Gauss Seidel Method

The linear system equations are solved using the Gauss Seidel method. The German scientists Carl Friedrich Gauss and Philipp Ludwig Siedel are honoured by the method’s names. Iterative methods are used to solve n linear equations with unknown variables. This approach is very straightforward and is used in digital computers for computation.

The gauss-iteration method has been modified by the gauss-seidel method.The number of iterations is decreased by this modification. The number of iterations in this method is reduced by the value of the unknown right away; the calculated value only replaces the earlier value at the end of the iteration.The gauss-seidel methods converge much more quickly as a result than the Gauss methods. In contrast to the Gauss method, the Gauss seidel method requires a lot fewer iterations to reach the solution.

Let us understand the Gauss-Seidel Method with the help of an example. So Consider the total current entering the k^th bus of an ‘n’ bus system is given by the equation shown below.

The complex power injected into the k^th bus is given as

The complex conjugate of the above equation becomes

Elimination of I_k from the equation (1) and (4) gives

Therefore, the voltage at any bus ‘k’ where P_k and Q_k are specified is given by the equation shown below.

Equation (6) shown above is the major part of the iterative algorithm.

At the bus 2, the equation becomes

At the bus 3, the equation becomes

Now for the k^th bus, the voltage at the (r + 1)^th iteration is given by the equation shown below.

In the above equation, the quantities P_k, Q_k, Y_kk and Y_ki are known, and they do not vary during the iteration cycle.

Now the value of C_k and D_k are shown below, which is computed in the beginning, and it is used in every iteration step.

So For the k^th bus, the voltage at the (r + 1) ^th iteration can be written as shown below.

Acceleration Factors in Gauss Seidel Method

In the Gauss Seidel method, a significant number of iterations are necessary to reach the desired convergence. By applying the acceleration factor to the solution discovered after each iteration, the rate of convergence can be accelerated. So The acceleration factor is a multiplier that improves voltage correction between two subsequent iterations.

Let us consider the acceleration Factor for the i^th bus.

• V_i^(r) is the value of the voltage at the r^th iteration.
• V_i^{(r + 1)} is the value of the voltage at the (r + 1)^th iteration.
• V_{i( accelerated)}^{(r + 1)} is the accelerated new value of the voltage at the (r+ 1) ^th iteration.
• r is the iteration count
• α is the accelerating factor

Then,

Thus, after calculating V_i^{(r + 1)} at ( r + 1)^th iteration, we calculate the value of new estimated bus voltage V_{i( accelerated)}^{(r + 1)} and this new value replaces the previously calculated value.  For real and imaginary components of the voltage different accelerating factors are used.

If V_i is resolved into real and imaginary components as

If α and β are the acceleration factor associated with a_i and b_i then the equation becomes as shown below.

So The parameters of the system determine the acceleration factor’s specific value. So For most systems, the ideal value of typically falls between the ranges of 1.2 and 1.6.