Determination of positions of fuel assembly elements

Abstract

A method for determining positions of elements of fuel assemblies arranged in a nuclear vessel is described herein. According to an implementation, the method involves capturing a plurality of images of a nuclear vessel and using the plurality of images to estimate a first set of positions of S-holes of a fuel assembly of the nuclear vessel. The method further involves determining a value representative of differences between: (a) the distances from the estimated set of positions to a location on a face of the fuel assembly and (b) known actual distances between the S-holes and the location on the face of the fuel assembly.

Claims

1. A method for determining positions of elements of fuel assemblies arranged in a nuclear vessel, the method comprising the following steps: capturing a plurality of images of a nuclear vessel; using the plurality of images to estimate a first set of positions of S holes of a fuel assembly of the nuclear vessel; determining a value representative of differences between the distances from the estimated set of positions to a location on a face of the fuel assembly and known actual distances between the S-holes and the location on the face of the fuel assembly; selecting coordinates for the location that minimizes the determined value; estimating a second set of positions for the S-holes based on the selected coordinates; and lowering an upper internals assembly onto the fuel assembly so that projecting pins of the upper internals assembly align with positions of the S-holes consistent with the selected coordinates of the location.

2. The method according to claim 1, wherein the location is the location of a center of the face of the fuel assembly.

3. The method according to claim 2, wherein the coordinates comprise three coordinates identifying the face of said fuel assembly in a plane, the three coordinates comprising two coordinates for the center of the face and an angle of rotation of the face relative to an axis of the plane, the method further comprising repeating the using, determining, and selecting steps for each of a plurality of fuel assemblies of the nuclear vessel.

4. The method according to claim 2, further comprising: repeating the using, determining, and selecting steps for each of a plurality of fuel assemblies of the nuclear vessel, wherein, when selecting coordinates for the location, the determined value is minimized while ensuring that there is at least a minimum distance between the centers of two of the fuel assemblies and/or between the center of one of the two fuel assemblies and an end of another of the two fuel assemblies.

5. The method according to claim 4, wherein the minimum distance is a function of the respective widths of the two fuel assemblies.

6. The method according to claim 1, further comprising: repeating the using, determining, and selecting steps for each of a plurality of fuel assemblies of the nuclear vessel, wherein, when selecting coordinates for the location, the determined value is minimized while ensuring that there is at most a maximum distance between S-holes of two fuel assemblies, the maximum distance being determined by the geometry of the nuclear vessel.

7. The method according to claim 1, further comprising minimizing the value at least twice for at least two initializations of coordinates to be selected.

8. The method according to claim 1, wherein determining a value comprises calculating the value using a least squares technique.

9. The method according to claim 1, said method further comprising the steps of: comparing the second set of positions with nominal positions of the S-holes of the fuel assemblies; and if the difference between any of the second set of positions and a corresponding nominal position of said S-holes is greater than a predetermined threshold, adjusting the position of said fuel assembly.

10. A non-transitory computer readable storage product with a program stored thereon, said program comprising instructions executable by a processor to carry out the steps of the method according to claim 1.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Other features and advantages of the invention will become apparent from the following detailed description and from the accompanying drawings in which:

(2) FIG. 1 shows the allowed tolerance for the difference between the nominal position and the estimated position of an S hole, based on the uncertainty associated with the method for determining current positions of the S hole;

(3) FIG. 2 shows a section in a vertical plane (x, z) of a nuclear reactor vessel comprising a plurality of fuel assemblies;

(4) FIG. 3 is a diagram representing the steps of a method according to some embodiments;

(5) FIG. 4 shows a section in a horizontal plane (x, y) of a fuel assembly;

(6) FIG. 5 shows a section in a horizontal plane (x, y) of two fuel assemblies where the positions were identified by a prior art method in a case where there is an overlap in the previously estimated positions;

(7) FIG. 6 shows a section in a horizontal plane (x, y) of two fuel assemblies;

(8) FIG. 7 shows a section in a horizontal plane (x, y) of a nuclear reactor vessel comprising a plurality of fuel assemblies;

(9) FIG. 8 illustrates a device according to some embodiments of the invention.

