Enhancing Lithium Titanate Battery Charging: Investigating the Benefits of Open-Circuit Voltage Feedback

Abstract

In applications where it is crucial that a battery is recharged from the partially discharged state in the minimum time, it is crucial to honor the technological constraints related to maximum safe battery terminal voltage and maximum continuous charging current prescribed by the battery cell manufacturer. To this end, this contribution outlines the design and comprehensive simulation analysis of an adaptive battery charging system relying on battery open-circuit voltage estimation in real time. A pseudo-random binary sequence test signal and model reference adaptive system are used for the estimation of lithium titanate battery cell electrical circuit model parameters, with the design methodology based on the Lyapunov stability criterion. The proposed adaptive charger is assessed against the conventional constant-current/constant-voltage charging system. The effectiveness of the real-time parameter estimator, along with both the adaptive and traditional charging systems for the lithium titanate battery cell, is validated through simulations and experiments on a dedicated battery test bench.

1. Introduction

Lithium secondary batteries represent the currently most advanced electrochemical battery technology. With their calendar life of up to 15 years and specific costs around 145 EUR/kWh, lithium batteries typically outperform alternatives such as hydrogen fuel cells, especially considering their round-trip efficiencies [1]. Thus, lithium-based batteries have found many applications that help with the “greening” of conventional transportation and energy systems. For example, the shipping industry is exploring using lithium-ion batteries either alone or in combination with supercapacitors for onboard energy storage, aiming to reduce greenhouse gas emissions of freight ships during their stays in ports [2]. In road transportation, hybrid electric and fully electric vehicles are becoming more widespread, with advanced fuel cell electric vehicles using auxiliary battery energy storage to extend the driving range [3]. Naturally, efficient thermal management systems need to be implemented to preserve the useful battery life and ensure safe operation in electric vehicles and other applications [4]. For example, the study presented in [3] has shown an effective way of using the battery/fuel cell thermal management system to accommodate additional riding comfort by using the battery/fuel cell waste heat for passenger cabin heating in winter.

The increased use of renewable energy sources requires efficient energy storage solutions to address the intermittent nature of renewables and ensure a stable energy supply. In this sense, batteries offer very flexible energy storage solutions, which can be highly modular and easily scaled and characterized by high round-trip efficiency and fast charging/discharging dynamics [5]. By integrating battery technology with hydrogen storage and solar power systems, significant enhancement of the overall hydrogen production system can be achieved, thus further contributing to the sustainable energy transition [6]. Other applications of advanced lithium batteries may include hybrid electric aircraft, wherein hydrogen fuel cells and battery hybrid systems are being explored for emission reduction in aviation [7]. Lithium-ion batteries are also being integrated into commercial buildings with photovoltaic systems for peak load shaving, for price arbitrage, and to increase the energy autonomy through increased self-consumption [8].

Lithium titanate oxide (LTO) is gaining popularity as an anode material in advanced batteries due to its ability to withstand a high number of charging/discharging cycles, rather fast charging capabilities, and good low-temperature performance [9], as well as its resistance to mechanical straining during high-rate discharging [10]. Additionally, utilization of LTO anodes in sodium-based batteries potentially offers cost advantages compared to lithium-based battery technologies [11]. Currently, LTO battery systems exhibit superior safety features, which are attributed to their enhanced thermal stability. Furthermore, LTO batteries offer a favorable combination of rather high power density (enabling high charge/discharge rates) and a significantly longer cycle life compared to other battery technologies [12]. They are characterized by minimal capacity fade, and their long cycle life (over 20,000 charging/discharging cycles) is attributed to the stable spinel structure of LTO [10], which minimizes volume changes during cycling and mitigates the formation of dendrites and lithium plating, a major cause of degradation in conventional lithium-ion batteries [13]. Moreover, according to the authors of [14], LTO cells’ charging and discharging rate capabilities make them well suited for hybrid power applications in electrified vehicle powertrains. Hence, they are also frequently considered as rapid response components in hybrid energy storage applications, such as the one proposed in [12], wherein lithium iron phosphate (LiFePO₄) and LTO batteries are used in complementary roles, with LiFePO₄ batteries being used for prolonged steady power delivery and LTO batteries handling pulsed power loads. However, in any such energy management application, the information about actual battery state-of-charge (SoC) is key for timely battery recharging after prolonged discharging operation and may also be useful during the charging process to enhance its precision and reduce the recharging time.

All batteries exhibit nonlinear terminal voltage vs. SoC behavior during the charging and discharging processes, particularly near the highly discharged and fully charged states [15], which can lead to state-of-charge estimation inaccuracy if the battery model used for SoC monitoring does not accurately capture these phenomena. Thus, it is crucial to develop highly accurate battery models that can predict battery behavior in real time and facilitate advanced energy management applications [16]. Advanced multi-scale models that capture electrochemical reactions, side reactions, and heat generation may offer such functionalities, i.e., they can be used for optimizing battery design and performance [17], but these models can also be rather complex. Hence, simpler battery models are typically used for battery SoC monitoring. The simplest mathematical models used for SoC monitoring are the so-called Coulomb counting [18] and battery terminal voltage and current monitoring [19], with both methods also applicable for battery parameter estimation [20]. However, these methods of SoC estimation are challenging due to the need for a high-precision current sensor in the former case and the narrow range of battery voltage variations in the latter case [18,21]. To improve battery SoC tracking, a combined approach is used that is based on a battery equivalent circuit model and battery terminal voltage and current measurements [19]. This approach typically involves a state estimator framework such as Kalman filtering [22]. Because the battery model is typically nonlinear, methodologies based on the unscented Kalman filter [21] or extended Kalman filter (EKF) [23] should be used to build the SoC estimation framework. Kalman filtering has been implemented for LTO cell SoC estimation in [24], but it has also been shown to be sensitive to battery model parameterization errors.

