Reaching the optimal values of several indicators simultaneously is usually impossible. The solution for such difficulties is the implementation of the requirement to support a number of indicators at a level no lower than acceptable, and to direct the selected indicator to its extremal value. By selecting different set of indicators, one can come to duality formulations of optimization problems. They have the same set of optimal solutions for all possible values of limiting levels [7].

However, in real-world problems the decision-making process subject cannot set numerical values of the limiting levels. As a way out of this situation, it is the required to seek solutions that are unimprovable for any of the criteria without worsening at least one of the other (V. Pareto principle). Just these decisions, which are considering with the multi-criteria decision theory, are of practical interest.

It should be noted that the improvable (dominated, inefficient) points are located within the feasible solution set. It is possible to shift from these points while remaining within the feasible solution set, into other points which are better in at least one criterion and not worse than the given point in other criteria. On the contrary, any improving shifts from the unimprovable point falls outside the boundary of the feasible solution set.

It should be noted that usually there are a lot of unimprovable points. Ultimately, one single point must be chosen either expertly, or formally, by taking some additional hypotheses about the preferences. But these additional hypotheses are not of a general character, in contrast to the original principle of unimprovability proposed by V. Pareto.

In the optimization theory, enough approaches have been developed to find effective solutions. One of them is the criteria constraints method, a universal method which is correct both for problems with convex and non-convex sets of feasibility.

The main idea of this method is based on the fact that all but one criteria values are limited, and the only criterion for selection is extremalized (in this case it is minimized):

(4)

The limiting levels are the problem parameters. Setting them at all possible values of the ranges from absolute minimum to absolute maximum of the corresponding figure in the original set of feasibility, it is possible to obtain all the efficient solutions.

This set of methods is based on the following heuristic argument: the best vector should be chosen as a vector which is placed in the criteria space closer than all other feasible vectors to some ideal vector, or to a whole set of them. The metric variation for measuring distances in the criteria space leads to a whole family of similar methods which, however, may lead to different results.

Let us briefly describe the principle of the method. Suppose there is a set of M criteria, each of which is desirable to be minimized in the set of possible solutions X. In accordance with the methodology of this approach, it is considered that in the criteria space Rm a nonempty set U is given, which is usually called the set of ideal (the best or utopian) vectors. Typically, this set is unfeasible. In addition, a metric should be set in the criteria space. A solution will be declared optimal if the following equation is realized for it:

(5)

which means that the estimate corresponding to the best solution , should be located as close as possible to the set of ideal estimates . Most often, for the solution of applied problems, any metric can be used from the following parametric family:

(6)

where

Varying the parameter vector аs, the importance of different criteria is taken into account. A greater importance is attached to the parameter vector component which meets the greater value criterion. It should be noted that the use of the above-mentioned parametric family metric does not always lead to Pareto optimal vectors.

To implement the criteria constraints and the goal programming methods, it is necessary to be able to enter additional information into the model, which represents both limiting levels and the ideal set. MESSAGE generates an MPS file, which is a standardized notation of a linear programming problem, entering the optimization block. To implement the described methods, this file must be corrected. A specially designed module ConCriM allows making necessary changes in order to implement these methods (Figure 1).

The reasonable goals method was designed by Dorodnicyn Computing Centre of the Russian Academy of Sciences. The basis of the method is visualization of many possible feasible vectors by imaging it through bi-criterion slices. The use of the method is mainly intended for computationally complex cases of infinite number of possible solutions and vectors [5].

One of the goal programming method’s weak points lies in the fact that the ideal vector is given without taking into account the real system capabilities. Therefore, the attainable values of figures, even the closest to a given ideal, may prove to be far from it. The given method seeks to overcome this shortcoming. In accordance with this method, a set of practically feasible vectors is accessible for decision-makers in a clear, easy for perception form. One can choose a certain compromise among them.

