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Application of an experimental design methodology for economic parameter analysis in an open market environment
Electric Power Systems Research 73 (2005) 61–66
Application of an experimental design methodology for economic parameter analysis in an open market environment D. Moitre∗ , F. Magnago, J. Martinez, M. Galetto National University of Rio Cuarto, University Campus, Ruta 36, Km 601, Rio Cuarto, Cordoba 5800, Argentina Received 1 October 2003; accepted 6 May 2004 Available online 21 September 2004
Abstract Statistical data analysis, through the utilization of experimental design, is a powerful tool commonly used in different areas where uncertainty is a problem, and a great amount of data is available for the analysis. After 10 years of open market experience, the data recorded during the last decade from the electric markets suggest that experimental design can be useful for electrical market data analysis. In this paper, the impact of different factors on the determination of economic parameters such as the spot price in an electric market is analyzed. The analysis is based on the experimental design methodology. The implementation of the proposed methodology is illustrated using simple examples from the Argentinien electric market. © 2004 Elsevier B.V. All rights reserved. Keywords: Experimental design; Electric energy market; ANOVA methodology; Spot price estimation
1. Introduction During the last decade, several electric market structures have changed all over the world from a vertical integrated monopoly to a vertical and horizontal segmented market. This process has been observed in countries such as Chile (1982), the United Kingdom (1989), Argentina (1992), Norway (1992), Central American countries (1997), and the United States (1998) [1]. As a result, two important issues can be highlighted taking into account over 10 years of open market experience in the countries that led themselves to those changes and from the available data. The first item to be considered is uncertainty with an open market; uncertainty increases with regards to power system structure, development, and price determination. The second one is the great amount of data available in databases. These two issues make the statistical data analysis, through the utilization of experimental design, a powerful tool for electrical market data analysis that can serve as a base for a second generation of reforms in the ∗
Corresponding author. Tel.: +54 358 4676 495; fax: +54 358 4676 495. E-mail address: [email protected] (D. Moitre).
0378-7796/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2004.05.005
electric market sector. Experimental design is a set of tests carried out on process or systems where inputs are changed in order to observe the output response and their relation with inputs. The main objective of the experimental design is to determine which variables are most influential on the output response, that is, to determine the parameters which lead to the best possible output response [2,3]. Initial experimental designs were applied in the area of agronomy and chemistry for the selection of the most efficient factor combination as described in [4–6], respectively. Later, the electronic industry used this methodology for the development of products and process [7]. Thus, experimental design becomes one of the most important tools in the statistical analysis of data in several areas, allowing companies to improve within an aggressive competitive environment. Recently, the idea of implementing experimental design in the power system area has been suggested. In [8], authors present a technique based on the analysis of the variance to calculate the contribution of generators to an electric market. Another implementation of the analysis of the variance method to detect the most influential factors on the level of line congestion is shown in [9,10]. The aim of this paper is to analyze the impact of different
62
D. Moitre et al. / Electric Power Systems Research 73 (2005) 61–66
factors on the determination of economic parameters such as the spot price in an electric market based on the experimental design methodology. First, the mathematical formulation of factorial experiments is revised; then, the implementation of this technique on the spot price determination is shown, and finally to illustrate the proposed method, a study case is presented.
