Cost related reliability evaluation of bulk power systems

Electrical Power and Energy Systems 23 (2001) 99±112

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Cost related reliability evaluation of bulk power systems R. Billinton*, W. Zhang Power Systems Research Group, College of Engineering, University of Saskatchewan, 57 Campus Drive, Saskatoon, Saskatchewan, Canada S7N 5A9 Received 2 February 1999; revised 18 April 2000; accepted 6 June 2000

Abstract Economics plays a major role in the application of reliability concepts and the attainment of an acceptable level of reliability. Inadequate reliability of electric power supply ultimately costs the customers much more than good reliability. It is therefore important to determine the optimal reliability level at which the reliability investment achieves the best results in reducing the customer damage costs due to power supply interruptions. This paper presents a technique to calculate cost related reliability indices of a composite system at the load points and for the overall system. These cost related reliability indices are calculated in the form of both annualized and annual values. System studies conducted on two reliability test systems are presented, which provide insight into the variation of the indices with different system factors. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: Power systems; Reliability; Adequacy; Cost

1. Introduction

expressed as,

Cost/bene®t analysis of composite power systems is becoming an essential factor in the determination of system reinforcement and expansion projects, due to the impact of electric utility deregulation and market competition. Power system planning is traditionally based on deterministic criteria. A generally used criterion is that the loss of any single generating unit or transmission line should not cause load interruption. This criterion does not explicitly consider the probability of component failures and the value of service to customers. It can therefore result in overbuilt systems due to low probability events. The criterion provides no economic input to the cost associated with a particular expansion plan in terms of the value of service provided to customers. Deterministic approaches are not suf®ciently responsive to con¯icting factors in the emerging competitive power supply environment. Fig. 1 visually illustrates the concepts of reliability cost/ bene®t analysis. System reliability normally increases with investment cost. On the other hand, the customer damage cost decreases as the reliability level increases. The total cost is the sum of the project and customer damage costs. This total cost exhibits a minimum, at which an optimum or target level of reliability is achieved. The cost/bene®t approach uses the total cost as a basis for ranking the system expansion alternatives [1±6]. The approach can be

Minimize : Total cost ˆ Investment cost

* Corresponding author. Tel.: 11-306-966-5280; fax: 11-306-966-5205.

1 Customer damage cost:

…1†

where the investment cost includes the capital cost and the operation/maintenance cost, and the customer damage cost re¯ects the value of unsupplied energy. The investment cost is basically deterministic in nature and can be obtained using well-established methods. The customer damage cost is conceptually the aggregated value the customers are willing to pay to avoid load interruptions or voltage standard violations, and is a function of interruption frequency, duration, load lost, location, and other social effects. In some cases the customer damage costs are tangible, with inherent dollar values; in other instances the costs are intangible and subjective, depending upon the type and timing of interruptions and the consumers affected. The calculation of the customer damage cost is a necessary and complex task in reliability cost/bene®t analysis. The technique to calculate the customer damage cost and the application of the technique in system studies on the RBTS [7] and IEEE-RTS [8] are developed and illustrated in this paper. The data for calculating the customer damage costs used in this paper come from the mail surveys conducted by the Power System Research Group at the University of Saskatchewan [9]. The concept of customer

0142-0615/01/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S 0142-061 5(00)00046-6

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R. Billinton, W. Zhang / Electrical Power and Energy Systems 23 (2001) 99±112

Fig. 1. Investment, damage and total cost as a function of system reliability.

damage functions (CDF) is introduced in the following section.

2. Customer damage function A CDF provides the interruption cost versus interruption duration for a speci®ed group of customers. This section shows interruption cost data in the form of sector CDF collected through mail surveys conducted by the Power System Research Group at the University of Saskatchewan. The section illustrates the derivation of composite CCDF for both the overall system and individual load points in the RBTS and the IEEE-RTS. Seven sectors, using the Standard Industrial Classi®cation (SIC) scheme of Statistics Canada, were identi®ed for data collection. The seven sectors are large user (peak demand . 5 MW), industrial, commercial, agriculture, residential, government and institutions and of®ce buildings. Table 1 shows the seven-sector CDF expressed in 1987 dollars. The seven-sector CDF are graphically shown in Fig. 2. The CCDF represents the total interruption cost as a function of the interruption duration for the combined customers in a particular service area or at a speci®c bus. The CCDF for a service area is obtained by weighting the sector CDF by the customer load composition for that area. The custoTable 1 Sector CDF expressed in $/kW User sector

