Reliability Analysis of a PCM–Liquid Hybrid Battery Thermal Management System for Electric Vehicles

Abstract

Electric vehicles (EVs) utilize batteries that generate thermal energy during charging and discharging processes. Inadequate heat management can result in thermal runaway, which is marked by a rapid and uncontrolled increase in battery temperature and may lead to fire or explosion. Battery Thermal Management Systems (BTMSs) are implemented to reduce the risk of thermal runaway by maintaining battery temperature within a defined safe operating range. Hybrid BTMS configurations integrate multiple cooling methods to enhance operational effectiveness and reliability, surpassing the performance of conventional liquid coolant systems. This work investigates the enhanced efficiency of an integrated phase change material (PCM)–liquid hybrid approach evaluated under severe 3C discharge conditions over a 10,000 h operational reliability window. Our study carefully investigates the system-level reliability using a Functional Fault Tree Analysis (FTA) framework to classify top, intermediate, and basic events. Multi-physics and probabilistic evaluation results demonstrate that the hybrid system achieves a 5 °C reduction in peak cell temperature and a 3 °C improvement in spatial temperature uniformity compared to standard liquid cooling. Reliability assessment establishes a system top-event failure probability of 6% over the 10,000 h window, identifying the coolant pump as the primary failure bottleneck with an individual contribution of 27%. These quantitative insights advance our understanding of hybrid safety architectures, providing essential baseline metrics for future electric vehicle thermal management development.

1. Introduction

Electric vehicles have become increasingly popular in recent years, as they offer a more environmentally friendly and sustainable alternative to traditional gasoline-powered vehicles. This shift towards EVs is driven by a number of factors, including the need to address climate change, reduce air pollution, and conserve fossil fuel resources. One of the key factors that has enabled the rise of EVs is the advancement of battery technology. Researchers and engineers have developed new types of batteries that are more efficient, lightweight, and energy-dense than previous generations. This has made it possible to create EVs that can travel longer distances on a single charge and are more affordable to purchase and operate. The adoption of EVs is also being accelerated by government policies and incentives. A number of countries have created financial incentives for consumers buying EVs, and some have even prohibited the sale of new gasoline-powered vehicles. This policy support is making EVs more affordable for consumers and accelerating the transition to zero emission transportation systems. Overall, the arrival of EVs is a good thing for the environment and society in general. EVs provide a more sustainable and efficient way of traveling and can help us to become less dependent on fossil fuels. With battery technology improving and the price of EVs dropping, we are going to see more individuals make the switch to electric vehicles in the next few years.

Table 1. Nomenclature of top and intermediate events.

Name	Description
$F_{12}^1$	Improper heat transfer to PCM
$F_{21}^1$	Allow current to flow when coolant temperature is low
$F_{51}^3$	Heat exchanger unable to perform
$F_{31}^3$	Absorb heat and reduce the coolant temperature
$F_{21}^3$	Reduce coolant temperature through heat exchanger
$F_{11}^3$	Develop coolant flow at high pressure
$F_{31}^2$	Send transducer signal to pump to vary the flow rate
$F_{21}^2$	Send control signals to transducer to vary the flow rate
$F_{11}^2$	Send temperature signal to controller

provides Nomenclature of top and intermediate events.

With the increasing adoption of EVs, thermal challenges associated with their batteries have become more evident. Lithium-ion batteries in EVs experience efficiency losses at extreme temperatures, electrode degradation at elevated heat, and safety risks such as thermal runaway, all of which significantly impact vehicle performance, reliability, cost, and safety. Maintaining optimal battery temperature is essential for efficient charging and discharging, highlighting the critical importance of effective EV battery thermal management [1]. Multiple studies have investigated Battery Thermal Management Systems (BTMSs) for EVs to address these challenges and propose solutions.

Thermal management strategies encompass various methodologies, including air cooling, liquid cooling, phase change materials (PCMs), heat pipes, and refrigerated cooling. Air and liquid cooling techniques have achieved significant commercial adoption [2,3]. However, ongoing research aims to enhance BTMS to address the diverse and evolving requirements of electric and hybrid vehicles.

Electric vehicles employ several cooling methods, each presenting distinct challenges. Air cooling is often insufficient, particularly for high-performance EVs or in hot climates, leading to uneven temperature distribution and potential overheating during sustained high-power operations such as fast charging. Liquid cooling is effective but increases system complexity, weight, and maintenance requirements. Phase change material (PCM) cooling offers energy efficiency but is limited to specific temperature ranges and heat loads. Hybrid cooling systems, which combine PCM and conventional methods, address these limitations. In such systems, liquid cooling manages continuous heat removal, while PCM absorbs peak thermal loads. This integration enables rapid thermal recovery and optimal performance under varying thermal demands, providing a balanced and efficient solution for EV battery cooling.

This study investigates a Hybrid Battery Thermal Management System (HBTMS) that integrates phase change material (PCM) and liquid coolant to improve the cooling performance of Battery Thermal Management Systems (BTMSs). This work investigates the higher efficiency of such a system, surpassing the capabilities of conventional liquid cooling, by combining these technologies in a hybrid approach. The study also includes a comprehensive reliability analysis, employing Functional Fault Tree Analysis (FTA) to classify top, intermediate, and basic events, further enhancing our understanding of BTMS performance. The paper is arranged as follows: Section 2 deals with the literature survey. Section 3 deals with the Battery Thermal Management System, while Section 4 is the analysis of the PCM and liquid coolant Hybrid Battery Thermal Management System. Section 5 focuses on reliability analysis, where FTA and procedures for the Functional Event Digraph are explained. Section 6 is a blockwise detailed analysis for each assembly, and Section 7 is the collective analysis. Section 8 is the discussion of the developed model and FTA of the developed model, and Section 9 is the conclusion.

2. Literature Survey

Electric vehicles (EVs) employ electric motors, providing a more environmentally friendly option compared to conventional internal combustion engines [1]. There exist four primary classifications: Battery Electric Vehicles (BEVs), Hybrid Electric Vehicles (HEVs), Plug-In Hybrid Electric Vehicles (PHEVs), and Fuel Cell Electric Vehicles (FCEVs) [2].

Battery storage, specifically in the context of electric vehicles (EVs), plays a crucial role in the domain of sustainable energy advancement. The importance of energy management, particularly in relation to operational temperatures, is crucial in determining the performance of batteries. Nevertheless, it is important to note that all these EVs have thermal runaway problems, which consequently require the implementation of a Battery Thermal Management System (BTMS) to ensure consistent and dependable performance. BTMSs play a crucial role in maintaining ideal battery temperatures, thereby improving both efficiency and safety in a wide range of vehicle categories, including compact vehicles, SUVs, and pickups [3].

A variety of Battery Thermal Management Systems (BTMSs) are available. Among these, phase change materials (PCMs) have received significant attention due to their unique ability to absorb and release latent heat during phase transitions. When integrated into BTMSs, PCMs serve as thermal buffers that efficiently regulate temperature fluctuations within battery packs. This approach addresses the challenge of maintaining stable temperatures and reducing the risk of overheating, which is critical for ensuring the safety and optimal performance of lithium-ion batteries [4,5].

Researchers have examined hybrid thermal management systems that incorporate multiple cooling techniques. For example, the combination of forced convection with phase change material (PCM) cooling has exhibited enhanced thermal efficiency, particularly under conditions of elevated discharge rates. Hybrid systems of this nature have the inherent benefit of effectively utilizing both active and passive cooling methods, thereby offering a resilient approach to the management of thermal dynamics in lithium-ion batteries [4].

Moreover, researchers have investigated the incorporation of phase change materials (PCMs) with various substances, including metal foam and graphite matrices, in order to improve thermal conductivity and enhance heat transmission capacities. These technological breakthroughs play a crucial role in the development of efficient and reliable Battery Thermal Management Systems (BTMSs), which are important for facilitating the sustainable expansion of electric and hybrid vehicles within the dynamic energy environment [6,7,8].

Reliability in EVs pertains to the vehicle’s capacity to fulfill its intended purpose without malfunctions or disruptions over time. Ensuring the reliability of EVs is crucial as it directly impacts safety, performance, and cost considerations. A reliable EV can minimize maintenance expenses and extend the vehicle’s longevity, ultimately delivering a more favorable return on investment for the owner [9,10,11,12]. Various reliability tools are present in the literature, including the Markov model, Monte Carlo simulation, state space model, Petri nets, Fault Tree Analysis, etc. The field of risk analysis has seen significant research and examination of Fault Tree Analysis (FTA) approaches, with a particular focus on their uses in relation to safety and economically important assets. Fault Tree Analysis (FTA) comprises advanced modeling methodologies, including conventional fault trees and their enhanced variants, such as dynamic, repairable, and extended fault trees. These methods are designed to evaluate hazards associated with critical systems and assets both qualitatively and quantitatively. FTA is particularly appropriate for analyzing EVs due to the complexity and interconnectivity of their systems. The high voltages and intricate electrical components present in EVs introduce significant safety challenges [13].

FTA provides a structured framework for analyzing system complexities, enabling engineers to identify specific failure paths and underlying causes. Visual representations support the understanding of probable failure scenarios, thereby improving stakeholders’ ability to allocate resources efficiently. Additionally, FTA integrates with risk management systems, allowing for the calculation of probabilities associated with each event in the tree. The methodology also incorporates historical data, facilitating the analysis of past failures to prevent future incidents. Recognized and endorsed by regulatory authorities, FTA is a critical tool for electric vehicle (EV) manufacturers, ensuring compliance with safety regulations and supporting comprehensive reliability assessments. Advances in technology, such as Model Based Dependability Analysis (MBDA) and the integration of machine learning and data mining techniques, further strengthen the theoretical foundation of risk analysis. Interdisciplinary approaches not only refine methodology but also play a crucial role in improving the reliability, dependability, and safety of complex systems, underscoring the ongoing commitment to excellence in risk assessment [14].

Efficient thermal management is essential for modern electric and hybrid vehicles, as temperature regulation is critical to battery performance and longevity. Literature demonstrates that the reliability of electric vehicles is significantly influenced by the effectiveness of their Battery Thermal Management Systems (BTMSs). Therefore, understanding the structure, functions, and hybrid configurations of BTMSs is vital for maintaining stability, safety, and consistent performance under diverse operating conditions. The following section provides a comprehensive analysis of BTMS concepts, operational mechanisms, and their importance in maintaining optimal battery temperatures for reliable vehicle operation.

