Field-Programmable Gate Array Implementation of Fast Detection Method for Power Grid Faults ^†

Abstract

We implemented a power grid fault detection scheme based on adaptive frequency tracking and an iterative discrete Fourier transform. We designed and verified the scheme on a Field-Programmable Gate Array (FPGA) development board, leveraging the inherent advantages of hardware parallel processing to overcome the speed and real-time limitations of traditional general-purpose processor-based solutions. The simulation results demonstrate that the FPGA implementation validates the effectiveness of the algorithm, achieving excellent performance in terms of resource utilization and power consumption. These findings provide a solid technical foundation for the development of specialized and miniaturized power grid protection devices.

1. Introduction

Accurate phasor estimation of current and voltage signals is paramount for the rapid fault detection and precise isolation required in modern power systems. When a short-circuit fault occurs, the current signal typically comprises a fundamental frequency component, harmonic components, and a significant exponentially decaying DC offset component. While the fundamental and harmonic components are periodic, the exponentially decaying DC offset is non-periodic, posing substantial challenges to traditional measurement techniques.

The discrete Fourier transform (DFT) is the most widely adopted method for phasor extraction due to its harmonic filtering capabilities. However, the presence of an exponentially decaying DC offset violates the periodicity assumption of the DFT algorithm. Since the DC offset component is a non-periodic signal and its frequency spectrum includes all frequencies, it cannot be completely removed during the standard phasor estimation process. This results in spectral leakage, which severely degrades the accuracy of the fundamental phasor magnitude and phase calculations. Consequently, effectively suppressing the adverse influence of the exponentially decaying DC offset to accurately track system parameters has become a primary objective in digital relay protection.

Many techniques have been proposed to eliminate the adverse effects of the DC offset component. These can be generally categorized into filtering methods and algorithmic modifications. For instance, ref. [1] introduced a novel discrete Fourier algorithm to improve computational accuracy, while Ref. [2] explored an optimal recursive Fourier-transform algorithm to enhance the signal-to-noise ratio. In the realm of digital filtering, ref. [3] proposed a specific filter design to remove attenuated DC components, and ref. [4] developed an accurate full-cycle Fourier algorithm. These approaches have contributed to the field of harmonic detection and DC rejection, offering varying degrees of error reduction.

However, existing methods often exhibit limitations when applied to real-time microcomputer-based protection systems. Related algorithms suffer from heavy computational burdens or require data windows longer than one cycle, leading to undesirable delays in the protection system’s response time [1,2,3,4]. Furthermore, methods based on mimic filters or pre-set models often fail to eliminate the DC offset if the actual time constant differs from the pre-known or modeled value. Modern power grids demand technologies that prioritize both computational efficiency and rapid dynamic response.

To address these shortcomings, we developed a rapid fault detection method that integrates an attenuation DC offset elimination strategy with a sliding-window iterative DFT algorithm. This proposed method organically combines DC elimination functionality [5] with real-time computational speed [6] and robust phasor-tracking capabilities [7]. Whereas previous studies compromise speed for accuracy, this approach resolves the latency and complexity issues, enabling fast and accurate fault detection even in the presence of decaying DC offsets.

The remainder of this article is organized as follows. Section 2 elaborates on the mathematical principles, including fault initiation rules, the derivation of the DC attenuation elimination, and the frequency estimation algorithms. Section 3 summarizes the logic flow of the proposed fault detection scheme. Section 4 presents the simulation results, demonstrating the algorithm’s performance under various conditions and comparing it with standard methods. Finally, Section 5 provides concluding remarks and explores future application prospects in smart grid protection.

2. Theoretical Foundations

2.1. DC Attenuation Component Removal

The traditional DFT algorithm is based on the periodicity assumption of the signal. The existence of the attenuated DC component causes spectrum leakage, which seriously affects the accuracy of the phase vector extraction. In order to eliminate this component, we used the time-domain repetition characteristic of the periodic signal to carry out the differential processing [Equation (1)].

$$x(t)=A+B_0\cdot e^{-t/\tau }+{\displaystyle {\sum }_{j=1}^K}{C}_j\mathit{sin}({\omega }_{j}t+{\phi }_j)$$

(1)

The signal difference at any instant with a lag of one fundamental period is expressed as

$$x(t)-x(t-T)=\left[A+B_0\cdot e^{-t/\tau }+{\displaystyle {\sum }_{j=1}^K}{C}_j\mathit{sin}({\omega }_{j}t+{\phi }_j)\right]-\left[A+B_0\cdot e^{-(t-T)/\tau }+{\displaystyle {\sum }_{j=1}^K}{C}_j\mathit{sin}\left({\omega }_j\right(t-T)+{\phi }_j)\right].$$

