Reliability-based robust design optimization of gap ...

Engineering Optimization

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Reliability-based robust design optimization of gap size of annular nuclear fuels using kriging and inverse distance weighting methods Jaehyeok Doh, Younghoon Kim & Jongsoo Lee To cite this article: Jaehyeok Doh, Younghoon Kim & Jongsoo Lee (2018): Reliability-based robust design optimization of gap size of annular nuclear fuels using kriging and inverse distance weighting methods, Engineering Optimization, DOI: 10.1080/0305215X.2018.1428316 To link to this article: https://doi.org/10.1080/0305215X.2018.1428316

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ENGINEERING OPTIMIZATION, 2018 https://doi.org/10.1080/0305215X.2018.1428316

Reliability-based robust design optimization of gap size of annular nuclear fuels using kriging and inverse distance weighting methods Jaehyeok Doh, Younghoon Kim and Jongsoo Lee School of Mechanical Engineering, Yonsei University, Seoul, Republic of Korea ABSTRACT

ARTICLE HISTORY

In this study, the design optimization of the gap size of annular nuclear fuels used in pressurized water reactors (PWRs) was performed. For this, thermoelastic–plasticity–creep (TEPC) analysis of PWR annular fuels was carried out using an in-house code to investigate the performance of nuclear fuels. Surrogate models based on the kriging and inverse distance weighting models were generated using computational performance data based on optimal Latin hypercube design. Using these surrogate models, the gap size of PWR annular fuel was deterministically optimized using the microgenetic algorithm to improve the heat transfer efficiency and maintain a lower level of stress. Reliability-based design optimization and reliabilitybased robust design optimization were conducted to satisfy target reliability and secure the robustness of the PWRs’ performance. The optimal gap size was validated through TEPC analysis and the optimum solutions were compared according to the approximate method and reliability index.

Received 4 April 2017 Accepted 20 December 2017

Nomenclature α E ν ρ T σc σt Q h k CP xi f gi u d (x, xi ) R μ(μ∗f , μXi )

Coefficient of thermal expansion (1/K) Young’s modulus (MPa) Poisson’s ratio Density (Mg/mm3 ) Temperature (K) Compressive hoop stress (MPa) Tensile hoop stress (MPa) Heat generation (mJ/mm3 -s) Convection (mJ/s-mm2 -K) Thermal conductivity (mJ/s-mm-K) Specific heat capacity (mJ/Mg-K) Design variables (inner gap: x1 ; outer gap: x2 ) Objective function Constraint function Interpolated value Distance function Entropy [in Equation (1)] Mean (objective function, design variables)

CONTACT Jongsoo Lee

[email protected]

© 2018 Informa UK Limited, trading as Taylor & Francis Group

KEYWORDS

Pressurized water reactor (PWR); annular nuclear fuel; thermoelastic–plasticity–creep (TEPC); inverse distance weighting (IDW); reliability-based robust design optimization (RBRDO)

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σ (σ f *, σ Xi ) wi RI (β) Rt

Deviation (objective function, design variables) Weighting factors Reliability index Target reliability

1. Introduction A uniform temperature distribution can be obtained by uniformly distributing the coated particles within the core of a nuclear reactor. The role of the cladding is to protect the nuclear fuels from mechanical failure and to prevent the fission product from being released to the outside of the coating layer. It not only protects the coated particles from chemical corrosion due to the impurities in the coolant, but also permits smooth conduction of heat between the coated particles and the reactor coolant. Thus, the structural health of the nuclear fuel is maintained both during normal operation and in the event of an accident. It has been reported that the maximum temperature of annular nuclear fuel is lower than that of conventional solid nuclear fuel; when operating a nuclear plant, safety concerns regarding accidents are high (Rowinski, White, and Zhao 2015). The annular nuclear fuels in pressurized water reactors (PWRs) are shown in Figure 1. The nuclear energy is generated by a series of fission reactions, and during this process, a high-temperature heat load and cooling water under high pressure are applied to the nuclear fuel. To evaluate nuclear fuels, steady-state heat transfer and stress analysis codes have been developed in the USA, France and Japan. However, there is no code to evaluate the status of a reactivity-initiated accident (RIA) or a loss of coolant accident (LOCA). It is difficult to apply the swelling and densification states of nuclear fuels in commercial programs. Therefore, a large-scale thermoelastic evaluation code is needed to prevent accidents in nuclear reactors (Ichikawa, Fujishiro, and Kawasaki 1989; Lamarsh and Baratta 2001; Vitanza 2006). Researchers have made considerable efforts to develop the finite element (FE) code for analysing the behaviour of the complex phenomena in nuclear fuels, because there are limitations to the experiments that can be performed. Pandey and Sarkar (2011) reported that prediction of the thermomechanical behaviour of fuel elements in a nuclear power plant is very important for the design, and for preventing the failure of the element during operation. They performed structural analysis of a fuel element in a nuclear power plant under operating conditions. The universal method of verifying the safety of a nuclear fuel rod is to simulate it using a fuel performance code. This simulates the nuclear fuel rod behaviour by considering the gap conductance using three-dimensional gap elements. Kang et al. (2016) simulated a nuclear fuel rod and compared the simulation results with experimental results. Kwon, Shim, and Song (2016) performed research on the behaviour of nuclear fuel rods during accidents. This research was very sophisticated and involved complex tasks related to thermal, elastic, plastic and creep phenomena. The nuclear reactor reloading pattern optimization model is solved using mixed-integer nonlinear optimization techniques. They treat the algorithmic extension using dedicated cuts in the mixedinteger nonlinear optimization algorithm, which push the optimization towards solutions where the