To avoid issues associated with battery off-line parameterization errors altogether, that is, to avoid recording detailed static maps of all battery equivalent circuit model parameters whose dependence on battery state-of-charge, charging/discharging current, and temperature needs to be precisely determined beforehand [25], an alternative on-line parameter estimation approach can be used instead [22]. Moreover, such an on-line parameter estimation approach may also be useful for monitoring the battery state-of-health (SoH), which is typically indicated by the increase in the battery internal resistance and decrease in its charge capacity [22] over the battery exploitation period. Such a parameter estimator-based approach can utilize the a priori known open-circuit voltage vs. state-of-charge (OCV vs. SoC) characteristic for indirectly determining the battery state-of-charge during the charging process, which may be subsequently used to improve the effectiveness of battery charging [15]. A suitable estimator, such as the extended Kalman filter [26] or one based on the System Reference Adaptive Model (SRAM) approach [27], also requires sufficient excitation to guarantee the estimated parameter convergence [28] and, consequently, the accurate capturing of the battery OCV value [26,27]. Thus, in real-time operation, a suitable test signal such as a pseudo-random binary sequence (PRBS) is typically overlaid with the battery current reference within the framework of a battery charging control system [27].

As was previously shown in [27] for the case of an advanced lithium iron phosphate (LiFePO₄) battery cell, using OCV feedback based on on-line battery model parameter estimation to augment the conventional charging control strategy may result in a notable reduction in the recharging time interval compared to conventional charging when the battery cell is recharged from deeply discharged state. The main advantage of SRAM estimator-based OCV feedback, as shown in [27], is (i) guaranteed stability of the battery equivalent circuit model parameter estimator and (ii) the ability to simply retrofit such an adaptive charging control system onto the conventional CCCV charging systems. The more sophisticated charging control strategies have been researched recently, such as those based on model predictive control (MPC) [29] and artificial neural networks [30], with even more advanced machine learning systems being researched in the context of battery management systems (BMSs), as shown in references [31,32]. According to the comprehensive review by Adaikkappan et al. [33], the main advantages of model-based adaptive battery control systems are a better trade-off between accuracy and computational load compared to artificial intelligence and machine learning based approaches. Moreover, it is also highly desirable that the state of the parameter estimator within the battery control system should be asymptotically stable to avoid issues with estimator convergence and robustness [34]. The Lyapunov stability criterion can be used for the model parameter adaptation within the adaptive charging control system. The Lyapunov stability criterion is a powerful method used to prove the stability of nonlinear dynamic systems without needing to explicitly solve the system’s differential equations. This makes it highly suitable for guaranteeing the convergence of adaptive estimators such as the one used in this work. Hence, it would be worthwhile to investigate the effectiveness of such an advanced adaptive battery charging strategy for the state-of-the-art lithium titanate battery cells as well s for a wider range of operating modes (charging currents) and initial battery charge states.

Having the above discussion in mind, the hypothesis of this paper is that, using indirect real-time information about battery SoC, provided by the SRAM estimator of battery OCV, a significant improvement of the charging process effectiveness can be achieved for a wide range of operating conditions. Thus, this paper presents a detailed simulation and experimental comparison between the conventional constant-current/constant-voltage (CCCV) charging method and an adaptive control strategy that utilizes the OCV-based feedback, utilizing the SRAM parameter estimator whose convergence is assured by the Lyapunov stability criterion [27]. Even though the CCCV-OCV methodology was introduced by the authors in a preliminary simulation study [27] for a LiFePO₄ cell, the contributions of this paper are the comprehensive simulation and experimental investigation of this adaptive strategy for LTO battery cells and the verification of the SRAM-based parameter estimator over a wide range of LTO battery cell operating conditions. This contribution also quantifies the performance gains across a wide range of initial charge states and charging currents, thus establishing the practical operational benefits and limitations of the OCV-feedback approach for the advanced LTO battery chemistry.

This paper is organized as follows. Section 2 presents the dynamic models of the battery cell and battery charger based on a step-down (buck) power converter, outlines the conventional and the adaptive charging control system, and presents the design of the OCV estimator using the SRAM approach and Lyapunov stability theory from [27]. Results of comprehensive simulation and experimental analyses for the considered control system structures are given in Section 3, along with a brief discussion of results. Concluding remarks are provided in Section 4.

2. Materials and Methods

This section outlines the dynamic model of a lithium titanate (LTO) battery cell and the battery charger topology, followed by the battery charging control system design.

2.1. Battery Equivalent Circuit Model and Buck-Converter Low-Level Current Control Loop

Figure 1 illustrates the low-level battery charging system, incorporating a buck (step-down) DC/DC power converter and a battery cell dynamic model. The buck converter in this study features the battery current feedback control system (using the current measurement i_bs) with embedded battery current controller, wherein the current controller commands an appropriate voltage reference to the buck converter [35]. In this way, precise adjustment of battery current can be facilitated from the superimposed control level corresponding to the battery terminal voltage control loop in conventional charging control systems ([35], see Section 2.3). The current controller performs pulse-width modulation (PWM) of the DC power supply voltage u_dc by means of the semiconductor commutating elements (MOSFET Q and the anti-parallel diode D), and the resulting output voltage u_c feeds the low-pass filtering inductor element (with inductance L_c and resistance R_c) between the power converter and the battery, thus adjusting the battery current i_b.

Typically, the power converter current control loop dynamics can be approximated by a first-order lag dynamic term [35]:

$$G_{ei}\left(s\right)={\displaystyle \frac{i_b\left(\mathrm{s}\right)}{i_{bR}\left(\mathrm{s}\right)}}={\displaystyle \frac{1}{T_{ei}s+1}}$$

(1)

where i_b is the battery current, i_bR is the current reference (wherein the current target i_bR is supplied by the superimposed voltage controller), T_ei is the equivalent time constant, and s is the so-called Laplace variable corresponding to the complex frequency.

The battery cell equivalent circuit model in Figure 1 comprises an ideal battery open-circuit voltage source U_oc, an equivalent parallel resistance–capacitance (RC) network modeling electrolyte polarization effects (polarization resistance R_p and capacitance C_p), and the equivalent series resistance R_b. This model can be described by the following voltage dynamic relationship [35]:

$$u_b\left(s\right)=i_b\left(s\right)R_b+{\displaystyle \frac{R_p{i}_b\left(s\right)}{{\tau }_{p}s+1}}+U_{oc}\left(s\right)$$

(2)

with τ_p = R_pC_p being the polarization dynamics equivalent time constant.

All parameters of the above model may depend on the battery temperature ϑ_b, battery current i_b, and its state-of-charge (SoC) ξ, which is defined in the following manner [35]:

$$\xi ={\displaystyle \frac{1}{Q_b}}\int i_{b}dt$$

(3)

where Q_b is the battery charge capacity, which may also depend on the battery current i_b [36].

2.2. Battery Cell Model Parameters

Experimental characterization of a commercial 30 Ah/2.4 V LTO battery cell [37] was performed using a dedicated battery test bed [38], and the results are presented in Figure 2 and Figure 3. It should be noted that the results of experimental characterization of battery equivalent circuit model parameters have not included their temperature dependence, because all experiments were conducted under constant temperature conditions [38]. Note also that, according to [14], the open-circuit voltage (OCV) is primarily determined by the chemistries of the positive and negative electrode and changes in their degrees of lithiation and is frequently modeled as a function of SoC alone [25]. Hence, the OCV variations with respect to temperature variations have not been considered herein. A more comprehensive characterization would require precise control of ambient temperature by using a thermal chamber, which is beyond the scope of this work.

Figure 2 illustrates the open-circuit voltage (OCV) vs. battery SoC curve. While the OCV exhibits an approximately linear relationship with SoC at mid-range values, significant nonlinearity is observed at both extremes of the SoC spectrum (SoC → 0% and SoC → 100%). To account for this nonlinear behavior, experimentally recorded OCV vs. SoC data points (indicated by red circles in Figure 2) were interpolated using cubic splines to generate a smooth OCV(SoC) curve [38]. Figure 3 depicts two-dimensional (2D) maps of the battery equivalent circuit model parameters R_b(ξ, i_b), R_p(ξ, i_b), and τ_p(ξ, i_b) [38]. As indicated in Figure 3a,b, both R_b(ξ, i_b) and R_p(ξ, i_b) maintain relatively low values (between 1 mΩ and 2 mΩ) within the mid-SoC range. However, these resistances perceptibly increase as the battery approaches deep discharge regime (SoC → 0) or full-charge state (SoC → 100%), reaching approximately 3 mΩ in the case of the R_b parameter and 6 mΩ for the R_p parameter. Furthermore, the polarization time constant, τ_p(ξ, i_b), exhibits a strong dependence on both the SoC and battery current, with values ranging from 2 s to 12 s over the examined battery operating range (Figure 3c).

Using the experimentally determined battery cell parameter maps from Figure 2 and Figure 3, the battery simulation model in Figure 4 has been built based on a mathematical representation of Equations (1) and (2) with battery current as the model input. This simulation model is implemented in MATLAB/Simulink (R2022a) using the ode45 solver with variable integration step and maximum step size of 0.01 s, and it was thoroughly verified against experimentally obtained charging data, as shown in [23].

2.3. Battery Charging Control Systems Under Investigation

Block diagrams of battery charging control strategies considered in this work, based on the cascade control system arrangement, are shown in Figure 5. Both concepts feature a superimposed control level that commands the inner current control loop and implements effective limiting of current reference through controller output saturation [39]. This cascade control approach also has the advantages of a modular control system structure and straightforward switching between constant-current and constant-voltage charging regimes, as illustrated in previous studies [27,35].

Figure 5a illustrates the control system structure for the conventional constant-current/constant-voltage (CCCV) battery charging strategy. This strategy, designated CCCV-VL, employs a battery terminal voltage-limiting PI controller as the superimposed control element. Such a control system configuration represents a standard battery charging solution and may be considered a benchmark case for more advanced control strategies, as indicated in references [40,41]. Within this control system layout, the current command (i_bR) is given as i_bR = I_max + i_blim, wherein the former (I_max) is the maximum charging current in the constant-current phase, and the latter (i_blim) represents the output of the voltage-limiting controller, activated when the battery voltage measurement (u_bm) surpasses the predefined voltage limit (u_blim), as indicated by the dead-zone block within the voltage-limiting controller in Figure 5a. At this transition point, the constant-voltage charging regime starts, wherein the battery current asymptotically tends to zero, while the terminal voltage asymptotically converges toward the desired OCV value U_oc(ξ). The latter directly corresponds to the desired battery SoC, as discussed in [27,35].

Figure 5b shows the adaptive control system using OCV feedback (the so-called CCCV-OCV charging strategy) [27]. In this control system arrangement, the principal open-circuit voltage (U_oc) PI controller commands the current reference i_bR and limits the current to I_max during charging. The SRAM-based estimator provides the battery OCV feedback ${\widehat{U}}_{oc}$, using measurements u_bs and i_bs from the battery voltage and current sensors. These battery voltage and current measurement signals are low-pass filtered by first-order filters characterized by the lag time constant T_fm. The SRAM estimator also requires persistent and ample excitation to facilitate accurate parameter estimation. This excitation is realized by superimposing a pseudo-random binary sequence (PRBS) test signal (∆i_bR) to the overall battery current reference i_bR + i_blim. The battery voltage-limiting PI controller now has a safety function; that is, it keeps the battery voltage below the limit value u_bmax via its current command i_blim. Hence, in this arrangement, the battery voltage target value u_bmax can be preset independently of the open-circuit voltage target U_ocR [27]. In fact, one of the key differences between the CCCV-VL and CCCV-OCV charging strategies is the use of different battery terminal voltage limit values within the voltage-limiting PI controller (see Figure 5a,b). Namely, by using a less conservative (i.e., larger) battery terminal voltage target value u_bmax within the CCCV-OCV strategy compared to the CCCV-VL voltage target value u_blim (i.e., u_bmax > u_blim), the CCCV-OCV strategy would allow the charging system to operate in the constant-current (CC) phase for a longer period of time, thus speeding up the charging process and shortening the charging duration. Moreover, the adaptive (CCCV-OCV) strategy represents a retrofitting solution aimed at enhancing the performance of conventional (CCCV-VL) chargers since the superimposed U_oc PI controller output augments the output of the battery terminal voltage-limiting PI controller, thus making this solution well suited for modernization of existing conventional chargers by means of simple software upgrades.

2.4. Open-Circuit Voltage Estimator

The battery model parameter estimator minimizes the parameter estimation error within the framework of the System Reference Adaptive Model (SRAM), whose adaptation (i.e., parameter estimation) is characterized by globally asymptotic convergence according to the Lyapunov stability criterion [28]. The SRAM-based parameter estimator design utilizes the battery model in Equation (2), which is rewritten in the following normalized form [27]:

$$U_0[u_{bn}(s)-U_{ocn}(s)]=I_0{\displaystyle \frac{b_1^{*}s+b_0^{*}}{s+a_0}}{i}_{bn}\left(s\right)$$

(4)

where u_bn, U_ocn, and i_bn are normalized battery variables corresponding to terminal voltage, open-circuit voltage, and battery current, and U₀ and I₀ are battery voltage and current rated values.

After rearranging, the above model reads as follows:

$$u_{bn}(s)={\displaystyle \frac{b_{1}s+b_0}{s+a_0}}{i}_{bn}(s)+U_{ocn}(s)$$

(5)

where the coefficients b₁, b₀, and a₀ in Equation (5) are defined as follows:

$$b_1=R_b{\displaystyle \frac{I_0}{U_0}},\dots b_0={\displaystyle \frac{R_b+R_p}{{\tau }_p}}{\displaystyle \frac{I_0}{U_0}},\dots a_0={\displaystyle \frac{1}{{\tau }_p}}$$

(6)

A block diagram representation of the SRAM parameter estimator is shown in Figure 6. To implement the SRAM parameter estimator, the time derivatives of the process model input and output (battery current i_b and battery terminal voltage u_b) need to be provided by using simple first-order filters, as shown in Figure 6 and elaborated in [27]. As shown in [28], Lyapunov’s stability criterion is used to derive the parameter update law such that the SRAM model tracking error e_m asymptotically converges to zero. In particular, the SRAM estimator shown in Figure 6 is globally asymptotically stable for the following choice of estimator parameter update gains K₁, K₂, K₃, and K₄: K₁ > 0, K₂ > 0, K₃ > 0, and K₄ > 0. Its convergence is assured under the ample and persistent excitation conditions [28]. To implement the SRAM parameter estimator, the time derivatives of the process model input and output (battery current i_b and battery terminal voltage u_b) need to be provided by using simple first-order filters characterized by lag time constant T_f, as shown in Figure 6 and elaborated in [27].

Assuming ample and persistent excitation of the SRAM estimator, the OCV estimate ${\widehat{U}}_{oc}$ should converge toward the actual OCV value $U_{oc}\left(s\right)$, albeit delayed due to the inner dynamics of the parameter estimator (characterized by the equivalent lag T_ee):

$${\widehat{U}}_{oc}\left(s\right)\approx {\displaystyle \frac{1}{T_{ee}s+1}}{U}_{oc}\left(s\right)$$

(7)

Since the relationship between the OCV and SoC is nonlinear, it should be linearized in the vicinity of SoC operating point ξ₀ to facilitate linear OCV controller design. Linearization yields the following approximate model of the OCV feedback (see Equation (3)):

$${\widehat{U}}_{oc}\left(s\right)\approx {\displaystyle \frac{K_{u\xi }}{T_{ee}s+1}}\cdot {\displaystyle \frac{1}{Q_{b}s}}\cdot i_b\left(s\right)$$

(8)

where K_uξ is the gradient of the OCV vs. SoC characteristic U_oc(ξ):

$$K_{u\xi }={\left.{\displaystyle \frac{\partial U_{oc}\left(\xi \right)}{\partial \xi }}\right|}_{\xi ={\xi }_0}$$

(9)

Note that the battery equivalent circuit model parameters are operating point-dependent, i.e., they are dependent on the battery SoC and current (according to the discussion in Section 2.2), but they may also notably depend on battery operating temperature, as indicated in [14]. Moreover, thermal stress may cause notable battery aging in the sense of accelerated degradation and parameter drift. However, the proposed SRAM-based parameter estimator is capable of and inherently suited to track slow-dynamics parameter variations, including those caused by gradual temperature variations during a single charging cycle.

2.5. Feedback Controller Tuning

The linear feedback controller tuning in this work is based on the damping optimum criterion [42], which is based on the following formulation of the closed-loop system characteristic polynomial:

$$A_c\left(s\right)=D_2^{n-1}{D}_3^{n-2}\dots D_n{T}_e^n{s}^n+\dots +D_2{T}_e^2{s}^2+T_{e}s+1$$

(10)

where T_e is the closed-loop system equivalent time constant, and D₂, D₃,…, D_n are the so-called characteristic ratios, which are set to 0.5 to achieve a well-damped closed-loop response [42].

According to the authors of [35], battery terminal voltage variations are primarily affected by the series resistance voltage drop due to slow polarization voltage and OCV dynamics. Based on this assumption, the following result is obtained with respect to terminal voltage-limiting PI controller tuning according to the damping optimum [35]:

$$T_{cl}=T_{el}\left(1-{\displaystyle \frac{D_{2u}{T}_{el}}{T_{\Sigma u}}}\right)$$

(11)

$$K_{cl}={\displaystyle \frac{1}{R_b}}\left({\displaystyle \frac{T_{\Sigma u}}{D_{2u}{T}_{el}}}-1\right)$$

(12)

where D_2u is the damping optimum characteristic ratio, and T_Σu = T_ei + T_fm is the equivalent lag, while the closed-loop equivalent time constant parameter T_el needs to satisfy the following inequality condition:

$$T_{el}<{\displaystyle \frac{T_{\Sigma u}}{D_{2u}}}$$

(13)

Under the assumption that the parameter estimator dynamics can be approximated by the linearized model given by Equations (8) and (9), the OCV controller design yields the following result using the damping optimum criterion [27]:

$$T_{c\xi }=T_{e\xi }\ge {\displaystyle \frac{T_{ee}+T_{\Sigma u}}{D_{2\xi }{D}_{3\xi }}}$$

(14)

$$K_{c\xi }={\displaystyle \frac{Q_b}{D_{2\xi }{T}_{e\xi }{K}_{u\xi }}}$$

(15)

where D_2ξ and D_3ξ are the damping optimum characteristic ratios in OCV controller design.

