Fuel loading pattern optimization of a pressurized water reactor using varying internal weights based particle swarm optimization

Fuel loading pattern optimization of a pressurized water reactor using varying internal weights based particle swarm optimization 1 corresp first-author Corresponding author aneelaz@pieas.edu.pk (Aneela Zameer Jaffery) Corresponding author aneelaz@pieas.edu.pk (Aneela Zameer Jaffery) aneelaz@pieas.edu.pk 2 1 3 Department of Computer & Information Sciences, Pakistan Institute of Engineering & Applied Sciences (PIEAS), Nilore, Islamabad 45650, Pakistan. Department of Computer & Information Sciences, Pakistan Institute of Engineering & Applied Sciences (PIEAS), Nilore, Islamabad 45650, Pakistan. Department of Physics & Applied Mathematics, Pakistan Institute of Engineering & Applied Sciences (PIEAS), Nilore, Islamabad 45650, Pakistan. Department of Physics & Applied Mathematics, Pakistan Institute of Engineering & Applied Sciences (PIEAS), Nilore, Islamabad 45650, Pakistan. Department of Nuclear Engineering, Pakistan Institute of Engineering & Applied Sciences (PIEAS), Nilore, Islamabad 45650, Pakistan. Department of Nuclear Engineering, Pakistan Institute of Engineering & Applied Sciences (PIEAS), Nilore, Islamabad 45650, Pakistan. Fuel reload pattern optimization is essential for attaining maximum fuel burnup for minimization of generation cost while minimizing power peaking factor (PPF). The aim of this work is to carryout detailed assessment of Particle Swarm Optimization (PSO) in the context of fuel reload pattern search. With astronomically large number of possible loading patterns, the main constraints are limiting local power peaking factor, fixed number of assemblies, fixed fuel enrichment, and burnable poison rods. In this work, initial loading pattern of fixed batches of fuel assemblies is optimized by using Particle Swarm Optimization technique employing novel feature of varying inertial weights with the objective function to obtain both flat power profile and cycle keff >1. For neutronics calculation, PSU-LEOPARD generated assembly depletion dependent group constants based ADD-files are used. The assembly data description file generated by PSU-LEOPARD is used as input cross-section library to MCRAC code, which computes normalized power profile of all fuel assemblies of PWR nuclear reactor core. The standard PSO with varying inertial weights is then employed to avoid trapping in local minima. A series of experiments have been conducted to obtain near optimal converged fuel loading pattern of 300MWe PWR Chashma reactor. The optimized loading pattern is found in good agreement with results found in literature. Hybrid scheme of PSO with Simulated Annealing has also been implemented and resulted in faster convergence. PWR loading pattern optimization PSO PPF in-core fuel management 1. Introduction 1 The fuel loading pattern optimization problem for nuclear reactors, is a combinatorial problem with the number of possible patterns increasing exponentially with the number of fuel assemblies of a nuclear reactor. Moreover, Loading Pattern Optimization (LPO) problem have characteristic high dimensionality and non-linearity with highly discontinuous search space. For a typical PWR core consisting of 121 fuel assemblies (FA), the possible number of loading patterns is of order ~10 200 . However, by invoking various symmetries in a nuclear reactor, this number reduces to ~10 25 , which is still a huge number. It is impossible to evaluate power profiles for all loading patterns (LPs) exhaustively. Therefore, recently computational intelligence techniques have been applied to obtain optimum core loading pattern, as discussed below [ 1 1 - 8 8 1-8 ]. Initially, the loading pattern (LP) problem was solved manually by the plant experts based on previous experience. Meneses introduced PSO to optimize loading pattern of Angra 1 nuclear power plant [ 1 1 ]. Their work showed that a standard PSO can be applied to discrete problem of fuel loading optimization and reported satisfactory convergence of loading pattern. Babazadah et. al applied PSO to solve multi objective problem of fuel loading to VVER nuclear power reactor [ 2 2 ]. They introduced a new discrete model for the position of particles in PSO and obtained optimized results without violating safety constraints. Zameer et. al performed initial loading optimization studies on loading pattern