Digital Twin Development of Li-Ion Batteries for Enhanced Energy Management in Microgrids

Proyecto Fin de Carrera Ingeniería de Telecomunicación Formato de Publicación de la Escuela Técnica Superior de Ingeniería Autor: F. Javier Payán Somet Tutor: Juan José Murillo Fuentes Dep. Teoría de la Señal y Comunicaciones Escuela Técnica Superior de Ingeniería Universidad de Sevilla Sevilla, 2013 Proyecto Fin de Máster Máster en Ingeniería Aeronáutica Digital Twin Development of Li-Ion Batteries for Enhanced Energy Management in Microgrids Autor: Miguel Ruiz López Tutor: Carlos Bordons Alba . Dpto. Ingeniería de Sistemas y Automática Escuela Técnica Superior de Ingeniería Universidad de Sevilla Sevilla, 2024 Proyecto Fin de Máster Máster en Ingeniería Aeronáutica Digital Twin Development of Li-Ion Batteries for Enhanced Energy Management in Microgrids Autor: Miguel Ruiz López Tutor: Carlos Bordons Alba Catedrático de Universidad Dpto. Ingeniería de Sistemas y Automática Escuela Técnica Superior de Ingeniería Universidad de Sevilla Sevilla, 2024 Proyecto Fin de Máster: Digital Twin Development of Li-Ion Batteries for Enhanced Energy Management in Microgrids Autor: Miguel Ruiz López Tutor: Carlos Bordons Alba El tribunal nombrado para juzgar el trabajo arriba indicado, compuesto por los siguientes profesores: Presidente: Vocal/es: Secretario: acuerdan otorgarle la calificación de: El Secretario del Tribunal Fecha: Agradecimientos H ay muchas personas que merecen ser mencionadas en este momento. Estoy convencido de que dejaré atrás a más de una, pero estoy seguro de que mis amigos me excusarán. A mis padres, que han sido mi pilar incondicional. No estaría escribiendo estas líneas si no fuera por su apoyo en todo momento. A mi tía Luisa, que siempre ha sido como una segunda madre para mí. A mis hermanos, por estar a mi lado cuando más los he necesitado. A Clara, mi compañera de viaje. Sin ti, no sería quien soy. Gracias también a tu familia por recibirme siempre con los brazos abiertos. A mis amigos, por su constante apoyo a lo largo de este camino. Gracias por cada vivencia compartida y por hacer de mi tiempo un regalo invaluable. A ENGREEN, y especialmente a mis compañeros de batalla. Vuestra desinteresada ayuda ha sido fundamental para hacer de este trabajo una realidad. Javier, sin ti no hubiera sido posible. Finalmente, a mi tutor, Carlos. Gracias por brindarme la oportunidad de trabajar en este proyecto tan especial y por ser un guía excepcional. Tu fe en mí y tus sabios consejos han sido cruciales. A todos, de corazón, ¡gracias! Miguel Ruiz López Máster en Ingeniería Aeronáutica Sevilla, 2024 I Resumen H oy en día, las microrredes representan un paradigma clave en la lucha contra el cambio climático, facilitando la integración de energías renovables en los sistemas energéticos. La gestión eficiente de la operación y mantenimiento de microrredes es un desafío complejo, que requiere soluciones innovadoras. El avance de tecnologías emergentes como la inteligencia artificial y el análisis de datos permite aprovechar la gran cantidad de información generada por estos sistemas para abordar este reto. En este contexto, el concepto de Gemelo Digital se presenta como una solución ideal para integrar estas tecnologías de manera eficiente y coherente. A diferencia de un Modelo Digital, el Gemelo Digital establece un flujo bidireccional de datos con su contraparte física, lo que le permite mantenerse actualizado y ejercer influencia sobre su funcionamiento. Este trabajo se centra en el desarrollo de un Gemelo Digital para una batería de ion de litio, estableciendo así el monitoreo y la visualización en tiempo real de la entidad física, así como la provisión de servicios de simulación. Esto resulta altamente beneficioso para mejorar la gestión energética en microrredes y se espera que la solución propuesta contribuya a optimizar su control. En la literatura, los diferentes componentes relacionados con el desarrollo de un Gemelo Digital de una batería han sido tratados con diversos niveles de profundidad. Sin embargo, según el conocimiento del autor, son escasos los estudios que integran de manera coherente todos estos elementos dentro del marco del Gemelo Digital. Este trabajo busca ofrecer una visión integral de cómo estas áreas interactúan y se basa en principios bien establecidos del concepto de Gemelo Digital para entender los requisitos, necesidades y desafíos que plantea el desarrollo de un Gemelo Digital eficiente. Este proyecto presenta dos contribuciones clave: i) propone una arquitectura para la implementación de un Gemelo Digital de una batería de ion de litio, detallando su implementación mediante herramientas de código abierto para facilitar su replicación por otros investigadores; y ii) diseña experimentos que validan el Gemelo Digital, justificando la validez de los procedimientos planteados. De esta forma, el trabajo busca mejorar la gestión energética en microrredes mediante soluciones digitales innovadoras, abriendo la puerta a futuras investigaciones y aplicaciones prácticas en el campo, y estableciendo una base sólida para desarrollos posteriores. III 2Chapter 1. Introduction This work is closely aligned with several of these research lines. Specifically, it focuses on the development of digital solutions, among other objectives, for the modelling of complex systems, applying artificial intelligence and data science techniques. The ultimate goal is to improve energy management in microgrids. 1.1.2 The Microgrid Concept To mitigate climate change, a transition from fossil-fueled energy systems to ones with a significant share of Renewable Energy Sources (RESs) is essential. However, this shift presents significant challenges, primarily due to the inherent intermittency of RESs and the uncertainties associated with user demand [47]. In response, the concept of microgrids has emerged as a promising approach to address the challenges of integrating Distributed Energy Resources (DERs), including RESs, into power systems. Microgrids offer a decentralized solution to technical problems, reducing the need for complex central coordination [50]. The microgrid concept was first introduced by Lasseter in [52], defining it as an aggregation of loads and microsources operating together as a single system to provide both power and heat. This definition was later adopted by the Consortium for Electric Reliability Technology Solutions (CERTS) [51]. A key aspect of this concept is that the microgrid is perceived by the main grid as a single controllable entity, responding to appropriate control signals [50]. In Lasseter’s work [52], microsources refer to both Distributed Generation (DG) units and Energy Storage Systems (ESSs). Additionally, he introduced the energy manager and distributed protection devices as essential components for the effective operation of a microgrid. According to [50], a microgrid consists of a coordinated cluster of loads, DG units, and ESSs, designed to reliably supply electricity. It is connected to the host power system at the distribution level through a single point of connection, known as the Point of Common Coupling (PCC). The Energy Management System (EMS) and protection devices play a critical role in ensuring the reliable, secure, and economical operation of the microgrid, whether it is operating in grid-connected mode or in stand-alone mode. Figure 1.1 illustrates a typical microgrid with various types of generation and storage systems. Figure 1.1 Example of a microgrid [47]. 1.1.3 Energy Storage in Microgrids Energy storage is a fundamental pillar of the microgrid (MG) concept, playing a critical role in balancing supply and demand. There are various energy storage technologies available, each with its own advantages and disadvantages depending on factors such as economic cost, energy and power ratings, autonomy, response time, lifespan, and degradation issues [47]. ESSs are essential in compensating for the imbalances introduced by RESs, providing MGs with valuable energy buffering capabilities. The specific field of application for an ESS depends on its buffering characteristics, particularly its power rating and discharge time (see Figure 1.2). For instance, ESSs with fast response times can be used to stabilize the grid by addressing power quality issues. In contrast, high-capacity ESSs are better suited for applications requiring longer discharge times, such as compensating for the intermittency of RESs or smoothing peak demand through a process known as 1.1 Context 3 peak shaving (see Figure 1.3). This latter application is of particular interest in this work, which focuses on improving energy management strategies that rely on Battery Energy Storage Systems. Figure 1.2 Typical power rating, discharge time, and fields of application of several ESSs [47]. Figure 1.3 Illustrative example of storage based-peak shaving [49]. 1.1.4 Battery Energy Storage Systems A Battery Energy Storage System (BESS) is an electrochemical device used to store electrical energy, consisting of one or more voltaic cells. Each cell has two half-cells with electrodes (positive and negative) 4Chapter 1. Introduction connected by an electrolyte containing ions. Through a redox reaction, the battery is charged by reducing cations (adding electrons) and oxidizing anions (removing electrons), and during discharge, this process is reversed (see Figure 1.4). An electromotive force is generated at each half-cell, and the difference between them defines the cell’s overall voltage. Open circuit voltage refers to the voltage when no current flows, while the voltage decreases during discharge due to internal resistance and increases when charging. Figure 1.4 Schematic diagram of Lithium-Ion battery [60]. Several battery technologies are used in MGs [47]. Lead–acid batteries, while mature and relatively inexpensive, are less efficient and heavier. Redox Flow Batteries offer advantages like power and capacity decoupling, and a long cycle life. However, lithium-ion batteries are the most widespread technology in MGs, due to their extensive use in various sectors, including renewable energy systems. Lithium-ion batteries consist of a lithium metal oxide cathode, a carbon anode, and an electrolyte containing lithium ions. 1.1.5 The Digital Twin Concept The earliest application of the digital twin (DT) concept can be traced back to NASA’s Apollo Program. For this mission, NASA built two identical spacecraft—one on Earth and the other in space. The Earth-bound spacecraft mirrored the state of the one in