Test: Gauss-Siedel Method - Electrical Engineering (EE) MCQ

• The reason the Gauss-Seidel method is commonly known as the successive displacement method is that the second unknown is determined from the first unknown in the current iteration, the third unknown is determined from the first and second unknowns, etc.
• In the G-S method of power flow problem, the number of iterations depends on tolerance.

Gauss-Seidel method:
Consider a 4-bus sample power system as shown in the figure, BUS 1 is considered a slack bus, where voltage magnitude and its angle are known.

In this case w = 4 and slack bus s = 1. From  we can write





In the Gauss-Seidel method, the new calculated voltage at (K + 1) i.e. V ^{(K + 1)} immediately replaces V (p) and is used in the solution of the subsequent equations.
Therefore, the above set of equations can be written in iterative form, i.e.,




Note that bus 1 is the slack bus.
Under normal operating conditions, the voltage magnitude of buses is in the neighborhood of 1.0 per unit or close to the voltage magnitude of the slack bus.
Therefore, an initial starting voltage of (1.0 + j 0.0) for unknown voltages is satisfactory, and the converged solution correlates with the actual operating states.

• Newton Raphson method has more computation time per iteration as compared to the Gauss Siedel method.
• Only the Gauss Siedel method has a problem in convergence for a system with long radial lines.

Comparison of two load flow methods is given below.

Load flow study:

• Load flow study determines the operating state of the system for a given loading.
• Load flow solves a set of simultaneous non-linear algebraic power equations for the two unknown variables (|V| and ∠δ) at each node in a system.
• The output of the load flow analysis is the voltage and phase angle, real and reactive power (both sides in each line), line losses, and slack bus power.
• Gauss seidel, Newton Raphson, and Fast decoupled load flow method are the different load flow methods.
• The number of iterations required for convergence of a load flow algorithm increases significantly with the increase of the number of buses with G-S load flow algorithm.
• The fast decoupled load flow method gives an approximate load flow solution because it uses several assumptions. Accuracy depends on the power mismatch vector tolerance.
• The fast decoupled load flow method is an extension of the Newton-Raphson method formulated in polar coordinates with certain approximations, which results in a fast algorithm for load flow solution.
• The fast decoupled method requires a greater number of iterations than the Newton-Raphson method.

No. of non-zero elements = no. of diagonal element + 2 (no. of line)
4000 = 500 + 2 (lines)

The given equations can be written as
10V₁ + 2V₂ = 10
⇒ V₁ = 1/10 [10 − 2V₂]
4V₁ + 7V₂ = 15
⇒ V₂ = 1/7 [15 − 4V₁]
Iteration 1:
Put V₂ = 0
⇒ V₁ = 1/10 [10 − 0] = 1

Gauss Seidel method:

• Gauss-Seidel method of power flow problem is an iterative method used to solve a system of linear equations.
• This method is very simple and uses digital computers for computing.
• In this method as we are using simple algebraical equations so that the calculation time for each iteration is less.

Disadvantages:

• Though it can be applied to any matrix with non zero diagonal elements, the convergence is guaranteed if the matrix is either strictly diagonally dominant or symmetric and positive definite.
• More number of iterations are required so that it has slow convergence.
• Initial approximate guessing value is required for convergence.
• The choice of slack bus affects convergence.
• It is not applicable to the large power system networks.
• It requires an accelerating factor for convergence. The accelerating factor is used for reducing the number of iterations in the Gauss-Seidel method by multplying voltage at each bus with the acceleration factor.
• The value of the accelerating factor is around 1.6 to 1.8.

If the reference bus is changed in two load flow runs with same system data and power obtained for reference bus taken as specified P and Q.
So, the bus is taken as load bus.
So, the loses will be same but the complex bus voltage will change.

Gauss Seidel method:

• Gauss-Seidel method of power flow problem is an iterative method used to solve a system of linear equations.
• This method is very simple and uses digital computers for computing.
• In this method as we are using simple algebraical equations so that the calculation time for each iteration is less.

Disadvantages:

1. Though it can be applied to any matrix with non zero diagonal elements, the convergence is guaranteed if the matrix is either strictly diagonally dominant or symmetric and positive definite.
2. More number of iterations are required so that it has slow convergence.
3. Initial approximate guessing value is required for convergence.
4. The choice of slack bus affects convergence.
5. It is not applicable to the large power system networks.
6. It requires an accelerating factor for convergence. The accelerating factor is used for reducing the number of iterations in the Gauss-Seidel method by multplying voltage at each bus with the acceleration factor.
7. The value of the accelerating factor is around 1.6 to 1.8.

Gauss Seidel Method Question 2 Detailed Solution

• The reason the Gauss-Seidel method is commonly known as the successive displacement method is that the second unknown is determined from the first unknown in the current iteration, the third unknown is determined from the first and second unknowns, etc.
• In the G-S method of power flow problem, the number of iterations depends on the number of buses.

​​​Gauss-Seidel method:

Consider a 4-bus sample power system as shown in the figure, BUS 1 is considered a slack bus, where voltage magnitude and its angle are known.

In this case w = 4 and slack bus s = 1. From  we can write


In the Gauss-Seidel method, the new calculated voltage at (K + 1) i.e. V ^{(K + 1)} immediately replaces V (p) and is used in the solution of the subsequent equations.

Therefore, the above set of equations can be written in iterative form, i.e.,

 

Note that bus 1 is the slack bus.

Under normal operating conditions, the voltage magnitude of buses is in the neighborhood of 1.0 per unit or close to the voltage magnitude of the slack bus.

Therefore, an initial starting voltage of (1.0 + j 0.0) for unknown voltages is satisfactory, and the converged solution correlates with the actual operating states.

Disadvantages of the GS method:

• Increase the number of iterations directly with the increased number of buses.
• The slow rate of convergence and thus, a large number of iterations.
• Effect on convergence due to choice of slack bus.