A Rigid Cuckoo Search Algorithm for Solving Short-Term Hydrothermal Scheduling Problem
Abstract
:1. Introduction
2. Methods
2.1. Mathematical Model of the Hydrothermal System
2.2. Cuckoo Search Algorithm and Lévy Flights
- Concept 1. Individually all cuckoos produce one single egg at a time that regards a proposed solution and randomly throws its egg up into the wanted nest among the set number of possible host nests.
- Concept 2. The egg of high quality thrown up in the best nests regards a better solution transferred to the subsequent generation.
- Concept 3. The possible number of host nests is constant, and the probability that the host bird can find a nest is indicated by the probability constant, Pa, with range [0,1]. Hence, it may either discard the egg or leave this nest and then build a new nest entirely in a different place.
2.3. Rigid Cuckoo Search Algorithm
2.4. Implementation of RCSA on a Hydrothermal System
- Step-1.
- In the StHS problem, the influential variables such as the release rate of water for the whole plants for several hours and thermal unit production for the entire period are chosen irregularly within the operating limitations. The storage capacity of every reservoir has been estimated using Equation (4), the generation of hydro plants has been calculated using Equation (3). Subsequently, the thermal power generation has been computed by applying Equation (2). The population of the host nest (NE) has been explained as:
- Step-2.
- Set the production number.
- Step-3.
- Compute the objective function using Equation (17). With the equation of restraints, many restrictions irregularly have been limited. Then, enhanced fuel cost has been calculated as Equation (19).
- Step-4.
- The modern solution has been created by using Levy flights. The new solution’s computation has been built in the preceding best nest by using Levy flights. For this technique, the optimal way for levy flights has been computed by Yang XS’s contribution, Deb S [18]. The new solution has been presented in Equation (20)
- Step-5.
- The effect of the detection of an alien egg in a nest of a host bird with Pa’s possibility produces a new solution for the problem comparable with the Levy flights. The new solution has been computed as following Equations (25)–(27):The increment value of has been defined by
- Step-6.
- The technique ends if modern production gives the maximum production number.
3. Results and Discussion
3.1. Parameter Selection
3.2. Obtained Results
3.3. Proposed System Validation
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
F | Gross cost of production |
Fi(PTj) | Production cost for PTj |
PTj | Production of power for the thermal unit at period j |
PH(i, j) | Production of power for hydro unit i at period j |
Pdj | System load demand at period j |
qij | Water release rate of hydro unit i at period j |
rij | Stream rate into the storage reservoir of the hydro plant at a period |
sij | Spillage of the reservoir at a period |
Xij | Storage volume of hydro plant i at period j |
Xi0 | Initial reservoir storage of hydro plant |
i | Number of units |
nh, np | Maximum number of unit hydro/thermal |
j | Number of scheduling intervals |
NH, NP | Maximum number of scheduling periods of hydro/thermal |
NC | Total number of constraints |
Rj | Resource constraint |
IT | Total iteration numbers |
IC | Current iterations |
λk | Penalty value for kth |
Viok | Violation amount of kth constraint |
ψ | Distribution factor |
Φ | Gamma distribution function |
m | Time period |
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Pa | Min Cost ($) | Avg. Cost ($) | Max Cost ($) | Std. dev. ($) | CPU (s) |
---|---|---|---|---|---|
0.1 | 709,932.115 | 709,995.586 | 710,655.745 | 327.189 | 17.9 |
0.2 | 709,922.44 | 710,768.874 | 711,384.563 | 599.382 | 17.9 |
0.3 | 709,911.445 | 709,994.412 | 710,989.321 | 489.733 | 17.8 |
0.4 | 709,866.727 | 709,745.236 | 709,999.741 | 103.937 | 18.1 |
0.5 | 709,886.651 | 709,887.698 | 709,988.258 | 47.653 | 18.6 |
0.6 | 709,862.129 | 709,899.987 | 709,991.951 | 54.512 | 18.5 |
0.7 | 709,862.027 | 709,878.852 | 709,996.159 | 59.661 | 18.2 |
0.8 | 709,901.478 | 709,910.357 | 710,920.357 | 478.225 | 18.3 |
0.9 | 709,902.685 | 709,901.753 | 710,901.753 | 471.185 | 18.7 |
m | PDm | Vm (acre-ft) | qm (acre-ft/hr) | Psm (MW) | Phm (MW) |
---|---|---|---|---|---|
