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Modelling electricity demand with representative load curves
Energy 24 (1999) 219–230
Modelling electricity demand with representative load curves P. Balachandra*, Vijay Chandru Department of Management Studies, Indian Institute of Science, Bangalore 560012, India Received 21 November 1997
Abstract Models for electricity planning require inclusion of demand. Depending on the type of planning, the demand is usually represented as an annual demand for electricity (GWh), a peak demand (MW) or in the form of annual load–duration curves. The demand for electricity varies with the seasons, economic activities, etc. Existing schemes do not capture the dynamics of demand variations that are important for planning. For this purpose, we introduce the concept of representative load curves (RLCs). Advantages of RLCs are demonstrated in a case study for the state of Karnataka in India. Multiple discriminant analysis is used to cluster the 365 daily load curves for 1993–94 into nine RLCs. Further analyses of these RLCs help to identify important factors, namely, seasonal, industrial, agricultural, and residential (water heating and aircooling) demand variations besides rationing by the utility. 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction Electricity-supply planning requires efficient management of existing electric power systems, as well as rationalization of decisions concerning new capacity additions. The form in which we represent the demand for electricity is an important aspect in developing models for electricity planning. The major difficulty in modelling demand arises from the complexity created by its high variability. Variations in demand during a day are mainly due to changes in the level of electricity-driven activities at different times. Daily variations through the year are due to seasonal factors (like weather, temperature, rainfall, etc.), level of industrial and agricultural activities and other causes such as holidays, festivals, etc. It would clearly be advantageous to have models that capture all of these variations. * Corresponding author. Fax: 91-80-3341683; e-mail: [email protected] 0360-5442/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 0 - 5 4 4 2 ( 9 8 ) 0 0 0 9 6 - 6
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P. Balachandra, V. Chandru / Energy 24 (1999) 219–230
In the context of macro-level electricity planning, the demand is represented in terms of annual energy (GWh) and peak power (MW) demand [1,2]. In most cases, the representation depends on the type of planning and its requisite level of accuracy. For long-term expansion planning, the accuracy of the representation is compromised in annual [3–6] or normalized load–duration curves [7–9] to minimize complexities. In short-term planning [10,11], accuracy of the representation is very important and the demand is normally represented in terms of actual or forecast daily load curves. For planning, the annual load–duration curve is typically partitioned into (a) base, (b) cycling (intermediate), (c) daily peak, and (d) seasonal peak loads [12]. Fig. 1 shows such a partition developed from hourly load data for Karnataka during 1993–94. Using this curve, models may be developed to schedule various generation facilities to service base, intermediate, daily peak, and seasonal peak loads. In addition to the listed types of planning, electricity systems constrained by both power and capital shortages require effective supply–demand matching. Due to severe shortages of supply, large electricity demands remain unmet. Through supply interruptions and power rationing, util-
Fig. 1.
The annual load–duration curve for 1993–94.
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ities (e.g. State Electricity Boards in India) avoid supplying part of the electricity demanded by consumers. Thus, managing the system requires planning for non-supply (i.e. failure to meet demands by resorting to curtailments) and supply of electricity. In order to plan in advance for both the supply and non-supply of electricity, utilities need to forecast the levels of demand and shortages and their times of occurrence in terms of hour of the day and day of the year. With this type of planning, the modelled demand has to encompass all variations. Estimates of electricity demand that are based on annual or normalized load–duration curves tend to be highly approximate but are adequate for long term strategic planning. The disadvantages of using these annual load–duration curves are: (i) information on the time of occurrence (hour, day) of a particular level of demand is not available; (ii) seasonal variations in demand levels are not known; (iii) information on non-seasonal variations in demand level due to factors like changes in the level of industrial or agricultural activities is also unavailable. We could use 365 daily load curves to represent the annual demand as 8760 input data points. This would lead to models that are intractable. For example, solving optimization models for planning would require excessive computational resources even with advanced software. Also, policy interpretations would be difficult to unravel from the reams of output. In summary, planners require representations of demand that capture both temporal and structural variations that depend on economic activities such as agricultural, industrial production and commerce. To include these variations, the planning horizon has to be at least 1 year. This tradeoff between adequate planning horizon and model complexity led us to the concept of RLCs. 2. Representative load curves An RLC is a typical daily load curve which represents a group of load curves exhibiting similar demand patterns. We believe that RLCs are useful for planning in resource constrained electricity systems and in situations where it is required to know the time variations in demand (e.g. supply– demand matching, seasonal scheduling of hydro plants and maintenance scheduling). RLCs are also used to identify the factors influencing variations in demand and to form some approximate indicators of these influences. The first step in developing RLCs is to group load curves on the basis of some similarities. To form different groups, a classification procedure needs to be used. Traditionally, multiple discriminant analysis is used to classify the individual samples into various groups and verify the correctness of the groupings. With multiple discriminant analysis, an observation is assigned to one or more groups on the basis of value. It serves to identify important variables to distinguish among groups and develop a procedure to predict group membership for new cases [13]. For the analysis, 24 hours of the day are used as 24 variables and 365 days of the year as the number of samples. Normally, in any statistical analysis, the variables measure different attributes of the study system and are commonly assumed to be independent of each other. However, in the present case, the 24 variables measure the same attribute; demand for electricity at different time periods (hours of the day). Therefore, one cannot expect these variables to be independent of each other (for example, in the case study presented below, the correlation coefficients vary between a high of 0.98 to a low of 0.60). However, the objective here is not to identify the variables that have maximum discriminating power but simply to classify the samples into various
