Accurate Lifetime Design of Critical Mechanical Equipment for Clean-Energy Generation in the Context of Carbon Neutrality

Run-Zi Wang; Wen-Rui Nie; Chuanyang Lu; Zhengyang Zhang; Yipu Xu; Yutaka S. Sato; Hideo Miura; Xian-Cheng Zhang; Shan-Tung Tu

doi:10.1016/j.eng.2025.09.029

Engineering ›› 2026, Vol. 61 ›› Issue (6) :317 -332. DOI: 10.1016/j.eng.2025.09.029

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Accurate Lifetime Design of Critical Mechanical Equipment for Clean-Energy Generation in the Context of Carbon Neutrality

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Abstract

In the context of carbon neutrality, the large-scale commercialization of clean-energy generation demands an innovative engineering paradigm that can ensure the reliability and durability of the associated critical mechanical equipment. In this study, structural-integrity challenges that are encountered during the clean-energy transition were investigated, and advancements in accurate lifetime-design methodologies were explored. This study addressed the complexities of multi-mode damage interactions and demonstrated the effects of such interactions on the critical mechanical equipment. By tracking the evolution of lifetime-design approaches, the fundamental aspects of damage-driven lifetime-design methodologies were determined. A case study that involved creep-fatigue-oxidation interactions demonstrated the simplicity and high accuracy of the modeling methodology that was developed during this study for industrial applications. To evaluate the carbon-reduction benefits that are associated with lifetime extension, a three-level quantitative criterion, which links prediction scatter, extension potential, and net emissions reduction, was developed. Hierarchical Bayesian modeling was also implemented to capture multi-level uncertainties across various regions and energy types, thereby providing probabilistic insights into diverse operational scenarios. In the future, accurate lifetime design is expected to be integrated into a full-chain technical tetrahedron for structural-integrity evaluations; thus, it will redefine the role of engineering in the design, manufacture, operation, and maintenance of mechanical equipment that is critical for a sustainable future.

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Mechanical equipment / Clean energy / Multiple damage interactions / Lifetime design / Carbon reduction

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Run-Zi Wang, Wen-Rui Nie, Chuanyang Lu, Zhengyang Zhang, Yipu Xu, Yutaka S. Sato, Hideo Miura, Xian-Cheng Zhang, Shan-Tung Tu. Accurate Lifetime Design of Critical Mechanical Equipment for Clean-Energy Generation in the Context of Carbon Neutrality. Engineering, 2026, 61 (6) : 317-332 DOI:10.1016/j.eng.2025.09.029

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1. Introduction

Global warming has progressed and affected the Earth's climate since the Industrial Revolution. Natural variability and human activities are both recognized as contributing factors to this continuous warming trend. CO₂ emissions, the global temperature, the rise in the sea level, and extreme weather events are key indicators of global climate change. Fig. 1 demonstrates that these indicators have notably increased since 1900, and particularly large increases have occurred within the last two decades. Because CO₂ emissions are caused by industrialization and economic growth, which are reliant on fossil fuels, they are primarily responsible for more than half of the observed warming [1]. Global CO₂ emissions have surged by approximately 50% since 2000, and the total amount of emissions that have been produced since 1900 has exceeded 1.5 trillion tonnes. Because greenhouse gases trap heat that would otherwise radiate from the Earth into space, the global temperature has risen by more than 1 °C since 1900. A substantial portion of the warming that has been observed has occurred after 1980. In addition, the global sea level has risen at an average rate of 2 mm per year, which led to a total increase of more than 200 mm since 1900. A continuous rise in the sea level will threaten human populations and damage coastal infrastructures. Furthermore, the occurrence of extreme weather events, such as simultaneous droughts and heatwaves, has also increased dramatically; the frequency of such events has increased by more than 140-fold since 1950 [2]. In response, various international agreements, such as the United Nations Framework Convention on Climate Change (1992), United Nations Framework Convention (1992), the Kyoto Protocol (1997), the Paris Agreement (2015), and the Glasgow Climate Convention (2022), have promoted carbon neutrality according to the principle of “common but differentiated responsibilities” [3].

Carbon neutrality, which is a central tenet of international efforts, is based on balancing carbon emissions with an equivalent amount of carbon uptake, as shown in Fig. 1(a). The achievement of carbon neutrality encompasses two critical processes: carbon sequestration and carbon reduction. Movement toward carbon neutrality is guided by strategic plans that outline the procedures by which net-zero emissions can be achieved and that emphasize the development and deployment of low-carbon techniques, many of which are still in their developmental stages [4]. According to the Carbon Neutral Energy Pathway Map [5], nearly half of the current emission-reduction strategies are expected to achieve net-zero emissions by 2050. The energy-source combination that has been adopted by China (Fig. 1(b)), which is expected to continue to evolve according to a pre-set strategy until 2060, represents an effective carbon-reduction strategy [6]. This evolution involves a significant transition from fossil fuels to clean-energy sources, such as wind, solar, and nuclear power systems. By 2060, these three sources are projected to constitute approximately 66% of the national energy production; this percentage represents a significant increase from the 10% of 2025.

One of the challenges associated with clean-energy equipment is that the equipment is often subjected to multi-mode damage interactions (i.e., creep-fatigue, oxidation-fatigue, and corrosion-fatigue interactions) due to harsh operating environments and long design lives. Such interactions present severe structural-integrity challenges. Structural integrity encompasses a comprehensive range of subjects [7], among which lifetime design serves as an essential link. The purposes of lifetime design include the revelation of the damage mechanisms at the material level, the determination of life-prediction models and strength criteria at the geometric level, the development of lifetime-assessment procedures at the component level, and the ultimate implementation of full-life safety-assurance strategies for mechanical products [8]. It ensures that structures and components not only comply with design specifications but also maintain their integrity throughout their entire life cycles. Predictive accuracy is essential to sustainable operations and reliable lifetime extension [9]; however, it introduces new challenges, such as the quantitative characterization of interactive damage-evolution trends and the adaptability to clean-energy generation techniques [10,11]. To address these challenges, advancements are needed in four key areas: multi-objective testing methods, multi-scale computational frameworks, multi-module collaborative optimizations, and multi-disciplinary management platforms. The goal of advanced multi-objective testing methods, which are exemplified by high-throughput [12] and miniaturized specimen techniques [13], is not only to elucidate the mechanisms of multi-mode damage interactions but also to provide a comprehensive understanding of material behaviors under harsh loading conditions. Multi-scale computational frameworks, which encompass component-scale life assessments, macro-scale damage evaluations, meso-scale energy dissipation, micro-scale dislocation dynamics, and molecular-scale quantum mechanics and molecular mechanics, are comprehensive tools that can guide engineering design and enable damage mechanisms to be understood from multiple perspectives. The primary innovation of multi-module collaborative optimization strategies lies in damage identification that is based on the divide-and-conquer principle. The feedback mechanisms of such strategies are essential modules by which active life-design paradigms can be achieved. Multi-disciplinary management platforms, which integrate expertise from diverse domains, including mechanics, environmental science, materials science, and informatics domains [14], can be supported by sophisticated software packages that act as carriers, thereby enabling engineers to transform multiple damage evolutions of critical components into actionable lifetime-management strategies [15].

