Keywords

1 Introduction

Nowadays, the applications of battery energy storage system are increasing at a rapid rate with its utilization in many aspects such as power bank for mobile phone [7], transportation through electric vehicle [18], energy backup for microgrid [16], etc. The power output of the battery system needs to be controlled as per system requirement. The main challenges associated while dealing with battery energy storage system is estimation of SoC and determination of battery voltage for a particular charged condition [2, 12, 22]. The terminal voltage of the battery is usually determined by portraying the characteristics between no-load voltage (symbolized as \({v}_{ oc}\)) and SoC [22]. The particular characteristics of a typical Li-ion battery is shown with a polynomial expression [25].

The motivation behind the present work is to accurately define the no-load voltage versus SoC relationship such that the problem of voltage determination is alleviated. Specifically, ascertaining the coefficients of polynomial expression relating no-load voltage and SoC. The problem of uncertainty in the parameter is overcome through adaptive control. Adaptive control is the suitable option for designing the controller to achieve the best possible operation under uncertain and varying system conditions [5]. In general, the design of controller involves maximizing the performance index. However, the conventional control does not have the potential to adapt to system uncertainties and variations. In this aspect, adaptive controllers possess an edge over other techniques. It involves adaptive updation of control parameter in the presence of uncertainity and/or model variances and thereby enabling the system to achieve desired response [21]. Adaptive control can also be employed along with parameter estimation [1]. There exist adaptive techniques such as Model Reference Adaptive Control (MRAC), Inverse Adaptive Control (IAC), etc., in designing the controller [4, 6, 10, 19, 20]. As stated earlier, the considered model of Li-ion battery consists of an uncertain parameter which can be estimated by employing adaptive control. In particular, Model Reference Adaptive Control (MRAC) is employed to estimate the uncertain parameter. Here, the dynamics of the working battery model will be in accordance to the dynamics of the reference battery model.

The rest of the document is organized as follows. Modeling of lithium-ion battery is explained in Sect. 2. Section 3 deals with the adaptive control law developed to meet the objective of parameter validation. Simulation results are verified in Sect. 4. Finally, concluding remarks are drawn in Sect. 5.

2 Modeling of Lithium-Ion Battery

A typical lithium-ion battery can be modeled in many ways [3, 9, 15, 26, 27]. Modeling of battery storage system can be carried out in a simpler manner through equivalent circuit representation [17]. In the present work, equivalent circuit representation is followed up to second order [8] where a dependent voltage source \({v}_{ oc}\) is placed in series with a source resistance \({R}_{ s}\) and two RC combinations as shown in Fig. 1. Here, the dependent voltage source (i.e., open-circuit voltage of battery \({v}_{ oc}\)) is treated as the function of SoC. The mathematical relation between \({v}_{ oc}\) and SoC is usually shown with a second-order polynomial expression. However, the aforesaid relation is found to be linear in certain region [17]. For simplicity, only linear region is considered in the present work. The open-circuit voltage is expressed as

$$\begin{aligned} {v}_{ oc}={a}_1 {x}_\mathrm{soc} + {a}_0. \end{aligned}$$
(1)

In other words, the characteristics between these variables is taken as a straight line with slope as ‘\({a}_1\)’ and y-intercept as ‘\({a}_0\)’. The state variables of the considered battery are \({x}_\mathrm{soc}\) (SoC), voltage across capacitors \({C}_1\) and \({C}_2\) (symbolized as \({v}_1\) and \({v}_2\)). The variation of SoC with time is governed by the following expression.

$$\begin{aligned} {x}_\mathrm{soc}({t})={x}_\mathrm{soc}(0)-\frac{1}{{Q}}\int \limits _{0}^\mathrm{t} {i}_{ t}{\hbox {d}t} \end{aligned}$$
(2)

where, ‘Q’ and ‘\({x}_\mathrm{soc}(0)\)’ refers to rating of the battery in Ampere hour or Coulomb and initial state of charge, respectively.

