Method for determining a coefficient of reactivity of a nuclear reactor

Abstract

The present description relates to a method (200) for determining a coefficient of reactivity (αl) of a nuclear reactor, the method comprising: - acquiring (202) a signal of temporally-serial reactivity data and signals of temporally-serial parameter data; - transforming (204) the temporally-serial data signals into a reactivity frequency signal and parameter frequency signals; - identifying (206) at least one parameter for which the frequency signal is correlated in a bandwidth with the frequency signal of another parameter and / or at least one parameter for which the frequency signal is uncorrelated in the bandwidth with the reactivity frequency signal; - deducing the coefficient of reactivity by linear regression (208) of the reactivity frequency signal explained by the frequency signals of the parameters, except for the at least one identified parameter.

Description

DESCRIPTION Method for determining the reactivity coefficient of a nuclear reactor This application claims priority from French patent application FR2411163 filed on October 15, 2024, entitled “Method for determining the reactivity coefficient of a nuclear reactor,” which is considered an integral part of this description within the limits provided by law. Technical field

[0001] This description relates generally to nuclear reactors, and in particular to the reactivity of nuclear reactors.

[0002] This description relates specifically to the determination of a reactivity coefficient, that is, a coefficient giving the change in reactivity relative to the change in an operating parameter of a nuclear reactor. Prior technique

[0003] A nuclear reactor comprises a core made up of fuel elements containing energetic fissile materials. The core is contained within a circuit called the primary circuit. A heat transfer fluid circulates within the core, from which it removes the thermal energy released by the fissile material's fission and transports the heat via a heat exchanger to a circuit called the secondary circuit, which typically includes a turbine and a generator and which generates electricity.

[0004] In water-cooled reactors, the coolant is water, which also acts as a neutron moderator, or moderator, in the core. A moderator is a substance that slows down neutron electrons. neutrons without absorbing them too much. The heat transfer fluid also generally serves to maintain the core temperature at a value compatible with the properties of the materials from which the core is made.

[0005] In pressurized water reactors (PWRs), the heat exchanger is integrated into a steam generator, which is generally part of the primary circuit. The steam generator transfers all or part of the heat from the primary circuit's heat transfer fluid to the water in the secondary circuit, transforming it into steam.

[0006] The primary circuit can be seen as a set of two systems coupled to each other: the core and the steam generator.

[0007] During power operation, the stability of this system is intrinsically ensured by the reactivity response ρ of the core to the temperature T of the heat transfer fluid, which acts as a neutron moderator, or moderator. In core physics, this response is expressed as a negative value of a reactivity coefficient C T at the temperature T of the heat transfer fluid.

[0008] Reactivity, expressed in pcm, per hundred thousand, is defined as a quantity that allows us to measure the deviation of a core from criticality.

[0009] Criticality characterizes a state of a medium or system where the production of neutrons by fission is likely to sustain a nuclear chain reaction. Criticality is quantified by an effective multiplication factor k eff which is less than, equal to, or greater than 1 depending on whether the medium or system considered is respectively subcritical, critical, or supercritical. For example, k eff is defined as a ratio of the total number of neutrons produced in a given fissile medium over an interval of time, to the total number of neutrons lost by absorption and leakage during the same time interval, all other things being equal.

[0010] The reactivity coefficient C is defined Ttemperature is defined as the change in reactivity ρ relative to the change in temperature T, generally the temperature of the heat transfer fluid, i.e.:

[0012] Thus, for a core with a reactivity coefficient C T negative, the response to an increase in temperature T results in a decrease in reactivity ρ.

[0013] In the rest of the document, the reactivity coefficient C T at temperature T can be designated, for simplicity, as the temperature coefficient.

[0014] More generally, a reactivity coefficient with respect to a parameter V (variable), which can be a parameter other than the temperature of the heat transfer fluid, is defined as the partial derivative of the reactivity with respect to this parameter, i.e.:

[0016] To date, the temperature coefficient C Tis measured accurately only in so-called isothermal and no-power configurations, for which the temperature is uniform over the entire core and steam generator, and other reactor parameters, such as primary circuit pressure, pump flow rates, moderator boron titer, are stable.

[0017] For example, there is a family of methods called time-series data / signal acquisition for determining a reactivity coefficient. These methods consist of conducting dedicated test campaigns, during In these tests, the reactor is shut down under critical conditions, with the core isothermal and at zero power. The reactivity coefficient is determined using temperature drift protocols. These protocols involve slowly drifting the core temperature in temperature steps, starting from a critical state at zero power. For each step, core criticality is restored by movements of the control mechanisms and / or by titrating the moderator with boron. Time-series signals, typically representing reactivity and temperature, are acquired during these operations. These protocols estimate the effects of temperature on reactivity by determining the effects of the control mechanisms and / or the moderator's boron titration on the return to core criticality at each temperature step.

[0018] Zero power means a neutron flux level (and therefore power) lower than the neutron flux level known as the "Doppler threshold", the Doppler threshold being the neutron flux level from which, during a divergence, the counter-reactions due to the temperature rise of the fuel are visible on the rate of increase of the neutron flux level.

[0019] These methods apply to a reactor at zero power. Indeed, during power operation, it is difficult to define an experimental protocol in which the core would be stabilized for a sufficient time at a given moderator temperature, without the other control parameters being modified.

[0020] An example of a time-series data acquisition method involves: - placing the reactor in a critical initial state, with a neutron flux level below the Doppler threshold and in having stabilized the core control parameters at defined protocol conditions, for example by adjusting the temperature and / or pressure of the moderator, the boron concentration of the moderator, the position of the control mechanisms; - the initial state being reached, acquire and archive the signals in time series with clocks synchronized between different measuring instruments, typically detectors or sensors of neutron flux, temperature, position of the control mechanisms, boron concentration (titration), pressure in the primary circuit (primary pressure); - modify the temperature of the moderator in predetermined temperature steps, the control parameters being adjusted to place the core in the critical state corresponding to each new temperature;- verify the stability of the critical state and temperature of the moderator at these control parameters, as well as, for example, the constancy of the boron concentration, primary pressure, and control mechanisms.

[0021] This family of methods more generally includes all methods which consist of implementing an experimental protocol requiring: - putting the reactor in a predetermined initial state, at zero power; - acquiring signals in time series; - making variations of a parameter, for example a temperature, whose influence on reactivity we wish to measure; - remaining in this state until the stability of this state and of all other parameters of the reactor, which would potentially influence the reactivity, is assured.

[0022] The time-series signals must then be interpreted and processed, a process that can be lengthy and tedious, and may even require additional tests or measurements.

[0023] These test protocols are therefore relatively long and restrictive for reactor operating schedules, with signal processing carried out later and which can also be lengthy.

[0024] Furthermore, these time series acquisition methods have the drawback of generating a significant amount of measurement noise that interferes with the signals being processed. Reducing this measurement noise may require the use of alternative experimental protocols to stabilize the parameters generating the noise, which could impose new constraints on reactor operating schedules.

[0025] During power operation, the reactor state in which the reactivity coefficient C is controlled T has an influence on the stability of the heart's reactivity, the reactivity coefficient C Tcannot be measured using the methods described above.

[0026] There is another family of methods called Fourier or Laplace transforms of signals, or neutron noise analysis methods, to determine a reactivity coefficient on a nuclear reactor in power.

[0027] For example, methods for estimating the temperature coefficient of reactivity in reactors operating at full power are developed using neutron noise analysis, particularly in boiling water reactors (E. LAGGIARD, J. RUNKEL, Evaluation of the moderator temperature coefficient of reactivity in a PWR by means of noise analysis. Ann. Nucl. Energy 1997, vol. 24, no. 5, 411–417), and in pressurized water reactors (O. AGUILAR, G. POR, Monitoring temperature reactivity). coefficient by noise method in an NPP at full power, Ann. Nucl. Energy 1987, vol 14 no. 10, 521-526). Patent application JPH0384496A (Method and instrument for measuring temperature coefficient of moderator of nuclear reactor) describes such a method.

[0028] In these methods, it is the heart's impulse response to temperature that is estimated. To do this, these estimation methods involve applying cross-power spectral density (CPSD) and auto-power spectral density (APSD) intercorrelation operators to neutron flux, reactivity, and temperature signals. The ratio of these operators allows for an estimation of the temperature coefficient C T .

