Second-Law-Constrained Model Decomposition and Real-Time Simulation of Electrical–Thermal Coupling Systems in Nuclear Power Plants

High- and medium-voltage electrical–thermal coupling systems in nuclear power plants encompass multiple interacting physical processes, including electromagnetic transients, energy transfer, and thermodynamic dissipation [1, 2]. These systems exhibit pronounced multiscale characteristics, strong nonlinear interactions, and severe numerical stiffness. Real-time simulation serves as a core technological instrument for safety assessment, fault mechanism analysis, and control strategy validation in nuclear power plant systems [3-5], directly determining operational reliability and the effectiveness of disturbance mitigation. Current research on real-time simulation of complex energy systems has largely concentrated on numerical algorithm optimization [6, 7] and computational architecture enhancement [8, 9]. However, the fundamental physical constraints imposed by irreversible processes are frequently overlooked. As a consequence, deviations may arise between the simulation result and actual physical evolution. Non-equilibrium thermodynamics, as the foundational theory describing irreversible energy conversion and dissipation processes [10, 11], enables quantitative characterization of entropy variation, entropy production, and the evolution of energy quality within complex systems, offering a fundamental physical basis for the modeling and simulation of complex coupled systems. Reconstruction of the simulation paradigm around non-equilibrium thermodynamic principles not only enhances physical fidelity and intrinsic stability but also promotes a transition from mathematically driven simulation frameworks to physics-driven methodologies. Such a transformation is of substantial scientific significance and engineering value for the accurate simulation and safe operation of electrical–thermal coupling systems in nuclear power plants.

Existing studies on real-time simulation of electrical–thermal coupling systems predominantly rely on mathematical stiffness analysis as the principal criterion for model decomposition [12]. Decomposition boundaries are typically aligned with fixed physical interfaces [13], limiting adaptability to dynamically evolving disturbances. Boundary data exchange between subsystems often lacks unified physical constraints, which may induce distortion in energy transfer and thermodynamic inconsistency [14, 15]. Furthermore, the selection of simulation time steps is generally governed by numerical error control strategies, rather than by the intrinsic thermodynamic evolution of the system. Consequently, coordinated optimization between real-time performance and computational fidelity remains difficult to achieve [16]. Under such conditions, thermodynamic consistency cannot be rigorously guaranteed throughout the simulation process. When subjected to strong disturbances or far-from-equilibrium operating regimes, numerical instability and simulation failure may readily occur.

Four fundamental scientific challenges remain unresolved within existing real-time simulation frameworks. First, numerical stability in simulation and thermodynamic stability of the system are treated as independent constructs, and no unified mathematical representation or constraint framework has been established to couple them coherently [17]. Second, model decomposition of electrical–thermal coupling systems is conducted primarily on the basis of mathematical criteria, while quantitative decision metrics derived from non-equilibrium thermodynamics are largely absent [18]. Third, data exchange across heterogeneous subsystems does not systematically adhere to the physical laws governing irreversible processes, thereby preventing rigorous enforcement of entropy balance and energy transfer [19, 20]. Fourth, the simulation time scale is not aligned with the intrinsic thermodynamic relaxation characteristics of the system, making it difficult to satisfy the demands of refined simulation under far-from-equilibrium disturbances [21].

To address these limitations, the conventional paradigm in which the second law of thermodynamics is treated merely as a posteriori evaluation criterion is transcended. Instead, it is elevated to the status of a first-principles foundation and central organizing logic for real-time simulation. A non-equilibrium thermodynamics-driven real-time simulation methodology for high- and medium-voltage electrical–thermal coupling systems in nuclear power plants is thereby established. The principal innovations are manifested in three aspects. First, a dynamic model decomposition method based on entropy production rate spectrum analysis is developed, enabling adaptive identification of coupling boundaries and domain-partitioned solution of subsystem models. Second, an entropy-preserving data interaction mechanism grounded in the maximum entropy production principle is proposed to ensure physical consistency in energy and entropy transfer across heterogeneous subsystems. Third, a nonlinear variable-step solution strategy oriented toward thermodynamic time is designed, allowing dynamic matching between simulation step size and the intrinsic thermodynamic relaxation characteristics of the system. Through these developments, numerical stability and thermodynamic stability are unified at the physical level, and the stiffness challenges inherent in strongly coupled multiscale systems are fundamentally mitigated.