Recent reliability studies of EV battery systems have employed Dynamic Fault Trees (DFTs), Bayesian Networks (BNs), and quantitative Fault Tree Analysis (FTA) to evaluate failure propagation and system-level risk. DFT models are effective in representing sequence-dependent failures and dynamic interactions; however, they require extensive failure-state information and high computational effort. Bayesian Networks can incorporate uncertainty and update failure probabilities using operational data, but their effectiveness depends on the availability of large datasets, which are often unavailable for emerging hybrid BTMS architectures. Quantitative FTA provides probabilistic assessment of failure events but typically focuses on component-level failures without explicitly capturing functional interactions among thermal management subsystems. In contrast, the present work adopts a Functional Cause Analysis framework integrated with Functional Event Digraphs (FEDs) and FTA. This approach explicitly maps functional dependencies among PCM modules, liquid cooling circuits, sensors, controllers, and actuators before constructing the fault tree. Consequently, the proposed methodology enables systematic identification of failure propagation paths in a PCM–liquid hybrid BTMS while maintaining lower computational complexity and improved interpretability [8].

3. Battery Thermal Management System (BTMS)

A BTMS is an important component in EVs that controls the battery pack’s temperature, a crucial factor influencing both performance and longevity. In the process of operation, batteries generate heat from electrochemical reactions, internal resistance, and heat dissipation to the surroundings. Excessive heat can result in reduced performance, degradation, and even fire risks, while low temperatures also impair efficiency and reliability [15].

For lithium-ion chemistries such as lithium nickel manganese cobalt oxide (NMC), a nominal temperature envelope of 20 °C to 45 °C is widely targeted in application literature to minimize capacity fade and internal resistance growth. However, this optimal operating window must be considered with caution, as exact safe thermal boundaries fluctuate significantly depending on the specific cell form factor, chemical sub-generation, C-rate severity, and local environmental conditions [10]. The BTMS prevents issues such as thermal runaway, thereby ensuring continuous operation under ideal conditions and mitigating accelerated deterioration. BTMS solutions are available in active, passive, and hybrid configurations, utilizing specialized radiators and condensers specifically designed for these applications [16].

The BTMS operates at typical vehicle voltages ranging from 450 to 800 V DC, with additional low voltage options at 12 or 24 V DC. It plays a critical role in ensuring the safety and efficiency of electric vehicles. Key benefits include enhanced battery performance, which results in increased driving range and faster charging, as well as prolonged battery life by preventing excessive heat accumulation and reducing fire risks. Therefore, an effective BTMS is essential for the safe and efficient operation of EVs, contributing to improved performance, extended lifespan, and minimized fire hazards [17].

Types of BTMSs

BTMSs help to keep the battery temperature within an optimal range, which is important for performance and longevity. BTMS can be categorized in various ways, including passive vs. active, parallel vs. series, heating vs. cooling, liquid vs. air, and thermoelectric cooler (TEC) vs. direct refrigerant cooling [11].

Passive BTMSs do not use any external energy to control the battery temperature. Instead, they rely on natural convection or PCMs to remove heat. Passive BTMSs are simpler and less expensive than active BTMSs, but they are also less effective at controlling the battery temperature.

Active BTMSs use external energy, such as fans or pumps, to circulate a coolant through the battery pack and remove heat. Active BTMSs are more effective at controlling the battery temperature, but they also require more energy and are more complex and expensive.

Common types of BTMSs include [12]:

• Air cooling: This is the simplest and most common type of BTMS. It uses fans to circulate air around the battery pack, which helps to remove heat. Air cooling is a passive BTMS.
• Liquid cooling: This type of BTMS uses a liquid coolant to remove heat from the battery pack. The coolant is circulated through a series of channels in the battery pack and then cooled outside the vehicle. Liquid cooling is an active BTMS [9].
• Phase change material (PCM) cooling: This type of BTMS uses PCMs to store and release heat. The PCM is placed in contact with the battery cells and absorbs heat when the battery temperature gets too high. The PCM then releases the heat when the battery temperature gets too low. PCM cooling can be either active or passive [13].
• Heat pipe cooling: This type of BTMS uses heat pipes to transfer heat from the battery cells to a heat sink outside the battery pack. Heat pipes are passive devices that use evaporation and condensation to transfer heat [14].
• Refrigerant cooling: This type of BTMS uses a refrigerant to remove heat from the battery pack. The refrigerant is circulated through a series of coils in the battery pack and then cooled in a condenser outside the vehicle. Refrigerant cooling is an active BTMS.

Researchers are also developing new BTMS technologies, such as:

• Thermoelectric cooling: This type of BTMS uses thermoelectric devices to transfer heat from the battery pack to a heat sink. Thermoelectric devices are passive devices that use the Peltier effect to transfer heat.
• Evaporative cooling: This type of BTMS uses evaporation to remove heat from the battery pack. Evaporative cooling is a passive BTMS that is well-suited for EVs that operate in hot climates.

The type of BTMS used in an EV depends on a number of factors, such as the size and type of battery pack, the vehicle’s operating environment, and the desired performance and cost.

Active BTMSs are more common in high-performance EVs and EVs that operate in extreme environments. Passive BTMSs are more common in lower-cost EVs and EVs that operate in milder environments.

4. Analysis of Liquid Cooling and PCM–Liquid Hybrid Battery Thermal Management System

Effective thermal management of batteries is essential for maintaining optimal performance, reliability, and safety in EVs. Hybrid systems that use phase change materials (PCMs) with liquid coolants represent one of the most widely researched configurations among advanced BTMS technologies. Each method has its own pros and limitations when it comes to thermal regulation, operating efficiency, and design that affect how well an EV’s battery system works as a whole. This section provides a comparative examination, commencing with traditional liquid cooling systems and progressing to the development of PCM-assisted hybrid designs to mitigate the shortcomings of single-mode cooling.

4.1. Analysis of Liquid Cooling Battery Thermal Management System

A thermal management system that depends solely on liquid coolant may be regarded as less effective for electric vehicle applications owing to various inherent limitations. Although liquid cooling is highly effective for continuous heat removal through convective heat transfer, its thermal buffering capability during rapid transient heat generation is limited. Under conditions such as fast charging or high-power operation, localized temperature spikes may develop before the cooling loop can fully dissipate the generated heat. In contrast, phase change materials (PCMs) absorb transient thermal peaks through latent heat storage, helping to suppress temperature fluctuations and reduce hotspot formation. Therefore, integrating PCMs with liquid cooling can improve temperature uniformity and enhance overall thermal management performance.

Temperature regulation becomes increasingly complex in the absence of PCMs, particularly when there are rapid fluctuations in heat generation rates. Such conditions can lead to uneven cooling within the battery pack and the formation of hotspots, which negatively affect battery health and performance. Furthermore, when operating at sustained high power, liquid cooling systems alone may be insufficient to maintain optimal temperatures, increasing the risk of overheating. Additionally, these systems are significantly dependent on operational components, including pumps and blowers, for effective coolant circulation. Because the system needs active cooling, it uses more energy and may lose efficiency. It is also more likely to break down mechanically or electrically. Figure 1 shows a block diagram of a Liquid Cooling Battery Thermal Management System.

Hybrid thermal management solutions that combine active and passive cooling techniques are necessary because, although liquid cooling is an efficient way to continuously remove heat, it has limitations in terms of energy economy, stability, and dependability. Consequently, the following section details a liquid–PCM cooling-based hybrid battery thermal management system.

It should be noted that while active liquid cooling provides continuous heat removal via fluid transport, a pure passive PCM operates primarily as a temporary latent heat buffer. To ensure analytical rigor, the baseline performance comparison between the two systems was conducted under strictly identical boundary conditions, including equivalent thermal heat loads, coolant flow rates, channel geometries, and control parameters.

4.2. PCM and Liquid Coolant Hybrid Battery Thermal Management System

The BTMS employed in EVs utilizing a PCM and liquid coolant represents a hybrid approach integrating the strengths of both passive and active cooling methods [18,19,20]. A PCM-based BTMS is attractive due to its high energy density and isothermal energy exchange. Nevertheless, pure PCMs have constraints. Liquid cooling, an active system, is more efficient than air cooling and can heat the battery pack in cold weather. Combining a PCM and liquid coolant improves temperature distribution by using the PCM to address localized hot spots, while liquid cooling extracts most of the battery-generated heat under steady state conditions. A PCM and liquid coolant can be configured in parallel or series [21].

In this hybrid BTMS, a PCM, a material that transitions between solid and liquid states at specific temperatures, absorbs heat as the battery temperature rises, melting and transferring heat to the liquid coolant circulating through channels in the PCM. When the temperature drops, the PCM releases heat and solidifies [22]. This hybrid approach efficiently manages temperature, enhancing performance, range, and charging speed while extending battery lifespan and reducing fire risks. Although more intricate and costly than traditional systems, hybrid BTMSs using PCMs and liquid coolant offer advantages in terms of performance, longevity, and safety.

Thermal energy can be stored in three main ways: latent heat, sensible heat, and chemical energy. Latent heat and sensible heat are the most common, and latent heat can store more energy per unit volume than sensible heat. PCMs are used to store heat in a variety of applications, including heat pumps, solar engineering, and spacecraft thermal control. PCMs can be used to regulate the temperature of EV and HEV battery packs and ensure that the temperature is uniform throughout. One of the most important factors to consider when selecting a PCM for this application is its melting point, as the phase change process occurs at a constant temperature.

Figure 2 shows a basic diagram of a thermal management system (TMS) for a lithium-ion (Li-ion) battery. A pump circulates liquid coolant through the battery, and the direction of flow is controlled based on the temperature of the coolant at the exit of the battery. There are two main ways to reject heat from the hot coolant: through a liquid heat exchanger (radiator) or through an evaporator using refrigerant. A heater may be needed to keep the cell temperature within the optimal operating range under cold conditions. PCMs are placed between the cells in battery modules to absorb heat from the battery. The temperature of the cell increases until it reaches the melting point of the PCM, and further heat from the battery leads to a phase change without any further increase in temperature. Although PCMs can stabilize the temperature of the cells locally, heat still needs to be removed from the module matrix to maintain sustained performance. A significant amount of energy from the battery itself is traditionally used to reject heat from the fluid loop through active cooling systems. This parasitic load is significantly reduced in this study by utilizing a unique dual-stage configuration combining an inter-cell PCM with an external PCM tank connected directly to the primary cooling cycle. Specifically, the proposed architecture combines a primary inter-cell PCM matrix (wrapped circumferentially around individual 21,700 cells for immediate hotspot buffering) with a secondary, external downstream PCM tank assembly ($a_3^3$). The hot fluid exiting the battery modules passes through this external PCM tank, where the secondary phase change matrix absorbs residual thermal energy from the liquid coolant before it reaches the heat exchanger. This heat is eventually dissipated to the atmosphere via convection, drastically lowering the peak refrigeration load on the vehicle’s active components, balancing fluid loop temperatures, and expanding driving range [16].