2. Factorial experiments
the sum of squares of the error. Then, the tests described by Eqs. (2) and (3) are based on a comparison between the independent estimates of σ 2 provided by the division of each term of SST by their degree of freedom, known as mean square: SSA a−1 SSB MSB = b−1 MSA =
SSAB (a − 1)(b − 1) SSE MSE = ab(n − 1)
(6)
MSAB =
Factorial experiments are widely used within the area of experimental design when experiments involve several factors and the study of their joint effect on a response is necessary [11]; the simplest type includes two factors: factor A with a levels and factor B with b levels. The hypotheses under which results are analyzed are the following: • fixed factors; • full randomized design; • normality assumptions are satisfied. The observations can be described by the following statistical lineal model: yij = µ + αi + βj + (αβ)ij + εijk
(1)
where µ is the total media, αi denotes the effect of the ilevel of the first factor, βj the effect of the j-level of the second factor, (αβ)ij the interaction effect between the i-level of the first factor,and the j-level of the second factor, and εijk the experimental error. Repeating the experiment n times, there will be nab observations, which are assumed normally distributed, media µij and variance σ 2 . The objective of the factorial design is to determine the effect of the first and second factor for which the following tests of hypothesis are performed: H0 : α1 = α2 = · · · = αa = 0 H1 : at least one αi = 0
(2)
H0 : β1 = β2 = · · · = βb = 0 H1 : at least one βj = 0
(3)
Furthermore, it is very important to determine the interaction between both factors; therefore, the following test of hypothesis is formulated: H0 : (αβ)11 = (αβ)2 = · · · = (αβ)ab = 0 H1 : at least one (αβ)ij = 0
(4)
For a given data set, the total sum of squares SST is defined as follows: SST = SSA + SSB + SSAB + SSE
(5)
where SSA is the sum of squares for the main effect A, SSB is the sum of squares for the main effect B, SSAB is the sum of squares of the interactions between factors, and SSE is
The effect of a factor is defined by the variations in the factor’s level. This is called main effect because it refers to the primary factors in the study. In some experiments, the factors may have different level response, when this occurs, and there is a considerable interaction between factors, the corresponding main effects will have no meaning since a significant interaction can masks main effects. Assuming fixed factors A and B, the mean square expected values are: bn E(MSA ) = σ 2 +
a i=1
α2i
a−1 b βj2 an
E(MSB ) = σ 2 +
E(MSAB ) = σ 2 +
j=1
b−1 b a (αβ)2ij n i=1 j=1
(a − 1)(b − 1)
E(MSE ) = σ 2 In order to perform the test described by Eqs. (2)–(4), mean square values are divided by the mean square error: MSA MSE MSB FB = MSE MSAB FAB = MSE FA =
(7)
These ratios follow an F distribution with two degree of freedoms; one degree of freedom corresponds to the numerator term and the other corresponds to the denominator, in this case, ab(n − 1). The critical region is located in the upper tail of the distribution, as shown in Fig. 1. The test procedure is arranged in an analysis of variance (ANOVA) table, shown in Table 1. Rejecting the null hypothesis H0 and H0 implies that there are differences between the means, although the exact nature of the differences is not specified. In this situation, further comparisons between groups through the implementation of multiple comparison techniques may be useful. This
D. Moitre et al. / Electric Power Systems Research 73 (2005) 61–66
63
Fig. 1. F Distribution. Fig. 2. Residuals February 2002. Table 1 ANOVA Source of variation
Sum of squares
Degree of freedom
Mean square
F
A B AB Error
SSA SSB SSAB SSE
a−1 b−1 (a − l)(b − l) ab(n − 1)
MSA MSB MSAB MSE
FA FB FAB
Total
SST
abn − 1
technique is very useful when a great amount of data is available and a scientific decision is needed, such is the case, for example, when an economic parameter determination or new regulatory reforms are needed. Next section will show the application of these techniques to determine the most influential factors in the economic parameters for an electric power market.
diction of market behavior, for example, to identify the value of the economic parameters in different days of the week or different hours within a day. To illustrate this, let us consider that the spot price is determined every hour during 24 h for 7 days a week. These prices can be considered as the result of a bifactorial experiment where the first factor is the hour of the day analyzed at 24 levels, and the second factor is the day of the week analyzed at seven levels, four observations per cell are selected. Each factor is previously determined, hence, a fixed factor model can be selected. In order to show the proposed technique, the hourly spot price data of the Argentinean system is taken as an example. Data correspond to the month of February 2002 [12]. The first step is to verify if the data satisfy the statistical lineal model hypothesis of Eq. (1). The error, also referred as residual, is defined as the difference between the observation value yijk and the estimated value yijk .
