energies Article
An Irregular Current Injection Islanding Detection Method Based on an Improved Impedance Measurement Scheme Menghua Liu 1, * , Wei Zhao 1 , Qing Wang 2 , Songling Huang 1 1 2
*
and Kunpeng Shi 1
Department of Electrical Engineering, Tsinghua University, Beijing 100084, China;
[email protected] (W.Z.);
[email protected] (S.H.);
[email protected] (K.S.) Department of Engineering, Durham University, Durham DH1 3LE, UK;
[email protected] Correspondence:
[email protected]; Tel.: +86-010-6277-3070
Received: 17 August 2018; Accepted: 8 September 2018; Published: 17 September 2018
Abstract: One class of islanding detection methods, known as impedance measurement-based methods and voltage change monitoring-based methods, are implemented through injecting irregular currents into the network, for which reason they are defined in this paper as irregular current injection methods. This paper indicates that such methods may be affected by distributed generation (DG) unit cut-in events. Although the network impedance change can still be used as a judgment basis for islanding detection, the general impedance measurement scheme cannot separate island events from DG unit cut-in events in multi-DG operation. In view of this, this paper proposes a new islanding detection method based on an improved impedance measurement scheme, i.e., dynamic impedance measurement, which will not be affected by DG unit cut-in events and can further assist some other equipment in islanding detection. The simulations and experiments verify the stated advantages of the new islanding detection method. Keywords: assisting islanding detection; distributed generation (DG) unit cut-in; impedance measurement; islanding detection; multi-DG
1. Introduction Due to the vigorous upward trend in the need for clean energy, Distributed generation (DG) has flourished, and its related technologies have attracted more and more attention. As an essential feature of DG, islanding detection has been the topic of many studies. A series of islanding detection methods have been developed, which are generally divided into remote methods and local methods. The local methods in turn include active methods and passive methods. The active methods attract more studies and applications due to their low cost and reliability. The active methods that are based on impedance measurement and voltage change monitoring are collectively referred to as irregular current injection islanding detection methods in this paper, as all of them are realized by injecting irregular currents, e.g., high frequency currents, into a network, and this paper will focus on such methods. With regard to the irregular current injection methods, the judgment on an island event is based on the changes of the irregular voltages excited by the injected irregular currents. On the basis of this idea, the irregular voltage changes were directly monitored in [1–8], whereas the network impedance was further calculated in [9–11] and the unbalance factor of the terminal voltage of DG units was calculated in [12]. Moreover, the injected irregular currents are also different in these references. High frequency currents were adopted in [1–4,9–11]; asymmetric sub-harmonics current was used in [5]; a pulse current was adopted in [6]; and negative sequence currents were adopted in [7,8,12]. In contrast, irregular voltage injection was proposed in [13–17] for islanding detection.
Energies 2018, 11, 2474; doi:10.3390/en11092474
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frequency currents were adopted in [1–4,9–11]; asymmetric sub-harmonics current was used in [5]; a Energies 2018, 11, 2474 2 of 18 pulse current was adopted in [6]; and negative sequence currents were adopted in [7,8,12]. In contrast, irregular voltage injection was proposed in [13–17] for islanding detection. In effective. TheThe methods proposed in In addition, addition,many manyother othertypes typesofofmethods methodsare arealso alsohighly highly effective. methods proposed [18,19] belong to another classic type of of islanding detection in [18,19] belong to another classic type islanding detectionmethods, methods,i.e., i.e.,frequency frequencyshift shiftmethods, methods, which perturbed current phase and reactive current, respectively; the Sandia Frequency Shift which perturbed current phase and reactive current, respectively; the Sandia Frequency Shift(SFS) (SFS) method methodwas wasimproved improvedin in[20], [20], which which is is aa classic classic active active method, method, by by decreasing decreasingthe the positive positivefeedback feedback gain the effect effecton onsystem systemstability; stability;ananidea idea grid stiffness measurement proposed in gain to to reduce reduce the ofof grid stiffness measurement waswas proposed in [21], [21], by which an island is identified according to the stiffness change; the method in [22] periodically by which an island is identified according to the stiffness change; the method in [22] periodically perturbed the reactive