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# Fault-line selection and fault-type recognition in DC systems based on graph theory

- Yan Xu
^{1}, - Jingyan Liu
^{1}Email author and - Yuan Fu
^{1}

**3**:27

https://doi.org/10.1186/s41601-018-0098-9

© The Author(s) 2018

**Received:**11 January 2018**Accepted:**18 July 2018**Published:**28 August 2018

## Abstract

When a fault occurs in a DC system, the fault current rises rapidly with no zero-crossing point which makes fault-line selection and fault-type identification difficult. In this paper, an online detection and protection method based on graph theory, namely the “double D method”, is proposed for fault-line selection and fault-type identification in DC systems. In the proposed method, the entire distribution network is visualized as a “map” with vertices representing the line convergence points and edges representing the connection lines. A network topology matrix “D” is formed by detecting the current directions as the current directions are altered following a fault, whereas the current directions at the ends of non-fault lines remain the same. In order to prevent misjudgment problems arising from power flow reversal, the rates of change of the fault currents are used to further determine whether a fault has occurred and the “double D method” is introduced to identify the fault type. Simulations results with different fault types verify the effectiveness and reliability of the proposed method.

## Keywords

- Network topology matrix
- Double
*D*method - Fault-line selection
- Fault-type identification
- Graph theory

## 1 Introduction

When large amount of renewable energy is connected to a traditional power grid, it becomes increasingly difficult for the grid to handle problems such as grid stability. Therefore, it is necessary to upgrade the traditional technologies. DC power systems have recently attracted significant attentions owing to their advantages such as high flexibility, controllability of power and improving power angle stability [1]. In city distribution networks, DC loads such as electric cars and lighting systems are developing and expanding rapidly. In view of this, the construction of DC distribution networks can not only reduce the number of power conversion links but also improve the power supply efficiency. In general, voltage-source converters (VSC) can have the two-level, three-level, or modular multilevel converter (MMC) topologies. The modulation principles of the two-level and three-level converters are the same, whereas two-level converters with the advantages of mature technology, simple control, and a low cost, have greater application prospects than MMC in distribution networks. Faults occurred in a DC line within a VSC based multi-terminal DC distribution network will have wider influences on the whole system than in a two-terminal DC transmission system. Therefore, to maintain safe operation, it is of great importance to study the fault-selection and fault-type recognition methods for DC lines to detect and remove the fault quickly.

Two main methods are used for fault location in DC power grids, i.e. the traveling wave method and the fault analysis method [2, 3]. Considering the similar physical natures of AC and DC lines, most studies have focused on application of fault location principles of AC lines to DC lines. The current fault location methods used in DC line fault location devices in commercial operations still predominantly use the traveling wave method [4–9]. By using the difference in the transmission time between the fault position and measuring point detected by the transient traveling wave, the fault location can be obtained. This method is not influenced by the type of line, fault impedance, fault type or system parameters on either side and thus has a relatively high location precision [10–12]. However, some technical problems that are difficult to overcome still exist. The identification and calibration of the wave head are highly dependent on the competence of the operating persons and therefore, it is difficult to realize automation. Moreover, the traveling wave fault location system requires a high sampling rate to accurately control the positioning error to within a certain range. With a limited wave amplitude, it is difficult to calibrate the starting point of the wave head accurately which can result in inaccurate fault location.

To solve the problems listed above, reference [13] proposed a fault location method for a DC line based on non-traveling wave principle using the electrical quantities of a single terminal to realize fault location. In theory, the amount of calculations required is relatively small and the results have good accuracy. However, it is prone to converter regulation on the opposite side. Moreover, the method was set up based on the hypothesis that the fault currents of the DC lines are provided by the rectifier alone, which is not true in practical systems. Reference [14] proposed a non-traveling wave principle and a fault location method of DC lines using only the electrical quantities of a single terminal. The advantages of this method, such as the relatively low requirements on sampling rate and minimal influence of transition resistance are attractive. However, the calculation process is rather complex resulting in unsatisfactory location accuracy in practical applications. A fault location principle for high-voltage direct current (HVDC) transmission lines was proposed in [15] using the natural frequency of the current. Using the electrical quantities of only a single terminal, it realized fault location with high speed and accuracy but required a system with high sampling frequency.

