# Comparative of Power Calculation Methods for Single-Phase Systems under Sinusoidal and Non-Sinusoidal Operation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Natural Frame Methods for Single-Phase Systems

#### 2.1.1. Time Delay Method

#### 2.1.2. Time Delay Integrator Method

#### 2.1.3. All-Pass Filter Method

#### 2.1.4. State Observer Method

_{1}, x

_{2}are the state variables.

_{1}= s

_{2}= −10ω and the same criterion will be used. Thus, after clearing the characteristic equation for $a$ $y$ $b$, the following constant values are obtained in (6):

#### 2.1.5. Hilbert Transform Method

#### 2.1.6. Method Using Mathematical Operations

#### 2.2. Power Calculation in the αβ0 Stationary Reference Frame

#### 2.3. Power Calculation in the dq0 Rotating Reference Frame

## 3. Experimental Results and Discussion

^{®}are exported and processed in MATLAB. At the end of this section, the results of the calculation time used by the different methods in the DSC are shown.

#### 3.1. Experimental Validation: Active and Reactive Power Calculation for an RL Load

#### 3.2. Experimental Validation: Active and Reactive Power Calculation for a Non-Linear Load

#### 3.3. Calculation Time of Active and Reactive Power

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 6.**Power calculation by Math Operations with a filter. Active power calculation (top). Reactive Power calculation (down).

**Figure 9.**Voltage (green) and current (orange) signals: (

**a**) fundamental; (

**b**) fundamental + 3rd harmonic; (

**c**) fundamental + 3rd + 5th harmonic; (

**d**) fundamental + 3rd + 5th + 7th harmonic.

**Figure 11.**Power calculation methods at the fundamental frequency: (

**a**) active power; (

**b**) reactive power.

**Figure 12.**Power calculation at fundamental frequency + 3rd harmonic: (

**a**) active power; (

**b**) reactive power.

**Figure 13.**Power calculation at fundamental frequency + 3rd harmonic + 5th harmonic: (

**a**) active power; (

**b**) reactive power.

**Figure 14.**Power calculation at fundamental frequency + 3rd harmonic + 5th harmonic + 7th harmonic: (

**a**) active power; (

**b**) reactive power.

**Figure 17.**Reactive power calculation with all methods at the fundamental frequency—non-linear load.

Feature | Parameter | Values |
---|---|---|

f | fundamental frequency | 60 Hz |

fsw | switching frequency | 1080 Hz |

fs | Sampling frequency | 20160 Hz |

Lf | filter inductance | 2.7 mH |

Cf | filter capacitance | 9.4 µF |

L | inductive load | 18.8 mH |

fc | non-linear load crest factor | 2.9 |

R_{L} | resistive load | 64.2 Ω |

Method/Component | State Observer | αβ Method | DQ Method | All-Pass Filter | ||||

P(W) | Q(VAR) | P(W) | Q(VAR) | P(W) | Q(VAR) | P(W) | Q(VAR) | |

F | 187.5 | 47.19 | 186.3 | 44.61 | 186.7 | 46.86 | 186.8 | 46.76 |

F + 3rd | 193.9 | 49.82 | 193.6 | 48.90 | 194.3 | 52.30 | 193.8 | 43.20 |

F + 3rd + 5th | 200.0 | 41.43 | 206.3 | 55.37 | 206.4 | 56.77 | 207.2 | 39.88 |

F + 3rd + 5th + 7th | 217.1 | 21.41 | 217.3 | 61.92 | 217.5 | 60.75 | 217.7 | 34.86 |

Method/Component | Mathematical Operations | αβ Methodwith Filter | Mathematical Operations with Filter | Fluke 1730 Measurement | ||||

P(W) | Q(VAR) | P(W) | Q(VAR) | P(W) | Q(VAR) | P(W) | Q(VAR) | |

F | 185.2 | 46.85 | 186.3 | 44.87 | 185.0 | 46.64 | 190.5 | 55.04 |

F + 3rd | 227.9 | 64.33 | 194.0 | 49.26 | 228.5 | 64.73 | 190.5 | 89.40 |

F + 3rd + 5th | 309.4 | 83.72 | 206.6 | 55.09 | 307.8 | 84.03 | 190.5 | 124.5 |

F + 3rd + 5th + 7th | 405.6 | 100.5 | 217.6 | 61.42 | 405.1 | 100.5 | 190.5 | 154.7 |

