Probabilistic evaluation of steam generator tube degradation with statistical sampling and artificial intelligence

Abstract

A method of maintaining tubes of a nuclear reactor steam generator includes performing location-specific statistical sampling to generate a distinct set of wear parameters for each of a plurality of locations of the tubes, each distinct set of wear parameters describing wear over time for one of the locations; analyzing each location of each tube based on the distinct set of wear parameters and mechanical properties for each location of each tube by computing a performance criteria probability for each location of each tube; calculating a computed performance criteria probability for the specified steam generator as a whole from the computed performance criteria for each location of each tube at a future time; and upon a determination that the computed performance criteria probability for the specified steam generator reaches or passes a predetermined threshold, identifying tubes to be plugged based the performance criteria probabilities for the locations.

Description

PROBABILISTIC EVALUATION OF STEAM GENERATOR TUBE DEGRADATION WITH STATISTICAL SAMPLING AND ARTIFICIAL INTELLIGENCE

[0001] The present disclosure relates generally to nuclear power plants and more specifically to a method of assessing the degradation of steam generator tubes.BACKGROUND

[0002] Steam generator tube wear is a stochastic failure mechanism that, if left unchecked, would lead to a loss of integrity of the reactor coolant system. Wear is monitored with periodic inspection, and a tube with wear is plugged if the measured wear is too large. Plugged tubes reduce the heat carrying capacity of the steam generator, which can limit the economic lifetime of the component.

[0003] Current industry practice is to use statistical sampling techniques to estimate wear for a large population of tube-to-support locations (e.g., 100,000 locations) by sampling from the measured wear rates of tubes in that population within the previous operating cycles or from available operating experience of similar model steam generators.SUMMARY

[0004] The current industry practice does not account for the specific wear behaviors of individual wear locations. More specifically, many tube locations will tend to have a “wear-in” period, where wear follows a logarithmic profile - initially a high wear rate, but then the wear rate slows down as the wear progresses. Further, with the statistical sampling approach, the wear rate for a tube location that is in low wear rate part of its life can be assigned a high wear rate from a different location within the same population. This statistical mixing and matching of wear from different locations tends to on average over-estimate the number of tubes that are predicted to have high amounts of wear. This leads to unnecessary plugging of tubes, based on these predictions, which is an unfavorable economic impact.

[0005] The present disclosure accounts uses a location-specific statistical sampling to account for the specific wear behaviors of individual wear locations for guiding tube plugging over the lifetime of the steam generator.

[0006] A method of maintaining tubes of a specified steam generator of a pressurized water reactor is provided. The method includes inputting historical measurements for a plurality of locations of a plurality of tubes of at least one steam generator, the at least one steam generator including the specified steam generator and / or at least one further steam generator; performing location-specific statistical sampling of the historical measurements to generate a distinct set of wear parameters for each of a plurality of locations of the tubes of the specified steam generator, each distinct set of wear parameters describing wear over time for one of the locations; analyzing each location of each tube based on the distinct set of wear parameters and mechanical properties for each location of each tube by computing a performance criteria probability for each location of each tube; calculating a computed performance criteria probability for the specified steam generator as a whole from the computed performance criteria for each location of each tube at a future time, and comparing to a predetermined threshold; upon a determination that the computed performance criteria probability for the specified steam generator reaches or passes the predetermined threshold, identifying tubes to be plugged based on a temporal and spatial distribution of the performance criteria probabilities for the locations to achieve an acceptable value for the performance criteria probability for the specified steam generator at the future time; and plugging the identified tubes to achieve the acceptable value for the performance criteria probability for the specified steam generator at the future time.

[0007] In examples, the performing of location-specific statistical sampling of the historical measurements to generate a distinct set of wear parameters for each of a plurality of locations of the tubes of the specified steam generator includes: analyzing an entire operating history of measured wear for each location in the specified steam generator.

[0008] In examples, the performing of location-specific statistical sampling of the historical measurements to generate a distinct set of wear parameters for each of a plurality of locations of the tubes of the specified steam generator includes: selecting a parametric mathematical function to describe wear as function of time at each of the locations; generating, for each location, optimal values for location-specific wear coefficients defining the wear parameters as a function of time and a covariance matrix defining a covariance of the location-specific wear coefficients; using the parametric mathematical function, the optimal values for the location-specific wearcoefficients and the covariance matrix for each location to statistically sample wear parameters at each location to generate the distinct set of wear parameters for each of the plurality of locations.

