---
title: Synthetic Batch-Level Pebble Measurements
url: https://www.emergentmind.com/topics/synthetic-batch-level-pebble-measurements
type: topic
---

# Synthetic Batch-Level Pebble Measurements

Synthetic batch-level pebble measurements are low-dimensional, noise-augmented constructs engineered to proxy the aggregate physical observables available from on-line gamma spectroscopy in pebble bed reactors (PBRs). Designed for use as features in machine learning models, particularly long short-term memory (LSTM) architectures, these measurements summarize the discharged pebble composition and activity over specified time windows. Their primary purpose is to provide physically-interpretable, computationally tractable input for predictive modeling of reactivity and flux evolution, notably under operational scenarios where direct, fully resolved measurement or simulation of each pebble is infeasible.

## 1. Formal Definitions and Physical Assumptions

The construction of synthetic batch-level pebble measurements relies on strict physical and statistical assumptions to reduce the dimensionality of the problem while maintaining sensitivity to the phenomena of interest.

### 1.1 Discharged-Pebble Burnup Distribution

At each depletion timestep, typically corresponding to ≈ 6.525 days (around 25,000 pebbles), discharged pebbles are binned into $G$ burnup intervals (nominally 12), uniform in percentage of fissions per initial metal atom (%FIMA). The true burnup $B$ of pebbles within bin $g$ is modeled as a Normal distribution:
$$
B \sim \mathcal{N}(\mu = \bar{b}_g, \sigma^2 = (c\, \bar{b}_g)^2)
$$
where $\bar{b}_g$ is the volume-averaged central burnup of bin $g$, and $c = 0.02288$, calibrated against high-fidelity (HxF) full-pebble depletion simulations. The fraction removed above a discard threshold $T$ is then:
$$
f_\text{removed} = P(B > T) = \int_T^\infty \frac{1}{\sqrt{2\pi}\, \sigma} \exp\left[ -\frac{(b-\mu)^2}{2\sigma^2} \right] db
$$

### 1.2 Volume-Averaged Nuclide Concentrations

For each burnup bin at discharge, average nuclide concentrations $\bar{C}_g^x$ are computed as pebble count-weighted means:
$$
\bar{C}_g^x = \sum_{i \in g} \left( \frac{N_i}{N_g} \right) C_i^x
$$
These bulk materials are then used as proxies for groupwise Serpent transport and depletion calculations, reducing the requirement for explicit tracking of millions of individual pebbles.

### 1.3 Defined Batch-Level Features

Within every timestep, five classes of dependent features are extracted:

1. **Burnup-bin counts** $\{N_b: b = 1 \dots 9\}$: Number of discharged pebbles in nine fixed %FIMA intervals (0–22.5% FIMA).
2. **Average total discharge burnup** $\bar{B}^\text{total}$: Mean burnup across all discharged pebbles.
3. **Number of discarded pebbles** $D$: Computed by summing the expected number of pebbles in each bin with $B_g > T$.
4. **Average last-pass burnup by radial zone** $B_r^\text{last}$: For $r = 1 \dots 4$, mean of pebbles' last radial pass burnup.
5. **Measurement noise model**: Zero-mean Gaussian noise is added to each synthetic feature, with standard deviations determined by a prescribed Mean Absolute Percentage Error (MAPE).

| Feature type                       | MAPE (%) |
|------------------------------------|----------|
| Burnup-bin counts                  | 5        |
| Average discharge burnup           | 2.5      |
| Number discarded pebbles           | 5        |
| Avg. last-pass BU by radial zone   | 10       |

The noise model is intended to emulate practical errors in gamma counting and inventory pathway assignment.

## 2. Generation Practices and Calibration Procedures

The procedure for producing synthetic batch-level measurements is predicated on a zone-based simulation of the reactor core using PEARLSim, combined with binning and noise augmentation strategies.

### 2.1 Zone-Based Core Simulation

The 280 MW gFHR core is decomposed into 4 radial and 10 axial zones, yielding up to 480 unique material regions across 12 burnup groups. Serpent 2.2.0 calculates, at each step, $k_\text{eff}$, power/flux meshes, and burnup group inventories. Pebble movement is modeled as an axial shift per interval, with top-zone pebbles rebinned upon discharge using the described statistical model for burnup.

### 2.2 Calibration Against High-Fidelity Simulation

The $\sigma$-factor $c$ is empirically determined by adjusting the width of the assumed Normal in burnup space such that the aggregate discard histogram at threshold $T = 180$ MWd/kgHM matches the reference curve from HxF simulations. This empirical anchoring is necessary to compensate for model error induced by the reduction to a zoned, batch-level formulation.

### 2.3 Observable Feature Extraction

At each batch (25,000 pebbles per window):

- Populate $N_b$ for 9 burnup bins.
- Compute $\bar{B}^\text{total}$, $D$, and $\{B_r^\text{last}\}$ using batch means and bin statistics.
- Add Gaussian noise per the feature-specific MAPE, scaled in accord with the measurement error model.

