Newton Raphson Method MCQ Quiz in తెలుగు - Objective Question with Answer for Newton Raphson Method - ముఫ్త్ [PDF] డౌన్‌లోడ్ కరెన్

The order of the Jacobian matrix (with one slack bus)

= (2n – 2 – m) × (2n – 2 – m)

where n = number of buses

m = number of buses whose voltage magnitude is specified

Total number of buses (n) = 10

Number of slack buses = 1 (G₁)

G₃, G₄ are generator buses.

Number of voltage controlled load buses = 2 (L₃, L₄)

G₂ acts as load bus

Number of load buses = 4 (L₁, L₂, L₅, L₆)

The number of non-linear equations required = 2 × number of bases - (number of load buses) - (number of voltage controlled load buses)

= 2n - 2 - 4

= 2 (10) - 2 - 4 = 20 - 2 - 4 = 14

Concept:

In the Newton-Raphson method,

The size of a Jacobian matrix = (2n – m – 2) × (2n – m – 2)

Where n = Number of buses

m = Effective number of pv buses or generator buses

Effective number of generator bus = Number of generator given - 1 (slack Bus)

Calculation:

Given that,

n = 20

Effective generator buses, m = 1 - 1 = 0

Size of Jacobian matrix = (2(20) – 0 – 2) × (2(20) – 0 – 2)

= 38 × 38

Concept:

According to Newton - Raphson Method:

• Newton-Raphson method has quadratic convergence,i.e order of convergence = 2
• Newton-Raphson method converges more rapidly than the other methods.
• The method can be used for solving algebraic and transcendental equations and it can also be used when the roots are complex.
• The iteration formula is,

Calculation:

We have, 

⇒ f(x) = x³ - x - 1 at x₀ = 1

⇒ f'(x) = 3x² - 1

⇒ 

⇒ X₁ = 

⇒ X₁ = 

⇒ X₁ = 

⇒ X₁ = 

⇒ X₁ = 1.5

∴ The first iteration by the Newton-Raphson method of the equation, f(x) = x3 - x - 1 by taking the initial guess as x0 = 1  is 1.5

Jacobian for the power flow equation:

Now, given that:  

Set of power flow equation:

Now  

= 100 [-1 + 4x₂ – x₃]

= 100 [-1 – x₂ + 4x₃]

= -100 x₃

Now, Jacobian of this power flow equation is

Explanation:

Comparison of Newton-Raphson (NR) and Gauss-Seidel (GS) Methods in Load Flow Analysis

Correct Option: Option 1: Convergence characteristic of the NR methods are not affected by selection of slack bus

Detailed Explanation:

The Newton-Raphson (NR) method is widely regarded as superior to the Gauss-Seidel (GS) method in solving load flow equations in power systems. This superiority arises from several factors, the most significant of which is the characteristic described in Option 1.

In load flow analysis, the slack bus is a reference bus used to balance the active and reactive power in the system. The selection of the slack bus is an important step in setting up the power flow equations. However, the NR method's convergence characteristics are largely independent of the choice of the slack bus. This means that the performance of the NR method does not degrade or change significantly based on which bus is chosen as the slack bus. This robustness is a significant advantage in practical scenarios where the system configuration may vary, and the selection of the slack bus may be somewhat arbitrary.

The NR method achieves this robustness through its mathematical approach. It uses a quadratic convergence technique, which ensures that the solution converges rapidly near the correct solution. This behavior is not influenced by the slack bus because the NR method solves the non-linear equations of the power flow problem by iteratively linearizing them using the Jacobian matrix. The iterative updates are based on the mismatch equations (power mismatches) and are not directly dependent on the specific choice of the slack bus.

In contrast, the GS method's convergence characteristics can be affected by the choice of the slack bus. The GS method is a sequential iteration technique that updates bus voltages one at a time. The order in which buses are updated and the numerical values used in the calculations can influence the convergence rate. This sensitivity to the slack bus and other factors often makes the GS method less robust and slower to converge compared to the NR method, especially in large and complex power systems.

Therefore, Option 1 correctly highlights one of the primary reasons why the NR method is superior to the GS method in load flow analysis.

Additional Information:

To further understand the comparison, let’s analyze the other options:

Option 2: The number of iterations required in the NR method is not independent of the size of the system.

This statement is incorrect. In fact, the number of iterations required in the NR method is largely independent of the size of the system. The NR method typically converges in a small number of iterations (around 3-5), regardless of whether the system is small or large. This is due to its quadratic convergence property, which ensures rapid convergence near the solution. In contrast, the GS method's number of iterations increases with the size and complexity of the system, making it less efficient for large systems.

Option 3: Time taken to perform one iteration in the NR method is less when compared to the GS method.

This statement is also incorrect. The NR method involves the computation and inversion of the Jacobian matrix in each iteration, which is computationally intensive and time-consuming. As a result, the time taken for one iteration in the NR method is significantly higher than that in the GS method. However, the NR method compensates for this by requiring far fewer iterations to converge, making it overall more efficient in terms of total computation time for large systems.

Option 4: The number of iterations required in the NR method is more than compared to that in the GS method.

This statement is incorrect. The NR method typically requires fewer iterations to converge compared to the GS method. The GS method is a first-order iterative technique and converges linearly, which means it requires many iterations to achieve an acceptable level of accuracy, especially for large or ill-conditioned systems. In contrast, the NR method's quadratic convergence ensures that it reaches the solution in fewer iterations.

Conclusion:

The Newton-Raphson method is superior to the Gauss-Seidel method in load flow analysis because its convergence characteristics are not affected by the selection of the slack bus, as correctly stated in Option 1. This robustness, combined with its rapid convergence and scalability, makes the NR method the preferred choice for solving load flow problems in modern power systems, despite its higher computational cost per iteration.

The unknown quantities of generator bus are: Q, δ

The unknown quantities of load bus are: |V|, δ.

Given that, three generator buses are converted into load bus.

As voltage angles are unknown in both the type of buses, the number of unknown voltage angles remains same.

The number of voltage magnitudes increases by three.

The size of Jacobian matrix = (2n - 2 - m) × (2n - 2 - m)

Where n is number of buses

M is number of voltage controlled buses

Size of Jacobian matrix = (2 (30) - 2 - 3) × (2 (30) - 2 - 3)

= 55 × 55

Number of buses (n) = 16

Generator busses (m) = 4

Order of Jacobian matrix = (2n − 2 − m) × (2n − 2 − m) = (2n − 2 − m) × (2n − 2 − m)

= (32 − 2 − 4) × (32 − 2 − 4) = 26 × 26

∴ Jacobian matrix = 26 × 26

Size = (2(250) – 2 – 15 – 20 – 10)

= (500 – 2 – 15 – 20 – 10)

= 453