Implementation of Reliability Centered Asset Management method on Power Systems

IN

DEGREE PROJECT ELECTRICAL ENGINEERING, SECOND CYCLE, 30 CREDITS

,

STOCKHOLM SWEDEN 2017

Implementation of Reliability

Centered Asset Management

method on Power Systems

YU ZHANG

Implementation of Reliability Centered

Asset Management method on Power

Systems

Yu Zhang

Master of Electric Power Engineering Thesis 2017 KTH School of Electrical Engineering

Osquldas väg 10 SE-100 44 Stockholm

Master of Science Thesis 2017

Implementation of Reliability Centered

Asset Management method on Power

Systems

Yu ZHANG Approved 2017-Examiner Patrik Hilber Supervisor Ebrahim Shayesteh Commissioner {Name} Contact person {Name}

Abstrakt

Kapitalförvaltning har inom alla områdem blivit allt viktigare, speciellt inom elkraftsteknik. Det beror i huvudsak av två orsaker. Den första är stor

investeringskostnad, vilket inkluderar design, konstruktion, utrustning och underhåll. Den andra är den höga straffavgiften för system operatören vid elavbrott. Dessutom, på grund av den nyligen avreglerade elmarknaden, så fäster elföretagen mer

uppmärksamhet på investerings och underhållskostnader. En av deras huvudmål är att maximera underhållsprestandan. Så utmaningen för operatörerna är att leverera tillförlitlig elkraft till kunder, samtidigt vara kostnadseffektiva mot leveratörer. Reliability Centered Asset Management (RCAM) är bland de bästa metoderna för att lösa detta problem. En enklare RCAM metod är introducerad först i denna rapport. Modellen inkluderar en underhållsstrategi-definition, underhållskostnad-kalkyl och en

optimiserings modell. Grundad på denna enklare modell, andra förbättringar är tillagda och en ny modell är föreslagen. Förbättringarna inrymmer en ny underhållsstrategi, ökad felfrekvens och en ny målfunktion. Den nya modellen tillhandahåller också en tidsbaserad underhållsplan.

Abstract

Asset management is getting increasingly important in nearly all fields, especially in the electric power engineering. It is mainly due to the following two reasons. First is the high investment cost include the design cost, construction cost, equipment cost and the high maintenance cost. Another reason is that there is always a high penalty fee for the system operator if an interruption happened in the system. Besides, due to the deregulation of electricity market in these years, the electricity utilities are paying more attentions to the investment and maintenance cost. And one of their main goals is to maximize the maintenance performance. So the challenge for the systems is to provide high-reliability power to the customs and meanwhile be cost-effective for the suppliers. Reliability Centered Asset Management (RCAM) is one of the best

methods to solve this problem.

The basic RCAM method is introduced first in this thesis. The model includes the maintenance strategy definition, the maintenance cost calculation and an optimization model. Based on the basic model some improvements are added and a new model is proposed. The improvements include the new improvement maintenance strategy, increasing failure rate and a new objective function. The new model is also able to provide a time-based maintenance plan.

The simulation is done to a Swedish distribution system-Birka system by GAMS. The results and a sensitivity analysis is presented. A maintenance strategy for 58

components and in 120 months is finally found. The impact on the changing failure rate is also shown for the whole peroid.

Key words:

Asset management, Reliability centered maintenance (RCM), Reliability centered asset management (RCAM), Multi-objective optimization, Maintenance optimization, Power system maintenance.

Acknowledgement

Two years’ master life is coming to the end and a new journey in my life is going to start. The two years’ study life is hard and full with the help from the teachers and friends. By the time I am finishing my master thesis, there are a lot of people I want to thank to.

First I would like to thank KTH, that provided me a great studying environment and learning experience. I learned a lot from KTH in this two years, not only professional knowledge but also a lot of knowledge for life which can help me a lot in my future when I step into society.

Then I would give thanks to my supervisor Ebrahim Shayesteh, who gave me this opportunity to work on this thesis and gave me continuous supports during the whole process of the master thesis. Ebrahim Shayesteh is very responsible and very patient with my every single question. He also gave me a lot of suggestions and

encouragement. Besides I also would like to thank my examiner Patrik Hilber, who give me the main guidance for the project and a lot of suggestions during the mid-presentation.

I would also give appreciate to my families and friends. Thanks to my parents for supporting my two years’ study in Sweden. Thank you for your pay out to raise me up and it is my time to pay back in the future and best wishes to you. I need to give a special appreciate to my girlfriend Yang Xu, who give me a lot of help and encouragement in my life during this two years.

Last but not least, I would give thanks to all the professors and classmates I met in the two years’ study. They also helped me a lot with my study, I learned a lot of useful knowledge from them.

Contents

ABSTRAKT ... I ABSTRACT ... III ACKNOWLEDGEMENT ... V CHAPTER 1 ... 1 INTRODUCTION ... 1 1.1 BACKGROUND ... 1 1.2 AIM AND OBJECTIVES ... 1 1.3 ETHICAL ASPECTS: ... 2 1.4 OVERVIEW OF THE REPORT ... 3 CHAPTER 2 ... 5 THEORY AND LITERATURE REVIEW ... 5 2.1 ASSET MANAGEMENT ... 5 2.2 MAINTENANCE STRATEGIES ... 5 2.3 POWER SYSTEM RELIABILITY BASIC ... 6 2.3.1 Power system reliability definition ... 6 2.4 Power system reliability evaluation ... 6 2.4.1 Basic reliability indices ... 6 2.4.2 The system performance indices [16] [12] [35] ... 7 2.5 RELIABILITY CENTERED MAINTENANCE (RCM) ... 9 2.6 RELIABILITY CENTERED ASSET MANAGEMENT (RCAM) ... 10 2.6 SUMMARY OF THE LITERATURE REVIEW ... 11 CHAPTER 3 ... 13 METHODOLOGY ... 13 3.1 COMPONENT RELIABILITY IMPORTANCE CALCULATION ... 13 3.1.1 Interruption cost index (Hazard rate index) IH ... 13 3.1.2 The maintenance potential IMP ... 15 3.1.3 Simulation based IM ... 16 3.2 MAINTENANCE STRATEGY DEFINITION ... 16

3.2.2 Corrective Maintenance (CM) ... 17

3.2.3 The connection between PM and CM ... 17

3.2.4 The Maintenance cost calculation ... 17

3.3 DECISION MAKING PROCESS ... 19

3.4 THE BASIC RCAM MODEL ... 19

3.4.1 Structure and description of the basic model ... 19

3.4.2 Formulation of the basic optimization model ... 20

3.4.3 The flowchart of the basic model ... 21

3.5 THE NEW IMPROVED RCAM MODEL ... 21

3.5.1 Applied Improvements ... 22 3.5.2 Formulation of new optimization model ... 24 3.5.3 The model of Delta total maintenance cost (∆"#) ... 25 3.5.4 The model of Total maintenance cost ("#) ... 26 3.5.5 The flowchart of the new RCAM model ... 28 3.5.6 Other indices and calculations ... 29 3.6 THE SOFTWARE GAMS ... 29 3.7 SUMMARY ... 31 CHAPTER 4 ... 32 THE SIMULATION AND RESULTS ... 32

