Jurnal RIA

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230 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2005

Distributed Processing of Reliability IndexAssessment and Reliability-Based Network

Reconfiguration in Power Distribution SystemsFangxing Li , Member, IEEE

Abstract—Parallel and distributed processing has been broadlyapplied to scientific and engineering computing, including variousaspects of power system analysis. This paper first presents a dis-tributed processing approach of reliability index assessment (RIA)for distribution systems. Then, this paper proposes a balancedtask partition approach to achieve better efficiency. Next, the dis-tributed processing of RIA is applied to reliability-based networkreconfiguration (NR), which employs an algorithm combininglocal search and simulated annealing to optimize system reliability.

Testing results are presented to demonstrate the speeded executionof RIA and NR with distributed processing.

Index Terms—Network reconfiguration, parallel and distributedprocessing, power distribution systems, reliability index assess-ment, scalability, simulated annealing.

I. INTRODUCTION

PARALLEL and distributed processing [1], [2] has signif-

icant contributions to scientific and engineering computa-

tions, especially for time-critical or time-consuming tasks. Par-

allel processing is usually carried out in dedicated multiproces-

sors with a global clock and shared memory, while distributed

processing is usually carried out at multiple workstations orcomputers connected to a network without a central clock and

shared memory. In distributed processing, message passing is a

common technique to share information since there is no shared

memory. Although the performance of networked computers is

not as competitive as a dedicated parallel computer, networked

computers are less expensive and more broadly available such as

in local area networks (LANs). As such, distributed processing

is sometimes referred to as the low-end parallel processing.

It should be noted that the term “parallel” is used occasionally

in this paper to indicate the concurrent execution of a computing

task. The discussion in this paper is essentially based on dis-

tributed processing. Especially, like many other distributed pro-cessing approaches, the proposed approach employs the mes-

sage-passing scheme for data sharing among collaborating pro-

cessors.

Parallel and distributed processing has been applied to power

system computing in various areas [3]–[12], such as load

flows [4]–[7], optimal power flows [8], [9], state estimation

[10], contingency analysis [11], and reliability evaluation for

Manuscript received May 16, 2004. Paper no. TPWRS-00643-2003.The author is with ABB Inc., Raleigh, NC 27606 USA (e-mail:

[email protected]; [email protected]).Digital Object Identifier 10.1109/TPWRS.2004.841231

generation and transmission systems by Monte Carlo simu-

lation [12]. These previous works can be classified into two

categories: applications of parallel processing [4]–[6], [12]

and applications of distributed processing [7]–[12]. It appears

that more recent works focus on distributed processing. This

is probably due to the latest development in network hardware

and software, which makes distributed processing faster, more

broadly available, and easier-to-implement than before.

This paper presents distributed processing schemes of relia-bility index assessment (RIA) and reliability-based network re-

configuration (NR) for distribution systems. Here, the radial dis-

tribution system is addressed, since the majority of U.S. distri-

bution systems are radial. In addition, system-level reliability

indices are addressed because they are the primary reliability

concerns of utilities. The discussion shows that RIA can be

easily de-coupled and executed in parallel among different pro-

cessors to achieve speedup in wall-clock running time. The dis-

cussion also shows NR is mainly composed of iterative runs of

RIA. Therefore, NR can be executed in parallel based on the

distributed processing of RIA. The proposed implementation

of distributed processing considers the unbalanced computingability of different processors, a typical feature of heterogeneous

computers connected to a LAN or a similar network.

This paper is organized as follows. Section II presents a con-

troller-worker model based on message passing to share the

data and coordinate the activity of different processors. Sec-

tion III first discusses the principle of an analytical approach to

assess distribution reliability and why it is highly parallelizable;

then presents a coarse-grained distributed processing scheme for

RIA. Section IV discussesthe balanced task partition among dif-

ferent processors in order to achieve better performance and ef-

ficiency. Section V presents and discusses the testing results for

distributed processing of RIA. Section VI applies the distributed

processing of RIA to reliability-based NR, which employs an

annealed local search and RIA. Test results are also provided.