## 3. Encoding for LSTM Model Input

The constructed synthetic measurements are concatenated with other known or controlled parameters to form the input feature vectors for recurrent neural network training.

### 3.1 Input Vector Specification

Each timestep $t$ yields a 21-dimensional feature vector:

- User-controlled: fraction graphite, total core power, control rod depth, circulation rate ($\Delta t$), burnup threshold.
- Synthetic batch features: $\bar{B}^\text{total}$, $D$, $N_1$–$N_9$, $B_1^\text{last}$–$B_4^\text{last}$, and average power/pebble.

### 3.2 Time Series Assembly and Training

Input sequences are windowed to 8 timesteps, yielding an input tensor of shape: batch\_size × 8 × 21. Six LSTM models, trained in parallel, target specific outputs: excess reactivity ($\rho$), five principal components (PCs) each for power and flux profiles, and the next-timestep versions of the dependent input features themselves for operational forecasting.

## 4. Feature Importance and Predictive Sensitivity

Feature relevance for predictive model accuracy is established using a permutation-importance methodology.

### 4.1 Permutation Testing Protocol

For each trained model, baseline mean absolute error (MAE) is computed; features are independently permuted and MAE recomputed. The difference $\Delta$MAE provides a quantitative importance metric for each feature.

### 4.2 Reactivity Forecasting Insights

For the primary reactivity model, the three features with highest impact on MAE are: fuel-insertion fraction, $\bar{B}^\text{total}$, and $D$. High-burnup bins (7–9) also demonstrate elevated importance, in alignment with delayed-propagation reactivity phenomena in PBRs. Last-pass radial burnup attributes register low importance, plausibly due to collinearity with other bulk burnup metrics.

### 4.3 Dependency Patterns in Forecasting

Model predictions for next-timestep average discharge burnup are governed predominantly by the immediate prior value (self-predictability). Likewise, next-step discard count predictions hinge on the populations in high-burnup bins and the set discard threshold $T$. Last-pass radial burnup is most sensitive to total core power and average power per pebble, reflecting flux distribution effects over short windows.

## 5. Limitations, Sources of Uncertainty, and Recommendations

Although the synthetic batch-level approach delivers tractable and interpretable proxies for physical observations, several important limitations and sources of uncertainty are recognized.

### 5.1 Impact of Coarse Zoning and Binning

Restricting the decomposition to 4 radial and 12 burnup groups obscures pebble-level variance; higher fidelity (finer zoning or pebble-wise depletion) would capture more detailed behavior at ≥10–100× computational cost.

### 5.2 Assumption of Normal Distribution

The discard-burnup distribution is approximated as Normal, whereas empirical distributions may deviate, even after calibration via HxF data, introducing residual bias.

### 5.3 Noise Model Simplifications

The imposed Gaussian noise, scaled by fixed MAPE, offers at best an approximation of practical sensor error, omitting potential count-rate non-Poisson characteristics and pathway-specific biases.

### 5.4 Methodological Recommendations

For accuracy:

- Increase tallying efforts (histories, cycles) in Serpent calculations to suppress mesh-induced statistical variability, especially for higher-order PCs.
- Average over multiple random shuffles per timestep to mitigate shuffle variance in $k_\text{eff}$.
- Increase spatial and burnup binning granularity or calibrate bin widths using hybrid HxF→PEARLSim pipelines.
- Investigate alternative recurrent units (e.g., GRU) or physics-driven preprocessing (e.g., reduced-order $k_\infty$).

For deployment:

- Periodically retrain the synthetic model with gold-standard reference measurements (HxF, gamma counting) to correct drift.
- Utilize online permutation-importance metrics to monitor feature reliability and detect measurement drift (e.g., burnup-bin sensor bias).

## 6. Context, Significance, and Applications

Synthetic batch-level pebble measurements, as implemented in Kolaja et al. (2025) [2511.05118], serve as a working compromise between detailed, computationally prohibitive modeling and the need for physically rooted observables for data-driven prediction. Several key implications emerge:

- The reduction to batch-level, noise-augmented proxies permits efficient LSTM training and operation without requiring full-core pebble tracking.
- The retained dependency of model accuracy on physically interpretable quantities (e.g., high-burnup bin population, mean discharged burnup) confirms that the approach respects the dominant physical drivers of reactivity evolution.
- The framework’s explicit allowance for user-controlled and measured (uncertain) input features aligns with practical online monitoring and operation in advanced reactor systems.

This suggests that the batch-level synthetic measurement methodology provides a viable and realistic path for the incorporation of machine-learning-based forecasting into PBR operational support, so long as limitations regarding aggregation, noise representation, and bin calibration are systematically addressed.

Source: https://www.emergentmind.com/topics/synthetic-batch-level-pebble-measurements