[0009] In examples, the generating, for each location, optimal values for wear coefficients defining the wear parameters as a function of time and a covariance matrix defining a covariance of the wear coefficients includes: for each location, if there is a sufficient number of historical wear measurements for the location, using location specific extrapolated wear curves defining the wear parameters as function of time, where the location specific extrapolated wear curves includes location-specific wear coefficients and covariance computed by regression to the historical wear measurements for the location, to statistically sample a population of extrapolated future wear curves to generate the distinct set of wear parameters for each of the locations having the sufficient number of historical wear measurements; and for each location, if there is an insufficient number of historical wear measurements for the location or there is no wear, an initiation time of the wear is predicted and then projections of wear rate are made using an artificial intelligence model to generate the distinct set of wear parameters for each of the locations having the insufficient number of historical wear measurement.

[0010] In examples, the parametric mathematical function is a logarithmic function, a power-law function or a hyperbolic function.

[0011] In examples, if there is an insufficient number of historical wear measurements for the location, the initiation time of the wear is predicted from the insufficient number of historical wear measurements.

[0012] In examples, if there is no wear at a location, the location is assigned a time in the future at which wear is predicted to initiate.

[0013] In examples, future wear initiation times are projected by combination of user input and artificial intelligence models training on past wear initiations.

[0014] In examples, the artificial intelligence model is fit to the location-specific wear coefficients and a covariance of the location-specific wear coefficients, using regressor variables for each location.

[0015] In examples, the regressor variables include geometrical coordinates in the specified steam generator of each location.

[0016] In examples, the regressor variables further include deposit mapping data, steam generator thermal hydraulic data, mechanical properties of tubes, mechanical properties of tube support plates.

[0017] In examples, the regressor variables are analytically computed parameters.

[0018] In examples, the analytically computed parameters include fluid cross flow velocities, temperature, water quality, and / or vibration amplitudes.

[0019] In examples, the wear parameters include wear rate as a function 1, wherein the computed performance criteria is a burst pressure at the location.

[0020] A method of identifying tubes of a specified steam generator of a pressurized water reactor for plugging is also provided. The method includes inputting historical measurements for a plurality of locations of a plurality of tubes of at least one steam generator, the at least one steam generator including the specified steam generator and / or at least one further steam generator; performing location-specific statistical sampling of the historical measurements to generate a distinct set of wear parameters for each of a plurality of locations of the tubes of the specified steam generator, each distinct set of wear parameters describing wear over time for one of the locations; analyzing each location of each tube based on the distinct set of wear parameters and mechanical properties for each location of each tube by computing a performance criteria probability for each location of each tube; calculating a computed performance criteria probability for the specified steam generator as a whole from the computed performance criteria for each location of each tube at a future time, and comparing to a predetermined threshold; upon a determination that the computed performance criteria probability for the specified steam generator reaches or passes the predetermined threshold, identifying tubes to be plugged based on a temporal and spatial distribution of the performance criteria probabilities for the locations to achieve an acceptable value for the performance criteria probability for the specified steam generator at the future time.

[0021] A computer program product, disposed on a non-transitory computer readable media, for identifying tubes of a specified steam generator of a pressurized water reactor for plugging, the computer program product including computer executable process steps operable to control a computer to implement the steps of the method of identifying tubes of a specified steam generator of a pressurized water reactor for plugging.BRIEF DESCRIPTION OF THE DRAWINGS

[0022] The present invention is shown with respect to the drawings in which:

[0023] Fig. 1 shows a lower part of a recirculating steam generator of a pressurized water nuclear reactor that contains exemplary tubes discussed in the present disclosure;

[0024] Fig. 2a shows a method of maintaining tubes of a specified steam generator of a pressurized water reactor;

[0025] Fig. 2b shows a flow chart illustrating further details of the steps of the method illustrated in Fig. 2a; and

[0026] Fig. 3 shows an example of a distinct set of wear parameters for one location of a steam generator tube.DETAILED DESCRIPTION

[0010] The present disclosure accounts for specific wear behaviors of individual wear locations by applying significant computing resources to perform the wear calculation for each potential wear location in the steam generator, using location-specific statistical sampling, machine learning, and user input to perform a probabilistic assessment of tube wear and tube plugging over the lifetime of the steam generator.

