Power system reliability

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					Introduction: The concept of reliability is very old. A device is said to be reliable if it performs its work satisfactorily. However such vague and qualitative notions of reliability are not entirely adequate. A quantitative evaluation of reliability is necessary if it has to be integrated into engineering design and applications. The applications of probability techniques provide a quantitative prediction of system performance so that the reliability levels of alternate proposals can be compared along with the cost. Reliability is the probability of a device performing its function adequately for the intended period of time under specified operating conditions. This definition consists of four basic parts: probability, adequate performance, time and operating conditions. Thus we can quantitatively define reliability as the probability that a device will perform its intended function during a specified period of time under stated conditions. It is the resistance to failure of a device or system. It is the probability for any given design or process to execute within the anticipated operational or design margin for a specified period of time and under expect operating conditions with a minimum amount of stoppage due to a design or to process errors, when working under normal operations through the assume design life of a product or service. Power system reliability security refers to the quality of system (voltage, frequency, etc.) whereas power system reliability refers to the probability of maintaining secure power system. So the term reliability and security are different in this sense. Mathematically, this may be expressed as,

where is the failure probability density function and t is the length of the period of time (which is assumed to start from time zero).

Approaches of Probability:

Approaches of Reliability

Deterministic Approach

Probabilistic Approach


1) Deterministic Approach: It is the conventional approach used for small power system with simple uses like lightning and isolated system. a) Percentage margin or capacity reserve margin (CRM) CRM= fixed percentage of installed capacity (IC) b) Loss of largest unit: Capacity reserve= capacity of largest unit (CLU) c) Loss of largest unit+ margin: Capacity reserve = CLU+ x* IC Value of x ranges from 5 to 15%.

2) Probabilistic Approach Reliability indices are used to predict reliability. • Mean time to failure (MTTF) • Mean time between failure (MTBF) • Forced Outage Rate (FOR) • Loss of Load Probability (LOLP) • Expected energy not supplied (EENS) • Customer supply interruption index Requirements for reliability • Plan for preventive maintenance which reduce the probability of failure. • Plan and schedule additional generating units. • Construct new transmission lines and upgrade the existing ones. • Upgrade equipments in power system • Repair any repairable items in minimum possible time so that your reliability will be maintained at desired level.

Properties of Reliability: 1) 0 ≤ R(t) ≤ 1 Verification: R(t) = n/ n+m When n=0, R(t)= 0 When n=N, R(t)= 1 2) R(0) =1 & R(∞) = 0 Verification: R(t)= e-λt R(0)= 1, R(∞)= 0


3) R (t) = e-λt Since R(t) is decreasing function of time as discussed earlier in eq. (xiv)

Bath-tub curve: Fig. shows the hazard function typical of many devices. Reliability curve has shape of bath-tub.Total life of a component consists of three distinct sections: a) Debugging period: It is the initial period or the period of infant mortality; the failure of component may be high due to errors in design and careless manufacture.

Failure Rate Debugging period Normal Operating Period

Wear-out period

b) Useful life or normal operating period: This period occurs after the debugging period. There is less failure rate due to correction of error in design and manufacture in the debugging period. There may be occurrence of only chance failures.

Fig: Bath-tub curve.


c) Old-age or wear-out period: It occurs after the normal operating period. The failure rate increases here. Repair may be costly in this period, so it would be better to replace the product by new one. It is possible to select a probability distribution for each of these three regions from Weibull distribution which can be expressed as λ(t) = K t β-1 for t>0 where λ(t) is the hazard or failure rate and K and β are constants. If β > 1, the failure rate increases with time (wear out stage). If β =1, the failure rate is constant (useful life period). If 0< β>1, the failure rate decreases (debugging region). Power system components can be kept within useful life period for a pretty long time by careful and proper periodic maintenance.


