The present disclosure relates to the regulation of operating parameters of a nuclear reactor.
Given the large share of nuclear power in the French energy mix, the massive introduction of renewable energy sources (wind and solar) into the electricity grid induces an additional need for flexibility in energy production, thus affecting nuclear reactors. Flexibility reflects the capacity of an electricity production unit to adapt its production. Renewable energies have the particularity of being intermittent over time or dependent upon climatic uncertainties. In the absence of a mass energy storage system, this increase in flexibility induces an increased need for adjusting the power made available by traditional generators, including nuclear reactors in France.
In addition, the regulation of the operating parameters of a nuclear reactor must be done in accordance with very restrictive multi-objective specifications. In particular, the regulatory systems must minimize the variations of the operating parameters and minimize the stresses on the actuators. At present, the regulatory systems of nuclear reactors are based on PIDs. However, they can only take these last constraints into account in a very imperfect manner.
In this context and according to a first aspect, an aim of the present disclosure is to provide a method of regulating the operating parameters of a nuclear reactor offering additional flexibility, while allowing compliance with very restrictive multi-objective specifications.
To this end, the present disclosure relates to a method of regulating operating parameters of a nuclear reactor, these operating parameters comprising at least the average temperature of the core and the axial power imbalance, the method comprising the following steps:
The regulation process may also have one or more of the characteristics below, considered individually or in any technically feasible combination:
u
K
=K
p(s)y1+Ki(s)y2 with y1=y and y2=z
where Kp and K1 are gain matrices, s is the Laplace variable, y being an output deviation vector between the current value of the output vector and the reference value of the output vector, z being a deviation vector of operating parameters between the current value of the vector of operating parameters to be checked and the reference value of the vector of operating parameters to be checked, and uK being the vector of the corrective values of the commands;
to be observed at least for a disturbance which is a power step demanded from the turbine(s) of ±P% of a nominal power PN of the nuclear reactor, P being between 5 and 15%;
to be observed at least for a disturbance which is a power step requested from the turbine(s) of ±P% of a nominal power PN of the nuclear reactor, P being between 5 and 15%;
where K denotes the gain matrices Kp and Ki, Ω denotes the set of gain matrices stabilizing the loop system, Tz→d
where Cb is the concentration of neutron poison in the or each primary circuit; uQ is the command to increase the concentration of neutron poison in the primary circuit resulting from a command for injecting neutron poison at a flow rate Qbor,
According to a second aspect, the present disclosure relates to a nuclear reactor comprising:
The nuclear reactor may also have one or more of the characteristics below, considered individually or in any technically feasible combination:
u
K
=Kp(s)y1+Ki(s)y2 with y1=y and y2=z
where Kp and K1 are gain matrices, s the Laplace variable, y being an output deviation vector between the current value of the output vector and the reference value of the output vector, z being a deviation vector of operating parameters between the current value of the vector of the operating parameters to be checked and the reference value of the vector of the operating parameters to be checked, and uK being the vector of the corrective values of the commands.
Other characteristics and advantages of the present disclosure will emerge from the detailed description which is given below, by way of indication and in no way limiting, with reference to the appended figures, among which:
The method represented schematically in
The nuclear reactor 1 comprises:
The neutron poison is typically boron. The diluent fluid is typically water.
Typically, the primary circuit 10 comprises one or more loops, each with a hot branch and a cold branch.
The nuclear reactor 1 comprises one or more turbines 17 supplied with steam by the primary circuit 10.
The nuclear reactor 1 is typically a PWR (Pressurized Water Reactor). It comprises a steam generator 19 for each loop of the primary circuit 10. Each loop connects the tank 3 in a closed circuit to a primary side of the associated steam generator 19. Furthermore, the nuclear reactor 1 comprises a secondary circuit 21 connecting in a closed circuit, a secondary side of the or each steam generator 19 to the associated turbine 17. Turbine 17 drives an alternator 23.
As a variant, the primary heat transfer fluid directly may drive each turbine.
The nuclear reactor 1 also includes a set of 25 control operating parameters of the core, typically called Core Control. The regulation assembly 25 comprises, for example, an information processing unit, formed by a processor and a memory associated with the processor (not shown). As a variant, the regulation assembly 25 is produced in the form of programmable logic components such as FGPAs (Field-Programmable Gate Array), or else in the form of dedicated integrated circuits such as ASICs (Application-Specific Integrated Circuit).
The regulation assembly 25 is configured to move the clusters 7 by functional groups. The clusters of the same group are moved together, and are always all in the same insertion position.
The groups are advantageously divided into one or more sets. The distribution of the groups and their use depend on the control mode of the nuclear reactor.
In certain control modes, several groups are brought together in a first set.
In this case, the groups of the first set are typically moved sequentially. By this is meant that they are inserted one after the other, with an overlap which is predetermined and as described below. Alternatively, the overlap may be variable.
According to an alternative embodiment, all the groups of the first set are moved together. This is understood to mean that they are all always in the same insertion position and move together.
In other control modes, the first set has only one group.
In some control modes, other groups are brought together in a second set.
In this case, the groups of the second subset are typically moved together.
In other control modes, all the groups are gathered in the first set, and there is no second set.
In still other control modes, certain groups are gathered in a third set in addition to the first and second sets.
In all cases, the regulation assembly moves the groups of the same assembly in a coordinated manner (sequentially, jointly, etc.). The regulation assembly moves the groups of the or each assembly to control the operation of the reactor, in particular to regulate the operating parameters.
Several examples of control mode are detailed below, inspired respectively by modes T, G and A.
In a control mode inspired by T mode, particularly suited to the European Pressurized Reactor (EPR), the groups are divided into two sets:
The first set is particularly well suited for checking the average temperature Tm. The second set Hbank is particularly well suited for checking the axial offset AO.
The first and second sets are of variable composition, as described below.