[0011] Fig. 1 shows a lower part of a recirculating steam generator 10 of a pressurized water nuclear reactor. The steam generator 10, as is conventional, comprises a pressure shell 12 of substantially cylindrical shape inside which a bundle wrapper 14 is arranged co-axially containing a tube bundle 16 of the steam generator 10. While a specific recirculating steam generat or is shownin Fig. 1, the present disclosure can be applied to any nuclear power steam generator, including once-through steam generators or steam generators helical tubes.

[0012] The tube bundle 16 is formed of a very high number of tubes 18 bent in a U-shape, each comprising two straight branches whose ends are engaged in and secured to a tube sheet 20 fixed to the lower part of the pressure shell 12. Inside the tube bundle wrapper 14, at successive positions over the height of the bundle, tube support plates 22 are provided to hold in position the branches of the tubes 18 of the bundle 16 to prevent tubes 18 from vibrating when the steam generator 10 is in operation. Each of the tube support plates 22 is pierced with a network of openings similar to the network of openings passing through the tube sheet 20 in which the ends of the tubes 18 of the bundle 26 are secured. The straight branches of the tubes 18 of the bundle are engaged in the aligned openings of tube support plates 22 spaced apart in the longitudinal direction of the tube 18.

[0027] Fig. 2a shows a method 200 of maintaining tubes of a specified steam generator of a pressurized water reactor. The method 200 includes a step 202 of inputting historical measurements for a plurality of locations of a plurality of tubes of at least one steam generator. The wear measurements can be taken via eddy current probe examination in a known manner. The at least one steam generator can include the specified steam generator and / or at least one further steam generator. In an advantageous example, the at least one steam generator can include the specified steam generator and a plurality of further steam generators. The further steam generators can be the same model as the specified steam generator or can be different models than the specified steam generator. If the further steam generators are of different models than the specified steam generator, then the historical measurements are preprocessed, for example using a machine learning and / or physics-based simulations, to extrapolate general wear trends that are applicable to different design variants, before being used in step 204.

[0028] For a physics-based simulation, this preprocessing can take into consideration vibrationamplitude and secondary fluid characteristics, which can include temperature, densities, axial and cross-flow velocities, and steam quality, as regression variables used to correlate wear at a specific location in one steam generator design to locations in one or more further steam generator designs.

[0029] For machine learning, an artificial intelligence model can be trained to predict wear as a function of these physics-based simulation parameters and historical measurements across a plurality of steam generators. In other words, the physics-based simulation parameters and historical measurements are used as inputs in training, and wear parameters are outputs.

[0030] The method 200 next includes a step 204 of performing location-specific statistical sampling of the historical measurements to generate a distinct set of wear parameters for each of a plurality of locations of the tubes of the specified steam generator. Each distinct set of wear parameters describes wear over time for one of the locations of the tubes of the specified steam generator. The wear results from relative motion between the tubes and each other (tube-to-tube) or between the tubes and surrounding structure (tube-to-structure), including tube support plates. The wear can be from flow-induced vibration of the tubes, which can cause wear in the form of fretting that occurs at the contact area between materials under load and subject to relative motion.

[0031] An example of a distinct set of wear parameters for one location is shown in Fig. 3, which illustrates a plot of wear as a percentage of tube through-wall(%TW) thickness with respect to time, as measured in effective full power years (EFPY) for a plurality of random simulations for a single location. The percentage of tube wear indicates a percentage of the tube that has worn away at the location shown in Fig. 3 over time. The plot of Fig. 3 indicates that the wear rate is higher between years five and ten, then levels off. In the example of Fig. 3, there have been six historical wear measurements - illustrated by dots - taken at the location of Fig. 3, which is at a tube-support interface of the steam generator. Fig. 3 also shows a plurality of lines representing a mathematical function with location-specific coefficients derived from the historical wear measurements. In the example of Fig. 3, the mathematical function is a logarithmic function. In other examples, the mathematical function can be a power-law function or a hyperbolic function. The thickest line in Fig. 3 is a location-specific extrapolated wear curve that illustrates a best fit line through the historical measurements.

