Tes: Newton Raphson Method - Electrical Engineering (EE) MCQ

Newton Raphson Method:

• The Newton-Raphson method is an advanced iterative method for generating appropriate solution steps to a real solution of a given nonlinear equation.
• The iterative formula used in the NR method is:
  

Number of buses in the system = N
Number of PQ buses = NL
Number of slack buses = 1
Number of PV buses = N - 1 - N_L
Newton Raphson method for load flow study in polar form

 

The submatrix M relates between [ΔQ] and [Δδ]
Number of elements in ΔQ vector = Number of known Q 
Number of elements in ΔQ vector = Number of PQ buses = N_L
Number of elements in Δδ vector = Number of unknown δ = N - 1
Size of matrix M = NL × (N - 1)
Size of other sub-matrix:
H = (N - 1) × (N - 1)
S = (N - 1) ×  N_L
R = N_L × N_L 

• 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 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.

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.

Number of buses (n) = 100
Generator busses (m) = 10
Order of Jacobian matrix =(2n − 1 − m) × (2n − 1 − m)
 = (200 − 1 − 10) × (200 − 1 − 10)
= 189 × 189

Difference between the Gauss-Seidel method and Newton-Raphson method:

As accuracy is more Newton-Raphson method have assured load flow solution

Concept:
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

Calculation:
Given that,
n = 10, m = 2
The order of the Jacobian matrix (with one slack bus) = (20 – 2 – 2) × (20 – 2 – 2) = 16 × 16

Size of Jacobian matrix = (2n – m – 2) × (2n – m - 2)
Where n = number of buses
m = number of pv buses
Given that,
Size of Jacobian matrix = 100 × 100
Number of PV buses (m) = 20
⇒ (2n – m - 2) = 100
⇒ 2n – 20 – 2 = 100
⇒ n = 61

Concept:
The minimum number of simultaneous equations will be
E = (2 × number of Load bus) + number of Generator buses
Calculation:
Given,
The number of load buses = 150
The number of generator buses = 32
Minimum no. of simultaneous equations
E = (2 × 150) + 32 = 332

Total number of bases (n) = 10
Number of slack bases = 1 (G₁)
G₃, G₄ are generator bases.
Number of voltage controlled load bases = 2 (L₃, L₄)
G₂ acts as load bas
Number of load bases = 4 (L₁, L₂, L₅, L₆)
The number of non-linear equations required = 2 × number of bases - (number of load bases) - (number of voltage controlled load bases)
= 2n - 2 - 4
= 2 (10) - 2 - 4 = 20 - 2 - 4 = 14