Large users Industrial Commercial Agricultural Residential Govt and Inst. Of®ce bldg

Interruption duration 1 min

20 min

1h

4h

8h

1.005 1.625 0.381 0.060 0.001 0.044 4.778

1.508 3.868 2.969 0.343 0.093 0.369 9.878

2.225 9.085 8.552 0.649 0.482 1.492 21.065

3.968 25.163 31.317 2.064 4.914 6.558 68.830

8.240 55.808 83.008 4.120 15.690 26.040 119.160

mer load compositions in terms of peak load and energy consumption percentages must be known in order to obtain the CCDF for the combined customers. The annual peak load percentage is usually used for weighting short durations (below 1 h) and the annual energy consumption percentage is used for weighting the longer durations (equal and above 1 h). Table 2 shows the assumed load compositions in terms of the annual peak load and energy consumption percentages for both the RBTS and the IEEE-RTS. The CCDF for the service area is obtained as shown in Table 3 and is shown graphically in Fig. 3. The CCDF at each load bus can be calculated in the same way as that for a service area if the sector peak load and energy consumption percentages at every load bus are known. This is usually done by ®rst assigning the peak loads for the different sectors at every load bus in the system and then calculating the sector peak load and energy consumption percentages at each load bus. The sector allocation at the load buses in sector peak loads must be chosen in such a way that meets the following two conditions: 8 X Sector peak at bus k ˆ Peak load at bus k > > < All sectors X > > Bus peak of sector m ˆ System peak of sector m : All buses

…2† The peak load percentage of a given sector at bus k can be simply calculated as follows: Sector peak load percentage at bus k ˆ

Sector peak load at bus k £ 100 Total peak load at bus k

…3†

There can be many ways of allocating the sector energy consumption at the various buses. An often-used approach is to assume that the sector load factor (L.F.), which is the ratio of the sector average load over the peak load, is constant at the various load buses or for the entire system service area. The sector energy consumption allocation to the buses can then be calculated using the sector peak load allocation at the buses and the sector LF. The L.F. of a given sector can be calculated from the system L.F., the sector peak load percentage and the sector energy consumption percentage as follows: Sector L:F: ˆ

Sector energy percentage £ System L:F: Sector peak load percentage …4†

The energy consumption percentage of a given sector at bus

R. Billinton, W. Zhang / Electrical Power and Energy Systems 23 (2001) 99±112

101

Fig. 2. Sector customer damage functions ($/kW).

k can be calculated as follows: …Sector energy consumption percentage at bus k† ˆ

Sector L:F: £ Sector peak load at bus k X £ 100 Sector L:F: £ Sector peak load at bus k All sectors

…5† The same load model has been used for both the RBTS [7] and the IEEE-RTS [8], for which the system L.F. is 61.40%. The seven-sector L.F. can be calculated using the given system L.F., the sector peak load and energy consumption percentages shown in Table 2 and Eq. (4). The results are shown in Table 4. The sector peak load allocations at the RBTS load buses are shown in Table 5. The seven-sector peak load percentages at each load bus can be calculated using the data given in Table 5 and Eq. (3). The results are shown in Table 6. The seven-sector energy consumption percentages at each load bus can be calculated using the data shown in Tables 4 and 5, and Eq. (5). The results are shown in Table 7. The CCDF at every load bus in the RBTS are calculated by weighting the user sector costs given in Table 1 for each interruption duration by the peak load and energy consumption percentages shown in Tables 6 and 7. The peak load percentages given in Table 6 are the weighting factors for Table 2 Assumed load compositions User sector

Sector peak (%)

Sector energy (%)

Large users Industrial Commercial Agricultural Residential Govt and Inst. Of®ce bldg Total