Section VII concludes the paper.

II. CONTROLLER-WORKER MODEL

FOR DISTRIBUTED PROCESSING

Distributed processing has two fundamental units: computa-

tion and communication. Computation is referred to as the CPU

activity to carry out the actual computing task, while commu-

nication is referred to as the overhead activity to transfer data

or share information among different collaborating processors.

Although communication is overhead indeed (and is desired to

0885-8950/$20.00 © 2005 IEEE



LI: DISTRIBUTED PROCESSING OF RIA AND RELIABILITY-BASED NR 231

be minimum), it is normally a necessary part for distributed pro-

cessing due to the lack of shared memory.

This paper employs the controller-worker model [7] to co-

ordinate computation and communication in distributed pro-

cessing. As described in [7], the model is comprised of two types

of processes, controllers and workers, which play different roles

in the message-passing scheme.A controller is a process with the following responsibilities:

• to accept user input;

• to assign initial data to workers;

• to invoke workers to execute tasks;

• to send/receive intermediate data to/from workers;

• to do system-wide, nonintensive calculations;

• to terminate the algorithm at the appropriate time and no-

tify workers to stop.

A worker is a process with the following responsibilities:

• to receive initial data from a controller;

• to do intensive calculations upon a controller’s request;

• to send/receive intermediate data to/from a controller;• to stop when notified.

In the approach here, there is only one controller, while there

are multiple workers. The controller is mainly responsible for

coordinating all workers and maybe some nonintensive compu-

tations, while workers are responsible for intensive computa-

tions [7]. This model implies a coarse-grained distributed pro-

cessing, in which parallelism occurs at the level of subroutine

or group of subroutines.

III. DISTRIBUTED PROCESSING OF RIA

This work assumes N-1 contingencies. Further, only perma-

nent component faults are considered in this section for sim-

plicity and illustration.

A survey of 205 U.S. utilities [13] showed that five relia-

bility indices, SAIFI, SAIDI, MAIFI , CAIDI, and ASAI (or

), have been popularly employed in the utili-

ties. The survey also mentioned other four less popular indices,

CAIFI, CTAIDI, ASIFI, and ASIDI. This section takes SAIFI

as an example to illustrate why the calculation of a reliability

index can be divided into many parallelizable steps. The Ap-

pendix briefly illustrates that the other system reliability indices

can be assessed in a similar way.

The purpose of RIA for distribution systems is to model each

system contingency and compute the reliability impact of each

contingency. It may be carried out through various approaches,

such as analytical approach based on component contributions

[14], [15], failure mode and effect analysis (FMEA) [16], Monte

Carlo simulation [17], [18], Markov modeling [19], and other

practical approaches [20]. This work employs the analytical ap-

proach in the previous work [14], [15] to evaluate reliability in-

dices such as SAIFI, SAIDI, MAIFI , etc. Fig. 1 briefly illus-

trates the input and output of RIA analysis. Appendix C also

provides the fundamental principles for the analytical simula-

tion of RIA. More detailed algorithms can be found in [15].

With the assumption of N-1 contingencies, a reliability indexcan be viewed as the sum of contribution from each contingency

Fig. 1. Input and output of RIA.

(the failure on a component). Taking SAIFI as an example, this

can be expressed as

(1)

where

contribution to SAIFI from component ;

total number of components.

Since SAIFI is defined as the number of customer inter-

ruptions that an average customer experiences during a year,

is the number of customer interruptions at an average

customer caused by failures on component . Hence,

can be written as [18]

(2)

where

failure rate per year of component ;

number of customers experiencing sustained interrup-

tion due to a failure of component ;

total number of customers.

