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This work takes a look on the importance and role of probability concept in the design of transmission lines. The reliability of transmission line is clearly a function of the maximum loads that may be imposed over the useful life of the lines. These loads are, more often than not, caused by the extreme wind atmospheric events to which the line is exposed. In the current design calculation of wind load for overhead transmission and distribution lines the annual or weekly extreme wind is always assumed to act at a right angle to the line. This is a conservative assumption, which may result in unnecessarily expensive designs with the reliability values much higher than required. A new method for analyzing line loadings, taking into account wind directions was used in carrying out this work. The approach was based on separating extreme wind data into eight convenient line orientations and non-directional. The probability distribution functions of the wind speeds were obtained for each of the line orientations and that of the non-directional. Weather data obtained from the meteorological department of Nigeria Civil Aviation Enugu were used to determine the loading of the wind speeds on the conductor. Reliability indices were computed for the eight line orientations and that of the non-directional. Results show that the mean values of the non-directional wind and loadings are always greater than the corresponding values in directional cases. Thus, the saving in the line construction costs could be significant if the effect of wind directions were taken into account in the calculation of the design wind load.




Title Page                                                                                       i                                            ii                                                                                         ii

Approval Page                                                                               ii

Dedication                                                                                      iii

Acknowledgement                                                                         iv                                          vi

Abstract                                                                                          vi

Table of Contents                                                                          viii                                           x

List of symbols                                                                               xiii

List of figures                                                                                 xiv                                         xiii

List of Tables                                                                                 xvi

List of Computer Application Programmes                                 xviii


1.0    Background of Study                                                           1

1.1    Statement of Problem                                                         11

1.2    Objective of Study                                                               13

1.3    Scope of Study                                                                    14

1.4    Limitation of Study                                                               14


2.0    Role of Probability Methods in Power System Analysis  16

2.1    Historical Perspective                                                          20

2.2    Problems Involving Consideration of Uncertainty            23

2.3    Brief Review of Power System Reliability Evaluation      24

2.4     Reliability Assessment Techniques                                  25

2.5   Measuring Service Quality-Performance Indices              25

2.6    Contingencies Selection and Ranking                              27

2.7    Performance Index (pi) Method                                          27

2.8    Screening Method                                                               27

2.9    Hybrid Contingency Selection and Ranking Method       28

2.10  Reliability Evaluation Method                                             28

2.11  State-Space Method of Reliability Evaluation                     30

2.12    Monte Carlo Method of Reliability Evaluation                 36

2.12.1   Monte Carlo Simulation Process                                   37

2.12.2   Non-Sequential Method: State-Space Approach         39

2.12.3   Non-Sequential Method: State Transition

Sampling Approach                                                         40

2.12.4   Sequential: State Sampling Approach                          41

2.13                                                                                                 Logic Diagram Method of Reliability Evaluation   42

2.13.1 Series RBD System                                                          44

2.13.2 Parallel RBD System                                                         45

2.14   Fault Tree Method of Reliability Method                          45

2.14.1 Minimal Cut-Set Method                                                   48

2.15  Probability Distribution in Reliability Evaluation                50

2.15.1 Derivation of the Poison Process                                    50

2.15.2 Binomial Distribution                                                          53

2.15.3 Weibull Distribution                                                            54

2.15.4 The Gama Distribution                                                      57

2.15.5 The Normal Distribution                                                    59

2.15.6 The Log-Normal Distribution                                            59


3.0    Markov Chain                                                                       61

3.1    Markov Properties                                                               63

3.1.1             Variations                                                                 64

3.1.2             Reducibility                                                               66

3.1.3             Periodicity                                                                 67

3.1.4             Recurrence                                                              68

3.1.5             Ergodicity                                                                  70

3.1.6             Steady State Analysis and Limiting Distribution   70

3.1.7             Steady State Analysis and the Time-Inhomogeneous

Markov                                                                                  72

3.1.8             Finite State-Space                                                   73

3.1.9             Time-Homogeneous Markov Chain with a Finite

State-Space                                                                         73

