# Research on Distribution Network Security Analysis Based onK(N-1+k) Criterion and Rank Preference Optimal Ordering

Authors: Wang Zengping; Yao Yuhai; Yu Junya; Yun Chentai
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

The objective of security analysis is to find the weak link of distribution network. TheK(N-1+k) criterion based on the characteristics of distribution network was proposed for distribution network security analysis. According to the K(N-1+k) criterion, the electrical devices in the contingency set can be classified into two kinds. The first kind meets the requirement of the K(N-1+k) criterion. The rank preference optimal ordering (RPOO) was proposed to evaluate the damage degree of power system. The second kind does not meet the requirement of the K(N-1+k) criterion, and it is the weak link of distribution network. A numerical experiment shows that the method is efficient and feasible, and the proposed method can provide assistant decision-making for safety precautions.


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## Body

## 1. Introduction

With the rapid development of distribution management system (DMS), distribution network security analysis (DNSA) is a vital part of DMS and has attracted wide attention [1].Compared to the study of transmission network security analysis (TNSA), the study of DNSA is still relatively less. Distribution network is designed in closed loop and operated in open loop [2, 3]. Transmission network has no limit on operation in open loop [4, 5]. The TNSA is based on the N-1 criterion [6–8]. The essence of the N-1 criterion is uninterrupted power supply, when one of electrical devices is fault. There are lots of switches in distribution network for the failure isolation and restoration. When permanent fault occurs in the distribution network, the short-term power off is inevitable in the nonfault and out-of-service areas. In this perspective, the DNSA cannot simply copy the N-1 criterion of TNSA.Traditional DNSA [9, 10] only focused on finding out the electrical devices which dissatisfy the requirement of the security analysis criterion. And it is not to evaluate the damage degree caused by the fault of these electrical devices. Once an electrical device is fault, it will bring a harmful effect on power system. DNSA should be to evaluate the damage degree after a fault occurs and to further find out the potential weakness in power grid.This paper presented theK(N-1+k) criterion for DNSA. According to the K(N-1+k) criterion, the rank preference optimal ordering method (RPOO) was introduced to evaluate damage degree, and it can identify the potential weakness in distribution network.


## 2.K(N-1+k) Criterion

Considering the characteristics of the distribution network, study [11] presented the N-1+1 criterion for DNSA. The N is the number of electrical equipment in power system. The −1 means that one of electrical devices is fault and isolation. The +1 means that one time of switching operation is needed for supply restoration. However many studies [12–14] illustrate that numbers of switching operation are required for supply restoration. Thus, the N-1+1 criterion for DNSA is not fit to the practical engineering.The analysis of DNSA is based on the given operation mode. It is supposed that the electrical devices are fault one by one. Under the condition of the system operation constraint, the DNSA is to evaluate the safety of distribution network and to find out the weaknesses in the system by judging whether power supply can be resumed completely.TheN-1+k criterion is improved based on the N-1+1 criterion. When one of the electrical devices is fault and it has been detected and isolated, in order to restore power in the nonfault and out-of-service areas, k times of switching operation are need.Because of the variable structure characteristic of distribution network, the restoration scheme is not always single butK. In the perspective of the security, greater value of K is desired.Based on theN-1+k criterion, the K(N-1+k) criterion is proposed further. The K is the number of feasible restoration schemes which meet the requirement of the N-1+k criterion.


## 3. Evaluation Indexes of DNSA

The main purpose of DNSA is to discover the potential weakness. According to theK(N-1+k) criterion, the electrical devices in the contingency set can be classified into two kinds. The first kind meets the requirement of the K(N-1+k) criterion. It will cause negative effects on the power system when this kind of the electrical devices is fault. The evaluation of negative effects can find the weakness in the system. The second kind does not meet the requirement of the K(N-1+k) criterion, and it is the weak link in power system. For this kind of the electrical devices, the customers that cannot be served need more attention.


