# System-level monitoring and diagnosis of starvation faults in solid oxide electrolyzers

Zaman Yazbeck, Federico Bribiesca-Argomedo, Minh Tu Pham, Bertrand Morel, Vincent Dimitriou

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Zaman Yazbeck, Federico Bribiesca-Argomedo, Minh Tu Pham, Bertrand Morel, Vincent Dimitriou. System-level monitoring and diagnosis of starvation faults in solid oxide electrolyzers. International Journal of Hydrogen Energy, 2026, 198, pp.152808. 10.1016/j.ijhydene.2025.152808. hal-05390256

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# System-level monitoring and diagnosis of starvation faults in solid oxide electrolyzers☆

Zaman Yazbeck<sup>a,b,c</sup>,<sup>*D</sup>, Federico Bribiesca-Argomedo<sup>a,D</sup>, Minh Tu Pham<sup>a,D</sup>, Bertrand Morel<sup>c</sup>,
Vincent Dimitriou<sup>b</sup>

## ARTICLE INFO

**Keywords:**

Solid oxide electrolyzer cell

State estimation

Fault detection

Square-root unscented Kalman filter

Adaptive threshold

## ABSTRACT

This paper presents a model-based methodology for real-time fault detection in Solid Oxide Cells (SOCs), focusing on fuel-side starvation in electrolysis mode. Steam starvation occurs when consumption exceeds supply, while hydrogen starvation arises from an insufficient inlet fraction; both risking electrode degradation. With limited measurements, a lumped model with a square-root Unscented Kalman Filter is adopted to estimate gas partial pressures, stack temperature, and deviations from the nominal inlet fuel flow rate, using stack voltage, outlet air temperature, and outlet hydrogen flow rate measurements. Faults are flagged when the estimated states exceed adaptive thresholds derived from nominal predictions via moving-window filter. For quantification, the observer is augmented with two states capturing deviations in inlet steam and hydrogen flow rates. Furthermore, the total resistance is calculated: it increases under steam starvation and remains nearly unchanged under hydrogen starvation. Experimental validation confirms accurate, real-time detection and identification, helping achieve safer SOEC operation.

## 1. Introduction

The global challenge of reducing carbon emissions, driven by the goals of sustainable development [1], has intensified research efforts to develop alternative technologies for the conversion and storage of energy in a more sustainable manner [2]. In this context, electrolyzers play a vital role in enabling the conversion of renewable energy into storable gases such as hydrogen and methane [3]. At the same time, fuel cells provide a cleaner alternative to traditional power generation methods by directly converting chemical energy into electricity, thereby reducing Carbon emission. These technologies can also be integrated with other systems, such as coal-based power plants [4], propulsion systems [5], and multigeneration systems based on renewable energy [6,7].

Among these technologies, Solid Oxide Cells (SOCs) have gained significant attention due to their high efficiency, fuel flexibility, and ability to operate at high temperatures (600–1000 °C) [8]. Unlike other electrochemical devices such as proton exchange membrane or alkaline cells, SOCs rely on a solid ceramic electrolyte, which enables operation with a wide range of fuels [9], including hydrogen and hydrocarbons [10], while maintaining high electrical efficiency [11].

A key advantage of SOCs is their ability to operate in both electrolysis (SOEC) and fuel cell (SOFC) modes, enabling efficient energy storage and conversion within the same system [12].

Despite their advantages, widespread deployment of SOCs remains limited because of high manufacturing costs and durability concerns. To address these challenges, research efforts have focused on developing alternative cathode, electrolyte, and anode materials to improve long-term stability. Although advances in material properties have contributed to improving the SOC performance [13], they alone are not sufficient to ensure long-term reliability. Equally important is the optimization and precise control of operating conditions, as improper management can lead to severe degradation mechanisms and a significant reduction of lifetime.

One of the major challenges in SOC operation is preventing or detecting starvation-related faults, which can lead to severe performance degradation. In electrolysis mode, steam starvation that can be caused by pump malfunctions, leakages and other system related failures can cause thermal stresses and irreversible electrolyte reduction [14]. While in fuel cell mode, hydrogen starvation causes Nickel reoxidation of the

<footnote><sup>a</sup> INSA Lyon, Université Claude Bernard Lyon 1, Ecole Centrale de Lyon, CNRS, Ampère, UMR5005, Villeurbanne, 69621, France</footnote>

<footnote><sup>b</sup> GENVIA SAS, Plaine Saint Pierre, Béziers, 34500, France</footnote>

<footnote><sup>c</sup> CEA/LITEN, Université Grenoble Alpes, 17 rue des Martyrs, CEDEX 9, Grenoble, 38054, France</footnote>

<footnote>☆ This document is the results of the research project funded by GENVIA.</footnote>

<footnote>* Corresponding author at: GENVIA SAS, Plaine Saint Pierre, Béziers, 34500, France.
E-mail address: zaman.yazbeck@genvia.com (Z. Yazbeck).</footnote>
## Nomenclature

### Symbols

<table><tr><td>Δ<i>H</i><sub>r</sub></td><td>Enthalpy of formation</td></tr><tr><td><i>n</i></td><td>Molar flow rate</td></tr><tr><td>η</td><td>Overpotential</td></tr><tr><td>σ<sub>SB</sub></td><td>Stefan-Boltzmann constant</td></tr><tr><td><i>A</i><sub>stack</sub></td><td>Surface area of the stack</td></tr><tr><td><i>C</i><sub>p<sub>s</sub></sub></td><td>Average Stack heat Capacity</td></tr><tr><td><i>d</i></td><td>Electrode Thickness</td></tr><tr><td><i>D</i><sup>eff</sup><sub>i</sub></td><td>Effective diffusion coefficient, <i>i</i> ∈ {H<sub>2</sub>, H<sub>2</sub>O, O<sub>2</sub>, N<sub>2</sub>}</td></tr><tr><td><i>E</i><sub>act</sub></td><td>Activation energy</td></tr><tr><td><i>F</i></td><td>Faraday constant</td></tr><tr><td><i>h</i><sub>i</sub></td><td>Enthalpy, <i>i</i> ∈ {H<sub>2</sub>, H<sub>2</sub>O, O<sub>2</sub>, N<sub>2</sub>}</td></tr><tr><td><i>I</i></td><td>Current density</td></tr><tr><td><i>m</i><sub>s</sub></td><td>Stack mass</td></tr><tr><td><i>P</i></td><td>Pressure</td></tr><tr><td><i>p</i><sub>i</sub></td><td>Partial pressure, <i>i</i> ∈ {H<sub>2</sub>, H<sub>2</sub>O, O<sub>2</sub>, N<sub>2</sub>}</td></tr><tr><td><i>R</i></td><td>Universal gas constant</td></tr><tr><td><i>T</i><sub>turn</sub></td><td>Furnace temperature</td></tr><tr><td><i>T</i><sub>s</sub></td><td>Stack temperature</td></tr><tr><td><i>V</i><sub>cell</sub></td><td>Cell voltage</td></tr><tr><td><i>V</i><sub>ch</sub></td><td>Volume of the channel</td></tr><tr><td><i>V</i><sub>ernst</sub></td><td>Nernst voltage</td></tr><tr><td><i>V</i><sub>s</sub></td><td>Stack voltage</td></tr><tr><td><i>y</i><sup>in</sup><sub>i</sub></td><td>Inlet molar fraction, <i>i</i> ∈ {H<sub>2</sub>, H<sub>2</sub>O, O<sub>2</sub>, N<sub>2</sub>}</td></tr></table>

### Acronyms

<table><tr><td>EIS</td><td>Electrochemical Impedance Spectroscopy</td></tr><tr><td>EKF</td><td>Extended Kalman Filter</td></tr><tr><td>PRBS</td><td>Pseudo-Random Binary Sequence</td></tr><tr><td>SOC</td><td>Solid Oxide Cell</td></tr><tr><td>SOEC</td><td>Solid Oxide Electrolyzer Cell</td></tr><tr><td>SOES</td><td>Solid Oxide Electrolyzer Stack</td></tr><tr><td>SOFC</td><td>Solid Oxide Fuel Cell</td></tr><tr><td>SR-UKF</td><td>Square-Root Unscented Kalman Filter</td></tr><tr><td>SRU</td><td>Single Repeating Unit</td></tr><tr><td>UKF</td><td>Unscented Kalman Filter</td></tr></table>

### Functions and Operators

<table><tr><td>E{}</td><td>Expected value</td></tr><tr><td>chol</td><td>Cholesky factorization</td></tr><tr><td>cholupdate</td><td>Rank 1 update to Cholesky factorization</td></tr><tr><td>qr</td><td>QR factorization</td></tr></table>

hydrogen electrode and air starvation can lead to undesired morphological changes [15,16]. Therefore, real-time monitoring and diagnosis of faults are crucial for ensuring the reliability and longevity of solid oxide cells. Accurate fault detection enables timely intervention, preventing irreversible damage to the stack. By minimizing the occurrence and severity of faults through precise diagnosis and effective control, serious degradation can be avoided, ultimately enhancing the durability and performance of the SOC stack [17].

Existing fault detection approaches for SOFC and SOEC systems can be categorized into three main types: model-based, data-driven, and hybrid methods that integrate both strategies [18]. Although data-driven methods have shown promising results, they often require extensive datasets, typically generated through simulations due to the difficulty

of obtaining failure data from operating SOC systems in electrolysis or fuel cell mode without risking irreversible damage.

Data-driven approaches based on experimental data have been explored, though they remain challenging due to limited access to real faulty conditions. Zheng et al. [19] proposed a Principal Component Analysis (PCA) for dimensionality reduction with a Support Vector Machine (SVM) classifier, applied to the detection of air leakage and fuel starvation from SOFC experiments. Similarly, Wu et al. [20] employed a least-squares support vector machine (LS-SVM) for SOFC fault diagnosis, then used hidden semi-Markov models (HSMMs) to estimate the remaining useful life (RUL) under the diagnosed fault conditions.

To address limitations in data availability, simulation-generated data have been also employed in various studies. For instance, a higher-order pseudo-2D model of SOFC was utilized to train an Artificial Neural Network (ANN) and k-nearest neighbors (KNN) models for detecting hydrogen starvation faults, with an additional ANN regressor estimating the affected area of Nickel electrode [21]. Moreover, Zhang et al. [22] developed a deep neural network outperforming SVM and PCA-SVM models to classify power stack degradation faults, validated using simulated system data. Similarly, a COMSOL-based physics model simulated SOFC failures for Electrochemical Impedance Spectroscopy (EIS) diagnostics, enabling a SVM model to classify failure types under diverse operating conditions [23]. Signal-based methods, such as wavelet transforms, have also been used to decompose SOFC voltage signals, enhancing fault detection through extracted features [24]. Techniques like total harmonic distortion provide further non-physical model alternatives [16]. Other approaches, relying on single-cell voltage measurement can classify SOFC faults based on time and frequency domain characteristics with different classifiers [25].

Despite the advancements in non model-based and data-driven methods, their application in real-time monitoring remains limited due to the need for large datasets and substantial computational resources, making real-time implementation challenging for SOEC systems.

In contrast, model-based methods rely on a physical model derived from first principles or system identification techniques to explicitly describe system behavior and the interactions between the variables [26]. Fault detection is achieved by monitoring the coherence between estimated and measured process variables, making this approach more interpretable, computationally efficient, and suitable for real-time SOFC fault diagnosis [18,26].

A model-based diagnosis approach typically involves developing a nominal system model, which reconstructs expected signal values during operation. By analyzing residuals, any significant deviation between the predictions of the model and measured signals serves as an indicator of faults, allowing for early diagnosis and preventive actions.

Several model-based techniques have been explored in the literature. Polverino et al. [27] developed a dynamic SOFC system model combined with Fault Tree Analysis to identify fault-symptom relationships. They simulated five faults, generated residuals, and used thresholds to build a refined version of Fault Signature Matrix in [28] for fault detection in both the stack and balance-of-plant components.

Gallo et al. [29] developed a fault mitigation strategy that integrates the health state of the stack in fuel cell mode through EIS measurement for real-time fault detection and isolation. Their approach utilizes a proportional-integral derivative (PID) controller-based estimator to evaluate fault severity, demonstrated through a hydrogen leakage scenario, and enables adaptive mitigation strategies using PID controller.

