Interior search algorithm for optimal power flow with non-smooth cost functions

Figures & data

Table

while any stop criteria are not satisfied

find the $x_{gb}^j$

for i = 1 to n

if xgb

Apply Equation $x_{gb}^j=x_{gb}^{j-1}+r_n*\mathit{\lambda }$

else if r1 > α

Apply Equation $x_i^j=L{B}^j+(U{B}^j-L{B}^j)*r_2$

else

Apply Equation $x_{m,i}^j=r_2{x}_i^{j-1}+(1-r_3)*x_{gb}^j$

Apply Equation $x_i^j=2x_{m,i}^j-x_i^{j-1}$

end if

Check the boundaries except for decomposition elements.

end for

for i = 1 to n

Evaluate $f(x_i^j)$

Apply Equation $x_i^j=\left\{\begin{array}{cc}{x}_i^j\hfill & f(x_i^j)<f(x_i^{j-1})\hfill \\ x_i^{j-1}\hfill & \text{Else}\hfill \end{array}\right.$

end for

end while

Figure 1. System voltage profile improvement.

Figure 2. Minimization of fuel cost for Cases 1, 3, 5 and 9 using algorithms.

Table 1. Prohibited zones for the IEEE 30-bus test system

Bus	Prohibited zones
	Zone 1		Zone 2
	Min	Max	Min	Max
1	55	66	80	120
2	21	24	45	55
5	30	36	–	–
8	25	30	–	–
11	25	28	–	–
13	24	30	–	–

Table 2. Optimal results for Case 1 through Case 8

Control variable	Case 1	Case 2	Case 3	Case 4	Case 5	Case 6	Case 7	Case 8
PG1 (MW)	177.124	174.21	139.852	140.114	198.66	197.97	177.34	198.967
PG2 (MW)	48.9332	48.909	54.9959	54.9996	26.361	29.5042	45.00	25.93146
PG5 (MW)	21.3175	21.808	22.525	27.3675	16.568	16.2901	21.31	16.96848
PG8 (MW)	21.0006	25.412	34.8961	30.8941	10	10.0161	21.036	10.00089
PG11 (MW)	11.8605	11.932	18.0515	15.3592	10	10.0009	11.876	10.00077
PG13 (MW)	11.86	11.86	19.6782	23.0195	12.14	11.8677	11.86	11.86047
V1 (p.u)	1.1	1.0121	1.09642	1.00848	1.1	1.01667	1.1	1.1
V2 (p.u)	1.08651	1.0151	1.08066	1.01431	1.0826	1.02147	1.0854	1.08292
V5 (p.u)	1.05916	1.0084	1.05201	1.00702	1.0537	1.00844	1.0576	1.054499
V8 (p.u)	1.06733	1.0295	1.06035	1.03022	1.0605	1.02803	1.0644	1.061925
V11 (p.u)	1.1	1.0107	1.09922	1.01115	1.1	1.00725	1.1	1.098131
V13 (p.u)	1.08773	0.986	1.08825	0.97892	1.0932	0.98491	1.0947	1.093675
T6–9	1.06831	0.9694	0.99603	1.04552	0.9749	1.01175	1.0307	1.029261
T6–10	1.04151	0.9778	1.03433	1.07346	1.0273	1.00364	1.0055	1.047304
T4–12	1.00613	1.0361	1.03891	1.07783	0.9936	1.0378	0.9997	1.060403
T28–27	0.97014	1.055	0.99883	1.0311	0.9665	1.00065	0.9949	0.95653
Qc10 (Mvar)	4.99986	4.9998	1.76102	4.99683	4.8511	4.99965	4.9996	4.998825
Qc12 (Mvar)	4.99982	4.9999	2.72987	4.9996	4.9989	4.99999	4.6449	4.995023
Qc15 (Mvar)	2.86865	0.0109	4.08933	2.87864	3.0656	0.0198	0.0001	0.93368
Qc17 (Mvar)	3.67783	5	2.5313	4.99964	2,2085	4.99993	2.8502	4.59062
Qc20 (Mvar)	4.98223	0.0043	4.52856	0.82387	4.7414	1.50242	4.9985	4.999738
Qc21 (Mvar)	4.6327	5	3.29318	4.98853	4.8079	4.99998	4.412	4.019706
Qc23 (Mvar)	4.99987	4.9996	4.08493	4.99989	4.9987	4.99996	4.3356	4.998595
Qc24 (Mvar)	2.94032	5	4.422	4.99989	3.1112	4.9999	3.7032	2.84681
Qc29 (Mvar)	3.00097	1.801	2.21434	2.35222	0.8395	1.3373	2.7485	1.855214
Fuel cost ($/h)	799.2776	807.6408	646.2532	653.7671	823.1379	832.7090	799.3089	823.1673
Ploss (MW)	8.6959	10.7312	6.5982	8.3545	11.3341	14.2489	8.7113	11.3291
VD	1.8462	0.1273	1.6456	0.1286	1.7637	0.1288	1.8083	1.7768

