# CHAPTER 2.3: STEADY STATE CHARACTERISTICS AND PERFORMANCE OF TRANSMISSION LINES¶

## Example 2.3.1, Page number 127-128¶

In :
#Variable declaration
P = 2.0*10**6           #Power delivered(W)
V_r = 33.0*10**3        #Receiving end voltage(V)
PF_r = 0.8              #Receiving end lagging power factor
R = 10.0                #Total resistance of the line(ohm)
X = 18.0                #Total inductive resistance of the line(ohm)

#Calculation
#Case(i)
I = P/(V_r*PF_r)                     #Line current(A)
sin_phi_r = (1-PF_r**2)**0.5         #Sinφ_R
V_s = V_r+I*R*PF_r+I*X*sin_phi_r     #Sending end voltage(V)
reg = (V_s-V_r)/V_r*100              #Voltage regulation(%)
#Case(ii)
PF_s  = (V_r*PF_r+I*R)/V_s           #Sending end lagging power factor
#Case(iii)
loss = I**2*R                        #Losses(W)
P_s = P+loss                         #Sending end power(W)
n = P/P_s*100                        #Transmission efficiency(%)

#Result
print('Case(i)  : Percentage voltage regulation = %.3f percent' %reg)
print('Case(ii) : Sending end power factor = %.2f (lag)' %PF_s)
print('Case(iii): Transmission efficiency, η = %.2f percent' %n)
print('\nNOTE: ERROR: pf is 0.8 and not 0.9 as mentioned in the textbook problem statement')

Case(i)  : Percentage voltage regulation = 4.316 percent
Case(ii) : Sending end power factor = 0.79 (lag)
Case(iii): Transmission efficiency, η = 97.21 percent

NOTE: ERROR: pf is 0.8 and not 0.9 as mentioned in the textbook problem statement


## Example 2.3.2, Page number 128-129¶

In :
#Variable declaration
l = 10.0                 #Length(km)
V_s = 11.0*10**3         #Sending end voltage(V)
P = 1000.0*10**3         #Load delivered at receiving end(W)
PF_r = 0.8               #Receiving end lagging power factor
r = 0.5                  #Resistance of each conductor(ohm/km)
x = 0.56                 #Reactance of each conductor(ohm/km)

#Calculation
#Case(a)
R = r*l                             #Resistance per phase(ohm)
X = x*l                             #Reactance per phase(ohm)
E_s = V_s/3**0.5                    #Phase voltage(V)
I = P/(3**0.5*V_s*PF_r)             #Line current(A)
#Case(b)
sin_phi_r = (1-PF_r**2)**0.5        #Sinφ_R
E_r = E_s-I*R*PF_r-I*X*sin_phi_r    #Receiving end voltage(V)
E_r_ll = 3**0.5*E_r/1000            #Receiving end line to line voltage(kV)
#Case(c)
loss = 3*I**2*R                     #Loss in the transmission line(W)
P_s = P+loss                        #Sending end power(W)
n = P/P_s*100                       #Transmission efficiency(%)
#Alternate method
Z = R**2+X**2
Q = 1.0*P*sin_phi_r/(3*PF_r)        #Reactive load delivered(VAR/phase)
A = (V_s**2/3.0)-2*(P_A*R+Q*X)      #Constant
B = (1/9.0)*P**2*Z/PF_r**2          #Constant
const = (A**2-4*B)**0.5             #sqrt(A^2-4B)
E_r_A = ((A+const)/2)**0.5/1000.0   #Receiving end voltage(kV/phase)
E_r_A_ll = 3**0.5*E_r_A             #Receiving end line-line voltage(kV)
I_A = P/(3**0.5*E_r_A_ll*1000*PF_r) #Line current(A)
loss_A = 3*I_A**2*R                 #Loss in the transmission line(W)
P_s_A = P+loss_A                    #Sending end power(W)
n_A = P/P_s_A*100                   #Transmission efficiency(%)

#Result
print('Case(a): Line current, |I| = %.1f A' %I)
print('Case(b): Receiving end voltage, E_r = %.f V (line-to-neutral) = %.2f kV (line-to-line)' %(E_r,E_r_ll))
print('Case(c): Efficiency of transmission = %.2f percent' %n)
print('\nAlternative solution by mixed condition:')
print('Case(a): Line current, |I| = %.1f A' %I_A)
print('Case(b): Receiving end voltage, E_r = %.3f kV/phase = %.2f kV (line-line)' %(E_r_A,E_r_A_ll))
print('Case(c): Efficiency of transmission = %.2f percent' %n_A)

Case(a): Line current, |I| = 65.6 A
Case(b): Receiving end voltage, E_r = 5868 V (line-to-neutral) = 10.16 kV (line-to-line)
Case(c): Efficiency of transmission = 93.93 percent

Alternative solution by mixed condition:
Case(a): Line current, |I| = 71.6 A
Case(b): Receiving end voltage, E_r = 5.823 kV/phase = 10.09 kV (line-line)
Case(c): Efficiency of transmission = 92.87 percent


## Example 2.3.3, Page number 129¶

In :
#Variable declaration
I = 200.0               #Line current(A)
PF_r = 0.8              #Receiving end lagging power factor
R = 0.6                 #Total resistance of the line(ohm)
X = 1.0                 #Total inductive resistance of the line(ohm)
n  = 0.93               #Efficiency(%)

#Calculation
V_r = 3*I**2*R/((3*I*PF_r/n)-3*I*PF_r)   #Receiving end phase voltage(V)
sin_phi_r = (1-PF_r**2)**0.5             #Sinφ_R
V_s = V_r+I*R*PF_r+I*X*sin_phi_r         #Sending end voltage(V)
V_s_ll = 3**0.5*V_s                      #Sending end line voltage(V)

#Result
print('Sending end voltage, V_s(line-line)  = %.2f V' %V_s_ll)

Sending end voltage, V_s(line-line)  = 3825.85 V


## Example 2.3.4, Page number 129¶

In :
#Variable declaration
P = 15.0*10**6            #Load delivered at receiving end(W)
PF_r = 0.85               #Receiving end lagging power factor
r = 0.905                 #Resistance of each conductor(ohm/km)
V_r = 132.0*10**3         #Receiving end voltage(V)
loss_per = 7.5/100        #Loss

#Calculation
loss = loss_per*P             #Losses in line(W)
I = P/(3**0.5*V_r*PF_r)       #Line current(A)
l = loss/(3*I**2*r)           #Length of line(km)

#Result
print('Distance over which load is delivered, l = %.2f km' %l)

Distance over which load is delivered, l = 69.55 km


## Example 2.3.5, Page number 130¶

In :
import math

#Variable declaration
f = 50.0                 #Frequency(Hz)
l = 20.0                 #Length(km)
P = 5.0*10**6            #Load delivered at receiving end(W)
PF_r = 0.8               #Receiving end lagging power factor
r = 0.02                 #Resistance of each conductor(ohm/km)
L = 0.65*10**-3          #Inductance of each conductor(H/km)
E_r = 10.0*10**3         #Receiving end voltage(V)

#Calculation
R = r*l                                     #Resistance per phase(ohm)
X = 2*math.pi*f*L*l                         #Reactance per phase(ohm)
#Case(a)
I = P/(E_r*PF_r)                            #Line current(A)
sin_phi_r = (1-PF_r**2)**0.5                #Sinφ_R
E_s = E_r+I*R*PF_r+I*X*sin_phi_r            #Sending end voltage(V)
E_s_kV = E_s/1000.0                         #Sending end voltage(kV)
reg = (E_s-E_r)/E_r*100                     #Voltage regulation(%)
#Case(b)
reg_new = reg/2                             #New regulation(%)
E_s_new = (reg_new/100)*E_r+E_r             #New value of sending end voltage(V)
tan_phi_r1 = ((E_s_new-E_r)*(E_r/P)-R)/X    #tanφ_r1
phi_r1d = phi_r1*180/math.pi                #φ_r1(degree)
PF_r1 = math.cos(phi_r1)                    #Lagging power factor of receiving end
sin_phi_r1 = (1-PF_r1**2)**0.5              #Sinφ_r1
I_R_new = P/(E_r*PF_r1)                     #New line current(A)
I_R = I_R_new*complex(PF_r1,-sin_phi_r1)
I_c = I_R-I*complex(PF_r,-sin_phi_r)        #Capacitive current(A)
I_C = I_c.imag                              #Imaginary part of Capacitive current(A)
c = I_C/(2*math.pi*f*E_r)*10.0**6           #Capacitance(µF)
#Case(c)
loss_1 = I**2*R                             #Loss(W)
n_1 = P/(P+loss_1)*100                      #Transmission efficiency(%)
loss_2 = I_R_new**2*R                       #Loss(W)
n_2 = P/(P+loss_2)*100                      #Transmission efficiency(%)

#Result
print('Case(a): Sending end voltage, E_s = %.2f kV' %E_s_kV)
print('         Voltage regulation of the line = %.1f percent' %reg)
print('Case(b): Value of capacitors to be placed in parallel with load, c = %.2f µF' %c)
print('Case(c): Transmission efficiency in part(a), η_1 = %.2f percent' %n_1)
print('         Transmission efficiency in part(b), η_2 = %.1f percent' %n_2)

Case(a): Sending end voltage, E_s = 11.73 kV
Voltage regulation of the line = 17.3 percent
Case(b): Value of capacitors to be placed in parallel with load, c = 67.48 µF
Case(c): Transmission efficiency in part(a), η_1 = 96.97 percent
Transmission efficiency in part(b), η_2 = 97.8 percent


## Example 2.3.6, Page number 130-131¶

In :
import math
import cmath

#Variable declaration
f = 50.0                                             #Frequency(Hz)
l = 10.0                                             #Line length(km)
P = 316.8*10**3                                      #Load side power(W)
PF_r = 0.8                                           #Load side power factor
E_r = 3.3*10**3                                      #Load bus voltage(V)