Combining (1) and (2), we have

(3)

Since is a constant and is related to the component itself,

the above equation shows that the key to calculate SAIFI is to

calculate . Since stands for the number of customers expe-

riencing sustained interruptions due to a failure of component

, it can be evaluated by identifying which components will be

de-energized based on the system topology, protection scheme,

and restoration. Although the evaluation of may be compli-

cated, it is certain that the evaluation of is independent of

. In other words, two processors can evaluate and

, respectively, in parallel, if both processors know the input of

the system data. Hence, evaluation of SAIFI contributions from

component and , or and , can be carried out

at different processors independently.

As shown in the Appendix, similar conclusions can be easily

drawn that the value of other indices is the sum of contributionsfrom all individual components, or can be directly obtained as a




function of other indices. Hence, the evaluation of the compo-

nent contributions can be carried out independently and concur-

rently.

With this simple, but very important feature of reliability in-

dices, a distributed processing approach for RIA is proposed.

This approach employs the controller-worker model presented

in the previous section. Assume there is a unique controller andworkers. Also assume the system has components. The ap-

proach is described as follows.

1) The controller reads in all input data including system

topology, component reliability rates, etc.

2) The controller sends the input data to all workers.

3) The controller requests each worker to calculate the reli-

ability index contributions from randomly selected

components. For instance, with the assumption that the

component ID is randomly distributed, the first worker

computes reliability index contributions for components

1 to ; the second worker computes reliability index

contributions for components to ; and so

on.

4) Each worker concurrently evaluates the reliability index

contributions for its assigned components.

5) Each worker sends the reliability index contributions of

its assigned components to the controller.

6) The controller adds up all reliability index contributions

to obtain the system reliability index.

The proposed coarse-grained distributed algorithm is highly

parallelized because of the following two reasons.

• The calculations of reliability index contributions among

different workers are highly independent. A worker does

not need any further communication with others once the

actual calculation starts.• The communication is kept minimum. The major commu-

nication occurs only at the very beginning to distribute the

input data (system topology, component reliability data,

etc) to workers. The collection of reliability index contri-

butions calculated by workers needs much less communi-

cation time.

It is noteworthy to mention that although this work assumes

N-1 contingencies, higher order contingencies can be applied

as well based on the component (contingency) contribution ap-

proach. The reason is that the basic unit of this approach is to

evaluate the impact or contribution to reliability indices from

each contingency event. Because each contingency event canbe assessed independently, RIA can be carried out in parallel.

Due to the same reason, similar parallelization approach can be

applied to meshed network RIA, although more implementation

efforts are needed than for radial system RIA.

IV. BALANCED TASK PARTITION WITH THE CONSIDERATION

OF COMPUTING ABILITY

Parallel or distributed processing usually partitions a task into

many small units processed at each individual participating pro-

cessors. It is a common practice that the task is equally parti-

tioned among processors, with the ideal assumption of a bal-

anced computing environment. That is, all processors can finishthe same amount of computing work in the same time duration.

Fig. 2. Partition of a task withs

independent steps.

This is typically true for parallel processing carried out in a ded-

icated supercomputer. However, sometimes the ideal environ-

ment may not be available, especially in a “loosely coupled” dis-

tributed processing environment like heterogeneous networked

computers. In this case, the computing power of different com-

puters may vary. An equal partition may lead some inef ficiency

and task balancing needs to be considered for higher ef ficiency.

Assume a task consisting of 3000 steps, computationally

identical and logically independent, will be carried out at two

different processors, P1 and P2. If running separately, P1 can

complete the task in 100 s while P2 in 50 s. Ignoring overhead,

an equal task partition (1500:1500) leads P1 to complete its

assignment in 50 s and P2 in 25 s. This causes P2 to be idle for

25 s. The overall wall-clock running time is 50 s.

To fully utilize the computing power, P2 should receive more

tasks than P1. If 1000 steps are assigned to P1 and 2000 steps

to P2, then each processor will finish its assignment in 33.3 s,

or 16.7 s less than the previous equal task partition. It can be

easily proved that this 1000:2000 partition is the most ef ficient

partition.