3. Economic parameter determination
εijk = yijk − yijk
The electric power market includes various commercial and financial agreements through contracts of different types and duration and short-term transactions. In a centralized market, the nucleus of these agreements is the spot market in which the electric energy is valued and commercialized. This task is carried out by the Independent System Operator (ISO). The prices of the spot market in general are based on shortterm marginal costs of energy, either based on actual variable generation costs or the value of the bid. Let us show as an example the Argentine market, where the reform started in 1992, therefore the market spot price data are available from this year, taken every hour a day and 7 days a week. Moreover, no modification has been done to the regulation since then, although attempts have been made in 2001. Hence, these data can be useful to make a response surface for different economic parameters, which can become the basis for pre-
Fig. 2 shows the accumulative distribution residuals of February 2002, the histogram is represented in Fig. 3. The residuals are normally distributed with zero mean; therefore, its representation should be a line. However, there are residuals that are bigger than the rest of the residuals, which are called outliers. Since the presence of one outlier may alter the analysis of the results, before any analysis, the origin of these outliers should be determined. In this example, negative residuals are related to lower spot prices due to over offer of hydro generation or highperformance thermal units such as combined cycle units. Residuals that are positive and sensibly higher than the others are related to higher spot prices due to outage of generator units or transmission lines. The deviation of these residuals does not misrepresent the lineal analysis and the corresponding conclusions.
(8)
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D. Moitre et al. / Electric Power Systems Research 73 (2005) 61–66
Fig. 3. Histogram February 2002. Fig. 4. Day spot price comparison.
Next the AN OVA table is drawn for the data set. Table 2 shows the results for the data from February 2002. From Table 2 the following decision are made: Reject H0 and conclude those hours of the day influencing the spot price. Reject H0 and conclude that the day of the week that affects the spot price. Do not reject H0 and conclude that there is no interaction between the hour of the day and the day of the week where the spot price is calculated during the winter season considered. In addition, the following P values are obtained: Phours = 0
Pdays = 0
Phours & days = 0.9999
(9)
These probabilities quantify the significance of the level of decisions. For example, the probability that the first and the second conclusions are wrong is almost zero, whereas the probability for the third conclusion to be correct is near one. The next step is to determine if there are days of the week or hours of the day for which the spot price is considerably different. This is obtained using multiple comparisons based on the test method proposed by Tukey and Kramer, with a trust degree of 95%. The following figures illustrate the obtained results. Fig. 4 shows the spot prices calculated for each day of the week. In this figure, it can be observed that the mean spot prices for Sunday (number 3) is remarkably lower than the mean spot prices calculated on the other days. This is a logic conclusion since it is a non-working day. Fig. 5 shows
the results for different hours of a day. It is observed that the mean spot price is significantly greater from 8 to 11 p.m. where the residential load increases. Moreover, it is possible to observe that during 7 a.m. and 7 p.m., the price is uniform which is compatible with the industrial load of the system. Now, the estimation is calculated applying Minimum Square Error (MSE) to the lineal model described in Eq. (10): ij yˆ ij = µ ˆ + αˆ i + βˆ j + αβ
where yˆ ij is the spot price for day i and hour j, µ ˆ is the monthly average spot price, αˆ i the estimation of the effect of ij is the day i, βˆ j the estimation of the effect of hour j, and αβ estimation of the interaction between day i and hour j. Results are shown on Figs. 6 and 7, numerical data are described in Appendix A. Figs. 6 and 7 show the effect of the day and the hour respectively. The graphics illustrate the mean response to the different levels of these factors with
Table 2 ANOVA, February 2002 Source of variation Hours Days Hours and days Error Total
Sum of squares
Degree of freedom
4425.8 1241.2 1375.3 8551.7
23 6 138 504
15593.9
671
Mean square 192.424 206.864 9.966 16.968
(10)
F 11.34 12.19 0.59
Fig. 5. Hourly spot price comparison.
D. Moitre et al. / Electric Power Systems Research 73 (2005) 61–66
65
determination of economic parameters such as the spot price in an electric market.
