# 100 Years of Symmetrical Components

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Formulations of the Fortescue Transformation

**T**is invertible, and the inverse matrix is:

#### 2.1. Symmetrical Components for Polyphase Systems

#### 2.2. Instantaneous Symmetrical Components

**T**is the same. The ISCs are complex quantities, used with their real and imaginary parts for the analysis. Starting from the phase voltages in the time domain ${v}_{\mathrm{abc}}\left(t\right)={\left[{v}_{a}\left(t\right),{v}_{b}\left(t\right),{v}_{c}\left(t\right)\right]}^{\mathrm{T}}$, the transformed voltages are defined as ${v}_{\mathrm{s}}\left(t\right)={\left[{\overline{v}}_{(+)}\left(t\right),{\overline{v}}_{(-)}\left(t\right),{v}_{\left(0\right)}\left(t\right)\right]}^{\mathrm{T}}=T\text{}{v}_{\mathrm{abc}}\left(t\right)$, where the components ${\overline{v}}_{(+)}\left(t\right)$ and ${\overline{v}}_{(-)}\left(t\right)$ contain complex numbers, while the component ${v}_{\left(0\right)}\left(t\right)$ contains real numbers. Moreover, by definition the term ${\overline{v}}_{(-)}\left(t\right)={\overline{v}}_{(+)}^{*}\left(t\right)$, so that the knowledge of the components ${\overline{v}}_{(+)}\left(t\right)$ and ${v}_{\left(0\right)}\left(t\right)$ is sufficient to represent the ISCs [26]. The use of the complex operator $\overline{\alpha}$ has also been replaced by a shift in the time domain corresponding to $2\mathsf{\pi}/3$, in order to avoid the use of complex quantities to deal with time-domain waveforms [27].

#### 2.3. Generalized Symmetrical Components

## 3. Applications to Power and Distribution System Analysis

**A**describes a relation

**c**=

**A b**in the original space, the Fortescue transformation of the vectors

**b**and

**c**into ${b}_{\mathrm{s}}=Tb$ and ${c}_{\mathrm{s}}=Tc$ yields ${T}^{-1}{c}_{\mathrm{s}}=A{T}^{-1}{b}_{\mathrm{s}}$, from which:

## 4. Applications to Harmonic Analysis

#### 4.1. Symmetrical Components and Harmonics

_{1}= 1/T, and fundamental radian frequency ω

_{1}= 2πf

_{1}) is equal to the periodic waveforms appearing in the other two phases with a time shift of T/3 and 2T/3, respectively. This definition is generally valid also if the waveform is non-sinusoidal. The waveform distortion generates harmonics, and for the balanced three-phase system there is a simple relation between the harmonic order h = 0, 1, …, ∞ and the sequence that represents the phase voltages or currents (Table 2, in which the details of the sequences are also presented up to the harmonic order h = 40). This regularity is quite interesting and has been used to explain a number of phenomena appearing in the electrical systems. In particular, the harmonic orders 3h are called triplen harmonics and correspond to a zero sequence with the three phasors superposed with each other. In a balanced three-phase system with neutral, the triplen harmonics are the only components contributing to the neutral current. However, if the three-phase system is not balanced, the regularity indicated vanishes, and components of different sequences may appear at each harmonic order h > 0 [45]. These aspects have driven the researchers to look for the possibility of representing the balanced or unbalanced system conditions by using the measurements of the phasors gathered from a distorted periodic waveform at different harmonic orders. In particular, some links have been constructed between the transformation matrix used to calculate the symmetrical components and the determination of suitable components. For example, two components (denoted as first unbalance and second unbalance) have been introduced in [46] by applying different transformation matrices to the various harmonic orders. The system unbalance has then been obtained by combining these components. Conversely, in the Symmetrical Component Based (SCB) approach presented in [47] the balance, unbalance and distortion components are determined by using the same transformation matrix from phase quantities into symmetrical components. These components are calculated by summing the squares of the RMS values, because of the orthogonality of the components at different harmonic orders.

