Various detection techniques are proposed, encompassing local control, remote control, artificial intelligence, and signal processing techniques. Remote-control methods, such as PLCC, SCADA, and transfer trip, provide highly reliable islanding detection by eliminating the non-detection zone (NDZ). However, these methods are cost-prohibitive and impractical for smaller systems due to extensive infrastructure requirements. Signal processing techniques, including Fourier and Wavelet transforms, analyse time and frequency-domain variations in voltage and current. While they offer precision, these methods demand sophisticated hardware, and significant computational resources, and still struggle to eliminate NDZ issues. Computational intelligence techniques, such as ANN, SVM, and ANFIS deliver high accuracy and adaptability to complex grid configurations but are computationally intensive, require large training datasets, and introduce implementation intricacies [5]. Remote-control methods lack localized monitoring of parameters like voltage and frequency, necessitating the adoption of local-control techniques for effective islanding detection. Local control methods are further categorized into active, passive, and hybrid techniques. Active methods inject disturbances at the point of common coupling (PCC) to detect grid disconnection, providing high reliability but degrading power quality and encountering challenges in multi-inverter setups. Hybrid methods combine active and passive approaches to enhance accuracy and minimize NDZ, but they impact power quality and exhibit significant complexity in large-scale system implementations [6]. This article primarily focuses on reviewing passive methods, where the primary challenge lies in detecting islanding within the NDZ. Several methods have been proposed to address this issue by enabling detection within a few cycles of islanding inception. One proposed method monitors the rate of change of kinetic energy over reactive power (ROCOKORP) [7], which calculates a novel index based on the kinetic energy of distributed generators (DGs). However, this method relies on accurate parameter estimation and involves potential computational complexity in dynamic grid scenarios. Another approach combines passive low-threshold settings with active three-phase static RC load activation [8], aiming to eliminate the NDZ. The complexity of this method lies in optimizing threshold settings and load configurations to ensure accurate detection while avoiding misoperations during non-islanding events. The passive communication-based technique in [9] leverages phase angle differences of superimposed impedance at DG and PCC ends via fast communication links. While effective, it depends on secure communication channels that may experience delays or failures under adverse network conditions. A multi-SOGI-based passive detection method [10] extracts harmonic signatures for virtual impedance calculation but suffers from dependency on accurate harmonic data, high computational requirements, and reduced effectiveness in distorted or dynamic grids. Another passive algorithm [11] utilizes the rate of change of potential energy as an index, achieving nearly zero NDZ, but faces sensitivity to noise in frequency and ROCOF calculations, reliance on precise parameter estimation, and potential computational inefficiency in dynamic conditions. In [12], an adaptive ROCOF-based technique dynamically calculates frequency and ROCOF using PLL settings, reducing NDZ to zero. However, it relies heavily on precise PLL configurations, exhibits sensitivity to transient disturbances, and encounters delays in frequency tracking during rapid grid changes, along with scalability challenges in complex microgrids. Similarly, the $\mu$ PMU-based islanding detection method in [13] achieves zero NDZ by monitoring voltage and current phasors. Despite its effectiveness, it relies on precise measurements, and high-resolution data, and is sensitive to noise and disturbances, making it economically unviable for widespread implementation. Furthermore, directly monitoring the operating point may not reliably detect islanding, especially in the non-detection zone (NDZ), where variations are minimal. In contrast, the BH curve provides deeper insight into magnetic characteristics, capturing subtle changes in inductance, flux, and harmonic distortion. Unlike voltage and frequency, which can remain stable, core saturation and flux behavior shift noticeably when the grid is lost. By analyzing alienation and correlation indices from successive BH curves, islanding can be detected even in conditions where traditional methods fail.

To overcome all the drawbacks discussed in the literature, this article proposes a novel islanding detection method using the magnetic characteristics (BH curves) derived from the current through the grid-side and inverter-side filter inductors, as well as the voltage across them. BH curves provide valuable insights into the magnetic behavior of grid-connected PV inverters and serve as a foundation for developing effective islanding detection techniques to enhance the reliability and safety of grid-connected renewable energy systems.The BH curves obtained over successive cycles are converted into gray scale images and compared to calculate the alienation coefficients [14]. These coefficients are used to compute an index, which is then compared against a threshold to detect islanding events. The proposed algorithm successfully detects islanding within the time limits prescribed by various international standards.

The rest of the paper is organized as follows: Section II discusses the $\mathrm{{BH}}$ curves in detail, including their mathematical analysis and potential application in islanding detection. Section III presents the test system discussion. Section IV elaborates on the proposed algorithm. Section V discusses the results, including the thresholds and a comparison of the outcomes, with the conclusion presented in Section VI.