reactivecurrent current and confirmed is a frequency undulation in an island perturbed the and confirmed that that therethere is a frequency undulation in an island condition, condition, which is seen as a criterion of islanding; a hybrid method based on the SFS method and which is seen as a criterion of islanding; a hybrid method based on the SFS method and rate of change rate of change of frequency (ROCOF) relay is proposed in [23], by which a smaller non-detection zone of frequency (ROCOF) relay is proposed in [23], by which a smaller non-detection zone (NDZ) is (NDZ) is achieved; two methods passive methods were reported in [24] and [25], respectively, the former of achieved; two passive were reported in [24] and [25], respectively, the former of which which employed the signal processing theory and measured the network impedance passively, and employed the signal processing theory and measured the network impedance passively, and the latter the latter of which estimated theand voltage and impedance parameters ofThevenin-like a linear Thevenin-like of which estimated the voltage impedance parameters of a linear model tomodel detect to wavelet packet transform used [26] to for islanding andetect island;ana island; waveleta packet transform was usedwas in [26] to in extract theextract signalthe for signal islanding detection; detection; theinmethod in a[27] was amethod, remote method, which took advantage of the phasor measurement the method [27] was remote which took advantage of the phasor measurement unit to unit to the collect the evidence of anand island; and remote communication was also utilized in [28] the to collect evidence of an island; remote communication was also utilized in [28] to facilitate facilitate the performance of islanding detection. performance of islanding detection. Although the methods methodsproposed proposed in the above literature have declared been declared some Although the in the above literature have been reliable,reliable, some practical practical exist forirregular existingcurrent irregular current injectionone methods, of which is the problemsproblems still exist still for existing injection methods, of whichone is the misjudgment misjudgment problem resulting from DG unit cut-in events. The existing methods are based on the problem resulting from DG unit cut-in events. The existing methods are based on the surge of measured surge of measured irregularboth voltage. However, both island events and cut-in large power unit cut-in irregular voltage. However, island events and large power DG unit events DG can bring about events can bring about such surge. The existing methods cannot distinguish between these two events such surge. The existing methods cannot distinguish between these two events by means of the general by means of measurement the general impedance measurement scheme,measurement. i.e., static impedance measurement. The impedance scheme, i.e., static impedance The following will briefly following introduce this problem: introduce will this briefly problem:
(a) Introduction Introduction of of irregular irregular current current injection injection methods: methods: Inverter-based Inverter-based DG DG isis currently currently very very (a) popular, and thus the analyses in this paper are based on this type of DG. An inverter-based DG popular, and thus the analyses in this paper are based on this type of DG. An inverter-based DG diagram is shown in Figure 1. diagram is shown in Figure 1.
Figure1.1.Inverter-based Inverter-baseddistributed distributedgeneration generation(DG). (DG). Figure
In aa grid-connected grid-connected condition condition(i.e., (i.e., with with the the switch switch SSggturned turnedon), on),the theequivalent equivalentinput inputnetwork network In impedance seen from the DG unit (hereinafter referred to as network impedance) is: impedance seen from the DG unit (hereinafter referred eq(grid) = Zg//Zl ZZ eq(grid) = Zg //Zl
where Rgg ++ jωL Since |Z |Zgg||is where Z Zgg and and Z Zll are are grid grid impedance impedance (i.e., (i.e., R jωLgg))and andload load impedance, impedance, respectively. respectively. Since is much smaller than |Z l | over a certain frequency range [5,29], |Z eq(grid) | is close to |Z g |. much smaller than |Zl | over a certain frequency range [5,29], |Zeq(grid) | is close to |Zg |. In Inan anisland island condition condition (i.e., (i.e., with with the the switch switch SSggturned turnedoff), off),the thenetwork networkimpedance impedance is: is:
Zeq(island) Zeq(island) = Zl = Zl According to the analyses above, Inequality (1) is established: According to the analyses above, Inequality (1) is established:
|Zeq(grid) | < |Zeq(island) |
(1)
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|Zeq(grid)| < |Zeq(island)|
