A computer implemented method for controlling a pressurized water reactor

Abstract

The present invention relates to a computer implemented method for controlling a pressurized water reactor, comprising Receiving (1000) a load schedule (62) for a next predetermined period of time for the electrical power output; Receiving (1010) a boration or dilution injection schedule (60) for the next predetermined period of time; Receiving (1020) current measured process values; Simulating (1030), based on the received current process values, the received load schedule (62), and the received boration / dilution injection schedule (60), the process variables of the pressurized water reactor for the next predetermined period of time, wherein the simulating includes: dividing the core into a plurality of meshes; calculating a fission rate and the process values for each of the meshes and each of predefined time points within the next predetermined period of time, wherein the fission rate is calculated based on a reactor model for each mesh, the reactor model being based on an inhour equation defining a relation between the reactivity and the reactor period as a variable.

Description

[0001]A computer implemented method for controlling a pressurized water reactor The present invention concerns a computer implemented method for controlling a pressurized water reactor. WO2020 / 224764 A1 and WO2023 / 151786 A1 disclose methods for governing a pressurized water nuclear reactor based on prediction of reactor process values. CN114970293 discloses coarse mesh diffusion coefficient calculation method based on two-dimensional one-dimensional coupling Fourier analysis. Known reactor simulations solve differential equations numerically using e.g. forward and / or backward Euler methods, which is slow. Object of the invention is to provide a fast prediction of the operation of a nuclear power plant, in order to improve the operation safety, efficiency, and plannability of the nuclear power plant and to support flexible power operation. According to one aspect, a computer implemented method for controlling a pressurized water reactor, comprising - Receiving a load schedule for a next predetermined period of time for the electrical power output; - Receiving a boration / dilution injection schedule for the next predetermined period of time; - Receiving current measured process values; - Simulating, based on the received current process values, the received load schedule, and the received boration / dilution injection schedule, the process variables of the pressurized water reactor for the next predetermined period of time, wherein the simulating includes: dividing the core into a plurality of meshes; calculating a fission rate and the process values for each of the meshes and each of predefined time points within the next predetermined period of time, wherein the fission rate is calculated based on a reactor model for each mesh, the reactor model being based on an inhour equation defining a relation between the reactivity and the reactor period as a variable. Further embodiments may relate to one or more of the following features, which may be combined in any technical feasible combination: ^ the measured process values include: a coolant inlet temperature, a coolant outlet temperature, a live steam pressure, axial offset, a thermal power of the reactor core, power control bank positions, positions of heavy and grey banks, in-core neutron fluxes, ex-core neutron fluxes, boron concentration, reaction rates, heating powers, average coolant temperature, fuel temperatures and / or coolant temperatures; ^ the method further comprising determining non-measurable process values, in particular reaction rates, heating powers, fuel temperatures, xenon concentrations, and / or an iodine concentrations, based on the measured process values, wherein the simulating is further based on the determined non-measurable process values; ^ the fission rate and the process values for each of the meshes for the next predetermined period of time are calculated iteratively regarding the whole array of the values, wherein a calculation of each particular value of the array do not use the same value from the previous run; ^ the number of meshes is between 8 and 16, in particular about 12; ^ the method further comprises receiving an insertion schedule for heavy banks and / or grey banks for the next predetermined period of time, wherein the simulating is further based on the received insertion schedule for heavy and / or grey banks; ^ the method further comprises the step of: - verifying whether the simulated process values remain within a predetermined operation domain; and - operating the pressurized water reactor using the boration / dilution injection schedule, the insertion schedule for heavy and / or grey banks, and / or the load schedule; ^ the fuel is uranium, plutonium, or a mixture of uranium and plutonium. ^ the absolute value of the power gradient is below 15% / min, in particular below 10% / min; ^ the next predetermined period of time is at least 2 h, in particular at least 12 h, for example 24 h;^ the inhour equation is defined by the following equation:+Δρ^^ = ^^^^^^ or by the following equation+ Δρ^^ = ^^^^^^ − ^^^^2^^ , with ^^∞,^^ being the absorption reactivity of the m-th mesh,Δρmbeing the reactivity change due to the neutron migration, ^^^^being the mesh period, and X and Y being constant values depending on the fuel; ^ for a uranium core of the pressurized water reactor X= 0.098 s and Y=3.5731 s2, s being the unit seconds; ^ for each mesh a quadratic or cubic equation is provided, based on the reactor model; ^ the quadratic or cubic equation is with ^^^^,^^being the fission rate for a mesh, index n being the number of the prediction step, index m being the number of the mesh in the core, with , and ^^^^,^^which are not dependent on the fission rate from the same mesh and same prediction step, in particular being not dependent from fission rate with indices m and n from previous run ; and / or ^ the quadratic or cubic equation is with ^^^^,^^being the fission rate for a mesh, index n being the number of the prediction step, index m being the number of the mesh in the core, with , ^^^^,^^, ^^^^,^^, and ^^^^,^^which are not