Module for the Reliability Computer Program RADPOW - DiVA

Development of a Simulation Module for the Reliability Computer Program RADPOW

´ JOHAN SETREUS

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

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

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 contributions 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 RADPOW and the NEPLAN software.

iii

Sammanfattning Detta examensarbete beskriver hur en Monte Carlo simulering (MCS) kan användas för tillförlitlighetsanalys av ett eldistributionssystem. Metoden har implementerats i verktyget RADPOW som nu kan utföra både analytiska och numeriska beräkningar. Angreppssättet för att utveckla denna MCS metod i RADPOW innefattade följande aktiviteter: • Vidareutvecklade av RADPOW med införandet av ett grafiskt användargränssnitt för Windows. • Utveckling och implementering av en iterativ analytisk metod för känslighetsanalys av eldistributionssystem i RADPOW. • Utveckling och implementering av MCS metoden i RADPOW, vilken placerades i en fristående modul kallad Sim. Den implementerade MCS metoden har validerats i en jämförande studie innefattande två testsystem med datorprogrammet NEPLAN. Resultat från denna studie visar att MCS metoden ger samma resultat som den analytiska metoden i RADPOW och det kommersiella verktyget NEPLAN.

v

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

vii

Contents Abstract

iii

Sammanfattning

v

Acknowledgements 1

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1 1 2 2 3

System Reliability 2.1 Introduction . . . . . . 2.2 Definitions . . . . . . . 2.3 Test System . . . . . . 2.4 Reliability Indices . . . 2.5 Distribution Functions

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5 5 7 9 12 13

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17 17 17 19 21 34 35

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Sensitivity analysis routine 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Sensitivity analysis with random disturbance . . . . . . . . . . . .

43 43 43

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Monte Carlo Simulation Method for RADPOW 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Simulation Method . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 47

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Introduction 1.1 Background 1.2 Objective . 1.3 Approach . 1.4 Outline . .

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RADPOW 3.1 Introduction . . . . . . . . . . . . . 3.2 Overview of RADPOW . . . . . . . 3.3 Reliability Evaluation in RADPOW 3.4 RADPOW_1999 version . . . . . . 3.5 RADPOW_1999_PF version . . . . 3.6 RADPOW_2006 version . . . . . .

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5.3 5.4

Implementation in RADPOW . . . . . . . . . . . . . . . . . . . . Approximations and Weaknesses in Method . . . . . . . . . . . .

6 Comparative Studie of the Methods 6.1 Introduction . . . . . . . . . . . . . . . . . . . . 6.2 Test System 1 . . . . . . . . . . . . . . . . . . . 6.3 Birka System . . . . . . . . . . . . . . . . . . . 6.4 Validation of the Simulation method in RADPOW 6.5 Sensitivity Analysis Routine . . . . . . . . . . .

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55 55 55 57 61 64

7 Closure 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Discussion and Future Work . . . . . . . . . . . . . . . . . . . .

73 73 73

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

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75 75 77 78 79

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

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Chapter 1

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 simulations 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. 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 distribution systems [3].

1.3 Approach The first step in this study was to implement the already existing version of RADPOW 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, programmed 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].

1.4. Outline

1.4

3

Outline

Chapter 2 first defines important terms and abbreviations that are used in this thesis, 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 interacts 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 implemented 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.

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 techniques 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. 5

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 random 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) =

Z x −∞

fX (t)dt

(2.2)

If the function fX (x) exists and applies to the function in Equation 2.2, then X is a continuous random variable of the distribution. The function fX (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 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. Therefor the functions can be donated FX (x) = F (t) and fX (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.

2.2. Definitions

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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 , ..., xn , is defined as n 1X x= xj n j=1

(2.7)

Definition 2.7 The variance σ 2 , of the output set x, is defined as: σ(x)2 =

n 1 X (xj − x)2 n − 1 j=1

(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.

8


Failure outage

Damaging Fault

Non-Damaging Fault

Two models of failure

Two models of restoration

Permanent forced outages

Passive event

Active event

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].

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. • rr - Repair restoration time is the time it takes to make the component operational by repairing it. • rp - Replacement restoration time is the time it take to replace a component to make it operational. • rs - Switching restoration time is the time it takes for a manual or automatic switching 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 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.

10

Chapter 2. System Reliability c1

c3

c15

c9

B3

c10

c5 LP5

c7 B4

B6

B1 c11

c12

c16

c18 c2

c6

c4 c8

c17

c13

c14

LP6 B2

B5

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

bus

transformer

supply point

breaker

disconnector

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 typical 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.

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 engineer 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 assumed to be 1 hour. The engineer then replaces the breaker and connects it to the grid which takes the time rp , 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 rs + rr = 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 disconnector and then close it is here denoted rs as defined in Section 2.2.2. The switching time rs is 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 rs 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

12

Chapter 2. System Reliability 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 rs , here assumed to be 1 hour. The reparation of the bus then takes the time rr , 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 rs + rr = 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: • 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 [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, Ni represents the number of customers in load point i:

2.5. Distribution Functions

13

• System Average Interruption Frequency Index (SAIFI) [int/yr,cust]: P Ni λi SAIF I = P

(2.9)

Ni

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

(2.10)

Ni

• Customer Average Interruption Duration Index (CAIDI) [h/int]: P SAIDI Ni Ui = CAIDI = P

Ni λi

SAIF I

(2.11)

• Average Energy Not Supplied per customer served (AENS) [kWh/yr,cust]: P LOEi AEN S = P

(2.12)

Ni

• Average Service Availability Index (ASAI): P

ASAI =

2.5

Ni · 8760 − Ui Ni P Ni · 8760

(2.13)

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 information 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 0

for t ≥ 0, m > 0 otherwise

(2.14)

14

Chapter 2. System Reliability (

FX (x) =

0 1 − e−x/m

for x < 0 for x ≥ 0

(2.15)

Both (2.14) and (2.15) are illustrated in Figure 2.4 together with the survivor function, 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 ranDensity function fX(t)

Distribution function FX(t)

0.25 1 0.2 0.8 fX(t)

FX(t)

0.15 0.1

0.4

0.05 0

0.6

0.2

0

10

20

0

30

0

10

t Reliability function RX(t)

30

Failure rate function λ(t)

1

0.5

0.8

0.4

0.6

0.3 λ(t)

RX(t)

20 t

0.4

0.2

0.2

0.1

0

0

10

20 t

30

0

0

10

20

30

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 to be stated. Given a uniform stochastic variabel U ∈ U (0, 1), the stochastic variable Y = FY−1 (U ) have the distribution function FY (x), and Y is thereby exponentially distributed. For the exponential distribution the inverse can be solved straight forward and is as follows: 1 Y = − ln U (2.16) λ

2.5. Distribution Functions

2.5.2

15

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, σ 2 ) has the following density and distribution functions: 1 2 2 fX (x) = √ e−(x−µ) /2σ σ 2π 1 FX (x) = √ σ 2π

Z x −∞

e−(y−µ)

(2.17)

2 /2σ 2

dy

(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 Density function fX(x)

Distribution function FX(x)

0.4 1

0.35 0.3

0.8

FX(x)

fX(x)

0.25 0.2 0.15

0.6

0.4

0.1 0.2 0.05 0 −4

−2

0 x

2

4

0 −4

−2

0 x

2

4

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 q distributed random variable X ∈ N (0, 1) is then calculated as X = V1

−2 ln R R

16


Given X ∈ N (0, 1) the following relation can be used to produce a normal distributed variable Y ∈ N (µ, σ 2 ): Y = µ + σX

(2.19)

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 disadvantages. 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 calculates 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 17

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. System data: • Network topology

• Component reliability Input

Output RADPOW

• Customer and power data

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]

• 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.

3.3. Reliability Evaluation in RADPOW

19

RADPOW_1999

Comp

Netw

Branch

Lpind

Sind

Mincut

Minpath

Abreak

Aafail

Comp

Netw

Branch

Lpind

Sind

Mincut

Minpath

Abreak

Aafail

Loadflow

Comp

Netw

Branch

Lpind

Sind

Mincut

Minpath

Abreak

Aafail

Loadflow

RADPOW_1999_PF

RADPOW_2006

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.

20

Chapter 3. RADPOW

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

Author Lina Bertling Ying He Philippe Rosett Johan Setréus

Developed modules Mincut, Abreak, Aafail and Lpind Minpath, Netw, Branch, Comp and Sind Loadflow Sim

Year 1999

Main references [1], [8]

1999

[13], [8]

2000 2006

[14] 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. System Data

Network Model

Assign each LPs the events that lead to failure for that LP Simulation method

Analytical method Calculate the reliability indices for each LP with formulas

Make a large number of random experiments to see how these affect LPs reliability

Calculate the reliability for the system

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

3.4. RADPOW_1999 version

21

RADPOW Analytical methods Analytical calculation

Sensitivity analysis routine

Simulation method

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 analytical 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

22

Chapter 3. RADPOW

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 LP5 LP6

Minimal 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] [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].