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In many cases, the difference between |Zeq(grid)| and |Zeq(island)| is obvious. Accordingly, a In many cases, surge the difference between |Zeq(grid) | |Zeq(island) | is obvious. Accordingly, network impedance can be used as the indicator ofand an island just formed. a network impedance surge can be used as the indicator of an island just formed. The network impedance is calculated by means of the irregular voltages and currents extracted The network impedance is calculated by means of the irregular voltages and currents from the measured voltages and currents at the terminal of the DG units. The types of theextracted irregular from the measured voltages and currents at the terminal of the DG units. The types of thesequence, irregular voltages and currents can be of a high frequency positive sequence, low frequency positive voltages and currents can be of a highfundamental frequency positive sequence, lowsequence frequencyand positive sequence, high frequency negative sequence, frequency negative low frequency high frequency negative sequence, fundamental frequency low frequency negative sequence. Such voltages and currents inherent in thenegative grid aresequence small andand uncontrollable. To negative sequence. Such voltages and currents inherent in the grid are small and uncontrollable. measure the network impedance accurately, such voltages or currents are injected by the DG units. To the network impedance accurately,are such voltages or currents injected from by thewhich DG units. By measure comparison, irregular current injections more common in the are literature, the By comparison, irregular current injections are more common in the literature, from which the irregular irregular current injection islanding detection method derives. current islanding detectionmethod methodinvolves derives. injecting irregular currents into a network, and Theinjection irregular current injection irregular injection method involves irregular currents intothe a network, then The the changes of current the resultant irregular voltage at the injecting terminal of a DG unit can reflect network and then the changes of the resultant irregular voltage at the terminal of a DG unit can the impedance changes. Consequently, considering the relationship shown in Inequality (1), reflect there will network impedance changes. Consequently, considering the relationship shown in Inequality (1), be a surge of the irregular voltage (magnitude) when an island event occurs. From this, the changes there will quantities, be a surge ofincluding the irregular (magnitude) when an island eventdistortion occurs. From this,and the of some the voltage irregular voltages, the total harmonic (THD) changes of some quantities, including the irregular voltages, the total harmonic distortion (THD) and unbalance factor of the terminal voltages of DG units, the network impedance, and the correlation unbalance factor of thecurrents terminaland voltages of DG units, the network and thechanges, correlation between the irregular voltages, which in fact reflect theimpedance, irregular voltage can between the irregular currents and voltages, which in fact reflect the irregular voltage changes, can be be used as the judgment basis of island events. used The as the judgment basis of island events. above analysis indicates that for an irregular current injection method the criterion of an The above analysis indicates that for an irregular injection method the criterion of an island event is essentially the irregular voltage surge.current Therefore, any event causing an irregular island event is essentially the irregular voltage surge. Therefore, any event causing an irregular voltage voltage surge in a grid-connected condition may result in a nuisance tripping. surge in a grid-connected condition may result in a nuisance tripping. (b) DG DG Unit Unit Cut-in: Cut-in: Figure Figure 22 shows shows aa scenario scenario where where there there are are nn running running DG DG units units and and all all the the (b) presented currents/voltages are the same type of irregular currents/voltages. When DG n+1 unit is cutpresented currents/voltages are the same type of irregular currents/voltages. When DGn+1 unit •
•
•
•
I DG(n +1)I DG I agg , which in,cut-in, the irregular currentcurrent will accumulated to the former current current is the irregular be accumulated to the aggregate former aggregate I agg , (n+be 1) will will thereby cause an increase of the irregular voltagevoltage Uir. TheUincrease of Uir isof comparable to what which will thereby cause an increase of the irregular Uir is comparable ir . The increase occurs anafter island event event if IDG(n+1) is large enough. Considering that that generally therethere is no to whatafter occurs an island if IDG(n+1) is large enough. Considering generally is communication between thethe DG units, the no communication between DG units, theformer formerrunning runningDG DGunits unitscannot cannot distinguish distinguish this event from an an island island event, event, for for which which reason reason aa misjudgment misjudgment is is inevitable. inevitable. from
Figure Figure 2. 2. Irregular Irregular currents currents and and irregular irregular voltages. voltages.
