System Parameters
Base MVA100 MVA
Convergence tolerance ε0.001 pu
Line Impedances (pu)
Bus Data
Bus 1: Slack bus |V|=1.05∠0°
Bus 2: PV bus (generator)
Bus 3: PQ bus (load)
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Bus 1 voltage [pu∠°]
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Bus 2 voltage [pu∠°]
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Bus 3 voltage [pu∠°]
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System losses [MW]
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Iterations
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Convergence
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Line 1→2 P [MW]
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Line 1→3 P [MW]
Gauss-Seidel Load Flow Formula
Each bus voltage is updated iteratively from the Y-bus matrix:
$$V_i^{(k+1)} = \frac{1}{Y_{ii}}\left[\frac{P_i - jQ_i}{(V_i^{(k)})^*} - \sum_{j \neq i} Y_{ij} V_j\right]$$Convergence criterion: $\max_i |V_i^{(k+1)} - V_i^{(k)}| < \varepsilon$
Line power flow: $S_{ij} = V_i \cdot I_{ij}^* = V_i(V_i - V_j)^* y_{ij}^*$
CAE Applications: Core algorithm of power system analysis (PSS/E, PowerFactory). Applied to smart grid design, distributed generation interconnection studies, and transformer tap optimization. Large-scale systems use Newton-Raphson or Fast Decoupled methods.