Universal power-efficiency trade-off in battery charging

Abstract

Designing efficient and fast-charging batteries is an important goal in the field of energy, crucial for upgrading new energy vehicles and portable electronic devices such as smartphones. Here, we incorporate the concept of finite-time thermodynamics into studying the resistor-capacitor (RC) series circuit and obtain the time-dependence of charging efficiency and charging power. Through this exploration, essential thermodynamic constraints governing the charging process, including the trade-off relation between charging power and efficiency, are obtained. Moreover, we reveal the lower bound for charging time and the corresponding optimal charging strategy, and further demonstrate the power-efficiency trade-off relation in such an optimized strategy. Our findings shed new light on seeking optimal battery charging methods with nonequilibrium thermodynamics.

Graphical abstract

Keywords

battery charging / charging strategy / finite-time thermodynamics / optimization

1 Introduction

Batteries function as a primary energy source for a wide range of mobile electronic devices, including smartphones and electric vehicles. They provide users with convenience and portability, simultaneously diminishing reliance on traditional fossil fuels and reducing greenhouse gas emissions [1]. Battery-based products have seen significant development in recent years, among which products offering fast charging and extended battery life are particularly favored by industries and consumers [2]. Therefore, improving charging speed and efficiency, enhancing energy and power density, and prolonging lifespan are essential considerations within battery design [3].

Charging remains a primary concern in state-of-the-art battery research, as the demand for fast charging inevitably leads to challenges such as overheating, battery degradation and reduced lifespan [4]. Massive efforts from various fields have been dedicated to optimizing the trade-off between charging time and these challenges. Novel battery designs such as lithium-ion batteries [5], sodium-ion batteries [6] and solid-state batteries [7] have been proposed to facilitate fast charging while boosting safety and long-term reliability. In the realm of materials science, advanced battery materials, employed in the anode, cathode and electrolyte, are developed to improve charging performance [8–11]. Internal and external battery thermal management systems are also devised to cope with overheating during fast charging [12, 13].

Meanwhile, analyzing optimal charging strategies based on current battery technologies is a critical focus in battery charging research [14–16]. Basic charging strategies such as constant current (CC) and constant voltage (CV) fail to seek a balance within charging time, overheating and battery degradation [17]. Considering all these factors, diverse charging strategies have been put forward from an engineering standpoint, such as constant current constant voltage (CCCV), multistage constant current (MSCC), pulsed charging and boost charging [18, 19]. Various optimization methods, such as evolutionary algorithm [20–22], Taguchi approach [23–25], integer linear programming [26], model predictive control [27–29], reinforcement learning [30–32] and analytical derivation [33, 34], have been proposed to find out the optimal charge pattern (OCP) for these strategies.

Although numerical results and experimental evidence have strongly validated the effectiveness of the charging strategies and optimization methods mentioned above [35–38], to the best of our knowledge, a universal and comprehensive theoretical analysis for general charging processes remains conspicuously absent. This lacuna is of particular concern as it leaves the fundamental physical constraints governing the charging process and the optimization boundaries pertinent to diverse charging strategies shrouded in mystery. From a thermodynamic perspective, fast charging requires a large amount of charge transfer per unit time, which unavoidably increases heat dissipation due to large currents [39]. On the contrary, in order to improve charging efficiency, the charging process should minimize heat dissipation which cannot be stored as energy in the battery [40]. This dichotomy unequivocally suggests the existence of a thermodynamic trade-off relation between the battery charging efficiency and the charging power.

In this study, we draw inspiration from finite-time thermodynamics [41–43] to study the simplest battery charging model, a resistor-capacitor (RC) series circuit. The time dependence of energy conversion in such a circuit is obtained, and we unveil typical thermodynamic constraints during the charging process. Furthermore, for practical purposes, in the cases of i) minimizing charging time and ii) balancing charging power and efficiency, we determine the corresponding OCP for MSCC strategy. This study provides a theoretical framework for the optimization of the battery charging process.

The rest of this paper is organized as follows. In Section 2 we demonstrate the power-efficiency trade-off relation in the CV charging process. The maximum charging power and the corresponding efficiency at maximum charging power are obtained. Furthermore, we study the performance of the MSCC charging in Section 3. The minimal charging time and the power-efficiency trade-off in the optimal MSCC strategy are given in Section 3.1 and Section 3.2, respectively. We conclude this work in Section 4.

2 Power-efficiency trade-off for CV charging

We start with the simplest charging model. As shown in Fig.1(a), an RC circuit with a resistor of R and a capacitor of C is charged under constant charging voltage \mathcal{E}. Intuitively, a higher charging current leads to more charge accumulation per unit time on the battery, indicating higher charging power. However, the thermal effects of high currents reduce the charging power. This indicates that there is a trade-off between charging power P and efficiency \eta.

The accumulated amount of battery charge q=q(t) satisfies \mathrm{d}q=I\mathrm{d}t, where I is the current flow from the source to the battery. Considering that the current and potential difference during the charging process satisfy Ohm’s law I(t)= [\mathcal{E}-V (t)]/R, the battery charge follows as

(1){\displaystyle \frac{\mathrm{d}q}{\mathrm{d}t}}=C{\displaystyle \frac{\mathrm{d}V}{\mathrm{d}t}}= {\displaystyle \frac{\mathcal{E}-V(t)}{R}}.

Hence, we obtain the voltage and charging current of the battery respectively as

(2)V(t)=\mathcal{E}-\left(\mathcal{E}-V_1\right)\exp \left(-{\displaystyle \frac{t}{CR}}\right),

and

(3)I(t)={\displaystyle \frac{\mathcal{E}-V_1}{R}}\exp \left(-{\displaystyle \frac{t}{CR}}\right),

where V_1 is the initial voltage of the battery. Therefore, the output energy of the source W_{\mathrm{o}\mathrm{u}\mathrm{t}}(\tau )=\mathcal{E}{\int }_0^{\tau }I\mathrm{d}t and the dissipated energy due to circuit heating Q_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}(\tau )={\int }_0^{\tau }{I}^{2}R\mathrm{d}t are explicitly written as

(4)W_{\mathrm{o}\mathrm{u}\mathrm{t}}(\tilde{\tau })= \mathcal{W}\lambda \left[1-\exp \left(-\tilde{\tau }\right)\right],

and

(5)Q_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}(\tilde{\tau })={\displaystyle \frac{\mathcal{W}{\lambda }^2}{2}}\left[1-\exp \left(-2\tilde{\tau }\right)\right].

Here, the dimensionless time \tilde{\tau }\equiv \tau / (CR), \mathcal{W}\equiv \underset{\tilde{\tau }\to \mathrm{\infty }}{lim}{W}_{\mathrm{o}\mathrm{u}\mathrm{t}}=C{\mathcal{E}}^2, and \lambda \equiv 1- V_1/\mathcal{E} characterizes the the ratio of initial charging voltage to full voltage. The amount of electrical energy stored in the battery W_{\mathrm{i}\mathrm{n}}(\tilde{\tau }), according to the first law of thermodynamics W_{\mathrm{i}\mathrm{n}}= W_{\mathrm{o}\mathrm{u}\mathrm{t}}-Q_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}, is obtained as

(6)W_{\mathrm{i}\mathrm{n}}(\tilde{\tau })=\mathcal{W}\lambda \left[1-\exp \left(-\tilde{\tau }\right)\right]\left[1-\lambda {\displaystyle \frac{1+\exp \left(-\tilde{\tau }\right)}{2}}\right].

To quantify the thermodynamic performance of the charging process, we introduce the charging efficiency \eta \equiv W_{\mathrm{i}\mathrm{n}}/W_{\mathrm{o}\mathrm{u}\mathrm{t}}, charging power P\equiv W_{\mathrm{i}\mathrm{n}}/\tau, and charging percentage \theta \equiv W_{\mathrm{i}\mathrm{n}}/W_{\mathrm{i}\mathrm{n}}(\tilde{\tau }\to \mathrm{\infty } ), which explicitly read

(7)\eta =1-\lambda {\displaystyle \frac{1+\exp \left(-\tilde{\tau }\right)}{2}},

(8)P={\displaystyle \frac{\mathcal{W}\lambda \left[1-\exp \left(-\tilde{\tau }\right)\right]\left[1-\lambda {\displaystyle \frac{1+\exp \left(-\tilde{\tau }\right)}{2}}\right]}{CR\tilde{\tau }}},

and

(9)\theta =1- {\displaystyle \frac{2\exp \left(-\tilde{\tau }\right)}{2-\lambda }},

respectively.

By using Eq. (7) to eliminate \tilde{\tau } in Eq. (8), we reveal that the trade-off between power P and efficiency \eta during constant current charging is given by

(10){\displaystyle \frac{P}{{\mathcal{E}}^2{R}^{-1}}}={\displaystyle \frac{2\left(\lambda -1+\eta \right)\eta }{\ln \left[\lambda /\left(2-2\eta -\lambda \right)\right]}} ,

where \lambda \equiv 1-V_1/\mathcal{E}. We plot this relation in Fig.1(b) as the red solid curve with \lambda = 1 (V_1=0). As shown in the figure, P and \eta are not always monotonically dependent on each other. In certain regions, an increase in efficiency is accompanied by a decrease in power, which leads us to call it the trade-off relation. It is worth mentioning that a similar power-efficiency trade-off has received extensive attention in the studies on finite-time heat engines [43]. Although the trade-off relation of batteries presented here and those of heat engines have a similar tendency, their specific forms differ significantly [44–49], which is caused by the different energy dissipation behaviors in the heat transfer process and the charging process.

According to the above trade-off, the maximum charging power {\eta }_{\mathrm{M}\mathrm{C}\mathrm{P}} is achieved when \mathrm{\partial }P/\mathrm{\partial }\eta =0, namely,

(11)\left(\lambda -1+2\eta \right)\ln {\displaystyle \frac{\lambda }{2-2\eta -\lambda }} -{\displaystyle \frac{2\eta \left(\lambda -1+\eta \right)}{2-2\eta -\lambda }} =0.

This transcendental equation determines the efficiency at maximum charging power, {\eta }_{\mathrm{M}\mathrm{C}\mathrm{P}}, as a function of \lambda, as shown in Fig.1(c). For \lambda = 1, one has {\eta }_{\mathrm{M}\mathrm{C}\mathrm{P}}\approx 0.358 and P_{\mathrm{m}\mathrm{a}\mathrm{x}}\approx 0.204 {\mathcal{E}}^2{R}^{-1}.

3 Optimization of the MSCC charging

For an ideal RC circuit system without extra constraints, the most energy efficient and time saving way for charging is to apply constant current (CC) on it [50], where the battery charge and voltage increase linearly with charging time. However, that is not the case for real-world batteries, where an upper battery voltage limit V_{\mathrm{m}} exists to ensure the safety and lifespan of the battery. Therefore, several charging strategies have been proposed to charge batteries quickly and safely [18, 19]. Among these strategies, multistage constant current (MSCC) is an emerging charging strategy that reduces charging time, enhances charging efficiency and extends battery lifespan [51]. Meanwhile, CC charging, an elementary strategy alongside CV charging, emerges as the minimal case of MSCC with charging steps N=1, enabling unified analysis of both strategies. While MSCC strategy is optimized mainly by numerical and experimental methods, in this part, we take MSCC strategy as an example to derive its OCP and minimum charging time, and further illustrate the power-efficiency trade-off in such real-world charging strategy.

A typical five-step MSCC charging process for an RC circuit, as illustrated in Fig.2, consists of multiple CC charging steps with decreasing current amplitudes. In each CC charging step, battery and charging voltage increase until the step is terminated upon reaching the charging cut-off voltage V_{\mathrm{m}}, after which the next CC step with lower current follows. Charging is terminated when charging voltage reaches V_{\mathrm{m}} in the final CC step, with the final battery voltage denoted as V_{\mathrm{f}}. Number of CC charging steps N, selection of charging current in each step and battery characteristics have great influence on the charging performance and need to be optimized.

Suppose that the initial voltage of the capacitor and charging current in the ith stage (i=1,2,\cdots ,N) are V_i and I_i, the accumulated charge \mathrm{\Delta } q_i and charging time \mathrm{\Delta }{t}_i of the ith stage can be written respectively as

(12)\mathrm{\Delta }{q}_i= C(V_{\mathrm{m}}- V_i-I_{i}R ),

(13)\mathrm{\Delta }{t}_i= {\displaystyle \frac{C}{I_i}}( V_{\mathrm{m}}-V_i)-CR .

Therefore, the total charge change \mathrm{\Delta }q and charging time \tau are respectively obtained as

(14)\mathrm{\Delta }q= {\displaystyle \sum _{i=1}^N\mathrm{\Delta }{q}_i=C(V_{\mathrm{f}}-V_1),}

and

(15)\tau ={\displaystyle \sum _{i=1}^N\mathrm{\Delta }{t}_i={\displaystyle \frac{C}{I_1}}(V_{\mathrm{m}}-V_1)-CR+{\displaystyle \sum _{i=2}^{N}CR\left({\displaystyle \frac{I_{i-1}}{I_i}}-1\right),}}

where V_{\mathrm{m}}= V_i+I_{i-1}R for (i\ne 1) has been used. Obviously, \mathrm{\Delta }q is a state function as it is completely determined once the initial and final voltage of the battery is given; while \tau is a process function since it depends on the charging current at each step during the charging process. Physically, finding the optimal strategy for fast charging means minimizing \tau for given \mathrm{\Delta } q, akin to the typical equilibrium thermodynamic task of minimizing irreversibility (characterized by irreversible entropy generation or irreversible work) when the initial and final states of the thermodynamic process are specified [52–55].

3.1 Minimizing the charging time in MSCC

To minimize charging time \tau, it is required that \mathrm{\partial }\tau /\mathrm{\partial }{I}_i=0,{\mathrm{\partial }}^2\tau /\mathrm{\partial }{I}_i^2>0 for each i\ne N, while I_N=(V_{\mathrm{m}}-V_{\mathrm{f}})/R remains constant. For i=2,3 ,\cdots ,N- 1, the optimal value of charging current I_i is obtained as

(16)I_i=\sqrt{I_{i-1}{I}_{i+1}} ,

which is consistent with the result obtained by Khan and Choi [33]. It follows from Eq. (16) that \ln I_i forms an arithmetic sequence since \ln I_i=(\ln I_{i-1}+\ln I_{i+1})/2. Therefore, we have

(17)I_i={\alpha }^{N-i} I_N,

where I_N=( V_{\mathrm{m}}- V_{\mathrm{f}})/R, and \alpha is an undetermined coefficient that depends on the free parameter I_1. However, it was ignored in Ref. [33] that the charging current in the first step I_1 can also be optimized. Using \mathrm{\partial }\tau /\mathrm{\partial }{I}_1=0 and {\mathrm{\partial }}^2\tau /\mathrm{\partial }{I}_1^2 >0, the optimal value of I_1 satisfies

(18)-{\displaystyle \frac{C}{I_1^2}}( V_{\mathrm{m}}-V_1)+ {\displaystyle \frac{CR}{I_2}}=0.

Then, combining Eqs. (18) and (17) yields

(19)\alpha ={\left({\displaystyle \frac{V_{\mathrm{m}}-V_1}{V_{\mathrm{m}}-V_{\mathrm{f}}}}\right)}^{\frac{1}{N}},

and thus the charging current in each step I_i is obtained as

(20)I_i={\displaystyle \frac{1}{R}}(V_{\mathrm{m}}-V_1{)}^{\frac{N-i}{N}} (V_{\mathrm{m}}-V_{\mathrm{f}}{)}^{\frac{i}{N}},

providing the OCP for MSCC strategy while showing a positive correlation with N.

By combining Eq. (20) with Eq. (15) and defining k\equiv (V_{\mathrm{m}}- V_1)/(V_{\mathrm{m}}- V_{\mathrm{f}})={\alpha }^N, the minimum total charging time {\tau }_{\mathrm{m}\mathrm{i}\mathrm{n}} is obtained explicitly as

(21){\tau }_{\mathrm{m}\mathrm{i}\mathrm{n}}=NC R(k^{\frac{1}{N}}-1),

where the charging time of each step, CR(k^{1/N}-1), is identical. Note that the charging power P=W_{\mathrm{i}\mathrm{n}}/\tau is a monotonically decreasing function of \tau as the input energy W_{\mathrm{i}\mathrm{n}}=C(V_{\mathrm{f}}^2- V_1^2)/2 is a constant value for a given battery. Therefore, minimum charging time {\tau }_{\mathrm{m}\mathrm{i}\mathrm{n}} results in maximum power P_{\mathrm{m}\mathrm{a}\mathrm{x}}, which means that the battery is charging under efficiency at maximum charging power {\eta }_{\mathrm{M}\mathrm{C}\mathrm{P}} within the proposed OCP. Specifically, for a practical case with a set of experimental parameters, N=5, C=11030\mathrm{F}, R=0.068\mathrm{\Omega }, V_1=3.35 \mathrm{V}, V_{\mathrm{f}}=4.20\mathrm{V} and V_{\mathrm{m}}= 4.22\mathrm{V} [33], the minimum charging time given by Eq. (21) is 4482 s, which is a 14 \mathrm{\%} reduction compared to the charging time 5224 s of the charging pattern proposed in Ref. [33].

According to Eq. (21), {\tilde{\tau }}_{\mathrm{m}\mathrm{i}\mathrm{n}}\equiv {\tau }_{\mathrm{m}\mathrm{i}\mathrm{n}}/(CR ) as a function of charging steps N is plotted in Fig.3(a). The monotonically decreasing behavior of \tilde{\tau } with respect to N is observed, while the reduction in \tilde{\tau } becomes less pronounced when N is sufficiently large. This is consistent with previous numerical and experimental studies [26, 33, 56, 57] which also demonstrate such an anti-correlation between charging time and number of charging steps. We remark here that a larger value of N is not always advantageous in real-world scenarios. Specifically, a higher N necessitates a more complex charger control circuitry, increasing costs and generating a larger initial current I_1 [as given by Eq. (20)] that could potentially result in battery overheating and degradation [4, 51]. However, incongruent with the cost incurred, the reduction in charging time is only marginal. Given these pros and cons, it is recommended to choose a moderate N in practical applications tailored to specific requirements.

The dependence of \tilde{\tau } on k is illustrated in Fig.3(b). Lower battery initial voltage V_1 and higher final voltage V_{\mathrm{f}} result in the increase of k, which increases charging time \tilde{\tau }. This is also consistent with former studies. In the large N regime of N\gg 1, retaining the first order of N^{-1} in Eq. (21), we find

(22){\tau }_{\mathrm{m}\mathrm{i}\mathrm{n}}\simeq CR\ln k+CR {\displaystyle \frac{{\ln }^{2}k}{2N}},

which shows the 1/ N-scaling as illustrated with the black dashed curve in Fig.3(a). When N\to \mathrm{\infty }, the lower bound CR\ln k for {\tau }_{\mathrm{m}\mathrm{i}\mathrm{n}} is obtained. Remarkably, as N\to \mathrm{\infty }, MSCC strategy transitions into CV strategy which we have discussed previously, indicating that CV strategy theoretically emerges as the fastest charging strategy within the constraints of charging cut-off voltage. In this case, constant charging voltage is exactly the cut-off voltage V_{\mathrm{m}}, while the current decreases exponentially with an initial value of I_1=(V_{\mathrm{m}}-V_1)/R [as given by Eq. (20)]. Nonetheless, such initial current is relatively high for most real-world batteries, leading to overheating and irreversible damage [2, 58]. This elucidates the reason for the limited utilization of CV strategy in practical scenarios. In this sense, it is crucial to consider both charging time indicated by charging power P and heat generation indicated by charging efficiency \eta when optimizing a charging process.

3.2 Power-efficiency trade-off in optimal MSCC strategy

Previous discussions suggest that the optimal MSCC strategy contains an underlying power-efficiency trade-off relation, akin to what we have uncovered in CV strategy. In this section, the power-efficiency trade-off in the optimal MSCC strategy is derived and illustrated in detail.

Electrical energy stored in the battery during MSCC charging is defined as

(23)W_{\mathrm{i}\mathrm{n}}={\displaystyle \frac{C}{2}}(V_{\mathrm{f}}^2-V_1^2).

Using Eqs. (13) and (20), dissipated energy due to thermal effects of current Q_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}=\sum _{i=1}^N{I}_i^{2}R\mathrm{\Delta } t_i is obtained as

(24)Q_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}=C(V_{\mathrm{m}}-V_1{)}^2\left[{\displaystyle \frac{k^2}{(1-k{)}^{-\frac{2}{N}}-1}}+{\displaystyle \frac{1}{(1-k{)}^{-\frac{1}{N}}+1}}\right].

It follows from Eqs. (21), (23) and (24) that the charging power in the optimal MSCC strategy is

(25)P={\displaystyle \frac{W_{\mathrm{i}\mathrm{n}}}{{\tau }_{\mathrm{m}\mathrm{i}\mathrm{n}}}}={\displaystyle \frac{V_{\mathrm{f}}^2-V_1^2}{2RN(k^{\frac{1}{N}}-1)}},

and the corresponding efficiency is

(26)\eta ={\displaystyle \frac{W_{\mathrm{i}\mathrm{n}}}{W_{\mathrm{i}\mathrm{n}}+Q_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}}}={\left[1+2{\displaystyle \frac{(V_{\mathrm{m}}-V_{\mathrm{f}})(k+1)}{(V_{\mathrm{f}}+V_1)(k^{\frac{1}{N}}+1)}}\right]}^{-1}.

Using Eqs. (25) and (26), the trade-off relations between P and \eta with different N are plotted in Fig.4(a). As shown in this figure, the maximum power P_{\mathrm{m}\mathrm{a}\mathrm{x}} and the efficiency at maximum power {\eta }_{\mathrm{E}\mathrm{M}\mathrm{P}} (circle marks) increase with charging steps N. As a cross-check of this analytical result, we employ trust region method [59] and Scipy [60] to numerically solve maximized power P for given \eta and obtain results showing good consistency with Fig.4(a) (see the Appendix).

It is further observed in Fig.4(a) that the order of limits P\to 0 and N\to \mathrm{\infty } seems to affect the value of \eta. This can be explained with Eq. (25) and Eq. (26):

(27)\{\begin{array}{l}\underset{N\to \mathrm{\infty }}{lim}\underset{V_{\mathrm{f}}\to V_{\mathrm{m}}}{lim}P=0\\ \underset{V_{\mathrm{f}}\to V_{\mathrm{m}}}{lim}\underset{N\to \mathrm{\infty }}{lim}P=0\end{array},

and

(28)\{\begin{array}{l}\underset{N\to \mathrm{\infty }}{lim}\underset{V_{\mathrm{f}}\to V_{\mathrm{m}}}{lim}\eta =1\\ \underset{V_{\mathrm{f}}\to V_{\mathrm{m}}}{lim}\underset{N\to \mathrm{\infty }}{lim}\eta ={\displaystyle \frac{1+V_1/V_{\mathrm{m}}}{2}}\end{array},

which together yields, in the case associated with Fig.4(a) with V_1=0,

(29)\{\begin{array}{l}\underset{N\to \mathrm{\infty }}{lim}\underset{P\to 0}{lim}\eta =1\\ \underset{P\to 0}{lim}\underset{N\to \mathrm{\infty }}{lim}\eta =1/2\end{array}.

When one takes the limit of N\to \mathrm{\infty } first, the P- \eta trade-off (red solid curve) is exactly the P- \eta trade-off for the CV process as illustrated in Fig.1(b), showing the consistency of our results.

Real-world applications focus mainly on maximizing charging power P (that is, minimizing charging time \tau), as well as reducing energy dissipation (i.e., enhancing charging efficiency \eta) for safety and battery lifespan concerns. Thus, we demonstrate the \eta as a function of N for given P in Fig.4(b) as a reference. When P is relatively small, \eta is monotonically functioned with N. As P gradually increases, an overall decrease in \eta is seen for all N values, while \eta decreases faster for smaller N values. Thus, there exists an optimal N for efficiency at maximum power {\eta }_{\mathrm{E}\mathrm{M}\mathrm{P}}, where the optimal N value increases with the increase of target P. For a given target P in real-world applications, it is recommended that the battery be charged with the charging steps N determined by {\eta }_{\mathrm{E}\mathrm{M}\mathrm{P}}, which minimizes heat dissipation and complexity of the charger circuit.

We would like to mention here that when N=1, MSCC is exactly the CC strategy; while MSCC becomes CV strategy when N\to \mathrm{\infty }, which means that MSCC serves as a bridge between the two elementary charging strategies, CC and CV. The exact N-dependent current, power and efficiency we present in Eqs. (20), (25) and (26) continuously demonstrate the transition from CC to CV. We further stress that this power-efficiency trade-off in battery charging is universal, even though we only systematically illustrated it in CV and MSCC strategy. A series of other advanced charging strategies have been proposed in recent studies, such as CCCV, multistage constant current-constant voltage (MCC-CV), boost charging and pulse charging [4, 18]. Despite their distinct implementations, they fundamentally combine basic CV and CC strategies (where CC is equivalent to MSCC with N=1) in specific sequences. As we have systematically analyzed these two elementary strategies, our theoretical framework enables universal derivation of power-efficiency trade-off relation for any charging strategy through specific superposition.

4 Summary

In this paper, we have analyzed the battery charging process in a typical RC series from a finite-time thermodynamic perspective, unveiling the fundamental thermodynamic trade-off relation between charging power and efficiency. We demonstrate such power-efficiency trade-off in different fundamental real-word charging strategies such as CV, MSCC and CC (equivalent to MSCC when N=1), which verifies the universality of our results. In addition, the OCP of MSCC strategy is analytically derived, which agrees well with previous studies via experimental or numerical approaches. While the industries focus on developing and optimizing specific techniques for fast charging and thermal management, our study unveils fundamental thermodynamic constraints that serve as stepping stones to understand the physical nature and optimization limit for battery charging. Our proposed method can be integrated with real-world charging systems or combined with specific optimization methods to achieve enhanced charging power and efficiency while respecting such thermodynamic constraints.

In future study, the theoretical method proposed in this paper may assist in designing and optimizing novel charging strategies that achieve a better balance between charging power and heat dissipation. Further efforts will focus on the power-efficiency trade-off in more complex and higher order equivalent circuit models (e.g., n-RC model, Thevenin model, Randles circuit, pseudo-two-dimensional model) [61], which might bridge the gap between idealized RC model and industrial battery systems. Moreover, wireless charging has emerged as a prevalent charging approach, affording users remarkable convenience and outstanding portability. However, it does come at a cost, as it generally exhibits lower charging power and efficiency when contrasted with traditional wired charging methods. Pinpointing the precise trade-off relation between power and efficiency within the wireless charging procedures is a challenging task, which urgently demands further in-depth investigation.

References

[1] W. Liu, T. Placke, and K. Chau, Overview of batteries and battery management for electric vehicles, Energy Rep. 8, 4058 (2022)

[2] Q. Lin, J. Wang, R. Xiong, W. Shen, and H. He, Towards a smarter battery management system: A critical review on optimal charging methods of lithium-ion batteries, Energy 183, 220 (2019)

[3] Q. Wang, B. Jiang, B. Li, and Y. Yan, A critical review of thermal management models and solutions of lithium-ion batteries for the development of pure electric vehicles, Renew. Sustainable Energy Rev. 64, 106 (2016)

[4] G. Ji, L. He, T. Wu, and G. Cui, The design of fast charging strategy for lithium-ion batteries and intelligent application: A comprehensive review, Appl. Energy 377, 124538 (2025)

[5] C. P. Grey and D. S. Hall, Prospects for lithium-ion batteries and beyond — a 2030 vision, Nat. Commun. 11(1), 6279 (2020)

[6] R. Usiskin, Y. Lu, J. Popovic, M. Law, P. Balaya, Y. S. Hu, and J. Maier, Fundamentals, status and promise of sodium-based batteries, Nat. Rev. Mater. 6(11), 1020 (2021)

[7] J. Janek and W. G. Zeier, Challenges in speeding up solid-state battery development, Nat. Energy 8(3), 230 (2023)

[8] Y. Liu, Y. Zhu, and Y. Cui, Challenges and opportunities towards fast-charging battery materials, Nat. Energy 4(7), 540 (2019)

[9] R. Borah, F. Hughson, J. Johnston, and T. Nann, On battery materials and methods, Mater. Today Adv. 6, 100046 (2020)

[10] J. He, J. Meng, and Y. Huang, Challenges and recent progress in fast-charging lithium-ion battery materials, J. Power Sources 570, 232965 (2023)

[11] Z. He, Y. Huang, H. Liu, Z. Geng, Y. Li, S. Li, W. Deng, G. Zou, H. Hou, and X. Ji, Anode materials for fast charging sodium-ion batteries, Nano Energy 129, 109996 (2024)

[12] A. Kumar Thakur, R. Sathyamurthy, R. Velraj, R. Saidur, A. Pandey, Z. Ma, P. Singh, S. K. Hazra, S. Wafa Sharshir, R. Prabakaran, S. C. Kim, S. Panchal, and H. M. Ali, A state-of-the art review on advancing battery thermal management systems for fast-charging, Appl. Therm. Eng. 226, 120303 (2023)

[13] J. Liu, S. Yadav, M. Salman, S. Chavan, and S. C. Kim, Review of thermal coupled battery models and parameter identification for lithium-ion battery heat generation in EV battery thermal management system, Int. J. Heat Mass Transf. 218, 124748 (2024)

