# Adaptive Constant-Current/Constant-Voltage Charging of a Battery Cell Based on Cell Open-Circuit Voltage Estimation

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

_{4}) batteries offer a good trade off regarding power and energy density and operational safety for a moderate energy storage-specific cost (i.e., cost per kilowatt-hour) [18], along with their maturity and high durability (battery cycle life) [19]. With reference [20] reporting over 2000 charging/discharging cycles characterized by average discharge depths of 80% per cycle, their ability to withstand many demanding charging/discharging cycles makes LiFePO

_{4}batteries good candidates for the demanding tasks of the future electrified transport [19].

#### 1.2. State-of-the-Art Methods in Battery Charging

#### 1.3. Research Gap and the Proposed Solution

_{4}cell presented in [50].

## 2. Materials and Methods

#### 2.1. Battery Cell-Equivalent Circuit Dynamic Model

_{oc}, electrolyte polarization effects modeled by an equivalent parallel RC circuit (with resistance R

_{p}and capacitance C

_{p}parameters), and an equivalent series resistance R

_{b}[54]. The abovementioned results in the following linear time-invariant (LTI) model between voltage u

_{b}and current i

_{b}, described by the following Laplace s-domain formulation:

_{s}is the series resistance voltage drop, u

_{p}is the polarization voltage drop, and τ

_{p}= R

_{p}C

_{p}is the polarization dynamics equivalent time constant.

_{b}, battery current i

_{b}, and its state of charge ξ, which is defined in the following manner:

_{b}is the battery charge capacity.

_{4}battery cell (see [20]) with respect to state-of-charge ξ for the case of battery charging, which were originally recorded in [52] and further elaborated upon for the purpose of a battery charging study conducted in [50]. Averaged profiles of battery polarization and equivalent series resistance R

_{p}(ξ) and R

_{b}(ξ) are shown in Figure 1a,b, whereas Figure 1c shows the OCV vs. SoC characteristic (U

_{oc}(ξ) curve). Variations in these parameters are more emphasized for boundary state-of-charge ξ values, i.e., for the case of deeply discharged (ξ ≈ 0) and fully-charged battery cell (ξ ≈ 1). As indicated in [52], all the aforementioned parameters show only a mild dependence on average battery current values, while the polarization equivalent time constant is estimated to τ

_{p}= 24 s (see Figure 1d) and does not show a significant variation with the operating point [52].

#### 2.2. Damping Optimum Criterion

_{e}is the closed-loop system equivalent time constant and D

_{2}, D

_{3}, …, D

_{n}are the so-called characteristic ratios. In the optimal case D

_{i}= 0.5 (i = 2 … n), the closed-loop system of any order n has a quasi-aperiodic step response characterized by an overshoot of approximately 6% (resembling a second-order system with damping ratio ζ = 0.71) and the approximate rise time (1.8–2.1)·T

_{e}. For larger T

_{e}value choices, the dominant closed-loop modes were characterized by a slower response, which generally improved the control system robustness and decreased the noise sensitivity.

#### 2.3. Conventional and Adaptive Cascade Control Systems for Battery Charging

_{bR}as a sum of the maximum charging current I

_{max}used during the constant-current stage of the charging process and the negative-current command i

_{blim}from the superimposed voltage-limiting controller. The voltage-limiting controller becomes activated only when the battery terminal voltage measurement u

_{bs}exceeds the preset battery voltage limit value u

_{blim}(dead-zone block in Figure 3a). As indicated in [50], the battery terminal voltage-limiting value also effectively determines the battery SoC at the end of charging, because it directly relates to the battery OCV as the charging current i

_{b}approaches zero (see Equation (1) and the U

_{oc}(ξ) characteristic in Figure 1c). Note that this battery terminal voltage-limiting value is less than the maximum safe battery terminal voltage value of 3.65 V obtained from the manufacturer’s technical specifications [20]. This is performed so that the CCCV-VL control strategy asymptotically approaches the fully-charged state without excursions towards the maximum safe terminal voltage (and thus to avoid possible overcharging).