_{I}) generalises the classical Total Harmonic Distortion (THD) indicator (calculated for each individual waveform) to provide a single indicator for the unbalanced three-phase system. The distortion component of the phase current is divided by the phase current components at fundamental frequency:

_{I}) generalises the classical unbalance indicator (defined as the ratio between the negative and the positive components at fundamental frequency) to take into account distorted currents:

#### 4.2. Extension to Interharmonics

^{k}H and $\Delta {f}_{h}={10}^{k}\text{}\Delta {\stackrel{\u02c7}{f}}_{z}$ are satisfied for any integer k = 1, 2, …, ∞, the application of the SCB approach (with a single transformation matrix) to the interharmonic orders z = 0, 1, …, ${N}_{z}$ assigns to the frequencies ${f}_{h}$ the same sequences provided by the application of the SCB approach to the harmonic orders h = 0, 1, 2, …, H (Table 3).

## 5. Recent Applications of the Fortescue Transformation

#### 5.1. Electrical Machines

#### 5.2. Distribution Systems with Distributed Energy Resources

_{grid}with the PLL output θ

_{PLL}, and determines the phase angle error θ

_{error}), a loop filter (i.e., a low-pass filter) that provides the radian frequency ω

_{a}, and the phase angle generator (a voltage controlled oscillator) that gives the phase angle θ

_{PLL}. Various PLL types and their applications have been reviewed in [88] and [89]. One of the main issues for the PLL is the failure to operate properly in the presence of unbalanced grid faults, because of the presence of negative sequence components, and with possible harmonics. The techniques used to compensate the effects of unbalanced grid voltages are typically based on the addition of a pre-filtering stage in the voltage phase angle detector. An example is the Decoupled Double Synchronous Reference Frame PLL [90], where the grid voltage is first converted into separate positive and negative synchronous reference frames, then the voltages at the positive and negative sequences are extracted from the corresponding frame and are used to determine the grid voltage phase angle through a PLL. An enhanced version of the Decoupled Double Synchronous Reference Frame has been presented in [91], in order to control the positive sequence and negative sequence active and reactive powers independently of each other. Another solution to avoid the erroneous response of the PLL in the presence of negative-sequence components has been presented in [27], by making the sequence components available both as phasors in the frequency domain and as sinusoidal signals in the time domain. Analytical formulas have been proposed in [92] to analyse the effects of unbalance, harmonics, and interharmonics on the PLL. Further improvements of the PLL dynamics under unbalanced faults can be obtained by using Loop Filter Modification (LFM) methods, in which the loop filter block is modified or tuned depending on the fault or disturbance, in many different ways [89]. The negative-sequence components have been filtered from the unbalanced network voltages in [93], where an EPLL has been used to reduce the higher-order harmonics in converters connected to unbalanced networks. Furthermore, the PLL proposed in [94] is able to reject the negative sequence component, at fundamental frequency, the DC offset component, and the other harmonic components in the three-phase voltages.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Language | Terms Used for the Sequences |
---|---|

English | positive, negative, zero |

Italian | diretta, inversa, omopolare |

French | direct, indirect, homopolaire |

Portuguese | directa, inversa, homopolar |

Romanian | directă, inversă, homopolară |

Spanish | directa, inversa, homopolar |

**Table 2.**Sequences for the different harmonic orders in a balanced three-phase system, h = 0, 1, …, ∞.

Harmonic Order | Radian Frequency | Sequence | |||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

3h − 2 | (3h − 2) ω_{1} | positive (+) | |||||||||||||||||||||||||||||||||||||||

3h − 1 | (3h − 1) ω_{1} | negative (-) | |||||||||||||||||||||||||||||||||||||||

3h | 3h ω_{1} | zero (0) | |||||||||||||||||||||||||||||||||||||||

Harmonic (h) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |

Sequence (h) | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + |

**Table 3.**Sequences for the harmonic and interharmonic orders under the consistency conditions (symbols: + for positive sequence, − for negative sequence, and 0 for zero sequence).

Harmonic order h | 0 | 1 | 2 | 3 | 4 | ||||||||||||||||||||||||||||||||||||

Interharmonic z | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |

Sequence (z) | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + | − | 0 | + |

Sequence (h) | 0 | + | − | 0 | + |

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**MDPI and ACS Style**

Chicco, G.; Mazza, A.
100 Years of Symmetrical Components. *Energies* **2019**, *12*, 450.
https://doi.org/10.3390/en12030450

**AMA Style**

Chicco G, Mazza A.
100 Years of Symmetrical Components. *Energies*. 2019; 12(3):450.
https://doi.org/10.3390/en12030450

**Chicago/Turabian Style**

Chicco, Gianfranco, and Andrea Mazza.
2019. "100 Years of Symmetrical Components" *Energies* 12, no. 3: 450.
https://doi.org/10.3390/en12030450