(c) As shown shown in in Figure Figure 2, 2, if if one one of of the running DG DG units units is is cut-out, cut-out, as as above, (c) DG DG Unit Unit Cut-out: Cut-out: As the running above, IIagg and U will decrease, which may not affect the islanding detection. agg and Uirirwill decrease, which may not affect the islanding detection. (d) Load Cut-in/Cut-out: For a strong grid, the grid impedance is much smaller than the load (d) Load Cut-in/Cut-out: For a strong has grid, theinfluence grid impedance is much impedance, smaller thanfor the load impedance, and thus a load cut-in/cut-out little on the network which impedance, and thus a loadand cut-in/cut-out has littletoinfluence on based the network impedance, for which reason Uir has little change there is no response it. A study on a weak grid is much more reason U ir has little change and there is no response to it. A study based on a weak grid is much more complex, which is not the interest of this paper and will not be analyzed. complex, which is not the interest of this paper and will not be analyzed. In conclusion, DG unit cut-in events may affect the irregular current injection methods, especially when large irregular currents are injected by the DG unit cut-in, by which a misjudgment is very likely to happen. On account of this, this paper will study this problem and propose a new method based on an improved impedance measurement scheme.
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This paper is organized as follows: Section 2 reveals the disadvantage of general impedance measurement scheme; Section 3 proposes a new islanding detection method; the simulations in Section 4 and experiments in Section 5 verify the performance of the proposed method; and finally, conclusions are drawn. 2. Impedance Measurement By observing Figure 2 it can be found that the network impedance does not change when a DG unit is cut-in in a grid-connected condition, whereas it is just the opposite when an island event occurs. In this case, why not use the network impedance change to determine an island? Theoretically, this is entirely possible, but the DG units have been unable to detect the network impedance in multi-DG operation, which includes the situation where a new DG unit has been cut-in. The analysis is as follows. As shown in Figure 2, the DG units share the terminal voltage. Considering that |Zg | is much •
smaller than |Zl |, in a grid-connected condition, U ir can be expressed as: •
•
•
•
•
U ir = ( I DG1 + . . . + I DG i + . . . + I DG n ) Zeq(grid) + U s_ir
(2)
•
where U s_ir is the inherent harmonic voltage in the source (see Figure 2). Since Us_ir is relatively small, the following analysis will ignore it. For the DGi (0 < i< n + 1) unit, the general approach is to calculate the static impedance as the network impedance, as shown below: •
Zs =
U ir •
I DG i By substituting Equation (2) into the above equation, the initial static impedance detected by DGi unit is: • • • • I DG1 + . . . + I DG(i−1) + I DG(i+1) + . . . + I DG n Zs0 = (1 + ) Zeq(grid) (3) • I DG i •
Obviously, even though U s_ir is not considered, the static impedance is the network impedance only in single-DG operation, not in multi-DG operation where it should be larger than the network impedance and the measured value of each DG unit is different. If a new unit called DGn+1 unit is cut-in at this time, the static impedance will be: •
Zs_cutin = (1 +
•
•
•
•
I DG1 + . . . + I DG(i−1) + I DG(i+1) + . . . + I DG n + I DG(n+1) •
) Zeq(grid)
(4)
I DG i Therefore, the relationship below can be derived from Equations (3) and (4): |Zs_cutin | > |Zs0 |
(5)
On the other hand, if an island event occurs in the network shown in Figure 2, as above, the static impedance will be: •
Zs_isl = (1 +
•
•
•
I DG1 + . . . + I DG(i−1) + I DG(i+1) + . . . + I DG n •
) Zeq(island)
(6)
I DG i By combining Expressions (1), (3) and (6), the relationship below is true: |Zs_isl | > |Zs0 |
(7)
|Zs_isl| > |Zs0|
(7)
Inequalities (5) and (7) reveal that there will be an impedance surge after both a DG unit cut-in Energies 2018, 11, 2474 5 of 18 event and an island event. Consequently, existing irregular current injection methods may treat both the two events as island events for they only deem a static impedance surge as the islanding criterion. This misjudgment is exactly problem withwill thebeexisting methods. this paper will Inequalities (5) and the (7) reveal that there an impedance surge For afterthis bothreason, a DG unit cut-in event and an island event. Consequently, existing irregular current injection methods may treat both propose an improved impedance measurement scheme to distinguish between these two events. the two events as island events for they only deem a static impedance surge as the islanding criterion. This misjudgment is exactly the problem with the existing methods. For this reason, this paper will 3. New Irregular Current Injection Islanding Detection Method propose an improved impedance measurement scheme to distinguish between these two events.