#Calculation
I_r = P/(E_r*PF_r)*cmath.exp(1j*-math.acos(PF_r))    #Line current(A)
sin_phi_r = (1-PF_r**2)**0.5                         #Sinφ_R
E_s = E_r+I_r*Z_line                                 #Sending end voltage(V)
reg = (abs(E_s)-abs(E_r))/abs(E_r)*100               #Voltage regulation(%)
R = Z_line.real                                      #Resistance of the load line(ohm)
loss = abs(I_r)**2*R                                 #Loss in the transmission line(W)
loss_kW = loss/1000.0                                #Loss in the transmission line(kW)
P_s = P+loss                                         #Sending end power(W)
angle_Er_Es = cmath.phase(E_s)*180/math.pi           #Angle between V_r and V_s(°)
angle_Er_Ir = math.acos(PF_r)*180/math.pi            #Angle between V_r and I_r(°)
angle_Es_Is = angle_Er_Es+angle_Er_Ir                #Angle between V_s and I_s(°)
PF_s = math.cos(angle_Es_Is*math.pi/180)             #Sending end power factor

#Result
print('Voltage regulation = %.2f percent' %reg)
print('Sending end voltage, E_s = %.f∠%.1f° V' %(abs(E_s),cmath.phase(E_s)*180/math.pi))
print('Line loss = %.f kW' %loss_kW)
print('Sending end power factor = %.2f ' %PF_s)

Voltage regulation = 16.94 percent
Sending end voltage, E_s = 3859∠3.5° V
Line loss = 36 kW
Sending end power factor = 0.76


## Example 2.3.7, Page number 132-133¶

In :
import math
import cmath

#Variable declaration
V_s = 66.0                    #Voltage(kV)
f = 50.0                      #Frequency(Hz)
l = 150.0                     #Line length(km)
r = 0.25                      #Resistance of each conductor(ohm/km)
x = 0.5                       #Inductive reactance of each conductor(ohm/km)

#Calculation
#Case(a)
R = r*l                                          #Total resistance(ohm)
X = x*l                                          #Inductive reactance(ohm)
Y = y*l                                          #Capacitive resistance(s)
Y_2 = Y/2                                        #1/2 of Capacitive resistance(s)
#Case(b)
Z = complex(R,X)                                 #Total impedance(ohm)
A = 1+(Y*cmath.exp(1j*90.0*math.pi/180)*Z/2)     #Line constant

#Result
print('Case(a): Total resistance, R = %.1f ohm' %R)
print('         Inductive reactance, X = %.1f ohm' %X)
print('         Capacitive resistance, Y = %.1e s' %Y)
print('         Capacitive resistance, Y/2 = %.1e s' %Y_2)

Case(a): Total resistance, R = 37.5 ohm
Inductive reactance, X = 75.0 ohm
Capacitive resistance, Y = 6.0e-04 s
Capacitive resistance, Y/2 = 3.0e-04 s
Case(b): Receiving end voltage at no-load, V_R = 67.51 kV


## Example 2.3.8, Page number 133-134¶

In :
import math
import cmath

#Variable declaration
f = 50.0                      #Frequency(Hz)
V_r = 132.0*10**3             #Line voltage at receiving end(V)
L = 100.0                     #Line length(km)
r = 0.17                      #Resistance(ohm/km/phase)
l = 1.1*10**-3                #Inductance(H/km/phase)
c = 0.0082*10**-6             #Capacitance(F/km/phase)
P_L = 70.0*10**6              #Load at receiving end(W)
PF_r = 0.8                    #Lagging load power factor

#Calculation
E_r = V_r/3**0.5                                                   #Receiving end phase voltage(V)
I_r = P_L/(3**0.5*V_r*PF_r)*cmath.exp(1j*-math.acos(PF_r))         #Receiving end current(A)
R = r*L                                                            #Total resistance(ohm/phase)
X = 2*math.pi*f*l*L                                                #Inductive reactance(ohm/phase)
Z = complex(R,X)                                                   #Total impedance(ohm/phase)
Y = 2*math.pi*f*c*cmath.exp(1j*90.0*math.pi/180)/L                 #Shunt admittance of line(mho/phase)
E = E_r+I_r*(Z/2)                                                  #Voltage across shunt admittance(V/phase)
I_s = I_r+E*Y                                                      #Sending end current(A)
E_s = E+I_s*(Z/2)                                                  #Sending end voltage(V/phase)
E_s_ll = 3**0.5*abs(E_s)/1000                                      #Sending end line to line voltage(kV)
angle_Er_Es = cmath.phase(E_s)*180/math.pi                         #Angle between E_r and V_s(°)
angle_Er_Is = cmath.phase(I_s)*180/math.pi                         #Angle between E_r and I_s(°)
angle_Es_Is = angle_Er_Es-angle_Er_Is                              #Angle between E_s and I_s(°)
PF_s = math.cos(angle_Es_Is*math.pi/180)                           #Sending end power factor

#Result
print('Voltage at sending end, E_s = %.2f∠%.2f° V/phase = %.f kV (line-to-line)' %(abs(E_s),cmath.phase(E_s)*180/math.pi,E_s_ll))
print('Current at sending end, I_s = %.1f∠%.1f° A' %(abs(I_s),cmath.phase(I_s)*180/math.pi))
print('Sending end Power factor = %.3f (lagging)' %PF_s)

Voltage at sending end, E_s = 89599.59∠4.27° V/phase = 155 kV (line-to-line)
Current at sending end, I_s = 382.7∠-36.9° A
Sending end Power factor = 0.753 (lagging)


## Example 2.3.9, Page number 134¶

In :
import math
import cmath

#Variable declaration
f = 50.0                      #Frequency(Hz)
E_r = 66.0*10**3              #Line voltage at receiving end(V)
l = 120.0                     #Line length(km)
r = 0.1                       #Resistance(ohm/km/phase)
x = 0.3                       #Inductive reactance(ohm/km/phase)
y = 0.04*10**-4               #Capacitive susceptance(S/km/phase)
P_L = 10.0*10**6              #Load at receiving end(W)
PF_r = 0.8                    #Lagging load power factor

#Calculation
R = r*l                                                            #Total resistance(ohm/phase)
X = x*l                                                            #Inductive reactance(ohm/phase)
Y = y*l                                                            #Susceptance(mho)
Z = complex(R,X)                                                   #Total impedance(ohm/phase)
V_r = E_r/3**0.5                                                   #Receiving end phase voltage(V)
V_1 = V_r+I_r*(Z/2)                                                #Voltage across capacitor(V)
I_c = 1j*Y*V_1                                                     #Charging current(A)
I_s = I_r+I_c                                                      #Sending end current(A)
V_s = V_1+I_s*(Z/2)                                                #Sending end voltage(V/phase)
V_s_ll = 3**0.5*abs(V_s)/1000.0                                    #Sending end line to line voltage(kV)
angle_Vr_Vs = cmath.phase(V_s)*180/math.pi                         #Angle between V_r and V_s(°)
angle_Vr_Is = cmath.phase(I_s)*180/math.pi                         #Angle between V_r and I_s(°)
angle_Vs_Is = angle_Vr_Vs-angle_Vr_Is                              #Angle between V_s and I_s(°)
PF_s = math.cos(angle_Vs_Is*math.pi/180)                           #Sending end power factor
P_s = 3*abs(V_s*I_s)*PF_s                                          #Sending end power(W)
n = P_L/P_s*100                                                    #Transmission efficiency(%)

#Result
print('Sending end voltage, |V_s| = %.f V/phase = %.3f V (line-to-line)' %(abs(V_s),V_s_ll))
print('Sending end current, |I_s| = %.2f A' %abs(I_s))
print('Transmission efficiency = %.2f percent' %n)
print('\nNOTE: ERROR: Calculation mistake in finding sending end power factor')
print('      Changes in the obtained answer from that of textbook is due to more precision')

Sending end voltage, |V_s| = 41243 V/phase = 71.435 V (line-to-line)
Sending end current, |I_s| = 98.57 A
Transmission efficiency = 96.25 percent

NOTE: ERROR: Calculation mistake in finding sending end power factor
Changes in the obtained answer from that of textbook is due to more precision


## Example 2.3.10, Page number 135¶

In :
import math
import cmath

#Variable declaration
f = 50.0                      #Frequency(Hz)
l = 125.0                     #Line length(km)
P_r = 40.0*10**6              #Load at receiving end(VA)
V_r = 110.0*10**3             #Line voltage at receiving end(V)
PF_r = 0.8                    #Lagging load power factor
R = 11.0                      #Resistance(ohm/phase)
X = 38.0                      #Inductive reactance(ohm/phase)
Y = 3.0*10**-4                #Capacitive susceptance(S)

#Calculation
#Case(i)
E_r = V_r/3**0.5                                                   #Receiving end phase voltage(V)
Z = complex(R,X)                                                   #Total impedance(ohm/phase)
I_c1 = E_r*(Y/2)*cmath.exp(1j*90.0*math.pi/180)                    #Current through shunt admittance at receiving end(A)
I = I_r+I_c1                                                       #Current through series impedance(A)
E_s = I*Z+E_r                                                      #Voltage across shunt admittance at sending end(V)
E_s_ll = 3**0.5*E_s/1000.0                                         #Line to line voltage at sending end(kV)
I_c2 = E_s*(Y/2)*cmath.exp(1j*90.0*math.pi/180)                    #Current through shunt admittance at sending end(A)
#Case(ii)
I_s = I_c2+I_r                                                     #Sending end current(A)
angle_Er_Es = cmath.phase(E_s)*180/math.pi                         #Angle between E_r and E_s(°)
angle_Er_Is = cmath.phase(I_s)*180/math.pi                         #Angle between E_r and I_s(°)
angle_Es_Is = angle_Er_Es-angle_Er_Is                              #Angle between E_s and I_s(°)
PF_s = math.cos(angle_Es_Is*math.pi/180)                           #Sending end power factor

#Result
print('Case(i) : Line to line voltage at sending end, E_s = %.f kV' %(abs(E_s_ll)))
print('Case(ii): Sending end power factor = %.3f ' %PF_s)
print('\nNOTE: Answer in the textbook are incomplete')

Case(i) : Line to line voltage at sending end, E_s = 121 kV
Case(ii): Sending end power factor = 0.779

NOTE: Answer in the textbook are incomplete


## Example 2.3.11, Page number 135-137¶

In :
import math
import cmath

#Variable declaration
f = 50.0                      #Frequency(Hz)
R = 28.0                      #Resistance(ohm/phase)
X = 63.0                      #Inductive reactance(ohm/phase)
Y = 4.0*10**-4                #Capacitive susceptance(mho)
P_r = 75.0*10**6              #Load at receiving end(VA)
PF_r = 0.8                    #Lagging load power factor
V_r = 132.0*10**3             #Line voltage at receiving end(V)

#Calculation
#Case(i) Nominal T method
Z = complex(R,X)                                                   #Total impedance(ohm/phase)
E_r = V_r/3**0.5                                                   #Receiving end phase voltage(V)
I_r = P_r/(3**0.5*V_r)*cmath.exp(1j*-math.acos(PF_r))              #Line current at receiving end(A)
E = E_r+I_r*(Z/2)
I_c = 1j*Y*E                                                       #Capacitive current(A)
I_s = I_r+I_c                                                      #Sending end current(A)
v_drop = I_s*(Z/2)                                                 #Voltage drop(V)
E_s = E+I_s*(Z/2)                                                  #Sending end voltage(V)
E_s_kV = E_s/1000.0                                                #Sending end voltage(kV)
E_s_ll= 3**0.5*abs(E_s)                                            #Sending end line voltage(V)
E_s_llkV = E_s_ll/1000.0                                           #Sending end line voltage(kV)
angle_Er_Es = cmath.phase(E_s)*180/math.pi                         #Angle between E_r and E_s(°)
angle_Er_Is = cmath.phase(I_s)*180/math.pi                         #Angle between E_r and I_s(°)
angle_Es_Is = angle_Er_Es-angle_Er_Is                              #Angle between E_s and I_s(°)
PF_s = math.cos(angle_Es_Is*math.pi/180)                           #Sending end power factor
P_s = 3**0.5*E_s_ll*abs(I_s)*PF_s                                  #Power at sending end(W)
reg = (abs(E_s_ll)-V_r)/V_r*100                                    #Regulation(%)
n = (P_r*PF_r)/P_s*100                                             #Transmission efficiency(%)
#Case(ii) Nominal π method
I_c2 = E_r*(1j*Y/2)                                                #Current through shunt admittance at receiving end(A)
I = I_r+I_c2                                                       #Line current(A)
E_s_p = E_r+I*Z                                                    #Sending end voltage(V)
E_s_pkV = E_s_p/1000.0                                             #Sending end voltage(kV)
E_s_pll = 3**0.5*abs(E_s_p)                                        #Sending end line voltage(V)
E_s_pllkV = E_s_pll/1000.0                                         #Sending end line voltage(kV)
I_c1 = E_s_p*(1j*Y/2)                                              #Current through shunt admittance at sending end(A)
I_s_p = I+I_c1                                                     #Sending end current(A)
angle_Er_Esp = cmath.phase(E_s)*180/math.pi                        #Angle between E_r and E_s(°)
angle_Er_Isp = cmath.phase(I_s)*180/math.pi                        #Angle between E_r and I_s(°)
angle_Es_Isp = angle_Er_Esp-angle_Er_Isp                           #Angle between E_s and I_s(°)
PF_s_p = math.cos(angle_Es_Isp*math.pi/180)                        #Sending end power factor
P_s_p = 3**0.5*E_s_pll*abs(I_s_p)*PF_s_p                           #Power at sending end(W)
reg_p = (abs(E_s_pll)-V_r)/V_r*100                                 #Regulation(%)
n_p = (P_r*PF_r)/P_s_p*100                                         #Transmission efficiency(%)

#Result
print('(i) Nominal T method')
print('Case(a): Voltage at sending end, E_s = %.2f∠%.2f° kV = %.1f kV (line-to-line)' %(abs(E_s_kV),cmath.phase(E_s_kV)*180/math.pi,E_s_llkV))
print('Case(b): Sending end current, I_s = %.1f∠%.2f° A' %(abs(I_s),cmath.phase(I_s)*180/math.pi))
print('Case(c): Power factor at sending end = %.4f (lagging)' %PF_s)
print('Case(d): Regulation = %.2f percent' %reg)
print('Case(e): Efficiency of transmission = %.2f percent' %n)
print('\n(ii) Nominal π method')
print('Case(a): Voltage at sending end, E_s = %.2f∠%.2f° kV = %.1f kV (line-to-line)' %(abs(E_s_pkV),cmath.phase(E_s_pkV)*180/math.pi,E_s_pllkV))
print('Case(b): Sending end current, I_s = %.1f∠%.2f° A' %(abs(I_s_p),cmath.phase(I_s_p)*180/math.pi))
print('Case(c): Power factor at sending end = %.4f (lagging)' %PF_s_p)
print('Case(d): Regulation = %.2f percent' %reg_p)
print('Case(e): Efficiency of transmission = %.2f percent' %n_p)
print('\nNOTE: Changes in the obtained answer from that of textbook is due to more precision here and more approximation in textbook')

(i) Nominal T method
Case(a): Voltage at sending end, E_s = 95.53∠6.87° kV = 165.5 kV (line-to-line)
Case(b): Sending end current, I_s = 306.7∠-31.97° A
Case(c): Power factor at sending end = 0.7789 (lagging)
Case(d): Regulation = 25.35 percent
Case(e): Efficiency of transmission = 87.63 percent

(ii) Nominal π method
Case(a): Voltage at sending end, E_s = 95.69∠6.87° kV = 165.7 kV (line-to-line)
Case(b): Sending end current, I_s = 306.8∠-32.00° A
Case(c): Power factor at sending end = 0.7789 (lagging)
Case(d): Regulation = 25.55 percent
Case(e): Efficiency of transmission = 87.48 percent

NOTE: Changes in the obtained answer from that of textbook is due to more precision here and more approximation in textbook


## Example 2.3.12, Page number 143¶

In :
#Variable declaration
E_s = 275.0                 #Sending end voltage(kV)
f = 50.0                    #Frequency(Hz)
l = 400.0                   #Line length(km)
x = 0.05                    #Inductive reactance(ohm/km)
y = 3.0*10**-6              #Line charging susceptance(S/km)
r = 0.0                     #Lossless line

#Calculation
#Case(a)
R = r*l                     #Total resistance(ohm/phase)
X = x*l                     #Inductive reactance(ohm/phase)
Y = y*l                     #Susceptance(mho)
Z = complex(R,X)            #Total impedance(ohm/phase)
A = 1+(Y*Z/2)*1j            #Line constant
E_r = E_s/abs(A)            #Receiving end voltage at no load(kV)
#case(b)
Z_0 = (X/Y)**0.5            #Load at receiving end(ohm)
#Case(c)
Z_0_new = 1.2*Z_0           #New load at receiving station(ohm)

#Result
print('Case(a): Receiving end voltage on open circuit = %.1f kV' %E_r)
print('Case(b): Load at receiving end for flat voltage profile on line, Z_0 = %.1f Ω' %Z_0)
print('Case(c): Distributed inductive reactance of the line is to be increased as, Loading for new voltage profile = %.2f Ω' %Z_0_new)

Case(a): Receiving end voltage on open circuit = 278.3 kV
Case(b): Load at receiving end for flat voltage profile on line, Z_0 = 129.1 Ω
Case(c): Distributed inductive reactance of the line is to be increased as, Loading for new voltage profile = 154.92 Ω


## Example 2.3.13, Page number 143-144¶

In :
import math
import cmath

#Variable declaration
V_r = 220.0*10**3           #Receiving end voltage(V)
Z = complex(20,100)         #Impedance(ohm/phase)
I_r = 300.0                 #Receiving end current(A)
PF_r = 0.9                  #Lagging power factor

#Calculation
V_2 = V_r/3**0.5                           #Receiving end phase voltage(V)
I_2 = I_r*cmath.exp(1j*-math.acos(PF_r))   #Receiving end current(A)
I_C2 = (Y/2)*V_2                           #Capacitive current at receiving end(A)
I = I_2+I_C2
V_1 = V_2+I*Z                              #Voltage across shunt admittance at sending end(V)
V_1kV = V_1/1000.0                         #Voltage across shunt admittance at sending end(kV)
I_C1 = (Y/2)*V_1                           #Capacitive current at sending end(A)
I_1 = I_C1+I_2                             #Sending end current(A)

#Result
print('Sending end voltage, V_1 = %.2f∠%.2f° kV' %(abs(V_1kV),cmath.phase(V_1kV)*180/math.pi))
print('Sending end current, I_1 = %.3f∠%.4f° A' %(abs(I_1),cmath.phase(I_1)*180/math.pi))

Sending end voltage, V_1 = 141.49∠10.45° kV
Sending end current, I_1 = 264.353∠-13.3849° A


## Example 2.3.14, Page number 144¶

In :
import math
import cmath

#Variable declaration
f = 50.0                      #Frequency(Hz)
r = 0.1                       #Resistance(ohm/km)
l = 1.4*10**-3                #Inductance(H/km)
c = 8.0*10**-9                #Capacitance(F/km)
g = 4.0*10**-8                #conductance(mho/km)
V_r = 400.0                   #Receiving end voltage(kV)
x = 200.0                     #Length of line(km)

#Calculation
V_2 = V_r/3**0.5                         #Receiving end phase voltage(kV)
z = r+1j*2*math.pi*f*l                   #Total impedance(ohm/km)
y = g+1j*2*math.pi*f*c                   #Total susceptance(mho/km)
Z_c = (z/y)**0.5                         #Surge impedance(ohm)
gamma = (z*y)**0.5                       #γ
#Case(i)
V_0_plus = V_2/2                         #Incident voltage to neutral at receiving end(kV)
#Case(ii)
V_0_minus = V_2/2                        #Reflected voltage to neutral at receiving end(kV)
#Case(iii)
gamma_l = gamma*x                        #γl
V_1_plus = (V_2/2)*cmath.exp(gamma_l)    #Incident voltage to neutral at 200 km from receiving end(kV)
V_1_minus = (V_2/2)*cmath.exp(-gamma_l)  #Reflected voltage to neutral at 200 km from receiving end(kV)
#Case(iv)
V_1 = V_1_plus+V_1_minus                 #Resultant voltage to neutral(kV)
V_L = abs(V_1)                           #Resultant voltage to neutral(kV)
V_L_ll = 3**0.5*V_L                      #Line to line voltage at 200 km from receiving end(kV)

