Optimal Generation Scheduling in Hydro-Power Plants with the Coral Reefs Optimization Algorithm
Abstract
:1. Introduction
2. Materials and Methods
2.1. Problem Definition and Generation Scheduling Modelling
2.1.1. Hydraulic Losses Model
2.1.2. Adjustment of the Model from Data
2.1.3. Optimization Modelling
- The power demand produced (()) can be considered approximately equal to the power demand requested (D), with an acceptable error ().
- The water discharge boundaries for each turbine-generator ( and ).
- The electric power boundaries ( and ) for each turbine-generator in operating zones ().
- Each turbine-generator has only one operating zone: on or off (, on (1) or off (0)).
2.2. Evolutionary Meta-Heuristic Proposed: The Coral Reefs Optimization Algorithm
- (1)
- Initialization. The algorithm is initialized by assigning random candidate solutions to a random number of squares of the grid, leaving the rest empty. Each solution is labeled with the problem’s objective function (i.e. the healthy function).
- (2)
- Reef formation. It is an iterative process that takes place over iterations. At each iteration several operators or search procedures are applied to emulate the corals’ reproduction in the reef. Therefore, new candidate solutions are obtained and try to settle on the reef. If the square of the reef found (i,j) is free, the coral settles in the hole. If the square is occupied, the candidate solution fights with the settled solution and the one with higher health function stays in the hole. If the new candidate cannot settle, it tries to find a new square, but after attempts without finding a place to grow, the candidate solution is discarded.
- (3)
- Predation. Once settlement of new corals has taken place, a predation phase may occur with probability . Should predation happen, a percentage of the reef has preyed and solutions previously settled are lost. Thus leaving holes for new candidate solutions (with bad health functions) from other areas of the search space, to enter the reef (and escape from local minima).
- (4)
- Stop. if halting criteria are satisfied; otherwise go to step (2) for the next cycle.The best individual in the reef is considered to be the final solution to the problem.
- Harmony Search mutation (HS): Mutation is recreating the Harmony Search algorithm [43]. New candidate is obtained: using the same values of the component from other reefs’ coral, with a probability HMCR (Harmony Memory Considering Rate); or performing slight modifications to the candidate, with a probability PAR (Pitch Adjusting Rate) ;
- Differential Evolution mutation (DE): Mutation implementing a Differential Evolution algorithm [44]. New candidates are obtained by combining a reef’s coral with a perturbation vector . Two different strategies have been used to find the perturbation:DEB: (using the best), andDER: (on a random basis)(where F stands for the evolution factor that weights the perturbation amplitude, in our case );
- Differential Evolution mutation with crossover (DEX): Mutation recreating the Differential Evolution algorithm [44]. New candidates are obtained after a crossover is applied between a coral from the reef and a perturbation vector generated by DER.
- Gaussian mutation (GM): The new candidate is obtained by applying gaussian mutation to a coral from the reef (, where ) is a random number following the Gaussian distribution. The Gaussian probability density function is:
3. Experiments and Results
3.1. Case Study and Parameter Settings
3.2. Daily Dispatch Programming Experiment
3.3. Hourly Dispatch Programming Experiment
4. Discussion
- visually we can see differences among the true means of and the rest of the algorithms, favouring version, but cannot perceive the overlap with others;
- it is not possible to conclude if a statistical significant difference of the means computed of RGA, , , , and algorithms exists (or not).
5. Conclusions
- Proposal of an effective mathematical modeling to power electric dispatch in HPP. The proposed model extends the electric power equation (Equation (9)), bringing modularity to the obtaining of (using regression fit) and (fluid mechanics’ model) terms. This modeling has been described and discussed in detail, being easily replicable.
- Inclusion of a hydraulic losses model in the mathematical modeling, to bring greater reality to the dynamics of the electric dispatch problem. In this sense, net water head () was obtained (Equation. (7)).
- Proposal and testing of the CRO algorithm using a combination of search operators or search operators independently. In this work, the latter has concluded on best results when the CRO with Gaussian mutation was used, outperforming real encoded genetic algorithms and differential evolution.
- Proposal of a statistical inference methodology to compare results from the meta-heuristics has been applied. To test the samples’ normality, the Kolmogorov-Smirnov test was implemented, showing that the behaviour using CRO(4) and DE(best/1/bin) algorithms is non-normal. Also, Kruskal-Wallis test was used to determine the differences among the mean ranks, resulting in statistically significant difference between the algorithms results. Finally, Tukey test was performed, confirming that CRO(4) configurations generate solutions with higher fitness function values.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Coefficient | Estimation | RMSE | p-Value |
---|---|---|---|
0.1463 | <0.0001 | ||
0.018076 | <0.0001 | ||
0.0050502 | <0.0001 | ||
−3.5254e-05 | <0.0001 | ||
−0.00112337 | <0.0001 | ||
−1.4507e-05 | <0.0001 |
D | RGA | DEB | DER | |||
---|---|---|---|---|---|---|
255 | 0.40 | 0.71 | 0.70 | −0.36 | 1.19 | 0.79 |
265 | 1.20 | 1.68 | 1.75 | −0.25 | 0.93 | 1.87 |
275 | 2.49 | 2.66 | 2.49 | 4.91 | 3.61 | 2.76 |
285 | 2.98 | 3.46 | 3.43 | 2.01 | 3.11 | 3.59 |
295 | 2.14 | 4.12 | 3.88 | 4.78 | 4.32 | 4.30 |
305 | 3.43 | 4.62 | 4.25 | 3.52 | 5.53 | 4.68 |
315 | 2.95 | 4.93 | 4.78 | 3.06 | 4.85 | 5.07 |
325 | 3.03 | 4.57 | 4.76 | 3.66 | 6.23 | 5.22 |
335 | 3.50 | 4.54 | 4.03 | 2.01 | 4.49 | 5.02 |
345 | 3.09 | 4.35 | 4.51 | 3.50 | 3.27 | 4.63 |
355 | 3.27 | 3.72 | 3.73 | 8.36 | 2.78 | 3.88 |
365 | 1.17 | 2.64 | 2.49 | 3.20 | 2.33 | 2.87 |
365 | 1.35 | 2.58 | 2.69 | 3.20 | 3.06 | 2.87 |
355 | 3.37 | 3.19 | 3.78 | 8.36 | 3.74 | 3.91 |
345 | 4.10 | 4.47 | 4.52 | 3.50 | 4.32 | 4.63 |
335 | 4.46 | 4.81 | 5.11 | 2.01 | 4.81 | 5.06 |
325 | 3.59 | 5.04 | 5.02 | 3.66 | 6.21 | 5.19 |
315 | 3.17 | 4.45 | 4.97 | 3.06 | 5.71 | 5.09 |
305 | 3.66 | 4.62 | 4.56 | 3.52 | 5.29 | 4.75 |
295 | 2.67 | 4.06 | 4.18 | 4.78 | 4.27 | 4.22 |
285 | 2.99 | 3.46 | 3.46 | 2.29 | 3.59 | 3.58 |
275 | 2.33 | 2.66 | 2.58 | 4.91 | 3.60 | 2.82 |
265 | 1.37 | 1.74 | 1.82 | 2.10 | 1.22 | 1.85 |
255 | 0.61 | 0.71 | 0.70 | −0.36 | 1.48 | 0.84 |
ASW (m/s) | 63.3 | 83.8 | 84.2 | 79.4 | 89.9 | 89.5 |
ASW (L/day) | 227.9 mi | 301.7 mi | 303.1 mi | 285.9 mi | 323.8 mi | 322.2 mi |
D | ||||||
255 | 0.85 | 1.59 | 0.69 | 1.42 | 1.68 | 0.71 |
265 | 1.73 | 2.83 | 1.88 | 2.80 | 2.69 | 1.72 |
275 | 2.71 | 3.76 | 2.70 | 3.52 | 3.69 | 2.83 |
285 | 3.06 | 4.65 | 3.37 | 4.55 | 4.55 | 3.47 |
295 | 4.23 | 5.33 | 4.16 | 5.10 | 5.30 | 4.26 |
305 | 4.05 | 5.94 | 4.55 | 5.06 | 5.75 | 4.79 |
315 | 5.04 | 6.37 | 4.80 | 6.08 | 5.96 | 4.90 |
325 | 4.90 | 6.39 | 5.15 | 5.06 | 5.93 | 4.89 |
335 | 5.02 | 6.24 | 4.85 | 6.12 | 5.95 | 4.78 |
345 | 4.57 | 5.78 | 4.62 | 5.63 | 4.89 | 4.62 |
355 | 3.73 | 5.33 | 3.70 | 4.92 | 2.50 | 4.04 |
365 | 2.71 | 4.10 | 2.88 | 3.94 | 3.74 | 2.84 |