30.0 14.0 10.0 4.0 34.0 6.0 2.0 100.0

31.0 19.0 9.0 2.5 31.0 5.5 2.0 100.0

the 1 and 20 min durations and the energy consumption percentages given in Table 7 are the weighting factors for the 1, 4 and 8 h durations. The results are given in Table 8 and are shown graphically in Fig. 4. The seven-sector peak load allocations for the IEEE-RTS load buses are shown in Table 9. The seven-sector peak load percentages at each load bus are obtained using Eq. (3) and are shown in Table 10. The seven-sector energy consumption percentages at each load bus are calculated using the data shown in Tables 4 and 9, and Eq. (5). The results are shown in Table 11. The CCDF at every IEEE-RTS load bus are calculated by weighting the user sector costs given in Table 1 for each interruption duration by the peak load and energy assumption percentages shown in Tables 10 and 11. The results are given in Table 12 and are shown graphically in Fig. 5. 3. Expected customer damage costs This section illustrates the algorithm used for the calculation of the expected cost of customer interruptions (ECOST), or expected customer damage cost, at a speci®ed system service area or at a load bus. The equation for calculating the ECOST is developed from the expected energy not supplied (EENS) index [10] and the composite CDF illustrated in Section 2. The EENS is de®ned as: X psi Lc si 8760 MW h=year; EENS ˆ si[F

where F is the set of system failure states in which load curtailments occur, psi, the probability of existence of outage state si, and Lc si, the load curtailed in MW at a speci®c bus or for overall system in system state si. The above equation can be applied in adequacy evaluations of both the overall system and the system load points. In the overall system adequacy evaluation, the F is the set of the system states in which load curtailments occur in the

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R. Billinton, W. Zhang / Electrical Power and Energy Systems 23 (2001) 99±112

Table 3 CCDF in ($/kW) for the test systems RBTS service area

Large users Industrial Commercial Agricultural Residential Govt. and Inst. Of®ce bldg. P Total customer cost, ($/kW)

Interruption duration 1 min

20 min

1h

4h

8h

1.005 £ 0.30 1.625 £ 0.14 0.381 £ 0.10 0.060 £ 0.04 0.001 £ 0.34 0.044 £ 0.06 4.778 £ 0.02 0.67

1.508 £ 0.30 3.868 £ 0.14 2.969 £ 0.10 0.343 £ 0.04 0.093 £ 0.34 0.369 £ 0.06 9.878 £ 0.02 1.56

2.225 £ 0.31 9.085 £ 0.19 8.552 £ 0.09 0.649 £ 0.025 0.482 £ 0.31 1.492 £ 0.055 21.065 £ 0.02 3.85

3.968 £ 0.31 25.163 £ 0.19 31.317 £ 0.09 2.064 £ 0.025 4.914 £ 0.31 6.558 £ 0.055 68.830 £ 0.02 12.14

8.240 £ 0.31 55.808 £ 0.19 83.008 £ 0.09 4.120 £ 0.025 15.690 £ 0.31 26.040 £ 0.055 119.160 £ 0.02 29.41

system and the Lc si is the total load curtailed in the system in system state si. If system load point adequacy evaluation is conducted, the F is the set of system states in which load curtailments occur at a speci®ed system load point and the Lc si is the load curtailed at the load point in system state si. The EENS equation can be rewritten in the following form: X psi …msi 1 lsi †dsi Lc si MW h=year; EENS ˆ si[F

where m si is the total repair rates of the failed components in system state si, l si is the total failure rates of the operating components in si, and dsi ˆ 8760=…msi 1 lsi † h; which is the expected duration at system state si. The ECOST can be calculated by replacing the dsi in the EENS equation with the cost c(dsi) of the energy not supplied during the load loss event si. The c(dsi) is given by the duration dsi and the CCDF for the system service area or load bus. The equation for the ECOST is as follows: X psi …msi 1 lsi †c…dsi †Lc si k$=year; …6† ECOST ˆ si[F

where c(dsi) is measured in $/kW or k$/MW. The ECOST calculated by Eq. (6) when based on a single

constant load level is designated as an annualized value. The annual ECOST can be evaluated using a stepped load model [11]. The annual ECOST are however evaluated here using a direct approach, in which the hourly load duration curve is directly incorporated in the calculation. In this direct approach, the available capacity at the system state si is ®rst obtained and then combined with the load duration curve to obtain the expected load curtailment. The equation for the EENS using the direct approach is: 0 1 X X @psi EENS ˆ …Lj 2 Csi †A MW h=year si[S

