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Development of a Simulation Module forthe Reliability Computer Program

RADPOW

JOHAN SETREUS

Master’s Degree ProjectStockholm, Sweden 2006

DEVELOPMENT OF A SIMULATION MODULE FOR

THE RELIABILITY COMPUTER PROGRAM

RADPOW

Master Thesisby Johan Setréus

Master Thesis written at KTH, the Royal Institute of Technology, 2006,School of Electrical Engineering

Supervisor: Lina Bertling, KTH School of Electrical EngineeringExaminer: Lina Bertling, KTH School of Electrical Engineering

XR-E-ETK 2006:010

Abstract

This master thesis describes an implementation of a Monte Carlo Simulation(MCS) method for reliability assessment of electrical distribution systems. Themethod has been implemented in the reliability assessment tool RADPOW whichnow is able to perform both analytical and simulation evaluations. The main con-tributions within this thesis includes the following activities;

• Further development of RADPOW by the introducing of a graphical userinterface for Windows.

• Development and implementation of an analytical sensitivity analysis rou-tine for RADPOW.

• Development and implementation of a sequential MCS method in RADPOWin a stand alone module referred to as Sim.

The implemented MCS method has been validated in a comparable study for twocase systems by a commercial software NEPLAN. Results shows that the imple-mented MCS method provides the same results as the analytical method in RAD-POW and the NEPLAN software.

iii

Sammanfattning

Detta examensarbete beskriver hur en Monte Carlo simulering (MCS) kan an-vändas för tillförlitlighetsanalys av ett eldistributionssystem. Metoden har imple-menterats i verktyget RADPOW som nu kan utföra både analytiska och numeriskaberäkningar. Angreppssättet för att utveckla denna MCS metod i RADPOW in-nefattade följande aktiviteter:

• Vidareutvecklade av RADPOW med införandet av ett grafiskt användar-gränssnitt för Windows.

• Utveckling och implementering av en iterativ analytisk metod för känslighet-sanalys av eldistributionssystem i RADPOW.

• Utveckling och implementering av MCS metoden i RADPOW, vilken plac-erades i en fristående modul kallad Sim.

Den implementerade MCS metoden har validerats i en jämförande studie innefat-tande två testsystem med datorprogrammet NEPLAN. Resultat från denna studievisar att MCS metoden ger samma resultat som den analytiska metoden i RAD-POW och det kommersiella verktyget NEPLAN.

v

Acknowledgements

First I would like to thank my examiner at the Royal Institute of Technology LinaBertling for taking her time and giving me support and encouragement during myproject.

Furthermore, I would like to thank Carl Johan Wallnerström for many rewardingdiscussions involving technical issues and aspects of all natures in world. I alsoappreciate the input and help I received from PhD student Patrik Hilber concerningthe simulation method.

Finally, I wish to thank my family and my beloved Lisa for supporting me dur-ing my work. Thank you.

Johan SetréusStockholm, June 2006

vii

Contents

Abstract iii

Sammanfattning v

Acknowledgements vii

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 System Reliability 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Test System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Reliability Indices . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . 13

3 RADPOW 173.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Overview of RADPOW . . . . . . . . . . . . . . . . . . . . . . . 173.3 Reliability Evaluation in RADPOW . . . . . . . . . . . . . . . . 193.4 RADPOW_1999 version . . . . . . . . . . . . . . . . . . . . . . 213.5 RADPOW_1999_PF version . . . . . . . . . . . . . . . . . . . . 343.6 RADPOW_2006 version . . . . . . . . . . . . . . . . . . . . . . 35

4 Sensitivity analysis routine 434.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Sensitivity analysis with random disturbance . . . . . . . . . . . . 43

5 Monte Carlo Simulation Method for RADPOW 475.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2 Simulation Method . . . . . . . . . . . . . . . . . . . . . . . . . 47

ix

5.3 Implementation in RADPOW . . . . . . . . . . . . . . . . . . . . 515.4 Approximations and Weaknesses in Method . . . . . . . . . . . . 52

6 Comparative Studie of the Methods 556.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.2 Test System 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.3 Birka System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.4 Validation of the Simulation method in RADPOW . . . . . . . . . 616.5 Sensitivity Analysis Routine . . . . . . . . . . . . . . . . . . . . 64

7 Closure 737.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.2 Discussion and Future Work . . . . . . . . . . . . . . . . . . . . 73

A Input Data File for RADPOW 75A.1 Network topology data . . . . . . . . . . . . . . . . . . . . . . . 75A.2 Customer data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77A.3 Component reliability data . . . . . . . . . . . . . . . . . . . . . 78A.4 Load flow data . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

B Test System Input Files for RADPOW 81B.1 Test System 1 Input Data File . . . . . . . . . . . . . . . . . . . . 81B.2 Birka System Input Data File . . . . . . . . . . . . . . . . . . . . 83

x

Chapter 1

Introduction

1.1 Background

A central part in the planning of distribution systems, which becomes even moreimportant in today’s de-regulated electrical power system, is preventive mainte-nance (PM). This is the planned and scheduled maintenance that aims to postponeor reduce failures of a system. Electrical distribution system operators (DSO) havechanged their organization and the pressure to reduce operational and maintenancecosts is already being felt. The driving forces are changing from technical factorsto economic and business factors and cost-effective PM is required. Consequently,there is an interest from DSOs to incorporate strategies for cost-effective mainte-nance. Reliability Centred Maintenance (RCM) is such a strategy where mainte-nance of system components is related to the improvement in system reliability.The RCM method has been further developed in the reliability-centred asset man-agement method (RCAM) [1] to provide a quantitative relationship between PM ofassets and the total cost of maintenance [2].

In the search of the best possible asset management strategy for electrical dis-tribution system it is essential to know the importance of the involved components.Each component is assigned performance indices that correspond for the overallreliability of supply. The indices can be used for prioritization of components; oneexample is to determine where maintenance actions will have the greatest effect.One way to perform such analysis is to evaluate the amount of interruptions a cer-tain component causes the system. A simulation approach of this kind of analysisenables us to develop models with a deeper level of detail for larger systems in amore straightforward manner compared to the analytical approach.

This thesis presents an implemented method for performing Monte Carlo sim-ulations on a power system in order to evaluate the system reliability with a nu-merical measurement. This method can then easily be extended to be used forprioritization of components and eventually also produce a distribution of resultsfrom which the mean, variance and other statistical measures can be computed.

1

2 Chapter 1. Introduction

1.2 Objective

The main objective of the thesis is to develop a simulation module for RADPOW,a computer program developed for system reliability analysis of power distributionsystems [3].

1.3 Approach

The first step in this study was to implement the already existing version of RAD-POW in a graphical user interface in Windows. This did not only provide an userfriendly interface, it also made it easier to validate the results in the development ofthe implemented simulation method. It also provided a deeper level of understand-ing for the different algorithms and methods already developed in the analyticalmetod in RADPOW. The graphical interface was put into practice by a number ofnew graphical modules, developed in Borland C++.

The second step was to implement an iterative analytical routine in RADPOWfor sensitivity analysis. This analytical sensitivity approach provided valuableknowledge of the generation of random numbers from various distributions, whichwere necessary in the work with the simulation approach. The analytical methodin RADPOW, in combination with this sensitivity routine, provided a distributionof the resulting system reliability indices including the mean values and variancesof the the samples. In order to validate the results from the sensitivity analysisroutine, the mean values of the system indices for two different test systems, werecompared with both RADPOW and the commercial reliability tool NEPLAN.

The final, third step, was to use the algoritms and methods for random num-ber generation in the development and implementation of the simulation method inRADPOW. The basic approach in the method was first tested in MATLAB. Thenthe implementation in RADPOW was made in a stand alone module Sim, pro-grammed in C++. The results from the simulation method in RADPOW was thenvalidated by comparing the results from two test systems with the results from boththe analytical method in RADPOW and NEPLAN.

This master thesis has also resulted in an article that has been presented forpublish at the Nordic conference on Nordic Distribution and Asset Management(NORDAC) in Stockholm, August 2006 [4].

1.4. Outline 3

1.4 Outline

Chapter 2 first defines important terms and abbreviations that are used in this the-sis, then the main evaluation methods and techniques used in RADPOW and in thisthesis are described.

Chapter 3 gives an insight to the basic functions of RADPOW, describing howthe evaluation methods are implemented in different modules and how these inter-acts with each other.

Chapter 4 describes the iterative analytical method developed by the author. Thismethod uses the analytical method developed in RADPOW to perform a sensitivityanalysis of a power system.

Chapter 5 describes the Monte Carlo Simulation method developed and imple-mented for RADPOW by the author.

Chapter 6 validates the simulation method and the iterative analytical method withthe analytical evaluation method in RADPOW and the commercial reliability pro-gram NEPLAN. Two different test systems are used for the validation.

Chapter 7 summarizes the results obtained and discusses the future work.

Chapter 2

System Reliability

This chapter first defines important terms and abbreviations that are used in thisthesis. Then the reader gets an introduction to the evaluation methods and tech-niques that are used in RADPOW, a system reliability computer program describedin Chapter 3.

2.1 Introduction

Reliability is the ability of a an item to perform a required function, under givenenvironmental and operational conditions and for a stated period of time [5]. In thisdefinition the term item is used to donate a component, subsystem or a system ofcomponents, depending on the certain reliability level that is going to be studied.These two reliability levels are referred to as system and component reliabilityrespectively.

2.1.1 System reliability

A system consists of one or more subsystems, each interconnected and each havinginterconnected components, in order to perform its required function. A systemcan be everything from a single machine, consisting of a number of components,or a interconnected network of the same machine, now considered as a component.There is no limit in the way an item can be considered as a system, it all dependson the specific situation.

The reliability of a system denotes the relationship between the required per-formance and its achieved performance [5]. The use of a probabilistic model ofthe system deals with this relation and gives a measurement of the system relia-bility, given its components reliability. For this purpose the characteristics of thesystem’s components needs to be known and well studied to determine the overallsystem reliability. In this thesis it as been considered that all the components in asystem are uncorrelated of each other, and thereby each component can be studiedand modeled separately.

5

6 Chapter 2. System Reliability

2.1.2 Component reliability

Based on experience and failure data for a certain component, its characteristicsin terms of reliability can be modeled. To describe the reliability of a component,there is a number of mathematical functions that can be used. The most importantare defined as:

Definition 2.1 The distribution function for the continuous one-dimensional ran-dom variable X is defined by

FX(x) = P (X < x),−∞ < x < ∞ (2.1)

The distribution function is evaluated by an integration as follows

FX(x) =∫ x

−∞fX(t)dt (2.2)

If the function fX(x) exists and applies to the function in Equation 2.2, then X is acontinuous random variable of the distribution. The function fX(x) is then calledthe density function of X [1][6].

Definition 2.2 The density function for the continuous one-dimensional randomvariable X is defined by

fX(x) =dFX

dx(2.3)

for all values of x where fX(x) is continuous [1][6].

In this thesis the main interest is the lifetime evaluation for a component. Thereforthe functions can be donated FX(x) = F (t) and fX(x) = f(t), where t is the timein e.g. years.

Definition 2.3 The reliability function, R(t), which also is called the survivalprobability function, is defined by

R(t) = P (T ≤ t) = 1− F (t) (2.4)

The consensus of this is that R(t) is the probability that the component does notfail in the time interval (0, t], or, in other words, the probability that the componentsurvives the time interval (0, t] and is still functioning at time t [5].

Definition 2.4 The failure rate function, λ(t), is defined by

λ(t) =f(t)R(t)

(2.5)

This function describes the components tendency to fail, in failure per time unit,for t ≥ 0.

2.2. Definitions 7

Definition 2.5 The mean time to failure (MTTF) donates the expected time to failand is defined as

MTTF = E(T ) =∫ ∞

0tf(t)dt =

∫ ∞

0R(t)dt (2.6)

For an actual measurement of the above functions output data, there are twoequations that are of interest in this thesis.

Definition 2.6 The mean value x for a set of output variables of X ,x = x1, ..., xn, is defined as

x =1n

n∑

j=1

xj (2.7)

Definition 2.7 The variance σ2, of the output set x, is defined as:

σ(x)2 =1

n− 1

n∑

j=1

(xj − x)2 (2.8)

Here the standard deviation of the result is donated σ.The functions stated above are applicable for any continuous variable X . In

Section 2.5 distribution functions used for modeling component reliability are pre-sented.

2.2 Definitions and Abbreviations

The following basic definitions are used in this thesis:

Definition 2.8 Functional failures is the ability of an item or equipment to fulfilone or more of its functions [7].

Definition 2.9 Failure modes are events that cause functional failures [7].

Definition 2.10 Reliability is the ability of an item to perform a required function,under given environmental and operational conditions and for a stated period oftime [5].

2.2.1 Failures

If a functioning electrical power system breaks down and can not deliver electricpower to some or all of its customer, an interruption of supply has occurred. In thisthesis the interruption of supply are referred to as failure or outage of the systemand the cause of this can be structured as in Figure 2.1. Failures in a system can bedivided into two categories; damaging faults and non-damaging faults [1]. Outagesdue to damaging faults are referred to as permanent forced outages and these faultsare caused by either an active or an passive failure.


Failure -outage

Damaging FaultTwo models of failure

Non-Damaging FaultTwo models of restoration

Passive event

Active event

Permanent forced

outages

Transient forced

outages

Temporary forced

outages

Automatic switching

Manual switching or fuse

replacement

Figure 2.1: Causes of failures [1].

Definition 2.11 An active failure is a failure of an item that causes the operationof the protection devices around it [1].

Protection devices are in this case breakers or fuses which, if functioning, trip(opens) and isolates the failure.

Definition 2.12 A passive failure is a failure of an item that does not causes theoperation of the protection devices around it [1].

When a permanent failure has occurred the component is restored by repairing orreplacing it. Passive failures occurs normally in open circuits or in inadvertentopening of breakers.

The second category of failures, non-damaging faults, are outages caused bythe protection devices. These outages are categorized into transient and temporaryforced outages, depending on the restoration of the fault. If a protection deviceare restored automatically, the outage time are negligible and therefore transientforced. Other types of protection devices needs to be restored manually, eithermechanically or by replacement of a fuse. These types of action takes time and theoutage are therefore called temporary forced outage.

When an active failure event occur and the protection devices around it opens,this may lead to outages in several load points associated with these devices. Theseevents are not caused by an damaging fault directly and are therefor referred to asadditional active failures [1].

2.3. Test System 9

Definition 2.13 An additional active failure is a failure mode that occur when anactive failure of an item causes the interruption of other items in the system [1].

2.2.2 Restoration Time

Depending on the failure and the action taking place to restore the failure, therestoration time for an outage can be categorized. The different types of restorationtimes for outages are defined below.

• rr - Repair restoration time is the time it takes to make the component oper-ational by repairing it.

• rp - Replacement restoration time is the time it take to replace a componentto make it operational.

• rs - Switching restoration time is the time it takes for a manual or automaticswitching device to isolate the failure.

• rc - Re-closure restoration time for a protection device.

All these restoration times are used in the RADPOW model [1].If repair of an component takes longer time than the replacement of it, the

later choice are normally considered. In this thesis and in the computer programRADPOW [1] in Chapter 3 it has been assumed that the shortest restoration timealways are the considered one, independent of other aspects as e.g. economical.

2.3 Test System

A test system with different failure events is used in this thesis to better understandthe different definitions mentioned in the previous section. This will illustrate thegeneral function of a distribution system and its components. In Chapter 3 the testsystem is used for reliability analysis with the tool RADPOW.

