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

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_eq in parallel with a historic current i_hist (if it is not a pure resistor).
Non-linearity is represented normally as a characteristic curve approximated by successive linear segments. Due to the non-linearity, R_eq is no longer constant and needs to be recalculated for each time step.
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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