Solving the Real Power Limitations in the Dynamic Economic Dispatch of Large-Scale Thermal Power Units under the Effects of Valve-Point Loading and Ramp-Rate Limitations
Abstract
:1. Introduction
- (1)
- Few NTOTs are used for the DED of large-scale TPUs, e.g., 1000 TPUs, that consider the effects of VPL with RRLs. The current study covers the lack in this field.
- (2)
- A novel algorithm, namely, a multi-gradient particle swarm optimization (MG-PSO) algorithm, is applied an optimization tool to solve the DED of 10 TPUs without/with transmission network loss; 100, 500 and, 1000 TPUs under VPL effects; and RRLs’ real power limitations.
- (3)
- The power generating scheduling of online TPUs is updated every 1 h for the duration of a day in a real-time operation to meet the load demands and real power limitations.
- (4)
- The generation is estimated to meet the load demand at any time during the period of a day. This procedure can maintain PGS stability and forecast the periodic maintenance schedule of the TPUs subjected to a massive change in their generation during the period of a day.
- (5)
- The research study provides details about the generation volume of each TPU over a 24 h period and which one is subjected to considerable changes during this period.
- (6)
- Most NTOTs in the literature often suffer from premature convergence and dimensionality when applied to a large-scale DED of real power (more than 500 TPUs) with many local minima due to the effects of VPL and RRLs. However, the MG-PSO algorithm was positively applied to alleviate particles’ convergence stagnancy even with 1000 TPUs throughout the solution’s inspections.
- (7)
- The approach used in the current research study is different from that adopted in our previous study [62,63]. In [62,63], the MG-PSO algorithm was used to solve the static ED for 1 h only of small and medium power systems, i.e., 6, 13, and 15 TPUs, under several power constraints, such as prohibited operating zones, RRLs, and VPL effects. However, in this study, the MG-PSO algorithm is used to solve the DED of large-scale TPUs, i.e., 100, 500, and 1000 TPUs, under the effects of VPL with RRLs during the 24 h period. It is a more complicated multi-objective function.
- (8)
- The GPSO-w algorithm is improved by using different negative gradient variations to prevent the best particle inside a swarm from falling in the local minimum and ensuring its escape.
- (9)
- With widespread simulated tests for the DED of medium-scale and large-scale TPU PGSs, the MG-PSO algorithm’s superiority over various competitive NTOTs, including the GPSO-w algorithm, in terms of performance measures has been demonstrated based on various numerous performance measures, such as fitness values, convergence rate, and consistency.
- (10)
- An illustrative example demonstrating the effects of VPL and RRLs on the OFC of two TPUs for 1 h is shown in Section 4.3.
- (11)
- This investigation provides useful technical references for economic dispatch operators to update their PGS programs to achieve economic benefits.
2. Mathematical Formation of the DED of Real Power
3. GPSO-w Algorithm
- (1)
- Every particle moving in the search space adapts its moving path based on two guides, namely, , its own experience, and , the best particle experience among the M particles.
- (2)
- M particles look for a solution, that is, global optimum; every particle gains the information from its and . Thus, a particle employs the best experience among the M particles while selecting the as its neighbor’s best experience.
- (3)
- GPSO-w is denoted as global PSO algorithm; the position of each particle is influenced by the particle in the whole population inside a swarm.
