KR20040038190A - Determination of Shape Annealing Method Matrix by Contrainted Regularization Method in Nuclear Power Plants - Google Patents

Abstract

PURPOSE: A method for deciding an excore detector SAM(Shape Annealing Matrix) of a nuclear power plant using regularization and inverse matrix constraints is provided to perform stably an SAM calculation process regardless of measuring errors and enhance the efficiency of the nuclear power plant by correcting only an SAM calculation subroutine. CONSTITUTION: A method for deciding an excore detector SAM of a nuclear power plant using regularization and inverse matrix constraints includes a process for determining the SAM. The SAM is determined by optimizing a regularization problem having linear constraints. The regularization problem having linear constraints is expressed by a predetermined mathematical expression.

Description

Determination of Shape Annealing Method Matrix by Contrainted Regularization Method in Nuclear Power Plants

The present invention relates to a core protection system of a nuclear power plant, and more particularly, to a method for determining a set value of an off-site instrument that is part of the core protection system.

Nuclear power plant designers pre-calculate and provide information on the maximum tongue output that the fuel can withstand, and the operator operates without exceeding the maximum allowable power to maintain the integrity of the fuel.

The output distribution in the core consists of a radial output distribution and an axial output distribution. The radial output distribution is determined at the design stage according to the fuel assembly loading model having an appropriate distribution of a burnable absorber. However, the axial output distribution is difficult to control due to the xenon transition phenomena, and the nuclear power plant has set an operating allowance area for the axial output distribution to maintain nuclear fuel integrity. That is, the accuracy of the out-of-counter instrument is very important as it limits the allowable region to the axial output deviation range. Therefore, the set value (SAM: Shape Annealing Matrix (SAM)) is determined and installed by using the data measured at 20 to 80% output to calculate the output distribution of the nuclear reactor at the beginning of each cycle. Drive.

In Korea's standard nuclear power plant, three types of off-site instruments, such as a safety channel, a control channel, and a startup channel, are installed outside the reactor vessel. Among these, the safety channel excore detector system consists of four independent channels, and each channel consists of three subchannels of TOP, MID, and BOT. have. The safety channel out-of-the-meter instrument signal is very important because it is input to the core protection system (Core Protection Calculator, CPC) and used to determine the core's output and output distribution after the setpoint is multiplied. 1A and 1B are a plan view and a side view schematically showing a reactor and a three-channel off-site instrument of a Korean standard nuclear power plant. In Fig. 1A, 11 represents a safe channel out-of-counter instrument and 12 represents a start-up or control channel out-of-counter instrument. In FIG. 1B, 13 is a reactor core, 14 is a shroud, 15 is a barrel, 16 is a vessel, 17 is a cavity streaming shield, 18 is a resin, 19 denotes an excore detector and 20 denotes a cavity wall. The core protection operator converts the upper, middle, and lower out-of-band instrument signals to the output distribution of the core divided into three parts: upper, middle, and lower. 2 conceptually illustrates the meaning of a SAM. In Fig. 2, the relationship between the core mean output 21, the outer edge portion 22, and the outer furnace measuring instrument 23 which detects the output of the core through the outer edge portion 22 is shown.

Out-of-surface instrumentation signals depend mainly on the output of the fuel assembly located outside the core. Thus, the physically out-of-core instrument signal has a more physical correlation with the core outer power distribution than the core average power distribution. On the other hand, in the core protection calculator, it is assumed that the upper, middle, and lower outputs of the outer core have a linear relationship with the upper, middle, and lower signals of the external measuring instrument. A 3x3 matrix representing the linearity of the liver is defined as SAM. That is, it is assumed that the upper, middle, and lower outputs P _i of the outer periphery have a relation as shown in Equation 1 below with the upper, middle, and lower signals D _i of the outside instrument.

