Differential influence of instruments in nuclear core activity evaluation by data assimilation
Bertrand Bouriquet, Jean-Philippe Argaud, Patrick Erhard, Sébastien Massart, Angélique Ponçot, Sophie Ricci, Olivier Thual
DDifferential influence of instruments in nuclearcore activity evaluation by data assimilation
Bertrand Bouriquet ∗ Jean-Philippe Argaud , Patrick Erhard S´ebastien Massart Ang´elique Pon¸cot Sophie Ricci Olivier Thual , October 29, 2018
Abstract
The global activity fields of a nuclear core can be reconstructed us-ing data assimilation. Data assimilation allows to combine measurementsfrom instruments, and information from a model, to evaluate the bestpossible activity within the core. We present and apply a specific pro-cedure which evaluates this influence by adding or removing instrumentsin a given measurement network (possibly empty). The study of variousnetwork configurations of instruments in the nuclear core establishes thatinfluence of the instruments depends both on the independant instrumen-tation location and on the chosen network.
Keywords:
Data assimilation, neutronic, activities reconstruction,nuclear in-core measurements, data acquisition network
Data assimilation has been widely developed in earth sciences and especiallymeteorology. Introducing data assimilation was an important step to improveweather forecasts [1, 2, 3]. Such a technique is now used operationally in allmeteorological office. The efficiency of data assimilation for field reconstructionhas already been demonstrated in several articles in meteorology [4, 5, 6].The purpose is to use this technique to make an optimal reconstruction ofthe activity field in a nuclear core using both measurements and informationcoming from a numerical model. This approach was also used for nuclear coreneutronic state evaluation, and particularly nuclear activity field [7, 8].In [8], the authors demonstrate how this method is tolerant to instrumentloss and that this effect is related to the instrument repartition. The aim of thepresent article is to go further in the understanding of the instruments locationand accuracy effect, and focus more specifically on the individual contribution ∗ [email protected] Sciences de l’Univers au CERFACS, URA CERFACS/CNRS No 1875, 42 avenue GaspardCoriolis, F-31057 Toulouse Cedex 01 - France Electricit´e de France, 1 avenue du G´en´eral de Gaulle, F-92141 Clamart Cedex - France Universit´e de Toulouse, INPT, UPS, IMFT, All´ee Camille Soula, F-31400 Toulouse -France a r X i v : . [ phy s i c s . d a t a - a n ] J un f each instrument. To achieve this goal, the method is based on the statisticaloccurrence of instruments with respect to the global quality of the reconstructionassociated.The next section describes the general data assimilation concepts that areused all along this article. Then the parametrisation of data assimilation com-ponents are presented in details. From those bases of data assimilation, thespecific methodology used to track the individual influence of an instrumenton data assimilation is described. Finally, using all previous information, aninvestigation is done on the instrument location and error modelling effect onthe influence of instrument in the core is done. Here are briefly introduced the useful data assimilation key points to understandtheir use as applied here [9, 10, 11]. But data assimilation is a wider domainand these techniques are for example the keys of the nowadays meteorologicaloperational forecast [12]. This is through advanced data assimilation methodsthat the weather forecast has been drastically improved during the last 30 years.Those techniques use all the available data, such as satellite measurements, aswell as sophisticated numerical models.The ultimate goal of data assimilation methods is to estimate the inaccessibletrue value of the system state, x t where the t index stands for ”true state”in the so colled ”control space”. The basic idea of most of data assimilationmethod is to combine information from an a priori on the state of the system(usually called x b , with b for ”background”), and measurements (referenced as y o ). The background is usually the result of numerical simulations but canalso be derived from any a priori knowledge. The result of data assimilation iscalled the analysis, denoted by x a , and it is an estimation of the true state x t researched.The control and observation spaces are not necessary the same, a bridgebetween them needs to be build. This is the observation operator H that trans-form values from the space of the background to the space of observations. Forour data assimilation purpose we will use its linearisation H around the obser-vation values. The inverse operation to go from space of observations to spaceof the background is given by the transpose H T of H .Two other ingredients are necessary. The first one is the covariance matrixof observation errors, defined as R = E [( y o − H ( x t )) . ( y o − H ( x t )) T ] where E [ . ]is the mathematical expectation. It can be obtained from the known errors onunbiased measurements which means E [ y o − H ( x t )] = 0. The second