23

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 LP5 LP6

Minimal paths [1 7 3 9 15 10 5] [1 7 3 11 16 12 5] [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 LP5 LP6

3.4.2

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

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].

24

Chapter 3. RADPOW

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

n

2

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 X

λi

(3.1)

λi ri

(3.2)

i=1 n X

Us =

i=1

Pn

rs =

i=1 λi ri

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 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 =

λ1 λ2 (r1 + r2 ) ≈ λ1 λ2 (r1 + r2 ) = λ1 (λ2 r1 ) + λ2 (λ1 r2 ) 1 + λ1 r1 + λ2 r2

(3.5)

r12 =

r1 r2 r1 + r2

(3.6)


25

U12 = λ12 r12 ≈ λ1 λ2 r1 r2

(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] y x y y x y x y x x y x λxy 12 = λ1 (λ2 r1 ) + λ2 (λ1 r2 ) + λ1 (λ2 r1 ) + λ2 (λ1 r2 )

(3.8)

rx ry λx (λy1 r2x ) rx ry λx1 (λy2 r1x ) · x 1 2 y + 2 xy · x2 1 y + xy λ12 r1 + r2 λ12 r2 + r1 y x y y y x y x λ1 (λ2 r1 ) r1 r2 λ2 (λ1 r2 ) r2x r1y · + · λxy r1x + r2y λxy r2x + r1y 12 12

rsxy =

(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]: m x m m x m λxm 12 = λ1 (λ2 r1 ) + λ2 (λ1 r2 )

(3.10)

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

3.4.3

x x λx1 (λm r1x r2m λx2 (λm r2x r1m 2 r1 ) 1 r2 ) · + · λxm r1x + r2m λxm r2x + r1m 12 12

(3.11)

Reliability Evaluation of Load Point Indices

The reliability for the load points, the load point indices, are calculated in RADPOW_1999 by summarizing the different failure rate contributors, which are [1]: • λc1 lp - single failure events from minimal cut sets of first order, • λc2 lp - overlapping failure events from minimal cut sets of second order, • λa1 lp - additional active failures from single failure events, and • λas lp - additional active failures with the probability of non-functioning protection devices. The reliability for the load point lp are then calculated as [1] c2 a1 as λlp = λc1 lp + λlp + λlp + λlp

(3.12)

c1 c2 a1 as Ulp = Ulp + Ulp + Ulp + Ulp

(3.13)

rlp =

Ulp λlp

(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:

26

Chapter 3. RADPOW Minimal cut sets of first order

c1 lp

Minimal cut sets of second order

c1

U lp

c2 lp

Additional active failures

a1 lp

c2

U lp

lp

Additional active failures with stuck probability

a1

U lp

as lp

as

U lp

U lp

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] λc1 lp =

n X

(λp,i + λte,i + λtr,i )

(3.15)

i=1

If there is a normally open path for the load point, the unavailability are defined as [1] c1 Ulp =

n X i=1

(λp,i · r + λte,i · rc,i )

(3.16)


27

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 is the case, the restoration time is equal to the replace or repair time rr/p,i of the failed component i. Here rr/p stands 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 rs 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 c1 Ulp =

n X

(λp,i · rr/p,i + λte,i · rc,i )

(3.18)

i=1

The restoration time of first order failures are given by c1 rlp

=

c1 Ulp

λc1 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] λc2 lp = 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 (λp2 r1p ) + λp2 (λp1 r2p )

(3.21)

r1p r2p r1p + r2p

(3.22)

Upp = λpp rpp

(3.23)

rpp =

For the terms λpm and λtem , with a maintenance outage followed by an permanent or a temporary failure, Equation 3.10 and 3.11 are used [1]: p m m p m λpm = λm 1 (λ2 r1 ) + λ2 (λ1 r2 )

rpm

p p λp1 (λm r1p r2m λp2 (λm r2p r1m 2 r1 ) 1 r2 ) = · p + · p λpm r1 + r2m λpm r2 + r1m 12 12

(3.24)

(3.25)

28

Chapter 3. RADPOW

and the unavailability is Upm = λpm rpm

(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]: p p te p te p te te p te λpte = λp1 (λte 2 r1 ) + λ2 (λ1 r2 ) + λ1 (λ2 r1 ) + λ2 (λ1 r2 ) p p λp2 (λte λp1 (λte r1p r2te r2p r1te 1 r2 ) 2 r1 ) + + · · r1p + r2te r2p + r1te λpte λpte 12 12 p te p te λte r1p r2te r2p r1te λte 1 (λ2 r1 ) 2 (λ1 r2 ) · · + r1p + r2te r2p + r1te λpte λpte 12 12 and the unavailability is

(3.27)

rpte =

Upte = λpte rpte

(3.28)

(3.29)

If there is a normally open path for the load point, the unavailability for the second order failures are defined as c2 0 0 0 0 0 Ulp = λpp · rpp + λpm · rpm + λpte · rpte + λtete · rrc + λtem · rtem , (3.30) 0 are the combined restoration time and where the rxy 0 rrc = (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 c2 Ulp = 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: c2 rlp =

c2 Ulp

(3.33)

λc2 lp

Additional active Failures of first order The contribution from the additional active failures are evaluated as [1] λa1 lp =

n X

(λa,i + λte,i + λtr,i )

(3.34)

i=1 a1 Ulp =

n X

(λa,i · rs,i + λte,i · rc,i )

(3.35)

,

(3.36)

i=1 a1 rlp =

a1 Ulp

λa1 lp

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


29

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 nonfunctioning of a breaker or a fuse, when it is stuck and can not open the circuit. The following equations are used to evaluate the contribution from the stuck probability in n active failures: λas lp =

n X

λa,i · Ps,i

(3.37)

i=1 as Ulp

=

n X

λa,i · rs,i

(3.38)

i=1 as rlp =

3.4.4

as Ulp λas lp

(3.39)

Implemented Method in RADPOW_1999

Figure 3.8 shows the overall data flow between the modules for the implemented analytical method in RADPOW. In the RADPOW_1999 version, the Loadfile box and 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 system and load point indices proceeds the following steps [1]: 1. The system data are transferred from the input data file *.radpow to the Loadfile routine that chops the information in the file into ten different sections. 2. Depending on the given data section, the modules Netw, Branch and Comp reads 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 the module 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 as output data. 8. For each load point that is going to be analyzed the steps 5 to 7 are performed. 9. The system indices are evaluated with the Sind module and with the load point indices and the customer data as input data.

30

Chapter 3. RADPOW

Data file *.radpow

Loadfile

Radpow.cpp

RADPOW_2006

Netw Branch Comp

Minpath

Abreak

Mincut

Aafail

Lpind

Loadflow

Sind

Output data

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 code logical and more understandable [1][13]. In reality a module consist of two files in C++, an cpp-file and a h-file. In the later one, the variables and methods are stated and these are then implemented further in the cpp-file. The advantage of working with modules is the ease of expanding the program with other functionalities or to use the modules separately in other projects. The main characteristic for a module in RADPOW is its independently to other modules and files involved. Branch, Comp and Netw The three modules Branch, Comp (Component) and Netw (Network) have been developed to read network and component data from an input text file and then store it in internal data containers which easily can be accessed by the other modules.


