# A Two-Stage Algorithm to Estimate the Fundamental Frequency of Asynchronously Sampled Signals in Power Systems

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## Abstract

**:**

## 1. Introduction

## 2. Two-Stage Algorithm for Estimating a Fundamental Frequency

_{f}(rad/s), amplitude A

_{f}(V or A), and phase θ

_{f}(rad), the N

_{M}-point data uniformly sampled with a sampling time Δt can be described in discrete time steps as follows:

_{0}(rad/s) is a nominal frequency and N

_{0}is the number of samples per cycle at ω

_{0}.

#### 2.1. Tuned Sine Filtering Followed by Time-Domain Interpolation

_{f}, respectively. Some minor calculations yield:

#### 2.2. Modified Curve Fitting with an Unknown Frequency

_{f}, S

_{f}, and ω

_{f}in Equation (6) respectively, the sine-filtered signal of Equation (6) can be expressed as:

#### 2.3. Frequency Estimation Procedure

## 3. Performance Evaluation

#### 3.1. Computer Simulations

#### 3.1.1. Number of Cycles in the Data Window

#### 3.1.2. Noise Level

#### 3.1.3. Harmonics

#### 3.1.4. Fluctuating Harmonic Component

_{k}is the frequency of fluctuation, which was set to vary randomly within 7 ± 2 Hz; A

_{k}is the amplitude of the fluctuating harmonic component, which was set to 0.5; and B

_{k}is the depth of fluctuation, which was set to 0.1. The other parameters were identical to those used in Section 3.1.3, except that here the integer multiplier S was set to 1. The order of the fluctuating harmonic component was varied from 2 to 10, and the maximum error of frequency estimation after 1000 repetitions for each fluctuating harmonic order is given in Figure 5; comparison between Figure 4 and Figure 5 shows that the maximum errors for most of the algorithms are significantly worse when one of the harmonics fluctuates. Performance slightly improves as the fluctuating harmonic order increases. This is most likely because spectral leakage from a fluctuating harmonic closer to the fundamental frequency has a greater effect on the estimate.

_{k}was varied from 0 to 1.0 in 0.1 increments. The maximum error of frequency estimation after 1000 repetitions for each modulation depth is shown in Figure 6. Similar to the effect seen when changing the fluctuating harmonic order, an increase in spectral leakage due to greater modulation depth has an adverse effect on the performance of the PSFEi, TDIS and two-stage algorithms. The 4PSF algorithm is more influenced by THD level and a fluctuating harmonic component does not have a significant effect on its performance.

**Figure 6.**Maximum error of frequency estimation with increasing modulation depth of the third harmonic.

#### 3.1.5. Fluctuating Inter-Harmonic Component

_{x}was varied from 0 to 0.7 in 0.1 increments. The maximum error of frequency estimation after 1000 repetitions for each of the inter-harmonic amplitude is given in Figure 7. At each repetition, index x was randomly varied from 1.5 to 10. The contamination of the fundamental component caused by the inter-harmonic component leads to increasing estimation errors for the TDIS, PSFEi, and two-stage algorithms with increasing inter-harmonic amplitude. Again, a fluctuating inter-harmonic component does not have a significant effect on the performance of the 4PSF algorithm.

#### 3.1.6. Computational Burden

Algorithm | Figure number | ||||||
---|---|---|---|---|---|---|---|

2 | 3 | 4 | 5 | 6 | 7 | ||

4PSF | 8.3337 | 8.2824 | 8.1042 | 8.3044 | 8.2134 | 8.5206 | |

PSFEi | 12.388 | 11.279 | 10.444 | 11.102 | 10.710 | 11.762 | |

TDIS | 463.98 | 465.78 | 459.23 | 466.76 | 462.63 | 481.13 | |

Two-stage | 254.12 | 348.07 | 331.07 | 341.41 | 339.56 | 355.14 |

#### 3.2. Hardware Implementation

No. | Device Name |
---|---|

1 | Electrical power quality calibrator (Fluke 6105A) |

2 | Waveform generator (Agilent 33500B) |

3,4 | Digitizing multi-meter (Agilent 3458A) |

5 | Current shunt (Fluke A40B) |

6 | Host computer |

**Table 2.**Maximum errors of frequency estimation with a fundamental frequency of 60 Hz (10

^{−6}× Hz/Hz).

Voltage | H_{1} = 110 V, θ_{1} = 0 | H_{1} = 220V, θ_{1} = 0 | |||||||
---|---|---|---|---|---|---|---|---|---|

Harmonic | H_{3} = 10% | H_{3} = 10% | H_{49} = 10% | H_{49} = 10% | H_{3} = 10% | H_{3} = 10% | H_{49} = 10% | H_{49} = 10% | |

θ_{3} = 0 | θ_{3} = π | θ_{49} = 0 | θ_{49} = π | θ_{3} = 0 | θ_{3} = π | θ_{49} = 0 | θ_{49} = π | ||

4PSF | 6.6318 | 12.492 | 9.9227 | 9.6065 | 8.7664 | 11.733 | 9.5059 | 9.7659 | |

PSFEi | 9.4811 | 9.9378 | 9.9649 | 9.6864 | 9.6831 | 9.4355 | 9.8214 | 9.9948 | |

TDIS | 11.623 | 8.4567 | 9.3250 | 9.1283 | 10.472 | 9.7417 | 9.0150 | 9.0917 | |

Two-stage | 9.5266 | 9.8951 | 9.9366 | 9.6903 | 9.5992 | 9.5416 | 9.7377 | 9.9564 |

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Moon, J.-H.; Kang, S.-H.; Ryu, D.-H.; Chang, J.-L.; Nam, S.-R.
A Two-Stage Algorithm to Estimate the Fundamental Frequency of Asynchronously Sampled Signals in Power Systems. *Energies* **2015**, *8*, 9282-9295.
https://doi.org/10.3390/en8099282

**AMA Style**

Moon J-H, Kang S-H, Ryu D-H, Chang J-L, Nam S-R.
A Two-Stage Algorithm to Estimate the Fundamental Frequency of Asynchronously Sampled Signals in Power Systems. *Energies*. 2015; 8(9):9282-9295.
https://doi.org/10.3390/en8099282

**Chicago/Turabian Style**

Moon, Joon-Hyuck, Sang-Hee Kang, Dong-Hun Ryu, Jae-Lim Chang, and Soon-Ryul Nam.
2015. "A Two-Stage Algorithm to Estimate the Fundamental Frequency of Asynchronously Sampled Signals in Power Systems" *Energies* 8, no. 9: 9282-9295.
https://doi.org/10.3390/en8099282