GMHD -3D Power Reconstruction

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The Recommended Publication for Citing

1. Dongmin Yun, Deokjung Lee*, “GMDH-based 3-D Reactor Power Reconstruction for Increment of Operation Margin of MDNBR Calculation in Core Monitoring System,” KNS Spring Meeting, Jeju, South Korea, May 18-19 (2023). download paper 

Introduction

The Core Operating Limits Supervisory System (COLSS) is an important component of commercial reactor core monitoring systems (CMS), developed by Combustion Engineering, Inc.. It collects reactor coolant measurements and in-core neutron detector signals and calculates multiple core safety parameters in real-time. The COLSS conservatively calculates lumped one-dimensional axial power distribution and multiplies penalties to estimate safety parameter. Ivakhnenko developed the General Method of Data Handling (GMDH) that is the machine learning algorithm to build regression model. This study aims to model the 3-D Assembly Power Distribution (APD) and increase the margin of the most critical safety parameter of the CMS, the minimum Departure from Nucleate Boiling Ratio (MDNBR), by replacing the conservative penalty with model uncertainty. The training data for GMDH are produced using 3-D whole-core two step code STREAM/RAST-K, which has been developed in UNIST. The methods and results of each procedure, including input data acquisition, GMDH training, and uncertainty evaluation, have been explained. Two GMDH models have been developed: one for the 3-D assembly power distribution and the other for the hot-pin's power distribution (HPD). This paper also explains various ways to apply these regression models on the COLSS to increase the operational margin of MDNBR.

Feature

• The GMDH algorithm builds least-squared fitted trained parameters as a high-order polynomial model.

• The polynomial model is reconstructed from ICI signal to 3-D power distribution.

 - Input    : 45x5 ICI signal

 - Output : 177 x 24 Power distribution

• - The uncertainty quantification of the fitted polynomial model has been implemented

Methods

Data generation

      This section describes the script for perturbing RAST-K input files and extracting output data to train the GMDH model. The script operates in a Linux environment.

1. Input File Perturbation

The input file perturbation is conducted in the following sequence:

2. Scenario Specification (User Input)

The user can configure scenarios such as reactor type and unit, power range, measurement signal noise, burnup step, control rod asymmetry probability, selection of transient or steady-state operation, and control rod movement during transient operation. Training data will be randomly sampled from the input parameters within the user-selected scenario

3. Core Power Sampling within the Specified Power Range

4. Setting the Power-Dependent Insertion Limit (PDIL) for Control Rods

For each reactor type, the PDIL values specified in the NDR documentation are interpolated based on power levels

5. Sampling Control Rod Positions between Uninserted State and PDIL

6. Sampling Asymmetric Control Rod Positions

Once the positions of control rod banks are sampled in step (1.1.1.4), the position of one specific control rod assembly within the bank is sampled again based on the probability of control rod asymmetry, allowing for some cases with control rod asymmetry to be included in the training dataset

7. Perturbation Information Storage

To maintain consistent perturbations for a given scenario, the perturbation process from steps (1.1.1.2) through (1.1.1.4) is saved. When reproducing data, the stored perturbation information is utilized.

GMHD Method

1. Combinatorial GMHD

The Combinatorial GMDH algorithm (COMBI), introduced by Ivakhnenko, is the most fundamental GMDH algorithm. Given X-Y data with n variables across mmm cases, the data is divided into training samples (T-group) and validation samples (V-group) to form a linear system. Using Gauss or Cholesky Least Squares Estimation, the algorithm predicts Y. The predicted Y is then compared with the sampled data Y to define absolute error as the external criterion, and it searches for the Y that minimizes this error. For each input variable Xi, a coefficient vector a is generated, constructing the model as a polynomial. Since COMBI calculates the coefficient vector a for all input variables in a single layer, it has limitations on the available data volume mmm due to memory constraints

2. Multilayer Iterational Algorithms

The Multilayer Iterational GMDH algorithm (MIA), also introduced by Ivakhnenko, resembles the forward propagation structure of an artificial neural network. The method of determining the coefficient vector a is similar to COMBI. However, MIA defines partial polynomials as quadratic polynomials, where the output variable Y has dimensions of an (m×1) matrix, the coefficient vector a is (p×1), and the input variable X is an (m×p) matrix with fixed dimensions. By constructing multiple smaller layers and independently calculating the coefficient vectors for each partial polynomial, MIA can accommodate larger input data, enabling training with extensive datasets. This makes MIA suitable for use in reactor monitoring systems with numerous scenario cases. Figure 1 illustrates the MIA's layer-by-layer forward propagation structure, showing the partial polynomials and coefficient vectors at each layer. Since the partial polynomials are quadratic, the GMDH polynomial model expands to a degree k polynomial at each i-th layer 

GMHD scheme

Brief Results

3-D Power Reconstruction

Uncertainty Quantification