Cell modelling for state-of-charge and state-of-health estimation

Treball de Fi Màster Master’s degree in Electric Power Systems and Drives Cell Modelling for State-of-Charge and State-of-Health Estimation Report Author: Antonia Rodríguez Moreno Supervisor: Francisco Díaz González Call: September 2023 Escola Tècnica Superior d’Enginyeria Industrial de Barcelona ii Resum Resum La correcta modelització de bateries és clau per a una gestió eficaç de l’energia, ja que té un paperfonamental tant en l’allargament de la durada de la bateria com en la garantia de la seva seguretat operativa. Dins del camp de la tecnologia de cel·les de bateries de ions de liti (Li-ion), l’estimació precisa de l’estat de càrrega (SOC) i l’estat de salut (SOH) és fonamental per aconseguir un rendiment òptim. Mitjançant la navegació per aquestes tècniques, es pretén aclarir els reptes, oportunitats i avenços en el camp de la modelització de bateries de ions de liti. El marc de modelització proposat s’ha validat utilitzant dades experimentals obtingudes de cel·les cilíndriques de Li-NMC (Lithium Nickel, Manganès, Cobalt Oxide). Es van dur a terme diversos procediments de prova en diferents condicions per caracteritzar el seu rendiment. L’anàlisi d’aquestes proves també és un aspecte important d’aquesta tesi, ja que explica el comportament de les cel·les Li-NMC i proporciona informació sobre les seves característiques de rendiment en diversos escenaris. Basant-se en la bibliografia previa, aquesta tesi de màster revisa els fonaments de les cel·les secundàries, amb un enfocament particular en les cel·les de bateries Li-NMC. Dins de l’estructura de cel·les es discuteixen diversos elements, oferint una comprensió detallada de les seves propietats i les seves funcions. A més, aquesta tesi aborda els mecanismes d’envelliment, examinant factors com l’envelliment del calendari i l’envelliment del cicle. A més a més, proporciona una visió general de diferents enfocaments de modelització de cel·les de bateries, que cobreixen models de circuits equivalents (ECM), models electroquímics i models basats en dades. Laselecciódelsmodelsitècniquess’habasatenelsrecursosdisponiblesis’alineen amb l’objectiu de proporcionar solucions fermes per a la correcta modelització de les bateries. En aquest context, s’han desenvolupat dos ECM diferents. Un es basa principalment en el perfil de càrrega dinàmica, que permet el monitoratge en línia, mentre que l’altre es basa en l’espectroscòpia d’impedància electroquímica (EIS), un mètode fora de línia. A més, es realitza una anàlisi diferencial, queestableixunarelacióentreelrendimentdurant unaprova decàrrega ielsmecanismes d’envelliment. Cell Modelling for State-of-Charge and State-of-Health Estimation iii Resumen La precisión en la modelización de las baterías es fundamental para gestionar eficazmente la energía, prolongar su vida útil y garantizar su seguridad operativa. En el campo de la tecnología de celdas de batería de iones de litio (Li-ion), la estimación precisa del estado de carga (SOC) y el estado de salud (SOH) es esencial para lograr un rendimiento óptimo. Este trabajo se adentra en una exploración de las diversas técnicas destinadas a alcanzar este objetivo. Al examinar estas técnicas, se busca esclarecer los desafíos, oportunidades y avances en el ámbito de la modelización de celdas Li-ion. El marco de modelización propuesto se valida utilizando datos experimentales obtenidos de celdas cilíndricas de Li-NMC (Óxido de Litio Níquel Manganeso Cobalto). Se llevaron a cabo diversos tests en diferentes condiciones para caracterizar su comportamiento. El análisis de estos ensayos también es un aspecto crucial de este trabajo, ya que proporciona una visión de las características en diversos escenarios. Este trabajo revisa los fundamentos de las baterías secundarias, con un enfoque especial en las celdas de baterías Li-NMC. Se analizan varios elementos dentro de la estructura de las celdas, ofreciendo una comprensión detallada de sus propiedades y funciones. Además, esta tesis aborda los mecanismos de envejecimiento, examinando factores como el envejecimiento por tiempo (calendario) y el envejecimiento cíclico. También ofrece una visión general de los diferentes enfoques de modelización de las celdas de batería, que incluyen modelos de circuito equivalente (MCE), modelos electroquímicos y modelos basados en datos. La selección de los modelos y técnicas empleadas se basa en los recursos disponibles y se ajusta al objetivo de proporcionar soluciones sólidas para una modelización precisa de las baterías. En este contexto, se han desarrollado dos ECM diferentes. Uno de ellos se basa principalmente en el perfil dinámico de pulsos, lo que permite la monitorización on-line, mientras que el otro se basa en la espectroscopia de impedancia electroquímica (EIS), un método off-line. Además, se lleva a cabo un análisis diferencial, estableciendo una relación entre el resultado durante las pruebas de carga y los mecanismos de envejecimiento. iv Abstract Abstract Accurate battery modelling is the key to effective energy management, as it plays an important role in extending battery lifespan and ensuring operational safety. Within the field of lithiumion (Li-ion) battery cell technology, precise estimation of state-of-charge (SOC) and state-of- health (SOH) is fundamental for achieving optimal performance. This thesis embarks on an exploration of the different techniques for estate and health monitoring. By navigating these techniques, it is intended to clarify the challenges, opportunities and advances in the field of lithium-ion battery cell modelling. The proposed modelling framework is validated using experimental data obtained from cylindrical Li-NMC (Lithium Nickel Manganese Cobalt Oxide) cells. Diverse test procedures were conducted under different conditions to characterize their performance. The analysis of these tests is also an important aspect of this thesis, as it sheds light on the behaviour of Li-NMC cells and provides insight into their performance characteristics in various scenarios. Departingfrom theliterature, this Master’s thesis reviewsthefundamentalsofsecondarybatteries, with a particular focus on Li-NMC battery cells. Within the cell structure, various elements are discussed, offering a detailed understanding of their properties and their roles. Moreover, this thesis addresses the ageing mechanisms, examining factors such as calendar ageing, and cycle ageing. Additionally, it provides an overview of different battery cell modelling approaches, covering equivalent circuit models (ECM), electrochemical models, and data-driven models. The selection of the models and techniques are based on the available resources and it aligns with the aim of providing robust solutions for accurate battery modelling. In this context, two different ECMs have been developed. One is primarily based on a dynamic load profile, enabling online monitoring, while the other relies on Electrochemical Impedance Spectroscopy (EIS), an offline method. Furthermore, a differential analysis is conducted, establishing a relationship between the performance during charging tests and ageing mechanisms. Cell Modelling for State-of-Charge and State-of-Health Estimation v List of Symbols ASystem matrix. AACross-sectional area. BTransition matrix. CNNominal capacity. CxCapacitance of x element. cWDrag coefficient. DStraight-way matrix. ∆GEquilibrium constant of the chemical reaction. E0Standard Electrode Potential. eFactor for rotational masses. FFaraday constant. Freq Driving resistances. FAAerodynamic drag. FCClimbing resistance. FIAcceleration resistance. FRRolling resistance. gGravitational acceleration. ICurrent. KFaraday constant. LxInductance of x element. mLMass of load. mVMass of vehicle. NNumber of cells. PCovariance matrix. QProcess noise matrix. RUniversal gas constant. RxResistance of x element. TTemperature. tTime. UVoltage. Umax Maximum voltage. Umin Maximum voltage. UVoltage. Uref Nominal voltage. ZImpedance. αAngle of inclination. ηEfficiency. ωWarburg coefficient . σAngular frequency. τxTime constant of the element x. ρDensity. vi Abbreviations and Symbols Abbreviations AC Alternating Current AI Artificial Intelligence BMS Battery Management System BOL Begin of Life CC Constant Current CEI Cathode-Electrolyte Interface CRM Critical Raw Material CV Constant Voltage CHA Charge CL Conductivity Loss CTS Cell Test System DAT Differential Analysis Test DC Direct Current DCH Discharge DOD Depth of Discharge DVA Differential Voltage Analysis ECM Equivalent Circuit Model EESS Electrochemical Energy Storage System EIS Electrochemical Impedance Spectroscopy EKF Extended Kalman Filter EOL End of Life ES Energy Storage ESS Energy Storage System Cell Modelling for State-of-Charge and State-of-Health Estimation vii ETSEIB Escola Tècnica Superior d’Enginyeria Industrial Barcelona EV Electric Vehicle GPR Gaussian Process Regression HPPC Hybrid Pulse Power Characterization ICA Incremental Capacity Analysis KF Kalman Filter LAM Loss of Active Material LCA Life Cycle Assessment LCO Lithium Cobalt Oxide LIB Lithium-Ion Battery LIC Lithium-Ion Cell LLI Loss of Lithium Inventory LFP Lithium Iron Phosphate LMO Lithium Manganese Oxide ML Machine Learning NCA Nickel Cobalt Aluminum NMC Nickel Manganese Cobalt NN Neural Network OCV Open Circuit Voltage PAU Pause PNGV Partnership for a New Generation of Vehicles redox Reduction-Oxidation RMSE Root Mean Square Error RPT Reference Parameter Test RUL Remaining Useful Life RVM Relevance Vector Machines SBL Sparse Bayesian Learning viii Abbreviations and Symbols SEI Solid-Electrolyte Interface SHE Standard Hydrogen Electrode SOC State of Charge SOH State of Health SVM Support Vector Machines THWS Technische Hochschule Würzburg-Schweinfurt TTZ-EMO Technologietransferzentrum Elektromobilität UPC Universitat Politècnica de Catalunya VAT Value Added Tax WLTP Worldwide Harmonized Light Vehicles Test Procedure Cell Modelling for State-of-Charge and State-of-Health Estimation 1 1. Preface Energy has emerged as a critical concern nowadays, and its relevance continues to grow. Simultaneously, the importance of energy storage has risen in tandem, affecting a variety of applications ranging from power systems to electric vehicles. As a consequence, Energy Storage (ES) has attracted significant attention as a relevant subject of study. Research centers in this field have diverse aims, including advancing scientific understanding, improving efficiency, ensuring safety, and exploring new battery applications. This impulse towards innovation is undoubtedly one of the driving forces behind the choice of this field of research. The choice to focus on LICs as the selected Energy Storage System (ESS) naturally aligned with my background in chemical engineering and my experience in battery testing. In addition, participation in the Energy Storage course at Escola Tècnica Superior d’Enginyeria Industrial Barcelona (ETSEIB) further solidified my understanding of cell behavior. Essentially, this project was due to a combination of circumstances that manifested an interest in delving into a comprehension of battery cell performance and modelling. 