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Evaluation of the marginal outage costs in composite generation and transmission systems
ELSEVIER
Electric Power Systems Research 31 (1994) 79-86
ELECTRIC POWER SW$?BI'I'I$ RBSBflROH
Evaluation of the marginal outage costs in composite generation and transmission systems R. Ghajar, R. Billinton Power Systems Research Group, University of Saskatchewan, Saskatoon, Sask., STN OWO, Canada
Received 7 January 1994; accepted 15 April 1994 Abstract
Evaluation of the marginal outage costs in generation systems using quantitative power system reliability techniques has been recently demonstrated. It was shown that the marginal outage cost varies with the operating reserve and the lead time into the future. This paper is concerned with the variation of the marginal outage costs over space because the location of loads with respect to the overall system topology containing both generation and transmission facilities has an impact on these costs. The effects of selected pertinent factors on the marginal outage cost in composite systems are also presented. The proposed method and sensitivity studies are illustrated using the IEEE Reliability Test System. Keywords: Spot pricing; Marginal outage costs; Reliability evaluation; Power system economics
1. Introduction
Spot pricing and other similar tariffs have been proposed as rational mechanisms of load management [1-4]. In addition, recent trends towards deregulation of the electric utility industry require flexible methods for setting electricity prices. Although the ultimate goal of any spot pricing scheme is to calculate and provide electricity spot prices at individual customer load points in the system, it is impossible to accomplish this in a practical system. Instead, spot prices are estimated for groups of customers within defined geographic areas or at bulk load points. Since the marginal outage costs are important components of electricity spot prices, methods capable of forecasting these costs for electric power systems must be developed. Marginal outage costs quantify the incremental economic effects of load changes on the reliability of electric service. A method capable of forecasting the marginal outage costs in generating systems using quantitative power system reliability techniques has been recently developed [1, 2]. The proposed method relates the marginal outage cost to the characteristics of generating units, the load model and the lead time used in the analysis. This method was illustrated by application to the IEEE Reliability Test System (IEEE-RTS) [5]. 0378-7796/94/$07.00 © 1994 Elsevier ScienceS.A. All rights reserved SSDI 0378-7796(94)00874-4
In addition to varying with the characteristics of generating units and the load level, marginal outage costs also vary over space, as the location of loads relative to generators and the existing transmission system constraints significantly affect costs [3, 4]. The evaluation of marginal outage cost in composite systems requires a thorough investigation of the adequacy of these systems using quantitative reliability methods [6-10]. An important factor in composite systems is the relationship between generation and transmission elements and how outages of these facilities affect the performance of the system. A substantial amount of research work in the area of composite system reliability evaluation has been done at the University of Saskatchewan [11-13] and a computer program called C O M R E L (COMposite system RELiability Evaluation) has been developed. This paper provides a brief description of the program and illustrates its utilization in calculating the marginal outage costs in composite systems by application to the IEEE-RTS [5]. In addition to the basic results, a number of sensitivity analyses are presented in order to show the impact of simple approximations on the marginal outage cost. The results from these studies show how simplified models can be used to calculate the marginal outage costs in composite systems. Such models have numerous practical applications because most utilities do not currently use
80
R. Ghajar, R. Billinton /Electric" Power Systems Research 31 (1994) 79-86
detailed composite system reliability models in the operating environment. base case system
2. Description of the COMREL program The C O M R E L program is based on analytical concepts of reliability evaluation and employs the contingency enumeration technique for the assessment of composite systems. The program can handle independent outages as well as common mode events and station-originating outages when required. Only independent outages, however, are considered in the analyses reported in this paper. The C O M R E L program is equipped with three network solution techniques for analyzing system contingencies: the transportation model [14], the DC load flow algorithm [15] and the AC load flow algorithm [16, 17]. Any one of the solution techniques can be selected for evaluating the system performance depending on the prescribed set of system failure criteria. The basic structure of the contingency enumeration algorithm used in C O M R E L is illustrated by the flowchart in Fig. 1. Addition features of the C O M R E L program can be found in Refs. [11-13]. In the contingency enumeration approach shown in Fig. 1, a contingency is selected and tested to asertain whether it causes any immediate system problem such as a circuit overload or bus voltage out of limits. If it does not, a new contingency is selected and tested. The occurrence of a system problem may by itself be entered as a failure. In some cases, however, it is possible to adjust generation or phase shifters to relieve overloads and to adjust transformer taps to bring bus voltages back within the acceptable range. A system failure is therefore recorded when the remedial actions, short of curtailing customer loads, are insufficient to eliminate the system problems. The severity of such system problems is assessed by calculating the amount and location of load curtailment necessary to eliminate the problem. In this way, it is possible to compute area or bus reliability indices that measure the frequency, duration and amount of expected load curtailment. These indices can then be used to calculate the marginal outage costs at the bulk customer load points of the system, as described in the following section of this paper.
