CN109063233B - A Monte Carlo method for evaluating nuclide content versus k eff Method for uncertainty influence - Google Patents
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- 238000000034 method Methods 0.000 title claims abstract description 47
- 238000000342 Monte Carlo simulation Methods 0.000 title claims abstract description 28
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- 238000009826 distribution Methods 0.000 claims abstract description 25
- 238000005070 sampling Methods 0.000 claims abstract description 22
- 238000012795 verification Methods 0.000 claims abstract description 8
- 238000012614 Monte-Carlo sampling Methods 0.000 claims abstract description 5
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Abstract
The invention belongs to the technical field of nuclear safety evaluation, and relates to a method for evaluating the nuclide content versus k by using a Monte Carlo method eff Uncertainty affects the method. The saidThe method of (2) is based on a Monte Carlo method, and comprises the following steps: (1) experimental verification: verifying experimental data by using a burnup simulation program, and comparing the measured value of the nuclide component content given in each experiment with the calculated value of the nuclide component content simulated by the burnup program; (2) Determining distribution of verification results of each nuclide and adjusting the standard deviation of the average value; (3) performing Monte Carlo sampling calculation; (4) carrying out statistical analysis on the sampling calculation result; (5) determining the total uncertainty. The Monte Carlo method is used for evaluating the nuclide content to k eff The uncertainty influence method can enable the calculation of the critical limit value in the nuclear critical analysis to be more accurate.
Description
Technical Field
The invention belongs to the technical field of nuclear safety evaluation, and relates to a method for evaluating the nuclide content versus k by using a Monte Carlo method eff Uncertainty affects the method.
Background
In critical analysis and critical design of a nuclear critical system, an effective increment factor k eff Playing a very important role, the evaluation of the uncertainty is of great importance for the determination of the critical security limit of the core.
Effective increment factor k eff Sources of uncertainty of (c) include mainly:
(1) Uncertainty of a calculation method and a calculation process, including uncertainty of burnup calculation and critical calculation;
(2) Uncertainty of the model, including simplification of the model, uncertainty of manufacturing tolerances, and the like;
(3) Uncertainty in the data, including nuclear cross-section data, uncertainty in nuclear material loading, and the like.
Wherein in critical analysis using burnup belief technique, nuclear critical system k is caused by uncertainty of nuclide composition eff There has been no good method of assessing uncertainty of (c).
Since the Monte Carlo sampling method is a method widely applied to the uncertainty evaluation field, the state published JF 1059.2-2012 "evaluation of measurement uncertainty by Monte Carlo method". However, due to the specificity of the nuclear critical system, the long calculation time consumption of the traditional Monte Carlo method, high requirement on conservation of calculation and the like, the traditional Monte Carlo method evaluation flow is required to be improved so as to meet the applicability requirement of uncertainty evaluation of the nuclear critical system.
Disclosure of Invention
The invention aims to provide a method for evaluating the nuclide content to k by using the Monte Carlo method eff The uncertainty influence method can enable the calculation of the critical limit value in the nuclear critical analysis to be more accurate.
To achieve this object, in a basic embodiment, the present invention provides a method for assessing the nuclide content versus k using the Monte Carlo method eff A method of uncertainty influence, said method being based on a monte carlo method, comprising the steps of:
(1) And (3) experimental verification: verifying experimental data by using a burnup simulation program, and comparing the measured value of the nuclide component content given in each experimental data with the calculated value of the nuclide component content simulated by the burnup program;
(2) Determining distribution of verification results of each nuclide and adjusting the standard deviation of the average value;
(3) Carrying out Monte Carlo sampling calculation;
(4) Carrying out statistical analysis on the sampling calculation result;
(5) The total uncertainty is determined.
In a preferred embodiment, the present invention provides a method for assessing nuclide content versus k using the Monte Carlo method eff The uncertainty influence method, wherein in the step (1), the nuclide component content measurement value and the kernel are calculated by adopting the following formulaRatio of calculated values of element content:
wherein:
In a more preferred embodiment, the present invention provides a method for assessing nuclide content versus k using the Monte Carlo method eff The uncertainty influence method, wherein in the step (1), the standard deviation of the average value and the average value of the ratio is calculated by adopting the following formula:
wherein:
N n sample number for the nth species;
σ' n is the standard deviation of the average value of the ratio.
In a preferred embodiment, the present invention provides a method for assessing nuclide content versus k using the Monte Carlo method eff Method of uncertainty influence, wherein in step (2), a normal distribution is used
As the nuclide distribution information, the average value of each nuclide sample is used as the average value of the normal distribution, wherein +.>
Is the average value of the ratio; sigma (sigma) Xn Is the adjusted value of the standard deviation sigma' n of the ratio average value.
