Phase error monitoring in this system involves comparing the GNSS’s Pulse Per Second (PPS) signal with the local Numerically-Controlled Oscillator (NCO) clock signal to measure their phase difference. This process can be expressed as follows (Eq. 1). e t = ϕ N C O t − ϕ G N S S t (1) where ϕ N C O t is the phase of the NCO-generated signal, and ϕ G N S S t is the phase of the GNSS PPS signal at time t . This continuous monitoring of e t provides a dynamic measure of the synchronization status between the local NCO and the GNSS clock.

This process determines the synchronization deviation between the two signals, which is the basis for subsequent frequency adjustment. The phase error signal is obtained by calculating this phase difference, where the precise GNSS PPS signal acts as a reference for synchronization against the local NCO signal.

The PI (Proportional-Integral) controller is essential in our method for fine-tuning the frequency of the Numerically-Controlled Oscillator (NCO) in response to phase errors. Its key role is to dynamically adjust the NCO’s frequency to reduce phase discrepancies, thereby aligning more accurately with the GNSS standard. The controller’s output, u t , which dictates the frequency adjustment for the NCO, is calculated as follows (Eq. 2): u t = K p ⋅ e t + K i ∫ 0 t e τ d τ (2)

Here, K p and K i are the proportional and integral gains of the controller, respectively.

The proportional gain, K p , is responsible for providing a rapid response to immediate phase errors, allowing the system to quickly react to sudden changes in phase alignment. It acts on the current value of the phase error e t , where e t is the real-time phase difference between the NCO and GNSS signals.

The integral gain, K i , on the other hand, focuses on long-term precision and stability. It integrates the phase error over time, effectively addressing any cumulative or drift errors. This aspect of the controller is essential for maintaining consistent synchronization over extended periods, as it compensates for ongoing small discrepancies that might be overlooked by the proportional component alone.

The output frequency of the Numerically-Controlled Oscillator (NCO) is adjusted based on a calculated control word, typically referred to as the Frequency Control Word (FCW). The relationship governing the output frequency f out of the NCO is expressed as follows (Eq. 3): f out = FCW × f clk 2 n (3)

Here, f clk represents the frequency of the clock driving the NCO, and n stands for the number of bits in the NCO’s phase accumulator. The FCW is the key parameter that determines the frequency at which the NCO operates. By modifying the FCW, we can change the output frequency of the NCO to either increase or decrease in relation to the clock frequency and the resolution offered by the phase accumulator.

The control logic within the NCO takes the desired frequency adjustment output from the PI controller u t and translates it into an appropriate FCW. This word is then used to set the frequency of the NCO, effectively syncing the local oscillator with the GNSS PPS signal. As the PI controller responds to the phase error e t , the FCW is dynamically updated, ensuring continuous phase alignment and maintaining synchronization accuracy.

The aim of the recursive DFT is to reduce the computational load required to compute the DFT at each sampling point. It is based on the fact that when the signal sequence x n receives a new sample, the new DFT result Eq. 4 X ′ k can be directly updated from the previous DFT result X k without recalculating the entire summation. X ′ k = ∑ n = 1 N x n ⋅ e − j 2 π N k n = e − j 2 π N k ∑ n = 0 N − 1 x n ⋅ e − j 2 π N k n − x n − N + x n = e − j 2 π N k ⋅ X k − x n − N + x n (4)

Here, X ′ k is the updated DFT value, while X k is the previously computed DFT value. Through this approach, we can recursively update the DFT, significantly reducing the computational effort.

Based on the results of the recursive DFT, the amplitude, phase and frequency of the signal can be estimated. These three parameters constitute the so-called synchronized phasor.

The amplitude A of the phasor can be calculated as follows (Eq. 5): A = 2 N X k (5)

This equation considers the amplitude normalization factor 2 N (since only half of the spectrum is used for real signals).

The phase angle θ can be calculated by taking the arctangent of the ratio of the imaginary part to the real part of X k as follows (Eq. 6): θ = ⁡ arctan Im X k Re X k (6)

The frequency f can be determined by the least squares method. Firstly, phase unwrapping is applied to the calculated phase values to eliminate discontinuities, producing a smooth phase sequence. This sequence, θ n , is then modeled as a linear function of time with the equation as follows (Eq. 7): θ n = 2 π f n T + ϕ (7) where f is the frequency of the signal, T is the sampling period, and ϕ is the initial phase.

Then, the frequency of the signal can be estimated using the following least squares method formula as follow Eq. 8: f = ∑ n n − n ¯ θ n − θ ¯ 2 π T ∑ n n − n ¯ 2 (8) where n ¯ and θ ¯ are the mean values of the sample indices and the unwrapped phase values, respectively.

