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# UCM | Line Model

**Line Model**

**Conversion from 3 coupled phases into 3 decoupled modes**

In a three-phase line, each phase is mutually coupled with other phases. To facilitate the line simulation, the EMTP uses a technique to convert a mutually coupled three-phase line into 3 decoupled, single-phase lines representing 3 decoupled modes.

**In HYPERSIM®, lines are modeled as follows:**

- Get three phase node voltages at each terminal
*K*and*M*, - Convert three phase voltages into three-mode voltages:

For an unbalanced line, **T _{v}** is the solution of the following equation:

Where lambda is a diagonal matrix containing the eigenvalues of the matrix product ZY.

For a balanced line:

Calculate three mode currents using a single phase line equation (represent each mode).

Convert three modes current into three-phase currents.

Inject phase current to terminal nodes *K* and *M*.

**Model of single-phase line**

Each mode (0, 1, 2) can then be represented as shown in **. The current sources in this figure are given as:

where

- m is the mode number (0, 1, 2)
- Z
_{c }is the line characteristic impedance - tau is the transmission delay along the line and
- R
_{T }is the line loss.

A similar expression can also be derived for* J _{mM}*, the current source at terminal

*M*.

Therefore a single line model is reduced at each terminal to an equivalent circuit composed of a current source in parallel with a resistor. The actual current source *J _{mK}* at the terminal

*K*and

*J*at terminal

_{mM}*M*depend only on the past voltages and past currents at both terminals. Equivalent at each terminal can therefore be calculated in parallel.

Model for frequency dependent lines uses the same philosophy but with a more complex equivalent.

**Equivalent circuit of one mode on a transmission line**

Each half (left and right) of the line equivalents will be converted from mode to phase form and incorporated into the corresponding substation equation to be solved with other elements.

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