Development of a Simulation Module for the Reliability Computer Program RADPOW

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Development of a Simulation Module for the Reliability Computer Program RADPOW

JOHAN SETR´ EUS

Master’s Degree Project

Stockholm, Sweden 2006

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D EVELOPMENT OF A S IMULATION MODULE FOR THE RELIABILITY COMPUTER PROGRAM

RADPOW

Master Thesis by Johan Setréus

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

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

XR-E-ETK 2006:010

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Abstract

This master thesis describes an implementation of a Monte Carlo Simulation (MCS) method for reliability assessment of electrical distribution systems. The method has been implemented in the reliability assessment tool RADPOW which now 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 user interface for Windows.

• Development and implementation of an analytical sensitivity analysis routine for RADPOW.

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

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

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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 numeriska beräkningar. 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 studie visar att MCS metoden ger samma resultat som den analytiska metoden i RAD- POW och det kommersiella verktyget NEPLAN.

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Acknowledgements

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

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

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

Johan Setréus

Stockholm, June 2006

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Introduction

1.1 Background

A central part in the planning of distribution systems, which becomes even more important in today’s de-regulated electrical power system, is preventive maintenance (PM). This is the planned and scheduled maintenance that aims to postpone or reduce failures of a system. Electrical distribution system operators (DSO) have changed their organization and the pressure to reduce operational and maintenance costs is already being felt. The driving forces are changing from technical factors to economic and business factors and cost-effective PM is required. Consequently, there is an interest from DSOs to incorporate strategies for cost-effective maintenance. Reliability Centred Maintenance (RCM) is such a strategy where maintenance of system components is related to the improvement in system reliability.

The RCM method has been further developed in the reliability-centred asset management method (RCAM) [1] to provide a quantitative relationship between PM of assets and the total cost of maintenance [2].

In the search of the best possible asset management strategy for electrical distribution system it is essential to know the importance of the involved components.

Each component is assigned performance indices that correspond for the overall reliability of supply. The indices can be used for prioritization of components; one example 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 certain component causes the system. A simulation approach of this kind of analysis enables us to develop models with a deeper level of detail for larger systems in a more 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 numerical measurement. This method can then easily be extended to be used for prioritization of components and eventually also produce a distribution of results from which the mean, variance and other statistical measures can be computed.

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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 distribution systems [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 user friendly interface, it also made it easier to validate the results in the development of the implemented simulation method. It also provided a deeper level of understanding for the different algorithms and methods already developed in the analytical metod in RADPOW. The graphical interface was put into practice by a number of new graphical modules, developed in Borland C++.

The second step was to implement an iterative analytical routine in RADPOW for sensitivity analysis. This analytical sensitivity approach provided valuable knowledge of the generation of random numbers from various distributions, which were necessary in the work with the simulation approach. The analytical method in RADPOW, in combination with this sensitivity routine, provided a distribution of the resulting system reliability indices including the mean values and variances of the the samples. In order to validate the results from the sensitivity analysis routine, the mean values of the system indices for two different test systems, were compared with both RADPOW and the commercial reliability tool NEPLAN.

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

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

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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 this thesis are described.

Chapter 3 gives an insight to the basic functions of RADPOW, describing how the 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. This method uses the analytical method developed in RADPOW to perform a sensitivity analysis 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 with the analytical evaluation method in RADPOW and the commercial reliability program NEPLAN. Two different test systems are used for the validation.

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

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Chapter 2

System Reliability

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

2.1 Introduction

Reliability is the ability of a an item to perform a required function, under given environmental and operational conditions and for a stated period of time [5]. In this definition the term item is used to donate a component, subsystem or a system of components, depending on the certain reliability level that is going to be studied.

These two reliability levels are referred to as system and component reliability respectively.

2.1.1 System reliability

A system consists of one or more subsystems, each interconnected and each having interconnected components, in order to perform its required function. A system can 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 depends on the specific situation.