BH curves, also known as hysteresis loops, graphically represent the variation between the magnetic flux density(B)and the magnetic field strength(H)in a magnetic material. They are widely used in the field of magnetism and in electromagnetic devices such as transformers, inductors, and magnetic sensors. The theory behind BH curves is rooted in the behavior of magnetic materials when subjected to an external magnetic field. When a magnetic material is exposed to a magnetic field, it becomes magnetized, with its magnetic domains aligning in the direction of the applied field. The relationship between the magnetic flux density(B)and the magnetic field strength(H) during this process is plotted on a graph, resulting in a hysteresis loop, as illustrated in Fig. 1.

As the magnetic field strength(H)increases, the magnetic flux density(B)increases proportionately until the material reaches saturation. At this point, the proportional increment of $B$ ceases, and the curve becomes flat at the edges, corresponding to ${B}_{\text{sat }}$. When the magnetic field strength(H)is reduced to zero, the flux density(B)does not return to zero. Instead, the magnetic material retains some residual magnetism, a property known as remanence (B_r). To remove the magnetism B=0 from the magnetic material, the magnetic field strength(H)must be applied in the reverse direction until it reaches ${H}_{\mathrm{c}}$ on the $x$ -axis. This value represents the coercivity of the magnetic material. The phenomenon of saturation is similarly observed in the reverse direction.

The potential application of BH curves for islanding detection in grid-connected PV systems lies in their ability to capture changes in the magnetic properties of components such as transformers or inductors during islanding events. Grid-connected PV systems typically include magnetic components like transformers and inductors, which play critical roles in power conversion and distribution. These components exhibit specific magnetic properties characterized by BH curves. During normal grid-connected operation, the magnetic properties of these components remain relatively stable. However, when an islanding event occurs and the grid becomes isolated, the operating conditions of the PV system and its connected components may change. These changes can affect the magnetic properties of the components, leading to alterations in their BH curves. Variations in load conditions, solar irradiance, or grid parameters during islanding events can induce changes in magnetic flux density(B)or magnetic field strength(H), resulting in shifts or distortions in the BH curves.

Monitoring the BH curves of critical components in real-time enables the detection of abnormal changes that indicate islanding conditions. Deviations from the expected BH curve characteristics can signal the presence of islanding and trigger appropriate protective measures. Overall, BH curve analysis offers a novel approach to islanding detection in grid-connected PV systems by leveraging the magnetic properties of system components and serves as a primary motivation for this work.

The BH curve of an inductor is obtained from the measured values of the voltage $V\left(i\right)$ and the current $I\left(i\right)$ through it. The BH curve [15] for the ${k}_{\text{th }}$ cycle is obtained from the following steps:

Obtain the mean voltage ${V}_{\text{mean }, k}$

where $i = 1,2,\ldots, q$ samples.

Compute the normalized voltage waveform ${V}_{k}\left(t\right)$ by subtracting ${V}_{\text{mean }, k}$

Perform continuous integration of the voltage waveform to obtain the magnetic flux at each data sample over the entire cycle $k$.

Calculate the magnetic flux density ${B}_{k}\left(t\right)$

where, N and s are the number of turns and the toroidal area of cross section.

Normalize the flux density to obtain only the changes in the flux density by finding the mean flux density ${B}_{\text{mean }, k}$ is obtained as

Compute the normalized flux density as

Determine the DC flux density which determines the operating point of the transformer or inductor in an in-circuit operation as ${B}_{\mathrm{{dc}}, k}$ and given by

where, ${L}_{k},{I}_{\text{ave }, k}$ are the inductance and the average value of the current. The value of inductance is computed as the ratio of the integral of voltage data to current data, given by

Obtain the magnetic flux density vector ${B}_{kf}\left(t\right)$ as

Obtain the magnetic field strength $H$ data by

where, N, ml are the number of turns and the length of the magnetic path respectively which are given as input based on the inductor physical data.

This process is repeated for all number of cycles from the acquired data. The flowchart in Fig. 2 illustrates the step-by-step procedure for obtaining the BH curve in a real-time application.

The correlation coefficient measures the similarity between two patterns. Correlation coefficient(R)of two dimensional matrices M and N of size $p \times q$ is shown in (11).

where, $\bar{M}$ and $\bar{N}$ are the mean of the matrices. The measure of uncorrelated patterns can be obtained by alienation coefficient

The BH curve obtained is converted into a two-dimensional matrix of size $p \times q$ (gray scale image). The matrices corresponding to islanding and non-islanding patterns must have the same size. If the patterns are exactly the same then $R = 1$ and $A = 0$ and if the patterns are completely different, then $R = 0$ and $A = 1$. Thus, the values of alienation and correlation coefficients can vary between 0 and 1.

