Many papers have tried to address this issue in their own unique ways. A common method seen is a polygonal representation of the boundary. With these polygons, Guirguis et al. uses the nearest edge to define an equivalent circular constraint. Risco et al. uses the nearest edge of the polygon to define the signed distance function. Note that both of these method’s use of the nearest edge in their formulation, which means that the constraint function is non-differentiable when there are two edges of equal distance away from the wind turbine. Additionally, there is a trade-off between computational cost and accuracy in the constraint function with respect to the number of edges in the polygon. The unique approach by Reddy poses an interpolation problem using a support vector domain description to describe the boundaries in an optimal way. This approach is continuous and differentiable, but the report does not provide any accuracy and scaling studies, so it is unknown how it may perform with increasingly complex boundaries.To the author’s knowledge, no other wind farm boundary constraint has taken the unique implicit surface reconstruction perspective as was described in Section 2.1. Implicit surface reconstruction constructs a level set function that represents the wind farm boundary by its zero level set. The inputs to most surface reconstruction methods is a point cloud, which is easily obtainable from the polygonal representations of previous studies. Note that Reddy’s SVDD method is very similar to the interpolation formulations in surface reconstruction. In this way, SVDD may suffer from the aforementioned limitations, including blobby reconstructions and significant computational expense for large scale SVDDs . The SDF is useful when the feasible zones are disconnected.
In gradient-based optimization, it is not possible for wind turbines to move across disconnected domains in a continuous way, however, drain trays for plants a relaxation approach to the boundary constraints can allow for wind turbines to move freely across regions until the true constraint is represented. The relaxation approach is shown to work well in avoiding turbines getting stuck within disconnected domains. With the SDF approximation described in Chapter 3, the relaxation approach is easily repeatable using the surface reconstruction method. We identify three of these frameworks relevant to the work within this thesis: PyWake, TOPFARM, and FLORIS. PyWake is an open-source Python library that contains many engineering wake models to calculate the annual energy production of wind farms. TOPFARM is an Open MDAO-based Python library that performs gradient-based optimization on wind farms using PyWake as its backend for modeling turbine wakes and calculating annual energy production. FLORIS is an open-source Python library that is controls-focused framework that models the wakes of yawed turbines and supports gradient free and gradient-based optimization. Gradient-based optimization within FLORIS is at alimited capacity, as it only supports automatic differentiation on some models. Otherwise, the finite difference method is used for derivative calculation. The finite difference method is less accurate and requires many more model evaluations. This section, in part, is currently being prepared for submission for publication of the material. The authors of this work are Anugrah Jo Joshy, Ryan C. Dunn, and John T. Hwang. The thesis author was a contributor to this material.We now apply our formulation to a number of geometric shapes involved in novel aircraft design. Aircraft design optimization is a long standing problem and has been the subject of recent interest in problems involving geometric non-interference constraints, e.g., the layout optimization of air cargo and aerodynamic shape optimization.
To enable gradient-based design optimization involving these constraints, a new generic method is required to represent numerous components within an aircraft’s design. We recognize the potential for our formulation and demonstrate its capabilities by conducting an experiment. In this experiment, we apply our formulation and quantify the resultant errors of five geometric shapes commonly associated with aircraft design. The geometries we model include a fuselage and a wing from a novel electric vertical take-off and landing concept vehicle, a human avatar, a luggage case, and a rectangular prism representing a battery pack within the wing. A visualization of these components in a feasible design configuration is illustrated in Fig. 4.7. Table 4.3 tabulates the on-surface error of the energy minimized LSF for each geometry. We observe that the smallest relative on-surface error is of the smooth fuselage shape, while the largest relative error is of the human avatar. We note from this example that geometries with features reasonably proportioned to their minimum bounding box diagonal are easier to represent using our method, hence our formulation poorly represents small scale features of the human avatar while it can represent the smooth fuselage very well. We observe that the bounding boxes of the fuselage, wing, and battery pack are poorly proportioned, yet they do not result in an increase of relative error compared to other geometries. However, their longer minimum bounding box diagonals will result in larger absolute errors.We now apply our method for enforcing geometric non-interference constraints to a medical robot design problem involving concentric tube robots . CTRs are composed of two or more long and slender pre-curved tubes made of super elastic materials. They can be designed to reach points in a large region of interest by rotating and translating the tubes relative to each other at their bases.