For degradation assessment, Dolenc [30] introduced an integrated method based on area-specific resistance (ASR). Their approach combines a lumped stack model with two Kalman filters: an Unscented Kalman Filter (UKF) for estimating total resistance and temperature as states and a standard Kalman Filter (KF), assumed to be linear, operating at a slower time scale to track the gradual variation of total resistance. The identified ASR drift is then used with a Monte Carlo simulation to predict the remaining useful life of the stack.
Observer-based approaches have also been used for SOFC fault detection, leveraging state observers to estimate system states and detect discrepancies indicative of faults. Vijay et al. [31] proposed a nonlinear observer-based strategy that employs two adaptive observers that rely on voltage, temperature and outlet molar fractions of hydrogen, steam, and oxygen as measured outputs with the input vector that presents the nominal operating conditions to detect and classify five different SOFC faults. The methodology was validated through simulations, demonstrating its effectiveness in fault detection and classification. Wu et al. [32] developed a state observer-based method for SOFC fuel input fault detection, utilizing outlet air temperature measurements instead of internal sensors. Their approach tracks stack chamber temperature variations, effectively identifying input flow sensor faults and fuel supply issues, while filtering disturbances for real-time diagnosis. Xu et al. [26] introduced a real-time fault diagnosis scheme for gas leakage detection in SOFC, integrating a UKF with adaptive covariance matching and an adaptive threshold generator. By reconstructing air and fuel leakage states from outlet fuel and air temperatures as output measurements, their approach improves fault diagnosis robustness through dynamically adjusting thresholds.

In this paper, a real-time fault diagnosis scheme based on a square-root unscented Kalman Filter (SR-UKF) is proposed to monitor and detect starvation-related faults in SOCs that can operate in electrolysis or fuel cell mode. Unlike the standard UKF, which propagates the covariance matrix directly, the SR-UKF propagates its Cholesky factor, thereby avoiding repeated Cholesky decompositions, improving numerical stability, and maintaining a positive-semidefinite covariance throughout the filtering process [33]. To validate the proposed approach, an experimental SOEC system operating in electrolysis mode is considered, with outlet air temperature, voltage, and outlet hydrogen flow rate used as the measured outputs. Based on these signals, the SR-UKF estimates and reconstructs the partial pressures of hydrogen, steam, and oxygen, the stack temperature, and the fuel input flow rate to the system.

The contributions of this work are as follows:

* Minimal reliance on sensor instrumentation is used: voltage, outlet air temperature, and outlet hydrogen flow rate while additional or different measurements are used in previous approaches [26,31] requiring more intrusive measurements.

* Direct estimation of key internal states including partial pressures and fuel input rate, enabling fault detection through deviation relative to a nominal model.

* Design of adaptive thresholds using a moving-window filter to reduce false alarms.

* Use of an SR-UKF formulation, which improves numerical stability compared to the standard UKF used in other model-based SOFC fault diagnosis schemes [26].

* Study of Hydrogen starvation (insufficient inlet H₂ fraction) in electrolysis mode.

In this work, two fault scenarios are analyzed using experimental data in electrolysis mode:

1. Steam starvation: caused by a sudden reduction in the steam inlet flow.

2. Hydrogen starvation: A small amount of hydrogen is added alongside steam to prevent nickel reoxidation at the stack inlet during electrolysis operation; a decrease in this hydrogen supply is considered as a fault.

This methodology is also applicable to fuel cell operation, enabling the detection of air starvation in addition to hydrogen starvation, where the consumption rate of hydrogen to produce electricity exceeds its supply. By accurately reconstructing key internal states and fuel flow conditions, the proposed SR-UKF framework enhances fault diagnosis reliability, ensuring early intervention and preventing irreversible degradation.

In this paper, a lumped zero-dimensional model, or system-level model, able to reconstruct partial pressures and inlet-flow deviations and detect fuel-side starvation has been chosen to represent the system behavior. The use of multi-dimensional distributed-parameter models, while useful to simulate local phenomena, is not adapted to the real-time reconstruction of internal states when only a few measurements are available, which is the main scenario considered in this study.

The overall objective of the results presented here is to enable the stack/system-level monitoring under limited sensing; we do not attempt cell or segment level localization. The lumped model, together with a SR-UKF, is shown to be capable of recovering internal partial pressures and inlet-flow deviations, providing enough information to detect and quantify starvation faults at the stack level. The proposed method uses only few, commonly available sensors, and does not require distributed measurements. Other approaches exist in the literature that exploit multiple sensors, for example external thermocouple arrays, in order to reconstruct some distributed quantities (such as the stack temperature field). While these approaches are able to detect and localize leakage hotspots in real time, they consistently require numerous added sensors and customized hardware, e.g. [34], which is outside of the scope of the current paper.

The paper is organized as follows: Section 2 introduces the dynamic modeling of an SOEC, and parameter estimation to calibrate the model. Section 3 describes the faults, while Section 4 presents the fault detection methodology along with its experimental validation. Finally, Section 5 concludes the paper.

## 2. Lumped SOES dynamic model

In SOEC hydrogen H₂(g) and oxygen O₂(g) gases are produced through an electrochemical reaction driven by a superheated steam flowing over the cathode with a sufficient potential difference applied across two electrodes (anode and cathode). At the cathode, gaseous water molecules are reduced at the interface between the cathode and the electrolyte, producing hydrogen gas and oxygen ions. These oxygen ions are transported through the electrolyte, due to diffusion phenomena and electric potential differences, and are then oxidized on the anode to form Oxygen gas [35]. The produced hydrogen at the cathode remains mixed with the residual steam. The corresponding half-cell and overall reactions are expressed as follows:

$$ \text{Cathode: } H_2O_{(g)} + 2e^- \rightarrow H_2_{(g)} + O^{2-} $$

$$ \text{Anode: } O^{2-} \rightarrow \frac{1}{2} O_{2(g)} + 2e^{-} $$

$$ \text{Overall: } H_2O_{(g)} \rightarrow \frac{1}{2} O_{2(g)} + H_2_{(g)} $$

It is essential to understand and analyze the behavior of a Solid Oxide Electrolyzer Stack (SOES), which consists of multiple cells connected in series. Thus, it is important to develop a model that can accurately describe its behavior under different operating conditions. The model is formulated from mass and energy balances, incorporating thermodynamic, electrochemical, and transport phenomena within the cell. In this work, we adopt a lumped, zero-dimensional or system-level formulation to match the limited online sensing; distributed-parameter models would require local measurements (flow rates, temperatures...) that are not available in our setup. Thus, the formulation follows the standard assumptions in SOEC/SOFC modeling:

1. The pressure drop along the gas channels has been assumed to be negligible: atmospheric condition has been considered [35].

2. Ideal gas behavior is assumed due to the high operating temperature and atmospheric pressure [36].

3. The stack is modeled as a single repeating unit (SRU), assuming uniform behavior across all cells for scalability, so the stack voltage is equal to the number of cells multiplied by the cell voltage.
4. The outlet air temperature is representative of the overall stack temperature, consolidating the anode and cathode temperatures into a single value, denoted $T_s$.

