The Development of a Model-Based Methodology to Implement a Fused Health Indicator for a Solid Oxide Fuel Cell

Abstract

Hydrogen-based technologies are growing, thanks to recent advancements in systems such as fuel cells and electrolyzers. The present work aims to develop a methodology for the definition of a fused health indicator to monitor the operating and health conditions of a solid oxide fuel cell system. A suitable degradation model was built to yield four trendable output indicators, which were subsequently merged to create the fused health indicator. Subsequently, the assessment of off-design conditions and two realistic scenarios (leakage and constant excess of air working regime) was carried out. The health indicator has proved suitable for fault detection, prognostic applications, control strategy improvement, and health management. In particular, the methodology has underlined the necessity of making the control strategy adaptive with respect to degradation. Through this approach, it is observed that reducing the solid oxide fuel cell temperature difference by 10 °C can result in a 1.2% increase in lifetime. In contrast, the leakage simulation reveals a decrease of about 10.5% in the health state after 100 h, resulting in about a 21% lower end-of-life.

1. Introduction

The current era sees humanity facing the consequences of growing global industrialization and the overuse of fossil fuels, which have led to numerous environmental disasters, including global warming. The Sixth Assessment Report of the Intergovernmental Panel on Climate Change (IPCC) estimated that the global surface temperature increased by 1.09 °C between 1850 and 2020 [1], and this caused serious and critical damage to the environment [2]. The main driver of climate change is greenhouse gas release; in fact, from 1850 to 2022, the global average atmospheric CO₂ concentration has risen to 419 ppm [2], despite a temporary decrease in carbon emissions during the COVID-19 pandemic [3]. Specifically, 72% of global greenhouse gas emissions are produced by energy-related activities: electricity and heating are responsible for about 31% of them, while the transportation field contributes about 15% [4]. To agree on how to intensify global action to address the climate crisis, the United Nations Framework Convention on Climate Change (UNFCCC) has met annually at the Conference of the Parties (COP) since 1995 [5]. Several studies have been carried out to determine the optimal strategy for decarbonization and address climate change [6]. International oil companies should redesign their business models to accommodate a low-carbon future [7]. Renewable energy sources are an important way to tackle the climate challenge, but they should be sustainable to guarantee that they meet the energy needs of future generations [8]. Hydrogen represents a promising energy vector for facing climate change and reducing CO₂ emissions [9]. Its lower heating value (LHV) is about 120 kJ/g, the greatest gravimetric energy density among all known substances [10]. The “Clean Hydrogen Partnership” is an Institutionalized European Partnership responsible for conducting research and innovation activities on hydrogen and fuel cell (FC) technologies, aiming to enhance the whole clean hydrogen application chain. In this regard, this partnership has prepared the “Strategic Research and Innovation Agenda” (SRIA), which is a document in the form of a series of interrelated technology development roadmaps [11]. Hydrogen can be used in applications such as a hydrogenation agent in industry, a direct anode fuel in FCs, and a raw material in combustion processes [12]. Mass-market commercialization of hydrogen-based technologies has various challenges that must be faced to successfully establish a significantly scaled-up hydrogen ecosystem across the European Union (EU) in the coming decade [11]. FCs have garnered prominence driven by the need for clean energy, dwindling fossil fuel reserves, and the potential to generate electricity silently. For mobile battery-powered devices, FCs can be efficiently utilized, whether it is a portable power tool that needs a power of a few hundred watts or a mobile phone that requires a power of a few watts [13]. The scalable and highly modular nature of the generation system makes it easy to identify and replace failed cells within a stack, thus leading to lower maintenance costs. Finally, depending on the cell type, there is some fuel flexibility. In terms of problems, the costs of FCs for stationary power generation (Euro/Wh) are still elevated, and FCs are not suitable to replace fossil fuel-based technologies. The long-term performance and durability of many FC technologies, particularly high-temperature ones, are still unknown. In addition, the usage of low-temperature FCs in the automotive sector is restricted by the challenge of storing an adequate amount of hydrogen in small tanks and the flammability and potential explosive nature of hydrogen [14]. As stated in the SRIA, to make FCs competitive with existing technologies, an important research area is the understanding of their performance and durability mechanisms. This work focuses on the solid oxide fuel cell system (SOFCS) and the monitoring of its operational and health conditions.

1.1. State of the Art

Solid oxide fuel cells (SOFCs) have excellent energy efficiency, close to 60 percent, and long-term performance stability. They use a rigid, nonporous ceramic compound as electrolyte and work at very high temperatures ranging between 800 °C and 1000 °C [14]. Yttrium Stabilized Zirconia (YSZ) is the predominantly used solid oxide electrolyte, allowing maximum fuel efficiency to be achieved at around 1000 °C [15]. The high-quality waste heat can be recovered and used to drive a gas turbine-based cogeneration system, virtually increasing electrical efficiency by up to 80%. Moreover, there is the possibility of starting the electrochemical reaction simply by providing air and avoiding the noble metals employed as catalysts; this, in turn, lowers the costs associated with the FC manufacturing process [14]. Furthermore, SOFCs enable the internal reforming of fuels such as natural gas, methanol, gasoline, diesel, etc. [16], thereby preventing the use of extremely expensive and complex external reformers and facilitating fuel reserves [17]. However, the disadvantage of high temperatures is that thermal management within this type of FC becomes very complicated, so these cells can rarely be embedded in portable power generation systems [14]. Worldwide deployment of SOFCs is hindered mainly by problems of durability, reliability, and high manufacturing costs [18]. In the control field, most of the work is dedicated to enhancing the whole durability and availability of SOFCs [19]. The rigid working conditions result in different degradation processes [20], resulting from complex interactions between elements, making long-term stability challenging [21]. In fact, in high-temperature operating environments, both material changes and chemical reactions between them lead to the performance deterioration of SOFC stack components [22]. Sanaz Zarabi Golkhatmi et al. classified the main degradation processes according to the component of the cell or stack in which they occur [21]. Based on their work, it is possible to identify

• Cathode degradation mechanisms: poisoning, microstructural damage, and thermal-chemical stress;
• Electrolyte degradation mechanisms: contamination, dopant migration, and mechanical breakdown;
• Anode degradation mechanisms: microstructural changes, delamination, coking, and poisoning;
• Interconnects’ degradation mechanisms: corrosion, chromium vaporization, and mechanical failures;
• Sealants’ degradation mechanisms: mechanical breakdown, leakage, corrosion, and poisoning.

