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Book Remaining Useful Life Prediction of Lithium Ion Batteries Under Different Temperature and Discharge Current Based on Piecewise Random Coefficient Models

Download or read book Remaining Useful Life Prediction of Lithium Ion Batteries Under Different Temperature and Discharge Current Based on Piecewise Random Coefficient Models written by and published by . This book was released on 2021 with total page 128 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Book Rapid Remaining useful life Prediction of Li ion Batteries Using Image based Machine Learning

Download or read book Rapid Remaining useful life Prediction of Li ion Batteries Using Image based Machine Learning written by Jonathan Couture and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: With the increased integration of lithium-ion batteries in our everyday lives, accurate and reliable battery management systems have become an imperative aspect of the well-being of our everyday electronics. This thesis proposes the use of novel machine learning methodologies to predict the remaining-useful-life (RUL) of lithium-ion batteries reliably, accurately, and swiftly. Firstly, a method that prides itself on being publicly available, and which can be easily implemented alongside existing methodology, is proposed to increase the prediction accuracy of the conventional health indicator methodology by 6.72% by using images of data curves as inputs. Subsequently, a more in-depth machine learning model is presented which managed to considerably outperform the current literature in terms of speed, accuracy, and reliability, achieving an RUL prediction accuracy of 90.85%. These proposed methodologies have a wide range of applications, from fault diagnostics, state-of-charge, and state-of-health prediction, to other, more complex, regression applications.

Book Modeling and Simulation of Lithium ion Power Battery Thermal Management

Download or read book Modeling and Simulation of Lithium ion Power Battery Thermal Management written by Junqiu Li and published by Springer Nature. This book was released on 2022-05-09 with total page 343 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on the thermal management technology of lithium-ion batteries for vehicles. It introduces the charging and discharging temperature characteristics of lithium-ion batteries for vehicles, the method for modeling heat generation of lithium-ion batteries, experimental research and simulation on air-cooled and liquid-cooled heat dissipation of lithium-ion batteries, lithium-ion battery heating method based on PTC and wide-line metal film, self-heating using sinusoidal alternating current. This book is mainly for practitioners in the new energy vehicle industry, and it is suitable for reading and reference by researchers and engineering technicians in related fields such as new energy vehicles, thermal management and batteries. It can also be used as a reference book for undergraduates and graduate students in energy and power, electric vehicles, batteries and other related majors.

Book Multidimensional Lithium Ion Battery Status Monitoring

Download or read book Multidimensional Lithium Ion Battery Status Monitoring written by Shunli Wang and published by CRC Press. This book was released on 2022-12-28 with total page 355 pages. Available in PDF, EPUB and Kindle. Book excerpt: Multidimensional Lithium-Ion Battery Status Monitoring focuses on equivalent circuit modeling, parameter identification, and state estimation in lithium-ion battery power applications. It explores the requirements of high-power lithium-ion batteries for new energy vehicles and systematically describes the key technologies in core state estimation based on battery equivalent modeling and parameter identification methods of lithium-ion batteries, providing a technical reference for the design and application of power lithium-ion battery management systems. Reviews Li-ion battery characteristics and applications. Covers battery equivalent modeling, including electrical circuit modeling and parameter identification theory Discusses battery state estimation methods, including state of charge estimation, state of energy prediction, state of power evaluation, state of health estimation, and cycle life estimation Introduces equivalent modeling and state estimation algorithms that can be applied to new energy measurement and control in large-scale energy storage Includes a large number of examples and case studies This book has been developed as a reference for researchers and advanced students in energy and electrical engineering.

