Parametric evaluation of an SMR design domain

Annals of Nuclear Energy xxx (2015) xxx–xxx

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Parametric evaluation of an SMR design domain Ahmad Al Rashdan ⇑, Pavel Tsvetkov Department of Nuclear Engineering, Texas A&M University, 3133 TAMU, College Station, TX 77843, United States

a r t i c l e

i n f o

Article history: Received 11 February 2015 Received in revised form 6 July 2015 Accepted 7 July 2015 Available online xxxx Keywords: Nuclear reactor design SMR AP1000 Sensitivity analysis Factorial design Design of experiments

a b s t r a c t The complex and coupled behavior of the variables that affect the currently developing generation IV reactors and the increase of interest in Small Modular Reactors are major incentives to seek efficient design methods. Instead of the brute force design methodology, often applied to the analysis of nuclear reactors, other more economical and systematic methods should be applied. In a complex system with a high number of variables, it often is necessary to develop models to describe the behavior of each performance characteristic with respect to a set of variables and their interactions. If expensive experiments are needed to develop these models, the number of experiments should be minimized. This can be achieved by designing the experiments to be performed. Once the system’s models are developed, they can be used to decouple the variables’ effects on the performance characteristics and simplify the optimization process. In this article, three screening and sensitivity analysis methods are applied to decouple and understand the effects of fourteen variables on six performance characteristics of a Small Modular Reactor’s design, based on the Advanced Passive (AP-1000) reactor. The application of these methods facilitated the determination of the most important parameters of each performance characteristic, and resulted in several distinctive findings. These findings are applicable to any Light Water Small Modular Reactor that falls within the variables’ design domain. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The development of nuclear reactors to utilize the energy of the atomic fission for power generation was driven by simple yet innovative physical concepts. From the time of the first reactor, the Chicago Pile (Needell, 1983), to the currently developing Generation IV reactors (GIF, 2014), the complexity of reactors has increased and continues to increase rapidly. In the early stages of nuclear reactors’ development, the lack of computational power and, in some cases, the lack of sufficient knowledge of the physical behaviors of various aspects of the reactor prevented the generation of what today would be considered an optimal design. However, such reactors’ early designs still provide the basis for the current reactors, and similarities are apparent when comparing the current designs to older designs (e.g. Westinghouse, 2011; Schulz, 2006). The complex and coupled behavior of the variables that impact the currently developing Generation IV reactors and Small Modular Reactors (SMRs) (IAEA, 2012) are major incentives to seeking efficient design methods. The experience gained from the design and operation of earlier reactors is being reviewed and utilized in the ⇑ Corresponding author. Cell: +1 979 422 4264. E-mail address: [email protected] (A. Al Rashdan).

development of these new reactors (see examples from IAEA, 2007a; Konomura and Ichimiya, 2007). However, as new design domains are being explored, economical and systematic design methods are required. The application of modern optimization methods to various aspects of the reactors’ design has received an increased amount of attention in recent years (e.g. Espinosa-Paredes and Guzmán, 2011; Castillo et al., 2005; Hirano et al., 1997). Nevertheless, optimization methods often are not thoroughly described, a brute force approach is applied, or a comprehensive trial and error process is followed (e.g. Cole and Bonin, 2007; Fujimoto et al., 2004; Munkhbat and Obara, 2013; Paquette and Bonin, 2011; Kloosterman et al., 2001; Patel and Tsvetkov, 2013). An iterative response-driven design process is sometimes followed (e.g. Kugo et al., 1997). The coupling of variables in the design of nuclear reactors makes the efforts of trial and error very inefficient. An integrated approach that targets the coupling effect of several variables on several performance characteristics is not commonly applied to the design of nuclear reactors. A sample of the number of variables and performance characteristics involved in a typical reactor’s design is shown in Fig. 1, obtained from Al Rashdan (2014). These design variables, along with others (IAEA, 2007b), vary significantly from one reactor to another. If two levels of variation are set for each of the 19 shown

http://dx.doi.org/10.1016/j.anucene.2015.07.013 0306-4549/Ó 2015 Elsevier Ltd. All rights reserved.

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Nomenclature AP1000 x BOL Keff

aFT ad cm DKeff DBARs EOL FFD g IFBA K MWth

Advanced Passive 1000 reactor binary state of either 0 or 1 Beginning Of Life BOL of effective multiplication factor BOL fuel temperature reactivity coefficient BOL water density reactivity coefficient centimeter depletion drop of effective multiplication factor discrete burnable absorber rods end of life Fractional Factorial Design gram Integral Fuel Burnable Absorber kelvin MegaWatt thermal power

variables, 219 = 524,288 combinations are possible. If a mathematical model is desired to describe the behavior of a performance characteristic with respect to these variables and their interactions, all of these combinations must be performed. However, performing such a large number of experiments is not practical, especially when such experiments are expensive. In the design of nuclear reactors, experiments, including computational experiments, are often expensive. As a result, instead of dealing with a large number of variables, designers often shortlist variables, with or without a clear evaluation process. For example, the method of Phenomenon Identification and Ranking Tables (PIRTs), is a common, systematic shortlisting method based on experts’ evaluations (e.g. USNRC, 2004, 2008). It is also possible to complement experimentation efforts by using theory-based modeling (e.g. Palmiotti et al., 2009). If it is not possible to reduce the number of variables or experiments, an efficient method of design of experiments is usually applied. The design of experiments is a well-developed widely discussed subject in literature and books, and is used to address under-determined systems. A system is considered to be under-determined when the actual number of experiments performed, m, is smaller than the number of possible combinations n = 2N, where N is the number of variables. Over-determined systems include replicated experiments and apply regression methods (see Sheather, 2008 for details on regression methods). It is necessary for under-determined systems either to make valid assumptions to constrain the system’s excessive degree of freedom, or to use existing relevant knowledge of the system. Such assumptions are usually used in screening and sensitivity analysis methods. Screening and sensitivity analysis methods are evaluation methods used in the design of experiments. Screening is the science of locating important variables within a set containing a high number of variables. Sensitivity analyses explore the variables parametric models, focusing on exploring parameters rather than isolating important variables. This analysis uses three of these methods to decouple and understand the effects of fourteen variables on six performance characteristics of a Small Modular Reactor’s design that is based on the Advanced Passive (AP-1000) reactor (Westinghouse, 2011). The majority of applied screening and sensitivity analyses rely on orthogonal arrays, such as the Fractional Factorial Designs (FFDs) described in NIST (2003). Orthogonal arrays enforce an assumption regarding the significance of the interactions to a certain order. Since a designer cannot know if this assumption is valid, the norm is to seek a few orders of interactions and assume that the higher orders are negligible. This

MCNP MHIV MSIV N NRMSE b PPF RFS y

Monte Carlo N-Particle transport code Monotonic Highly Interacting Variables Monotonic Sparsely Interacting Variables number of variables Normal Root Mean Square Error parameter BOL power peaking factor BOL Ratio of Fast Spectrum performance characteristic or response q reactivity RMSE Root Mean Square Error SMR Small Modular Reactor Eth threshold energy LWSMR Light Water Small Modular Reactor x variable

is inefficient, because it assumes that the orders of interactions of all the variables are equally important, when in reality they are not. In addition, exploring very low orders of interactions results in inaccurate results. The main disadvantage of orthogonal arrays is that they do not provide information regarding whether the performed resolution is sufficient to produce an accurate model. As a result, the methods used in this analysis are the Monotonic Sparsely Interacting Variables (MSIV) and the Monotonic Highly Interacting Variables (MHIV) methods of Al Rashdan (2014), which are adaptive methods that overcome these limitations. The MSIV and MHIV methods are used to determine the level of exploration of interactions for each variable with respect to a defined threshold, and thus significantly improve the efficiency of experiments. The methods utilize various degrees of knowledge regarding the directions of variables’ effects on a response, and thus significantly simplify the designer’s input. The methods assume that the direction of the effects of the active variables tend to be monotonic. This regularity was seldom investigated in literature, and was proven to be valid when the system regularities of Bergquist et al. (2011) and Li et al. (2006) are present, as explained in Al Rashdan (2014). In addition, the methods evaluated the validity of this assumption at an early stage of their application. Both MSIV and MHIV align the variables’ effects on a performance characteristic in one direction. They begin by calculating the evaluation criteria, referred to as integrated odd or even parameters, in order to identify which variables are important, interacting, and monotonic. The integrated odd or even parameters are the summation of the odd or even order of parameters, which will be explained in a later section. If the variables were monotonic and highly interacting, the MHIV method was applied. If they were monotonic but weakly interacting, the MSIV method was applied. If they were not monotonic, the FFD was applied. The MHIV method breaks every group of interacting variables into smaller groups of non-interacting variables by recursively isolating the most interacting variables. For example, a group of seven highly interacting variables will be broken into two groups of six variables at the two states of the seventh (most interacting) variable. This process is repeated until all of the child groups contain no interacting variables. A first order linear model is then applied to each child group. The MSIV method separates the interacting and non-interacting variables into two groups. It applies any existing screening or sensitivity analysis method to the non-interacting variables. With regard to the interacting variables’ group, the method finds the integrated odd and even parameters for the next

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3

Fig. 1. Example variables and performance characteristics used in the design of nuclear reactors.

order of parameters. Ideally, if all the interacting variables only interact with other interacting variables, the summation of the next order of integrated odd parameters should equal the lower order of integrated even parameters. A significant difference between these two values for any variable indicates that the variable is cross-interacting with the non-interacting variables’ group. For every cross interacting variable, the non-interacting variable is spanned into two groups at the two states of this variable. If the MHIV and MSIV methods cannot be applied due to a weak monotonic behavior of variables, the widely used but less efficient FFDs are applied.