For example, the control clusters 7 are grouped into 5 groups P1 to P5.
As shown in
The sequence would be the same if the control clusters were grouped into a different number of clusters.
Thus, the groups of the first set Pbank are moved sequentially. The groups of the second Hbank set are moved together.
The term “position of the first Pbank set” is understood here to mean the cumulative position of the groups belonging to the first set. This position is denoted Pbank.
For example, the position is calculated using the following equation:
P
bank=min(P4, 214)+min(P3, 214)+min(P2, 214)+P1
where P1, P2, P3 and P4 denote the positions of groups P1 to P4 respectively. The value 214 is chosen for a mid-core position of the cluster groups. The position is expressed in number of extraction steps from the maximum insertion position of the groups
By position of the second set Hbank is meant here the position of group P5, which is never integrated into the first Pbank set.
In a second control mode, inspired by G mode, the groups are divided into two sets:
the first set includes a single group, the R group, formed of black clusters, i.e. very absorbent clusters;
the second set, called PCG for Power Compensation Group, is made up of groups G1 and G2 (G for gray clusters) and groups N1 and N2 (N for black clusters). Gray clusters are relatively less absorbent than black clusters.
The groups of this second subset are inserted sequentially. Advantageously, they are inserted as a function of the electrical power demanded from the turbine.
In this second control mode, the AO is advantageously controlled mainly by injections of neutron poison or of diluent.
In a third control mode, inspired by mode A, the functional groups are all grouped together in the first set, here designated by the acronym DCBA.
This is typically made up of four groups A, B, C and D, which fit sequentially like the Pbank.
The first set is particularly well suited for controlling the average temperature Tm.
In this third control mode, the AO is advantageously controlled mainly by injections of neutron poison or of diluent fluid.
The operating parameters to be controlled include at least the average temperature of the core Tm and the axial power AO imbalance.
The average core temperature Tm is defined here as being the average between Tf and Tc, Tf and Tc being the temperatures of the primary coolant at the outlet of the core 3 and at the inlet of the core 3, i.e. at the hot and cold branches 13 and 11.
When the primary circuit has several loops, the average temperature of the hot and cold branches of the primary circuits is considered for example.
The axial power imbalance AO is expressed using the following relation:
AO=(FH−FB)/(FH+FB)
where FH and FB are the neutron fluxes respectively in the upper and lower parts of the core.
Advantageously, the operating parameters to be checked further include the position Pbank of the groups of the first set.
This is typically the case at least for the first mode of controlling the reactor.
This amounts to controlling the operating parameter Pmax, i.e. the maximum power that may be reached by rapidly extracting the groups to their maximum extraction position.
In fact, the Pmax is advantageously translated into a reference position of the Pbank group making it possible to compensate for the power fault. Controlling the Pmax therefore amounts to controlling the position of the Pbank group according to an insertion profile determined as a function of the power of the core and the power to which the operator wants to be able to return. For example, a Pmax of 100% PN means that the position of the Pbank group makes it possible to return to 100% PN only by its extraction.
For the second and third modes of controlling the reactor, the position of the groups of the first set is not typically part of the operating parameters to be controlled. These only include Tm and AO.
The regulation process takes into account at least one input and several outputs, as shown in
An input is defined either as a predicted path, or as an additional constraint or a constraint modification applied to the control process.
The at least one input is typically a power demanded from the turbine(s) of the nuclear reactor.
Typically, the power supplied by the turbine(s) 17 of the nuclear reactor comprises two components: a programmed power DU, according to a predetermined program, and a power disturbance dp. The programmed power is, for example, predetermined for a period of at least one day. The power disturbance corresponds, for example, to an adjustment made in the operation of the primary circuit, or of the secondary circuit, to a charge reserve step, etc.
According to an alternative embodiment, the term power demanded from the turbines or nuclear power reactor is understood to mean the programmed power DU.
According to another variant embodiment, the term “power requested from the turbine(s) of the nuclear reactor” is understood to mean the power supplied DP, with DP=DU+dp.
The at least one alternative input includes one or more of the inputs below, in addition to, or in place of, the power demanded from the turbine(s):
This list is not exhaustive.
The outputs preferably include, in addition to the operating parameters, the temperature Tc of the primary coolant in the hot branch 13, and the thermal power of the core Pk.
To allow regulation, the nuclear reactor 1 comprises:
The nuclear reactor 1 comprises a control system equipped with a set of sensors making it possible to access the current values of the following quantities: Tc, Tf, AO, the power of the core PK, and Pbank.
The control system may also provide DU and is equipped with sensors for accessing DP.
The acquisition unit 27 is configured to acquire the current value of the power demanded from the turbine(s) directly from the control system.
The acquisition unit 29 is configured to acquire the current values of certain outputs directly from the control system, in particular Tc, AO, PK, and Pbank. The acquisition unit 29 is configured to calculate the current values of the other outputs from values supplied by the control system, in particular Tm.
The acquisition units 27 and 29 are, for example, modules of the regulation assembly 25 or directly inform the regulation assembly 25.
In the regulation process, the operating parameters are regulated by giving commands to actuators.
These commands advantageously comprise at least one rate of movement Vbarres of the control clusters, and at least one injection rate of neutron poison or of dilutent fluid.
In certain control modes (the first control mode for example), the at least one rate of movement Vbarres of the control clusters typically comprises the rate of movement of the groups of the first Pbank set and the rate of movement of the groups of the second Hbank set.
These rates correspond to the derivatives over time of the position of the first Pbank set as defined above and of the position of the second Hbank set as defined above.
These rates vPbank and vHbank are respectively noted.
In other control modes (the second and third control modes, for example), the at least one rate Vbarres of movement of the control clusters typically corresponds to the rate of movement of the groups of the first set.
