Exponential active anti-islanding method and device

Abstract

A device and method based on an active anti-islanding technique for Distributed Power Generator Systems. The present invention is based on the Sandia Voltage Shift (SVS) technique, which includes a small Non-Detection Zone (NDZ) and by an acceptable solution to the tradeoff between the output power quality and the effectiveness of islanding detection. The present invention has the advantage to improve the NDZ and to reduce the anti-islanding detection times. This is due to the exponential-product modification made in the positive feedback to inject current, thereby making the response faster than SVS. Additionally, a self-adaptive gain is considered to achieve a low Total Harmonic Distortion (THD) at different power levels.

Claims

1. A system with a controller for detecting islanding at a point of common coupling between a source and a utility grid, the controller of the system comprising: a root mean square function; an exponential function; an absolute function; a sign function; an adaptive gain; a voltage sensor at the point of common coupling measures a voltage signal; a reference amplitude, a comparator; wherein a peak voltage amplitude is calculated by the root mean square of said voltage signal; wherein said absolute value function is an input of the exponential function, which is summed by −1 producing a first signal, wherein the sign function multiplies the first signal, producing an exponential current perturbation signal, wherein said exponential current perturbation signal is added to a reference current of an inverter current control; wherein the adaptive gain is in function of an injected power, a root mean square voltage at the point of common coupling, wherein said comparator has a low limit value and a high limit value; and wherein if the peak voltage amplitude is under the low limit value or above the high limit value, then a disconnection signal is generated to turn off the source of power linked by an inverter, otherwise, if the peak voltage amplitude is above the low limit value and under the high limit value, then the inverter remains injecting current to the utility grid.

2. The system according to claim 1, wherein the reference amplitude is subtracted to the peak voltage amplitude to produce a voltage error which is multiplied by the adaptive gain, producing a second signal, wherein said second signal is an input of the absolute function, and the sign function.

3. A method for detecting an islanding at a point of common coupling between a source of power and a utility grid, the method comprising the following step of: a) measuring a voltage signal of the utility grid at the point of common coupling; b) determining a peak voltage amplitude by means of a root mean square of the voltage at the point of common coupling; c) determining a voltage error by means of subtracting the reference amplitude from the peak voltage amplitude; d) determining an adaptive gain and multiplying with the voltage error; e) determining an absolute value of the multiplication of the adaptive gain and the voltage error, determining an exponential value of the absolute value, adding −1 to the exponential value, determining a sign value of the multiplication of the adaptive gain and the voltage error, multiplying the sign value with the exponential value −1; f) adding an exponential current perturbation signal to a reference current of an inverter current control; g) comparing the peak voltage amplitude with a low limit value and with a high limit value; and wherein the exponential current perturbation is produced by the multiplication of the exponential value −1, and the sign value; wherein the adaptive gain is in function of an injection power, a root mean square voltage at the point of common coupling, and a reference amplitude; wherein if the peak voltage amplitude is under the low limit value or is over the high limit value the utility is shut down and an inverter is turned off, otherwise, if the peak voltage amplitude remains above the low limit value and the high limit value the utility grid is still operating.

4. The method according to claim 3, further comprising the following step of updating the injected power by an exponential current perturbation signal.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 shows a flowchart of the method. The voltage at the PCC (Point of Common Coupling) is measured (S102), then the peak voltage amplitude of the voltage at the PCC is calculated (S104). At this time, the peak voltage amplitude is compared with a reference to get an error voltage (S106), and an optimal Gain is calculated in function of the injected power (S108). After that, an exponential current perturbation is calculated in function of the optimal Gain and the error voltage (S110), and it is added to the injected reference current by the inverter to the grid (S112). The injected power is updated as a result of the exponential current perturbation (S114). Finally, if the peak voltage is over or under a voltage threshold (S116), this means that the electric grid is turned off and the inverter must be shut down (S118), otherwise, if the peak voltage remains in the normal operation range, that means the electric grid is still operating.

(2) FIG. 2 shows a system for detecting islanding 200, wherein 202 is the source of power DPGS (Distributed Power Generator System), 204 is the Inverter, 206 is the Transformer, 208 is the electric grid, 210 is the Point of Common Coupling (PCC), 212 is the exponential current perturbation signal I.sub.ref dis, 214 is the inverter current controller, 216 is the current reference I.sub.ref, 218 is the voltage sensor 220 is the root mean square voltage of the sensed voltage V.sub.o at PPC, 222 is the peak voltage amplitude V.sub.p, 224 is the voltage error V.sub.e, 226 is the disconnection signal, 228 is comparator the Over-Under Voltage (OUV) threshold, 230 is the voltage peak reference V.sub.p ref, 232 is the dinamical gain k.sub.e that is in function of the injected power, 234 is the absolute function, 236 is the Sign function, 238 is the exponential function, 240 is a multiplier, 242 is the V.sub.eK.sub.e signal.