[0032] A location-specific statistical sampling of a population of extrapolated future wear curves can be performed to generate a distinct set of wear parameters, and the thinner lines represent the distinct set of wear parameters for the location. In the example of Fig. 3, there are 100 thinnerlines, with each line representing one wear parameter (%TW) over time. In the example of Fig. 3, each thinner line represents a different trial in a Monte Carlo simulation for the location. As shown in Fig. 3, two of the hundred thinner lines (2%) exceed a 40% wear rate by forty EFPY, which is significant because if a measured tube location has a wear rate greater than 40% during a scheduled maintenance inspection under existing industry standards, the tube is plugged.

[0033] The plot of the wear parameters in the example of Fig. 3 is the logarithmic function: f(x) = A * log (t - tO), where: t = time in EPFY,A, and tO are the fitting coefficients, A describes the wear rate, tO describes the time, in EPFY, at which wear initiates.

[0034] In the example of Fig. 3, as shown at the top of the graph, the location is defined by cylindrical coordinates including the elevation position, and the radial position and the angle with respect to a center axis of the steam generator. Also shown in Fig. 3 is the root mean square of the historical wear measurements in Fig. 3, which at 1.07 can be considered a statistically acceptable goodness of fit. Historical measurements having a root mean square above a predetermined value can be excluded from generating a population of extrapolated future wear curves, and instead the population of extrapolated future wear curves can be generated by the artificial intelligence model in substep 204e, discussed below, when the root mean square is above a predetermined value. In one example, the predetermined value can be 3.0. In the example of Fig. 3, the A coefficient has a value of 9.7 and a standard deviation of 0.41, while the tO coefficient has a value of 5.8 and a standard deviation of 0.47.

[0035] The method 200 next includes a step 206 of analyzing each location of each tube based on the distinct set of wear parameters and mechanical properties for each location of each tube by computing performance criteria for each location of each tube. In one example, the performance criteria is a burst pressure for the tube at the location based on the wear parameters and mechanical properties. The mechanical properties can include properties that characterize the strength, such as yield strength or tensile strength. For example, if the percentage of tubewear is predicted to be 45% at a future time, the burst pressure - i.e., the pressure at which the tube is predicted to burst at the location - is computed based on this percentage of tube wear and its corresponding mechanical properties. The burst pressure can be for example calculated using the wear parameters and the mechanical properties yield strength and tensile strength using the standards set forth in the EPRI Steam Generator Management Program: Steam Generator Degradation Specific Management Flaw Handbook, Revision 2.

[0036] For each location, the distinct set of wear parameters from the population of extrapolated future wear curves is considered as a whole to compute a performance criteria probability for the location. For example, if 2 out of 100 statistical samples are predicted to have a burst pressure that exceeds a predetermined threshold at a future time, the performance criteria probability is the probability of survival of the tube (i.e., a probability that the tube will not burst, which equals 1 minus the probability of burst) and it is 0.98 at the future time.

[0037] The method 200 next includes a step 208 of calculating a computed performance criteria probability for the steam generator as a whole from the computed performance criteria for each location of each tube at a future time and comparing the computed performance criteria for the steam generator as a whole to a predetermined threshold. For example, if the performance criteria probability for each location is the probability of survival, the probably of survival for the steam generator as a whole is computed by multiplying the probability of survival for each location times each other. As discussed below with respect to Fig. 2b., if the probability of survival for the steam generator as a whole is below a predetermined value, at least one the tubes needs to be plugged to alter the probability of survival to above the predetermined value.

[0038] The method 200 next includes a step 210 of, upon a determination that the computed performance criteria probability for the specified steam generator reaches or passes the predetermined threshold, identifying tubes to be plugged based on a temporal and spatial distribution of the computed performance criteria probabilities for the locations to achieve an acceptable value for the performance criteria probability for the steam generator as a whole at the future time. The temporal aspect can be based upon the time of the computed performance criteria probabilities of the tubes in comparison to scheduled maintenance operations of the steam generator. For example, if a scheduled maintenance operation is approaching, step 210can include determining whether the computed performance criteria probabilities of the tubes exceed the performance criteria probability in the time period between the scheduled maintenance operation and a next expected scheduled maintenance operation. In another example, the determination can be made during shutdown, with the results either leading to plugging during the current shutdown or being pushed back to a future shutdown.