one is thecovariance matrix of background errors, defined as B = E [( x b − x t ) . ( x b − x t ) T ].It represents the error on the a priori state, assuming it to be unbiaised followingthe E [ x b − x t ] = 0 no biais property. There are many ways to get this a priori state and background error matrices. However, those matrix are commonlythe output of a model and an evaluation of accuracy, or the result of expertknowledge.It can be proved, within this formalism, that the Best Unbiased LinearEstimator (BLUE) x a , under the linear and static assumptions, is given by thefollowing equation: x a = x b + K (cid:0) y o − H x b (cid:1) , (1)2here K is the gain matrix: K = BH T ( HBH T + R ) − . (2)Moreover, we can get the analysis error covariance matrix A , characterising theanalysis errors x a − x t . This matrix can be expressed from K as: A = ( I − KH ) B , (3)where I is the identity matrix.It is worth noting that solving Equation 1 is, if the probability distributionis Gaussian, equivalent to minimise the following function J ( x ), x a being theoptimal solution: J ( x ) = ( x − x b ) T B − ( x − x b ) + (cid:0) y o − Hx (cid:1) T R − (cid:0) y o − Hx (cid:1) . (4)This minimisation is known in data assimilation as 3D-Var methodology [9]. The framework of this study is the standard configuration of a 900 MWe nu-clear Pressurized Water Reactor (PWR900). To perform data assimilation, bothsimulation code and data are needed. For the simulation code, the EDF experi-mental calculation code for nuclear core COCAGNE in a standard configurationis used. The description of the basic features of this model are done in Section3.1.To have a good understanding of the instrumentation effect on nuclear ac-tivity reconstruction, various kind of configurations are studied, even some thatdo not exist operationally and so cannot be tested experimentally. For thatpurpose, synthetic data are used, that allows to have an homogeneous approachall along the document. Synthetic data is generated from a model simulation,filtered thought an instrument model and noised according to a predefined mea-surement error density function (Gaussian type).In the present case, we study the activity field reconstruction. An horizontalslice of a PWR900 core is represented on the Figure 1. There is a total of 157assemblies within this core. Among those assemblies, 50 are instrumented withMobiles Fissions Chambers (MFC). Those assemblies are divided verticaly in 29vertical levels. Thus, the size of the control x to be estimated is 4553 (157 × y o is 1450 (50 × The aim of a neutronic code like COCAGNE is to evaluate the neutronic activityfield and all associated values within the nuclear core. This field depend on theposition in the core and on the neutron energy. To do such an evaluation,the population of neutrons are divided in several groups of energy. In thepresent case only two groups are taken into account giving the neutronic fluxΦ = (Φ , Φ ) (even if the present code have no limit for the group number).3 position y po s i t i on
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
Figure 1: The positions of MFC instruments in the nuclear core are localisedin assemblies in black within the horizontal slice of the core. The assemblieswithout instrument are marked in white and the reflector, out of the reactivecore, is in gray.The material properties depend on the position in the core, as the neutronicflux Φ, identified by solving two-group diffusion equations described by: (cid:40) − div( D grad Φ ) + (Σ a + Σ r )Φ = 1 k (cid:16) ν Σ f Φ + ν Σ f Φ (cid:17) − div( D grad Φ ) + Σ a Φ − Σ r Φ = 0 , (5)where k is the effective neutron multiplication factor, all the quantities and thederivatives (except k ) depend on the position in the core, 1 and 2 are the groupindexes, Σ r is the scattering cross section from group 1 to group 2, and foreach group, Φ is the neutron flux, Σ a is the absorption cross section, D is thediffusion coefficient, ν Σ f is the corrected fission cross section.The cross sections also depend implicitly on the concentration of boron,which is a substance added in the water used for the primary circuit to controlthe neutronic fission reaction, throught a feedback supplementary model. Thismodel takes into account the temperature of the materials and of the neutronmoderator, given by external thermal and thermo-hydraulic models. A detaileddescription of the core physic and numerical solving can be found in reference[13].The overall numerical resolution consists in searching for boron concentrationsuch that the eigenvalue k is equal to 1, which means that the nuclear powerproduction is stable and self-sustaining. It is named critical boron concentrationcomputation. 4he activity in the core is obtained through a combination of the fluxes Φ =(Φ , Φ ), given on the chosen mesh of the core. Using homogeneous materials foreach assembly (for example 157 in a PWR900 reactor), and choosing a verticalmesh compatible with the core (usually 29 vertical levels), this result in a fieldof activity of size 157 ×
