- Email: [email protected]
Importance of uncertainty quantification in nuclear fuel behaviour modelling and simulation
Nuclear Engineering and Design 355 (2019) 110311
Contents lists available at ScienceDirect
Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Importance of uncertainty quantification in nuclear fuel behaviour modelling and simulation
T
A. Bouloré CEA, DEN, DEC, Saint Paul lez Durance 13108, France
A R T I C LE I N FO
A B S T R A C T
Keywords: Uncertainty Nuclear fuel Modelling Safety analysis
Fuel performance codes describe the physical phenomena occurring in fuel during its irradiation in steady-state and transient conditions. In this paper, the principles of the modelling of nuclear fuel behaviour are presented. Many reasons show that it is also important to consider uncertainties in the fuel modelling process. It can be experimental reasons (validation database and material properties determination), uncertainties in the modelling process itself (lack of knowledge for example). This paper also presents several applications of uncertainty quantification and sensitivity analysis applied to fuel behaviour modelling. The first example is about the validation process of the fuel performance code, which results in a quantification of the uncertainty of the calculated quantities. The second application is about the use of the code in safety analysis to determine probabilities of failure considering epistemic and random uncertainties. In the last application, we show how important it is to quantify modelling uncertainties in the calibration process of complex physical models.
1. Introduction The aim of a fuel performance code is to describe the physical phenomena occurring in the fuel during its irradiation in reactor and to calculate the evolution of some physical quantities (like geometry, inner pressure, stresses, and temperature) which can be used directly in safety assessment of the fuel. In most cases, a fuel performance code is made of several coupled models, each of them describing one particular physical phenomenon (thermal transfer, fission gas behaviour, mechanical behaviour, cladding corrosion, etc.). Fuel performance codes are deterministic codes, but a lot of data (material properties, experimental results, etc.) used in these codes are uncertain. In this paper, we try to show how important it is to take into account these uncertainties in the use of a fuel performance code and what can be the consequences. 2. Nuclear fuel behaviour modelling In PWR, nuclear fuel is made of UO2 or (U,Pu)O2 pellets stacked in a cladding tube made of zirconium alloy (Bailly et al., 1999). Fission nuclear reactions in the pellet generate heat which is evacuated to the coolant outside the cladding. But because of the low thermal conductivity of UO2, the effective temperature at the center of the pellet is significantly higher than at the surface. This temperature gradient causes differential thermal strain between the center and the edge of
the pellet which results in pellet cracking. During the first power increase (Fig. 1), the gap width decreases because of differential thermal expansion between the fuel pellet and the cladding, and because of relocation of fuel pellet towards the cladding. In the early stages of irradiation in nuclear power plant (Fig. 1a and b), phenomena such as densification (pore removal), cladding creep and fuel swelling occur, which result in the gap closure at a burnup about 20 GWd/tM. Above a certain temperature which depends on burnup, gaseous fission products contribute to fission gas bubbles precipitation growth (gaseous swelling). Fission gas is also released in the plenum of the fuel rod by a coalescence/percolation mechanism (Fig. 1c). This release increases the inner rod pressure. Generally, this phenomenon occurs at the center of the pellet in PWR conditions. Simultaneously, near the edge of the pellet, epithermal capture of neutron by U238 atoms produce a higher concentration of plutonium than in the rest of the pellet. This leads to a local higher burnup (about twice than the rest of the pellet). As this burnup accumulation occurs at low temperature (below the default recovery temperature), irradiation defaults concentration rises and a local restructuring occurs at the rim of the pellet (so called “rim effect”). The new microstructure called High Burnup Structure (HBS) is characterized by a very small grain size, and a very high porosity which contains a large quantity of fission gas that is only partly released during steady state irradiation (Fig. 1d). This zone extends to about 200 µm from the edge of the pellet (Fig. 2). In
E-mail address: [email protected]. https://doi.org/10.1016/j.nucengdes.2019.110311 Received 18 December 2018; Received in revised form 19 June 2019; Accepted 21 August 2019 0029-5493/ © 2019 Elsevier B.V. All rights reserved.
Nuclear Engineering and Design 355 (2019) 110311
A. Bouloré
Fig. 1. Different stages of fuel behaviour in steady state conditions.
Fig. 3. Fuel cracking and creep during power ramp.
element structural integrity (cladding failure risk and loss of integrity of the fissile column in particular). So, the quantities of interest which have to be calculated by fuel performance codes are then: Fig. 2. High burnup structure.
- Temperature - Fuel and cladding mechanical behaviour (strains and stresses) - Fission products behaviour
case of accident conditions like LOss of Coolant Accident for example (LOCA), the effective temperature in this zone becomes much higher than in normal operation, and most of the fission products are still retained in the material in intergranular position. At high temperature intergranular fission products are easily released, and increase a lot the inner pressure of the fuel rod. In off normal conditions (power ramps), high temperature in the central part of the pellet induces fuel thermal strain, gaseous fuel swelling, fuel creep which loads the cladding leading to its plastic strain. Gas release in the plenum allows a little decreasing of the stresses. A more important cracking is observed in the outer parts of the fuel pellet (Fig. 3). In LOCA (LOss of Coolant Accident) conditions, the temperature of the cladding increases, which results in clad ballooning because of the cladding creep and of the internal pressure in the rod. As the fuel temperature increases as well, and because of the decreasing of the pressure contact between the cladding and the pellet, fuel fragmentation can occur in high burnup fuel. In RIA (Reactivity Initiated Accident) conditions, the temperature in the fuel increases rapidly, particularly in the outer parts of the pellet, which results in a high swelling rate of the pellet. Important fuel swelling and cladding creep lead to a ballooning of the cladding which may crack. All the phenomena listed may have a significant impact on the fuel
All these quantities have to be calculated simultaneously, because all the physical phenomena listed above are strongly coupled. The thermal model solves the heat transfer equation:
ρ · c p·
⎯⎯⎯⎯⎯⎯→ dT = div (λ· grad T ) + pv dt
(1)
where ρ is the density, cp and λ respectively the specific heat and the thermal conductivity of the material, and pv is the volume power generated by the fission. It models the heat transfer from the pellet to the fuel-to-clad gap, then through the cladding and finally in the coolant. As fuel thermal conductivity is quite low, the temperature in the fuel is high, and most physical phenomena occurring in the fuel are driven by temperature. Regarding safety criteria, temperature has a direct impact on fuel melting assessment, but also on fuel rod internal pressure (mainly via fission gas release mechanism). A good knowledge of the strains and stresses in the fuel element is important as cladding is the first barrier for fission products. Some physical phenomena, such as fission gas bubbles growth depend on the mechanical state of the fuel pellet. Cracking behaviour also depends a lot on the mechanical state. As gaseous swelling and the mechanical problem are strongly coupled, they have to be resolved simultaneously, 2
Nuclear Engineering and Design 355 (2019) 110311
A. Bouloré
Fig. 4. Coupling scheme of ALCYONE fuel performance code (Marelle et al., 2016) in the PLEIADES numerical platform.