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groups. Since the purpose is limited, we have assumed that discriminant analysis is a reasonable heuristic for our classification. We used SPSS (Statistical Package for Social Sciences) software to carry out the multiple discriminant analysis. 3. The case study of Karnataka In Karnataka, generation and distribution of electricity are managed by the Karnataka Power Corporation Ltd (KPCL) and the Karnataka Electricity Board (KEB), respectively. The total installed capacity in the state is about 3470 MW, consisting of 2502 MW of hydro and 968 MW of thermal. In addition, the state has a share of about 615 MW from the central generating stations (under the control of the Government of India). The financial year 1993–94 (1 April 1993–31 March 1994) is significant because it was a year when supply potential exceeded demand for most of the year. This demand was therefore almost exactly what was supplied and the latter has been recorded. This was the reason for choosing 1993–94 as the reference year. The RLCs were developed using the following steps: 1. The inputs for discriminant analysis were the 24 variables (hours of the day), the 365 samples (days of the year), and the 8760 data points (load levels) of 1993–94. 2. Initially, 365 load curves were grouped according to the calendar months of the year. Thus, 12 groups of load curves were obtained. Discriminant analysis was performed by assuming equal prior probabilities of classification among the groups. 3. The individual, case by case, classification results were used for reclassification of the individual load curves using the following criteria: (a) the load curves belonging to the start (first days) of the group and found to be misclassified were regrouped with the previous group; (b) similarly the load curves belonging to the last days of the group and found to be misclassified were regrouped with the next group; (c) stray load curves that were found to be misclassified were ignored for reclassification. They were retained in the same group. 4. After reclassification of the load curves, nine groups of load curves were formed. Again, discriminant analysis was performed after giving prior probabilities of classification among different groups. 5. The group means for the different variables gave the representative values for those variables. The final RLCs were obtained from these group mean load levels. 3.1. Results of the multiple discriminant analysis The results of groupwise classification are given in Table 1 with details of both actual groupings given as input to the discriminant analysis and predicted group membership. Overall, the result indicates that 87.12 per cent of the load curves are classified correctly. Different statistics were used to test the suitability of the discriminant analysis. First two discriminant functions alone explain about 89 per cent (65.21 and 24.01) of the total variance. The large eigenvalues (8.70, 3.20) and small values of Wilks lambda (0.0074, 0.0717) in the case of the first few functions indicate the effectiveness of the discriminant functions in distinguishing between the groups. The large magnitudes of the canonical correlations (0.95, 0.87) indicate the high degree of association
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Table 1 Classification results by discriminant analysis. The percentage of correctly classified cases is 87.12 per cent Actual Group
No. of cases
Predicted group membership
1 1
37
2
26
3
18
4
103
5
25
6
56
7
37
8
25
9
38
36 97.3% 1 3.8% 0 0.0% 0 0.0% 0 0.0% 0 0.0% 0 0.0% 1 4.0% 0 0.0%
2
3
0 0.0% 23 88.5% 0 0.0% 2 1.9% 0 0.0% 0 0.0% 0 0.0% 0 0.0% 0 0.0%
0 0.0% 0 0.0% 16 88.9% 1 1.0% 0 0.0% 1 1.8% 0 0.0% 0 0.0% 0 0.0%
4 0 0.0% 2 7.7% 2 11.1% 84 81.6% 0 0.0% 7 12.5% 2 5.4% 0 0.0% 0 0.0%
5 0 0.0% 0 0.0% 0 0.0% 1 1.0% 25 100.0% 5 8.9% 0 0.0% 0 0.0% 0 0.0%
6 0 0.0% 0 0.0% 0 0.0% 10 9.7% 0 0.0% 40 71.4% 1 2.7% 0 0.0% 0 0.0%
7 0 0.0% 0 0.0% 0 0.0% 4 3.9% 0 0.0% 3 5.4% 34 91.9% 1 4.0% 0 0.0%
8 0 0.0% 0 0.0% 0 0.0% 1 1.0% 0 0.0% 0 0.0% 0 0.0% 23 92.0% 1 2.6%
9 1 2.7% 0 0.0% 0 0.0% 0 0.0% 0 0.0% 0 0.0% 0 0.0% 0 0.0% 37 97.4%
between the discriminant scores and the groups. The small values of Wilks lambda also indicate that inter-group variability is greater than variability within the group. 3.2. Representativeness of the RLCs The estimated average hourly load levels for the nine RLCs are given in Table 2. From the table, we can see that these nine RLCs can be grouped to represent the three seasons, summer, monsoon and winter of the year. RLCs 1, 2, 8 and 9 represent the summer, 3 and 4 represent the monsoon, and 5, 6, and 7 represent the winter season of 1993–94. All nine RLCs are shown in Fig. 2. It may be observed from the figure and the table that the RLCs represent different levels of demand. RLC 9 represents the period (22 February to 31 March 1994) during which the hourly demand levels are the highest. RLC 8 represents the second highest demand period (28 January to 21 February 1994). The lowest levels of hourly demand occurred during the period represented by RLC 5 (2–28 October 1993). The periods represented by RLCs 1 and 2 are when scheduled rationing measures like power and energy cuts were imposed on the industrial sector [14]. The impact of these measures can be clearly observed from the figure in the form of substantial reduction in the demand levels. However, when RLCs 1 and 2 are compared with those of 8 and 9, the need for these rationing measures seem questionable since the system had met higher demand levels during 1993–94. The error frequency distribution curves were developed to determine the accuracy of the RLCs
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Table 2 Hourly demand (in MW) for different RLCs from 93 to 94 Hours
RLC—1 RLC—2 RLC—3 RLC—4 RLC—5 RLC—6 (1 April to (8 May to (3 June to (21 June (2 Oct. to (27 Oct. 7 May) 2 June) 20 June) to 1 Oct.) 26 Oct.) to 21 Dec.)