Table 3. Optimal results for Case 9 through Case 11

Control variable (p.u.)	Case 9	Case 10	Case 11
P1 (p.u)	140.297965	157.331371	143.284824
P2 (p.u)	87.6032363	99.9998803	82.6737671
P3 (p.u)	44.7563272	42.7420443	45.3545803
P6 (p.u)	75.1485934	5.0982276	73.5050975
P8 (p.u)	459.99558	456.982543	461.334812
P9 (p.u)	95.9585096	96.472764	97.1573258
P13 (p.u)	362.149974	407.889603	362.561839
V1 (p.u)	1.03776821	1.01256065	1.03893849
V2 (p.u)	1.04215958	1.01401062	1.04252178
V3 (p.u)	1.03201601	1.00277245	1.03216019
V6 (p.u)	1.04688409	1.00529369	1.05108243
V8 (p.u)	1.06311898	1.00706659	1.06558728
V9 (p.u)	1.04143849	0.98281995	1.04086891
V13 (p.u)	1.0248601	1.00525444	1.02408603
T4–18	0.92646197	0.99631524	1.00526435
T4–18	0.93987275	0.98594086	0.96707268
T21–20	0.98669423	1.0185223	1.04337586
T24–25	0.99145917	0.94690761	0.97749794
T24–25	1.00138985	0.9499337	0.97352597
T24–26	1.01906685	1.03538627	0.96829392
T7–29	1.02246784	0.9963351	1.0188147
T34–32	0.99930487	1.0768902	1.06084883
T11–41	1.0367947	1.01744033	0.93391552
T15–45	1.03502693	0.91745731	0.96557376
T14–46	1.02917518	0.94443448	0.90735034
T10–51	0.9269382	1.06381535	0.95767122
T13–49	0.98087975	1.03201152	0.98827504
T11–43	0.92916229	0.95763177	0.96139549
T40–56	1.07384495	1.00908972	0.98846446
T39–57	0.95416665	0.9638778	1.06407405
T9–55	0.92324215	1.02381055	0.97052031
QC18	10.2325736	7.2915012	10.0362723
QC25	12.8902838	15.520675	12.8718025
QC53	11.197684	10.4411106	11.5559707
Fuel cost ($/h)	41,676.9466	41,939.7706	41,705.2941
Ploss (MW)	15.1102	15.7164	15.0722
VD	2.3863	0.9931	2.4171

Table 4. Control parameters of the related algorithms used in the tests

Algorithm	Value	Description	Description of method
BBO	pmutate = 0	Mutation probability	Biogeography-based optimization
	pmodify = 1	Habitat modification probability
	pmutate = 0.005	Initial mutation probability
DE	F = 0.8	Differential weight	Differential evolution
	Cr = 0.8905	Crossover probability
HS	HMS = 25	Harmony memory size	Harmony Search
	nNew = 20	Number of new harmonies
	HMCR = 0.9	Harmony memory consideration rate
	PAR = 0.1	Pitch adjustment rate
	FW_damp = 0.995	Fret width damp ratio
WDO	RT = 3	RT coefficient.	Wind driven optimization
	g = 0.2	Gravitational constant
	alp = 0.4	Constants in the update equation
	c = 0.4	Coriolis effect
	maxV = 0.3	Maximum allowed speed
ISA	α = 0.25		Interior search algorithm
Common parameters	Population size = 40 for each cases
	Maximum number of iterations = 300