4.1 THE DESCRIPTION OF THE STUDIED SYSTEM – BIRKA SYSTEM ... 32

4.1.1 Network introduction ... 32 4.1.2 The input Data ... 34 4.2 THE SIMULATION RESULTS ... 37 4.2.1 The first stage - The system reliability level ... 37 4.2.2 The second stage - The maintenance strategy level ... 49 4.3 DISCUSSION OF SIMULATION RESULTS ... 55 4.4 SENSITIVITY ANALYSIS ... 55 4.4.1 Maximal number of PM maintenance action per month ... 56 4.4.2 The Corrective Maintenance Cost ... 60 4.4.3 The Preventive Maintenance Cost ... 64 4.5 SUMMARY ... 68 CHAPTER 5 ... 69 CONCLUSION ... 69

CHAPTER 6 ... 71 FUTURE WORK ... 71 REFERENCE ... 72 APPENDIX I ... 79 APPENDIX II ... 107

Chapter 1

Introduction

1.1 Background

Asset management is getting increasingly important in nearly all fields, especially in the electric power engineering. It is mainly due to the following two reasons. First is the high investment cost include the design cost, construction cost, equipment cost and the high maintenance cost. Another reason is that there is always a high penalty fee for the system operator if an interruption happened in the system. Besides, due to the deregulation of electricity market in these years, the electricity utilities are paying more attentions to the investment and maintenance cost [1]. And one of their main goals is to maximize the maintenance performance [58]. So the challenge for the systems is to provide high-reliability power to the customs and meanwhile be cost-effective for the suppliers. Reliability Center Asset Management (RCAM) is one of the best methods to solve this problem. RCAM is an advanced method that provides quantitative methods for power system asset management. In other words, the output of this method is an optimum plan for the maintenance of the power system, which minimizing the total cost and maximizing the system reliability [2]. Although many applications have been studied in power system area in [3]-[8]. But there is still no well-defined model for the implementation of RCAM in this area. The work of this thesis is based on Ebrahim’s previous work in [9] which proposed an RCAM algorithm for electric power system. Based on that model some improvements are applied in the new proposed model in this thesis.

1.2 Aim and objectives

1. Propose an algorithm and functions that are needed to apply the RCAM method in power system.

3. Improve the basic algorithm and GAMS [38] code in different aspects. a) Dynamic programming by updating the importance indices.

b) Includes the increasing failure rate by time. Because the failure rate of the electric component is changing anytime. A function about the relation between failure rate and time will be proposed and applied to the model. c) Provides a time-based maintenance plan. The basic model it will just tell us

how much is the optimum maintenance level for each component, but it cannot say which component has more priority for performing the maintenance.

4. Write a simple GAMS code based on the proposed algorithm.

1.3 Ethical aspects:

The study on reliability centered asset management (RCAM) have a lot of benefits not only bring profit to the system operator, but also for the whole society in many

aspects like the environment and the customers. First, the better maintenance plan can provide the customers with high quality power. Secondly, the better maintenance plan will make sure that the equipment is in a better status. It saves the environments in another way. For example, if there is not a good maintenance plan for a transformer, there is more possibility for it to be in a bad status. There will be more hidden danger to the environments. The transformer may release some toxic gas which will be quite bad to the environment. The same problem may also happen to a circuit breaker or other equipment. Thirdly, to have a better maintenance plan can also extend the life of an equipment. Then it not only saved the money but also saved the environment, as we saved the resource to produce the new equipment and reduced the noxious gas we produced during the production. Besides, a smart maintenance plan also save a lot of human resource, then in another way to reduce the waste of energy and the emission of CO2 .

1.4 Overview of the report

This report is mainly centered on the optimization-based approach for implementing the RCAM method in electric power system. The report is divided into the following 6 parts.

Chapter 1 Introduction: In this part a brief introduction of the background of the topic RCAM is given at the start. And then the problem definition and motivation is presented. The aim and objectives of the thesis is introduced at last.

Chapter 2 Theory and Literature review: First, this part gives a more detailed introduction of the concepts that connected with this thesis. It is mainly in the following five aspects: the asset management, maintenance strategies, the power system reliability, the component reliability indices, RCM and RCAM. Then a literature review on the previous work of the area asset management, maintenance optimization and RCAM is done. The literatures are summarized in a table at last. Chapter 3 Methodology: In this chapter, the methodology of the thesis is introduced. The basic RCMA model is first explained in detail. It includes three main parts: the component importance calculation, the maintenance strategy definition and the optimization model. Then the new improved model is proposed. The applied improvements are explained. The improvements are in following aspects: dynamic programming, new PM maintenance strategy definition, time based maintenance plan, include the increasing failure rate, and consider the system reliability. Then the structure of the improved RCAM model and the formulations are presented. In the improved method, two optimization model is introduced, one use the total cost as the objective function and the other use the delta total cost as the objective function. The simulation software GAMS is introduced briefly at last.

Chapter 4 The simulation and results: This part presents the simulation results of the improved RCAM model on Swedish Birka system. The results include both the TC model and ∆TC Model. Also a sensitivity analysis is done on some uncertain parameters. A short discussion is also presented at the end of this chapter.

Chapter 5 The conclusion: Based on the previous results, both general conclusions and specific conclusions are concluded in this chapter.

Chapter 6 The future work: The suggestions about the future work is given here. It includes some improvements and how to continue the work in the future.

Chapter 2

Theory and Literature Review

2.1 Asset management

Asset management (AM) is one of the most important topics in electric industry because of the increasing cost stress on utilities. AM is a method about decision making. It is a systematic approach that aims to find an optimal decision on assets meanwhile considering the risk [10].

Asset is a wide concept, in a company a lot of things can be referred as asset, for example, the equipment, the employees, the customer base and so on. The asset we considered in the method we proposed in this thesis is the physical asset i.e. the electric equipment [11] [1].

Generally, the goal for a company that performs asset management is to maximize the profit. So there are some actions that related with asset management. In this thesis, we mainly consider the maintenance actions on the electric component and aim to find out an optimal maintenance strategy for the components in power system.

2.2 Maintenance strategies

Maintenance is defined as the actions we take to fulfill a required performance in a component by maintaining it to a more correct function or a previous level of function (if the previous function is optimum) [12]. Maintenance actions can be classified in several ways like in paper [13] and [14]. In this thesis, one of them is considered, in which the maintenance actions are divided into two groups: preventive maintenance (PM) and corrective maintenance (CM).

• Preventive Maintenance (PM)

Preventive maintenance is also called scheduled maintenance, which is carried out periodically. The preventive maintenance aims to reduce the probability that the component failed, by checking or replacing some parts. This kind of maintenance is

applied before a component fails. Generally, the preventive maintenance is pre-scheduled.

• Corrective Maintenance (CM)

Corrective maintenance aims to renew the component performance by repairing it after it failed. Corrective maintenance is costly and always tried to be avoided by the operator.

2.3 Power system reliability basic

2.3.1 Power system reliability definition

Reliability is a wide concept. The reliability of system is always referred as the ability of the system to perform its designed function. There is a widely accepted definition given in [15], “Reliability is the probability of a device or system performing its function adequately, for the period intended, under the operating conditions intended.” For electric power system, the function is to provide electricity to its customers continuously with an acceptable quality.