Since this study was primarily focused on the large-scale commercialization of clean-energy generation, the development of a novel theoretical framework that can quantify the carbon-reduction benefits of clean-energy generation was pursued, as illustrated in Fig. 1(c). The pursuit of this framework, which provides a quantitative evaluation of the carbon-reduction benefits that are associated with accurate life-predictions for clean-energy equipment, is the primary contribution of this work. In the future, accurate lifetime design will be integrated into a comprehensive technical framework that is referred to as a “full-chain technical tetrahedron,” which will seamlessly connect essential elements of structural integrity, including material design, reliability-centered manufacturing, and operation and maintenance (O&M), as illustrated in Fig. 1(d).

This study addressed the scientific and engineering challenges associated with accurate lifetime design of the critical mechanical equipment of clean-energy systems. Section 2 explores the intricacies of multi-mode damage interactions and emphasizes their impacts on a variety of critical mechanical equipment used for clean-energy generation. Section 3 discusses the damage-driven lifetime-design methodology, which was based on the main operational features of clean-energy equipment that was developed during this study. A case-study validation is presented to demonstrate its high predictive accuracy and practical applicability. Section 4 discusses the development of an innovative theoretical framework that can accurately calculate and validate carbon emission-reduction benefits. It emphasizes the importance of accurate lifetime design to the achievement of carbon-reduction objectives. Section 5 presents a probabilistic model that was built to compare the decarbonization potential across diverse energy systems and national contexts. Both conclusions and perspectives are provided in Section 6. The primary contributions of this work include the development of a deterministic-probabilistic-coupled evaluation framework that can quantify carbon-reduction benefits with respect to the lifetime prediction accuracy. This advancement may enhance structural reliability and quantify environmental contributions within the broader context of carbon neutrality.

2. Multi-mode damage interactions in critical mechanical equipment

Reduced carbon emissions and enhanced energy efficiency require that the mechanical equipment utilized by clean-energy generation systems be subjected to higher operating temperatures, greater working pressures, and harsher service environments than equipment that is employed in fossil fuel-based energy systems. However, a significant challenge that is associated with the mechanical equipment powered by clean energy is the synchronous accumulation of diverse types of damage, which results in unexpected system failures that may occur without obvious signs. This section introduces the critical mechanical equipment used in wind, solar, and nuclear energy-generation systems, along with the multi-mode damage interactions within the equipment.

The development of modern wind turbines, which are exemplified by the 55 kW turbine that was introduced in 1980, represents a significant advance due to its associated cost reductions and achievement of large-scale energy generation [16]. An advanced wind-turbine system primarily consists of a power module, blades, a control and communications system, a nacelle, a drive train, and auxiliary systems [17], as shown in Fig. 2. The turbine blades, which are the most critical and expensive components of the system, possess high failure probabilities of approximately 25%; therefore, they influence the efficiency and reliability of the entire system. The fatigue phenomenon caused by fluctuating wind conditions is often accelerated by aerodynamic asymmetry and yaw misalignment, which result in rapid damage accumulation [18]. Additional factors, such as ice accumulation and dirt infiltration, can further intensify the blade wear and thereby accelerate the aging process and increase the maintenance requirements. Furthermore, power modules, which encompass a variety of components, such as inverters, generators, and rotor hubs, are also prone to mechanical, electrical, and cooling-system failures; their principal sources of damage are fatigue, corrosion, and vibration.

The concentrating solar power (CSP) technique is a vital alternative for clean-energy generation [19]. It operates by concentrating solar radiation onto a small area, thereby heating a heat-transfer fluid. This energy is subsequently transferred to a thermal-energy storage system and is ultimately delivered to a turbine in the form of a fluid, thereby facilitating a continuous and stable power supply [20]. The critical components of the CSP technique include a solar field with heliostat structures and a central receiver or tower. However, prolonged operation of the central receiver, which is subjected to thermal stresses from corrosive molten salts at elevated temperatures, introduces potential risks of central-receiver, pump, or piping failures, as well as corrosion damage to the materials from which these components are constructed [21]. Such failures can lead to serviceability deteriorations, which reduce the power-generation efficiency and result in unscheduled outages.

Nuclear energy is a critical and large-scale alternative to fossil-fuel energy, and it can provide a stable and reliable power supply. It requires fission reactions that release controlled energy to produce continuous heat that can be used to generate steam and drive turbine generator systems. Rapid advancements have been made in nuclear-energy generation; these include the development of standardized, large-scale plants, as well as the development of Generation III reactors, which can enhance safety and capacity. The Generation IV reactor, which has been recognized for its safety and economic viability, is actively being promoted and will enter the demonstration stage by 2030 [22]. The key components of a nuclear system include a steam generator, a reactor pressure vessel, and fuel cladding. The reactor pressure vessel contains the controlled fission reactions of the fuel rods. It is consistently subjected to irradiation damage during the nuclear reactions, oxidation damage due to the harsh operational environment, and creep damage, which is caused by prolonged steady-state operation.

The prominent features of damage mechanisms that are shared among various types of mechanical equipment are the simultaneous exposure of the equipment to diverse modes of damage. Although extensive research has been conducted regarding individual damage modes, such as creep, fatigue, corrosion, oxidation, and erosion, the understanding of the complexities of multi-mode damage interactions and their multi-scale characteristics is incomplete. Acquiring knowledge that can mitigate this deficiency is essential if the design and service requirements of advanced energy systems are to be met. It should be emphasized that multi-mode damage interactions are not merely simple superpositions of single damage modes. The nonlinear nature of these interactions frequently results in accelerated rates of damage accumulation [23]. Therefore, a primary scientific challenge associated with clean-energy generation is accurate descriptions of the multi-scale, spatiotemporal evolutions of the microstructures, damage accumulation, and defects of mechanical structures, particularly those subjected to the harsh service environments typical of clean-energy generation techniques.

Multi-mode damage interactions predominantly encompass creep-fatigue, corrosion-fatigue, and erosion-fatigue interactions. These interactions encapsulate most of the failure modes that are encountered in the structural materials of major clean-energy generation systems. As shown in Fig. 2, creep-fatigue interactions shorten the surface fatigue-crack initiation cycle and accelerate the rate of subsequent crack propagation, while the influence of fatigue on creep reduces the creep ductility of the material itself [24]. Components that often experience creep-fatigue interactions include reactor pressure vessels, nuclear-system piping, and solar reflectors and frames. Next, corrosion-fatigue interactions are some of the predominant damage mechanisms that are observed in offshore-turbine blades, solar-support structures, and nuclear steam generators. In contrast to stress-assisted corrosion, corrosion-fatigue interactions can occur in any polycrystalline materials [25] when a combination of a corrosive medium and alternating stress is present. Erosion-fatigue interactions represent a form of failure in which the erosion and fatigue damage mechanisms primarily act together on metallic surfaces. These interactions are reflected in variations in the rate of mass loss [26]. They typically occur in cases of premature failure of several types of components, such as nacelle and tower components, solar-panel surfaces, and nuclear heat exchangers [27]. Finally, complex interactions, such as creep-fatigue-oxidation interactions, further complicate the degradation mechanisms by the introduction of chemical reactions. These multifaceted damage modes emphasize the need for interdisciplinary approaches so that the lifetimes of critical mechanical equipment can be modeled and predicted under real-world service conditions.

