Loading pattern optimization for a PWR using Multi-Swarm Moth Flame Optimization Method with Predator

Figures & data

Figure 1. A fuel loading pattern and k-infinity vector.

Figure 2. Core geometry used in optimization calculations.

Figure 3. Candidates of next moth position.

Figure 4. Generation of candidate moth and assignment of actual fuel inventory.

Figure 5. Flowchart of MFO with predator (MFO-P).

Figure 6. Flowchart of MSMFO-P.

Table 1. Fuel inventory used for optimization calculations.

U235 Enrichment (wt%)	Burnable Poison	Burnup (GWd/t)	Number of Fuel Assemblies
4	-	34.7	1
4	-	34.7	4
4	-	34.7	4
4	-	32.2	4
4	-	32.0	8
4	-	28.9	4
4	-	27.9	8
4	-	23.8	8
4	-	23.0	4
4	-	19.5	4
4	-	19.0	8
4	-	18.9	8
4	-	18.4	8
4	-	16.8	4
4	-	16.8	4
4	-	12.6	4
4	-	12.6	4
4	-	11.3	8
4	-	10.2	8
4	-	0.0	8
4	-	0.0	8
4	-	0.0	8
4	Gd	0.0	4
4	Gd	0.0	8
4	Gd	0.0	8
4	Gd	0.0	8
Total			157

Table 2. Calculation conditions used in the sensitivity analysis.

Case Number	NumMoth	Fdtype	FuncType	CoeffD	NumFlame	EatMoth	ViewRange	BornCoeff
0	100	1	1	1	100	10	0.5	10
1	1000	1	1	1	100	10	0.5	10
2	100	2	1	1	100	10	0.5	10
3	100	1	2	1	100	10	0.5	10
4	100	1	1	0.5	100	10	0.5	10
5	100	1	1	10	100	10	0.5	10
6	100	1	1	1	10	10	0.5	10
7	100	1	1	1	50	10	0.5	10
8	100	1	1	1	100	1	0.5	10
9	100	1	1	1	100	50	0.5	10
10	100	1	1	1	100	10	0.1	10
11	100	1	1	1	100	10	1	10
12	100	1	1	1	100	10	0.5	0.5
13	100	1	1	1	100	10	0.5	1

Figure 7. Result of sensitivity analysis of MSMFO-P.

Table 3. Calculation conditions of the conventional optimization methods.

SA	Initial temperature	1
	Cooling rate	0.99029/1000	calculations
	Number of assemblies in one swap	1–3
	Number of assembly swap	1–3
Hybrid GA	Number of populations	60
	Number of survived candidates after selection	50
	Number of binary swaps for random mutaion	5
	Tournament size	5
	Probability of random mutation	0.1
	Probability of crossover	0.99
PSO	Number of particles in one generation	500
	c1 in Equation (9)(9) $v_{ij}\left(t+1\right)=w^t{v}_{ij}\left(t\right)+c_1{r}_1\left(t\right)\left(p_{ij}\left(t\right)-x_{ij}\left(t\right)\right)+c_2{r}_2\left(t\right)\left(p_{gj}\left(t\right)-x_{ij}\left(t\right)\right)$(9)	2.4
	c2 in Equation (9)(9) $v_{ij}\left(t+1\right)=w^t{v}_{ij}\left(t\right)+c_1{r}_1\left(t\right)\left(p_{ij}\left(t\right)-x_{ij}\left(t\right)\right)+c_2{r}_2\left(t\right)\left(p_{gj}\left(t\right)-x_{ij}\left(t\right)\right)$(9)	2.4
	w_max in Equation (11)(11) $w^t=\frac{w_{max}-w_{min}}{t_{max}}t$(11)	0.9
	w_min in Equation (11)(11) $w^t=\frac{w_{max}-w_{min}}{t_{max}}t$(11)	0.4

Table 4. Optimization results of MSMFO-P.

Initial random seed	Objective function value [-]	Power peaking factor [-]	Cycle length [Gwd/t]	Assembly mximum burnup [Gwd/t]
1	−0.9888	1.3954	19.7769	47.9607
2	−0.9884	1.3988	19.7673	47.1154
3	−1.0038	1.3881	20.0765	47.5023
4	−0.9921	1.3981	19.8427	47.6035
5	−0.9998	1.3976	19.9952	47.1899
6	−0.9834	1.3944	19.6671	47.9761
7	−1.0041	1.3993	20.0819	47.6675
8	−0.9873	1.3974	19.7462	44.8057
9	−0.9927	1.3909	19.8532	47.4170
10	−1.0053	1.3889	20.1061	47.0931
Average (within limits)	−0.9946	1.3949	19.8913	47.2331
Standard deviation (within limits)	0.0080	0.0042	0.1603	0.9093

Table 5. Optimization results of SA.

