Reliability evaluation of distribution systems considering optimal restoration sequence and variable restoration times P. Wang and W. Li Abstract: Switching sequence, available restoration resources and customer reliability requirements have an important impact on customer interruption times in distribution system failure restoration. An optimal restoration sequence based on the minimum interruption cost of customers has been proposed and incorporated in the reliability evaluation of distribution systems. A faulttraversal algorithm has been used to dynamically trace the affected areas of failure events and the switches involved for the failure isolation and supply restoration. Multi-state models for sequential, parallel and hybrid switching processes have been proposed to determine variable restoration times of different load points. A test distribution system is used to illustrate the technique.
1
Introduction
Restoration time is a very important parameter in the reliability evaluation of distribution systems. For a distribution system, which consists of a wide variety of components connected in a complex configuration and operating in different modes, supply restoration is a quite complicated process that involves fault isolation, switch allocation, decision-making, switching and repairing actions. Restoration time for each individual load point depends not only on the physical systems such as system configuration, the types and locations of switches and alternative supplies, but also on the speed of decisionmaking for restoration, available manpower for the restoration and customer reliability requirement. Therefore restoration times for different load points can be quite different for a particular failure. The techniques used for the service restoration have been investigated and implemented in distribution system operation [1 – 4]. The main focus of these techniques is to optimise the restoration sequence to reduce restoration times. However, these restoration techniques have seldom been implemented in the reliability evaluation of distribution systems. Most conventional techniques [5– 7] for the reliability evaluation of distribution systems are based on the minimum cut set approach, the failure mode and effect analysis and the network-equivalent technique [7]. In these techniques, the fixed switching time, repair time or replacement time are used to approximately determine failure duration of each load point without considering switching sequences. The impact of the two-stage service restoration on distribution system reliability is considered in [8]. In [9], all load points are divided into limited classes, and the fixed restoration time for a given class is used in reliability evaluation. # The Institution of Engineering and Technology 2007 doi:10.1049/iet-gtd:20060451 Paper first received 28th October 2006 and in revised form 18th January 2007 P. Wang is with the Nanyang Technological University, Singapore 639798, Singapore W. Li is with the Harbin Institute of Technology (HIT), Harbin 150001, China E-mail:
[email protected]
688
However, restoration times for customers in a practical distribution system depend on many factors and sometimes cannot be presented by the limited classes. Decision-making process, available manpower for switching and customer reliability requirements have not been considered in these techniques. The optimal switching sequence based on customer willingness to pay for the reliability also has not been considered. This paper proposes a generalised technique for reliability evaluation of distribution systems to incorporate the optimal restoration sequence. A fault-traversal technique [9], including parent-search, offspring-search and breadth-first-search, is applied to dynamically trace the affected area, downstream and upstream switches. The multi-state models have been proposed to determine the times for the service restorations with sequential, parallel and hybrid switching. The parallel, sequential and hybrid restoration processes may involve single and multiple switching actions. Customer reliability requirements in terms of customer interruption costs are considered in the determination of restoration sequence. An optimal restoration sequence of the affected areas based on the degree of system automation, available manpower and customer reliability requirements has been incorporated in the reliability evaluation technique. Restoration times of different load points are determined based on the optimal restoration sequence and multi-state models for the restoration. The network capacity constraints after system reconfiguration are also considered using an improved ac power flow technique. A test distribution system is analysed to illustrate the technique. 2
Restoration processes and models
Switching/repairing models [5] used in existing techniques for the reliability evaluation of distribution systems and switch stations are usually the two-state model as shown in Fig. 1 or the three-state model as shown in Fig. 2. In these figures, O, S and R represent operating, switching and repairing states, respectively, and l, m and ls are component failure rate, repair (or replacement) rate and switching rate, respectively. These models are approximate IET Gener. Transm. Distrib., 2007, 1, (4), pp. 688 – 695
Fig. 1 Two-state repairing model
because the complicated switching actions are either ignored as shown in Fig. 1 or are simply presented using a single switching state S as shown in Fig. 2. However, when a failure occurs in a distribution system, the system usually experiences a series of processes of fault isolation, determination of affected area, switch allocation, decision-making for the service restoration, switching and repairing/replacing actions [9]. Restoration times for load points after a failure may be different due to the failure location, the number and type of switches involved, and the available manpower for the restoration and the switching sequences. Some load points can be restored using automatic switching actions, whereas some may require manual switching actions and others may wait until the failure component is repaired or replaced by spare element. Some load points may require more switching actions than others, which depends on the system configuration and the fault location. The speeds of switching and repairing actions depend on the degree of system automation and available switching resources. Even for the same customer, the interruption time changes with the location of the failure. The processes involved in the restoration are usually classified into the followings.
