# A Protection Method Based on Feature Cosine and Differential Scheme for Microgrid

Authors: Lai Lei; Cong Wang; Jie Gao; Jinjin Zhao; Xiaowei Wang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The fault current level of microgrid is different between islanded mode and grid connected mode. This situation degrades the performance of traditional overcurrent protection schemes. Hence, this paper proposes a protection method based on feature cosine and differential scheme. Firstly, feature cosine is proposed; it employs ellipse equation and minimum least squares to quantify the united behavior about voltage and current. Secondly, fault current direction and feature cosine are analyzed when fault occurs at different locations of a typical microgrid, and then the difference of feature cosine between faulty and healthy section locations is obtained. Thirdly, based on feature cosine and differential scheme, the differential direction is defined and utilized to detect faulty section location. Lastly, various time domain simulation case studies, including different microgrid operation modes, grounding resistances, faulty types, faulty section locations, and noise influence, are conducted and demonstrate that the proposed protection has high accuracy.


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## Body

## 1. Introduction

Due to climate concern, technological advancements, and government incentives, microgrids are emerging as a low-medium voltage energy network which include various types of distributed generation (DG) such as solar, wind, and geothermal [1, 2]. Some advantages of the presence of microgrids include reduction of power delivery loss, reduced greenhouse gases, and enhancement of power reliability.However, the emergence of microgrids involves a lot of technical challenges, e.g., voltage and frequency control, stability problems, power quality, and so on, because microgrid operation, control, and protection are typically different from those of traditional power networks [3–5]. One of the major issues in microgrid is the microgrid protection. Different reasons responsible for this are as follows: (i) power flow may become bidirectional when DGs are integrated into the microgrid; (ii) fault current levels may be limited to 2~3 times the inverter-rated current; (iii) the microgrid may connect or disconnect DGs; (iv) the microgrid has two operation modes including islanded and grid connected modes; (v) zero and negative sequence currents cannot be neglected in normal conditions since single- and three-phase unbalanced loads are embedded into microgrids.At present, scholars have proposed various methods to protect microgrids. Reference [6] proposes a three-stage approach to microgrid protection including offline analyses, online calculation, and real-time protection, which is based on establishing dynamic security model in microgrid. Reference [7] employs hysteresis control and maximum current RMS control to limit the fault current; it can make fault current have a percentage of fifth harmonic which is utilized to build protection strategy, but this method is only suitable for islanded microgrid. Reference [8] investigates protection issues related to LV microgrids and elaborates LV microgrid protection concepts comprehensively. Voltage and frequency relays are used as a point of common coupling relay, and directional overcurrent (DOC) relay is utilized for feeder protection. DOC takes a long time to operate. Besides, with its operation, all of the feeder elements should be tripped. In [9, 10], sequence components, e.g., positive and negative sequence components, are employed to design the protection, but it can make relay coordination become more complex because of multiple settings. Reference [11] has proposed a novel fault location method based on the maximum oscillation magnitude of the transient voltage signal, but this method does not consider the three-phase microgrid. The d-q wavelet packet transform-based digital protection has been implemented to investigate several transient disturbances occurring in different parts of the microgrid system [12]; it suffers a problem about choosing suitable wavelet basis functions. Two time-frequency-based differential schemes, which can avoid the selection of wavelet basis functions, are presented for microgrid protection; one is the Hilbert-Huang transform [13], and the other is the generalized S-transform [14]. In these two works, how to find effective threshold is necessary. This problem can be avoided by some statistical classifiers; for instance, [15, 16] have used the three-phase voltage amplitude, voltage total harmonic distortions, and other 19 electrical features to build classifiers including Decision Tree and Naive Bayes, to detect fault and achieve protection; [17] has utilized combined wavelet transform and decision tree to detect and classify the fault; however, the protection strategies in [15, 16] are only designed for islanded microgrid, and the unbalanced loads situation is not considered in [17]. Reference [18] presents a data-mining-based intelligent differential protection scheme for the microgrid, but it uses 7 feature components, for example, the differential rate-of-change of frequency, the differential reactive power change with time and so on, but the shortcoming of this method is that it has much data needed to solve including many feature components and high sampling rates. This problem can also be found in [15–17].Thus, compared with other differential methods, the contributions of this paper are as follows:(1) This paper extracts the fault feature through solving the parameters of ellipse equation based on the current and voltage fault component. It can only use 5 sampling points to determine the parameters of ellipse which are employed to acquire polarity relationship between the busbar voltage and current.(2) As for detection criteria, this paper uses ±1 to define the feature direction of busbar. It can decrease the difficulty of communication and avoid threshold selection for different operation modes of microgrid.The rest of this paper is organized as follows. Section2 employs ellipse equation and minimum least squares to acquire feature cosine to describe the united behavior about voltage and current. The microgrid structure and the reason why the feature cosine can be used are shown in Section 3. Section 4 discusses the change of feature cosine when fault occurs at different locations. Section 5 presents how to use the feature cosine to obtain the differential feature direction to detect faulty section location. Section 6 shows the performance of proposed method including different microgrid operation modes, grounding resistances, and faulty section locations. Moreover, the condition of unbalanced loads is taken into consideration.


## 2. Feature Cosine of One Phase in a Bus

### 2.1. Mathematical Background

The voltage and current signals in one phase of a bus at timet can be written as(1)ut=U0sin⁡ωt(2)it=I0sin⁡ωt-ϕwhere the angleϕ is the delay betweeni(t) andu(t).U0 andI0 are the peak values ofu(t) andi(t), respectively.ω is the angular velocity,ω=2πf. In China system,f=50 Hz. cos⁡ϕ is defined as the feature cosine.As for a bus, Figure1 shows the united behavior of the voltage and current sine signals for one phase at different conditions. This behavior can be modeled using a conic section mathematical equation or, more specifically, the equation of the ellipse.Figure 1
Elliptical behavior of a phase in a bus for different situations. (a) Elliptical behavior of a phase in a bus for different values ofϕ and (b) elliptical behavior of a phase in a bus for different values ofI0.
(a) (b)By applying (5), the Cartesian form for the equation of the bus ellipse is defined as (4) [19].(3)sin2ωt+cos2ωt=1(4)x2t-2cos⁡ϕxtyt+y2t-sin2ϕ=0(5)xt=utU0=sin⁡ωt(6)yt=itI0=sin⁡ωt-ϕThe equation of the ellipse, for any value ofU0 andI0, is given by (7)I02u2t-2U0I0cos⁡ϕutit+U02i2t-U02I02sin2ϕ=0Hence, as for a bus, using the geometric properties of (7) can monitor the behavior of voltage and current signals for each phase and identify the moment when potential faults have occurred; besides, this paper proposes that the parameters of the ellipse can be seen as an index to indicate the faulty section location, which is elaborated in the following sections.


### 2.2. Feature Cosine

According to [20], an ellipse section defined in R2 represents a set of points whose coordinates satisfy the general equation (8), the form of (8) is equivalent to (7), and (8) can be seen as a fitting model to obtain the parameters of (7).(8)ax2+bxy+cy2+dx+ey+f=0with the parametersa orb orc ≠ 0.Then, (7) and (8) can both be rewritten in matrix form as(9)fx=αx+1(10)a=abcdex=x1x2x3x4x5T=x2xyy2xyT=u2tutiti2tutitTBased on the minimum least squares, a cost functionJa can be written as (11)Jaa,⋯,e=12∑i=1nfxi-02Hence, the parameters estimation problem is transformed into a simpler problem of finding a local minimizer forJa; this problem can be solved by Levenberg-Marquardt algorithm, and it is described in [16, 17].Lastly, the parameters of equation (7) can be obtained from Levenberg-Marquardt algorithm.(12)a=a∗=a∗b∗c∗d∗e∗And the feature cosinecos⁡ϕ can be shown in (13), cos⁡ϕ can represent the phase relation between the voltage and current, and it is mentioned that cos⁡ϕ for each phase is a constant even in the condition of unbalanced loads if a microgrid operates normally. cos⁡ϕ is the index mentioned in Section 2.1. Section 3 will illustrate why cos⁡ϕ can be employed to detect microgrid faulty location.(13)cosϕ=-b∗2a∗·c∗


