The results of multiple optimizations are shown in Table 6.4, where the design space is gradually reduced. According to prior studies, positive yaw misalignment angles reduce the fatigue loads and increase the overall lifespan of the turbine blades opposed to negative yaw misalignment angles. As a result, the optimization study favors design spaces with more positive yaw misalignment angles. It is shown that the SR optimization is much faster than our gradient-based optimizer, despite having a good initialization. However, by using the gradient-based optimizer, a higher AEP is able to be achieved for approximately an order of magnitude increase in optimization time. In a previous study, it was shown that the optimization time of a gradient-based optimizer to same problem as problem 5.27 was approximately 100x slower than SR. However, the gradient-based optimizer used was SLSQP using finite difference to calculate derivatives, which is much less efficient than our model. The increase in AEP by using our gradient-based model is less significant for smaller design spaces, as the yaw bounds only achieves an increase of 0.145 GWh opposed to the yaw bounds increase of 1.360 GWh in the optimal result. We attribute this to the fact that the SR discretization of the design space is held constant, bud drying rack so as the design space reduces in size so too does the step size between yaw angle samples. With a smaller step size, SR is expected to find a result closer to the optimal result.
The optimization result for the yaw bounds is shown in Fig. 6.5.The Hollandse Kust site VI is an off-shore wind farm zone in the Dutch North Sea. Using a publicly released metocean survey, the HKW site’s windrose and weibull data are presented in Fig. B.14 and Table B.2, respectively. Note that Nwd = 12 and Nws = 6 are used in this optimization study. The HKW site is planned to be commissioned in 2026 with 54 wind turbines and a total capacity of 756 MW. As a result of the predicted size, we use the IEA37 15MW off-shore reference wind turbine to perform a layout optimization. The parameters for this turbine are in Table 5.1 and Cp and power curves in Fig. B.1. We note that in such a large optimization problem the optimization tolerance was almost never met. Therefore, the optimizations were terminated when the AEP did not increase by any significant margins O and the constraints were satisfied. The HKW wind farm boundary is defined by sets of disconnected and irregularly shaped polygons shown in Fig. B.15. The exclusion zones represent shipwrecks, unexploded ordinances, shipping lanes, steep seabed gradients, and other regions where the placement of a wind turbine is infeasible. To begin a layout optimization study on this site, we must represent these boundaries in a continuous and smooth way for gradient-based optimization. We apply the aforementioned method of geometric non-interference constraints given by Algorithm 1. The resultant values from solving the energy minimization problem are in Table 6.5, and a visualization of the zero contour is given in Fig. B.16.
Notably, the accuracy of the boundary constraint representation has an on-surface root-mean-square error of 6.29 meters, which is on the same order of magnitude of the tower’s diameter.In order to quantify the computational speedup of the new boundary constraint model, constraint evaluations were ran for the TOPFARM model and our new model. Running an optimization with the same initial starts would produce a different optimal layout because the constraint functions are inherently different, despite the wake models being identical. As a result, it is only appropriate to highlight the differences in a single model evaluation. With 54 wind turbines and NΓ = 80,710, a single model evaluation in TOPFARM takes 1.060 seconds, and our new model takes 0.034 seconds. The timing results are the average timings of 10 model evaluations. Given this result, the significant increase in model evaluation time makes it significantly more computationally expensive to solve a layout optimization problem with this boundary in TOPFARM. As the number of model evaluations required for optimization, typically hundreds to thousands, the additional computational expense of using TOPFARM becomes impractical. As a result, we strongly recommend using the new boundary constraint formulation for wind farm layout optimization. In order to optimize the wind farm, two measures were taken to avoid local minima. The first method is the implementation of the aforementioned WEC model to decrease thelocal minima in the flow fields. As done in the optimization study presented in Sec. 6.2, the optimization is completed with ζ = 4 first, then with ζ = 1.
