Method for determining the reactivity coefficient of a nuclear reactor

Abstract

Method for determining a reactivity coefficient of a nuclear reactor This description relates to a method for determining (200) a reactivity coefficient (αl) of a nuclear reactor, the method comprising: - a time series data acquisition step (202) including 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 transformation step (204) of the time series data into frequency signals including a frequency signal of reactivity and frequency signals of the parameters;- an identification step (206) of at least one parameter from the parameter set whose frequency signal is correlated, within a bandwidth, 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 not correlated, within the bandwidth, with the reactivity frequency signal; - the deduction of the reactivity coefficient (αl) by a linear regression step (208) of the reactivity frequency signal explained by the frequency signals of the parameters, except for the at least one identified parameter. Figure for the abstract: Fig. 2;

Description

Title of the invention: Method for determining the reactivity coefficient of a nuclear reactor technical field

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

[0002] The present description relates particularly to the determination of a reactivity coefficient, that is to say a coefficient giving the variation of reactivity referred to the variation of an operating parameter of a nuclear reactor. Previous 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 in the core, from which it removes the thermal energy released by the fission of the fissile materials and transports the heat via a heat exchanger to a circuit called the secondary circuit, which typically includes a turbine and an alternator 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. It should be noted that a moderator is a substance that slows down neutrons without absorbing too much of them. The coolant also generally serves to maintain the core temperature at a level compatible with the properties of the materials from which the core is made.

[0005] In pressurized water reactors (PWRs), the heat exchanger is part of a steam generator, which is generally part of the primary circuit. The steam generator transfers all or part of the heat from the heat transfer fluid in the primary circuit to the water in the secondary circuit to transform 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 assembly is intrinsically ensured by the reactivity response p 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 by a negative value of a reactivity coefficient CT at the temperature T of the heat transfer fluid.

[0008] Reactivity, expressed in pcm, per hundred thousand, is defined as a quantity allowing measurement of 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 keff, 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, keff is defined as the ratio of the total number of neutrons produced in a given fissile medium during a time interval 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 CT at temperature is defined as the change in reactivity p relative to the change in temperature T, generally the temperature of the heat transfer fluid, i.e.:

[0011] r ^T~ <1T

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

[0013] In the remainder of the document, the reactivity coefficient CT at temperature T can be referred to, for simplification, as the temperature coefficient.

[0014] More generally, a reactivity coefficient with respect to a parameter V (variable), which may 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.:

[0015] r _d£_ c V “ dV

[0016] To date, the temperature coefficient CT is only accurately measured 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, and moderator boron content, are stable.

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

[0018] By zero power, we mean a level of neutron flux (and therefore of power) lower than the level of neutron flux called the "Doppler threshold", the Doppler threshold being the level of neutron flux 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, in 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 consists of: - placing the reactor in an initial critical state, with a neutron flux level below the Doppler threshold and having stabilized the core control parameters under defined protocol conditions, for example by adjusting the temperature and / or pressure of the moderator, the boron content of the moderator, the position of the control mechanisms; - once the initial state is 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 control mechanisms, concentration (titration) of boron, pressure in the primary circuit (primary pressure); - modify the moderator temperature by predetermined temperature steps, with the control parameters 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 content, 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; - to acquire signals in time series; - to make variations of a parameter, for example a temperature, whose influence on reactivity we wish to measure; - remain in this state until the stability of this state and of all other reactor parameters, which could potentially influence reactivity, is assured.

[0022] The time series signals must then be interpreted and processed, this processing being potentially long and tedious, or even leading to 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 to be processed. Reducing this measurement noise may require the use of other experimental protocols to stabilize the parameters generating the measurement noise, which may impose new constraints on reactor operating schedules.

[0025] In power operation, a reactor state in which control of the reactivity coefficient CT influences the stability of core reactivity, the reactivity coefficient CT cannot be measured in the methods described above.

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

[0027] For example, methods for estimating the temperature coefficient on reactors at full power are carried out using neutron noise analysis, particularly in boiling 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 this end, these estimation methods consist of applying cross-correlation operators, of the CPSD type (Cross-Power Spectral Density), and auto-correlation operators, of the APSD type (Auto-Power Spectral Density), to neutron flux, or reactivity, and temperature signals. The ratio of these operators allows for an estimation of the temperature coefficient CT.

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

[0030] Ci / p(t) =]Qhe(T).9(tT)dr

[0031] Where p(t) is the heart reactivity as a function of time, 0(t) is the moderator temperature as a function of time, and hu(t) is the impulse response of the heart, or of the primary circuit.