## 2.1. Mathematical Background

The voltage and current signals in one phase of a bus at timet can be written as(1)ut=U0sin⁡ωt(2)it=I0sin⁡ωt-ϕwhere the angleϕ is the delay betweeni(t) andu(t).U0 andI0 are the peak values ofu(t) andi(t), respectively.ω is the angular velocity,ω=2πf. In China system,f=50 Hz. cos⁡ϕ is defined as the feature cosine.As for a bus, Figure1 shows the united behavior of the voltage and current sine signals for one phase at different conditions. This behavior can be modeled using a conic section mathematical equation or, more specifically, the equation of the ellipse.Figure 1
Elliptical behavior of a phase in a bus for different situations. (a) Elliptical behavior of a phase in a bus for different values ofϕ and (b) elliptical behavior of a phase in a bus for different values ofI0.
(a) (b)By applying (5), the Cartesian form for the equation of the bus ellipse is defined as (4) [19].(3)sin2ωt+cos2ωt=1(4)x2t-2cos⁡ϕxtyt+y2t-sin2ϕ=0(5)xt=utU0=sin⁡ωt(6)yt=itI0=sin⁡ωt-ϕThe equation of the ellipse, for any value ofU0 andI0, is given by (7)I02u2t-2U0I0cos⁡ϕutit+U02i2t-U02I02sin2ϕ=0Hence, as for a bus, using the geometric properties of (7) can monitor the behavior of voltage and current signals for each phase and identify the moment when potential faults have occurred; besides, this paper proposes that the parameters of the ellipse can be seen as an index to indicate the faulty section location, which is elaborated in the following sections.


## 2.2. Feature Cosine

According to [20], an ellipse section defined in R2 represents a set of points whose coordinates satisfy the general equation (8), the form of (8) is equivalent to (7), and (8) can be seen as a fitting model to obtain the parameters of (7).(8)ax2+bxy+cy2+dx+ey+f=0with the parametersa orb orc ≠ 0.Then, (7) and (8) can both be rewritten in matrix form as(9)fx=αx+1(10)a=abcdex=x1x2x3x4x5T=x2xyy2xyT=u2tutiti2tutitTBased on the minimum least squares, a cost functionJa can be written as (11)Jaa,⋯,e=12∑i=1nfxi-02Hence, the parameters estimation problem is transformed into a simpler problem of finding a local minimizer forJa; this problem can be solved by Levenberg-Marquardt algorithm, and it is described in [16, 17].Lastly, the parameters of equation (7) can be obtained from Levenberg-Marquardt algorithm.(12)a=a∗=a∗b∗c∗d∗e∗And the feature cosinecos⁡ϕ can be shown in (13), cos⁡ϕ can represent the phase relation between the voltage and current, and it is mentioned that cos⁡ϕ for each phase is a constant even in the condition of unbalanced loads if a microgrid operates normally. cos⁡ϕ is the index mentioned in Section 2.1. Section 3 will illustrate why cos⁡ϕ can be employed to detect microgrid faulty location.(13)cosϕ=-b∗2a∗·c∗


## 3. Feature Cosine Utilization Reason

### 3.1. Microgrid Topology

A modified microgrid based on the topology of CERTS microgrid model [21] is shown in Figure 2(a) and established in Matlab/Simulink. This microgrid is a part of the power distribution system, and it is connected to the utility grid using a transformer 0.4/10kV. Each distributed generation (DG) is controlled by improved droop controller [22, 23]. Table 1 shows each load capacity. The system parameters are given: the power electronic device is the IGBT; the capacities of DGs are all 80kVA; switch frequency is 6kHz; the DC voltage is 800V; Line: resistance and inductance are 0.642 Ω/km and 0.083H/km; the length of Line 1, Line 2, and Line 3 are 0.3km, 0.2km, and 0.5km, respectively.Table 1
The capacity of loads.
Load Name Active power in kW Reactive power in kVar L-1 45 15 L-2 45 22.5 L-3 35 10 L-4 45 15Figure 2
Single line diagram and fault current of microgrid. (a) Single line diagram; (b) fault current by DG1 at LLLG; (c) fault current by DG2 at LG.
(a) (b) (c)


### 3.2. Reason

The fault current-limiting strategy is applied for each DG according to [24, 25]. For example, when the microgrid occurs, three-line-ground (LLLG) fault or single-line-ground (LG) fault at 0.2s, whose faulty location is Line 2 of Figures 2(a), 2(b), and 2(c), shows that the fault current-limiting strategy, after 0.21s, can effectively limit each DG fault current into 2 times of the rated current; besides, this strategy can restrain the harmonics and make the frequency of output current and voltage equal to 50Hz. This is the reason why this paper can use the feature cosine to detect the faulty section.


## 3.1. Microgrid Topology

A modified microgrid based on the topology of CERTS microgrid model [21] is shown in Figure 2(a) and established in Matlab/Simulink. This microgrid is a part of the power distribution system, and it is connected to the utility grid using a transformer 0.4/10kV. Each distributed generation (DG) is controlled by improved droop controller [22, 23]. Table 1 shows each load capacity. The system parameters are given: the power electronic device is the IGBT; the capacities of DGs are all 80kVA; switch frequency is 6kHz; the DC voltage is 800V; Line: resistance and inductance are 0.642 Ω/km and 0.083H/km; the length of Line 1, Line 2, and Line 3 are 0.3km, 0.2km, and 0.5km, respectively.Table 1
The capacity of loads.
Load Name Active power in kW Reactive power in kVar L-1 45 15 L-2 45 22.5 L-3 35 10 L-4 45 15Figure 2
Single line diagram and fault current of microgrid. (a) Single line diagram; (b) fault current by DG1 at LLLG; (c) fault current by DG2 at LG.
(a) (b) (c)


## 3.2. Reason

The fault current-limiting strategy is applied for each DG according to [24, 25]. For example, when the microgrid occurs, three-line-ground (LLLG) fault or single-line-ground (LG) fault at 0.2s, whose faulty location is Line 2 of Figures 2(a), 2(b), and 2(c), shows that the fault current-limiting strategy, after 0.21s, can effectively limit each DG fault current into 2 times of the rated current; besides, this strategy can restrain the harmonics and make the frequency of output current and voltage equal to 50Hz. This is the reason why this paper can use the feature cosine to detect the faulty section.


## 4. The Analysis of Microgrid Fault Feature

### 4.1. Fault at Line 1

Based on the analysis method from [26], this paper focuses on the current direction. When short-circuit fault occurs at Line 1 of Figure 2(a), B1, B2, B3, B4, and B5 are bus. Taking B1 and B2 as an example, Figure 3 shows the current direction of B1 and B2, which is suitable for the islanded mode and grid connected mode.Figure 3
The current direction of B1 and B2 when fault occurs at Line 1. (a) DGs supply power for downstream load; (b) DGs supply power for upstream load.
(a) (b)When DGs supply power for downstream load shown in Figure3(a), phase ϕ1 and ϕ2 are shown in (14); this is because loads in microgrid are generally inductive loads.(14)ϕ1=arg⁡U˙1-arg⁡I˙1>0∘ϕ2=arg⁡U˙2-arg⁡I˙2>0∘whereU˙1 and U˙2 are the voltage of B1 and B2. I˙1 and I˙2 are the current of B1 and B2. arg⁡(U˙1) means the voltage phase of B1. arg⁡(I˙1) means the current phase of B1.After fault, because the grounding impedance of fault point is obviously less than the load impedance [22], as for current direction, B1 and B2 are both from bus to fault point. Moreover, the voltage phases of B1 and B2 are not changed. Based on this, (15) and (16) can be expressed and present that cos⁡(ϕ1f) is positive and cos⁡(ϕ2f) is negative when the fault occurs at Line 1.(15)ϕ1f=arg⁡U˙1f-arg⁡I˙1f≈arg⁡U˙1-arg⁡I˙1>0∘ϕ2f=arg⁡U˙2f-arg⁡I˙2f≈arg⁡U˙2-arg⁡I˙2-180∘<-90∘(16)cos⁡ϕ1f≈cos⁡arg⁡U˙1-arg⁡I˙1>0cos⁡ϕ2f≈cos⁡arg⁡U˙2-arg⁡I˙2-180∘<0where ϕ1f and (ϕ2f) are the postfault phase of B1 and B2.When the DGs supply power for upstream load, the current directions at B1 and B2 are shown in Figure 3(b). Through analyzing, when the fault occurs at Line 1, cos⁡(ϕ1f) is positive and cos⁡ϕ2f is negative.