29 = 4553 that cover all the core. H The H observation operator is the composition of a selection and of a normali-sation procedure. The selection procedure extracts the values corresponding toeffective measurement among the values of the model space. The normalisationprocedure is a scaling of the value with respect to the geometry and power ofthe core. The overall operation is non linear. However, with a range of valuecompatible with assimilation procedure, we can calculate the linear associatedoperator H . This observation matrix is a (4553 × The B matrix represents the covariance between the spatial errors for the thebackground. The B matrix is estimated as the product of a correlation matrix C by a scaling factor to set variances.The correlation C matrix is built using a positive function that defines thecorrelations between instruments with respect to a pseudo-distance in modelspace. Positive functions have the property (via Bochner theorem) to buildsymmetric defined positive matrix when they are used as matrix generator [14,15]. In the present case, Second Order Auto-Regressive (SOAR) function isused to prescribe the C matrix. In such a function, the amount of correlationdepends from the euclidean distance between spatial points. The radial andvertical correlation length ( L r and L z respectively, associated to the radial r coordinate and the vertical z coordinate) have different values, which means weare dealing with a global pseudo euclidean distance. The used function can beexpressed as follow: C ( r, z ) = (cid:18) rL r (cid:19) (cid:18) | z | L z (cid:19) exp (cid:18) − rL r − | z | L z (cid:19) . (6)The matrix obtained by the above Equation 6 is a correlation matrix. It isthen multiplied by a suitable variance coefficient to get covariance matrix. Thiscoefficient is obtained by statistical study of difference between model and mea-surements in real case. In real cases, this value is set around a few percent. Inour case, the size of the B matrix is related to the size of model space so it is(4553 × The observation error covariance matrix R is approximated by a diagonal ma-trix. It means we assume here that no significant correlation exists between themeasurement errors of the MFC. The usual modelling consists on taking the di-agonal values as a percentage of the observation values. This can be expressedas: R jj = (cid:0) α y oj (cid:1) , ∀ j (7)5n our case, the α parameter is fixed according to the accuracy of the measure-ment and the representative error associated to the instrument, and is the samefor all the diagonal coefficients.This hypothesis of error dependence of the measure amplitude is an usualone. Such an modelling error means that location with a large signal are theone with the higher error. In the following, a constant observation variance isalso used. This leads to define the R matrix diagonal according to the followingequation: R jj = β , ∀ j (8)The size of the R matrix is related to the size of observation space, so it is(1450 × To evaluate the quality of individual instruments with respect to the estimationof the core activity field, a reference state is needed. This state, denoted by x aref ,corresponds to the data assimilation analysis using all the available instruments.If x a denotes an analysis experiment performed with less instruments than themaximum available ones, we denote by (cid:15) = || x a − x aref || the norm of its differencewith the reference analysis. As x aref is the best available analysis, we expect thatthe norm of the difference (cid:15) to increase as the number of instrument decreases.The norm (cid:15) is thus a measure of the quality of a given experiment.We consider a set P of p experiments, for instance all those for which a givennumber of instruments are removed. If n is the maximum number of availableinstruments and k the fixed number of removed instruments, there are p = C kn experiments in P . We then denote by (cid:15) i , for i = 1 , ..., p , the norms of thedifferences for each experiment with the reference analysis. We can thus plotthe histogram of the (cid:15) i series. For instance, the histograms corresponding todifferent choices of k can be compared. The average of the (cid:15) i series increaseswhen p increases, since the quality of the analyses decreases when instrumentsare removed. Its root mean square provides information on the homogeneity ofthe instrument quality for the field reconstruction.Dealing with the (cid:15) i series can provide information about individual instru-ment. If we select the (cid:15) i values above a fixed threshold, for instance the 10%highest values, we can point out, for each corresponding experiment i , the setof instruments which has not been removed. For each instrument, we can counthow many times it appears in such an experiment and compare this numberto the ones obtained with the other instruments. We can thus assess that aninstrument appearing a great number of times in experiments leading to thesehigh values of (cid:15) i is of poor quality compared to an instrument which seldomappears for these low quality experiments. First of all, the behaviour of (cid:15) i for two extreme cases is examined: when 2instruments are removed ( i.e.