3. Advances in lifetime design methodologies

3.1. Evolution of lifetime design methodologies

According to a technical report from the National Academy of Sciences of the United States of America [28], over the past decades, the persistent endeavor to shift from an “empirical” paradigm to an “accurate” paradigm has led to substantial efforts in the exploration of new frontiers in various engineering fields. The evolutionary processes of design methodologies reflect the evolving complexity of engineering demands and analytical capabilities, as illustrated in Fig. 3.

Before 2010, design research could be divided into two distinct eras in which either “strength designs” or “durability designs” were dominant. Although the strength-design methodologies rely on pure phenomenology, most of the theories evolved beyond the notion of infinite life; rather, they shifted toward safe-life design before 1980. For instance, for cases of cyclic loading, the design principle involves the use of design stresses for fatigue-strength calculations that exceed the fatigue limit; thus, the predetermined lives of the relevant components are ensured. Classical strength criteria that have been determined using this approach remain a vital basis for structural-integrity assessments in contemporary practice. As engineering operations encountered an increasing number of challenges, the design emphasis transitioned to durability-design methodologies, which were in use for nearly 30 years (1980-2010). In this era, it was recognized that the endurance of materials subjected to high temperatures and harsh environmental influences was as crucial as the initial strength criteria. This period marked the advent of a more dynamic understanding of material behavior. It was characterized by a focus on understanding material degradation with time and enhancing the accuracy of predictive tools with physical implications [29,30].

From another perspective, the concept of full lifetime typically encompasses both the crack-initiation and crack-propagation periods. The study of damage mechanics primarily addresses scientific issues that occur prior to the crack-initiation life, whereas the study of fracture mechanics focuses on the crack-propagation life or with defects in the initial state. Generally, the evolution trend in analysis techniques has supported advances in theoretical approaches, with no exceptions for either damage mechanics or fracture mechanics, as shown in Fig. 3. In the durability-design era (before 2010), the synchrony in the development of both damage-mechanics and fracture-mechanics techniques is particularly notable. This is evident in theoretical modifications that concern several features of interest, such as high temperatures, probabilistic statistics, and computational mechanics. A more representative example is the rapid enhancement of multi-physics finite element analysis (MP-FEA) techniques that occurred in the 2000s; this enhancement has simultaneously propelled the domains of both multi-scale and multi-field approaches in damage-mechanics and fracture-mechanics research [31].

As the 21st century progressed, the effects of multi-mode damage interactions on both materials and structures, particularly within the realm of clean-energy generation, became increasingly evident. Scholars and engineers have recognized that traditional strength and durability methodologies may not be adequate for the complexities that are introduced by these interactions. In this era, many scholars have attempted to utilize damage variables, rather than empirical variables, to address the emerging fracture-mechanics challenges. Meanwhile, the study of damage mechanics has progressed even further than that of fracture mechanics (Fig. 3), increasingly focusing on interdisciplinary integration, which is exemplified by the incorporation of thermodynamics into energy-based damage theory (EB-DT) [32] and the inclusion of manufacturing principles in anti-damage design theory (AT-DT) [33]. After 2020, the exponential growth of machine learning analysis (MLA) and surrogate modeling analysis (SMA) markedly enhanced the computational efficiency of multi-physics simulations [34,35]. This progress has been instrumental to the initiation of the formulation of data-driven fracture theories (DD-FTs) [36] and hybrid-driven damage theories (HD-DTs) [37]. Consequently, the subsequent era of “damage-driven lifetime design” is defined in this review, which consolidates the overall development trends in extensive damage-related research that have occurred since 2010.

3.2. Fundamentals of damage-driven lifetime design

In traditional practice, engineers have often relied on standardized formulas and experience-based codes to estimate the service lifetimes of mechanical equipment [38], [39], [40]. However, as mentioned in Section 2, the complexity of multi-mode damage interactions has exposed the limits of empirical methods and has motivated a shift toward more physics-based strategies.

A flowchart comparison between traditional empirical-based lifetime design methodologies and the proposed damage-driven approach is illustrated in Fig. 4. In traditional methods, the design cycle or hour typically begins with the sequential definition of the loading waveform, analyses of the heat transfer and stresses, and utilization of experimental data. Engineers use the S-N curve, the Langer curve, the Larson-Miller curve, or codified formulas to estimate the life, and they apply safety factors to account for uncertainties [41]. After several empirical trial-and-error iterations, the final service lifetime is determined for the mechanical equipment, as shown on the left side of Fig. 4(a). In contrast, in the damage-driven design flow, the relevant damage modes are identified from the operating conditions [42]. Accurate damage-driven life-predictions are the most important means by which the waste of resources caused by excessive and unjustified conservatism can be overcome. This principle will be illustrated in detail later. Rather than using lumped safety factors, engineers assign a damage variable to each mechanism and define how it evolves under the given conditions [43]. The resultant criteria not only ensure high predictive accuracy but also ensure flexibility and safety for a variety of objects. Typically, these criteria can be seamlessly integrated with probabilistic damage evaluations and damage- and physics-based reliability assessments [44]. Iterations are then performed in the design process according to reliability-based adaption, and the design parameters that affect the external factors are clearly adjusted, as shown on the right side of Fig. 4(a). In essence, damage-driven lifetime design is an interdisciplinary paradigm in which design decisions are based upon physical damage depictions and condition-specific life-predictions rather than on purely empirical safety factors or lifetime curves.

An example of damage identification in a representative piece of rotating equipment used in clean-energy generation systems is demonstrated in Fig. 4(b). Different components experience various degradation phenomena and their interactions. For example, fatigue damage may accumulate in stress-concentration regions of blades and disks due to cyclic loading. If the blades and disks operate at high temperatures with long exposure times, they can also suffer creep and oxidation damage [45]. In addition, corrosion, erosion, and contamination inevitably occur in components that are exposed to more severe environments, such as those in combustion chambers and exhaust liners [46]. As illustrated in Fig. 4(b), engineers are expected to create damage maps for the equipment; in such maps, each critical position is linked with the dominant damage mechanisms. This application-oriented step ensures that no significant failure mode is overlooked. As such, cataloging of the damage modes enables the design to incorporate appropriate models and countermeasures for each [47]. The result of this process is an in-depth understanding of where and how the equipment might degrade during service, thus providing the foundation for a subsequent robust life-prediction.