Method/Component | State Observer | αβ Method | DQ Method | All-Pass Filter | |||||||||

P % | Q % | P % | Q % | P % | Q % | P % | Q % | ||||||

F | 3.130 | 14.267 | 3.712 | 18.93 | 3.519 | 14.85 | 3.467 | 15.04 | |||||

F + 3rd | 0.191 | 44.26 | 0.048 | 45.23 | 0.387 | 41.49 | 0.126 | 51.65 | |||||

F + 3rd + 5th | 6.477 | 66.71 | 6.587 | 55.55 | 6.648 | 54.40 | 7.068 | 67.96 | |||||

F + 3rd + 5th + 7th | 12.20 | 86.16 | 12.30 | 59.98 | 12.41 | 60.7 | 12.50 | 67.96 | |||||

Method/Component | MathematicalOperations | αβ Methodwith Filter | Mathematical Operations with Filter | ||||||||||

P % | Q % | P % | Q % | P % | Q % | ||||||||

F | 4.314 | 14.8 | 3.745 | 18.47 | 4.373 | 15.25 | |||||||

F + 3rd | 17.76 | 28.04 | 0.261 | 44.89 | 18.07 | 27.59 | |||||||

F + 3rd + 5th | 59.8 | 32.75 | 6.789 | 55.75 | 59.05 | 32.50 | |||||||

F + 3rd + 5th + 7th | 109.5 | 35.00 | 12.46 | 60.30 | 109.30 | 34.99 |

State Observer | αβ Method | DQ Method | All-Pass Filter | ||||

P(W) | Q(VAR) | P(W) | Q(VAR) | P(W) | Q(VAR) | P(W) | Q(VAR) |

202.1 | 14.81 | 194.9 | 129.3 | 199.9 | 19.09 | 199.4 | 27.17 |

MathematicalOperations | αβ Methodwith filter | Mathematical Operations with filter | Fluke 1730 Measurement | ||||

P(W) | Q(VAR) | P(W) | Q(VAR) | P(W) | Q(VAR) | P(W) | Q(VAR) |

131.7 | 2.113 | 198.7 | 127.7 | 148.2 | 3.201 | 199.3 | 220.6 |

Mathematical.Operations | αβ Methodwith Filter | Mathematical Operationswith Filter | |||||

P % | Q % | P % | Q % | P % | Q % | ||

23.20 | 107.7 | 2.970 | 371.2 | 23.20 | 111.8 | ||

State Observer | αβ Method | DQ Method | All-Pass Filter | ||||

P % | Q % | P % | Q % | P % | Q % | P % | Q % |

4.740 | 45.35 | 1.000 | 377.0 | 3.590 | 29.58 | 3.350 | 0.220 |

Method | Time |
---|---|

All-pass filter | $5.2\mathsf{\mu}$s |

State Observer | $5.2\mathsf{\mu}$s |

Mathematical Operations | $8.4\mathsf{\mu}$s |

αβ method | $15.6\mathsf{\mu}$s |

DQ method | $12\mathsf{\mu}$s |

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**MDPI and ACS Style**

Chica Leal, A.d.J.; Trujillo Rodríguez, C.L.; Santamaria, F.
Comparative of Power Calculation Methods for Single-Phase Systems under Sinusoidal and Non-Sinusoidal Operation. *Energies* **2020**, *13*, 4322.
https://doi.org/10.3390/en13174322

**AMA Style**

Chica Leal AdJ, Trujillo Rodríguez CL, Santamaria F.
Comparative of Power Calculation Methods for Single-Phase Systems under Sinusoidal and Non-Sinusoidal Operation. *Energies*. 2020; 13(17):4322.
https://doi.org/10.3390/en13174322

**Chicago/Turabian Style**

Chica Leal, Alonso de Jesús, César Leonardo Trujillo Rodríguez, and Francisco Santamaria.
2020. "Comparative of Power Calculation Methods for Single-Phase Systems under Sinusoidal and Non-Sinusoidal Operation" *Energies* 13, no. 17: 4322.
https://doi.org/10.3390/en13174322