(3) FIGS. 3a-c. show gain k.sub.e in function of the injected power for SVS and EA.sup.2T with a THD.sub.i max=5% according to IEEE 1547 Standard. V.sub.o is fixed in 220 V.sub.rms ph-ph. FIG. 3a shows 0-80 kW Power range. FIG. 3b shows 0-8 kW Power range. FIG. 3c shows 0-0.8 kW Power range.

(4) FIGS. 4a-b show islanding detection times comparison of SVS and EA.sup.2T. FIG. 4a shows a Perspective 1 and FIG. 4b shows a perspective 2. Power evaluation range: 1-8 kW. Gain k.sub.e evaluation range: 0.01-0.08 (according to FIG. 3 (b)). Islanding detection times are in function of injected power and the gain k.sub.e. V.sub.o is fixed in 220 V.sub.rms ph-ph.

(5) FIGS. 5a-b show a current Injection Performance of EA.sup.2T in Steady State in Normal Operation Voltage Range (88%-110% of V.sub.o) according to Std. 1547 and Std. 929-2000 for six-pulse converters. FIG. 5a shows perspective 1. FIG. 5b shows a perspective 2. Power evaluation range: 1-8 kW. Grid voltage V.sub.o evaluation range: 88-110%.

(6) FIGS. 6a-c show an experimental test, Case 1. Performance of EA.sup.2T and SVS anti-islanding techniques when there is no current injection to the grid. FIG. 6a shows an Electric diagram. FIG. 6b shows EA.sup.2T. FIG. 6c shows SVS. Voltage at PCC (up), and islanding and detection flags (down).

(7) FIGS. 7a-c show an experimental test, Case 2. Performance of EA.sup.2T and SVS anti-islanding techniques when the DPGS and the grid are feeding the local load. FIG. 7a shows an electric diagram. FIG. 7b shows EA.sup.2T. FIG. 7c shows SVS. Voltage at PCC (up), and islanding and detection flags (down).

(8) FIGS. 8a-c show an experimental test, Case 3. Performance of EA.sup.2T and SVS anti-islanding techniques when the DPGS is injecting current to the grid and to the local load. FIG. 8a shows an electric diagram. FIG. 8b shows EA.sup.2T. FIG. 8c shows SVS. Voltage at PCC (up), and islanding and detection flags (down).

(9) FIGS. 9a-c show an experimental test, Case 4. Performance of EA.sup.2T and SVS anti-islanding techniques when the DPGS is injecting current to the grid and to the local nonlinear load FIG. 9a shows an electric diagram. FIG. 9b shows EA.sup.2T. FIG. 9c shows SVS. Voltage at PCC (up), and islanding and detection flags (down).

(10) FIG. 10 shows a robustness of the EA.sup.2T technique vs voltage harmonics components with THD of 9%. Three-phase distorted voltage, R1, R2, R3 (150 V/div, 10 ms/div), and three-phase balanced currents (5 A/div, 10 ms/div).

DETAILED DESCRIPTION OF THE INVENTION

(11) In a further embodiment, the invention comprises a method to detect if the electric grid is turned off. The method is described in FIG. 1 with a flowchart of nine steps. First, the voltage at the PCC (Point of Common Coupling) is measured (S102), then the peak voltage amplitude of the voltage at the PCC is calculated (S104). At this time, the peak voltage amplitude is compared with a reference to get an error voltage (S106), and an optimal Gain is calculated in function of the injected power (S108). After that, an exponential current perturbation is calculated in function of the optimal Gain and the error voltage (S110), and it is added to the injected reference current by the inverter to the grid (S112). Now, the injected power is updated as a result of the exponential current perturbation (S114). Finally, if the peak voltage is over or under a voltage threshold (S116), this means that the electric grid is turned off and the inverter must be shut down (S118), otherwise, if the peak voltage remains in the normal operation range, that means the electric grid is still operating.

(12) In a detailed embodiment, said method for detecting if the electric grid is turned off, comprises the steps of:

(13) a) Measure the voltage at the PCC (Point of Common Coupling);

(14) b) calculate the peak voltage amplitude of the voltage at the PCC

(15) c) compare the peak voltage amplitude with a reference to get an error voltage

(16) d) calculate an optimal Gain in function of the injected power,

(17) e) calculate an exponential current perturbation in function of the optimal Gain and the error voltage, adding to the injected reference current by the inverter to the grid.

(18) In a further embodiment said method further comprises updating the injected power by an exponential current perturbation.