### 4.2. Fault at Line 2

When the fault occurs at Line 2 of Figure2(a), the current direction of B1 and B2 is shown in Figure 4.Figure 4
The current direction of B1 and B2 when fault occurs at Line 2. (a) DGs supply power for downstream load; (b) DGs supply power for upstream load.
(a) (b)When the DGs supply power for downstream loads shown in Figure4(a)and the fault occurs at Line 2, the current directions, the current directions of B1 and B2 are both from bus to fault point because the load impedance is bigger than the grounding impedance [22]. cos⁡(ϕ1f) and cos⁡ϕ2f can be shown in (17)cos⁡ϕ1f≈cos⁡arg⁡U˙1-arg⁡I˙1>0cos⁡ϕ2f≈cos⁡arg⁡U˙2-arg⁡I˙2>0Similarly, if the DGs supply power for downstream load shown in Figure4(b), cos⁡ϕ1f and cos⁡ϕ2f are both positive when fault occurs at Line 2.


### 4.3. Fault at Line 3

The current direction of B1 and B2 is shown in Figure 5 when the fault occurs at Line 3 of Figure 2(a). There are two situations as follows because the distance is relatively far from fault point to B2.Figure 5
The current direction of B1 and B2 when fault occurs at Line 3. (a) DGs supply power for downstream load when the effect is serious; (b) DGs supply power for downstream load when the effect is mild; (c) DGs supply power for upstream load.
(a) (b) (c)(I) The grounding impedance is relatively small; that is, the effect of fault point on B2 is serious. And the current direction of B1 and B2 will be both from bus to fault point; these directions are opposite to that in prefault and are shown in Figure 5(a). (18) shows their feature cosine; they are both negative.(18)cos⁡ϕ1f≈cos⁡arg⁡U˙1-arg⁡I˙1-180∘<0cos⁡ϕ2f≈cos⁡arg⁡U˙2-arg⁡I˙2-180∘<0(II) When the effect of fault point on B2 is mild, the current direction of B1 and B2 can be maintained; these are same as that in prefault shown in Figure 5(b). The cos⁡ϕ1f and cos⁡ϕ2f are both positive.Similarly, if the DGs supply power for upstream load shown in Figure5(c), there are also two situations according to the effect of fault point on B2. And cos⁡ϕ1f and cos⁡ϕ2f will be both negative in these two situations.In summary, no matter how the initial current direction changes in bus,cos⁡ϕ1f is positive and cos⁡ϕ2f is negative when the fault occurs at the section between B1 and B2. Moreover, there are two situations when the fault does not occur at the area between B1 and B2: (1) cos⁡ϕ1f and cos⁡ϕ2f are both positive and (2) cos⁡ϕ1f and cos⁡ϕ2f are both negative. These can be clearly shown in Table 2 where “+” is positive, “–” is negative. The difference of feature cosine can be utilized to detect the faulty section in the next section.Table 2
The direction of feature cosine of B1 and B2.
Faulty part B1 B2 Line1 + –  Line2 + + – –  Line3 + + – –


## 4.1. Fault at Line 1

Based on the analysis method from [26], this paper focuses on the current direction. When short-circuit fault occurs at Line 1 of Figure 2(a), B1, B2, B3, B4, and B5 are bus. Taking B1 and B2 as an example, Figure 3 shows the current direction of B1 and B2, which is suitable for the islanded mode and grid connected mode.Figure 3
The current direction of B1 and B2 when fault occurs at Line 1. (a) DGs supply power for downstream load; (b) DGs supply power for upstream load.
(a) (b)When DGs supply power for downstream load shown in Figure3(a), phase ϕ1 and ϕ2 are shown in (14); this is because loads in microgrid are generally inductive loads.(14)ϕ1=arg⁡U˙1-arg⁡I˙1>0∘ϕ2=arg⁡U˙2-arg⁡I˙2>0∘whereU˙1 and U˙2 are the voltage of B1 and B2. I˙1 and I˙2 are the current of B1 and B2. arg⁡(U˙1) means the voltage phase of B1. arg⁡(I˙1) means the current phase of B1.After fault, because the grounding impedance of fault point is obviously less than the load impedance [22], as for current direction, B1 and B2 are both from bus to fault point. Moreover, the voltage phases of B1 and B2 are not changed. Based on this, (15) and (16) can be expressed and present that cos⁡(ϕ1f) is positive and cos⁡(ϕ2f) is negative when the fault occurs at Line 1.(15)ϕ1f=arg⁡U˙1f-arg⁡I˙1f≈arg⁡U˙1-arg⁡I˙1>0∘ϕ2f=arg⁡U˙2f-arg⁡I˙2f≈arg⁡U˙2-arg⁡I˙2-180∘<-90∘(16)cos⁡ϕ1f≈cos⁡arg⁡U˙1-arg⁡I˙1>0cos⁡ϕ2f≈cos⁡arg⁡U˙2-arg⁡I˙2-180∘<0where ϕ1f and (ϕ2f) are the postfault phase of B1 and B2.When the DGs supply power for upstream load, the current directions at B1 and B2 are shown in Figure 3(b). Through analyzing, when the fault occurs at Line 1, cos⁡(ϕ1f) is positive and cos⁡ϕ2f is negative.


## 4.2. Fault at Line 2

When the fault occurs at Line 2 of Figure2(a), the current direction of B1 and B2 is shown in Figure 4.Figure 4
The current direction of B1 and B2 when fault occurs at Line 2. (a) DGs supply power for downstream load; (b) DGs supply power for upstream load.
(a) (b)When the DGs supply power for downstream loads shown in Figure4(a)and the fault occurs at Line 2, the current directions, the current directions of B1 and B2 are both from bus to fault point because the load impedance is bigger than the grounding impedance [22]. cos⁡(ϕ1f) and cos⁡ϕ2f can be shown in (17)cos⁡ϕ1f≈cos⁡arg⁡U˙1-arg⁡I˙1>0cos⁡ϕ2f≈cos⁡arg⁡U˙2-arg⁡I˙2>0Similarly, if the DGs supply power for downstream load shown in Figure4(b), cos⁡ϕ1f and cos⁡ϕ2f are both positive when fault occurs at Line 2.


## 4.3. Fault at Line 3

The current direction of B1 and B2 is shown in Figure 5 when the fault occurs at Line 3 of Figure 2(a). There are two situations as follows because the distance is relatively far from fault point to B2.Figure 5
The current direction of B1 and B2 when fault occurs at Line 3. (a) DGs supply power for downstream load when the effect is serious; (b) DGs supply power for downstream load when the effect is mild; (c) DGs supply power for upstream load.
(a) (b) (c)(I) The grounding impedance is relatively small; that is, the effect of fault point on B2 is serious. And the current direction of B1 and B2 will be both from bus to fault point; these directions are opposite to that in prefault and are shown in Figure 5(a). (18) shows their feature cosine; they are both negative.(18)cos⁡ϕ1f≈cos⁡arg⁡U˙1-arg⁡I˙1-180∘<0cos⁡ϕ2f≈cos⁡arg⁡U˙2-arg⁡I˙2-180∘<0(II) When the effect of fault point on B2 is mild, the current direction of B1 and B2 can be maintained; these are same as that in prefault shown in Figure 5(b). The cos⁡ϕ1f and cos⁡ϕ2f are both positive.Similarly, if the DGs supply power for upstream load shown in Figure5(c), there are also two situations according to the effect of fault point on B2. And cos⁡ϕ1f and cos⁡ϕ2f will be both negative in these two situations.In summary, no matter how the initial current direction changes in bus,cos⁡ϕ1f is positive and cos⁡ϕ2f is negative when the fault occurs at the section between B1 and B2. Moreover, there are two situations when the fault does not occur at the area between B1 and B2: (1) cos⁡ϕ1f and cos⁡ϕ2f are both positive and (2) cos⁡ϕ1f and cos⁡ϕ2f are both negative. These can be clearly shown in Table 2 where “+” is positive, “–” is negative. The difference of feature cosine can be utilized to detect the faulty section in the next section.Table 2
The direction of feature cosine of B1 and B2.
Faulty part B1 B2 Line1 + –  Line2 + + – –  Line3 + + – –