The remainder of this study is organized below. In Section 2, the second law of thermodynamics is reformulated as the first-principles foundation of the simulation framework, and a thermodynamically constrained manifold together with a unified stability characterization framework is constructed. Section 3 introduces the dynamic model decomposition method based on entropy production rate spectrum analysis, with explicit decomposition criteria and boundary identification algorithms. Section 4 establishes the entropy-preserving data interaction mechanism derived from the maximum entropy production principle and formulates the cross-domain subsystem interaction protocol. Section 5 presents the nonlinear variable-step solution strategy based on thermodynamic time and develops a collaborative simulation architecture for heterogeneous platforms. Section 6 provides validation under representative nuclear power plant operating conditions, and comparative analyses are conducted to demonstrate the performance advantages of the proposed methodology. Section 7 discusses the theoretical contributions, practical implications, and potential extensions. Section 8 concludes with a synthesis of the principal findings and core conclusions.

For the high- and medium-voltage electrical–thermal coupling system of a nuclear power plant, a distributed entropy production rate model is established. Analytical expressions for the local entropy production rate of key components are explicitly formulated, providing a quantitative foundation for constructing the entropy production rate spectrum. For the reactor core, the local entropy production rate is dominated by the irreversibility associated with nuclear heat release and convective heat transfer to the coolant. The corresponding analytical expression is written as:

${{\text{ }\!\!\sigma\!\!\text{ }}_{\text{core}}}\text{=}\frac{{{q}''}}{{{\text{T}}_{\text{fuel}}}}\text{-}\frac{{{q}''}}{{{\text{T}}_{\text{cool}}}}$                      (1)

where, q′′ denotes the heat flux at the fuel rod surface, T_fuel represents the fuel cladding temperature, and T_cool is the bulk coolant temperature. For the steam generator, entropy production arises from both heat transfer and flow dissipation. It is expressed as:

${{\text{ }\!\!\sigma\!\!\text{ }}_{\text{sg}}}\text{=}\frac{\text{Q}}{{{\text{T}}_{\text{1}}}}\text{-}\frac{\text{Q}}{{{\text{T}}_{\text{2}}}}\text{+}\frac{\text{ }\!\!\Delta\!\!\text{ }{{\text{P}}_{{{\dot{m}}}}}}{{{\text{T}}_{\text{avg}}}}$                     (2)

where, Q denotes the heat transfer rate, T₁ and T₂ are the average temperatures of the primary and secondary working fluids, respectively, ΔP represents the pressure drop, ${\dot{m}}$ is the mass flow rate, and T_avg is the average fluid temperature. For the high- and medium-pressure turbine stages, entropy production is governed by the irreversibility of steam expansion, i.e.,

${{\text{ }\!\!\sigma\!\!\text{ }}_{\text{turb}}}{=\dot{m}(}{{\text{s}}_{\text{1}}}\text{-}{{\text{s}}_{\text{2}}}\text{)-}\frac{{{{{\dot{W}}}}_{\text{act}}}}{{{\text{T}}_{\text{turb}}}}$                      (3)

where, s₁ and s₂ denote the specific entropy at the turbine inlet and outlet, respectively, ${{{\dot{W}}}_{\text{act}}}$ denotes the actual output power, and T_turb represents the average operating temperature of the steam turbine. The entropy production rate of the generator corresponds to electromagnetic and mechanical losses, and is expressed as σ_gen = P_loss/T_gen, where T_gen is the average stator temperature of the generator. The entire system is treated as a distributed parameter system. The control volumes of each component are discretized using the finite volume method, and the spatiotemporal evolution of the entropy production rate is obtained by solving the following governing equation:

$\frac{\partial \text{ }\!\!\sigma\!\!\text{ }}{\partial \text{t}}\text{+}\nabla \cdot \text{( }\!\!\sigma\!\!\text{ u)=}{{{\dot{ }\!\!\sigma\!\!\text{ }}}_{\text{source}}}$                       (4)

where, u denotes the working fluid velocity and ${{{\dot{ }\!\!\sigma\!\!\text{ }}}_{\text{source}}}$ represents the entropy production source term. Through this formulation, the entropy production rate spectrum of the entire system is constructed, enabling quantitative characterization of irreversible dissipation intensity across different regions. Figure 3 presents a schematic of the system entropy production rate spectrum and the principle of dynamic boundary layer identification.