(19) Finally, as a result, if the peak voltage is over or under a voltage threshold, this means that the electric grid is turned off and the inverter must be shut down, otherwise, if the peak voltage remains in the normal operation range, that means the electric grid is still operating.

(20) Furthermore, the invention comprises an anti-islanding control system to detect if the electric grid is turned off. The system is depicted in FIG. 2 and comprises of the source of power (Distributed Power Generator System—DPGS) (202), and an inverter (204) which is interconnected by a transformer (206) to the grid (208). Moreover, it is comprised by a sensor to get the voltage signal V.sub.o (218) at the PCC (Point of Common Coupling) (210), after that, the block (220) gets the peak voltage value V.sub.p (222) of V.sub.o, applying the next equation: V.sub.p=√{square root over (2)}V.sub.rms. Then, the signal V.sub.p (222) is compared with a voltage peak reference V.sub.p ref (230) and the voltage error V.sub.e (224) is generated to be multiplied by the optimal Gain k.sub.e. This optimal Gain k.sub.e (232) is calculated by the function (8), which is in function of the injected power P.sub.DPGS in order to warranty a maximal THD.sub.i (5%) in the current permitted by Standard 1547. Afterwards, the voltage error V.sub.e (224) multiplied by the optimal Gain k.sub.e is processed in parallel by sign function (236), and by an absolute (234) and exponential function (238). This process can be formally written as the equation (7). The calculated signal (Shaded area) is the exponential current perturbation I.sub.ref dis (212), which is added to the current reference I.sub.ref (216) of the inverter current control (214). Finally, the voltage signal V.sub.p (222) is compared with an over/under threshold (228). If the peak voltage signal V.sub.p (222) is out of this normal operation range, then a disconnection signal (226) is activated to turn off the inverter, otherwise, if the peak voltage signal V.sub.p (222) is in this normal operation range, then the inverter remains injecting current to the grid (208).

(21) Thus, the anti-islanding control system detects if the electric grid is turned off, comprising a source of power (Distributed Power Generator System—DPGS) (202), an inverter (204) which is interconnected by a transformer (206) to the grid (208), wherein a voltage signal (218) is measured in (210) to get the peak voltage value V.sub.p (222) with the block (220). The peak voltage value V.sub.p (222) is modified by a second signal (230) therefore generating a third signal V.sub.e (224) which is multiplied by the optimal Gain k.sub.e, wherein the control system is modulated by (232) and it is calculated by the function

(22) $k_e=\frac{\ln \left(0.05\frac{P_{DPGS}}{3V_o}+1\right)}{0.12V_{p\mathrm{ref}}}$
which is in function of the injected power P.sub.DPGS, wherein the voltage error V.sub.e (224) multiplied by the optimal Gain k.sub.e is processed in parallel by sign function (236), and by an absolute (234) and exponential function (238).

(23) In a preferred embodiment, the control system further comprises an exponential current perturbation I.sub.ref dis (212), which is added to the current reference I.sub.ref (216) of the inverter current control (214); wherein the optimal Gain k.sub.e (232) warrants a maximal THD.sub.i (5%) in the current; wherein said current is the maximal permitted by Standard 1547; wherein the voltage signal V.sub.p (222) is compared with an over/under threshold (228). Finally, if the voltage signal V.sub.p (222) is out of this normal operation range, then a disconnection signal (226) is activated to turn off the inverter, otherwise, if the voltage signal V.sub.p (222) is in this normal operation range, then the inverter remains injecting current to the grid (208). In a further embodiment the system comprises a sensor to get the voltage signal V.sub.o (218) at the PCC (Point of Common Coupling) (210); wherein the block (220) gets the peak voltage value of V.sub.o, which is calculated by V.sub.p=√{square root over (2)}V.sub.rms.

(24) In a preferred embodiment, the system of the invention comprises a modification block in the linear positive feedback of the normal SVS, making an exponential-product adjustment to inject current into the grid. This means, eq. (5) is modified in order to have exponential behavior. This proposed variation is shown in FIG. 2, in the shaded area, and it is composed by a sign, exponential, and an absolute function in order to have the next equation:
I.sub.ref dis=[e.sup.|k.sup.e.sup.V.sup.e|−1]sign(k.sub.eV.sub.e)   (7)

(25) It can be deduced from FIG. 2 and (7), that if the measured voltage at PCC increases/decreases, the current at the output of the inverter will rise/reduce at an exponential rate until the voltage reaches the OUV threshold, having a better islanding detection time than SVS, which has linear feedback. It is important to say that increasing the gain k.sub.e in the SVS affects the system performance at the steady-state while this problem is minimized in the proposed invention EA.sup.2T. This result is achieved due to the self-adaptive gain k.sub.e, which depends on the produced power by the inverter P.sub.DPGS, the maximal deviation of the voltage V.sub.o (in a normal operation at the PCC is 88%≤V.sub.o≤110%), and the maximal THD.sub.i (5%) permitted by Standard 1547. Consequently, the gain k.sub.e can be obtained from (7) and written in function of P.sub.DPGS, V.sub.o, and the voltage peak reference V.sub.p ref:

(26) $\begin{array}{cc}{k}_e=\frac{\ln \left(0.05\frac{P_{DPGS}}{3V_o}+1\right)}{0.12V_{p\mathrm{ref}}}& \left(8\right)\end{array}$

(27) Thus, the gain k.sub.e is not a constant as in SVS but it is in function of the injected power in order to reduce the disturbance current component at low power. It also improves the detection ability of the island fault at high power and reduces the detection time.

(28) The sensitivity analysis of gain k.sub.e is provided in function of islanding detection times, the power quality, and the effectiveness of the islanding detection. Specifically, the gain k.sub.e is chosen in such a way that the injected current to the grid has a THD.sub.i max of less than 5%, according to IEEE 1547 Standard. Moreover, the gain k.sub.e should have a small NDZ, enough to trigger the anti-islanding mode when an islanding case occurs. These two points should be taken into consideration in order to guarantee a smooth power injection to the grid.

(29) In this context, a simulation in MATLAB platform has been developed taking into account the equations (6)-(8) and the maximal THD.sub.i according to IEEE 1547 Standard. Therefore, charts of the gain k.sub.e as a function of the injected power for SVS and EA.sup.2T are depicted in FIG. 3. This is, for a given injected power P, a certain gain k.sub.e must be selected in equations (6) and (7) in order to preserve a THD.sub.i in the injected current of less than 5%, either for SVS (blue line) or EA.sup.2T (red line) depicted in FIG. 3. The values of the gain k.sub.e for SVS and EA.sup.2T in an injected power range of 0-80 kW can be seen in FIG. 3 (a). All values below the blue line and red line (SVS and EA.sup.2T, respectively) mean that the injected current has a THD.sub.i less than 5%. It is important to note that the gain k.sub.e for SVS is directly proportional to the injected power, while for EA.sup.2T is logarithmically proportional to the injected power (See eq. (8)), due to the exponential function used in the EA.sup.2T. This logarithmic behavior of the gain k.sub.e for EA.sup.2T makes a lower rate of change in the gain k.sub.e as compared to SVS. It can be seen the important difference in the rate of growth of the gain k.sub.e for SVS (linear growth) and EA.sup.2T (logarithmic growth) in FIG. 3 (a). This behavior is less noticeable in smaller injected power ranges as it can be seen in FIG. 3 (b) and FIG. 3 (c), which are in an injected power range of 0-8 kW and 0-0.8 kW, respectively.

(30) Furthermore, interconnected system DPGS-electric grid is modeled in MATLAB Simulink as the arrangement shown in FIG. 2 for SVS and EA.sup.2T. Different cases are simulated to register the islanding detection times as a function of different values of gain k.sub.e (8) according to different values of the injected power. In this context, it can be seen in FIG. 4 that the islanding detection time is in function of the gain k.sub.e and the injected power. These two variables are in a range of k.sub.e=0.01-0.08, and a range of the injected power of 1-8 kW as it is registered in FIG. 3 (b). Also, it can be seen in two different perspectives in FIG. 4 that the EA.sup.2T has lower islanding detection times as compared to SVS. The greater the gain k.sub.e and the injected power are, the shorter the islanding detection times are in comparison with SVS. It is important to notice that the maximum difference of islanding detection times is when the gain k.sub.e and the injected power are the highest value, this is, 0.08 and 8 kW, respectively. As a result, FIG. 4 (a) and (b) show that the SVS tends to have higher times, while the EA.sup.2T tends to keep lower detection times for higher values of the gain k.sub.e and the injected power.

(31) For multiple DPGS, many different scenarios should be considered to acquire a relationship among the gain k.sub.e, the number of DPGS, and the power generated individually. In this sense, there is a deep study for multiple DPGS using active AI techniques (Vahedi, H., Karrari, M.: “Adaptive fuzzy Sandia frequency-shift method for islanding protection of inverter-based distributed generation”IEEE Trans. Power Deliv., 2013, 28, (1), pp. 84-92), which depicts that the gain k.sub.e of every DPGS should be proportional by a factor of the fraction of the power that each DPGS shares. Certainly, a deeper study should be conducted for multiple DPGS using EA.sup.2T. Therefore, multiple DPGS with EA.sup.2T is out of the scope in this patent.

