CHAPTER 2.8: CORONA

Example 2.8.1, Page number 227

In [1]:
import cmath

#Variable declaration
d = 30.0/10          #Diameter of conductor(cm)
delta = 0.95         #Air density factor
m = 0.95             #Irregularity factor
E = 230.0            #Line voltage(kV)
g_0 = 30.0/2**0.5    #Breakdown strength of air(kV/cm)

#Calculation
E_0 = E/3**0.5                             #Disruptive critical voltage(kV)
r = d/2.0                                  #Radius of conductor(cm)
D = cmath.exp(E_0/(m*delta*g_0*r))*r/100   #Minimum spacing between conductors(m)

#Result
print('Minimum spacing between conductors, D = %.3f m' %abs(D))
print('\nNOTE: Changes in obtained answer from that of textbook due to precision')

Minimum spacing between conductors, D = 1.529 m

NOTE: Changes in obtained answer from that of textbook due to precision

Example 2.8.2, Page number 227-228

In [1]:
import math

#Variable declaration
V = 220.0            #Operating line voltage(kV)
f = 50.0             #Frequency(Hz)
d = 1.5              #Diameter of conductor(cm)
D = 300.0            #Distance b/w conductor(cm)
delta = 1.05         #Air density factor
g_0 = 21.1           #Breakdown strength of air(kV/cm)
m = 1.0              #Irregularity factor

#Calculation
E = V/3**0.5                                           #Phase voltage(kV)
r = d/2.0                                              #Radius of conductor(cm)
E_0 = m*g_0*delta*r*math.log(D/r)                      #Disruptive critical voltage to neutral(kV/phase)
E_0_ll = 3**0.5*E_0                                    #Line-to-line Disruptive critical voltage(kV)
P = 244.0*10**-5*(f+25)/delta*(r/D)**0.5*(E-E_0)**2    #Corona loss(kW/km/phase)
P_total = P*3.0                                        #Corona loss(kW/km)

#Result
print('Critical disruptive voltage, E_0 = %.2f kV/phase = %.2f kV (line-to-line)' %(E_0,E_0_ll))
print('Corona loss, P = %.2f kW/km' %P_total)
print('\nNOTE: ERROR: Calculation mistake in the final answer in textbook')

Critical disruptive voltage, E_0 = 99.56 kV/phase = 172.44 kV (line-to-line)
Corona loss, P = 19.72 kW/km

NOTE: ERROR: Calculation mistake in the final answer in textbook

Example 2.8.3, Page number 228

In [1]:
import math

#Variable declaration
V = 132.0            #Operating line voltage(kV)
f = 50.0             #Frequency(Hz)
d = 1.17             #Diameter of conductor(cm)
D = 300.0            #Distance b/w conductor(cm)
m = 0.96             #Irregularity factor
b = 72.0             #Barometric pressure(cm)
t = 20.0             #Temperature(°C)

#Calculation
delta = 3.92*b/(273.0+t)                                         #Air density factor
r = d/2.0                                                        #Radius of conductor(cm)
E_0 = 21.1*m*delta*r*math.log(D/r)                               #Critical disruptive voltage for fair weather condition(kV/phase)
E_0_foul = 0.8*E_0                                               #Critical disruptive voltage for foul weather(kV/phase)
E = V/3**0.5                                                     #Phase voltage(kV)
P_fair = 244.0*10**-5*(f+25)/delta*(r/D)**0.5*(E-E_0)**2         #Corona loss for fair weather condition(kW/km/phase)
P_foul = 244.0*10**-5*(f+25)/delta*(r/D)**0.5*(E-E_0_foul)**2    #Corona loss for foul weather condition(kW/km/phase)

#Result
print('Corona loss in fair weather, P = %.3f kW/km/phase' %P_fair)
print('Corona loss in foul weather, P = %.3f kW/km/phase' %P_foul)

Corona loss in fair weather, P = 0.208 kW/km/phase
Corona loss in foul weather, P = 3.102 kW/km/phase