2. Design domain specifications The design specifications of the Light Water Small Modular Reactors (LWSMRs) are mostly based on the existing fleet of operating light water reactors. In this analysis, a LWSMR was designed based on the design specifications of the AP1000 with a proposed reduction of power to a range of 100–300 MWth. The scaling down of such a reactor is a complex problem influencing several performance characteristics. Designing a large set of performance characteristics to fall within a desired range is a significant challenge because such

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Table 1 Performance characteristics and their desired values.

Table 3 Number of particles in the experiments and order of magnitude of the errors for the six performance characteristics.

Performance characteristics

Desired values

BOL effective multiplication factor BOL flux spectrum in fuel and cladding BOL fuel temperature reactivity coefficient

Between 1.15 and 1.2 As thermal as possible Negative and high in magnitude Positive and high in magnitude As close as possible to one BOL Keff 1

BOL water density reactivity coefficient BOL power peaking factor Depletion drop of effective multiplication factor

characteristics interact with one another and often conflict. This study determined the behaviors of six performance characteristics with respect to fourteen variables in order to find the most important variables and interactions, and to develop a better understanding of the system’s behavior. The performance characteristics of interest were the Beginning Of Life (BOL) effective multiplication factor (Keff), BOL flux spectrum in the fuel and cladding (referred to here as RFS for Ratio of Fast Spectrum), BOL fuel temperature reactivity coefficient (aFT ), BOL water density reactivity coefficient (ad ), BOL power peaking factor (PPF) and depletion drop of the effective multiplication factor (DK eff ). The desired ranges of the performance characteristics are shown in Table 1. The variables of interest and their design limits are shown in Table 2. Since one of the main objectives of SMRs is to reduce the size and power of the core, these criteria were used in the definition of the design limits. For abbreviation purposes, the indices of the variables listed in Table 2 were used instead of the full variable names. The least demanding experiments model that can be used to evaluate a performance characteristic’s dependence on a set of variables is a linear model. In a linear model, the two end points

Table 2 Design variables and their defined ranges. Index Variables Core level 1 No of active core assemblies (cm) 2 Active fuel height (cm) 3 Core radial surrounding water (cm) 4 Enrichments (%) 5 6 7

Fuel pin level 11 IFBA and fuel pellet diameter (cm) 12 IFBA and fuel rod gap thickness (cm) 13 IFBA and fuel rod clad thickness (cm) 14 Discrete burnable absorber area (cm2) *

Max*

AP1000**

37

73

157

85.0 24.96

256.0 49.91

426.7 49.91

2.35, 3.4, 4.45 3.4, 4.45, 4.95 Power (MWth) 100 300 3 Moderator density (g/cm ) 0.716 (305C 0.813 (243C 15.5 MPa) 7.2 MPa) Fuel temperature (K) 600 900

Assembly level 8 No of rods per assembly 9 Rod pitch (cm) 10 No of DBARs and control rods per assembly

**

Min*

2.35, 3.4, 4.45 3415 0.716 (305C 15.5 MPa) 293.15 (cold)

11  11 1.051 9

13  13 1.260 21

17  17 1.260 25

0.680

0.819

0.819

0.0065

0.011

0.008

0.046

0.057

0.057

0.094

0.187

0.375

Rounded to the number of shown decimals. Based on actual and estimated values.

Performance characteristic

Keff

RFS

ad

aFT

PPF

DKeff

Number of particles in millions 2 2 2 32 4 1 Error order of magnitude* 0.000x 0.00x 0.000x 0.0001x 0.0x 0.000x *

x is a decimal number.

of each variable can be used to define the link between the performance characteristic and the variable. If the interactions of variables are to be considered, the model is only linear with respect to single variables. If all interactions are to be explored, an exponentially increasing number of experiments is needed. The model developed in this application is: E½yðx1 ; x2 ;x3 ; . .. :;xn Þ ¼ b0 þ

N X i¼1

bi xi þ

N X N N X N X N X X bij xi xj þ bijk xi xj xk þ . .. i¼1 j>i

i¼1 j>i k>i

ð1Þ

where y is the performance characteristic, x is the state of a variable, i, j, and k are variable indices, b is a parameter, and N is the number of variables. 3. Simulation model The tool used to simulate the reactor’s design was the Monte Carlo N-Particle transport code MCNP5 (X-5 Monte Carlo Team, 2003). The number of particles used for each Monte Carlo experiment depended on the sensitivity of the performance characteristic on the Monte Carlo error; these are shown in Table 3. The MCNP model was developed using the basic design structure of the AP-1000. The core layouts of the assemblies were determined by variable 1, and are shown in Appendix A. The assemblies’ layouts are determined by variables 8 and 10, and are as shown in Appendix B. The assemblies contained three types of uranium dioxide fuel material. Each had a different enrichment that is controlled by variable 4, and dimensions controlled by variables 11, 12 and 13. The materials used in the model are listed in Appendix C. In addition to the radial helium plenums and clad, the fuel was also axially surrounded by an axial clad and axial plenums proportional to the volume of the fuel element. The model also contained three types of Integral Fuel Burnable Absorber (IFBA) rods. Each had a different enrichment controlled by variable 4, and dimensions controlled by variables 11, 12, and 13. The IFBA had a thin absorber coating of boron and zirconium isotopes. The discrete burnable poison rod consisted of layers of absorber, stainless steel 304, helium and clad as shown in Fig. 2. The dimensions of the discrete burnable poison rod were controlled by variable 14. The main simplifications of the model can be summarized as:  The vessel was represented by a non-reflecting boundary. Any neutron traveling beyond the water radial boundary specified by variable 3 and the axial water boundaries of 1 m above and below the active core was terminated.  A homogeneous time dependent distribution of every material was assumed throughout the depletion cycles. The time steps of core depletion were set at once a year for five years. The first year was split into four durations of 90 days each.  Supporting structural elements such as brackets and instrumentation channels were assumed to be negligible, and therefor were not represented in the model.

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Absorber

Clad

SS304

Clad

Absorber

He

Fig. 2. Examples of two sizes of discrete burnable absorber areas.

4.1. Directional alignment of performance characteristic

Table 4 Definition of the density’s binary states for 0% and 50% voids. State/void (%)

0% void (reference case)

50% void at saturation

Density state low (g/cm3) Density state high (g/cm3)

0.716 (305C at 15.5 MPa)

0.34815 (50% void at 15.5 MPa)

0.813 (243C at 7.2 MPa)

0.38695 (50% void at 7.2 MPa)

Table 5 Definition of fuel temperature’s binary states for low and high temperatures. State/temperature

Low (reference case)

High

Temperature state low (K) Temperature state high (K)

600 900

2500 2500

This step aligned the low end of the performance characteristic with the potential saturation plateau, if one was expected to exist. This was applied to each performance characteristic as follows. Keff: If the nuclear reactor design fell into the under-moderated region, the effective multiplication factor was theoretically limited by a zero at its low end, and by the under-to-over moderation plateau at its high end. Since two plateaus limited Keff, its direction was unchanged. RFS: The neutron’s energy spectrum of light water reactors fell mostly in the thermal region. This implied that a saturation behavior could be expected as the spectrum was increasingly thermalized. Since the saturation plateau is to be aligned with the bottom end of the performance characteristic, the performance characteristic was defined as reversely proportional to the thermal end of the spectrum. Thus:

R Emax P4  While control rod thimbles were represented, the control rods themselves were not represented in the model. 4. Analysis and results This section applies the MSIV, MHIV, and FFDs methods. The following sections presents a step by step description of the MSIV and MHIV methods’ application.