Basic Concepts of Reliabiliy: 1)Two state model A simple representation of the TU life history of a repairable unit TU Uptime (during the useful life period) is shown in the figure. After the unit has been operation for Repair some time a fault develops and Period Operating repair work is carried out. During Period TD the period of repair the unit is TD Down shut down or is in down state. Time After the repair, the unit is switched on again and another Fig: Two state model operating period starts. Then another fault develops and so on. Thus the life history of the unit consists of ‘up’ periods when the unit is operating and ‘down’ periods when the unit is under repair. These ‘up’ and ‘down’ periods alternate each other. Failure rate and Repair Rate: The number of failures per unit time of a component is known as failure rate, denoted by λ. Similarly, the number of repair per unit time is known as repair rate, denoted by µ.Both λ and µ are assumed to be constant for most of the system. The life time distribution of a component with failure rate is exponential. The converse of this is also true, i.e, failure rate associated with exponential distribution is constant. A component with exponential life time distribution has a constant probability of failure in any interval dt, i.e., the probability of failure in any interval t to t+dt is constant if the component is operating at time t (irrespective of value of t) . Only exponential distribution has this property. 2) Probability density function: Probability of non-failure in the interval 0 to t is the reliability R(t) and the probability of failure is the unreliability Q(t). R(t)+ Q(t) = 1 (i) Consider N is the no. of components tested repeatedly during the time t, among which n components survived the test whereas m components failed the test. Probability of survival = R(t) = n / N = n / (n+m) (ii)

We get, dR(t)/dt = -1/N dm/dt (iii) As dt approaches zero, we get instantaneous probability, i.e., probability density function f(t) .


dR(t)/dt = - f(t) From eq. (iii), dm/dt = -N dR(t)/dt dn/dt = d[N-m]/dt = -dm(t)/dt dm/dt = -N dR(t)/dt 1/n dm/dt = -N/n dR(t)/dt = λ(t)


(v) (vi) (vii) (viii)

where λ(t) is the instantataneous failure rate or hazard rate. From eqs. (ii) and (viii) we get, λ(t) = -1/R(t) dR(t)/dt (ix) From eqs. (iv) and (ix), λ(t) = f(t)/R(t) If failure rate is independent of time, λ(t) is written as λ. 3) Probability of survival and failure: The probability of a component surviving a time t in a constant failure environment is denoted by R(t) and is given by R(t)= e-λt The failure density function f(t) is f(t) = -dR(t)/dt = λ e-λt Q(t), the probability of failure in time t is given by Q(t) = 0 ∫ t f (t) dt = 0 ∫ t λ e-λt dt = 1- e-λt The probalbility of survival in time t is R(t) R(t) =





∫ ∞ f (t) dt = 0 ∫ ∞ λ e-λt dt = e-λt


If λt << 1, then Q(t) = λt and R(t) = 1- λt


Figure shows a variation of f(t) with t. This is known as exponential failure density function f(t).

f (t)

t Fig: Exponential failure density function

4) State space diagram:


Normal State

Failed State

µ Figure shows the state space diagram of a single repairable component for which the failure and repair rates are characterized by exponential distributions. The transition rate


from normal state to failed state is λ whereas the transition rate from failed state to normal state is µ. Reliability indices 1) Mean Time To Failure (MTTF) It isan estimate of the average, or mean time until a design's or component's first failure, or disruptionin the operation of the product, process, procedure, or design occurs. MTTF assumes that the product cannot be repaired and the product cannot resume any of it’s normal operations. Mathematically, MTTF=

Normal State

Failed State

∫ ∞ t f t (t) dt

MTTF = 1/ λ



Mean Time Between Failures (MTBF) MTBF includes the repair time of the designs or components. If a design or component works for an extended time, then it fails, is repaired in a reasonable amount of time, and then once again works amount of time that it is in operational condition. MTBF is the average time between failures to include the average repair time, or MTTR. It is typically small amount longer than