Example 2.8.4, Page number 228-229

In [1]:
import math

#Variable declaration
V = 110.0            #Operating line voltage(kV)
f = 50.0             #Frequency(Hz)
l = 175.0            #Line length(km)
d = 1.0              #Diameter of conductor(cm)
D = 300.0            #Distance b/w conductor(cm)
t = 26.0             #Temperature(°C)
b = 74.0             #Barometric pressure(cm)
m = 0.85             #Irregularity factor
m_v_local = 0.72     #Roughness factor for local corona
m_v_gen = 0.82       #Roughness factor for general corona

#Calculation
delta = 3.92*b/(273.0+t)                                                  #Air density factor
r = d/2.0                                                                 #Radius of conductor(cm)
E_0 = 21.1*m*delta*r*math.log(D/r)                                        #Critical disruptive voltage(kV) rms
E_v_local = 21.1*m_v_local*delta*r*(1+(0.3/(delta*r)**0.5))*math.log(D/r) #Critical disruptive voltage for local corona(kV) rms
E_v_gen = 21.1*m_v_gen*delta*r*(1+(0.3/(delta*r)**0.5))*math.log(D/r)     #Critical disruptive voltage for general corona(kV) rms
E = V/3**0.5                                                              #Phase voltage(kV)
#Case(i)
P_c_i = 244.0*10**-5*(f+25)/delta*(r/D)**0.5*(E-E_0)**2                   #Peek's formula for fair weather condition(kW/km/phase)
P_c_i_total = P_c_i*l*3                                                   #Total power loss(kW)
#Case(ii)
P_c_ii = 244.0*10**-5*(f+25)/delta*(r/D)**0.5*(E-0.8*E_0)**2              #Peek's formula for stormy condition(kW/km/phase)
P_c_ii_total = P_c_ii*l*3                                                 #Total power loss(kW)
#Case(iii)
F_iii = 0.0713                                                            #From text depending on E/E_0
P_c_iii = 21.0*10**-6*f*E**2*F_iii/(math.log10(D/r))**2                   #Peterson's formula for fair condition(kW/km/phase)
P_c_iii_total = P_c_iii*l*3                                               #Total power loss(kW)
#Case(iv)
F_iv = 0.3945                                                             #From text depending on E/E_0
P_c_iv = 21.0*10**-6*f*E**2*F_iv/(math.log10(D/r))**2                     #Peterson's formula for stormy condition(kW/km/phase)
P_c_iv_total = P_c_iv*l*3                                                 #Total power loss(kW)

#Result
print('Case(i)  : Power loss due to corona using Peek formula for fair weather condition, P_c  = %.3f kW/km/phase' %P_c_i)
print('           Total corona loss in fair weather condition using Peek formula = %.1f kW' %P_c_i_total)
print('Case(ii) : Power loss due to corona using Peek formula for stormy weather condition, P_c  = %.2f kW/km/phase' %P_c_ii)
print('           Total corona loss in stormy condition using Peek formula = %.f kW' %P_c_ii_total)
print('Case(iii): Power loss due to corona using Peterson formula for fair weather condition, P_c  = %.4f kW/km/phase' %P_c_iii)
print('           Total corona loss in fair condition using Peterson formula = %.2f kW' %P_c_iii_total)
print('Case(iii): Power loss due to corona using Peterson formula for fair weather condition, P_c  = %.4f kW/km/phase' %P_c_iv)
print('           Total corona loss in stormy condition using Peterson formula = %.1f kW' %P_c_iv_total)
print('\nNOTE: ERROR: Calculation mistake in the final answer in textbook')

Case(i)  : Power loss due to corona using Peek formula for fair weather condition, P_c  = 0.475 kW/km/phase
           Total corona loss in fair weather condition using Peek formula = 249.5 kW
Case(ii) : Power loss due to corona using Peek formula for stormy weather condition, P_c  = 2.78 kW/km/phase
           Total corona loss in stormy condition using Peek formula = 1457 kW
Case(iii): Power loss due to corona using Peterson formula for fair weather condition, P_c  = 0.0391 kW/km/phase
           Total corona loss in fair condition using Peterson formula = 20.54 kW
Case(iii): Power loss due to corona using Peterson formula for fair weather condition, P_c  = 0.2165 kW/km/phase
           Total corona loss in stormy condition using Peterson formula = 113.6 kW