3.1.10 Reversible Markov Chain                                                 76

3.1.11 Bernoulli Scheme                                                              78

3.1.12 General State-Space                                                        78

3.2    Wind Directional Model                                                       79

3.3    Wind Speed Mode                                                               81

3.4    Design of Line Span                                                           83

3.5    Extreme Value Distribution                                                 94

3.6    Type 1 Asymptotic Extreme Value Distribution                96

  1. 7 Transverse Wind Load on Conductor 98

3.8    The Method of Moment                                                      99

3.9    Design Line Considerations                                               125

3.9.1 Line Supports                                                                       126

3.10 Conductor Spacing                                                             128

3.11  Conductor Clearances                                                        129

3.12  Span Length                                                                        129

3.13  Tension                                                                                 131

3.14  Sag in Overhead lines                                                        132

3.15  Conductor Sag and Tension                                              135

3.16  Calculation of Sag                                                               135

3.17 Methods for Constrained Optimization                              139

3.18 Methods for Solving Constrained Optimization Problems   140


4.0    Reliability Indices Computation                                          149

4.1    Basic Postulations                                                               150

4.2    Reliability Indices Sample Calculation using the

State-Space Method                                                           150

4.3    Monte Carlo Simulation Method                                         152

4.4    Accuracy of Monte Carlo Simulation                                 157

4.5    Constrained Optimization Operation Time                        160

4.6    Comparison of the Run-Time for Different Constraint     160



5.0    Conclusions                                                                         164

5.1    Principal Findings                                                                165

5.2    Contribution                                                                          168

5.3    Recommendation                                                                168

          References                                                                        169

          Appendix A                                                                         174

          Appendix B                                                                         176





1.0 Background of Study              

Transmission lines form a vital component of any electric power system.  They carry electrical energy from generating stations to the consumers. Most power system interruptions are caused by failures of the transmission lines. Occasionally, however, a line can fail because of mechanical problems affecting its components, which are most often caused by extreme atmospheric conditions such as strong winds, and varying temperatures [1].

The goal of a transmission line designer is to build an efficient system which carries energy between any two locations. It is useful to consider the line system as being made of several subsystems, themselves made up of components as shown in Table 1.1. As can be seen in Table 1.1, a line system is a collection of many individually designed components that work together to produce an economic whole. During its life, a line will be subjected to an almost infinite variety of climatic events such as windstorms and varying temperatures. When a line is subjected to a climatic event, it responds in the form of stresses and displacements in each of its components. That response is called the load effect, which depends not only on the climatic event but also on the actual configuration of the line. The traditional transmission line design process involves the sizing of components to withstand a few carefully selected combinations of wind velocity, and temperature. Once the components have the strength to withstand the loads produced by few selected combinations, they are expected to have sufficient strength to resist most of the severe climatic events to which they will be subjected [2].

          Table 1.1   Transmission Line Systems, Sub- Systems                                        and Components

System Sub-Systems Components


Conductors and Ground Wire Cables, Spices
Connection Hardware Insulators, Clamp
Support Structure Member and Connection
Foundation Structure and Soil

 Source [2]

When there is a requirement for the transportation of large amounts of electric power between two given points, electric power utility often build several transmission circuits along the same right-of-way (ROW). Economy (the cost of land) and public demands are the main reasons for this practice. From the reliability point of view, however, it would be much more expedient to have the circuits built along different ROWs, since this would certainly reduce the chance of all circuits being lost at the same time as a result of the same extreme atmospheric event. The potential loss of all circuits -on a common ROW is of great interest to power system planners and analysts. Quite often the permanent loss of conductors comes about through high winds. Since the occurrence and the strength of extreme wind events is a random phenomenon, the frequency of ROW losses caused by extreme winds can be properly assessed only in probabilistic terms [3].

Reliability and availability evaluation of a system can help answer questions like “How reliable will the system be during its operating life?” and/or “What is the probability that the system will be operating as compared to out of service?” System failures occur in a random manner and failure phenomena can be described in probabilistic terms. Fundamental reliability and availability evaluations depend on probability theory [4].

Reliability is defined as “the probability of a device performing its purpose adequately for the period of time intended under the operating conditions encountered” [5]. The probability is the most significant index of reliability but there are many parameters used and calculated. The term reliability is frequently used as a generic term describing the other indices. These indices are related to each other and there is no single all-purpose reliability formula or technique to cover the evaluation.

The following are examples of these other indices [5]:

  • The expected number of failures that will occur in a specific period of time
  • The average time between failures
  • The expected loss of service capacity due to failure
  • The average outage duration or downtime of a system
  • The steady-state availability of a system

The approaches taken and the resulting formula should always be connected with an understanding of the assumptions made in the area of reliability evaluation. Attention must be paid to the validation of the reliability analysis and prediction to avoid significant errors or omissions.

