CHAPTER 3 ANALYSIS OF RELIABILITY AND PROBABILITY MEASURES

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1 27 CHAPTER 3 ANALYSIS OF RELIABILITY AND PROBABILITY MEASURES 3.1 INTRODUCTION The express purpose of this research is to assimilate reliability and its associated probabilistic variables into the Unit Commitment problem (UCP). While the deterministic analysis of the UCP is in itself a tough proposition, the schematic is fairly straightforward and follows the method set by the operator. It is based on the specific procedure followed by the power system utility. Alternatively, it is dependent on a specific hybrid method formulated by a theoretician. The last mentioned group is drawn from Electrical Engineers, Mathematicians, Statisticians and Power system analysts. However, the adoption of a probabilistic approach to the same problem requires a smattering of reliability and probability basics. This section attempts to provide the necessary background necessary for a better understanding. 3.2 REVIEW OF PROBABILITY FUNCTIONS Cumulative Distribution Function The Cumulative Distribution Function (CDF) of a random variable X is defined as the probability of the event (X x).

2 28 That is, the probability of the random variable X takes on a value in the set (-, x). In terms of the underlying sample space, the CDF is the probability of the event {ζ:x(ζ) x}. The event {X x} and its probability varies as x is varied; in other words is a function of the variable x. The CDF is a convenient way of specifying the probability all semi-infinite intervals of the real line in the form. The axioms of probability and their corollaries imply that the CDF has the following properties. i. ii. m iii. m iv. is a non decreasing function of x, that is, if a < b, then v. is continuous from the right, that is, for h > 0, m Probability Density Function derivative of ; The probability density function of X (PDF) is defined as the The PDF is a more useful way of specifying the information contained in the cumulative distribution function. It represents the density of probability at the point x in the following sense. The probability that X is in a small interval in the vicinity of x {x X x + h} is

3 29 If the CDF has a derivative at x, then as h becomes very small, Thus represents the density of probability at the point x in the sense that the probability that X is in a small interval in the vicinity of x is approximately The derivative of the CDF, when it exists, is positive since the CDF is a non decreasing function of x. Thus It is represented graphically by Figures 3.1 and 3.2. Figure 3.1 General probability distribution function

4 30 Figure 3.2 Specific probability distribution function Equations (3.4) and (3.5) provide an alternative approach to specifying the probabilities involving the random variable X. The nonnegative function, the probability density function specifies the probabilities of events of the form X falls in a small interval of width about the point as shown in Figure 3.1. The probabilities of events involving X are then expressed in terms of the PDF by adding the probabilities of intervals of width. As the widths of the intervals approach zero, we obtain an integral in terms of the PDF. To cite an example, the probability of an interval [, ] is given by The probability of an interval is therefore the area under in that interval, as shown in Figure 3.2. The CDF of X can be obtained by integrating the PDF.

5 31 In essence, the PDF completely specifies the behavior of the continuous random variable. In a real-life situation, the variables and Thus are replaced by the time variables. 3.3 REVIEW OF RELIABILITY FUNCTIONS Probability Density and Cumulative Distribution Functions in Time Domain The lifetime of a non-repairable component lasts until a catastrophic failure occurs. Hence, the most important quantity to describe such a component is its life time, a random variable which, in turn is described by its probability distribution. The cumulative probability distribution function (CDF) of, is defined by The corresponding Probability Density Function (PDF), is given by m

6 Reliability and Hazard Functions At this point, we introduce the classical definition of reliability of a component/equipment/system. It is defined as the probability that the function is performed adequately, for the period of time intended, under the intended operating conditions. This definition is however drafted differently for repairable systems. To convert this definition into a quantitative statement, assume that the system is operated in an environment for which it was designed and next define the lifetime so that it is terminated when the performance of the component is no longer adequate. By the above definition, the reliability of a component is expressed as where is defined as the mission time. Since is a function of, the reliability function is defined as Comparing Equation (3.14) with Equation (3.10), For small values, the hazard function, is defined as the probability that a component that has survived until time t will fail in the next interval. More formally,

7 33 m [ (, ) ] m The conditional probability element in Equation (3.16) can be expanded by using Equation (3.17). Thus [ ] Following which Thus, Relationship Among the Cardinal Variables The four cardinal variables(pdf, CDF, hazard function and the reliability function) are inter-related and are detailed in Table 3.1.

8 34 Table 3.1 Relation between cardinal variables These cardinal functions are considered sufficient to generate case-specific reliability indices that are required in most practical situations. 3.4 IMPORTANT CONTINUOUS DISTRIBUTIONS The following lists some of the important continuous distributions that are used frequently. These are defined through their PDF s and used by choice whenever the results of an experiment appear to fit a particular distribution Exponential Distribution scale factor. The exponential distribution is defined by the following PDF with {,, It is graphically represented in Figure 3.3

9 35 The corresponding CDF is {,, It is graphically represented in Figure 3.4 The mean and variance f T (t) λ Area t Figure 3.3 Exponential PDF F T (t) e λt t Figure 3.4 Exponential CDF

10 Weibull Distribution by the following. The Weibull distribution is a two-parameter distribution, described All of the above are valid for else, they equal zero. The two parameters are, the scale parameter and Their effects are illustrated in Figures 3.5 and 3.6., the shape parameter. Figure 3.5 Effect of shape parameter on the Weibull distribution

11 37 Figure 3.6 Effect of scale parameter on the Weibull distribution The mean of the Weibull distribution is given by The variance is given by { } Where is the Gamma function, as defined for > 0 The shapes of the associated hazard functions depend on the value of ; If > 1 - is an increasing function of t, If 1 - = = constant (Reduces to the exponential function) If - is a decreasing function of t`

12 38 The significance of the Weibull distribution lies in the fact that it can be made to fit a wide range of hazard functions, as dictated by experience or experiments Gamma Distribution Another two-parameter distribution used in reliability studies is the Gamma distribution. It assumes significance on account of its use in the Weibull distribution. The two parameters are α, the shape parameter and, the scale parameter. The PDF is represented for by For It is graphically represented in Figure 3.7 for different values of α. The mean and variance of the distribution is given by and Figure 3.7 Effect of scale parameter on the Gamma distribution

13 39 The corresponding CDF is represented for the case when α is an integer. This case is also called the Erlang distribution. [ ( )] The hazard function can exhibit a wide variety of shapes similar to those observed in the case of the Weibull distribution Normal Distribution The normal distribution is a two-parameter distribution. Its PDF is expressed directly in terms of its mean and variance. The well-known bell-shaped graph of the normal PDF, symmetrical around the mean and having a peak value of is detailed in Figure 3.8. Figure 3.8 PDF of the normal distribution

14 40 The normal distribution is an important one, though less frequently used in reliability studies than the other distributions. It is not as suitable as the exponential or Weibull distribution for reliability applications, because a normal random variable can assume negative values whereas lifetimes, repair times among other such variables cannot take on such values.