Algorithm 1. Pseudocode of the global particle swarm optimization with the inertia weight (w) (GPSO-w) algorithm learning strategy |
|
End |
4. MG-PSO Algorithm
4.1. MG-PSO Algorithm Learning Strategy
4.2. MG-PSO Algorithm Structure
Algorithm 2. Pseudocode of the natural learning mechanism of the multi-gradient particle swarm optimization (MG-PSO) algorithm |
Let f(x) be the objective function to be minimized Select Determine and using Equations (19) and (20), respectively Initialization: Iteration, t = 0 Obtain using Equations (10)–(13) Begin exploration stage for (begin of episode k) Determine using Equation (22) for Determine using Equation (25) for Update the particle’s velocity and position vectors as follows: Update as follows: end i loop Obtain as follows: end t loop Obtain and end k loop (end of episode k) for Obtain end k loop Obtain new search space (neighborhood) by taking and of each element of End exploration stage Begin exploitation stage Use the new search space Initialization: Iteration, t = 1 for Update for Determine using Equation (24) for Update the particle’s velocity and position vectors as follows: Update as follows: end i loop Obtain as follows: end t loop Optimum solution = Optimum value = End exploitation stage End MG-PSO algorithm |
4.3. Illustrative Example
5. Case Studies and Simulation Results
- (1)
- Case study #1: 10-TPU PGS without/with consideration of . This case study has 240 (24 × 10) decision variables.
- (2)
- Case study #2: 100-TPU PGS by repeating the PGS of case study #1 10 times. This case study has 2400 (24 × 100) decision variables.
- (3)
- Case study #3: 500-TPU PGS by repeating the PGS of case study #1 50 times. This case study has 12,000 (24 × 500) decision variables.
- (4)
- Case study #4: 1000-TPU PGS by repeating the PGS of case study #1 500 times. This case study has 24,000 (24 × 1000) decision variables.
5.1. Performance and Fitness Measures
- (1)
- Every NTOT is employed with independent runs;
- (2)
- Every NTOT is employed with iterations;
- (3)
- The values in USD were determined by Equation (1) under its real power limitations Equations (4) to (9) over independent runs at every t iteration;
- (4)
- is the values in USD over independent runs;
- (5)
- is the values in USD over independent runs;
- (6)
- is the values in USD over independent runs;
- (7)
- is the values in USD over independent runs;
- (8)
- The is determined by the overall time implemented by the NTOT after a convergence over independent runs.
5.2. Case Study #1: 10-TPU PGS without/with Consideration of Transmission Network Loss
5.3. Case Studies #2 to #4: 100-, 500-, and 1000-TPU PGS