### 3.1. The Evaluation Indexes for the First Kind

For the devices that meet the requirement of theK(N-1+k) criterion, the evaluation indexes of DNSA are as follows.Thef1 is the amount of restorable loads in nonfault and out-of-service areas, and it is proportional to the short-term power failure:
(1)f1=∑i∈APi,
where A is the collection of restorable nodes and Pi is the active power of the ith node.Thef2 is the time of switching operation for service restoration and it represents the cost of restoration process:
(2)f2=k,
where k is the time of switching operation.Thef3 is the load rate and itdescribes the maximum load rate of the electrical device in distribution network:
(3)f3=max⁡i∈B(IiINi),
where Ii is the current at the ith feeder; INi is the rated current of the ith feeder; B is the collection of feeders.Thef4 is the power loss of the power grid:
(4)f4=∑i∈BPiloss,
where Piloss is the ith feeder power loss.Thef5 is the lowest bus voltage in the distribution network after service restoration. For the quality of power supply, the f5 should be as high as possible:
(5)f5=min⁡i∈Ω(Vi),
where Vi is the voltage of the ith bus and Ω is the collection of the buses.Thef6 is the number of the feasible restoration schemes:
(6)f6=K,
where K is the number of the feasible restoration schemes.According to the indexes above, thef1 and f2 are cost type index, and the f3,f4,f5, and f6 are benefit type index. The smaller index value of cost type is better, and the larger index value of cost type is better.The analytic hierarchy process (AHP) [15] is applied to determine the weight of each index. The f1 and f2 are cost index. The f3 and f6 are reliability index. The f4 and f5 are economy index. The hierarchy model is shown as Figure 1.Figure 1
The hierarchy model of the first kind.


### 3.2. The Evaluation Indexes for the Second Kind

For these devices that do not meet the requirement of theK(N-1+k) criterion, the situation of power supply is the main concern. The f1, f2, and f7 are used to evaluate the security of distribution network, and they are cost type index.Thef7 is the amount of unrecovered loads in nonfault and out-of-service areas:
(7)f7=∑i∈A′Pi,
where A′ is the collection of unrecovered buses.The hierarchy model is shown as Figure2.Figure 2
The hierarchy model of the second kind.


## 3.1. The Evaluation Indexes for the First Kind

For the devices that meet the requirement of theK(N-1+k) criterion, the evaluation indexes of DNSA are as follows.Thef1 is the amount of restorable loads in nonfault and out-of-service areas, and it is proportional to the short-term power failure:
(1)f1=∑i∈APi,
where A is the collection of restorable nodes and Pi is the active power of the ith node.Thef2 is the time of switching operation for service restoration and it represents the cost of restoration process:
(2)f2=k,
where k is the time of switching operation.Thef3 is the load rate and itdescribes the maximum load rate of the electrical device in distribution network:
(3)f3=max⁡i∈B(IiINi),
where Ii is the current at the ith feeder; INi is the rated current of the ith feeder; B is the collection of feeders.Thef4 is the power loss of the power grid:
(4)f4=∑i∈BPiloss,
where Piloss is the ith feeder power loss.Thef5 is the lowest bus voltage in the distribution network after service restoration. For the quality of power supply, the f5 should be as high as possible:
(5)f5=min⁡i∈Ω(Vi),
where Vi is the voltage of the ith bus and Ω is the collection of the buses.Thef6 is the number of the feasible restoration schemes:
(6)f6=K,
where K is the number of the feasible restoration schemes.According to the indexes above, thef1 and f2 are cost type index, and the f3,f4,f5, and f6 are benefit type index. The smaller index value of cost type is better, and the larger index value of cost type is better.The analytic hierarchy process (AHP) [15] is applied to determine the weight of each index. The f1 and f2 are cost index. The f3 and f6 are reliability index. The f4 and f5 are economy index. The hierarchy model is shown as Figure 1.Figure 1
The hierarchy model of the first kind.


## 3.2. The Evaluation Indexes for the Second Kind

For these devices that do not meet the requirement of theK(N-1+k) criterion, the situation of power supply is the main concern. The f1, f2, and f7 are used to evaluate the security of distribution network, and they are cost type index.Thef7 is the amount of unrecovered loads in nonfault and out-of-service areas:
(7)f7=∑i∈A′Pi,
where A′ is the collection of unrecovered buses.The hierarchy model is shown as Figure2.Figure 2
The hierarchy model of the second kind.


## 4. The Evaluation Method for DNSA

The rank preference optimal ordering method (RPOO) [16] was introduced for DNSA. On the multiobjective decision-making method, the RPOO is simple and practical, and it can get the obvious differences among alternatives.