3.1. Improved Impedance Measurement Scheme 3. New Irregular Current Injection Islanding Detection Method Section 2 has pointed that Uir Scheme will increase after both a DG unit cut-in event and an island 3.1. Improved Impedanceout Measurement event. If the Section DG units the output irregular currents by a and certain percentage, 2 hasimmediately pointed out thatincrease Uir will increase after both a DG unit cut-in event an island • event. If the DG units immediately increase the output irregular currents by a certain percentage,
noted as k I DG i , •when Uir goes into a steady state, Uir will increase again. The whole process is shown as k I DG i , when Uir goes into a steady state, Uir will increase again. The whole process is shown in Figurenoted 3, where Uir starts to increase at t1 until it stabilizes at Uir1 at t2; then the DG units increase in Figure 3, where Uir starts to increase at t1 until it stabilizes at Uir1 at t2 ; then the DG units increase the irregular currents at t3, atwhich is followed increase of U ir until Uir stabilizes at Uir2 at t4. the irregular currents t , which is followedby by aa new new increase of U until U stabilizes at U at t . 3
ir
ir
ir2
4
Figure 3. Changes of U in the improved impedance measurement scheme.
Figure 3. Changes of Uir inir the improved impedance measurement scheme. In this way, an improved impedance measurement scheme using dynamic impedance will be
In this way, an below. improved impedance measurement scheme demonstrated A dynamic impedance Zd can be expressed as: using dynamic impedance will be demonstrated below. A dynamic impedance Zd can be• expressed as: • Zd =•
Zd =
U ir2 − U ir1 •
•
(8)
U ir2k−I DG Uiir1
(8)
•
3.1.1. A DG Unit Cut-In Event in a Grid-Connected Condition
k I DG i
According to Figures 2 and 3, it is supposed that the DGn+1 unit is cut-in at t1 , and then the equations below are true:
3.1.1. A DG Unit Cut-In Event in a Grid-Connected Condition •
•
•
•
•
U ir1 = ( I DG1 + . . . + I DG i + . . . + I DG n + I DG(n+1) ) Zeq(grid) (9) According to Figures 2 and 3, it is supposed that the DGn+1 unit is cut-in at t1, and then the equations below are true: •
•
•
•
•
U ir2 = ((1 + k) I DG1 + . . . + (1 + k) I DG i + . . . + (1 + k) I DG n + I DG(n+1) ) Zeq(grid) •
•
•
•
•
Accordingly,U byir1referencing can obtain a dynamic impedance: = ( I DG1 +Equation ... + I DG(8)i +the...DG + iIunit DG n + I DG(n +1) ) Z
(9)
eq(grid)
•
•
U ir2 =
•
•
•
•
I DG1 + . . . + I•DG(i−1) + I DG(i+1) + . .•. + I DG n• (1 + Zeq(grid) ((1Z+d_cutin k ) I=DG1 + ... + (1 + k ) I DG i +•I ... + (1 + k ) I DG n + I)DG( n +1) ) Z eq(grid) DG i
By comparing the above equation with Equation (3), we have:
Accordingly, by referencing Equation (8) the DGi unit can obtain a dynamic impedance: |Zd_cutin | = |Zs0 |
(10)
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3.1.2. An Island Event As above, it is supposed that an island event occurs at t1 , and the following equations are established: • • • • U ir1 = ( I DG1 + . . . + I DG i + . . . + I DG n ) Zeq(island) (11) •
•
•
•
U ir2 = ((1 + k) I DG1 + . . . + (1 + k) I DG i + . . . + (1 + k) I DG n ) Zeq(island) The dynamic impedance is: •
Zd_isl = (1 +
•
•
•
I DG1 + . . . + I DG(i−1) + I DG(i+1) + . . . + I DG n •
) Zeq(island)