The program module ParSAM (Pareto Set Approximation Module) for an energy planning software MESSAGE is designed for automation of the approximation and visualization of Pareto set as well as for accounting of uncertainties in system parameters.

With the ParSAM, a user can create a base project set of linear programming problems, which are based on the criteria linear convolution method, and the solution of which conforms to the Pareto efficiency condition. With the created initial data set, the module allows serial calculations, processing the calculation results and presenting them in tabular and graphical form. All the sources, intermediate, and the resulting files are compatible with MESSAGE environment file formats. This allows, if required, making the necessary adjustments through MESSAGE environment.

The basic functionality of the module (Figure 2):

forming a set of matrices for solving a parametric linear programming problem;

forming a set of files for post-optimization processing of calculation results;

controlling the MESSAGE modules to improve efficiency of the Pareto set approximation;

different ways of weights generation for the Pareto set approximation;

different ways of the Pareto set visualization.

Information on the Pareto set can be used to estimate the marginal rate of substitution, which can effectively support the selection of the most preferred criteria point in multi-objective problems.

MESSAGE software, representing a flexible environment for specification of the energy systems, is not adapted for the description of nuclear power system features. This is due to the fact that the demand for services both at front-end and back-end stages of the NFC is determined by the characteristics of nuclear reactor and its fuel supply. All other system parameters can be calculated for certain values of these variables and a minimum set of additional ones. At the same time, working directly in the MESSAGE environment requires an independent input of these parameters, which makes the work with the model difficult and fallible.

For this reason, a nuclear power systems specification interface was specifically created for energy planning package MESSAGE—NESI (Nuclear Energy System Interface) (Figure 3). This interface can be used by specialists to facilitate the innovative nuclear power system specification.

The basic functionality of the module:

specification of nuclear power system with regard to possible parameter changes over time;

NFC process chain visualization;

Possibility of integration with neutronics calculation programs;

flexible control function for the MESSAGE’s program components, in order to improve the calculations efficiency.

Currently, a set of basic calculation cases for once-through and closed NFC has also been developed, which allows calculating the NFC material flows and is intended for the modeling of steady-state and developing nuclear power systems.

According to the general methodology of modeling in the system analysis framework, it is necessary to identify the following typical stages: 1) generation of the constraint equations reflecting the actual “physical” limitations; 2) the determination of exogenously defined objective functionals reflecting the concept of efficiency of the relevant object functioning; and 3) if necessary, equations that close the system.

The initial system of constraint equations represents in an aggregated differential form the process of nuclear reactors fueling in a particular region (the capacity increase must be supply by fuel of a given form and amount)

(7)

where I = 1,…,n—reactor type number; j = 1,…,m—region number; k = 1,…,l—fuel type number.

The left side of the equation determines the demand for fuel of given form. The demand is composed by the needs to provide new capacities (first component) and to replace burnt-up fuel in the nuclear reactors in operation (second component). The right side of the equation sets an offer for a certain fuel type. It is determined by the existing fuel supply structure of nuclear power system in the relevant region, and by the possibility to import fuel components from other regions. From now on the following conventional signs are used: —initial fuel loading into the reactor, —the annual refueling of reactor. The functional dependence of the equation parameters on time takes into account possible technology updating.

Let us introduce the following notation: N(t),Nc(t),Nd(t)—are respectively installed capacity of reactors in year t, the total capacity of reactors put into operation by the year t, and the total decommission of reactors by the year t. These values are related by:

(8)

where [·]—the integer part.

Taking this fact into account is necessary for correct modeling of the real need for the introduction of new nuclear reactor capacity, caused not only by increasing demand for nuclear power, but also by decommission of worn-out reactors.

Structural and organizational features of the NFC can be taken into account as additional equations and constraints. In particular, the following conditions can be considered: the nuclear materials supply from other regions and other types of reactors; delays on various stages and capacities of NFC facilities; property changes in fuel materials and possible technological modifications of the NFC facilities in the nuclear power system deployment.

