A control rod drop reliability analysis method based on random collocation perturbation

Abstract

A reliability analysis method for control rod drop based on random collocation perturbation is disclosed, belonging to the field of reliability analysis. First, based on the control rod drop structure, an uncertain drop dynamics model is constructed, and the random variables, random variable vectors, and their distributions during the drop process are determined. Second, considering the random variables, the statistical characteristic values ​​of the drop time are calculated using the random collocation perturbation method based on the uncertain drop dynamics model. Third, using the statistical characteristic values, the probability density function of the drop time is obtained based on the maximum entropy principle. Finally, based on reliability theory, the reliability index of the control rod drop time is calculated. This invention performs reliability analysis based on the calculated probability density function of the control rod drop time, which significantly improves computational efficiency; it fills the gap in the research on reliability analysis of control rod drop when random variables are present; and compared with other reliability analysis methods, it is more efficient in calculating reliability indices.

Description

Technical Field

[0001] This invention belongs to the field of reliability analysis and relates to a method for reliability analysis of control rod drop based on random point perturbation. Background Technology

[0002] Nuclear power generation plays a crucial role in achieving modern energy transition goals due to its extremely low carbon emissions and continuous, stable power output throughout its entire lifecycle. However, a reactor runaway accident poses a serious threat to the environment and public safety. In a reactor, control rods, as the core actuators of the emergency shutdown system, directly determine the success of the shutdown process if they can quickly and reliably descend to their designated positions during an accident.

[0003] In practical engineering, the control rod dropping process is affected by numerous random variables, including but not limited to: uncertainties in geometric parameters caused by manufacturing tolerances, random fluctuations in coolant flow and temperature fields within the reactor core, and uncertainties in parameters obtained through experimental fitting. These random variables cause the dropping response (such as total dropping time, buffer velocity, etc.) to no longer be a single deterministic value, but rather a random response with a specific probability distribution. Using the extreme values ​​(minimum and maximum values) or nominal values ​​of random variables for calculations makes it difficult to accurately describe the statistical distribution and reliability of the dropping response, potentially leading to overly conservative designs or underestimation of potential risks. Therefore, conducting reliability analysis of the control rod dropping process—that is, quantifying the probability that it can still meet performance requirements under random variable conditions—has become an indispensable part of reactor safety assessment and optimization design.

[0004] Currently, some progress has been made in the analysis methods for control rod drop behavior. Chinese invention patent "A Calculation Method for Control Rod Drop Considering Contact Collision" (application number 202211692654.6) uses multibody dynamics analysis to comprehensively consider the collision and friction process between the control rod and the guide tube, thus more realistically simulating the drop dynamics. Chinese invention patent "A Three-Dimensional Fluid-Structure Coupling Simulation Method and Device for Control Rod Drive Line Drop Behavior" (application number 202510909365.4) uses high-fidelity three-dimensional fluid-structure interaction simulation technology, combined with modal synthesis and the Hertzian contact model, to achieve a precise simulation of fluid resistance and structural deformation during the control rod drop process. However, these methods mainly focus on deterministic parameters and fail to systematically quantify the comprehensive impact of the aforementioned random variables on the reliability of the drop.

[0005] In summary, while existing technologies have continuously deepened their deterministic physical modeling, they cannot solve the reliability analysis problem caused by random variables, constituting a current technological gap. To address the problems of existing technologies, this invention proposes a reliability analysis method for stochastic collocation perturbations in the control rod drop problem. The aim is to establish an efficient and accurate probabilistic analysis tool, providing core technical support for the probabilistic safety design, tolerance allocation, and robustness optimization of control rod drive lines. Summary of the Invention

[0006] The purpose of this invention is to overcome the shortcomings of existing technologies by proposing a reliability analysis method for control rod drop based on random collocation perturbation. Specifically, it is a method for reliability analysis of the control rod drop process in nuclear reactors based on random collocation perturbation. This invention employs the fifth-order random collocation perturbation method (ACSP), selects key collocation points, and utilizes perturbation expansion to efficiently calculate the statistical moments of the response, thereby obtaining the probability density function of the control rod drop time for reliability analysis, achieving a one- to two-order improvement in computational efficiency. This invention fills the gap in research on the reliability analysis of control rod drop when random variables are present. Furthermore, compared with other reliability analysis methods, this control rod drop reliability analysis method is more efficient in calculating reliability indicators.

