An Electro-Thermal Model based fast optimal charging strategy for Li-ion batteries

1.   Introduction

Driven by the urge to comply with the clean air acts enacted by the governments around the world, the automotive manufacturers are moving toward electrification of their fleets, because powertrain electrification contributes significantly toward making cars clean and environmentally friendly ^[1], and lithium-ion batteries have emerged as the only plausible solution for driving powertrain electrification to the next level. There is also an ever-increasing demand for compact, low cost batteries for numerous applications ranging from portable and hand-held electronic devices to large electrical appliances. Over the last three decades, lithium-ion batteries have dominated both consumer electronics and electric vehicle markets, thanks to their falling price and continuous improvement of their energy density ^[2]. However, they still suffer from two major drawbacks, namely, long charging times and limited capacity ^[3].

While material scientists are working on energy density improvement technologies ^[4], other researchers are focusing on the development of charging infrastructures and fast charging strategies, keeping into consideration capacity degradation due to battery aging and charging cycles ^[5]. It is well known that both charging methodology and operating temperature have significant effects on battery lifetime and battery performance ^[6], which give rise to the need for a robust battery management system (BMS) with an optimized charging strategy. To ensure a safe operation of Li-ion batteries, a BMS controls and monitors the state of each cell within the battery pack ^[7]. Besides measuring current, voltage, and temperature, a BMS also controls cooling, cell balancing, and power limits of the cells. This assures safe battery operation, as well as good performance and durability.

Before committing to the design phase of a BMS, it is essential to develop an accurate battery model. Such a model allows the prediction of long-term battery behavior as well as its performance. Various types of battery models proposed in the literature to date can be classified according to their accuracy and complexity. Some authors categorize them into physical, empirical and abstract (analogous) models ^[8]. While physical models offer a high level of accuracy by capturing the key features of the physical process, they have high computational time complexity, which can be problematic, especially if the model is required to run in real-time. Empirical models, on the other hand, provide a poor, simplistic representation of the system since they rely on experimental data instead of a mathematical representation of the system. Finally, abstract (analogous) models provide a trade-off between accuracy and simplicity. Although different types of analogous battery models can be constructed, electrical equivalent circuit (EEC) models are most popular and have found widespread use in BMS and other applications ^[8].

Furthermore, a BMS requires an optimum battery charging strategy to improve a battery's charge time without compromising its performance. According to Zhang et al. ^[9], battery charging strategies can be classified into four broad categories, namely, methods based on the improvement of the battery materials ^[10,11], polarization-based methods ^[12,13], methods based on improvement of the charging current ^{[14,15,16,17]}, and methods based on the improvement of battery models ^{[18,19,20,21,22]}. All these methods are characterized by their relative advantages and disadvantages. For example, methods based on improvement of battery materials seem promising as researches are actively experimenting with different electrode and electrolyte materials to develop next-generation Li-ion batteries for improving specific energy and reducing the cost. However, an extensive amount of research and experimentation is needed to prove their safety and reliability for automotive and other applications. Similarly, the polarization-based methods are also of interest in the future because they provide a promising alternative, but they require further development and are more complicated compared to the methods based on improvements of charging current and battery models. The methods belonging to the last two categories are of special interest here, because the charging scheme presented in this paper is also based on similar principles.