Lj .Csi

where S is the set of all investigated system states, Lj, the hourly peak load in one year at a speci®c load bus in system state si, and Csi, is the available capacity at the speci®c load bus in system state si. The equation for the annual ECOST can be derived from the annual EENS equation in the same way as that for the annualized ECOST. In this situation, as the system state si is not a complete failure state using a single constant load, the expected system state failure duration df si is used for the interruption cost calculation. The annual ECOST is given

Fig. 3. The composite customer damage function ($/kW).

R. Billinton, W. Zhang / Electrical Power and Energy Systems 23 (2001) 99±112 Table 4 Load factors of the seven sectors

Table 6 Sector peak load percentages for the RBTS

User sector

Sector peak (%)

Sector energy (%)

Sector L.F. (%)

Large users Industrial Commercial Agricultural Residential Govt and Inst. Of®ce bldg

30.0 14.0 10.0 4.0 34.0 6.0 2.0

31.0 19.0 9.0 2.5 31.0 5.5 2.0

63.45 83.33 55.26 38.38 55.98 56.28 61.40

User sector

Peak load percentages Bus 2

Large users Industrial Commercial Agricultural Residential Govt. and Inst. Of®ce bldg. Total

by, ECOST 0 1 X X @psi …msi 1 lsi †c…df si † …Lj 2 Csi †A=8760 k$=year ˆ Lj .Csi

si[S

103

17.50 18.50 36.25 27.75 100.0

Bus 3

Bus 4

65.29 3.59 5.53

40.75 11.75

23.41

47.50

2.18 100.0

100.0

Bus 5

18.50 44.50 27.75 9.25 100.0

Bus 6 15.25 8.50 37.00 39.25

100.0

System 30.0 14.0 10.0 4.7 34.0 6.0 2.0 100.0

was used for the analysis of the system states. A pass 1 load curtailment policy was used for the RBTS evaluations and a pass 2 load curtailment policy was used for the IEEE-RTS assessments.

…7†

P

where df si ˆ Lj .Csi 1=…msi 1 lsi † is the hours in which the load is greater than the available capacity at a speci®c load bus in system state si.An often used index in reliability cost analysis is the interrupted energy assessment rate (IEAR), which is calculated as the ratio of the ECOST and the EENS at either the load buses or for the overall system, as shown below:

4. Evaluation of the RBTS

The IEAR is a convenient and readily understandable index, which provides a monetary evaluation of energy de®ciencies at the load buses and for the overall system from a customer damage cost point of view. The ECOST and the IEAR analyses were applied to the RBTS and the IEEERTS. The results of these analyses are shown in the following two sections and provide insight into the variation of the ECOST and the IEAR due to different system factors. System states of up to four generators, up to three transmission lines and up to three combined generators and transmission lines on outage were investigated in both the RBTS and the IEEE-RTS evaluations. A d.c. load ¯ow technique

The total probability of the investigated RBTS states is 0.99997350 with the given outage level de®nition of 4G 1 3L. Fig. 6 shows the ECOST at the load buses for three different load levels. The graph clearly shows that the ECOST at buses 2, 3 and 4 decreases rapidly with the reduction in the load level. This tendency, however, does not apply to bus 6. This is because the failures at buses 2, 3 and 4 are mainly due to generation capacity de®ciencies, while failures at bus 6 are dominated by outage of the single transmission line, which directly supplies bus 6. Fig. 7 shows the overall system ECOST at the different load levels. There is a multiplication factor above the bar at the 185 MW load level, which indicates that the actual ECOST is the indicated value multiplied, by the factor. The graph shows that the overall system ECOST decreases rapidly when the load level reduces from 185 to 148 MW and does not vary signi®cantly when the load level decreases from 148 to 74 MW. This indicates that generation capacity de®ciency is a major factor in the customer damage cost at a high system load level. The transmission

