scheme, is shown in Figure 3. The procedure is briefly summarized as below. First, the battery voltage can be expressed as:

**Figure 3.** Adaptive control scheme in Simulink/MATLAB.

$$\begin{split} \dot{\upsilon}\_{b} &= \dot{\upsilon}\_{\text{oc}} - \dot{R}\_{s} \dot{\dot{i}}\_{b} - R\_{s} \dot{\dot{i}}\_{b} - \dot{\upsilon}\_{c} \\ &= \sigma(\upsilon\_{\text{oc}}) \text{x} - \sigma(R\_{S}) \text{x} \dot{i}\_{b} - R\_{s} \dot{i}\_{b} - \dot{\upsilon}\_{c} \end{split} \tag{1}$$

Here,

$$\sigma(p) = \begin{bmatrix} \frac{\partial p}{\partial S \partial \mathcal{C}} & \frac{\partial p}{\partial T} \ \frac{\partial p}{\partial h} \end{bmatrix} \tag{2}$$

$$\mathbf{x}^T = \left[\frac{\partial \text{SoC}}{\partial t} \begin{array}{c} \frac{\partial T}{\partial t} \end{array} \frac{\partial \text{h}}{\partial t} \right] \tag{3}$$

Here, *vb*, *voc*, *RS*, *RT*, . *ib*, and *vc* represent voltage of battery, open-circuit voltage, first-order and second-order IRs, battery current, and voltage drop across RC circuit, respectively, based on the electrical circuit model (ECM) shown in Figure 3. The projected parameters *vb* and *RS* are assumed as functions of *SOC, T*, and *h*, and change with time. In addition, Equation (1) can be simplified with the following assumptions:

(1) Small deviation of every battery's discharge is assumed, compared to rated useful capacity, thus ∂*SoC*/∂*t* ≈ 0.

(2) In normal operating conditions, deviation of cell temperature, *T* is slow by using a fan, thus ∂*T*/∂*t* ≈ 0.

(3) A long-time usage history, h, is performed, thus ∂*h*/∂*t* ≈ 0.

Accordingly, it follows that Equation (3) ≈ 0; furthermore, Equation (1) can be rewritten as:

$$\begin{split} \dot{\upsilon}\_{b} = \left( \frac{\partial \upsilon\_{\rm oc}}{\partial \text{SoC}} \frac{\partial \text{SoC}}{\partial t} + \frac{\partial \upsilon\_{\rm oc}}{\partial T} \frac{\partial T}{\partial t} \right) - \left( \frac{\partial R\_{S}}{\partial \text{SoC}} \frac{\partial \text{SoC}}{\partial t} + \frac{\partial R\_{S}}{\partial T} \frac{\partial T}{\partial t} \right) \dot{\iota}\_{b} \\ - \frac{1}{C\_{I}R\_{t}} \upsilon\_{b} - R\_{s} \dot{\iota}\_{b} - \frac{R\_{I} + R\_{s}}{C\_{I}R\_{t}} \dot{\iota}\_{b} + \frac{\upsilon\_{\rm oc}}{C\_{I}R\_{t}} \end{split} \tag{4}$$

Equation (4) can also be written as a vector form.

$$
\dot{w}\_b = \boldsymbol{\theta}^T \boldsymbol{X} \tag{5}
$$

$$\boldsymbol{\Theta}^{T} = \begin{bmatrix} \boldsymbol{\theta}\_{1} \ \boldsymbol{\theta}\_{2} \ \boldsymbol{\theta}\_{3} \ \boldsymbol{\theta}\_{4} \end{bmatrix}^{T} = \begin{bmatrix} \boldsymbol{R}\_{s} \ \frac{\boldsymbol{R}\_{s} + \boldsymbol{R}\_{t}}{\boldsymbol{C}\_{t} \boldsymbol{R}\_{t}} \ \frac{1}{\boldsymbol{C}\_{t} \boldsymbol{R}\_{t}} \ \frac{\boldsymbol{v}\_{0:}}{\boldsymbol{C}\_{t} \boldsymbol{R}\_{t}} \end{bmatrix} \tag{6}$$

$$X^T = \begin{bmatrix} \vdots \\ -\dot{i}\_b - i\_b - \upsilon\_b \ 1 \end{bmatrix} \tag{7}$$

Equation (6) is rewritten as Equation (8) with respect to every estimated state . *v*ˆ*b*

$$
\psi\_b = \boldsymbol{\Phi}^T \boldsymbol{\mathcal{X}} + \boldsymbol{\mu} \tag{8}
$$

where *<sup>X</sup>*<sup>ˆ</sup> *<sup>T</sup>* <sup>=</sup> [<sup>−</sup> . *ib* − *ib* − *v*ˆ*<sup>b</sup>* 1], u is adjustable for input parameters. θ is estimated results of target parameters.

The adaptive control algorithm is based on tracking input signals to modify the target parameters in control states, so that the convergent criteria are satisfied as below:

$$\lim\_{t \to \infty} e = \lim\_{t \to \infty} (v\_b(t) - v\_b(t)) = 0 \tag{9}$$

Here, Rt and Ct represent first-order IR and capacitance, respectively, based on ECM. If the thermal effect of temperature is not considered, ∂T/∂t ≈ 0 is obtained.

The adaptive control algorithm shown in Figure 3 is introduced to optimize the unknown target parameters of Rs, Rt, and OCV in Equation (4). A filtering process is used to improve the measured noise and enhance the estimation reliability. In the estimation process, IR is a sensitized parameter of lifecycle, hence it is used to indicate the deviation of lifecycle. An OCV curve related to the battery's useful capacity is applied to estimate the remained capacity of SOC. The algorithm is discretized, and embedded in BMS.

### **4. Setup of Test Bench**

.

There are limitations of an electrochemical battery's performance. The charge/discharge response of a lithium-ion battery is much slower than a UC. Therefore, for reusing a lithium-ion battery, a high-power UC can support sudden peak current, and extend the life of an LIB. To take a simple parallel combination between an RLIB and a UC, the ultracapacitor is operated as a dc-side buffer for supporting peak current. In case of using a DC-DC converter, it allows more flexible management between an RLIB and an ultracapacitor. However, it is not competitive in reusing cost. Consequently, a simple hybrid parallel connection is established in this study. PWM control is applied for adjusting the duty ratio of an RLIB. A UC connected with the BMS directly is to improve the life of the LIB, by restricting the voltage drop or DOD. RLIB packs with rated voltage of 52 V and 50.4 V are employed, respectively, as shown in Figure 4a. The first pack of 52 V uses a commercialized BMS for reference, and the other one of 50.4 V uses an in-house BMS. Figure 4b shows the developed circuit of the BMS with a PWM control in this study. If the duty ratio per unit time of battery is selected as 40%, then the UC's load becomes 60%.

**Figure 4.** (**a**) Two battery packs in test bench (52 V and 50.4 V); (**b**) circuit of BMS with pulse-width modulation (PWM) control.