Book Degradation Mechanisms and Lifetime Prediction for Lithium Ion Batteries    A Control Perspective

Download or read book Degradation Mechanisms and Lifetime Prediction for Lithium Ion Batteries A Control Perspective written by and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Predictive models of Li-ion battery lifetime must consider a multiplicity of electrochemical, thermal, and mechanical degradation modes experienced by batteries in application environments. To complicate matters, Li-ion batteries can experience different degradation trajectories that depend on storage and cycling history of the application environment. Rates of degradation are controlled by factors such as temperature history, electrochemical operating window, and charge/discharge rate. We present a generalized battery life prognostic model framework for battery systems design and control. The model framework consists of trial functions that are statistically regressed to Li-ion cell life datasets wherein the cells have been aged under different levels of stress. Degradation mechanisms and rate laws dependent on temperature, storage, and cycling condition are regressed to the data, with multiple model hypotheses evaluated and the best model down-selected based on statistics. The resulting life prognostic model, implemented in state variable form, is extensible to arbitrary real-world scenarios. The model is applicable in real-time control algorithms to maximize battery life and performance. We discuss efforts to reduce lifetime prediction error and accommodate its inevitable impact in controller design.

Book Lifetime Prediction and Simulation Models of Different Energy Storage Devices

Download or read book Lifetime Prediction and Simulation Models of Different Energy Storage Devices written by Julia Kowal and published by MDPI. This book was released on 2020-11-13 with total page 92 pages. Available in PDF, EPUB and Kindle. Book excerpt: Energy storage is one of the most important enablers for the transformation to a sustainable energy supply with greater mobility. For vehicles, but also for many stationary applications, the batteries used for energy storage are very flexible but also have a rather limited lifetime compared to other storage principles. This Special Issue is a collection of articles that collectively address the following questions: What are the factors influencing the aging of different energy storage technologies? How can we extend the lifetime of storage systems? How can the aging of an energy storage be detected and predicted? When do we have to exchange the storage device? The articles cover lithium-ion batteries, supercaps, and flywheels.

Book Fuzzy Filter Based State of Energy Estimation for Lithium Ion Batteries

Download or read book Fuzzy Filter Based State of Energy Estimation for Lithium Ion Batteries written by Shunli Wang and published by Cambridge Scholars Publishing. This book was released on 2024-03-21 with total page 141 pages. Available in PDF, EPUB and Kindle. Book excerpt: Awareness of the safety issues of lithium-ion batteries is crucial in the development of new energy technologies, and real-time and high-precision State of Energy (SOE) estimation is not only a prerequisite for battery safety, but also serves as the basis for predicting the remaining driving range of electric vehicles and aircrafts. In order to achieve real-time and accurate estimation of the energy state of lithium-ion batteries, this book improves the calculation method of the open-circuit voltage in the traditional second-order RC equivalent circuit model. It also combines a fuzzy controller and a dual-weighted multi-innovation algorithm to optimize the traditional Centralized Kalman Filter (CKF) algorithm in terms of the aspects of convergence speed, estimation accuracy, and algorithm robustness. This enables the precise estimation of SOE and the maximum available energy. The content of this book provides theoretical support for the development of new energy initiatives.

Book Advances in Lithium Ion Batteries for Electric Vehicles

Download or read book Advances in Lithium Ion Batteries for Electric Vehicles written by Haifeng Dai and published by Elsevier. This book was released on 2024-03 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt: Advances in Lithium-Ion Batteries for Electric Vehicles: Degradation Mechanism, Health Estimation, and Lifetime Prediction examines the electrochemical nature of lithium-ion batteries, including battery degradation mechanisms and how to manage the battery state of health (SOH) to meet the demand for sustainable development of electric vehicles. With extensive case studies, methods and applications, the book provides practical, step-by-step guidance on battery tests, degradation mechanisms, and modeling and management strategies. The book begins with an overview of Li-ion battery aging and battery aging tests before discussing battery degradation mechanisms and methods for analysis. Further methods are then presented for battery state of health estimation and battery lifetime prediction, providing a range of case studies and techniques. The book concludes with a thorough examination of lifetime management strategies for electric vehicles, making it an essential resource for students, researchers, and engineers needing a range of approaches to tackle battery degradation in electric vehicles.