31

This simplifies the work for the other modules, in the way that there is no need to implement methods for reading data from text files in each module. The number of input files in RADPOW_1999 and RADPOW_1999_PF were eight respective ten separate text files, but in this thesis an effort has been made to gather these files into one file containing all data for the system. The work of developing a filesystem and how the text file is defined is described in Section 3.6.3. The modules are further described in [13]. The data processed by these three modules, which defines the system network modell, 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 different components. 3. Customer data - Describes the loads and number of customers in each load point. 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 topology into minimal paths from supply points to each load point in terms of component numbers. For the definition of minimal paths see Definition 3.3. As shown in Figure 3.9, the algorithm also indicates whereas a specific minimal path is normally closed (n/c) or normally open (n/o). The method and algorithm for obtaining the minimal paths are described in [13]. Mincut Netw Branch Comp

Input

Minpath

Output

Aafail

Testsystem 1a LP5 n/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] LP6 n/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 1b LP5 n/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] LP6 n/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 and indicates 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 and second order for each load point in terms of component numbers. For the definition

32

Chapter 3. RADPOW

of minimal cut set see Definition 3.1. Figure 3.10 shows how the module uses the minimal paths given from the Minpath module and branch data from the Branch module as input. The output from the module is for each load point a vector with the minimal cut sets, a vector with normally open minimal paths (see Definition 3.4) and the present load point number. The algorithm is described in [1] and its implementation in [8]. Netw Branch Comp

Aafail Input

Minpath

Testsystem 1a LP5 Minimal 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]

Mincut

Output

Lpind

Testsystem 1b LP5 Minimal 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 vector n/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 vector n/a

Figure 3.10: With the minimal paths for each load point, the Mincut module deduces the minimal 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 isolate the component and thereby the failure. The module Abreak (Associated breakers) deduces which breakers or fuses that are associated to a specific component in the system. The input data required by the algorithm includes the minimal paths to all load points, network topology data for the system and component data. The output is a list with all components, each with a vector containing the associated breakers and fuses for the component. The algorithm is described in [1] and its implementation in [8]. Aafail The module Aafail (additional active failures) deduces the additional active failure modes for each load point. These are components that are not included in the normal minimal cut set for the specific load point, but still if not functioning, cause the break down of the load point ,as defined in Definition 2.13. The algorithm is described in [1] and its implementation in [8]. There are two types of additional active failures defined in RADPOW [1]:


Minpath

33

Input

Abreak

Component 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Output

Aafail

Testsystem 1a Testsystem 1b [7] [7] [8] [8] [7, 9, 11] [7, 9, 11] [8, 13] [8, 13] [10, 12, 14] [10, 12] [10, 12, 14] [14] [9, 11] [9, 11] [13] [13] [7, 10, 11 ] [7, 10, 11 ] [9, 12, 14] [9, 12] [7, 9, 12] [7, 9, 12] [10, 11, 14] [10, 11] [8, 14] [8, 14] [10, 12, 13] [13] [9, 10] [9, 10] [11, 12] [11, 12] [13, 14] [13, 14] [10, 12, 14] []

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

• First order - Failure modes caused by a failed component and the trip of its associated breakers or fuses. • Second order - Failure modes caused by two failures. First a component fails and then its associated protection device is stuck and fails to isolate the failure.

Abreak

Netw Branch Comp

Input

Aafail

Output

Lpind

Mincut

Testsystem 1a LP5 Additonal active failures [9, 10, 11, 12, 15+9, 15+10, 16+11, 16+12]

Testsystem 1b LP5 Additonal active failures [6, 10, 12, 14, 18, 15+10, 16+12, 17+14, 13+14, 9+10, 11+12]

LP6 Additonal active failures -

LP6 Additonal 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 load point using the equations described in Section 3.4.3. The input data for each load point to the module is the minimal cut set vectors, the additional active failures

34

Chapter 3. RADPOW

vectors and the component reliability data. The results of the evolution are presented as the load point indices described in Section 2.4.1. Figure 3.13 shows the data flow to and from the Lpind module. The module is described in [1] and its implementation in [8]. Aafail

Netw Branch Comp

Input

Lpind

Output

Sind

Mincut For 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 separately.

Sind Given the indices for each load point, the Sind (system indices) module evaluates the system indices for the power distribution system. These system indices are defined in Section 2.4.2. The method and algorithm are described in [13]. Netw Branch Comp Input Lpind

Sind

Output

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 calculations and has the extension of one new module, Loadflow, and two new input data files pflo and pfrx. In the main file for this version there is an user defined parameter


35

that makes it possible to turn off the load flow calculations. By doing this RADPOW_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 and load point to deduce if any still functioning branches in the system are overloaded and 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 load point. The results from the module are the changes in reliability for the specified load point, and these are then added to the results from Lpind [14]. The running of this module is optional, which is the reason why this block is drawn with broken line 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 the Loadflow module was not used in this thesis. In Section 3.6.8 this module’s identified weaknesses that were found during this project are described. The Loadflow module is described in detail in [14].

3.6

RADPOW_2006 version

The RADPOW_2006 version has been developed by the author and contains the same core of RADPOW as the RADPOW_1999 and the RADPOW_1999_PF versions. The extensions is a graphical user interface, a new filesystem and two new methods of calculating the system indices, described in Chapters 4 and 5. One of these methods, the Monte Carlo Simulation (MCS) method, has been placed in a stand alone module referred to as Sim. The other routines that has been developed are referred to as files or routines because these does not fulfill the requirement of generality 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 describe the system analytical with mathematical equations, without making large approximations. In these cases a computer simulation can be preferable. In all simulation methods the model is tested with a series of experiments to see how it reacts on different events. The result from the experiments are then collected and evaluated. Normally it takes a large number of experiments to find solutions of a problem, and that is why this method is time consuming. When performing a simulation over time, stochastic samples from a given probability distribution are used to create different events. This simulation process are referred to as stochastic simulation and this is actually a statistical sampling experiment with the model [11]. Because one of the central problems in stochastic simulation is how to generate random numbers from different distributions, the

36

Chapter 3. RADPOW

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 code for a project involved in the development of the atomic bomb. The name comes from the gamling casinos at the city of Monte Carlo in Monaco [11]. There are generally two ways of performing MCS; random or sequential simulation. In the random approach the time intervalls are chosen randomly, while as in the sequential the intervalls are chosen in chronological order. In reliability studies it is most likely that one time intervall depends on previous ones, and therefor the sequential 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 different random events in order to perform a simulation of the model. The times for the different random events need to be generated and for this an appropriate random number generator has to be used. Generating Random Numbers The generation of random numbers is the most essential aspect in Monte Carlo simulation. In order to make a realistic simulation of the system, all states of the system needs to be performed, and this means adequate input of random events. When it come to generate random numbers there are both software and hardware generators available. The algorithms that these uses are divided into deterministic and non-deterministic methods. The non-deterministic algorithms are usually called pseudo-random generators, which is the algorithm used in this thesis. The name pseudo reveals that this is not a true random generator mathematically, and that is because it repeat itself after a large sequence of numbers. The generator itself are initiated with a number, a seed, which sets the starting point of the sequence. The same initiating seed at two different occasions will always give the same sequence of numbers and therefor the seed also has to be random. One way of solving this problem is to initiate the generator with the current time in seconds, which is easy to implement. A adequate random generator has to fulfil the following aspects: • It has to represent the chosen distribution, which normally is the uniform distribution. • 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 itself are large enough, the pseudo-random generator are ideal for MCS because of its speed in calculation.


37

Random generator algorithms normally produces uniform distributed numbers in the interval (0, 1). These uniform distributed numbers can then be transformed to represent an arbitrary distribution as described below [12]. Generating Random Numbers from arbitrary Distributions If an inverse, FY−1 (t), exists to the distribution function FY (t), the inverse transform 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 transformed to the distribution FY . For the exponential distributed function the inverse exists as showed in Equation 2.16, and hence the inverse transform method can be used. But for other, more complicated functions, as the normal distributed function, there are no analytical way of solving the inverse and therefor an iterative method of generating random numbers has to be used, as described in Section 2.5.2. Transform with F-1y

Generate a uniform random number

Random number with the new distribution

1

Y=F-1y(X)

x

X~U(0,1)

0

Y~Fy(t)

y

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

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, several help files has been developed within this thesis project. In Figure 3.16 a schematic picture of the new files are shown. Note that the sim file is identical with the Sim module. Of the files in the picture, Radpow, Sim and Random are for the calculation methods, Loadfile for the data input in RADPOW and the others for the graphical user interface. The arrows between some of the files in the figure symbolizes the data flow. The Radpow file represent several different modules as described in plot.cpp

about.cpp

loadfile.cpp sim.cpp

Project.cpp

Main.cpp

radpow.cpp

random.cpp ind.cpp

newfile.cpp

info.cpp

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

38

Chapter 3. RADPOW

Section 3.4.4 and illustrated in Figure 3.8. Radpow The Radpow module contains the actual core of the program RADPOW. It combines the different modules described in Section 3.4.5 and has accessors and methods to initiate and perform a calculation of a system. The calculation can either be performed by an analytical method or by a simulation method. The results returned from this module are • the system and load point indices and • extended information about the system, such as the minimal cut sets or the minimal paths. Sim The main objective of this thesis has been to develop a module for a simulation approach for the evaluation of the system indices of a power distribution system. The simulation method and its implementation, the Sim module, are described in Chapter 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++ random number 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 random numbers from the Exponential or Normal distribution, given µ and σ. The purpose of implementing the random number generator in a stand alone module is for the ease of developing the random generator further with more advanced methods of producing random numbers. File System with Loadfile In the RADPOW_1999_PF version of RADPOW the system input data were separated 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. In addition, information of these text files needed to be entered for the internal data structure 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 going to be implemented or changed in the text files, the core of RADPOW had to be recompiled in order to perform the calculations. To overcome this, the file Loadfile were developed by the author. This routine contributes with the following abilities:


39

• 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 in RADPOW. • The Loadfile evaluate the size of the system with the given input data file. By adding this ability, there is no need to recompile the source code of RADPOW each time the system is changed. The ten different data files used in RADPOW_1999_PF has simply been pasted into one file, each into different sections. Each section consists of a header containing the name of the section, followed by the specific data. The data input file for RADPOW and its ten sections with the data parameters are described in Appendix A.