EXAMPLES

Example 1

Performance of the Invention in Steady State in Normal Operation Voltage Range

(32) An analysis of the invention performance in the steady-state in normal operating voltage range has been performed. For this examination, the interconnected system DPGS-electric grid shown in FIG. 2 has been modeled in MATLAB Simulink in a steady state operation. The THD.sub.i of the injected current has been calculated in function of the Power and the voltage magnitude in a range of 88%-110% of the voltage grid V.sub.o, this is, in a range of 193.6 V.sub.rms ph-ph-242 V.sub.rms ph-ph. Moreover, the power evaluation range for the evaluated system shown in FIG. 2 is 1-8 kW and the values of L.sub.f and its parasite resistance are 7 mH and 1Ω, respectively.

(33) As a result, in FIG. 5 is shown the behavior of the THD.sub.i of the injected current (z-axis) in function of the injected power to the grid (x-axis) and the voltage magnitude (y-axis). It can be seen in FIG. 5 (a) (Perspective 1) in the red circle that the THD.sub.i remains under 3% for voltage magnitudes between 100%-110% (220 V.sub.rms ph-ph-242 V.sub.rms ph-ph). On the other hand, for voltage magnitudes between 88%-100% (193.6 V.sub.rms ph-ph-220 V.sub.rms ph-ph) the calculated THD.sub.i has an increasing behavior till 4.6% for an injected power of more than 5 kW, as it is depicted in FIG. 5 (b) (Perspective 2) in the red circle. This is, the less voltage magnitude is, the more THD.sub.i is generated in the injected current. This is due to the decreasing level to absorb perturbations by the grid when the voltage level is lower than the normal magnitude level.

(34) Hence, in order to ensure a THD.sub.i lower than 5% when the V.sub.o is varying over the normal voltage operation, the gain k.sub.e is chosen smaller than any value below of the red line in FIG. 3, depending on the injected power value. This selection of the gain k.sub.e will extend the detection time in a small percentage when a fault occurs, depending on how far away the assigned gain value k.sub.e is. It is important to mention that the gain k.sub.e has a lower limit value before the anti-islanding technique stops to detect fault conditions. This value depends greatly on the electrical noise in the voltage sensor and the control board resolution. Therefore, it should be fixed by experimental data.

Example 2

Experimental Test of the Invention

(35) Experimental tests of the method in FIG. 1 and the system in FIG. 2 are presented below to validate the high performance of the invention versus the normal technique SVS. The present invention EA.sup.2T and the SVS techniques were implemented in a Single Board of National Instruments based on an FPGA Xilinx Spartan-6 LX45 at 2.5 MHz. Also, the main inverter controller implemented in simulation and experimental tests is a digital controller, where the errors between the sensed currents on the PCC and sinusoidal references stated by the voltage grid are minimized by a single band hysteresis current controller, which is best known for robustness, fast error tracking, good dynamic response, and ease of implementation (Singh, J. K., Behera, R. K.: “Hysteresis Current Controllers for Grid Connected Inverter: Review and Experimental Implementation,” in “Proceedings of 2018 IEEE International Conference on Power Electronics, Drives and Energy Systems, PEDES 2018” (Institute of Electrical and Electronics Engineers Inc., 2018)).

(36) In this context, it is important to highlight that these experimental tests were made in a three-phase two-stage conventional inverter connected to a local load, and a programmable ac power source Chroma 61700 which emulates the utility grid. The DPGS is emulated by a Genesys™ 2U 5 kW Programmable DC Power Supply and the inductors L.sub.f and parasite resistances are 7 mH and 1Ω, respectively. Moreover, the current and voltage sensors (current transducer LA55-P/SP1 and voltage transducer LV25-P) have response times of 40 us and 1 us, respectively. It can be seen that these response times are much faster than the detection times showed in the previous section, which are in the order of milliseconds. Consequently, the dynamic response of the current and voltage sensors is almost instantaneous and can be omitted for simplification purposes.

(37) After the experimental setup explanation, the experimental tests are divided into four cases according to the current flow among the DPGS-Inverter, Load, and Grid (See e.g. FIG. 6 (a)). Furthermore, every case shows the three-phase voltages at the PCC, and the islanding interruption and detection Boolean flags at the bottom of every graphic (See e.g. FIG. 6 (b) and (c)). In this sense, the ac power source Chroma 61700 (Emulated grid) activates a bit flag to know that the islanding fault has occurred (Indicated in blue at the bottom of e.g. FIG. 6 (b) and (c)). Besides, the power quantities of every case are depicted in Table 1, whereas the current graphics have been omitted to reduce the space in the patent due to the fact that they have similar waveforms of the voltages since a linear load was employed, except for case 4.