The usefulness of a BTMS directly affects the reliability and durability of an EV. Advanced hybrid BTMS technologies that combine PCM and liquid cooling improve thermal management and reduce temperature fluctuations that can degrade battery performance. Reliability analysis is essential to confirm proper system operation throughout the EV’s lifecycle. By assessing the durability and efficiency of BTMS components, reliability studies help identify potential failure points, optimize system design, and enhance vehicle safety and performance.

The structural and reliability analysis is anchored upon a high-performance EV battery module architecture. The reference system consists of cylindrical 21,700 Lithium-ion cells arranged in a 10S6P module configuration, with a composite phase change material matrix (1.8 kg mass per module) wrapped circumferentially around the cells to buffer peak thermal loads at a 45 °C phase change threshold. Active heat transport is handled via mini-channel aluminum cold plates integrated within the cell matrix, operating at a variable coolant flow rate up to 1.5 L/min. The fluid loop is driven by a 12V DC centrifugal pump and a 1.2 kW micro-channel radiator heat exchanger. System regulation is governed by a proportional control strategy that monitors cell temperatures via localized sensors to dynamically adjust pump rates, keeping the battery pack within its optimal operating boundaries of 20 °C to 45 °C.

The design parameters presented in this study are not intended to represent a specific commercial EV battery pack. Instead, they define a representative reference architecture for reliability modeling purposes. The selected 21,700 cylindrical cells and 10S6P module configuration are based on battery module arrangements commonly reported in EV thermal management literature. The PCM mass (1.8 kg per module), coolant flow rate (up to 1.5 L/min), 12 V DC circulation pump, and 1.2 kW micro-channel radiator were chosen from design ranges reported in previous studies on hybrid PCM–liquid battery thermal management systems and commercially available automotive cooling components [17]. These parameters were adopted to establish a realistic and technically consistent framework for the functional reliability analysis [18]. Since the objective of this work is structural reliability assessment rather than thermal performance optimization, the analysis focuses on the functional interactions and failure propagation pathways of BTMS components rather than the sensitivity of specific thermal parameters [23].

5. Reliability Analysis of BTMS

Reliability analysis is a fundamental approach used to evaluate the probability that a system or component will perform its intended function under specified conditions and within defined timeframes [24]. In the context of EVs, reliability analysis examines critical components such as the battery, motor, and power electronics to ensure compliance with safety, durability, and efficiency standards [25]. The process encompasses component identification, performance data collection, mathematical modeling to estimate failure probabilities and lifespans, experimental evaluation of models, and the development of strategies to enhance reliability. Reliability analysis is a continuous process throughout the EV life cycle, spanning design, development, production, and operation. This approach ensures that EVs adhere to required safety, durability, and efficiency standards, thereby providing consumers with reliable and cost-effective transportation solutions [26].

The reliability analysis of Battery Thermal Management Systems (BTMSs) can be performed using various methodologies, including Failure Modes and Effects Analysis (FMEA), Monte Carlo simulations, Markov modeling, and Fault Tree Analysis (FTA). FMEA systematically identifies potential failure modes, their causes, and effects, enabling proactive risk mitigation. Monte Carlo simulations provide probabilistic insights into system behavior under uncertainty, while Markov models analyze state transitions and failure probabilities over time. Among these approaches, FTA is particularly suitable for this research because it offers a systematic and visual framework for identifying the root causes of system failures. By employing Boolean logic to represent failure paths, FTA enables a clear depiction of how specific component failures contribute to overall system dysfunction. Consequently, FTA serves as an effective tool for assessing BTMS reliability, identifying weaknesses, and implementing preventive measures to enhance safety and efficiency. While Markov models and Dynamic Fault Trees (DFTs) capture sequence-dependent failures, they introduce high computational complexity that is unnecessary for the steady-state Boolean interactions analyzed here. Standard FTA is uniquely appropriate because it yields clear, interpretable, and computationally efficient minimal cut sets directly linking localized component losses to systemic thermal consequences.

5.1. Fault Tree Analysis (FTA) of BTMS

FTA is a graphical technique used to identify the causes of a system failure [19]. It starts with the top event, which is the system failure, and then breaks it down into smaller and smaller events until the basic events are reached. Basic events are events that cannot be broken down further, such as a component failure [20].

FTA uses Boolean logic to combine the events. Boolean logic is a type of logic that uses the operators AND, OR, and NOT to combine statements. The events in an FTA are combined using Boolean logic to show how they can lead to the top event [21].

FTA is a top-down approach, meaning that it starts with the top event and then works its way down to the basic events. This is in contrast to other failure analysis techniques, such as bottom-up approaches, which start with the basic events and then work their way up to the top event.

FTA is a powerful tool that can be used to identify the causes of system failures and to assess the probability of those failures occurring. It is used in a variety of industries, including safety engineering, reliability engineering, and aerospace engineering [22].

Traditional Fault Tree Analysis (FTA) is a well-established tool in reliability engineering; however, its standard application typically relies on static, manual top-down failure mapping that does not account for dynamic subsystem dependencies. The novelty of the framework presented in this section offers an original adaptation of a structured Functional Cause Analysis (FCA) methodology, specifically designed for the multi-physics (coupled thermal and electrical) boundary conditions of a hybrid PCM–liquid Battery Thermal Management System (BTMS). By systematically mapping the complex feedforward and feedback loops across eight distinct operational assemblies (Section 6), this approach addresses the gap between qualitative physical degradation and systemic structural vulnerabilities. It enables the isolation of how minor operational deviations in electrical regulation can propagate into critical thermal failure paths, a capability not present in conventional standalone cooling reliability models.

The integrated research framework operates via a sequential multi-physics and probabilistic coupling architecture. First, the transient Thermal Model determines cell-to-cell temperature profiles and localized hotspot points across varied operational C-rates. Second, these physical temperature thresholds serve as boundary triggers for the causal paths mapped in the Functional Event Digraph (FED). Finally, the sequential failure chains identified by the FED are formally translated into Boolean logic gates within the Fault Tree Analysis (FTA) framework, allowing physical thermal stress to directly drive the quantitative system-level reliability and degradation curves.

The composition of a complex system encompasses its individual components and the interconnections between them. Understanding the structure of the system involves identifying and labeling its parts at different hierarchical levels. For example, a system (S) may have subsystems, assemblies, and components. A subsystem can be denoted as $S_i$ (i = 1, 2, …, N), its $m^{th}$ assembly as $a_m^i$ (m = 1, 2, …, M), and its $k^{th}$ component as $c_k^{im}$ (k = 1, 2, …, L). Depending on the system’s complexity, additional classifications of elements may be introduced [23].

System functions can be broadly classified into three types: overall, primary, and secondary [24]. The overall function signifies the desired output of the system or subsystem. For instance, the overall function of a battery thermal management system might be “maintaining the battery temperature within specified limits.” The primary function, conversely, represents the main purpose of a system element, whether it’s a component, assembly, or subsystem. For example, the primary function of an electric vehicle battery is to “store and supply electrical energy to the vehicle.” To fulfill the primary function, the support of a secondary function is necessary. For instance, the secondary function of the battery during charging or discharging could be “transferring heat to PCM.”

System functions are designated according to the system’s structure as described earlier. Let ‘F’ denote the system’s overall function, with ${\mathrm{F}}_{Si}$ (i = 1, 2, …, N) representing the overall function of the $i^{th}$ subsystem. The $p^{th}$ primary function of the $m^{th}$ assembly in the ith subsystem is denoted by ${\mathrm{F}}_{mp}^i$ (m = 1, 2, …, M; p = 1), while the $s^{th}$ secondary function is denoted by ${\mathrm{F}}_{ms}^i$ (m = 1, 2, …, M; s = 2, 3, …, P). Similarly, the $p^{th}$ primary function of the $k^{th}$ component of the mth assembly in the $i^{th}$ subsystem is represented by ${\mathrm{F}}_{kp}^{mi}$ (k = 1, 2, …, L; p = 1), and the $s^{th}$ secondary function is represented by ${\mathrm{F}}_{ks}^{mi}$ (k = 1, 2, …, L; s = 2, 3, …, Q). Each assembly or component possesses a primary function supported by one or more secondary functions. For simplicity, ‘p’ is represented as ‘1,’ and ‘s’ ranges from 2, 3, 4, and so on [25]. For analytical clarity and to avoid formatting ambiguity, these multi-indexed structural relations are simplified throughout the text, figures, and tables into direct, three-digit alphanumeric function codes (e.g., $F_{12}^1,F_{21}^1,F_{51}^3$), where the index uniquely represents the designated assembly and its corresponding operational path.

An event is a specific situation [25]. A functional event is an event that is related to a particular function. This section presents the concept of a Function Event Diagram (FED), which is created by identifying important operations, such as instances of operational failure, and the connections between all components. These connections are evaluated using input and output operations. An output function is an action that is the primary objective of a component. An input function is an action that supports the output function. A single component can have multiple input and output functions. If a component fails to carry out its designated function, the incident leading to this failure is called a functional failure or functional loss occurrence. Functional failures can be minor, moderate, or major, and can sometimes lead to the breakdown of the entire system [26].

This paper presents a functional cause analysis methodology for liquid-cooled battery thermal management systems. The methodology extends the approach proposed by Lapp and Powers [27] and further extended by Loganathan and Gandhi [26], which utilizes fault tree concepts. Physically, a gain value of 1 represents a linear, moderate operational deviation, whereas a value of 10 indicates an uncontrollable, drastic functional loss. Mathematically, let $x_i$ represent the state variable deviation at an input node, and y represent the state variable deviation at a downstream output node. The causal propagation along a directed edge is governed by the structural transfer function equation:

$$y={\displaystyle \sum _{i=1}^n}\left(G_{ix}\cdot x_i\right)$$

where $G_{ix}\in \left\{-10,-\mathrm{1,0},+1,+10\right\}$ represents the signed digraph edge gain. To map these continuous directional gains directly into standard Boolean fault-tree events, a functional threshold mapping operator ($\Phi$) is introduced. A downstream functional failure event ($E_y$) occurs when the absolute deviation crossing into the node exceeds its permissible operational limit ($ϵ$). Because the individual assembly failure paths originate from separate physical domains and operate without hardware redundancy, any single input deviation satisfying $\left|G_{ix}\cdot x_i\right|\ge ϵ$ acts as a sufficient condition to trigger the downstream loss. This multi-variable causal constraint is formally translated into an OR gate configuration using the Boolean reduction mapping operator:

$$E_y={\displaystyle \underset{i=1}{\overset{n}{\bigvee }}}\left(\left|G_{ix}\cdot x_i\right|\ge ϵ\right)=E_{x1}\vee E_{x2}\vee \dots \vee E_{xn}$$

This mathematical threshold mapping provides a rigorous algebraic basis for converting signed, multi-physics digraph gains into clear, quantitative Boolean structures.