[0029] These methods are based in particular on the following relationship:

[0031] Where ρ(t) is the core reactivity as a function of time, θ(t) is the moderator temperature as a function of time, and h θ (t) is the impulse response of the heart, or of the primary circuit.

[0032] Using Laplace or Fourier transforms, a transfer function H θ representing the transform of the system's impulse response can be estimated by the following two theoretically identical formulations:

[0033]

[0034] and ^^^^^^^^^^^^^^^^(^^^^^^^^)

[0035] ^^^^ ^^^^2 =^^^^^^^^^^^^^^^^ ( ^^^^^^^^,^^^^^^^^ )

[0036] However, this type of estimation method appears to lack numerical stability in real-world applications. In particular, the figures in the two publications cited above show little stability in the estimated transfer function. Furthermore, this method does not allow for a clear determination of the uncertainty in the estimated temperature coefficient, especially in the transfer function H. θ .

[0037] Furthermore, this type of estimation method requires a fairly long listening time to perform a neutron noise analysis. Measurement estimates can therefore only be obtained with a time lag, resulting in a near real-time measurement. However, during this listening period for neutron noise on an operating reactor, numerous parameters are likely to change. Their influence should therefore be taken into account and measured during the neutron noise analysis.

[0038] However, estimating a transfer function assumes that there is only one input signal for a single output. Therefore, this type of method, based on transfer function estimation, does not allow for the simultaneous estimation of other parameters influencing the reactivity coefficient without signal corrections based on prior knowledge or calculations.

[0039] Thus, in this type of estimation method, we find an estimate of the reactivity coefficient without measurement uncertainty, or confidence index, based on the assumption that other parameters do not influence the measurement of the reactivity coefficient.

[0040] There is a need for a method of determining a reactivity coefficient of a nuclear reactor that allows for the determination, or estimation, of a reactivity coefficient in reactor power operating configurations, i.e., under real conditions of use of the reactivity coefficient, including conditions under which the core is not isothermal.

[0041] It would be advantageous if the determination method allowed the determination of the reactivity coefficient at any parameter that can influence the reactivity of the reactor, which is not necessarily limited to a temperature.

[0042] It would be advantageous if the determination method allowed for obtaining a reactivity coefficient with very good accuracy.

[0043] It would be advantageous if the determination method allowed for the determination of several reactivity coefficients, each a function of a reactor parameter, for example a parameter that is not necessarily a temperature, and for example to determine if the different parameters are linked to each other in their influence on reactivity.

[0044] It would be advantageous if the determination method allowed for the calculation of measurement uncertainties for each determined reactivity coefficient. In particular, it would be advantageous if the determination method allowed for the estimation of uncertainties regarding the influence of the various parameters on reactivity.

[0045] It would be advantageous if the determination method could take advantage of existing measuring instruments in the nuclear reactor, i.e., if it did not require adding other measuring instruments.

[0046] It would be advantageous if the determination method did not disrupt the operation of the nuclear reactor and did not require shutting down a nuclear reactor during dedicated test campaigns. Summary of the invention

[0047] One embodiment overcomes all or part of the drawbacks of known methods for determining a reactivity coefficient.

[0048] An embodiment provides a method for determining, by a processing device, a reactivity coefficient of a nuclear reactor, the method comprising: - a step of acquiring time series data, and of storing time series data in a memory, said time series data comprising a time series data signal of nuclear reactor reactivity, and time series data signals of a set of parameters, the parameters being parameters of the nuclear reactor likely to have an influence on reactivity;- a step of frequency transformation of the time series data, so as to transform the time series data into frequency signals, the frequency signals comprising a reactivity frequency signal obtained by the frequency transformation of the reactivity time series data signal and frequency signals of the parameters obtained by the frequency transformations of the time series data signals of the set of parameters; - a step of identifying at least one parameter of the set of parameters whose frequency signal is correlated, in a bandwidth, with another frequency signal of another parameter of the set of parameters and / or at least one parameter of the set of parameters whose frequency signal is not correlated, in the bandwidth, with the reactivity frequency signal;- the deduction of the reactivity coefficient by a linear regression step of the reactivity frequency signal explained by the frequency signals of the parameters, except for at least one identified parameter.;

[0049] According to one embodiment, the identification step includes a frequency signal correlation step of the parameters, so as to identify whether at least one parameter of the parameter set has a frequency signal correlated, in the bandwidth, with another frequency signal of another parameter of the parameter set.

[0050] According to one embodiment, the correlation step includes the determination of correlation coefficients of the frequency signals of the parameters, two frequency signals of two parameters being correlated if the correlation coefficient of the two frequency signals of the two parameters is greater than a first threshold, for example at least one of the two parameters whose frequency signals are correlated is identified and / or the frequency signal of at least one of the two parameters whose frequency signals are correlated is removed.

[0051] According to one embodiment, the identification step includes a variance analysis step of the reactivity frequency signal explained by the frequency signals of the parameters, so as to identify whether at least one parameter of the parameter set has a frequency signal that is not correlated, in the bandwidth, with the reactivity frequency signal.

[0052] According to one embodiment, the analysis of variance step includes determining the variance proportions of the parameter frequency signals to the reactivity frequency signal. A parameter frequency signal is considered uncorrelated with the reactivity frequency signal if the variance proportion of the parameter frequency signal is less than a second threshold, for example, the parameter whose frequency signal is uncorrelated with The frequency signal of reactivity is identified and / or the frequency signal of said parameter is removed.

[0053] According to one embodiment, the identification step comprises: - a step of organizing the reactivity frequency signal in the bandwidth into a response vector, and the frequency signals of the parameters in the bandwidth into a matrix of explanatory variables, such that each column of the matrix of explanatory variables includes the frequency signal of one and the same parameter; - a step of reducing the dimension of the matrix of explanatory variables, so as to form a reduced matrix of explanatory variables, purged of at least one frequency signal of at least one identified parameter; the linear regression step being a linear regression of the response vector explained by the values ​​of the reduced matrix of explanatory variables, so as to estimate a parameter vector, such that: ^^^^ = ^^^^.where Y is the response vector, U is the reduced matrix of explanatory variables, and A is the parameter vector; the reactivity coefficient is deduced from the parameter vector.

[0054] According to one embodiment, the correlation step is a correlation of the columns of the matrix of explanatory variables.

[0055] According to one embodiment, the correlation step comprises constructing a correlation matrix from the matrix of explanatory variables using the following operation: ... whose extra-diagonal elements represent the correlation coefficients.

[0056] According to one embodiment, the analysis of variance step is performed using a linear regression model of the response vector explained by the values ​​of the matrix of explanatory variables.

[0057] According to one embodiment, the parameter vector is estimated by a least squares method, where a mean least squares estimator of the parameter vector is given by the equation: where  is the average estimator and U is the reduced matrix of explanatory variables.

[0058] According to one embodiment, the reactivity coefficient comprises at least one reactivity coefficient for at least one parameter, each reactivity coefficient being estimated by the formula: where α l is the reactivity coefficient and  l corresponds to a line l of the mean estimator Â.

[0059] According to one embodiment, an uncertainty value of the reactivity coefficient is estimated in the form of a variance coefficient, where the variance coefficient is a diagonal element of a covariance matrix of the mean estimator.

[0060] According to one embodiment, the acquisition step includes: - a time series data sampling step over an acquisition period; and / or - an iso-cadence time series data step at a common frequency.

[0061] According to one embodiment, the parameters include at least one temperature and / or pressure measured in the nuclear reactor while operating at power, for example: - a core outlet temperature (TSC); - a core inlet temperature (TEC); - a steam generator outlet temperature (TSGV); and / or - a primary pressure at the core outlet (PP).

[0062] According to a particular embodiment, at least one measured temperature corresponds to the temperature of a heat transfer fluid circulating in the nuclear reactor, for example the heat transfer fluid is also a neutron moderator fluid.

[0063] According to one embodiment, the acquisition step includes a reactivity transformation step of a time series data signal of neutron flux, measured in the nuclear reactor operating at power, into the time series data signal of reactivity, for example the reactivity transformation step implements a one-point kinetic equation inversion method.

[0064] One embodiment provides a processing device configured to implement the method described above.

[0065] One embodiment provides a system for determining a reactivity coefficient comprising: - the treatment device described above; - a reactivity measurement instrument connected to the treatment device; and - parameter measurement instruments connected to the treatment device.