_{oc}) PI controller commands an appropriate current reference i

_{bR}(again limited to the maximum current value I

_{max}during charging) based on the estimated battery OCV value (Û

_{oc}signal in Figure 3b). The OCV feedback is provided herein by the SRAM estimator whose operation requires persistent excitation, introduced through a pseudo-random binary sequence (PRBS) test signal Δi

_{bR}(see e.g., [57]) that is superimposed to the battery overall current reference (i

_{bR}+ i

_{blim}) resulting from joint actions of dual controllers. The auxiliary battery terminal voltage-limiting controller presently has the safety function of maintaining the battery terminal voltage below the limit value u

_{blim}, which can be preset independently of the battery OCV target U

_{ocR}. From the standpoint of the control system design, it was assumed that the dynamics of the battery OCV estimate with respect to the battery charging current i

_{b}could be approximated by a first-order lag plus integral term dynamics with the equivalent time constant T

_{ee}and the open-circuit voltage vs. state-of-charge static map U

_{oc}(ξ), as indicated in the lowermost portion in Figure 3b. Since the proposed modification only superimposed an additional control level to the existing conventional CCCV control strategy without changing the underlying control system structure, it made the modification of the charging strategy easily retrofitted to the existing CCCV charging control system. Again, the battery limit voltage u

_{blim}was set to a value lower than the maximum safe terminal voltage of 3.65 V declared in the manufacturer’s technical data [20] for safety reasons.

#### 2.4. Feedback Controller Tuning Rules

_{oc}and polarization voltage u

_{p}dynamics were treated as slow “disturbances”, so that “fast” battery terminal voltage dynamic variations were primarily affected by the series resistance voltage drop [28]. The fast dynamics of the inner current control loop (approximated by a fast first-order lag term with its equivalent time constant T

_{ei}) and the voltage sensor (with its lag T

_{fm}) are lumped together and represented by the equivalent first-order lag term with the time constant T

_{Σu}= T

_{ei}+ T

_{fm}:

_{Σu}may also incorporate the time discretization (sampling) related lag inherent to the use of a discrete time (digital) controller [58].

_{el}[50]:

_{b}in expression (6), i.e., R

_{b}= max(R

_{b}(ξ)).

_{ee}, which could be added to the inner current control loop lag T

_{ei}, similar to the terminal voltage-limiting controller design presented above (see Equation (4)). The overall lag in the OCV controller design would then be T

_{ee}+ T

_{ei}. Moreover, it was also assumed that the nonlinear OCV vs. SoC characteristic U

_{oc}(ξ) was continuously differentiable over its domain of definition, so that its gradient (equivalent gain) in the vicinity of the battery SoC operating point ξ

_{0}can be expressed as follows:

_{ξ}in Equation (10) to its maximum value, i.e., K

_{ξ}= max(K

_{ξ}(ξ)), assures a robust charging system closed-loop operation under the superimposed (supervisory) open-circuit voltage control.

_{ee}parameter and OCV gradient parameter (gain K

_{ξ}) used in the open-circuit voltage PI controller design were chosen to be sufficiently large (see discussion regarding damping optimum criterion equivalent time constant choice in the previous subsection). Similarly, the series resistance parameter R

_{b}in the battery terminal voltage controller design should also be set to the maximum anticipated value in order to facilitate the robust tuning of the dominant closed-loop dynamics (see subsequent subsections). The less-dominant dynamics (i.e., the very slow OCV variations and polarization voltage dynamics) were treated herein as a slow-disturbance term in the battery terminal voltage measurement, which was effectively dealt with by the integral action of the terminal voltage PI controller.

_{el}being rather small. On the other hand, the SRAM estimator-based superimposed feedback loop ought to be characterized by slow control action (narrow bandwidth) due to the equivalent time constant of the OCV feedback loop T

_{eu}being rather large, owing to relatively large SRAM-based parameter estimator lag T

_{ee}(see next section).

#### 2.5. Parameter Estimator Design Based on the Lyapunov Stability Theory

_{bn}, U

_{ocn}, and i

_{bn}represent the normalized battery terminal voltage, battery OCV, and battery current, respectively, and U

_{0}and I

_{0}are battery voltage and current normalization factors, respectively.

_{1}, b

_{0}, and a

_{0}:

_{0m}, b

_{1m}, b

_{0m}, and w

_{m}are updated online, as shown in Figure 4a.

