Shadowing effect using MCNP
Motivation
In nuclear reactors, the rod worth of control rods is not a fixed value. Instead, it is influenced by the shadowing effect, where control rods interact with each other. This phenomenon means that the effectiveness of one control rod can be reduced or "shadowed" by the presence and position of other nearby control rods. Current control rod calibration methods, including the Stable Period Method used at the Nuclear Engineering Teaching Laboratory (NETL), typically do not fully account for this complex shadowing effect.
Method
To address the limitations of traditional calibration methods, we secured 20,000 MCNP samples by systematically varying the positions of different control rods. From these extensive samples, a neural network was trained. Evaluation of the trained model allowed us to obtain continuous integral and differential rod worth curves. Additionally, we trained residuals using polynomial regression and Gaussian process regression, and these bias-corrected results are represented in the interactive graphs below.
Interactive Graphs
Explore the shadowing effect by moving the sliding bars below. For instance, when all control rods are initially at 0, the graph displays the differential rod worth as only the control rod corresponding to the title changes its position from 0 to 960, while the other three control rods remain at 0. If you adjust the control rod positions using the sliding bars or the up/down arrows on the right (e.g., all set to 480), the graph will then show the differential rod worth as the titled control rod varies from 0 to 960, with the other three control rods fixed at 480. Furthermore, the checkboxes below allow you to interactively view the actual control rod calibration graph measured at NETL, alongside the bias-corrected results from various models.
Model Explanations
NETL: NETL represents the rod worth measured at NETL using the Period method, assuming no shadowing effect. The Period method is a standard rod calibration technique: once the reactor is brought to criticality, approximately 20 cents of positive reactivity is inserted, and the subsequent increase in reactor power is monitored. Assuming that the differential rod worth follows a sinusoidal curve, three parameters can be fitted to the observed data, providing the basis for determining the control rod worth.
MCNP: MCNP is the result of training a Feedforward neural network that learned keff values from 20,000 randomly selected samples of control rod heights. This neural network has 3 hidden layers (256, 128, and 64 neurons respectively) and was trained using the ELU activation function. Early stopping was used to prevent overfitting.
Bias corrected (MCNP+offset): This model is a simple bias correction model that adds a constant offset to the MCNP results.
Bias corrected (MCNP+MLR): Any simulation is bound to show differences from actual measured values. Therefore, we created other models to correct the neural network trained from MCNP results using available measured values. MLR stands for Multiple Linear Regression, a simple linear model with 4 inputs.
Differential control rod worth
Integral control rod worth
Integral rod worth calculator
| Integral rod worth (¢) | Eigenvalue / reactivity | |||||||
|---|---|---|---|---|---|---|---|---|
| Tran | Shim1 | Shim2 | Reg | Total | keff | ρ | ρ ($) | |
| NETL | ||||||||
| MCNP | ||||||||
| MCNP+offset | ||||||||
| MCNP+MLS | ||||||||
Rod Heights during Calibration
| Calibration Set | Tran | Shim1 | Shim2 | Reg | Period (real-time) | Period (stop watch) |
|---|