operation, allowing engineers to simulate and evaluate strategies for safe operation. This setup enabled rapid adjustments and modifications to match the real-time conditions of the spacecraft in space. Although these early events occurred long ago, the DT concept as we know it today was first proposed by Michael Grieves in his research on Product Lifecycle Management (PLM). Grieves introduced the concept as a key element of his Conceptual Ideal for PLM [54] (see Figure 1.5). From its inception, the DT was intended to interact with the real-world entity throughout its entire lifecycle—including planning, design, manufacturing, operation, disposal, and recycling—helping to reduce inefficiencies. While NASA’s simulators were focused on the operational phase rather than managing the full lifecycle, they laid the groundwork for the broader applications of DTs seen today. Grieves’ initial idea in [54] was later refined and eventually formalized as the Digital Twin Model (DTM) [53]. Even in its early form, the model contained the key elements that define a digital twin. The physical entity, often referred to as the physical twin, represents the real-world subject matter of the digital twin, which 1.1 Context 5 could include not only tangible objects such as systems, machines, and infrastructure, but also intangible elements like organizations, supply chains, or processes such as manufacturing steps or weather events [55]. The corresponding digital twin is a data-driven virtual representation of the physical entity, synchronized with it at a defined frequency and level of detail [55]. A major advantage of this virtual space is the ability to create subspaces where different scenarios can be tested and simulated. Lastly, the synchronization mechanism ensures continuous, bidirectional data flow between the physical and digital twins. This mechanism allows real-time changes in the physical twin to be reflected in the digital model and enables the physical entity to be adjusted based on the insights gained from the virtual model to achieve desired outcomes [55]. Figure 1.5 First Digital Twin Model concept [54]. It is also important to clarify the distinction between the related concepts of the digital model and digital shadow, and their relation to the digital twin. The key difference among these terms lies in their synchronization with the physical twin [57]. A digital twin features bidirectional synchronization with the real-world entity across its lifecycle, while a digital shadow allows only one-way synchronization, from the physical entity to the digital counterpart. In contrast, a digital model does not have continuous synchronization; while it may be derived from real-world data, it remains static and does not automatically update over time (see Figure 1.6). Figure 1.6 Data flow in a digital model, a digital shadow and a digital twin [57]. 6Chapter 1. Introduction 1.2 Motivation The integration of DTs in microgrids represents a paradigm shift in energy management, control, and system resilience. As the world increasingly embraces renewable energy sources (RESs) and decentralized energy generation (DG), the complexity of managing these systems escalates. DTs offer innovative solutions to tackle these challenges, enhancing operational efficiency, predictive capabilities, and strategic planning throughout the MG lifecycle. Forecasting is a key application of DTs in MGs, effectively addressing uncertainties associated with RESs generation, electricity market prices, and consumer behaviour. By incorporating historical, realtime operational, and environmental data from sensors, DTs improve RESs forecasting, leading to accurate predictions of power generation across various timescales [61]. This capability enhances energy supply reliability, reduces the necessity for oversized equipment, and facilitates the effective participation of RESs in ancillary services, thus ensuring stable MG operations [62]. Furthermore, DTs analyze consumer and prosumer behaviour related to Vehicle-to-Grid (V2G) systems by processing metering data for load profiling and simulating responses to pricing. This analysis enhances MG efficiency and grid stability by utilizing electric vehicle batteries as energy storage during peak demand [62]. Additionally, DTs are essential for predicting electricity market prices, which depend on the complex interactions among market players and load dynamics [62]. In liberalized markets, prices are determined by balancing generator offers with consumer bids, accompanied by penalties for real-time imbalances [47]. DTs model these dynamics, enabling real-time adjustments and optimizing energy procurement strategies for MG operators and prosumers. Fault detection,system integrity, and operational resilience in MGs are significantly enhanced by DTs, addressing the unique challenges associated with integrating DERs and dual operational modes [63]. Traditional synchronous generators differ from RESs connected via power electronic converters, which may experience rapid dynamic responses and exhibit weaker fault tolerance, often leading to self-protection and off-grid behaviour during disturbances. DTs provide real-time monitoring and predictive capabilities that enable simulations of transient destabilization mechanisms and assessments of system resilience [61]. They support efficient fault diagnosis by continuously comparing system performance against reference behaviours, minimizing downtime, and improving reliability. Furthermore, DTs enhance security measures by identifying and simulating potential attack scenarios, enabling proactive responses to threats in both physical and cyber layers [62]. By improving situational awareness through advanced analytics, DTs help anticipate risks and bolster fault recovery and resilience in MGs. Predictive maintenance in MGs is also revolutionized by DTs, which continuously monitor the State of Health (SOH) of assets and predict their Remaining Useful Life (RUL). Utilizing intelligent sensors and data transmission systems, DTs replicate power equipment in virtual environments, allowing realtime performance analysis, fault identification, and state evaluations. This data-driven approach optimizes maintenance schedules and reduces costs compared to traditional periodic strategies [62]. Moreover, DTs can simulate potential failures using real and synthetic data, enhancing accuracy, especially in remote or hazardous locations [63]. This capability enables operators to mitigate risks, extend the life of critical components, and prevent unexpected breakdowns, ultimately improving reliability while minimizing crew exposure to hazardous conditions [63]. While the applications discussed primarily pertain to the operational phase of MGs, it is crucial to recognize that DTs interact with real entities throughout their entire lifecycle. Thus, their application extends to the design,development, and expansion planning or policy-making [62]. Returning to operational applications, DTs play a pivotal role in optimizing control within MGs. Microgrid control systems must address various perspectives and timescales. Rapid electrical control of phase, frequency, and voltage must occur within sub-second intervals, while unit commitment, economic dispatch, demand-side optimization, and energy exchanges with the utility grid are managed over longer timescales—ranging from minutes to days or even months. Consequently, a hierarchical control structure is often implemented [47] (see Figure 1.7). This work primarily focuses on enhancing energy management, which corresponds to these longer timescales. For instance, DTs can significantly improve controller design and Model Predictive Control (MPC) applications [53]. By synchronizing real-time data with simulation models, DTs facilitate realistic experimentation and validation of control strategies in virtual environments, reducing costs and risks. They enable automatic adjustments of control plans based on operational conditions, enhancing adaptability and robustness. In the context of MPC, DTs provide updated system information, improving real-time decision-making and enabling better management of constraints. This integration enhances performance and 1.3 Project Objectives and Scope Definition 7 stability in dynamic environments, underscoring the transformative potential of DTs in advanced control applications. Figure 1.7 Hierarchical control levels of a microgrid [47]. 1.3 Project Objectives and Scope Definition As outlined in Section 1.1.1, the overall objective of this project is to contribute to the development of digital solutions that enhance energy management in microgrids through advanced modelling techniques. By leveraging artificial intelligence and data science, this work aims to improve energy efficiency and reliability in microgrids. The Digital Twin Model (DTM) was identified as the ideal framework to achieve this, and the role of Energy Storage Systems, particularly Battery Energy Storage Systems, in energy management has been discussed. Therefore, this work focuses on the development of a Digital Twin for a lithium-ion battery, thereby establishing real-time monitoring and visualization of the physical entity, as well as providing simulation services. These features are expected to significantly enhance energy management practices in microgrids, ultimately contributing to optimized control of these systems. In alignment with the DTM concept described earlier, a schematic representation of the DTM proposed for this work is shown in Figure 1.8. To meet the overarching objective, several sub-objectives have been identified: • Design and implementation of the DT architecture, including how information is transmitted between elements and where data is stored. •Establishment of the physical twin: planning and installation of an experimental setup. • Definition of the virtual entity that mirrors the real system, including a review of battery modelling approaches and selection of the most appropriate method aligned with the DT objectives. • Development of the synchronization mechanism: enabling real-time data from the physical system to be captured and reflected in the digital model. • Implementation of monitoring and visualization, as well as simulation capabilities, as integrated services. •Validation of the DT’s predictive capabilities (i.e., simulation services) through experimental data. 8Chapter 1. Introduction Figure 1.8 Schematic of the Digital Twin Model proposed for this work. By achieving these objectives, this work aims to make two key contributions: i) to propose an architecture for implementing a Digital Twin of a lithium-ion battery, detailing its implementation using open-source tools to facilitate its replication by other researchers; and ii) to design experiments to validate the Digital Twin, justifying the validity of the proposed procedures. As mentioned in 1.2, the developed DT could be directly applied to enhance controller design, allowing for realistic experimentation and validation of control strategies in virtual environments, thereby reducing costs and risks. However, this particular application is beyond the current scope of this work. The natural continuation of this research would involve integrating the Digital Twin with control mechanisms, influencing battery operation in real-world scenarios. In summary, this project achieves several significant milestones, providing valuable insights and tools for energy management in microgrids while laying the groundwork for further research and development. 