48 remain) and when only 2 instruments remain.In both cases, P sets contain p = 1225 case. This corresponds to all possiblecombinations to C = C = 1225. On Figure 2 are presented the distributions6 N u m be r o f c a s e s ||xa-xaref|| Distributions of ||xa-xaref||
Figure 2: Norm distribution for all the cases where only two instruments aresuppressed or remain in the instrument network on a REP900 reactorof the (cid:15) i ’s for the cases where two instruments are lost ( i.e
48 remain) or only2 remain.On Figure 2, large differences between the two distributions can be noticed.The shifting of the mean value between those two extreme cases is logical asavailable information is dramatically changing. Thus the difference of (cid:15) i be-tween the two case is changing a lot on the mean. However, the shape of thedistribution is also changing. The distribution is a very sharp one when 48 in-struments remain, whereas when only 2 instruments remain the distribution isbroader. The broadening of the distribution shows that the instruments do nothave the same influence on the activity field reconstruction. If all instrumentshave had the same effect on data assimilation, they would have been equivalent.Suppressing whatever instrument would then change the mean value of the dis-tribution of the norm but not the shape. Ideally, this distribution shape shouldbe a very sharp peak. In the present case the distribution when 48 instrumentsremain could be a good candidate. Thus if all the instruments where equiv-alent, only a translation between the distributions of 48 instruments remainsand 2 instruments remains would have been observed. However, transformationbetween the two distributions proves that the instruments are not all equivalent.From that quality measurement, the aim is now to determine the effect ofeach instrument on the data assimilation results. To archive this goal a detailedstudy of the distribution of the (cid:15) i ’s is done. Several networks of instrumentswithin the core are considered to obtain an overall understanding of the influenceof instruments. 7 .1 Standard PWR900 instruments repartition The first case is a standard PWR900, with data assimilation done as describedpreviously. In this case, the measurement error is proportional to the measureitself as described by Equation 7.Here are studied the scenarios when only two instruments remain. In suchcase, the histogram of the (cid:15) i ’s is rather broad, as show in Figure 2. Largedistributions (as the one of 2 instruments lost) ensures a better separation ofthe different classes of instruments that can be present within a given slice inthe (cid:15) i ’s values. In a very narrow distribution, as the one where 48 instrumentsremains of Figure 2, confusion between the different classes associated to a slicein (cid:15) i value is more likely.To quantify the results within a slice of (cid:15) i values, first the whole distributionis investigated when no selection on (cid:15) i is done. In this case, the distributionof instruments occurrences as defined in Section 4 will be flat. The interestingpoint for further comparison is its amplitude of 49 occurrences. This value ofamplitude represents 2450 = 1225 × (cid:15) i values.The subset of 10% scenarios that give the highest values is selected to studythe influence of instruments among (cid:15) i distribution. Assuming an equal influenceof all instruments in the 49 cases, and considering the 10% highest values of (cid:15) i ,the occurence of each instrument should have only a mean value of 4 . . • the hypothesis taken here is to have some measurement error that areproportional to the absolute value of the measurement, as given by Equa-tion 7. This implies that implicit error is higher at location when activityis stronger. • the instruments repartition is complex. This has the benefit to work incases that are close to real one. However in the same time it leads to a8 N u m be r o f o cc u r en c e s Instrument number
Instruments occurences
Figure 3: Histogram of the occurrence of the instruments in the 10% scenariosthat give the highest values of (cid:15) i . Each value in abscisse are corresponding toa reference number of one instrument. In ordinate is shown the occurrence ofeach instrument within remaining instruments list.difficult interpretation.Thus more theoretical and idealistic configurations need to be consideratedto get rid of those both difficulties points and to answer better the question. To avoid both difficulties presented in previous section, the following instru-ments configuration is taken: • a repartition of the instruments within the core following a Cartesian map.The location of the instruments within such a configuration is reported onthe Figure 5. Within this configuration, only 40 MFC are available. Thisnumber of instruments is a bit lower than the 50 of the standard PWR900case presented in Figure 1. It leads to an overall difference in instrumentsdensity and quality of initial reconstruction. • Measurement errors are taken as constant, as described by Equation 8.If only the hypothesis on geometry presented above is kept, the overall con-clusion of this section is not changed at all. This second hypothesis on mea-surement error modelling, whatever is the instrument location choice, has loweffect on the results. 9