The neutron poison or diluent fluid injection rate is typically expressed as a rate of change in the neutron poison concentration in the primary coolant denoted uQ. In other cases, it is expressed in the form of a mass flow rate injected into the primary coolant, denoted Qbor or Qdil.
The commands are produced by the regulation assembly 25, which transmits them to the actuators. The actuators are the drive mechanisms 9 of the clusters, and the injection circuit 15 of neutron poison or diluent fluid.
The regulation process is designed to comply with specifications, i.e. a plurality of objectives. These constraints will be described only for the first control mode.
The Core Control is constrained by an authorized operating domain in which the deviations of the operating parameters from their references must be maintained. This domain is defined by the Limiting Condition Operation (LCO), i.e. the upper and lower limits for each controlled operating parameter (namely, Tm, AO and Pbank) with respect to their references.
We define these references by:
Tm,ref the average temperature reference
AOref the AO reference
Pbank,ref the position reference of the Pbank group
The deviation of the parameters to be checked aganst their references is given by:
ΔTm the average temperature deviation from its reference.
ΔAO the deviation of AO from its reference.
ΔPbankthe position difference of Pbank from its reference.
We then define the limits of the field of operation such as:
ΔTm ∈[−ΔTmmax, ΔTmmax]
ΔAO ∈[−ΔAOmax, ΔAOmax]
ΔPbank ∈[−ΔPbankmax, ΔPbankmax]
Typical values given for these parameters are for example:
ΔTmmax1.5° C.
ΔAOmax=5% AO
ΔPbankmax=30 steps
The Core Control must help realize the flexibility demands on the reactor, for example:
Adaptation to the variation in daily demand (load monitoring)
Load ramps: +5%/min PN between 25% and 100% of PN
Adaptation to real-time demand variation (frequency control)
Primary: ±5% PN at 1%/s
Secondary: ±5% PNat 1%/min
Adaptation to network disturbances (rotating reserve)
Step: ±10% PN between 30% and 100% PN
The purpose of Core Control is to keep the outputs to be controlled within the authorized operating range defined above, regardless of the power variations specified above.
The maximum authorized demands are defined as follows:
Saturation of the control clusters in positions:
Minimum position: Pbarresmin=0 steps extracted
Maximum position: P barresmax=410 steps extracted
Minimum speed: Vbarresmax=4 steps/min
Maximum speed: Vbarresmax=75 steps/min
Maximum and minimum rates of neutron poison injection/dilution:
Min. neutron poison flow rate: Qbormin 322 kg/s
Max. neutron poison flow rate: Qbormax=3 kg/s
Min. diluent fluid flow rate: Qdilmin=1 kg/s
Max. flow rate of diluting fluid: Qdilmax=10 kg/s
Min. concentration: Cbmin=0 ppm
The regulator must have guarantees of robustness:
Minimum modulus margin: Mm=0.5
The references of the outputs to be controlled depend on the operation of the core. They are defined as follows:
The reference temperature Tm,ref is a function of the power of the nuclear reactor. It is read directly on a predetermined curve, as a function of the power requested from the turbine(s). Here we consider the current value of the at least one input.
The reference axial power imbalance AOref is updated periodically, for example every month, to take into account the exhaustion of the core. It is supplied directly by the operator of the nuclear reactor and is considered constant between two updates.
The reference position Pbank,ref of the Pbank sub-assembly is a function of the power of the nuclear reactor. It is read directly on a predetermined curve, as a function of the power demanded from the turbine(s). This reference position is given in cumulative position.
The method of regulating the operating parameters will now be detailed, for the first control mode.
This process was designed to take into account the fact that the control of a nuclear reactor core has specific features.
The reactor has disparate dynamics, i.e. both slow and fast. The dynamics linked to xenon are very slow (of the order of an hour) while those linked to power and temperature are rather fast (of the order of ten seconds).
Over the entire operating domain, the behavior of the core is highly non-linear, mainly due to the insertion of the cluster groups into the core. Between the maximum power (100% PN) and an intermediate power (e.g. 60% PN), the effect of the groups of clusters on the various operating parameters to be controlled changes greatly. We even observe a reversal of the effects of the actuators in certain areas of the core.
The neutron poison actuator considered in the problem of control has a significant delay: 300 seconds. Although the system that one seeks to control is relatively slow (overall order of magnitude: 10 seconds), a delay of this magnitude is significant for the regulation.
The specifications set out above contain a large number of constraints, including time constraints that it is often difficult to take into account by traditional control techniques.
Currently, each electricity production unit receives a daily load variation program. This program is therefore known in advance. However, we are not currently taking advantage of this signal to predict future commands. In the present control method, it is possible to consider the future variations of this signal for the development of the command.
To respond to the difficulties specified above, the regulation method implements a hierarchical control strategy.
The method comprises the following steps:
acquisition of the current value DU, DP of the at least one input;
acquisition of the current value Y of the output vector;
development of a reference value Yref of the vector of the outputs, using the current value DU, DP of the at least one input signal;
development of a vector US of values of the commands of the nuclear reactor by a supervisor 31 implementing a predictive control algorithm, using at least said current value DU, DP of the at least one input and the current value Y of the vector of outputs;
development of a vector uK of corrective values of the nuclear reactor controls by a regulator 33 implementing a sequenced gain control algorithm, using the current value Y of the vector of the outputs and the reference value Yref of the vector of the outputs;
development of a vector U of corrected values of the commands of the nuclear reactor, using the vector US of the values of the commands produced by the supervisor 31 and the vector uK of corrective values of the commands produced by the regulator 33;
regulation of the operating parameters of the nuclear reactor, by controlling actuators using the vector uK of corrected values of the controls.
As described above, the at least one input is typically the power demanded from the turbine(s). This typically corresponds to the programmed power, supplied, for example, by the load monitoring program known in advance.