Normal State

Failed State



Mean up time, MU = 0 ∫ t (d Fu(t)/dt) dt = 1/ λ Mean Down Time, MD = 1/µ where, µ = constant repair rate. MTBF = Mu + MD = 1/ λ + 1/µ = (µ+λ) / (µ λ)

TU Uptime


TD Down Time


Availability Availability is defined as the ratio of the expected value of the uptime of a system to the aggregate of the expected values of up and down time. It is the ratio of the total time a functional unit is capable of being used during a given interval to the length of the interval. Mathematically, A = MU / (MU+ MD) = µ / (λ + µ) Proportion of time in the long run that it is not ready for service, Ā = 1- A = λ / (λ + µ) Forced Outage Rate It is defined as the percentage of time that a given point in the supply chain is nonfunctional due to forced outages. It is used when calculating the overall reliability of an energy delivery system. It sets the expected level of unplanned outages, which result in a partial or complete loss of generating capability for a certain period of time. FOR is not a rate at all but rather an estimator for a probability. Mathematically, FOR = FOH/(FOH+SH) where, Forced outage hours (FOH) is the number of hours a unit was in an unplanned outage state. Service hours (SH) is the number of hours a unit was in the in-service state. It does not include reserve shutdown hours. FOR= PR/ (PR+ PN) where, PR= probability of being in repair state PN=probability of being in normal state.


FOR= 1/(µ/λ+1) = Ā Loss of Load Probability (LOLP) It is a measure of the probability that a system demand will exceed capacity during a given period. It is often expressed as the estimated number of days over a long period, frequently 10 years or the life of the system. LOLP is the method to calculate power system (electrical network) reliability. It combines the probability that certain load could occur with the probability that certain amount of generation could deliver it. LOLP for n identical units A capacity outage means loss of generation and may or may not lead to a loss of load depending on the reserve generating capacity and the system load at the time of capacity outage. Pg = C(n,g) (Ā)g (A)n-g where, Pg = probability of outage of g units out of n units Capacity outage probability table for such case looks like:

No. of units out (g) 0 1 …. N

Capacity available (MW) max …. …. 0

Outage Probability, Pg …. … …. ….. Σ=1

In the above table, there is maximum capacity available at no. of units out=0 and minimum at N no. of units out.


Availability of thermal generating units of different sizes in India during the year 2003-2004 MW rating 500 250 200/210 140/150 120 110 100 70/85 62.5/67.5 60 50/57.5 20/40 Nonavailability due to Non availability due to planned maintenace % force outage , % 6.88 3.09 6.03 11.77 12.93 12.74 18.57 4.99 11.05 34.7 42.36 12 8.59 4.62 6.29 6.05 18.35 34.66 20.35 14.12 21.6 9.35 8.91 11.78 37.7 9.48 Operating availability , % 88.5 90.62 87.92 69.88 52.41 66.91 67.31 73.41 79.6 56.39 49.86 50.3 81.93

The above table shows the data of different size plants in India. We have, Operating availability= 100% - nonavailability due to planned maintainance – non availability due to forced outage Operating availability of larger unit should be higher than the lower one as shown in figure.

Effect of maintenance can be included in LOLP calculation by modifying the load characteristics

Load MW

Modified curve

Load MW Modified curve

Original curve Capacity on Maintenance

Original curve

Time Time

First figure shows the effect of LOLP calculation when a constant capacity is on maintenance throughout the period. Second figures shows the effect of LOLP calculation when the maintenance is done in light load period. The effect of maintenance can be included in LOLP calculations by modifying the load characteristics. If a certain capacity is to be out for maintenance throughout the year, the actual load duration curve is shifted by a margin equal to the capacity on maintenance. Then the LOLP calculations are done with the modified load duration curve. In many systems, the maintenance work is done only during light load period and not throughout the period. In such cases the load duration curve is shifted up only for the duration of maintenance. During the remaining period of the year the original load duration curve is valid.