Incoming and outgoing flows can be defined in two ways. Firstly, additional equations can be introduced for linking the definable variables and ultimately closing the initial fuel balance system. In general, the model will appear as a set of differential equations with retarded argument.

Secondly, if the objective functional is set, a part of the flow can be defined by solving the corresponding optimal control problem, whereas the others are solved by the closing system of equations. Various constraints on the variables’ acceptable region of models or functionals calculated with their help, allows taking into account the system constraints directly. Thus, the calculations will not lead into a non-physical solution region.

Thirdly, this simulation method can be effectively integrated with the described below method described below of the parameter space investigation that allows solving the nuclear power system structure optimization problem in the multi-objective formulation.

The parameter space investigation method was suggested by I.M. Sobol and R.B. Statnikov, who demonstrated its wide opportunities in solving engineering optimization and optimal control problems [8,9]. The method is intended to solve multi-objective nonlinear programming problems and is based on the idea of random search, where so-called low-discrepancy sequences are used as the random numbers.

The parameter space investigation method consists of three phases:

formation of test tables—that is, a set of calculated values of the criteria for a given set of testing points that satisfy a given system of equations and system of constraints;

selection of criteria constraints and choice of solutions to satisfy them;

checking the solvability of the problem and formation of an effective solutions set.

Figure 4 shows the flow-chart and related software components that implement an integrated approach based on the system dynamics and parameter space investigation methods.

The developed methodology allows taking into account nonlinear constraints and nonlinear functionals on model variables, to approximate the Pareto set in the case of multi-objective problems. The internal mechanism of alternatives selection helps identify the most effective directions; and the dynamic character of the model assists in evaluation of the transition to the asymptotic scenarios.

The described integrated approach is implemented as a MEDNES software system (Multi-objective Evaluator of Developing Nuclear Energy Systems). MEDNES is designed to solve multi-objective problems, e.g., structure optimization, parameters identification, control, etc. MEDNES is intended for construction of the feasible solutions set during interactive dialogues between expert and computer. MEDNES is applicable for analyzing the dependence of criteria on the parameters; on the criteria and constraints correction.

The number of variable parameters depends on sample points generator used. Different sample point generators can be used. Currently, implemented are the generators of Holton, Sobol and Faure low-discrepancy sequences [10]. The standard pseudorandom number generators and uniform sequences can also be used.

The expert chooses the most preferred solution on a basis of feasible set and Pareto set analysis. MEDNES includes the following analysis tools:

test tables;

feasible and Pareto-optimal solutions table;

tables of vectors not satisfying functional constraints;

“criteria-parameters” and “criteria-criteria” plots reflecting the criteria sensitivity to parameters and criteria dependence on the criteria.

Because the multi-objective problems with high parameter vector dimensionality require a lot of computer time, the possibility is provided to run MEDNES in parallel mode.

The method of criteria constraints implemented in a software module ConCriM was used to optimize the structure of the global nuclear power taking into account regional restrictions and presence of fast reactors in a closed U-Pu-Th fuel cycle in its structure. As indicators, which are restricted and determine finally the choice of a solution from a set of effective solutions, fraction of fast and heavy water reactors, natural uranium reserves, the accumulated amount of spent nuclear fuel and plutonium were selected.

In the framework of MESSAGE following options of global nuclear power with closed NFC were considered (Figure 5):

(1)    U-Pu closed NFC with LWR, HWR and their advanced prototypes as well as fast breeder and burner reactors;

(2)    U-Pu-Th closed NFC with fast breeders with Th blankets and Pu cores.

The latter case corresponds to so-called mixed NFC with the fuel exchange between the uranium thermal and fast reactors with plutonium cores and thorium blankets. In such a fuel cycle of fast reactors, keeping plutonium fuel in the core, to produce 233U in its thorium blankets to start-up and recharge uranium thermal reactors, which, in turn, working in uranium fuel from a mixture of 233 and 238 uranium isotopes, accumulating plutonium to start-up new fast reactors.

