Читать книгу Mathematical Programming for Power Systems Operation - Alejandro Garcés Ruiz - Страница 41

Consider a small photovoltaic system formed by three solar panels A, B, and C, placed as depicted in Figure 2.6. Each solar system has a power electronic converter that requires to be connected to a common point E before transmitted to the final user in D. 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:

(2.35)

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:

(2.36)

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:

(2.37)

then,

(2.38)

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.