Book Long Term Health State Estimation of Energy Storage Lithium Ion Battery Packs

Download or read book Long Term Health State Estimation of Energy Storage Lithium Ion Battery Packs written by Qi Huang and published by Springer Nature. This book was released on 2023-08-18 with total page 101 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book investigates in detail long-term health state estimation technology of energy storage systems, assessing its potential use to replace common filtering methods that constructs by equivalent circuit model with a data-driven method combined with electrochemical modeling, which can reflect the battery internal characteristics, the battery degradation modes, and the battery pack health state. Studies on long-term health state estimation have attracted engineers and scientists from various disciplines, such as electrical engineering, materials, automation, energy, and chemical engineering. Pursuing a holistic approach, the book establishes a fundamental framework for this topic, while emphasizing the importance of extraction for health indicators and the significant influence of electrochemical modeling and data-driven issues in the design and optimization of health state estimation in energy storage systems. The book is intended for undergraduate and graduate students who are interested in new energy measurement and control technology, researchers investigating energy storage systems, and structure/circuit design engineers working on energy storage cell and pack.

Book Mathematical Modeling of Lithium Ion Batteries and Cells

Download or read book Mathematical Modeling of Lithium Ion Batteries and Cells written by V. Subramanian and published by The Electrochemical Society. This book was released on 2012 with total page 37 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Book Experimental Investigation and Modeling of Lithium ion Battery Cells and Packs for Electric Vehicles

Download or read book Experimental Investigation and Modeling of Lithium ion Battery Cells and Packs for Electric Vehicles written by Satyam Panchal and published by . This book was released on 2016 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The greatest challenge in the production of future generation electric and hybrid vehicle (EV and HEV) technology is the control and management of operating temperatures and heat generation. Vehicle performance, reliability and ultimately consumer market adoption are dependent on the successful design of the thermal management system. In addition, accurate battery thermal models capable of predicting the behavior of lithium-ion batteries under various operating conditions are necessary. Therefore, this work presents the thermal characterization of a prismatic lithium-ion battery cell and pack comprised of LiFePO4 electrode material. Thermal characterization is performed via experiments that enable the development of an empirical battery thermal model. This work starts with the design and development of an apparatus to measure the surface temperature profiles, heat flux, and heat generation from a lithium-ion battery cell and pack at different discharge rates of 1C, 2C, 3C, and 4C and varying operating temperature/boundary conditions (BCs) of 5oC, 15°C, 25°C, and 35°C for water cooling and ~22°C for air cooling. For this, a large sized prismatic LiFePO4 battery is cooled by two cold plates and nineteen thermocouples and three heat flux sensors are applied to the battery at distributed locations. The experimental results show that the temperature distribution is greatly affected by both the discharge rate and BCs. The developed experimental facility can be used for the measurement of heat generation from any prismatic battery, regardless of chemistry. In addition, thermal images are obtained at different discharge rates to enable visualization of the temperature distribution. In the second part of the research, an empirical battery thermal model is developed at the above mentioned discharge rates and varying BCs based on the acquired data using a neural network approach. The simulated data from the developed model is validated with experimental data in terms of the discharge temperature, discharge voltage, heat flux profiles, and the rate of heat generation profile. It is noted that the lowest temperature is 7.11°C observed for 1C-5°C and the highest temperature is observed to be 41.11°C at the end of discharge for 4C-35°C for cell level testing. The proposed battery thermal model can be used for any kind of Lithium-ion battery. An example of this use is demonstrated by validating the thermal performance of a realistic drive cycle collected from an EV at different environment temperatures. In the third part of the research, an electrochemical battery thermal model is developed for a large sized prismatic lithium-ion battery under different C-rates. This model is based on the principles of transport phenomena, electrochemistry, and thermodynamics presented by coupled nonlinear partial differential equations (PDEs) in x, r, and t. The developed model is validated with an experimental data and IR imaging obtained for this particular battery. It is seen that the surface temperature increases faster at a higher discharge rate and a higher temperature distribution is noted near electrodes. In the fourth part of the research, temperature and velocity contours are studied using a computational approach for mini-channel cold plates used for a water cooled large sized prismatic lithium-ion battery at different C-rates and BCs. Computationally, a high-fidelity turbulence model is also developed using ANSYS Fluent for a mini-channel cold plate, and the simulated data are then validated with the experimental data for temperature profiles. The present results show that increased discharge rates and increased operating temperature results in increased temperature at the cold plates. In the last part of this research, a battery degradation model of a lithium-ion battery, using real world drive cycles collected from an EV, is presented. For this, a data logger is installed in the EV and real world drive cycle data are collected. The vehicle is driven in the province of Ontario, Canada, and several drive cycles were recorded over a three-month period. A Thevenin battery model is developed in MATLAB along with an empirical degradation model. The model is validated in terms of voltage and state of charge (SOC) for all collected drive cycles. The presented model closely estimates the profiles observed in the experimental data. Data collected from the drive cycles show that a 4.60% capacity fade occurred over 3 months of driving.

