Solution of combined economic and emission dispatch problems of power systems without penalty

Figures & data

Figure 1. Valve point loading effect.

Table 1. Penalty factor calculation

Penalty factor types	Penalty factor formula
$p_{f_{j,max-max}}$	=$\frac{FC(P_j^{max})}{E(P_j^{max})}$
$p_{f_{j,min-min}}$	=$\frac{FC(P_j^{min})}{E(P_j^{min})}$
$p_{f_{j,max-min}}$	=$\frac{FC(P_j^{max})}{E(P_j^{min})}$
$p_{f_{j,min-max}}$	=$\frac{FC(P_j^{min})}{E(P_j^{max})}$
$p_{f_{j,average}}$	=$\frac{p_{f_{j,max-max}}+p_{f_{j,min-min}}+p_{f_{j,max-min}}+p_{f_{j,min-max}}}{4}$
$p_{f_{j,com}}$	=$\frac{p_{f_{j,average}}}{numberofgenerators}$

Table 2. The parameter for the OWP-based OMF algorithm

	NFilter	NNeighbor	NIntensification
Test System 1& 2	9	3	2
Test System 3	12	3	2

Table 3. Best achieved solutions for test system I: 6 generators ($P_d=2.834MW$)

Units	ELD	EMD	CEED
PG1 (MW)	0.109719	0.390482	0.23286
PG2 (MW)	0.299766	0.493212	0.535011
PG3 (MW)	0.524298	0.502448	0.42019
PG4 (MW)	1.016198	0.453322	0.880278
PG5 (MW)	0.524298	0.502448	0.35909
PG6 (MW)	0.359719	0.492085	0.406568
Total generation (pu)	2.834	2.834	2.834
Fuel cost ($/h)	600.111385	633.284828	611.121863
Emission (ton/h)	0.205005	0.186125	0.195626
Total Cost ($/h)	–	–	465.697239

Table 4. Comparison of the minimum FC in ELD, minimum E in EMD, and minimum TC in CEED obtained by difirent algorithms for test system I: 6 generators

Algorithms	Fuel cost minimization	Emission minimization	Best compromise solution
	FC ($/h)	E (ton/h)	TC ($/h)	FC ($/h)	E (ton/h)
GSA (Jiang et al. 2015b)	602.2311	0.1954	1792.9181	618.4426	0.1981
DE (Niknam, Mojarrad, and Firouzi 2013)	601.3428	0.194217	NR	NR	NR
SOA (Benasla, Belmadani, and Rahli 2014)	600.986	0.18729*	1821.914	624.604	0.18708
MOHS (Sivasubramani and Swarup 2011)	600.6909	0.1947	NR	608.7367	0.2023
GAEPSO (Jiang et al. 2015b)	600.2978	0.1942	1788.4129	615.3802	0.1978
MDE (Niknam, Mojarrad, and Firouzi 2013)	600.173	0.194208	NR	NR	NR
MSA (Jevtic et al. 2017)	600.11141	0.19420	NR	606.80105	0.20329
PSOGSA (Radosavljević 2016)	600.11141	0.194203	NR	606.79829	0.203289
Tribe-MDE (Niknam, Mojarrad, and Firouzi 2013)	600.1114	0.194202	NR	NR	NR
OWP-based OMF	600.1113	0.186125	465.697239	611.121863	0.195626

*Emission calculated based on reported power generation and emission coefficients in (Benasla, Belmadani, and Rahli 2014) is equal to 0.195978.

Figure 2. Comparison of the results for $P_d=2.834MW$.

Figure 3. Ranking of OWP-based OMF and the compared algorithms by the Friedman’s test on test system I.

Figure 4. Ranking of OWP-based OMF and the compared algorithms by the Quade’s test on test system I.

Table 5. Statistical results obtained over 30 runs with OWP-based OMF for test system 1

	Min	Max	Mean	Standard	Q1	Median	Q3	Time (s)
ELD	600.1114	600.1125	600.111503	0.000283	600.1114	600.1114	600.1114	0.094
EMD	0.186125	0.187046	0.186183	0.000200	0.1861	0.1861	0.186133	0.093
CEED	465.696222	465.699553	465.69757	0.000780	465.697	465.697	465.6981	0.031

Figure 5. Box plot-distributed results for ELD, EMD and CEED problems for test system I.