Fig. 1
figure 1

Equivalent circuit representation of Li-ion battery

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The state equations of the battery are as follows:

$$\begin{aligned} \dot{{x}}_\mathrm{soc}= & {} \frac{-1}{{Q}}{i}_{ t} \\ \dot{{v}}_1= & {} \frac{-1}{{R}_1 {C}_1}{v}_1 + \frac{1}{{C}_1}{i}_{ t} \\ \dot{{v}}_2= & {} \frac{-1}{{R}_2{C}_2}{v}_2 + \frac{1}{{C}_2}{i}_{ t}. \end{aligned}$$

The terminal voltage equation of the battery is shown below.

$$\begin{aligned} {v}_{ t}={a}_1{x}_\mathrm{soc}+{a}_0-{v}_1-{v}_2-{R}_{ s} {i}_{ t}. \end{aligned}$$
(3)

The above equation holds good in discharging mode of operation, i.e., battery acting as a source. The same equation in charging mode of operation can be obtained by reversing the direction of terminal current. The particular battery system can be modeled by taking terminal current and voltages as input and output variables, respectively. The described system consists of the uncertain parameters \({a}_0\) and \({a}_1\) in the output. In this paper, the adaptive control law is developed to address the uncertainty pertaining to \({a}_1\), assuming \({a}_0\) is a known parameter. The equations can be rearranged by choosing a new state \({x}_1 \) given by

$$\begin{aligned} {x}_1 = {a}_1 {x}_\mathrm{soc}. \end{aligned}$$
(4)

In realizing the same system with new variables, the uncertainty can be seen in state \( x_1 \) rather than in the output. For simplicity, consider \( v_1 = x_2 \) and \( v_2 = x_3 \). For the terminal current of u(t), the state equations are as follows:

$$\begin{aligned} \dot{{x}}_{1}= & {} \frac{-{a}_1}{{Q}}{u}({t}) \nonumber \\ \dot{{x}}_2= & {} \frac{-1}{{R}_1{C}_1}{x}_2 + \frac{1}{{C}_1}{u}({t}) \nonumber \\ \dot{{x}}_3= & {} \frac{-1}{{R}_2 {C}_2}{x}_3 + \frac{1}{{C}_2}{u}({t}) \nonumber \\ {y}({t})= & {} {a}_0 + {x}_1 - {x}_2 - {x}_3 - {R}_{ s} {u}({t}). \end{aligned}$$
(5)

In many applications, each state variable may not be directly measurable [24]. In the working system, SoC of Li-ion battery cannot be measured directly. The other state variables are insignificant in the present context as the physical phenomena of battery is realized in the form of equivalent circuit. Hence, a state estimator like Kalman filter is used with rational assumptions [12, 26]. It is an algorithm which estimates the system states where few (or all) states are not directly measurable. It requires process noise co-variance data along with state-space model of the system [11].

3 Design of the Adaptive Feedback Control Law

A reference system is chosen such that the system has desired characteristics and it is defined as follows:

$$\begin{aligned} \dot{\hat{{x}}}_1= & {} \frac{-\hat{{a}}_1}{{Q}}{r}({t}) \nonumber \\ \dot{\hat{{x}}}_2= & {} \frac{-1}{{R}_1{C}_1}\hat{{x}}_2 + \frac{1}{{C}_1}{r}({t}) \nonumber \\ \dot{\hat{{x}}}_3= & {} \frac{-1}{{R}_2{C}_2} \hat{{x}}_3+\frac{1}{{C}_2}{r}({t}) \nonumber \\ {y}_{{r}}= & {} {a}_0 + \hat{{x}}_1 -\hat{{x}}_2-\hat{{x}}_3-{R}_{ s}{r}({t}), \end{aligned}$$
(6)

where \(\hat{{a}}_1\) is the estimated parameter in the reference system, r(t) is the reference signal and the reference states are \(\hat{{x}}_{ i}, {i} = 1, 2, 3 \).

Constraint over parameter estimation. The parameter estimate \( \hat{{a}}_1 \) is to be updated from an adaptive law and should stay in prespecified regions (that is, \( \hat{{a}}_1> \) 0, since actual parameter \({a}_1 \) is greater than zero) needed for implementation of the adaptive control. This constraint is built with the understanding that the terminal voltage available is more when SoC is more.

