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Fig. 5. The reconstructed RL (dash) in comparison with the actual data (Line) for different time intervals. The periodogram analysis of the original series and Eigen vectors tells one which frequency must be considered. If the periodograms of the Eigen vector have sharp spark(s) around some frequencies, then the corresponding Eigen triples must be regarded as those related to the signal component [29]. Fig. 4 depicts the periodogram of the paired Eigen triples (2, 3), (4, 5), (6, 7), (8, 9), (10, 11), (12, 13), (14, 15), (18, 19), and (24, 25) of the RL series. From Fig. 4, all the above-mentioned Eigen triples should be regarded as the selected Eigen triples in the grouping step in conjunction with another Eigen triple (the first Eigen triple) that we need to reconstruct the series. In SSA forecasting, the model is expressed by a Linear Recurrent Formula (LRF). This LRF applied to the last L— 1 terms of the initial

Figure 5 The reconstructed RL (dash) in comparison with the actual data (Line) for different time intervals. The periodogram analysis of the original series and Eigen vectors tells one which frequency must be considered. If the periodograms of the Eigen vector have sharp spark(s) around some frequencies, then the corresponding Eigen triples must be regarded as those related to the signal component [29]. Fig. 4 depicts the periodogram of the paired Eigen triples (2, 3), (4, 5), (6, 7), (8, 9), (10, 11), (12, 13), (14, 15), (18, 19), and (24, 25) of the RL series. From Fig. 4, all the above-mentioned Eigen triples should be regarded as the selected Eigen triples in the grouping step in conjunction with another Eigen triple (the first Eigen triple) that we need to reconstruct the series. In SSA forecasting, the model is expressed by a Linear Recurrent Formula (LRF). This LRF applied to the last L— 1 terms of the initial

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The expression in Eq. (4) corresponds to averaging the elements along diagonals i+j=k+2. This diagonal averaging, applied to a resultant matrixx;,, produces a time series Y, of length T and thus the initial series Yr is decomposed into the sum of M series. Calling the decomposed series as Y, it will be derived as: Where, the trajectory matrix X is represented as a sum of M resultant matrices. Next, diagonal averaging transfers each matrix X;,,n = 1,...,M into a time series. Let X be a (L x K) matrix with elements xj. Make L* = min(L, K), KY = max(L,K). Lets define xij = xj, if L<K; and xj =x;, otherwise. Diagonal averaging transfers matrix X to a series go,...,g7_1 following the formula:
The expression in Eq. (4) corresponds to averaging the elements along diagonals i+j=k+2. This diagonal averaging, applied to a resultant matrixx;,, produces a time series Y, of length T and thus the initial series Yr is decomposed into the sum of M series. Calling the decomposed series as Y, it will be derived as: Where, the trajectory matrix X is represented as a sum of M resultant matrices. Next, diagonal averaging transfers each matrix X;,,n = 1,...,M into a time series. Let X be a (L x K) matrix with elements xj. Make L* = min(L, K), KY = max(L,K). Lets define xij = xj, if L<K; and xj =x;, otherwise. Diagonal averaging transfers matrix X to a series go,...,g7_1 following the formula:
Fig. 1. (a) The experimental hourly required load (RL) of Iran electricity network for 844 days, (b) some parts of Fig. 1(a) magnified. The experimental data is the RL time series of Iran electricity market in MWh for 844 days from 24 September 2005 up to 15 Dec.
Fig. 1. (a) The experimental hourly required load (RL) of Iran electricity network for 844 days, (b) some parts of Fig. 1(a) magnified. The experimental data is the RL time series of Iran electricity market in MWh for 844 days from 24 September 2005 up to 15 Dec.
Fig. 2. Principal components related to the first 36 Eigen triples
Fig. 2. Principal components related to the first 36 Eigen triples
Fig. 3. Logarithms of the 48 Eigen values. In SSA analysis, there is some extra information, which helps us to identify of the useful Eigen triples of the SVD of the trajectory
Fig. 3. Logarithms of the 48 Eigen values. In SSA analysis, there is some extra information, which helps us to identify of the useful Eigen triples of the SVD of the trajectory
Logarithms of the 48 Eigen values versus the Eigen value number. Table 1
Logarithms of the 48 Eigen values versus the Eigen value number. Table 1
Fig. 4. Periodograms of the paired Eigen triples for RL series. Usually every harmonic component with a different frequency produces two Eigen triples with close singular values. These Eigen pairs will be clearer if T, L and K are sufficiently large. Another useful insight is provided by checking breaks in the Eigen value spectra. As arule, a pure noise series produces a slowly decreasing sequence of singular values. Therefore, explicit plateau in the Eigen value spectra prompt the ordinal numbers of the paired Eigen triples [26]. Fig. 3 depicts the plot of the logarithms of the 48 singular values for the RL series where the values of the singular value logarithms versus the Eigen value numbers are listed in Table 1 for more
Fig. 4. Periodograms of the paired Eigen triples for RL series. Usually every harmonic component with a different frequency produces two Eigen triples with close singular values. These Eigen pairs will be clearer if T, L and K are sufficiently large. Another useful insight is provided by checking breaks in the Eigen value spectra. As arule, a pure noise series produces a slowly decreasing sequence of singular values. Therefore, explicit plateau in the Eigen value spectra prompt the ordinal numbers of the paired Eigen triples [26]. Fig. 3 depicts the plot of the logarithms of the 48 singular values for the RL series where the values of the singular value logarithms versus the Eigen value numbers are listed in Table 1 for more
Where, in Eq. (10), 0 < a, 6 < 1,a+ 6 = 1 are constants which should be chosen and Yy7c1,1)is the first element of Yrc which has been subtracted from it to avoid continuity violation. A proper choice for
Where, in Eq. (10), 0 < a, 6 < 1,a+ 6 = 1 are constants which should be chosen and Yy7c1,1)is the first element of Yrc which has been subtracted from it to avoid continuity violation. A proper choice for
DME, and DPE of four weeks for the forecasted RL time series. Table 2
DME, and DPE of four weeks for the forecasted RL time series. Table 2
Fig. 7. Comparison of forecasted RL with its actual value for: (a) the February week, (b) The May week, (c) The August week, and (d) The November Week.
Fig. 7. Comparison of forecasted RL with its actual value for: (a) the February week, (b) The May week, (c) The August week, and (d) The November Week.
WME and WPE of four weeks for the forecasted RL time series. References
WME and WPE of four weeks for the forecasted RL time series. References
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Mechanical EngineeringElectricity MarketElectricity MarketsInterdisciplinary Engineering
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