A SOFC voltage control method based on deep operator neural network

Abstract

This invention relates to the field of SOFC voltage control technology, and particularly to a SOFC voltage control method based on a deep operator neural network, comprising: S1, establishing a dynamic physical model of the SOFC; S2, performing simulation based on the dynamic physical model to form a training dataset; S3, constructing a deep operator neural network prediction model to replace the forward prediction process based on the dynamic physical model; S4, training the deep operator neural network prediction model using the training dataset; S5, using the deep operator neural network prediction model trained in S4 as a predictor to construct a model predictive control optimization framework; S6, obtaining the current system state of the SOFC, calling the optimization framework in S5, and solving the corresponding finite-time domain optimization problem based on the current system state. This method can significantly reduce the online computational burden of model predictive control while ensuring high-precision prediction capability, achieving high-precision, fast-response, robust, and real-time tracking control of the SOFC output voltage while satisfying operational constraints.

Description

Technical Field

[0001] This invention relates to the field of SOFC voltage control technology, and in particular to an SOFC voltage control method based on deep operator neural networks. Background Technology

[0002] As the global energy structure accelerates its transformation from traditional fossil fuels to clean and low-carbon energy sources, power systems are placing higher demands on energy conversion technologies that are efficient, clean, and flexible. Solid oxide fuel cells (SOFCs), due to their high energy conversion efficiency, strong fuel adaptability, and near-zero emissions, are considered a key power generation unit in distributed energy systems, microgrids, and new power systems. However, SOFCs exhibit significant strong nonlinearity, time-varying dynamic characteristics, and deep multiphysics coupling during actual operation, resulting in complex behaviors such as high sensitivity of their output voltage to operating variables like fuel flow rate, gas partial pressure, and temperature. This makes the system's dynamic response complex and difficult to model accurately.

[0003] To achieve stable and accurate tracking of SOFC output voltage, traditional control strategies such as proportional-integral-derivative (PID) control, sliding mode control, fuzzy control, or adaptive control have been widely explored. Among these, PID control is commonly used due to its simple structure and ease of engineering implementation, but its fixed parameters are difficult to adapt to the nonlinear dynamic changes of SOFC under different loads, start-up, or variable operating conditions, leading to performance degradation. While sliding mode control has strong robustness, it is prone to high-frequency chattering in practical applications, affecting system lifespan and power quality. Although fuzzy control and adaptive control can handle parameter uncertainty to some extent, their rule base construction relies on expert experience, lacks a global optimization mechanism, and is difficult to explicitly handle input / state constraints, making it difficult to balance control accuracy, dynamic response speed, and safety in complex operating scenarios.

[0004] In recent years, Model Predictive Control (MPC) has demonstrated significant advantages in energy system control due to its ability to explicitly handle multivariable coupling, operational constraints, and optimization objectives. The core of MPC lies in its ability to predict the output over a finite time domain based on the system model at each sampling time. By solving a constrained optimization problem, the optimal control sequence is obtained, and only the initial control input is applied to the system, repeating the process at the next time step. This mechanism is naturally suitable for complex systems like SOFCs (Single-Input Multiple-Output Systems) with operational constraints (such as fuel flow limits and temperature safety boundaries). However, the strong nonlinear dynamic characteristics of SOFCs mean that their high-fidelity physical models are typically a set of complex partial differential-algebraic equations (PDAEs). Repeatedly calling such models for forward prediction within the MPC framework incurs extremely high online computational burdens. Especially when the prediction time domain is long or the sampling period is short, the solution time for the optimization problem often fails to meet real-time control requirements, severely limiting the engineering application of MPC in SOFC voltage control.

[0005] The root of the above problems lies in two aspects: firstly, the high-dimensional nonlinear dynamics of SOFC systems are difficult to accurately characterize using traditional linear or simplified models, resulting in limited performance of MPC control based on simplified models; secondly, if a high-precision physical model is directly embedded into the MPC, the computational complexity of online optimization increases exponentially, making real-time closed-loop control difficult to achieve. Therefore, how to significantly reduce the computational overhead of the forward prediction stage in the MPC while ensuring prediction accuracy has become a key technical bottleneck for improving the real-time performance and robustness of SOFC voltage control. The difficulty in solving this problem lies in achieving a delicate balance between model accuracy, computational efficiency, and control performance—neither sacrificing the ability to represent the complex dynamics of SOFCs nor failing to meet the real-time requirement of millisecond-level control cycles, which poses extremely high challenges to both modeling methods and control architectures.

[0006] Therefore, how to significantly reduce the online computational burden of model predictive control while ensuring high-precision prediction capability, so as to achieve high-precision, fast-response, robust, and operationally constrained real-time tracking control of SOFC output voltage, has become an urgent problem to be solved. Summary of the Invention

[0007] To address the aforementioned shortcomings of existing technologies, the present invention aims to provide a SOFC voltage control method based on deep operator neural networks. This method can significantly reduce the online computational burden of model predictive control while ensuring high-precision prediction capabilities, thereby achieving high-precision, fast-response, robust, and real-time tracking control of SOFC output voltage that meets operational constraints.

[0008] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows:

[0009] A SOFC voltage control method based on deep operator neural networks includes the following steps:

[0010] S1. Establish a dynamic physical model of SOFC to describe the dynamic relationship between fuel flow rate, gas partial pressure and output voltage, and output the corresponding system state sequence and output voltage sequence through simulation calculation based on the given control input sequence and initial system state.

[0011] S2. Simulation is performed based on the dynamic physical model of S1, and multiple sets of simulation data are collected to form a training dataset.