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

[0033] tj cpsdWV APSDÇôff)

[0034] and

[0035] jj APsaôp) ~ CPSDiôO / ip)

[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 Hu transfer function.

[0037] Furthermore, this type of estimation method requires a fairly long listening time to perform a neutron noise analysis. The measurement estimates can therefore only be obtained with a time lag, which constitutes a quasi-real-time measurement. However, during this listening time 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 preliminary 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 for determining a reactivity coefficient of a nuclear reactor which 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, in particular conditions under which the core is not isothermal.

[0041] It would be advantageous if the determination method made it possible to determine 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 made it possible to obtain a reactivity coefficient with very good accuracy.

[0043] It would be advantageous if the determination method made it possible to determine 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 for the determination method to allow for the determination of measurement uncertainties for each determined reactivity coefficient. In particular, it would be advantageous for the determination method to allow for the estimation of uncertainties regarding the influence of the different parameters on the reactivity.

[0045] It would be advantageous if the determination method made use of existing measuring instruments in the nuclear reactor, i.e., it did not require the addition of 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 blocking a nuclear reactor during dedicated test campaigns. Summary of the invention

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

[0048] One embodiment provides a method for determining, by a processing device, a reactivity coefficient of a nuclear reactor, the method comprising: - a time series data acquisition step, and time series data storage 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 nuclear reactor parameters likely to have an influence on reactivity; - a frequency transformation step 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 parameter frequency signals obtained by the frequency transformations of the time series data signals of all the parameters; - a step of identifying at least one parameter from the set of parameters whose frequency signal is correlated, within a bandwidth, with another signal in frequency of another parameter from the set of parameters and / or of at least one parameter from 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 step of linear regression 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 set of parameters has a frequency signal correlated, in the bandwidth, with another frequency signal of another parameter of the set of parameters.

[0050] According to one embodiment, the correlation step includes determining the 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 set of parameters has a frequency signal that is not correlated, in the bandwidth, with the reactivity frequency signal.

[0052] According to one embodiment, the variance analysis step includes determining the proportions of variance of the frequency signals of the parameters on the frequency signal of reactivity, a frequency signal of a parameter not being correlated with the frequency signal of reactivity if the proportion of variance of the frequency signal of said parameter is less than a second threshold, for example the parameter whose frequency signal is not correlated 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, so 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 is a linear regression of the response vector explained by the values ​​of the reduced matrix of explanatory variables, in order to estimate a parameter vector, such that: y= u. a where Y is the response vector, U is the reduced matrix of explanatory variables and A is the parameter vector; the reactivity coefficient being 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 the construction of a correlation matrix from the matrix of explanatory variables according to the following operation: CORR^X'X where X is the matrix of explanatory variables and CORR is the correlation matrix, which is a symmetric matrix whose extra-diagonal elements represent the correlation coefficients.

[0056] According to one embodiment, the analysis of variance step is carried out 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: ifUY where  is the average estimator and U is the reduced matrix of variables explanatory.

[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: = || A-||5î^m( A>) ) °where ai is the reactivity coefficient and Âi corresponds to line 1 of the average estimator Â.

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

[0060] According to one embodiment, the acquisition step comprises: - a time series data sampling step over an acquisition period; and / or - a step of iso-cadence of the data in time series 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 (SGT); and / or - a primary pressure at the heart's 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 accompanying figures, among which:

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

[0069] [Fig.1B] represents an example of an embodiment of the processing device of [Fig.1A];

[0070] [Fig.2] illustrates in a simplified way a method for determining a reactivity coefficient according to an embodiment;

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

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

[0073] Fig. 5A, Fig. 5B, Fig. 5C and Fig. 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 the implementation methods

[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 embodiments have been shown and are detailed. In particular, not all the components of a nuclear reactor are detailed, nor is the operation of the nuclear reactor. Furthermore, not all the details of the calculations are given, as they are within the grasp of a person skilled in the art.

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

[0077] In the following description, when reference is made to absolute position qualifiers, such as the terms "front", "back", "top", "bottom", "left", "right", etc., or relative position qualifiers, such as the terms "above", "below", "superior", "inferior", etc., or to orientation qualifiers, such as the terms "horizontal", "vertical", etc., reference is made, unless otherwise specified, to the orientation of the figures.

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

[0079] Throughout this description, references to a flux, unless otherwise specified, referencing a neutron flux, generally mean a neutron flux. In general, flux denotes a measure of neutron flux, which characterizes a neutron reaction rate, and which is generally determined by a counting rate measured by a neutron detector.