Fig. 4(c) illustrates the core procedure by which the damage-driven methodology predicts the lifetime, generally building upon the foundation of engineering damage theory, which introduces internal damage variables to quantitatively represent the physical degradation and capacity exhaustion [48]. As mentioned previously, each position, X, is associated with a subset of damage modes: X → D₁, …, D_n, where n represents the damage index. For example, each damage mode is represented by an evolution trend, in which a differential equation or an incremental rule updates the damage variable as a function of another variable, such as the time or the load. This theory is generally written according to Eq. (1):

(1)$D_{n}=f\left(\dot{D}_{n}, \widetilde{P}_{n}, t, N\right)$

where Ḋ_n represents the damage accumulation rate and $\widetilde{P}_{n}$ is the damage variable, which serves as the status parameter. t represents the time experienced by the damage, N represents the number of cycles, and f(·) denotes a general functional relationship to calculate accumulated damage. It should be emphasized that the practical form may involve complexities beyond those of this generalized form. The nonlinearity of the multi-mode damage summation is represented by high-dimension damage-interaction diagrams, as shown in Fig. 4(c). Such a diagram provides a unified rule that maps out failure envelopes for combined damage states, ensuring that multiple damage variables are not simply added linearly but are evaluated in a physics-informed manner [49]. It can be generally expressed by Eq. (2):

(2)$f\left(D_{1}, \ldots, D_{n}\right)=1$

Damage-interaction diagrams allow engineers to graphically and analytically capture how one damage mechanism might accelerate or hinder another, leading to more accurate life estimates under complex service conditions. In the damage-driven design approach, the damage evolutions of all the locations are evaluated in parallel, so the progression of multi-mode damage with time or continuous cycling can be tracked at specific sites [23]. By applying the interaction diagram to the damage state of each location, the predicted life for each critical region can be determined. Moreover, when considering random factors, such as material uncertainty or loading variations, the development of reliability-based life-prediction and probabilistic evaluation methods for mechanical structures is essential. Accordingly, the correlations between the failures of multiple locations should be taken into account. In the authors’ previous work, the cumulative damage-damage threshold interference principle was proposed and applied to reliability analyses of critical high-temperature structures [50]. In these analyses, it was observed that the lifetime of an entire structure with a high reliability level was shorter than that of any individual critical location. This finding demonstrated the effects of multiple failure modes and weak regions in the context of damage-driven lifetime design.

3.3. Validation with a case study involving a creep-fatigue-oxidation interaction

A combined creep-fatigue-oxidation interaction is next introduced as a representative case study to emphasize the modeling transparency and prediction accuracy of the damage-driven lifetime design methodology.

As mentioned in Section 2, many high-temperature rotating components in modern clean-energy equipment are subjected to repeated start-stop thermal cycles along with high centrifugal stresses in oxidizing combustion-gas atmospheres. Creep-fatigue-oxidation interactions that exhibit remarkable microscopic degradation features are prevalent, as shown in Fig. 5(a). It has been reported that many failures of such equipment, ranging from approximately 30% to more than 50% of the total failures, can be attributed to this trio of interacting damage modes [51]. Moving toward Fig. 5(b), the creep-fatigue-oxidation damage mechanism operates through a complex and synergistic combination of deformation processes, which occur under high-temperature cyclic loading with dwell periods [52,53]. Fatigue cycling initiates microstructural damage, such as slip bands and microcracks. Meanwhile, creep contributes to grain-boundary deformation and the nucleation of voids. Oxygen ingress simultaneously promotes the formation of oxide scales along exposed surfaces and the advancement of crack fronts. These mechanisms progressively interact with one another. The oxide formation leads to embrittlement along the crack paths, while the cavities caused by creep tend to merge with the propagating fatigue cracks. The result is accelerated intergranular-crack growth and eventual failure. This combined evolution of various damage modes provides a convincing explanation for unexpected component failure, which is often underestimated by models that consider only a single safety factor [54].

Four candidate life-prediction models were considered for this case-study validation: the frequency modified-damage function (FM-DF) [55], the strain energy-density exhaustion (SEDE) model [56], the generalized SEDE model (GSEDE) [57], and the GSEDE with multi-axial treatments (GSEDE-M) [8]. To ensure consistent comparability, all four models employed the “strain energy density” as the damage variable. The primary differences between these models lie in the number and nature of the driving mechanisms that were considered. Specifically, the FM-DF, SEDE, GSEDE, and GSEDE-M models represented a single-driven model, a dual-driven model, a triple-driven model, and a triple-driven model with a multi-axial stress state, respectively.

The single-driven FM-DF model is an extension of the Mansion-Coffin equation, which considers the time-dependent loading waveforms [55]. It can be expressed by Eq. (3):

(3)$\Delta w_{\text {in }} \cdot N^{a_{1}} \cdot \gamma_{\text {eff }}^{a_{1}\left(b_{1}-1\right)}=\text { Constant }$

where Δ_win is the inelastic strain energy-density range, γ_eff represents the effective loading frequency, and a₁, a₂, b₁, and b₂ are material parameters. The dual-driven SEDE model was developed for cyclic loads with tensile dwell periods. It is a typical life-prediction model that is based on the linear damage summation (LDS) rule [32], and it is expressed by Eq. (4):

(4)$\begin{aligned} d_{\mathrm{f}} & =\frac{1}{a_{2} \cdot \Delta w_{\text {in }}^{-b_{2}}} \\ d_{\mathrm{c}} & =\int_{0}^{t_{\mathrm{h}}}\left[\frac{M_{2}-N_{2} \cdot \log (1+t)}{\min \left(\varphi_{2} \cdot w_{\text {in }}^{n_{2}}, w_{\mathrm{f}, \text { trans }}\right) \cdot(1+t)}-\frac{M_{2}-N_{2} \cdot \log (1+t)}{w_{\mathrm{f}, \text { trans }} \cdot(1+t)}\right] \mathrm{d} t \\ N & =\frac{1}{d_{\mathrm{f}}+d_{\mathrm{c}}} \end{aligned}$

In Eq. (4), d_f and d_c are the fatigue and tensile creep damage per cycle, respectively; M₂ and N₂ are the transitional equations related to the cyclic deformation behaviors; ẇ_in and w_f,trans represent the inelastic strain energy-density rate and the transition failure strain energy density, respectively; t_h is the hold time in one cycle; and ϕ₂ and n₂ are material parameters. The triple-driven GSEDE model was developed for the additional calculations associated with oxidation damage and the generalized loading waveforms. Therefore, both Eqs. (5a) and (5b) could be inherited into this model, and the oxidation damage could be superimposed on the result [57]. The GSEDE model can be expressed by Eqs. (5a) and (5b):

(5a)$d_{\mathrm{o}}=\frac{r_{\text {cross }}^{2}-\left(r_{\text {cross }}-x_{a}-\sqrt{k_{\mathrm{p}} t}\right)^{2}}{r_{\text {cross }}^{2}} \cdot \frac{\int_{0}^{x_{\mathrm{p}}+x_{\mathrm{a}}} \delta(x) \mathrm{d} x}{\delta_{\text {max }} \cdot\left(x_{\mathrm{p}}+x_{\mathrm{a}}\right)}$

(5b)$N=\frac{1}{d_{\mathrm{f}}+d_{\mathrm{c}}+d_{\mathrm{o}}}$

where d_o is oxidation damage per cycle; x_p and x_a represent the depth of the oxide layer and the thickness of the oxidation-affected zone, respectively; r_crossis the radius of the cross-section; δ(x) and δ_max are the depth-related oxidation damage indicator and the upper platform, respectively; and k_p denotes the parabolic rate-control constant. The GSEDE-M life-prediction model differs from the other three models, in that its damage-mode superposition is achieved from a fundamentally different conceptual perspective. Specifically, this model uses finite element analysis to address complex conditions, such as stress concentrations [8], ratcheting deformation [58], and non-proportional loading [59]. For example, while retaining the GSEDE equations, the GSEDE-M model is enhanced by the application of the critical-plane method and the incorporation of the Wen-Tu factor [60]. Thus, it enables structural-level lifetime design under realistic service conditions.