[0012] S3. Construct a deep operator neural network prediction model to learn the nonlinear mapping relationship from the future control input sequence and the current system state to the future output voltage trajectory, in order to replace the forward prediction process based on the dynamic physical model.

[0013] S4. Train the deep operator neural network prediction model using the training dataset from S2.

[0014] S5. Using the deep operator neural network prediction model trained in S4 as the predictor, construct a model predictive control optimization framework to construct a finite-time optimization problem with the future N-step control input sequence as the optimization variable, given the current system state. The objective of the finite-time optimization problem is to minimize the tracking error between the future output voltage trajectory predicted by the deep operator neural network prediction model and the reference voltage trajectory, while minimizing the incremental change of the control input sequence.

[0015] S6. During system operation, repeatedly perform the following operations: obtain the current system state of SOFC, call the optimization framework of S5, solve the corresponding finite-time domain optimization problem based on the current system state, apply the obtained optimal control input to the system, and realize real-time tracking control of the output voltage.

[0016] Compared with the prior art, the present invention has the following advantages:

[0017] 1. It can significantly reduce online computational complexity and improve control real-time performance. Traditional MPC in SOFC control relies on high-dimensional nonlinear physical models for rolling prediction. Each optimization step requires multiple calls to the complex differential equation solver, resulting in long computation time and difficulty in meeting real-time control requirements. This solution uses offline-trained deep operator neural networks to replace the online physical simulation process, transforming the originally time-consuming numerical integral prediction into efficient neural network forward inference, greatly compressing the prediction time in the MPC optimization loop, thus effectively solving the problem of insufficient real-time performance of MPC in strongly nonlinear systems.

[0018] 2. It maintains high-precision dynamic prediction capabilities while taking into account nonlinearity and multivariate coupling characteristics. Deep operator neural networks have the ability to learn the nonlinear mapping between input and output functions, accurately capturing the complex dynamic relationships between fuel flow, gas partial pressure, and output voltage in SOFC systems. Compared to traditional linearized models or simplified empirical models, this method fully preserves the original nonlinearity and coupling characteristics of the system without sacrificing prediction accuracy, providing a more reliable prediction basis for subsequent optimization.

[0019] 3. It enables multi-objective collaborative optimization, enhancing system operational safety and stability. When constructing the MPC optimization framework, it considers not only the tracking error of the output voltage to the reference trajectory but also minimizes the incremental change in the control input, effectively suppressing drastic fluctuations in the control signal. Compared to traditional PID or sliding mode control, which only focuses on error feedback while neglecting control smoothness, this method reduces the risk of actuator wear while ensuring tracking performance. It also naturally supports the explicit embedding of operational constraints such as control amplitude and rate of change, improving the overall robustness and safety of the system.

[0020] 4. This approach eliminates the reliance on online access to precise mechanistic models, improving engineering applicability. It decouples complex physical modeling from control design: high-fidelity models are used only for offline data generation and network training, while only lightweight neural network models are needed for actual operation. This avoids the difficulties of deploying complex differential equation solvers on embedded controllers or edge devices, significantly enhancing the deployability and practicality of advanced control strategies in resource-constrained industrial scenarios.

[0021] In summary, this method can significantly reduce the online computational burden of model predictive control while ensuring high-precision prediction capability, thereby achieving high-precision, fast-response, robust, and operationally-constrained real-time tracking control of SOFC output voltage.

[0022] Preferably, in step S1, the dynamic physical model includes:

[0023] ;

[0024] in, This indicates the input hydrogen flow rate. Indicates fuel flow rate. Indicates the conversion factor; Let represent the response time constant of hydrogen gas in the reaction chamber, used to construct a first-order inertial dynamic model, and s be the Laplace transform operator. The denominator of the transfer function that constitutes the dynamic process of gas partial pressure;

[0025] ;

[0026] in, Indicates the first Inlet flow rate of each component Indicates the first The outlet flow rate of each component, Indicates the flow rate involved in the reaction; , and These represent the universal gas constant, the reaction chamber temperature, and the reaction chamber volume, respectively. This represents the instantaneous partial pressure of the i-th gas component within the anode or cathode reaction chamber;

[0027] ;

[0028] ;

[0029] ;

[0030] in, , , These represent the partial pressures of hydrogen, oxygen, and water vapor, respectively. and These represent the molar constant of the hydrogen valve and the response time constant of the hydrogen flow rate, respectively. and These represent the molar constant of the oxygen valve and the response time constant of the oxygen flow rate, respectively. and These represent the molar constant of the steam valve and the response time constant of the steam flow rate, respectively. Represents the reaction rate constant; This indicates the operating current of the SOFC;

[0031]

[0032]

[0033]

[0034]

[0035]

[0036] in, Indicates the output voltage of the fuel cell stack; Represents the ideal standard electric potential. Indicates the maximum limiting current. and These represent the Tafel constant and the Tafel slope, respectively. Indicates ohmic loss; Represents electric potential; Denotes Faraday's constant; Indicates ohmic loss; Indicates activation loss; Indicates concentration loss; N represents the number of SOFC cells connected in series.

[0037] This setup achieves the following: 1. A high-fidelity dynamic model capable of resolving and embedding electrochemical mechanisms is constructed, supporting accurate data generation and closed-loop control design. Compared to existing methods that rely solely on black-box identification or simplified steady-state models, the physical model employed in this scheme explicitly couples gas flow dynamics, thermodynamic equilibrium, and electrochemical polarization mechanisms, accurately reproducing the voltage dynamic response of SOFCs under varying operating conditions. This ensures strong physical consistency in the simulation data generated based on this model, providing high-quality supervisory signals for the learning of deep operator neural networks, avoiding the propagation of prediction biases caused by model distortion, and fundamentally guaranteeing the reliability and generalization capability of subsequent MPC control.