[0007] To achieve the above objectives, the technical solution adopted by the present invention is as follows: a method for reliability analysis of control rod drop based on random point perturbation, the method comprising the following steps:

[0008] The first step is to construct an uncertain drop rod dynamics model based on the existing control rod drop structure, and to determine the random variables, random variable vectors, and their distributions during the drop process. Specifically:

[0009] Step 1.1: The control rod lowering structure mainly consists of a control rod, a guide tube, a drive mechanism, and other components. During the lowering process, the control rod is mainly subjected to its own weight, buoyancy, fluid resistance, and the forces exerted by the drive mechanism and other components. Its dynamic equation is:

[0010] (1)

[0011] in, To control the quality of the rod; The acceleration of the control rod during its descent, and the time of descent. Related; It is the acceleration due to gravity; This refers to the buoyancy force experienced by the control rod during its descent. The fluid resistance experienced by the control rod during its descent; The control rod is subjected to forces from the drive mechanism and other components during its descent.

[0012] Based on the characteristics of the existing control rod descent structure, calculate the gravity during the control rod's descent. ,buoyancy Fluid resistance and the forces acting on the drive mechanism and other components .

[0013] Furthermore, other components of the control rod's falling structure include springs and gear sets, all of which are conventional and known components.

[0014] Step 1.2, based on the actual control rod drop structure, determine the random variables affecting the control rod drop time as follows: Let there be n random variables, and assume that these random variables follow a normal distribution. Represent these random variables as a vector. , where, definition A vector of random variables The first in A random variable, The mean vector of the corresponding random variable is... ,definition Represents arbitrary variables The mean, The standard deviation vector of the corresponding random variable is: ,definition Represents random variables standard deviation .

[0015] Considering the influence of random variables, the dynamic equation of the control rod falling under uncertainty is established according to equation (1), and the expression is:

[0016] (2)

[0017] in, To account for the quality of the control rods that take into account the influence of random variables; To account for the buoyancy force experienced by the control rod during its descent, taking into account the influence of random variables; The fluid resistance experienced by the control rod during its descent was taken into account, taking into account the influence of random variables. The control rod is subjected to forces from the drive mechanism and other components during its descent, taking into account the influence of random variables.

[0018] Step 1.3, convert the random variable vector Convert to a vector of random variables Each zero-mean random variable is defined therein. Corresponding to a random variable The expression is:

[0019] (3)

[0020] From equation (2), we can obtain the random variable. The standard deviation is ,in In vector form, it is .

[0021] Step 1.4, convert the random variable vector Substituting these values ​​into equation (2) yields the uncertain dropping rod dynamics model.

[0022] Step 2: Considering random variables, based on the uncertain drop bar dynamics model from Step 1, the statistical characteristic values ​​of the drop bar time are calculated using the stochastic collocation perturbation method. These statistical characteristic values ​​include the initial drop bar time. First-order center distance, second-order center distance, third-order center distance, and fourth-order center distance; specifically:

[0023] Step 2.1, first convert the random variable vector Each random variable in The value is defined as 0, and substituted into the uncertain drop bar dynamics model in step 1.4 to calculate the initial drop bar time. .

[0024] Step 2.2, construct the random variable vector The first set of sample points is:

[0025] (4)

[0026] in, For the first The value of each random variable is A vector whose other random variables have a value of 0; For the first The value of each random variable is A vector whose other random variables have a value of 0; For the first The value of each random variable is A vector whose other random variables have a value of 0; For the first The value of each random variable is A vector whose other random variables have a value of 0; Represents a zero-mean random variable Expectation to the fourth power; Represents a zero-mean random variable The expectation of the fourth power.

[0027] vector of random variables The first set of sample points , , , As input vectors, these are substituted into the uncertain drop bar dynamics model in step 1.4 to obtain the drop bar time. , , , ,in, for The calculated drop time, for The calculated drop time, for The calculated drop time, for The calculated drop time of the rod.

[0028] Step 2.3, Construct a vector of random variables The second sample point set is:

[0029] (5)

[0030] in, For the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0; For the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0; For the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0; For the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0.

[0031] vector of random variables The second set of sample points , , , Substituting these values ​​into the uncertain drop bar dynamics model obtained in step 1.4, respectively, yields the corresponding drop bar times. , , and ,in, for The calculated drop time, for The calculated drop time, for The calculated drop time, for The calculated drop time of the rod.