DETAILED DESCRIPTION

(10) FIG. 2 shows a vessel 20 of a nuclear reactor comprising a plurality of fuel assemblies 21, each assembly 21 having an associated index, i for example, between 1 and n.sub.A where n.sub.A is the total number of fuel assemblies in the vessel 20. The number n.sub.A of fuel assemblies 21.i and their nominal positions are preferably predefined. As is represented in FIG. 2, the fuel assemblies 21 are separated by an inter-assembly gap. A three-dimensional coordinate system (x, y, z) is defined and FIG. 2 shows a cross-section within plane (x, z). A cover of the vessel 20, not represented in FIG. 2, is provided for covering the vessel 20 and the UIA then engage with the S holes of the fuel assemblies 21. In the description that follows, each fuel assembly is considered to have two S holes. However, no restriction is placed on the number of S holes of a fuel assembly 21. Alternatively, it is possible to identify the locations of fuel assembly elements other than S holes. S holes are discussed in the following by way of example.

(11) The dimensions of the vessel can also be known (possibly with some uncertainty, for example equal to 1 mm), and a vessel structure will be discussed further below (with reference to FIG. 6).

(12) FIG. 3 is a flowchart illustrating the steps of a method according to some embodiments of the invention.

(13) In step 301, a first set of previously estimated positions of the S holes of fuel assemblies 21.i is obtained. No restrictions are placed on the method for obtaining the first set of previously estimated positions. For example, any known method may be employed. A videocamera or camera may be used for this purpose, moved about above the vessel within a spatial plane at constant height z in order to capture a video or a set of images which can be used to determine the first set of previously estimated positions. In addition, the method described in patent application FR1150655 may for example be applied for restoring the captured images.

(14) The present invention provides for obtaining one or more previously estimated positions for each S hole. The capture of a single video or a single set of images of the vessel can be used to infer several different estimates for the estimated position of an S hole. These estimates are added to the first set.

(15) Each previously estimated position of an S hole comprises an abscissa, an ordinate, and possibly an elevation, indexed by the index of the assembly comprising the S hole. It is also necessary to differentiate the two S holes of a same fuel assembly. For this purpose, referring to FIG. 4, a fuel assembly 21.i comprises a first S hole 41.1 located at the lower right and a second S hole 41.2 located at the top left. The number of S holes per fuel assembly 21.i and their relative positions in the fuel assembly are provided in FIG. 4 for illustrative purposes only. It is, of course, possible to change the relative positions of the S holes, or to add or subtract S holes.

(16) The previously estimated positions of the first set are imprecise in prior art techniques, as they do not take into account predefined geometric constraints of the vessel and/or of the fuel assemblies.

(17) The distance between the centers of two S holes of the same fuel assembly is referred to as the center-to-center distance. Due to the low quality of the captured images, the previously estimated positions of two S holes of a same fuel assembly 21.i may be separated by a distance which is different from the center-to-center distance of the fuel assembly 21.i, although the fuel assembly is not deformable (given the low mechanical stresses it is subjected to). In addition, it is possible for some fuel assemblies 21.8 to overlap according to the previously estimated positions, which is not possible from a practical point of view, or for some fuel assemblies 21.i to occupy a larger space than what is defined by the internal structure of the tank.

(18) The present invention proposes optimizing the first set of previously estimated positions in order to obtain a second set of optimized estimated positions that satisfy at least one of the geometric constraints defined above.

(19) In step 302, a criterion representative of the distances between the previously estimated positions of the first set and the possible positions of the elements is determined, the possible positions satisfying the predefined geometric constraint(s).

(20) For example, as detailed in the following, the criterion in question may be a least squares criterion.

(21) In order to integrate the geometric constraint in which the center-to-center distance is constant, the positions of two S holes of a same fuel assembly 21.i are expressed relative to the center of that assembly. Referring again to FIG. 4, the fuel assembly 21.i is square with sides L, and has a center 40. The inclination of the fuel assembly relative to the x-axis is indicated by the angle .sub.i. The position of S hole 41.1 is identified by the coordinates of the center 42.1 of S hole 41.1 and the position of S hole 41.2 is identified by the coordinates of the center 42.2 of S hole 41.2. The center-to-center distance is denoted D below.

