A nuclear power plant is as detailed a modern marvel of comprehensive engineering as any on the planet. Nuclear reactor core design is the central design element of a nuclear power plant. The lattice design is the basic element of the nuclear reactor design. In the modern commercial LWR, the BWR lattice design is more complicated than for pressurised water reactors (PWRs) since it has features such as void dependence, dense neutron poison fuel pin distribution, and partial fuel pin length design. Many research efforts [1, 2] have attempted to apply perturbation theory to BWR lattice design and optimization. Although the major theoretical challenges of determining the amount of neutron poison distributed in fuel pins have been resolved by using transport theory or the Monte Carlo method instead of diffusion theory, they have remained unsolved for perturbation theory until now.

Perturbation theory, like other approximation theories such as sensitivity analysis [3, 4], has been studied and successfully used as an approximation method in science and engineering for centuries. For example, in quantum mechanics [5] or quantum chemistry [6], the first-order perturbation theory is an extremely important method for describing and solving some quantum systems that are very difficult to accurately solve, if not impossible. Another example is the neutron transport equation [7]. With the adjoint function, perturbation theory allows the reactivity, the effective multiplication factor and the multiplication rate constant to be estimated by using the first-order approximation if the change is small. (The adjoint function aims to account for a neutron’s importance to reactivity calculations based on its position and other factors). However, because of complexity, higher-than-first-order approximations remain very difficult to achieve in most cases, especially for a large number of variables. If the change is not small, the first-order approximation cannot be used—even with the adjoint function—because large changes of variables will disqualify the original reference solution, which can no longer be assumed to be unperturbed to these large changes. Therefore, one has to set up a new reference case defined to have a small change relative to the case of interest. During optimization, the solution of the new reference case often is not available, yet an exact calculation for the new reference case needs to be performed.

For cases of a single variable, such as ordinary differential equations, whether linear or non-linear, homogeneous or inhomogeneous, including some special integrals etc, the authors of [8] has summarized in detail the perturbation theory and methods with various approximations. However, modern engineering design and optimization always involves a large number of variables and large-scale variations. These methods are not applicable [3, 4] since they are too complicated to iterate among large number of variables, especially for the interactions of these variables. This is the reason why perturbation theory has not widely been used for modern engineering designs.

As computers developed and were applied extensively to science and technology in the last century, performing accurate large-scale numerical calculations has become routine for complicated engineering design and optimization. Because these large-scale numerical calculations take a long time, and because manual optimization usually needs many case-calculations performed, modern engineering design and optimization still remains a big challenge. However modern computers have made it possible to evaluate and use the first- or second-order of perturbation theory with large numbers of variables. A lot of research has taken place for modern engineering design and optimization with perturbation theory [1, 9]. Of course, during optimization a large range of change often occurs and the perturbation theory cannot be used without frequently changing and recalculating the reference cases. Also it is not an easy task to identify a new reference case in advance because there is no clear path to achieve the design goal during optimization. This paper introduces a new combined micro-parameter and macro-parameter perturbation (CMM perturbation theory) to resolve these issues.

For optimization of design, both speed and accuracy are important. In nuclear engineering, a basic element of reactor design [10] is fuel lattice design [7]. In a LWR, a reactor core consists of hundreds of fuel assemblies which are usually divided into several groups, such as three groups or more. Each group has the same structural parameters, fuel enrichment distribution, and so on. For a BWR, each group also has the same initial enrichment distribution and neutron poison (Gadolinium, or Gd) weight distribution among fuel elements denoted as fuel pins, and these distributions are different between several (from 3 to 6 regions, or more) vertical regions of a fuel assembly. Therefore, each region usually has its unique heterogeneous distributions horizontally, but is vertically uniform for each fuel element. The cross-section of an assembly and half-gap (water, control blade or both) between neighbouring assemblies consist of a unique structure, denoted as a lattice. Because of their vertically-uniform distributions, lattice designs will only deal horizontally with the two-dimension neutron transport equation, and hundreds of variables (about 100 fuel pins’ enrichments and their neutron poison weights, and so on). It is common to design more than 10 different lattices for a BWR core.

In our reloading design and optimization for nuclear power plants, engineers often start a new lattice design with a fixed lattice type, that is, a fixed geometry, such as 10X10, 9X9 or ATRUM10 etc. and then change each fuel pin’s (or fuel rod’s) enrichment and Gd weight, and perform lattice calculations with qualified lattice codes such as CASMO4 to see if their characteristic performances meet the design criteria. This process will be iterated many times, including the iteration between lattice and core, until all the lattices meet the criteria required from 3D core design, and 3D core design achieves the goal of the nuclear power plant. This takes weeks or months.