I DG i In the same way, by comparing the above equation with Equations (3) and (6), there is: |Zd_isl | = |Zs_isl | > |Zs0 |
(12)
On the basis of the analyses above, it is found that although the improved impedance measurement scheme still cannot detect the network impedance, the relationship between the initial static impedance and the dynamic impedance, i.e., Expressions (10) and (12), can be used to distinguish island events from DG unit cut-in events. The proposed method in this paper exploits this advantage and can thereby solve the mentioned problem with existing irregular current injection methods. In other words, for a DG unit, an island can be determined only when the relationship below is established. |Zd | > |Zs0 | Additionally, in reality, a DG unit does not have to wait until the Uir becomes stable to calculate the |Zd |. When the command of increasing the irregular current is issued, i.e., t3 in Figure 3, it can •
begin to continuously calculate according to Equation (8), where the U ir2 is replaced by the latest •
measured U ir . Although the resulting |Zd | may be inaccurate, it can be used for judgment like the true value. More importantly, the islanding detection will be faster by following this mode. 3.2. Irregular Current and Signal Extraction Generally, to mitigate the adverse influence of some other factors on islanding detection, the type of the injected irregular current should be different from that in the network as much as possible. Thus, this paper injects negative sequence second harmonic currents into a three-phase network. To minimize the damage to power quality, comply with the relevant grid codes and take into account the control accuracy of a DG unit, initially second harmonic currents that are equivalent to 0.5% of the rated current of the DG unit are injected, and when calculating the dynamic impedance, k is set to 0.6 to make the increased second harmonic currents remain within the 1% limit [30]. Meanwhile, a strict restriction on the injected irregular currents can bring about a more transient current/voltage transition, i.e., smaller t2 −t1 and t4 −t3 in Figure 3, which is conducive to the rapidity of an islanding detection. To calculate the impedance, the magnitude of the negative sequence second harmonic voltages is needed, which is extracted by the algorithm shown in Figure 4a. According to this algorithm, first, the terminal voltages of a DG unit are converted from the three-phase stationary reference frame to the two-phase double synchronous frequency reverse rotating (i.e., clockwise) reference frame, i.e., ABC to dq(−2ω) reference frame, which is shown in Figure 4b and can be specifically expressed as: ud uq
!
=
2 3
cos(−2ωt) − sin(−2ωt)
cos(−2ωt − 2π/3) − sin(−2ωt − 2π/3)
cos(−2ωt + 2π/3) − sin(−2ωt + 2π/3)
!
ua ub uc
specifically expressed as:
ua ud 2 cos(−2t ) cos(−2t − 2π / 3) cos(−2t + 2π / 3) u = uq 3 − sin(−2t ) − sin(−2t − 2π / 3) − sin(−2t + 2π / 3) b Energies 2018, 11, 2474 uc 7 of 18 where ω is the fundamental frequency. where ω is the fundamental frequency.
(a)
(b)
4. Negative sequence second harmonic voltages extraction. (a) Signal (b) to ABC to FigureFigure 4. Negative sequence second harmonic voltages signalsignal extraction. (a) Signal flow; flow; (b) ABC dqreference (−2ω) reference frame transformation. dq(−2ω) frame transformation.