[0039] The spatial aspect can be based on the location of the tubes and the corresponding functional performance in the operation of the steam generator. In particular, the tubes having the highest computed performance criteria probabilities are considered, along with the location of each of the tubes having the highest computed performance criteria probabilities. For example, if ten tubes having the highest computed performance criteria probabilities are considered, and all of these locations have approximately (i .e., + / - 5%) the same computed performance criteria probability, and plugging five of these tubes will bring the probability of survival for the steam generator as a whole below the predetermined value, the five tubes with the least functional responsibility in the steam generator are selected for plugging.

[0040] The method 200 lastly includes a step 212 of plugging the tube or tubes including the locations identified in step 210 to achieve the acceptable value for the performance criteria probability for the steam generator at the future time.

[0041] Fig. 2b shows a flow chart illustrating further details of the steps of method 200, in particular step 204.

[0042] As shown in Fig. 2b, step 204 of method 200 includes a first substep 204a of choosing a parametric mathematical function to describe wear as function of time at each of the locations. An example of the parametric mathematical function is the logarithmic function discussed above with respect to Fig. 3.

[0043] Step 204 then includes a substep 204b of determining, for each location, if there is a sufficient number of historical wear measurements for the location, optimal values for wear coefficients defining the wear parameters as a function of time and a covariance of the wear coefficients. The wear coefficients and their covariance are computed by regression to the historical wear measurements for each location. For example, the coefficients A, tO discussedabove with respect to Fig. 3 are optimized by performing a least squares regression, which results in the best fit line in Fig. 3. In particular, a covariance matrix can be gathered that includes the uncertainty in A, the uncertainty in tO, correlation between A and tO, and the root mean square error, which as discussed above with respect to Fig. 3.

[0044] The scatter in these wear curves is a function of the goodness-of-fit for the parametric mathematical function The goodness-of-fit is used to select locations to be include as training data for the Al models. For example, if a location has a high value for root mean square error, it may be deprioritized or eliminated from the artificial training data used in substep 204d.

[0045] Step 204 also includes a substep 204c in which regressor variables for every location in the steam generator are gathered. The regressor variables can include the geometrical coordinates in the steam generator of each location. The regressor variables can also include: -other inspection data, including deposit mapping data;-steam generator thermal hydraulic data from the power plant computer, including feedwater flow rates and steam flow rates;- mechanical properties of the tubes, mechanical properties of the tube support plates and / or-analytically computed physics parameters, including fluid cross flow velocities, temperature, water quality, and vibration amplitudes.In other examples, the regressor variables can include any feature that, based on engineering principals and experience, has an effect on or is at least partially correlated to tube wear rate.

[0046] Step 204 also includes a substep 204d in which artificial intelligence models are trained using the regressor variables for each wear location gathered in substep 204c as inputs and coefficients generated in substep 204b, from the locations with sufficient historical wear measurements, as outputs to define a parametric mathematical function for each location. In one example, an ensemble of supervised and unsupervised machine learning models are used. Other examples may use any manner of other techniques, such as artificial neural networks of any form useful for time series prediction, or convolutional models useful to geometric location information. These artificial intelligence models are trained to predict wear parameters, which can include wear rate as a function of time and wear initiation as a function of location, for eachindividual location as a function of the regressor variables. Predicting predict wear parameters in such a manner allows a location-specific wear rate to be determined, which is not possible in present state-of-the-art and decreases conservatism in long-tailed population distribution of wear rates. In other words, a location-specific wear rate can prevent low wear rates locations from being penalized by other high wear rate locations in steam generator, as is the case with the present state-of-the-art.

[0047] Step 204 also includes a substep 204e where for each location, if there is an insufficient number of historical wear measurements for the location or there is no detected wear, an initiation time of the wear is predicted and then projections of wear rate are made using the artificial intelligence model from substep 204d by inputting regressor variables for each specific location into the artificial intelligence model. For both locations with insufficient number of historical wear measurements and locations with no wear, the artificial intelligence model used to project wear rate is fit to the wear coefficients and their covariance, using regressor variables for each location.

