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Low-fidelity and high-fidelity data

The LF model was made in the US NRC codes, Purdue Advanced Reactor Core Simulator (PARCS)19. This model consists of three different fuel bundles labeled each with varying uranium enrichment and gadolinia concentration. The model includes 560 fuel bundles encircled by reflectors. Along with the radial setup, there are 26 axial planes made up of 24 fuel nodes, plus a node of reflectors at the top and bottom planes.

In this work, the model was made in quarter symmetry to save computational time and further reduce the data complexity20. The symmetry was conducted in the radial direction only. The axial discretization was explicitly modeled from bottom to top of the reactor, from reflector to reflector. This is because BWRs axial variation is not symmetrical axially, so it is required to model it in sufficient detail. Based on this description, the boundary condition was set to be reflective in the west and north of the radial core and vacuum (zero incoming neutron currents) for the other directions.

For developing the ML model, the depletion steps were reduced to 12 steps, from the typical 3040 depletion steps. The PARCS cross-section library was generated using CASMO-4 for fuel lattices and reflectors. The library includes group constants from eight lattice simulations over control rod positions, coolant density, and fuel temperature. Lattices were simulated at 23 kW/g of heavy metal power density to a burnup of 50 GWd/MT of initial heavy metal.

The HF data were collected using Serpent21 Monte Carlo simulations. The model was created to reproduce PARCS solutions on the same core conditions but with higher resolutions and using the state-of-the-art simulation approach. This means no diffusion approximation and continuous energy neutron transport was modeled in detailed geometry structures. Each Serpent calculation was run on 500,000 particles, 500 active cycles, and 100 inactive cycles. The other simulation settings were also optimized for depletion calculations.

The reactor model used in this work is based on cycle 1 of the Edwin Hatch Unit 1 nuclear power plant. The power plant, located near Baxley, Georgia, is a boiling water reactor of the BWR-4 design, developed by General Electric, with a net electrical output of approximately 876 MWe and 2436 MWth of thermal output. Since its commissioning in 1975, Unit 1 has operated with a core design containing uranium dioxide fuel assemblies, utilizing a direct cycle where water boils within the reactor vessel to generate steam that drives turbines.

The specification of cycle 1 of Hatch reactor unit 1 is presented in Table 5. While it is a commercial, large power plant, Hatch 1 is not as large as a typical 1,000 GWe LWR. Some BWR designs also have about 700-800 assemblies. Nevertheless, due to the availability of the core design for this work, it is generally viable to use this model as a test case.

There are 560 fuel bundles the size of a 7(times)7 GE lattice in the Hatch 1 Cycle 1 model. Out of the number of fuel bundles in the cycle 1 core, there are three different types of fuels with varying enrichments and burnable absorbers. Using the procedures in running the Serpent model, high-resolution simulations were obtained as shown in the geometry representation in Fig. 6. In the figure, different colors represent different material definitions in Serpent. Because of how the materials were defined individually, the color scheme shown also varied from pin to pin and assembly to assembly. The individual material definition in the pin level was required to capture the isotopic concentration and instantaneous state variables at different fuel exposures and core conditions.

Geometry representation of the full-size BWR core modeled in Serpent. Images were generated by the Serpent geometry plotter.

There are 2400 data points collected as samples for this work with various combinations of control blade patterns and core flow rates and 12 different burnup steps. These data points are translated from 200 independent cycle runs for both PARCS and Serpent to provide LF and HF simulation data, respectively. The collected data were processed into a single HDF5 file.

The data processing parts are performed through data split procedures and data normalization. The data is separated into different sets, with a training-validation-test ratio of 70:15:15. The training data is used to teach the network, the validation data to tune hyperparameters and prevent overfitting, and the test data to evaluate the models generalization performance on unseen data. From the 2400 data points (200 cycles), the dataset was separated into:

Train Dataset: 140 runs or 1680 data points

Validation Dataset: 30 runs or 360 data points

Test Dataset: 30 runs or 360 data points

The data splitting process was not conducted randomly, but based on the average control blade position in a cycle run. Figure 7 presents the distribution of the average control rod inserted in the reactor. The maximum number of steps is 48 for fully withdrawn blades. In the plot, it can be inferred that the test data have the lowest average CR position (largest insertion), followed by the validation set, and the train data have the highest average CR position (smallest insertion).

Train-validation-test data split based on average control blade position in the BWR core. Image was generated using Python Matplotlib Library.

The CR-based splitting for the dataset has the purpose of demonstrating the generalization of the model on out-of-sample CR position data. On the other hand, random splitting is not preferred for small datasets, like this problem as the ML model tends to overfit (or imitate) the data. The fixed (CR-based) splitting process used here ensures that the model can perform well on data with a different distribution than the training dataset.

After splitting the data, normalization of the data is important for the ML model to ensure data integrity and avoid anomalies. In this context, the data processing employs Min-Max scaling, a common normalization technique, to rescale the features to a range [0, 1]. This is achieved by subtracting the minimum value of each feature and then dividing by the range of that feature. The scaling is conducted to fit the training data using the MinMaxScaler class from the scikit-learn package then apply the same scaling to the validation and testing data.