[0032] Step 2.4, based on the theory of collocational perturbation, in the random variable vector Under the influence of the first-order center distance for calculating the drop time of the bar As shown in formula (6):

[0033] (6)

[0034] in, A vector of random variables Drop time under influence;

[0035] (7)

[0036] Step 2.5, Construct a vector of random variables The third set of sample points, wherein the third set of sample points is:

[0037] ; ; (8)

[0038] in, For the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0; , These are undetermined constants in the calculation process.

[0039] vector of random variables The third set of sample points Substituting this into the uncertain drop bar dynamics model obtained in step 1.4, we obtain the drop bar time. ,remember .

[0040] Step 2.6, based on the theory of collocational perturbation, in the random variable vector Under the influence of the second-order center distance for calculating the drop time of the rod. As shown in formula (9):

[0041] (9)

[0042] in, Represents a vector of random variables The expected value of the square of the time of fall of the bar; Represents a vector of random variables The expected impact on the drop bar time; Represents a vector of random variables Drop time under influence; Indicates the initial drop time;

[0043] (10)

[0044] in, Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model of step 1.4, we obtain the drop bar time.

[0045] Step 2.7, based on the theory of collocational perturbation, in the random variable vector Under the influence of the drop time, the third-order center distance is calculated. As shown in formula (11):

[0046] (11)

[0047] in, Represents a vector of random variables The expected value of the cube that affects the falling time of the bar;

[0048] (12)

[0049] in, Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model of step 1.4, we obtain the drop bar time.

[0050] Step 2.8, based on the theory of collocational perturbation, in the random variable vector Under the influence of the fourth-order center distance for calculating the drop time of the rod. As shown in formula (13):

[0051] (13)

[0052] in, Represents a vector of random variables The expected value of the fourth power affecting the falling time of the bar;

[0053] (14)

[0054] in, Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model of step 1.4, we obtain the drop bar time.

[0055] Step 3: Using the first-order central moment of the drop time obtained in Step 2.4 and the higher-order central moments of the drop time obtained in Steps 2.6-2.8, where the higher-order central moments of the drop time include the second-order central moments of the drop time (Step 2.6), the third-order central moments of the drop time (Step 2.7), and the fourth-order central moments of the drop time (Step 2.8), the probability density function of the drop time is obtained based on the maximum entropy principle; specifically:

[0056] Step 3.1, calculate the first, second, third, and fourth order central moments of the drop bar time. As a constraint, among which... ,get:

[0057] (15)

[0058] in, Time of bat drop The probability density function. Considering the normalization condition, we have .

[0059] Step 3.2: Construct the Lagrange function based on the maximum entropy principle. Introduce Lagrange multipliers. , , , , Construct the Lagrange function :

[0060] (16)

[0061] in, Indicates the time of bat drop The probability density function; Represents the normalization constraint coefficient; Indicates the first-order central moment constraint coefficient; Indicates the second-order central moment constraint coefficient; Indicates the third-order central moment constraint coefficient; Indicates the fourth-order central moment constraint coefficient; This represents the average drop time of the bar obtained in step 2.1; Represents the drop time variable;

[0062] Step 3.3, Lagrange function pairs Taking the partial derivative, we get:

[0063] (17)

[0064] Then, construct the equivalent constraint form:

[0065] (18)

[0066] in, Indicates normalization constraints; This represents a first-order central moment constraint. This represents a second-order central moment constraint; This represents a third-order central moment constraint; This represents a fourth-order central moment constraint;

[0067] Solving the above constraints using Newton's method yields the following results. , , , , Substituting this into formula (17), we obtain the probability density function of the drop time.

[0068] Step 4: Based on reliability theory, calculate the reliability index of the control rod drop time; specifically:

[0069] Step 4.1: Set the upper limit of the allowable drop time. ;

[0070] Step 4.2, calculate the failure probability ;

[0071] (19)

[0072] according to The size of the drop bar system is used to assess its reliability.

[0073] Furthermore, the permitted upper limit Determined based on the characteristics of the control rod's falling structure.

[0074] Compared with the prior art, the present invention has the following beneficial effects: (1) The control rod drop reliability analysis method proposed in the present invention considers the influence of a variety of random variables, and compared with other drop time solutions, it can better assess the safety of the control rod drop time.

[0075] (2) The control rod drop reliability analysis method based on random point perturbation proposed in this invention requires fewer sample points and has higher accuracy than the traditional uncertainty analysis method, thus reducing the amount of computation required for reliability analysis.

