Risk-Based Resource Allocation for Distribution System Maintenance

Risk-Based Resource Allocation for Distribution System Maintenance
PSERC
Risk-Based Resource Allocation
for Distribution System Maintenance
Final Project Report
Power Systems Engineering Research Center
A National Science Foundation
Industry/University Cooperative Research Center
since 1996
Power Systems Engineering Research Center
Risk-Based Resource Allocation
for Distribution System Maintenance
Final Project Report
Project Team
Ward Jewell, Project Leader, Joseph Warner
Wichita State University
James McCalley, Yuan Li
Sree Rama Kumar Yeddanapudi
Iowa State University
PSERC Publication 06-26
August 2006
Information about this Project
For information about this project contact:
Ward Jewell
Professor
Wichita State University
Department of Electrical and Computer
Engineering
Wichita, Kansas 67260-0044
Phone: 316-978-6240
Fax: 316-978-5408
Email: [email protected]
James D. McCalley
Professor
Iowa State University
Department of Electrical and Computer
Engineering
Ames, Iowa 50011
Phone: 515-294-4844
Fax: 515-294-4263
Email: [email protected]
Power Systems Engineering Research Center
This is a project report from the Power Systems Engineering Research Center (PSERC).
PSERC is a multi-university center conducting research on challenges facing the electric
power industry and educating the next generation of power engineers. More information
about PSERC can be found at the Center’s website: http://www.pserc.org.
For additional information, contact:
Power Systems Engineering Research Center
Arizona State University
577 Engineering Research Center
Box 878606
Tempe, AZ 85287-8606
Phone: 480-965-1643
FAX: 480-965-0745
Notice Concerning Copyright Material
PSERC members are given permission to copy without fee all or part of this publication
for internal use, if appropriate attribution is given to this document as the source material.
This report is available for downloading from the PSERC website.
© 2006 Wichita State University and Iowa State University. All rights reserved.
Acknowledgements
This is the final report for the Power Systems Engineering Research Center (PSERC)
research project titled “Risk-Based Maintenance Resource Allocation for Distribution
System Reliability Enhancement” (PSERC project T-24). We express our appreciation
for the support provided by PSERC’s industrial members and by the National Science
Foundation’s Industry/University Cooperative Research Center program.
We are particularly grateful to MidAmerican Energy, the National Rural Electric
Cooperative Association, and the Minnesota Valley Cooperative Light and Power
Association for supplying data used in some parts of this project.
i
Executive Summary
Distribution systems are maintenance-intensive so maintenance budgets are a
substantial share of costs for distribution businesses. Maintenance budgets are high, in
part, due to the size of the systems and the number of people that it takes to properly
maintain the system to achieve the appropriate reliability level. Operations and
maintenance (O&M) budgets can be reduced through improved efficiency. However, of
concern is the effect of such budget reductions on a distribution business’ ability to keep
its system operating at the desired reliability level. To meet customer needs for affordable
and reliable service while complying with regulatory requirements, with limited budgets,
it is necessary to find tools and techniques that, when coupled with a sound asset
management policy, can be used to optimally maintain distribution systems. Such a
policy also extends equipment life to avoid or defer costly capital investments resulting
from poor equipment maintenance.
In this project, we have developed a comprehensive and cost-effective maintenance
allocation and scheduling system, and have implemented it in software tools. These tools
assist in answering three concerns commonly faced by an asset manager:
1. How to identify and justify the resources needed for managing the assets of the
entire system.
2. How to allocate the available resources to different maintenance programs.
3. How to select a set of maintenance tasks to be performed within each
maintenance program.
Our system allocates resources and schedules maintenance tasks to optimize system
reliability by maximizing risk-reduction achieved from those tasks. It uses information
obtained from inspection and monitoring to determine the state of the system. Available
maintenance tasks are identified, and the risk reduction provided by each is computed.
The risk reduction for each task is based on the condition of the component being
serviced, the task’s effect on improving the component’s condition of equipment, and the
resulting improvement in reliability indices. The tasks are prioritized, subject to
constraints on available resources, using an optimization technique combining integer
programming, Lagrange relaxation, and dynamic programming. For this initial
development work, the maintenance tasks incorporated so far are associated with wood
poles, reclosers, and vegetation management of distribution line right of way. Actually,
maintaining these particular assets represents a large percentage of maintenance budgets;
furthermore, outages of these assets can significantly affect system reliability. This work
can be adapted to most types of distribution equipment.
The essential elements of the maintenance allocation and scheduling system include:
1. Failure mode identification: Taxonomies are essential in identifying the effects of
maintenance tasks on failure rates. We determined the taxonomies of failure
modes together with maintenance tasks that address those failure modes.
2. Failure rate estimation: Failure rate reductions provided by each maintenance task
were used to optimize the allocation of maintenance resources. Methods were
developed for estimating the probabilistic failure rate for wood poles and
ii
reclosers using condition measurements obtained from either continuous
monitoring or from periodic inspection and testing. These methods also estimate
the reduction in failure rate by maintenance task for each component.
3. Risk reduction due to maintenance: Using information on failure rate and its
reduction by maintenance task, risk reduction was estimated with a reliability
assessment tool developed in this project.
4. Maintenance task selection and prioritization: Risk reduction estimates form a
pool of candidate maintenance tasks, along with their resource requirements. The
system selects and prioritizes maintenance tasks based on the risk reduction
obtained. Constraints on the optimization include the maintenance budget and
level of labor resources. An integer programming optimization technique was
developed for the selection and prioritization of candidate maintenance tasks.
A degradation-path model to estimate failure probability and probability reduction
was developed. This model was applied to wood poles to predict individual pole failure
probability based on condition measurements that represent degradation in the pole’s
residual strength.
A condition assessment technique was developed for reclosers. A check sheet for
evaluating a recloser’s condition, either in the field or in the shop, is provided. The
condition score is then correlated with historical data to provide an estimate of the
recloser’s failure rate. Maintenance changes the recloser’s condition and thus its failure
rate. Similar techniques can be applied to other distribution system components.
Research-grade software from this research includes:
1. Reliability evaluation tool: A predictive reliability evaluation tool was developed
in Excel. It is used to compute system reliability levels. This software tool also
computes sensitivities of reliability metrics to maintenance tasks. When combined
with estimates of failure rate reduction obtained from maintenance tasks, the tool
computes the risk reduction associated with maintenance. The output of this tool
is one input to an optimizer tool that selects and prioritizes maintenance tasks.
2. Optimizer tool: An optimizer processes the inputs of (1) candidate maintenance
tasks, (2) effect of each task on reliability (i.e., risk reduction), (3) financial and
labor resources needed for each maintenance task, and (4) available resources.
The program selects and prioritizes maintenance tasks for the budget cycle.
Follow-on work to this project is needed. The reliability and inspection models
developed should be further expanded, verified, and then adapted to other distribution
equipment. Specifically, the wood pole degradation path model should be validated for
other components with complex failure processes, such as switches and transformers.
Similarly, the inspection methods developed for reclosers should be applied to other
components, and the resulting failure rate estimates should be verified.
The problem formulation should be further enhanced by considering scheduling
issues involved in equipment maintenance. The result will provide a schedule of planned
maintenance for a budget period.
The optimal resource allocation strategy sacrifices some accuracy to solve the largescale problem. Further research involving other optimization techniques will help
improve the accuracy of the solution.
iii
Table of Contents
1
2
3
4
Introduction................................................................................................................. 1
1.1 Asset Management Problem ................................................................................ 1
1.2 State of the Art in Power System Maintenance ................................................... 2
1.2.1 Corrective Maintenance ........................................................................... 2
1.2.2 Time-Based Preventive Maintenance ...................................................... 2
1.2.3 Condition-Based Preventive Maintenance............................................... 3
1.2.4 Reliability-Centered Maintenance (RCM)............................................... 3
1.2.5 Risk-Based Preventive Maintenance ....................................................... 3
1.3 Risk-Based Allocation of Distribution System Maintenance Resources............. 4
1.3.1 Definition of Risk .................................................................................... 5
Maintenance Practices .............................................................................................. 10
2.1 Reclosers ............................................................................................................ 10
2.1.1 Failure Modes ........................................................................................ 10
2.1.2 Maintenance Practices ........................................................................... 10
2.2 Vegetation .......................................................................................................... 11
2.2.1 Failure Modes ........................................................................................ 11
2.2.2 Maintenance Actions.............................................................................. 11
2.2.3 Inspection Methods................................................................................ 12
2.2.4 Factors Influencing Failure Rates .......................................................... 12
2.2.5 Vegetation Condition and Modeling ...................................................... 12
2.3 Wood Poles ........................................................................................................ 13
2.3.1 Decay of Wood Poles ............................................................................. 13
2.3.2 Detection and Measurement of Decay................................................... 13
2.3.3 Maintenance Practice ............................................................................. 13
Failure Rate Estimation ............................................................................................ 14
3.1 Recloser.............................................................................................................. 14
3.1.1 Condition Assessment............................................................................ 14
3.1.2 Failure Rate Calculation ........................................................................ 16
3.1.3 Effects of Maintenance .......................................................................... 18
3.1.4 Example ................................................................................................. 18
3.1.5 Summary ................................................................................................ 21
3.2 Vegetation .......................................................................................................... 22
3.3 Wood Poles ........................................................................................................ 24
3.3.1 Degradation Path Model Approach Basis .............................................. 24
3.3.2 Degradation Path Model ........................................................................ 25
3.3.3 Illustration .............................................................................................. 27
Reliability Evaluation for Distribution Systems....................................................... 33
4.1 Parameters Used in Reliability Modeling of Distribution System Equipment .. 33
4.1.1 Permanent Failure Rate ( p).................................................................. 33
4.1.2 Mean Time to Repair (MTTR)............................................................... 33
4.1.3 Protection Reliability (PR)..................................................................... 34
4.1.4 Reclose Reliability (RR) ........................................................................ 34
4.1.5 Switching Reliability (SR)..................................................................... 34
4.1.6 Mean Time to Switch (MTTS)............................................................... 34
iv
4.1.7 Probability of Failure (PF) ..................................................................... 34
4.2 Models Used in the Program.............................................................................. 36
4.2.1 Overhead Line and Underground Cable Segments................................ 36
4.2.2 Fuses, Reclosers, and Breakers.............................................................. 36
4.2.3 Switches ................................................................................................. 36
4.2.4 Sectionalizers ......................................................................................... 37
4.2.5 Equivalent Component........................................................................... 37
4.3 System Response to Outages ............................................................................. 37
4.3.1 Circuit Breaker....................................................................................... 37
4.3.2 Fuse with No Upstream Recloser .......................................................... 38
4.3.3 Recloser.................................................................................................. 38
4.3.4 Fuse with Upstream Recloser ................................................................ 39
4.3.5 Sectionalizer with Upstream Recloser ................................................... 40
4.3.6 Switching ............................................................................................... 42
4.4 Analytical Reliability Evaluation....................................................................... 44
4.5 Regulatory Penalty Risk Evaluation .................................................................. 45
4.6 Validation of Reliability Assessment Tool ......................................................... 52
5
Optimization ............................................................................................................. 55
5.1 Problem Statement ............................................................................................. 55
5.2 Possible Solution Methods for Task Selection Subproblem .............................. 57
5.2.1 Prioritization Method ............................................................................. 57
5.2.2 Branch-and-Bound Method ................................................................... 57
5.2.3 ELPR-LRH Method ............................................................................... 58
5.3 Solution Methods for Budget Planning Subproblem ......................................... 59
5.4 Summary ............................................................................................................ 60
6
Illustration................................................................................................................. 62
6.1 Historical Reliability Evaluation........................................................................ 63
6.2 Predictive Analysis............................................................................................. 63
6.2.1 Failure and Repair Parameter Estimation for Predictive Analysis......... 64
6.2.2 Results.................................................................................................... 65
6.2.3 Discussion .............................................................................................. 66
6.3 Computation of Risk Reduction......................................................................... 67
6.3.1 Recloser Maintenance............................................................................ 68
6.3.2 Wood Pole Maintenance ........................................................................ 69
6.3.3 Tree-Trimming Maintenance ................................................................. 71
6.4 Optimization ...................................................................................................... 72
6.4.1 Three Level Questions ........................................................................... 72
6.4.2 Labor Sensitivity Analysis ..................................................................... 74
7
Conclusions............................................................................................................... 75
7.1 Summary ............................................................................................................ 75
7.2 Conclusions........................................................................................................ 76
7.3 Further Work ...................................................................................................... 76
References......................................................................................................................... 77
Appendix A: Distribution Reliability Metrics .................................................................. 79
Appendix B: User Manual for Reliability Evaluation Tool.............................................. 82
Appendix C: User Manual for the Optimizer ................................................................... 95
v
List of Tables
Table 3.1: Recloser score sheet......................................................................................... 14
Table 3.2: Condition of typical failed recloser ................................................................. 20
Table 3.3: Score for recloser in average condition ........................................................... 20
Table 3.4: Score for recently maintained recloser ............................................................ 21
Table 3.5: Population predictions ..................................................................................... 31
Table 3.6: Estimate of failure rate..................................................................................... 31
Table 3.7: Estimate of maintenance effect........................................................................ 32
Table 4.1: Protection response of circuit breaker ............................................................. 38
Table 4.2: Protection response of fuse.............................................................................. 38
Table 4.3: Protection response of recloser........................................................................ 39
Table 4.4: Protection response of fuse with upstream recloser ........................................ 40
Table 4.5: Protection response of sectionalizer with upstream recloser........................... 42
Table 4.6: Switching response for upstream isolation ...................................................... 43
Table 4.7: Switching response for downstream isolation ................................................. 44
Table 4.8: Customer data .................................................................................................. 54
Table 4.9: Lengths of feeder section................................................................................. 54
Table 4.10: Reliability indices for the ieee- reliability test system, bus 2 ........................ 54
Table 5.1. Risk reduction vs. budget................................................................................. 56
Table 5.2. Decision variable code table ............................................................................ 56
Table 6.1: Overall historical reliability indices for distribution system ........................... 63
Table 6.2: Historical indices—overhead, underground, and device failures.................... 63
Table 6.3: Historical indices—outages caused by miscellaneous failures ....................... 63
Table 6.4: Reliability parameter estimates for overhead lines and underground cables .. 64
Table 6.5: Reliability parameter estimates for protective and switching devices............. 65
Table 6.6: Adjusted failure parameters for overhead and underground line segments .... 66
Table 6.7: Reliability indices using adjusted failure rates and repair times ..................... 66
Table 6.8: Reliability indices using adjusted failure rates and equivalent component..... 66
Table 6.9: Failure modes and corresponding maintenance activities ............................... 68
Table B.1: Feeder Topology Spreadsheets ....................................................................... 82
vi
List of Figures
Figure 1.1: Reliability benefit obtained from various resource-allocation levels............... 1
Figure 1.2: Risk-based resource allocation for distribution systems. ................................. 5
Figure 3.1: Recloser score vs. failure rate......................................................................... 19
Figure 3.2: Computing contingency probability reductions. ............................................ 22
Figure 3.3: Flow chart of degradation model approach.................................................... 25
Figure 3.4: Number of poles at every age of the decayed population. ............................. 27
Figure 3.5: Decayed population Lspi(t) plot..................................................................... 28
Figure 3.6 The average degradation level at every age. ................................................... 28
Figure 3.7: Percentage of decayed pole at every age........................................................ 29
Figure 3.8: Hazard function of decayed poles. ................................................................. 30
Figure 4.1: Variation in SAIFI before maintenance of wood pole. .................................. 48
Figure 4.2: Variation in SAIFI after maintenance of wood pole. ..................................... 48
Figure 4.3: Variation in SAIFI before tree-trimming. ...................................................... 49
Figure 4.4: Variation in SAIFI after tree-trimming. ......................................................... 49
Figure 4.5: Variation in SAIFI before recloser maintenance............................................ 51
Figure 4.6: Variation in SAIFI after recloser maintenance............................................... 51
Figure 4.7: Variation in SAIDI before recloser maintenance. .......................................... 52
Figure 4.8: Variation in SAIDI after recloser maintenance.............................................. 52
Figure 4.9: IEEE- reliability test system [36], bus 2......................................................... 53
Figure 5.1: Flowchart of ELPR-LRH optimization method. ............................................ 58
Figure 5.2: Reliability benefit vs. budget.......................................................................... 60
Figure 5.3: Resource splitting curve for different categories. .......................................... 60
Figure 6.1: Risk-based resource allocation implementation............................................. 62
Figure 6.2: Risk reduction due to maintenance on reclosers. ........................................... 69
Figure 6.3: Risk reduction obtained due to wood pole maintenance................................ 71
Figure 6.4: Risk reduction due to tree-trimming at a feeder level. ................................... 72
Figure 6.5: Budget vs. risk reduction................................................................................ 73
Figure 6.6: Budget-splitting curve for different tasks....................................................... 73
Figure 6.7: Labor sensitivity. ............................................................................................ 74
Figure C.1. Input file of pole candidate tasks. .................................................................. 95
Figure C.2. Risk reduction versus budget table (ptable.m)............................................... 96
Figure C.3. Task selection table (trim_select.m for tree-trimming). ................................ 96
vii
1 Introduction
Both inspection and maintenance of equipment are a critical part of utility
expenditures. It is important to ensure that every dollar spent helps improve the reliability
and performance of the system. As illustrated in Figure 1.1, the ideal budgetary allocation
results when the greatest benefit is obtained for every dollar spent. In this case, the
benefit is reliability improvement indicated by relevant reliability indices.
From the asset manager’s point of view, a resource allocation within the indicated
region in Figure 1.1 is desirable because it is the resource allocation for which the ratio of
benefit to allocation is greatest. For larger resource allocations, this ratio falls off, and
within the organization, the strength of the argument for obtaining such resource
allocations diminishes.
This chapter describes, in detail, the challenge that asset managers face in maintaining
their different distribution system assets. A brief review of common utility maintenance
practices and their impact on reliability is discussed. A risk-based method of allocating
maintenance resources is then proposed. Because of the limited resources available to this
project, the methodology developed is limited to reclosers, vegetation, and wood poles.
The methods developed for these can be adapted to other distribution equipment.
Figure 1.1: Reliability benefit obtained from various resource-allocation levels.
1.1 Asset Management Problem
Asset managers allocate resources among various maintenance activities. They are
constrained by limited monetary and labor resources available for a broad array of
maintenance activities. This presents a set of challenges to the asset manager that can be
broadly classified into three categories.
The first is how to identify and justify the resources needed for asset management.
Usually once a year, each asset manager must make a case for the financial and human
resources required to manage equipment for which he/she is responsible. His/her
1
argument is best made in terms of the benefit obtained from the resources allocated. This
establishes the total resources available to each asset manager.
Each manager must then decide how to allocate the available resources to different
maintenance programs. 1 This secondary resource allocation distributes available
resources from the first allocation to the different asset management programs. For this,
the asset manager must understand how the total benefit from all programs changes as
resources are shifted from one program to another.
The third problem is to select a set of maintenance projects2 to be completed within
each program, constrained by the secondary budgetary allocation. A solution to this
problem allows the asset manager to compare the benefits of the different maintenance
tasks available within a program and choose the best options depending on the resources
available.
Apart from the above three issues, there may be a situation where certain parts of the
system need to be maintained due to safety or regulatory requirements, regardless of the
reliability benefit obtained. Such obligatory tasks also have to be addressed.
In order to find a comprehensive solution to each of the above problems, asset
managers need tools to assess the benefit obtained from each maintenance task. Once that
is determined, the corresponding cost and labor requirements can be used to judge the
usefulness of the activity and prioritize accordingly.
1.2 State of the Art in Power System Maintenance
Before proposing a solution to the asset management problem presented in Section
1.1, a brief review of the state of the art in power system maintenance will be presented.
Maintenance of a component reduces its failure rate and, thereby, the frequency and
duration of interruptions experienced by customers. Utilities follow different procedures
[1] or strategies to maintain different kinds of equipment. These maintenance practices
can be broadly classified into two categories: corrective maintenance and preventive
maintenance.
1.2.1
Corrective Maintenance
Also known as the run-to-failure strategy, corrective maintenance involves no
maintenance of equipment until it fails. Once a component fails, it is replaced with a new
or repaired component. This strategy can be disastrous in terms of reliability and can
result in costly regulatory penalties. Most utilities have evolved from this method and use
one or more of the following preventive maintenance strategies.
1.2.2
Time-Based Preventive Maintenance
Unlike corrective maintenance, preventive maintenance is done on equipment before
a failure occurs, thus improving its condition and increasing the time before its next
failure. In time-based preventive maintenance, a fixed time period is associated with each
piece of equipment, after which it is replaced or maintained. This period is based on
1
Program: A budgetary category within the asset management group. Programs are typically identified by a
geographical region or type of equipment, e.g., tree-trimming, recloser maintenance, wood pole maintenance for a city
or county, etc.
2
Projects: A set of tasks within a particular program, e.g., tree-trimming for three feeders in a city, wood pole
maintenance (that may comprise of reinforcement or replacement) for ten segments, etc.