## 5. Protection Methodology

Firstly, it is mentioned that the relays at the ends of a bus are considered and not at the ends of the line; this is because it can reduce the number of relays being used; however, the circuit breakers need to be installed at the ends of the line for isolating a fault.Then, it is presented how to obtain the feature direction (D) of each bus through feature cosine.Step 1.
u n w t, un+1wt, inwt, and in+1wt are retrieved at Bn and Bn+1, unwt and inwt are the normalized voltage and current of phase-w of Bn, un+1wt and in+1wt are the normalized voltage and current of phase-w of Bn+1, w is a or b or c,n is the number of bus, and n is equal to 1, 2 and 4.Step 2.
As forBn, according to Levenberg-Marquardt algorithm, the parameters of bus ellipse equation αnw can be calculated. αnw can represent the behavior of voltage and current signals.(19)anw=anwbnwcnwdnwenwStep 3.
Similarly, as forBn+1, the parameters of line ellipse equation an+1w can be obtained.(20)an+1w=an+1wbn+1wcn+1wdn+1wen+1wStep 4.
cos ⁡ ( ϕ n w ) and cos⁡(ϕn+1w) are calculated, cos⁡(ϕnw) and cos⁡(ϕn+1w) are the feature cosine of Bn and Bn+1.(21)cos⁡ϕnw=-bnw2anwcnw(22)cos⁡ϕn+1w=-bn+1w2an+1wcn+1wStep 5.
D n w, Dn+1w, and Dcnw can be obtained by the following equations. Dnw and Dn+1w, which can show the fault current direction of bus, are the feature direction of Bn and Bn+1, and Dcnw is the differential direction.(23)Dnw=1,cos⁡ϕnw>0-1,cos⁡ϕnw<0,Dn+1w=1,cos⁡ϕn+1w>0-1,cos⁡ϕn+1w<0(24)Dcnw=Dn+1w-DnwLastly, ifDcnw<0, the section between Bn and Bn+1 is judged as faulty section; if Dcnw≧0, the section between Bn and Bn+1 is judged as healthy section. Taking B1 and B2 as an example, the procedures of this paper’s proposed method can be shown clearly in Figure 6.Figure 6
The procedures of this paper’s proposed method.


## 6. Simulation Results

In order to prove the effectiveness of the proposed method in this paper, different fault conditions are simulated through Matlab/Simulink for the microgrid shown in Figure2(a), and conditions for the fault study are explained as follows.(1) Faulty resistances (Rf): 0.01, 0.1, 0.2, 0.3, 0.5, 1, 5, 10 Ω.(2) Different faulty types (Ft): single-line-ground (LG), double-line-ground (LLG), line-line (LL), and three-line-ground (LLLG), where LG(a) means the single-line-ground whose faulty phase is phase-a.(3) Faulty section (FL): Line 1, Line 2, and Line 3.(4) Different microgrid operation modes (Om): islanded and grid connected mode.(5) Unbalanced loads: each phase capacity of load is changed.(6) Noise influence: the signal to noise ratio (SNR) is set as 50db, 20db, and 5db.


### 6.1. Different Grounding Resistances

When the microgrid shown in Figure2(a) is operated in islanded mode, LG fault is supposed to happen in Line 1 at t=0.2s; that is, faulty section is between B1 and B2 labelled B1-B2. Faulty conditions include that faulty phase is phase-a and grounding resistances are set to 0.01Ω, 0.5Ω, 0.1Ω, 2Ω, and 10Ω, respectively. Then Table 3 gives the feature cosine of each bus and judgment results after implementing this paper’s method;P is phase sequence.Table 3
The feature cosine of each bus and results with different grounding resistances.
R f P [B1, B2, B3, B4, B5] EachP  result Finalresult 0.01Ω a [0.954, -0.990, -0.982, -0.859, -0.729] B1- B2 B1-B2 b [0.998, 0.999, 0.999, -1.007, -1.009] healthy c [0.999, 0.999, 0.999, -0.997, -0.995] healthy  0.1Ω a [0.961, -0.986, -0.978, -0.904, -0.797] B1 - B2 B1-B2 b [0.998, 0.998, 0.998, -0.991, -0.987] healthy c [0.998, 0.998, 0.999, -0.992, -0.989] healthy  0.5Ω a [0.999, -1.000, -0.999, -0.997, -0.997] B1 - B2 B1 - B2 b [-0.891, -0.894, -0.896, 0.720, 0.699] healthy c [-0.958, -0.960, -0.962, 0.925, 0.919] healthy  2Ω a [0.936, -0.979, -0.960, -0.897, -0.767] B1 - B2 B1-B2 b [0.999, 0.999, 1.000, -0.994, -0.993] healthy c [1.000, 1.000, 1.001, -0.996, -0.994] healthy  10Ω a [0.927, -0.974, -0.952, -0.888, -0.752] B1 - B2 B1-B2 b [0.999, 0.999, 0.999, -0.995, -0.993] healthy c [0.999, 0.999, 1.000, -0.998, -0.997] healthyTake grounding resistance 0.5Ω as an example, Figure7 gives the voltage and current of B1 and B2, and Figure 8 shows the bus ellipses of each phase of B1 and B2. Figure 7 illustrates that the faulty phase voltage direction of B1 is same as that of B2 and the faulty phase current direction of B1 is opposite to that of B2, which is illustrated obviously in Figure 8.Figure 7
The voltage and current of B1 and B2 when fault occurs at Line 1. (a) B1 voltage; (b) B2 voltage; (c) B1 current; (d) B2 current.
(a)
B1 voltage (b)
B2 voltage (c)
B1 current (d)
B2 currentFigure 8
The bus ellipse of B1 and B2 for different phases. (a) Phase-a; (b) phase-b; (c) phase-c.Then, as for phase-a, from Table3 and this paper’s method, the feature cosines from B1 to B5 are 0.999, -1.000, -0.999, -0.997, and -0.997, respectively. So, the feature directions from B1 to B5 are 1, -1, -1, -1, and -1, respectively; that is, their differential directions are -2, 0, 0, 0, respectively. Similarly, as for phase-b and phase-c, their differential directions are all 0. Hence, the faulty section is between B1 and B2; this result is consistent with the actual fault. Lastly, from Table 3 we can find that this paper’s method can detect fault part effectively in different grounding resistances.


### 6.2. Different Faulty Types

When the microgrid is operated in islanded mode, different faulty types, including LG(b), LLG(ab), LL(bc), and LLLG, are supposed to happen in Line 1 at 0.2s with grounding resistance 0.1Ω; that is, faulty section is B1-B2. Then, Table 4 gives the feature cosine of each bus and judgment results; it shows that the proposed method cannot be influenced by different faulty types.Table 4
The feature cosine of each bus and judgment results with different faulty types.
F t P [B1, B2, B3, B4, B5] EachP  result Finalresult LG(b) a [1.003, 1.003, 1.003, -1.008, -1.012] healthy B1-B2 b [0.928, -0.915, -0.871, -1.100, -1.084] B1-B2 c [1.002, 1.002, 1.002, -0.991, -0.9867] B1-B2  LLG(ab) a [0.957, -0.946, -0.915, -0.994, -0.970] B1-B2 B1-B2 b [0.672, -0.508, -0.380, -0.829, -0.614] B1-B2 c [0.998, 0.997, 0.997, 1.004, 1.006] healthy  LL(bc) a [0.401, 0.631, 0.715, 0.113, 0.897] healthy B1-B2 b [0.953, -0.941, -0.958, -0.977, -0.900] B1-B2 c [0.956, -0.943, -0.957, -0.978, -0.909] B1-B2  LLL a [0.836, -0.636, -0.473, -0.949, -0.869] B1-B2 B1-B2 b [0.609, -0.696, -0.497, -0.852, -0.642] B1-B2 c [0.999, -0.866, -0.770, -1.040, -1.010] B1-B2


