. 2023 Dec 5;13:21446. doi: 10.1038/s41598-023-48479-6

Table 21.

Optimum control variables for the 118-bus grid to optimize the multi-objective function.

Control variables	AHA	HHO	mAHA	RUN	SCA	SMA	TSA	WOA	MJAYA⁶⁷
$P_{G 1}$ (MW)	18.07	72.32	40.66	40.46	19.31	40.47	33.42	19.48	50.23
$P_{G 4}$ (MW)	18.071783	72.31595	40.660591	40.457111	19.306794	40.471011	33.424649	19.483865	4.81
$P_{G 6}$ (MW)	44.017731	1.3831367	21.161305	29.748183	28.931331	40.027392	17.045858	70.156137	64.46
$P_{G 8}$ (MW)	23.028437	81.752214	56.376741	19.026106	28.86892	10.923141	36.726754	39.982318	3.11
$P_{G 10}$ (MW)	50.231	1.3969462	22.048955	30.930415	78.949703	8.6516309	43.33107	20.321959	167.05
$P_{G 12}$ (MW)	283.31041	415.21215	310.44926	327.00309	73.558177	348.01603	92.220054	127.19647	53.18
$P_{G 15}$ (MW)	75.832547	113.12214	75.735244	93.401035	110.43887	65.671559	68.455398	110.32746	34.90
$P_{G 18}$ (MW)	34.478531	75.164004	41.036352	72.06681	66.207553	41.286114	22.448149	69.978491	23.65
$P_{G 19}$ (MW)	47.828237	10.979219	47.061548	37.051758	43.1937	38.342663	40.74003	6.8214424	71.86
$P_{G 24}$ (MW)	42.169521	66.963842	23.310636	57.427369	39.387832	35.712421	12.443041	57.700429	33.08
$P_{G 25}$ (MW)	35.954214	69.730894	0.0654419	30.369156	27.861816	24.80807	85.12492	62.282004	200.06
$P_{G 26}$ (MW)	142.32751	113.92602	129.4088	160.38569	220.6471	179.92375	258.24186	126.31629	207.52
$P_{G 27}$ (MW)	225.49923	69.067183	226.09857	83.586667	143.45843	156.85726	256.05459	201.30856	25.87
$P_{G 31}$ (MW)	30.428814	38.968158	36.673222	57.004756	50.587049	31.358738	66.319929	30.224141	10.65
$P_{G 32}$ (MW)	9.7118704	1.4478344	7.1768536	7.2451877	40.627324	7.7847454	48.558999	7.6722874	76.65
$P_{G 34}$ (MW)	10.154722	73.567255	5.3586903	29.325702	31.317443	44.768643	100	38.491182	61.91
$P_{G 36}$ (MW)	41.634698	31.871083	51.2431	35.79511	40.424286	50.022292	45.200328	2.1616249	33.29
$P_{G 40}$ (MW)	41.492364	3.2768603	37.302258	38.516879	79.210986	4.9493626	100	1.8804106	58.63
$P_{G 42}$ (MW)	28.086789	30.238652	0.1279117	29.299195	90.865591	27.072279	100	36.768382	65.83
$P_{G 46}$ (MW)	44.170228	20.950867	34.607258	65.1051	49.196018	29.311559	45.466769	16.783503	20.39
$P_{G 49}$ (MW)	25.479106	55.049438	21.463662	35.100865	49.519175	19.159563	30.349455	19.998876	219.98
$P_{G 54}$ (MW)	171.03288	152.74633	172.85744	138.82457	120.32187	152.00955	138.22158	112.96108	76.98
$P_{G 55}$ (MW)	46.632883	63.884832	57.206871	33.155876	27.620499	57.726208	116.18878	85.52397	50.00
$P_{G 56}$ (MW)	43.696699	74.4453	3.78E-20	37.843095	31.805611	28.190215	5.17E+00	5.75E+01	51.73
$P_{G 59}$ (MW)	39.762611	42.846159	99.950554	70.855249	66.012402	34.325836	100	28.925178	132..86