Clear comparisons of the complexity of the models, which were achieved using both absolute and relative effective degrees of freedom (EDF), are presented in Fig. 5(c). In Fig. 5(c-i), as more damage modes are incorporated, the EDF increases in a gradual and controlled manner. In Fig. 5(c-ii) presents relative EDF values across various modeling categories. Damage-driven life-prediction models exhibit controlled complexity that increases approximately linearly with the number of damage modes. In contrast, the modeling category that is based on fully coupled differential equations (FCDEs) often exhibits exponential growth in complexity due to nonlinear couplings and differential interactions [61]. The MLA-based modeling category often involves hidden layers of complexity because these models suffer from implicit hyperparameters and uncontrolled or inflated EDFs [62]. The relative EDFs of FEM-induced models (which rely on LDS, as shown in Fig. 5(c)), such as the GSEDE-M model, are slightly higher than those of LDS models. However, these increases are always essential for advancements toward structural-level life-predictions. Even under complex service conditions that involve creep, fatigue, and oxidation, the EDF remains explicitly defined and computationally tractable. In general, this characteristic ensures that damage-driven lifetime-design methodologies remain transparent and interpretable, which enables their application to highly complex service conditions.

The life-prediction accuracy of the damage-driven lifetime design methodology was also validated, as shown in Fig. 5(d). The FM-DF, SEDE, GSEDE, and GSEDE-M models were used for the validation. As the model evolved from the FM-DF model to the SEDE model, and finally to the GSEDE model, for the uniaxial cases (Figs. 5(d-i)-(d-iii), which include superalloys, Cr-Mo steels, and stainless-steels, respectively), the predicted-versus-measured experimental-life point cluster progressively moved toward the ±1.5 error bands. Therefore, the life-prediction scatter for the GSEDE model could be defined as S = 1.5. Since the experiments were designed to reflect the primary creep, fatigue, and oxidation damage modes, the GSEDE model served as the representative model carrier for the damage-driven lifetime-design methodology. Furthermore, when typical multi-axial stress states caused by geometric discontinuities and multi-axial loading were considered, the prediction-accuracy indicators of the GSEDE-M model moved toward the ±3 error bands due to the presence of more disturbance factors (Figs. 5(d-iv)-(d-vi). As previously discussed, this extension forms a necessary bridge from theoretical models to practical engineering designs. Therefore, the prediction accuracy of the “damage-driven lifetime-design methodology” is well supported, and a life-prediction scatter less than 5 could be reasonably adopted for subsequent quantitative analyses of carbon-reduction benefits.

4. Net carbon-reduction benefits: Calculations and validation

4.1. Motivation

The achievement of carbon neutrality necessitates the widespread deployment of reliable clean-energy systems, in which mature nuclear energy has a particularly vital role. Currently, more than two-thirds of the 442 nuclear reactors that are in operation around the world have been in service for more than 30 years, and they are approaching or have surpassed their originally intended design lifetimes of approximately 40 years [63]. These statistics emphasize a critical issue: the divergence between initial design lifetimes and actual service times. This issue has profound implications for effective lifetime-extension strategies and decarbonization potential.

Lifetime-extension techniques offer practical and environmentally favorable methods of enabling prolonged reactor operation without the high carbon footprint that is associated with the construction of new facilities. Therefore, they allow major infrastructure investments to be deferred while substantially reducing life-cycle emissions, in addition to offering cost-effective pathways to carbon reduction. Empirical evidence has already demonstrated the feasibility of such lifetime extensions, and operating licenses have been granted for up to 60 years in some regions. In addition, exploratory initiatives have even extended the design horizon to 100 years [64]. However, the carbon-reduction benefits of lifetime extensions are not guaranteed. High scatter may undermine safety margins, increase operational risks, and necessitate frequent interventions that offset potential environmental gains.

Therefore, there is an urgent need for the development of a quantitative assessment framework that links prediction accuracy, design decisions, and net carbon benefits. This framework must not only accommodate the technological scalability of energy systems but must also capture the associated balance between operational emissions and system reliability. Accordingly, a structured approach was developed to support scalable evaluations and informed decision-making.

4.2. Three-level evaluation criterion roadmap

The theoretical perspective presented in Section 3 indicates that the damage-driven lifetime-design methodology has the potential to significantly reduce prediction scatters, thereby enabling more reliable and efficient lifetime extensions. However, lifetime extensions can be both advantageous and disadvantageous in the context of carbon mitigation. Importantly, extending the service lives of energy systems prevents the substantial emissions and resource use that are associated with new infrastructure. However, lifetime extensions require additional operational and maintenance activities, which also generate incremental emissions. This compromise between marginal utility (i.e., emissions avoided by service extensions) and marginal emissions (i.e., emissions added by keeping aging systems in operation) complicates the net evaluation of carbon benefits [65]. Therefore, a structured three-level framework that quantifies the net carbon-reduction benefit, E_net, with respect to the life-prediction scatter, S, is proposed, as shown in Fig. 6(a).

(1) Level I: Determination of the relationship between the lifetime-extension factor and the life-prediction scatter. The primary objective at this level was the determination of a mathematical relationship between the lifetime-extension factor (α) and S. This relationship can be generally denoted by α = f₁(S) (Fig. 6(a)), and serves as the foundational component by which the potential carbon-reduction benefits enabled by accurate lifetime extensions of clean-energy systems are assessed. The lifetime-extension factor is defined by the ratio N_s/N_d, where N_s represents the extended lifetime and N_d represents the initial lifetime. This definition provides a straightforward yet effective method of guiding engineering decisions concerning system upgrades or license renewals.

To construct a reasonable form of the α = f₁(S) relationship, two principles had to be addressed. First, the relationship must reflect a diminishing marginal utility, where improvements in the life-prediction accuracy should yield increasingly limited gains in the service-lifetime extension. Second, it should exhibit smooth and continuous behavior that is consistent with the gradual and nonlinear evolution of lifetime-design methodologies over time. Therefore, the logistic function in Eq. (6) was recommended [66]:

(6)$\alpha(S)=\frac{A}{1+\left(S / S_{0}\right)^{p}}+\alpha_{\min }$

In Eq. (6), A represents the asymptotic saturation level of the extension factor, p controls the steepness of the curve, S₀ is the inflection point, and α_min is the minimum allowable extension factor under low-accuracy conditions.

To understand the empirical behavior of this relationship, one may refer to representative datasets from the long-term operation of clean-energy systems. As an illustrative case, the Dresden Nuclear Station in the United States demonstrates how lifetime-extension efforts have historically progressed from an initial 40-year design life to projected service durations of up to 100 years [67]. This evolution corresponds to a significant reduction in S, which decreased from approximately 20 years in the initial phase to approximately 5 years under advanced methodologies. Table 1 summarizes these phases, not as isolated data points but as broader trends in clean-energy infrastructure, where comparable extensions have been pursued under increasingly stringent reliability criteria [22,68].