2.4 Power system reliability evaluation

The power system reliability can be evaluated in mathematical way by which reliability is normally accessed by ‘probability’. So some system reliability indices have been widely used in this area. The is subchapter will give a brief introduction of some basic reliability indices for component and load points which are also called reliability measures, and some system reliability indices.

2.4.1 Basic reliability indices

There are some basic indices used in electric power system as the measure of reliability for a component or load point.

Failure rate is an index, which shows the frequency that a component or a system load point fails. It is defined as the failure per unit time which is normally failure per year [f/yr] for electric power components.

The failure rate of a load point is calculated from the failure rate of the components connected on that load point.

The calculation is as follows: For a series connection:

$_& = $₍

(

For a parallel connection:

b) Mean Time to repair (MTTR)

Mean time to repair (MTTR) is also called the average repair time, it is the time between the point that the component is failed and the point that it return to the normal condition and back to work.

c) Unavailability

Unavailability is defined as the probability that a component works in a correct condition during a given time, the unit hour per year is normally used. The unavailability is expressed as: ) =₊ *

,-*

Where r is the average repair time [yr/f] $ is the failure rate [f/yr]

As $ ⋅ / ≪ 1, so it can also be approximately expressed as ) = $ ∙ /

2.4.2 The system performance indices [16] [12] [35]

a) System Average Interruption Frequency Index (SAIFI)

The SAIFI is defined as the total customer interruptions during a period divided by the total number of the customer served. The unit is [f/yr, customer].

34565 =789:; <=>?@/ 8A B=C98>@/ D<9@//=E9D8<C 789:; <=>?@/ 8A B=C98>@/ C@/F@G =

H( ∗ $(

where,

$₍ is the failure rate of each load point

H( is the number of customer served at load point i

b) System Average Interruption Duration Index (SAIDI)

SAIDI is the total hour of customer interruption duration per year divided by the total customer served. The unit is [h/yr, customer]

345J5 =789:; B=C98>@/ D<9@//=E9D8< G=/:9D8<C 789:; <=>?@/ 8A B=C98>@/ C@/F@G =

H₍ ∗ )₍ H₍ where )( is the expected outage of load point i.

c) Customer Average Interruption Frequency Index (CAIFI)

The CAIFI is defined as the total number of customer interruptions divided by the number of affected customers. The unit is [f/yr, customer]

K4565 =789:; <=>?@/ 8A B=C98>@/ D<9@//=E9D8<C 789:; <=>?@/ 8A :AA@B9@G B=C98>@/C =

H₍∗ $₍ H_L( where N_NO is the number of affected customers

d) Customer Average Interruption Duration Index (CAIDI)

CAIDI is calculated as the total hour of customer interruption durations per year divided by the total number of customer interruptions per year and with the unit [h/int]

K45J5 =789:; B=C98>@/ D<9@//=E9D8< G=/:9D8<C

789:; <=>?@/ 8A :AA@B9@G B=C98>@/C =

H(∗ )(

H₍∗ $₍

e) Customer total average interruption duration index (CTAIDI)

The CTAIDI is the total hour of customer interruption duration per year divided by the sum of affected customers. The unit is [h/yr, customer]

K745J5 = 789:; ℎ8=/ 8A B=C98>@/ D<9@//=E9D8<C 989:; <=>?@/ 8A :AA@B9@G B=C98>@/C =

H₍ ∗ )₍ H_L( f) Average energy not supplied per customer served (AENS)

Besides these there are some more system performance indices like Average energy not supplied per customer served (AENS) which is related with energy and is defined as:

4QH3 = 789:; @<@/RS <89 C@/F@G 789:; <=>?@/ 8A B=C98>@/ C@/F@G g) Average Service Availability Index (ASAI)

ASAI is a index that shows the available time that customers get supplied. 4345 = F:D;:?;@ C@/FDB@ ℎ8=/C

7ℎ@ G@>:<G@G C@/FDB@ ℎ8=/C

2.5 Reliability centered maintenance (RCM)

This subchapter will give a brief introduction of the RCM method include the concepts of RCM, the background and further the developing of this method in electric power engineering.

Reliability Centered Maintenance (RCM) is a systemic method aims to find an optimal maintenance plan which is cost-effective. This method is focusing on the reliability aspects and the output of the is systemic effort. The main methodology is to find out the balance between the preventive maintenance and corrective maintenance to achieve the cost-effectiveness [17].

The method was first proposed by F.Stanley Nowlan and Howard F.Heap, and was first applied to the aircraft industry in 1960s when the Boeing 747series aircraft was created [18]. Then with some successful maintenance results, the method attracted more interest and was further improved. In 1970s the RCM concept was defined by US Department of Commerce. Then in 1978 the first detailed description of this method was published. After that the method has been widely used in several areas, like nuclear industry, chemical industry, shipping, oil and gas industry and some small size companies. In electric power engineering area, the RCM method was first introduced to nuclear power industry by Electric Power Research Institute (EPRI) in 1980s. And with the development these years, the RCM method has been widely used by the electricity utilities to solve the maintenance planning problem.

The RCM topic has been studied in the previous years. And several literatures on this topic have been studied when preparing this thesis. A brief introduction about RCM is given in paper [19] and a RCM structure is presented. Paper [20] implemented the RCAM approach to a substation in the transmission system of Slovenia. And get the results shows that the maintenance cost reduced 25% when provide the same quality of power. The implementation of RCM method is also applied to the transmission components in [21]. In this paper, RCM method is used to find the optimal maintenance strategy among several possible maintenance scenarios. And the final result prove that the maintenance is more cost-effective than the traditional maintenance plan. Besides the RCM method is also applied to the wind turbines. In [22], the method is applied to a wind turbine from Vesta. [23] and [24] propose a system approach for the turbine blade and gearbox respectively.

2.6 Reliability centered asset management (RCAM)

Reliability centered asset management is the method proposed based on the

application of RCM [27]. The RCM model can provide a cost-effective model more from the reliability point of view. The idea of RCAM method is to find an optimal balance between the customer requirement for the power quality at a suitable price and the appreciate refund on the investment [25]. It tried to find a qualitative relationship between the component reliability and system maintenance [26]. The deregulation of the electricity market also promoted the execution of the RCAM method.

The RCAM concept was first proposed in [27]by Lina Bertling, and a lot of work about the RCAM in power system has been done up till now. In [28] a case study of RCAM has been done to the Mexican Sub-Transmission. The result of this study includes the technical index and the economic index and are presented for 55 year. Another study is done to the Turkish transmission system [29], it proposed a RCAM model to improve the transmission reliability for the circuit breakers, transmission lines and the transformers. Paper [26] presented a research on the component reliability and life model for RCAM method and proposed a maintenance model for power distribution system. Besides the transmission and distribution system, the RCAM maintenance model is also studied for the wind power system in [30]and [23].

transformers in [40]. Even through, the study of RCAM in power system has been done a lot earlier, there are still no adequate implementation of this method on power system. This paper aims to propose an approach to fill this gap.

