In recent years, Artificial Intelligence, particularly Machine Learning, has achieved remarkable success in solving complex problems. However, this progress has also revealed the emergence of unexpected, poorly understood, and elusive phenomena that characterize the behavior of machine intelligence and learning processes. These phenomena often challenge researchers to interpret them within the boundaries of existing Machine Learning theoretical frameworks, thereby motivating the development of new and more comprehensive theoretical foundations. One such phenomenon, known as grokking, refers to the sudden and substantial improvement in a model's performance following a prolonged period of stagnant or even regressive learning. In this paper, we argue that it is possible to provide insights into grokking by leveraging the existing theoretical foundations of Machine Learning, in particular concepts from Statistical Learning Theory, such as norm-based and stability-based generalization bounds. We further show how these theories can help reconcile the phenomenon of grokking with established principles of learning and generalization. Furthermore, we demonstrate the practical applicability of these insights through concrete examples.

Modern power grids are becoming increasingly complex with the integration of heterogeneous distributed energy resources, underscoring the need for accurate and efficient Power Flow Analysis to ensure stability, reliability, and market operations. Existing methods generally rely on iterative numerical techniques (INT) or machine learning (ML). While INT is physically consistent and highly accurate, it can be computationally expensive and vulnerable to slow or non-convergence. ML methods offer faster solutions but often require extensive data, suffer from limited extrapolation capabilities, and lack physical consistency. Physics-informed ML (PIML) bridges these gaps by embedding domain knowledge before, during, and after training. However, current PIML approaches typically do not leverage this full range of opportunities. In this paper, we propose a novel PIML framework for Power Flow Analysis that integrates physical insights at all three stages (pre, in, and post-processing) to achieve superior accuracy and efficiency. Notably, we introduce a new post-processing technique that partitions the power network into its mesh and radial components: the mesh portion is handled via PIML, while the radial portion is efficiently solved with a convex optimization approach informed by the PIML outputs. This approach is efficient with radial topologies, especially in power distribution networks where the radial part is predominant. Experiments on realistic power networks demonstrate that our method outperforms state-of-the-art approaches in both accuracy and computational performance.

As Polymer Electrolyte Membrane Fuel Cells (PEMFCs) emerge as a promising technology for transport decarbonization, the development of durability assessment protocols tailored to specific applications, such as maritime operations, is becoming relevant for the identification of stressors and lifetime enhancement. This study presents a preliminary experimental campaign aimed at introducing a methodology to assess the degradation of PEMFCs subjected to Accelerated Stress Test (AST). In particular, the methodology encompasses the utilization of electrochemical characterization and, in this work, the fuel cell operating profile has been chosen to mimic the operation of a small passenger vessel. The tests were carried out on two single Membrane Electrode Assemblies (MEAs) for 500 h. One membrane was subjected to the AST, and a second sample, tested under constant load operation, served as a reference. Periodic electrochemical characterization was conducted to assess performance degradation through polarization curves, electrochemical impedance spectroscopy, and cyclic voltammetry. The electrochemical analysis of degradation was conducted through a dual-method approach combining model-free and model-based methods for impedance analysis, as well as catalyst active area evaluation from voltammograms. Results show that the dynamic operation characteristic of the passenger ferry increases degradation compared to constant operation, evidenced by increased ohmic and interfacial resistances and losses in catalyst active area. This work provides a framework for developing application-specific durability protocols, enriched with multi-method diagnostic approaches to assess PEMFC degradation under realistic maritime conditions. Such methodologies support the development of durability enhancement strategies tailored to maritime applications, allowing a broader application of the technology in the sector.

As navies and maritime organisations transition towards low-emission propulsion systems, spark-ignited (SI) gas engines capable of operating on sustainable, low-reactivity fuels are gaining renewed interest. These engines, while offering potential for fossil-free operation, present significant challenges under transient conditions due to complex interactions between throttle control, fuel regulation, and combustion stability. Accurate dynamic modelling is critical to integrate these engines into resilient naval power systems and to support the development of advanced control strategies. This study evaluates several high-fidelity mean value first principle engine modelling (MVFPEM) approaches for simulating the dynamic gas path behaviour of a large, high-speed, SI marine engine under rapid load changes. Models with varying levels of complexity, including simplified and full turbocharger implementations and different gas path volume resolutions, were calibrated using a single measurement campaign and validated against measured transient data. Several methods for turbocharger performance mapping (Stapersma, Casey & Robinson, and Jensen) were evaluated for their applicability in predicting the engine behaviour in dynamic operating scenarios. The results highlight that models incorporating three control volumes and full turbocharger dynamics achieve the highest accuracy, particularly during rapid load increases and recovery phases. Simplified models fail to capture turbocharger inertia and pressure transients, limiting their applicability to investigate naval propulsion or electric power generation plant behaviour under transient load conditions. This work provides guidance on selecting and validating engine models for marine applications and reinforces the role of high-fidelity MVFPEMs in the design and simulation of future naval energy systems.