Table 6. Best achieved solutions for test system II: 10 generators ($P_d=2000MW$)

Units	ELD	EMD	CEED
PG1 (MW)	55	55	55
PG2 (MW)	80	80	80
PG3 (MW)	106.8771	80.9248	81.1085
PG4 (MW)	100.7023	81.1019	80.7725
PG5 (MW)	81.5370	160	160
PG6 (MW)	82.9221	240	240
PG7 (MW)	300	294.5495	291.5934
PG8 (MW)	340	297.6624	296.0321
PG9 (MW)	470	396.3406	398.8523
PG10 (MW)	470	396.0266	398.3150
Total generation (pu)	2087.0387	2081.6060	2081.6740
Power Loss	87.0386	81.6057	81.6790
Fuel cost ($/h)	111497.6407	116411.9936	116391.8270
Emission (ton/h)	4572.4396	3932.2538	3932.4035
Total Cost ($/h)	–	–	320946.3799

Table 7. Comparison of the minimum FC in ELD, minimum E in EMD, and minimum TC in CEED obtained by difirent algorithms for test system II: 10 generators

Algorithms	Fuel cost minimization	Emission minimization	Best compromise solution
	FC ($/h)	E (ton/h)	TC ($/h)	FC ($/h)	E (ton/h)	PL
DE (Basu 2011)	111500	3923.4*	NR	NR	NR	NR
TLBO (Roy and Bhui 2013)	111500	3932.2	NR	113471	4113.5	83.8459
QOTLBO (Roy and Bhui 2013)	111498	3932.2	NR	113460	4110.2	83.8433
SMA (Ahmed et al. 2021)	NR	NR	NR	113490	4108.6	83.89
KKO (Srivastava and Das 2020)	NR	NR	NR	113481	3982.85	84.17
WOA (Rex, Edwin, and Annrose 2019)	NR	NR	321220	113610	3989.5	82.9669
OWP-based OMF	111497.6407	3932.2538	320946.3799	116391.8270	3932.4035	81.6790

*Emission calculated based on reported power generation and emission coefficients in (Basu 2011) is equal to 3932.417288.

Figure 6. Comparison of the results for $P_{loss}$.

Figure 7. Ranking of OWP-OMF and the compared algorithms by the Friedman’s test on test system II.

Figure 8. Ranking of OWP-OMF and the compared algorithms by the Quade’s test on test system II.

Table 8. Statistical results obtained over 30 runs with OWP-based OMF for test system 2

	Min	Max	Mean	Standard	Q1	Median	Q3	Time (s)
ELD	111497.640	111497.642	111497.64	0.00039	111497.6414	111497.6416	111497.6418	0.031
EMD	3932.253	3932.255	3932.2542	0.00032	3932.2539	3932.2540	3932.2543	0.016
CEED	320946.384	320946.385	320946.39	0.00021	320946.384	320946.385	320946.3852	0.032

Figure 9. Box plot-distributed results for ELD, EMD and CEED problems for test system II.

Table 9. Best achieved solutions for test system III: 40 generators ($P_d=10500MW$)