_{0}> 0 is assumed), the time derivative of the process model input (battery current i

_{b}) needs to be calculated. This can be achieved by utilizing a simple first-order state-variable filter (see [57]) of the normalized battery current i

_{bn}, as shown in Figure 4a. The same type of filter was applied to the normalized battery voltage u

_{bn}in order to match the time lag of the filtered current signal i

_{bf}and its time derivative di

_{bf}/dt, thus resulting in the final dynamic model with an estimated battery voltage u

_{bm}regarded as the output and the filtered current i

_{bf}regarded as the input of the from the adaptive model within the SRAM estimator:

_{0}> 0), the first term on the right-hand side of Equation (26) is always negative; therefore, the estimator convergence would be satisfied if and only if the remaining dynamic terms multiplying the parameter estimator errors on the right-hand side of Equation (26) are zero, namely, if [53]:

_{1}, b

_{0}, a

_{0}, and w are assumed to be constant or very slowly varying, i.e., ${\dot{\tilde{b}}}_{1}=-{\dot{b}}_{1m}$, ${\dot{\tilde{b}}}_{0}=-{\dot{b}}_{0m}$, ${\dot{\tilde{a}}}_{0}=-{\dot{a}}_{0m}$, and $\dot{\tilde{w}}=-{\dot{w}}_{m}$ are valid, the model adaptation law is given in the following final form (also shown in Figure 4a):

_{1}> 0, K

_{2}> 0, K

_{3}> 0 and K

_{4}> 0 [53], and whose convergence is assured under ample and persistent excitation conditions [57].

_{b}, polarization resistance R

_{p}, and polarization time constant τ

_{p}) are subsequently reconstructed, as shown in Figure 4b, based on the following relationships derived from (14):

_{m}and parameter a

_{0m}within the adaptive model, the OCV estimate Û

_{oc}is reconstructed in the open-loop manner (Figure 4b) and is used (after de-normalization) as OCV feedback within the CCCV-OCV strategy:

_{0m}, b

_{1m}, b

_{0m}, and w

_{m}) because these parameters were all positive-valued within the battery model, as well as the lower value saturation of parameter a

_{0m}because it could not take a zero value (a

_{0m}> 0). Finally, the reconstructed physical parameters of the battery model were low-pass filtered by means of first-order filters (all with the time constant T

_{fp}) to obtain smooth (noise-free) parameter estimates.

## 3. Simulation Results

_{4}battery cell implemented within the MATLAB/Simulink environment.

#### 3.1. Control System Robustness Analysis

_{ξ}(see Equation (8)) for the particular LiFePO

_{4}battery cell (cf. Figure 1c), which was used in the linearization of the OCV control loop based on the SRAM parameter estimator. Clearly, the OCV vs. SoC gradient value adopts on a wide range of values, thus affecting the overall closed-loop dynamics of the OCV control loop. Figure 5b,c show the root locus plots of the OCV control loop without a voltage-limiting controller being activated for the maximum and minimum values of OCV vs. SoC curve gradient K

_{ξ}, respectively, with the open-circuit voltage PI controller tuned with the maximum K

_{ξ}gradient value to facilitate robust closed-loop behavior, as indicated in Section 3.3. The comparative root locus plots show that the resulting closed-loop systems are characterized by well-damped closed-loop behavior, with the poles of the linearized closed-loop system being characterized by favorable damping ratios (ζ ≈ 0.5 or higher), even in the case of the notable mismatch of the parameter estimator equivalent lag T

_{ee}(±50 of the nominal value). Finally, the stability of the closed-loop system with both OCV and voltage-limiting PI controllers was also analyzed by means of comparative root locus plots, which are shown in Figure 5d for the case of the robust tuning of both controllers (with maximum values of K

_{ξ}and R

_{b}parameters) and no variations in the parameter estimator equivalent lag T

_{ee}from the nominal value. The root locus plots in Figure 5d indicate that the introduction of the auxiliary voltage-limiting control loop does not significantly affect the overall closed-loop system stability. Thus, it may be surmised that the proposed dual-controller closed-loop system is quite robust to process parameter variations.

#### 3.2. Illustration of Parameter Estimator Convergence

_{b}) was fed to the battery LTI model and used to provide the SRAM-based parameter estimator with ample excitation. A PRBS signal with the amplitude of a 20 A peak-to-peak, sampling period T

_{PRBS}= 8 s, depth of 6 bits (sequence length 2

^{6}= 64 distinct states before repetition), and a DC offset of 70 A was used herein, similar to the case of online parameter estimation during charging control.

