Minimal cut sets are the basis of reliability analysis using analytical techniques. At the present stage, minimal cut sets are mainly obtained by dealing with minimal path sets, which involves cumbersome steps and slower operational speed. The speed of reliability analysis is limited by that of calculating minimal cut sets. In consideration of the characteristics of microgrid, a hierarchical approach for fast calculating minimal cut sets is proposed in this paper. Firstly, an equivalent principle is proposed to convert topology structure into network node diagram; then grades of nodes are designated based on their original connection and the breadthfirst search approach; afterwards,
A microgrid is a small electrical power generation and distribution system consisting integrally of distributed generation, loads, energy storage devices, converters, monitoring, and protection devices [
Compared to the microgrid, the traditional grid is mainly characterized with high safety margin and strong selfhealing, so it can be regarded as a repairable system. Based on the statistical data or empirical data [
The approach proposed in this paper focuses on isolated microgrid. The isolated microgrid will have lower safety margin and antirisk ability when it loses the support of the traditional grid. Considering the actual operation, many faults cannot be eliminated in time, and the isolated microgrid inclines to be an unrepairable system in essence. Therefore, the minimal cut sets are used as core in the reliability analysis method of the unrepairable system. The calculation of minimal cut sets directly restricts the speed and efficiency of reliability analysis, and a lot of works have been done in the face of bottleneck problem.
Reference [
There are three ways to get the minimal cut sets nowadays [
In consideration of the characteristics of unrepairable and simple structure in the microgrid, this paper presents a quick way to calculate minimal cut sets, that is, hierarchical approach. First of all, an applicable equivalent principle is proposed that convert structure of power system into network node diagram based on graph theory. Nodes are classified and ranked by breadthfirst search; then the total minimal cut sets are traversed out by replacing high graded nodes with low graded nodes. Taking the electrical power system in type A380 more electric aircraft as an example, this paper verifies the correctness and superiority of the proposed method by comparisons.
Different from the traditional method from minimal path sets to minimal cut sets, the hierarchical approach designates grades for nodes; then all nodes are traversed in terms of their grades vertically to get the minimal cut sets. The overall flow diagram of the hierarchical approach is shown in Figure
The overall flow diagram of the hierarchical approach.
In this step, topology structure is transformed into network node diagram. Generally speaking, for a large grid, the follow guidelines must be obeyed: the network node diagram should reflect connections and working status of the original topology structure; the network node diagram should be as simple and easy as possible to enable making full use of the functions of computers.
However, there are many differences between microgrid and traditional grid. As electrical components in a microgrid feature with unrepairable, simple structure, and low concurrency, it is possible to find a more proper and simple equivalent principle when converting topology structure into network node diagram by considering such specifications. Hence, the concrete rules adopted are as follows:
All components, such as generator, transformer rectifier, and DCbus, are replaced by nodes.
Arcs between nodes only reflect their connection relationship and direction of power flow. Any arc connecting a node standing for a component controlling energy flow is directed arc, such as transformer rectifiers and diodes; otherwise, it is undirected.
Any singleline bidirection connected node is permitted to be absorbed by other nodes jointed with multiline or by directed arcs.
Set a virtual node as source of power injected into the entire microgrid. Power is injected directly to nodes standing for generators or batteries from virtual source. Load nodes are taken as trap nodes consuming all power energy, and their arcs do not consume any power.
Different from the mainstream equivalent rules, this principle has two particular improvements combining with features of microgrid:
All components in microgrid are represented as nodes because of their simple connection relationship, and an arc just represents a relationship existing between two nodes. So, it needs not to classify and discuss nodes and arcs.
Only critical nodes stand for the reliability of an overall system, and irrelevant nodes are allowed to be extensively absorbed. Nodes outside research range or with unidirectional flow of energy can be replaced with relevant nodes surrounding since they have few impact on critical nodes. This rule can greatly reduce the total number of nodes in the system and increase operation speed.
This procedure is to designate grades for nodes in a network node diagram by considering the directions of energy flow using breadthfirst search and then to build
This section mainly includes how to create
Grades of nodes are designated by breadthfirst search: the length of an arc is recorded as 1. For a specific node, its distance to the source node equals the number of arcs between them, which is defined as its grade. In this way, all nodes in network node diagram have their own grades (
The source node is a virtual node which grade is set as 0. Grades of nodes connected with directed arcs are adjusted in this way that a directed arc is pointed from node at high grade (small value) to node at low grade. The nodes with the same grades can be connected by undirected arcs to indicate that flow energy is undirected.
Row vectors of a
Obviously, a leaf node could be root of nodes at lower level. On the one hand, the
Vertical traversing is to search for all the combination of all nodes. According to grades of nodes, root nodes are replaced with leaf nodes in vertical direction so as to find all free combination among different grades. To prevent that different root nodes have the same leaf node, minimized testing must be performed to ensure all minimal cut sets obtained are different.
Figure
Flowchat for obtaining minimal cut sets on the basis of
The variable
All possible cut sets are stored in the
This new cut set will be put into the
It is said that a newly formed vector is minimized if leaf nodes of any arbitrary node are not completely contained by its previous vector. This method of testing minimization is only applicable for a new vector obtained each time.
Figure
A radial network diagram.
The standard of comparison is the time required to complete all the calculation. It is supposed that a basic operation (it refers to a logical operation or search for a node) costs
The time of mainstream method is
It mainly consists of two parts: the minimal path set is obtained by depthfirst search, costing
The firstorder minimal cut set can be directly obtained, and the secondorder minimal cut set is obtained by operation “OR” on nodes corresponding to any two paths. Similarly, third, fourth, and
Equation (
The proposed approach costs
The hierarchical approach includes two parts: set up grades by using breadthfirst search, costing
The time taken by the two methods is compared, as shown in
The mainstream method needs more time than the hierarchical approach, when
Electrical power systems of more electric aircraft are characterized by unrepairable, simple connection, low concurrency, and high redundancy. In consideration of these characters and the topologies, they can be regarded as typical isolated microgrids. The electrical power system of Airbus 380, a representative of more electric aircraft, is taken as a case to test the approach presented, whose electrical structures are shown in Figure
Power system structure sketch of A380.
The overall electrical system is roughly divided into four main channels: E1 channel is powered by AC generators G1 and G2; E2 channel by AC generators G3 and G4; E3 channel by RAT (Ram Air Turbine) and static converters; and APU (Auxiliary Power Unit) channel associated with APU startup.
A380 utilizes variable frequency generating technology, so the ACbuses operate independently, while the DCbuses run in parallel. This parallelcluster combines the advantages of operation of independent and parallel: on the one side, the DC system in parallel could ensure uninterrupted power supply; on the other hand, independent operation could effectively prevent the occurrences of cascading failure.
According to the equivalent principle presented in Section
Network nodes diagram.
The power is generated from the virtual source node v1 and the flows into ACbuses (v8, v9, v10, and v11) through generators (v3, v4, v6, and v7). Node v2 represents that power is generated from RAT into ACbuses for keeping the key equipment running on emergency. Node v5 represents APU. Node v20 represents ACbus for key equipment, while node v21 represents static converter or BCRU, depending on the direction of energy flow. Power electronics interface, representation of nodes v13, v14, 15, v16, v25, v27, and v29, is used to connect different buses. DCbuses are represented by nodes v24, v26, v28, and v30.
As mentioned in the equivalent principle, a portion of irrelevant loads can be absorbed by power supply nodes nearby. For example, the AC loads are absorbed by nodes 9, 10, and 11 represented by ACbus.
As indicated in Section
Network node diagram with node grades.
As indicated in Section
Entire minimal cut sets.
Node 20  

1  v20 
2  v8v11v24v2 
3  v8v11v26v2 
4  v8v11v21v2 
5  v8v11v25v2 
6  v8v24v2v22 
7  v8v26v2v22 
8  v8v2v22v21 
9  v8v2v22v25 
10  v11v24v2v12 
11  v11v26v2v12 
12  v11v2v12v21 
13  v11v2v12v25 
14  v24v2v12v22 
15  v26v2v12v22 
16  v2v12v22v21 
17  v212v22v25 
Power system of more electric aircraft is taken as an example to be equivalent to network node diagram with 30 nodes. The approach is in comparison with mainstream on platform of MATLAB about operational time and result. There is the same result between two methods. Except the artificial step of equivalence, minimal cut sets are got after calculating minimal path sets detailed in Table
Minimal path sets from source to node 20.
Length of path  Node 20 

3  v1v2v20 


5  v1v3v8v12v20 
v1v7v11v22v20  


7  v1v5v18v13v8v12v20 
v1v5v18v16v11v22v20  


9  v1v4v9v14v18v13v8v12v20 
v1v6v10v15v18v13v8v12v20  
v1vv7v11v16v18v13v8v12v20  
v1v3v8v13v18v16v11v22v20  
v1v4v9v14v18v16v11v22v20  
v1v6v10v15v18v16v11v22v20  
v1v4v9v17v26v25v24v21v20  


11  v1v5v18v14v9v17v26v25v24v21v20 
v1v6v10v19v28v27v26v25v24v21v20  


13  v1v3v8v13v18v14v9v17v26v25v24v21v20 
v1v6v10v15v18v14v9v17v26v25v24v21v20  
v1v7v11v16v18v14v9v17v26v25v24v21v20  
v1v5v18v15v10v19v28v27v26v25v24v21v20  
v1v7v11v23v30v29v28v27v26v25v24v21v20  


15  v1v3v8v13v18v15v10v19v28v27v26v25v24v21v20 
v1v4v9v14v18v15v10v19v28v27v26v25v24v21v20  
v1v7v11v16v18v15v10v19v28v27v26v25v24v21v20  
v1v5v18v16v11v23v30v29v28v27v26v25v24v21v20  


17  v1v3v8v13v18v16v11v23v30v29v28v27v26v25v24v21v20 
v1v4v9v14v18v16v11v23v30v29v28v27v26v25v24v21v20  
v1v6v10v15v18v16v11v23v30v29v28v27v26v25v24v21v20 
The superiorities of this method were verified in the field of a microgrid, and this method would have a faster speed and wider application on structure with more nodes.
Minimal cut sets play important roles in reliability assessment. This paper presents a hierarchical approach for fast calculating minimal cut sets of a microgird. This approach simplifies the steps to get minimal cut sets and increases operation speed. Quick calculation of minimal cut sets can improve the speed of reliability analysis; it also can reduce memory footprint during the procedure of reliability analysis by avoiding minimal path sets.
A topological equivalence principle for microgrid is also proposed, which potentially advances the pace of study on reliability assessment. Meanwhile, the approach proposed broadens applications of the breadthfirst search and enhances the feasibility of reliability assessment based on minimal cut sets. This approach can also be extended and applied to all forms of microgrids, which is provided as a supplementary method to the existing reliability assessment system.
The authors declare that they have no conflicts of interest.
The authors would like to acknowledge financial support from National Natural Science Foundation of China (U1533126).