#Result
print('Case(i)  : Incident voltage to neutral at receiving end, V_0_plus = %.1f∠%.f° kV' %(abs(V_0_plus),cmath.phase(V_0_plus)*180/math.pi))
print('Case(ii) : Reflected voltage to neutral at receiving end, V_0_minus = %.1f∠%.f° kV' %(abs(V_0_minus),cmath.phase(V_0_minus)*180/math.pi))
print('Case(iii): Incident voltage to neutral at 200 km from receiving end, V_1_plus = (%.3f+%.2fj) kV' %(V_1_plus.real,V_1_plus.imag))
print('Case(iv) : Resultant voltage to neutral at 200 km from receiving end, V_L = %.2f kV' %V_L)
print('           Line to line voltage at 200 km from receiving end = %.2f kV' %V_L_ll)

Case(i)  : Incident voltage to neutral at receiving end, V_0_plus = 115.5∠0° kV
Case(ii) : Reflected voltage to neutral at receiving end, V_0_minus = 115.5∠0° kV
Case(iii): Incident voltage to neutral at 200 km from receiving end, V_1_plus = (115.808+24.86j) kV
Case(iv) : Resultant voltage to neutral at 200 km from receiving end, V_L = 225.87 kV
Line to line voltage at 200 km from receiving end = 391.23 kV


## Example 2.3.15, Page number 145¶

In :
import math
import cmath

#Variable declaration
f = 50.0                      #Frequency(Hz)
L = 200.0                     #Line length(km)
l = 1.20*10**-3               #Inductance(H/km)
c = 8.0*10**-9                #Capacitance(F/km)
r = 0.15                      #Resistance(ohm/km)
g = 0.0                       #Conductance(mho/km)

#Calculation
z = r+1j*2*math.pi*f*l                  #Total impedance(ohm/km)
Z = z*L                                 #Total impedance(ohm)
y = g+1j*2*math.pi*f*c                  #Total susceptance(mho/km)
Y = y*L                                 #Total susceptance(mho/km)
gamma_l = (Z*Y)**0.5                    #γl
alpha_l = gamma_l.real                  #αl
beta_l = gamma_l.imag                   #βl
Z_c = (Z/Y)**0.5                        #Surge impedance(ohm)
A = cmath.cosh(gamma_l)                 #Constant
B = Z_c*cmath.sinh(gamma_l)             #Constant(ohm)
C = (1/Z_c)*cmath.sinh(gamma_l)         #Constant(S)
D = A                                   #Constant

#Result
print('A = D = %.3f∠%.2f° ' %(abs(A),cmath.phase(A)*180/math.pi))
print('B = %.2f∠%.3f° Ω' %(abs(B),cmath.phase(B)*180/math.pi))
print('C = %.2e∠%.3f° S' %(abs(C),cmath.phase(C)*180/math.pi))

A = D = 0.981∠0.44°
B = 80.64∠68.447° Ω
C = 4.99e-04∠90.144° S


## Example 2.3.16, Page number 145-146¶

In :
import math
import cmath

#Variable declaration
V_r = 132.0*10**3           #Receiving end voltage(V)
f = 50.0                    #Frequency(Hz)
L = 200.0                   #Line length(km)
l = 1.3*10**-3              #Inductance(H/km)
c = 9.0*10**-9              #Capacitance(F/km)
r = 0.2                     #Resistance(ohm/km)
g = 0.0                     #Conductance(mho/km)
PF_r = 0.8                  #Lagging power factor at receiving end

#Calculation
z = r+1j*2*math.pi*f*l                              #Total impedance(ohm/km)
y = g+1j*2*math.pi*f*c                              #Total susceptance(mho/km)
Z_c = (z/y)**0.5                                    #Surge impedance(ohm)
gamma = (z*y)**0.5                                  #γ
gamma_l = gamma*L                                   #γl
cosh_gl = cmath.cosh(gamma_l)                       #cosh γl
sinh_gl = cmath.sinh(gamma_l)                       #sinh γl
V_2 = V_r/(3**0.5)                                  #Receiving end phase voltage(V)
I_2 = P_r/(3*V_2)*cmath.exp(1j*-math.acos(PF_r))    #Line current(A)
V_1 = V_2*cosh_gl+I_2*Z_c*sinh_gl                   #Sending end voltage(V)
V_1kV = V_1/1000.0                                  #Sending end voltage(kV)
I_1 = (V_2/Z_c)*sinh_gl+I_2*cosh_gl                 #Sending end current(A)
angle_V2_V1 = cmath.phase(V_1)*180/math.pi          #Angle between V_2 and V_1(°)
angle_V2_I1 = cmath.phase(I_1)*180/math.pi          #Angle between V_2 and I_1(°)
angle_V1_I1 = angle_V2_V1-angle_V2_I1               #Angle between V_1 and I_1(°)
PF_s = math.cos(angle_V1_I1*math.pi/180)            #Sending end power factor
P_1 = 3*abs(V_1*I_1)*PF_s                           #Sending end power(W)
P_2 = P_r*PF_r                                      #Receiving end power(W)
n = P_2/P_1*100                                     #Efficiency

#Result
print('Sending end voltage, V_1 = %.3f∠%.4f° kV per phase' %(abs(V_1kV),cmath.phase(V_1kV)*180/math.pi))
print('Sending end current, I_1 = %.3f∠%.2f° A' %(abs(I_1),cmath.phase(I_1)*180/math.pi))
print('Power factor = %.3f ' %PF_s)
print('Efficiency, η = %.2f percent' %n)

Sending end voltage, V_1 = 92.532∠6.1382° kV per phase
Sending end current, I_1 = 191.395∠-25.85° A
Power factor = 0.848
Efficiency, η = 88.77 percent


## Example 2.3.17, Page number 147-148¶

In :
import math
import cmath

#Variable declaration
f = 50.0                    #Frequency(Hz)
L = 160.0                   #Line length(km)
r = 0.15                    #Resistance(ohm/km/phase)
l = 1.2*10**-3              #Inductance(H/km/phase)
c = 0.008*10**-6            #Capacitance(F/km/phase)
g = 0.0                     #Conductance(mho/km/phase)

#Calculation
#Case(i) Using convergent series(Complex angles) method
z = r+1j*2*math.pi*f*l                    #Impedance(ohm/km)
Z = z*L                                   #Total series impedance(ohm)
Y = y*L                                   #Total shunt admittance(S)
A = 1+(Y*Z/2)+((Y*Z)**2/24)               #Constant
B = Z*(1+(Y*Z/6)+((Y*Z)**2/120))          #Constant(ohm)
C = Y*(1+(Y*Z/6)+((Y*Z)**2/120))          #Constant(mho)
D = A                                     #Constant
#Case(ii) Using convergent series(Real angles) method
gamma_l = (Z*Y)**0.5                      #γl
alpha_l = gamma_l.real                    #αl
beta_l = gamma_l.imag                     #βl
Z_c = (Z/Y)**0.5                          #Surge impedance(ohm)
A_2 = cmath.cosh(gamma_l)                 #Constant
B_2 = Z_c*cmath.sinh(gamma_l)             #Constant(ohm)
C_2 = (1/Z_c)*cmath.sinh(gamma_l)         #Constant(mho)
D_2 = A_2                                 #Constant

#Result
print('Case(i): Using convergent series(Complex Angles) method')
print('A = D = %.3f∠%.1f° ' %(abs(A),cmath.phase(A)*180/math.pi))
print('B = %.f∠%.1f° ohm' %(abs(B),cmath.phase(B)*180/math.pi))
print('C = %.4f∠%.1f° mho' %(abs(C),cmath.phase(C)*180/math.pi))
print('\nCase(ii): Using convergent series(Real Angles) method')
print('A = D = %.3f∠%.1f° ' %(abs(A_2),cmath.phase(A_2)*180/math.pi))
print('B = %.1f∠%.1f° ohm' %(abs(B_2),cmath.phase(B_2)*180/math.pi))
print('C = %.4f∠%.1f° S' %(abs(C_2),cmath.phase(C_2)*180/math.pi))
print('\nNOTE: Slight change in obtained answer from that of textbook is due to more precision')

Case(i): Using convergent series(Complex Angles) method
A = D = 0.988∠0.3°
B = 65∠68.4° ohm
C = 0.0004∠90.1° mho

Case(ii): Using convergent series(Real Angles) method
A = D = 0.988∠0.3°
B = 64.7∠68.4° ohm
C = 0.0004∠90.1° S

NOTE: Slight change in obtained answer from that of textbook is due to more precision


## Example 2.3.18, Page number 148¶

In :
import math
import cmath

#Variable declaration
V_r = 220.0*10**3                    #Line voltage at receiving end(V)
Z = complex(40,200)                  #Impedance per phase(ohm)
I_r = 200.0                          #Receiving end current(A)
PF_r = 0.95                          #Lagging power factor

#Calculation
#Case(a)
A = 1+(Y*Z/2)+((Y*Z)**2/24)                               #Constant
B = Z*(1+(Y*Z/6)+((Y*Z)**2/120)+((Y*Z)**3/5040))          #Constant(ohm)
C = Y*(1+(Y*Z/6)+((Y*Z)**2/120)+((Y*Z)**3/5040))          #Constant(mho)
D = A                                                     #Constant
E_r = V_r/3**0.5                                          #Receiving end phase voltage(V)
I_r1 = I_r*cmath.exp(1j*-math.acos(PF_r))                 #Line current(A)
E_s = A*E_r+B*I_r1                                        #Sending end voltage(V)
E_s_ll = 3**0.5*E_s/1000.0                                #Sending end line voltage(kV)
#Case(b)
I_s = C*E_r+D*I_r1                                        #Sending end current(A)