Subtracting (5) from (6) and defining \( \tilde{{a}}_1(t) = \hat{{a}}_1({t}) - {a}_1({t})\), we obtain the error signals \({e}_{ i} = \hat{{x}}_{ i} - {x}_{ i}, {i} = 1, 2, 3 \) as

$$\begin{aligned} \dot{{e}}_1= & {} \frac{\hat{{a}}_1}{{Q}}({u}-{r}) - \frac{\tilde{{a}}_1}{{Q}}{u} \nonumber \\ \dot{{e}}_2= & {} \frac{-1}{{R}_1{C}_1}{e}_2 - \frac{1}{{C}_1}({u}-{r}) \nonumber \\ \dot{{e}}_3= & {} \frac{-1}{{R}_2{C}_2}{e}_3 - \frac{1}{{C}_2}({u}-{r}). \end{aligned}$$
(7)

We define the adaptive update law for the parameter estimate as

$$\begin{aligned} \dot{\hat{{a}}}_1= & {} \frac{{p}_1}{\gamma {Q}}{e}_1 {u}, \end{aligned}$$
(8)

and choose adaptive control law u(t) given by

$$\begin{aligned} {u}({t})= & {} \big (\frac{-{p}_1{e}_1({t})}{{Q}}\hat{{a}}_1({t}) + \frac{{p}_2{e}_2({t})}{{C}_1} + \frac{{p}_3{e}_3({t})}{{C}_2} \big ) + {r}({t}), \end{aligned}$$
(9)

where \({p}_{ i}, {i} = 1, 2, 3\) are the constants which could be adjusted in order to obtain the desired response.

Lemma 1

The adaptive control scheme with control law u(t) in the (9) updated by the adaptive law in (8), when applied to linear error dynamics (7) ensures that the all closed loop signals within the system are bounded and the tracking error asymptotically approaches zero, that is, \(\lim _{{t}\rightarrow \infty } {e}_{ i}({t}) = 0, {i} = 1, 2, 3\) [13, 14, 23].

Proof

We define a positive definite function V as follows :

$$\begin{aligned} {V}= & {} \frac{1}{2} \sum _{{i}=1}^{3} {p}_{ i}{e}_{ i}^2 + \frac{1}{2}\gamma _1\tilde{{a}}^2_1 \end{aligned}$$
(10)

where \({p}_{ i} > 0, {i} = 1, 2, 3\) and \(\gamma > 0\), and we ascertain the dynamical system for asymptotic stability. After differentiating V in (10), we get

$$\begin{aligned} \dot{{V}}= & {} {p}_1{e}_1\dot{{e}}_1 + {p}_2{e}_2\dot{{e}}_2 + {p}_3{e}_3\dot{{e}}_3 + \gamma _1\tilde{{a}}_1\dot{\tilde{{a}}}_1 \end{aligned}$$
(11)

Since \( \tilde{{a}}_1({t}) = \hat{{a}}_1({t}) - {a}_1({t}) \), so by differentiating \(\tilde{{a}}_1({t})\), we obtain

$$\begin{aligned} \dot{\tilde{{a}}}_1({t})= & {} \dot{\hat{{a}}}_1({t}) - \dot{{a}}_1({t}) \end{aligned}$$

where \(\dot{{a}}_1\) vanishes as \({a}_1\) is constant term. Now, substituting the values of \( \dot{e}_i(t) \) from (7) and \( \dot{\tilde{a}}_1 \) from above in Eq. (11), we have

$$\begin{aligned} \dot{{V}} = \frac{p_1\hat{{a}}_1}{{Q}}{e}_1({u}-{r}) - \frac{{p}_1\tilde{{a}}_1}{{Q}}{e}_1{u} - \frac{{p}_2}{{R}_1{C}_1}{e}^2_2 + \frac{{p}_2}{{C}_1}{e}_2({r}-{u}) \nonumber \\ - \frac{{p}_3}{{R}_2{C}_2}{e}_3^2 + \frac{{p}_3}{{C}_2}{e}_3({r}-{u}) + \gamma _1\tilde{{a}}_1\dot{\hat{{a}}}_1. \end{aligned}$$
(12)

On substituting \( \dot{\hat{a}}_1 \) from (8), the following expressions can be obtained:

$$\begin{aligned} \dot{{V}} = \frac{{p}_1\hat{{a}}_1}{{Q}}{e}_1({u}-{r}) - \frac{{p}_2}{{R}_1{C}_1}{e}^2_2 + \frac{{p}_2}{{C}_1}{e}_2({r}-{u}) \nonumber \\ - \frac{{p}_3}{{R}_2{C}_2}{e}_3^2 + \frac{{p}_3}{{C}_2}{e}_3({r}-{u}). \end{aligned}$$
(13)