E

m¼1

RFS ¼ R thE P4 th 0

m¼1

uðE; mÞdE

ð2Þ

uðE; mÞdE

where uðE; mÞ is the neutrons’ flux at energy E of the material m. The index m was used to find the sum of the spectrum of one cladding material and three fuel materials with different enrichments. Eth is the threshold energy used to separate the thermal and fast spectrums. A value of 0.05 eV was chosen. This value was found to maximize the sensitivity of the ratio on design changes. Higher

Table 6 Variables’ directions of effect.* Variable index

Variable

Keff

RFS

ad

aFT

PPF

DKeff

1 2 3 4 5 6 7 8 9 10 11 12 13 14

No of fuel assemblies Active fuel height Core radial surrounding water Enrichments Power Moderator density Fuel temperature lib No of rods per assembly Rod Pitch No of DBARs per assembly IFBA and fuel pellet diameter IFBA and fuel rod gap thickness IFBA and fuel rod clad thickness Discrete burnable absorber area

+ + + + = +  + +     

+ + + + =   +  + + + + +

    =  +   + ** + + +

    =   +  + + + + +

+ +  + = + +  +  +   +

   + + +   +     

(a) (b) (c) (d) (e)

* The positive sign corresponds to the alignment of the variable’s high state to 1 and low state to 0. The negative sign corresponds to the alignment of the high state to 0 and the low state to 1. The equal sign indicates that the variable state has no effect on the performance characteristic. ** Assumed to be negative, but found to be positive in the end result.

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Odd Parameters

Even Parameters

Odd Parameters

60

0.14

50

0.12

Even Parameters

0.1

40

0.08

30

0.06

20

0.04

10

0.02

0 5 3 7 c a 8 d 1 2 6 b 4 e 9

-10

0 5 e 3 7 c d 8 a 6 1 2 4 b 9

eff

Odd Parameters

Even Parameters

Odd Parameters

4000

800

3000

600

2000

400

1000

200

0

Even Parameters

0 5 7 c d 3 6 2 4 e a 9 8 1 b

-1000

5 2 1 3 a c 8 d 6 e 4 7 b 9 -200

αd Odd Parameters

Even Parameters

15

Odd Parameters

Even Parameters

60 50

10

40 30

5

20 10

0

0 5 6 c d 9 3 7 8 1 4 e 2 b a

-5

-10

3 c a d 7 e 6 8 4 1 2 5 9 b

-20 eff Fig. 3. Integrated odd and even parameters at the main level of the performance characteristics.

Table 7 Thresholds of important integrated even parameters. Performance characteristic Threshold

Keff 3

RFS 0.0027

ad

aFT

240

39

PPF 2

DKeff 8.5

Eth values produced very small ratios with magnitudes that are of the same order as the result’s error. Emax is the neutron’s maximum energy, and was set at 20 MeV. ad: Since water represents a medium for moderation, it was expected that if the reactor fell into the under-moderated region,

the reactivity would drop as the density is decreased. As ad approached zero, the effect of the density change became weak. Thus, it was expected that ad was limited by a plateau at its low end. Since the saturation plateau was to be aligned with the bottom end of the performance characteristic, the performance characteristic was defined as:

ad ¼ qref  q0:5Sat ¼

ðK eff ;ref  1Þ ðK eff ;0:5Sat  1Þ  K eff K eff ;0:5Sat

ð3Þ

where q is the reactivity. The range of ad that could be investigated was the full range of the density (0–100% void). However, since the

Table 8 Categorized effects of variables. Performance characteristic

Keff

RFS

ad

aFT

PPF

DKeff

Most important interacting variables Important interacting variables Non-interacting variables

49e 6b 12378acd

9b 12468ad 357ce

18b 2469ae 357cd

79b 46de 12358ac

2ab 134789cde 56

9b 124568a 37cd

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123456789abcde

469be (2nd order)

469 (Locally Interacting) 123578acd

be (Cross Interacting) e b Fig. 4. Flow chart of the recursive exploration of Keff using the MSIV method.

123456789abcde

128ad

9b=00

46=xx

9b=01

18ad

357ce

9b=10

18ad

246=xxx

246=xx

9b=11

8a

1246d

124=011

124=000

6d

124=001

6d

124=010

124=100

6d

124=101

6d

124=110

124=111

6d=xx

6d=xx

6d

6d

Fig. 5. Flow chart of the recursive exploration of RFS using the MHIV method.

uncertainty of this coefficient increases as the void increases (Jatuff et al., 2009), ad was investigated from a 0% void, representing normal operational state, to a 50% void of the saturation state at the same pressure. Since density was one of the fourteen variables analyzed in the LWSMR design, the 50% saturation density was different for each state. The densities used are summarized in Table 4. aFT: The main cause of the negative effect of temperature on is the Doppler broadening of the resonance region in the cross sections’ energy spectrum. Since the fuel used in the LWSMR was uranium oxide, thermal expansion and other phenomena that could cause temperature-related reactivity changes could be neglected because they were insignificant (Allison et al., 1993). The range of aFT in this example was expected to vary from around zero to a high negative value. As a result, aFT was expected to have a plateau at its high end of zero, and needed to be reversed. Since the saturation plateau was to be aligned with the low end of the performance characteristic, the performance characteristic was defined as: 







aFT ¼  q2500 K  qref ¼ qref  q2500 K ¼



   K eff ;ref  1 K eff ;2500 K  1  K eff K eff ;2500 K

ð4Þ

The low, or reference, end of the range of aFT was assumed to have a value between 600 K and 900 K. The high end was chosen based on the maximum temperature in the cross section libraries

in ENDF of MCNP. This was found to be 2500 K (X-5 Monte Carlo Team, 2003; NNDC, 2011). The fuel temperatures used are summarized in Table 5. PPF: As the power peaking was reduced, the PPF approached a value of one. This was considered to be a theoretical limit since only an infinitely large core could have a PPF value of one. On the other hand, as the core got smaller, it was expected that the PPF value would increase, due to a very steep power profile. Accordingly, the PPF value was limited by a plateau at its low end, which was aligned with the bottom end of the performance characteristic. Thus, the performance characteristic was defined as:

PPF ¼

Power Max Power Av g

ð5Þ

The core was split into 100 spatial elements consisting of ten axial elements and ten radial elements. The PPF was defined as the ratio of the power in the highest element to the average of all 100 elements. DKeff: Since the LWSMR was a thermal reactor and had a conversion ratio lower than one, DKeff could not be negative after five years of operation. DKeff was thus expected to have a plateau at one end and remain unbounded at the other. Since the saturation

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123456789abcde

18b=000

269ae

4=x

18b =100

18b =001

269e

4=x

18b =101

269e

4a=xx

4a=xx

6

18b =010

357cd

6

49ae

18b =011

26

49ae

4a =00

9e

4a =01

9e

4a =10

9e

4a=11

9e

18b =110

18b =111

4a=xx

49ae

4a =00

9e

4a =01

9e

4a =00

9e

4a =01

9e

4a =10

9e

4a=11

9e

4a =10

9e

4a=11

9e

Fig. 6. Flow chart of the recursive exploration of ad using the MHIV method.

plateau was to be aligned with the low end of the performance characteristic, the performance characteristic was defined as:

DK eff ¼ K eff ;BOL  K eff ;5y

ð6Þ

values of RFS, ad, aFT, PPF and DKeff occurred at the variables’ states of 1111  001011111, 0000  010010111, 0000  001011111, 1101  110101001, and 00011100100000 respectively.