The RCAM model proposed in this thesis is to provide a maintenance plan considering the reliability importance of each component. By considering the component importance, the maintenance plan will be created according to the prioritization of each component. The maintenance plan includes the maintenance level and the maintenance time schedule. By the RCAM method, a more

cost-effective maintenance plan will be found.

2.6 Summary of the literature review

The literature review of this these is done to several topics that related with RCAM. The literature reviews can be classified into the following groups according to the key word. And is summarized in Table 1.1

Table 1.1 The summary of literatures

RCM method

Introduction and framework: [43], [17], Power distribution system: [44],[55] Wind power system: [1], [2], [6], [56] Transmission and distribution system: [20], [41],

[21],[19], [45] Distribution system: [42], Asset management method [46], [47], [48] Component reliability modeling [49], [20], [50] Maintenance Optimization, Maintenance

Management. RCAM method [17], [26], [28], [30], [40] Importance of components, critical component identification. [34], [25], [57], Multi-Objective Optimization [58]

Chapter 3

Methodology

In this chapter the RCAM model and the simulation process will be described. Need to mention that this thesis is based on Ebrahim’s work. So the model proposed by Ebrahim.S in [31] is used as the basic model of this thesis. The formulation of the basic model and the theory will be introduced and explained to give a detailed understanding of the method. Then the improvements applied based on the basic algorithm will be presented. A general introduction of the software GAMS will be given at last.

3.1 Component reliability importance calculation

One of the main differences between RCM and RCAM method is the RCAM method is that a component importance index(CRI) is included in the RCAM model to connect the power system reliability with the maintenance of each component. Component reliability importance indices show the importance of the components. The index can prioritize the components, to show where preventive maintenance is the most effective [32]. There are several kinds of traditional CRIs, e.g. the Birnbaum’s reliability importance IB_{, Fussell-Vesely’s measure of importance I}FV_,

Criticality Importance ICR _{, and improvement potential I}IP _{. These four classical}

indices have been introduced briefly in paper [32]. Based on these indices, several indices are proposed in [32] and [33] i.e. the IM, IMP, IH. A brief introduction of these indexes will be given and one of them is used in the proposed model.

3.1.1 Interruption cost index (Hazard rate index)

_IH

The Hazard rate index _IH_{is proposed based on the concept of the Birnbaum’s}

reliability importance IB_{. The Hazard rate index I}H

aims to relate the interruption

assessment of multi-objective networks compared with IB. The probabilities are taken place by the total interruption cost as the measure of system reliability. The definition of the index is

5₍T₌UK&

U$₍, D ∈ 1,2, ⋯ , 5 (3.1)

In the basic model the calculation of Hazard index is as follows: K&= $^⋅ _^⋅ `^+ B^⋅ `^⋅ /^ b ^c+ (3.2) 5₍T ₌UK& U$₍, D ∈ 1,2, ⋯ , 5 (3.3) Where;

Ie _{The hazard rate index of component i $/f}

C_k The total interruption cost of the system $/yr λ_O The failure rate of component i f/yr

λ_o The failure rate of load point l f/yr r_o The total repair time at load point l h/f P_o The electrical power at load point l (kW) k_o Cost constant at load point l $/f ⋅ kW C_o cost constant at load point l $/kWh I The total number of components

L The total number of load points in the system

In order to calculate the in the model, a sensitivity analysis is preformed with a very small adjustment ∆$ to component failure rate. When the ∆$ is applied, it will result in a change in the failure rate in each load point in the network, which will

H I

finally lead to a change to the total interruption cost. Then the 5₍T _{can be calculated}

approximately by the following equation:

5₍T ₌K& $(+ ∆$( − K&($()

∆$₍ ≈

UK& U$₍ (3.4)

The result will finally show us that how the total system interruption cost is influenced by the failure rate of one specific component. In other words, it shows the importance of each component for the total system.

3.1.2 The maintenance potential

IMP

With the index _IH

introduced above, the total interruption cost caused by each

component can be approximately calculated as below in equation (3.5) and (3.6). K( ≈ 5(T$( (3.5)

K&≈ K( (

(3.6)

This contribution of each component is defined as the maintenance potential. Maintenance potential is referred as the system interruption cost reduction of each component in case that the component is in perfect condition which means no failure. It can also be defined as the total interruption that will be caused due to the failure of the studied component in one year. So the definition can be expressed as below:

5{| _{= K} & 1, $ − K& $ (3.7) 5{| _{= 5} (T$( (3.8) where

5₍T _{[$/yr] is the Hazard index defined above in 3.1.1,}

Need to mention that the definition is based on the assumption that the relationship between K_& and $₍ is linear.

3.1.3 Simulation based

IM

Based on the concepts of IH and IMP, the simulation based index IM is proposed in paper [32]. IM is also called the simulation based index [35], which is derived from the simulation of interruption cost calculation. The index is calculated by computing the total interruption cost of the finally causing component. The component will be considered as the responsible component. Then the total accumulated cost of this component will be divided by the total simulation time and the interruption cost per unit time will be got. The definition is as below:

5{ ₌ (

7 (3.9)

According to paper [32], these indices are calculated by similar principles, and some of them can even be calculated from each other. So _IH_{is simply chosen in the}

proposed model.

3.2 Maintenance strategy definition

Besides the CRI, the potential maintenance strategy of each component is another important input to the proposed RCAM model. The maintenance strategy is defined for each component in the system, and several maintenance levels are defined with different impact on the reliability of each component.

In this model, the impact on the reliability is defined as the variation on the failure of each component. Two kinds of maintenance, preventive maintenance and corrective maintenance, are considered. The PM and CM maintenance has been introduced in the previous section, here we will introduce the PM and CM strategies that defined in the basic model.

3.2.1 Preventive Maintenance (CM)

In the maintenance strategy of the basic model, 11 PM maintenance level is defined as shown in the figure below. In this model the PM maintenance action is increasing with maintenance level.

Table 3.1 Maintenance level definition of the basic model

Maintenance level 1 2 3 4 5 6 7 8 9 10 11

Number of PM actions -5 -4 -3 -2 -1 0 1 2 3 4 5

3.2.2 Corrective Maintenance (CM)

In this model, the cost of corrective maintenance is assumed 10 times to the preventive maintenance [58].

3.2.3 The connection between PM and CM

The preventive maintenance and corrective maintenance are connected and impacted by each other. As defined above, in the maintenance strategy model, increased PM actions will decrease the failure rate of each component by different levels which will lead a decreasing number of CM actions. As a result, the PM cost and CM cost will increase and decrease respectively. Similarly, a decreased PM action will increase the CM action and the costs will be reduced or increased respectively.

3.2.4 The Maintenance cost calculation

The maintenance cost in the basic model is calculated through the following equations.