(2)

By using the periodicity of the harmonic and the cancellation characteristic of constant quantity, all periodic components and constant DC components in Equation (2) are eliminated, and only the difference information of the attenuated DC component is retained as follows.

$$\Delta x(t,T_1)=B_0\mathit{exp}\left(-{\displaystyle \frac{t}{\tau }}\right)\left(1-\mathit{exp}\left({\displaystyle \frac{T_1}{\tau }}\right)\right)$$

(3)

According to Equation (3), we select two adjacent $t_k, t_{k+1}$ sampling moments and their corresponding pre-sampling points one cycle earlier $x_{k-N}, x_{k+1-N}$ to establish a system of equations, as shown in (4):

$$\left\{\begin{array}{cc}\hfill x_k-x_{k-N}& =B_0\cdot \left(\mathit{exp}\left(-{\displaystyle \frac{k\Delta t_s}{\tau }}\right)-\mathit{exp}\left(-{\displaystyle \frac{(k-N)\Delta t_s}{\tau }}\right)\right)\hfill \\ \hfill x_{k+1}-x_{k+1-N}& =B_0\cdot \left(\mathit{exp}\left(-{\displaystyle \frac{(k+1)\Delta t_s}{\tau }}\right)-\mathit{exp}\left(-{\displaystyle \frac{(k+1-N)\Delta t_s}{\tau }}\right)\right)\hfill \end{array}\right.$$

(4)

To simplify the calculation, a single-step attenuation factor is introduced, as shown in Equation (5).

$${\rho }_s=\mathit{exp}(-\Delta t_s/\tau )$$

(5)

Given $\mathit{exp}(-N\Delta t_s/\tau )={\rho }_s^N$, substituting into Equation (5) yields Equation (6).

$$\left\{\begin{array}{cc}\hfill B_k& =B_0\cdot {\rho }_s^k\cdot (1-{\rho }_s^N)\hfill \\ \hfill B_{k+1}& =B_0\cdot {\rho }_s^{k+1}\cdot (1-{\rho }_s^N)\hfill \end{array}\right.$$

(6)

where $B_m=x_m-x_{m-N}$.

By applying the two equations in Equation (6), the attenuation factor is solved using Equation (7), and the initial amplitude of the attenuated DC is calculated using Equation (8).

$${\rho }_s={\displaystyle \frac{x_{k+1}-x_{k+1-N}}{x_k-x_{k-N}}}$$

(7)

$$B_0={\displaystyle \frac{x_k-x_{k-N}}{{\rho }_s^k(1-{\rho }_s^N)}}$$

(8)

After calculating ${\rho }_s$ and $B_0$ using Equations (7) and (8), the instantaneous value of the attenuated DC component at any sampling point is obtained using Equation (9).

$$d_{k,est}=B_0\cdot ({\rho }_s{)}^k$$

(9)

Finally, the corrected signal sequence is obtained by subtracting the estimated DC component from the original sampled data, as shown in Equation (10).

$$x_{k,corr}=x_k-d_{k,est}$$

(10)

The DFT of the corrected sequence can be used to obtain the high-precision fundamental phase vector.

2.2. Frequency Estimation and Rate Measurement Algorithm

In microprocessor-based protection, the full-cycle Fourier algorithm extracts the fundamental positive sequence component. With the sampling frequency $F_S$ set as an integer multiple of the rated frequency $F_N$ (${\omega }_N$), the DFT phasor $V$ remains stationary when $F_{act}=F_N$. However, any frequency deviation ($\mathsf{\Delta }F=F_{act}-F_N$) creates a mismatch between the actual ($\omega {\Omega }_{act}$) and reference (${\omega }_N$) angular frequencies, causing the phasor $V$ to rotate when the $\Delta F>0$ vector rotates counterclockwise. When $\Delta F<0$, the vector rotates clockwise. The frequency deviation of the system is deduced accurately by using the rotation velocity of the phasor on the complex plane.