It should be noted that the tuning of the voltage-limiting PI controller is dependent on the battery series resistance, assuming the series resistance effect is the dominant contribution to the battery terminal voltage dynamics. Note also that the above controller tuning rules assume a priori known values of key tuning parameters, i.e., the battery series resistance parameter R_b in the case of the voltage-limiting PI controller and OCV vs. SoC curve gradient K_uξ (see Equation (9)) in the case of the U_oc PI controller. In order to simplify the control system implementation and to provide robust tuning of the aforementioned controllers, the voltage-limiting PI controller and the U_oc PI controller have been tuned for the maximum anticipated value of battery series resistance and the gradient of the OCV vs. SoC characteristic, respectively, thus resulting in robust tuning characterized by lower values of PI controller proportional gains (cf. Equations (12) and (15), respectively). Naturally, such tuning is not optimal in the sense of controller response speed, but this should not be critical because the battery voltage and OCV vary rather slowly with battery SoC, especially in the low-to-mid SoC range (see Figure 2). In the final stages of the charging process, characterized by the largest values of the series resistance parameter R_b and the OCV vs. SoC curve gradient K_uξ, the behavior of the charging control system may be considered near-optimal because the tuning of individual controllers would then be nearly matched to the actual values of series resistance and OCV vs. the SoC curve gradient.

3. Results

The proposed charging strategies have been assessed through comparative successive simulations for different initial SoC and average charging current limit values. Follow-up experimental verification carried out on the dedicated battery test setup is used to strengthen the key findings of the simulation study.

3.1. Results of Comparative Simulation Assessment of Proposed Strategies

Simulation verification of the two proposed battery charging control systems has been conducted by utilizing the previously developed MATLAB/Simulink model of the LTO battery cell [38]. Key parameters used within the simulation model are listed in

Table 1. Key parameters of the simulation model and charging scenarios.

Parameter	Value
Battery and current sensor time constant Tfm	5 ms
LTO battery cell charge capacity Qb	30 Ah
Battery charging current 1C rated value	30 A
Charging strategy current maximum values Imax used in the simulation study	30 A, 24 A, 18 A, 12 A (1C, 0.8C, 0.6C, 0.4C)
Charging strategy turn-off current Imin	0.3 A (0.01C)
PRBS test signal peak-to-peak amplitude	8 A (±4 A)
Battery state-of-charge initial conditions ξ0	20%, 40%, 60%, 80%
CCCV-VL battery terminal voltage limit ublim	2.68 V
CCCV-OCV battery terminal voltage limit ubmax	2.72 V
Battery OCV target value UocR	2.68 V
Current control loop lag Tei	21.8 ms
Voltage-limiting controller gain Kcl	111.11
Voltage-limiting controller time constant Tcl	5.5 ms
OCV controller gain Kcu	2870
OCV controller time constant Tcu	120 s
SRAM model adaptation gain K1	0.05
SRAM model adaptation gain K2	0.05
SRAM model adaptation gain K3	0.0001
SRAM model adaptation gain K4	0.1
SRAM model input filter time constant Tf	1.0 s

. Table 1 also lists the key parameters of the proposed charging control strategies and the SRAM-based parameter estimator. Moreover, for both battery charging strategies (i.e., OCV controller and battery terminal voltage controller; see Figure 5a,b), the charging process is characterized by two parameters: current limit I_max and the end-of-charging threshold I_min, which is used to terminate the charging process in the constant-voltage charging regime when i_b < I_min. The battery terminal voltage limit u_bmax = 2.72 V is used within the CCCV-OCV strategy as a precaution against battery over-voltage, whereas the CCCV-VL strategy defines the battery voltage target value u_blim = 2.68 V, which is below the maximum battery terminal voltage value, which corresponds to battery terminal voltage at the fully charged state (SoC = 100%) and idling conditions, i.e., when the charging current reaches zero (u_b = U_oc in this case, as discussed earlier).

It should be noted herein that the original experimental characterization of the battery model has been carried out up to 24 A for power converter operational safety reasons (as explained in Section 2.2). Since in the presented simulation study, the battery charging process is carried out with average charging currents reaching 30 A (Table 1), and the battery model parameters for the operating point characterized by a 30 A average charging current (1C rate) have been extrapolated using the parameter maps recorded at 18 A and 24 A as the basis for linear extrapolation.

Figure 7, Figure 8, Figure 9 and Figure 10 show the comparative results of successive simulations for the benchmark CCCV-VL charging strategy and the adaptive CCCV-OCV charging strategy for different initial values of battery SoC and the charging current upper limit I_max within the battery voltage-limiting controller in the former case and the OCV controller in the latter case. The simulation traces in Figure 7, Figure 8, Figure 9 and Figure 10 show that, in all the simulated scenarios, the CCCV-OCV adaptive charging control system is characterized by reaching the end-of-charging condition (i_b < I_min) noticeably faster compared to the benchmark CCCV-VL strategy, as previously reported in [27]. This is primarily due to prolonged constant-current operation commanded by the superimposed OCV controller within the CCCV-OCV system and greater allowance for battery terminal voltage increase compared to CCCV-VL, which is designed to asymptotically approach the steady-state condition characterized by the final terminal voltage value at idling (i_b = 0). In particular, the less conservative voltage-limiting PI controller voltage target value within the CCCV-OCV strategy (u_bmax = 2.72 V) compared to the voltage target value of the CCCV-VL strategy (u_blim = 2.68 V) effectively prolongs the duration of the constant-current (CC) phase, thus resulting in a shorter re-charging time interval in the case of the CCCV-OCV strategy. The key performance indices of both control strategies (charging time T_chg and final state-of-charge ξ_fin) have been summarized in

Table 2. Comparative charging times T_chg and final state-of-charge values ξ_fin for different state-of-charge initial conditions ξ₀ and charging strategy current maximum values I_max used in simulations.