K₍|{ _{= Å}

(|{∙ Ç(|{, D ∈ 1,2, … , 5 (3.10)

K₍Ö{ _{= Ü}

Where;

K₍|{ _{Cost of preventive maintenance of component D}

K₍Ö{ _{Cost of corrective maintenance of component D}

$₍ The failure rate of component D

Ç₍|{ _{Total numbers of preventive maintenance for component D}

Å₍|{ _{Preventive maintenance cost for component D}

Ü₍Ö{ _{Corrective maintenance cost for component D}

The variation in the cost of PM and CM due to the change in the maintenance strategy is calculated as follows: áK₍|{ _{= Å} (|{∙ (Ç((àâ− Ç(äãâ), D ∈ 1,2, … , 5 (3.12) áK₍Ö{= Ü₍Ö{∙ $àãå₍ − $₍ç^ä , D ∈ 1,2, … , 5 (3.13) $àãå₍ = $ç^ä₍ ∙ 2(é_èèêëíé_èìîë)_{, D ∈ 1,2, … , 5 (3.14)} Where,

áK₍|{ _{Variation in the cost of preventive maintenance of component D}

áK₍Ö{ _{Variation in the cost of corrective maintenance of component D}

$₍ç^ä _{The initial failure rate of component D}

$₍àãå _{The new failure rate of component D}

Ç₍(àâ _{Number of additional PM for component D}

3.3 Decision Making process

This part is the main part of the whole model, which helps to find the optimum solution of the maintenance strategy of each component by considering the system reliability and the total maintenance cost.

A maintenance optimization model is proposed to achieve this goal with the input values calculated in the previous steps. The result of CRI calculation and the maintenance cost calculation are as the input to the optimization model, the output is the optimal salutation of the maintenance strategy for all the components.

The whole model and the formulations will be presented and explained detailed in the following section.

3.4 The Basic RCAM Model

3.4.1 Structure and description of the basic model

The structure of the proposed model is shown below as the flowchart in Figure 3.1 As shown in Figure 3.1, there are three main parts in the model: 1) the component reliability importance calculation, 2) maintenance strategy definition, 3) decision making process.

At the start, the initial failure rate and repair time will be given as the input values to the model. The power system importance index calculation for each load point will be carried out and the results are the input to the CRI (IH) calculation. The average hazard index (5_ïT_{) will be calculated to show the importance of each component and}

be used to calculate the interruption cost in the following part.

Then the second part is about the maintenance strategy. The maintenance strategy of each component is defined as above. The effect of each maintenance strategy on the failure will be calculated as equation (3.14). The cost of each maintenance strategy will be calculated by equation (3.10) to (3.13). These values will be the input to the optimization model.

The last part of the model is the decision making process. With the input values calculated above, the appropriate maintenance strategy will be get as the output of the optimization model. The proposed optimization model is presented below as equation (3.15) to equation (3.20).

The objective function is to find the optimal maintenance strategy, which minimize the total cost meanwhile maximize the power system reliability. The total cost is consisted of three parts in the model, the CM maintenance cost, PM maintenance cost and the interruption cost. The calculation of total CM cost and PM cost is formulated as equation (3.17) to equation (3.18), which is similar as described above in equation (3.10) to equation (3.14). The interruption cost is calculated by equation (3.19). It is also the connection between the total cost and the system reliability.

3.4.2 Formulation of the basic optimization model

As described above the model is formulated as below >D< _é è,ñ èêë_,é è,ñìîë,éèóòì ∆7K( ô (c+ (3.15) s.t. ∆7K_(,å= ∆K_(,åÖ{ _{+ ∆K} (,å|{ + ∆K(,åôÖ D ∈ 1,2, … , 5 ö ∈ 1,2, … , õ (3.16) ∆K₍Ö{= ∆$(∙ Ü(Ö{, D ∈ 1,2, … , 5 (3.17) ∆K₍|{ = > ∙ Å₍|{∙ Ç_(,ú(àâ− Ç_(,úäãâ , D ∈ 1,2, … , 5 { úc+ (3.18) ∆K₍ôÖ _{= ∆$} (∙ 5ïT, D ∈ 1,2, … , 5 $(àãå = $(ç^ä∙ Ç(,úäãâ∙ 2ú + { úc+ Ç₍ùéä+ Ç_(,ú(àâ_{∙ 2}íú { úc+ , D ∈ 1,2, … , 5 (3.19) ∆$₍ = $₍àãå_{− $} ( ç^ä_{, D ∈ 1,2, … , 5} Ç_(,úäãâ ₊ { úc+ Ç(ùéä+ {úc+ Ç(,ú(àâ = 1, D ∈ 1,2, … , 5 (3.20) x_O,üO†°_{, x} O,ü ¢£°_{, x} O§•¢are binary variables.

3.4.3 The flowchart of the basic model

Start

1) Input the initial failure rate and repair time.

2) The reliability index calculation for each load point.

3) Calculate the hazard rate index IH _{for each component.}

4) Calculate the average hazard rate index IH _{for each component.}

Components Reliability Importance calculation

Maintenance Strategy Definition

1) Define the maintenance strategy for each component.

2) Calculate the maintenance strategy impact on the failure rate of each component and consider the increasing failure rate.

3) Calculate the maintenance cost (PM and CM) of each maintenance strategy.

1) The hazard rate index and defined maintenance strategy will be as the input value to the optimization model.

.

2) Run the optimization model to find the optimal maintenance strategy for each component for the time interval. .

3) Output the optimal maintenance strategy for each component in the system.

.

END Decision Making Process

Start

1) Input the initial failure rate and repair time.

2) The reliability index calculation for each load point.

3) Calculate the hazard rate index IH _{for each component.}

4) Calculate the average hazard rate index IH _{for each component.}

Components Reliability Importance calculation

Maintenance Strategy Definition

1) Define the maintenance strategy for each component.

2) Calculate the maintenance strategy impact on the failure rate of each component and consider the increasing failure rate.

3) Calculate the maintenance cost (PM and CM) of each maintenance strategy.

1) The hazard rate index and defined maintenance strategy will be as the input value to the optimization model.

.

2) Run the optimization model to find the optimal maintenance strategy for each component for the time interval. .

3) Output the optimal maintenance strategy for each component in the system.

.

END Decision Making Process

Figure 3.1 The flowchart of basic RCAM model [9]

3.5 The new improved RCAM model

Even though the basic model presented above can already solve the maintenance optimization problem, there are still some defects can be improved further to make it

more connected with the reality. So based on the basic model introduced above some improvements have been applied. The new improved model will be introduced in this subsection.

3.5.1 Applied Improvements

a) Dynamic programming

In the basic model, the optimal solution of the maintenance strategy has been found. But it can just find the optimal solution according to the initial component importance. By adding this improvement, the new model is enabled to find the final optimal solution by updating the importance indices several times until the accuracy of Δ$ reach the requirement.

b) New PM level definition

There are several PM levels defined in the basic model. But as in reality, the preventive action is very limited. So in the new improved model, the maintenance strategy is defined in another way to fit practical situation. Four level of PM actions are defined as below according to the study in [37], the minor maintenance, medium maintenance, major maintenance, and total replacement. Different maintenance level will decrease the failure rate back to 90%, 60%, 20% or remain the same to the initial failure rate. The detailed model is shown as below in Table 3.2

Table 3.2 The new maintenance level definition

Maintenance Level Maintenance Action Failure rate after PM

Minor maintenance $100 Remain the same

Medium maintenance $700 Decreased to 90%

Major maintenance $6,000 Decreased to 60%

Replacement $75,000 Decreased to 20%

Minor maintenance is defined as the general checking which means just go and check the status of the component, or do some small adjustment e.g. lubrication. The cost is assumed to be around $100. The medium maintenance is defined as the replacements

the actions like the overhaul of the component with the cost $6,000. The last maintenance level is defined as the replacement of the whole component, and the cost is assumed as $75,000 [37].

And the cost is different for preventive maintenance actions, the corrective maintenance is assumed to be $20,450 which is 10 times of the average cost of the preventive maintenance [35].

c) Provide a time-based maintenance plan

In the basic the RCAM model, the maintenance strategy is solved and assumed to be applied to all the components together. But in reality, the maintenance actions are very costly and take time. Thus, it is not possible to maintenance all the components once together according to the maintenance strategy.

So in the new model, it is possible to solve a time-based maintenance plan. The new time-based maintenance plan presents a maintenance plan for a long-time interval according to the components importance.

Based on that a limit is also added to the possible maintenance actions per month. In the model, it is defined as at most five maintenance actions can be applied during a month. So the components with higher importance will be maintenance first. And the maintenance to all the components in the system will be done in the defined time interval.

d) Considering the increasing failure rate

In the basic model, the failure rate of the components is assumed to be fixed if no change in the preventive maintenance is done. But as we know, the failure rate of the component is changing over time in reality. And as explained above that the maintenance is going to be done in a long-time period. So the increasing failure rate will have a significant influence on the component importance during this period and changing the final time-based maintenance strategy.

Failure rate of a component is considered as a function of time in the improved model to make the final result more reliable and accurate.

The basic RCAM model aims to find an optimal maintenance strategy by considering the component reliability importance and the total maintenance cost. But there is a problem that it did not consider the system reliability. In another word, it cannot not grantee the system reliability after the maintenance.

To improve this problem, the system reliability index SAIFI is included in the new model. To control the proportion of PM cost and CM cost, a weight factor ≠ for CM and (1-≠) for PM is introduced. So we can change the factor ≠ to change the importance of the CM cost and PM cost in order to further control the share of PM and CM to improve the system reliability until the system reliability meets the requirement. The new flowchart after we consider the system reliability is shown in figure 3.2. Start Run the RCAM model Calculate the SAIFI Check if the SAIFI meets the requairment END YES NO Reduce the weight factor ω Start Run the RCAM model Calculate the SAIFI Check if the SAIFI meets the requairment END YES NO Reduce the weight factor ω

Figure 3.2 The flowchart after considering the system reliability

3.5.2 Formulation of new optimization model

function with the basic model, which minimize the variation in the total cost due to the implementation of different maintenance strategies for each component (we call it ∆TC model in this report). The other one is new proposed, which change the total delta cost to total cost (we call it TC model in this report). As both of the functions are correct in theory, and it is hard to say which objective function is more correct and reasonable. So both of them are considered and applied in the simulation for further study. The two models are listed in the following sections.

3.5.3 The model of Delta total maintenance cost (

∆"#)

Objective function >D<_éè,ñ,Ø ∆7K(,å ∞ åc+ ô (c+ (3.21) Subject to: ∆7K_(,å = 1 − ≠ ∗ ∆K_(,åÖ{ _{+ ≠ ∗ ∆K} (,å|{ + ∆K(,åôÖ D ∈ 1,2, … , 5 , ∈ 1,2, … , õ (3.22) ∆K_(,åÖ{ _{= ∆$} (,å∙ Ü(Ö{, D ∈ 1,2, … , 5 ö ∈ 1,2, … , õ (3.23) ∆K_(,å|{ _{= Ç} (,ú,å∙ Å(,ú, D ∈ 1,2, … , 5 { úc+ (3.24) ∆K_(,åôÖ _{= ∆$} (,å∙ 5ï,åT , D ∈ 1,2, … , 5 (3.25) $_(,åàãå _{= $} (,åí+ àãå _{∙ A} (àâ+ Ç(,ú,å∙ ($(,úù(é− { úc+ $_(,å±L*(L≤(çà_{), D ∈ 1,2, … , 5 , w > 1 (3.26)} $_(,åàãå_{= $} ( ç^ä_{+ Ç} (,ú,å∙ ($(,úù(é− { úc+ $_(,å±L*(L≤(çà_{), D ∈ 1,2, … , 5 , w = 1 (3.27)} ∆$(,å = $(,åàãå− $(,åí+àãå ∙ A(àâ, D ∈ 1,2, … , 5 , ö > 1 3.28 ∆$_(,å = $_(,åàãå_{− $} ( ç^ä_{, D ∈ 1,2, … , 5 , ö = 1 3.29} Ç_(,ú,å= 1 { úc+ , D ∈ 1,2, … , 5 (3.30) ∞ åc+

Ç_(,ú,å ≤ <_^(ú(≤ ô (c+ , ∀>, ∀ö (3.31) $_(,å±L*(L≤(çà _{= $} (∗ (A(àâ)åí+ (3.46) Ç_(,ú,å DC ?D<:/S F:/D:?;@ Where

≠ is the weight factor balancing the PM and CM costs

<_^(ú(≤ is the maximal number of PM actions which can be done in one month

Equation 3.22 to 3.25 is the calculation of total delta cost, delta PM cost, delta CM cost and delta interruption cost. Equation 3.31 is the limit to the maintenance actions in each month.

3.5.4 The model of Total maintenance cost (

"#)

Objective function >D<_éè,ñ,Ø 7K_(,å ∞ åc+ ô (c+ (3.32) Subject to: 7K_(,å = 1 − ≠ ∗ K_(,åÖ{ _{+ ≠ ∗ K} (,å|{ + K(,åôÖ D ∈ 1,2, … , 5 , ö ∈ 1,2, … , õ (3.33) K_(,åÖ{ _{= $} (,å∙ Ü(Ö{, D ∈ 1,2, … , 5 ö ∈ 1,2, … , õ (3.34) K_(,å|{ _{= Ç} (,ú,å∙ Å(,ú, D ∈ 1,2, … , 5 { úc+ (3.35) K_(,åôÖ _{= ∆$} (,å∙ 5ï,åT , + KåôÖ∑ D ∈ 1,2, … , 5 (3.36) $_(,åàãå _{= $} (,åí+ àãå _{∙ A} (àâ+ Ç(,ú,å∙ ($(,úù(é− { úc+ $_(,å±L*(L≤(çà_{), D ∈ 1,2, … , 5 , w > 1 (3.37)}

$_(,åàãå_{= $} ( ç^ä_{+ Ç} (,ú,å∙ ($(,úù(é− { úc+ $_(,å±L*(L≤(çà_{), D ∈ 1,2, … , 5 , w = 1 (3.38)} ∆$(,å = $(,åàãå− $(,åí+àãå ∙ A(àâ, D ∈ 1,2, … , 5 , ö > 1 3.39 ∆$_(,å = $_(,åàãå_{− $} ( ç^ä_{, D ∈ 1,2, … , 5 , ö = 1 3.40} Ç_(,ú,å= 1 { úc+ , D ∈ 1,2, … , 5 (3.41) ∞ åc+ Ç_(,ú,å ≤ <_^(ú(≤ ô (c+ , ∀>, ∀ö (3.42) KåôÖ∑ = $^∏ ∗ (_^∏∗ E^∏ + B^∏ ∗ E^∏∗ /^∏ b∏ ^∏c+ ) (3.43) $_(,å±L*(L≤(çà _{= $} (∗ (A(àâ)åí+ (3.45) Ç_(,ú,å DC ?D<:/S F:/D:?;@ Where

≠ is the weight factor balancing the PM and CM costs _^∏is the cost constant at each load point lp ($/f, kW)

B_^∏is the cost constant at each load point lp ($/kWh) E_^∏is the power at each load point

Here equation 3.33 is the total cost, equation 3.34 to equation 3.36 are used to calculate the PM cost, CM cost and interruption cost respectively. Need to mention that the equation 3.36 is main change we applied in this model. In equation 3.36, we calculate the intimal interruption cost and plus it with the delta interruption cost and assume that it is the total interruption cost. The initial cost is calculated by the equation 3.43.