Mathematically, reliability, often denoted as R(t), is the probability that a system will be successfully operating during the mission time t:

          R(t) = P(t),     ³ 0 –        –        –        –        –        –        1.1

Where is a random variable denoting the time to failure. In other words, reliability is the probability that the value of the random variable is greater than the mission time t.

Probability of failure, F(t), is defined as the probability that the system will fail by time t:

F(t) = P(£ t),     ³ 0 –        –        –        –        –        –        1.2

In other words, F(t) is the failure distribution function, which is often called the cumulative failure distribution function. The reliability function is also known as the survival function. Hence,

R(t) = 1 – F(t)      –        –        –        –        –        –        –        1.3

The derivative of F(t), therefore, gives a function that is equivalent to the probability density function, and this is called the failure density function, f(t), where

–        –        –        –        –        –        1.4

Or, if we integrate both sides of Equation (1.4),

            –        –        –        –        –        –        –        1.5


              –        –        –        –        1.6

In the case of discrete random variables, the integrals in Equations (1.5) and (1.6) can be replaced by summations.

A hypothetical failure density function is shown in Figure 1.1, where the values of F(t) and R(t) are illustrated by the two appropriately shaded areas. F(t) and R(t) are the areas under their respective portions of the curve.


Figure 1.1: Hypothetical failure density function, F(t) = probability of failure by time tR(t) = probability of survival by time t.


The failure density function shows the probability of the system failing at any given point in time. Because the sum of all the probabilities must be 1 (or 100%), we know the area under the curve must be 1. The probability of failing by time is thus the sum of the probabilities of failing from = 0 until time t, which is the integral of f(t) evaluated between 0 and t. Reliability is the probability that the system did not fail by time and is, thus, the remainder of the area under the curve, or the area from time to infinity. This is the same as the integral of f(t) evaluated from to infinity.

Mean time to failure (MTTF) is the expected (average) time that the system is likely to operate successfully before a failure occurs.

By definition, the mean or expected value of a random variable is the integral from negative infinity to infinity of the product of the random variable and its probability density function. Thus, to calculate mean time to failure we can use Equation (1.7), where f(t) is the probability density function and is time. We can limit the integral to values of that are zero or greater, since no failures can occur prior to starting the system at time = 0.

                –        –        –        –        –        –        1.7

Substituting for f(t) using Equation (1.4),


Equation (1.7) then becomes


–        –        –        –        1.8

The first term in Equation (1.8) equals zero at both limits. It is zero when is zero precisely because is zero, and it is zero when is infinite because the probability of the component continuing to work (i.e., surviving) forever is zero. This leaves the MTTF function as

                  –        –        –        –        –        –        1.9

In terms of failure, the hazard rate is a measure of the rate at which failures occur. It is defined as the probability that a failure occurs in a time interval [t1t2], given that no failure has occurred prior to t1, the beginning of the interval. The probability that a system fails in a given time interval [t1t2] can be expressed in terms of the reliability function as

Where f(t) is again the failure density function. Thus, the failure rate can be derived as

–        –        –        –        –        –        –        1.10

If we redefine the interval as [t+ ∆t], Equation (1.10) becomes


The hazard function is defined as the limit of the failure rate as the interval approaches zero. Thus, the hazard function h(t) is the instantaneous failure rate, and is defined by

                –        1.11

Integrating both sides and noticing the right side is the definition

of the natural logarithm, ln, of R(t) yields



             –        –        –        –        –        1.12

For the special case where h(t) is a constant and independent of time, Equation (1.12) simplifies to

          R(t) = e–ht             –        –        –        –        –        –        –        1.13

This special case is known as the exponential failure distribution. It is customary in this case to use l to represent the constant failure rate, yielding the equation

          R(t) = elt   –        –        –        –        –        –        –        –        1.14

Figure 1.2 shows the hazard rate curve, also known as a bathtub curve

Infant Mortality Period
Constant Failure Rate
End of Service Life

Figure 1.2: Bathtub Curve

Reliability is a measure of successful system operation over a period of time or during a mission. During the mission time, no failure is allowed. Availability is a measure that allows for a system to be repaired when failures occur. Availability is defined as the probability that the system is in normal operation. Availability (A) is a measure of successful operation for repairable systems. Mathematically,



Or, because the system is “up” between failures,

                         –        –        –        –        –        –        1.15

where MTTR stands for mean time to repair.