(22) The abscissa of center 40 of the fuel assembly is denoted x.sub.i.sup.C and its ordinate is denoted y.sub.i.sup.C.

(23) For the triplet (x.sub.i.sup.C, y.sub.i.sup.C, .sub.i.sup.C), the coordinates of center 42.2 of S hole 41.2, defining its position, are given by:

(24) $\begin{array}{cc}{x}_i^H=x_i^C-\frac{D}{2}\cos \left(-\frac{}{4}+{}_i\right);& \left(1\right)\\ y_i^H=y_i^C-\frac{D}{2}\sin \left(-\frac{}{4}+{}_i\right).& \left(2\right)\end{array}$

(25) Similarly, the coordinates of center 42.1 of S hole 41.1, defining its position, are given by:

(26) $\begin{array}{cc}{x}_i^B=x_i^C+\frac{D}{2}\cos \left(-\frac{}{4}+{}_i\right);& \left(3\right)\\ y_i^B=y_i^C+\frac{D}{2}\sin \left(-\frac{}{4}+{}_i\right).& \left(4\right)\end{array}$

(27) By identifying the position of the S holes 41.1 and 41.2 as a function of the center 40 of the fuel assembly 21.i that comprises them, the geometric constraint on the center-to-center distance of the assemblies 21.i is taken into account.

(28) For the following discussion, the previously estimated positions of the first set resulting from step 301 are indicated by the following notations: n.sub.i.sup.H (respectively n.sub.i.sup.B) is the number of previously estimated positions for center 42.2 (respectively 42.1) of S hole 41.2 (respectively 41.1) located at the top left (respectively the bottom right) of fuel assembly 21.i of index i. n.sub.i.sup.H and n.sub.i.sup.B are greater than or equal to 1; for the j.sup.th previously estimated position for S hole 41.2 of fuel assembly 21.i, j ranging between 1 and n.sub.i.sup.H, (x.sub.i,j.sup.H,y.sub.i,h.sup.H) are the coordinates defining the previously estimated position; for the j.sup.th previously estimated position for S hole 41.1 of fuel assembly 21.i, j ranging between 1 and n.sub.i.sup.B, (x.sub.i,j.sup.B, y.sub.i,j.sup.B) are the coordinates defining the previously estimated position.

(29) From these data, a criterion C in the least squares sense can be defined in step 302 as follows:

(30) $C=\underset{i=1}{\overset{n_A}{.Math.}}\left(\underset{j=1}{\overset{n_i^H}{.Math.}}{.Math..Math.\begin{array}{c}{x}_i^C-\frac{D}{2}\cos \left(-\frac{}{4}+{}_i\right)-x_{i,j}^H\\ y_i^C-\frac{D}{2}\sin \left(-\frac{}{4}+{}_i\right)-y_{i,j}^H\end{array}.Math..Math.}^2+\underset{j=1}{\overset{n_i^B}{.Math.}}{.Math..Math.\begin{array}{c}{x}_i^C+\frac{D}{2}\cos \left(-\frac{}{4}+{}_i\right)-x_{i,j}^B\\ y_i^C+\frac{D}{2}\sin \left(-\frac{}{4}+{}_i\right)-y_{i,j}^B\end{array}.Math..Math.}^2\right)$

(31) No restriction is placed on the norm considered above.

(32) As detailed below, no restrictions are placed on the criterion considered, criterion C being given as an illustration. Using criterion C, it is possible to vary parameters and x.sub.i.sup.C, y.sub.i.sup.C and .sub.i in order to minimize the criterion. Parameters and x.sub.i.sup.C, y.sub.i.sup.C and .sub.i that minimize criterion C are obtained in step 303.