3. Proposed method for calculating the marginal outage costs in composite systems The marginal outage cost in a composite system is defined as the change in that system expected outage cost due to an incremental change in load at a customer load point [4, 18]. This concept is formulated in (1) by adding the incremental expected outage costs, caused by a load increase at bus k, of all the load buses in the system:
analysis
V
next condngency
yes
I
I
n~~uc~m;ila~ice syes rlo
Fig. 1. Flowchartof the COMRELprogram. nbus i--1
M O C , = ~' IEAR, x AEUEk,
$/kW
(1)
where MOCk is the marginal outage cost at load bus k, IEARi is the interrupted energy assessment rate at load bus i (S/kWh), AEUEki is the incremental expected unserved energy at load bus i caused by a load increment at bus k (MWh/MW), and nbus is the total number of load buses in the system. The IEAR in generation systems is an expected value of the cost of unserved energy resulting from the inadequacy of the generation system. This idea of utilizing expected values has also been used to calculate practical IEAR estimates for the individual load buses in a composite generation and transmission system adequacy assessment [19]. This work was done using the C O M R E L program. For each outage contingency, the system state is scrutinized and, if necessary, appropriate corrective actions are taken. A system failure is recorded when corrective actions, short of curtailing customer loads, are unable to eliminate the system problem. The severity of a failure is evaluated by calculating the frequency, duration, magnitude and lo-
R. Ghajar, R. Billinton / Electric Power Systems Research 31 (1994) 79-86
cation of load curtailment. For each contingency j that leads to load curtailment at a load bus i, the variables generated by C O M R E L are the magnitude Lij (MW) of load curtailment, the frequency fj (occurrences/year) and the duration dj (hours) of the contingency j. The expected unserved energy at bus i due to all the contingencies that lead to load curtailment, EUEi, is given by (2), where NC is the total number of contingencies that lead to power interruptions at bus i,
Bus 17 Bus
22
Bus 23
Bus 16
r Bus 19
NC
EUEi = Z L~.Jjdj MWh/year
81
Bus
(2)
14 Bus 13
j=l
The interruption cost to customers at bus i of an outage of duration dj, c2(dj), can be obtained from the composite customer damage function of that bus, CCDFg. This function represents the interruption costs of all the customers at bus i as a function of the interruption duration [19]. The expected cost of power interruptions to customers at bus i for all contingencies, ECOSTi, is given by (3) and the IEAR at bus i, IEARi, is calculated using (4):
230 kV Bus 11 m l l a m l l m
BUS9 [
m~/amlam~Bus 12
" P - ' v ' i ' ~ Bust° Bu~6
II
Bus 4
Bus 5 NC
ECOSTi = ~ L i S j c j ( d j )
x occ./year x $/kW (3)
MW
Bus 8
138 kV
j=l cable IEAR
i -
~?--c1Li~jcj(4) )~?__c1 L ~ j f j 4
S/kWh
(4)
The second component required for calculating the marginal outage cost is the incremental expected unserved energy resulting from a load increment at bus k, AEUE~. This variable can be calculated by taking the difference between two EUE values that are evaluated at incrementally different load levels. The expected unserved energy at bus i, E U G , can be calculated using (2). If the load at bus k is increased by 1 MW, the following expression can be used to calculate the new expected unserved energy at bus i, E U G : NC'
EUE~ = ~ LJ'jd}
MWh/year
(5)
j=l
where the variables L~j, f~, d~ and NC' have the same definitions as those given in (2). Given the values of EUEi and EUE'i, the incremental expected unserved energy at bus i caused by a load increment at bus k can be calculated using (6). The factor of 1/8760 is employed to express the results on an hourly rather than annual basis:
AEUEki =
US61(NO=
~xj21Z;J~.dlj
NCLijfJ4 )
-- j:IZ
(6)
The utilization of the proposed method for calculating the marginal outage costs in composite systems is illustrated in the following section of this paper by application to the IEEE-RTS.
E-Bus1
BUS7
Fig. 2. Single-line diagram of the IEEE-RTS.
4. System studies The single-line diagram of the IEEE-RTS is given in Fig. 2. This system has 17 buses which have loads connected to them. The remaining buses are either free buses or generator buses without connected load. The generation and transmission system data for the IEEERTS are given in Ref. [5] and the composite customer damage function for each load bus in the system is given in Table A1 in the Appendix of this paper. These functions were used together with the reliability data of the IEEE-RTS in C O M R E L to calculate the IEAR at each load bus using (4). The results from these calculations are summarized in Table A2 in the Appendix. In order to facilitate the comparison between the marginal outage cost profiles at the load buses, six categories of load buses have been chosen according to their type, voltage level, and location relative to a generating station. This classification helps in comparing the marginal outage cost profiles of buses falling into one class with those of buses falling into other classes. The buses in the six categories are as follows. (a) 138 kV buses (south region):
Class 1. Buses having local generation (buses 1, 2 and 7)
82
R. Ghajar, R. Billinton /Electric Power Systems Research 31 (1994) 79-86
Class 2. Buses that are one line away from generating stations and connected with two lines (buses 4, 5 and 6) Class 3. Buses that are one line away from generating stations and terminated with three or more lines (buses 3 and 8) Class 4. Buses that are two lines away from generating stations (buses 9 and 10)
11~ t
~
~
overallmarginal outagecostprofile
(b) 230 kV buses (north region):
Class 5. Buses having local generation (buses 13, 15, 16 and 18) Class 6. Buses with no local generation (buses 14, 19 and 20) On the basis of the above classification, a number of marginal outage cost profiles corresponding to each class were calculated. These profiles are plotted as a function of the system's operating reserve in Fig. 3 which shows that the differences between them are negligible. This observation is significant because a single marginal outage cost profile can be used regardless of the location of the load increment within the system. It is recognized that this observation may be peculiar to the IEEE-RTS and therefore it should not be applied to other systems unless detailed studies of the specific system are performed. The magnitude of the marginal outage cost in composite systems largely depends on the contributions from the individual load buses. The contribution of each bus is a function of the sensitivity of the bus to load increases in the system and its IEAR. In order to determine the contribution of the buses in each class to the overall marginal outage cost, the highest contributions from each class are compared with the overall marginal outage cost for a load increment at bus 4 in
l° I 1
.1 ¸ o
~ .01. E
Bus# 1 (Class 1) Bus # 4 (Class2) Bus# 8(Class3) Bus # 10 (Class4) Bus # 13 (Class5) Bus # 19 (Class6)
* .001. ~ • •
. 0 0 0 1
•
100
,
i
300
•
.
i
•
500
'~ •
•
l
700
•
.
i
900
•
•
i
1100
.
.
1300
operatingreserve(MW) Fig. 3. Marginal outage cost profiles of selected buses in the IEEERTS.
.00001 100
300
500 700 900 operatingreserve(MW)
1100
1300
Fig. 4. Contribution of selected buses in the various classes to the marginal outage cost of bus 4 in the IEEE-RTS.
Fig. 4. It can be clearly seen from this Figure that buses in class 5 (230 kV north region) have the highest contribution while those in class 4 (138 kV south region) have the lowest contribution. This result is very important because the buses in class 5 are located far away from bus 4 and therefore it was expected that there contribution would be minimal. In addition, the IEAR values of the buses in class 5 are very small (Table A2). Therefore, it is concluded that the high unreliability of the buses in class 5 is responsible for the bulk of their contributions to the marginal outage cost of the composite system.