dependent on the fission rate from the same mesh and same prediction step, in particular ^^^^,^^, ^^^^,^^, , and ^^^^,^^being not dependent from fission rate with indices m and n from previous run . According to a further aspect, a computer program product is provided, which, when the program is executed by a computer, causes the computer to carry out the computer implemented method of one of the preceding claims. According to an embodiment, the computer program product may be provided on a physical software product, for example a hard disc, a solid-state disc, a CD-ROM, a DVD, comprising the program. According to another aspect, a data carrier signal carrying the computer program product according to an embodiment disclosed herein is provided. According to another aspect, a computer-readable storage medium is provided comprising instructions which, when executed by a computer, causes the computer to carry out the computer implemented method according to embodiment disclosed herein. According to a further aspect, a data processing system is provided, in particular a device, comprising means for carrying out the computer implemented method according to an embodiment disclosed herein. Embodiments are also directed to the system for carrying out the disclosed methods steps and in particular including apparatus parts and / or devices for performing described method steps. The method steps may be performed by way of hardware components, firmware, software, a computer programmed by appropriate software, by FPGA (Field Programmable Gate Array), by any combination thereof or in any other manner. In particular, for electronic and / or software means, each of the above listed terms means and encompasses electronic circuits or parts thereof, as well as stored, embedded or running software codes, firmware, configware, and / or routines, algorithms, or complete programs, suitably designed for achieving the technical result and / or the functional performances for which such means are devised. The method according to the invention provides an ultrafast predictive plant simulation. Further advantages, features, aspects, and details are evident from the dependent claims, the description, and the drawings. The accompanying drawings relate to embodiments of the invention and are described in the following: Figure 1 shows schematically a nuclear power plant ; and Figure 2 shows schematically a flow chart of a method according to an embodiment of the invention. Figure 3 shows the calculation sequence for subscripted process variables; and Figure 4 shows dependences between subscripted variables. FIG. 1 provides a simplified schematic overview of a nuclear power plant 1 with a pressurized water reactor (PWR) 3. The PWR comprises a reactor core 5 having a reactor pressure vessel (RPV) 7. During operation, the heat produced by the reactor core 5 within the reactor pressure vessel 7 is transferred by a primary cooling medium, for example water, circulating in a primary cooling circuit 10 under the driving force of a reactor coolant pump (RCP) 12. The primary cooling circuit includes one or more steam generators 14, in which the heat of the primary cooling medium is transferred to a secondary cooling medium circulating in a secondary cooling circuit 16, thereby evaporating the secondary cooling medium. The primary cooling medium is then led again into the reactor core 5 by the RCP 12. The steam produced by the one or more steam generators 14 drives a steam turbine 18, which is coupled to an electrical generator 20 to generate electricity. The generated electricity is fed into an electrical grid 22. After passing through the steam turbine 18 the steam is condensed in at least one condenser 24 and then provided again into the at least one steam generator 14 by at least one feedwater pump 26. A feedwater tank 28 within the secondary cooling circuit 16 can be used, in some embodiments, as a compensating reservoir. In some embodiments, the flow rate of steam passing into the steam turbine 18 may be controlled by one or more turbine valves 30 in a steam feed line 32 between the at least one steam generator 14 and the at least one steam turbine 18 in the secondary cooling circuit 16. In some special situations, for example, plant start-up, turbine trip, etc., there is excess steam, which is directly guided from the at least one steam generator 14 to the at least one condenser 24 via a bypass line 34 comprising one or more bypass valves 36. The one or more turbine valves 30 and the one or more bypass valves 36 are controlled by a turbine controller 38 and / or a bypass controller 40. The turbine controller 38 and / or a bypass controller 40 use as parameters, in particular the steam pressure p in the steam feed line 32, the rotational speed n of the steam turbine 18, and / or the electrical power P output by the electrical generator 20. For monitoring the pressurized water reactor 3, in some embodiments there are provided a plurality of detectors 42 within the reactor core 5 for continuous measurement of the neutron flux density and its spatial distribution. They are also called in-core detectors 42. According to an embodiment, eight times six detectors are provided in a so-called SPND (self-powered neutron detector) lances. Each lance includes six detectors, which are distributed along their longitudinal length along the height of the reactor core 5. The lances are provided in different fuel assemblies. Further, there are ex-core neutron flux detectors 43 provided outside the reactor pressure vessel 7. The power of the reactor core 3 is controlled in particular via a number of power control rods 41, which can be inserted into the reactor core 5. The power control rods 41 absorb neutrons and depending on the insertion depth, the power production of the nuclear reactor can be controlled, for example because they influence the neutron flux within the reactor. Usually, the control rods in pressurized water reactors are grouped into control assemblies. The rods of a single control assembly are driven by a single rod drive mechanism and move together within a single fuel assembly. In particular, a plurality, for example four or six symmetrically located control assemblies form a control group. The groups are further assigned