Reliability of series and parallel systems a) Series system: All the components need to be in up state to make the the system in up state. When subsystems with reliability R1,R2 , R3,……….Rn are connected in series, then the overall reliability of the system is given by Rs = R1* R2 * R3*……….Rn = e-λ1t * e-λ2t *…….* e-λnt

b) Parallel System: Degree of system reliability depends upon the degree of redundancy in the system in c case of parallel system. Let n components having reliabilities R1,R2 , R3,……….Rn are connected in parallel, the probability of all the n components being in failed state at any time =(1- R1)*(1- R2 )* (1-R3)*………(1-Rn) If availability of only one component for the system success then, reliability of the system is given by, RP = 1-(1- R1)*(1- R2 )* (1-R3)*………(1-Rn) Probability of success in parallel system is higher than that in series system.


Reliability Block diagram:

Fig: Combination of series and parallel connection of the subsystems. In the above diagram subsystems B are connected in parallel whereas C, D are connected in series. Reliability Planning: The electrical power system consists of generation, transmission and distribution systems. The ultimate goal in reliability analysis is the evaluation of the entire system so that overall reliability measures can be obtained. However this is rather too ambitious and difficult task because of a very large number of components and complex interconnections. Therefore, it is more prudent to calculate reliability measures separately for the generation, transmission and distribution systems. An added advantage of this approach is that the weakest links and components can be spotted more easily and remedial measures taken for reliability improvement. Reliability improvement is possible through the use of better components and provision of redundancy. Redundancy in generation means additional generation capacity. Redundancy in transmission means that ties between stations and load centres should be stronger. This extra transmission capacity can be used to avoid overloading during usual operating conditions. Redundancy in distribution systems means duplication of certain components and use of better bus schemes. Distribution systems have a large number of components in series and a number of switching devices. A system with perfect switching devices is most desirable. However most of the switching devices are rather imperfect and are responsible for most of the unreliability. Preparation of Reliability models: The preparation of a proper analytical model for reliability studies requires the consideration of many factors some of which are as under: a) Composition of system: The model depends on the number of and types of components and their interconnection. 12

b) System failure criteria: The failure criteria depends on the part of the system being analyzed. It is usually necessary to carry out the failure effect analysis to define the system failure criteria. In one system the continuity of supply, irrespective of voltage available, may be sufficient. In another study a lower than a specified voltage may mean unreliability. c) Assumptions in modeling: The assumptions largely determine the method of analysis. The assumptions should be realistic but should not be liberal as to give wrong results. The assumptions are needed in many diverse areas, e.g. weather model, system states to be neglected in the study, load conditions, common mode failure, maintenance,etc. d) Selection and preparation of model: The analytical model should be realistic as well as simple. It must take into account all the important generation and load conditions as well as weather effects and possible interconnections. The various steps, in reliability study are: 1. Define the system 2. Define the failure 3. Make suitable assumptions 4. Prepare the system model 5. Carry out failure effect analysis 6. Evaluate reliability indices 7. Analyze the results 8. Prepare reliability improvement plans A reliability study can be undertaken only when a proper and reliable data bank of failure rates and repair times for various components is available.


CONCLUSION: We were able to study about the concept of the reliability of power system. We studied about the new concepts of reliability apart from our syllabus like bath-tub curve, series and parallel connection and effect of maintenance on LOLP calculation. We also studied about the data analysis of the availability of generating units in case of India. We were also familiar with the assumptions used in the reliability theory and the difference between reliability and security. The different approaches of probability was discussed and its mathematical calculation too. The LOLP calculation of n identical units was also studied. The reliability planning and preparation of reliability models were also studied in detai


BIBLIOGRAPHY        

Electric energy system management. (August 1999,Vinay Kumar Bhandari) Power System Analysis and Design (2005, B.R. Gupta) World wide web (internet) www.wikipedia.org www.google.com www.hotmail.com Generation of Electrical energy (September 2007, B.R. Gupta) Electrical Engineering (1999, Theresa)


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