### 6.3. Different Faulty Section Locations

The islanded microgrids are simulated at 0.2s with different faulty section locations, including B1-B2, B2-B3, and B4-B5. The fault conditions include fault type LG whose faulty phase is phase-c or phase-a; grounding resistances is 0.3Ω or 10Ω. Then, Table 5 illustrates that this paper’s method can detect different fault parts. It is mentioned that the feature cosine is consist with the analysis in Section 4.3 when the grounding resistance is 10Ω and faulty section location is B4-B5.Table 5
The feature cosine of each bus and judgment results in different faulty section locations.
F t  Rf  FL P [B1, B2, B3, B4, B5] EachP  result Finalresult LG(c) a [0.998, 0.999, 0.999, -0.988, -0.985] healthy B1-B2 0.3Ω b [0.998, 0.999, 0.999, -0.990, -0.987] healthy B1-B2 c [0.883, -0.940, -0.906, -0.797, -0.656] B1-B2  LG(c) a [0.985, 0.990, 0.992, -0.972, -0.963] healthy B2-B3 0.3Ω b [0.988, 0.992, 0.993, -0.974, -0.965] healthy B2-B3 c [0.766, 0.922, -0.979, -0.739, -0.627] B2-B3  LG(c) a [-0.821, -0.806, -0.796, 0.835, 0.850] healthy B4-B5 0.3Ω b [-0.847, -0.833, -0.825, 0.847, 0.861] healthy B4-B5 c [-0.618, -0.472, -0.374, 0.866, -0.9461] B4-B5  LG(a) a [0.909, -0.956, -0.922, -0.525, -0.345] B1-B2 B1-B2 10Ω b [0.998, 0.998, 0.998, -0.996, -0.994] healthy B1-B2 c [0.999, 1.000, 1.000, -0.993, -0.992] healthy  LG(a) a [0.544, 0.780, -0.993, -0.486, -0.324] B2-B3 B2-B3 10Ω b [0.993, 0.995, 0.996, -0.985, -0.980] healthy B2-B3 c [0.994, 0.996, 0.997, -0.992, -0.989] healthy  LG(a) a [-0.210, -0.105, -0.034, 0.795, -0.985] B4-B5 B4-B5 10Ω b [-0.923, -0.909, -0.899, 0.942, 0.953] healthy B4-B5 c [-0.922, -0.910, -0.899, 0.931, 0.939] healthy


### 6.4. Grid Connected Operation Mode

When the microgrid is operated at grid connected mode, different faults are supposed to happen including different faulty section locations, types, and grounding resistances. Table6 gives the judgments and shows that this paper’s method can detect different faults in grid connected operation mode; besides, the protection scheme cannot be changed compared with islanded mode.Table 6
The feature cosine of each bus and judgment results in grid connected operation mode.
F t  Rf  FL P [B1, B2, B3, B4, B5] EachP  result Finalresult LG(a) a [0.054, -0.540, -0.271, -0.985, -0.974] B1-B2 B1-B2 0.01Ω b [-0.5926, 1.000, 1.000, 0.4800, 0.340] healthy B1-B2 c [-0.443, -0.013, 0.842, 0.256, 0.421] healthy  LG(b) a [0.191, 0.780, 0.901, 0.768, 0.822] healthy B1-B2 0.2Ω b [0.507, -0.909, -0.840, -0.984, -0.981] B1-B2 B1-B2 c [0.181, 0.764, 0.895, 0.713, 0.782] healthy  LLG(ac) a [0.107, 0.999, -0.829, -0.656, -0.021] B2-B3 B2-B3 0.5Ω b [0.112, 0.875, 0.940, 0.709, 0.759] healthy B2-B3 c [0.108, 0.989, -0.839, -0.899, -0.545] B2-B3  LLG(bc) a [-0.559, 0.856, 0.943, -0.005, 0.228] B2-B3 B2-B3 5Ω b [0.649, 0.974, -0.701, -0.967, -0.431] B2-B3 B2-B3 c [0.637, 0.973, -0.749, -0.961,0.307] healthy  LLL a [-0.948, -0.870, -0.585, 0.053, -0.319] B4-B5 B4-B5 10Ω b [-0.935, -0.843, -0.515, 0.619, -0.302] B4-B5 B4-B5 c [-0.987, -0.969, -0.193, 0.575, -0.515] B4-B5


### 6.5. Unbalanced Loads

In order to prove the effectiveness of proposed method in the condition of the unbalanced loads, the capacity of load L-2 in Figure2(a) is set as follows: phase-a: the active power and reactive power are 19kW and 8kW; phase-b: the active power and reactive power are 17kW and 7kW; phase-c: the active power and reactive power are 10kW and 8kW.When different faults are supposed to happen in the microgrid, Table7 gives the feature cosine of each bus and judgment results. When faulty condition is that faulty section location is B1-B2, grounding resistance is 0.1Ω, operation mode is islanded, and faulty type is LG (a), Figure 9 is the current of B2 and the bus ellipses of B1 and B2. From Figure 9 and Table 7, this paper method can achieve the faulty section location detection in the condition of the unbalanced loads.Table 7
The feature cosine of each bus and judgment results in unbalanced loads.
F t  Rf  FL  Om P [B1, B2, B3, B4, B5] EachP  result Finalresult LG(a) a [-0.999, 0.999, 0.999, 0.995, 0.997] B1-B2 B1-B2 0.1Ω b [0.653, 0.667, 0.675, -0.867, -0.851] healthy B1-B2 c [0.840, 0.852, 0.859, -0.911, -0.902] healthy islanded  LL(ab) a [-0.969, -0.834, 0.948, 0.951, 0.984] B2-B3 B2-B3 0.1Ω b [-0.718, -0.043, -0.856, 0.909, 0.931] healthy B2-B3 c [0.948, 0.949, 0.949, -0.999, -0.998] B2-B3 islanded  LLLG a [0.994, 0.999, 1.000, -0.991, 0.985] B4-B5 B4-B5 0.1Ω b [0.936, 0.979, 0.987, -1.021, 1.095] B4-B5 B4-B5 c [1.095, 1.023, 1.006, -0.991, 0.966] B4-B5 islanded  LG(b) a [-0.167, -0.191, -0.208, 0.115, 0.109] healthy B1-B2 1Ω b [0.999, -0.980, -0.980, -0.137, -0.169] B1-B2 B1-B2 c [-0.774, -0.780, -0.785, -0.296, -0.298] healthy Grid  LLG(ab) a [0.958, 0.735, -0.921, -0.456, -0.557] B2-B3 B2-B3 1Ω b [0.837, 0.341, 0.798, -0.993, -0.997] healthy Grid c [-0.960, -0.961, -0.962, 0.588, 0.582] healthy B2-B3  LLLG a [-0.836, -0.880, -0.899, 0.997, -0.994] B4-B5 B4-B5 1Ω b [-0.840, -0.887, -0.912, 0.993, -1.038] B4-B5 Grid c [-0.933, -0.952, -0.964, 0.994, -0.899] B4-B5 B4-B5Figure 9
The B2 current and bus ellipse for B1 and B2; (a) B2 current, (b) bus ellipse of three phases.
(a) (b)


### 6.6. Noise Influence

In fact, the elliptical representation of the behavior of both voltage and current signals should consider the existence of the noise, which is inherent in the processes of generation, transmission, or even disturbances related to digital recorders. Hence, in order to implement the proposed method effectively, the wavelet denoising method is employed [27]. It is embedded in Step 1 of Section 5.When different faults with different SNR including 50db and 5db are supposed to happen in the microgrid, Table8 shows that this paper method after adding wavelet denoising method can detect faulty section locations in different SNR including 50db and 5db.Table 8
The feature cosine of each bus and judgment results in different SNR.
(a)
SNR=50db
F t  Rf  FL  Om P [B1, B2, B3, B4, B5] EachP  result Finalresult LG(a) a [0.907, -0.945, -0.909, -0.527, -0.352] healthy B1-B2 5Ω b [0.999, 0.999, 0.999, -0.995, -0.993] B1-B2 B1-B2 c [0.999, 0.999, 0.999, -0.996, -0.995] healthy islanded  LLG(ab) a [0.535, 0.920, -0.064, -0.899, 0.073] B2-B3 B2-B3 1Ω b [0.754, 0.939, 0.039, -0.938, -0.726] healthy B2-B3 c [0.995, 1.000, -1.000, -0.964, -0.940] B2-B3 islanded  LLLG a [-0.641, 0.450, 0.275, -0.283, 0.377] B4-B5 B4-B5 5Ω b [-0.967, 0.913, 0.768, -1.000, 0.964] B4-B5 B4-B5 c [0.699, 0.677, 0.647, -0.858, 1.000] B4-B5 Grid  (b)
SNR=5db
F t  Rf  FL  Om P [B1, B2, B3, B4, B5] EachP  result Finalresult LG(a) a [0.907, -0.945, -0.909, -0.527, -0.352] healthy B1-B2 5Ω b [0.999, 0.999, 0.999, -0.995, -0.993] B1-B2 B1-B2 c [0.999, 0.999, 0.999, -0.996, -0.996] healthy islanded  LLG(ab) a [0.535, 0.920, -0.064, -0.899, 0.073] B2-B3 B2-B3 1Ω b [0.754, 0.939, -0.100, -0.958, -0.726] healthy B2-B3 c [0.995, 1.000, 1.000, -0.974, -0.940] B2-B3 islanded  LLLG a [-0.643, -0.450, -0.275, 0.283, -0.377] B4-B5 B4-B5 5Ω b [-0.970, -0.913, -0.768, 1.000, -0.964] B4-B5 B4-B5 c [-0.699, -0.677, -0.647, 0.858, -1.000] B4-B5 Grid