2 2.Introduction 2. Introduction In an era where energy storage solutions are essential to address the changing needs of renewable energy integration, electric mobility and grid stability, it is indispensable to understand the complexities of battery behaviour. The increased use of Li-ion batteries across diverse sectors has contributed to a notable reduction in their cost over the past years. Achieving this reduction has required a concentrated effort across multiple disciplines. The goal has been to engineer batteries that are not only safer and more durable but also energy and power-dense, all while becoming more cost-effective. Initially, research primarily focused on improving battery chemistry. However, recent years have witnessed incremental enhancements in battery performance achieved through the reengineering of manufacturing processes and innovations in models that enhance our comprehension of battery performance during real-world usage [1]. 2.1 Motivation The current energy scenario has created a significant need for efficient energy storage systems. As the penetration of renewable energy sources increases, the challenge of accurately forecasting the energy they generate emphasizes the need for adaptable solutions, such as pumped hydro storage and advanced battery banks. Simultaneously, the electrification of transportation, particularly the remarkable evolution of Electric Vehicles (EVs), has driven the forefront of development. At the centre of this evolution is the application of LICs as the main energy source for EVs. The significance of making accurate predictions of the Remaining Useful Life (RUL) of batteries extends to a number of fields, contributing to the effective performance management of a range of devices or systems, from EVs to smartphones. This innovative approach not only helps refine production processes and resource allocation, but also contributes to the advancement of battery design and performance. In this context, the longevity and performance of energy storage solutions are linked to the efficacy of Battery Management System (BMS). Achieving optimal control, monitoring and protection of battery cells depends on a deep understanding of their behaviour. This requires meticulous characterisation of battery cells, an effort that brings to light the complex interaction of factors governing performance, efficiency and durability. By unlocking these complexities, researchers and engineers can open the way to improved battery technologies that are ready to meet the challenges of energy storage, integration of renewable energy and sustainable transport. Cell Modelling for State-of-Charge and State-of-Health Estimation 3 2.2 Scope This thesis centers on the characterization of battery cells, taking into account both SOC and SOH. The objective is to understand the behavior of a LIC, particularly cells based on NMC chemistry, through various methodologies to predict its performance. While the primary emphasis is on the selected LIC, this study will also provide an overview of different battery technologies. Additionally, diverse battery cell models have been explored, and a subset has been chosen for comparative analysis. 2.3 Prerequisites The Master’s thesis was conducted in Bad Neustadt an der Saale, Germany, at the facilities of Technologietransferzentrum Elektromobilität (TTZ-EMO) [2]. The collaboration between Universitat Politècnica de Catalunya(UPC)andTechnische Hochschule Würzburg-Schweinfurt (THWS) facilitated the use of BaSyTec Cell Test System (CTS) and the Gamry Reference 3000 for test execution (further details regarding their usage are provided in the chapter Test Procedure Analysis and Results). TTZ-EMO specializes in research and development projects related to electrical power engineering, drive technology, and electromobility. To initiate the project, certain administrative requirements had to be met to ensure access and work within their facility. Despite these initial challenges, the cooperation and support of the centre made it possible to resolve administrative issues quickly. The commitment and hospitality of the centre allowed for a smooth transition, taking advantage of its state-of-the-art laboratories and battery systems research group. This experience has proven invaluable for gaining insights into battery cell performance and modelling. 2.4 Objectives The project aims to enhance the understanding of battery cell behavior and its variations concerning SOC and SOH. This comprehension serves as a fundamental step in gathering relevant information for performance improvement and life extension through the BMS. Four cylindrical cells were available for this study, and the following objectives have been defined to achieve the project goal: 1. Review the state-of-the-art regarding LIC and the ageing mechanisms. 2. Develop a comprehensive test procedure capable of capturing all relevant parameters, facilitating the reproducibility of performance and modelling. 3. Analysis of the test procedure outcomes and establish a consistent methodology for parameter acquisition and identification. 4. Evaluate the applicability and effectiveness of various battery cell models. 4 3.Electrochemical Energy Storage systems 3. Electrochemical Energy Storage systems During a chemical reaction, a change in potential energy occurs, and in certain reactions, this change manifests as electrical energy. The field of electrochemistry focuses on the analysis and understanding of interactions and transformations at the atomic scale, where chemical energy converts into electrical energy, and vice versa. Consequently, the extraction and storage of electrical power become feasible through these processes [3]. The voltage or current provides an additional degree of freedom, which is an undeniable advantage of electrochemistry. Therefore it is feasible to predict electrochemical device behaviour and improve the performance by modifying the energy of the active material in a controlled way to create highly selective reactivity [4]. The emergence of Electrochemical Energy Storage Systems (EESSs) was driven by the desire to profit this inherent quality of electrochemical systems. They can be classified in terms of rechargeability in primary and secondary systems. The first consists of systems that typically cannot be recharged, while the latter focuses on systems that are rechargeable. This resport is centered on secondary systems, more precisely on LICs based on NMC chemistry. Further details about this topic will be explored in the next chapter, State-of-the-Art of NMC Li-ion Cell. 3.1 Electrochemical reaction The electrochemical reaction is fundamentally a charge(electron)-transfer reaction occurring at an interface where electroactive materials are involved. In the context of secondary systems, due to their rechargeable nature, these reactions are also reversible. The general scheme of a half-reaction can be illustrated as follows: Ox +ne−reduction −−−−−−⇀ ↽−−−−−− oxidation Red (3.1) The component Ox is reduced (reduction half-reaction) by gaining nelectrons, resulting in the formation of the component Red. The reversible reaction is oxidation, where nfree electrons are generated. Often, such reactions involve two species that exchange electrons, leading to the term Reduction-Oxidation (redox) reactions. All participants involved have an oxidation number or state, representing their capacity to gain, lose, or share electrons [4]. The energy generated to do electrical work depends on both the cell potential and the number the electrons produced during the electrochemical reaction. The standard cell potential of the cell (Eo) can be represented by the following equation: Eo cell =Eo reduction −Eo oxidation (3.2) The standard cell potential (Eo) is set using the Standard Hydrogen Electrode (SHE), which is designated as 0 Vand serves as the fundamental reference for comparing other Eovalues. For lithium-ion cells (LIC), the specific value is Li/Li+= 3.04V. Cell Modelling for State-of-Charge and State-of-Health Estimation 5 Moreover, the voltage or cell potential is intricately linked to the Gibbs free energy (∆G). This relationship is established through the knowledge of the number of moles of electrons (n), the electrode potential (E), and the utilization of the Faraday constant (F= 96485,C/mol): ∆G=−nFE (3.3) The determination of the free energy can be calculated by the equilibrium constant of the chemical reaction (K), the universal gas constant (R), the temperature (T) and the standard value ∆G0. This relationship is expressed by the equation: ∆G= ∆G0−RT ln K(3.4) From the principlesdiscussedearlier, the Nernstequationemerges, providing information about cell potential: E=Eo−RT nF ln K(3.5) This equation establishes a direct link between electrochemical measurements and thermodynamic properties. It provides valuable understanding of how the cell potential evolves in response to the cell reaction [3]. Beyond thermodynamics, there are others parameters, which are also important and have to be consider during these reactions: •Kinetics must be also favourable in order to the reaction occur. This term is presented in Nernst equation via the equilibrium constant of the chemical reaction K. •Mass transport takes placeinelectrochemical systems through three forms: diffusion, convection, and migration. One of the most important factors for performing effective electrochemical operations are transport processes. Subsequently the thermodynamic and kinetic components of electrochemistry are not feasible until reactants can be transported efficiently to and from an electrode. Given their significance, mass transport challenges are often a central consideration in the development of electrochemical systems. •The electrochemical double layer affects in the performance of the reaction because this can cause the buildup of ions or contaminants. This accumulation can constrain operational voltage and device lifespan [5]. •Ionic and electronic resistance lead to losses, frequently manifesting as ohmic losses. Many factors are involved during an electrochemical reaction. Designing a new system necessitates a comprehensive consideration of all these factors to ensure optimal performance and longevity. 