[0080] In the following description, when reference is made to a reactor, it refers, unless otherwise specified, to a nuclear reactor.

[0081] In the following description, when reference is made to a determination method, unless otherwise specified, it refers to a method for determining, or estimating, a reactivity coefficient of a nuclear reactor with respect to one parameter, or to several reactivity coefficients with respect to 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 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 may be referred to as the moderator fluid 103.

[0084] The nuclear reactor 100 further 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 of [Fig.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 from a secondary circuit to transform the latter 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 adapted to circulate water from the secondary circuit between an outlet 105a of the steam generator 105, through which the (vaporized) water from the secondary circuit exits the steam generator 105, and an inlet 105b of the steam generator 105, through which the water from the secondary circuit enters the steam generator 105 again, passing through the alternator 108.

[0091] The 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 (TSGV); - a pressure sensor 112 positioned in the tank 102, for example at the outlet of the core 101, to measure a primary pressure (PP); - a temperature sensor 113 positioned in the tank 102 at the inlet of the core 101 to measure a core inlet temperature (TEC); - a temperature sensor 114 positioned in the tank 102 at the outlet of the core 101 to measure a core outlet temperature (TSC), which is also the GV inlet temperature; - a neutron detector 115 positioned outside the core 101, for example around the vessel 102, allowing the neutron flux (NF) level in the core to be monitored.

[0092] Other measuring instruments, not shown in [Fig. 1A], may be included in the nuclear reactor 100 in a manner known to those skilled in the art. Furthermore, the locations of the various measuring instruments are indicated in [Fig. 1A] for illustrative purposes. For example, the neutron detector is not necessarily positioned outside the vessel and the core; it may be positioned inside the vessel, or even inside the core. Moreover, there may be several neutron detectors, for example, one outside the core or the vessel and another inside the core or the 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 allow, for example, the quantification of an ATC variation of the temperature in the core 101 (ATc=TSC-TEC), an ATGV variation of the temperature in the GV 105 (ATGv=TSC-TSGV). They also allow the determination of an average temperature TM in the core which can be estimated by the formula TM= (TEC+TSQ / 2.

[0096] In power operation, when the energy extracted in the secondary circuit by the alternator 108 is constant and the parameters of the reactor 100 are stabilized, the following phenomena, or phenomenology (Table 1), can be considered in a simplified manner: - we start from time t=t0, where TSC=tsc0, TEC=tec0, TSGV=tsgv0, PP=pp0, and NF=nf0; - at t=ti greater than t0, a power demand of the alternator 108 leads to a decrease in the enthalpy of the secondary of the GV 105, which results in an increase in ATGv: at this instant, the temperature TSC being fixed, it is the temperature TSGV which decreases, taking a value tsgvi less than tsgv0; - at t=t2 greater than tb 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 temperature of the core 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 power of the core; - at t=t4 greater than t3, the increase in core power leads to an increase in ATC: the TEC temperature being fixed, it is the TSC temperature that increases, this temperature increase also resulting in an increase in the primary pressure PP which takes a value pp4 greater than pp0, the latter can be balanced by the level of the pressurizer.

[0097] [Tables 1] t TEC TSC TSGV PP NF to tec0 tsc0 tsgv0 PPo nf0 tl tec0 tsc0 tsgv i <tsgv0 PPo nf0 t2 tec 2<tec0 -tsgVj tsc0 tsgVi PPo nf0 tec 2~tsgvi tsc0 tSgVi PPo nf 3> nf0 t4 tec 2~tsgvi tse 4>tSC0 tSgVi PP 4>PPo nf3

[0098] It can be noted that, in this example, the reactivity response following the power demand at t=t3 is observed on the neutron flux NF later at t=t3 and on the core exit temperature TSC even later at t=t4. A delay in the dynamics between the TSGV, NF, and TSC signals is therefore expected. The use of Z-transforms (Fourier or Laplace) of the signals, described later, advantageously allows this delay to be taken into account through the phases. Indeed, the use of Z-transforms makes it possible 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 allows us to obtain a coefficient that takes into account these phase shifts, 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 of determining a reactivity coefficient, and to deliver one (or more) reactivity coefficient(s) ai, from the data from the measurements of the 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 further includes, connected to the processor P: - an instruction memory (INSTR_MEMORY) including the instructions of the computer program; - an input / output interface (I / O INTERFACE) allowing, in particular, the retrieval of data from measurements taken by the measuring instruments of 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 nuclear reactor 100; and - a data storage (DATA_MEMORY), allowing for example the storage of determined or estimated reactivity coefficients.