the failed components and the service restoration to the affected load points through network reconfigurations. The speed of switching process depends on the number of switches involved, the types and locations of these switches, available resources to open or close the switches and the switching sequences such as parallel, sequential and the combination of both. Therefore the total switching time for different load point may be different. For a sequential switching, the total switching time is the summation of switching times of all switches involved X TSs ¼ TSk (1) k[N
where TSk is the switching time for switch k and N the set of switches involved in switching actions. For a parallel switching, the total switching time is the longest one of all the switching times TSp ¼ max(TS1 , TS2 , . . . , TSN )
For the combination of parallel and sequential switching, the total switching time is TSh ¼ TSs þ TSp
Fault isolation process
When a failure occurs, the associated protection equipment in the protection zone will operate to interrupt all load points affected. The duration (Ti) of this process is very short and can be ignored in the total restoration time. 2.2
Decision-making process
After a fault, the related fault information and system configuration data are collected and analysed to find the location of the fault and to determine the affected area and the available switches for the restoration. The load flow analysis is performed after the reconfiguration, and the optimal restoration sequence is determined based on available restoration resources and customer reliability requirements using the optimisation technique. The duration of decision-making (Td) depends on the software used and the complexity of the failure. 2.3
Switching actions
Switching process involves various switching actions. These switching actions usually include the isolation of
Fig. 2 Three-state switching and repairing model IET Gener. Transm. Distrib., Vol. 1, No. 4, July 2007
(3)
It can be seen from the analysis that switching time varies with the switches involved and switching sequences used and cannot be presented using the limited values for the fixed classes [9]. 2.4
2.1
(2)
Repairing and replacing process
The repair time Tr usually depends on the type of the failed component and the available repairing resource. Different components may require different repair times. Repair time for a component may also be different for different failures. In reliability evaluation, the average repair time is used for each component. Repair time for a component is longer than its switching or replacing time. Therefore the failed component, sometimes, is replaced using spare component. It is assumed, in this paper, that the repairing or replacement follows the switching process. 2.5
Restoration models
Considering the processes involved, multi-state models for the service restoration of a load point have been developed for parallel, sequential and hybrid switching. The multistate model for the service restoration of a load point with sequential switching action is shown in Fig. 3. Fi and D in the figure are the fault isolation and decisionmaking states, respectively. Si is the state for switching action i. N is the number of switches involved in the switching actions. lsk ¼ 1/Tsk , li ¼ 1/Ti , l ¼ 1/T, ld ¼ 1/Td and m¼ 1/Tr are the switching rate, fault isolating rate, failure rate, the rate of decision-making for the restoration
Fig. 3 Multi-state model with sequential switching 689
the total restoration times after failure event h will not include Thr . 2.6
Fig. 4
Multi-state model with parallel switching
and repair & replacement rate, respectively. The sequential switching is usually used in distribution systems with manual switches and limited restoration manpower for manual switching. The total restoration time for load point j after failure event h for sequential switching actions is X Thj ¼ Thi þ Thd þ Thr þ TSk (4) k[Nhj
where Nhj is the number of switches involved in the switching actions. Fig. 4 shows the multi-state model for the service restoration with parallel switching actions. In a practical distribution system, the parallel switching can be implemented in the area with the automated switches and sufficient manpower for manual switching. For the manual switching, the parallel switching means that the repairing teams are sent to the switch locations simultaneously. The switching time in this case is mainly the time used for the manpower to reach the switch locations. The total restoration time for load point j after failure event h for parallel switching actions is Thj ¼ Thi þ Thd þ Thr þ max{TS1 , TS2 , . . . , TSN }
(5)
hj
Fig. 5 shows the multi-state model with hybrid (sequential/parallel) switching actions. The total restoration time for load point j after failure event h for the hybrid switching actions is X Thj ¼ Thi þ Thd þ Thr þ TSk þ max{TS1 , TS2 , . . . , TSN } k[Nhj
Optimal switching sequence