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

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6 Chapter 2. System Reliability

2.1.2 Component reliability

Based on experience and failure data for a certain component, its characteristics in 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 important are defined as:

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

F_X(x) = P (X < x), −∞ < x < ∞ (2.1)

The distribution function is evaluated by an integration as follows F_X(x) =

Z _x

−∞f_X(t)dt (2.2)

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

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

f_X(x) = dF_X

dx (2.3)

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

In this thesis the main interest is the lifetime evaluation for a component. Therefor the functions can be donated F_X(x) = F (t) and f_X(x) = f (t), where t is the time in e.g. years.

Definition 2.3 The reliability function, R(t), which also is called the survival probability 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 not fail in the time interval (0, t], or, in other words, the probability that the component survives 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.

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2.2. Definitions 7

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

M T T F = E(T ) = Z _∞

0 tf (t)dt = Z _∞

0 R(t)dt (2.6)

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

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

x = 1 n

Xn j=1

x_j (2.7)

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

σ(x)²= 1 n − 1

Xn j=1

(x_j− 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 presented.

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 fulfil one 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 of time [5].

2.2.1 Failures

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

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Failure - outage

Damaging Fault

Two models of failure

Non-Damaging Fault

Two 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 operation of 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 the operation of the protection devices around it [1].

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

The second category of failures, non-damaging faults, are outages caused by the protection devices. These outages are categorized into transient and temporary forced outages, depending on the restoration of the fault. If a protection device are restored automatically, the outage time are negligible and therefore transient forced. Other types of protection devices needs to be restored manually, either mechanically or by replacement of a fuse. These types of action takes time and the outage 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. These events are not caused by an damaging fault directly and are therefor referred to as additional active failures [1].

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2.3. Test System 9

Definition 2.13 An additional active failure is a failure mode that occur when an active 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, the restoration time for an outage can be categorized. The different types of restoration times for outages are defined below.

• r_r- Repair restoration time is the time it takes to make the component operational by repairing it.

• r_p - Replacement restoration time is the time it take to replace a component to make it operational.

• r_s- Switching restoration time is the time it takes for a manual or automatic switching device to isolate the failure.

• r_c- 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 program RADPOW [1] in Chapter 3 it has been assumed that the shortest restoration time always 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 understand the different definitions mentioned in the previous section. This will illustrate the general function of a distribution system and its components. In Chapter 3 the test system 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]. The system has been divided into two separate cases which are referred to as Test System 1a and Test System 1b. The only difference between these systems is that the disconnector are normally open in 1a and considered as a closed point in 1b. The symbols used in Figure 2.2 are defined in Figure 2.3. Test System 1 consists of standard components used in distribution systems and features two load points and two supply points. It has a number of breakers to isolate failures and one disconnector which can transfer load if closed. The branches, indicated by a B in Figure 2.2, are sections of components connected in series. These are used in the reliability computer program RADPOW, described in Chapter 3.

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c2 c4

c8 c13 c17

c14

c18 c10

c12 c9

c11 c16

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 breaker

bus 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’s components 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 understanding for the definitions defined earlier in Section 2.2.

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

Failure events on Test system 1a

Disconnector c18 is normally open for these two separate events.

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2.3. Test System 11

• Permanent Outage caused by a passive fault

A passive fault, caused by an software error, strike breaker c14 which opens without a reason. None of the protection devices are triggered. The system operator gets aware of the outage in LP6 immediately and his first idea is to close disconnector c18 manually, but when he looks at his monitor he sees that it is stuck and can not be closed. He then decides that the best way to solve the problem is to replace breaker c14 and consequently he order an engineer to go to the spot and start the replacement of the component. When the engineer arrives to the spot he first disconnect the breaker electrically from 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 r_s as defined in Section 2.2.2. The switching time r_s is in this event assumed to be 1 hour. The engineer then replaces the breaker and connects it to the grid which takes the time r_p, here assumed to be 5 hours. The customers in LP6 are affected by the permanent outage during the time it takes to restore breaker c14 which is r_s+ r_r= 1 + 5 = 6 hours.