Low-Temperature Polymer Electrolyte Membrane Fuel Cells (LT-PEMFCs) have recently emerged as a promising solution for sustainable ship energy systems. However, enhancing durability is essential to enable their broader adoption in the maritime sector. Durability enhancement depends on a thorough understanding of degradation mechanisms and accurate prognostics, both of which are highly application-specific. The current literature lacks a comprehensive understanding of LT-PEMFC degradation under maritime operating conditions and its integration into reliable prognostic models. To address this gap, this review provides an overview of LT-PEMFC durability and prognostic models from the perspective of maritime applications. Through a comparative analysis of studies across various sectors, we identify and discuss maritime-specific degradation drivers, including ship load profiles, sodium chloride contamination, vibrations, and wave-induced inclinations. Building on this analysis, we critically evaluate existing prognostic models and their suitability for lifetime prediction in maritime applications. This review proposes durability enhancement strategies based on current knowledge and highlights key research gaps requiring further investigation. In addition, it outlines promising prognostic methodologies and identifies technical challenges for application to maritime LT-PEMFCs. In this way, this work lays the foundation for enhancing LT-PEMFC durability in maritime environments and supporting its broader adoption for zero-emission ships.

Batteries have emerged as a promising solution across diverse vessel segments, offering benefits in operational efficiency, cost reduction, and emissions reduction. This study investigates the specific requirements of batteries onboard 7 vessel types, such as tugboats, ferries, cruise ships, yachts, fishing vessel, vessels with cranes, and dynamic positioning vessels, through an in-depth analysis of load profiles and operational needs. By identifying 24 potential operational requirements, ranging from battery electric operation to silent operations and load smoothing, a mixed-integer linear programming model is used to optimize the power and energy allocation for each requirement. This framework enables a generalization of battery requirements for various vessel segments and enables the assessment of three lithium-ion battery chemistries: Lithium Iron Phosphate, Nickel Manganese Cobalt Oxide, and Lithium Titanate Oxide. The results indicate that different vessel types prioritize either high energy density batteries or those capable of delivering high power relative to energy capacity. To guide battery selection, a decision tree is presented that matches battery types with specific vessel needs. Lithium Titanate Oxide batteries are well-suited for applications requiring frequent, high power cycles, especially where fast charging is needed. Lithium Iron Phosphate batteries are best for energy-intensive operations, while Nickel Manganese Cobalt Oxide batteries perform well in both high power and high energy applications. This study offers a practical approach, an inventory of battery requirements, and guidance on selecting the chemistries best suited to various vessel types and operational needs.

Relying on a traffic simulator is often necessary when multiple traffic conditions need to be replicated, either to predict specific quantities of particular interest or to assess more general network properties, such as efficiency or resilience, under different scenarios. While potentially delivering a high level of fidelity, producing such a simulation may come at a prohibitive computational cost when it is applied in a real context, making this approach unsuitable for real-time applications in most cases. In this regard, the goal of the present work is twofold. Firstly, we aim to surrogate a traffic simulator with a data-driven approach in order to produce real-time traffic predictions that are also sufficiently accurate and effective. Secondly, we want to determine the minimum amount of data, i.e., the smallest number of sensors deployed on the network, that still allow to obtain predictions within a predefined bound. The effectiveness of the approach is evaluated on the full-scale urban network of Rapallo, Italy, in which we employ the AIMSUN NEXT simulator targeting the morning peak hours, i.e. between 7:00 a.m. and 9:00 a.m. In the paper, multiple state-of-the art ML algorithms are tested to assess their effectiveness as surrogate models under the considered problem.