The feature vector, which includes the alienation coefficient, the ratio of alienation and correlation index, and the cumulative index, is obtained both before and after the islanding event. The absolute maximum value post-islanding or disturbance, calculated for a half cycle or quarter cycle, is selected as the islanding index. Islanding indices are computed from the feature vectors for various cases, such as R -load, ${RL}$ - load, and variations in $P$ and q. The minimum value of all the islanding indices is defined as the “Islanding index-min,” which equals the minimum of the absolute maximum values from the half cycle after islanding across all islanding cases. Similarly, the absolute maximum value (or islanding index) is also obtained for non-islanding events, such as load switching or capacitor switching. The maximum value of these indices is referred to as the “Islanding index-max”. For normal cases, the same half-cycle post-disturbance time is considered, and the absolute maximum value is calculated. Thus, Islanding index-max is defined as the maximum absolute value from the half cycle after disturbances in non-islanding cases. The threshold is determined by taking the average of Islanding index-min and Islanding index-max. Therefore, the threshold is approximately calculated as

The proposed algorithm is evaluated by considering non-islanding events such as load switching and DG tripping. Improperly designed islanding detection algorithms may fail to filter transient disturbances, potentially triggering false islanding detection alarms. Sudden changes in grid parameters caused by load switching can erroneously activate the detection mechanism, resulting in unnecessary disconnection of the PV system from the grid. In the test scenario, load switching is introduced by opening a breaker at $1\mathrm{\;s}$. The variation in the indices are shown in Fig. 13 （ ${\Delta P} = {\Delta Q} = 0\%$). It is observed that the index increases after the load switching event remains well within the threshold. After the tripping or disconnection of the solar inverter, the index decreased sharply. Notably, only actual islanding events result in the index exceeding the threshold, facilitating straightforward detection when the load is a parallel RLC load. The variations of the cumulative indices under different power mismatch conditions such as ${\Delta P} = 0\%$ and ${\Delta Q} = + 5\%$, ${\Delta P} = 0\%$ and ${\Delta Q} = - 5\%,{\Delta P} = + 5\%$ and ${\Delta Q} = 0\%$ and ${\Delta P} =$ $- 5\%$ and ${\Delta Q} = 0\%$ are shown in Figs. 14-17, respectively.

The proposed algorithm is tested by changing the type of the loads. Three load combinations, such as RLC, RL and RC are considered to replicate resonance condition, positive reactive power, and negative reactive power respectively. The values of $R = {15.11\Omega }, L = {48.86}\mathrm{{mH}}$ and $C = {210\mu }\mathrm{F}$ are chosen. The variation in the indices are shown in Fig. 18. It can be observed that the proposed algorithm successfully detects islanding in the presence of all types of load combinations.Additionally, the detection time is longer for RLC load when compared to RL and $\mathrm{{RC}}$ loads due to resonance.

The proposed algorithm is tested as per the IEC 62116 standard, which accounts for active and reactive power mismatches. Under normal conditions, power discrepancies are balanced by the grid. However, when the grid is disconnected in islanding mode, changes in active power are reflected as voltage fluctuations, while changes in reactive power manifest as over/under frequency deviations. The worst-case scenario arises when the active and reactive powers of the load and the inverter are perfectly matched, a condition referred to as the non-detection zone (NDZ). Under this scenario, the differences in active and reactive power are ${\Delta P} = {P}_{\text{inv }} - {P}_{\text{load }} = 0$ and ${\Delta Q}$ $= {Q}_{\text{inv }} - {Q}_{\text{load }} = 0$. The values of $R = {15.11\Omega }, L = {48.86}\mathrm{{mH}}$ and $C = {210\mu }\mathrm{F}$ are chosen to obtain the resonance condition at ${50}\mathrm{\;{Hz}}$ for the inverter capacity of ${3.5}\mathrm{{kVA}}$. An incremental variations in active and reactive powers are maintained as + or $- 5\%$ along with the NDZ condition. The time of detection is enumerated in the Table II. It can be observe that the proposed cumulative index demonstrates shorter average detection times compared to either the grid-side alienation index or the inverter-side alienation index when evaluated independently.

The detection time of the proposed algorithm is evaluated against various international standards. It is observed that the detection time is below the minimum of ${0.2}\mathrm{\;s}$ according to Table II. Using the cumulative index the detection can be achieved in less than 3 cycles for NDZ.