An original quantitative criterion for coupling stiffness based on the magnitude of the entropy production rate gradient is proposed, thereby replacing conventional eigenvalue-based mathematical stiffness diagnostics with a physically grounded measure. The coupling stiffness K is defined as the magnitude of the spatial gradient of the local entropy production rate:

$K=\nabla \text{ }\!\!\sigma\!\!\text{ }=\sqrt{{{\left( \frac{\partial \text{ }\!\!\sigma\!\!\text{ }}{\partial \text{x}} \right)}^{\text{2}}}\text{+}{{\left( \frac{\partial \text{ }\!\!\sigma\!\!\text{ }}{\partial \text{y}} \right)}^{\text{2}}}\text{+}{{\left( \frac{\partial \text{ }\!\!\sigma\!\!\text{ }}{\partial \text{z}} \right)}^{\text{2}}}}$                                                              (5)

where, σ denotes the local entropy production rate, and x, y, z represent the spatial coordinates. The core innovation of this criterion lies in the direct linkage established between coupling stiffness and irreversible dissipation. A smaller value of K indicates a relatively smooth spatial distribution of entropy production, weaker interaction among physical processes, and proximity to local thermodynamic equilibrium. Conversely, a larger value of K signifies pronounced spatial variation in entropy production, intensified irreversibility in energy conversion between electrical and thermal domains, and strong coupling behavior. Through this formulation, the decision basis for model decomposition is elevated from a purely mathematical level to a physical level. As a result, the partitioning result remains consistent with the intrinsic thermodynamic evolution of the system, thereby preventing the physical distortions that may arise from conventional mathematically enforced decomposition.

A dynamic boundary layer adaptive identification algorithm is further constructed. By combining threshold discrimination of coupling stiffness with entropy wavefront tracking, real-time dynamic updating of decomposition boundaries is achieved. The critical coupling stiffness threshold K_cr is determined by fitting steady-state operational data of the system, satisfying ${{\text{K}}_{\text{cr}}}\text{=k}\cdot {{{\bar{K}}}_{\text{steady}}}$, where k is a correction coefficient (typically within the range 1.2–1.5), and K_steady denotes the average coupling stiffness under steady-state conditions. Based on this criterion, the system state is classified into near-equilibrium regimes (K≤K_cr) and far-from-equilibrium regimes (K>K_cr). When disturbances occur, regions exhibiting abrupt variation in entropy production propagate in the form of entropy waves. The propagation velocity of the entropy wavefront is defined as:

${{\text{v}}_{\text{wave}}}\text{=}\frac{\partial \text{ }\!\!\sigma\!\!\text{ /}\partial \text{t}}{\|\nabla \text{ }\!\!\sigma\!\!\text{ }\|}$                      (6)

The entropy wavefront position is tracked in real time by integrating the velocity:

$\text{r(t)=}{{\text{r}}_{\text{0}}}\text{+}\mathop{\int }_{\text{0}}^{\text{t}}{{\text{v}}_{\text{wave}}}\text{dt}$                    (7)

where, r₀ denotes the initial disturbance location. At each simulation step, the global distribution of coupling stiffness is evaluated, and the entropy wavefront is identified as a dynamic boundary layer. In this manner, the decomposition boundary is adaptively updated in synchrony with disturbance propagation, ensuring that partition interfaces remain precisely aligned with regions of concentrated dissipation.