Based on the above analysis, the balanced task partition with

computing ability adjusted for two processors is illustrated in

Fig. 2. The weighting factorsof task partition for each processor,and , are given by

(4)

(5)

where

time to complete the task with a single processor P1;

time to complete the task with a single processor P2.

Hence, each processor finishes its own assignment in

s for 1 p.u. task.

Equations (4) and (5) can be extended to multiple processors.Equation (6) shows the weighting factor for the th processor. It

can be easily verified that each processor completes its assign-

ment simultaneously in

(6)

where

weighting factor for processor ;

time to complete the task with a single processor .

The above discussion is based on two assumptions.

1) Each step within a task is independent.2) Each step is computationally equal.




The first assumption is true for the RIA task because each

RIA step, i.e., the evaluations of the contribution to a reliability

index from each individual component, is independent.

The second assumption may not be precisely true for the RIA

algorithm because the evaluation of the reliability index contri-

bution from a component depends on connectivity, protection,

restoration, etc. For example, it takes less CPU time to eval-uate SAIFI contribution from a component in a feeder with 50

components than a component in a feeder with 100 components.

However, the second assumption shall be roughly true for a large

distribution system in the scale of thousands of components, be-

cause each processor is assigned a large number of randomly

selected components. Hence, an average step carried out at one

processor shall be computationally equal to an average step at

another processor.

The above discussion makes it apparent that Step 3 in the

distributed processing of RIA should be modified slightly as

follows:

The th processor should be responsible for randomly

selected components instead of components.

It should be noted that there are alternative ways for task bal-

ancing such as dynamic assignment, which balances the compu-

tation among processors at run-time based on assignment-com-

pletion loops [11]. However, in general, it requires more syn-

chronization and communication overhead. It is more suitable

for a dynamic task with computation time very unpredictable

or in a wide range. Since the computation-dominant RIA appli-

cation completes in a stable manner (in steps of component

contribution evaluation), the proposed approach tends to be ef-

fective for RIA application. The reason is that there is no mul-

tiple, iterative run-time communication involved, because each

worker receives its total assignment at the start of the algorithm.Implementation is also simplified. Meanwhile, ef ficiency can be

ensured with the task partition adjusted by weighting factors.

V. TEST RESULTS OF RIA DISTRIBUTED PROCESSING

A. Test Systems, Environment, and Procedure

This section presents testing results of the distributed pro-

cessing of RIA. Here the five popular system-level reliability

indices, SAIFI, SAIDI, MAIFI , CAIDI, and ASAI [13], are

considered. The tests are explored in several actual distribution

systems with sizes from 3835 components to 11 026 compo-

nents. This is shown in Table I.The test environment is a Local Area Network with up to 6

computers, each with Pentium III processor. The speeds of the

6 processors vary from 450 MHz to 1 GHz. The network speed

is 10 M bps. The memory sizes vary from 128 to 256 MB. The

distributed RIA processing is implemented in Visual C++ with

TCP/IP protocol for message passing.

The testing procedure is described as follows. First, the test

is examined for the RIA algorithm in a sequential processing

mode. That is, the RIA algorithm is executed in a single pro-

cessor for each of the six processors. The running time of each

individual processor will be used to compute the weighting

factor for balanced task partition based on (6). Then, the RIA is

executed in the distributed processing mode with balanced task partition in 2, 3, 4, 5, and 6 processors, respectively. The results

TABLE ISIZES OF FIVE TEST SYSTEMS

TABLE IITIME (SECONDS) TO COMPLETE A SEQUENTIAL RIA AT EACH

INDIVIDUAL MACHINE

of scalability are presented. Here the scalability is measured as

speedup and ef ficiency, defined by the following equations [1],

[2]:

(7)

(8)

where

wall-clock time to complete the best sequential algo-

rithm on a single processor;

wall-clock time to complete the parallel or distributed

algorithm on multiple processors;

number of processors.The above speedup definition does not indicate which pro-

cessor is associated with. This implies each processor is iden-

tical for typical parallel processing environment. Since this work

is carried out in processors with different computing capability,

is taken as the average of the sequential RIA running time at

all participating processors.