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

[0103] In the following description, examples of a method for determining one (or more) reactivity coefficient(s) ai implemented by the treatment device 120 will be described. 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 p, deduced from the measurement of neutron flux NF, and to estimate at least one reactivity coefficient at at least one parameter from the set of parameters.

[0104] Figure [Fig.2] illustrates in a simplified manner a method for determining a reactivity coefficient ai according to one embodiment.

[0105] The determination method 200 comprises: - a step 202 (DATA ACQUISITION AND STORAGE) of time-series data and storage of time-series data in a memory: these time series data include a time series data signal of nuclear reactor reactivity, and time series data signals of a set of parameters, the parameters being nuclear reactor parameters that may influence reactivity; - a transformation step 204 (FREQUENCY TRANSFORMED) of the time series data into frequency: the time series data are transformed into frequency signals, the frequency signals comprising a reactivity frequency signal (transform of the data signal into a time series of reactivity) and parameter frequency signals (transforms of the data signals into time series of all 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 set of parameters whose frequency signal is uncorrelated with the reactivity frequency signal; and - the deduction of the reactivity coefficient ai by a linear regression step 208 (LINEAR REGRESSION) of the reactivity frequency signal 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 estimation of the reactivity coefficient, as explained later.

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

[0108] Fig. 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 Fig. 3B.

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

[0110] An acquisition step 301A allows for the acquisition and storage of signals 310, 315, in the form of time series data from measurements taken by various measuring instruments: - a time series 311 of GV 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 (TEC(t)) measurements by temperature sensor 113; - a time series 314 of core outlet temperature (TSC(t)) measurements 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 (p(t)). This reactivity transformation can be performed using a reactivity meter. It may consist of, or include, an inversion of the point kinetic equations, for example, at 6 or 8 precursor groups. This reactivity transformation step thus makes it possible to acquire 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, typically a few sliding hours, for example 5 to 6 hours, is defined, 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 they are not necessarily timed in a regular manner.

[0116] Thus, we can foresee an iso-timing step 303 (ISO-TIMING OF SIGNALS) which consists of (re)sampling and timing the signals in time series at a common frequency, and this, for example, in order 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 may, for example, be between 50 Hz and 500 Hz, for example be equal to 100 Hz.

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

[0120] Step 303 can be included in the acquisition step 202 or in the transformation step 204 of [Fig.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 data in iso-clocked time series into frequency signals. This 304 transformation step can correspond to an example of a frequency transformation in the 204 transformation step of [Fig. 2].

[0122] Instead of a Fourier transform, a Laplace transform can be performed. For sampled signals, this can be referred to as a Z-transform (f(t)F(Z)).

[0123] Consequently, 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 parameter TSC 334 (TSC(Z)), and the time series reactivity signal 325 (p(t)) is transformed into a frequency reactivity signal 335 (p(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 signals 310, 315, 325 (T) are processed, and after transformation step 304, frequency signals 330, 335 (Z) are processed.

[0124] The reactivity frequency signal 335 will feed a response vector Y in the adjustment process 340 described later in relation to [Fig. 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, for example 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 the temperature change and the 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 conditioned by the time phase shift observed between a variation in the temperature TSC and the variation in the reactivity p. As an illustration, if this phase shift is 20 seconds, periods much greater than 20 seconds can be defined, for example 10 times greater, i.e. frequencies less than 1 / 200 s 1 (Hz).

[0129] According to an example, the lower limit is equal to approximately 1 / (10*60), corresponding to a period of approximately 10 minutes, i.e. 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 [Fig.3B].

[0131] The example of the fitting process 340 illustrated in [Fig.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 may be included in, or correspond to, identification step 206 of [Fig. 2]. Linear regression step 345 may correspond to an example of linear regression step 208 of [Fig. 2].

[0134] We start from the frequency signals of the parameters 330 (TSGV(Z), PP(Z), TEC(Z), TSC(Z)) on the bandwidth [Za ; Zb] selected in step 305, and from the frequency signal of reactivity 335 (p(Z)) on the bandwidth [Za ; Zb] 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 [Za; Zb], each of which comprises several data points for several frequencies in the bandwidth. The matrix X is conditioned so that each column vector of this matrix X comprises 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 represented in simplified form as follows:

[0137] TSGV (Z a ) PP(Z a ) TSC(Z a ) TEC(Z a ) ' D= TSGV (Z a+ i) PP(Z a+l ) TSC^) TEC(Z a+ï ) . TSGV(Z b ) PP(Z h ) TSC(Z b ) TEC(Z b ) ,