Even on modern computers running a qualified lattice code such as CASMO4 [11], which combines time-dependent burn-up calculation and time-independent two-dimension neutron transport equation, a BWR lattice case usually takes about 2.5 hours to calculate. For optimization, hundreds of cases need be solved. Can we use perturbation theory for design and optimization to achieve both requirements of speed and accuracy? There are many papers [1, 2] published in this area. It seems very difficult to fully meet both requirements, especially for the burnable absorber (Gd) distribution; adding or removing Gd pins is always considered a big disturb. To resolve these issues, CMM perturbation theory makes two major assumptions that are reasonable both physically and mathematically.

First, it considers all independent design variables as micro-parameters. For example, if a BWR 10X10 lattice has 100 fuel pins, then 100 fuel enrichments and 100 burnable poison (Gd) weights for this type of lattice have to be designed. (For convenience, a BWR 10X10 lattice design is used as an example throughout this paper.) Therefore, these 200 independent design variables are called micro- or local parameters. Under the fixed lattice geometry, these micro-parameters themselves determine the entire performance of the lattice. Mathematically, the lattice performance factors, such as reaction rate, reactivity Kinf, maximum pin power and so on, are functions of micro-parameters, and are called performance functions in this paper. Traditionally, perturbation theory can use the derivatives of these functions with respect to each micro-parameter as the change rates to estimate any small design change. Since these functions are very complicated, the change rates have to be numerically calculated [1]. In this paper, this is called micro-parameter perturbation. These micro-parameters are truly independent, and are basic.

However, as is well-known, there always are some global design parameters that are more important to the performance of the entire fuel lattice, and they are functions of micro-parameters. For example, the average enrichment and the average Gd weight of the lattice have global impacts and are functions of micro-parameters in the above example. These global design parameters depend on micro-parameters, and usually one knows their explicit dependent functions, and they are usually very simple, such as the average enrichment, E, and the average Gd weight, W, and total number of Gd pins, G, in BWR 10X10 lattices:

where M is 100 and ei is the ith fuel pin’s enrichment, and wi is the ith pin’s Gd weight. In order to differentiate them from micro-parameter based performance functions, these global design parameters are called macro-parameters in this paper. Furthermore, for optimization, the required performance of an optimised lattice are called design targets.

Second, CMM perturbation theory assumes that all the important macro-parameters are independent design variables, but with explicit dependent functions as the all-time and all-space conditions that will guarantee that they are physically and mathematically correct. For example, the average enrichment E can be chosen to equal to 4.5% independently during optimization, and then the valid sets of ei must meet one of the all-time and all-space conditions:

Or vice versa, if a set of fuel pin enrichments (ei) is given, the macro-parameter E must be equal to:

This approach is similar to the idea that a valid solution must meet boundary conditions and initial conditions.

In BWR lattice design, each fuel pin’s enrichment and Gd weight are micro-parameters, but the average lattice enrichment, the average Gd weight and the total number of Gd fuel pins are macro-parameters. The design targets, such as reactivity Kinf and maximum pin power of a lattice, are now functions of micro-parameters and macro-parameters, but with their all-time and all-space conditions together, which means that any further mathematical or perturbation operations must meet these all-time and all-space conditions. In so doing, introducing macro-parameters into performance functions as independent variables won’t alter the physical problem.

Third, CMM perturbation theory then assumes that the macro-parameter perturbation takes into account all the higher orders and interactions remaining from the first-order micro-parameter perturbation. Therefore, CMM perturbation theory can achieve accuracy comparable to exact solutions and cover all design ranges, and not only for small changes, since it allows macro-parameters’ (global) variations.

With CMM perturbation theory, not only can a fast simulator be built without frequently changing and recalculating the reference cases, but also the new perturbation theory can provide information for design since the perturbation coefficient of each parameter is the rate of change of the target with respect to it. Thus, this theory will accelerate the optimization process, and perform an optimization of design parameters one-by-one for the best results [12], especially with the macro information from macro-parameters, something human engineers cannot achieve. With this CMM perturbation theory, a BWR fuel assembly lattice design and optimization code package (BALO/FLS) for most types of BWR fuel assembly lattices has successfully been built at Areva.