This section also presents an illustrative test to verify the per-

formance improvement with balanced task partition. The test is

performed in the mid-sized system SYS-3.

B. Test Results

Table II shows the running time in seconds to complete a se-

quential RIA algorithm with a single processor. It shows that

these networked computers have different computing power.

This is used for balanced task partition in the distributed pro-

cessing.

Tests for distributed processing are carried out for five sce-

narios with 2–6 processors working in parallel. The processor

assignment is as follows.

Two processors: P1, P2.

Three processors: P1, P2, P3.

Four processors: P1, P2, P3, P4.

Five processors: P1, P2, P3, P4, P5.

Six processors: P1, P2, P3, P4, P5, P6.

In the distributed processing scenario, each processor runsa worker process. In addition, one of the processors runs the




TABLE IIISPEEDUP OF DISTRIBUTED PROCESSING OF RIA

Fig. 3. Ef ficiency of distributed processing of RIA.

controller process as well. The purpose is to fully utilize the

computing ability of that processor, because a controller is not

computationally intensive while a worker is. In fact, when the

worker is calculating the reliability index contribution, the con-

troller is essentially idle except waiting for the feedbacks from

all workers. Typically, the most powerful processor is selected

to host the controller and a worker.

Table III and Fig. 3 present the speedup and ef ficiency, respec-

tively, of the distributed processing of RIA. It is shown that the

distributed processing can speed the wall-clock execution time

of RIA. It is also shown that, in general, the speedup and the

ef ficiency increase when the system size increases. When the

number of involved processors increases up to 6, the speedup

keeps increasing at a considerable rate, although the ef ficiency

drops.

C. Tests for Equal and Balanced Task Partitions

A test run of distributed RIA processing with the equal task

partition is carried out for comparison against the balanced task partition. The test is executed in the mid-sized system SYS-3

for illustrative purpose.

In Table IV, the second row shows the speedup with equal

task partition, while the third row shows the speedup with bal-

anced task partition. The last row shows the improved speedup

in percentage. It is clearly demonstrated that the balanced task

partition improves the ef ficiency of the distributed processing

for RIA, if the processors have various computing capabilities.

VI. DISTRIBUTED PROCESSING OF RELIABILITY-BASED

NRETWORK RECONFIGURATION

The distributed processing for RIA proposed in Section III,together with the balanced task partition, can be easily applied to

TABLE IVSPEEDUP OF DISTRIBUTED PROCESSING WITH TWO PARTITIONS FOR

THE SYS-3 SYSTEM

reliability-based optimization such as NR. NR is accomplished

by closing normally open switches and opening normally closed

switches. Since the NR is usually very time-consuming (from

tens of minutes to hours), the application of distributed pro-

cessing may be very beneficial to improve the wall-clock run-

ning time.

Reliability-based NR is to identify the optimal network

configuration without any financial investment to achieve the

highest possible reliability. Although various formulations may

be applied to NR, this work takes a weighted sum of multiple

reliability indices as the objective function [15], [21]

(9)

The constraints could be component loadings and voltage

drop limits. Penalties can be added into the objective function if

constraints are violated [21].

Since NR is a nonlinear, noncontinuous optimization

problem, many combinatory techniques like local search, tabu

search, simulated annealing and genetic algorithm have been

applied to NR. This paper employs an approach presented

in [15] and [21], which combines local search and simulated

annealing [22], to demonstrate that the previous distributed

processing of RIA can be easily applied to NR. The traditional,

sequential algorithm is described in details in previous works

[15], [21]. Here it is sketched as follows.

1) Set initial parameters for simulated annealing such as the

starting temperature (T) and the annealing rate (R).

2) Identify the objective function of the initial config-

uration.

3) Set the above obj as the temporary best objective, OBJ.