1.4 Structure of the work This document, following the conclusion of this introductory chapter, will be structured as follows: • First, a detailed explanation of the established experimental setup will be provided, including the architecture developed for the Digital Twin Model and the specifics of its implementation. Additionally, the monitoring and visualization services created will be presented. • Next, a review of the state of the art regarding various approaches to battery modelling will be conducted. This review will inform the definition of the virtual representation to be used in this project, considering the intended objectives. • Once the virtual representation is established, the procedure designed for its synchronization with the physical entity will be explained in detail. • Subsequently, the results obtained from a series of experiments designed to establish the initial Digital Twin will be presented, validating the entire process by utilizing the implemented simulation services. • Finally, the conclusions drawn throughout the project will be summarized, along with potential directions for future work. 2 Experimental Setup and Architecture of the Digital Twin Model In this chapter, we provide a detailed explanation of the established experimental setup, including the architecture developed for the Digital Twin Model and the specifics of its implementation. Additionally, we present the monitoring and visualization services created for this project. 2.1 Experimental setup 2.1.1 Components and layout First, a brief description of each one of the elements that constitute the experimental setup will be provided, as well as their functions in it. The core subject of this work is the Lithium-ion battery pack under study. Rather than focusing on a single battery, this work deals with a complete battery pack. Specifically, we use a pack of 15 Pylontech H48050 Lithium-ion batteries, managed by a Pylontech SC1000-100S Battery Management System (BMS). The BMS ensures safe and reliable operation by continuously monitoring the batteries, preventing issues such as overcurrent or overheating. It also serves as the primary system interface for obtaining the required data from the Battery Energy Storage System (BESS). The full BESS is shown in Figure 2.1, and its relevant specifications are listed in Table 2.1. Table 2.1 Relevant specifications of the BESS under study. Specification Value Unit Nominal Capacity 50 Ah Nominal Voltage 720 V Voltage Range 652.5−810 V Maximum Continuous Current 25 A Depth of Discharge 90 % The system also includes DC/DC converters to control current and adjust voltage ranges between the devices. In this work, we use a set of four 8k 8380i Epic Power converters (Figure 2.2). The relevant specifications for the converter pack, focusing on the BESS side, are summarized in Table 2.2. These specifications ensure that the full operational range of the BESS can be studied. Table 2.2 Relevant specifications for the Epic Power converter pack on the BESS side. Specification Value Unit Voltage Range 650-848 V Maximum Continuous Power 32 kW Maximum Continuous Current 44 A 9 10 Chapter 2. Experimental Setup and Architecture of the Digital Twin Model Figure 2.1 Pylontech BESS under study. Figure 2.2 Epic Power DC/DC converters connected to the BESS. 2.1 Experimental setup 11 To emulate the grid, a ITECH IT6018B-1500-40 bidirectional programmable DC power supply will be connected to the opposite side of the converter pack, as shown in Figure 2.3. This device allows for realistic simulation of various scenarios, testing the system’s reliability and efficiency under different circumstances. While the device can simulate solar panels, fuel cells, and custom load profiles, in this project, its primary function is to maintain a stable voltage level. It will provide or consume the required power so that the battery charges or discharges at the desired current level imposed by the converter pack. The device is modular, capable of both isolated and coordinated operation, although a single module is sufficient to study the full operational range of the BESS. The relevant specifications are listed in Table 2.3. Table 2.3 Specifications of the IT6018B-1500-40 power supply. Specification Value Unit Voltage Range 0-1500 V Maximum Power 18 kW Maximum Current 40 A Figure 2.3 IT6018B-1500-40 bidirectional DC power supply. To coordinate, monitor, and control all these devices at a low level, the Beckhoff CX-5140 Programmable Logic Controller (PLC) is used, as shown in Figure 2.4. PLCs are similar to standard PCs in some ways, sharing components such as a power supply, a Central Processing Unit (CPU), and inputs and outputs (I/O). However, PLCs are specifically designed for industrial environments, offering a robust and versatile solution for the control and automation of processes. Finally, a workstation available in the laboratory was used to communicate higher-level commands to the PLC. These commands include starting the operation, loading profiles, and stopping the system. The workstation also receives system information from the PLC and hosts the Digital Twin System (DTS). 2.1.2 Set up process The first step of the setup process is to perform the electrical installation. This process can be divided into the following sub-steps: • Battery and Converter Rack Assembly: The assembly of the racks is a fundamental task that involves the secure and proper physical arrangement of both the batteries and the converters. High-resistance racks were chosen to ensure stability and safety for all components. Special attention was given to the arrangement of the batteries to ensure optimal airflow and temperature conditions for their operation. 18 Chapter 2. Experimental Setup and Architecture of the Digital Twin Model Figure 2.10 Partial view of the dashboard created in Grafana, used for monitoring and visualization services (III). Figure 2.11 Picture taken showing how the sunlight hits directly on the BESS at a specific time of day. 3 Defining the Virtual Entity: Approaches to Battery modelling In this chapter, we aim to define the virtual entity by selecting an appropriate modelling approach for the battery. We will briefly review various battery modelling methods found in the literature, enabling a thorough comparison. Our focus will be on identifying the advantages and disadvantages, as well as the application cases, of each method, allowing us to confidently choose the most suitable model for our needs. We seek an approach that strikes a compelling balance between accuracy and computational efficiency for real-time system-level control applications. Moreover, this approach must be highly flexible and easily adjustable to accommodate a wide range of effects. 3.1 Main battery modelling approaches Based on their modelling principles, battery models are generally categorized into four main types: electrochemical models, equivalent circuit models (ECMs), mathematical models, and data-driven models [46]. Battery models can also be classified based on the level of physical insight they provide, falling into three categories: white, gray, and black box models [48] (see Figure 3.1). Electrochemical models, which offer the highest level of physical detail, are classified as white box models. Following these are ECMs and mathematical models, which can be considered gray box models due to their intermediate level of physical insight. Finally, data-driven models, which lack physical insight, are categorized as black box models. In this section, a general overview of the presented categories will be provided. 3.1.1 Electrochemical Models Electrochemical models explicitly represent the chemical processes within a battery, making them the most accurate and informative. These models can provide "virtual" measurements that are not obtainable in practice and offer insights into crucial aspects such as aging or temperature effects, which are often overlooked by other models. However, this accuracy comes at the cost of significant computational demand. For instance, simulating a single charge-discharge cycle using a detailed electrochemical model can take several hours if model reduction techniques are not applied. The Lumped Parameter Model (LPM) represents early physics-based battery models [45]. It simplifies the battery into a system characterized by a small set of differential algebraic equations (DAEs) with averaged properties across the cell, which do not account for spatial variations within the battery’s components. Basic DAEs fall short in capturing the complex electrochemical processes in modern batteries, such as Lithium-Ion cells. To address this, Newman, Doyle, and Fuller developed the pseudo-2D (P2D) model [39]–[44], also known as the Doyle-Fuller-Newman model, which integrates Porous Electrode Theory and Concentrated Solution Theory. This model uses partial differential equations (PDEs) to describe the concentration and potential gradients within the electrodes and electrolyte, accounting for spatial variations in properties. A major drawback of electrochemical models is the difficulty in obtaining the actual values for their numerous parameters. Moreover, even with accurate parameter values, the computational demands of these models make them challenging for real-time or control system applications. Consequently, significant research 19 20 Chapter 3. Defining the Virtual Entity: Approaches to Battery modelling Figure 3.1 Schema of modelling categories and their classification into white, gray, and black box models. focuses on reducing their computational complexity [45]. The most mature simplified model based on the P2D model is the Single Particle Model (SPM), which simplifies the electrode to a single particle [15] (see Figure 3.2). Another promising method is the application of fractional-order calculus for model-order reduction. Fractional electrochemical models significantly reduce the computational burden associated with high-order physics-based models while still capturing key battery characteristics. Additionally, by utilizing fractional-order differentiation, these models require fewer parameters, which can still retain physical significance [29]. Figure 3.2 Structure of the pseudo-2D (P2D) model and SPM [45]. 