2.3.1 Test System 1

The test system shown in Figure 2.2 are referred to as Test System 1 [1][8]. Thesystem has been divided into two separate cases which are referred to as Test System1a and Test System 1b. The only difference between these systems is that thedisconnector are normally open in 1a and considered as a closed point in 1b. Thesymbols used in Figure 2.2 are defined in Figure 2.3. Test System 1 consistsof standard components used in distribution systems and features two load pointsand two supply points. It has a number of breakers to isolate failures and onedisconnector which can transfer load if closed. The branches, indicated by a B inFigure 2.2, are sections of components connected in series. These are used in thereliability computer program RADPOW, described in Chapter 3.


c2 c4

c8 c13c17

c14

c18

c10

c12

c9

c11c16

c15

c7

c1 c3 c5

LP5

LP6

c6

B2 B5

B1

B4

B3

B6

Figure 2.2: Test System 1, with components c, and branches B [1].

transformer breakerbus disconnector

supply point load point

Figure 2.3: Symbols used in Test system 1 and in this thesis in general [1].

2.3.2 Protection devices

There are generally three types of protection devises in a power system; Breakers,Disconnecters and Fuses. The main purpose of these is to protect the system’scomponents and isolate upcoming failures in the system.

2.3.3 Example events

For better understanding of the distribution system and its components, some typi-cal fault events have been listed below. These events will also give a better under-standing for the definitions defined earlier in Section 2.2.

At the first scenario Test System 1a is used and the disconnector c18 is thennormally open. After these events Test System 1b is studied. All events are assumedto be independent of each other and all components are assumed to be functioningfrom the beginning.

Failure events on Test system 1a

Disconnector c18 is normally open for these two separate events.

2.3. Test System 11

• Permanent Outage caused by a passive faultA passive fault, caused by an software error, strike breaker c14 which openswithout a reason. None of the protection devices are triggered. The systemoperator gets aware of the outage in LP6 immediately and his first idea is toclose disconnector c18 manually, but when he looks at his monitor he seesthat it is stuck and can not be closed. He then decides that the best way tosolve the problem is to replace breaker c14 and consequently he order anengineer to go to the spot and start the replacement of the component. Whenthe engineer arrives to the spot he first disconnect the breaker electricallyfrom the grid and then starts the replacement. The time it takes for the en-gineer to get to the spot and then disconnect the component is denoted rs

as defined in Section 2.2.2. The switching time rs is in this event assumedto be 1 hour. The engineer then replaces the breaker and connects it to thegrid which takes the time rp, here assumed to be 5 hours. The customers inLP6 are affected by the permanent outage during the time it takes to restorebreaker c14 which is rs + rr = 1 + 5 = 6 hours.

• Temporary forced outageAn extremely large demand for power in LP6 forces the system operator toremotely open breaker c8 immediately, due to the risk of overloading thetransformer c17. The customers in LP6 is suffering an interruption of supplyand the system operator decides to close the disconnector c18 manually andtherefor he orders an engineer to go to the spot. The engineer arrives to thespot and closes the switch. The time for the engineer to get to the discon-nector and then close it is here denoted rs as defined in Section 2.2.2. Theswitching time rs is in this event assumed to be 1 hour. The customers inLP6 are affected by the temporary forced outage during the time it takes toclose the disconnector which is 1 hour.

Failure events on Test system 1b

Disconnector c18 is considered to be functioning as a closed point for these twoseparate events.

• Permanent Outage caused by an active faultA rainy weather causes an active fault on bus c5 due to inadequate housing.The fault are automatically isolated by the breakers c10, c12 and c14, whichmeans that both LP5 and LP6 is suffering an interruption of supply. Thesystem operator decides to repair bus c5 and open disconnector c18 for therestoration of LP6 and consequently he order an engineer to go to bus c5 andanother to c18 for the manually switching. The customers in LP6 are affectedduring the time it takes to open c18 and then reclose breaker c14 whichis made remotely by the system operator. The switching time rs for thedisconnector is assumed to be 1 hour and the customers in LP6 are thereforsuffering a temporary outage, lasting for 1 hour. For the customers in LP5


the failure in bus c5 affects them until the c5 is repaired. The reparation of c5starts after the engineer has arrived and disconnected the bus from the gridwhich takes the time rs, here assumed to be 1 hour. The reparation of thebus then takes the time rr, which is assumed to be 2 hours. The customers inLP5 are affected by the permanent outage during the time it takes to restorethe bus c5, which takes rs + rr = 1 + 2 = 3 hours.

• Transient OutageAn active fault occurs on transformer c17, caused by a ground failure. Thefault triggers the breakers c13 and c14, which isolates the fault. The failureleads to very short voltage drop in LP6, which still gets power delivered viabus c5. This is a transient outage which outage time are negligible. Thesystem operator later decides to replace transformer c17 and then reclose thebreakers c13 and c14.

2.4 Reliability Indices

The reliability indices gives a quantitative measurement of the reliability in the loadpoints or in the overall system. The indices that are used in this thesis, and in thecomputer program RADPOW discussed in Chapter 3, corresponds to general usedindices in literature [1].

2.4.1 Load Point Indices

The indices used for measuring the reliability in a load point i (lpi) are:

• λlpi[f/yr] = Expected failure rate per year

• Ulpi[h/yr] = The annual unavailability in hours per year

• rlpi[h/f ] = Expected outage duration for a failure

• LOElpi[kWh/yr] = The average loss of energy per year

These indices are evaluated for each load point in the system by using the meth-ods described in Section 3.4.2, given the component reliability parameters for thesystem described in Section 2.1.2.

2.4.2 System Indices

Based on the load point indices, the performance of the systems ability to deliverenergy to its customers, can be evaluated to system indices. These indices canbe divided into two groups; Customer-weighted and capacity-weighted [9]. In thesystem indices listed below, Ni represents the number of customers in load point i:

2.5. Distribution Functions 13

• System Average Interruption Frequency Index (SAIFI) [int/yr,cust]:

SAIFI =∑

Niλi∑Ni

(2.9)

• System Average Interruption Duration Index (SAIDI) [h/yr,cust]:

SAIDI =∑

NiUi∑Ni

(2.10)

• Customer Average Interruption Duration Index (CAIDI) [h/int]:

CAIDI =∑

NiUi∑Niλi

=SAIDI

SAIFI(2.11)

• Average Energy Not Supplied per customer served (AENS) [kWh/yr,cust]:

AENS =∑

LOEi∑Ni

(2.12)

• Average Service Availability Index (ASAI):

ASAI =∑

Ni · 8760− UiNi∑Ni · 8760

(2.13)

2.5 Distribution Functions

The Exponential and the Normal distributions have been used in this thesis for themodeling of power systems. They are only briefly presented here, more informa-tion can be found in [6] and [5].

2.5.1 The Exponential Distribution

In reliability analysis the exponential distribution is the most common model todescribe the lifetime of an item. The reasons for this is due to its mathematicalsimplicity and that the model are suitable for many different items or situations.An exponentially-distributed variable T ∈ Exp(m) has the following density anddistribution functions:

f(t) =

{(1/m) · e−t/m for t ≥ 0,m > 00 otherwise

(2.14)


FX(x) =

{0 for x < 01− e−x/m for x ≥ 0

(2.15)

Both (2.14) and (2.15) are illustrated in Figure 2.4 together with the survivor func-tion, R(t), and the failure rate function λ(t) = 1/m. According to this the failurerate of an exponentially distributed item is constant and thereby independent oftime. This implies that it is memoryless and can be considered as good as newat any time when still functioning. This also implies that it is no meaning to re-place a still functioning component in preventive maintenance if its failure rate isexponential modeled. A constant failure rate is normally a good assumption for anitem during its useful life period. In order to generate exponential distributed ran-

0 10 20 300

0.05

0.1

0.15

0.2

0.25

Density function fX(t)

f X(t

)

t0 10 20 30

0

0.2

0.4

0.6

0.8

1

Distribution function FX(t)

FX(t

)

t

0 10 20 300

0.2

0.4

0.6

0.8

1

Reliability function RX(t)

RX(t

)

t0 10 20 30

0

0.1

0.2

0.3

0.4

0.5Failure rate function λ(t)

λ(t)

t

Figure 2.4: Different functions of an exponential distributed variable T ∈ Exp(4)

dom numbers, the inverse of the exponential distribution function FX(x) needs tobe stated. Given a uniform stochastic variabel U ∈ U(0, 1), the stochastic variableY = F−1

Y (U) have the distribution function FY (x), and Y is thereby exponentiallydistributed. For the exponential distribution the inverse can be solved straight for-ward and is as follows:

Y = − 1λ

ln U (2.16)

2.5. Distribution Functions 15

2.5.2 The Normal Distribution

To describe an uncertainty in a measured or statistical evaluated parameter of anitem, the normal distribution can be used. A normal distributed random variableX ∈ N(µ, σ2) has the following density and distribution functions:

fX(x) =1

σ√

2πe−(x−µ)2/2σ2

(2.17)

FX(x) =1

σ√

2π

∫ x

−∞e−(y−µ)2/2σ2

dy (2.18)

If µ = 0 and σ = 1, the distribution X ∈ N(0, 1) is called the standard normaldistribution and the density and distribution functions are then donated ϕ(x) andΦ(x) respectively. These are both illustrated in Figure 2.5. An inverse formula to

−4 −2 0 2 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Density function fX(x)

f X(x

)

x−4 −2 0 2 40

0.2

0.4

0.6

0.8

1

Distribution function FX(x)

FX(x

)

x

Figure 2.5: Density and distribution function for a standard normal distributed variableX ∈ N(0, 1)

the normal distributed distribution function, Φ−1(x), does not exists but a varietyof methods can be used to generate a normal distributed variabel X ∈ N(0, 1).The method used in this thesis is the Box-Muller transform[10] in polar form. Thealgorithm proceeds according to the following steps:

1. Generate two independently uniform distributed variabels U1, U2 ∈ U [0, 1].

2. Scale U1 and U2 to V1 and V2 respectively, to the uniform distribution ∼U [−1, 1].

3. Let the variable R be defined by R = V 21 + V 2

2 . If R = 0 or R > 1 startover at step1, otherwise proceed.

4. The normal distributed random variable X ∈ N(0, 1) is then calculated as

X = V1

√−2 ln R

R


Given X ∈ N(0, 1) the following relation can be used to produce a normal dis-tributed variable Y ∈ N(µ, σ2):

Y = µ + σX (2.19)

Chapter 3

RADPOW

This chapter describes the function of RADPOW, a computer program developedfor system reliability analysis of power distribution systems.

3.1 Introduction

The reliability computer program RADPOW was first developed by Lina Bertlingand Ying He at the Department of Electrical Engineering, KTH, as a part of theirPhD projects during the years 1997-2002 [1][13]. The name RADPOW is an ab-breviation for Reliability Assessment of Distribution Power Systems, and as thename reveals, the program is developed for analysis of electric power distributionsystems. For this purpose there already exists a number of programs, developedboth for commercial and research use, but each having their advantages and dis-advantages. One of the main purpose of the development of RADPOW, were theneed of a tool in the research of RCM [1] and Automation [13]. Other aspects ofcreating a completely new tool were to build up new expertise and understandingfor different methods in the field.

The program has been developed in the computer language C++ which is anobject oriented software. The code is written in the C++ standard from 1999.

Originally the method for evolution of a power system in RADPOW was an-alytical and the purpose of this thesis is to develop a new module that also willallow RADPOW to make simulation calculations [3]. The proposed method andimplementation of the simulation approach in RADPOW are discussed in Chapter5. In the following sections the different modules in the program and the overallpicture of RADPOW and its versions are described briefly.

3.2 Overview of RADPOW

Given the data for a specific electric power distribution system, RADPOW calcu-lates the load point indices and system indices. Figure 3.1 shows the function ofRADPOW. Given the relation between the components, the reliability data for the

17

18 Chapter 3. RADPOW

components, customer data and power flow data, RADPOW presents the resultsincluding the reliability indices for each load point and the overall system indices.These indices are defined in Section 2.4.1.

RADPOW

Input Output

System Indices:• SAIFI [int/yr and cust.]• SAIDI [h/yr and cust.]• CAIDI [h/int.]• AENS [kWh/yr and cust.]• ASAILoad Point Indices:• [f/yr]• U [h/yr]• r [h/f]• L [kW]• LOE [kWh/yr]

System data:• Network topology

• Component reliability

• Customer and power data

• Load flow constraints

Figure 3.1: General function of RADPOW showing the required input data and the results.The user is able to choose whether or not the load flow constraints are considered in thecalculations.

The input data for the system are defined in a standard text file with a syntaxdescribed in Section 3.6.3 and in Appendix A. The output are presented to the usereither directly on the screen or as a text file for further analysis in other computerprograms, e.g. MATLAB.

3.2.1 The versions of RADPOW and its contributors

The first development of RADPOW, by Lina Bertling and Ying He, resulted in theversion referred to as RADPOW_1999, named by the final year of development.Figure 3.2 shows this version at the top, with the involved modules in the program.The method for the evaluation of a system is based on a analytical approach. Then amaster thesis project, made by Philippe Rosett, resulted in a new improved versionof RADPOW_1999 referred to as RADPOW_1999_PF [14][15]. This version alsoresulted in a new module, Loadflow, which considers the load flow constrains inthe model, and adds the result to the analytical calculations.

This thesis has resulted in a third version referred to as RADPOW_2006. Thisversion uses the modules from the two earlier versions together with a numberof newly developed modules in order to (i) implement a simulation method, (ii)implement a iterative analytical method and (iii) develop a graphical user interfacefor RADPOW.

Table 3.1 summarizes the involved developers and contributors to RADPOW,and also shows which modules each author have developed and implemented. Inthe following sections of this thesis the name RADPOW are considered as the latestversion, RADPOW_2006, if nothing else is mention.

3.3. Reliability Evaluation in RADPOW 19

Mincut Minpath Abreak Aafail

Lpind SindNetw BranchComp

RADPOW_1999



RADPOW_1999_PF

Loadflow



RADPOW_2006

Loadflow

Sim

Figure 3.2: The development of modules in RADPOW have resulted in three differentversions.

3.3 Reliability Evaluation in RADPOW

In reliability analysis the first step is always, as in all mathematical analysis, tomake a representativ model of the real system that is going to be studied. Whenthe model has been formulated, one can solve the desired problem with this model.The evolution of the problem can be achieved either by an analytical approachor an numerical approach. The analytical approach usually solves the problem di-rectly with mathematical formulas, whereas the numerical approach uses numericalmethods. Two special types of numerical methods are simulation and the MonteCarlo methods which uses random experiments to find a solution of a problem.There are slightly differences between the definitions of these words, and in thisthesis these are both referred to as Monte Carlo Simulation (MCS), which is thesimulation approach.


Table 3.1: The different modules has been developed by four different persons at the schoolof Electrical Engineering, KTH, Sweden.

Author Developed modules Year Main referencesLina Bertling Mincut, Abreak, Aafail

and Lpind1999 [1], [8]

Ying He Minpath, Netw, Branch,Comp and Sind

1999 [13], [8]

Philippe Rosett Loadflow 2000 [14]Johan Setréus Sim 2006 Section 3.6, [16]

In this thesis both the analytical and simulation approach has been adoptedto make an comparative studie. The flowchart in Figure 3.3 shows the overallmethodic used.