### 4.1. The Rank Preference Optimal Ordering Method

Them schemes are needed to be evaluated, and each scheme has n evaluation indexes.The evaluation matrixR(m×n) is as follows:
(8)R=[fij]m×n,
where fij is the actual value of the jth index related to the ith evaluation scheme.The index can be divided into two categories, the benefit type and the cost type. The higher the index value of the benefit type is, the better it is. And the cost type is just opposite. Theh intervals are formed from the maximum index value to the minimum index value. The h is larger, and the different index value is easier to fall into the different intervals.Theh intervals of the benefit type are divided as follows:
(9)1[min⁡j⁡{fij},min⁡j⁡{fij}+1hdj)⋮⋮k[min⁡j⁡{fij}+k-1hdj,min⁡j⁡{fij}+khdj)⋮⋮n[min⁡j⁡{fij}+h-1hdj,max⁡j⁡{fij}].Theh intervals of the cost type are divided as follows:
(10)1[min⁡j⁡{fij}+h-1hdj,max⁡j⁡{fij}]⋮⋮k[min⁡j⁡{fij}+k-1hdj,min⁡j⁡{fij}+khdj)⋮⋮n[min⁡j⁡{fij},min⁡j⁡{fij}+1hdj).Thedj is defined as follows:
(11)dj=max⁡j⁡{fij}-min⁡j⁡{fij},
where the maxj{fij} and minj{fij} denote the maximum and minimum value of the jth index related to all schemes. The dj is the difference between the maximum and minimum value of the jth index.TheG is the index rank matrix and is defined as follows:
(12)G=[gij]m×n,
where the gij is the rank value of the jth index related to the ith evaluation scheme.The index optimal number,ailj, describes the dominant degree which is the comparison of the jth indexrelated to the ith scheme and the jth index related to the lth scheme, and it is defined as follows:
(13)ailj={1gij-glj=h-1⋮⋮h-12(h-1)-kgij-glj=k⋮⋮h-12(h-1)-1gij-glj=10.5gij=glj0other.To solve the practical multiobjective decision-making problem, the relative importance of each index needs to be taken into account. The weight of index is as follows:(14)∑j=1nwj=1,
where the wj is the weight of the jth index and wj∈[0,1].The optimal number,ail, describes the dominant degree which is the comparison of the ith scheme and the lth scheme, and it is defined as follows:
(15)ail={∑j=1nwjailji≠l,0i=l.The total optimal number (TON) describes the dominant degree which is the comparison of theith scheme and the other schemes. Larger TON is better. The TON is defined as follows:
(16)Ti=∑l=1mail,
where Ti is the TON of the ith scheme.


### 4.2. The Steps of the RPOO

The steps of the RPOO method are shown as follows.Step
1. Calculate the index values of each scheme, and obtain the evaluation matrix.Step
2. Determine the weights and the h, and get the index rank matrix.Step
3. According to the index rank matrix, get the index optimal number.Step
4. According to the index optimal number and the weight, get the optimal number.Step
5. Calculate the TON of each scheme according to the optimal number.


### 4.3. The Steps of the DNSA

The steps of the DNSA are shown as follows.Step
1. Establish the contingency set, and obtain the weights of the evaluation index based on the AHP.Step
2. Assume that the electrical devices in the contingency set are fault one by one, and get, respectively, the feasible restoration scheme and the best restoration scheme by the service restoration algorithm [17].The paper [17] used the nondominated sorting genetic algorithm-II to solve the service restoration. The feasible restoration scheme is the Pareto optimal solution, and the best restoration scheme is got by the preference knowledge described in the paper [17].Step
3. Calculate the index values of DNSA based on the best restoration scheme and the feasible restoration scheme.Step
4. According to the K(N-1+k) criterion, the electrical devices in the contingency set are classified into two kinds. The first kind meets the requirement of the K(N-1+k) criterion. The second kind does not meet the requirement of the K(N-1+k) criterion.Step
5. Evaluate the two kinds of electrical equipment, respectively, by using the RPOO.