2
analysis of failure statistics and may use trial-and-error methods, expert judgment, or
more analytical methods to estimate the optimal frequency of maintenance that is both
economical and reliable at acceptable levels. The use of fixed-time period replacements,
however, can lead to sub-optimal use of assets and unnecessary maintenance of
equipment. Such strategies do not compensate for different conditions that identical
components may experience on a system.
1.2.3
Condition-Based Preventive Maintenance
Condition-based preventive maintenance better allocates resources by using
information regarding the current state of equipment to determine when and what kind of
maintenance needs to be done. These methods require inspection and monitoring to
estimate the piece of equipment’s condition and its remaining useful life before
maintenance. Examples include dissolved gas tests for transformer oil, recloser operation
counters, and visual inspection of feeders for vegetation growth. Condition information is
used to predict the probability of component failure and the maintenance that is needed to
prevent failure. Relative to time-based maintenance, condition-based methods typically
extend the interval between successive maintenances and, therefore, reduce maintenance
costs [2]. This method is restricted, however, to equipment whose cost of failure
outweighs the inspection and monitoring costs incurred. Improved testing, monitoring,
inspection, and data collection methods are needed to accurately predict the state of many
power system components.
Condition-based maintenance uses information from equipment inspection and
monitoring to estimate the condition of the equipment and schedule maintenance. The
method does not, however, consider the effects of component failure or quantify the
benefits of preventing failures. Decisions are made solely based on equipment condition,
not its relative importance.
1.2.4
Reliability-Centered Maintenance (RCM)
Reliability-centered maintenance (RCM) is a preventive strategy that is being used
increasingly by utilities. In this method, condition-based measurements are used to
determine the various components that require maintenance. Maintenance projects are
then ranked according to their effect on improving selected criteria. One or more
reliability indices are usually chosen as the criterion, and maintenance projects are carried
out to achieve desired target levels. While traditional maintenance programs, such as
vegetation management, recloser maintenance, and maintenance of sectionalizing
devices, are considered as discrete and unrelated programs, RCM provides a method to
integrate a variety of programs and tasks with a single global objective of improving
system performance [3].
1.2.5
Risk-Based Preventive Maintenance
Risk-based preventive maintenance methods further advance RCM [2]. Failure
probabilities estimated by condition monitoring methods, along with the failure effects
quantified by RCM methods, are used to determine the risk associated with failure of a
particular piece of equipment. This risk is combined with the financial and human
resource requirements to prioritize maintenance projects in order to maximize risk
reduction.
3
For transmission systems, risk is defined as the time-dependent product of the
probability of equipment failure and the consequence of its failure [2]. The consequence
of failure is the quantified effect of equipment outage, such as overload of equipment,
cascading failures, and low voltage. Risk-based maintenance is thus a form of RCM, with
the following specific attributes when applied to transmission systems [2]:
a. Condition information is used to estimate equipment failure probability.
b. Failure consequences are estimated and used in prioritizing maintenance tasks.
c. Equipment failure probability and consequence at any particular time are
combined into a single metric called “risk.”
d. Equipment risk may be accumulated over a time interval (e.g., a year or several
years) on an hour-by-hour basis to provide a cumulative risk associated with each
piece of equipment.
e. The prioritization (and thus selection) of maintenance tasks is based on the
amount of reduction in cumulative risk achieved by each task.
Selection and scheduling of maintenance tasks are performed at the same time (using
optimization algorithms), since the amount of reduction in cumulative risk depends on
the time when a maintenance task is implemented.
1.3 Risk-Based Allocation of Distribution System Maintenance
Resources
The objective of this work is to develop a similar risk-based strategy to allocate
maintenance resources and prioritize maintenance projects for distribution system assets.
The work will also provide a solution to the asset management problem discussed in
Section 1.1. To do this, certain important differences between transmission and
distribution need to be understood before extending the method to distribution systems.
First, unlike transmission systems, which are highly networked, most distribution
systems are radial. Hence, the effects of an outage are localized, and the chance of
cascading outages is very small. Furthermore, maintenance scheduled in one area can be
assumed to be independent of the conditions in another region of the system. This is not
the case in transmission systems, where maintenance of a component in one transmission
region may restrict a task in another region due to stability constraints.
Second, distribution systems have a much larger number of components than
transmission systems. The consequence of failure in most distribution components is thus
lower than that in transmission components. This implies a large number of decision
variables (candidate maintenance tasks) from which to choose and, hence, the need for
optimization techniques that can suitably handle them.
Furthermore, the conditions in a distribution system are relatively constant or
predictable compared to those in a transmission network, which can be highly dependent
on such variables as network topology, loading, and equipment outages due to
maintenance and environmental conditions.
This results in an important distinction in the nature of failure consequences. The
consequence of failure of a specific transmission component is time varying and
influences the short-term (hourly) as well as long-term (yearly) reliability indices. The
failure consequence of a distribution component tends to be constant and thus can be well
represented using the long-term or yearly indices.
4
Figure 1.2 provides an outline of steps involved in the risk-based resource allocation
strategy as it applies to distribution systems. Historical outage data and condition
measurements are used to develop models that can predict equipment failure rates. The
failure models are used to estimate how much each maintenance task will reduce a
component’s failure rate. The effects of a failure are then related to changes in reliability
indices. Failure rate reduction and the associated change in indices are used to compute
the risk reduction associated with each maintenance task. Finally, tasks are selected and
scheduled to maximize risk reduction subject to the resources available.
DATA ACQUISITION
Develop failure models for
individual components
Evaluate failure rate reduction for
each maintenance activity
Compute risk reduction for each
maintenance task
Optimize
Figure 1.2: Risk-based resource allocation for distribution systems.
1.3.1
Definition of Risk
Every piece of equipment in the distribution system has a finite life, with failure
probability that tends to increase with time. Maintenance improves the condition of
equipment and thus reduces its likelihood of failure. In defining risk, the following
effects of equipment failure will be considered:
a. Customer satisfaction, in terms of the expected number and duration of outages.
b. Revenue lost by the utility due to energy not served.
c. Cost to replace or repair failed equipment.
d. Regulatory or contractual penalties paid by the utility due to missed reliability
targets.
Repair and switching times for each component are assumed to be constant, and the
distribution network configuration is considered fixed. This allows the reliability effects
of each component to be expressed as linear contributions to the overall system indices
[4]. These effects are expressed [4], [5] as follows.
5
1.3.1.1 Effect on customer satisfaction
SAIFI, the system average interruption frequency index, is defined as
SAIFI = total number of customer interruptions / total number of customers
served
for a given time period. For time period Δt, as a function of failure rate λk,l, for failure
mode l of component k, the contribution of failure mode l of component k to the system
SAIFI is
SAIFI (t | k , l ) = λk ,l .Δt ⋅
nk ,l
N
(1.1)
with units of average number of interruptions per customer in time period Δt. The system
SAIFI is the sum of these individual SAIFI contributions over all components k and
failure modes l.
SAIDI, the system average interruption duration frequency index, is defined as
SAIDI = sum of customer interruption durations / total number of customers
served
for a given time period. The contribution of failure mode l of component k to the system
SAIDI is
n k ,l
SAIDI (t | k , l ) = λ k ,l .Δ t ⋅
∑d
j =1
j
N
(1.2)
with units of average hours of interruptions per customer in time Δt. The system SAIDI is
the sum of the individual SAIDI contributions over all components k and failure modes l.
1.3.1.2 Revenue lost by the utility
nk , l
ENS (t | k , l ) = λk ,l .Δ t ⋅ ∑ Pj d j
(1.3)
j =1
1.3.1.3 Cost of equipment failure
DevRisk (t | k , l ) = λk ,l .Δt ⋅ Cost (k , l )
(1.4)
1.3.1.4 Regulatory penalties due to violation of regulatory limits
The effects expressed by equations (1.1) to (1.4) can be directly computed using
standard analytical methods [6-9]. However, due to increased regulatory monitoring of
reliability indices, it may be necessary for utilities to also estimate the risk of paying
penalties that might arise from missed reliability targets. In such scenarios, it becomes
necessary to estimate not only the average reliability indices for the system but also the
variability in the indices [11] due to events that have low probability of occurrence with
6
substantially high penalties. The risk of penalties associated with each component may be
defined as shown in equations (1.5) and (1.6).
⎧⎪ ∞
⎫⎪
PBRF (t | k , l ) = ⎨ ∫ PBR ( SAIFI ). f ( SAIFI (t | k , l )) d ( SAIFI (t | k , l )) ⎬.Δt
⎪⎩TF
⎪⎭
(1.5)
⎧⎪ ∞
⎫⎪
PBRD(t | k , l ) = ⎨ ∫ PBR( SAIDI ). f ( SAIDI (t | k , l ))d ( SAIDI (t | k , l ))⎬.Δt
⎪⎩TD
⎪⎭
(1.6)
where:
• λk,l is the failure rate of component ‘k’ due to a maintainable failure mode ‘l.’
• Δt is the time interval under consideration.
• N is the total number of customers served.
• nk,l is the number of customers affected due to failure of component ‘k’ in mode
‘l.’
• dj is the duration of the interruption seen by the ‘j’th customer due to failure of
component ‘k’ in mode ‘l.’
• Pj is the load connected at point ‘j.’
• Cost (k,l) is the cost of failure for component ‘k’ in mode ‘l.’
• PBR(SAIFI) is a performance-based penalty for a SAIFI violation beyond
threshold TF.
• PBR(SAIDI) is a performance based penalty for a SAIDI violation beyond
threshold TD.
• f ( SAIDI (t | k , l )) is a probability distribution of SAIDI obtained by non-sequential
Monte Carlo simulation for component ‘k’ in failure mode ‘l.’
• f ( SAIFI (t | k , l )) is a probability distribution of SAIFI obtained by non-sequential
Monte Carlo simulation for component ‘k’ in failure mode ‘l.’
The time interval ‘Δt’ is assumed to be one year, so it can be removed from equations
(1.1) to (1.6). As discussed in Section 1.3, the consequence of a component’s failure is
assumed to be constant throughout the year. Unless the component is maintained, the
component’s failure rate is also assumed to be constant throughout the year. This
removes scheduling from the optimization problem, and leaves allocation of resources to
programs and selection of maintenance tasks for the year.
Furthermore, the subscript ‘l,’ indicating the maintainable failure mode, can also be
dropped without loss of generality, assuming that each of the equations (1.1) to (1.6)
represents the consequences of equipment failure due to a single maintainable failure
mode. Thus, the simplified expressions for the risk associated with each component can
be correspondingly written, as shown in equations (1.7) to (1.12):
SAIFI (k ) = λ ( k ).
nk
N
(1.7)
nk
SAIDI
(k ) = λ ( k ).
7
∑
j =1
N
d
j
(1.8)
nk
ENS (k ) = λ ( k ).∑ P j d j
(1.9)
j =1
DevRisk (k ) = λ (k ).Cost (k )
PBRF (k ) =
(1.10)
∞
∫ PBR ( SAIFI ). f (SAIFI (k ))d (SAIFI (k ))
(1.11)
TF
PBRD(k ) =
∞
∫ PBR(SAIDI ). f (SAIDI (k ))d (SAIDI (k ))
(1.12)
TD
The consequence of equipment failure may be expressed as the sum of the quantities
defined by equations (1.7) to (1.12). This sum comprises the risk associated with a
component’s failure. The risk associated with a component varies with its failure
probability. If, during the time period under consideration, the failure rate of the
component remains constant and is sufficiently low, the failure probability in equations
(1.7) to (1.12) can be replaced with the failure rate of the component.
Maintenance reduces the failure rate of a component and thus the risk associated with
its failure. The following expressions then can be used to define the effect of maintenance
on a component:
ΔSAIFI (k ) = SAIFI B (k ) − SAIFI A (k) = (λ B (k ) − λ A (k ) ) ⋅
nk
Δ SAIDI
Δ PBRF (k ) =
(k ) =
SAIDI
B ( k ) − SAIDI
j =1
nk
dj
= Δ λ ( k ).
N
nk
nk
j =1
j =1
(1.13)
∑d
j =1
N
j
(1.14)
ΔENS (k ) = (λ B ( k ) − λ A ( k ) ).∑ Pj d j = Δλ ( k ).∑ Pj d j
(1.15)
ΔDevRisk (k ) = (λB (k ) − λ A (k ) ).Cost (k ) = Δλ (k ).Cost (k )
(1.16)
∞
∫ PBR ( SAIFI ). f (SAIFI
TF
A ( k ) = (λ B ( k ) − λ A ( k ) ) ⋅
∑
n
nk
= Δλ (k ) ⋅ k
N
N
∞
B
(k ))d (SAIFI (k )) − ∫ PBR ( SAIFI ). f (SAIFI A (k ))d (SAIFI (k ))
TF
(1.17)
∞
∞
TD
TD
ΔPBRD (k ) = ∫ PBR ( SAIDI ). f (SAIDI B (k ))d (SAIDI (k )) − ∫ PBR ( SAIDI ). f (SAIDI A (k ))d (SAIDI (k ))
(1.18)
The subscripts ‘B’ and ‘A’ used in equations (1.13) to (1.18) correspond to the state
of the component before and after maintenance, respectively. Thus, the overall risk
reduction obtained from maintaining a component ‘k’ can be written as a linear
combination of each of factors, as shown in equation (1.19).
8
Customer satisfaction
revenue
6444
447444448 6Lost
47
48
ΔRisk (k ) = α 1 . ΔSAIFI (k ) + α 2 .ΔSAIDI (k ) + α 3 .ΔENS (k )
Cost of component failure
gulatory penalties
6447448 6444Re4
47444448
+ α 4 .ΔDevRisk (k ) + α 5 .ΔPBRF (k ) + α 6 .ΔPBRD(k )
(1.19)
The coefficients (αi) in equation (1.19) correspond to weights that an asset manager
assigns to the different factors based on their relative importance or confidence in their
accuracy. By choosing the units appropriately for the coefficients (αi), the overall risk
reduction associated with a component’s failure can be represented by a single monetary
value.
9
2 Maintenance Practices
This chapter reviews common maintenance practices for the following distribution
components included in the methodology developed in this project: reclosers, vegetation,
and wood poles.
2.1 Reclosers
Reclosers are very reliable devices that seldom fail. When failures do occur, however,
they can lead to widespread outages and damage that significantly affect reliability
indices and costs. Thus, many utilities use time-based preventive maintenance for
reclosers, scheduling maintenance for all reclosers on the system every three to five
years. Reducing the frequency of maintenance by using a risk-based methodology may
significantly reduce recloser maintenance costs.
2.1.1
Failure Modes
Failure of reclosers can occur in four different modes:
a. Failure to open.
b. Failure to close/reclose.
c. False trip.
d. Failure to lockout.
Most failures are caused by improper settings. Causes of recloser failure fit within these
four modes, and most can result in more than one type of failure mode.
Causes of recloser failure can be classified as follows:
a. Mechanical moving parts, including linkage, plunger, and contacts.
b. Electrical insulation, including bushings, stringers, and oil.
c. Structural, which addresses the integrity of the tank.
d. Improper setting or placement of a recloser.
e. Electronic, for electronic reclosers.
Preventive maintenance is performed to reduce the probability that these will occur.
2.1.2
Maintenance Practices
All but the very simplest recloser maintenance must be done in a shop; therefore, the
recloser must be removed from service. When a recloser is removed, another recently
serviced or new one is installed in its place. Since removal is costly, most utilities
perform a standard maintenance procedure on each recloser that comes into the shop. The
procedure returns the recloser to a serviced condition and reduces its failure rate.
During service of reclosers, oil is replaced or filtered. Mechanical parts, bushings, and
stringers are inspected and replaced if they are damaged or excessively worn. Contacts
are inspected for wear and replaced if needed. Insulation is tested to reduce the likelihood
of internal or external recloser faults. Structural maintenance includes removing rust and
repainting the tank can to a specified thickness of paint to reduce the effects of weather.
When maintenance is complete, the recloser is tested to ensure that it is operating in
accordance with its specified time curves. It is then returned to the warehouse for
installation when needed.
10
2.2 Vegetation
Vegetation-related failures are a large contributor to distribution system interruptions.
Utilities spend sizeable portions of their maintenance budgets controlling vegetation.
Because of the high cost, utilities must assess the effectiveness of their vegetation
maintenance programs.
2.2.1
Failure Modes
Tree growth into power distribution lines is less of a factor in distribution outages that
it is in transmission. Most utility tree-trimming programs are effective in keeping
growing vegetation away from distribution lines. Tree growth causes about 20 percent of
sustained distribution outages, most of which are of short duration. Growth-related
failures are maintainable and can be effectively controlled through regular tree-trimming
[11].
Tree failures occur when branches or entire trees break and come into contact with
the power-carrying conductors, resulting in short-circuited or downed conductors. Trees
outside the actual right-of-way may fail and cause outages, which makes maintenance
more difficult because utilities have limited authority outside of this area. Tree failure
causes about 40 percent of all sustained distribution outages. These faults are often more
severe and take longer to repair.
Some tree failures are preventable and thus maintainable, that is, if the tree shows
external signs of decay or degradation. If identified, these failures can be corrected by
providing structural support or removing dead or weak branches. Other tree failures, such
as those caused by severe weather, may cause extensive damage to the distribution
network. Such failures, which account for about 40 percent of all tree-related outages
[12], are not maintainable.
2.2.2
Maintenance Actions
Corrective maintenance refers to repair activities done to restore the system after a
fault. Crews are dispatched to locate the fault and remove the branch or tree from the
circuit. They should also clear any overhang that may come in contact with the lines in
the near future. Such maintenance is local and is aimed at restoring service to customers
in the shortest possible time.
Preventive vegetation maintenance is done before a failure actually occurs and may
include the following:
• Tree-trimming, which is the most common vegetation maintenance activity. Most
utilities follow a three- to six-year cycle of trimming, whereby a specialized crew
identifies vegetation overgrowth and trims to prescribed standards.
• Tree growth regulators. These are chemical agents used to slow vegetation growth
rates and are typically used after trimming to slow regrowth.
• Tree removal. Utilities also remove trees that threaten the system, sometimes
replacing them with shorter, slower-growing species.
• Spacer and tree cables. Insulated overhead conductors are used in areas requiring
higher reliability and in regions where accessing the right-of-way is difficult.
These cables allow vegetation to grow closer to the conductors and reduce the
number of outages.
11
2.2.3
Inspection Methods
To identify areas where tree-related outages are likely to occur and to determine the
proximity of trees to conductors, utilities have inspection programs to assess vegetation
near their circuits. Vegetation is inspected visually, often midway between two treetrimming cycles. Remote sensing and laser imagery, e.g., light detection and ranging
(LiDAR), are also used.
Some utilities also have inspection activities that extend beyond the right-of-way.
These hazard tree programs identify trees that are likely to fail and determine the
maintenance needed, including reinforcement or replacement, to avoid failures.
2.2.4
Factors Influencing Failure Rates
A feeder’s vegetation-related failure rates are influenced by the following factors
[13]:
a. Length of overhead lines.
b. Local density of vegetation, measured in number of trees per mile.
c. Growth and regrowth rates of different species of vegetation.
d. Climate, weather, and other environmental factors.
Since these factors may vary significantly among feeders, it is appropriate to model each
individual feeder’s failure rate, if data is available.
2.2.5
Vegetation Condition and Modeling
Overhead feeders are repairable systems, and vegetation-related failures are a
recurring process. When failures occur, repairs restore the system to a working state.
Repair or maintenance decreases the failure rate of the system. However, the system
tends to deteriorate as vegetation regrows and clearances decrease. This causes the
number of tree-related outages to increase with time, thereby increasing the failure rate.
Vegetation management decreases, but does not totally eliminate, vegetation-related
failures. The number of tree-related outages occurring in a unit of time may be used as a
measure to estimate the state of the system. If this value is higher than a specified limit,
the feeder may inspected to identify areas of maintenance. A low value indicates that no
maintenance is required. This method can also be used to determine if the current
trimming cycle is adequate.
Information that utilities maintain about tree-related outages is generally obtained
from an outage management system. Such information may include the location, date,
and time when the outage occurred, the time it took to repair the problem, the number of
customers interrupted, and the time when service was restored. There is often, however,
no information about the failure mode or maintenance performed.
Parametric failure rate models require information on each of these factors for
individual feeders. Because such information is often not available, the use of nonparametric models may be necessary. Non-parametric models only require historical
outage information and information about when the feeder was trimmed.
12
2.3 Wood Poles
Wood poles keep energized conductors and equipment away from the public and the
ground, and maintain separation between conductors. Poles also serve as a support
platform for equipment such as capacitors, regulators, and reclosers [14].
2.3.1
Decay of Wood Poles
Wood poles decay both internally and externally. Most decay is just below ground
level, where moisture, temperature, air, and absence of direct sunlight are most favorable
to the growth of fungi. This portion of the pole is also hidden from view and is close to its
natural breaking point under strain. Thus, it is the most critical part of the pole and
warrants special inspection and maintenance.
Wood pole failures usually occur as a result of physical stress such as wind, ice, or
vehicle impact. The tendency of a pole to fail under such stress is related to the strength
of the pole at ground level, where almost 90 percent of pole failures occur [15].
