Thermodynamic Evolution and Operational State Detection in Solid Oxide Fuel Cell Systems: Modeling and Analysis

2.1 Hybrid-Dimensional thermodynamic dynamic modeling

To achieve accurate characterization of thermodynamic characteristic evolution and online state detection for SOFC systems, it is necessary to construct a dynamic model that balances modeling accuracy and computational efficiency, addressing the core challenge that traditional modeling methods are difficult to apply online. The proposed hybrid-dimensional thermodynamic dynamic model adopts a differentiated modeling strategy, designing the architecture according to the differences in thermodynamic characteristics between the stack and the auxiliary system: the stack is discretized in one dimension along the fuel and oxidant flow directions, divided into 8–12 independent control volumes to accurately capture the temperature gradient along the flow direction; the auxiliary system adopts zero-dimensional lumped modeling, and achieves heat exchange coupling with the stack through a thermal network. The design basis of this architecture originates from system thermodynamic characteristic analysis: the temperature gradient along the flow direction inside the stack directly dominates the distribution and evolution of irreversible entropy production, thereby affecting the system degradation process, while the internal temperature gradient of the auxiliary system is relatively small, so adopting zero-dimensional lumped modeling can simplify the model structure while ensuring computational accuracy. The number of discrete control volumes is determined based on thermal impedance theory; 8–12 nodes can fully capture the temperature gradient while avoiding computational redundancy caused by excessive discretization, laying the foundation for the online application of the model. Figure 1 shows the overall architecture of the hybrid-dimensional thermodynamic dynamic modeling for SOFC.

Figure 1. Architecture of the hybrid-dimensional thermodynamic dynamic modeling for solid oxide fuel cells (SOFCs)

The energy transfer and conversion process of the stack is the core of thermodynamic characteristic evolution. Based on the law of conservation of energy for control volumes, gas-phase and solid-phase energy equations are established respectively to accurately describe the thermodynamic behavior of the stack. Taking the cathode as an example, the gas-phase energy equation considers the dynamic characteristics of the one-dimensional discrete control volume, expressed as:

${{\text{ }\!\!\rho\!\!\text{ }}_{\text{c,i}}}{{\text{c}}_{\text{p,c}}}{{\text{A}}_{\text{c}}}\frac{\partial {{\text{T}}_{\text{c,i}}}}{\partial \text{t}}\text{=}{{{\dot{m}}}_{\text{c,i-1}}}{{\text{c}}_{\text{p,c}}}{{\text{T}}_{\text{c,i-1}}}\text{-}{{ {\dot{m}}}_{\text{c,i}}}{{\text{c}}_{\text{p,c}}}{{\text{T}}_{\text{c,i}}} \text{+}{{\text{h}}_{\text{cv,c,i}}}{{\text{A}}_{\text{cv,c,i}}}\text{(}{{\text{T}}_{\text{s,i}}}\text{-}{{\text{T}}_{\text{c,i}}}\text{)}$                       (1)

where, ${{\text{ }\!\!\rho\!\!\text{ }}_{\text{c,i}}}$ is the gas density of the cathode in the i-th control volume, ${{\text{c}}_{\text{p,c}}}$ is the constant-pressure specific heat capacity of the cathode gas, ${{\text{A}}_{\text{c}}}$ is the cathode flow cross-sectional area, ${{\text{T}}_{\text{c,i}}}$ is the cathode gas temperature of the i-th control volume, ${{{\dot{m}}}_{\text{c,i}}}$ is the cathode gas mass flow rate of the i-th control volume, ${{\text{h}}_{\text{cv,c,i}}}$ is the convective heat transfer coefficient between the cathode gas and the solid phase, ${{\text{A}}_{\text{cv,c,i}}}$ is the heat transfer area of the cathode in the i-th control volume, and ${{\text{T}}_{\text{s,i}}}$ is the solid-phase temperature of the stack in the i-th control volume. Through discretization, this equation enables the independent solution of the gas temperature in each control volume, accurately capturing temperature gradient variations along the flow direction, and overcoming the limitation of traditional lumped parameter models that cannot characterize local thermodynamic characteristics. The solid-phase energy equation embeds electrochemical reaction heat and ohmic heat terms to achieve deep coupling of electrochemical and thermodynamic processes, expressed as:

${{\text{m}}_{\text{s,i}}}{{\text{c}}_{\text{p,s}}}\frac{\partial {{\text{T}}_{\text{s,i}}}}{\partial \text{t}}\text{=}{{\text{h}}_{\text{cv,c,i}}}{{\text{A}}_{\text{cv,c,i}}}\text{(}{{\text{T}}_{\text{c,i}}}\text{-}{{\text{T}}_{\text{s,i}}}\text{)} \text{+}{{\text{h}}_{\text{cv,a,i}}}{{\text{A}}_{\text{cv,a,i}}}\text{(}{{\text{T}}_{\text{a,i}}}\text{-}{{\text{T}}_{\text{s,i}}}\text{)+}{{{\dot{Q}}}_{\text{rxn,i}}}\text{+}{{{\dot{Q}}}_{\text{ohm,i}}}$                      (2)

where, ${{\text{m}}_{\text{s,i}}}$ is the solid-phase mass of the stack in the i-th control volume, ${{\text{c}}_{\text{p,s}}}$ is the constant-pressure specific heat capacity of the solid phase, ${{\text{T}}_{\text{a,i}}}$ is the anode gas temperature of the i-th control volume, ${{{\dot{Q}}}_{\text{rxn,i}}}$ is the electrochemical reaction heat of the i-th control volume, and ${{{\dot{Q}}}_{\text{ohm,i}}}$ is the ohmic heat loss of the i-th control volume. The embedding of reaction heat and ohmic heat enables accurate characterization of energy conversion and losses caused by electrochemical processes, clarifying the sources of thermodynamic irreversibility and providing a basis for subsequent entropy production rate calculations.

As an important support for thermodynamic modeling, the electrochemical sub-model adopts a semi-empirical form to balance modeling accuracy and computational efficiency, with emphasis on introducing local temperature dependence to improve model adaptability. The Nernst voltage, as the theoretical open-circuit voltage of the stack, is calculated considering the local solid-phase temperature and gas partial pressures of the i-th control volume, expressed as:

${{\text{E}}_{\text{Nernst,i}}}\text{=}\frac{\text{R}{{\text{T}}_{\text{s,i}}}}{\text{2F}}\text{ln}\left( \frac{{{\text{P}}_{{{\text{H}}_{\text{2}}}\text{,i}}}\text{P}_{{{\text{O}}_{\text{2}}}\text{,i}}^{\text{0}\text{.5}}}{{{\text{P}}_{{{\text{H}}_{\text{2}}}\text{O,i}}}} \right)$                          (3)

where, R is the universal gas constant, F is Faraday’s constant, P_H2,_i, P_O,_i, P_H2O,_i are the partial pressures of hydrogen, oxygen, and water vapor in the i-th control volume, respectively. The activation overpotential is calculated based on the double polarization theory, embedding the local solid-phase temperature to reflect the effect of temperature on reaction kinetics, expressed as:

${{\text{ }\!\!\eta\!\!\text{ }}_{\text{act,i}}}\text{=}\frac{\text{R}{{\text{T}}_{\text{s,i}}}}{\text{2F}}\left[ \text{sin}{{\text{h}}^{\text{-1}}}\left( \frac{{{\text{j}}_{\text{i}}}}{\text{2}{{\text{j}}_{\text{0,a,i}}}} \right)\text{+sin}{{\text{h}}^{\text{-1}}}\left( \frac{{{\text{j}}_{\text{i}}}}{\text{2}{{\text{j}}_{\text{0,c,i}}}} \right) \right]$                                 (4)

where, ${{\text{j}}_{\text{i}}}$ is the current density of the i-th control volume, ${{\text{j}}_{\text{0,a,i}}}$ and ${{\text{j}}_{\text{0,c,i}}}$ are the exchange current densities of the anode and cathode in the i-th control volume, respectively. Combining ohmic overpotential and concentration overpotential, the unit voltage of the i-th control volume can be expressed as:

${{\text{V}}_{\text{i}}}\text{=}{{\text{E}}_{\text{Nernst,i}}}\text{-}{{\text{ }\!\!\eta\!\!\text{ }}_{\text{act,i}}}\text{-}{{\text{ }\!\!\eta\!\!\text{ }}_{\text{ohm,i}}}\text{-}{{\text{ }\!\!\eta\!\!\text{ }}_{\text{conc,i}}}$                            (5)

where, ${{\text{ }\!\!\eta\!\!\text{ }}_{\text{ohm,i}}}$ is the ohmic overpotential of the i-th control volume, and ${{\text{ }\!\!\eta\!\!\text{ }}_{\text{conc,i}}}$ is the concentration overpotential of the i-th control volume. The introduction of local temperature dependence enables the electrochemical model to accurately match the thermodynamic state of each control volume, avoiding the prediction deviation caused by the use of average temperature in traditional models, and further improving the overall accuracy of thermodynamic modeling.

