Electrical Machines: Solved Assignments and MATLAB Code
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Added on 2023/04/25
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This document provides solved assignments and MATLAB code for Electrical Machines. It covers topics such as voltage regulation, maximum power angle, and complex power of network. Suitable for students of Electrical Engineering.
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Running head: ELECTRICAL MACHINES ELECTRICAL MACHINES Name of the Student Name of the University Author Note
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1ELECTRICAL MACHINES 8.9: a)Given that, Generator poles P = 4, frequency f = 50 Hz. Generator p.u. reactance Xpu = 0.85. Generator rating = 40 MVA and generator rated voltage = 26 kV. Given, that the field current of the generator is adjusted from 0.75 to 1.5 times of the rated voltage. Hence, the induced voltage is varied from 26*0.75 = 19.5 volts to 26*1.5 = 39 volts. Now, the generator power delivered is P = MVA rating * power factor = 40*(10^6)*cosθ Now, X = Xbase*Xpu. Xbase = ((rated voltage in kV)^2)/MVA = ((26)^2)/40 = 16.9 Ω. Hence, X = 16.9*0.85 = 14.365 Ω. Now, line reactance Xl = 0.5*8 = 4 Ω. Hence, total reactance = 18.365 Ω. Line resistance R = 0.07*8 = 0.56 Ω. Hence, Z = sqrt(R^2 + X^2) = sqrt(0.56^2 + 18.365^2) = 18.3735 Ω. Now, 40*(10^6)*cosθ =(inducedvoltage)2 Z= (line voltage)^2/18.3735 Voltage regulation = (induced voltage – network voltage)/network voltage
2ELECTRICAL MACHINES MATLAB code: rated_volt = 26e3; ind_volt = (0.75:0.1:1.5).*rated_volt; % induced voltage xpu = 0.85; mva = 40e6; xbase = rated_volt^2/mva; power = ((ind_volt).^2)./(18.3735); pf = power./mva; figure(1) subplot(2,1,1) plot(pf(1:3),ind_volt(1:3),'r-') xlabel('power factor') ylabel('induced voltage in volts') net_volt = 24e3; % network voltage volt_regu = (ind_volt - net_volt)./net_volt; leading_pf = linspace(0.5,1,length(volt_regu)); subplot(2,1,2) plot(leading_pf,volt_regu) xlabel('leading power factor')
3ELECTRICAL MACHINES ylabel('voltage regulation') 0.50.550.60.650.70.750.80.85 power factor 2 2.2 2.4 2.6 induced voltage in volts104 0.50.550.60.650.70.750.80.850.90.951 leading power factor -0.2 0 0.2 0.4 0.6 voltage regulation Hence, 10% voltage regulation corresponds to 0.8 power factor (exactly 0.814). b)Now, the maximum power angle is calculated by the following method. The maximum steady state power P = EV/X Here, E = voltage behind the direct axis synchronous reactance or excitation voltage V = voltage of the network = 24 kV. X = Xd + Xl Where, Xd = direct-axis synchronous generator reactance and Xl = line reactance. Now, Xd = Xbase*Xpu. Xbase = ((rated voltage in kV)^2)/MVA = ((26)^2)/40 = 16.9 Ω. Hence, Xd = 16.9*0.85 = 14.365 Ω.
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4ELECTRICAL MACHINES Now, Xl = 0.5*8 = 4 Ω. Hence, X = 14.365 + 4 = 18.365 Ω. MATLAB code: rated_volt = 26e3; ind_volt = (0.75:0.1:1.5).*rated_volt; vnet = 24e3; X = 18.365; max_power = (24e3.*ind_volt)/18.365; plot(ind_volt,max_power) xlabel('induced voltage in volts') ylabel('max power in watts') Plot: 1.822.22.42.62.833.23.43.63.8 induced voltage in volts104 2.5 3 3.5 4 4.5 5 max power in watts 107
5ELECTRICAL MACHINES Now, if the excited voltage goes beyond 19 kV then the rating of generator will exceed more than 40 MVA. Hence, the excitation voltage that can deliver maximum power is 19 KV. 8.12: a) The one line diagram of the problem is shown below. Generator rating = 380 MVA. Xgpu = 1.2 pu. Transformer rating = 480 MVA. Primary side/secondary side voltage = 22 kV/340 kV. % Xtrfr= 15% Hence, the transformers are step up transformers. Line length = 45 miles. Line impedance = 0.07 + j0.5 Ω/mi. Hence, the total line impedance = (0.07 + j0.5)*45 = 3.15 + j*22.5 Ω. Now, the base voltage is considered to be 22 kV. Hence, Xgbase = (kV)^2 /MVA = 22^2/(380) = 1.2737 Ω. Hence, Xg = Xgbase* Xgpu = 1.2737*1.2 = 1.5284 Ω.
6ELECTRICAL MACHINES Now, the reactance of the transformer is connected to the secondary side. Hence, the voltage across reactance is 340 kV. Now, % X = ((kVA)*X)/(10*(kV)^2) = ((480*10^3)*X)/(10*(340^2)) 15 = ((480*10^3)*X)/(10*(340^2)) X = ((10*(340^2))*15)/ (480*10^3) X= 36.13 Ω. Hence, transformer reactance Xtrfr = 36.13 Ω. b)Now, given that pf = 0.88 lagging. Hence, power P = MVA*pf = 380*0.88 = 334.4 MW. Now, Generator current Ig = 334.4 MW / 22kV Now, excitation voltage E = V + jIg*X = 22 kV + Ig*j1.5284 = 41.6 kV∠63.9° c)Absolute value of excitation voltage = 41.6 kV. Now, transmitted power to the network is given by P = ((E*V)sinδ)/X. V = network voltage = 21 kV. X = 1.5284 Ω. Hence, P = (41.6*21*10^6)/ 24.0284 = 571.58sinδ MW Now, in MATLAB the transmitted power vs power angle is plotted. MATLAB code: delta = 0:1:90;
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7ELECTRICAL MACHINES P = 571.58*sind(delta); plot(delta,P) xlabel('power angle in degrees') ylabel('transmitted power in MW') Plot: 0102030405060708090 power angle in degrees 0 100 200 300 400 500 600 transmitted power in MW Hence, from the above plot it can be seen that when power transmitted is 350 MW then the power angle is approximately 50 degrees.
8ELECTRICAL MACHINES 8.5: Generator rating = 250 MVA. Generator line voltage = 24 kV. % reactance = 125%. Now, % reactance = ((kVA)*X)/(10*(kV)^2) = ((250*10^3)*X)/(10*24^2) X = (125*(10*24^2))/(250*10^3) = 2.88 Ω. Network voltage V = 27 kV. The line to neutral voltage or phase voltage = 24/sqrt(3) = 13.8564 kV. Excitation voltage or induced voltage E = 2* line to neutral voltage = 2*13.8564 = 27.7128 kV. Phase angle δ between network and induced voltage = 55 degrees. Now, the complex power of network is given by, Snetwork = Pe + jQe Where,Pe=(E∗V)sinδ X(1)is the real or active power And,Qe=(E∗V)cosδ X–V2 X(2)is the reactive power Now, putting the values calculated above in (1) and (2) we get, Pe=373MVAandQe= -1.49 MVA.