Appendix A Table A.1 Days
Fig. 6. Effect of the day.
respect to the average spot price µ, ˆ which for this example is: $ µ ˆ = 12.6326 MWh From Fig. 6, it is possible to conclude that the spot price increases from Monday to Thursday and decreases on Friday and Saturday, reaching the lowest value on Sunday. The analysis suggests that days from Monday to Thursday can be grouped as working days, Friday can be considered as an atypical day. For the operation programming, days are classified as normal days (Tuesday to Friday), non-working day (Sunday), part-time working day (Saturday) and Monday as an atypical day, which differs from the statistical analysis. From Fig. 7, it can be inferred that the spot price increases from 8 to 11 p.m. with a peak at 9 p.m. then decreases until a minimum is reached at 7 a.m. Taking into account that for the operation the hours are classified as: peak from 7 to 12 p.m., valley from 12 p.m. to 6 a.m. and rest from 6 a.m. to 7 p.m., the statistical analysis arrived at the same conclusions. Therefore, from the analysis of the effect of the hours, it can be seen that the observations are consistent with the classification.
4. Conclusions The implementation of the experimental design and analysis to observe the effect of different parameters on the spot price calculation has been carried out. This technique has been applied to the Argentinian electric market. A simple example has been selected to illustrate the application of the methodology proposed, reaching the same conclusions as experts do. Results suggest that this statistical tool may become very useful to evaluate the impact of different factors on the
Fig. 7. Effect of the hours.
Day effect ␣I
[$/MWh]
Friday [␣1 ]
−0.2208
Saturday [␣2 ]
−0.7588
Sunday [␣3 ]
−2.5776
Monday [␣4 ]
0.0549 0.2808
Tuesday [␣5 ] Wednesday [␣6 ]
1.1694
Thursday [␣7 ]
2.051
Table A.2 Hours Hour effect (a.m.), β1
[$/MWh]
Hour effect (p.m.), β1
[$/MWh]
01:00 [β2 ] 02:00 [β3 ]
−0.168
01:00 [β13 ]
0.312
−1.184
02:00 [β14 ]
03:00 [β4 ] 04:00 [β5 ]
−2.0972
03:00 [β15 ]
−0.2026 −0.2026
−2.2583
04:00 [β16 ]
05:00 [β6 ] 06:00 [β7 ]
−2.4247
05:00 [β17 ]
0.348 −0.3447
−2.213
06:00 [β18 ]
−0.2737
07:00 [β8 ]
−2.893 −1.7826
07:00 [β19 ] 08:00 [β20 ]
−0.2133
08:00 [β9 ] 09:00 [β1 ]
−1.0158
09:00 [β21 ]
10:00 [β10 ] 11:00 [β11 ]
−0.6444
10:00 [β22 ]
9.7549 4.607
−0.433
11:00 [β23 ]
1.3717
12:00 [β12 ]
0.196
12:00 [β24 ]
0.417
2.0395
References [1] H. Rudnick, J. Zolezzi, Electric sector deregula-tion and restructuring in Latin America: lessons to be learn and possible waysforward, IEE Proceedings Generation, Transmission and Distribution, vol. 148 (2) 2001. [2] D. Montgomery, Design and Analysis of Experiments, Wiley, NY, 1991. [3] N.L. Frigon, D. Mathews, Practical Guide to Experimental Design, Wiley, New York, 1996. [4] C.R. Hicks, K.V. Turner .Jr., Fundamental Concepts in the Design of Experiments, Oxford University Press, Oxford, UK, 1999. [5] R. Morar, A. Iuga, L. Dascalescu, A. Samuila, Factors which influence the insulationmetal electroseparation, J. Electrostatics 30 (1993) 403–412.
