Wind Farm Optimization

This paper will formulate integer programs to determine the optimal positions of wind turbines within a wind farm. The formulations were based on variations of the vertex packing problem. Three formulations are presented, which seek to maximize power generated in accordance with constraints based on the number of turbines, turbine proximity, and turbine interference. These were in the form of budget, clique, and edge constraints. Results were promising, with turbines exhibiting a tendency to concentrate in areas of high elevation and avoid situations where downstream interference would be significant. 1. Development of the integer programs Three (mixed) integer programming models are presented. The first two integer program models are vertex packing problems, while the third MIP model is a Generalized Vertex Packing Problem (GVP) [1]. The GVP problem was introduced by Hanif D. Sherali and J. Cole Smith [2]. In these formulations, G = (V, E) denotes a graph with vertices V and edges E VxV. The set E is set of vertex pairs between which there exists some relationship. In our case, the vertices V correspond to the locations where turbines can be positioned, and the edges E represent relationships between the vertices, such as turbine proximity and interference. ⊆ An appreciation of the relationship between the physical domain and the graph on which the (mixed) integer programs are based is crucial to understanding the material that follows. The graph is based on an orthogonal grid that is superimposed onto the physical topography. The intersection points of this grid represent the vertices in our graph. The vertex packing problem will thus involve selecting the combination of vertices, or grid points, which generates the most power. 2. Modeling turbine proximity The first integer program formulation enforced a minimal separation distance between turbines to ensure the blades did not physically clash with one another. The term proximity shall define the area immediately surrounding a turbine in which no other turbine can be built. The grid points that lie within this area are a function of the turbine radius and the physical distance between the intersection points in our grid. Figure 2.1 demonstrates that a turbine centered on the solid vertex will eliminate the surrounding vertices as potential locations. That is, the vertices connected to the solid vertex by an edge are too close to accommodate another turbine. Those vertices that are not connected to the solid vertex do not impinge on the space required by this turbine. Figure 2.1: Vertex proximity constraint. To model this proximity requirement we construct the graph G by considering each vertex in turn, and placing an edge between this vertex i and any vertex j, where the position occupied by j violates the area required by a turbine positioned at i. For example, in Figure 2.1 an edge would exist between the solid vertex and every surrounding vertex, as shown by the lines. This constraint can be formulated mathematically as: E j i x x j i ∈ ∀ ≤ + ) , ( , 1 An edge constraint of this form will exist between every vertex in our graph and any other vertex that lies within the required separation distance. In the physical model, this corresponds to a pair of grid points existing too close for a turbine to be located at both positions. The above “weak edge” formulation can be improved by considering a larger subset of vertices affected by turbine proximity. The structure of the edges on G, as well as the relationship between four neighboring vertices Q, is shown in Figure 2.2.