Degradation phenomena are difficult to assess because long-term research is necessary, and operating conditions, such as current density, temperature, fuel impurities, and so on, influence the process [21]. Indeed, they reduce FC performance over time, particularly impacting the output power of a specific fuel input during SOFC functioning. Several parameters, both explicit and implicit, can be considered to measure degradation. The degradation rate (DR) is designed as the rate of deterioration of an FC’s performance over time, as measured by a specified index [23]. Forecasting the remaining useful life (RUL) of an FC holds significant value for several reasons [19]:

• To help the control system engineer in adopting strategies to extend the achievable lifespan;
• To reduce the cost of electricity [24];
• To prepare for the system overhaul on time, thereby reducing energy production downtime and maximizing system availability.

Predicting the RUL of a specific system is very complicated. In the scientific literature, this topic has been the subject of attention in various areas, from mechanical systems, such as bearings (both with similarity analysis [25] and data-driven methodologies [26]), to electrochemical systems, such as proton-exchange membrane FCs (PEMFCs) [27] and batteries (either by investigating dynamic working conditions [28] or through deep learning approaches [29]). Nevertheless, there is relatively little research on the health condition monitoring of SOFCSs. Jingxuan Peng et al. [22] studied the performance of SOFCs, highlighting that the prediction of performance degradation can help in the implementation of prognosis measures to achieve an extended RUL. The most common approaches to prediction include model-based, data-driven, electrochemical impedance spectroscopy-based, and image-based methods. Xiaolong Wu et al. [30] built an Elman neural network to forecast SOFCS faults through a data-driven approach. P. Polverino et al. [18] designed a model-based method to improve fault diagnosis and isolation capabilities in SOFCs, utilizing sensor data and system models to extend the information provided. B. Dolenc et al. [19] presented an integrated approach for the state of health (SoH) evaluation depending on a stack’s Ohmic area-specific resistance (ASR). The RUL was predicted using a drift model based on ASR degradation, validating it as a suitable SoH metric for SOFC stacks. Mumin Rao et al. [31] proposed a deep learning model for predicting the SOFCS state. Their model utilized a two-layer Long Short-Term Memory (LSTM) network structure that supported inputs of several sequence features and allowed system states’ forecasting in multiple steps with experimental data from the SOFCS. M. Gallo et al. [32] developed a model-based approach to mitigate FC system faults, with the aim of deriving performance indicators for all system components. The same author, in another work [33], built an online natural aging estimating algorithm integrated with electrochemical impedance spectroscopy (EIS)-based diagnosis for RUL evaluation and an enhanced diagnostic method to detect and isolate failures. ShanJen Cheng et al. [34] investigated three types of dynamic neural network (DNN) models to forecast degradation trends and assess the RUL: the neural network autoregressive (NNARX) model with external inputs, the neural network autoregressive moving average (NNARMAX) model with external inputs, and the neural network output error (NNOE). Their analysis revealed that the NNARX model is more accurate and provides a prediction of RUL outperforming the NNARMAX and NNOE models in terms of relative errors. Chuang Sheng et al. [35] conducted a comparison between an LSTM network–recursive method for RUL predictions and model-based Kalman filtering (KF) prognostics, showing that the LSTM network is more adequate for RUL forecasting than the KF algorithm.

1.2. Contributions and Objectives of the Present Work

The current work is concerned with the development of a model-based methodology for the definition of a fused health indicator (HI) to monitor the operating and health conditions of an SOFCS. This methodology combines multiple degradation-sensitive variables into a single metric, thereby effectively capturing the health status of an SOFCS. Due to the lack of experimental data, a system degradation model was built under the assumptions of constant power operating conditions and linear voltage DR. Although the assumption of a linear DR may not fully capture the complex, nonlinear degradation phenomena inherent in SOFCs, it provided a crucial initial framework due to the limited availability of comprehensive experimental data. This simplification facilitated the development of fundamental models and enhanced computational efficiency, providing a necessary foundation for further investigation. The DR is introduced within the multi-input multi-output (MIMO) model, thus enabling the simulation of system outputs under conditions of degradation. By comparing these with corresponding nominal (i.e., non-degraded) values, the model identifies four key indicators for monitoring trends, which are then aggregated into an overall HI. This approach is a novel method for tracking SOFCS health conditions that does not rely on direct experimental data, thereby enabling effective monitoring, even in the absence of extensive data. The constructed HI was validated across a range of operational scenarios, including variations in current density, temperature gradients, and two realistic applications involving air leakage and a constant excess of air (λ_air) working regime. A comparison of the proposed method with existing methodologies reported in the literature, as discussed in Section 1.1, reveals several distinguishing features. Existing methodologies often rely on single indicators, limited operating scenarios, or purely data-driven approaches. In contrast, the proposed method integrates multiple degradation-sensitive variables into a fused HI and evaluates its performance across diverse and realistic operational scenarios. This methodology enables comprehensive monitoring, fault detection, RUL prediction, and adaptive control, thereby highlighting its versatility and potential for broader applications across different FC technologies. This versatility demonstrates the HI’s ability to consistently monitor performance in both typical and potentially severe operating conditions, highlighting its value for operational contexts involving SOFCSs and enhancing its applicability in dynamic and challenging environments. The HI is designed for a range of applications, including the following:

• Fault detection;
• Prognostic applications, i.e., RUL prediction;
• Improvement of the control system;
• Optimization of operating conditions;
• Health condition management and maintenance improvement.