<Equation 1>

Equation 1 is basically an approximation, and the actual physics should be expressed in a more complex relationship. Therefore, it is necessary to determine the SAM that provides the minimum error for the various out-of-core output distributions and corresponding out-of-core instrument signals. In other words, various core output distributions and off-the-counter instrument outputs are needed to determine the SAM. Currently, in the case of Korean standard nuclear power plants, SAM is determined using only measured data, and necessary measurement data are obtained by different methods depending on fuel cycle. First, in the case of the initial core, the SAM is determined by using xenon vibration at 20% and 50% of the cycle time. The use of xenon vibration in the initial core is because the output distribution of the core is very similar regardless of the output level. The use of xenon vibration takes about two to three days due to its nature. In the case of reloading core, SAM increases the core output from 20% output to 80% output while maintaining the evaporation rate of about 3% / hour through the Fast Power Ascension Test (FPA). Determine. In the reloading core, the axial power distribution varies greatly with power due to axial inhomogeneities in fuel assembly combustion. Therefore, in the reloading core, the SAM is determined using only 30 to 70 measurement data acquired during the FPA process without using xenon vibration. Each measurement data is composed of furnace instrument output of the power distribution of the reactor core of a particular moment and at that time, and is calculated by the outside core power distributions P _i is a computer program.

As such, if a variety of specific data are given through xenon vibration or FPA, SAM is determined according to the least-squares method. Assuming that N data are given, for each core protection operator channel, each element of SAM is solved using the least-square method as shown in <Equation 2>.

<Equation 2>

here, Is,

i means channels A, B, C, and D, and superscript means N measurement data.

However, the existing SAM determination method has the following characteristics and problems.

First, a satisfactory SAM cannot be obtained every cycle. In other words, when using the existing SAM determination method, the accuracy of the SAM varies depending on the cycle, and the properties of the determined SAM may vary greatly depending on the core protection operator channel. The main reason for this phenomenon is mainly due to the noise signal included in the measurement data.

Second, the output distribution is relatively accurate at the beginning of the cycle, but the error of the output distribution increases as the combustion degree of the core increases. This is because the SAM is determined at the beginning of the cycle and applied to the overall combustion degree. Therefore, at the beginning of the cycle, it is necessary to determine a SAM that can sufficiently reflect the physical characteristics of the off-the-shelf instrument, but there is no solution for this.

SUMMARY OF THE INVENTION The present invention has been made to solve this problem, and improves the accuracy of the SAM decision calculation that calculates the axial third core output distribution with the out-of-measuring instrument signal, and provides the solution of the region with a physical meaning all the time so that It aims to improve core safety by reducing the reload cycle start time of nuclear power plants and improving the reliability of the core protection system by reducing and performing stable calculations.

1A and 1B are a plan view and a side view schematically showing a reactor and a three-channel off-site instrument of a Korean standard nuclear power plant,

2 conceptually illustrates the meaning of SAM,

3 and 4 are the error comparison of the axial output distribution calculated using the present SAM and the present invention for the case 1, 2, and

5 and 6 show the comparison results of tracking the error of the axial output distribution calculated using the existing SAM and the present invention for cases 1 and 3 from the reactor cycle beginning to the end of the cycle.

Explanation of symbols on the main parts of the drawing

11: Safety channel out-of-band instrument 12: Start or control channel out-of-band instrument

13: reactor 14: shroud

15: cylinder 16: container

17: shield 18: resin

19: off-site instrument 20: cavity wall

In order to achieve this object, the present invention provides a generalization problem with a linear constraint. Determination of out-of-measured instrument SAM using equation optimization method. Here, in the case of a three-level out-of-band instrument, the regularization factor λ is selected from 1 × 10 ⁻³ to 1 × 10 ⁻¹ , and the limiting matrix G is to be.

Hereinafter, the present invention will be described in detail.

First, in order to reduce the influence of measurement noise, the least-square method using regularization is introduced as follows.

The solution of <Equation 2> is obtained by solving the least-squares problem as in <Equation 3>.

<Equation 3>

Here, ₂ means quadratic norm.

At this time, if the matrix (A ^T A) is ill-conditioned, that is, the condition number (≡∥ (A ^T A) ∥ · ∥ (A ^T A) ^-1 ∥) is very large, the calculated solution It is very sensitive to perturbation. In other words, even in the presence of very small measurement noise, the area of solution cannot be predicted at all, and in many cases, the result is a region having no physical meaning. If the matrix A has a perfect invertible condition, the condition number is 1, and the larger the ill-conditioned property, the greater the increase in the singular matrix.