to control banks. A PWR reactor has for example four so called P-banks consisting of one control group each. These P-banks 41 are usually moved individually and automatically by coolant average temperature controller or by live steam controller 56, depending on the embodiment. All groups, which are not assigned to P-banks are usually consolidated into one large (heavy) bank called H-bank 47. In some embodiments further control banks called grey banks 45 are provided. Grey rods, forming the grey banks, have smaller absorption reactivity comparing to the normal (black) rods and are used in some embodiments to control the reactor power. Due to the lower absorption reactivity, grey banks have less influence on the power density distribution in the core. In most embodiments, having grey banks, the grey banks are used to reduce the reactor power in large steps and are either completely withdrawn or completely inserted. The heavy bank (H-bank) 47 is typically moved as a whole and is used for shutting down of the reactor and in this case is completely withdrawn from the active zone during the normal operation of the nuclear reactor. In some advanced control concepts, the heavy bank is used also to rectify the power density distribution in the active zone. For this purpose, the H-bank is slightly inserted into the upper part of the active zone and can be slightly moved by a so-called power density axial offset (AO) controller (not shown). Long term modification of the reactivity, in particular due to xenon poisoning and fuel depletion is controlled by amending the boron concentration by injection of boric acid (boration) and / or demineralized water (dilution) into the primary cooling circuit 10. Boron within the primary cooling circuit 10 acts as a neutron absorber. Thus, with a higher concentration of boric acid the reactivity of the reactor core 5 and consequently its power decreases. To increase the reactivity, demineralized water is added to the primary cooling circuit 10 in order to reduce the concentration of boric acid and thus to increase the reactivity. According to an embodiment, the pressurized water reactor 3 comprises at least one first pump 44 (boration pump) to inject boric acid and at least one second pump 46 (dilution pump) to inject demineralized water into the primary cooling circuit 10 and thus also into the reactor pressure vessel 7. The amount of demineralized water and / or boric acid can be controlled using a boration valve 48, a dilution valve 50, and / or the pumps 44, 46. According to embodiments, which may be combined with other embodiments disclosed herein, an injection controller 52 is provided, which controls the operation of the pumps 44, 46, and / or the valves 48, 50. Further, the nuclear power plant 1 comprises a controller 54 for the start-up of the pressurized water reactor 3. Such a controller is called Φ-controller or neutron flux controller, which controls the neutron flux Φ, typically measured by one or more ex-core detectors 43. In particular, in dependence on the measured neutron flux, the power control rods 41 (or power control banks or P-Banks) are moved in or out of the reactor core 5. They may be also moved to any intermediate positions. The thermal power of the reactor core 5 can be obtained from the difference between the measured temperature T2 of the primary coolant medium at the outlet of the reactor pressure vessel 7 and the measured temperature T1 of the primary coolant medium at the inlet of the reactor pressure vessel 7. Fission power can be obtained from the neutron flux, in particular measured by the detectors 42 and / or 43. The average reactor coolant temperature (ACT) represents an average of primary coolant medium inlet and outlet temperatures T1, T2. Alternatively, live steam pressure p in the secondary cooling circuit 16 can be taken instead of the ACT as a variable to be controlled as explained here-below. According to embodiments, there is provided reactor power controller 56, for example in form of an average coolant temperature (ACT) controller or live steam pressure (LSP) controller, responsible for power operation, in particular after start-up. The reactor power controller 56 relies on measured values for the temperatures of the primary cooling medium, in particular an average coolant temperature (ACT) derived from the primary coolant medium inlet temperature T1 and the primary coolant medium outlet temperature T2 with respect to the reactor core 5. In another embodiment, additionally or alternatively, the live steam pressure p (LSP) in steam feed line 32 may be used. In particular, in dependence on the measured average coolant temperature ACT and / or live steam pressure p, the power control rods 41 are moved automatically into or out of the reactor core 5. They may be also moved to any intermediate positions. According to embodiments, the electrical power of the nuclear power plant 1 measured at the generator 20 is controlled by the turbine controller 38 via the turbine inlet valves 30. The power control rods 41 or P-banks are then moved by the reactor power controller 56 in order to adapt the power of the pressurized water reactor 3 to the power required by the generator 20. ACT and / or LSP are used thereby as an indicator of power imbalance. According to the present invention, further, the nuclear power plant comprises predictive device 58. The device 58 is adapted to receive the current measured process values, to receive a load schedule 62 for the next predetermined period of time, for example 24 hours, for the electrical power output, is adapted to receive a boration / dilution injection schedule 60 for the next predetermined period of time, and is optionally adapted to receive an insertion schedule 64 for heavy and / or grey banks, when available, for the next predetermined period of time. The boration / dilution injection schedule 60 details the planned boration / dilution injection for the next predetermined period of time and can be executed manually or automatically. It is also possible to process insertion schedules 64 for heavy banks 47 and / or grey banks 45 similar to the boration / dilution injection schedule, if the nuclear power plant possesses such moveable banks. The measurable process values include