Load Flow Studies MCQ Quiz - Objective Question with Answer for Load Flow Studies - Download Free PDF

Explanation:

Gauss-Seidel Method vs. Newton-Raphson Method

The Gauss-Seidel method and the Newton-Raphson method are both iterative techniques used for solving systems of nonlinear equations. Each method has its own advantages and disadvantages, depending on the specific problem and context in which they are applied. Below, we will delve into the detailed analysis of the Gauss-Seidel method and the Newton-Raphson method, and why option 4 is the correct answer to the question regarding the advantages of the Gauss-Seidel method over the Newton-Raphson method.

Gauss-Seidel Method:

The Gauss-Seidel method is an iterative technique for solving a system of linear equations. It is a relaxation method that improves the solution iteratively by considering each equation one at a time and using the latest available values for the variables.

• Single iteration of computation is faster: The Gauss-Seidel method often requires less computational effort per iteration compared to the Newton-Raphson method because it deals with linear equations and updates the solution incrementally.
• Ease of programming: The Gauss-Seidel method is simpler to implement programmatically as it involves straightforward iterative updates without the need for complex derivative calculations.
• Most efficient use of core memory: The Gauss-Seidel method can be more memory efficient as it updates the solution in place and does not require storage of large matrices or additional data structures.

Newton-Raphson Method:

The Newton-Raphson method is a powerful and widely-used iterative technique for solving systems of nonlinear equations. It uses the first-order Taylor series expansion to approximate the solution and requires the computation of Jacobian matrices and their inverses.

• Quadratic convergence: The Newton-Raphson method typically converges faster than the Gauss-Seidel method because it has a quadratic rate of convergence, meaning that the error decreases quadratically as the iterations proceed.
• Robustness: The Newton-Raphson method is more robust for solving nonlinear equations as it can handle more complex and nonlinear systems effectively.

Correct Option Analysis:

The correct option is:

Option 4: Number of iterations required is lesser.

This option is NOT an advantage of the Gauss-Seidel method over the Newton-Raphson method. In fact, the Newton-Raphson method generally requires fewer iterations to converge to a solution compared to the Gauss-Seidel method. This is due to the quadratic convergence property of the Newton-Raphson method, which allows it to reach the solution more quickly as the iterations progress.

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

Option 1: Single iteration of computation is faster.

This is an advantage of the Gauss-Seidel method. The computational effort per iteration is generally lower in the Gauss-Seidel method because it deals with linear equations and updates the solution incrementally without the need for complex derivative calculations.

Option 2: Ease of programming.

This is also an advantage of the Gauss-Seidel method. The method is simpler to implement programmatically, as it involves straightforward iterative updates and does not require the calculation of Jacobian matrices or their inverses, which can be complex and computationally intensive.

Option 3: Most efficient use of core memory.

This is another advantage of the Gauss-Seidel method. It can be more memory efficient as it updates the solution in place and does not require storage of large matrices or additional data structures, making it suitable for problems with limited memory resources.

Conclusion:

In summary, the Gauss-Seidel method is advantageous in terms of computational speed per iteration, ease of programming, and memory efficiency. However, it generally requires more iterations to converge to a solution compared to the Newton-Raphson method, which is known for its faster convergence due to its quadratic convergence property. Therefore, the statement that the number of iterations required is lesser is NOT an advantage of the Gauss-Seidel method over the Newton-Raphson method, making option 4 the correct answer.

Explanation:

To determine the size of the Jacobian matrix in a power system network, we need to understand the components involved. In this network, we have:

• Total buses (n) = 40
• Voltage-controlled buses (PV buses) = 12
• Generator buses (also PV buses) = 5

In power system analysis, the Jacobian matrix is used to solve the power flow equations using the Newton-Raphson method. The size of the Jacobian matrix is determined based on the number of variables in the system. These variables are primarily the voltage magnitudes and angles at different buses, excluding the reference bus (slack bus).

Steps to determine the size of the Jacobian matrix:

• First, identify the total number of buses (n).
• Determine the number of PV buses (voltage-controlled buses).
• Subtract the reference bus (slack bus) from the total number of buses to find the number of unknown variables.