2.3.2
Detection and Measurement of Decay
Nondestructive evaluation methods estimate the effective area of the pole cross
section at the ground line. Visual inspection is ineffective, since it will not reveal internal
decay or decay below this point. Other approaches vary in accuracy and cost. These
include acoustic [16] and resistance force [17], sometimes combined with measurements
of humidity [18]. Another simple but cost-effective approach is to remove external decay
and assess the internal decay by drilling into the pole. This project assumes
measurements based on this approach.
2.3.3
Maintenance Practice
The primary maintenance on wood poles is ground line treatment [19], which can
provide an economical extension of a pole’s physical life. Ground line treatment is
recommended under the following conditions:
• Whenever a pole is inspected and the decay is not so far advanced that the pole
must be replaced.
• Whenever a pole over five years old is reset.
• Whenever a used pole is installed as a replacement.
Ground line treatment consists of removing the external decay, followed by
application of a preservative paste or grease. Then the treated section is wrapped, and the
dirt around the pole is replaced.
A decayed pole can be stubbed, whereby the decayed section is simply cut off, if the
remaining portion is long enough, strong enough, and in good enough condition.
Stubbing costs one-third to one-half the cost of replacing a pole. If stubbing is not
possible, the pole must be replaced when its residual strength is below applicable
standards.
13
3 Failure Rate Estimation
This chapter presents models that estimate the failure rates of components, as well as
the failure rate reduction achieved by preventive maintenance tasks.
3.1 Recloser
This section discusses the methodology for determining the condition of a recloser
while in service. This condition data is then used to estimate the recloser’s failure rate.
3.1.1
Condition Assessment
The methodology begins by assessing the condition of the recloser. A scoring sheet
that itemizes relevant failure causes is shown in Table 3.1. Each of the criteria on the
score sheet contributes to the reliability of a recloser, and most can be improved by
preventive maintenance. Those that cannot be improved are still relevant in determining
the recloser’s condition. These include the age of the recloser, which can only be
improved by replacing the recloser, and the duty cycle rate and environmental factor,
both of which are a function of placement on the distribution system rather than any
maintenance performed.
For some reclosers, each of the criteria can be evaluated while the recloser is in
service or from prior maintenance records. A large part of the cost of recloser
maintenance is removing it from service, and removing a recloser from service to assess
it, without performing maintenance, is never cost-effective. Components that can never
be assessed in service are therefore omitted from Table 3.1.
Table 3.1: Recloser score sheet
Score (0 - 1)
Weight
Criteria
Age of Oil
Duty Cycle Rate
Environmental Factor
Oil Dielectric Strength
Condition of Contacts
Age of Recloser
Experience with this Recloser Type
Condition of Tank
Sum
Weighted Average
14
Pre-Maintenance
Scoring criteria are as follows:
• Age of Oil – The oil in a recloser is the most important dielectric in the unit,
especially if the contacts are not in a vacuum. The oil helps extinguish arcs as
contacts open and close, keeps arcs from occurring between other electrical
conductors within the recloser, lubricates most of the moving parts, and is used to
raise the trip piston after operation. The average expected life of oil is three years.
Oil age thus provides a rough estimate of the oil’s dielectric strength without
removing the recloser from service.
• Duty Cycle Rate – Duty cycle is a measure of the use a recloser has experienced
since its last maintenance and is one of the most important criteria for determining
when maintenance should be performed again. Duty cycle is a combination of the
number of interruptions the recloser has performed, and either the percent of rated
interrupting current or the circuit X/R value. NEMA has defined a standard duty
cycle for distribution class reclosers [20]. Constant monitoring of every recloser’s
duty cycle is impractical, so an alternate criterion, duty cycle rate, is defined. To
calculate the duty cycle rate, the number of faults a recloser will see per year in a
certain location is determined from the historical data used to calculate the
utility’s SAIFI index. The value of system X/R at the recloser location is
determined from system data. Then the NEMA standard duty cycle definitions
give the number of operations per duty cycle for that location. Dividing the
operations/cycle by the expected operations/ year gives the duty cycle in
years/cycle for a recloser at that location. Duty cycle is then compared to expected
oil life, whereby duty cycle rate equals the expected duty cycle divided by the
expected oil life. This score is high for a high expected remaining duty cycle. If
the score is greater than one, then the expected duty cycle is longer than the
expected oil life, and the score is entered as one. This score is a function of
recloser location on the system and not of the actual recloser condition.
• Environment Factor – This criterion is for reclosers in locations that require more
frequent maintenance. It consists of a combination of recloser placement and
environmental effects on the physical condition of the recloser. For example, a
recloser bank protecting a feeder along a coastline will experience air with a much
higher salt content than one located farther inland. The salt may cause the
dielectric strength of the recloser oil to fall below standards much sooner than
normal. This criterion addresses such conditions.
• Oil Dielectric Strength – This score is important if the utility’s recloser
maintenance includes filtering the oil instead of replacing it. The score should be
given as the difference between the post-maintenance oil dielectric strength,
which is measured as part of maintenance, and the minimum allowable oil
dielectric strength, divided by the difference between the new and minimum oil
dielectric strengths.
• Condition of Contacts – This score is given as a percentage of remaining useful
contact life.
• Age of Recloser – This is important because, as with all machines, reclosers
become less reliable and fail with age. However, reclosers have proven to last for
many years, and age has not been shown to be a reliable predictor of failure.
Recloser age should still be monitored, though, as one indicator of condition.
15
•
•
Experience with this Recloser Type – This criterion is used to differentiate among
failure rates for different manufacturers or models, types, and sizes of reclosers.
Condition of Tank – If a tank has excessive damage, either from nature or
handling, the recloser may need maintenance before it is justified by the other
factors.
3.1.1.1 Scoring
Recloser assessment begins with selecting the criteria for a particular recloser, as
shown in Table 3.1. Different recloser types and models, for example, will use different
criteria. Each recloser’s score will then be normalized by dividing the score by the
maximum possible for the scored criteria. For example, evaluating contacts and oil
dielectric strength for many reclosers requires removal from service. These criteria will
not be included in the assessment or maximum possible score for those reclosers.
The score for each item is per unit of the remaining state of the recloser criterion. For
example, if the contacts are 60 percent of their original size, their score would be 0.60. A
recloser that has completed 75 percent of its recommended duty cycle would have a duty
cycle score of 1.00 – 0.75 = 0.25, indicating the remaining 25 percent of its duty cycle.
The resulting condition score, between 0 and 1, is denoted as xcs.
3.1.1.2 Weighting
The weight column in Table 3.1 represents the influence that a particular condition
actually has on the failure rate of a recloser. Weights will be determined in practice by
the combined opinion of manufacturers, utility engineers, and field personnel. Certain
items are utility dependent, such as the environment factor inspection item.
3.1.2
Failure Rate Calculation
To relate a recloser’s condition score to its numerical failure rate, historical failure
rate data from a number of systems were compiled for various power system components,
including reclosers [21]. From this data, the best, worst, and average failure rates for each
component were calculated. The resulting values for reclosers are as follows:
λ (0 ) = 0.0025 (Best)
λ (1 / 2 ) = 0.015 (Average)
λ (1) = 0.060 (Worst)
If no historical data exists for the system to be modeled, then these values can be
used. If, however, historical data is available for the system, then that data can, and
should, be used to determine recloser failure-rate statistics for that system. Equation (3.1)
[22] demonstrates how a system wide average recloser failure rate is calculated:
λ (1/2 ) =
Total number of recloser failures
(Number of reclosers ) x (Time period
)
(3.1)
Ideally, the number of reclosers should be constant over the time period; a failure rate
should be calculated for each such period, and then the failure rate for the entire period
calculated from these values. The calculations are complicated by the inherent reliability
and low failure rates of reclosers. The accuracy of the calculation thus depends on the
16
availability of such data and the time period over which it is available. Some utilities
already have systems in place to collect data that can be used to track component
reliability. Those that do not must use the best available data while gathering the needed
information.
Also from the available data, the lowest and highest failure rates for reclosers on the
system become the best, λ(0), and worst, λ(1), historical failure rates. If the calculated
values are not judged to be accurate, then the published [21] values should be used.
From the historical failure rates, coefficients A, B, and C are calculated using
equation (3.2) [21]:
λ (1 / 2) − λ (0)]2
[
A=
λ (1) − 2λ (1 / 2) + λ (0)
⎛ λ (1 / 2) + A − λ (0) ⎞
B = 2 ln⎜
⎟
A
⎠
⎝
C = λ ( 0) − A
(3.2)
These coefficients are recalculated periodically as data becomes available. Equation (3.3)
then estimates the failure rate for an individual recloser, based on the coefficients and its
condition [21]:
λ ( x ) = Ae B*x + C
(3.3)
where λ(x) is the recloser’s failure rate, and x is a modified condition score that is
calculated from the check sheet score xcs using equation (3.4):
x
1−
xcs − x1
x0 − xcs
x0 − x1
x0 − x1
(3.4)
If xcs is used directly, then a recloser would need a score of xcs = 1 to be assigned the best
failure rate on the system, and a score of xcs = 0 to be assigned the worst. A recloser with
xcs = 0 would have completely failed every condition with a score of zero, which is not
practical. Instead, the best and worst scores on the system should relate to the best and
worst historical failure rates. Thus, x1 is the worst recloser condition score recorded on
the system, and x0 is the best. The resulting value is subtracted from 1, because a high xcs
indicates a low failure rate, and a high x in equation (3.3) must represent a high failure
rate.
Equation 3.4 will produce values that are negative when a recloser score xcs is greater
than the previous best score or greater than one when xcs is less that the previous worst
score. When this occurs, xcs replaces the previous best x1 or worst x0 historical score, as
shown in equations 3.5 and 3.6. Then x for the recloser is recalculated with the new
values as follows:
If x < 0, then xcs is the updated x0
(3.5)
If x > 1, then xcs is the updated x1
(3.6)
17
3.1.3
Effects of Maintenance
The maintenance tasks associated with each criterion on the score sheet are assumed
to bring the score for that criterion to a predetermined value; this may be 1 or something
less than 1. New post-maintenance coefficients, equation (3.2), and a new failure rate,
equation (3.3), are calculated. Reliability indices for the system being simulated are then
computed using the evaluation tool discussed in Chapter 4 and Appendix B. The
calculated failure rates should then be calibrated so that the indices correlate with
historical indices. A least-squares approach has been suggested for this [21], using the
method of gradient descent.
3.1.4
Example
Six years of outage data were obtained from a utility and are used to illustrate the
recloser assessment method. Out of 341 reclosers on the system, 23 recloser failures
occurred during a 6.44-year period. Equation (3.1) produces an average failure rate λ(1/2)
of
λ (1/2) =
23
= 0.010473
341 * 6 . 44
The best and worst failure rates, λ(0) and λ(1), respectively, are then calculated. Each
recloser on the system failed either zero times or one time during the six-year period.
This gives failure rates of
λ(0) = 0 failure / 6.44 years = 0.00000
λ(1) = 1 failure / 6.44 years = 0.15528
These are too low and too high, respectively, to be practical; therefore, the following
published failure rates [21] are used for the best and worst values: 
λ(0) = 0.0025
λ(1/2) = 0.010478021
λ(1) = 0.060
Next, the A, B, and C coefficients are calculated using equation (3.2) to be
A = 0.0015321
B = 3.6514524
C = 0.0009679
The resulting equation (3.3) is
λ ( x) := 0.0015321 ⋅ e
3.6514524 ⋅x
+ 0.0009679
and the relationship of assessment score to failure rate is shown in Figure 3.1.
18
0.06
0.05
0.04
λ ( x) 0.03
0.02
0.01
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
Figure 3.1: Recloser score vs. failure rate.
Best and worst historical scores for reclosers on the system were not available;
therefore, the worst (x1) and best (x0) scores were assumed to be 0.31 and 0.95,
respectively. Table 3.2 then shows actual scores for a recloser that failed while in service.
It was considerably past its expected duty cycle. Its condition score xcs was 0.392.
Equation (3.4) corrects this to
x
x
0.95 − 0.932
0.95 − 0.31
0.872
which results in a failure rate, equation (3.3), of
λ ( 0.872) = 0.038
The low condition score, 0.392, as expected, produced a higher-than-average failure
rate.
19
Table 3.2: Condition of typical failed recloser
Score (0 - 1)
Weight
Criteria
Pre-Maintenance
Age of Oil
20
0
Duty Cycle Rate
20
0.5
Environmental Factor
20
N/A
Oil Dielectric Strength
15
N/A
Condition of Contacts
15
N/A
Age of Recloser
10
0.65
Experience with this Recloser Type
10
0.7
Condition of Tank
5
0.4
Sum
Weighted Average
65
25.5
0.392307692
Next, a recloser in near-average condition was scored, as shown in Table 3.3.
Table 3.3: Score for recloser in average condition
Score (0 - 1)
Weight
Criteria
Pre-Maintenance
Age of Oil
20
0.33
Duty Cycle Rate
20
0.9
Environmental Factor
20
N/A
Oil Dielectric Strength
15
N/A
Condition of Contacts
15
N/A
Age of Recloser
10
0.65
Experience with this Recloser Type
10
0.9
Condition of Tank
5
0.65
Sum
Weighted Average
65
43.35
0.666923077
This score produced the estimated failure rate of
20
⎛ 0.95 − 0.667 ⎞ = 0.00867
⎟
⎝ 0.95 − 0.31 ⎠
λ⎜
which is close to the system average failure rate of 0.01048.
Finally, Table 3.4 shows scores for a relatively new recloser that underwent
scheduled maintenance about a year before it was scored. This is indicated by the age of
the oil in the recloser. The recloser is not expected to complete a duty cycle before the oil
is due to be changed again.
Table 3.4: Score for recently maintained recloser
Score (0 - 1)
Weight
Criterion
Pre-Maintenance
Age of Oil
20
0.66
Duty Cycle Rate
20
1
Environmental Factor
20
N/A
Oil Dielectric Strength
15
N/A
Condition of Contacts
15
N/A
Age of Recloser
10
0.95
Experience with this Recloser Type
10
0.9
Condition of Tank
5
0.85
Sum
Weighted Average
65
55.95
0.860769231
The estimated failure rate for this recloser is
⎛ 0.95 − .861 ⎞
⎟
⎝ 0.95 − 0.31 ⎠
λ⎜
= 0.00351
which is close to 0.0025, the best failure rate previously found on the system.
A condition score, xcs, of 0.30, produces an equation (3.4) score, x, of 1.015. This
score of 0.30 replaces (equation (3.6)) the previous historical low score of 0.31, and the
recloser is assigned the lowest historical failure rate on the system.
3.1.5
Summary
This methodology allows for quantifiable assessment of a recloser’s condition. The
assessment is designed to be done in the field without removing the recloser from service.
The assessment score is converted to an estimated failure rate, which is based on
historical data.
21
The assessment criteria are directly related to maintenance tasks that may be
performed on the recloser. Each maintenance task will increase the score for the
associated criteria, resulting in a lower calculated failure rate. This method can be
adapted to other power system components.
3.2 Vegetation
Another approach to computing failure probabilities is illustrated in Figure 3.2, based
on a multi-state Markov probability model, where each of the J states is represented as a
deterioration level. Boundary conditions separating J states of deterioration in component
k are defined in terms of the measurements ck(t), using the deterioration function g(ck(t)).
The deterioration function returns a deterioration level j identified by dj-1<g(ck(t))<dj,
where the last state j=J represents the failed state. State J need not represent the relatively
rare catastrophic failure, for which very little data is typically available. Rather, state J
represents a set of measurement values for which engineering judgment indicates the
component should be removed from service. The particular representation in Figure 3.2
shows J=4 deterioration levels, and deterioration level j can be reached only from
deterioration level j-1. However, the model is flexible so that any number of deterioration
levels can be represented, and if data indicates that transitions occur between nonconsecutive states (e.g., state 1 to state 3), then the model can accommodate this easily.
The transition from level 4 to level 1 stochastically represents the effects of maintenance,
and if the decision problem is whether to maintain or not (a deterministic result of the
problem), then we would set μ41=0. The steps to implementing this approach are
described as follows.
μ41
Historical
Data c(t)
t=1,…,T
Statistical
Processing
Most Recent
Observation
c(T+1)
Level 1 λ12 Level 2 λ23 Level 3 λ34 Level 4
(minor)
(major)
(new)
(failed)
λjk
Deterioration
Function
g(c(T+1))
Figure 3.2: Computing contingency probability reductions.
a. Deterioration function: The deterioration function, denoted by g(ck), may be an
analytical expression, if one is available, or it may be a set of rules encoded as a program,
likely consisting of a nested set of if-then statements that returns a scalar assessment
value. For the model of Figure 3.2, the assessment value would be a deterioration level 1,
2, 3, or 4.
This represents a flexible and practical way of connecting our approach to the wealth
of existing knowledge and experience contained in the industry relative to interpreting
22
condition monitoring measurements. Often, such rules depend not only on the
measurements ck(t) but also on the rates of change in such measurements. These rules,
together with expertise provided by industry advisors, are used to develop the
deterioration functions. For example, a comprehensive compilation of such rules for
transformers [23] provides 62 different measurements for characterizing 23 transformer
failure modes. Examples, and some of the failure modes they detect, include dissolved
gas analysis results on main tank oil (indicating insulation deterioration, deterioration of
cooling system, or oil pump failure) and load tap changer oil (indicating oil dielectric
weakening), thermography testing (indicating magnetic circuit overheating or bushing
overheating), ultrasonic testing (indicating oil pump failure), partial discharge testing
(indicating magnetic circuit overheating), and winding and oil temperature measurements
(indicating deterioration of the transformer cooling system).
Little has been published on correlating equipment deterioration with operating
histories, a fact that stems from the difficulty in obtaining and merging operating and
condition data in ways that properly characterize deterioration. Statistical modeling and
analysis can be used to capture such trends, however. For vegetation, probabilistic
vegetation failure rate models developed in [13, 24], are used to capture deterioration in
this failure mode.
b. Transition intensities: Transition intensities between the various states of the model
can be obtained from life histories of multiple units of the same manufacturer and model.
In the case of Figure 3.2, λ12, λ23, and λ34 are needed. Consider a set of condition
measurements c(t)=[c1(t),c2(t),…,cK(t)] for K similar components taken over an extended
period of time t=0,1,…,T. For component i, the deterioration function is used to compute
the deterioration level indicated by each measurement. This gives the time the component
spent in deterioration level j. The mean of the durations for all components is then used
as the estimated time spent in state j. Reasonable estimates of the desired transition
intensities are obtained by inverting these mean duration times.
Transition intensities computed in this way capture deterioration in the state of the
equipment, but they do not capture variations in equipment failure propensity as a
function of loading or environmental conditions. To do this, one needs to model the
dependency of the transition intensities on these parameters. However, component
loading and environmental histories typically reside in database systems (such as control
center historians) distinct from component condition histories. This requires a significant
effort in data integration.
c. Desired failure probability: For a particular set of transition intensities, the
transition probability matrix for the case represented by Figure 3.2 is given by equation
(3.7). The state probability vector gives the probability that a component is in any
particular deterioration level at a given time and is denoted by p(hT)=[p1(hT) p2(hT)
p3(hT) p4(hT)], where h = 1,2,3,…, and T is the time step. If at time t=0 the component
resides in deterioration level 1, then the initial state probability vector is p(0)=[1 0 0 0].
The probability of finding the component in any deterioration level at time hT is then
given by p(hT)=p(0)Ph. Given that at time t=0 the component’s deterioration level is
known, this provides the probability of residing in the failed state in any future time
interval. We denote this failure probability for component k as pk(c). This probability is a
function of the time-dependent physical condition of the equipment c(t).
23
0
0 ⎤
λ12
⎡1 − λ12
⎢ 0
1 − λ 23
λ 23
0 ⎥
⎥
P=⎢
0
1 − λ34
λ34 ⎥
⎢ 0
⎢
⎥
0
0
1 − μ 41 ⎦
⎣ μ 41
(3.7)
In addition to failure probability, this model provides the ability to predict
maintenance-induced probability reduction and expected time to failure, metrics that are
important for a number of decision problems. If a particular maintenance task results in
renewing a component to deterioration level 1, for example, then if the component is in
deterioration level 3, the probability reduction for maintenance task m, Δp(m,k), is given
by the last element of the 1×4 row vector resulting from the calculation:
[1 0 0 0]P-[0 0 1 0]P=[1 0 -1 0]P
The expected time to failure is captured by computing first-passage times. Firstpassage time is the expected value of the time a process will take to transition from a
given state j to another state i, under the assumption that the process begins in state j. The
remaining life of the component is estimated from this computation. The method of
computing first-passage times is provided in [25], [26] discusses this issue from a power
system reliability perspective.
3.3 Wood Poles
Wood poles form the backbone of most overhead distribution circuits. Their purpose
is to keep conductors and equipment away from the public and the ground, and to
maintain separation between conductors. Poles also serve as a support platform for
equipment such as regulators and reclosers [14].
For most utilities, the wood pole is one of the most ubiquitous assets, and different
maintenance strategies for them can result in significant cost differences. Maintenance
planning can be more cost-effective if pole degradation can be predicted. Such predictive
capabilities provide the ability to estimate the number of required replacements in the
next budget cycle. Pole-specific prediction provides the ability to determine which poles
are most likely to need replacement. If degradation information can be transformed to
probabilistic failure indicators, e.g., probability of failure and time to failure, then the
effect of wood pole maintenance on these indices can be evaluated. These failure
indicators can then be used in system-level decision tools, such as reliability evaluation
programs, to compare different maintenance-related resource allocations among regions,
components, and types of maintenance. This section describes conversion of wood pole
condition data gathered from the field into predictive functions and illustrates the use of
these functions in developing probabilistic failure indicators.