[0066] One embodiment provides a computer program comprising instructions for implementing the method described above, when the program is executed by a processing device. Brief description of the drawings

[0067] These features and advantages, as well as others, will be described in detail in the following description of particular embodiments, given by way of non-limiting example, in relation to the attached figures, among which:

[0068] Figure 1A schematically represents a nuclear reactor and a treatment device according to one embodiment;

[0069] Figure 1B represents an example of the implementation of the processing device of Figure 1A;

[0070] Figure 2 illustrates in a simplified way a method for determining a reactivity coefficient according to one embodiment;

[0071] Figure 3A and Figure 3B illustrate an example of the implementation of a method for determining a reactivity coefficient;

[0072] Figure 4A, Figure 4B, and Figure 4C illustrate time-series reactivity signals for different time windows in a nuclear reactor operating at power; and

[0073] Figures 5A, 5B, 5C, and 5D illustrate frequency signals of reactivity, core outlet temperature, steam generator outlet temperature, and primary pressure in a nuclear reactor operating at power. Description of embodiments

[0074] The same elements have been designated by the same reference numerals in the different figures. In particular, structural and / or functional elements common to the different embodiments may have the same reference numerals and may have identical structural, dimensional and material properties.

[0075] For the sake of clarity, only the steps and elements necessary for understanding the described implementation methods have been shown and detailed. In particular, not all components of a nuclear reactor are detailed, nor is the operation of the nuclear reactor. Furthermore, not all the details of the calculations are provided, as they are within the grasp of a person skilled in the art.

[0076] Unless otherwise specified, when referring to two connected elements, this means directly connected without any intermediate elements other than conductors, and when referring to two coupled elements, this means that these two elements can be connected or linked through one or more other elements.

[0077] In the description that follows, when referring to absolute positional qualifiers, such as the terms "front", "back", "top", "bottom", "left", "right", etc., or relative positional qualifiers, such as the terms "above", "below", "superior", "inferior", etc., or to orientational qualifiers, such as the terms "horizontal", "vertical", etc., unless otherwise specified, it refers to the orientation of the figures.

[0078] Unless otherwise specified, the expressions "approximately", "roughly", "approximately", and "on the order of" mean to within 10% or 10°, preferably to within 5% or 5°.

[0079] Throughout this description, when a flux is referred to, unless otherwise specified, it refers to a neutron flux. In general, flux denotes a measure of neutron flux, which characterizes a neutron reaction rate and is usually determined by a counting rate measured by a neutron detector.

[0080] In the description that follows, when a reactor is referred to, unless otherwise specified, it refers to a nuclear reactor.

[0081] In the following description, when a method of determination is referred to, unless otherwise specified, as a method for determining, or estimating, a reactivity coefficient of a nuclear reactor with one parameter, or several reactivity coefficients with several different parameters. In the remainder of the description, a parameter that may influence reactivity may also be represented, for example in the space of complex numbers, by an explanatory variable, or shorthand variable.

[0082] Figure 1A schematically represents a nuclear reactor 100 and a treatment device 120 according to one embodiment. The nuclear reactor 100 is typically a pressurized water reactor (PWR).

[0083] The nuclear reactor 100 comprises a core 101 (CO) in a vessel 102, or primary reactor vessel. The vessel 102 contains a heat transfer fluid 103 which is pressurized within the vessel 102 by a pressurizer 104. The heat transfer fluid 103 is typically water in the case of a PWR, where it also acts as a neutron moderator. In the following description, it will be referred to as the moderator fluid 103.

[0084] The nuclear reactor 100 also includes a steam generator 105 (SG) connected to the vessel 102 via a circuit 106.

[0085] The circuit 106 is for example a closed circuit which is adapted to circulate the heat transfer fluid 103 between an outlet 102a of the tank 102, through which the heat transfer fluid 103 exits the tank 102, and an inlet 102b of the tank 102, through which the heat transfer fluid 103 re-enters the tank 102, passing through the steam generator 105, as indicated by the arrows along the circuit 106.

[0086] In the example in Figure 1A, the moderating fluid 103 flows from the bottom of the heart 101 to the top of the heart 101, as indicated by the arrow in the heart 101.

[0087] The circuit 106 is connected to a circulation pump 107 adapted to circulate the moderator fluid 103 between the outlet 102a and the inlet 102b of the tank 102.

[0088] By heat exchange, the steam generator 105 transfers calories from the heat transfer fluid 103 to water in a secondary circuit to transform it into steam.

[0089] The secondary circuit includes an alternator 108 connected to the steam generator 105 by a circuit 109 in which the water from the secondary circuit circulates.

[0090] The circuit 109 is for example a closed circuit which is adapted to circulate the water of the secondary circuit, between an outlet 105a of the steam generator 105, through which the (vaporized) water of the secondary circuit exits the steam generator 105, and an inlet 105b of the steam generator 105, through which the water of the secondary circuit re-enters the steam generator 105, via the alternator 108.

[0091] Nuclear reactor 100 includes various measuring instruments, in particular: - a temperature sensor 111 positioned on the circuit 106 at the outlet of the GV 105, to measure a GV outlet temperature (GVAT); - a pressure sensor 112 positioned in the vessel 102, for example at the outlet of the core 101, to measure a primary pressure (PP); - a temperature sensor 113 positioned in the vessel 102 at the inlet of the core 101 to measure a core inlet temperature (CIT); - a temperature sensor 114 positioned in the vessel 102 at the outlet of the core 101 to measure a core outlet temperature (CIT), which is also the GV inlet temperature; - a neutron detector 115 positioned outside the core 101, for example around the vessel 102, allowing monitoring of the neutron flux level (NF) in the core.

[0092] Other measuring instruments, not shown in Figure 1A, may be included in nuclear reactor 100 in a manner known to those skilled in the art. Furthermore, the locations of the various measuring instruments are indicated in Figure 1A for guidance purposes. For example, the neutron detector is not necessarily positioned outside the reactor vessel and core; it may be located inside the vessel or even within the core. Moreover, there may be several neutron detectors, for example, one outside the core or vessel and another inside the core or vessel.

[0093] For example, the measured temperatures correspond to the temperature of the moderator fluid 103 at the different locations where the temperature sensors are positioned.

[0094] When operating at power, the temperature of the moderator fluid 103 rises when it passes through the core 101 and decreases when it passes through the steam generator 105.

[0095] Temperature sensors, for example, allow us to quantify a variation ΔT C of the temperature in the heart 101 (ΔT C (TSC-TEC), a variation ΔT GV of the temperature in the GV 105 (ΔT GV =TSC-TSGV). They also allow us to determine an average temperature TM in the core which can be estimated by the formula TM= (TEC+TSC) / 2.

[0096] During power operation, when the energy extracted from the secondary circuit by alternator 108 is constant and the parameters of reactor 100 are stabilized, the following phenomena, or phenomenology (Table 1), can be considered in a simplified manner: - starting from time t=t0, where TSC=tsc0, TEC=tec0, TSGV=tsgv0, PP=pp0, and NF=nf0; - at t=t1 greater than t0, a power demand on alternator 108 leads to a decrease in the enthalpy of the secondary of GV 105, which results in an increase in ΔT GV- at this instant, with the temperature TSC fixed, it is the temperature TSGV that decreases, taking a value tsgv1 lower than tsgv0; - at t=t2 greater than t1, the moderator fluid 103 at the outlet of the GV 105 circulates in the pump 107 and is injected at the inlet of the core 101: the temperature TEC takes a value tec2 lower than tec0; the average core temperature TM will also decrease, which will lead to an increase in reactivity as indicated below; - at t=t3 greater than t2, the moderator fluid 103 at a lower temperature than at t0 (before the loop) passes through the core 101; the reactivity increases and an increase in the neutron flux level NF is observed, which takes a value nf3 greater than nf0; the increase in reactivity leads to an increase in the core power; - at t=t4 greater than t3, the increase in heart power leads to an increase in ΔT CSince the TEC temperature is fixed, it is the TSC temperature that increases; this temperature increase also results in an increase in the primary pressure PP, which takes on a value pp4 greater than pp0, the latter can be balanced by the level of the pressurizer.

[0097] [Table 1] t TEC TSC TSGV PP NF t0tec0tsc0tsgv0pp0nf0t1tec0tsc0tsgv1 <tsgv0pp0nf0t2tec 2< tec0tsc0tsgv1pp0nf0~tsgv1t3tec2~tsgv1tsc0tsgv1pp0nf3>nf0t4tec2~tsgv1tsc4>tsc0tsgv1pp4>pp0nf3

[0098] It can be noted that, in this example, the reactivity response following the power demand at t=t1 is observed on the neutron flux NF later at t=t3 and on the core exit temperature TSC even later at t=t4. We therefore expect to observe a delay in the dynamics between the TSGV, NF, and TSC signals. The use of Z-transforms (Fourier or Laplace) of the signals, described later, advantageously allows us to account for this delay through the phases. Indeed, the use of Z-transforms allows us to decompose the signal into magnitude (amplitude) and phase (or phase shift) and to process these two pieces of information (magnitude and phase) together, which makes it possible to obtain a coefficient that takes these phase shifts into account without underestimating the coefficient.

[0099] The processing device 120 is connected to the temperature and pressure sensors 111, 112, 113, 114 and to the neutron detector 115. The processing device 120 is adapted to implement the method for determining a reactivity coefficient, and to deliver one (or more) reactivity coefficient(s) α l , based on data from measurements taken by measuring instruments.

[0100] Figure 1B represents an example of an embodiment of the processing device 120 of Figure 1A. In this example, the processing device 120 is based on a processor P. The processor P is adapted to execute a computer program comprising instructions for implementing the determination method described below.

[0101] The processing device 120 also includes, connected to the processor P: - an instruction memory (INSTR_MEMORY) containing the instructions of the computer program; - an input / output interface (I / O INTERFACE) allowing in particular the retrieval of data from the measurements of the measuring instruments of the nuclear reactor 100; - a user interface (USER INTERFACE), which allows for example the processing device 120 to interface with a user, for example an operator of the nuclear reactor 100; and - a data storage (DATA_MEMORY), allowing for example the storage of the determined or estimated reactivity coefficients.

[0102] This example is not exhaustive, and a person skilled in the art may consider other processing devices. For example, it could be a hardware-based processing device, such as an application-specific integrated circuit (ASIC) or a field-programmable gate array (FPGA).

[0103] In the following description, we will describe examples of methods for determining one (or more) reactivity coefficient(s) α l implemented by the treatment device 120. The examples described aim to determine the influence of a set of parameters, including the different temperatures TEC, TSC, TSGV and the pressure PP, on the reactivity ρ, deduced from the neutron flux measurement NF, and to estimate at least one reactivity coefficient to at least one parameter from the set of parameters.

[0104] Figure 2 illustrates in a simplified way a method for determining a reactivity coefficient α l according to a particular embodiment.

[0105] The determination method 200 comprises: - a step 202 (ACQUISITION DATA STORAGE) of time series data and of storing the time series data in a memory: this time series data includes a time series data signal of the reactivity of the nuclear reactor, and time series data signals of a set of parameters, the parameters being parameters of the nuclear reactor likely to have an influence on the reactivity; - a step 204 (FREQUENCY TRANSFORM) of the time series data into frequency: the time series data is transformed into frequency signals, the frequency signals comprising a frequency signal of reactivity (transform of the time series data signal of reactivity) and frequency signals of the parameters (transforms of the time series data signals of the set of parameters);- an identification step 206 (IDENTIFICATION) configured to identify, within a defined bandwidth: at least one parameter from the parameter set whose frequency signal is correlated with another frequency signal from another parameter in the parameter set; and / or at least one parameter from the parameter set whose frequency signal is not correlated with the reactivity frequency signal; and; - the deduction of the reactivity coefficient α l by a linear regression step 208 (LINEAR REGRESSION) of the frequency signal of reactivity explained by the frequency signals of all parameters except at least one identified parameter.

[0106] In other words, linear regression 208 is performed on a reduced set of parameters, purged of the parameter(s) identified in the identification step 206. This allows for a more precise estimate of the reactivity coefficient, as explained later.

[0107] Figure 3A and Figure 3B illustrate an example of the implementation of a method for determining a reactivity coefficient.

[0108] Figure 3A illustrates in particular a phase of signal acquisition and pre-processing, which is followed by a phase, or process, of adjustment 340 illustrated in more detail in Figure 3B.

[0109] The illustrated 300 determination method includes the steps described below.

[0110] An acquisition step 301A acquires and stores signals 310, 315 in the form of time-series data from measurements taken by various measuring instruments: - a time series 311 of steam generator outlet temperature measurements (TSGV(t)) by temperature sensor 111; - a time series 312 of primary pressure measurements (PP(t)) by pressure sensor 112; - a time series 313 of core inlet temperature measurements (TEC(t)) by temperature sensor 113; - a time series 314 of core outlet temperature measurements (TSC(t)) by temperature sensor 114; and - a time series 315 of neutron flux measurements (NF(t)) by neutron detector 115.

[0111] Temperature and pressure correspond to a set of parameters.

[0112] In a reactivity transformation step 301B (FLUX-REACTIVITY TRANSFORMATION), the neutron flux time series signal 315 (NF(t)) is transformed into a reactivity time series signal 325 (ρ(t)). This reactivity transformation can be performed using a reactivity meter. It may consist of, or include, an inversion of the kinetic equations, for example, at 6 or 8 precursor groups. This reactivity transformation step thus allows the acquisition of reactivity time series data.

[0113] In a sampling step 302 (TIME WINDOW ACQUISITION), from the time series 310, 325, a sliding period of a defined duration is defined, typically a few sliding hours, for example 5 to 6 hours, constituting a time acquisition window, on which the time series 310, 325 can be stored.

[0114] The duration of the acquisition time window can condition the lowest frequencies, or the slowest transients, which can be observed and processed later by transformations in the frequency space.

[0115] The different time series are not necessarily timed with the same frequencies, or even necessarily timed in a regular manner.

[0116] Thus, we can anticipate an iso-timing step 303 (ISO-TIMING OF SIGNALS) which consists of (re)sampling and timing the signals in series time-domain at a common frequency, and this, for example, to be able to perform a Fast Fourier Transform (FFT) on the signals.

[0117] The choice of this common frequency can condition the highest frequencies, or the fastest transients, which can be observed and processed later by transformations in the frequency space.

[0118] The common frequency can, for example, be between 50 Hz and 500 Hz, for example be equal to 100 Hz.

[0119] Steps 301A and 301B and 302 may be included in, or correspond to, acquisition step 202 of Figure 2.

[0120] Step 303 can be included in the acquisition step 202 or in the transformation step 204 of Figure 2, or even be an intermediate step between the acquisition step 202 and the transformation step 204.

[0121] In a 304 transformation step (FFT), a Fourier transform is performed on the time-series data in frequency domains. This 304 transformation step can be considered an example of a frequency transformation in the 204 transformation step shown in Figure 2.

[0122] Instead of a Fourier transform, we can perform a Laplace transform. For sampled signals, we can speak of a Z-transform (f(t)^F(Z)).

[0123] Therefore, the time series signals 310 (TSGV(t), PP(t), TEC(t), TSC(t)) are transformed into frequency signals of the parameters 330: a frequency signal of the parameter TSGV 331 (TSGV(Z)), a frequency signal of the parameter PP 332 (PP(Z)), a frequency signal of the parameter TEC 333 (TEC(Z)) and a frequency signal of the The parameter TSC 334 (TSC(Z)) is used, and the time-series signal with reactivity 325 (ρ(t)) is transformed into a frequency-series signal with reactivity 335 (ρ(Z)). The frequency signals 330 and 335 are each represented by a function having a real part and an imaginary part. Up to transformation step 304, time-series signals 310, 315, and 325 (T) are processed, and after transformation step 304, frequency signals 330 and 335 (Z) are processed.

[0124] The reactivity frequency signal 335 will feed a response vector Y in the adjustment process 340 described later in connection with Figure 3B, while the parameter frequency signals 330 will feed a matrix of explanatory variables X in the adjustment process 340. Each explanatory variable corresponds to a parameter. Each explanatory variable is expressed as a complex number: thus, for the signal of each parameter, we have the real part and the imaginary part of the explanatory variable.

[0125] In selection step 305 (BANDWIDTH SELECTION), a common bandwidth is selected for the frequency-transformed signals. The lower and upper bounds of the bandwidth can, for example, be selected from the dynamics of the reactor phenomenology, such as the moderator fluid velocity, the duration of a moderator fluid loop in the circuit between the core / vessel and the steam generator, and / or the phase shift between a temperature change and a reactivity change.

[0126] The lower bound can be conditioned by the sampling step 302. The lower bound in frequency is preferably much greater than the inverse of the time acquisition window defined in the sampling step 302.

