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

Table 18.

Optimum control variables for the 118-bus grid to reduce the fuel cost.

Control variables	AHA	HHO	mAHA	RUN	SCA	SMA	TSA	WOA	SP-DE³⁸
$P_{G 1}$ (MW)	73.97791	4.499385	22.34	28.51	32.61	62.48	26.51	75.48	30.0317
$P_{G 4}$ (MW)	73.97791	4.499385	22.344972	28.512002	32.613835	62.479978	26.507791	75.4781	30.0143
$P_{G 6}$ (MW)	45.97772	73.17094	26.185102	26.860476	91.853307	26.809622	1.8723375	70.738237	30.0122
$P_{G 8}$ (MW)	29.93431	35.65567	39.501458	16.474302	29.166119	30.453661	95.23658	43.234597	30.0052
$P_{G 10}$ (MW)	33.18618	31.77735	34.305901	23.590872	10.268544	43.827001	52.250983	13.845696	317.0191
$P_{G 12}$ (MW)	251.1501	187.4009	351.80698	352.67081	286.41893	326.38048	120.60999	465.36772	66.8229
$P_{G 15}$ (MW)	58.22862	29.47036	65.718688	63.100463	137.34071	83.13546	68.04316	7.167916	30.0162
$P_{G 18}$ (MW)	13.41684	21.87009	0.616095	59.648963	14.099157	33.990822	68.346459	78.719897	30.0063
$P_{G 19}$ (MW)	32.04036	39.65482	31.222613	56.384484	8.2057148	50.448569	91.710876	70.658032	30.0495
$P_{G 24}$ (MW)	19.10174	56.57999	42.215128	55.01974	1.6884434	19.54139	36.000053	28.267784	30.0111
$P_{G 25}$ (MW)	13.98576	46.38767	25.078764	63.230667	17.746971	30.412801	37.796965	61.962804	152.1726
$P_{G 26}$ (MW)	189.4145	174.1855	138.94574	186.26184	21.815462	142.72163	99.04144	233.12736	220.8106
$P_{G 27}$ (MW)	211.7612	243.0143	222.3762	242.06373	118.33172	186.70857	37.118078	152.35096	30.0364
$P_{G 31}$ (MW)	31.05151	6.349815	37.725412	40.554279	4.7761776	10.488302	23.902022	23.015401	32.1004
$P_{G 32}$ (MW)	10.93366	25.30334	6.0960596	35.119706	90.605037	8.1223853	89.120842	43.180497	30.0143
$P_{G 34}$ (MW)	14.73455	82.59279	34.132257	32.856941	2.0278425	41.702618	11.831651	32.969058	30.0024
$P_{G 36}$ (MW)	43.42933	79.18123	42.748537	21.21653	69.961238	56.899056	28.454377	49.000841	30.0132
$P_{G 40}$ (MW)	42.09941	61.41026	30.536958	25.977006	94.098715	40.134857	40.323222	63.140272	30.0216
$P_{G 42}$ (MW)	70.17412	12.86707	37.884417	38.35962	90.957297	39.956996	6.0282521	6.3599397	30.038
$P_{G 46}$ (MW)	71.26958	32.4453	36.026695	53.479184	51.516614	29.959632	30.384363	21.133322	35.7003
$P_{G 49}$ (MW)	17.09888	13.58845	19.311066	36.102997	12.384684	20.20281	39.982041	34.514193	162.3848
$P_{G 54}$ (MW)	135.2571	172.7611	137.37011	162.0917	68.470381	167.90036	114.96766	171.56946	44.6599
$P_{G 55}$ (MW)	50.64071	53.12128	56.939724	23.583623	22.256192	59.888237	148	54.126573	30.0485
$P_{G 56}$ (MW)	52.70551	4.66097	37.779088	39.84767	67.238865	60.219718	1.1644984	56.361017	30.0079
$P_{G 59}$ (MW)	49.23414	18.42636	44.549145	17.564006	27.886048	55.356403	86.401461	3.5725475	125.3306