The identified HI is a comprehensive indicator that consolidates information from a multitude of variables, including voltage, net power, cathode temperature difference, and excess air levels. This integration of several variables within a unified framework facilitates enhanced SOFCS condition monitoring, enabling more precise fault detection and greater adaptability in system control. While the proposed methodology has been developed for SOFCs, its applicability to other fuel cell technologies, such as PEMFCs and molten carbonate fuel cells (MCFCs), is an important consideration. Although the fundamental approach of constructing a fused HI by combining multiple degradation-sensitive variables remains valid across different fuel cell types, certain modifications would be necessary to account for the distinct operational characteristics and degradation mechanisms of these systems. PEMFCs operate at lower temperatures and are particularly sensitive to membrane dehydration, catalyst degradation, and water management issues [36]. Consequently, in an HI framework tailored for PEMFCs, additional degradation indicators, such as membrane resistance, catalyst activity, and water content distribution, should be incorporated. Unlike SOFCs, where degradation is largely associated with high-temperature material stresses, PEMFC degradation is often accelerated by transient load variations and startup/shutdown cycles. Therefore, a PEMFC-specific HI would need to account for dynamic operating conditions more explicitly. For MCFCs, degradation mechanisms differ significantly from those of SOFCs and PEMFCs: MCFCs typically suffer from electrolyte loss, electrode poisoning, and carbonate migration [37]. Since the primary degradation parameter in MCFCs is often electrolyte depletion rather than voltage decay, the HI formulation would need to integrate indicators related to electrolyte retention and conductivity. Additionally, since MCFCs are designed for continuous operation, the HI should be adapted to scenarios where transient conditions are less frequent and long-term material degradation is a primary concern. While the core methodology of HI construction remains transferable, fuel cell-specific adjustments are required to ensure accurate degradation tracking and health assessment. The proposed methodology underlines the significance of adapting control strategies in response to degradation. Based on real-time health assessments derived from the fused HI, a practical implementation of an adaptive control strategy would involve integrating a degradation-aware control algorithm that continuously monitors system health and modifies key control setpoints accordingly. This could be achieved through a combination of real-time diagnostic models and predictive maintenance strategies. For instance, as degradation progresses, the control system could dynamically adjust the air-to-fuel ratio, optimize temperature management, or regulate power output to mitigate the adverse effects of aging components. Furthermore, it demonstrates the potential of leveraging the HI as a feedback indicator to sustain or enhance SOFCS health under varying operational conditions. This adaptive approach represents a significant advancement, as it enables HI to directly influence control strategies, thereby enhancing both operational efficiency and system lifetime. Although the methodology was initially developed using simulated data, it has been designed for future integration with experimental data, thus highlighting its potential for real-world applications and greater diagnostic accuracy. Specifically, the integration of data from real-world SOFC installations and experimental testing across various operating conditions will enable a comprehensive evaluation of the methodology’s robustness. Following the application of the model to experimental data, its reliability can be assessed, and the methodology can be refined to account for a broader range of operating conditions, including the effects of temperature, pressure, and other variables on the HI and the system’s RUL. This flexibility facilitates enhanced reliability and precision in practical SOFCS monitoring and management. Therefore, the work provides a robust foundation for future advancements in SOFCS monitoring, particularly in terms of facilitating the integration of real-time diagnosis, adaptive control, and predictive maintenance in practical fuel cell systems.

1.3. Manuscript Organization

This manuscript is organized in the following way: Section 1 introduces the State of the Art and outlines the contributions of the present work; Section 2 defines the proposed methodology, focusing on nominal working conditions characterized by a specific set of input and output values; Section 3 examines the behavior of the HI in design applications and in realistic scenarios, while Section 4 provides a summary of the research performed and remarks on possible applications and future developments.

2. Adopted Methodology

The approach adopted in this study consists of several phases, first to define the HI and then to implement it, as can be seen in Figure 1a, which illustrates a synoptic map with the various steps followed. Referring to an SOFCS whose balance-of-plant is presented in Figure 1b, the model was developed in a MATLAB^® and Simulink^® R2023a environment, assuming constant-power operating conditions.

This work employs a high-level approach to facilitate the implementation of digital twin and real-time monitoring, thereby avoiding an excessive focus on highly specific technical details. The methodology is designed to prioritize flexibility and adaptability, thus ensuring its applicability to real-time, large-scale systems. This broad focus is essential to avoid the constraints associated with excessive technical specifications, which could limit scalability and generalizability. The proposed model offers a robust framework tailored to the demands of practical applications, achieved through the adoption of simplicity without compromising functionality. It is a three-input, seven-output model, and the nominal conditions considered are given in

Table 1. An Overview of the multi-input, multi-output (MIMO) model structure and parameter configuration.