[0138] The matrix D, and in particular the real and imaginary parts of the frequency signals TSGV(Z), PP(Z), TEC(Z), TSC(Z), can be organized, and the matrices Dre=Re(D) and Dim=Im(D) can be defined in the set of real numbers, corresponding respectively to the real and imaginary parts of the complex number matrix D. Starting from the matrices Dre and Dim, we can thus condition the matrix of variables explanatory X in the following way:

[0139] Dre ^im D. n L ^im ^re j

[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 (p(Z)) over the bandwidth [Za ; Zb ], that is to say, the signal is organized into a reactivity frequency of 335, which includes Several data points for multiple frequencies within the bandwidth are represented as a vector. The signal, with reactivity frequency p(Z), comprises a real part Re(p(Z)) and an imaginary part Im(p(Z)). The response vector Y is conditioned as follows:

[0142]

[0143] This conditioning of the matrix of explanatory variables X and of the response vector Y advantageously allows obtaining 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 several neutron detectors and several 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 what follows as there are columns in the matrix Y.

[0145] A correlation step 342 (CORR(X)) can then be performed. In the correlation step 342, correlations between the frequency signals of the different parameters. For example, correlations are performed between the columns of the matrix of explanatory variables X.

[0146] For example, the operation X'*X is performed to obtain a correlation matrix CORR, represented below in tabular form (Table 2):

[0147] [Tables2] 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 to say the values ​​a, b, c, d, e, f. These values ​​represent correlations of the frequency signals of the parameters, that is to say 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 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 0, the core outlet temperature TSV and the primary pressure PP are considered to be correlated, and the frequency signals of either TSV or PP can then be suppressed. For example, if c is greater than 0, the GV outlet temperature TSGV and the core inlet temperature TEC are considered to be 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 the 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] One can also carry out 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 carried out 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 frequency signal of reactivity 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 qj2 of the vector Y explained by the explanatory variables ranked from most influential to least influential.

[0157] To these estimates of variance proportions T]j2 can be associated p-values, that is, probabilities that the contributions are zero, and a parameter can be considered not to have a significant influence on the reactivity if the p-value of the frequency contribution of the signal of this parameter is greater than a threshold y. For example, the threshold y is equal to about 0.001.

[0158] A threshold X can be defined for the proportions of variance q2 below which the frequency signal of a parameter is considered to have no influence on the frequency signal of reactivity.

[0159] For example, if we consider that the threshold X 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 y) and that the threshold X respects the formula:

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

[0162] An example of calculating the proportions of variance q2 is described below.

[0163] The total variance (SST) is estimated as follows: [°164] SST=||y-ÿ||2

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

[0166] The residual variance (SSE) is estimated as follows: [°167] SSE= ||y_ÿ||2

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

[0169] ÿ = XÆ

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

[0171] Y =

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

[0173] X'^XX^XY

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

[0175] ssx=sst-sse= ||ry|| 2

[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 first explanatory variable is implemented:

[0178] Y^A'.

[0179] Linear regression models containing other explanatory variables are performed by adding another explanatory variable each time:

[0180] Y = X}_jÂ\.j

[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 (ST), the primary pressure (PP), and the core inlet temperature (CIT), in that order or in another order. Alternatively, the first explanatory variable is the core outlet temperature (ST), 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 in another order.

[0183] After each jth linear regression model performed, the variance SSRj explained by the addition of the jth explanatory variable is calculated as follows:

[0184] çç» . _ |l ÿ||“

[0185] Where Yj are the values ​​predicted by a linear regression reduced to the j other explanatory variables. Yj could also be denoted ŸL j.

[0186] The proportion of variance q2 explained by the explanatory variable j is then expressed as follows:

[0187] o ^7 - SST

[0188] Other variants of analysis of variance may be considered by a person skilled in the art. 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 ways of obtaining this order may be considered by a person skilled in the art.

[0189] Following the correlation step 342 and the analysis of variance step 343, a resolution step 344 (RESOLUTION OF X ENTRIES) can be performed. This step is a decision-making process for removing explanatory variables from the matrix X, corresponding to parameters. A reduced matrix of explanatory variables U is obtained, 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 having the most contribution to the variance.