$P_{G 61}$ (MW)	147.5841	133.5096	120.61706	149.9252	150.36156	123.40271	50.0175	165.77374	124.1197
$P_{G 62}$ (MW)	169.4893	199.4909	108.57273	106.47964	218.8781	137.04383	165.83336	65.599081	30.0168
$P_{G 65}$ (MW)	6.41605	73.08044	40.944572	31.579011	34.657683	1.9314348	68.574207	26.620347	289.6489
$P_{G 66}$ (MW)	255.7686	171.7997	292.77812	196.30856	117.12508	240.05332	62.712269	152.8487	289.1504
$P_{G 69}$ (MW)	213.1634	398.099	296.83693	185.15846	35.520406	243.04034	174.06136	148.16923	0
$P_{G 70}$ (MW)	31.56371	37.11601	24.484254	27.322567	15.456538	22.941071	93.180001	59.337525	30.0116
$P_{G 72}$ (MW)	27.08306	25.62866	0	73.768372	67.391792	4.1033667	100	49.359205	30.0013
$P_{G 73}$ (MW)	27.37227	18.52833	37.737553	44.226564	12.025639	46.130819	1.6457489	5.7959714	30.0033
$P_{G 74}$ (MW)	43.74254	41.95115	55.615479	28.002771	65.39176	38.237159	48.966833	84.863694	30.0088
$P_{G 76}$ (MW)	54.42703	29.86474	35.924267	10.819407	85.272013	52.486266	85.510698	4.1194451	30.0074
$P_{G 77}$ (MW)	52.81395	45.55287	26.572018	73.661641	18.892959	26.213381	95.38631	58.854267	30.0141
$P_{G 80}$ (MW)	357.4542	82.12018	360.85279	180.68852	247.831	324.2164	426.67001	49.283144	350.9989
$P_{G 85}$ (MW)	21.17923	7.799365	2.58E−07	7.77E−01	3.57E+01	3.00E+01	3.78E+01	3.45E+01	30.0087
$P_{G 87}$ (MW)	2.631196	13.35761	2.2740058	3.9666089	20.423257	6.7902426	0.4226002	9.3148947	31.2015
$P_{G 89}$ (MW)	311.4639	209.0417	403.60339	399.63584	358.45128	364.83383	32.719446	404.48601	379.9452
$P_{G 90}$ (MW)	32.7639	53.09261	0.5130444	17.555897	90.922614	15.081347	100	24.944836	30.0443
$P_{G 91}$ (MW)	30.05628	23.15726	39.702145	20.049802	23.480439	28.269296	60.073161	35.553433	30.021
$P_{G 92}$ (MW)	57.33347	80.33754	32.488743	73.277493	47.547086	30.462968	59.311775	46.100697	30.0162
$P_{G 99}$ (MW)	31.61704	40.58355	3.9487838	70.02128	21.549019	2.3238609	29.622289	17.458742	30.0027
$P_{G 100}$ (MW)	178.3839	306.9364	155.93808	149.7165	200.49516	197.60342	230.92259	156.79878	177.1013
$P_{G 103}$ (MW)	51.27641	50.5826	43.078587	39.795366	123.81709	32.568123	14.427038	93.309792	42.0053
$P_{G 104}$ (MW)	9.151841	18.9915	37.806812	16.613335	91.82221	47.154856	99.813121	5.1890923	30.0088
$P_{G 105}$ (MW)	34.58507	43.9233	25.50231	19.099752	0.05311	36.200227	86.769711	33.537418	30.0022
$P_{G 107}$ (MW)	7.48512	5.020658	41.267475	44.434903	37.644208	27.766235	3.4558507	30.806867	30.013
$P_{G 110}$ (MW)	54.41779	35.85211	32.907949	67.016359	89.230502	26.802435	7.7373735	9.6659355	30.0043
$P_{G 111}$ (MW)	33.39558	89.46353	32.974757	32.350711	16.981895	33.565675	48.89309	39.344939	40.8014