In this test, the nominal sampling rate and the actual sampling rate were set to 1,200 Hz and 1,199.991 Hz, respectively. The test signal was a 50 Hz sine wave with 60 dB white noise. The results are shown in Figure 2.

In traditional sampling methods, Sampling Time Error (STE) accumulates over time, reaching up to 2 × 10⁻³ V, and can only be cleared with the arrival of the next Pulse Per Second (PPS) signal. In contrast, the method discussed herein significantly mitigates the cumulative effect of STE in the sampling data, ensuring it does not exceed 1 × 10⁻⁵ V per second. By comparison, software-based resampling synchronization methods also effectively reduce STE, maintaining it below 2 × 10⁻⁵ V. However, these methods demand substantial computational resources from the core processor, rendering them impractical for operation in small-scale devices when measuring three-phase information simultaneously. This comparison highlights the advantages of the proposed method in reducing STE with greater efficiency and lower computational demand, offering a viable solution for high-precision measurements in constrained environments.

The frequency offset testing, as mandated by the IEEE C37.118.1 standard, is crucial for assessing the adaptability and accuracy of synchronization algorithms and devices within the context of modern electrical grids. This testing, performed through a signal generator producing signals across a range from 45 Hz to 55 Hz, addresses the increasing frequency fluctuations attributed to the rising integration of renewable energy sources. Such fluctuations pose challenges to grid stability and reliability. The expanded frequency range for testing is essential to ensure that our algorithms and devices can handle the broader variability seen in grids with a high proportion of renewable energy, where traditional stability mechanisms are often stretched to their limits. This testing is therefore vital for applications in contemporary energy networks, which are increasingly characterized by significant renewable energy contributions and the resultant frequency variations.

The results, as depicted in the accompanying Figure 3, clearly demonstrate that devices employing the proposed PI-controlled algorithm consistently met the standard requirement of a Total Vector Error (TVE) less than 1% across all test frequencies. In contrast, devices using the traditional method exhibited a consistently higher TVE, exceeding 1%. This discrepancy is attributed to the cumulative effect of sampling time errors inherent in the traditional approach, underscoring the enhanced accuracy and reliability of the PI-controlled method in handling frequency variations.

With the continuous increase in the dynamic nature of grid signal, there is a gradual rise in the bandwidth requirements for dynamic measurements. According to the specifications of IEEE C37.118.1 standard, the bandwidth for synchronous phasor measurements should be determined by applying sine amplitude and phase modulation scanning inputs, while ensuring that the Total Vector Error (TVE) does not exceed 3%. The input of the positive sequence signal can be calculated using the Eq. 9. X 1 = X m 1 + k x ⁡ cos   ω t × ⁡ cos   ω 0 t + k a ⁡ cos   ω t − π (9)

Here, X m represents the amplitude of the input signal, ω 0 is the nominal power system frequency, ω denotes the modulation frequency in radians per second, k x is the amplitude modulation factor, and k a is the phase angle modulation factor. This implies that the amplitude or phase angle of the test signal is no longer a fixed value, but undergoes sinusoidal transformations. In this test, the modulation frequency ω ranges from 0 to 5 Hz with an increment of 0.5 Hz. The coefficients k x and k a in the equation are set to 0 or 0.1, depending on the experiment conducted. Figure 4 illustrates the waveforms after amplitude and phase modulation.

This study conducts tests on the signal using both traditional methods and PI control, evaluating the measurement errors in frequency and phase angle at different modulation frequencies. The test results, as shown in Figure 5, reveal that at a modulation frequency of 5 Hz, the traditional method exhibits total harmonic distortion (TVE) of 4.38% and 3.23% in amplitude modulation and phase modulation tests, respectively, exceeding the limits specified in the IEEE C37.118.1 standard. In contrast, in both tests, the maximum TVE value with PI control is 2.94%, still complying with the standard requirements. Therefore, the results indicate that the proposed PI control method outperforms traditional methods in this study.

The method introduced was implemented in a Phasor Measurement Unit (PMU) and compared with the most advanced synchronization measurement device in the distribution network side, the Universal Grid Analyzer (UGA). As shown in Figure 6, the testing results for frequency and phase indicate that the precision of the proposed method closely matches that of the UGA, with a frequency relative error of ±0.5 mHz and a phase relative error of 0.05°. This demonstrates the method’s excellent accuracy. Additionally, the proposed method does not require extra computational overhead, making it suitable for scenarios demanding low power consumption or cost control.