[0038] 2. The model structure is adapted to the operator learning paradigm, facilitating decoupling between offline training and online deployment. All dynamic equations are given in Laplace domain (s-domain) form, and key variables are represented as input flows. Rational fractional functions inherently possess a mapping structure from input function to output function, which highly aligns with the learning objective of deep operator neural networks—namely, learning the operator mapping from control input sequences to voltage trajectories. This modeling approach preserves physical interpretability while creating favorable conditions for efficient neural network training, significantly outperforming traditional state-space models that require discretization of the state space.

[0039] Preferably, in step S5, the objective function of the finite-time optimization problem is:

[0040] ;

[0041] in, This indicates the predicted future output voltage trajectory; Indicates the reference voltage trajectory; This indicates the incremental change in the control input sequence; This represents the weighted square norm of the output tracking error, where the weight matrix Q is a positive real number; Let R represent the weighted square norm of the control input increment, where the weight matrix R is a positive real number.

[0042] This setup, by introducing a control increment weighting term, effectively suppresses actuator oscillations and high-frequency system disturbances, improving control stability and hardware compatibility. Compared to the traditional MPC objective function that only minimizes the tracking error, this scheme explicitly adds a quadratic penalty term to the control increment. This is equivalent to imposing a control smoothness constraint during optimization. This design can significantly reduce control signal jitter caused by prediction noise or model mismatch in deep operator networks, and avoid mechanical / thermal stress damage to SOFC electrode structures and sealing materials caused by frequent valve opening and closing or current surges. At the same time, this strategy can naturally achieve incremental adjustment of control actions without additional hard constraints, making it more suitable for fuel cell systems with stringent dynamic stability requirements than methods such as sliding mode control.

[0043] 2. The adjustability of the weight matrices Q and R endows the controller with flexible performance trade-offs, adapting to the operational needs of multiple scenarios. By adjusting the relative values ​​of Q and R, a continuous trade-off can be made between tracking accuracy and control economy / safety: increasing Q enhances tracking performance, suitable for rapid load tracking conditions; increasing R prioritizes control stability, suitable for disturbance-sensitive scenarios such as start-stop transitions or weak power grid support. This weighted design based on clearly defined physical meaning is superior to PID or fuzzy rules that rely on experience tuning, giving the controller the potential for adaptive optimization for different operating modes.

[0044] Preferably, in step S5, the constraints of the finite-time optimization problem include:

[0045] ;

[0046] ;

[0047] ;

[0048] ;

[0049] ;

[0050] in, The physical model function of the system; This represents the predicted system state at the (i+1)th future step at sampling time k; This represents the predicted future system state at sampling time k; express The corresponding predictive control input; express The corresponding predictive control input, This represents the predicted system state at the (i-1)th future step at sampling time k; This represents the mapping function output by the deep neural network. The generated voltage prediction value; , These represent the physically feasible range of the control inputs; The table controls the upper limit of the amplitude constraint for the increment.

[0051] This setup, through multi-level hard constraint embedding, ensures that the optimal solution strictly meets the physical limits and operational safety requirements of SOFC. Compared to traditional methods that rely solely on soft penalties for the objective function, this scheme explicitly incorporates key constraints such as control amplitude, rate of change, and state evolution into the optimization problem as hard constraints, fundamentally preventing the emergence of control solutions that violate the equipment safety boundaries. Especially for electrochemical systems like SOFC that are sensitive to operating windows, this design effectively avoids optimal but infeasible control commands caused by model approximation errors or external disturbances, significantly improving the engineering reliability of the closed-loop system.

[0052] Preferably, in step S2, each set of simulation data includes: an initial system state, a future control input sequence, and a future output voltage trajectory calculated by the dynamic physical model based on the initial system state and the future control input sequence.

[0053] This setup, by systematically traversing different initial states and control sequences, can efficiently generate simulation datasets covering a wide operating domain. Compared to traditional methods that rely solely on limited experimental data, this strategy can fully stimulate the nonlinear characteristics of the system in the offline phase, enabling the trained operator network to possess stronger dynamic capture capabilities and laying the foundation for stable tracking of MPC under varying operating conditions and strong disturbances.

[0054] Preferably, in step S3, the deep operator neural network prediction model includes a branch network and a backbone network; the deep operator neural network prediction model is configured to: encode the future control input sequence of the input through the branch network to obtain the branch network coefficients, encode the current system state and prediction step size index of the input through the backbone network to obtain the backbone network basis function, and perform a weighted summation of the branch network coefficients and the backbone network basis function to generate a predicted value of the future output voltage trajectory.

[0055] This setup achieves two key advantages: 1) Decoupling the modeling of control inputs and system state influences, significantly improving network generalization ability and training efficiency. Traditional end-to-end networks need to simultaneously learn a high-dimensional joint mapping between the input sequence and the initial state, making them susceptible to data distribution shifts. This approach, however, models the control action and system response separately: branch networks learn only how the control sequence modulates the response intensity, while the backbone network learns only the inherent dynamic modes of the system under different states. This decoupling makes the model more robust to changes in initial state or control strategy. For example, when system aging causes a slow drift in polarization characteristics, only the backbone network needs fine-tuning to adapt to the new basis functions, without retraining the entire network. Furthermore, due to strong parameter sharing, the number of samples required for training is significantly reduced, accelerating convergence.