3.3.1
Degradation Path Model Approach Basis
Wood pole failures occur as a result of physical stresses such as wind, ice, and
vehicle impact. The tendency of a pole to fail under such stress is usually related to the
strength of the pole at the ground line, where almost 90 percent of pole failures occur
[15]. Therefore, the most useful indicator of wood pole condition is its residual strength
at the ground line.
24
This strength is usually measured as the effective area of the cross section at the wood
pole ground line. A number of approaches for doing this vary in accuracy and
implementation cost. Some approaches described in the literature include acoustic [16],
resistance force [17], and combining measurements of resistance force and humidity [18].
Another simple but cost-effective approach is to remove external decay and assess the
internal decay pocket by drilling. This method is assumed here.
Figure 3.3 provides a flow chart of the degradation path model approach to convert
such condition measurements into probabilistic failure indicators. After obtaining the
condition history (1), the component degradation path model (2) is determined, and the
lifetime analysis (3) is performed using the actual failure data, or the extrapolated failure
data from the degradation path model. These two procedures provide the population
degradation path model (4) and the age-based hazard function (5), which are then mapped
point by point to get the condition-based failure rate(6), and then the time to failure and
the effect of maintenance are estimated. This model is data-driven; more data and better
data result in better models and ultimately better decision-making.
2
Obtain component
degradation path
Condition
History
3
Lifetime
analysis
1
4
6
Obtain population
degradation path
5
Hazard
function
Map condition to the failure rate &
estimate the effect of preventive
maintenance activity
Figure 3.3: Flow chart of degradation model approach.
3.3.2
Degradation Path Model
Let Rsgi(t) represent the residual strength, in units of N/mm2, at the ground line of
wood pole i, as a function of time t. Because different poles have different initial
strengths, the residual strength is normalized as
Lspi(t)=1-Rsgi(t)/Rsgi(0)
(3.8)
where Lspi(t) represents the lost strength percentage for pole i at time t.
Wood poles decay continuously; therefore, Lspi(t) is non-decreasing over time. If all
poles were identical and operated under exactly the same conditions and in exactly the
same environment, they would have the same degradation path. But of course, there is a
degree of variability in some or all of these factors. This variability in turn causes
variability in the degradation path. While different poles have different degradation paths,
the general degradation path formed will be quite similar from pole to pole. The
degradation-path model thus represents the degradation path of a particular wood pole
over time as
Lspi(t)=g(t;βi0,βi1,…, βin)
(3.9)
where t>0, and βi0,βi1,…, βin are the time regression coefficients for pole i. In general, the
form of g may be linear, polynomial, or exponential in the coefficients. Condition data
25
from the field are used to obtain the coefficients of this degradation path model. Two
kinds of nondestructive measurement data can be used in this model. The best kind
involves measurements for multiple poles taken over multiple time instances. Such
measurements provide the ability to obtain pole-specific degradation functions. More
common, though, is the kind that involves measurements for multiple poles taken at
approximately the same time, resulting in a single measurement per pole. Although such
data are inferior multiple measurements, they may still be used to characterize
degradation functions and, from those, to extract probabilistic failure indicators [27],
[28], [29].
Degradation Leading to Failure
Loading on a wood pole varies with time as weather conditions (mainly wind and ice)
change, so the model should include these conditions [27]. It is possible to use force
analysis based on weather modeling to obtain a statistical load model [17], but in this
report a simpler model is used. The National Electric Safety Code requires that a pole be
rejected when 33 percent of its strength is lost [18]. Based on this requirement, a pole is
assumed to fail when its strength falls below a given percentage of its initial strength,
denoted by fp (failure percentage), to which a value of 33 percent is assigned.
After obtaining a group of Lspi(t) curves, interpolation (when the poles have lost
more than 33 percent of their original strength) or extrapolation is used to obtain the
random variable lifetime (LT). The lifetime distribution cumulative function F(t), and the
hazard function H(t) can be obtained by standard statistical methods [28].
Variability in the degradation level across a pole population at a particular age t is
best described by a distribution. This distribution is denoted as:
Lspd(t)~dist{Lspm(t), Lspe(t)}
(3.10)
where
Lspd(t)is the degradation distribution at age t.
Lspm(t) is the mean of the distribution at age t.
Lspe(t) is the standard deviation of the distribution at age t.
At each age t, the mean Lspm(t) is mapped to the hazard function H(t) for the decayed
population; that is, if the lost strength percentage of pole i at an age t, Lspi(t), equals the
lost strength percentage population mean at some age t1, Lspm(t1), then the pole i failure
rate equals H(t1). This ensures that the condition-based failure rate can be estimated.
After the mean at every age is fit, the following expression is obtained for Lspm(t):
Lspm(t)= Ф(t; α0, α1,…, αn)
(3.11)
where t>0, and α0, α1,…, αn are the time regression coefficients.
The failure probability for any pole of age t, defined as P(T<t), is given by the
probability that the random variable Lspd(t) exceeds fp, according to
F(t)= P(T≤t)=P(Lspd(t)>fp)
26
(3.12)
Effect of Preventive Maintenance
A maintenance activity on a component subject to degradation may renew the
component to a less degraded state, slow the future rate of degradation, or both. For
example, a wood pole may be treated by chemical material to slow decay. The effect on
degradation can be quantified in the failure rate and time to failure. It is important to note
that failure rate and time to failure are population averages.
The lost strength percentage before maintenance is Lspi(tc)=Lspm(t0)→H(t0), and the
lost strength percentage after maintenance is Lspi(tc)=Lspm(t1)→H(t1). The failure rate
reduction, Δh, is
Δh= H(t1)- H(t0)
(3.13)
and the increase in time to failure, ΔTTF, is
ΔTTF= t0 - t1
(3.14)
This degradation path model can be used to estimate the condition-based failure rate,
failure rate reduction, and increase in time to failure. This information is highly useful in
asset management decision making. These procedures are illustrated in Section 3.3.3,
together with an application of budget planning and maintenance task selection.
3.3.3
Illustration
Field data consisting of age, initial strength, and one residual strength measurement
per pole were obtained for a group of wood poles. The total pole population includes
13,940 poles ranging in age from 1 to 79 years with a mean age of 30 years.
Measurements indicated that of the total population, 1,163 poles (8 percent) had begun to
decay. These are referred to as the decayed population, ranging in age from 5 to 67 years
with a mean age of 37. Figure 3.4 shows the distribution of the number of poles at each
age for the decayed population.
120
Number of poles
100
80
60
40
20
0
10
20
30
40
50
60
Age
Figure 3.4: Number of poles at every age of the decayed population.
27
Obtaining the Degradation Path Model
Figure 3.5 plots the lost strength percentage for each pole as a function of pole age t
for the decayed population. Each point represents a specific pole’s degradation level at its
given age.
1
0.9
Strength Loss Percentage
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
50
60
70
Age
Figure 3.5: Decayed population Lspi(t) plot.
From the data illustrated in Figure 3.5, for each age, the average lost strength
percentage was computed using the lost strength percentages for all poles of the given
age. The resulting averages are plotted against pole age in Figure 3.6.
1
0.9
Strength Loss Percentage
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
50
60
70
Age
Figure 3.6 The average degradation level at every age.
Figure 3.6 indicates that the average degradation trend of the population is nearly a
straight line. Therefore, as shown in equation (3.4), the degradation path for the decayed
population is represented using a linear model of the lost strength percentage mean
Lspm(t)=a1*t-a2
(3.15)
where the random variable a1 is called the mean strength loss rate. After removal of
several outliers, regression is used to obtain a1=0.014418, and a2=0.10683.
28
Equation (3.15) characterizes well the lost strength percentage for a pole once it is
known that the pole has begun to decay. However, as indicated previously, the number of
decayed poles is only eight percent of the total population. For very new poles, the
percentage of decayed poles is expected to be significantly less than eight percent, and
for very old poles, it is expected to be significantly more than eight percent. Figure 3.7
shows a plot of the percentage of decayed poles in the total population as a function of
age.
Percentage of the Decayed Poles
0.5
0.4
0.3
0.2
0.1
0
10
20
30
40
Age
50
60
70
Figure 3.7: Percentage of decayed pole at every age.
Figure 3.7 indicates that the percentage of decayed poles increases almost linearly
with age beginning at about ten years. Therefore, after removing several outliers, linear
regression is again used to obtain a linear model of the percentage of decayed poles as a
function of pole age:
Per(t)=0.004*t-0.04
(3.16)
Per(t) can also be interpreted as the probability of decay at age t. This information is
useful for predicting the number of decayed poles in a system as a function of time.
Figure 3.4 confirms the observation from Figure 3.7 that very few poles begin
deteriorating until about ten years old. We also observe from Figure 3.7 and equation
(3.16) that the percentage of decayed poles grows with time, indicating that the time at
which a pole actually begins to decay is a random variable. We call this random variable
the penetration age and represent it as b. The reason the penetration age almost always
exceeds ten years is due to the chemical treatment applied to each pole prior to
installation. This treatment resists decay very well until it is penetrated, at which time the
degradation process begins and continues from then on. By inspecting the number of
poles having a minimal but non-zero level of strength loss in Figure 2.5, it can be seen
that the penetration time ranges from 10 years to about 55 years. This variability is due to
the quality of the pretreatment and the pole location and environment.
From Figure 3.6 and equation (3.15), the mean strength loss rate is calculated as
a1=0.014418. From Figure 3.7 and equation (3.16), the penetration age b is identified as a
random variable. Therefore, for a given value of b, the mean lost strength percentage is
expressed as a function of pole age as
Lspm(t)=a1×(t-b)
29
(3.17)
Because there is only one measurement per pole (and the population degradation rate
is used to predict the degradation of each pole), a1 is fixed and b is a random variable.
Implied is that, while the age at which decay begins is unknown, once it begins, the pole
will decay at the rate of a1.
Estimation of Failure Rate
The transformation of equation (3.8) and the measurements are used to interpolate or
extrapolate the lifetime of the decayed pole population. After comparing several different
distributions, the Weibull distribution is selected, giving a hazard function having the
form
H(t)=(β/η)*(t/ η)( β-1)
(3.18)
The parameters are determined using the maximum likelihood method [28], resulting in
β=4.6676 and η=50.6090, which is shown in Figure 3.8.
To obtain the failure rate, the degradation (Lspi(t)) of the pole is measured. The
‘condition age’ is the age ta, where Lspm(ta)=Lspi(t), and t is the actual age of the pole.
Lspm(ta) is found from equation (3.15) and substituted into equation. (3.18) to get the
failure rate. Failure rate reduction and time to failure increase can then be calculated
using equations (3.5) and (3.6).
0.18
0.16
0.14
Hazard rate
0.12
0.1
0.08
0.06
0.04
0.02
0
0
10
20
30
Age
40
50
60
Figure 3.8: Hazard function of decayed poles.
Wood Pole Asset Management Decision Making
Budget Planning
The above information can facilitate asset management decisions for wood poles. An
asset manager, in planning financial resources for the next year, must answer the
following two questions: How many poles need to be replaced? How many poles need to
be treated?
To predict the failed number of poles and thus the number of poles to replace, the
strength loss rate a1 is used to estimate the degradation level of the decayed pole in the
future. For example, if a pole i has Lspi=0.3, it is estimated to reach a strength loss
reduction of 0.33 (and therefore fail) within (0.33-0.3)/0.014418 = 2.08 years.
30
To predict failure times for poles not yet decaying, the randomness at which healthy
poles join the decayed population must be accounted for. Equation (3.16) predicts the
number of decayed poles. The age distribution of the poles moves forward along the ageaxis in the next year, meaning more poles are decaying at that time. Table 3.5 presented
the decayed pole percentage, expected number of failed poles, expected number of poles
needing chemical treatment for future years 2006 and 2015, and the condition history of
2005, the current year. This data, together with replacement and treatment costs, facilitate
development of condition-driven budgets by the asset manager.
Table 3.5: Population predictions
Year
Decayed Pole Percentage
Failed Number of Poles*
Poles Need Treatment
2005
8.156%
541
622
2006
8.479%
549
633
2015
12.012%
644
1030
* Failure does not imply that the pole falls but rather, as defined in Section 3.1, that the strength loss reduction
percentage exceeds 33 percent.
Maintenance Tasks Selection
Budget constraints often require asset managers to prioritize maintenance tasks.
Useful indicators in this process for poles are the lost strength percentage, condition age,
and failure rate of each pole. Table 3.6 provides this information together with the actual
age for four selected poles. It is interesting that poles 3 and 4, although nearly the same
age, have significantly different condition ages and corresponding failure rates.
Table 3.6: Estimate of failure rate
Pole
Age
Lspi(age)
1
10
0
Condition Age
(years)
0
Failure Rate
(failures/year)
0
2
17
0.1025
14.5
0.001
3
39
0.0615
11.7
0.0004
4
42
0.2929
27.7
0.01
Similarly, the effect of maintenance can be estimated. Replacement is assumed to
entirely renew the pole, whereas treatment delays further decay by five years but does not
improve the condition of the pole. Therefore, both actions result in an increase in time-tofailure, but the effects on failure rate of the two actions are different; while replacement
causes immediate failure rate reduction, the failure rate reduction from treatment is not
incurred until the next and following years when the treated pole’s failure rate remains
fixed but the untreated pole’s failure rate continues to increase.
For poles without decay in the current year, equation (3.9) is used to estimate the
probability of decay in the next year. For example, for pole 1, which is ten years old
without decay, the next year’s failure rate is Per(11)*H(1)=0.004*5.2*10-8=2*10-10, and
its time to failure increase is Per(11)*5=0.004*5=0.02. For this pole, replacement and
treatment have the same effect, because this pole is in good condition to start. For
decayed poles, such as pole 2, replacement will renew the pole, so the failure rate
reduction is H(14.5)=0.01, and the increase in time to failure is its condition age of 14.5
31
years. Treatment will stop the decay, and the failure rate reduction seen in the next year is
H(15.5)-H(14.5)=0.000; the increase in time to failure is 5. These procedures were
applied to poles 1 through 4, with results summarized in Table 3.7. The results are
reasonable: maintenance activities on healthy poles have almost no effect but result in
significant benefit on the most decayed poles.
Table 3.7: Estimate of maintenance effect
Pole
Age
Failure Rate Reduction
(failures/year)
replace
2*10
-10
treatment
(per year)
2*10-10
Time to Failure Increase
(years)
replace
treatment
0.02
0.02
1
10
2
17
0.01
0.0003
14.5
5
3
39
0.0004
0.00015
11.7
5
4
42
0.01
0.0014
27.7
5
In selecting maintenance tasks, an asset manager should consider not only the effect
on failure rate and time to failure but also the consequences of failure in terms of the
effect of each candidate maintenance task on system reliability indices such as SAIDI and
SAIFI. A risk-based approach has been developed to address this issue [30].
32
4 Reliability Evaluation for Distribution Systems
The objective of this chapter is to describe the development of a predictive reliability
assessment tool that computes the reliability indices of a system and estimates the risk
reduction associated with maintaining each component in the system. The tool, whose
user manual in Appendix B, is used to determine the relative importance of each
component and prioritize maintenance resource allocation. Section 4.1 defines the various
parameters used to describe the reliability of distribution system equipment. A
description of the various components modeled in the analysis is provided in Section 4.2.
Section 4.3 describes the different states associated with protective and switching
equipment in response to a sustained or permanent fault, while Section 4.4 describes the
algorithm used for analytical reliability evaluation. Section 4.5 describes a Monte Carlo
integration method to compute the risk associated with regulatory penalties imposed due
to violations of SAIFI and SAIDI limits.
4.1 Parameters Used in Reliability Modeling of Distribution System
Equipment
The various components modeled in the distribution system include overhead and
underground line segments, protective devices (fuses, reclosers, circuit breakers,
sectionalizers) that operate when a fault occurs, and switching devices that are used to
reconfigure the system after the fault is cleared. In order to develop a predictive
reliability analysis method, mathematical equivalents for each component in the system
are required to represent their failure and repair characteristics. Since the indices used to
compute the risk reduction are associated with the effects of sustained or permanent
faults, temporary faults are excluded from the analysis. The following is a list of
parameters used to describe the reliability of distribution system equipment [1], [32].
4.1.1
Permanent Failure Rate (λp)
The permanent failure rate is a measure of the expected number of sustained or
permanent outages of a component in a fixed duration of time, usually one year.
Permanent faults require operating protective devices to clear them. Customers
downstream of the protective devices are interrupted and experience an outage duration
equivalent to the time taken to repair the fault, or determined by the switching and
sectionalizing actions done after the fault is interrupted.
4.1.2
Mean Time to Repair (MTTR)
The mean time to repair (MTTR) a component is the expected time required to repair
a permanent fault occurring on the component. It includes the time it takes to identify the
failed component, travel to the fault location, isolate the fault, and carry out repairs
before service is restored. The MTTR for protective and switching devices, however, is
the expected duration of repair for a component that fails to operate in response to a fault.
33
4.1.3
Protection Reliability (PR)
Protection reliability (PR) of a protective device is the conditional probability that the
protective device will operate to clear a fault. In other words, it is the probability of
successful operation contingent on a fault occurring. Thus, it is a quantity between zero
and one, where the value of one represents 100 percent successful operation upon the
occurrence of a fault. Failure to operate may arise from mechanical breakdown or
improper settings.
4.1.4
Reclose Reliability (RR)
Reclosers have the ability to repeat open/close actions during a fault. These actions
allow time for temporary faults to clear, thus avoiding long outages, and results in
reclosers failing in two mutually exclusive modes: failure to open and failure to close.
Both are conditional on a fault occurring downstream of the recloser. Reclose reliability
(RR) is the probability that a recloser will successfully reclose after it successfully opens
in response to a fault.
Failure to open and failure to close are the modes of failure on demand. Protective
devices can also fail due to inadvertent operation [33] when there is no fault, but this is
not considered here. Inadvertent operation is usually due to a problem in device
coordination and is comparatively rare.
4.1.5
Switching Reliability (SR)
Also called the probability of successful switching, the switching reliability (SR)
parameter defines the conditional probability that a switching action takes place
successfully as a part of the fault isolation scheme. Switching actions may not occur
because of mechanical failures, inability to locate a switch, failure of a crew to operate a
switch, or conditions such as overloading of feeders. Thus, SR is the probability that
switching is performed based on the occurrence of a fault whose MTTR is greater than
the time taken to perform the switching operations.
4.1.6
Mean Time to Switch (MTTS)
Mean time to switch (MTTS) represents the average time to operate a switch and
isolate a faulted area. This includes both the time to identify the faulted area and the time
to operate the switch. In the case of manually operated switches, it also includes the time
taken to travel to the switch’s location.
4.1.7
Probability of Failure (PF)
In order to compute the reliability metrics for protective and switching devices like
fuses, reclosers, and switches, the average number of times the device was expected to
operate and the number of times it was successful are required. Since such data may not
be available, because such records often are not maintained, device parameters like
protection reliability, reclose reliability, and switching reliability may be approximated
from the available data.
An approximate value for a devices’ probability of failure (PF) is estimated by using
the ratio of the number of failures of a particular type of device to its number of
operations. Thus, a device’s probably of PF may be defined as
34
PF =
Number of device failures
Total number of device operations
(4.1)
The total number of device operations in equation (4.1) includes the number of times
the device successfully operated and the number of times it failed. Using this failure
probability, the reliability metrics for various protective and switching devices can be
estimated. In the case of fuses, sectionalizers, and substation breakers, whose primary
mode of failure is failure to open, PR is simply the complement of PF, as shown in
equation (4.2). A similar expression can be used compute the switching reliability of
switches, as shown in equation (4.3).
PR = 1 − PF
(4.2)
SR = 1 − PF
(4.3)
Special consideration needs to be given to reclosers. Since these devices have more
than one mode of failure, both the protection reliability and reclose reliability must be
computed. If data distinguishing the failure modes is not available, PR and RR can be
estimated by using PF. If ‘A’ is the event that the recloser fails to open in the event of a
fault, and ‘B’ is the event that the recloser fails to reclose, then a recloser has four
different states of operation:
A:
Recloser opens when a fault occurs, with a probability of P( A ) = PR
(4.4)
A : Recloser fails to open when a fault occurs, with a probability of
P( A) = 1 − PR
(4.5)
B A : Recloser successfully recloses after opening on a fault, with a probability
of P ( B A ) = RR
(4.6)
B A : Recloser fails to reclose after clearing a fault, with a probability of
P( B A ) = 1 − RR
(4.7)
The events described by equations (4.6) and (4.7) depend on the successful opening
of a recloser during a fault. From the four possible states of a recloser, it can be observed
that those represented by equations (4.5) and (4.7) correspond to recloser failures. Since
these are mutually exclusive (a recloser can either fail to open or fail to reclose once
opened, but not both), the total probability of recloser failure can be written as
PF = P( A) + P( B ∩ A )
(4.8)
This can be further simplified as
PF = P( A) + P( B A ) * P( A ) = (1 − PR) + (1 − RR) * PR
(4.9)
Assuming the probability a recloser opens in response to a fault is equal to the
probability that it recloses after clearing a fault (PR = RR), and rearranging the above
35
expression, the values of protection reliability and reclose reliability can be obtained as
PR = RR = 1 − PF
(4.10)
4.2 Models Used in the Program
The parameters defined in Section 4.1 are now used in this section to describe the
reliability characteristics of distribution system components.
