Symmetry and Complexity

Symmetry and complexity are the focus of a selection of outstanding papers, ranging from pure Mathematics and Physics to Computer Science and Engineering applications. This collection is based around fundamental problems arising from different fields, but all of them have the same task, i.e. breaking the complexity by the symmetry. In particular, in this Issue, there is an interesting paper dealing with circular multilevel systems in the frequency domain, where the analysis in the frequency domain gives a simple view of the system. Searching for symmetry in fractional oscillators or the analysis of symmetrical nanotubes are also some important contributions to this Special Issue. More papers, dealing with intelligent prognostics of degradation trajectories for rotating machinery in engineering applications or the analysis of Laplacian spectra for categorical product networks, show how this subject is interdisciplinary, i.e. ranging from theory to applications. In particular, the papers by Lee, based on the dynamics of trapped solitary waves for special differential equations, demonstrate how theory can help us to handle a practical problem. In this collection of papers, although encompassing various different fields, particular attention has been paid to the common task wherein the complexity is being broken by the search for symmetry

Hydrodynamic responses and power efficiency analyses of an oscillating wave surge converter under different simulated PTO strategies

Experimental investigation on the power performance of a bottom hinged oscillating wave surge converters (OWSC) with different power take-off (PTO) damping strategies (provided by a generic PTO simulation platform) are conducted in regular and irregular waves. The hydrodynamic performance of the OWSC under different damping modes, in regular waves and irregular waves, is observed. For regular waves, the effects of the main influential parameters (including the incident wave height, wave frequency, phase difference between the buoy velocity and wave elevation) on the output power were quantitatively studied. Six damping coefficients of the linear PTO damping is examined under constant incident wave height, and increasing wave frequencies and an output power curve along wave frequency are presented for each input gain of the PTO simulation platform in both linear damping mode and nonlinear damping mode. Additionally, the best coefficient or input gain is obtained for both linear or nonlinear PTO damping mode in different wave conditions. The phase difference between the buoy velocity and wave elevation of the OWSC model in irregular waves has the same trend as that in regular waves. The output electricity in the JONSWAP spectrum is found to be (approximately 300%) higher than that in a user-defined spectrum for the same wave parameters. However, nonlinear PTO strategies have no distinct advantage in the amount of electricity output but have better stability and broader damping range

An innovative generic platform to simulate real-time PTO damping forces for ocean energy converters based on SIL method

This paper proposes a generic PTO (power-take-off) simulation platform which can be used to predict how devices perform in wave conditions when a simulated real-time linear or non-linear PTO damping forces is employed. The experimental platform could be used to investigate the maximum power output of wave converters(WECs) without constructing a physical PTO system and complex control strategies at the design stage of a WEC, thus making it efficient and inexpensive to explore different PTO solutions. For this purpose, a software-in-the-loop (SIL) simulation method is adopted which uses an innovative control loop running on an inexpensive real-time controller coupled to a DC motor which simulates the PTO damping torque. To calibrate the proposed PTO simulation platform, 1349 drop tests are carried out. A series of relationship curves and corresponding equations are drawn for both the linear and non-linear PTO cases. Moreover, correlation curves for input gains and the produced damping force coefficients are provided. The correlation indicates the PTO simulation platform's capacity of simulating linear PTO can reach 40–220 and can reach 10–70 for quadratic damping in terms of damping force coefficient. To investigate the accuracy of the platform, uncertainty analyses are also carried out in good details. The calibrating tests and uncertainty analyses indicate that the proposed experimental platform can be used to overcome many of the limitations in modelling PTO systems at laboratory scale to simulate both real-time linear and quadratic PTO damping forces

Software-in-the-loop applications for improved physical model tests of ocean renewable energy devices using artificial intelligence