When solving a discrete ill-posed problem, several sub-optimal solutions may exist depending on the solution constraints and the following considerations are required.

First, the effect of measurement noise must be taken into account when the condition number of the matrix (A ^T A) is large.

Second, replacing the matrix A ^T A with a well-conditioned matrix derived from it does not necessarily lead to a meaningful (useful) solution.

Third, prudence is required when applying additional constraints.

The introduction of the regularization method is to reduce the influence of measurement noise even when the condition number of the matrix (A ^T A) is very large.

The least-square solution of <Equation 3> can be written as in <Equation 4> using SVD (Singular Value Decomposition).

<Equation 4>

Where p is the number of nonzero singular values, U _p consists of the first column p of U, and V _p consists of the first p column of V. And, Is a diagonal matrix where each element is the reciprocal of nonzero non-normal values of the matrix A ^T A. vector The elements of are simply the dot product of the first p columns of the matrix Ub and can be written as

<Equation 5>

Upper Vette If you multiply by, you get

<Equation 6>

Finally, multiplying the above vector by Vp, we can find χ _LS by least-squares as shown in <Equation 7>.

<Equation 7>

The above equation for least-squares solution is very useful. This is because it shows how small singular values have a big influence on the least-squares solution in the presence of measurement noise. Where there is random noise, even if true data is orthogonal to U _i Is likely to be nonzero. At this time, when there is a very small singular value σ _i characteristic of the ill-conditioned problem Is divided by very small values, and after the singular vector V _i is multiplied, the least-squares solution is mixed with the large component in the direction V _i . It can be said that the so-called Picard condition is satisfied when the value approaches zero first compared to the singular value σ _i . This is an indication that the pseudo-inverse solution is not sensitive to measurement noise. If the Picard condition is not satisfied, the solution is very sensitive to the noise of the data and is likely to give a solution away from the exact area of the solution.

As a method of minimizing the effects of measurement noise, consider the damped least squares problem as shown in Equation 8 by introducing the normalization factor λ.

<Equation 8>

This is the general and best known regularization form known as Tikhonov regularization.

<Equation 9> is equivalent to <Equation 8>.

<Equation 9>

The solution of this least squares problems can be obtained by solving the normal equation

<Equation 10>

If this is simplified, it is as follows.

<Equation 11>

In terms of SVD, <Equation 11> can be written as

<Equation 12>

It is simplified as in Equation 13 and has the only solution given by Equation 14.

<Equation 13>

<Equation 14>

here, It is referred to, and the filter factor, σ _i »λ surface, f _i is about 1, σ _i« λ If, f _i is almost zero. Between 0 and 1 it serves to reduce the contribution of small singular values to the solution.

It is highly desirable that this information can be used to reconstruct the solution if the behavior of the solution is known in practice. In other words, using the appropriate side constraints can help you find the correct solution.

The following equation represents a normalization problem with linear constraints.

<Equation 15>

If G = I and d = 0 for non-zero x, the damped least squares problem is converted into the following equation (16).

<Equation 16>

Here, the matrix A and the vector b are converted as shown in Equation 17 below.

<Eq. 17>

In addition, if H = A ^T A and g = -A ^T b, the problem of <Equation 16> is the same as the following <Equation 18>.

<Equation 18>

This is a well known constrained quadratic programming (QP) problem, and there are well known methods for solving such problems as the active set method.

Numerical simulation results of the SAM calculations provide that the regularization factor λ is always stable in the range of 1 × 10 ^-3 to 1 × 10 ^-1 , especially in the range of 1 × 10 ^-3 to 1 × 10 ^-2 .

The SAM matrix represents the inverse of the axial spatial weighs of neutron transport, which is very simplified from the outer periphery of the core to the off-surface instrumentation region. Therefore, the determination of the 3x3 SAM matrix is a typical ill-posed least square problem, and Equation 1 is a simplified transport kernel under the assumption that the 3-level reactor output distribution is a linear sum of the 3-level out-of-band instrument signals. Similar to the first Fredholm integration equation of the transport kernel, it is well known that its inverse problem is a typical ill-posed least square problem.