#Result
print('Case(a): Sending end voltage, E_s = %.1f∠%.2f° kV (line-to-line)' %(abs(E_s_ll),cmath.phase(E_s_ll)*180/math.pi))
print('Case(b): Sending end current, I_s = %.1f∠%.2f° A' %(abs(I_s),cmath.phase(I_s)*180/math.pi))
print('\nNOTE: ERROR: Z = (40+j200)Ω, not Z=(60+j200)Ω as given in problem statement')
print('      Changes in obtained answer from that of textbook is due to more precision')

Case(a): Sending end voltage, E_s = 229.7∠16.46° kV (line-to-line)
Case(b): Sending end current, I_s = 209.8∠39.42° A

NOTE: ERROR: Z = (40+j200)Ω, not Z=(60+j200)Ω as given in problem statement
Changes in obtained answer from that of textbook is due to more precision


## Example 2.3.19, Page number 148-149¶

In :
import math
import cmath

#Variable declaration
V_r = 220.0*10**3                    #Line voltage at receiving end(V)
Z = complex(40,200)                  #Impedance per phase(ohm)
I_R = 200.0                          #Receiving end current(A)
PF_r = 0.95                          #Lagging power factor

#Calculation
#Case(i) Nominal π method
#Case(a)
E_r = V_r/3**0.5                             #Receiving end phase voltage(V)
I_r = I_R*cmath.exp(1j*-math.acos(PF_r))     #Line current(A)
I_c2 = Y_2*E_r                               #Current through shunt admittance at receiving end(A)
I = I_r+I_c2                                 #Current through impedance(A)
IZ_drop = I*Z                                #Voltage drop(V)
E_s = E_r+IZ_drop                            #Sending end voltage(V)
E_s_kV = E_s/1000.0                          #Sending end voltage(kV)
#Case(b)
I_c1 = E_s*Y_2                               #Current through shunt admittance at sending end(A)
I_s = I+I_c1                                 #Sending end current(A)
#Case(ii) Nominal T method
#Case(a)
I_r_Z2 = I_r*Z/2                             #Voltage drop at receiving end(V)
E = E_r+I_r_Z2                               #Voltage(V)
I_c = Y*E                                    #Current through shunt admittance(A)
I_s_2 = I_c+I_r                              #Sending end current(A)
I_s_Z2 = I_s_2*(Z/2)                         #Voltage drop at sending end(V)
E_s_2 = I_s_Z2+E                             #Sending end voltage(V)
E_s_2kV = E_s_2/1000.0                       #Sending end voltage(kV)

#Result
print('Case(i): Nominal π method')
print('         Case(a): Sending end voltage, E_s = %.1f∠%.2f° kV' %(abs(E_s_kV),cmath.phase(E_s_kV)*180/math.pi))
print('         Case(b): Sending end current, I_s = %.1f∠%.2f° A' %(abs(I_s),cmath.phase(I_s)*180/math.pi))
print('Case(ii): Nominal T method')
print('         Case(a): Sending end voltage, E_s = %.1f∠%.2f° kV' %(abs(E_s_2kV),cmath.phase(E_s_2kV)*180/math.pi))
print('         Case(b): Sending end current, I_s = %.1f∠%.2f° A' %(abs(I_s_2),cmath.phase(I_s_2)*180/math.pi))
print('\nThe results are tabulated below')
print('________________________________________________________')
print('METHOD            E_s(kV)                I_s(A)')
print('________________________________________________________')
print('Rigorous        √3*132.6∠16.46°        209.8∠39.42°')
print('Nominal π       √3*%.1f∠%.2f°        %.1f∠%.2f°' %(abs(E_s_kV),cmath.phase(E_s_kV)*180/math.pi,abs(I_s),cmath.phase(I_s)*180/math.pi))
print('Nominal T       √3*%.1f∠%.2f°        %.1f∠%.2f°' %(abs(E_s_2kV),cmath.phase(E_s_2kV)*180/math.pi,abs(I_s_2),cmath.phase(I_s_2)*180/math.pi))

Case(i): Nominal π method
Case(a): Sending end voltage, E_s = 134.0∠17.07° kV
Case(b): Sending end current, I_s = 205.8∠38.76° A
Case(ii): Nominal T method
Case(a): Sending end voltage, E_s = 131.3∠16.34° kV
Case(b): Sending end current, I_s = 217.2∠41.22° A

The results are tabulated below
________________________________________________________
METHOD            E_s(kV)                I_s(A)
________________________________________________________
Rigorous        √3*132.6∠16.46°        209.8∠39.42°
Nominal π       √3*134.0∠17.07°        205.8∠38.76°
Nominal T       √3*131.3∠16.34°        217.2∠41.22°


## Example 2.3.20, Page number 149-153¶

In :
import math
import cmath

#Variable declaration
f = 50.0                  #Frequency(Hz)
L = 280.0                 #Line length(km)
Z = complex(35,140)       #Series impedance(ohm)
P_r = 40.0*10**6          #Power delivered(W)
V_r = 220.0*10**3         #Voltage at receiving end(V)
PF_r = 0.9                #Lagging power factor