[0048] If there is an insufficient number of historical wear measurements for the location, the initiation time of the wear is predicted from the insufficient number of historical wear measurements. As noted above, the projections of wear rate are then made using the artificial intelligence model from substep 204d. This advantageously allows the limited historical information about that wear location to be useful, while filling in the unknowns with artificial intelligence model to generate wear parameters, such as those shown for example in Fig. 3, for the wear location. In other words, for some locations with historical wear measurements, but of an insufficient quantity or value (e.g., using the root mean square technique discussed with respect to Figs. 2a, 3) to determine coefficients and their covariance, the initiation time is determined from the wear measurements, and then the artificial intelligence model from substep 204d is used to determine the wear rate.

[0049] If there is no wear at a location, the location is assigned a time in the future at which wear is predicted to initiate. In one example, future wear initiation times are projected by a combination of user input and artificial intelligence models training on past wear initiations to create a heat map of the most likely locations for wear. The projection of future wear initiationtimes can include a first substep of creating a mathematical function that describes the relative probability of initiation for each location. This function may be created by fitting an artificial intelligence model to historical measurements of wear initiation. Next, in a second substep, a function is created that describes the probability of initiation as a function of time. The function that describes the probability of initiation as a function of time may be created by fitting an artificial intelligence model to historical measurements of wear initiation. This function can be modified by a Probability of Detection (POD) model. Then, in a third substep, these two functions - the function that describes the relative probability of initiation for each location and the function describes the probability of initiation as a function of time - are combined to sample an initiation time for each location in the steam generator.

[0050] In comparison to the present state-of-the-art, future wear initiation time projections based in part on user input can advantageously allow for abnormal perturbations to be specified in wear initiations as a function of time. In particular, future wear initiation time projections based in part on user input can advantageously allow for explicit analysis of assumed changes in wear initiations due to potential implementation of a power uprate. For example, a physics simulation of a power uprate may show that the boiling region will shift within the steam generator. This shift can be explicitly applied to the function that describes the relative probability of initiation for each location. In a second example, operating experience may show that the rate of new flaw initiations initially increases after implementation of a power uprate, and then a rate of new flaw initiations decreases in the future. The function that describes the probability of initiation as a function of time can be modified to account for this change in expected behavior.

[0051] Further, by decoupling of wear rate versus time and wear location allows the implementation of a POD model where assumptions can be made about the number of wear locations that were missed by the non-destructive examination. This is accomplished by adjusting the function that describes the probability of initiation as a function of time.

[0052] Location-specific wear initiation time projections are not possible in present state-of-the art, and such projections increase the accuracy of wear rate predictions for future wear locations. Future wear locations are likely to initiate in locations of with high wear rate projections, which is an important behavior to capture for safety purposes. Further, for steam generators withoutprior wear history, the artificial intelligence models from other comparable units can be used to make realistic projections and plugging decisions, based on knowledge derived from operating experience at other units. For example a comparable unit can be a steam generator with the same tube materials, same tube support structure, and / or manufactured with the same tolerances.

[0053] Next, in a substep 204f, the applicable substep 204b or 204e is used to select wear parameters for each location in the steam generator.

[0054] As a last substep 204g of step 204, the wear parameters, in the form of a wear rate curve such as that shown in Fig. 3 describing wear as a function of time, are sampled for each tube-to- support location (e.g., -240,000 locations per steam generator) for an appropriate number of Monte Carlo trials (e g., 100).

[0055] Specifically, the parametric mathematical functions, along with the corresponding optimal coefficients and their covariance, generated in substeps 204b and 204e and selected in substep 204f, are used to statistically sample, for example via a Monte Carlo simulation, a population of future wear curves describing wear as a function of time for each location.

[0056] An example of the results of step is shown by the thinner wear curves in Fig. 3. Referring back to the example of Fig. 3, the optimum coefficients (A, tO) and the covariance matrix (standard deviation of A, standard deviation of tO, correlation between A and tO) are used to generate a statistical sample, in the form of a population of wear curves, for each location.