3.5.5 The flowchart of the new RCAM model

Start

1) Input the initial failure rate and repair time.

2) The reliability index calculation for each load point.

3) Calculate the hazard rate index IH _{for each component.}

4) Calculate the average hazard rate index IH _{for each component.}

Components Reliability Importance calculation

Maintenance Strategy Definition

1) Define the maintenance strategy for each component.

2) Calculate the maintenance strategy impact on the failure rate of each component and consider the increasing failure rate.

3) Calculate the maintenance cost (PM and CM) of each maintenance strategy.

Decision Making Process

1 ) The hazard rate index and defined maintenance strategy will be as the input value to the optimization model.

2 ) Run the optimization model to find the optimal maintenance strategy for each component for the time interval.

3) Dynamic programming by checking if the accuracy index meet the requirement.

No

4) Output the optimal maintenance strategy for each component in the system. .

Refresh the failure rate and recalculate the Hazard index

Yes

END

Start

1) Input the initial failure rate and repair time.

2) The reliability index calculation for each load point.

3) Calculate the hazard rate index IH _{for each component.}

4) Calculate the average hazard rate index IH _{for each component.}

Components Reliability Importance calculation

Maintenance Strategy Definition

1) Define the maintenance strategy for each component.

2) Calculate the maintenance strategy impact on the failure rate of each component and consider the increasing failure rate.

3) Calculate the maintenance cost (PM and CM) of each maintenance strategy.

Decision Making Process

1 ) The hazard rate index and defined maintenance strategy will be as the input value to the optimization model.

2 ) Run the optimization model to find the optimal maintenance strategy for each component for the time interval.

3) Dynamic programming by checking if the accuracy index meet the requirement.

No

4) Output the optimal maintenance strategy for each component in the system. .

Refresh the failure rate and recalculate the Hazard index

Yes

END

3.5.6 Other indices and calculations

Besides the calculations and indices above, there are some other indices that are defined and used in the model.

a) The index for dynamic programming

To realize the dynamic programming, an index is introduced to control the program. The index is defined as the average difference between the failure rate of each component before and after maintenance. It is expressed as the following equation. Δ$_ï =3=> 8A 9ℎ@ GDAA@/@<B@ ?@9ö@@< A:D;=/@ /:9@ 8A @:Bℎ B8>E8<@<9 989:; <=>?@/ 8A B8>E8<@<9C Δ$ï = ($₍àãå_{− $} ( ç^ä₎ ô ( 5 (3.43) So at the start of the model an initial value will be given as a standard of Δ$_ï, at the end of the model the Δ$_ï will be calculated. The model will keep updating the component reliability importance and solving the optimal maintenance strategy again and again until the Δ$ï meets the given standard.

b) The average Hazard Index in the new model

As it is solving a time-based maintenance plan in the new model, the Hazard Index of each component is changing every month during the maintenance intervals. So the average Hazard Index needed to be calculated every month instead of whole interval. The average hazard index is expressed as 5_ï,åT _{and calculated as follows:}

5_ï,åT ₌K& $(,åí++ ∆$( − K&($(,åí+)

∆$₍ (3. 44)

3.6 The software GAMS

The simulation of this thesis is performed on the software GAMS. GAMS is short for General Algebraic Modeling System, which is an advanced modeling program for mathematical modeling and optimization problems.

GAMS is designed for several types of problems, especially for linear, nonlinear and mixed integer problems. There are some basic types of model [38]: Mixed Integer Linear programs (MIP), Mixed Integer Nonlinear Programs (MINLP), Mixed Complementarity Problems (MCP), Mathematical Programs with Equilibrium Constraints (MPEC), Constrained Nonlinear Systems (CNS) and Extended Mathematical Programming (EMP). The model in this thesis is solved by MIP. • Basic features of GAMS

1. It is an advanced modeling software which is scalable. 2. It is individualized for large-scale and complex modeling. 3. It has a very simple structure which is easy to follow.

4. The errors will be checked by the system and possible reasons will be listed. 5. It is able to exchange data with other software like Excel and Matlab. • Basic structure of a GAMS model [39]:

The GAMS model is consisted of five parts which is Set, Data input, Variables, Equations, and Solver.

1. Set is the fundamental blocks in a GAMS model which makes it easier to state and read the model.

2. Data input is the block to input the data that going to be used in the model. There are three basic forms: Scalar, Parameter and Table.

3. Then the next part Variable is the declaration and manipulation of the variables used in the GAMS model.

4. Equations part is to define and declare the equations in the Model. 5. Then the last part is to choose the right solver to solve the model.

The GAMS of the model of this thesis is attached in the Appendix for the readers to further understand the model.

3.7 Summary

This chapter is mainly consisted of two parts. The first part was to introduced the basic RCAM model, the second part was the introduction of the improvements and the new model.

First, the basic calculations of some indices were introduced. Then the maintenance strategy was defined. The optimization model and formulation of the basic model was further introduced. After that, a flowchart of the basic model was presented to show the model structure.

Then in the second part, the improvements were presented first. Then the improved optimization formulation was listed. Following this a flowchart of the new model was provided. Ultimately the software GAMS is introduced.

Chapter 4

The Simulation and Results

After finishing the GAMS programming of the improved model, all the simulation results are presented in this chapter. The studied system will be introduced first. Then the result of the simulation on the system reliability (the change of SAIFI index and the maintenance cost with different weight factor π) and the final maintenance strategy will be presented. Following is a sensitivity analysis on the simulation results with respects to the assumed parameters in the model. A discussion on the results will be given at the end.

4.1 The description of the studied System – Birka System

4.1.1 Network introduction

The case study of this model is performed on a small distribution system called Birka System, see figure 4.1, which is located at the southern part of Sweden [12].