4) For each tie switch:

a) identify a new configuration by performing a tie switch

shift;b) perform an RIA to calculate the new objective function,

obj’, for the new configuration;

c) generate a random number ;

d) check whether the new configuration is acceptable or

not: if or , then ’;

otherwise, shift the tie switch to its original position;

e) Go to 4a) to perform switch shift for remaining adjacent

switches.

5) Repeat 4) for all tie switches.

6) Change annealing factor by setting .

7) If there is no change of OBJ since last change of T, stop.

Otherwise, go to step 4).

As the above description shows, the NR algorithm requiresmultiple runs of Step 4), especially Step 4b) to complete a RIA




run for each new configuration. Normally, there may be hun-

dreds, even thousands, of new configurations to examine in large

systems. Even though network simplification may be employed

to reduce the running time to some extent, the NR algorithm

is still very time-consuming. Therefore, it is very beneficial if

distributed processing can be applied to NR to reduce the wall-

clock running time.

Practical tests show that the repeated RIA executions of Step

4b) typically consume more than 98% CPU time of the overall

NR algorithm. Hence, the key to parallelize NR is to parallelize

the RIA executions. With the annealing algorithm, it is very dif-

ficult to carry out the multiple instances of RIA in parallel. This

is because a new configuration is determined only after the RIA

of the previous configuration is completed. The acceptance or

rejection of the previous configuration, together with the pertur-

bation mechanism, may affect the new configuration. However,

there is a simple yet effective approach to parallelize the sequen-

tial version of NR. The approach is to parallelize each individual

RIA using the distributed RIA processing presented previously,

rather than to parallelize different RIA instances. As such, thefollowing three principles are proposed for the distributed pro-

cessing of NR.

• The controller knows the global parameters like T, R, and

OBJ. All steps except 4b) will be sequentially carried out

at the controller without the involvement of the workers.

• The responsibility of the workers is to collaboratively

complete Step 4b), the RIA, in parallel for each new con-

figuration. The workers know the system data in order to

complete RIA and do not have any knowledge about the

global parameters for NR. Here, balanced task partition is

considered in the distributed processing of RIA.

• The controller needs to transfer the bulk system data (com-ponents, topology, protection schemes, etc) to the workers

in the first RIA run. In the following RIA run, the con-

troller only needs to notify the workers about the topo-

logical changes after each new configuration is identified.

Hence, the workers will be able to complete another RIA

run for the new configuration without unnecessary bulk

data transfer.

The above distributed processing of NR can be easily imple-

mented by reusing the previous approach of RIA distributed pro-

cessing. Good ef ficiency is expected even though Step 4b) is the

only step carried out in parallel because it dominates over the

other steps in CPU consumption.

It should be noted that the annealed local search algorithm is

selected for illustrative purpose. Other optimization algorithms

like a genetic algorithm may work for NR as well, but it is

likely that reliability-based optimization depends on the RIA al-

gorithm to evaluate the system reliability for each new config-

uration. Therefore, the proposed distributed processing for NR

may be applied to other similar reliability-based optimization

algorithms.

The distributed processing of reliability-based NR is imple-

mented as an extension of the sequential version of the pre-

vious works [15], [21]. Then, It is tested at the mid-sized system

SYS-3 for demonstrative purpose. To achieve acceptable so-

lution quality without unnecessary computation time, an an-nealing rate of 0.9 and a starting temperature of 0.3 are selected.

TABLE VSCALABILITY OF DISTRIBUTED PROCESSING FOR NR ON THE SYSTEM SYS-3

The description of the distributed processing implies that ex-ecutions of NR may yield different results and require different

number of RIA runs because of the random number generation

in Step 4c). To examine the ef ficiency of distributed processing,

the completion time of a NR test is normalized to each RIA run.

It should be noted that if the random numbers are identically

generated among different NR tests, the result of each NR test

should be the same regardless of distributed processing or not.

The number of RIA runswithin each NRtestshould be the same,

too.