**Micro-parameters **

Design variables, such as xi (i=1, 2, ∙∙∙∙I), are defined as micro-parameters in CMM perturbation theory. Usually, there are multiple groups of independent micro-parameters. Engineers usually change the values of these micro-parameters or their distributions to discover if the target design values or curves are met. If not, engineers will redesign and test them again. Usually this is a long iterative process, continuing until all design criteria are met. If the design functions are continuous functions of micro-parameters and all the derivatives including mixed derivatives exist, the multiple-variable Taylor series can be used for micro-parameter perturbation with cut-offs at certain orders, such as first-order, second-order, and so on. Therefore, as is well-known, a cut-off error will exist, and the accuracy will depend on functions’ behaviours and the order, and the variations of multiple variables. For large numbers of variables, it is desired to only take the first order. Otherwise, too many derivatives have to be calculated. For example, in a 10X10 lattice, the first order will have 200 partial derivatives at least. In BALO/FLS, the micro-parameter perturbation is the first order, and the cut-off errors will be taken into account by macro-parameter perturbation as discussed in next section.

For engineering design, some discrete variables (such as integers) often occur, and then it is not possible to use the concept of multiple Taylor series, which rely on the continuous nature of variables to calculate derivatives. However, as long as the rates of change of the design functions with respect to these discrete variables are calculable as in BALO/FLS, they can be used instead of derivatives; indeed, the derivative is actually the same concept as a rate of change.

**Macro-parameters**

The micro-parameter perturbation, as the first-order approximation, takes into account individual impacts of micro-parameters on design function, locally and globally. The superposition of each individual impact may be a good approximation for local impact but not for global impact, since it does not consider their collective action—their interactions. Actually the global performance related to their collective impact is most determined by the global design parameters; macro-parameters. From a physics viewpoint they are much more important than each individual micro-parameter. For example, Kinf at end of life (EOC) is almost uniquely determined not by the individual pin’s enrichment, but by the average enrichment of a lattice. Despite their importance, these important macro-parameters have never been introduced into functions and perturbation theory as independent variables, because they physically depend on micro-parameters.

Traditionally, a function is always defined mathematically with respect to its independent variables. The impact of any macro-parameter on the function is automatically included through all these independent variables, but it is very complicated. For example, each pin’s enrichment is included in a time-dependent 2D neutron transport equation through all kinds of nuclear macro cross-sections. Their impacts on the function of this equation, individually and collectively (or locally and globally) are described by this equation. However, it is difficult to separate and identify each individual impact and their corrective impact. But now for design and optimization with perturbation theory, it is desirable that we estimate each individual impact and their corrective impact separately. It is reasonable to assume that each macro-parameter represents the interactions among micro-parameters in the corresponding group. Although it is impossible to analytically separate macro-parameters from micro-parameters in the function because the function consists only of micro-parameters, it is possible numerically to separate, to display and to evaluate approximately the macro-parameter’s impact, since the individual micro-parameter impact is calculable and much smaller than the macro-parameter’s.

**CMM perturbation theory applied**

CMM perturbation theory offers a result with no cut-off errors of perturbation coefficients by definition of macro-parameter perturbation coefficients within each group. However, there is still a need for a lot of numerical calculations that may create numerical errors, depending on how all the reference points of multiple spatial dimensions are set up. This is especially true for the interactions among the groups, and the kind of numerical method that is used. Although these can be improved with better numerical methods, what is more important is physics experience. Scientific applications are a kind of art. Using physics experience to choose the best numerical algorithms and reduce unnecessary perturbations is very important. It is particularly true for modern, complicated, large-scale engineering systems where analysis is difficult and massive numerical results can be confusing without physics experience. Today, it is often a good and necessary practice for engineers to judge the numerical results based on physics concepts, and then evaluate the results numerically.

Another concern is how accurate the macro perturbation coefficients are when applied to the neighbours of the reference points and how big the step length of reference points for each parameter can be—especially for the macro-parameters—to achieve the required accuracy. Actually, this is dependent on each engineering system considered. However, this theory has not only resolved the issues related with Gd distribution optimization, but also proved to be accurate and efficient for BWR lattice design and optimization, as described below.

The problem of nuclear fuel lattice design is actually to solve a time-dependent two-dimensional neutron transport equation [7]. Under the given geometry, all the components’ sizes of lattice are fixed. Different geometry designs create different lattice types, such as 9X9, 10X10 etc. Engineers routinely change the enrichment distribution and Gd weight distributions among all the fuel pins for a given type of lattice, which is a fixed geometry. To do so requires changing neutron macro cross sections, and then solving the time-dependent two-dimensional neutron transport equation for flux. After flux is obtained for a case, a post-calculation is performed for all necessary information, such as reactivity, pin power and so on. The neutron transport equation is time-dependent, and so is its solution. For a given power profile, the time-dependence converts to burn-up dependence. There are three kinds of major curves that are important: reactivity vs. burn-up, maximum pin power vs. burn-up, and Feff or K-factor vs. burn-up for different voids, such as 0%, 40% and 80%. These are called the major three nuclear lattice characteristic curves at Areva. Feff, or K-factor, is the CPR (critical power ratio) correlation factor [13], and different companies may use different names or different type of correlations. In Areva’s BALO/FLS, these correlations are built in, and Feff or K-factor is calculated and optimized for different types of lattices.

