## Outline

- Abstract
- Keywords
- Introduction
- Mathematical Derivation of the Proposed Method
- Improvement of the Proposed Method for Power Quality Disturbances
- Case Studies
- Case 1: Ideal Test System
- Case 2: Test System with Harmonic Source
- Case 3: Non-Ideal Test System
- Case 4: The Sinusoidal Signals Mixed with Noise
- Conclusions
- References

## رئوس مطالب

- چکیده
- مقدمه
- مشتقات ریاضی روش پیشنهادی
- مطالعات موردی
- نتیجه گیری ها

## Abstract

The accurately and fast estimation of Phase Difference (PD) is required between the voltage and current of an AC electrical power system to calculate the Power Factor (PF) for defining how effectively the electrical energy is converted into the useful form. Many complex methods based on difficult mathematical equations are presented by the researchers to estimate the PD. In this study, a new and simple algorithm derived by using the trigonometric functions is proposed for PD estimation to calculate PF of a power system. With this method, the fast-time and unaffected by distorted sinusoids of PD estimation are carried out by decreasing the number of mathematical equations. The performance of the proposed method is evaluated under the various system conditions by performing the simulation case studies. The results of these studies are given to verify its effectiveness under the distorted system conditions.

**Keywords**: Phase difference - Power distortion - Power factor - Power quality

## Conclusions

In this study, a new and simple algorithm is proposed for PD between two sinusoidal signals having a same frequency to calculate the accurately and fast estimation of PF. The proposed method is based on the simple trigonometric functions instead of the complex mathematical equations according to the other methods. Its theory is given by supporting mathematical derivation and is improved by considering PQD.

The performance of the proposed method is evaluated under the various system conditions by performing simulation case studies. In the first case study, the ideal system having the inductive load is formed by changing the breakers positions. The value of u between V and I is easily realized by the readers in Fig. 4 due that there are no any harmonics at the system variables. In the second and third studies, the systems having the current harmonics and the line-ground faults are form to see the negative effects of the harmonics and faults, respectively. In these case studies, u can be calculated with numerical methods due to the harmonics. Also, the negative effects of the faults are easily realized by the readers in Fig. 6. According to the results, PD and PF are correctly calculated by proposed method at these studies. In the fourth study, the two sinusoidal signals are mixed with noises having different frequencies and amplitude. The values of PD and PF are calculated with small errors (less than 0.015%) ignored by the users.

The simulation results show that the proposed method correctly and quickly calculates the values of PD and PF. In addition, the implementation of its algorithm can be easily integrated to the micro-controllers and the digital signal processors instead of FFT and the other methods due that the less space is required in the memory of these devices. The other advantage of the proposed method is the inputs (u and y) of the fourth-part illustrated in Fig. 2. The values of u and y represent the instant values of the active and reactive powers of the test system, respectively. Hence, the proposed method can be easily integrated to the system’s power calculation besides the extraction of PD and PF. However, the fluctuation can be occurred on the curve of u and y if the input signals contain the high harmonic levels. If the time averages of u and y are separately taken to remove the ripples, their constant values are obtained at the fourth-part.