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[6] A. Iuga, R. Morar, A. Samuila, L. Dascalescu, Electrostatic separation of metals and plastics from granular industrial wastes, in: Proceedings of the IEESci. Measurement Technology, vol. 148, March 2001, pp. 47–54. [7] R.-S. Guo, E. Sachs, Modeling, optimization and control of spatial uniformity in manufacturing processes, IEEE Trans. Semicond. Manuf. Vol. 6 (1) (1993) 41–57. [8] D. Moitre, S.- Brignone, M.- Carnero, M. Pontin, Effect of the installed capacity of generation on the price of the energy contracts for the distributors in a wholesale competitive electric energy market, in: Proceedings of the IASTED International Conference, Power and Energy Systems, Las Vegas, USA, 1999, pp. 402–405.
[9] S. Deladreue, F. Brouaye, P. Bastard, L. Peligry, Application of ANOVA methodology to the uncertain management in power system planning in an open market environment, in: Proceedings of the Power Technology Conference, Porto, Portugal, 2001, pp. 4–8. [10] S. Deladreue, F. Brouaye, P. Bastard, L. Peligry, Using two multivariate methods for line congestion study in transmission system under uncertainty, IEEE Trans. Power Syst. 18 (1) (2003) 353–358. [11] F. Yates, Design and analysis of factorial experiment, Technical Communication no. 53, Imperial Bureau of Soil Sciences, London, UK, 1993. [12] www.cammesa.com.ar.
Application of an experimental design methodology for economic parameter analysis in an open market environment D. Moitre∗ , F. Magnago, J. Martinez, M. Galetto National University of Rio Cuarto, University Campus, Ruta 36, Km 601, Rio Cuarto, Cordoba 5800, Argentina Received 1 October 2003; accepted 6 May 2004 Available online 21 September 2004
Abstract Statistical data analysis, through the utilization of experimental design, is a powerful tool commonly used in different areas where uncertainty is a problem, and a great amount of data is available for the analysis. After 10 years of open market experience, the data recorded during the last decade from the electric markets suggest that experimental design can be useful for electrical market data analysis. In this paper, the impact of different factors on the determination of economic parameters such as the spot price in an electric market is analyzed. The analysis is based on the experimental design methodology. The implementation of the proposed methodology is illustrated using simple examples from the Argentinien electric market. © 2004 Elsevier B.V. All rights reserved. Keywords: Experimental design; Electric energy market; ANOVA methodology; Spot price estimation
1. Introduction During the last decade, several electric market structures have changed all over the world from a vertical integrated monopoly to a vertical and horizontal segmented market. This process has been observed in countries such as Chile (1982), the United Kingdom (1989), Argentina (1992), Norway (1992), Central American countries (1997), and the United States (1998) [1]. As a result, two important issues can be highlighted taking into account over 10 years of open market experience in the countries that led themselves to those changes and from the available data. The first item to be considered is uncertainty with an open market; uncertainty increases with regards to power system structure, development, and price determination. The second one is the great amount of data available in databases. These two issues make the statistical data analysis, through the utilization of experimental design, a powerful tool for electrical market data analysis that can serve as a base for a second generation of reforms in the ∗
Corresponding author. Tel.: +54 358 4676 495; fax: +54 358 4676 495. E-mail address: [email protected] (D. Moitre).
0378-7796/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2004.05.005
electric market sector. Experimental design is a set of tests carried out on process or systems where inputs are changed in order to observe the output response and their relation with inputs. The main objective of the experimental design is to determine which variables are most influential on the output response, that is, to determine the parameters which lead to the best possible output response [2,3]. Initial experimental designs were applied in the area of agronomy and chemistry for the selection of the most efficient factor combination as described in [4–6], respectively. Later, the electronic industry used this methodology for the development of products and process [7]. Thus, experimental design becomes one of the most important tools in the statistical analysis of data in several areas, allowing companies to improve within an aggressive competitive environment. Recently, the idea of implementing experimental design in the power system area has been suggested. In [8], authors present a technique based on the analysis of the variance to calculate the contribution of generators to an electric market. Another implementation of the analysis of the variance method to detect the most influential factors on the level of line congestion is shown in [9,10]. The aim of this paper is to analyze the impact of different
62
D. Moitre et al. / Electric Power Systems Research 73 (2005) 61–66
factors on the determination of economic parameters such as the spot price in an electric market based on the experimental design methodology. First, the mathematical formulation of factorial experiments is revised; then, the implementation of this technique on the spot price determination is shown, and finally to illustrate the proposed method, a study case is presented.