### 6.7. Looped Microgrid

When different faults are supposed to happen in the looped microgrid shown in Figure10, Table 9 gives the feature cosine of each bus and judgment results. From Table 9, we can know that this paper method can achieve the faulty section location detection in different faulty conditions for looped microgrid.Table 9
The feature cosine of each bus and judgment results for the looped microgrid.
F t  Rf  FL  Om P [B1, B2, B3, B4,B5] EachP  result Finalresult LG(a) a [0.965, -0.973, -0.949, -0.790, -0.741] B1- B2 B1- B2 0.1Ω b [-1.102, -1.103, -1.103, -1.098, -1.100] healthy B1-B2 c [-1.043, -1.044, -1.044, -1.040, -1.041] healthy islanded  LLG(ab) a [0.932, 0.978, -0.929, 0.958, 0.980] B2- B3 B2- B3 0.1Ω b [0.535, 0.763, -0.500, 0.657, 0.788] B2- B3 B2-B3 c [0.997, 0.998, 0.998, 0.997, 0.998] healthy islanded  LLLG a [-0.906, -0.896, -0.887, 0.691, -0.673] B4- B5 B4- B5 0.1Ω b [-0.654, -0.612, -0.583, 0.561, -0.480] B4- B5 B4-B5 c [-1.032, -1.032, -1.031, 0.918, -0.932] B4- B5 islanded  LG(b) a [0.534, -0.953, -0.912, 0.386, 0.939] B1- B2 B1- B2 1Ω b [1.275, 0.767, 0.933, 1.273, 0.933] healthy B1-B2 c [0.647, 0.764, 0.928, 0.645, 0.928] healthy Grid  LLG(ac) a [0.393, 0.935, -0.788, 0.673, 0.938] B2- B3 B2- B3 1Ω b [-0.357, 0.880, 1.007, -0.357, 1.006] healthy Grid c [0.357, 0.953, -0.856, 0.724, 0.937] B2- B3 B2-B3  LLLG a [0.551, 0.746, 0.819, 0.531, -0.831] B4- B5 B4- B5 1Ω b [0.442, 0.588, 0.662, 0.292, -0.503] B4- B5 Grid c [0.907, 1.039, 1.067, 1.006, -1.051] B4- B5 B4-B5Figure 10
The looped microgrid.


### 6.8. Sample Microgrid

When different faults are supposed to happen in the looped microgrid shown in Figure11, Table 10 gives the feature cosine of each bus and judgment results. From Table 10, we can know that this paper method can achieve the faulty section location detection in different faulty conditions for the microgrid whose nodes are with only loads.Table 10
The feature cosine of each bus and judgment results for the sample microgrid.
F t  Rf  FL  Om P [B1, B2, B3, B4,B5] EachP  result Finalresult LG(a) a [0.762, -0.981, -0.980, -0.976, -0.973] B1- B2 B1- B2 0.3Ω b [-0.988, -0.987, -0.987, -0.981, -0.973] healthy B1-B2 c [-0.987, -0.987, -0.987, -0.981, -0.971] healthy islanded  LLG(ab) a [1.087, 1.006, -0.973, -0.990, -0.988] B2- B3 B2- B3 0.3Ω b [0.310, 0.689, -0.969, -0.950, -0.944] B2- B3 B2-B3 c [-0.984, -0.984, -0.983, -0.976, -0.974] healthy islanded  LLL a [-0.978, -0.977, -0.977, 0.427, -0.965] B4- B5 B4- B5 / b [-0.973, -0.972, -0.971, 0.291, -0.956] B4- B5 B4-B5 c [-0.971, -0.970, -0.970, 0.420, -0.956] B4- B5 islanded  LG(b) a [-0.990, -0.990, -0.989, -0.981, -0.978] healthy B1- B2 3Ω b [0.071, -1.027, -1.027, -1.152, -1.159] B1- B2 B1-B2 c [-0.989, -0.989, -0.989, -0.981, -0.978] healthy Grid  LLG(ac) a [0.435, 0.966, -0.979, -0.949, -0.946] B2- B3 B2- B3 3Ω b [-0.976, -0.976, -0.976, -0.965, -0.961] healthy Grid c [0.563, 0.967, -0.979, -0.952, -0.945] B2- B3 B2-B3  LLLG a [-0.977, -0.976, -0.976, 0.283, -0.965] B4- B5 B4- B5 3Ω b [-0.965, -0.964, -0.963, 0.246, -0.956] B4- B5 Grid c [-0.962, -0.962, -0.961, 0.447, -0.956] B4- B5 B4-B5Figure 11
Sample microgrid whose nodes are with only loads.


## 6.1. Different Grounding Resistances

When the microgrid shown in Figure2(a) is operated in islanded mode, LG fault is supposed to happen in Line 1 at t=0.2s; that is, faulty section is between B1 and B2 labelled B1-B2. Faulty conditions include that faulty phase is phase-a and grounding resistances are set to 0.01Ω, 0.5Ω, 0.1Ω, 2Ω, and 10Ω, respectively. Then Table 3 gives the feature cosine of each bus and judgment results after implementing this paper’s method;P is phase sequence.Table 3
The feature cosine of each bus and results with different grounding resistances.
R f P [B1, B2, B3, B4, B5] EachP  result Finalresult 0.01Ω a [0.954, -0.990, -0.982, -0.859, -0.729] B1- B2 B1-B2 b [0.998, 0.999, 0.999, -1.007, -1.009] healthy c [0.999, 0.999, 0.999, -0.997, -0.995] healthy  0.1Ω a [0.961, -0.986, -0.978, -0.904, -0.797] B1 - B2 B1-B2 b [0.998, 0.998, 0.998, -0.991, -0.987] healthy c [0.998, 0.998, 0.999, -0.992, -0.989] healthy  0.5Ω a [0.999, -1.000, -0.999, -0.997, -0.997] B1 - B2 B1 - B2 b [-0.891, -0.894, -0.896, 0.720, 0.699] healthy c [-0.958, -0.960, -0.962, 0.925, 0.919] healthy  2Ω a [0.936, -0.979, -0.960, -0.897, -0.767] B1 - B2 B1-B2 b [0.999, 0.999, 1.000, -0.994, -0.993] healthy c [1.000, 1.000, 1.001, -0.996, -0.994] healthy  10Ω a [0.927, -0.974, -0.952, -0.888, -0.752] B1 - B2 B1-B2 b [0.999, 0.999, 0.999, -0.995, -0.993] healthy c [0.999, 0.999, 1.000, -0.998, -0.997] healthyTake grounding resistance 0.5Ω as an example, Figure7 gives the voltage and current of B1 and B2, and Figure 8 shows the bus ellipses of each phase of B1 and B2. Figure 7 illustrates that the faulty phase voltage direction of B1 is same as that of B2 and the faulty phase current direction of B1 is opposite to that of B2, which is illustrated obviously in Figure 8.Figure 7
The voltage and current of B1 and B2 when fault occurs at Line 1. (a) B1 voltage; (b) B2 voltage; (c) B1 current; (d) B2 current.
(a)
B1 voltage (b)
B2 voltage (c)
B1 current (d)
B2 currentFigure 8
The bus ellipse of B1 and B2 for different phases. (a) Phase-a; (b) phase-b; (c) phase-c.Then, as for phase-a, from Table3 and this paper’s method, the feature cosines from B1 to B5 are 0.999, -1.000, -0.999, -0.997, and -0.997, respectively. So, the feature directions from B1 to B5 are 1, -1, -1, -1, and -1, respectively; that is, their differential directions are -2, 0, 0, 0, respectively. Similarly, as for phase-b and phase-c, their differential directions are all 0. Hence, the faulty section is between B1 and B2; this result is consistent with the actual fault. Lastly, from Table 3 we can find that this paper’s method can detect fault part effectively in different grounding resistances.