• Temporary forced outage

An extremely large demand for power in LP6 forces the system operator to remotely open breaker c8 immediately, due to the risk of overloading the transformer c17. The customers in LP6 is suffering an interruption of supply and the system operator decides to close the disconnector c18 manually and therefor he orders an engineer to go to the spot. The engineer arrives to the spot and closes the switch. The time for the engineer to get to the discon- nector and then close it is here denoted r_sas defined in Section 2.2.2. The switching time r_sis in this event assumed to be 1 hour. The customers in LP6 are affected by the temporary forced outage during the time it takes to close 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 two separate events.

• Permanent Outage caused by an active fault

A 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, which means that both LP5 and LP6 is suffering an interruption of supply. The system operator decides to repair bus c5 and open disconnector c18 for the restoration of LP6 and consequently he order an engineer to go to bus c5 and another to c18 for the manually switching. The customers in LP6 are affected during the time it takes to open c18 and then reclose breaker c14 which is made remotely by the system operator. The switching time r_s for the disconnector is assumed to be 1 hour and the customers in LP6 are therefor suffering a temporary outage, lasting for 1 hour. For the customers in LP5

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the failure in bus c5 affects them until the c5 is repaired. The reparation of c5 starts after the engineer has arrived and disconnected the bus from the grid which takes the time r_s, here assumed to be 1 hour. The reparation of the bus then takes the time r_r, which is assumed to be 2 hours. The customers in LP5 are affected by the permanent outage during the time it takes to restore the bus c5, which takes r_s+ r_r= 1 + 2 = 3 hours.

• Transient Outage

An active fault occurs on transformer c17, caused by a ground failure. The fault triggers the breakers c13 and c14, which isolates the fault. The failure leads to very short voltage drop in LP6, which still gets power delivered via bus c5. This is a transient outage which outage time are negligible. The system operator later decides to replace transformer c17 and then reclose the breakers c13 and c14.

2.4 Reliability Indices

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

2.4.1 Load Point Indices

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

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

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

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

• LOE_lpi[kW h/yr] = The average loss of energy per year

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

2.4.2 System Indices

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

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2.5. Distribution Functions 13

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

SAIF I =

PN_iλ_i

PN_i (2.9)

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

SAIDI =

PN_iU_i

PN_i (2.10)

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

CAIDI =

PN_iU_i

PN_iλ_i = SAIDI

SAIF I (2.11)

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

AEN S =

PLOE_i

PN_i (2.12)

• Average Service Availability Index (ASAI):

ASAI =

PN_i· 8760 − U_iN_i

PN_i· 8760 (2.13)

2.5 Distribution Functions

The Exponential and the Normal distributions have been used in this thesis for the modeling 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 to describe the lifetime of an item. The reasons for this is due to its mathematical simplicity and that the model are suitable for many different items or situations.

An exponentially-distributed variable T ∈ Exp(m) has the following density and distribution functions:

f (t) =

( (1/m) · e^−t/m for t ≥ 0, m > 0

0 otherwise (2.14)

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F_X(x) =

( 0 for x < 0

1 − 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 failure rate of an exponentially distributed item is constant and thereby independent of time. This implies that it is memoryless and can be considered as good as new at any time when still functioning. This also implies that it is no meaning to replace a still functioning component in preventive maintenance if its failure rate is exponential modeled. A constant failure rate is normally a good assumption for an item during its useful life period. In order to generate exponential distributed ran-

0 10 20 30

0 0.05 0.1 0.15 0.2 0.25

Density function f X(t)

fX(t)

t

0 10 20 30

0 0.2 0.4 0.6 0.8 1

Distribution function F X(t)

FX(t)

t

0 10 20 30

0 0.2 0.4 0.6 0.8 1

Reliability function R_X(t)

RX(t)

t

0 10 20 30

0 0.1 0.2 0.3 0.4 0.5

Failure 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 F_X(x) needs to be stated. Given a uniform stochastic variabel U ∈ U (0, 1), the stochastic variable Y = F_Y⁻¹(U ) have the distribution function F_Y(x), and Y is thereby exponentially distributed. For the exponential distribution the inverse can be solved straight for- ward and is as follows:

Y = −1

λln U (2.16)

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2.5. Distribution Functions 15

2.5.2 The Normal Distribution

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

f_X(x) = 1 σ√

2πe^−(x−µ)²^/2σ² (2.17)

F_X(x) = 1 σ√

2π Z _x

−∞e^−(y−µ)²^/2σ²dy (2.18)

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

−4 −2 0 2 4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Density function f X(x)

fX(x)

x

−4 −2 0 2 4

0 0.2 0.4 0.6 0.8 1

Distribution function F X(x)

FX(x)

x

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

the normal distributed distribution function, Φ⁻¹(x), does not exists but a variety of 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. The algorithm 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 = V₁²+ V₂². If R = 0 or R > 1 start over at step1, otherwise proceed.

4. The normal distributed random variable X ∈ N (0, 1) is then calculated as X = V₁

q−2 ln R R

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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)

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Chapter 3

RADPOW

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

3.1 Introduction

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

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

Originally the method for evolution of a power system in RADPOW was analytical and the purpose of this thesis is to develop a new module that also will allow RADPOW to make simulation calculations [3]. The proposed method and implementation of the simulation approach in RADPOW are discussed in Chapter 5. In the following sections the different modules in the program and the overall picture 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 of RADPOW. Given the relation between the components, the reliability data for the

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18 Chapter 3. RADPOW

components, customer data and power flow data, RADPOW presents the results including 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.]

• ASAI

Load 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 the calculations.

The input data for the system are defined in a standard text file with a syntax described in Section 3.6.3 and in Appendix A. The output are presented to the user either directly on the screen or as a text file for further analysis in other computer programs, 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 the version 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 a master thesis project, made by Philippe Rosett, resulted in a new improved version of RADPOW_1999 referred to as RADPOW_1999_PF [14][15]. This version also resulted in a new module, Loadflow, which considers the load flow constrains in the model, and adds the result to the analytical calculations.

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

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

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3.3. Reliability Evaluation in RADPOW 19

Mincut Minpath Abreak Aafail

Lpind Sind

Netw Branch

Comp RADPOW_1999

Lpind Sind

Netw Branch

Comp RADPOW_1999_PF

Loadflow

Lpind Sind

Netw Branch

Comp RADPOW_2006

Loadflow

Sim

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

3.3 Reliability Evaluation in RADPOW

In reliability analysis the first step is always, as in all mathematical analysis, to make a representativ model of the real system that is going to be studied. When the 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 approach or an numerical approach. The analytical approach usually solves the problem directly with mathematical formulas, whereas the numerical approach uses numerical methods. Two special types of numerical methods are simulation and the Monte Carlo methods which uses random experiments to find a solution of a problem.

There are slightly differences between the definitions of these words, and in this thesis these are both referred to as Monte Carlo Simulation (MCS), which is the simulation approach.

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

Author Developed modules Year Main references Lina Bertling Mincut, Abreak, Aafail

and Lpind

1999 [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 adopted to make an comparative studie. The flowchart in Figure 3.3 shows the overall methodic 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 method Analytical 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 used in order to determine the system and load point indices. A symbolic picture of these are 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

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3.4. RADPOW_1999 version 21

RADPOW

Analytical methods

Simulation method Analytical

calculation

Sensitivity analysis

routine

1 2 3

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

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

2. The sensitivity analysis routine uses the analytical method consecutive times with random input values. The resulting indices are the same as in method 1, but with a standard deviation measurement for the results. This method is described in Chapter 4.

3. The simulation method makes a large number of experiments on the system and then evaluates these. This method, further described in Chapter 5, gives the 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 the simulation 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 component if this not cause a system failure.

3.4 RADPOW_1999 version

In the analytical method, equations for evaluation of the reliability of the system can 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 and Markovian modeling [1]. Of these two, the Network model is the easiest method to implement, specially for larger systems. In Markovian modeling each state of the system and the transitions between these needs to be defined. This means that the size of the model grows exponentially with the number of components in the

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system, which makes it hard to use for larger systems. For smaller subsystems in a larger system, Markovian modeling can be used for approximations of overlapping failures, in cooperative with the Network modeling, which is described in Section 3.4.2. The Markovian modeling method and teori are not described any further in this 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 is considered. The model describes the behavior of the system if one or more of its components fails to fulfill its function. These different failure modes for the system are described by the minimal cut set.