Among the methods based on improvement of charging current, the older ones are of heuristic nature and lack mathematical rationality and basis. On the other hand, the newer methods, as exemplified by ^{[14,15,16,17]}, are based on designing optimum current profiles that can lower charging time and prolong battery life. For instance, Guo et al. ^[14] proposed an EEC model based method of designing an optimum nonlinear current profile using genetic algorithm as an optimization tool. The proposed method is potentially useful for reducing charging time and prolonging cycle life. However, one of the drawbacks of the proposed method is that it uses a highly nonlinear current profile and furthermore, it may not be suitable for time-critical applications because genetic algorithm has a relatively high computational time complexity. The other methods ^[15,16,17] can be regarded as variants of multistage constant current charging (MCC) charging strategies. For instance, Vo et al. ^[15] proposed a Taguchi method to search for an optimum MCC charging current profile. Compared with the traditional constant-current constant-voltage (CCCV) method, it can reduce the charging time, decrease the temperature variation, and improve energy efficiency. However, the computational time complexity of the proposed method is relatively high and therefore, it is not suitable for time-critical applications. Contrary to passive charging techniques, which utilize a fixed pre-set charging profile, an adaptive charging current would adapt with the variation of internal battery parameter changes during its cycle life. Such techniques can be broadly classified into two categories, namely, experiment-based methods and model-based methods, combined with some optimization technique ^[16,17]. Chen et al. ^[16] proposed a method searching for the optimal charging current pattern of a MCC current profile based on an EEC model and the grey wolf optimization technique. The proposed method is potentially useful for simultaneously reducing the charging loss and shortening the total charging time. In addition, this method can significantly reduce the temperature rise, and improve charging efficiency and battery life cycle. Jiang et al. ^[17] also proposed a hybrid charging strategy with adaptive current control, which is called CC-CC-CV strategy. It can also reduce the maximum temperature rise and energy losses.

Among the methods based on improvement of battery models ^{[18,19,20,21,22]}, most of the methods proposed to date use either simplified electrochemical models ^[18,19,20] or EEC models ^[21,22] to develop fast charging strategies. Liu et al. ^[18] posed the problem of fast charging as a dynamic optimization problem that attempts to optimize the charging current trajectory of the battery to meet a desired target state of charge (SOC) subject to a side reaction constraint governing the rate of degradation via lithium-ion plating. They used a physics based model and employed Gauss pseudospectral method [GPM] to optimize the charging current trajectory. Xu et al. ^[19] proposed an optimal MCC charging strategy that not only minimizes the charging time but also attempts to minimize capacity fade due to solid electrolyte interphase (SEI) increase, decrease lithium plating, and reduce the temperature rise. The proposed method is based on an electrochemical battery model and optimization method employed is dynamic programming (DP). A similar approach was also proposed by Liu et al. ^[20], but they used genetic algorithm (GA) as the optimization tool. All the charging strategies mentioned above are interesting and useful for certain applications, but one of their major drawbacks stems from the fact that both the electrochemical models as well as the proposed optimization methods are computationally intensive and not suitable for time-critical applications that require fast search for optimal charge patterns. In contrast to the above methods, the EEC model based methods, as exemplified by ^[21,22], are much simpler to implement and therefore, potentially useful for time-critical applications. Both Hu et al. ^[21] and Zhang et al. ^[22] proposed EEC model based fast charging strategies that are based on solving a multi-objective optimization problem for simultaneously minimizing both charging time and charging losses. Their main difference lies in the choice of the optimization tool. Hu et al. ^[21] used large scale sequential quadratic programming (SQP) method, whereas Zhang et al. employed dynamic programming (DP). However, both SQP and DP are computationally intensive methods that are not suitable for time-critical applications, and there exists a need for developing faster optimization strategies.

From the above discussion, it is clear that EEC models are best suited for developing fast charging strategies that are going to be useful for time-critical applications. While minimizing charging time, however, attention must also be paid to battery safety issues, such as temperature rise and lithium plating. Thus, an EEC model needs to be augmented by ancillary models that account for some of the battery safety issues. Since Li-ion batteries are prone to thermal runaway, preventing excessive temperature rise during charging is deemed most critical. In fact, excessive temperature rise during charge/discharge operations can also cause capacity degradation, premature aging, and temperature related stresses. In view of this, we focus on developing a fast charging algorithm that not only minimizes the battery charging time, but also restricts its temperature rise. To achieve this dual objective, the charging algorithm proposed here utilizes an electro-thermal battery model that consists of an EEC battery model augmented with a thermal model. Such a coupled electro-thermal model can predict the battery's terminal voltage as well as the internal temperature of the battery ^[23,24]. Thus, such a model is useful in the quest for an optimal charge pattern, while restricting the battery's temperature within safe limits, thus ensuring operational safety and reducing stresses on the battery.

Having selected an electro-thermal battery model as the backbone of our fast charging strategy, the next important questions concern the selection of a charge current profile as well as selection of a thermal model. Both issues are discussed in details in Sections 2 and 3 below, but just to summarize the main principle, we would like to mention here that that a MCC charge pattern is chosen in this study because numerous researchers have found that compared to traditional CCCV charging method, which used to be popular in the past, MCC decreases total charging time and also reduces temperature rise ^[25,26]. Finally, as for the thermal model, an analytical-numerical (ANM) thermal model is adopted in this study because this model eliminates the need for measuring the core temperature of the battery, which is thought to be a hazardous task ^[27]. An ANM thermal model allows estimation of the core temperature instead of actually measuring it. Furthermore, just like an EEC model, an ANM thermal model offers a good trade-off between modeling accuracy and the ease of implementation.

In this paper, we present a multistage constant current (MCC) charge pattern optimization strategy that minimizes the total charge time of a Li-ion battery, while limiting its temperature rise within safe limits specified by the user. The main contributions here are two fold, namely, presentation of a fast optimum charging strategy that has very low computational time complexity compared to the existing charging strategies, and at the same time, restricting the temperature rise of the battery to a safe level specified by the user. The organization of this paper is as follows. Section 2 summarizes the basic principle of MCC charging method. Section 3 presents a brief overview of an electro-thermal model of a Li-ion battery. The problem of battery charge pattern optimization is discussed in Section 4, which also introduces the proposed fast charging algorithm. Section 5 presents the results of some simulation studies. Finally, some concluding remarks are provided in section 6.

2.   Multistage constant current charging

Multistage Constant Current (MCC) charging method is the focus of this study because a number of researchers have shown its advantages, such as shorter charging time, lower temperature rise, and extended battery life ^[25]. Also, compared to the conventional constant current constant voltage (CCCV) charging method, MCC has been proven to be less stressful and safer ^[26]. The conventional CCCV charging method consists of two phases, a constant current (CC) phase where the battery is charged using a constant current until a cutoff voltage is reached, which is followed by a constant voltage (CV) phase where the battery maintains that cutoff voltage until the charging current reaches the charging current reaches a near zero. Two key factors of CCCV are the charging current, ${I}_{cc}$, during CC phase and the cutoff voltage, ${V}_{cv}$, during CV phase. The cutoff voltage is typically chosen in accordance with the nominal cell voltage. For conventional Li-ion batteries, ${V}_{cv}$is chosen to be approximately 4.0 V. However, it should be noted that during CV mode, a constant stress is continuously applied to the battery while maintaining that cutoff voltage.

Contrary to CCCV, a multistage constant current (MCC) charging profile consists of N (≥ 2) constant current charging stages with different current levels until the battery's cutoff voltage is reached. Typically, the charging current starts from a high initial value, which is reduced gradually in a stepwise fashion for each successive stage when a preset switching condition is met. In other words, the successive charging currents can be denoted by ${I}_{k}$, 1 $\le k \le$N and ${I}_{k}>{I}_{k+1}$. Since CV mode is not utilized during MCC charging, it presents a significant advantage over conventional CCCV. However, the performance of a MCC charging method is governed by many factors, such as the number of stages, $N$, charging currents, ${I}_{k}$, 1 $\le k \le$N, and the total charging time.

3.   Integrated Electro-Thermal Model of a Li-ion battery

The integrated electro-thermal model studied here is based on a coupled electrical equivalent circuit (EEC) model and a thermal battery model. The EEC model is used to predict the battery's terminal voltage during charge or discharge operations, while the thermal model estimates the temperature rise of the battery ^[23]. First, we discuss the EEC model and then summarize the thermal model.

3.1.   Electrical Equivalent Circuit (EEC) Model

A simplified form of the voltage-current response circuit of a popular EEC model ^[28] is shown in Figure 2 below. This second order EEC battery model, characterized by five lumped parameters, has been widely used in literature to model the voltage-current relationship of a Li-ion battery during charge and discharge operations.

This circuit models the transient response of the battery when connected to an external load. The voltage-current response of a battery due to a step change in the load current exhibits both instantaneous characteristics as well as short-term and long-term transient characteristics. The instantaneous response is modeled by a series resistor, R₀, which basically represents the internal series resistance of the battery, whereas the short-term and long-term transients are modeled by a pair of parallel RC circuits, (R₁, C₁) and (R₂, C₂). Also, ${V}_{ocv}$ represents the battery's open circuit voltage, which is a voltage-controlled voltage source dependent on the $SOC$.

The equations governing the circuit operation are given by:

where ${C}_{n}$ denotes the battery capacity, I denotes the current, V_T is the terminal voltage and $f\left(SOC\right)$ is a known nonlinear function that models the relationship between open circuit voltage, ${V}_{ocv}$, and state of charge ($SOC$).

3.1.1.   Remark 1

Although V_OCV is primarily a function of SOC, as pointed out by a number of researchers recently ^[30,31], it also depends (to a much lesser extent) on temperature. In fact, this temperature coefficient of voltage (TCV) is only of the order 0.1 mV/℃ for Li-carbon anodes ^[31], but still for a high capacity battery, such as a 75 AH battery studied in ^[30], it may cause a V_OCV difference of about 25 mV for a temperature variation from 0 ℃ to 45 ℃. Since this study focusses on a low capacity (2.3 AH) battery, and the temperature variation is typically restricted to below 10 ℃, we ignore the temperature coefficient of voltage here and assume V_OCV is essentially a function of SOC. However, the results presented here can easily be extended to high capacity batteries and battery packs by adding a correction term due to TCV.

A state space representation of the circuit is given by the following equations:

where ${V}_{1}$ and ${V}_{2}$ denote the state variables, which represent the voltages across the capacitors, ${C}_{1}$ and ${C}_{2}$, respectively. It is also important to note that the parameters of the EEC model are time-varying and dependent on both temperature and $SOC$.

3.2.   Thermal Model of a Li-ion battery

During charging of a Li-ion battery, heat is generated due to both Joule heating in the internal resistance of the battery and the entropic changes resulting from the associated chemical reactions. Thus, it is important to monitor the temperature of a battery during charging operations, because failure to do so may result in capacity degradation, premature aging, temperature related stresses, and other safety concerns. Since battery performance is highly dependent on temperature, it is important to limit the rise of temperature during battery operations. Battery thermal models have been found to be very useful for this purpose, because such models can be used to estimate the temperature rise of a battery during charging.

A wide variety of battery thermal models, suitable for different applications, have been proposed in the literature ^[32]. They differ from each other in their level of complexity, accuracy and computational cost. These models can be broadly divided into three categories, namely, lumped parameter models (LPM), finite element models (FEM), and analytical-numerical models (ANM). The LPM models are useful for fast simulation, but suffer from limited accuracy, whereas FEM models are highly accurate, but have a high (computational) time complexity, which precludes their usage in time-constrained applications. The ANM models, on the other hand, offer a reasonable trade-off between simulation accuracy and (computational) time complexity. In view of above, a simplified ANM thermal model has been adopted in this study.

A simplified ANM thermal model of a Li-ion battery is depicted in Figure 3 below ^[23,24]. A state space representation of this model is given by:

where ${C}_{c}$ and ${C}_{s}$ denote the lumped heat capacities of the core and casing, respectively. Also, ${R}_{c}$ and ${R}_{u}$ denote the heat conduction and convection resistances, respectively, and ${T}_{f}, {T}_{c}$ and ${T}_{s}$ denote the ambient, core, and surface temperatures, respectively. Moreover, $Q$ is the heat generated during the chemical reaction of the battery, which includes the Joule heating due to over-potential and the heat generated due to entropy change. It can be expressed as:

However, according to ^[24], the effect of the entropic heat generation is often omitted for simplicity as its contribution to the overall heat generation is relatively small.

Also, the mean temperature, ${T}_{m}$, of the battery can be written as:

3.3.   Inter-coupling between the EEC and Thermal Models

An inter-coupling between the EEC and the thermal models is formed when the heat generated by the electrical model, $Q$, serves as an input to the thermal model. The core and surface temperatures are predicted by the thermal model, and its mean value, ${T}_{m}$, is used to update the values of the EEC model parameters ^[23]. The state vector, X, input, U, and output, Y, of the integrated electro-thermal model can be expressed as:

4.   MCC charge pattern optimization strategy

A MCC charge pattern optimization method is presented in this section. Here the objective of optimization is to find a suitable charging pattern that minimizes the total charging time, while limiting the battery's temperature rise. The charge pattern optimization problem can be stated as follows.