_{b}and terminal voltage u

_{b}) from the idle state (initial polarization voltage u

_{p}(0) = 0 V) are shown in Figure 6a. These results show that there is an initial lag in the battery terminal voltage response due to polarization voltage dynamics (τ

_{p}= 24 s), after which the battery voltage establishes a stable limit-cycle due to periodic excitation by the PRBS test signal. Figure 6b shows the comparative results of battery model parameter estimation using the SRAM approach, where the first-order state-variable filter was used to extract the current time derivative and match the battery voltage lag (Figure 4a) has the time constant T

_{f}= 1 s. Filtering of the estimated parameters was also performed by first-order low-pass filters with the time constant T

_{fp}= 1 s (see Figure 4b) to smooth out the parameter estimates. The estimated parameter traces of R

_{b}, R

_{p}, and τ

_{p}showed good accuracy values for the final estimate (they were initially mismatched with respect to actual values by 10%) with rapid initial convergence, and subsequently slower convergence when reaching the actual values. Top-right plot in Figure 6b shows the response of the OCV estimate from the initial state (set to zero within the parameter estimator). The speed of convergence of the OCV, which represents an unknown offset parameter within the SRAM model, depends on the choice of the update gains K

_{1}, … K

_{4}. These are obviously a trade-off between the response speed of the OCV estimate and the parameter estimator response damping. The middle case (K

_{1}= 5·10

^{−3}, K

_{2}= K

_{3}= 10

^{−6}, K

_{4}= 5·10

^{−4}) was used in the subsequent tuning of the parameter estimator within the adaptive charging control system. Based on the properties of the OCV estimate response in Figure 6b, the equivalent estimator lag (time constant T

_{ee}) was approximately estimated to T

_{ee}≈ 1 min (60 s), and this value was used in OCV PI controller tuning according to Equations (9) and (10).

#### 3.3. Charging Control Strategy Results

_{4}battery cell [20] implemented within the MATLAB/Simulink environment, with key parameters of both simulation scenarios listed in Table 2, whereas the parameters of the control and estimation strategies used in the comparative simulation analysis are listed in Table 3. The presented simulation scenario corresponded to battery cell recharging from the initial deeply discharged state (characterized by battery SoC of 20 %) to a fully-charged state (with battery SoC target set to 100%). This represents a kind of “ideal” case study (see e.g., [30]), which is intended to illustrate the advantages of the adaptive charging scheme (CCCV-OCV) proposed herein. Naturally, in real-life applications related to EV battery charging, the goal would be to recharge the battery up to 80–90% to avoid a constant-voltage operating regime characterized by low charging-current values and relatively long durations with respect to additional charge gain compared to the constant-current charging regime [32]. It should also be noted that the battery terminal voltage-limiting values u

_{blim}for CCCV-VL and CCCV-OCV control strategies were not chosen to be the same. This discrepancy was the direct consequence of the CCCV-OCV strategy having an additional degree of freedom in controlling the battery terminal voltage through the additional OCV estimation-based feedback control action. More precisely, the battery terminal voltage limit u

_{blim}and the OCV target U

_{ocR}were effectively decoupled within the CCCV-OCV control strategy, whereas, in the case of the CCCV-VL control strategy, the same battery terminal voltage limit also served as indirect OCV target (which was only valid when battery current approached zero). Of course, the battery limit voltage values for both the CCCV-VL and CCCV-OCV control strategies were set to lower values (u

_{blim}= 3.4 V in the former case and u

_{blim}= 3.5 V in the latter case) compared to the maximum safe terminal voltage of 3.65 V listed in the manufacturer’s technical data [20], which would also be performed in actual applications for safety reasons.