[0127] The upper bound can be conditioned at least partially by the iso-timing step 303. Other parameters can condition the upper bound, as explained later.

[0128] For example, for a reactivity coefficient that is a function of the core outlet temperature (TSC), the upper frequency limit can be determined by the time phase shift observed between a change in TSC temperature and the change in reactivity ρ. As an illustration, if this phase shift is 20 seconds, periods much longer than 20 seconds can be defined, for example, 10 times longer, i.e., frequencies below 1 / 200 s. -1 (Hz).

[0129] As an example, the lower bound is approximately 1 / (10*60), corresponding to a period of approximately 10 minutes, or 1 / 600 s -1 (Hz), and the upper limit is equal to approximately 1 / 300 s -1 (Hz), corresponding to a period of approximately 5 minutes.

[0130] The selection step 305 is followed by the adjustment process 340 detailed below in relation to figure 3B.

[0131] The example of the 340 fitting process illustrated in Figure 3B and described below includes several steps consisting of shaping matrices of explanatory variables from the frequency signals 330, 335 (step 341), reducing the dimensionality of the parameters, and of the explanatory variables, if parameters are correlated, or do not have a significant influence on the reactivity (steps 342, 343, 344), and then performing a linear regression (step 345).

[0132] It is through this adjustment process that one or more reactivity coefficients can be estimated, and for example, a statistical law can be established for each estimated reactivity coefficient.

[0133] Steps 341, 342, 343 and 344 can be included in, or correspond to, identification step 206 in Figure 2. Linear regression step 345 can correspond to an example of linear regression step 208 in Figure 2.

[0134] We start from the frequency signals of the parameters 330 (TSGV(Z), PP(Z), TEC(Z), TSC(Z)) over the bandwidth [Z a ; Z b ] selected in step 305, and the signal in reactivity frequency 335 (ρ(Z)) over the bandwidth [Z a ; Z b ] selected in step 305. All these frequency signals are each represented by a function having a real part and an imaginary part.

[0135] In a construction step 341 (X, Y), or organization step, a matrix of explanatory variables X is constructed from the frequency signals of the parameters 330 over the bandwidth [Z a ; Z b], which each include several data points for several frequencies in the bandwidth. The matrix X is conditioned so that each column vector of this matrix X includes the frequency signal (real and imaginary parts) of one and the same parameter.

[0136] We begin by constructing a matrix of complex numbers D, which we represent in a simplified way as follows:

[0138] We can organize the matrix D, and in particular the real and imaginary parts of the frequency signals TSGV(Z), PP(Z), TEC(Z), TSC(Z), and define the matrices D re =Re(D) and D im =Im(D) in the set of real numbers, corresponding respectively to the real and imaginary parts of the matrix of complex numbers D. Therefore D matrices re and D im We can thus condition the matrix of explanatory variables X in the following way:

[0140] The frequency signal data of the same parameter are arranged in the same column of the complex number matrix D, and also of the explanatory variable matrix X.

[0141] In construction step 341, a response vector Y is also constructed from the reactivity frequency signal 335 (ρ(Z)) over the bandwidth [Z a ; Z b That is, the reactivity frequency signal 335, which includes several data points for several frequencies in the bandwidth, is organized into a vector. The reactivity frequency signal ρ(Z) comprises a real part Re(ρ(Z)) and an imaginary part Im(ρ(Z)). The response vector Y is conditioned as follows:

[0142]

[0143] This conditioning of the matrix of explanatory variables X and the response vector Y advantageously allows us to obtain statistical laws of the reactivity coefficients, with typically estimators, and standard deviations (allowing us to provide uncertainties on the estimated reactivity coefficients), as explained in the steps described below.

[0144] This example is based on the use of a single neutron detector and a reactivity signal. It would be possible to have multiple neutron detectors and multiple reactivity signals arranged in a matrix Y, instead of a vector. A person skilled in the art will be able to perform as many linear regressions in the following steps as there are columns in the matrix Y.

[0145] We can then perform a correlation step 342 (CORR(X)). In the correlation step 342, we determine correlations between the frequency signals of the different parameters. For example, we perform correlations between the columns of the matrix of explanatory variables X.

[0146] For example, we perform operation X -1 X, to obtain a CORR correlation matrix, represented below in tabular form (Table 2):

[0147] [Table 2] CORR TSGV PP TSC TEC TSGV 1 abc PP a 1 de TSC bd 1 ​​f TEC cef 1

[0148] This CORR correlation matrix is ​​symmetric and contains 1s on the diagonal. We are particularly interested in the values ​​that are not on the diagonal (off-diagonal values), that is, the values ​​a, b, c, d, e, f. These values ​​represent correlations of the frequency signals of the parameters, that is, correlations between the frequency signals of the different parameters TSGV, PP, TEC, TSC.

[0149] A threshold δ can be defined for the correlation values ​​a, b, c, d, e, f above which the frequency signals of two parameters are considered to be correlated. If a correlation value is greater than the threshold δ, one of the two parameters whose frequency signals are correlated can be identified, or even the frequency signal of one of the two parameters whose frequency signals are correlated can be suppressed. For example, the threshold δ is between between 0.80 and 0.95, or between 0.85 and 0.95, for example, is equal to approximately 0.9.

[0150] For example, if d is greater than δ, the core outlet temperature TSV and the primary pressure PP are considered correlated, and the frequency signals of either TSV or PP can then be suppressed. Similarly, if c is greater than δ, the steam generator outlet temperature TSGV and the core inlet temperature TEC are considered correlated, and the frequency signals of either TSGV or TEC can then be suppressed. The same reasoning can be applied to the correlation values ​​a, b, e, and f, and their corresponding explanatory variables.

[0151] If two parameters are correlated, the frequency signal of at least one of these two parameters can be removed from the adjustment process, for example during a resolution step 344 described later.

[0152] We can also perform an analysis of variance step 343 (ANOVA (Y~X)), or ANOVA, from the English ANalyse Of VAriance.

[0153] The order of the correlation steps 342 and analysis of variance steps 343, if both steps are implemented, is indifferent.

[0154] This analysis of variance can be performed from a complex-valued linear regression model (Y~X), or analysis of variance of the linear regression model of the response vector Y explained by the values ​​of the matrix of explanatory variables X.

[0155] This analysis of variance step 343 is configured to determine a level of influence, or contribution, of the frequency signal 330 of each of the parameters on the reactivity frequency signal 335, and thus the level of influence of each parameter on the reactivity.

[0156] The levels of influence of the different parameters can be expressed as proportions of variance η j 2 of the vector Y explained by the explanatory variables ranked from most influential to least influential.

[0157] To these estimates of variance proportions η j 2 p-values, i.e., probabilities that the contributions are zero, can be associated with a parameter, and a parameter can be considered to have no significant influence on reactivity if the p-value of the frequency contribution of the signal from that parameter is greater than a threshold γ. For example, the threshold γ is approximately 0.001.

[0158] We can define a threshold λ for the proportions of variance η 2 j below which we consider that the frequency signal of a parameter has no influence on the frequency signal of reactivity.

[0159] For example, if we consider that the threshold λ is equal to 0.90, we will only be interested in the first k most influential explanatory variables such that their p-value is less than 0.001 (threshold γ) and that the threshold λ respects the formula:

[0160] ^^^^ ≤ ∑ 2 ^^^^<^^^ ^^^^^^^^^

[0161] We can thus identify at least one parameter whose frequency signal has little or no influence on the reactivity, or even remove the frequency signal of this parameter, for example during the resolution step 344 described later.

[0162] The following describes an example of calculating the proportions of variance η 2 j.

[0163] The total variance (SST) is estimated as follows:

[0164] ^^^^^^^^^^^^ = ‖^^^^ − ^�^^^‖2

[0165] where Ӯ is the scalar corresponding to the average of the elements of the vector Y.

[0166] The residual variance (SSE) is estimated as follows: 2

[0167] ^^^^^^^^^^^^ =�^^^^ − ^�^^^�

[0168] where Ŷ are the values ​​predicted by the regression estimators:

[0169] ^�^^^ = ^^^^.^�^^^′

[0170] A' is the parameter vector used to solve the equation:

[0171] Y = ^^^^.^^^^′

[0172] Â' is the mean least squares estimator of A', calculated as follows:

[0173]

[0174] The total variance SST and the residual variance SSE allow us to express the variance explained by the explanatory variables (SSR) according to the equation:

[0175]

[0176] To perform the ANOVA analysis of variance, several linear regressions can be carried out successively.