3.2 Cell Components In general, the EESSs are composed of at least two electrodes connected by an ion-conducting electrolyte, being the electrochemical interface the contact zone between them. A separator is typically also present in a battery cell. Each part has a fundamental role. The performance 6 3.Electrochemical Energy Storage systems of the battery cell will be influenced by the properties of these components, depending on the chemistryof electrodematerials, the stability ofthe electrolyte andthe interactions amongthem. Figure 3.1: Schematic of a Li-ion battery cell [6] 3.2.1 Electrode As illustrated in Figure 3.1, the core structure of these systems comprises a positive electrode, also known as the cathode, and a negative electrode, usually known as the anode. These electrodes are connected to their respective current collectors to enable efficient electron flow. The fundamental purpose of electrodes within this system is to enable the exchange of charges, facilitating the electrochemical reactions. An electrode, by definition, stands as a key element in which various electrochemical transformations take place. In this context, the anode serves as the site for oxidation reactions, while the cathode emerges as the element where reduction reactions occur. In electrochemical cells, the anode and cathode roles switch during charging and discharging, reflecting the change in current direction. To prevent confusion, it is better to use "positive" and "negative" for electrode labels based on their potential. However, conventionally, batteries still refer to the negative electrode as the anode and the positive electrode as the cathode, focusing on discharge behavior [7]. It is important to recognize that the cell potential or voltage is highly influenced by the nature of the electrodes themselves. The properties of a material with a given chemical composition are strongly determined by its microstructure, crystal arrangement, and electronic configuration. While the fundamental nature of the chosen materials determines the electrochemical properties of electrodes, changes in their microstructures can occur due to different synthesis or processing methods and conditions [8]. Desirable combinations of anode and cathode materials are characterized by their capacity to produce lightweight cells with elevated voltage and storage capacity. However, it is worth noting that achieving such combinations may encounter challenges due to factors such as the handling complexity of materials, reactivity with other cell elements, manufacturing intricacies, polarization tendencies, and the potential cost constraints associated with certain materials. Cell Modelling for State-of-Charge and State-of-Health Estimation 7 Achieving the optimal balance between raw material availability, electrochemical performance, cost-effectiveness, and environmental considerations is imperative when designing ideal electrode particles [9]. 3.2.2 Electrolyte The electrolyte plays a important role as selective barrier, allowing the passage of ionic species while impeding the flow of electronic charge. Essentially, the electrolyte is an electronic insulator [10]. It is important to note that the reaction involves not only ions but also electrically neutral atoms. The electrolyte serves as the medium that facilitates ion transport between the cathode and anode of a cell. Electrolytes are often associated with liquid solutions containing dissolved salts, acids, or alkalis. These components are essential for enabling ionic conduction. However, it is worth noting that many batteries, including conventional types, utilize solid electrolytes. Key properties of electrolytes that merit consideration include [11]: •Viscosity, which is related to the ionic mobility. •Dielectric constant of solvents. •Ionic conductivity. •Electrochemical stability window. •Chemical and thermal stability over an extended temperature and voltage range. •Non-reactivity characteristic towards other battery components. •Sustainability, cost-effectiveness and safety. 3.2.3 Separator The battery separator is a polymer-based element with selective ion-conducting characteristics, primarily attributed to its porosity. It serves as a barrier between the two pairs of electrodes and plays a pivotal role in facilitating mass transfer within the battery cell. Apart from porosity and ion conductivity, the separator material must possess two essential properties: electrochemical inertness and chemical stability. This quality prevents undesirable chemical reactions between the separator and other components, contributing to the longevity and stability of the cell. The concept of ion conductivity within the separator is closely linked to its tortuosity, which quantifies the complexity of the pathway of the ions through the porous structure. Lower tortuosity values indicate a more direct route for ions, but this often accompanies a higher level of porosity, making the separator structure more delicate and fragile [12]. Furthermore, the separator should possess adequate mechanical strength to maintain its structural integrity and 8 3.Electrochemical Energy Storage systems prevent physical damage that could lead to short circuits or the closure of pores, thereby impeding ion transport [13]. 3.3 Battery Cell Performance Metrics The characteristics of a battery cell are typically defined by a set of parameters that provide valuable insights into its performance and capabilities. Some of these parameters are typically included in the datasheet of the battery cell, which serves as a reference for users, engineers, and designers. Here are the key parameters: •Cut-off voltages (Umin/Umax) indicate the voltage limits at which the discharge or charge process must be interrupted in order to prevent overcharging or over-discharging. These values, set by the manufacturer, help to guarantee the safety of the battery. •Nominal or ratedcapacity(CN) refers tothe quantityofcharge, in Ah, iscapabletodeliver. This measurement takes into consideration variables like load, temperature, and cut-off voltages. •The energy of a battery is another way to give information about the capacity. It is calculated as the product of the capacity (Ah) and voltage. This measurement is expressed in watt-hours (Wh). Related terms, such as specific energy (or gravimetric energy density), which measures energy in relation to mass (Wh/kg), and energy density (or volumetric energy density), which quantifies energy concerning volume (Wh/l), are commonly used in battery characterization. Batteries with higher specific energy and energy density values can store more energy in a smaller or lighter form factor, making them ideal for applications where space and weight constraints are critical for optimal performance. •Specific power, with a unit of watts per kilogram (W/kg), represents the power output a battery can deliver relative to its mass. This parameter is crucial for applications requiring rapid power delivery. •The efficiency (η) is calculated by dividing the energy released during discharging by the energy stored during charging. It offers insights into the effectiveness of energy conversion. 3.4 Battery Chemistries Since the appearance of EESSs, different technologies have been developing to meet diverse energy storage needs. Each chemistry possesses unique characteristics that impact its performance, efficiency, and application suitability. The following sections are an overview of common battery chemistries that find utility in various sectors. Cell Modelling for State-of-Charge and State-of-Health Estimation 9 3.4.1 Lead-Acid Batteries Lead-acid batteries have a long history of use in a wide range of applications, from small-scale to large-scale, due to their reliability, cost-effectiveness, and ease of manufacturing. Today, a significant portion of the global battery market relies on this technology, making it one of the most well-known electrochemical systems. In these batteries, the negative electrode consists of sponge lead (Pb), while the positive electrode is made of lead dioxide (PbO2), both immersed in an electrolyte solution of sulfuric acid( H2SO4). A separator is placed between the electrodes to prevent short-circuits and enhance mechanical stability [10]. This combination of materials and design principles has made lead-acid batteries a versatile and reliable choice for a wide range of applications, from automotive to backup power systems. The chemical reactions that occur within a lead-acid battery are as follows: Half-reaction anode: Pb + HSO4–−−⇀ ↽−− PbSO4+ H++ 2e– Half-reaction cathode: PbO2+ 3H++ HSO4–+ 2e–−−⇀ ↽−− PbSO4+ 2H2O Overall reaction PbO2+ Pb + 2H++ 2HSO4–DCH −−−⇀ ↽−−− CHA 2PbSO4+ 2H2O The nominal voltage of this type of cell is typically around 2.5 Vat 25°C. During discharge, the electrolyte reacts with both electrodes, leading to the production of water. As the discharge progresses, the concentration of protons gradually decreases, creating a relationship between the density of the electrolyte or pH and the voltage of the battery [14]. Lead-acid battery chemistry faces three significant challenges: sulfation, relatively low energy density, and poor cyclability performance. Over time, efforts have been made to address these issues, resulting in various types of lead-acid batteries with improved performance and characteristics. Figure 3.2: Schematic illustration of a Lead-acid battery cell [15] 3.4.2 Nickel-Based Batteries Nickel-based batteries, which include Nickel-Cadmium (Ni −Cd) and Nickel-metal hydride (Ni −MH) batteries, have emerged as direct competitors to lead-acid batteries due to their similar characteristics, but with some differences. These batteries typically operate at a lower voltage, around 1.3 V. They offer a long cycle life and can function effectively across a wide temperature range [14]. 10 3.Electrochemical Energy Storage systems Ni-Cd Batteries Ni-Cd batteries consist of a positive nickel electrode, a negative cadmium electrode, and an alkaline electrolyte, typically a potassium hydroxide solution (KOH). Half-reaction anode: Cd + 2OH–−−⇀ ↽−− Cd(OH)2+ 2e– Half-reaction cathode: NiOOH + 2H2O + 2e–−−⇀ ↽−− 2Ni(OH)2+ 2OH– Overall reaction Cd + NiOOH + 2 H2ODCH −−−⇀ ↽−−− CHA Cd(OH)2+ 2Ni(OH)2 Ni-Cd batteries possess several advantages beyond those common to Ni-based batteries. They exhibit tolerance to deep discharges and overcharge operations and can be stored at any SOC without suffering irreversible damage. Another notable characteristic is the memory effect, which allows for the recovery of capacity through one or several cycles of complete charge and discharge processes [10]. Ni-MH Batteries Ni-MH batteries were developed for high-performance applications, featuring a significant difference from their predecessors in the form of a negative electrode composed of an alloy, which permits the exchange of hydrogen. Half-reaction anode: MH + OH–−−⇀ ↽−− M + H2O + e– Half-reaction cathode: NiOOH + H2O + e–−−⇀ ↽−− Ni(OH)2+ OH– Overall reaction MH + NiOOH DCH −−−⇀ ↽−−− CHA M + Ni(OH)2 Ni-MH batteries offer advantages such as high power density and the absence of cadmium, making them environmentally friendly. Additionally, the risk of dendritic short-circuits is minimal due to the low solubility of cell compounds in the electrolyte [10]. 