These characteristics make them ideal for minimally invasive surgeries where a surgeon can operate on a small region of interest with high dexterity through actuation at the base. In the foundational works of Sears and Dupont and Webster et al., expressions for the shape and tip position of the CTR are derived with respect to the robot’s geometric and control variables. Bergeles et al.use these expressions to perform gradient-free optimization of the CTR’s geometric and control variables with anatomical constraints. These anatomical constraints, i.e., geometric non-interference constraints, enforce that the CTR does not interfere with the anatomy during operation. Recent work by Lin et al. shows that gradient based optimization enables an efficient and scalable solution to simultaneously optimize the large set of the tube’s geometric and control variables while enforcing anatomical constraints. The experiment we now present follows the workflow of Lin et al., however, using our new formulation for representing the anatomical constraint function. The presented workflow involves the solution of multiple optimization problems, including an initial path planning problem, and the geometric design and control of the CTR . The path planning problem solves for a parametric 3D curve that represents an optimal collision-free path to the surgical site within the anatomy. Then, points along this path serve as inputs to the geometric design and control optimization of the CTR, which involve a kinematic model of the robot. In both subproblems, the non-interference constraints are enforced by evaluating a discrete set of points along the path or physical CTR to ensure that no points lie outside of the anatomy. We begin our experiment with an investigation in the heart anatomy which represents the non-interference constraint of the problem. The initial oriented point cloud of the heart is obtained from segmentation and 3D reconstruction by magnetic resonance imaging scans. Due to the limited machine accuracy, error introduced by aligning multiple scans, and normal approximation, the oriented point cloud is noisy, nonuniform, and contains poorly oriented normals. We perform a simple and necessary smoothing step on this point cloud as illustrated in Fig. 4.8. Although less precise at capturing small scale features, 4 x 8 grow tray the smoothing step assists our method in reconstructing a smooth zero contour for constraint representation.The smooth representation has relative errors 3.1×10−3 and 1.9×10−2 compared to the original noisy representation. The error in our energy minimized function obtained from the smoothed heart model is tabulated in Table 4.4. We observe that the on-surface RMS and max error of our representation is an order of magnitude less than the error introduced by the smoothing step. This implies that our representation of the smooth model is no worse than the smoothing step itself. We see that our method generates a function with a reliable zero level set of the smooth heart geometry, with an on-surface RMS error of 2.1×10−4 .
This error is lower compared to all the other examples in Table 4.2, and we attribute this to the smoothness of the heart geometry. We also note that the max on- and off-surface absolute errors of our representation are of the same order as the diameter of the CTR itself, typically 0.5-2.0 mm.Models of the wind turbine are necessary for characterizing the flow field in its wakes and for the power production of an individual turbine. For the analytical wake models and the calculation of annual energy production, we solely need a wind turbine to provide coefficient of thrust and power production . For simplicity and computational efficiency, the wind turbine model is simplified to a surrogate model where CT and P are simply functions of the wind speed experienced at the rotor. We note that any increase to the fidelity of the model will improve the power and CT accuracy, however the analytical wake models used in this study do not consider varying thrust/power at different radial sections of the blades. We do recommend a more accurate turbine model in the future work that will account for structural loading and fatigue on the blade structures. The two wind turbines considered in this study are the National Renewable Energy Laboratory 5MW open-source turbine model, and the International Energy Agency Wind Task 37 15MW turbine model. These wind turbines were selected for their size and viability in off-shore wind farms on small and large scale. The power and coefficient of thrust curves for the NREL 5MW and IEA37 15MW reference wind turbines are shown in Fig. 5.1 and Fig. B.1, respectively. The parameters of each wind turbine are also tabulated in Table 5.1. Note the significant diameter size.While this thesis does not provide validation with ground-truth, verification is provided to the existing models within PyWake and FLORIS. The major assumptions with this model depends on the accuracy of models individually. Validation studies are important and would be a valuable future work to this thesis. A study on flexible and stiff blades found that low fidelity models have significant discrepancies at low wind speeds and small inter-turbine spacing. Additionally, we rely heavily on the assumption an accurate thrust coefficient curve for the given wind turbine. An example to verify the Bastankhah wake model in a simple 3 turbine row is done in Fig. B.3. The error in the AEP calculated from our model on the order of machine precision, confirming the model’s accuracy. Derivative calculation is automatically done in CSDL, but for additional validation it is verified with finite differencing during optimization, and is confirmed to be on the order of the step length of the finite difference O. A verification of the Gaussian wake model in a simple 3 turbine row is done in Fig. B.4. The net AEP calculated from our model has a normalized error on the order of machine precision, confirming the model’s accuracy. Derivative calculation is automatically done in CSDL, but for additional validation it is confirmed to be on the order of the step length of the finite difference O. This chapter, in part, is currently being prepared for submission for publication of the material. The authors of this work are Anugrah Jo Joshy, Ryan C. Dunn, and John T. Hwang. The thesis author was a contributor to this material.In this optimization study, we perform a yaw misalignment optimization with the Lillgrund site. We utilize FLORIS’s built in boolean optimization tool called Serial Refine in order to initialize our optimization. Serial Refine reduces the design space by considering wind turbines in order of upwind to downwind, and recursively discretizes the design space and checks the model evaluations for the maximum value. In this way, it is a gradient-free optimizer, and it has shown to be very fast at finding a near-optimal solution and not susceptible to local minima. Because SR is fast and robust to local minima, we may apply it to get a good initial guess for our optimization. Our new model can take this suboptimal initial guess and improve it further using gradient-based optimization by considering yaw as a fully continuous variable. The yaw optimization problem is shown in problem 5.27, and the Gaussian wake model was used with the Crespo and Hern´andez turbulence intensity model. Using the Lillgrund site, the optimization problem has a total of 3,456 design variables.