Table V shows the speedup of the distributed processing of

NR. It appears that the scalability here is higher than that of the

distributed processing of RIA. The reason is that less relative

overhead is involved in distributed processing of NR. Detailedexplanations are given as follows.

The NR algorithm contains many iterative RIA runs. The ini-

tial bulk data transfer (Step 2 in the distributed processing of

RIA in Section III) is necessary only in the first RIA run. In the

follow-up runs, only a small amount of information needs to be

transferred from the controller to the workers. The information

is the change of network topology such as which switches are

opened or closed. There is no need to re-send the whole system

information. Therefore, the initial data transfer, which is a con-

siderable overhead for RIA algorithm, is negligible in the NR

algorithm. Hence, it is apparent that the higher speedup should

be observed in the distributed processing of NR algorithm.

VII. CONCLUSION

The RIA for distribution systems can be carried out in par-

allel, because the contributions to a reliability index from dif-

ferent components are independent. A balanced task partition

scheme may be applied to improve ef ficiency, especially in a

distributed processing environment where the computing abili-

ties of participating processors are usually different.

The distributed processing of RIA can be applied to a NR al-

gorithm, which is to optimize the system reliability based on an-

nealed local search. Since the kernel of the NR algorithm mainly

consists of multiple RIA runs, it can be easily implemented on

top of the distributed processing of RIA.

The proposed distributed processing of RIA and NR should

be practically beneficial if applied to utility distribution systems,

which may have tens, or even hundreds, of thousands of compo-

nents. It should be especially attractive for NR application since

NR adds a dimension of computational complexity on top of

RIA.

APPENDIX

A. System Reliability Indices

The analytical approach considers a reliability index as the

sum of contributions from failures of individual componentsduring the course of a year. This Appendix gives the following




equations as a formal method to calculate reliability indices.

These equations are expected to make it easy to understand why

the analytical approach is parallelizable.

SAIFI, SAIDI, and MAIFI can be calculated with

(A1)–(A3). Equation (3) is repeated here as (A1) for com-

pleteness. These three equations can also be found in [18]

(A1)

(A2)

(A3)

where

failure rate of component ;

number of customers experiencing sustained interrup-

tion due to a failure of component ;

sustained interruption durations for all customers due

to a failure of component ;

sustained interruption duration for customer due to a

failure of component ;

number of customers experiencing temporary interrup-tion (event) due to a failure of component ;

total number of customers;

total number of components.

Three other popular reliability indices, CAIDI, ASAI, and

ASUI, can be directly derived from SAIFI and SAIDI. They are

given as follows:

(A4)

(A5)

(A6)

Some less popular reliability indices, such as CAIFI,

CTAIDI, ASIFI, and ASIDI [13], can still be obtained using

the contribution approach. They are given by

(A7)

(A8)

TABLE AIOCCURRENCE OF INTERRUPTION AT LOAD POINTS DUE TO N-1 CONTINGENCIES

(A9)

(A10)

where

total number of customers experiencing at least 1 in-

terruption per year;

total connected kVA served;

interrupted kVA due to a failure of component ;

interrupted kVA weighted by interruption hours due to

a failure of component ;

interrupted kVA for customer due to a failure of com-ponent . (Note: ).

Other important reliability indices, though not shown in [13],

include Expected Energy Not Served (EENS) and Average En-

ergy Not Served (AENS). They can be obtained with

(A11)

(A12)

where is the power factor of customer .

B. Load Point Reliability Indices

Although this work does not address load point reliability in-

dices [16], [17], they can be calculated with slight refinement

of the analytical approach based on component contributions.

The refinement is as follows: when a component failure is ana-

lyzed, a temporary record is kept for each load point to identify:

1) whether it will be interrupted and 2) the duration of the inter-

ruption.

Table AI shows a simple example of the impact to load points

from component failures under N-1 contingencies. If a load

point experiences an interruption after a component fails, the

corresponding entry is set to “1”. Otherwise, it is set to “0” if

no interruption at the load point. Hence, the annual interruptionrate at each load point, , is the sum of its column, weighted




by component failure rate. If the values are replaced with in-

terruption durations, then we can obtain the annual load point

interruption duration , which is the weighted column sum.