5.4. MG-PSO Algorithm with Several Fitness Measures
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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TPUi | |||||||||
---|---|---|---|---|---|---|---|---|---|
TPU1 | 0.00043 | 21.60 | 958.29 | 450 | 0.041 | 150 | 470 | 80 | 80 |
TPU2 | 0.00063 | 21.05 | 1313.60 | 600 | 0.036 | 135 | 460 | 80 | 80 |
t | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
1036 | 1110 | 1258 | 1406 | 1480 | 1628 | 1702 | 1776 | 1924 | 2072 | 2146 | 2220 | |
t | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
2072 | 1924 | 1776 | 1554 | 1480 | 1628 | 1776 | 2072 | 1924 | 1628 | 1332 | 1184 |
TPUi | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|
0.00043 | 0.00063 | 0.00039 | 0.00070 | 0.00079 | 0.00056 | 0.00211 | 0.00480 | 0.10908 | 0.00951 | |
21.60 | 21.05 | 20.81 | 23.90 | 21.62 | 17.87 | 16.51 | 23.23 | 19.58 | 22.54 | |
958.20 | 1313.6 | 604.97 | 471.60 | 480.29 | 601.75 | 502.70 | 639.40 | 455.60 | 692.40 | |
450 | 600 | 320 | 260 | 280 | 310 | 300 | 340 | 270 | 380 | |
0.041 | 0.036 | 0.028 | 0.052 | 0.063 | 0.048 | 0.086 | 0.082 | 0.098 | 0.094 | |
470 | 460 | 340 | 300 | 243 | 160 | 130 | 120 | 80 | 55 | |
150 | 135 | 73 | 60 | 73 | 57 | 20 | 47 | 20 | 55 | |
80 | 80 | 80 | 50 | 50 | 50 | 30 | 30 | 30 | 30 | |
80 | 80 | 80 | 50 | 50 | 50 | 30 | 30 | 30 | 30 |
Parameters | Exploration Stage | Exploitation Stage | |
---|---|---|---|
0.3 | 0.3 | 0.3 | |
c1, c2 | 2.05 | 2.05 | 2.05 |
150 | 150 | 350 | |
0.80 | 0.80 | 0.35 | |
0.10 | 0.20 | 0.20 | |
−4.66 × 10−3 | −3.99 × 10−3 | −4.28 × 10−4 |
t | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 150.02 | 135.00 | 194.93 | 60.00 | 122.00 | 122.96 | 129.00 | 47.00 | 20.04 | 55.00 | 28,184.9284 | 9.23 |
2 | 150.02 | 135.05 | 268.96 | 60.05 | 122.02 | 122.85 | 129.00 | 47.00 | 20.00 | 55.00 | 29,738.3371 | 9.12 |
3 | 226.62 | 215.00 | 309.57 | 60.00 | 73.01 | 122.53 | 129.00 | 47.24 | 20.00 | 55.00 | 32,884.0400 | 9.21 |
4 | 303.02 | 222.03 | 324.00 | 60.75 | 122.37 | 122.81 | 129.00 | 47.00 | 20.00 | 55.00 | 36,118.9614 | 9.23 |
5 | 379.42 | 302.00 | 291.00 | 60.04 | 73.17 | 122.59 | 129.65 | 47.00 | 20.10 | 55.00 | 37,738.0667 | 9.47 |
6 | 456.00 | 309.17 | 306.48 | 60.44 | 122.12 | 122.76 | 129.00 | 47.00 | 20.00 | 55.00 | 40,942.9400 | 9.47 |
7 | 456.00 | 309.00 | 308.99 | 80.98 | 172.00 | 123.99 | 129.00 | 47.00 | 20.00 | 55.00 | 42,607.1703 | 9.72 |
8 | 456.00 | 309.06 | 309.00 | 126.99 | 172.99 | 152.00 | 129.00 | 47.00 | 20.00 | 55.00 | 44,217.7972 | 9.82 |
9 | 456.00 | 389.00 | 306.00 | 176.96 | 222.04 | 122.99 | 129.00 | 47.00 | 20.00 | 55.00 | 47,653.4300 | 9.89 |