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

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

For the function of a specific load point each minimal cut set for the load point has to functioning. In logical terms this is an AND statement. If the definition in 3.1 is applied to Test System 1a, described in Section 2.3, the resulting minimal cut sets for 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. LP6 has only minimal cut sets of first order.

The minimal cut sets are used for the evolution of each load points reliability indices. In the analytical method in RADPOW a load point driven approach has been adopted. This means that all failure events for each load point are considered in turn and consequently that the load point indices for each load point are evaluated separately with help from the minimal cut sets. When implementing a general algorithm for deducing all the minimal cut sets in a system, it is easier to first 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 guarantees the operation of the system. A path is minimal when it cannot be reduced any further and still remain a path [1].

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For the function of a specific load point it is enough for one of its minimal path to be functioning, that is all the path’s components are functioning. In logical terms this is an AND statement for the components within a path and an OR statement for all the paths belonging to a specific load point. For Test System 1a the minimal paths 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 be operational for the functioning of the load point.

In order to model a normally open disconnector, which can be closed and transfer 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 be used as an alternative route for power.

In Test System 1a the disconnector c18 can be closed to transfer power between the two load points. The normally open paths for each load point are shown in table 3.4. Test System 1b does not have any normally open paths because of the closed point in c18.

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

Load point Normally open paths LP5 [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 of the load point indices. The minimal cut sets of first order represents a serial system of components and the second order a serial system with the two components in each set in parallel. The formulas presented here are the ones used in the computer reliability program RADPOW. The program and some of the formulas are developed, amongst others, by [1].

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The first order minimal cut sets represents a serial reliability system which are shown 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 λ_s=

Xn i=1

λ_i (3.1)

U_s= Xn i=1

λ_ir_i (3.2)

r_s= P_n

i=1λ_ir_i λ_s = U_s

λ_s (3.3)

LOE_s= U_s· P_s (3.4)

, where P_sis the average capacity demand (kW) in the serial reliability system.

The second order minimal cut sets represents a parallel system with two components in which it is sufficient for at least one of the components to be functioning for 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 event has occurred. If the two failure types is of the same kind the reliability for the parallel 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)

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U₁₂= λ₁₂r₁₂≈ λ₁λ₂r₁r₂ (3.7)

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

λ^xy₁₂ = λ^x₁(λ^y₂r^x₁) + λ^x₂(λ^y₁r₂^x) + λ^y₁(λ^x₂r^y₁) + λ^y₂(λ^x₁r₂^y) (3.8) r_s^xy = λ^x₁(λ^y₂r^x₁)

λ^xy₁₂ · r^x₁r₂^y

r^x₁ + r₂^y +λ^x₂(λ^y₁r^x₂)

λ^xy₁₂ · r₂^xr₁^y r₂^x+ r₁^y + λ^y₁(λ^x₂r^y₁)

λ^xy₁₂ · r^x₁r^y₂

r^x₁ + r^y₂ +λ^y₂(λ^x₁r₂^y)

λ^xy₁₂ · r₂^xr₁^y

r₂^x+ r₁^y (3.9) There is one constraint, when it come to the scheduled maintenance, that makes these equations smaller and more practical. The constraint is that no operator would never ever take a component out for maintenance if this would cause a system failure. The failure rate for two overlapping events, where the first is the scheduled maintenance (m) and the next is a failure (x), can be described as [1]:

λ^xm₁₂ = λ^m₁ (λ^x₂r₁^m) + λ^m₂ (λ^x₁r^m₂ ) (3.10) The equation for the restoration time for this types of overlapping events are [1]:

r_s^xm= λ^x₁(λ^m₂ r₁^x)

λ^xm₁₂ · r^x₁r₂^m

r^x₁ + r₂^m +λ^x₂(λ^m₁ r₂^x)

λ^xm₁₂ · r^x₂r₁^m

r^x₂ + r₁^m (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]:

• λ^c1_lp - single failure events from minimal cut sets of first order,

• λ^c2_lp - overlapping failure events from minimal cut sets of second order,

• λ^a1_lp - additional active failures from single failure events, and

• λ^as_lp - additional active failures with the probability of non-functioning protection devices.