Figure 4.1 The structure of Birka System [12]

As shown in the figure above, the model is distributed from station SP which is 220/110KV to the Liljeholmen station which is 33/11KV. There are three load points shown in the figure which are SJ (Stockholm Subway), LH11 and HD (Högalid). The SJ and HD are two 33KV feeders, and the LH11 which is from component 28 to 35 includes about thirty 11KV outgoing feeders. In each load, there are different amounts of customers, demand and interruption cost. The detailed information will be presented in next subchapter. There are totally 58 components in this system and can be classified into four types, which are Circuit Breaker, transmission line, transformers and bus bar. Different components are expressed by different symbol in the figure. Each component is individual and has different failure rate and repair time. 29

c40 c41 c42 c14 SJ c2 c3 c4 c5 c6 c7 c10 c11 c12 c13 c8 c9 c30 c31 c32 c33 c34 c35 LH11 c27 _c29 c28 c23 c26 c25 c24 c19 c20 c21 c22 c15 c16 c17 c18 c37 c38 c39 c43 c44 c45 c36 HD c46 c47 c48 c49 c50 c51 c52 c53 c54 c55 c56 c57 c58 Sp c1 33 kV 110 kV 220 kV 11 kV

Figure 3.1. The studied Birka system [3], note that components 28-35 are “average components” representing in total 32 outgoing feeders.

3.5.2 Customer interruption costs

Costs for interruptions on an aggregated system level are one of the major factors for the indices proposed in this thesis. For electric power systems these costs are usually referred to as interruption costs and represent the cost at different load points in the system. The resulting system interruption cost for the whole network is used as a measure of total system reliability performance and in the calculation of the importance indices. The expected yearly load point interruption costs are in this thesis based on the number of interruptions and the total

interruption duration and on node specific interruption cost parameters (kL and cL), as follows:

( )

s L L L L L L

L

C =

∑

λ k P +c P r [€/yr] (3.10)

where Cs is the total interruption cost for the system, PL [kW] average power, λL [f/yr] and rL

4.1.2 The input Data

As explained above in Chapter 3, the failure rate and repair time of each component, and the load points information are the main inputs to the RCAM model. Table 4.1 below is the value of failure rate and repair time for each component.

Besides the simulation period is ten years which is 120 months. And the accuracy of ∆$ is set as 0.001. The number of maintenance actions can be done in one month is assumed as 5.

Table 4.1 The failure rate and repair time of each component Component Number Failure Rate(f/yr) Repair Time(h/f)

1 0.00964 8760 2 0.0087 8760 3 0.0261 365 4 0.0087 365 5 0.07012 52.14285714 6 0.0205 365 7 0.00089 8760 8 0.0087 8760 9 0.0261 365 10 0.0087 365 11 0.07031 52.14285714 12 0.0205 365 13 0.00089 8760 14 0.00964 8760 15 0.00089 8760 16 0.00028 182.5

17 0.01989 365 18 0.00243 8760 19 0.00089 8760 20 0.00028 182.5 21 0.01989 365 22 0.00243 8760 23 0.00089 8760 24 0.00028 182.5 25 0.01989 365 26 0.00243 8760 27 0.00867 8760 28 0.00243 365 29 1.00E-08 1.00E+06 30 0.10069 8760 31 0.10069 8760 32 1.00E-08 1.00E+06 33 0.00331 182.5 34 0.0134 2190 35 1.00E-08 1.00E+06 36 1.00E-08 1.00E+06 37 0.00089 8760 38 0.02291 182.5 39 0.00089 8760 40 0.00089 8760 41 0.02285 182.5

42 0.00089 8760 43 0.00089 8760 44 0.02265 182.5 45 0.00089 8760 46 1.00E-08 1.00E+06 47 0.00089 365 48 0.00964 8760 49 1.00E-08 1.00E+06 50 0.00089 8760 51 0.00863 182.5 52 0.00089 8760 53 0.00089 8760 54 0.00837 182.5 55 0.00089 8760 56 1.00E-08 1.00E+06 57 0.00089 365 58 0.00964 8760

Besides the failure rate and repair time, the load information of each load point is also an important input for the calculation of Hazard index. The customer number, total demand, interruption cost and cost of energy not served are shown below in Table 4.2.

Table 4.2 The load point data Load point Interruption

Cost ($/f, kWh) Cost of ENS ($/kWh) Total Demand (kW) Number of customer SJ 1.14 7.11 800 1

LH11 0.73 3.07 24600 14300

HD 0.42 1.64 23000 23400

Besides there are some assumptions to the parameter in the simulation. First is the number of maximal maintenance number in each month, it is assumed as 5. Then the failure rate increasing rate is assumed as 1% per month.

4.2 The simulation results

With the input data given above, the simulation results will be presented in the following section. After we considering the system reliability, the results are obtained in two stages. We define these two stage as system reliability level and maintenance strategy level. In the first stage, the output results are the value of system reliability index SAIFI for different weight factor ≠, CM cost, PM cost and total maintenance cost. After we get the results in first stage, an analysis will be done and the final weight factor will be chosen by considering the system requirement and the cost. Then in the second stage, after we set the system index and the weight factor, the final maintenance strategy will be got. And as we have considered two kind of optimization models, so the result of each stage are consisted of two parts.

4.2.1 The first stage - The system reliability level

In the simulation of this stage, the weight factor ≠ is set as 0 to 1 at a step of 0.02. So there are totally 50 groups of results. The results of the simulation will be listed in the following figures.

4.2.1.1 With the objective function of

∆"#

The results are presented below in Table 4.3. There are 7 columns of data in the table. To show the to show the relation between them and make the results clearer, six figures are made according to the data. The figures are shown below from Figure 4.1 to Figure 4.6.

Simulation Scenarios π SAIFI ∆#ª,º #Ω _∫# ª,º æ# _∆# ª,º øΩ _∆"# 1 1 0.099695234 0 0 5800 5800 2 0.98 0.099252762 -16189.21589 -164170.8669 8200 -172160.0828 3 0.96 0.099252762 -16189.21589 -164170.8669 8200 -172160.0828 4 0.94 0.09765755 -16326.99804 -229784.0856 11800 -234311.0836 5 0.92 0.097210643 -16360.88835 -265751.5638 14200 -267912.4522 6 0.9 0.097210643 -16839.92291 -271637.3225 14800 -273677.2454 7 0.88 0.097210643 -17991.35622 -280046.2973 16000 -282037.6535 8 0.86 0.097210643 -18524.31261 -292038.5772 17800 -292762.8898 9 0.84 0.087982929 -18285.39384 -313529.827 21400 -310415.2208 10 0.82 0.087982929 -18285.39384 -313529.827 21400 -310415.2208 11 0.8 0.087982929 -18285.39384 -313529.827 21400 -310415.2208 12 0.78 0.087982929 -18285.39384 -313529.827 21400 -310415.2208 13 0.76 0.087982929 -18285.39384 -313529.827 21400 -310415.2208 14 0.74 0.087982929 -18285.39384 -313529.827 21400 -310415.2208 15 0.72 0.087982929 -18285.39384 -313529.827 21400 -310415.2208 16 0.7 0.087982929 -18393.99127 -314864.1322 22000 -311258.1234 17 0.68 0.087982929 -18393.99127 -314864.1322 22000 -311258.1234 18 0.66 0.087982929 -18393.99127 -314864.1322 22000 -311258.1234 19 0.64 0.087982929 -18393.99127 -314864.1322 22000 -311258.1234 20 0.62 0.087982929 -18473.71688 -315843.6976 22600 -311717.4145 21 0.6 0.087928665 -20542.20063 -333264.4161 35000 -318806.6167 22 0.58 0.087928665 -20542.20063 -333264.4161 35000 -318806.6167 23 0.56 0.087928665 -20542.20063 -333264.4161 35000 -318806.6167