$P_{G 61}$ (MW)	129.52251	116.12683	76.658674	166.07081	193.92364	92.479551	242.27976	157.45117	120.23
$P_{G 62}$ (MW)	103.05631	36.190765	138.68366	114.25976	95.542923	135.76617	41.185466	166.30814	32.06
$P_{G 65}$ (MW)	39.369363	25.349497	23.846489	55.672231	28.136807	21.024433	78.657468	21.241019	240.04
$P_{G 66}$ (MW)	298.75103	262.11184	341.99049	199.24584	72.566588	275.76712	1.7263742	191.18238	170.77
$P_{G 69}$ (MW)	284.11136	36.256137	283.98504	220.86191	276.87909	286.24061	318.82676	296.49002	342.23
$P_{G 70}$ (MW)	24.420847	58.530444	54.851824	53.777007	0.7200069	9.8963742	27.838283	41.434461	47.94
$P_{G 72}$ (MW)	35.806337	13.273314	23.263764	40.71018	23.429675	29.263315	36.307227	32.33344	55.09
$P_{G 73}$ (MW)	22.949981	32.829655	34.343879	58.593999	38.352759	19.58988	68.374703	63.458191	54.80
$P_{G 74}$ (MW)	61.679556	34.968754	28.480787	26.208667	44.383678	68.625131	56.835762	4.283734	45.34
$P_{G 76}$ (MW)	72.271418	86.228001	49.190982	34.899477	34.458189	64.626361	54.049394	56.806653	53.51
$P_{G 77}$ (MW)	22.968618	21.404284	2.17E−05	26.042217	66.256809	35.426756	4.72E+01	2.82E+01	48.16
$P_{G 80}$ (MW)	309.36318	385.2597	329.82833	298.5911	366.15764	384.16613	32.524339	385.46092	332.42
$P_{G 85}$ (MW)	2.69E+01	6.52E+00	2.50E−02	5.51E+01	6.81E+01	5.11E+01	4.76E+01	4.52E+01	41.58
$P_{G 87}$ (MW)	10.411067	1.5478632	3.0227718	16.559781	77.918609	5.1554427	44.379518	4.7550816	1.90
$P_{G 89}$ (MW)	310.416	327.15864	406.90523	323.12488	39.590487	373.11708	68.420195	329.78795	210.71
$P_{G 90}$ (MW)	28.647952	30.358655	20.943072	31.490746	65.53683	14.564027	63.223173	5.6993441	23.19
$P_{G 91}$ (MW)	36.594081	67.953096	0.5205968	34.553001	1.3003347	9.2421047	55.935017	61.352129	45.22
$P_{G 92}$ (MW)	54.511709	1.3701922	21.364362	34.566542	92.135365	49.228727	20.0611	37.708013	41.92
$P_{G 99}$ (MW)	52.752491	14.950853	32.600003	43.432928	46.268455	20.68691	88.030205	67.258764	9.86
$P_{G 100}$ (MW)	133.53893	177.73846	192.59437	145.93183	240.75949	179.14495	232.19815	176.67993	166.02
$P_{G 103}$ (MW)	22.090747	89.441876	37.813647	82.066965	7.9019222	55.986594	0.7737086	59.304231	72.61
$P_{G 104}$ (MW)	14.273216	77.872717	39.411678	24.811601	17.394399	29.288941	2.2671886	65.607451	62.06
$P_{G 105}$ (MW)	32.027813	62.589341	28.136218	10.817359	19.18079	39.169855	85.048863	10.402192	56.43
$P_{G 107}$ (MW)	42.421291	56.626913	54.491021	32.472667	71.086143	28.126646	81.344505	45.104053	42.68
$P_{G 110}$ (MW)	39.133718	30.492124	25.65837	20.107864	2.9827881	13.937075	64.539857	27.406063	45.99
$P_{G 111}$ (MW)	46.772592	19.666794	34.048531	62.964561	67.095646	35.003227	70.964034	94.498892	21.59
