In this section, proposed algorithms validated on 23 classic unconstrained benchmark functions (CUBFs) listed in Table 1 (\({f}_{1}\)- \({f}_{7}\) are unimodal, \({f}_{8}\)- \({f}_{13}\) are multimodal and \({f}_{14}\)- \({f}_{23}\) are fixed-dimension). Then utilized to solve 5 small (structural optimization) and 1 large (economic load dispatch) scale engineering design optimization problems described in Table 2 and Table 3 respectively. The bracket operator penalty (Deb 1995) is picked in present study due to its higher efficiency. It is fine-tuned and fixed at R=\(1{e}^{03}\) for presented algorithms. All simulation was piloted on CPU @ 2.30GHz with 4 GB RAM, Intel (R) Core (TM) i5-2350 and simulated in C language. In each table, overall best values are emphasized with bold of matching algorithms. In all experiment stopping criteria, independent run and population size of presented algorithms are taken as same or the least of comparative methods for fair judgement.
4.1 on CUBFs
The created result by suggested approaches on 23 UBFs is equated with-
- classical algorithms: HHO (Heidari et al. 2019) & EO (Faramarzi et al. 2019)
- PSO variants: HEPSO (Mahmoodabadi et al. 2014) & RPSOLF (Yan et al. 2017)
- DE variants: JADE (Zhang and Sanderson 2009) & SHADE (Tanabe and Fukunaga 2013)
- hybrid variants: FAPSO (Xia et al. 2018) & PSOSCALF (Chegini et al. 2018)
The parameters of all above equated and projected methods are registered in Table 4. The relative performance of each method recorded in Table 5 as average value (avg), standard deviation (std.) and ranking (rank) over 30 independent runs of the objective function values.
Table 4
Presented and compared methods parameter setting for CUBFs
Methods
|
Factor
|
Population Size
|
Stopping criterion
|
Run
|
Term
|
Values
|
HHO (Heidari et al. 2019)
|
escaping energy
|
\(E<0.5\), \(E\ge 0.5\)
|
30
|
500
|
30
|
EO (Faramarzi et al. 2019)
|
a1, a2 & GP
|
{1, 1.5, 2, 2.5, 3}, {0.1, 0.5, 1, 1.5, 2} & (0.1., 0.25, 0.5, 0.75, 0.9}
|
30
|
500
|
30
|
HEPSO (Mahmoodabadi et al. 2014)
|
PC & PB
|
0.95 & 0.02
|
50
|
500
|
30
|
RPSOLF (Yan et al. 2017)
|
w, c1, c2, c3, β & ε
|
0.55, 1.49, 1.49, 1.5 & 0.99
|
50
|
500
|
30
|
JADE (Zhang and Sanderson 2009)
|
Fi & CRi,
|
randni (µCR, 0.1) & randci (µF, 0.1)
|
50
|
1000
|
30
|
SHADE (Tanabe and Fukunaga 2013)
|
Pbest & Arc rate
|
0.1 & 2
|
30
|
500
|
30
|
FAPSO (Xia et al. 2018)
|
-
|
-
|
50
|
5000
|
30
|
PSOSCALF (Chegini et al. 2018)
|
wmin, wmax, c1min, c1max, c2min, c2max & β
|
0.4, 0.9, 0.5, 2.5, 0.5, 2.5 & 1.5
|
50
|
500
|
30
|
ihPSODE
|
presented
|
-
|
-
|
30
|
500
|
30
|
nDE
|
τ
|
[1.5, 2.2]
|
30
|
500
|
30
|
nPSO
|
-
|
-
|
30
|
500
|
30
|
Table 5
Experimental results on CUBFs
Function
|
Stat
|
Algorithms
|
Classical methods
|
PSO alternatives
|
DE alternatives
|
Hybrid alternatives
|
Projected methodologies
|
HHO
|
EO
|
HEPSO
|
RPSOLF
|
JADE
|
SHADE
|
PSOSCALF
|
FAPSO
|
nPSO
|
nDE
|
ihPSODE
|
f_1
|
avg
|
2.03e + 00
|
3.32e-40
|
16.26772
|
5.065e-269
|
1.87e-31
|
1.42e-09
|
1.11014e-20
|
2.87e − 127
|
0
|
0
|
0
|
std
|
4.04e − 01
|
6.78e-40
|
10.01293
|
0
|
6.43e-31
|
3.09e-09
|
1.83289E-20
|
1.76e − 127
|
0
|
0
|
0
|
rank
|
9
|
4
|
8
|
2
|
5
|
7
|
6
|
3
|
1
|
1
|
1
|
f_2
|
avg
|
1.70e + 00
|
7.12e-23
|
1.28424
|
1.000e-134
|
2.79e-15
|
0.0087
|
4.09460E-11
|
1.02e-17
|
0
|
0
|
0
|
std
|
7.37e − 02
|
6.36e-23
|
0.41611
|
3.753e-134
|
9.51e-15
|
0.0213
|
5.68981E-11
|
1.43e-17
|
0
|
0
|
0
|
rank
|
9
|
3
|
8
|
2
|
4
|
7
|
6
|
5
|
1
|
1
|
1
|
f_3
|
avg
|
1.17e + 02
|
8.06e-09
|
7.423e + 03
|
7.791e-249
|
1.10e-03
|
15.4352
|
2.16858E-12
|
1.68e-11
|
0.17e-129
|
0
|
0
|
std
|
5.28e + 00
|
1.60e-08
|
7.423e + 03
|
0
|
5.14e-03
|
9.9489
|
1.03815E-11
|
2.49e-11
|
1.67e-131
|
0
|
0
|
rank
|
9
|
6
|
10
|
2
|
7
|
8
|
4
|
5
|
3
|
1
|
1
|
f_4
|
avg
|
2.05e + 00
|
5.39e-10
|
23.95145
|
1.937e-157
|
1.66e-03
|
0.9796
|
8.47410E-08
|
4.09e + 03
|
8.35e-098
|
3.28e-101
|
0
|
std
|
7.40e − 02
|
1.38e-09
|
7.71460
|
1.061e-156
|
1.98e-03
|
0.7995
|
1.23324E-07
|
6.53e + 02
|
7.08e-098
|
7.12e-103
|
0
|
rank
|
9
|
5
|
10
|
2
|
7
|
8
|
6
|
11
|
4
|
3
|
1
|
f_5
|
avg
|
2.95e + 00
|
2.53e + 01
|
2.380e + 03
|
27.42672
|
1.18e + 01
|
24.4743
|
21.97646
|
6.55e-11
|
4.85e-012
|
1.25e-021
|
2.21e-033
|
std
|
8.36e − 02
|
0.16e + 00
|
1.852e + 03
|
0.24848
|
1.57e + 01
|
11.2080
|
0.54774
|
1.99e-11
|
3.21e-012
|
1.85e-023
|
3.72e-037
|
rank
|
9
|
7
|
11
|
8
|
10
|
6
|
5
|
4
|
3
|
2
|
1
|
f_6
|
avg
|
2.49e + 00
|
8.29e-06
|
21.55405
|
2.98244
|
4.59e-31
|
5.31e-10
|
7.13998E-12
|
2.37e-12
|
1.75e-032
|
0
|
0
|
std
|
8.25e − 02
|
5.02e-06
|
9.33263
|
0.23250
|
1.65e-30
|
6.35e-10
|
3.65884E-11
|
1.84e-13
|
9.16e-035
|
0
|
0
|
rank
|
9
|
8
|
6
|
10
|
3
|
7
|
5
|
4
|
2
|
1
|
1
|
f_7
|
avg
|
8.20e + 00
|
1.17e-02
|
0.12982
|
0.00104
|
6.49e-03
|
0.0235
|
0.00012
|
0
|
5.11e-001
|
2.19e-003
|
1.07e-003
|
std
|
1.69e − 01
|
6.54e-04
|
0.09727
|
7.644e-04
|
2.48e-03
|
0.0088
|
0.00010
|
0
|
1.70e-002
|
1.10e-004
|
1.40e-005
|
rank
|
11
|
6
|
10
|
7
|
5
|
9
|
4
|
1
|
8
|
3
|
2
|
f_8
|
avg
|
4.86e + 00
|
-9016.34
|
-2.139e + 03
|
-3.254e + 03
|
-1.24E + 04
|
-11713.1
|
-12569.48
|
2.48e-11
|
-6.37e + 004
|
-1.25e + 004
|
-1.25e + 004
|
std
|
1.03e + 00
|
595.1113
|
8.282e + 02
|
2.860e + 02
|
1.27e + 02
|
230.49
|
2.39996e-07
|
6.44e-12
|
2.10E-001
|
1.07e-017
|
0
|
rank
|
4
|
6
|
5
|
7
|
1
|
8
|
1
|
2
|
9
|
1
|
1
|
f_9
|
avg
|
3.77e + 00
|
0
|
42.00118
|
0
|
1.71e-04
|
8.5332
|
0
|
0
|
0
|
0
|
0
|
std
|
8.87e − 01
|
0
|
7.08632
|
0
|
1.52e-04
|
2.1959
|
0
|
0
|
0
|
0
|
0
|
rank
|
3
|
1
|
5
|
1
|
2
|
4
|
1
|
1
|
1
|
1
|
1
|
f-10
|
avg
|
3.75e + 00
|
8.34e-14
|
2.83842
|
4.085e-15
|
1.31e-14
|
0.3957
|
2.24609e-11
|
4.86e-15
|
1.97e-014
|
1.44e-015
|
2.88e-016
|
std
|
8.75e − 01
|
2.53e-14
|
0.66134
|
1.084e-15
|
2.46e-14
|
0.5868
|
2.33542e-11
|
1.74e-15
|
0
|
0
|
0
|
rank
|
8
|
7
|
9
|
3
|
5
|
10
|
8
|
4
|
6
|
2
|
1
|
f_11
|
avg
|
4.17e + 00
|
0
|
1.16858
|
0.00e + 00
|
2.87e-03
|
0.0048
|
0
|
1.74e-16
|
3.37e-111
|
0
|
0
|
std
|
5.56e − 01
|
0
|
0.12602
|
0.00e + 00
|
7.85e-03
|
0.0077
|
0
|
3.60e-16
|
1.11e-119
|
0
|
0
|
rank
|
7
|
1
|
6
|
1
|
4
|
5
|
1
|
3
|
2
|
1
|
1
|
f_12
|
avg
|
1.90e + 01
|
7.97e-07
|
0.47856
|
0.26157
|
1.73e-02
|
0.0346
|
8.46465e-14
|
1.57e-32
|
3.34E-002
|
1.05E-032
|
3.73E-033
|
std
|
3.31e + 00
|
7.69e-07
|
0.22623
|
0.03386
|
7.74e-02
|
0.0875
|
2.79106e-13
|
0
|
1.02e-004
|
2.77e-034
|
3.18e-034
|
rank
|
11
|
5
|
10
|
9
|
6
|
8
|
4
|
3
|
7
|
2
|
1
|
f_13
|
avg
|
1.89e + 01
|
0.029295
|
1.85056
|
2.05282
|
5.45e-24
|
7.32e-04
|
0.00399
|
1.58e-32
|
9.30e-004
|
2.09e-021
|
1.58e-032
|
std
|
1.56e + 00
|
0.035271
|
0.65246
|
0.16579
|
2.58e-23
|
0.0028
|
0.00928
|
0
|
3.71e-004
|
3.27e-023
|
1.89e-043
|
rank
|
10
|
7
|
8
|
9
|
2
|
4
|
6
|
1
|
5
|
3
|
1
|
f_14
|
avg
|
9.98e − 01
|
0.99800
|
0.99800
|
1.54064
|
9.98e-01
|
0.998004
|
1.13027
|
9.98e-001
|
9.98e-001
|
9.98e-001
|
9.98e-001