## 6.2. Different Faulty Types

When the microgrid is operated in islanded mode, different faulty types, including LG(b), LLG(ab), LL(bc), and LLLG, are supposed to happen in Line 1 at 0.2s with grounding resistance 0.1Ω; that is, faulty section is B1-B2. Then, Table 4 gives the feature cosine of each bus and judgment results; it shows that the proposed method cannot be influenced by different faulty types.Table 4
The feature cosine of each bus and judgment results with different faulty types.
F t P [B1, B2, B3, B4, B5] EachP  result Finalresult LG(b) a [1.003, 1.003, 1.003, -1.008, -1.012] healthy B1-B2 b [0.928, -0.915, -0.871, -1.100, -1.084] B1-B2 c [1.002, 1.002, 1.002, -0.991, -0.9867] B1-B2  LLG(ab) a [0.957, -0.946, -0.915, -0.994, -0.970] B1-B2 B1-B2 b [0.672, -0.508, -0.380, -0.829, -0.614] B1-B2 c [0.998, 0.997, 0.997, 1.004, 1.006] healthy  LL(bc) a [0.401, 0.631, 0.715, 0.113, 0.897] healthy B1-B2 b [0.953, -0.941, -0.958, -0.977, -0.900] B1-B2 c [0.956, -0.943, -0.957, -0.978, -0.909] B1-B2  LLL a [0.836, -0.636, -0.473, -0.949, -0.869] B1-B2 B1-B2 b [0.609, -0.696, -0.497, -0.852, -0.642] B1-B2 c [0.999, -0.866, -0.770, -1.040, -1.010] B1-B2


## 6.3. Different Faulty Section Locations

The islanded microgrids are simulated at 0.2s with different faulty section locations, including B1-B2, B2-B3, and B4-B5. The fault conditions include fault type LG whose faulty phase is phase-c or phase-a; grounding resistances is 0.3Ω or 10Ω. Then, Table 5 illustrates that this paper’s method can detect different fault parts. It is mentioned that the feature cosine is consist with the analysis in Section 4.3 when the grounding resistance is 10Ω and faulty section location is B4-B5.Table 5
The feature cosine of each bus and judgment results in different faulty section locations.
F t  Rf  FL P [B1, B2, B3, B4, B5] EachP  result Finalresult LG(c) a [0.998, 0.999, 0.999, -0.988, -0.985] healthy B1-B2 0.3Ω b [0.998, 0.999, 0.999, -0.990, -0.987] healthy B1-B2 c [0.883, -0.940, -0.906, -0.797, -0.656] B1-B2  LG(c) a [0.985, 0.990, 0.992, -0.972, -0.963] healthy B2-B3 0.3Ω b [0.988, 0.992, 0.993, -0.974, -0.965] healthy B2-B3 c [0.766, 0.922, -0.979, -0.739, -0.627] B2-B3  LG(c) a [-0.821, -0.806, -0.796, 0.835, 0.850] healthy B4-B5 0.3Ω b [-0.847, -0.833, -0.825, 0.847, 0.861] healthy B4-B5 c [-0.618, -0.472, -0.374, 0.866, -0.9461] B4-B5  LG(a) a [0.909, -0.956, -0.922, -0.525, -0.345] B1-B2 B1-B2 10Ω b [0.998, 0.998, 0.998, -0.996, -0.994] healthy B1-B2 c [0.999, 1.000, 1.000, -0.993, -0.992] healthy  LG(a) a [0.544, 0.780, -0.993, -0.486, -0.324] B2-B3 B2-B3 10Ω b [0.993, 0.995, 0.996, -0.985, -0.980] healthy B2-B3 c [0.994, 0.996, 0.997, -0.992, -0.989] healthy  LG(a) a [-0.210, -0.105, -0.034, 0.795, -0.985] B4-B5 B4-B5 10Ω b [-0.923, -0.909, -0.899, 0.942, 0.953] healthy B4-B5 c [-0.922, -0.910, -0.899, 0.931, 0.939] healthy


## 6.4. Grid Connected Operation Mode

When the microgrid is operated at grid connected mode, different faults are supposed to happen including different faulty section locations, types, and grounding resistances. Table6 gives the judgments and shows that this paper’s method can detect different faults in grid connected operation mode; besides, the protection scheme cannot be changed compared with islanded mode.Table 6
The feature cosine of each bus and judgment results in grid connected operation mode.
F t  Rf  FL P [B1, B2, B3, B4, B5] EachP  result Finalresult LG(a) a [0.054, -0.540, -0.271, -0.985, -0.974] B1-B2 B1-B2 0.01Ω b [-0.5926, 1.000, 1.000, 0.4800, 0.340] healthy B1-B2 c [-0.443, -0.013, 0.842, 0.256, 0.421] healthy  LG(b) a [0.191, 0.780, 0.901, 0.768, 0.822] healthy B1-B2 0.2Ω b [0.507, -0.909, -0.840, -0.984, -0.981] B1-B2 B1-B2 c [0.181, 0.764, 0.895, 0.713, 0.782] healthy  LLG(ac) a [0.107, 0.999, -0.829, -0.656, -0.021] B2-B3 B2-B3 0.5Ω b [0.112, 0.875, 0.940, 0.709, 0.759] healthy B2-B3 c [0.108, 0.989, -0.839, -0.899, -0.545] B2-B3  LLG(bc) a [-0.559, 0.856, 0.943, -0.005, 0.228] B2-B3 B2-B3 5Ω b [0.649, 0.974, -0.701, -0.967, -0.431] B2-B3 B2-B3 c [0.637, 0.973, -0.749, -0.961,0.307] healthy  LLL a [-0.948, -0.870, -0.585, 0.053, -0.319] B4-B5 B4-B5 10Ω b [-0.935, -0.843, -0.515, 0.619, -0.302] B4-B5 B4-B5 c [-0.987, -0.969, -0.193, 0.575, -0.515] B4-B5


## 6.5. Unbalanced Loads

In order to prove the effectiveness of proposed method in the condition of the unbalanced loads, the capacity of load L-2 in Figure2(a) is set as follows: phase-a: the active power and reactive power are 19kW and 8kW; phase-b: the active power and reactive power are 17kW and 7kW; phase-c: the active power and reactive power are 10kW and 8kW.When different faults are supposed to happen in the microgrid, Table7 gives the feature cosine of each bus and judgment results. When faulty condition is that faulty section location is B1-B2, grounding resistance is 0.1Ω, operation mode is islanded, and faulty type is LG (a), Figure 9 is the current of B2 and the bus ellipses of B1 and B2. From Figure 9 and Table 7, this paper method can achieve the faulty section location detection in the condition of the unbalanced loads.Table 7
The feature cosine of each bus and judgment results in unbalanced loads.
F t  Rf  FL  Om P [B1, B2, B3, B4, B5] EachP  result Finalresult LG(a) a [-0.999, 0.999, 0.999, 0.995, 0.997] B1-B2 B1-B2 0.1Ω b [0.653, 0.667, 0.675, -0.867, -0.851] healthy B1-B2 c [0.840, 0.852, 0.859, -0.911, -0.902] healthy islanded  LL(ab) a [-0.969, -0.834, 0.948, 0.951, 0.984] B2-B3 B2-B3 0.1Ω b [-0.718, -0.043, -0.856, 0.909, 0.931] healthy B2-B3 c [0.948, 0.949, 0.949, -0.999, -0.998] B2-B3 islanded  LLLG a [0.994, 0.999, 1.000, -0.991, 0.985] B4-B5 B4-B5 0.1Ω b [0.936, 0.979, 0.987, -1.021, 1.095] B4-B5 B4-B5 c [1.095, 1.023, 1.006, -0.991, 0.966] B4-B5 islanded  LG(b) a [-0.167, -0.191, -0.208, 0.115, 0.109] healthy B1-B2 1Ω b [0.999, -0.980, -0.980, -0.137, -0.169] B1-B2 B1-B2 c [-0.774, -0.780, -0.785, -0.296, -0.298] healthy Grid  LLG(ab) a [0.958, 0.735, -0.921, -0.456, -0.557] B2-B3 B2-B3 1Ω b [0.837, 0.341, 0.798, -0.993, -0.997] healthy Grid c [-0.960, -0.961, -0.962, 0.588, 0.582] healthy B2-B3  LLLG a [-0.836, -0.880, -0.899, 0.997, -0.994] B4-B5 B4-B5 1Ω b [-0.840, -0.887, -0.912, 0.993, -1.038] B4-B5 Grid c [-0.933, -0.952, -0.964, 0.994, -0.899] B4-B5 B4-B5Figure 9
The B2 current and bus ellipse for B1 and B2; (a) B2 current, (b) bus ellipse of three phases.
(a) (b)