[0056] 2. Naturally compatible with the MPC rolling optimization framework, supporting efficient online prediction and real-time control deployment. Since the prediction output is a linear combination of coefficients and basis functions, once the backbone network's forward computation is complete, subsequent predictions of any control sequence only require rerunning the lightweight branch network and performing dot product operations. The computational complexity is far lower than repeatedly calling fully connected networks or physical simulations. When the predicted trajectories of different control candidate sequences need to be evaluated multiple times in each step of the MPC optimization, this structure can achieve millisecond-level fast evaluation, meeting the stringent real-time requirements of SOFC systems and providing computational power assurance for high-frequency rolling optimization.

[0057] Preferably, based on the input pre-system state and future control input sequence The deep operator neural network prediction model obtains the future output trajectory. : ;

[0058] Wherein, the predicted output voltage trajectory in step i We obtain it from the following formula:

[0059] ;

[0060] Where p is a positive integer, representing the number of basis functions, i.e., the latent dimension; This represents the future control input sequence of the branch network. The branch network coefficients obtained through encoding; This indicates the current system state of the backbone network in response to input. The prediction step size index i is encoded to obtain the backbone network basis functions.

[0061] With this setup, the prediction result is decomposed into the sum of the product of the system's inherent dynamic modes and the control modulation intensity, such that each term... It can correspond to specific physical processes. Compared to black-box fully connected networks, this structure facilitates the analysis of basis functions. The timing characteristics or coefficients The sensitivity to control variables can be used to diagnose whether the model behavior conforms to the electrochemical mechanism. In SOFC control, if a certain basis function exhibits a decay characteristic consistent with the diffusion time constant of the anolyte gas, it can be verified that the model captures the real physical process, thus enhancing the engineering credibility.

[0062] Preferably, S6 includes:

[0063] S61. At any sampling time k, obtain the current system state x of the SOFC system. k ; Call the S5 optimization framework, with x k Solve the corresponding finite-time optimization problem for the input to obtain the optimal control input sequence. ;Will The first control quantity in Apply to the SOFC system and proceed to the next sampling time k+1;

[0064] S62. At the next sampling time k+1, based on the acquired system state x k+1 The optimization problem of S61 is solved repeatedly and the control quantity is applied to achieve real-time tracking control of the SOFC output voltage.

[0065] This setup not only makes the rolling execution scheme a standard procedure for MPC, but also, due to its co-design with a high-performance predictive model, enables robust, real-time, and high-precision closed-loop control capabilities in highly nonlinear, slow-dynamic systems.

[0066] Preferably, the current system state x k This includes the partial pressure of hydrogen at the anode, the partial pressure of oxygen at the cathode, the partial pressure of water vapor at the anode, and the flow rate of hydrogen.

[0067] Preferably, in step S4, when training the deep operator neural network prediction model, an iterative optimization algorithm based on backpropagation is used to minimize the mean square error between the predicted future output voltage trajectory and the actual simulated output voltage trajectory corresponding to each group of data in the training dataset as the loss function for iterative optimization. Attached Figure Description

[0068] To make the objectives, technical solutions, and advantages of the invention clearer, the invention will now be described in further detail with reference to the accompanying drawings, wherein:

[0069] Figure 1 This is a flowchart of the method;

[0070] Figure 2 This is a schematic diagram of the deep neural operator architecture in Example 1. Detailed Implementation

[0071] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0072] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, not all of them. Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to represent selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0073] It should be noted that similar reference numerals and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures. In the description of this invention, it should be noted that the terms "center," "upper," "lower," "left," "right," "vertical," "horizontal," "inner," and "outer," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the figures, or the orientation or positional relationship commonly used when the product is in use. They are only for the convenience of describing the invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on the invention. Furthermore, the terms "first," "second," and "third," etc., are only used to distinguish descriptions and should not be construed as indicating or implying relative importance. In addition, the terms "horizontal," "vertical," etc., do not indicate that the component is required to be absolutely horizontal or suspended, but can be slightly tilted. For example, "horizontal" simply means that its direction is more horizontal than "vertical," and does not mean that the structure must be completely horizontal, but can be slightly tilted. In the description of this invention, it should also be noted that, unless otherwise explicitly specified and limited, the terms "set," "install," "connect," and "link" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0074] Example 1

[0075] like Figure 1 As shown, this invention provides a SOFC voltage control method based on a deep operator neural network, comprising the following steps:

[0076] S1. Establish a dynamic physical model of SOFC to describe the dynamic relationship between fuel flow rate, gas partial pressure and output voltage, and output the corresponding system state sequence and output voltage sequence through simulation calculation based on the given control input sequence and initial system state.

[0077] To facilitate a better understanding by those skilled in the art, the following explanation is provided.

