2.1 CHASNUPP 300 MWe core Design Specifications

This research work has been carried out by using design data of CHASNUPP unit-1. It is a two-loop PWR reactor having 300MWe net power. Its core consists of 121 fuel assemblies having identical mechanical design but different levels of enrichment. There are three different enrichment zones. These are 2.4%, 2.67%, 3.0% by weight enriched 41, 40, 40 FAs, respectively. Fuel pellets are composed of sintered UO₂. Each fuel assembly is made of 225 pins arranged in a 15×15 rectangular array. The detailed design data table of CHASNUPP is given in Table-1.

PWR core of CHASNUPP reactor is symmetric. It has 1/4^th symmetry as well as 1/8^th symmetry. Figure 1(a) shows a two-dimensional cross-sectional view of the core with a layout of FAs of three different enrichments. Burnable poison (BP) rods used in different fuel assemblies are also mentioned here. For the same reactor, either 16, 4a, 4b, 2 or no BP-rods are inserted in fuel rod bundles i.e. each FA is equipped with no BP rods or 2 BP rods, or 16 BP rods or 4a or 4b. It depends on the core design. It is another constraint in our problem. In other words, number of BP rods of each type is fixed in whole core. Design specification of CHASHNUP core unit-1 is given in Table 1.

Figure 1Full-screen View

(Color online) (a) 2D cross-sectional view of CHASNUPP-I core showing different enrichment zones; (b) Block diagram of neutronics calculations and particle swarm optimization process

2.2 Power Profile Evaluation

The detailed neutronics calculations have been carried out using the available reactor physics codes. PSU-LEOPARD has been used for generation of macroscopic group-constants whereas; MCRAC has been used for power profile evaluation. The description of these codes is given below.

3.1 Particle Swarm Optimization

In 1995, Eberhart & Kennedy proposed Particle Swarm Optimization technique to solve non-linear continuous functions [15]. They were inspired by social behavior of organisms such as birds flocking and fish schooling. Every particle of PSO is a candidate solution and moves towards global minima based on its own velocity, previous knowledge and experience of whole swarm. The swarm explores the search space initially and exploits a specific area based on its basic equations.

Particle Swarm Optimization is based on social behavior of organisms like bird flocks and fish schooling. It is an efficient population based meta-heuristic technique. In this technique, a swarm of birds or particles is introduced in the search space of the problem. Each bird is characterized by its velocity and position and is a candidate for a global minima position. The position of each particle is updated based on its own best and social best fitness position as well as its own inertial velocity [16]. This swarm of birds moves based on basic equations of PSO as given below:

where i= [1, 2,..., n] and j= [1, 2,..., d]. c₁ and c₂ are known as acceleration constants. These are positive numbers and their value depends on nature of optimization problem. r1 and r2 are random numbers between [0 1], and w is inertial weightage factor. It gives weightage to the previous knowledge of particles.

Each bird has dimensions equal to the number of unknown variables in the problem. Therefore, one has to randomly initialize a swarm of n particles which is dispersed in d-dimensional search space. Each i^th particle has a position corresponding to each d-dimensional as x_i(t)=[x_i1,x_i2,…,x_id] and velocity of each i^th particle for all d-dimensions as v_i(t)=[v_i1,v_i2,…,v_id], where t shows the number of current iteration. The first term of velocity equation V_ij is inertial component or previous velocity of particle. It controls the drastic change in particle’s velocity. Second term is cognitive component of nostalgia of particle; Y^t_ij is the personal best position of each particle. It indicates previous best position obtained by that particle. During travelling through search space, each bird stores position of variables associated with best cost function that is lowest one. This value is known as local best position of the swarm. Another best is the global best position, which is the lowest of all local best positions. Third term indicates global best and is known as social component of the swarm. G^t_ij indicates global best position of whole swarm. It evaluates performance relative to whole swarm [16].

The standard PSO equations accelerate the swarm to explore for global minimum fitness value in the search space. For continuous PSO formulation there are mainly two basic approaches namely inertia weight approach (IWA) and constricted factor approach (CFA). In the present study, IWA is used and inertia weight w^t decreases linearly during the iteration. The governing equation is given below:

where w_max is initial weight, w_min is final weight, t_max is maximum number of iterations and t is the current iteration number. It was observed by researchers that w_max= 0.9 and w_min= 0.4 [17] are independent of the nature of a problem. Moreover, initially setting w to a high value performs extensive exploration and gradually decreasing it to w_min=0.4 would make it quite exploitative system afterwards. Therefore, w_max=0.9 and w_min=0.4 are chosen in this study. Basically, if we compare PSO with hybrid GA(SA), then we can say that GA is used for exploration while SA uses exploitation, therefore their hybrid produces good convergence results. In the present study, PSO is inherently capable of utilizing exploration and exploitation due to cognitive (second term of Eq.(1)) and social (third term of Eq.(1)) terms. Variation of inertial weights is playing this vital role in which initially exploration has more weightage and with iterations exploitations takes over [18].

3.2 PSO Parameter Dependence

PSO is highly dependent on its parameters and its efficiency is greatly affected by the choice of its parameters values. To optimize parameters, we have tested different values; to check which perfectly suits the current problem.

3.3 Fitness Function

In this problem, LPO based on flattening of power profile is performed with the constraint that k_eff value must be greater than one. Power peaking factor is consequently minimized and one can have more burnup of fuel, while satisfying safety constraints. As normalized power of all FAs throughout the 1/8^th core is determined, there is a need to minimize curvature of this power profile. The objective or cost function 'J’ for this case can be written as:

P_a is normalized power and dim is number of fuel assemblies present in the core. For assembly power normalization procedure, the average power produced per fuel assembly for the entire core has been used. In this study, it is ensured that reactor remains critical. Consequently, k_eff should be greater than one, has been used as constraint condition.

3.4 Discretization using Random Keys

In this work, a variant of the basic PSO developed by Eberhart & Kennedy is utilized to solve this combinatorial problem [15]. In the standard PSO, position of particles is continuous, which has to be converted into discrete numbers that corresponds to fuel assemblies. Consequently, Particle swarm optimization with random keys is employed here for this conversion [16].

In case of PWR, each particle of the swarm is representing a fuel loading pattern. The random keys model encodes and decodes a continuous vector with real numbers. The random keys model proposed by Bean made use of Single Machine Scheduling Problem approach. In PSO with random keys method, the continuous updated position of particles obtained from PSO is sorted out. The original position index number of each entry in the sorted array is thus a discrete number corresponding to fuel assemblies. For example, a continuous key sequence of 21 positions is shown in Fig.2 (a). Since, 0.151 is the lowest number and it is in the twentieth (20^th) place so first entry of decoding sequence will be 20. In this way, discretization of this position vector for all particles is carried out into integers. Furthermore, these indices are used to choose FAs in a specific order without changing their fixed composition.

Figure 2Full-screen View

(Color online) (a) Conversion of Continuous Position to Discrete Position for 21 FAs; (b) Assembly wise profile of PWR CHASNUPP reactor at core mid-plane cross section

3.5 PSO with Random Keys applied to Nuclear LP Optimization problem

In the current LPO task, there are a total of 121 FAs in PWR reactor core but design is such that power distribution should be symmetric around the reactor. This reduces complexity of the optimization problem. As mentioned earlier, the reactor core is 1/8^th symmetric. Figure 2(b) shows both four- fold symmetry as well as eight-fold symmetry. So, the problem is simplified and left with only 21 FAs to be permutated in 1/8^th part of the reactor core.

In the PSO with random keys algorithm, the position and velocity of all particles are randomly initialized. The position of particles is initiated such that it almost covers the whole search space of twenty one fuel assemblies. Then, the initial fitness of all particles is evaluated after converting continuous positions into integers ranging from 1 to 21.

The local best and global best are stored. After this, main loop of PSO is started, in which fitness of all particles is evaluated. New global best position of the swarm and local best positions of all the particles are then computed and saved. The velocity and positions are updated based on decreasing inertial weight approach till stopping criteria is met. Following stopping criteria have been set:

· Maximum number of generations is met;

· The fitness value is achieved (or lesser than 10^-4);

· Cost function does not improve for further 50 generations.

A detailed flowchart of this process is shown in Fig.3(a).

Figure 3Full-screen View

(Color online) (a) Schematic diagram of proposed FL-Inw-PSO using MCRAC code; (b) Variation of Reactor’s Multiplication factor value with the PSO iteration count

4.1 Swarm Size based Analysis of FL-inw-PSO

The number of particles in a swarm is very important. It depends on nature of problem. If swarm size is increased, initial diversity of swarm is enhanced provided that uniform initialization scheme is used covering the whole search space. In this work, different swarm sizes are tested in order to reach an optimum number.

4.2 Multiplication factor versus iterations of FL-inw-PSO

Figure 3(b) shows trend of the multiplication factor versus iterations for the best optimized LP for all swarms. It shows multiplication factor value variations at the end of cycle of the first core loading. In the middle, there is a peak indicating that fuel is not effectively burnt and one can extract more energy even at EOC. After some iteration, converged pattern is reached, as there is a constant curve of k_eff.