[0076] (3) This invention solves the problem of control rod drop reliability analysis by using the random perturbation point allocation method, filling the gap in the research on control rod drop reliability analysis. Attached Figure Description

[0077] Figure 1 This is a flowchart of a control rod reliability analysis method based on random point perturbation.

[0078] Figure 2 This is a simplified diagram of the nuclear reactor control rod drive circuit at the initial moment in the embodiment of the invention;

[0079] In the figure: 1 Base; 2 Spring; 3 First Gear; 4 Second Gear; 5 Third Gear; 6 Fourth Gear; 7 Drive Rod; 8 First Connecting Part; 9 Second Connecting Part; 10 Absorbing Rod; 11 Conduit; 12 Conduit Side Hole; 13 Reduction Section; 14 Reduction Section Bottom Hole. Detailed Implementation

[0080] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments. These embodiments are based on the technical solution of the present invention and provide detailed implementation methods and specific operating procedures. However, the scope of protection of the present invention is not limited to the following embodiments.

[0081] refer to Figure 1 The flowchart of the control rod drop reliability analysis method based on random point perturbation is shown below. Figure 2 Reliability analysis was performed on the control rod drop model shown. Figure 2 This is a simplified diagram of the nuclear reactor control rod drive circuit at the initial moment in this embodiment of the invention. The drive circuit mainly consists of three parts: a drive line moving component, a drive line fixing component, and other components. The drive line moving component, from top to bottom, consists of: a drive rod 7, a first connecting part 8, a second connecting part 9, and an absorber rod 10. The drive line fixing component, from top to bottom, consists of: a base 1, a guide tube 11, a guide tube side hole 12, a deceleration section 13, and a deceleration section bottom hole 14. The other components, from top to bottom, consist of: a spring 2, a first gear 3, a second gear 4, a third gear 5, and a fourth gear 6. Specifically, the spring connects the base 1 to the drive rod 7, and the first gear 3, second gear 4, third gear 5, and fourth gear 6 mesh sequentially, with the fourth gear 6 in direct contact with the drive rod 7. Initially, the spring at the top of the moving component is compressed, used to accelerate the falling of the moving component and ensure it reaches the bottom of the guide tube within a specified time. The gears on the side of the drive rod provide resistance to the moving component, preventing it from colliding and causing structural damage due to excessive speed during its descent. This embodiment includes the following steps:

[0082] The first step is to construct an uncertain drop rod dynamics model based on the existing control rod drop structure, and to determine the random variables, random variable vectors, and their distributions during the drop process. Specifically:

[0083] Step 1.1: The control rod lowering structure mainly consists of a control rod, a guide tube, a drive mechanism, and other components. During the lowering process, the control rod is mainly subjected to its own weight, buoyancy, fluid resistance, and the forces exerted by the drive mechanism and other components. Its dynamic equation is:

[0084] (1)

[0085] in, To control the quality of the rod; The acceleration of the control rod during its descent, and the time of descent. Related; It is the acceleration due to gravity; This refers to the buoyancy force experienced by the control rod during its descent. The fluid resistance experienced by the control rod during its descent; The control rod is subjected to forces from the drive mechanism and other components during its descent.

[0086] Based on the characteristics of the existing control rod descent structure, calculate the gravity during the control rod's descent. ,buoyancy Fluid resistance and the forces acting on the drive mechanism and other components .

[0087] Other components of the control rod's falling structure include springs and gear sets, all of which are conventional and known components.

[0088] Step 1.2, based on the actual control rod drop structure, determine the random variables affecting the control rod drop time as follows: Let there be n random variables, and assume that these random variables follow a normal distribution. Represent these random variables as a vector. , where, definition A vector of random variables The first in A random variable, The mean vector of the corresponding random variable is... ,definition Represents arbitrary variables The mean of the random variable. The standard deviation vector of the corresponding random variable is... ,definition Represents random variables standard deviation .

[0089] Considering the influence of random variables, the dynamic equation of the control rod falling under uncertainty is established according to equation (1), and the expression is:

[0090] (2)

[0091] in, To account for the quality of the control rods that take into account the influence of random variables; To account for the buoyancy force experienced by the control rod during its descent, taking into account the influence of random variables; The fluid resistance experienced by the control rod during its descent was taken into account, taking into account the influence of random variables. The control rod is subjected to forces from the drive mechanism and other components during its descent, taking into account the influence of random variables.