Position in x P o s i t i on i n y Distribution of occurence within the core
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
Figure 4: Occurrence of the instruments as a function of their position within core. The cases kept for doing this histogram are the ones within the 10%highest values of the (cid:15) i distribution.Using both above hypothesis for the calculation, the same study as the onedone for standard configuration, is done here. The analysis is done accordingto method described in Section 4. In the present case the value of x aref is notthe same as in the standard case described in Section 5.1. This new value of x aref is calculated assuming all the 40 instruments are localised according toFigure 5. Within such a configuration there are C = 780 possible scenarioswhen 2 instruments remain, so all configurations are evaluated.If all the cases are taken into account, an uniform distribution is obtained.The amplitude of this distribution is 39. This value represent 1560 = 780 × .
0. This valueis a reference to compare with the values obtained in occurrence distributionwithin a slice of value of x a − x aref .As in Section 5.1, the ensemble of scenarios that leads to the 10% highest x a − x aref values are investigated. The occurrence of the instruments for thescenarios giving the 10% highest x a − x aref values are plotted in Figure 6.The peaks in Figure 6 within the distribution prove that the instruments arenot equivalent, even in the case of a regular repartition of instruments accordingto a Cartesian grid. This confirms that all instruments do not have the sameeffect, whatever is the chosen network. Another point is the quasi symmetryof the distribution with respect to the mean of the chosen numbering. Thenumbering of the instruments is done from right to left and from top to bottom.Which means that the instruments numbered 1 at the top left is equivalent to10 position y po s i t i on
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
Figure 5: The MFC instruments within the nuclear core are localised in assem-blies in black within the horizontal slice of the core. The assemblies withoutinstrument are marked in white and the reflector is in gray.the one numbered 40 at the bottom right. This classification lead to equivalencebetween the instruments numbered from 1 to 20 with the corresponding one,respectively, between 40 and 21. Thus symmetry imposed by numbering appearsin the histogram. Moreover, even if a fixed value of the measurement error isimposed, this do not eliminate the effect of location on the instruments. Thislocation effect is then rather dominant with respect to the error effect. To havea better view on a plan of those instruments, their spatial locations within thecore are represented in Figure 7.Figure 7 shows the instruments that have a low contribution to the fieldreconstruction with data assimilation. Those instruments are mainely localisedin the centre of the core. In this case this interpretation is easy as the in-struments location are distributed regularly as shown in Figure 5. This centrallocation of the instruments confirms what was seen in Figure 6 about the centralinstruments locations.Beyond this global similitude, several differences are notable comparing Fig-ure 7 and Figure 4. In Figure 7, the most represented instrument is locatedin the top middle of the core centre. However, in Figure 6, this instrument islocated on the right side of the core. This effect of location changing, on themost represented instrument, can be attributed to the global difference betweenavailable instruments positions as measurement error repartition and assemblytechnical specifications hypothesis can be excluded. No effect of the modellingof the error respect to Equation 7 or Equation 8 can be seen whatever is used.The repetition of the assembly is exactly the same as in standard case and the11 N u m be r o f o cc u r en c e s Instrument number
Instruments occurences