4.1. Comparison between the marginal outage costs of the generation and composite systems The IEEE-RTS is a large power system with many generating units and transmission lines that can affect the value of the marginal outage cost. A comparison between the marginal outage cost profile of the generation system [ 1] and that of the composite system caused by a load increment at bus 4 is made in Fig. 5. It can be seen from this Figure that the two profiles are relatively close. Since the generation system reliability data and the basic customer outage cost data used in the analyses are essentially unchanged, the difference between the two profiles can be attributed to the reliability techniques employed. In an effort to identify the contributions of the generation and transmission systems to the marginal outage cost of a composite system, the marginal outage cost profile caused by a load increment at bus 4 was recalculated using a fully reliable generation system (transmission contribution) and a fully reliable transmission system (generation contribution). The results from these studies are compared with the overall corn-
R. Ghajar, R. Billinton / Electric Power Systems Research 31 (1994) 79-86
10
~
5. Sensitivity analyses
I ----4n-- composite system M.O.C. profile
.01.
"~ .001.
.0000) i • 100
•
, • 300
-
, • 500
-
, . 700
.
. . 900
.
.
. . 1100
1300
operating reserve (MW) Fig. 5. C o m p a r i s o n o f the m a r g i n a l o u t a g e cost profile o f bus 4 in the I E E E - R T S with the g e n e r a t i o n s y s t e m profile.
lO
I
composite system M.O.C. profile ---o-- cont. of generation system .
-
83
[
The method proposed in Section 3 of this paper for calculating the marginal outage cost in composite systems can be considered to be reasonably comprehensive. This technique entails considering every possible contingency and examining it to see if the corresponding system problem will lead to load curtailment. Comprehensive evaluation of composite system reliability is very time consuming when applied to large power systems. In practical system studies, a number of approximations are normally made to the exact method in order to reduce the computing time. These approximations are done because of lack of computational tools, lack of data, or in order to meet the stringent turnaround time constraints of the operation environment. This section compares the marginal outage costs calculated using a number of approximation with the exact values shown in Fig. 3. The observations drawn from these analyses can be used to make recommendations for practical implementations of the proposed method of calculating the marginal outage costs in composite systems•
5. I. Effect of load flow solutions .01
•~ .OOl
.0001
.00001 100
•
300
500
700
900
1100
1300
operating reserve (MW) Fig. 6. C o n t r i b u t i o n s o f the g e n e r a t i o n a n d t r a n s m i s s i o n systems to the m a r g i n a l o u t a g e cost o f bus 4 in the I E E E - R T S .
posite system profile in Fig. 6 which shows that the contribution of the generation system is much larger than that of the transmission system. The contribution of the transmission system is negligible for most operating reserves except when the system is heavily loaded (i.e. low operating reserve). The discontinuities in this profile indicate that the contribution of the transmission system at these operating reserves is either zero or negligible. The differences between the overall marginal outage cost profile and the contribution of the generation system are due to the limitations placed on the generation system model used in composite system studies (i.e. only up to four overlapping independent outages of generating units). It is expected, however, that these differences will decrease if higher order contingencies are considered.
The adequacy appraisal of a bulk power system entails the solution of a network configuration under selected outage situations. A number of solution techniques, depending on the adequacy criteria employed and the intent behind the studies, are available to analyze the adequacy of a power system. In composite system reliability evaluation, load flows must be repeated for each examined state in the process and the efficiency of the entire evaluation depends a great deal upon the load flow algorithm employed. The three basic analytical techniques that are usually employed in composite generation and transmission adequacy studies are the network flow method [ 14], the DC load flow methods [15] and the AC load flow methods [16, 17]. In general, line overloads can be estimated from less accurate load flows such as the network flow and the DC load flow methods• The COMREL program has the capability of utilizing any one of the three load flow methods in quantitative adequacy assessment of composite systems [ 11 - 13]. The purpose of this study is to ascertain the impact of these solutions techniques on the marginal outage cost of the IEEE-RTS. The results from all three methods are compared in Fig. 7 which shows that the marginal outage cost profiles calculated using the AC and DC load flow methods are not profoundly different and, for all practical purposes, the results from the network flow method are acceptable when compared with the AC load flow method.
84
R. Ghajar, R. Billinton / E l e c t r i c Power Systems Research 31 (1994) 7 9 - 8 6
10
10
1,
z i .01
.1 .01
E .001,
.001
"'-'~ *
.0001 100
300
500 700 900 operating reserve (MW)
load curtailmentphilosophy Pass 1 * Pass 2 .-..-o.---- Pass 3
AC load flow DC load flow network flow method
.0001.
1100
1300
100
300
500 700 900 operating reserve (MW)
1100
1300
Fig. 7. Effect of load flow solutions on the marginal outage cost profile of bus 4 in the IEEE-RTS.
Fig. 8. Effect of the load curtailment philosophy on the marginal outage cost profile of bus 4 in the IEEE-RTS.
5.2. Effect of the load curtailment philosophy
load increments at various buses have similar shapes. Comparison between the marginal outage cost profiles of the generation system and the composite system shows that the two profiles are relatively close. The contribution of the generation system calculated using composite system reliability techniques is found to be much more significant than that of the transmission system. The contribution of the transmission system becomes more noticeable, however, when the system is heavily loaded. The exact method of calculating the marginal outage cost in composite generation and transmission systems is very time consuming and therefore it may not be suitable for applications in the operation environment. This paper presents a number of approximations that reflect the impact of changes in selected pertinent factors on the marginal outage cost profile of the IEEERTS. The results from the sensitivity studies show that the marginal outage cost at a load bus is largely unaffected by the network flow solution method and the load curtailment philosophy used in the composite system reliability program.
The curtailment of load at the customer load points in the event of a deficiency in the generation capacity can be decided in a number of ways depending on the relative priority assigned to each of the major load centers. For the purposes of composite system reliability evaluation, the load at each load bus is usually designated as a curtailable and firm load. The impact of a system disturbance that results in swing-bus overload (i.e. an indication of capacity deficiency in the system) can be confined to a small or large region of the system. If the relative importance of the load at a bus is such that the film load at the customer load point will not be curtailed unless it is inevitable, then it is quite obvious that more buses in the system will experience load curtailment. These provisions have been made in the load curtailment philosophy algorithm utilized in the C O M R E L program by defining three different load curtailment passes [ 12]. The effects of these three passes on the marginal outage cost profile of the IEEE-RTS are examined in this study. The results from this study are shown graphically in Fig. 8. It can be seen from this Figure that the marginal outage cost of the IEEE-RTS is not affected by the number of load curtailment passes.
