Utilisation of quantitative reliability concepts in evaluating the marginal outage costs of electric generating systems

Reliability Engineering and System Safety 46 (1994) 93-100 ~) 1994 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0951-8320/94/$'/.00

ELSEVIER

Utilisation of quantitative reliability concepts in evaluating the marginal outage costs of electric generating systems Raymond Ghajar & Roy Billinton Power Systems Research Group, University of Saskatchewan, Saskatoon, Saskatchewan, Canada, S7N OWO

Marginal outage costs are an important component of electricity spot prices. This paper describes a methodology based on quantitative power system reliability concepts for calculating these costs in electric generating systems. The proposed method involves the calculation of the incremental expected unserved energy at a given operating reserve level and lead time and the multiplication of this value by the average cost of unserved energy of the generating system. An extension of the proposed method is applied to interconnected generating systems in order to calculate the impact of assistance from neighbouring systems on the marginal outage cost profile of the assisted system. This method is based on the equivalent assisting unit approach. The methods discussed in this paper are illustrated by calculating the marginal outage cost profile of a small educational test system and by examining the effect of selected modelling assumptions and parameters to see how simplified representations can be used to approximate the results obtained using more detailed reliability models.

vary from season to season, day to day or hour to hour even though there are marked differences in instantaneous costs. Under this pricing regime, a utility sets the average price of electrical energy in a way that covers the utility's costs and profits. The customers are therefore insensitive to the varying costs of electrical energy and have no economic incentives to adjust their consumption to take advantage of low-cost periods or to avoid usage during high-cost periods. Spot prices, on the other hand, are capable of achieving significant gains in short-term efficiency. These gains can be to the benefit of both the utility and its customers and the incorporation of one or more real time elements into a tariff makes it more responsive to utility and customer needs. 3-5 In general, spot prices are explicit functions of a number of random variables that change over time and space. 3 The components of a spot price can be grouped in two different categories: marginal operating costs and marginal outage costs. Marginal operating costs are generally defined as the additional fuel costs that are incurred in serving an incremental load, where these additional costs may partly arise from line losses and off-economic dispatch. Marginal outage costs are defined as the customer outage costs

1 INTRODUCTION A major utility concern is to find realistic and acceptable procedures for reducing demand at times of critical load conditions. These conditions may arise infrequently in emergency situations but more regularly at peak load levels. Of equal concern are surplus periods where low cost energy is available but cannot be marketed because the price offered to customers does not reflect the abundance of the resource. Thus, both the utility and the customer fail to benefit from these time-varying conditions. Against this background, one can focus on the inherent rigidity in the tariffs most commonly used by electric utilities. This rigidity is the result of calculating and posting the tariffs up to a year in advance of the time of consumption. Therefore, they must be developed based upon the best estimates of the expected values of key variables that are likely to prevail in the future. More often than not, actual conditions turn out to be considerably different than those predicted when the tariffs were calculated a year or more earlier. 1'2 Traditional pricing of electricity for the vast majority of customers is based on the 'average' cost of generation, transmission and distribution. It does not 93

94

Raymond Ghajar, Roy BiUinton

related to both capacity shortages and network capacity constraints that are incurred in serving an incremental load. In principle, the marginal operating costs needed for spot pricing can be estimated using economic dispatch models. Such models determine the output levels of all generating units in a system by minimising the production costs subject to transmission constraints, system security and operating reliability constraints. This paper is concerned with the development of methodologies for calculating the marginal outage costs in isolated and interconnected generating systems using quantitative power system reliability concepts. The derived methodologies are illustrated using an educational test system designated as the RBTS. 6

is presented in Ref. 8 and a brief description of the F & D approach is given here to illustrate the salient features. The estimation of the 1EAR involves the generation of a capacity margin model 9 which indicates the severity, frequency and duration of the expected negative margin states. This model can be used in conjunction with the composite CDF (CCDF) for the service area to estimate the lEAR. The generation model is developed from the capacities, forced outage rates, failure rates and repair rates of the generating units. An exact-state load model is utilized which represents the actual daily system load cycle by a sequence of discrete load levels. 6 The total loss of energy expectation (LOEE) for the estimated loss of load events within the period of study is given by N