4.3. Determination of integrated parameters of variables 4.2. Directional alignment of variables This step represents the designer’s input into the evaluation process. If the variables were aligned in one direction, it was expected that the regularities studied in Bergquist et al., (2011) and Li et al., (2006) would reduce the confounded interactions, as was explained in Al Rashdan (2014). The variables’ directions of effect on the performance characteristics are shown in Table 6, and are justified for all performance characteristics in Appendix D. The unknown directions of effect were found using low accuracy Monte Carlo experiments. According to the table, the maximum Keff occurred at the variables’ state of 1111  101100000. Thus, the variables were aligned to all be high at this combination for Keff. If a variable was found in the desired direction of effect, its range definition was sustained. On the other hand, if a variable’s direction of effect was to be reversed, the range definition of the variable was set to a high state at the initial 0 state. For example, the high state of variable 1 on Keff remained at its initial state 1, but the high state of variable 7 on Keff was set to the state of 0. A variable with an x effect had no effect, so it could be defined in either direction. The alignment was applied to all performance characteristics. The maximum

In this step, the first order integrated odd and even parameters were found using the definitions provided in Cotter (1979) as explained in Al Rashdan (2014). The first order integrated odd parameters are the sums of all the odd parameters of Eq. (1), (i.e., bi, bijk, bijkl, etc.), and the first order integrated even parameters are the sums of all the even parameters (i.e. bij, bijkl, etc.). The alignment of the variables in the previous step caused the signs of all the parameters to be biased in the positive or negative direction, thereby reducing the probability of parameters canceling or confounding each other when summed. Due to the nature of the hierarchy of the parameters, the first order integrated odd parameters were mostly based on the first order parameter, bi, while the integrated even parameters were mostly based on the second order parameter, bij. Thus, the integrated odd parameters were used as an indication of the variable’s importance, while the integrated even parameters were an indication of the variable’s degree of interaction. Monotonic variables had a magnitude of integrated even parameters that was smaller than that of the integrated odd parameters. The degree of the variables monotonic behavior was dependent upon the difference between these two values (i.e. bij  bi for highly monotonic variables, bij < bi for monotonic

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123456789abcde

9b=00

14568

9b=01

168

37cd

9b=10

2a=xx

245a

1245a

2a =00

2a =01

25 =00

45=xx

45=xx

1a=xx

2a=11

25 =10

45=xx

4=x

2a =10

4

5=x

125=000

68

9b=11

68

1245a

4

25 =01

1a

4=x

1a

25=11

14

a=x

4

125=001

4a

125=010

4a

125=011

4a

4

125=101

4a

125=110

4a

125=111

4

a=x

125=100

a=x

a=x Fig. 7. Flow chart of the recursive exploration of DKeff using the MHIV method.

123456789abcde

12358ac

79b=000

46de

79b =001

46de

79b =010

46de

79b =011

46de

79b =000

46de

79b =001

46de

79b =010

46de

79b =011

46de

Fig. 8. Flow chart of the recursive exploration aFT of using the MHIV method.

variables, bij > bi for non-monotonic variables, and bij  bi for highly non-monotonic variables). The integrated parameters of the six performance characteristics were sorted according to the strength of the integrated even parameters and are presented in Fig. 3. The units used were 103 for Keff and DKeff, pcm for ad and, 100 for RFS, and 102 for PPF.

4.4. Categorization of variables In Fig. 3, the integrated odd and even parameters of every variable are compared to a threshold that is set according to the desired number of experiments, and the strength of integrated odd and even parameters. The selected thresholds of important

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A. Al Rashdan, P. Tsvetkov / Annals of Nuclear Energy xxx (2015) xxx–xxx Table 9 Evaluation of second order interactions of 469be of Keff. Variable 1st order integrated even parameters

2nd order summation of integrated odd parameters

Difference

4 6 9 b e

8.2188 2.3363 17.4438 14.6000 1.2088

0.4463 1.7338 1.8763 8.0200 8.3163

7.7725 4.0700 19.3200 6.6150 9.5250

Table 10 Heat map of the percentage of main effects on the six performance characteristics.

Index '1' '2' '3' '4' '5' '6' '7' '8' '9' 'a' 'b' 'c' 'd' 'e'

Variable

eff

No of Active Core Assemblies Active Fuel Height Core Radial Surrounding Water Enrichments Power Moderator Density Fuel Temperature No of Rods per Assembly Rod Pitch No of DBARs per Assembly IFBA and Fuel Pellet Diameter IFBA and Fuel Gap Thickness IFBA and Fuel Clad Thickness Discrete Burnable Abs. Area

8.25 8.66 0.00 15.10 0.00 6.12 -1.47 9.17 23.79 -11.42 -8.76 -0.44 -1.19 -7.80

integrated even parameters are shown in Table 7. According to the defined thresholds, the variables were categorized into important interacting, important non-interacting and non-important variables. The variables’ categorized effects on each performance characteristic are shown in Table 8. 4.5. Variables interactions exploration From Fig. 3, it is clear that the integrated even parameters were below the integrated odd parameters for the five performance characteristics. This implied that the directions of effect of the variables on these performance characteristics were monotonic. Since most of the variables affecting Keff were non-interacting, the MSIV method was applied. The remaining monotonic performance characteristics RFS, ad, aFT, and DKeff were analyzed using the MHIV method. The PPF values showed a strong non-monotonic behavior, and thus a 256 resolution V FFD was used. The details of the adaptive methods’ application can be found in Al Rashdan (2014), and are summarized using the exploration flow

eff

1.69 3.13 0.38 11.53 0.00 -5.24 -0.44 2.38 -29.95 -4.28 28.51 0.86 0.97 1.45

-16.75 -15.52 -1.06 -4.86 0.01 -4.88 1.44 -8.78 -24.03 2.71 8.12 0.57 2.08 3.15

-1.10 -1.51 -0.39 -5.97 0.01 -6.37 -17.63 1.14 -33.23 -1.12 22.45 0.68 2.32 3.43

7.17 8.35 -0.32 10.96 0.00 3.31 -0.20 7.98 12.08 -6.70 -3.59 -0.22 -0.88 -6.07

-10.18 -17.13 -0.06 7.28 17.76 3.24 -0.73 -3.44 16.41 -3.59 -17.03 -0.08 -0.64 -2.81

charts in Figs. 4–8. Fig. 4 demonstrates the application of the MSIV on Keff. In this application, variables 469be were categorized as interacting, and variables 123578acd as non-interacting. As a result, the second order integrated odd and even parameters of 469be were found. The second or higher order integrated odd and even parameters followed the same architecture as the first order parameters, but with the odd parameters beginning from the desired order. For example, the second order integrated odd parameters could be found from the sum of all the odd parameters of Eq. (1) with respect to the starting second order (i.e., bij, bijkl, etc.), and the second order integrated even parameters could be found from the sum of all the parameters with respect to the starting second order (i.e. bijk, bijklm, etc.). The summation of the second order integrated odd parameters of every interacting variable was compared to the first order integrated even parameters. If the interacting variables only interacted with one another, these values should have been identical. However, the results shown in Table 9 demonstrated that variables b and e cross interacted with the non-interacting variables. This implied that some of the

Table 11 Categorized main effects on the six performance characteristics. Index

Variable

Keff

RFS

ad

aFT

PPF

DKeff

‘1’ ‘2’ ‘3’ ‘4’ ‘5’ ‘6’ ‘7’ ‘8’ ‘9’ ‘a’ ‘b’ ‘c’ ‘d’ ‘e’

No of active core assemblies Active fuel height Core radial surrounding water Enrichments Power Moderator density Fuel temperature No of rods per assembly Rod pitch No of DBARs per assembly IFBA and fuel pellet diameter IFBA and fuel gap thickness IFBA and fuel clad thickness Discrete burnable abs. area

M M

W W

W W

M M

M S

S

M

S S W W

M

M

M W M VS M M

M

M S W VS W VS

W

W VS W VS

W W M VS W M

M S W

M M M W

W S W S

W M

W

W W

W W

M

W

VS, Very Strong (>20); S, Strong (15–20); M, Moderate (5–15); W, Weak (1–5); Blank Very Weak (<1).

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A. Al Rashdan, P. Tsvetkov / Annals of Nuclear Energy xxx (2015) xxx–xxx

11

Table 12 Heat map of the most important interactions of the six performance characteristics by percentage.