The SAM is basically a kind of linear transfer operator that connects the core output to the off-site instrument, which basically represents the physical characteristics of the off-site instrument. Therefore, there are some important physical and mathematical properties that must be satisfied for SAM to accurately represent the relationship between the core and the off-site instrument. The characteristics of SAM can be identified mainly from the inverse matrix point of view. By definition, a SAM is an operator that converts out-of- instrument measurements into core outputs. Thus, fundamentally, SAM corresponds to expressing the physical phenomenon of the response of an off-site instrument in the opposite direction. Therefore, in order to accurately understand the characteristics of the SAM, it is necessary to analyze the characteristics of the inverse SAM (ISAM) of the SAM representing the physical phenomenon.

ISAM basically describes how the off-site instrument reacts to a given core output distribution. ISAM has the following characteristics depending on the physical characteristics of the off-road instrument. First, all elements must be positive. This is because the neutrons in the core have a probability of reacting in the off-site instrument, regardless of their location. In particular, from the point of view of contributions to off-the-counter instruments in three-core cores, such as ISAM, each area clearly has a non-zero importance.

Another characteristic of ISAM is that the elements that make up each row must satisfy a certain inequality. When the element of ISAM is called T _ij , there is an inequality between these elements.

<Equation 19>

here, to be.

The above inequality is based on the characteristics of the out-of-field instrument: Basically, the signal from the off-site instrument can be generated when the neutrons generated in the core reach the instrument through various barriers such as fuel, moderator, barrel, and pressure vessel. Therefore, the neutrons generated near the instrument are more likely to react in the instrument than neutrons generated farther. Because of these physical properties, the above inequality holds. As can be seen from Equation 19, ISAM has the characteristics of so-called Diagonal Dominance. Of course, the characteristics of the ISAM described above are those assuming an ideal ISAM. In general, the measured ISAM does not satisfy all the characteristics of the ISAM presented above because of problems with measurement data or limitations of the determination method itself. However, in order for SAM to be optimal, it must basically satisfy the characteristics of the above ISAM.

This characteristic of ISAM means that diagonal element S _ij must be positive and satisfy Diagonal Dominance from SAM point of view.

The diagonal dominance of the SAM can be evaluated in two aspects. First, as defined, the diagonal of a column must be greater than the sum of the absolute values of the remaining non-diagonal elements. In this respect, diagonal dominance is characteristic for each off-channel instrument subchannel.

In this regard, constraints can be placed on the SAM matrix. In Equation 19, S ₁₂ , one of the upper diagonal elements, is calculated, such that S ₁₂ = T ₁₃ T ₃₂ -T ₁₂ T ₃₃ <0. Because T ₁₃ <T ₁₂ to T _{32 <} T ₃₃ , S ₁₂ is always a negative value. Similarly, S ₂₁ , S ₃₂ and S ₁₃ should always be negative. This constraint can always be assured because it is a physical constraint that must be met by an off-the-shelf instrument viewing the output of the furnace.

Therefore, in the calculation of the SAM matrix, each constraint matrix for the upper, middle, and lower parts of the measuring instrument becomes as follows.

(Eq. 20)

In the case of the current Korean standard nuclear power plant, the diagonal dominance of the SAM matrix is evaluated in terms of the so-called Test Value (T _v ). T _v is the sum of the absolute values of all elements of the matrix given by the product of the diagonals of SAM and ISAM times SAM, and according to the current procedure, 3 <T _v <6 relations are satisfied. The T _v value is determined according to the characteristics of the off-chamber instrument, and a particularly influential factor is the distance between the off-chamber instrument and the pressure vessel. If the out-of-war instrument is very close to the pressure vessel, most of the out-of-war instrument signal will be generated by neutrons leaking from the core adjacent to the out-of-war instrument. In other words, the contribution of the area to the out-of-core instrument from the instrument to the top or bottom of the core rapidly decreases. However, if the out-of-core instrument is considerably far from the pressure vessel, the core's importance will not be very sensitive depending on the height of the core. It can be said that the upper limit and the lower limit of the T _v value are determined based on the physical phenomenon. As a result, the T _v value may vary depending on the design of the out-of-band instrument. In the case of Korea's standard nuclear power plant, the experience so far indicates that the optimum T _v value is near 4.0. That is, when the SAM is optimally determined and the error in the core protection operator output distribution is small, the T _v value is similar to 4.0. That is, it is most preferable that the T _v value is similar to 4.0 and satisfies the constraint G of <Equation 20>, and the method proposed in the present invention results in satisfying this.