Input	Output
Current density Maximum temperature difference Fuel utilization factor	Voltage Net power Net efficiency Cathode inlet temperature Anode inlet temperature Outlet temperature Excess of air

. Setting the input values of current density J [A/cm²], the maximum temperature difference between the SOFC outlet temperature and the SOFC inlet temperature ∆T_max [°C], and the fuel utilization factor UF, the output values of voltage V [V], net power P_net [kW], net efficiency η_net, cathode inlet temperature T_in,cat [°C], anode inlet temperature T_in,an [°C], outlet temperature T_out [°C], and excess of air λ_air were derived. The input parameters were selected for their direct impact on the electrochemical and thermal behavior of the FC, while the output variables were chosen to capture critical operational states and provide the essential information for defining the fused HI. This selection process enables the model to effectively monitor system performance and track degradation trends, even in the absence of extensive experimental data. Building on this framework, the SOFC system is modeled as a multi-input, multi-output structure that incorporates degradation phenomena. This enables the identification of key degradation-sensitive indicators through a comparison of degraded and nominal operating conditions. A constrained optimization process was then carried out to combine these indicators into a single, robust diagnostic variable, leading to the definition of a fused HI. To test the HI’s performance and reliability, the model was subjected to off-design and fault scenarios. This enabled exploration of significant operating conditions and assessment of the indicator’s effectiveness.

The implemented proportional integral (PI) control system acts on the excess of air to pursue the ∆T_max target. It assumes a constant fuel cell inlet temperature of 700 °C.

Once the nominal condition is fixed, degradation is introduced through a DR. Generally, it is typically quantified as voltage decay rate per 1000 h [21]. Equation (1) shows one of the most common DR definitions:

$$DR =\left|{\displaystyle \frac{V\left(J,t\right)-V\left(J,t =0\right)}{V\left(J,t =0\right)}}\right|\times {\displaystyle \frac{1000}{t}}\times 100\mathrm{\%},$$

(1)

where $V\left(J,t =0\right)$ is the initial cell voltage, in V, at current density $J$, in A/cm², and $V\left(J,t\right)$ is the cell voltage measured at a current density $J$ after the operation time $t$ (in hours).

A DR on the voltage of 1%/1000 h was added, in accordance with values sourced from the literature. Using this parameter, it has thus been possible to derive the degraded voltage values $V_d\left(t\right)$, as expressed by Equation (2):

$$V_d\left(t\right)=\left(1-DR\right)\cdot V_{100\mathrm{\%}}$$

(2)

where $V_{100\mathrm{\%}}$ is the nominal voltage value, equal to 0.784 V, as indicated in

Table 2. As described in Section 2, this is a 3-input–7-output model. The constant power nominal conditions are set out below. They refer to the initial condition of the simulated SOFCS, i.e., in the absence of degradation.

Input		Output
Current density Maximum temperature difference Fuel utilization factor	0.2 [A/cm2] 125 [°C] 0.7	Voltage Net power Net efficiency Cathode inlet temperature Anode inlet temperature Outlet temperature Excess of air	0.784 [V] 2.042 [kW] 0.417 696.17 [°C] 705.21 [°C] 825.23 [°C] 4.72

, and $DR$ is given by Equation (2) and assumed to be 1%/1000 h as previously stated. Fixing the nominal condition in terms of inputs, the degraded output values corresponding to $V_d\left(t\right)$, given by Equation (2), were recorded every 500 h. Starting with this data, the goal was to construct a dimensionless fused HI in the following form:

$$HI\left(t\right) = \mathit{X}\left(\mathit{t}\right)\cdot w$$

(3)

in which $\mathit{X}$ is the matrix of the trendable indicators, and $w$ is the vector of the weights, both defined hereafter in Equations (8) and (9), respectively.

Not all output variables have contributed to determining the four trend indicators. For those that provide similar information, such as net power and net efficiency, only one was chosen. Regarding the thermal aspect, the focus was on the thermal gradient across the cathode, which is critical for thermal management. In addition, voltage and excess of air were taken into account. Consequently, 4 trendable indicators were determined: one relative to voltage, one relative to net power, one relative to cathode temperature difference, and one relative to excess of air, determined in Equations (4)–(7), respectively. All these ratios were computed by dividing the degraded value by the nominal one, and they were combined to generate the $\mathit{X}$ matrix, whose dimensions are [n_r × 4], where n_r is the number of rows or observations, corresponding to the instants of time considered. Figure 2a demonstrates that time-induced degradation results in a decrease in $H{I}_V$ and $H{I}_{P_{net}}$ and a rise in $H{I}_{\Delta T}$ and $H{I}_{{\lambda }_{air}}$:

$$H{I}_V\left(t\right)={\displaystyle \frac{V_d\left(t\right)}{V_{100\mathrm{\%}}}}$$

(4)

$$H{I}_{P_{net}}\left(t\right)={\displaystyle \frac{P_{ne{t}_d}\left(t\right)}{P_{ne{t}_{100\mathrm{\%}}}}}$$

(5)

$$H{I}_{\Delta T}\left(t\right)={\displaystyle \frac{T_{ou{t}_d}\left(t\right)-T_{in,ca{t}_d}\left(t\right)}{T_{ou{t}_{100\mathrm{\%}}}-T_{in,ca{t}_{100\mathrm{\%}}}}}$$

(6)

$$H{I}_{{\lambda }_{air}}\left(t\right)={\displaystyle \frac{{\lambda }_{ai{r}_d}\left(t\right)}{{\lambda }_{ai{r}_{100\mathrm{\%}}}}}$$

(7)

in which $V_{100\mathrm{\%}}$, $P_{ne{t}_{100\mathrm{\%}}}$, $T_{in,ca{t}_{100\mathrm{\%}}}$, $T_{ou{t}_{100\mathrm{\%}}}$, and ${\lambda }_{ai{r}_{100\mathrm{\%}}}$ correspond to the nominal data given in Table 2, whereas $V_d\left(t\right)$, $P_{ne{t}_d}\left(t\right)$, $T_{in,ca{t}_d}\left(t\right)$, $T_{ou{t}_d}\left(t\right)$, and ${\lambda }_{ai{r}_d}\left(t\right)$ are the values that deteriorate over time, denoted by the subscript $d$.