$P_{G 112}$ (MW)	8.857481	70.67769	45.518618	40.794676	94.263477	33.818006	34.885385	38.906668	30.0166
$P_{G 113}$ (MW)	24.46341	57.18604	0	32.034234	8.6023708	29.474881	76.7596	36.94084	30.0223
$P_{G 116}$ (MW)	35.4573	7.701292	44.746955	21.343602	89.693115	20.858996	53.187439	76.391852	30.0052
$V_{1}$ (pu)	0.967659	1.001713	0.9649205	1.0016137	0.9874263	1.0033598	0.9829249	1.0500917	0.9871
$V_{4}$ (pu)	0.970687	1.001713	0.9871647	1.0290477	0.9513892	1.0311074	0.94	1.0534883	1.0153
$V_{6}$ (pu)	0.98612	1.001713	0.9802136	1.022922	0.9970753	1.0246972	0.9608864	1.0532677	1.008
$V_{8}$ (pu)	1.002829	1.001713	1.0209111	1.0009079	1.0452373	1.0149989	0.94	1.0482726	1.0388
$V_{10}$ (pu)	0.998449	1.001713	1.0315085	1.0310317	1.0343418	1.0320938	0.94	1.053621	1.0494
$V_{12}$ (pu)	0.983058	1.001713	0.9873455	1.0201567	0.9692529	1.0113408	0.9944888	1.0497367	1.002
$V_{15}$ (pu)	0.974677	1.001713	1.001113	1.0014277	1.0519176	0.9967199	0.9847773	1.0506516	0.9988
$V_{18}$ (pu)	0.979009	1.001713	1.0053845	1.0046698	1.0563817	0.9978715	0.94	1.0496127	0.9989
$V_{19}$ (pu)	0.973143	1.001713	0.9943712	1.0017399	1.0539285	0.996978	0.94	1.0503201	0.9986
$V_{24}$ (pu)	0.995082	1.001713	0.981033	1.0083117	0.9458489	1.0130014	1.0589681	1.0537513	1.015
$V_{25}$ (pu)	0.977966	1.001713	0.9827997	1.0072629	1.0403332	1.0249363	0.9616315	1.0535148	1.0298
$V_{26}$ (pu)	0.964085	1.001713	0.9726292	1.0189158	0.9981863	1.0002045	0.9977761	1.0536214	1.0744
$V_{27}$ (pu)	0.973053	1.001713	0.9824432	1.0050693	0.9779946	1.0171781	1.06	1.0535553	1.0045
$V_{31}$ (pu)	0.982447	1.001713	1.0411387	1.0225398	0.94	0.9999279	1.06	1.0510996	0.9992
$V_{32}$ (pu)	0.979873	1.001713	1.0035496	1.0025787	0.9970108	1.0117008	1.06	1.0536976	1.0045
$V_{34}$ (pu)	0.971057	1.001713	0.9899964	1.0162582	1.0576962	1.0153464	1.0339476	1.0537348	1.0141
$V_{36}$ (pu)	0.969313	1.001713	0.9848331	1.0153129	1.0565053	1.0133968	1.0261699	1.0536101	1.0105
$V_{40}$ (pu)	0.976783	1.001713	0.9839382	1.0327823	1.0341608	1.0008319	0.94	1.0527091	1.0001
$V_{42}$ (pu)	0.975279	1.001713	0.9748858	1.0136824	0.9918544	1.0054377	0.94	1.0499071	1.0081
$V_{46}$ (pu)	0.979741	1.001713	1.0084533	1.0189781	1.0341575	1.0114349	0.9483151	1.0466944	1.0316
$V_{49}$ (pu)	0.966236	1.001713	1.0126129	0.9989289	1.0241897	0.996714	0.9642479	1.0500917	1.0429
$V_{54}$ (pu)	0.981281	1.001713	1.0009026	0.9840295	0.9522491	1.0007592	0.94	1.055891	1.0217
$V_{55}$ (pu)	0.976437	1.001713	0.9890595	0.9843174	0.9623507	1.0017023	0.94	1.0528088	1.0215
$V_{56}$ (pu)	0.976024	1.001713	0.9930601	0.9824469	0.9659387	0.9994556	0.94	1.0520192	1.0215
$V_{59}$ (pu)	0.974589	1.001713	0.9624873	1.0079295	0.9987788	1.0207273	1.013766	1.0537264	1.0424
