Improved Three-Phase AFD Islanding Detection Based on Digital Control and Non-Detection Zone Elimination

Abstract

This paper proposes an improved active frequency drift (AFD) islanding detection method of a three-phase inverter. Different than the existing single-phase AFD method, in the three-phase system, the disturbance is added to the phase angle, which takes part in coordinate transformation calculations. Thus, the frequency drift of the three-phase reference current can be realized by the disturbance of the phase angle. It is unnecessary to add frequency drift to each phase respectively with the proposed method, which can simplify the control system. Furthermore, the non-detection zone can be eliminated by updating certain parameters; therefore, the detection method is sensitive. In this paper, the application of the proposed method in a digital control system is discussed in detail. The non-detection zone is analyzed, and an elimination method is proposed. Finally, the simulation and experimental results are given to verify the theoretical analysis.

1. Introduction

The three-phase grid-connected inverter plays an important role in the field of distributed power generation [1]. The islanding phenomenon would cause confusion in AC frequency, phase, and amplitude. Islanding detection is necessary for avoiding damage to electrical equipment and personal injury [2,3]. Islanding detection methods can be divided into passive and active methods. The passive methods detect the voltages, currents, and phases of the inverter. If one or more of them are not in the allowed range, islanding can be detected [4,5,6,7,8]. The non-detection zone (NDZ) is the main disadvantage of the passive methods. For solving the problem, an islanding detection method based on feature recognition technology is proposed in [5]. The features of the inverter system are identified. The vector model of the inverter features is established instead of the limited parameters in the traditional passive methods. Thus, the NDZ can be eliminated. However, comprehensive conditions should be taken into consideration. The feature recognition of the inverter system is complex.

The application of active islanding detection methods is more common. The active methods can be divided into grid variable variation-based methods and impedance estimation-based methods [9,10]. The principle and classification of the impedance estimation-based methods are discussed in [10]. In these methods, islanding is detected according to the variations at the power converter output impedance. These methods have the advantages of low current distortion and high detection speed [11,12,13,14,15]. However, false judgment and a complex control strategy limit the application of these methods. The grid variable variation-based methods can detect the islanding effect of a grid response on a small disturbance [16,17,18,19,20,21,22,23]. The methods based on frequency and voltage drift are very popular. However, to eliminate the NDZ, additional disturbance should be added. The current distortion cannot be avoided.

To decrease current distortion and eliminate the NDZ, a simplified power control strategy with islanding detection for a three-phase grid-connected photovoltaic inverter is proposed in [11]. Harmonic current is injected as the disturbance, which results in harmonic voltage on the local load. By transforming these voltages into the rotating reference frame, the DC component of these voltages can be detected. A low-pass filter is applied in this method, which would cause a time delay and the islanding would not be detected in time. A systematic phase-locked loop (PLL) modeling and design approach for evaluating the frequency-based islanding detection method is proposed in [19]. An input injection signal is added to the PLL. This method is sensitive to the control parameters. The system stability may be threatened. In [20], an active method based on voltage positive feedback is given. The method uses a sliding mode controller to detect the islanding effect of the photovoltaic system with string inverter. The active power perturbation has been applied only in deceleration mode, considering the disturbance definition in the inverter voltage control loop. In [21], a high-performance islanding search sequence technique is proposed. The output of the inverter can be seen as a voltage source. This method does not apply to an inverter in which the output is a current source.

According to the above analysis, this paper proposes an improved active frequency drift (AFD) method, which applies to the three-phase grid-connected inverter. The traditional AFD method can only be used for the single-phase inverter and there is an NDZ problem [24,25,26]. The existing improved methods, such as automatic phase-shift and AFD with positive feedback, cannot be used in the three-phase systems [27,28,29,30]. The proposed method can eliminate the NDZ, and the current distortion is low. The frequency disturbance is added to the output of the PLL. Thus, the phase angle, which takes part in the coordinate transformation, tends to change the frequency of the current reference. Furthermore, the NDZ can be totally eliminated by certain parameter updating.

This paper is organized as follows. In Section 2, the principle of the proposed method is introduced. Section 3 analyzes the cause of NDZ. In Section 4, the improved method, which can eliminate the NDZ, is given. Finally, Section 5 shows the simulation and experimental results of the proposed islanding detection method to verify the theoretical analysis.

2. Three-Phase AFD Method

2.1. Principle of the Three-Phase AFD Method

Figure 1 shows the principle of the three-phase AFD method. In Figure 1a, e_a, e_b, and e_c are the grid voltages. Z_a, Z_b and Z_c are the line impedances. In the islanding detection, they can be ignored. Thus, e_a, e_b, e_c and the voltages e_as, e_bs, e_cs are the same. In the space vector pulse width modulation (SVPWM) control, the phase angle θ₀, which takes part in the coordinate transformation, is generated by PLL [31]. In the dashed box, the disturbed angle θ₀′ takes part in the coordinate transformation from i_q^*, i_d^* to i_dra^*, i_drb^*, i_drc^*. θ₀ takes part in the coordinate transformation from i_dra^*, i_drb^*, i_drc^* to i_drq^*, i_drd^*. The three-phase current reference i_dra^*, i_drb^*, i_drc^* is shown in Figure 1b,c. θ₀′ does not take part in other coordinate transformations to avoid additional disturbances.

In Figure 1b,c the waveforms from top to bottom are, respectively, the current reference without disturbance, the phase angle, and the current reference with disturbance. The disturbance can increase or decrease the current frequency after the inverter is disconnected from the grid. θ₀ is the output of the PLL. It changes from −π to π with the grid frequency. θ₀′ is the phase angle after AFD. It changes from −π to π with the disturbed frequency. To correct the frequency when the inverter is connected to the grid, zero-crossing correction (ZCC) is necessary.

In Figure 1b, the disturbance tends to increase the current frequency. ZCC operates when θ₀′ is 0 and θ₀ is less than 0. In this condition, θ₀′ does not change from 0 until θ₀ reaches 0. The slope of θ₀′ is higher than that of θ₀. At time t₃, θ₀′ finishes a liner-changed cycle. During t₃ to t₄, θ₀′ keeps 0. At time t₄, θ₀ finishes a liner-changed cycle, which is just the grid cycle T. After t₄, θ₀′ and θ₀ begin to increase from 0 with different slopes.

Similarly, in Figure 1c, the disturbance tends to decrease the current frequency. ZCC operates when θ₀ is 0 and θ₀′ is less than 0. At the zero-crossing time, θ₀′ is enforced as 0.

2.2. AFD Control Strategy in the Digital Control System

According to Figure 1, the aim of the phase angle disturbance is to realize frequency drift. When the inverter works normally, the frequency achieved by the PLL is the grid frequency. In each grid cycle, the frequency disturbance is a fixed value. It is imposed on the grid frequency and will not be accumulated. After the inverter is disconnected from the grid, ZCC still works. If the local load is resistive, the phases of the AC current and voltage are the same. The zero-crossing times of θ₀′ and θ₀ are the same. ZCC will not force the frequency correction of θ₀′. Thus, the current frequency changes with the disturbance. In addition, the voltage frequency will change and it can be detected by the PLL. In the next cycle, the disturbance will be accumulated.

In the digitally-controlled three-phase inverter system, the frequency is calculated by the slope of the phase angle, which can be expressed as follows:

$$f=\frac{{\omega }_0}{2\pi }=\frac{1}{2\pi }\frac{d{\theta }_0}{dt}=\frac{1}{2\pi }\frac{\Delta \theta }{\Delta t}$$

(1)

In the digital controller, the phase angle θ₀ is calculated in every switching cycle. T_c is the switching cycle. Thus, Δt is T_c and Δθ is the difference of θ₀ in the two adjacent calculations. The simplest filtering algorithm is applied. In fact, Δθ is the average value. It can be expressed as follows:

$$\Delta \theta =\frac{1}{k}{\displaystyle \sum _{i=1}^k\Delta {\theta }_i}$$

(2)

where k is the number of calculations.

Figure 2 shows the change of the phase angle after the inverter gets disconnected from the grid at t₀. The local load is resistive. The grid frequency is f₀. The disturbance is Δf₀. If the disturbance tends to increase the current frequency, Δf₀ is larger than 0. On the contrary, Δf₀ is less than 0. In each k times of the switching cycle, the average value Δθ and the AC voltage frequency are calculated. In the next k times of the switching cycle, the disturbance Δf₀ is added to the AC voltage frequency. The disturbed frequency is the frequency of the current reference. As a result, the disturbance cycle is kT_c. Before t₀, the AC voltage frequency is always f₀. Therefore, the frequency of the current reference is always f₀ + Δf₀.

Because of ZCC, the real current frequency is still f₀. After t₀, the AC voltage frequency changes with the disturbance. y is the number of disturbance cycles after t₀. At time t₀, y is 1. In the area where y is 1, the AC voltage frequency is not the grid frequency. According to (1), in this cycle, the AC voltage frequency can be expressed as follows:

$$f_{OFF(y=1)}=\frac{(w-1)f_0+(k-w+1)(f_0+\Delta f_0)}{k}\begin{array}{cc}& \left(1\le w\le k\right)\end{array}$$

(3)

where w is the number of switching cycles at t₀. It is larger than or equal to 1 and less than or equal to k. Thus, the range of f_OFF(y=1) is [f₀ + Δf₀/k, f₀ + Δf₀]. If f_OFF(y=1) is still in the allowed range, the disturbance Δf₀ would be added to f_OFF(y=1) instead of f₀. In the next disturbance cycle, the frequencies of the AC current and voltage both are

$$f_{OFF(y=2)}=f_{OFF(y=1)}+\Delta f_0$$

(4)

The frequency accumulation does not stop until the AC voltage frequency exceeds the allowed range. However, in some certain conditions, NDZ would occur, which will be analyzed in the next section.

2.3. Comparison between the Existing Method and the Proposed Method

In [24], a typical single-phase AFD method is described in detail. A current waveform distortion to the original reference current of the inverter has been injected to realize the frequency drift. Taking the condition where the disturbance tends to increase the current frequency, for example, Figure 3 shows the application of the existing method in the three-phase system. The waveforms from top to bottom respectively are the current reference without disturbance, the injected waveform, and the current reference with disturbance. By introducing a zero conduction time at the end of each cycle, the phase angle of the fundamental component of the current is shifted. Comparing Figure 3 and Figure 1b, it can be seen clearly that frequency drift should be added to each phase respectively with the existing method. With the proposed method, the control system can be simplified.

3. Analysis of the Non-Detection Zone

3.1. Detection Process and Parameter Calculations

The local load can be seen as a parallel structure of the resistance, inductance and capacitance, which can be defined as RLC parallel load. The impedance angle of it can be expressed as follows:

$${\phi }_{load}=\arctan Q_{RLC}\left(\frac{f_r}{f}-\frac{f}{f_r}\right)$$

(5)

where Q_RLC is the quality factor and f_r is the resonant frequency. They can be expressed as follows:

$$Q_{RLC}=\frac{R}{2\pi f_{r}L}=2\pi R{f}_{r}C=R\sqrt{\frac{C}{L}}$$

(6)

$$f_r=\frac{1}{2\pi \sqrt{LC}}$$

(7)

If the AC voltage frequency f is the same as f_r, φ_load is 0 and the local load is resistive. If f is less than f_r, φ_load is positive and the local load is inductive. If f is larger than f_r, φ_load is negative and the local load is capacitive.

According to Figure 1, when the inverter works normally, the phase angle of the reference current in the (i + 1) switching cycle can be expressed as follows:

$${\theta }_{i+1}{}^{\prime }={\theta {}^{\prime }}_i+2\pi \left(f_0+\Delta f_0\right)T_c$$

(8)

The AC current phase difference between the two adjacent switching cycles is

$$\Delta {\theta }_i{}^{\prime }=\Delta {\theta }_i+2\pi \Delta f_0{T}_c\begin{array}{cc}& \left(1\le i\le k\right)\end{array}$$

(9)

In Figure 1, the difference between θ₀′ and θ₀ is Δσ. Δσ changes from 0 to the maximum value in each grid cycle. At the end of the cycle, Δσ is forced to 0 again because of ZCC. The range of Δσ can be expressed as follows:

$$0\le \Delta \sigma \le \frac{2\pi \Delta f_0}{f_0+\Delta f_0}$$

(10)

Because of the digital filtering, the calculation of the AC voltage frequency needs k times of the switching cycle; i.e., the AC voltage frequency is updated at the end of each disturbance cycle. After the inverter is disconnected from the gird at time t₀, the phase difference between the AC voltage and current is determined by the impedance angle φ_load. In the first switching cycle after t₀, the phase angle of the AC current is θ_c + Δθ_i′, while the phase angle of the AC voltage is θ_c + Δθ_i′ + φ_load(y=1). The AC voltage phase difference between this and the previous switching cycles is as follows:

$$\Delta {\theta }_w=\Delta {\theta }_i{}^{\prime }+{\phi }_{load(y=1)}+\Delta \sigma \begin{array}{cc}& (i=w)\end{array}$$

(11)

According to (2) and (11), the average AC voltage phase difference of the adjacent switching cycles in the condition that y is 1 can be expressed as follows:

$$\Delta {\theta }_{OFF(y=1)}=\frac{1}{k}({\displaystyle \sum _{i=1}^{w-1}\Delta {\theta }_i}+{\displaystyle \sum _{i=w}^k\Delta {\theta }_i{}^{\prime }}+{\phi }_{load(y=1)}+\Delta \sigma )$$

(12)

Figure 4 shows the relationship among Δθ, Δθ_OFF, and the inverter working conditions. Δθ is 2πT_cf₀. It is clamped by the grid frequency. The allowed range is [2πT_cf_0min, 2πT_cf_0max]. Δθ_OFF is influenced by the local load. The inverter does not stop working until Δθ_OFF exceeds the range [2πT_cf_0min, 2πT_cf_0max]. If the inverter has approached the steady state and Δθ_OFF is still in the allowed range, the islanding cannot be detected.

Taking (9) into (12), Δθ_OFF(y=1) can be simplified as follows:

$$\Delta {\theta }_{OFF(y=1)}={\theta }_{initial}+\frac{{\phi }_{load(y=1)}}{k}$$

(13)

where θ_initial and φ_load(y=1) can be expressed as follows:

$${\theta }_{initial}=\Delta \theta +\frac{2\pi \Delta f_0{T}_c(k-w+1)+\Delta \sigma }{k}\begin{array}{cc}& (1\le w\le k)\end{array}$$

(14)

$${\phi }_{load\left(y=1\right)}=\arctan Q_{RLC}\left(\frac{f_r}{f_0+\Delta f_0}-\frac{f_0+\Delta f_0}{f_r}\right)$$

(15)

After the parameters Δθ_OFF(y=1), Δθ_OFF(y=2), and Δθ_OFF(y=3) are calculated in turn, the general expression can be given as

$$\Delta {\theta }_{OFF\left(y=n+1\right)}=\Delta {\theta }_{OFF\left(y=n\right)}+2\pi \Delta f_0{T}_c+\frac{B_p}{k}{\phi }_{load\left(y=n+1\right)}$$

(16)

where B_p represents the phase changing condition of the AC voltage in the area that y is n + 1. If the phase of the AC voltage changes suddenly, B_p is 1; otherwise, B_p is 0. φ_load(y=n+1) is the phase angle of mutation. It can be expressed as follows:

$${\phi }_{load\left(y=n+1\right)}=\arctan Q_{RLC}\left(\frac{f_r}{f_n+\Delta f_0}-\frac{f_n+\Delta f_0}{f_r}\right)$$

(17)

where f_n is the AC voltage frequency calculated in the previous disturbance cycle. It can be expressed as follows:

$$f_n=\frac{\Delta {\theta }_{OFF\left(y=n\right)}}{2\pi T_c}$$

(18)

3.2. Cause of the Non-Detection Zone

Figure 5 shows the change of θ₀ and θ₀′ in the condition where the disturbance tends to increase the current frequency. The inverter gets disconnected from the grid at time t₀. According to the principle of ZCC, when θ₀′ reaches 0 and θ₀ is less than 0, θ₀′ does not stop increasing until θ₀ reaches 0. After t₀, θ₀ is the phase angle of the AC voltage and θ₀′ is the phase angle of the AC current. In Figure 5a, the local load is inductive. θ₀ is ahead of θ₀′. It is impossible that θ₀′ is 0 while θ₀ is less than 0. Therefore, θ₀′ will not be corrected at the zero-crossing time. In the area that y is 1, θ₀ changes suddenly and B_p is 1. According to (13), Δθ_OFF(y=1) is larger than Δθ. It means that the calculated frequency of the AC voltage increases. In the areas that y is larger than 1, B_p is always 0. According to (16), Δθ_OFF(y=n+1) is always larger than Δθ_OFF(y=n). It means that the frequency disturbance would be accumulated. The islanding can be quickly detected in a few disturbance cycles. The NDZ will not occur.

In Figure 5b, the local load is capacitive. In the area that y is 1, θ₀ changes suddenly and B_p is 1. The frequency of the AC current is f₀ + Δf₀. According to (13), it is possible that Δθ_OFF(y=1) = Δθ. Thus, the calculated AC voltage frequency in this disturbance cycle is still the grid frequency f₀. In the area that y is 2, the frequency of the AC current is still f₀ + Δf₀. B_p is 0. According to (16), Δθ_OFF(y=2) is larger than Δθ_OFF(y=1). The calculated AC voltage frequency in this disturbance cycle is f₀ + Δf₀. If the frequency is still in the allowed range, the inverter keeps on working. In the area that y is 3, the frequency of the AC current f₃ is f₀ + 2Δf₀. The change value of θ₀ from π to −π is 2π. The effect of this change is not sudden. Therefore, B_p is 0. Δθ_OFF(y=3) is larger than Δθ_OFF(y=2). In the area that y is 4, θ₀′ is corrected by ZCC. B_p is 1. According to (16), Δθ_OFF(y=4) is less than Δθ_OFF(y=3). The difference between them is |φ_load(y=4)/k|. Furthermore, the decrease of θ₀ in the area that y is 4 is larger than the area that y is 1. f₃ is larger than f₀. According to (15) and (17), φ_load(y=4) is larger than φ_load(y=1). It is possible that Δθ_OFF(y=4) is the same as Δθ; i.e., the disturbance accumulation would be offset by the lagging of the AC voltage phase angle. In the next zero-crossing time, the same condition would happen again and the islanding would not be detected.

Considering the common situation, in the area that y is x, Δθ_OFF(y=x) is the same as Δθ. In the process of y = 1 to y = x, B_p is 1 in the conditions that y respectively is 1 and x. In the other areas, B_p is 0. According to (16), the cumulative calculation of Δθ_OFF(y=x) can be expressed as follows:

$$\Delta {\theta }_{OFF\left(y=x-1\right)}={\theta }_{initial}+\frac{{\phi }_{load\left(y=1\right)}}{k}+2\pi \Delta f_0{T}_c\left(x-2\right)$$

(19)

$$\Delta {\theta }_{OFF(y=x)}=\Delta {\theta }_{OFF(y=x-1)}+2\pi \Delta f_0{T}_c+\frac{{\phi }_{load(y=x)}}{k}=\Delta \theta =2\pi f_0{T}_c$$

(20)

where φ_load(y=x) can be expressed as

$${\phi }_{load(y=x)}=\arctan Q_{RLC}(\frac{2\pi T_c{f}_r}{\Delta {\theta }_{OFF(y=x-1)}+2\pi T_c\Delta f_0}-\frac{\Delta {\theta }_{OFF(y=x-1)}+2\pi T_c\Delta f_0}{2\pi T_c{f}_r})$$

(21)

The requirement of the NDZ is that the cumulative disturbance cannot exceed the allowed range. In the process of y = 1 to y = x, the disturbance has been accumulated x − 2 times. Therefore, the requirement of x is

$$x-2<\frac{f_{0\max }-f_{0\min }}{\Delta f_0}$$

(22)

Another requirement of x is that the AC current phased angle should be corrected by the ZCC, which can be expressed as follows:

$$2\pi k{T}_c{\displaystyle \sum _{n=1}^x{f}_n>2\pi }$$

(23)

Then, (22) and (23) can be combined and simplified as follows:

$$\{\begin{array}{l}x<2+\frac{f_{0\max }-f_{0\min }}{\Delta f_0}\\ x>\frac{\Delta f_0-2f_0+\sqrt{{(2f_0-\Delta f_0)}^2-8\Delta f_0{}^2+8\Delta f_0/T_{c}k}}{2\Delta f_0}\end{array}$$

(24)

The left term should be less than the right term. As a result, if the NDZ occurs, the range of the disturbance Δf₀ can be expressed as follows:

$$\{\begin{array}{l}\Delta f_0<\frac{-g_f-\sqrt{g_f{}^2-64f_0(f_{0\max }-f_{0\min })}}{16}\\ \Delta f_0>\frac{-g_f+\sqrt{g_f{}^2-64f_0(f_{0\max }-f_{0\min })}}{16}\end{array}$$

(25)

where g_f can be expressed as follows:

$$g_f=3\left(f_{0\max }-f_{0\min }\right)+8f_0+4/T_{c}k$$

(26)

According to the standard GBT 15945-2008, f_0max is 50.5 Hz and f_0min is 49.5 Hz. To improve the detection speed, kT_c should be shorter than T₀. The switching frequency is 5 kHz. Thus, k should be less than 100. In this paper, k is 20, f₀ is 50 Hz, and T_c is 1/5000 s. According to (25), if the NDZ occurs, Δf₀ should be less than 0.167 Hz or larger than 74.45 Hz.

If Δf₀ is 0.1 Hz, according to (24), the range of x is 4.97 to 12. Islanding is more likely to happen when the difference between Δθ_OFF(y=x) and Δθ is less. In this condition, Δσ is 0, w is k, and x is the minimum integer, which is 5. Thus, the NDZ can be achieved, which is the shadow area in Figure 6. Similarly, in conditions where the disturbance tends to decrease the current frequency—when the local load is inductive—the NDZ would occur.

4. Elimination of the Non-Detection Zone

According to the above analysis, to eliminate the NDZ, the range of Δf₀ can be set as follows:

$$\{\begin{array}{l}\Delta f_0<\frac{-g_f+\sqrt{g_f{}^2-64f_0(f_{0\max }-f_{0\min })}}{16}\\ \Delta f_0>\frac{-g_f-\sqrt{g_f{}^2-64f_0(f_{0\max }-f_{0\min })}}{16}\end{array}$$

(27)

With the mentioned parameters, if Δf₀ is 0.2 Hz, the NDZ can be avoided. However, the larger Δf₀ is, the higher the current distortion is. For solving the problem, the NDZ elimination method has been proposed. The change of θ₀ with and without the elimination method is shown in Figure 7.

The AC voltage phase difference between the two adjacent switching cycles is calculated every switching cycle. The allowed range is [2πT_cf_0min, 2πT_cf_0max]. If it exceeds the allowed range, i would be cleared to 0. If i increases to k, the average value of Δθ is calculated. According to (1), the voltage frequency can be achieved. With this method, the voltage frequency can be the same as the current frequency. Thus, the NDZ can be eliminated. The essence of this method is to avoid the areas in which B_p is 1. Figure 7 corresponds to the area where y is 4, which is mentioned in Figure 5b. θ₀ has a sudden change in this area. Δθ_n+1′ is the AC voltage phase difference between the two adjacent switching cycles when i is n. It is less than 2πT_cf_0min. With the NDZ elimination method, i would be cleared to 0. As a result, the shadow area can be avoided. The calculated AC voltage frequency will not decrease by the sudden change of the phase angle.

Similarly, in the condition where the disturbance tends to decrease the current frequency, when Δθ_n+1′ is larger than 2πT_cf_0max, i is cleared to 0 and the NDZ can be eliminated.

5. Simulation and Experimental Verification

An 18 kVA three-phase inverter was established based on MATLAB/Simulink. The grid is 220 V/380 V/50 Hz. The switching frequency is 5 kHz.

Figure 8 shows the simulation results with the inductive load. The frequency disturbance is 0.5 Hz. It tends to increase the current frequency. The parameters of the RLC load are 8 Ω, 0.5 mH, and 10 mF. The resonant frequency of it is 71.2 Hz, and k is 20.

The waveforms from top to bottom are the AC voltage and current, the real frequency of the AC voltage, the calculation frequency of the AC voltage, the reference of the current frequency, and the off-grid signal, respectively. The inverter becomes disconnected from the grid at the time 0.05 s. Because the load is inductive, the disturbance will aggravate the frequency variation after the inverter is disconnected from the grid. According to the analysis, Figure 5a, and the Formula (16), the calculation frequency increases quickly. The islanding can be detected in a few disturbance cycles and the NDZ will not occur. The off-grid signal changes to 0 when the calculation frequency exceeds the allowed range. The inverter stops working.

Figure 9 shows the simulation results with the capacitive load. The parameters of the RLC load are 8 Ω, 1.03 mH, and 10 mF. The resonant frequency of it is 49.6 Hz. The quality factor of the load is 25. According to the theoretical analysis, NDZ will not occur when the frequency disturbance is 0.5 Hz. Therefore, the disturbance is 0.1 Hz. The waveforms from top to bottom are the AC voltage and current, the real frequency of the AC voltage, the calculation frequency of the AC voltage, the reference of the current frequency, and the off-grid signal, respectively. The inverter becomes disconnected from the grid at the time 0.05 s. The calculated frequency in the area that y = n is just the real frequency in the area that y = n − 1. In the area that y is 5, ZCC happens and the reference of the current frequency in the next disturbance cycle would decrease. The islanding happens, which is shown in Figure 9a. In Figure 9b, the NDZ estimation method is applied. The shadow area in Figure 9b is the cause of the NDZ. By detecting the phase angle change and clearing the counter, the shadow area can be avoided, and the islanding can be detected.

A principle prototype is established. The parameters are the same as that of the simulation. Figure 10 shows the connection mode of the inverter system.

Figure 11 shows the experimental results with the inductive load. The frequency disturbance is 0.1 Hz. The parameters of the RLC load are the same as that of Figure 8. The waveforms respectively are the AC current and voltage. The inverter becomes disconnected from the grid at time t₀. Because the load is inductive, the disturbance will aggravate the frequency variation after the inverter is disconnected from the grid. The islanding can be quickly detected. At this time, the inverter stops working and the reliability of the system is guaranteed. The result is consistent with the simulation result that is shown in Figure 8.

Figure 12 shows the experimental results with the capacitive load. The frequency disturbance is 0.1 Hz. The parameters of the RLC load are the same as that of Figure 9. The islanding cannot be detected without the NDZ estimation method. In Figure 9b, the NDZ estimation method is added and the islanding can be detected quickly. The result is consistent with the simulation result that is shown in Figure 9.

Figure 13 shows the experimental results when the inverter is connected to the grid. In Figure 13a, the waveforms are the AC voltage, the disturbed phase angle, and the AC current, respectively. The disturbance is 0.1 Hz. When the phase angle crosses zero, the corresponding current waveform remains for a short time. Figure 13b shows the harmonic spectrum of the AC voltage and current. The filter with the inductance-capacitance- inductance structure, which is defined as LCL filter, is applied and their parameters are 0.9 mH, 1.5 mH, and 20 μF, respectively. In phase a, the Total Harmonic Distortion (THD) of the current is 3.8%. It can be seen that the current harmonic distortion is low when the inverter is working normally, which satisfies the requirement of the grid connection.

Table 1. The phase angles and THDs of the three-phase current.

Parameters	iag	ibg	icg
Phase Angle/°	−3.0	237.5	117.5
THD/%	3.8	4.7	4.8

shows the phase angles and THDs of the three-phase current when the inverter is connected to the grid. In phases b and c, the THDs are 4.7% and 4.8%, respectively. They are both in the allowed range.

Figure 14 shows the change of the three-phase THDs with different frequency disturbances. The higher the frequency disturbance, the higher are the THDs. With the proposed method, the distortion of phase a is less than that of the other phases.

Table 2. The response speed with different load conditions.

No.	R/Ω	L/mH	C/mF	Response Speed/s
1	8	0.5	10	0.006
2	8	1.01	10	0.012
3	8	1.02	10	0.015
4	8	1.03	10	0.028
5	8	1.1	10	0.009

shows the response speed of the proposed detection method with different load conditions. It can be seen that the islanding can be quickly detected.

6. Conclusions

This paper proposes an improved AFD method with NDZ elimination, which can be used in the three-phase inverter system based on digital control. The calculated voltage frequency is updated every disturbance cycle and the disturbance can be a small value with the NDZ elimination method. The current distortion is low when the inverter works normally. The proposed method is sensitive and it has a higher detection speed. Furthermore, the disturbance only needs to be added to the phase angle in the proposed method instead of adding disturbances to all of the three phases; thus, the control system can be simplified. Compared to the impedance estimation-based methods, false judgment is less likely to happen.