(38) TABLE-US-00001 TABLE 1 Experimental Test Power Flow. Case DPGS Power Load Power Grid Power 1 1000 W 1000 W 0 W 2  300 W 1000 W 700 W 3 1300 W 1000 W −300 W (Injected) 4 1000 W  250 W −750 W (Injected)

(39) 1.1. Case 1

(40) Case 1 is when there is no current injection to the Grid. Certainly, it is the most critical case and is made to verify the NDZ of the proposed invention since the power consumed by the load must be exactly the same as the power produced by the DPGS. Therefore, the current flows just from the DPGS-Inverter to the Load, as it can be seen in FIG. 6 (a). The DPGS-inverter generates 1 kW (±0.05%). Consequently, the self-adaptive gain k.sub.e (Eq. 8) for EA.sup.2T assigns the value of 0.009. While the gain for SVS is manually selected to a value of 0.01 according to FIG. 3 to maintain the THD.sub.i less than 5%. The increase of the THD.sub.i in the transient gap is a sign that the contribution of the exponential current perturbation I.sub.ref dis is present in the current reference I.sub.ref since the islanding occurs until the technique detects the fault. Furthermore, it can be seen in FIG. 6 (b) and (c) the velocity of EA.sup.2T and SVS techniques in the transient gap, respectively. As a result, the EA.sup.2T takes 7 ms to detect the islanding condition, while, the SVS takes a longer time of 27 ms to detect the fault (See FIG. 6 (c)). That is more than three times the EA.sup.2T detection time. Consequently, this performance agrees with equations (6) and (7) since the EA.sup.2T has an exponential behavior, while SVS has a linear performance until the threshold is triggered.

(41) 1.2. Case 2

(42) Case 2 is shown in FIG. 7 (a). This case is when the DPGS-Inverter and the Grid share the power generation to feed the local Load. For this experiment, the DPGS-inverter generates 0.3 kW (±0.05%) and the grid produces 0.7 kW (±0.1%) to feed a local load of 1 kW. According to eq. (8), the self-adaptive gain k.sub.e takes a value of 0.0028 for EA.sup.2T. Meanwhile, for SVS is 0.0029 to preserve the THD.sub.i less than 5% as it is depicted in FIG. 3. As a result, FIG. 7 (b) shows a very short transient with a duration of 2.1 ms. Certainly, the transient has a bigger perturbation than in case 1, but smaller than the SVS depicted in FIG. 7 (c), which has a longer detection time of 6.5 ms with a pronounced perturbation caused by the contribution of the exponential current perturbation I.sub.ref dis. It is important to highlight that in this case, the voltage signals were getting smaller in amplitude until the lower threshold was triggered in both techniques at different velocities: exponential (EA.sup.2T) and linear (SVS) rate. In particular, for EA.sup.2T, this mechanism is possible due to the Sign and Absolute function, i.e. the Sign function gives the negative sign to “exp(|k.sub.eV.sub.e|)−1”, for a negative error V.sub.e for this case.

(43) 1.3. Case 3

(44) The FIG. 8 (a) shows the Case 3 where the DPGS-Inverter injects current to the Grid and also feeds the local Load. In this case, the DPGS-inverter produces 1.3 kW (±0.05%), of which 0.3 kW are injected to the Grid and 1 kW is consumed by the local Load. Therefore, the gain k.sub.e for SVS is manually selected to a value of 0.013 according to FIG. 3 to maintain the THD.sub.i less than 5% while the self-adaptive gain (Eq. 8) for EA.sup.2T gives the value of 0.011. FIG. 8 (b) and (c) show that the upper threshold was triggered by the increase of the voltage signals because V.sub.p was bigger than V.sub.p ref and hence, V.sub.e was positive in the first moment of the islanding condition. Therefore, it can be seen a clear distortion in the signals with a drastic difference of velocities in the transient, exponential (EA.sup.2T), and linear rate (SVS). As a result, the EA.sup.2T technique has a detection time of 3 ms, whereas the SVS technique has a longer detection time of 15.2 ms.

(45) 1.4. Case 4

(46) Finally, Case 4 is shown in FIG. 9 (a). In this case, the DPGS-Inverter produces 1 kW (±0.05%), injects 0.75 kW to the Grid and also feeds a 0.25 kW local Nonlinear Load, which is a conventional three-phase rectifier. Both techniques have a similar behavior than Case 3 because the power flow is the same but they produce smaller detection times and more distortion in the transient gap. In this particular case, the self-adaptive gain k.sub.e (eq. 8) for EA.sup.2T takes the value of 0.009. For SVS, the value is manually selected to 0.01 according to FIG. 3, maintaining an acceptable THD.sub.i of less than 5%. As a result, it can be seen in FIG. 9 (b) that EA.sup.2T has a detection time of 2.1 ms; however, the SVS has a longer detection time after the islanding condition with 6.1 ms (See FIG. 9 (c)).