The model is solved using the Crank–Nicolson implicit time-marching scheme, which combines stability and computational accuracy and can effectively capture the thermal relaxation process of the stack. Considering that the system thermal time constant is 10–100 s, the time step is set to 0.1–1 s to ensure accurate tracking of the dynamic evolution of thermodynamic characteristics. During numerical solving, the finite difference method is used to discretize the governing equations, and matrix solving is employed to update parameters such as temperature and voltage in each control volume simultaneously, significantly improving computational efficiency. Tests show that the single-step solution time of the model is less than 10 ms. Compared with the minute-level solution time of traditional three-dimensional CFD models, the computational efficiency is improved by two orders of magnitude. It not only retains the ability to accurately capture key thermodynamic characteristics such as temperature gradients but also meets the real-time requirements of online state detection, achieving an effective balance between modeling accuracy and computational efficiency.

2.2 Entropy production rate decomposition and evolution characterization

Based on the temperature, current density, and overpotential distributions solved by the hybrid-dimensional thermodynamic dynamic model established above, combined with the fundamental theory of nonequilibrium thermodynamics, various types of irreversible losses within the system can be quantitatively evaluated. Figure 2 illustrates the decomposition of the SOFC “entropy fingerprint” and its correlation mechanism with multi-source measurable variables. According to the physical formation mechanisms of energy dissipation, the total entropy production rate across the entire fuel cell domain is divided into four mutually independent components, corresponding respectively to the electrochemical reaction activation process, charge transport process, species diffusion and mass transfer process, and gas–solid interface heat transfer process, thus completely covering all irreversible thermodynamic behaviors during system operation. The total entropy production rate of the system can be uniformly expressed as:

${{ {\dot{S}}}_{\text{gen}}}\text{=}{{ {\dot{S}}}_{\text{act}}}\text{+}{{ {\dot{S}}}_{\text{ohm}}}\text{+}{{ {\dot{S}}}_{\text{conc}}}\text{+}{{ {\dot{S}}}_{\text{therm}}}$                       (6)

In this equation, ${{ {\dot{S}}}_{\text{gen}}}$ is the total entropy production rate of the whole system, ${{ {\dot{S}}}_{\text{act}}}$ is the entropy production rate due to electrochemical reaction activation, ${{ {\dot{S}}}_{\text{ohm}}}$ is the entropy production rate caused by charge transport (ohmic loss), ${{ {\dot{S}}}_{\text{conc}}}$ is the entropy production rate arising from species mass transfer (concentration overpotential), and ${{ {\dot{S}}}_{\text{therm}}}$ is the entropy production rate due to gas–solid heat conduction. This decomposition strictly follows the local entropy balance criterion and enables segment-by-segment statistics of irreversible losses along the flow direction. Compared with global averaging calculation forms, it can accurately restore the spatial distribution pattern of entropy production inside the stack.

Figure 2. Mechanism diagram of Solid oxide fuel cell (SOFC) “entropy fingerprint” decomposition and correlation with multi-source measurable variables

On the basis of clarifying the entropy production component division, explicit mathematical relationships between each entropy production component and the system’s measured operating parameters are established, realizing the direct coupling of thermodynamic loss indicators with engineering-measurable physical quantities. The ohmic entropy production rate generated in the charge transport process takes the discrete control volume as the calculation unit, fully considering local temperature and current distribution differences. Its calculation expression is:

${{ {\dot{S}}}_{\text{ohm}}}\text{=}\mathop{\sum }_{\text{i=1}}^{\text{N}}\frac{\text{j}_{\text{i}}^{\text{2}}{{\text{A}}_{\text{i}}}{{\text{R}}_{\text{ohm,i}}}}{{{\text{T}}_{\text{s,i}}}}$                         (7)

In practical engineering applications, the equivalent global ohmic resistance can be obtained via electrochemical impedance spectroscopy testing, and simplified solutions can be completed by combining the average solid-phase temperature, namely:

${{ {\dot{S}}}_{\text{ohm}}}\approx \frac{{{\text{I}}^{\text{2}}}{{\text{R}}_{\text{ohm}}}}{{{\text{T}}_{\text{avg}}}}$                  (8)

where, N is the total number of discrete control volumes along the flow direction of the stack, ${{\text{j}}_{\text{i}}}$ is the local current density of the i-th control volume, ${{\text{A}}_{\text{i}}}$ is the effective reaction area of the corresponding control volume, ${{\text{R}}_{\text{ohm,i}}}$ is the equivalent ohmic impedance of the local region, ${{\text{T}}_{\text{s,i}}}$ is the solid-phase operating temperature of the control volume, I is the total output current of the stack, ${{\text{R}}_{\text{ohm}}}$ is the global equivalent ohmic resistance of the stack, and ${{\text{T}}_{\text{avg}}}$ is the average solid-phase temperature of the stack.

The concentration entropy production rate caused by species mass transfer is jointly determined by the local current density and concentration polarization loss, and can be directly linked to inlet flow rate and composition concentration regulation parameters. Its expression is:

${{ {\dot{S}}}_{\text{conc}}}\text{=}\mathop{\sum }_{\text{i=1}}^{\text{N}}\frac{{{\text{j}}_{\text{i}}}{{\text{A}}_{\text{i}}}\text{ }\!\!|\!\!\text{ }{{\text{ }\!\!\eta\!\!\text{ }}_{\text{conc,i}}}\text{ }\!\!|\!\!\text{ }}{{{\text{T}}_{\text{s,i}}}}$                     (9)

where, ${{\text{ }\!\!\eta\!\!\text{ }}_{\text{conc,i}}}$ is the concentration polarization overpotential inside the i-th control volume. The thermal conduction entropy production rate formed by the temperature difference between the gas and solid phases is solved using distributed temperature acquisition data, fully accounting for the irreversible heat loss caused by convection heat transfer at the anode and cathode sides. Its specific form is:

${{ {\dot{S}}}_{\text{therm}}}\text{=}\mathop{\sum }_{\text{i=1}}^{\text{N}}\left[ \frac{{{{ {\dot{Q}}}}_{\text{conv,c,i}}}}{{{\text{T}}_{\text{g,c,i}}}}\left( \text{1-}\frac{{{\text{T}}_{\text{g,c,i}}}}{{{\text{T}}_{\text{s,i}}}} \right)\text{+}\frac{{{{ {\dot{Q}}}}_{\text{conv,a,i}}}}{{{\text{T}}_{\text{g,a,i}}}}\left( \text{1-}\frac{{{\text{T}}_{\text{g,a,i}}}}{{{\text{T}}_{\text{s,i}}}} \right) \right]$                           (10)

where, ${{ {\dot{Q}}}_{\text{conv,c,i}}}$ and ${{ {\dot{Q}}}_{\text{conv,a,i}}}$ are the convective heat transfer quantities on the cathode and anode sides, respectively; T_g,c,i and T_g,a,i are the temperatures of the cathode and anode gas media, respectively. Through the above correlation forms, entropy production parameters that originally could only be solved offline using mechanism models can now be calculated in real time using conventional on-site monitoring equipment, greatly improving the engineering practicability of thermodynamic loss analysis.

Under different operating conditions, the proportion levels and variation trends of various entropy production components exhibit significantly differentiated characteristics. During the start-up heating stage, the large temperature difference between gas and solid makes heat conduction–related entropy production dominant. During rapid load switching, sudden changes in current density simultaneously cause fluctuations in ohmic loss and mass transfer loss. In the process of long-term continuous operation, material aging and structural degradation promote persistent unidirectional changes in specific types of entropy production. Slowly increasing electrolyte internal impedance continuously raises the overall proportion of ohmic entropy production, decreasing electrode catalytic activity steadily increases activation entropy production, and dust deposition and clogging in fuel channels directly aggravate irreversible mass transfer losses. By relying on long-term operational data statistics to analyze the composition ratio and time-series fluctuation patterns of various entropy production components, unique system operational characteristic patterns can be formed to distinguish different internal degradation forms and complete refined identification of the system’s internal health status.