[0191] Step 342, step 343, and / or step 344 may constitute a dimensionality reduction step for the matrix of explanatory variables X. This reduction step may 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) may be performed, which removes 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 an overfitting problem during the linear regression step described below. Without this reduction, reactivity coefficients could be obtained for parameters that are highly correlated with each other, potentially resulting in redundant influence on reactivity and thus less accurate.In summary, reducing the dimensionality of the matrix of explanatory variables 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, i.e., 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 the equation to be solved:

[0194] Y = UA

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

[0196] In Fourier space, or frequency space (Z), the reactivity p(Z) is considered as a linear composition of parameters with reactivity coefficients ai constants, where the index 1 corresponds to the different parameters, in this example TSGV, PP, TEC and TSV, i.e.:

[0197] p(Z) - aTSGVTSGV (Z) + aPPPP(Z) +arscTSC(Z) +aTECTEC(Z)

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

[0199] We start from the complex-valued response vector ¥=¥re+iYim and the reduced matrix of complex-valued explanatory variables U=Ure+iUim

[0200] We seek to estimate the parameter vector A=Are+iAim consisting of the coefficients Ai, where the index 1 corresponds to the different explanatory variables / different parameters, so as to solve the equation:

[0201] Y-e+— (Ure + iL) (Are + iAim)

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

[0203] In this example, the parameter vector A includes elements ATGsv, APP, Atsc and Atec which have values ​​in the space of complex numbers.

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

[0205] Y re = U re HAS re - U^ m A^ m Y. =TJ. A Az im im^re ' 17 retint

[0206] The estimation of the parameters A can be carried out by an ordinary least squares method, i.e. non-complex number least squares method, by composing the following matrices:

[0207]

[0208] y x re y. tm Y U -U- 1 U- U L un ure i U ra ™re . ^im. HAS The mean least squares estimator  of A becomes:

[0209] a^CUUT^UY

[0210] Where the average estimator A consists of a real part and an imaginary part, either : [oziii

[0212] Estimators a{ of the reactivity coefficients ai are obtained by the following formula:

[0213] ||Aj|5fgW( Arg^A;) )

[0214] where || Âi || is the modulus of row 1 of the mean estimator Âb. Arg(Âi) is the argument of the mean estimator Âi: the argument represents the phase shift between the effect of parameter 1 and the response of the reactivity signal. The use of Sign(Arg(Âi)) allows us to estimate whether the coefficient ai is positive or negative, that is, whether parameter 1 has a positive or negative effect on the reactivity.

[0215]

[0216] Furthermore, it should be noted that, by construction, the vectors Ure and -Uim AI of matrix U are orthogonal, and thus non-collinear. These vectors are not not collinear, this allows us to estimate the real and imaginary components of the parameters of the mean estimator Â. Correlations of the real components And imaginary values ​​of the same explanatory variable are zero, and thus the components real and imaginary variance <y2 de l’estimateur a{ d’un même coefficient de The reactivity values ​​ai are identical. These observations allow us to establish that the statistical distributions of the estimators at of the reactivity coefficients ai are Rice(Oi,Vi) distributions with standard deviation Oi and modulus vb where: v / = HaII Thus, the described determination method allows for uncertainties in the reactivity coefficients ab. These uncertainties can be expressed in the form of variance coefficients. <y2.

[0217]

[0218]

[0219]

[0220]

[0221]

[0222]

[0223]

[0224]

[0225]

[0226]

[0227]

[0228]

[0229] For example, the variance coefficients <y2 sont les éléments diagonaux de la matrice de covariance Cov(Â) de l’estimateur moyen Â. The covariance matrix Cov(Â) of the mean estimator  can be estimated as follows: Cov(A) = ^(W)4 with the scalar, estimator of the variance of the residuals, or residual variance: ty res ■> ■> ^res— np with n the number of lines of Y, and p the number of lines of Â. 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. The covariance matrix Cov(Â) is a diagonal matrix. Indeed, the matrix U is defined as follows: u= FU re . im -U- 1 im D We obtain U D mid And so: U im Ure ■ U re . ~Uim U im ]\U V re D u im -U im u re . U 2 +u 2 re T 17 )m TJ. TJ -TJ IJ. u re u im -u re u. an n tm T-tl rj2 Ure^ Uim

[0230]

[0231] Either :

[0232]

[0233] So : yuuy'

[0234] It follows that the covariance matrix Cov(Â) thus obtained is indeed a diagonal matrix, whose diagonal coefficients are equal and whose extradiagonal coefficients are zero:

[0235] Cw^A^^UV)'^

[0236] The determination method makes it possible to determine, and / or estimate, reactivity coefficients in power operating configurations of a nuclear reactor, without disrupting operating schedules, for example, without shutting down the nuclear reactor during dedicated test campaigns. It simply requires retrieving data from measuring instruments that are generally already installed in the nuclear reactor. These measurements can be taken during reactor power operation. Thus, the determination method makes use of existing measuring instruments in the nuclear reactor; that is, it does not require adding any other measuring instruments.