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 is completely generated by Borland C++.

40

Chapter 3. RADPOW

The Main file contains methods that executes when an action is made by the user, which normally is a press on a button. The different actions starts one of the following methods that may use other files or modules: • Analytical calculation of the system indices in the file Radpow, after the system has been initiated. • Simulation of a system in the file Radpow in order to determine the system indices and its deviations. • Iterative analytical calculations with a disturbance in the input data in order to determine the deviation of the system indices. This is performed with the file Radpow. • Graphical presentation of the results of an iterative calculation or an simulation 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 minimal paths, 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 calculation in a graphical window. The used indices are the same as the ones described in Section 2.4.2. Info Given a system that has been initiated or calculated, the Info file shows a graphical window with extended information of the system. The window opens with the button Extended Info in RADPOW. The information presented in this window includes • 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 the system has been defined correctly in the input data file for the system.


41

Plot Given a number of consecutive calculations of the system indices, the Plot file sorts the results into intervalls and counts the hits in each interval for each index. The hits in these intervals are then plotted in a graph in the plot window. A algorithm determines a predefined interval length, which can be changed by the user. The user 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 version RADPOW_2006, the file Newfile has been developed. As described earlier in this section the system data in RADPOW_1999_PF were separated into ten different files. The file Newfile contains a window that lets the user define which files that are 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 authors of RADPOW and contact information. The window opens with About in the Help menu.

3.6.5

Identified Weaknesses and Corrections

When working with the development of the simulation module in RADPOW, the output data from each module were checked for different test systems. In this work a number of failures and weaknesses in the program were found and corrected and a 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. For larger systems like the Birka System, described in Section 6.3, the conversion from minimal paths to minimal cut sets of second order was incorrect. The bug causing this problem was found in the method mc2vf in the Mincut module. The cause and 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 breakers as described in Section 3.4.5. Figure 3.18 shows a simple test system with two load points that is used to illustrate the upcoming weakness. If an active failure occurs in the components c5 or c6 the breaker c3 will trip and both the load points

42

Chapter 3. RADPOW

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

c4

c1 c2

c5

c3

LP4

LP6

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

3.6.8 Loadflow The Loadflow module was developed in an old C++ version which were not compatible with the C++ standard. Even if a quite large effort has been made in this thesis to convert the code to the C++ standard, the module did not work properly for larger system as singularities occurred. The module was not used in the analysis in 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 specific component in the ctype-section, eg. BR220, but this type is missing in the cereliasection, RADPOW does not rise a warning to the user, instead it uses an arbitrary component 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 implemented for RADPOW by the author. The method and its implementation in RADPOW are then validated in Chapter 6.

4.1

Introduction

A major part of reliability analysis of power systems is the acquire of accurate component reliability and system data. If reliability data for the system and its components are available, it is normally uncertainties in this data due to measuring uncertainties, small population of similar components or poor documentation routines. Sometimes reliability data for a certain component does not exists at all and then standard values from tables with perhaps large uncertainties has to be used. All these uncertainties in input data affects the output results, which for RADPOW is the load point and system indices. In order to derive these uncertainties in the output 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 a large number of analytical calculations with a randomized input parameters. The already existing analytical method in RADPOW has been used for the calculations in the implemented routine that performs these iterative calculations. In RADPOW there are a number of different input parameters that could be studied in a sensitivity analysis. These input parameters can be divided into three major categories [1]: 1. Component reliability data including failure rates, restoration times, maintenance 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 category one, the parameters for the component reliability, has been studied. These eleven component parameters are loaded by RADPOW from the Cerelia-section in the input 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 in a measured or statistical evaluated parameter of an item. The normal distribution is described in Section 2.5.2. Figure 4.1 illustrates how each input parameter for the 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 BoxDeviation of parameters for components, d [%] Generate a normal distributed number Parameter value, p from input data file

Value of one parameter for one component

Expected value of parameter, E[X] = p Deviation,

= E[x] * d -

E[X]

+

Figure 4.1: For each parameter in all components in the system a random value is generated based 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 generates a random number X ∈ N [0, 1] which with Equation 2.19 is used to generate the generalized 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 calculated as Y = µ + σX = p + p · d · X

(4.1)

where Y ∈ N [µ, σ 2 ].

4.2.2 Method and Implementation The iterative method used for the sensitivity analysis has been implemented in the overall Main file described in Section 3.6.4. The implemented method uses the analytical calculation method described in Section 3.4.4, which is implemented in the Radpow file. Figure 4.2 shows the overall method, with the required input data and the output results. The overall algorithm to deduce the deviation of the system indices contains the following steps.

4.2. Sensitivity analysis with random disturbance

45

Component data #iterations Input ++i

Generate random parameters for each component

Run RADPOW Analytical method

Store results

no

Deviation in component parameters [%] 1

, r, sw, ... n

SAIFI, SAIDI, CAIDI, AENS, ASAI SAIFI SAIDI CAIDI AENS ASAI x x x x x

#iteration > i yes Evaluate results

Output

- Mean vaules of indices - Max and min - Deviation

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 parameters for each component type. Get the number of iterations and the deviation for all parameters from the user. 2. For each parameter in each component, generate the new value of the parameter, given the specified value in file and the chosen deviation of component as illustrated in Figure 4.1. 3. Start the analytical method in RADPOW with this specific component configuration 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 evaluate the mean, max and min value and the deviation.

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

Chapter 5

Monte Carlo Simulation Method for RADPOW This chapter describes the simulation method developed and implemented in RADPOW by the author. The method and its implementation are then validated in Chapter 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 difficult or even impossible to describe the system analytical with mathematical equations, without making large approximations. In these cases a computer simulation can be adapted. In all simulation methods the model is tested with a series of experiments to see how it reacts on different events. The result from the experiments are then collected and evaluated. Normally it takes a large number of experiments to find solutions of a problem, and that is why this method is time consuming. In order to develop a simulation method for RADPOW the Monte Carlo Simulation (MCS) technique has been adopted and implemented in a new module referred to as the Sim module.

5.2 5.2.1

Simulation Method Component states

In order to make a simulation analysis on a power distribution system, the components involved in the system needs to be studied. As the system consists of several interconnected components, each having a probability to fail, one has to deduce how a specific component affect the system and its load points, given the status of the component. In the simulation method in this thesis, each component has been assigned an integer defining the present status of the component. Three different component 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 a time to failure (TTF), which will affect the component in the future. When the TTF has been reached, the component suffers an active failure and needs to be disconnected in order to restore the component. According to the definitions in Section 2.2.2, 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 from the beginning. System Data

Network Model

Assign each LPs the events that lead to failure for that LP Simulation method

Analytical method Calculate the reliability indices for each LP with formulas

Make a large number of random experiments to see how these affect LPs reliability

Calculate the reliability for the system

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-pointdriven approach, which means that all possible failures for each load point are deduced separately. This deduction results in the minimal cut set of first and second order and the additional active failures (see Definition 2.13), which are available for each load point. The simulation method developed for RADPOW uses an eventdriven approach, which means that all failure events are treated separately to see the 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 load points that are affected, and this is achieved by the already deduced minimal cut sets and the additional active failures for each load point. Figure 5.1, from Chapter 2, shows how the simulation method interacts with the already developed analytical method in RADPOW. Depending on the component state described previously in Section 5.2.1, a failed component with state 1 or 2 will affect the load point differently. If a component is in state 1 it affects all the load points having this component included in its minimal cut sets of first order or in the additional active failures. On the other hand, if the component is in state 2, it only affects the load points having this component in its minimal cut sets. For the second order failures, both state 1 and 2 for each of the two components will affect the load points having these components in its minimal cut sets of second order.