- Material properties: in many cases, material properties are measured directly by specific experimental techniques, and very often, the experimental data are scattered. There is an uncertainty due to the technique itself, but also an uncertainty linked to the representativeness of the chosen sample. It is the case for experimental data of fuel thermal conductivity, fresh or irradiated (ex. Fig. 6). - Experimental conditions used for validation: a typical quantity which is uncertain is the power at which the rod is irradiated during an experimental irradiation. And it is a parameter of first order of all the schemes in fuel performance codes. - Experimental results which are used to validate/calibrate the different models of the fuel performance codes. For example: Fission gas release measurement which is an integral quantity used to validate the fission gas behaviour model. Temperature measurements in irradiation devices also have an uncertainty, and this quantity is used to validate the global thermal model of the code. - Model parameters. Some model parameters can be measured directly (like material properties mentioned above), but very often, specific experiments are conducted which allow a determination of the parameters using inverse method. It means that a specific model is built to extract basic data from the experiment. A good example of such parameter is diffusion coefficients. In this case, the uncertainty is linked to both experimental data/conditions uncertainties, and to the degree of simplification of the model.
which means in a global convergence scheme. An example of fuel performance code that illustrates all these couplings is ALCYONE (Fig. 4), the multidimensional fuel performance code developed and validated by CEA (Commissariat à l’Énergie Atomique) (Marelle et al., 2016). In the 1D scheme (Fig. 5), the fissile stack is represented by slices which are meshed radially. It is designed to model a whole fuel rod in irradiation conditions. 2D and 3D schemes are developed to have a more detailed evaluation of the mechanical behaviour, specifically for the situations of pellet-cladding interaction (PCI). The validation (Struzik and Marelle, 2012) of this code is done as well at the macroscopic scale (integral results, inner pressure in the rod, dimension changes of the whole rod, etc.) as well as at a microscopic scale (comparison of fission gas retention to microprobe and SIMS (Secondary Ion Mass Spectrometry) measurements after irradiation for example).
• •
3. Why uncertainty analysis in fuel modelling? In the frame of nuclear fuel behaviour modelling and simulation, due to the difficulties associated to measures under irradiation, it is very often not possible to obtain a reasonable value of material properties by direct measurements. Then the determination of the model physical parameters is very often not direct which induces numerous sources of uncertainty coming from different topics. 3
Nuclear Engineering and Design 355 (2019) 110311
A. Bouloré
Fig. 5. Different representations of the fuel element (1D, 2D and 3D) in ALCYONE.
Sensitivity analysis helps to determine which uncertain parameter or data is the most sensitive on the outputs of the fuel performance code. It helps then to determine which uncertain inputs have to be more precisely known (reduced uncertainty). The second topic is safety analysis. As the fuel performance code is a best-estimate code, uncertainty quantification/sensitivity analysis and probabilistic approaches allow taking into account the uncertainties on inputs and model parameters to make sure that even with some uncertainties, the safety criterion is still verified. It is shown later in the paper that the criterion verification can be replaced by a probability assessment in those cases.
- Modelling uncertainty is more difficult to represent and quantify. In fact a model is always a simplified mathematical representation of a complex physical phenomenon. Assessing the modelling uncertainty consists in a quantification of the error made when simplifying the representation. It is a very difficult task. Two main objectives can be listed to justify the need of uncertainty analysis. First, it is a way to improve the degree of confidence of the fuel performance code by demonstrating that in the global simulation/ modelling process, uncertainties on the calculated quantities are being quantified. In this objective, propagation on input uncertainties in the code is important in the global validation process of the code (showing that the experimental results used for validation are in the range of uncertainty of the calculated results (Struzik and Marelle, 2012).
4. General methodology The methodology used for about 10 years is presented in detail in
6
Thermal conductivity (W/m/K)
5,5 5
batch n°1
batch n°2
batch n°3
batch n°4
4,5 4 3,5 3 2,5 2 1,5 1 500
1000
1500
2000
2500
Temperature (K) Fig. 6. Scattering of experimental data for fuel thermal conductivity. 4
3000
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Fig. 7. General methodology recommended for UQ/SA. 450
Morris sensitivity analysis method (Morris, 1991; Campolongo and Saltelli, 1997) can be used to reduce the number of uncertain parameter. It is a one-at-a-time method which does not require a large set of calculations. The Morris sensitivity indices may depend on the experimental conditions (for example gap open or closed) and the method has to be applied for the experimental conditions of the case. After the uncertainty modelling phase, uncertainty propagation can be performed by Monte Carlo sampling and direct calls to the fuel performance code. The size of the sample depends on the computation resources available. The case presented is the GRIMOX2 experiment. GRIMOX2 was an experiment conducted in the SILOE nuclear test reactor in Grenoble during the nineties (Caillot and Delette, 1998). It had been performed in order to study the thermal behaviour of MOX fuel compared to the UO2 fuel in the beginning of irradiation. For that purpose a short fuel rod of PWR diametrical geometry including MOX and UO2 pellets was irradiated in a boiling capsule pressurized to 13 MPa placed at the edge of the SILOE reactor core. At both ends of the experimental rod, the pellets were drilled in order to introduce a thermocouple. So this device allowed the measurement of the centerline temperature of both fresh fuel, UO2 and MOX, during an irradiation up to 0.5 at%. The time duration of the experiment was about 4000 h (Fig. 8). The uncertainty on temperature measurement announced by the operator is ± 10 °C. The experimental fissile power was determined by the mean of neutron flux detector during the experiment and it was cross checked by quantitative gamma spectrometry after irradiation. The order of magnitude of power level is comparable to the PWR one in standard and incidental conditions. Although two types of fuel had been tested in this experiment this paper only deals with the UO2 stack. In the case presented, 10,000 simulations of the experiment have been performed and the set of results can be used to determine the uncertainty on the calculated temperature. All experimental data are in the 95% confidence interval of the calculated distributions (Fig. 9), so it validates the modelling of the thermal behaviour of the fuel element. A focus will be done in the presentation on the propagation of this uncertainty on temperature in the other models of the code. The problem is more complex than what it looks like, as all the models are coupled in the fuel performance code, and all models need a calibration phase to make the final code a best-estimate code.