The state characterization system formed based on temporal evolution laws can effectively fill the lag problem in discrimination inherent in traditional external characteristic monitoring methods. When the fuel cell output voltage shows obvious decay, the internal irreversible losses of the system are often already at a relatively advanced stage, whereas directional shifts in entropy production components can capture internal degradation trends in advance before obvious changes occur in the macro electrical characteristics of the equipment. With this characterization method, normal operating intervals and abnormal degradation intervals can be clearly defined, while precisely distinguishing single-degradation types from compound-degradation conditions. This provides comprehensive, physically supported feature evidence for the subsequent construction of nonlinear state observation models and early operational anomaly warning.

2.3 Design of entropy production observer based on square-root unscented Kalman filter

The entropy production rate and its components, as core indicators characterizing the irreversible evolution of SOFC systems, cannot be directly measured by existing test equipment and require the construction of a nonlinear observer to realize soft measurement. The strong nonlinear characteristics present during system operation make traditional linear observers prone to errors introduced by linearization approximations, leading to divergent estimation results and making it difficult to meet the requirements of high-precision online monitoring. To address this, an entropy production observer based on the SR-UKF is constructed. Through state-space modeling and multi-source data fusion, stable and accurate estimation of entropy production rates and their components is achieved. Figure 3 shows the flowchart of the nonlinear entropy production observer and fault detection based on the dual time-scale SR-UKF.

Figure 3. Flowchart of the nonlinear entropy production observer and fault detection based on the dual time-scale Square-Root Unscented Kalman Filter (SR-UKF)

The construction of the state-space model is based on the system’s thermodynamic and electrochemical characteristics, with a focus on incorporating entropy production–related variables into the state vector to enable direct real-time estimation of irreversibility. The state equation and observation equation are expressed as:

${{\text{x}}_{\text{k+1}}}\text{=f(}{{\text{x}}_{\text{k}}}\text{,}{{\text{u}}_{\text{k}}}\text{,}{{\text{ }\!\!\theta\!\!\text{ }}_{\text{k}}}\text{)+}{{\text{w}}_{\text{k}}}$                       (11)

${{\text{y}}_{\text{k}}}\text{=h(}{{\text{x}}_{\text{k}}}\text{,}{{\text{u}}_{\text{k}}}\text{)+}{{\text{v}}_{\text{k}}}$                     (12)

where, ${{\text{x}}_{\text{k}}}$ is the state vector at time step k, containing the total entropy production rate, individual entropy production components, solid-phase temperatures of each stack control volume, anode/cathode gas partial pressures, and key degradation parameters, thus comprehensively covering the thermodynamic and health states of the system; ${{\text{u}}_{\text{k}}}$ is the input vector, including controllable operating parameters such as inlet flow rates, current load, and ambient temperature; ${{\text{ }\!\!\theta\!\!\text{ }}_{\text{k}}}$ is the model parameter vector; ${{\text{w}}_{\text{k}}}$ is the process noise, following a Gaussian distribution with mean 0 and covariance $\text{Q}$; ${{\text{y}}_{\text{k}}}$ is the observation vector, composed of measurable parameters such as voltage, current, and temperatures at monitoring points; and ${{\text{v}}_{\text{k}}}$ is the observation noise, following a Gaussian distribution with mean 0 and covariance $\text{R}$. Incorporating entropy production variables directly into the state vector breaks the limitation of traditional observers that can only estimate macroscopic parameters, enabling direct tracking of irreversible evolution processes.

The observer adopts the SR-UKF algorithm, optimizing the covariance matrix update process through square-root decomposition to enhance numerical stability while reducing computational complexity. The core steps of the algorithm begin with Sigma point generation. Based on the state estimate and the square root of the covariance matrix, a set of Sigma points is generated to represent the probability distribution of the state vector, specifically:

${{\text{X}}^{\text{(0)}}}\text{=}{{ {\hat{x}}}_{\text{k }\!\!|\!\!\text{ k}}}\text{, }\!\!~\!\!\text{ }{{\text{W}}^{\text{(0)}}}\text{=}\frac{\text{ }\!\!\kappa\!\!\text{ }}{{{\text{n}}_{\text{x}}}\text{+ }\!\!\kappa\!\!\text{ }}$                    (13)