3.1.2 Mathematical Models Mathematical models are often regarded as simplifications of electrochemical models, where the battery terminal voltage is expressed as a mathematical function of the SOC and current. The state of charge SOC represents the current charge level of the battery relative to its nominal capacity and is defined as: 3.1 Main battery modelling approaches 21 SOC =Qremaining Cnom ×100% (3.1) where Qremaining is the remaining capacity of the battery, and Cnom is the nominal capacity. For mathematical models, the SOC is typically computed using the Coulomb Counting Method (CCM), which integrates the current over time to estimate the charge remaining in the battery. The basic equation for SOC using the CCM is given by: SOC(t) = SOC(t0)−1 Crated Zt t0 I(t′)dt′(3.2) where SOC(t0) is the initial SOC , Cnom is the nominal capacity of the battery, and I(t′) is the current flowing into or out of the battery at time t′ , considered negative for charging and positive for discharging by convention. From (3.2) , several limitations can be identified. The accuracy of SOC calculations can be compromised by [76]: •Initial SOC: This method requires initial SOC to start with. •Current measurement error: Inaccuracies in the current sensor readings can lead to drift over time •Current integration error: The method used to approximate the integral can affect the results [76] • Non-modelled effects: Self-discharge, charging/discharging losses or capacity fade due to aging can affect the prediction accuracy [77]. These limitations underscore the need for complementary approaches to enhance accuracy in real-time SOC estimation, with OCV −SOC lookup tables, model-based, and data-driven methods being particularly popular [78]. Mathematical models, while valuable, typically use empirical or semi-empirical approaches, which, while simplifying the physical processes, can limit their generality and accuracy across different operating conditions. The classical mathematical models in the literature include the Shepherd model, the Unnewehr universal model, and the Nernst model [38] (see Table 3.1). Table 3.1 Classical mathematical models. U is the terminal voltage, R0 is the internal resistance, I is the discharging current, SOC represents the battery state of charge and E0 , K , K1 and K2 are constants. Model Equation Shepherd Model U=E0−R0I−K/SOC Unnewehr Universal Model U=E0−R0I−K SOC Nernst Model U=E0−R0I+K1ln SOC +K2ln(1−SOC) Modifying and hybridizing these basic models can yield more accurate representations and allow for the modelling of more complex phenomena. For instance, the original Shepherd model includes a non-linear term, K I/SOC [35], which closely mimics real battery behaviour but introduces algebraic loops and simulation instability. To mitigate this issue, the Modified Shepherd Model was introduced in [37]: U=E0−R0I−KQ Q−it +Ae−Bit ,(3.3) where U is the terminal voltage, R0 is the internal resistance, I is the discharging current, E0 is a constant voltage, K is the polarization voltage, Q is the battery capacity, A and B are, respectively, the exponential zone amplitude and time constant inverse (see Figure 3.3), and it is the extracted capacity. Thus, the polarization term can be rewritten in terms of the SOC of the battery as follows: U=E0−R0I−K/SOC +Ae−Bit .(3.4) If the voltage drop in the exponential zone is too rapid to match observed experimental data, the exponential term in (3.4) can be omitted, resulting in the expressions given in Table 3.1. Furthermore, [37] provides a method to determine model parameters using the manufacturer’s discharge curves, making these models particularly useful in early design phases when limited information about the battery is available. 22 Chapter 3. Defining the Virtual Entity: Approaches to Battery modelling Figure 3.3 Typical discharge curve [37]. The previous model relies solely on SOC as a state variable to represent voltage behaviour, which limits its effectiveness to steady-state conditions (i.e., constant current). To address this, [36] proposed extensions to the model for variable charging or discharging currents, particularly for Lithium-Ion batteries: U=     E0−R0I−KQ Q−it (it +i∗)+Ae−Bit ,if I>0(discharging) E0−R0I−KQ Q−it it −KQ it −0.1Qi∗+Ae−Bit ,if I<0(charging) (3.5) where i∗ is the low-frequency current dynamics that a low pass filter can obtain. Although these modifications extend the model’s applicability, they still simplify the physical processes compared to more comprehensive models, such as electrochemical models. Additionally, these basic models can be hybridized to enhance their capabilities. For instance, [34] combines several models into the following expression: U=E0−R0I−K1/SOC −K2SOC +K3ln SOC +K4ln(1−SOC).(3.6) Advanced mathematical models can represent more complex phenomena, such as temperature effects, hysteresis, or aging. However, incorporating these effects significantly increases the computational burden, and integrating them is more challenging compared to electrochemical models, where such factors are inherently included in the model’s structure. In general, mathematical models are simple and computationally efficient, but their accuracy and flexibility are limited. Therefore, these models are most appropriate in applications where simplicity and rapid simulations are prioritized over high accuracy. However, due to their limitations, these models are often complemented by more detailed models during later stages of battery design and analysis. 3.1.3 Equivalent Circuit Models (ECMs) ECMs attempt to model the electrochemical behaviour of a battery using electrical components. These models are generally categorized into two main types: Integer-Order Models (IOMs) and Fractional-Order Models (FOMs). IOMs employ standard integer-order elements, which are simpler and more widely used, often addressing basic time-domain responses. Conversely, FOMs utilize fractional-order elements to model more intricate battery dynamics and frequency responses with greater accuracy, though this comes with the trade-off of higher computational complexity. Three common IOMs found in the literature are the Rint model, the Thevenin models, and the PNGV models [45], [46], [48]. These models are illustrated in Figure 3.4, along with their corresponding expressions. The simplest among these is the Rint model, which consists of a voltage source representing the Open- Circuit Voltage ( OCV ) dependent on the SOC of the battery, and a resistor representing the internal resistance R0 of the battery (see Figure 3.4a). As with mathematical models, the SOC is typically calculated using the CCM. 3.1 Main battery modelling approaches 23 (a) Rint model . (b) Thevenin model . (c) PNGV models . Figure 3.4 Three common Integer-Order ECMs and their expressions. The Thevenin models extend the Rint model by adding n resistance-capacitance ( RC ) networks (see Figure 3.4b). These RC pairs model the voltage response at different time scales, reflecting the time-dependent effects of charge depletion and recovery [45]. The Partnership for a New Generation of Vehicles (PNGV) models build upon the Thevenin models by incorporating an additional capacitor C0 , which represents the bulk energy storage capability of the battery (see Figure 3.4c). This additional component impacts how the terminal voltage behaves over longer periods or during slow charge/discharge cycles and partly accounts for hysteresis effects [30]. As discussed previously for electrochemical models, fractional-order calculus can be applied to ECMs to capture more complex dynamics. Compared to the previously mentioned IOMs, FOMs generalize the ideal capacitors by incorporating fractional-order capacitive elements, often referred to as constant phase elements (CPEs) [15]. This approach allows FOMs to better represent the frequency response of the battery. Electrochemical Impedance Spectroscopy (EIS) is a powerful tool used to analyze a battery’s internal characteristics by measuring its impedance across a range of frequencies. The results are often presented as a Nyquist plot, which shows the real and imaginary components of impedance. This plot helps to visualize different processes occurring within the battery, such as charge transfer and diffusion. Figure 3.5 displays the Nyquist plot of a typical Lithium-Ion cell’s impedance spectrum alongside a representative FOM. The high-frequency section is usually represented through an ohmic resistance, and 24 Chapter 3. Defining the Virtual Entity: Approaches to Battery modelling sometimes an inductance is introduced [30]. The mid-frequency region, associated to the electrochemical double-layer and charge-transfer reactions, is captured using resistance-CPE pairs. The low-frequency straight line, resulting from the diffusion in solic electrodes, is typically modelled with a single CPE, often referred to as Warburg impedance [29]. Figure 3.5 Nyquist plot of a typical Lithium-Ion cell’s impedance spectrum alongside a representative FOM (modified from [29]). All of these models can be modified so that their parameters depend on the internal state of the battery (such as SOC and temperature) and the operating conditions (such as charge/discharge cycles and C-rate). Additionally, certain variations of ECMs can model additional phenomena, such as the hysteresis effect. However, this complexity is not always beneficial. For instance, in [33], the inclusion of hysteresis modelling did not improve the accuracy of 1RC and 2RC models due to the increased number of unknown parameters, which led to overfitting and ultimately reduced model reliability. Similarly, choosing between different ECMs requires careful consideration; increased model complexity does not always result in more accurate or robust models. Introducing these complexities also raises the computational cost, so a balance must be struck between accuracy, reliability, flexibility, and computational efficiency for each specific application. In general, ECMs provide a flexible approach to battery modelling by balancing accuracy and computational efficiency. The choice between IOMs, FOMs, or their variants depends on the specific needs of the application. IOMs are straightforward and ideal for real-time applications with limited resources, offering sufficient 3.2 Comparison and approach selection 25 accuracy for many use cases. FOMs, though more complex, capture intricate battery dynamics at the cost of increased computational demands. Ultimately, selecting the right ECM involves finding the right balance between model complexity, accuracy, and computational cost to meet the application’s requirements. 