[0057] Next, as discussed above, step 206 is performed, which includes analyzing each location of each tube based on the distinct set of wear parameters and other mechanical properties for each location of each tube by computing performance criteria for each location of each tube. Mechanical properties, including yield strength and tensile strength, can be sampled, and then burst pressure can be calculated as a function of time, taking into account the size of the wear and the mechanical properties.

[0058] Referring to the example of Fig. 3, the each wear curve from a location-specific population is analyzed in combination with the corresponding mechanical properties for the location to determine the burst pressure for each location. Then, the probability of survival is calculated for each location.

[0059] After step 206, as noted above, step 208 can include calculating a computed performance criteria probability for the steam generator as a whole from the computed performance criteria for each location of each tube at a future time, and comparing the computed performance criteria for the steam generator as a whole to a predetermined threshold.

[0060] For example, the probably of survival for the steam generator as a whole can be computed by multiplying the probability of survival for each location times each other. Referring to the simplistic example in Table 1 below, if the probability of survival is required to be 0.95 or above, a plugging operation must be performed because multiplying the probability of survival for location leads to a value below the predetermined threshold of 0.95.Table 1

[0061] The method 200 next includes a step 210 of identifying tubes to be plugged based on a temporal and spatial distribution of the computed performance criteria probabilities for the locations to achieve an acceptable performance criteria probability for the steam generator as a whole.

[0062] Referring back to Table 1, Locations 2, 5, and 6 are functional locations. If these locations are plugged, the probability of survival is brought from 0.913 to 0.951. Thus, theperformance criteria probability of the steam generator as a whole can be brought to the predetermined threshold without plugging functional locations that have more of an impact on the operation of the steam generator than non-functional locations.

[0063] Method 200 advantageously allows tubes to be plugged in a less conservative manner than state of the art techniques. 40% TW is currently the required plugging threshold, but plants are at times having to plug at less than this amount for preventative reasons to get desired inspection intervals. For example, in the current state of the art, all flaws with a measured depth of 30% sometimes need to be preventatively plugged in order to defer the subsequent examination to a desired long interval (e.g., 54 months between exams), based on the wear rates projected by sampling from the measured wear rates of tubes in that population within the previous operating cycles or from available operating experience of similar model steam generators. However, when using method 200, where a distinct set of wear parameters is generated for each of a plurality of locations of the tubes of the specified steam generator to calculate a location-specific wear rate, the same 54 month re-exam interval can be justified with plugging only tubes with flaws have a measured depth of 40%. This can prolong the life of the steam generator compared to conventional techniques because less tubes may be plugged.

[0064] In the preceding specification, the present disclosure has been described with reference to specific exemplary embodiments and examples thereof. It will, however, be evident that various modifications and changes may be made thereto without departing from the broader spirit and scope of present disclosure as set forth in the claims that follow, including combination of examples where technically possible. The specification and drawings are accordingly to be regarded in an illustrative manner rather than a restrictive sense.

Claims

WHAT IS CLAIMED IS:

1. A method of maintaining tubes of a specified steam generator of a pressurized water reactor, the method comprising: inputting historical measurements for a plurality of locations of a plurality of tubes of at least one steam generator, the at least one steam generator including the specified steam generator and / or at least one further steam generator; performing location-specific statistical sampling of the historical measurements to generate a distinct set of wear parameters for each of a plurality of locations of the tubes of the specified steam generator, each distinct set of wear parameters describing wear over time for one of the locations; analyzing each location of each tube based on the distinct set of wear parameters and mechanical properties for each location of each tube by computing a performance criteria probability for each location of each tube; calculating a computed performance criteria probability for the specified steam generator as a whole from the computed performance criteria for each location of each tube at a future time, and comparing to a predetermined threshold; upon a determination that the computed performance criteria probability for the specified steam generator reaches or passes the predetermined threshold, identifying tubes to be plugged based on a temporal and spatial distribution of the performance criteria probabilities for the locations to achieve an acceptable value for the performance criteria probability for the specified steam generator at the future time; and plugging the identified tubes to achieve the acceptable value for the performance criteria probability for the specified steam generator at the future time.

2. The method as recited in claim 1, wherein the performing of location-specific statistical sampling of the historical measurements to generate a distinct set of wear parameters for each of a plurality of locations of the tubes of the specified steam generator includes: analyzing an entire operating history of measured wear for each location in the specified steam generator.