Fitting the logistic model in Eq. (6) to such representative data yielded A = 1, α_min = 1.5, S₀ = 8.5, and p = 3; these parameter values resulted in an S-shaped curve with an excellent fit (R² = 0.93), as shown in Fig. 6(b). This curve reveals three characteristic stages in the development of lifetime-design methodologies. The first, which is referred to as the strength-design stage (prior to 1980 in Fig. 3), primarily focused on the determination of conservative safety margins that were based on short-term mechanical strength, while minimal attention was given to long-term operational performance. During this early phase, N_d was generally limited to 30-40 years. Due to insufficient understanding of material-degradation mechanisms and the lack of reliable monitoring tools, the value of S usually exceeded 20 (Fig. 6(b)). The second stage marked a transition toward durability design (1980-2010 in Fig. 3). This shift was facilitated by the integration of enhanced material monitoring and preventive maintenance strategies, which enabled moderate service-lifetime extensions. However, during this period, prediction capabilities were still constrained by limited insight into the underlying damage mechanisms. The third and most advanced stage can be defined as damage-driven design (after 2010 in Fig. 3). It is expected to be adopted in critical infrastructure, such as Gen IV nuclear-energy systems. As a result, the life-prediction accuracy theoretically improves and the S values typically fall below 5. In this stage, the marginal gains from further lifetime extensions begin to plateau, indicating that the exploitable performance potential of the materials will be fully utilized.

(2) Level II: quantification of the carbon-reduction benefit and carbon-emission cost. At this level, the focus first shifted to quantification of the carbon-reduction benefit, E_b, which can be generally expressed by E_b = f₂(α), as shown in Fig. 6(a). It is directly influenced by the actual in-service durations of clean-energy generation systems. If it is assumed that the annual carbon-reduction capacity of a given clean-energy system remains constant, then E_b can be expressed by Eq. (7):

(7)$E_{\mathrm{b}}(\alpha)=C_{\mathrm{b}} \cdot N_{\mathrm{d}} \cdot \alpha$

where C_b is the carbon-reduction coefficient specific to the energy-generation system. However, extensions in the operational lifetimes of energy systems also involve additional carbon-emission costs, E_c, which comprise both fixed and variable components. The variable portion of E_c typically follows a saturation growth pattern, and it represents the O&M emissions per unit of annual electricity generation. Although O&M-related emissions increase with time, the rates of increase gradually decrease and eventually stabilize near an upper limit. This behavior reflects the practical reality that aging systems demand increasingly intensive interventions, which ultimately plateau due to physical or economic constraints. Therefore, an appropriate representation of the annual evolution of O&M-related carbon costs can be expressed by Eq. (8) [69]:

(8)$E_{\mathrm{c}}(\alpha)=E_{\mathrm{c}, \text { fixed }}+\sum_{i=1}^{\alpha N_{\mathrm{d}}} \frac{C_{\mathrm{c}}}{1+\mathrm{e}^{-k_{\mathrm{c} i} i}}$

where E_c,fixed denotes the one-time fixed carbon emissions that are associated with licensing procedures and C_c is the maximum annual O&M carbon cost. ($1+\mathrm{e}^{-k_{\mathrm{c}} i}$) represents the annually varying carbon-emission cost. k_c is the annually varying coefficient of carbon-emission cost, i is the service year index, and e is the natural constant.

(3) Level III: calculation of the net carbon-reduction benefit. By combining Levels I and II, E_net was formulated. Its expression is provided in Eq. (9):

(9)$E_{\mathrm{net}}(S)=\theta\left[E_{\mathrm{b}}(\alpha(S))-E_{\mathrm{c}}(\alpha(S))\right]$

where θ is the energy-adjustment coefficient, which accounts for variations in the generation characteristics, such as the capacity factor, the dispatchability, and the life-cycle emissions, across various clean-energy systems. The reliability and material degradation of aging systems affect the operational carbon costs; thus, θ is related to the reliability factor (RF), and the material mass loss (ML):

(10)$\begin{array}{l} \theta=\frac{1}{1-\mathrm{RF}} \cdot \frac{1}{1-\mathrm{ML}} \\ \mathrm{RF}=1-\frac{\mathrm{FOH}}{\mathrm{PH}} \\ \mathrm{ML}=1-\frac{\mathrm{RM}}{\mathrm{TM}} \end{array}$

In Eq. (10), FOH represents the forced-outage hours, PH is the total operational hours [70], RM is the remaining structural mass, and TM denotes the initial total mass of the system. Therefore, E_net can be explicitly rewritten as Eq. (11):

(11)$E_{\text {net }}(\alpha)=\theta \cdot E_{\mathrm{b}}(\alpha)-\theta \cdot E_{\mathrm{c}, \text { fixed }}-\theta \cdot \sum_{i=1}^{\alpha N_{\mathrm{d}}} \frac{C_{\mathrm{c}}}{1+\mathrm{e}^{-k_{\mathrm{c}} i}}$

The application of Eq. (11) to engineering decision-making can be simplified under practical conditions. When α falls within a moderate range and O&M-related emissions increase slowly, E_c can be linearly approximated according to Eq. (12):

(12)$E_{\mathrm{c}}(\alpha) \approx C_{\mathrm{c}} \cdot N_{\mathrm{d}} \cdot \alpha+E_{\mathrm{c}, \text { fixed }}$

Then, E_net can be rewritten as Eq. (13):

(13)$E_{\text {net }}(\alpha) \approx \theta \cdot\left(C_{\mathrm{b}}-C_{\mathrm{c}}\right) \cdot N_{\mathrm{d}} \cdot \alpha-\theta \cdot E_{\mathrm{c}, \text { fixed }}$

4.3. Case studies with various clean-energy generation techniques

This section presents a set of illustrative case studies that are based on Eq. (13) to demonstrate the real-world applicability of the accurate lifetime-design methodology. These case studies encompass nuclear, solar photovoltaic (PV), and onshore wind power generation under consistent energy-output conditions. Spain was selected as an illustrative national case due to its nuclear infrastructure profile and ongoing policy debates, while comparative data from other countries were used to generalize the findings [71,72]. Spain currently relies on seven nuclear reactors, which were commissioned in the 1980s, to generate more than 20% of its electricity. As these units approach the ends of their initial design lifespans, the decision whether to extend their lifetimes or decommission them becomes increasingly critical. In this paper, the insights from Spain’s experience are situated within a broader global context, in which energy systems with diverse reliability profiles, cost structures, and design philosophies face similar challenges and opportunities. To facilitate comparisons across a variety of technologies and countries, Table 2 was constructed to summarize the representative parameters that were used to estimate the emission-reduction performance under a uniform annual generation capacity of 1 TW·h. This comparison includes clean-energy sources and a natural gas combined-cycle (NGCC) plant, which serves as the emission-intensive baseline. The selected parameters emphasize the systemic characteristics that influence lifetime-extension outcomes. These values were obtained from multiple empirical sources and are intended to reflect typical, rather than technology-specific, operating conditions.

For this analysis, a carbon price of 50 euros per ton, which was proposed in the 2030 distributed generation scenario by the World Environment Organization [73], was adopted. By applying this carbon price to the model parameters, the levelized costs of the clean-energy generation techniques and the emission-reduction benefits were easily converted into carbon-emission equivalents (tCO₂eq) for quantitative comparison. To validate the proposed quantitative evaluation framework, a series of scenario analyses were performed, as shown in Fig. 7. In these analyses, theoretical evaluations were combined with real-world cases. Fig. 7(a) shows that the NGCC system accumulated a substantial O&M-related E_c value throughout its operational lifespan, yet it yielded no corresponding E_b. In contrast, all the clean-energy technologies that were evaluated consistently exhibited positive E_net values across the full ranges of their service lifetime extensions. When coupled with a high-precision predictive methodology, lifetime extension is an effective mechanism by which the decarbonization potential of an energy system can be enhanced, as shown in Fig. 7(b). For example, when the damage-driven design methodology is used with S = 2, the E_net values of the wind, solar, and nuclear technologies increased significantly, with gains of approximately five-fold, four-fold, and two-fold, respectively, relative to the value obtained by using conventional design strategies with S > 20.