${{\text{X}}^{\text{(j)}}}\text{=}{{ {\hat{x}}}_{\text{k }\!\!|\!\!\text{ k}}}\text{ }\!\!\pm\!\!\text{ }{{\left( \sqrt{\text{(}{{\text{n}}_{\text{x}}}\text{+ }\!\!\kappa\!\!\text{ )}{{\text{P}}_{\text{k }\!\!|\!\!\text{ k}}}} \right)}_{\text{j}}}\text{, }\!\!~\!\!\text{ }{{\text{W}}^{\text{(j)}}}\text{=}\frac{\text{1}}{\text{2(}{{\text{n}}_{\text{x}}}\text{+ }\!\!\kappa\!\!\text{ )}}$                   (14)

where, ${{ {\hat{x}}}_{\text{k }\!\!|\!\!\text{ k}}}$ is the state estimate at time k, ${{\text{P}}_{\text{k }\!\!|\!\!\text{ k}}}$ is the state estimation covariance matrix at time k, ${{\text{n}}_{\text{x}}}$ is the dimension of the state vector, $\text{ }\!\!\kappa\!\!\text{ }$ is a scaling factor chosen as $\text{3-}{{\text{n}}_{\text{x}}}$ to ensure numerical stability, and ${{\text{W}}^{\text{(j)}}}$ is the weight of the Sigma point. State prediction is performed through the nonlinear propagation of Sigma points, with the prediction given by:

${{ {\hat{x}}}_{\text{k+1 }\!\!|\!\!\text{ k}}}\text{=}\mathop{\sum }_{\text{j=0}}^{\text{2}{{\text{n}}_{\text{x}}}}{{\text{W}}^{\text{(j)}}}\text{X}_{\text{k+1 }\!\!|\!\!\text{ k}}^{\text{(j)}}$                       (15)

In the observation update stage, the deviation between the observed value and the predicted observation is calculated, the Kalman gain is solved, and the state estimate and covariance matrix are updated. The Kalman gain is expressed as:

${{\text{K}}_{\text{k+1}}}\text{=}{{\text{P}}_{\text{xy}}}\text{P}_{\text{yy}}^{\text{-1}}$                          (16)

where, ${{\text{P}}_{\text{xy}}}$ is the cross-covariance matrix between the state prediction and the observation prediction, and ${{\text{P}}_{\text{yy}}}$ is the covariance matrix of the observation prediction. The introduction of square-root decomposition effectively prevents the covariance matrix from becoming non-positive definite, adapts to the strong nonlinearity of the system, and solves the divergence problem common in traditional linear observers.

A dual time-scale data fusion strategy is adopted to improve observation accuracy, realizing an organic combination of high-frequency real-time tracking and low-frequency precise calibration. At the high-frequency level, conventional monitoring data such as voltage, current, and temperature are collected every second, and the observer estimates the entropy production rate in real time, ensuring the timeliness of online monitoring. At the low-frequency level, electrochemical impedance spectroscopy tests are triggered every 5–30 minutes to extract the system’s equivalent ohmic resistance, which is then used to calibrate the estimated results of the ohmic entropy production rate. The measurement uncertainty of the electrochemical impedance spectroscopy test is controlled within ±2%, and the covariance ${{\text{R}}_{\text{EIS}}}$ for the calibration process is set accordingly to ensure calibration accuracy. Based on the observed entropy production components, entropy production ratio indicators are defined for fault detection. Taking the ohmic entropy production ratio as an example, its expression is:

${{\text{r}}_{\text{ohm}}}\text{=}\frac{{{{ {\dot{S}}}}_{\text{ohm}}}}{{{{ {\dot{S}}}}_{\text{ohm,0}}}}$                          (17)

where, ${{ {\dot{S}}}_{\text{ohm,0}}}$ is the baseline value of the ohmic entropy production rate during the initial normal operation period. The 95% confidence interval for each entropy production ratio is determined through statistical analysis, and the control limit is set to 1.3. When a certain entropy production ratio exceeds the control limit, the system is judged to have experienced corresponding type of degradation or fault, and an early warning signal is issued, realizing precise monitoring and anomaly warning of the system’s operational state.

2.4 Physics-informed neural network-assisted hybrid degradation model

Although the thermodynamic mechanism model established earlier can fully reproduce the entropy production evolution law of the fuel cell under conventional operating conditions, it is difficult to achieve high-precision characterization for complex degradation behaviors involving the coupling of multiple factors such as electrode microstructure evolution and interface impurity poisoning, relying solely on physical equations. Pure data-driven models, despite having strong data-fitting capabilities, tend to deviate from basic thermodynamic principles, and their inference results are prone to unreasonable phenomena that violate the laws of energy conservation and entropy increase. Therefore, a hybrid degradation modeling framework combining mechanism models and physics-informed neural networks is constructed. The deterministic thermodynamic mechanism model is used to build the main framework of system loss evolution, and the neural network is employed to compensate for modeling deviations caused by complex degradation processes, effectively solving the state prediction problem under the condition of scarce measured time-series degradation data samples. Figure 4 shows the topology of the Physics-Informed Neural Network (PINN)-assisted hybrid degradation model embedded with thermodynamic hard constraints.