a coolant inlet temperature (T1), a coolant outlet temperature (T2), a live steam pressure (p), axial offset (AO), a thermal power of the reactor core, power control rod positions, in-core neutron fluxes, ex-core neutron fluxes, boron concentration, generator power, average coolant temperature, etc. For example, the values are provided by the controllers 38, 40, 54, 56, and / or detectors 42, 43. The device may also receive the temperatures (T1, T2), live steam pressure (p) directly from sensors. The process values can also be received from a process computer (not shown). The device 58 is adapted to determine non-measurable process values, in particular reaction rates, heating powers, fuel temperatures, xenon concentrations and / or an iodine concentrations, based on the measured process values. The reaction rates include the fission rate and may also include other rates like betta decay rates. The device 58 is adapted to interactively support an operator for the compilation of the boration / dilution schedule 60 and optionally an insertion schedule for heavy and grey banks 64 using a prediction of plant process values, in particular the measurable process values and / or the non-measurable process values, within predetermined period of time, for example 24 hours. According to embodiments, the load schedule 62, the boration / dilution schedule 60, and / or the insertion schedule for heavy and / or grey banks 64 can be, for example, applied to the plant manually by the operator to control the electrical power set point within the turbine controller 38 as well as actuators in order to operate the dilution valve 50, the pumps 44, 46, and / or to move heavy banks 47 and / or grey banks 45. In another embodiment, the turbine controller 38, the injection controller 52, and / or a position controller for heavy banks 47 and / or grey banks 45 are adapted to apply load schedule, the boration / dilution schedule 60, and / or the insertion schedule for heavy and / or grey banks 64 automatically to control actuators in order to operate the dilution valve 50, the pumps 44, 46, the turbine valves 30, and / or to move heavy and grey banks. Arbitrary combinations of manual and automatic control are possible. The method will be further explained in conjunction with Figure 2. In a step 1000, a load schedule 62, representing a plan for the electrical power output for a next predetermined period of time, in particular stored at a working place of the turbine operator, is transferred to the device 58. A load schedule 62 for the next predetermined period of time means that the load schedule is provided for a period between the current point in time until a predetermined future point in time. For example, the next predetermined period of time is at least 12 h, for example 24 h. The draft of the load schedule 62 may be received from a grid dispatcher, for example ones a day and may be edited and approved by the plant operator. After the approval the load schedule will be transferred from a working place of the turbine operator to the device 58. This automatic transfer can be repeated for example every 10 to 100 ms to apply the schedule changes in real time. In step 1010, a boration / dilution injection schedule 60 and, optionally, an insertion schedule for heavy and / or grey banks 64 are received, in particular by the device 58, for the next predetermined period of time. For example, boration / dilution injection schedule 60 and the insertion schedule 64 may be received by the device 58 every 10 to 100 ms. In some embodiments, the boration / dilution injection schedule 60 and, optionally, an insertion schedule for heavy and / or grey banks 64 can be edited by an operator. In step 1020, current measured process values are received. For example, the current measured process values are received from sensors and / or existing I&C (instrumentation and control) devices, or from the process computer. For example, the current process values are received or updated every 5 to 20 ms. According to embodiment, non- measurable process values are determined, in particular by the device 58 based on the measured process values. The non-measurable process values are for example reaction rates, heating powers, fuel temperatures, xenon concentrations, and / or an iodine concentrations. The non-measurable process values may be also determined in step 1030, in particular during the simulation. Steps 1000 to 1020 may have also another sequence. In the next step 1030, the device 58 is simulating the process values for the next predetermined period of time, for example 24 h based on the received current process values, the received load schedule 62, the received boration / dilution injection schedule 60 and optionally, insertion schedule for heavy and grey banks 64. The simulating includes the dividing the core into a plurality of meshes, in particular according to Avery’s method, as disclosed in R. L. Avery “Theory of Coupled Reactors” Proc.2nd Int. Conf. Peaceful Uses of At. Energy, Vol.12, 182 (1958). According to embodiments, the number of meshes, is between 8 and 16, in particular about 12. For example, each mesh has substantially the same shape. According to embodiments, each mesh corresponds to layer of the core 5 of the nuclear reactor 3. According to this embodiment, the meshes are stacked upon each other. In an example, a mesh has a substantial cylindrical shape. For example, each mesh has the same height. To describe the reactor core dynamics, in particular according to Avery’s method, each mesh is considered as a single reactor, wherein the single reactors are coupled by neutron exchange and coolant flow. For each mesh, the kinetic equations will be considered: For each mesh there is a system of seven coupled first order linear differential equations for neutron concentration ^^(^^) and concentration of the delayed neutron precursors ^^^^(^^), wherein m is number of mesh, ^^^^ , is the change of the neutron concentration in a mesh,^^∞ is the absorption reactivity with ^^∞ , and Λ is the prompt neutron generationtime with Λ ≡ (^^ ∙ ^^ ∙ , β is the total fraction of the delayed neutrons, ^^ = ,^^^^(^^) is the neutron density in the m-th mesh , ^^^^ is a fraction, and is the decay constantof the i-th group (as neutron