**Define the Gauss Seidel Method**

The Gauss-Seidel method is an iterative method in which the voltage of each node is calculated in turn using the most up to date voltages for the other nodes. The voltage calculated is that which gives the correct solution for the node considered, based upon the data specified and the previously calculated voltages of the other nodes. The method is equally applicable to both a.c. and d.c. circuits.

Consider the three busbar system shown in Figure.

Using Nodal analysis, the current fed into each bus is given as:

I_{1} = Y_{11} V_{1} + Y_{12} V_{2} + Y_{13} V_{3}

I_{2} = Y_{21} V_{1} + Y_{22 }V_{2} + Y_{23} V_{3}

I_{3} = Y_{31} V_{1} + Y_{32} V_{2} + Y_{33} V_{3}

or, in matrix form,

[I] = [Y] [V]

where Y_{nn} is node n self admittance (sum of all the admittances connected to node n).

Y_{nm} is the mutual admittance between nodes n and m (sum of the admittances connecting node n to node m)

Note that: (i) all mutual admittance terms have a negative sign,

(ii) all injected currents are positive.

Recalling that S = P + j Q = V I *

In general, for an n busbar system, the current fed into busbar k is given as:

Therefore, the voltage at busbar k is given as:

Where V_{k} and V_{i} are the voltages at nodes 'k' and 'i', respectively,

Y_{kk} is node 'k' self admittance,

Y_{ki} is the mutual admittance between nodes 'k' and 'i',

P_{k} - jQ_{k} is the total power flowing into node 'k'.