365 | 2.95 | 4.12 | 2.88 | 4.18 | 3.99 | 2.83 |
355 | 3.86 | 5.13 | 3.70 | 5.14 | 3.63 | 5.88 |
345 | 4.59 | 6.10 | 4.60 | 5.82 | 5.30 | 4.59 |
335 | 5.02 | 6.32 | 5.03 | 6.22 | 6.27 | 5.12 |
325 | 5.25 | 6.52 | 5.37 | 5.86 | 6.28 | 5.13 |
315 | 5.14 | 6.21 | 5.10 | 6.09 | 6.19 | 5.31 |
305 | 4.49 | 5.94 | 4.55 | 5.44 | 5.83 | 4.85 |
295 | 4.25 | 5.33 | 4.35 | 5.29 | 5.34 | 4.37 |
285 | 3.54 | 4.70 | 3.57 | 4.64 | 4.62 | 3.68 |
275 | 2.82 | 3.69 | 2.70 | 3.62 | 3.75 | 2.64 |
265 | 1.66 | 2.83 | 1.88 | 2.80 | 2.74 | 1.86 |
255 | 0.79 | 1.74 | 0.69 | 1.80 | 1.71 | 0.71 |
ASW (m/s) | 87.0 | 116.9 | 87.8 | 111.1 | 108.3 | 90.8 |
ASW (L/day) | 313.2 mi | 420.9 mi | 315.9 mi | 399.9 mi | 389.8 mi | 326.9 mi |
| | |||||
---|---|---|---|---|---|
UN | () | (m/s) | (%) | () | () |
1 | 54.6 | 109.00 | 0.93 | 54.76 | 0.24 |
2 | 55.7 | 111.00 | 0.93 | 54.81 | 0.19 |
3 | 54.5 | 108.73 | 0.93 | 54.81 | 0.19 |
4 | 53.2 | 106.01 | 0.93 | 54.82 | 0.18 |
5 | 53.7 | 107.02 | 0.93 | 54.81 | 0.19 |
6 | 53.3 | 106.3 | 0.93 | 54.80 | 0.20 |
SUM | 324.99 | 648.05 | Flow in SCM: 653.1 (m/s) | ||
SUB | −0.01 | 5.05 | Productivity index: 0.50148 | ||
| | |||||
UN | () | (m/s) | (%) | () | () |
1 | 54.6 | 109.00 | 0.93 | 54.76 | 0.24 |
2 | 54.2 | 108.00 | 0.93 | 54.8 | 0.20 |
3 | 53.9 | 107.42 | 0.93 | 54.81 | 0.19 |
4 | 53 | 105.63 | 0.93 | 54.82 | 0.18 |
5 | 54.2 | 108.06 | 0.93 | 54.84 | 0.18 |
6 | 53.1 | 109.92 | 0.93 | 54.84 | 0.18 |
SUM | 325.01 | 648.04 | Flow in SCM: 653.1 (m/s) | ||
SUB | +0.01 | 5.15 | Productivity index: 0.50152 | ||
| | |||||
UN | () | (m/s) | (%) | () | () |
1 | 53.6 | 107 | 0.93 | 54.77 | 0.23 |
2 | 54.7 | 109 | 0.93 | 54.8 | 0.20 |
3 | 54.3 | 108.2 | 0.93 | 54.81 | 0.19 |
4 | 53.9 | 107.4 | 0.93 | 54.81 | 0.19 |
5 | 53.7 | 107.4 | 0.93 | 54.82 | 0.18 |
6 | 54.7 | 109.02 | 0.93 | 54.81 | 0.19 |
SUM | 324.92 | 647.78 | Flow in SCM: 653.1 (m/s) | ||
SUB | −0.08 | 5.32 | Productivity index: 0.50155 |
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Marcelino, C.G.; Camacho-Gómez, C.; Jiménez-Fernández, S.; Salcedo-Sanz, S. Optimal Generation Scheduling in Hydro-Power Plants with the Coral Reefs Optimization Algorithm. Energies 2021, 14, 2443. https://doi.org/10.3390/en14092443
Marcelino CG, Camacho-Gómez C, Jiménez-Fernández S, Salcedo-Sanz S. Optimal Generation Scheduling in Hydro-Power Plants with the Coral Reefs Optimization Algorithm. Energies. 2021; 14(9):2443. https://doi.org/10.3390/en14092443
Chicago/Turabian StyleMarcelino, Carolina Gil, Carlos Camacho-Gómez, Silvia Jiménez-Fernández, and Sancho Salcedo-Sanz. 2021. "Optimal Generation Scheduling in Hydro-Power Plants with the Coral Reefs Optimization Algorithm" Energies 14, no. 9: 2443. https://doi.org/10.3390/en14092443