The process starts with an undesirable event (top event) and ends with the identification of the root causes. Traditional fault trees are created manually and implicitly incorporate system structure, while Lapp and Powers’ approach explicitly incorporates system structure and is suitable for computer-assisted fault tree creation.

5.2. Quantitative Reliability Formulation

The top event of the functional fault tree is explicitly defined as a functional failure, specifically the improper heat transfer to the PCM ($F_{12}^1$), which serves as a functional precursor to system-level thermal boundary violation rather than an immediate physical over-temperature state. All underlying component failure rates (${\lambda }_i$) and basic-event probabilities ($Q_i$) utilized in the calculation procedures are strictly anchored in industry-standard reliability databases, specifically MIL-HDBK-217F [28] for electrical components and OREDA [29] for fluid-loop sub-assemblies, assuming independent, stationary constant hazard distributions.

To quantify the top-event probability ($P_{Top}$), defined as improper heat transfer to the PCM ($F_{12}^1$), which inevitably causes pack temperatures to exceed safe operational limits, standard Boolean reduction of the functional fault tree is executed using the cut sets derived from the logic gates. Under the assumption of statistically independent basic events, the system reliability function $R_{sys}\left(t\right)$ and individual component reliability $R_i\left(t\right)$ follow a constant failure rate (${\lambda }_i$) exponential distribution model:

$$R_i\left(t\right)= e^{-{\lambda }_{i}t}$$

The probability of failure (unreliability) for each basic event over time is expressed as:

$$Q_i\left(t\right)=1-R_i\left(t\right)=1-e^{-{\lambda }_{i}t}$$

For the primary series-configured subsystems governed by OR logic gates (such as the coolant pump loop assembly), the top-event failure probability calculation incorporates the cumulative risk of these basic events:

$$P_{Top}=1-{\displaystyle \prod _{i=1}^n}(1-Q_i)$$

Mathematically, the top event ($F_{12}^1$) is governed by an OR logic gate configuration combining the primary intermediate events. This independent series formulation is reasonably justified for the reference non-redundant architecture of the baseline BTMS layout, which functions without active hardware redundancy (i.e., operating via a single coolant pump, a single controller, a single radiator loop, and localized sensor networks). While the system exhibits complex multi-domain feedback and feedforward interactions during normal regulation states, its reliability boundary dictates that the complete functional failure of any single primary component acts as a direct, unmitigated single point of failure. If any individual component experiences a catastrophic functional loss, the primary active heat dissipation loop is breached, immediately triggering the top event ($F_{12}^1$). Consequently, under standard probabilistic risk assessment (PRA) principles, treating these non-redundant, physically separate assemblies as statistically independent series events connected via Boolean OR logic provides a mathematically rigorous and appropriately conservative baseline for system-level unreliability calculation.

The Mean Time Between Failures (MTBF) for individual components is mathematically defined as the reciprocal of the constant hourly failure rate:

$${MTBF}_i={\displaystyle \frac{1}{{\lambda }_i}}$$

By applying these probabilistic formulations to the developed fault tree structure, the quantitative influence of individual basic event probabilities ($Q_i$) on the overall systemic failure risk ($P_{Top}$) is evaluated. In this framework, the basic event probabilities are treated as parametric variables to dynamically map system-level vulnerability.

To ensure absolute transparency in the quantitative reliability metrics, the hourly baseline failure rates (${\lambda }_i$) were extracted using specific environmental and operational stress factor models prescribed by the respective standards. For electronic and sensor components modeled under MIL-HDBK-217F (Notice 2), the operational failure rate is governed by the part stress formulation:

$${\lambda }_p={\lambda }_b\cdot {\pi }_T\cdot {\pi }_Q\cdot {\pi }_E$$

where ${\lambda }_b$ represents the base failure rate, ${\pi }_T$ is the temperature acceleration factor adjusted for a nominal operating cell reference temperature of 45 °C (318.15 K), ${\pi }_Q$ represents the quality factor fixed at a commercial-grade baseline (${\pi }_Q=2.0$), and ${\pi }_E$ is the environmental factor anchored under the Ground Mobile ($G_M$) classification (${\pi }_E=4.0$) to realistically simulate passenger electric vehicle vibration and shock profiles. For fluid loop and mechanical assemblies (such as the coolant pump and radiator channels), failure boundaries were extracted directly from the OREDA (Offshore & Onshore Reliability Data) Handbook, utilizing the ‘Pumps/Centrifugal’ and ‘Heat Exchangers/Plate-Fin’ entries under stable liquid-circulation operational envelopes. The constant hazard rates assume an independent stationary process, where individual component unreliability values scale lineally across the designated lifecycle time domain as parameterized in Section 8.3.

6. Blockwise Detailed Analysis for Each Assembly

Digraph models are employed to comprehend the structure and functions of a system or process. These models are created for each component or unit, taking into account the relationships between their input–output functions and variations. Within the digraph, nodes represent component or assembly functions and specific failure types, while directed edges indicate connections among functional events or failures, denoted by numerical values: 0 (no deviation), 1 (moderate deviation), and 10 (uncontrollable or drastic deviation). The deviation or gain sign, which indicates an increase or decrease, is assigned as ‘+’ or ‘−‘ based on the direction of the second parameter’s deviation relative to the first. For example, if a moderate and opposing deviation exists in the function of the third assembly compared to the second one, a gain of ‘(−1)’ is assigned to the connecting edge. In cases of drastic deviation, the edge is assigned ‘(−10)’. This approach creates digraph models that illustrate the relationships between function events and functional failure events. These digraph models are then combined to generate a digraph for the entire system or process. The resulting system digraph facilitates the development of fault trees by identifying a top event node and considering the connected nodes or events that act as inputs. Similar to fault tree diagrams that begin with a top or undesired event and trace back to root causes, functional cause analysis trees start with a top or undesired functional event and conclude with the root or contributing functional causes.

The diagram in Figure 1 illustrates the EV battery thermal management system employing PCMs and liquid coolant. Additionally,

Table 2. Subsystems and assemblies.

Subsystem	Assembly/Assemblies
$S_1,$ coolant flow subsystem	$a_1^3$ and $a_4^3$ (Pump) $a_2^3$ and $a_5^3$ (Heat Exchanger) $a_3^3$ (PCM tank)
$S_2,$ relay and electrical energy storage subsystem	$a_2^1$ (Relay), $a_1^1$ (Battery Pack)
$S_3,$ signal sensing and control subsystem	$a_1^2$ and $a_4^2$ (Temperature Sensor) $a_2^2$ and $a_5^2$ (Temperature Controller) $a_3^2$ and $a_6^2$ (Transducer)

provides a detailed breakdown of its subsystems and assemblies.

The input and output functions, as well as potential function loss events or functional failures, for every assembly within the battery thermal management system are determined through a thorough analysis of their structural and operational attributes. Additionally, individual diagrams for each assembly are created and formulated as part of this process.

6.1. Battery Pack with PCM Modeling

In the case of the battery pack, Figure 3 illustrates its input functions, both primary and secondary output functions, as well as events related to functional failures. Let’s consider the input function “Allow the current to flow when coolant temperature is low ($F_{21}^1$)” and its impact on the output functions, namely “Store and supply electrical energy to vehicle ($F_{11}^1$)” (primary function) and “Transfer heat to PCM ($F_{12}^1$)” (secondary function). When the current flowing in increases, it moderately affects the storage of electrical energy in the same direction. If the input current decreases, less energy is stored, whereas an increase in current results in more energy being stored for supply. Consequently, a value of (+1) is assigned to the connection between ($F_{21}^1$) and ($F_{11}^1$) to signify this relationship. Simultaneously, as the current inflow rises, the battery pack begins to heat up, necessitating the removal of more heat by the coolant to maintain a suitable temperature. This requires the transfer of heat to the PCM of the battery pack. Thus, a value of (+1) is also assigned to the connection between ($F_{21}^1$) and ($F_{12}^1$) to indicate this association.

It is important to clarify that while the diagram explicitly labels function $F_{21}^1$ as ‘Allow the current to flow when coolant temperature is low’ for simplicity, this physically represents a multi-tiered BMS thermal protection interlock and dynamic current derating protocol. In a realistic electric vehicle architecture, traction current is governed by driver load demand and inverter switching states. However, function $F_{21}^1$ acts as the control evaluation path where the BMS continuously communicates maximum permissible charge and discharge current boundaries over the vehicle CAN bus based on localized cell temperatures. If the coolant loop fails and temperatures exceed safe operational envelopes, the BMS executes a safety override by first throttling the current limits (thermal derating) to minimize internal Joule heating ($I^{2}R$), and ultimately commanding the high-voltage contactors to open if critical thermal runaway thresholds are breached. This models the critical functional dependency where electrical powertrain loop execution is strictly constrained by the operational health of the cooling subsystem.

Failure analysis using developed models:

• PCM Leakage: The escape of PCM from a battery pack, known as PCM leakage, occurs when the material unintentionally leaks out. Several factors, including manufacturing defects or physical damage, can cause this leakage. When PCM leaks, it undermines the efficiency of the thermal management system, possibly resulting in overheating problems within the battery pack. Swift action is necessary to address leaks promptly, ensuring the safety and optimal operation of the battery system.
• PCM inadequate thermal conductivity: Insufficient thermal conductivity in a Battery Pack with PCM shows the PCM’s inefficiency in conducting heat. This limits its ability to regulate the battery pack’s temperature adequately. Poor thermal conductivity can cause uneven temperature distribution, leading to overheating in certain regions of the battery pack. This threatens the integrity of the battery cells and the overall safety of the system.
• Transients during running of a vehicle: When a vehicle is operating, transients are defined as abrupt shifts in power or load. These fluctuations can disrupt the performance of a battery pack containing PCM. Transients may cause inconsistent thermal management, resulting in sudden temperature spikes or drops within the battery pack. Such variations can place additional strain on battery cells, negatively impacting their performance, lifespan, and safety. Therefore, implementing robust thermal control mechanisms is essential to mitigate the effects of transients on battery systems.