Table 5 Sector peak load allocation for the RBTS

Table 7 Sector energy consumption percentages for the RBTS

User sector

User sector

IEAR ˆ

ECOST …$=kW h† EENS

Large users Industrial Commercial Agricultural Residential Govt. and Inst. Of®ce bldg. Total

…8†

Peak load allocation (MW) Bus 2

Bus 3

Bus 4

3.50 3.70

55.50 3.05 4.70

16.30 4.70

7.25 5.55

19.90

19.00

20.00

85.00

2.18 0.0

Bus 5

3.70 8.90 5.55 1.85 20.00

Bus 6 3.05 1.70 7.40 7.85

20.00

System 55.50 25.90 18.50 7.40 62.90 11.10 3.70 185.00

Energy consumption percentages Bus 2

Large users Industrial Commercial Agricultural Residential Govt. and Inst. Of®ce bldg. Total

24.02 16.84 33.42 25.72 100.0

Bus 3

Bus 4

66.90 4.83 4.94

50.65 9.69

21.17

39.66

2.16 100.0

100.0

Bus 5

18.12 44.14 27.68 10.06 100.0

Bus 6 23.72 8.77 26.50 41.01

100.0

System 31.0 19.0 9.0 2.5 31.0 5.5 2.0 100.0

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R. Billinton, W. Zhang / Electrical Power and Energy Systems 23 (2001) 99±112

Table 8 CCDF for the RBTS load buses ($/kW) System bus

Bus 2 Bus 3 Bus 4 Bus 5 Bus 6

Interruption duration 1 min

20 min

1h

4h

8h

0.367 0.840 0.707 0.525 0.303

1.362 1.524 1.969 1.607 1.006

4.167 2.906 5.621 4.295 3.274

14.646 7.941 17.727 16.585 11.276

39.322 18.198 42.530 41.163 28.041

line capacity constraints become the major factor when the system load level is low. The graph also shows that customer damage cost due to generation capacity de®ciencies is larger than that due to the transmission line capacity constraints.

Fig. 8 shows the IEAR at the load buses for three different load levels. The graph shows that the IEAR for the RBTS is different at each load bus, but does not change signi®cantly with the selected load levels. There is no IEAR value shown at bus 2 when the load is at the 148 MW level, as in this case both the EENS and the ECOST are effectively zero. Fig. 9 shows the overall system IEAR at different load levels. The graph further indicates that the IEAR index is quite stable with respect to load level variations. Fig. 10 shows the total annual system ECOST and the distribution of the ECOST at the load buses. The annual system ECOST, which is 565.1 k$/year, is much smaller than the annualized value at the peak load level, which is 6632.9 k$/year. The graph also shows that the annual ECOST at bus 6 is the major contribution to the total annual RBTS ECOST and contributes 78% of the system

Fig. 4. CCDF for the RBTS load buses ($/kW). Table 9 Sector peak load allocation for the IEEE-RTS Buses

Sector peak load allocation in MW Large users

1 2 3 4 5 6 7 8 9 10 13 14 15 16 18 19 20

85.50 42.75 42.75 85.47 213.75 42.75 188.20 110.97 42.86

Sum

855.00

Industrial 39.90 59.80 19.90 39.95 39.95 19.90 39.95 59.80 39.95 39.90

399.00

Commercial 14.25 14.25 14.25 14.25 14.25 14.25 14.25 28.55 8.50 14.25 28.55 5.60 34.50 14.25 22.55 14.25 14.25 285.00

Agricultural

11.45 11.45 22.70 33.80 17.90 16.70

114.00

Residential

Government

36.85 48.45 94.50 25.55 36.85 67.50 48.10 94.05 41.50 80.15 80.15 62.98 54.50 25.90 62.40 55.78 53.79

17.00 34.30

969.00

171.00

Of®ce

34.20 2.85 25.65

2.85 5.70

25.65

11.40

17.10

14.25 19.95

17.10 57.00

Total 108.00 97.00 180.00 74.00 71.00 136.00 125.00 171.00 175.00 195.00 265.00 194.00 317.00 100.00 333.00 181.00 128.00 2850.00

