Hybrid Grey Wolf Optimizer Based Optimal Capacitor Placement in Radial Distribution Systems

Figures & data

Figure 1. One-line diagram of a RDS with shunt capacitor $Q_c.$

Figure 2. Hierarchy of a pack of grey wolves [29].

TABLE 1. Assumed capacitor sizes and cost/(kVAR-year).

j	1	2	3	4	5
Capacitor size, Qc (kVAR)	150	300	450	600	750
Cost ($/kVAR-year)	0.500	0.350	0.253	0.220	0.276
j	6	7	8	9	10
Capacitor size, Qc (kVAR)	900	1050	1200	1350	1500
Cost ($/kVAR-year)	0.183	0.228	0.170	0.207	0.201

Figure 3. One-line diagram of the 34-bus RDS.

TABLE 2. Simulation results and comparison for the 34-bus RDS.

Description	Base Case	GA [34]	ACO [9]	ET [35]	MMS [26]	SAMHBMO [36]	HGWO
Total loss(kW)	221.73	164.96	164.508	161.33	160.92	159.709	160.47
Vmin (p.u)	0.9416	0.9478	0.9501	0.9506	0.9506	0.9501	0.9505
CLocation (Bus no.)	–	5, 9, 12, 22, 26	9, 19, 25	8, 18, 25	8, 12, 18, 21, 24, 25, 28	9, 16, 20, 23	8, 18, 23
CSize (kVAR)	–	300, 300, 300, 600, 300	450, 450, 1050	1050, 750, 750	150, 450, 450, 600, 150, 450, 300	600, 600, 600, 600	750, 900, 750
Total kVAR	–	1,800	1,950	2,550	2,550	2,400	2,400
Cost of capacitor ($/(kVAR-year))		552.00	467.1	653.54	728.55	528	578.7
Cost of kW loss ($/year)	37,250.64	27,713.28	27,637.344	27,103.44	27,034.56	26,831.11	26,958.96
Total cost		28,265.28	28,104.44	27,756.84	27,763.11	27,359.11	27,537.66
Net savings ($/year)		8,985.36	9,146.20	9,493.8	9,487.53	9,890.43	9,712.98
Savings (%)		24.12	24.55	25.49	25.47	26.55	26.08
Worst cost ($/year)	--	NA	NA	NA	NA	NA	27,603.00
Best cost ($/year)	--	NA	NA	NA	NA	NA	27,537.66
Mean cost ($/year)	--	NA	NA	NA	NA	NA	27,538.50
Std. deviation	--	NA	NA	NA	NA	NA	30.1568

NA = not available.

Figure 4. Voltage profile of the 34-bus RDS without and with capacitor compensation.

Figure 5. Typical convergence curves of the GWO and HGWO for the 34-bus RDS.

Figure 6. One-line diagram of the 69-bus RDS.

TABLE 3. Simulation results and comparison for the 69-bus RDS.

Description	Base Case	Heuristic method [17]	DSA [18]	TLBO [21]	GSA [11]	IIV [27]	HGWO
Total loss (kW)	225	148.48	147.00	146.35	145.9	145.22	145.22
Vmin (p.u)	0.9092	0.9305	0.9317	0.9313	0.9511	0.9308	0.9308
CLocation (Bus no.)	–	8, 58, 60	61, 15, 60	12, 61, 64	26,13,15	11, 17, 61	11, 17, 61
CSize (kVAR)	–	600, 150, 1050	900, 450, 450	600, 1050,150	150, 150, 1050	300, 300, 1200	300,300,1200
Total kVAR	–	1800	1800	1800	1350	1800	1800
Cost of capacitor ($/(kVAR-year))	–	446.4	392.4	446.4	389.4	414	414
Cost of kW loss ($/year)	37,800	24,944.64	24,696	24,586.8	24,511.20	24,396.96	24,396.96
Total cost		25,391.04	25,088.4	25,033.2	24,900.6	24,810.96	24,810.96
Net savings ($/year)		12,408.96	12,711.6	12,766.8	12,899.4	12,989.04	12,989.04
Savings (%)		32.83	33.63	33.77	34.13	34.36	34.36
Worst cost ($/year)	--	NA	NA	NA**	NA	NA	25,253.00
Best cost ($/year)	--	NA	NA	NA**	NA	NA	24,810.96
Mean cost ($/year)	--	NA	NA	NA**	NA	NA	25,014.0
Std. deviation	--	NA	NA	NA**	NA	NA	121.611

NA = not available. NA**= not applicable.