According to GB/T3359-2008 ' determination of statistical treatment and interpretation tolerance interval of data ', the confidence coefficient of variance unknowns under normal distribution is 95% (the confidence coefficient can be selected according to the requirement, 99% or 95% is generally selected, the invention will be illustrated by taking 95% as an example), and the tolerance coefficient of coverage is 68.3% (the coverage is 68.3% which is the inclusion probability of normal distribution standard deviation and cannot be modified at will) is used for sigma ' n Adjusting to obtain
Where k4 (n; 0.683; 0.95) represents a double-sided tolerance coefficient with degree of freedom n, confidence 95%, coverage 68.3% for which the variance is unknown in a normal distribution.
In a preferred embodiment, the present invention provides a method for assessing nuclide content versus k using the Monte Carlo method eff In the method of uncertainty influence, in the step (3), the content of different nuclides is sampled according to the nuclide distribution determined in the step (2), and the formula of the sampling method is as follows:
wherein:
c is the nuclide content used in critical calculation;
c n the content of the nth species simulated for the burnup calculation program;
In a preferred embodiment, the present invention provides a method for assessing nuclide content versus k using the Monte Carlo method eff In the method of uncertainty influence, in the step (3), the content of the nuclide after sampling is subjected to critical calculation to obtain effective increment factors under each sampling state
Wherein i represents different sampling points; simultaneously, the nuclide components simulated by the burnup calculation program are directly subjected to critical calculation without adjustment, and k is obtained by calculation eff Defined as k eff-REF 。
In a preferred embodiment, the present invention provides a method for assessing nuclide content versus k using the Monte Carlo method eff Method of uncertainty influence, wherein in step (4) sample N is calculated c Effective increment factor k of times (which should be adjusted according to different critical systems and is recommended to be more than 500 times) eff Average value of (2)
And standard deviation->
The calculation formulas are respectively as follows:
wherein:
is an effective increment factor for the i-th sampling point.
In a more preferred embodiment, the present invention provides a method for assessing nuclide content versus k using the Monte Carlo method eff Method for uncertainty influence, wherein in step (4), in calculating the effective increment factor k eff Average value of (2)
And standard deviation
On the basis of (1) further adopting the following formula to calculate k respectively eff Deviation k eff bias and k eff Uncertainty of deviation k eff unc：
Wherein:
k eff-REF k obtained by directly carrying out critical calculation on nuclide components simulated by the burnup calculation program without adjustment eff ；
k 3 (n; 0.95) is a single-sided tolerance coefficient with 95% coverage for 95% with 95% confidence that the variance is unknown in normal distribution.
In a preferred embodiment, the present invention provides a method of assessing by the Monte Carlo methodNuclide content vs k eff A method of uncertainty influence, wherein in step (5), the total uncertainty is calculated using the formula:
wherein:
k eff bias is an effective increment factor k eff Deviation;
k eff un is an effective increment factor k eff Uncertainty of the deviation;
k eff-REF k obtained by directly carrying out critical calculation on nuclide components simulated by the burnup calculation program without adjustment eff 。
The invention has the beneficial effects that the Monte Carlo method is utilized to evaluate the nuclide content to k eff The uncertainty influence method can enable the calculation of the critical limit value in the nuclear critical analysis to be more accurate.
The invention aims at the nuclear species component content pair critical system k simulated by the burnup calculation program in the critical analysis of the spent fuel storage grillwork eff The evaluation of the influence is combined with the characteristics and requirements of critical system calculation. The evaluation result obtained by the evaluation method provided by the invention has the characteristics of both the requirement of the Monte Carlo method on the sampling times and the time consumption of critical calculation, and simultaneously considers the conservation of the calculation result, thereby having important application value for improving the uncertainty evaluation method.
Drawings
FIG. 1 is an exemplary Monte Carlo method of the present invention for assessing nuclide content versus k eff A flow chart of a method of uncertainty influence.
FIG. 2 is a schematic cross-sectional view of a spent fuel storage rack storage unit used in the assessment method of the present invention as exemplified in the detailed description. As shown, the storage unit comprises a nuclear fuel storage unit outer wall 1 at the peripheral edge and a fuel assembly storage area 3 at the inside, wherein a fuel assembly 4 is stored in the fuel assembly storage area 3, and a neutron poison 2 is arranged between the fuel assembly storage area 3 and the nuclear fuel storage unit outer wall 1.