Alternatively, it is the real power of the turbine, denoted here DP, given by the following equation: DP=Dudp.
The at least one alternative input includes one or more of the inputs listed above, in addition to or instead of the power demanded from the turbine(s).
Advantageously, the reference value Yref of the vector of the outputs is determined only from the programmed power Du. The reference value Yref is therefore not modified by the power variations considered to be random, i.e. given by dp.
The reference value Yref of the vector of the outputs is as follows:
Y
ref=(Tc,ref Tm,ref AOref PK
zTc,ref is the reference hot branch temperature. It is read on a predetermined curve, giving Tc,ref directly as a function of the current value of the at least one input.
Tm,ref, AOref, and Pbank,ref are determined as described above.
PK
The vector of the corrected commands U, i.e. the commands given to the actuators, are obtained by adding the vector US of the values of the commands produced by the supervisor 31 and the vector uK of the corrective values of the commands produced by the regulator 33: U=US+uK
The current value Y of the output vector is as follows:
Y=(Tc Tm AO PK Pbank)T
Y is obtained as described above.
The supervisor 33 considers the vector y as input, defined as being the difference between the current value Y of the vector of the outputs and the reference value Yref of the vector of the outputs:
y=Y−Y
ref, with y=(δTc δTm δAO δPK δPbank)T
The sequenced gain regulator 33 addresses the following issues:
Ensure close control of the system by ensuring good performance a priori around each operating point, in particular for the rejection of disturbances linked to frequency adjustment.
Control the reactor over the entire operating range by adapting the gains that make it up as the operation progresses.
Guarantee robustness (multi-objective approach), locally, around the operating points.
Take into account a large number of command constraints, imposed in the specifications.
However, for systems exhibiting large non-linearities as is the case here, a sequenced gain regulator may show poor performance. In fact, it is synthesized at each operating point on the basis of a linearized model. However, the use of a linearized model may lack representativeness of the global non-linear model. The disadvantage is that the trajectories taken by this regulator may then be far from the optimal path of the overall behavior.
The sequenced gain regulator is a structured regulator, preferably of the multivariable PI type.
The sequencing of the regulator is advantageously in the position of the first Pbank_sub-set. In other words, the regulator 31 comprises a set of linear regulators, each determined for a predetermined operating point, i.e. for a predetermined insertion position of the first Pbank set.
The nuclear reactor model used to synthesize linear regulators, also called LTI (Linear Time Invariant), is a non-linear point model, linearized around predetermined operating points. It does not model xenon. In fact, since xenon is very slow in the face of variations in other states, the multi-objective regulator will not have the task of anticipating it, this task being dedicated to the supervisor.
In addition, it is synthesized on a set of local LTI regulators. It therefore does not benefit from the good representativeness of the non-linear model for large power amplitude variations (e.g. load variations).
The supervisor 31 implements a predictive control algorithm using the same non-linear point model of the reactor as that used to synthesize the linear regulators of the regulator 33. This control technique responds to a large number of the issues mentioned above:
It combines both the ability to control the system whatever its dynamics (slow: xenon, and/or fast: temperature) by using a potentially non-linear model of the system.
It allows the taking into account of delays, even long ones, in particular the injection of neutron poison.
In addition, knowing the load monitoring program in advance, it anticipates the behavior by calculating the optimal paths given the program.
Finally, the paths of the regulator will be optimized over the overall behavior of the system and not locally as is the case for a regulator with sequenced gains alone.
However, like any finite horizon non-linear predictive control algorithm:
It does not have any robustness guarantees.
It calculates fixed commands over a defined time horizon called ‘no sampling’. Depending on the calculation capacities available, the prediction horizon and the complexity of the model used, this sampling interval may be large. In this case, the supervisor does not have the capacity to adapt its commands to reject any unplanned disturbances. In fact, if these disturbances are faster than the sampling step, the supervisor, having fixed commands on this step, will not be able to adapt its commands sufficiently quickly to reject the disturbances. In this case, it is the variations in power due to the frequency setting that may pose a problem. In fact, the latter are random and fast.
Thus, a central idea of the present disclosure is to combine a predictive control algorithm and a multi-objective regulator with sequenced gains. The advantages of the one make it possible to compensate at least in part for the weaknesses of the other, as highlighted in the table below.
Thus, the sequenced gain control algorithm of the regulator 33 comprises a plurality of linear regulators, each determined for a specific operating point of the nuclear reactor.
Said operating points being staggered to cover a power range of the nuclear reactor going from 25% to 100% of a nominal power of the nuclear reactor.
In the first control mode, each operating point is characterized by a determined insertion position Pbank of the first set of groups, typically exclusively characterized by the position Pbank.
Alternatively, each operating point may be characterized by one or more of the physical parameters below, in addition to, or in place of, the determined insertion position Pbank of the first set of groups:
the power level of the reactor;
the insertion position of the Hbank group;
the boron concentration;
the temperatures of the primary coolant at the inlet and outlet of the core;
the primary fluid flow.
Each linear regulator is expressed, for example, in the following form, illustrated in
u
K
=K(s)·y=Kp(s)y1+Ki(s)y2 with y1=y and y2=z
where Kp and K1 are gain matrices, s the Laplace variable, y being the output difference vector between the current value Y of the output vector and the reference value Yref of the output vector, z being an operating parameter deviation vector between the current value of the vector of operating parameters to be checked and the reference value of the vector of operating parameters to be checked, and ukbeing the vector of the corrective values of the commands.