#Calculation
R = Z.real                                                     #Resistance of the line(ohm)
#Case(a)
I_r_a = P_r/(3**0.5*V_r*PF_r)*cmath.exp(1j*-math.acos(PF_r))   #Receiving end current(A)
I_s_a = I_r_a                                                  #Sending end current(A)
V_r_a = V_r/3**0.5                                             #Phase voltage at receiving end(V)
V_s_a = V_r_a+I_r_a*Z                                          #Sending end voltage(V)
V_s_a_ll = 3**0.5*V_s_a                                        #Sending end line voltage(V)
V_s_a_llkv = V_s_a_ll/1000.0                                   #Sending end line voltage(kV)
reg_a = (abs(V_s_a_ll)-V_r)/V_r*100                            #Voltage regulation(%)
loss_a = 3*abs(I_r_a)**2*R                                     #Line loss(W)
input_a = P_r+loss_a                                           #Input to line(W)
n_a = P_r/input_a*100                                          #Efficiency of transmission(%)
A_a = 1.0                                                      #Constant
B_a = Z                                                        #Constant(ohm)
C_a = 0                                                        #Constant(mho)
D_a = A_a                                                      #Constant
#Case(b)
V_b = V_r_a+I_r_a*Z/2                                  #Voltage drop across shunt admittance(V)
I_c_b = Y*V_b                                          #Current through shunt admittance(A)
I_s_b = I_r_a+I_c_b                                    #Sending end current(A)
V_s_b = V_b+I_s_b*Z/2                                  #Sending end voltage(V)
V_s_b_ll = 3**0.5*V_s_b                                #Sending end line voltage(V)
V_s_b_llkv = V_s_b_ll/1000.0                           #Sending end line voltage(kV)
angle_V_Is_b = cmath.phase(I_s_b)*180/math.pi          #Angle between V_r and I_s_b(°)
angle_V_Vs_b = cmath.phase(V_s_b)*180/math.pi          #Angle between V_r and V_s_b(°)
angle_Is_Vs_b = angle_V_Is_b-angle_V_Vs_b              #Angle between V_s_b and I_s_b(°)
PF_s_b = math.cos(angle_Is_Vs_b*math.pi/180)           #Sending end power factor
P_s_b = 3**0.5*abs(V_s_b_ll*I_s_b)*PF_s_b              #Sending end power(W)
n_b = P_r/P_s_b*100                                    #Efficiency of transmission(%)
reg_b = (abs(V_s_b_ll)-V_r)/V_r*100                    #Voltage regulation(%)
A_b = 1+(1.0/2)*Y*Z                                    #Constant
B_b = Z*(1+(1.0/4)*Y*Z)                                #Constant(ohm)
C_b =Y                                                 #Constant(mho)
D_b = A_b                                              #Constant
#Alternative solution for case(b)
V_s_ba = A_b*V_r_a+B_b*I_r_a                           #Sending end voltage(V)
V_s_ba_ll = 3**0.5*V_s_ba                              #Sending end line voltage(V)
V_s_ba_llkv = V_s_ba_ll/1000.0                         #Sending end line voltage(kV)
I_s_ba = C_b*V_r_a+D_b*I_r_a                           #Sending end current(A)
angle_V_Is_ba = cmath.phase(I_s_ba)*180/math.pi        #Angle between V_r and I_s_b(°)
angle_V_Vs_ba = cmath.phase(V_s_ba)*180/math.pi        #Angle between V_r and V_s_b(°)
angle_Is_Vs_ba = angle_V_Is_ba-angle_V_Vs_ba           #Angle between V_s_b and I_s_b(°)
PF_s_ba = math.cos(angle_Is_Vs_ba*math.pi/180)         #Sending end power factor
P_s_ba = 3**0.5*abs(V_s_ba_ll*I_s_ba)*PF_s_ba          #Sending end power(W)
n_ba = P_r/P_s_ba*100                                  #Efficiency of transmission(%)
reg_ba = (abs(V_s_ba_ll)-V_r)/V_r*100                  #Voltage regulation(%)
#Case(c)
I_c2_c = Y/2.0*V_r_a                              #Current through shunt admittance at receiving end(A)
I_c = I_r_a+I_c2_c                                #Current through impedance(A)
V_s_c = V_r_a+I_c*Z                               #Sending end voltage(V)
V_s_c_ll = 3**0.5*V_s_c                           #Sending end line voltage(V)
V_s_c_llkv = V_s_c_ll/1000.0                      #Sending end line voltage(kV)
I_c1_c = V_s_c*Y/2.0                              #Current through shunt admittance at sending end(A)
I_s_c = I_c+I_c1_c                                #Sending end current(A)
angle_V_Is_c = cmath.phase(I_s_c)*180/math.pi     #Angle between V_r and I_s_c(°)
angle_V_Vs_c = cmath.phase(V_s_c)*180/math.pi     #Angle between V_r and V_s_c(°)
angle_Is_Vs_c = angle_V_Is_c-angle_V_Vs_c         #Angle between V_s_c and I_s_c(°)
PF_s_c = math.cos(angle_Is_Vs_c*math.pi/180)      #Sending end power factor
P_s_c = 3**0.5*abs(V_s_c_ll*I_s_c)*PF_s_c         #Sending end power(W)
n_c = P_r/P_s_c*100                               #Efficiency of transmission(%)
reg_c = (abs(V_s_c_ll)-V_r)/V_r*100               #Voltage regulation(%)
A_c = 1+(1.0/2)*Y*Z                               #Constant
B_c = Z                                           #Constant(ohm)
C_c =Y*(1+(1.0/4)*Y*Z)                            #Constant(mho)
D_c = A_c                                         #Constant
#Alternative solution for case(c)
V_s_ca = A_c*V_r_a+B_c*I_r_a                      #Sending end voltage(V)
V_s_ca_ll = 3**0.5*V_s_ca                         #Sending end line voltage(V)
V_s_ca_llkv = V_s_ca_ll/1000.0                    #Sending end line voltage(kV)
I_s_ca = C_c*V_r_a+D_c*I_r_a                      #Sending end current(A)
angle_V_Is_ca = cmath.phase(I_s_ca)*180/math.pi   #Angle between V_r and I_s_c(°)
angle_V_Vs_ca = cmath.phase(V_s_ca)*180/math.pi   #Angle between V_r and V_s_c(°)
angle_Is_Vs_ca = angle_V_Is_ca-angle_V_Vs_ca      #Angle between V_s_b and I_s_c(°)
PF_s_ca = math.cos(angle_Is_Vs_ca*math.pi/180)    #Sending end power factor
P_s_ca = 3**0.5*abs(V_s_ca_ll*I_s_ca)*PF_s_ca     #Sending end power(W)
n_ca = P_r/P_s_ca*100                             #Efficiency of transmission(%)
reg_ca = (abs(V_s_ca_ll)-V_r)/V_r*100             #Voltage regulation(%)
#Case(d).(i)
gamma_l = (Y*Z)**0.5                                                #γl
Z_c = (Z/Y)**0.5                                                    #Surge impedance(ohm)
V_s_d1 = V_r_a*cmath.cosh(gamma_l)+I_r_a*Z_c*cmath.sinh(gamma_l)    #Sending end voltage(V)
V_s_d1_ll = 3**0.5*V_s_d1                                           #Sending end line voltage(V)
V_s_d1_llkv = V_s_d1_ll/1000.0                                      #Sending end line voltage(kV)
I_s_d1 = V_r_a/Z_c*cmath.sinh(gamma_l)+I_r_a*cmath.cosh(gamma_l)    #Sending end current(A)
angle_V_Is_d1 = cmath.phase(I_s_d1)*180/math.pi                     #Angle between V_r and I_s_d(°)
angle_V_Vs_d1 = cmath.phase(V_s_d1)*180/math.pi                     #Angle between V_r and V_s_d(°)
angle_Is_Vs_d1 = angle_V_Is_d1-angle_V_Vs_d1                        #Angle between V_s_d and I_s_d(°)
PF_s_d1 = math.cos(angle_Is_Vs_d1*math.pi/180)                      #Sending end power factor
P_s_d1 = 3**0.5*abs(V_s_d1_ll*I_s_d1)*PF_s_d1                       #Sending end power(W)
n_d1 = P_r/P_s_d1*100                                               #Efficiency of transmission(%)
reg_d1 = (abs(V_s_d1_ll)-V_r)/V_r*100                               #Voltage regulation(%)
A_d1 = cmath.cosh(gamma_l)                                          #Constant
B_d1 = Z_c*cmath.sinh(gamma_l)                                      #Constant(ohm)
C_d1 = (1/Z_c)*cmath.sinh(gamma_l)                                  #Constant(mho)
D_d1 = A_d1                                                         #Constant
#Case(d).(ii)
A_d2 = (1+(Y*Z/2)+((Y*Z)**2/24.0))                       #Constant
B_d2 = Z*(1+(Y*Z/6)+((Y*Z)**2/120))                      #Constant(ohm)
C_d2 = Y*(1+(Y*Z/6)+((Y*Z)**2/120))                      #Constant(mho)
D_d2 = A_d2                                              #Constant
V_s_d2 = A_d2*V_r_a+B_d2*I_r_a                           #Sending end voltage(V)
V_s_d2_ll = 3**0.5*V_s_d2                                #Sending end line voltage(V)
V_s_d2_llkv = V_s_d2_ll/1000.0                           #Sending end line voltage(kV)
I_s_d2 = C_d2*V_r_a+D_d2*I_r_a                           #Sending end current(A)
angle_V_Is_d2 = cmath.phase(I_s_d2)*180/math.pi          #Angle between V_r and I_s_d(°)
angle_V_Vs_d2 = cmath.phase(V_s_d2)*180/math.pi          #Angle between V_r and V_s_d(°)
angle_Is_Vs_d2 = angle_V_Is_d2-angle_V_Vs_d2             #Angle between V_s_d and I_s_d(°)
PF_s_d2 = math.cos(angle_Is_Vs_d2*math.pi/180)           #Sending end power factor
P_s_d2 = 3**0.5*abs(V_s_d2_ll*I_s_d2)*PF_s_d2            #Sending end power(W)
n_d2 = P_r/P_s_d2*100                                    #Efficiency of transmission(%)
reg_d2 = (abs(V_s_d2_ll)-V_r)/V_r*100                    #Voltage regulation(%)

#Result
print('Case(a): Short line approximation')
print('Sending end voltage, V_s = %.1f∠%.1f° kV (line-to-line)' %(abs(V_s_a_llkv),cmath.phase(V_s_a_llkv)*180/math.pi))
print('Voltage regulation = %.1f percent' %reg_a)
print('Transmission efficiency, η = %.1f percent' %n_a)
print('A = D = %.f ' %A_a)
print('B = %.1f∠%.1f° ohm' %(abs(B_a),cmath.phase(B_a)*180/math.pi))
print('C = %.f ' %C_a)
print('\nCase(b): Nominal T method approximation')
print('Sending end voltage, V_s = %.1f∠%.1f° kV (line-to-line)' %(abs(V_s_b_llkv),cmath.phase(V_s_b_llkv)*180/math.pi))
print('Voltage regulation = %.2f percent' %reg_b)
print('Transmission efficiency, η = %.1f percent' %n_b)
print('A = D = %.3f∠%.2f° ' %(abs(A_b),cmath.phase(A_b)*180/math.pi))
print('B = %.1f∠%.1f° ohm' %(abs(B_b),cmath.phase(B_b)*180/math.pi))
print('C = %.2e∠%.f° S' %(abs(C_b),cmath.phase(C_b)*180/math.pi))
print('\tALTERNATIVE SOLUTION:')
print('\tSending end voltage, V_s = %.1f∠%.1f° kV (line-to-line)' %(abs(V_s_ba_llkv),cmath.phase(V_s_ba_llkv)*180/math.pi))
print('\tVoltage regulation = %.2f percent' %reg_ba)
print('\tTransmission efficiency, η = %.1f percent' %n_ba)
print('\tA = D = %.3f∠%.2f° ' %(abs(A_b),cmath.phase(A_b)*180/math.pi))
print('\tB = %.1f∠%.1f° ohm' %(abs(B_b),cmath.phase(B_b)*180/math.pi))
print('\tC = %.2e∠%.f° S' %(abs(C_b),cmath.phase(C_b)*180/math.pi))
print('\nCase(c): Nominal π method approximation')
print('Sending end voltage, V_s = %.f∠%.1f° kV (line-to-line)' %(abs(V_s_c_llkv),cmath.phase(V_s_c_llkv)*180/math.pi))
print('Voltage regulation = %.2f percent' %reg_c)
print('Transmission efficiency, η = %.1f percent' %n_c)
print('A = D = %.3f∠%.2f° ' %(abs(A_c),cmath.phase(A_c)*180/math.pi))
print('B = %.1f∠%.1f° ohm' %(abs(B_c),cmath.phase(B_c)*180/math.pi))
print('C = %.2e∠%.1f° mho' %(abs(C_c),cmath.phase(C_c)*180/math.pi))
print('\tALTERNATIVE SOLUTION:')
print('\tSending end voltage, V_s = %.1f∠%.1f° kV (line-to-line)' %(abs(V_s_ca_llkv),cmath.phase(V_s_ca_llkv)*180/math.pi))
print('\tVoltage regulation = %.2f percent' %reg_ca)
print('\tTransmission efficiency, η = %.1f percent' %n_ca)
print('\tA = D = %.3f∠%.2f° ' %(abs(A_c),cmath.phase(A_c)*180/math.pi))
print('\tB = %.1f∠%.1f° ohm' %(abs(B_c),cmath.phase(B_c)*180/math.pi))
print('\tC = %.2e∠%.f° S' %(abs(C_c),cmath.phase(C_c)*180/math.pi))
print('\nCase(d): Long Line Rigorous Solution')
print(' Case(i): Using Convergent Series (Real Angles) Method')
print(' Sending end voltage, V_s = %.f∠%.1f° kV (line-to-line)' %(abs(V_s_d1_llkv),cmath.phase(V_s_d1_llkv)*180/math.pi))
print(' Voltage regulation = %.2f percent' %reg_d1)
print(' Transmission efficiency, η = %.1f percent' %n_d1)
print(' A = D = %.3f∠%.2f° ' %(abs(A_d1),cmath.phase(A_d1)*180/math.pi))
print(' B = %.f∠%.1f° ohm' %(abs(B_d1),cmath.phase(B_d1)*180/math.pi))
print(' C = %.2e∠%.1f° mho' %(abs(C_d1),cmath.phase(C_d1)*180/math.pi))
print('\n Case(ii): Using Convergent Series (Complex Angles) Method')
print(' Sending end voltage, V_s = %.f∠%.1f° kV (line-to-line)' %(abs(V_s_d2_llkv),cmath.phase(V_s_d2_llkv)*180/math.pi))
print(' Voltage regulation = %.2f percent' %reg_d2)
print(' Transmission efficiency, η = %.1f percent' %n_d2)
print(' A = D = %.3f∠%.2f° ' %(abs(A_d2),cmath.phase(A_d2)*180/math.pi))
print(' B = %.1f∠%.1f° ohm' %(abs(B_d2),cmath.phase(B_d2)*180/math.pi))
print(' C = %.2e∠%.1f° mho' %(abs(C_d2),cmath.phase(C_d2)*180/math.pi))
print('\nNOTE: Changes in obtained answer from that of textbook is due to more precision')