4.1.   Optimization problem 1

Find optimal values of $\left\{\mathrm{N}, \mathrm{ }{Δ\mathrm{t}}_{\mathrm{k}}\; \mathrm{a}\mathrm{n}\mathrm{d}\; \mathrm{ }{\mathrm{I}}_{\mathrm{k}}, \mathrm{ }1\le \mathrm{k}\le \mathrm{N}\right\}$ that minimize the total charge time, ${\mathrm{t}}_{\mathrm{N}}$, subject to the constrains,

where $N$ denotes the number of stages, ${I}_{k}$ is the charging current and ${t}_{k}$ denotes the charge termination time for stage k, and ${Δt}_{k} = {t}_{k}-{t}_{k-1}$ represents the charging duration for stage k. Also, ${ΔT}_{k}$ and ${ΔSOC}_{k}$ denote the temperature-rise and SOC gain for stage k, ${ΔT}_{max}$ is the maximum allowable temperature rise of the battery during charging, and ${SOC}_{Goal}$ denotes the targeted SOC level.

The above optimization problem is difficult to solve unless some simplifying assumptions are introduced. In fact, most researchers assume that N is known a priori, and the allowable values of charge current levels, I_k, are assumed to be known fractions or multiples of the C-rate charge current, such as 0.5C, 1C, 1.5C, etc. ^[18,21,27]. Thus, a simplified version of the above problem can be expressed as:

4.1.1.   Optimization problem 2

Find optimal values of {${\mathrm{t}}_{\mathrm{k}}$, ${\mathrm{I}}_{\mathrm{k}}$, 1 ≤ k ≤ N} that minimize the total charge time, ${\mathrm{t}}_{\mathrm{N}}$, subject to the constraints (11a–11b).

Many optimization techniques can be applied to find a solution to the above Optimization Problem 2, including a multi-objective optimization algorithm, such as minimax or goal programming, or an evolutionary optimization method, such as genetic algorithm (GA) ^[33]. In fact, GA has been used by several researchers ^[18,21]. However, the solutions provided by GA should not be considered as optimal, but only quasi-optimal due to the following facts:

- The number of stages $N$ is assumed to be known a priori;

- The presumed charging profile may not be optimal;

- A solution provided by GA is not guaranteed to be the global optimum. It can at best be a very good local optimal solution.

Furthermore, although GA can provide a good solution, it is rather slow to execute because it requires numerous random searches to find such a solution. Thus, GA and other evolutionary algorithms are not suitable for fast charging applications.

It is clear from the above discussion that the development of a fast charging algorithm calls for further simplification of Optimization Problem 2. To simplify the problem, we retain constraint (11b) as it is, but replace constraint (11a) by a new one related to temperature-rise in each stage, and define a simplified optimization problem as:

4.1.2.   Optimization problem 3

For a given set of charge currents, {${\mathrm{I}}_{\mathrm{k}}$, 1 ≤ k ≤ N}, find optimal values of {${\mathrm{t}}_{\mathrm{k}}$, 1 ≤ k ≤ N} that minimize the total charge time, ${\mathrm{t}}_{\mathrm{N}}$, subject to the constraints:

where $Δ{T}_{k, max}$ represents the maximum (permitted) temperature rise for stage k, 1 ≤ k ≤ N.

It may be pointed out that Optimization Problem 3 constitutes a new idea, which, at the cost of introducing some additional prior information, i.e., knowledge of ${I}_{k}and Δ{T}_{k, max}$, 1 ≤ k ≤ N, significantly reduces the time complexity of the optimization problem. The algorithm proposed below solves Optimization Problem 3.

4.2.   Proposed fast charging algorithm

Notice that an optimal solution of Optimization Problem 3 would require a multivariable global minimization technique. However, the available global minimization algorithms are time-consuming and never guaranteed to yield the global optimal solution. In view of this, since the primary goal of this study is to develop a fast and safe charging strategy for time-constrained applications, we propose to use a quasi-optimal search strategy consisting of successive single variable searches to find the optimal values of {${t}_{k}$, 1 ≤ k ≤ N}. Specifically, for the ${k}^{th}$ charging stage, we find the optimal value of the charge termination time, ${t}_{k}$, by using a technique similar to interval-halving method ^[33], which is described below. Before describing the algorithm, first we introduce a few notations that are shown in Table 2 below.

