Читать книгу Mathematical Programming for Power Systems Operation - Alejandro Garcés Ruiz - Страница 48

1 Find the highest and lowest point, of the set given by the intersection of the cylinder x2 + y2 ≤ 1 with the plane x + y + z = 1, as shown in Figure 2.8.Figure 2.8 Intersection of an affine space with a cylinder.

2 What is the new value of zmax and zmin, if the cylinder increases its radius in a small value, that is, if the radius changes from (r = 1) to (r = 1 + Δr) (Consider the interpretation of the Lagrange multipliers).

3 The following algebraic equation gives the mechanical power in a wind turbine: (2.59)where P is the power extracted from the wind; ρ is the air density; Cp is the performance coefficient or power coefficient; λ is the tip speed ratio; v is the wind velocity, and A is the area covered by the rotor (see [15] for details). Determine the value of λ that produce maximum efficiency if the performance coefficient is given by (Equation 2.60): (2.60)Use the gradient method, starting from λ = 10 and a step of t = 0.1. Hint: use the module SymPy to obtain the expression of the gradient.

4 Solve the following optimization problem using the gradient method: (2.61)Depart from the point (0, 0) and use a fixed step t = 0.8. Repeat the problem with a fixed step t = 1.1. Show a plot of convergence.

5 Solve the following optimization problem using the gradient method. (2.62)where 1n is a column vector of size n, with all entries equal to 1; b is a column vector such that bk = kn2; and H is a symmetric matrix of size n × n constructed in the following way: hkm = (m + k) / 2 if k ≠ m and hkm = n2 + n if k = m. Show the convergence of the method for different steps t and starting from an initial point x = 0. Use n = 10, n = 100, and n = 1000. All index k or m starts in zero.

6 Show that Euclidean, Manhattan, and uniform norms fulfill the four conditions to be considered a norm.

7 Consider a modified version of Example 2.6, where the position of the common point E must be such that xE = yE. Solve this optimization problem using Newton’s method.

8 Solve the problem of Item 4 with the following constraint (use Newton’s method): (2.63)

9 Solve problem of Item 5 including the following constraint (use Newton’s method): (2.64)

10 Newton’s method can be used to solve unconstrained optimization problems. Solve the following problem using Newton’s method and compare the convergence rate and the solution with the gradient method. (2.65)