2 E V A L U A T I O N OF THE MARGINAL O U T A G E COSTS IN ISOLATED GENERATING SYSTEMS

Marginal outage costs are included in electricity spot prices because rates should depend upon t h e extent to which expected customer outage costs change as load changes. Although, the majority of outages are caused by factors that are unrelated to load levels, loads do affect the characteristics of outages (e.g. frequency, duration, unserved energy, etc.) and consequently their associated customer economic costs. The problem of estimating the economic costs incurred by customers due to power outages has been discussed quite extensively in the literature. Reference 7 provides a comprehensive background on the evolution of the methodologies used to estimate the economic costs of power outages incurred by customers. Marginal outage costs are expected quantities that depend upon two major factors: (i) the customer economic costs that accompany various outage levels; and (ii) the effects of load changes on the probabilities that these costs will actually be incurred. The customer economic costs resulting from power outages can be evaluated using a number of methodologies. One method considered to yield acceptable results is the customer survey method where customers are surveyed to estimate their economic losses and to create customer damage functions (CDFs) for each customer group. The CDFs developed using the survey method can be used to calculate an average cost of unserved energy for each customer group and for the entire service area. Such a cost factor was developed at the University of Saskatchewan and is designated as the interrupted energy assessment rate (IEAR) expressed in (S/kWh). The detailed description of the concepts involved in calculating an 1EAR using a basic frequency and duration ( F & D ) approach or Monte Carlo simulation

Total LOEE = ~ mifidi (kWh/day)

(1)

i=1

where m~ is the margin state capacity for load loss event i (kW), f~ is the frequency for load loss event i (occ/day), d~ is the duration for load loss event i (h), and N is the total number of load loss events. The cost ci(d~) of the energy not supplied during load loss event i can be obtained from the duration di and the CCDF for the given service area. The total expected cost of all the system load curtailment events is given by N

Total cost = ~ mifici(di) (S/day)

(2)

i=1

The IEAR for the service area is calculated as the ratio of the total cost and the total LOEE as shown below: N

Y~ mi~ci(di) Estimated l E A R -

i=l N

(S/kWh)

(3)

i=1

Sensitivity analyses conducted in Ref. 8 show that the IEAR is quite stable and does not vary significantly with peak load, exposure factor and other pertinent system operating considerations. The second component required for calculating the marginal outage cost incorporates the probabilities of capacity outages and the system load demand in the form of a quantitative risk index. The most suitable risk index for the evaluation of the marginal outage cost is the expected unserved energy (EUE) index which can be calculated using the basic and well-known recursive technique. 9 The effect of load changes on the E U E can be measured by taking the difference between two E U E values that are evaluated at incrementally different load levels (e.g. a load increase of 1 MW). The marginal outage cost can

Evaluating marginal outage costs then be calculated as follows:

principle of the matrix multiplication method is given

MOC = 1EAR × AEUE

($/kW)

(4)

where MOC is the marginal outage cost, IEAR is the interrupted energy assessment rate (S/kWh), and AEUE is the incremental expected unserved energy due to an incremental change in load (MWh/MW). A simple expression for calculating AEUE directly from the capacity outage probability table (COPT) of the system has recently been developed ~° and is given by AEUE = P(K) (MWh/MW) (5) where K is defined such that C - X(K)<_ L. C is the installed capacity of the system, X(K) is a discrete outage level in the COPT, L is the load level and P(K) is the cumulative probability of a capacity outage greater than or equal to X(K). It is clear from eqn (5) that the incremental expected unserved energy can be calculated directly from the capacity outage probability table. A basic generating unit parameter used in creating the COPT is the forced outage rate (FOR). This parameter provides an estimate of the probability of the unit being forced out of service at some distant time in the future. The FOR for a two-state generating unit model is given by the following equation: FOR =