5.2.3

Algorithm

The MCS algorithm used in this thesis is described in Figure 5.2. The algorithm proceeds the following steps, as numbered in Figure 5.2. 1. The input data consist of the number of samples (N ), simulation time (T stop), component reliability data for the system, the minimal cut set vectors, normally open paths and the additional active failure vectors for each load 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 (T tot) 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 using the 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 interval is donated ∆t. During this intervall the system is stable. Count up the total time (T tot) with ∆t and decrease all the times for the components in the system with ∆t. 4. Check how the LP:s were affected during this interval and with the current states of the components. Use the minimal cut sets and the additional active failure 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 possible count up the outage time for the LP with the switching time for the disconnector. If normally open paths do not exists, add the total interval length to the outage time for the LP. Then check if the LP suffered an outage the interval 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 a new one (depending on its previous). The state for the component follows

50

Chapter 5. Monte Carlo Simulation Method for RADPOW Input data

0

2. All components are functioning. Generate time to failure for each component

n = n +1

TTF3

TTF10 TTF8

TTF1

Stop time

years Ttot = 0 Current time

0

3. Jump to the next event Ttot = Ttot + t

TTF3

TTF10 TTF8

TTF1

years

Ttot = Ttot + t t Current time

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

4. How are the LPs affected during the time interval and with the current component status?

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.

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

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

6.Update the component status after the interval. Produce a new time for component, depending on next status 0, 1 or 2. 0 7a. New TTF is generated

no

1 7b. SW-time from data

0

TTF3

TTF10 TTF8

TTF1

SW3 years

2 7c. Rep-time from data

Current time

8. Ttot > Tstop ? yes

9. n == N ? no yes

10. Evaluate the outages for each load point and each iteration

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 easy to determine the next state. 7a. If the component were in state 2 in the interval, it is now functioning, and hence its new state is set to 0. A new random TTF for the component is generated. 7b. If the component were in state 0, its new state is set to 1. The time for isolating the component, the switching time, is set to the present static value from the input data. 7c. If the component were in state 1, its new state is set to 2. The time for repair or replacement of the component, is set to the present static value from the input data. It has been assumed that the component always is replaced if this option is available.

5.3. Implementation in RADPOW

51

8. If the time for simulation, T stop, is reached, stop the current simulation and save the outage data for each LP. If not, proceed at step 3. 9. If the current number of simulated samples, n, equals the predetermined samples, N , the total simulation phase is finished. If not, restart from step 2 and count up n by one. 10. Evaluate the data from all samples, with the system indices and its standard deviations 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 and then the statistical data are evaluated for all the samples, with the mean values and variances of the samples. The chosen simulation time in a simulation analysis of a power system depends on the complexity of the system and the required accuracy of the output results. In the proposed method the total simulation time consists of the chosen number of simulation samples and the sample time in years. In order to determine these input parameters one has to see how the output results converges for the specific system with different sample lengths in years. These parameters depends on the size of the system and the reliability data for the components; the smaller probabilities for the components to fail, the longer sample times are needed to comprehend the system behavior. If some events happens very occasionally, but have a large impact on the system, a large number of samples or a long sample time is needed for the simulation. 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 module Sim. The Sim module works together with the other modules in RADPOW as shown in the flow chart in Figure 5.3. The minimal cut sets and the additional active failures are deduced in the same way as in the analytical method described in Section 3.4.4. A comparison of the flow chart in Figure 5.2 and the flow chart for the analytical method in Figure 3.8, shows that the only differences is the exchange of the Lpind module with the Sim module. The input data to the Sim module is besides the minimal cut sets and the additional active failures, also the normally open paths and the component reliability data. The output data from the module are then delivered either directly to the user as load point indies, or to the Sind module for evaluation of the system indices.

52

Chapter 5. Monte Carlo Simulation Method for RADPOW

Input data

Data file *.radpow

#iterations = N Sim time = Tstop

Radpow.cpp

Loadfile

Netw Branch Comp

Minpath

Abreak

Mincut

Aafail

Sim

Sind

Output 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 limitations for 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 not considered. 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 one hour. Besides these limits in the method, there are only failure modes of first and second order included in the model as the minimal cut sets do not include failure modes of higher order.

Chapter 6

Comparative Studie of the Methods This chapter validates the results from the sensitivity analysis routine and the simulation method, by comparing the results from two different test systems and computer analysis program.

6.1

Introduction

In order to validate the results from the developed simulation method and the sensitivity analysis routine for RADPOW, two test systems have been used. These systems are then evaluated with different methods and computer programs to compare and validate the results. First the two test systems, referred to as Test System 1 and the Birka System are presented in detail with the necessary data. These both systems has been implemented and analyzed in RADPOW and in the commercial tool NEPLAN. NEPLAN is an electric power analyzer which has been developed by the BCP group in Switzerland. This software package is used mainly for transmission and distribution system analysis [18]. The analys work and implementation in NEPLAN has been performed in companion 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 the various methods in this thesis. This system is the same as presented in Section 2.3 and origins from [1] and [8]. Test System 1 is divided into Test System 1a and Test System 1b, with the only difference that the first is considered to have a normally open disconnector in c18, and the later a closed point in c18. The system is described in detail in Section 2.3. 55

56

Chapter 6. Comparative Studie of the Methods c1

c3

c15

c9

B3

c10

c5 LP5

c7 B4

B6

B1 c11

c16

c12 c18

c2

c6

c4 c8

c17

c13

c14

LP6 B2

B5

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. Additional data such as the customer type; industrial, residential or commercial have not been included in this table because these data do not affect the results in this studie. Table 6.1: Customer and power data for the two load points in Test System 1 [8].

Load point LP5 LP6

Number of customers 100 80

Active power [kW/cust] 4.0 5.0

6.2.2 Component Reliability Data Table 6.2 presents the component reliability data for Test System 1. Changes with the source data in [8] and the data presented here has been made and these are as follows: 1. All the permanent failures have been considered to be active and consequently the active failure rate has been set to the same value as the permanent. 2. The replacement of components is not considered. 3. The stuck probability for the breakers is set to zero and is thereby not considered.

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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

6.2.3

Type Bus Bus Bus Bus Bus Bus Break Break Break Break Break Break Break Break Transf Transf Transf Discon

λpermanent [f/yr] 0.001 0.001 0.001 0.001 0.001 0.001 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.015 0.015 0.015 0.002

λactive [f/yr] 0.001 0.001 0.001 0.001 0.001 0.001 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.015 0.015 0.015 0.002

rrepair [h] 2 2 2 2 2 2 24 24 24 24 24 24 24 24 15 15 15 4

rreplace [h] -

Input Data File

Appendix B shows the input data file for Test System 1. The file presented in this appendix includes the changes made in the component reliability data, as described in 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 system belonging to Fortum (previously by Birka Energy). The system model was first presented by [1] in a maintenance and reliability studie with the RCM methodology. Figure 6.2 shows a network model for the Birka System, including one supply point and three load points. The sources of input data for the model and estimations and 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 distribution system where it has been deduced from.

58

Chapter 6. Comparative Studie of the Methods

c1 c2

c8

c3

c9 c4

c10

B1

B2 c5

bus

transformer

breaker

fuse

c11 line

c6

c12 c7

supply point

load point

c13

c14 B15

c49 c50

c15

c19

c54 B17 c55

c51 B16 c52

c20 B3

B4 c21

c17 B18

c57

c18

B5 c25

c22

c27

c58

c24

B6

B10

c36

c53 c16

c56

c23

B11 c38

c37 B12 c11

c40 B13 c41

c43

c39

c42

c45

c46

B14

c26

c44

c47

c48

c28

LP58

LP48 c29 B7

B8 c30

c31

c32 c33

B9

c34 c35 LP35

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

6.3. Birka System

6.3.1

59

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 has not 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 LP35 LP48 LP58

6.3.2

Number of customers 447 23400 1

Active power [kW/cust] 1.7203 0.9829 0.80

Component Reliability Data

Table 6.4 presents the component reliability data for the Birka System. Changes with the source data in [1] and the data presented here has been made and these are as follows: 1. The passive failure failure rate has been set to zero for all components. According 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 the temporary 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 structure of this file is described in Appendix A. Changes form earlier versions of the Birka System, used in [1], have been stated in the previous section. Besides the previous changes in the input data file one major change were made from the source file provided by [1]. In the source, the component type BR220 was not defined, although two components, c2 and c8, had been specified to this type. RADPOW does not rise a warning for this, instead the two components are 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 this problem was to change component c2 and c8 to the component type BR110, which according to [1] has exactly the same component reliability data as BR220.