## 4.1. The Rank Preference Optimal Ordering Method

Them schemes are needed to be evaluated, and each scheme has n evaluation indexes.The evaluation matrixR(m×n) is as follows:
(8)R=[fij]m×n,
where fij is the actual value of the jth index related to the ith evaluation scheme.The index can be divided into two categories, the benefit type and the cost type. The higher the index value of the benefit type is, the better it is. And the cost type is just opposite. Theh intervals are formed from the maximum index value to the minimum index value. The h is larger, and the different index value is easier to fall into the different intervals.Theh intervals of the benefit type are divided as follows:
(9)1[min⁡j⁡{fij},min⁡j⁡{fij}+1hdj)⋮⋮k[min⁡j⁡{fij}+k-1hdj,min⁡j⁡{fij}+khdj)⋮⋮n[min⁡j⁡{fij}+h-1hdj,max⁡j⁡{fij}].Theh intervals of the cost type are divided as follows:
(10)1[min⁡j⁡{fij}+h-1hdj,max⁡j⁡{fij}]⋮⋮k[min⁡j⁡{fij}+k-1hdj,min⁡j⁡{fij}+khdj)⋮⋮n[min⁡j⁡{fij},min⁡j⁡{fij}+1hdj).Thedj is defined as follows:
(11)dj=max⁡j⁡{fij}-min⁡j⁡{fij},
where the maxj{fij} and minj{fij} denote the maximum and minimum value of the jth index related to all schemes. The dj is the difference between the maximum and minimum value of the jth index.TheG is the index rank matrix and is defined as follows:
(12)G=[gij]m×n,
where the gij is the rank value of the jth index related to the ith evaluation scheme.The index optimal number,ailj, describes the dominant degree which is the comparison of the jth indexrelated to the ith scheme and the jth index related to the lth scheme, and it is defined as follows:
(13)ailj={1gij-glj=h-1⋮⋮h-12(h-1)-kgij-glj=k⋮⋮h-12(h-1)-1gij-glj=10.5gij=glj0other.To solve the practical multiobjective decision-making problem, the relative importance of each index needs to be taken into account. The weight of index is as follows:(14)∑j=1nwj=1,
where the wj is the weight of the jth index and wj∈[0,1].The optimal number,ail, describes the dominant degree which is the comparison of the ith scheme and the lth scheme, and it is defined as follows:
(15)ail={∑j=1nwjailji≠l,0i=l.The total optimal number (TON) describes the dominant degree which is the comparison of theith scheme and the other schemes. Larger TON is better. The TON is defined as follows:
(16)Ti=∑l=1mail,
where Ti is the TON of the ith scheme.


## 4.2. The Steps of the RPOO

The steps of the RPOO method are shown as follows.Step
1. Calculate the index values of each scheme, and obtain the evaluation matrix.Step
2. Determine the weights and the h, and get the index rank matrix.Step
3. According to the index rank matrix, get the index optimal number.Step
4. According to the index optimal number and the weight, get the optimal number.Step
5. Calculate the TON of each scheme according to the optimal number.


## 4.3. The Steps of the DNSA

The steps of the DNSA are shown as follows.Step
1. Establish the contingency set, and obtain the weights of the evaluation index based on the AHP.Step
2. Assume that the electrical devices in the contingency set are fault one by one, and get, respectively, the feasible restoration scheme and the best restoration scheme by the service restoration algorithm [17].The paper [17] used the nondominated sorting genetic algorithm-II to solve the service restoration. The feasible restoration scheme is the Pareto optimal solution, and the best restoration scheme is got by the preference knowledge described in the paper [17].Step
3. Calculate the index values of DNSA based on the best restoration scheme and the feasible restoration scheme.Step
4. According to the K(N-1+k) criterion, the electrical devices in the contingency set are classified into two kinds. The first kind meets the requirement of the K(N-1+k) criterion. The second kind does not meet the requirement of the K(N-1+k) criterion.Step
5. Evaluate the two kinds of electrical equipment, respectively, by using the RPOO.