Parameter Index '9b' '2a' '18' '8a' '79' '25' '59' 'ae' '29' '2e' '2b' '7b' '259' '28' '125b' '1259b' '49' '4b' '8b' '14' 'ab' '24' '19' '125' '18a' '1259' '29b' '1b' '5b' '12' '59b' '129' '15' '159' '19b' '45' '9a' '89' '189' '6b' 'be' '169' '69' '25b' '7e' '46' '2ae' '1ae' '68' '7d' '8e' '4a' '16' '7bc' '28b' '9ab' '28a' '18b' '1d' '9e'

eff

7.04 0.00 -1.20 0.00 0.00 0.00 0.00 -2.39 0.00 -0.04 -0.02 -0.20 0.00 0.00 0.00 0.00 1.21 -1.79 -1.00 0.00 2.34 0.00 0.00 0.00 0.00 0.00 0.00 -0.12 0.00 1.08 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.42 0.49 0.00 -1.37 0.00 -0.01 0.16 0.00 0.05 0.00 0.00 0.71 0.00 0.00 0.00 0.00 0.00 0.00 0.09 0.01 1.03

eff

-9.52 -0.01 0.00 0.00 0.00 0.01 0.00 0.00 -1.68 0.01 1.66 0.00 -0.01 0.01 0.00 0.00 -2.40 2.36 1.23 0.18 -1.87 0.57 -1.38 0.00 0.00 0.00 -0.98 0.97 0.00 0.05 0.00 -0.04 0.01 0.00 -0.68 0.01 1.65 -1.16 0.00 -0.93 0.00 0.18 0.93 0.01 0.00 -0.37 0.00 0.00 -0.01 -0.01 0.00 -0.01 -0.18 0.00 0.01 0.87 0.00 0.00 0.07 0.00

variables assumed to be non-interacting were actually interacting, and the confounded interactions of these variables caused them to be falsely evaluated. To find these confounded interactions, the non-interacting variables were sub-grouped at the states of

2.42 -0.01 2.29 -0.53 -0.01 -0.01 -0.01 0.77 0.00 -0.01 1.33 0.01 0.00 0.76 0.00 0.00 0.76 -1.55 1.72 0.51 -1.38 0.02 0.95 0.00 -0.04 0.00 0.00 1.39 0.01 0.87 -0.01 0.00 -0.01 0.00 -0.56 -0.03 -0.37 0.72 -0.11 1.52 -1.46 0.00 0.00 -0.01 0.01 0.02 0.00 -0.22 0.34 0.01 -0.53 -0.22 0.86 0.01 0.03 0.35 0.00 -0.13 -0.01 0.00

-6.73 0.01 0.01 0.00 4.05 -0.01 0.00 0.00 0.00 0.00 0.01 -2.81 0.00 0.01 0.01 0.00 1.16 -1.07 -0.01 0.00 -0.01 0.00 0.00 0.01 0.01 0.00 0.00 0.01 0.01 -0.01 0.00 0.00 -0.01 0.00 0.00 0.00 0.00 0.00 0.00 -0.89 0.67 0.00 0.56 0.00 -0.42 0.00 0.01 0.01 0.00 -0.24 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 -0.93

3.08 -5.73 -4.65 4.44 0.10 0.00 0.00 -3.80 -1.86 -3.07 3.00 -0.12 0.00 2.55 0.00 0.00 1.66 -0.54 -0.82 2.34 1.60 2.28 -2.18 0.00 -2.08 0.00 -0.40 0.32 0.00 0.99 0.00 -0.54 0.00 0.00 0.25 0.00 -0.34 0.53 -1.53 1.04 -0.26 -1.37 -0.52 0.00 1.33 1.31 -1.29 -1.26 1.26 1.26 1.24 1.21 -1.16 -1.15 -1.14 -1.14 1.12 1.11 -1.09 0.63

-1.52 -0.42 0.00 -0.03 0.01 4.03 3.97 -0.03 -3.47 -0.02 1.61 -0.02 2.70 -0.02 -2.43 -2.42 0.43 -1.07 2.35 -0.20 0.82 -1.59 -1.58 2.10 0.00 2.07 -2.05 1.96 -1.94 -1.91 1.88 -1.88 1.81 1.78 -1.73 1.67 -0.23 -1.59 0.01 -0.56 -0.02 -0.01 0.16 -1.36 -0.01 0.01 -0.01 -0.01 0.00 -0.01 0.00 0.10 0.00 0.00 0.00 0.04 -0.01 0.00 0.00 0.01

variables b and e. The subgroups of the non-interacting variables at be = 11, be = 01, be = 10 and be = 00 were evaluated using FFDs, and the FFD models were used to project the performance characteristics for the complete domain of the variables. In the

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A. Al Rashdan, P. Tsvetkov / Annals of Nuclear Energy xxx (2015) xxx–xxx

Table 13 Categorized most important interactions of the six performance characteristics. Parameter Index

Keff

RFS

ad

aFT

PPF

DKeff

‘9b’ ‘2a’ ‘18’ ‘8a’ ‘79’ ‘25’ ‘59’ ‘ae’ ‘29’ ‘2e’ ‘2b’ ‘7b’ ‘259’ ‘28’ ‘125b’ ‘1259b’ ‘49’ ‘4b’ ‘8b’ ‘14’ ‘ab’ ‘24’ ‘19’ ‘125’ ‘18a’ ‘1259’ ‘29b’ ‘1b’ ‘5b’ ‘12’ ‘59b’ ‘129’ ‘15’ ‘159’ ‘19b’ ‘45’ ‘9a’ ‘89’ ‘189’ ‘6b’ ‘be’ ‘169’ ‘69’ ‘25b’ ‘7e’ ‘46’ ‘2ae’ ‘1ae’ ‘68’ ‘7d’ ‘8e’ ‘4a’ ‘16’ ‘7bc’ ‘28b’ ‘9ab’ ‘28a’ ‘18b’ ‘1d’ ‘9e’

M

M

W

M

W M W W

W

W

W W

W W W

W W W W

W W

W

W W

W W W W W W W W

W W W

W W

W

W

W

W W

W W W W W W W

W

W W W

W W W W W W W W W W W W

W W

W W

W W W

W W

W W W W W W W W W W W W W W W W W W W

VS, Very Strong (>20); S, Strong (15–20); M, Moderate (5–15); W, Weak (1–5), Blank Very Weak (<1).

Table 14 Summary of applied methods and accuracy of results.

ad

Performance characteristic

Keff

RFS

Applied method

98 MSIV 256 FFD 9.747 1.98

130 168 MHIV MHIV 128 FFD 16 FFD

Evaluation set RMSE NRMSE (%)

0.008032 462.3 1.19 1.85

aFT

PPF

90 256 FFD MHIV 16 FFD 292 MHV 67.4 6.52 1.67 4.74

DKeff 154 MHIV 16 FFD 24.4 3.67

Table 15 Number and order of parameters for the highest thirty parameters of the six performance characteristics. Parameter order

Keff

RFS

ad

aFT

PPF

DKeff

1st 2nd 3rd 4th 5th

12 13 3 1 1

11 15 4 0 0

12 16 2 0 0

13 14 3 0 0

9 16 5 0 0

10 10 7 2 1

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A. Al Rashdan, P. Tsvetkov / Annals of Nuclear Energy xxx (2015) xxx–xxx

Variable 1=0 W

W

W

W

W

W

W

W

W

W

W

W

UH2

UH1

UH2

W

W

W

W

W

UL2

UM2

UL1

UM2

UL2

W

W

W

UH2

UM2

UL1

UM1

UL1

UM2

UH2

W

W

UH1

UL1

UM1

UL1

UM1

UL1

UH1

W

W

UH2

UM2

UL1

UM1

UL1

UM2

UH2

W

W

W

UL2

UM2

UL1

UM2

UL2

W

W

W

W

W

UH2

UH1

UH2

W

W

W

W

W

W

W

W

W

W

W

W

Variable 1=1 W

W

W

W

W

W

W

W

W

W

W

W

W

W

W

W

W

W

W

UH1

W

W

W

W

W

W

W

W

W

W

UH2

UH2

UL1

UH2

UH2

W

W

W

W

W

W

W

UH2

UM2

UL1

UM1

UL1

UM2

UH2

W

W

W

W

W

UH2

UM2

UL1

UM1

UL1

UM1

UL1

UM2

UH2

W

W

W

W

UH2

UL1

UM1

UL1

UM1

UL1

UM1

UL1

UH2

W

W

W

UH1

UL1

UM1

UL1

UM1

UL1

UM1

UL1

UM1

UL1

UH1

W

W

W

UH2

UL1

UM1

UL1

UM1

UL1

UM1

UL1

UH2

W

W

W

W

UH2

UM2

UL1

UM1

UL1

UM1

UL1

UM2

UH2

W

W

W

W

W

UH2

UM2

UL1

UM1

UL1

UM2

UH2

W

W

W

W

W

W

W

UH2

UH2

UL1

UH2

UH2

W

W

W

W

W

W

W

W

W

W

UH1

W

W

W

W

W

W

W

W

W

W

W

W

W

W

W

W

W

W

W

UL1

Uranium at Low Enrichment Type 1

UH1

Uranium at High Enrichment Type 1

UL2

Uranium at Low Enrichment Type 2

UH2

Uranium at High Enrichment Type 2

UM1

Uranium at Medium Enrichment Type 1

W

Water

UM2

Uranium at Medium Enrichment Type 2 Fig. 9. Core layouts.