In contemporary industrial applications, predictive models have been pivotal in bolstering production efficiency, product quality, scalability, and cost-effectiveness while promoting sustainability. These predictive models can be constructed solely based on domain-specific knowledge, exclusively on observational data, or by amalgamating both approaches. They are commonly referred to as Full-, Zero-, or Partial-knowledge-based predictive models, respectively. Full-knowledge based models are highly explainable, point-wise accurate, data efficient, computationally demanding in the prediction phase to achieve high accuracy. Zero-knowledge based models are poorly explainable, data hungry, highly accurate on average, computationally demanding in the construction phase but inexpensive the prediction phase. Partial-knowledge based models focus on taking the best of the two worlds. To maintain a focused scope and avoid redundancy with existing literature, our review will primarily delve into the key industrial applications within a narrow context, encompassing sectors such as extraction, chemical processes, manufacturing, transportation, energy, and construction. Our approach entails several steps. Initially, we conducted a meta-review to pinpoint gaps in previously published surveys on Full-, Zero-, and Partial-knowledge-based predictive models for industrial applications. Subsequently, we present a formal analysis of the subject matter, supplemented with illustrative examples to offer valuable insights. The core of our work comprises a review of existing research categorized by specific industrial applications. Finally, we outline the unresolved challenges and future prospects in this burgeoning field of research. We contend that our work serves as a valuable resource, catering to the needs of both young researchers seeking a solid foundation for their studies, industrial practitioners aiming to grasp core concepts and applications, and senior researchers seeking potential real-world applications for their findings.

Hybrid propulsion is considered a reliable alternative to solely mechanical or electrical propulsion for enhanced ship energy performance. Nevertheless, an increased number of components and interconnections results in more complex ship design problems. The automotive and aviation industries already examine new designs on predefined driving and flying cycles. However, new ships are still assessed on one design point with the regulated Energy Efficiency Design Index (EEDI). Its limited consideration of calm water conditions and installed rated power is characterised as insufficient, if not dangerous. A design methodology that accounts for operational and environmental uncertainty is lacking. This paper proposes a design optimisation framework for the topology selection and sizing of hybrid propulsion systems integrating probability distributions of actual sailing profiles from continuous monitoring. The methodology is demonstrated on the ‘Holland’ class ocean patrol vessels of the Royal Netherlands Navy. Its multi-objective consideration examines a wide design space from an environmental, financial, and technical perspective, solving the mixed-integer nonlinear programming (MINLP) problem with a multi-starting scheme that combines a genetic algorithm and interior point method. The low computational cost is achieved by integrating a state-of-the-art digital twin approach leveraging data-driven and first-principle modelling. The results demonstrate feasible improvements of approximately 4 % for carbon intensity and 11 % for operational expenditure by increasing the size of the electrical motors. The exact configuration and percentage improvement are sensitive to actual operational and environmental conditions, while calm water conditions tend to overestimate savings. Consequently, the use of actual sailing profiles is recommended for more accurate life-cycle predictions.

To reduce emissions, methanol is a favorable carbon-neutrally-producible alternative fuel, which can substitute gasoline in direct-injection spark-ignition (DISI) engines. Robust DISI engine operation relies on a consistent air-fuel mixture. To understand the physical processes that characterize the mixture formation, predictive computational fluid dynamics (CFD) simulations are used to improve the understanding, operation, and emissions of these engines. However, using an alternative fuel such as methanol often poses challenges to the CFD simulations’ validity due to the alteration of the fuel properties. This study presents the validation of a CFD modeling approach that can be applied to the predictive modeling of DISI methanol engines. Our methodology uses Lagrangian-Eulerian methods to model the methanol eight-hole counter-bore style Spray M injector from the Engine Combustion Network (ECN). We used the Spray M1 condition, which represents a late-injection spray under a high ambient pressure and temperature environment. For the present study, we employed both a Reynolds Averaged Navier Stokes (RANS) and a Large Eddy Simulation (LES) turbulence approach in CONVERGE-CFD. To validate our models, we used the projected liquid volume (PLV) maps generated by the tomographic liquid volume fraction (LVF) based on methanol. Subsequently, we tuned our models based on the corresponding numerical predictions of the liquid penetration and LVF distributions. The results demonstrated that both the RANS and LES models could replicate the spray morphology and liquid length. While the RANS model was unable to fully capture the complex phenomena of spray collapse and sweeping in the methanol multi-hole spray, the LES model effectively reproduced these behaviors without excessive tuning effort.

