Shows the single-line diagram of a three-bus system which has 2 generators and a

Question

Shows the single-line diagram of a three-bus system which has 2 generators and a load bus. As an operational engineer in a utility company, you are responsible to ensure that the power flow on the system is adequate for any base case or contingency loading situation. The magnitude of voltage at slack bus is adjusted to 1.01 pu while the voltage magnitude at bus 3 is fixed at 1.05 pu. A load consisting of 250 MW and 100 Mvar is taken from bus 2. The bus data and line data arc given in the Table 1 and Table 2 respectively. Line impedances are marked in per unit on a 100 MVA base and the line charging susceptances are neglected. Design the above system using MiPower software and simulate the power flow solution to verify the results using Gauss-Seidel, Newton-Raphson and Fast Decoupled method. Differentiate these three methods for power flow solution in terms of coordinates, arithmetic operations, number iterations, convergence, accuracy, applications and reliability.

Explanation / Answer

Let us assume that an n -bus power system contains a total np number of P-Q buses while the number of P-V (generator) buses be ng such that n = np + ng + 1. Bus-1 is assumed to be the slack bus. We shall further use the mismatch equations of Pi and Qi given in (4.9) and (4.10) respectively. The approach to Newton-Raphson load flow is similar to that of solving a system of nonlinear equations using the Newton-Raphson method: At each iteration we have to form a Jacobian matrix and solve for the corrections from an equation of the type given in (4.27). For the load flow problem, this equation is of the form

(4.30)

where the Jacobian matrix is divided into submatrices as

(4.31)

It can be seen that the size of the Jacobian matrix is ( n + np 1) x ( n + np 1). For example for the 5-bus problem of Fig. 4.1 this matrix will be of the size (7 x 7). The dimensions of the submatrices are as follows:

J11: (n - 1) ´ (n - 1), J12: (n - 1) ´ np, J21: np ´ (n - 1) and J22: np ´ np

The submatrices are

(4.32)

(4.33)

(4.34)

Load Flow by Gauss-Seidel Method

Updating Load Bus Voltages
Updating P-V Bus Voltages
Convergence of the Algorithm

The basic power flow equations (4.6) and (4.7) are nonlinear. In an n -bus power system, let the number of P-Q buses be np and the number of P-V (generator) buses be ng such that n = np + ng + 1. Both voltage magnitudes and angles of the P-Q buses and voltage angles of the P-V buses are unknown making a total number of 2np + ng quantities to be determined. Amongst the known quantities are 2np numbers of real and reactive powers of the P-Q buses, 2ng numbers of real powers and voltage magnitudes of the P-V buses and voltage magnitude and angle of the slack bus. Therefore there are sufficient numbers of known quantities to obtain a solution of the load flow problem. However, it is rather difficult to obtain a set of closed form equations from (4.6) and (4.7). We therefore have to resort to obtain iterative solutions of the load flow problem.

At the beginning of an iterative method, a set of values for the unknown quantities are chosen. These are then updated at each iteration. The process continues till errors between all the known and actual quantities reduce below a pre-specified value. In the Gauss-Seidel load flow we denote the initial voltage of the i th bus by Vi(0) , i = 2, ... , n . This should read as the voltage of the i th bus at the 0th iteration, or initial guess. Similarly this voltage after the first iteration will be denoted by Vi(1) . In this Gauss-Seidel load flow the load buses and voltage controlled buses are treated differently. However in both these type of buses we use the complex power equation given in (4.5) for updating the voltages. Knowing the real and reactive power injected at any bus we can expand (4.5) as

(4.11)

We can rewrite (4.11) as

(4.12)

In this fashion the voltages of all the buses are updated. We shall outline this procedure with the help of the system of Fig. 4.1, with the system data given in Tables 4.1 to 4.3. It is to be noted that the real and reactive powers are given respectively in MW and MVAr. However they are converted into per unit quantities where a base of 100 MVA is chosen.

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