Thus, the matrix $\mathbf{X}$ in Equation (8) and the vector $w$ in equation (9) are structured as follows, both expressed as dimensionless groups:

$$X =\left[\begin{array}{cccc}H{I}_V\left(1\right)& H{I}_{P_{net}}\left(1\right)& H{I}_{\Delta T}\left(1\right)& H{I}_{{\lambda }_{air}}\left(1\right)\\ ⋮& ⋮& ⋮& ⋮\\ H{I}_V\left(nr\right)& H{I}_{P_{net}}\left(nr\right)& H{I}_{\Delta T}\left(nr\right)& H{I}_{{\lambda }_{air}}\left(nr\right)\end{array}\right]$$

(8)

$$w=\left[\begin{array}{c}{w}_1\\ w_2\\ w_3\\ w_4\end{array}\right]$$

(9)

Since the degraded output values were taken every 500 h from time zero to the final time of 40,000 h, n_r is equal to 81. Similarly to what has been performed in a hypothetical ideal, the HI function was selected as the target in a constrained optimization process, with the aim of determining the vector of weights $w$. The HI function should be compliant with the prognostic criteria of monotonicity, trendability, and prognosability and should be simple to implement. As highlighted in the linear function, it is a suitable candidate for meeting the above aspects and in light of the linearity hypotheses that were assumed in the current degradation model. In the optimization procedure, a linear HI function was thus selected as the objective function. Specifically, a decreasing function from 1 to 0 was chosen, assuming an SOFC lifetime of 40,000 h, a reference value for industrial-scale electricity cogeneration, and it is represented by (10):

$$HI\left(t\right)={\displaystyle \frac{-1}{\mathrm{40,000}}}\cdot t+1$$

(10)

The constrained optimization algorithm was implemented in the MATLAB environment, adopting a supervised approach: indicators exhibiting a decreasing trend were assigned positive initial weights, while those demonstrating an increasing trend were assigned negative initial weights. The optimization was carried out with the interior-point algorithm in MATLAB^®’s fmincon function, which was selected for its robustness and efficiency in solving nonlinear constrained optimization problems, such as those addressed in this study. The solver’s objective was to minimize the root mean square error $e_{RMS}$ under the constraints reported in Equation (12). Therefore, optimization allows for the determination of the weights associated with the 4 indicators to be merged and seeks to minimize the $e_{RMS}$ (11) thus defined:

$$e_{RMS}=\sqrt{{\displaystyle \frac{1}{nr}}\cdot \sum _{i =1}^{nr}(H{I}_i-\sum _{j =1}^4{X}_{i,j}\cdot w_j{)}^2}$$

(11)

where $w_1$ is the weight associated with voltage, $w_2$ with net power, $w_3$ with cathode temperature difference and $w_4$ with an excess of air. Furthermore, constraints were imposed so that the product $X_{i,j}\cdot w_j$ is equal to 1 when $t$ = 0 and is equal to 0 when $t$ = 40,000 h. Below, the imposed limitations are reported:

$$\left\{\begin{array}{l}0.2\le w_1\le 0.8\\ 0.2\le w_2\le 0.8\\ -0.8\le w_3\le -0.2\\ -0.8\le w_4\le -0.2\\ {\displaystyle \sum _{j =1}^4}{X}_{1,j}\cdot w_j=1\\ {\displaystyle \sum _{j =1}^4}{X}_{nr,j}\cdot w_j=0\end{array}\right.$$

(12)

The procedure to minimize $e_{RMS}$ given in Equation (11), according to the constraints listed in (12), led to the determination of the weights shown in

Table 3. The weights relative to the 4 trend indicators constituting the X_matrix are listed. They are derived from the constrained optimization algorithm to minimize e_RMS, provided by Equation (11), according to the constraints given in (12).

Weight	Value
Voltage	0.686
Net power	0.8
Cathode temperature difference	0.2
Excess of air	0.286

. The HI time trend is derived from the application of Equation (3) to the simulated indicators to be monitored in the form of the X_matrix (8) and their relative weights, resulting in the pattern shown in Figure 2b.

As demonstrated in Figure 2, the fused HI exhibits the capacity to integrate and synthesize the primary information derived from the SOFCS. While individual HIs (i.e., $H{I}_V$, $H{I}_{P_{net}}$, $H{I}_{\Delta T}$, and $H{I}_{{\lambda }_{air}}$) capture deviations of a single variable with respect to its nominal value, the fused HI combines them into a unified metric. In this manner, it not only reflects the contribution of each monitored parameter but also highlights their interdependence, thereby providing a more reliable and comprehensive representation of the overall system condition. This methodological approach enables the fused HI to discern degradation phenomena or abnormal behaviors that may not be evident when observing single indicators in isolation, thereby offering a comprehensive perspective on the SOFC health state.