## 5. Case Studying

In order to verify the performance of the proposed method, a model [18] as Figure 3 is used. The solid line represents the on state of the switch, and the dashed line represents the off state of the switch. The system holds 16 feeders, and each feeder has switch. Its rated voltage is 23 kV. It is assumed that the rated current of each feeder is 700 A. The sum of the loads is 28.7 + j17.3 MVA. With the operation condition shown in Figure 3, the network loss is 511.44 kW. And the current of feeders is showed as follows: 1–4 (394.13 A), 2–8 (691.61 A), and 3–13 (223.53 A).Figure 3
IEEE distribution system diagram with 16 feeders.It is assumed that the contingency set comprises all 16 feeders. And theh is 100.The weight of each index is obtained by the method of AHP. The judgment matrix and the weights of the second kind are shown in Table1.Table 1
Judgment matrix and weight for the security analysis.
Security analysis
f
1
f
2
f
7
Weight
f
1
1
3
1/7
0.1549
f
2
1/3
1
1/9
0.0685
f
7
7
9
1
0.7766The judgment matrixes and the weights of the first kind are shown in Tables2, 3, 4, 5, and 6.Table 2
Judgment matrix for the security analysis.
Security analysis
C
E
R
C
1
3
1/5
E
1/3
1
3
R
5
1/3
1Table 3
Judgment matrix for the cost.
Cost
f
1
f
2
f
1
1
1/3
f
2
3
1Table 4
Judgment matrix for the economy.
Economy
f
4
f
5
f
4
1
5
f
5
1/5
1Table 5
Judgment matrix for the reliability.
Reliability
f
3
f
6
f
3
1
3
f
6
1/3
1Table 6
Weight for the security analysis.
Index
f
1
f
2
f
3
f
4
f
5
f
6
Weight
0.0748
0.2245
0.2762
0.2772
0.0554
0.0921It is assumed that the electrical devices in the contingency set are fault one by one, and the restoration algorithm [17] is used to get the corresponding solutions. According to the K(N-1+k) criterion, the electrical devices in the contingency set were classified into two kinds. And the RPOO was introduced to evaluate the damage effect of the two kinds.Table7 shows the second kind. This kind does not meet the requirement of the K(N-1+k) criterion. When a fault occurs on 9–12, there are no switches connecting with the nonfault and out-of-service areas, node 12, so the load of node 12 cannot be transformed to other feeders. When one of 4-5, 13-14, and 2–8 is fault, there is not enough usable amount of transmission capacity to transform the load of the nonfault and out-of-service areas. So, malfunction of these electrical devices reduces the reliability of power supply and should be taken seriously.Table 7
The electrical devices of the second kind.
Electrical device
The best restoration scheme
f
1/MW
f
2
f
7/MW
TON
4-5
4-5
0
0
3
1.3787
5–11
7–16
10–14
2–8
2–8
9.6
4
4
0.3883
9–12
8-9
7–16
9–12
5–11
0
0
4
0.8351
7–16
10–14
9–12
13-14
5–11
0
0
1
2.5825
7–16
10–14
13-14From Table7 we can see that the fault of 2–8 causes the most negative impacts on the distribution network than the others. When 2–8 is fault, 9.6 MW loads are power failure temporarily, and 4 MW loads cannot be restored. And the TON of the 2–8 is the smallest. So, the TON can reflect damage degree caused by the malfunction of the electrical devices, and the TON is in inverse proportion to the damage effect of the distribution network.Table8 shows the first kind. This kind meets the requirement of the K(N-1+k) criterion.Table 8
The electrical devices of the first kind.
Electrical device
The best restoration scheme
f
1/MW
f
2
f
3
f
4/kW
f
5
f
6
TON
1–4
1–4
8.5
1
0.9880
945.24
0.9348
3
2.2305
5–11
10–14
4–6
4–6
3.5
3
0.9125
551.52
0.9714
4
2.8580
5–11
8–10
6-7
6-7
1.5
3
0.9125
497.54
0.9714
3
3.8306
5–11
8–10
8–10
5–11
1
1
0.9125
483.86
0.9714
2
5.3212
8–10
7–16
8-9
4–6
10.1
3
0.9792
703.46
0.9578
2
1.6052
8-9
10–14
9–11
9–11
0.6
1
0.9574
493.15
0.9693
2
4.2470
10–14
7–16
3–13
9–11
5.1
3
0.9574
658.84
0.9662
2
2.1315
10–14
3–13
13–15
5–11
3.1
3
0.9125
539.82
0.9714
3
3.2484
8–10
13–15
15-16
5–11
2.1
3
0.9125
508.02
0.9714
3
3.4830
8–10
15-16Form the indexes shown in Table8, the fault of 8-9 causes the most negative impacts on the distribution network than the others. When 8–10 is fault, the operation condition is better than the others. And the TON reveals the trend.The TON can get the obvious differences among alternatives, and it can help engineers to find weak points of the distribution network.


## 6. Conclusion

TheK(N-1+k) criterion was proposed for distribution network security analysis, and it is suitable for the radial structure of distribution network. The RPOO was introduced to the multi-index decision-making, and it can get the obvious differences among alternatives. The total optimal number can reflect damage degree caused by the fault of electrical device in the contingency set. The validity and effectiveness of the proposed approach are verified with a numerical example. The proposed approach can point out the weakness of the distribution network and provide guidance for the distribution network transformation and the security defense.



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