Network Graph Laplacian-Based Sensor Projection in System-Level Modeling of Liquid-Metal Loop

Abstract

Accurate modeling of nuclear reactor transients is important for improving operation and maintenance planning strategies in advanced reactors. An adequate understanding of the connectivity between various components and sensor signals can provide the means to project plant information even where sensors are not operational. This work presents a method for projecting temperature signals using network graph Laplacian and support vector regression (SVR).

A system model of a scaled liquid-metal-cooled reactor experiment facility is used to generate thermal-hydraulic data simulating cold-shock transients into the reactor plenum. Two different flow rates are investigated, 0.03 kg/s and 0.06 kg/s, describing the different degrees of nonlinear disturbances in two cases. The temperatures at 72 locations within the system are used to test the temperature prediction model. The inter-connectivity of these points is described using the network graph Laplacian, which describes a location in the loop as a node and then comprises a weighted matrix of connections between each of these nodes. Additional nodes, called ghost nodes, are used to model the heat transfer of the loop with the surroundings. The graph Laplacian and corresponding nodal sensing data are then used to train a kernel, which can inform the impact of various phenomena at one location in the plant on phenomena at other locations in the plant.

The sensing data and graph network of these nodes are used to construct a surrogate model of the liquid-metal loop. This model is then used to predict the behavior at certain nodes where the training data are not provided to the model. These model predictions are compared against the test data for the two different inlet flow rates. When optimized, the average error of the simulated temperature data remained below 5% when 3 of the 72 nodes are predicted using the graph-based SVR model. As the number of unknown sensors is increased, the root-mean-square error increases slightly but still remains below 2% when 24 of the 72 nodes are unknown or tested. The test or unknown sensor location plays a larger role than the number of unknowns, with sensor locations near ghost nodes and near the outlet pipes in the plenum having the largest error, with a maximum recorded error of 7%.

The objective of the graph-based SVR model is to not only capture the temperature or field variable accurately but also to capture the relative connectivity between the simulated sensors. The correlation coefficient matrix provides a scaled reference for the correlation between temperatures at two different node locations. The calculated correlation coefficient for the simulated temperature data and actual temperature data is within 5% for most of the system, with a maximum relative error of 15%.

Keywords:

• System-level modeling
• digital twins
• sensor projection
• graph Laplacian
• machine learning

Nomenclature

A	=	= cross-section area
b	=	= bias vectors
C	=	= regularization parameter
Cp	=	= heat capacity at constant pressure
d	=	= diameter
e	=	= unit vector
F	=	= volume force
f	=	= friction factor
h	=	= heat transfer coefficient
K	=	= kernel matrix
k	=	= thermal conductivity
L	=	= graph Laplacian
l	=	= length
Q	=	= heat
r	=	= correlation coefficient
S	=	= data set
SD	=	= standard deviation
T	=	= temperature
t	=	= time
u	=	= velocity field
w	=	= weight vector
Z	=	= pipe wall parameter

Greek

$\mathit{\gamma }$	=	= ghost node weighting factor
$\mathit{\theta }$	=	= unknown temperature
$\mathit{\lambda }$	=	= eigenvalue
$\mathit{\mu }$	=	= mean value
$\mathit{\xi }$	=	= slack variable
$\mathit{\epsilon }$	=	= insensitive loss parameter
$\mathit{\rho }$	=	= density
$\mathit{\sigma }$	=	= spatial constant
$\mathit{\tau }$	=	= temporal constant

Subscripts and Superscripts

0	=	= time of sensor failure
D	=	= Darcy
eff	=	= effective
ext	=	= exterior
g	=	= ghost
h	=	= hydraulic
hx	=	= heat exchanger
i	=	= node index
j	=	= node index

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by U.S. Nuclear Regulatory Commission’s Research & Development Program [award no. 31310021M0044].