## Fundamental symmetrical components

Normally the electric power system operates in a balanced three-phase sinusoidal steady-state mode. Disturbances, for example a fault or short circuit, lead to a unbalanced condition.

By using the method of the symmetrical components it is possible to transform any unbalanced threephase system into 3 separated sets of balanced three-phase components, the positive, negative and zero sequence.

The main advantage of the symmetrical components is that it makes life (and calculation) much easier. In case of a fault or short circuit the unbalanced system can be easily transformed into symmetrical components, wherewith the calculations can be done straight forward. In the end the results are transformed back into the „real-life“ phase voltages and currents.

In general a 3-phase system can be displayed and described as following:

A balanced 3-phase system may look like this: same RMS-value for all line voltages and currents, and a 120° phase shift between each of them.

In order to explain the basic idea of the symmetrical components, the first step is to define the operator „a“ as unit vector with an phase angle of 120° or 2pi/3.

So the voltages can be described in different ways now:

## Calculation of zero-sequence system

In an symmetrical system the following equation is valid:

In a real system this sum is not equal to zero. A voltage difference occurs:

This voltage difference divided through 3 represents the so called zero-sequence system:

The zero-sequence systems for the three phases (u10, u20, u30) have the same aplitude and phase. Therefore, only value for the zero-sequence system "U_0" will be shown.

The calculation of the zero-sequence current is analogue to this procedure.

HINT: If you multiply the currents for the zero-sequences system with 3 (=I_0 x 3) you will get the current of the neutral line U_N.

## Calculation of positive-sequence system

The positive sequence system has the same rotating direction as the original system (right). This means it will have the same rotating direction of an electrical machine connected to the grid.

As the values of the positive-sequence system for all three phases have the same amplitude (now they are symmetrical) and an phase shift of exactly 120°, it's enough to show one value. The value for the positive-sequence system in Dewesoft X is called „U_1“.

## Calculation of negative-sequence system

The negative sequence system has the opposite rotating direction as the original system (left). This means it will rotate in opposite direction of an electrical machine connected to the grid.

As the values of the negative-sequence system for all three phases have the same amplitude (now they are symmetrical) and an phase shift of exactly 120°, it's enough to show one value. The value for the negative-sequence system in Dewesoft X is called „U_2“.

## Matrix of zero, positive and negative-sequence system

According to the following equations the phase voltages and currents are transformed into the symmetrical components. The result are three balanced 3-phase systems, the positive (U ^{1}, I^{1}), negative (U ^{2}, I^{2}) and zero sequence (U^{0}, I^{0}).

NOTE: The basic values of symmetrical components (U_0, U_1, U_2, I_0, I_1, I_2) are calculated for each harmonic and added up geometrically.

As you can see in the next pictures, a unbalanced system can be composed by using the positive, negative and zero symmetrical components. The following picture shows an screenshot of Dewesoft X showing the real system:

The following picture shows an screenshot of Dewesoft X showing the three systems (positive, negative and zero) of the symmetrical components:

This screen is provided by Kurt Stranner (KNG Netz GmbH).

Out of the parameters of the symmetrical components (positive-, negative- zero- sequence) the original system can easily be rebuild:

The following variables are calculated in Dewesoft X and shows the components of the zero- and negative-sequence system compared to the positive-sequence system (for total and fundamental harmonic).

## Extended positive sequence parameters (according to IEC 614000)

The following calculations are based on Annex C of IEC 61400-21.

Based on the measured phase voltages and currents, the fundamental's Fourier coefficients are calculated over one fundamental cycle T as first step.

It is important to mention that the index a stands for the line voltage L_{1}. The coefficients for L_{2} (ub) and L _{3} (uc) as well as the coeffiecients for the currents (ia, ib, ic) are calculated exactly the same. Furthermore f _{1} is the frequency of the fundamental. The RMS value of the fundamental line voltage is:

## Extended negative sequence parameters (according to IEC 614000)

## Extended zero sequence parameters (according to IEC 614000)