Calculation:

The total number of buses (n) = 40.

Out of these, 12 are PV buses. Since the generator buses are a subset of the PV buses, we do not double-count them. Thus, we have:

• PV buses = 12 (including generator buses)
• PQ buses (load buses) = Total buses - PV buses - 1 (slack bus)
• PQ buses = 40 - 12 - 1 = 27

In power flow analysis, the Jacobian matrix is split into four submatrices:

• J1: Partial derivatives of active power with respect to voltage angles (θ).
• J2: Partial derivatives of active power with respect to voltage magnitudes (V).
• J3: Partial derivatives of reactive power with respect to voltage angles (θ).
• J4: Partial derivatives of reactive power with respect to voltage magnitudes (V).

The size of the Jacobian matrix is determined by the number of equations and the number of variables. For a system with (n-1) buses, we have:

• n-1 voltage angle equations (excluding the reference bus).
• n-m-1 voltage magnitude equations, where m is the number of PV buses.

Thus, the total number of variables (unknowns) is the sum of the voltage angles and magnitudes:

• Number of voltage angle variables = n - 1 = 40 - 1 = 39
• Number of voltage magnitude variables = n - m - 1 = 40 - 12 - 1 = 27

The size of the Jacobian matrix is the sum of these variables:

• Total number of variables = 39 + 27 = 66

Therefore, the size of the Jacobian matrix is 66 * 66.

Correct Option Analysis:

The correct option is:

Option 3: 66 * 66

This option correctly represents the size of the Jacobian matrix for the given power system network. The matrix accounts for all the voltage angles and magnitudes that need to be solved in the power flow analysis.

Additional Information

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

Option 1: 46 * 46

This option is incorrect because it underestimates the total number of variables. The size of the Jacobian matrix should account for both the voltage angles and magnitudes, which result in a larger matrix.

Option 2: 23 * 23

This option is incorrect as it grossly underestimates the number of variables in the system. It does not take into account the total number of buses and the necessary voltage magnitudes and angles.

Option 4: 21 * 21

This option is incorrect as it also underestimates the number of variables. The calculation of the Jacobian matrix size should consider all unknown voltage angles and magnitudes, leading to a larger matrix size.

Conclusion:

Understanding the structure and calculation of the Jacobian matrix is essential for solving power flow equations in electrical networks. The correct size of the Jacobian matrix for a network with 40 buses, including 12 PV buses, is 66 * 66, as it accounts for all the necessary voltage angles and magnitudes. This ensures accurate and efficient analysis of the power system.

Bus-bar Ratings:

A bus-bar is an electrical conductor used to collect and distribute electrical power. It plays a crucial role in electrical power systems, ensuring efficient distribution of electricity. The correct rating of a bus-bar is essential for the safety and reliability of the electrical system.

Factors Determining Bus-bar Ratings:

• Current: The bus-bar must be rated for the maximum current it is expected to carry. This ensures that the bus-bar can handle the load without overheating or causing damage to the system.
• Voltage: The bus-bar must be rated for the maximum voltage of the system. Proper insulation and clearance distances must be maintained to prevent electrical arcing and ensure safe operation.
• Frequency: Although not as critical as current and voltage, frequency can affect the performance of the bus-bar, especially in systems with high harmonic content. The bus-bar must be designed to handle the operating frequency of the system.
• Short Circuit Current: The bus-bar must be capable of withstanding the mechanical and thermal stresses caused by short circuit conditions. This includes the peak short circuit current and the duration for which the bus-bar can sustain this current without damage.

Conclusion:

The correct rating of a bus-bar encompasses current, voltage, frequency, and short circuit current. Ensuring that the bus-bar is appropriately rated for these parameters is critical for the safe and reliable operation of the electrical system. Therefore, the correct option is option 4.

The quantities specified for a different type of buses shown in the below table.

Explanation:

A bus in a power system is a line at which the several components of the power system like generators, loads, and feeders, etc., are connected.

The buses in a power system are associated with four quantities, these quantities are the following:

• The magnitude of the voltage
• Phase angle
• Active power
• Reactive power
   

In the load flow studies, two variables are known, and the other two have to be determined.