precursors can be consolidated into 6 groups, where the difference of the groups are the decay constants), ^^^^is the concentration of the delayed neutron precursor from the i-th group, describes neutron sources and drains as leakage rate, emergence rate caused by the neutron influx from other meshes, the source rate, including spontaneous emission and artificial neutron sources, Σ^^is the macroscopic neutron absorption cross section, is the macroscopic fission cross section, ^^ is the average number of neutrons emitted by a single fission, ^^ is the neutron velocity, ^^ being the time. Due to neutron migration from one mesh to another, which is included in the term as leakage and influx, kinetic equations for single meshes become coupled with each other. In case of a 12-mesh model, a system of 12 x 7 = 84 differential equations is obtained. Using the neutron migration theory one can calculate rates of leakage and influx for each mesh and put these rates into (1). On this way one obtains for internal meshes m=2..11from (1): and for boundary meshes b=1 and b=12, where the boundary meshes are in this example the uppermost and lowermost meshes:with ^^ = ℎ2 , being a neutron migration constant, ^^ = is a neutron leakage constant,with ^^2being the migration length, which is for a PWR about 6.6 cm, h being the heightof a cylindrical mesh, ^^^^ , and ^^^^ are the source rate of the m-th or b-th mesh, ^^^^ and^^^^ are form factors of axial power distribution for internal and boundary meshesrespectively: Formfactors are dimensionless scalars depending only on the curvature of the axial distribution of neutron density in the reactor core. The above terms can be simplified, considering the reactor core during normal power operation. In such a case the reactivity changes and the changes of the curvature of the axial power distribution are much slower than the reaction of neutron density on them and the neutron sources can be neglected. Then, the following equation for m = 1..12 can be obtained: with being absorption reactivity, Δ^^^^reactivity term caused by neutron between meshes and axial neutron leakage (reactivity change due to neutron migration).For internal meshes Δ^^^^ = ^^ ∙ ^^^^ and for boundary meshes Δ^^ ^^^^ = 2 ^^^^ − ^^ ∙ (2 − ^^^^) .Equations (7) and (2) compose a system of 7 x 12 = 84 first order differential equations for ^^(^^) and ^^^^(^^) : Under the assumption that the reactivity changes and the changes of the curvature of the axial power distribution are much slower than the reaction of neutron density on them, and power changes in a pressurized water reactor PWR core during normal operation are slow compared to the decay of the delayed neutron precursors, one can consider the system (8, 9) as a system of linear differential equations with constant coefficients. Such kind of differential equations is known to have exponential solutions. So, one can look for solutions in following form: with ^^^^known as reactor period, which is in the present case a mesh period. Inserting this expected solution form into the system (8, 9) and processing the equations one obtains a system of 12 coupled algebraical equations for ^^^^. These equations are called inhour equations: The coupling is given by the terms Δ^^^^describing the neutron exchange between the meshes. To simplify numerical calculations equations (12) can be rewritten as follows: with ^^^^ = ^^^^⁄ ^^ . For example, in case uranium is taken as fuel (235U), the terms for and ^^^^can be looked up in a table in the book “Nuclear Reactor Physics, Weston M. Stacey, Second Edition, 2007, ISBN 978-3-527-40679-1: i ^^^^[^^−1] ^^^^1 0.0124 0.033 2 0.0305 0.219 3 0.111 0.196 4 0.301 0.395 5 1.14 0.115 6 3.01 0.042 Table 1 Further, equation (13) can be rewritten as follows: where is known as effective neutron generation time determined as follows: In order to further simplify the inhour equation, the reactor power gradient during normal operation is assumed to remain within the region from -10% / min up to +10% / min, where the percentage value is taken from the current power and not from the rated power.Using values Λ = 6 ∙ 10−5 and ^^ = 0.0075 , which are typical for PWR, several numericalcalculations of (14) and (15) are performed with the power gradients as variable. Power^^ [^^] ^^−1 [^^−1] ^^^^^^^^(^^) [^^] ^^∞,^^ + Δ^^^^ [^^^^^^]gradient -10% / min -600 −1.6̅x10-3 0.1043 -17.38-5% / min -1200 −8.3̅x10-4 0.1009 -8.41-1% / min -6000 −1.6̅x10-4 0.09841 -1.640.0% / min Infinity 0.0 0.09782 0.0 1% / min +6000 1.6̅x104 0.09724 1.625% / min +1200 8.3̅x10-4 0.09500 7.9210% / min +600 1.6̅x103 0.09236 15.39Table 2 The numerical values of Table 2 show a nearly perfect linear dependence between^^−1 and in the region of -10% / min to +10% / min for reactor power gradient, or inthe present case, reactor mesh power gradient. This fact allows a linear regression: where ^^^^^^^^ (0) = 0.098 ^^ and ^^ = −3.5731 ^^2 .It follows thereof, in this example, the simplified inhour equation is as follows: ^^^^^^^^ ^^ ∙ ^^−1^^ ^^ ^^2 If other fuels are used, the numerical values may be slightly different. Then, the inhour equation is with X and Y being constants, which slightly depend on the fuel. The values can be deduced from tables corresponding to the table 1 for other fuels. For example, the239Pu values are rather similar to the one of table 1. The expression (18) defines a relation between the mesh reactivity, in particular the sum of the absorption reactivity and the reactivity change due to neutron migration, and (only) the mesh period as variable, wherein the inhour equation is simplified under the condition that the absolute value of the power gradient being within a predetermined region . Numerical values X and Y are calculated for a particular fuel. For example, in the above case, the fuel is uranium and / or the absolute value of the power gradient is below 15% / min, in particular below 10% / min. The inhour equation can be further simplified, when considering that the quadradic term in equation (17) in case of power gradient below ±10% / min, is less than 10-5(1.0 pcm) and in the most cases less than one step of the control rods. Thus, the quadratic term can be neglected: The effective neutron generation time can be defined as X, as for the equation (18). Then, in the simplified inhour equation, the reactivity of a mesh, in particular the sum of the absorption reactivity and the reactivity change due to neutron migration, is inverse proportional to the reactor (mesh) period. The mesh period ^^^^can be also defined as follows: where ^^^^is the fission rate in the m-th mesh. The reactor model can be solved by determining the fission rate. If the fission rate as a function of time and location is known, all other process values in the reactor can be calculated. The location is given by the number of mesh m, and for computation of the time dependency discretization of the next predetermined period of time is needed. For example, the next 24 h can be divided, as an example, into 1440 prediction steps 1 min each. These prediction steps can be numerated by an index n = 0, 1, 2, 3, …1439, 1440. In this case the fission rate ^^ will be calculated for 12 points in space and for 1441 point in time. All together 12 x 1441 = 17292 points. The derivative in (20) can be determined as: with ∆^^ time step of prediction, in present example 1 min. Inserting (21) into (20) one obtains with ^^ and being values, which do not depend on the fission rate in the m from the prediction step n. The values ^^1,^^,^^and can be determined from equation According to equations (3) to (7) and using proportionality between neutron flux and fission rate, Δ^^^^,^^can be represented as with ^^2,^^,^^and ^^2,^^,^^being values, which do not depend on the fission rate in the mesh m from the prediction step n. In other words, ^^2,^^,^^and ^^2,^^,^^do not contain ^^^^,^^. The absorption reactivity ^^∞,^^,^^depends on temperatures of fuel and coolant. Temperature gains in a mesh are proportional to thermal power of the mesh, which is linear function of the fission rate in the mesh. According to this consideration, the absorption reactivity in a mesh can be represented as a linear function of the fission rate in the mesh: ^^∞,^^,^^ = ^^^^,^^ ∙ ^^^^,^^ + ^^3,^^,^^ , (24)with and ^^3,^^,^^being values, which do not depend on the fission rate in the mesh m from the prediction step n. In other words, and ^^3,^^,^^do not contain ^^^^,^^. The equation can be determined by using the Tailor expansion of ^^∞,^^,^^(^^^^,^^) up to the first power of ^^^^,^^. The coefficient ^^ can be also called reactivity coefficient and ^^ is the residual reactivity. When inserting (22-24) into (19), the reactor model equation will take the form: By introducing and and multiplying all terms by ^^^^,^^the reactor model equation (25) will take the form: not explicitly dependent on ^^^^,^^In other words, ^^ , So, a quadratic equation is obtained, which can be simply solved by using the quadratic formula: The number of quadratic equations corresponds to the number of meshes multiplied by the number of prediction steps, in the present case 17292 equations, whereas the meshes are connected with each other due to the neutron transport and prediction steps are connected with each other by the inhour equation describing the mesh dynamics. These connections are given by the coefficients ^^ and ^^ , which depend on the fission rates in neighbor meshes and in the same mesh but in previous prediction step, see FIG.4. According to an embodiment, keeping the quadratic term in (18), a cubic equation may be obtained for ^^^^,^^: with ^^^^,^^being the fission rate for a mesh, index n being the number of the prediction step, index m being the number of the mesh in the core, with which are not dependent on the fission rate from the same mesh and same prediction step, in particular being not dependent from fission rate with indices m and n from previous iteration As ^^^^ = ^^^^ ∙ ^^^^ ∙ ^^^^ , where ^^^^ is the neutron flux in the m-th mesh, ^^^^ is themicroscopic fission cross section, and ^^^^ is number of the fissile nuclei in the m-th mesh,^^^^ = ^^^^ ∙ ^^ , where ^^^^ is the neutron concentration in the m-th mesh, and ^^ is theneutron velocity, alternatively the neutron concentration or neutron flux may be used instead of fission rate as independent variable in the equations (28), (29) and (30). Then, according to an embodiment, based on the derived reactor model, a fission rate and the process values for each of the meshes and each of the prediction steps within predetermined period of time are calculated, see FIG.3. The calculation of 17292 values for fission rate will be performed starting from the bottom of the reactor core (m=12) to the top (m=1) and from the time point ‘now’ (n=0) into the future up to the end of the predetermined period of time (n=1440). The m loop is nested inside the n loop. In one run an array of 17292 values for ^^^^,^^is calculated. All other values of the process variables are calculated using this array. Then, the acquisition modules 1000 – 1020 will be processed again and the calculation repeats starting again from n=0 m=12. One run of such calculation, which corresponds to one loop in FIG.3, takes for example 1 ms. The device 58 runs in an infinite loop for calculating the fission rate, so that the algorithm of 58 can be considered as an anytime algorithm. Although the calculation method is iterative regarding the ^^^^,^^array, since each run uses the calculation result of previous run, each single calculation of fission rate ^^^^,^^, however, do not use the fission rate value with the same indices. The coefficients a, b, and c in the equation (29) contain ^^^^−1,^^, ^^^^,^^−1, ^^^^,^^+1, see FIG.4. With the calculation sequence described above, the only value, which is taken from the previous run is , see FIG.4. The dependence of ^^^^,^^on ^^^^,^^−1is, however, weak comparing to other dependences. This fact is responsible for very fast convergency of the whole array, which makes the simulation process substantial faster comparing to other known methods. The result converges very fast after a limited number of runs. For example, latest after 3 runs, numbers with a sufficient precision are achieved. For example, the inherent precision of a floating- point numbers in the device 58 is achieved after 20 runs. The term run is used instead of iteration, as process values and the schedules acquired during steps 1000-1020 are also (slightly) changing from one run to the next. The main process