|
std
|
9.23e − 01
|
1.54e-16
|
9.219e-17
|
1.84429
|
0
|
5.83e-17
|
0.50338
|
1.27e-08
|
0
|
0
|
0
|
rank
|
1
|
1
|
1
|
3
|
1
|
1
|
2
|
1
|
1
|
1
|
1
|
f_15
|
avg
|
3.89e − 04
|
0.00239
|
6.404e-04
|
0.00171
|
3.01e-03
|
0.002374
|
3.13244E-04
|
3.95e-04
|
3.99e-004
|
3.83e-004
|
3.02e-004
|
std
|
1.96e − 04
|
0.00609
|
2.801e-04
|
0.00508
|
6.92e-03
|
0.0061
|
2.17489E-05
|
6.02e-08
|
2.96e-007
|
9.23e-012
|
2.38e-018
|
rank
|
2
|
11
|
7
|
9
|
8
|
10
|
3
|
5
|
6
|
4
|
1
|
f_16
|
avg
|
−1.029e + 00
|
-1.03161
|
-1.03161
|
-1.03161
|
-1.03e + 00
|
-1.03162
|
-1.0316
|
−1.03e + 00
|
−1.03e + 000
|
−1.03e + 000
|
−1.03e + 000
|
std
|
6.69e − 16
|
6.04e-16
|
3.554e-15
|
1.650e-05
|
6.78e-16
|
6.51e-16
|
4.40244E-16
|
0
|
0
|
0
|
0
|
rank
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
f_17
|
avg
|
3.971e − 01
|
0.397887
|
0.39787
|
0.39837
|
3.98e-01
|
0.397887
|
0.39788
|
3.98e-001
|
3.98e-001
|
3.98e-001
|
3.98e-001
|
std
|
2.539e − 06
|
0
|
6.594e-13
|
5.267e-04
|
0
|
3.24e-16
|
3.66527E-15
|
0
|
0
|
0
|
0
|
rank
|
1
|
2
|
2
|
1
|
1
|
2
|
2
|
1
|
1
|
1
|
1
|
f_18
|
avg
|
3.000e + 00
|
3.000e + 00
|
0.65246
|
3.00002
|
3.000e + 00
|
3.000e + 00
|
3.000e + 00
|
3.00e + 000
|
3.00e + 000
|
3.00e + 000
|
3.00e + 000
|
std
|
0
|
1.56e-15
|
5.146e-11
|
1.658e-05
|
1.82e-15
|
1.87e-15
|
5.96540e-13
|
5.31e-016
|
1.33e-018
|
0
|
0
|
rank
|
1
|
1
|
2
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
f_19
|
avg
|
−3.86e + 00
|
-3.86278
|
-3.86278
|
-3.85923
|
-3.86e + 00
|
-3.86278
|
-3.86278
|
−3.87e + 000
|
−3.86e + 000
|
−3.86e + 000
|
−3.86e + 000
|
std
|
2.44e − 03
|
2.59e-15
|
1.008E-13
|
0.00283
|
2.71e-15
|
2.69e-15
|
8.31755e-15
|
2.11e-004
|
3.36e-021
|
0
|
0
|
rank
|
1
|
1
|
1
|
3
|
1
|
1
|
1
|
2
|
1
|
1
|
1
|
f_20
|
avg
|
−3.322
|
-3.2687
|
-3.31803
|
-3.10441
|
-3.29e + 00
|
-3.27047
|
-3.27168
|
-3.29e + 000
|
-3.27e + 000
|
-3.32e + 000
|
-3.32e + 000
|
std
|
0.137406
|
0.05701
|
0.02170
|
0.15760
|
5.11e-02
|
0.0599
|
0.06371
|
0
|
0
|
0
|
0
|
rank
|
1
|
6
|
2
|
7
|
4
|
5
|
5
|
3
|
5
|
1
|
1
|
f_21
|
avg
|
−10. 1531
|
-8.55481
|
-10.15319
|
-4.76171
|
-9.14e + 00
|
-9.2343
|
-10.15319
|
-7.51e + 00
|
-9.87e + 001
|
-1.01e + 001
|
-1.01e + 001
|
std
|
0.885673
|
2.76377
|
2.680e-05
|
0.73723
|
2.06e + 00
|
1.3969
|
4.46227e-15
|
1.21e-01
|
1.07e-011
|
0
|
0
|
rank
|
1
|
5
|
1
|
6
|
4
|
3
|
1
|
3
|
2
|
1
|
1
|
f_22
|
avg
|
−10.4015
|
-9.3353
|
-10.39978
|
-4.81927
|
-9.88e + 00
|
-10.2809
|
-10.40294
|
-6.04e + 01
|
-9.87e + 000
|
-1.04e + 001
|
-1.04e + 001
|
std
|
1.352375
|
2.43834
|
0.01728
|
0.75699
|
1.61e + 00
|
1.3995
|
1.80672e-15
|
2.14e-01
|
2.08e-002
|
2.91e-012
|
1.10e-016
|
rank
|
1
|
5
|
1
|
7
|
3
|
2
|
1
|
6
|
4
|
1
|
1
|
f_23
|
avg
|
−10.5364
|
-9.63655
|
-10.53640
|
-5.06376
|
-1.03e + 01
|
63.333
|
-10.53640
|
-5.64e + 00
|
-9.13e + 000
|
-1.05e + 001
|
-1.05e + 001
|
std
|
0.927655
|
2.38811
|
5.845e-07
|
0.82968
|
1.40e + 00
|
80.872
|
4.84794e-15
|
5.70e-02
|
2.05e-003
|
1.07e-021
|
0
|
rank
|
1
|
3
|
1
|
6
|
2
|
7
|
1
|
5
|
4
|
1
|
1
|
Sum of rank
|
119
|
102
|
125
|
107
|
87
|
124
|
75
|
75
|
78
|
35
|
24
|
Average
|
5.17
|
4.43
|
5.43
|
4.65
|
3.78
|
5.39
|
3.26
|
3.26
|
3.39
|
1.52
|
1.04
|
Overall rank
|
8
|
6
|
10
|
7
|
5
|
9
|
3
|
3
|
4
|
2
|
1
|
It would be distinguished that from Table 5, the average objective function values of presented algorithms (nPSO, nDE and ihPSODE) are better and/or equal than other compared classical and different variants of DE and PSO with hybrid variants for maximum CUBFs. Ultimately, suggested ihPSODE, nPSO and nDE yields less std. for most of the UBFs which designates their stability. Additionally, all methods are separately ranked in Table 5 based on average objective function values and determined that ihPSODE ranked 1st, nDE ranked 2nd and nPSO ranked 4th successively. Similarly, in Table 5 average and whole rank of suggested methods versus others are calculated which shows ihPSODE, nDE and nPSO are superior to others. Additionally, supremacy of projected methods is statistically certified over other methods from Wilcoxon Signed Rank (WSR) test and one-tailed t-test (Das and Parouha 2015). The results of these test are termed in Table 6 on CUBFs and determined that the suggested methods perform in case of –
- t-test – “a”: significantly better or “a+”: highly significance than other
- WSR test – “+”: better or “≈”: equally performances in most of cases
- p-values: for the majority of runs, less values accomplish that the results are trustworthy
Table 6
Statistical evaluations of presented versus other methods for CUBFs
versus
|
Standards
|
Methods
|
Classical methods
|
PSO alternatives
|
DE alternatives
|
Hybrid alternatives
|
Projected methodologies
|
|
|
HHO
|
EO
|
HEPSO
|
RPSOLF
|
JADE
|
SHADE
|
PSOSCALF
|
FAPSO
|
nDE
|
ihPSODE
|
nPSO
|
Better
|
11
|
21
|
20
|
19
|
20
|
13
|
19
|
15
|
0
|
0
|
Equal
|
4
|
2
|
2
|
2
|
2
|
9
|
3
|
4
|
7
|
8
|
Worst
|
8
|
0
|
1
|
2
|
1
|
1
|
1
|
4
|
16
|
15
|
R+
|
293
|
387
|
312
|
323
|
335
|
305
|
382
|
300
|
350
|
400
|
R−
|
172
|
78
|
153
|
142
|
130
|
160
|
83
|
165
|
115
|
65
|
p-value
|
5.1e-09
|
5.3e-10
|
5.7e-10
|
5.1e-09
|
6.2e-10
|
4.6e-08
|
5.6e-10
|
5.8e-10
|
6.2e-09
|
5.3e-10
|
t-test
|
a
|
a
|
a
|
a+
|
a
|
a+
|
a+
|
a+
|
a
|
a+
|
Decision
|
≈
|
≈
|
≈
|
+
|
+
|
+
|
+
|
+
|
+
|
+
|
versus
|
|
HHO
|
EO
|
HEPSO
|
RPSOLF
|
JADE
|
SHADE
|
PSOSCALF
|
FAPSO
|
nPSO
|
ihPSODE
|
nDE
|
Better
|
15
|
21
|
21
|
21
|
20
|
13
|
18
|
14
|
15
|
0
|
Equal
|
8
|
2
|
2
|
2
|
3
|
9
|
5
|
9
|
8
|
16
|
Worst
|
0
|
0
|
0
|
0
|
0
|
1
|
0
|
0
|
0
|
7
|
R+
|
416
|
313
|
329
|
465
|
345
|
355
|
323
|
377
|
342
|
315
|
R−
|
49
|
152
|
136
|
79
|
120
|
130
|
142
|
88
|
123
|
150
|
p-value
|
5.6e-10
|
5.2e-10
|
6.2e-10
|
6.9e-07
|
8.2e-10
|
5.8e-10
|
4.3e-09
|
6.2e-11
|
5.1E-10
|
6.9e-07
|
t-test
|
a
|
a
|
a
|
a
|
a
|
a+
|
a
|
a+
|
a+
|
a+
|
Decision
|
+
|
+
|
+
|
≈
|
+
|
≈
|
+
|
≈
|
+
|
+
|
versus
|
|
HHO
|
EO
|
HEPSO
|
RPSOLF
|
JADE
|
SHADE
|
PSOSCALF
|
FAPSO
|
nDE
|
nPSO
|
ihPSODE
|
Better
|
14
|
0
|
20
|
20
|
20
|
13
|
15
|
14
|
8
|
15
|
Equal
|
7
|
6
|
3
|
3
|
3
|
10
|
8
|
9
|
15
|
8
|
Worst
|
2
|
17
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
R+
|
321
|
294
|
330
|
313
|
329
|
367
|
377
|
293
|
323
|
304
|
R−
|
144
|
171
|
135
|
152
|
136
|
98
|
88
|
172
|
142
|
161
|
p-value
|
6.2e-10
|
5.1e-10
|
5.1e-07
|
5.1e-10
|
4.6e-08
|
5.7e-10
|
5.3e-08
|
6.2e-09
|
4.6e-10
|
5.7e-07
|
t-test
|
a
|
a
|
a
|
a+
|
a
|
a+
|
a+
|
a
|
a+
|
a
|
Decision
|
+
|
+
|
+
|
≈
|
≈
|
+
|
+
|
≈
|
+
|
+
|
To demonstrate that proposed algorithms has a satisfied convergence speed, the convergence curves (objective function values versus iterations) over eight (f_1, f_5, f_6, f_7, f_8, f_9, f_10 and f_11) 30-D CUBFs are plotted in Fig. 4(a-h) with others. From Fig. 4(a-h), it can find that ihPSODE, nDE and nPSO has quicker convergence speed than equated algorithms on most functions.