Average Linear heat Rate (W/cm)
400 350 300 250 200 150 100 50 0 0
500
1000
1500
2000
2500
3000
3500
4000
Time (h)
Fig. 8. Irradiation history of the GRIMOX2 experiment.
(De Rocquigny et al., 2008). The general scheme of this recommendation is printed on Fig. 7. This method is decomposed in 4 steps: problem specification, modelling of the uncertainties, uncertainty propagation phase and eventually sensitivity analysis if required. The first two steps are very often the most difficult; the last two require large computing capacities.
5. Examples of applications 5.1. Code validation: example on fuel thermal behaviour To quantify the uncertainty on the temperature calculated by the fuel performance code ALCYONE, uncertainty on the following inputs are taken into account: - Fuel thermal conductivity - Radial distribution of power which also leads to the HBS restructuring - Power (or Linear Heat Rate) generated by the fuel rod - Thermal transfer through the pellet-to-clad gap Different methods can be used to model these inputs as random variables (expert opinion, inverse methods, iterative methods…). But some of these uncertain input depend on a lot of parameters, and the 5
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Fig. 9. Validation taking into account uncertainties.
5.2. Safety analysis; application of reliability methods to fuel melting The second example is about the safety analysis topic. The objective is to determine at which power, a melting of the fuel pellet could occur:
Pfusion = max{P,
max(Tfuel ) < Tmelting }
(2)
As there are uncertainties on the inputs and parameters used to calculate the fuel temperature, and also on the knowledge of the melting temperature of the irradiated fuel, the problem of threshold determination is turned into the determination of the probability of melting for a given target power:
P (Tfuel > Tmelting )
(3)
The same general methodology is applied to quantify this probability and two types of methods (maybe more) can be used. Monte Carlo direct sampling requires a large set of simulations but allows an evaluation of the uncertainty on the calculated probability (confidence interval). Reliability methods like FORM/SORM (First Order and Second Order Reliability Method) (Du, 2005) can also be used and require much less runs of the code (Fig. 10). Using FORM/SORM we can evaluate the probability of fuel melting, but also the contribution of each uncertain input to the probability of melting (kind of sensitivity analysis). But it assumes that the code behaviour is “regular” around the domain of failure, because it needs to evaluate second derivatives which are very sensitive to the precision of calculation and convergence of the global scheme (Struzik et al., 2017).
Fig. 10. FORM/SORM methods.
fission gas behaviour model (Bouloré et al., 2015). Initially, the model (Jomard et al., 2014) contains about 40 numerical or physical scalar parameters to be adjusted. To reduce the computation time during the first iterative calibration phase, preliminary to the calibration process, a fast sensitivity analysis has been performed using the Morris method (Campolongo and Saltelli, 1997). From this analysis, only 7 scalar parameters are still to be calibrated because they have a direct effect on the result of the model (FGR: fission gas release): diffusion coefficient of xenon enhanced by irradiation (2 parameters), efficiency of the bubble nucleation, resolution of gas atoms from nanometric bubbles, resolution of gas atoms from precipitated bubbles, rate of accumulation of dislocations, and a parameter involved in the fraction of bubbles connected to porosity. The first automatic method used for calibration is an iterative method based on the SIMPLEX algorithm which is used to minimize a scalar criterion. On each experiment of the calibration database, the objective is to minimize the quadratic error between calculation and experimental value for FGR (fission gas release). So the scalar criterion is defined as:
5.3. Model/code calibration In the fuel behaviour simulation process, a lot of models contain fitting parameters. In order to make the code a “best-estimate” code, all these parameters have to be adjusted on a set of experiments called calibration database (including the associated experimental uncertainties) covering as far as possible the domain on which the model is going to be used in the future. This is also part of the global validation process. As most of the time, the experimental quantities used for calibration have an uncertainty, the statistic and probabilistic approaches can help in the optimization process to determine the uncertainty on the fitted model parameters due to the uncertainty of the calibration database. In the following, an example is given for the calibration process of a 6
Nuclear Engineering and Design 355 (2019) 110311
A. Bouloré
crit =
n
∑i =0
1 (FGRicalc − FGRiexp )2 σi2
(4)
where σi is the uncertainty on the experimental values for experiment i, “exp” indicates experimental values and “calc” the results of the code. Probabilistic approaches can also be used (Bachoc et al., 2014). The first one is the “Weighted probabilistic least-square approach”. Let y (x) = f(x,β) be a model for which a set of experimental data yexp(xi), i = 1…n, is available. Let β be the vector of p parameters to be adjusted, n the number of experiments available (xi, i = 1…n) in the calibration database. The β vector is obtained by minimizing the quadratic error between experiment and calculation, and taking into account the uncertainty on each experiment σi as weights. The solution given by least-square method is then: n
βLS = arg min ∑
i=0
1 (y (x i ) − yexp(xi ) )2 σi2
Fig. 11. Results of the different calibration methods.
(5)
If we assume that the model is a linear function of the parameters around a reference value βref, the solution given by least square method is:
βLS = βref + (H T C −1H )−1H T C −1 (yexp − yref )
- Experimental data used to build material properties, or to calibrate models - Experimental conditions, - Model parameters, etc.