3.4.3 Sodium-Sulfur Batteries In Sodium-Sulfur Batteries, high-temperature molten salt technology is employed, with the battery operating at temperatures around 300°C. The electrolyte is ceramic, and the negative electrode consists of molten sodium, while the positive electrode comprises molten sulfur. During the discharge, polysulfides are generated. As the battery discharges further and more energy is released, lower-order polysulfides are produced. Consequently, the stoichiometry of the reaction varies with the voltage [14]. Half-reaction anode: Na −−⇀ ↽−− Na++ e– Half-reaction cathode: nS + 2 Na++ 2e–−−⇀ ↽−− Na2Sn(4 ≥n≥8) Special containers thermally insulate the cells, requiring initial heating before becoming selfmaintaining. Due to the high reactivity of the sodium with water, these containers must shield Cell Modelling for State-of-Charge and State-of-Health Estimation 17 Another crucial element of the cell system is the interphase, a layer that forms during the first charging cycles. Referred to as the SEI, this thin layer is a thin layer composed of a combination of inorganic and organic compounds. These compounds are derived from the decomposition of electrolyte components. Comprising solid phases with electrolytic behavior, it is located between the electrode and electrolyte. The SEI plays a critical role in regulating the movement of Li+ions from the electrolyte to the electrode (ionic conduction), influencing the speed at which electrode materials can absorb lithium and with volume changes. This process impacts factors like cell ageing, electrode protection, and overall performance. Researchers sometimes differentiate between SEI for the anode and Cathode-Electrolyte Interface (CEI) for the cathode, with the former being more extensively studied [32] [33]. Figure 4.4: Evolution of SEI layer [34] 4.2 Cell Housing LICs have a hermetic encapsulation. The interior of the cell is not in direct contact with the external environment. Modern lithium-ion cells have housing and packaging mainly made of metal. Only metal are capable of stopping moisture from entering the cell (hydrolysis trigger of the conducting salt LiPF6into hydrogen fluoride (HF)) and preventing of solvent loss by diffusion [20]. Figure 4.5: Cell housing: (b) cylindrical cell; (c) prismatic cell; (d) coin cell; (e) pouch cell. [28] 18 4.State-of-the-Art of NMC Li-ion Cell The most common commercial LICs configurations used nowadays are coin, cylindrical, prismatic, and pouch (Figure 4.5). However, it is the last three that are used for large-capacity applications. Several electrode pairs are stacked together or one electrode pair is twisted and coiled, depending on the rated capacity. There are a few different methods to obtain the internal structure desired by adapting to the housing, as illustrated in the figure below: Figure 4.6: Inner structure of lithium-ion cell [20] In cylindrical cells, the external poles can be located on the same side or on opposite sides. A separator layer electrically isolates the electrode stack and the poles from the housing. On the one hand, it is possible to incorporate a safety vent and a current interrupter to prevent short circuits. In addition, its metal housing is quite robust making it durable. However, it has one significantdrawback, whichis the asymmetrictemperatureprofile with prominentgradients.As a result, internal resistance varies across the cell. In relation to prismatic cells, the electrodes are coiled (flat jelly roll) or layered (stacking) one on top of another after being split into many pairs by separator layers. Furthermore, collector lugs inside the cell housing connect the electrodes pair connections and they are electrically insulated. For the housing, steel or aluminium alloys are employed. There is less temperature gradient in the cell volume because the electrode stack is wider than it is thick. The reduced energy density of these cells is a disadvantage. The coffee bag design is another name for the pouch format. The electrode stack is layered like prismatic cells. In this case a plastic-laminated aluminum foil bag acts as the shell, being this one a advantage because of mass reduction. The heat capacity is lower and for that reason it is easier to cool the cell. This design is less robust, so the mechanical construction is an issue. Additionally, what are known as heat seal joints are used to seal the shell. These joints might serve less as barriers and permit the flow of moisture with the environment [20]. The cylindrical cells are characterised by having better energy densities. Prismatic and pouch cells are frequently employed due to their lower module-level dead volume and greater design flexibility. Prismatic and pouch-type batteries can also be easily customised for specific uses [28]. Cell Modelling for State-of-Charge and State-of-Health Estimation 19 4.3 Cell Ageing Every cell has an optimal operational window, and the Battery Management System (BMS) should be designed to ensure that the battery operates within this window at all times. Deviations from this optimal range can lead to various degradation mechanisms taking effect. These effects are categorized in three groups: Conductivity Loss (CL), Loss of Active Material (LAM) and Loss of Lithium Inventory (LLI). If there is a malfunction of the battery operates outside this window, different degradation mechanisms can be initiated. The degradation mechanisms specific to LICs are illustrated in the figure below: Figure 4.7: Representation of the Safe Operating Area for LIC [35] Even within the optimal range, control and management remain necessary to achieve peak performance. This is where the cell diagnosis and estimation of SOH play a crucial role to ensure the battery safety. Therefore, understanding the ageing mechanisms to prolong the lifespan of the battery is vital. As mentioned in Redondo-Iglesias et al. (2018) [36], ageing mechanisms can be affected by calendar or cycling ageing. Calendar ageing occurs when a cell is in a resting condition, while cycling ageing occurs during cycles of charge and discharge. These mechanisms lead to transformations within the cell components, degrading both the capacity and power capability of the battery. The effects can range from capacity fade, increment of the impedance, thermal runaway, among others. Regarding the anode, the key factor influencing ageing is the SEI formation. This phenomenon serves a dual role: it provides protection while also contributing to accelerated ageing due to the occurrence of lithium metal plating. Lithium plating, an undesired secondary reaction, consists of the deposition of lithium metal on the surface of the anode. This process can potentially lead to the creation of dendrites, which are structures that could pierce through the separator, posing safety risks within the cell. 20 4.State-of-the-Art of NMC Li-ion Cell Moving to cathode materials, they can experience a LAM due to structural changes during cycling, chemical decomposition or dissolution reactions, and modifications in surface films. Other factors that contribute to the ageing include degradation of contact with inactive components, metal dissolution, electrolyte decomposition, particle cracking and collector corrosion [37]. Figure 4.8: Cause and effect of degradation mechanisms [38] Cell Modelling for State-of-Charge and State-of-Health Estimation 21 5. Cell Characterization and Modelling Battery characterization refers to the comprehensive understanding of the battery behavior and performance. It involves conducting various tests, measurements, and analyses under different conditions. Gathering this data provides insights into the capabilities, limitations, and overall quality of the battery. This process is essential for developing accurate battery models that enable the estimation of SOC and SOH, which are crucial for effective battery management and prediction of remaining battery life. 5.1 Definitions Certain terms are essential to clarify as they have a relevance in battery testing, characterization, analyzing performance and lifecycle evaluation. •The C-rate is a measure used to express the rate of current flow in a cell. It quantifies how fast a battery is charged or discharged normalized to its nominal capacity (CN). C-rate =Current CN =1 time (5.1) In the case of a 2 Ah battery cell, discharging at a 2C rate would mean a current of 4 A. This discharge would take 0.5 hours to complete. •Charge and discharge procedures can be carried out by means of two fundamental methods: –Constant Current (CC) is a charging or discharging process where the current rate is constant. –Constant Voltage (CV) is the mode where the battery or cell voltage remains constant while the current gradually changes in order to maintain the voltage level. Figure 5.1: Charging procedure [39] 22 5.Cell Characterization and Modelling •SOC is the available capacity (C(t)) related of the rated capacity (CN). SOC =C(t) CN (5.2) When considering the current (i), SOC can be also calculated considering the Coulomb efficiency (η) with the following equation [40]: SOC =SOC0−1 CNZt 0 ηi(t)dt (5.3) •Depth of Discharge (DOD) refers to the proportion (Cd) of the total capacity (CN) that has been discharged from a fully charged battery [41]. DOD =Cd CN (5.4) This parameter is often expressed in relation to SOC: DOD = 1 −SOC (5.5) •SOH is the ratio between the capacity of a battery or cell at a specific time (CN(t)) and its initial rated capacity (CN(t0)). When SOH is less than 100%, it indicates that the capacity of the battery has degraded over time. SOH =CN(t) CN(t0)(5.6) According to Wang et al. ([42], it is feasible to express the (SOH) as a function of resistance: SOH =REOL −Rcur REOL −Rnew (5.7) In the provided equation, Rcur represents the current internal resistance, while Rnew and REOL denote the internal resistance values of a new cell or battery and an EOL cell, respectively. •End of Life (EOL) refers to the stage where the performance of the battery has substantially declined, and its capacity is no longer adequate for the intended application or purpose. 5.2 Battery Characterization Techniques This section explores various methods employed to comprehend cell behavior and translate it into a model. Cell Modelling for State-of-Charge and State-of-Health Estimation 23 5.2.1 Capacity Test A capacity test is a procedure, which helps to monitor both the SOC and the SOH of a battery cell. This test requires the cell to be subjected to specific test designed to quantify its charge storage capacity under a defined C-rate and temperature. Figure 5.2: Typical capacity discharge test profile at different temperatures and various C-rates. A - 1C. B - 2C [43] . The key steps in a capacity test typically include: •Understanding cell condition: Sometimes it is not possible to determine whether the cell is completely charged or not. In such cases, two options are typically considered: –Precharge: This involves charging the cell to a known SOC (100%) to establish a consistent starting point for the subsequent test steps. –Remaining capacity test: This involves testing the capacity of the cell from its current SOC to Umin. However, this test is primarily informative and is followed by a full recharge of the cell. •Discharge: This step consists of discharging the cell under CC until it reaches Umin (Figure 5.2). •Charge: the typical procedure involves applying CC until Umax is reached. Following this, the charging mode shifts to CV to maintain the charge, continuing until a minimum current rate is achieved (Figure 5.1). Between these test phases, it is common to include rest periods that allow the voltage of the cell to stabilize. 