RLC—7 RLC—8 RLC—9 (22 Dec. (28 Jan. to (22 Feb. to 27 Jan.) 21 Feb.) to 31 March)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1977 1974 1967 1969 2064 2268 2281 2246 2182 2195 2148 2137 2066 2060 2024 2064 2148 2214 2323 2427 2419 2363 2200 2080
1741 1722 1706 1742 1868 2171 2520 2621 2556 2493 2399 2302 2235 2217 2236 2295 2354 2464 2647 2635 2550 2323 2004 1820
1604 1577 1574 1550 1638 1890 2057 2077 2024 1983 1951 1953 1942 1913 1875 1898 1963 2002 2131 2282 2186 2053 1841 1700
1343 1312 1309 1331 1424 1635 1936 1987 1914 1791 1773 1718 1683 1631 1611 1629 1682 1718 1893 2043 1967 1842 1535 1419
1610 1594 1580 1616 1705 2001 2327 2366 2288 2197 2083 2043 1994 1949 1945 1956 2033 2156 2362 2410 2321 2126 1835 1686
1286 1242 1222 1245 1343 1576 1953 1961 1879 1800 1725 1663 1626 1591 1563 1587 1664 1937 2129 2039 1879 1665 1464 1321
1484 1465 1467 1482 1615 1927 2249 2313 2224 2134 2052 1987 1945 1909 1928 1957 2054 2270 2400 2349 2217 1966 1676 1554
2098 2070 2044 2070 2192 2502 2717 2730 2654 2612 2504 2446 2455 2368 2409 2427 2499 2538 2736 2793 2755 2604 2369 2149
2312 2298 2299 2310 2406 2574 2735 2725 2660 2574 2491 2456 2395 2346 2391 2403 2473 2516 2700 2742 2766 2688 2550 2374
with respect to the actual load curves. The percentage error is determined by the ratio of the difference between the actual and representative load, and the actual. It is given by et ⫽ (ALCt ⫺ RLCt) ⫻ 100/ALCt where et is the percentage error of estimation at hour t, ALCt is the actual load (MW) at t, and RLCt is the representative load (MW) at t. Fig. 3 shows the error frequency distribution curve for all the 8760 hourly loads of the year and it may be observed that the distribution approximately follows the normal distribution with a mean of ⫺ 0.55 and a standard deviation of 7.67. Further analysis of the error frequency distribution shows that the percentage error of representation of about 60 per cent of the hourly loads falls within ⫾ 5 per cent. About 86 per cent falls within ⫾ 10 percentage error. In this type of representation, it will be advantageous to have slightly overestimated representative hourly loads (compared to actual loads) rather than underestimates since the slack will always be useful for planning in the case of constrained electricity systems. In the case of about 83 per cent of the hourly loads, the percentage error is ⱕ 5 per cent and it is ⱕ 10 per cent for about 94 per cent of the hourly loads.
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Fig. 2. RLCs for the whole year 1993–94.
3.3. Factors responsible for forming distinctive RLCs These nine groups of load curves represent classes with significant differences in the mean and variance. It appears that there are some significant factors which explain the occurrence of different levels of demand observed during the year. They are: (i) seasonal variation in demand (e.g. monsoon versus summer), (ii) variations in agricultural demand (electricity for irrigation pumpsets) which depends on agricultural operations, cropping pattern, etc., (iii) variations in industrial demand, (iv) rationing measures like power and energy cuts and load shedding adopted
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Fig. 3.
Error distribution curve.
to reduce demand, and (v) changes in the level of demand for water heating and air cooling in residential and commercial sectors. A simple analysis of the results was carried out to emphasize the indicators of above factors. The analysis starts with the ranking of nine RLCs on base (minimum level of demand during the day), intermediate (cycling demand), morning peak, evening peak and average demand and also on the basis of contributions to total demand by base and intermediate loads, morning and evening peak loads. These nine ranks were further reduced to three levels—high (H), medium (M) and low (L)—for each of the above attributes by clubbing together three ranks in each level (Table 3). This has made it possible to explain why the nine RLCs represent distinct groups of load curves. Based on the information obtained from the table an attempt was made to link the variations in demand for electricity from different sectors to the variations in the RLCs. The analysis clearly showed that the factors mentioned earlier are the main causes for significant differences in the nine RLCs. By combining the information provided in Table 3 and seasonal variations in demand
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Table 3 Grouping details of different demand levels and contributions to the total load (H: high; M: medium; L: low) RLCs
Base
Intermediate Morning demand peak demand
Evening peak demand
Average demand
Intermediate Morning load peak load
Evening peak load
RLC—1 RLC—2 RLC—3 RLC—4 RLC—5 RLC—6 RLC—7 RLC—8 RLC—9
H M L M L L M H H
M L L M L M H H H
M L L M L M H H H
M L L M L M H H H
L L M M H H H M L
M H L M H H L M L
M L L M L M H H H
L L L H M H M M H
levels, approximate indicators were obtained. The following are two significant examples of such an analysis: 1. The levels (H, M, L) of base and intermediate demand are indicative of the levels of industrial and agricultural demand. For instance, an H–H combination indicates that demand from both industrial and agricultural sectors are high while an L–L indicates the opposite. Other combinations like H–M, M–H, L–M, etc., for base and intermediate demand indicate significant changes in demand from one of the two sectors or both. 2. The level of contributions to the total load by morning peak and intermediate load are indicative of the demand for water heating in the residential and commercial sectors. Combinations like H–H, H–M, M–H indicate high demand for water heating. This can be observed during monsoon and winter seasons. Out of the nine RLCs, 1 and 2 are influenced by power and energy cuts on the consumption of electricity by the industrial sector. In the case of RLC 1, the level of base demand is H and that of intermediate demand is M. This indicates substantially high demand from the agricultural and low demand from the industrial sector. However, in the case of RLC 2, the level of base demand is M and that of intermediate demand is L. This shows substantial reduction in demand from the agricultural sector while demand from the industrial sector was forced to remain almost at the same level due to rationing measures. Table 4 contains the details that describe the effect of these and other factors on the individual RLCs. The modelling of time-varying demand by the RLCs for the electricity system of Karnataka has provided some new insights. Though the contributions of various factors in determining the level of demand are not quantifiable through this approach, the influence of these factors has been clearly established. In addition, the RLCs represent with reasonable accuracy, the varying demand levels that existed in 1993–94 within and across the seasons. The analysis has shown that even from an aggregate demand analysis like the one carried out in this work, it is possible to bring out the influences of some important consumer categories and end-uses in determining the hourly demand pattern.