Experimental research in laboratory is a necessary and useful method to explore the full potential of a device. Because it does not only require much less money than the prototype at sea test, it also provides more reliable results compared to numerical simulations. Hence, it is significantly vital to make accurate model tests of the concerned ocean renewable energy (ORE) devices possible. For this reason, this study for a PhD degree has been finished and a thesis, therefore, is produced. There is a need for a method to provide linear or nonlinear real-time power-take-off forces to the wave energy converting mechanism in the water during the experiment. More urgently, it is essential to overcome the discrepancy caused by following Froude-scaling law and Reynold-scaling law in the test of a model-scaled FOWT. Two applications for WECs and FOWTs are proposed separately, to meet the challenges.;Following the conceptual design of the software-in-the-loop (SIL) application for a WEC, an innovative generic platform, which can explicitly provide a real-time PTO damping force in terms of either linear or non-linear (at different scales) is developed and characterised by 1349 drop tests. Subsequent physical model tests of a OWSC WEC device are carried out. The power efficiency of the OWSC WEC device under different PTO strategies is then estimated based on the analysis of experimental results. The best linear damping in regular waves is driven by gaining 80 in the control function, while 160 for nonlinear PTO damping. Furthermore, it is revealed that nonlinear PTOs have no distinct advantage in the amount of electricity output, but can lead to better stability and broader damping range. Following the conceptual design of an AI-based hybrid testing application for a FOWT system, a prediction module of the rotor thrust is needed to be estimated and optimised in the first place. For this reason, a considerable amount of simulations under various conditions are carried out by fully-coupled computation software, and the results obtained are used to train an artificial intelligence structure. Then a prediction module which depends on five inputs, and gives one output rotor thrust, is estimated mathematically. The mathematical module is converted to the control function in the program in a controller to execute it in real-time tests. Therefore, the AI machine is sometimes referred to as the SIL application for FOWTs, which consists of a prediction module obtained by AI training, a controller, and the program in the controller. The AI machine is the key component to implement the AI-based real-time hybrid model (AIReaTHM) testing methodology.;As one of the highlights in the present study, the AIReaTHM testing rig is developed, and bench tests are carried out with a manoeuvrable motion simulator. The comprehensive testing results are analysed for three purposes: 1, validating the AIReaTHM testing methodology. 2, assessing the influences of wind speed, wind turbulence intensity, wave spectrum, input hydrodynamic motions on rotor thrust are reflected by the SIL application.3, evaluating the systematic uncertainty in the testing rig, which is to be compensated by further improving the testing system. The effect of the surge frequency, wave spectrum and wind models have on the targeted thrust is discussed. The time delay in the testing system is identified as within 0.1s, and the overall uncertainty from the testing rig is 5-15KN (the minimum rotor thrust is 508KN, hence the uncertainty is 0.98%-2.95% in percentage) when compared to the AI prediction.;The testing rig developed is further applied to a 1:73 model of a Hywind floating wind turbine. 4 testing campaigns are carried out, and 303 independent tests are conducted. Testing results with the real-time rotor thrust provided by the AI-based software-in-the-loop application are compared with the other three comparative testing patterns. They are tests with a constant rotor thrust, without any rotor thrust, with AI predicted rotor thrust but without wave inputs, and in only wave conditions respectively. The performance of the rotor thrust obtained by the AI prediction agrees well with the benchmark testing results. Then, the hydrodynamic responses of the model are compared among those four testing patterns, for both regular wave tests and irregular wave tests in terms of time histories, RAOs, statistical analysis, and spectral analysis. The RAOs of the model under three testing patterns are given for regular wave tests. The hydrodynamic response revealed that the AIReaTHM is better than applying a constant rotor thrust atop of the model, though further improvement is required to meet realistic response. In the final chapter, conclusions are drawn and original contribution of this PhD study is outlined. Besides, a few points concerning future work are addressed.Experimental research in laboratory is a necessary and useful method to explore the full potential of a device. Because it does not only require much less money than the prototype at sea test, it also provides more reliable results compared to numerical simulations. Hence, it is significantly vital to make accurate model tests of the concerned ocean renewable energy (ORE) devices possible. For this reason, this study for a PhD degree has been finished and a thesis, therefore, is produced. There is a need for a method to provide linear or nonlinear real-time power-take-off forces to the wave energy converting mechanism in the water during the experiment. More urgently, it is essential to overcome the discrepancy caused by following Froude-scaling law and Reynold-scaling law in the test of a model-scaled FOWT. Two applications for WECs and FOWTs are proposed separately, to meet the challenges.;Following the conceptual design of the software-in-the-loop (SIL) application for a WEC, an innovative generic platform, which can explicitly provide a real-time PTO damping force in terms of either linear or non-linear (at different scales) is developed and characterised by 1349 drop tests. Subsequent physical model tests of a OWSC WEC device are carried out. The power efficiency of the OWSC WEC device under different PTO strategies is then estimated based on the analysis of experimental results. The best linear damping in regular waves is driven by gaining 80 in the control function, while 160 for nonlinear PTO damping. Furthermore, it is revealed that nonlinear PTOs have no distinct advantage in the amount of electricity output, but can lead to better stability and broader damping range. Following the conceptual design of an AI-based hybrid testing application for a FOWT system, a prediction module of the rotor thrust is needed to be estimated and optimised in the first place. For this reason, a considerable amount of simulations under various conditions are carried out by fully-coupled computation software, and the results obtained are used to train an artificial intelligence structure. Then a prediction module which depends on five inputs, and gives one output rotor thrust, is estimated mathematically. The mathematical module is converted to the control function in the program in a controller to execute it in real-time tests. Therefore, the AI machine is sometimes referred to as the SIL application for FOWTs, which consists of a prediction module obtained by AI training, a controller, and the program in the controller. The AI machine is the key component to implement the AI-based real-time hybrid model (AIReaTHM) testing methodology.;As one of the highlights in the present study, the AIReaTHM testing rig is developed, and bench tests are carried out with a manoeuvrable motion simulator. The comprehensive testing results are analysed for three purposes: 1, validating the AIReaTHM testing methodology. 2, assessing the influences of wind speed, wind turbulence intensity, wave spectrum, input hydrodynamic motions on rotor thrust are reflected by the SIL application.3, evaluating the systematic uncertainty in the testing rig, which is to be compensated by further improving the testing system. The effect of the surge frequency, wave spectrum and wind models have on the targeted thrust is discussed. The time delay in the testing system is identified as within 0.1s, and the overall uncertainty from the testing rig is 5-15KN (the minimum rotor thrust is 508KN, hence the uncertainty is 0.98%-2.95% in percentage) when compared to the AI prediction.;The testing rig developed is further applied to a 1:73 model of a Hywind floating wind turbine. 4 testing campaigns are carried out, and 303 independent tests are conducted. Testing results with the real-time rotor thrust provided by the AI-based software-in-the-loop application are compared with the other three comparative testing patterns. They are tests with a constant rotor thrust, without any rotor thrust, with AI predicted rotor thrust but without wave inputs, and in only wave conditions respectively. The performance of the rotor thrust obtained by the AI prediction agrees well with the benchmark testing results. Then, the hydrodynamic responses of the model are compared among those four testing patterns, for both regular wave tests and irregular wave tests in terms of time histories, RAOs, statistical analysis, and spectral analysis. The RAOs of the model under three testing patterns are given for regular wave tests. The hydrodynamic response revealed that the AIReaTHM is better than applying a constant rotor thrust atop of the model, though further improvement is required to meet realistic response. In the final chapter, conclusions are drawn and original contribution of this PhD study is outlined. Besides, a few points concerning future work are addressed