In this study, the DC-fault transient processes of DC power distribution systems are analyzed. Methods of fault-line selection and fault-type identification in a DC system are proposed based on graph theory. Fault-line selection is mainly accomplished by the combination of judging the current flow and extracting the di/dt fault feature.

## 2 Discussion

### 2.1 Analysis of DC line fault

#### 2.1.1 Distribution network topology

### 2.2 Analysis of pole-to-pole fault

In Fig. 3, *C* is the DC capacitance, *V*_{dc} is the capacitor voltage, *R* is the line resistance, *L* is the line inductance and *I*_{L} is the current flowing through the line. *VD*_{i}(*i* = 1,2,...6) denotes an insulated gate bipolar transistor in a circulating bridge. *L*_{s} represents the AC side inductance. *I*_{sabc} indicates the current flowing through the inductor on the AC side.

*t*

_{0}, the initial values of the voltage and current are assumed to be

*V*

_{0}and

*I*

_{0}, respectively. The line resistance is usually small and thus, \( R<2\sqrt{L/C} \). Solving (1) gives the transient expressions for the voltage

*V*

_{dc}and current

*I*

_{L}as

### 2.3 Analysis of pole-to-ground fault

*R*

_{f}is the grounding resistance.

## 3 Methods

### 3.1 Fault-line selection based on graph theory

#### 3.1.1 Basics of graph theory

Definition 1: A graph consists of several different vertices and edges connecting the different vertices [19].

The constituent elements of a graph are the points which are often called vertices, and the edges which are ordered pairs of vertices. If two vertices can be connected by an edge, the two have adjacent relations. The chart can generally be shown as *G* = (*V*, *E*) , where *V* = [*v*_{1}, *v*_{2}, …, *v*_{n}] is the collection of all vertices, and *E* = [*e*_{1}, *e*_{2,}…, *e*_{n}] is a multiple subset of the Cartesian product whose elements are directed edges and are simply called edges.

*G*= (

*V*,

*E*) is a digraph with no margin. When

*V*= [

*v*

_{1},

*v*

_{2}, …,

*v*

_{n}], the N × N matrix

*D*= [

*d*

_{ij}] is called the adjacency matrix of the

**G**graph, denoted as

**D**[

**G**]. The adjacency matrix is a matrix representing the relationship between the vertices as

### 3.2 Directed topology description of distribution network

Irrespective of the type of the network structure, the system is composed of switches, overhead lines and a variety of other devices. Thus, the network composition can be simplified and the directional topology of the distribution network can be obtained.

According to Definition 1, the intersection of the distribution network lines can be seen as a vertex and the feeder between two adjacent vertices as an edge. Thus, the distribution network can be seen as a map. The connection relationship between the vertex and feeder in the distribution network can be described by the definition of the graph, and the topology of the distribution network can be described by the adjacency matrix in Definition 2.

When a fault occurs, the direction of the current flowing through the distribution feeder will change suddenly. Therefore, the feeder of the distribution network can be viewed as a directional edge. Assuming that the number of nodes in the distribution network is *n*, the distribution network (hereafter referred to as the network topology matrix) can be described by an *n*-order adjacency matrix D. If the direction of the edge is vertex *i* pointing to vertex *j*, the element *d*_{ij} = 1 in matrix D. If the direction of the edge is vertex *j* pointing to vertex *i*, the element *d*_{ij} = − 1 in matrix D. The remaining elements are all zero and the end nodes are all set to zero. That is, the positive direction of each current transformer is defined as the vertex pointing to the line.

*L*

_{1}and a trailing end

*L*

_{2}. The first end of the secondary coil is

*K*

_{1}and the other end is

*K*

_{2}. The positive direction of the ammeter is indicated by the black arrow in Fig. 7a and “*” indicates the same polarity of the current transformer.