(6)
where H is the matrix of the derivatives of the code with respect to β and C the diagonal matrix given by:
hij =
These uncertainties can have a significant effect on the interpretation of the calculated results, especially if the quantity of interest is a threshold. In the paper, we have tried to illustrate the different types of sources of uncertainty, and the different steps where uncertainty quantification is important in the simulation process: calibration, validation, code usage. Whatever the application of the simulation tool (experiment interpretation, safety analysis), it looks essential now to give the results of the simulation plus uncertainties. In the safety analysis for example, instead of determining a maximum power without melting of the fuel by conservative method, the introduction of probability distributions for the uncertain parameters and inputs leads to an evaluation of the probability of failure as a function of the target power (Struzik et al., 2017). This is more precise information. In fuel modelling and simulation domain, there are two main issues regarding uncertainty quantification and sensitivity analysis. The first one is about input uncertainty modelling. It is particularly challenging to model material properties and mechanical behaviour laws in terms of uncertainty. The example presented in this paper about fresh fuel thermal conductivity is quite simple, but it can be much more difficult when it is about creep behaviour of UO2 or Zircaloy cladding, or when it is about the evolution of these properties in irradiation conditions, as very few experimental data are available to build statistical models. The second one is about the propagation of the uncertainties in the global coupling scheme of a fuel performance code. In fact, as illustrated in this paper, it is possible to quantify the uncertainties of the outputs of one model, taking into account the uncertainties of the inputs of this model. But in most physical models (fission gas behaviour model for example), there are some model parameters, which have to be calibrated in the first step of the validation process. This calibration is made by determining the best value of these parameters, which make the results of the model fit the experimental data. Methodologies exist to assess the uncertainties of these parameters due to the uncertainty of the experimental database, as illustrated in this paper. However, the calibration is often done model by model, considering that all the outputs of the other models are deterministic. But in reality, the calibration of the model parameters depend on the uncertainty of the other models of the coupling scheme. An example of simultaneous calibration in a simple case has been presented in (Robertson et al., 2018), but a more systematic and theoretical approach has to be developed. Only then, a global uncertainty quantification of the whole fuel performance code can be performed.
∂f (x i ) 1 , and Cij−1 = 2 δij ∂βj σi
If we assume that the differences between experiments and results of the model are represented by a vector of random variables with a null average and the covariance matrix C, the probabilistic estimation β of β and its covariance matrix Qpost are is given by (Bachoc et al., 2014):
β = βref + (H T C −1H )−1H T C −1 (yexp − yref )
(7)
Qpost = (H T C −1H )−1
(8)
In particular, the diagonal terms of the Qpost matrix give an estimation of the variance of the parameters considered as random variables. The last approach we have tested considers that the real β vector is a particular value of a vector of random variables whose distribution is known a priori. This mathematical modelling is representative of a lack of knowledge more than a stochastic variability. In this work, we use a Gaussian distribution for the Bayesian modelling of β (mean βprior = βref and covariance matrix Qprior). Then the posterior distribution of β given the observations yexp is Gaussian with mean vector βpost and covariance matrix Qpost (Bachoc et al., 2014): −1 βpost = βref + (Qprior + H T C −1H )−1H T C −1 (yexp − yref )
Qpost =
−1 (Qprior
+H
T
C −1H )−1
(9) (10)
The diagonal terms of Qpost give an estimation of the variance of the parameters. In the following, the Qprior matrix will be chosen diagonal. All these approaches have been tested (simple probabilistic leastsquare, Bayesian with linear assumption), all of them giving satisfactory results for the calibration process (Fig. 11), and an estimation of the covariance matrix of the vector of model parameters, which represents the uncertainty on the parameters related to the experimental uncertainty (Bouloré et al., 2015). Recent work is conducted to use a non-linear Bayesian approach. The method itself is still under theoretical development (Damblin and Gaillard, 2018). 6. Conclusion A lot of data used for fuel behaviour simulation are uncertain: 7
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Declaration of Competing Interest
Proc. AEN-NEA Seminar on Thermal Performance of High Burn-up LWR Fuel. Campolongo, F., Saltelli, A., 1997. Sensitivity analysis of an environmental model: an application of different analysis methods. Reliab. Eng. Syst. Saf. 57, 49–69. Damblin, G., Gaillard, P., 2018. A bayesian framework for quantifying the uncertainty of physical models integrated into thermal-hydraulic computer codes. ANS Best Estimate Plus Uncertainty International Conference (BEPU 2018). De Rocquigny, E., Devictor, N., Tarantola, S., 2008. Uncertainty in Industrial Practice: A Guide to Quantitative Uncertainty Management. John Wiley and Sons. Du, Xiaoping, 2005. FORM/SORM. Probabilistic Engineering Design course, Ch. 7. University of Missouri – Rolla. Jomard, G., et al., 2014. CARACAS: an industrial model for the description of fission gas behavior in LWR-UO2 fuel. Proc. WRFPM 2014. Marelle, V., et al., 2016. New developments in ALCYONE 2.0 fuel performance code. Proc. TopFuel 2016. Morris, M.D., 1991. Factorial sampling plans for preliminary computational experiments. Technometrics 33, 161–174. Robertson, G., et al., 2018. Bayesian inverse uncertainty quantification for fuel performance modelling. Keynote lecture ANS Best Estimate Plus Uncertainty International Conference (BEPU 2018). Struzik, C., et al., 2017. Determination of the fuel melting probability in MTR using reliability methods. Proc. WRFPM “TopFuel” 2017. Struzik, C., Marelle, V., 2012. Validation of fuel performance CEA code ALCYONE, scheme 1D, on extensive database. Proc. TopFuel 2012.
There is no conflict of interest. Acknowledgement I would like to thank Christine Struzik, Bruno Michel, Jean-Marc Martinez and Fabrice Gaudier for their direct or indirect contribution to this paper. References Bachoc, F., et al., 2014. Calibration and improved prediction of computer models by universal Kriging. Nucl. Sci. Eng. 176, 81–97. Bailly, H., et al., 1999. The Nuclear Fuel of Pressurized Water Reactors and Fast Reactors Design and Behaviour. TEC&DOC, Intercept Ltd. Bouloré, A., et al., 2015. Contribution of probabilistic approaches for the calibration of a fission gas release model. ANS Transactions, ANS Annual Meeting 2015. Caillot, L., Delette, G., 1998. Phenomena affecting mechanical and thermal behaviour of PWR fuel rods at beginning of life: some recent experiment and modeling results.