6. Conclusions This paper presents a method for calculating the marginal outage costs at bulk customer load points in a composite generation and transmission system. The application of the proposed method to the IEEE-RTS shows that the marginal outage cost profiles caused by
Appendix The most basic requirements for calculating the interrupted energy assessment rates at bulk customer load points are the composite customer damage functions (CCDFs) at each load bus in the system. These functions can be calculated by assigning different sectors to specific load buses in the system. A number of sectors have been allocated to the load buses in the IEEE-RTS resulting in the CCDFs given in Table A1. The procedure used to calculate the CCDFs is well documented in Ref. [19].
R. Ghajar, R. Billinton /Electric Power Systems Research 31 (1994) 79-86 Table A1 CCDF for each load bus in the IEEE-RTS System bus no.
1 2 3 4 5 6 7 8 9 10 13 14 15 16 18 19 20
Interruption duration ($/kW in 1987) 1 min
20 min
1 hour
4 hours
8 hours
0.658 0.072 0.574 0.094 0.532 0.623 0.574 0.340 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 1.729 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 5.026 5.057 3.650 2.577 3.801 5.094 3.707 3.423 2.521 4.471 2.183 2.095
17.489 9.327 15.387 10.697 16.973 16.446 16.075 13.688 7.544 11.885 16.113 10.511 9.731 8.264 12.791 6.204 7.561
43.213 29.131 37.241 33.295 42.202 39.144 38.669 36.024 16.948 28.888 37.804 24.746 21.766 22.923 27.704 15.741 21.400
Table A2 IEAR values for each load bus in the IEEE-RTS Load bus no.
Classification
1EAR (S/kWh)
1 2 3 4 5 6 7 8 9 10 13 14 15 16 18 19 20
class class class class class class class class class class class class class class class class class
6.20 4.89 5.30 5.62 6.11 5.50 5.41 5.40 2.30 4.14 5.39 3.41 3.01 3.54 3.75 2.29 3.64
1 1 3 2 2 2 1 3 4 4 5 6 5 5 5 6 6
The IEAR value at each load bus of the IEEE-RTS was calculated using the following modeling assumptions and parameters in C O M R E U (1) The peak load of the IEEE-RTS was set at 2850 MW. (2) A single-step load model is used in the analysis. The effect of a multistep load model can be included in composite system reliability studies in order to produce more representative annual IEAR values at the expense of computation time. (3) All load buses were assumed to have 20% curtailable load. (4) Passl load curtailment philosophy was used.
85
(5) The AC load flow technique was used. (6) Contingency enumeration of up to the following number of contingencies was used: (i) four or less generating units were examined; (ii) three or less transmission lines were examined; and (iii) up to two generating units and one line and one generating unit and two lines were considered. (7) The effects of station-originating outages were not considered. (8) The effects of common-mode outages were not considered. The results from the above calculations are given in Table A2.
References [1] R. Billinton and R. Ghajar, Evaluation of the marginal outage costs of generating systems for the purposes of spot pricing, 1EEE Trans. Power Syst., 9 (1994) 68 75. [2] R. Ghajar and R. Billinton, Comparison of alternative techniques for evaluating the marginal outage costs of generating systems, IEEE Trans. Power Syst., 8 (1993) 1550-1556. [3] F.C. Schweppe, M.C. Caramanis, R.D. Tabors and R.E. Bohn, Spot Pricing o f Electricity, Kluwer, Boston, MA, 1988. [4] A.P. Sanghvi, Flexible strategies for load/demand management using dynamic pricing, IEEE Trans. Power Syst., 4 (1989) 83 93. [5] IEEE Committee Rep., IEEE Reliability Test System, IEEE Trans. Power Appar. Syst., PAS-98 (1979)2047-2054. [6] IEEE Committee Rep., Bibliography on the application of probability methods in power system reliability evaluation: 1971-1977, IEEE Trans. Power Appar. Syst., PAS-97 (1978) 2235 -2245. [7] R.N. Allan, R. Billinton and S.H. Lee, Bibliography on the application of probability methods in power system reliability evaluation: 1977-1982, IEEE Trans. Power Appar. Syst., PAS103 (1984) 275-282. [8] R.N. Allan, R. Billinton, S.M. Shahidehpour and C. Singh, Bibliography on the application of probability methods in power system reliability evaluation: 1982 1987, IEEE Trans. Power Appar. Syst., 3 (1988) 1555-1564. [9] M.Th. Schilling, R. Billinton, A.M. Leite de Silva and M.A. E1-Kady, Bibliography on composite system reliability: 1964-1988, IEEE Trans. Power Syst., 4 (1989) 11221132. [10] R.N. Allan, R. Billinton, A.M. Breipohl and C.H. Grigg, Bibliography on the application of probability methods in power system reliability evaluation: 1987-1991, IEEE Trans. Power Syst., 9 (1994) 41 49. [11] T.K.P. Medicherla and R. Billinton, Overall approach to the reliability evaluation of composite generation and transmission systems, l E E Proc. C, 127 (1980) 72-81. [12] M.S. Sachdev, T.K.P. Medicherla and R. Billinton, Generation rescheduling and load shedding to alleviate line overloads--analysis, IEEE Trans. Power Appar. Syst., PAS-98 (1979) 1876 1884. [13] R. Billinton and S. Kumar, Pertinent factors in the adequacy assessment of composite generation and transmission system, Proc. Can. Electr. Assoc., 25 (1986) Paper No. 86-SP141. [14] L.R. Ford and D.R. Fulkerson, Flows in Networks, Princeton University Press, Princeton, MJ, 1962.
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R. Ghajar, R. Billinton /Electric Power Systems Research 31 (1994) 79-86
[15] Electric Power Research Institute, Transmission System Reliability Methods Mathematieal Methods, Computing Methods and Results, EPRI Rep. No. EL-2126, Power Technologies Inc., New York, July 1982. [16] B. Stagg, and A.H. El-Abiad, Computer Methods in Power System Analysis, McGraw-Hill, New York, 1968. [ 17] B. Scott and O. Alsac, Fast decoupled load flow, IEEE Trans.
Power Appar. Syst., PAS-93 (1974) 859 869. [18] L.D. Kirsch, R.L. Sullivan, T.A. Flaim, J.J. Hipius and M.G. Krantz, Developing marginal costs for real-time pricing, IEEE Trans. Power Syst., 3 (1988) 1133-1138. [19] J. Oteng-Adjei and R. Billinton, Evaluation of interrupted energy assessment rates in composite systems, IEEE Trans. Power Syst., 5(1990) 1317 1323.