[0092] Step 1.3, convert the random variable vector Convert to a vector of random variables Each zero-mean random variable is defined therein. Corresponding to a random variable The expression is:

[0093] (3)

[0094] From equation (2), we can obtain the random variable. The standard deviation is ,in In vector form, it is .

[0095] Step 1.4, convert the random variable vector Substituting these values ​​into equation (2) yields the uncertain falling rod dynamics model.

[0096] Step 2: Considering random variables, the statistical characteristic value of the drop time is calculated using the random collocation perturbation method. This statistical characteristic value includes the initial drop time. First-order center distance, second-order center distance, third-order center distance, and fourth-order center distance; specifically:

[0097] Step 2.1, first convert the random variable vector Each random variable in The value is defined as 0, and substituted into the uncertain drop bar dynamics model in step 1.4 to calculate the initial drop bar time. .

[0098] Step 2.2, construct the random variable vector The first set of sample points is:

[0099] (4)

[0100] in, For the first The value of each random variable is A vector whose other random variables have a value of 0; For the first The value of each random variable is A vector whose other random variables have a value of 0; For the first The value of each random variable is A vector whose other random variables have a value of 0; For the first The value of each random variable is A vector whose other random variables have a value of 0; Represents a zero-mean random variable Expectation to the fourth power; Represents a zero-mean random variable The expectation of the fourth power.

[0101] vector of random variables The first set of sample points , , , As input vectors, these are substituted into the uncertain drop bar dynamics model in step 1.4 to obtain the drop bar time. , , , ,in, for The calculated drop time, for The calculated drop time, for The calculated drop time, for The calculated drop time of the rod.

[0102] Step 2.3, Construct a vector of random variables The second sample point set is:

[0103] (5)

[0104] in, For the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0; For the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0; For the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0; For the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0.

[0105] vector of random variables The second set of sample points , , , Substituting these values ​​into the uncertain drop bar dynamics model obtained in step 1.4, respectively, yields the corresponding drop bar times. , , and ,in, for The calculated drop time, for The calculated drop time, for The calculated drop time, for The calculated drop time of the rod.

[0106] Step 2.4, based on the theory of collocational perturbation, in the random variable vector Under the influence of the first-order center distance for calculating the drop time of the bar As shown in formula (6):

[0107] (6)

[0108] in, A vector of random variables Drop time under influence;

[0109] (7)

[0110] Step 2.5, Construct a vector of random variables The third set of sample points, wherein the third set of sample points is:

[0111] ; ; (8)

[0112] in, For the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0; , These are undetermined constants in the calculation process.

[0113] vector of random variables The third set of sample points Substituting this into the uncertain drop bar dynamics model obtained in step 1.4, we obtain the drop bar time. ,remember .

[0114] Step 2.6, based on the theory of collocational perturbation, in the random variable vector Under the influence of the second-order center distance for calculating the drop time of the rod. As shown in formula (9):

[0115] (9)

[0116] in, Represents a vector of random variables The expected value of the square of the time of fall of the bar; Represents a vector of random variables The expected impact on the drop bar time; Represents a vector of random variables Drop time under influence; Indicates the initial drop time;

[0117] (10)

[0118] in, Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model of step 1.4, we obtain the drop bar time.

[0119] Step 2.7, based on the theory of collocational perturbation, in the random variable vector Under the influence of the drop time, the third-order center distance is calculated. As shown in formula (30):

[0120] (30)

[0121] in, Represents a vector of random variables The expected value of the cube that affects the falling time of the bar;

[0122] (12)

[0123] in, Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain dropping rod dynamics model, we obtain the dropping time.

[0124] Step 2.8, based on the theory of collocational perturbation, in the random variable vector Under the influence of the fourth-order center distance for calculating the drop time of the rod. As shown in formula (32):

[0125] (13)

[0126] in, Represents a vector of random variables The expected value of the fourth power affecting the falling time of the bar;

[0127] (14)

[0128] in, Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain drop bar dynamics model in step 1.4, we obtain the drop bar time; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. Substituting this into the uncertain dropping rod dynamics model, we obtain the dropping time.