Initial random seed	Objective function value [-]	Power peaking factor [-]	Cycle length [Gwd/t]	Assembly maximum burnup [Gwd/t]
1	−0.9709	1.3980	19.4185	45.8646
2	−0.9731	1.3876	19.4614	47.6073
3	−0.9555	1.3832	19.1109	47.6174
4	−0.9631	1.3803	19.2613	47.4817
5	−0.9661	1.3978	19.3221	47.3424
6	−0.9708	1.3932	19.4156	47.3677
7	−0.9581	1.3870	19.1625	46.3525
8	−0.9583	1.3804	19.1663	46.2128
9	−0.9544	1.3906	19.0888	47.9333
10	−0.9553	1.3960	19.1054	46.9831
Average (within limits)	−0.9626	1.3894	19.2513	47.0763
Standard deviation (within limits)	0.0072	0.0068	0.1440	0.6973

Table 6. Optimization results of Hybrid GA.

Initial random seed	Objective function value [-]	Power peaking factor [-]	Cycle length [Gwd/t]	Assembly maximum burnup [Gwd/t]
1	−0.9553	1.3959	19.1064	47.2429
2	−0.9871	1.39777	19.7429	45.6215
3	−0.9343	1.39846	18.6858	45.3398
4	−0.9601	1.39242	19.2019	47.1136
5	−0.9549	1.39954	19.0987	43.4144
6	−0.9603	1.39387	19.2069	46.8826
7	−0.9606	1.38456	19.2122	47.8374
8	−0.9585	1.3996	19.1704	46.6201
9	−0.9754	1.39984	19.5072	45.7462
10	−0.9832	1.39638	19.6642	45.4915
Average (within limits)	−0.9630	1.3958	19.2597	46.1310
Standard deviation (within limits)	0.0154	0.0047	0.3080	1.2798

Table 7. Optimization results of PSO.

Initial random seed	Objective function value [-]	Power peaking factor [-]	Cycle length [Gwd/t]	Assembly maximum burnup [Gwd/t]
1	−0.9534	1.3926	19.0684	47.1146
2	−0.9290	1.3969	18.5804	46.9788
3	−0.9243	1.3995	18.4853	47.1263
4	19.0132	1.4071	19.7507	47.8684
5	−0.9602	1.3965	19.2050	47.8062
6	−0.9593	1.3788	19.1855	47.7289
7	−0.9774	1.3998	19.5486	44.8491
8	−0.9599	1.3998	19.1975	47.6917
9	−0.9556	1.3939	19.1121	45.6608
10	−0.9218	1.3904	18.4361	46.2739
Average (within limits)	−0.9490	1.3942	18.9799	46.8034
Standard deviation (within limits)	0.0193	0.0067	0.3855	1.0179

Figure 8. Comparison of objective function values satisfying limits.

Figure 9. Example of generated loading patterns by (a) MSMFO-P, (b) SA, (c) Hybrid GA, (d) PSO. Value in the assembly indicates k-infinity.

Table 8. Number of L3P loading patterns in which fresh fuel assemblies without Gd are loaded core inboard in 10 trials.

MSMFO-P	SA	HybridGA	PSO
10	5	4	5

Table 9. Average and standard deviations of the L3P indexes in the last generation of MFO, MFO-P, MSMFO, and MSMFO-P.

	MFO	MFO-P	MSMFO	MSMFO-P
Average	112.62	145.06	161.59	182.33
Standard deviation	34.93	36.12	22.47	20.11
Relative standard deviation	0.31	0.25	0.14	0.11

Table 10. Average and standard deviations of the objective function of MFO, MFO-P, MSMFO, and MSMFO-P.

	MFO	MFO-P	MSMFO	MSMFO-P
Average	−0.9608	−0.9773	−0.9782	−0.9917
Standard deviation	0.0169	0.0152	0.0112	0.0101
Relative standard deviation	0.0176	0.0156	0.0115	0.0102

Figure 10. Averages of maximum L3P index of each method.