Figure 6: Histogram of the occurrence of the instruments in the 10% scenariosthat give the highest values of (cid:15) i . Each value in abscisse is corresponding to areference number of one instrument. In ordinate is shown the occurrence of theinstruments within remaining instruments list.assembly technical characteristics are not correlated directly to the results.The conclusion from that point is that locations of the most influential in-struments are dominated by the instrumental network pattern chosen for thecore. Moreover, the precise location of the most represented instrument can beput in light precisely using statistics on the occurrence of each instrument inspecific slice of (cid:15) i norm values. In the study presented in Section 5.1 and 5.2,both are assuming that two instruments are added to a non instrumented core.This kind of hypothesis is rather strong and do not allows to conclude in a moregeneral case.From present step, it seems that the instruments located at the centre arethe most represented in the 10% highest (cid:15) i values. In this sense, they lead tothe worth reconstruction of the core, which seem very paradoxal. To under-stand that point, it is necessary to keep in mind that within a data assimilationprocedure, the measurement provided by an instrument have influence withina rather large radius around the measure itself. This comes from the construc-tion on the B matrix that is presented in Equation 6. Typically, this matrix isconstructed with an influence length of few assemblies. Thus, the improvementthrough data assimilation is driven by the few measurements that can generatedtoo large modification, that are not optimum on close assemblies. This does notlead to a overall quality improvement of the final analysis.Such an effect means that it is necessary to have more instruments in thestarting network to make a better balance between information they provide.12 Position in x P o s i t i on i n y Distribution of occurence within the core
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
Figure 7: Occurrence of the instruments as a function of their position withinthe core. The cases kept for doing this histogram are the ones within the 10%highest values of the (cid:15) i distribution.To confirm this hypothesis, the amplitude of the highest peak in Figure 6 giveenough information. This plot correspond to the occurrence of a given instru-ments in the 10% higher value of x a − x aref norm. On Figure 6 maximum peakvalue is 30. This value is 23% lower than the flat limit when no selection is doneat 39. This mean that in 23% of the cases, there are a counter balancing ofthe overconfidenced information given by one instrument. Thus, the influenceof other instruments in the network must be considered. The aim of this section is to comfort conclusions on the influence of startinginstrumental network configuration obtained in Section 5.1 and 5.2. To keepthe advantage of the flat error and regular distribution new case is build onthose basis. The assumption that the core is half instrumented will be done inthis part. This means that one half will be considered as fixed one. Then onlyinstruments coming from the other half are added.To keep the regularity of the distribution, instruments are separated in thefixed or removable categories alternatively using as basis the regular distribu-tion presented in Section 5.2. Thus, two categories of 20 instruments each areobtained. The location of the instruments for each category is presented on Fig-ure 8. This process to build classes, even if it leads to some asymmetry, allowson overall to keep the convenient regularity features of the initial repartition.Thus, an addition of 2 instruments from the removable set is done to the13igure 8: The MFC instruments within the nuclear core are localised in assem-blies in black with in the horizontal slice of the core. The assemblies withoutinstrument are marked in white and the reflector is in gray. The left picture isfor the instruments that are always kept. The right figure represent instrumentsthat may be suppressed.set of 20 fixed instruments. As only 20 instruments remains the number ofpossible scenarios is smaller than in Section 5.1 and 5.2 remain. In this case,there are only C cases which means 190 possibilities. As expected, the normdistribution of x a − x aref is a narrower that the previous case. This distribution,contrary to the one presented for two instruments remaining in Figure 2, israther symmetric. Still the distribution is board enough to make a cut over the10% higher values of the distribution.To get a clear view of the instruments, we will make occurrence histogramtaking into account only the two instruments that are added. However we willstill working in the framework of all instruments set. This new configurationwill be analysed according to the method used in Section 5.1 and 5.2.In the present case only half of the instruments are considered. Thus theplot of occurrence on all the studied scenarios is a sampling function as onlythe locations with a removable instrument are considered. The frequency ofthis sampling function is 1 / × x a − x aref distribu-tion that are investigated. The results of the instruments occurrence histogramare plotted on Figure 9. For this 10% subset of instruments, the hypothesisof independence of instruments would leads to an histogram of occurrence as asampling function of frequency 1 / . .