The methodology is designed to provide adaptability across various configurations (e.g., planar, tubular, and micro-tubular) and power sizes of SOFCs, thus enhancing the versatility of the technology for diverse applications. Moreover, the proposed approach is multipurpose and can be applied in various operational contexts, thereby enabling its application across a range of operating conditions, including different temperatures. This flexibility is essential, as it allows the same approach to be applied even when conditions vary, such as during shifts to different temperature settings. The versatility of the methodology is further supported by a validation process based on semi-experimental data from a previous study. This data played a key role in verifying the accuracy and reliability of the model. Specifically, the model was tested against real-world datasets and successfully replicated observed system behaviors, thereby demonstrating its ability to align with experimental outcomes. While this initial validation confirms the methodology’s potential, further refinement and broader validation are necessary. Future work will integrate additional experimental datasets under diverse operating conditions to achieve these goals. This will strengthen the accuracy and generalizability of the proposed approach. In summary, the HI is constructed by combining four key trendable indicators (i.e., voltage, net power, cathode temperature difference, and an excess of air) through a constrained optimization process. This process assigns specific weights to each indicator to minimize the root mean square error with respect to a predefined target HI function, ensuring monotonicity and trendability. The overall HI functions as a single, dimensionless metric that effectively captures the overall health state of the SOFC system. The integration of information from multiple variables enables the HI to facilitate real-time monitoring of electrochemical performance, thermal behavior, and system efficiency. The system’s versatility enables its application in off-design conditions and fault scenarios, thereby providing a robust tool for real-time system diagnostics, prognostics, and the optimization of control strategies. This comprehensive approach enhances the ability to track degradation trends and predict remaining useful life, even in the absence of extensive experimental datasets.

3. Analysis of Off-Design Applications and Realistic Scenarios

Once the methodology for constructing the HI was defined (Section 2), its behavior was observed in off-design applications, varying the current density J (Section 3.1) and the maximum temperature difference ∆T_max (Section 3.2), and in two realistic scenarios, namely, the leakage case (Section 3.4) and the working regime with constant excess of air (Section 3.5).

3.1. Current Density Variation

In the first step, the effect of the change in current density J was thoroughly examined on the HI. Starting from the nominal condition of $J = 0.2$ (A/cm²), the HI definition methodology described in the previous Section 2 was implemented for J values of 0.3 (A/cm²), 0.4 (A/cm²), and 0.15 (A/cm²). When J increases to 0.3 (A/cm²) and 0.4 (A/cm²), the HI moves to an improved end-of-life condition; conversely, lowering J to 0.15 (A/cm²), the HI forecasts a shorter RUL. This is shown in Figure 3a, where the blue line represents the HI under nominal conditions, referring to the trend in Figure 2b, while the red line indicates the condition at J reduced to 0.15 (A/cm²), the turquoise line is for J of 0.3 (A/cm²), and the green line is for J equal to 0.4 (A/cm²). The reason for this behavior is found in the reductions of HI_∆T (Figure 3b) and HI_λair (Figure 3c) as J increases, whilst HI_V and HI_Pnet have not deviated substantially from the nominal condition.

It was expected that at higher currents, there would be greater losses and thus a shorter lifetime, but Figure 3 reveals the opposite result. This is because, in addition to current density, the temperature difference is involved, which increases when J decreases, leading to greater dissipation, as evidenced in Figure 3b. However, the stimulation performed on the system by varying J led to an observation based on the cathode temperature difference time pattern shown in Figure 4 and correlated with the HI prognostic response in Figure 3a. When J varies, different outlet temperature control maps come into play, resulting in different responses: those at higher currents work better because they are closer to the controller’s design condition, i.e., ∆T_max in Table 2. The control system was designed without taking degradation into account; so, under all operating conditions with different J, the temperature difference at the cathode side increases over time as the system degrades, as outlined in Figure 4. When the DR, defined in Equation (1), was introduced into the model via Equation (2), the outlet temperature increased due to induced dissipation, but the controller acted by increasing the excess of air to keep T_out close to the reference value, which is the nominal value in Table 2. The growth in λ_air led to a reduction in the cathode inlet temperature because there was more air flow exchanging with the post-burner gases, and this caused the cathode temperature difference to increase. This phenomenon occurs regardless of the J value, due to the presence of degradation in the system, which reflects in a reduction of the cathode inlet temperature, and is more pronounced at lower currents, as highlighted in Figure 4. In conclusion, the defined methodology has pointed out the need for a more robust control strategy, which considers the effects of the DR on the temperatures involved. Therefore, by adopting this methodology for prognostic purposes, it was seen how it can also be useful in highlighting possible problems in the control system, suggesting enhancements to be made.

3.2. Maximum Temperature Difference Variation

The impact of maximum temperature difference variations on HI was investigated. As illustrated in Figure 5, an increase in ∆T_max at 130 °C and a decrease at 115 °C were considered. The change in HI is mainly due to the variation of HI_∆T, whose weight, provided by the constrained optimization algorithm described in Section 2, is negative, as reported in Table 3. By raising the maximum temperature difference, there is an increase in HI_∆T, defined by Equation (6), so that overall HI, obtained by Equation (3), decreases and leads to a lower end-of-life; vice versa, it increases, as depicted in Figure 5. This is explained by the presence of higher thermal stress, resulting in greater dissipation and thus a shorter lifespan. In this regard, Figure 5 suggests that it is possible to have a 1.2% increase in RUL with a 10 °C reduction of the SOFC temperature difference (i.e., from 125 °C to 115 °C); the black line representing this operating condition reaches zero after more than 40,000 h, as indicated in Table 3. It is important to note that, given a baseline end-of-life fixed at 40,000 operating hours, this reduction in temperature difference corresponds to an increase of approximately 480 h: this value is calculated by applying the established degradation model, which links the cathode temperature difference to the rate of degradation and, consequently, to the RUL of the SOFC.

3.3. Off-Design Applications Overview

Table 4. The following is an overview of the results obtained using the HI definition methodology, described in Section 2, for different off-design applications with prognostic purposes. It allows for the assessment of end-of-life variations as inputs change.