Altogether, it may have decided that the performance of suggested ihPSODE, nDE and nPSO methods are superior and/or equal to other intelligent compared optimization methods on most test functions. However, among three projected algorithms ihPSODE considered as an extremely effective and efficient method.
4.2 on SSEDOPs and LSEDOP
The results of proposed ihPSODE, nPSO and nDE algorithms on 5 small scale engineering design optimization problems (SSEDOPs) are equated with GA (Davis 1991), PSO (Kennedy and Eberhart 1995), DE (Storn and Price 1997), HS (Geem et al. 2001), SBM (Akhtar 2002), SAC (Ray and Liew 2003), IPSO (He et al. 2004), FSA (Hedar and Fukushima 2006), ABC (Karaboga and Basturk 2007), CDE (Huang et al. 2007), modified DE (Montes et al. 2007), CPSO (He and Wang 2007), DSS-MDE (Zhang et al. 2008), CS (Yang and Deb 2009), GSA (Rashedi et al. 2009), QPSO (Coelho 2010), ACO (Kaveh and Talatahari 2010), KH (Gandomi and Alavi 2012), WCA (Eskandar et al. 2012), MBA (Sadollah et al. 2013), GWO (Mirjalili et al. 2014), AFA (Baykasoglu and Ozsoydan 2015), DA (Mirjalili 2016), SCA (Mirjalili 2016), MVO (Mirjalili et al. 2016), SHO (Dhiman and Kumar 2017), EPO (Dhiman and Kumar 2018), EO (Faramarzi et al. 2019), CSDE (Zhang et al. 2019), CSKH (Basset 2019), GSA-GA (Garg 2019), PSO-OPS (Isiet and Gadala 2020) and PSOSCANMS (Fakhouri et al. 2020). In experiment 25- independent run, 1500 iterations - stopping criteria and 30- population size of presented algorithms is taken same as comparative algorithms for fair comparison. Rest parameter of presented algorithms as similar as above. The relative results of presented methods with others on corresponding SSEDOPs are presented in Table 7 (for SSEDOP-1), Table 8 (for SSEDOP-2), Table 9 (for SSEDOP-3), Table 10 (for SSEDOP-4) and Table 11 (for SSEDOP-5) in terms of optimal values for variables, best, worst, mean and std. of the objective function values over 25- independent run.
Table 7
Comparative results for SSEDOP-1
Methods
|
Optimal values for variables
|
best
|
worst
|
mean
|
std.
|
H (\({x}_{1}\))
|
L(\({x}_{2}\))
|
T(\({x}_{3}\))
|
b(\({x}_{4}\))
|
GA (Davis 1991)
|
0.164171
|
4.032541
|
10.00000
|
0.223647
|
1.8739710
|
2.320125
|
2.1192400
|
0.0348200
|
PSO (Kennedy and Eberhart 1995)
|
0.197411
|
3.315061
|
10.00000
|
0.201395
|
1.8203950
|
3.048231
|
2.2303100
|
0.3245250
|
SBM (Akhtar 2002)
|
0.240700
|
6.485100
|
8.239900
|
0.249700
|
2.4426000
|
2.631500
|
2.5215000
|
0.0221840
|
IPSO (He et al. 2004)
|
0.244400
|
6.217500
|
8.291500
|
0.244400
|
2.3810000
|
2.311000
|
2.3819000
|
0.0052300
|
FSA (Hedar and Fukushima 2006)
|
0.244400
|
6.125800
|
8.293900
|
0.244400
|
2.3811000
|
2.488900
|
2.4041000
|
0.0321940
|
CPSO (He and Wang 2007)
|
0.202400
|
3.544200
|
9.048210
|
0.205700
|
1.7280000
|
1.782143
|
1.7488310
|
0.0129260
|
ABC (Karaboga and Basturk 2007)
|
0.205730
|
3.470489
|
9.036624
|
0.205730
|
1.7248520
|
1.734852
|
1.7419130
|
0.0310000
|
CDE (Huang et al. 2007)
|
0.203100
|
3.543000
|
9.033500
|
0.206200
|
1.7335000
|
1.824105
|
1.7681580
|
0.0221940
|
CS (Yang and Deb 2009)
|
0.201500
|
3.562000
|
9.041400
|
0.205700
|
1.7312065
|
1.878656
|
2.3455793
|
0.2677989
|
GWO (Mirjalili et al. 2014)
|
0.205678
|
3.475403
|
9.036964
|
0.206229
|
1.7269950
|
1.727128
|
1.7275640
|
0.0011570
|
SCA (Mirjalili 2016)
|
0.244400
|
6.238000
|
8.288600
|
0.244600
|
2.3854000
|
6.399600
|
3.2551000
|
0.9590000
|
MVO (Mirjalili et al. 2016)
|
0.205611
|
3.472103
|
9.040931
|
0.205709
|
1.7254720
|
1.741651
|
1.7296800
|
0.0048660
|
DA (Mirjalili 2016)
|
0.194288
|
3.466810
|
9.045430
|
0.205695
|
1.7080800
|
1.940760
|
2.5210600
|
0.2502340
|
SHO (Dhiman and Kumar 2017)
|
0.205563
|
3.474846
|
9.035799
|
0.205811
|
1.7256610
|
1.726064
|
1.7258280
|
0.0002870
|
EPO (Dhiman and Kumar 2018)
|
0.205411
|
3.472341
|
9.035215
|
0.201153
|
1.7235890
|
1.727211
|
1.7251240
|
0.0043250
|
EO (Faramarzi et al. 2019)
|
0.205700
|
3.470500
|
9.036640
|
0.205700
|
1.7248530
|
1.736725
|
1.7264820
|
0.0032570
|
nPSO
|
0.193710
|
3.485710
|
8.278520
|
0.222410
|
1.7248600
|
1.726980
|
1.7354200
|
0.0003540
|
nDE
|
0.194070
|
3.325470
|
8.278830
|
0.204570
|
1.7235500
|
1.728740
|
1.7262500
|
0.0002840
|
ihPSODE
|
0.184280
|
3.266410
|
8.241330
|
0.204580
|
1.6978200
|
1.723210
|
1.7242100
|
0.0001320
|
Table 8
Comparative results for SSEDOP-2
Methods
|
Optimal values for variables
|
best
|
worst
|
mean
|
std.
|
A1(\({x}_{1}\))
|
A2(\({x}_{2}\))
|
PSO (Kennedy and Eberhart 1995)
|
7.803e-01
|
4.330e-01
|
264.543754826635
|
2.70864524844245e + 05
|
264.775374387455
|
1.85577512704186e + 00
|
DE (Storn and Price 1997)
|
7.887e-01
|
4.080e-01
|
263.148352624688
|
65 535
|
263.411272728312
|
1.05476883610644e − 01
|
SAC (Ray and Liew 2003)
|
0.7886210370
|
0.4084013340
|
263.8958466
|
263.96975
|
263.9033
|
1.26e-02
|
DSS-MDE (Zhang et al. 2008)
|
0.7886751359
|
0.4082482868
|
263.8958434
|
263.8958498
|
263.8958436
|
9.72e-07
|
CS (Yang and Deb 2009)
|
7.357e-01
|
5.945e-01
|
263.602007628033
|
5.23525611223402e + 12
|
263.671445662651
|
1.63380787023616e − 02
|
KH (Gandomi and Alavi 2012)
|
7.885e-01
|
4.088e-01
|
2.639e + 02
|
2.650e + 02
|
2.639e + 02
|
1.658e-01
|
MBA (Sadollah et al. 2013)
|
7.886e-01
|
4.086e-01
|
2.639e + 02
|
2.639e + 02
|
2.639e + 02
|
3.930e-03
|
CSDE (Zhang et al. 2019)
|
7.886e-01
|
4.082e-01
|
263.148352124271
|
65535
|
263.148352318831
|
1.44060040776154e − 08
|
GSA-GA (Garg 2019)
|
0.788676171219
|
0.408245358456
|
263.8958433
|
263.8958459
|
263.8958437
|
5.34e-07
|
PSO-OPS (Isiet and Gadala 2020)
|
7.886e-01
|
4.082e-01
|
2.639 e + 02
|
2.639e + 02
|
2.639e + 02
|
1.354e-03
|
nPSO
|
0.780327
|
0.408291
|
262.6263
|
263.7845
|
263.7845
|
1.0358e-08
|
nDE
|
0.780315
|
0.408217
|
263.4263
|
263.8497
|
263.8497
|
1.2458e-08
|
ihPSODE
|
0.788526
|
0.408452
|
261.1438
|
262.9796
|
262.5782
|
2.51420e-09
|
Table 9
Comparative results for SSEDOP-3
Methods
|
Optimal values for variables
|
best
|
worst
|
mean
|
std.