[0078] SOFC generator principle

[0079] A typical SOFC structure consists of a porous anode and a porous cathode, separated by a dense solid ceramic electrolyte. This electrolyte efficiently conducts negatively charged ions at high temperatures. (During operation, oxygen molecules on the cathode side gain electrons from the external circuit and are reduced to oxygen ions. These oxygen ions then migrate through the electrolyte to the anode side, where they undergo an electrochemical reaction with hydrogen to produce water, simultaneously releasing electrons back into the external circuit, thus achieving a direct conversion of chemical energy into electrical energy. The corresponding electrochemical reaction is shown below:)

[0080] Negative level:

[0081] anode:

[0082] SOFC dynamic model construction

[0083] To reduce operating costs in practical industrial applications, hydrogen is typically produced through a methane reforming process. The dynamic characteristics of this fuel processing can be approximated by a first-order transfer function, the mathematical expression of which is shown below:

[0084]

[0085] in, This indicates the input hydrogen flow rate. Indicates fuel flow rate. Indicates the conversion factor; Let represent the response time constant of hydrogen gas in the reaction chamber, used to construct a first-order inertial dynamic model, and s be the Laplace transform operator. The denominator of the transfer function that constitutes the dynamic process of gas partial pressure;

[0086] The first in fuel cell stack The mass balance equation for each component can be established based on its inlet flow rate, outlet flow rate, and outlet molar concentration, and its expression is as follows:

[0087]

[0088] in, Indicates the first Inlet flow rate of each component Indicates the first The outlet flow rate of each component, Indicates the flow rate involved in the reaction; , and These represent the universal gas constant, the reaction chamber temperature, and the reaction chamber volume, respectively. This represents the instantaneous partial pressure of the i-th gas component within the anode or cathode reaction chamber;

[0089] The transfer function of hydrogen partial pressure in the anode pipe is as follows:

[0090]

[0091] Similarly, by performing a component balance analysis on oxygen and water vapor, the following set of equations can be obtained:

[0092]

[0093]

[0094] in, , , These represent the partial pressures of hydrogen, oxygen, and water vapor, respectively. and These represent the molar constant of the hydrogen valve and the response time constant of the hydrogen flow rate, respectively. and These represent the molar constant of the oxygen valve and the response time constant of the oxygen flow rate, respectively. and These represent the molar constant of the steam valve and the response time constant of the steam flow rate, respectively. Represents the reaction rate constant; This indicates the operating current of the SOFC.

[0095] The output voltage of the fuel cell stack is modeled by combining the Nernst equation and Ohm's law, while also considering Ohmic loss, concentration gradient loss, and activation loss. Its expression is shown below:

[0096]

[0097]

[0098]

[0099]

[0100]

[0101] in, Indicates the output voltage of the fuel cell stack; Represents the ideal standard electric potential. Indicates the maximum limiting current. and These represent the Tafel constant and the Tafel slope, respectively. Indicates ohmic loss; Represents electric potential; Denotes Faraday's constant; Indicates ohmic loss; Indicates activation loss; Indicates concentration loss; N represents the number of SOFC cells connected in series.

[0102] Compared to existing methods that rely solely on black-box identification or simplified steady-state models, the physical model employed in this scheme explicitly couples gas flow dynamics, thermodynamic equilibrium, and electrochemical polarization mechanisms, accurately reproducing the voltage dynamic response of SOFCs under varying operating conditions. This ensures strong physical consistency in the simulation data generated based on this model, providing high-quality supervision signals for the learning of deep operator neural networks, avoiding the propagation of prediction biases caused by model distortion, and fundamentally guaranteeing the reliability and generalization ability of subsequent MPC control. Furthermore, all dynamic equations are given in Laplace domain (s-domain) form, and the key variable is represented by the input flow rate. Rational fractional functions inherently possess a mapping structure from input function to output function, which highly aligns with the learning objective of deep operator neural networks—namely, learning the operator mapping from control input sequences to voltage trajectories. This modeling approach preserves physical interpretability while creating favorable conditions for efficient neural network training, significantly outperforming traditional state-space models that require discretization of the state space.

[0103] S2. Based on the dynamic physical model of S1, simulations are performed, and multiple sets of simulation data are collected to form a training dataset.

[0104] In practice, each set of simulation data includes: an initial system state, a future control input sequence, and a future output voltage trajectory calculated by the dynamic physical model based on the initial system state and the future control input sequence. In this way, by systematically traversing different initial states and control sequences, a simulation dataset covering a wide operating domain can be efficiently generated. Compared to traditional methods that rely solely on limited experimental data, this strategy can fully stimulate the nonlinear characteristics of the system in the offline phase, enabling the trained operator network to possess stronger dynamic capture capabilities and laying the foundation for stable tracking of MPC under varying operating conditions and strong disturbances.

[0105] S3. Construct a deep operator neural network prediction model to learn the nonlinear mapping relationship from the future control input sequence and the current system state to the future output voltage trajectory, replacing the forward prediction process based on a dynamic physical model. The deep neural operator architecture is as follows: Figure 2As shown.

[0106] In specific implementation, the deep operator neural network prediction model includes a branch network and a backbone network. The deep operator neural network prediction model is configured as follows: the future control input sequence is encoded through the branch network to obtain the branch network coefficients; the current system state and prediction step size index are encoded through the backbone network to obtain the backbone network basis function; and the branch network coefficients and the backbone network basis function are weighted and summed to generate the predicted value of the future output voltage trajectory.

[0107] Traditional end-to-end networks require simultaneous learning of a high-dimensional joint mapping between the input sequence and the initial state, making them susceptible to data distribution shifts. This approach, however, models the control action and system response separately: the branch network learns only how the control sequence modulates the response intensity, while the backbone network learns only the inherent dynamic modes of the system under different states. This decoupling makes the model more robust to changes in initial state or control strategy. For example, when system aging causes a slow drift in polarization characteristics, only the backbone network needs fine-tuning to adapt to the new basis functions, without retraining the entire network. Furthermore, due to strong parameter sharing, the number of samples required for training is significantly reduced, accelerating convergence. In addition, since the predicted output is a linear combination of coefficients and basis functions, once the backbone network's forward computation is complete, subsequent predictions of any control sequence only require rerunning the lightweight branch network and performing dot product operations, resulting in computational complexity far lower than repeatedly calling fully connected networks or physical simulations. In each optimization step of MPC, where the predicted trajectories of different control candidate sequences need to be evaluated multiple times, this structure can achieve millisecond-level rapid evaluation, meeting the stringent real-time requirements of SOFC systems and providing computational power guarantees for high-frequency rolling optimization.