5.2.2 OCV-SOC curve Another essential parameter required for modelling a battery cell is the OCV-SOC curve. The OCV represents the voltage of the cell when there is no current flow through it. This curve 24 5.Cell Characterization and Modelling basically represents the relationship between OCV and SOC. It is important to know that the OCV-SOC curve changes over time because of cell ageing, so the degradation should be also considered [44]. The low-rate current method and the static method are typically the accepted techniques for solving the OCV-SOC curve. These methods take into account the hysteresis effect that can be observed in battery cells, where the OCV-SOC curve is not symmetric during charge and discharge processes. The low-rate current method requires charging or discharging the cell from 100% of SOC until Umin (considered 0% SOC). On the other hand, the static approach depends on the cell being at rest for a long time (often more than 3 hours) after achieving different SOC levels [45]. Figure 5.3: Methods for obtaining OCV-SOC curve 5.2.3 Temperature Effects on Battery Behavior Temperature has a important effect on the behavior of battery cells. For LIBs, there exists an operational temperature window that is typically considered safe, from -20°C to 60°C. Outside this temperature range, the degradation mechanisms become more pronounced, significantly increasing the risk of safety issues. At lower temperatures, LIBs chemical activity and charge-transfer velocity decrease. This slowdown extends to the ionic conductivity in the electrolytes and the lithium-ion diffusivity within the electrodes. These effects can lead to a decrease in energy and power capacity, and in some cases, failure. On the other hand, high temperatures can also negatively affect the performance and longevity of batteries. Some of the effects oof the degradation at high temperature is loss of capacity and power. LLI and LAM at high temperature contribute to decreased the capacity and increased internal resistance, leading to power loss. Moreover, when thermal runaway occurs in a battery operating at high temperatures, self-ignition or even explosion can be the results [46]. Effective thermal management and cooling strategies are critical in for this kind of batteries. Cell Modelling for State-of-Charge and State-of-Health Estimation 25 5.2.4 Hybrid Pulse Characterization HPPC test, developed as part of the Partnership for a New Generation of Vehicles (PNGV) program, is designed to evaluate the dynamic power capabilities of a battery or cell within its usable charge and voltage range. This test involves a series of discharging and charging pulses and aims to evaluate how a battery performs in terms of power output and recovery, particularly under various charge and voltage conditions [47]. The duration and relaxation time of these pulses can be adjusted to meet specific modelling requirements or to simulate real-world scenarios more accurately. Figure 5.4: Pulse text at determined SOC During the discharging and charging pulses of the HPPC test, there is a noticeable voltage drop, which is primarily attributed to the internal resistance (R0) of the cell. Figure 5.5: Voltage DCH pulse plot For instance, Figure 5.5 represent the voltage behavior during the discharging pulse. The voltage drop from V0to V1and the voltage increase from V2to V3are indicative of the internal 26 5.Cell Characterization and Modelling resistance impact on the voltage profile. Therefore it is possible to have a approximation of this resistance considering the current Ithrough the following equation [48]: R0=(V0−V1)+(V3−V2) 2I(5.8) However, this method may introduce some errors, because normally the data acquisition speed of the test is relatively slow to have a exact value. This test is executed using direct current Direct Current (DC) pulses. Typically, the test profiles consist of a sequence comprising a discharge pulse, followed by a relaxation time, a charging pulse, and another relaxation time. Each of these phases is defined by specific pulse current magnitude and duration. It is important to keep the duration within reasonable limits to avoid significant disruptions to the SOC [49]. Collecting voltage response data from the HPPC test enables the development of an ECM. This model aids in evaluating the condition and performance of the cell throughout its operational life [50]. 5.2.5 Electrochemical Impedance Spectroscopy EIS is a non-destructive technique, which is useful for systems with numerous impedance elements. It is a matter of applying a small sinusoidal perturbation to the system while measuring the resulting cell response. The impedance can be evaluated by changing the Alternating Current (AC) frequency fbetween 10mHz and 100 kHz. The perturbation can either be galvanostatic, utilizing current, or potentiostatic, employing voltage. The impedance Zcan be calculated through the relation between the voltage and the current when AC voltage is applied. Incorporating the amplitudes of voltage and current ( ˆ Uand ˆ I, respectively), the angular frequency ω, the phase shift θthe following equation is formulated [51]: Z=u(t) i(t)=ˆ U·sin(ωt) ˆ I·cos(ωt −θ)=|Z| · sin(ωt) cos(ωt −θ)(1) Finally, it is possible to obtain the real Zre and imaginary part Zim from the impedance: Zre =|Z| · cos(θ)(2) Zim =|Z| · sin(θ)(3) The Nyquist plot is generated using this two parameters, with the real part of the impedance Zre plotted on the x-axis and the negative of the imaginary part of the impedance −Zim on the y-axis. Cell Modelling for State-of-Charge and State-of-Health Estimation 33 5.3.2 Electrical Circuit based Model In essence, a LIC is constructed from several integral components: a cathode, an anode, a separator, current collectors, and an electrolyte. The movement of electrons and lithium ions occurs within each of these components, each characterized by its own electrical attributes. Consequently, these elements of the battery cell, together with the double layer formed at their interfaces, can be represented by an equivalent circuit. As illustrated in Figure 5.12, considering all the individual elements would lead to an equivalent circuit that demands a significant amount of computational resources. The objective is to use an optimal ECM that is physically relevant and reduces the number of variables to a minimum. This is because a larger number of variables adds more complexity to the model, which in turn increases the risk of errors and inaccuracies. Figure 5.12: Entire circuit model of a lithium-ion battery [57] These models utilize a mix of voltage and current sources, resistors, capacitors, and other components. The most basic arrangement involves just an OCV element and internal resistance (R0). Figure 5.13: Simple ECM Moving beyond this initial but somewhat imprecise model, the ECM can be made as intricate as needed. The Thevenin-based model expands its scope by incorporating supplementary components, such as parallel resistor-capacitor modules, to replicate a response of the battery to load 34 5.Cell Characterization and Modelling variations at a particular SOC, representing the mass transport and double layer effect [42]. Additionally, the option of including a resistor in parallel with the OCV allows for the simulation of self-discharge phenomena. Figure 5.14: Thevenin-based ECM On the other hand, the Impedance model, which combines the previous model with impedance element. These models find extensive application in conjunction with EIS data. However, it is important to note that Impedance models lack the ability to predict DC responses or estimate battery lifespan, as they are limited to constant SOC and temperature conditions [68]. Figure 5.15: EIS-based ECM In conclusion, equivalent circuit models offer advantages over electrochemical models due to their feasibility for real-time implementation and simplified nature. However, it is important to note that equivalent circuit models are more susceptible to errors as they rely on numerous approximations. 5.3.3 Alternative Models A wide array of models have been developed to meet various challenges. With the passage of time, several alternative battery models have surfaced, offering fresh perspectives on battery behavior. These include stochastic models, transmission line models, and more prominently, data-driven methods [65]. The latter, which heavily incorporates machine learning, has become a prevalent trend in battery modelling. These methods utilize monitoring data to create predictive models that capture degradation relationships. While these approaches are easier to put into practice, they necessitate a sufficient amount of training data and meticulous validation to establish their suitability and robustness for particular applications. Cell Modelling for State-of-Charge and State-of-Health Estimation 35 5.4 Model Selection and Estimation Process This section delves into the review of selected models for SOC and SOH estimation, covering the methods of EIS and HPPC. In addition, this chapter will provide information on the development of models. Each of these methods has unique characteristics that make them suitable for certain applications, while posing challenges for others. For example, the HPPC method excels at determining the dynamic power capabilities of a battery cell by incorporating charge and discharge pulses into test profile. On the other hand, EIS takes advantage of the fact that as battery capacity degrades, impedance increases, resulting in distinct dynamics in various frequency ranges in the test. Although EIS proves to be a valuable tool for ageing and fault detection, it is important to note that these experimental methods are currently restricted to off-line identification of SOH and SOC. 5.4.1 Equivalent Circuit Model for HPPC The chosen ECM is commonly referred to as the 2RC circuit. It consists of an OCV element, an ohmic or internal resistance R0, and two parallel combinations of a resistor and capacitor (R1,R2,C1, and C2) connected in series. This selection is based on its extensive use [69, 70]. Moreover, Figure 5.16, created by Fotouhi et al. (2017) [71], supports this choice. The figure illustrates the normalised voltage prediction error and time taken by the different models. On the basis of this assessment, the 2RC model appears to be a more suitable candidate for the 2RC model. Figure 5.16: Normalized cell voltage prediction error and time using different models. [71] In fact, the incorporation of these specific components in the model is due to the intention to reproduce the characteristic shape, which can be observed in the graphical representation of the charging and discharging current pulse behavior (Figure 5.17). For this modeland duetothe methodology appliedfor estimating the SOC, the modelequations mustberepresentedin discrete form. All equationsare based onthe workof Campestrini(2017) 36 5.Cell Characterization and Modelling Figure 5.17: Relationship between the cell response and the elements from ECM. Based on [71] [72] and Wang et al. (2022) [48]. By applying Kirchhoff’s voltage and current law to the circuit elements, the voltage Ucell and the current Ican be expressed as follows: Ucell =UOCV +U0+U1+U2(5.20) I(t) = iCx(t) + iRx(t)(5.21) Being xone of the two parallel resistor-capacitor compounds. The current, which flows through any capacitor iCxand resistor iRxcan be expressed as: iCx(t) = Cx·dUx(t) dt =Cx·˙ Ux(5.22) iRx(t) = Ux(t) Rx (5.23) Substituting the previous current equation and clearing the derivative of the voltage Ux: ˙ Ux=−1 Rx·Cx ·Ux(t) + 1 Cx ·I(t)(5.24) The SOC value is possible to calculate as follow: ˙ SOC =1 CN ·I(t)(5.25) With these equation, it is possible to form a system with the following general state-space notation: ˙x(t) = Atx(t) + Btu(t)(5.26) Cell Modelling for State-of-Charge and State-of-Health Estimation 37 y(t) = Htx(t) + Dtu(t)(5.27) Considering that x(t)is the voltage and u(t)the current, the equation for 2RC model in continuous state-space notation results:    ˙ U1(t) ˙ U2(t) ˙ SOC(t)   =    −1 R1C10 0 0−1 R2C20 0 0 0     ·   U1(t) U2(t) SOC(t)   +   1/C1 1/C1 1/CN   I(t)(5.28) U(t) = h1 1 UOC SOC(t)i   U1(t) U2(t) SOC(t)   +R0I(t)(5.29) The equation can be transformed into discrete mode, with all parameters depending on discrete quantities k, resulting in the following general equations: ˙xx+1 =Akxk+Bkuk(5.30) yk=Hkxk+Dkuk(5.31) After applying the Laplace transformation to Atto obtain the fundamental matrix Ak, and integrating Btand multiplying it by the fundamental matrix to obtain the transition matrix Bk, the resulting system equation is as follows: xk+1 =     e−∆t R1C10 0 0e−∆t R2C20 0 0 1      ·xk+     R1(1 −e−∆t R1C1) R1(1 −e−∆t R2C2) ∆t Cact      uk(5.32) yk=1 1 UOC(xn+1,k) xn+1,k huki+R0uk(5.33) Where ∆tis the sample time, ktime step and xk= [U1, U2, SOCt]t. This system of equations can be written in terms of the value of the time constant (τ), knowing that: τx=RxCx(5.34) Estimation SOH To estimate the SOH of the battery cell, the process commences with the voltage and R0. The initial voltage value provides information about SOC since all tests start after the cell has rested for some time. By referring to the OCV-SOC curve, we can determine the SOC at the beginning of the test. The estimation of R0can be done using the formula 5.8. Through parameter estimation, a dataset is available to link 3 parameters: SOC, R0and SOH. When both SOC and SOH are available, the estimation of SOH becomes achievable using the MATLAB function ’fitlm’, which is employed for linear regression modelling. This method has demonstrated effectiveness in estimating the SOH parameter with the data available. 