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# UCM | Substation Modeling

- In each substation, there are passive components interpreted as RLC elements which can be linear or non-linear, circuit-breakers, different kinds of generation interpreted as voltage and current sources equipped with control systems.
- Machines and motors are considered sources with control systems.
- Besides control systems, other equipments are power elements working at the power system level voltages and currents.
- Power elements of a substation are interconnected together via nodes. Power elements are not simulated sequentially one by one but rather simultaneously all together in a single equation called the node equation:

**YV=I**

- where
**Y**is the substation admittance matrix,**V**is a vector of node voltages and**I**is a vector of node currents (currents injected to nodes). - Control systems are modeled using the block diagram principle, either under graphic form (HYPERSIM ® block diagram and Simulink block diagram) or coded in C/C++. Their inputs can be node voltages and currents while their outputs can be used to control sources and switches.

**RLC element**

**Trapezoidal integration**

HYPERSIM as EMTP, uses the trapezoidal integration technique, it means that:

is evaluated as

where T is the calculation time step. By the same rule, the derivative

is approximated as a difference deduced from the equation

**L Branch**

For an L branch connected between node k and node m, the following equation is applied:

**OR**

Using the trapezoidal integration rule given by the equation above, we get

with

As shown, an *L* branch is equivalent to resistor *R _{eq} *in parallel to historic current source

*i*. One can see that the historic current depends only on the voltage and current values of the previous step.

_{hist}For a fixed inductor, *L* is constant, therefore *R _{eq} *is constant and need not be recalculated at every time step.

**C branch**

The current of a C branch connected between nodes *k* and *m* is given by

Replace the derivative by the difference equation from page 7 and we get:

or

with

The C branch is also equivalent to a resistor given by the equation above, in parallel with a historic current source as shown. Here again, for a fixed capacitor, *R _{eq} *is constant. The historic current is recalculated at each time step using voltages and currents from the previous time step.

**Branch of RLC Combination**

For branches of different combinations of RLC elements, one can always write down the voltage-current relationship, replace integrals by **eq. “” on page 489, derivative by eq. “” on page 489 (what is this?) **, and get more or less complex forms of a equivalent resistor and a historic current.

**Current & Voltage Sources**

- A current source
*i*flowing from node k to node m has the effect of removing a current*i*from node k and adding a current*i*to node*m*. - Voltage sources with output impedances are converted into current sources in parallel with the same impedance using Thevenin-Norton conversion.

**Non-linear Elements**

- Non-linear elements are treated as RLC elements, and are also equivalent to a resistor
*R*in parallel with a historic current_{eq}*i*(if it is not a pure resistor)._{hist} - Non-linearity is represented normally as a characteristic curve approximated by successive linear segments. Due to the non-linearity,
*R*is no longer constant and needs to be recalculated for each time step._{eq} - Theoretically, this must be done based on the conditions of the actual time step, but HYPERSIM® does it based on conditions obtained at the last time step because the current results are not yet available.
- The change from one segment to the next can be one time-step delayed.
- It is therefore a good practice to define the non-linearity characteristic with a smooth changing (more points where there are more changes and vice versa) to avoid searching.

**HYPERSIM® works rather with the conductance**

**and splits it as follows:**

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.

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