_{blim}set to 3.4 V, which corresponds to the fully-charged battery OCV, i.e., U

_{oc}(ξ = 100%). The state-of-charge simulation trace in the top plot in Figure 7 shows that the battery SoC practically reaches the fully charged state at the end of the charging process (ξ ≈ 99.6%), where the small error of final SoC is attributed to the end-of-charging condition, i.e., battery current value dropping below the lower threshold I

_{min}= 5 A (upper middle plot in Figure 7). The battery charging current simulation traces show that the constant-current regime characterized by the upper current limit I

_{max}= 70 A is maintained only during the first 21.5 min. The remainder of the charging process (ending at t = 97.7 min) is conducted in the constant voltage regime (lower middle plot in Figure 7), with the battery terminal voltage limited to u

_{blim}= 3.4 V. The OCV trace also confirmed that the battery asymptotically approached the fully charged state with the final reached value U

_{oc}= 3.38 V.

_{bR}with the peak-to-peak amplitude of 20 A, sampling period T

_{PRBS}= 8 s, and 6-bit depth. The CCCV-OCV control strategy was characterized by the battery OCV target U

_{ocR}= 3.4 V and terminal voltage limit u

_{blim}= 3.5 V, thus allowing for an additional degree of freedom during the recharging process. The OCV estimate provided by the SRAM approach can follow the actual OCV value quite accurately after the initial response transient, and results in the close matching of the target OCV (U

_{ocR}) at the end of the charging process. Moreover, the proposed adaptive charging CCCV-OCV strategy results in an effective speed-up of the charging process when compared to the CCCV-VL benchmark case due to the constant-current regime being maintained for longer than 62.6 min. This, in turn, results in the final battery SoC (99.9%) being reached within 74.3 min (top plot in Figure 8), which corresponds to approximately 23.9% speed-up of the overall charging process, with the battery terminal voltage limit u

_{blim}= 3.5 V being maintained only in the final phase of the charging process (11.7 min overall). These results clearly indicate that, by using the proposed dual-controller structure, along with the adaptive OCV estimator, a more effective battery charging process can be facilitated compared to the conventional CCCV-VL approach.

## 4. Discussion of Obtained Results and Comparison with Other Methods

## 5. Conclusions

_{4}) battery cell. The simulation results sho that the CCCV-OCV control strategy has a distinct advantage over the more traditional CCCV-VL charging strategy due to the former having an additional degree of freedom through its simultaneous control of battery OCV and independent battery terminal voltage limitation. The CCCV-OCV strategy was able to: (i) maintain a constant-current charging regime three-fold longer compared to the conventional charging strategy and (ii) reduce the charging duration by about 23.9% compared to the conventional (CCCV-VL) strategy. The latter result is in agreement with the previous findings related to the utilization of a state estimator-based superimposed feedback loop with the conventional charging system from [50]. The proposed CCCV-OCV control strategy was apparently less effective in terms of charging speed-up, when compared to offline optimization-based battery charging systems. However, its modular design and straightforward tuning makes it suitable for retrofitting conventional CCCV control systems. The proposed CCCV-OCV control strategy was also shown to be robust to variations in the OCV vs. SoC curve gradient (process model gain) and parameter estimator equivalent lag.

_{4}and lithium-titanate (LTO) chemistries.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**Charging control system based on state-of-charge estimation with state-of-charge and battery terminal voltage-limiting controllers (

**a**) and principal block diagram of an EKF-based battery state-of-charge estimator (

**b**).

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**Figure 1.**Battery cell series resistance (

**a**), polarization resistance (

**b**), and open-circuit voltage (

**c**) with respect to state of charge and polarization time constant identification experiment (

**d**).

**Figure 2.**Charging control strategy flowcharts: conventional strategy (

**a**) and novel control strategy based on OCV estimation (

**b**).

**Figure 3.**Cascade control systems used for battery charging: battery terminal voltage-limiting superimposed controller (

**a**) and open-circuit voltage/battery terminal voltage-limiting dual superimposed controllers (

**b**).

**Figure 4.**Battery model parameter estimator based on SRAM approach (

**a**) and reconstruction of battery internal and polarization resistance, polarization time constant, and open-circuit voltage (

**b**).

**Figure 5.**Battery cell OCV vs. SoC curve gradient K

_{ξ}(

**a**), comparative root locus plots of linearized OCV feedback loop tuned for maximum value of gradient K

_{ξ}(

**b**), minimum value of gradient K

_{ξ}(

**c**) for different SRAM estimator lags, and comparative root locus plots of charging control systems without and with active voltage-limiting loop (

**d**).

**Figure 6.**Simulation results of linear time-invariant battery model with respect to PRBS shape of battery current signal (

**a**) and illustration of SRAM estimator convergence with respect to different choices of update gains (

**b**).

**Figure 7.**Simulation results of CCCV-VL charging strategy for battery terminal voltage limit set to u

_{blim}= 3.4 V.

