Gauss–Seidel and Newton–Raphson Load Flow Methods

LOAD FLOW ANALYSIS – OVERVIEW

Purpose

Load flow (power flow) analysis is used to determine:

• Bus voltages (magnitude and angle)
• Active power (P)
• Reactive power (Q)
• Power flows in transmission lines

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Basic Load Flow Equations

For bus i:

Active power:

$$P_i = \sum_{j=1}^{n} |V_i||V_j| (Y_{ij}) \cos(\theta_{ij} + \delta_j - \delta_i)$$

Reactive power:

$$Q_i = - \sum_{j=1}^{n} |V_i||V_j| (Y_{ij}) \sin(\theta_{ij} + \delta_j - \delta_i)$$

These equations are nonlinear → solved using iterative methods.

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TYPES OF BUSES (VERY IMPORTANT FOR GATE)

1. Slack Bus

Given:

$$|V|, \delta$$

Unknown:

$$P, Q$$

Purpose:
Balances system losses.

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2. PV Bus (Generator Bus)

Given:

$$P, |V|$$

Unknown:

$$Q, \delta$$

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3. PQ Bus (Load Bus)

Given:

$$P, Q$$

Unknown:

$$|V|, \delta$$

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GAUSS–SEIDEL LOAD FLOW METHOD

Basic Principle

Uses iterative voltage updating.

Each bus voltage is updated using latest available values.

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Voltage Update Equation (MOST IMPORTANT)

For PQ bus:

$$V_i^{k+1} = \frac{1}{Y_{ii}} \left[ \frac{P_i - jQ_i}{(V_i^k)^*} - \sum_{j \ne i} Y_{ij}$$

Where:

k = iteration number

= complex conjugate

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Steps of Gauss-Seidel Method

Step 1: Assume initial voltages
Usually:

$$V = 1 \angle 0°$$

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Step 2: Update voltages bus by bus

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Step 3: Continue until convergence

Condition:

$$|V_i^{k+1} - V_i^k| < \epsilon$$

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For PV Bus

Reactive power calculated first:

$$Q_i = - Im \left[ V_i^* \sum Y_{ij}V_j \right]$$

Then voltage magnitude is fixed.

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Acceleration Factor (IMPORTANT FOR GATE)

To improve convergence:

$$V_i^{new} = V_i^{old} + \alpha (V_i^{calculated} - V_i^{old})$$

Typical value:

$$\alpha = 1.6$$

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Advantages of Gauss-Seidel

Simple

Less memory required

Easy to program

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Disadvantages

Slow convergence

Not suitable for large systems

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Convergence Speed

Slow

Iterations required: 50–200

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NEWTON–RAPHSON LOAD FLOW METHOD

MOST IMPORTANT METHOD FOR GATE

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Basic Principle

Uses Taylor series expansion.

Linearizes nonlinear equations.

Uses Jacobian matrix.

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Power Mismatch Equations

$$\Delta P = P_{specified} - P_{calculated}$$ $$\Delta Q = Q_{specified} - Q_{calculated}$$

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Matrix Form

$$\begin{bmatrix} \Delta P \\ \Delta Q \end{bmatrix} = \begin{bmatrix} J_1 & J_2 \\ J_3 & J_4 \end{bmatrix} \begin{bmatrix} \Delta \delta \\ \Delta |V| \end{bmatrix}$$

Where:

J = Jacobian matrix

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Jacobian Matrix Structure

$$J = \begin{bmatrix} \frac{\partial P}{\partial \delta} & \frac{\partial P}{\partial V} \\ \frac{\partial Q}{\partial \delta} & \frac{\partial Q}{\partial V} \end{bmatrix}$$

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Steps of Newton-Raphson Method

Step 1: Assume initial voltages

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Step 2: Calculate P and Q

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Step 3: Find mismatch

$$\Delta P, \Delta Q$$

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Step 4: Form Jacobian matrix

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Step 5: Solve:

$$\Delta X = J^{-1} \Delta P$$

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Step 6: Update voltages

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Step 7: Repeat until convergence

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Convergence Condition

$$\Delta P, \Delta Q < tolerance$$

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COMPARISON TABLE (VERY IMPORTANT FOR GATE)

Parameter	Gauss-Seidel	Newton-Raphson
Convergence speed	Slow	Very fast
Iterations	High	Low
Accuracy	Medium	High
Memory requirement	Low	High
Computation per iteration	Low	High
Total computation time	High	Low
Suitable for large systems	No	Yes
Convergence type	Linear	Quadratic

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CONVERGENCE CHARACTERISTICS (IMPORTANT)

Gauss-Seidel:

Linear convergence

Newton-Raphson:

Quadratic convergence (very fast)

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NUMBER OF EQUATIONS (GATE FAVORITE)

For n bus system:

Slack bus: 1

PV bus: m

PQ bus: n-m-1

Unknowns:

$$(n-1) \text{ angles}$$ $$(n-m-1) \text{ voltage magnitudes}$$

Total unknowns:

$$2n-m-2$$

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ADVANTAGES OF NEWTON-RAPHSON

Fast convergence

Highly accurate

Suitable for large systems

Less number of iterations

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DISADVANTAGES

Complex programming

High memory requirement

Jacobian calculation required

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GATE EXAM IMPORTANT POINTS

MOST IMPORTANT:

Newton-Raphson → fastest method

Gauss-Seidel → slowest method

Newton-Raphson → quadratic convergence

Gauss-Seidel → linear convergence

Slack bus → balances losses

PV bus → voltage magnitude fixed

PQ bus → load bus