Figure 1. (a) Solid nuclear fuel; (b) annular nuclear fuel.

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local power peaks in parts of the core are avoided (Quist Arie and Roos 2001). In other research, the gap size of PWR annular nuclear fuels was optimized to minimize the maximum average temperature. For this, FE analysis was repeatedly conducted to obtain the online performance data without using approximate models. However, the disadvantage of this method is that analysis time and cost are required to obtain optimal solutions (Kwon et al. 2015). In the reliability analysis and reliability-based design optimization (RBDO) of roadway minimum radius design based on vehicle dynamics, the probability of an accident is evaluated using the first order reliability method and numerical studies are conducted using a single-unit truck model (Shin and Lee 2015). The RBDO is a technique used for engineering design when uncertainty is being considered. A typical RBDO problem can be formulated as a stochastic optimization model, where the performance of a system is optimized and the reliability requirements are treated as constraints. The single-loop deterministic method for RBDO is proposed to reduce the computational effort of RBDO without sacrificing much accuracy (Li et al. 2013). Reliability-based robust design optimization (RBRDO) deals with two objectives of structural design methodologies subject to various uncertainties. The reliability constraints deal with the probability of failures, while the robustness minimizes the product quality loss. In general, the product quality loss is described using the first two statistical moments (Youn, Choi, and Yi 2005). The highlight of this study is the use of design optimization to secure the robustness and reliability of the annular nuclear fuel performance in nuclear power plants. The proper surrogate model was proposed by comparing the accuracy of the surrogate models created using the biased nonlinear sample data and the results of approximate optimization using kriging and IDW. In this study, the PWR annular fuels have wider surfaces than conventional solid fuels, as this improves the heat transfer efficiency. This ensures a lower surface temperature for the nuclear fuel, compared to conventional solid nuclear fuels, thus providing some extra margin for unexpected transient situations, such as an RIA or a LOCA. A thermoelastic–plasticity–creep (TEPC) analysis was conducted on the complex thermal–mechanical phenomena of PWR annular fuels, with an in-house code developed using Visual FORTRAN. Surrogate models of the objective and constraint functions were generated using the computational performance data based on an optimal Latin hypercube design (OLHD). To generate the surrogate models of constraint functions, the kriging and inverse distance weighting (IDW) methods were used. The accuracy of the approximate models was evaluated through the root mean square error (RMSE) and R2 value. Using these models, the gap size of the PWR annular fuels was deterministically optimized using the micro-genetic algorithm (MGA) to increase the heat transfer efficiency and maintain a lower level of stress. RBDO and the RBRDO were conducted to take into consideration the uncertainty of design variables. The optimal gap size was validated by TEPC analysis of the PWR annular fuels, and the obtained optimum solutions were compared according to the approximate method and reliability index (RI).

2. TEPC analysis of PWR annular nuclear fuels In general, the converged solutions are iteratively computed through the fine displacement and load incremental method in nonlinear FE analysis. In the case of nuclear fuels, the conventional increment method is not suitable for FE solutions, owing to the concurrent interactions of complex material behaviours, such as thermoelastic–plasticity and creep effects. To compute the temperature distribution and tensile–compressive hoop stress due to the complex material behaviours using two-dimensional axisymmetric modelling of annular nuclear fuels under steady-state and transient conditions depending on time and temperature, in previous research an in-house code was developed using Visual FORTRAN (Kwon et al. 2011; Kim, Yang, and Koo 2013). This FE code was verified with commercial packages such as ADINA, Marc and NISA-II. The developed nonlinear FE code used the effective stress function algorithm (Kojic and Bathe 1987) for TEPC analysis. This algorithm has the advantage that the complex analysis of the thermoelastic–plasticity and creep behaviours can be converged into one variable (Bathe 1996). In the