3.1.4 Data-Driven Models Data-Driven models avoid any level of physical insight and instead rely on large datasets to predict behaviour through the use of machine learning techniques. The typical process for creating a data-driven model involves several stages: data collection, preprocessing, model selection, training, validation, and deployment. Additionally, within a DT framework, these models must be continuously updated throughout the system’s lifetime. Common techniques used are Neural Network (NN) models and Support Vector Machine (SVM) models [15], [25], [27], [28], [46]. One of the key advantages of these models is that, by abstracting themselves from the battery’s internal mechanisms, they can adapt to a wide variety of battery chemistries, configurations, and operating conditions. This makes them particularly valuable in applications where obtaining detailed physical parameters is challenging or impractical. Moreover, data-driven models are known for their capability of handling nonlinearities, although effectively managing these complexities often requires sophisticated and well-tuned models, such as Deep Neural Networks (DNNs). However, the accuracy of these models depends heavily on the quality and quantity of the datasets on which they are trained. Insufficient or biased data can significantly hinder model performance, impacting their ability to generalize effectively. Without diverse and comprehensive training data, these models may struggle to adapt to unseen conditions or new battery chemistries, limiting their effectiveness in dynamic or evolving applications. Additionally, data-driven models often lack interpretability, which poses a significant drawback in applications where understanding the decision-making process is crucial, such as in autonomous systems, including our own. 3.2 Comparison and approach selection Table 3.2 summarizes the main advantages, disadvantages, and typical application cases of the different battery models discussed in this section. While no single model is perfect, selecting the most appropriate one for each application is crucial. At this stage, it is important to restate our objectives and clearly define the requirements for the modelling approach. Our goal is to create an effective virtual representation of the DT, with the primary focus on accurately predicting system-level behaviour. Additionally, to be useful for control purposes, the DT model must deliver real-time performance. While electrochemical models provide high accuracy by simulating a battery’s internal states, these details are not relevant to our objectives. As such, the increased computational complexity associated with these models is not justified, especially given our real-time performance requirement. In the context of DTs, where large volumes of data can be collected from the physical system, data-driven methods present a promising alternative. However, for this work, the required volume and quality of data were not obtained. This, combined with the typical lack of interpretability of data-driven models, makes them less suitable for our current needs. Nonetheless, future research could explore pure data-driven methods or hybrid approaches, once sufficient data is available. Mathematical models are computationally inexpensive, simple, and easy to tune, making them ideal for early-stage design and analysis where rapid simulation is prioritized over accuracy. However, their limited precision and inability to capture complex phenomena pose significant limitations for fully exploiting the capabilities of the DT. For our application, ECMs are preferred due to their balanced trade-off between accuracy, flexibility, and computational efficiency. ECMs provide the system-level modelling we require while maintaining reasonable accuracy and real-time performance. Additionally, ECMs allow for easy integration of more complex phenomena, making them the best option for this work. Following this reasoning, IOMs are favored over Fractional-Order Models FOMs. Although FOMs offer a more accurate representation of battery behaviour, they introduce significant computational complexity due to the nature of fractional calculus. Moreover, characterizing the battery’s frequency response through EIS is a requirement for FOMs, further adding to the complexity. Thus, IOMs are chosen for this study, striking a balance between accuracy, simplicity, and faster processing times. 26 Chapter 3. Defining the Virtual Entity: Approaches to Battery modelling Table 3.2 Comparison of battery model types including main advantages, disadvantages, and typical application cases. Model Electrochemical Mathematical ECM Data-Driven Advantages - Provides high accuracy with detailed insights into internal processes - Models complex phenomena inherently - Simple and computationally efficient - Basic variants require minimal data - Easy to understand and implement - Balances accuracy, flexibility, and computational efficiency - Capable of handling complex, non-linear relationships - Does not require physical insight Disadvantages - High computational cost - Requires extensive knowledge of battery chemistry and processes - Limited accuracy and flexibility - Struggles with nonlinearities and transient effects - Complexity can lead to instability - Limited ability to model internal processes - Accuracy is highly dependent on the quality of the training dataset - Labor-intensive preprocessing and tuning required Application case - In-depth research and development - High accuracy is required, and real-time performance is not critical - Early stages of design and analysis - Simplicity and rapid simulations are priority - Real-time control in dynamic environments - Balance between efficiency and accuracy is required - Data is unavailable and modelling complex relationships is necessary - Interpretability issues Among the various IOMs, the Thevenin models are particularly notable for their widespread use in literature. They are favored for their relatively low complexity, good flexibility, and ability to capture battery dynamics across multiple timescales, all while maintaining manageable computational costs. This makes the Thevenin model an ideal choice for our work. Although increasing the number of RC pairs in the Thevenin model can enhance accuracy, it also risks overfitting [15], which can lead to model instability. Studies, such as [33], demonstrate that while a 2RC model offers greater accuracy compared to a 1RC model, the 1RC model is more reliable. Further, [32] shows that the 2RC model provides the best balance between computational complexity and modelling error. Meanwhile, [31] compares various battery models and concludes that the 1RC model is well-suited for lithium nickel manganese cobalt oxide (LiNMC) cells, while a 1RC model with hysteresis is better for lithium iron phosphate (LiFePO4) batteries. This explains why 1RC and 2RC Thevenin models are the most commonly cited in the literature. As discussed, each RC pair in Thevenin models captures the dynamics associated with a specific frequency range. The 2RC model, for instance, can capture both the fast, short-term dynamics of charge-transfer phenomena and the slower diffusion effects that occur over a much longer timescale. In a discharging process, the voltage drop due to internal resistance and the short-term RC pair dynamics vanishes after a few minutes (e.g., at tm ), while the terminal voltage reaches equilibrium only after several hours (e.g., at th ). This process is illustrated in Figure 3.6. In contrast, a standard 1RC model is unable to capture these long-term dynamics, leading to inaccuracies in cases where load profiles involve infrequent long relaxation periods [26]. On the other hand, increasing the model order also increases computational complexity and can introduce instabilities in parameter identification. To address this, [26] proposes a trade-off solution by encapsulating both the conventional OCV and the long-term RC pair dynamics into a dynamic OCV . This dynamic OCV becomes dependent not only on SOC but also on time, and can be derived from the characteristic times tmand th[26]. For this work, however, a further simplification of the model is adopted. In cases where highly dynamic load profiles are used and discharge times do not exceed tm , the dynamic OCV can again be considered static but set to the terminal voltage at tm rather than at th , which would correspond to the conventional OCV definition. While this OCV remains static in terms of time, it still depends on whether the battery is in charging or discharging conditions, as well as on SOC . This simplification limits the model’s validity to 3.2 Comparison and approach selection 27 Figure 3.6 Dynamic OCV concept [26]. scenarios with highly dynamic profiles, such as those encountered in electric vehicles or in our case, energy management in microgrids. Thus, the 1RC Thevenin model with the modified OCV described here will be used as the virtual representation for this work. From Figure 3.4b, the circuit schematic and voltage equation are straightforward. The specific circuit schematic can be seen in Figure 3.7, and the terminal voltage is computed as: U=OCV −R0I−URC,1(3.7) where ˙ URC,1=−1 R1C1 URC,1+1 C1 I(3.8) Figure 3.7 Schematic of the 1RC Thevenin Model. In the tests conducted for this work, which will be detailed in Chapter 5, temperature remains nearly constant. This stability is largely due to the cooling system integrated into the BESS and managed by the BMS, which regulates the operating temperature to ensure optimal performance. As a result, the effects of temperature will be neglected in this study. Consequently, temperature dynamics will not be modelled, nor will their