Figure 7a shows the current flow in the current transformer under normal system operation. As there is no significant change in the primary current, no current is induced on the secondary side and the ammeter does not register any current. The element in matrix D is thus “1” at this time. When the current at the detection point changes, the primary current of the current transformer undergoes two processes. Firstly, the flow of *I*_{1} from the *L*_{1}terminal rapidly decreases to zero. Then, the direction of *I*_{1} changes and the current flows from the *L*_{2} terminal rapidly increases. The direction of the current induced on the secondary side shown in Fig. 7b and c, is opposite to the positive direction of the current meter and the sign of the current flowing through the current meter is thus negative. At this time, the element in matrix D is “− 1.”

### 3.3 Matrix fault criterion and fault detection

*a*and

*b*, in matrix D,

*d*

_{ab}=

*d*

_{ba}= 1. The fault information matrix S is introduced as

*s*

_{ij}=

*d*

_{ij}+

*d*

_{ji}and the fault criterion is given as

*n*represents the number of vertices.

When the system is in normal operation, all the elements in the **S** matrix will be 0. When a fault occurs, the elements in the faulty line will be 2 while the remaining elements will be 0. The fault information matrix can show whether the system is faulty and determine the location of the faulty line.

The proposed method based on graph theory forms a network topology matrix. When the logical elements are determined, the running states are represented by 0, 1 and − 1. The matrix generation speed is fast, the sparseness is strong and the operation speed is also fast.

### 3.4 Shortcomings and improvements of the proposed method

#### 3.4.1 Preventing misjudgment and extracting the degree of failure di/dt

_{kc}, the additional fault criterion is

*i*/d

*t*is calculated by the difference method. As long as the sampling time interval is properly controlled, the positioning accuracy can be guaranteed. The sampling interval can be selected from 20 to 100 μs depending on the transient characteristics of the DC system. The value of d

*i*/d

*t*at time

*t*

_{c}can be approximated using the backward difference method as

*k*is the sampling instant,

*i*(

*k*) and

*i*(

*k*− 1) are the sampled currents at the k

^{th}and (k-1)

^{th}instants, respectively, and Δ

*t*is the sampling interval.

When S is an all-zero matrix, it is not necessary to use (11) because the change of the current direction is a necessary and insufficient condition for fault occurrence. When S is a non-all-zero matrix, the line corresponding to the non-all-zero elements is confirmed first, and the d*i/*d*t* is then detected by the current transformer of the line. The fault can then be judged using (11).

- (1)
Normal operation: the system has neither a fault nor power reversal. The criterion is that the fault information matrix

**S**is an all-zero matrix. - (2)
Fault state: the system has at least one line failure. The criteria are that

**S**is a non-all-zero matrix and the current change rate of the line determined by the non-zero elements satisfies di/dt ≥ 1/2b_{kc}. - (3)
Reversal of trend: the system has no fault but there are lines in the state of reversal of trend. The criteria are that

**S**is a non-all-zero matrix and the current change rate of the line determined by the non-zero elements satisfies di/dt < 1/2b_{kc}.

The matrix algorithm can judge the specific lines that may be faulty according to the current direction. The method of calculating the current increment can only judge the occurrence of the fault rather than the specific position. Therefore, the combination of the two can be more accurate.

#### 3.4.2 Adding the function of fault-type recognition and introducing the “double **D**” method

**D**will not change and the state will be mistaken for normal operation. In order to detect the failure of the negative line, the negative line network topology matrix

**D**’ is introduced. It is hereby stated that the positive directions of the current transformers in matrices

**D**and

**D**’ are the same. When the system is in normal operation, the two elements in the same position in matrices

**D**and

**D**’ are opposite of each other. Two types of fault diagrams are shown in Fig. 8, where i and j represent both ends of the same line and the ends may be the power, the load, or a meeting point. The specific process to identify the fault type is described below.