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developed FE code, the von Mises yield condition and the isotropic hardening condition are applied to the yield function. The creep behaviour is also described using the power creep model. There are three types of gap state. One is a non-contact state in which there exists a initial gap between the pellet and the cladding. The second type is a state of sticking of the gap in which the gap starts to contact owing to thermal expansion and coolant pressure. This occurs when the axial contact force is smaller than the frictional force by the radial contact force. Lastly, the opposite case to the second type is a state of slip. In this study, the contact problem between the pellet and the cladding was analysed by applying a frictional contact at the gap, with an isoparametric eight-node element under the generalized plane strain condition (Anthoine 1997; Marchal, Campos, and Garnier 2009). 2.1. FE modelling and boundary conditions The geometry of annular nuclear fuels is illustrated in Figure 2. The FE model was generated using the two-dimensional axisymmetric model with eight-node isoparametric elements. In the case of high-order elements, the isoparametric variables can map the curvilinear and Cartesian coordinates by interpolation. The isoparametric element which has the curve and curvature boundary is useful for the geometric shape representation. Moreover, the FE model was generated using an identical number of high-order elements (eight nodes), owing to the time and cost considerations for TEPC analysis. The pellet is subjected to heat generation. The surface of the cladding is subjected to heat convection by the coolant under a coolant pressure of 15.5 MPa. Furthermore, the inner and outer gaps are subjected to a gas pressure of 3.15 MPa. The boundary condition is illustrated in Figure 3. 2.2. Mechanical and thermal properties of the material The annular nuclear fuels, which are comprised of the pellet and the cladding, have a complex behaviour under high pressure and temperature, such as thermal expansion due to nuclear fission and oxidation by the coolant. The material properties applied to the TEPC analysis were obtained from the material model based on experiments. To account for the change in properties with temperature, during the TEPC analysis, user-specified material properties were applied with continuous functions in the developed FE code. The property of specific heat has high nonlinearity at the interval of the specific temperature. Considering this nonlinearity, the TEPC analysis was performed using the initial mechanical and thermal material properties of nuclear fuels obtained from the Korea Atomic Energy Research Institute (KAERI). The initial material properties are listed in Table 1.

Figure 2. Two-dimensional axisymmetric geometry of annular fuels.

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Figure 3. Boundary conditions and finite element model of annular fuels. Table 1. Mechanical and thermal material properties. α

E

ν

ρ

0.6721e − 5 0.9106e − 5

0.7952e + 5 0.1803e + 6

0.3567 0.3160

6.5500e−9 1.0412e − 8

Cp

k

0.3258e + 9 0.3138e + 9 0.3138e + 9

0.1290e + 2 0.3440e + 0 0.3448e + 1

Mechanical Cladding Pellet Thermal

T∞

Cladding Gap Pellet

562.7 – –

Q – 629.8

h 100 – –

2.3. Results of the TEPC analysis The TEPC analysis was conducted with a conventional gap size (inner gap: 0.07 mm; outer gap: 0.06 mm). The initial gaps due to tension and compression behaviour of nuclear fuels decrease with time. In the analysis, the perfect contact occurs when the relative displacement of the inner and outer gap nodes is 0. The outer and inner gaps between the pellet and cladding were in perfect contact at 403,610 s and 903,610 s, respectively, as shown in Figure 4. The tensile hoop stress is caused by the thermal

Inner gap (x1)=0.07 (mm) Outer gap (x2)=0.06 (mm)

Gap size (mm)

0.10

Inner gap Outer gap

0.08 0.06 0.04 0.02 0.00 0

200000 400000 600000 800000

Time (sec.) Figure 4. Gap size variation according to time.

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expansion due to the nuclear fission of the pellet. As the outer gap is smaller than the inner gap, the outer gap is contacted before the inner gap. It can be seen that the gap size decreases as a result of the initial thermal expansion and creep effect. The maximum tensile hoop stress of the inner pellet was 94.68 MPa. The maximum compressive hoop stress of the outer cladding was 113.18 MPa. If the outer gap is contacted in advance, both the tensile hoop stress of the inner cladding and the compressive hoop stress of the outer cladding decrease (Figure 5). Furthermore, the tensile stress of the outer pellet is decreased and the compressive hoop stress of the inner pellet is increased (Figure 6).