9. Such a huge amplitude peak signs one moretime the non equivalence of all the instruments within the data assimilationprocedure. No more symmetry exists in the histogram comparing to Figure 6and this is even asymmetry that appears. This an interesting change as theinstruments selection method should keep some how this symmetry. This sign14 N u m be r o f o cc u r en c e s Instrument number
Instruments occurences
Figure 9: Histogram of the occurrenceof instruments for the 10% highest val-ues of the (cid:15) i distribution. Each valuein abscisse are corresponding to a refer-ence number of one instrument. In or-dinate is shown the occurrence of theinstruments within remaining instru-ments list. Position in x P o s i t i on i n y Distribution of occurence within the core
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
Figure 10: Occurrence of the instru-ments as a function of their positionwith in core. The cases kept for do-ing this histogram are the ones withinthe 10% highest values of the (cid:15) i distri-bution.an influence of the existing instruments network on data assimilation results.The precise spatial location of the instruments is reported in Figure 10.Comparing the location of the instruments in Figure 10 respect to the case ofregular repartition for the same kind of slice in x a − x aref presented in Figure 7several differences are obvious.The most interesting point is that in the present case presented in Figure 9a peak arise at the very bottom of the core. Thus, the MFC that got thelowest influence on reconstruction by data assimilation are in this case not anymore localised in the centre of the core. This show one more time a significantdifference with results obtained especially in Section 5.2 that is dealing with avery close initial configuration.Even if the same instrument locations are used originally, those results arevery different from the ones presented in Figure 7. Thus we see that both theoriginal present instrumentation as well as the location of the instruments havean effect on the activity field reconstruction quality.Examining together Figures 4, 10 and 7, we can notice that the instrumentsthat are the least important often seem to be localised in area where the densityof instruments present or potentially available is fairly large. One of the purpose of this study was to make a study of the influence of the in-struments within a data assimilation procedure to reconstruct activity in nuclearcore. The quality of reconstruction is evaluated thought the norm of x a − x aref for all the possible combination of two instruments addition that is the best casein terms of statistic and analysis of the results.Focusing on the instruments leading to the 10% highest value of (cid:15) i wheninstruments are added to a non instrumented core of PWR900, it can be noticedthat they are not equivalent. It appears that within a distribution that is15upposed to be flat if all instrument play the same role, some instruments havehigher occurrence than other.This effect can be understood focusing on a Cartesian regular repartitionof the instruments. Within this regular configuration, it was shown that theinstruments localised in the centre are the least important for activity fieldreconstruction with data assimilation. This proves that the instruments locationis the most important factor that impact the quality reconstruction with dataassimilation.To better understand the effect of adding instruments to a system, we chooseto do do the same procedure starting from a core with a half instrumented regu-lar distribution. Through this study, it can be demonstrated that the locationsin the center of the least influential instruments is not a general rule.Within those studies, it is determined, empirically, that the influence ofthe instruments is related to the density of the present or potentially availableinstruments around this location.The determination of the worst instruments (respectively the best) to addto an measurement system, to improve data assimilation with it, is dependingon several parameters.The starting instrumental configuration to which instruments need to beadded play a fundamental role. It was proven here that results are very differentwhen instruments are added to a non instrumented system than to a partially(half) instrumented one. Such a non equivalence respect to starting point provesthat building a complete instrumental system cannot be done iteratively. Sucha system will have limited efficiency as each step is dependant of the previousand not of the global situation.This imply that, in order to build an optimal measurement network in anuclear core, it is necessary to be able to take into account all the instrumentsglobally.Developing tools and diagnostic for a determination of such optimal networkis then complex. This is especially true when a lot of measurement are needed.Moreover such a goal can only be achieved through advanced mathematicalstudy and powerful computing usage. References [1] David F. Parrish and John C. Derber. The national meteorological center’sspectral statistical interpolation analysis system.
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