YBus

Ybus, short for bus admittance matrix, is a fundamental construct in power system analysis. It is a complex N×N matrix that relates the complex current injections at each bus to the bus voltages, via the equation I = Ybus · V, where I is the vector of bus currents and V is the vector of bus voltages. Ybus is central to AC power-flow studies and small-signal analyses.

Construction and interpretation: For a network with N buses, Ybus is built by summing the admittances of all elements connected to each bus. For every branch between buses m and n with series admittance y = 1/z and any shunt admittance y_sh at the ends, the standard updates are:

- Y_mm += y + y_sh

- Y_nn += y + y_sh

- Y_mn += -y

- Y_nm += -y

If a branch has no explicit shunt, y_sh is zero. When multiple branches connect the same pair of buses, their series admittances add, and the corresponding off-diagonal term is the negative of the total mutual admittance. The diagonal terms accumulate all admittances connected to a bus, including line charging and shunt elements. In symmetric, passive networks the matrix is complex and symmetric; losses and line charging are reflected in the real and imaginary parts.

Applications and variants: Ybus is widely used in Newton-Raphson AC power-flow methods, transient and small-signal analyses, and network reductions. In per-unit systems, Ybus is typically stored in sparse form to handle large networks. A reduced Ybus is formed by eliminating the reference (slack) bus, yielding a smaller system for voltage or injection calculations. Ybus accuracy depends on the network model, including line conductance, reactance, transformer tap settings, and shunt elements.