Depending on the quantity to be specified the buses are classified into three categories as follow:

The table shown below shows the types of buses and the associated known and unknown value.

Generation Bus  or Voltage Control Bus:

• This bus is also called the P-V bus.
• on this bus, the voltage magnitude corresponding to generate voltage and true or active power P corresponding to its rating are specified.
• Voltage magnitude is maintained constant at a specified value by injection of reactive power.
• The reactive power generation (Q) and phase angle (δ) of the voltage is to be computed.
   

Load Bus:

• This is also called the P-Q bus
• at this bus, the active and reactive power is injected into the network.
• The magnitude and phase angle of the voltage is to be computed.
• Here the active power P and reactive power Q are specified, and the load bus voltage can be permitted within a tolerable value, i.e., 5 %.
• The phase angle of the voltage, i.e.δ is not very important for the load.
   

Slack, Swing, or Reference Bus:

• Slack bus in a power system absorbs or emits active or reactive power from the power system.
• The slack bus does not carry any load.
• At this bus, the magnitude and phase angle of the voltage is specified.
• The phase angle of the voltage is usually set equal to zero.

Load flow analysis:

Explanation:

A bus in a power system is a line at which the several components of the power system like generators, loads, and feeders, etc., are connected.

The buses in a power system are associated with four quantities, these quantities are the following:

• The magnitude of the voltage
• Phase angle
• Active power
• Reactive power
   

In the load flow studies, two variable are known, and the other two is to determined.

Depends on the quantity to be specified the buses are classified into three categories as follow:

The table shown below shows the types of buses and the associated known and unknown value.

Generation Bus  or Voltage Control Bus:

• This bus is also called the P-V bus.
• on this bus, the voltage magnitude corresponding to generate voltage and true or active power P corresponding to its rating are specified.
• Voltage magnitude is maintained constant at a specified value by injection of reactive power.
• The reactive power generation Q and phase angle δ of the voltage is to be computed.
   

Load Bus:

• This is also called the P-Q bus
• at this bus, the active and reactive power is injected into the network.
• The magnitude and phase angle of the voltage is to be computed.
• Here the active power P and reactive power Q are specified, and the load bus voltage can be permitted within a tolerable value, i.e., 5 %.
• The phase angle of the voltage, i.e.δ is not very important for the load.
   

Slack, Swing, or Reference Bus:

• Slack bus in a power system absorbs or emits active or reactive power from the power system.
• The slack bus does not carry any load.
• At this bus, the magnitude and phase angle of the voltage is specified.
• The phase angle of the voltage is usually set equal to zero.

Given that:

Power system has 300 buses

n = total buses = 300

buses connected with generator = 20

buses with generator except slack bus (G) = 20 – 1 = 19

buses which controlled reactive power (Q) = 25

buses with fined shunt capacitor (C) = 15

Now, load buses = 300 – 20 – 25 – 15 = 240

Size of matrix = (2n – m - 2) × (2n – m - 2)

Where, m = total no. of Pv buses

m = Voltage controlled bus = Q + G

⇒ m = 25 + 19 = 44

(∵ fixed capacitor supply constant reactive power therefore it is considered as load buses)

Now size of matrix = (2n – m - 2) × (2n – m - 2)

= (2 × 300 – 44 - 2) × (2 × 300 – 44 - 2)

Classification of Buses:

The power system is nothing but the interconnection of the various bus.

Each of these buses is associated with four electrical parameters namely voltage with magnitude and phase angle, active power, and reactive power.