Book A Hybrid Prognostic Approach for Battery Health Monitoring and Remaining useful life Prediction

Download or read book A Hybrid Prognostic Approach for Battery Health Monitoring and Remaining useful life Prediction written by Mohamed Ahwiadi and published by . This book was released on 2020 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Lithium-ion (Li-ion) batteries are commonly used in various industrial and domestic applications, such as portable communication devices, medical equipment, and electric vehicles. However, the Li-ion battery performance degrades over time due to the aging phenomenon, which may lead to system performance degradation or even safety issues, especially in vehicle and industrial applications. Reliable battery health monitoring and prognostics systems are extremely useful for improving battery performance, to diagnose the battery's state-of-health (SOH), and to predict its remaining-useful-life (RUL). In general, it is challenging to accurately track the battery's nonlinear degradation features as battery degradation parameters are almost inaccessible to measure using general sensors. In addition, a battery is an electro-chemical system whose properties vary with variations in environmental and operating conditions. Although there are some techniques proposed in the literature for battery SOH estimation and RUL analysis, these techniques have clear limitations in applications, due to reasons such as lack of proper representation of the posterior probability density functions to capture and model the nonlinear dynamic system of Li-ion batteries. In addition, these techniques cannot effectively deal with the time-varying system properties, especially for long-term predictions. To tackle these problems, a novel hybrid prognostic framework has been developed in this PhD work for battery SOH monitoring and RUL prediction. It integrates two new models: the model-based filtering method and the evolving fuzzy rule-based prediction technique. The strategy is to propose and use more efficient techniques in each module to improve processing, accuracy and reliability. Firstly, a newly enhanced mutated particle filter technique is proposed to enhance the performance of particle filter technique and improve the modeling accuracy of the battery system's degradation process. It consists of three novel aspects: an enhanced mutation approach, a selection scheme, and an outlier detection method. Secondly, an adaptive evolving fuzzy technique is suggested for long-term time series forecasting. It has a novel error-assessment method to control the fuzzy cluster/rule generation process-also, a new optimization technique to enhance incremental learning and improve modeling efficiency. Finally, a new hybrid prognostic framework integrates the merits of both proposed techniques to capture the underlying physics of the battery systems for its SOH estimation, and improve the prognosis of dynamic system for long-term prediction of Li-ion battery RUL. The effectiveness of the proposed techniques is verified through simulation tests using some commonly used-benchmark models and battery databases in this field, such as the one from the National Aeronautics and Space Administration (NASA) Ames Prognostic Center of Excellence. Test results have shown that the proposed hybrid prognostics framework can effectively capture the battery SOH degradation process, and can accurately predict its RUL.

Book Investigation Into the Effect of Thermal Management on the Capacity Fade of Lithium ion Batteries