The model is general and applies to both fuel cell and electrolysis modes. In fuel cell mode, a positive current is used, while in electrolysis mode, the current is negative to reflect the reverse electrochemical reaction.

Consequently, the observer provides system-level detection and quantification of fuel-side starvation but does not localize starvation at the cell or segment level.

## 2.1. Thermal balance equation

The thermal balance equation describing the temperature dynamics of the stack is expressed as follows [37]:

$$m_s C_{p_s} \frac{dT_s}{dt} = \dot{E}^{in} - \dot{E}^{out} - IV_s + N_{cells} r_{H_2O} \Delta \dot{H}_r + Q^{rad} \quad (1)$$

where $V_s$ is the stack voltage presented in the next subsection $\dot{E}^{in/out}$ denote the inlet and outlet energy flows of the stack respectively, further decomposed as:

$$\dot{E}^{\alpha} = \dot{n}_{j}^{a} \sum_{i} y_{i}^{a} h_{i}(T_{j}^{a}), \quad (2)$$

where $i \in \{H_2, H_2O, O_2, N_2\}$, $j \in \{fuel, air\}$, $\alpha \in \{in, out\}$, and $r_i$ represents the produced/consumed flows described according to Faraday's law [35] as:

$$r_{H_2} = -r_{H_2O} = 2r_{O_2} = -\frac{I}{2F} \quad (3)$$

The term $Q^{rad}$ represents the radiative heat exchange between the stack and the furnace [38], given by:

$$Q^{rad} = A_{stack} \sigma_{SB} (T_{furn}^{4} - T_{s}^{4}) \quad (4)$$

## 2.2. Electrochemical model

According to the third modeling assumption presented at the beginning of the section, considering the voltage to be homogeneous across all cells, the stack voltage $V_s$ is given by the cell voltage, $V_{cell}$, multiplied by the number of cells in the stack, $N_{cell}$, as follows:

$$V_s = N_{cell} V_{cell} = N_{cell}(V_{nernst} - \eta_{ohm} - \eta_{act} - \eta_{conc}) \quad (5)$$

* Nernst voltage is [35]:

$$V_{nernst} = E^0(T_s) + \frac{RT_s}{2F} \ln \left( \frac{p_{H_2} p_{O_2}^{0.5}}{p_{H_2O}} \right) \quad (6)$$

Where $E^0$ refers to the Gibbs free-energy change of the reaction.

* Ohmic loss can be expressed by [35]:

$$\eta_{ohm} = I R_s \exp \left( \frac{E_{ohm}^{act}}{RT_s} \right) \quad (7)$$

where $R_s$ is the internal resistance, and $E_{ohm}^{act}$ is the activation energy, fitted from experimental data.

* Activation Overpotential is calculated using the Butler-Volmer equations [35]:

$$\eta_{act} = \frac{RT_s}{F} \left[ \sinh^{-1} \left( \frac{I}{2I_0^{an}} \right) + \sinh^{-1} \left( \frac{I}{2I_0^{cath}} \right) \right] \quad (8)$$

Where the anodic and cathodic exchange current densities, $I_0^{an}$ and $I_0^{cath}$, respectively follow the Arrhenius's law [35]:

$$I_0^{cath} = \gamma_{cath} p_{H_2}^{\zeta} p_{H_2O}^{\theta} \exp \left( -\frac{E_{act}^{cath}}{RT_s} \right) \quad (9)$$

$$I_0^{an} = \gamma_{an} p_{O_2}^{\kappa} \exp \left( -\frac{E_{act}^{an}}{RT_s} \right) \quad (10)$$

The exponents $\zeta$, $\theta$, and $\kappa$ are fixed to be equal to 0.066, 0.81, and 0.15 respectively according to [39]. While $\gamma_{an}$ and $\gamma_{cath}$ and the activation energies for corresponding reactions, $E_{act}^{an}$ and $E_{act}^{cath}$, are identified from experimental data

* Concentration Overpotential arises due to the reduction in the concentration of the reactants at the electrodes under operation [35]:

$$\eta_{conc} = \frac{RT_s}{2F} \ln \left( \frac{p_{H_2}^{TPB} p_{H_2O}}{p_{H_2} p_{H_2O}^{TPB}} \right) + \frac{RT_s}{4F} \ln \left( \frac{p_{O_2}^{TPB}}{p_{O_2}} \right) \quad (11)$$

$$p_{H_2O}^{TPB} = p_{H_2O} + \frac{RT_s}{2F} \frac{d_{cath}}{D_{H_2O}^{eff} P_{fuel}} I \quad (12)$$

$$p_{H_2}^{TPB} = p_{H_2} - \frac{RT_s}{2F} \frac{d_{cath}}{D_{H_2O}^{eff} P_{fuel}} I \quad (13)$$

$$p_{O_2}^{TPB} = 1 - (1 - p_{O_2}) \exp \left( \frac{RT_s}{4F} \frac{d_{an}}{D_{O_2}^{eff} P_{air}} I \right) \quad (14)$$

The partial pressures at the electrode-electrolyte interface, $p_i^{TPB}$, are calculated using the Dusty-Gas Model (DGM) for multicomponent transport in the porous electrode, where the effective diffusion coefficient, $D_i^{eff}$, accounts for both binary and Knudsen diffusion. More details on the calculation can be found in [35].

## 2.3. Mass balance

According to the law of conservation of mass, the expression for the partial pressure ($p_i$) of component $i$ in the channels is given by:

$$\frac{V_{ch}}{RT_s} \frac{d p_i}{dt} = \dot{n}_j^{in} y_i^{in} - \dot{n}_j^{out} y_i^{out} \pm r_i \quad (15)$$

for $i \in \{H_2, H_2O, O_2\}$, $j \in \{fuel, air\}$. The output flow rate for the fuel and air side is calculated as follows:

$$\dot{n}_j^{out} = \dot{n}_j^{in} + \sum r_i \quad (16)$$

More details can be found in [36].

## 2.4. Model calibration

Accurate parameter estimation is essential to ensure the alignment between the model and the experimental data provided by CEA.<fnref n="1" /> To achieve this, a series of current-voltage (I-V) measurements were performed in electrolysis mode on the experimental test bench to excite the system and estimate key parameters for model calibration. Additionally, Electrochemical Impedance Spectroscopy (EIS) was conducted using Pseudo-Random Binary Sequence (PRBS) signals. Various sensors recorded the inlet and outlet temperatures and pressures of the fuel and air, the stack voltage, the inlet flow rates of the gases, the current, the oven temperature and the produced hydrogen flow rate providing a comprehensive data set for validation. The need for parameter fitting is common in electrochemical models, as seen in studies on batteries and other fuel cell systems [40]. In this work, parameter estimation was carried out by minimizing the mean square error between the experimental voltage and the model output. A truncated sensitivity matrix in a Gauss-Newton optimization method was employed, as described in [41], allowing the update of parameters with the highest sensitivity to the output. Fig. 1 illustrates the validation of the calibrated model against experimental data. The mean stack voltage and stack temperature exhibit strong agreement with the measured values. The close match between the model and experimental results demonstrates the effectiveness of the proposed calibration approach. The estimated parameters are summarized in Table 1.