60


Table 6.4: Component reliability data for the Birka System [1]. Component 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

Type Bus Break Transf Break Cable Transf Break Break Transf Break Cable Transf Break Bus Break Cable Transf Break Break Cable Transf Break Break Cable Transf Break Bus Break Bus Cable Cable Bus Transf Fuse Bus Bus Break Cable Break Break Cable Break Break Cable Break Bus Break Bus Bus Break Cable Break Break Cable Break Bus Break Bus

λpermanent [f/yr] 0.00964 0.00870 0.02610 0.00870 0.07012 0.02050 0.00089 0.00870 0.02610 0.00870 0.07031 0.02050 0.00089 0.00964 0.00089 0.00028 0.01989 0.00243 0.00089 0.00028 0.01989 0.00243 0.00089 0.00028 0.01989 0.00243 0.00867 0.00243 0.0 0.10069 0.10069 0.0 0.00331 0.01340 0.0 0.0 0.00089 0.02291 0.00089 0.00089 0.02285 0.00089 0.00089 0.02265 0.00089 0.0 0.00089 0.00964 0.0 0.00089 0.00863 0.00089 0.00089 0.00837 0.00089 0.0 0.00089 0.00964

λactive [f/yr] 0.00964 0.00870 0.02610 0.00870 0.07012 0.02050 0.00089 0.00870 0.02610 0.00870 0.07031 0.02050 0.00089 0.00964 0.00089 0.00028 0.01989 0.00243 0.00089 0.00028 0.01989 0.00243 0.00089 0.00028 0.01989 0.00243 0.00867 0.00243 0.0 0.10069 0.10069 0.0 0.00331 0.01340 0.0 0.0 0.00089 0.02291 0.00089 0.00089 0.02285 0.00089 0.00089 0.02265 0.00089 0.0 0.00089 0.00964 0.0 0.00089 0.00863 0.00089 0.00089 0.00837 0.00089 0.0 0.00089 0.00964

rrepair [h] 1 168 504 168 168 504 72 168 504 168 168 504 72 1 72 48 504 48 72 48 504 48 72 48 504 48 1 48 0 6 6 0 48 4 0 0 72 48 72 72 48 72 72 48 72 0 72 1 0 72 48 72 72 48 72 0 72 1

rreplace [h] 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 -

6.4. Validation of the Simulation method in RADPOW

6.4

61

Validation of the Simulation method in RADPOW

In order to validate the results from the simulation method, both the analytical part of RADPOW and the commercial tool NEPLAN has been used. The two test systems presented earlier in this chapter has been analyzed with these three different methods and the results has then been compared. In the simulation, 10000 samples with a length of 1000 years have been performed for both the systems. This sample time is of course not realistic, but a lower value of this result in under estimations of both outage frequency and duration, because of the assumption that all components are functioning from the beginning and the relatively high component reliability. The simulation parameters met the required accuracy of three decimals in the results. The results converged to the same values, within three decimals, when performing a number of simulations with these parameters. The total computation time where about 30 seconds on a normal computer for the Birka System.

6.4.1

Test System 1

Table 6.5 shows the failure rates per year for the two load points in Test System 1. The results from the simulation method in RADPOW are compared both to the analytical 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 [f/yr] Test System 1a LP5 LP6 Test System 1b LP5 LP6

RADPOW Simulation

RADPOW Analytical

NEPLAN

0.102 0.078

0.103 0.078

0.103 0.078

0.064 0.064

0.064 0.064

0.064 0.064

failure rates from the different methods reveals that the results from the simulation method is accurate. A comment to the result for Test System 1a, would be that the redundancy in transformators for LP5 increases the failure rate instead of decreasing it. LP6 has a single transformator feeding it and has a lower failure rate. This is due to the extra failure rates that are added by the additional breakers involved of the double feeding of LP5. In Test System 1b the failure rates in LP5 and LP6 are identical because of the closed point c18 connecting these. Table 6.6 presents the unavailability for the two load points in hours per year for Test System 1. These results are also very accurate when comparing the differ-

62


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

Load point [h/yr] Test System 1a LP5 LP6 Test System 1b LP5 LP6

RADPOW Simulation

RADPOW Analytical

NEPLAN

0.103 0.079

0.104 0.079

0.104 0.079

0.065 0.065

0.065 0.065

0.065 0.065

ent methods. The unavailability for the load points in Test System 1a and 1b are almost equal or slightly higher than the failure rates, which means that the restoration time for the load point is around one hour. These results are logical; for Test system 1a the disconnector is closed in one hour and for Test System 1b the affected component is switched off within one hour and then gets power from the common busbar c18. The slightly higher value than one hour is explained by the interruptions caused in busbars c5 and c6, which is repaired within two hours. The load point indices are evaluated by the computer programs to system indices with the additional customer data in Table 6.1. Table 6.7 presents the system indices for Test System 1. As shown in the table, the results are very accurate, and there are only minor differences. Table 6.7: Evaluated system indices for Test System 1. Both RADPOW and NEPLAN has been used to validate the results.

System Indices Test System 1a SAIFI[int/yr.cu] SAIDI[h/yr.cu] CAIDI[h/int] AENS[kWh/yr.cu] ASAI Test System 1b SAIFI[int/yr.cu] SAIDI[h/yr.cu] CAIDI[h/int] AENS[kWh/yr.cu] ASAI

RADPOW Simulation

RADPOW Analytical

NEPLAN

0.091 0.093 1.013 0.405 0.99999

0.092 0.093 1.011 0.407 0.99999

0.091 0.092 1.012 0.407 0.99999

0.064 0.065 1.017 0.288 0.99999

0.064 0.065 1.017 0.289 0.99999

0.064 0.065 1.017 0.289 0.99999

6.4. Validation of the Simulation method in RADPOW

6.4.2

63

Birka System

Table 6.8 shows the failure rates per year for the three load points in the Birka System. The results from the simulation method in RADPOW are compared both to the analytical results in RADPOW and in NEPLAN. The failure rate is significant at LP35 compared with the other two load points. The average failure rate for a LP35 customer is about 0.28 failures/year. For an LP48 or LP58 customer the failure 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 [f/yr] LP35 LP48 LP58

RADPOW Simulation 0.282 0.059 0.058

RADPOW Analytical 0.282 0.059 0.058

NEPLAN 0.278 0.057 0.056

The results in failure rates are equal for the simulation method and the analytical method in RADPOW. For NEPLAN there is a minor difference in results and this is probably explained by the different evaluation method used compared to the analytical method in RADPOW. The fact that the function of fuses is not included in the simulation model (see Section 5.4), does not seem to affect the results for LP35. Table 6.9 presents the unavailability for the load points in hours per year. The results for the unavailability are almost equal for the simulation as for the analytical method 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 [h/yr] LP35 LP48 LP58

RADPOW Simulation 0.474 0.099 0.098

RADPOW Analytical 0.475 0.100 0.099

NEPLAN 0.470 0.098 0.096

The load point indices are evaluated by the programs to system indices with the additional customer data in Table 6.3. Table 6.10 presents the system indices for the Birka system. The evaluated system indices are almost equal when comparing the different methods. CAIDI differs a bit more than the other indices, which is explained by the fact that this index depends both on the failure rates and unavailabilities for the load points.

64


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

System Indices SAIFI[int/yr.cu] SAIDI[h/yr.cu] CAIDI[h/int] AENS[kWh/yr.cu] ASAI

RADPOW Simulation 0.063 0.105 1.671 0.110 0.99999

RADPOW Analytical 0.063 0.107 1.683 0.111 0.99999

NEPLAN 0.062 0.104 1.703 0.113 0.99999

6.4.3 Conclusions The results from the comparative studie of the two test systems clearly shows that the results from the implemented simulation module are valid. Only minor differences are present when the results are compared with the analytical method in RADPOW and with NEPLAN. Although the simulation module is valid for the two test systems presented in this chapter, one has to keep in mind that a number of approximations and simplifications are involved in the simulation method, as described in Section 5.4. The both test systems has been modified, as described earlier in this chapter, in order to not take these parameters into account.

6.5 Sensitivity Analysis Routine The system indices from the evaluation methods in previously version of RADPOW are average values. There will be variations in these indices depending on the accuracy in the input parameters for the components. As described in Chapter 4 the parameters for each component are assumed to be normal distributed and in this analysis a deviation of 10% for all components are assumed. For both of the test systems the mean values of the method has been compared to the results in the previous section. The deviation of the indices has not been validated 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, performed by 5000 samples with different component values. The maximum and minimum values from these analysis are specific and will differ between two different analysis. The number of decimals are chosen with respect to the convergence of each index. Several analysis with different number of samples has been performed to validate the consistency within these number of decimals. The results for the mean 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. The uncertainty is 10% in all component parameters.