[0108] S4. Train the deep operator neural network prediction model using the training dataset from S2.

[0109] In practice, when training the deep operator neural network prediction model, an iterative optimization algorithm based on backpropagation is used to minimize the mean square error between the predicted future output voltage trajectory and the actual simulated output voltage trajectory corresponding to each group of data in the training dataset as the loss function for iterative optimization.

[0110] S5. Using the deep operator neural network prediction model trained in S4 as the predictor, construct a model predictive control optimization framework to construct a finite-time optimization problem with the control input sequence of the next N steps as the optimization variable, given the current system state. The objective of the finite-time optimization problem is to minimize the tracking error between the future output voltage trajectory predicted by the deep operator neural network prediction model and the reference voltage trajectory, while minimizing the incremental change of the control input sequence.

[0111] To facilitate a better understanding of the technical content of S3-S5 of this method by those skilled in the art, the following explanation is provided.

[0112] Model predictive control

[0113] This paper combines Model Predictive Control (MPC) with the SOFC dynamic model to provide an advanced and systematic closed-loop control strategy for SOFC systems characterized by strong nonlinearity, time-varying parameters, and multivariable coupling. Its core mechanism lies in using the system dynamic model to describe the real-time dynamic evolution of the system and, based on the current system state, making rolling predictions of the system behavior over a future prediction time domain. Simultaneously, it generates the optimal control sequence online by solving a finite-time domain optimization problem containing process constraints. This method not only explicitly handles state and input constraints during system operation, ensuring system safety, but also proactively adapts to dynamic changes caused by factors such as changes in operating conditions, load disturbances, and system aging. Therefore, this method can achieve dynamic coordinated optimization of multiple objectives (such as power output and operating efficiency) in SOFC systems, and is an important theoretical tool for improving their dynamic performance, robustness, and service life.

[0114] In model predictive control, at each sampling time... The control input is calculated by solving a finite-time optimal control problem based on system prediction. Given the current system state... MPC in length is Minimize the following cost function in the prediction time domain:

[0115]

[0116] in, This indicates the predicted future output voltage trajectory; Indicates the reference voltage trajectory; This indicates the incremental change in the control input sequence; This represents the weighted square norm of the output tracking error, where the weight matrix Q is a positive real number; Let R represent the weighted square norm of the control input increment, where the weight matrix R is a positive real number.

[0117] Simultaneously satisfy the following constraints:

[0118]

[0119]

[0120]

[0121]

[0122] in, The physical model function of the system; This represents the predicted system state at the (i+1)th future step at sampling time k; This represents the predicted future system state at sampling time k; express The corresponding predictive control input; express The corresponding predictive control input, This represents the predicted system state at the (i-1)th future step at sampling time k; This represents the mapping function output by the deep neural network. The generated voltage prediction value; , These represent the physically feasible range of the control inputs; The table controls the upper limit of the amplitude constraint for the increment.

[0123] In model predictive control, the optimal control sequence contains only the first control input. It is applied to the system. At the next sampling time, the optimization problem is solved again using the updated system state measurements.

[0124] Regarding deep operator neural networks

[0125] To address the low computational efficiency of traditional model predictive control methods, this approach introduces deep operator neural networks to approximate the nonlinear input-output operators of the system. First, to describe the input-output evolution relationship of the dynamic system in predictive control, predictive operators are introduced. The operator Current system status and future control input sequence Mapped to the corresponding future output trajectory ,Right now:

[0126]

[0127] To achieve an efficient data-driven representation of this nonlinear operator, DeepONet is introduced to support the operator. Parameterized representation is performed. DeepONet consists of branch networks and a backbone network. The branch networks are used to control the input sequence. Encode and generate a set of coefficients. The backbone network, on the other hand, monitors the system status. and prediction step size index Encode and output a set of basis functions. Based on this decomposition structure, the first... The predicted output of the step can be represented as a weighted combination of the branch network output and the backbone network output, as shown in the following expression:

[0128]

[0129] in, This represents the latent dimension (i.e., the number of basis functions). This expression is used for the prediction operator. A computable alternative model is provided, which can be easily embedded into the optimization framework of Model Predictive Control (MPC) to achieve fast online trajectory prediction.

[0130] In the traditional MPC method, output prediction relies on the recursive computation of the nonlinear state transition equation and the output equation, i.e., equations (14)–(15). In this process, the future state… and output It is based on the system dynamics model and output function The result is obtained through step-by-step recursive calculation. This recursive prediction process constitutes the main bottleneck in the nonlinear MPC calculation of the SOFC system.

[0131] In contrast, in the control framework proposed in this invention, the prediction operator based on DeepONet learned in equation (18) is utilized. It can directly display the current system state. and future control input sequence Mapped to predicted output trajectory This replaces the repeated calculations of equations (14) and (15). Therefore, the MPC optimization problem in equation (13) still retains the original cost function structure and the input constraints in equations (16)–(17), while the prediction part based on the physical model is completely replaced by a data-driven alternative model. Through this reconstruction method, the online computational burden can be significantly reduced, and fast rolling time-domain control can be achieved without reducing the prediction accuracy.

