Symmetry-Breaking for Airflow Control Optimization of an Oscillating-Water-Column System
Abstract
:1. Introduction
2. Model Statement
2.1. Wave Surface Dynamics
2.2. Capture Chamber Model
2.3. Wells Turbine Model
2.4. Doubly Fed Induction Generator Model
3. Airflow Control Strategy
3.1. Stalling Behavior Problem
3.2. Control Problem Formulation
4. Symmetry-Breaking Constraints
- Variable symmetry where permuting variables are solution invariant defined as:
- Value symmetry where permuting solution values are solution invariant defined as:
- Variable/value symmetry where both variables and values permutation is solution invariant defined as:
5. Particle Swarm Optimization
Algorithm 1 PSO algorithm |
1. Define parameters of PSO algorithm: , , , , , . |
2. Arbitrarily initialize the swarm particles’ positions and velocities . |
3. Assess this initial population and calculate and . |
4. Increment the iteration k and updated the position of every particle of the flock by means of the update Equations (24)–(26). |
5. Assess the associated fitness values |
(i) if then and , |
(ii) if then and . |
where represents the best previous fitness of the particle and represents the best fitness in the whole swarm. |
6. If the stopping criterion is met, the program finishes and stops searching with the obtained solution . Otherwise, go back to step 4. |
6. Results and Discussion
6.1. Optimization Results
6.2. OWC Performance
7. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
Nomenclature
Wavelength, amplitude, and height (m) | |
Sea depth and wave surface elevation (m) | |
Wave period (s) and wave frequency (rad/s) | |
g | Acceleration gravity (m/s) |
Pressure drop (Pa) | |
Capture chamber inner width and length (m) | |
Capture chamber volume (m) and flow rate (m/s) | |
Atmospheric density (kg/m) and airflow speed (m/s) | |
Blade chord length, blade span and turbine diameter (m) | |
Blade number, pole number, wave number, and turbine constant | |
Electromagnetic and turbine torques (N·m) | |
J | Turbo-generator inertia (kg·m) |
Torque, power and flow coefficients | |
Stator and rotor resistances () | |
Stator and rotor inductances (H) | |
Stator and rotor currents (A) | |
Stator and rotor flux (Wb) | |
Stator and rotor rotational speed (rad/s) | |
e | Error between the reference variable and measured variable |
u | Control signal obtained from the PID controller |
Proportional, integral, and derivative gains of the PID controller | |
P | Optimization problem |
Optimal solution for the problem | |
f | Cost function |
Penalty function | |
Scaling penalty parameter for the jth constraint | |
Number of constraints | |
Symmetry permutation | |
∑ | Set of symmetry permutations |
Swarm particles positions and velocities | |
Cognitive and social scaling factors | |
w | Inertia factor |
Best positions formerly found for the ith particle and the entire swarm at the kth iteration |
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Capture Chamber | Wells Turbine | DFIG Generator | |
---|---|---|---|
= 4.5 m | n = 5 | = 0.5968 | = 18.45 kW |
= 4.3 m | b = 0.21 m | = 0.6258 | = 400 V |
=1.19 kg/m | l = 0.165 m | = 0.0003495 H | = 50 Hz |
= 1029 kg/m | r = 0.375 m | = 0.324 H | |
a = 0.4417 m | = 0.324 H |
SBC | Cost Function | CPU Time (s) | |||||
---|---|---|---|---|---|---|---|
Worst | Mean | Best | S.D. | Worst | Mean | Best | |
without | 11.7433 | 8.3576 | 8.0927 | 1.6928 | 7091 | 6745 | 6492 |
with | 11.2061 | 8.3324 | 8.0243 | 1.4491 | 5876 | 5604 | 5328 |
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M’zoughi, F.; Garrido, I.; Garrido, A.J. Symmetry-Breaking for Airflow Control Optimization of an Oscillating-Water-Column System. Symmetry 2020, 12, 895. https://doi.org/10.3390/sym12060895
M’zoughi F, Garrido I, Garrido AJ. Symmetry-Breaking for Airflow Control Optimization of an Oscillating-Water-Column System. Symmetry. 2020; 12(6):895. https://doi.org/10.3390/sym12060895
Chicago/Turabian StyleM’zoughi, Fares, Izaskun Garrido, and Aitor J. Garrido. 2020. "Symmetry-Breaking for Airflow Control Optimization of an Oscillating-Water-Column System" Symmetry 12, no. 6: 895. https://doi.org/10.3390/sym12060895
APA StyleM’zoughi, F., Garrido, I., & Garrido, A. J. (2020). Symmetry-Breaking for Airflow Control Optimization of an Oscillating-Water-Column System. Symmetry, 12(6), 895. https://doi.org/10.3390/sym12060895