Switching sequence depends on the degree of system automation and available manpower for the manual switches. In a deregulated power system, the reliability for a customer also depends on its willingness to pay. Some customers may want to pay more for a high reliability and others may want to pay less for a low reliability. Customer reliability requirements can also be incorporated in the determination of the switching sequence. There may be many restoration sequences after considering these factors. Therefore the optimal switching sequence has to be determined based on available switching resources and customer reliability requirements. This will be discussed in the following section. 3
Determination of variable restoration times
From the above discussions, the determination of restoration times for different load points is a very complicated procedure which includes the following steps: 1. to determine the affected area and load points, 2. to identify the repair sub-area, upstream sub-areas, downstream sub-areas and the switches involved in each sub-area, 3. to determine the optimal switching sequence for the restoration and 4. to calculate the restoration time for each load point based on the optimal switching sequence and the processes involved. The procedure for determining the variable restoration times of load points in the reliability evaluation of distribution systems can be illustrated using a small complex distribution system shown in Fig. 6. Assume that disconnect switches S1 – S11 are normally closed, tie-switches K1 – K3 are normally opened, S1 is an automated switch and the remaining switches are manually operated. Manpower can only be used to do parallel switching involving two switches in this example. 3.1
Affected areas and load points
hj
(6)
The affected load points after a failure can be determined based on the system configuration and protection scheme using the parent-search technique. When a failure occurs in a distribution system, upstream protection devices will
It should be noticed that all the multi-state models include the repairing/replacing process. For the load points that do not involve the repairing/replacing process,
Fig. 5 690
Multi-state model with sequential/parallel switching
Fig. 6 Radial distribution network with different sub-areas IET Gener. Transm. Distrib., Vol. 1, No. 4, July 2007
operate to clear the fault. In this technique, the search starts from the parent node of the failed component, and the upstream component connected to the parent node is searched and examined. If the component is a breaker, the search stops. If the component is not a breaker, the upstream search continues until a breaker is found. The downstream load points interrupted by the first upstream breaker will suffer a failure and they are the affected load points. The search procedure can be explained using the distribution system shown in Fig. 6. If a failure occurs in feeder 15 between nodes 15 and 16, the parent node is 15. The upstream component connected to node 15 is searched and examined. Because the component is a feeder, the search continues to find its parent node which is node 14. The search stops when the first upstream breaker B2 is found. The affected load points interrupted by breaker B2 include load points 9– 35.
Table 1: Sector interruption cost ($/kW) User sector
Interruption cost 1 min.
20 min
1.005
1.508
2.225
1.625
3.868
9.085
25.16
55.81
commercial
0.381
2.969
8.552
31.32
83.01
agricultural
0.060
0.343
0.649
2.064
residential
0.001
0.093
0.482
4.914
15.69
govt. & inst.
0.044
0.369
1.492
6.558
26.04
office
4.778
9.878
3.3
Optimal restoration sequence
The customer reliability requirements are very important for determining optimal restoration sequence. It is usually difficult to directly evaluate customer reliability requirements. Customer reliability requirements can be indirectly assessed using customer damage functions obtained from the survey to different customers. A standard industrial classification can be used to divide customers into large user, industrial, commercial, agriculture, residential, government & institutions and office & buildings categories. Postal surveys conducted by the University of Saskatchewan have been utilised to estimate the customer interruption losses for the seven different customer sectors [10, 11]. The surveys show the cost of an interruption depends on the type of customer interrupted and on the magnitude and the duration of the interruption. The survey data have been analysed to give the sector customer damage functions (SCDFs) which are used as the customer interruption cost models. The SCDFs for different customers are shown in Table 1. The customer damage functions have been used in [12 – 14] to calculate reliability worth and unreliability cost for different customers. The objective of the problem is to select the optimal restoration sequence for sub-areas under IET Gener. Transm. Distrib., Vol. 1, No. 4, July 2007
21.06
3.968
480 min.