[0132] S6. During system operation, repeatedly perform the following operations: obtain the current system state of SOFC, call the optimization framework of S5, solve the corresponding finite-time domain optimization problem based on the current system state, apply the obtained optimal control input to the system, and realize real-time tracking control of the output voltage.

[0133] In specific implementation, S6 includes:

[0134] S61. At any sampling time k, obtain the current system state x of the SOFC system. k ; Call the S5 optimization framework, with x k Solve the corresponding finite-time optimization problem for the input to obtain the optimal control input sequence. ;Will The first control quantity in Apply to the SOFC system and proceed to the next sampling time k+1;

[0135] S62. At the next sampling time k+1, based on the acquired system state x k+1 The optimization problem of S61 is solved repeatedly and the control quantity is applied to achieve real-time tracking control of the SOFC output voltage.

[0136] Wherein, the current system state x k This includes the partial pressure of hydrogen at the anode, the partial pressure of oxygen at the cathode, the partial pressure of water vapor at the anode, and the flow rate of hydrogen.

[0137] Traditional MPC (Multi-Level Policing) in SOFC (Self-Controlled Fiber-Road) control relies on high-dimensional nonlinear physical models for rolling predictions. Each optimization step requires multiple calls to the complex differential equation solver, resulting in long computation times and difficulty in meeting real-time control requirements. This solution utilizes an offline-trained deep operator neural network to replace the online physical simulation process, transforming the time-consuming numerical integral prediction into efficient neural network forward inference. This significantly reduces the prediction time in the MPC optimization loop, effectively solving the problem of insufficient real-time performance in strongly nonlinear systems. Furthermore, the deep operator neural network has the ability to learn the nonlinear mapping between input and output functions, accurately capturing the complex dynamic relationships between fuel flow rate, gas partial pressure, and output voltage in SOFC systems. Compared to traditional linearized models or simplified empirical models, this method fully preserves the original nonlinear and coupling characteristics of the system without sacrificing prediction accuracy, providing a more reliable prediction basis for subsequent optimizations.

[0138] This method, when constructing the MPC optimization framework, not only considers the tracking error of the output voltage to the reference trajectory but also minimizes the incremental change of the control input, effectively suppressing drastic fluctuations in the control signal. Compared to traditional PID or sliding mode control, which only focuses on error feedback while ignoring control smoothness, this method reduces the risk of actuator wear while ensuring tracking performance and naturally supports the explicit embedding of operational constraints such as the amplitude and rate of change of the control quantity, improving the overall robustness and safety of the system. Furthermore, this scheme decouples complex physical modeling from control design: the high-fidelity model is only used for offline data generation and network training, while only a lightweight neural network model is needed in the actual operation phase. This avoids the difficulty of deploying complex differential equation solvers on embedded controllers or edge devices, significantly improving the deployability and practicality of advanced control strategies in resource-constrained industrial scenarios.

[0139] This method can significantly reduce the online computational burden of model predictive control while ensuring high-precision prediction capability, thereby achieving high-precision, fast-response, robust, and real-time tracking control of SOFC output voltage that meets operational constraints.

[0140] Example 2

[0141] To better illustrate the effectiveness of this method, the following explanation is provided.

[0142] To verify the effectiveness and real-time feasibility of the DeepONet–MPC control strategy proposed in this paper for SOFC voltage tracking, a closed-loop simulation platform was built based on the SOFC dynamic model driven by physical mechanisms.

[0143] under constant load current Under the test conditions, fuel flow rate As a controlled input variable, the system state is: The system output is the terminal voltage. It is determined by the Nernst potential and polarization loss, and is described by equations (8)–(12). The sampling time is set to The predicted time domain length is .

[0144] All simulations were performed in Python 3.12. The training and inference processes of DeepONet were implemented using PyTorch, while the online model predictive control problem was built on the CasaADi platform. The related nonlinear programming problem was solved using the IPOPT interior-point optimization algorithm. This implementation allows for comparison of real-time computational performance between the proposed DeepONet-MPC method and traditional MPC frameworks.

[0145] Under the same operating conditions, the computation time and tracking error of the two methods were statistically analyzed, as shown in Table 1. The method proposed in this patent can obtain more accurate tracking results with only a shorter computation time.

[0146] Table 1 Comparison of computation time between this method and traditional methods

[0147] .

[0148] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit the technical solutions. Those skilled in the art should understand that any modifications or equivalent substitutions to the technical solutions of the present invention without departing from the spirit and scope of the present invention should be covered within the scope of the claims of the present invention.

Claims

1. A SOFC voltage control method based on deep operator neural networks, characterized in that, Includes the following steps: S1. Establish a dynamic physical model of SOFC to describe the dynamic relationship between fuel flow rate, gas partial pressure and output voltage, and output the corresponding system state sequence and output voltage sequence through simulation calculation based on the given control input sequence and initial system state. S2. Simulation is performed based on the dynamic physical model of S1, and multiple sets of simulation data are collected to form a training dataset. S3. Construct a deep operator neural network prediction model to learn the nonlinear mapping relationship from the future control input sequence and the current system state to the future output voltage trajectory, in order to replace the forward prediction process based on the dynamic physical model. S4. Train the deep operator neural network prediction model using the training dataset from S2. S5. Using the deep operator neural network prediction model trained in S4 as the predictor, construct a model predictive control optimization framework to construct a finite-time optimization problem with the future N-step control input sequence as the optimization variable, given the current system state. The objective of the finite-time optimization problem is to minimize the tracking error between the future output voltage trajectory predicted by the deep operator neural network prediction model and the reference voltage trajectory, while minimizing the incremental change of the control input sequence. S6. During system operation, repeatedly perform the following operations: obtain the current system state of SOFC, call the optimization framework of S5, solve the corresponding finite-time domain optimization problem based on the current system state, apply the obtained optimal control input to the system, and realize real-time tracking control of the output voltage.