|
Ts (\({x}_{1}\))
|
Th(\({x}_{2}\))
|
R(\({x}_{3}\))
|
L(\({x}_{4}\))
|
GA (Davis 1991)
|
0.752362
|
0.399540
|
40.452514
|
198.00268
|
5890.3279
|
7005.7500
|
6264.0053
|
496.128
|
PSO (Kennedy and Eberhart 1995)
|
0.8125
|
0.4375
|
42.091266
|
176.7465
|
5891.3879
|
6531.5032
|
7394.5879
|
534.119
|
DE (Storn and Price 1997)
|
0.8125
|
0.4375
|
42.098411
|
176.637690
|
6074.6231
|
6751.5312
|
6619.0083
|
358.799
|
HS (Geem et al. 2001)
|
1.099523
|
0.906579
|
44.456397
|
179.65887
|
6550.0230
|
8005.4397
|
6643.9870
|
657.523
|
IPSO (He et al. 2004)
|
0.812500
|
0.437500
|
42.098445
|
176.6365950
|
6059.7143
|
6251.5312
|
6289.92881
|
305.78
|
ABC (Karaboga and Basturk 2007)
|
0.812500
|
0.437500
|
42.098446
|
176.636596
|
6059.714339
|
6650.5102
|
6245.308144
|
205.000
|
CDE (Huang et al. 2007)
|
0.8125
|
0.4375
|
42.098411
|
176.63769
|
6059.7340
|
6085.2303
|
6371.0455
|
43.0130
|
CPSO (He and Wang 2007)
|
0.8125
|
0.4375
|
42.091266
|
176.746500
|
6061.0777
|
6147.1332
|
6363.8041
|
86.4545
|
CS (Yang and Deb 2009)
|
0.812500
|
0.437500
|
42.0984456
|
176.6363595
|
6059.714
|
6495.3470
|
6447.7360
|
502.693
|
ACO (Kaveh and Talatahari 2010)
|
0.8125
|
0.4375
|
42.103624
|
176.572656
|
6014.6231
|
6651.5312
|
6219.0083
|
423.524
|
GWO (GWO (Mirjalili et al. 2014)
|
0.8125
|
0.4345
|
42.089181
|
176.75731
|
5889.3689
|
5894.6238
|
5891.5247
|
013.910
|
DA (Mirjalili 2016)
|
0.782825
|
0.384649
|
40.3196
|
200
|
5923.11
|
222536
|
21342.2
|
470.44
|
EO (Faramarzi et al. 2019)
|
0.7781
|
0.3846
|
40.319619
|
199.99999
|
6059.7143
|
6668.114
|
7544.4925
|
566.24
|
CSDE (Zhang et al. 2019)
|
0.8125
|
0.4.375
|
42.10
|
176.6
|
6059.7133
|
1.528E + 22
|
6261.4178
|
263.6758
|
CSKH (Basset 2019)
|
0.7781686
|
0.3846491
|
40.3196187
|
200.0000
|
5885.332773
|
5885.486467
|
5885.382053
|
0. 049080
|
nPSO
|
0.81239
|
0.43748
|
40.39457
|
179.79881
|
5885.56134
|
5885.86612
|
5885.43434
|
0.910214
|
nDE
|
0.81241
|
0.43729
|
40.31749
|
179.85887
|
5883.42511
|
5883.78501
|
5883.23448
|
0.013023
|
ihPSODE
|
0.81250
|
0.43750
|
40.31962
|
179.65887
|
5882.43871
|
5882.83421
|
5882.83421
|
0.011289
|
Table 10
Comparative results for SSEDOP-4
Methods
|
Optimal values for variables
|
Optimal cost
|
best
|
worst
|
mean
|
b
|
m
|
p
|
ll
|
l2
|
d1
|
d2
|
GA (Davis 1991)
|
3.510253
|
0.7
|
17
|
8.35
|
7.8
|
3.362201
|
5.287723
|
3067.561
|
3313.199
|
3186.523
|
17.1186
|
PSO (Kennedy and Eberhart 1995)
|
3.500019
|
0.7
|
17
|
8.3
|
7.8
|
3.352412
|
5.286715
|
3005.763
|
3211.174
|
3105.252
|
79.6381
|
HS (Geem et al. 2001)
|
3.520124
|
0.7
|
17
|
8.37
|
7.8
|
3.366970
|
5.288719
|
3029.002
|
3619.465
|
3295.329
|
57.0235
|
GSA (Rashedi et al. 2009)
|
3.600000
|
0.7
|
17
|
8.3
|
7.8
|
3.369658
|
5.289224
|
3051.120
|
3363.873
|
3170.334
|
92.5726
|
CS (Yang and Deb 2009)
|
3.501
|
0.700
|
0.700e + 00
|
7.6057
|
7.818
|
3.352
|
5.287
|
3.001 e + 03
|
3.009e + 03
|
3.007e + 03
|
4.968e + 00
|
KH (Gandomi and Alavi 2012)
|
3.500
|
0.700
|
0.700e + 00
|
7.3667
|
7.823
|
3.350
|
5.287
|
2.997 e + 03
|
3.011e + 03
|
3.006e + 03
|
2.638e + 00
|
MBA (Sadollah et al. 2013)
|
3.500
|
0.700
|
0.700e + 00
|
7.3007
|
7.716
|
3.350
|
5.287
|
2.994 e + 03
|
3.000e + 03
|
2.997e + 03
|
1.560e + 00
|
GWO (Mirjalili et al. 2014)
|
3.506690
|
0.7
|
17
|
7.380933
|
7.815726
|
3.357847
|
5.286768
|
3001.288
|
3008.752
|
3005.845
|
5.83794
|
AFA (Baykasoglu and Ozsoydan 2015)
|
3.500
|
0.700
|
0.700e + 00
|
7.3027
|
7.8007
|
3.350
|
5.287
|
2.996 e + 03
|
2.997e + 03
|
2.996e + 03
|
9.000 e-02
|
MVO (Mirjalili et al. 2016)
|
3.508502
|
0.7
|
17
|
7.3
|
7.8
|
3.358073
|
5.286777
|
3002.928
|
3060.958
|
3028.841
|
13.0186
|
SCA (Mirjalili 2016)
|
3.508755
|
0.7
|
17
|
7.3
|
7.8
|
3.461020
|
5.289213
|
3030.563
|
3104.779
|
3065.917
|
18.0742
|
SHO (Dhiman and Kumar 2017)
|
3.50159
|
0.7
|
17
|
7.3
|
7.8
|
3.35127
|
5.28874
|
2998.5507
|
3003.889
|
2999.640
|
1.93193
|
EPO (Dhiman and Kumar 2018)
|
3.50123
|
0.7
|
17
|
7.3
|
7.8
|
3.33421
|
5.26536
|
2994.2472
|
2999.092
|
2997.482
|
1.78091
|
nPSO
|
3.500
|
0.7
|
17
|
7.37980
|
7.8
|
3.349
|
5.2869
|
2992.1242
|
2995.0214
|
2991.4756
|
0.17951
|
nDE
|
3.500
|
0.7
|
17
|
7.38089
|
7.8
|
3.351
|
5.2888
|
2994.2442
|
2997.1845
|
2995.9547
|
0.08791
|
ihPSODE
|
3.500
|
0.7
|
17
|
7.38091
|
7.8
|
3.350
|
5.2869
|
2990.3582
|
2993.2145
|
2992.5481
|
0.01693
|
Table 11
Comparative results for SSEDOP-5
Methods
|
Optimal values for variables
|
Optimal cost
|
best
|
worst
|
mean
|
d(\({x}_{1}\))
|
D(\({x}_{2}\))
|
N(\({x}_{3}\))
|
PSO (Kennedy and Eberhart 1995)
|
0.051728
|
0.357644
|
11.244543
|
0.016508
|
0.015234
|
0.049161
|
4.027e-04
|
DE (Storn and Price 1997)
|
0.051609
|
0.354714
|
11.410831
|
0.012693
|
0.020034
|
0.012744
|
3.680e-05
|
HS (Geem et al. 2001)
|
0.051154
|
0.349871
|
12.076432
|
0.012776352
|
0.015214230
|
0.013069872
|
0.000375
|
CDE (Huang et al. 2007)
|
0.051609
|
0.354714
|
11.410831
|
0.012624
|
0.012914
|
0.012830
|
5.29e-05
|
CPSO (He and Wang 2007)
|
0.051728
|
0.357644
|
11.244543
|
0.012674
|
0.012924
|
0.012730
|
5.19e-05
|
modified DE (Montes et al. 2007)
|
0.051688
|
0.356692
|
11.290483
|
0.012665
|
0.012654
|
0.012666
|
2.0e-06
|
GSA (Rashedi et al. 2009)
|
0.050276
|
0.323680
|
13.525410
|
0.012873881
|
0.014211731
|
0.013438871
|
0.000287
|
QPSO (Coelho 2010)
|
0.051515
|
0.352529
|
11.538862
|
0.012665
|
0.017759
|
0.013524
|
1.268e-03
|
WCA (Eskandar et al. 2012)
|
0.0517208702
|
0.3579276279
|
11.1912042488
|
0.012630231
|
0.017722009
|
0.013388089
|
1.0864e-03
|
GWO (Mirjalili et al. 2014)
|
0.05169
|
0.356737
|
11.28885
|
0.012666
|
0.012654
|
0.012741
|
1.281e-05
|
AFA (Baykasoglu and Ozsoydan 2015)
|
5.167E-02
|
3.562E-01
|
1.132E + 01
|
0.012670
|
0.012710
|
0.012680
|
1.281e-05
|
SCA (Mirjalili 2016)
|
0.052160
|
368159
|
10.648442
|
0.012669
|
0.016717
|
0.012922
|
5.92e-04
|
SHO (Dhiman and Kumar 2017)
|
0.051144
|
0.343751
|
12.0955
|
0.012674000
|
0.012715185
|
0.012684106
|
0.000027
|
EPO (Dhiman and Kumar 2018)
|
0.051087
|
0.342908
|
12.0898
|
0.012658
|
0.126553
|
0.012754
|
2.02e-06
|
EO (Faramarzi et al. 2019)
|
0.0516199100
|
0.355054381
|
11.387967
|
0.012666
|
0.013997
|
0.013017
|
3.91e-04
|
PSOSCANMS (Fakhouri et al. 2020)
|
0.05072
|
0.334801
|
10.79431
|
0.012676
|
0.013977
|
0.013817
|
3.92e-04
|
nPSO
|
0.05163
|
0.34792
|
11.49631
|
0.012660
|
0.012660
|
0.128451
|
2.12e-06
|
nDE
|
0.05017
|
0.35023
|
11.45272
|
0.012475
|
0.012652
|
0.012664
|
1.12e-06
|
ihPSODE
|
0.05012
|
0.32843
|
11.25213
|
0.012873881
|
0.014211731
|
0.013438871
|
0.000287
|
It is very clear that from these tables, the presented nPSO, nDE and ihPSODE methods produce improved and/or similar results on all small scale engineering design optimization problems. Eventually, for all cases proposed nPSO, nDE and ihPSODE yields less std. which describes their constancy. The convergence charts of all presented and best non-proposed methods (to avoid complicacy) is plotted and depicted in Fig. 5(a-e) for all SSEDOPs. From this figures, it is clearly pictured that presented methods converge earlier than others. Therefore, presented methods are computationally capable.