Network Model

System Data

Assign each LPs the events that lead to failure

for that LP

Calculate the reliability indices for each LP with

formulas

Calculate the reliability for the system

Make a large number of random experiments to see

how these affect LPs reliability

Simulation methodAnalytical method

Figure 3.3: Flow chart for the analytical and simulation method used in this thesis.

3.3.1 Evolution methods

In RADPOW there are three different reliability evolution methods that can be usedin order to determine the system and load point indices. A symbolic picture of theseare shown in Figure 3.4. The three different methods, as numbered in Figure 3.4,has the following main properties:

1. The analytical calculation method is the original method and this evaluates

3.4. RADPOW_1999 version 21

RADPOW

Analytical methods

Simulation method

Analytical calculation

Sensitivityanalysisroutine

1 2 3

Figure 3.4: The three evolution methods that can be used in RADPOW. (Method two andthree in figure are developed within this thesis).

the system with the formulas described in Section 3.4.2. The modules in-volved for this method are described in Section 3.4.4.

2. The sensitivity analysis routine uses the analytical method consecutive timeswith random input values. The resulting indices are the same as in method1, but with a standard deviation measurement for the results. This method isdescribed in Chapter 4.

3. The simulation method makes a large number of experiments on the systemand then evaluates these. This method, further described in Chapter 5, givesthe system and load point indices as output result.

3.3.2 Approximations and Assumptions

In RADPOW the following approximations has been used in the analytical and thesimulation evaluation methods of the system model [1].

• Only minimal cut sets of the first and second order are considered.

• The outage time for transient failures are negligible.

• It has been assumed that scheduled maintenance only are applied to a com-ponent if this not cause a system failure.

3.4 RADPOW_1999 version

In the analytical method, equations for evaluation of the reliability of the systemcan be used directly to the model. There are several techniques used for analyti-cal evolution and two of these that generally are used are Network modeling andMarkovian modeling [1]. Of these two, the Network model is the easiest methodto implement, specially for larger systems. In Markovian modeling each state ofthe system and the transitions between these needs to be defined. This means thatthe size of the model grows exponentially with the number of components in the


system, which makes it hard to use for larger systems. For smaller subsystems in alarger system, Markovian modeling can be used for approximations of overlappingfailures, in cooperative with the Network modeling, which is described in Section3.4.2. The Markovian modeling method and teori are not described any further inthis thesis. For further reading about Markovian modeling see [5].

3.4.1 Network Modeling

In network modeling the relationship between the system and its components isconsidered. The model describes the behavior of the system if one or more of itscomponents fails to fulfill its function. These different failure modes for the systemare described by the minimal cut set.

Definition 3.1 A cut set is a set of components which upon failure, cause a failureof the system. A cut set is minimal when it cannot be reduced any further and stillremain a cut set [1].

Definition 3.2 The number of different failure events in a minimal cut set is calledthe order of the cut set [5].

For the function of a specific load point each minimal cut set for the load point hasto functioning. In logical terms this is an AND statement. If the definition in 3.1 isapplied to Test System 1a, described in Section 2.3, the resulting minimal cut setsfor the load points are as in table 3.2. LP5 has four minimal cut sets of first order

Table 3.2: Minimal cut set vectors for Test System 1a.

Load point Minimal cut set vector

LP5[1, 7, 3, 5, 9+11, 9+16, 9+12, 15+11,15+16, 15+12, 10+11, 10+16, 10+12]

LP6 [4, 13, 17, 14, 6, 2, 8]

and nine minimal cut sets of second order, according to the definition in 3.2. LP6has only minimal cut sets of first order.

The minimal cut sets are used for the evolution of each load points reliabilityindices. In the analytical method in RADPOW a load point driven approach hasbeen adopted. This means that all failure events for each load point are consid-ered in turn and consequently that the load point indices for each load point areevaluated separately with help from the minimal cut sets. When implementing ageneral algorithm for deducing all the minimal cut sets in a system, it is easier tofirst deduce the minimal paths, and then convert these to minimal cut sets.

Definition 3.3 A path is a set of components that when all operating guaranteesthe operation of the system. A path is minimal when it cannot be reduced anyfurther and still remain a path [1].


For the function of a specific load point it is enough for one of its minimal path tobe functioning, that is all the path’s components are functioning. In logical termsthis is an AND statement for the components within a path and an OR statementfor all the paths belonging to a specific load point. For Test System 1a the minimalpaths are showed in table 3.3. LP5 has two minimal paths and if at least one of these

Table 3.3: Minimal paths for Test System 1a.

Load point Minimal paths

LP5[1 7 3 9 15 10 5][1 7 3 11 16 12 5]

LP6 [2 8 4 13 17 14 6]

are functioning, having its components operational, the load point is functioning.For LP6, only having one minimal path, each component in this path has to beoperational for the functioning of the load point.

In order to model a normally open disconnector, which can be closed and trans-fer power in alternative routes, a normally open path is used.

Definition 3.4 A normally open path is a minimal path that, if operational, can beused as an alternative route for power.

In Test System 1a the disconnector c18 can be closed to transfer power between thetwo load points. The normally open paths for each load point are shown in table3.4. Test System 1b does not have any normally open paths because of the closedpoint in c18.

Table 3.4: Normally open paths for Test System 1a.

Load point Normally open pathsLP5 [5 18 6 14 17 13 4 8 2]

LP6[6 18 5 10 15 9 3 7 1][6 18 5 12 16 11 3 7 1]

3.4.2 Reliability Evolution of Serial and Parallel Systems

As mentioned before the minimal cut sets are used for the reliability evolution ofthe load point indices. The minimal cut sets of first order represents a serial systemof components and the second order a serial system with the two components ineach set in parallel. The formulas presented here are the ones used in the com-puter reliability program RADPOW. The program and some of the formulas aredeveloped, amongst others, by [1].


The first order minimal cut sets represents a serial reliability system which areshown in Figure 3.5. Figure 3.5 is in accordance with the definition for minimal cut

1 2 n

Figure 3.5: Serial reliability system with n components.

sets; all components needs to be operating in order for the function of the system.The reliability of a serial system having n components is evaluated as

λs =n∑

i=1

λi (3.1)

Us =n∑

i=1

λiri (3.2)

rs =∑n

i=1 λiri

λs=

Us

λs(3.3)

LOEs = Us · Ps (3.4)

, where Ps is the average capacity demand (kW) in the serial reliability system.The second order minimal cut sets represents a parallel system with two com-

ponents in which it is sufficient for at least one of the components to be functioningfor the functioning of the system. The parallel system is shown in Figure 3.6. If one

1

2

Figure 3.6: Parallel reliability system with 2 components.

of these components fail when the other is non-operational an overlapping eventhas occurred. If the two failure types is of the same kind the reliability for theparallel system can be evaluated as

λ12 =λ1λ2(r1 + r2)

1 + λ1r1 + λ2r2≈ λ1λ2(r1 + r2) = λ1(λ2r1) + λ2(λ1r2) (3.5)

r12 =r1r2

r1 + r2(3.6)


U12 = λ12r12 ≈ λ1λ2r1r2 (3.7)

However, in analysis of a power system there can be different failure types forthe two components and this makes it more complicated. The formulas for twooverlapping failure events of different types, x and y, are [1]

λxy12 = λx

1(λy2r

x1 ) + λx

2(λy1r

x2 ) + λy

1(λx2ry

1) + λy2(λ

x1ry

2) (3.8)

rxys =

λx1(λy

2rx1 )

λxy12

· rx1ry

2

rx1 + ry

2

+λx

2(λy1r

x2 )

λxy12

· rx2ry

1

rx2 + ry

1

+

λy1(λ

x2ry

1)λxy

12

· rx1ry

2

rx1 + ry

2

+λy

2(λx1ry

2)λxy

12

· rx2ry

1

rx2 + ry

1

(3.9)

There is one constraint, when it come to the scheduled maintenance, that makesthese equations smaller and more practical. The constraint is that no operatorwould never ever take a component out for maintenance if this would cause a sys-tem failure. The failure rate for two overlapping events, where the first is thescheduled maintenance (m) and the next is a failure (x), can be described as [1]:

λxm12 = λm

1 (λx2rm

1 ) + λm2 (λx

1rm2 ) (3.10)

The equation for the restoration time for this types of overlapping events are [1]:

rxms =

λx1(λm

2 rx1 )

λxm12

· rx1rm

2

rx1 + rm

2

+λx

2(λm1 rx

2 )λxm

12

· rx2rm

1

rx2 + rm

1

(3.11)

3.4.3 Reliability Evaluation of Load Point Indices

The reliability for the load points, the load point indices, are calculated in RAD-POW_1999 by summarizing the different failure rate contributors, which are [1]:

• λc1lp - single failure events from minimal cut sets of first order,

• λc2lp - overlapping failure events from minimal cut sets of second order,

• λa1lp - additional active failures from single failure events, and

• λaslp - additional active failures with the probability of non-functioning pro-

tection devices.

The reliability for the load point lp are then calculated as [1]

λlp = λc1lp + λc2

lp + λa1lp + λas

lp (3.12)

Ulp = U c1lp + U c2

lp + Ua1lp + Uas

lp (3.13)

rlp =Ulp

λlp(3.14)

In this thesis there are four different failure events considered. These can besingle or overlapping events which abbreviation and explanation are stated below.For single failure events:


lpc1

lpc1

U lpc2

lpc2

U lpa1

lpa1

U lpas

lpas

U

lp lpU

Minimal cut sets of first order

Minimal cut sets of second order

Additional active failures

Additional active failures with stuck

probability

Figure 3.7: The load point indices are calculated from four different contributors.

• p - permanent failure

• te - temporary failure

• m - maintenance outage

• tr - transient failure

And for the overlapping failure events the single failures can be combined to:

• pp - two overlapping permanent failures

• tete - two overlapping temporary failures

• pte - overlapping permanent and temporary failures

• pm - maintenance outage and then a permanent failure

Minimal Cut Sets of First Order

The failure rate from the single failure events are evaluated from the minimal cutsets vector of first order as [1]

λc1lp =

n∑

i=1

(λp,i + λte,i + λtr,i) (3.15)

If there is a normally open path for the load point, the unavailability are defined as[1]

U c1lp =

n∑

i=1

(λp,i · r + λte,i · rc,i) (3.16)


where the restoration time r is defined as [1]

r = (1− P ) · rs + P · rr/p,i (3.17)

and where P is the probability that the normally open path cannot be used. If that isthe case, the restoration time is equal to the replace or repair time rr/p,i of the failedcomponent i. Here rr/p stands for either repair or replacement restoration time. Ifthe replacement time are greater than zero, this is always chosen before repairwhich normally takes longer time. If a open path is available, and are functioningwith a probability of (1−P ), the restoration time is equal to the switching time rs

because of the re-closure of the disconnector.If there are no normally open paths for the load point to be used, the unavail-

ability is given by

U c1lp =

n∑

i=1

(λp,i · rr/p,i + λte,i · rc,i) (3.18)

The restoration time of first order failures are given by

rc1lp =

U c1lp

λc1lp

(3.19)

Minimal Cut Sets of Second Order

The failure rate from the overlapping failure events are evaluated from the secondorder minimal cut sets vector as [1]

λc2lp = λpp + λpm + λpte + λtete + λtem (3.20)

For the two overlapping permanent failures, the Equations 3.5 to 3.7 are used[1]:

λpp = λp1(λ

p2r

p1) + λp

2(λp1r

p2) (3.21)

rpp =rp1r

p2

rp1 + rp

2

(3.22)

Upp = λpprpp (3.23)

For the terms λpm and λtem, with a maintenance outage followed by an perma-nent or a temporary failure, Equation 3.10 and 3.11 are used [1]:

λpm = λm1 (λp

2rm1 ) + λm

2 (λp1r

m2 ) (3.24)

rpm =λp

1(λm2 rp

1)λpm

12

· rp1r

m2

rp1 + rm

2

+λp

2(λm1 rp

2)λpm

12

· rp2r

m1

rp2 + rm

1

(3.25)


and the unavailability is

Upm = λpmrpm (3.26)

The same equations are used for λtem.For the term λpte, with a temporary failure and a permanent failure overlapping,

the Equations 3.8 and 3.9 are used [1]:

λpte = λp1(λ

te2 rp

1) + λp2(λ

te1 rp

2) + λte1 (λp

2rte1 ) + λte

2 (λp1r

te2 ) (3.27)

rpte =λp

1(λte2 rp

1)λpte

12

· rp1r

te2

rp1 + rte

2

+λp

2(λte1 rp

2)λpte

12

· rp2r

te1

rp2 + rte

1

+

λte1 (λp

2rte1 )

λpte12

· rp1r

te2

rp1 + rte

2

+λte

2 (λp1r

te2 )

λpte12

· rp2r

te1

rp2 + rte

1

(3.28)

and the unavailability is

Upte = λpterpte (3.29)

If there is a normally open path for the load point, the unavailability for thesecond order failures are defined as

U c2lp = λpp · r′pp + λpm · r′pm + λpte · r′pte + λtete · r′rc + λtem · r′tem, (3.30)

and where the r′xy are the combined restoration time

r′rc = (1− P ) · rs + P · rxy,i. (3.31)

As in Equation 3.17, P is the probability for the open path to be functioning.If there are no normally open path for the load point, the unavailability are

defined as

U c2lp = λpp · rpp + λpm · rpm + λpte · rpte + λtete · rc + λtem · rtem. (3.32)

The restoration time for the second order minimal cut sets are evaluated by:

rc2lp =

U c2lp

λc2lp

(3.33)

Additional active Failures of first order

The contribution from the additional active failures are evaluated as [1]

λa1lp =

n∑

i=1

(λa,i + λte,i + λtr,i) (3.34)

Ua1lp =

n∑

i=1

(λa,i · rs,i + λte,i · rc,i) (3.35)

ra1lp =

Ua1lp

λa1lp

, (3.36)

where i is the component number that cause the active failure and n is the totalnumber of active failures for the load point.


Additional active Failures with Stuck probability

If an additional active failure occurs in a component i, its associated breakers will,if functioning, isolate the failure. But there is a probability, Ps,i, for the non-functioning of a breaker or a fuse, when it is stuck and can not open the circuit. Thefollowing equations are used to evaluate the contribution from the stuck probabilityin n active failures:

λaslp =

n∑

i=1

λa,i · Ps,i (3.37)

Uaslp =

n∑

i=1

λa,i · rs,i (3.38)

raslp =

Uaslp

λaslp

(3.39)

3.4.4 Implemented Method in RADPOW_1999

Figure 3.8 shows the overall data flow between the modules for the implementedanalytical method in RADPOW. In the RADPOW_1999 version, the Loadfile boxand the input data file in the figure are represented by eight separate text files,which is described in Section 3.6.3. The algorithm for the evaluation of the systemand load point indices proceeds the following steps [1]:

1. The system data are transferred from the input data file *.radpow to theLoadfile routine that chops the information in the file into ten different sec-tions.

2. Depending on the given data section, the modules Netw, Branch and Compreads the data into data containers that are accessible for the other modules.

3. The Minpath module deduces the minimal paths for all load points.

4. For each component the associated breakers or fuses are deduced by themodule Abreak.

5. Minimal cut sets are deduced with the Mincut module.

6. Additional active failures are deduced with the Aafail module.

7. The load point indices are evaluated with the Lpind module and saved asoutput data.

8. For each load point that is going to be analyzed the steps 5 to 7 are per-formed.

9. The system indices are evaluated with the Sind module and with the loadpoint indices and the customer data as input data.


Mincut Aafail

Lpind

Minpath

Sind

NetwBranchComp

Data file*.radpow

Abreak

Loadfile

Loadflow

Output data

Radpow.cpp

RADPOW_2006

Figure 3.8: Data flow between the modules in RADPOW [1].