A A+/z

(6)

where A is the failure rate of the generating unit (failures/h), and/z is the repair rate of the generating unit (repairs/h). The F O R as defined by eqn (6) cannot be used to calculate AEUE in the short term as it describes the unavailability of generating units in the steady state. A parameter capable of doing this is the time-dependent unavailability of a generating unit which is also referred to as the outage replacement rate (ORR). This statistic is a function of the lead time t into the future. The equation of ORR(t) for a two-state model is given by eqn (7). 1~ It can be seen from this equation that the steady state value ( t ~ o o ) of ORR(t) is (Z/h +/x) which is the forced outage rate of a generating unit. A

ORR(t) - - - - ~ A+tz

95

Ae -('~+'~)t

h+/z

(7)

The evaluation of ORR(t) for a two-state model is relatively simple because the time dependent probabilities can be easily derived. As the number of states increases, however, it becomes increasingly more difficult to derive the time-dependent probabilities. In these cases, the probabilities can be evaluated using the matrix multiplication method 1~ which is normally used to evaluate the probabilities of a discrete Markov chain after n equal intervals of time. The basic

by 11

P(n) = e(o) × P"

(8)

where W is the stochastic transitional probability matrix, P(n) is the vector of the time dependent probabilities, P(o) is the vector of the initial values of the states probabilities, and n is the number of time intervals at which the state probabilities are to be evaluated. The expression of the incremental expected unserved energy derived in this paper is used together with an appropriate IEAR value to calculate the marginal outage cost of the RBTS generating system.

3 APPLICATION TO AN ISOLATED RBTS GENERATING SYSTEM

Marginal outage cost evaluation involves timedependent analysis as opposed to the conventional steady state analysis utilised in generating capacity planning. This introduces an additional dimension and complexity into the calculations. The application of the proposed methodology of calculating the marginal outage cost to an isolated generating system is illustrated in this section using the RBTS. The marginal outage cost is calculated at different load points and plotted versus the operating reserves corresponding to these load points. The generation system data for the RBTS are given in Ref. 6. The IEAR value calculated using eqn (3) is 3.6 S/kWh. The utilisation of a constant IEAR to calculate the marginal outage cost is not a limitation of the proposed method. The value of 3-6 S/kWh is used to illustrate the procedure. Given sufficient detailed cost of interruption data, the 1EAR value is tailored to the system under consideration and could even be time-dependent given the supporting data. The marginal outage cost profile of the RBTS is shown in Fig. 1. It can be seen from this figure that the marginal outage cost is equal to the 1EAR when the system is deficient or has no operating reserves and it decreases as the operating reserve becomes more plentiful. The steps in the profile are due to the discrete nature of the COPT and the size of the RBTS. It is expected that a larger system would have fewer steps and therefore a smoother marginal outage cost profile. In addition to varying with the operating reserve, the marginal outage cost also varies with the lead time which is equal to the duration of time between the hour being forecast and the time the forecast is made. The variation of the marginal outage cost with the lead time can be illustrated by calculating the marginal outage cost as a function of the lead time for a selected operating reserve level. Figure 2 depicts the

96

Raymond Ghajar, Roy Billinton 10'

Opera~g Resecve < 0

8

.01

.001 -25

io

20

3'5

65

so

Operating Reserve (MW)

Fig. 1. Variation of the marginal outage cost of the RBTS as a function of the operating reserve. 1-

~

Fig. 3. Variation of the marginal outage cost of the RBTS with the operating reserve and the lead time.