experiments’ setup language, this is equivalent to setting variables b and e at each combination and performing an FFD on the non-interacting variables. Figs. 5–8 demonstrate applying the MHIV method to four performance characteristics. The application in Fig. 5 will be explained in this section for illustration. From the previous evaluation of the integrated odd and even parameters of RFS, it was found that the 9 and b variables interacted the most. The variables highlighted in grey indicate the isolated variables at every level of the exploration, which were 357ce at this level. Four subgroups were established for all the possible combinations of the 9b variables. In the experiments setup language, this is equivalent to setting the 9 and b variables at each combination, and targeting the other interacting variables at that combination. In each of these subgroups the first order integrated odd and even parameters of the other interacting 12468ad variables were found. The 46 variables were found to be interacting within subgroup 9b = 00, and the 128ad variables were found to be non-interacting. As a result, four

subgroups representing the states of the 46 variables were created in subgroup 9b = 00. In each of the 46 variables’ subgroups, no interacting variables existed and a first order model of parameters could be built using only the already-found integrated odd parameters of all the variables. The same recursive spanning was applied to the remaining subgroups of 9b. 4.6. Projection and generation of results After a model was developed for every level of the subgroups, the performance characteristics were projected at all levels, and the global model of Eq. (1) was determined. The percentage of Main Effects (MEs) of the fourteen variables on the six performance characteristics are shown in Table 10. The percentage ME was calculated using:

MEð%Þ ¼ 100

2bi ðyMax  yMin Þ

ð7Þ

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A. Al Rashdan, P. Tsvetkov / Annals of Nuclear Energy xxx (2015) xxx–xxx

F F F F F F F F F F F F F

F F F F F F F F F F F F F

F F F F

F F F F

Variable 8=0 Variable 10=0 F F F F F F F F F F F F F F F F F C F F F F C F F F C F

F F F F

F F F F

F F F F

F F F C

F F F F

Variable 8=0 Variable 10=1 F F F F F F F C F C F C F F F F F F F F F C F C F C F C

F F F F

F F

F F

F C

F F

F F

F C

F F

F F

F C

F F

F F

F F

F C

F F

F C

F F

F C

F F

F C

F F

F C

F F

F F F F

F F F F

F F F F

F C F F

F F F F

F F C F

F F F F

F C F F

F F F F

F F F F

F F F F

F F F F

F C F F

F F F F

F C F C

F F F F

F C F C

F F F F

F C F C

F F F F

F C F F

F F F F

F

F F F F F F F Variable 8=1 Variable 10=0 F F F F F F F F F F F F F F F F F C F F F C F F F F F C F F F F F F F F F F F F F F F F F C F F F F F F F F F F F F F F F F F C F F F F F C F F F C F F F F F F F F F F F F F F F F F

F

F

F

F

F

F

F

F F F F F F C F F F F F F

F F F F F F F F F F F F F

F F F F F F F F F F F F F

F F F F F F F F F F F F F

F F F F C F F F C F F F F

F F C F F F F F F F C F F

F F F F F F C F F F F F F

F F F F F F C F F F F F F

F C

F F F F F F F Variable 8=1 Variable 10=1 F F F F F F F C F F F C F F F F F F F F C F F C F F F F C F F F C F F F F F F F F F F F C F F C F F F F F F F F C F F F C F F F F C F F F F F F F F F F C C F F F C F F F F F F F F F

F F F F C F F F C F F F F

F F F F F F F F F F F F F

Fuel Rod at Low Enrichment Control Rod Thimble Fig. 10. Assembly layouts of uranium at low enrichment type 1 (UL1).

I I

I F

Variable 8=0 Variable 10=0 F F F F F F F I F F F F F F F F

I I

F F

F F

F F

Variable 8=0 Variable 10=1 F F F F F F F C I C I C F F

F F

F F F

F F F

F F F

F C F

F F I

C F F

F F I

F C F

F F F

F F F

F F F

F F F

F C I

F F F

F C F

F F F

F C F

F F F

F C F

F F F

F C I

F F F

F F F F

F F F F

C F F F

F F C F

F I F F

C F F C

F I F F

F F C F

C F F F

F F F F

F F F F

F F F F

C I C F

F F F F

C F C F

F F F F

C F C F

F F F F

C F C F

F F F F

C I C F

F F F F

I I

F F

F I

I I

F F

F F

F F

C F

F F F F F F C F

I F F F F F F F

I I F F F F F F

I F F F F F F F

F F F F C F F F

F F C F F F I F

F F F I F F C F

F F F F F

F F F F I

F F F I I

F F F F I

C F F F F

F F C F F

F I F F F

I I F F F F F F

I F F F F F F F

F F F F F F C F

F F F F F F F I F F F F F F Variable 8=1 Variable 10=0 F F F F F F F F F F F F F F F F F C F F F C F F F F F C F I F I F I F F F F F F F F F I F C F I F F F F F F F F

F F F I I

F F F F I

F F F F F

F C F F F

I F F F F

F F F F F

I F C F F

F F F F F

I F F F F

F C F F F

F

Fuel Rod at Low Enrichment

I C

IFBA Rod at Low Enrichment Control Rod Thimble

I C I C F F F F F F F F F F Variable 8=1 Variable 10=1 F F F F F F F C F F F C F F F F I F F F C F F C F F I F C F F F C F F F F F F F F F F F C F F C I F F F F F F F C F F C F

F F F F F

F C I F F

F F F F F

C F F C F

F I F F F

F F C F F

F F F F C F F F

I F F F F F F F

C F F F F

F F F F I

Fig. 11. Assembly layouts of uranium at low enrichment type 2 (UL2).

Please cite this article in press as: Al Rashdan, A., Tsvetkov, P. Parametric evaluation of an SMR design domain. Ann. Nucl. Energy (2015), http://dx.doi.org/ 10.1016/j.anucene.2015.07.013

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A. Al Rashdan, P. Tsvetkov / Annals of Nuclear Energy xxx (2015) xxx–xxx

which is a measurement of the effect of the variable on the performance characteristic, as compared to the span of the projected performance characteristic. For example the ME value of 8.25% implies that the effect of changing the state of variable 1, which is the number of active core assemblies, from the low state to the high state listed in Table 2 would result in a 8.25% increase of Keff, in addition to the effect of the interactions. These values were categorized into strengths of effect and are listed in Table 11. The categorization thresholds are presented in the table’s legend. According to the categorization, the 8.25% effect of the number of active core assemblies on Keff represents a medium strength of effect, and thus is represented with M for Medium. The same was applied to the most significant interactions, which are shown in Tables 12 and 13. For example, the effect of interaction 9b, which represents the rod pitch by the pellet diameter resulted in a 7.045% increase of Keff if either of the variables changed from a low to a high state. This was in addition to the effect of the main effects and other interactions.

4.7. Validation This step describes the evaluation of the projected results with respect to the validation set. In order to ensure that the validation set was as spread out as possible throughout the domain of experiments, the validation set selected was an FFD set of experiments. The validation was performed by comparing each method’s projected results with the experiments performed via the other method, using the Root Mean Square Error (RMSE):

I I F F F F F F F F F I I

I F F F F F I F F F F F I

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN m 2 ^ i¼1 ðyi  yi Þ RMSE ¼ Nm

ð8Þ

^i is the projected perwhere Nm is the number of validation points, y formance characteristic, and yi is the performed performance char^i . The number of acteristic at the same state of variables as y experiments, method, evaluation set of experiments and error are presented in Table 14. The table shows that the Normalized Root Mean Square Error (NRMSE) fell below 5% for all of the performance characteristics, and below 2% for four of them. 5. Discussion of results From the number and order of the highest thirty parameters affecting the performance characteristics listed in Table 15, it is evident that interactions play a significant role in the design of nuclear reactors. The LWSMR design required an exploration of various depths of interactions depending upon the considered performance characteristic; however required a minimum level of exploration of third order interactions in all performance characteristics. The weak sparsity and relatively small magnitude of the first order parameters indicated that neglecting the non-linearity of single variables did not produce a significant error. It is also evident from the results that as the size of the LWSMR was varied, the directions of the variables’ effects on the performance characteristics established a balance. For example, the response of Keff to variables 1 and 2 could be balanced by variables b and a. This balance enabled the designer to easily

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F F F F I F F I F F F F F F Variable 8=1 Variable 10=0 F F F F F F F F F F I F F F F I I B I I F B I F F F I B I F F I F F I F F F I F F F F I I C I I F F F F I F F F I F F I F F I B I F F F I B F I I B I I F F F F I F F F F F F F F F F

F F I F I I B I I F I F F

F B I I I B F

I I F I F I I

B I I C I I B

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F B I I I B F

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Fuel Rod at Medium Enrichment

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IFBA Rod at Medium Enrichment Discrete Burnable Absorber Control Rod Thimble

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F B F B F F I I F I F F I I Variable 8=1 Variable 10=1 F F I F F F F B I F I B I F F F F F F I B I F B F I F I B F I F B I F F I F I F F F I F C F I B F F I F I F F F B F I F B I F I F B F I F I F F F F F I B B I F I B I F F F I F F F F

I F F I B I F I B I F F I

I I F F F F I F F F F I I

Fig. 12. Assembly layouts of uranium at medium enrichment type 1 (UM1).