3.4.5 Modules

RADPOW has been developed in modules that are used to make the program codelogical and more understandable [1][13]. In reality a module consist of two files inC++, an cpp-file and a h-file. In the later one, the variables and methods are statedand these are then implemented further in the cpp-file. The advantage of workingwith modules is the ease of expanding the program with other functionalities or touse the modules separately in other projects. The main characteristic for a modulein RADPOW is its independently to other modules and files involved.

Branch, Comp and Netw

The three modules Branch, Comp (Component) and Netw (Network) have beendeveloped to read network and component data from an input text file and then storeit in internal data containers which easily can be accessed by the other modules.


This simplifies the work for the other modules, in the way that there is no need toimplement methods for reading data from text files in each module. The number ofinput files in RADPOW_1999 and RADPOW_1999_PF were eight respective tenseparate text files, but in this thesis an effort has been made to gather these files intoone file containing all data for the system. The work of developing a filesystem andhow the text file is defined is described in Section 3.6.3. The modules are furtherdescribed in [13].

The data processed by these three modules, which defines the system networkmodell, can be separated into four different types [1]:

1. Component data - Includes component reliability data, repair times etc.

2. Network Topology - Describes the interconnections between the differentcomponents.

3. Customer data - Describes the loads and number of customers in each loadpoint.

4. Load flow - Data for restrictions in power flow levels to load points.

Minpath

The module Minpath (Minimal paths) deduces the data given in the network topol-ogy into minimal paths from supply points to each load point in terms of compo-nent numbers. For the definition of minimal paths see Definition 3.3. As shownin Figure 3.9, the algorithm also indicates whereas a specific minimal path is nor-mally closed (n/c) or normally open (n/o). The method and algorithm for obtainingthe minimal paths are described in [13].

MinpathOutputInputNetw

BranchComp

Mincut

Aafail

Testsystem 1aLP5n/c [1 7 3 9 15 10 5]n/c [1 7 3 11 16 12 5]n/o [2 8 4 13 17 14 6 18 5]LP6n/c [2 8 4 13 17 14 6]n/o [1 7 3 9 15 10 5 18 6]n/o [1 7 3 11 16 12 5 18 6]

Testsystem 1bLP5n/c [1 7 3 9 15 10 5]n/c [1 7 3 11 16 12 5]n/c [2 8 4 13 17 14 6 18 5]LP6n/c [2 8 4 13 17 14 6]n/c [1 7 3 9 15 10 5 18 6]n/c [1 7 3 11 16 12 5 18 6]

Figure 3.9: The Minpath module deduces all the minimal paths for each load point andindicates if it is normally closed (n/c) or normally open (n/o).

Mincut

The module Mincut (Minimal cut set) deduces the minimal cut sets of first andsecond order for each load point in terms of component numbers. For the definition


of minimal cut set see Definition 3.1. Figure 3.10 shows how the module uses theminimal paths given from the Minpath module and branch data from the Branchmodule as input. The output from the module is for each load point a vector withthe minimal cut sets, a vector with normally open minimal paths (see Definition3.4) and the present load point number. The algorithm is described in [1] and itsimplementation in [8].

MincutOutputInput

NetwBranchComp

Aafail

Lpind

Testsystem 1aLP5Minimal cut set vector[1, 7, 3, 5, 9+11, 9+16, 9+12, 15+11, 15+16, 15+12, 10+11, 10+16, 10+12]Normally open path vector[2 8 4 13 17 14 6 18 5]

LP6[2, 8, 4, 13, 17, 14, 6]Normally open path vectors[1 7 3 9 15 10 5 18 6][1 7 3 11 16 12 5 18 6]

Minpath

Testsystem 1bLP5Minimal cut set vector[5, 1+18, 1+6, 1+14, 1+17, 1+13, 1+4, 1+8, 1+2, 7+18, 7+6, 7+14, 7+17, 7+13, 7+4, 7+8, 7+2, 3+18, 3+6, 3+14, 3+17, 3+13, 3+4, 3+8, 3+2]Normally open path vectorn/a

LP6[6, 2+18, 2+5, 2+3, 2+7, 2+1, 8+18, 8+5, 8+3, 8+7, 8+1, 4+18, 4+5, 4+3, 4+7, 4+1, 13+18, 13+5, 13+3, 13+7, 13+1, 17+18, 17+5, 17+3, 17+7, 17+1, 14+18, 14+5, 14+3, 14+7, 14+1]Normally open path vectorn/a

Figure 3.10: With the minimal paths for each load point, the Mincut module deduces theminimal cut sets of first and second order for each load point.

Abreak

If a component fail the closest breakers or fuses, if such exists, will trip and isolatethe component and thereby the failure. The module Abreak (Associated breakers)deduces which breakers or fuses that are associated to a specific component in thesystem. The input data required by the algorithm includes the minimal paths toall load points, network topology data for the system and component data. Theoutput is a list with all components, each with a vector containing the associatedbreakers and fuses for the component. The algorithm is described in [1] and itsimplementation in [8].

Aafail

The module Aafail (additional active failures) deduces the additional active failuremodes for each load point. These are components that are not included in thenormal minimal cut set for the specific load point, but still if not functioning, causethe break down of the load point ,as defined in Definition 2.13. The algorithm isdescribed in [1] and its implementation in [8]. There are two types of additionalactive failures defined in RADPOW [1]:


AbreakOutputInput

Minpath Aafail

Component123456789

101112131415161718

Testsystem 1a[7][8]

[7, 9, 11][8, 13]

[10, 12][14]

[9, 11][13]

[7, 10, 11 ][9, 12]

[7, 9, 12][10, 11][8, 14]

[13][9, 10]

[11, 12][13, 14]

[ ]

Testsystem 1b[7][8]

[7, 9, 11][8, 13]

[10, 12, 14][10, 12, 14]

[9, 11][13]

[7, 10, 11 ][9, 12, 14][7, 9, 12]

[10, 11, 14][8, 14]

[10, 12, 13][9, 10]

[11, 12][13, 14]

[10, 12, 14]

Figure 3.11: The Abreak module deduces each component’s associated breaker which, iffunctioning, will trip if the component suffers a fault.

• First order - Failure modes caused by a failed component and the trip of itsassociated breakers or fuses.

• Second order - Failure modes caused by two failures. First a componentfails and then its associated protection device is stuck and fails to isolate thefailure.

AafailOutputInputNetw

BranchComp

Abreak

Lpind

Mincut

Testsystem 1aLP5Additonal active failures[9, 10, 11, 12, 15+9, 15+10, 16+11, 16+12]

LP6Additonal active failures-

Testsystem 1bLP5Additonal active failures[6, 10, 12, 14, 18, 15+10, 16+12, 17+14, 13+14, 9+10, 11+12]

LP6Additonal active failures[5, 10, 12, 14, 18, 15+10, 16+12, 17+14, 13+14, 9+10, 11+12]

Figure 3.12: The Aafail module deduces the additional active failures for each load point.

Lpind

The Lpind (load point indices) evaluates the the reliability indices for each loadpoint using the equations described in Section 3.4.3. The input data for each loadpoint to the module is the minimal cut set vectors, the additional active failures


vectors and the component reliability data. The results of the evolution are pre-sented as the load point indices described in Section 2.4.1. Figure 3.13 shows thedata flow to and from the Lpind module. The module is described in [1] and itsimplementation in [8].

LpindOutput

SindInputNetw

BranchComp

Aafail

MincutFor each load point i,Load Point Indices:• lpi [f/yr] , expected failure rate per year• Ulpi [h/yr], the annual unavailability in hours per year• rlpi [h/f], expected outage duration for a failure• Llpi [kW], the average load• LOElpi [kWh/yr], the average loss of energy per year

Figure 3.13: The Lpind module evaluates the reliability indices for each load point sepa-rately.

Sind

Given the indices for each load point, the Sind (system indices) module evaluatesthe system indices for the power distribution system. These system indices aredefined in Section 2.4.2. The method and algorithm are described in [13].

SindOutput

Input

Lpind

NetwBranchComp

Loadflow

Optional System Indices:• SAIFI [int/yr and cust.]• SAIDI [h/yr and cust.]• CAIDI [h/int.]• AENS [kWh/yr and cust.]• ASAI

Figure 3.14: The Sind module evaluates the system indices.

3.5 RADPOW_1999_PF version

The RADPOW_1999_PF version has the ability to perform load flow calculationsand has the extension of one new module, Loadflow, and two new input data filespflo and pfrx. In the main file for this version there is an user defined parameter


that makes it possible to turn off the load flow calculations. By doing this RAD-POW_1999_PF works exactly as the RADPOW_1999 version.

3.5.1 Loadflow module

The Loadflow (load flow) module performs load flow analysis for each failure andload point to deduce if any still functioning branches in the system are overloadedand can not be used. Input data to the module are branch and component data,voltage levels in per unit (p.u) at buses and the minimal cut sets for each loadpoint. The results from the module are the changes in reliability for the specifiedload point, and these are then added to the results from Lpind [14]. The running ofthis module is optional, which is the reason why this block is drawn with brokenline in Figures 3.8 and 3.14.

Difficulties of translating functions and methods from the old C++ standard,led to errors when running RADPOW with this module. Due to these errors theLoadflow module was not used in this thesis. In Section 3.6.8 this module’s iden-tified weaknesses that were found during this project are described. The Loadflowmodule is described in detail in [14].

3.6 RADPOW_2006 version

The RADPOW_2006 version has been developed by the author and contains thesame core of RADPOW as the RADPOW_1999 and the RADPOW_1999_PF ver-sions. The extensions is a graphical user interface, a new filesystem and two newmethods of calculating the system indices, described in Chapters 4 and 5. One ofthese methods, the Monte Carlo Simulation (MCS) method, has been placed in astand alone module referred to as Sim. The other routines that has been developedare referred to as files or routines because these does not fulfill the requirement ofgenerality that a module has to fulfill.

3.6.1 Simulation Method

If the system model gets to complex it may be difficult or even impossible to de-scribe the system analytical with mathematical equations, without making largeapproximations. In these cases a computer simulation can be preferable. In allsimulation methods the model is tested with a series of experiments to see how itreacts on different events. The result from the experiments are then collected andevaluated. Normally it takes a large number of experiments to find solutions of aproblem, and that is why this method is time consuming.

When performing a simulation over time, stochastic samples from a givenprobability distribution are used to create different events. This simulation processare referred to as stochastic simulation and this is actually a statistical samplingexperiment with the model [11]. Because one of the central problems in stochas-tic simulation is how to generate random numbers from different distributions, the


simulation method are commonly referred to as Monte Carlo Simulation (MCS)[1].

3.6.2 Monte Carlo Method

The term "Monte Carlo" was first introduced during World War II as a secret codefor a project involved in the development of the atomic bomb. The name comesfrom the gamling casinos at the city of Monte Carlo in Monaco [11].

There are generally two ways of performing MCS; random or sequential simu-lation. In the random approach the time intervalls are chosen randomly, while as inthe sequential the intervalls are chosen in chronological order. In reliability studiesit is most likely that one time intervall depends on previous ones, and therefor thesequential simulation method normally is adapted and also used in this thesis.

As mention before one of the key issues in MCS is how to generate differentrandom events in order to perform a simulation of the model. The times for thedifferent random events need to be generated and for this an appropriate randomnumber generator has to be used.

Generating Random Numbers

The generation of random numbers is the most essential aspect in Monte Carlosimulation. In order to make a realistic simulation of the system, all states of thesystem needs to be performed, and this means adequate input of random events.

When it come to generate random numbers there are both software and hard-ware generators available. The algorithms that these uses are divided into determin-istic and non-deterministic methods. The non-deterministic algorithms are usuallycalled pseudo-random generators, which is the algorithm used in this thesis. Thename pseudo reveals that this is not a true random generator mathematically, andthat is because it repeat itself after a large sequence of numbers. The generatoritself are initiated with a number, a seed, which sets the starting point of the se-quence. The same initiating seed at two different occasions will always give thesame sequence of numbers and therefor the seed also has to be random. One wayof solving this problem is to initiate the generator with the current time in sec-onds, which is easy to implement. A adequate random generator has to fulfil thefollowing aspects:

• It has to represent the chosen distribution, which normally is the uniformdistribution.

• A random number should not be correlated to previous ones.

• It should not occur any trends or periodicity in a sequence of numbers.

If these aspects are fulfilled and the the sequence of numbers before it repeat itselfare large enough, the pseudo-random generator are ideal for MCS because of itsspeed in calculation.


Random generator algorithms normally produces uniform distributed numbersin the interval (0, 1). These uniform distributed numbers can then be transformedto represent an arbitrary distribution as described below [12].

Generating Random Numbers from arbitrary Distributions

If an inverse, F−1Y (t), exists to the distribution function FY (t), the inverse trans-

form method can be used to generate random numbers with the distribution FY (t).Figure 3.15 shows how the uniform distributed numbers X ∈ U(0, 1) are trans-formed to the distribution FY . For the exponential distributed function the inverseexists as showed in Equation 2.16, and hence the inverse transform method can beused. But for other, more complicated functions, as the normal distributed function,there are no analytical way of solving the inverse and therefor an iterative methodof generating random numbers has to be used, as described in Section 2.5.2.

X~U(0,1)

Generate a uniform random number

Y=F-1y(X)

0

1

x

y

Transform with F-1y

Y~Fy(t)

Random number with the new distribution

Figure 3.15: Inverse transform method is used to generate random numbers of arbitrarydistribution.

3.6.3 New files

The graphical user interface has been implemented in one main file called Main.In order make the machine code in the program logical and understandable, severalhelp files has been developed within this thesis project. In Figure 3.16 a schematicpicture of the new files are shown. Note that the sim file is identical with the Simmodule. Of the files in the picture, Radpow, Sim and Random are for the calculationmethods, Loadfile for the data input in RADPOW and the others for the graphicaluser interface. The arrows between some of the files in the figure symbolizes thedata flow. The Radpow file represent several different modules as described in

Main.cppProject.cpp

about.cpp

radpow.cpp

ind.cpp info.cpp

plot.cpp

newfile.cpp

random.cpp

sim.cpp

loadfile.cpp

Figure 3.16: The different modules working together within the graphical interface.


Section 3.4.4 and illustrated in Figure 3.8.

Radpow

The Radpow module contains the actual core of the program RADPOW. It com-bines the different modules described in Section 3.4.5 and has accessors and meth-ods to initiate and perform a calculation of a system. The calculation can either beperformed by an analytical method or by a simulation method. The results returnedfrom this module are

• the system and load point indices and

• extended information about the system, such as the minimal cut sets or theminimal paths.

Sim

The main objective of this thesis has been to develop a module for a simulationapproach for the evaluation of the system indices of a power distribution system.The simulation method and its implementation, the Sim module, are described inChapter 5.

Random

When performing MCS, it is essential to produce random numbers of good quality,as described in Section 3.6.2. The Random file uses the standard C++ randomnumber generator to produce uniform distributed random numbers in the [0..1]interval. The methods described in Section 3.6.2 are then used to produce randomnumbers from the Exponential or Normal distribution, given µ and σ.

The purpose of implementing the random number generator in a stand alonemodule is for the ease of developing the random generator further with more ad-vanced methods of producing random numbers.