Electric Power Systems Research 31 (1994) 79-86
ELECTRIC POWER SW$?BI'I'I$ RBSBflROH
Evaluation of the marginal outage costs in composite generation and transmission systems R. Ghajar, R. Billinton Power Systems Research Group, University of Saskatchewan, Saskatoon, Sask., STN OWO, Canada
Received 7 January 1994; accepted 15 April 1994 Abstract
Evaluation of the marginal outage costs in generation systems using quantitative power system reliability techniques has been recently demonstrated. It was shown that the marginal outage cost varies with the operating reserve and the lead time into the future. This paper is concerned with the variation of the marginal outage costs over space because the location of loads with respect to the overall system topology containing both generation and transmission facilities has an impact on these costs. The effects of selected pertinent factors on the marginal outage cost in composite systems are also presented. The proposed method and sensitivity studies are illustrated using the IEEE Reliability Test System. Keywords: Spot pricing; Marginal outage costs; Reliability evaluation; Power system economics
1. Introduction
Spot pricing and other similar tariffs have been proposed as rational mechanisms of load management [1-4]. In addition, recent trends towards deregulation of the electric utility industry require flexible methods for setting electricity prices. Although the ultimate goal of any spot pricing scheme is to calculate and provide electricity spot prices at individual customer load points in the system, it is impossible to accomplish this in a practical system. Instead, spot prices are estimated for groups of customers within defined geographic areas or at bulk load points. Since the marginal outage costs are important components of electricity spot prices, methods capable of forecasting these costs for electric power systems must be developed. Marginal outage costs quantify the incremental economic effects of load changes on the reliability of electric service. A method capable of forecasting the marginal outage costs in generating systems using quantitative power system reliability techniques has been recently developed [1, 2]. The proposed method relates the marginal outage cost to the characteristics of generating units, the load model and the lead time used in the analysis. This method was illustrated by application to the IEEE Reliability Test System (IEEE-RTS) [5]. 0378-7796/94/$07.00 © 1994 Elsevier ScienceS.A. All rights reserved SSDI 0378-7796(94)00874-4
In addition to varying with the characteristics of generating units and the load level, marginal outage costs also vary over space, as the location of loads relative to generators and the existing transmission system constraints significantly affect costs [3, 4]. The evaluation of marginal outage cost in composite systems requires a thorough investigation of the adequacy of these systems using quantitative reliability methods [6-10]. An important factor in composite systems is the relationship between generation and transmission elements and how outages of these facilities affect the performance of the system. A substantial amount of research work in the area of composite system reliability evaluation has been done at the University of Saskatchewan [11-13] and a computer program called C O M R E L (COMposite system RELiability Evaluation) has been developed. This paper provides a brief description of the program and illustrates its utilization in calculating the marginal outage costs in composite systems by application to the IEEE-RTS [5]. In addition to the basic results, a number of sensitivity analyses are presented in order to show the impact of simple approximations on the marginal outage cost. The results from these studies show how simplified models can be used to calculate the marginal outage costs in composite systems. Such models have numerous practical applications because most utilities do not currently use
80
R. Ghajar, R. Billinton /Electric" Power Systems Research 31 (1994) 79-86
detailed composite system reliability models in the operating environment. base case system
2. Description of the COMREL program The C O M R E L program is based on analytical concepts of reliability evaluation and employs the contingency enumeration technique for the assessment of composite systems. The program can handle independent outages as well as common mode events and station-originating outages when required. Only independent outages, however, are considered in the analyses reported in this paper. The C O M R E L program is equipped with three network solution techniques for analyzing system contingencies: the transportation model [14], the DC load flow algorithm [15] and the AC load flow algorithm [16, 17]. Any one of the solution techniques can be selected for evaluating the system performance depending on the prescribed set of system failure criteria. The basic structure of the contingency enumeration algorithm used in C O M R E L is illustrated by the flowchart in Fig. 1. Addition features of the C O M R E L program can be found in Refs. [11-13]. In the contingency enumeration approach shown in Fig. 1, a contingency is selected and tested to asertain whether it causes any immediate system problem such as a circuit overload or bus voltage out of limits. If it does not, a new contingency is selected and tested. The occurrence of a system problem may by itself be entered as a failure. In some cases, however, it is possible to adjust generation or phase shifters to relieve overloads and to adjust transformer taps to bring bus voltages back within the acceptable range. A system failure is therefore recorded when the remedial actions, short of curtailing customer loads, are insufficient to eliminate the system problems. The severity of such system problems is assessed by calculating the amount and location of load curtailment necessary to eliminate the problem. In this way, it is possible to compute area or bus reliability indices that measure the frequency, duration and amount of expected load curtailment. These indices can then be used to calculate the marginal outage costs at the bulk customer load points of the system, as described in the following section of this paper.