industrial
68.83
8.240
4.120
119.2
minimum total system unreliability cost X
Lj CDFj (dj )
(7)
j[U
Determination of restoration sub-areas
Restoration sub-areas and the involved switches can be determined during the search procedure using a faulttraversal technique. In this technique, a distribution system is represented using a dynamic tree data structure. Nodes and branches in a distribution system are related to each other through parent/offspring relationship of nodes. Using the breadth-first-search technique, the search starts from the failed component, and the adjacent nodes in all directions are traversed sequentially. The first switch or the end of feeder in each direction is found to form a switch set S which covers repair area. The switches in S can be further divided into the upstream and downstream switches based on their locations. Secondly, the automated switches in each up- and downstream are searched. The end of each downstream feeder and tie switches are also searched. Different sub-areas are determined by these switches. For the fault shown in Fig. 6, the repair sub-area I is covered by the switch set fS3 , S4 , S8g [ S. There are two upstream sub-areas II and V, and two downstream subareas III and IV. Sub-area V covers the nodes which are not included in other sub-areas.
240 min.
larger user
min COSTi ¼ 3.2
60 min.
where COSTi is the system interruption cost due to failure i, U the set of affected load points, Lj the average load of load point j, dj represents the interruption duration and CDFj(dj) the per unit customer interruption cost for duration dj . It should be noticed that the customer interruption cost for a customer also depends on the time of the interruption. The effect of an interruption at daytime on a customer may be quite different with that at nighttime. Therefore more detailed time-dependent CDFs should be provided in order to include the time effect in the optimal restoration. A practical distribution system can be divided into few sub-areas after a failure occurs. The optimal restoration sequence of the sub-areas can be determined based on the involved switches, manpower for switching and customer reliability requirements of load points in each sub-area using optimisation techniques. In this paper, the restoration process can be classified into two categories based on the types of switches in each sub-area: 1. For the sub-areas with only automated switches, the service can be first restored in parallel. 2. For the sub-areas with both automated and manual switches, the COSTi of each sub-area should be calculated and compared with other sub-areas. The sub-area with the maximum COSTi is restored first followed by the sub-area with the less COSTi . The sub-area with least customer interruption cost should be restored lastly. For the fault shown in Fig. 6, the optimal sequence restores the sub-area V first by opening switch S1 and reenergising breaker B2 . For other sub-areas, the switching sequence depends on customer reliability requirements and available manpower. It is assumed that the customers in sub-area IV are willing to pay more followed by sub-areas II and III. Thus, the restoration sequence is sub-area IV followed by II and III. It should be noticed that the optimal restoration based on customer reliability requirements can only be implemented for non-isolated (restorable) customers when the system has the corresponding switching infrastructure. 3.4
Calculation of restoration times
Restoration times of load points for the failure of feeder 15 can be easily calculated based on the restoration sequence of the sub-areas and the involved switches in each sub-area. 691
For the load points in sub-area V, the restoration time is TV ¼ T15i þ T15d þ TS1
(8)
For the load points in sub-area IV, S4 is opened and K3 is closed in parallel. The restoration time for these load points is TIV ¼ T15i þ T15d þ max{TS4 , TK3 }
(9)
The restoration time for the load points in sub-areas II and III is after sub-area IV. To restore the service for the load points in sub-area II, S3 is opened and automated switch S1 is re-closed, the interruption time for these load points is given as TII ¼ TIV þ TS1 þ TS3
(10)
To restore the service for the load points in sub-area III, S8 is opened and K2 is closed, the interruption time for these load points is given as TIII ¼ TIV þ TS8 þ TK2
692
Reliability indices
The following three basic reliability indices of failure rate lj , average annual outage time Uj and average outage time rj are calculated for each individual load point j X lj ¼ li (13) i[M
Uj ¼
X
li Tji
(14)
i[M
rj ¼
Uj lj
(15)
(12)
It is assumed in the above analysis that the alternative supply can supply all the loads after system reconfiguration. However, the restoration of downstream supply also
Fig. 7
3.5
(11)
For the load points in sub-area I, the interruption time TI ¼ min{TII , TIII } þ T15r þ TS3
depends on the capacity of alternative supply and voltage levels at the nodes after reconfiguration. For each downstream sub-area connected with alternative supply, it should be analysed using ac load flow techniques [9, 15– 17] to examine network violations. If the alternative supply cannot supply some of downstream load points, the restoration time for these load points should be calculated using (12). The load flow technique has been incorporated to determine these load points.
where li is the failure rate of element i, M the number of components which affect load point j and Tji the restoration time of load point j after the failure of component i and can be determined using the proposed technique.