- (1)
Normal operation, recorded as (0, 0). The two elements at the same position in matrices

**D**and**D**’ are opposite of each other and are non-zero. From Fig. 8a, it can be seen that*d*_{ij}= 1,*d*_{ji}= − 1 in matrix**D**and*d*'_{ji}= 1,*d*'_{ij}= − 1 in matrix**D**’. The criteria are that, during normal operation,*d*_{ij}and*d*_{ji}are non-zero and opposite of each other, and same for*d*'_{ji}and*d*'_{ij}. - (2)
Pole-to-ground fault. Positive pole-to-ground fault and negative pole-to-ground fault are marked as (1, 0) and (0, 1), respectively. The two elements in matrix

**D**or**D**’ represent the current directions of the fault occurrence line are 1 and the other elements are not affected. From Fig. 8b, it is known that*d*_{ij}=*d*_{ji}= 1 in matrix**D**and*d*'_{ji}= 1 and*d*'_{ij}= − 1 in matrix**D**’. The criteria are that when the positive line suffers a pole-to-ground fault,*d*_{ij}=*d*_{ji}= 1 and*d*'_{ji}and*d*'_{ij}are opposite of each other and are non-zero. When the negative line has a pole-to-ground fault,*d*_{ij}=*d*_{ji}= 1 and*d*_{ij}and*d*_{ij}are opposite of each other and are non-zero. - (3)
Pole-to-pole fault, recorded as (1, 1). The two elements in matrix

**D**are “1” indicating that the current directions at both ends of the faulty line constitute the vertex pointing to the line and the two elements in matrix**D**‘are “− 1″. From Fig. 8c, it is known that*d*_{ij}=*d*_{ji}= 1 in matrix**D**and*d*'_{ji}= 1,*d*'_{ij}= − 1 in matrix**D**’. The criteria are:*d*_{ij}=*d*_{ji}= 1 and*d*'_{ji}=*d*'_{ij}= − 1 in the faulty line.

This method is applicable to online monitoring and protection system of DC distribution networks. The algorithm first selects the lines that may be faulty in the system according to the elements in the S matrix and then uses the current change rate to detect the specific faulty lines. Finally, the double ‘D’ method is used to detect the fault type. A further explanation for Fig. 9 is given below. Matrix D’ is the same as matrix D in the second rectangular box and matrix D is used as an example here.

## 4 Results

### 4.1 Simulation analysis

#### 4.1.1 Simulation system

*I*

_{mk}(

*m, k*refers to the current flowing from vertex

*m*to vertex

*k*in the specified positive direction). Similarly, the current of each current transformer on the negative line is expressed as

*II*

_{mk}. Figure 11 shows the current representation of the four current transformers between vertices 1 and 2. The simulation time is 0.2 s. A fault occurs at 0.05 s at the midpoint of the line connecting vertices 1 and 2.

Parameters of the model

DC bus voltage | 500 V |
---|---|

Converter outlet capacitance | 2 mF |

Wind power unit | 20 kW/220 V permanent magnet wind turbines |

Rated wind speed is 12 m/s | |

Rated speed is 75 r/min | |

G-VSC rated power | 20 kW |

Battery rated power | Bi-DC rated power is 20 kW |

Load unit rated power | 20 kW |

Line resistance | 0.0139 Ω/km |

Line inductance | 0.159 mH/km |

Line length | 1 km |

### 4.2 Simulation results analysis

- A.
*Case study 1: Normal operation*

- B.
*Case study 2: Pole-to-pole fault condition*

It can be seen that the two matrices are both non-zero matrices with the non-zero elements present in *S*_{12} and *S*_{21}. It can be judged that the fault occurs in the line between vertexes 1 and 2, and that both the positive and negative lines have faults.

*I*

_{12}and

*I*

_{21}are both 1.43 × 10

^{6}A/s. The positive and negative fault currents are of the same magnitude but of opposite directions. The change rates of fault currents

*II*

_{12}and

*II*

_{21}are both 1.43 × 10

^{6}A/s. According to (3), the calculated value of 1/2

*b*

_{kc}is 1.36 × 10

^{6}A/s which satisfies (11).