4.3 Effect of Swarm size on Fitness behavior

The propose FL-inw-PSO code has been executed for a number of population sizes. Figure 4(a) represent evolutionary progress plots of PSO algorithm convergence for population sizes 100, 300 and 500. It can be observed that in this case, population size of 500 shows the best convergence. It indicates fitness behavior with generations (time) for different swarms. As swarm size is increased, there is a greater probability of avoiding local minima. It can be seen from Figures 4(a) that global minimum is found at 37^th iterations. After this, the whole swarms moves towards this minimum value. For small swarm size, a value of 3.8164 is found. For medium sized swarm, fitness value is further refined to 3.7622.

Figure 4Full-screen View

(Color online) (a) Evolutionary progress plot during convergence of PSO algorithm; (b) Convergence by linearly varying inertial weights scheme for independent runs

It can be observed from Table 3 that for large sized swarm, best fitness value of 3.5679 is achieved with corresponding k_eff as 1.0007. These results obtained by applying the proposed FL-inw-PSO to LPO of nuclear reactor core are comparable to those obtained by existing optimization technique. The best initial loading pattern of Chashma nuclear power plant with flattest power profile is achieved by FL-inw-PSO technique.

A statistical analysis is conducted by executing the same code for ten (10) independent runs for each population size of 100, 300 and 500 and the resultant minimum fitness achieved is shown in Fig.4(b). It can be observed from this graph that the proposed algorithm, FL-inw-PSO, remains stable under independent experiments.

4.4 Normalized power profile

This converged pattern obtained by FL-inw-PSO has flattest power profile. Fig.5(a) shows assembly-wise normalized power profile of 1/8th core of the Chashma nuclear reactor which is compared with previously published results with optimization by genetic algorithms and hybrid approach of genetic algorithms and simulated annealing. This shows that power of all the FAs is nearly approaching to one while maximum peaking value is 1.19. It also demonstrates a good comparison with the other techniques. There are 3D representations of the random and optimized assembly power profiles for reload pattern the nuclear reactor core in Fig.5(b). These simulations for power profile flattening were carried out without employing any fuel management options such as control rod positioning or changes in boron concentrations etc. This is standard procedure already widely used as in various cited research works [19].

Figure 5Full-screen View

(Color online) (a) Normalized power profile of converged LP using FL-inw-PSO; (b) 3D view of assembly profile of the initial and optimized loading pattern of the nuclear reactor core

4.5 Power peaking factor variation versus iterations

The variation of power peaking factor with iterations is presented in Figure 6(a). Number of particles for this simulation was chosen to be 300. It achieved a lowest peaking factor for which corresponding fitness value was not so good. Therefore, swarm moved towards best fitness for which power peaking is 1.19.

Figure 6Full-screen View

(a) Power peaking factor variations with PSO iterations; (b) Fitness convergence by linearly varying inertial weights using PSO-SA algorithm

4.6 Parameter setting of PSO

Selection of appropriate parameters for PSO is an important task [20], for our problem, tuning of parameters, c₁ and c₂ is required for one inertia weight scheme. Various values of these parameters have been tried and tested against fitness value and convergence as listed in Table 4. This variation includes value of c₁ in range 0.4-0.7 while that of c₂ in range 1.1-1.7. The best fitness is achieved for set of c₁ and c₂ as (0.5, 1.4) and (0.7, 1.4). All these experiments were executed for 500 number of particles and 100 generations keeping linearly decreasing inertia weight scheme in all cases of proposed FL-inw-PSO.

4.7 Variation of inertial weight schemes

Different inertia weight schemes have been implemented to study the convergence behavior of fitness function, namely linearly decreasing inertia weights, random inertia weights, and SA based inertial weights taken from [21]. Results are presented in Table 5. All these runs have been carried out using same parameters of PSO and for population size of 100 for study purpose only.

4.8 Comparison of varying inertial weights approach with standard PSO

In this work, continuous PSO has been converted into discrete PSO by using Random Key approach as discussed earlier. Along with that, varying inertial weights approach has been taken into account, which generated converged results which are in good agreement with GA(SA) hybrid approach based results. Without inertial weight variation, results were not satisfactory and convergence at some other minima was observed.

4.9 PSO hybrid with Simulated Annealing

Due to slower convergence of PSO random key with varying inertial weights algorithm, simulated annealing (SA) algorithm is added and thereby implemented this hybrid PSO-SA technique. Basically in this hybrid scheme, PSO is used for exploration and SA is employed for exploitation. SA algorithm is called when there is no further improvement in the fitness function value for a sufficient number of generations. PSO-SA scheme has generated good results with faster convergence and achieving the global optimal value more frequently than PSO alone. Fitness convergence results are presented in Figure 6(b).