March 13, 2015

Fig. 1: Fission Process in Nuclear Reactors, (Source:
Wikimedia
Commons) |

In the fission process in nuclear reactors, neutrons collide with U-235 atoms causing them to split into other products such as Kr-92 and Ba-141 while also releasing 2.5 new neutrons as seen in Fig. 1. This starts a chain of fission reactions that releases energy with each new fission event. To manage this chain reaction, control rods are used to absorb the neutrons and reduce new events where neutrons are created. Control rods are an integral part of nuclear reactors as they can be adjusted to control the amount of fission that occurs, which, in turn, control the amount of energy produced by the reactor. [1]

Different materials are used as control rods such as Cd-113 and B-10 which have thermal absorption cross-sections of 20,000 barns and 3,800 barns, respectively. Cadmium is usually used in the form of control rods while boron is used as control rods or is incorporated in water-cooled reactors in a soluble form. [2]

In this report, I will present two different methods to control the rods and thus the reactor's power generation. The first method is the conventional method that is currently used, it involves solving differential equations to calculate the neutron flux wanted and adjusting the parameter accordingly. The second method utilizes Machine learning techniques, which uses statistical methods to control the parameters needed to control the reactor's power generation.

Nuclear reactors produce power in a rate proportional to the number of fission processes (or equivalently, neutron formation) occurring per unit time. The neutron formation rate of change is given by [1,2]

d Φ dt |
= | (k-1) Φ τ |

where Φ is the neutron density, τ is the native neutron amplification rate, and k is the effective neutron multiplication factor, defined as [1]

k | = | (# of fission events in a given
generation) (# of fission events in previous generation) |

The value of k needed to have a steady state reactor that produces meaningful energy is 1 in which case the reactor is said to be in critical condition. A k value of less than 1 gives a subcritical reactor that cannot produce sustainable chain reactions. A k value greater than 1 produced a supercritical reaction in which its energy grows exponentially. [1]

Since it is very challenging to sustain a k value of 1, control rods are used to adjust for the time varying neutron flux as they can absorb any additional neutrons that would make the reactor unstable. [1]

Using feedback control systems and computer programs to solve the above equations, nuclear power plants operators can monitor the situation such that the safety requirements can be maintained. [3]

Machine learning is a statistical technique that has been proven in both classification and regression problems. There are many different machine learning algorithms such as linear regression, weighted linear regression, and Support Vector Machines (SVMs). The idea behind using Machine learning to use measured data to build models that can be used to classify or estimate certain problems without the need to write specific programs. [4] Machine learning has proved to be applicable in many areas including vehicle control and dynamics models which are usually constructed by solving differential equations similar to those governing nuclear reactors. [4]

In their paper, Bae et al. present a technique to use SVMs to estimate the power that is generated in a reactor based of different features. The SVM models are constructed with training data and later implemented on testing data from Multipurpose Analyzer for Static and Transient Effects of Reactor (MASTER). The models were then validated on the first fuel cycle of the Yonggwang Nuclear Power Plant Uni 3. [4]

Models are constructed by mapping a certain set of
features (commonly called inputs) with one certain value or
classification (commonly called outputs) using different statistical
techniques. The models are optimized in a way to avoid both under
fitting and over fitting of training data. Such optimization helps in
providing a general model that would give high accuracy for any set of
testing data. The models were constructed using 12 features: Reactor
power (%), Inlet temperature (°C), Pressure (bar), Mass flow rate
(kg m^{-3} sec^{-1}), Axial shape index, R1 through R5
control rod position (1 feature each) P control rod position, and SPND
signals (3 axial positions of core center). The output of the model was
the power peaking factor. [4]

The results presented in the paper show that when the
constructed model was used on the testing data, we obtained an average
RMS error of about 0.15% - giving very accurate results relative to
those found by MASTER. Bae *et al.* state that the prediction
intervals were very small and that "SVM regression models are accurate
enough for use in core protection and monitoring that uses power peaking
factor." [4]

This report presents two different methods to monitor and control reactor energy production and stability. The first method is presents the conventional method that is currently used. On the other hand, the second method explores the use of Machine Learning - a form of artificial intelligence in which a computer uses statistical methods to determine the power of the reactor.

The first method already has a proven record of safety since it is the conventional method used to control nuclear reactors. On the other hand, the second method shows that there is a potential to use statistical techniques to monitor and control nuclear reactors - a technique developed by a relatively new field of machine learning.

Despite the good results achieved by Bae *et
al.*, The question of long term reliability in an actual reactor
still need to be addressed. Will this method work well with other types
of reactors? Or will the results still give the same accuracy range for
the next cycles in the reactor or if some parameters change? Can we make
the model more accurate if we change some of the SVM parameters? All
these questions need to be researched and investigated extensively
before such technique can be adapted.

© Mahmood Alhusseini. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] J. Grayson, "Control Rods in Nuclear Reactors," Physics 241, Stanford Univerisity, winter 2011.

[2] D. Bodansky, *Nuclear Energy: Principles,
Practices, and Prospects, 2nd Ed.* (Springer, 2008).

[3] H. Mansour, H. M. Saad, and M. Aziz, "Analysis of
Reactivity - Initiated Accident for Control Rods Ejection," Journal
of Nuclear and Particle Physics **3**, 45 (2013)

[4] I.-H. Bae *et al.*, "Estimation of the Power
Peaking Factor in a Nuclear Reactor Using Support Vector Machines and
Uncertainty Analysis," Nucl. Eng. Technol. **41**, 1181 (2009).