1 April to 7 May Summer 1993
8 May to 2 June Summer 1993
3 June to 20 June Onset of 1993 monsoon
21 June to 1 October 1993
RLC—1
RLC—2
RLC—3
RLC—4
Peak monsoon
Seasonal
Low demand
Reduction in demand
Normal demand from the irigation pumpsets
High demand from the irrigation pumpsets
Agricultural
Normal demand
Normal demand
Reduction in demand due to cuts on consumption
Reduction in demand due to cuts on consumption
Industrial
Reduction in air cooling demand and slight increase in water heating demand Low air cooling and high water heating demand
High air cooling and low water heating demand
High air cooling and low water heating demand
Commercial
Factors responsible for variations in demand for electricity
Representative period
RLCs
Table 4 Factors responsible for variations in RLCs
High power and energy cuts
High power and energy cuts
Rationing measures
Reduction in No cuts air cooling demand and slight increase in water heating demand Low air No cuts cooling and high water heating demand
High air cooling and low water heating demand
High air cooling demand and low water heating demand
Residential
High base demand, medium level intermediate, morning and evening peak, and average demand Medium level of base demand, low level of intermediate, average, morning and evening peak demand Low level of base, intermediate, morning peak, evening peak and average demand Medium level of base, intermediate, morning peak, evening peak and average demand
Medium base, intermediate and evening peak loads, high morning peak load
Medium base load, low intermediate load and morning peak load, high evening peak load Low base load, medium intermediate load, low morning and evening peak loads
High base load, low intermediate load and morning peak load, medium evening peak load
Level of various Contributions types of demand from various types of loads
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28 January to 21 Beginning High demand Febraury 1994 of summer
22 Febraury to 31 March 1994
RLC—8
RLC—9
Summer
Winter
22 December 1993 to 27 January 1994
RLC—7
Normal demand
High demand
Increase in demand
Highest demand due to increased activities (finacial year ending)
Significant increase in demand (high demand for meeting the targets)
Increase in demand
Slight increase Normal in demand demand
27 October to 21 Winter December 1993
Lowest demand
RLC—6
Beginning of winter
2 to 26 October 1993
RLC—5
Table 4 Continued
High air cooling and low water heating demand
Increase in air cooling and reduction in water heating demand
Low air cooling and high water heating demand
Low air cooling and high water heating demand
Low air cooling and high water heating demand
No cuts
No cuts
No cuts
High air cooling and low water heating demand
No cuts
Increase in air No cuts cooling and reduction in water heating demand
Low air cooling and high water heating demand
Low air cooling and high water heating demand
Low air cooling and high water heating demand
Low level of base, intermediate, morning and evening peak and average demand Low level of base demand, medium level of intermediate, morning and evening peak and average demand Medium level of base demand, high level of intermediate, morning and evening peak, and average demand High level of base, intermediate, morning and evening peak, and average demand High level of base, intermediate, morning and evening peak, and average demand High base load, low intermediate and evening peak load, high moring peak load
High base load, medium intermediate, morning and evening peak load
Medium base load, high intermediate load, medium morning peak load and low evening peak load
Low base load, high intermediate, morning and evening peak load
Low base load, high intermediate and evening peak load and medium morning peak load
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4. Conclusion Multiple discriminant analysis was used to cluster 365 load curves of 1993–94 for Karnataka into nine RLCs, which adequately model the varying demand. Further analyses of these RLCs identified important factors such as seasonal, industrial, agricultural, and residential (water heating and air-cooling) demand variations, as well as utility rationing. The RLCs are superior to annual load–duration curves in terms of providing information about daily and seasonal demand variations and approximate time of occurrence of a particular demand level. This information is useful in planning for electricity shortages, maintenance scheduling and seasonal scheduling of hydroelectric plants. These advantages of RLCs have been realized by embedding them in tractable planning models for resource-constrained electricity systems [15]. Acknowledgements We owe a deep debt of gratitude to the late B.G. Raghavendra, who contributed substantially during the initial stages of this work. References [1] Reddy AKN, Sumithra GD, Balachandra P, D’Sa A. Economic and Political Weekly 1991;XXVI(14):891; XXVI(15):983. [2] Wang E, Jaraiedi M, Torries TF. Energy Economics 1996;18:49. [3] Turvey R, Anderson D. In: Electricity economics: essays and case studies. A World Bank research publication. London: The Johns Hopkins University Press, 1977:245. [4] Scherer CR. In: Estimating electric power system marginal costs, in contributions to economic analysis. Amsterdam: North-Holland, 1977:67. [5] Murphy FH, Wang ZX. Naval Research Logistics 1993;40:451. [6] Hobbs BF. European Journal of Operations Research 1995;83:1. [7] Petersen ER. Management Science 1973;20:656. [8] Zahavi J. Journal of Operational Research Society 1980;31:367. [9] Wenyuan L, Billinton R. IEEE Transactions on Power Systems 1993;8:628. [10] Tong SK, Shahidehpour SM. IEEE Transactions on Power Systems 1990;5:665. [11] Wong KP, Wong YW. IEEE Transactions on Power Systems 1996;11:128. [12] Soyster AL, Eynon RT. The conceptual basis of the electric utility sub-model of project independence evaluation system. In: Energy modelling studies and conservation, Proceedings of a Seminar of the United Nations Economic Commission for Europe, Washington DC, 24–28 March 1980. Oxford: Pergamon Press. [13] Overall JE, Klett JC. Applied multivariate analysis. New York: McGraw Hill, 1972. [14] Annual Administration Report– ⫺ 1993–94. Cauvery Bhavan, Bangalore, India: Karnataka Electricity Board (KEB), 1994. [15] Balachandra P. Electricity supply–demand matching: an integrated approach. M.Sc. (Engg.) thesis, Department of Management Studies, Indian Institute of Science, Bangalore, India, 1997.