Furthermore, following unit test system of LSEDOP i.e. economic load dispatch (ELD) problem with and without valve point loading effects are solved using presented algorithms and the results are compared with the modern methods.
Table 12
Unit Test Systems (UTSs)
|
Description
|
UTS-1: 3-unit test system
(Sinha et al. 2003)
|
it involves valve-point effects with 850 MW total demand.
|
UTS-2: 6-unit test system
(Gaing 2003)
|
it contains 1263 MW total demand with constraints ramp-rate limit, prohibited operating zone and transmission losses.
|
UTS-3: 15-unit test system
(Gaing 2003)
|
it includes 2630 MW total demand with constraints prohibited operating zone and ramp-rate limits.
|
UTS-4: 40-unit test system
(Sinha et al. 2003)
|
it involves 10500 MW total demand and valve-point effects.
|
UTS-5: 140-unit test system
(Dos Santos Coelho et al. 2014)
|
it contains 49342 MW total demand with constraints prohibited operating zone, valve-point loading effects and ramp-rate limits.
|
The results created by presented methods on above considered diverse test systems of ELD problem are equated with other modern algorithms. These compared algorithms are listed as follows: GA (Davis 1991), PSO (Kennedy and Eberhart 1995), DE (Storn and Price 1997), IPSO (He et al. 2004), NCS (Kuo 2008), BCO (Chokpanyasuwan et al. 2009), EHM (Kasmaei and Nejad 2011), IABC (Aydın et al. 2011), IPSO-TVAC (Mohammadi et al. 2012), MGSO (Zare et al. 2012), 𝜽-PSO (Hosseinnezhad and Babaei 2013), DEPSO (Sayah and Hamouda 2013), DHS (Wang and Li 2013), MPSO-TVAC (Abdullah et al. 2014), MPSO (Basu 2015), DPD (Parouha and Das 2016), THS (Mohammed et al. 2016) and MTVPSO (Parouha 2019). So as to observe the competence of the presented nPSO, nDE and ihPSODE between compared methods, smallest values of maximum number of iterations (1000), population size (30) and independent runs (30) have been considered. The relative simulation results of presented and equated methods are reported in Table 13 (for UTS-1), Table 14 (for UTS-2), Table 15 (for UTS-3), Table 16 (for UTS-4) and Table 17 (for UTS-5) over 30 runs.
As per these tables the best cost created by presented - (i) nPSO: for UTS-1, UTS-2, UTS-3, UTS-4 and UTS-5 are 8234.07173 ($/hr), 15441.1093 ($/hr), 32542.8820 ($/hr), 121404.5378 ($/hr) and 1560436.88 ($/hr) respectively, (ii) nDE: for UTS-1, UTS-2, UTS-3, UTS-4 and UTS-5 are 8234.07173 ($/hr), 15441.1095 ($/hr), 32542.7820 ($/hr), 121405.7384 ($/hr) and 1560435.84 ($/hr) respectively and (iii) ihPSODE for for UTS-1, UTS-2, UTS-3, UTS-4 and UTS-5 are 8234.07173 ($/hr), 15440.1084 ($/hr), 32542.7320 ($/hr), 121403.5454 ($/hr) and 1560434.75 ($/hr) separately.
These reported cost results for all UTSs shows that the presented methods succeed in finding the best solution then others. Likewise, the mean fuel cost, maximum fuel cost, standard deviation and mean time for each UTSs are also noted in the same tables. It is noteworthy that the presented methods can still produce best solutions with low standard deviations and adequate time. It indicates that presented methods has better convergence, greater stability, robustness compared to others.
Table 13
Simulation results for UTS-1
Power Output (MW)
|
Equaled Algorithms
|
Presented algorithms
|
GA
(Davis 1991)
|
PSO
(Kennedy and Eberhart 1995)
|
DE
(Storn and Price 1997)
|
MGSO
(Zare et al. 2012)
|
THS
(Mohammed et al. 2016)
|
MTVPSO
(Parouha 2019)
|
nPSO
|
nDE
|
ihPSODE
|
1
|
300.00
|
300.270
|
300.27
|
300.27
|
300
|
300.27
|
300.27
|
300.27
|
300.26
|
2
|
400.00
|
400.000
|
400.00
|
400.00
|
400
|
400.00
|
400.00
|
400.00
|
400.00
|
3
|
150.00
|
149.730
|
149.73
|
149.74
|
149.7331
|
149.73
|
149.73
|
149.73
|
149.74
|
Total power (MW)
|
850
|
850.000
|
850.00
|
850
|
850
|
850.00
|
850.00
|
850.00
|
850.00
|
Min cost ($/hr)
|
8234.60
|
8234.07173
|
8234.07173
|
8234.07
|
8234.07
|
8234.07173
|
8234.07173
|
8234.07173
|
8234.07173
|
Mean cost ($/hr)
|
8236.75
|
8235.979526
|
8234.823309
|
8237.85
|
8234.55
|
8234.07173
|
8234.07173
|
8234.07173
|
8234.07173
|
Max cost ($/hr)
|
8239.99
|
8241.587522
|
8241.587522
|
8240.54
|
8236.87
|
8234.07173
|
8234.07173
|
8234.07173
|
8234.07173
|
Std.
|
3.581
|
3.21860
|
2.293281
|
5.26
|
1.8020
|
1.85012e − 12
|
2.5845e-13
|
1.4856e-13
|
1.2548e-14
|
Mean time (s)
|
2.272
|
4.370
|
2.061
|
2.85
|
1.48
|
1.289
|
1.299
|
1.287
|
1.254
|
Table 14
Simulation results for UTS-2
Power Output (MW)
|
Equated Algorithms
|
Presented algorithms
|
GA
(Davis 1991)
|
PSO
(Kennedy and Eberhart 1995)
|
MTVPSO
(Parouha 2019)
|
IPSO
(He et al. 2004)
|
NCS
(Kuo 2008)
|
BCO
(Chokpanyasuwan et al. 2009)
|
EHM
(Kasmaei and Nejad 2011)
|
IPSO-TVAC
(Mohammadi et al. 2012)
|
𝜽-PSO
(Hosseinnezhad and Babaei 2013)
|
MPSO-TVAC
(Abdullah et al. 2014)
|
nPSO
|
nDE
|
ihPSODE
|
1
|
474.8066
|
446.986
|
451.5204
|
449.802
|
446.71
|
444.9513
|
449.1546
|
447.5840
|
447.3555
|
448.170
|
451.5204
|
451.5384
|
451.5204
|
2
|
178.6363
|
170.196
|
172.1750
|
171.042
|
173.01
|
173.8016
|
173.0613
|
173.2010
|
173.2577
|
173.291
|
172.1750
|
172.1250
|
172.1750
|
3
|
262.2089
|
252.902
|
258.4186
|
250.865
|
265.00
|
263.3943
|
266.0092
|
263.3310
|
263.3848
|
263.145
|
258.4186
|
259.3497
|
258.4186
|
4
|
134.2826
|
150.000
|
140.6441
|
150.000
|
139.00
|
138.6992
|
127.1203
|
138.8520
|
139.0440
|
138.714
|
140.6441
|
138.5341
|
140.6441
|
5
|
151.9039
|
178.780
|
162.0797
|
159.347
|
165.23
|
167.9755
|
174.2603
|
165.3280
|
165.3317
|
165.960
|
162.0797
|
158.0893
|
162.0797
|
6
|
74.18120
|
77.0850
|
90.34150
|
94.6330
|
86.780
|
87.16640
|
85.87770
|
87.15000
|
87.05930
|
86.6910
|
90.34150
|
90.24050
|
90.34150
|
Total power (MW)
|
1276.03
|
1275.95
|
1275.1795
|
1275.69
|
1275.7
|
1275.9882
|
1275.4834
|
1275.4460
|
1275.433
|
1275.97
|
1275.1395
|
1275.1668
|
1275.1345
|
Power loss (MW)
|
13.0217
|
12.95
|
12.1795
|
12.69
|
12.733
|
12.9864
|
12.4834
|
12.4460
|
12.4429
|
12.97
|
12.1395
|
12.1668
|
12.1345
|
Power balance (MW)
|
0.0083
|
0.000
|
0.000
|
0.000
|
0.0048
|
0.0018
|
0.0
|
0.000
|
−0.0099
|
0.000
|
0.000
|
0.000
|
0.000
|
Min cost ($/hr)
|
15459.00
|
15450.00
|
15441.1084
|
15453.50
|
15447.00
|
15450.031
|
15441.5974
|
15443.063
|
15442.9411
|
15449.91
|
15441.1093
|
15441.1095
|
15440.1084
|
Mean cost ($/hr)
|
15461.35
|
15454.00
|
15441.1087
|
15462.59
|
15448.58
|
15452.257
|
15442.8547
|
15443.582
|
15442.9419
|
15450.17
|
15441.1098
|
15441.1097
|
15440.2087
|
Max cost ($/hr)
|
15485.87
|
15492.00
|
15441.1104
|
15468.48
|
15449.85
|
15455.458
|
15446.5874
|
155445.114
|
15442.9500
|
15451.57
|
15441.1101
|
15441.1100
|
15441.1104
|
Std.