File System with Loadfile

In the RADPOW_1999_PF version of RADPOW the system input data were sep-arated in ten different files, each containing different information of the system.The names of these files were defined as text strings in the core of RADPOW. Inaddition, information of these text files needed to be entered for the internal datastructure in RADPOW, such as the number of load points, number of components,number of supply points and the number of branches. If a new system was goingto be implemented or changed in the text files, the core of RADPOW had to berecompiled in order to perform the calculations. To overcome this, the file Loadfilewere developed by the author. This routine contributes with the following abilities:


• All the input data from the ten data files are merged into one single file.This makes it easier for the user when a system is defined and handled inRADPOW.

• The Loadfile evaluate the size of the system with the given input data file. Byadding this ability, there is no need to recompile the source code of RAD-POW each time the system is changed.

The ten different data files used in RADPOW_1999_PF has simply been pastedinto one file, each into different sections. Each section consists of a header con-taining the name of the section, followed by the specific data. The data input file forRADPOW and its ten sections with the data parameters are described in AppendixA.

3.6.4 New files for the Graphical User Interface

The graphical user interface has been made in Borland C++ Builder version 6.0(Build 10.161) for Windows. Figure 3.17 shows the start window of RADPOW.Figure 3.16 includes the files for the graphical user interface.

Figure 3.17: The graphical user interface in RADPOW for Windows.

Main and Project

The Project file starts and initiates the Main file window. The code in this file iscompletely generated by Borland C++.


The Main file contains methods that executes when an action is made by theuser, which normally is a press on a button. The different actions starts one of thefollowing methods that may use other files or modules:

• Analytical calculation of the system indices in the file Radpow, after thesystem has been initiated.

• Simulation of a system in the file Radpow in order to determine the systemindices and its deviations.

• Iterative analytical calculations with a disturbance in the input data in orderto determine the deviation of the system indices. This is performed with thefile Radpow.

• Graphical presentation of the results of an iterative calculation or an simula-tion with the Plot file.

• Presentation of the system and load point indices with the Ind file.

• Extended information of the system, with the minimal cut sets and minimalpaths, presented with the Info file.

• Open procedure of a file and initiation of a system in the file Radpow.

Ind

The Ind file presents the system and load point indices from the latest calculationin a graphical window. The used indices are the same as the ones described inSection 2.4.2.

Info

Given a system that has been initiated or calculated, the Info file shows a graph-ical window with extended information of the system. The window opens withthe button Extended Info in RADPOW. The information presented in this windowincludes

• minimal cut sets of first and second order,

• additional active failures of first and second order and

• minimal paths for each load point.

The information given can be used in educationally purposes or to find out if thesystem has been defined correctly in the input data file for the system.


Plot

Given a number of consecutive calculations of the system indices, the Plot file sortsthe results into intervalls and counts the hits in each interval for each index. Thehits in these intervals are then plotted in a graph in the plot window. A algorithmdetermines a predefined interval length, which can be changed by the user. Theuser can also chose to save the plot as an image file with Save as in the File menu.

Newfile

In order to convert the input data files from RADPOW_1999_PF to the new versionRADPOW_2006, the file Newfile has been developed. As described earlier in thissection the system data in RADPOW_1999_PF were separated into ten differentfiles. The file Newfile contains a window that lets the user define which files thatare going to be merged into one single file that are appropriate for RADPOW_2006.This conversion window is reachable from the Import files in the File menu.

About

The file About contains a window that, if opened, shows the name of the authorsof RADPOW and contact information. The window opens with About in the Helpmenu.

3.6.5 Identified Weaknesses and Corrections

When working with the development of the simulation module in RADPOW, theoutput data from each module were checked for different test systems. In this worka number of failures and weaknesses in the program were found and corrected anda list of these are listed below.

3.6.6 Minimal Cuts of second order

The minimal cut sets are deduced from the minimal paths for each load point. Forlarger systems like the Birka System, described in Section 6.3, the conversion fromminimal paths to minimal cut sets of second order was incorrect. The bug causingthis problem was found in the method mc2vf in the Mincut module. The causeand correction in the code are described in [16].

3.6.7 Additional Active Failures

The module Aafail deduces additional failures that occurs due to tripping of break-ers as described in Section 3.4.5. Figure 3.18 shows a simple test system with twoload points that is used to illustrate the upcoming weakness. If an active failureoccurs in the components c5 or c6 the breaker c3 will trip and both the load points


will be out of power. This scenario is correct modeled in RADPOW, as the addi-tional active failures from the module Aafail are c5 and c6 for LP4. The outputresults for this test system and with RADPOW are correct after a validation. Nowsuppose that the breaker c3 is removed from the system. Due to the lack of break-ers in the system, no additional active failures are generated and this means thatsome of the failure modes are not considered in the calculation of the model. Ifc5 is affected by an short circuit this will not only cause an outage in LP6, as inRADPOW, but also in LP4. This weakness is normally not a problem as breakersnormally exists as in Figure 3.18 or close to the supply point.

c2c3

LP4

c4c1

c5

LP6

c6

Figure 3.18: A simple test system to illustrate a found weakness in RADPOW that occurswhen no breakers are used.

3.6.8 Loadflow

The Loadflow module was developed in an old C++ version which were not com-patible with the C++ standard. Even if a quite large effort has been made in thisthesis to convert the code to the C++ standard, the module did not work properlyfor larger system as singularities occurred. The module was not used in the analysisin this thesis.

3.6.9 Data File

During the analysis studie of the Birka System in this thesis, described in Chapter 6,a weakness in RADPOW were found. If a component type is defined for a specificcomponent in the ctype-section, eg. BR220, but this type is missing in the cerelia-section, RADPOW does not rise a warning to the user, instead it uses an arbitrarycomponent type resulting in incorrectness of the results.

The input data file is described further in Appendix A.

Chapter 4

Sensitivity analysis routine

This chapter describes the analytical sensitivity analysis routine that were imple-mented for RADPOW by the author. The method and its implementation in RAD-POW are then validated in Chapter 6.

4.1 Introduction

A major part of reliability analysis of power systems is the acquire of accuratecomponent reliability and system data. If reliability data for the system and itscomponents are available, it is normally uncertainties in this data due to measuringuncertainties, small population of similar components or poor documentation rou-tines. Sometimes reliability data for a certain component does not exists at all andthen standard values from tables with perhaps large uncertainties has to be used.All these uncertainties in input data affects the output results, which for RADPOWis the load point and system indices. In order to derive these uncertainties in theoutput results a sensitivity analysis, giving the expected deviations, can be obtain.

4.2 Sensitivity analysis with random disturbance

The deviation of the system indices has in this thesis been studied by performing alarge number of analytical calculations with a randomized input parameters. Thealready existing analytical method in RADPOW has been used for the calculationsin the implemented routine that performs these iterative calculations.

In RADPOW there are a number of different input parameters that could bestudied in a sensitivity analysis. These input parameters can be divided into threemajor categories [1]:

1. Component reliability data including failure rates, restoration times, mainte-nance rates and maintenance outage times.

2. Customer and power data for each load point.

43

44 Chapter 4. Sensitivity analysis routine

3. Power flow data between the buses in the system.

In the routine described for the sensitivity analysis in this thesis only categoryone, the parameters for the component reliability, has been studied. These elevencomponent parameters are loaded by RADPOW from the Cerelia-section in theinput data file, described in Appendix A.

4.2.1 Disturbance in Component Data

The normal distribution has been used in this thesis to describe an uncertainty ina measured or statistical evaluated parameter of an item. The normal distributionis described in Section 2.5.2. Figure 4.1 illustrates how each input parameter forthe components are randomly disturbed and then used in the analytical calculation.In order to generate the normal distributed random number in Figure 4.1 the Box-

Expected value of parameter, E[X] = p

Deviation, = E[x] * d

Deviation of parameters for components, d [%]

Generate a normal distributed number

E[X]- +

Value of one parameter for one componentParameter value, p

from input data file

Figure 4.1: For each parameter in all components in the system a random value is generatedbased on the chosen uncertainty or deviation and the value from the input file.

Muller method described in Section 2.5.2 has been applied. This method generatesa random number X ∈ N [0, 1] which with Equation 2.19 is used to generate thegeneralized random number Y with input parameters p and d showed in Figure 4.1.The output random number with these two input parameters can the be calculatedas

Y = µ + σX = p + p · d ·X (4.1)

where Y ∈ N [µ, σ2].

4.2.2 Method and Implementation

The iterative method used for the sensitivity analysis has been implemented in theoverall Main file described in Section 3.6.4. The implemented method uses theanalytical calculation method described in Section 3.4.4, which is implemented inthe Radpow file. Figure 4.2 shows the overall method, with the required input dataand the output results. The overall algorithm to deduce the deviation of the systemindices contains the following steps.

4.2. Sensitivity analysis with random disturbance 45

Generate random

parameters for each

component

Run RADPOWAnalytical method

Store results

#iteration > i

Evaluate results

Deviation in component parameters [%]Input

no

yes

Output

SAIFI, SAIDI, CAIDI, AENS, ASAI

#iterations

Component data

SAIFI SAIDI CAIDI AENS ASAI x x x x x

1

n

, r, sw, ...

- Mean vaules of indices- Max and min- Deviation

++i

Figure 4.2: The general method for the sensitivity analysis, as implemented in RADPOW.

1. Load the component data from the Cerelia-section, specifying the parame-ters for each component type. Get the number of iterations and the deviationfor all parameters from the user.

2. For each parameter in each component, generate the new value of the param-eter, given the specified value in file and the chosen deviation of componentas illustrated in Figure 4.1.

3. Start the analytical method in RADPOW with this specific component con-figuration and calculate the standard system indices.

4. Store the results from this calculation in a matrix.

5. If the present number of cumulative iterations, i, is less than the specified,start over from step 2. If not, stop the iteration process.

46 Chapter 4. Sensitivity analysis routine

6. Evaluate the results from the stored data in the matrix. For each index eval-uate the mean, max and min value and the deviation.

The implemented method in the Main file interacts with the Radpow and Randomfiles during its iteration process. When the iteration process has been finished, thedata is statistical evaluated and the results are then presented to the user either onthe screen or in the file aAnalysis.out. The file is placed in the same folderas RADPOW and contains the five system indices ordered in columns for eachiteration. There is also an option for the user to plot the resulting distributions ofthe different indices with the Plot file, described in Section 3.6.4.

Chapter 5

Monte Carlo Simulation Methodfor RADPOW

This chapter describes the simulation method developed and implemented in RAD-POW by the author. The method and its implementation are then validated in Chap-ter 6 by the output results for two different systems.

5.1 Introduction

If the model of a power distribution system gets too complex it may be difficultor even impossible to describe the system analytical with mathematical equations,without making large approximations. In these cases a computer simulation can beadapted. In all simulation methods the model is tested with a series of experimentsto see how it reacts on different events. The result from the experiments are thencollected and evaluated. Normally it takes a large number of experiments to findsolutions of a problem, and that is why this method is time consuming.

In order to develop a simulation method for RADPOW the Monte Carlo Sim-ulation (MCS) technique has been adopted and implemented in a new module re-ferred to as the Sim module.

5.2 Simulation Method

5.2.1 Component states

In order to make a simulation analysis on a power distribution system, the compo-nents involved in the system needs to be studied. As the system consists of severalinterconnected components, each having a probability to fail, one has to deducehow a specific component affect the system and its load points, given the status ofthe component. In the simulation method in this thesis, each component has beenassigned an integer defining the present status of the component. Three differentcomponent states are used:

47

48 Chapter 5. Monte Carlo Simulation Method for RADPOW

State 0 - The component is functioning.

State 1 - The component suffers an active failure.

State 2 - The component is being repaired or replaced.

At the first state the component is functioning, but it have already been donated atime to failure (TTF), which will affect the component in the future. When the TTFhas been reached, the component suffers an active failure and needs to be discon-nected in order to restore the component. According to the definitions in Section2.2.2, this operation takes the time rs before the component has been switched off.Then the component is repaired ,or , if possible, replaced during the time rr and rp

respectively.Only active failures have been considered, and hence each component follows

the sequence 0-1-2-0-1.., if it is assumed that all components are functioning fromthe beginning.

Network Model

System Data

Assign each LPs the events that lead to failure

for that LP

Calculate the reliability indices for each LP with

formulas

Calculate the reliability for the system

Make a large number of random experiments to see

how these affect LPs reliability

Simulation methodAnalytical method

Figure 5.1: Flow chart for the analytical and simulation method used in this thesis.

5.2.2 Event-driven approach

As described in Section 3.3, the analytical method in RADPOW uses a load-point-driven approach, which means that all possible failures for each load point arededuced separately. This deduction results in the minimal cut set of first and secondorder and the additional active failures (see Definition 2.13), which are available foreach load point. The simulation method developed for RADPOW uses an event-driven approach, which means that all failure events are treated separately to seethe effect of the failure on the whole system by identifying the affected load points

5.2. Simulation Method 49

[1]. This means when a component fails, the method has to deduce which loadpoints that are affected, and this is achieved by the already deduced minimal cutsets and the additional active failures for each load point. Figure 5.1, from Chapter2, shows how the simulation method interacts with the already developed analyticalmethod in RADPOW.

Depending on the component state described previously in Section 5.2.1, afailed component with state 1 or 2 will affect the load point differently. If a com-ponent is in state 1 it affects all the load points having this component included inits minimal cut sets of first order or in the additional active failures. On the otherhand, if the component is in state 2, it only affects the load points having this com-ponent in its minimal cut sets. For the second order failures, both state 1 and 2 foreach of the two components will affect the load points having these components inits minimal cut sets of second order.

5.2.3 Algorithm

The MCS algorithm used in this thesis is described in Figure 5.2. The algorithmproceeds the following steps, as numbered in Figure 5.2.

1. The input data consist of the number of samples (N ), simulation time(Tstop), component reliability data for the system, the minimal cut set vec-tors, normally open paths and the additional active failure vectors for eachload point (LP). The present iteration number, n, is set to n = 0.

2. All components are set to be functioning, which means state 0. The total time(Ttot) is set to 0. The LP:s number of failures and outage times are reset.Then generate a time to failure (TTF), in years, for each component usingthe exponential random generator and with the component data as input.

3. Go to the next event; that is the event with the shortest time and the intervalis donated ∆t. During this intervall the system is stable. Count up the totaltime (Ttot) with ∆t and decrease all the times for the components in thesystem with ∆t.

4. Check how the LP:s were affected during this interval and with the currentstates of the components. Use the minimal cut sets and the additional activefailure vectors for this purpose.

5. If a LP is affected, check if there is any alternative paths, still functioning,that can be used by closing a normally open disconnector. If this is possiblecount up the outage time for the LP with the switching time for the discon-nector. If normally open paths do not exists, add the total interval lengthto the outage time for the LP. Then check if the LP suffered an outage theinterval before and if not increase the number of failures for this LP.

6. At the event, the present component is changing its component state to anew one (depending on its previous). The state for the component follows


2. All components are functioning.

Generate time to failure for each component

0

years

TTF3 TTF10 TTF8 TTF1

Input data

3. Jump to the next event

Ttot = Ttot + t

0

years


Ttot = 0

Ttot = Ttot + tt

4. How are the LPs affected during the time

interval and with the current component

status?

1. Check if any components with status 1 are included in the additional active failures of the LP.

2. Check if any components with status 1 or 2 are included in the minimal cut sets of first and second order of the LP.

3. Check if there is any open paths that can be closed with disconnectors.

5. If LP is affected:1.Add a failure to LP (if functioning before).2.Add the outage time t to LP

6.Update the component status after the interval. Produce a new time for component, depending on next status 0, 1 or 2.