[0177] A first linear regression model with only one explanatory variable is created:

[0178] ^^^^ = ^^^^1.^�^^^′ 1

[0179] Some j èmeLinear regression models containing j other explanatory variables are created by adding another explanatory variable each time:

[0180]

[0181] Thus, j can successively take the value 2, then 3, then 4 etc.

[0182] For example, the first explanatory variable is the steam generator outlet temperature (SGAT), and the j other variables are successively the core outlet temperature (STT), the primary pressure (PP), and the core inlet temperature (CIT), in that order or another order. Alternatively, the first explanatory variable is the core outlet temperature (STT), and the j other variables are successively the steam generator outlet temperature (SGAT), the primary pressure (PP), and the core inlet temperature (CIT), in that order or another order.

[0183] After each j ème Once the linear regression model is created, the SSR variance is calculated. jexplained by the addition of the j ème The explanatory variable is thus:

[0184]

[0185] Where Ŷ j are the values ​​predicted by a linear regression reduced to the j other explanatory variables. j could also be noted as Ŷ 1…j .

[0186] The proportion of variance η 2 j explained by the explanatory variable j is then expressed as follows:

[0187]

[0188] Other variations of analysis of variance can be considered by the expert. Since the objective of ANOVA is to find the order in which explanatory variables are added that results in the most influential variables being added first, several methods for achieving this order can be considered by the expert.

[0189] Following the correlation step 342 and the analysis of variance step 343, a resolution step 344 (RESOLUTION OF X INTEGERS) can be performed. This is a decision-making process for removing explanatory variables from the matrix X, corresponding to parameters. We obtain a reduced matrix of explanatory variables U constructed from the explanatory variables that have not been removed.

[0190] For example, if two parameters are determined to be correlated during the correlation step 342, the ANOVA step 343 can allow us to decide to keep, among the two correlated parameters, the parameter with the most contribution to the variance.

[0191] Step 342, step 343, and / or step 344 can constitute a dimensionality reduction step for the matrix of explanatory variables X. This reduction step can be replaced by one or more steps, or by any other method, that reduces the dimensionality of the matrix of explanatory variables X, in particular to remove at least one of the correlated explanatory variables, while preserving the separate effects of each of the explanatory variables on the response. For example, a principal component analysis (PCA) can be performed, which allows for the removal of at least one of the correlated explanatory variables.

[0192] Reducing the dimensionality of the explanatory variable matrix X allows for more accurate reactivity coefficients. Indeed, removing explanatory variables that are correlated with each other and / or have a weak influence on reactivity avoids overfitting during the linear regression step described below. Without this reduction, reactivity coefficients could be obtained for parameters that are highly correlated, potentially resulting in redundant influence on reactivity and thus less accurate. In summary, reducing the dimensionality of the explanatory variable matrix X improves the stability of the estimators of these reactivity coefficients, and the accuracy of determining these reactivity coefficients, for example with the smallest possible p-value.

[0193] In a linear regression step 345 (LINEAR REGRESSION (Y~U)), a complex-valued linear regression is performed using the response vector Y and the reduced matrix of explanatory variables U; that is, a linear regression of the vector Y explained by the values ​​of U. This linear regression aims to estimate a parameter vector A that allows us to solve the equation:

[0194] Y = ^^^^.^^^^

[0195] The linear regression step 345 is, for example, carried out in the following way.

[0196] In Fourier space, or frequency space (Z), we consider the reactivity ρ(Z) as a linear composition of parameters with reactivity coefficients α lconstants, where the index l corresponds to the different parameters, in this example TSGV, PP, TEC and TSV, i.e.:

[0197] ^^^^(^^^^) = ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^(^^^^) + ^^^^^^^^^^^^^^^^^^^^(^^^^) + ^^^^^^^^^^^^^^^^^^^^^^^^^^^^(^^^^) + ^^^^^^^^^^^^^^^^^^^^^^^^^^^^(^^^^)

[0198] As a reminder, Z-transforms are functions with a real part and an imaginary part. The coefficients α l are complex numbers, and can be estimated by complex-valued linear regression as follows.

[0199] We start from the complex-valued response vector Y=Y re +iY im and the reduced matrix of complex-valued explanatory variables U=U re +iU im.

[0200] We seek to estimate the parameter vector A=A re +iA im, consisting of the coefficients A l , where the index l corresponds to the different explanatory variables / different parameters, so as to solve the equation:

[0201]

[0202] and thus, to find the reactivity coefficients α lto the different parameters.

[0203] In this example, the parameter vector A includes elements A TGSV , HAS PP , HAS TSC and A TEC which have values ​​in the space of complex numbers.

[0204] By separating the real and imaginary parts, we can obtain the following system:

[0205]

[0206] The estimation of the parameters A can be carried out using an ordinary least squares method, that is, a non-complex number least squares method, by composing the following matrices: ^^^^ −^^^^ ^^^^

[0207] � ^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^ ^^^^^^^^ ^^^^ � =�^^^^ ^^^^ � .�^^ � ^^ ^^^^^^^^ ^^^^^^^^ ^^^^^^^^ ^^^^^^^^^^^^ ^^^^ ^^^^

[0208] The mean least-squares estimator  of A becomes: − 1^^^^ ^^^^

[0209] ^̂^^^ =� ^^^^ ^^^^� ^^^^ ^^^^

[0210] Where the average estimator  consists of a real part and an imaginary part, i.e.: ��

[0211] ^̂^^^ = ^^^^ + ^^^^^^^^^^^^^^^^ ^^^^^^^^

[0212] On obtient des estimateurs ^^�^^ des coefficients de^^^^ Reactivity α is given by the following formula: l

[0213]

[0214] where ║Â║ is the modulus of line l of the estimator l mean Â, Arg(Â) is the argument of the mean estimator Â: l l l the argument represents the phase shift between the effect of the parameter l and the reactivity signal response. The use of Sign(Arg( l )) allows us to estimate if the coefficient α l is positive or negative, that is, whether the parameter has a positive or negative effect on reactivity.

[0215] Furthermore, it should be noted that, by construction, the v ecteurs the matrices U are orthogonal, and Thus, these vectors are not collinear. Since these vectors are not collinear, this allows us to estimate the real and imaginary components of the parameters of the mean estimator Â. The correlations of the real and imaginary components of the same explanatory variable are zero, and thus the real and imaginary components of the variance σ l 2 from the estimator ^^^^ ^^^^ of the same reactivity coefficient α l are identical. These observations allow us to establish that the statistical laws of the estimators ^^�^^ ^^^^ reactivity coefficients α l are Rice's laws(σ l ,ν l ) of standard deviation σ l and of modulus ν l , where: ^^^^^^^^ =�^�^^^^^^^�

[0216] Thus, the described determination method allows for uncertainties in the reactivity coefficients α. These uncertainties can be expressed in the form of l variance coefficients σl 2 .

[0217] For example, the variance coefficients σ l 2 are the diagonal elements of the covariance matrix Cov(Â) of the mean estimator Â.

[0218] The covariance matrix Cov(Â) of the mean estimator  can be estimated as follows: −1

[0219] ^^^^^^^^^^^^�^̂^^^� = ^^^�^2 � ^^^^ ^^^^^^^^^^^^ ^^^^ ^^^^�

[0220] avec ^^^�^2 le scalaire, estimateur de la variance des^^^^^^^^^^^^ residuals, or residual variance:

[0222] with n the number of lines of Y, and p the number of lines of Â.

[0223] In the case of complex-valued linear regression, the parameter vector A, the response vector Y and the explanatory variables being decomposed into real and imaginary parts, n and p are found to be doubled.

[0224] The covariance matrix Cov(Â) is a diagonal matrix. Indeed, the matrix U is defined as follows:

[0225]

[0226] We obtain:

[0227] ^^^^ ^^^^ =� ^^^^^^^^^^^^ ^^^^^^^^^^^^ −^^^^ ^^^^� ^^^^^^^^ ^^^^^^^^

[0228] And so:

[0229]

[0230] Either :

[0231]

[0232] So :

[0233]

[0234] We deduce that the covariance matrix Cov(Â) thus obtained is indeed a diagonal matrix, whose diagonal coefficients are equal and whose extra-diagonal coefficients are zero:

[0235]

[0236] The determination method allows for the calculation and / or estimation of reactivity coefficients in nuclear reactor operating configurations at power, without disrupting operational schedules, for example, without shutting down the reactor during dedicated testing campaigns. It simply requires retrieving data from measuring instruments that are generally already installed in the reactor. These measurements can be taken during reactor operation at power. Thus, the determination method leverages existing measuring instruments in the nuclear reactor; it does not require the addition of any other measuring instruments.