	ξ0 = 20%	ξ0 = 40%	ξ0 = 60%	ξ0 = 80%
CCCV-VL with Imax = 30 A (1C)	Tchg = 59.12 min ξfin = 99.94%	Tchg = 45.72 min ξfin = 99.94%	Tchg = 32.32 min ξfin = 99.94%	Tchg = 18.92 min ξfin = 99.94%
CCCV-OCV with Imax = 30 A (1C)	Tchg = 54.38 min ξfin = 99.84%	Tchg = 40.52 min ξfin = 99.81%	Tchg = 27.17 min ξfin = 99.82%	Tchg = 14.08 min ξfin = 99.83%
CCCV-VL with Imax = 24 A (0.8C)	Tchg = 72.33 min ξfin = 99.94%	Tchg = 55.58 min ξfin = 99.94%	Tchg = 38.83 min ξfin = 99.94%	Tchg = 22.08 min ξfin = 99.94%
CCCV-OCV with Imax = 24 A (0.8C)	Tchg = 66.95 min ξfin = 99.80%	Tchg = 50.25 min ξfin = 99.78%	Tchg = 33.53 min ξfin = 99.76%	Tchg = 16.95 min ξfin = 99.69%
CCCV-VL with Imax = 18 A (0.6C)	Tchg = 94.37 min ξfin = 99.94%	Tchg = 72.04 min ξfin = 99.94%	Tchg = 49.71 min ξfin = 99.94%	Tchg = 27.37 min ξfin = 99.94%
CCCV-OCV with Imax = 18 A (0.6C)	Tchg = 89.18 min ξfin = 99.82%	Tchg = 66.87 min ξfin = 99.84%	Tchg = 44.85 min ξfin = 99.85%	Tchg = 22.52 min ξfin = 99.83%
CCCV-VL with Imax = 12 A (0.4C)	Tchg = 138.64 min ξfin = 99.94%	Tchg = 105.15 min ξfin = 99.94%	Tchg = 71.64 min ξfin = 99.94%	Tchg = 38.14 ξfin = 99.94%
CCCV-OCV with Imax = 12 A (0.4C)	Tchg = 134.12 min ξfin = 99.93%	Tchg = 100.52 min ξfin = 99.89%	Tchg = 66.92 min ξfin = 99.87%	Tchg = 33.32 min ξfin = 99.86%

for the considered simulation parameters (i.e., initial state-of-charge values ξ₀ and maximum current I_max commanded by the control strategy).

Table 2 presents the charging time comparative data for both the CCCV-OCV and CCCV-VL charging strategies (Figure 7, Figure 8, Figure 9 and Figure 10), along with the corresponding relative speedup and final SoC difference at the termination of the charging process. Based on the data presented in Table 2, the charging process relative speedup of the CCCV-OCV strategy compared to the benchmark CCCV-VL control strategy is defined as follows:

$${\displaystyle \frac{\Delta T_{chg}}{T_{chg,VL}}}=\left(1-{\displaystyle \frac{T_{chg,OCV}}{T_{chg,VL}}}\right)\cdot 100\%$$

(16)

where ∆T_chg = T_chg,VL − T_chg,OCV is the charging time difference between the benchmark CCCV-VL strategy and the adaptive CCCV-OCV strategy, and T_chg,VL, and T_chg,OCV are CCCV-VL and CCCV-OCV strategy charging times, respectively.

The final state-of-charge difference (SoC mismatch) between the final SoC results of the proposed charging strategies is given by the following:

$$\Delta {\xi }_{fin}={\xi }_{fin,VL}-{\xi }_{fin,OCV}$$

(17)

where ∆ξ_fin is the final state-of-charge difference between the CCCV-VL and CCCV-OCV strategies, while ξ_fin,VL and ξ_fin,OCV are the final SoC values obtained by the CCCV-VL and CCCV-OCV charging strategies, respectively.

The data from Table 2 are also visually represented in Figure 11 (with warmer colors in 3D graphs signifying higher values, and vice versa). The results demonstrate a consistent reduction in charging time when employing the adaptive (CCCV-OCV) strategy compared to the benchmark conventional (CCCV-VL) strategy. Furthermore, the observed speedup is the most emphasized (up to 25%) under conditions of higher current command limits (I_max values) and higher initial battery SoC (ξ₀) values. It should be noted that the adaptive strategy consistently reaches a somewhat lower final SoC value ξ_fin when compared to the benchmark strategy, which is due to a more abrupt current drop in the case of the former, thus reaching the end-of-charging current limit (and threshold condition i_b < I_min is satisfied more quickly in that case). However, the final state-of-charge difference ∆ξ_fin between the charging strategies is consistently less than 0.25%, which points to the favorable performance of the CCCV-OCV strategy in the final stage of the charging process. It should be noted that the observed consistent undercharging effect encountered when the adaptive CCCV-OCV control strategy is used instead of the conventional CCCV-VL strategy should not be an issue in practical applications, especially when opportunity-based charging is considered. Namely, in those types of scenarios, the reduction in charging time can be far more valuable than adding the last fraction of a percent of the SoC to the battery charge, because the observed undercharging effect that is consistently less than 0.25% of the SoC obtained by conventional charging would result in only 0.25% less energy available from the battery after the charging is completed. As explained above, consistent undercharging encountered with the CCCV-OCV adaptive control strategy is a feature that results from the more aggressive current drop at the end of the charging, which causes the turn-off threshold (I_min) to be met sooner compared to the case of the conventional CCCV-VL charging strategy.