.01

.001 mP

~

5,~V 35MW 65 MW

*' .0001

"

12

" 23

"

" 34

" 45 Tune

"

" 56

"

" 67

"

" 78

(hours)

accurate representation of the power system, 9 which, in turn, will result in better estimates of the system's marginal outage costs. In order to show how multi-state modelling can affect the marginal outage cost of a power system, the two 40 MW thermal units of the RBTS were assigned a 50% derated state as given in Ref. 6. The results from this study are compared to the base case (i.e. two-state model) in Fig. 4 for a lead time of 10 h which is a typical value for calculating day-ahead spot prices. It can be seen from this figure that the marginal outage cost profile calculated using the three-state models is lower than that of the base case. The difference between the two profiles increases as the operating reserve increases.

Fig. 2. Variation of the marginal outage cost of the R B T S as a function of the lead time.

10

reliabilltymodel • •

IF.AR

variation of the marginal outage cost with the lead time for three operating reserves. It can be seen from this figure that each profile tends towards its respective steady state value as the lead time increases. It is clear from Figs 1 and 2, the marginal outage cost varies with both the operating reserve level and the lead time. These variations can be combined into a single three-dimensional surface plot that describes the relationship between the marginal outage cost and both the operating reserve and the lead time. Such a plot is shown in Fig. 3 for the RBTS. 3.1 Effect o f modelling derated states

The utilisation of nmlti-state models to represent large generating units in reliability studies results in a more

2-slate 3-state

8"d .01

.001 -25.0"

"-12.5"

0~0 12.5 " "2;.0 Operating Reserve (MW)

"

"37'.5

"

"-~f~.0

Fig. 4. Effect of using a three-state model to represent the large units in the RBTS on the marginal outage cost profile.

Evaluating marginal outage costs 10

1o 1

mladatd dcviatim 1EAR -

"

.1

•

0%

*

3%

----o---

97

cai~ity

m / •

OMW

,

20MW

x

60MW

5%

40MW

1

g ~

.1

~3

I

.01

.01 .001

.001

. -25

•

, -10

•

.

, 5

.

.

.

. 20

.

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-

•

35

, 50

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, 65

-

.0001

- ' 80

Operating R e s e r v e ( M W )

-25

-15

-5

5

15

25

35

45

55

65

OperatingReserve(,ww)

Fig. 5. Effect of load forecast uncertainty on the marginal outage cost profile of the RBTS.

Fig. 6. Effect of removing units for maintenance on the marginal outage cost profile of the RBTS.

3.2 Effects of load forecast uncertainty

In order to show the impact on the marginal outage cost profile of the RBTS of removing units for maintenance, a study was done where units of different sizes were removed for maintenance and the marginal outage costs calculated using the rest of the units in the system. The results of this study are compared with the base case (no units on maintenance) in Fig. 6. It can be seen from this figure that the marginal outage cost profile is not significantly affected by the removal of the 20 MW unit. As the capacity on maintenance increases, however, the marginal outage cost becomes slightly lower than the base case for most operating reserve levels. Therefore, it can be concluded from this study that a number of marginal outage cost profiles pre-calculated using selected combinations of generating units can be used to approximate the margifial outage cost of a practical system for the purposes of spot pricing.

In the previous studies reported in this paper, it has been assumed that the peak load is known with a probability of 1.0. This is extremely unlikely in actual practice as the forecast can be described by a probability distribution whose parameters can be determined from past experience, future load modelling and possible subjective evaluation. 9 In order to show the impact of the load forecast uncertainty on the marginal outage cost profile of the RBTS, a number of studies were conducted using a seven-step approximation of the normal distribution and various standard deviations. The results for standard deviations of 3 and 5% are compared to the base case (i.e. standard deviation of 0%) in Fig. 5. It can be seen from this figure that the marginal outage cost generally increases as the standard deviation increases. In addition, the load forecast uncertainty has a smoothing effect on the profile because the marginal outage cost values are expected values calculated using seven different load levels.