4.2.1
Overhead Line and Underground Cable Segments
Overhead lines and underground cables are modeled as repairable systems. Faults
occur on segments that are isolated by protective and switching devices before carrying
out repairs. The parameters used to describe segments are as follows: permanent failure
rate per mile (λp), mean time to repair a fault (MTTR), average cost per failure (COF),
which is the cost of repair that results from a failure, and length of the segment.
4.2.2
Fuses, Reclosers, and Breakers
Protective devices are assumed to be located at the upstream node of a segment and
are normally closed. Fuses, reclosers, breakers, and sectionalizers are modeled using their
probability of failure, protection reliability, reclose reliability, and MTTR.
4.2.3
Switches
Switches are used to reconfigure a system after a fault is interrupted. Normally closed
(NC) switches are located at the source, or upstream, node of a segment. Normally open
(NO) switches are located on the load, or downstream, node of a segment. Switches
cannot interrupt fault currents, so their PR is set to zero.
Switches are modeled by switching reliability and mean time to switch. When a fault
occurs, the upstream protective device responds to interrupt the fault. If there were no
switches between the faulted segment and the upstream device, the time of outage for all
customers downstream of the device is the MTTR of the faulted segment. However, if
there is a switch between the faulted segment and the device, it may be beneficial to
operate the switch, so that the interrupting device can be reset to restore service to some
customers. Switching in this case requires two operations:
• Open the upstream switch nearest to the faulted segment, with a time of MTTSswi.
• Close the previously open device, with a time of MTTSdev for that device.
The total time to complete this sequence is estimated as
MTTSseq = MTTSswi + MTTS swi − MTTS dev
(4.11)
This equation assumes that all switch and device MTTS values represent base-to-station
travel time, so |MTTSswi – MTTSdev| is the travel time between the switch and the
protective device.
The switching reliability of the protective device represents the probability that the
device will actually be reset when desired. A value other than 1 means overloading and
mechanical failures.
36
4.2.4
Sectionalizers
Sectionalizers are used in conjunction with an upstream recloser in cases where fast
switching to restore load is required. When a fault occurs, the upstream recloser opens
and recloses, allowing sufficient time for the fault to clear. After a predefined number of
such operations, the recloser remains open long enough for the sectionalizer to open and
then reclose again. This allows customers between the sectionalizer and recloser to avoid
a sustained outage. Sectionalizers are modeled using PR, MTTR, MTTS, and SR.
4.2.5
Equivalent Component
In addition to the failures of overhead lines, underground cables and protective
equipment, outages also occur for many other reasons, including the following:
a. Transmission outages.
b. Public acts, such as accidents, vandalism, and accidental dig-ins.
c. Failure of other equipment, including lightning arrestors, capacitor banks,
transformers, and many others.
d. Utility errors.
Since it would be inappropriate to attribute such outages to any of the equipment
listed in subsections 4.2.1 to 4.2.4, they are represented as a separate component with
characteristics equivalent to the components and failure modes that are not modeled. This
equivalent component will be included at the beginning of every feeder. It represents the
combined effects of all other outage causes. Similar to an overhead or underground line
segment, the equivalent component is modeled using permanent failure rate (λp) and
MTTR.
4.3 System Response to Outages
When an outage occurs in a distribution system, the system transitions into one or
more states based on events that happen after the outage. This begins with the nearest
upstream protective device sensing a fault and operating to isolate it. After a crew is
dispatched to repair the fault, the nearest switch upstream of the fault is opened by the
crew. This isolates the fewest possible customers while the rest are restored by reclosing
the protective device. Further restoration is possible by opening switches downstream of
the fault and closing tie switches to connect the isolated region to another feeder or
another region of the same feeder. Once the failed equipment is repaired, the system
returns to its original state and remains there until the next fault.
In this section, the response of protective and switching devices to a sustained fault
are described along with their equivalent models.
4.3.1
Circuit Breaker
A circuit breaker is described by its protection reliability, mean time to repair
(MTTRB), switching reliability, and mean time to switch. For a fault with failure rate (λ)
occurring downstream of the breaker, its probability of operating successfully and
clearing the fault is PR. The repair time of the faulted segment is MTTR, so all the
customers downstream of the breaker experience an outage with frequency (PR * λ) and
duration of MTTR.
37
If the breaker fails to operate, the next upstream protective device is expected to
operate and clear the fault. The number of customers interrupted is determined by the
upstream device. The frequency of such events is given by (1-PR)* λ, and the duration of
interruption is MTTR+MTTRB. The probability of the upstream device failing will be
neglected in this analysis because the probability of multiple breaker failures is very low.
If LP and LS refer to the load interrupted when the primary and the secondary (backup)
protection device operate, the outcomes of the two states are shown in Table 4.1.
Table 4.1: Protection response of circuit breaker
Circuit breaker successfully clears a fault with failure rate (λ) and repair time (MTTR)
Frequency
PR * λ
Duration
MTTR
Customers interrupted
Downstream of breaker
Expected cost of failure
PR * λ * COF, where COF is cost of outage on faulted line segment
Expected energy interrupted
PR * λ * MTTR * LP
Circuit breaker fails to clear a fault with failure rate (λ) and repair time (MTTR)
Frequency
(1-PR)* λ
Duration
(MTTR+MTTRB)
Customers interrupted
Downstream of backup (next upstream) protective device
(1 - PR) * λ * (COF + COFD), where COFD is cost associated with
Expected cost of failure
breaker failure
Expected energy interrupted
(1 - PR) * λ * (MTTR + MTTRB) * LS
4.3.2
Fuse with No Upstream Recloser
A fuse that is not coordinated with an upstream recloser responds like a breaker to a
sustained fault. Hence, the states shown in Table 4.2 apply to a fuse.
Table 4.2: Protection response of fuse
Fuse successfully clears a fault with failure rate (λ) and repair time (MTTR)
Frequency
PR * λ
Duration
MTTR
Customers interrupted
Downstream of fuse
Expected cost of failure
PR * λ * COF, where COF is cost of outage on faulted line segment
Expected energy interrupted
PR * λ * MTTR * LP
Fuse fails to clear a fault with failure rate (λ) and repair time (MTTR)
Frequency
(1 - PR) * λ
Duration
(MTTR + MTTRF), where MTTRF is average repair time for a fuse
Customers interrupted
Downstream of backup protective device
(1 - PR) * λ * (COF + COFD), where COFD is cost associated with fuse
Expected cost of failure
failure
Expected energy interrupted
(1 - PR) * λ * (MTTR + MTTRF) * LS
4.3.3
Recloser
As a primary protective device, a recloser’s response to a sustained fault is similar to
that of a breaker. In the event of a permanent fault, the recloser is expected to open and
lockout. Recloser failure occurs when the recloser fails to open during a fault. For a
38
downstream fault with failure rate (λ) and repair time (MTTR), the states shown in Table
4.3 occur for a recloser with protection reliability (PR) and repair time (MTTRR).
Table 4.3: Protection response of recloser
Recloser successfully clears a fault with failure rate (λ) and repair time (MTTR)
Frequency
PR * λ
Duration
MTTR
Customers interrupted
Downstream of recloser
Expected cost of failure
PR * λ * COF, where COF is cost of outage on faulted line segment
Expected energy interrupted
PR * λ * MTTR * LP
Recloser fails to clear a fault with failure rate (λ) and repair time (MTTR)
Frequency
(1 - PR) * λ
Duration
(MTTR + MTTRR)
Customers interrupted
Downstream of backup protective device
(1 - PR) * λ * (COF + COFD), where COFD is cost associated with
Expected cost of failure
recloser failure
Expected energy interrupted
(1 - PR) * λ * (MTTR + MTTRR) * LS
4.3.4
Fuse with Upstream Recloser
Fuses are coordinated with upstream reclosers to allow temporary faults to clear by
opening and reclosing, thus saving the fuse. The fuse is described by its protection
reliability (PRF) and repair time (MTTRF), and the recloser is described by its protection
reliability (PRR), reclose reliability (RRR), and repair time (MTTRR). For a permanent
fault downstream from the fuse, with fault failure rate λ and repair time MTTR, the
following mutually exclusive events can occur:
a. Recloser opens and recloses, and the fuse opens, causing an outage to customers
downstream of the fuse. The frequency of such a situation is given by (PRF* RRR
* PRR * λ), and the duration of interruption is MTTR.
b. Recloser opens and recloses, but the fuse fails to open. The recloser locks out,
interrupting customers downstream of the recloser. The frequency of such a
situation is given by ((1 - PRF) * RRR * PRR2 * λ), and the duration of the
interruption is (MTTR + MTTRF).
c. Recloser opens and recloses, the fuse fails to open, and the recloser fails to open,
causing the fault to be interrupted by the next device upstream of the recloser and
interruptions to all customers downstream of that device. The frequency of
occurrence of this event is ((1 - PRR)*(1 - PRF) * RRR * PRR * λ), and the duration
of interruption is (MTTR + MTTRF + MTTRR). The probability of this event,
however, is very low and will be neglected in this analysis.
d. Recloser opens but fails to reclose, causing interruptions to customers
downstream of the recloser. The frequency of this event is ((1 - RRR) * PRR * λ),
with an outage duration of (MTTR + MTTRR).
e. Recloser fails to open and the fuse opens, interrupting customers downstream of
the fuse. The frequency of the event is given by: (PRF * (1 - PRR) * λ), while the
duration of outage experienced by the customers downstream of the fuse is
MTTR. This is an event when the failure of the recloser goes unnoticed since the
fuse successfully operates to clear the fault. It must be included in the analysis,
however, to completely describe the coordinated fuse/recloser combination.
39
f. Recloser fails to open, and the fuse fails to open, resulting in the fault being
cleared by the next device upstream of the recloser. The frequency of this
occurrence is given by ((1 - PRF) * (1 - PRR) * λ), which is a very low and will be
neglected in this analysis. The interruption duration is (MTTR+MTTRF+MTTRR).
These six states completely describe a fuse coordinated with a recloser when a
sustained fault occurs downstream of the fuse. These events are summarized in Table 4.4.
Table 4.4: Protection response of fuse with upstream recloser
Recloser opens and recloses, and fuse clears the fault with failure rate (λ) and repair time (MTTR)
Frequency
(PRR*RRR*PRF*λ)
Duration
MTTR
Customers interrupted
Downstream of fuse
Expected cost of failure
PRF*RRR*PRR *λ*COF (COF is cost of outage on faulted line segment)
Expected energy interrupted
PRF*RRR*PRR *λ*MTTR*LP (Lp is load downstream of fuse)
Recloser opens and recloses, fuse fails to clear the fault, and recloser opens to clear the fault with
failure rate (λ) and repair time (MTTR)
Frequency
((1-PRF)*RRR*PRR2*λ)
Duration
(MTTR+MTTRF)
Customers interrupted
Downstream of recloser
((1-PRF)*RRR*PRR2*λ)*(COF+COFD), where COFD is cost associated
Expected cost of failure
with failed fuse.
((1-PRF)*RRR*PRR2*λ)*(MTTR+MTTRF)*LS, where LS is load
Expected energy interrupted
downstream of recloser.
Recloser opens and recloses, fuse fails to open, and recloser fails to open to clear the fault; this
event is not modeled due to very low probability
Recloser opens but fails to reclose, causing outage to all downstream customers
Frequency
((1-RRR)*PRR*λ)
Duration
(MTTR + MTTRR)
Customers interrupted
Downstream of recloser
((1-RRR)*PRR*λ)*(COF+COFD), where COFD is cost associated with
Expected cost of failure
failed recloser.
((1-RRR)*PRR*λ)*(MTTR+MTTRR)*LS, where LS is load downstream
Expected energy interrupted
of recloser
Recloser fails to open in response to the fault, and fuse opens to clear the fault
Frequency
(PRF*(1-PRR)*λ)
Duration
MTTR
Customers interrupted
Downstream of fuse
Expected cost of failure
(PRF*(1-PRR)*λ)*(COF)
Expected energy interrupted
(PRF*(1-PRR)*λ)*(MTTR)*LP, where LP is load downstream of fuse
Recloser fails to open and fuse fails to open; this event is not modeled due to very low probability
4.3.5
Sectionalizer with Upstream Recloser
Sectionalizers are switches that are coordinated with an upstream recloser. In
response to a sustained fault, they open while the recloser is open to isolate the fault. If
the sectionalizer fails to open, the coordinated recloser opens again, locking out to isolate
the fault. A sectionalizer is described by its protection reliability (PRS) and repair time
(MTTRS). When a permanent fault with failure rate (λ) and repair time (MTTR) occurs
downstream of the sectionalizer, the following mutually exclusive events can happen:
40
a. Recloser opens to clear the fault and sectionalizer opens to isolate the faulted
segment. The frequency of such an event is (PRR * PRS * RRR * λ), with an outage
duration of MTTR to the customers downstream of the sectionalizer.
b. Recloser opens, but the sectionalizer fails to open. The recloser opens and locks
out, interrupting customers downstream of the recloser. The frequency of such an
event is ((PRR) 2 * (1-PRS) * λ) with an outage duration of (MTTR+ MTTRS).
c. Recloser opens and sectionalizer opens, but the recloser fails to reclose, causing
sustained interruptions to all customers downstream of the recloser. The
frequency of this event is: (PRR * PRS * (1 - RRR) * λ), with interruption duration
of (MTTR + MTTRR).
d. Recloser opens but sectionalizer fails to open. The recloser then recloses and fails
to open again. The fault is interrupted by the backup protection device upstream
of the recloser. This event has a very low frequency of occurrence
e. ((1 - PRR) * (1 - PRS) * PRR * λ), and is neglected in this analysis. The duration of
interruption for this event is (MTTR + MTTRS + MTTRR).
f. Recloser fails to open, so the fault is cleared by the backup protective device. All
customers downstream of the backup device are interrupted. The frequency of this
event is ((1 - PRR) * λ), while the duration of interruption is MTTR + MTTRR.
41
Table 4.5: Protection response of sectionalizer with upstream recloser
Recloser opens, sectionalizer opens to isolate the fault, and recloser recloses
(PRR* PRS*RRR* λ)
MTTR
Downstream of sectionalizer
(PRR* PRS*RRR* λ)*COF (COF is cost of outage on faulted line
Expected cost of failure
segment)
Expected energy interrupted (PRR* PRS*RRR* λ)*MTTR*LP (Lp is load downstream of sectionalizer)
Recloser opens, sectionalizer fails to open, and recloser opens again and locks out to clear the fault
Frequency
((1-PRS)*PRR2*λ)
Duration
(MTTR+MTTRS)
Customers interrupted
Downstream of recloser
Frequency
Duration
Customers interrupted
((1-PRS)*PRR2*λ)*(COF+COFD), where COFD is cost associated with
failed sectionalizer
(((1-PRS)*PRR2*λ)*(MTTR+MTTRS)*LS, where LS is load downstream
Expected energy interrupted
of recloser
Recloser opens, sectionalizer fails to open, and recloser fails to open again to clear the fault. This
event is not modeled due to very low probability
Recloser opens, sectionalizer opens to isolate the fault, but recloser fails to reclose, causing outage
to all downstream customers
Frequency
((1-RRR)* PRS *PRR*λ)
Duration
(MTTR + MTTRR)
Customers interrupted
Downstream of recloser
((1-RRR)* PRS *PRR*λ)*(COF+COFD), where COFD is cost associated
Expected cost of failure
with failed recloser
((1-RRR)* PRS *PRR*λ)*(MTTR+MTTRR)*LS, where LS is load
Expected energy interrupted
downstream of recloser
Recloser fails to open
Frequency
(1-PRR)*λ
Duration
MTTR+MTTRR, where MTTRR is recloser’s expected repair time
Customers interrupted
Downstream of backup protection device
((1-PRR)*λ)*(COF+COFD), where COFD is failure cost associated with
Expected cost of failure
failed recloser
((1-PRR)*λ)*(MTTR+ MTTRR)*LT, where LT is load downstream of
Expected energy interrupted
backup device
Expected cost of failure
4.3.6
Switching
After these devices operate, distribution circuits are switched to isolate the faulted
portion of the system and restore power to as many customers as possible in the shortest
possible time. In the previous sections, protective devices and their fault responses were
modeled. In this section, two such switching modes are modeled: upstream isolation and
backfeeding, or downstream, isolation.
4.3.6.1 Upstream Isolation
In an upstream isolation scheme, the upstream switch nearest to the fault is opened
after the fault is interrupted. The interrupting device is then reset (closed). This reduces
the outage duration to those customers between the switch and interrupting device. If
there is no upstream switch, or if it is not opened, all customers downstream of the
interrupting device will experience an outage duration equal to the time taken to repair
42
the fault. If SRD and SRS are the switching reliabilities of the device that interrupted the
fault and the upstream switch operated to isolate the faulted area, respectively, then
SRseq= SRD * SRS represents the switching reliability of the switching sequence. The
equivalent switching time required to open the switch and close the protective device is
as follows:
MTTS seq = MTTS swi + MTTS swi − MTTS dev
The customers that are restored by switching experience an equivalent outage duration
given by
MOTseq = SRseq * MTTS seq + (1 − SRseq ) * MTTR hours
where MTTR is the time taken to repair the fault. Table 4.6 describes the states
associated with upstream switching done after a sustained fault.
Table 4.6: Switching response for upstream isolation
Switching sequence is successful
Frequency
Interruption duration for customers downstream of switch
Interruption duration for customers downstream of
protective device and upstream of switch
Expected energy restored (switching restores some of
load that was interrupted by the protective device)
Switching sequence fails
Frequency
Interruption duration for customers downstream of switch
Interruption duration for customers downstream of
protective device and upstream of switch
Expected energy restored
SRseqλ
MTTR
MTTSseq
λ∗SRseq* MTTSseq*Lswi, where Lswi is load
restored by switching
(1-SRseq) λ
MTTR
MTTR
None
4.3.6.2 Backfeeding
Another method of reducing the outage duration experienced by customers during a
sustained interruption is through backfeeding. Normally open switches are closed, while
normally closed switches are opened, to provide alternative paths for service to
customers. Thus, when a sustained outage occurs, the nearest NC switch downstream of
the fault is opened, and an NO switch located further downstream of the circuit is closed,
restoring power to the segments in between the switch pair. NO switches may connect to
other parts of the faulted feeder or to an adjacent feeder when closed. The expressions for
the expected outage duration and energy restored are similar to those for upstream
switching.
If SRNO and SRNC are the switching reliabilities of the NO switch that is closed and
the NC switch that is opened, respectively, then SRseq= SRNC*SRNO represents the
switching reliability of the sequence to restore service to customers downstream of the
NC switch,. The equivalent switching time is
MTTS seq = MTTS NC + MTTS NC − MTTS NO
The outage duration for restored customers is
43
MOTseq = SRseq * MTTS seq + (1 − SRseq ) * MTTR hours
where MTTR is the time to repair the fault. Table 4.7 describes the states associated with
downstream switching done after a sustained fault.
Table 4.7: Switching response for downstream isolation
Switching sequence is successful
Frequency
Interruption duration for customers upstream of NC switch
Interruption duration for customers restored downstream of
NC switch
Expected energy restored (switching restores some load
interrupted by protective device)
Switching sequence fails
Frequency
Interruption duration for customers upstream of NC switch
Interruption duration for customers downstream of NC
switch
Expected energy restored
SRseqλ
MTTR
MTTSseq
λ∗SRseq* MTTSseq*Lswi, where Lswi is
load restored by switching
(1-SRseq) λ
MTTR
MTTR
None
4.4 Analytical Reliability Evaluation
Analytical evaluation of reliability is a predictive method in which each contingency
is simulated, and its effect on each of component in the system is determined and
weighted by the probability of the contingency. This gives the expected (average) values
for the frequency and duration of outages caused by each contingency. The expected cost
of equipment failure and expected energy not served are also computed, as required by
the formulation for risk reduction of Section 1.3.1.
An enumerative analysis algorithm, in which the failure consequences of each
component are weighted by its failure probability, is used to compute system reliability
indices. The following is a brief description of the algorithm:
a. For a feeder, determine the protective and switching device locations, and the
number of customers and load interrupted when each operates in response to a
fault.
b. Select a contingency and evaluate its outage effects (number of customers
interrupted, duration of interruption, energy not served, and cost of the failure) on
the feeder by determining the following:
(1) The device that interrupts the fault.
(2) Switching actions that reconfigure the feeder and restore some customers.
c. Weight the outage effects of the contingency by the probability of its occurrence
and update the outage effects on the feeder.
d. If the component failed is an overhead line segment or a recloser, determine its
sensitivity to maintenance by the following:
(1) For a line segment, recompute the number of customers affected, customer
hours interrupted, energy not served and cost of failure using the failure rate
of the overhead line segment under consideration after maintenance.