R. Billinton, W. Zhang / Electrical Power and Energy Systems 23 (2001) 99±112

105

Table 10 Sector peak load percentages for the IEEE-RTS Buses

Sector peak load in % Large user

1 2 3 4 5 6 7 8 9 10 13 14 15 16 18 19 20

Industrial

Commercial

36.94

13.20 14.69 7.92 19.26 20.07 10.48 11.40 16.70 4.86 7.31 10.77 2.89 10.88 14.25 6.77 7.87 11.14

33.22

48.86 21.92 16.13 44.06 67.43 42.75 56.52 61.31 33.48

28.03 29.37 31.96 11.64 20.49 22.57 20.59 11.98

Agricultural

6.36 8.42 18.16 19.31 9.18 6.30

Residential

Government

34.12 49.95 52.50 34.53 51.90 49.63 38.48 55.00 23.71 41.10 30.25 32.46 17.19 25.90 18.74 30.82 42.02

15.74 35.36

Of®ce

46.21 2.10 15.00

1.66 3.26

9.68

4.30

17.10

4.50 5.99

13.36

Total 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

5. Evaluation of the IEEE-RTS

annual ECOST. The second major contribution comes from bus 3 which provides 12% of the total system ECOST and the third major contribution comes from bus 4 with 10% of the total system ECOST. The annual ECOST at buses 2 and 5 are very small. Fig. 11 shows the annual IEAR at the load buses and for the overall system. The graph shows that the annual system IEAR is smaller than the annualized system IEAR and the annual IEAR at the load buses are totally different from the annualized values. This indicates that although the annualized IEAR is relatively stable with load level variations, the annual IEAR are quite different from the annualized values.

The total probability of the investigated system states is 0.98581246 for the IEEE-RTS at the given outage level of 4G 1 3L. This probability is not as large as the value for the RBTS studies. This is mainly due to the larger size of the IEEE-RTS. When the investigated system states are extended to 5G 1 3L, the probability of the total system states increases to 0.99411566. The annual ECOST and IEAR based on the 5G 1 3L system state investigation are also provided in this section. Fig. 12 shows the ECOST at the load buses for two different load levels. The graph shows that the ECOST

Table 11 Sector energy consumption percentages for the IEEE-RTS Buses

Sector energy consumption (%) Large user

1 2 3 4 5 6 7 8 9 10 13 14 15 16 18 19 20

Industrial

Commercial

46.61

11.05 14.5 6.85 19.01 17.47 9.26 10.25 15.59 4.76 6.56 9.53 2.46 9.83 13.32 5.87 7.19 10.53

43.33

54.99 22.6 16.39 43.09 69.93 45.88 56.25 64.3 36.35

36.78 39.12 43.35 16.39 27.74 30.13 26.44 15.66

Agriculture

3.82 5.16 11.34 13.15 5.72 3.87

Residential

Government

28.92 49.95 46 34.53 45.75 44.4 35.06 52.03 23.55 37.38 27.12 28.01 15.73 24.52 16.45 28.51 40.25

13.42 35.55

Of®ce

46.46 2.06 14.27

1.72 3.55

8.73

4.23

16.28 12.87

4.51 5.77

Total 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

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R. Billinton, W. Zhang / Electrical Power and Energy Systems 23 (2001) 99±112

Table 12 CCDF for the IEEE-RTS load buses ($/kW) Load buses

1 2 3 4 5 6 7 8 9 10 13 14 15 16 18 19 20

Interruption duration 1 min

20 min

1h

4h

8h

0.658 0.072 0.574 0.094 0.532 0.623 0.574 0.339 0.677 0.587 0.784 0.789 0.934 0.492 1.075 0.646 0.385

1.911 0.613 1.591 0.774 1.728 1.730 1.673 1.217 1.291 1.410 1.946 1.577 1.800 1.155 2.126 1.187 0.924

5.519 2.011 4.769 2.485 5.056 5.027 5.058 3.648 2.577 3.801 5.094 3.706 3.422 2.521 4.471 2.183 2.095