Hoop stress (MPa)

150

Inner gap (x1)=0.07 (mm) Outer gap (x2)=0.06 (mm)

100 50 0 -50

Inner cladding

-100

Outer cladding

-150 0

200000 400000 600000 800000

Time (sec.) Figure 5. Hoop stress variation of cladding according to time.

Hoop stress (MPa)

150

Inner gap (x1)=0.07 (mm) Outer gap (x2)=0.06 (mm)

100 50 0 Inner pellet Outer pellet

-50 0

200000 400000 600000 800000 Time (sec.)

Figure 6. Hoop stress variation of pellet according to time.

Figure 7. Temperature according to radius.

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The maximum temperature of annular nuclear fuels is 835.5 K, at the centre of the pellet (Figure 7). It can be seen that the maximum temperature at the centre of the pellet needs to be decreased. Moreover, the maximum temperature of the nuclear fuel depends on the contact time of the gaps. With faster gap contact, the maximum temperature of the internal pellet will be decreased owing to conduction heat transfer. There is a considerable temperature gradient between the pellet and cladding owing to heat convection caused by the application of gas pressure at the inner–outer gap and coolant at the surface of the cladding. As there is a difference in the contact time of the gaps according to the gap size, the maximum temperature of the pellets is also different. The performance index can be obtained quantitatively through TEPC analysis of the annular nuclear fuels.

3. Design of experiments by optimal Latin hypercube design The design of experiments (DOE) is the one of the key factors determining the success of surrogate modelling. Over the past few decades, researchers have developed a number of sampling techniques for different purposes in surrogate modelling, to ensure accuracy and efficiency. The Latin hypercube design (LHD) (McKay, Beckman, and Conover 1979; Wang, Beeson, and Wiggs 2006) creates a matrix of n rows representing the level of design variables and k columns representing the number of design variables. In this method, the test points are arranged by making each column a sequence of different integers. This is advantageous in that the implementation is simple and there is no loss of information due to the overlapping of the test points, since they can be made through permutations of different integers. Even though the results can be obtained using this method, the test points are not evenly dispersed, which may result in poor filling properties. To overcome this drawback, the OLHD method can be used, which involves selecting the experimental points that satisfy the properties of LHD and maximize the entropy (Park 1994), as shown in Equation (1): Maximize |R|

(1)

xi ∈L

where L is the set of test points that satisfy the properties of LHD. Therefore, OLHD can be used to find the most evenly distributed experimental points among several Latin hypercubes. Yao et al. (2013) analysed the suitability of several popular interpolation methods for complex terrains and proposed an optimal method. Four spatial interpolation methods—ordinary kriging, IDW, linear regression and regression kriging—were used for modelling. The performance of each method was assessed quantitatively. The prediction accuracy of the models in complex terrain differed significantly between these methods. The OLHD method is widely used for random sampling, so it is suitable for the kriging method. This DOE method was used in this study to obtain the performance data through TEPC analysis.

x2: Outer gap (mm)

1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

x1: Inner gap (mm) Figure 8. Non-uniform sample data by optimal Latin hypercube design in the design domain.

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However, OLHD is a method of extracting sample data evenly and is a design feature that is not realistic. To take into account the uncertainty of the surrogate model that occurs in practical engineering problems, biased sample data were added to create a surrogate model based on unequal data (Figure 8). In this study, the number of sample data was first determined to be 40 (between 0.01 mm and 1 mm), considering the gap size of the nuclear fuel. After investigating the performance data, the sampling points were added, with 40 points from 0.01 mm to 0.1 mm to decrease the maximum temperature of the pellet. The total number of sample data was 80.

4. Surrogate modelling Surrogate modelling for approximate design optimization has been widely used in engineering to approximate computationally intensive simulation models for design purposes. Kriging (Simpson et al. 2001; Martin and Simpson 2004; Zhao 2011) is used more for the fitting of data obtained from larger experimental fields than for data from lower order polynomial regression. In addition, it is effective in solving multi-variable, nonlinear and complex design problems. The kriging model is composed of a global model and a localized deviation. IDW is a type of deterministic method for multivariate interpolation with a known scattered set of points. The values assigned to unknown points are calculated with a weighted average of the values available at the known points (Zimmerman et al. 1999; Mueller et al. 2004). It is generally used in the interpolation method for non-grid data, in which there is no tendency. This method is mainly used to determine the contour line of geographical data. When the performance data have a high nonlinearity, depending on the design variables, the IDW interpolation method is suitable as the approximate method for design optimization. In this study, the DOE-based TEPC analysis was conducted to obtain the performance data. To obtain temperature data, a surrogate model was generated using the response surface method (RSM), because the nonlinearity of the data was lower than that of the stress data. The surrogate models of the tensile–compressive hoop stress were generated using conventional kriging and IDW methods, considering the nonlinearity of the data for approximate design optimization. In Figure 9, the accuracy of kriging and IDW models is represented with RMSE (Doh, Lee, and Lee 2016). The number of training data was set as 64 and the number of test data was set as 16 to generate surrogate models. The accuracy of the RSM was also evaluated in terms of R2 values. Although the global RMSE value was relatively high, these surrogate models were used because of the high accuracy of the area of interest of constraint conditions. The constraint function was generated using biased sample data with strong nonlinearity. The RMSE of IDW was higher than that of kriging. From this, it was judged appropriate to use kriging to generate a global surrogate model for biased nonlinear sample data. In the case of IDW, the accuracy is high with a high density of sample data in the design domain, and low for a low density of data. In addition, the accuracy of the IDW method depends on the size of the neighbourhood, the number of neighbours and the distance between neighbours (Setianto and Triandini 2013).

