The Power Flow Problem and Solution

Power flow problem analysis is fundamental for determining real and reactive power flows in network components, such as transmission lines, transformers, and loads. The power system's single-line diagram provides data on the bus, transmission line, and transformer. Each bus k in the system is characterized by four key variables: voltage magnitude V_k​, phase angle δ_k​, real power P_k​, and reactive power Q_k​. Two of these four variables are inputs, while the power flow program computes the remaining. The power delivered to bus k can be expressed in terms of generator and load components:

Based on their operational characteristics, buses within the power system are categorized into three types: swing bus, load (PQ) Bus, and voltage-controlled bus. The Swing Bus has a voltage magnitude close to 1.0 per unit and a phase angle of zero degrees. At load (PQ) Bus, the real and reactive power are specified, while the voltage magnitude and phase angle are unknown. For a voltage-controlled bus, the real power and voltage magnitude are given.

The current equations for a network are expressed in terms of admittance matrices:

Where I is the vector of source currents injected, and V  is the vector of bus voltage. For each bus k, the current and the complex power is:

The two main iterative methods for solving the power flow problem are Gauss-Seidel and Newton-Raphson. The Gauss-Seidel method iteratively solves nodal equations, recalculating current for load buses using known power values and adjusting reactive power for voltage-controlled buses until convergence. The Newton-Raphson method linearizes power flow equations and uses the Jacobian matrix for voltage corrections, generally converging faster and more suitable for large systems. These iterative methods are fundamental for ensuring the power system operates within its specified parameters, maintaining stability and efficiency across the network.