|
0.0078
|
0.0025
|
0.0031
|
0.84
|
0.00145
|
0.0069
|
0.00046
|
0.00255
|
0.0015
|
0.37
|
0.0029
|
0.0029
|
0.0021
|
Mean time (s)
|
41.58
|
14.89
|
0.32
|
1.25
|
7.58
|
3.10
|
0.32
|
0.89
|
5.4429
|
1.68
|
0.32
|
0.30
|
0.28
|
Table 15
Simulation results for UTS-3
Power Output (MW)
|
Equated Algorithms
|
Presented algorithms
|
GA
(Davis 1991)
|
PSO
(Kennedy and Eberhart 1995)
|
NCS
(Kuo 2008)
|
BCO
(Chokpanyasuwan et al. 2009)
|
EHM
(Kasmaei and Nejad 2011)
|
𝜽-PSO
(Hosseinnezhad and Babaei 2013)
|
DEPSO
(Sayah and Hamouda 2013)
|
MPSO-TVAC
(Abdullah et al. 2014)
|
DPD
(Parouha and Das 2016)
|
MTVPSO
(Parouha 2019)
|
nPSO
|
nDE
|
ihPSODE
|
1
|
415.3108
|
455.00
|
455.00
|
452.9151
|
455.0000
|
455.00
|
455.00
|
455.00
|
454.9999
|
454.9812
|
454.8823
|
454.8923
|
454.9993
|
2
|
359.7206
|
380.00
|
380.00
|
358.8547
|
380.0000
|
380.00
|
420.00
|
380.00
|
454.9999
|
455.0000
|
455.0000
|
455.0000
|
455.0000
|
3
|
104.4250
|
130.00
|
130.00
|
127.6452
|
130.0000
|
130.00
|
130.00
|
130.00
|
130.0000
|
130.0000
|
130.0000
|
130.0000
|
130.0000
|
4
|
74.98530
|
130.00
|
130.00
|
128.4156
|
130.0000
|
130.00
|
130.00
|
130.00
|
130.0000
|
130.0000
|
130.0000
|
130.0000
|
130.0000
|
5
|
380.2844
|
154.42
|
170.00
|
276.0158
|
170.0000
|
170.00
|
270.00
|
170.00
|
234.2005
|
235.5844
|
235.6334
|
235.5944
|
235.5334
|
6
|
426.7902
|
460.00
|
460.00
|
429.9371
|
460.0000
|
460.00
|
460.00
|
459.99
|
460.0000
|
460.0000
|
460.0000
|
460.0000
|
460.0000
|
7
|
341.3164
|
430.00
|
430.00
|
437.8152
|
430.0000
|
430.00
|
430.00
|
430.00
|
464.9999
|
465.0000
|
460.0000
|
459.0000
|
460.0000
|
8
|
124.7867
|
60.000
|
60.000
|
62.84580
|
90.14947
|
75.0139
|
60.000
|
72.600
|
60.00000
|
60.00000
|
60.00000
|
61.00000
|
60.00000
|
9
|
133.1445
|
74.270
|
71.050
|
59.53430
|
37.75777
|
55.8293
|
25.000
|
58.320
|
25.00000
|
25.00000
|
26.00000
|
24.00000
|
25.00000
|
10
|
89.25670
|
160.00
|
159.85
|
96.72150
|
160.0000
|
160.00
|
62.000
|
159.73
|
30.99387
|
28.98670
|
27.99970
|
28.67770
|
27.87570
|
11
|
60.05720
|
80.000
|
80.000
|
75.21180
|
80.00000
|
80.000
|
80.000
|
80.000
|
76.70138
|
76.83570
|
76.19870
|
75.99870
|
75.18870
|
12
|
49.99980
|
79.600
|
80.000
|
78.45220
|
80.00000
|
80.000
|
80.000
|
80.000
|
79.99999
|
80.00000
|
80.00000
|
80.00000
|
80.00000
|
13
|
38.77130
|
25.000
|
25.000
|
35.51180
|
25.00000
|
25.0012
|
25.000
|
25.010
|
25.00000
|
25.00000
|
25.00000
|
26.00000
|
25.00000
|
14
|
41.94250
|
27.590
|
15.000
|
19.15490
|
15.00000
|
15.0000
|
15.000
|
15.000
|
15.00000
|
15.00000
|
15.00000
|
14.00000
|
15.00000
|
15
|
22.64450
|
15.000
|
15.000
|
21.01430
|
15.00000
|
15.0181
|
15.000
|
15.000
|
15.00000
|
15.00000
|
15.00000
|
15.00000
|
15.00000
|
Total power (MW)
|
2630.000
|
2660.88
|
2660.9
|
2660.0483
|
2657.90724
|
2660.8625
|
2657.96
|
2660.66
|
2656.89544
|
2656.3882
|
2656.3592
|
2656.3712
|
2656.3398
|
Power loss (MW)
|
38.2782
|
30.88
|
30.908
|
29.4073
|
27.90724
|
30.8699
|
27.976
|
30.66
|
26.89544
|
26.3882
|
26.3592
|
26.3712
|
26.3398
|
Power balance (MW)
|
0.1218
|
0.00000
|
0.0000
|
0.641
|
0.00000
|
−0.0074
|
0.010
|
0.00000
|
0.00000
|
0.00000
|
0.00000
|
0.00000
|
0.00000
|
Min cost ($/hr)
|
33113
|
32731.96
|
32708
|
32714.265
|
32672.9595
|
32706.6048
|
32588.81
|
32704.47
|
32548.5857
|
32542.7320
|
32542.8820
|
32542.7820
|
32542.7320
|
Mean cost ($/hr)
|
33254
|
33039.00
|
32854
|
32725.251
|
32699.2548
|
32709.3196
|
32591.49
|
32705.00
|
32556.6793
|
32550.9885
|
32550.8885
|
32550.9885
|
32550.8885
|
Max cost ($/hr)
|
33285
|
33331.00
|
32975
|
32755.369
|
32708.8547
|
32739.4865
|
32588.99
|
32728.99
|
32564.4051
|
32562.4861
|
32559.8761
|
32562.4861
|
32558.9861
|
Std.
|
1.598
|
2.8475
|
6.2541
|
2.4584
|
3.5841
|
7.31400
|
4.02
|
3.510
|
2.095632
|
2.04619
|
1.94319
|
1.89464
|
1.0461
|
Mean time (s)
|
49.31
|
26.590
|
12.790
|
12.420
|
0.3680
|
11.7380
|
1.96
|
12.78
|
1.985940
|
0.1900
|
0.1890
|
0.185
|
0.183
|
Table 16
Simulation results for UTS-4
Power Output (MW)
|
Equated Algorithms
|
Presented algorithms
|
IABC
(Aydın et al. 2011)
|
DHS
(Wang and Li 2013)
|
DEPSO
(Sayah and Hamouda 2013)
|
MPSO
(Basu 2015)
|
DPD
(Parouha and Das 2016)
|
THS
(Mohammed et al. 2016)
|
MTVPSO
(Parouha 2019)
|
nPSO
|
nDE
|
ihPSODE
|
1
|
110.8067
|
110.7998
|
110.802
|
111.3021
|
111.7629
|
114
|
110.7998
|
110.7898
|
110.7898
|
110.7898
|
2
|
110.8163
|
110.7998
|
110.801
|
110.8937
|
111.6926
|
113.3808
|
110.7998
|
110.7918
|
110.7998
|
110.7998
|
3
|
97.4000
|
97.3999
|
97.400
|
97.4024
|
97.40940
|
97.4102
|
97.3999
|
97.3898
|
97.3898
|
97.3899
|
4
|
179.7330
|
179.7331
|
179.733
|
179.7417
|
179.7721
|
179.7357
|
179.7331
|
179.7331
|
179.7331
|
179.7331
|
5
|
87.8133
|
87.7999
|
87.800
|
96.2717
|
88.30690
|
96.9973
|
87.7999
|
87.7999
|
87.7999
|
87.7999
|
6
|
139.9999
|
140.0000
|
140.000
|
140.0000
|
139.9833
|
140.0000
|
140.0000
|
140.0000
|
140.0000
|
140.0000
|
7
|
259.5996
|
259.5997
|
259.600
|
259.5998
|
259.7218
|
259.6047
|
259.5997
|
259.5997
|
259.5997
|
259.5997
|
8
|
284.6008
|
284.5997
|
284.600
|
284.6047
|
284.7273
|
284.6041
|
284.5997
|
284.5987
|
284.5997
|
284.5897
|
9
|
284.5997
|
284.5997
|
284.600
|
284.6048
|
284.6157
|
284.6018
|
284.5997
|
284.5977
|
284.5777
|
284.5797
|
10
|
130.0000
|
130.0000
|
130.000
|
130.0020
|
130.0583
|
130
|
130.0000
|
130.0000
|
130.0000
|
130.0000
|
11
|
168.7998
|
94.0000
|
94.000
|
168.7993
|
168.7990
|
168.8034
|
94.0000
|
94.0000
|
94.0000
|
94.0000
|
12
|
94.0000
|
94.0000
|
94.000
|
94.0019
|
168.7894
|
214.7619
|
94.0000
|
94.0000
|
94.0000
|
94.0000
|
13
|
125.0000
|
214.7598
|
214.760
|
214.7600
|
214.7593
|
394.2794
|
214.7598
|
214.7898
|
214.7898
|
214.7898
|
14
|
400.0000
|
394.2794
|
394.279
|
394.2799
|
304.5391
|
304.5215
|
394.2794
|
394.2594
|
394.2594
|
394.2594
|
15
|
394.2791
|
394.2794
|
394.279
|
394.2789
|
394.2707
|
304.5209
|
394.2794
|
394.2397
|
394.2394
|
394.2398
|
16
|
394.2793
|
394.2794
|
394.279
|
304.5202
|
394.2713
|
489.2841
|
394.2794
|
394.2496
|
394.2494
|
394.2493
|
17
|
489.2796
|
489.2794
|
489.279
|
489.2798
|
489.2894
|
489.2891
|
489.2794
|
489.2397
|
489.2394
|
489.2394
|
18
|
489.2794
|
489.2794
|
489.279
|
489.2823
|
489.3177
|
511.2813
|
489.2794
|
489.2198
|
489.2194
|
489.2197
|
19
|
511.2792
|
511.2794
|
511.279
|
511.2796
|
511.2724
|
511.2790
|
511.2794
|
511.2394
|
511.2394
|
511.2399
|
20
|
511.2793
|
511.2794
|
511.279
|
511.2823
|
511.2800
|
523.2838
|
511.2794
|
511.2891
|
511.2994
|
511.2997
|
21
|
523.2798
|
523.2794
|
523.279
|
523.2799
|
523.3291
|
523.2819
|
523.2794
|
523.2716
|
523.2894
|
523.2896
|
22
|
523.2793
|
523.2794
|
523.279
|
523.2794
|
523.2992
|
523.2779
|
523.2794
|
523.2815
|
523.2894
|
523.2893
|
23
|
523.2793
|
523.2794
|
523.279
|
523.2815
|
523.3545
|
523.2801
|
523.2794
|
523.2608
|
523.2694
|
523.2694
|
24
|
523.2793
|
523.2794
|
523.279
|
523.2800
|
523.2793
|
523.2824
|
523.2794
|
523.2716
|
523.2894
|
523.2897
|
25
|
523.2793
|
523.2794
|
523.279
|
523.2799
|
523.3890
|
523.2799
|
523.2794
|
523.2688
|
523.2694
|
523.2698
|
26
|
523.2793
|
523.2794
|
523.279
|
523.2812
|
523.2776
|
523.2799
|
523.2794
|
523.2899
|
523.2794
|
523.2797
|
27
|
10.0000
|
10.0000
|
10.000
|
10.0010
|
10.00000
|
10.00000
|
10.0000
|
10.0000
|
10.0000
|
10.0000
|
28
|
10.0000
|
10.0000
|
10.000
|
10.0002
|
10.00000
|
10.00000
|
10.0000
|
10.0000
|
10.0000
|
10.0000
|
29
|
10.0000
|
10.0000
|
10.000
|
10.0021
|
10.00000
|
10.00000
|
10.0000
|
10.0000
|
10.0000
|
10.0000
|
30
|
91.4006
|
87.7999
|
87.800
|
88.0447
|
88.72800
|
96.99000
|
87.7999
|
88.0000
|
88.0000
|
88.0000
|
31
|
189.9999
|
190.0000
|
190.000
|
189.9997
|
189.9811
|
190.0000
|
190.0000
|
189.8998
|
189.8999
|
189.8999
|
32
|
189.9999
|
190.0000
|
190.000
|
189.9996
|
189.9982
|
190.0000
|
190.0000
|
190.0000
|
190.0000
|
190.0000
|
33
|
190.0000
|
190.0000
|
190.000
|
189.9999
|
189.9795
|
190.0000
|
190.0000
|
190.0000
|
190.0000
|
190.0000
|
34
|
164.7997
|
164.7998
|
164.800
|
164.8005
|
164.8962
|
164.8838
|
164.7998
|
165.7298
|
165.7198
|
165.7198
|
35
|
199.9978
|
200.0000
|
194.395
|
199.9950
|
165.0916
|
200.000
|
200.0000
|
199.9888
|
199.8888
|
199.8888
|
36
|
199.9999
|
194.3978
|
200.000
|
199.9953
|
165.2998
|
200.000
|
194.3978
|
195.3817
|
195.3977
|
195.3977
|
37
|
110.0000
|
110.0000
|
110.000
|
109.9987
|
109.9827
|
110.000
|
110.0000
|
109.9815
|
109.9975
|
109.9975
|
38
|
109.9998
|
110.0000
|
110.000
|
109.9997
|
109.9824
|
110.000
|
110.0000
|
110.9150
|
110.9850
|
110.9850
|
39
|
110.0000
|
110.0000
|
110.000
|
109.9990
|
109.9634
|
110.000
|
110.0000
|
110.0000
|
110.0000
|
110.0000
|
40
|
511.2793
|
511.2794
|
511.279
|
511.2957
|
511.5279
|
511.2795
|
511.2794
|
511.3819
|
511.3799
|
511.3799
|
Total power (MW)
|
10500.00
|
10500.00
|
10500.00
|
10500.00
|
10500.00
|
10500.00
|
10500.00
|
10500.00
|
10500.00
|
10500.00
|
Min cost ($/hr)
|
121491.2751
|
121403.5355
|
121412.56
|
121379.43
|
121410.5355
|
121,467.44
|
121403.5355
|
121404.5378
|
121405.7384
|
121403.5454
|
Mean cost ($/hr)
|
121539.4175
|
121410.5967
|
121419.31
|
121384.43
|
121412.5729
|
121,524.26
|
121410.5967
|
121410.5977
|
121409.5829
|
121410.4658
|
Max cost ($/hr)
|
121582.3865
|
121417.2274
|
121468.25
|
121391.07
|
121441.9027
|
121,598.65
|
121417.2274
|
121416.3319
|
121415.2314
|
121413.9827
|
Std.