17.490 9.327 15.388 10.697 16.974 16.450 16.075 13.683 7.545 11.886 16.113 10.510 9.730 8.264 12.791 6.204 7.562

43.217 29.131 37.242 33.296 42.206 39.152 38.669 36.017 16.949 28.889 37.804 24.743 21.764 22.924 27.704 15.740 21.403

Fig. 17 shows the annual IEAR at the load buses and for the overall system. The graph shows that the annual system IEAR is smaller than the annualized system IEAR and the annual IEAR at the load buses are quite different from the annualized IEAR values. This further indicates that although the annualized IEAR is relatively stable with load level variation, the annual IEAR are different from the annualized values. The annual ECOST and IEAR are shown in Figs. 18 and 19, respectively, for the studies considering the expanded system states (5G 1 3L). Fig. 18 shows that the total system ECOST is now 50,234.0 $k/year, which is 1.28 times the value of 39,356.6 k$/y obtained when 4G 1 3L system states were investigated. There are no signi®cant variations in the distribution of the load bus ECOST due to the extended system state investigation. Fig. 19 shows that the IEAR are changed slightly compared with those shown in Fig. 17. 6. Conclusions

decreases tremendously at all the load buses when the load level reduces by only 10%. This indicates that generation capacity de®ciency is a crucial factor in the customer damage costs at the load buses at the system peak load level. Fig. 13 shows the overall system ECOST at different load levels. The system ECOST at the 2850 MW load level shown in the graph is only one quarter of the actual value, as the actual value is too large to be placed on the graph. The graph shows that the overall system ECOST rapidly decreases when the load level drops from 2850 to 1995 MW. This indicates that the ECOST of the IEEERTS is mainly due to generation capacity de®ciencies and that transmission line capacity constraints are virtually negligible. Fig. 14 shows the variation in the IEAR at the load buses for two different load levels. The graph shows that while the IEAR of the IEEE-RTS varies at the different load buses, it does not vary signi®cantly with load level. Fig. 15 shows the overall system IEAR for different load levels. The graph further indicates that the IEAR index is quite stable with variation in the load levels. Fig. 16 shows the total annual system ECOST and the distribution of the annual ECOST at the load buses. The annual system ECOST, which is 39,356.6 k$/year, is much smaller than the annualized value at the peak load level, which is 763,259.2 k$/year. The graph shows that the annual ECOST at bus 18 is the major contributor to the total annual system ECOST and provides 65% of the total annual system ECOST. The second major contribution comes from bus 20, which provides 12% of the total annual system ECOST. The third major contribution comes from bus 10 with 7% of the total annual system ECOST. The total ECOST for the remaining 14 load buses contribute 16% of the total annual system ECOST.

This paper describes the basic concepts required for composite system reliability cost/bene®t analysis. The basic approach is to minimize a total cost composed of the overall investment cost and the customer damage cost. The investment cost is deterministic in nature and can be obtained using well-established methods. The customer damage cost is probabilistic and is conceptually the aggregated value the customers are willing to pay in order to avoid load interruptions or voltage standard violations. The customer damage cost is a function of interruption frequency, duration, load lost, location, and other societal effects. The paper illustrates the calculation of the ECOST (expected customer damage cost) and the related IEAR (interrupted energy assessment rate) indices in composite power systems. The basic data used to calculate the ECOST and IEAR come from the mail surveys conducted by the Power System Research Group at the University of Saskatchewan. The ECOST and the IEAR of the RBTS and the IEEE-RTS were calculated at both the load points and for the overall systems under various conditions. In these studies, both annualized and annual indices were calculated. The results are illustrated, compared and analyzed in the paper to provide insight into the variation of the ECOST and the IEAR with different system factors. The results of the system studies show that the ECOST decreases rapidly with reduction in load level for those load buses at which possible failures are mainly due to generation de®ciencies. The ECOST can be very large and very sensitive to system load level variations, when the system load level is relatively high. The results also show that the annual ECOST at the load buses have a very different distribution than do the annualized values. The system studies show that the annualized IEAR index does not

R. Billinton, W. Zhang / Electrical Power and Energy Systems 23 (2001) 99±112

Fig. 5. CCDF for the IEEE-RTS load buses ($/kW).