*d*

_{12}= 1,

*d*

_{21}= 1 ,

*d*’

_{12}= − 1 and

*d*’

_{21}= − 1. According to Fig. 10, a pole-to-pole fault occurs on the line between the vertices 1 and 2.

- C.
*Case study 3: Pole-to-ground fault condition*

It can be seen that the matrix S**′** is an all-zero matrix but S is a non-all-zero matrix. *S*_{12} and *S*_{21} are non-zero elements and it is possible to determine that the location of the fault is at the positive pole of the line connecting vertex 1 and vertex 2.

^{4}A/s and 6.25 × 10

^{4}A/s for

*I*

_{12}and

*I*

_{21}, respectively. The calculated value of 1/2

*b*

_{kc}according to (7) is 1.4× 10

^{4}A/s which satisfies (11).

The adjacency matrices D and D’ show that *d*_{12} = 1, *d*_{21} = 1, *d*’_{12} = 0 and *d*’_{21} = 0. Thus, according to Fig. 10, a pole-to-ground short-circuit fault is detected on the line between vertices 1 and 2.

Because the fault point is at the midpoint of the line, the parameters become symmetric. Thus, in Fig. 12, currents *I*_{12} and *I*_{21} overlap, so as currents *II*_{12} and *II*_{21}. In Fig. 13, currents *II*_{12} and *II*_{21} also coincide.

## 5 Conclusions

For selecting the fault line, the entire DC system is seen as a “map” by placing current transformers on both sides of the apex. The network topology matrix method and current rate of change are combined to obtain dual judgment to avoid the problem caused by no current zero crossing during a DC fault and to ensure the reliability of the detection.

For determining fault occurrence, the fault type needs to be identified. The differences between the fault information matrices are analyzed for different types of faults and the “double D method” is introduced to obtain the true validity of the method. The proposed method is suitable for a variety of topologies and fault types, and is easy to implement. Although it does require communication devices which increase the costs, with the development of DC distribution networks, communication devices will become essential from the dispatch point of view.

## Declarations

### Acknowledgements

Thanks to the guidance and help of Mr. Wang Yi, thanks for the help of brothers and sisters in the format of the essay, blessing the teachers to work smoothly and blessing the brothers and sisters to have a bright future.

### Funding

Thanks for the financial support from the following fund projects: Project Supported by National Natural Science Foundation of China (51607070).

### Availability of data and materials

Please contact author for data requests.

### The innovation

This study presents methods for selecting the faulty line and identifying the fault type in a DC power distribution network, using the graph theory. The entire distribution network is considered to be a graph with the nodes represented by the vertices and connection lines represented by the edges. Current transformers help identify the current directions and current rates, which can be used for the fault-line selection and fault-type identification. Simulations verify the efficiency of the proposed methods. We believe that our study makes a significant contribution to the literature because it is easy to implement and can be applied to a large variety of devices and fault types.

### Authors’ contributions

XY is in charge of the conception of this article. He proposed to introduce the knowledge of graph theory into the fault line selection of DC distribution network and the structure of the whole article is controlled. LJY applied specific theoretical knowledge to the DC distribution network fault, and analyzed in detail the fault diagnosis method for different fault types, and drafted this article. FY is responsible for the simulation part of this article. All authors read and approved the final manuscript.

### Authors’ information

**Y. Xu** (1976-), male, Associate Professor, Major in new energy, power system relay protection.

**J. Y. Liu** (1993-), female, Master graduate student, Major in fault characteristics and protection technology of flexible DC transmission and distribution.

**Y. Fu** (1982-), male, lecturer, Engaged in wind power control technology research.

### Competing interests

This manuscript has not been published or presented elsewhere in part or in entirety and is not under consideration by another journal. We have read and understood your journal’s policies, and we believe that neither the manuscript nor the study violates any of these. The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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