**Figure 8.**Simulation results of the adaptive CCCV-OCV charging strategy for battery open-circuit voltage target set to U

_{ocR}= 3.4 V and terminal voltage limit set to u

_{blim}= 3.5 V.

**Figure 9.**Comparative responses of conventional charging, charging based on SoC estimator, and charging based on the open-circuit voltage (parameter) estimator.

Parameter | Value |
---|---|

Battery model series resistance R_{b} | 0.7 mΩ |

Battery model polarization resistance R_{p} | 1.0 mΩ |

Battery model polarization time constant τ_{p} | 24 s |

Battery model open-circuit voltage U_{oc} | 3.2 V |

Parameter | Value |
---|---|

Inner current control-loop lag T_{ei} | 20 ms |

Current/voltage sensor time constant T_{fm} | 5 ms |

LiFePO_{4} battery cell charge capacity Q_{b} | 100 Ah |

Battery charging current upper limit I_{max} | 70 A |

Battery charging turn-off current I_{min} | 5 A |

Initial battery state of charge ξ_{0} | 20 % |

Target battery state of charge ξ_{R} | 100 % |

CCCV-VL battery limit voltage value u_{blim} | 3.4 V |

CCCV-OCV open-circuit voltage target value U_{ocR} | 3.4 V |

CCCV-OCV battery limit voltage value u_{blim} | 3.5 V |

Parameter | Value |
---|---|

Voltage-limiting PI controller proportional gain K_{cl} | 157.8 |

Voltage-limiting PI controller integral time constant T_{cl} | 5.4 ms |

OCV PI controller proportional gain K_{cu} | 16,325 |

OCV PI controller integral time constant T_{cu} | 44.1 s |

SRAM parameter estimator gain K_{1} | 5·10^{−3} |

SRAM parameter estimator gain K_{2} | 10^{−6} |

SRAM parameter estimator gain K_{3} | 10^{−6} |

SRAM parameter estimator gain K_{4} | 5·10^{−4} |

SRAM parameter estimator pre-filtering time constant T_{f} | 1 s |

SRAM parameter estimator post-filtering time constant T_{pf} | 5 s |

Voltage normalization parameter U_{0} | 3.2 V |

Current normalization parameter I_{0} | 100 A |

Control strategy sampling period T | 4 ms |

Charging Method and Reference | Speed-Up vs. Conventional Charging |
---|---|

Fuzzy logic charging controller [34] | 13.2% |

MCC charging with offline optimization [37] | 19.1% to 29.7% |

MPC with moving horizon estimation [35] | 17.5% |

GA-based offline optimization [36] | 39.7% |

MCC charging with LMS estimator of OCV [41] | 39.3% |

EKF-based SoC estimation plus CCCV (CCCV-SoC) [50] | 25.0% |

SRAM-based OCV estimation plus CCCV (CCCV-OCV) | 23.9% |

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## Share and Cite

**MDPI and ACS Style**

Pavković, D.; Kasać, J.; Krznar, M.; Cipek, M.
Adaptive Constant-Current/Constant-Voltage Charging of a Battery Cell Based on Cell Open-Circuit Voltage Estimation. *World Electr. Veh. J.* **2023**, *14*, 155.
https://doi.org/10.3390/wevj14060155

**AMA Style**

Pavković D, Kasać J, Krznar M, Cipek M.
Adaptive Constant-Current/Constant-Voltage Charging of a Battery Cell Based on Cell Open-Circuit Voltage Estimation. *World Electric Vehicle Journal*. 2023; 14(6):155.
https://doi.org/10.3390/wevj14060155

**Chicago/Turabian Style**

Pavković, Danijel, Josip Kasać, Matija Krznar, and Mihael Cipek.
2023. "Adaptive Constant-Current/Constant-Voltage Charging of a Battery Cell Based on Cell Open-Circuit Voltage Estimation" *World Electric Vehicle Journal* 14, no. 6: 155.
https://doi.org/10.3390/wevj14060155