8. Ttot > Tstop ?

7a. New TTF is generated

7b. SW-time from data

7c. Rep-time from data

0 1 2

9. n == N ?

0

years


SW3

Current time

Current time

Current time

Stop time

yes

no

yesno 10. Evaluate the

outages for each load point and each iteration

n = n +1

Figure 5.2: The overall algorithm used in the simulation method in RADPOW.

the sequence 0, 1, 2, 0, 1..., as described in Section 5.2.1, and hence it is easyto determine the next state.

7a. If the component were in state 2 in the interval, it is now functioning, andhence its new state is set to 0. A new random TTF for the component isgenerated.

7b. If the component were in state 0, its new state is set to 1. The time forisolating the component, the switching time, is set to the present static valuefrom the input data.

7c. If the component were in state 1, its new state is set to 2. The time for repairor replacement of the component, is set to the present static value from theinput data. It has been assumed that the component always is replaced if thisoption is available.

5.3. Implementation in RADPOW 51

8. If the time for simulation, Tstop, is reached, stop the current simulation andsave the outage data for each LP. If not, proceed at step 3.

9. If the current number of simulated samples, n, equals the predeterminedsamples, N , the total simulation phase is finished. If not, restart from step 2and count up n by one.

10. Evaluate the data from all samples, with the system indices and its standarddeviations as results.

The output results consist of the load point and system indices for the system.These are for each sample evaluated with the formulas presented in Section 2.4 andthen the statistical data are evaluated for all the samples, with the mean values andvariances of the samples.

The chosen simulation time in a simulation analysis of a power system dependson the complexity of the system and the required accuracy of the output results. Inthe proposed method the total simulation time consists of the chosen number ofsimulation samples and the sample time in years. In order to determine these inputparameters one has to see how the output results converges for the specific systemwith different sample lengths in years. These parameters depends on the size ofthe system and the reliability data for the components; the smaller probabilitiesfor the components to fail, the longer sample times are needed to comprehend thesystem behavior. If some events happens very occasionally, but have a large impacton the system, a large number of samples or a long sample time is needed for thesimulation. Systems having this property are referred to as duogen systems [12][9].

5.3 Implementation in RADPOW

The simulation method in RADPOW has been implemented in the new moduleSim. The Sim module works together with the other modules in RADPOW asshown in the flow chart in Figure 5.3. The minimal cut sets and the additionalactive failures are deduced in the same way as in the analytical method describedin Section 3.4.4. A comparison of the flow chart in Figure 5.2 and the flow chart forthe analytical method in Figure 3.8, shows that the only differences is the exchangeof the Lpind module with the Sim module. The input data to the Sim module isbesides the minimal cut sets and the additional active failures, also the normallyopen paths and the component reliability data. The output data from the moduleare then delivered either directly to the user as load point indies, or to the Sindmodule for evaluation of the system indices.


Mincut Aafail

Sim

Minpath

Sind

NetwBranchComp

Data file*.radpow

Abreak

Loadfile

Output data

Radpow.cpp

#iterations = NSim time = TstopInput data

Figure 5.3: The Sim module with the simulation method as implemented in RADPOW.

5.4 Approximations and Weaknesses in Method

Simplifications in the developed method has been adopted due to time limitationsfor the master thesis project. These simplifications has been listed below.

1. Passive faults are not included in the method.

2. Temporary and transient failures are not considered.

3. Outages caused by maintenance followed by an overlapping failure are notconsidered.

4. The non functioning of breakers, the stuck probability, are not considered.

5.4. Approximations and Weaknesses in Method 53

5. The function of fuses are not implemented in the method.

6. The switching time for the normally open disconnector has been set to onehour.

Besides these limits in the method, there are only failure modes of first and secondorder included in the model as the minimal cut sets do not include failure modes ofhigher order.

Chapter 6

Comparative Studie of theMethods

This chapter validates the results from the sensitivity analysis routine and the sim-ulation method, by comparing the results from two different test systems and com-puter analysis program.

6.1 Introduction

In order to validate the results from the developed simulation method and the sen-sitivity analysis routine for RADPOW, two test systems have been used. Thesesystems are then evaluated with different methods and computer programs to com-pare and validate the results.

First the two test systems, referred to as Test System 1 and the Birka Systemare presented in detail with the necessary data. These both systems has been im-plemented and analyzed in RADPOW and in the commercial tool NEPLAN. NE-PLAN is an electric power analyzer which has been developed by the BCP groupin Switzerland. This software package is used mainly for transmission and distri-bution system analysis [18].

The analys work and implementation in NEPLAN has been performed in com-panion with Shima Mousavi Gargari and is described further in [18].

6.2 Test System 1

Figure 6.1 shows Test System 1 that has been used to compare the results from thevarious methods in this thesis. This system is the same as presented in Section2.3 and origins from [1] and [8]. Test System 1 is divided into Test System 1aand Test System 1b, with the only difference that the first is considered to have anormally open disconnector in c18, and the later a closed point in c18. The systemis described in detail in Section 2.3.

55

56 Chapter 6. Comparative Studie of the Methods

c2 c4

c8 c13c17

c14

c18

c10

c12

c9

c11c16

c15

c7

c1 c3 c5

LP5

LP6

c6

B2 B5

B1

B4

B3

B6

Figure 6.1: Test System 1, with components c, and branches B [1].

6.2.1 Load Point Data

Table 6.1 specifies the customer and load point data for the two load points. Ad-ditional data such as the customer type; industrial, residential or commercial havenot been included in this table because these data do not affect the results in thisstudie.

Table 6.1: Customer and power data for the two load points in Test System 1 [8].

Load point Number of Active powercustomers [kW/cust]

LP5 100 4.0LP6 80 5.0

6.2.2 Component Reliability Data

Table 6.2 presents the component reliability data for Test System 1. Changes withthe source data in [8] and the data presented here has been made and these are asfollows:

1. All the permanent failures have been considered to be active and conse-quently the active failure rate has been set to the same value as the per-manent.

2. The replacement of components is not considered.

3. The stuck probability for the breakers is set to zero and is thereby not con-sidered.

6.3. Birka System 57

4. The temporary failure rates for all components have been set to zero.

5. The intensity for maintenance has been set to zero for all the components.

Table 6.2: Component reliability data for Test System 1. Source [8].

Component Type λpermanent[f/yr] λactive[f/yr] rrepair[h] rreplace[h]1 Bus 0.001 0.001 2 -2 Bus 0.001 0.001 2 -3 Bus 0.001 0.001 2 -4 Bus 0.001 0.001 2 -5 Bus 0.001 0.001 2 -6 Bus 0.001 0.001 2 -7 Break 0.020 0.020 24 -8 Break 0.020 0.020 24 -9 Break 0.020 0.020 24 -10 Break 0.020 0.020 24 -11 Break 0.020 0.020 24 -12 Break 0.020 0.020 24 -13 Break 0.020 0.020 24 -14 Break 0.020 0.020 24 -15 Transf 0.015 0.015 15 -16 Transf 0.015 0.015 15 -17 Transf 0.015 0.015 15 -18 Discon 0.002 0.002 4 -

6.2.3 Input Data File

Appendix B shows the input data file for Test System 1. The file presented in thisappendix includes the changes made in the component reliability data, as describedin the previous section. The structure of this file is described in Appendix A.

6.3 Birka System

The Birka System is a model for a part of the Stockholm City distribution systembelonging to Fortum (previously by Birka Energy). The system model was firstpresented by [1] in a maintenance and reliability studie with the RCM methodol-ogy. Figure 6.2 shows a network model for the Birka System, including one supplypoint and three load points. The sources of input data for the model and estimationsand approximations that has been undertaken are described in [1], pages 198-209.In this thesis I have chosen to just present the model, and not the real distributionsystem where it has been deduced from.


c1

c3 c9

c11c5

c6 c12

c14

c20c16 c24

c27

c17 c21 c25

c31c30

c29

c32

c33

c35

c34

c54c51

c49

LP35

c56

c58

LP58

c11c38

c36

c46

c48

LP48

c41

c50 c53

c52 c55

c57 c47

c37 c40 c43

c39 c42 c45

c28

c22 c26

c23c19

c7 c13

c2 c8

c4 c10

transformer breakerbus

supply point

load point

line

c15

c18

c44

fuseB1 B2

B15

B16 B17

B18

B3 B4 B5

B6

B7 B8

B9

B10

B11 B12 B13

B14

Figure 6.2: The Birka System, a model for a part of the Stockholm city distribution system[1].

6.3. Birka System 59

6.3.1 Load Point Data

The customer and load point data for the three load points are specified in Table 6.3.Additional data such as the customer type; industrial, residential or commercial hasnot been included in this table because these does not affect the result in this studie.

Table 6.3: Customer and power data for the three load points in the Birka System [1].

Load point Number of Active powercustomers [kW/cust]

LP35 447 1.7203LP48 23400 0.9829LP58 1 0.80

6.3.2 Component Reliability Data

Table 6.4 presents the component reliability data for the Birka System. Changeswith the source data in [1] and the data presented here has been made and these areas follows:

1. The passive failure failure rate has been set to zero for all components. Ac-cording to the definition of permanent failures, λpermanent = λactive +λpassive, this means that the permanent failure rate equals the active.

2. The stuck probability for the breakers has been set to zero.

3. The time for recovery after a temporary fault has been set to zero. Since thetemporary failure rate already is zero, this does not affect the results.

6.3.3 Input Data File for RADPOW

The Birka System data file for RADPOW is included in Appendix B. The structureof this file is described in Appendix A. Changes form earlier versions of the BirkaSystem, used in [1], have been stated in the previous section.

Besides the previous changes in the input data file one major change were madefrom the source file provided by [1]. In the source, the component type BR220was not defined, although two components, c2 and c8, had been specified to thistype. RADPOW does not rise a warning for this, instead the two componentsare specified to an arbitrary component type, which leads to errors in the results.This weakness in RADPOW is described in Section 3.6.5. The solution to thisproblem was to change component c2 and c8 to the component type BR110, whichaccording to [1] has exactly the same component reliability data as BR220.


Table 6.4: Component reliability data for the Birka System [1].

Component Type λpermanent[f/yr] λactive[f/yr] rrepair[h] rreplace [h]1 Bus 0.00964 0.00964 1 -2 Break 0.00870 0.00870 168 243 Transf 0.02610 0.02610 504 244 Break 0.00870 0.00870 168 245 Cable 0.07012 0.07012 168 -6 Transf 0.02050 0.02050 504 247 Break 0.00089 0.00089 72 248 Break 0.00870 0.00870 168 249 Transf 0.02610 0.02610 504 24

10 Break 0.00870 0.00870 168 2411 Cable 0.07031 0.07031 168 -12 Transf 0.02050 0.02050 504 2413 Break 0.00089 0.00089 72 2414 Bus 0.00964 0.00964 1 -15 Break 0.00089 0.00089 72 2416 Cable 0.00028 0.00028 48 -17 Transf 0.01989 0.01989 504 2418 Break 0.00243 0.00243 48 2419 Break 0.00089 0.00089 72 2420 Cable 0.00028 0.00028 48 -21 Transf 0.01989 0.01989 504 2422 Break 0.00243 0.00243 48 2423 Break 0.00089 0.00089 72 2424 Cable 0.00028 0.00028 48 -25 Transf 0.01989 0.01989 504 2426 Break 0.00243 0.00243 48 2427 Bus 0.00867 0.00867 1 -28 Break 0.00243 0.00243 48 2429 Bus 0.0 0.0 0 -30 Cable 0.10069 0.10069 6 -31 Cable 0.10069 0.10069 6 -32 Bus 0.0 0.0 0 -33 Transf 0.00331 0.00331 48 2434 Fuse 0.01340 0.01340 4 -35 Bus 0.0 0.0 0 -36 Bus 0.0 0.0 0 -37 Break 0.00089 0.00089 72 2438 Cable 0.02291 0.02291 48 -39 Break 0.00089 0.00089 72 2440 Break 0.00089 0.00089 72 2441 Cable 0.02285 0.02285 48 -42 Break 0.00089 0.00089 72 2443 Break 0.00089 0.00089 72 2444 Cable 0.02265 0.02265 48 -45 Break 0.00089 0.00089 72 2446 Bus 0.0 0.0 0 -47 Break 0.00089 0.00089 72 2448 Bus 0.00964 0.00964 1 -49 Bus 0.0 0.0 0 -50 Break 0.00089 0.00089 72 2451 Cable 0.00863 0.00863 48 -52 Break 0.00089 0.00089 72 2453 Break 0.00089 0.00089 72 2454 Cable 0.00837 0.00837 48 -55 Break 0.00089 0.00089 72 2456 Bus 0.0 0.0 0 -57 Break 0.00089 0.00089 72 2458 Bus 0.00964 0.00964 1 -

6.4. Validation of the Simulation method in RADPOW 61

6.4 Validation of the Simulation method in RADPOW

In order to validate the results from the simulation method, both the analyticalpart of RADPOW and the commercial tool NEPLAN has been used. The twotest systems presented earlier in this chapter has been analyzed with these threedifferent methods and the results has then been compared.

In the simulation, 10000 samples with a length of 1000 years have been per-formed for both the systems. This sample time is of course not realistic, but a lowervalue of this result in under estimations of both outage frequency and duration, be-cause of the assumption that all components are functioning from the beginningand the relatively high component reliability. The simulation parameters met therequired accuracy of three decimals in the results. The results converged to thesame values, within three decimals, when performing a number of simulationswith these parameters.

The total computation time where about 30 seconds on a normal computer forthe Birka System.

6.4.1 Test System 1

Table 6.5 shows the failure rates per year for the two load points in Test System1. The results from the simulation method in RADPOW are compared both to theanalytical results in RADPOW and in NEPLAN. A comparison of the evaluated

Table 6.5: Failure rates per year for the load points in Test System 1.

Load point RADPOW RADPOW NEPLAN[f/yr] Simulation AnalyticalTest System 1aLP5 0.102 0.103 0.103LP6 0.078 0.078 0.078Test System 1bLP5 0.064 0.064 0.064LP6 0.064 0.064 0.064

failure rates from the different methods reveals that the results from the simulationmethod is accurate. A comment to the result for Test System 1a, would be that theredundancy in transformators for LP5 increases the failure rate instead of decreas-ing it. LP6 has a single transformator feeding it and has a lower failure rate. Thisis due to the extra failure rates that are added by the additional breakers involvedof the double feeding of LP5. In Test System 1b the failure rates in LP5 and LP6are identical because of the closed point c18 connecting these.

Table 6.6 presents the unavailability for the two load points in hours per yearfor Test System 1. These results are also very accurate when comparing the differ-


Table 6.6: Unavailability in hours per year for the load points in Test System 1.

Load point RADPOW RADPOW NEPLAN[h/yr] Simulation AnalyticalTest System 1aLP5 0.103 0.104 0.104LP6 0.079 0.079 0.079Test System 1bLP5 0.065 0.065 0.065LP6 0.065 0.065 0.065

ent methods. The unavailability for the load points in Test System 1a and 1b arealmost equal or slightly higher than the failure rates, which means that the restora-tion time for the load point is around one hour. These results are logical; for Testsystem 1a the disconnector is closed in one hour and for Test System 1b the af-fected component is switched off within one hour and then gets power from thecommon busbar c18. The slightly higher value than one hour is explained by theinterruptions caused in busbars c5 and c6, which is repaired within two hours.

The load point indices are evaluated by the computer programs to system in-dices with the additional customer data in Table 6.1. Table 6.7 presents the systemindices for Test System 1. As shown in the table, the results are very accurate, andthere are only minor differences.