[14] S. Paul, A. M. Schlaffer, and J. A. Nossek, Optimal charging of capacitors, IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 47(7), 1009 (2000)

[15] Y. Perrin, A. Galisultanov, H. Fanet, and G. Pillonnet, Optimal charging of nonlinear capacitors, IEEE Trans. Power Electron. 34(6), 5023 (2019)

[16] S. Hemavathi and A. Shinisha, A study on trends and developments in electric vehicle charging technologies, J. Energy Storage 52, 105013 (2022)

[17] J. M. Tarascon and M. Armand, Issues and challenges facing rechargeable lithium batteries, Nature 414(6861), 359 (2001)

[18] A. Tomaszewska, Z. Chu, X. Feng, S. O’kane, X. Liu, J. Chen, C. Ji, E. Endler, R. Li, L. Liu, Y. Li, S. Zheng, S. Vetterlein, M. Gao, J. Du, M. Parkes, M. Ouyang, M. Marinescu, G. Offer, and B. Wu, Lithium-ion battery fast charging: A review, ETransportation 1, 100011 (2019)

[19] Q. Lin, J. Wang, R. Xiong, W. Shen, and H. He, Towards a smarter battery management system: A critical review on optimal charging methods of lithium-ion batteries, Energy 183, 220 (2019)

[20] Y. H. Liu, J. H. Teng, and Y. C. Lin, Search for an optimal rapid charging pattern for lithium-ion batteries using ant colony system algorithm, IEEE Trans. Ind. Electron. 52(5), 1328 (2005)

[21] S. C. Wang and Y. H. Liu, A PSO-based fuzzy-controlled searching for the optimal charge pattern of Li-ion batteries, IEEE Trans. Ind. Electron. 62(5), 2983 (2015)

[22] W. Du, J. Ma, and W. Yin, Orderly charging strategy of electric vehicle based on improved PSO algorithm, Energy 271, 127088 (2023)

[23] T. T. Vo, X. Chen, W. Shen, and A. Kapoor, New charging strategy for lithium-ion batteries based on the integration of Taguchi method and state of charge estimation, J. Power Sources 273, 413 (2015)

[24] C. H. Lee, C. Y. Hsu, S. H. Hsu, and J. A. Jiang, Effect of weighting strategies on Taguchi-based optimization of the four-stage constant current charge pattern, IEEE Trans. Aerosp. Electron. Syst. 57(5), 2704 (2021)

[25] L. Jiang, Y. Li, Y. Huang, J. Yu, X. Qiao, Y. Wang, C. Huang, and Y. Cao, Optimization of multi-stage constant current charging pattern based on Taguchi method for Li-ion battery, Appl. Energy 259, 114148 (2020)

[26] L. R. Dung,J. H. Yen, ILP-based algorithm for lithium-ion battery charging profile, in: 2010 IEEE International Symposium on Industrial Electronics, 2010, pp 2286–2291

[27] Y. Wang, C. Zhou, and Z. Chen, Optimization of battery charging strategy based on nonlinear model predictive control, Energy 241, 122877 (2022)

[28] C. Zou, C. Manzie, and D. Nesic, Model predictive control for lithium-ion battery optimal charging, IEEE/ASME Trans. Mechatron. 23(2), 947 (2018)

[29] N. Tian, H. Fang, and Y. Wang, Real-time optimal lithium-ion battery charging based on explicit model predictive control, IEEE Trans. Industr. Inform. 17(2), 1318 (2021)

[30] S. Park, A. Pozzi, M. Whitmeyer, H. Perez, A. Kandel, G. Kim, Y. Choi, W. T. Joe, D. M. Raimondo, and S. Moura, A deep reinforcement learning framework for fast charging of Li-ion batteries, IEEE Trans. Transp. Electrif. 8(2), 2770 (2022)

[31] H. M. Abdullah, A. Gastli, and L. Ben-Brahim, Reinforcement learning based EV charging management systems – a review, IEEE Access 9, 41506 (2021)

[32] M. H. Abbasi, Z. Arjmandzadeh, J. Zhang, B. Xu, and V. Krovi, Deep reinforcement learning based fast charging and thermal management optimization of an electric vehicle battery pack, J. Energy Storage 95, 112466 (2024)

[33] A. B. Khan and W. Choi, Optimal charge pattern for the high-performance multistage constant current charge method for the Li-ion batteries, IEEE Trans. Energ. Convers. 33(3), 1132 (2018)

[34] A. B. Khan,V. L. Pham,T. T. Nguyen,W. Choi, Multistage constant-current charging method for Li-ion batteries, in: 2016 IEEE Transportation Electrification Conference and Expo, Asia-Pacific (ITEC Asia-Pacific), 2016, pp 381–385

[35] P. Keil and A. Jossen, Charging protocols for lithium-ion batteries and their impact on cycle life — an experimental study with different 18650 high-power cells, J. Energy Storage 6, 125 (2016)

[36] M. Ye, H. Gong, R. Xiong, and H. Mu, Research on the battery charging strategy with charging and temperature rising control awareness, IEEE Access 6, 64193 (2018)