Download or read book Investigation Into the Effect of Thermal Management on the Capacity Fade of Lithium ion Batteries written by Andrew Carnovale and published by . This book was released on 2016 with total page 96 pages. Available in PDF, EPUB and Kindle. Book excerpt: The popularity of electric (and hybrid) vehicles has raised the importance of effective thermal management for lithium-ion batteries, both to prevent thermal runaway leading to a fire hazard, and to minimize capacity fade for longer lifetime. In this research, the focus was on the effect of thermal management on the capacity fade of lithium-ion batteries. A battery thermal management system will impact the battery operation through its temperature, thermal gradient and history, as well as the cell-to-cell temperature variations in a battery module. This study employed AutoLionST, a software for the analysis of lithium-ion batteries, to better understand capacity fade of lithium-ion batteries, complemented by the experimental investigation. Experimental capacity fade data for a lithium-ion battery cycled under isothermal, 1C charge/discharge conditions was measured first, which was used to validate the numerical model. Then the software's ability to model degradation at moderate to lower temperatures of around 20°C was investigated with simulation of battery capacity under isothermal conditions for a variety of operating temperatures. The next phase of the study modeled battery capacity fade under a variety of different operating conditions. In the first set of simulations, three different base temperatures, constant discharge rates, and heat transfer coefficients were considered. In the second set of simulations, a fixed-time drive cycle was used as the load case to model a typical day's worth of driving, while varying the base temperature, charge voltage, and heat transfer coefficient. These simulations were repeated considering regenerative braking. It was found that temperature has the largest direct impact on the capacity fade which is expected based on prior sutdies. Further, it was found that thermal management does have a significant impact on capacity fade, as effective thermal management is capable of preventing significant battery temperature rise. As concluded from the constant discharge rate simulations, effective thermal management is most crucial at high discharge rates, which will result in high heat generation. It was also concluded from both constant discharge rate and drive cycle simulations, that thermal management is much more effective at preventing capacity fade at battery temperatures close to 20°C. In the drive cycle simulations, using the same discharge profile, there is a much more significant spread in battery capacity between high and low heat transfer coefficients for a lower base temperature (20°C) compared to higher base temperatures (35°C and 50°C). As well, it was shown that using a lower charge voltage will result in slightly less capacity fade over cycling. Additionally, using regenerative braking makes it more realistic to use lower charge voltages, since the battery pack can be recharged during operation, thereby increasing driving range, while preventing increased capacity fade. The final phase showed that effective thermal management would be even more imperative for more intense and realistic driving styles. It was shown that different driving styles can result in significant rises in heat generation and hence battery temperature. From previous conclusions this implies that much intense driving (high acceleration) can result in a higher need for effective thermal management.

Book Experimental Aging and Lifetime Prediction in Grid Applications for Large Format Commercial Li Ion Batteries

Download or read book Experimental Aging and Lifetime Prediction in Grid Applications for Large Format Commercial Li Ion Batteries written by and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Due to the growth of electric vehicle and stationary energy storage markets, the production and use of lithium-ion batteries has grown exponentially in recent years. For many of these applications, large-format lithium-ion batteries are being utilized, as large cells have less inactive material relative to their energy capacity and require fewer electrical connections to assemble into packs. And especially for stationary energy storage systems, where energy delivered is the only revenue source, the economics of these battery systems is highly dependent on cell lifetime. However, testing of large-format lithium-ion batteries is time consuming and requires high current channels and large testing chambers, making information on the performance of commercial, large-format lithium-ion batteries hard to come by. Here, accelerated aging test data from four commercial large-format lithium-ion batteries is reported. These batteries span both NMC-Gr and LFP-Gr cell chemistries, pouch and prismatic formats, and a range of cell designs with varying power capabilities. Accelerated aging test results are analyzed to examine both cell performance, in terms of efficiency and thermal response under load, as well as cell lifetime. Cell thermal response is characterized by measuring temperature during cycle aging, which is used to calculated a normalized thermal resistance value that may help estimate both cell cooling needs or to help extrapolate aging test results to different thermal environments. Cell lifetime is evaluated qualitatively, considering simply the average calendar and cycle life across a range of conditions, as well as quantitatively, using statistical modeling and machine-learning methods to identify predictive aging models from the accelerated aging data. These predictive aging models are then used to investigate cell sensitivities to stressors, such as cycling temperature, voltage window, and C-rate, as well as to predict cell lifetime in various stationary storage applications. Results from this work show that cell lifetime and sensitivity to aging conditions varies substantially across commercial cells, necessitating testing for specific cell formats to make quantitative lifetime predictions. That being said, all commercial cells tested here are predicted to reach at least 10-year lifetimes for stationary storage applications. Based on the aging test results and modeling, some cells are expected to be relatively insensitive to temperature and use-case, making them suited for simple use cases with little or no thermal management and simple controls, while the lifetime of other cells could be extended to 20+ years if operated with thermal management and degradation-aware controls.