4 EVALUATION OF THE MARGINAL OUTAGE COSTS IN INTERCONNECTED GENERATING SYSTEMS

3.3 Effect of removing units for maintenance

The marginal outage cost for a particular operating reserve and lead time is calculated by constructing the capacity outage probability table at that lead time and extracting the appropriate cumulative probability from it to multiply by the lEAR. The construction of the COPT for every hour in the forecast period is very time consuming. Therefore, if it can be shown that the marginal outage costs are not affected greatly by removing a few units for maintenance, then a number of pre-calculated COPTs can be used for any day-type regardless of which units are on maintenance that day.

The marginal outage costs calculated in Section 3 of this paper are associated with isolated generating systems. The basic method used to calculate these costs is extended to interconnected generating systems in this section. The proposed method is based on the equivalent assisting unit approach and is used to calculate the impact of assistance from neighbouring systems on the marginal outage costs. The evaluation of the marginal outage costs in interconnected generating systems is performed using the same basic techniques available for assessing the reliability of these systems. The reliability assessment

Raymond Ghajar, Roy Billinton

98

of interconnected generating systems can be done using analytical techniques9'~2 or Monte Carlo simulation. The main advantage of Monte Carlo simulation lies in its ability to include a large number of operating considerations in the model. It requires, however, a large amount of computing time and storage before reasonable confidence in the results can be obtained. Therefore, it should not be used if analytical techniques are available. Most analytical techniques available for interconnected generating systems reliability studies can also be used to calculate the marginal outage costs of these systems. In this paper, the equivalent assisting unit method 9 is used to illustrate the impact of interconnection assistance and tie-line modelling on the marginal outage cost of two interconnected generating systems. This method represents the assistance between two systems in terms of an equivalent multi-state unit 9 which describes the ability of one system to accommodate capacity deficiencies in the other. The maximum assistance available from the assisting system corresponds to the 'no outage state' of that system. The capacity assistance level for a particular outage state in the assisting system is given by the minimum of the tie-line capacity and the available system reserve at that outage state. All capacity assistance levels greater than or equal to the tie-line capacity are replaced by one assistance level which is equal to the tie-line capacity. This capacity assistance table can then be converted into a capacity model of an equivalent multi-state unit 9 which is added to the existing capacity model of the assisted system. The marginal outage cost of the assisted system is then calculated as if it is a single area system using the methodology described in Section 2. The application of the equivalent assisting unit method to the evaluation of the marginal outage cost of two interconnected generating systems is presented by considering the two systems shown in Fig. 7. System A is designated as the assisted system and system B as the assisting system. In addition, it has been assumed that any negative capacity margin in the assisting system does not affect the assisted system. Positive capacity margins in either system do, however, increase the operating reserve (i.e. decrease the marginal outage cost) 1° of the other system. In order to illustrate the two system interconnection case, two RBTS test systems are assumed to be interconnected by a perfect tie-line with infinite Tie fines SystemA

i ) I I

System B

Fig. 7. Two RBTS generating systems interconnected by a number of tie-lines.

I0

maximum ~ i s ~ c e

mmmmm

•

0MW

)

20MW

I

60MW 80 ~

"7..'.0001

-100

-80

-60 -40 -20 0 20 40 Operating reserve of isolated area (MW)

60

80

Fig. 8. Effect of assistance from system B on the marginal outage cost profile of system A.

