Admittance matrix

In power engineering, nodal admittance matrix (or just admittance matrix) or Y Matrix or Ybus is an N x N matrix describing a power system with N buses. It represents the nodal admittance of the buses in a power system. In realistic systems which contain thousands of buses, the Y matrix is quite sparse. Each bus in a real power system is usually connected to only a few other buses through the transmission lines. The Y Matrix is also one of the data requirements needed to formulate a power flow study.

Electric power transmission needs optimization in order to determine the necessary real and reactive power flows in a system for a given set of loads, as well as the voltages and currents in the system. Power flow studies are used not only to analyze current power flow situations, but also to plan ahead for anticipated disturbances to the system, such as the loss of a transmission line to maintenance and repairs. The power flow study would determine whether or not the system could continue functioning properly without the transmission line. Only computer simulation allows the complex handling required in power flow analysis because in most realistic situations the system is very complex and extensive and would be impractical to solve by hand. The Y Matrix is a tool in that domain. It provides a method of systematically reducing a complex system to a matrix than can be solved by a computer program. The equations used to construct the Y matrix come from the application of Kirchhoff’s current law and Kirchhoff’s voltage law to a circuit with steady-state sinusoidal operation. These laws give us that the sum of currents entering a node in the circuit is zero, and the sum of voltages around a closed loop starting and ending at a node is also zero. These principles are applied to all the nodes in a power flow system and thereby determine the elements of the admittance matrix, which represents the admittance relationships between nodes, which then determine the voltages, currents and power flows in the system.

Starting from the single line diagram of a power system, there are three main steps before writing the equations that form the ${\displaystyle Y}$ Matrix. First, the single line diagram is converted to an impedance diagram. Next, all voltage sources are converted to their equivalent current source representations. From here, the impedance diagram is then converted to an admittance diagram. Following these three steps, the admittance matrix can be created in a straightforward manner: For an admittance diagram with ${\displaystyle N}$ buses, the admittance between the bus in consideration, k, and another bus, i, connected to k, can be described by ${\displaystyle y_{ik}=g_{ik}+jb_{ik}}$. The term ${\displaystyle y_{k}}$ should be introduced here; this term accounts for the admittance of linear loads connected to bus ${\displaystyle k}$ as well as the admittance-to-ground at bus ${\displaystyle k}$. The general mathematical expression follows:

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