5. Deterministic design optimization Surrogate models were generated using the OLHD-based performance data obtained from the TEPC analysis of annular nuclear fuels. Using identical performance data, kriging and IDW models were generated for the constraint function of the tensile–compressive hoop stress. The objective function of maximum temperature was generated by RSM because the sample data do not have a high nonlinearity. To perform deterministic design optimization (DDO), the MGA embedded in the commercial package PIAnO (2016) was applied for the global approximate design optimization. When applying the kriging and IDW models, the optimal solutions obtained were compared with each other and the deterministic optimal solution was verified by TEPC analysis.

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Predict comp. stress (MPa)

-100

RMSE-Kriging: 14.24 RMSE-IDW: 36.30 Test data: 16 -200 Training data: 64

-300

IDW Kriging Actual value

Predict tens. stress (MPa)

-400 -400

500 400

-300 -200 Actual comp. stress (MPa) (a)

-100

RMSE-Kriging: 15.98 RMSE-IDW: 73.76 Test data: 16 Training data: 64

300 IDW Kriging Actual value

200 100 100

200 300 400 Actual tens. stress (MPa)

500

(b)

Predict Max. temperature (K)

2500

2

R = 0.998

2000

1500 RSM Actual value

1000

1000 1500 2000 Actual Max. temperature (K)

2500

(c) Figure 9. Accuracy of surrogate models with optimal Latin hypercube design: (a) compressive hoop stress; (b) tensile hoop stress; (c) maximum temperature. RMSE = root mean square error; IDW = inverse distance weighting.

5.1. Formulation of the DDO problem To perform the design optimization of the annular nuclear fuel gap size, the objective function, constraint function and design variables were determined through the formulation of the optimization problem following Equation (2). The objective function was applied to minimize the maximum temperature of the internal pellet in accordance with the gap size. A tensile–compressive strength less than

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150 MPa was applied to the constraint functions to satisfy a safety factor of 3 or more, considering the yield strength of the pellet as 460 MPa. In addition, the constraint condition was applied assuming that the summation of the inner and outer gap size is more than 0.1 mm, considering manufacturing constraints. The initial design variables (inner gap: x1 ; outer gap: x2 ) were set as [x1 , x2 ] = [0.5, 0.5] (mm), with 0.01 mm and 1 mm being the upper and lower bounds, respectively. The unit of the objective function is the absolute temperature in Kelvin (K). Minimize fDDO (x1 , x2 ) Maximum temperature (K)

(2)

Subject to gσc ≥ −150 MPa gσt ≤ 150 MPa gl = x1 + x2 ≥ 0.1 mm 0.01 ≤ x1 ≤ 1inner gap (mm) 0.01 ≤ x2 ≤ 1outer gap (mm) 5.2. Results of deterministic optimization In this study, the DDO of annular nuclear fuel gaps was performed. The MGA for global optimization was used to consider the nonlinearity of the surrogate model. The population number was 10 and the number of generations was 200. The optimal solutions obtained using the kriging and IDW models were [x1 = 0.072 mm, x2 = 0.03 mm] and [x1 = 0.027 mm, x2 = 0.072 mm], respectively. The deterministic optimal solution was verified by TEPC analysis, as shown in Table 2. It was found that the deterministic optimal solutions were different when applying the kriging and IDW models under identical initial values of the design variables. The temperature objective functions (fDDO−kriging , fDDO−IDW ) were minimized at the centre of the internal pellets. They decreased by 51.9% and 52.1%, respectively, compared with the initial values. On the other hand, the constraint functions of the maximum tensile–compressive hoop stress represented a lower absolute difference in the stress owing to offset effects in the IDW than in the kriging model. From the TEPC analysis, it was found that the maximum relative error between the approximate stress value and the actual tensile–compressive hoop stress obtained by the kriging model was 1.315%, but this result violates the tensile hoop stress constraint conditions (gσ t ≤ 150 MPa). On the other hand, the IDW model satisfied all the constraints with a maximum relative error of 2.630%. These results show that the optimal solutions were different according to the different surrogate modelling methods. Table 2. Results of deterministic design optimization (DDO) with kriging and inverse distance weighting (IDW) meta-models. Optimal solutions DDO x 1 (mm) x 2 (mm) fDDO = μ∗f (K) gσ c (MPa) gσ t (MPa) gl (mm)