3. Proposed method for calculating the marginal outage costs in composite systems The marginal outage cost in a composite system is defined as the change in that system expected outage cost due to an incremental change in load at a customer load point [4, 18]. This concept is formulated in (1) by adding the incremental expected outage costs, caused by a load increase at bus k, of all the load buses in the system:
analysis
V
next condngency
yes
I
I
n~~uc~m;ila~ice syes rlo
Fig. 1. Flowchartof the COMRELprogram. nbus i--1
M O C , = ~' IEAR, x AEUEk,
$/kW
(1)
where MOCk is the marginal outage cost at load bus k, IEARi is the interrupted energy assessment rate at load bus i (S/kWh), AEUEki is the incremental expected unserved energy at load bus i caused by a load increment at bus k (MWh/MW), and nbus is the total number of load buses in the system. The IEAR in generation systems is an expected value of the cost of unserved energy resulting from the inadequacy of the generation system. This idea of utilizing expected values has also been used to calculate practical IEAR estimates for the individual load buses in a composite generation and transmission system adequacy assessment [19]. This work was done using the C O M R E L program. For each outage contingency, the system state is scrutinized and, if necessary, appropriate corrective actions are taken. A system failure is recorded when corrective actions, short of curtailing customer loads, are unable to eliminate the system problem. The severity of a failure is evaluated by calculating the frequency, duration, magnitude and lo-
R. Ghajar, R. Billinton / Electric Power Systems Research 31 (1994) 79-86
cation of load curtailment. For each contingency j that leads to load curtailment at a load bus i, the variables generated by C O M R E L are the magnitude Lij (MW) of load curtailment, the frequency fj (occurrences/year) and the duration dj (hours) of the contingency j. The expected unserved energy at bus i due to all the contingencies that lead to load curtailment, EUEi, is given by (2), where NC is the total number of contingencies that lead to power interruptions at bus i,
Bus 17 Bus
22
Bus 23
Bus 16
r Bus 19
NC
EUEi = Z L~.Jjdj MWh/year
81
Bus
(2)
14 Bus 13
j=l
The interruption cost to customers at bus i of an outage of duration dj, c2(dj), can be obtained from the composite customer damage function of that bus, CCDFg. This function represents the interruption costs of all the customers at bus i as a function of the interruption duration [19]. The expected cost of power interruptions to customers at bus i for all contingencies, ECOSTi, is given by (3) and the IEAR at bus i, IEARi, is calculated using (4):
230 kV Bus 11 m l l a m l l m
BUS9 [
m~/amlam~Bus 12
" P - ' v ' i ' ~ Bust° Bu~6
II
Bus 4
Bus 5 NC
ECOSTi = ~ L i S j c j ( d j )
x occ./year x $/kW (3)
MW
Bus 8
138 kV
j=l cable IEAR
i -
~?--c1Li~jcj(4) )~?__c1 L ~ j f j 4
S/kWh
(4)
The second component required for calculating the marginal outage cost is the incremental expected unserved energy resulting from a load increment at bus k, AEUE~. This variable can be calculated by taking the difference between two EUE values that are evaluated at incrementally different load levels. The expected unserved energy at bus i, E U G , can be calculated using (2). If the load at bus k is increased by 1 MW, the following expression can be used to calculate the new expected unserved energy at bus i, E U G : NC'
EUE~ = ~ LJ'jd}
MWh/year
(5)
j=l
where the variables L~j, f~, d~ and NC' have the same definitions as those given in (2). Given the values of EUEi and EUE'i, the incremental expected unserved energy at bus i caused by a load increment at bus k can be calculated using (6). The factor of 1/8760 is employed to express the results on an hourly rather than annual basis:
AEUEki =
US61(NO=
~xj21Z;J~.dlj
NCLijfJ4 )
-- j:IZ
(6)
The utilization of the proposed method for calculating the marginal outage costs in composite systems is illustrated in the following section of this paper by application to the IEEE-RTS.
E-Bus1
BUS7
Fig. 2. Single-line diagram of the IEEE-RTS.
4. System studies The single-line diagram of the IEEE-RTS is given in Fig. 2. This system has 17 buses which have loads connected to them. The remaining buses are either free buses or generator buses without connected load. The generation and transmission system data for the IEEERTS are given in Ref. [5] and the composite customer damage function for each load bus in the system is given in Table A1 in the Appendix of this paper. These functions were used together with the reliability data of the IEEE-RTS in C O M R E L to calculate the IEAR at each load bus using (4). The results from these calculations are summarized in Table A2 in the Appendix. In order to facilitate the comparison between the marginal outage cost profiles at the load buses, six categories of load buses have been chosen according to their type, voltage level, and location relative to a generating station. This classification helps in comparing the marginal outage cost profiles of buses falling into one class with those of buses falling into other classes. The buses in the six categories are as follows. (a) 138 kV buses (south region):
Class 1. Buses having local generation (buses 1, 2 and 7)
82
R. Ghajar, R. Billinton /Electric Power Systems Research 31 (1994) 79-86
Class 2. Buses that are one line away from generating stations and connected with two lines (buses 4, 5 and 6) Class 3. Buses that are one line away from generating stations and terminated with three or more lines (buses 3 and 8) Class 4. Buses that are two lines away from generating stations (buses 9 and 10)
11~ t
~
~
overallmarginal outagecostprofile
(b) 230 kV buses (north region):
Class 5. Buses having local generation (buses 13, 15, 16 and 18) Class 6. Buses with no local generation (buses 14, 19 and 20) On the basis of the above classification, a number of marginal outage cost profiles corresponding to each class were calculated. These profiles are plotted as a function of the system's operating reserve in Fig. 3 which shows that the differences between them are negligible. This observation is significant because a single marginal outage cost profile can be used regardless of the location of the load increment within the system. It is recognized that this observation may be peculiar to the IEEE-RTS and therefore it should not be applied to other systems unless detailed studies of the specific system are performed. The magnitude of the marginal outage cost in composite systems largely depends on the contributions from the individual load buses. The contribution of each bus is a function of the sensitivity of the bus to load increases in the system and its IEAR. In order to determine the contribution of the buses in each class to the overall marginal outage cost, the highest contributions from each class are compared with the overall marginal outage cost for a load increment at bus 4 in
l° I 1
.1 ¸ o
~ .01. E
Bus# 1 (Class 1) Bus # 4 (Class2) Bus# 8(Class3) Bus # 10 (Class4) Bus # 13 (Class5) Bus # 19 (Class6)
* .001. ~ • •
. 0 0 0 1
•
100
,
i
300
•
.
i
•
500
'~ •
•
l
700
•
.
i
900
•
•
i
1100
.
.
1300
operatingreserve(MW) Fig. 3. Marginal outage cost profiles of selected buses in the IEEERTS.
.00001 100
300
500 700 900 operatingreserve(MW)
1100
1300
Fig. 4. Contribution of selected buses in the various classes to the marginal outage cost of bus 4 in the IEEE-RTS.
Fig. 4. It can be clearly seen from this Figure that buses in class 5 (230 kV north region) have the highest contribution while those in class 4 (138 kV south region) have the lowest contribution. This result is very important because the buses in class 5 are located far away from bus 4 and therefore it was expected that there contribution would be minimal. In addition, the IEAR values of the buses in class 5 are very small (Table A2). Therefore, it is concluded that the high unreliability of the buses in class 5 is responsible for the bulk of their contributions to the marginal outage cost of the composite system.
