The nodal admittance matrix of a power system is a form of
Laplacian matrix of the nodal admittance diagram of the power system, which is derived by the application of
Kirchhoff's laws to the admittance diagram of the power system. Starting from the
single line diagram of a power system, the nodal admittance diagram is derived by: • replacing each line in the diagram with its equivalent admittance, and • converting all voltage sources to their equivalent current source. Consider an admittance graph with N buses. The
vector of bus voltages, V, is an N \times 1 vector where V_{k} is the voltage of bus k, and
vector of bus current injections, I, is an N \times 1 vector where I_{k} is the cumulative current injected at bus k by all loads and sources connected to the bus. The admittance between buses k and i is a complex number y_{ki}, and is the sum of the admittance of all lines connecting busses k and i. The admittance between the bus i and ground is y_{k}, and is the sum of the admittance of all the loads connected to bus k. Consider the
current injection, I_{k}, into bus k. Applying
Kirchhoff's current law : I_{k} = \sum_{i=1, 2, \ldots, N} I_{ki} where I_{ki} is the current from bus k to bus i for k \neq i and I_{kk} is the current from bus k to ground through the bus load. Applying
Ohm's law to the admittance diagram, the
bus voltages and the line and load currents are linked by the relation : I_{ki} = \begin{cases} V_{k} {y_{k}}, & \mbox{if} \quad i = k \\ ( V_{k} - V_{i} ) y_{ki}, & \mbox{if} \quad i \neq k. \end{cases} Therefore, : I_{k} = \sum_{i=1, 2, \ldots, N \atop i \neq k} { ( V_{k} - V_{i} ) y_{ki} } + V_{k} y_{k} = V_{k} \left( y_{k} + \sum_{i=1, 2, \ldots, N \atop i \neq k} y_{ki} \right) - \sum_{i=1, 2, \ldots, N \atop i \neq k} V_{i} y_{ki} This relation can be written succinctly in matrix form using the admittance matrix. The nodal admittance matrix Y is a N \times N matrix such that bus voltage and current injection satisfy Ohm's law : Y V = I in vector format. The entries of Y are then determined by the equations for the
current injections into buses, resulting in : Y_{kj} = \begin{cases} y_{k} + \sum_{i=1, 2, \ldots, N \atop i \neq k} {y_{ki}}, & \mbox{if} \quad k = j \\ -y_{kj}, & \mbox{if} \quad k \neq j. \end{cases} As an example, consider the admittance diagram of a fully connected three bus network of figure 1. The admittance matrix derived from the three bus network in the figure is: : Y = \begin{pmatrix} y_{1} + y_{12} + y_{13} & -y_{12} & -y_{13} \\ -y_{12} & y_{2} + y_{12} + y_{23} & -y_{23} \\ -y_{13} & -y_{23} & y_{3} + y_{13} + y_{23} \\ \end{pmatrix} The diagonal entries Y_{11}, Y_{22}, ..., Y_{nn} are called the
self-admittances of the network nodes. The non-diagonal entries are the
mutual admittances of the nodes corresponding to the subscripts of the entry. The admittance matrix Y is typically a
symmetric matrix as Y_{ki} = Y_{ik}. However, extensions of the line model may make Y asymmetrical. For instance, modeling phase-shifting transformers, results in a
Hermitian admittance matrix. == Applications ==