(2) For a recloser, determine the same quantities for reduced failure
probabilities or improved PR and RR due to maintenance.
44
The difference in the outage effects before and after maintenance can then be used
to determine the risk reduction associated with maintenance.
e. Repeat steps 2 to 4 until all contingencies have been simulated.
In step 4, the change in reliability indices for reclosers and wood poles is performed
for every component on the system, thus identifying the risk reduction for every recloser
and wood pole on the feeder. Vegetation maintenance is assumed to be done on an entire
feeder, so the risk reduction for vegetation growth is computed for the entire feeder.
Maintenance performed on a pole influences the failure rate of the overhead line
segment it supports. Hence, the risk reduction due to maintenance on a particular pole is
determined by the sensitivity of the reliability indices to maintenance on the
corresponding segment. Reliability indices are linearly dependent on the failure rate of
the pole, so sensitivities can be computed using the difference in indices before and after
maintenance.
It can also be shown that the reliability indices of a feeder change linearly with
respect to the vegetation-related failure rate of the feeder. If the failure rate of all
overhead line segments is changed by the same proportion, which corresponds to the
failure rate reduction due to vegetation maintenance, then the change in reliability indices
can be predicted using a linear relationship.
For a recloser, however, two parameters quantify its reliability [34]: protection
reliability and reclose reliability. Furthermore, a recloser may have to function in one or
more of the following ways during a sustained fault:
a. As a primary protective device in the event of a fault occurring directly
downstream,
b. In conjunction with a downstream fuse for a fault downstream of the fuse.
c. In conjunction with a sectionalizer, interrupting the fault and then reclosing after
the sectionalizer opens.
The reliability indices are not linearly related to PR and RR, and can be deduced from
expressions in Table 4.3, Table 4.4, and Table 4.5. The reliability indices, however, can
be approximated with less than 5 percent error, as varying linearly, by assuming that the
recloser’s PR and RR are equal.
If reliability indices vary linearly with failure rates, the risk reduction associated with
maintaining each component can be obtained by computing the reliability indices before
and after maintenance.
4.5 Regulatory Penalty Risk Evaluation
The definition of risk in Section 1.3.1 includes regulatory penalties. A method to
determine the regulatory penalty risk associated with the failure of a component is
described here.
The computation of regulatory risk uses information obtained from the analytical
evaluation. The reliability indices, SAIDI and SAIFI, and the failure rates, computed
before and after maintenance of a component, are used as inputs. Because the indices are
assumed to be linear, the equations of the straight lines for SAIFI and SAIDI as a
function of the component failure rate can be determined.
For wood poles and vegetation, a vector of random numbers ‘un’ is created in which
‘n’ represents the number of years the simulation is carried out. Assuming a Poisson
distribution for the number of times a component fails in a given year, the number of
45
failures in each of the ‘n’ years can be determined by solving equation (4.12), where x(i)
is the number of failures in year ‘i’:
u n (i ) =
λ x (i ) e − λ
x(i )!
(4.12)
By using the number of times the component fails in a particular year, instead of its
failure rate, SAIFI and SAIDI can be computed using the linear relationships between the
indices and the failure rate of the component. Thus, for a set of random numbers drawn
for a component, the number of times it fails each year and the corresponding SAIFI(k)
and SAIDI(k) indices for each of the years can be determined. Since the numbers drawn
are random, SAIFI(k) and SAIDI(k) are also random. Similarly, random variations of
SAIFI(k) and SAIDI(k) can be determined using the failure rate of the component after
maintenance. Since SAIFI(k) and SAIDI(k) are randomly distributed with unknown
distributions, statistical methods can be used to suitably fit parametric equations that
represent their distributions. But this is a cumbersome process and not an attractive
solution, especially if the number of components is very large. If there are m components
in a system, 4m curve-fitting procedures (for SAIFI and SAIDI, before and after
maintenance) would be needed to evaluate the expected risk.
Using the Monte Carlo integration [35] instead, the complex integral defined in
equation (3.11) can be reduced to a more convenient summation, as shown in equation
(4.13).
∞
PBRF (k ) = ∫ PBR( SAIFI ). f (SAIFI (k ))d (SAIFI (k ))
TF
≈
(4.13)
n
1
∑ PBR(SAIFI ( x k (i))
n i =1
A similar expression can be developed for SAIDI.
These expressions are evaluated for each component before and after maintenance to
determine the reduction in regulatory penalty risk obtained by maintaining each
component. To draw a comparison between the curve-fitting method proposed in the
literature [11] and this Monte Carlo integration, the lognormal distribution is used as
suggested in [11]. However, the distributions were highly skewed, and the lognormal fit
does not accurately represent the risk of events with low probability and high
consequences, especially for equipment with very low failure rates. The fit improves for
higher failure rates, but variability in the reliability indices also increases. This provides
erroneous estimates for the risk of penalties associated with component failure.
Because the Monte Carlo integration method does not make an assumption about the
distribution of variability in the indices, it represents low probability events with greater
accuracy. The Monte Carlo integration method was also found to take about 30 to 40
percent less computation time than the conventional curve-fitting method. The drawback
of the Monte Carlo technique, however, is that the accuracy of solution depends on the
length of the simulation period, and it requires a very large sample to achieve significant
accuracy.
The method to compute the regulatory penalty risk reduction for wood poles and
vegetation-related outages is summarized as follows:
46
a. Input the failure rate of the component before and after maintenance, and the
resulting SAIFI and SAIDI values before and after maintenance.
b. Determine the straight line equations for SAIFI and SAIDI as functions of the
failure rate of the component.
c. Specify the number of years to simulate.
d. Determine the number of times the component fails each year by drawing uniform
random numbers for each of the simulated years, as in equation (4.12).
e. Determine SAIFI and SAIDI for each of the simulated years by using the linear
relationship between the failure rate and the reliability index, replacing the failure
rate with the number of times the component fails during each year.
f. Express the variability in SAIFI (or SAIDI) indices due to the variability in the
component’s failure each year as a probability distribution.
g. Compute risk as an expectation of the probability distribution and the penalty
curve PBR(SAIFI), as in equation (4.14), where PBR(SAIFI) is a piecewise linear
function that describes the penalty as a function of the SAIFI for the year.
∞
PBRF (k ) = ∫ PBR( SAIFI ). f (SAIFI (k ))d (SAIFI (k ))
(4.14)
TF
h. Keep in mind that equation (4.14) gives the risk of penalty before maintenance.
Repeating the computation using the failure rate after maintenance provides the
risk of penalty after maintenance. Risk reduction is the difference in risk before
and after maintenance.
To illustrate the variability in SAIFI from maintaining wood poles and vegetation,
probability plots obtained using this method are shown in Figure 4.1 to Figure 4.4. For
this illustration, it was assumed that the failure rate of the component is halved after
maintenance. Similar plots can also be obtained for SAIDI.
47
Figure 4.1: Variation in SAIFI before maintenance of wood pole.
Figure 4.2: Variation in SAIFI after maintenance of wood pole.
48
Figure 4.3: Variation in SAIFI before tree-trimming.
Figure 4.4: Variation in SAIFI after tree-trimming.
These figures illustrate that maintenance not only improves the expected values of the
indices but also reduces the risk of low probability, high consequence events.
For reclosers, Section 4.4 shows that the reliability indices are approximately linearly
dependent on protection reliability or reclose reliability. Unlike the wood pole or
vegetation sensitivities, which are functions of failure rate, the recloser’s PR is a
probability between zero and one. Hence, the risk of penalties associated with recloser
49
failure can be simulated using a Bernoulli distribution of parameter PR. For each year
simulated, PR is determined in the following manner:
a. Either assume a fixed number of faults per year. Or, if the distribution of the
number of faults occurring downstream of the recloser is known, randomly
generate the number of faults per year.
b. Generate Bernoulli-distributed random numbers, either zero or one, for parameter
PR, to represent recloser failure or operation for each of the fault.
c. Remember that the average number of times the recloser operates is the PR for the
specified year.
Using the estimated PR for each simulated year, and the straight line equations for
SAIFI and SAIDI as functions of PR, the reliability indices are computed. Using the
Monte Carlo integration of equation (4.13), the corresponding risks of penalties before
and after maintenance are computed. This method is summarized as follows:
a. Input PR before and after maintenance, and the resulting reliability indices, SAIFI
and SAIDI, before and after maintenance.
b. Determine the straight line equations for SAIFI and SAIDI as functions of PR.
c. Specify the number of years to simulate.
d. Determine recloser PR for each year:
(1) Assume a fixed number of faults per year. Or, if the distribution of the
number of faults occurring downstream of the recloser is known, randomly
generate the number of faults per year.
(2) Generate Bernoulli-distributed random numbers, either zero or one, for
parameter PR, to represent recloser failure or operation for each of the fault.
(3) Keep in the mind, that the average number of times the recloser operates is
the PR for the specified year.
e. For the estimated PR, calculate the annual reliability indices SAIFI and SAIDI
using their straight line equations.
f. Express the annual variability in SAIFI and SAIDI as probability distribution
functions.
g. Using equation (4.14), calculate the risk of penalty before maintenance. Repeat
for the risk of penalty after maintenance.
h. Remember that risk reduction is the difference in risk before and after
maintenance.
To illustrate the variability in SAIFI and SAIDI from maintaining reclosers,
probability plots obtained using this method are shown in figures 4.5 through 4.8.
The method developed for wood poles and vegetation-related failures can also be
applied to failures of overhead and underground conductors. The method for reclosers
can be similarly extended to other protective equipment, such as circuit breakers and
fuses.
50
Figure 4.5: Variation in SAIFI before recloser maintenance.
Figure 4.6: Variation in SAIFI after recloser maintenance.
51
Figure 4.7: Variation in SAIDI before recloser maintenance.
Figure 4.8: Variation in SAIDI after recloser maintenance.
4.6 Validation of Reliability Assessment Tool
The reliability evaluation tools developed in Sections 4.2 to 4.6 are validated using
the IEEE test system [36]. Figure 4.9 shows the test system of four radial distribution
feeders. The number of customers and corresponding loads connected to each load point
are shown in Table 4.8. Table 4.9 provides the lengths of each feeder section. The
52
overhead line failure rate is 0.065 failures/km-yr, and the average repair time is five hours.
The transformers in the system are modeled as lines, with a failure rate of 0.015
failures/year and average repair time of 200 hours. Switching time (MTTS) is assumed to
be one hour. All protective and switching devices are assumed to operate with 100
percent reliability. Transformers and corresponding line segments are reduced to a single
equivalent component using the failure rate and repair time expressions for series
connected components [31].
Figure 4.9: IEEE- reliability test system [36], bus 2.
53
Table 4.8: Customer data
Feeder
Load Points
1
1
1
2
2
3
3
3
3
4
4
4
4
1-3
4-5
6-7
8
9
10-11
12
13-14
15
16
17-19
20-21
22
Load/Point
(kW)
535
566
454
1000
1150
535
450
566
454
454
450
566
454
Number of
Customers/Points
210
1
10
1
1
210
200
1
10
10
200
1
10
Customer Type
Residential
Institution
Commercial
Industrial
Industrial
Residential
Residential
Institution
Commercial
Commercial
Residential
Institution
Commercial
Table 4.9: Lengths of feeder section
Length (km)
0.60
0.75
0.80
Feeder Section Numbers
2 6 10 14 17 21 25 28 30 34
1 4 7 9 12 16 19 22 24 27 29 32 35
3 5 8 11 13 15 18 20 23 26 31 33 36
As shown in Table 4.10, the reliability indices computed by the reliability evaluation
tool exactly match those provided in [36]. Further validation was also performed on two
actual distribution feeders using the results obtained from the reliability evaluation
software, DRIVe, developed by EPRI and Iowa State University. The reliability indices
computed by the reliability tool were found to be in close agreement with those predicted
by DRIVe.
Table 4.10: Reliability indices for the ieee- reliability test system, bus 2
Feeder #
1
2
3
4
System
*
SAIFI
(customers/year)
Predicted
RBTS*
0.248
0.248
0.140
0.140
0.250
0.250
0.247
0.247
0.248
0.248
SAIDI
(customer hours/year)
Predicted
RBTS*
3.618
3.620
0.523
0.520
3.624
3.620
3.605
3.610
3.613
3.610
Reliability indices given in [36].
54
ENS
(kWh/year)
Predicted
RBTS*
13172.06
13172
1122.06
1122
11203.20
11203
12248.36
12248
37745.68
37746
5 Optimization
This chapter describes a risk reduction optimization problem and its corresponding
solution. The reliability evaluation tool developed in Chapter 4 first computes the risk
reduction introduced by each candidate maintenance task. Then the results for each task
are combined with the resource consumed, resulting in triplets comprised of the candidate
task risk reduction, financial cost, and labor cost. These triplets are the inputs to the
optimizer. The optimization problem is presented in Section 5.1. Possible solution
methods are summarized in Section 5.2. Finally, Section 5.3 describes the solution
method selected and implemented in this project.
5.1 Problem Statement
In the problem statement, the following terms are used:
P is the number of maintenance categories.
p=1,…P is the index over the set of categories.
Np is the number of candidate components within category p.
k=1,…,Np is the index over the set of candidate components within category p.
Mk is the number of maintenance tasks for component k.
l=1,…,Mk is the index over the set of maintenance activities for component k.
The risk reduction related to each candidate preventive maintenance task is ΔRisk(k,l).
Resource requirements for each task are represented by cost, Cost(k,l), and labor,
Labor(k,l). Therefore, each task it is associated with a triplet: {ΔRisk(k,l), Cost(k,l), and
Labor(k,l)}. For every task, variable Iselect(k,l) reflects whether the task is selected (1) or
not (0). The triplets are input to the optimizer, which identifies the values of Iselect(k,l)
for all tasks that maximize the risk reduction subject to the resource constraints.
Budget(p) is the budget assigned to maintenance category p. TotLabor(p) is the available
labor, in person-hours, in maintenance category p. Totbudget is the total budget for all
categories.
The optimization formulation has two steps. The first is the task selection
subproblem, selecting tasks within resource constraints in each maintenance category.
The second is the budget planning subproblem, allocating budgeted resources to the
maintenance tasks.
The formulation of the task selection subproblem is as follows:
∑ ∑
Max :
Np
Mk
k =1
l =1
ΔRisk (k , l )Iselect(k , l )
(5.1)
Subject to the following constraints:
∑ ∑
Np
Mk
k =1
l =1
∑ ∑
Sel (k , l )Cost (k , l ) ≤ Budget( p)
Np
Mk
k =1
l =1
∑
Mk
l =1
Sel (k , l )Labor (k , l ) ≤ TotLabor( p )
Iselect(k , l ) ≤ 1, k = 1,2,...N
55
(5.2)
(5.3)
(5.4)
The objective, equation (5.1), is to maximize the total risk reduction. The constraint,
equation (5.2) represents the budget constraint, and equation (5.3) represents the available
labor resource constraint. The constraint represented by equation (5.4) indicates that each
component is maintained at most once during the time frame.
This task selection subproblem is a low-level formulation of the maintenance
optimization problem for a specific category p of maintenance tasks. Its results are used
in a higher-level problem. Task selection is solved repeatedly, increasing the budget each
time by a specified increment until all candidate tasks in each category are selected, or
available resources are exhausted.
Task selection is repeated for each category p=1.,,,P, resulting in a risk-reduction vs.
budget table for each category, as illustrated with example data in Table 5.1. Tasks are
selected, as illustrated by the binary strings in Table 5.2, where element k of a string
indicates whether task k is performed (1) or not (0). Each cell in Table 5.2 corresponds to
the cell in the same position in Table 5.1.
Table 5.1. Risk reduction vs. budget
Risk Reduction from Different
Categories
Budget
($1,000)
TreeWood Pole
Recloser
Trimming
0
0
0
0
1
4.2
3.75
2.25
…
…
…
…
9
19.8
13.5
14.4
10
20.7
13.5
15
…
…
…
…
Table 5.2. Decision variable code table
Profit from Different Categories
Budget
Tree($1,000) Wood Pole Recloser
Trimming
0
000…000 000…000
000…000
1
010…100 101…001
100…010
…
…
…
…
9
100…110
110...101
010…101
10
110…011 111…100
110…110
…
…
…
…
Cells in the risk-reduction table are denoted by Cat_ΔRisk(i,xi), corresponding to
category i with budget allocation xi. To obtain the maximum risk-reduction within the
total budget constraint, the budget planning subproblem is solved. This subproblem is
formulated as follows:
Max :
∑
c
i =1
Cat _ ΔRist (i, xi )
56
(5.5)
Subject to the following constraints:
∑
C
xi ≤ TotBudget
(5.6)
xi = 0,1,...∀i = 1,...
(5.7)
i =1
5.2 Possible Solution Methods for Task Selection Subproblem
The task selection subproblem is an integer programming problem that is known for
its difficulty. Three different solution methods were tried: prioritization, branch and
bound, and enhanced linear programming relaxation (ELPR) with the Lagrangean
relaxation plus heuristic method (LRH). All three are summarized here, and the ELPRLRH method is selected because it provides a better solution without a significant
increase in computation time.
5.2.1
Prioritization Method
For optimization using prioritization, the cost-effectiveness ratio index is defined as
R=
ΔRisk (k , l )
Cost (k , l )
(5.8)
The prioritization algorithm is as follows:
a. Obtain R for each candidate task.
b. Rank all candidate tasks by R.
c. For maintenance tasks on the same component, select the task with the highest
ranking and eliminate all others from the list.
d. Select tasks from the top of the ranking list until the cost limit is reached,
available labor resources are used up, or the reliability target is reached. This
algorithm is very fast and easy to perform, but there is no elegant way to apply
constraints (5.3) and (5.4).
5.2.2
Branch-and-Bound Method
Most integer programming optimizations are solved with the branch-and-bound
method. This technique is a mature and robust algorithm capable of solving integer
programming problems to optimality. In the general case, the algorithm begins with a
linear program identical to the original integer program, except that all variables are
relaxed to be real. The problem is solved, and then one variable is selected. Two
additional problems are formed—one with the selected variable constrained to be zero
and the other with the selected variable constrained to be one. The two problems are
solved, and the one with the best objective value is selected as the next branching point.
From this point, a new variable is selected, and two new problems are again formed—one
with the selected variable constrained to be zero and the other with the selected variable
constrained to be one. At this stage, then, the two new problems have two variables that
are constrained to be integers. The process continues until a branching point is reached
where there are no more real-valued variables. The algorithm terminates at this point.
57
Computation time for the branch-and-bound method increases exponentially with
problem size, and it will not solve a large-scale problem, like the task selection problem,
in a reasonable time. Sometimes stop criteria are introduced, such as errors between
upper and lower bounds, or maximum number of nodes searched. But it is still a timeconsuming method for large-scale problems.
5.2.3
ELPR-LRH Method
The method used in this project to solve the task selection subproblem combines the
enhanced linear programming relaxation method with the Lagrangean relaxation plus
heuristic method. ELPR ignores the 0-1 integer constraint while introducing a new
constraint, 0≤xi≤1. Lagrange relaxation retains the 0-1 integer constraint but relaxes all
other resource constraints; LRH improves Lagrange relaxation with a heuristic. Figure
5.1 illustrates the ELPR-LRH algorithm.
Begin
Solve ELPR
Is solution
integer?
Y
Stop
N
Solve LR & Heuristics (LRH)
Finish
Figure 5.1: Flowchart of ELPR-LRH optimization method.
The task selection sub-problem is represented in standard integer programming form as
Max : z = cx
(5.9)
Ax ≤ b
(5.10)
x ∈ {0,1}
(5.11)
Subject to the following constraints:
The ELPR is solved using the general linear programming algorithm. If the solution is
an integer, then the solution is optimal, and the algorithm stops. If the solution is not an
integer, the following LRH is solved as follows:
Max : z(λ ) = (c − λA)x + λb = c′x + λb
Subject to the following constraint:
58
(5.12)
x ∈ {0,1}
(5.13)
In equation (5.12), c is the reliability risk-reduction benefit and c’ is the net benefit
after deducting the cost of the resources. The optimality criterion for equation (5.12) is
simple: if the net benefit of a task is positive, then select it. The difficulty in solving the
problem represented by equations 5.12 and 5.13 is obtaining the Lagrange multiplier λ.
One approach suggested in the literature is the subgradient method [37], [38], but this is
computationally intensive, and can experience convergence problems. Therefore the
ELPR’s optimal dual solution is used, which provides a good estimate of the Lagrange
multiplier, as indicated in [39], and confirmed by numerical experiments performed in
this project. This approximate method is a fast and stable way to obtain the Lagrange
multiplier.
Sometimes the Lagrangean solution is infeasible or not good enough, e.g., too many
tasks are selected; therefore, some heuristic methods are used to improve the Lagrange
solution. For this project, a simple heuristic is performed after the solution of the
Lagrange relaxation: if it is infeasible, then the least net benefit decision variable is
removed until all the constraints are satisfied. If there is residual resource left, then the
largest net benefit decision variable among the unselected group is chosen until a
constraint is violated.