(33) For example, to solve the problem of minimizing criterion C, one approximation consists of considering the desired values of .sub.i to be very close to 0. The expression of criterion C can then be linearized and written in matrix form:

(34) $C=\underset{i=1}{\overset{n_A}{.Math.}}\left(\underset{j=1}{\overset{n_i^H}{.Math.}}{.Math..Math.M_H{X}_i-Y_{i,j}^H.Math..Math.}^2+\underset{j=1}{\overset{n_i^B}{.Math.}}{.Math..Math.M_B{X}_i-Y_{i,j}^B.Math..Math.}^2\right)$

(35) where:

(36) $X_i=\left[\begin{array}{c}{x}_i^C\\ y_i^C\\ {}_i\end{array}\right];$$M_H=\left[\begin{array}{ccc}1& 0& -\frac{D}{2\sqrt{2}}\\ 0& 1& -\frac{D}{2\sqrt{2}}\end{array}\right];M_B=\left[\begin{array}{ccc}1& 0& \frac{D}{2\sqrt{2}}\\ 0& 1& \frac{D}{2\sqrt{2}}\end{array}\right];$$Y_{i,j}^H=\left[\begin{array}{c}{x}_{i,j}^H+\frac{D}{2\sqrt{2}}\\ y_{i,j}^H-\frac{D}{2\sqrt{2}}\end{array}\right]\mathrm{and}{Y}_{i,j}^B=\left[\begin{array}{c}{x}_{i,j}^B-\frac{D}{2\sqrt{2}}\\ y_{i,j}^B+\frac{D}{2\sqrt{2}}\end{array}\right].$

(37) In order to accelerate convergence of the algorithm to a criterion minimizer, the various parameters

(38) $X_i=\left[\begin{array}{c}{x}_i^C\\ y_i^C\\ {}_i\end{array}\right]$
of the criterion are optimized together. For this purpose, the set of parameters, for i ranging between 1 and n.sub.A, can be grouped in the same vector X of size 3n.sub.A.

(39) Ignoring the constant terms, the criterion expression is then written in the form:

(40) $C=\frac{1}{2}{X}^T\mathrm{HX}+f^{T}X.$

(41) It is possible to take into account other geometric constraints in order to constrain the criterion C minimization problem.

(42) A second geometric constraint to consider concerns the non-overlapping of fuel assemblies 21.i. FIG. 5 shows a top view (cross-section in the horizontal plane (x, y)) of two fuel assemblies 21.i and 21.k having positions identified by a method of the prior art, in a case where there is overlap according to the previously estimated positions.

(43) The prior art methods for estimating the positions of S holes do not take into account the geometry of the fuel assemblies, as the centers 42.1.i and 42.2.i of the S holes of the first fuel assembly 21.i and the centers 42.1.k and 42.2.k of the S holes of the second fuel assembly 21.h can be located as shown in FIG. 5. Such estimates are inconsistent because the assemblies 21.i and 21.k overlap. The present invention therefore proposes taking the geometry of the fuel assemblies 21.i and 21.k into account.

(44) Considering two adjacent fuel assemblies 21.i and 21.k in the vessel 20, the following five inequality constraints can be expressed: i) the distance, according to the L1 norm (sum of the absolute value of differences between respective abscissae and respective ordinates), between center 40 of fuel assembly 21.i and the two nearest corners of fuel assembly 21.k must be greater than

(45) $\frac{L}{\sqrt{2}},$
L indicating the width of fuel assembly 21.i (all fuel assemblies are considered to have the same width). FIG. 6 shows a top view (cross-section in a horizontal plane (x,y)) of two fuel assemblies 21.i and 21.k. An orthonormal system is defined where the origin is center 40.i of fuel assembly 21.i and the axes are denoted X.sub.i and Y.sub.i (such that axis Xi passes through centers 42.1.i and 42.2.i of the S holes of fuel assembly 21.i). The position of a corner 50.k of fuel assembly 21.k is identified by the coordinates (X.sub.i,k.sup.{circumflex over (1)}, Y.sub.i,k.sup.{circumflex over (1)}) in this orthonormal system. In this system, the first of the three geometric constraints of no overlapping is written as:

(46) $.Math.X_{i,k}^{\hat{1}}.Math.+.Math.Y_{i,k}^{\hat{1}}.Math.\frac{L}{\sqrt{2}}.$ To return to the original coordinate system (axes x, y and z), a translation and rotation must be applied:

(47) 0$.Math.\left(x_{i,k}^{\hat{1}}-x_i^C\right)\cos \left(-\frac{}{4}+{}_i\right)+\left(y_{i,k}^{\hat{1}}-y_i^C\right)\sin \left(-\frac{}{4}+{}_i\right).Math.+.Math.-\left(x_{i,k}^{\hat{1}}-x_i^C\right)\cos \left(-\frac{}{4}+{}_i\right)+\left(y_{i,k}^{\hat{1}}-y_i^C\right)\cos \left(-\frac{}{4}+{}_i\right).Math.\frac{L}{\sqrt{2}}$ where x.sub.i,j.sup.{circumflex over (1)} and y.sub.i,j.sup.{circumflex over (1)} respectively indicate the abscissa and ordinate of corner 50.k in the initial coordinate system comprising the axes x, y, and z. Similarly, an additional inequality is obtained for a second corner 51.k of fuel assembly 21.k. ii) the same approach as in i) can be applied by switching the indices i and k: the distance according to the L1 norm between center 40.k of fuel assembly 21.k and the two nearest corners of fuel assembly 21.i must be greater than

(48) $\frac{L}{\sqrt{2}},$
which provides two additional inequalities. iii) In addition, the Euclidean distance between the centers of fuel assemblies 21.i and 21.k must be greater than L:
{square root over ((x.sub.i.sup.Cx.sub.k.sup.C).sup.2+(y.sub.i.sup.Cy.sub.k.sup.C).sup.2)}L.

(49) A third geometric constraint to be taken into account concerns the dimensions of the vessel 20 and the space occupied by the set of fuel assemblies 21. In order to take this geometric constraint into account, it is possible to take into account only the fuel assemblies 21 peripherally located within the vessel 20.

(50) The following description distinguishes between geometric constraints along the y-axis (ordinates) and geometric constraints along the x-axis (abscissae). FIG. 7 illustrates geometric constraints according to the y-axis.

(51) FIG. 7 shows one geometry of a vessel 20 comprising a plurality of fuel assemblies 21 arranged in rows and columns. Among the fuel assemblies 21, some fuel assemblies 60 which, for each column, occupy extreme positions on the y axis are shaded in gray. One will note that this set of fuel assemblies 60 can be separated into a first subset having vertices (and therefore S holes) of respective maximum ordinates and a second subset having vertices (and therefore S holes) of respective minimum ordinates.

(52) It is thus possible to define a geometric constraint, between each pair of fuel assemblies containing a fuel assembly from the first subset and a fuel assembly from the second subset.

(53) For example, consider the fuel assemblies labeled 21.m (first subset) and 21.n (second subset). Fuel assembly 21.m comprises two vertices 61.m where the ordinates (denoted y.sub.m.sup.H.sup.1 and y.sub.m.sup.H.sup.2) are the maximum ordinates for the column of fuel assemblies comprising fuel assembly 21.m. Fuel assembly 21.n comprises two vertices 61.n where the ordinates (denoted y.sub.n.sup.B.sup.1 and y.sub.n.sup.B.sup.2) are the minimum ordinates for the column of fuel assemblies comprising fuel assembly 21.n.

(54) The geometry of the vessel is used to find the maximum distance 62 between the two extremes of the vessel corresponding to the positions of vertices 61.m and 61.n. By projecting the distance 62 along the y axis of ordinates, a distance D.sub.m,n is obtained. The distance D.sub.m,n, representing the maximum difference between the ordinates of vertices 61.m and 61.n, provides four new constraints:
y.sub.m.sup.H.sup.1y.sub.n.sup.B.sup.1D.sub.m,n;
y.sub.m.sup.H.sup.2y.sub.n.sup.B.sup.1D.sub.m,n;
y.sub.m.sup.H.sup.1y.sub.n.sup.B.sup.1D.sub.m,n;
y.sub.m.sup.H.sup.2y.sub.n.sup.B.sup.2D.sub.m,n;

(55) As mentioned above, the geometry of the vessel may be known with some uncertainty, for example 1 mm, in which case this uncertainty is added to the distance D.sub.m,n.

