UCM | Switches

Switches

Switches such as circuit-breakers and power electronic switches are simulated as resistors with two values: one small value (milliohms) for closing state and one large value (meghoms) for opening state. To know which resistor value to use, HYPERSIM® needs to know the state of switch prior to the calculation of Y matrix.

Generalized Power Element

Each branch element described above is connected between node k and m of the substation and is equivalent to a resistor

in parallel to a current source

A generalized power element is simply a "black box" connected to nodes k, m, n etc. in the substation. It is equivalent to a conductance (inverse of resistance) square matrix

and a vector of historic current

Their dimensions are the number of nodes of the element.

of the matrix

is its self conductance at node k while

is the mutual conductance between node k and node m.

In case of a non-linear element, Y_eq is split into two parts:

where Y_ini is the fixed part used as initial condition and Y_add is the varying part used to update the conductance according to its operating point.

Therefore, all branches are also generalized elements connected to two nodes in the substation. Linear branches have Y_add = 0 and Y_ini = Y_eq

All power elements, fixed or non-linear, are equivalent to resistors in parallel with historic currents. In the node equation

therefore becomes a pure conductance matrix, I is the algebraic sum of currents, including source currents and historic currents, injected into nodes; currents entering a node have a + sign, currents leaving have a - sign.

Solution of Substation's Node Equation

Y has dimension n x m where n is the number of nodes in the substation (excluding the ground node), V and I are vectors of dimension n. A three-phase bus bar corresponds to 3 nodes.

As with each individual element, HYPERSIM splits Y into a fixed part and a varying part:

is formed by the fixed parts Y_ini of all elements and Y_ADD is formed by varying parts Y_add of all elements.

The simulation of all power elements in a substation required to solve the node equation YV=I where Y and I are known eqs. to and V is the node voltage vector to be found.

HYPERSIM solves the node equation using the LDU conversion: Y=LDU

where L is a lower triangular matrix with all values in the upper triangular matrix equal to 0 and all elements on the diagonal equal to 1, D is a diagonal matrix, U is an upper triangular matrix with all values in the lower triangle equal to 0 and all elements on the diagonal equal to 1. For example, if the matrix Y has a dimension 3 times 3, the LDU conversion will have the following form:

The LDU conversion is chosen instead of the LU conversion because when Y is symmetrical (which is the case for most of the substation's node equation), U=L^T. HYPERSIM will use this advantage if it exists to save the time needed to compute U.

The node equation YV=I now becomes LDUV=I

This is solved in two steps: first, replace J=DUV and solve for J in:

LJ = I
using the forward substitution (because L is lower triangular). Then solve for V in

DUV = J

using the backward substitution (because DU is an upper triangular matrix).

LDU decomposition

The LDU conversion is done based on the pivot technique used to nullify a particular element in a matrix.

Only the final formulas are listed here:

Elements in lower triangular:

Elements in upper triangular:

For i<j.

At any time step, if Y does not change from the previous step, the LDU decomposition does not need to be redone.

If Y changes due to either switching or a change of behavior of some non-linear elements, then the LDU decomposition must be recalculated.

In HYPERSIM, for saving space, matrices L, D, and U are stored on a single Y_LDU matrix, as shown, for an example of 3 modes. Elements of D are on the diagonal, and elements of L and U (except their diagonal) are respectively in the lower left and upper right triangular.

Example of a

matrix and a part of it which needs to be recalculated if a resistor connected to node 2 changes

All equivalent resistors of power elements contributing to Y are connected either between a node k and the ground node or between two nodes k and m. According to previous equations, if a resistor connected to node k changes its value, this changes at least the value of

Therefore, D_ii, L_ij and U_ij, for all i and

have to be re-evaluated. In the Y_LDU matrix shown here, this corresponds to elements in the square block below and on the right starting from

k=2 in this example)

For the purpose of saving computation time, HYPERSIM orders the Y and Y_LDU matrices such as nodes connected to varying elements, groups them together and corresponds to the lower right part of these matrices.

Sparseness of Y Matrix

The $Y$ matrix is normally sparse, meaning that many elements are zero. To avoid calculations with zero elements, HYPERSIM keeps a matrix $YF$ indicating the state of each element of

Y: