**4. Improved Islanding Detection Method Based on the Proposed Solution**

According to the previous section, the cancellation problem occurs by using two perturbations in the opposite directions. The main meaning of the solution is how to prevent the occurrence of cancellation.

First, the beginning of the proposed IDM uses the perturbation signal in the range *n* = 0–1.1. As the range of perturbation signal includes values over and below the normal value (*n* = 1), the injected signal cancellation caused by the perturbation signals in opposite directions can occur.

Second, to prevent the occurrence of cancellation, the perturbation signal value is changed so that the opposite signals cannot occur.

Therefore, the solution for the cancellation problem is to change the perturbation signal values in the range *n* = 0–1.1 to *n* = 0–1, and in the normal condition from *n* = 0.9–1.1 to *n* = 0.9–1. By changing this value, the injected signal cancellation is eliminated. Similar to the Equations (3)–(15), the solution is explained below:

The injected signal cancellation occurs when:

$$2\mathfrak{n}\_0 = \mathfrak{n}\_1 + \mathfrak{n}\_2 \tag{16}$$

By using the new perturbation signal (*n*<sup>0</sup> = 1, *n*<sup>1</sup> = 1, and *n*<sup>2</sup> = 0.9), Equation (16) becomes

$$2 \neq 1.9\tag{17}$$

Based on Equation (17), the fluctuation of DC-link voltage caused by the perturbation signals from PV1 and PV2 cannot cancel each other. Thus, the proposed solution can solve the injected signal cancellation.

*Appl. Sci.* **2019**, *9*, 4054

By generalizing this solution with *n<sup>k</sup>* perturbation signals (*k* = 1 ÷ ∞), the injected signal cancellation occurs when:

$$
\Delta v\_{1d\mathcal{c}}(t) + \Delta v\_{2d\mathcal{c}}(t) + \dots + \Delta v\_{kd\mathcal{c}}(t) = 0 \tag{18}
$$

or

$$n\_0 - n\_1 + n\_0 - n\_2 + \dots + n\_0 - n\_k = 0\tag{19}$$

Finally,

$$kn\_0 = \sum\_{k=1}^{\infty} n\_k \tag{20}$$

As *n*<sup>0</sup> = 1 (without perturbation signal) and *n<sup>k</sup>* ≤ 1, Equation (20) is correct when all perturbation signal values are 1 (without perturbation signal case).

With other values of the perturbation signals (with perturbation signal), Equation (20) becomes

$$k n\_0 > \sum\_{k=1}^{\infty} n\_k \tag{21}$$

For this reason, the injected signal cancellation is eliminated by using the proposed solution.

The flowchart in Figure 7 explains the detailed procedure of the proposed solution. The solution is verified by a step-by-step simulation.

**Figure 7.** Flowchart of the improved IDM program (*n*<sup>0</sup> is the perturbation factor at the beginning, ∆*V* is the voltage deviation, *Eps* is the abnormal event value, *dP* is the rate of change of output power).

The testing scenarios in Table 3 are the worst case (the power of DGs and the load are balance), and the procedure is similar to the previous testing procedure.


**Table 3.** Multi-PV operation scenario.

The model parameters are almost similar to the previous testing, only the new perturbation factors are different. The perturbation factors at normal condition in cancellation case are *n* = 1–0.9 for PV1 and *n* = 0.9–1 for PV2.

In addition, the eight-PV scenario also tests to verify the effect of the improved IDM. The parameters of the eight-PV scenario are as below:


As shown in the results in Figure 8, the injected signal cancellation occurs in two-PV scenario. With the improved islanding detection method based on the proposed solution, the islanding condition is detected after 40 ms when the islanding occurs. The solution is effective in two-PV scenario.

**Figure 8.** Solution result in two-PV scenario.

The results of three-PV scenario are shown in Figure 9 and 10. In Figure 9, the islanding detection method can not detect the islanding phenomenon when injected signal cancellation occurs.

The results in Figure 10 show that the islanding condition is detected after 64 ms when the islanding occurs by applying the proposed solution to the improved islanding detection method in the injected signal cancellation case.

The four-PV scenario has the similar results, the islanding condition can be detected after 48 ms when the islanding occurs by using the improved islanding detection method, but it cannot be detected by using the original one. The results are shown in Figures 11 and 12.

**Figure 11.** Cancellation result in four-PV scenario.

**Figure 12.** Solution result in four-PV scenario.

The eight-PV scenario result is shown in Figure 13. The improved islanding detection method by applying the proposed solution can detect the islanding condition after 48 ms.

**Figure 13.** Solution result in eight-PV scenario.

Based on the simulation results and mathematical explanations, the proposed solution eliminates the injected signal cancellation by using the improved islanding detection method based on the proposed solution in two-PV, three-PV, four-PV, and eight-PV scenarios.

Moreover, the injected signal cancellation is eliminated not only in the specified cases but also in the general case by using the improved islanding detection method. This result shows the achievement and efficiency of the proposed solution.