6.2. Temperature Sensor Modeling

In the case of a temperature sensor, its input, output and functional failure events are shown in Figure 4. Its input function is “Transfer heat to PCM ($F_{12}^1$)” and its output function is “Send temperature signal to controller ($F_{11}^2)$”. The failure events are Inaction or delay, Sensor failure or open circuit, and Environmental factors.

Failure analysis using developed model:

• Inaction or delay: A temperature sensor’s inaction or delay refers to its inability to transmit correct measurements or react quickly to temperature changes. Signal delays, electrical faults and calibration problems could cause this. Such failures interfere with temperature regulation, putting various applications at risk of overheating or inadequate cooling.
• Sensor failure or open circuit: When a temperature sensor fails or encounters an open circuit, it signifies a significant problem where the sensor becomes non-operational or experiences a break in the circuit. As a result, the sensor loses its ability to accurately measure temperature. This failure can result in inaccurate temperature readings, which have the potential to create safety risks or cause malfunctions within different applications.
• Environmental factors: Temperature sensors can be influenced by the environment. Extreme temperatures, humidity, corrosive substances, and contaminants can impair sensor performance. High humidity or moisture ingress can cause sensor malfunction or drift.

6.3. Temperature Controller Modeling

In the case of a temperature controller, its input, output and functional failure events are shown in Figure 5. Its input function is “Send temperature signal to controller ($F_{11}^2),$” and its output function is “Send control signal to transducer to vary the flow rate ($F_{21}^2).$” The failure events are Internal short circuit, Communication failure, and Controller inaction or delay.

Failure analysis using developed model:

• Internal short circuit: An internal short circuit in a temperature controller is a significant malfunction that occurs when the electrical pathways within the controller are compromised, resulting in a direct connection between components. This compromises the controller’s capability to regulate temperature effectively, which can lead to unpredictable or excessively high/low temperature outputs. These failures can be particularly risky in systems where precise temperature control is vital, such as industrial processes or electronic devices.
• Communication failure: Temperature controllers that establish communication with external devices or systems, for example, through digital interfaces or protocols, may encounter issues with communication. These problems can interfere with the transmission of temperature data or control signals, ultimately resulting in imprecise temperature regulation.
• Controller inaction or delay: When a temperature controller exhibits inaction or delay, it means that the controller does not promptly respond to temperature changes or encounters delays in effectively regulating the temperature. As a result, this can result in ineffective temperature control, which can cause fluctuations or insufficient heating/cooling within systems. These delays can be attributed to various factors such as sensor problems, programming errors, or electrical faults, all of which have an impact on the overall performance of the device.

6.4. Transducer Modeling

In the case of a transducer, its input, output and functional failure events are shown in Figure 6. Its input function is “Send control signal to transducer to vary the flow rate ($F_{21}^2),$” and its output function is “Send transducer signal to pump to vary the flow rate ($F_{31}^2).$” The failure events are Transducer broken or inactive and drift.

Failure analysis using developed model:

• Transducer broken or inactive: When a transducer is broken or inactive, it means that the device is no longer functional or has suffered damage. This malfunction hampers its ability to accurately convert signals, resulting in incorrect data or signal loss. In applications such as sensors or communication systems, this problem can impede proper functioning and adversely affect the overall performance of devices or processes.
• Drift: Over time, transducers have the potential to undergo drift, which refers to a gradual deviation of their output from the actual value. This drift can be caused by factors such as aging, environmental influences, or alterations in the characteristics of the sensor. Consequently, measurement errors may arise as a result.

6.5. Pump Modeling

In the case of a pump, its input, output and functional failure events are shown in Figure 7. Its input function is “Send transducer signal to pump to vary the flow rate ($F_{31}^2),$” and its output function is “Develop coolant flow at high pressure ($F_{11}^3)$.” The failure events are Pump shutdown, Blockage or clogging and Bearing failure.

Failure analysis using the developed model:

• Pump shutdown: Pump shutdown occurs when the pump stops working. This can happen due to a variety of factors, such as power outages, mechanical problems, or sensor malfunctions. When a pump fails, it can no longer circulate fluids, which can disrupt processes that rely on continuous fluid flow, such as cooling systems or industrial operations.
• Blockage or clogging: The presence of foreign particles, debris, or contaminants in the pumped medium can lead to blockages or clogs in the pump, impeller, or intake. This obstruction can limit the flow, raise the pressure, and potentially harm the components of the pump.
• Bearing failure: Pump bearings play a crucial role in supporting the rotating shaft and facilitating smooth operation. Bearing failure can happen as a result of insufficient lubrication, excessive load, misalignment, or wear that accumulates over time. This failure can result in heightened friction, vibration, and eventually lead to the failure of the pump.

6.6. AC Heat Exchanger Modeling

In the case of an AC heat exchanger, its input, output and functional failure events are shown in Figure 8. Its input function is “Develop coolant flow at high pressure ($F_{11}^3),$” and its output function is “Reduce coolant temperature through Heat Exchanger ($F_{21}^3)$.” The failure events are Flow maldistribution, Fouling and scaling and Erosion.

Failure analysis using the developed model:

• Flow maldistribution: When fluid or gas does not flow evenly through a heat exchanger, it can cause flow maldistribution. This can lead to hot spots in some areas, less heat transfer, more pressure drop, and worse overall performance.
• Fouling and scaling: Fouling and scaling are two common problems that can affect heat exchangers. Fouling occurs when impurities in the fluid buildup on the heat exchanger surfaces, forming a layer that reduces heat transfer efficiency. Scaling is caused by mineral deposits from hard water. Fouling and scaling reduce heat exchanger efficiency by insulating its surfaces. Regular cleaning and maintenance are necessary to prevent these issues and ensure optimal performance.
• Erosion: Erosion of heat exchanger surfaces occurs when high-velocity fluids interact with the material. Contributing factors include impurities in the fluid, uneven flow distribution, and suboptimal design. Erosion reduces heat transfer efficiency, increases the risk of leaks, and may result in mechanical failure.

6.7. PCM Tank Modeling

In the case of a PCM tank, its input, output and functional failure events are shown in Figure 9. Its input function is “Reduce coolant temperature through Heat Exchanger ($F_{21}^3),$” and its output function is “Absorb heat and reduce the coolant temperature ($F_{31}^3).$” The failure events are PCM leakage, PCM degradation and Inadequate thermal insulation.

Failure analysis using the developed model:

• PCM leakage: PCM tanks can leak for a variety of reasons, including mechanical damage, worn-out seals, and manufacturing defects. Leaks can cause a loss of PCM, reducing the tank’s ability to store heat and rendering it ineffective.
• PCM degradation: The materials used to make PCM tanks can break down over time, especially if they are exposed to high temperatures or corrosive substances. This can weaken the tank and cause it to crack, leak, or even fail completely.
• Inadequate thermal insulation: PCM tanks need to be well-insulated to keep the temperature difference between the inside and outside of the tank as high as possible and prevent too much heat from escaping. If the insulation is not good enough or has been damaged, heat can leak out, making the PCM tank less efficient and effective.

6.8. Relay Modeling

In the case of a relay, its input, output and functional failure events are shown in Figure 10. Its input function is “Heat exchanger unable to perform ($F_{51}^3),$” and its output function is “Allow current to flow when coolant temperature is low ($F_{21}^1).$” The failure events are Short circuit, High current inflow and Relay inaction or delay.

To clarify the causal loop logic in Figure 10, while the diagram directly labels the relay’s input function as “Heat exchanger unable to perform” ($F_{51}^3$), this represents a system-level protective boundary sequence rather than an unmediated physical connection. In the actual vehicle hardware architecture, the relay is strictly controlled by the central BMS protection logic. If the heat exchanger fails to reject heat ($F_{51}^3$), the resulting thermal spike is captured by fluid sensors and processed by the BMS software algorithm. The BMS logic then issues a digital execution command to pull down the relay coil voltage, causing the contactors to trip. This macro-level protective sequence is modeled as a direct functional dependency to illustrate how a thermal component failure cascades through the processing logic to force an immediate electrical system shutdown response.

Failure analysis using the developed model:

• Short circuit: A short circuit in the relay coil might happen because of insulation breakdown or mechanical damage, causing issues such as coil overheating, failure to energize or de-energize correctly, or potential damage to the driving circuitry.
• High current inflow: A high current inflow in a relay signifies a scenario where an excessive electric current passes through the relay. This occurrence can be a result of overloads or short circuits in the devices connected to it. When a relay is subjected to currents beyond its capacity, it can overheat, causing damage to the relay and potentially triggering electrical failures in the system it operates within.
• Relay inaction or delay: Relay inaction or delay refers to a situation where the relay doesn’t react promptly to electrical signals or encounters delays. This problem can arise from various factors, including mechanical issues or weak connections. When a relay fails to act properly, it disrupts the electrical circuit. This can cause failures in systems that depend on precise and timely switching, such as control systems or safety mechanisms.

7. Collective Analysis

After analyzing each assembly to understand how input functions and failure events affect output functions, the researchers combined the assembly diagrams to create a function event digraph for the BTMS, as shown in Figure 11. The BTMS incorporates various types of loops, including feedback and feedforward loops. A feedback loop begins and ends at the same point, like a robotic arm positioning system. Feedback loops can be classified as negative or positive, depending on the product of their standard gains. If the product is negative, the loop is negative feedback, which helps to stabilize the system. If the product is positive, the loop is positive feedback, which can amplify signals and lead to instability.

In a feedforward loop, one node influences another node through different paths, with either the same gain (positive feedforward) or the opposite gain (negative feedforward). Feedforward loops are often used to control various components in complex systems. They are important for determining the reliability of a system because any functional changes in one part of a control loop can quickly spread to the rest of the loop. This can affect the performance of other system components.

8. Discussion on Developed Model

The functional event digraph models developed for the eight assemblies of the hybrid BTMS show in great detail how input functions, output functions, and failure events work together in the system. Each digraph shows how changes spread through the system by utilizing directed edges with numerical gain values. This approach facilitates the systematic identification of functional vulnerabilities. The model offers a comprehensive understanding of how localized failures may influence the overall reliability of the H-BTMS by analyzing each assembly individually and subsequently integrating them into a complete system-level digraph.