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

λ_lp = λ^c1_lp + λ^c2_lp + λâ1_lp + λâs_lp (3.12) U_lp = U_lp^c1+ U_lp^c2+ U_lpâ1+ U_lpâs (3.13) r_lp = U_lp

λ_lp (3.14)

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

For single failure events:

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lp c1

lp

Uc1 ^c2^lp U^c2^lp lp a1

lp

Ua1 lp as

lp

Uas

lp Ulp 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 cut sets vector of first order as [1]

λ^c1_lp = Xn i=1

(λ_p,i+ λ_te,i+ λ_tr,i) (3.15)

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

U_lp^c1 = Xn i=1

(λ_p,i· r + λ_te,i· r_c,i) (3.16)

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where the restoration time r is defined as [1]

r = (1 − P ) · r_s+ P · r_r/p,i (3.17)

and where P is the probability that the normally open path cannot be used. If that is the case, the restoration time is equal to the replace or repair time r_r/p,iof the failed component i. Here r_r/pstands for either repair or replacement restoration time. If the replacement time are greater than zero, this is always chosen before repair which normally takes longer time. If a open path is available, and are functioning with a probability of (1 − P ), the restoration time is equal to the switching time r_s because of the re-closure of the disconnector.

If there are no normally open paths for the load point to be used, the unavailability is given by

U_lp^c1= Xn i=1

(λ_p,i· r_r/p,i+ λ_te,i· r_c,i) (3.18)

The restoration time of first order failures are given by r_lp^c1= U_lp^c1

λ^c1_lp (3.19)

Minimal Cut Sets of Second Order

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

λ^c2_lp = λ_pp+ λ_pm+ λ_pte+ λ_tete+ λ_tem (3.20) For the two overlapping permanent failures, the Equations 3.5 to 3.7 are used [1]:

λ_pp= λ^p₁(λ^p₂r^p₁) + λ^p₂(λ^p₁r^p₂) (3.21)

r_pp= r^p₁r^p₂

r^p₁+ r^p₂ (3.22)

U_pp= λ_ppr_pp (3.23)

For the terms λ_pmand λ_tem, with a maintenance outage followed by an permanent or a temporary failure, Equation 3.10 and 3.11 are used [1]:

λ_pm= λ^m₁ (λ^p₂r₁^m) + λ^m₂ (λ^p₁r₂^m) (3.24)

r_pm= λ^p₁(λ^m₂ r₁^p)

λ^pm₁₂ · r^p₁r^m₂

r^p₁+ r^m₂ +λ^p₂(λ^m₁ r^p₂)

λ^pm₁₂ · r^p₂r^m₁

r^p₂+ r^m₁ (3.25)

Development of a Simulation Module for the Reliability Computer Program RADPOW

Development of a Simulation Module for the Reliability Computer Program RADPOW

JOHAN SETR´ EUS

Master’s Degree Project

Stockholm, Sweden 2006

D EVELOPMENT OF A S IMULATION MODULE FOR THE RELIABILITY COMPUTER PROGRAM

RADPOW

Master Thesis by Johan Setréus

Abstract

Sammanfattning

Acknowledgements

Contents

Chapter 1

Introduction

1.1 Background

1.2 Objective

1.3 Approach

1.4 Outline

Chapter 2

System Reliability

2.1 Introduction

2.2 Definitions and Abbreviations

Failure - outage

2.3 Test System

2.4 Reliability Indices

2.5 Distribution Functions

Chapter 3

RADPOW

3.1 Introduction

3.2 Overview of RADPOW

3.3 Reliability Evaluation in RADPOW

3.4 RADPOW_1999 version