variable or value, which is calculated in step 1030 is fission rate ^^ as a function of coordinate m and time n, see equation (29) and FIG 3. Having the array ^^^^,^^, all other process values of the process variables will be calculated. Spatial distribution of fission rate allows among others, calculation of axial power distribution and axial offset. The chosen prediction time steps within the predetermined period of time, defining the time resolution of the prediction, are for example between 20 seconds and 5 minutes, for example between 30 seconds and 2 minutes, for example about 1 minute. With a prediction step of 1 min, predictive simulation for 24 h requires 1441 prediction points. The number of prediction points is predefined and can be adapted in order to obtain sufficient resolution and accuracy. According to an embodiment a prediction of 24 h of operation, i.e. the next predetermined period of time is 24 h, needs about 1 ms for one run. After for example 3 runs which takes for example 3 ms, sufficient precision is achieved. In step 1040 process values calculated for the next predetermined period of time, for example 24 h, will be outputted and graphically displayed on the operator screen 66. Furthermore, it is verified whether the simulated process values remain within a predetermined operation domain. For example, in case if the predicted process values at some time point in the future leave the predetermined operation domain, the boration / dilution schedule and optionally an insertion schedule for heavy and grey banks can be adapted manually or automatically to avoid exceeding the limits of the operation domain during the predetermined period of time. For automatic procession of the schedules for example, a device like one, described in WO2020 / 224764 A1 may be used. According to embodiments, one run of 1010 - 1020 – 1030 takes about 1 ms for a PC processor, so every 1 ms a new set of process values for 12 meshes and 1441 time points will be calculated and outputted. This computation rate allows the operator to work interactively with the device. So, by manual adaptation of the load schedule, and / or boration / dilution schedule, and / or the insertion schedule for heavy and grey banks, the operator will see the impact of his adaptation immediately on the screen 66. Furthermore, this computation rate allows to use the method and the device 58 as a solver for an AI application like the one disclosed in WO2020 / 224764 A1. In this case, the adaptation of the boration / dilution schedule and / or optionally the insertion schedule for heavy and grey banks will be performed automatically using reinforcement learning process. Optionally, the module 1050 can automatically apply the schedules to the plant: the boration / dilution injection schedule, heavy and grey banks insertion schedule and / or the load schedule. For example, the injection controller 52 can be adapted to perform the boration and dilution automatically according to the boration / dilution injection schedule, and the turbine controller 38 can be adapted to follow the load schedule automatically. In some embodiments also the position of the heavy 47 or / and grey rod banks 45 may be adapted based on the corresponding insertion schedule. Otherwise, the schedules can be applied to the plant manually by an operator. The invention disclosure enables a more plannable operation of a nuclear power plant, which is in particular important for load follow operations. Ultrafast predictive simulation (24 h of operation in about 1 ms) allows the reactor operator to work interactive with the device, to adapt schedules for electricity production, boration / dilution, and possibly insertion of heavy and grey banks, instantly observing on the screen 66 the effect of his planning on the reactor and plant behavior within the prediction horizon. The invention disclosure enables on this way, for example, to avoid exceeding the limits of the operation domain in the predicted future increasing operation predictivity, reliability and safety. According to the invention, the differential equations are solved analytically, allowing ultrafast numerical calculation within a digital device and thus an instant interaction with the operator. Furthermore, the ultrafast numerical calculation can be used as a solver for devices as disclosed in WO2020 / 224764 A1 and WO2023 / 151786 A1. List of reference signs: 1 nuclear power plant 3 pressurized water reactor 5 reactor core 7 reactor pressure vessel 10 primary cooling circuit 12 reactor coolant pump 14 steam generator 16 secondary cooling circuit 18 steam turbine 20 electric generator 22 electrical grid 24 condenser 26 feedwater pump 28 feedwater tank 30 turbine valves 32 steam feed line 34 bypass line 36 bypass valves 38 turbine controller 40 bypass controller 41 power control banks (P-banks) 42 in-core detectors 43 ex-core detectors 44 boration pump 45 grey banks (G-banks) 46 dilution pump 47 heavy banks (H-banks) 48 boration valve 50 dilution valve 52 injection controller 54 neutron flux controller 56 reactor power controller 58 device 60 boration or dilution injection schedule 62 load schedule 64 insertion schedule for heavy and / or grey banks 66 screenList of variables:^^ fraction of the delayed neutrons^^^^ fraction of the i-th delayed neutrons group^^^^ concentration of the delayed neutron precursor from the i-th group^^ neutron migration constant^^ form factor of the axial power distribution^^ neutron flux^^^^^^^^ effective neutron generation time^^ neutron leakage constant^^^^ decay constant of the i-th groupΛ prompt generation time^^ index for the number of mesh^^ index for the number of the time step of prediction^^ number of fissile nuclei^^ average number of neutrons emitted in a single fission^^^^(^^) neutron density in the m-th mesh^^^^,^^ fission rate in the m-th mesh for the n-th prediction step^^∞ absorption reactivity∆^^^^ reactivity change due to neutron migration^^^^ microscopic fission cross sectionΣ^^ macroscopic fission cross sectionΣ^^ macroscopic neutron absorption cross section^^, ^^^^, ^^^^ source rates^^ mesh period, reactor period^^ timeΔ^^ time step for the predictive simulation^^ neutron velocityX constant, depending on the fuel Y constant, depending on the fuel