[37] W. Xie, X. Liu, R. He, Y. Li, X. Gao, X. Li, Z. Peng, S. Feng, X. Feng, and S. Yang, Challenges and opportunities toward fast-charging of lithium-ion batteries, J. Energy Storage 32, 101837 (2020)

[38] B. Bose, A. Garg, B. Panigrahi, and J. Kim, Study on Li-ion battery fast charging strategies: Review, challenges and proposed charging framework, J. Energy Storage 55, 105507 (2022)

[39] J. Chen, Optimization on the charging process of a capacitor, Int. J. Electron. 88(2), 145 (2001)

[40] S. Xia and L. Chen, Theoretical and experimental investigation of optimal capacitor charging process in RC circuit, Eur. Phys. J. Plus 132(5), 235 (2017)

[41] B. Andresen, P. Salamon, and R. S. Berry, Thermodynamics in finite time: Extremals for imperfect heat engines, J. Chem. Phys. 66(4), 1571 (1977)

[42] R. S. Berry,P. Salamon,B. Andresen (Eds.), Finite-Time Thermodynamics, MDPI-Multidisciplinary Digital Publishing Institute, 2022

[43] Y. H. Ma,X. H. Zhao, Finite-time thermodynamics: A journey beginning with optimizing heat engines, arXiv: 2024)

[44] L. Chen and Z. Yan, The effect of heat-transfer law on performance of a two-heat-source endoreversible cycle, J. Chem. Phys. 90(7), 3740 (1989)

[45] V. Holubec and A. Ryabov, Efficiency at and near maximum power of low-dissipation heat engines, Phys. Rev. E 92(5), 052125 (2015)

[46] N. Shiraishi, K. Saito, and H. Tasaki, Universal trade-off relation between power and efficiency for heat engines, Phys. Rev. Lett. 117(19), 190601 (2016)

[47] Y. H. Ma, D. Xu, H. Dong, and C. P. Sun, Universal constraint for efficiency and power of a low-dissipation heat engine, Phys. Rev. E 98(4), 042112 (2018)

[48] R. X. Zhai, F. M. Cui, Y. H. Ma, C. P. Sun, and H. Dong, Experimental test of power-efficiency trade-off in a finite-time Carnot cycle, Phys. Rev. E 107(4), L042101 (2023)

[49] Y. H. Ma,C. Fu, Unified approach to power-efficiency trade-off of generic thermal machines, arXiv: 2024)

[50] D. Wang, The most energy efficient way to charge the capacitor in a RC circuit, Phys. Educ. 52(6), 065019 (2017)

[51] M. U. Tahir, A. Sangwongwanich, D. I. Stroe, and F. Blaabjerg, Overview of multi-stage charging strategies for Li-ion batteries, J. Energy Chem. 84, 228 (2023)

[52] Y. H. Ma, D. Xu, H. Dong, and C. P. Sun, Optimal operating protocol to achieve efficiency at maximum power of heat engines, Phys. Rev. E 98(2), 022133 (2018)

[53] Y. H. Ma, R. X. Zhai, J. Chen, C. P. Sun, and H. Dong, Experimental test of the 1/τ-scaling entropy generation in finite-time thermodynamics, Phys. Rev. Lett. 125(21), 210601 (2020)

[54] G. Li, J. F. Chen, C. P. Sun, and H. Dong, Geodesic path for the minimal energy cost in shortcuts to isothermality, Phys. Rev. Lett. 128(23), 230603 (2022)

[55] Y. H. Ma, J. F. Chen, C. P. Sun, and H. Dong, Minimal energy cost to initialize a bit with tolerable error, Phys. Rev. E 106(3), 034112 (2022)

[56] G. J. Chen, Y. H. Liu, S. C. Wang, Y. F. Luo, and Z. Z. Yang, Searching for the optimal current pattern based on Greywolf optimizer and equivalent circuit model of Li-ion batteries, J. Energy Storage 33, 101933 (2021)

[57] H. Min, W. Sun, X. Li, D. Guo, Y. Yu, T. Zhu, and Z. Zhao, Research on the optimal charging strategy for Li-ion batteries based on multi-objective optimization, Energies 10(5), 709 (2017)

[58] A. Al-Haj Hussein and I. Batarseh, A review of charging algorithms for nickel and lithium battery chargers, IEEE Trans. Vehicular Technol. 60(3), 830 (2011)

[59] A. R. Conn,N. I. M. Gould,P. L. Toint, Trust region methods, MPACS-Mathematical Programming Advisory Committee, Society for Industrial and Applied Mathematics, 2000

[60] Virtanen , R. Gommers, T. E. Oliphant, M. Haberland, and T. Reddy, . SciPy 1.0: Fundamental algorithms for scientific computing in python, Nat. Methods 17(3), 261 (2020)

[61] X. Ding, D. Zhang, J. Cheng, B. Wang, and P. C. K. Luk, An improved Thevenin model of lithium-ion battery with high accuracy for electric vehicles, Appl. Energy 254, 113615 (2019)

RIGHTS & PERMISSIONS

Higher Education Press