2. Factorial experiments
the sum of squares of the error. Then, the tests described by Eqs. (2) and (3) are based on a comparison between the independent estimates of σ 2 provided by the division of each term of SST by their degree of freedom, known as mean square: SSA a−1 SSB MSB = b−1 MSA =
SSAB (a − 1)(b − 1) SSE MSE = ab(n − 1)
(6)
MSAB =
Factorial experiments are widely used within the area of experimental design when experiments involve several factors and the study of their joint effect on a response is necessary [11]; the simplest type includes two factors: factor A with a levels and factor B with b levels. The hypotheses under which results are analyzed are the following: • fixed factors; • full randomized design; • normality assumptions are satisfied. The observations can be described by the following statistical lineal model: yij = µ + αi + βj + (αβ)ij + εijk
(1)
where µ is the total media, αi denotes the effect of the ilevel of the first factor, βj the effect of the j-level of the second factor, (αβ)ij the interaction effect between the i-level of the first factor,and the j-level of the second factor, and εijk the experimental error. Repeating the experiment n times, there will be nab observations, which are assumed normally distributed, media µij and variance σ 2 . The objective of the factorial design is to determine the effect of the first and second factor for which the following tests of hypothesis are performed: H0 : α1 = α2 = · · · = αa = 0 H1 : at least one αi = 0
(2)
H0 : β1 = β2 = · · · = βb = 0 H1 : at least one βj = 0
(3)
Furthermore, it is very important to determine the interaction between both factors; therefore, the following test of hypothesis is formulated: H0 : (αβ)11 = (αβ)2 = · · · = (αβ)ab = 0 H1 : at least one (αβ)ij = 0
(4)
For a given data set, the total sum of squares SST is defined as follows: SST = SSA + SSB + SSAB + SSE
(5)
where SSA is the sum of squares for the main effect A, SSB is the sum of squares for the main effect B, SSAB is the sum of squares of the interactions between factors, and SSE is
The effect of a factor is defined by the variations in the factor’s level. This is called main effect because it refers to the primary factors in the study. In some experiments, the factors may have different level response, when this occurs, and there is a considerable interaction between factors, the corresponding main effects will have no meaning since a significant interaction can masks main effects. Assuming fixed factors A and B, the mean square expected values are: bn E(MSA ) = σ 2 +
a i=1
α2i
a−1 b βj2 an
E(MSB ) = σ 2 +
E(MSAB ) = σ 2 +
j=1
b−1 b a (αβ)2ij n i=1 j=1
(a − 1)(b − 1)
E(MSE ) = σ 2 In order to perform the test described by Eqs. (2)–(4), mean square values are divided by the mean square error: MSA MSE MSB FB = MSE MSAB FAB = MSE FA =
(7)
These ratios follow an F distribution with two degree of freedoms; one degree of freedom corresponds to the numerator term and the other corresponds to the denominator, in this case, ab(n − 1). The critical region is located in the upper tail of the distribution, as shown in Fig. 1. The test procedure is arranged in an analysis of variance (ANOVA) table, shown in Table 1. Rejecting the null hypothesis H0 and H0 implies that there are differences between the means, although the exact nature of the differences is not specified. In this situation, further comparisons between groups through the implementation of multiple comparison techniques may be useful. This
D. Moitre et al. / Electric Power Systems Research 73 (2005) 61–66
63
Fig. 1. F Distribution. Fig. 2. Residuals February 2002. Table 1 ANOVA Source of variation
Sum of squares
Degree of freedom
Mean square
F
A B AB Error
SSA SSB SSAB SSE
a−1 b−1 (a − l)(b − l) ab(n − 1)
MSA MSB MSAB MSE
FA FB FAB
Total
SST
abn − 1
technique is very useful when a great amount of data is available and a scientific decision is needed, such is the case, for example, when an economic parameter determination or new regulatory reforms are needed. Next section will show the application of these techniques to determine the most influential factors in the economic parameters for an electric power market.