(47) Summarizing the four cases of study, it can be seen that in all the tests, the NDZ has been improved and a THD.sub.v˜4% has been maintained in the grid-connected operation. Moreover, the detection time has been reduced and the waveform of the transient response is maintained as in SVS. In all cases, reducing a potential damage to the load.

(48) Certainly, standards for Grid-Connected systems indicate a maximum of 2 seconds for the detection time; however, it is always a good practice to achieve the detection in a faster way to prevent malfunction in the equipment that is still connected in the grid.

(49) It is important to notice that a false positive may occur due to variations of the grid voltage and not necessarily for disconnection, but certainly, the permitted region is determined by the standards. For this reason, a robustness test of the EA.sup.2T technique vs voltage harmonics components have been made in FIG. 10. It can be seen that the current injection is still working, even with a distorted voltage grid of THD.sub.v=9% causing a THD.sub.i in the third and fifth harmonic of less than 4%. Clearly, this distortion is in the limits of the recommended in IEEE Standard 929-2000 for six-pulse converters.

(50) Although the results of the proposed invention versus the SVS are satisfactory. It is important to compare the method to other anti-islanding techniques, which is described in the next section.

Example 3

Comparison of the Invention with Other Anti-Islanding Techniques

(51) In Table 2, the performance of the proposed invention is compared to the most representative active anti-islanding techniques which are the Reactive Power Variation Method (RPVM) (Jo, J., Cha, H.: “Performance of anti-Islanding of an improved reactive power variation method based on positive feedfback,” in “2017 IEEE Energy Conversion Congress and Exposition, ECCE 2017” (Institute of Electrical and Electronics Engineers Inc., 2017), pp. 4761-4765), the Adaptive Reactive Power Control (ARPC) (Chen, X., Wang, X., Jian, J., Tan, Z., Li, Y., Crossley, P.: “Novel islanding detection method for inverter-based distributed generators based on adaptive reactive power control”J. Eng., 2019, 2019, (17), pp. 3890-03894), and the Active Cross-Correlation Anti-islanding Scheme (AC.sup.2AS) (Voglitsis, D., Papanikolaou, N. P., Kyritsis, A. C.: “Active Cross-Correlation Anti-Islanding Scheme for PV Module-Integrated Converters in the Prospect of High Penetration Levels and Weak Grid Conditions”IEEE Trans. Power Electron., 2019, 34, (3), pp. 2258-2274). All the compared techniques are classified as active techniques with small variations among them. It can be seen a brief description of every active technique, including the proposed invention in the second row of Table 2. In addition, the most important parameters of every technique are presented in the lower part of the table.

(52) In addition, the third row of Table 2 summarizes the detection times of the compared techniques. It is important to note that the fastest detection time is registered by the proposed EA.sup.2T technique with a range of 2.1-7 ms. It is followed by the SVS with detection times between 6.1-27 ms. RPVM and ARPC have reported detection times of 53-150 ms and 48.3-276 ms, respectively. Finally, the AC.sup.2AS reported detection times of more than 400 ms, which is the slowest method of the compared techniques.

(53) Next, the fourth row of Table 2 depicts a qualitative comparison of the NDZ among the active anti-islanding techniques. According to FIGS. 3 and 4, EA.sup.2T has a smaller NDZ than SVS and it is in function of the feedback self-adaptive gain k.sub.e. Furthermore, the RPVM and the ARPC methods have a low and a very low NDZ, respectively. This is in the most critical case when the power injection is the same as the consumed load power (P.sub.inj=P.sub.Load), and the impedance Z.sub.R is equal to the impedance Z.sub.LC. Also, the AC.sup.2AS has a very low NDZ, which is in function of grid weakness.

(54) Moreover, it is shown in the fifth row of Table 2 that any of the compared active anti-islanding technique needs a current sensor to detect the islanding condition.

(55) One important characteristic of the active techniques is their perturbation intensity to the grid which is shown in the sixth row of Table 2. According to FIG. 3, the EA.sup.2T and SVS have a very low power quality distortion, which is less than 5% of THD.sub.i. Even so, RPVM and ARPC have a variation of the power of the inverter P.sub.inv of ±5%, and −5% and +7%, respectively, which is greater than the values used in EA.sup.2T and SVS. On the other hand, AC.sup.2AS injects a small perturbation of 1% of the second harmonic component.

(56) Furthermore, it is important to highlight the complexity comparison shown in the seventh row of Table 2. It can be seen that EA.sup.2T and SVS are simple and easy to implement. However, RPVM, ARPC, and AC.sup.2AS are more complex than the previous methods due to the utilization of more complex mathematical operations.