## 6.6. Noise Influence

In fact, the elliptical representation of the behavior of both voltage and current signals should consider the existence of the noise, which is inherent in the processes of generation, transmission, or even disturbances related to digital recorders. Hence, in order to implement the proposed method effectively, the wavelet denoising method is employed [27]. It is embedded in Step 1 of Section 5.When different faults with different SNR including 50db and 5db are supposed to happen in the microgrid, Table8 shows that this paper method after adding wavelet denoising method can detect faulty section locations in different SNR including 50db and 5db.Table 8
The feature cosine of each bus and judgment results in different SNR.
(a)
SNR=50db
F t  Rf  FL  Om P [B1, B2, B3, B4, B5] EachP  result Finalresult LG(a) a [0.907, -0.945, -0.909, -0.527, -0.352] healthy B1-B2 5Ω b [0.999, 0.999, 0.999, -0.995, -0.993] B1-B2 B1-B2 c [0.999, 0.999, 0.999, -0.996, -0.995] healthy islanded  LLG(ab) a [0.535, 0.920, -0.064, -0.899, 0.073] B2-B3 B2-B3 1Ω b [0.754, 0.939, 0.039, -0.938, -0.726] healthy B2-B3 c [0.995, 1.000, -1.000, -0.964, -0.940] B2-B3 islanded  LLLG a [-0.641, 0.450, 0.275, -0.283, 0.377] B4-B5 B4-B5 5Ω b [-0.967, 0.913, 0.768, -1.000, 0.964] B4-B5 B4-B5 c [0.699, 0.677, 0.647, -0.858, 1.000] B4-B5 Grid  (b)
SNR=5db
F t  Rf  FL  Om P [B1, B2, B3, B4, B5] EachP  result Finalresult LG(a) a [0.907, -0.945, -0.909, -0.527, -0.352] healthy B1-B2 5Ω b [0.999, 0.999, 0.999, -0.995, -0.993] B1-B2 B1-B2 c [0.999, 0.999, 0.999, -0.996, -0.996] healthy islanded  LLG(ab) a [0.535, 0.920, -0.064, -0.899, 0.073] B2-B3 B2-B3 1Ω b [0.754, 0.939, -0.100, -0.958, -0.726] healthy B2-B3 c [0.995, 1.000, 1.000, -0.974, -0.940] B2-B3 islanded  LLLG a [-0.643, -0.450, -0.275, 0.283, -0.377] B4-B5 B4-B5 5Ω b [-0.970, -0.913, -0.768, 1.000, -0.964] B4-B5 B4-B5 c [-0.699, -0.677, -0.647, 0.858, -1.000] B4-B5 Grid


## 6.7. Looped Microgrid

When different faults are supposed to happen in the looped microgrid shown in Figure10, Table 9 gives the feature cosine of each bus and judgment results. From Table 9, we can know that this paper method can achieve the faulty section location detection in different faulty conditions for looped microgrid.Table 9
The feature cosine of each bus and judgment results for the looped microgrid.
F t  Rf  FL  Om P [B1, B2, B3, B4,B5] EachP  result Finalresult LG(a) a [0.965, -0.973, -0.949, -0.790, -0.741] B1- B2 B1- B2 0.1Ω b [-1.102, -1.103, -1.103, -1.098, -1.100] healthy B1-B2 c [-1.043, -1.044, -1.044, -1.040, -1.041] healthy islanded  LLG(ab) a [0.932, 0.978, -0.929, 0.958, 0.980] B2- B3 B2- B3 0.1Ω b [0.535, 0.763, -0.500, 0.657, 0.788] B2- B3 B2-B3 c [0.997, 0.998, 0.998, 0.997, 0.998] healthy islanded  LLLG a [-0.906, -0.896, -0.887, 0.691, -0.673] B4- B5 B4- B5 0.1Ω b [-0.654, -0.612, -0.583, 0.561, -0.480] B4- B5 B4-B5 c [-1.032, -1.032, -1.031, 0.918, -0.932] B4- B5 islanded  LG(b) a [0.534, -0.953, -0.912, 0.386, 0.939] B1- B2 B1- B2 1Ω b [1.275, 0.767, 0.933, 1.273, 0.933] healthy B1-B2 c [0.647, 0.764, 0.928, 0.645, 0.928] healthy Grid  LLG(ac) a [0.393, 0.935, -0.788, 0.673, 0.938] B2- B3 B2- B3 1Ω b [-0.357, 0.880, 1.007, -0.357, 1.006] healthy Grid c [0.357, 0.953, -0.856, 0.724, 0.937] B2- B3 B2-B3  LLLG a [0.551, 0.746, 0.819, 0.531, -0.831] B4- B5 B4- B5 1Ω b [0.442, 0.588, 0.662, 0.292, -0.503] B4- B5 Grid c [0.907, 1.039, 1.067, 1.006, -1.051] B4- B5 B4-B5Figure 10
The looped microgrid.


## 6.8. Sample Microgrid

When different faults are supposed to happen in the looped microgrid shown in Figure11, Table 10 gives the feature cosine of each bus and judgment results. From Table 10, we can know that this paper method can achieve the faulty section location detection in different faulty conditions for the microgrid whose nodes are with only loads.Table 10
The feature cosine of each bus and judgment results for the sample microgrid.
F t  Rf  FL  Om P [B1, B2, B3, B4,B5] EachP  result Finalresult LG(a) a [0.762, -0.981, -0.980, -0.976, -0.973] B1- B2 B1- B2 0.3Ω b [-0.988, -0.987, -0.987, -0.981, -0.973] healthy B1-B2 c [-0.987, -0.987, -0.987, -0.981, -0.971] healthy islanded  LLG(ab) a [1.087, 1.006, -0.973, -0.990, -0.988] B2- B3 B2- B3 0.3Ω b [0.310, 0.689, -0.969, -0.950, -0.944] B2- B3 B2-B3 c [-0.984, -0.984, -0.983, -0.976, -0.974] healthy islanded  LLL a [-0.978, -0.977, -0.977, 0.427, -0.965] B4- B5 B4- B5 / b [-0.973, -0.972, -0.971, 0.291, -0.956] B4- B5 B4-B5 c [-0.971, -0.970, -0.970, 0.420, -0.956] B4- B5 islanded  LG(b) a [-0.990, -0.990, -0.989, -0.981, -0.978] healthy B1- B2 3Ω b [0.071, -1.027, -1.027, -1.152, -1.159] B1- B2 B1-B2 c [-0.989, -0.989, -0.989, -0.981, -0.978] healthy Grid  LLG(ac) a [0.435, 0.966, -0.979, -0.949, -0.946] B2- B3 B2- B3 3Ω b [-0.976, -0.976, -0.976, -0.965, -0.961] healthy Grid c [0.563, 0.967, -0.979, -0.952, -0.945] B2- B3 B2-B3  LLLG a [-0.977, -0.976, -0.976, 0.283, -0.965] B4- B5 B4- B5 3Ω b [-0.965, -0.964, -0.963, 0.246, -0.956] B4- B5 Grid c [-0.962, -0.962, -0.961, 0.447, -0.956] B4- B5 B4-B5Figure 11
Sample microgrid whose nodes are with only loads.


## 7. Discussion

When different faults, including different faulty resistances and faulty section, occur in microgrid at 0.2s, Figure12 shows the fault current peak value of B1 whose time length is from 0.21s~0.23s.Figure 12
Fault current peak value of B1 in different Rf. (a) 0.01Ω; (b) 1Ω; (c) 10Ω.
(a) (b) (c)It is shown in Figure10 that the B1 fault current in grid connected mode is higher compared with the islanded mode; besides, the B1 fault current decreases for the faults occurring on the lines away from the B1. So, as for fault current, there is a difference between grid connected and islanded modes; besides, if the microgrid has unbalanced loads, its fault current for each phase is obviously different. It can be concluded that most commercially used overcurrent relay with a preset threshold to operate in grid connected mode may fail or may take a longer time to operate in islanded mode [2]. Hence, [14] has different thresholds to detect fault when the microgrid is operated in different modes. However, [14] did not consider the condition of unbalanced load and will make the protection scheme more complex. Hence, in this paper, the protection method, which has the same threshold in different operation mode and unbalanced load, is proposed and superior to [14].


## 8. Conclusion

Based on the geometric representation of one phase in a bus and the analysis of feature cosine of different buses, this paper proposes a protection method to detect the faulty section location using the differential direction. The proposed method uses comprehensively every phase voltage and current to calculate the differential direction; this method is found to work effectively under various case studies as discussed in this paper including different grounding resistances and faulty types. The advantage of the proposed method is that it is not needed to change the threshold to detect the faulty section location when the microgrid operation mode is changed and has unbalanced loads.



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*Source: 7248072-2019-03-10.xml*