Based on the dynamically identified boundary layers, a rigorous domain-adaptive modeling rule is established to achieve coordinated optimization between simulation accuracy and computational efficiency. Within near-equilibrium regions, the system approaches local thermodynamic equilibrium, and thermodynamic forces and fluxes approximately satisfy the Onsager reciprocal relations. Reduced-order models are therefore employed. On the thermal side, a lumped-parameter model is adopted, and the entropy production rate is simplified as:

${{\text{ }\!\!\sigma\!\!\text{ }}_{\text{lump}}}\text{=}\frac{\sum {{{{\dot{Q}}}}_{\text{i}}}}{{{\text{T}}_{\text{avg}}}}\text{-}\frac{\sum {{{{\dot{W}}}}_{\text{i}}}}{{{\text{T}}_{\text{avg}}}}$                     (8)

On the electrical side, a quasi-steady-state model is employed. High-frequency electromagnetic transient components are neglected, and only the evolution of the fundamental frequency component is computed. Within far-from-equilibrium regions, nonlinear effects become significant, and the Onsager reciprocal relations are no longer applicable. High-fidelity models are therefore adopted. On the thermal side, a reduced computational fluid dynamics formulation is applied, in which the three-dimensional Navier–Stokes equations coupled with the energy equation are solved to obtain the spatial distribution of entropy production. On the electrical side, a full-order electromagnetic transient model is employed. The Park transformation equations are solved, and entropy production associated with electromagnetic losses is computed accordingly. The adaptive rule dynamically matches the thermodynamic state of the system with model complexity in real time. Under the global entropy production non-negativity constraint, computational efficiency is maximized, providing high-efficiency and high-precision model support for real-time simulation.

An FPGA–CPU heterogeneous collaborative real-time solution architecture is constructed to fully exploit the complementary performance advantages of the two computing platforms, enabling efficient parallel solution of the electrical and thermal subsystems. On the electrical side, the parallel pipelined architecture of the FPGA is utilized to execute full-order electromagnetic transient simulations. A generator electromagnetic transient model is established based on the Park transformation equations, and hardware-level logic implementation enables sub-microsecond solution cycles. In this manner, abrupt entropy production variations induced by electromagnetic losses are captured. On the thermal side, a multi-core CPU parallel architecture is employed to execute a hybrid solution strategy combining a simplified computational fluid dynamics model and lumped-parameter models. The thermal system is partitioned into multiple component-level computational units, and a synchronized parallel solution is achieved through multi-core scheduling. This strategy ensures an effective balance between computational accuracy and efficiency. Data interaction between the heterogeneous platforms is realized through a high-speed communication bus, guaranteeing coordinated solution progress between the electrical and thermal subsystems. The FPGA–CPU heterogeneous collaborative real-time solution architecture is illustrated in Figure 4.

A thermodynamic clock manager governed by global entropy variation is designed to replace traditional CPU clock-cycle-based scheduling logic. Through this mechanism, simulation progression is deeply aligned with thermodynamic evolution. The global entropy variation ΔS is defined as the integral of the system-wide entropy production rate over a single simulation step:

$\text{ }\!\!\Delta\!\!\text{ S=}\mathop{\int }_{\text{t}}^{\text{t+h}}{{{\dot{S}}}_{\text{gen}}}\text{dt}$               (10)

where, ${{{\dot{S}}}_{\text{gen}}}$ denotes the global entropy production rate and h represents the current simulation time step. The thermodynamic clock manager computes ΔS in real time. When ΔS reaches a prescribed threshold ΔS₀, subsystem synchronization and boundary data exchange are triggered. If the threshold is not reached, the current step is dynamically extended until the entropy variation constraint is satisfied. By placing thermodynamic evolution at the center of the scheduling logic, the simulation trajectory remains intrinsically consistent with the physical evolution of the system, thereby avoiding the decoupling between simulation time and physical processes that may arise under conventional CPU-clock-driven scheduling.

A coordinated assurance mechanism for real-time performance and physical fidelity is further established to achieve optimal balance between these two objectives. This mechanism operates through three synergistic layers. Dynamic adaptation between step-size control and thermodynamic relaxation time, ensuring high-resolution simulation under far-from-equilibrium conditions and high computational efficiency in near-equilibrium regions. Heterogeneous task allocation, whereby the FPGA is dedicated to high-frequency electromagnetic transient computation and the CPU to parallel thermal computation, thereby maximizing computational resource utilization. Real-time verification and feedback adjustment, in which step-size suitability and entropy production consistency are continuously monitored. When deviations in real-time performance or thermodynamic consistency are detected, the step-size adjustment coefficient α and heterogeneous scheduling priorities are automatically modified. A real-time error ε and a physical fidelity deviation δ are defined, subject to the constraints ε≤10 μs and δ≤5%. Through closed-loop feedback regulation, coordinated optimization of real-time performance and thermodynamic fidelity is achieved. This integrated framework provides robust technical support for real-time simulation of electrical–thermal coupling systems in nuclear power plants.