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A. Al Rashdan, P. Tsvetkov / Annals of Nuclear Energy xxx (2015) xxx–xxx

evaluate and mitigate the consequences of changing a variable at an early stage of the design. The results reveal that the variables’ monotonic behavior is a significantly present physical behavior in the design of nuclear reactors. In this study, it was used in the analysis of five of the six performance characteristics, and resulted in a relatively accurate model for each, after a relatively small number of experiments. The resulting models can now be used to project the six performance characteristics of any design with the variables falling within the defined domain. The projection’s error is known from the validation step. If validation is not performed, the error order of magnitude can instead be estimated from the defined threshold or the applied method’s specific error. The dependence of the performance characteristics on the variables can be summarized as:

K eff ¼ f ð1; 2; 4; 6; 8; 9; a; b; e; Þ

ð9Þ

RFS ¼ f ð4; 6; 9; bÞ

ð10Þ

ad ¼ f ð1; 2; 8; 9; bÞ

ð11Þ

aFT ¼ f ð4; 6; 7; 9; bÞ

ð12Þ

PPF ¼ f ð1; 2; 4; 8; 9; a; eÞ

ð13Þ

DK eff ¼ f ð1; 2; 4; 5; 9; bÞ

ð14Þ

I I F F F F F F F F F I I

I F F F F F I F F F F F I

These equations decouple the variables’ effects, and should be used to simplify optimization efforts. For example, RFS was found to depend upon four variables instead of fourteen variables, which is an enormous reduction in the number of optimization variables. Variables 3, c and d, representing radial surrounding water, gap thickness, and clad thickness, were found to have weak or very weak effects on all performance characteristics, and thereby could be omitted from the list of variables in the optimization process. The weak effect of the radial surrounding water indicates that it is sufficient to sustain a good neutrons economy and small fluence on the vessel. Since the current system has fourteen variables and six performance characteristics, it is currently underdetermined, and additional performance characteristics could be introduced. However, since variables 3, c and d were found to be insignificant for any of the six performance characteristics, the current actual degree of freedom of the system is 14-6-3 = 5. The found parameters indicate that variable 9, representing the rod pitch, was the most important variable, since it significantly affected all of the considered performance characteristics. In order to change its effect, it is possible to use b, representing pellet diameter, as it will increase the effect of 9 in three performance characteristics while reducing the effect of 9 in the other three. It is also possible to change the effect of 9 on specific performance characteristics (i.e., DKeff by using 59, and aFT by using 79). Variable b, representing pellet diameter, was the second most important variable. Its effect was moderately coupled with 9, representing the rod pitch, and weakly coupled with other variables. Interaction

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F F F F I F F I F F F F F F Variable 8=1 Variable 10=0 F F F F F F F F F F I F F F F F F B F F F B I F F F I B I F F F F F I F F F F F F F F F F C F F F F F F F F F F I F F F F F I B I F F F I B F F F B F F F F F F I F F F F F F F F F F

F F F F F F B F F F F F F

F B I F I B F

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F I F F F I F

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Fuel Rod at Medium Enrichment

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IFBA Rod at Medium Enrichment Discrete Burnable Absorber Control Rod Thimble

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F F B I F F I F F I B F F

F F I F F F B F F F I F F

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F B F B F F F F F F I F F I Variable 8=1 Variable 10=1 I F F F I F F B F F F B F F F F I F F I B F F B F F F I B I F I B F F I F F F I F F F F C F F B I I F F F I F F B I F I B F F F F B F F F I F F I F F I B B F F F B F F I F F F I F F

F F F F B F F F B F F F F

I F F F I F F F I F F F I

Fig. 13. Assembly layouts of uranium at medium enrichment type 2 (UM2).

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A. Al Rashdan, P. Tsvetkov / Annals of Nuclear Energy xxx (2015) xxx–xxx

I I F F F F F F F F F I I

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Variable 8=0 Variable 10=0 F F F F F F F I F F F I F F F F

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F F F F I F F I F F F F F F Variable 8=1 Variable 10=0 F F F F F F F F F F I F F F F I I B I I F C I F F F I B I F F I F F I F F F I F F F F I I C I I F F F F I F F F I F F I F F I C I F F F I C F I I C I I F F F F I F F F F F F F F F F

F F I F I I C I I F I F F

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I F F I B I F I C I F F I

I I F F F F F F F F F I I

Fig. 14. Assembly layouts of uranium at high enrichment type 1 (UH1).

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I F I F I F F I I

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I F F I

I I F F

C I F I C I F F I

F I F I F F F I I

Fig. 15. Assembly layouts of uranium at high enrichment type 2 (UH2).

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A. Al Rashdan, P. Tsvetkov / Annals of Nuclear Energy xxx (2015) xxx–xxx

Table 16 Materials fractions and densities of the MCNP model. Material

Isotope

Weight fraction

Isotope

Weight fraction

Uranium dioxide (three materials) Helium Zirc4

Variables 4 and 7 He-3 Zr-90 Zr-91 Zr-92 Zr-94 Zr-96 Sn-112 Sn-114 Sn-115 Sn-116 Sn-117 B-10 B-11 Zr-90 Zr-91 Fe-54 Fe-56 Fe-57 Fe-58 Cr-50 Cr-52 Cr-53 Cr-54 Ni-58 Ni-60 Ni-61 B-10 B-11 O-16 H-1 H-1

0.00000137 0.50539300 0.11021400 0.16846400 0.17072400 0.02750400 0.00014100 0.00009600 0.00004900 0.00210800 0.00111400 0.01870000 0.17130000 0.41674500 0.09088200 0.03996500 0.62736800 0.01448900 0.00192800 0.00825600 0.15919900 0.01805200 0.00449400 0.06467300 0.02491200 0.00108300 0.00700000 0.03190000 0.55220000 0.11100000 0.11100000

He-4 Sn-118 Sn-119 Sn-120 Sn-122 Sn-124 Fe-54 Fe-56 Fe-57 Fe-58

0.99999863 0.00351200 0.00124600 0.00472400 0.00067100 0.00084000 0.00018700 0.00293600 0.00006800 0.00000900

Zr-92 Zr-94 Zr-96

0.13891500 0.14077800 0.02268000

5.42000

Ni-62 Ni-64 Mg-24 Mg-25 Mg-26 Si-28 Si-29 Si-30 C-0 P-31

0.00345300 0.00087900 0.01579800 0.00200000 0.00220200 0.00922300 0.00046800 0.00030900 0.00080000 0.00045000

7.94000

Si-28 Si-29 Si-30 O-16 O-16

0.37720000 0.01910000 0.01260000 0.88900000 0.88900000

2.29900

IFBA absorber

Discrete burnable rod SS304

Discrete burnable rod absorber

In-core water Core surrounding water

Density (g/cm3) 10.4700000 0.00016 6.56000

Variable 6 Variable 6

Table 17 Justification of variables’ directions of effect on Keff.

*

Index

Variable

Justification

Direction of effect on Keff

1 2 3

No of fuel assemblies Active fuel height Core radial surrounding water

+ + +(+)*

4 5 6

Enrichments (%) Power Moderator density

7 8 9

Fuel temperature lib No of rods per assembly Rod pitch

10 (a) 11 (b)

No of DBARs per assembly IFBA and fuel pellet diameter

12 (c)

IFBA and fuel rod gap thickness

13 (d)

IFBA and fuel rod clad thickness

14 (e)

Discrete burnable absorber area

Larger core causes lower leakage ratio to other reaction rates Larger core causes lower leakage ratio to other reaction rates Higher reflection by surrounding water Higher capture in surrounding water Higher ratio of fission to absorption in fuel No effect Higher moderation Higher water capture Higher absorption in the epithermal energy region Lower ratio of absorber to fuel Higher moderation Higher water capture Higher ratio of absorber to fuel More fissile material Lower water capture Lower water moderation Lower water moderation Lower water capture Lower water capture but higher clad, stronger, capture Lower water moderation Higher ratio of absorber to fuel

+ = +(+)*  + +(+)*  ++()

+()* – 

These variables can be non-monotonic, but assuming the design will fall mostly in the under-moderated region, they are assumed to follow the moderation effect.