Modelling electricity demand with representative load curves P. Balachandra*, Vijay Chandru Department of Management Studies, Indian Institute of Science, Bangalore 560012, India Received 21 November 1997
Abstract Models for electricity planning require inclusion of demand. Depending on the type of planning, the demand is usually represented as an annual demand for electricity (GWh), a peak demand (MW) or in the form of annual load–duration curves. The demand for electricity varies with the seasons, economic activities, etc. Existing schemes do not capture the dynamics of demand variations that are important for planning. For this purpose, we introduce the concept of representative load curves (RLCs). Advantages of RLCs are demonstrated in a case study for the state of Karnataka in India. Multiple discriminant analysis is used to cluster the 365 daily load curves for 1993–94 into nine RLCs. Further analyses of these RLCs help to identify important factors, namely, seasonal, industrial, agricultural, and residential (water heating and aircooling) demand variations besides rationing by the utility. 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction Electricity-supply planning requires efficient management of existing electric power systems, as well as rationalization of decisions concerning new capacity additions. The form in which we represent the demand for electricity is an important aspect in developing models for electricity planning. The major difficulty in modelling demand arises from the complexity created by its high variability. Variations in demand during a day are mainly due to changes in the level of electricity-driven activities at different times. Daily variations through the year are due to seasonal factors (like weather, temperature, rainfall, etc.), level of industrial and agricultural activities and other causes such as holidays, festivals, etc. It would clearly be advantageous to have models that capture all of these variations. * Corresponding author. Fax: 91-80-3341683; e-mail: [email protected] 0360-5442/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 0 - 5 4 4 2 ( 9 8 ) 0 0 0 9 6 - 6
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In the context of macro-level electricity planning, the demand is represented in terms of annual energy (GWh) and peak power (MW) demand [1,2]. In most cases, the representation depends on the type of planning and its requisite level of accuracy. For long-term expansion planning, the accuracy of the representation is compromised in annual [3–6] or normalized load–duration curves [7–9] to minimize complexities. In short-term planning [10,11], accuracy of the representation is very important and the demand is normally represented in terms of actual or forecast daily load curves. For planning, the annual load–duration curve is typically partitioned into (a) base, (b) cycling (intermediate), (c) daily peak, and (d) seasonal peak loads [12]. Fig. 1 shows such a partition developed from hourly load data for Karnataka during 1993–94. Using this curve, models may be developed to schedule various generation facilities to service base, intermediate, daily peak, and seasonal peak loads. In addition to the listed types of planning, electricity systems constrained by both power and capital shortages require effective supply–demand matching. Due to severe shortages of supply, large electricity demands remain unmet. Through supply interruptions and power rationing, util-
Fig. 1.
The annual load–duration curve for 1993–94.
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221
ities (e.g. State Electricity Boards in India) avoid supplying part of the electricity demanded by consumers. Thus, managing the system requires planning for non-supply (i.e. failure to meet demands by resorting to curtailments) and supply of electricity. In order to plan in advance for both the supply and non-supply of electricity, utilities need to forecast the levels of demand and shortages and their times of occurrence in terms of hour of the day and day of the year. With this type of planning, the modelled demand has to encompass all variations. Estimates of electricity demand that are based on annual or normalized load–duration curves tend to be highly approximate but are adequate for long term strategic planning. The disadvantages of using these annual load–duration curves are: (i) information on the time of occurrence (hour, day) of a particular level of demand is not available; (ii) seasonal variations in demand levels are not known; (iii) information on non-seasonal variations in demand level due to factors like changes in the level of industrial or agricultural activities is also unavailable. We could use 365 daily load curves to represent the annual demand as 8760 input data points. This would lead to models that are intractable. For example, solving optimization models for planning would require excessive computational resources even with advanced software. Also, policy interpretations would be difficult to unravel from the reams of output. In summary, planners require representations of demand that capture both temporal and structural variations that depend on economic activities such as agricultural, industrial production and commerce. To include these variations, the planning horizon has to be at least 1 year. This tradeoff between adequate planning horizon and model complexity led us to the concept of RLCs. 2. Representative load curves An RLC is a typical daily load curve which represents a group of load curves exhibiting similar demand patterns. We believe that RLCs are useful for planning in resource constrained electricity systems and in situations where it is required to know the time variations in demand (e.g. supply– demand matching, seasonal scheduling of hydro plants and maintenance scheduling). RLCs are also used to identify the factors influencing variations in demand and to form some approximate indicators of these influences. The first step in developing RLCs is to group load curves on the basis of some similarities. To form different groups, a classification procedure needs to be used. Traditionally, multiple discriminant analysis is used to classify the individual samples into various groups and verify the correctness of the groupings. With multiple discriminant analysis, an observation is assigned to one or more groups on the basis of value. It serves to identify important variables to distinguish among groups and develop a procedure to predict group membership for new cases [13]. For the analysis, 24 hours of the day are used as 24 variables and 365 days of the year as the number of samples. Normally, in any statistical analysis, the variables measure different attributes of the study system and are commonly assumed to be independent of each other. However, in the present case, the 24 variables measure the same attribute; demand for electricity at different time periods (hours of the day). Therefore, one cannot expect these variables to be independent of each other (for example, in the case study presented below, the correlation coefficients vary between a high of 0.98 to a low of 0.60). However, the objective here is not to identify the variables that have maximum discriminating power but simply to classify the samples into various