2. The SOFC voltage control method based on deep operator neural networks as described in claim 1, characterized in that, In step S1, the dynamic physical model includes: ； in, This indicates the input hydrogen flow rate. Indicates fuel flow rate. Indicates the conversion factor; Let represent the response time constant of hydrogen gas in the reaction chamber, used to construct a first-order inertial dynamic model, and s be the Laplace transform operator. The denominator of the transfer function that constitutes the dynamic process of gas partial pressure; ； in, Indicates the first Inlet flow rate of each component Indicates the first The outlet flow rate of each component Indicates the flow rate involved in the reaction; , and These represent the universal gas constant, the reaction chamber temperature, and the reaction chamber volume, respectively. This represents the instantaneous partial pressure of the i-th gas component within the anode or cathode reaction chamber; ； ； ； in, , , These represent the partial pressures of hydrogen, oxygen, and water vapor, respectively. and These represent the molar constant of the hydrogen valve and the response time constant of the hydrogen flow rate, respectively. and These represent the molar constant of the oxygen valve and the response time constant of the oxygen flow rate, respectively. and These represent the molar constant of the steam valve and the response time constant of the steam flow rate, respectively. Represents the reaction rate constant; This indicates the operating current of the SOFC; in, Indicates the output voltage of the fuel cell stack; Represents the ideal standard electric potential. Indicates the maximum limiting current. and These represent the Tafel constant and the Tafel slope, respectively. Indicates ohmic loss; Represents electric potential; Denotes Faraday's constant; Indicates ohmic loss; Indicates activation loss; Indicates concentration loss; N represents the number of SOFC cells connected in series.

3. The SOFC voltage control method based on deep operator neural networks as described in claim 1, characterized in that, In step S5, the objective function of the finite-time optimization problem is: ； in, This indicates the predicted future output voltage trajectory; Indicates the reference voltage trajectory; This indicates the incremental change in the control input sequence; This represents the weighted square norm of the output tracking error, where the weight matrix Q is a positive real number; Let R represent the weighted square norm of the control input increment, where the weight matrix R is a positive real number.

4. The SOFC voltage control method based on deep operator neural networks as described in claim 3, characterized in that, In step S5, the constraints of the finite-time optimization problem include: ； ； ； ； ； in, The physical model function of the system; This represents the predicted system state at the (i+1)th future step at sampling time k; This represents the predicted future system state at sampling time k; express The corresponding predictive control input; express The corresponding predictive control input, This represents the predicted system state at the (i-1)th future step at sampling time k; This represents the mapping function output by the deep neural network. The generated voltage prediction value; , These represent the physically feasible range of the control inputs; The table controls the upper limit of the amplitude constraint for the increment.

5. The SOFC voltage control method based on deep operator neural networks as described in claim 1, characterized in that, In step S2, each set of simulation data includes: an initial system state, a future control input sequence, and a future output voltage trajectory calculated by the dynamic physical model based on the initial system state and the future control input sequence.

6. The SOFC voltage control method based on deep operator neural networks as described in claim 5, characterized in that, In step S3, the deep operator neural network prediction model includes a branch network and a backbone network. The deep operator neural network prediction model is configured to: encode the future control input sequence of the input through the branch network to obtain the branch network coefficients; encode the current system state and prediction step size index of the input through the backbone network to obtain the backbone network basis function; and perform a weighted summation of the branch network coefficients and the backbone network basis function to generate the predicted value of the future output voltage trajectory.

7. The SOFC voltage control method based on deep operator neural networks as described in claim 6, characterized in that, Based on the pre-system state of the input and future control input sequence The deep operator neural network prediction model obtains the future output trajectory. : ; Wherein, the predicted output voltage trajectory in step i We obtain it from the following formula: ； Where p is a positive integer, representing the number of basis functions, i.e., the latent dimension; This represents the future control input sequence of the branch network. The branch network coefficients obtained through encoding; This indicates the current system state of the backbone network in response to input. The prediction step size index i is encoded to obtain the backbone network basis functions.

8. The SOFC voltage control method based on deep operator neural networks as described in claim 1, characterized in that, S6 include: S61. At any sampling time k, obtain the current system state x of the SOFC system. k ; Call the S5 optimization framework, with x k Solve the corresponding finite-time optimization problem for the input to obtain the optimal control input sequence. ;Will The first control quantity in Apply to the SOFC system and proceed to the next sampling time k+1; S62. At the next sampling time k+1, based on the acquired system state x k+1 The optimization problem of S61 is solved repeatedly and the control quantity is applied to achieve real-time tracking control of the SOFC output voltage.

9. The SOFC voltage control method based on deep operator neural networks as described in claim 8, characterized in that, The current system state x k This includes the partial pressure of hydrogen at the anode, the partial pressure of oxygen at the cathode, the partial pressure of water vapor at the anode, and the flow rate of hydrogen.

10. The SOFC voltage control method based on deep operator neural networks as described in claim 1, characterized in that, In step S4, when training the deep operator neural network prediction model, an iterative optimization algorithm based on backpropagation is used to minimize the mean square error between the predicted future output voltage trajectory and the actual simulated output voltage trajectory corresponding to each group of data in the training dataset as the loss function for iterative optimization.

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