4.1. Comparison between the marginal outage costs of the generation and composite systems The IEEE-RTS is a large power system with many generating units and transmission lines that can affect the value of the marginal outage cost. A comparison between the marginal outage cost profile of the generation system [ 1] and that of the composite system caused by a load increment at bus 4 is made in Fig. 5. It can be seen from this Figure that the two profiles are relatively close. Since the generation system reliability data and the basic customer outage cost data used in the analyses are essentially unchanged, the difference between the two profiles can be attributed to the reliability techniques employed. In an effort to identify the contributions of the generation and transmission systems to the marginal outage cost of a composite system, the marginal outage cost profile caused by a load increment at bus 4 was recalculated using a fully reliable generation system (transmission contribution) and a fully reliable transmission system (generation contribution). The results from these studies are compared with the overall corn-
R. Ghajar, R. Billinton / Electric Power Systems Research 31 (1994) 79-86
10
~
5. Sensitivity analyses
I ----4n-- composite system M.O.C. profile
.01.
"~ .001.
.0000) i • 100
•
, • 300
-
, • 500
-
, . 700
.
. . 900
.
.
. . 1100
1300
operating reserve (MW) Fig. 5. C o m p a r i s o n o f the m a r g i n a l o u t a g e cost profile o f bus 4 in the I E E E - R T S with the g e n e r a t i o n s y s t e m profile.
lO
I
composite system M.O.C. profile ---o-- cont. of generation system .
-
83
[
The method proposed in Section 3 of this paper for calculating the marginal outage cost in composite systems can be considered to be reasonably comprehensive. This technique entails considering every possible contingency and examining it to see if the corresponding system problem will lead to load curtailment. Comprehensive evaluation of composite system reliability is very time consuming when applied to large power systems. In practical system studies, a number of approximations are normally made to the exact method in order to reduce the computing time. These approximations are done because of lack of computational tools, lack of data, or in order to meet the stringent turnaround time constraints of the operation environment. This section compares the marginal outage costs calculated using a number of approximation with the exact values shown in Fig. 3. The observations drawn from these analyses can be used to make recommendations for practical implementations of the proposed method of calculating the marginal outage costs in composite systems•
5. I. Effect of load flow solutions .01
•~ .OOl
.0001
.00001 100
•
300
500
700
900
1100
1300
operating reserve (MW) Fig. 6. C o n t r i b u t i o n s o f the g e n e r a t i o n a n d t r a n s m i s s i o n systems to the m a r g i n a l o u t a g e cost o f bus 4 in the I E E E - R T S .
posite system profile in Fig. 6 which shows that the contribution of the generation system is much larger than that of the transmission system. The contribution of the transmission system is negligible for most operating reserves except when the system is heavily loaded (i.e. low operating reserve). The discontinuities in this profile indicate that the contribution of the transmission system at these operating reserves is either zero or negligible. The differences between the overall marginal outage cost profile and the contribution of the generation system are due to the limitations placed on the generation system model used in composite system studies (i.e. only up to four overlapping independent outages of generating units). It is expected, however, that these differences will decrease if higher order contingencies are considered.
The adequacy appraisal of a bulk power system entails the solution of a network configuration under selected outage situations. A number of solution techniques, depending on the adequacy criteria employed and the intent behind the studies, are available to analyze the adequacy of a power system. In composite system reliability evaluation, load flows must be repeated for each examined state in the process and the efficiency of the entire evaluation depends a great deal upon the load flow algorithm employed. The three basic analytical techniques that are usually employed in composite generation and transmission adequacy studies are the network flow method [ 14], the DC load flow methods [15] and the AC load flow methods [16, 17]. In general, line overloads can be estimated from less accurate load flows such as the network flow and the DC load flow methods• The COMREL program has the capability of utilizing any one of the three load flow methods in quantitative adequacy assessment of composite systems [ 11 - 13]. The purpose of this study is to ascertain the impact of these solutions techniques on the marginal outage cost of the IEEE-RTS. The results from all three methods are compared in Fig. 7 which shows that the marginal outage cost profiles calculated using the AC and DC load flow methods are not profoundly different and, for all practical purposes, the results from the network flow method are acceptable when compared with the AC load flow method.
84
R. Ghajar, R. Billinton / E l e c t r i c Power Systems Research 31 (1994) 7 9 - 8 6
10
10
1,
z i .01
.1 .01
E .001,
.001
"'-'~ *
.0001 100
300
500 700 900 operating reserve (MW)
load curtailmentphilosophy Pass 1 * Pass 2 .-..-o.---- Pass 3
AC load flow DC load flow network flow method
.0001.
1100
1300
100
300
500 700 900 operating reserve (MW)
1100
1300
Fig. 7. Effect of load flow solutions on the marginal outage cost profile of bus 4 in the IEEE-RTS.
Fig. 8. Effect of the load curtailment philosophy on the marginal outage cost profile of bus 4 in the IEEE-RTS.
5.2. Effect of the load curtailment philosophy
load increments at various buses have similar shapes. Comparison between the marginal outage cost profiles of the generation system and the composite system shows that the two profiles are relatively close. The contribution of the generation system calculated using composite system reliability techniques is found to be much more significant than that of the transmission system. The contribution of the transmission system becomes more noticeable, however, when the system is heavily loaded. The exact method of calculating the marginal outage cost in composite generation and transmission systems is very time consuming and therefore it may not be suitable for applications in the operation environment. This paper presents a number of approximations that reflect the impact of changes in selected pertinent factors on the marginal outage cost profile of the IEEERTS. The results from the sensitivity studies show that the marginal outage cost at a load bus is largely unaffected by the network flow solution method and the load curtailment philosophy used in the composite system reliability program.
The curtailment of load at the customer load points in the event of a deficiency in the generation capacity can be decided in a number of ways depending on the relative priority assigned to each of the major load centers. For the purposes of composite system reliability evaluation, the load at each load bus is usually designated as a curtailable and firm load. The impact of a system disturbance that results in swing-bus overload (i.e. an indication of capacity deficiency in the system) can be confined to a small or large region of the system. If the relative importance of the load at a bus is such that the film load at the customer load point will not be curtailed unless it is inevitable, then it is quite obvious that more buses in the system will experience load curtailment. These provisions have been made in the load curtailment philosophy algorithm utilized in the C O M R E L program by defining three different load curtailment passes [ 12]. The effects of these three passes on the marginal outage cost profile of the IEEE-RTS are examined in this study. The results from this study are shown graphically in Fig. 8. It can be seen from this Figure that the marginal outage cost of the IEEE-RTS is not affected by the number of load curtailment passes.