Test distribution system IET Gener. Transm. Distrib., Vol. 1, No. 4, July 2007
Other reliability indices of load points, such as EENS and ECOST, and system reliability indices, such as SAIFI, SAIDI, ASAI, EENS and ECOST, can easily be calculated based on the basic load point indices [5]. 4
Procedure of evaluation algorithm
The procedure to calculate reliability indices of distribution systems using the proposed technique includes the following steps: 1. input data, 2. consider the failure of component i, 3. determine the affected area and load points using the search techniques, 4. add the failure rate li to the load points affected, 5. determine the sub-areas and the involved switches using the fault-traversal technique, 6. calculate the unreliability cost of each sub-area, 7. determine the optimal switching schedule for the service restoration, 8. calculate the restoration times for the load points in each sub-area using the related multi-state models, 9. calculate load point index Uji ¼ liTji , 10. go to step 11 if all components have been considered; otherwise, go to step 2, 11. calculate rj for each load point j and 12. calculate system indices. 5
System studies
A computer program has been developed using the proposed technique. A modified distribution system connected to Bus 6 of RBTS system [18] has been evaluated using the program. The distribution system is shown in Fig. 7. Three alternative supplies have been added to feeders F5, F6 and F7, respectively. All the tie-switches (TS1 – TS4) are assumed to be normally opened and the others are normally closed. The time for decision-making is assumed to be 0.2 h. Per unit cost for the 480th minute is used for the duration larger than 480 min. Three cases have been tested using the developed technique and the results are presented and discussed in this section. 5.1
Case 1
Assumed that all the switches have the switching time of 1 h and the parallel switching process is used. Reliability indices for load points are presented in Table 2. Reliability indices for the system and feeders are presented in Table 3. 5.2
Case 2
Assumed that all switches have the switching time of 1h and sequential switching process is used. The optimal restoration sequence has been used in the restoration. The indices for the load points, feeders and system are calculated. The system and feeder reliability indices for scheme A (F7 ! F5 ! F6) are presented in Table 4. Table 5 shows system reliability indices for the two restoration schemes A and B (F6 ! F5 ! F7). It can be seen that the system total interruption cost for scheme A is less than that for scheme B. Therefore the optimal restoration sequence has to be selected in system restoration and to be included in reliability evaluation. IET Gener. Transm. Distrib., Vol. 1, No. 4, July 2007
Table 2: Reliability indices for the load point Load
l, f/yr
U, hr/yr
EENS, MW h/yr
ECOST, $/yr
point 1
0.338
1.02
0.181526
274.65
2
0.351
1.03
0.185586
278.75
3
0.348
1.06
0.187275
284.83
4
0.338
0.99
0.179834
271.03
5
0.348
1.01
0.217667
325.86
6
0.338
1.01
0.218659
331.32
7
0.377
1.07
0.177419
261.71
8
0.381
1.11
0.184203
275.83
9
0.381
1.06
0.188429
277.70
10
0.368
0.98
0.162749
236.188
11
0.377
1.11
0.200041
300.15
12
0.368
1.03