(56) Four inequalities, such as those presented above, are obtained for each pair of fuel assemblies containing a fuel assembly of the first subset and a fuel assembly of the second subset.

(57) Similar geometric constraints are obtained along the horizontal axis: the fuel assemblies 21 concerned are the fuel assemblies located at the left and right extremes (along the x axis).

(58) All the inequalities defined above, based on geometric constraints related to the space occupied by the fuel assemblies relative to the vessel and on geometric constraints of fuel assemblies not overlapping, can be placed in the following form: R.sub.k(X)0.

(59) Thus, from the criterion C and the set of inequalities R.sub.k(X)0 defined above, the problem of constrained optimization can be solved by means of an interior point algorithm. Such an algorithm is mentioned for illustration only; the present invention is not restricted to the use of this algorithm alone.

(60) In particular, it is possible to use the MATLAB fmincon function by indicating the criterion (C) calculation function and its gradient, as well as the functions associated with the geometric constraints (R.sub.k(X)).

(61) The above algorithm is a local optimization method. However, the set of feasible solutions is not convex. For a given initialization of the parameters corresponding to possible positions of the S holes, it is possible not to converge to a solution that generally minimizes criterion C.

(62) The invention therefore proposes running the optimization procedure for different initializations. Among the various solutions obtained, the one corresponding to the smallest value of C after convergence is retained.

(63) In practice, it is possible to generate the various initializations by applying minor disruptions to the nominal position of the fuel assemblies 21 (shifting the centers 40 of the fuel assemblies 21, changing the angle of orientation .sub.i of the fuel assemblies 21, reducing the gap between assemblies), so that the geometric constraints are satisfied in the parameter initialization.

(64) An alternative solution for initializing the algorithm consists of solving the optimization problem by setting the angles .sub.i to zero and varying only the position of the centers 40 of the fuel assemblies 21. Note that in practice, the angle of orientation .sub.i of the fuel assemblies is very close to zero, and by setting 0, to zero the convergence to the desired solution is accelerated dramatically while a high level of accuracy is maintained.

(65) By using the parameters and .sub.i that minimize criterion C and complying with the geometric constraints R.sub.k(X), step 304 provides a second set of optimized estimated positions of the S holes 41 (entirely defined by parameters x.sub.i.sup.C, y.sub.i.sup.C and .sub.i and by relations (1), (2), (3) and (4), for i ranging between 1 and n.sub.A). The optimized estimated positions of the second set satisfy the geometric criterion of center-to-center distance because the optimized estimated positions of the S holes are obtained directly from the parameters x.sub.i.sup.C, y.sub.i.sup.C and .sub.i resulting from minimization of criterion C and from the geometry of the fuel assemblies 21. Compliance is also ensured with the constraints of non-overlapping and the space occupied by the fuel assemblies 21.i by constraining the criterion optimization using the set of inequalities R.sub.k(X).

(66) The invention is not restricted to a simultaneous consideration of the three geometric constraints outlined above. Indeed, the invention may take into account other geometric constraints, or may take into account any one or two of the three constraints presented above.

(67) In step 305, the optimized estimated positions of the S holes 41 in the second set are compared with the respective nominal positions of the S holes 41. Reusing the notations of FIG. 1, in the case where, for an S hole 41, the circle 13 centered on the optimized estimated position having a radius equal to the uncertainty of the method extends beyond the circle 11 centered on the nominal position and having a radius equal to the tolerance (for example 8.3 mm), an alert is generated in step 307 so that the fuel assembly or assemblies comprising the S hole(s) 41 concerned can be replaced.

(68) Otherwise, the vessel cover can be closed in step 306.

(69) FIG. 8 illustrates a device 70 according to some embodiments of the invention.

(70) The device comprises a unit 71 for obtaining a first set of previously estimated positions of fuel assembly elements (S holes, for example). As described above, no restrictions are placed on the means for acquiring the first set; a known method can be implemented.