[0237] The determination method can be implemented in a processing device, such as the processing device 120 of [Fig. 1B], connected to the measuring instruments of the nuclear reactor, such as the sensors and detector 111, 112, 113, 114, 115 of [Fig. 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 can be made to determine reactivity coefficients for several parameters of the nuclear reactor, for example not limited to a temperature, and for example to determine whether the different parameters are linked to each other in their influence on reactivity.

[0240] The description shows that a method for determining the uncertainties in the determined reactivity coefficients is available. In particular, the method for determining the uncertainties allows them to be quantified. Reactivity coefficients for different parameters are quantified, meaning the uncertainties in the influence of these parameters on reactivity are assessed. This allows for a confidence level to be provided 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 CT(aT), i.e., 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 incident and accident scenario studies.

[0245] Moreover, in power operation, the temperature coefficient CT being a parameter whose estimation requires coupling of several Scientific Computing Tools (SCT), such as neutronics, thermo-hydraulic and thermal tools, it constitutes a quantity of interest for the validation on experimental data of coupled physics models.

[0246] Since the temperature coefficient CT is influenced in particular by core wear, the position of the reactor control and piloting mechanisms, the temperature of the moderator and fuel, as well as by the reactor power level, knowledge of it allows an operator to have a complementary observation of the state of the core and the reactor, and for example, to carry out finer control.

[0247] More generally, reactivity coefficients can be used to ensure the reactivity stability of a nuclear reactor during its power operation. For example, reactivity coefficients can be used to act more safely and stably on nuclear reactor parameters, such as various temperatures and primary pressure during its 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 adjustment process 340.

[0249] In this example, the core inlet temperature (TEC) parameter was not taken into account. It could be estimated as a first approximation that this parameter was correlated with the steam generator outlet temperature (SGAT).

[0250] Construction step 341

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

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

[0253] Each time-series reactivity signal is deduced from the measurements of a neutron flux detector (NF). This can be reactor 100 of [Fig.1A] and neutron detector 115 of [Fig.1A].

[0254] Oscillations of the reactivity signal can be observed in [Fig. 4C] for (sliding) periods between 5 and 10 minutes, i.e., 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'), i.e., [0.00167-0.00333] Hz(s').

[0255] Fig. 5A, Fig. 5B, Fig. 5C and Fig. 5D illustrate frequency signals (s1) 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 (p), core outlet temperature (TSC), steam generator outlet temperature (TSGV), and primary pressure (PP).

[0257] It can be seen that in the four figures 5A to 5D, a peak in the frequency reactivity signal is found in the bandwidth [0.00167-0.00333] s1, indicated by the arrow between the vertical dashed lines, corresponding to the period range [5-10] minutes identified in [Fig. 4C]. Peaks are also found for each of the TSC, TSGV and PP signals in the same bandwidth.

[0258] We can thus select the bandwidth [Za ; Zb] equal to [0.00167 ; 0.00333] s1. For example, we obtain a set of N frequency values ​​over the bandwidth [0.00167 ; 0.00333], N being, for example, approximately 30. We can thus construct the response vector Y from the data (N data for N frequencies) of the frequency reactivity signal p(Z) over the bandwidth [0.00167 ; 0.00333], and the complex number matrix D from the data (N data per parameter for N frequencies) for each of the frequency signals TSGV(Z), PP(Z), TSC(Z) over the bandwidth [0.00167 ; 0.00333]. We then construct the matrix of explanatory variables X from the matrix of complex numbers D.

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

[0260] Correlation step 342

[0261] The operation X'*X is performed to obtain the CORR matrix shown below. in tabular form (Table 3):

[0262] [Tables3] 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] p (z) = aTsc.TSC (Z) + aTSGV.TSGV (Z) + aPP.PP(z)

[0267] Where p(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 aTSC, aTSGV, and aPP are complex coefficients representing the reactivity coefficients of the respective parameters TSC, TSGV, and PP.

[0268] Linear regression with complex variables allows us to estimate the complex coefficients aTSc, aT sgv and aPP and the analysis of variance is presented in the table below (Table 4).