The different vectors are composed as follows:
We therefore define the matrices Kp and K1 as follows:
The method comprises a step of obtaining linear regulators, comprising for each linear regulator the following sub-steps:
development of a linearized model of the nuclear reactor by linearization of a non-linear model of the nuclear reactor at the corresponding operating point, the linearized model relating:
determination of operating constraints of the nuclear reactor to be respected for predetermined disturbances dp of the at least one input or predetermined disturbances dU of the vector U of the corrected values of the commands, or of the predetermined disturbances dy of the deviation vector of the outputs y;
translation of each operating constraint into a digital condition to be respected for a transfer function between:
the disturbance dp of at least one input DU, or the disturbance dU of the corrected command vector U, or the disturbance dy of the deviation vector of the outputs y, on the one hand, and
either the difference between the current value of one of the operating parameters and the reference value of said operating parameter, or the difference between the current value of one of the outputs and the reference value of said output, or one of the corrective values of the commands-on the other hand;
determination of the gains of the gain matrices Kp and K1, said gains being determined by an optimization algorithm so as to stabilize at least the looped system for the corresponding insertion position and to satisfy the digital conditions corresponding to all the operating constraints.
The optimization algorithm is typically non-smooth.
The non-linear model of the nuclear reactor is as follows:
The delay on the neutron poison is expressed by the following relation:
u
U
τAO AO time constant
τbc Hot branch time constant
τbf Cold branch time constant
τco Core time constant
τGV GV time constant
c Concentration of precursors in the core
Cb Boron concentration
CP Specific heat capacity of the water in the primary circuit
Kn Power conversion coefficient
n Neutron density in the core
QP Primary circuit water flow
TcGV Hot branch temperature at the GV
TfGV Cold branch temperature at the GV level
ρ Reactivity
The linearization is carried out according to any suitable method, for example by carrying out a Taylor expansion of the equations comprising non-linearities around the operating points in question.
The linearized model around an operating point looks like this:
with the coefficients in bold identified at each operating point.
These equations constitute the GLPV model shown in
We approximate the neutron poison injection delay by a Laguerre approximant defined as follows:
where s is the Laplace variable; h is the value of the delay; n is an integer, h is typically between 100 and 500 seconds, preferably between 200 and 400 seconds, and may be, for example, 300 seconds, n is typically between 3 and 15 and may be chosen, for example, equal to 9.
Combined with the LPV model GLPV, these equations constitute the model GR of
The GLPV model ma be expressed in the following form:
with the state x is defined by
x=(Tc Tf AO ρdop n c TcGV TfGV Pbank Hbank Cb)
the input vector is defined by
u=(vPbank vHbank uQ)T
the disturbance vector is defined by d=Pturb.
The disturbances and the constraints to be repected are those of the specifications, defined above.
The translation of the operating constraints into digital conditions that may be used to determine the gain matrices involves, in particular, translating a time constraint into a frequency constraint. To do this, we seek an approximant of the maximum amplitude of an output signal y of a transfer function
in response to a known demand. We then use the norm of the transfer Ty→d, characterizing the worst case gain thereof. From this gain, the maximum amplitude of the output signal is characterized as a function of the amplitude of the square wave signal for a defined stabilized initial state.
Usually, taking into account temporal criteria is a difficult issue for control problems. In the present case, it is a major constraint of the control problem. The specifications specify certain time constraints that must be respected. For example, in any scenario, the average temperature should not exceed 1.5° C. i.e. ∀t, |δTm|1.5° C.
In the present disclosure, the maximum overshoot of certain parameters is characterized under a known demand, considering a stabilized initial state. The possible demands are given by the specifications: ramps at 5%PN/min, frequency adjustment or even power step up to 10%PN. According to Framatome's expertise, the most penalizing scenario (the one which constrains the regulation the most) is a power step demanded from the turbine(s) of ±P% of a nominal power PN of the nuclear reactor, P being between 5 and 15%, P being for example 10%.
The assumption is therefore made here that if the looped system respects the specifications for the scenario: power step of P% PN, it respects the control objectives in all the cases in question.
Let us consider an excitation signal dP% the signal representing the most penalizing case (power step of P%). This signal is approximated by a square wave signal of period T and amplitude ΔPmax=P% PN, assumed to be sufficiently representative of the square wave. Such a signal has been shown in
Let us break down the Fourier series signal dp%, denoted s. We obtain the known result for a following niche:
and ΔPmax the amplitude of the signal. In order to simplify, we will limit the study to the first three harmonics of the signal supposed to be sufficiently representative. Let s be the Fourier decomposition at the third harmonic. We then have:
Assuming a zero initial state (y(t=0)=0), the amplitude of the output signal y is equal to the sum of the amplitudes of the harmonics multiplied by the gain of the transfer evaluated at the pulse of each harmonic.
The maximum amplification of the input signal s by the worst-case transfer function provides a good approximant of the maximum amplitude ymax of the output signal y. We then have:
∥Ty→d∥∞×smax˜ymax
with smax the maximum amplitude of s. Now, since is the Fourier series decomposition of the signal dP%, we have s˜dP% and it follows that: smaxΔPmax. Since we are looking for Dmax so that ymax≤Dmax, we find:
∥Ty→d∥∞×ΔPmax≤Dmax
This equation is used subsequently for the expression of the various criteria reflecting the constraints on the command. It is thanks to this that we reformulate the requirements in mathematical criteria.
where Tu
The following operating constraint is preferably also taken into account to obtain the linear regulator:
a disturbance dp which is a power step demanded from the turbine(s) of ±P% of a nominal power PN of the nuclear reactor, P being between 5 and 15%, causes a minimum variation of the operating parameters.
Said constraint is translated into the following numerical condition:
where K denotes the gain matrices Kp and Ki, Ω denotes the set of gain matrices stabilizing the looped system, Tz→d
As specified above, z=(δTm δAO δPbank)T is the vector of variation of the operating parameters to be controlled.
Typically, Wz the frequency weighting matrix is defined by:
The weightings chosen for each channel are defined by:
They are defined as follows:
where K1, K2, K3, τ1, τ2 and τ3 are predetermined coefficients.