10 | 456.00 | 396.00 | 299.00 | 226.93 | 222.05 | 160.99 | 129.00 | 77.00 | 50.00 | 55.00 | 51,082.4354 | 9.16 |
11 | 256.01 | 396.01 | 341.00 | 248.76 | 222.37 | 160.24 | 129.58 | 85.00 | 52.00 | 55.00 | 52,760.8264 | 9.42 |
12 | 456.03 | 460.00 | 300.99 | 298.43 | 222.53 | 160.99 | 129.00 | 85.00 | 52.00 | 55.00 | 54,507.9729 | 9.10 |
13 | 456.00 | 396.00 | 298.00 | 248.49 | 222.50 | 160.00 | 129.00 | 85.00 | 22.00 | 55.00 | 51,001.1804 | 9.36 |
14 | 456.76 | 396.00 | 287.99 | 198.88 | 172.06 | 122.99 | 129.28 | 85.00 | 20.00 | 55.00 | 47,808.6845 | 9.15 |
15 | 379.99 | 396.00 | 284.00 | 180.99 | 122.99 | 123.00 | 129.00 | 85.00 | 20.00 | 55.00 | 44,544.4962 | 9.40 |
16 | 303.00 | 316.00 | 318.00 | 131.00 | 74.00 | 123.00 | 129.00 | 85.00 | 20.00 | 55.00 | 39,562.9241 | 9.20 |
17 | 226.00 | 310.00 | 288.99 | 120.99 | 122.00 | 122.99 | 129.00 | 85.00 | 20.00 | 55.00 | 37,930.0339 | 9.45 |
18 | 303.15 | 309.00 | 310.00 | 121.00 | 172.84 | 123.00 | 129.00 | 85.00 | 20.00 | 55.00 | 41,163.4372 | 9.80 |
19 | 379.01 | 389.00 | 301.99 | 120.98 | 173.00 | 123.00 | 129.00 | 85.00 | 20.00 | 55.00 | 44,378.0508 | 9.61 |
20 | 457.00 | 460.00 | 313.00 | 171.00 | 223.00 | 123.00 | 130.00 | 86.00 | 21.00 | 56.00 | 50,781.9900 | 9.45 |
21 | 456.20 | 396.00 | 315.99 | 121.00 | 222.74 | 123.00 | 129.04 | 85.00 | 20.00 | 55.00 | 47,611.3299 | 9.54 |
22 | 379.81 | 316.09 | 275.99 | 70.99 | 172.99 | 122.99 | 129.09 | 85.00 | 20.00 | 55.00 | 41,101.1436 | 9.35 |
23 | 303.00 | 236.00 | 197.00 | 61.00 | 123.00 | 123.00 | 129.00 | 85.00 | 20.00 | 55.00 | 34,693.3884 | 9.31 |
24 | 226.90 | 222.44 | 189.93 | 60.14 | 73.00 | 122.13 | 129.00 | 85.00 | 20.00 | 55.10 | 31,476.8566 | 9.34 |
Total | 1,010,490.42 USD | 225.80 s |
No. | NTOTs | σ | ||||
---|---|---|---|---|---|---|
1 | EP [9] | 1,048,638.00 | NA | NA | NA | 902.94 |
2 | GPSO-w [10] | 1,027,679.00 | 1,034,340.00 | 1,031,716.00 | NA | NA |
3 | EAPSO [17] | 1,018,510.00 | 1,019,302.00 | 1,018,710.00 | NA | 37.50 |
4 | TLA [18] | 1,019,925.00 | 1,021,118.00 | 1,020,411.00 | NA | 2.94 |
5 | ICA [18] | 1,018,467.00 | 1,021,796.00 | 1,019,291.00 | 693.487 | NA |
6 | GA [20] | 1,033,481.00 | 1,042,606.00 | 1,038,014.00 | NA | NA |
7 | ABC [41] | 1,021,576.00 | 1,024,316.00 | 1,022,686.00 | NA | 156.18 |
8 | CSO [52] | 1,017,660.00 | 1,019,286.00 | 1,018,710.00 | 302.3103 | 57.66 |
9 | HIGA [54] | 1,108,473.00 | 1,022,284.00 | 1,019,328.00 | NA | 211.8 |
10 | CSDE [57] | 1,023,432.00 | 1,027,634.00 | 1,026,475.00 | NA | 18.00 |
11 | CDBCO [58] | 1,021,500.00 | NA | 1,024,300.00 | NA | 40.20 |
12 | MG-PSO | 1,010,490.42 | 1,010,490.47 | 1,010,490.45 | 0.0104 | 225.80 |
t | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
1036 | 1110 | 1258 | 1406 | 1480 | 1628 | 1702 | 1766 | 1924 | 2072 | 2146 | 2220 | |
1035.95 | 1109.95 | 1257.97 | 1405.98 | 1479.97 | 1627.97 | 1701.96 | 1777.04 | 1923.99 | 2017.97 | 2145.97 | 2219.97 | |
t | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
2072 | 1924 | 1776 | 1554 | 1480 | 1628 | 1776 | 2072 | 1924 | 1628 | 1332 | 1184 | |
2171.99 | 1923.96 | 1775.97 | 1554.00 | 1479.97 | 1627.99 | 1775.98 | 2071.76 | 1923.97 | 1627.95 | 1332.00 | 1183.64 |
Bji | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|
1 | 8.70 | 0.43 | −4.61 | 0.36 | 0.32 | −0.66 | 0.96 | −1.60 | 0.80 | −0.10 |
2 | 0.43 | 8.30 | −0.97 | 0.22 | 0.75 | −0.28 | 5.04 | 1.70 | 0.54 | 7.20 |
3 | −4.61 | −0.97 | 9.00 | −2.00 | 0.63 | 3.00 | 1.70 | −4.30 | 3.10 | −2.00 |
4 | 0.36 | 0.22 | −2.00 | 5.30 | 0.47 | 2.62 | −1.96 | 2.10 | 0.67 | 1.80 |
5 | 0.32 | 0.75 | 0.63 | 0.47 | 8.60 | −0.80 | 0.37 | 0.72 | −0.90 | 0.69 |
6 | −0.66 | −0.28 | 3.00 | 2.62 | −0.80 | 11.80 | −4.90 | 0.30 | 3.00 | −3.00 |
7 | 0.96 | 5.04 | 1.70 | −1.96 | 0.37 | −4.90 | 8.24 | −0.90 | 5.90 | −0.60 |
8 | −1.60 | 1.70 | −4.30 | 2.10 | 0.72 | 0.30 | −0.90 | 1.20 | −0.96 | 0.56 |
9 | 0.80 | 0.54 | 3.10 | 0.67 | −0.90 | 3.00 | 5.90 | −0.96 | 0.93 | −0.30 |
10 | −0.10 | 7.20 | −2.00 | 1.80 | 0.69 | −3.00 | −0.60 | 0.56 | −0.30 | 0.99 |
t | Pl,t | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 150.00 | 135.00 | 207.00 | 60.12 | 122.00 | 122.99 | 129.0 | 47.00 | 20.00 | 55.00 | 1048.12 | 12.12 | 1036 | 0.00 | 28,439.90 | 9.46 |
2 | 150.21 | 135.01 | 282.54 | 60.99 | 122.01 | 122.87 | 129.0 | 47.00 | 20.00 | 55.00 | 1124.63 | 14.63 | 1110 | 0.00 | 30,060.63 | 9.22 |
3 | 226.11 | 135.00 | 308.00 | 60.35 | 172.42 | 123.00 | 129.0 | 47.00 | 20.00 | 55.00 | 1275.88 | 17.88 | 1258 | 0.00 | 33,338.67 | 9.31 |
4 | 303.00 | 215.00 | 305.00 | 60.20 | 172.21 | 122.95 | 129.0 | 47.00 | 20.00 | 55.00 | 1429.36 | 23.37 | 1406 | 0.00 | 36,637.72 | 9.33 |
5 | 379.81 | 222.21 | 297.63 | 60.00 | 172.71 | 122.42 | 129.5 | 47.00 | 20.00 | 55.00 | 1506.28 | 26.28 | 1480 | 0.00 | 38,296.74 | 9.57 |
6 | 456.50 | 222.20 | 305.04 | 80.06 | 222.61 | 122.56 | 129.5 | 47.00 | 20.00 | 55.00 | 1660.47 | 32.47 | 1628 | 0.00 | 41,715.72 | 9.57 |
7 | 456.50 | 302.27 | 312.27 | 120.41 | 172.70 | 122.41 | 129.5 | 47.00 | 20.00 | 55.00 | 1738.06 | 36.06 | 1702 | 0.00 | 43,450.29 | 9.82 |
8 | 456.50 | 309.27 | 299.55 | 170.45 | 172.70 | 122.61 | 129.52 | 77.00 | 20.00 | 55.00 | 1812.60 | 36.60 | 1776 | 0.00 | 45,261.46 | 9.92 |
9 | 456.50 | 389.50 | 299.57 | 189.50 | 222.61 | 122.43 | 129.5 | 85.30 | 20.00 | 55.00 | 1969.90 | 45.90 | 1924 | 0.00 | 48,733.35 | 9.69 |
10 | 456.50 | 396.80 | 325.60 | 239.50 | 222.91 | 160.00 | 129.5 | 115.30 | 20.00 | 55.00 | 2121.11 | 49.11 | 2072 | 0.00 | 52,202.09 | 9.26 |
11 | 458.40 | 396.80 | 340.15 | 289.50 | 227.60 | 160.02 | 129.5 | 120.00 | 20.00 | 55.00 | 2196.97 | 50.97 | 2146 | 0.00 | 53,836.89 | 9.52 |
12 | 456.50 | 460.00 | 325.14 | 300.00 | 222.61 | 160.00 | 129.5 | 120.00 | 50.00 | 55.00 | 2278.75 | 58.75 | 2220 | 0.00 | 55,806.50 | 9.19 |
13 | 456.50 | 396.80 | 298.55 | 300.00 | 222.60 | 122.42 | 129.5 | 120.00 | 20.00 | 55.00 | 2121.36 | 49.36 | 2072 | 0.00 | 52,386.68 | 9.39 |
14 | 456.50 | 316.80 | 302.45 | 250.00 | 222.60 | 122.40 | 129.5 | 90.00 | 20.00 | 55.00 | 1965.25 | 41.25 | 1924 | 0.00 | 48,807.00 | 9.25 |
15 | 379.80 | 309.50 | 294.85 | 241.12 | 172.76 | 122.46 | 129.5 | 85.30 | 20.00 | 55.00 | 1810.28 | 34.28 | 1776 | 0.00 | 45,390.70 | 9.49 |
16 | 303.25 | 229.50 | 318.48 | 191.24 | 122.83 | 122.40 | 129.5 | 85.30 | 20.01 | 55.00 | 1577.50 | 23.50 | 1554 | 0.00 | 40,201.11 | 9.28 |
17 | 226.60 | 222.20 | 287.67 | 180.76 | 172.70 | 122.68 | 129.50 | 85.30 | 20.00 | 55.00 | 1502.41 | 22.41 | 1480 | 0.00 | 38,564.89 | 9.55 |
18 | 303.25 | 222.20 | 312.74 | 180.80 | 222.60 | 123.52 | 129.50 | 85.30 | 20.00 | 55.00 | 1654.90 | 26.90 | 1628 | 0.00 | 41,874.71 | 9.89 |
19 | 379.80 | 302.20 | 313.17 | 180.80 | 222.60 | 122.56 | 129.50 | 85.30 | 20.00 | 55.00 | 1810.92 | 34.92 | 1776 | 0.00 | 45,252.41 | 9.70 |
20 | 456.50 | 382.20 | 340.17 | 230.81 | 230.40 | 160.01 | 129.50 | 115.30 | 20.00 | 55.00 | 2119.89 | 47.89 | 2072 | 0.00 | 52,004.90 | 9.51 |
21 | 456.50 | 396.90 | 301.41 | 180.80 | 222.60 | 122.40 | 129.50 | 85.30 | 20.00 | 55.00 | 1970.40 | 46.40 | 1924 | 0.00 | 48,721.06 | 9.54 |
22 | 379.80 | 316.80 | 278.34 | 130.83 | 172.70 | 122.62 | 129.50 | 55.30 | 20.00 | 55.00 | 1660.89 | 32.89 | 1628 | 0.00 | 41,817.63 | 9.45 |
23 | 303.20 | 236.80 | 198.30 | 118.30 | 122.80 | 122.40 | 129.96 | 47.00 | 20.00 | 55.00 | 1353.76 | 21.76 | 1332 | 0.00 | 35,306.36 | 9.39 |
24 | 226.60 | 222.20 | 184.72 | 120.40 | 73.01 | 122.50 | 129.50 | 47.00 | 20.00 | 55.00 | 1200.92 | 16.92 | 1184 | 0.00 | 31,853.95 | 9.39 |
Total | 802.62 MW | 1,029,961.36 USD | 227.7 s |
No. | NTOTS | σ | |||||
---|---|---|---|---|---|---|---|
1 | AIS [10] | 1,045,715.00 | 1,048,431.00 | 1,047,050.00 | NA | 835.62 | 1858.20 |
2 | GA [10] | 1,052,251.00 | 1,062,511.00 | 1,058,041.00 | NA | NA | 206.64 |
3 | GPSO-w [10] | 1,048,410.00 | 1,057,170.00 | 1,052,092.00 | NA | NA | 245.58 |
4 | CDBCO [14] | 1,042,900.00 | 1,043,626.00 | 1,044,700.00 | NA | 839.31 | 91.80 |