As illustrated by the results in Table 2 and Figure 11c, the charging current magnitude during the constant-current (CC) stage has a direct impact on the performance of the adaptive charging strategy (CCCV-OCV). The relative charging speedup is most pronounced at higher charging currents and a higher initial SoC, reaching about 25% in the case of a 1C charging current and an initial SoC of 80%. However, the charging speedup effect is less emphasized at lower constant-current stage currents and for a lower initial SoC; that is, the charging speedup diminishes to only 3.3% in the case of a 0.4C charging current and an initial SoC of 20%. Note also that the simulation study did not include the case of a 6 A charging current (0.2C rate) because the total charging time for both strategies would be rather long (about 5 h), so this scenario is omitted from the analysis to maintain focus on the key features of the proposed adaptive control strategy. The charging speedup trends observed in Figure 11c indicate that the time-saving benefits of the CCCV-OCV strategy diminish as the charging currents become lower; e.g., at 12 A (0.4C), the response speedup is already minimal. Therefore, it may be inferred that, at 6 A, the charging times for both strategies would be nearly identical. Note also that the above criteria for assessing the battery charging system’s performance have been chosen based on the practical battery limitations, i.e., battery operation within the envelope defined by the maximum continuous battery current (30 A) and maximum terminal voltage (2.9 V), according to the manufacturer’s technical data in [37].

The presented simulation results for the case of an LTO battery cell and a wide range of charging conditions agree well with the results presented in the previous study [27] considering a single case scenario characterized by low initial battery SoC (ξ₀ = 20%) and a single charging rate (0.7C) for a LiFePO₄ battery cell. Namely, the study conducted in [27] has yielded about a 23% speedup of the battery charging response with the adaptive charging strategy (CCCV-OCV) compared to the case of conventional charging (CCCV-VL). Consequently, the results presented herein also confirm that the adaptive (CCCV-OCV) control strategy can be applied to heterogenous battery chemistries and is determined by a narrow set of technological parameters, i.e., the battery charging current limit I_max and battery voltage limit u_bmax, battery open-circuit voltage (OCV), and battery internal resistance R_b characteristics, which are required for parameterization and tuning of the battery voltage control systems within the adaptive control strategy.

As indicated by results in Figure 7, Figure 8, Figure 9 and Figure 10, both the CCCV-VL and CCCV-OCV charging strategies maintain constant charging rates during the initial constant-current (CC) charging phase due to their current references being effectively limited by the superimposed voltage controllers, with the CCCV-OCV strategy also featuring a PRBS-based test signal in the current reference path for the SRAM-based battery model parameter estimator. The second constant-voltage (CV) phase typically lasts much less than the constant-current phase and is characterized by much lower charging rates. Therefore, it may be inferred that the bulk of battery charging heat losses occur during the constant-current charging phase. Figure 12 shows the cumulative heat losses W_L = ∫P_Ldt for both charging strategies and all considered simulation scenarios (shown in Figure 7, Figure 8, Figure 9 and Figure 10), wherein the battery’s irreversible power losses (corresponding to resistive heat losses) are estimated based on charging current and polarization voltage within the battery model as follows:

$$P_L=i_b^2{R}_b\left(i_b,\xi \right)+u_p^2/R_p\left(i_b,\xi \right)$$

(18)

Results in Figure 12a,b show that both charging strategies result in similar total generated heat at the end of charging, with the CCCV-OCV charging strategy generating only slightly more waste heat compared to the CCCV-VL charging strategy. This small discrepancy is likely due to the PRBS test signal, which increases the current magnitude and, in turn, increases the resistive power losses when compared to the CCCV-VL strategy, which does not use a test signal. These results point to similar performance of both charging strategies with respect to battery heat losses, whereas the CCCV-OCV charging strategy has been shown to be more effective with respect to battery charging time (see discussion related to Figure 11). Consequently, the CCCV-OCV charging strategy would have the same effect on battery temperature-related aging effects, battery state-of-health, and its cycling stability. Therefore, it would likely result in a similar expected battery lifetime when compared to the conventional CCCV-VL strategy.

Finally, the simulation durations with respect to the SoC initial condition and charging current limit value are shown in Figure 13. It is evident that longer simulation durations are obtained for the cases of longer simulated charging processes and vice versa, wherein the simulation length is determined by the initial battery SoC value and charging current limit, as shown in Figure 11. In this sense, the correlation between the battery charging time in simulations and the simulation duration itself is evident from plots in Figure 11a,b and Figure 13a,b.

3.2. Results of Experimental Verification

Experimental verification of the proposed charging concept has been carried out on the battery test setup [38], whose photograph is shown in Figure 14. The description and key parameters of the battery test bed components are listed in

Table 3. Specifications of test setup components from Figure 14.

No.	Description	Technical Specifications
1	Control computer with acquisition and control cards	Single-core Pentium 4 CPU (3 GHz clock frequency) and 4 GByte RAM used as control computer and for data acquisition. Acquisition and control cards: Advantech (Taipei, Taiwan) PCL 812 PG cards with 16 analogue input channels and 2 analogue output channels (12-bit resolution each) [43]
2	DC/DC power converter	SIGLENT (Shenzhen, China) SPS 5041X 30 A/40 V/360 W controllable/programmable DC power source used as current-controlled battery charging power converter [44]
3	Lithium titanate battery cell	ELERIX (Oldham, Manchester, UK) EX-30TK battery cell 30 Ah/2.4 V with 1C (nominal) charging rate of 30 A [37]
4	Reverse flow blocking diode	RURG80100 (80 A average rectified current/1000 V blocking voltage) used as reverse current blocking diode [45]
5	Circuit breaker	B 32 breaker type (medium speed/32 A RMS tripping current)
6	Auxiliary 24 Vdc power supply	Adjustable stabilized laboratory DC power source 0–30 V/0–3 A for supplying the isolation amplifier
7	Isolation amplifier	PR Electronics (Rønde, Denmark), Isolated Converter Type 3104 (0…5 V input/0…5 V output) [46], which is used to galvanically isolate the battery terminal current measurement

. Unlike the “idealized” simulation scenarios, the experimental scenarios include realistic measurement noise and non-ideal sensor characteristics, which can be encountered in practical applications. In all experiments, the initial and the final battery terminal voltages at idling are measured by means of a high-precision voltmeter to obtain a precise initial and final SoC estimate based on battery OCV vs. SoC characteristics (Figure 2).