Event	Cause	End Life
J = 0.2 [A/cm2] ∆Tmax = 125 [°C] UF = 0.7	Nominal condition	40,000 h
J = 0.15 [A/cm2]	Current density decrease	39,565 h
J = 0.3 [A/cm2]	Current density increase	41,349 h
J = 0.4 [A/cm2]	Current density increase	42,182 h
∆Tmax = 115 [°C]	Max temperature difference decrease	40,475 h
∆Tmax = 130 [°C]	Max temperature difference increase	39,662 h

provides a summary of previous evaluations under off-design conditions and highlights the potential of the HI as a tool for prognostic applications. The last column shows how end-of-life varies as operating conditions change. The three variations in current density, explained in Section 3.1, reveal the need to improve the control strategy to account for the effect of degradation on cathode inlet temperature. Furthermore, the methodology delineated in Section 2 underscores the imperative of implementing the strategy with respect to operating conditions, clearly also taking into account the application. For instance, in Section 3.2, it was demonstrated that working at ∆T_max of 115 °C rather than 125 °C could allow an increase in lifetime of about 1.2%.

As demonstrated in Table 4, the findings suggest that established control policies may potentially lead to degradation outcomes. For instance, when the current density decreases, a fixed controller may not adjust the thermal management adequately, potentially leading to higher cathode inlet temperatures and accelerated degradation. Conversely, a reduction in ∆T_max under otherwise similar conditions enables the system to achieve a longer end-of-life, thereby demonstrating that controller adaptation can mitigate the negative effects of degradation. These observations underscore the importance of integrating degradation-aware or adaptive control strategies when using the HI for real-time monitoring and prognostics, ensuring that the indicator reflects the true health of the system under dynamic operating conditions. To further validate the proposed methodology, a sensitivity analysis was performed to evaluate the approach’s robustness, considering three key factors:

• The degradation rate exhibited a variation of ±20% from its nominal value.
• The measurement noise was introduced by adding Gaussian noise with standard deviations of up to 5% of the signal amplitude.
• The operating conditions, which included variations in load profiles and temperature, were also considered.

The findings indicate that the methodology maintains consistent performance across these scenarios, exhibiting only minor deviations in accuracy. This outcome serves to validate its reliability under realistic uncertainties.

3.4. Leakage Fault

The hypothesis introduced considers that leakage may occur at a certain time instant, for example, attributed to the presence of a hole in the air compressor. This results in a reduction in excess of air, so the compressor must send more air to meet the same demand and compensate for the loss. The leakage model was realized through an excess air reduction factor f, set equal to 0.8, and the descriptive equation for this fault is as follows:

$${\lambda }_{ai{r}_{d,f}}\left(t>t_f\right) = {\lambda }_{air,d}\left(t\right)·f$$

(13)

where t_f is the time in hours at which the failure appears, and ${\lambda }_{ai{r}_{d,f}}\left(t>t_f\right)$ are the excess of air values affected by the leakage, and f is the fault factor, both of which are dimensionless parameters.

Considering $t_f$ equals to 5000 h, the response of the HI to a variation in operating conditions introduced by the presence of leakage was assessed over a time interval of 100 h. Figure 6 shows the HI in two cases: the blue line represents the nominal condition, which is presented in Figure 2b, while the red line indicates the condition with leakage introduced by Equation (13). Before $t_f$, the two lines are perfectly overlapped, then when the fault occurs, the HI moves to a lower condition than the nominal case, thus representing a deterioration in the health state of the cell. Looking at Figure 6a,b, which show the individual indicators involved in the leakage fault, it can be noted that this behavior is due to changes in net power, which decreases as compressor power increases, and excess air, which increases to compensate for leakage. In fact, HI_V and HI_∆T, defined in Equations (4) and (6), respectively, do not change with respect to the nominal scenario, except for a short trigger at $t_f$ caused by the leakage. The loss of air due to the hole forces the compressor to send more air to supply the stack with the same excess of air as before the fault. This means that in order to obtain the same ${\lambda }_{air}$ in the stack, the ${\lambda }_{ai{r}_{d,comp}}$ relative to the compressor must increase when the leakage is detected, as shown by $H{I}_{{\lambda }_{air}}$ in Figure 3c. Consequently, this leads to an increase in the power absorbed by the compressor, reducing net power. In the prognostic view, the red line in Figure 6a, representative of the leakage simulation through Equation (13), reveals a 21% reduction in lifetime due to a decrease in HI of about 10.5% after 100 h of loss.

3.5. Constant Excess of Air Working Regime

Another proposed scenario involves an intervention induced by operating conditions to assess the implications in a working regime with constant excess air. In the literature, various studies explore the effects of operating at constant excess air while considering changes in other operating variables [39]. In this scenario, λ_air is initially only influenced by degradation, and then at a certain time t_f, the compressor is supposed to start working with a fixed excess of air. Equation (14) represents the simplified model that was employed:

$${\lambda }_{ai{r}_{d,f}}\left(t>t_f\right) = {\lambda }_{air,d}\left(t = t_f\right) = constant$$

(14)

in which t_f is the instant of time when ${\lambda }_{air,d}$ is the fixed constant, again placed equal to 5000 h per the leakage simulation described in the previous Section 3.4. Initially, the ${\lambda }_{air,d}$ increases due to the dissipation introduced into the system through the DR with Equation (2). Then, a time interval of 100 h with constant ${\lambda }_{air,d}\left(t = 5000\mathrm{h}\right) = 5.12$ was evaluated. Figure 7a depicts the HI in the nominal case, corresponding to the scenario shown in Figure 2b, and in the constant ${\lambda }_{air}$ working regime.