capacity. The generation models of both systems are identical and defined in Ref. 6. The value of the lEAR is assumed to be 3.6 S/kWh. A range of studies were performed using selected lead times. The studies reported in this paper were done using a lead time of 10 h. The object of the study is to show the effect of the assistance available from system B (assisting system) on the marginal outage cost profile of system A (assisted system). The maximum assistance available from system B equals the operating reserve in that system since the tie-line is assumed to have an infinite capacity. The study was carried out by varying the maximum assistance available from system B and calculating the corresponding marginal outage cost profile of system A. The resulting profile is expressed as a function of the operating reserve of the isolated system A in Fig. 8. This expression highlights the shift in the marginal outage cost profile of system A caused by the assistance from system B. It can be seen from Fig. 8 that the marginal outage cost of the isolated system A (i.e. no assistance) is equal to the IEAR when the system has no operating reserve or when it is reserve deficient and it decreases as the operating reserve increases. If some assistance is available to system A, however, the marginal outage cost becomes equal to the 1EAR for operating reserves that are less than or equal to the maximum assistance and decreases otherwise• In general, it can be observed that, for operating reserves greater than the absolute value of the maximum assistance, the marginal outage cost of the interconnected system A decreases as the maximum assistance available from system B increases. The amount of the reduction is a function of the generation and load models in both systems and the reserve sharing policy employed.

Evaluating marginal outage costs

1EAR

approximate the results from more detailed reliability models.

0MW e .t

20 Atilt 40MW 60MW

n

80MW

5 CONCLUSION

m .~

.01.

.001 •

.oool

-80

-60

-40 -20 o 20 40 Opcra~g reserve of isolated area (MW)

60

99

80

Fig. 9. Effect of tie-line capacity on the marginal outage cost profile of system A using a maximum assistance from system B of 55 MW.

4.1 Effect of tie-line capacity The effect of tie-line capacity on the marginal outage cost of system A is illustrated by varying the tie capacity from 0 to 80 MW in steps of 20 MW. The maximum assistance available from system B is assumed to be 55 MW. The results from this study are shown in Fig. 9. This figure shows the decrease in the marginal outage cost profile of system A which occurs due to the increase in the tie-line capacity between the two systems. The marginal outage cost profile converges to a limiting value which represents the minimum profile that system A can attain under these conditions. The marginal outage cost profile at this point is designated as the 'infinite tie capacity profile' as there will be no further decrease in the marginal outage cost with the addition of further tie capacity. The infinite tie capacity value is directly related to the maximum assistance available from system B. This is easily seen in this example since any tie capacity greater than 55 MW does not contribute significantly to the reduction of the marginal outage cost of system A. The assistance available from neighbouring systems depends on the operating reserves in the assisting system, the interconnection limitations and the type of reserve sharing agreement existing between the systems. These factors are all interlinked in regard to their impact on the marginal outage cost of an interconnected group of generating systems. Their individual impacts can be examined by varying one factor at a time and considering the change in the marginal outage cost of the assisted system. The results from these studies can then be used to show how simplified representations can be used to

This paper describes a methodology based on quantitative reliability assessment techniques for the calculation of the marginal outage costs of generating systems. The proposed methodology calculates the marginal outage cost at a given load level by multiplying the incremental unserved energy of the system at that load level by an average cost of unserved energy. A method was proposed for the evaluation of the incremental unserved energy directly from the capacity outage probability table. The average cost of unserved energy is represented by the interrupted energy assessment rate of the system. The proposed method was applied to the RBTS. The marginal outage cost profile of this test system was calculated as a function of the operating reserve level and the lead time. It was found that the marginal outage cost is equal to the lEAR when the system has no reserves or when it is reserve deficient. As the operating reserves become more plentiful, the marginal outage cost decreases. The variation of the marginal outage cost with the lead time shows that the marginal outage cost for a given operating reserve increases as the lead time increases. A number of sensitivity analyses were conducted to show the effect of selected modelling assumptions on the marginal outage cost profile of the RBTS. The inclusion of generating unit derated states did not have a large impact on the marginal outage cost profile of the RBTS. However, if a large number of units should be represented using multi-state models, it is expected that the marginal outage cost profile will be significantly different. It was also found that the marginal outage cost is generally higher when load forecast uncertainty is included in the model. The magnitude of the increase depends on the standard deviation of the load forecast distribution. Finally, a study of the sensitivity of the marginal outage cost profile to the number of units on line was conducted. The purpose of the study was to see if a number of pre-calculated marginal outage cost profiles can be used regardless of which units are unavailable due to scheduled maintenance on a particular day. The results from the study show that the marginal outage cost profile of the RBTS is not greatly affected by the number of units removed for maintenance particularly when the capacity of such units is small relative to the installed capacity of the system. In practical system studies, such variations will be minimal if the installed capacity of the system is large and the capacities of most generating units are relatively small compared to the installed capacity of the system.