<footnote n="1">¹ Commissariat à l'énergie atomique et aux énergies alternatives (French Alternative Energies and Atomic Energy Commission).</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1. Validation of the model outputs against measured data from the experimental test bench.</figcaption></figure>

<table><caption>**Table 1**
Estimated parameters.</caption><thead><tr><th>Parameter</th><th>Value</th><th>Unit</th></tr></thead><tbody><tr><td>R<sub>s</sub></td><td>0.22</td><td>Ω cm<sup>2</sup></td></tr><tr><td>E<sup>ohm</sup><sub>act</sub></td><td>120e3</td><td>J/mol</td></tr><tr><td>E<sup>an</sup><sub>act</sub></td><td>100e3</td><td>J/mol</td></tr><tr><td>E<sup>cath</sup><sub>act</sub></td><td>100e3</td><td>J/mol</td></tr><tr><td>γ<sub>an</sub></td><td>1.05e5</td><td>A/cm<sup>2</sup></td></tr><tr><td>γ<sub>cath</sub></td><td>2.5e5</td><td>A/cm<sup>2</sup></td></tr></tbody></table>

### 3. Fault description

In electrolysis mode, steam starvation is a common fault that can arise in the stack due to both external and internal causes. External ones include leakage issues, mass flow controller malfunctions, or other system-related failures. Internally, the fault can result from design limitations or extreme operating conditions, which may affect cells and seal integrity. Indeed, the high operating temperatures and thermal gradients within the cell induce thermal stresses due to the differing coefficients of thermal expansion [17].

Moreover, in electrolysis mode, a small fraction of hydrogen is intentionally mixed with steam and supplied to the stack to prevent nickel reoxidation at the inlet of the stack, particularly under low-current conditions when hydrogen production is minimal [42]. An interruption in this hydrogen supply is considered a fault scenario, referred to as hydrogen starvation in electrolysis mode.

Detecting and distinguishing these two types of starvation faults in electrolysis mode is essential for fault correction. This study addresses the problem by employing an observer that reconstructs the partial pressures of the gases in the channels affected by starvation and estimates the actual inlet fuel flow rates at the stack inlet.

Using available experimental data, we investigate both fault scenarios. Hydrogen starvation is induced by setting the hydrogen flow to zero, resulting in a hydrogen molar fraction of zero. Steam starvation is implemented by reducing the overall fuel flow rate, thus lowering both the hydrogen and steam inlet flow rates, while maintaining the same

molar fractions. Under normal operating conditions, the fuel supply to the stack remains constant. The observer, utilizing stack voltage, air outlet temperature, and outlet hydrogen flow rate measurements, reconstructs the actual fuel flows (inlet hydrogen and steam flow rates) and gas partial pressures. When an anomaly occurs, deviations from nominal conditions indicate a fault. To isolate and quantify the detected fault in hydrogen and steam starvation case, the deviation from their nominal values is reconstructed by the observer. Finally, the total resistance is calculated to show how the low-frequency resistance responds to these faults during impedance measurements, which may differ from the true low-frequency resistance due to aging. A significant variation in resistance indicates steam starvation, while hydrogen starvation has little to no effect on the total resistance. The corresponding equations and results will be presented in the next section.

### 4. Fault detection methodology

The SOEC mathematical model derived in the previous section can be rewritten in a state-space representation as follows:

$$ 
\begin{aligned}
\dot{x} &= f(x, u) \\
y &= h(x, u)
\end{aligned} 
\qquad (17) 
$$

Where $f(.)$ and $h(.)$ are the states and output functions respectively.

The state vector $x$ is defined as:

$$ x = [p_{\text{H}_2} \ p_{\text{H}_2\text{O}} \ p_{\text{O}_2} \ T_s]^T $$

The input vector as:

$$ u = [\dot{n}_{\text{fuel}}^{\text{in,nom}} \ y_{\text{H}_2}^{\text{in,nom}} \ y_{\text{H}_2\text{O}}^{\text{in,nom}} \ I \ \dot{n}_{\text{air}}^{\text{in,nom}} \ T_{\text{oven}} \ T_{\text{fuel}}^{\text{in}} \ T_{\text{air}}^{\text{in}}]^T $$

Where

$$ \dot{n}_{\text{fuel}}^{\text{in,nom}} = \dot{n}_{\text{H}_2}^{\text{in,nom}} + \dot{n}_{\text{H}_2\text{O}}^{\text{in,nom}} $$

$$ y_{\text{H}_2}^{\text{in,nom}} = \frac{\dot{n}_{\text{H}_2}^{\text{in,nom}}}{\dot{n}_{\text{fuel}}^{\text{in,nom}}} $$

$$ y_{\text{H}_2\text{O}}^{\text{in,nom}} = \frac{\dot{n}_{\text{H}_2\text{O}}^{\text{in,nom}}}{\dot{n}_{\text{fuel}}^{\text{in,nom}}} $$

$$ y_{\text{H}_2\text{O}}^{\text{in,nom}} = \frac{\dot{n}_{\text{H}_2\text{O}}^{\text{in,nom}}}{\dot{n}_{\text{fuel}}^{\text{in,nom}}} = 1 - y_{\text{H}_2}^{\text{in,nom}} $$
<figure><img src="image_4.png" /><figcaption>Fig. 2. Starvation detection scheme.</figcaption></figure>

The available measured outputs are:

$$y = [V_s \ T_s]^T$$

These measurements are sufficient to reconstruct the partial pressures of hydrogen, steam, and oxygen, as well as the stack temperature even in faulty conditions. They also enable the detection of faults on either the fuel or oxygen side. In the case of a fault on the fuel side, the partial pressures of hydrogen and steam will be both affected. Consequently, these partial pressures can be reconstructed, and the fault can be detected. Subsequently, the total resistance can be estimated, as detailed in Section 4.3, to distinguish whether the fault corresponds to hydrogen or steam starvation.