System Indices Test System 1a SAIFI[int/yr.cu] SAIDI[h/yr.cu] CAIDI[h/int] AENS[kWh/yr.cu] ASAI Test System 1b SAIFI[int/yr.cu] SAIDI[h/yr.cu] CAIDI[h/int] AENS[kWh/yr.cu] ASAI

Mean value

Max sample

Min sample

Deviation σ

0.092 0.093 1.01 0.407 0.999990

0.112 0.136 1.38 0.592 0.999993

0.070 0.059 0.69 0.259 0.999984

0.0059 0.010 0.090 0.044 1.2×10−6

0.064 0.065 1.02 0.289 0.99999

0.089 0.098 1.39 0.434 0.999996

0.041 0.039 0.69 0.174 0.999989

0.0059 0.0085 0.095 0.038 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 component parameters are the same.

66


SAIFI (int/yr.cust) 350 300

Hits/interval

250 200 150 100 50

0.067 0.071

0.076

0.081

0.086 0.091 0.096 (int/yr.cust)

0.101

0.106

0.111

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

SAIDI (h/yr.cust) 220 200 180

Hits/interval

160 140 120 100 80 60 40 20 0.056 0.063

0.072

0.081

0.090 0.099 (h/yr.cust)

0.108

0.117

0.126

0.135

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


67

CAIDI (h/int) 240 220 200 180 Hits/interval

160 140 120 100 80 60 40 20 0.670 0.740

0.820

0.900

0.980

1.060 (h/int)

1.140

1.220

1.300

1.380

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

AENS (kWh/yr.cust) 240 220 200 Hits/interval

180 160 140 120 100 80 60 40 20 0.245

0.280

0.320

0.360

0.400 0.440 (kWh/yr.cust)

0.480

0.520

0.560

0.600

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

68



Hits/interval

160 140 120 100 80 60 40 20 0.056 0.063

0.072

0.081

0.090 0.099 (h/yr.cust)

0.108

0.117

0.126

0.135

Figure 6.7: Distribution of 5000 samples of ASAI, for Test System 1a with 10% deviation in 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, performed by 4000 samples with different component values. The maximum and minimum values from these analysis are specific and will differ between two different analysis. The number of decimals are chosen with respect to the convergence of each index. Several analysis with different number of samples has been performed to 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 Indices SAIFI[int/yr.cu] SAIDI[h/yr.cu] CAIDI[h/int] AENS[kWh/yr.cu] ASAI

Mean value 0.063 0.107 1.68 0.111 0.99999

Max sample 0.077 0.136 2.10 0.140 0.999990

Min sample 0.050 0.085 1.37 0.090 0.999985

Deviation σ 0.0036 0.0069 0.10 0.0070 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 component parameters are the same.


69

SAIFI (int/yr.cust) 220 200 180

Hits/interval

160 140 120 100 80 60 40 20 0.0490

0.0525

0.0565

0.0605 0.0645 (int/yr.cust)

0.0685

0.0725

0.0765

Figure 6.8: Distribution of 4000 samples of SAIFI, performed on the Birka System with 10% deviation in all component parameters. The interval for each bar is 0.0005.


Hits/interval

180 160 140 120 100 80 60 40 20 0.083 0.088

0.094

0.100

0.106 0.112 (h/yr.cust)

0.118

0.124

0.130

0.136

Figure 6.9: Distribution of 4000 samples of SAIDI, performed on the Birka System with 10% deviation in all component parameters. The interval for each bar is 0.001.

70


CAIDI (h/int) 160 140

Hits/interval

120 100 80 60 40 20

1.350 1.420

1.500

1.580

1.660

1.750 (h/int)

1.830

1.910

2.000

2.090

Figure 6.10: Distribution of 4000 samples of CAIDI, performed on the Birka System with 10% deviation in all component parameters. The interval for each bar is 0.01.

AENS (kWh/yr.cust) 120 110 100

Hits/interval

90 80 70 60 50 40 30 20 10 0.0885 0.0940

0.1010

0.1080

0.1150 0.1220 (kWh/yr.cust)

0.1290

0.1360

Figure 6.11: Distribution of 4000 samples of AENS, performed on the Birka System with 10% deviation in all component parameters. The interval for each bar is 0.0005.


71

ASAI 240 220 200

Hits/interval

180 160 140 120 100 80 60 40 20 0.99998420

0.99998540

0.99998683 0.99998826 probability

0.99998969

Figure 6.12: Distribution of 4000 samples of ASAI, performed on the Birka System with 10% 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 evaluating the reliability indices for a distribution system. The conclusion of this thesis is that the implemented MCS method in RADPOW provides the same results as the analytical method in RADPOW and the NEPLAN software for two different case systems. There is no reason to believe that it is not functioning for a general system and therefor the method can be used for reliability assessment of power distribution system and has a potential to be developed further to incorporate general life time distributions for the components in the system. Another aspect that can be developed in a more straightforward manner is prioritization of components e.g. to determine where the maintenance actions will have the greatest effect. The implemented analytical sensitivity analysis routine that also is presented in this thesis gives a quantitative measurement of the uncertainty in the system indices, given the uncertainty of the component values. Given these distribution of the system indices the variation and other statistical measures can be computed as shown.

7.2

Discussion and Future Work

Computation time is an issue that usually is hold against simulations in order to get appropriate results that converges. The MCS described in this paper uses the most basic sample strategy referred to as simple sampling, which needs a relatively large number of samples to receive a sufficiently accurate result. There are techniques for reduction of calculation times without loss of precision, variance reduction techniques as stratified sampling and weighted sampling as examples, but still it is costly in terms of computation time. However, there are situations when analytical methods are not suitable to use because of the difficulties to model the problem analytical without making too large approximations. In these situations the 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 deterioration. Future possible development of the MCS method in RADPOW is to include the ability to handle temporary and transient failure rates for the components and eventually also incorporate the stuck probability for breakers. Another possible development is to introduce the Loadflow module with the Sim module, which makes 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 a deeper understanding e.g. how different second order failure events occurs or how the repair or replacement of components are dealt with when there are constraints in the work force. The MCS method can also easily be extended to be used for prioritization of components; one example is how to determine where the maintenance action will have the greatest effect. Taken further this prioritization can be used in the optimization of maintenance from a system reliability perspective, which is one of the major 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 in one 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 RADPOW_1999_PF version, as described in Section 3.6.3. In the earlier version, the data used for input for RADPOW were separated into ten different files. These files has been merged into one file containing ten different sections, each having exactly the same data, but with a header describing the section. These section headers are then used by RADPOW to identify where the different types of data in the file are present. This appendix describes the different sections in the file and how the system 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 in Appendix B and provides the reader with examples which may be valuable in the further reading of this appendix. The network model of these two systems are found in Figure 2.2 and 6.2 respectively.

A.1

Network topology data

These sections describes the basic structure of the system. The network topology data is described by the branch data, the location of supply and load points, normally open components and the type of each component. The topology of a system is in RADPOW defined by how a number of branches are connected to each other. A branch is in RADPOW defined as a number of components connected in series which starts and ends with a bus bar. The components in 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 common 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 each branch 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 the branch. 4. The unidirectional (Unid) parameter defines if power is permitted to flow in one direction or both directions. If this parameter is set to 0, the power through the branch is permitted to flow in any direction. If the parameter is set to a non-zero value the power is only permitted to flow from the sending end (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 and each separated by a blank space. 3. The end of the branch (EOB) parameter is set to −1 for each branch. This parameter 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 completely reliable net. The component number of these bus bars are separated by a blank space.

A.1.4 nlp-section In the nlp-section the component number of the bus bars with a load point connected 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 nnopsection. These components are normally disconnecters that can be closed in case of an outage.

A.2. Customer data

A.1.6

77

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 with the 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 consequently the number of the LP is the component number of that busbar. The load points that are going to be analyzed are then entered in the nlp-section. Data about the customer type can be defined in the ncuspow-section, but these data are not used by RADPOW for the standard reliability analysis.

A.2.1

ncuspow-section

The following customer and power data are specified for each load point, and in the following order: 1. The load point number (lpno), which is identical to the component number of 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 components 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 the following order: 1. A text string identifying the name of the component type (tno). This name are 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 time interval the component are unconnected until it has ben replaced or repaired 12. The replacement time (trep) in hours for the component. If this parameter is zero, a replacement of the component is not an option. If replacement is available, this option is always chosen before repair restoration. 13. The probability of switching devices being stuck (sprob).