impact on other system behaviours be analyzed. Finally, the SOC will be calculated using the CCM. To address the known limitations of the standard CCM (3.2) , we propose introducing a parameter that adjusts the calculated SOC to match the actual SOC values provided by the BMS: 34 Chapter 4. Synchronization Mechanism Here, εk=yk−xT kˆ θk−1 is the a priori error, Pk is the covariance matrix of the parameter estimates, and Kk is the gain vector defined as: Kk=Pk−1xk 1+xT kPk−1xk (4.15) While effective in static environments, the basic RLS algorithm suffers from data saturation [25]. If the input data is sufficiently exciting, the elements of P will converge to zero, causing the parameter estimates to become fixed and unresponsive to new data [2]. As a result, the algorithm is not well-suited for tracking time-varying parameters. To overcome this limitation, various strategies have been developed for handling dynamic scenarios. A common modification involves introducing an exponential forgetting factor 0<λ<1 , which progressively reduces the influence of older data, allowing the model to adapt to more recent changes. This is achieved by modifying the cost function (4.13) as follows: J= k ∑ i=1 λk−i(yi−xT iθi)2(4.16) The RLS algorithm with exponential forgetting takes the form:    ˆ θk=ˆ θk−1+Kkεk Pk=1 λ[Pk−1−KkxT kPk−1](4.17) The gain vector is updated as: Kk=Pk−1xk λ+xT kPk−1xk (4.18) Instead of using the forgetting factor λdirectly, an equivalent memory horizon defined by N=1 1−λ(4.19) can serve as a more intuitive tuning parameter, representing the effective memory length of the algorithm. First introduced in [14] and later in [2], a general forgetting algorithm was proposed, from which many common forgetting methods can be derived as special cases. The algorithm consists of two primary steps: •Time update (ˆ θk,k−1=ˆ θk−1,k−1 Pk,k−1=F{Pk−1,k−1}(4.20) •Measurement update (ˆ θk,k=ˆ θk,k−1+Pk,kxk[yk−xT kθk,k−1] P−1 k,k=P−1 k,k−1+xkxT k (4.21) Note that the notation used here aligns with that of the Kalman Filter. The key distinction between various forgetting algorithms lies in the choice of the function F{·} for the time update of P . This function must ensure that Pk,k−1 remains symmetric and positive definite, while Pk,k−1−Pk−1,k−1 must be positive semidefinite, since no new information is added during the time update step [14]. For example, the RLS algorithm with exponential forgetting can be derived from this general framework by setting F{Pk−1,k−1}=Pk−1,k−1/λ(4.22) Indeed, by substituting (4.20) with this specific choice of F{·} into (4.21) , and denoting ˆ θk,k≡ˆ θk and Pk,k≡Pk, we obtain the following: 4.1 Parameter identification methodology 35 (ˆ θk=ˆ θk−1+Pkxkεk P−1 k=P−1 k−1λ+xkxT k (4.23) To avoid the computational complexity of matrix inversion, the matrix inversion lemma can be applied: (A+UCV)−1=A−1−A−1U(C−1+VA−1U)−1VA−1(4.24) where A=P−1 k−1λ , U=xk , V=xT k and C=I1 (a 1×1 identity matrix). In this manner, (4.23) can be rewritten as    ˆ θk=ˆ θk−1+Pkxkεk Pk=1 λPk−1−Pk−1xk λ+xT kPk−1xk xT kPk−1(4.25) By applying the definition of the gain vector in (4.18) , we can directly derive (4.17) . It is important to note that (4.21) should not be implemented directly. Instead, using a square root or U-D factorization algorithm is recommended to avoid numerical issues, as suggested in [3]. When employing the exponential forgetting method (4.17) with a constant forgetting factor, another issue arises in scenarios of insufficient excitation. Since information is continuously forgotten, periods with little or no data can lead to unbounded growth of the covariance matrix. Given that the adjustment of parameter estimates is proportional to P , the algorithms become highly sensitive to noise. This phenomenon is referred to in the literature as blow-up or covariance wind-up [2], [13], [14]. To address this problem, several methods have been proposed that aim to maintain tracking capability and prevent unbounded growth of P . One widely adopted approach, suggested by Fortescue in [12], involves the use of a time-varying forgetting factor to stabilize the measure of information content over time. This time-varying law is given by λk=1−ε2 k Σ0(1+xT kPk−1xk)(4.26) Here, Σ0=σ2 0N0 represents the constant measure of information content, where σ2 0 is the expected measurement noise variance based on system knowledge, and N0 is a nominal memory length that influences the filter’s adaptation speed when used as a tuning parameter [12]. With this method, the forgetting factor approaches one when little information is available and prediction errors are small, while it decreases in the opposite scenario. Additionally, imposing a lower bound λk≥λmin can prevent the forgetting factor from becoming too small, which could induce instability. The main limitations of using (4.22) for the time update are twofold. First, older information is uniformly discarded across all directions. Consequently, if the information flow is confined to a subspace, the corresponding eigenvalues of P may grow unbounded, resulting in blow-up. Second, there are scenarios where prior knowledge indicates that certain parameters exhibit more rapid time variations than others. The current schema does not accommodate this information [14]. A substantial body of literature has proposed more sophisticated variations of the RLS algorithm to mitigate these issues, including directional forgetting algorithms [4], [5], [7] and multiple forgetting factor algorithms [2], [9]–[11], [13], [14]. Some of these algorithms have been successfully applied to identify parameters in EMCs. For example, [26] combined the multiple forgetting factor algorithm from [11] with a modified version of Fortescue’s method to estimate parameters of a 1RC Thevenin model, aiming to account for the differing dynamics of the parameters. Similarly, [8] utilized a multiple forgetting factor algorithm inspired by [9], where the values for the forgetting factors were determined using Particle Swarm Optimization (PSO). This work considers the directional forgetting algorithm proposed by Cao et al. in [7]. This algorithm utilizes an orthogonal decomposition of the information matrix Rk=P−1 k along the exciting direction, enabling old data to be forgotten selectively in directions where it can be replaced by new information. The time update equation for the covariance matrix in this algorithm is given by Pk,k−1=Pk−1,k−1+1−λk λk xkxT k xT kRk−1xk (4.27) To complete the algorithm, the update equation for the information matrix Rkis defined as 36 Chapter 4. Synchronization Mechanism Rk= [I−Mk]Rk−1+xkxT k(4.28) where      Mk= (1−λk)Rk−1xkxT k xT kRk−1xk if |xk|>δ Mk=0if |xk| ≤ δ (4.29) Here, δ serves as a threshold, ensuring that decomposition occurs only when |xk| exceeds the dead zone [7]. This parameter can be estimated based on the noise level in the data. To compute the forgetting factor λk at each time step, we will utilize Fortescue’s method (4.26) , ensuring that λk≥λmin . For estimating σ0 , we will calculate the exponentially weighted variance of the a priori error, as implemented in [79], allowing for an adaptive and dynamic σ0rather than relying on a fixed value. Another important consideration is that not all input-output data are equally suitable for accurate parameter identification. In [6], a 1RC Thevenin model is used to illustrate how the accuracy of various identification scenarios is affected by factors such as voltage noise, current amplitude, and frequency. While the current profile is generally uncontrollable during normal operation, this observation offers valuable insights into the potential challenges of parameter identification. Autoregressive exogenous model (ARX) of the terminal voltage Before proceeding, it is necessary to establish a model of the form (4.12) to apply the RLS algorithm. To achieve this, we will follow the procedure outlined in [1], which details the method for the generic nRC Thevenin model. Specifically, our focus will be on the case where n=1. First, the electrical behaviour of the 1RC Thevenin model can be expressed in the frequency domain as follows: U(s) = OCV(s)−I(s)R0+R1 1+R1C1s(4.30) Defining E(s) = U(s)−OCV (s), the transfer function can be represented as: G(s) = E(s) I(s)=−R0+R1+R0R1C1s 1+R1C1s(4.31) Using the bilinear transformation: s=2 Ts 1−z−1 1+z−1(4.32) where zis the discretization operator and Tsis the sampling time, (4.31) can be rewritten as: G(z−1) = a2+a3z−1 1−a1z−1(4.33) Here, the coefficients a1,a2and a3are defined as:                    a1=−Ts−2R1C1 Ts+2R1C1 a2=−R0Ts+R1Ts+2R0R1C1 Ts+2R1C1 a3=−R0Ts+R1Ts−2R0R1C1 Ts+2R1C1 (4.34) From (4.33), we derive the equation Ek=a1Ek−1+a2Ik+a3Ik−1(4.35) The OCV is significantly influenced by the SOC , temperature, and usage history. However, if we consider small energy consumption relative to the usable capacity and a properly designed battery cooling system, 4.2 Parameter learning methodology 37 the variations in these factors over a sample time can be neglected [1]. This assumption holds in our case, allowing us to approximate OCV(k)≈OCV(k−1). Consequently, (4.35) can be expressed as: Uk= (1−a1)OCV(k)+a1Uk−1+a2Ik+a3Ik−1(4.36) Finally, defining the output y=Uk , the regressor vector as xk= [1Uk−1IkIk−1] , and the parameter vector as θk= [a0a1a2a3]with a0= (1−a1)OCV(k), we can write: yk=xT kθk(4.37) Once the RLS algorithm has been applied and the parameter vector obtained, the original parameters of the Thevenin model — OCV , R0 , R1 and C1 — can be determined from (4.34) . This leads to the following relationships:                                OCV =a0 1−a1 R0=a3−a2 1+a1 R1=−2(a3+a1a2) 1−a2 1 C1=−Ts 4 (1+a1)2 a3+a1a2 (4.38) 4.2 Parameter learning methodology 4.2.1 Batch vs online learning In machine learning,batch learning and online learning (also called stream or incremental learning) represent two different approaches to training models. • Batch Learning: This approach trains models using all available data at once. It works well in situations where the entire dataset is accessible at the start. However, batch learning can be memoryintensive, particularly with large datasets, since it requires loading all data into memory for each training cycle. Once trained, the model doesn’t change until retraining with a new dataset. • Online Learning: In contrast, online learning is well-suited for scenarios where data arrives incrementally, such as real-time applications (e.g., stock price predictions or sensor data streams). Instead of using the full dataset at once, online learning updates the model one observation at a time as data arrives. This makes it more efficient in terms of memory and allows the model to adapt continuously to new data, without needing full retraining. A significant advantage of online learning is its ability to manage concept drift. Concept drift occurs when the statistical properties of a target variable, or the relationship between input features and the target variable, change over time. If a model is not updated to reflect these changes, its performance may degrade. Figure 4.3 illustrates the four types of concept drift, categorized based on the patterns of data distribution changes [83]. To maintain accuracy over time, online learning algorithms must incorporate techniques specifically designed to address concept drift. However, this adaptation also presents challenges. Online learning makes it difficult to store and utilize prior observations for adjusting future updates, as the model primarily relies on recent data. Additionally, standard practices in machine learning, such as standardization and validation of model performance, must be restructured to suit online learning, necessitating the computation of running statistics and metrics [81]. The River Python library [80] provides a robust framework for efficiently implementing online learning techniques. This work will leverage the tools offered by this library to facilitate the application of online learning methods. 