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Per-unit method of solving of 3-phase problemsFor the system shown in Figure 4, draw the electric circuit or reactance diagram, with all reactances marked in per-unit (p.u.) values, and find the generator terminal voltage assuming both motors operating at 12 kV, three-quarters load, and unity power factor. GeneratorTransformers(each)Motor AMotor BTransmissionLine13.8kV25,000 kVA15,000 kVA10,000 kVA–25,000 kVA 3-phase13.2/69 kV13.0 kV13.0 kV–X” = 15 percentX L = 15 percentX” = 15 percentX” = 15 percentX = 65 Ω. The base voltage of the transmission line is then determined by the turns ratio of the connecting transformer:(13.8 kV)(69 kV / 13.2 kV) = 72.136 kVThe base voltage of the motors is determined likewise but with the 72.136 kV value, thus:(72.136 kV)(13.2 kV / 69 kV) = 13.8 kVThe selected base S value remains constant throughout the system, but the base voltage is 13.8 kV at the generator and at the motors, and 72.136 kV on the transmission line.2. Calculate the Generator ReactanceNo calculation is necessary for correcting the value of the generator reactance because it is given as 0.15 p.u. (15 percent), based on 25,000 kVA and 13.8 kV.
If a different S base were used in this problem, then a correction would be necessary as shown for the transmission line, and power transformers.3. Calculate the Transformer ReactanceIt is necessary to make a correction when the transformer nameplate reactance is used because the calculated operation is at a different voltage, 13.8 kV / 72.136 kV instead of 13.2 kV / 69 kV. Use the equation for correction: per-unit reactance:(nameplate per-unit reactance) (base kVA/nameplate kVA) (nameplate kV/base kV) 2 =(0.11) (25,000/25,000) (13.2/13.8) 2 = 0.101 p.u.This applies to each transformer.4. Calculate the Transmission-Line ReactanceUse the equation:. X per unit = (ohms reactance)(base kVA)/(1000)(base kV) 2 =. X per unit = (65) (25,000)/(1000)(72.1) 2 = 0.313 p.u.5. Calculate the Reactance of the MotorsCorrections need to be made in the nameplate ratings of both motors because of differences of ratings in kVA and kV as compared with those selected for calculations in this problem.
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Use the correcting equation from Step 3, above. Thus, expressed in per unit, the combined motor current is obtained by using the equation:I per unit = per-unit power/per-unit voltage = 0.75/0.87 = 0.862 ∠0° p.u.8. Calculate the Generator Terminal VoltageThe voltage at the generator terminals is:. V G = V motor + the voltage drop through transformers and transmission line. V G = 0.87 ∠0° + 0.862 ∠0°(j0.101 + j0.313 + j0.101). V G = 0.87 + j0.444 = 0.977 ∠27.03° p.u.In order to obtain the actual voltage, multiply the per-unit voltage by the base voltage at the generator.
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Thus,. V G = (0.977 ∠27.03°) (13.8 kV) = 13.48 ∠27.03° kVRelated CalculationsIn the solution of these problems, the selection of base voltage and apparent power are arbitrary. However, the base voltage in each section of the circuit must be related in accordance with.