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groups. Since the purpose is limited, we have assumed that discriminant analysis is a reasonable heuristic for our classification. We used SPSS (Statistical Package for Social Sciences) software to carry out the multiple discriminant analysis. 3. The case study of Karnataka In Karnataka, generation and distribution of electricity are managed by the Karnataka Power Corporation Ltd (KPCL) and the Karnataka Electricity Board (KEB), respectively. The total installed capacity in the state is about 3470 MW, consisting of 2502 MW of hydro and 968 MW of thermal. In addition, the state has a share of about 615 MW from the central generating stations (under the control of the Government of India). The financial year 1993–94 (1 April 1993–31 March 1994) is significant because it was a year when supply potential exceeded demand for most of the year. This demand was therefore almost exactly what was supplied and the latter has been recorded. This was the reason for choosing 1993–94 as the reference year. The RLCs were developed using the following steps: 1. The inputs for discriminant analysis were the 24 variables (hours of the day), the 365 samples (days of the year), and the 8760 data points (load levels) of 1993–94. 2. Initially, 365 load curves were grouped according to the calendar months of the year. Thus, 12 groups of load curves were obtained. Discriminant analysis was performed by assuming equal prior probabilities of classification among the groups. 3. The individual, case by case, classification results were used for reclassification of the individual load curves using the following criteria: (a) the load curves belonging to the start (first days) of the group and found to be misclassified were regrouped with the previous group; (b) similarly the load curves belonging to the last days of the group and found to be misclassified were regrouped with the next group; (c) stray load curves that were found to be misclassified were ignored for reclassification. They were retained in the same group. 4. After reclassification of the load curves, nine groups of load curves were formed. Again, discriminant analysis was performed after giving prior probabilities of classification among different groups. 5. The group means for the different variables gave the representative values for those variables. The final RLCs were obtained from these group mean load levels. 3.1. Results of the multiple discriminant analysis The results of groupwise classification are given in Table 1 with details of both actual groupings given as input to the discriminant analysis and predicted group membership. Overall, the result indicates that 87.12 per cent of the load curves are classified correctly. Different statistics were used to test the suitability of the discriminant analysis. First two discriminant functions alone explain about 89 per cent (65.21 and 24.01) of the total variance. The large eigenvalues (8.70, 3.20) and small values of Wilks lambda (0.0074, 0.0717) in the case of the first few functions indicate the effectiveness of the discriminant functions in distinguishing between the groups. The large magnitudes of the canonical correlations (0.95, 0.87) indicate the high degree of association
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Table 1 Classification results by discriminant analysis. The percentage of correctly classified cases is 87.12 per cent Actual Group
No. of cases
Predicted group membership
1 1
37
2
26
3
18
4
103
5
25
6
56
7
37
8
25
9
38
36 97.3% 1 3.8% 0 0.0% 0 0.0% 0 0.0% 0 0.0% 0 0.0% 1 4.0% 0 0.0%
2
3
0 0.0% 23 88.5% 0 0.0% 2 1.9% 0 0.0% 0 0.0% 0 0.0% 0 0.0% 0 0.0%
0 0.0% 0 0.0% 16 88.9% 1 1.0% 0 0.0% 1 1.8% 0 0.0% 0 0.0% 0 0.0%
4 0 0.0% 2 7.7% 2 11.1% 84 81.6% 0 0.0% 7 12.5% 2 5.4% 0 0.0% 0 0.0%
5 0 0.0% 0 0.0% 0 0.0% 1 1.0% 25 100.0% 5 8.9% 0 0.0% 0 0.0% 0 0.0%
6 0 0.0% 0 0.0% 0 0.0% 10 9.7% 0 0.0% 40 71.4% 1 2.7% 0 0.0% 0 0.0%
7 0 0.0% 0 0.0% 0 0.0% 4 3.9% 0 0.0% 3 5.4% 34 91.9% 1 4.0% 0 0.0%
8 0 0.0% 0 0.0% 0 0.0% 1 1.0% 0 0.0% 0 0.0% 0 0.0% 23 92.0% 1 2.6%
9 1 2.7% 0 0.0% 0 0.0% 0 0.0% 0 0.0% 0 0.0% 0 0.0% 0 0.0% 37 97.4%
between the discriminant scores and the groups. The small values of Wilks lambda also indicate that inter-group variability is greater than variability within the group. 3.2. Representativeness of the RLCs The estimated average hourly load levels for the nine RLCs are given in Table 2. From the table, we can see that these nine RLCs can be grouped to represent the three seasons, summer, monsoon and winter of the year. RLCs 1, 2, 8 and 9 represent the summer, 3 and 4 represent the monsoon, and 5, 6, and 7 represent the winter season of 1993–94. All nine RLCs are shown in Fig. 2. It may be observed from the figure and the table that the RLCs represent different levels of demand. RLC 9 represents the period (22 February to 31 March 1994) during which the hourly demand levels are the highest. RLC 8 represents the second highest demand period (28 January to 21 February 1994). The lowest levels of hourly demand occurred during the period represented by RLC 5 (2–28 October 1993). The periods represented by RLCs 1 and 2 are when scheduled rationing measures like power and energy cuts were imposed on the industrial sector [14]. The impact of these measures can be clearly observed from the figure in the form of substantial reduction in the demand levels. However, when RLCs 1 and 2 are compared with those of 8 and 9, the need for these rationing measures seem questionable since the system had met higher demand levels during 1993–94. The error frequency distribution curves were developed to determine the accuracy of the RLCs
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Table 2 Hourly demand (in MW) for different RLCs from 93 to 94 Hours
RLC—1 RLC—2 RLC—3 RLC—4 RLC—5 RLC—6 (1 April to (8 May to (3 June to (21 June (2 Oct. to (27 Oct. 7 May) 2 June) 20 June) to 1 Oct.) 26 Oct.) to 21 Dec.)