However, to quantify this fault, an additional measurement is required, the outlet hydrogen flow rate. Since this measurement is available in the test bench used, the measured output becomes:

$$y = [V_s \ T_s \hat{n}_{H_2}^{out}]^T$$

To specifically address starvation, which is directly related to the real inlet fuel flow rate fed to the stack, two additional state variables, $\epsilon_1$ and $\epsilon_2$, are introduced in the state vector of the observer. These variables represent the deviation of the real inlet steam and hydrogen flow rate respectively from their nominal value, where:

$$\begin{aligned} \dot{n}_{H_2O}^{in,real} &= \dot{n}_{H_2O}^{in,nom} \cdot (1 - \epsilon_1) \\ \dot{n}_{H_2}^{in,real} &= \dot{n}_{H_2}^{in,nom} \cdot (1 - \epsilon_2) \end{aligned} \qquad (18)$$

where $\dot{n}_{H_2O}^{in,nom}$ and $\dot{n}_{H_2}^{in,nom}$ are the nominal inlet steam and hydrogen molar flow rates respectively, and $\epsilon_{1/2}$ quantifies the deviation from nominal values due to faults. By incorporating these states, we gain insight into the actual fuel supply during faulty conditions, which is crucial for distinguishing starvation-related issues. Additionally, this information is essential for fault-tolerant control, a key aspect of the overall health management strategy.

Thus, the observer extended state vector is now:

$$\hat{x} = [p_{H_2} \ p_{H_2O} \ p_{O_2} \ T_s \ \epsilon_1 \ \epsilon_2]^T$$

To verify fault occurrence, the estimated state vector is compared to a reference that represents the nominal behavior of the system, as shown in Fig. 2 and introduced to a decision block to automatically detect the type of fault using adaptive thresholds. The reference states are computed using the SOEC model under nominal operating conditions. By doing so, deviations in $\epsilon_{1/2}$ indicate fuel flow discrepancies (steam, hydrogen, or both inlet flow rates), while the reconstructed partial pressures provide additional insights into starvation conditions.

## 4.1. Square-root unscented Kalman filter

The primary objective of Kalman filters is to iteratively estimate $x_k$ based on the history of measurements, represented as $Y_k = (y_1, y_2, ..., y_k)$, which follows the posterior density function $p(x_k|Y_k)$. Under the

assumption that both the process noise $v(t) \sim N(0, Q)$ and the measurement noise $w(t) \sim N(0, R)$ are Gaussian, and that the system dynamics are linear, the Kalman filter represents the posterior density $p(x_k|Y_k)$ using its mean and covariance, enabling state estimation. However, for nonlinear systems, direct parametrization becomes impractical, requiring approximation techniques. A common approach is the Extended Kalman Filter (EKF), which applies local linearization around the predicted mean. Despite its utility, the EKF struggles with highly nonlinear systems, potentially degrading estimation performance.

To overcome the limitations of local linearization, the Unscented Kalman Filter (UKF) and square-root Unscented Kalman Filter (SR-UKF) approximate $p(x_k|Y_k)$ using a set of deterministically chosen weighted sample points, denoted as $\chi$ in the Algorithm in [33]. The weights of these sample points, $W_i^{(m)}$ and $W_i^{(c)}$ for $i \in \{0, ..., 2n\}$, are determined using the scaling parameters $\lambda = n(\alpha^2 - 1)$, and $\gamma = \sqrt{n+\lambda}$, where $n$ represents the state-space dimension. The parameter $\alpha$ controls the dispersion of sample points around the estimated mean $\hat{x}$ and is typically chosen within $10^{-4} \le \alpha \le 1$. The parameter $\beta$ is a positive scalar used to incorporate prior knowledge of the distribution of $x$ [33]. When the weighted sample points undergo a nonlinear transformation, represented as $f(.)$, the mean and covariance of the transformed distribution can be estimated [33]. Compared to the UKF, the SR-UKF enhances numerical stability by avoiding the need for a Cholesky decomposition at each iteration. Instead, it propagates the square root factors through the Kalman iteration and employs an orthogonal transformation (e.g., QR), to compute covariance matrices. Through this algorithm (see Section 4.1.1), state estimates $\hat{x}_k$ and prior distributions are obtained.

### 4.1.1. Algorithm

Select the sigma point set weights $W_i^{(m)}$ and $W_i^{(c)}$, the weight coefficients for mean and error covariance respectively, according to the following equations:

$$\begin{aligned} W_0^{(m)} &= \frac{\lambda}{n + \lambda} & \text{for } i = 0; \\ W_0^{(c)} &= W_0^{(m)} + (1 - \alpha^2 + \beta) & \text{for } i = 0; \\ W_i^{(c)} &= W_i^{(m)} = \frac{1}{2(n + \lambda)} & \text{for } i \neq 0; \end{aligned} \qquad (19)$$

where we set $\alpha = 0.1$, and $\beta = 2$ [33].

**Initialization:**

$$\hat{x}_0 = \mathbb{E}[x_0], \quad S_0 = \text{chol} \{\mathbb{E}[(x_0 - \hat{x}_0)(x_0 - \hat{x}_0)^T]\} \qquad (20)$$

Where the initial condition of the partial pressure states are set to be equal to the inlet gas composition, and zero for $\epsilon_{1/2}$ which indicates that the fuel flow rate used is set to its nominal condition:

$$\hat{x}_0 = [y_{H_2}^{in} \ y_{H_2O}^{in} \ y_{O_2}^{in} \ T_{s,i=0} \ 0 \ 0]^T.$$

The covariance matrices used in this work are as follows:

* $S_0$ – Cholesky factor of covariance matrix of the initial state estimate:

$$S_0 = \text{diag}(2e - 2, \ 2e - 2, \ 2.5e - 3, \ 5e - 1, \ 1e - 4, \ 1e - 3)$$

* $Q$ – Process noise covariance matrix:

$$Q = \text{diag}(2e - 4, \ 3e - 4, \ 1e - 4, \ 1e - 1, \ 5e - 5, \ 5e - 2)$$

* $R$ – Measurement noise covariance matrix:

$$R = \text{diag}(1e - 1, \ 5e - 7, \ 1e - 8)$$

### Step 1: Time update

* Calculate the sigma points:

$$\chi_{k-1}^{-} = [\hat{x}_{k-1} \ \hat{x}_{k-1} + \gamma S_{k-1}^{i} \ \hat{x}_{k-1} - \gamma S_{k-1}^{i}] \qquad (21)$$

Where $S_{k-1}^i$ is the $i$th column of $S_{k-1}$.
• Propagate sigma points:

$$ \chi_{k|k-1}^* = F(\chi_{k-1}^{-}, u_{k-1}) \qquad (22) $$

Where F is the discrete-time version of the function of f in Eq. (17)

• Compute predicted mean:

$$ \hat{\chi}_k^{-} = \sum_{i=0}^{2n} W_i^{(m)} \chi_{i,k}^{*} |_{k-1} \qquad (23) $$

• Compute predicted covariance:

$$ \begin{aligned} S_{x_k} &= qr \left\{ \left[ \sqrt{W_1^{(c)}} (\chi_{1:2n,k|k-1}^* - \hat{\chi}_k^{-}) , \sqrt{Q} \right] \right\} \\ S_{x_k} &= cholupdate \left\{ S_{x_k} , (\chi_{0,k}^* - \hat{\chi}_k^{-}) , W_0^{(c)} \right\} \end{aligned} \qquad (24) $$

Where $\sqrt{Q}$ is the square root of the diagonal matrix Q.