$P_{G 112}$ (MW)	34.307033	1.5027623	43.304354	57.455894	89.57808	52.389674	82.141071	29.356489	54.83
$P_{G 113}$ (MW)	18.859456	41.155896	30.000843	10.499888	11.97101	25.836736	1.6570369	18.929262	56.99
$P_{G 116}$ (MW)	46.641469	31.761899	33.175967	28.198132	65.744664	16.191412	3.9474418	9.8490344	6.02
$V_{1}$ (pu)	0.9755947	1.015563	0.972368	0.9678034	1.0307964	0.9818775	0.94	1.0028985	0.98
$V_{4}$ (pu)	1.0037545	1.0107167	0.9915827	0.9984153	0.9796535	1.0184511	1.06	1.0075327	1.00
$V_{6}$ (pu)	0.9930883	1.0107404	0.9790376	1.0009824	1.0226205	0.9968005	1.06	1.0033096	1.01
$V_{8}$ (pu)	1.001848	1.0107353	0.9838811	1.0139053	1.0582519	1.0103585	0.94	1.0141423	0.98
$V_{10}$ (pu)	1.0130388	1.0107393	0.9833819	1.0223518	1.0394668	1.0206254	0.9559686	1.0029636	1.00
$V_{12}$ (pu)	0.9990578	1.0107102	0.990659	0.9957636	1.0391086	0.9901969	1.06	1.0032827	1.00
$V_{15}$ (pu)	1.0077276	1.0107208	0.9749551	0.9931806	1.0234861	1.0025749	1.0158677	1.0015763	0.99
$V_{18}$ (pu)	1.0175008	1.0107335	0.9817632	0.9904964	0.95216	1.0139521	1.06	1.0012431	0.99
$V_{19}$ (pu)	1.0072259	1.0120977	0.9754394	0.9877359	0.9690685	1.0051226	1.06	1.0033828	0.99
$V_{24}$ (pu)	0.9868438	1.0107109	0.9991512	0.9667762	1.0223629	1.0339587	1.06	1.0033885	0.99
$V_{25}$ (pu)	1.0004621	1.0106932	0.9859443	1.0053381	0.9674887	0.9995287	0.9797086	1.0048057	1.02
$V_{26}$ (pu)	1.0239666	1.0125367	0.9742331	0.9842713	0.9504169	0.9964773	0.9672845	1.0023007	0.99
$V_{27}$ (pu)	1.0099687	1.0107504	1.0254128	0.9988385	0.9938304	0.9926038	1.0599755	1.0075133	1.00
$V_{31}$ (pu)	1.0117357	1.0107365	0.9987288	1.0062992	1.0118899	1.0042962	1.0425167	1.003698	1.01
$V_{32}$ (pu)	1.0055948	1.0107625	1.0082739	0.9960715	0.9767095	0.9926645	1.06	1.0008036	0.99
$V_{34}$ (pu)	1.0093233	1.0107218	0.9900721	1.002729	0.989945	0.9904552	1.0275926	1.0063636	1.01
$V_{36}$ (pu)	1.0043946	1.0107173	0.9847753	0.9997235	0.9935939	0.9845946	1.0580623	1.0019018	1.01
$V_{40}$ (pu)	1.0145578	1.0107049	0.9824566	0.9950745	1.0268603	0.9980806	0.94	1.004353	0.99
$V_{42}$ (pu)	1.0097933	1.0107262	0.9826538	0.9708156	1.0484933	0.9990177	0.9400494	1.0030512	1.00
$V_{46}$ (pu)	0.9765032	1.015024	0.9423184	0.990603	0.9614848	1.0110713	1.06	1.0130523	0.99
$V_{49}$ (pu)	1.0087427	1.0107596	0.9522425	1.0006226	1.043314	0.9994073	1.06	1.006489	1.03
$V_{54}$ (pu)	1.0186545	1.010725	0.9854332	0.9805281	0.946709	0.9990618	0.94	1.0034572	1.04
$V_{55}$ (pu)	1.0164577	1.0107313	0.9794105	0.9777562	0.9729364	0.9967516	0.94	1.003249	1.03
$V_{56}$ (pu)	1.0152935	1.0107518	0.9812836	0.9779015	0.9571829	0.9960331	0.94	1.0017566	1.03
$V_{59}$ (pu)	1.0120328	1.0140715	1.0033543	0.9883887	1.008423	0.9934477	0.94	1.0028546	1.00