The step-by-step setting of restrictions, reflecting the system requirements to nuclear power (such as total amount of natural uranium, spent nuclear fuel, the locations of fast reactors) have defined the following structure based on the solution of optimization problem on the criterion of minimizing the total discounted costs (Figure 6):

once-through uranium NFC with no restrictions on uranium resources (I);

closed U-Pu NFC with and without restrictions on the locations of fast reactors and the limitations on the amount of available natural uranium (II, III);

closed U-Pu-Th NFC with restrictions on the location of fast reactors and the amount of available natural uranium (IV).

The study results have proven that it is difficult, or even impossible with certain reactor technologies, to meet the set of constraints that reflect the system requirements for a nuclear power system, taking into account the regional restrictions, for unlimited time. In respect to this type of system constraints, the nuclear power system structure becomes extremely sensitive to the reactor technology types and loses its ability to adapt to changing external conditions, while meeting the set of conflicting system constraints at the same time. In such a system, each technology plays an important role. If one of them is lost, there will be no optimal solution for a given set of constraints. In this study a compromise could be found in transition to U-Pu-Th NFC.

At the same time, the system must remain flexible enough to adapt to changing external conditions, while meeting a set of system constraints. To balance the conflicting factors in such a system, it seems appropriate to implement specialized multifunction innovative reactor systems, whose tasks must be determined depending on the particular stage of development.

Using the reasonable goals method, the studies on Russian nuclear power with fast reactors with different breeding parameters were carried out to establish a compromise development strategy on a set of conflicting criteria (Figure 7).

The ParSAM module was used to perform a multi-objective optimization of Russian nuclear power structure on the criteria of minimizing total discounted costs, total consumption of natural uranium, total amount of spent nuclear fuel and capacities of “sensitive” NFC technologies. For this set of criteria, a set of effective solutions was built demonstrating the upper productive limits of a given reactor technologies mix, and allowing choosing a compromise strategy of Russian nuclear power development on the set of criteria mentioned above.

For example, Figure 8 shows the so called alternative profiles of the four non-dominated solutions demonstrating the changes in integral indicator values for the last year’s forecasting for four possible nuclear power structures. As can be seen from Figure 8, the improvement of an indicator is achieved by the deterioration of another, which corresponds to the condition of Pareto optimality.

The estimates performed showed the principal opportunity to solve the problem of finding a perspective nuclear power structure compromised from a set of indicators. It should be noted that a more diverse structure of nuclear power formed based on different reactor types allows getting closer to the desirable compromise goal. This approach, in contrast to the method of criteria restrictions, provides an opportunity to directly assess trade-offs between conflicting indicators.

The results show that different structures of developing nuclear power systems based on considered NFC are comparable by the indicators: “total amount of fissile materials in NFC” and “potential productivity of fissionable materials”. It should be noted that the improvement of one indicator is achieved by the worsening of another. This suggests that it is impossible to make definitive judgments about the prospect of a nuclear power structure and the NFC type from the non-proliferation viewpoint based on material flow assessment, without a detailed analysis of the proliferation scenarios and specification of acting national and international systems of nuclear security and nonproliferation regime management.

The values of levelized cost for the entire development program calculated taking into account the uncertainty in unit costs for goods and services of NFC and reactors investment costs, in accordance with data collected from set international sources [15,16] for two different scenario assumptions, are presented in Table 1. The uncertainty assessment of levelized cost was performed by using the GRS method implemented in the ParSAM module.

The quantitative analysis performed shows that the considered scenarios are statistically indistinguishable (90% confidence intervals of uncertainty of the levelized cost for various scenarios overlap). This suggests that it is impossible to make definitive judgments about the prospects of one or other structure on economic indicators, taking into account the existing uncertainties in the unit cost data.