Mathematical Model Derivation for Frequency Measurement

• Relationship between frequency deviation and phase angle rate

The time-domain of the actual input signal is expressed as Equation (11).

$$x\left(t\right)=A_m\mathit{sin}({\omega }_{act.}t+{\varphi }_0),$$

(11)

where $A_m$ is the amplitude, ${\omega }_{act.}$ is the actual angular frequency, and ${\varphi }_0$ is the initial phase angle. In a rotating synchronous coordinate system, the observed phase angle ${\phi }_P\left(t\right)$ is defined as the total phase minus the synchronous rotation component.

$${\phi }_P\left(t\right)={\varphi }_0+({\omega }_{act.}-{\omega }_N)t={\varphi }_0+\Delta \omega \cdot t$$

(12)

By differentiating Equation (12), we obtained the rate of change of the phase angle.

$${\dot{\phi }}_P\left(t\right)={\omega }_{act.}-{\omega }_N=\Delta \omega$$

(13)

The angular frequency is converted into frequency, and the expression for frequency deviation ΔF (14) is derived from Equation (13).

$$\Delta F={\displaystyle \frac{1}{2\pi }}{\dot{\phi }}_P(t)$$

(14)

The equation shows that the first-order time derivative of the phase angle ${\phi }_P\left(t\right)$ is strictly proportional to the frequency deviation. In the digital microcomputer system, the finite difference method is used to approximate the continuous differential. Let the time interval between two measurements be $\Delta t_{samp}=t_k-t_{k-1}$; the equation to calculate the actual system frequency ${\mathrm{F}}_{\mathrm{a}\mathrm{c}\mathrm{t}.}$ is defined as Equation (15).

$$F_{act.}=F_N+{\displaystyle \frac{\Delta {\phi }_P}{2\pi \Delta t_{samp}}},$$

(15)

where $\Delta {\phi }_P={\phi }_{P,k}-{\phi }_{P,k-1}$.

2.3. Iterative DFT Algorithm

Given the system sampling frequency ${\mathrm{f}}_{\mathrm{s}}$ and the grid fundamental frequency $\mathrm{f}$, the number of sampling points per fundamental cycle $N=f_s/f$. For the discrete time series $x\left(n\right)$, the DFT transform result $X_k\left(n\right)$ of its k-th harmonic component at time $\mathrm{n}$ is given by (16):

$$X_k(n)={\displaystyle {\sum }_{i=0}^{N-1}}x(n-N+1+i)e^{-j{\displaystyle \frac{2\pi ki}{N}}}$$

(16)

To avoid repeating $N$ times multiplication and addition operations for each new window, the sliding-window property is utilized for recursive calculation. When the sampling window slides from time $n-1$ to time $n$, the latest sample point $x\left(n\right)$ enters the window while the oldest sample point $x(n-N)$ exits. Based on the linear time-shift property of DFT, the spectral value $S\left(n\right)$ at time $n$ can be rapidly iteratively derived from the result $S(n-1)$ at time $n-1$. Based on the algorithm proposed in the literature, the iterative process quantity $S\left(n\right)$ is defined, with its complex-form recursive relationship shown in Equation (17).

$$S(n)=\left[S(n-1)+x\left(n\right)-x(n-N)\right]\cdot W_N^{-k},$$

(17)

where the rotation iteration factor $W_N^{-k}=e^{j\frac{2\pi k}{N}}=\mathit{cos}(\frac{2\pi k}{N})+j\mathit{sin}(\frac{2\pi k}{N})$.

The physical principle of this formula is to first update the time-domain data difference, followed by frequency-domain rotation correction. Specifically, the algorithm adds the difference between the old and new data to the previous moment’s result and then multiplies it by a fixed complex constant. This reduces the computational complexity of single-harmonic detection from $O\left(N\right)$ to $O\left(1\right)$, significantly improving computational efficiency.

In practical digital systems, the aforementioned complex number operations are typically decomposed into independent iterations of the real part $S_a$ and the imaginary part $S_b$, as expressed in Equations (18) and (19).

$$S_a(n)=[S_a(n-1)+x(n)-x(n-N)]\mathit{cos}({\displaystyle \frac{2\pi k}{N}})-[S_b(n-1)+x(n)-x(n-N)]\mathit{sin}({\displaystyle \frac{2\pi k}{N}})$$

(18)

$$S_b(n)=[S_b(n-1)+x(n)-x(n-N)]\mathit{sin}({\displaystyle \frac{2\pi k}{N}})+[S_a(n-1)+x(n)-x(n-N)]\mathit{cos}({\displaystyle \frac{2\pi k}{N}})$$

(19)

3. Proposed Method

To detect the system fault quickly, a starting criterion based on the linear relationship of the sinusoidal sampling value is adopted in this study. This method uses the waveform characteristics of the voltage or current signals in the steady-state operation of the power system and identifies the fault by verifying the constraint relationship between the continuous sampling points. When the system is in a normal steady state, the sampled signal is approximated as a pure fundamental sine wave and is shown in Equations (20) and (21).