The second method is the implementation of relaxation to the boundary constraints. The constraints are first enforced to be within 250 meters, then within 25 meters, then fully enforced to be within the wind farm boundaries. A visualization of the 250 meters boundary 25 meters boundary, and zero boundary are shown in Fig B.17. In total, the optimization problem is completed by solving four sequential optimizations described in Table 6.6. The application of these two methods represent the state-of-the-art methodologies to avoid local minima within gradient-based wind farm layout optimization, and are still subject to local minima in practice.We conduct a wind farm layout optimization using Nt = 54 for the HKW site. The initialization is shown in Fig. B.18, the result of the first optimization in Fig. B.19, the result of the second optimization in Fig. B.20, the result of the third optimization in Fig. B.21, and the result of the last optimization in Fig. B.22, with annual energy production and the median distance to the boundary shown for all optimization iterations. The optimal layout found from the four sequential optimizations is shown in Fig. 6.6. The error of the wind direction discretization with respect to the 1◦ discretization is 2.24%. The 360 wind direction bins were approximated using linear interpolation. Note that many wind turbines lie on the outer-most boundary, maximizing their distance from one another. Additionally, note that many wind turbines are on disconnected zones from one another. This result is difficult to reproduce in gradient-based optimization without a boundary constraint relaxation approach. The AEP was increased by 6.82% from a random initialization that violated the boundary constraint within 250 meters. No meaningful conclusions can be made about improvement of AEP because it was subject to a random initialization. Using a heuristic-based or gradient-free approach to achieve a better initial guess is of high priority for future work. Nonetheless, the result represents the capability of using gradient-based optimization on a wind farm layout optimization problem.In addition, we conduct an optimization study varying the number of wind turbines at the HKW site and tabulated the results in Table 6.7. For each data point, the sequential optimization was completed with their own random initializations. Due to the randomness of the initialization, vertical grow rack system no conclusive differences were drawn based on optimization time, as the quality of the initial start plays an important role to the number of model evaluations required to optimize. For reference, the scaling of one model evaluation and sensitivity analysis times are shown in Fig. 6.1. Based on the results of this study, is unclear that the addition of more wind turbines than the baseline Nt = 54 will harm the efficiency of the overall wind farm. As shown in Table 6.7, the amount of AEP produced on a per turbine basis remains steady with the largest difference between 46 and 70 wind turbines. The incorrect conclusion drawn from this data is that increasing the number of wind turbines will increase the AEP of the wind farm without significant loss in efficiency due to wakes. In reality, this result represents the limitation of the models in our implementation in capturing diminishing returns. A better model would consider levelized cost of energy , increased fatigue loading based on wake interactions on the turbine blades, and other models to capture to the lifecycle of the wind farm. This should be addressed in future work.Two topics were explored in this thesis.
The first topic was concerned with developing a new scalable geometric non-interference constraint formulation. In Sec. 1.1, we consolidated the terminology used in prior literature and call this category of constraints ‘geometric non-interference constraints’. Additionally, we framed the set of optimization problems with geometric non-interference constraints into three groups: layout optimization, shape optimization, and optimal path planning problems. Section 2.1 reviewed the existing geometric non-interference constraint formulations in gradient-based optimization and contextualized our formulation within the field of surface reconstruction. In Chapter 3, we drew upon ideas from surface reconstruction techniques to construct our constraint formulation. Our formulation is based on an approximation of the signed distance function generated by solving an energy minimization problem for the values of the B-spline control points. Chapter 4 presented accuracy and scaling studies with our formulation. The second topic was concerned with investigating gradient-based wind farm optimization problems. In Sec. 1.2, we introduced the scope of optimization and its importance to wind farm design in combating climate change. Section 2.2 reviewed the literature that conducted studies on wind farm optimization. A clear gap in these studies was identified within gradient-based optimization, and particularly, the continuous and smooth representation of wind farm boundary constraints. In Chapter 5, we presented the models used to represent the turbines, wakes, and optimization model to conduct the optimization studies. Chapter 6 presented the results of scaling and optimization studies, including hub heights, yaw misalignment, and a layout optimization problem.The first topic of this thesis contributes a new formulation for representing geometric non-interference constraints in gradient-based optimization. This formulation involves a scalable, smooth, and fast-to-evaluate constraint function that approximates the local signed distance to a geometric shape. The use of B-spline functions is key to our formulation being scalable, smooth, and fast-to-evaluate. We showed that our formulation achieves a level of accuracy on the same order of magnitude as surface reconstruction methods used in computer graphics. Additionally, our formulation scales better in accuracy, up to a certain limit, and computational time with respect to the number of points sampled on the geometric shape NΓ compared to previous non-interference constraint formulations used by the optimization community. Our resultant computational speed is on the order of 10−6 seconds per point as measured on a modern desktop workstation, entirely independent of the number of sample points NΓ. The method results in a 78% and 56% speedup in optimization time for a path planning and design subproblem, respectively, for an existing concentric tube robot gradient-based design optimization problem. The second topic of this thesis investigates the use of gradient-based optimization for wind farm layout design. We investigated three subproblems to wind farm optimization problems including turbine hub heights, yaw misalignment control, and wind farm layout optimization. In the optimization of hub heights, we identified consistent patterns as previous works, however were subject to local minima. In the optimization of yawmisalignment angles, an increase of 1.360 GWh AEP was achieved for only an order a magnitude increase in optimization time, compared to a gradient-free optimizer. It was shown that the new geometric non-interference constraint formulation provides an efficient way to enforce complex wind farm boundary constraints. Model evaluations with the new constraint function were 30x faster than previously in an industry leading optimization framework. In addition, the constraint formulation is well-suited for relaxation approaches, as it provides an approximate distance to the boundary. We investigated the vulnerabilities of our model including the lack of diminishing returns when adding more wind turbines, local minima in the results, and inaccuracies in the annual energy production calculation due to the wind direction discretization.We identify multiple directions for future work for the first topic. Adaptive octrees with B-splines can represent small-scale features such as edges and sharp corners more accurately. Using octrees for discretization instead of using a uniform grid can clearly yield faster and more accurate solutions in problems where any of the modeled geometries remain constant during optimization iterations, e.g., the CTR or wind farm layout optimization problems. However, it is worth restating that when geometries evolve during optimization, rediscretizing surfaces using octrees in each optimization iteration becomes unreasonably expensive, and we only recommend a uniform discretization in such cases.