diction of market behavior, for example, to identify the value of the economic parameters in different days of the week or different hours within a day. To illustrate this, let us consider that the spot price is determined every hour during 24 h for 7 days a week. These prices can be considered as the result of a bifactorial experiment where the first factor is the hour of the day analyzed at 24 levels, and the second factor is the day of the week analyzed at seven levels, four observations per cell are selected. Each factor is previously determined, hence, a fixed factor model can be selected. In order to show the proposed technique, the hourly spot price data of the Argentinean system is taken as an example. Data correspond to the month of February 2002 [12]. The first step is to verify if the data satisfy the statistical lineal model hypothesis of Eq. (1). The error, also referred as residual, is defined as the difference between the observation value yijk and the estimated value yijk .
3. Economic parameter determination
εijk = yijk − yijk
The electric power market includes various commercial and financial agreements through contracts of different types and duration and short-term transactions. In a centralized market, the nucleus of these agreements is the spot market in which the electric energy is valued and commercialized. This task is carried out by the Independent System Operator (ISO). The prices of the spot market in general are based on shortterm marginal costs of energy, either based on actual variable generation costs or the value of the bid. Let us show as an example the Argentine market, where the reform started in 1992, therefore the market spot price data are available from this year, taken every hour a day and 7 days a week. Moreover, no modification has been done to the regulation since then, although attempts have been made in 2001. Hence, these data can be useful to make a response surface for different economic parameters, which can become the basis for pre-
Fig. 2 shows the accumulative distribution residuals of February 2002, the histogram is represented in Fig. 3. The residuals are normally distributed with zero mean; therefore, its representation should be a line. However, there are residuals that are bigger than the rest of the residuals, which are called outliers. Since the presence of one outlier may alter the analysis of the results, before any analysis, the origin of these outliers should be determined. In this example, negative residuals are related to lower spot prices due to over offer of hydro generation or highperformance thermal units such as combined cycle units. Residuals that are positive and sensibly higher than the others are related to higher spot prices due to outage of generator units or transmission lines. The deviation of these residuals does not misrepresent the lineal analysis and the corresponding conclusions.
(8)
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D. Moitre et al. / Electric Power Systems Research 73 (2005) 61–66
Fig. 3. Histogram February 2002. Fig. 4. Day spot price comparison.
Next the AN OVA table is drawn for the data set. Table 2 shows the results for the data from February 2002. From Table 2 the following decision are made: Reject H0 and conclude those hours of the day influencing the spot price. Reject H0 and conclude that the day of the week that affects the spot price. Do not reject H0 and conclude that there is no interaction between the hour of the day and the day of the week where the spot price is calculated during the winter season considered. In addition, the following P values are obtained: Phours = 0
Pdays = 0
Phours & days = 0.9999
(9)
These probabilities quantify the significance of the level of decisions. For example, the probability that the first and the second conclusions are wrong is almost zero, whereas the probability for the third conclusion to be correct is near one. The next step is to determine if there are days of the week or hours of the day for which the spot price is considerably different. This is obtained using multiple comparisons based on the test method proposed by Tukey and Kramer, with a trust degree of 95%. The following figures illustrate the obtained results. Fig. 4 shows the spot prices calculated for each day of the week. In this figure, it can be observed that the mean spot prices for Sunday (number 3) is remarkably lower than the mean spot prices calculated on the other days. This is a logic conclusion since it is a non-working day. Fig. 5 shows
the results for different hours of a day. It is observed that the mean spot price is significantly greater from 8 to 11 p.m. where the residential load increases. Moreover, it is possible to observe that during 7 a.m. and 7 p.m., the price is uniform which is compatible with the industrial load of the system. Now, the estimation is calculated applying Minimum Square Error (MSE) to the lineal model described in Eq. (10): ij yˆ ij = µ ˆ + αˆ i + βˆ j + αβ
where yˆ ij is the spot price for day i and hour j, µ ˆ is the monthly average spot price, αˆ i the estimation of the effect of ij is the day i, βˆ j the estimation of the effect of hour j, and αβ estimation of the interaction between day i and hour j. Results are shown on Figs. 6 and 7, numerical data are described in Appendix A. Figs. 6 and 7 show the effect of the day and the hour respectively. The graphics illustrate the mean response to the different levels of these factors with
Table 2 ANOVA, February 2002 Source of variation Hours Days Hours and days Error Total
Sum of squares
Degree of freedom
4425.8 1241.2 1375.3 8551.7
23 6 138 504
15593.9
671
Mean square 192.424 206.864 9.966 16.968
(10)
F 11.34 12.19 0.59
Fig. 5. Hourly spot price comparison.