(57) Finally, the proposed invention can be compared also with remote based islanding recognition techniques, which can eliminate the NDZ completely. According to (Das, P., Ghore, S., Biswal, M.: “Comparative assessment of various islanding detection methods for AC and DC microgrid,” in “2020 1st International Conference on Power, Control and Computing Technologies, ICPC2T 2020” (Institute of Electrical and Electronics Engineers Inc., 2020), pp. 396-400), the PLCC and the SCADA systems have null NDZ and zero error detection ratio; however, the detection times are 200 ms for PLCC and 100 ms-300 ms for SCADA under optimized conditions.

(58) This is at least fourteen times slower than the worst detection time registered for the proposed invention EA.sup.2T (7 ms for case 1). Moreover, due to the high installation cost in comparison with active techniques, it is not viable for small scale DPGSs.

(59) TABLE-US-00002 TABLE 2 Comparison summary of different anti-islanding techniques. INVENTION EA.sup.2T .sup.a SVS .sup.b RPVM .sup.c ARPC .sup.d AC.sup.2AS .sup.e Classification Exponential Active Active Power Reactive Power Adaptive Reactive Modified Power Variation Variation Variation Power Control Incremental Conductance Description The inverter has The inverter It consists of two The voltage The scheme faster response in responds to small parts. The first part variation and periodically injects voltage with changes in voltage has a fundamental correlation factor a second-order corresponding with corresponding amplitude of between reactive harmonic current exponential changes in current ±5% P.sub.inv, and the power disturbance component of low changes in current that are sufficient to second part has a and frequency magnitude and that are sufficient cause a further positive feedback variation are the evaluates grid to cause a further change in the same using a frequency criteria for the response by means change in the direction deviation adaptive of correlation same direction disturbance slope Detection 2.1-7 ms .sup.f 6.1-27 ms .sup.f 53-150 ms .sup.f 48.3-276 ms .sup.f <400 ms time NDZ Lower than SVS. Depends on Low .sup.g Very low .sup.g Very low .sup.h See FIG. 4. feedback gain k.sub.e AC current No no no no no sensor Power quality Very low. Very low. According ±5% P.sub.inv −5% P.sub.inv, +7% P.sub.inv 1% of the second distortion According to to Standard IEEE harmonic Standard IEEE 1547 (<5% THD) component. 1547 (<5% THD) According to IEC 61727 Complexity Low. Exponential Very low. Linear Medium. It is Medium. It is Medium. It has to positive feedback positive feedback designed for based on reactive compute mainly the gain gain synchronous power and cross-correlation reference frame, frequency algorithm and a adding more variation functions cost function transformation matrix to solve .sup.a Exponential Active Anti-islanding Technique. Self-adaptive gain k.sub.e according to (8); .sup.b Sandia Voltage Shift. Gain k.sub.e must be changed manually according to FIG. 3; .sup.c Reactive Power Variation Method based on Positive Feedback. K.sub.RPV = 0.24 (Constant gain), P.sub.inv = 1.3 kW, V.sub.grid = 220 V.sub.rms, f.sub.o = 60 Kz, Q.sub.f = 2.5 (Quality factor), PF = 0.9975 (Power Factor) (Jo, J., Cha, H.: “Performance of anti-Islanding of an improved reactive power variation method based on positive feedfback,” in “2017 IEEE Energy Conversion Congress and Exposition, ECCE 2017” (Institute of Electrical and Electronics Engineers Inc., 2017), pp. 4761-4765); .sup.d Adaptive Reactive Power Control. |ΔV| ≥ 0.04 p.u. (Voltage fluctuation), C.sub.f (Correlation factor in function of the power quality distortion, fundamental frequency and voltage variation), Q.sub.f = 2.5 (Chen, X., Wang, X., Jian, J., Tan, Z., Li, Y., Crossley, P.: “Novel islanding detection method for inverter-based distributed generators based on adaptive reactive power control” J. Eng., 2019, 2019, (17), pp. 3890-3894); .sup.e Active Cross-Correlation Anti-islanding Scheme. DPF = 1 (Displacement Power Factor), x.sub.r = 0.165 (Reactance-resistance ratio at PCC), φ.sub.ref = −62° (Phase angle of V.sub.ref), CC.sub.Base = 1018 (Cross-correlation index of two periodical signals) (Voglitsis, D., Papanikolaou, N. P., Kyritsis, A. C.: “Active Cross-Correlation Anti-Islanding Scheme for PV Module-Integrated Converters in the Prospect of High Penetration Levels and Weak Grid Conditions” IEEE Trans. Power Electron., 2019, 34, (3), pp. 2258-2274); .sup.f Under voltage nominal conditions. V.sub.grid = 220 V.sub.rms; .sup.g Critical case P.sub.inj = P.sub.Load, Z.sub.R = Z.sub.LC; .sup.h NDZ is function of grid weakness.