In the equations above, ∥ ∥∞ symbolizes the H∞norm, while ∥ ∥2 symbolizes the H2 norm. These normes are defined, for example, in J. M. Maciejowski. Multivariable Feedback Design. Addison-Wesley, 1989.
At least for the first control mode, the following operating constraint is taken into account to obtain the linear regulator
the difference δPbank between the current value Pbank and the reference value Pbank,ref is between −ΔPbankmax and ΔPbankmax.
Said operating constraint is translated into the following digital condition:
∥WP·TδP
Furthermore, in the first control mode, the operating constraint for the rate of movement of the clusters is
is the transfer function between said power step and rate of movement of the sets Pbank and Hbank, with
The determination of the gains of the gain matrices Kp and Kiis performed using a non-smooth optimization method, well suited to solving control issues. The tool used is for example Systune from Matlab. This tool has a complete integrated environment making it possible to express the constraints H2 and H∞ as described above.
According to an advantageous variant, at a determined Pbank insertion position, the gains of the gain matrices Kp and K1 are determined by the optimization algorithm so as to stabilize the looped system for said determined insertion positions, while stabilizing the looped systems for at least the two neighboring determined insertion positions, and while satisfying the digital conditions corresponding to all the operating constraints (see
For example, the gains of the gain matrices Kp and K1 are determined by the optimization algorithm so as to stabilize the looped system for said determined insertion position, and to stabilize the looped systems for the four positions of determined insertion adjacent to the determined insertion position, on each side thereof.
In fact, the interpolation of LTI regulators synthesized at different operating points may pose certain difficulties, in particular when the synthesis of two regulators with two neighboring operating points does not provide regulators sufficiently close to one another. In this case, the regulator interpolated between these two operating points does not necessarily constitute a viable solution, either because it may lead to instability, or else it induces too sudden transients when the coefficients are changed.
To solve this problem, the determination of the gains at a given operating point consists of a multi-model approach by zone, as indicated above. The resulting regulator will check the constraints and will be optimal for all the models in question.
In addition, the controller optimized at one operating point is used to initiate optimization at the neighboring operating point.
Finally, the variation of the coefficients of the regulator matrices is constrained, so as to keep its coefficients sufficiently close to those of the initial regulator (i.e. the neighboring regulator).
The regulator 33 alone, for a scenario corresponding to a power ramp from 100% PN to 60% PN at 5% PN/min, makes it possible to obtain the following performances:
|ΔTmmax|=0.56° C.≤1.5° C.; |ΔAOmax|=5.06% AO˜5% AO; |ΔPbankmax|=14.3 PE≤30 PE
These performances are correct for Tm and Pbank, but are at the authorized limit for AO.
The supervisor 31 will now be described.
The model used for the supervisor is the non-linear model described above. The interests of this model are that:
The effect of xenon is modeled.
The real delay of boron is modeled.
More precision than the linearized model.
The model equations are given above. We then define the model used by the supervisor as follows:
{dot over (x)}
S
=F
S(xS, US)
y
S
=G
S(xS, US)
with xS the state, US the signal of the reference commands calculated by the supervisor, yS the outputs, FS and GS the functions defining the development of the state and the output as a function of the inputs and the state. Using the same notations as before, we have:)
x
S=(Th Tc AO ρdop n c TcGV TfGV Pbank Hbank Cb)T
y
S=(Tc Tm AO PK Pbank)T
and
U
S=(VPbank VHbank UQ)T
with VPbank, VHbank and UQ the reference commands for each actuator calculated by the supervisor.
Unlike the linearized model which uses a Laguerre approximation to represent the neutron poison delay, the latter is considered this time as follows:
∀t,
where UQ denotes the neutron poison control and hborethe neutron poison delay. For example, the delay may be 300 seconds.
Typically, the following parameters are chosen for the supervisor setting:
We define by USj the vector of the orders on the step j (i.e. with time j.TS) calculated by the algorithm of optimization such that:
═j≥1, USj=(VPbankj VHbankj UQj)T
The base of functions used for the commands is the base of the piecewise constant functions. This means that the commands will be constant over the sampling period, and discontinuous from one sampling step to the next.
At each sampling step, the supervisor optimizes Nc values, defining the commands on each channel (Pbank, Hbank and boron flow). We set the matrix of the commands calculated with the step of sampling such that:
=[USj . . . USj+N
At each sampling step, we will therefore have:
The set of coefficients of the matrix therefore correspond to the optimization decision variables. Finally, at each sampling step j, the first noted computed command USj is then applied. Then, the control signal applied to the process, denoted US, evolves continuously according to the following relation:
∀t ∈[j.TS, (j+1).TS[, US(t)=USk
The objective function in question for the supervisor is based on the performance objectives defined above. The goal of the supervisor will be to calculate the reference commands for the Pbank, Hbank and neutron poison actuators, minimizing the deviation of the parameters to be checked from their respective references.
The parameters to be checked are the same as regulator 33:
The average temperature noted Tm.
The axial power distribution denoted AO.
The position of the first Pbank subset denoted Pbank
The references of these parameters are given by Tm,ref, AOref et Pbank,ref. We then set δTm, δAO and δPbank the deviations from the respective references of the parameters to be checked such that:
δTm=Tm−Tm,ref
δAO=AO−AOref
δPbank=Pbank−Pbank,ref
The objective function is constructed as follows:
J=J
U
+J
Z
We define here ∥f∥2,[a,b]=√{square root over (∫ab|f(t)|2dt)}
To simplify writing, we denote ∥f∥2=∥f∥2,[0,N
The criterion for the control JZ is then defined as follows:
J
Z
=K
T
∥δT
m∥2+KAO∥δAO∥2+KP
And the criterion is defined as follows:
i. J
U
=K
P
·∥VP
bank
f∥2+KH
With KP
The criterion defined by the equations above, translated in the discrete field, with the matrix of the commands defined above, is expressed in the following form:
J()=Σi=j+1j+N
In addition, we will denote zSj the vector of the deviations of the outputs to be checked, evaluated at the sampling step (i.e. at time j.TS), in the discrete domain, such that:
z
S
j
=δT
m(j.TS) δAO(j.TS) δPbank(j.TS)]T
The constraints imposed on the supervisor, through the command issue, arise from the requirements formulated in the specifications concerning the demands upon command. The interest of the supervisor 31, compared to the regulator 33 is that it has the capacity to explicitly take into account formal time constraints, whether they relate to the inputs or to the state variables: here the position constraints and maximum speed of the groups of clusters as well as the maximum flow rates of di luti on/b ori cati on.