5.3 Solution Methods for Budget Planning Subproblem
The ELPR-LRH algorithm solves the basic integer programming problem, which
addresses the third question posed in the introduction, Section 1.1: to select a set of
maintenance projects to be completed within each program, constrained by the budget
allocation. The dynamic programming (DP) method is used to answer the budget
planning subproblem, which addresses the first and second problems in Section 1.1: how
to identify and justify the resources needed for asset management, and how to allocate the
available resources to the different maintenance programs. DP is chosen because it
provides not only the optimal policy but also the optimal policies of the subproblems
[40].
Illustrations of typical results obtained from these algorithms are given in Figures 5.2
and 5.3. The curve in Figure 5.2 provides the asset manager with a direct view of the
relationship between the reliability risk-reduction benefit and the maintenance budget. It
allows a solution to the first problem regarding the total maintenance budget.
The DP solution also gives the optimal allocation of the budget among different
categories, as shown in Figure 5.3. For example, with a total budget of $12,000, the
manager can allocate $3,000 to the recloser category, $4,000 to the wood pole category,
and $5,000 to the tree-trimming category. This addresses the second problem. Finally,
using the information in Table 5.2, the manager decides which tasks should be performed
to solve the third problem.
59
50
45
Risk-reduction
40
35
30
25
20
15
10
5
0
0
5
10
15
20
25
30
Budget(Thousand$)
Figure 5.2: Reliability benefit vs. budget.
Resource allocation(Thousand$)
12
Wood Pole
Recloser
Tree Trimming
10
8
6
4
2
0
0
5
10
15
20
25
30
Budget(Thousand$)
Figure 5.3: Resource splitting curve for different categories.
5.4 Summary
The techniques described in Chapter 5 provide the optimal budget for all assets, the
allocation of the budget to different maintenance categories, and the selection of projects
within those categories. Different maintenance categories with different characteristics,
e.g., some needing scheduled outages and load transfers, while others do not, can be
addressed with new categories with special constraints. Additional maintenance
60
categories to account for other types of equipment can also be added. If maintenance for a
period is to focus on certain categories, these can be weighted to bias task selection to
those categories. If contractors perform certain types of maintenance for a company, this
strategy can provide guidance to the asset manager on appropriate contractor pricing.
61
6 Illustration
Data from an actual distribution system is used in this chapter to illustrate the
proposed risk-based resource allocation strategy developed in the previous chapters. The
system has 66 feeders, each classified as either urban or rural, and is divided into three
operating regions.
Figure 6.1 shows the steps to implement this method. Historical outage data and
network topology, including load and customer information, obtained from the utility’s
outage management system (OMS), are the primary inputs. They are used to compute the
historical reliability indices, and to develop the statistical models to predict the failure
characteristics of distribution equipment and evaluate the effects of maintenance. They
also provide the basis for modeling various distribution components for the predictive
reliability evaluation, and a reference or benchmark for the predictive reliability
evaluation method.
The predictive reliability evaluation is followed by the computation of risk reductions
for various maintenance tasks, which are then optimized to provide the asset manager
with solutions to resource allocation problems. This chapter describes the results obtained
from the historical analysis, predictive analysis, and risk reduction computation (shaded
boxes in Figure 6.1) for the example feeder.
Develop Statistical Failure
Models for Individual
Components
Historical Outage and
Maintenance Data
Network Topology including
Load and Customer Data
Historical Reliability
Evaluation
Predictive Reliability
Evaluation and Failure Rate
Adjustment
Compute Risk Reduction for
Each Maintenance Task
Optimization
Figure 6.1: Risk-based resource allocation implementation.
62
6.1 Historical Reliability Evaluation
Outage history is used to calculate SAIFI and SAIDI [41] for a five-year period from
2000 to 2004. They provide a reference to compare with those obtained from the
predictive analysis. The outage history analysis also forms the basis for the average
failure rate and outage duration estimation used in the predictive analysis presented in
later sections. Storm-related extreme weather events are excluded. Outages of less than
one minute are classified as temporary, while those of one minute or longer are classified
as sustained. Sustained outages are further classified into two categories:
a. Outages caused by overhead or underground line failures, including those caused
by weather, vegetation, animals, overloads, and component failure, including
switches, reclosers, fuses, sectionalizers, and substation breakers.
b. Outages due to other causes, including transmission failure, public interference,
failures of lightning arrestors and distribution transformers, utility maintenance
personnel errors, and other events.
Table 6.1 summarizes the historical reliability indices for all causes of outage. Tables
6.2 and 6.3 present the indices by category 1 and 2. Table 6.1 values are the sum of the
corresponding indices in Table 6.2 and Table 6.3.
Table 6.1: Overall historical reliability indices for distribution system
Region
Total
Region 1
Region 2
Region 3
SAIFI (2000-2004)
1.489
1.625
1.280
1.315
SAIDI (2000-2004)
1.919 Hours
2.047 Hours
1.676 Hours
1.857 Hours
Table 6.2: Historical indices—overhead, underground, and device failures
Region
Total
Region 1
Region 2
Region 3
SAIFI (2000-2004)
0.869
0.836
0.952
0.839
SAIDI (2000-2004)
1.316 Hours
1.358 Hours
1.339 Hours
1.074 Hours
Table 6.3: Historical indices—outages caused by miscellaneous failures
Region
Total
Region 1
Region 2
Region 3
SAIFI (2000-2004)
0.620
0.789
0.328
0.476
SAIDI (2000-2004)
0.603 Hours
0.689 Hours
0.337 Hours
0.783 Hours
6.2 Predictive Analysis
Before predicted indices are used, they must correlate with historical indices.
Predicted indices are calibrated by adjusting component failure rates and repair times.
63
6.2.1
Failure and Repair Parameter Estimation for Predictive Analysis
To predict the reliability indices of a distribution system, the assessment tool must
have values for failure rates and mean time to repair for each component modeled:
overhead lines, underground cables, fuses, reclosers, breakers, sectionalizers and
switches. To estimate the failure rate of overhead lines, the total number of sustained
overhead outages observed during the 2000–04 period was divided by the number of
overhead circuit miles and the number of years, as shown in equation (6.1).The average
repair time is computed from the repair times for each of the sustained faults during the
same period, as shown in equation (6.2).
A similar procedure was followed for under-ground cables. Table 6.4 summarizes the
estimated average failure rates and repair times for overhead lines and underground
cables.
λp =
Total number of sustained interruptions
Total circuit miles * number of years
MTTR =
faults / mi-yr
(6.1)
Total repair time
Total number of sustained interruptions
(6.2)
Table 6.4: Reliability parameter estimates for overhead lines and underground cables
Component
Category
Urban
Overhead Line
Rural
Underground Cable
Urban
Rural
Phase
3-Phase
2-Phase
1-Phase
3-Phase
2-Phase
1-Phase
3-Phase
2-Phase
1-Phase
3-Phase
2-Phase
1-Phase
Average Failure Rate
(failures/mile-year)
0.0632
0.8043
2.1268
0.0983
0.1323
0.4543
0.0672
6.7692
0.3922
0.0116
0.0000
0.4156
MTTR (hours)
1.75
1.75
2.00
1.75
2.50
2.00
3.25
0.50
3.75
2.15
0.00
3.50
The reliability parameters for protective and switching devices is ideally computed
from the average number of times the device is expected to operate and the number of
times it is successful. However, these were not available in the outage database. Instead,
the failure probability was estimated from available data using equation (6.3).
PF =
Number of device failures
Total number of device operations
(6.3)
Reliability metrics for protective and switching devices can be estimated from PF.
For fuses, sectionalizers, and substation breakers, whose primary mode of failure is fail to
open during a fault, protection reliability is the complement of PF, as described in
Section 4.1.7 and shown in equation (6.4).
64
PR = 1 − PF
(6.4)
Similarly a switch’s switching reliability is estimated by equation (6.5).
SR = 1 − PF
(6.5)
For reclosers, two failure modes are possible: failure to open, and failure to reclose.
Protection reliability and reclose reliability are estimated as described in Section 4.1.7
and shown in equation (6.6).
PR = RR = 1 − PF
(6.6)
The calculated reliability measures for protective and switching devices are tabulated
in Table 6.5. The mean time to switch for each is assumed to be one hour, based on the
field experience of the switching personnel. Because the switching times of the protective
devices are not known, it is assumed that the MTTSdev is one hour, the same as the
MTTSswi. For this example, it is assumed that all switching failures are due to a switch
failing to do the intended operation. Protective device reliability is then 100 percent, and
the probability for the switching sequence is SR of the switch alone.
Table 6.5: Reliability parameter estimates for protective and switching devices
Component
Fuse
Recloser
Switch
Sectionalizer
Substation Breaker
6.2.2
Category
Protection
Reliability (PR)
MTTR
(hours)
Urban
Rural
Urban
Rural
Urban
Rural
Urban
Rural
Urban
Rural
0.970
0.950
1.000
0.975
0.000
0.000
1.000
0.950
0.988
0.985
1.13
1.50
1.75
1.75
2.00
1.50
1.75
1.75
1.75
2.00
Reclose
Reliability
(RR)
0.000
0.000
1.000
0.975
0.000
0.000
0.000
0.000
0.000
0.000
Switching
Reliability (SR)
1.00
1.00
1.00
1.00
0.60
0.73
1.00
1.00
1.00
1.00
Results
Using the values listed in Table 6.4 and Table 6.5, predictive analysis was performed.
However, the indices predicted did not agree with those from the historical analysis, as
shown in Table 6.2. To correct this, the component failure rates and mean time to repair
values are adjusted in a two-stage process [1].
First, the failure rates for the urban and rural regions are varied linearly (in this case,
decreased), keeping the repair times constant, until the predicted SAIFI values matched
those from the historical analysis. Then the MTTR of the urban and rural regions are
varied until the predicted SAIDI values nearly equal those from the historical analysis.
The adjusted failure rates and the repair times of overhead and underground lines are
shown in Table 6.6.
In this procedure, the protective and switching device parameters in Table 6.5 were
held constant for two reasons. First, the utility personnel involved in the project were
confident that the reliability estimates for protective and switching devices were close to
65
the actual values experienced in the system. Second, it was observed that the predicted
indices were not very sensitive to variations in protective and switching parameters, as
shown in Table 6.5. This is confirmed in a published sensitivity analysis [42].
The predicted indices obtained using the adjusted failure rates and repair times are
summarized in Table 6.7. These are very close to the historical indices in Table 6.2.
Table 6.6: Adjusted failure parameters for overhead and underground line segments
Component
Category
Urban
Overhead Line
Rural
Underground Cable
Urban
Rural
Phase
3-Phase
2-Phase
1-Phase
3-Phase
2-Phase
1-Phase
3-Phase
2-Phase
1-Phase
3-Phase
2-Phase
1-Phase
Average Failure Rate
(failures/mile-year)
0.0155
0.2010
0.5317
0.1180
0.1590
0.5450
0.0168
1.6923
0.0980
0.0140
0.0000
0.4990
MTTR (hours)
1.25
1.25
1.50
1.00
1.75
1.25
2.75
0.50
3.25
1.50
0.00
2.75
To include the miscellaneous failures of Table 6.3, the failure rate and repair time of
the equivalent component at the sending end of every feeder is adjusted until the
predicted indices are nearly equal to total indices in Table 6.1. For the test feeder,
equivalent component failure rates of 0.4 faults/year for urban and 0.7 faults/year for
rural, and repair times of 1.25 hours for urban and 1.00 hours for rural provide the
indices in Table 6.7, which are in close agreement with those in Table 6.1.
Table 6.7: Reliability indices using adjusted failure rates and repair times
Region
Total
Region 1
Region 2
Region 3
SAIFI
Predicted
Historical
0.901
0.869
0.892
0.836
0.910
0.952
0.917
0.839
SAIDI
Predicted
Historical
1.337
1.316
1.409
1.358
1.231
1.339
1.318
1.074
Table 6.8: Reliability indices using adjusted failure rates and equivalent component
Region
Total
Region 1
Region 2
Region 3
6.2.3
SAIFI
Predicted
Historical
1.428
1.489
1.534
1.625
1.298
1.280
1.305
1.315
SAIDI
Predicted
Historical
1.930
1.919
2.082
2.047
1.723
1.676
1.810
1.857
Discussion
Calibration of failure rates and repair time for lines and cables is a multidimensional
problem, since each component has two adjustable parameters. Because only two system
indices per region (SAIFI and SAIDI) are computed, the problem is under-constrained;
66
therefore, more than one set of parameter values can yield historical indices. Reference
[1] suggests a least-squares approach using the method of gradient descent to determine
one or more such values. Combining such an approach with the experience and judgment
of utility engineers may be most effective.
The extent of adjustment needed to the input data is influenced by various factors. An
important one is the accuracy of the initial estimates. For the example system, the
adjustment to the estimates for the urban regions is higher than for the rural regions.
Because the estimated average failure rate and repair times for the rural region are
determined over a large area, with about 1,400 miles of conductors, they tend to better
represent the actual observed indices. In contrast, the urban regions are estimated over a
much smaller region (about 170 conductor-miles) and, hence, are less accurate. Similar
conclusions can also be drawn from the initial estimated failure rates in Table 5.4. The
estimated failure rate for urban two-phase underground cables is unusually high. The
initial estimate in this case is influenced by the short cable length rather than the number
of outages. The 0.03-mile cable had one outage during the five-year period, resulting in a
failure rate estimate of 6.7692 failures/mile-year. Also, the number of outages on the
three-phase system was lower than the one-phase and two-phase outages, which explains
the lower three-phase failure rates.
The predictive analysis estimates steady-state, long-term, reliability indices, while the
historical indices reflect recent performance, which may not be representative of the
steady-state values. Hence, the predicted reliability indices tend to be closer to those from
the historical analysis when a longer period of outage data is used.
The granularity, or extent of modeling, also influences the predicted indices.
Predictive analysis using individual failure rates for three-phase, two-phase single-phase
lines resulted in estimates much closer to historical values than when one failure rate was
used for all lines regardless of phase.
In summary, predicted indices are closer to historical indices when the system is
modeled at a sufficient level of granularity, and the input failure estimates accurately
reflect each component’s tendency to fail. Thus, it is important to first validate input data
with historical indices as described here.
6.3 Computation of Risk Reduction
In the previous section, the predictive reliability algorithm was implemented on the
example system, and estimated failure parameters were adjusted to ensure that the
predicted indices match historical indices. The predictions thus correlate well with the
actual system reliability, and can be used to calculate the benefits of maintenance and
associated risk reduction. In this section, risk-reduction computations are performed for
the example system. The results are then provided to an optimizer to allocate resources,
as discussed in Section 1.1.
Table 6.9 lists the available maintenance tasks. Three categories are considered: wood
poles, reclosers, and tree-trimming. Each category has its own labor pools. There are
5,026 wood poles on the system, so there are 5,026 candidate tasks in the wood pole
category. There are 252 candidate tasks in the recloser category and 66 candidate treetrimming tasks. This produces a total of 5,344 triplets. The risk reduction introduced by
each is calculated, and the financial and labor costs are obtained. These are input to the
optimizer.
67
Table 6.9: Failure modes and corresponding maintenance activities
Contingency
Distribution
line outage
Failure Modes
Tree contact
Pole failure
Maintenance Activity
Tree-trimming
Pole treatment and
replacement
Maintenance Level
Feeder-based
Segment-based
Cost of Failure
$500/ outage
$200
Recloser
failure
Failure to open
and failure to
reclose
Minor maintenance,
major maintenance, and
replacement
Component-based
$25,000
Regulatory penalties are assumed to be as follows:
• $25,000 if a feeder SAIFI exceeds 3.0 sustained outages/customer-year
• $75,000 if a feeder SAIDI value exceeds 3.5 h/customer-year
Coefficients for the various contributing factors are assumed to be as follows:
100.00
• Customer satisfaction
• Lost Revenue
10.00
• Cost of component failure 1.00
• Regulator penalties
0.01
Each utility will specify these coefficients to represent the relative importance, or the
relative confidence in the values computed, for each. The total risk reduction obtained
from maintaining a component is given by equation (6.7).
Customer satisfaction
revenue
6444
447444448 6Lost
47
48
ΔRisk (k ) = 100.(ΔSAIFI (k ) + ΔSAIDI (k )) + 10.ΔENS (k )
Cost of component failure
gulatory penalties
64
4744
8
6444Re4
47444448
+ ΔDevRisk (k ) + 0.01.(ΔPBRF (k ) + ΔPBRD(k ))
6.3.1
(6.7)
Recloser Maintenance
No statistical failure models were found for reclosers, so a simpler deterministic
approach is used for risk-reduction calculations. As shown in Table 6.9, three different
activities are considered for recloser maintenance: minor maintenance (oil change), major
maintenance (recalibration), and replacement. Reclosers are modeled by their protection
reliability and reclose reliability. As discussed previously, these are assumed to be equal,
which gives a linear relationship between reliability indices and PR.
In this example, recloser reliability before maintenance is assumed to be the average
value estimated by the predictive analysis developed in Section 6.2.1. In reality, however,
each recloser will have its own PR. These differences can be modeled using statistical
models of Section 3.1 to determine each recloser’s failure rate.
PR after maintenance is also assumed to be deterministic. After minor maintenance,
PR is assumed to be improved by 0.005. Similarly, major maintenance improves PR by
0.0125, and replacement improves it by 0.025. Full implementation of the recloser model
of Section 3.1 will require incorporation of the model into the software, which is beyond
the scope of this project.
68
Figure 6.2 illustrates the risk reduction obtained from recloser maintenance. Three
levels of maintenance—minor, major, and replacement—of five reclosers were chosen to
demonstrate the benefits of maintenance versus their cost and labor resource
requirements. To simplify the example, reclosers were assumed to be in identical
condition before maintenance. This figure shows that despite identical initial conditions,
the risk and corresponding risk reduction obtained from maintenance vary significantly
for the five reclosers and thus should be included when prioritizing maintenance tasks.
The figure also shows that the risk reduction obtained from a lower-level maintenance
task can be greater than that obtained from a higher-level, more expensive maintenance
task on another recloser. This demonstrates the importance of using these decisionmaking methods to optimize the use of available resources.
Figure 6.2: Risk reduction due to maintenance on reclosers.
6.3.2
Wood Pole Maintenance
The risk-reduction computation for wood poles is done for each pole. Data was not
available for the example system, however, so pole maintenance was computed by line
segment, and segments needing maintenance were chosen randomly. Degradation in a
pole’s mechanical strength on each segment was drawn from a uniform random generator
that produces values between 0 and 0.3. A 0.3 value, or 30 percent degradation,
represents pole failure.
Regression expressions [43] estimate failure rates before maintenance:
69
con _ age =
h=
con + a 2
years
a1
b
* (con _ age)b −1
b
a
(6.8)
(6.9)
where
• ‘con’ is the degradation in pole mechanical strength.
•
‘con_age’ is the conditional age estimate for the pole, based on its condition.
•
‘a1’ and ‘a2’ are linear regression coefficients [43] that determine the relationship
between the degradation level ‘con’ and the condition age as follows:
o a1 = 0.014418
o a2 = 0.10683
•
‘h’ is the pole failure rate derived from the Weibull hazard function shown in
equation (6.9) with these parameters [43]:
o a= 50.6090
o b= 4.6676
As indicated in [43], two separate maintenance activities are considered for wood
poles. Pole reinforcement is assumed to reduce a pole’s failure rate to 1/4th its value
before maintenance. Pole replacement reduces the pole’s failure rate to that of a new
pole. The risk reduction associated with pole reinforcement and replacement is then
calculated as described in Section 4.4.
Figure 6.3 illustrates the risk reduction obtained from maintaining wood poles,
considering reinforcement and replacement. It may seem surprising that the risk reduction
associated with maintaining wood poles is lower that the expenses involved in
maintaining them. However, it may be noted that the average life span of a typical wood
pole extends typically in tens of years while that of a recloser is only a few years. Since,
the formulation of risk looks at the potential benefits of maintenance over the next year, it
is unable to capture the benefits of maintenance done on wood poles.
70
Figure 6.3: Risk reduction obtained due to wood pole maintenance.
6.3.3
Tree-Trimming Maintenance
When data such as tree-density and precipitation are available, existing models [13]
can estimate the vegetation-related failure rate for each feeder. These were not available
for the example system. Instead, it was assumed that 35 percent of overhead failures were
caused by vegetation. Thus, the total overhead failure rate of each feeder is multiplied by
0.35 to obtain the vegetation-related failure rate before maintenance. This failure rate is
then reduced to 40 percent of its original value to get the failure rate after maintenance.
Figure 6.4 illustrates the potential benefits obtained by implementing tree-trimming
programs on distribution feeders. It may be noted that the cost of maintenance in this case
is proportional to the length of the feeder. The cost of performing tree-trimming in the
case of Feeder 2 is nearly twice that of Feeder 3, even while the risk reduction obtained
may be comparable.
71
Figure 6.4: Risk reduction due to tree-trimming at a feeder level.
6.4 Optimization
6.4.1
Three Level Questions
The optimization procedure solves the task selection problem first and then the
budget planning problem. It is common practice in industry, however, to first set the total
maintenance budget and then distribute that budget to the maintenance categories. The
solution for the example system is presented in that order. More examples are presented
in the Optimizer User Manual in Appendix C.