38 Chapter 4. Synchronization Mechanism Figure 4.3 Types of concept drift [82]. 4.2.2 Adaptative Random Forest regression for online learning Ensemble methods and neural networks are two of the most popular machine learning techniques used for regression tasks in the presence of concept drift [85]. Ensemble learners, in particular, are frequently favored for learning from evolving data streams due to their ability to achieve high performance with minimal optimization. One of their key strengths lies in their flexibility, as they enable the dynamic addition, updating, resetting, or removal of individual learners, making them highly adaptable to environments where data characteristics change over time [86]. In this work, the Adaptive Random Forest (ARF) technique is employed. Initially introduced in [86] for classification tasks and later extended to regression in [87], ARF has proven to be effective in handling evolving data streams. Specifically, the implementation used here is based on the version available in the River library [88], where Hoeffding Trees are used as the base learners, instead of the Fast Incremental Model Trees with Drift Detection (FIMT-DD) employed in the original ARF framework [87]. Moreover, this version introduces an enhanced strategy for monitoring predictions and detecting concept drift more efficiently [88]. The decision to use ARF over neural network-based algorithms is informed by several factors, primarily related to the requirements of this specific application. Neural networks generally require extensive tuning, larger datasets, and more complex architectures, which can lead to substantial overhead in both training time and computational resources. In contrast, ARF is designed to efficiently handle evolving data with minimal optimization, making it an ideal choice for scenarios where the data distribution shifts frequently. Additionally, in this application, the relatively small number of features and the simplicity of the relationships between them make ARF particularly well-suited. While neural networks can model complex, nonlinear relationships, ARF’s ensemble structure is more efficient for modelling simpler dependencies without the additional resource burden. This combination of adaptability, computational efficiency, and alignment with the characteristics of the data makes ARF the optimal choice for this work. The Adaptive Random Forest technique is grounded in several key principles [87]: •The base learner is a regression tree. While FIMT-DD was analyzed in [87], Hoeffding Trees will be utilized for this work. • Final predictions are made by averaging outputs, with optional weighting based on the performance of individual trees. 4.2 Parameter learning methodology 39 • Diversity is introduced into the forest by training trees on different subsets of data and limiting the split decisions to randomly selected subsets of features from the original input space. • Model updates rely on both internal and external drift detectors for each tree, with new trees growing in the background when a warning is detected. In this context, the ADWIN (ADaptive WINdowing) algorithm [84] was used in [87], and it will also be used in this work. We will now explore these aspects in more detail. Hoeffding Trees are the online equivalent of decision trees typically used in batch learning, designed to efficiently learn from large-scale data streams. Figure 4.4 illustrates an example decision tree structure for predicting internal resistance, R0. Figure 4.4 Schematic of a decision tree for internal resistance R0prediction. The algorithm takes advantage of the Hoeffding bound, a statistical property that allows for confident node-splitting decisions based on a relatively small subset of data: ε=sR2ln 1/δ 2n(4.39) where •ε is the Hoeffding bound, representing the probabilistic bound between the observed mean and the true mean. •Ris the range of the random variable. •δis the probability that the true mean falls within the bound. •nrepresents the number of observations in the sample. This bound ensures that the algorithm’s decisions converge to those it would have made if it had access to the full dataset. Like traditional decision trees, online trees can also suffer from overfitting. To mitigate this, model complexity is controlled by limiting tree depth (see Figure 4.4) or constraining memory usage [89]. In 40 Chapter 4. Synchronization Mechanism ARF, this is further handled by using an ensemble of trees trained on resampled data, similar to the bagging technique, and by restricting each tree’s access to a random subset of features at each node [86]. This introduces diversity, enabling the final prediction to be made through voting. Figure 4.5 depicts the ensemble prediction process. Figure 4.5 Schematic of an ensemble of trees, describing the prediction process. To address concept drift, ARF incorporates drift detectors to dynamically update both individual trees and the ensemble structure as needed. In this work, the ADWIN algorithm, as described in [90], is employed for this purpose. ADWIN [84] is a drift detection algorithm that maintains a dynamic window of recent observations. This window adjusts its size automatically based on data stability, expanding when data appears stable and contracting when drift is detected. Figure 4.6 illustrates how ADWIN reacts to different types of concept drift. Figure 4.6 Output of the ADWIN algorithm under different types of concept drift [84]. Left: Abrupt change. Right: Gradual changes. W denotes the dynamic window size; µt represents the true expected value at time t, and ˆ µWcorresponds to the observed average within the window. ADWIN manages concept drift by utilizing the Hoeffding bound, which offers a probabilistic limit on the discrepancy between the observed mean of a sample and the true mean. This enables ADWIN to evaluate whether the difference between older and more recent data within the window is statistically significant. If the difference exceeds a defined threshold, signaling potential concept drift, ADWIN responds by reducing the window size, discarding the older data to adapt to the new data distribution. 4.2 Parameter learning methodology 41 The ARF implementation used here [88] provides the flexibility to tune several hyperparameters, organized into distinct groups. For instance, parameters related to the base learner include tree-splitting criteria and pruning strategies. Ensemble-related settings cover aspects like the number of trees, prediction aggregation methods, bagging parameters, and feature selection for splitting. Furthermore, drift detection mechanisms and their associated parameters can also be adjusted to better suit the specific data characteristics. 5 Validation of the Digital Twin: Insights from Experimental Results In this chapter, the goal is to validate the proposed methodologies and, ultimately, the Digital Twin framework. First, the experimental setup designed for this validation will be described. Next, the developed methodologies will be applied, and the results will be analyzed and discussed. Additionally, the simulation capabilities of the resulting Digital Twin will be assessed. 5.1 Experiments description Although the proposed methodology is designed for application during normal battery operation, a series of experiments were designed to evaluate its performance. Standard pulsed charge and discharge tests were selected for this purpose. In each test, the current magnitude and direction remain consistent for each pulse, and this is repeated across the 10% −90% SOC range, which is constrained by the BMS. Both charge and discharge current values of 10A and 20A were used, corresponding to 0.2 and 0.4 C-rates, respectively. These values were chosen to represent typical operating conditions, as the BMS limits the C-rate to 0.5. The duration of the pulses and relaxation periods was carefully chosen to capture sufficient system dynamics, while also minimizing significant SOC variation within a pulse. This ensures the relationship between SOC and the identified parameters can be analyzed effectively. Figures 5.1 and 5.2 illustrate the terminal voltage, current, and SOC from a sample experiment. Due to resource constraints, including limited access to experimental equipment and facilities, the tests could not be conducted continuously across the full SOC range. Consequently, the experiments were executed in segments. Figure 5.3 demonstrates the process of constructing a full SOC range experiment from individual segments. The four complete SOC range tests, corresponding to different combinations of operating conditions, are shown in Figure 5.4. In all figures, k denotes the time step with respect to the start of the experiment, with a sample time of Ts=1second. This notation will also be used in the subsequent figures. 