Slack Bus:

• Slack Bus is also known as Swing or Reference Bus.
• One of the generator buses is considered as a slack bus.
• The slack bus does not exist in real rather it is assumed for the consideration of losses occurring during power transmission.
• Actually, there exist only two buses in the power system, Load Bus and Generator Bus for which active power is specified.
• Since active power delivered by Generator Bus and consumed by Load Bus differ
• This means that a power loss equal to the difference between Generator Bus P and Load Bus P is occurring.
• This loss can only be calculated after the solution of Load Flow.
• Therefore to supply power loss, an extra generator bus is considered for which bus magnitude and voltage are specified, and active power and reactive power are to be calculated.
• This active power of slack bus is the equivalent power loss occurring in different systems.
• Generally, the phase angle of the slack bus is taken for reference for the entire load flow solution. Therefore this bus is also called Reference Bus.​

Load Bus:

• For load bus, real power P and reactive power Q are known but the magnitude and phase angle of bus voltage is unknown.
• It is desired to find the bus voltage using load flow analysis. At load bus, voltage is allowed to vary within some specified limit.

Generator Bus:

• Generator Bus is also called a voltage-controlled bus.
• The generator is connected to this bus.
• Therefore the bus voltage corresponding to generation voltage and active power generation corresponding to generator rating is specified for this bus.

The quantities specified for different type of buses shown in the below table.

• The most relevant information needed to specify P at PV buses is the solution of economic load dispatch.
• Economic load dispatch is a precursor to the load flow study (LFS), i.e. to perform the LFS the first step is to perform the economic load dispatch.
• The economic load dispatch means the real and reactive power of the generator varies within certain limits and fulfills the load demand with less fuel cost.
• The economic scheduling of the generators aims to guarantee at all times the optimum combination of the generator connected to the system to supply the load demand.
• The economic load dispatch problem involves two separate steps. These are the online load dispatch and unit commitment.
• The unit commitment selects that unit which will anticipate the load of the system over the required period at minimum cost.
• The online load dispatch distributes the load among the generating unit which is parallel to the system in such a manner as to reduce the total cost of supplying. It also fulfills the minute to the minute requirement of the system.

Slack bus:

• In load flow studies, it is assumed that the losses in the transmission network can be supplied by one of the generators instead of shared by all the generators so that it is easy to calculate the loss.
• The generator which supplies the entire transmission line losses and also shares the demand is called the slack bus. The remaining generators are only sharing the demand.
• Power for slack bus = Total power going into the system - Total p going out of the system + transmission line losses.
• This difference is called slack, and hence the name 'Slack Bus'.
• When the system is having a greater number of buses the calculations can begin at one of the bus. The bus where the calculations are started is called as a Reference bus.

To make possible variations in real and reactive powers of the slack bus during the iterative process to be of a small percentage of its generating capacity, the bus connected to the largest generating station is normally selected as the slack bus. Therefore, slack bus has to be a generator bus.

Important Points:

The quantities specified for different type of buses shown in the below table.

Concept:

In the Newton-Raphson method,

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

Where n = number of buses

m = number of pv buses or generator buses

Calculation:

Given that,

Slack bus = 1

Generator buses = 8

Load buses = 41

Size of Jacobian matrix = (2(50) – 8 – 2) × (2n – 8 – 2)

= 90 × 90

Load flow analysis

In a power system network, there are three types of buses i.e.slack bus, load bus, and generator bus.

Each bus is associated with 4 quantities, two are known and two are unknown quantities.

Hence, the correct answer is option 1.

In the Gauss-Seidel method of load flow analysis, the number of iterations required to reach convergence are dependent on the size of the system, and increases with the increase in the size of the system.

Important Points:

Bus Admittance Matrix:

• In a power system, Bus Admittance Matrix represents the nodal admittances of the various buses.
• Admittance matrix is used to analyze the data that is needed in the load or a power flow study of the buses.

​Incidence matrix:

• It is the matrix that gives a relation between the branches and nodes.
• The rows of the incidence matrix [A] represent the number of nodes and the column of the matrix represents the number of branches in the given graph.
• If there are ‘n’ number of rows in a given incidence matrix[A], that means in a graph there are ‘n’ number of nodes. 
• Similarly, if there are ‘b’ numbers of columns in that given incidence matrix[A], that means in that graph there are ‘b’ number of branches.
• We can construct the incidence matrix for the directed graph. We can draw a graph with the help of the incidence matrix.
• It explains the topology of the network.
• The algebraic sum of elements of all the columns is zero.
• The rank of the incidence matrix is (n–1).
• The determinant of the incidence matrix of a closed loop is zero.