The electrification of ship power systems plays a center role in the mobility transition towards sustainable transport solutions. It allows the integration of various power sources, energy storage systems, and intermittent generation. The integration of an increasing number of components with distinct characteristics shapes the notion of a shipboard microgrid which benefits from a modular approach in its design to reduce costs and uncertainties. DC distribution facilitates the modular design by simplifying the control, and, combined with power electronics interfaces, increases the controllability of power flows in the system. To handle the increasing system complexity, this work proposes a distributed and predictive control approach, addressing the modular topology of future shipboard power systems and leveraging load power forecasting. Investigations show that a distributed, predictive energy management reaches a similar performance as a centralized implementation. For a modular shipboard power system, the proposed method decreases both fuel and degradation costs with increasing performance gains for longer prediction horizons.

The maritime sector aims to achieve short and medium-term sustainability targets through the conversion of Internal Combustion Engines to methanol operation. For small to medium sized engines, Port Fuel Injection (PFI) is the most viable injection method to achieve this conversion. However, the knowledge of the behaviour of methanol in combustion engines, particularly its spray characteristics under PFI conditions, is limited. To better understand liquid methanol sprays, this paper studies the injection of methanol in marine PFI conditions through Computational Fluid Dynamics (CFD) modelling. The CFD models use the Lagrangian-Eulerian (LE) coupling method within the Reynolds Averaged Navier Stokes (RANS) turbulence framework. Numerical results were validated using dedicated methanol experiments from the literature for both high and low injection pressures. Subsequently, this predictive CFD framework was used in a number of different injection pressures with scaled injection quantities that represent marine applications. Moreover, we demonstrated that high injection pressure improves atomisation and, thus, evaporation prior to wall impingement. This work strongly contributes to our understanding of marine PFI methanol engines by modelling fuel quantities relevant for ship applications. Our approach can be implemented in full engine simulations to solve evaporation challenges often found in small-bore methanol marine engines.

Methanol is considered an alternative fuel for the shipping decarbonisation, the properties of which, however, impact the marine dual-fuel engines ignition and combustion characteristics, especially at low load conditions. This study aims at parametrically optimising a marine dual-fuel engine operating with methanol high energy fraction at low loads to achieve knock-free combustion with the highest efficiency and lowest emissions. Computational Fluid Dynamics (CFD) modelling in the CONVERGE software is employed for the investigated large-bore marine four stroke engine considering four injection strategies including single, two stage and stratified injection. The Reynolds Averaged Navier Stokes (RANS) approach is employed to represent turbulence, the Lagrangian-Eulerian approach is used for the spray formation, and the SAGE detailed chemistry solver is used for modelling combustion. The CFD model was first developed and validated for the engine diesel mode. Subsequently, the validated model was expanded to accommodate the direct injection (DI) of both methanol and diesel fuels. Parametric runs are performed considering the compression ratio (CR) in the range 14–17 and the temperature range at inlet valve closing (T_IVC) 360–400 K. The results reveal that acceptable combustion efficiency and high thermal efficiency are achieved with CR and T_IVC above 17 and 380 K respectively for single injection, above 16 and 380 K respectively for double injection, as well as above 14 and 360 K respectively for stratified injection. Stratified injection is proposed to improve engine performance and reduce NOx emissions. This study provides insights to achieve stable and efficient operation of methanol-fuelled marine engines at low loads, and as such it contributes to the maritime industry decarbonisation.

The maritime industry increasingly adopts hybrid fuel cell systems to reduce emissions and improve energy efficiency. This chapter examines the current state-of-the-art energy management strategies (EMS) for hybrid fuel cell applications in ships. It provides an in-depth analysis of various strategies, including rule-based, optimization-based, and learning-based approaches, highlighting their benefits, challenges, and real-world applications. The review begins with an overview of hybrid fuel cell systems, their configurations, and control strategies, followed by a detailed examination of EMS. Rule-based strategies are discussed in terms of their simplicity and effectiveness in dynamic marine environments. Optimization-based strategies are evaluated for their ability to enhance system and performance through advanced computational techniques. Learning-based strategies, particularly those leveraging machine learning and reinforcement learning, are explored for their potential to adapt to varying operational conditions. The chapter concludes by identifying the technical, economic, and regulatory challenges facing the adoption of these strategies and proposing future research directions.