After 5000 h, the two lines no longer overlap, and zooming in on this area, it can be noticed that the HI is brought to a higher level and with a smaller derivative; hence, there is an enhancement in the health state because the system degrades more slowly. Observing the single indicators, the one related to excess air (Figure 7e) remains constant when ${\lambda }_{air,d}$ is fixed, as can be deduced from Equation (7), and this triggers variations in voltage (Figure 7a) and net power (Figure 7c), whose indicators, expressed by Equations (4) and (5), respectively, are raised to a higher level and with a less leaning derivative. Instead, there are no substantial variations on the temperature difference, as evident from Figure 7d, since both T_out and T_in,cat increase in relation to their corresponding nominal values, thus keeping HI_∆T (t), defined in Equation (6), almost unaltered from its nominal condition. Therefore, the methodology presented in Section 2 points out that this intervention can improve the HI by 0.15% within 100 h.

A final remark should be made regarding the constant excess of air operating regime. As stated in the literature [39], keeping the excess air constant results in an increase in the FC outlet temperature. Generally, the operating temperature of an SOFC varies within the range 823–1273 K [40]. Thereby, it might be appropriate to link the HI to an outlet temperature measurement indicator $\varphi$ in order to detect the potential occurrence of an excessively high temperature [39]. $\varphi \left(t\right)$ is defined as the difference between the constant excess air outlet temperature $T_{out,{\lambda }_{cost}}\left(t\right)$ and the outlet temperature under degraded conditions $T_{out,d}\left(t\right)$:

$$\varphi \left(t\right) = T_{out,{\lambda }_{cost}}\left(t\right)-T_{out,d}\left(t\right)$$

(15)

The indicator $\varphi \left(t\right)$ is expressed in degrees Celsius (°C). In this study, a threshold of 1 °C was set: if $\varphi \left(t\right)$ exceeds this value, the detection logic sets a fault flag, indicating that the SOFC outlet temperature has deviated by more than 1 °C from the nominal condition. This threshold was selected as a conservative value to capture early deviations while avoiding false alarms. In the present work, an increase in outlet temperature was found by fixing ${\lambda }_{air}$ (Figure 7e) for 100 h. However, the parameter $\varphi \left(t\right)$ does not exceed the set threshold, as shown in Figure 8, so no fault detection occurred through the HI.

To further validate this detection strategy, a case study was performed by fixing the λ_air constant for the entire operating period after the fault occurrence. In this condition, the $\varphi$ indicator gradually increases and reaches the 1 °C threshold after approximately 250 h, triggering the warning flag. This finding serves to validate the efficacy of the proposed thermal drift metric in identifying incipient anomalies within the outlet temperature profile. This approach underscores the potential of $\varphi$ as a complementary diagnostic instrument, demonstrating its efficacy in averting degradation phenomena associated with unfavorable thermal dynamics_.

4. Concluding Remarks

The current research activity seeks to develop a methodology that involves the fusion of four trendable indicators, obtained from an SOFCS degradation model, through a supervised constrained optimization process with the aim of defining a fused HI. After establishing the procedure for the implementation of the HI, the behavior of this methodology was examined in off-design applications and two realistic scenarios, namely, a leakage fault and a working regime with constant excess of air. It is crucial to highlight that the proposed methodology was implemented under realistic and plausible operational conditions, with the objective of analyzing scenarios that closely align with real-world applications. By varying the current density value for prognostic purposes, the defined methodology reveals the need for a more robust control strategy in order to consider the effects of the DR on the cathode inlet temperature. Instead, reducing the SOFC temperature difference by 10 °C can provide a 1.2% increase in lifespan. By adopting this methodology in realistic scenarios, the potential of the HI is highlighted as a tool to monitor the SOFCS health state, which can improve or worsen depending on the working mode and the possible presence of faults. In fact, in the case of leakage, the HI is reduced by approximately 10.5% compared to the nominal case because the compressor must send more air to compensate for the loss, thus absorbing more power. In comparison, by working at constant ${\lambda }_{air}$, there is an improvement in the health condition, and the HI is raised to a 0.15% higher level after 100 h, with a less sloping derivative. In this situation, however, it is recommended that the outlet temperature be monitored to avoid an excessive rise, employing the HI as a fault detection tool. The proposed research developed a single HI that consolidates information from multiple variables, including voltage, net power, cathode temperature, and air excess, thus simplifying the monitoring of complex systems. This versatile HI offers several advantages, such as supporting fault detection, prognostic applications, and improved control strategies. The potential application of this technology lies in SOFC–micro gas turbine systems, where the HI can effectively monitor both units, thereby facilitating predictive maintenance and dynamic load sharing. This, in turn, enhances overall efficiency and extends the lifespan of components. It is imperative to acknowledge that the methodology was validated under the assumption of a linear degradation rate. Given the inherently nonlinear nature of SOFC degradation, this simplification may potentially impact the accuracy of the HI under more realistic conditions. Subsequent studies will tackle this issue by integrating nonlinear and adaptive degradation models to enhance the reliability of predictions. While the present study focused on SOFCs, the methodology is general and can be extended to other fuel cell technologies, such as PEMFCs and alkaline FCs, by adapting the degradation indicators and fusion process to each system’s specific failure mechanisms and operating conditions. From an implementation standpoint, the proposed HI can be directly integrated into the built-in control system, leveraging existing sensor data for voltage, temperature, power, and air excess. The absence of necessity for additional or costly hardware makes the methodology cost-effective and practical. Once integrated, the HI can support predictive control strategies, enabling adaptive adjustments to operating conditions and facilitating real-time monitoring and maintenance planning to enhance system efficiency and extend component lifetime. Subsequent advancements will include experimental validation and the implementation of adaptive monitoring strategies to enhance predictive capabilities under dynamic operating conditions.