On a distinctive feature of problems of calculating time-average characteristics of nuclear reactor optimal control sets

This article deals with a feature of problems of calculating time-average characteristics of nuclear reactor optimal control sets. The operation of a nuclear reactor during threatened period is considered. The optimal control search problem is analysed. The xenon poisoning causes limitations on the variety of statements of the problem of calculating time-average characteristics of a set of optimal reactor power off controls. The level of xenon poisoning is limited. There is a problem of choosing an appropriate segment of the time axis to ensure that optimal control problem is consistent. Two procedures of estimation of the duration of this segment are considered. Two estimations as functions of the xenon limitation were plot. Boundaries of the interval of averaging are defined more precisely.


Introduction
The present advancement of nuclear power allows building nuclear power plants in areas of unstable climatic conditions or with an increased risk of seismic activity, as well as regions of political instability and terroristic threats [1,2].
Such power plants may undergo force majeure circumstances, which cause abrupt shutdown of nuclear reactors and their consequent idleness. The consumers of energy produced by the reactor may suffer damages induced by an unexpected shutdown and outage of the chief source of power.
The time when force majeure situation occurs can not be predicted precisely. There is usually a threatened period, which is declared beforehand, given by its duration and probability distribution for the event time. When the threatened event comes, nuclear reactors are shut down. Shutdowns at certain states of a reactor may cause long-term outages due to very high levels of xenon poisoning [2]. Taking the above into consideration, the problem of the search of the best reactor power control set during the threatened period is analyzed in this article.
Consider the time interval between the threatened period's start and the time of the reactor shutdown. The problem is to find the optimal control function, defined on this interval, which satisfies the conditions and minimizes the cost functional.
The cost functional is defined by some parameter, which depends on the state of the system and the control function. For example, the power generation loss before the reactor shutdown, as well as the reactivity margin or the concentration of xenon-135 in the reactor core at the time of the shutdown, etc. may be used as such parameters.
As the reactor control is impossible in case of exceeding the marginal xenon-135 concentration level inside the reactor core [3,4], the limitation of xenon-135 concentration before the reactor MPMM2016 IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 788 (2017) 012060 doi:10.1088/1742-6596/788/1/012060 1 shutdown is introduced. Also, as it was mentioned above, the reactor shutdown at some set of states may cause a long-term reactor outage. The long-term reactor outage due to xenon poisoning may cause damage to the infrastructure and industry. That is why the condition for the values of both iodine-135 and xenon-135 at the end of process is introduced in order to limit the concentration of xenon-135 after the reactor shutdown. Finally, it is stated, that after the reactor shutdown occurs, there is an additional time span, reserved for the elimination of the accident's consequences, when the reactor can't operate. Thus, the optimization problem statement conforms to the problem of optimal control on a limited interval with the limitation of xenon-135 concentration before shutdown and both iodine-135 and xenon-135 at the time of shutdown.
The threatened event time is random; it defines the duration of the process and the control function, which corresponds to the duration value. In practice, for the problem stated the optimal process can't be determined unless the period of control is given. Nevertheless, if the optimal control search problem is resolved in advance for any process duration value, the functions derived and their characteristics may be useful for applied control modes development.
However, there is a set of process duration values, for which the solution to the problem given doesn't exist.

Statement and proof
That is why issues occur, that are related to problems, which require existence of optimal control for any value of process duration. Problems of calculating time-averaged characteristics of optimal control are of that type. In the time axis, originating simultaneously with the start of the reactor control process, there exists the time interval, containing instants, when the reactor shutdown doesn't satisfy the conditions. Further this interval is denoted by τ  . The origin of this interval coincides with the origin of the time axis, the ending is defined by the parameters of the system and the limitations stated.
One-group point model of a nuclear reactor [5] was used in the research. The process of xenon poisoning is described by the system of differential equations [6] (1) Then for the case of the nuclear reactor after having been shut down, the system takes the following form Two ways of estimating the length of the interval were considered. The first way corresponds with the lower estimation.
Let be the origin of the threatened period and be the instant of the reactor shutdown. In these terms, if and the transients are described by (2). Let Φ be the set of admissible control functions is the time span of the control process. The set of states of the model (1) is represented by the quadrant O . Let A be the set of probable model states at time  , given Let B be the set of acceptable states at 1 t t  that is the end of the control process. In these terms, if , the problem is inconsistent. This is to be proved by the following.
To investigate the properties of the set B , consider the function ) , ( x i F over the quadrant Q ; the function is defined by (2) solution properties as follows. Let be the solution of (2) given initial conditions The equation describing the phase curve for the system (2) has the form: Let ) (i f be the right-hand member of this equation; note also that . Then the equiscalar lines of , , then there exists a circular area , then the construction of the set c B may be based on the following reasoning. In the coordinate system ixh O the tangential plane defined by the equation O phase plane projection of the intersection line between the tangential plane and the plane lim x h  may be used as a part of the boundary of the set c B (line Γ in figure 2). The sides of the polygon c A , estimating the set A , are formed by K estimations for the solutions of (1) denoted by  As results of the research carried two ways of estimation of the time interval observed were developed and applied and the dependencies of the estimations as functions of xenon limitation value were plot. The results obtained may be used in further research of the improvements to the problem statement considered in this article.
In particular, the following researches may include the problems of averaging the reactor control characteristics over the threatened period, which point of origin is known in advance.