8
Contents lists available at ScienceDirect
Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Importance of uncertainty quantification in nuclear fuel behaviour modelling and simulation
T
A. Bouloré CEA, DEN, DEC, Saint Paul lez Durance 13108, France
A R T I C LE I N FO
A B S T R A C T
Keywords: Uncertainty Nuclear fuel Modelling Safety analysis
Fuel performance codes describe the physical phenomena occurring in fuel during its irradiation in steady-state and transient conditions. In this paper, the principles of the modelling of nuclear fuel behaviour are presented. Many reasons show that it is also important to consider uncertainties in the fuel modelling process. It can be experimental reasons (validation database and material properties determination), uncertainties in the modelling process itself (lack of knowledge for example). This paper also presents several applications of uncertainty quantification and sensitivity analysis applied to fuel behaviour modelling. The first example is about the validation process of the fuel performance code, which results in a quantification of the uncertainty of the calculated quantities. The second application is about the use of the code in safety analysis to determine probabilities of failure considering epistemic and random uncertainties. In the last application, we show how important it is to quantify modelling uncertainties in the calibration process of complex physical models.
1. Introduction The aim of a fuel performance code is to describe the physical phenomena occurring in the fuel during its irradiation in reactor and to calculate the evolution of some physical quantities (like geometry, inner pressure, stresses, and temperature) which can be used directly in safety assessment of the fuel. In most cases, a fuel performance code is made of several coupled models, each of them describing one particular physical phenomenon (thermal transfer, fission gas behaviour, mechanical behaviour, cladding corrosion, etc.). Fuel performance codes are deterministic codes, but a lot of data (material properties, experimental results, etc.) used in these codes are uncertain. In this paper, we try to show how important it is to take into account these uncertainties in the use of a fuel performance code and what can be the consequences. 2. Nuclear fuel behaviour modelling In PWR, nuclear fuel is made of UO2 or (U,Pu)O2 pellets stacked in a cladding tube made of zirconium alloy (Bailly et al., 1999). Fission nuclear reactions in the pellet generate heat which is evacuated to the coolant outside the cladding. But because of the low thermal conductivity of UO2, the effective temperature at the center of the pellet is significantly higher than at the surface. This temperature gradient causes differential thermal strain between the center and the edge of
the pellet which results in pellet cracking. During the first power increase (Fig. 1), the gap width decreases because of differential thermal expansion between the fuel pellet and the cladding, and because of relocation of fuel pellet towards the cladding. In the early stages of irradiation in nuclear power plant (Fig. 1a and b), phenomena such as densification (pore removal), cladding creep and fuel swelling occur, which result in the gap closure at a burnup about 20 GWd/tM. Above a certain temperature which depends on burnup, gaseous fission products contribute to fission gas bubbles precipitation growth (gaseous swelling). Fission gas is also released in the plenum of the fuel rod by a coalescence/percolation mechanism (Fig. 1c). This release increases the inner rod pressure. Generally, this phenomenon occurs at the center of the pellet in PWR conditions. Simultaneously, near the edge of the pellet, epithermal capture of neutron by U238 atoms produce a higher concentration of plutonium than in the rest of the pellet. This leads to a local higher burnup (about twice than the rest of the pellet). As this burnup accumulation occurs at low temperature (below the default recovery temperature), irradiation defaults concentration rises and a local restructuring occurs at the rim of the pellet (so called “rim effect”). The new microstructure called High Burnup Structure (HBS) is characterized by a very small grain size, and a very high porosity which contains a large quantity of fission gas that is only partly released during steady state irradiation (Fig. 1d). This zone extends to about 200 µm from the edge of the pellet (Fig. 2). In
E-mail address: [email protected]. https://doi.org/10.1016/j.nucengdes.2019.110311 Received 18 December 2018; Received in revised form 19 June 2019; Accepted 21 August 2019 0029-5493/ © 2019 Elsevier B.V. All rights reserved.
Nuclear Engineering and Design 355 (2019) 110311
A. Bouloré
Fig. 1. Different stages of fuel behaviour in steady state conditions.
Fig. 3. Fuel cracking and creep during power ramp.
element structural integrity (cladding failure risk and loss of integrity of the fissile column in particular). So, the quantities of interest which have to be calculated by fuel performance codes are then: Fig. 2. High burnup structure.
- Temperature - Fuel and cladding mechanical behaviour (strains and stresses) - Fission products behaviour
case of accident conditions like LOss of Coolant Accident for example (LOCA), the effective temperature in this zone becomes much higher than in normal operation, and most of the fission products are still retained in the material in intergranular position. At high temperature intergranular fission products are easily released, and increase a lot the inner pressure of the fuel rod. In off normal conditions (power ramps), high temperature in the central part of the pellet induces fuel thermal strain, gaseous fuel swelling, fuel creep which loads the cladding leading to its plastic strain. Gas release in the plenum allows a little decreasing of the stresses. A more important cracking is observed in the outer parts of the fuel pellet (Fig. 3). In LOCA (LOss of Coolant Accident) conditions, the temperature of the cladding increases, which results in clad ballooning because of the cladding creep and of the internal pressure in the rod. As the fuel temperature increases as well, and because of the decreasing of the pressure contact between the cladding and the pellet, fuel fragmentation can occur in high burnup fuel. In RIA (Reactivity Initiated Accident) conditions, the temperature in the fuel increases rapidly, particularly in the outer parts of the pellet, which results in a high swelling rate of the pellet. Important fuel swelling and cladding creep lead to a ballooning of the cladding which may crack. All the phenomena listed may have a significant impact on the fuel
All these quantities have to be calculated simultaneously, because all the physical phenomena listed above are strongly coupled. The thermal model solves the heat transfer equation:
ρ · c p·
⎯⎯⎯⎯⎯⎯→ dT = div (λ· grad T ) + pv dt
(1)
where ρ is the density, cp and λ respectively the specific heat and the thermal conductivity of the material, and pv is the volume power generated by the fission. It models the heat transfer from the pellet to the fuel-to-clad gap, then through the cladding and finally in the coolant. As fuel thermal conductivity is quite low, the temperature in the fuel is high, and most physical phenomena occurring in the fuel are driven by temperature. Regarding safety criteria, temperature has a direct impact on fuel melting assessment, but also on fuel rod internal pressure (mainly via fission gas release mechanism). A good knowledge of the strains and stresses in the fuel element is important as cladding is the first barrier for fission products. Some physical phenomena, such as fission gas bubbles growth depend on the mechanical state of the fuel pellet. Cracking behaviour also depends a lot on the mechanical state. As gaseous swelling and the mechanical problem are strongly coupled, they have to be resolved simultaneously, 2
Nuclear Engineering and Design 355 (2019) 110311
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Fig. 4. Coupling scheme of ALCYONE fuel performance code (Marelle et al., 2016) in the PLEIADES numerical platform.
