Newton Raphson Method MCQ Quiz in मल्याळम - Objective Question with Answer for Newton Raphson Method - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

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

\(= \left( {2\left( {15} \right) - 2 - 4} \right) \times \left( {2\left( {15} \right) - 2 - 4} \right) = 24 \times 24\)

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

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,

\(X_{(n \ + \ 1)} = X_n - \frac{f(X_n)}{f'(X_n)}\)

Calculation:

We have, 

⇒ f(x) = x³ - x - 1 at x₀ = 1

⇒ f'(x) = 3x² - 1

⇒ \(x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}\)

⇒ X₁ = \(1 - \frac{(1)^3 \ - \ 1 \ -1}{3(1)^2 \ - \ 1}\)

⇒ X₁ = \(1 - \frac{-1}{2}\)

⇒ X₁ = \(1 + \frac{1}{2}\)

⇒ X₁ = \(\frac{3}{2}\)

⇒ 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:

\(\frac{{\partial t}}{{\partial {x_n}}} = \left[ {\begin{array}{*{20}{c}} {\frac{{\partial {f_1}}}{{\partial {x_1}}}}& \ldots &{\frac{{\partial {f_1}}}{{\partial {x_n}}}}\\ \vdots & \ddots & \vdots \\ {\frac{{\partial {f_m}}}{{\partial {x_1}}}}& \ldots &{\frac{{\partial {f_m}}}{{\partial {x_n}}}} \end{array}} \right]\) 

Now, given that: \(x = \left[ {\begin{array}{*{20}{c}} {{x_2}}\\ {{x_3}} \end{array}} \right]\) 

Set of power flow equation:

\({f_1} = 1.0 - 100\;{x_2} + 200x_2^2 - 100\;{x_2}{x_3}\) 

\({f_2} = 0.5 - 100\;{x_3} - 100{x_3}{x_2} + 200\;x_3^2\) 

Now \(\frac{{\partial {f_1}}}{{\partial {x_2}}} = - 100 + 400{x_2} - 100\;{x_3}\) 

= 100 [-1 + 4x₂ – x₃]

\(\frac{{\partial {f_2}}}{{\partial {x_3}}} = - 100\;{x_2}\) 

\(\frac{{\partial {f_2}}}{{\partial {x_3}}} = - 100 - 100{x_2} + 400{x_3}\) 

= 100 [-1 – x₂ + 4x₃]

\(\frac{{\partial {f_2}}}{{\partial {x_2}}} = - 100{x_3} + 0\) 

= -100 x₃

Now, Jacobian of this power flow equation is

\(\frac{{\partial f}}{{\partial x}} = \left[ {\begin{array}{*{20}{c}} {\frac{{\partial {f_1}}}{{\partial {x_2}}}}&{\frac{{\partial {f_1}}}{{\partial {x_3}}}}\\ {\frac{{\partial {f_2}}}{{\partial {x_2}}}}&{\frac{{\partial {f_2}}}{{\partial {x_3}}}} \end{array}} \right]\) 

\( = \left[ {\begin{array}{*{20}{c}} {100\left( { - 1 + 4{x_2} - {x_3}} \right)}&{ - 100{x_2}}\\ { - 100{x_3}}&{100\left[ { - 1 - {x_2} + 4{x_3}} \right]} \end{array}} \right]\) 

\(\frac{{\partial f}}{{\partial x}} = 100\left[ {\begin{array}{*{20}{c}} { - 1 + 4{x_2} - {x_3}}&{ - {x_2}}\\ { - {x_3}}&{ - 1 - {x_2} + 4{x_3}} \end{array}} \right]\)

Explanation:

Newton-Raphson Method in Load Flow Study

The Newton-Raphson method is a widely used numerical technique for solving non-linear algebraic equations, and it plays a crucial role in load flow studies of power systems. The method is iterative and converges quickly under normal operating conditions, making it a preferred choice for solving power flow problems in large and complex networks.

