forked from byuflowlab/Jensen3D
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Jensen.py
204 lines (176 loc) · 8.58 KB
/
Jensen.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
import numpy as np
import scipy as sp
import matplotlib.pyplot as plt
from math import pi
from mpl_toolkits.mplot3d import Axes3D
import mpl_toolkits.mplot3d.art3d as art3d
from matplotlib.patches import Circle
def Jensen_Wake_Model(x):
nTurbs = (len(x)-2)/2
xHAWT = x[0:nTurbs]
yHAWT = x[nTurbs:nTurbs*2]
zTall = x[len(x)-2]
zShort = x[len(x)-1]
nTall = nTurbs/2
zHAWT = np.zeros(nTurbs)
zHAWT[0:nTall] = zTall
zHAWT[nTall:nTurbs] = zShort
theta = 0.1
alpha = sp.tan(theta)
rho = 1.1716
a = 1. / 3.
Cp = 4.*a*(1-a)**2.
# nTurbines = len(xin)/2.
r_0 = np.ones(nTurbs)*63.2
U_velocity = 8.
U_direction = pi/3.5
"Make the graphic for the turbines and wakes"
# jensen_plot(x, y, r_0, alpha, U_direction_radians)
"Calculate power from each turbine"
return jensen_power(xHAWT, yHAWT, zHAWT, r_0, alpha, a, U_velocity, rho, Cp, U_direction)
# plt.show()
#Determine how much of the turbine is in the wake of the other turbines
def overlap(x, xdown, y, ydown, z, zdown, r, rdown, alpha):
overlap_fraction = np.zeros(np.size(x))
#print "x: ", x
#print "y: ", y
#print "z: ", z
for i in range(0, np.size(x)):
#define dx as the upstream x coordinate - the downstream x coordinate then rotate according to wind direction
dx = xdown - x[i]
#define dy as the upstream y coordinate - the downstream y coordinate then rotate according to wind direction
dy = abs(ydown - y[i])
dz = abs(zdown - z[i])
d = sp.sqrt(dy**2.+dz**2.)
R = r[i]+dx*alpha #The radius of the wake depending how far it is from the turbine
A = rdown**2*pi #The area of the turbine
if dx > 0:
#if d <= R-rdown:
# overlap_fraction[i] = 1 #if the turbine is completely in the wake, overlap is 1, or 100%
if d == 0:
#print "Area of turbine: ", A
#print "Area of wake: ", pi*R**2
if A <= pi*R**2:
overlap_fraction[i] = 1.
else:
overlap_fraction[i] = pi*R**2/A
elif d >= R+rdown:
overlap_fraction[i] = 0 #if none of it touches the wake, the overlap is 0
else:
#if part is in and part is out of the wake, the overlap fraction is defied by the overlap area/rotor area
overlap_area = rdown**2.*sp.arccos((d**2.+rdown**2.-R**2.)/(2.0*d*rdown))+R**2.*sp.arccos((d**2.+R**2.-rdown**2.)/(2.0*d*R))-0.5*sp.sqrt((-d+rdown+R)*(d+rdown-R)*(d-rdown+R)*(d+rdown+R))
overlap_fraction[i] = overlap_area/A
else:
overlap_fraction[i] = 0 #turbines cannot be affected by any wakes that start downstream from them
# print overlap_fraction
return overlap_fraction #retrun the n x n matrix of how each turbine is affected by all of the others
#for example [0, 0.5]
#[0, 0] means that the first turbine (row one) has half of its area in the
#wake of the second turbine (row two). The overlap_fraction on the second
#turbine is zero, so we can conclude that it is upstream of the first
#Jensen wake decay to determine the total velocity deficit at each turbine
def loss(r_0, a, alpha, x_focus, x, y_focus, y, overlap):
loss = np.zeros(np.size(x))
loss_squared = np.zeros(np.size(x))
dx = np.zeros(len(x))
dy = np.zeros(len(y))
for i in range(0, np.size(x)):
dx = x_focus-x[i]
dy = abs(y_focus-y[i])
R = r_0[i]+dx*alpha
if dx > 0:
loss[i] = overlap[i]*2.*a*(r_0[i]/(r_0[i]+alpha*(dx)))**2*0.5*(np.cos(-dy*pi/(R+4*r_0[i])))
loss_squared[i] = loss[i]**2
else:
loss[i] = 0
loss_squared[i] = 0
total_loss = sp.sqrt(np.sum(loss_squared))
return total_loss
def jensen_power(x, y, z, r_0, alpha, a, U_velocity, rho, Cp, U_direction):
"Effective velocity at each turbine"
x_r, y_r = rotate(x, y, U_direction)
"""print "x: ", x
print "y: ", y
print "x_r: ", x_r
print "y_r: ", y_r"""
V = np.zeros([np.size(x)])
total_loss = np.zeros([np.size(x)])