1 | 1199 | 101,897 | 1832 | 892 | 300 |
2 | 1497 | 85,959 | 3328 | 892 | 600 |
3 | 1098 | 93,847 | 1340 | 892 | 200 |
4 | 1795 | 59,998 | 4817 | 892 | 900 |
5 | 948 | 70,428 | 1124 | 783 | 158 |
6 | 1289 | 59,998 | 2863 | 783 | 509 |
Hours | Water Discharge (×104 m3) | Ps (MW) | |||
---|---|---|---|---|---|
Plant 1 | Plant 2 | Plant 3 | Plant 4 | ||
1 | 9.4884 | 6.1377 | 26.3477 | 13.1009 | 1036.249 |
2 | 9.4025 | 6.3104 | 25.2088 | 13.1024 | 1066.243 |
3 | 9.197 | 6.256 | 24.504 | 13.1035 | 1049.837 |
4 | 8.8096 | 6.3038 | 23.8832 | 13.1003 | 996.409 |
5 | 8.5269 | 6.2552 | 22.7287 | 13.1009 | 975.264 |
6 | 8.3314 | 6.4278 | 21.9238 | 13.1007 | 1076.828 |
7 | 8.4101 | 6.821 | 206226 | 13.1005 | 1294.982 |
8 | 8.6081 | 7.2309 | 19.2428 | 13.1023 | 1623.787 |
9 | 8.8072 | 7.3387 | 18.5775 | 13.1077 | 1849.655 |
10 | 8.465 | 7.5783 | 17.8006 | 13.1163 | 1917.3778 |
11 | 8.3429 | 7.7186 | 17.0931 | 13 | 1816.169 |
12 | 8.4121 | 7.8032 | 16.9969 | 13.4302 | 1885.493 |
13 | 8.2444 | 7.8624 | 16.3144 | 14.6742 | 1786.317 |
14 | 8.1224 | 8.0001 | 15.439 | 15.9371 | 1737.617 |
15 | 7.9417 | 8.1501 | 14.3596 | 17.2956 | 1652.815 |
16 | 7.8102 | 8.5518 | 12.9066 | 18.4244 | 1580.968 |
17 | 7.8153 | 8.9344 | 11.4353 | 20.0065 | 1630.883 |
18 | 7.6626 | 9.3193 | 10.1008 | 21.4157 | 1639.432 |
19 | 7.672 | 9.9655 | 10.1316 | 23.0205 | 1739.29 |
20 | 7.5945 | 10.5252 | 10.1043 | 24.2843 | 1787.795 |
21 | 7.3908 | 11.1875 | 12.1618 | 25.0985 | 1761.32 |
22 | 7.586 | 12.2327 | 12.747 | 25.0636 | 1659.735 |
23 | 7.5513 | 13.1386 | 13.2403 | 25.0994 | 1411.135 |
24 | 7.4103 | 14.2533 | 13.6662 | 25.0972 | 1177.636 |
Techniques | Min Cost ($) | Average Cost ($) | Max Cost ($) | CPU Time (s) |
---|---|---|---|---|
GA [4] | 942,600 | 946,609.1 | 951,087 | 1920 |
EGA [5] | 934,727.00 | 936,058.00 | 937,339.00 | --- |
FEP [6] | 930,267.92 | 930,897.44 | 931,396.81 | 1911.2 |
CEP [6] | 930,166.25 | 930,373.23 | 930,927.01 | 2292.1 |
IFEP [6] | 930,129.82 | 930,290.13 | 930,881.92 | 1033.2 |
PSO [5] | 928,878.00 | 933,085.00 | 938,012.00 | --- |
CSA-Lévy [26] | 927,934.23 | 927,980.45 | 928,000.66 | 79.08 |
CSA Cauchy [26] | 927,967.66 | 927,981.49 | 927,992.53 | 81.30 |
CSA Gauss [26] | 927,957.26 | 927,978.911 | 928,003.23 | 85.75 |
APSO [7] | 926,151.54 | --- | --- | --- |
EPSO [5] | 922,904.00 | 923,527.00 | 924,808.00 | --- |
MDE [8] | 922,555.44 | --- | --- | --- |
IPSO [9] | 922,553.49 | --- | --- | --- |
MAPSO [7] | 922,421.66 | 922,544.00 | 923,508.00 | --- |
TLBO [10] | 922,373.39 | 922,462.24 | 922,873.81 | --- |
CSA [20] | 913,945.87 | 917,624.024 | 921,994.25 | --- |
RIFEP [12] | 709,862.05 | --- | --- | --- |
GS [13] | 709,877.38 | --- | --- | --- |
SA [14] | 709,874.36 | --- | --- | 901 |
ORCSA–Lévy flight [11] | 709,862.048 | --- | --- | 18 |
ORCSA–Cauchy [11] | 709,862.048 | --- | --- | 18 |
Proposed RCSA | 709,862.027 | --- | --- | 17 |
Techniques | Min Cost ($) | Average Cost ($) | Max Cost ($) | CPU Time (s) |
---|---|---|---|---|
SA [27] | 47,306 | -- | -- | -- |
CEP [27] | 45,466 | -- | -- | -- |
CEP-IFS [25] | 45,036.00 | -- | -- | -- |
PSO [27] | 44,740 | -- | -- | -- |
DE [28] | 44,526.1 | -- | -- | 200 |
MDE [28] | 42,611.14 | -- | -- | 125 |
CSR [15] | 42,440 | -- | -- | 109 |
TLBO [10] | 42,385.88 | 42,407.23 | 42,441.36 | -- |
HDE [28] | 42,337.3 | -- | -- | 48 |
SQP [17] | 42,120.02 | -- | -- | 625.07 |
KHA [16] | 41,926 | -- | -- | -- |
MHDE [28] | 41,856.5 | -- | -- | 31 |
CSA [25] | 41,046.897 | -- | -- | 94.4 |
Proposed RCSA | 41,013.09 | 41,401.5 | 41,789.9 | 17 |
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Zheyuan, C.; Hammid, A.T.; Kareem, A.N.; Jiang, M.; Mohammed, M.N.; Kumar, N.M. A Rigid Cuckoo Search Algorithm for Solving Short-Term Hydrothermal Scheduling Problem. Sustainability 2021, 13, 4277. https://doi.org/10.3390/su13084277
Zheyuan C, Hammid AT, Kareem AN, Jiang M, Mohammed MN, Kumar NM. A Rigid Cuckoo Search Algorithm for Solving Short-Term Hydrothermal Scheduling Problem. Sustainability. 2021; 13(8):4277. https://doi.org/10.3390/su13084277
Chicago/Turabian StyleZheyuan, Cui, Ali Thaeer Hammid, Ali Noori Kareem, Mingxin Jiang, Muamer N. Mohammed, and Nallapaneni Manoj Kumar. 2021. "A Rigid Cuckoo Search Algorithm for Solving Short-Term Hydrothermal Scheduling Problem" Sustainability 13, no. 8: 4277. https://doi.org/10.3390/su13084277