From the above discussion, a normal lattice design, for example 10X10, involves 100 variables of enrichment and 100 variables of Gd weight, totaling 200 variables, and the number of results or response parameters is huge: there are 70 burn-up points from BOC to EOL, there are at least three voids, plus three kinds of curves. In total, the lattice performance comprises 70X3X3, or 630 response parameters.

The authors of [7] have fully and analytically discussed and summarized the first- order perturbation and its application in nuclear engineering for small changes, together with adjoint operator and variation methods. However, there are at least three major reasons that the first-order perturbation theory cannot be applied for lattice design and optimization. First, during design and optimization all the variables will have a large swing, that is, large changes, especially for Gd fuel pins. Second, the reference calculations will be very frequent and time-consuming, sometimes even the reference point cannot be easily predefined. Third, some resultant parameters such as Feff [14] or K-factor [13] are partly related to neutron flux and partly related to thermal hydraulics and are thus more difficult to correlate with analytical perturbation formulas.

All the derivatives, that is, perturbation coefficients, are pre-calculated at each reference point and stored in the perturbation library. All the pre-calculations can be performed with any qualified lattice code, such as CASMO4 used in BALO/FLS, to ensure the required accuracy and to meet all the lattice design requirements. It usually takes a couple of weeks for the hundreds of reference cases for this kind of perturbation library to be worked out. However, once the library is built, then it can be used forever, and added to if necessary [12, 15, 16, 17]. Different types of lattices, such as different fuel pitches and channels, need to build different libraries. If one wants to generally optimize lattice design with geometry dimensions as independent variables, another group of micro variables should be added. Since the geometry dimensions are just a few variables, and not a large number as enrichments are, they do not need to be dealt with using CMM theory. At Areva, the different perturbation libraries have been built for the different types of lattices to deal with these dimension issues.

**Results, analysis and conclusions**

CMM perturbation theory has been successfully applied to build a tool version of BALO/FLS at Areva. FLS is a Fast Lattice Simulator, and BALO is a BWR Assembly Lattice Optimizer. A CMM perturbation library (Plib) for different kinds of lattices, including C-Lattice, D-Lattice, full-length lattice, a partial-length lattice, A10 lattice, 9X9 lattice and a delta lattice has been built with direct calculations of CASMO4 [11]. Also, a GUI, ALAADIN [18], was developed and modified for graphic application of BALO/FLS. FLS can run a case for three kinds of nuclear characteristic curves just in six seconds compared with 180 minutes or more of 2D neutron transport codes, such as CASMO4, or APOLLO2-A [19]. The accuracy for Kinf is within 0.001 for different voids from BOC to EOL, and 0.01 for maximum pin power, Feff and K factor, which meet the industrial design criteria at Areva. To take advantage of CMM perturbation theory, which provides all necessary intelligence information on both micro and macro scales, and offers a super-fast speed, BALO was developed into three basic models with either Feff or K-factor: fitting optimization model, eSAVING model, and eSAFETY model to address optimization goals such as minimizing costs [12, 15, 16, 17].

From the above analysis, CMM perturbation theory can successfully be used for BWR lattice design where a large number of variables and large-scale variations are involved; and where the macro-parameter perturbations are extremely important as global and collective impacts. As shown above, all micro perturbation and macro perturbations are just first-order calculations plus some necessary mixed partials with no iterations. These calculations are super-fast with modern computers and good algorithms. In addition, CMM perturbation theory can provide all the change rates with respect to macro-parameters, as well as the change rate with respect to each micro-parameter. That information will provide very important intelligence to perform optimization globally and locally, parameter by parameter. In BALO and FLS, both macro and micro information are used to achieve enrichment saving (eSAVING model) and to improve safety margin (eSAFETY model), which cannot be done manually.

Finally, the author would like to point out that CMM perturbation theory may have great potential for applications in other disciplines of engineering where there are many micro variables, and first-order approximation of perturbation theory can be used for small changes. However, it is still a big task to identify the macro-parameters and derive individual algorithms for each discipline of engineering. Apparently, further study on CMM perturbation theory for different areas, especially for non-linear operators and cases with resonance interaction, is needed.

Author Info:

Albert G Gu, formerly of AREVA NP Inc, Richland, WA.

This article was first published in the May 2012 issue of Nuclear Engineering International

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Mathematical overview of this work