The digraph for the battery pack with phase change material illustrates the influence of electrical input flow on thermal output. The model demonstrates that changes in incoming current directly affect both energy storage and heat transfer to the PCM, with moderate variations represented by gains of +1. The presence of PCM introduces additional complexity through temperature-dependent behavior, as indicated by secondary functional connections. The model delineates how failure paths such as PCM leakage, low thermal conductivity, and transient thermal disturbances impact both thermal buffering and electrical performance.

The temperature sensor digraph depicts the conversion of thermal data into electrical temperature signals. There is a direct correlation between the heat generated by the battery and the accuracy of the transmitted signal. The model identifies critical failure modes, including inaction or delay, open-circuit faults, and environmental interference, all of which can compromise the accuracy of temperature data transmitted to the controller. Because the BTMS requires accurate and timely sensing for safe operation, these failures significantly affect subsequent system components.

The controller digraph illustrates the process by which input temperature signals are analyzed to generate a control signal for coolant flow regulation. The model emphasizes the controller’s role as the central decision-making unit within the BTMS. Functional deviations, such as internal short circuits, communication failures, or delayed actuation, can propagate rapidly and may result in mismatched coolant flow conditions. These failures directly compromise the regulation cycle and can intensify thermal variations within the battery pack.

The transducer digraph depicts the transformation of control commands into signals for mechanical flow rate regulation. Moderate and significant deviations in this conversion directly affect pump performance. The model identifies failure modes such as transducer damage and drift, highlighting how signal inaccuracies impair coolant circulation. As the transducer serves as the interface between the controller and the pump, any deviations in this component can result in considerable instability in coolant flow regulation.

The pump diagram illustrates the transformation of electrical control signals into high-pressure coolant flow. The model demonstrates how variations in input control signals influence coolant velocity and pressure. Failure events such as pump shutdowns, clogging, or bearing malfunctions propagate rapidly to downstream components, thereby reducing cooling efficiency. The digraph indicates that pump failures result in both immediate and cascading consequences throughout the BTMS, establishing it as one of the most critical assemblies within the system.

The heat exchanger diagram illustrates the cooling process of high-temperature coolant before it re-enters the battery cell. Functional connections indicate that variations in coolant flow directly affect heat rejection efficiency. Failure modes such as fouling, flow maldistribution, and erosion cause substantial deviations in the heat exchanger’s capacity to reduce coolant temperature. The model emphasizes that deterioration of the heat exchanger can undermine steady-state cooling and increase the likelihood of thermal instability throughout the entire system.

The PCM tank digraph depicts the interaction between cooled coolant and latent heat absorption processes. The model reveals how variations in coolant temperature affect the melting and solidification of the PCM. Critical failure events, including PCM leakage, material degradation, and insufficient insulation, substantially reduce the tank’s thermal buffering capacity. The digraph emphasizes that PCM-related malfunctions produce cascading effects due to the PCM’s central role in maintaining temperature uniformity and thermal stability.

The relay digraph depicts how the relay enables or restricts current flow in response to coolant temperature. Functional deviations, such as short circuits, overload-induced current surges, or delayed switching, affect both electrical supply continuity and thermal performance. The relay digraph demonstrates that electrical components can induce thermal effects, underscoring the substantial interaction between electrical and thermal subsystems in BTMS operation.

After developing distinct digraphs for each of the eight assemblies, they were integrated to create a comprehensive functional event digraph for the hybrid BTMS. The integrated directed graph includes many feedback and feedforward loops that show how deviations propagate through subsystems. Negative and positive feedback loops indicate stabilization or amplification tendencies. Feedforward pathways highlight direct channels of influence between assemblies. This system-level digraph formed the structural foundation for developing the fault tree, helping identify the main event and trace its contributing functional causes. The integrated model offers a comprehensive perspective on the interaction complexities within the hybrid BTMS and facilitates a more precise reliability assessment.

The integrated digraph offers a well-defined basis for recognizing the essential functional failures that impact system behavior. Utilizing these relationships, the fault tree was constructed by translating the functional dependencies into failure events and logical linkages.

Mathematically, the top event ($F_{12}^1$) is governed by an OR logic gate configuration combining the primary intermediate events. The corresponding Boolean expression is defined as

$$F_{12}^1=F_{21}^1 {\vee F}_{51}^3 \vee F_{31}^3 \vee F_{21}^3 \vee F_{11}^3 \vee F_{31}^2 \vee F_{21}^2\vee F_{11}^2$$

Because all primary intermediate assemblies operate in a non-redundant series configuration within the primary cooling path, the first-order minimal cut sets consist of the individual basic events themselves. Consequently, the top-event probability calculation directly aggregates these independent failure paths as defined in the formulation of Section 5.2.

The Fault Tree is prepared by selecting “Improper heat transfer to PCM” ($F_{12}^1$) as the top event or undesired event. If heat is not transferred to the PCM, the battery will heat and cause failure. It is caused due to the function “Allow the current to flow when coolant temperature is low ($F_{21}^1$)” and the three failures, i.e., PCM inadequate thermal conductivity, transients during vehicle operation, and PCM leakage. The Fault Tree diagram is created using the Function Event Digraph. Figure 12 shows the Fault Tree Diagram of the proposed BTMS. The top event, intermediate event and basic events are described in

Table 3. Top, intermediate and basic events.

Top Event	Intermediate Event	Basic Event
Improper heat transfer to PCM ($F_{12}^1$)	Allow current to flow when coolant temperature is low ($F_{21}^1$)	PCM inadequate thermal conductivity
		Transients during running of vehicle
		PCM leakage
	Heat exchanger unable to perform ($F_{51}^3$)	Short circuit
		High current inflow
		Relay inaction or delay
	Absorb heat and reduce the coolant temperature ($F_{31}^3$)	Erosion
		Fouling and scaling
		Flow maldistribution
	Reduce coolant temperature through heat exchanger ($F_{21}^3$)	PCM leakage
		PCM degradation
		Inadequate thermal insulation
	Develop coolant flow at high pressure ($F_{11}^3$)	Fouling and scaling
		Flow maldistribution
		Erosion
	Send transducer signal to pump to vary the flow rate ($F_{31}^2$)	Pump shutdown
		Blocking or clogging
		Bearing failure
	Send control signals to transducer to vary the flow rate ($F_{21}^2$)	Transducer broken or inactive
		Drift
	Send temperature signal to controller ($F_{11}^2)$	Internal short circuit
		Communication failure
		Controller inaction or delay

.

It is important to emphasize that within this Functional Fault Tree Analysis (Functional FTA) framework, the top event is defined as a functional failure (“Improper heat transfer to PCM,” $F_{12}^1$) rather than a final physical consequence. This functional loss serves as a functional precursor to system-level thermal boundary violation, as the inability to transfer heat to the phase change material inevitably leads to the battery temperature exceeding its safe operating limits.

The alignment of intermediate events and basic events in Table 3 directly corresponds to the localized failure paths of each independent fluidic and electrical assembly. This structure maps how individual component degradation cascades to impair the primary subsystem functions modeled in the fault tree hierarchy.

8.1. Thermal Performance: Liquid BTMS vs. PCM–Liquid Hybrid BTMS

A high-demand electric vehicle operating scenario, such as a 3C-rate discharge or fast charging, was simulated to compare the thermal responses of a conventional liquid-cooled BTMS and the proposed PCM–liquid hybrid BTMS. Figure 13 presents the battery cell temperature as a function of time for both systems, with an initial temperature of 25 °C. In the liquid BTMS system, the cell temperature increases steadily, reaching approximately 55 °C after 600 s of high-power operation.

In contrast, the hybrid system’s temperature curve stabilizes when the PCM begins to melt at its designated phase-change threshold of approximately 45 °C, as indicated by a dashed line. This process buffers heat absorption. The hybrid system stabilizes when the local cell hotspot temperatures reach approximately 45 °C, triggering the latent heat phase change threshold of the PCM, which effectively absorbs the peak thermal load and maintains the overall pack average temperature near 40 °C. In comparison, the liquid BTMS system permits a peak temperature exceeding 50 °C. This results in an approximate 15 °C, or 27%, reduction in peak cell temperature for the hybrid design.

Consequently, the time required for localized cell hotspots to reach the 45 °C phase change activation threshold exceeds 600 s for the hybrid system, significantly delaying thermal saturation compared to standalone configurations while keeping the bulk pack average temperature near 40 °C. Figure 14 provides additional quantification of these advantages: the hybrid BTMS achieves a much lower peak cell temperature and, importantly, reduces the cell-to-cell temperature spread—a key indicator of temperature uniformity—to approximately 3 °C, compared to about 8 °C in the liquid BTMS system after 10 min. In summary, under high thermal loads, the hybrid PCM–liquid BTMS outperforms pure liquid cooling by maintaining cell temperatures within safe limits and delivering good thermal uniformity across the battery pack.

In contrast, the hybrid system’s temperature curve stabilizes when the PCM begins to melt at its designated phase-change threshold of approximately 45 °C, as indicated by a dashed line. This process buffers heat absorption. To ensure analytical clarity, the localized thermal metrics are defined based on the spatial nodes of the numerical model: (i) Localized Cell Hotspots (or Peak Pack Temperature, $T_{max}$) represent the absolute maximum temperature recorded at the central, highly constrained core spatial nodes of the cell-PCM matrix where heat dissipation pathways are longest; (ii) Pack Average Temperature ($T_{avg}$) represents the volume-weighted average computed across all fluid, solid cell, and material domains using the numerical integration method $T_{avg}={\displaystyle \frac{1}{V}}\iiint T\hspace{0.17em}dV$. The hybrid system stabilizes when the local cell hotspot temperatures reach approximately 45 °C, triggering the latent heat phase change threshold of the PCM, which effectively absorbs the peak thermal load and maintains the overall pack average temperature near 40 °C. In comparison, the liquid BTMS system permits a peak temperature exceeding 50 °C. This results in an approximate 15 °C, or 27%, reduction in peak cell temperature for the hybrid design. Consequently, the time required for localized cell hotspots to reach the 45 °C phase change activation threshold exceeds 600 s for the hybrid system, significantly delaying thermal saturation compared to standalone configurations while keeping the bulk pack average temperature near 40 °C.