$$x\left(t\right)=A cos(\omega t+\varphi ),$$

(20)

where $A$ is the amplitude, $\omega$ is the angular frequency, and $\varphi$ is the initial phase angle.

$$\begin{gathered} X_0=A\mathit{cos}(\omega t-\omega \Delta t+\varphi ) \\ {X}_1=A\mathit{cos}(\omega t+\varphi ) \\ {X}_2=A\mathit{cos}(\omega t+\omega \Delta t+\varphi ) \end{gathered}$$

(21)

The triangle identity is used to check that the sum of the two sampling values yields.

$$X_0+X_2=2A\mathit{cos}(\omega t+\varphi \left)\mathit{cos}(\omega \Delta t\right)$$

(22)

The intermediate sampling value $X_1=A\mathit{cos}(\omega t+\varphi )$, and then the linear constraint relationship between three consecutive sampling values is obtained using Equation (23).

$$X_0+X_2=2X_1\mathit{cos}(\omega \Delta T)$$

(23)

The constructed equilibrium equation is shown in Equation (24).

$$\epsilon =\left|2\mathit{cos}(\omega \Delta T)\cdot X_1-X_0-X_2\right|$$

(24)

Therefore, by monitoring whether the calculated value exceeds a preset threshold, a fault detection procedure can be rapidly activated. The fault detection method has the following characteristics:

• When the system is operating normally, the actual frequency can be accurately calculated using two cycles of sampling data.
• This method can effectively filter out DC attenuation components when the system encounters line faults.
• Since fault currents inherently contain DC attenuation components, they can easily cause calculation errors between the fundamental wave and its harmonics in a spectral distribution. This method, however, effectively filters out DC attenuation components, thereby reducing the error in system parameter calculations to an extremely low proportion.
• This method uses an iterative calculation approach, which can be fully embedded in hardware for real-time detection.

Figure 1 shows the flowchart of the fault detection method proposed.

Figure 1. Flowchart of fault detection method.

4. Performance Evaluation

We conducted direct frequency calculations for various signal waveforms and compared them with the frequency calculation method presented in this paper. The comparison sample waveforms are illustrated in Table 1. A pure AC waveform is provided to simulate the system’s normal operating state. Then, a DC attenuation component is introduced at any given time point, with its parameters set to typical values. The complete calculation and comparison results are presented in Figure 2, demonstrating the superiority of this method.

Table 1. Frequency estimation error table.

Actual Frequency (Hz)	Estimate Frequency (Hz)	Error Value (%)	Actual Frequency (Hz)	Estimate Frequency (Hz)	Error Value (%)
58.00	58.06	0.10205	60.20	60.20	0.00111
58.20	58.20	0.08444	60.40	60.40	0.00441
58.40	58.44	0.06769	60.60	60.61	0.00987
58.60	58.63	0.05198	60.80	60.81	0.01741
58.80	58.82	0.03897	61.00	61.02	0.02193
59.00	59.02	0.02230	61.20	61.17	0.03831
59.20	59.16	0.01759	61.40	61.43	0.05142
59.40	59.41	0.00995	61.60	61.64	0.06607
59.60	59.60	0.00444	61.80	61.85	0.08209
59.80	59.80	0.00111	62.00	62.06	0.09921

Figure 2. Comparison of DC attenuation before and after removal.

5. Conclusions

Given that fault currents frequently inflict severe damage on power systems, rapid fault identification, accurate calculation of fault voltage and current values, and precise determination of system parameters have become critical challenges in modern protection schemes. After evaluating accuracy, timeliness, and computational complexity, the authors propose a hardware-integrated method for real-time parameter computation. This approach encompasses calculating the true system frequency, activating the fault detection mode, computing and filtering DC attenuation components, and performing iterative DFT for both fundamental and harmonic components. The experimental results demonstrate that the workflow can determine the system frequency within two cycles under normal operation and compute precise AC fault current parameters within one cycle plus two additional sampling points following fault occurrence.

The increasing integration of renewable energy generation and AC/DC conversion equipment in future power systems necessitates faster response times from protection devices. This heightens the challenge of reducing fault detection latency and optimizing parameter calculations. To more effectively pinpoint fault occurrence and enable practical circuit implementation, the moving-window time–frequency analysis method emerges as a promising research direction. Moreover, the widespread deployment of synchronous phasor measurement units in smart grids provides real-time system parameters and status data. Future efforts should focus on combining these real-time monitoring capabilities with operational data from smart grid measurement units to establish a robust foundation for rapid and reliable fault detection.