FIG. 3 is a schematic diagram of the calculation results of the evaluation method of the present invention as exemplified in the detailed description.
Detailed Description
The following describes the embodiments of the present invention further with reference to the drawings.
Exemplary of the present invention for assessing nuclide content versus k using Monte Carlo method eff The flow of the method of uncertainty influence is shown in fig. 1 and comprises the following steps.
(1) Experiment verification
And verifying experimental data by using a burnup simulation program, and comparing the measured value of the nuclide component content given in each experimental data with the calculated value of the nuclide component content simulated by the burnup program.
Calculating the ratio of the measured value of the nuclide component to the calculated value of the nuclide component by adopting the following formula:
wherein:
The average value of the ratio and the standard deviation of the average value of the ratio are calculated by adopting the following formula:
wherein:
N n sample number for the nth species;
σ' n is the standard deviation of the average value of the ratio.
(2) Determining distribution of verification results of each nuclide and adjusting average standard deviation
In sampling using the Monte Carlo method, it is first necessary to determine
Is a normal distribution +.>
As the distribution information thereof, a sample average value was used as the average value of the normal distribution. According to GB/T3359-2008 'determination of statistical treatment and interpretation tolerance interval of data', the confidence of variance unknowns under normal distribution is 95% (the confidence can be selected according to the requirement, 99% or 95% is generally selected, the invention will be illustrated by taking 95% as an example), the coverage is 68.3% (the coverage is 68.3% positive
Inclusion probability of state distribution standard deviation, not arbitrarily modifiable) double-sided tolerance coefficient pair σ' n Adjusting to obtain
Wherein k is 4 (n; 0.683; 0.95) means a double-sided tolerance coefficient with degree of freedom n, confidence 95%, coverage 68.3% for which variance is unknown under normal distribution.
(3) Monte Carlo sampling calculation
Sampling the content of different nuclides according to the nuclide distribution determined in the step (2), wherein the sampling method comprises the following formula:
wherein:
c is the nuclide content used in critical calculation;
c n the content of the nth species simulated for the burnup calculation program;
Critical calculation is carried out on the nuclide content after sampling to obtain effective increment factors under each sampling state
Wherein i represents different sampling points; simultaneously, the nuclide components simulated by the burnup calculation program are directly subjected to critical calculation without adjustment, and k is obtained by calculation eff Defined as k eff-REF 。
(4) Statistical analysis of the sampled calculation
Calculate sample N c Effective increment factor k of times (which should be adjusted according to different critical systems and is recommended to be more than 500 times) eff Average value of (2)
And standard deviation->
The calculation formulas are respectively as follows:
wherein:
is an effective increment factor for the i-th sampling point.
On the basis, the following formulas are further adopted to calculate k respectively eff Deviation k eff bias and k eff Uncertainty of deviation k eff unc：
Wherein:
k eff-REF k obtained by directly carrying out critical calculation on nuclide components simulated by the burnup calculation program without adjustment eff ；
k 3 (n; 0.95) is a single-sided tolerance coefficient with 95% coverage for 95% with 95% confidence that the variance is unknown in normal distribution. Confidence and coverage should be chosen as desired, here by way of example 95%.
(5) Determining total uncertainty
The total uncertainty is calculated using the following formula:
wherein:
k eff bias is an effective increment factor k eff Deviation;
k eff un is an effective increment factor k eff Uncertainty of the deviation;
k eff-REF k obtained by directly carrying out critical calculation on nuclide components simulated by the burnup calculation program without adjustment eff 。
The above exemplary Monte Carlo method of the present invention is used to evaluate the nuclide content versus k eff The application of the uncertainty-affected method is exemplified as follows.
According to the steps (1) and (2), the experimental data of Calvert Cliffs 1, H.B. Robinson 2, takahama 3 and TMI 1 are simulated by using a CASMO5 program (CASMO 5 is a two-dimensional component burnup calculation program developed by Studik corporation in U.S., and the nuclide density of various isotopes of the component under different operation histories can be given), and 27 kinds of actinides and main fission products commonly used in the burnup trust technology are totally used 234 U、 235 U、 236 U、 238 U、 237 Np、 238 Pu、 239 Pu、 240 Pu、 241 Pu、 242 Pu、 241 Am、 243 Am、 95 Mo、 99 Tc、 101 Ru、 103 Rh、 109 Ag、 133 Cs、 147 Sm、 149 Sm、 150 Sm、 151 Sm、 152 Sm、 143 Nd、 145 Nd、 153 Eu and 155 gd) was analyzed. Average value X of different nuclides n Uncertainty sigma of each nuclide n As shown in table 1 below.