|
0.00548
|
4.80
|
0.00584
|
2.3685
|
0.437608
|
36.7026
|
0.001714
|
0.001714
|
0.0037608
|
0.0001546
|
Mean time (s)
|
1.950
|
1.32
|
7.895
|
5.43
|
21.76485
|
13.5846
|
11.32101
|
12.78485
|
11.12101
|
11.02356
|
Table 17
Simulation results for UTS-5
Power Output(MW)
|
Equaled Algorithms
|
Presented algorithms
|
|
Equaled Algorithms
|
Presented algorithms
|
DHS
(Wang and Li 2013)
|
MPSO
(Basu 2015)
|
MTVPSO
(Parouha 2019)
|
nPSO
|
nDE
|
ihPSODE
|
|
DHS
(Wang and Li 2013)
|
MPSO
(Basu 2015)
|
MTVPSO
(Parouha 2019)
|
nPSO
|
nDE
|
ihPSODE
|
1
|
119.0000
|
116.2514
|
116.2518
|
116.2418
|
116.2528
|
116.2528
|
71
|
500.0000
|
143.3803
|
143.3803
|
143.8778
|
143.3804
|
143.4785
|
2
|
119.0000
|
188.8395
|
188.8395
|
188.8295
|
188.8375
|
188.8345
|
72
|
500.0000
|
388.3236
|
388.3236
|
388.1248
|
388.7854
|
388.3785
|
3
|
164.0000
|
189.9734
|
189.9734
|
189.9834
|
189.9734
|
189.9724
|
73
|
241.0000
|
202.9482
|
202.9482
|
202.7854
|
202.1245
|
202.7785
|
4
|
164.0000
|
189.8984
|
189.8984
|
189.8984
|
189.8974
|
189.8974
|
74
|
241.0000
|
176.0783
|
176.0783
|
176.0745
|
176.7854
|
176.0785
|
5
|
190.0000
|
168.6455
|
168.6455
|
168.6655
|
168.6455
|
168.6445
|
75
|
241.0000
|
175.8523
|
175.8523
|
175.8545
|
175.4561
|
175.8785
|
6
|
190.0000
|
187.3990
|
187.3990
|
187.3890
|
187.3910
|
187.3940
|
76
|
241.0000
|
177.8040
|
177.8040
|
177.8045
|
177.8125
|
177.8874
|
7
|
190.0000
|
489.9976
|
489.9976
|
489.9976
|
489.9977
|
489.9946
|
77
|
774.0000
|
179.5410
|
179.5410
|
179.5411
|
179.4856
|
179.5254
|
8
|
190.0000
|
489.8645
|
489.8645
|
489.8745
|
489.8645
|
489.8645
|
78
|
774.0000
|
330.6797
|
330.6797
|
330.6745
|
330.6785
|
330.6258
|
9
|
168.5398
|
495.9917
|
495.9917
|
495.9817
|
495.9918
|
495.9997
|
79
|
769.0000
|
530.9939
|
530.9939
|
530.9488
|
530.9987
|
530.9358
|
10
|
168.5398
|
495.9000
|
495.9000
|
495.9100
|
495.7854
|
495.9040
|
80
|
769.0000
|
530.9852
|
530.9852
|
530.9878
|
530.9258
|
530.9154
|
11
|
190.0000
|
495.9999
|
495.9999
|
495.9299
|
495.7896
|
495.9949
|
81
|
3.0000
|
260.0529
|
260.0529
|
260.0787
|
260.7854
|
260.0535
|
12
|
190.0000
|
495.9418
|
495.9418
|
495.9318
|
495.9408
|
495.9408
|
82
|
82 3.0000
|
56.0472
|
56.04720
|
56.04748
|
56.07854
|
56.04715
|
13
|
490.0000
|
505.9503
|
505.9503
|
505.9803
|
505.9513
|
505.9513
|
83
|
83 3.0000
|
115.2868
|
115.2868
|
115.2487
|
115.2987
|
115.2154
|
14
|
490.0000
|
508.9842
|
508.9842
|
508.9242
|
508.9822
|
508.9832
|
84
|
84 3.0000
|
115.0050
|
115.0050
|
115.1452
|
115.1545
|
115.0785
|
15
|
490.0000
|
505.9854
|
505.9854
|
505.9454
|
505.9814
|
505.9844
|
85
|
250.0000
|
115.3836
|
115.3836
|
115.3987
|
115.3886
|
115.3834
|
16
|
490.0000
|
504.9535
|
504.9535
|
504.9435
|
504.9635
|
504.9525
|
86
|
250.0000
|
207.0000
|
207.0000
|
207.0078
|
207.0000
|
207.0788
|
17
|
496.0000
|
505.9841
|
505.9841
|
505.9941
|
505.9741
|
505.9881
|
87
|
250.0000
|
207.0965
|
207.0965
|
207.0965
|
207.0785
|
207.0985
|
18
|
496.0000
|
505.9365
|
505.9365
|
505.9265
|
505.9465
|
505.9365
|
88
|
250.0000
|
175.1027
|
175.1027
|
175.1125
|
175.1485
|
175.1078
|
19
|
496.0000
|
504.9788
|
504.9788
|
504.9688
|
504.9888
|
504.9748
|
89
|
250.0000
|
175.1656
|
175.1656
|
175.1145
|
175.1678
|
175.1678
|
20
|
496.0000
|
504.9618
|
504.9618
|
504.9718
|
504.9418
|
504.9628
|
90
|
250.0000
|
182.1361
|
182.1361
|
182.1361
|
182.1378
|
182.1354
|
21
|
496.0000
|
504.9769
|
504.9769
|
504.9869
|
504.9369
|
504.9779
|
91
|
250.0000
|
175.1669
|
175.1669
|
175.1785
|
175.1754
|
175.1678
|
22
|
496.0000
|
504.9081
|
504.9081
|
504.9281
|
504.9181
|
504.9091
|
92
|
250.0000
|
579.9855
|
579.9855
|
579.9452
|
579.9785
|
579.9854
|
23
|
496.0000
|
504.9514
|
504.9514
|
504.9614
|
504.9814
|
504.9524
|
93
|
250.0000
|
645.0000
|
645.0000
|
645.0089
|
645.0754
|
645.0999
|
24
|
496.0000
|
504.9742
|
504.9742
|
504.9442
|
504.9642
|
504.9742
|
94
|
250.0000
|
983.9400
|
983.9400
|
983.9200
|
983.9487
|
983.9485
|
25
|
506.0000
|
536.9944
|
536.9944
|
536.9844
|
536.9844
|
536.9934
|
95
|
250.000
|
977.9925
|
977.9925
|
977.9785
|
977.9925
|
977.9978
|
26
|
506.0000
|
536.9804
|
536.9804
|
536.9704
|
536.9704
|
536.9814
|
96
|
250.0000
|
681.9838
|
681.9838
|
681.9025
|
681.9895
|
681.9825
|
27
|
509.0000
|
548.9866
|
548.9866
|
548.9666
|
548.9466
|
548.9867
|
97
|
250.0000
|
719.9953
|
719.9953
|
719.9954
|
719.9975
|
719.9985
|
28
|
509.0000
|
548.9698
|
548.9698
|
548.9498
|
548.9398
|
548.9699
|
98
|
250.0000
|
718.0000
|
718.0000
|
718.0854
|
718.0045
|
718.0897
|
29
|
506.0000
|
500.9609
|
500.9609
|
500.9309
|
500.9509
|
500.9604
|
99
|
250.0000
|
719.9553
|
719.9553
|
719.9785
|
719.9553
|
719.9558
|
30
|
506.0000
|
500.8584
|
500.8584
|
500.8984
|
500.8484
|
500.8585
|
100
|
250.0000
|
963.8138
|
963.8138
|
963.8250
|
963.8125
|
963.8145
|
31
|
505.0000
|
505.9982
|
505.9982
|
505.9482
|
505.8982
|
505.9987
|
101
|
165.0000
|
957.9252
|
957.9252
|
957.9152
|
957.9245
|
957.9285
|
32
|
505.0000
|
505.9890
|
505.9890
|
505.9990
|
505.8890
|
505.9899
|
102
|
165.0000
|
1006.9391
|
1006.9391
|
1006.9321
|
1006.9311
|
1006.9381
|
33
|
506.0000
|
505.9872
|
505.9872
|
505.9472
|
505.8872
|
505.9472
|
103
|
165.0000
|
1005.9547
|
1005.9547
|
1005.9547
|
1005.9547
|
1005.9527
|
34
|
506.0000
|
505.8398
|
505.8398
|
505.8398
|
505.9398
|
505.8394
|
104
|
165.0000
|
1012.9951
|
1012.9951
|
1012.9971
|
1012.9781
|
1012.9981
|
35
|
506.0000
|
499.9759
|
499.9759
|
499.9759
|
499.8759
|
499.9554
|
105
|
165.0000
|
1019.9532
|
1019.9532
|
1019.9512
|
1019.9572
|
1019.9512
|
36
|
506.0000
|
499.8309
|
499.8309
|
499.8801
|
499.8325
|
499.7854
|
106
|
165.0000
|
953.9784
|
953.9784
|
953.9784
|
953.9778
|
953.9785
|
37
|
505.0000
|
240.9712
|
240.9712
|
240.9718
|
240.9745
|
240.8454
|
107
|
165.0000
|
951.9893
|
951.9893
|
951.9893
|
951.9898
|
951.9847
|
38
|
505.0000
|
240.9668
|
240.9668
|