Another frequently used term is mean time between failures (MTBF). Like MTTF and MTTRMTBF is an expected value of the random variable time between failures. Mathematically,

          MTBF = MTTF + MTTR

If MTTR can be reduced, availability will increase. A system in which failures are rapidly diagnosed and recovered is more desirable than a system that has a lower failure rate but the failures take a longer time to be detected, isolated, and recovered.

Figure 1.3 shows pictorially the relationship between MTBFMTTR, and MTTF. From the figure it is easy to see that MTBF is the sum of MTTF and MTTR.

2nd Failure Occurs
Repair Completed
Failure Occurs

Figure 1.3. MTTR, MTBF, and MTTF.

Downtime is an index associated with the service unavailability. Downtime is typically measured in minutes per service year:

Downtime = 525,960 × (1 – Availability) min/yr        –        1.16

where 525,960 is number of minutes in a year.

Other indices are also taken into consideration during reliability evaluations; the above are the major ones.

1.1 Statement of Problem

          Prior to the 1960s, the reliability of proposed power systems transmission lines was often estimated by extrapolating the experience obtained from existing systems and using rule- of- thumb methods to forecast the reliability of the new system .

During the 1960’s, considerable work was performed in the field of power system transmission line reliability and some excellent papers were published. The most significant publications were two company papers by a group of Westinghouse Electric Corporation and Public Service and Gas Company. These papers introduced the concept of a fluctuating environment to describe the failure rate of transmission system with the assumption that the annual or weekly extreme wind always act at a right angle to the line. It is therefore intuitively obvious that both the loading and the resultant forces on the line conductor are greater when the wind blows in the direction normal to the line than when they come from any other directions. This is a conservative assumption, which may result in unnecessary expensive designs with reliability values much higher than required.

The reliability of a transmission line can be viewed as the probability of its satisfactory performance according to some performance function under specific service and extreme conditions within a stated time period. In estimating this probability, system uncertainties are modelled using random variables with mean values and probability distribution functions. The reliability of transmission line is therefore, a function of the maximum loads that may be imposed over the useful life of the line. These loads are more often than not, caused by the extreme atmospheric events to which the line is exposed. Because the extreme wind speed and direction are impossible to predict exactly, and any prediction is subject to uncertainties, the reliability of the lines may be assumed only in terms of the probability that the available strength will be adequate to withstand the life time maximum load.

1.2 Objective of Study

          The study is an assessment of wind induced electric transmission line loads on a probabilistic basis. Specifically, the study will determine:

  • The reliability indices such as:
  • the frequency of right-of-way losses
  • the mean duration time of an outage
  • the down time of the line
  • the coefficient of variation of the line which includes finding the mean and the standard deviation of the wind loads
  • the availability of the line and
  • the unavailability of the line
  • The Optimum Span length of the transmission line with due consideration of the line direction of the extreme wind speed loads




1.3 Scope of Study

The scope of this project is limited to the state of the art in reliability assessment of electric power systems. The developed techniques enable probabilistic risk assessment. The project therefore, places emphasis on the reliability assessment addressing the issues of component reliability as well as system reliability.

Proposed methodology used for the work was based on the Markov model. The State-space and Monte Carlo Simulation techniques were used for the reliability computations and their comparisons were made. The optimum span-length was achieved with the help of constrained optimization methods (penalty function, Lagrange multiplier, Quadratic programming and the gradient projection using equality and inequality constraints.

Finally, the study is limited to overhead transmission line from 11KV and above.

1.4 Limitation of Study

For the purpose of this research work, the following uncertainties were considered.

  • Uncertainty about the transformation of wind velocity into forces on the line components.
  • Uncertainty about the drag coefficient.
  • Uncertainty about the dynamic wind response of the line.
  • Uncertainty about the actual span length versus the design span.


Table 1.2   Classifications of Extreme Wind Speeds

Scale Speed(km/hr) Path length (km) Expected Damage
F0 64 – 115 0.5 – 1.6 Light
F1 116 -179 1.7 – 5.0 Moderate
F2 180 – 251 5.1 – 16.0 Considerate
F3 252 – 330 16.1 – 50.0 Severe
F4 331 – 416 50.1 – 159.0 Devastating
F5 417 – 509 160.0 – 507.0 Incredible

          Source [6]


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