TABLE 1 average values of different nuclides and uncertainty of each nuclide
Establishing a critical calculation model as shown in FIG. 2 according to the steps (3) and (4)Performing 1 critical calculation based on the simulated nuclide component of CASMO5 at 40GWD/tU to obtain k eff-REF 0.8557. The critical calculation uses the MONK10A program (MONK 10A program is a large three-dimensional Monte Carlo method particle transport program developed by ANSSWERS corporation, england, and is widely applied to critical calculation) and then samples nuclide components 1000 times, namely 1000 times of critical calculation, and statistical analysis is carried out on the results, and the results are shown in figure 3. In FIG. 3, k eff Upper limit and average k eff For values ending at number, e.g. number 200 is the corresponding k for the first 200 calculations eff Upper limit and average k eff . Calculated 1000 times
k eff bias，/>
k eff unc。
As described in step (5), due to k eff bias is greater than 0, and the total uncertainty is calculated:
k eff unc/k eff-REF ＝0.48％
it will be apparent to those skilled in the art that various modifications and variations can be made to the present invention without departing from the spirit or scope of the invention. Thus, it is intended that the present invention also include such modifications and alterations insofar as they come within the scope of the appended claims or the equivalents thereof. The above embodiments are merely illustrative of the present invention, and the present invention may be embodied in other specific forms or with other specific forms without departing from the spirit or essential characteristics thereof. The described embodiments are, therefore, to be considered in all respects as illustrative and not restrictive. The scope of the invention should be indicated by the appended claims, and any changes that are equivalent to the intent and scope of the claims are intended to be encompassed within the scope of the invention.
Claims (4)
1. A Monte Carlo method for evaluating nuclide content versus k eff A method of uncertainty influence, characterized in that said method is based on a monte carlo method, comprising the steps of:
(1) And (3) experimental verification: verifying experimental data by using a burnup simulation program, and comparing the measured value of the nuclide component content given in each experimental data with the calculated value of the nuclide component content simulated by the burnup program;
(2) Determining distribution of verification results of each nuclide and adjusting the standard deviation of the average value;
(3) Carrying out Monte Carlo sampling calculation;
(4) Carrying out statistical analysis on the sampling calculation result;
(5) Determining a total uncertainty;
in the step (1), the ratio of the measured value of the nuclide component to the calculated value of the nuclide component is calculated by adopting the following formula:
wherein:
in the step (2), normal distribution is adopted
As the nuclide distribution information, the average value of each nuclide sample is used as the average value of the normal distribution, wherein +.>
Is the average value of the ratio; />
Is the standard deviation sigma 'of the average value of the ratio' n The adjusted value; in the step (3), sampling the content of different nuclides according to the nuclide distribution determined in the step (2), wherein the sampling method has the formula:
wherein:
c is the nuclide content used in critical calculation;
c n the content of the nth species simulated for the burnup calculation program;
in step (4), sample N is calculated c The next effective increment factor k eff Average value of (2)
And standard deviation->
The calculation formulas are respectively as follows:
wherein:
an effective increment factor for the ith sampling point;
in step (5), the total uncertainty is calculated using the following formula:
wherein:
k eff bias is an effective increment factor k eff Deviation;
k eff un is an effective increment factor k eff Uncertainty of the deviation;
k eff-REF k obtained by directly carrying out critical calculation on nuclide components simulated by the burnup calculation program without adjustment eff 。
2. The method of claim 1, wherein in step (1), the average value of the ratio to the standard deviation of the average value of the ratio is calculated using the following formula:
wherein:
N n sample number for the nth species;
σ′ n is the standard deviation of the average value of the ratio;
3. the method according to claim 1, characterized in that: in the step (3), the content of the nuclide after sampling is subjected to critical calculation to obtain effective increment factors under each sampling state
Wherein i represents different sampling points; simultaneously, the nuclide components simulated by the burnup calculation program are directly subjected to critical calculation without adjustment, and k is obtained by calculation eff Defined as k eff-REF 。
4. The method according to claim 1, wherein in step (4), in calculating the effective increment factor k eff Average value of (2)
And standard deviation->
On the basis of (1) further adopting the following formula to calculate k respectively eff Deviation k eff bias and k eff Uncertainty of deviation k eff unc：
Wherein:
k eff-REF k obtained by directly carrying out critical calculation on nuclide components simulated by the burnup calculation program without adjustment eff ；
k 3 (n; 0.95) is a single-sided tolerance coefficient with 95% coverage for 95% with 95% confidence that the variance is unknown in normal distribution.
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