Figure 6.5 displays the calculated risk reduction for the example system versus the
total maintenance budget, which is the solution to the budget-planning problem. Each
increment of the budget is allocated to the activities that produce maximum risk
reduction. This plot addresses the Level 1 question of how much to spend on all
maintenance activities. The key to answering this question is the amount of risk reduction
obtained for each increment of maintenance spending, which is the slope of the Figure
6.5 curve (ΔRisk_reduction/Δbudget) at a given budget level. As spending increases, the
slope decreases. The asset manager can identify a ratio below which no further
maintenance spending is justified. For example, Figure 6.5 indicates that the reliability
benefit per additional dollar spent is low when the total budget increases beyond $1,000k.
The ratio is much better for maintenance budgets up to about $500k.
72
Figure 6.5: Budget vs. risk reduction.
Figure 6.6 displays resource allocation vs. total budget, which is the result of solving
the Level 1 problem for various values of total budget. This identifies the optimal budget
split among maintenance categories. For example, the figure indicates that for a budget of
$500k, maximum risk reduction is achieved by allocating about $240k each to recloser
maintenance and tree-trimming, and only about $20k to wood pole maintenance. Wood
pole maintenance spending should remain relatively small for budgets below about
$850k. Then wood pole spending increases for budgets exceeding $850k, when risk
reduction from additional spending on recloser maintenance or tree-trimming is minimal.
Figure 6.6: Budget-splitting curve for different tasks.
73
6.4.2
Labor Sensitivity Analysis
Labor constraints are included in Figure 6.5 and Figure 6.6. It is also useful to modify
the labor constraints to see if increased or decreased labor spending results in different
spending decisions. This information provides the basis for increasing or decreasing the
number of maintenance crews.
Figure 6.7 shows the results of this analysis. The lower curve is the same as Figure
6.5, with existing labor constraints. The middle curve reflects the addition of one new
recloser maintenance crew. The upper curve represents no labor constraints at all. The
curves show that additional labor resources provide no significant improvement until a
budget of about $300k is reached. If a desired ΔRisk_reduction/Δbudget occurs at a
budget level below $300k, then labor reduction may be in order. Likewise, if a desired
ΔRisk_reduction/Δbudget occurs for a higher budget level, then increased labor should be
considered. Complementary information may be obtained by plotting risk-reduction
against labor for a fixed monetary budget.
6
3.5
x 10
Risk-reduction($)
3
2.5
2
1.5
1
0.5
0
0
500
1000
budget(Thousand$)
Figure 6.7: Labor sensitivity.
74
1500
7 Conclusions
7.1 Summary
Distribution system reliability and asset management are assuming greater
importance as utilities try to control costs while maintaining service quality. Equipment
maintenance and associated reliability improvements are important not only to ensure
that equipment lasts as long as it should but also to ensure customer satisfaction and
retention, manage operating costs, and comply with relevant service quality regulations.
Advanced strategies like reliability-centered maintenance are being adopted by utilities to
manage their vast amounts of assets.
Preventive maintenance reduces the failure probability of a component and, hence,
reduces the risk due to failure of the component. Each preventive maintenance task has
both financial and labor costs. The objective of this project is to maximize risk reduction
obtained from maintenance, within the available budget and labor constraints.
In the work presented in this report, the following applies:
• Available reliability evaluation techniques are reviewed
• Utility maintenance strategies are reviewed.
• A risk-based resource allocation method is developed.
The proposed method uses information obtained from inspection and monitoring to
determine the state of the system. Available maintenance tasks are identified, and the risk
reduction provided by each is computed. The risk reduction for each task is based on the
condition of the component being serviced, the task’s effect on improving the
component’s condition of equipment, and the resulting improvement in reliability indices.
The tasks are prioritized subject to the constraints on available resources using an
optimization technique combining integer programming, Lagrange relaxation, and
dynamic programming. The method is demonstrated in Section 6 using data from an
actual distribution system.
This method assists in answering the three concerns commonly faced by an asset
manager:
1. How to identify and justify the resources needed for managing the assets of the
entire system.
2. How to allocate the available resources to different maintenance programs.
3. How to select a set of maintenance tasks to be performed within each
maintenance program.
A degradation-path model to estimate failure probability and probability reduction
was developed. This model was applied to wood poles to predict individual pole failure
probability based on condition measurements that represent degradation in the pole’s
residual strength.
A condition assessment technique was developed for reclosers. A check sheet for
evaluating a recloser’s condition, either in the field or in the shop, is provided. The
condition score is then correlated with historical data to provide an estimate of the
recloser’s failure rate. Maintenance changes the recloser’s condition and thus its failure
rate. Similar techniques can be applied to other distribution system components.
75
7.2 Conclusions
The risk-based approach to maintenance scheduling integrates information obtained
from inspection and monitoring of distribution system assets with their failure
characteristics to estimate failure rates. The failure rate combined with the consequences
of failure determines risk. Computing the risk reduction provided by each maintenance
task allows the asset manager to weigh the benefits against the cost. This method
combines maintenance activities into a single objective of maximizing system
performance with minimal allocation of resources. It is a comprehensive strategy that
results in optimal utilization of resources and enhanced system performance.
A reliability evaluation tool was developed in Chapter 4. It is deliverable in the form
of a spreadsheet that can be used to estimate the reliability of a feeder and the benefits of
reliability improvement schemes such as automation, introduction of reclosers or
sectionalizing devices, and feeder reconfiguration. The User Manual for the spreadsheet
is included as Appendix B to this report.
To include all distribution maintenance activities, both failure models and inspection
methods are needed for each component under consideration. This method is highly dataintensive and requires detailed information about the equipment’s condition obtained
from inspection, monitoring, and maintenance records. Some of the data is available in a
utility database, but these are usually designed for bookkeeping and inventory, and are
not readily accessible for failure analysis. The problem may be compounded by
insufficient or inaccurate data or significant changes to the network configuration.
Because risk-reduction computations depend on accurate component failure-rate
estimations, the accuracy of the entire method depends on the quality of condition and
historical data available. But with the correct data, even though the number of devices in
a distribution system is very large, the strategy can be effectively used to solve the
resource allocation problem in a reasonable time and with sufficient accuracy.
7.3 Further Work
The reliability and inspection models developed should be further expanded, verified,
and then adapted to other distribution equipment. Specifically, the wood pole degradation
path model should be validated for other components with complex failure processes,
such as switches and transformers. Similarly, the inspection methods developed for
reclosers should be applied to other components, and the resulting failure rate estimates
should be verified.
The problem formulation should be further enhanced by considering scheduling
issues involved in equipment maintenance. The result will provide a schedule of planned
maintenance for a budget period.
The optimal resource allocation strategy sacrifices some accuracy to solve the largescale problem. Further research involving other optimization techniques will help
improve the accuracy of the solution obtained.
76
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78
Appendix A: Distribution Reliability Metrics
A brief review of some of the standard definitions and indices used in distribution
reliability evaluation are provided in this appendix. The definitions provided here are
found in the current Draft Guide for Electric Power Distribution Reliability Indices IEEE
P1366-2003 [44].
Definitions
1. Connected Load: The connected transformer kVA, peak load, or metered demand on
the circuit or portion of circuit that is interrupted.
2. Interrupting Device: An interrupting device interrupts the flow of power, usually in
response to a fault. Restoration of service or disconnection of loads can be
accomplished by manual, automatic, or motor-operated methods. Some interrupting
devices include: transmission circuit breakers, feeder breakers, line reclosers, fuses,
sectionalizers and motor-operated switches.
3. Interruption: The loss of service to one or more customers connected to the
distribution portion of the system as the result of one or more component outages.
4. Interruption Duration: The time period from the initiation of an interruption to a
customer until service has been restored to that customer.
5. Momentary Interruption: A single operation of an interrupting device that results in
a voltage zero.
6. Momentary Interruption Event: An interruption of duration limited to the period
required to restore service by an interrupting device. Switching operations must be
completed within a specified time of 5 minutes or less. If a reclosing device operates
multiple times within 5 minutes of the first operation, then all of the momentary
interruptions are classified as a momentary interruption event.
7. Outage: The state of a component when it is not available to perform its intended
function due to some event directly associated with that component. An outage may
or may not cause an interruption of service to customers, depending on system
configuration.
8. Planned Interruption: Loss of electric power that results when a component is
deliberately taken out of service at a selected time, usually for the purposes of
construction, preventative maintenance or repair. If it is possible to avoid the
interruption, then it is classified as a planned interruption.
9. Planned Outage: The state of a component when it is not available to perform its
intended function due to a planned event directly associated with that component.
10. Sustained Interruption: Any interruption not classified as a part of a momentary
event, which would be any interruption lasting more than five minutes.
Reliability Indices
The most common reliability indices used by utilities are SAIDI, SAIFI, CAIDI, and
ASAI [31]. Most of them are based on averages of customer reliability that weight each
customer equally. The following are five of the most common reliability indices used for
79
distribution systems as defined in the IEEE Guide for Electric Power Distribution
Reliability Indices (IEEE 1366-2003):
1. System Average Interruption Frequency Index (SAIFI):
The system average interruption frequency index (SAIFI) indicates how often the
average customer experiences a sustained interruption over a predefined period of time
for a given area in the system:
SAIFI =
Total number of customers interrupted
Total number of customers served
(A.1)
For a fixed number of customers, the only way to improve SAIFI is to reduce the number
of sustained interruptions.
2. System Average Interruption Duration Index (SAIDI):
System average interruption duration index (SAIDI) indicates the total duration of
interruption for the average customer during a predefined period of time, commonly
measured in customer minutes or customer hours of interruption:
SAIDI =
Total duration of all customer interruptions
Total number of customers served
(A.2)
SAIDI can be improved by reducing the number of interruptions or the duration of the
interruptions.
3. Customer Average Interruption Duration Index (CAIDI):
The customer average interruption duration index (CAIDI) represents the average
time taken to restore service to the customers:
CAIDI =
Total duration of all customer interruptions
Total number of customers interrupted
(A.3)
CAIDI can be improved by reducing the length of interruptions by faster crew response
time and repair times.
4. Average Service Availability Index (ASAI):
The average service availability index (ASAI) represents the fraction of time that a
customer has received power during the defined reporting period. Higher ASAI values
reflect higher levels of reliability [31]:
ASAI =
Customer hours service availability
Customer hours service demand
80
(A.4)
5. Customer Average Interruption Frequency Index (CAIFI):
The customer average interruption frequency index (CAIFI) gives the average
frequency of sustained interruptions for those customers experiencing interruptions (each
customer is counted only once, regardless of how many interruptions they experienced):
CAIFI =
Total number of customer interruptions
Total number of customers interrupted
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(A.5)
Appendix B: User Manual for Reliability Evaluation
Tool
The following document describes in detail the usage of the reliability evaluation tool
discussed in Chapter 4.
The spreadsheet was developed in Microsoft Excel. To use it, macros must be
enabled. In Excel, go to Tools>Macro> Security and set the Security Level to “low.”
Click “OK.” Close Excel and restart it. Then go to Tools> Customize> Toolbars and
check the box for Visual Basic. Then add the resulting VBA toolbar to the tools:
The reliability evaluation software comes as a zip file that contains the following
files:
1. A folder named “Backup” with the updated Visual Basic macros and user-forms.
2. Spreadsheets for six individual feeders: Feeder1, Feeder2….Feeder6. Each
describes the topology of one feeder in the distribution system being analyzed.
Table B.1 describes the layout of these spreadsheets. To create or add new feeders
to the existing system of six feeders, copies of any of the feeder files can be used.
3. Special function files:
a. Automate_Control.xls: provides the interface and platform for performing
reliability evaluation on the entire distribution system. It can also be used to
assign failure and repair parameters to various feeders (urban or rural). Please
do not rename, delete or move this file from the folder.
b. Master_file.xls: Contains the default values of failure and repair parameters of
various distribution components. Sheet 1 stores all the default values for the
rural feeders. Sheet 2 stores the values for urban feeders. Please do not
rename, delete or move this file from the folder.
c. Results_summary.xls: Updates the results obtained from a reliability
evaluation done on the entire system. It also computes the sensitivities of the
various reliability indices when maintenance is done on a particular
component. Please do not rename, delete or move this file from the folder.
Table B.1: Feeder Topology Spreadsheets
One line of data for each feeder segment
Col. No. Column
Description
A
Segment#
Unique identifier for each segment of feeder; first segment for
each feeder is the equivalent component discussed in section
4.2.5.
B
From node #
Node number that marks beginning of segment
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C
To node #
Node number that marks end of segment
D
Upstream segment#
Upstream segment of segment
E
Protection zone
Column left blank (output of program)
F
Switching zone
Column left blank (output of program)
G
Ht above the ground
Column left blank (output of program)
H
Over/ under flag
Takes values "“Over"” or "“Under"” or "“-"” for overhead line
segments, underground cables, and equivalent segment,
respectively
I
Length
Length of line segment in kft (1000 ft)
J
Load A
Load connected to Phase A in kWh
K
Load B
Load connected to Phase B in kWh
L
Load C
Load connected to Phase C in kWh
M
No. of customers
Customers connected to segment
N
Permanent
before
O
Permanent failure rate after
Failure rate of segment after maintenance (failures/mile-year)
P
Cost of failure
Cost of failure on segment ($)
Q
MTTR
Average repair time of segment (hours)
R
Phase
Phase configuration of segment
S
Protective device #
For segments with protective device, contains unique identifier
for protective device
T
Segment#
Same as column A
U
Type
BRK/ FUS/ REC/ SWI/ SZL
V
Upstream segment #
For a “CLOSED” device, same as column D; for an “OPEN”
switch, same as column A
W
PR
Value between 0.00 and 1.00
X
RR
Value between 0.00 and 1.00
Y
MTTS
Average time to perform switching (hours)
Z
SR
Value between 0.00 and 1.00
AA
Status
CLOSED/ OPEN
AB
Close to node
For a “CLOSED” device, same as column B; for an “OPEN”
switch, same as column C
failure
rate
Failure rate of segment before maintenance (failures/mile-year)
83
AC
MTTR of device
Average repair time of device (hours)
AD
Cost of device failure
Cost of failure of device ($)
AE
AF
AG
AH
AI
AJ
AK
AL
Expected number of
customers affected before
maintenance
Expected number of
customers hours interrupted
before maintenance
Expected energy interrupted
before maintenance
Expected cost of failure
before maintenance
Expected number of
customers affected after
maintenance
Expected number of
customers hours interrupted
after maintenance
Expected energy interrupted
after maintenance
Expected cost of failure after
maintenance
AM
SAIFI(i)
AN
SAIDI(i)
AO
Delta SAIFI(i)
AP
Delta SAIDI(i)
AQ
Delta ENS(i)
AR
Delta COF(i)
AS
PRa(Recloser)
AT
RRa(Recloser)
AU
Delta SAIFI(i)
AV
Delta SAIDI(i)
AW
Delta ENS(i)
AX
Delta COF(i)
Column left blank (output of program)
Column left blank (output of program)
Column left blank (output of program)
Column left blank (output of program)
Column left blank (output of program)
Column left blank (output of program)
Column left blank (output of program)
Column left blank (output of program)
SAIFI of the feeder after maintenance done on segment ‘i’
(output of program)
SAIDI of the feeder after maintenance done on segment ‘i’
(output of program)
Improvement in SAIFI due to maintenance on segment ‘i’
(output of program)
Improvement in SAIDI due to maintenance on segment ‘i’
(output of program)
Improvement in energy not served (ENS) due to maintenance on
segment ‘i’; column left blank (output of program)
Reduction in cost of failure (COF) due to maintenance on
segment ‘i’; column left blank (output of program)
Recloser protection reliability after maintenance; value between
0.00 and 1.00 (only for reclosers)
Recloser reclose reliability after maintenance; value between
0.00 and 1.00 (only for reclosers)
Improvement in SAIFI due to maintenance done on recloser
(output of program)
Improvement in SAIDI due to maintenance on recloser (output
of program)
Improvement in ENS due to maintenance on recloser; column
left blank (output of program)
Reduction in COF due to maintenance on recloser; column left
blank (output of program)
84
NORMAL MODE OF OPERATION:
The basic mode of using the reliability evaluation tool is at the feeder level. The
following macros are available in this mode.
To assign default failure and repair parameters for each feeder:
Go to Tools>Macro> Macros, select rate_assign and press Run. Alternatively, press
[Ctrl + d].
Upon execution, failure and repair parameters are assigned to various distribution
system components, as shown in the following figures. The default values that appear are
drawn from the Master_File.xls. They can be changed in the interfaces shown.
Parameters for individual components and segments are changed directly in the
spreadsheet.
85
Using the line and cable interfaces shown, failure rates before and after maintenance,
mean time to repair (MTTR), and cost of failure (COF) can be input for three-phase, twophase and one-phase line segments. The failure rates are the average number of sustained
or permanent failures expected in a unit mile length. Values for failure probabilities
should be between 0.00 and 1.00. MTTR is entered in h, and COF has the units of US$.
Once all values are entered, they are applied to the feeder by clicking ‘OK.’ Clicking
‘Reset’ restores the default parameters stored in the Master_File.xls. ‘Cancel’ sets all
parameters to zero.
86
87
To enter data for reclosers, the user has two options. Option 1 requires the user to
input the probability that a recloser fails to open in the event of a fault, and the
probability that a recloser fails to reclose once it has cleared a fault. Also required are the
probabilities of failure after maintenance and the recloser’s MTTR and COF.
If information characterizing individual failure modes is not available (i.e., failures
cannot be classified as failure to open or reclose), Option 2 can be used to enter the
overall failure probability of the recloser.
88
The interface for fuses, circuit breakers, and sectionalizers is similar to that for
reclosers. These components have only one failure mode—failure to open in response to
a fault.
89
90
In the switch interface shown, switch failure probability is the probability that a
switch fails to switch when needed. The MTTR and mean time to switch (MTTS)
represent the average repair and switching times, in h.
The switching characteristics of protective devices are not included in the rate_assign
macro. The values MTTS and switching reliability (SR), the probability that a device is
switched when required, are entered in the spreadsheet in columns 25 and 26,
respectively.
91
To compute the reliability of the feeder:
Go to Tools>Macro> Macros, select reliability_evaluation, and press Run.
Alternatively, press [Ctrl + r]. This computes the reliability indices of the feeder and the
corresponding sensitivities due to maintenance. Outputs appear in the spreadsheet, as
indicated in Table B.1.
AUTOMATED MODE OF OPERATION:
In the automated mode, failure and repair parameters can be assigned to feeders
grouped as rural or urban feeders. Reliability evaluation can also be performed on the
entire system. These functions can be performed using the Automate_Control.xls
spreadsheet provided. Open the file Automate_Control.xls.
To assign default failure and repair parameters for all rural feeders:
Go to Tools>Macro>Macros, select default_values_rural and press Run.
Alternatively, press [Ctrl + r]. Enter the reliability parameters as they were entered in the
rate_assign macro.
92
To assign default failure and repair parameters for all urban feeders:
Go to Tools>Macro>Macros, select default_values_urban and press Run.
Alternatively, press [Ctrl + u]. Enter the reliability parameters as they were entered in the
rate_assign macro.
To compute the reliability of the entire system:
Go to Tools>Macro> Macros, select automate, and press Run. Alternatively, press
[Ctrl + a]. This requires the user to input the value of ‘1’ if reliability evaluation for the
entire system is desired and ‘0’ if not.
93
If ‘0’ is entered, the program execution is terminated and is followed by the message
below.
94
Appendix C: User Manual for the Optimizer
The optimizer is written in Matlab and compiled into executable file. The following
files are contained in a compressed file, RDRA.zip, which contains the following:
/RDRA
/RDRA/DATA/recl.dat
/RDRA/DATA/tree.dat
/RDRA/DATA/pole.dat
/RDRA/optconFiguredat
/RDRA/task.exe
/RDRA/budget.exe
//root directory
//recloser candidate triplets
//tree-trimming candidate triplets
//wood pole candidate triplets
//user configuration file
//solves the task selection subproblem
//solves the budget planning subproblem
In the optconFiguredat, the user can set the available resource, budget step size, and
maximum budget for the study. An example of this file is as follows:
unit=1000;
maxbudget=2000;
pole_labor=16000;
recl_labor=1600;
trim_labor=9600;
%UNIT=1000$
%so the total budget = maxbudget*unit
%person hour for wood pole category
%person hour for recloser category
%person hour for tree-trimming category
In the recloser (recl.dat), tree (tree.dat), and pole (pole.dat) data files, the inputs are
the candidate triplets: {risk-reduction, money cost, labor cost}.
Figure C.1. Input file of pole candidate tasks.
Each line of data has two triplets, corresponding to two levels of maintenance activities
for each device.
The two executable files, task.exe and budget.exe, solve the task selection and budget
planning subproblems. After running the task.exe file, there will appear two new .dat files
in /RDRA/data. One is the risk-reduction vs. budget table (ptable.m) and the other is the
corresponding task selection table (pole_Iselect.m for pole, etc.).
95
Figure C.2. Risk reduction versus budget table (ptable.m).
Figure C.3. Task selection table (trim_select.m for tree-trimming).
Budget.exe creates two files, which appear in the directory /RDRA/DATA/. One,
totrisk.dat, is the total risk-reduction vs. budget. The other, split.dat, is the budget split.
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