38 5.Cell Characterization and Modelling Estimation SOC through Kalman Filter The estimation of the SOC was performed using Kalman Filter (KF). This method is a prediction-correction method designed to optimally estimate the state of a linear dynamic system, even when there are measurement noises and uncertainties. Specifically, the method applied has been EKF a variant designed for estimating the state of nonlinear dynamic systems. The following steps, as outlined in Campestrini (2017) [72], describe the KF procedure: 1. Initial estimates: The process commences with the establishment of initial parameters for ˆx0and P0. 2. Prediction: The predicted state x− kand error covariance P− kat time kis given by: x− k=Ak−1x+ k−1+Bk−1uk−1(5.35) P− k=Ak−1P+ k−1AT k−1+Q(5.36) 3. Correction: This step is about the measurement update. Calculating the Kalman Gain K, it is possible to update the state x+ kand yk, and the error covariance P+ k. yk=Hkˆx− k+Dkuk(5.37) K=P− kHT k(HkP− kHT k+r)−1(5.38) x+ k=x− k+Kk(Uk−yk)(5.39) P+ k= (I−KkHk)P− k(5.40) In summary, the schematic figure below illustrates these steps. Figure 5.18: Complete picture of the Kalman Filter operation. Based on [73] 5.4.2 Equivalent Circuit Model based on Impedance As highlighted in Chapter 5.2.5, the Nyquist plot illustrates different regions, each characterized by specific behaviors that contain important information. These behaviors can be effectively translated into circuit elements, forming the basis for constructing an ECM. Cell Modelling for State-of-Charge and State-of-Health Estimation 39 For the purpose of creating a precise circuit model, it is essential to employ appropriate electrical components that effectively describe the system. In this report, the selection of the ECM is influenced by the study conducted by Eddahech et al. (2011) [74]. The authors noted that the ECM they employed obtained accurate results. The only modification made for this model involves the addition of a new element to account for behavior in the high-frequency region [75]. The proposed model comprises four components that are connected in series and they are related to a zone from Nyquist plot. Figure 5.19: Relationship between ECM elements used for the model (above) and Nyquist plot (below) zones. Based on [76] In order to build the ECM based on impedance results, it is essential to determine the total impedance Zt, which is the sum of all elements in series: Zt=ZL||RL+ZR0+ZR1||C1+ZR2||C2(5.41) Each element and its equation are described as follows: •Parallel inductor Land resistor RLcircuit: This configuration represents the inductance behaviour observed in the high-frequency region of the Nyquist plot. The impedance from this circuit follows the next equation: ZL||RL=R·ω2·L2 R2+ω2·L2+R2·ω·L R2+ω2·L2·i(5.42) •Ohmic resistance or internal resistance of the battery cell R0is represented by one resistor: ZR0=R0(5.43) •Two parallel resistor and capacitor circuit (Rx|| Cx): These components replicate the charge-transfer resistance and the double-layer capacitances, following the next expression [75]: ZRx||Cx=R 1 + R2·ω2·C2−i·R2·ω·C 1 + R2·ω2·C2(5.44) 40 5.Cell Characterization and Modelling •Warburg impedance ZW: this element is added in order to consider the diffusion phenomena in the low-frequency region. In this case the Warburg coefficient (σ) will be the resposible to fit this part. ZW=1 σW·(i·ω)0.5(5.45) Angular frequency ωcan be calculated from frequency (ω= 2 ·π·f). Estimation SOH and SOC To estimate the SOH and SOC of the battery cell, a method similar to the previous model was employed, utilizing the MATLAB function ’fitlm’ for linear regression modelling. By employing a dataset obtained from the parameter fitting and their relationship with SOH and SOC, this approach proves to be a good choice estimation. For data preparation, ECM parameters serve as inputs, while SOC and SOH values are target variables. The dataset is divided into training and testing subsets using a Cross-validation method (crossvalind). The estimation of SOH and SOC was carried out separately, one linear regression for each one. Even when working with a limited dataset, this MATLAB function demonstrates great performance in predicting SOH and SOC. This is particularly noteworthy considering the relatively narrow working range of SOH (from 100% to 94.8%) and the considerably wide intervals of SOC (at each 20%) between tests. Cell Modelling for State-of-Charge and State-of-Health Estimation 41 6. Test Procedure Analysis and Results In this chapter, an analysis of the testing is presented, focusing on the employed testing methodology to model and the derived results. The goal is to provide a deeper understanding of the performance of the selected LIC. It begins by introducing the chosen LIC and explaining in the test procedure used for data collection along with some insights extracted from the collected data. This chapter serves as a comprehensive guide to the research approach and provides an insight into the experimental process. 6.1 Studied Li-ion Cell: LG Chem INR21700 M50 The focus of this study is on the INR21700 M50 cell, a cylindrical LIC designed and manufactured by LG Chem, whose cathode is based on NMC chemistry. Figure 6.1: Photographs of LG Chem INR21700 M50 cell housing [77] The specifications, as provided in the cell datasheet, are detailed in Table 6.1. 42 6.Test Procedure Analysis and Results Table 6.1: Specifications of LG Chem Li-ion INR21700 M50 Cell [77] Item Value Cathode Material Li(Ni0.84Co0.10Mn0.06)O2NMC Anode Material Graphite with silica particles (graphite-SiOx) Weight 68 ±1 g Cell Dimension ≤21.10 mm diameter, ≤70.15 mm height Nominal Energy 18.20 Wh Nominal Voltage 3.63 V Maximum Voltage 4.2 ±0.05 V Minimum Voltage 2.5 V Maximum Charge Current 0.3C (0 25 °C) and 0.7C (25 50 °C) Maximum Discharge Current 0.2C (-30 -20 °C), 0.3C (-20 5 °C) and 1.5C (5 60 °C) Initial DC Resistance 30 ±6 mΩ 6.2 Test Procedure and Analysis The test procedure was designed to collect the essential data needed to analyse cell performance andsubsequentmodel development. Four samples were employed for this study, all maintained at a constant temperature of 25 °C. The emphasis in this report is in ageing modelling rather than temperature modelling. This process is intended to ensure the acquisition of accurate and reliable data, facilitating the construction of the models. The procedure involves several steps to ensure the quality of the information collected. 6.2.1 Test Plan The test plan has been designed taking into consideration the time spent in TTZ-EMO, resources available and modelling prerequisites. It consists of two consecutive cycles and each is composed of the following stages: 1. Reference Parameter Test (RPT) 2. WLTP 3. Cycling As illustrated in the Figure 6.2, the four cells followed almost identical procedures. The testing sequence began with an initial RPT, during which two out of the four cells additionally underwent an EIS. Subsequently, a WLTP and cycling were conducted, with the latter responsible for inducing the ageing process. To assess the cell status, a follow-up RPT was executed, followed by another round of WLTP and cycling. In particular, the final RPT mirrored the initial one, involving EIS measurements on the same set of cells. Cell Modelling for State-of-Charge and State-of-Health Estimation 49 6.2.4 Cycling and Cell Ageing The battery cells were tested through different conditions to analyse their degradation patterns. The four cells underwent through different discharge and charge rate and DOD as indicated by Table 6.8. Table 6.8: Cycling characteristics of each cell. Cell C-rate DOD Cell 1 1C 80% Cell 2 0.5C 60% Cell 3 0.5C 40% Cell 4 1C 40% Therefore, the test procedure applied for the ageing process follows the detailed structure below: Table 6.9: Generic cycling test procedure Step Instruction Parameter Abort criterion 1 DCH I = XCaAh = Y%b·QN U < 2.5 V 2 PAU t > 15 min 3 CHA I = XCcU > 4.2 V U = 4.2 V I < 0.020 A 4 PAU t > 15 min aX represents the C-rate related each cell bY is the DOD related each cell cThe C-rate is limited to 0.7C for charging Due to the effects of ageing mechanisms, there is a progressive increase in voltage drop across the cycles. Naturally, the extent of this voltage drop becomes more pronounced with higher DOD values and C-rates (Figure 6.10 and Table 6.10). 50 6.Test Procedure Analysis and Results Figure 6.10: Voltage drop of the DCH step cycle Table 6.10: Voltage drop from the first to final cycle. Cell Voltage drop (V) Cell 1 0.066 Cell 2 0.019 Cell 3 0.007 Cell 4 0.014 In Figure 6.10, eachcurve correspondstocombinations ofDODsandC-rates. The lower position of the curve for cell 1 can be attributed to its higher DOD and C-rate. This ageing phenomenon is not only evident during cycles but also becomes apparent during the resting the tests. •Capacity fade: Similar to the observed voltage drop, capacity fade appears to be more pronounced in cell 1, resulting in the lowest achieved SOH. Table 6.11: Capacity fade results from the cells LG Chem INR21700 m50 after 100 and 200 cycles Capacity (Ah) Cycles Cell 1 Cell 2 Cell 3 Cell 4 0 4.9563 4.9595 4.9777 4.9453 100 4.7815 4.8106 4.8684 4.8258 200 4.7010 4.7320 4.8073 4.7604 Cell Modelling for State-of-Charge and State-of-Health Estimation 51 Figure 6.11: Capacity fade across cycles •Changes in OCV-SOC curve: In Figure 6.12, the curves represent changes in cell 1. As can be observed, the changes are minimal, indicating that the cell is not ageing significantly enough to exhibit a clear alteration. Therefore, during the modelling process, the applied curve will remain at 100% SOC. Figure 6.12: OCV-SOC curves changes from 100% and 94.8% SOH •Changes EIS: At 40% SOC, the curve shifts to the right, while at 100%, the most significant change is observed in the diffusion zone. Meanwhile, at 0% SOC, the two semicircles become more distinct (Figure 6.13. 