## Fundamental symmetrical components

Normally an electric power system operates in a balanced three-phase sinusoidal steady-state mode. Disturbances, for example a fault or short circuit, lead to an unbalanced condition. As the following image depicts on the left-hand side is a balanced system with a symmetrical phase shift and equal vector distances. On the right-hand side is an unbalanced system with an unsymmetrical phase shift and uneven vector lengths.

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

The advantage of the symmetrical system lies in that the calculations are simplified. Should a fault arise or there is a short circuit in the system, an unbalanced system can be transformed into a balanced system with symmetrical components, where the system calculations can be done with the normal formulas that would be used in a balanced system. The calculated values are then transformed back to the unsymmetrical system (real-scenario) phase voltages and currents.

In general, a 3-phase system can be depicted and mathematically described such as on the following page:

A balanced 3-phase system looks like the image below with the 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 would be to define the operator „ “ as a unit vector with an phase angle of 120° or 2*pi/3.

The voltages can be described mathematically in different ways as the table below shows:

## Calculation of zero-sequence system

In an symmetrical system the following equation is valid:

In a real system the sum won't be zero. There will be a voltage difference as the following equation shows:

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

The zero-sequence systems for the three phases (u_{10}, u_{20}, u_{30}) all have the same amplitude and phase. Therefore, only the value for the zero-sequence system „U_0“ will be shown.

The calculation of the zero-sequence current is analog to that of the voltage equation.

When the currents of the zero-sequence system are multiplied by three (= I_0 ∙ 3) it will yield the current that is flowing in 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 that they are symmetrical) and has a phase shift of exactly 120°, it's adequate to show only 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 has a phase shift of exactly 120°, it's adequate 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}).

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 illustrated in the following images, an unbalanced system can be rectified using the positive, negative and zero symmetrical components. The image below depicts an unsymmetrical system as a screen shot taken from Dewesoft X .

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- and zero- sequence) the original system can be rebuilt easily, e.g.:

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 the total and the 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)