Table 5. Comparison of the ISA with BBO, DE, HS and WDA for solving different OPF problems

Cases		ISA	BBO	DE	HS	WDA
Case 1	Best	799.2775	802.6928	799.310915	799.5643	799.6444
	Mean	799.2845	803.6076	799.5600	799.9286	799.8434
	Worst	799.3042	804.3370	799.8091	800.8656	800.1186
	Std.	0.0092	0.6950	0.3523	0.6259	0.1889
Case 2	Best	934.9252	974.3532	936.5690	961.0799	944.7191
	Mean	936.8553	989.0107	937.6885	964.2001	961.5295
	Worst	938.5149	999.0186	939.5740	967.0019	971.4390
	Std.	1.2327	9.6062	1.3193	2.4810	10.3380
Case 3	Best	646.2532	652.5699	647.4818	649.1088	650.6093
	Mean	649.3892	658.5834	648.0807	650.6525	652.0016
	Worst	651.6315	670.1644	648.8991	653.3658	654.3714
	Std.	2.2964	6.8418	0.7338	1.6863	1.7328
Case 4	Best	787.3417	846.4735	795.3084	818.3512	815.5853
	Mean	790.9931	860.2433	799.5446	821.7694	824.2396
	Worst	794.5488	870.3910	804.2022	826.0212	838.2845
	Std.	2.9132	9.9976	4.4619	3.3621	8.9931
Case 5	Best	823.1379	831.8466	823.9152	824.1363	824.0153
	Mean	823.2163	834.0031	825.5778	824.5204	825.1159
	Worst	823.4845	835.9777	827.9075	824.8977	827.2251
	Std.	0.1503	1.7894	2.0781	0.3854	1.2406
Case 6	Best	961.4677	1,023.68943	962.5552	991.9074	967.322769
	Mean	963.7583	1,036.19247	963.6568	992.9245	–
	Worst	967.9412	1,053.55143	964.5832	994.3956	–
	Std.	1.9093	11.0394	0.8379	1.0680	–
Case 7	Best	799.3089	802.1666	799.5587	799.7293	799.7948
	Mean	799.3232	803.6673	799.8470	800.0358	800.0453
	Worst	799.3389	804.5599	799.9946	800.5751	800.6144
	Std.	0.0115	1.0633	0.2497	0.3739	0.3377
Case 8	Best	823.1673	832.5560	823.9212	824.7796	824.3581
	Mean	823.2726	835.9265	825.4197	825.6587	825.0148
	Worst	823.4481	840.7193	826.9149	826.3168	826.3213
	Std.	0.1277	3.1244	1.2885	0.6697	0.8115
Case 9	Best	41,676.9466	41,721.2464	41,681.295	41,693.35813	–
	Mean	41,677.4271	–	–	–	–
	Worst	41678,2442	–	–	–	–
	Std.	0.5527	–	–	–	–
Case 10	Best	160,946.6871	163,475.793	161,166.85	161,747.272	–
	Mean	160,949.508	–	161,196.302	–	–
	Worst	160,953.5771	–	161,284.657	–	–
	Std.	3.6104	–	58.9034	–	–
Case 11	Best	41,705.2941	41,764.8019	41,708.4914	41,758.4312	–
	Mean	41,706.0507	–	–	–	–
	Worst	41,710.2813	–	–	–	–
	Std.	1.5435	–	–	–	–
No. of NC^*		0	53	26	56	72
Infeasibility rate (%)		0	24	12	25.5	33

Note: NC^* refers to how many times this algorithm has not converge for this case.