The target parameters used here are the core eigenvalue (or (k_{textrm eff})) and power distribution. The ML model will provide the correction (via predicted errors) of the target parameters that can be used to obtain the predicted HF parameters of interest. The perturbed variables are the parameters that are varied and govern the data collection process and in ML modeling. In this case, the perturbed variables are summarized in Table 6.

In this work, a neural network architecture, called BWR-ComodoNet (Boiling Water ReactorCorrection Model for Diffusion SolverNetwork) is built which is based on the 3D2D convolutional neural network (CNN) architecture. This means that the spatial data in the input and output are processed according to their actual dimensions, which are 3D and 2D arrays. The scalar data are still processed using standard dense layers of neural networks.

The architecture of the BWR-ComodoNet is presented in Fig. 8. The three input features: core flow rate, control rod pattern, and nodal exposure enter three different channels of the network. The scalar parameter goes directly into the dense layer in the encoding process, while the 2D and 3D parameters enter the 2D and 3D CNN layers, respectively. The encoding processes end in the step where all channels are concatenated into one array and connected to dense layers.

Architecture of BWR-ComodoNet using 3D-2D CNN-based encoder-decoder neural networks. Image was generated using draw.io diagram application.

The decoding process follows the shape of the target data. In this case, the output will be both (k_{textrm eff}) error (scalar) and the 3D nodal power error. Since the quarter symmetry is used in the calculation, the 3D nodal power has the shape of (14,14,26) in the x,y, and z dimensions, respectively. BWR-ComodoNet outputs the predicted errors, so there is an additional post-processing step to add the LF data with the predicted error to obtain the predicted HF data.

The output parameters from the neural network model comprise errors in the effective neutron multiplication factor, (k_{eff}), and the errors in nodal power, which is quantified as:

$$begin{aligned} begin{array}{l} e_{k} = k_H-k_L \ vec {e}_{P} = vec {P}_H-vec {P}_L end{array} end{aligned}$$

(4)

Here, (e_k) denotes the error in (k_{eff}) and (vec {e}_{P}) represents the nodal power error vector. The subscripts H and L indicate high-fidelity and low-fidelity data, respectively. According to the equation, the predicted high-fidelity data can be determined by adding the error predictions from the machine learning model to the low-fidelity solutions22.

Given the predicted errors, (hat{e}_k) and (hat{vec {e}}_{P}), the predicted high-fidelity data, (k_H) and (vec {P}_H) is defined as:

$$begin{aligned} begin{array}{l} k_H = k_L + hat{e}_k = k_L + mathscr {N}_k(varvec{theta }, textbf{x}) \ vec {P}_H = vec {P}_L + hat{vec {e}}_{P} = vec {P}_L + mathscr {N}_P(varvec{theta }, textbf{x}) end{array} end{aligned}$$

(5)

where (mathscr {N}_k(varvec{theta }, textbf{x})) and (mathscr {N}_P(varvec{theta }, textbf{x})) are the neural networks for (k_{eff}) and power with optimized weights (varvec{theta }) and input features (textbf{x}). Although Eq. 5 appears to represent a linear combination of low-fidelity parameters and predicted errors, itis important to note that the neural network responsible for predicting the errors is inherently non-linear. As a result, the predicted error is expected to encapsulate the non-linear discrepancies between the low-fidelity and high-fidelity data.

The machine learning architecture for predicting reactor parameters is constructed using the TensorFlow Python library. The optimization of the model is performed through Bayesian Optimization, a technique that models the objective function, which in this case is to minimize validation loss, using a Gaussian Process (GP). This surrogate model is then used to efficiently optimize the function23. Hyperparameter tuning was conducted over 500 trials to determine the optimal configuration, including the number of layers and nodes, dropout values, and learning rates.

The activation function employed for all layers is the Rectified Linear Unit (ReLU), chosen for its effectiveness in introducing non-linearity without significant computational cost. The output layer utilizes a linear activation function to directly predict the target data.

Regularization is implemented through dropout layers to prevent overfitting and improve model generalizability. Additionally, early stopping is employed with a patience of 96 epochs, based on monitoring validation loss, to halt training if no improvement is observed. A learning rate schedule is also applied, reducing the learning rate by a factor of 0.1 every 100 epochs, starting with an initial rate. The training process is conducted with a maximum of 512 epochs and a batch size of 64, allowing for sufficient iterations to optimize the model while managing computational resources.

It is important to note that the direct ML model mentioned in the results, which directly outputs (k_{eff}) and nodal power, follows a different architecture and is independently optimized with distinct hyperparameters compared to the LF+ML model. This differentiation allows for tailored optimization to suit the specific objectives of each model.

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Integrating core physics and machine learning for improved parameter prediction in boiling water reactor operations ... - Nature.com