$V_{61}$ (pu)	0.979266	1.001713	0.9763314	1.0160096	0.9507834	1.0173327	1.0133727	1.0534964	1.0496
$V_{62}$ (pu)	0.97347	1.001713	0.9766384	1.0086118	0.94	1.0123299	1.0193777	1.0525318	1.0462
$V_{65}$ (pu)	0.993191	1.001713	0.9989158	0.9938653	1.030179	1.0148225	0.94	1.0534243	1.0623
$V_{66}$ (pu)	1.005867	1.001713	1.0150899	0.9990538	0.948453	1.0092482	0.94	1.053419	1.0593
$V_{69}$ (pu)	0.997562	1.001713	1.0443869	0.9941237	0.9811877	1.0171895	0.9872601	1.0500917	1.0389
$V_{70}$ (pu)	0.980913	1.001713	1.0072379	0.9971868	0.9987397	0.9923827	0.94	1.053458	1.0195
$V_{72}$ (pu)	0.984396	1.001713	0.9962062	1.0187389	0.9436571	1.0030257	1.06	1.0537036	1.0191
$V_{73}$ (pu)	0.984033	1.001713	0.9952997	1.0160158	0.9489485	1.0006245	0.94	1.0535951	1.0234
$V_{74}$ (pu)	0.97218	1.001713	0.9840221	0.9698298	1.06	0.9765873	0.9913531	1.0537809	1.0058
$V_{76}$ (pu)	0.961308	1.001713	0.9597947	0.9501527	0.9661673	0.9695291	0.94	1.0537658	0.9868
$V_{77}$ (pu)	0.974911	1.001713	0.9959514	0.9899145	0.9620065	0.9915929	1.0049576	1.0502009	1.013
$V_{80}$ (pu)	0.973564	1.001713	0.9991992	1.0058527	0.9687811	0.9999807	1.06	1.053419	1.0218
$V_{85}$ (pu)	0.986859	1.001713	0.9994004	0.9954768	1.0131234	1.0010699	0.94	1.053422	1.0242
$V_{87}$ (pu)	0.995449	1.001713	1.0461667	1.0036803	1.06	1.0306193	0.9941376	1.0511895	1.0432
$V_{89}$ (pu)	0.988563	1.001713	1.0103991	1.024946	1.0197158	1.0231108	0.94	1.0535838	1.0274
$V_{90}$ (pu)	0.989681	1.001713	1.0079355	0.9857261	1.0484171	1.0058755	1.0171342	1.0535094	1.0062
$V_{91}$ (pu)	1.00143	1.001713	0.9898406	1.002056	1.06	1.0020844	1.0579789	1.0537953	1.0074
$V_{92}$ (pu)	0.982589	1.001713	0.9837033	0.9957563	0.9657501	0.9990564	0.9658267	1.0500917	1.0153
$V_{99}$ (pu)	0.992646	1.001713	1.0045406	0.9986228	1.014638	0.9992424	1.0179871	1.0535517	1.0182
$V_{100}$ (pu)	1.001404	1.001713	1.0062797	1.0074172	0.9623508	1.0131239	0.9928849	1.0519887	1.0187
$V_{103}$ (pu)	1.006439	1.001713	1.0109474	1.0013357	0.9677742	1.0092176	0.9438118	1.0494301	1.0146
$V_{104}$ (pu)	0.987071	1.001713	0.9930689	0.9924871	1.0347805	1.0049059	0.997348	1.0536568	1.0067
$V_{105}$ (pu)	0.988016	1.001713	0.9923136	0.9982727	1.06	1.0054728	1.0293715	1.0537466	1.0063
$V_{107}$ (pu)	0.959298	1.001713	0.980022	1.0041706	1.0370517	1.0040741	1.06	1.0534344	0.9993
$V_{110}$ (pu)	0.992368	1.001713	1.0012863	1.0171262	1.0508403	1.0088357	1.0134337	1.0505183	1.0147
$V_{111}$ (pu)	0.981082	1.001713	1.0093074	1.0076076	0.983494	1.0052806	0.94	1.0506049	1.0247
$V_{112}$ (pu)	0.992586	1.001713	1.0118551	1.0271379	1.017392	1.0175045	1.06	1.053739	1.0046
