In order to test the proposed hybrid models, two simulation experiments are carried out based on the same original data as that used for ANN and SVM models’ forecasts. In the first experiment, data are selected from date January 1st, 2007 to June 30th, 2007 and classified into four partitions by FCM model. Data from date July 1st, 2007 to December 31st, 2007 are divided into five partitions by FCM model in the second experiment. In each of the partitions, SVM model is applied respectively to forecast half-hour electricity prices by taking the advantages of aggregated data information. Therefore, in this chapter, a hybrid forecasting model is developed by conjunctive use of FCM clustering algorithm and SVM algorithm in order to overcome the limitations of individual model and get a high degree of prediction accuracy.
MatLab is both a powerful numerical computing environment and programming language. For the experiment of data fuzzy partitioning, the software MatLab has been used to solve the problem.
The orange color in figure 5.4 shows the forecasting objective III of this section and the current procedure against the whole project structure.
Chapter 5. Hybrid Forecasting Models Based on FCM and SVM ‐ 67 ‐ Fig. 5.4 Objective III and current procedure
5.4.1 Forecasts of Case A
As mentioned before, data of case study A are half-hour UK spot power market prices in the first half year 2007. The total number of data is 7968. Those data are classified into four clusters, which are shown in different colors, such as green, red, yellow and blue in figure 5.5. The center of each cluster is marked in black color with a bigger size.
Chapter 5. Hybrid Forecasting Models Based on FCM and SVM ‐ 68 ‐ Electricity Prices (￡/MWh)
Time Periods
Fig. 5.5 FCM_4 clusters of case A
In each of the clusters, training data set is separated from testing data set, which is composed of points belonged to the forecasting time periods from June 16th, 2007 to June 30th, 2007.
Forecasting model SVM is applied on data clusters one by one and the algorithm for prediction is epsilon-SVR with kernel function RBF. During the cross-validation, folder V is set for 10. The best parameters of each training program are listed in table 5.1.
0 5 10 15 20 25 30 35 40 45 50 0 50 100 150 200 250
Chapter 5. Hybrid Forecasting Models Based on FCM and SVM ‐ 69 ‐ Clusters Number Clusters Color Points Number Best Parameters C P Q Cluster 1 Red 366 64.0 0.00390625 0.03125 Cluster 2 Yellow 2677 64.0 0.00390625 0.00390625
Cluster 3 Light blue 2205 64.0 0.00390625 0.00390625
Cluster 4 Light green 2720 32.0 0.00390625 0.00390625
Table 5.1 Best Parameters for each training program in case A
Reference prices and the forecast prices are listed in table E (a) (b) and (c) in appendix, respectively. Corresponding to the real price curves, the curves of forecast prices to the fifteen days (16th – 30th June 2007) are illustrated in figures 5.6 - 5.20.
Criterion MAPE is used to calculate the forecasting errors of proposed hybrid models. The calculation results are listed in table 5.2 and illustrated in figure 5.21.
Chapter 5. Hybrid Forecasting Models Based on FCM and SVM ‐ 70 ‐
Fig. 5.6 Hybrid model forecasting results on Jun 16th, 2007 Fig. 5.7 Hybrid model forecasting results on Jun 17th, 2007
Fig. 5.8 Hybrid model forecasting results on Jun 18th, 2007 Fig. 5.9 Hybrid model forecasting results on Jun 19th, 2007
Fig. 5.10 Hybrid model forecasting results on Jun 20th, 2007 Fig. 5.11 Hybrid model forecasting results on Jun 21st, 2007
Chapter 5. Hybrid Forecasting Models Based on FCM and SVM ‐ 71 ‐
Fig. 5.14 Hybrid model forecasting results on Jun 24th, 2007 Fig. 5.15 Hybrid model forecasting results on Jun 25th, 2007
Fig. 5.16 Hybrid model forecasting results on Jun 26th, 2007 Fig. 5.17 Hybrid model forecasting results on Jun 27th, 2007
Fig. 5.18 Hybrid model forecasting results on Jun 28th, 2007 Fig. 5.19 Hybrid model forecasting results on Jun 29th, 2007
Chapter 5. Hybrid Forecasting Models Based on FCM and SVM ‐ 72 ‐ Time Periods 1 2 3 4 5 6 MAPE 3.513 5.4 3.573 4.193 7.527 4.256 Time Periods 7 8 9 10 11 12 MAPE 5.785 0.734 1.215 4.226 3.118 5.425 Time Periods 13 14 15 16 17 18 MAPE 7.48 4.196 0.542 0.468 2.279 4.635 Time Periods 19 20 21 22 23 24 MAPE 4.677 4.88 7.523 9.345 11.159 14.401 Time Periods 25 26 27 28 29 30 MAPE 7.309 6.056 4.629 5.924 5.957 5.443 Time Periods 31 32 33 34 35 36 MAPE 6.394 6.876 7.293 4.877 4.911 5.192 Time Periods 37 38 39 40 41 42 MAPE 1.972 10.066 4.958 4.972 6.692 4.195 Time Periods 43 44 45 46 47 48 MAPE 1.858 1.739 0.689 3.768 0.155 0.209
Table 5.2 Hybrid models’ MAPE of 48 time periods (16th - 30th June 2007)
Chapter 5. Hybrid Forecasting Models Based on FCM and SVM ‐ 73 ‐ 5.4.2 Forecasts of Case B
Data of case study B are half-hour UK spot power market prices in the second half year 2007. The total number of data is 8064. From figure 5.22, it can be seen that the data are classified into five clusters shown in different colors, such as green, red, blue, purple and yellow. The bigger sizes of points in black color are the centers of clusters.