A.4. Load flow data

A.4

79

Load flow data

The RADPOW_1999_PF version introduced a new module, Loadflow (described in Section 3.5.1 ). In the earlier version, RADPOW_1999, a failure for a load point were based on the total loss of continuity (TLOC) criteria, which means that a load point is interrupted when all its paths between the supply points and the load point are disconnected. But there are situations when a path, or branch, are not capable to 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 an electrical model in RADPOW, and not only a probabilistic. For this purpose, a number of electrical parameters for each load point and branch needs to be defined in 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) and these 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 the ones 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 the branch. 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 for RADPOW B.1

Test System 1 Input Data File

The following included text is the input data file for the Test System 1a that were used in the studie for RADPOW. The only different for the Test System 1b data file compared to this one is the component number of normally open points, which in the nnop-section is 0 instead of 18. For changes from earlier versions of the Test System 1 data file, see Section 6.2. ************************** bend-section Brno SE RE Unid 1 1 3 0 2 2 4 0 3 3 5 0 4 3 5 0 5 4 6 0 6 5 6 0

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

EOB -1 -1 -1 -1 -1 -1

************************** nsp-section Idno of supply points 1 2

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

81

82

Chapter B. Test System Input Files for RADPOW

nlp-section Load points to be analysed 5 6 ************************** nnop-section Idno of normally open points 18 ************************** ncuspow-section lpno tnc icp rcp ccp tappc iapp 5 100 0.0 0.2 0.8 4.0 0.0 6 80 0.0 0.2 0.8 5.0 0.0

rapp 0.3 0.3

capp 0.7 0.7

trppc 2.0 2.5

irpp 0.0 0.0

rrpp 0.3 0.3

crpp 0.7 0.7

tswi 1.0 1.0 1.0 1.0

trepl 0.0 0.0 0.0 0.0

sprob 0.0 0.0 0.0 0.0

************************** ctype-section Idno Typeno 1 BUS1 2 BUS1 3 BUS1 4 BUS1 5 BUS1 6 BUS1 7 BREAK1 8 BREAK1 9 BREAK1 10 BREAK1 11 BREAK1 12 BREAK1 13 BREAK1 14 BREAK1 15 TRANSF1 16 TRANSF1 17 TRANSF1 18 DISCON1 19 BUS1 ************************** crelia-section tno tc frp fra BUS1 Bus 0.001 0.001 BREAK1 Break 0.020 0.020 TRANSF1 Transf 0.015 0.015 DISCON1 Discon 0.002 0.002

frte 0.00 0.00 0.00 0.00

frtr 0.0 0.0 0.0 0.0

frm 0.0 0.0 0.0 0.0

trep 2.0 24.0 15.0 4.0

************************** pflo-section lpno lapc(pu) lrpc(pu) ldcu 5 0.2 0.2 1 6 0.2 0.25 1 ************************** pfrx-section Brno SE RE R(pu) X(pu) 1 1 3 0.01 0.1 2 2 4 0.005 0.05 3 3 5 0.0 0.05 4 3 5 0.0 0.05 5 4 6 0.01 0.1 6 5 6 0.005 0.05

Max_current(pu) 0.8 0.8 0.4 0.4 0.8 0.8

tmain 0.0 0.0 0.0 0.0

trec 0.0 0.0 0.0 0.0

B.2. Birka System Input Data File

B.2

83

Birka System Input Data File

The following included text is the input data file for the Birka System that were used in the studie for RADPOW. For changes from earlier versions of the Birka System data file, see Section 6.3. ************************** bend-section Brno SE RE Unid 1 1 14 0 2 1 14 0 3 14 27 0 4 14 27 0 5 14 27 0 6 27 29 0 7 29 32 0 8 29 32 0 9 32 35 0 10 14 36 0 11 36 46 0 12 36 46 0 13 36 46 0 14 46 48 0 15 14 49 0 16 49 56 0 17 49 56 0 18 56 58 0

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

************************** nsp-section Idno of supply points 1 **************************

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

84


nlp-section Load points to be analysed 35 48 58

************************** nnop-section Idno of normally open points

************************** ncuspow-section lpno tnc icp rcp ccp tappc 35 447 0.25 0.5 0.25 1.7203 48 23400 0.1 0.8 0.1 0.9829 58 1 0 0 1 0.8

************************** ctype-section Idno Typeno 1 BU220 2 BR110 3 TR220 4 BR110 5 CA110a 6 TR110 7 BR33 8 BR110 9 TR220 10 BR110 11 CA110b 12 TR110 13 BR33 14 BU220 15 BR33 16 CALH33 17 TR33 18 BR11 19 BR33 20 CALH33 21 TR33 22 BR11 23 BR33 24 CALH33 25 TR33 26 BR11 27 BU11 28 BR11 29 BUSD 30 CALH11 31 CALH11 32 BUSD 33 TR11 34 FUSE 35 BU04 36 BUSD 37 BR33

iapp 0.4 0.2 0

rapp 0.4 0.7 0

capp trppc irpp 0.2 0.0 0.0 0.1 0.0 0.0 1 0.0 0.0

rrpp 0.0 0.0 0.0

crpp 0.0 0.0 0.0

B.2. Birka System Input Data File 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

85

CAHDa BR33 BR33 CAHDb BR33 BR33 CAHDc BR33 BUSD BR33 BU220 BUSD BR33 CASJa BR33 BR33 CASJb BR33 BUSD BR33 BU220

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

************************** pflo-section lpno lapc(pu) lrpc(pu) ldcu 35 0.4 0.2 1 48 0.4 0.2 1 58 0.4 0.2 1

86


************************** pfrx-section Brno SE RE R(pu) X(pu) Max_current(pu) 1 1 14 0.01 0.1 0.8 2 1 14 0.01 0.1 0.8 3 14 27 0.01 0.1 0.8 4 14 27 0.01 0.1 0.8 5 14 27 0.01 0.1 0.8 6 27 29 0.01 0.1 0.8 7 29 32 0.01 0.1 0.8 8 29 32 0.01 0.1 0.8 9 32 35 0.01 0.1 0.8 10 14 36 0.01 0.1 0.8 11 36 46 0.01 0.1 0.8 12 36 46 0.01 0.1 0.8 13 36 46 0.01 0.1 0.8 14 46 48 0.01 0.1 0.8 15 14 49 0.01 0.1 0.8 16 49 56 0.01 0.1 0.8 17 49 56 0.01 0.1 0.8 18 56 58 0.01 0.1 0.8

References [1] L. Bertling. Reliability Centred Maintenance for Electric Power Distribution Systems. 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 maintenance method for assessing the impact of maintenance in power distribution systems. IEEE Transactions on Power Systems, 2003. TPWRS-00271-2003.R3. [3] L. Bertling. Projektbeskrivning av examensarbete. Department of Electrical Engineering, KTH, January 2006. [4] J. Setréus, L. Bertling, S. Mousavi Gargari. Simulation method for reliability assessment of electrical distribution systems. To be published at the Nordic conference on Nordic Distribution and Asset Management (NORDAC), August 2006. [5] M. Rausand, A. Hoyland. System reliability theory: Models and statistic methods. John Wiley and Sons Inc., Norwegian University of Science and Technology, 2 edition, January 2004. ISBN 0-471-47133-X. [6] G. Blom. Sannolikhetsteori med tillämpningar C. Studentlitteratur AB, Sweden, 4 edition, 1989. ISBN 91-44-03594-2. [7] O. Wilhelmsson. Evolution of the introduction of rcm for hydro power generators at vattenfall vattenkraft. Master’s thesis, Department of Electrical Engineering, KTH, 2005. X-ETS/EEK-0517. [8] L. Bertling Y. He. The verification of the reliability assesment computer program 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 deviates. In The Annals of Mathematical Statistics, volume 29, pages 610–611. Tokyo, 1958. ISSN 0373-5990, 00203157. 87

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REFERENCES

[11] R.Y. Rubinstein. Simulation and the Monte Carlo Method. John Wiley And Sons 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 Control on Reliability of Power Distribution Systems. Doctoral dissertation, Department of Electrical Engineering, KTH, Stockholm, Sweden, September 2002. 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-EES0005. [15] L. Bertling. Status report for the computer program radpow. Technical report, Department of Electric Power Engineering, KTH, 2000. [16] J. Setréus. Logfile for development and revision of RADPOW. Department of Electrical Engineering, KTH, Maj 2006. [17] R. Billinton and R.N. Allan. Reliability Evaluation of Power Systems. Plenum publishing corporation, New York, 2 edition, 1994. ISBN 0-306-45259-6. [18] S. Mousavi Gargari. Reliability assessment of complex power systems and the use of the neplan tool. Master’s thesis, Department of Electrical Engineering, KTH, 2006. Triata: XR-E-ETK 2006:11.

Module for the Reliability Computer Program RADPOW - DiVA

Module for the Reliability Computer Program RADPOW - DiVA

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