Through deep integration of non-equilibrium thermodynamics and engineering real-time simulation, a paradigm shift in thermodynamics-driven numerical simulation has been achieved. The second law of thermodynamics, traditionally treated merely as a posteriori validation criterion, has been elevated to the status of a first-principles foundation and central organizing logic for real-time simulation of electrical–thermal coupling systems. By constructing a thermodynamically constrained manifold, a unified mathematical representation of numerical stability and thermodynamic stability has been established. The proposed original methodologies—including entropy production rate spectrum analysis, maximum entropy production principle–constrained interaction, and thermodynamic-time-driven variable-step control—systematically extend the application framework of non-equilibrium thermodynamics to the numerical evolution of distributed energy systems. The resulting paradigm is not restricted to a specific system architecture; rather, its fundamental criteria are applicable to a broad class of multiphysics coupled systems characterized by irreversible energy conversion. The universality of the second law of thermodynamics as a foundational physical constraint for simulation has thus been substantiated. Furthermore, a novel theoretical pathway has been opened for interdisciplinary research at the interface of non-equilibrium thermodynamics, computational mechanics, and numerical simulation.

The proposed methodology has been specifically developed to address the multiscale and strongly stiff characteristics of high- and medium-voltage electrical–thermal coupling systems in nuclear power plants. A practically deployable real-time simulation solution has thereby been formed, demonstrating substantial engineering value. The framework can be directly integrated into full-scope nuclear power plant real-time simulation platforms, enhancing both simulation fidelity and numerical robustness under steady-state and transient operating conditions. The method enables accurate reproduction of energy evolution and entropy production distribution under representative disturbance scenarios, such as grid short-circuit events and sudden steam load variations. Reliable thermodynamic data support is thereby provided for fault mechanism inversion and fault localization. In addition, a high-precision real-time validation environment can be established for coordinated control and fault-tolerant control strategies of nuclear generating units. By reducing on-site commissioning risks and associated costs, the framework offers significant support for safe, stable, and intelligent operation and maintenance of nuclear power plants.

The proposed methodology possesses clearly defined applicability boundaries and inherent limitations. Superior performance is maintained within the high- and medium-voltage sections of nuclear power plants under conventional energy conversion processes spanning near-equilibrium to far-from-equilibrium regimes. However, under extreme transient conditions characterized by intense phase transitions, strong multiphase flow coupling, or highly non-equilibrium shock phenomena, further refinement of local entropy production rate modeling is required. The performance of the framework depends on accurate real-time identification of key thermodynamic parameters and characteristic boundary temperatures of critical components. Measurement or estimation errors in these parameters may influence entropy production rate calculation and dynamic boundary identification accuracy. In addition, the present investigation focuses on electrical–thermal coupling systems in nuclear power plants. For complex multi-energy systems incorporating chemical energy storage, hydrogen energy conversion, and large-scale renewable integration, the entropy production rate models and interaction protocols would require adaptive reconstruction.

Future research will proceed along two principal directions. First, the thermodynamics-driven real-time simulation framework will be extended to multi-energy coupling scenarios, including wind–solar–nuclear–storage hybrid systems and integrated energy systems. In such contexts, entropy production rate formulations and dynamic coupling criteria across multiple energy forms will be reconstructed to enhance cross-scenario applicability. Second, deeper integration between non-equilibrium thermodynamics and digital twin technology will be pursued. Real-time entropy production monitoring and thermodynamic consistency verification will be embedded within the closed-loop evolution logic of digital twins, enabling both physical entities and virtual models to evolve in strict compliance with the second law of thermodynamics. Furthermore, machine learning techniques may be incorporated to enable rapid prediction of entropy production rate spectra and adaptive optimization of dynamic boundaries, thereby further improving the efficiency and precision of real-time simulation for complex coupled systems.