The transient thermal simulation under a 3C-rate discharge was conducted using ANSYS Fluent software. The system’s thermal behavior is governed by the localized energy conservation equation

$$\rho C_p{\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }t}}=\nabla \ast \left(k\nabla T\right)+q_{gen}$$

where the cell volumetric heat generation rate ($q_{gen}$) is computed using Bernardi’s equation:

$$q_{gen}={\displaystyle \frac{I}{V_{cell}}}[\left(V_{oc}-V\right)-T{\displaystyle \frac{d{U}_{oc}}{dT}}]$$

To ensure absolute reproducibility of the numerical results, the foundational electro-thermal and material parameters were strictly parameterized as follows: each commercial 21,700 cell features a nominal capacity of 5.0 Ah, an individual cell volume ($V_{\mathrm{cell}}$) of $2.42\times {10}^{-5}{ \mathrm{m}}^3$, a baseline internal resistance ($R_{\mathrm{int}}$) of $22 \mathrm{m}\mathsf{\Omega }$, and an empirical entropic heat coefficient (${\displaystyle \frac{d{U}_{\mathrm{oc}}}{dT}}$) fixed at −0.12 mV/K. The composite PCM matrix is modeled with a thermal conductivity (k) of $0.65 \mathrm{W}/\mathrm{m}\cdot \mathrm{K}$, a density ($\rho$) of $920{ \mathrm{kg}/\mathrm{m}}^3$, and a latent heat of fusion of 180 kJ/kg across a melting range of 42 °C to 45 °C. The active liquid loop utilizes a standard water–ethylene glycol (50:50) mixture as the coolant medium (density $\rho =1060{ \mathrm{kg}/\mathrm{m}}^3$, specific heat $C_p=3300 \mathrm{J}/\mathrm{kg}\cdot \mathrm{K}$, thermal conductivity $k=0.38 \mathrm{W}/\mathrm{m}\cdot \mathrm{K}$) entering at a constant initial boundary temperature of 25 °C with a convective heat transfer coefficient ($h=250{ \mathrm{W}/\mathrm{m}}^2\cdot \mathrm{K}$) applied at the mini-channel interface walls. Numerically, the spatial domain was discretized using a high-resolution, independent structured hexahedral mesh featuring approximately $4.2\times {10}^5$ elements. The transient calculation was executed using a second-order implicit solver scheme with a fixed time step ($\Delta t=0.1 \mathrm{s}$) over a 600 s total discharge window. Solution monitoring was governed by a strict convergence criterion requiring absolute scaled residuals to drop below ${10}^{-6}$ for the energy equation and ${10}^{-4}$ for the momentum and continuity equations at each individual time step.

8.2. Component Criticality Index (CCI) for Hybrid BTMS

The Component Criticality Index (${CCI}_i$) for each basic component is quantified mathematically using Birnbaum’s structural importance measure weighted by relative unreliability:

$${CCI}_i\left(t\right)={\displaystyle \frac{\mathsf{\partial }{P}_{Top}\left(t\right)}{\mathsf{\partial }{Q}_i\left(t\right)}}\ast {\displaystyle \frac{Q_i\left(t\right)}{P_{Top}\left(t\right)}}$$

where $P_{Top}\left(t\right)$ is the system top-event failure probability, and $Q_i\left(t\right)$ represents the individual basic-event unreliability calculated from the parametric failure rates (${\lambda }_i$) compiled in the Fault Tree model.

Figure 15 presents the CCI (Component Criticality Index) for the main assemblies of the Hybrid Battery Thermal Management System. The CCI measures each assembly’s contribution to failure propagation, leading to the top event, defined as improper heat transfer to the PCM.

The pump has the highest criticality index, highlighting its potential as a single-point failure in the cooling loop. The battery pack with PCM and the PCM tank also show elevated criticality because of their direct roles in thermal buffering and energy dissipation. The heat exchanger has moderate criticality, as it significantly affects coolant temperature before it reaches the PCM.

Control-oriented components, such as temperature sensors, controllers, transducers, and relays, have lower criticality indices. Their influence is mainly indirect, acting through signal processing and regulation rather than direct thermal effects.

8.3. Sensitivity Analysis of Hybrid BTMS Reliability

A sensitivity study identified which component’s reliability most influences the overall system failure probability (top-event). The failure probabilities of three components—the coolant pump, the phase change material (PCM; degradation/leakage), and the temperature sensor—were varied independently across a defined range. All other components were kept at baseline values from the literature. Figure 16 illustrates the response of the BTMS top-event probability to changes in each of these failure rates.

The results indicate that the system is most sensitive to the reliability of the coolant pump. For instance, increasing the pump’s failure probability from 0.02 to 0.04 (doubling from 2% to 4%) results in an approximate 30% increase in the total system failure probability (from approximately 6% to 8%). A further increase in pump failure probability to 0.06 (6%) nearly doubles the overall risk compared to the baseline. In contrast, a comparable relative increase in the sensor’s failure probability produces a more moderate rise, approximately half the effect observed for the pump.

The system’s sensitivity to the PCM failure (degradation) rate is comparatively minor. Even a substantial increase in PCM failure probability (for example, from 0.5% to 3% or higher) results in only a marginal rise in the top-event probability. These parametric trends confirm that pump reliability is the most influential factor affecting the hybrid BTMS’s safety performance, followed by the sensor, albeit to a lesser extent. Consequently, enhancing pump reliability or incorporating redundancy provides the greatest reduction in overall system failure risk, whereas moderate variations in PCM reliability have minimal impact.

To analyze the structural stability of the safety model, a localized boundary sensitivity check was performed by varying individual asset unreliability limits ($Q_{\mathrm{comp}}$).

Table 4. Sensitivity analysis of system ($P_{\mathrm{Top}}$) under varying component unreliability bounds.

Component Failure Probability (${\mathit{Q}}_{\mathbf{comp}}$)	System ${\mathit{P}}_{\mathbf{Top}}$ (Pump Variation Path)	System ${\mathit{P}}_{\mathbf{Top}}$ (Sensor Variation Path)	System ${\mathit{P}}_{\mathbf{Top}}$ (PCM Variation Path)
0.005	0.048	0.058	0.063
0.020	0.063	0.073	0.078
0.040	0.083	0.093	0.098
0.060	0.103	0.113	0.118

presents the resulting system-level failure probabilities ($P_{\mathrm{Top}}$) evaluated over the 10,000 h mission lifecycle.

To illustrate the Fault Tree probability propagation process, the top-event failure probability is evaluated using the simplified disjoint Boolean structure:

$$P_{Top}=1-{\displaystyle \prod _{j=1}^n}(1-Q_j)$$

The baseline component unreliability value $Q_{\mathrm{pump}}=0.02$ is rigorously derived from the database-anchored component failure rate (${\lambda }_{\mathrm{pump}}=2.02\times {10}^{-6} / \mathrm{h}$) over a 10,000 h mission time using the standard exponential unreliability model (${\lambda }_{\mathrm{pump}}=2.02\times {10}^{-6} / \mathrm{h}$).

When this baseline value is integrated with the cumulative product of all other independent background component reliability paths ($\prod \left(1-Q_{\mathrm{other}}\right)=0.9561$), the baseline system unreliability is calculated exactly as:

$$P_{\mathrm{Top}}=1-\left[\left(1-0.02\right)\times 0.9561\right]=0.063\hspace{1em}\left(6.3\%\right)$$

When the pump failure probability is parametrically isolated and increased to $Q_{\mathrm{pump}}=0.04$ under elevated operating stresses, the updated system top-event unreliability propagates linearly as:

$$P_{\mathrm{Top}}=1-\left[\left(1-0.04\right)\times 0.9561\right]=0.083\hspace{1em}\left(8.3\%\right)$$

Similarly, at the upper bound evaluation point where $Q_{\mathrm{pump}}=0.06$, the final system risk envelope shifts predictably to:

$$P_{\mathrm{Top}}=1-\left[\left(1-0.06\right)\times 0.9561\right]=0.103\hspace{1em}\left(10.3\%\right)$$

It is explicitly noted that this work represents a theoretical and computational modeling study, and direct physical prototyping or experimental testing was not conducted. To ensure rigorous computational verification and numerical stability, the baseline component failure rates were strictly anchored in globally recognized historical reliability databases (MIL-HDBK-217F and OREDA). Furthermore, the transient thermal performance response curves and spatial cell gradients were closely benchmarked against established semi-empirical trends found in hybrid thermal management literature. This comparative anchoring verifies that the multi-physics dependencies and probabilistic logic boundaries remain bounded, mathematically sound, and predictive within established engineering margins [28,29].

8.4. Reliability Degradation Behavior of Liquid and Hybrid BTMS

In addition to the Fault Tree-based reliability assessment, a time-domain reliability projection was conducted for both the conventional liquid BTMS and the proposed PCM–liquid hybrid BTMS. The time-domain reliability degradation curves illustrated in Figure 17 are modeled using a standard exponential reliability distribution governed by a constant hazard rate (${\lambda }_i$). The system-level time-dependent reliability function is calculated as $R_{sys}\left(t\right)={\prod }_{i=1}^n{e}^{-{\lambda }_{i}t}$ over an operational lifecycle time domain spanning 0 to 10,000 h, utilizing the baseline failure rate parameters defined in the preceding sections.

Figure 17 presents the variation in system reliability over operating time. The projected curves reveal a monotonic decline in reliability for both configurations; however, the hybrid BTMS demonstrates a consistently slower reduction compared to the liquid BTMS system. This outcome indicates that the integration of latent-heat buffering by PCM and continuous coolant-based heat rejection mitigates thermal stress accumulation within the cooling path, thereby enhancing long-term operational dependability. A persistent gap between the two curves is observed throughout the operating period, indicating that the hybrid architecture consistently demonstrates a higher probability of successful operation at all evaluated time points. This finding is consistent with earlier results presented in this manuscript, which document lower peak temperatures and enhanced thermal uniformity for the hybrid system compared to the liquid BTMS configuration. The reliability projection therefore provides a quantitative confirmation that the thermal advantages of the hybrid design are expected to yield greater long-term system robustness.

Quantitatively, the time-domain reliability benchmark reveals that the hybrid BTMS maintains a significantly higher operational probability over extended duty cycles compared to the conventional liquid BTMS, establishing a clear long-term dependability margin.

8.5. Analysis of Cell-to-Cell Temperature Distribution

Time-history temperature plots provide valuable insight into the transient response of a battery thermal management system (BTMS), but the spatial temperature distribution across battery cells is equally important. Localized hot spots can accelerate non-uniform aging and compromise pack reliability. The cell-to-cell temperature distribution profile illustrated in Figure 18 is derived directly from the numerical multi-physics simulation framework described in Section 4.2 and Section 8.1, utilizing the identical 10S6P module configuration, 21,700 cell dimensions, circumferential PCM boundary constraints, and 3C-rate discharge thermal generation models. Figure 18 compares temperature distributions across representative battery cells for both the liquid BTMS and the hybrid phase change material (PCM)–liquid BTMS. The liquid BTMS system shows a broader thermal spread, with higher temperatures near the central cells. In contrast, the hybrid system achieves a more uniform distribution with significantly less spatial variation.