Table 6.7: Evaluated system indices for Test System 1. Both RADPOW and NEPLAN hasbeen used to validate the results.

System RADPOW RADPOW NEPLANIndices Simulation AnalyticalTest System 1aSAIFI[int/yr.cu] 0.091 0.092 0.091SAIDI[h/yr.cu] 0.093 0.093 0.092CAIDI[h/int] 1.013 1.011 1.012AENS[kWh/yr.cu] 0.405 0.407 0.407ASAI 0.99999 0.99999 0.99999Test System 1bSAIFI[int/yr.cu] 0.064 0.064 0.064SAIDI[h/yr.cu] 0.065 0.065 0.065CAIDI[h/int] 1.017 1.017 1.017AENS[kWh/yr.cu] 0.288 0.289 0.289ASAI 0.99999 0.99999 0.99999

6.4. Validation of the Simulation method in RADPOW 63

6.4.2 Birka System

Table 6.8 shows the failure rates per year for the three load points in the BirkaSystem. The results from the simulation method in RADPOW are compared both tothe analytical results in RADPOW and in NEPLAN. The failure rate is significantat LP35 compared with the other two load points. The average failure rate fora LP35 customer is about 0.28 failures/year. For an LP48 or LP58 customer thefailure rate is only about 0.06 failures/year, almost five times lower than for LP35.

Table 6.8: Failure rates per year for the load points in the Birka System.

Load point RADPOW RADPOW NEPLAN[f/yr] Simulation AnalyticalLP35 0.282 0.282 0.278LP48 0.059 0.059 0.057LP58 0.058 0.058 0.056

The results in failure rates are equal for the simulation method and the analyt-ical method in RADPOW. For NEPLAN there is a minor difference in results andthis is probably explained by the different evaluation method used compared to theanalytical method in RADPOW. The fact that the function of fuses is not includedin the simulation model (see Section 5.4), does not seem to affect the results forLP35.

Table 6.9 presents the unavailability for the load points in hours per year. Theresults for the unavailability are almost equal for the simulation as for the analyticalmethod in RADPOW. NEPLAN differ again a minor from the other two methods.

Table 6.9: Unavailability for the load points in the Birka System.

Load point RADPOW RADPOW NEPLAN[h/yr] Simulation AnalyticalLP35 0.474 0.475 0.470LP48 0.099 0.100 0.098LP58 0.098 0.099 0.096

The load point indices are evaluated by the programs to system indices with theadditional customer data in Table 6.3. Table 6.10 presents the system indices forthe Birka system. The evaluated system indices are almost equal when comparingthe different methods. CAIDI differs a bit more than the other indices, which isexplained by the fact that this index depends both on the failure rates and unavail-abilities for the load points.


Table 6.10: Evaluated system indices for the Birka System. Both RADPOW and NEPLANhas been used to validate the results.

System RADPOW RADPOW NEPLANIndices Simulation AnalyticalSAIFI[int/yr.cu] 0.063 0.063 0.062SAIDI[h/yr.cu] 0.105 0.107 0.104CAIDI[h/int] 1.671 1.683 1.703AENS[kWh/yr.cu] 0.110 0.111 0.113ASAI 0.99999 0.99999 0.99999

6.4.3 Conclusions

The results from the comparative studie of the two test systems clearly shows thatthe results from the implemented simulation module are valid. Only minor dif-ferences are present when the results are compared with the analytical method inRADPOW and with NEPLAN.

Although the simulation module is valid for the two test systems presented inthis chapter, one has to keep in mind that a number of approximations and simpli-fications are involved in the simulation method, as described in Section 5.4. Theboth test systems has been modified, as described earlier in this chapter, in order tonot take these parameters into account.

6.5 Sensitivity Analysis Routine

The system indices from the evaluation methods in previously version of RAD-POW are average values. There will be variations in these indices depending onthe accuracy in the input parameters for the components. As described in Chapter4 the parameters for each component are assumed to be normal distributed and inthis analysis a deviation of 10% for all components are assumed.

For both of the test systems the mean values of the method has been comparedto the results in the previous section. The deviation of the indices has not beenvalidated by other methods and is left for further work.

6.5.1 Test System 1

Table 6.11 shows the results from the sensitivity analysis of Test System 1, per-formed by 5000 samples with different component values. The maximum and min-imum values from these analysis are specific and will differ between two differentanalysis. The number of decimals are chosen with respect to the convergence ofeach index. Several analysis with different number of samples has been performedto validate the consistency within these number of decimals. The results for themean values is in accordance with the results in Table 6.7.

6.5. Sensitivity Analysis Routine 65

Table 6.11: Results from a sensitivity analysis with 5000 samples from Test System 1. Theuncertainty is 10% in all component parameters.

System Mean Max Min DeviationIndices value sample sample σ

Test System 1aSAIFI[int/yr.cu] 0.092 0.112 0.070 0.0059SAIDI[h/yr.cu] 0.093 0.136 0.059 0.010CAIDI[h/int] 1.01 1.38 0.69 0.090AENS[kWh/yr.cu] 0.407 0.592 0.259 0.044ASAI 0.999990 0.999993 0.999984 1.2×10−6

Test System 1bSAIFI[int/yr.cu] 0.064 0.089 0.041 0.0059SAIDI[h/yr.cu] 0.065 0.098 0.039 0.0085CAIDI[h/int] 1.02 1.39 0.69 0.095AENS[kWh/yr.cu] 0.289 0.434 0.174 0.038ASAI 0.99999 0.999996 0.999989 9.7×10−7

Figures 6.3 to 6.7 shows the distribution of the 5000 samples for each index.These figures shows that the system indices are normal distributed if the componentparameters are the same.


SAIFI (int/yr.cust)

(int/yr.cust)0.067 0.071 0.076 0.081 0.086 0.091 0.096 0.101 0.106 0.111

Hits

/inte

rval

350

300

250

200

150

100

50

Figure 6.3: Distribution of 5000 samples of SAIFI, for Test System 1a with 10% deviationin all component parameters. The interval for each bar is 0.001.

SAIDI (h/yr.cust)

(h/yr.cust)0.056 0.063 0.072 0.081 0.090 0.099 0.108 0.117 0.126 0.135

Hits

/inte

rval

220

200

180

160

140

120

100

80

60

40

20

Figure 6.4: Distribution of 5000 samples of SAIDI, for Test System 1a with 10% deviationin all component parameters. The interval for each bar is 0.001.


CAIDI (h/int)

(h/int)0.670 0.740 0.820 0.900 0.980 1.060 1.140 1.220 1.300 1.380

Hits

/inte

rval

240

220

200

180

160

140

120

100

80

60

40

20

Figure 6.5: Distribution of 5000 samples of CAIDI, for Test System 1a with 10% deviationin all component parameters. The interval for each bar is 0.01.

AENS (kWh/yr.cust)

(kWh/yr.cust)0.245 0.280 0.320 0.360 0.400 0.440 0.480 0.520 0.560 0.600

Hits

/inte

rval

240

220

200

180

160

140

120

100

80

60

40

20

Figure 6.6: Distribution of 5000 samples of AENS, for Test System 1a with 10% deviationin all component parameters. The interval for each bar is 0.005.


SAIDI (h/yr.cust)

(h/yr.cust)0.056 0.063 0.072 0.081 0.090 0.099 0.108 0.117 0.126 0.135

Hits

/inte

rval

220

200

180

160

140

120

100

80

60

40

20

Figure 6.7: Distribution of 5000 samples of ASAI, for Test System 1a with 10% deviationin all component parameters. The interval for each bar is 1×10−7.

6.5.2 The Birka System

Table 6.12 shows the results from the sensitivity analysis of the Birka System, per-formed by 4000 samples with different component values. The maximum and min-imum values from these analysis are specific and will differ between two differentanalysis. The number of decimals are chosen with respect to the convergence ofeach index. Several analysis with different number of samples has been performedto validate the consistency within these number of decimals. The results for the

Table 6.12: Results from a sensitivity analysis with 4000 samples from the Birka system.The uncertainty is 10% in all component parameters.

System Mean Max Min DeviationIndices value sample sample σ

SAIFI[int/yr.cu] 0.063 0.077 0.050 0.0036SAIDI[h/yr.cu] 0.107 0.136 0.085 0.0069CAIDI[h/int] 1.68 2.10 1.37 0.10AENS[kWh/yr.cu] 0.111 0.140 0.090 0.0070ASAI 0.99999 0.999990 0.999985 7.9×10−7

mean values is in accordance with the results in Table 6.10.Figures 6.8 to 6.12 shows the distribution of the 4000 samples for each index.

These figures shows that the system indices are normal distributed if the componentparameters are the same.


SAIFI (int/yr.cust)

(int/yr.cust)0.0490 0.0525 0.0565 0.0605 0.0645 0.0685 0.0725 0.0765

Hits

/inte

rval

220

200

180

160

140

120

100

80

60

40

20

Figure 6.8: Distribution of 4000 samples of SAIFI, performed on the Birka System with10% deviation in all component parameters. The interval for each bar is 0.0005.

SAIDI (h/yr.cust)

(h/yr.cust)0.083 0.088 0.094 0.100 0.106 0.112 0.118 0.124 0.130 0.136

Hits

/inte

rval

240

220

200

180

160

140

120

100

80

60

40

20

Figure 6.9: Distribution of 4000 samples of SAIDI, performed on the Birka System with10% deviation in all component parameters. The interval for each bar is 0.001.


CAIDI (h/int)

(h/int)1.350 1.420 1.500 1.580 1.660 1.750 1.830 1.910 2.000 2.090

Hits

/inte

rval

160

140

120

100

80

60

40

20

Figure 6.10: Distribution of 4000 samples of CAIDI, performed on the Birka System with10% deviation in all component parameters. The interval for each bar is 0.01.

AENS (kWh/yr.cust)

(kWh/yr.cust)0.0885 0.0940 0.1010 0.1080 0.1150 0.1220 0.1290 0.1360

Hits

/inte

rval

120

110

100

90

80

70

60

50

40

30

20

10

Figure 6.11: Distribution of 4000 samples of AENS, performed on the Birka System with10% deviation in all component parameters. The interval for each bar is 0.0005.


ASAI

probability0.99998420 0.99998540 0.99998683 0.99998826 0.99998969

Hits

/inte

rval

240

220

200

180

160

140

120

100

80

60

40

20

Figure 6.12: Distribution of 4000 samples of ASAI, performed on the Birka System with10% deviation in all component parameters. The interval for each bar is 1×10−7.

Chapter 7

Closure

7.1 Conclusions

This thesis has presented a proposed method for a MCS approach of evaluatingthe reliability indices for a distribution system. The conclusion of this thesis isthat the implemented MCS method in RADPOW provides the same results as theanalytical method in RADPOW and the NEPLAN software for two different casesystems. There is no reason to believe that it is not functioning for a general systemand therefor the method can be used for reliability assessment of power distribu-tion system and has a potential to be developed further to incorporate general lifetime distributions for the components in the system. Another aspect that can bedeveloped in a more straightforward manner is prioritization of components e.g. todetermine where the maintenance actions will have the greatest effect.

The implemented analytical sensitivity analysis routine that also is presentedin this thesis gives a quantitative measurement of the uncertainty in the systemindices, given the uncertainty of the component values. Given these distribution ofthe system indices the variation and other statistical measures can be computed asshown.

7.2 Discussion and Future Work

Computation time is an issue that usually is hold against simulations in order toget appropriate results that converges. The MCS described in this paper uses themost basic sample strategy referred to as simple sampling, which needs a rela-tively large number of samples to receive a sufficiently accurate result. There aretechniques for reduction of calculation times without loss of precision, variance re-duction techniques as stratified sampling and weighted sampling as examples, butstill it is costly in terms of computation time. However, there are situations whenanalytical methods are not suitable to use because of the difficulties to model theproblem analytical without making too large approximations. In these situationsthe simulation approach is an alternative. In a simulation approach there is also

73

74 Chapter 7. Closure

possible to extend the model to handle general distributions of component deterio-ration.

Future possible development of the MCS method in RADPOW is to includethe ability to handle temporary and transient failure rates for the components andeventually also incorporate the stuck probability for breakers. Another possibledevelopment is to introduce the Loadflow module with the Sim module, whichmakes it possible to perform load flow calculations for each system state.

If the failures in the simulation are saved in a log file, the MCS provides adeeper understanding e.g. how different second order failure events occurs or howthe repair or replacement of components are dealt with when there are constraintsin the work force.

The MCS method can also easily be extended to be used for prioritization ofcomponents; one example is how to determine where the maintenance action willhave the greatest effect. Taken further this prioritization can be used in the opti-mization of maintenance from a system reliability perspective, which is one of themajor goals for asset management of electrical networks that is handled by RCAM.

Appendix A

Input Data File for RADPOW

The system model for a power system is with RADPOW defined by the user inone input data file, which is a normal text file with the file extension radpow.The filesystem for the RADPOW_2006 version has been changed since the RAD-POW_1999_PF version, as described in Section 3.6.3. In the earlier version, thedata used for input for RADPOW were separated into ten different files. These fileshas been merged into one file containing ten different sections, each having exactlythe same data, but with a header describing the section. These section headers arethen used by RADPOW to identify where the different types of data in the fileare present. This appendix describes the different sections in the file and how thesystem data are inserted correctly when defining a new system in RADPOW.

The input data file for Test System 1a and the Birka System are included inAppendix B and provides the reader with examples which may be valuable in thefurther reading of this appendix. The network model of these two systems arefound in Figure 2.2 and 6.2 respectively.

A.1 Network topology data

These sections describes the basic structure of the system. The network topologydata is described by the branch data, the location of supply and load points, nor-mally open components and the type of each component.

The topology of a system is in RADPOW defined by how a number of branchesare connected to each other. A branch is in RADPOW defined as a number of com-ponents connected in series which starts and ends with a bus bar. The componentsin a branch are each identified by its component number. The different branches,each identified by a branch number, are then connected to each other by their com-mon bus bars. The branches in Figure 6.2 are donated the prefix B.

75

76 Chapter A. Input Data File for RADPOW

A.1.1 bend-section

The bend-section describes how the different branches are connected. For eachbranch in the section, there are four different parameters separated by blank space:

1. The branch number (Brno) of the branch, defined as an integer from 1 to k,where k is the number of branches in the system.

2. The component number for the bus bar of the sending end (SE) of the branch.

3. The component number for the bus bar of the receiving end (RE) of thebranch.

4. The unidirectional (Unid) parameter defines if power is permitted to flowin one direction or both directions. If this parameter is set to 0, the powerthrough the branch is permitted to flow in any direction. If the parameter isset to a non-zero value the power is only permitted to flow from the sendingend (SE) to the receiving end (RE).

A.1.2 bcomp-section

The bcomp-section defines which components that are included in each branch.For each branch there are three different parameters:

1. The branch number (Brno) of the branch.

2. All the components in the branch, identified by the component number andeach separated by a blank space.

3. The end of the branch (EOB) parameter is set to −1 for each branch. Thisparameter is only for internal use in RADPOW.

A.1.3 nsp-section

The nsp-section defines at which bus bars there is a power supply from a com-pletely reliable net. The component number of these bus bars are separated by ablank space.

A.1.4 nlp-section

In the nlp-section the component number of the bus bars with a load point con-nected and are going to be analyzed are defined.

A.1.5 nnop-section

If a component is normally open the component number are entered in the nnop-section. These components are normally disconnecters that can be closed in caseof an outage.