5.2 Results 5.2.1 State of charge (SOC) modelling Parameter identification Here, the results of applying the Kalman Filter to estimate the SOC and identify parameters of the proposed SOC model are presented. Figures 5.5 through 5.8 illustrate the results for each experiment, while Figure 5.9 shows the identification of the parameter η across all four experiments. These results were obtained using the methodology outlined in Section 4.1.1, with R=0.25 , α=1.5 , and the process noise covariance matrix: Q=2×10−40 0 5 ×10−6(5.1) 43 50 Chapter 5. Validation of the Digital Twin: Insights from Experimental Results Figure 5.11 Results of the first phase of the training process to predict η. Figure 5.12 Example of dataset generation for the second phase of the training process to predict η. 5.2 Results 51 Figure 5.13 Results of the second phase of the training process to predict η. Figure 5.14 Evolution of the performance metric throughout the entire training process to predict η. 52 Chapter 5. Validation of the Digital Twin: Insights from Experimental Results 1. Predict the parameter value ηpred,k for the current predicted SOC , SOCpred,k , and experiment number ExpNum using the trained ARF model ARFη,trained: ηpred,k=ARFη,trained(SOCpred,k,ExpNum)(5.3) 2. Predict the SOC at the next time step k+1 using the SOC model (3.9) with the predicted parameter value ηpred,k: SOCpred,k+1=SOCpred,k−ηpred,kIkTs Crated (5.4) This procedure can be applied across all experiments for each training iteration, allowing the analysis of the model’s SOC predictive capabilities throughout the training process. The results of this analysis are presented in Figure 5.15, showing the SOC predictive error defined as SOCpred,k−SOCk , where SOCk is the continuous SOC derived previously. In the initial three iterations, corresponding to the synthetic dataset training, predictions diverge from the true values over time. Once training with the identified dataset begins, this drift diminishes with each iteration as the model adapts to the identified data. In most cases, drift is nearly eliminated within the nominal SOC range, while some room for improvement remains for extreme SOC values. This issue will be addressed as more data becomes available for these zones, enhancing model confidence and allowing for further splits to improve adjustments. The evolution of the predicted SOC MAE throughout the iterations is displayed in Figure 5.16, showcasing a gradual improvement in predictive capabilities once training with the identified dataset begins. Finally, an example illustrating the substantial reduction of drift is depicted in Figure 5.17. Figure 5.15 SOC predictive error evolution throughout the entire training process. 5.2 Results 53 Figure 5.16 MAE of the SOC predictions per iteration throughout the entire training process. Figure 5.17 Example illustrating the substantial reduction of drift achieved in SOC prediction. 54 Chapter 5. Validation of the Digital Twin: Insights from Experimental Results 5.2.2 Terminal voltage modelling Parameter identification This section presents the results of applying the Recursive Least Squares (RLS) filter to estimate the parameters of the 1RC Thevenin voltage model: OCV , R0 , R1 , and C1 . Before discussing the results, however, it is important to address an aspect related to the validity of the chosen approach. As stated in 3.2, the selected approach assumes a definition of the OCV parameter that limits its validity to conditions where long-term dynamics have developed and extended discharge times are absent. Therefore, the filter cannot be applied to all the data obtained in each experiment; a waiting period is needed to allow long-term dynamics to develop. Upon analyzing the data, 45 minutes is considered an acceptable estimate for this period. Furthermore, data points where the discharge time exceeds the point at which short-term dynamics are no longer significant must be removed. This time can be estimated as tm=3τ1=R1C1 , where the system reaches 95% of the steady-state response. From the identified values for R1 and C1 , it can be estimated that tm≈100 seconds in this case. By applying these two filters, a suitable dataset for the RLS filter can be extracted for each experiment, as shown in Figure 5.18. Without this preprocessing step, a different voltage model might be obtained that minimizes the error for all samples, instead of focusing on the region of interest where the model is valid (see Figure 5.19). Figure 5.18 Example of the extraction of suitable samples to feed the RLS Filter. Following this preprocessing, the RLS filter, as described in 4.1.2, can be applied. To simulate the filter’s behaviour if additional data were available beyond what was obtained in this study, the filter is applied 5 times to the available dataset. The results are shown in Figures 5.20-5.22. It can be observed that with each iteration, the filter extracts more information from the input-output data, resulting in increased stability of the parameter estimates. This behaviour will allow us to test the performance of the ARF algorithm in handling gradual drifts. It is worth noting that these results could be useful for continuous battery health monitoring and predictive maintenance. For example, it is well-known that the internal resistance R0 increases throughout a battery’s lifetime. By monitoring its evolution, informed decisions can be made to maximize battery health or assist in 5.2 Results 55 Figure 5.19 Example of the difference of considering the preprocessing step for the RLS Filter application. scheduling maintenance tasks. This was one of the motivations mentioned in the introductory chapter of this work for the development of Digital Twins. Online learning methodology As previously mentioned, the identified dataset of parameters will now be used to train one ARF model per parameter. Thanks to the gradual changes in the identified parameters due to the RLS filter’s convergence process, the ARF algorithm’s behaviour under gradual drifts will be tested. The same features used for predicting η will be employed here, along with identical settings for the ARF algorithm and the same performance metric. The evolution of the predictions made by each model after each iteration is shown in Figures 5.23-5.26, while the evolution of the performance metric throughout the training process for each model is depicted in Figure 5.27. From these figures, it can be concluded that the algorithm successfully handles gradual drift, achieving a high degree of fit to the parameter distribution in the final iteration. Similar to the SOC modelling, the evolution of the terminal voltage predictive capabilities during the training process can be assessed. In this case, once the terminal voltage is initialized to its known starting value, the procedure to simulate terminal voltage for each time step kis as follows: 1. Based on the SOC obtained through simulation, SOCpred,k , and the experiment number, ExpNum , use the corresponding trained ARF model to predict each parameter:          OCVpred,k=ARFOCV,trained(SOCpred,k,ExpNum) R0|pred,k=ARFR0,trained(SOCpred,k,ExpNum) R1|pred,k=ARFR1,trained(SOCpred,k,ExpNum) C1|pred,k=ARF C1,trained(SOCpred,k,ExpNum) (5.5) 2. With these parameter estimates, the terminal voltage at the next step can be calculated using the model defined in (3.7). 56 Chapter 5. Validation of the Digital Twin: Insights from Experimental Results Figure 5.20 Evolution of the parameters a0,a1,a2,a3throughout the identification process. Figure 5.21 Evolution of the a priori,a posteriori error and forgetting factor throughout the identification process. 5.2 Results 57 Figure 5.22 Evolution of the parameters OCV ,R0,R1,C1throughout the identification process. Figure 5.23 Evolution of the prediction made by the OCV model throughout the learning process. 58 Chapter 5. Validation of the Digital Twin: Insights from Experimental Results Figure 5.24 Evolution of the prediction made by the R0model throughout the learning process. Figure 5.25 Evolution of the prediction made by the R1model throughout the learning process. 5.2 Results 59 Figure 5.26 Evolution of the prediction made by the C1model throughout the learning process. Figure 5.27 Evolution of the performance metric throughout the entire training process for each model. 66 List of Figures 5.4 Terminal voltage evolution across all experiments performed 45 5.5 SOC model parameter identification for Experiment I 46 5.6 SOC model parameter identification for Experiment II 46 5.7 SOC model parameter identification for Experiment III 47 5.8 SOC model parameter identification for Experiment IV 47 5.9 SOC model parameter identification for all experiments combined 48 5.10 Example of synthetic dataset generation for the first phase of the training process to predict η49 5.11 Results of the first phase of the training process to predict η50 5.12 Example of dataset generation for the second phase of the training process to predict η50 5.13 Results of the second phase of the training process to predict η51 5.14 Evolution of the performance metric throughout the entire training process to predict η51 5.15 SOC predictive error evolution throughout the entire training process 52 5.16 MAE of the SOC predictions per iteration throughout the entire training process 53 5.17 Example illustrating the substantial reduction of drift achieved in SOC prediction 53 5.18 Example of the extraction of suitable samples to feed the RLS Filter 54 5.19 Example of the difference of considering the preprocessing step for the RLS Filter application 55 5.20 Evolution of the parameters a0,a1,a2,a3throughout the identification process 56 5.21 Evolution of the a priori,a posteriori error and forgetting factor throughout the identification process 56 5.22 Evolution of the parameters OCV,R0,R1,C1throughout the identification process 57 5.23 Evolution of the prediction made by the OCV model throughout the learning process 57 5.24 Evolution of the prediction made by the R0model throughout the learning process 58 5.25 Evolution of the prediction made by the R1model throughout the learning process 58 5.26 Evolution of the prediction made by the C1model throughout the learning process 59 5.27 Evolution of the performance metric throughout the entire training process for each model 59 5.28 Example of predicted voltage versus experimental data across iterations 60 5.29 Example of voltage error across iterations 61 5.30 Evolution of the voltage MAE across experiments throughout iterations 61 List of Tables 2.1 Relevant specifications of the BESS under study 9 2.2 Relevant specifications for the Epic Power converter pack on the BESS side 9 2.3 Specifications of the IT6018B-1500-40 power supply 11 3.1 Classical mathematical models. 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