0.213159
315.01
13
0.377
1.04
0.215574
316.56
14
0.251
0.92
0.430894
4151.70
15
0.245
1.07
1.753551
12157.61
16
0.249
1.28
1.159691
7938.73
17
0.251
1.62
0.759097
7412.70
18
1.179
6.78
1.124482
1805.68
19
1.179
6.78
1.225475
1967.86
20
1.179
6.78
1.695195
783.65
21
1.179
6.78
1.784666
825.01
22
1.179
6.78
1.40306
2253.02
23
1.207
7.03
1.166577
1877.53
24
1.214
7.00
2.13888
999.84
25
1.179
10.86
1.68809
2874.67
26
1.207
11.12
3.147112
1407.34
27
1.179
10.86
1.721765
2932.01
28
1.610
5.53
0.859362
1228.08
29
1.610
5.53
0.876505
1252.58
30
1.610
5.53
1.383053
675.96
31
1.831
7.56
1.175035
1766.44
32
1.868
7.9
1.523903
726.09
33
1.831
7.56
1.198476
1801.68
34
1.831
7.56
1.891096
903.48
35
1.831
7.56
1.990906
951.16
36
1.812
7.39
1.148729
1721.58
37
1.847
7.71
1.487196
709.64
38
1.812
7.39
2.092698
1001.23
39
1.812
7.39
1.171645
1755.92
40
1.812
7.39
2.259759
1081.15
Table 3: System and feeder reliability indices Indices
SAIFI,
SAIDI,
f/yr
h/yr
ASAI
EENS
ECOST
42.76902
68711.93
system
0.783
3.59
0.99959
F1
0.344
1.02
0.999884
1.170547
1766.44
F2
0.375
1.06
0.999879
1.341574
1983.14
F3
0.250
1.26
0.999856
4.103233
31660.75
F4
1.185
7.73
0.999118
F5
1.831
7.56
0.999137
7.779416
6148.85
F6
1.610
5.53
0.999369
3.11892
3156.62
F7
1.813
7.39
0.999156
8.160027
6269.52
17.0953
17726.62
693
Table 4: System and feeder reliability indices for Scheme A Indices
SAIFI,
SAIDI,
f/yr
h/yr
ASAI
EENS
ECOST
49.07815
75176.36
system
0.783
4.09
0.999533
F1
0.344
1.14
0.99987
1.315964
1981.27
F2
0.375
1.19
0.999864
1.523688
2252.19
F3
0.250
1.26
0.999856
4.103233
31660.75
F4
1.185
7.73
0.999118
F5
1.831
9.52
0.998913
9.779212
7764.69
F6
1.610
10.33
0.998821
5.825443
6552.37
F7
1.813
8.56
0.999023
9.435306
7238.47
17.0953
17726.62
Fig. 9 EENS of the load points for the three cases
Table 5: System reliability indices for the two schemes Indices
SAIFI,
SAIDI,
f/yr
h/yr
ASAI
EENS
ECOST
scheme A
0.783
scheme B
0.783
4.09
0.999533
49.07815
75176.36
4.09
0.999533
51.01041
76122.39
Fig. 10 ECOST of the load points for the three cases Table 6: System and feeder reliability indices for Scheme A Indices
SAIFI,
SAIDI,
f/yr
h/yr
ASAI
EENS
ECOST
6
system 0.783209 3.227943 0.999632 38.92357 0.343661 0.815478 0.999907
0.946006
1381.235
F2
0.375399 0.837584 0.999904
1.076547
1544.629
F3
0.25017
F4
1.184794 6.661124 0.99924
F5
1.831033 7.157034 0.999183
7.358561
6403.147
F6
1.61
0.999207
3.917544
4213.801
F7
1.812618 6.810334 0.999223
7.514689
6269.516
5.3
1.026012 0.999883
3.301755 24718.12 14.80847
17278.01
Case 3
In this case, all tie-switches are assumed to be operated sequentially with the switching time of 1 h and other switches can be operated by control centre in parallel with the switching time of 0.1 h. In this case, the hybrid switching models with the optimal switching sequence has to be used in the simulation. Table 6 shows system and feeder reliability indices for the restoration scheme A. The unavailability, EENS and ECOST for the three cases are shown in Figs. 8, 9 and 10, respectively. It can be seen from the figures that the multi-state switching models and restoration sequences have significant impact on load
Fig. 8 694
Conclusions
61808.46
F1
6.946
point reliability and should be considered in the reliability evaluation.