Figure 7. Voltage profile of the 69-bus RDS without and with capacitor compensation.

Figure 8. Typical convergence curves of the GWO and HGWO for the 69-bus RDS.

Figure 9. One-line diagram of the 119-bus RDS.

TABLE 4. Simulation results and comparison for the 119-bus RDS.

Description	Base Case	HSA-PABC [11]	Heuristic Approach [34]	ABC [7]	FPA [15]	Improved Harmony Search [25]	HGWO
Total loss (kW)	1,294.35	856.96	854.42	853.39	844.47	843.146	836.97
Vmin (p.u)	0.8688	0.9132	0.9107	0.9088	0.9002	0.9020	0.9089
CLocation (Bus no.)		99, 111, 111, 111, 86, 76, 108, 49, 71, 50	C at 15 locations	32, 35, 40, 50, 70, 73, 79, 105, 106, 109, 110	39, 43, 70, 74, 86, 91, 107, 109, 118	39,43,70, 74, 86, 91,107, 109,118	41,49,51,59,73, 78,96,106,110
CSize (kVAR)		1000, 1300, 1500, 550, 1350, 1450, 1350, 1500	9750 at 15 locations	850,1050,1300, 800, 550, 1300,1200,700 250,800,1200	1500, 600, 500, 1050, 900, 1500, 700, 500,1050	1500,1000, 1000, 1000, 900,1500 850,850,1200	1050,900,1050,1050, 1500,1500,1200, 1500, 1350
Total kVAR		10,000	9,750	10,000	8,300	9,800	11,100
Cost of capacitor ($/(kVAR-year))		NA	NA	NA	NA	NA	2,270.85
Cost of kW loss ($/year)	217,450.8	143,969.28	143,374.56	143,369.52	141,870.96	141,648.53	140,610.96
Total cost		NA	NA	NA	NA	NA	142,881.81
Net savings ($/year)		NA	NA	NA	NA	NA	74,568.99
Savings (%)		NA	NA	NA	NA	NA	34.29
Worst cost ($/year)	--	NA	NA	NA	NA	NA	151,014.96
Best cost ($/year)		NA	NA	NA	NA	NA	142,881.81
Mean cost ($/year)	--	NA	NA	NA	NA	NA	147,703.68
Std. deviation	--	NA	NA	NA	NA	NA	1,791.75

NA = not applicable

Figure 10. Voltage profile of the 119-bus RDS without and with capacitor compensation.

Figure 11. Sample convergence curves of the HGWO and GWO for the 119-bus RDS.

TABLE 5. Simulation results and comparison for the 22-bus practical RDS.

Description	DSA [18]	CBO [39]	TLBO [21]	HGWO
Total Loss(kW) Ppeak / Pnom. / Plight	24.89 / 9.66 / 2.39	32.93 / 9.78 / 2.59	24.43 / 9.31 / 2.39	24.19 / 9.37 / 2.37
Vmin (p.u)	0.9701 / 0.9824 / 0.9909	0.9621 / 0.9816 / 0.9924	0.9712 / 0.9822 / 0.9903	0.9702 / 0.9811 / 0.9915
CLocation(Bus no.) CSize(kVAR) Cpeak / Cnom. / Clight	4 150 / 150 / 0 13 450 / 300 / 150 16 300 / 150 / 150 17 150 / 150 / 0	18 150 / 300 / 150 20 150 / 150 / 150	9 150 / 150 / 150 14 300 / 150 / 0 17 150 / 150 / 150 20 300 / 150 / 0	5 150 / 150 / 0 12 150 / 150 / 150 15 300 / 150 / 0 19 300 / 150 / 150
Total optimal kVAR	1050	450	900	900
Cost of capacitor ($)	3150	1350	2700	2700
Cost of energy loss ($)	5575.59	6360.82	5423.33	5418.67
Best total cost ($)	8725.59	7710.82	8123.33	8118.67
Worst total cost ($)	--	--	8158.51	8187.37
Mean total cost ($)	--	--	8134.65	8138.17
Std. deviation	--	--	0.02058	0.1075

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