Electricity Prices (￡/MWh)
Time Periods
Fig. 5.22 FCM_5 clusters of case B
In each of the clusters, training data set is separated from testing data set whose points are belonged to the forecasting time periods, December 1st 2007 to December 15th 2007. As the same forecasting procedure in case study A, model SVM is applied on data clusters one by one in case study B. The algorithm for prediction is epsilon-SVR and the kernel function is RBF. The folder V in cross-validation is set for 10. The best parameters of each training program are listed in table 5.3.
0 5 10 15 20 25 30 35 40 45 50 0 100 200 300 400 500 600 700
Chapter 5. Hybrid Forecasting Models Based on FCM and SVM ‐ 74 ‐ Clusters Number Clusters Color Points Number Best Parameters C P Q
Cluster 1 Light green 3378 64.0 0.00390625 0.00390625
Cluster 2 Purple 209 64.0 0.5 0.0625
Cluster 3 Yellow 1362 64.0 0.00390625 0.00390625
Cluster 4 Light blue 20 NA NA NA
Cluster 5 Red 3095 64.0 0.00390625 0.00390625
Table 5.3 Best Parameters for each training program in case B
In case study B, the data of cluster 4 are the extremely volatile price spikes. Therefore, the twenty data points, as shown above in table 5.3, are computed automatically independent of the other data groups and are not used for the training programs, consequently the data noise is reduced with regard to the forecasting.
The forecasting results of case study B and reference data are listed in table F (a) (b) and (c) in appendix, respectively. Corresponding to real price curves, the curves of forecasting prices to the fifteen days, December 1st 2007 to December 15th 2007, are illustrated in figures 5.23 - 5.37.
The forecasting errors of proposed hybrid models are calculated by the same method
MAPE. The calculation results are listed in table 5.4 and illustrated in figure 5.38.
Chapter 5. Hybrid Forecasting Models Based on FCM and SVM ‐ 75 ‐
Fig. 5.23 Hybrid model forecasting results on Dec 1st, 2007 Fig. 5.24 Hybrid model forecasting results on Dec 2nd, 2007
Fig. 5.25 Hybrid model forecasting results on Dec 3rd, 2007 Fig. 5.26 Hybrid model forecasting results on Dec 4th, 2007
Fig. 5.27 Hybrid model forecasting results on Dec 5th, 2007 Fig. 5.28 Hybrid model forecasting results on Dec 6th, 2007
Chapter 5. Hybrid Forecasting Models Based on FCM and SVM ‐ 76 ‐
Fig. 5.31 Hybrid model forecasting results on Dec 9th, 2007 Fig. 5.32 Hybrid model forecasting results on Dec 10th, 2007
Fig. 5.33 Hybrid model forecasting results on Dec 11th, 2007 Fig. 5.34 Hybrid model forecasting results on Dec 12th, 2007
Fig. 5.35 Hybrid model forecasting results on Dec 13th, 2007 Fig. 5.36 Hybrid model forecasting results on Dec 14th, 2007
Chapter 5. Hybrid Forecasting Models Based on FCM and SVM ‐ 77 ‐ Time Periods 1 2 3 4 5 6 MAPE 1.375 8.315 3.803 9.24 7.57 2.733 Time Periods 7 8 9 10 11 12 MAPE 4.16 7.000 1.946 3.332 1.245 6.24 Time Periods 13 14 15 16 17 18 MAPE 3.791 1.832 1.19 0.658 6.056 6.367 Time Periods 19 20 21 22 23 24 MAPE 4.138 8.168 8.549 9.346 12.335 8.4 Time Periods 25 26 27 28 29 30 MAPE 8.564 5.679 6.453 5.549 3.402 1.958 Time Periods 31 32 33 34 35 36 MAPE 0.941 0.604 4.338 12.645 17.916 13.925 Time Periods 37 38 39 40 41 42 MAPE 4.884 3.11 3.392 3.245 2.425 1.357 Time Periods 43 44 45 46 47 48 MAPE 1.409 0.587 0.266 0.538 0.497 0.758
Table 5.4 Hybrid models’ MAPE of 48 time periods (1st -15th December 2007)
Fig. 5.38 Hybrid models’ MAPE of 48 time periods (1st -15th December 2007)
Chapter 5. Hybrid Forecasting Models Based on FCM and SVM ‐ 78 ‐
5.5 SUMMARY
In order to overcome the limitations of individual forecasting model, a hybrid model combining FCM clustering algorithm and SVM regression is proposed in this chapter. According to the value of power prices, thousands of price data are classified by the unsupervised learning algorithm of FCM clustering. SVM regression model is applied onto each cluster by taking the advantages of aggregated data information, which has less noise for the training programs. Based on the same experimental data as that used for ANN and SVM forecasting, two case studies are carried out to test the proposed hybrid model and the results show its feasibility. In order to demonstrate the advances of the improved forecasting model, a further comparison among ANN model, single SVM model and the hybrid model will be analyzed and shown in next chapter 6.
Chapter 6. Comparative Analysis of Different Modeling and Forecasting ‐ 79 ‐