- Material properties: in many cases, material properties are measured directly by specific experimental techniques, and very often, the experimental data are scattered. There is an uncertainty due to the technique itself, but also an uncertainty linked to the representativeness of the chosen sample. It is the case for experimental data of fuel thermal conductivity, fresh or irradiated (ex. Fig. 6). - Experimental conditions used for validation: a typical quantity which is uncertain is the power at which the rod is irradiated during an experimental irradiation. And it is a parameter of first order of all the schemes in fuel performance codes. - Experimental results which are used to validate/calibrate the different models of the fuel performance codes. For example: Fission gas release measurement which is an integral quantity used to validate the fission gas behaviour model. Temperature measurements in irradiation devices also have an uncertainty, and this quantity is used to validate the global thermal model of the code. - Model parameters. Some model parameters can be measured directly (like material properties mentioned above), but very often, specific experiments are conducted which allow a determination of the parameters using inverse method. It means that a specific model is built to extract basic data from the experiment. A good example of such parameter is diffusion coefficients. In this case, the uncertainty is linked to both experimental data/conditions uncertainties, and to the degree of simplification of the model.
which means in a global convergence scheme. An example of fuel performance code that illustrates all these couplings is ALCYONE (Fig. 4), the multidimensional fuel performance code developed and validated by CEA (Commissariat à l’Énergie Atomique) (Marelle et al., 2016). In the 1D scheme (Fig. 5), the fissile stack is represented by slices which are meshed radially. It is designed to model a whole fuel rod in irradiation conditions. 2D and 3D schemes are developed to have a more detailed evaluation of the mechanical behaviour, specifically for the situations of pellet-cladding interaction (PCI). The validation (Struzik and Marelle, 2012) of this code is done as well at the macroscopic scale (integral results, inner pressure in the rod, dimension changes of the whole rod, etc.) as well as at a microscopic scale (comparison of fission gas retention to microprobe and SIMS (Secondary Ion Mass Spectrometry) measurements after irradiation for example).
• •
3. Why uncertainty analysis in fuel modelling? In the frame of nuclear fuel behaviour modelling and simulation, due to the difficulties associated to measures under irradiation, it is very often not possible to obtain a reasonable value of material properties by direct measurements. Then the determination of the model physical parameters is very often not direct which induces numerous sources of uncertainty coming from different topics. 3
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Fig. 5. Different representations of the fuel element (1D, 2D and 3D) in ALCYONE.
Sensitivity analysis helps to determine which uncertain parameter or data is the most sensitive on the outputs of the fuel performance code. It helps then to determine which uncertain inputs have to be more precisely known (reduced uncertainty). The second topic is safety analysis. As the fuel performance code is a best-estimate code, uncertainty quantification/sensitivity analysis and probabilistic approaches allow taking into account the uncertainties on inputs and model parameters to make sure that even with some uncertainties, the safety criterion is still verified. It is shown later in the paper that the criterion verification can be replaced by a probability assessment in those cases.
- Modelling uncertainty is more difficult to represent and quantify. In fact a model is always a simplified mathematical representation of a complex physical phenomenon. Assessing the modelling uncertainty consists in a quantification of the error made when simplifying the representation. It is a very difficult task. Two main objectives can be listed to justify the need of uncertainty analysis. First, it is a way to improve the degree of confidence of the fuel performance code by demonstrating that in the global simulation/ modelling process, uncertainties on the calculated quantities are being quantified. In this objective, propagation on input uncertainties in the code is important in the global validation process of the code (showing that the experimental results used for validation are in the range of uncertainty of the calculated results (Struzik and Marelle, 2012).
4. General methodology The methodology used for about 10 years is presented in detail in
6
Thermal conductivity (W/m/K)
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batch n°1
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Fig. 7. General methodology recommended for UQ/SA. 450
Morris sensitivity analysis method (Morris, 1991; Campolongo and Saltelli, 1997) can be used to reduce the number of uncertain parameter. It is a one-at-a-time method which does not require a large set of calculations. The Morris sensitivity indices may depend on the experimental conditions (for example gap open or closed) and the method has to be applied for the experimental conditions of the case. After the uncertainty modelling phase, uncertainty propagation can be performed by Monte Carlo sampling and direct calls to the fuel performance code. The size of the sample depends on the computation resources available. The case presented is the GRIMOX2 experiment. GRIMOX2 was an experiment conducted in the SILOE nuclear test reactor in Grenoble during the nineties (Caillot and Delette, 1998). It had been performed in order to study the thermal behaviour of MOX fuel compared to the UO2 fuel in the beginning of irradiation. For that purpose a short fuel rod of PWR diametrical geometry including MOX and UO2 pellets was irradiated in a boiling capsule pressurized to 13 MPa placed at the edge of the SILOE reactor core. At both ends of the experimental rod, the pellets were drilled in order to introduce a thermocouple. So this device allowed the measurement of the centerline temperature of both fresh fuel, UO2 and MOX, during an irradiation up to 0.5 at%. The time duration of the experiment was about 4000 h (Fig. 8). The uncertainty on temperature measurement announced by the operator is ± 10 °C. The experimental fissile power was determined by the mean of neutron flux detector during the experiment and it was cross checked by quantitative gamma spectrometry after irradiation. The order of magnitude of power level is comparable to the PWR one in standard and incidental conditions. Although two types of fuel had been tested in this experiment this paper only deals with the UO2 stack. In the case presented, 10,000 simulations of the experiment have been performed and the set of results can be used to determine the uncertainty on the calculated temperature. All experimental data are in the 95% confidence interval of the calculated distributions (Fig. 9), so it validates the modelling of the thermal behaviour of the fuel element. A focus will be done in the presentation on the propagation of this uncertainty on temperature in the other models of the code. The problem is more complex than what it looks like, as all the models are coupled in the fuel performance code, and all models need a calibration phase to make the final code a best-estimate code.