6. Conclusions This paper presents a method for calculating the marginal outage costs at bulk customer load points in a composite generation and transmission system. The application of the proposed method to the IEEE-RTS shows that the marginal outage cost profiles caused by
Appendix The most basic requirements for calculating the interrupted energy assessment rates at bulk customer load points are the composite customer damage functions (CCDFs) at each load bus in the system. These functions can be calculated by assigning different sectors to specific load buses in the system. A number of sectors have been allocated to the load buses in the IEEE-RTS resulting in the CCDFs given in Table A1. The procedure used to calculate the CCDFs is well documented in Ref. [19].
R. Ghajar, R. Billinton /Electric Power Systems Research 31 (1994) 79-86 Table A1 CCDF for each load bus in the IEEE-RTS System bus no.
1 2 3 4 5 6 7 8 9 10 13 14 15 16 18 19 20
Interruption duration ($/kW in 1987) 1 min
20 min
1 hour
4 hours
8 hours
0.658 0.072 0.574 0.094 0.532 0.623 0.574 0.340 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 1.729 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 5.026 5.057 3.650 2.577 3.801 5.094 3.707 3.423 2.521 4.471 2.183 2.095
17.489 9.327 15.387 10.697 16.973 16.446 16.075 13.688 7.544 11.885 16.113 10.511 9.731 8.264 12.791 6.204 7.561
43.213 29.131 37.241 33.295 42.202 39.144 38.669 36.024 16.948 28.888 37.804 24.746 21.766 22.923 27.704 15.741 21.400
Table A2 IEAR values for each load bus in the IEEE-RTS Load bus no.
Classification
1EAR (S/kWh)
1 2 3 4 5 6 7 8 9 10 13 14 15 16 18 19 20
class class class class class class class class class class class class class class class class class
6.20 4.89 5.30 5.62 6.11 5.50 5.41 5.40 2.30 4.14 5.39 3.41 3.01 3.54 3.75 2.29 3.64
1 1 3 2 2 2 1 3 4 4 5 6 5 5 5 6 6
The IEAR value at each load bus of the IEEE-RTS was calculated using the following modeling assumptions and parameters in C O M R E U (1) The peak load of the IEEE-RTS was set at 2850 MW. (2) A single-step load model is used in the analysis. The effect of a multistep load model can be included in composite system reliability studies in order to produce more representative annual IEAR values at the expense of computation time. (3) All load buses were assumed to have 20% curtailable load. (4) Passl load curtailment philosophy was used.
85
(5) The AC load flow technique was used. (6) Contingency enumeration of up to the following number of contingencies was used: (i) four or less generating units were examined; (ii) three or less transmission lines were examined; and (iii) up to two generating units and one line and one generating unit and two lines were considered. (7) The effects of station-originating outages were not considered. (8) The effects of common-mode outages were not considered. The results from the above calculations are given in Table A2.
References [1] R. Billinton and R. Ghajar, Evaluation of the marginal outage costs of generating systems for the purposes of spot pricing, 1EEE Trans. Power Syst., 9 (1994) 68 75. [2] R. Ghajar and R. Billinton, Comparison of alternative techniques for evaluating the marginal outage costs of generating systems, IEEE Trans. Power Syst., 8 (1993) 1550-1556. [3] F.C. Schweppe, M.C. Caramanis, R.D. Tabors and R.E. Bohn, Spot Pricing o f Electricity, Kluwer, Boston, MA, 1988. [4] A.P. Sanghvi, Flexible strategies for load/demand management using dynamic pricing, IEEE Trans. Power Syst., 4 (1989) 83 93. [5] IEEE Committee Rep., IEEE Reliability Test System, IEEE Trans. Power Appar. Syst., PAS-98 (1979)2047-2054. [6] IEEE Committee Rep., Bibliography on the application of probability methods in power system reliability evaluation: 1971-1977, IEEE Trans. Power Appar. Syst., PAS-97 (1978) 2235 -2245. [7] R.N. Allan, R. Billinton and S.H. Lee, Bibliography on the application of probability methods in power system reliability evaluation: 1977-1982, IEEE Trans. Power Appar. Syst., PAS103 (1984) 275-282. [8] R.N. Allan, R. Billinton, S.M. Shahidehpour and C. Singh, Bibliography on the application of probability methods in power system reliability evaluation: 1982 1987, IEEE Trans. Power Appar. Syst., 3 (1988) 1555-1564. [9] M.Th. Schilling, R. Billinton, A.M. Leite de Silva and M.A. E1-Kady, Bibliography on composite system reliability: 1964-1988, IEEE Trans. Power Syst., 4 (1989) 11221132. [10] R.N. Allan, R. Billinton, A.M. Breipohl and C.H. Grigg, Bibliography on the application of probability methods in power system reliability evaluation: 1987-1991, IEEE Trans. Power Syst., 9 (1994) 41 49. [11] T.K.P. Medicherla and R. Billinton, Overall approach to the reliability evaluation of composite generation and transmission systems, l E E Proc. C, 127 (1980) 72-81. [12] M.S. Sachdev, T.K.P. Medicherla and R. Billinton, Generation rescheduling and load shedding to alleviate line overloads--analysis, IEEE Trans. Power Appar. Syst., PAS-98 (1979) 1876 1884. [13] R. Billinton and S. Kumar, Pertinent factors in the adequacy assessment of composite generation and transmission system, Proc. Can. Electr. Assoc., 25 (1986) Paper No. 86-SP141. [14] L.R. Ford and D.R. Fulkerson, Flows in Networks, Princeton University Press, Princeton, MJ, 1962.
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R. Ghajar, R. Billinton /Electric Power Systems Research 31 (1994) 79-86
[15] Electric Power Research Institute, Transmission System Reliability Methods Mathematieal Methods, Computing Methods and Results, EPRI Rep. No. EL-2126, Power Technologies Inc., New York, July 1982. [16] B. Stagg, and A.H. El-Abiad, Computer Methods in Power System Analysis, McGraw-Hill, New York, 1968. [ 17] B. Scott and O. Alsac, Fast decoupled load flow, IEEE Trans.
Power Appar. Syst., PAS-93 (1974) 859 869. [18] L.D. Kirsch, R.L. Sullivan, T.A. Flaim, J.J. Hipius and M.G. Krantz, Developing marginal costs for real-time pricing, IEEE Trans. Power Syst., 3 (1988) 1133-1138. [19] J. Oteng-Adjei and R. Billinton, Evaluation of interrupted energy assessment rates in composite systems, IEEE Trans. Power Syst., 5(1990) 1317 1323.