52 6.Test Procedure Analysis and Results Figure 6.13: Nyquist plots comparison at different SOH Following this procedure, the data acquired did not give the desired results. Despite having designed this procedure, the observed capacity fade is lower than expected. However, this result speaks positively of the performance of the cell. In the worst-case scenario, after 200 cycles (at 1C and 80% DOD), the cell has only experienced a 5.2% decrease in capacity. Cell Modelling for State-of-Charge and State-of-Health Estimation 53 7. Models Analysis and Results This chapter presents the results of the modelling based on HPPC and EIS. Both methods have been developed with MATLAB following the model equations detailed in Section 5.4. This enables the resolution of this system of equations using MATLAB, specifically the optimization technique used is nonlinear least-squares data-fitting method (lsqnonlin). It allows for accurately fitting the data to the established model. When performing optimization for parameter fitting for a ECM, the selection of appropriate upper and lower bounds, as well as initial guesses for the parameters, is crucial. In this context, the software Echem Analyst developed by Gamry Instruments and the work of Pietro(2022) [78] play significant roles. The last part is an analysis based on the two differential methods: DVA and ICA. According to Li (2018) [61], this method is superior to experimental and adaptive modelling techniques due to its computational efficiency and its ability to identify battery cell degradation. 7.1 SOC and SOH Estimation using HPPC This model was developed by creating an ECM adapted to the dynamic response of the HPPC. In addition, the capacity measurements and OCV-SOC curve are essential to properly characterise the behaviour of the battery cell within the model (Figure 7.1). Figure 7.1: Test data results requirements for modelling through HPPC 7.1.1 Results To initiate the modeling process, the test has been divided into different segments, each classified by SOC and SOH. Each of these segments represents the response of the cell to the test under different conditions. By applying the code based on discrete cell equations and optimization tools, it is possible to obtain all parameters and variations depending on the two variables mentioned above (Figure 7.2). The selection of these profiles with charge and discharge pulses was made because, despite some parts of the test where the model does not fit perfectly (normally the relaxation time), the 54 7.Models Analysis and Results Figure 7.2: Plot comparison between the data from HPPC and the model at different SOC and SOH parameters obtained are capable of being suitable for a wide range of test conditions. With this step, it is possible to understand how the parameters from ECM evolve with SOC and SOH: (a) Resistance values R0(b) Resistance values R1 (c) Time constant values τ1(d) Resistance values R2 (e) Time constant values τ2 Figure 7.3: Parameters variation from HPPC ECM related to SOC and SOH Cell Modelling for State-of-Charge and State-of-Health Estimation 55 The only parameter that distinctly differentiates with respect to SOC is the internal resistance R0. In contrast, for the other parameters, even though their curves share similar profiles on the same plot, there is no discernible trend or significant differentiation based on SOC. Regarding SOC, it appears that the parameter exhibits significant differentiation between 10% and 100% SOC values when compared to the rest of the range. After applying EKF and use the entire test sequence to check the response, the following plots are obtained: (a) (b) (c) (d) Figure 7.4: Results from ECM based on HPPC. (a) Estimated vs measurement voltage, (b) Error voltage, (c) Coulomb counting vs EKF SOC estimation, (d) SOC error The estimated SOH is a value of 0.98, from a new cell. The maximum error observed is 0.3 V at the end of the test. This discrepancy is not unexpected due to safety considerations. Specifically, the final pulse was limited to 10% SOC, and extrapolation under such conditions may introduce inaccuracies. In case of using the data from WLTP: 56 7.Models Analysis and Results (a) (b) (c) (d) Figure 7.5: Results from WLTP. (a) Estimated vs measurement voltage, (b) Error voltage, (c) Coulomb counting vs EKF SOC estimation, (d) SOC error In this scenario, a similar results are shown, where the higher error comes from last stage of the test. This is the data from a new cell and the estimated SOH is 0.99, so the linear regression has performed well in this specific test. Using EKF has proven to be a valuable tool. Due to its corrective nature, it contributes to data smoothing and filtering, reducing errors. It is beneficial for tasks such as interpolation and other forms of estimation. 7.2 SOC and SOH Estimation using EIS Estimating the SOC and SOH of a battery cell using EIS response involves employing ECM. This method allows us to gain insights into the electrochemical behavior status of the battery cell by interpreting the impedance data obtained through EIS measurements. By fitting the impedance data to the circuit elements of the ECM, we can extract information about the different components. For this purpose, additionally to EIS test data, the capacity information is also required (Figure 7.6) Cell Modelling for State-of-Charge and State-of-Health Estimation 57 Figure 7.6: Test data results requirements for modelling through EIS 7.2.1 Results As mentioned in Section 6.2.2, the EIS test was conducted across various SOC levels, both at the beginning and the conclusion of the complete testing procedure. This approach resulted in the acquisition of a dataset that encompasses information relating to both SOC and the SOH of the battery cell. By analyzing the impedance data obtained at different SOC levels and comparing them with the corresponding SOH, it is possible to identify patterns that permits the estimation of both parameters. By implementing the created code for this specific model, a dataset can be built, associating the value of each component of the circuit with its corresponding SOH and SOH. Consequently, for each instance of the EIS test carried out, the code will generate the resultant curve (Figure 7.7) alongside the computed parameters. Figure 7.7: Nyquist plot comparison between the data from EIS and the model at different SOC and SOH Indeed, the behavior of inductance and diffusion areas appears to present the most significant challenges when it comes to modeling. The impact of these challenges is relatively manageable since the estimation of both SOH and SOC does not have a strong influence these specific regions. However, it isimportant to note that incaseswhere amorein-depthstudyofdiffusionbehaviour is required, it is crucial to achieve a better fit. In such cases, further tuning of the model is essential to obtain accurate results. This highlights the need to adapt the complexity and parameters of the model to fit the specific aspects, ensuring a robust representation of the processes. So at the end it is possible to obtain the value of different parameters and build a dataset, according to the different test conditions. The evolution of this parameter is illustrated in Figure 58 7.Models Analysis and Results 7.8. As observed, the parallel R-L circuit does change so much, without relevant information in terms of SOH and SOC estimation, which made it irrelevant for this evaluation as indicated by Li(2021)[54] (a) Inductance values L(b) Resistance values RL (c) Resistance values R0(d) Diffusion coefficient σW (e) Resistance values R1(f) Capacitance values C1 (g) Resistance values R2(h) Capacitance values C2 Figure 7.8: Parameters variation from EIS ECM related to SOC and SOH It is evident that some of the parameters show similar shapes in the plotted data. In particular, Cell Modelling for State-of-Charge and State-of-Health Estimation 65 10. Environmental Assessment Working towards a sustainable future through the use of LIBs may seem attractive in theory; however, it is imperative to recognise that this technology is not without environmental repercussions. Some key considerations include: •Mining: LIBs account for approximately 40% of the global demand for Critical Raw Materials (CRMs) like lithium and cobalt [87]. The rising demand has led to an increase in mining activities, often resulting in habitat disturbance, soil and water pollution and other environmental problems. •Energy Intensive: Both the mining of raw materials and the manufacturing of LIBs are energy-intensive processes. This high energy demand contributes significantly to greenhouse gas emissions, stressing the need for more energy-efficient production methods [88]. •Transportation: Transporting LIBs involves the movement over long distances. •EOL: Some batteries may still have a portion of their capacity remaining and can be repurposed for less demanding applications, such as stationary energy storage. Normally, EVs batteries until 80% and 60% for stationary [88]. Extending the useful life of batteries before recycling them can improve overall sustainability. Addressing these issues is crucial to mitigate the environmental impacts associated with LIBs and work towards a more sustainable energy future. •Sustainable Sourcing: Responsible sourcing of raw materials, ethical mining practices, and efforts to minimize the environmental impact of extraction can reduce the negative consequences of mining activities. •Energy Efficiency: Implementing energy-efficient manufacturing processes can help reduce the carbon footprint of battery production. Innovations in electrode coating, drying, and assembly methods are examples of areas where energy efficiency can be improved. •Recycling and Reuse: Recycling cathode materials, such as cobalt, nickel, and lithium, can significantly reduce greenhouse gas emissions and secure access to CRMs. •Research and Innovation: Continued research and innovation are vital for developing greener battery technologies and recycling methods. However, it is imperative to emphasize that sustainability should be also a central focus, extending beyond battery technology itself. [88]. In conclusion, while Li-ion batteries are central to a more sustainable energy future, addressing the environmental challenges they present, requires a multifaceted approach. Sustainable sourcing, energy efficiency, recycling, and ongoing innovation are key pillars in ensuring that Li-ion technology contributes positively to the environmental goals. 