P = np.zeros([np.size(x)])
for i in range(0, np.size(x)):
A = r_0[i]**2*pi
overlap_fraction = overlap(x_r, x_r[i], y_r, y_r[i], z, z[i], r_0, r_0[i], alpha)
total_loss[i] = loss(r_0, a, alpha, x_r[i], x_r, y_r[i], y_r, overlap_fraction)
V = (1-total_loss[i])*U_velocity
P[i] = 0.5*rho*A*Cp*V**3
"Calculate Power from each turbine and the total"
P_total = np.sum(P)
return P_total
def jensen_plot(x, y, r_0, alpha, U_direction_radians):
#plt.plot([x], [y], 'ro', markersize=10)
wakes = np.linspace(0, 1000, num=101)
for i in range(0, np.size(y)):
turbine_y_top = y[i]+r_0
turbine_y_bottom = y[i]-r_0
turbine_x = [x[i], x[i], x[i]]
turbine_y = [turbine_y_bottom, y[i], turbine_y_top]
plt.plot(turbine_x, turbine_y, linewidth=2, c='r')
for j in range(1, np.size(wakes)):
wake_x = x[i]+wakes[j]
wake_top_y = y[i]+r_0+wakes[j]*alpha
wake_bottom_y = y[i]-r_0-wakes[j]*alpha
plt.plot(wake_x, wake_top_y, 'b.', markersize=2)
plt.plot(wake_x, wake_bottom_y, 'b.', markersize=2)
def rotate(x, y, U_direction_radians):
x_r = x*np.cos(U_direction_radians)-y*np.sin(U_direction_radians)
y_r = x*np.sin(U_direction_radians)+y*np.cos(U_direction_radians)
return x_r, y_r
if __name__ == '__main__':
"Define Variables"
x = np.array([0, 0, 0, 500, 500, 500, 1000, 1000, 1000]) #x coordinates of the turbines
y = np.array([0, 500, 1000, 0, 500, 1000, 0, 500, 1000]) #y coordinates of the turbines
# z = np.array([150, 150, 150, 250, 250, 250, 350, 500, 350]) #hub height of each turbine
# r_0 = np.array([40, 40, 40, 50, 50, 50, 60, 75, 60])
# r_0 = np.ones(len(x))*20
zTall = 150
zShort = 50
xin = np.hstack([x,y,zTall,zShort])
"0 degrees is coming from due North. +90 degrees means the wind is coming from due East, -90 from due West"
U_direction = -90.
U_direction_radians = (U_direction+90) * pi / 180.
#print U_direction_radians
# x_r, y_r = rotate(x, y, U_direction_radians)
# Jensen_Wake_Model(overlap, loss, jensen_power, jensen_plot, x, y, r_0, alpha, U_direction_radians)
# xin = np.hstack([x, y])
print Jensen_Wake_Model(xin)
ax = Axes3D(plt.gcf())
"""for i in range(len(x)):
ax.scatter(x[i], y[i], z[i], c = 'r', s=pi*r_0[i]**2, marker='.')
fillstyles = ('none')
#plt.axis(U_direction_radians)
xtemp = (x[i], x[i])
ytemp = (y[i], y[i])
ztemp = (0, z[i]-r_0[i])
ax.plot(xtemp, ytemp, ztemp, zdir='z', c='b', linewidth = 5.0)
ax.set_xlim([0, np.max(x)])
ax.set_ylim([0, np.max(y)])
ax.set_zlim([0, np.max(z)])
plt.show()"""
# wakes = np.linspace(0, 1000, num=101)
#
# plt.figure(2)
# for i in range(0, np.size(y)):
# turbine_x_top = ((wakes[0])*sp.cos(-U_direction_radians)-(r_0+alpha*wakes[0])*sp.sin(-U_direction_radians))+x[i]
# turbine_y_top = ((wakes[0])*sp.sin(-U_direction_radians)+(r_0+alpha*wakes[0])*sp.cos(-U_direction_radians))+y[i]
# turbine_x_bottom = ((wakes[0])*sp.cos(-U_direction_radians)-(-r_0-alpha*wakes[0])*sp.sin(-U_direction_radians))+x[i]
# turbine_y_bottom = ((wakes[0])*sp.sin(-U_direction_radians)+(-r_0-alpha*wakes[0])*sp.cos(-U_direction_radians))+y[i]
# turbine_x = [turbine_x_bottom, x[i], turbine_x_top]
# turbine_y = [turbine_y_bottom, y[i], turbine_y_top]
# plt.plot(turbine_x, turbine_y, linewidth=2, c='r')
# for j in range(1, np.size(wakes)):
# top_x = ((wakes[j])*sp.cos(-U_direction_radians)-(r_0+alpha*wakes[j])*sp.sin(-U_direction_radians))+x[i]
# top_y = ((wakes[j])*sp.sin(-U_direction_radians)+(r_0+alpha*wakes[j])*sp.cos(-U_direction_radians))+y[i]
# bottom_x = ((wakes[j])*sp.cos(-U_direction_radians)-(-r_0-alpha*wakes[j])*sp.sin(-U_direction_radians))+x[i]
# bottom_y = ((wakes[j])*sp.sin(-U_direction_radians)+(-r_0-alpha*wakes[j])*sp.cos(-U_direction_radians))+y[i]
# plt.plot([top_x], [top_y], 'b.', markersize=2)
# plt.plot([bottom_x], [bottom_y], 'b.', markersize=2)
# plt.show()