Case(a): Short line approximation
Sending end voltage, V_s = 239.7∠5.4° kV (line-to-line)
Voltage regulation = 9.0 percent
Transmission efficiency, η = 96.6 percent
A = D = 1
B = 144.3∠76.0° ohm
C = 0

Case(b): Nominal T method approximation
Sending end voltage, V_s = 225.0∠6.5° kV (line-to-line)
Voltage regulation = 2.28 percent
Transmission efficiency, η = 96.4 percent
A = D = 0.935∠1.00°
B = 139.6∠76.4° ohm
C = 9.30e-04∠90° S
ALTERNATIVE SOLUTION:
Sending end voltage, V_s = 225.0∠6.5° kV (line-to-line)
Voltage regulation = 2.28 percent
Transmission efficiency, η = 96.4 percent
A = D = 0.935∠1.00°
B = 139.6∠76.4° ohm
C = 9.30e-04∠90° S

Case(c): Nominal π method approximation
Sending end voltage, V_s = 226∠6.6° kV (line-to-line)
Voltage regulation = 2.67 percent
Transmission efficiency, η = 97.2 percent
A = D = 0.935∠1.00°
B = 144.3∠76.0° ohm
C = 9.00e-04∠90.5° mho
ALTERNATIVE SOLUTION:
Sending end voltage, V_s = 225.9∠6.6° kV (line-to-line)
Voltage regulation = 2.67 percent
Transmission efficiency, η = 97.2 percent
A = D = 0.935∠1.00°
B = 144.3∠76.0° ohm
C = 9.00e-04∠90° S

Case(d): Long Line Rigorous Solution
Case(i): Using Convergent Series (Real Angles) Method
Sending end voltage, V_s = 225∠6.5° kV (line-to-line)
Voltage regulation = 2.47 percent
Transmission efficiency, η = 97.0 percent
A = D = 0.936∠0.98°
B = 141∠76.3° ohm
C = 9.10e-04∠90.3° mho

Case(ii): Using Convergent Series (Complex Angles) Method
Sending end voltage, V_s = 225∠6.5° kV (line-to-line)
Voltage regulation = 2.47 percent
Transmission efficiency, η = 97.0 percent
A = D = 0.936∠0.97°
B = 141.2∠76.3° ohm
C = 9.10e-04∠90.3° mho

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


## Example 2.3.21, Page number 153¶

In :
import math
import cmath

#Variable declaration
V_r = 132.0*10**3                              #Line voltage at receiving end(V)
PF_r = 0.8                                     #Lagging power factor
A = 0.99*cmath.exp(1j*0.3*math.pi/180)         #Constant
B = 70.0*cmath.exp(1j*69.0*math.pi/180)        #Constant(ohms)
C = A                                          #Constant
D = 4.0*10**-4*cmath.exp(1j*90.0*math.pi/180)  #Constant

#Calculation
E_r = V_r/3**0.5                                          #Receiving end phase voltage(V)
I_r = P_L/(3**0.5*V_r)*cmath.exp(1j*-math.acos(PF_r))     #Line current(A)
E_s = A*E_r+B*I_r                                         #Sending end voltage(V)
E_s_llkV = 3**0.5*E_s/1000.0                              #Sending end line voltage(kV)
I_s = C*I_r+D*E_r                                         #Sending end current(A)
angle_Er_Es = cmath.phase(E_s)*180/math.pi                #Angle between E_r and E_s(°)
angle_Er_Is = cmath.phase(I_s)*180/math.pi                #Angle between E_r and I_s(°)
angle_Es_Is = angle_Er_Es-angle_Er_Is                     #Angle between E_s and I_s(°)
PF_s = math.cos(angle_Es_Is*math.pi/180)                  #Sending end power factor
P_s = 3*abs(E_s*I_s)*PF_s                                 #Sending end power(W)
P_skW = P_s/1000.0                                        #Sending end power(kW)
P_r = P_L*PF_r                                            #Receiving end power(W)
n = P_r/P_s*100                                           #Transmission efficiency(%)

#Result
print('Case(i)  : Sending end voltage, E_s = %.1f∠%.f° kV (line-to-line)' %(abs(E_s_llkV),cmath.phase(E_s_llkV)*180/math.pi))
print('Case(ii) : Sending end current, I_s = %.1f∠%.1f° A' %(abs(I_s),cmath.phase(I_s)*180/math.pi))
print('Case(iii): Sending end power, P_s = %.f kW' %(P_skW))
print('Case(iv) : Efficiency of transmission = %.2f percent' %n)
print('\nNOTE: Changes in obtained answer from that textbook is due to more precision')

Case(i)  : Sending end voltage, E_s = 151.5∠5° kV (line-to-line)
Case(ii) : Sending end current, I_s = 178.4∠-28.7° A
Case(iii): Sending end power, P_s = 38915 kW
Case(iv) : Efficiency of transmission = 92.51 percent

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


## Example 2.3.23, Page number 156¶

In :
import math
import cmath

#Variable declaration
A_1 = 0.98*cmath.exp(1j*2.0*math.pi/180)         #Constant of 1st line
B_1 = 28.0*cmath.exp(1j*69.0*math.pi/180)        #Constant of 1st line(ohms)
C_1 = 0.0002*cmath.exp(1j*88.0*math.pi/180)      #Constant of 1st line(mho)
D_1 = A_1                                        #Constant of 1st line
A_2 = 0.95*cmath.exp(1j*3.0*math.pi/180)         #Constant of 2nd line
B_2 = 40.0*cmath.exp(1j*85.0*math.pi/180)        #Constant of 2nd line(ohms)
C_2 = 0.0004*cmath.exp(1j*90.0*math.pi/180)      #Constant of 2nd line(mho)
D_2 = A_2                                        #Constant of 2nd line

#Calculation
A = A_1*A_2+B_1*C_2         #Constant
B = A_1*B_2+B_1*D_2         #Constant(ohm)
C = C_1*A_2+D_1*C_2         #Constant(mho)
D = C_1*B_2+D_1*D_2         #Constant

#Result
print('A = %.3f∠%.1f° ' %(abs(A),cmath.phase(A)*180/math.pi))
print('B = %.1f∠%.f° ohm' %(abs(B),cmath.phase(B)*180/math.pi))
print('C = %.6f∠%.1f° mho' %(abs(C),cmath.phase(C)*180/math.pi))
print('D = %.3f∠%.1f° ' %(abs(D),cmath.phase(D)*180/math.pi))

A = 0.921∠5.3°
B = 65.3∠81° ohm
C = 0.000582∠91.7° mho
D = 0.923∠5.1°


## Example 2.3.24, Page number 156-157¶

In :
import math
import cmath

#Variable declaration
A = 0.94*cmath.exp(1j*1.5*math.pi/180)         #Constant
B = 150.0*cmath.exp(1j*67.2*math.pi/180)       #Constant(ohm)
D = A                                          #Constant
Z_t = 100.0*cmath.exp(1j*70.0*math.pi/180)     #Series impedance(ohm)

#Calculation
C = (A*D-1)/B               #Constant(mho)
A_0 = A*(1+Y_t*Z_t)+B*Y_t   #Constant
B_0 = A*Z_t+B               #Constant(ohm)
C_0 = C*(1+Y_t*Z_t)+D*Y_t   #Constant(mho)
D_0 = C*Z_t+D               #Constant

#Result
print('A_0 = %.3f∠%.f° ' %(abs(A_0),cmath.phase(A_0)*180/math.pi))
print('B_0 = %.f∠%.1f° ohm' %(abs(B_0),cmath.phase(B_0)*180/math.pi))
print('C_0 = %.6f∠%.1f° mho' %(abs(C_0),cmath.phase(C_0)*180/math.pi))
print('D_0 = %.3f∠%.1f° ' %(abs(D_0),cmath.phase(D_0)*180/math.pi))
print('\nNOTE: Changes in obtained answer from that of textbook is due to more precision')

A_0 = 1.000∠1°
B_0 = 244∠68.9° ohm
C_0 = 0.000640∠85.7° mho
D_0 = 0.861∠3.4°

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


## Example 2.3.25, Page number 163¶

In :
import math

#Variable declaration
z = complex(0.2,0.6)     #Per phase impedance(ohm)
V_r = 6351.0             #Receiving end voltage per phase(V)
reg = 7.5/100.0          #Voltage regulation

#Calculation
V_s = (1+reg)*V_r                                  #Sending end voltage per phase(V)
R = z.real                                         #Resistance of the line(ohm)
X = z.imag                                         #Reactance of the line(ohm)
Z = (R**2+X**2)**0.5                               #Impedance per phase(ohm)
P_m = (V_r**2/Z)*((Z*V_s/V_r)-R)                   #Maximum power transmitted through line(W/phase)
P_m_MW = P_m/10**6                                 #Maximum power transmitted through line(MW/phase)
P_m_MWtotal = 3*P_m_MW                             #Total maximum power(MW)
Q = -(V_r**2*X)/Z**2                               #Reactive power per phase(Var)
Q_MW = Q/10**6                                     #Reactive power per phase(MVAR)
phi_r = math.atan(abs(Q_MW/P_m_MW))*180/math.pi    #Φ_r(°)
PF_r = math.cos(phi_r*math.pi/180)                 #Receiving end lagging PF
I = P_m/(V_r*PF_r)                                 #Current delivered(A)
I_KA = I/1000.0                                    #Current delivered(KA)
loss = 3*I**2*R                                    #Total line loss(W)
loss_MW = loss/10**6                               #Total line loss(MW)

#Result
print('Maximum power transmitted through the line, P_m = %.1f MW' %P_m_MWtotal)
print('Receiving end power factor = %.2f (lagging)' %PF_r)
print('Total line loss = %.2f MW' %loss_MW)