ThenThen the negative sequence second harmonic component will will be present as constants in thein the the negative sequence second harmonic component be present as constants rotating reference frameframe whilewhile the other components will be present as fluctuation quantities. Next,Next, rotating reference the other components will be present as fluctuation quantities. two low pass filters (LPFs) are used to filter out the fluctuation quantities; and finally, the magnitude of two low pass filters (LPFs) are used to filter out the fluctuation quantities; and finally, the magnitude the negative sequencesequence second harmonic voltages is obtained synthesizing the outputsthe of the filters. of the negative second harmonic voltages is by obtained by synthesizing outputs of the Additionally, in order not toinaggravate the to computational burden on the system, the magnitude of the the filters. Additionally, order not aggravate the computational burden on the system, negative sequence harmonic currents can harmonic be represented by the value rather magnitude of second the negative sequence second currents can reference be represented by the than reference extracted thethan actual currents. valuefrom rather extracted from the actual currents. 3.3. Assisting OtherOther Equipment to Detect an Island 3.3. Assisting Equipment to Detect an Island The equipment that can an island passively usually observes whether the harmonic The equipment thatonly can detect only detect an island passively usually observes whether the harmonic voltage or the frequency is out of their limits. To assist such equipment to detect an island, the DG unit voltage or the frequency is out of their limits. To assist such equipment to detect an island, the DG mentioned in this paper can reduce the restriction on I to boost the U , i.e., the harmonic voltage, DG unit mentioned in this paper can reduce the restriction on IDG toirboost the Uir, i.e., the harmonic once voltage, an islandonce is determined. can be set Ias an island isSpecifically, determined.IDG Specifically, DG follows: can be set as follows: IDGIDG = =5% 5% I1I,1,ififUUirir |Zs0 |, DG unit cut-in events will not be misjudged. Due to the lack of knowledge of Zeq(island) , no value for Zth can ensure that island events are not misjudged; however, a smaller |Zth | can reduce the probability of such misjudgment. To conclude, Zth can be set as: |Zth | = k2 |Zs0 | (16) where k2 > 1. To reduce the probability of misjudgment while taking into account the probable fluctuation of Zd that results from the fluctuation of Uir , a compromised k2 is set to 2 in this paper. 3.4.2. An Event Followed by a Large Increase of Uir Parts or all of the DG units employing the existing methods will judge this situation as an island event and then shut down themselves. The curve of Uir will be still similar to that in Figure 3 while Uir1 and Uir2 are different from that in the scenario presented in Section 3.4.1. With regard to the expressions of Uir1 and Uir2 , they can be obtained by removing the irregular currents corresponding to the DG units having been shut down from Equations (9), (11) and (13). However, for the dynamic impedance, Expressions (14) and (15) are still true. Thus, Zth and k2 can be set to Equation (16) and 2 as well. 3.5. Selection of k In Section 3.2, k has been set to 0.6. However, it can also be set to other values. As the proposal in Section 3.2, as a compromise, the initial irregular current is set to 0.5% of the rated current of a DG unit. To make the increased irregular current remain within the 1% limit, k should satisfy the relationship 0 < k < 1.
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In this case, k for each DG unit may be different. If so, Equation (14) must be rewritten as follows, where kDGi denotes the parameter k of DGi unit: kDG(i −1) • kDG(i +1) • kDG p • kDG1 • kDG i I DG1 + ... + kDG i I DG(i −1) + kDG i I DG(i +1) + ... + kDG i I DG p • I DG i • • kDG(i −1) kDG(i +1) • kDG p • kDG1 I + ... + I + DG(i −1) kDG i DG1 kDG i kDG i I DG(i +1) + ... + kDG i I DG p • I DG i
Zd_cutin = (1 + Zd_isl = (1 +
) Zeq(grid) (17)
) Zeq(island)
As mentioned in Section 3.4.1, DG unit cut-in events can be prevented from being misjudged. To maintain this good feature, the relationship below should be met: |Zd_cutin | < |Zth | = k2 |Zs0 | By combining Equations (3), (17) and the above inequality, the inequality below can be derived, where kDGj denotes the parameter k of DGj unit and 1 ≤ j ≤ p: kDGj /kDGi < k2 Considering 0 < k < 1, the following relationship can ensure that the above inequality is true: kDGi > 1/k2 Actually, unit DGi represents any DG unit. To be more general, the above relationship can be rewritten as follows. If the parameter k of a DG unit is set as follows, the reliability of the proposed method will increase. In this paper, k2 is set to 2 and k is set to 0.6, which meets the relationship below: k > 1/k2 3.6. Non-Detection Zone (NDZ) of the Proposed Method According to the analyses in the preceding subsections, an island event cannot be detected if there is: |Zd_isl | < |Zth | By combining Equations (3), (16), (17) and the above inequality, we have the relationship below: rI < k2 rZ
(18)
where rI and rZ are defined as follows and 0 < rZ
