Load Flow Analysis (Gauss-Seidel)

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Number of buses *

Total number of buses in the system

Base MVA *

System base power in MVA for per-unit calculations

Base voltage

System base voltage in kV

Frequency (Hz)

Convergence tolerance

Maximum voltage change for convergence

Maximum iterations

Maximum number of Gauss-Seidel iterations

Use SOR acceleration

Enable Successive Over-Relaxation acceleration

SOR acceleration factor

Over-relaxation factor (1.0-2.0, typical 1.0-1.2)

Enforce generator Q limits

Convert PV to PQ when reactive limits violated

Calculate line power flows

Calculate system losses

Note: Bus data and line data are complex arrays. Please edit these parameters using the advanced JSON editor in the main calculation form.

Bus data: Define bus type (Slack, PV, PQ), voltage specifications, generation, and load for each bus.

Line data: Define transmission line impedances (resistance, reactance) and shunt susceptance for each connection.

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Calc number

Tag number

Title

Description

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Company name

Project number

Prepared by

Project name

Client company

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Your Company

Calculation No.: Calc-001
Revision: A
Description:
Created By: Engineer Name

Project: Project Name
Project No.: P001
Client: Client Name

Load Flow Analysis - Gauss-Seidel Method

System: 3-bus network, base 100.0 MVA / 11.0 kV
Method: Gauss-Seidel (standard)
Converged in 10 iterations

Network Diagram

Bus types and solved voltages

Step 1: Y-bus Admittance Matrix

The bus admittance matrix is assembled from the line impedances (per-unit).

Formula: $$ y_{ij} = \frac{1}{z_{ij}} = \frac{1}{R_{ij} + jX_{ij}} $$

Formula: $$ Y_{ii} = \sum_{k \neq i} y_{ik} + y_{\text{shunt},i} $$

Formula: $$ Y_{ij} = -y_{ij} \quad (i \neq j) $$

Parameter	Symbol	Value
Self admittance	Y_{00}	6.2500 - j18.7500
Mutual admittance	Y_{01}	-5.0000 + j15.0000
Mutual admittance	Y_{02}	-1.2500 + j3.7500
Mutual admittance	Y_{10}	-5.0000 + j15.0000
Self admittance	Y_{11}	6.6667 - j20.0000
Mutual admittance	Y_{12}	-1.6667 + j5.0000
Mutual admittance	Y_{20}	-1.2500 + j3.7500
Mutual admittance	Y_{21}	-1.6667 + j5.0000
Self admittance	Y_{22}	2.9167 - j8.7500

Step 2: Gauss-Seidel Update Equation

For each non-slack bus i, with S_i = P_i + jQ_i the net injection. Buses j < i use the already-updated V_j^(k+1); buses j > i use V_j^(k).

Formula: $$ V_i^{(k+1)} = \frac{1}{Y_{ii}} \left( \frac{S_i^*}{V_i^{(k)*}} - \sum_{j \neq i} Y_{ij} V_j \right) $$

Step 3: PV Bus Voltage Magnitude Enforcement

PV (generator) buses: 1.

Formula: $$ V_i^{(k+1)} = |V_i|_{\text{spec}} \cdot e^{j \angle V_i^{\text{temp}}} $$

Where Q limits are enforced, a PV bus that hits a limit is held at that limit and treated as a PQ bus.

Formula: $$ Q_i^{(k+1)} = \text{Im}\left\{ V_i^{(k+1)} \left( \sum_j Y_{ij} V_j \right)^* \right\} $$

Step 4: Convergence Check

Tolerance ε = 1.00e-06. Converged in 10 iterations.

Formula: $$ \max_i \left| V_i^{(k+1)} - V_i^{(k)} \right| < \varepsilon $$

Step 5: Final Bus Voltages

Solved bus voltages (magnitude and angle, per-unit on the system base):

Parameter	Symbol	Value	Notes
Bus 0 (Slack)	V_{0}	1.0500 ∠ 0.000°	11.550 kV
Bus 1 (PV)	V_{1}	1.0400 ∠ 1.424°	11.440 kV
Bus 2 (PQ)	V_{2}	0.9501 ∠ -4.146°	10.451 kV

Step 6: Power Injection Verification

Actual injections are recalculated from the solved voltages and checked against the specified values (positive = generation into the bus).

Formula: $$ S_i = V_i \left( \sum_j Y_{ij} V_j \right)^* $$

Parameter	Symbol	Value	Notes
Bus 0 (Slack) real power	P_{0}	5.1904	specified 0.0000 MW, mismatch 5.19e+00 MW
Bus 0 (Slack) reactive power	Q_{0}	61.1329	specified 0.0000 MVAr
Bus 1 (PV) real power	P_{1}	100.0002	specified 100.0000 MW, mismatch 1.75e-04 MW
Bus 1 (PV) reactive power	Q_{1}	4.4388	specified 4.4388 MVAr
Bus 2 (PQ) real power	P_{2}	-100.0000	specified -100.0000 MW, mismatch 8.57e-07 MW
Bus 2 (PQ) reactive power	Q_{2}	-50.0000	specified -50.0000 MVAr

Step 7: Line Power Flows

Flow leaving bus i towards bus j; the branch loss is the sum of the flows measured at both ends.

Formula: $$ S_{ij} = V_i \, I_{ij}^* , \quad S_{\text{loss},ij} = S_{ij} + S_{ji} $$

Parameter	Symbol	Value	Notes
Line 0 → 1	S_{01}	-35.2955 + j29.8275	loss 0.3874 MW + j1.1622 MVAr
Line 0 → 2	S_{02}	40.4859 + j31.3054	loss 1.9005 MW + j5.7015 MVAr
Line 1 → 2	S_{12}	64.3173 + j33.1042	loss 2.9027 MW + j8.7081 MVAr

Step 8: System Summary

Total real power loss: 5.1906 MW
Total reactive power loss: 15.5718 MVAr
Convergence: Yes (10 iterations)

Standards Reference

IEEE Std 399: Power Systems Analysis (Brown Book)
AS/NZS 3000: Electrical installations

Created By
Engineer Name
Date: ..........................

Checked
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Date: ..........................

Approved
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Date: ..........................

Client Approved
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Date: ..........................

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