$V_{61}$ (pu)	1.0076086	1.0107254	1.0164649	0.9950227	0.9947235	0.989046	1.0326911	1.0028546	0.99
$V_{62}$ (pu)	1.0050346	1.0125743	1.0047093	0.9909107	0.9919765	0.9787503	1.0337894	1.0035799	0.98
$V_{65}$ (pu)	1.0222197	1.0107256	0.9992318	1.0017812	1.0078381	1.0072969	1.06	1.007566	1.00
$V_{66}$ (pu)	1.0138962	1.0154207	0.9946429	1.0052744	0.9559682	0.9866618	1.0239445	1.013851	1.03
$V_{69}$ (pu)	1.042357	1.010736	0.9939733	1.0393596	1.0242939	1.0412257	1.0575811	1.0025165	1.01
$V_{70}$ (pu)	1.0130808	1.0107575	0.9905907	1.0065353	1.0309907	1.0110704	1.0016933	1.0037228	1.03
$V_{72}$ (pu)	1.0025594	1.0107261	0.9707553	0.9784075	1.0013086	1.0209373	1.0190739	1.0102543	0.99
$V_{73}$ (pu)	1.0130594	1.0107785	1.0102506	1.0336309	0.9451181	1.0030994	0.94	1.0015605	1.06
$V_{74}$ (pu)	1.0085043	1.0106968	0.9692249	0.9772556	0.9775804	1.0073711	0.94	1.0067351	1.00
$V_{76}$ (pu)	0.9954803	1.0107309	0.9594846	0.9543715	1.0258366	0.9958658	0.9627969	1.0031265	0.98
$V_{77}$ (pu)	1.0111951	1.0129077	0.9983227	0.9908284	1.0078177	0.9962884	0.94	0.9992381	0.99
$V_{80}$ (pu)	1.0250285	1.0107211	1.0440151	1.0152545	0.9956994	0.9984555	0.94	1.003369	1.01
$V_{85}$ (pu)	1.0012245	1.0107397	1.0132613	0.9858348	0.9797477	0.9947601	0.9829425	1.0032828	0.99
$V_{87}$ (pu)	1.006652	1.01072	0.9837245	0.9966944	1.0102235	1.0505185	0.9452304	1.0024074	0.98
$V_{89}$ (pu)	0.9901991	1.0107344	1.0468375	0.9947819	1.0382378	0.9713792	1.06	1.0108695	1.00
$V_{90}$ (pu)	1.0032796	1.0119949	1.008365	1.0165073	0.9668342	0.9772183	0.94	1.0024562	1.01
$V_{91}$ (pu)	1.0307402	1.0107125	0.9969373	0.999607	0.9515844	1.0052582	1.06	1.0038764	0.99
$V_{92}$ (pu)	1.0129018	1.0107204	0.992354	0.9824628	0.9802555	0.9781234	0.94	1.0018415	0.98
$V_{99}$ (pu)	0.9913718	1.0109658	0.9936652	0.9991168	0.952732	0.9917847	0.94	1.003235	1.01
$V_{100}$ (pu)	1.0318508	1.0107373	0.9959108	0.9982472	0.9816469	0.9994647	1.0310818	1.0060341	0.99
$V_{103}$ (pu)	1.0155488	1.0112627	0.9959641	0.9983592	0.9889007	1.0100378	1.06	1.0038077	1.02
$V_{104}$ (pu)	1.0071165	1.0107182	0.9879583	0.9972382	0.9765162	0.9997485	1.0044333	1.0046284	1.02
$V_{105}$ (pu)	1.0090936	1.0107395	0.9909608	0.9966941	0.9510039	0.9969756	1.06	1.0046202	1.02
$V_{107}$ (pu)	1.0219929	1.0107689	1.011368	1.0028014	1.0448404	0.9979651	1.06	1.0035749	1.00
$V_{110}$ (pu)	1.0041077	1.0107085	0.981842	0.9874705	1.030782	0.9981363	1.0318386	1.0047424	0.99
$V_{111}$ (pu)	1.0098048	1.0123173	0.9967582	0.9881657	1.0354325	1.0010923	1.06	1.002754	0.99
$V_{112}$ (pu)	0.9896521	1.0107238	0.9678187	0.9986953	1.006486	1.0048775	0.9958937	1.0016868	1.03
$V_{113}$ (pu)	1.0162678	1.0107178	0.989258	0.9846165	1.0501106	1.0048607	0.9958676	1.0034255	1.01