[0237] The determination method can be implemented in a processing device, such as the processing device 120 of Figure 1B, connected to the measuring instruments of the nuclear reactor, such as the sensors and detector 111, 112, 113, 114, 115 of Figure 1A. The processing device is for example included in a system for determining reactivity coefficients.

[0238] The determination method is non-intrusive and allows for a virtually real-time display available to the reactor operator.

[0239] The description shows that a method of determination is available to determine reactivity coefficients for several parameters of the nuclear reactor, for example not limited to a temperature, and for example to determine if the different parameters are linked to each other in their influence on reactivity.

[0240] The description shows that a method of determination is available to estimate uncertainties on the determined reactivity coefficients. In particular, the determination method allows for the quantification of uncertainties in the reactivity coefficients for different parameters; that is, it quantifies the uncertainties in the influence of the different parameters on reactivity. This provides a confidence level for the estimated reactivity coefficients.

[0241] The various data used and obtained during the different stages of the determination method, for example the measurements of the sensors and detectors, the explanatory variables, and the reactivity coefficients and their uncertainties, are for example stored in a memory.

[0242] The determination method is, for example, implemented in a computer, for example in the form of software, or a computer program.

[0243] Reactivity coefficients, as well as uncertainties on estimated reactivity coefficients, can for example be used in the design, safety studies, and control, or more generally the operation, of nuclear reactors.

[0244] In particular, the temperature coefficient C T (α T The reactivity coefficient at a given temperature is a key parameter for the safety of a nuclear reactor, as it demonstrates the control and intrinsic stability of the reactor core. This key parameter is relevant during the reactor's nominal power operation, but also upstream in the analysis of incident and accident scenarios.

[0245] Furthermore, when operating at power, the temperature coefficient C Tbeing a parameter whose estimation requires coupling several Scientific Computing Tools (SCTs), such as neutronics, thermo-hydraulic and thermal tools, it constitutes a quantity of interest for the validation of coupled physics models on experimental data.

[0246] The temperature coefficient C T Being influenced in particular by core wear, the position of the reactor's control and piloting mechanisms, the temperature of the moderator and fuel, as well as by the reactor's power level, knowledge of this allows an operator to have a complementary observation of the state of the core and the reactor, and for example, to carry out finer piloting.

[0247] More generally, reactivity coefficients can be used to ensure the reactivity stability of a nuclear reactor during power operation. For example, reactivity coefficients can be used to control nuclear reactor parameters, such as temperature and primary pressure, more safely and stably during power operation. Reactivity coefficients can also be used to review, for example, to justify the reduction of potentially excessive design margins in a nuclear reactor, while ensuring its safety.

[0248] The following describes a step-by-step example of the implementation of the 340 adjustment process.

[0249] In this example, the core inlet temperature (TEC) parameter was not taken into account. It was initially estimated that this parameter was correlated with the steam generator outlet temperature (SGV).

[0250] Construction stage 341

[0251] Figure 4A, Figure 4B and Figure 4C illustrate time series signals of reactivity ρ (Reactivity (pcm)) for different time-hour windows (Time (hour)), in a nuclear reactor operating at power.

[0252] The time window in Figure 4B (approximately 3 hours 30 minutes) is smaller than that in Figure 4A (approximately 5 hours), and the time window in Figure 4C (approximately 20 minutes) is smaller than that in Figure 4B.

[0253] Each time-series reactivity signal is derived from measurements of a neutron flux detector (NF). This could be reactor 100 in Figure 1A and neutron detector 115 in Figure 1A.

[0254] Figure 4C shows oscillations of the reactivity signal for (sliding) periods between 5 and 10 minutes, or between 300 and 600 seconds, indicated by the arrows between the vertical dashed lines. This corresponds to a bandwidth of [1 / 600-1 / 300] Hz(s- 1 ), or [0.00167-0.00333] Hz(s -1 ).

[0255] Figure 5A, Figure 5B, Figure 5C and Figure 5D illustrate frequency signals (s -1 ) of reactivity, core outlet temperature, steam generator outlet temperature and primary pressure, in a nuclear reactor operating at power.

[0256] These signals correspond to the Z-transforms of the time series signals of reactivity (ρ), core outlet temperature (TSC), steam generator outlet temperature (TSGV), and primary pressure (PP).

[0257] We observe that in the four figures 5A to 5D, there is a peak in the frequency reactivity signal over the bandwidth [0.00167-0.00333] s -1 , identified by the arrow between the vertical dotted lines, corresponding to the period range [5-10] minutes identified in figure 4C. Peaks are also found for each of the TSC, TSGV and PP signals on the same bandwidth.

[0258] We can thus select the bandwidth [Z a ; Z b ] equal to [0.00167 ; 0.00333] s -1For example, we obtain a set of N frequency values ​​over the bandwidth [0.00167; 0.00333], where N is approximately 30. We can then construct the response vector Y from the data (N data points for N frequencies) of the frequency reactivity signal ρ(Z) over the bandwidth [0.00167; 0.00333], and the complex number matrix D from the data (N data points per parameter for N frequencies) for each of the frequency signals TSGV(Z), PP(Z), and TSC(Z) over the bandwidth [0.00167; 0.00333]. We then construct the matrix of explanatory variables X from the complex number matrix D.

[0259] We will evaluate the reactivity coefficients for the TSC, TSGV and PP parameters over this bandwidth, in power operation.

[0260] Correlation step 342

[0261] We perform operation X -1X, to obtain the CORR matrix represented below in tabular form (Table 3):

[0262] [Table 3] CORR TSC TSGV PP TSC 1.000 0.570 0.997 TSGV 0.570 1.000 0.559 PP 0.997 0.559 1.000

[0263] Table 3 shows that the variables TSC and PP are strongly correlated.

[0264] Analysis of Variance Step 343

[0265] An analysis of variance (ANOVA) is performed on the linear regression with complex variables using the following model:

[0266] ρ(Z) = ^^^^TSC. TSC(Z) + ^^^^TSGV. TSGV(Z) + ^^^^^^^^^^^^.^^^^^^^^(^^^^)

[0267] Where ρ(Z) is the Z-transform of the reactivity time-series signal constructed from the neutron flux detector (NFD) time-series signal, and TSC(Z), TSGV(Z), and PP(Z) are the Z-transforms of the temperature and pressure time-series signals constructed from the core outlet temperature, steam generator (SG) outlet temperature, and primary pressure sensors, respectively. The coefficients α TSC , α TSGV and α PP are complex coefficients that represent the reactivity coefficients of the respective parameters TSC, TSGV and PP.

[0268] Linear regression with complex variables allows us to estimate the complex coefficients α TSC , α TSGV and α PP and the analysis of variance is presented in the table below (Table 4).

[0269] [Table 4] Proportion of variance Re(TSC) 91.2% Re(TSGV) 3.6% Re(PP) 3.6% Im(TSC) 0.0% Im(TSGV) 0.1% Im(PP) 0.0% Residuals 1.5%

[0270] We observe that the real component of the TSC parameter contributes more than 91% to the variance (91% proportion of variance).

[0271] Given the strong correlation between TSC and PP observed in Table 3 and the significantly larger proportion of variance in the TSC parameter compared to that in the PP parameter observed in Table 4, we can decide to retain only the TSC parameter among the PP and TSC parameters. In other words, the PP parameter is an identified parameter.

[0272] Resolution step 344

[0273] We obtain a reduced matrix of explanatory variables U constructed from only the explanatory variables that have not been removed. In this example, as indicated above, between the parameters TSC and PP, we retain only the parameter TSC. Furthermore, we identify the parameter TSGV since we observe in Table 4 that it has very little influence on the reactivity signal. In this example, we retain only the parameter TSC, and the corresponding explanatory variable, in the reduced matrix of explanatory variables U.

[0274] Linear regression step 345

[0275] In this step of complex-valued linear regression from the response vector Y and the reduced matrix of explanatory variables U, we obtain a parameter vector A whose only real and imaginary components are those of A TSC , that's to say :

[0276] ρ(Z) = ^^^^TSC. TSC(Z)

[0277] The real and imaginary components of A TSCallow access to the reactivity coefficient α TSC .