[0269] [Tables4] 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% Residues 1.5%

[0270] It is observed 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 of the TSC parameter compared to that of the PP parameter observed in Table 4, it is possible to 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] A reduced matrix of explanatory variables U is obtained, constructed from only the explanatory variables that have not been removed. In this example, as indicated above, only the TSC parameter is retained between the TSC and PP parameters. Furthermore, the TSGV parameter is identified since it can be observed in Table 4 that it has very little influence on the reactivity signal. In this example, only the TSC parameter, and the corresponding explanatory variable, are retained 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 ATSC, that is:

[0276] p(Z)=ATSC.TSC(Z)

[0277] The real and imaginary components of ATSC allow access to the reactivity coefficient aTSc-

[0278] Estimates of the real and imaginary components of ATSC are presented in Table 5 below:

[0279] [Tables5] Parameter Mean Standard deviation (o) 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 aTSc reactivity coefficient in pcm / °C can be estimated using the formula:

[0281] aTSC = 11 Atsc | J Sign(Arg(ATSC))

[0282] and is therefore equal in this example:

[0283] aTSC~Rice (a - 0.75; v - - 26.5)

[0284] The p-value shows the probability that the aTSC reactivity coefficient 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 highly significant aTSC reactivity coefficient.

[0285] Various embodiments and variations have been described. Those skilled in the art will understand that certain features of these various embodiments and variations could be combined, and other variations will become apparent to those skilled in the art. 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 the estimation of a reactivity coefficient for any parameter that may influence reactivity. Reactivity can be described by a linear composition, or linear combination, of the reactivity coefficients for the parameters, each associated with its corresponding parameter.

[0286] In addition, the measuring instruments can be arranged differently from the arrangement in [Fig.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 embodiments and variants described is within the reach of a person skilled in the art, based on the functional indications given above.

Claims

Demands

1. Method for determining (200; 300), by a processing device (120), a reactivity coefficient (aj) of a nuclear reactor (100), the method comprising: - a step of acquiring (202; 301A, 301B, 302, 303) time-series data (310, 325), and storing 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 (Za; Zb), with another frequency signal of another parameter of the set of parameters and / or of 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 (ai) by a linear regression step (208; 345) of the reactivity frequency signal (335) explained by the frequency signals of the parameters (330), except for at least one identified parameter.;

2. A method according to claim 1, wherein the identification step comprises a frequency correlation step (342) of the parameter signals (330), so as to identify whether at least one parameter of the parameter set has a signal in frequency correlated, in the bandwidth, with another signal in frequency of another parameter of the set of parameters.

3. Method according to claim 2, wherein the correlation step (342) comprises 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 (ô), 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. Method according to any one of claims 1 to 3, wherein the identification step includes 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 uncorrelated, in the bandwidth, with the reactivity frequency signal.

5. Method according to claim 4, wherein the variance analysis step (343) comprises determining variance proportions (q,2) of the frequency signals of the different parameters on the reactivity frequency signal, 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 (X), 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. A method according to any one of claims 1 to 5, wherein the identification step (206; 341, 342, 343, 344) comprises: - an organization step (341) of the reactivity frequency signal (335) in the bandwidth (Za; Zb) into a response vector (Y), and of the frequency signals of the parameters (330) in the bandwidth into a matrix of explanatory variables (X), such that each column of the matrix of explanatory variables (X) includes 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 Y = UA where Y is the response vector, U is the reduced matrix of explanatory variables and A is the parameter vector; the reactivity coefficient (ai) being deduced from the parameter vector (A).

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

8. 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: CORR = XlX 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. Method according to claim 6 in its dependence on claim 4 or 5, wherein the analysis of variance step (343) is carried out from a linear regression model of the response vector (Y) explained by the values ​​of the matrix of explanatory variables (X).

10. A method according to any one of claims 6 to 9, wherein the parameter vector (A) is estimated by a least squares method, where a mean least squares estimator (Â) of the parameter vector (A) is given by the equation: A = ^UU^UY 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 (ai) comprises at least one reactivity coefficient to at least one parameter, each reactivity coefficient being estimated by the formula: where ai is the reactivity coefficient and Ai corresponds to a line 1 of the average estimator A.

12. Method according to claim 10 or 11, wherein an uncertainty value of the reactivity coefficient (aj is estimated in the form of a variance coefficient (Oi2), where the variance coefficient is a diagonal element of a covariance matrix (Cov(Â)) of the mean estimator (Â).

13. 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 operating at power, for example: - a core outlet temperature (CAT); - a core inlet temperature (CIT); - a steam generator outlet temperature (SGT); and / or - a primary outlet core 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 reactivity transformation step (301B) of a time-series data signal of neutron flux (315), measured in the nuclear reactor operating at power, into the time-series data signal of reactivity (325), for example the transformation step in reactivity implements a method of inverting the kinetic equations at a point.

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 treatment device (120) according to claim 16; - a reactivity measuring instrument (115) connected to the treatment device; and - parameter measuring instruments (111, 112, 113, 114) connected to the treatment 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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