[0129] Step 3: Using the first-order central moment of the drop time obtained in Step 2.4 and the higher-order central moments of the drop time obtained in Steps 2.6-2.8, where the higher-order central moments of the drop time include the second-order central moment of the drop time (Step 2.6), the third-order central moment of the drop time (Step 2.7), and the fourth-order central moment of the drop time (Step 2.8), the probability density function of the drop time is obtained based on the maximum entropy principle; specifically:

[0130] Step 3.1, calculate the first, second, third, and fourth order central moments of the drop bar time. As a constraint, among which... ,get:

[0131] (15)

[0132] in, Time of bat drop The probability density function. Considering the normalization condition, we have: .

[0133] Step 3.2: Construct the Lagrange function based on the maximum entropy principle. Introduce Lagrange multipliers. , , , , Construct the Lagrange function :

[0134] (16)

[0135] Step 3.3, Lagrange function pairs Taking the partial derivative, we get:

[0136] (17)

[0137] Then, construct the equivalent constraint form:

[0138] (18)

[0139] in, Indicates normalization constraints; This represents a first-order central moment constraint. This represents a second-order central moment constraint; This represents a third-order central moment constraint; This represents a fourth-order central moment constraint;

[0140] Solving the above constraints using Newton's method yields the following results. , , , , Substituting into formula (36), we obtain the probability density function of the drop time.

[0141] Step 4: Based on reliability theory, calculate the reliability index of the control rod drop time; specifically:

[0142] Step 4.1: Set the upper limit of the allowable drop time. =0.80 seconds;

[0143] Step 4.2, calculate the failure probability ;

[0144] (19)

[0145] Therefore, the failure probability of the dropping bar is 1.314%.

[0146] The above-described embodiments are merely illustrative of the implementation methods of the present invention, but should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the protection scope of the present invention.

Claims

1. A control rod drop reliability analysis method based on random collocation perturbation, characterized in that, The method for analyzing the reliability of the control rod drop includes the following steps: The first step is to construct an uncertain falling rod dynamics model based on the control rod falling structure, and to determine the random variables, random variable vectors and their distributions during the falling rod process; Second step: Based on the uncertainty falling rod dynamics model of the first step, the statistical characteristic values of falling rod time are calculated by using the random collocation perturbation method in the case of considering random variables, wherein the statistical characteristic values include the initial falling rod time first order central distance, second order central distance, third order central distance, fourth order central distance; in particular: Step 2.1, convert the random variable vector Each random variable in The value is defined as 0, and substituted into the uncertain drop bar dynamics model in step 1.4 to calculate the initial drop bar time. ; Step 2.2, Construct a vector of random variables The first set of sample points is: (4) in, For the first The value of each random variable is A vector whose other random variables have a value of 0; For the first The value of each random variable is A vector whose other random variables have a value of 0; For the first The value of each random variable is A vector whose other random variables have a value of 0; For the first The value of each random variable is A vector whose other random variables have a value of 0; Represents a zero-mean random variable Expectation to the fourth power; Represents a zero-mean random variable Expectation to the fourth power; vector of random variables The first set of sample points , , , As input vectors, these are substituted into the uncertain drop bar dynamics model in step 1.4 to obtain the drop bar time. , , , ,in, for The calculated drop time, for The calculated drop time, for The calculated drop time, for The calculated drop time; Step 2.3, Construct a vector of random variables The second sample point set is: (5) in, For the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0; For the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0; For the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0; For the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0; vector of random variables The second set of sample points , , , Substituting these values ​​into the uncertain drop bar dynamics model obtained in step 1.4, respectively, yields the corresponding drop bar times. , , and ,in, for The calculated drop time, for The calculated drop time, for The calculated drop time, for The calculated drop time; Step 2.4, in the random variable vector Under the influence of the first-order center distance for calculating the drop time of the bar As shown in formula (6): (6) in, A vector of random variables Drop time under influence; (7) Step 2.5, Construct a vector of random variables The third sample point set is: ； ； (8) in, For the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0; , These are constants to be determined during the calculation process; vector of random variables The third set of sample points Substituting this into the uncertain drop bar dynamics model obtained in step 1.4, we obtain the drop bar time. ,remember ; Step 2.6, based on the theory of collocational perturbation, in the random variable vector Under the influence of the second-order center distance for calculating the drop time of the rod. As shown in formula (9): (9) in, Represents a vector of random variables The expected value of the square of the time of fall of the bar; Represents a vector of random variables The expected impact on the drop bar time; Represents a vector of random variables Drop time under influence; Indicates the initial drop time; (10) in, Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is , No. The value of each random variable is A vector whose other random variables have a value of 0. ; Step 2.7, in the random variable vector Under the influence of the drop time, the third-order center distance is calculated. As shown in formula (11): (11) in, Represents a vector of random variables The expected value of the cube that affects the falling time of the bar; (12) in, Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. ; Step 2.8, based on the theory of collocational perturbation, in the random variable vector Under the influence of the fourth-order center distance for calculating the drop time of the rod. As shown in formula (13): (13) in, Represents a vector of random variables The expected value of the fourth power affecting the falling time of the bar; (14) in, Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. ; Indicates the first The value of each random variable is A vector whose other random variables have a value of 0. ; Step 3: Using the statistical characteristic values ​​obtained in Step 2, obtain the probability density function of the drop time based on the principle of maximum entropy; Step 4: Based on reliability theory, calculate the reliability index of the control rod drop time.