TEPC analysis

Initial

Kriging

IDW

Kriging

IDW

0.500 0.500 1681 −143.5 142.4 1.000

0.072 0.030 807.3 −148.9 123.3 0.102

0.027 0.072 804.4 −128.8 123.5 0.099

0.072 0.030 805.0 −150.2 121.7 0.102

0.027 0.072 803.9 −125.5 123.9 0.099

Note: TEPC = thermoelastic–plasticity–creep.

Relative error (%) Kriging

IDW

–

–

– 0.290 0.870 1.315 –

– 0.062 2.630 0.323 –

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6. Design optimization considering uncertainties Based on the deterministic approximate optimization results, RBDO and RBRDO were performed (Youn and Xi 2009) considering the uncertainty of the nuclear fuel gap size. The performance index tends to be sensitive to the gap size. In particular, the robustness of the temperature objective function needs to be guaranteed and the target reliability for the stress constraint condition with high nonlinearity must be satisfied. To solve this problem, an optimal design problem of the nuclear fuel gap size is formulated. In addition, RBDO and RBRDO were performed using a single-loop, singlevector (SLSV) algorithm (Jeong and Park 2017) embedded in the commercial package PIAnO. The SLSV algorithm was proposed to reduce the computational cost (Chen, Hasselman, and Neill 1997). This method obtains a single optimization loop and a single vector of design variables using the sensitivity of a constraint in the previous design cycle. Hence, it has a lower computational cost than the double-loop approach. The SLSV method estimates the most probable point (MPP) for each active probabilistic constraint using gradient information from the previous cycle. This gradient vector is the steepest descent direction at the previous MPP (Jeong and Park 2017). In the RBDO, the optimal solution is compared by changing the RI assuming that the probability density function of the design variables has a normal distribution. In the RBRDO, the mean and deviation of the objective function are calculated by DDO. The obtained optimal solutions are verified by comparing them with the solutions obtained by the RI and surrogate models through the TEPC analysis. 6.1. Reliability-based design optimization The purpose of this study is to optimize the gap size to increase the heat transfer efficiency of annular nuclear fuel, to secure the robustness of the heat transfer efficiency and to satisfy the target reliability of the stress constraint conditions. Based on the RBDO, the optimal gap size satisfying the reliability of the constraint condition was calculated. Therefore, the RBDO formulation [Equation (3)] is used as a sequential process for performing RBRDO. A stochastic approach can be taken to the uncertainty of the structure. For this probabilistic approach, uncertainties that should be considered in the design will be treated as statistical random variables. At this point, the limit state that affects the performance of the structure is defined. Calculating the probability of failure and the RI is called reliability analysis. Methods of reliability analysis include reliability index analysis, performance measure approach (Tu, Choi, and Park 1999), Monte Carlo simulation and SLSV. In this study, to conduct the RBDO of the gap size in PWR annular nuclear fuels, the optimization problem was formulated as in Equation (3). The optimal solution calculated by DDO was applied to the mean and initial values of the RBDO. The probability density function of the design variables was assumed to have a normal distribution as Xi ∼ N(m∗f , σ Xi = 0.005), and the scale of the gap size was considered for assuming the deviation. In addition, RBDO was performed using the SLSV algorithm for reliability indices 2 and 3. These optimal solutions were verified by TEPC analysis. Minimize fRBDO (μX1 , μX2 ) Maximum temperature (K) Subject to P(gσc ≥ −150 MPa) ≥ (−β) = Rt P(gσt ≤ 150 MPa) ≥ (−β) = Rt P(gl = μX1 + μX2 ≥ 0.1 mm) ≥ (−β) = Rt 0.01 ≤ μX1 ≤ 1inner gap (mm) 0.01 ≤ μX2 ≤ 1outer gap (mm)

(3)