and compared simulated annealing with genetic algorithm. Their studies showed that Genetic Algorithm becomes stagnated while finding LPs [ 3 3 ]. So, they proposed a hybrid Genetic algorithm and Simulated Annealing (GA-SA) technique which avoids stagnation at local minima and efficiently computes optimized LP. Augusto et. al also carried out research work using PSO and introduced a new technique of communication among particles known as dynamic topology PSO [ 4 4 ]. Their results were better as compared to the other meta-heuristic techniques presented in the literature. Khoshahval et. al also conducted study on LPO using PSO and found better results as compared to SA and GA [ 5 5 ]. On the other hand, Liu et. al proposed improved pivot particle swarm optimization method and obtained 9.6% increase in multiplication factor, while decreasing power peaking factor by 0.6% for first core loading of Daya Bay nuclear power plant [ 6 6 ]. Karahroudi et. al performed multi-objective optimization of first core of nuclear reactor by using genetic algorithm [ 7 7 ]. Similarly, Yadav et. al applied PSO to solve two cycles of a typical PWR core using PRISHA model [ 8 8 ]. They achieved reasonably low peaking factors for both cycles and found that results are insensitive to number of particles and number of iterations used in PSO. However, they had to perform a number of experiments to reach at best fitness parameter. Poursalehi et. al proposed Harmony Search (HS) algorithm to search optimal loading pattern for a PWR nuclear reactor [ 9 9 ]. Recently, Hill and Parks solved in-core fuel management by using Tabu search and outperformed the existing techniques for different objective functions [ 10 10 ]. The LPO problem typically includes maximization of fuel discharge burnup by improving effective multiplication factor of core, minimization of power peaking factor etc. This also includes that safety related factors of nuclear power plant are within permissible limits. The vital safety related factors of a plant typically include power peaking factor of core, critical heat flux and linear power density of fuel. The general problem of in-core fuel optimization depends on the fuel enrichment, number of burnable poison (BP) rods and fuel loading pattern. But fuel enrichment and BP rods configuration are determined before LPO. Generally, meta-heuristic optimization techniques work efficiently to achieve optimized solution in engineering applications and their hybrids with other algorithms may further enhance the efficiency and improve the quality of optimal solutions [ 11 11 ]. The loading pattern optimization is a combinatorial problem, while standard particle swarm optimization is continuous, therefore one may not apply standard PSO directly. One has to transform continuous PSO into discrete one. PSO is an efficient computational technique based on position and velocity of particles. It is less complex with fast searching speed. But its performance is greatly influenced by its parameters and it is often trapped in local minima. Due to these drawbacks and continuous nature of PSO, its use is not common for LP problems. However, researchers have applied PSO to LP problem by incorporating different changes in its standard equations. In this work, the problems of standard PSO when applied to LPO are being tackled by using random keys encoding and decoding schemes to convert continuous PSO into discrete one. In this study, a novel variant of PSO using Random Keys with linearly decreasing inertia weights for optimized fuel loading pattern weights (FL-inw-PSO) is proposed to search for possible flattest power profile. Various inertia weight schemes have been implemented and compared and hybrid algorithm of PSO with Simulated Annealing (SA) has also been implemented to further improve the efficiency and convergence. The rest of this paper is organized as follows: Section 2 gives brief description of the reference reactor used in this study. This section also discusses neutronics codes description and computational intelligence technique used in this work along with mathematical formulation of fitness function. In section 3, detailed results and analysis of multiplication factor, fitness function and power peaking factor versus number of iterations are discussed. Finally, concluding remarks of this work are given in Section 4. 