240.9668
|
240.9154
|
240.9596
|
108
|
165.0000
|
1005.9502
|
1005.9502
|
1005.9502
|
1005.9522
|
1005.957
|
39
|
505.0000
|
773.9998
|
773.9998
|
773.9198
|
773.9896
|
773.9154
|
109
|
180.0000
|
1012.9827
|
1012.9827
|
1012.9827
|
1012.9828
|
1012.9847
|
40
|
505.0000
|
768.9143
|
768.9143
|
767.9123
|
768.9145
|
768.9154
|
110
|
180.0000
|
1020.9864
|
1020.9864
|
1020.9864
|
1020.9887
|
1020.9874
|
41
|
505.0000
|
3.4842
|
3.484200
|
3.984201
|
3.484220
|
3.484295
|
111
|
180.0000
|
1014.9943
|
1014.9943
|
1014.9943
|
1014.9987
|
1014.9973
|
42
|
505.0000
|
3.0274
|
3.027400
|
3.087478
|
3.027401
|
3.027444
|
112
|
180.0000
|
94.0355
|
94.03550
|
94.03550
|
94.035485
|
94.78569
|
43
|
505.0000
|
186.7539
|
186.7539
|
186.5847
|
186.7525
|
186. 9636
|
113
|
103.0000
|
94.0373
|
94.03730
|
94.03730
|
94.037378
|
94.03737
|
44
|
505.0000
|
216.9286
|
216.9286
|
216.4896
|
216.9287
|
216.9255
|
114
|
103.0000
|
94.0173
|
94.01730
|
94.01730
|
94.017325
|
94.25732
|
45
|
505.0000
|
249.3928
|
249.3928
|
249.2541
|
249.3928
|
249.3924
|
115
|
198.0000
|
244.3547
|
244.3547
|
244.3547
|
244.35474
|
244.3547
|
46
|
505.0000
|
249.9292
|
249.9292
|
249.4856
|
249.9293
|
249.9299
|
116
|
198.0000
|
244.0590
|
244.0590
|
244.0590
|
244.05902
|
244.1590
|
47
|
505.0000
|
246.5461
|
246.5461
|
246.1596
|
246.5458
|
246.5445
|
117
|
312.0000
|
244.1444
|
244.1444
|
244.1444
|
244.1785
|
244.1844
|
48
|
505.0000
|
248.6987
|
248.6987
|
248.6562
|
248.6785
|
248.7854
|
118
|
312.0000
|
95.3255
|
95.32550
|
95.32550
|
95.32458
|
95.32150
|
49
|
537.0000
|
248.7121
|
248.7121
|
248.7486
|
248.7485
|
248.7854
|
119
|
308.5679
|
95.0186
|
95.01860
|
95.01860
|
95.01848
|
95.01660
|
50
|
537.0000
|
248.8088
|
248.8088
|
248.7854
|
248.8052
|
248.2541
|
120
|
308.5881
|
116.1636
|
116.1636
|
116.1636
|
116.1645
|
116.1536
|
51
|
537.0000
|
166.6126
|
166.6126
|
166.8526
|
166.6115
|
166.1548
|
121
|
163.0000
|
175.0352
|
175.0352
|
175.0352
|
175.0351
|
175.0152
|
52
|
537.0000
|
165.0625
|
165.0625
|
165.7425
|
165.0156
|
165.8651
|
122
|
163.0000
|
2.0064
|
2.00640
|
2.00640
|
2.00645
|
2.00646
|
53
|
549.0000
|
169.5018
|
169.5018
|
169.8918
|
169.5785
|
169.8745
|
123
|
95.0000
|
4.0402
|
4.04020
|
4.04020
|
4.04022
|
4.04021
|
54
|
549.0000
|
166.3101
|
166.3101
|
166.3101
|
166.3125
|
166.9658
|
124
|
95.0000
|
15.0993
|
15.0993
|
15.0993
|
15.0993
|
15.8992
|
55
|
549.0000
|
180.0956
|
180.0956
|
180.7856
|
180.0154
|
180.0956
|
125
|
511.0000
|
9.0055
|
9.00550
|
9.00550
|
9.00555
|
9.07550
|
56
|
549.0000
|
180.2748
|
180.2748
|
180.7848
|
180.2715
|
180.2254
|
126
|
511.0000
|
12.0655
|
12.0655
|
12.0655
|
12.0654
|
12.0555
|
57
|
501.0000
|
103.1771
|
103.1771
|
103.1521
|
103.1715
|
103.1741
|
127
|
511.0000
|
10.0080
|
10.0080
|
10.0080
|
10.0084
|
10.0180
|
58
|
501.0000
|
198.1757
|
198.1757
|
198.7847
|
198.1745
|
198.1351
|
128
|
511.0000
|
112.0382
|
112.0382
|
112.0382
|
112.038
|
112.1382
|
59
|
499.0000
|
309.9475
|
309.9475
|
309.9985
|
309.9479
|
309.9874
|
129
|
490.0000
|
4.0812
|
4.08120
|
4.08120
|
4.08123
|
4.08520
|
60
|
499.0000
|
293.8267
|
293.8267
|
293.8157
|
293.8214
|
293.8895
|
130
|
490.0000
|
5.0275
|
5.02750
|
5.02750
|
5.02750
|
5.02550
|
61
|
506.0000
|
163.5888
|
163.5888
|
163.5898
|
163.5145
|
163.5147
|
131
|
256.7772
|
5.0517
|
5.05170
|
5.05170
|
5.05145
|
5.05570
|
62
|
506.0000
|
95.3003
|
95.30030
|
95.30140
|
95.30152
|
95.30454
|
132
|
256.7802
|
50.1071
|
50.1071
|
50.1071
|
50.1458
|
50.1171
|
63
|
506.0000
|
168.1584
|
168.1584
|
168.1244
|
168.1598
|
168.1785
|
133
|
490.0000
|
5.0468
|
5.04680
|
5.04680
|
5.78542
|
5.04680
|
64
|
506.0000
|
166.0715
|
166.0715
|
166.0725
|
166.0025
|
166.0701
|
134
|
490.0000
|
42.0244
|
42.0244
|
42.0244
|
42.0785
|
42.0244
|
65
|
506.0000
|
486.8818
|
486.8818
|
486.8808
|
486.8085
|
486.8458
|
135
|
490.0000
|
42.0341
|
42.0341
|
42.0341
|
42.2548
|
42.0331
|
66
|
506.0000
|
205.5539
|
205.5539
|
205.5529
|
205.5025
|
205.7889
|
136
|
490.0000
|
41.0343
|
41.0343
|
41.0343
|
41.7854
|
41.0343
|
67
|
506.0000
|
483.1274
|
483.1274
|
483.1324
|
483.1215
|
483.1984
|
137
|
130.0000
|
17.0443
|
17.0443
|
17.0443
|
17.7869
|
17.0453
|
68
|
506.0000
|
486.0292
|
486.0292
|
486.1522
|
486.0785
|
486.1442
|
138
|
130.0000
|
7.1497
|
7.14970
|
7.14970
|
7.14948
|
7.14980
|
69
|
500.0000
|
130.2962
|
130.2962
|
130.2252
|
130.2902
|
130.2542
|
139
|
339.4395
|
7.0292
|
7.02920
|
7.02920
|
7.02921
|
7.02910
|
70
|
500.0000
|
339.3610
|
339.3610
|
339.3140
|
339.3678
|
339.3055
|
140
|
339.4395
|
26.4594
|
26.4594
|
26.4594
|
26.4684
|
26.4894
|
Total power(MW)
|
49342.0
|
49342
|
49342.000
|
49342.000
|
49342.000
|
49342.000
|
Min cost ($/hr)
|
1657944.86
|
1560436.0
|
1560436.71
|
1560436.88
|
1560435.84
|
1560434.75
|
Mean cost ($/hr)
|
1657944.86
|
1560445.0
|
1560446.22
|
1560446.78
|
1560443.84
|
1560440.89
|
Max cost ($/hr)
|
1657944.86
|
1560462.0
|
1560460.55
|
1560461.78
|
1560461.99
|
1560460.99
|
Std.
|
0.0005
|
0.000458
|
0.00003
|
0.000005
|
0.000004
|
0.0000003
|
Mean time (s)
|
4.60
|
18.431
|
17.036
|
6.788
|
5.541
|
3.252
|
The diagrams of convergence are plotted in Fig. 6(a-e) for UTS-1, UTS-2, UTS-3, UTS-4 and UTS-5 of presented and compared methods in terms of fuel cost and iterations. From this figure, it can be observed that presented methods (ihPSODE, nPSO and nDE) converge quicker to a better value. Also, these figures demonstrate that the supremacy of the presented algorithm in attaining minimum fuel cost compared to others with different demands. Therefore, presented algorithms are economically competent. At large, it can be stating that (from the all above result investigation) presented algorithms (nPSO, nDE and ihPSODE) are performing better and/or similar with others. Still, between three presented algorithms hybrid algorithm i.e. ihPSODE have greater capability.