Table 6. Convergence speed of the ISA algorithm

Case	Cut-point 1	Cut-point 2	Cut-point 3	Cut-point 4
Case 1	99.95	99.99	99.99	99.99
Case 2	98.88	99.69	99.93	99.99
Case 3	99.43	99.65	99.84	99.98
Case 4	97.45	99.31	99.97	99.99
Case 5	99.87	99.95	99.98	99.99
Case 6	97.43	99.5	99.88	99.99
Case 7	99.96	99.99	99.99	99.99
Case 8	99.92	99.97	99.99	99.99
Case 9	99.94	99.98	99.99	99.99
Case 10	99.61	99.98	99.99	99.99
Case 11	99.98	99.98	99.99	99.99
Average	99.3109	99.8173	99.9582	99.9891

Table 7. Some of results obtained by different algorithms

Methods	Case 1	Case 3	Case 5	Case 9	Method description
ISA	799.2776	646.2532	823.1379	41,676.947	Interior search algorithm
ICA (Ghasemi, Ghavidel, Rahmani, Roosta, & Falah, 2014)	801.7784	647.5687	826.1275	41,710.4933	Imperialist competitive algorithm
TLA (Ghasemi et al., 2014)	801.6524	647.4413	825.6608	41,686.7915	Teaching learning algorithm
MICA (Ghasemi et al., 2014)	801.0794	647.134	824.6749	41,683.048	Modified imperialist competitive algorithm
SFLA (Niknam, Narimani, & Azizipanah-Abarghooee, 2012)	801.97	–	825.9906		Shuffle frog leaping algorithm
GSA (Duman, Guvenc, Sonmez, & Yorukeren, 2012)	–	–	–	41,695.8717	Gravitational search algorithm
PSO (Abido, 2002b)	–	647.69	–	–	Particle swarm optimization
PSO (Narimani, Azizipanah-Abarghooee, Zoghdar-Moghadam-Shahrekohne, & Gholami, 2013)	801.89	649.41	–	–	Particle swarm optimization
PSOGSA (Taylor, Radosavljevi, Klimenta, & Jevti, 2015)	800.4985	–	824.7096	–	Hybrid PSO/GSA

Ghasemi, M., Ghavidel, S., Rahmani, S., Roosta, A., & Falah, H. (2014). A novel hybrid algorithm of imperialist competitive algorithm and teaching learning algorithm for optimal power flow problem with non-smooth cost functions. Engineering Applications of Artificial Intelligence, 29, 54–69.10.1016/j.engappai.2013.11.003

 Web of Science ®Google Scholar

Niknam, T., Narimani, M. R., & Azizipanah-Abarghooee, R. (2012). Anewhybrid algorithm for optimal power flow considering prohibited zones and valve point effect. Energy Conversion and Management, 58, 197–206.

 Web of Science ®Google Scholar

Duman, S., Guvenc, U., Sonmez, Y., & Yorukeren, N. (2012). Optimal power flow using gravitational search algorithm. Energy Conversion and Management, 59, 86–95.

 Web of Science ®Google Scholar

Abido, M. A. (2002b). Optimal power flow using particle swarm optimization. International Journal of Electrical Power & Energy Systems, 24, 563–571.10.1016/S0142-0615(01)00067-9

 Web of Science ®Google Scholar

Narimani, M. R., Azizipanah-Abarghooee, R., Zoghdar-Moghadam-Shahrekohne, B., & Gholami, K. (2013). A novel approach to multi-objective optimal power flow by a new hybrid optimization algorithm considering generator constraints and multi-fuel type. Energy, 49, 119–136.10.1016/j.energy.2012.09.031

 Web of Science ®Google Scholar

Taylor, P., Radosavljevi, J., Klimenta, D., & Jevti, M. (2015). Electric power components and systems optimal power flow using a hybrid optimization algorithm of particle swarm optimization and gravitational search algorithm, 0–13. doi:10.1080/15325008.2015.1061620

 Google Scholar