We then formulate the following constraints, deduced from the specifications:
A predictive control algorithm used in the present disclosure is as follows.
The model used to calculate the outputs of the model according to the commands (the decision variables), is a non-linear model. Given the constraints and the objective function defined previously, a non-linear under-constrained optimization algorithm is required for the resolution of such a problem. For example, a non-linear under-constrained optimization algorithm based on the interior point method is used, and as implemented by the function fmincon of Matlab.
The supervisor algorithm is presented in the form of a principle diagram in
At each sampling step, the supervisor receives the measurements Y from the system, the references Zref of the outputs to be checked, as well as the used turbine power profile DU. During this sampling step:
it updates the value of the initial state vector of the system at the step k denoted X0k (assuming here that the complete state is reconstructed),
from the input data, it calculates the sequence of the optimal commands by the function fmincon,
it applies from tj to tj+TS the first element USj of the command sequence calculated over the sampling period to the real system,
it memorizes the rest of the sequence of commands U0j+1=(USj+1 . . . USj+N
and finally it memorizes the applied neutron poison command (in step j) and updates the delayed neutron poison command .
The supervisor 31 alone, for a scenario corresponding to a power ramp from 100% PN to 60% PN at 5% PN/min, makes it possible to obtain the following performances:
|ΔTmmax|=0.82° C.≤1.5° C.; |ΔAOmax|=4.62% AO<5% AO; |ΔPbankmax|=22.97 PE≤30 PE
These performances are correct for Tm and Pbank, but are at the authorized limit for AO.
We can see in
The simulation results of the method of the present disclosure, implementing a hierarchical control comprising the supervisor and the multi-objective regulator with sequenced gains which will be called SMORC for Supervised Multi-Objective Regulator of the Core, will be described below.
The SMORC is simulated on the non-linear model of the reactor described above. In order to meet the actuator demand requirements, saturations will be introduced on them in accordance with the maximum demands defined in the specifications.
The behavior of the SMORC was first tested on a transient load at 5%PN/min from 100% PN to 60% PN followed by a load rise from 60% PN to 100% PN at the same speed. Secondly, it was tested for a power step of 10% PN. Finally, it was tested in the case of frequency adjustment superimposed on a transient load.
The SMORC simulation curves for a 100-60-100% PN at 5%PN/min power ramp are shown in
In this ramp scenario, the SMORC provides correct results against the control criteria. It may be seen that all the outputs to be regulated are kept in the authorized domain, defined by the LCOs, i.e. none of the outputs exceeds the maximum and minimum limits associated with it.
We note, first, a difference in mean temperature, AO and Pbank position of 0.19° C., 2.9% AO and 6.7PE respectively, or respectively 12%, 58% and 22% of the maximum authorized deviations for these variables. We note that the difference (in %) of the average temperature is lower than for the other variables, in particular thanks to the frequency weighting Wz
In addition, the xenon is completely compensated and even anticipated by the regulator by variations in boron. It should also be noted that the speeds of the actuators are saturated during the simulation. They therefore cannot exceed the physical limits of the real actuators.
Let us now compare the results of the SMORC with those of the supervisor alone. We note that all the deviations of the outputs to be regulated by the supervisor alone, on the same scenario, are higher than in the case of the SMORC; in particular because the SMORC benefits from close control, unlike the supervisor alone. We may therefore see here the interest of the proposed hierarchical architecture.
Taking into account the preceding elements, we may therefore say that the SMORC presents good performances in this scenario.
The SMORC simulation curves for a 100-90% PN power step are shown in
There is a difference in mean temperature, AO and Pbank position of 1.0° C., 5.09% AO and 19PE respectively.
The conclusions of these simulation results are similar to those of the previous section, namely:
The outputs to be regulated are all maintained in the authorized operating range in question. The overshoot for AO is very low and over a short period of time, which is largely tolerable
The static error is zero.
The actuator speeds and flow are saturated and therefore meet the associated requirements.
Boron compensates well for the variation in xenon concentration.
An appreciable behavior of SMORC is the insertion of Hbank during load variation to aid in average temperature control, followed by extraction to provide AO control.
However, we note that, in this scenario, the AO is closer to the authorized limits than in the case of the load ramp. This is explained by the scenario studied here being considered to be the most dimensioning in the issue in question.
Finally, taking into account the previous elements, the SMORC provides good results, in accordance with the specifications.
The SMORC simulation curves for a 100-70% PN at 5%PN/min ramp with frequency adjustment are shown in
We note a difference in average temperature, AO and Pbank position of respectively 0.47° C., 3.3% AO and 10.1PE
The conclusions of these simulation results are similar to those of the previous sections.
The specifications and the regulation method for the second control mode will now be described.
Only the points at which these specifications and this regulation method differ from those of the first embodiment will be detailed below.
The specifications do not include a criterion for the position deviation of the first set of groups.
The linear regulators of the sequenced gain control algorithm are again each determined for a specific operating point of the nuclear reactor. On the other hand, each operating point is characterized by the power demanded from the turbine(s), and typically only by this power.