These findings support the earlier discussion of thermal performance, showing that the hybrid BTMS reduces the maximum temperature spread from about 8 °C to 3 °C. This improved uniformity is due to the PCM’s thermal buffering, which absorbs localized heat and prevents hot spot formation, while the liquid loop removes the accumulated thermal load. Reducing the thermal gradient is essential for battery durability, as more uniform cell temperatures improve pack balancing, limit differential degradation, and enhance the overall reliability of the BTMS.

8.6. Impact of Operating Severity on Top-Event Probability

The impact of operating severity on failure behavior was evaluated by analyzing the top-event probability across various representative load levels. Figure 19 indicates that system failure probability increases with higher operating severity for both battery thermal management system (BTMS) architectures. The increase is substantially steeper for the Liquid BTMS, whereas the hybrid BTMS exhibits a more gradual rise in failure probability.

These results indicate that the hybrid design is better equipped to withstand aggressive duty cycles, such as high C-rate discharge and rapid charging. This observation is consistent with prior thermal analyses, which showed that the hybrid BTMS maintained a safer temperature range, while the Liquid BTMS exceeded the recommended operating threshold under severe loading. Therefore, the current analysis expands the discussion beyond temperature response and demonstrates that the hybrid BTMS provides a reliability advantage under demanding, real-world EV operating conditions.

Under maximum simulated operational severity (e.g., high C-rate discharge kinetics), the top-event failure probability of the conventional liquid system climbs at a substantially steeper rate, validating the superior, robust safety margins of the hybrid architecture under aggressive real-world loading conditions.

The causal link between the discharge C-rate and top-event probability ($P_{Top}$) is governed by temperature-dependent failure rate acceleration. Higher C-rates increase internal heat generation ($q_{gen}$), which elevates the system operating temperature (T). This thermal stress scales the baseline component failure rates using an Arrhenius acceleration factor, ${\lambda }_{updated}={\lambda }_{baseline}\cdot \exp \left[{\displaystyle \frac{-E_a}{R}}\left({\displaystyle \frac{1}{T}}-{\displaystyle \frac{1}{T_{ref}}}\right)\right]$. To ensure full parametric traceability across the evaluated temperature range of 25 °C to 55 °C, the universal gas constant is set to $R=8.314 \mathrm{J}/\mathrm{mol}\cdot \mathrm{K}$, and the baseline reference temperature is fixed at $T_{\mathrm{ref}}=298.15 \mathrm{K}$ (25 °C). The component-specific activation energy ($E_a$) values—which govern the slope of failure acceleration under thermal stress—are derived from historical empirical electronics packaging data and defined as $E_a=0.45 \mathrm{eV}$ (43.4 kJ/mol) for the BMS controller and temperature sensors, and $E_a=0.30 \mathrm{eV}$ (28.9 kJ/mol) for the mechanical and fluid loop sub-assemblies (coolant pump and radiator fan). These values mathematically govern the non-linear upward shift in component unreliability propagated through the fault tree structure at elevated discharge severities.

8.7. Pareto Analysis of Failure Contribution

In addition to the previously presented component criticality index (CCI), a Pareto-style ranking of component-level failure contributions was conducted to visualize the relative importance of each assembly in propagating the top event. Figure 20 shows that the coolant pump constitutes the largest share of the total failure burden, followed by the battery pack with phase change material (PCM), the PCM tank, and the heat exchanger (radiator). In contrast, the temperature sensor, temperature controller, transducer, and relay/contactor contribute substantially less to overall failures.

This ranking identifies the coolant pump as the component with the highest criticality index, while control-oriented components display lower direct thermal criticality. The Pareto representation improves interpretability by illustrating that a limited number of assemblies account for the majority of system risk. From a design standpoint, reliability improvement efforts should focus on the coolant pump, battery pack with PCM thermal interface, PCM tank containment, and heat exchanger (radiator) health, rather than allocating resources evenly across all subsystems.

The percentage contribution ($C_i$) of each component failure mode is calculated by normalizing its individual unreliability weight against the sum of all critical failures:

$$C_i=\left(Q_i/{\displaystyle \sum _{j=1}^n}{Q}_j\right)\times 100$$

The cumulative percentage ($C{P}_k$) is then computed sequentially as $C{P}_k={\displaystyle {\sum }_{i=1}^k}{C}_i$. The underlying raw contribution data represented in the Pareto analysis of Figure 20 is organized in

Table 5. Complete quantitative calculation path and cumulative Pareto risk contribution metrics (t = 10,000 h).

S. No.	Component/Assembly	Database Failure Rate (${\mathit{\lambda }}_{\mathit{i}},\mathit{p}\mathit{e}\mathit{r} {10}^6 \mathbf{h})$	Component Unreliability (${\mathit{Q}}_{\mathit{i}}=1-{\mathit{e}}^{-{\mathit{\lambda }}_{\mathit{i}}\mathit{t}}$)	Individual Pareto Contribution Formula	Failure Contribution Percentage (%)	Cumulative Percentage (%)
1	Coolant Pump	1.633	0.01620	${\displaystyle \frac{0.01620}{0.060}}\times 100$	27.0%	27.0%
2	Battery Pack with PCM	1.086	0.01080	${\displaystyle \frac{0.01080}{0.060}}\times 100$	18.0%	45.0%
3	PCM Tank	0.965	0.00960	${\displaystyle \frac{0.00960}{0.060}}\times 100$	16.0%	61.0%
4	Heat Exchanger (Radiator)	0.844	0.00840	${\displaystyle \frac{0.00840}{0.060}}\times 100$	14.0%	75.0%
5	Temperature Sensor	0.541	0.00540	${\displaystyle \frac{0.00540}{0.060}}\times 100$	9.0%	84.0%
6	Temperature Controller	0.421	0.00420	${\displaystyle \frac{0.00420}{0.060}}\times 100$	7.0%	91.0%
7	Transducer	0.300	0.00300	${\displaystyle \frac{0.00300}{0.060}}\times 100$	5.0%	96.0%
8	Relay/Contactor	0.240	0.00240	${\displaystyle \frac{0.00240}{0.060}}\times 100$	4.0%	100.0%
	System Total ($P_{\mathrm{Top}}$)	5.970	0.06000			100.0%

.

To evaluate the systemic risk mitigation achieved by the proposed design,

Table 6. System-level top-event unreliability comparison and mitigation metrics (t = 10,000 h).

Configuration/Metric	Conventional Liquid Cooling Layout (Baseline)	Proposed Hybrid PCM–Liquid Cooling Layout	Analytical Derivation Path & Multi-Physics Calculation Track
System Top-Event Unreliability ($P_{\mathrm{Top}}$)	11.40% (0.1140)	6.00% (0.0600)	Boolean reduction ($\bigvee Q_i$) of the Fault Tree Analysis framework driven by individual component unreliability vectors.
System Reliability ($R=1-P_{\mathrm{Top}})$	88.60% (0.8860)	94.00% (0.9400)	Direct mathematical complement of the calculated top-event unreliability function ($R=1-P_{\mathrm{Top}}$).
Relative System Unreliability Reduction	Baseline Value	47.37%	Calculated as: ${\displaystyle \frac{P_{\mathrm{Top}, \mathrm{Conv}}-P_{\mathrm{Top}, \mathrm{Hybrid}}}{P_{\mathrm{Top}, \mathrm{Conv}}}}={\displaystyle \frac{0.1140-0.0600}{0.1140}}\times 100\%$
Relative System Reliability Improvement	Baseline Value	6.09%	Calculated as: ${\displaystyle \frac{R_{\mathrm{Hybrid}}-R_{\mathrm{Conv}}}{R_{\mathrm{Conv}}}}={\displaystyle \frac{0.9400-0.8860}{0.8860}}\times 100\%$

provides a direct comparative overview of the quantitative safety indices between the conventional liquid cooling system and the proposed dual-stage hybrid PCM–liquid configuration over the designated 10,000 h operational profile.

The analytical tracking path driving these figures is derived from the direct coupling of the transient multi-physics thermal models and the Fault Tree Analysis framework. In the conventional liquid cooling architecture, numerical simulations show prolonged exposure to elevated hotspot temperatures during intensive 3C discharging due to the lack of latent-heat storage. When these elevated temperatures are passed to the Arrhenius stress model (Section 8.6), they inflate the hourly component hazard rates ($\mathit{\lambda }$), leading to an increased system-level top-event unreliability of 11.40%. Conversely, in the proposed hybrid design, the phase change material absorbs peak transient thermal spikes, keeping the system within stable operational limits. This effectively suppresses thermal acceleration factors, preventing hazard rate inflation and keeping the system unreliability at its nominal baseline of 6.00%. This translates to a 47.37% relative reduction in overall system unreliability, which corresponds mathematically to a true 6.09% relative improvement in system reliability (R).

A primary limitation of this study is the absence of direct experimental validation and physical prototyping. While the transient thermal distribution curves were rigorously benchmarked against established semi-empirical trends in literature, and the safety logic structures were verified using standardized reliability databases (MIL-HDBK-217F and OREDA), physical prototyping and hardware-in-the-loop (HIL) testing were outside the scope of this work. Consequently, this work stands as a theoretical and computational modeling study designed to establish a predictive framework, and future research will focus on physical experimental testing to validate these numerical margins under live environmental stresses.

9. Conclusions

This study successfully evaluated a hybrid liquid-PCM Battery Thermal Management System (BTMS) using a combined Functional Event Digraph and Fault Tree Analysis (FTA) framework. Multi-physics numerical simulations demonstrated that the integrated configuration successfully optimizes thermal uniformity, achieving a significant reduction in peak localized cell hotspot temperatures and suppressing transient thermal stresses compared to conventional liquid cooling baselines. Integrating a Functional Cause Analysis (FCA) digraph mapping with Boolean Fault Tree Analysis (FTA) provides a rigorous qualitative and quantitative safety assessment, establishing a clear mapping of total system top-event failure paths regarding improper heat transfer to the PCM over the designated lifecycle window. Furthermore, the systematic Pareto risk distribution successfully isolates the primary sub-assembly bottlenecks, identifying the coolant circulation pump and the external radiator heat exchanger as the leading critical failure contributors to overall system unreliability. These traceable trends prove that coupling localized latent-heat material buffering with active convective loops drastically mitigates thermal acceleration stresses, providing a verified engineering roadmap for high-reliability EV thermal architecture design.