The results of applying the present invention to the SAM determination process using the actual measurement data and the magnitude of the measurement error generated as the cycle proceeds are as follows.

Tables 1, 2 and 3 below are performance comparisons of SAMs calculated using the existing SAM determination method and the present invention.

Table 1 (Case 1)

end. In case of using the existing method

I. When using the present invention

In the case of Table 1, in case the conventional method is also relatively good fit, T _v value is similar to the 4.0 channel A, C, D is T _v is the value change but little change T _v is the channel B are the effects of the present invention 5.03 T _v was improved to 4.21.

<Table 2> (case 2)

end. In case of using the existing method

I. When using the present invention

Table 3 (Case 3)

end. In case of using the existing method

I. In case of using the present invention

Tables 2 and 3 show cases where the effects of the present invention are apparent. In the case of the existing invention of Tables 2 and 3, the result of the measurement noise shows that many negative values should not exist physically among the matrix elements of ISAM, and the inequality relation of <Equation 19> is not satisfied at all. T _{v is} also very far from the optimum value of 4.0, so if you use this value, the error increases greatly as the cycle progresses and you cannot use it for actual operation. Results using the present invention in Table 2, 3 is to eliminate the influence of measurement noise and satisfy the relation of inequality <Equation 19> _v T also shows a very close accurate calculation of the optimum value of 4.0.

3 and 4 are error comparisons of the axial output distribution calculated using the present SAM and the present invention for cases 1 and 2. Here, the error = Ax-b, b is the least-square solution, which shows almost the same results as the existing method to which the least-square method is applied. The calculation method of the present invention also provides an optimal value while providing a result that is not inconsistent with the existing method. it means.

5 and 6 show the comparison results of tracking the error of the axial output distribution calculated using the existing SAM and the present invention for cases 1 and 3 from the reactor cycle beginning to the end of the cycle. FIG. 5 shows that the correct SAM is determined through the existing method, and the accuracy is slightly improved when the calculation method of the present invention is applied. However, FIG. 6 illustrates the calculation of the present invention when the existing method does not provide the correct SAM. The method shows that the accuracy is greatly improved as the cycle progresses. This is because the calculation method of the present invention provides a physically meaningful SAM.

As can be seen from the above comparison, the effects of the present invention can be summarized as follows. First, the present invention provides accurate and stable SAM calculation results even when using data containing a large number of measurement errors. Therefore, it is possible to improve the utilization rate of the power plant by reducing the low output operation time by preventing the cycle start operation time delay, which often occurs when using existing methods such as re-measurement test due to inaccuracy of SAM calculation.

In addition, the present invention is because the data acquisition / calculation procedure for SAM determination is the same as the existing method, there is no technical difficulty in applying the present invention to the field can be used as it is, if you modify only the SAM calculation subroutine. No further procedure changes are required.

Claims (3)

In the method of determining the setpoint (SAM) of a multilevel off-road instrument as part of the core protection system of a nuclear power plant, Is a regularization problem with a linear constraint The out-of-measurement instrument SAM determination method, characterized in that for determining the SAM using the optimization method of the equation. The method of claim 1, wherein the regularization factor λ is selected from 1 × 10 ⁻³ to 1 × 10 ⁻¹ . The method of claim 2, wherein the restriction matrix G is In the three-level off-site instrument consisting of the upper, middle, and lower The off-road measuring instrument SAM determination method, characterized in that.