$V_{113}$ (pu)	0.977017	1.001713	1.0343396	1.0090427	0.9927719	1.0010442	0.94	1.0521195	1.0057
$V_{116}$ (pu)	0.99086	1.001713	0.9821816	0.9880097	0.9693682	1.0097187	0.94	1.0532565	1.0592
$T_{8}$ (8–5)	0.988728	0.987085	1.0402999	0.9968856	1.0847371	0.9895942	1.0031343	0.9397173	1.0148
$T_{32}$ (26–25)	0.9935602	1.017392	1.0116047	1.0109498	1.0121767	1.0008969	1.1	0.9993158	1.0978
$T_{36}$ (30–17)	0.9654374	0.987126	0.9212424	1.0308447	1.0846095	1.005469	0.9206088	0.9991882	1.0348
$T_{51}$ (38–37)	1.011300	0.980655	0.9773194	0.9989706	1.0960439	0.9862349	0.9	0.9902346	1.0107
$T_{93}$ (63–59)	1.017721	0.980662	0.977828	1.0045998	1.079463	0.9942505	0.9171906	0.9508028	0.9946
$T_{95}$ (64–61)	0.974791	0.980749	0.9704812	0.945104	1.0825113	0.9673068	0.9086571	0.9556937	1.0095
$T_{102}$ (65–66)	0.945034	0.988361	0.9901509	0.9827035	1.0987101	0.9847707	1.0273736	0.9907212	0.9771
$T_{107}$ (68–69)	0.946050	0.99266	0.9310724	0.9915445	1.0315702	0.9425921	1.0085836	0.9991271	0.9715
$T_{127}$ (81–80)	0.98313	0.981881	0.9656016	1.0031895	0.9388147	0.9731689	0.9	0.9715853	1.0214
$Q_{34}$ (MVAR)	10.83555	21.1929	6.8187763	15.778227	17.918252	13.791896	26.290058	14.655474	0.8808
$Q_{44}$ (MVAR)	12.18013	19.07619	21.659563	11.082027	13.41777	7.2540701	8.8815734	10.516174	5.768
$Q_{45}$ (MVAR)	7.836917	1.303665	15.489975	22.637872	24.056449	6.7121784	19.8684	15.802734	21.5888
$Q_{46}$ (MVAR)	7.829812	24.64004	15.945289	13.111509	5.7751418	13.340721	4.9661793	13.734777	10.9322
$Q_{48}$ (MVAR)	7.304086	1.9898	11.666063	9.8557041	4.1918716	13.132709	5.3761902	4.0110494	4.6786
$Q_{74}$ (MVAR)	8.509035	17.56941	0.0285094	16.534195	7.1184334	15.380355	19.6847	8.4288615	24.2029
$Q_{79}$ (MVAR)	9.962153	4.52566	4.8781181	12.43247	25.993238	15.329586	25.066682	12.658385	23.8787
$Q_{82}$ (MVAR)	19.55525	19.43275	14.373142	9.6701052	3.7858894	19.335016	1.4871303	5.6523165	23.5807
$Q_{83}$ (MVAR)	7.027822	13.91448	7.7951153	11.470982	20.036399	8.1539749	8.4579903	16.436302	20.4897
$Q_{105}$ (MVAR)	12.34095	21.1684	14.461638	21.634179	11.179999	16.01292	11.586491	3.4548903	13.4731
$Q_{107}$ (MVAR)	17.66833	1.404061	12.010001	5.1187294	3.4374936	11.949043	16.050517	13.731969	1.936
$Q_{110}$ (MVAR)	21.95423	15.71943	23.761681	5.4946062	2.2205707	8.9900653	12.051309	3.9038993	18.1676
Fuel cost ($/h)	134,460.2	148,691	132,849.31	137,230.17	405,883.6	133,149.81	351,542.7	147,566.87	135,055.7
Power losses (MW)	66.6784	81.3435	82.76452	86.409369	138.11708	64.87534	112.24476	93.439966	60.9596
Voltage deviation (pu)	1.73006	0.47199	1.0684904	0.8930666	2.3370508	0.635191	2.4379931	2.8862895	1.0715
Iterations time (s)	232.8176	7260.92	578.900	484.923	191.44367	218.561	286.73577	282.9444	–