Methanol is a promising alternative fuel, which can assist in reducing emissions in heavy-duty (HD) dual-fuel (DF) compression ignition (CI) engines. In medium and large bore marine engines, DF operation is achieved through either direct injection (DI) or port fuel injection (PFI) of methanol with diesel acting as a DI pilot fuel for ignition. However, the injection of methanol presents a significant challenge due to its high latent heat of vaporization and decreased lower heating value (LHV) compared to diesel. Therefore, for the same energy content operation, methanol requires around eight times the amount of heat to evaporate completely in comparison to diesel, which results in lower in-cylinder temperatures. This charge cooling effect leads to a strong negative temperature gradient influencing ignition and flame propagation. This paper aims to quantify the cooling effect of methanol in a heavy-duty dual-fuel direct injection compression ignition (DICI) engine environment. The presented methodology uses computational fluid dynamics (CFD) simulations to model methanol sprays with validation originating from the engine combustion network (ECN) Spray D experimental data. The CFD models operate within the Lagrangian–Eulerian framework in CONVERGE-CFD using the Reynolds Averaged Navier Stokes (RANS) turbulence modeling. Compared to diesel, injecting methanol with the same energy content exhibited up to 100 K more decreased temperature within the mixture. Consequently, this cooled mixture may pose challenges to combustion stability due to the intense temperature gradients. Nonetheless, lower mixture temperature decreases NOx emissions, which can prove beneficial for high methanol energy fractions in dual-fuel DICI engines.

A key factor towards zero-emission shipping is the adoption of electric propulsion with hybrid power sources. The heterogeneous power sources of modern electric vessels require optimal energy management systems, as conventional rule-based control in hybrid energy systems may result in suboptimal solutions with limited flexibility. Advanced optimal control strategies offer a promising avenue to address this issue. This paper presents a novel control strategy based on the Equivalent Consumption Minimization Strategy for a dual-fuel full-electric vessel operating with diesel engines and hydrogen fuel cells taking into account both fuel cost and NOx emissions. The effectiveness of the developed controllers is evaluated against a benchmark derived from state-of-the-art strategies in a simulation study using real-world data. The results highlight the controller's performance, as well as the operator's choice by selection of weights for the objectives. The proposed control strategy achieves nearly 2 % fuel savings compared to a single-objective rule-based controller. It also exploits the potential for up to 45 % reductions in NOx emissions. When both objectives are combined, the controller still delivers over 0.5 % fuel savings while reducing NOx emissions by nearly 15 %. If a financial cost is assigned to emissions, the total operational cost savings increase to more than 4 %.

Optimizing the hull form is crucial for enhancing vessel performance, requiring a careful exploration of design variations once an initial concept is chosen. To address this, a well-defined parameterization and search space must be established, followed by a numerical optimization process. This phase involves balancing three key requirements: improving performance indicators like resistance (and consequently emissions), ensuring physical plausibility (e.g., designs that meet stability standards), and maintaining computational efficiency. While existing methods partially address these challenges using surrogate models to optimize performance, conducting a-posteriori checks for physical plausibility, and reducing the search space a-priori, they fall short of fully integrating these aspects. In this work, we introduce a two-fold approach to address these challenges. First, we integrate a key physical plausibility criterion directly into the optimization loop: the International Maritime Organization (IMO) Intact Stability Code. For the first time, we use a surrogate model to include this constraint within the numerical optimization process. This enables the generation of designs that inherently meet stability requirements. Second, we reduce the computational demands by applying data-driven methods to limit the search space in a problem-specific manner. This targeted reduction reduces the computational load without compromising the overall optimization performance. We test our proposal by optimizing the KCS hull form. Our results demonstrate two main achievements. First, we find that current optimization pipelines fail to meet the IMO stability guidelines, whereas our method achieves compliance by design. Second, our approach reduces the computational burden by 30 % without sacrificing performance, representing a significant leap forward in practical hull design optimization.