Average Linear heat Rate (W/cm)
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Fig. 8. Irradiation history of the GRIMOX2 experiment.
(De Rocquigny et al., 2008). The general scheme of this recommendation is printed on Fig. 7. This method is decomposed in 4 steps: problem specification, modelling of the uncertainties, uncertainty propagation phase and eventually sensitivity analysis if required. The first two steps are very often the most difficult; the last two require large computing capacities.
5. Examples of applications 5.1. Code validation: example on fuel thermal behaviour To quantify the uncertainty on the temperature calculated by the fuel performance code ALCYONE, uncertainty on the following inputs are taken into account: - Fuel thermal conductivity - Radial distribution of power which also leads to the HBS restructuring - Power (or Linear Heat Rate) generated by the fuel rod - Thermal transfer through the pellet-to-clad gap Different methods can be used to model these inputs as random variables (expert opinion, inverse methods, iterative methods…). But some of these uncertain input depend on a lot of parameters, and the 5
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Fig. 9. Validation taking into account uncertainties.
5.2. Safety analysis; application of reliability methods to fuel melting The second example is about the safety analysis topic. The objective is to determine at which power, a melting of the fuel pellet could occur:
Pfusion = max{P,
max(Tfuel ) < Tmelting }
(2)
As there are uncertainties on the inputs and parameters used to calculate the fuel temperature, and also on the knowledge of the melting temperature of the irradiated fuel, the problem of threshold determination is turned into the determination of the probability of melting for a given target power:
P (Tfuel > Tmelting )
(3)
The same general methodology is applied to quantify this probability and two types of methods (maybe more) can be used. Monte Carlo direct sampling requires a large set of simulations but allows an evaluation of the uncertainty on the calculated probability (confidence interval). Reliability methods like FORM/SORM (First Order and Second Order Reliability Method) (Du, 2005) can also be used and require much less runs of the code (Fig. 10). Using FORM/SORM we can evaluate the probability of fuel melting, but also the contribution of each uncertain input to the probability of melting (kind of sensitivity analysis). But it assumes that the code behaviour is “regular” around the domain of failure, because it needs to evaluate second derivatives which are very sensitive to the precision of calculation and convergence of the global scheme (Struzik et al., 2017).
Fig. 10. FORM/SORM methods.
fission gas behaviour model (Bouloré et al., 2015). Initially, the model (Jomard et al., 2014) contains about 40 numerical or physical scalar parameters to be adjusted. To reduce the computation time during the first iterative calibration phase, preliminary to the calibration process, a fast sensitivity analysis has been performed using the Morris method (Campolongo and Saltelli, 1997). From this analysis, only 7 scalar parameters are still to be calibrated because they have a direct effect on the result of the model (FGR: fission gas release): diffusion coefficient of xenon enhanced by irradiation (2 parameters), efficiency of the bubble nucleation, resolution of gas atoms from nanometric bubbles, resolution of gas atoms from precipitated bubbles, rate of accumulation of dislocations, and a parameter involved in the fraction of bubbles connected to porosity. The first automatic method used for calibration is an iterative method based on the SIMPLEX algorithm which is used to minimize a scalar criterion. On each experiment of the calibration database, the objective is to minimize the quadratic error between calculation and experimental value for FGR (fission gas release). So the scalar criterion is defined as:
5.3. Model/code calibration In the fuel behaviour simulation process, a lot of models contain fitting parameters. In order to make the code a “best-estimate” code, all these parameters have to be adjusted on a set of experiments called calibration database (including the associated experimental uncertainties) covering as far as possible the domain on which the model is going to be used in the future. This is also part of the global validation process. As most of the time, the experimental quantities used for calibration have an uncertainty, the statistic and probabilistic approaches can help in the optimization process to determine the uncertainty on the fitted model parameters due to the uncertainty of the calibration database. In the following, an example is given for the calibration process of a 6
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crit =
n
∑i =0
1 (FGRicalc − FGRiexp )2 σi2
(4)
where σi is the uncertainty on the experimental values for experiment i, “exp” indicates experimental values and “calc” the results of the code. Probabilistic approaches can also be used (Bachoc et al., 2014). The first one is the “Weighted probabilistic least-square approach”. Let y (x) = f(x,β) be a model for which a set of experimental data yexp(xi), i = 1…n, is available. Let β be the vector of p parameters to be adjusted, n the number of experiments available (xi, i = 1…n) in the calibration database. The β vector is obtained by minimizing the quadratic error between experiment and calculation, and taking into account the uncertainty on each experiment σi as weights. The solution given by least-square method is then: n
βLS = arg min ∑
i=0
1 (y (x i ) − yexp(xi ) )2 σi2
Fig. 11. Results of the different calibration methods.
(5)
If we assume that the model is a linear function of the parameters around a reference value βref, the solution given by least square method is:
βLS = βref + (H T C −1H )−1H T C −1 (yexp − yref )
- Experimental data used to build material properties, or to calibrate models - Experimental conditions, - Model parameters, etc.