3. The method as recited in claim 1, wherein the performing of location-specific statistical sampling of the historical measurements to generate a distinct set of wear parameters for each of a plurality of locations of the tubes of the specified steam generator includes: selecting a parametric mathematical function to describe wear as function of time at each of the locations; generating, for each location, optimal values for location-specific wear coefficients defining the wear parameters as a function of time and a covariance matrix defining a covariance of the location-specific wear coefficients; using the parametric mathematical function, the optimal values for the location-specific wear coefficients and the covariance matrix for each location to statistically sample wear parameters at each location to generate the distinct set of wear parameters for each of the plurality of locations.

4. The method as recited in claim 3, wherein the generating, for each location, optimal values for wear coefficients defining the wear parameters as a function of time and a covariance matrix defining a covariance of the wear coefficients includes: for each location, if there is a sufficient number of historical wear measurements for the location, using location specific extrapolated wear curves defining the wear parameters as function of time, where the location specific extrapolated wear curves includes location-specific wear coefficients and covariance computed by regression to the historical wear measurements for the location, to statistically sample a population of extrapolated future wear curves to generate the distinct set of wear parameters for each of the locations having the sufficient number of historical wear measurements; and for each location, if there is an insufficient number of historical wear measurements for the location or there is no wear, an initiation time of the wear is predicted and then projections of wear rate are made using an artificial intelligence model to generate the distinct set of wear parameters for each of the locations having the insufficient number of historical wear measurement.

5. The method as recited in claim 3, wherein the parametric mathematical function is a logarithmic function, a power-law function or a hyperbolic function.

6. The method as recited in claim 4, wherein if there is an insufficient number of historical wear measurements for the location, the initiation time of the wear is predicted from the insufficient number of historical wear measurements.

7. The method as recited in claim 4, wherein if there is no wear at a location, the location is assigned a time in the future at which wear is predicted to initiate.

8. The method as recited in claim 7, wherein future wear initiation times are projected by combination of user input and artificial intelligence models training on past wear initiations.

9. The method as recited in claim 4, wherein the artificial intelligence model is fit to the locationspecific wear coefficients and a covariance of the location-specific wear coefficients, using regressor variables for each location.

10. The method as recited in claim 9, wherein the regressor variables include geometrical coordinates in the specified steam generator of each location.

11. The method as recited in claim 10, wherein the regressor variables further include deposit mapping data, steam generator thermal hydraulic data, mechanical properties of tubes, mechanical properties of tube support plates.

12. The method as recited in claim 10, wherein the regressor variables are analytically computed parameters.

13. The method as recited in claim 12, wherein the analytically computed parameters include fluid cross flow velocities, temperature, water quality, and / or vibration amplitudes.

14. The method as recited in claim 1, wherein the wear parameters include wear rate as a function of time and wear initiation as a function of the location.

15. The method as recited in claim 1 , wherein the computed performance criteria is a burst pressure at the location.

16. A method of identifying tubes of a specified steam generator of a pressurized water reactor for plugging, the method comprising: inputting historical measurements for a plurality of locations of a plurality of tubes of at least one steam generator, the at least one steam generator including the specified steam generator and / or at least one further steam generator; performing location-specific statistical sampling of the historical measurements to generate a distinct set of wear parameters for each of a plurality of locations of the tubes of the specified steam generator, each distinct set of wear parameters describing wear over time for one of the locations; analyzing each location of each tube based on the distinct set of wear parameters and mechanical properties for each location of each tube by computing a performance criteria probability for each location of each tube; calculating a computed performance criteria probability for the specified steam generator as a whole from the computed performance criteria for each location of each tube at a future time, and comparing to a predetermined threshold; upon a determination that the computed performance criteria probability for the specified steam generator reaches or passes the predetermined threshold, identifying tubes to be plugged based on a temporal and spatial distribution of the performance criteria probabilities for the locations to achieve an acceptable value for the performance criteria probability for the specified steam generator at the future time.

17. A computer program product, disposed on a non-transitory computer readable media, for identifying tubes of a specified steam generator of a pressurized water reactor for plugging, the computer program product including computer executable process steps operable to control a computer to implement the method of claim 16.

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