Unavailability of the load points for the three cases
This paper proposes a generalised technique for reliability evaluation of distribution systems considering the optimal switching sequence and variable restoration times. Multi-state models have been proposed to determine variable times for the restorations with sequential, parallel and hybrid switching. The formulation of the optimal switching sequence and solution method have been proposed for the service restoration of the affected areas based on involved switches, available manpower and customer reliability requirements. A computer program has been developed using the proposed technique. A test distribution system is employed to illustrate the technique. Numerical results show that different switching models will result in different customer and system reliabilities. The optimal restoration sequences can reduce the total system interruption cost. Therefore the proposed models and technique can be used in the reliability evaluation of distribution systems. Constant failure rates and repair rates have been used in the analysis. However, the failure rates will change with operating condition and the repair rates depend on the manpower for the repair. These effects should be considered in the future work. 7
References
1 Aoki, K., Nara, K., Itoh, M., Satoh, T., and Kuwabara, H.: ‘A new algorithm for service restoration in distribution systems’, IEEE Trans. Power Deliv., 1994, 4, (3), pp. 1832– 1839 2 Toune, S., Fudo, H., Genji, T., Fukuyama, Y., and Nakanishi, Y.: ‘Comparative study of modern heuristic algorithms to service restoration in distribution systems’, IEEE Trans. Power Deliv., 2002, 17, (1), pp. 173–181 3 Huang, C.M.: ‘Multiobjective service restoration of distribution systems using fuzzy cause– effect networks’, IEEE Trans. Power Syst., 2003, 18, (2), pp. 867– 874 4 Ucak, C., and Pahwa, A.: ‘An analytical approach for step-by-step restoration of distribution systems following extended outages’, IEEE Trans. Power Deliv., 1994, 9, (3), pp. 1717– 1723 5 Billinton, R., and Allan, R.N.: ‘Reliability evaluation of power systems’ (Plenum Press, New York, 1995, 2nd edn.) IET Gener. Transm. Distrib., Vol. 1, No. 4, July 2007
6 Billinton, R., and Wang, P.: ‘Teaching distribution system reliability evaluation using Monte Carlo simulation’, IEEE Trans. Power Syst., 1999, 14, (2), pp. 397 –403 7 Billinton, R., and Wang, P.: ‘Network-equivalent approach to distribution-system-reliability evaluation’, IEE Proc., Gener. Trans. Distrib, 1998, 145, (2), pp. 149– 153 8 Brown, R.E., and Hanson, A.P.: ‘Impact of two-stage service restoration on distribution reliability’, IEEE Trans. Power Syst., 2001, 16, (4), pp. 624 –629 9 Li, W., Wang, P., Li, Z., and Liu, Y.: ‘Reliability evaluation of complex radial distribution systems considering restoration sequence and network constraints’, IEEE Trans. Power Deliv., 2004, 19, (2), pp. 753–758 10 Wacker, G., Wojczynski, E., and Billinton, R.: ‘Customer damage resulting from electric service interruption’, GEA R&D Project 907, U131 Report, 1982 11 Wacker, G., Wojczynski, E., and Billinton, R.: ‘Comprehensive bibliography of electrical service interruption costs’, IEEE Trans. Power Appar. Syst., 1983, 102, (6), pp. 1831– 1837 12 Wang, P., and Billinton, R.: ‘Reliability cost/worth assessment of distribution system incorporating time varying weather conditions and restoration resources’, IEEE Trans. Power Deliv., 2002, 17, (1), pp. 260–265
IET Gener. Transm. Distrib., Vol. 1, No. 4, July 2007
13 Wang, P., and Billinton, R.: ‘Time sequential distribution system reliability worth analysis considering time varying load and cost models’, IEEE Trans. Power Deliv., 1999, 14, (3), pp. 1046–1051 14 Wang, P., and Billinton, R.: ‘Demand-side optimal selection of switching devices in radial distribution system planning’, IEE Proc., Gener. Trans. Distrib, 1998, 145, (4), pp. 409– 414 15 Zhang, F., and Cheng, C.: ‘A modified Newton method for radial distribution systems power flow’, IEEE Trans. Power Syst., 1997, 12, (1), pp. 389–397 16 Nanda, J., Srinivas, M.S., Sharma, M., Dey, S.S., and Lai, L.L.: ‘New findings on radial distribution system load flow algorithms’. Proc. IEEE Power Engineering Society Winter Meeting, 2000, vol. 2, pp. 1157–1161 17 Augugliaro, A., Dusonchet, L., Ippolito, M.G., and Riva Sanseverino, E.: ‘An efficient iterative method for load-flow solution in radial distribution networks’. Proc. IEEE Porto Power Tech. Conf., pp. 34–39 18 Allan, R.N., Billinton, R., Sjarief, I., Goel, L., and So, K.S.: ‘A reliability test system for education purposes – basic distribution system data and results’, IEEE Trans. Power Syst., 1991, 6, (2), pp. 813– 820
695