The commands are only the speed of the first set (group R), as well as the rate of injection of neutron poison or diluent fluid.
The controlled operating parameters only comprise the mean core temperature and the axial power imbalance.
The outputs may comprise, in addition to those indicated for the first control mode, the power demanded from the turbine(s).
The specifications and the regulation method for the third control mode will now be described.
Only the points by which these specifications and this regulation method differ from those of the first embodiment will be detailed below.
The specifications do not include a criterion for the position deviation of the first set of groups.
The commands are only the speed of the first set, as well as the rate of injection of neutron poison or diluent fluid.
The controlled operating parameters only comprise the mean core temperature and the axial power imbalance.
The outputs may comprise, in addition to those indicated for the first control mode, the power demanded from the turbine(s).
According to an alternative embodiment applicable to the three control modes, the sequencing of the regulator is carried out by taking into account other parameters, such as, for example, the rate of combustion (burn-up) or the cycles. In other words, the gains of the gain matrices Kp and K1 vary as a function of these parameters. To do this, we always proceed in the same way, by establishing linear models around predetermined operating points and by interpolating the gains. However, the operating points are no longer characterized by a single parameter, but by three parameters which vary. These parameters may be, for example, the position of Pbank, the rate of combustion (burn-up), and the cycle.
According to a second aspect, the present disclosure relates to the nuclear reactor 1 described above. This nuclear reactor comprises the core 5 and the regulation assembly 25 for regulating the operating parameters of the core, these operating parameters comprising at least the mean temperature of the core and the axial power imbalance.
The nuclear reactor 1 further comprises:
The regulation assembly 25 comprises:
The nuclear reactor 1 typically comprises:
a tank 3;
a core 5 comprising a plurality of nuclear fuel assemblies, placed in the tank 3;
clusters 7 for controlling the reactivity of the core 5, and mechanisms 9 configured to move each cluster 7 in the direction of insertion into the core 5 or in the direction of extraction out of the core 5;
a primary circuit 10 for cooling the core 5 in which a primary coolant circulates, comprising cold and hot branches 11, 13 pierced in the tank 3 and through which the primary coolant respectively enters the tank 3 and leaves the tank 3;
an injection circuit 15, configured to selectively inject a neutron poison or diluent fluid without neutron poison into the primary heat transfer fluid.
The neutron poison is typically boron. The diluent fluid is typically water.
In this case, the controls advantageously comprise at least one rate of movement of the control clusters and at least one injection rate of neutron poison or of diluting fluid.
Typically, the control assembly 25 is configured to move the clusters 7 in groups. The clusters of the same group are moved together, and are always all in the same insertion position.
The groups are advantageously divided into one or more sets. The distribution of the groups and their use depend on the control mode of the nuclear reactor.
In certain control modes, several groups are brought together in a first set.
In this case, the groups of the first set are typically moved sequentially. By this is meant that they are inserted one after the other, with an overlap which is predetermined, as described below. Alternatively, the overlap may be variable.
According to an alternative embodiment, all the groups of the first set are moved together. This is understood to mean that they are all always in the same insertion position and move together.
In other control modes, the first set has only one group.
In some control modes, other groups are brought together in a second set.
In this case, the groups of the second subset are typically moved together.
In other control modes, all the groups are gathered in the first set, and there is no second set.
Several examples of control mode will be detailed below, inspired respectively by modes T, G and A.
In a control mode inspired by T mode, particularly suited to the EPR (European Pressurized Reactor), the groups are divided into two sets:
the first set, called Pbank,
a second set, called Hbank.
The first set is particularly well suited for controlling the average temperature Tm. The second set Hbank is typically particularly well suited for controlling the axial offset AO.
The groups of the first Pbank set are moved sequentially. The groups of the second Hbank set are moved together.
In a second control mode, inspired by G mode, the groups are divided into two sets:
the first set comprises the R group, formed of black clusters, i.e. of very absorbent clusters;
the second set, called PCG for Power Compensation Group, is made up of groups G1 and G2 (G for gray clusters) and groups N1 and N2 (N for black clusters). Gray clusters are relatively less absorbent than black clusters.
The groups of this second subset are inserted sequentially. Advantageously, they are inserted as a function of the electrical power requested from the turbine.
In this second control mode, the AO is advantageously controlled mainly by injections of neutron poison or of diluent fluid.
In a third control mode, inspired by mode A, the functional groups are all grouped together in the first set, here designated by the acronym DCBA.
This is typically made up of four groups A, B, C and D, which are inserted sequentially like the Pbank.
The first set is particularly well suited for controlling the average temperature Tm.
In this third control mode, the AO is advantageously controlled mainly by injections of neutron poison or of diluent fluid.
In certain control modes (in the first control mode for example), the at least one rate of movement of the control clusters typically comprises the rate of movement of the groups of the first Pbank set and the rate of movement of the groups of the second Hbank set. These speeds are noted vPbank and vHbank respectively.
In other control modes (in the second and third control modes for example), the at least one rate of movement of the control clusters typically corresponds to the rate of movement of the groups of the first set.
The regulation assembly 25 is configured to implement the regulation method which has been described above.
In particular, the supervisor 31 and the regulator 33 are as described above with respect to the regulation method.
Modules 35 and 37 are also as described above.
The devices 25, 27 are as described above.
The operating parameter regulation module 39 is configured to send the commands to the actuators, these actuators being the mechanisms 9 for moving the clusters 7 and the injection circuit 15.
Preferably, the management of the groups is carried out as described above, in particular the distribution of the groups in the first and second sets, Pbank and Hbank, and the movement of the first and second sets.
Number | Date | Country | Kind |
---|---|---|---|
FR 1850867 | Feb 2018 | FR | national |
Filing Document | Filing Date | Country | Kind |
---|---|---|---|
PCT/EP2019/052543 | 2/1/2019 | WO | 00 |