RLC—7 RLC—8 RLC—9 (22 Dec. (28 Jan. to (22 Feb. to 27 Jan.) 21 Feb.) to 31 March)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1977 1974 1967 1969 2064 2268 2281 2246 2182 2195 2148 2137 2066 2060 2024 2064 2148 2214 2323 2427 2419 2363 2200 2080
1741 1722 1706 1742 1868 2171 2520 2621 2556 2493 2399 2302 2235 2217 2236 2295 2354 2464 2647 2635 2550 2323 2004 1820
1604 1577 1574 1550 1638 1890 2057 2077 2024 1983 1951 1953 1942 1913 1875 1898 1963 2002 2131 2282 2186 2053 1841 1700
1343 1312 1309 1331 1424 1635 1936 1987 1914 1791 1773 1718 1683 1631 1611 1629 1682 1718 1893 2043 1967 1842 1535 1419
1610 1594 1580 1616 1705 2001 2327 2366 2288 2197 2083 2043 1994 1949 1945 1956 2033 2156 2362 2410 2321 2126 1835 1686
1286 1242 1222 1245 1343 1576 1953 1961 1879 1800 1725 1663 1626 1591 1563 1587 1664 1937 2129 2039 1879 1665 1464 1321
1484 1465 1467 1482 1615 1927 2249 2313 2224 2134 2052 1987 1945 1909 1928 1957 2054 2270 2400 2349 2217 1966 1676 1554
2098 2070 2044 2070 2192 2502 2717 2730 2654 2612 2504 2446 2455 2368 2409 2427 2499 2538 2736 2793 2755 2604 2369 2149
2312 2298 2299 2310 2406 2574 2735 2725 2660 2574 2491 2456 2395 2346 2391 2403 2473 2516 2700 2742 2766 2688 2550 2374
with respect to the actual load curves. The percentage error is determined by the ratio of the difference between the actual and representative load, and the actual. It is given by et ⫽ (ALCt ⫺ RLCt) ⫻ 100/ALCt where et is the percentage error of estimation at hour t, ALCt is the actual load (MW) at t, and RLCt is the representative load (MW) at t. Fig. 3 shows the error frequency distribution curve for all the 8760 hourly loads of the year and it may be observed that the distribution approximately follows the normal distribution with a mean of ⫺ 0.55 and a standard deviation of 7.67. Further analysis of the error frequency distribution shows that the percentage error of representation of about 60 per cent of the hourly loads falls within ⫾ 5 per cent. About 86 per cent falls within ⫾ 10 percentage error. In this type of representation, it will be advantageous to have slightly overestimated representative hourly loads (compared to actual loads) rather than underestimates since the slack will always be useful for planning in the case of constrained electricity systems. In the case of about 83 per cent of the hourly loads, the percentage error is ⱕ 5 per cent and it is ⱕ 10 per cent for about 94 per cent of the hourly loads.
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Fig. 2. RLCs for the whole year 1993–94.
3.3. Factors responsible for forming distinctive RLCs These nine groups of load curves represent classes with significant differences in the mean and variance. It appears that there are some significant factors which explain the occurrence of different levels of demand observed during the year. They are: (i) seasonal variation in demand (e.g. monsoon versus summer), (ii) variations in agricultural demand (electricity for irrigation pumpsets) which depends on agricultural operations, cropping pattern, etc., (iii) variations in industrial demand, (iv) rationing measures like power and energy cuts and load shedding adopted
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Fig. 3.
Error distribution curve.
to reduce demand, and (v) changes in the level of demand for water heating and air cooling in residential and commercial sectors. A simple analysis of the results was carried out to emphasize the indicators of above factors. The analysis starts with the ranking of nine RLCs on base (minimum level of demand during the day), intermediate (cycling demand), morning peak, evening peak and average demand and also on the basis of contributions to total demand by base and intermediate loads, morning and evening peak loads. These nine ranks were further reduced to three levels—high (H), medium (M) and low (L)—for each of the above attributes by clubbing together three ranks in each level (Table 3). This has made it possible to explain why the nine RLCs represent distinct groups of load curves. Based on the information obtained from the table an attempt was made to link the variations in demand for electricity from different sectors to the variations in the RLCs. The analysis clearly showed that the factors mentioned earlier are the main causes for significant differences in the nine RLCs. By combining the information provided in Table 3 and seasonal variations in demand
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Table 3 Grouping details of different demand levels and contributions to the total load (H: high; M: medium; L: low) RLCs
Base
Intermediate Morning demand peak demand
Evening peak demand
Average demand
Intermediate Morning load peak load
Evening peak load
RLC—1 RLC—2 RLC—3 RLC—4 RLC—5 RLC—6 RLC—7 RLC—8 RLC—9
H M L M L L M H H
M L L M L M H H H
M L L M L M H H H
M L L M L M H H H
L L M M H H H M L
M H L M H H L M L
M L L M L M H H H
L L L H M H M M H
levels, approximate indicators were obtained. The following are two significant examples of such an analysis: 1. The levels (H, M, L) of base and intermediate demand are indicative of the levels of industrial and agricultural demand. For instance, an H–H combination indicates that demand from both industrial and agricultural sectors are high while an L–L indicates the opposite. Other combinations like H–M, M–H, L–M, etc., for base and intermediate demand indicate significant changes in demand from one of the two sectors or both. 2. The level of contributions to the total load by morning peak and intermediate load are indicative of the demand for water heating in the residential and commercial sectors. Combinations like H–H, H–M, M–H indicate high demand for water heating. This can be observed during monsoon and winter seasons. Out of the nine RLCs, 1 and 2 are influenced by power and energy cuts on the consumption of electricity by the industrial sector. In the case of RLC 1, the level of base demand is H and that of intermediate demand is M. This indicates substantially high demand from the agricultural and low demand from the industrial sector. However, in the case of RLC 2, the level of base demand is M and that of intermediate demand is L. This shows substantial reduction in demand from the agricultural sector while demand from the industrial sector was forced to remain almost at the same level due to rationing measures. Table 4 contains the details that describe the effect of these and other factors on the individual RLCs. The modelling of time-varying demand by the RLCs for the electricity system of Karnataka has provided some new insights. Though the contributions of various factors in determining the level of demand are not quantifiable through this approach, the influence of these factors has been clearly established. In addition, the RLCs represent with reasonable accuracy, the varying demand levels that existed in 1993–94 within and across the seasons. The analysis has shown that even from an aggregate demand analysis like the one carried out in this work, it is possible to bring out the influences of some important consumer categories and end-uses in determining the hourly demand pattern.