66 11.Social and Gender Equality Assessment 11. Social and Gender Equality Assessment The battery sector, like the engineering sector, has indeed faced challenges when it comes to gender and social equality. Historically, women have been absent from technical and leadership positions in the industry. Although there has been some improvement in increasing the presence of women, true gender equality has not yet been fully achieved. A recent study by Tsagkari(2022) [89] emphasizes the importance of considering gender issues throughout the entire lifecycle of energy projects, from planning to implementation. Despite an increase in the number of women involved in new projects during this energy transition, companies and organizations must pay more attention to gender equality to avoid a return to a male-dominated energy sector. The study highlights the crucial role that women can play as agents of change in the energy industry. It also draws attention to their underrepresentation in energy communities and the workforce. By addressing gender inequalities and encouraging an inclusive environment, the battery industry can take advantage of the diverse talents and perspectives of both men and women to drive innovation and progress in the field. Regarding social equality, remains a persistent challenge. In countries where social welfare is a top priority, cases of social injustice tend to be less frequent. However, a significant concern ariseswhenweconsiderthatmany componentsofbatteriesoriginatefromcountrieswhere individual rights may not be given importance. Mining exploitation required for these components often entails significant risks, some of which we may not be fully aware of (Figure 11.1). Figure 11.1: Supply of raw materials in 2020 [90] It is essential to carefully monitor each stage of battery production, paying particular attention to Social Life Cycle Assessment (LCA). This approach allows to understand of how each project impacts health and well-being, as well as its social implications. Within the battery industry, a comprehensive Social LCA can illuminate the social effects of elements extraction and battery manufacturing. Cell Modelling for State-of-Charge and State-of-Health Estimation 67 In summary, achieving gender and social equality in the battery sector is not only a matter of ethics but also essential for a sustainable and responsible approach to energy storage. As the industry evolves, it must prioritize gender and social considerations to create a more equitable and environmentally responsible future. 68 Conclusions Conclusions In conclusion, this research has provided valuable insights into the performance and degradation of LIBs, particularly focusing on the LG Chem INR21700 M50 cell. Considering the extensive literature review covering various secondary batteries, with a deeper immersion into Li-ion technology, specifically the NMC cell variant, it becomes evident that, despite continuous innovations in battery materials, the current emphasis lies in modelling and performance enhancements. A comprehensive test procedure was conducted to explore various methods for estimating SOC and SOH. Through an in-depth analysis of various testing techniques, including HPPC, EIS, DVA, and ICA, a understanding of how these batteries respond under diverse conditions was gained. Notably, the performance of the cell exceeded expectations, with a minimal capacity fade, which carries promising implications for the manufacturer, LG Chem. A significant contribution of this study is the development of an ECM tailored specifically for dynamic HPPC responses and EIS data. By fitting these models to experimental data, the effectiveness in capturing the dynamic behavior or the impedance of Li-ion batteries has been demonstrated. These modeling approaches provide practical tools for predicting battery performance and degradation across a range of operating conditions. In case of the model based on HPPC, using EKF has been a successful choice as it helps to minimise errors and obtain a good approximation. For the model based on EIS, semicircles have been the better fit. In case of further investigation of another aspect such as diffusion, this model should be reworked to obtain a better overall fit. However, it is crucial to acknowledge the limitations of this research. While the developed ECM offers a valuable framework, it does involve simplifications and assumptions. Further improvements and validation efforts are necessary to enhance its accuracy and reliability. Moreover, additional testing is required to obtain a more extensive dataset, which would enable the construction of a robust model applicable to a broader range of conditions. Similarly, the data obtained from the DAT point to the need for further study to delve deeper into the aging mechanisms. Future Work In the growing field of battery research, the abundance of models and tests opens up many exciting possibilities. Among them, thermal models are very promising. It is undeniable that in order to fully understand the performance of a battery, it is essential to take temperature into account. Another important area for future exploration is data acquisition. The creation of a larger dataset is essential, as it contributes to the development of highly accurate models. The quest Cell Modelling for State-of-Charge and State-of-Health Estimation 69 for greater accuracy through the accumulation of more data remains a central objective. Additionally, it is worth considering the integration of Artificial Intelligence (AI) and ML techniques into battery research. These technologies have the potential to analyze vast amounts of data and discover patterns and insights that might not be immediately apparent through traditional methods. Finally, an exciting area is the development of models for DVA and ICA. The prospect of unlocking the mechanisms of degradation within the cell through the charging process alone is impressive. 70 Acknowledgement Acknowledgement First and foremost, I would like to express my deep gratitude to my supervisor, Francisco Díaz González, who, alongside Daniel Ansgar Ackva, Daniel Montesinos-Miracle, Betina Römling, andAndreas Ziegler, madeitpossible for metoembark onmy journeyatTTZ-EMO anddevelop my Master’s thesis there. I thank them all for this incredible opportunity. I would also like to extend my thanks to David Oeser and Thiemo Hein for their invaluable technical assistance, insightful advice on my research topic, and their assistance during the testing phase. I have gained a wealth of knowledge during this collaboration, thanks to their expertise. On the other hand, I would like to give special thanks to Victor Oliveras, who, alongside Markus Hollas and Hans-Georg Schmedes, passed on to me their enthusiasm for the energy storage field. It all began with lead-acid batteries and Digatron, and I am truly appreciative of their guidance and patience. Additionally, I would like to acknowledge two other important figures during my work period in Barcelona, Victor Peraire and Enrique Grau, for their support and assistance in numerous ways. I also want to express my gratitude to my university colleagues, who have been my partners in this journey, offering their help and working side by side with me. I am grateful to my friends for their understanding during challenging times. A special thanks goes to Rebeca, Cristian, and Ana, for always being there, no matter what. Last but certainly not least, I would like to thank my parents, Rosa and Pepe, and my sister, Isabel, for their tireless support in all my decisions and their continuous encouragement to become a better person. Your belief in me has been a driving force in my life. Cell Modelling for State-of-Charge and State-of-Health Estimation 71 Bibliography [1] Chang-Hui Chen, Ferran Brosa Planella, Kieran O’Regan, Dominika Gastol, W. Dhammika Widanage, and Emma Kendrick. Development of experimental techniques for parameterization of multi-scale lithium-ion battery models. Journal of The Electrochemical Society, 167 (8):080534, 2020. doi: 10.1149/1945-7111/ab9050. [2] TTZ-EMO. https://ttz-emo.thws.de/, Accessed on 11.05.2023. [3] Cornelia Breitkopf and Karen Swider-Lyons (Eds.). Handbook of Electrochemical Energy. 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Journal of Energy Storage, 70:108034, 2023. doi: 10.1016/j.est.2023.108034. 81 Vt_Error = []; ik = length ( Current ); delta = diff( Time ); for k =1:1: ik U = Current (k); % A SOC = X(1); V1 = X(2); V2 = X(3); if k == 1 delta = 0; else delta = delta(k -1) ; end % functions for the current SOH & SOC R0 = F_R0 (SOH , SOC ); R1 = F_R1 (SOH , SOC ); R2 = F_R2 (SOH , SOC ); Tau_1 = F_tau1 (SOH , SOC ); Tau_2 = F_tau2 (SOH , SOC ); OCV = polyval ( SOCOCV , SOC ); % calculate the values of OCV a1 = exp (- delta / Tau_1 ); a2 = exp (- delta / Tau_2 ); b1 = R1 * (1 - exp (- delta / Tau_1 )); b2 = R2 * (1 - exp (- delta / Tau_2 )); y = OCV + R0*U + V1 + V2; dOCV = polyval ( dSOCOCV , SOC); C_x = [dOCV 1 1]; error_x = y_meas (k) - y; y_est = [ y_est ; y]; SOC_est = [ SOC_est ;X (1) ]; u_error = [u_error; error_x]; A = [1 0 0; 0 a1 0; 0 0 a2 ]; B = [( delta/Cn); b1; b2 ]; X = (A * X) + (B * U); P_x = (A * P_x * A') + Q_x; KalmanGain_x = (P_x ) * (C_x ') * ( inv (( C_x * P_x * C_x ') + ( R_x ))) ; 82 X = X + ( KalmanGain_x * Error_x ); P_x = (eye(n_x , n_x) - ( KalmanGain_x * C_x )) * P_x ; Q_x = KalmanGain_x * Error_x * KalmanGain_x '; end \ section *{ Model based in EIS } \ begin { lstlisting }[ style = Matlab - editor ] function [Zr , Zi ] = EIS_estimate ( params , omega ) % Circuit => s( p(L, R_l), R0 , p(R1 , C1), p(R2 , C2), W) L = params (1); R_l = params (2); R0 = params (3); R1 = params (4); C1 = params (5); R2 = params (6); C2 = params (7); W = params (8); % impedance from each branch Z_RL = RL_element (omega , L, R_l); Z_R0 = R_element (R0); Z_RC1 = RC_element (omega , R1 , C1); Z_RC2 = RC_element (omega , R2 , C2); Z_W = W_element (omega , W); Z = Z_RL + Z_R0 + Z_RC1 + Z_RC2 + Z_W; Zr = real(Z); Zi = imag(Z); function Z = RL_element (omega , L, R) Z =R .* omega .^2.* L ^2./( R ^2+ omega .^2.* L^2) +R ^2.* omega .* L ./( R ^2+ omega .^2.* L ^2) .*1 i; end function Z = R_element (R) Z = R; end function Z = RC_element (omega , R, C) Z = R./(1 + ( omega .* R .* C) .^2) - 1i .* omega .* R .^2 .* C ./ (1 + (omega .* R .* C) .^2); end function Z = W_element (omega , W) Z = 1./( W .*(1 i .* omega ) .^0.5) ; end 83 end SOH and SOC Estimation function [ estimatedSOC , estimatedSOH ] = EIS_estimation ( eis_parameters_table , inputParams ) % Extract input and output data from the sample data ecm_parameters = eis_parameters_table {: , 5: end }; soc_values = eis_parameters_table{:, 1}; soh_values = eis_parameters_table{:, 2}; % Split the dataset into training and testing sets rng (42) ; % 80% training , 20% testing [ trainInd , testInd ] = crossvalind ('HoldOut', size( eis_parameters_table , 1) , 0.2) ; X_train = ecm_parameters (trainInd , :); y_train_soc = soc_values ( trainInd ); y_train_soh = soh_values ( trainInd ); % Regression models for SOC and SOH regressor_soc = fitlm ( X_train , y_train_soc ); regressor_soh = fitlm ( X_train , y_train_soh ); % Estimation SOC and SOH estimatedSOC = predict ( regressor_soc , inputParams ); estimatedSOH = predict ( regressor_soh , inputParams ); % Ensure SOC and SOH remain between 0 and 1 if estimatedSOC <=0 || estimatedSOC >=1 estimatedSOC = max (min ( estimatedSOC , 1) , 0) ; elseif estimatedSOH <=0 || estimatedSOH >=1 estimatedSOH = max (min ( estimatedSOH , 1) , 0) ; end DAT Filtering function [V, dQdV , dVdQ , Q1 ]= DVA_ICA_Gaussian (Q, Volt , step) i=1; k=1; numV = numel ( Volt ); while i< numV j=1; dV(k)= Volt(i+j)-Volt(i); 84 V(k) = Volt (i); Q1(k) = Q(i); dQ(k)=Q(i+j)-Q(i); i=i+1; k=k+1; end dQdV = dQ ./ dV ; dVdQ = dV ./ dQ ; dQdV = gaussianFilter (dQdV , 50 , 0.5) ; dVdQ = gaussianFilter (dVdQ , 50 , 0.5) ; end function smoothed_data = gaussianFilter ( data , window_size , sigma ) filter = gausswin ( window_size , sigma ); filter = filter / sum(filter); smoothed_data = conv(data , filter , 'same'); end