Figure 15 shows the results of experimental verification of the proposed control strategies for the case of the upper charging current limit I_max = 18 A (0.6C) and two distinct initial SoC values ξ₀ = 4.2% (nearly fully discharged battery charge) and ξ₀ = 69.5% (medium-high initial SoC) determined from the battery idling voltage u₀ before the charging (as indicated in Figure 15a,b). Higher charging current limits (0.8C and 1C) were not considered in the experiments due to operational safety reasons (because they would result in high heat losses at the reverse-flow blocking diode). The key findings in terms of the charging process speedup and final SoC mismatch between CCCV-OCV and CCCV-VL strategies are summarized in

Table 4. Comparative performance indices analyzed charging control strategies obtained during experimental tests for different SoC initial conditions and current command within charging controller limited to I_max = 18 A (0.6C).

Initial Battery State	Final State CCCV-VL	Final State CCCV-OCV	CCCV-VL Charging Time Tchg,VL	CCCV-OCV Charging Time Tchg,OCV	Charging Speedup
ub0 = 2.072 V (ξ0 = 4.2%)	ubfin = 2.678 V (ξfin = 99.95%)	ubfin = 2.644 V (ξfin = 99.62%)	111.08 min	106.17 min	4.42%
ub0 = 2.272 V (ξ0 = 69.5%)	ubfin = 2.674 V (ξfin = 99.91%)	ubfin = 2.667 V (ξfin = 99.85%)	35.32 min	31.28 min	11.33%

. These experimental results confirm the key findings of the detailed simulation analysis conducted for a wider range of initial state-of-charge conditions and current command limit values. In particular, the more perceptible charging process speedup is obtained when applying the CCCV-OCV control strategy at higher initial SoC values, and vice versa, with relative speedups following the trends observed in simulations for similar operating regimes. The experimental results also confirm that the proposed adaptive control strategy can be applied under operating conditions characterized by realistic sensor noise conditions.

3.3. Discussion

Based on the comprehensive simulation and experimental data, the proposed adaptive CCCV-OCV charging strategy demonstrates a marked improvement in charging speedup over the conventional CCCV-VL method. Simulations conducted across a wide range of initial state-of-charge (SoC) values (20% to 80%) and charging currents (0.4C to 1C) consistently show that the CCCV-OCV strategy reduces charging time. The relative speedup is most significant at higher initial SoC values and greater charging currents, reaching up to 25%. This reduction in time comes with a minimal trade-off: a final SoC that is slightly lower than the benchmark strategy, though this difference remains consistently below 0.25%. Moreover, the cumulative ohmic and polarization heat losses for both strategies are comparable, indicating that the faster charging characteristic for the CCCV-OCV method should not negatively impact battery temperature or related aging effects.

The key findings from the simulations were subsequently reinforced by physical experiments. Tests were performed using an 18 A (0.6C) charging current at two distinct initial SoC levels: 4.2% and 69.5%. The experimental results directly mirror the simulation trends; a more significant charging speedup was observed at the higher initial SoC. Specifically, the CCCV-OCV strategy was 4.42% faster when starting from a low charge and 11.33% faster when starting from a medium-high charge. This validation confirms the practical viability of the adaptive (CCCV-OCV) charging approach.

Hence, the simulation and experimental investigation confirm that the CCCV-OCV strategy is a more effective method for charging LTO battery cells compared to the traditional CCCV-VL technique. The primary advantage is a consistent reduction in charging time, especially when the battery is only partially recharged, and without incurring a penalty in terms of excessive heat generation or a meaningful loss in the final state-of-charge. Good agreement between the extensive simulation data and the focused experimental tests validates the CCCV-OCV strategy’s favorable performance and suggests its applicability to various battery chemistries and practical charging scenarios.

4. Conclusions

This paper presented a detailed comparative analysis of two previously developed control system designs for constant-current/constant-voltage (CCCV) battery charging. Both approaches utilize a cascade control architecture with a common inner current control loop embedded within the charging power converter. Different implementations of the superimposed charging control level are outlined in this paper. These include the conventional charging control approach, designated CCCV-VL herein, which implements a battery terminal voltage-limiting control loop. The second approach, denoted as CCCV-OCV, utilizes a battery open-circuit voltage (OCV) control loop, incorporating an adaptive SRAM-based OCV estimator for feedback along with the auxiliary battery terminal voltage controller, which is now used to prevent the battery terminal voltage excursions during charging. The adaptive CCCV-OCV control strategy can be regarded as an extension of the conventional (CCCV-VL) charging strategy since it augments the output of the battery terminal voltage-limiting PI controller within the conventional charging strategy. Thus, the adaptive charging strategy can be applied to the existing conventional chargers by means of a simple control software upgrade.

A detailed simulation-based assessment has shown that the CCCV-OCV charging strategy is consistently characterized by faster battery recharging compared to the conventional benchmark CCCV-VL strategy. This is attributed to prolonged constant-current operation commanded by the charging controller and greater allowance on battery terminal voltage increase compared to the CCCV-VL strategy. The performance indices of the proposed charging control strategies indicate that the CCCV-OCV strategy can achieve the charging speedup with respect to the benchmark CCCV-VL strategy with minimal mismatch (less than 0.25%) in the final state-of-charge (after recharging is completed). Furthermore, a simulation assessment has revealed that more emphasized speedup (up to 25%) is generally obtained at higher charging current limit values and higher initial SoC values. Results from experimental tests of both proposed control strategies confirm the key findings of the simulation study. The key overall finding is that the adaptive CCCV-OCV strategy should generally be preferred when recharging the battery from a partially discharged state, especially when recharging with higher initial charging rates.

Future work will be directed toward a more detailed experimental examination of the presented CCCV-OCV control strategy, aimed at a more detailed experimental identification of charging speedup gains inherent to the use of the CCCV-OCV strategy compared to the CCCV-VL benchmark case. Future work may also include an analysis of cycling stability and related battery aging effects (such as charge capacity decline) for the considered CCCV-OCV strategy and its effects on control system robustness.