Figure 4. Topology of the Physics-Informed Neural Network (PINN)-assisted hybrid degradation model embedded with thermodynamic hard constraints

The physics-informed neural network constructed in this paper takes the system’s actual operating conditions and time-series information as input features. After multi-layer nonlinear mapping operations, it outputs entropy production deviation corrections, which are used for dynamic correction of the mechanism model’s solution results. The network training process no longer relies solely on measured data for fitting; instead, the fundamental laws of thermodynamics are embedded into the loss function to complete joint constrained training. The overall loss function is constructed as follows:

$\text{L=}{{\text{L}}_{\text{data}}}\text{+}{{\text{ }\!\!\lambda\!\!\text{ }}_{\text{1}}}{{\left| \text{1-}\frac{\mathop{\sum }_{\text{k}}\text{(}{{{ {\dot{S}}}}_{\text{k,mech}}}\text{+ }\!\!\Delta\!\!\text{ }{{{ {\dot{S}}}}_{\text{k}}}\text{)}}{\mathop{\sum }_{\text{k}}{{{ {\dot{S}}}}_{\text{k,mech}}}\text{+}\mathop{\sum }_{\text{k}}\text{ }\!\!\Delta\!\!\text{ }{{{ {\dot{S}}}}_{\text{k}}}} \right|}^{\text{2}}} \text{+}{{\text{ }\!\!\lambda\!\!\text{ }}_{\text{2}}}\mathop{\sum }_{\text{k}}\underset{~}{\mathop{\text{max}}}\,{{\text{(0,- }\!\!\Delta\!\!\text{ }{{ {\dot{S}}}_{\text{k}}}\text{)}}^{\text{2}}}$                           (18)

The first term is the measured data fitting loss, used to reduce the deviation between the model calculation results and the actual observation data; the second term is the global entropy increase constraint, ensuring that the total entropy production change trend of the system over the entire time range conforms to the basic laws of irreversible process thermodynamics; the third term is the non-negative constraint for each entropy production component, preventing corrected loss entropy production from taking negative values or other physically unrealistic calculation results. Parameters and are the weighting coefficients of the two physical constraints, which can be adaptively adjusted according to operating condition characteristics and model fitting requirements, ensuring data fitting accuracy while firmly maintaining the boundaries of physical mechanisms.

After completing network training, the correction output by the neural network is coupled and superimposed with the calculated values of the mechanism model to obtain real-time entropy production results that take into account both physical laws and actual degradation characteristics. The fusion expression is:

${{ {\dot{S}}}_{\text{gen,total}}}\text{=}{{ {\dot{S}}}_{\text{gen,mech}}}\text{+}{{\text{N}}_{\text{PINN}}}\text{(}{{\text{z}}_{\text{k}}}\text{)}$                       (19)

where, ${{\text{z}}_{\text{k}}}$ represents the set of input features of the current operating condition, and ${{\text{N}}_{\text{PINN}}}\text{(}{{\text{z}}_{\text{k}}}\text{)}$ is the entropy production correction vector output by the physics-informed neural network. During the operation of the entire detection system, this hybrid model is embedded into the state prediction step of the SR-UKF, and the corrected entropy production sequence is used to optimize the prior state estimation results, alleviating the prediction drift problem of a single mechanism model under long-term aging conditions, and further enhancing the state tracking capability of the observer in slow-varying degradation scenarios.

This hybrid modeling form realizes the deep integration of theoretical mechanisms and measured information, preserving both the clear physical logic and interpretability of traditional thermodynamic models—consistent with academic research norms in energy and thermodynamic system modeling—and fully leveraging the ability of data-driven methods to uncover hidden degradation patterns. Even in scenarios where only a small amount of valid labeled data remains after long-term continuous operation, the built-in physical constraints still ensure the rationality of the model inference results, further completing the full theoretical system from thermodynamic characteristic evolution analysis to online operational state detection.