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6.2. Reliability-based robust design optimization The RBRDO was performed to secure the robustness of the objective function and satisfy the target reliability of the constraint functions. In this study, depending on the gap size, the maximum temperature of the internal pellet is represented by a tendency towards high sensitivity. To secure the robustness of the performance, using the objective function obtained by RSM following Equation (4), the mean (μ∗f ) and deviation (σf∗ ) of the objective function were defined using Equations (5) and (6) (Lee and Park 2001). f (X) = 711.273 + 969.243X1 + 905.997X2 − 768.481X12 − 591.539X22 + 1472.89X1 X2

(4)

μf = f (μX1 , μX2 )

(5)

  n   ∂f (X) 2 · σX2i σf ≈  ∂X i μ i=1

(6)

The RBRDO problem is formulated in Equation (7). The temperature objective function in RBRDO is obtained by the summation of the ratio with the deterministic mean and deviation. The optimum design is obtained by assigning the weight (w1 , w2 ) as 0.5 in the objective function. The optimal solutions were compared and verified by TEPC analysis. Minimize fRBRDO = w1

μf σf + w2 ∗ Maximum temperature (K) ∗ μf σf

(7)

w1 + w2 = 1 Subject to P(gσc ≥ −150 MPa) ≥ (−β) = Rt P(gσt ≤ 150 MPa) ≥ (−β) = Rt P(gl = μX1 + μX2 > 0.1 mm) ≥ (−β) = Rt 0.01 ≤ μX1 ≤ 1inner gap (mm) 0.01 ≤ μX2 ≤ 1outer gap (mm) 6.3. Comparisons of optimal solutions The results of Cases 3 and 4, obtained using the IDW method, were found to satisfy the constraint functions. On the other hand, in Case 2, the tensile hoop stress was found to be in violation of the constraint [P(gσ t ≤ 150 MPa) ≥ (−β)]. This shows that the interpolation value of the kriging model differs from the IDW model, and the nonlinearity of the constraint functions is high. In addition, the deterministic optimal solutions are different from each other, and the local optimal solution is calculated based on the initial value of the RBDO. The results are summarized in Table 3. The maximum relative error of IDW is 3.42% higher than in the kriging model because the interpolation value of the IDW is affected by the distance of the sample data. This is because kriging is based on probability whereas IDW is a deterministic method. Kriging associates some probability with each prediction; hence, it provides not just a surface, but some measure of the accuracy of that surface. In addition, kriging forms weights from surrounding measured values to predict values in the design space. As with IDW interpolation, the closest measured values usually have the most influence. IDW uses a simple algorithm based on distance, but kriging weights come from the variance of the difference

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Table 3. Results of reliability-based design optimization (RBDO) with kriging and inverse distance weighting (IDW) models. Optimal solutions of RBDO Kriging

RBDO RBDO μX 1 (mm) μX 2 (mm) fRBDO (K) gσ c (MPa) gσ t (MPa) gl (mm)

IDW

Case 1

Case 2

Case 3

Case 4

RI (β) = 2 σ X 1,X2 = 0.005 0.072 0.042 818.7 −147.2 110.5 0.114

RI (β) = 3 σ X 1,X2 = 0.005 0.017 0.104 818.1 −144.3 206.6 0.121

RI (β) = 2 σ X 1,X2 = 0.005 0.026 0.088 813.1 −132.0 141.2 0.114

RI (β) = 3 σ X 1,X2 = 0.005 0.045 0.076 824.2 −122.8 100.9 0.121

Note: RI = reliability index.

between predicted and actual values. Thus, the kriging weights for the surrounding sampling points are more sophisticated than those in IDW. In Case 3, the reasonable optimal solutions of the gap size were obtained as x1 = 0.026 mm and x2 = 0.088 mm. The temperature objective function was 815.3 K and was 51.4% less than the initial maximum temperature. Furthermore, the constraint functions were satisfied as the lowest absolute difference between the tensile and compressive hoop stresses was 13.5 MPa. It should be noted that the stress equilibrium of nuclear fuels is maintained by offset effects. Table 4 shows the relative error between the optimal design result and the verification solution of the performance function. The robustness is considered by using the mean and deviation of the temperature objective function, and the optimal design is obtained to satisfy the target RI of the constraint functions. The RBRDO solutions obtained using the kriging and IDW models satisfy all the constraints, but the maximum relative error of the IDW model is higher than that of the kriging model. In Cases 1 and 3, the optimal solutions were similar, [x1 = 0.072 mm, x2 = 0.042 mm] and [x1 = 0.071 mm, x2 = 0.044 mm], and the temperature objective functions were 51.3% less than the initial maximum temperature. The results are summarized in Table 5. The absolute difference between tensile hoop stress and compressive hoop stress is 38.7 MPa (IDW) and 40.2 MPa (kriging), respectively. If the absolute difference in stress is small, the stress equilibrium of nuclear fuels is maintained by offset effects, thus increasing the life cycle of the nuclear fuels. Table 6 lists the relative errors between the optimal design result and the verification solution of the performance function.