[0278] Estimates of the real and imaginary components of A TSC are presented in the table below (Table 5):

[0279] [Table 5] Mean parameter Standard deviation (σ) p-value Re(TSC) -25.85 0.75 <2.00E-16 Im(TSC) -5.63 0.75 4.30E-10

[0280] As a reminder, the reactivity coefficient α TSC in pcm / °C can be estimated using the formula:

[0281]

[0282] and is therefore equal in this example: ^^^^^^^^^^^^

[0283] ^^^^^^^^^^^^^^^^~^^^^^^^^^^^^^^^^(^^^^ = 0,75 ; ^^^^ = −26,5 ) °^^^^

[0284] The p-value shows the probability that the reactivity coefficient α TSC is zero, meaning that the TSC has no influence on reactivity. In this example, we observe that the p-value is very low. We therefore obtain a reactivity coefficient α TSC very significant.

[0285] Various embodiments and variants have been described. Those skilled in the art will understand that some features of these various embodiments and variants could be combined, and other variants will become apparent to them. In particular, although the method examples are described with core outlet temperature, steam generator outlet temperature, primary pressure, and core inlet temperature as parameters, or explanatory variables, the method developed can be generalized to estimating a reactivity coefficient for any parameter that may influence reactivity. Reactivity can be described by a linear composition, or combination linear, reactivity coefficients to the parameters associated each with its parameter.

[0286] In addition, the measuring instruments can be arranged differently from the arrangement in Figure 1A.

[0287] Furthermore, although the preceding description relates to a pressurized water nuclear reactor, the embodiments can be applied to other types of nuclear reactors, such as boiling water reactors and fast spectrum reactors.

[0288] Finally, the practical implementation of the described methods and variants is within the reach of the person in the trade, based on the functional indications given above.

Claims

CLAIMS 1. Method for determining (200; 300), by a treatment device (120), a reactivity coefficient (α l) of a nuclear reactor (100), the method comprising: - an acquisition step (202; 301A, 301B, 302, 303) of time-series data (310, 325), and of storage of the time-series data in a memory, said time-series data comprising a time-series data signal of reactivity (325) of the nuclear reactor, and time-series data signals of a set of parameters (310), the parameters being parameters of the nuclear reactor likely to have an influence on the reactivity; - a transformation step (204;304) in frequency (Z) of the time series data (310, 325), so as to transform the time series data into frequency signals (330, 335), the frequency signals comprising a reactivity frequency signal (335) obtained by the frequency transformation of the reactivity time series data signal (325) and parameter frequency signals (330) obtained by the frequency transformations of the time series data signals of the set of parameters; - an identification step (206; 341, 342, 343, 344) of at least one parameter of the set of parameters whose frequency signal is correlated, in a bandwidth (Z; a ; Z b), with another frequency signal of another parameter from the parameter set and / or at least one parameter from the parameter set whose frequency signal is uncorrelated, in the bandwidth, with the reactivity frequency signal; - the deduction of the reactivity coefficient (α l ) by a linear regression step (208; 345) of the signal in reactivity frequency (335) explained by the frequency signals of the parameters (330), except for the at least one identified parameter.

2. Method according to claim 1, wherein the identification step comprises a correlation step (342) of the frequency signals of the parameters (330), so as to identify whether at least one parameter of the parameter set has a frequency signal correlated, in the bandwidth, with another frequency signal of another parameter of the parameter set. 3.A method according to claim 2, wherein the correlation step (342) comprises determining correlation coefficients of the frequency signals of the parameters, two frequency signals of two parameters being correlated if the correlation coefficient of the two frequency signals of the two parameters is greater than a first threshold (δ), for example, at least one of the two parameters whose frequency signals are correlated is identified and / or the frequency signal of at least one of the two parameters whose frequency signals are correlated is removed.

4. A method according to any one of claims 1 to 3, wherein the identification step comprises an analysis of variance step (343) of the reactivity frequency signal explained by the frequency signals of the parameters, so as to identify whether at least one parameter of the parameter set has a frequency signal that is not correlated, in the bandwidth, with the reactivity frequency signal.

5. Method according to claim 4, wherein the variance analysis step (343) comprises the determination of variance proportions (η. j 2 ) frequency signals of the different parameters on the frequency signal of reactivity, a frequency signal of a parameter not being correlated with the reactivity frequency signal if the variance proportion of the frequency signal of said parameter is less than a second threshold (λ), for example, the parameter whose frequency signal is not correlated with the reactivity frequency signal is identified and / or the frequency signal of said parameter is removed.

6. Method according to any one of claims 1 to 5, wherein the identification step (206; 341, 342, 343, 344) comprises: - a step of organizing (341) the reactivity frequency signal (335) in the bandwidth (Z a ; Z b) in a response vector (Y), and frequency signals of the parameters (330) in the bandwidth in a matrix of explanatory variables (X), such that each column of the matrix of explanatory variables (X) comprises the frequency signal of one and the same parameter; - a reduction step (342, 343, 344) of the dimension of the matrix of explanatory variables (X), so as to form a reduced matrix of explanatory variables (U), purged of at least one frequency signal of at least one identified parameter; the linear regression step (345) being a linear regression of the response vector (Y) explained by the values ​​of the reduced matrix of explanatory variables (U), so as to estimate a parameter vector (A), such that ^^^^ = ^^^^.^^^^ where Y is the response vector, U is the reduced matrix of explanatory variables and A is the parameter vector; the reactivity coefficient (α l ) being deduced from the parameter vector (A).

7. A method according to claim 6 in its dependence on claim 2 or 3, wherein the correlation step (342) is a correlation of the columns of the matrix of explanatory variables (X).

8. A method according to claim 7 in its dependence on claim 3, wherein the correlation step (342) comprises the construction of a correlation matrix (CORR) from the matrix of explanatory variables (X) according to the following operation: where X is the matrix of explanatory variables and CORR is the correlation matrix, which is a symmetric matrix whose off-diagonal elements represent the correlation coefficients.

9. A method according to claim 6 in its dependence on claim 4 or 5, wherein the analysis of variance step (343) is performed from a linear regression model of the response vector (Y) explained by the values ​​of the matrix of explanatory variables (X). 10.Method according to any one of claims 6 to 9, wherein the parameter vector (A) is estimated by a least squares method, where an average least squares estimator (Â) of the parameter vector (A) is given by the equation:. where  is the mean estimator, Y is the response vector, and U is the reduced matrix of explanatory variables.

11. Method according to claim 10, wherein the reactivity coefficient (α l ) includes at least one reactivity coefficient with at least one parameter, each reactivity coefficient being estimated by the formula: where α l is the reactivity coefficient and  l corresponds to a line l of the average estimator Â.

12. Method according to claim 10 or 11, wherein an uncertainty value of the reactivity coefficient (α l ) is estimated in the form of a variance coefficient (σ l 2), where the variance coefficient is a diagonal element of a covariance matrix (Cov(Â)) of the mean estimator (Â).

13. A method according to any one of claims 1 to 12, wherein the acquisition step comprises: - a sampling step (302) of the time series data over an acquisition period; and / or - an iso-timing step (303) of the time series data at a common frequency. 14.A method according to any one of claims 1 to 13, wherein the parameters include at least one temperature and / or pressure measured in the nuclear reactor during power operation, for example: - a core outlet temperature (CAT); - a core inlet temperature (CIT); - a steam generator outlet temperature (SGAT); and / or - a primary outlet pressure (POP); for example, the at least one measured temperature corresponds to the temperature of a heat transfer fluid (103) circulating in the nuclear reactor, for example the heat transfer fluid is also a neutron moderator fluid.

15. A method according to any one of claims 1 to 14, wherein the acquisition step comprises a step. transformation into a reactivity signal (301B) of a time-series data signal of neutron flux (315), measured in the nuclear reactor during power operation, into a time-series data signal of reactivity (325), for example, the transformation into reactivity step implements a one-point kinetic equation inversion method.

16. Processing device (120) configured to implement the method according to any one of claims 1 to 15.

17. System for determining a reactivity coefficient comprising: - the processing device (120) according to claim 16; - a reactivity measuring instrument (115) connected to the processing device; and - parameter measuring instruments (111, 112, 113, 114) connected to the processing device. 18.Computer program comprising instructions for implementing the method according to any one of claims 1 to 15, when the program is executed by a processing device (120).

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