100

Raymond Ghajar, Roy Billinton

The basic method used to calculate the marginal outage cost in single generating systems was extended to interconnected generating systems in this paper. The proposed method was applied to two interconnected generating systems. It was found that the marginal outage cost profile of the assisted system decreases as the maximum assistance available from the assisting system increases. The amount of the reduction is a function of the generation and load models in both systems and the reserve sharing policy employed. Since the ultimate goal of any spot pricing scheme is the evaluation of the marginal outage costs at the customer load points, work is currently underway at the University of Saskatchewan to develop a method for calculating these costs at the bulk system supply points. This work will require a quantitative assessment of the composite system reliability and therefore it will be done using a resident program which is well known and is the result of many years of research in the area of composite system reliability evaluation. This program is designated as C O M R E L 13'14 (COMposite system RELiability evaluation program). REFERENCES

1. Garcia, E. V. & Runnels, J. E., The utility perspective of spot pricing. IEEE Trans. PAS, PAS-104(6) (1985) 1391-3. 2. Sanghvi, A. P., Flexible strategies for load/demand management using dynamic pricing. IEEE Trans. PAS, PAS-4(1) (1989) 83-93.

3. Schweppe, F. C., Caramanis, M. C., Tabors, R. D. & Bohn, R. E., Spot Pricing of Electricity. Kluwer Academic Publishers, Boston, MA USA, 1988. 4. Sioshansi, F. P., The pros and cons of spot pricing--electric utility perspective. Energy Policy, August (1988) 353-8. 5. Kirsch, L. D., Sullivan, R. L., Flaim, T. A., Hipius, J. J. & Krantz, M. G., Developing marginal costs for real-time pricing. IEEE Trans. PWRS, 3(3) (1988) 1133-8. 6. Billinton, R. et al., A reliability test system for educational purposes--Basic data. IEEE Trans. PWRS, 4(3) (1989) 1238-44. 7. Tollefson, R., Billinton, R. & Wacker, G., Comprehensive bibliography on reliability worth and electrical service consumer interruption costs: 1980-1990. IEEE Trans. PWRS, 6(4) (1991) 1508-14. 8. Billinton, R., Oteng-Adjei, J. & Ghajar, R., Comparison of two alternate methods to establish an interrupted energy assessment rate. IEEE Trans. PWRS 2(3) (1987) 751-7. 9. Billinton, R. & Allan, R. N., Reliability Evaluation of Power Systems. Plenum Press, New York, USA, 1984. 10. Billinton, R. & Ghajar, R., Evaluation of the marginal outage costs of generating systems for the purposes of spot pricing. IEEE Trans. PWRS, 9(1) (1994) 68-75. 11. Billinton, R. & Allan, R. N., Reliability Evaluation of Engineering Systems: Concepts and Techniques. Plenum Press, New York, USA, 1983. 12. Pang, C. K. & Wood, A. J., Multi-area generation system reliability calculations. IEEE Trans. PAS, PAS-94(2) (1975) 508-17. 13. Medicherla, T. K. P. & Billinton, R., Overall approach to the reliability evaluation of composite generation and transmission systems. IEE Proc., 127(2) (1980) 72-81. 14. Kumar, S. & Billinton, R., Pertinent Factors in the Adequacy Assessment of Composite Generation and Transmission Systems. Canadian Electrical Association Paper No. 86-SP-141, Vol. 25, Pt 3, March 1986.