2. The method for reliability analysis of control rod drop based on random point perturbation according to claim 1, characterized in that, The first step is specifically as follows: Step 1.1, the control rod lowering structure includes a control rod, a guide tube, a drive mechanism, and other components, including springs and gear sets; the dynamic equation of the control rod lowering structure is: (1) in, To control the quality of the rod; The acceleration of the control rod during its descent, and the time of descent. Related; It is the acceleration due to gravity; This refers to the buoyancy force experienced by the control rod during its descent. The fluid resistance experienced by the control rod during its descent; The control rod is subjected to forces from the drive mechanism and other components during its descent. Step 1.2, based on the control rod's descent structure, determine the random variable affecting the control rod's descent time. There are 10 random variables, and it is assumed that the random variables follow a normal distribution; the random variables are represented in vector form as follows: , where, definition A vector of random variables The first in A random variable, The mean vector of the corresponding random variable is ,definition Represents random variables The mean, The standard deviation vector of the corresponding random variable is ,definition Represents random variables standard deviation ; Considering the influence of random variables, the dynamic equation of the control rod falling under uncertainty is established according to equation (1), and the expression is: (2) in, To account for the quality of the control rods that are affected by random variables; To account for the buoyancy force experienced by the control rod during its descent, taking into account the influence of random variables; To account for the fluid resistance experienced by the control rod during its descent, taking into account the influence of random variables; To account for the influence of random variables, the control rod is subjected to forces from the drive mechanism and other components during its descent. Step 1.3, convert the random variable vector Convert to a vector of random variables Each zero-mean random variable is defined therein. Corresponding to a random variable The expression is: (3) From equation (2), we can obtain the random variable. The standard deviation is ,in In vector form, it is ; Step 1.4, convert the random variable vector Substituting these values ​​into equation (2) yields the uncertain dropping rod dynamics model.

3. The method for reliability analysis of control rod drop based on random point perturbation according to claim 2, characterized in that, The third step is specifically as follows: Step 3.1, calculate the first, second, third, and fourth order central moments of the drop bar time. As a constraint, among which... ,get: (15) in, Time of bat drop The probability density function; considering the normalization condition, we have ; Step 3.2: Construct the Lagrange function based on the maximum entropy principle; introduce Lagrange multipliers. , , , , Construct the Lagrange function : (16) in, Indicates the time of bat drop The probability density function; Represents the normalization constraint coefficient; Indicates the first-order central moment constraint coefficient; Indicates the second-order central moment constraint coefficient; Indicates the third-order central moment constraint coefficient; This represents the fourth-order central moment constraint coefficient; This represents the initial drop time obtained in step 2.1; Represents the drop time variable; Step 3.3, Lagrange function pairs Taking the partial derivative, we get: (17) Then, construct the equivalent constraint form: (18) in, Indicates normalization constraints; This represents a first-order central moment constraint. This represents a second-order central moment constraint; This represents a third-order central moment constraint; This represents a fourth-order central moment constraint; Solving the above constraints using Newton's method yields the following results: , , , , Substituting this into formula (17), we obtain the probability density function of the drop time.

4. The method for reliability analysis of control rod drop based on random point perturbation according to claim 3, characterized in that, The fourth step is specifically as follows: Step 4.1: Set the upper limit of the allowable drop time. ; Step 4.2, calculate the failure probability ; (19) according to The size of the drop bar system is used to assess its reliability.

5. The method for reliability analysis of control rod drop based on random point perturbation according to claim 4, characterized in that, The allowed upper limit Determined based on the characteristics of the control rod's falling structure.

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