(71) The device 70 further comprises a first unit 72 for determining a criterion representative of the distances between the positions of the first set and parameters to be optimized, the parameters identifying possible positions of the elements, the possible positions satisfying a predefined geometric constraint. The criterion may be a criterion in the least squares sense, or some other criterion, as described below. The parameters to be optimized may be the parameters x.sub.i.sup.C, y.sub.i.sup.C and .sub.i of fuel assemblies 21.i, as described above.

(72) The device 70 may further comprise an optimization unit 73 for minimizing the criterion determined by the first determination unit 72, as per a predetermined norm, while satisfying the geometric constraints. As described above, the criterion may be minimized for different parameter initializations, and the smallest criterion value obtained after minimization can then be selected for deriving the parameters corresponding to the optimized estimated positions of a second set. For this purpose, a second determination unit 74 determines a second set of optimized estimated positions from the parameters for which the criterion is minimized.

(73) In addition, the device 70 may comprise a comparator unit 75 for comparing the optimized estimated positions of the second set with the nominal positions of fuel assembly elements, and a warning unit 76 which, if the difference between an optimized estimated position of a fuel assembly element and the nominal position of the element is greater than a predetermined threshold, issues an alert so that the current position of the fuel assembly in question can be adjusted.

(74) The invention is not restricted to the embodiments described above. It extends to other variants.

(75) For example, steps 302 and 303 were described for determining the parameters x.sub.i.sup.C, y.sub.i.sup.C and .sub.i of different fuel assemblies 21.i which minimize a constrained least squares criterion C. This assumes that the error in the position of each S hole follows a Gaussian law.

(76) However, the Gaussian model can be improved when modeling errors in the position of each S hole. The position error is usually small in amplitude unless there is a small number of S holes, in which case it is significantly larger. To take into account the isolated nature of some errors, the invention may use a different norm to define the following criterion C in step 302:

(77) $C^{}=\underset{i=1}{\overset{n_A}{.Math.}}\left(\begin{array}{c}\underset{j=1}{\overset{n_i^H}{.Math.}}\sqrt{{\left(x_i^C-\frac{D}{2}\cos \left(-\frac{}{4}+{}_i\right)-x_{i,j}^H\right)}^2+{\left(y_i^C-\frac{D}{2}\sin \left(-\frac{}{4}+{}_i\right)-y_{i,j}^H\right)}^2}+\\ \underset{j=1}{\overset{n_i^B}{.Math.}}\sqrt{{\left(x_i^C+\frac{D}{2}\cos \left(-\frac{}{4}+{}_i\right)-x_{i,j}^B\right)}^2+{\left(y_i^C+\frac{D}{2}\sin \left(-\frac{}{4}+{}_i\right)-y_{i,j}^B\right)}^2}\end{array}\right)$

(78) The disadvantage of this expression is that it is not differentiable (due to the non-differentiability of the square root of zero). The following approximation can then be used:

(79) $C^{}=\underset{i=1}{\overset{n_A}{.Math.}}\left(\begin{array}{c}\underset{j=1}{\overset{n_i^H}{.Math.}}\sqrt{{\left(x_i^C-\frac{D}{2}\cos \left(-\frac{}{4}+{}_i\right)-x_{i,j}^H\right)}^2+{\left(y_i^C-\frac{D}{2}\sin \left(-\frac{}{4}+{}_i\right)-y_{i,j}^H\right)}^2+{}^2}+\\ \underset{j=1}{\overset{n_i^B}{.Math.}}\sqrt{{\left(x_i^C+\frac{D}{2}\cos \left(-\frac{}{4}+{}_i\right)-x_{i,j}^B\right)}^2+{\left(y_i^C+\frac{D}{2}\sin \left(-\frac{}{4}+{}_i\right)-y_{i,j}^B\right)}^2+{}^2}\end{array}\right)$

(80) The value of is chosen to be very small compared to the desired level of precision, in order to minimize the impact of this approximation. For example, =0.05 mm can be chosen for a desired precision of 1 mm.

(81) The invention extends to the inclusion of other criteria.