2. Materials and Methods 1 In this section, description of the reference reactor, methods, neutronics codes, and PSO code used are discussed. 2.1 CHASNUPP 300 MWe core Design Specifications 2 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 UO 2 . 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 Table-1 . 1 1  Parameters 1 1 Value 1 1 Net power (MWe) 1 1 300 1 1 Number of FAs 1 1 121 1 1 Fuel assembly grid 1 1 15×15 1 1 Fuel rods per assembly 1 1 204 1 1 Control & BP rods guide thimbles per assembly 1 1 20 1 1 In-core instrumentation position per assembly 1 1 1 1 1 Core active length (mm) 1 1 2900 1 1 Equivalent core diameter (mm) 1 1 2486 1 1 Core Inlet temperature ( <math id="M1"><mrow><mo>°</mo><mtext>C</mtext></mrow></math> ) 1 1 288 1 1 Core Outlet temprature ( <math id="M2"><mrow><mo>°</mo><mtext>C</mtext></mrow></math> ) 1 1 315 1 1 Fuel material 1 1 UO 2 1 1 Clad material 1 1 Zr-4 1 1 Moderator 1 1 Water 1 1 Core active length (m) 1 1 2.9 1 1 Equivalent core diameter (m) 1 1 2.486 1 1 Uranium mass in first loading (Ton) 1 1 35.92 1 1 # of FAs corresponding to Enrichment (%) (2.4, 2.67, 3.0) 1 1 41,40,40 PWR core of CHASNUPP reactor is symmetric. It has 1/4 th symmetry as well as 1/8 th symmetry. Figure 1(a) 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 Table 1 . 2.2 Power Profile Evaluation 2 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. 2.2.1 PSU-LEOPARD 3 In this project, normalized power of fuel assemblies is evaluated by using industrial standard neutronic codes package developed at Pennsylvania state university named as Penn State Fuel Management Package (PFMP). It includes PSU-LEOPARD and Multiple Cycle Reactor Analysis Code (MCRAC) computer codes. PSU-LEOPARD is similar to LEOPARD with additional feature of polynomial representation of cross sections calculated by this code [ 12 12 , 13 13 ]. It consists of modified MUFT and SOFOCATE models. MUFT model calculates fast group constants by dividing energy range into 54 energy groups while its energy range is from 0 eV to 10 eV with 0.625 eV as boundary line between fast and thermal groups. SOFOCATE model which has 172 energy levels of thermal energy range computes thermal group constants. These constants are generated by using diffusion theory calculations and saved in form of polynomials as assembly data description file compatible with MCRAC code [ 13 13 , 14 14 ]. There are 21 FAs to be permutated whose corresponding ADD-files are listed in Table 2 Table 2 . 1 1 # of BP rods per assembly 1 1 Enrichment (w%) 1 1 ADD file 1 1 Fuel Bundle ID 1 1 - 1 1 2.4 1 1 101 1 1 1,3,5,8,10,14,16,18 1 1 16 1 1 2.4 1 1 103 1 1 2 1 1 - 1 1 2.67 1 1 105 1 1 12 1 1 16 1 1 2.67 1 1 110 1 1 4,9,11,15 1 1 4 1 1 2.67 1 1 115 1 1 6 1 1 - 1 1 3.0 1 1 120 1 1 7,13,20,21 1 1 2 1 1 3.0 1 1 125 1 1 17 1 1 4 1 1 3.0 1 1 130 1 1 19 1 1 - 1 1 Reflector 1 1 210 1 1 - 2.2.2 Multiple Cycle Reactor Analysis Code 3 The Multiple Cycle Reactor Analysis Code is a 2-D, two-group diffusion based code which is capable of automatic depletion and multi cycle analysis. Its input is cross sections in the form of polynomials. PSU-LEOPARD generates ADD-file compatible with MCRAC input. MCRAC performs detailed diffusion theory calculation when PSU-LEOPARD generated ADD-files and loading pattern of FAs as input is fed to it. It then generates assembly-wise normalized power profile of all FAs. 3. Particle Swarm optimization based Model 1 The methodology developed in this work is presented in block diagram of Figure 1(b) Figure 1(b) . At the start, the macroscopic group-constants are generated using PSU-LEOPARD for all types of fuel assemblies appearing in the core. A unique identity number of ADD-file is assigned to each fuel assembly such as 101, 103, 105 etc. The parameters appearing in the fitness function are evaluated using the MCRAC code. 3.1 Particle Swarm Optimization 2 In 1995, Eberhart & Kennedy proposed Particle Swarm Optimization technique to solve non-linear continuous functions [ 15 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 16 ]. This swarm of birds moves based on basic equations of PSO as given below: <math display="block" id="M3"><mrow><msubsup><mi>V</mi><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>=</mo><msup><mi>W</mi><mi>t</mi></msup><msubsup><mi>V</mi><mrow><mi>i</mi><mi>j</mi></mrow><mi>t</mi></msubsup><mo>+</mo><msub><mi>c</mi><mn>1</mn></msub><msubsup><mi>r</mi><mrow><mn>1</mn><mi>j</mi></mrow><mi>t</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow><mi>t</mi></msubsup><mo>−</mo><msubsup><mi>X</mi><mrow><mi>i</mi><mi>j</mi></mrow><mi>t</mi></msubsup></mrow><mo>)</mo></mrow><mo>+</mo><msub><mi>c</mi><mn>2</mn></msub><msubsup><mi>r</mi><mrow><mn>2</mn><mi>j</mi></mrow><mi>t</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>G</mi><mrow><mi>i</mi><mi>j</mi></mrow><mi>t</mi></msubsup><mo>−</mo><msubsup><mi>X</mi><mrow><mi>i</mi><mi>j</mi></mrow><mi>t</mi></msubsup></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math> <math display="block" id="M4"><mrow><msubsup><mi>X</mi><mi>i</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>=</mo><msubsup><mi>X</mi><mi>i</mi><mi>t</mi></msubsup><mo>+</mo><msubsup><mi>V</mi><mi>i</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>,</mo></mrow></math> where i= [1, 2,..., n] and j= [1, 2,..., d] . c 1 and c 2 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 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: <math display="block" id="M5"><mrow><msup><mi>W</mi><mi>t</mi></msup><mo>=</mo><msub><mi>W</mi><mrow><mtext>max</mtext></mrow></msub><mo>−</mo><mfrac><mrow><mrow><mo>(</mo><mrow><msub><mi>W</mi><mrow><mtext>max</mtext></mrow></msub><mo>−</mo><msub><mi>W</mi><mrow><mtext>min</mtext></mrow></msub></mrow><mo>)</mo></mrow></mrow><mrow><msub><mi>t</mi><mrow><mtext>max</mtext></mrow></msub></mrow></mfrac><mo>*</mo><mi>t</mi><mo>,</mo></mrow></math> 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 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 18 ]. 3.2 PSO Parameter Dependence 2 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 2 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: <math display="block" id="M6"><mtable columnalign="left"><mtr><mtd><mi>J</mi><mo>=</mo><munderover><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>a</mi><mo>−</mo><mn>1</mn></mrow><mrow><mtext>dim</mtext></mrow></munderover><msup><mrow><mo>(</mo><mrow><msub><mi>P</mi><mi>a</mi></msub><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup><msub><mi>k</mi><mrow><mtext>eff</mtext></mrow></msub><mo>></mo><mn>1</mn></mtd></mtr><mtr><mtd><mtext>Subject to constraint          </mtext><msub><mi>k</mi><mrow><mtext>eff</mtext></mrow></msub><mo>></mo><mn>1</mn></mtd></mtr></mtable></math> 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 2 In this work, a variant of the basic PSO developed by Eberhart & Kennedy is utilized to solve this combinatorial problem [ 15 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 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) 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. 3.5 PSO with Random Keys applied to Nuclear LP Optimization problem 2 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) 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) Fig.3(a) . 4. Results and Discussion 1 To analyze the performance of the proposed FL-inw-PSO, a series of experiments have been performed for small, medium, and large sized swarm with c 1 =0.7, c 2 =1.4, and with a decrease in the value of inertial factor w from 0.9 to 0.4 linearly. These experiments were performed within 100 to 200 generations. If number of generations is increased, swarm convergence probability is enhanced. The proposed FL-inw-PSO algorithm is implemented in MATLAB and has to run many times due to random position and velocity initialization as well as stochastic nature of the intelligent technique. The iterations are treated as converged if the value of cost function becomes less than 10 -4 or the value of cost function does not improve further for 50 generations. Ten runs for all the three swarm sizes have been carried out. Detailed results are given in Table 3 Table 3 . It shows best fitness value and corresponding k eff value for each experiment. The time taken by each run is also mentioned. 