• Resampling of the sigma point:

$$ \chi_{k|k-1} = \left[ \hat{\chi}_k^{-} \quad \hat{\chi}_k^{-} + \gamma S_{x_k}^{i} \quad \hat{\chi}_k^{-} - \gamma S_{x_k}^{i} \right] \qquad (25) $$

## Step 2: Measurement Update

• Compute predicted measurement:

$$ \begin{aligned} \mathcal{Y}_{k|k-1} &= h(\chi_{k|k-1}) \\ \hat{\mathcal{Y}}_k &= \sum_{i=0}^{2n} W_i^{(m)} \mathcal{Y}_{i,k|k-1} \end{aligned} \qquad (26) $$

• Compute innovation covariance:

$$ \begin{aligned} S_{\hat{\mathcal{Y}}_k} &= qr \left\{ \left[ \sqrt{W_1^{(c)}} (\mathcal{Y}_{1:2n,k} - \hat{\mathcal{Y}}_k^{-}) , \sqrt{R} \right] \right\} \\ S_{\hat{\mathcal{Y}}_k} &= cholupdate \left\{ S_{\hat{\mathcal{Y}}_k} , \mathcal{Y}_{0,k} - \hat{\mathcal{Y}}_k^{-} , W_0^{(c)} \right\} \end{aligned} \qquad (27) $$

Where $\sqrt{R}$ is the square root of the diagonal matrix R.

• Compute cross covariance:

$$ P_{x_k y_k} = \sum_{i=0}^{2n} W_c^{i} (\chi_{i,k|k-1} - \hat{\chi}_k^{-}) (\mathcal{Y}_{i,k|k-1} - \hat{\mathcal{Y}}_k^{-})^T \qquad (28) $$

• Compute Kalman gain:

$$ \mathcal{K}_k = P_{x_k y_k} (S_{y_k} S_{y_k}^T)^{-1} \qquad (29) $$

• Update state:

$$ \hat{\chi}_k = \hat{\chi}_k^{-} + \mathcal{K}_k (y_k - \hat{\mathcal{Y}}_k^{-}) \qquad (30) $$

• Update covariance:

$$ S_k = cholupdate \left\{ S_{x_k}, \mathcal{K}_k S_{\hat{\mathcal{Y}}_k}^{-1} \right\} \qquad (31) $$

## 4.2. Adaptive threshold for fault detection

Fault detection in electrochemical systems requires dynamic thresholding to distinguish between normal variations and faulty behavior while mitigating false alarms. A fixed threshold is often inadequate due to varying operating conditions, system uncertainties, process noise, and aging effects. To address this, we employ an adaptive threshold computed using a moving window filter approach.

Instead of directly monitoring the residuals between the measured and estimated outputs, the proposed method compares the states estimated by the observer with those computed by the model under nominal operating conditions. The partial pressures of hydrogen, steam, and oxygen are key parameters for detecting starvation, as starvation directly affects the local gas composition. Consequently, these partial

pressures serve as reliable indicators of fault conditions. The nominal states calculated using the model under nominal operating conditions represent the expected behavior in a fault-free scenario. Comparing the estimated states by the observer with these nominal values enables accurate fault detection, ensuring effective system monitoring.

## 4.3. Experimental studies

A moving window filter is then applied to compute a threshold that adjusts dynamically based on statistical properties of the nominal state evolution. The filtered state vector of the nominal system based on the moving window m, which is set to 120 samples in this study, is:

$$ \bar{\chi}_k = \frac{1}{m} \sum_{i=k-m+1}^{k} x_{nom_i} \qquad (32) $$

The variance of the state within the same moving window at the kth sample point is given by:

$$ \sigma_k^2 = \frac{1}{m-1} \sum_{i=k-m+1}^{k} (x_{nom_i} - \bar{x}_k)^2 \qquad (33) $$

According to statistical theory, the confidence interval for the mean is given by:

$$ P\{\bar{x} - z\sigma < \bar{x} < \bar{x} + z\sigma\} = 1 - \mu \qquad (34) $$

where $\mu$ represents the confidence level, and $z$ is the confidence coefficient [43]. The term $(1 - \mu)$ defines a confidence interval, which is typically selected within the range of 90% to 99%. In this study, the coefficient $z$ is set to 3 to account for noise perturbations (corresponding to 99.7% confidence interval under Gaussian noise). Based on this formulation, the adaptive threshold is determined as:

$$ Th_k = \bar{x}_k \pm 3\sigma_k \qquad (35) $$

where $Th_k$ represents the upper and lower limits of fault-free conditions and adapts dynamically to reflect changes in the actual operating conditions.

The choice of the moving window m directly influences the adaptability of the threshold:

• A small m results in a faster response, increasing the sensitivity to small changes but also making the system more susceptible to noise.

• A large m smooths variations and prevents excessive reactivity, but may reduce the responsiveness of fault detection.

To balance these effects, a weighted sum of past and present residual statistics is used to prevent excessive variations in threshold values [44]:

$$ Th_k = \bar{x}_k \pm 3\bar{\sigma}_k \qquad (36) $$

where the weighted mean and variance are computed as:

$$ \begin{aligned} \bar{\sigma}_k &= \rho\sigma_k + (1 - \rho)\sigma_{k-1}, \\ \bar{x}(k) &= \rho\bar{x}_k + (1 - \rho)\bar{x}_{k-1}. \end{aligned} \qquad (37) $$

Here, $\rho \in (0, 1)$ is a momentum parameter that controls the influence of past values on the threshold adaptation that is set to 0.9 in this study.

Thus, a fault is detected when the estimated state $\hat{x}(t)$ deviates from the nominal state $x_{nom}(t)$ beyond the adaptive threshold.