5 | ICA [16] | 1,040,758.00 | 1,043,173.00 | 1,041,664.00 | 603.758 | 848.79 | NA |
6 | CSO [52] | 1,038,320.00 | 1,042,518.00 | 1,039,374.00 | 395.673 | 802.68 | 88.86 |
7 | EBSO [53] | 1,038,915.00 | 1,039,272.00 | 1,039,188.00 | NA | NA | 13.20 |
8 | HIGA [54] | 1,041,088.00 | 1,042,516.00 | 1,041,218.00 | NA | 853.53 | NA |
9 | ECE [71] | 1,043,989.00 | NA | 1,044,470.00 | NA | NA | 228.0 |
10 | MG-PSO | 1,029,961.36 | 1,029,961.40 | 1,029,961.37 | 0.035 | 802.62 | 227.7 |
Parameters | Exploration Stage | Exploitation Stage | ||
---|---|---|---|---|
0.3 | 0.3 | 0.3 | 0.3 | |
c1, c2 | 2.05 | 2.05 | 2.05 | 2.05 |
150 | 150 | 150 | 350 | |
0.90 | 0.80 | 0.80 | 0.35 | |
0.05 | 0.10 | 0.20 | 0.20 | |
−5.66 × 10−3 | −4.66 × 10−3 | −3.99 × 10−3 | −4.28 × 10−4 |
No. | NTOTs | σ | ||||
---|---|---|---|---|---|---|
Case #3: 100 TPU | ||||||
1 | CSO [52] | 10,183,633.00 | 10,192,352.00 | 10,185,287.00 | 1323.00 | 297.00 |
2 | GA [55] | 10,908.741.00 | 11,987,675.00 | 11,584,628.00 | NA | 542.40 |
3 | GPSO-w [55] | 10,366.076.00 | 11,310,279.00 | 10,766,385.00 | NA | 353.40 |
4 | FA [55] | 10,197,269.00 | 11,216,243.00 | 10,419,457.00 | NA | 266.46 |
5 | SAFA [55] | 10,183,819.00 | 10,388,958.00 | 10,286,043.00 | NA | 84.60 |
6 | MG-PSO | 9,549,134.75 | 9,549,134.96 | 9,549,134.68 | 0.098 | 419.98 |
Case #4: 500 TPU | ||||||
1 | CSO [52] | 51.044,611.00 | 51,082,986.00 | 51,050,457.00 | 6067.00 | 534.60 |
2 | MG-PSO | 49,655,500.34 | 49,655,501.81 | 49,655,501.73 | 0.511 | 765.29 |
Case #5: 1000 TPU | ||||||
1 | CSO [52] | 102,122,060.00 | 102,186,925.00 | 102,133,104.00 | 9627.00 | 721.8 |
2 | MG-PSO | 99,311,000.68 | 99,311,003.63 | 99,311.001.47 | 1.022 | 903.34 |
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Al-Bahrani, L.; Seyedmahmoudian, M.; Horan, B.; Stojcevski, A. Solving the Real Power Limitations in the Dynamic Economic Dispatch of Large-Scale Thermal Power Units under the Effects of Valve-Point Loading and Ramp-Rate Limitations. Sustainability 2021, 13, 1274. https://doi.org/10.3390/su13031274
Al-Bahrani L, Seyedmahmoudian M, Horan B, Stojcevski A. Solving the Real Power Limitations in the Dynamic Economic Dispatch of Large-Scale Thermal Power Units under the Effects of Valve-Point Loading and Ramp-Rate Limitations. Sustainability. 2021; 13(3):1274. https://doi.org/10.3390/su13031274
Chicago/Turabian StyleAl-Bahrani, Loau, Mehdi Seyedmahmoudian, Ben Horan, and Alex Stojcevski. 2021. "Solving the Real Power Limitations in the Dynamic Economic Dispatch of Large-Scale Thermal Power Units under the Effects of Valve-Point Loading and Ramp-Rate Limitations" Sustainability 13, no. 3: 1274. https://doi.org/10.3390/su13031274