1 April to 7 May Summer 1993
8 May to 2 June Summer 1993
3 June to 20 June Onset of 1993 monsoon
21 June to 1 October 1993
RLC—1
RLC—2
RLC—3
RLC—4
Peak monsoon
Seasonal
Low demand
Reduction in demand
Normal demand from the irigation pumpsets
High demand from the irrigation pumpsets
Agricultural
Normal demand
Normal demand
Reduction in demand due to cuts on consumption
Reduction in demand due to cuts on consumption
Industrial
Reduction in air cooling demand and slight increase in water heating demand Low air cooling and high water heating demand
High air cooling and low water heating demand
High air cooling and low water heating demand
Commercial
Factors responsible for variations in demand for electricity
Representative period
RLCs
Table 4 Factors responsible for variations in RLCs
High power and energy cuts
High power and energy cuts
Rationing measures
Reduction in No cuts air cooling demand and slight increase in water heating demand Low air No cuts cooling and high water heating demand
High air cooling and low water heating demand
High air cooling demand and low water heating demand
Residential
High base demand, medium level intermediate, morning and evening peak, and average demand Medium level of base demand, low level of intermediate, average, morning and evening peak demand Low level of base, intermediate, morning peak, evening peak and average demand Medium level of base, intermediate, morning peak, evening peak and average demand
Medium base, intermediate and evening peak loads, high morning peak load
Medium base load, low intermediate load and morning peak load, high evening peak load Low base load, medium intermediate load, low morning and evening peak loads
High base load, low intermediate load and morning peak load, medium evening peak load
Level of various Contributions types of demand from various types of loads
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28 January to 21 Beginning High demand Febraury 1994 of summer
22 Febraury to 31 March 1994
RLC—8
RLC—9
Summer
Winter
22 December 1993 to 27 January 1994
RLC—7
Normal demand
High demand
Increase in demand
Highest demand due to increased activities (finacial year ending)
Significant increase in demand (high demand for meeting the targets)
Increase in demand
Slight increase Normal in demand demand
27 October to 21 Winter December 1993
Lowest demand
RLC—6
Beginning of winter
2 to 26 October 1993
RLC—5
Table 4 Continued
High air cooling and low water heating demand
Increase in air cooling and reduction in water heating demand
Low air cooling and high water heating demand
Low air cooling and high water heating demand
Low air cooling and high water heating demand
No cuts
No cuts
No cuts
High air cooling and low water heating demand
No cuts
Increase in air No cuts cooling and reduction in water heating demand
Low air cooling and high water heating demand
Low air cooling and high water heating demand
Low air cooling and high water heating demand
Low level of base, intermediate, morning and evening peak and average demand Low level of base demand, medium level of intermediate, morning and evening peak and average demand Medium level of base demand, high level of intermediate, morning and evening peak, and average demand High level of base, intermediate, morning and evening peak, and average demand High level of base, intermediate, morning and evening peak, and average demand High base load, low intermediate and evening peak load, high moring peak load
High base load, medium intermediate, morning and evening peak load
Medium base load, high intermediate load, medium morning peak load and low evening peak load
Low base load, high intermediate, morning and evening peak load
Low base load, high intermediate and evening peak load and medium morning peak load
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4. Conclusion Multiple discriminant analysis was used to cluster 365 load curves of 1993–94 for Karnataka into nine RLCs, which adequately model the varying demand. Further analyses of these RLCs identified important factors such as seasonal, industrial, agricultural, and residential (water heating and air-cooling) demand variations, as well as utility rationing. The RLCs are superior to annual load–duration curves in terms of providing information about daily and seasonal demand variations and approximate time of occurrence of a particular demand level. This information is useful in planning for electricity shortages, maintenance scheduling and seasonal scheduling of hydroelectric plants. These advantages of RLCs have been realized by embedding them in tractable planning models for resource-constrained electricity systems [15]. Acknowledgements We owe a deep debt of gratitude to the late B.G. Raghavendra, who contributed substantially during the initial stages of this work. References [1] Reddy AKN, Sumithra GD, Balachandra P, D’Sa A. Economic and Political Weekly 1991;XXVI(14):891; XXVI(15):983. [2] Wang E, Jaraiedi M, Torries TF. Energy Economics 1996;18:49. [3] Turvey R, Anderson D. In: Electricity economics: essays and case studies. A World Bank research publication. London: The Johns Hopkins University Press, 1977:245. [4] Scherer CR. In: Estimating electric power system marginal costs, in contributions to economic analysis. Amsterdam: North-Holland, 1977:67. [5] Murphy FH, Wang ZX. Naval Research Logistics 1993;40:451. [6] Hobbs BF. European Journal of Operations Research 1995;83:1. [7] Petersen ER. Management Science 1973;20:656. [8] Zahavi J. Journal of Operational Research Society 1980;31:367. [9] Wenyuan L, Billinton R. IEEE Transactions on Power Systems 1993;8:628. [10] Tong SK, Shahidehpour SM. IEEE Transactions on Power Systems 1990;5:665. [11] Wong KP, Wong YW. IEEE Transactions on Power Systems 1996;11:128. [12] Soyster AL, Eynon RT. The conceptual basis of the electric utility sub-model of project independence evaluation system. In: Energy modelling studies and conservation, Proceedings of a Seminar of the United Nations Economic Commission for Europe, Washington DC, 24–28 March 1980. Oxford: Pergamon Press. [13] Overall JE, Klett JC. Applied multivariate analysis. New York: McGraw Hill, 1972. [14] Annual Administration Report– ⫺ 1993–94. Cauvery Bhavan, Bangalore, India: Karnataka Electricity Board (KEB), 1994. [15] Balachandra P. Electricity supply–demand matching: an integrated approach. M.Sc. (Engg.) thesis, Department of Management Studies, Indian Institute of Science, Bangalore, India, 1997.