As mentioned before, to replicate steam starvation fault experimentally, we gradually reduced the total inlet fuel flow rate $n_{\text{fuel}}^{\text{in}}$ from its nominal value $n_{\text{fuel}}^{\text{nom}}$. The fuel consisted primarily of steam with a small fraction of hydrogen, and both flows were decreased simultaneously while maintaining roughly a constant molar fraction, as shown in Fig. 3. At the same time, the current was adjusted to maintain the thermoneutral voltage, ensuring high efficiency and preventing irreversible degradation [45].
<figure><img src="image_5.png" /><figcaption>Fig. 3. Operating conditions showing both estimated and nominal inlet total flow rates during the Steam Starvation scenario for comparison. The colored regions indicate the time windows where the fault is injected. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. Operating conditions showing both estimated and nominal inlet total flow rates during the Hydrogen Starvation scenario for comparison. The colored regions indicate the time windows where the fault is injected. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

Hydrogen starvation, defined as an insufficient inlet hydrogen fraction, poses a risk of nickel reoxidation, in particular at the inlet hydrogen electrode. To simulate this scenario, the hydrogen fraction was reduced from its nominal value of 0.1 to zero over a defined time window (highlighted in Fig. 4), while the current was held constant. Under these conditions, the voltage remains largely unaffected, whereas the outlet hydrogen flow rate, directly impacted by this fault, was included among the measured signals.

The fault detection algorithm was applied using measured stack voltage, temperature, and outlet hydrogen flow rate with inputs in nominal operating conditions. When a fault occurred, the reconstructed partial pressures exceeded their adaptive thresholds, confirming the onset of starvation. Additionally, the reconstructed inlet hydrogen and steam flow rates accurately reflected the actual inputs, which is valuable for health management strategies. Initially, estimated fault indicators remained within normal limits but began to diverge clearly as
<figure><img src="image_7.png" /><figcaption>Fig. 5. States estimation in Steam Starvation scenario and fault detection thresholds ($Th_{min/max}$) for each variable. The colored regions indicate the time windows where the fault is injected. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6. States estimation in Hydrogen Starvation scenario and fault detection thresholds ($Th_{min/max}$) for each variable. The colored regions indicate the time windows where the fault is injected. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

starvation progressed as shown in the highlighted windows in Figs. 5 and 6, demonstrating the ability of diagnostic scheme for early detection.

The experimental results validate the proposed fault detection methodology. The SR-UKF observer effectively identified deviations in fuel supply conditions, while the adaptive threshold minimizes false alarms. If implemented in a real SOC system, this approach could enable early intervention strategies, reduce degradation risks, and improve system reliability.

Adaptive thresholding is adjusted dynamically to process variations while maintaining sensitivity to faults. Unlike static thresholds, this method responds to evolving system conditions, making it particularly effective for detecting starvation faults in SOCs.

Figs. 5 and 6 illustrate the adaptive threshold applied in electrolysis mode, where partial pressures and inlet fuel flow rates are monitored. The methodology is also applicable to fuel cell mode, where hydrogen starvation can be detrimental for cells due to nickel reoxidation. In addition, air starvation can be also detected by monitoring oxygen partial

pressure. A fault is identified when the reconstructed oxygen pressure drops outside the adaptive thresholds, indicating an air starvation event.

The total resistance was derived from the estimated states as complementary information. This resistance, corresponding to the low-frequency resistance in the electrochemical impedance spectrum, which may differ from the true low-frequency resistance due to aging, is given by:

$$R_{\text{total}} = \frac{\partial V_{\text{cell}}}{\partial I} \qquad (38)$$

Although not used for fault identification, this resistance reflects the impact of partial pressure variations on electrochemical performance. In particular, steam starvation causes noticeable changes in overpotentials and a significant increase in total resistance. Fig. 7 illustrates this evolution, consistent with trends reported in the literature [23]. However, relying only on impedance-derived resistance may not be sufficient to detect hydrogen starvation or deviations in small inlet molar fraction in electrolysis mode, since voltage responses can remain
<figure><img src="image_9.png" /><figcaption>Fig. 7. Total resistance behavior in steam and hydrogen starvation scenarios, respectively.</figcaption></figure>

nearly unchanged. This highlights the importance of state estimation–based diagnostics, which enable more precise differentiation and fault interpretation.

## 5. Conclusion

This study presents a real-time fault detection approach for SOECs using a model-based observer. A square-root Unscented Kalman Filter (SR-UKF) estimates key states, including the partial pressures of hydrogen, steam, and oxygen, as well as the deviation of the actual fuel flow rate from its nominal value. The outlet hydrogen flow rate, directly affected by hydrogen starvation, is used as one of the measured outputs with stack voltage and air outlet temperature. Using these measurements with nominal inputs, the observer detects starvation faults by comparing estimated states with their nominal counterparts and is able to reconstruct the actual inlet flow rates. An adaptive threshold, computed using a moving window filter of the nominal states, enhances fault detection. In addition, the total resistance is calculated and increases in the case of steam starvation, while remaining nearly unchanged during hydrogen starvation in electrolysis mode, providing further insight into the nature of the fault.

Experimental validation confirms the effectiveness of the proposed method, demonstrating that fuel starvation can be detected with minimal measurements, making it suitable for implementation. The approach also extends to SOFCs operating in fuel cell mode, where air starvation can be identified by monitoring deviations in oxygen partial pressure from its thresholds.

Beyond fault detection, estimating the real input fuel flow rate is particularly beneficial for fault-tolerant control, enabling corrective actions to dynamically adjust fuel supply and prevent performance degradation. Additionally, voltage and temperature trends provide valuable insights into long-term aging effects: as internal resistance increases due to degradation, the stack experiences higher Joule heating, resulting in an overall rise in operating temperature and voltage. Future work will focus on refining the model to incorporate degradation dynamics or potential local measurements, when available.

### CRediT authorship contribution statement

**Zaman Yazbeck:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Federico Bribiesca-Argomedo:** Writing – review & editing, Visualization, Validation, Supervision, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Minh Tu Pham:** Writing – review & editing, Visualization, Validation, Supervision, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Bertrand Morel:** Writing – review & editing, Supervision, Conceptualization. **Vincent Dimitriou:** Resources, Project administration, Funding acquisition.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

This work was supported by GENVIA and the ANRT (Association Nationale de la Recherche et de la Technologie), France under the framework of a CIFRE thesis (n° 2022/0955) conducted in collaboration between GENVIA, CEA, and INSA Lyon.

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