(6)
where H is the matrix of the derivatives of the code with respect to β and C the diagonal matrix given by:
hij =
These uncertainties can have a significant effect on the interpretation of the calculated results, especially if the quantity of interest is a threshold. In the paper, we have tried to illustrate the different types of sources of uncertainty, and the different steps where uncertainty quantification is important in the simulation process: calibration, validation, code usage. Whatever the application of the simulation tool (experiment interpretation, safety analysis), it looks essential now to give the results of the simulation plus uncertainties. In the safety analysis for example, instead of determining a maximum power without melting of the fuel by conservative method, the introduction of probability distributions for the uncertain parameters and inputs leads to an evaluation of the probability of failure as a function of the target power (Struzik et al., 2017). This is more precise information. In fuel modelling and simulation domain, there are two main issues regarding uncertainty quantification and sensitivity analysis. The first one is about input uncertainty modelling. It is particularly challenging to model material properties and mechanical behaviour laws in terms of uncertainty. The example presented in this paper about fresh fuel thermal conductivity is quite simple, but it can be much more difficult when it is about creep behaviour of UO2 or Zircaloy cladding, or when it is about the evolution of these properties in irradiation conditions, as very few experimental data are available to build statistical models. The second one is about the propagation of the uncertainties in the global coupling scheme of a fuel performance code. In fact, as illustrated in this paper, it is possible to quantify the uncertainties of the outputs of one model, taking into account the uncertainties of the inputs of this model. But in most physical models (fission gas behaviour model for example), there are some model parameters, which have to be calibrated in the first step of the validation process. This calibration is made by determining the best value of these parameters, which make the results of the model fit the experimental data. Methodologies exist to assess the uncertainties of these parameters due to the uncertainty of the experimental database, as illustrated in this paper. However, the calibration is often done model by model, considering that all the outputs of the other models are deterministic. But in reality, the calibration of the model parameters depend on the uncertainty of the other models of the coupling scheme. An example of simultaneous calibration in a simple case has been presented in (Robertson et al., 2018), but a more systematic and theoretical approach has to be developed. Only then, a global uncertainty quantification of the whole fuel performance code can be performed.
∂f (x i ) 1 , and Cij−1 = 2 δij ∂βj σi
If we assume that the differences between experiments and results of the model are represented by a vector of random variables with a null average and the covariance matrix C, the probabilistic estimation β of β and its covariance matrix Qpost are is given by (Bachoc et al., 2014):
β = βref + (H T C −1H )−1H T C −1 (yexp − yref )
(7)
Qpost = (H T C −1H )−1
(8)
In particular, the diagonal terms of the Qpost matrix give an estimation of the variance of the parameters considered as random variables. The last approach we have tested considers that the real β vector is a particular value of a vector of random variables whose distribution is known a priori. This mathematical modelling is representative of a lack of knowledge more than a stochastic variability. In this work, we use a Gaussian distribution for the Bayesian modelling of β (mean βprior = βref and covariance matrix Qprior). Then the posterior distribution of β given the observations yexp is Gaussian with mean vector βpost and covariance matrix Qpost (Bachoc et al., 2014): −1 βpost = βref + (Qprior + H T C −1H )−1H T C −1 (yexp − yref )
Qpost =
−1 (Qprior
+H
T
C −1H )−1
(9) (10)
The diagonal terms of Qpost give an estimation of the variance of the parameters. In the following, the Qprior matrix will be chosen diagonal. All these approaches have been tested (simple probabilistic leastsquare, Bayesian with linear assumption), all of them giving satisfactory results for the calibration process (Fig. 11), and an estimation of the covariance matrix of the vector of model parameters, which represents the uncertainty on the parameters related to the experimental uncertainty (Bouloré et al., 2015). Recent work is conducted to use a non-linear Bayesian approach. The method itself is still under theoretical development (Damblin and Gaillard, 2018). 6. Conclusion A lot of data used for fuel behaviour simulation are uncertain: 7
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Declaration of Competing Interest
Proc. AEN-NEA Seminar on Thermal Performance of High Burn-up LWR Fuel. Campolongo, F., Saltelli, A., 1997. Sensitivity analysis of an environmental model: an application of different analysis methods. Reliab. Eng. Syst. Saf. 57, 49–69. Damblin, G., Gaillard, P., 2018. A bayesian framework for quantifying the uncertainty of physical models integrated into thermal-hydraulic computer codes. ANS Best Estimate Plus Uncertainty International Conference (BEPU 2018). De Rocquigny, E., Devictor, N., Tarantola, S., 2008. Uncertainty in Industrial Practice: A Guide to Quantitative Uncertainty Management. John Wiley and Sons. Du, Xiaoping, 2005. FORM/SORM. Probabilistic Engineering Design course, Ch. 7. University of Missouri – Rolla. Jomard, G., et al., 2014. CARACAS: an industrial model for the description of fission gas behavior in LWR-UO2 fuel. Proc. WRFPM 2014. Marelle, V., et al., 2016. New developments in ALCYONE 2.0 fuel performance code. Proc. TopFuel 2016. Morris, M.D., 1991. Factorial sampling plans for preliminary computational experiments. Technometrics 33, 161–174. Robertson, G., et al., 2018. Bayesian inverse uncertainty quantification for fuel performance modelling. Keynote lecture ANS Best Estimate Plus Uncertainty International Conference (BEPU 2018). Struzik, C., et al., 2017. Determination of the fuel melting probability in MTR using reliability methods. Proc. WRFPM “TopFuel” 2017. Struzik, C., Marelle, V., 2012. Validation of fuel performance CEA code ALCYONE, scheme 1D, on extensive database. Proc. TopFuel 2012.
There is no conflict of interest. Acknowledgement I would like to thank Christine Struzik, Bruno Michel, Jean-Marc Martinez and Fabrice Gaudier for their direct or indirect contribution to this paper. References Bachoc, F., et al., 2014. Calibration and improved prediction of computer models by universal Kriging. Nucl. Sci. Eng. 176, 81–97. Bailly, H., et al., 1999. The Nuclear Fuel of Pressurized Water Reactors and Fast Reactors Design and Behaviour. TEC&DOC, Intercept Ltd. Bouloré, A., et al., 2015. Contribution of probabilistic approaches for the calibration of a fission gas release model. ANS Transactions, ANS Annual Meeting 2015. Caillot, L., Delette, G., 1998. Phenomena affecting mechanical and thermal behaviour of PWR fuel rods at beginning of life: some recent experiment and modeling results.
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