9b, representing rod pitch and pellet diameter, was the most important interaction. This is consistent with the common design consideration of pitch and diameter ratio. The monotonically positive and negative effects of variables 9 and b, respectively, indicate that the reactor’s design domain fell completely in the under-moderated region. Other useful findings can be generated from the results. For example, the results demonstrate that except for DKeff, variables 1 and 2, representing the number of fuel assemblies and active fuel

height, had almost identical main effects on all of the performance characteristics. This behavior should enable the designer to increase either dimension and balance it by reducing the other. This, however, was not the behavior of their interactions. Another interesting finding was that the dependence of DKeff on 9 and b is as strong as its dependence on the mass of fuel in the core. This was probably due to the effect of these variables on the spectrum, and thus on the conversion of fissile materials of the reactor

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A. Al Rashdan, P. Tsvetkov / Annals of Nuclear Energy xxx (2015) xxx–xxx Table 18 Justification of variables’ directions of effect on RFS.

*

Index

Variable

Justification

Direction of effect on RFS

1 2 3

No of fuel assemblies Active fuel height Core radial surrounding water

+ + +(+)

4 5 6

Enrichments Power Moderator density

7 8 9

Fuel temperature lib No of rods per assembly Rod pitch

10 (a) 11 (b)

No of DBARs per assembly IFBA and fuel pellet diameter

12 (c)

IFBA and fuel rod gap thickness

13 (d)

IFBA and fuel rod clad thickness

14 (e)

Discrete burnable absorber area

Larger core causes higher fuel ratio to surrounding radial water Larger core causes higher fuel ratio to surrounding axial water Lower fuel ratio to surrounding radial water Higher fission at the high enrichment edge of the core Higher production of fast neutrons and absorption of thermal neutrons No effect Higher moderation Higher water capture Higher absorption in the epithermal energy region Lower absorber ratio to fuel Higher moderation Higher water capture Higher absorption of neutrons especially thermal neutrons Lower moderation Lower water capture Lower moderation Lower water capture Lower moderation Lower water capture Higher absorption of neutrons especially thermal neutrons

+ = +()*  + +()* + +(+)* +(+)* +(+)* +

These variables can be non-monotonic, but assuming the design will fall mostly in the under-moderated region, they are assumed to follow the moderation effect.

Table 19 Justification of variables’ directions of effect on ad. Index

Variable

Justification

Direction of effect on ad

1 2 3 4 5 6

No of fuel assemblies Active fuel height Core radial surrounding water Enrichments Power Moderator density

    = ++()

7 8 9 10 11 12 13 14

Fuel temperature lib No of rods per assembly rod pitch No of DBARs per assembly IFBA and fuel pellet diameter IFBA and fuel rod gap thickness IFBA and fuel rod clad thickness Discrete burnable absorber area

Harder spectrum reduces the consequence of water void increase Harder spectrum reduces the consequence of water void increase Harder spectrum reduces the consequence of water void increase Harder spectrum reduces the consequence of water void increase No effect More water increases the effect of its void Softer spectrum reduces the consequence of water void increase Higher absorption in the epithermal energy region Harder spectrum reduces the consequence of water void increase Softer spectrum reduces the consequence of water void increase Harder spectrum reduces the consequence of water void increase Harder spectrum reduces the consequence of water void increase Harder spectrum reduces the consequence of water void increase Harder spectrum reduces the consequence of water void increase Harder spectrum reduces the consequence of water void increase

(a) (b) (c) (d) (e)

+  +()  (+) (+) (+) (+)

Table 20 Justification of variables’ directions of effect on aFT. Index

Variable

Justification

Direction of effect on aFT

1 2 3 4 5 6 7 8 9 10 (a) 11 (b)

No of fuel assemblies Active fuel height Core radial surrounding water Enrichments Power Moderator density Fuel temperature lib No of rods per assembly Rod pitch No of DBARs per assembly IFBA and fuel pellet diameter

+() +() +() +() =   +  + ++

12 (c) 13 (d) 14 (e)

IFBA and fuel rod gap thickness IFBA and fuel rod clad thickness Discrete burnable absorber area

Harder spectrum increases the effect of Doppler broadening Harder spectrum increases the effect of Doppler broadening Harder spectrum increases the effect of Doppler broadening Harder spectrum increases the effect of Doppler broadening No effect Softer spectrum reduces the effect of Doppler broadening Softer spectrum reduces the effect of Doppler broadening Harder spectrum increases the effect of Doppler broadening Softer spectrum reduces the effect of Doppler broadening Harder spectrum increases the effect of Doppler broadening Harder spectrum increases the effect of Doppler broadening More fuel increases the effect of temperature Harder spectrum increases the effect of Doppler broadening Harder spectrum increases the effect of Doppler broadening Harder spectrum increases the effect of Doppler broadening

6. Conclusions This article demonstrated that the application of screening and sensitivity analysis methods at an early stage of the design of a

+ + +

complex nuclear reactor system can decouple the effects of variables on performance characteristics, enormously reduce the needed experimentation, systematize the design process, and significantly simplify the optimization efforts. It revealed behavioral

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Table 21 Justification of variables’ directions of effect on PPF. Index

Variable

Justification

Direction of effect on PPF

1 2 3 4 5 6 7 8

No of fuel assemblies Active fuel height Core radial surrounding water Enrichments Power Moderator density Fuel temperature lib No of rods per assembly

(+) (+)  + = + + +()

9

Rod pitch

10 (a) 11 (b)

No of DBARs per assembly IFBA and fuel pellet diameter

12 (c) 13 (d) 14 (e)

IFBA and fuel rod gap thickness IFBA and fuel rod clad thickness Discrete burnable absorber area (cm2)

Larger core causes lower PPF Larger core causes lower PPF More water increases reflection at the edges Higher enrichment causes higher heat production, especially at the center No effect Softer spectrum results in higher heat production, especially at the center Higher temperature causes higher absorption especially at the center Larger core causes lower PPF Smaller burnable absorber to fuel ratio causes higher PPF Softer spectrum causes higher heat production, especially at the center Larger core causes lower PPF Higher absorption, especially at the center Harder spectrum, especially at the center Higher heat production, especially at the center Harder spectrum, especially at the center Harder spectrum, especially at the center Higher absorption, especially at the center

+(+)  +(+)   (+)

Table 22 Justification of variables’ directions of effect on DKeff. Index

Variable

Justification

Direction of effect on DKeff

1 2 3 4 5 6 7 8 9 10 11 12 13 14

No of fuel assemblies Active fuel height Core radial surrounding water Enrichments Power Moderator density Fuel temperature lib No of rods per assembly Rod pitch No of DBARs per Assembly IFBA and fuel pellet diameter IFBA and fuel rod gap thickness IFBA and fuel rod clad thickness Discrete burnable absorber area

Higher fuel to power ratio Higher fuel to power ratio Unknown effect More fissile material Faster fuel depletion Thermalizing spectrum reduces breeding Unknown effect Higher fuel to power ratio Thermalizing spectrum reduces breeding Absorber depletes with time, and thus adds reactivity Higher fuel to power ratio Hardening spectrum increases breeding Hardening spectrum increases breeding Absorber depletes with time, and thus adds reactivity

  ?() (+) + + ?()  +     

(a) (b) (c) (d) (e)

characteristics of the system that were not identifiable through simple examination. Several methods with various assumptions could be applied to such systems. However this article demonstrated that assuming a monotonic behavior of variables proved to be a predominantly valid assumption for the LWSMR design. The applied methods utilized this assumption, and produced approximation models that were used to project the performance characteristics of any point in the design domain. These models can be combined with the introduction of additional performance characteristics to reduce the degree of freedom of the system, and narrow down the design domain.

Appendix A. Core layout See Fig. 9

Appendix B. Assemblies layout See Figs. 10–15

Appendix C. Materials fractions and densities See Tables 16

Appendix D. Justification of variables’ directions of effect

 

See Tables 17–22

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Please cite this article in press as: Al Rashdan, A., Tsvetkov, P. Parametric evaluation of an SMR design domain. Ann. Nucl. Energy (2015), http://dx.doi.org/ 10.1016/j.anucene.2015.07.013