The converters and the user’s location are fixed, but the common point E can be moved at will. The coordinates of the solar panels and the final user areA = (0, 40), B = (20, 70), C = (30, 0), and D = (100, 50) respectively.
Figure 2.6 A small photovoltaic system with three solar panels.
The cost of the cables is different since each cable carries different current. Our objective is to find the best position of E in order to minimize the total cost of the cable. Therefore, the following unconstrained optimization problem is formulated:
where costij¯ is the unitary cost of the cable that connects the point i and j, and ij¯ is the corresponding length.
The costs of the cables are costAE¯ = 12, costBE¯ = 13, costCE¯=11 and costDE¯=18 The distance between any two points U = (u0, u1) and V = (v0, v1) is given by the following expression:
This equation is required several times; thus, it is useful to define a function, as presented below:
import numpy as np A = (0,40) B = (20,70) C = (30,0) D = (100,50) def dist(U,V): return np.sqrt((U[0]-V[0])**2 + (U[1]-V[1])**2) P = [31,45] f = 12*dist(P,A) + 13*dist(P,B) + 11*dist(P,C) + 18*dist(P,D) print(f)
The function is evaluated in a pointP = (10, 10) to see its usage2. The value of the objective function is easily calculated as function of dist(U, V). Likewise, the gradient of f is defined as function of the gradient of dist(U, V) with V fixed, as presented below:
then,
These functions are easily defined in Python as follows:
def g_d(U,V): "gradient of the distance" return [U[0]-V[0],U[1]-V[1]]/dist(U,V) def grad_f(E): "gradient of the objective function" return 12*g_d(E,A)+13*g_d(E,B)+11*g_d(E,C)+18*g_d(E,D)
Now the gradient method consists in applying the iteration given by (Equation 2.30), as presented below:
t = 0.5
E = np.array([10,10])
for iter in range(50): E = E -t*grad_f(E) f = 12*dist(E,A) + 13*dist(E,B) + 11*dist(E,C) + 18*dist(E,D) print("Position:",E) print("Gradient",grad_f(E)) print("Cost",f)
In this case, t = 0.5 and a initial point E = (10, 10) with 50 iterations were enough to find the solution. The reader is invited to try with other values and analyze the effect on the algorithm’s convergence.
The step t is very important for the convergence of the gradient method. It can be constant or variable, according to a well-defined update rule. There are many variants of this algorithm, most of them with sophisticated ways to calculate this step3. A plot of ‖∇f‖ versus the number of iterations may be useful for determining the optimal value of t and showing the convergence rate of the algorithm, as presented in the next example. We expect a linear convergence for the gradient method, although the algorithm can lead to oscillations and even divergence if the parameter t is not selected carefully. Fortunately, there are modules in Python that make this work automatically.
Example 2.7
The convergence of the algorithm can be visualized by using the module MatplotLib as follows:
import matplotlib.pyplot as plt t = 0.5 conv = [] E = [10,10] for iter in range(50): E += - t*grad_f(E) conv += [np.linalg.norm(grad_f(E))] plt.semilogy(conv) plt.grid() plt.xlabel("Iteration") plt.ylabel("|Gradient|") plt.show()
The result of the script is shown in Figure 2.7. As expected, the convergence rate is linear; that is to say, the convergence plot describes almost a line in a semi-logarithmic scale. The value of ε can be used as convergence criteria (a gradient ‖∇f‖ ≤ 10−4 can be considered as the local optimum for this problem).
Figure 2.7 Convergence of the gradient method.
Notice that addition was simplified by the statement +=. In general, an statement such as x=x+1 is equivalent to x+=1. More details about this aspect are presented in Appendix C.
2.6 Lagrange multipliers
Reality imposes physical constraints into the problems and these constraints must be considered into the model. For example, an optimization