$V_{116}$ (pu)	0.9723242	1.0107233	0.970682	1.0144823	0.9813626	1.0239536	1.06	1.007034	0.99
$T_{8}$ (8–5)	0.9969683	0.974166	1.0028485	1.014071	0.9904684	0.9831192	0.9	1.0138265	1.00
$T_{32}$ (26–25)	1.0608599	0.9861295	1.0164473	0.991363	1.0432633	1.0214152	1.0421361	0.9927923	1.02
$T_{36}$ (30–17)	0.9858643	1.0307987	1.01755	0.9396623	0.9193303	0.9655758	0.9	0.9886933	1.02
$T_{51}$ (38–37)	0.9879601	0.9761415	0.9798519	0.965365	1.076739	1.0208299	0.9	1.0007664	0.95
$T_{93}$ (63–59)	1.0010503	0.9832594	0.9703067	0.9986791	1.0633998	0.9435669	1.1	1.0103129	1.01
$T_{95}$ (64–61)	1.0172656	1.0078413	0.973569	0.9556252	0.9078842	0.9608501	1.1	0.989212	0.99
$T_{102}$ (65–66)	0.9934653	0.9751778	1.0129403	0.9702031	1.0211318	1.0031683	1.0528714	0.9801001	0.99
$T_{107}$ (68–69)	0.9809535	1.0271104	0.9863273	0.9862177	1.0730421	1.0054443	0.9	1.0069097	0.98
$T_{127}$ (81–80)	1.061382	0.99588	0.9453683	0.9584389	1.0264662	0.9801847	1.1	0.9743607	0.96
$Q_{34}$ (MVAR)	14.853535	17.753942	11.183449	9.8051439	16.769534	10.749402	5.5676094	22.276187	13.97
$Q_{44}$ (MVAR)	8.6918036	19.665199	12.052902	6.9347835	2.7199204	17.254878	18.111223	18.063734	11.12
$Q_{45}$ (MVAR)	10.84757	1.2307905	11.55766	5.6683816	10.090073	15.948375	12.740237	21.518359	23.03
$Q_{46}$ (MVAR)	14.524919	26.204434	9.6965216	7.4479659	0.0223675	14.457601	2.4888498	9.5501911	16.69
$Q_{48}$ (MVAR)	11.628841	18.691988	13.917422	13.394523	24.976169	14.374375	26.25648	20.927619	19.83
$Q_{74}$ (MVAR)	14.437324	18.73354	13.96664	8.4089813	11.720403	17.894635	12.985434	16.288014	10.40
$Q_{79}$ (MVAR)	18.384129	10.75443	13.046392	21.573761	23.911704	8.6485609	7.2860588	16.674911	9.23
$Q_{82}$ (MVAR)	21.614886	16.891906	22.644904	8.5475088	29.984245	14.180096	22.253465	1.6265926	11.45
$Q_{83}$ (MVAR)	22.180259	18.090222	12.870701	16.550216	24.22214	19.879856	6.6839413	0.9938805	15.64
$Q_{105}$ (MVAR)	15.942382	11.075106	11.801482	7.3126552	18.910879	6.3138873	30	6.09791	16.78
$Q_{107}$ (MVAR)	16.104502	18.131344	15.399917	13.560577	8.2444647	20.712886	26.397513	9.0391943	27.74
$Q_{110}$ (MVAR)	12.902698	24.836424	9.4198206	7.4903267	2.0881328	13.281062	10.239639	12.192777	22.42
Objective function	134,581.11	147,663.18	133,257.99	137,402.63	431,355.38	133,921.61	431,849.5	143,003.58	140,575.3099
Fuel cost ($/h)	134,270.00	140,040.00	132,960.00	137,170.00	160,430.00	133,330.00	158,170.00	137,670.00	137,617.0912
Power losses (MW)	69.216576	68.925131	80.478981	66.058629	96.068573	82.104475	127.43309	79.604315	58.8779
Voltage deviation (pu)	0.5839816	0.5226177	1.2824171	0.9787408	1.2948132	0.9143519	2.7579362	0.5040643	0.7335
Iterations time (s)	197.1166	4716.907	618.2695	399.4292	204.7562	207.9040	219.67135	269.45299	–