Fig. 6. Expected customer damage costs for the RBTS load buses at variable load levels.

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Fig. 7. Expected customer damage costs for the overall RBTS at various load levels.

Fig. 8. Interrupted energy assessment rate for the RBTS load buses at variable load levels.

Fig. 9. Interrupted energy assessment rate for the overall RBTS at various load levels.

R. Billinton, W. Zhang / Electrical Power and Energy Systems 23 (2001) 99±112

Fig. 10. Distribution of the annual expected customer damage cost in the RBTS.

Fig. 11. Annual interrupted energy assessment rate for the overall RBTS and at the RBTS load buses.

Fig. 12. Expected customer damage costs for the IEEE-RTS load buses at variable load levels.

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Fig. 13. Expected customer damage costs for the overall IEEE-RTS at various load levels.

Fig. 14. Interrupted energy assessment rate for the IEEE-RTS load buses at variable load levels.

Fig. 15. Interrupted energy assessment rate for the overall IEEE-RTS at various load levels.

R. Billinton, W. Zhang / Electrical Power and Energy Systems 23 (2001) 99±112

Fig. 16. Distribution of the annual expected customer damage cost in the IEEE-RTS.

Fig. 17. Annual interrupted energy assessment rate for the overall IEEE-RTS and at the load buses.

Fig. 18. Distribution of the annual expected customer damage cost in the IEEE-RTS considering 5G 1 3L.

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Fig. 19. Annual interrupted energy assessment rate for the overall IEEE-RTS and at the load buses considering 5G 1 3L.

change signi®cantly with variation in system load level. This result was recognized in previous research work. A further result shown in this paper is that the actual annual IEAR values for the overall system and the load buses are not equal to the corresponding annualized values, although the annualized IEAR is stable with load level variation. The paper illustrates the essential techniques and philosophy of reliability cost/bene®t analysis in composite power systems. References [1] Udo V, Agarwal SK, Vojdani A, Harlacher MI. Balancing cost and reliability Ð a quantitative study at Atlantic electric. IEEE Trans Power Sys 1997;12(3):1103±8. [2] Li W, Billinton R. A minimum cost assessment method for composite generation and transmission system expansion planning. IEEE Trans Power Sys 1993;8(2):628±35. [3] Neudorf EG, Kiguel DL, Hamoud GA, Porreta B, Stephenson WM, Sparks RW, Logan DM, Bhavaraju MP, Billinton R, Garrison DL. Cost±bene®t analysis of power system reliability Ð two utility cases. IEEE Trans Power Sys 1995;10(3):1667±75.

[4] Dalton JD, Garrison DL, Fallon CM. Value-based reliability transmission planning. IEEE Trans Power Sys 1996;11(3):1400±8. [5] Sullivan M, Vardell T, Suddeth N, Vojdani A. Interruption costs, customer satisfaction and expectations for service reliability. IEEE Trans Power Sys 1996;11(2):989±95. [6] Vojdani AF, Williams RD, Gambel W, Li W, Eng L, Suddeth BN. Experience with application of reliability and value of service analysis in system planning. IEEE Trans Power Sys 1996;11(3):1489±96. [7] Billinton R, Kumar S, Chowdhury N, Chu K, Debnath K, Goel L, Khan E, Kos P, Nourbakhsh G, Oteng-Adjei J, Reliability A. Test system for educational purposes Ð basic data. IEEE Trans Power Sys 1988;4:1238±44. [8] IEEE PES Task Force, The IEEE Reliability Test System-1996, IEEE Publication, 96 WM 326-9 PWR S. [9] Tollefson G, Billinton R, Wacker G, Chan E, Aweya J. A Canadian customer survey to access power system reliability worth. IEEE Trans Power Sys 1994;2(1):443±50. [10] Billinton R, Allan RN. Reliability evaluation of power systems. 2nd ed. New York: Plenum Press, 1996. [11] Billinton R, Zhang W. Enhanced adequacy equivalent for composite power system reliability evaluation. IEE Proc Gener Transm Distrib 1996;143(5):420±6.