Table 4. Verification of reliability-based design optimization (RBDO) solution with thermoelastic–plasticity–creep (TEPC) analysis. TEPC analysis Kriging Performance functions μf (K) gσ c (MPa) gσ t (MPa)

Case 1 818.0 −148.6 109.7

Relative error (%) μf (K) gσ c (MPa) gσ t (MPa) Note: IDW = inverse distance weighting.

Case 1 0.086 0.942 0.729

IDW Case 2 816.6 −143.6 203.8 Case 2 0.184 0.487 1.374

Case 3 815.3 −132.7 146.2 Case 3 0.270 0.528 3.420

Case 4 827.2 −118.9 99.07 Case 4 0.363 3.280 1.847

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Table 5. Results of reliability-based robust design optimization (RBRDO) with kriging meta-models. Optimal solutions of RBRDO Kriging

IDW

Case 1 Case 2 μ∗f = 807.3, σf∗ = 6.651 RBRDO (w1 = 0.5, w2 = 0.5) μX 1 (mm) μX 2 (mm) fRBRDO μf (K) σ f (K) gσ c (MPa) gσ t (MPa) gl (mm)

RI (β) = 2 σ X 1,X2 = 0.005 0.072 0.042 1.008 818.7 6.659 −147.4 110.7 0.114

Case 3 Case 4 μ∗f = 804.4, σf∗ = 6.659

RI (β) = 3 σ X 1,X2 = 0.005

RI (β) = 2 σ X 1,X2 = 0.005

RI (β) = 3 σ X 1,X2 = 0.005

0.071 0.051 1.013 825.9 6.667 −143.4 102.3 0.122

0.071 0.044 1.009 817.9 6.660 −146.2 110.6 0.115

0.067 0.055 1.015 826.4 6.671 −141.6 102.4 0.122

Note: IDW = inverse distance weighting; RI = reliability index.

Table 6. Verification of reliability-based robust design optimization (RBRDO) with thermoelastic–plasticity–creep (TEPC) analysis. TEPC analysis Kriging Performance functions μf (K) gσ c (MPa) gσ t (MPa) Relative error (%) μf (K) gσ c (MPa) gσ t (MPa)

IDW

Case 1

Case 2

Case 3

Case 4

817.9 −148.6 109.9

826.1 −146.8 102.6

818.3 −148.1 107.9

828.0 −145.0 97.20

Case 1 0.098 0.808 0.728

Case 2 0.024 2.316 0.292

Case 3 0.049 1.283 2.441

Case 4 0.193 2.345 5.350

Note: IDW = inverse distance weighting.

7. Concluding remarks The approximate design optimization of gap sizes of PWR annular fuels was conducted considering uncertainties. The surrogate models of the objective and constraint functions were generated using the computational performance data based on OLHD. DDO of the gap size in PWR annular fuels was conducted using the MGA to improve the heat transfer efficiency and maintain a lower stress level. To consider uncertainties in the design variables, RBDO and RBRDO were also conducted. These optimum solutions were verified through TEPC analysis of PWR annular fuels. In this study, the temperature objective function was found through design optimization considering uncertainty and obtaining a conservative design about the objective and constraint functions. The obtained value was higher than that obtained using DDO. Moreover, the optimal solutions were different from those obtained in accordance with the approximate method. In addition, in the field of nuclear engineering, design optimization considering uncertainties contributes towards nuclear plant management and preventing accidents such as LOCA and RIA by performing the size optimization of nuclear fuels using the developed FE code. In future work, research into resilience-driven system design will be conducted, incorporating RBRDO with prognostics and health management (PHM) in the field of nuclear engineering design. The system design needs to sustain functionality by resisting (RBRDO) and recovering (PHM) uncertain events. The nuclear fuel system designed using RBRDO is also affected by downtime during its life. A resilience-driven system design study will be conducted to apply for maintenance and prediction of the remaining useful life in nuclear power plant operation.

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Disclosure statement No potential conflict of interest was reported by the authors.

Funding This research is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Science, ICT & Future Planning [grant number 2017R1A2B4009606]; and the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry & Energy (MOTIE) of the Republic of Korea [grant number 20163030024420].

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