Units	ELD	EMD	CEED	Units	ELD	EMD	CEED
PG1 (MW)	111.6961	114	114	PG21 (MW)	523.1440	439.4349	437.4749
PG2 (MW)	111.3125	114	114	PG22 (MW)	523.2872	439.4954	437.3899
PG3 (MW)	97.3397	120	120	PG23 (MW)	523.2862	439.6127	437.9355
PG4 (MW)	179.7161	169.3896	178.1443	PG24 (MW)	523.2782	439.7326	437.5263
PG5 (MW)	87.7900	97	97	PG25 (MW)	523.2517	440.1845	437.5633
PG6 (MW)	140	124.1845	129.9247	PG26 (MW)	523.2835	440.1239	437.6229
PG7 (MW)	259.5241	299.7437	300	PG27 (MW)	10	28.97960	19.5926
PG8 (MW)	284.5739	297.9777	299.8678	PG28 (MW)	10	28.9527	19.5290
PG9 (MW)	284.4869	297.2446	298.9768	PG29 (MW)	10	28.9676	19.5488
PG10 (MW)	130	130	130	PG30 (MW)	87.7587	97	97
PG11 (MW)	94	298.4228	307.6796	PG31 (MW)	190	172.3832	175.7052
PG12 (MW)	94	298.0930	307.0133	PG32 (MW)	190	172.3506	175.8870
PG13 (MW)	214.7560	433.5619	433.9764	PG33 (MW)	190	172.3367	175.1343
PG14 (MW)	394.2935	421.6761	408.9414	PG34 (MW)	164.7192	200	200
PG15 (MW)	394.1647	422.7523	411.9811	PG35 (MW)	194.3587	200	200
PG16 (MW)	394.2238	422.7974	411.9954	PG36 (MW)	199.6042	200	200
PG17 (MW)	489.2651	439.3725	452.0437	PG37 (MW)	109.9278	100.7880	104.1737
PG18 (MW)	489.2467	439.4338	452.1124	PG38 (MW)	110	100.8778	104.1864
PG19 (MW)	511.2367	439.3958	437.1739	PG39 (MW)	109.9198	100.8685	104.0048
PG20 (MW)	511.2989	439.4240	437.4983	PG40 (MW)	511.2548	439.4403	437.3952
	ELD		EMD		CEED
Total generation (pu)	10500		10500		10500
Fuel cost ($/h)	121308.4046		129855.4288		128595.9724
Emission (ton/h)	359798.2078		176682.3690		178557.2230
Total Cost ($/h)	–		–		95722.2719

Table 10. Comparison of the minimum FC in ELD, minimum E in EMD, and minimum TC in CEED obtained by difirent algorithms for test system III: 40 generators

Algorithms	Fuel cost minimization	Emission minimization	Best compromise solution
	FC ($/h)	E (ton/h)	TC ($/h)	FC ($/h)	E (ton/h)
CBA (Adarsh et al. 2016)	121412.5468	NR	NR	NR	NR
MOMVO (Sundaram 2020)	123410	178150	NR	125470	209910
MSSA (Elsayed et al. 2016)	121413.4686	NR	NR	NR	NR
CE-SQP (Subathra et al. 2015)	121412.88	NR	NR	NR	NR
QPSO (Meng et al. 2010)	121448.21	NR	NR	NR	NR
DE (Elsayed and El-Saadany 2015)	121412.68	NR	NR	NR	NR
SMA (Ahmed et al. 2021)	121413.0	NR	NR	NR	NR
NSGA-II.R (Kuk et al. 2021)	121414.50	176718.90	NR	125668.93	195668.93
QOTLBO (Roy and Bhui 2013)	121428.00	176682.50	NR	125161.00	206490.40
KKO (Srivastava and Das 2020)	NR	NR	NR	125852	210837
PSOGSA (Radosavljević 2016)	121461,429	176678,019	NR	128710,905	178591,659
WOA (Rex, Edwin, and Annrose 2019)	NR	NR	1510240	123644	213240
OWP-based OMF	121308.4046	176682.3690	95722.2719	128595.9724	178557.2230

Figure 10. Comparison of the results for $P_d=10500MW$.

Figure 11. Ranking of OWP-OMF and the compared algorithms by the Friedman’s test on test system III.

Figure 12. Ranking of OWP-OMF and the compared algorithms by the Quade’s test on test system III.

Table 11. Statistical results obtained over 30 runs with OWP-based OMF for test system 3

	Min	Max	Mean	Standard	Q1	Median	Q3	Time (s)
ELD	121307.08	121315.9	121312.97	1.741	121312.07	121312.80	121314.43	0.203
EMD	176682.20	176682.41	176682.34	0.043	176682.31	176682.33	176682.37	0.438
CEED	95722.01	95722.97	95722.39	0.200	95722.25	95722.39	95722.52	0.078

Figure 13. Box plot evaluation for ELD, EMD and CEED problems for test system III.

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