D. Moitre et al. / Electric Power Systems Research 73 (2005) 61–66
65
determination of economic parameters such as the spot price in an electric market.
Appendix A Table A.1 Days
Fig. 6. Effect of the day.
respect to the average spot price µ, ˆ which for this example is: $ µ ˆ = 12.6326 MWh From Fig. 6, it is possible to conclude that the spot price increases from Monday to Thursday and decreases on Friday and Saturday, reaching the lowest value on Sunday. The analysis suggests that days from Monday to Thursday can be grouped as working days, Friday can be considered as an atypical day. For the operation programming, days are classified as normal days (Tuesday to Friday), non-working day (Sunday), part-time working day (Saturday) and Monday as an atypical day, which differs from the statistical analysis. From Fig. 7, it can be inferred that the spot price increases from 8 to 11 p.m. with a peak at 9 p.m. then decreases until a minimum is reached at 7 a.m. Taking into account that for the operation the hours are classified as: peak from 7 to 12 p.m., valley from 12 p.m. to 6 a.m. and rest from 6 a.m. to 7 p.m., the statistical analysis arrived at the same conclusions. Therefore, from the analysis of the effect of the hours, it can be seen that the observations are consistent with the classification.
4. Conclusions The implementation of the experimental design and analysis to observe the effect of different parameters on the spot price calculation has been carried out. This technique has been applied to the Argentinian electric market. A simple example has been selected to illustrate the application of the methodology proposed, reaching the same conclusions as experts do. Results suggest that this statistical tool may become very useful to evaluate the impact of different factors on the
Fig. 7. Effect of the hours.
Day effect ␣I
[$/MWh]
Friday [␣1 ]
−0.2208
Saturday [␣2 ]
−0.7588
Sunday [␣3 ]
−2.5776
Monday [␣4 ]
0.0549 0.2808
Tuesday [␣5 ] Wednesday [␣6 ]
1.1694
Thursday [␣7 ]
2.051
Table A.2 Hours Hour effect (a.m.), β1
[$/MWh]
Hour effect (p.m.), β1
[$/MWh]
01:00 [β2 ] 02:00 [β3 ]
−0.168
01:00 [β13 ]
0.312
−1.184
02:00 [β14 ]
03:00 [β4 ] 04:00 [β5 ]
−2.0972
03:00 [β15 ]
−0.2026 −0.2026
−2.2583
04:00 [β16 ]
05:00 [β6 ] 06:00 [β7 ]
−2.4247
05:00 [β17 ]
0.348 −0.3447
−2.213
06:00 [β18 ]
−0.2737
07:00 [β8 ]
−2.893 −1.7826
07:00 [β19 ] 08:00 [β20 ]
−0.2133
08:00 [β9 ] 09:00 [β1 ]
−1.0158
09:00 [β21 ]
10:00 [β10 ] 11:00 [β11 ]
−0.6444
10:00 [β22 ]
9.7549 4.607
−0.433
11:00 [β23 ]
1.3717
12:00 [β12 ]
0.196
12:00 [β24 ]
0.417
2.0395
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