The average interruption time can be obtained as .

This is summarized as follows:

(A13)

(A14)

(A15)

where

if a failure of component causes an interruption at

load point ; otherwise 0;

interruption duration at load point if component

fails.It is noteworthy to mention that the sum of the row in

Table AI, weighted by the number of customers at each load

point shall be , which is the contribution factor used to

calculate SAIFI. If the values in the table are outage durations,

then the sum of the row weighted by the number of customers

is , which is used to calculate SAIDI.

Since such tables are intermediate results and consumes lots

of memory, it is not necessary to implement the tables. In the

actual implementation, it is only necessary to keep records of

the accumulated value of the weighted column (or row) sum for

load point reliability indices (or for system reliability indices as

done in this work).The interruption rate at load point can be employed to

calculate , the total number of customers experiencing at least

one interruption per year, which is used in (A7)–(A8). This is

given by

(A16)

where

total number of load points;

number of customers at load point .In the above equation, is the probability that load

point experiences at least one interruption per year. This is be-

cause a component with a constant interruption rate follows

a Poisson process [15], [18]. The probability density function of

its being interrupted times per year is given as follows:

(A17)

Hence, the probability of being interrupted at least once per

year is given by

(A18)

C. Description of the Analytical Approach of RIA Simulation

The algorithm of the analytical approach of RIA is outlined

as follows.

1) For a permanent fault at component C (repair time

):

a) Fault isolation: An upstream search is performed to find

the nearest protection or reclosing device, P, which op-

erates to isolate the fault.

b) Upstream restoration: If there is a switching device S

between P and C, S is opened in minutes

to restore components between P and S. All restored

components experience a (sustained) interruption of

minutes. If this is less than

the threshold of momentary interruption in the case that

S is automated, the interruption is classified as a mo-

mentary interruption event.

c) Downstream restoration: If there is an alternate power

source through a normally open switch (NOS) and

there is another normally closed switch (NCS) be-tween NOS and C, all components between the two

switches experience a (sustained) interruption of

minutes

due to downstream restoration. If the time is less than

the momentary interruption threshold because of au-

tomated switching, the interruption is classified as a

momentary interruption event.

d) All isolated and unrestored components experience a

sustained interruption of MTTR minutes.

Note:

2) For a temporary fault:

a) Fuse-saving: If a temporary fault can be cleared by

an upstream reclosing device, R, with fuse-saving

scheme, all components downstream of R experience

a momentary interruption event.

b) Fuse-clearing: If the there is no fuse-saving protection,

the fuse blows and isolates its downstream compo-

nents. The interrupted downstream components may

be restored through back-feed as described in 1c).

The above approach shall be applied to each component toidentify the interruption type and duration at each load/customer

point if the specific component fails. Hence, the contribution

factors to reliability from a component failure, such as

and so on, can be easily obtained.

The above approach can be extended to address more com-

plicated cases like operation failures, transfer switches, and dis-

tributed generations. Details can be found in [15].

ACKNOWLEDGMENT

The author is very grateful to Dr. R. E. Brown for his thoughts

and discussions in power distribution reliability issues. The au-thor would also like to thank the reviewers and the editor for




their comments and suggestions, which helped the author to re-

vise and improve this paper.

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FangxingLi (M’01) receivedthe B.S.E.E. andM.S.E.E.degrees fromSoutheastUniversity, Nanjing, China, in 1994and 1997, respectively, andthe Ph.D. degree

from Virginia Tech, Blacksburg, Virginia, in 2001.He is presently a Senior Consulting R&D Engineer at ABB Inc., Raleigh,

NC, where he specializes in computational methods and applications in powersystems, especially in power distribution analysis and energy marketsimulation.

Dr. Li is a member of Sigma Xi.

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