Book Fault Diagnosis and Failure Prognostics of Lithium ion Battery Based on Least Squares Support Vector Machine and Memory Particle Filter Framework

Download or read book Fault Diagnosis and Failure Prognostics of Lithium ion Battery Based on Least Squares Support Vector Machine and Memory Particle Filter Framework written by Mohammed Ali Lskaafi and published by . This book was released on 2015 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: A novel data driven approach is developed for fault diagnosis and remaining useful life (RUL) prognostics for lithium-ion batteries using Least Square Support Vector Machine (LS-SVM) and Memory-Particle Filter (M-PF). Unlike traditional data-driven models for capacity fault diagnosis and failure prognosis, which require multidimensional physical characteristics, the proposed algorithm uses only two variables: Energy Efficiency (EE), and Work Temperature. The aim of this novel framework is to improve the accuracy of incipient and abrupt faults diagnosis and failure prognosis. First, the LSSVM is used to generate residual signal based on capacity fade trends of the Li-ion batteries. Second, adaptive threshold model is developed based on several factors including input, output model error, disturbance, and drift parameter. The adaptive threshold is used to tackle the shortcoming of a fixed threshold. Third, the M-PF is proposed as the new method for failure prognostic to determine Remaining Useful Life (RUL). The M-PF is based on the assumption of the availability of real-time observation and historical data, where the historical failure data can be used instead of the physical failure model within the particle filter. The feasibility of the framework is validated using Li-ion battery prognostic data obtained from the National Aeronautic and Space Administration (NASA) Ames Prognostic Center of Excellence (PCoE). The experimental results show the following: (1) fewer data dimensions for the input data are required compared to traditional empirical models; (2) the proposed diagnostic approach provides an effective way of diagnosing Li-ion battery fault; (3) the proposed prognostic approach can predict the RUL of Li-ion batteries with small error, and has high prediction accuracy; and, (4) the proposed prognostic approach shows that historical failure data can be used instead of a physical failure model in the particle filter.

Book Dynamic Modeling and Thermal Characterization of Lithium ion Batteries

Download or read book Dynamic Modeling and Thermal Characterization of Lithium ion Batteries written by Khaled I. Alsharif and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Lithium-ion batteries have revolutionized our everyday lives by laying the foundation for a wireless, interconnected and fossil-fuel-free society. Additionally, the demand for Li-ion batteries has seen a dramatic increase, as the automotive industry shifts up a gear in its transition to electric vehicles. To optimize the power and energy that can be delivered by a battery, it is necessary to predict the behavior of the cell under different loading conditions. However, electrochemical cells are complicated energy storage systems with nonlinear voltage dynamics. There is a need for accurate dynamic modeling of the battery system to predict behavior over time when discharging. The study conducted in this work develops an intuitive model for electrochemical cells based on a mechanical analogy. The mechanical analogy is based on a three degree of freedom spring-mass-damper system which is decomposed into modal coordinates that represent the overall discharge as well as the mass transport and the double layer effect of the electrochemical cell. The dynamic system is used to estimate the cells terminal voltage, open-circuit voltage and the mass transfer and boundary layer effects. The modal parameters are determined by minimizing the error between the experimental and simulated time responses. Also, these estimated parameters are coupled with a thermal model to predict the temperature profiles of the lithium-ion batteries. To capture the dynamic voltage and temperature responses, hybrid pulse power characterization (HPPC) tests are conducted with added thermocouples to measure temperature. The coupled model estimated the voltage and temperature responses at various discharge rates within 2.15% and 0.40% standard deviation of the error. Additionally, to validate the functionality of the developed dynamic battery model in a real system, a battery pack is constructed and integrated with a brushless DC motor (BLDC) and a load. Moreover, because of the unique pole orientation that a BLDC motor possesses, it puts a pulsing dynamic load on the battery pack of the system. HPPC testing was conducted on the cell that is used in the battery pack to calibrate the model parameters. After the battery model is calibrated, the rotation experiment is conducted at which a battery pack is used to drive a benchtop BLDC motor with a magnetorheological brake as a programable load at varying running speeds. The voltage and current of the battery and the BLDC motor driver are recorded. Meanwhile, the speed and the torque of the motor are recorded. These data are compared to the predicted voltage of the battery pack using the mechanical analogy model. The model estimated the voltage response of a battery pack within 0.0385% standard deviation of the error.