New Challenges Arising in Engineering Problems with Fractional and Integer Order

Mathematical models have been frequently studied in recent decades, in order to obtain the deeper properties of real-world problems. In particular, if these problems, such as finance, soliton theory and health problems, as well as problems arising in applied science and so on, affect humans from all over the world, studying such problems is inevitable. In this sense, the first step in understanding such problems is the mathematical forms. This comes from modeling events observed in various fields of science, such as physics, chemistry, mechanics, electricity, biology, economy, mathematical applications, and control theory. Moreover, research done involving fractional ordinary or partial differential equations and other relevant topics relating to integer order have attracted the attention of experts from all over the world. Various methods have been presented and developed to solve such models numerically and analytically. Extracted results are generally in the form of numerical solutions, analytical solutions, approximate solutions and periodic properties. With the help of newly developed computational systems, experts have investigated and modeled such problems. Moreover, their graphical simulations have also been presented in the literature. Their graphical simulations, such as 2D, 3D and contour figures, have also been investigated to obtain more and deeper properties of the real world problem

Three Classes of Fractional Oscillators

This article addresses three classes of fractional oscillators named Class I, II and III. It is known that the solutions to fractional oscillators of Class I type are represented by the Mittag-Leffler functions. However, closed form solutions to fractional oscillators in Classes II and III are unknown. In this article, we present a theory of equivalent systems with respect to three classes of fractional oscillators. In methodology, we first transform fractional oscillators with constant coefficients to be linear 2-order oscillators with variable coefficients (variable mass and damping). Then, we derive the closed form solutions to three classes of fractional oscillators using elementary functions. The present theory of equivalent oscillators consists of the main highlights as follows. (1) Proposing three equivalent 2-order oscillation equations corresponding to three classes of fractional oscillators; (2) Presenting the closed form expressions of equivalent mass, equivalent damping, equivalent natural frequencies, equivalent damping ratio for each class of fractional oscillators; (3) Putting forward the closed form formulas of responses (free, impulse, unit step, frequency, sinusoidal) to each class of fractional oscillators; (4) Revealing the power laws of equivalent mass and equivalent damping for each class of fractional oscillators in terms of oscillation frequency; (5) Giving analytic expressions of the logarithmic decrements of three classes of fractional oscillators; (6) Representing the closed form representations of some of the generalized Mittag-Leffler functions with elementary functions. The present results suggest a novel theory of fractional oscillators. This may facilitate the application of the theory of fractional oscillators to practice