Claims

CLAIMS 1. Computer implemented method for controlling a pressurized water reactor, comprising - Receiving (1000) a load schedule (62) for a next predetermined period of time for the electrical power output; - Receiving (1010) a boration / dilution injection schedule (60) for the next predetermined period of time; - Receiving (1020) current measured process values; - Simulating (1030), based on the received current process values, the received load schedule (62), and the received boration / dilution injection schedule (60), the process variables of the pressurized water reactor for the next predetermined period of time, characterized in that the simulating includes: dividing the core into a plurality of meshes; calculating a fission rate and the process values for each of the meshes and each of predefined time points within the next predetermined period of time, wherein the fission rate is calculated based on a reactor model for each mesh, the reactor model being based on an inhour equation defining a relation between the reactivity and the reactor period as a variable.

2. Method according to claim 1 or 2, wherein the measured process values include: a coolant inlet temperature (T1), a coolant outlet temperature (T2), a live steam pressure (p), axial offset (AO), a thermal power of the reactor core, power control bank positions, positions of heavy and grey banks, in-core neutron fluxes, ex-core neutron fluxes, boron concentration, reaction rates, heating powers, fuel temperatures, average coolant temperature and / or coolant temperatures.

3. Method according to one of the preceding claims, further comprising determining non-measurable process values, in particular reaction rates, heating powers, fuel temperatures, xenon concentrations, and / or an iodine concentrations, based on the measured process values, wherein the simulating is further based on the determined non-measurable process values.

4. Method according to one of the preceding claims, wherein the fission rate and the process values for each of the meshes for the next predetermined period of time are calculated iteratively regarding the whole array of the values, wherein a calculation of each particular value of the array do not use the same value from the previous run.

5. Method according to one of the preceding claims, wherein the number of meshes is between 8 and 16, in particular about 12.

6. Method according to one of the preceding claims, further comprising receiving (1010) an insertion schedule for heavy banks and / or grey banks (64) for the next predetermined period of time, wherein the simulating (1030) is further based on the received insertion schedule for heavy and / or grey banks.

7. Method according to any one of the preceding claims, further comprising the step of: - verifying (1040) whether the simulated process values remain within a predetermined operation domain; and - operating (1050) the pressurized water reactor using the boration / dilution injection schedule, the insertion schedule for heavy and / or grey banks, and / or the load schedule.

8. Method according to one of the preceding claims, wherein the fuel is uranium, plutonium, or a mixture of uranium and plutonium.

9. Method according to one of the preceding claims, wherein the absolute value of the power gradient is below 15% / min, in particular below 10% / min.

10. Method according to one of the preceding claims, wherein the next predetermined period of time is at least 2 h, in particular at least 12 h, for example 24 h.

11. Method according to one of the preceding claims, wherein the inhour equation is defined by the following equation:^^∞,^^ + Δρ^^ = ^^^^^^or by the following equation^^∞,^^ + Δρ^^ =^^^^ − ^^^^2^^ , withbeing the absorption reactivity of the m-th mesh,Δ^^^^ being the reactivity change due to the neutron migration, ^^^^ being the meshperiod, and X and Y being constant values depending on the fuel.

12. Method according to claim 11, wherein, for a uranium core of the pressurized water reactor X= 0.098 s and Y=3.5731 s2, s being the unit seconds.

13. Method according to one of the preceding claims, wherein for each mesh a quadratic or cubic equation is provided, based on the reactor model.

14. Method according to claim 13, wherein the quadratic or cubic equation iswith ^^^^,^^being the fission rate for a mesh, index n being the number of the prediction step, index m being the number of the mesh in the core, with,which are not dependent on the fission rate from the same meshand same prediction step, in particularbeing not dependent from fission rate with indices m and n from previous run .

15. Method according to claim 13, wherein the quadratic or cubic equation iswith ^^^^,^^being the fission rate for a mesh, index n being the number of the prediction step, index m being the number of the mesh in the core, with,,and ^^ ^^,^^ which are not dependent on the fission rate from the same meshand same prediction step, in particular, ^^^^,^^, ^^^^,^^, and ^^^^,^^being not dependent from fission rate with indices m and n from previous run .

16. A computer program product comprising instructions, which, when the program is executed by a computer, causes the computer to carry out the computer implemented method of one of the preceding claims.

17. A data carrier signal carrying the computer program product of claim 16.

18. A computer-readable storage medium comprising instructions which, when executed by a computer, causes the computer to carry out the computer implemented method of one of the preceding claims 1 to 15.

19. A data processing system, in particular a device (58), comprising means for carrying out the computer implemented method of one of the preceding claims 1 to 15.

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