A.2. Customer data 77

A.1.6 ctype-section

In RADPOW the component reliability data are defined for each component type.In the ctype-section each component in system are given a component type withthe specified component type name which are defined in the cerelia-section.

A.2 Customer data

The customer and power and data are defined separately for each load point (LP)in the system. A load point is always connected to a busbar and consequentlythe number of the LP is the component number of that busbar. The load pointsthat are going to be analyzed are then entered in the nlp-section. Data about thecustomer type can be defined in the ncuspow-section, but these data are not usedby RADPOW for the standard reliability analysis.

A.2.1 ncuspow-section

The following customer and power data are specified for each load point, and inthe following order:

1. The load point number (lpno), which is identical to the component numberof the connected bus bar.

2. The total number of customers (tnc) in the load point.

3. The percentage of industrial customers (icp) in the load point.

4. The percentage of residential customers (icp) in the load point.

5. The percentage of commercial customers (icp) in the load point.

6. The total active power per customer (tappc) in kW.

7. The percentage of the active power consumed by the industrial customers(iapp) in the load point.

8. The percentage of the active power consumed by the residential customers(rapp) in the load point.

9. The percentage of the active power consumed by the commercial customers(capp) in the load point.

10. The total reactive power per customer (trppc) in kVar.

11. The percentage of the reactive power consumed by the industrial customers(irpp) in the load point.

12. The percentage of the active power consumed by the residential customers(rrpp) in the load point.

78 Chapter A. Input Data File for RADPOW

13. The percentage of the active power consumed by the commercial customers(crpp) in the load point.

A.3 Component reliability data

For each component type reliability data needs to be specified. The different com-ponents are then defined a specific type in the ctype-section.

A.3.1 cerelia-section

The component reliability data are specified for each type of component and in thefollowing order:

1. A text string identifying the name of the component type (tno). This nameare then used in the ctype-section.

2. The type of component (tc). The options for this parameter are Bus, Break,Cable, Fuse, Transf or Discon.

3. The permanent failure rate (frp) for the component.

4. The active failure rate (fra) for the component.

5. The temporary failure rate (frte) for the component.

6. The transient failure rate (frtr) for the component.

7. The maintenance outage rate (frm) for the component.

8. The repair time (trep) in hours for the component.

9. The time for maintenance outage in hours (tmain).

10. Time for recovery (trec) after a temporary fault in hours.

11. The switching time (tswi) for a failed component in hours. After this timeinterval the component are unconnected until it has ben replaced or repaired

12. The replacement time (trep) in hours for the component. If this parameteris zero, a replacement of the component is not an option. If replacement isavailable, this option is always chosen before repair restoration.

13. The probability of switching devices being stuck (sprob).

A.4. Load flow data 79

A.4 Load flow data

The RADPOW_1999_PF version introduced a new module, Loadflow (describedin Section 3.5.1 ). In the earlier version, RADPOW_1999, a failure for a load pointwere based on the total loss of continuity (TLOC) criteria, which means that a loadpoint is interrupted when all its paths between the supply points and the load pointare disconnected. But there are situations when a path, or branch, are not capableto feed the extra amount of load when other components in the system have failed.This criteria is called the partial loss of continuity (PLOC), which introduces anelectrical model in RADPOW, and not only a probabilistic. For this purpose, anumber of electrical parameters for each load point and branch needs to be definedin two different sections in the input data file.

A.4.1 pflo-section

For each load point, with the component number as defined in the ncuspow-section,the following data needs to be defined:

1. The active power consumption (lapc) in the load point in per unit (p.u).

2. The active power consumption (lrpc) in the load point in per unit (p.u).

3. The load duration curve for the load point, numbered with the integer 1 to 3.The load duration curve are either constant (1), sinusoid (2) or linear (3) andthese are defined in [14].

A.4.2 pfrx-section

For each branch in the system line data needs to specified in the following order:

1. The branch number (Brno) for the specific branch, which is identical to theones in the bend- and bcomp-sections.

2. The component number for the bus bar of the sending end (SE) of the branch.

3. The component number for the bus bar of the receiving end (RE) of thebranch.

4. The resistance (R) in the branch in per unit (p.u).

5. The reactance (X) in the branch in per unit (p.u).

6. The maximal current (Max_current) the branch can transfer in per unit(p.u).

Appendix B

Test System Input Files forRADPOW

B.1 Test System 1 Input Data File

The following included text is the input data file for the Test System 1a that wereused in the studie for RADPOW. The only different for the Test System 1b data filecompared to this one is the component number of normally open points, which inthe nnop-section is 0 instead of 18. For changes from earlier versions of the TestSystem 1 data file, see Section 6.2.

**************************bend-sectionBrno SE RE Unid1 1 3 02 2 4 03 3 5 04 3 5 05 4 6 06 5 6 0

**************************bcomp-sectionBrno Real components in branch EOB1 1 7 3 -12 2 8 4 -13 3 9 15 10 5 -14 3 11 16 12 5 -15 4 13 17 14 6 -16 5 18 6 -1

**************************nsp-sectionIdno of supply points1 2

**************************

81

82 Chapter B. Test System Input Files for RADPOW

nlp-sectionLoad points to be analysed5 6

**************************nnop-sectionIdno of normally open points18

**************************ncuspow-sectionlpno tnc icp rcp ccp tappc iapp rapp capp trppc irpp rrpp crpp5 100 0.0 0.2 0.8 4.0 0.0 0.3 0.7 2.0 0.0 0.3 0.76 80 0.0 0.2 0.8 5.0 0.0 0.3 0.7 2.5 0.0 0.3 0.7

**************************ctype-sectionIdno Typeno1 BUS12 BUS13 BUS14 BUS15 BUS16 BUS17 BREAK18 BREAK19 BREAK110 BREAK111 BREAK112 BREAK113 BREAK114 BREAK115 TRANSF116 TRANSF117 TRANSF118 DISCON119 BUS1

**************************crelia-sectiontno tc frp fra frte frtr frm trep tmain trec tswi trepl sprobBUS1 Bus 0.001 0.001 0.00 0.0 0.0 2.0 0.0 0.0 1.0 0.0 0.0BREAK1 Break 0.020 0.020 0.00 0.0 0.0 24.0 0.0 0.0 1.0 0.0 0.0TRANSF1 Transf 0.015 0.015 0.00 0.0 0.0 15.0 0.0 0.0 1.0 0.0 0.0DISCON1 Discon 0.002 0.002 0.00 0.0 0.0 4.0 0.0 0.0 1.0 0.0 0.0

**************************pflo-sectionlpno lapc(pu) lrpc(pu) ldcu5 0.2 0.2 16 0.2 0.25 1

**************************pfrx-sectionBrno SE RE R(pu) X(pu) Max_current(pu)1 1 3 0.01 0.1 0.82 2 4 0.005 0.05 0.83 3 5 0.0 0.05 0.44 3 5 0.0 0.05 0.45 4 6 0.01 0.1 0.86 5 6 0.005 0.05 0.8

B.2. Birka System Input Data File 83

B.2 Birka System Input Data File

The following included text is the input data file for the Birka System that wereused in the studie for RADPOW. For changes from earlier versions of the BirkaSystem data file, see Section 6.3.

**************************bend-sectionBrno SE RE Unid1 1 14 02 1 14 03 14 27 04 14 27 05 14 27 06 27 29 07 29 32 08 29 32 09 32 35 010 14 36 011 36 46 012 36 46 013 36 46 014 46 48 015 14 49 016 49 56 017 49 56 018 56 58 0

**************************bcomp-sectionBrno Real components in branch EOB1 1 2 3 4 5 6 7 14 -12 1 8 9 10 11 12 13 14 -13 14 15 16 17 18 27 -14 14 19 20 21 22 27 -15 14 23 24 25 26 27 -16 27 28 29 -17 29 30 32 -18 29 31 32 -19 32 33 34 35 -110 14 36 -111 36 37 38 39 46 -112 36 40 41 42 46 -113 36 43 44 45 46 -114 46 47 48 -115 14 49 -116 49 50 51 52 56 -117 49 53 54 55 56 -118 56 57 58 -1

**************************nsp-sectionIdno of supply points1

**************************


nlp-sectionLoad points to be analysed35 48 58

**************************nnop-sectionIdno of normally open points

**************************ncuspow-sectionlpno tnc icp rcp ccp tappc iapp rapp capp trppc irpp rrpp crpp35 447 0.25 0.5 0.25 1.7203 0.4 0.4 0.2 0.0 0.0 0.0 0.048 23400 0.1 0.8 0.1 0.9829 0.2 0.7 0.1 0.0 0.0 0.0 0.058 1 0 0 1 0.8 0 0 1 0.0 0.0 0.0 0.0

**************************ctype-sectionIdno Typeno1 BU2202 BR1103 TR2204 BR1105 CA110a6 TR1107 BR338 BR1109 TR22010 BR11011 CA110b12 TR11013 BR3314 BU22015 BR3316 CALH3317 TR3318 BR1119 BR3320 CALH3321 TR3322 BR1123 BR3324 CALH3325 TR3326 BR1127 BU1128 BR1129 BUSD30 CALH1131 CALH1132 BUSD33 TR1134 FUSE35 BU0436 BUSD37 BR33

B.2. Birka System Input Data File 85

38 CAHDa39 BR3340 BR3341 CAHDb42 BR3343 BR3344 CAHDc45 BR3346 BUSD47 BR3348 BU22049 BUSD50 BR3351 CASJa52 BR3353 BR3354 CASJb55 BR3356 BUSD57 BR3358 BU220

**************************crelia-sectiontno tc frp fra frte frtr frm trep tmain trec tswi trepl sprobBUSD Bus 0 0 0 0 0 0 0 0.0 1.0 0.0 0.0BU220 Bus 0.00964 0.00964 0 0 0 1 0 0.0 1.0 0.0 0.0BU11 Bus 0.00867 0.00867 0 0 0 1 0 0.0 1.0 0.0 0.0BU04 Bus 0 0 0 0 0 0 0 0.0 1.0 0.0 0.0BR110 Break 0.00870 0.00870 0 0 0 168 0 0.0 1.0 24.0 0.0BR33 Break 0.00089 0.00089 0 0 0 72 0 0.0 1.0 24.0 0.0BR11 Break 0.00243 0.00243 0 0 0 48 0 0.0 1.0 24.0 0.0CA110a Cable 0.07012 0.07012 0 0 0 168 0 0.0 1.0 0.0 0.0CA110b Cable 0.07031 0.07031 0 0 0 168 0 0.0 1.0 0.0 0.0CALH33 Cable 0.00028 0.00028 0 0 0 48 0 0.0 1.0 0.0 0.0CAHDa Cable 0.02291 0.02291 0 0 0 48 0 0.0 1.0 0.0 0.0CAHDb Cable 0.02285 0.02285 0 0 0 48 0 0.0 1.0 0.0 0.0CAHDc Cable 0.02265 0.02265 0 0 0 48 0 0.0 1.0 0.0 0.0CASJa Cable 0.00863 0.00863 0 0 0 48 0 0.0 1.0 0.0 0.0CASJb Cable 0.00837 0.00837 0 0 0 48 0 0.0 1.0 0.0 0.0CALH11 Cable 0.10069 0.10069 0 0 0 6 0 0.0 1.0 0.0 0.0FUSE Fuse 0.01340 0.01340 0 0 0 4 0 0.0 1.0 0.0 0.0TR220 Transf 0.02610 0.02610 0 0 0 504 0 0.0 1.0 24.0 0.0TR110 Transf 0.02050 0.02050 0 0 0 504 0 0.0 1.0 24.0 0.0TR33 Transf 0.01989 0.01989 0 0 0 504 0 0.0 1.0 24.0 0.0TR11 Transf 0.00331 0.00331 0 0 0 48 0 0.0 1.0 24.0 0.0

**************************pflo-sectionlpno lapc(pu) lrpc(pu) ldcu35 0.4 0.2 148 0.4 0.2 158 0.4 0.2 1


**************************pfrx-sectionBrno SE RE R(pu) X(pu) Max_current(pu)1 1 14 0.01 0.1 0.82 1 14 0.01 0.1 0.83 14 27 0.01 0.1 0.84 14 27 0.01 0.1 0.85 14 27 0.01 0.1 0.86 27 29 0.01 0.1 0.87 29 32 0.01 0.1 0.88 29 32 0.01 0.1 0.89 32 35 0.01 0.1 0.810 14 36 0.01 0.1 0.811 36 46 0.01 0.1 0.812 36 46 0.01 0.1 0.813 36 46 0.01 0.1 0.814 46 48 0.01 0.1 0.815 14 49 0.01 0.1 0.816 49 56 0.01 0.1 0.817 49 56 0.01 0.1 0.818 56 58 0.01 0.1 0.8

References

[1] L. Bertling. Reliability Centred Maintenance for Electric Power DistributionSystems. Doctoral dissertation, Department of Electrical Engineering, KTH,Stockholm, Sweden, August 2002. ISBN 91-7283-345-9.

[2] L. Bertling, R.N. Allan, R. Eriksson. A reliability-centred asset maintenancemethod for assessing the impact of maintenance in power distribution sys-tems. IEEE Transactions on Power Systems, 2003. TPWRS-00271-2003.R3.

[3] L. Bertling. Projektbeskrivning av examensarbete. Department of ElectricalEngineering, KTH, January 2006.

[4] J. Setréus, L. Bertling, S. Mousavi Gargari. Simulation method for reliabilityassessment of electrical distribution systems. To be published at the Nordicconference on Nordic Distribution and Asset Management (NORDAC), Au-gust 2006.

[5] M. Rausand, A. Hoyland. System reliability theory: Models and statisticmethods. John Wiley and Sons Inc., Norwegian University of Science andTechnology, 2 edition, January 2004. ISBN 0-471-47133-X.

[6] G. Blom. Sannolikhetsteori med tillämpningar C. Studentlitteratur AB, Swe-den, 4 edition, 1989. ISBN 91-44-03594-2.

[7] O. Wilhelmsson. Evolution of the introduction of rcm for hydro power gen-erators at vattenfall vattenkraft. Master’s thesis, Department of ElectricalEngineering, KTH, 2005. X-ETS/EEK-0517.

[8] L. Bertling Y. He. The verification of the reliability assesment computer pro-gram radpow. Technical report, Department of Electrical Power Engineering,KTH, 1998. A-EES-9808.

[9] T. Solver. Reliability in Performance-Based Regulation. Lic dissertation,Stockholm, Sweden, August 2005. ISBN 91-7178-136-6.

[10] G.E.P. Box, M.E. Muller. A note on the generation of random normal devi-ates. In The Annals of Mathematical Statistics, volume 29, pages 610–611.Tokyo, 1958. ISSN 0373-5990, 00203157.

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[11] R.Y. Rubinstein. Simulation and the Monte Carlo Method. John Wiley AndSons Ltd, Haifa, Israel, March 1981. ISBN 0-471-08917-6.

[12] M. Amelin. On Monte Carlo Simulation and Analysis of Electricity Markets.PhD thesis, Department of Electrical Engineering, KTH, Stockholm, Sweden,September 2004. ISBN 91-7283-850-7.

[13] Y. He. Modelling and Evaluating Effect of Automation, Protection, and Con-trol on Reliability of Power Distribution Systems. Doctoral dissertation, De-partment of Electrical Engineering, KTH, Stockholm, Sweden, September2002. ISBN 91-7283-351-3.

[14] P. Rosett. Power Flow Assessment for Reliability Evalution in RADPOW.Master’s thesis, Department of Electrical Engineering, KTH, 2000. B-EES-0005.

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