The oneline diagram of a three bus electrical grid is shown

The one-line diagram of a three bus electrical grid is shown below Bus number one is the slack bus with a voltage of V_t = 1 0 degree pu kV. Bus number three is a voltage control bus with enough reactive power generation capability to maintain its voltage magnitude at |V_3| = 1.00 pu kV. Bus number two is a load bus drawing a total power of S_D = 0.9 + j0.4 pu MVA. The transmission lines connecting the buses are modeled by series admittance as shown in the figure. a. Calculate the Y_bus matrix. b. We want to do a load flow using Newton-Raphson algorithm. How many unknown variables we have? Specify them Which one of these unknowns are calculated by iterations according to the Newton-Raphson algorithm9 How and when are the rest of unknowns calculated? c. Write the load flow equations needed to be used in the Newton-Raphson algorithm for calculating voltages. d. Substitute the numerical values available (except the first guess at part f) into equations of part c and write th

Solution

(a) The bus admittance matrix is given by

Y_BUS = [-j2.5 j2.5 0

j2.5 -j4.5 j2

0 j2 -j2 ]

(b) There are six unknown variables which are to be determined. They are

1) Real and Reactive power at the slack bus i.e., bus 1

2) Magnitude and Phase angle of voltage at the load bus i.e., bus 2

3) Phase angle of the voltage and Reactive power at the voltage controlled bus i.e, bus 3

Quantities in the serial numbers 2 and 3 are calculated by iterations. Real and Reactive power at the slack bus is calculated after the convergence of iterations.

(c) The load flow equations needed to calculate the voltages are given below:

V₂ = [(P₂ - jQ₂)/V₂ - Y₂₁V₁ - Y₂₃V₃ ] / Y₂₂

     Bus 3 is a regulated bus where V and P are specified. Reactive power is first computed as:

Q₃ = - imaginery part of [V₃^*(V₃Y₃₃ + V₁Y₃₁ + V₂Y₃₂)]

     V₃ = [(P₃ - jQ3)/V₃ - Y₃₁V₁ - Y₃₂V₂ ] / Y₃₃

(d) With values substituted, the above equations are:

V₂ = -(1/j4.5)*(-0.9-j0.4 -j2.5 -j2) = 1.0889 -j0.2

likewise the values are substituted in Q3 and V3