Fast charging algorithm

Step 1 (Initialization): Set ${SOC}_{Initial}$ and ${SOC}_{Goal}$. Choose $N$; ${I}_{k}$, $1\le k\le N$; and $Δ{T}_{k, max}$, $1\le k\le N.$ Initialize ${SOC}_{Accu} = {SOC}_{Initial};$ set $k = 1$.

Step 2: If (${SOC}_{Accu}\ge {SOC}_{Goal}), stop; else$ start a search for an optimal ${t}_{k}$; let ${t}_{k\mathrm{L}} = 0$ and calculate ${t}_{k\mathrm{U}}$ as the time needed to completely charge the battery using charging current ${I}_{k}$:

Step 3: Run thermal model to estimate the expected temperature-rise (if charged at current ${I}_{k}$ for a duration, ${t}_{kU}-{t}_{k-1}$), i.e., ΔT$({I}_{k}, {t}_{kU}-{t}_{k-1})$.

● If ΔT (${I}_{k}$, ${t}_{kU}-{t}_{k-1}$) ≤ $Δ{T}_{k, max}$

- set ${t}_{k} = {t}_{kU}$;

- simulate electro-thermal battery model for $t∈\mathrm{ }({t}_{k-1}, {t}_{k})$;

- ${ΔSOC}_{k} = {I}_{k}$(${t}_{k}-{t}_{k-1}); {SOC}_{Accu} = {SOC}_{Accu}+{ΔSOC}_{k};$

- set $k = k+1$;

- go back to step 2 to proceed to the next stage.

● else ${t}_{k}$ ϵ (${0, t}_{kU}$), which is the interval of uncertainty (IOU) for ${t}_{k}$;

- set ${t}_{kL} = 0$;

- go to Step 4 to search for an optimal value of ${t}_{k}$

Step 4: (Interval-halving search for an optimal value of ${t}_{k}$)

4A. Find the middle point of the current IOU, ${t}_{kM} = \frac{({t}_{kL}+{t}_{kU})}{2}$.

● If ${(t}_{kU}-{t}_{kL}) < tol$, stop further search for ${t}_{k}$;

- set ${t}_{k} = {t}_{kM}$;

- simulate electro-thermal battery model for $t∈\mathrm{ }({t}_{k-1}, {t}_{k})$;

- ${ΔSOC}_{k} = {I}_{k}$(${t}_{k}-{t}_{k-1}); {SOC}_{Accu} = {SOC}_{Accu}+{ΔSOC}_{k};$

- set $k = k+1$;

- go back to step 2 to proceed to the next stage.

● else proceed to step 4B below.

4B. Run thermal model to estimate the expected temperature rise (if charged at current ${I}_{k}$ for a duration, ${t}_{kM}-{t}_{k-1}$), i.e. ΔT$({I}_{k}, {t}_{kM}-{t}_{k-1})$.

● If ΔT$\left({I}_{k}, {t}_{kM}-{t}_{k-1}\right) < Δ{T}_{k, max}$, ${{{t}_{k}ϵ (t}_{kM}, t}_{kU})$, which is the new (reduced) IOU for ${t}_{k}$;

- set ${t}_{kL} = {t}_{kM},$ keep ${t}_{kU}$ unchanged;

- go back to step 4A to continue interval-halving search.

● else if ΔT$\left({I}_{k}, {t}_{kM}-{t}_{k-1}\right)>Δ{T}_{k, max}$, ${{{t}_{k}ϵ (t}_{kL}, t}_{kM})$, which is the new (reduced) IOU for ${t}_{k}$;

- set ${t}_{kU} = {t}_{kM}$, keep ${t}_{kL}$ unchanged;

- go back to step 4A to continue interval-halving search.

● else if ΔT$\left({I}_{k}, {t}_{kM}-{t}_{k-1}\right) = Δ{T}_{k, max}$,

- set ${t}_{k} = {t}_{kM}$;

- simulate electro-thermal battery model for $t∈\mathrm{ }({t}_{k-1}, {t}_{k})$;

- ${ΔSOC}_{k} = {I}_{k}$(${t}_{k}-{t}_{k-1}); {SOC}_{Accu} = {SOC}_{Accu}+{ΔSOC}_{k};$

- set $k = k+1$; go back to step 2 to proceed to the next stage.