Correct Option Analysis:

The correct option is:

Option 4: Use the estimated |V|(0) and δ(0) to calculate a total n number of injected real power \(\rm P_{calc}^{(0)}\) and equal number of real power mismatch ΔP(0).

This step does NOT hold true in the context of the Newton-Raphson method for load flow study. Let us analyze this:

• The Newton-Raphson method does not involve the calculation of the total number of injected real power (\(\rm P_{calc}\)) for all buses, as it primarily focuses on load buses (PQ buses) and the phase angles of the generator buses (PV buses).
• In a typical load flow analysis, the slack bus is used to balance the total system power, and its voltage magnitude and angle are predetermined. Therefore, the real power mismatch ΔP(0) is not calculated for all n buses.
• Instead, the method focuses on calculating mismatches only for load buses (PQ buses) and generator buses (PV buses) where power injections are not fixed.
• Hence, the calculation of real power mismatches (ΔP) for all buses, including the slack bus, is incorrect and does not align with the Newton-Raphson method.

Steps in the Newton-Raphson Method:

To clarify, let us review the key steps involved in the Newton-Raphson method for load flow analysis:

1. Initialization: Choose initial values for the voltage magnitudes |V|(0) for all load buses (PQ buses) and angles δ(0) for all buses except the slack bus. These are typically set to flat start values, e.g., |V| = 1 p.u. and δ = 0°.
2. Formulate the Jacobian Matrix: Use the initial estimates of |V|(0) and δ(0) to compute the Jacobian matrix J(0), which represents the partial derivatives of the power mismatches with respect to voltage magnitudes and angles.
3. Calculate Mismatches: Using the initial estimates, calculate the real power mismatch ΔP and reactive power mismatch ΔQ for PQ and PV buses based on the difference between specified power and calculated power (\(\rm P_{spec} - P_{calc}\) and \(\rm Q_{spec} - Q_{calc}\)).
4. Solve for Corrections: Solve the linearized set of equations ΔX = J⁻¹ × ΔP/Q to determine the corrections to voltage magnitudes and angles.
5. Update Estimates: Update the voltage magnitudes and angles using the corrections, i.e., |V|(k+1) = |V|(k) + Δ|V| and δ(k+1) = δ(k) + Δδ.
6. Convergence Check: Repeat steps 2–5 until the mismatches ΔP and ΔQ are within a predefined tolerance.

As seen above, the focus is on mismatches for PQ and PV buses, and not for all n buses, which validates why Option 4 is incorrect.

Additional Information

To further understand, let’s evaluate the other options:

Option 1: Use the estimated |V|(0) and δ(0) to formulate the Jacobian matrix J(0).

This is a correct step in the Newton-Raphson method. The Jacobian matrix plays a critical role as it contains the partial derivatives of the power mismatches with respect to voltage magnitudes and angles. It is formulated using the initial estimates of |V| and δ.

Option 2: Choose initial values of the voltage magnitude |V|(0) of all np load buses and n-1 angles δ(0) of the voltages of all the buses except the slack bus.

This is also a correct step. Initial values are required to begin the iterative process, and they are typically chosen as flat start values (|V| = 1 p.u., δ = 0°) for simplicity.

Option 3: Use the estimated |V|(0) and δ(0) to calculate a total np number of injected reactive power \(\rm Q_{calc}^{(0)}\) and equal number of reactive power mismatch ΔQ(0).

This is another correct step. For load buses (PQ buses), the reactive power mismatches ΔQ are calculated based on the difference between specified and calculated reactive power (\(\rm Q_{spec} - Q_{calc}\)).

Conclusion:

The Newton-Raphson method is an effective and widely used numerical technique for load flow analysis in power systems. Understanding the correct sequence of steps is crucial to its successful implementation. Option 4 is incorrect as it misrepresents the scope of the calculations involved, specifically suggesting the calculation of real power mismatches for all buses, which is not performed in this method. Instead, the focus is on PQ and PV buses, with the slack bus serving as the reference point for balancing power in the system.

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)

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