In addition, E_net is influenced not only by the type of energy source but also by the underlying design philosophy and the life-prediction accuracy. For example, nuclear energy systems typically feature high generation capacities and relatively low fixed costs, whereas wind and solar technologies often operate with lower capacity factors and comparatively higher fixed-cost structures. Fig. 7(b) demonstrates that the shift from strength-design methodologies to durability-design methodologies was a transition point. At this point, E_net of solar energy overtook that of wind energy; this result emphasizes the importance of tailoring predictive strategies to technology-specific characteristics. Importantly, the usefulness of these case studies extends beyond model validation, in that they also provide a foundation for scenario-based simulations that incorporate regional heterogeneity and technology-specific characteristics. Effectively capturing variability across countries and energy-system types requires a probabilistic, multi-level analytical framework that is capable of accommodating structural uncertainties and enabling robust decision-making in diverse deployment contexts.

5. Hierarchical Bayesian modeling of lifetime divergence across countries and energy types

Building upon the deterministic insights that were presented in Section 4, a hierarchical Bayesian modeling (HBM) framework was designed to account for nested, multi-level structures and enable interpretable probabilistic inference across diverse technologies and regional contexts [74]. This framework enables interpretable probabilistic inference and comparative analysis of lifetime divergence across countries and energy types.

5.1. Methodology

As previously mentioned, N_d is inherently stochastic due to external influences. To account for this characteristic, Bayesian inference was used to estimate the posterior distribution of N_d based on the observed samples. It was assumed that the true distribution of N_d followed a Weibull distribution, which is commonly used in lifetime analyses because it can model right-skewed and heavy-tailed data. The N_d distribution was therefore written using Eq. (14):

(14)$N_{\mathrm{d}} \sim \operatorname{Weibull}\left(\lambda_{0}, k_{0}\right)$

where λ₀ denotes the initial (population-level) scale parameter of the Weibull distribution, representing the characteristic life of the design-life variable N_d. It serves as the true or baseline parameter, whereas λ in the subsequent Bayesian inference represents its stochastic estimate derived from the observed dataset. It was assumed that there were a set of m independent design-life observations, L = [l₁, l₂, …, l_m] (m ∈ N⁺), which were drawn from a Weibull distribution with a scale parameter of λ and a shape parameter of k₀. Therefore, the likelihood function in Eq. (15) could be constructed:

(15)$p(L \mid \lambda)=\prod_{i=1}^{m} \frac{k_{0}}{\lambda}\left(\frac{l_{i}}{\lambda}\right)^{k_{0}-1} \exp \left(-\left(\frac{l_{i}}{\lambda}\right)^{k_{0}}\right)$

Because the Weibull distribution does not have a conjugate prior for the scale parameter, an inverse Gamma distribution was selected as the prior for λ. The inverse Gamma distribution and its prior density function are given in Eq. (16):

(16)$\begin{array}{l} \lambda \sim \text { Inverse }-\operatorname{Gamma}\left(\alpha_{0}, \beta_{0}\right) \\ p(\lambda)=\frac{\beta_{0}^{\alpha_{0}}}{\Gamma\left(\alpha_{0}\right)} \lambda^{-\alpha_{0}-1} \exp \left(-\frac{\beta_{0}}{\lambda}\right) \end{array}$

where α₀ and β₀ are the fitting parameters. The posterior distribution was obtained by multiplying the likelihood and the prior distributions, as shown in Eq. (17):

(17)$p(\lambda \mid L) \propto p(L \mid \lambda) \cdot p(\lambda)$

Given that the posterior did not admit a closed-form solution, Markov chain Monte Carlo (MCMC) techniques, such as the Metropolis-Hastings algorithm and the Hamiltonian Monte Carlo, were used to sample from it. For each posterior sample, λ⁽^s⁾, the corresponding index, s, of the Weibull distribution can be expressed by Eq. (18):

(18)$\begin{array}{l} \mathbb{E}\left[N_{\mathrm{d}}^{(s)}\right]=\lambda^{(s)} \cdot \Gamma\left(1+\frac{1}{k_{0}}\right) \operatorname{Var}\left[N_{\mathrm{d}}^{(s)}\right]=\left(\lambda^{(s)}\right)^{2}\left[\Gamma\left(1+\frac{2}{k_{0}}\right)-\Gamma^{2}\left(1+\frac{1}{k_{0}}\right)\right] \end{array}$

where $\mathbb{E}\left[N_{\mathrm{d}}^{(s)}\right]$ and $\operatorname{Var}\left[N_{\mathrm{d}}^{(s)}\right]$ are the expected value and the variance of $N_{\mathrm{d}}^{(s)}$, respectively, and Γ(·) is the Gamma function. By averaging these moments over all the posterior samples, estimates of the expected value and the variance of N_d could be obtained, along with credible intervals (CIs) that reflected the associated uncertainty. These probabilistic outputs were then propagated through the relationship that defined α and E_net; thus, complete posterior distributions were obtained for both quantities.

5.2. Hierarchical model structure

To reflect the real-world variability in the design life, N_d, across various countries and energy types, a three-level HBM that captured nested dependencies and supported probabilistic inference, accounting for both within- and between-group variations, was constructed. At the first level, referred to as the observation level, individual design-life data points, l_gem, were modeled as samples from a Weibull distribution, where the indices g, e, and m represent the country, energy type, and observation number, respectively. The distribution is defined in Eq. (19):

(19)$l_{g e m} \sim \text { Weibull }\left(\lambda_{g e}, k_{g e}\right)$

where λ_ge and k_ge are the scale and shape parameters, respectively, in terms of l_gem. The second level is referred to as the energy-type level. To account for variations in equipment configuration, material degradation, and operational environment across various energy types within the same country, both λ_ge and k_ge were treated as log-normally distributed random variables:

(20)$\log \lambda_{g e} \sim \operatorname{Normal}\left(\mu_{\lambda, g}, \sigma_{\lambda, g}^{2}\right), \log k_{g e} \sim \operatorname{Normal}\left(\mu_{k, g}, \sigma_{k, g}^{2}\right)$

In Eq. (20), μ is the mean parameter and σ² is the variance parameter for the log-normal prior. This level incorporates heterogeneity among various technologies within each national context; for example, a centralized nuclear infrastructure may be contrasted with decentralized solar PV systems.

At the third level, which is referred to as the country level, hyper-prior distributions are assigned to the country-specific means and variances. These priors, which can be expressed by Eq. (21), introduce broader assumptions regarding inter-country variations and provide regularization when data are sparse:

(21)$\begin{aligned} \mu_{\lambda, g} & \sim \operatorname{Normal}\left(\mu_{\lambda, g}, \sigma_{\lambda, g}^{2}\right), \sigma_{\lambda, g} \sim \operatorname{Half-Cauchy}(0,2) \\ \mu_{k, g} & \sim \operatorname{Normal}\left(\mu_{k, g}, \sigma_{k, g}^{2}\right), \sigma_{k, g} \sim \operatorname{Half-Cauchy}(0,2) \end{aligned}$

This hierarchical design ensures that the inference for each country benefits not only from its own data but also from global patterns. This is particularly important for countries with limited data availability; for such countries, national estimates can be informed by trends observed in countries with more mature energy systems. In addition to improving the statistical efficiency, this model structure facilitates structured sensitivity analyses. Decision-makers can explore the effects of variations in the reliability, degradation rates, and technological maturity at each level on the expected decarbonization outcomes and the associated uncertainties.