1 1 Swarm Info 2 1 N pop = 100 Iterations=100 2 1 N pop = 300 Iterations =100 2 1 N pop = 500 Iterations =100 1 1 Serial # 1 1 Fitness Value/ k eff 1 1 t e (min) 1 1 Fitness Value/ k eff 1 1 t e (min) 1 1 Fitness Value/k eff 1 1 t e (min) 1 1 1 1 1 5.7325/1.0107 1 1 8.13 1 1 4.0140/1.0051 1 1 24.89 1 1 4.3209/1.0038 1 1 39.62 1 1 2 1 1 4.7256/1.0044 1 1 8.09 1 1 3.8912 /1.0056 1 1 23.78 1 1 4.2632/1.0023 1 1 39.14 1 1 3 1 1 4.8326/1.0073 1 1 8.01 1 1 5.8276/1.0153 1 1 24.07 1 1 3.9308/1.0038 1 1 38.92 1 1 4 1 1 4.8342/1.0059 1 1 7.80 1 1 3.9612/1.0051 1 1 24.09 1 1 5.1318/1.0076 1 1 43.59 1 1 5 1 1 4.4150/1.0068 1 1 7.98 1 1 3.7622 /1.0034 1 1 24.43 1 1 3.5809/1.0007 1 1 39.70 1 1 6 1 1 5.7180/1.0077 1 1 7.20 1 1 4.7289/1.0098 1 1 23.14 1 1 3.72017/1.0049 1 1 39.28 1 1 7 1 1 4.9535/1.0130 1 1 7.80 1 1 4.4996/1.0089 1 1 23.18 1 1 3.8845/1.00624 1 1 40.03 1 1 8 1 1 3.8164/1.0029 1 1 7.82 1 1 4.6673/1.0122 1 1 24.05 1 1 3.5679/1.0007 1 1 41.71 1 1 9 1 1 4.6020/1.0021 1 1 7.84 1 1 4.2846/1.0071 1 1 25.90 1 1 4.6682/1.0066 1 1 39.23 1 1 10 1 1 5.3540/1.0022 1 1 7.81 1 1 4.3231/1.0036 1 1 25.86 1 1 3.9043/1.0041 1 1 42.95 1 1 Max. fitness 1 1 5.7182 1 1 - 1 1 5.8276 1 1 - 1 1 5.1318 1 1 - 1 1 Avg. Fitness 1 1 4.8985 1 1 - 1 1 4.3958 1 1 - 1 1 4.0973 1 1 - 1 1 Min. fitness 1 1 3.8164 1 1 - 1 1 3.7622 1 1 - 1 1 3.5679 1 1 - The average, maximum, and minimum fitness function values are quoted in Table3 Table3 .The flattest power profile obtained has a fitness value of 3.567925 and its corresponding multiplication factor at the end of cycle (EOC) is 1.0007. 4.1 Swarm Size based Analysis of FL-inw-PSO 2 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 2 Figure 3(b) 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 2 The propose FL-inw-PSO code has been executed for a number of population sizes. Figure 4(a) 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) 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. It can be observed from Table 3 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) 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 2 This converged pattern obtained by FL-inw-PSO has flattest power profile. Fig.5(a) 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) 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 19 ]. 4.5 Power peaking factor variation versus iterations 2 The variation of power peaking factor with iterations is presented in Figure 6(a) 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. 4.6 Parameter setting of PSO 2 Selection of appropriate parameters for PSO is an important task [ 20 20 ], for our problem, tuning of parameters, c 1 and c 2 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 Table 4 . This variation includes value of c 1 in range 0.4-0.7 while that of c 2 in range 1.1-1.7. The best fitness is achieved for set of c 1 and c 2 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. 1 1 Experiment No. 1 1 c 1 1 1 c 2 1 1 Fitness value 1 1 1 1 1 0.4 1 1 1.1 1 1 4.7502 1 1 2 1 1 0.4 1 1 1.2 1 1 4.0880 1 1 3 1 1 0.4 1 1 1.3 1 1 4.1321 1 1 4 1 1 0.4 1 1 1.4 1 1 3.6732 1 1 5 1 1 0.5 1 1 1.1 1 1 4.6092 1 1 6 1 1 0.5 1 1 1.2 1 1 4.7011 1 1 7 1 1 0.5 1 1 1.3 1 1 5.1428 1 1 8 1 1 0.5 1 1 1.4 1 1 3.5679 1 1 9 1 1 0.6 1 1 1.2 1 1 4.1599 1 1 10 1 1 0.6 1 1 1.3 1 1 3.9177 1 1 11 1 1 0.6 1 1 1.4 1 1 3.6730 1 1 12 1 1 0.7 1 1 1.2 1 1 4.1392 1 1 13 1 1 0.7 1 1 1.3 1 1 3.9904 1 1 14 1 1 0.7 1 1 1.4 1 1 3.5679 4.7 Variation of inertial weight schemes 2 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 21 ]. Results are presented in Table 5 Table 5 . All these runs have been carried out using same parameters of PSO and for population size of 100 for study purpose only. 1 1   1 1 Inertia weight schemes 1 1 Min fitness 1 1   1 1 1 1 1 Linearly decreasing inertia weights 1 1 3.8919 1 1 26 1 1 2 1 1 Random inertia weights 1 1 4.2445 1 1 378 1 1 3 1 1 Simulated Annealing inertia weights 1 1 4.52121 1 1 40 4.8 Comparison of varying inertial weights approach with standard PSO 2 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 2 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) Figure 6(b) . 5. Conclusion 1 In this study, a random key based PSO method with linearly varying inertia weight approach has been developed and implemented to optimize initial fuel loading pattern of a PWR reactor core for flat power profile. MCRAC is used to compute the normalized power profile of the reactor core using libraries generated by PSU-LEOPARD. ADD-files generated by PSU-LEOPARD and loading pattern of FAs are used as input to MCRAC, which creates the normalized power profile of reactor core. It was found that randomly initialized swarm converges rapidly as number of particles (from 100 to 500) and iterations are increased with computational cost. A series of experiments have been carried out to reach at best fitness value. Hybrid technique of PSO-SA has been implemented and found effective to speed up the convergence rate and to reduce computational cost. The obtained loading pattern is consistent with the existing techniques. 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