NOTE: ERROR: Calculation mistake in the final answer in textbook

Example 2.8.5, Page number 229

In [1]:
import cmath

#Variable declaration
V = 132.0            #Operating line voltage(kV)
f = 50.0             #Frequency(Hz)
dia = 1.956          #Diameter of conductor(cm)
v_c = 210.0          #Disrputive voltage(kV) 
g_0 = 30.0/2**0.5    #Breakdown strength of air(kV/cm)

#Calculation
r = dia/2.0                              #Radius of conductor(cm)
V_c = v_c/3**0.5                         #Disrputive voltage/phase(kV)
m_0 = 1.0                                #Irregularity factor
delta = 1.0                              #Air density factor
d = cmath.exp(V_c/(m_0*delta*g_0*r))*r   #Spacing between conductors(cm)

#Result
print('Spacing between the conductors, d = %.f cm' %abs(d))
print('\nNOTE: Changes in the obtained answer from that of textbook is due to precision')

Spacing between the conductors, d = 338 cm

NOTE: Changes in the obtained answer from that of textbook is due to precision

Example 2.8.6, Page number 229

In [1]:
#Variable declaration
P_c1 = 53.0           #Total corona loss(kW)
V_1 = 106.0           #Operating line voltage(kV)
P_c2 = 98.0           #Total corona loss(kW)
V_2 = 110.9           #Operating line voltage(kV)
V_3 = 113.0           #Operating line voltage(kV)

#Calculation
E_1 = V_1/3**0.5                        #Phase voltage(kV)
E_2 = V_2/3**0.5                        #Phase voltage(kV)
P_ratio = (P_c2/P_c1)**0.5
E_0 = (P_ratio*E_1-E_2)/(P_ratio-1)     #Disruptive critical voltage(kV)
E_3 = V_3/3**0.5                        #Phase voltage(kV)
W = ((E_3-E_0)/(E_1-E_0))**2*P_c1       #Corona loss at 113 kV(kW)

#Result
print('Disruptive critical voltage, E_0 = %.f kV' %E_0)
print('Corona loss at 113 kV, W = %.f kW' %W)

Disruptive critical voltage, E_0 = 53 kV
Corona loss at 113 kV, W = 121 kW

Example 2.8.7, Page number 229-230

In [1]:
import math

#Variable declaration
d = 3.0             #Diameter of conductor(cm)
e_r = 4.0           #Relative permittivity
d_1 = 3.5           #Internal diameter of porcelain bushing(cm)
d_2 = 9.0           #External diameter of porcelain bushing(cm)
V = 25.0            #Voltage b/w conductor and clamp(kV)

#Calculation
r = d/2.0                                                     #Radius of conductor(cm)
r_1 = d_1/2.0                                                 #Internal radius of porcelain bushing(cm)
r_2 = d_2/2.0                                                 #External radius of porcelain bushing(cm)
g_2max = r/(e_r*r_1)                                          #Maximum gradient of inner side of porcelain
g_1max = V/(r*math.log(r_1/r)+g_2max*r_1*math.log(r_2/r_1))   #Maximum gradient on surface of conductor(kV/cm)

#Result
print('Maximum gradient on surface of conductor, g_1max = %.2f kV/cm' %g_1max)
print('Since, gradient exceeds 21.1 kV/cm, corona will be present')

Maximum gradient on surface of conductor, g_1max = 42.71 kV/cm
Since, gradient exceeds 21.1 kV/cm, corona will be present

Example 2.8.8, Page number 230

In [1]:
import math

#Variable declaration
d = 2.0             #Diameter of conductor(cm)
D = 150.0           #Spacing b/w conductor(cm)
delta = 1.0         #Air density factor

#Calculation
r = d/2.0                           #Radius of conductor(cm)
V_d = 21.1*delta*r*math.log(D/r)    #Disruptive critical voltage(kV/phase)
V_d_ll = 3**0.5*V_d                 #Line voltage for commencing of corona(kV)

#Result
print('Line voltage for commencing of corona = %.2f kV' %V_d_ll)
print('\nNOTE: Solution is incomplete in textbook')

Line voltage for commencing of corona = 183.12 kV

NOTE: Solution is incomplete in textbook