In the above equation, consider \(V_1\) given by

$$\begin{aligned} {V}_1= & {} -\frac{{p}_3}{{R}_2{C}_2}{e}_3^2 - \frac{{p}_2}{{R}_1{C}_1}{e}^2_2. \end{aligned}$$
(14)

It can be seen that \({V}_1 \le 0 \), because \({V}_1\) carries negative definiteness in itself. Therefore,

$$\begin{aligned} \dot{{V}}= & {} {V}_1 + \frac{{p}_1\hat{{a}}_1}{{Q}}{e}_1({u}-{r}) - \frac{{{p}}_3}{{C}_2}{e}_3({u}-{r}) - \frac{{p}_2}{{C}_1} {e}_2({u}-{r}). \end{aligned}$$
(15)

Substituting the adaptive control law given by (9) in the above equation, we get

$$\begin{aligned} \dot{{V}} \le 0. \end{aligned}$$
(16)

This implies that, asymptotically, the defined function V will become zero. Hence we conclude \(\lim _{t\rightarrow \infty } e_i(t) = 0 \), \(i = 1, 2, 3 \). It can be noted that the asymptotic stability is not lost though the constraint described earlier is not imposed in the proof.

4 Case Study

In this section, simulation studies are presented to show the efficacy of the employed methodology. The specifications corresponding to Li-ion battery are shown in Table 1. The gains \({p}_1, {p}_2\) and \({p}_3\) are assigned with \(5 \times 10^{4}\), 1 and 1 respectively and \(\gamma \) as \(10^{-4} \). The optimal values of these parameters are selected by changing each of them in an iterative manner from a very low value to very high value. Table 2 shows the nominal values and range of the parameters \( {a}_0, {a}_1 \) and SoC [15, 17].

Table 1 Main specifications of the Li-ion Battery
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Table 2 Nominal and range of parameters for a typical Li-ion battery
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In this simulation, the considered Li-ion battery model is subjected to both charging and discharging modes of operation. Initially, battery is charged with a current 1.2 A for 300 s and then discharged with 0.8 A for next subsequent 300 s as shown in Fig. 2. The chosen lower limit for the estimated parameter for \({a}_1\) is 0.001. The applicability of adaptive control is verified by introducing a sudden disturbance in the parameter \({a}_1\) at 450th sec, which changes from initial 0.5–0.7. Although such a disturbance may be unrealistic, the control technique is designed to trace the particular parameter. It can be seen in Fig. 3 that the change in the actual parameter leads to the change in estimated parameter. The tracking error \({e}_1 \) is observed to settle at zero as shown in Fig. 4. Figures 5 and 6 shows the Terminal Voltage and the estimated SoC respectively. It is evident that both the quantities tend to increase during charging and decrease while discharging. The rate of change of voltage and SoC becomes negative after 300 s since the current in battery changed its direction. Also, it is noteworthy that there is no major change in the output voltage in the event of disturbance introduced in the parameter \({a}_1\) at 450th s. The reason behind this particular observation is due to the appropriate estimation of \({a}_1\) by adaptive controller.

Fig. 2
figure 2

Top: Control input signal; Bottom: Reference input signal

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Fig. 3
figure 3

Top: Parameter error, Bottom: Parameter estimate

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Fig. 4
figure 4

Tracking error of the state \({x}_1\)

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Fig. 5
figure 5

Terminal voltage of the battery

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Fig. 6
figure 6

SoC of the battery

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It is to be noted that the task of ascertaining terminal voltage is accomplished through proper estimation of uncertain parameter \({a}_1\). In other words, the terminal voltage of the battery can be determined by assigning the final estimated value of the parameter in the voltage equation shown in (1).

5 Concluding Remarks

In this paper, an adaptive control technique is presented against the problem of parameter uncertainty associated with Li-ion battery. The task of determining the terminal voltage of the battery is accomplished through validation of uncertain parameter with an assumption that SoC can be estimated. Moreover, the proposed technique is verified even under the variation of uncertain parameter in order to maintain the desired response. The methodology described may find a practical application in the testing and operation of Li-ion batteries.