Maximum power transmitted through the line, P_m = 91.8 MW
Receiving end power factor = 0.45 (lagging)
Total line loss = 68.39 MW


## Example 2.3.26, Page number 163-164¶

In :
import math
import cmath

#Variable declaration
L = 100.0              #Length of line(km)
PF_r = 1.0             #Receiving end Power factor
Z_c = 400.0            #Characteristic impedance(ohm)
V_s = 230.0            #Sending end voltage(kV)

#Calculation
beta_L_d = beta_L*180/math.pi               #(°)
A = math.cos(beta_L)                        #Constant
B = 1j*Z_c*math.sin(beta_L)                 #Constant
alpha_angle = cmath.phase(A)*180/math.pi    #α(°)
beta_angle = cmath.phase(B)*180/math.pi     #β(°)
V_r = V_s                                   #Receiving end voltage due to lossless line(kV)
P_max = (V_s*V_r/abs(B))-(abs(A)*V_r**2/abs(B))*math.cos((beta_angle-alpha_angle)*math.pi/180)   #Maximum power transferred(MW)

#Result
print('Maximum power that can be transferred to the load at receiving end, P_max = %.f MW' %P_max)
print('\nNOTE: Changes in obtained answer from that of textbook is due to more precision')

Maximum power that can be transferred to the load at receiving end, P_max = 1105 MW

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


## Example 2.3.27, Page number 168-170¶

In :
import math
import cmath

#Variable declaration
L = 160.0                                        #Length of line(km)
PF_r = 0.85                                      #Receiving end lagging power factor
E_r = 230.0*10**3                                #Receiving end voltage(V)
A = 0.9785*cmath.exp(1j*0.3*math.pi/180)         #Constant
B = 85.2*cmath.exp(1j*77.47*math.pi/180)         #Constant(ohm)
C = 0.000503*cmath.exp(1j*90.1*math.pi/180)      #Constant(mho)
D = A                                            #Constant

#Calculation
V_r = E_r/3**0.5                                  #Receiving end phase voltage(V)
I_r = P_r/(3**0.5*E_r*PF_r)                       #Line current(A)
#Case(a)
V_s_ph = E_r*85.2/V_r                             #Sending end voltage per phase(kV)
V_s = 3**0.5*V_s_ph                               #Sending end line voltage(kV)
I_s = 36.0*10**6/(V_s_ph*10**3)                   #Sending end current(A)
theta_s = 27.0                                    #θ_s(°)
PF_s = math.cos(theta_s*math.pi/180)              #Sending end power factor
regulation = (V_s-(E_r/1000.0))/(E_r/1000.0)*100  #Regulation
P_s = 3*V_s_ph*I_s*PF_s*1000.0                    #Sending end power(W)
n = P_r/P_s*100                                   #Efficiency(%)
#Case(b)
#Case(c)
power_phase = load_c/3.0                          #Power per phase(MW)
V_s_c = 238*85.2/(V_r/1000)                       #Voltage(kV)
drop_phase_c = V_s_c-133.0                        #Drop per phase(kV)
MVAR_phase_c = 16.0                               #MVAR per phase
MVAR_c = 3*MVAR_phase_c                           #Total MVAR required
#Case(d)
#Case(e)
P_e = 8.15*20                                     #Maximum power for one phase(MW)
P_e_total = 3*P_e                                 #Maximum power for all three phases(MW)
theta_e = 50.5                                    #θ_r(°)
PF_e = math.cos(theta_e*math.pi/180)              #Power factor

#Result
print('Case(a): Sending end voltage = %.f kV (line-to-line)' %V_s)
print('         Sending end current, |I_s| = %.f A' %I_s)
print('         Sending end power factor = %.2f ' %PF_s)
print('         Regulation = %.1f percent' %regulation)
print('         Efficiency of transmission line = %.1f percent' %n)
print('Case(c): Voltage drop per phase = %.1f kV' %drop_phase_c)
print('         Total MVAR required = %.f MVAR' %MVAR_c)
print('Case(d): Leading MVAR = %.f MVAR' %MVAR_3_d)
print('Case(e): Maximum power transmitted = %.f MW' %P_e_total)
print('         Corresponding power factor = %.3f (leading)' %PF_e)
print('\nNOTE: ERROR: PF = 0.85, not PF = 0.8 as given in textbook statement')
print('      All values are taken directly from textbook circle diagram')

Case(a): Sending end voltage = 256 kV (line-to-line)
Sending end current, |I_s| = 244 A
Sending end power factor = 0.89
Regulation = 11.1 percent
Efficiency of transmission line = 93.5 percent
Case(b): Load carried at 8 percent regulation = 75 MW
Case(c): Voltage drop per phase = 19.7 kV
Total MVAR required = 48 MVAR
Case(d): Leading MVAR = 690 MVAR
Case(e): Maximum power transmitted = 489 MW
Corresponding power factor = 0.636 (leading)

NOTE: ERROR: PF = 0.85, not PF = 0.8 as given in textbook statement
All values are taken directly from textbook circle diagram


## Example 2.3.28, Page number 170-172¶

In :
import math
import cmath

#Variable declaration
E_r = 220.0                                      #Receiving end voltage(kV)
PF_r = 0.8                                       #Receiving end lagging power factor
A = 0.9055*cmath.exp(1j*0.6*math.pi/180)         #Constant
B = 156.0*cmath.exp(1j*84.5*math.pi/180)         #Constant(ohm)
C = 0.00116*cmath.exp(1j*90.2*math.pi/180)       #Constant(mho)
D = A                                            #Constant

#Calculation
#Case(i)
P_base = E_r**2/abs(B)              #Base power(MW)
V_s_pu = 1.135                      #Sending end voltage(p.u)
V_s = V_s_pu*E_r                    #Sending end voltage(kV)
delta = 11.8                        #Phase shift between sending & receiving end voltage(°)
#Case(ii)
P_s_real = 0.249*P_base             #Sending end real power(MW)
P_s_reactive = 0.035*P_base         #Sending end reactive power(MW)
line_loss = P_s_real-P_r            #Line loss(MW)
#Case(iii)
P_s_max = 1.26*P_base               #Sending end maximum power(MW)
P_r_max = 1.035*P_base              #Receiving end maximum power(MW)
#Case(iv)
SPM_cap = 0.03*P_base               #SPM capacity to improve pf(MVAR)
#Case(v)
V_s_5 = 1.105*E_r                   #Sending end voltage(kV)
reg = (V_s_5-E_r)/E_r*100           #Voltage regulation(%)
reg_noSPM = (V_s-E_r)/E_r*100       #Voltage regulation without SPM(%)

#Result
print('Case(i)  : Sending end voltage = %.f kV' %V_s)
print('           Phase shift between sending end & receiving end voltage, δ = %.1f° ' %delta)
print('Case(ii) : Sending end real power = %.1f MW' %P_s_real)
print('           Sending end reactive power = %.2f MVAR' %P_s_reactive)
print('           Line loss = %.1f MW' %line_loss)
print('Case(iii): Maximum power transmitted from sending end = %.f MW' %P_s_max)
print('           Maximum power received at receiving end = %.2f MW' %P_r_max)
print('Case(iv) : SPM capacity to improve power factor = %.1f MVAR' %SPM_cap)
print('Case(v)  : Voltage regulation with SPM = %.1f percent' %reg)
print('           Voltage regulation without SPM = %.1f percent' %reg_noSPM)
print('           Regulation is improved from %.1f to %.1f percent with SPM installation' %(reg_noSPM,reg))
print('\nNOTE: Some values are taken directly from textbook circle diagram')

Case(i)  : Sending end voltage = 250 kV
Phase shift between sending end & receiving end voltage, δ = 11.8°
Case(ii) : Sending end real power = 77.3 MW
Sending end reactive power = 10.86 MVAR
Line loss = 2.3 MW
Case(iii): Maximum power transmitted from sending end = 391 MW
Maximum power received at receiving end = 321.12 MW
Case(iv) : SPM capacity to improve power factor = 9.3 MVAR
Case(v)  : Voltage regulation with SPM = 10.5 percent
Voltage regulation without SPM = 13.5 percent
Regulation is improved from 13.5 to 10.5 percent with SPM installation

NOTE: Some values are taken directly from textbook circle diagram


## Example 2.3.29, Page number 172-173¶

In :
import math
import cmath

#Variable declaration
V_s = 250.0                                      #Sending end voltage(kV)
V_r = 230.0                                      #Receiving end voltage(kV)
A = 0.9785*cmath.exp(1j*0.3*math.pi/180)         #Constant
B = 85.2*cmath.exp(1j*77.47*math.pi/180)         #Constant(ohm)
D = A                                            #Constant

#Calculation
cos_beta = math.cos(cmath.phase(B))       #cosβ
line_loss_1 = 0.2*20*2*cos_beta           #Line loss when 90MW is supplied(MW/phase)
line_loss_1_total = 3*line_loss_1         #Total line loss(MW)
line_loss_2 = 0.13*20*2*cos_beta          #Minimum line loss(MW/phase)
line_loss_2_total = 3*line_loss_2         #Total minimum line loss(MW)
power = 3*0.2*20                          #Receiving end power(MW)
PF = 0.174                                #Power factor

#Result
print('Line loss when 90MW is supplied to load at receiving end = %.2f MW/phase = %.2f MW (total)' %(line_loss_1,line_loss_1_total))
print('Minimum line loss = %.3f MW/phase = %.3f MW (total)' %(line_loss_2,line_loss_2_total))
print('Receiving end power = %.f MW' %power)
print('Power factor = %.3f ' %PF)
print('\nNOTE: Changes in obtained answer from that of textbook is due to more precision')
print('      Some values are taken directly from textbook circle diagram')

Line loss when 90MW is supplied to load at receiving end = 1.74 MW/phase = 5.21 MW (total)
Minimum line loss = 1.128 MW/phase = 3.384 MW (total)
Receiving end power = 12 MW
Power factor = 0.174

NOTE: Changes in obtained answer from that of textbook is due to more precision
Some values are taken directly from textbook circle diagram