5.3. Probabilistic assessments and cross-country insights

To support empirical grounding and facilitate interpretation, probabilistic assessments and cross-country insights were derived from the three-level HBM. Table 3 [75], [76], [77], [78], [79] summarizes the generation capacity and primary life-design parameters for each country-energy-type pair that was considered during this study. Country-level priors were obtained from commercial datasets, while the posterior distributions were generated using the HBM that was based on these priors and synthetic observation data. This integration ensured that the resulting posterior estimates reflected both empirical variability and the structured dependencies.

It also allows for the propagation of uncertainty through the full evaluation process. The corresponding probabilistic results of E_net as a function of S across various countries, which were obtained from the three-level HBM, are graphically presented in Fig. 8. The general pattern across various countries and energy types is similar to that depicted in Fig. 7(b). In the United States, the dominance of nuclear energy (Table 3) resulted in a narrow 95% CI, which is characterized by well-defined degradation behavior and narrow design tolerances [75], [76], [77], [78], [79]. In China, although the mixture of energy types is more diverse and is rapidly moving toward wind and solar technologies, the vast deployment scale and the large number of ongoing modeling improvements contribute to similarly high benefit estimates. In contrast, countries with emerging infrastructures, such as the Republic of Argentina and the Republic of South Africa, display lower mean values of E_net and wider 95% CIs. These results emphasize the need for enhanced predictive infrastructure and more robust design standards in developing regions.

The widths of the 95% CIs serve as means of visualizing the variations in the engineering capability, data quality, and reliability governance across the different countries. Specifically, the widths of the 95% CIs expanded as S increased. For example, in high-scatter regimes, where S > 20, the expected benefit sometimes approached zero or even became negative. This trend indicates that the marginal lifetime-extension value drops rapidly when the predictive reliability is low. Such conditions expose energy systems to elevated operational risks while reducing the environmental returns. This trend also reveals the potential hazards of extending service lifetimes without robust diagnostic capabilities, particularly in countries with limited monitoring infrastructure and less mature engineering practices. In contrast, countries with mature energy infrastructures exhibited narrow 95% CIs. This result could be attributed to several factors, such as accurate lifetime-design methodologies, long-term operational experience, and the implementation of advanced O&M procedures.

The effects of the energy type further compound these variations. Of the various energy technologies, nuclear systems exhibited the most stable and resilient E_net estimations across a wide range of S values. This robustness was attributed to centralized system architectures, relatively uniform material characteristics, and comprehensive operational records. In contrast, the wind and solar systems, especially those implemented in Germany and Spain (Table 3), exhibited heightened sensitivities to low prediction accuracies. The decentralized nature of their deployment, their exposure to heterogeneous environmental conditions, and challenges associated with ensuring uniformity across distributed assets introduced significant complexity into their lifetime estimations; thus, greater variance was produced in their decarbonization outcomes.

Overall, the HBM framework is central to systematic capture and explanation of such patterns. Its nested architecture facilitates statistical learning across related contexts by the incorporation of information for multiple countries and energy types. This capability is particularly advantageous when the framework is used in data-scarce regions; in such regions, national estimates can be refined using global and regional trends that are embedded in the hyperpriors of the model. Furthermore, the probabilistic nature of the model ensures that uncertainty is not only acknowledged but is rigorously quantified and transparently propagated through all the stages of the analysis. Finally, the reduction in the prediction scatter that is achieved by the damage-driven lifetime-design methodology enables the achievement of substantial, verifiable decarbonization outcomes. Consequently, the advances in accurate lifetime design for critical mechanical equipment should no longer be viewed as auxiliary improvements. Instead, they must be recognized as fundamental prerequisites to the development of climate-resilient energy infrastructure. Embedding these methodologies at the center of engineering practice is essential if the carbon-reduction potential of clean-energy technologies is to be fully utilized and if a sustainable transition toward global carbon neutrality is to be achieved.

6. Conclusions and perspectives

This paper demonstrates the importance of accurate lifetime design to management of the challenges associated with climate change and facilitation of the global transition to carbon neutrality. The need for advanced methodologies that can ensure the reliability and sustainability of critical mechanical equipment within clean-energy generation systems is emphasized. The necessity of damage-driven lifetime-design methodologies is demonstrated by discussion of the lifetime-design evolutionary process, fundamental explanations, and case validation. A deterministic-probabilistic pathway that can be used to quantify the net carbon-reduction benefit as a function of the life-prediction scatter is also proposed. The study produced three primary conclusions as follows.

(1) A damage-driven lifetime-design methodology that is tailored to clean-energy mechanical equipment subjected to multiple damage modes was developed. By the use of damage-variable evolution and damage-interaction diagrams, this methodology can reduce the life prediction scatter to less than 5; this result represents a significant improvement from the values produced by traditional empirical methodologies. A case study of a creep-fatigue-oxidation interaction demonstrated that the methodology could achieve both high prediction accuracy and model transparency.

(2) For the deterministic portion of the methodology, a three-level evaluation-criterion process was developed to quantify net carbon-reduction benefits as a function of the life prediction scatter. The analytical results showed that, when the damage-driven design methodology was used, the net carbon-reduction benefits for nuclear, wind, and solar systems could be improved by 200%-500% with respect to those obtained by traditional methods. Improving the prediction accuracy is considered to be a crucial factor that enables lifetime extension to produce environmental benefits.

(3) For the probabilistic portion of the methodology, the implementation of hierarchical Bayesian modeling confirmed that a lower prediction scatter led to wider CIs and more stable carbon-reduction benefits. Wider CIs, which were observed in most of the developing countries, resulted in lower carbon-reduction benefits. This result indicates that the accurate lifetime predictions can ensure globally equitable and reliable clean-energy transitions.

In light of the above, prospects focusing on embedding accurate lifetime design within a broader and more integrated engineering framework and maximizing its impact are presented. It is expected that accurate lifetime design will become a core element of the full-chain technical tetrahedron of structural integrity, which encompasses material design, lifetime design, reliability-centered manufacturing, and O&M. By linking failure criteria with multi-scale microstructural evolution and data-driven models, materials can be precisely engineered for durability and performance. Various manufacturing strategies, such as surface strengthening, defect-controlled additive processing, and optimized treatments, will further enhance damage resistance. Real-time health monitoring and predictive maintenance will extend service lives under variable conditions. This integrated paradigm is poised to reshape the way clean-energy equipment is designed, built, and maintained, and it supports a more sustainable engineering future. In addition, advancements in the lifetime design of critical mechanical equipment for clean-energy generation can be further enhanced through the continuation of investigations into a balance of the technological, economic, and environmental effects of accurate lifetime design; such investigations should consider manufacturing, long-term operation, reuse, remanufacturing, and recycling practices that align with circular economic principles throughout the life cycles of critical mechanical equipment. This impact would be enhanced by an expansion in the application of the proposed model to a broader range of energy sources and countries across the world.

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