Stabilization of the stochastic point reactor kinetic equation through power feedback

Abstract

Most basic models for the power (or equivalently, the neutron population) in a nuclear core consider the power as a function of time (with an energetic and spatial distribution) and lead to deterministic description of the reactor kinetics. While these models are of common use and are undoubtedly the main analytic tool in understanding the reactor kinetics, the true nature of the power in a reactor core is stochastic and should be considered as a stochastic process in time. The stochastic fluctuations of the power around the mean field (which is given by the deterministic models) are referred to as "reactor noise", and understanding them is a basic topic in nuclear science and engineering. Traditionally, most models for reactor noise consider a sub-critical core, reaching steady state after exposure to an external source. The focus on a sub-critical setting is driven by two main factors. First, from a practical point of view, measuring the power fluctuations in a sub-critical core (known as "noise experiments") has proven to be a very efficient tool for estimating the static and kinetic parameters of the core. Second, once we assume a critical setting, the current models become statistically unstable, while the mean field solution has a stationary solution, the variance tends to ∞ linearly in time. The instability of the stochastic models is a known problem, and it has been conjectured in the past that this (some what strange) increase in the variance-that is not observed in physical systems-can be restrained by power feedback. However, this conjecture was never proven. The outline of the present study is to present a stochastic analysis to the point reactor kinetics model, proving that once the reactivity has a negative feedback, it not only forces a specific steady-state solution (in terms of the mean field equation), but also prevents the variance to "explode", and the variance is bounded in time.

Patchy nuclear chain reactions

“…As noticed by Williams 1 , the variance-to-mean (V/M) ratio of the neutron population in the core would diverge linearly with time and a critical reactor would undergo a "critical catastrophe", namely unbounded power fluctuations (eventually leading to shutdown). In practice, feedback mechanisms such as the temperature effect at high power or operator-induced control rod movements at low power, supposedly quench the critical catastrophe 1,9,28 . The bounded behavior of the variance-to-mean ratio observed during the experiments therefore calls for a different modeling.…”

"Prediction the Performance of the Iaea 10 Mw MTR Research Reactor at Transient Startup"

“…The dynamic model is developed using MATLAB/SIMULINK as shown in Figure (1). This model is based on; reactor point kinetics, with six groups of delayed neutrons, thermal hydraulic, one dimensional heat conduction created with lumped parameters and the feedback due to the; fuel and coolant temperature effects.…”

“…In zero-power reactors, the neutron flux density must be controlled, since it is proportional to the reactor power [1]. Ordinarily, these reactors do not virtually contain (or contains very small) number of free neutrons during shut down.…”

Advances in the physics and thermohydraulics of nuclear reactors

“…A more "fundamental" approach-based on the true stochastic nature of power generation in a reactor core-describes neutron kinetics as a time stochastic process: "reactor noise"-Fig. 1 Kyoto University Critical Assembly in a subcritical configuration consisting of stochastic fluctuations of the power around the mean field (which is given by the deterministic models)-becomes then the basic topic for nuclear engineering [5]. However, stochastic models have well-known stability problems in that even if the mean field solution is stable, its variance grows linearly in time; this feature is not observed in real measurements and a conjecture is examined assuming that reactivity feedback provides the necessary stabilization mechanism.…”