The push to attain commercialization of the floating offshore wind industry and subsequently achieving net-zero carbon emission by the year 2050 requires the utilization of cutting-edge design and analyses techniques. Geometric design parameterization and optimization is an effective technique that can be employed in modelling and optimizing a Floating Offshore Wind Turbine (FOWT) substructure. It is an essential framework with the capability of innovative concept generation of platform types in the FEED design phase. This study addresses the conceptual design shape generation, multidisciplinary design analysis and optimization (MDAO) of spar variants FOWT substructure developed from the standard NREL OC3 spar. The methodology involves the use of non-uniform rational basis spline (NURBS) parameterization technique to generate design variants with the flexibility of varying the control points to facilitate varying geometric shapes due to the local propagation property of the NURBS curve. Design variables passed through the NURBS curves control points generates a robust and rich design space and the potential flow hydrodynamic analysis tool in the DNV SESAM suite is used to estimate the hydrodynamic response. The design and analysis phases are explored and exploited for optimal design solution based on specified objectives and constraints with the use of state-of-the-art derivative-free optimizers. The optimal designs were evaluated for three sets of FOWT static pitch angle constraints (5, 7 and 10) degrees, a positive ballast constraint for stability and a constraint on nacelle acceleration root mean square (RMS) value below 30 % of the gravitational acceleration. The single objective function considered in the study is to ensure a minimum mass of the steel material utilized in the design, which invariably leads to a reduction in cost of the substructure material used in fabrication. Achieving this single objective results in an altered geometric shape variants from the baseline OC3 spar substructure for all the three cases evaluated. Verification of the nacelle acceleration response in time domain was further evaluated for the three optimal design cases selected and compared with recommended standards which is below 0.3 g. Although, the nacelle acceleration for the optimal variants is more conservative in time domain assessment than the frequency domain assessment, the values are still below the recommended 0.3 g from standards. Also, the masses of selected optimal design for each constraint were compared to the standard OC3 case study. An observation made in this study is that as the static pitch angle of the FOWT system gets larger, the lower the mass of the optimal substructure and inherently the capital expenditure of the substructure. Finally, the selected optimized platforms were analysed with a non-linear, time domain approach to confirm the level of accuracy of the key response parameters obtained with the frequency-based approach.

In the scope of the energy transition, the maritime industry, still heavily relying on fossil fuels, is facing expectations to reduce its carbon output. Electrified shipboard power systems (SPSs) equipped with hydrogen fuel cells (FCs) and energy storage systems (ESSs) are a promising solution for the shift to zero-emission shipping. A remaining challenge is the efficient coordination of multiple parallel power generation and storage modules. This article proposes a modular approach to the power system control to offer a plug-and-play capability for multiple FCs and ESSs, facilitating a topology reconfiguration. Virtual impedance-based droop is implemented to achieve power sharing and load frequency decoupling in a decentralised architecture. An additional low-bandwidth communication is leveraged to enable parameter adaptation after a topology reconfiguration. The methodology is tested numerically with a short-sea cargo vessel serving as a case study. The local controllers are tuned to achieve load frequency decoupling between FCs and batteries matching the specified time constant. For a maneuvering power profile, the average FC power gradient could be decreased by 36%, limiting their degradation caused by dynamic operation, while increasing the depth-of-discharge of the batteries. The simulations further show that an adaptation of control parameters after a component fault can be used to maintain the system’s voltage dynamics. The voltage drop caused by a load step in a reconfigured system that disconnected one of two ESS could be reduced by 37.5% by control parameter adaptation.

Hybrid power systems are increasingly adopted onboard. Lithium-ion batteries now serve as a viable energy storage solution that enhances fuel efficiency and reduces the operating hours of main power units, thereby reducing operational expenses. However, integrating batteries onboard requires decision-making that accounts for diverse scenarios, including battery chemistry, variations in vessel operational profiles, and fluctuating fuel prices. To address these challenges, this study investigates whether battery sizing and scheduling of the power and energy management system require a scenario-based stochastic decision framework. Specifically, it examines how energy storage requirements are influenced by varying load profiles, whether the optimal battery size and power management strategy are affected by fuel price fluctuations, and how robust the overall strategy remains under operational uncertainties. A deterministic equivalent of a two-stage stochastic decision framework is introduced to incorporate these uncertainties, offering insights into the required battery technology, capacity, and correlated behavior of onboard energy management. Multiple scenarios are applied to a trailing suction hopper dredger, analyzing three load profiles with distinct variations in power demand. With reserve power constraints enforced, the optimal battery capacity remains fixed. However, when these constraints are relaxed, the optimal battery size becomes more sensitive to fuel price changes. In addition, the results showcase reduction in diesel engine operating hours—thereby lowering both fuel consumption and maintenance costs, demonstrating that these operational benefits depend not only on the battery's size but also on its available throughput, which allows for deeper cycling.