For better clarity, the above optimization procedure is also illustrated in the following flow chart.

5.   Results and discussion

The proposed quasi-optimal MCC charge pattern optimization strategy was simulated using an integrated electro-thermal battery model for a 2.3 AH A123-26650 Li-ion battery ^[23,34]. The electrical specifications and thermal model parameters of this battery that are used in this study, are summarized in Tables 3A and 3B below. This electro-thermal model has been experimentally verified to be reasonably accurate and used by a number of researchers for fast charging and other applications. Although the performance of the proposed optimization algorithm was evaluated by conducting an extensive set of simulation studies, the results of only four such studies are summarized here to illustrate its key properties.

In all the case studies presented here, the ambient temperature is assumed to be 25 ℃, ${SOC}_{Initial}$ is set to 0%, and the ${SOC}_{Goal}$ is set to 100%. Also, the number of MCC stages is chosen to be $N = 5.$

For the first sample case study, we choose maximum allowed overall temperature rise, ${ΔT}_{max} = 9$ ℃. The values of the charging current levels and the maximum allowed temperature rise for the five MCC stages are chosen to be as follows:

The simulation results, shown in blue color in Figures 6–9, indicate that the battery's core temperature rises to 33.9 ℃ and the total charge time is 1748 seconds.

For the second sample case study, we choose ${ΔT}_{max}$ = 7 ℃. The values of the charging current levels and the maximum allowed temperature rise for the five MCC stages are chosen to be as follows:

The simulation results, shown in red color in Figures 6–9, indicate that the battery's core temperature rises to 31.2 ℃ and the total charge time is 2340 seconds.

For the third sample case study, we choose ${ΔT}_{max}$ = 5 ℃. The values of the charging current levels and the maximum allowed temperature rise for the five MCC stages are chosen to be as follows:

The simulation results, shown in gray color in Figures 6–9, indicate that the battery's core temperature rises to 29.8 ℃ and the total charge time is 2780 seconds.

A comparison of the above three case studies illustrates the trade-off between ${ΔT}_{max}$ and total charge time ${t}_{N}$, namely, a gradual decrease of ${ΔT}_{max}$ results in a concomitant increase of total charge time. This is in accordance with our expectation, because a lower ${ΔT}_{max}$ would necessitate lower heat dissipation in the battery, or equivalently, lower charging currents, which would increase the total charge time.

In the last sample case study, we demonstrate that a significantly faster charging can indeed be achieved at the cost of a higher overall temperature rise. For this case study, we chose ${ΔT}_{max}$ = 12 ℃. The values of the charging current levels and the maximum allowed temperature rise for the five MCC stages are chosen to be as follows:

The simulation results, shown in yellow color in Figures 6–9, indicate that the total charge time is now reduced to 1450 seconds, at the cost of an overall core temperature rise to 36.5 ℃.

Finally, to allow a quick comparison of the achievable trade-offs between ${ΔT}_{max}$ and total charge time, ${t}_{N}$, a summary of the above four case studies is provided in Table 4 below.

6.   Conclusions

This paper presents an electro-thermal model based quasi-optimal multistage constant current charging algorithm for Li-ion batteries. The search for an optimal charge pattern is carried out on a stage by stage basis by restricting the temperature rise at each stage within safe limits. The algorithm consists of a sequence of successive one-dimensional searches for each stage, which is very easy to implement and suitable for fast charging applications. The proposed charging algorithm was simulated based on an integrated electro-thermal model of a 2.3 AH A123-26650 Li-ion battery, and the simulation results indicate that the algorithm performs as expected and can serve as an useful, easy-to-implement alternative to existing computationally intensive optimal charging strategies proposed by other researchers. However, as mentioned earlier, the proposed method only provides a quasi-optimal solution, because it is based on a sequential single variable search technique instead of a multivariable search method. Our on-going research is geared toward addressing the above shortfall.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Conflict of interest

The authors declare that there is no conflict of interest in this paper.