Load forecasting problem is a complex nonlinear problem linked with economic and weather factors. Long-term load forecasting provides useful information for maintenance scheduling, adequacy assessment, and limited energy resources for electrical power systems. Fuzzy time series forecasting models can be used for long-term load forecasting. However, the interval length has been chosen arbitrarily in the implementations of known fuzzy time series forecasting models, which has an important impact on the performance of these models. In this paper, a time-variant ratio multiobjective optimization fuzzy time series model (TV-RMOP) is proposed, and its performance is tested on the prediction of enrollment at the University of Alabama. Results clearly promote the forecasting accuracy as compared to the conventional models. A genetic algorithm is used to search for the length of intervals based on the training data while Pareto optimality theory provides the necessary conditions to identify an optimal one. The TV-RMOP model is applied for the long-term load forecasting in Shanghai of China.
Electric power load forecast has been a research topic for many decades and the accuracy of load forecast is crucial to electricity power industry due to its direct influence on generating planning. During the past years, there have been numerous attempts to improve the accuracy of load forecasting methods [
There are three different types of electric load forecasting depending on the time horizon and the operating decision that needs to be made: short-, medium-, and long-term forecasting. In general, long term forecasting includes prediction making for a range more than a year and is needed for power supply and delivery system. Therefore, the weather factor is not the main factor to affect the long-term load forecast, only using historical load data can predict the 1 year ahead load. Fuzzy time series [
Fuzzy time series has been widely studied for recent years for the aim of forecasting. The basic process of conventional model is shown in Figure
Two distinct prediction processes of fuzzy time series forecasting.
Conventional process
Our process
The rest of this paper is organized as follows. Section
The concepts of fuzzy time series are described as follows [
Let
Let
It is assumed that
Let
It is common in decision-making problems to have a set of parameters that play a dominant role in the process. It is also desirable for the decision made to be the best possible with respect to the available data. In the context of mathematics, the best decision coincides with the optimization of a function of the decision parameters, known as cost function or objective.
On the one hand, if the goal is to optimize a single cost function, then the problem is classified as single objective optimization. On the other hand, the process of simultaneously optimizing two or more conflicting objectives, subject to certain constraints, is identified as multiobjective optimization [
Such problems are called multiobjective optimization problems and formulated as follows:
Multiobjective optimization problems do not have a single global solution in contrast to the single objective problems. As a result, it is required to specify a number of criteria-conditions for which a set of solutions can be identified as optimal. In general, the optimal solutions for
Time series is divided into two parts. One part is used for training purpose while the other part is used for testing. The proposed model was applied to the training data to make estimation and then the predicted values were found for the test data. In the training phase, five measures are adopted to evaluate the model prediction accuracy with the intention to formulate the multiobjective problem. These measures are objective functions with parameters and defined as follows:
As shown in Figure
Proposed time-variant ratio multiobjective optimization fuzzy time series model.
Define the universe of discourse and intervals using equations defined in the upper-right dashed box. Where
Define fuzzy sets based on the universe of discourse and fuzzify the historical data.
Establish fuzzy relationships:
Forecast. Let
Calculate RMSE, MAE, MAPE, MAP, and THEIL, respectively. Each of them is a function with variable
In the training phase, the training samples (
In the test phase, an optimal solution is brought into the model, and the forecast value is determined in the same way as the training phase.
Genetic algorithm can be used to search for a solution of multiobjective optimization problems. The process is shown in the dashed box on the left in Figure
In order to show what we have achieved from the TV-RMOP model, we apply it to the enrollment of University of Alabama and then have compared the obtained results with those obtained from Chen’s method [
The enrollment data.
Year | Enrollments |
---|---|
1971 | 13055 |
1972 | 13563 |
1973 | 13867 |
1974 | 14696 |
1975 | 15460 |
1976 | 15311 |
1977 | 15603 |
1978 | 15861 |
1979 | 16807 |
1980 | 16919 |
1981 | 16388 |
1982 | 15433 |
1983 | 15497 |
1984 | 15145 |
1985 | 15163 |
1986 | 15984 |
1987 | 16859 |
1988 | 18150 |
1989 | 18970 |
1990 | 19328 |
1991 | 19337 |
1992 | 18876 |
The minimal value of
We use a MATLAB function “gamultiobj” to solve (
Pareto solutions of enrollment.
Initial | Ratio | RMSE | MAE | MAPE | MAP | THEIL |
---|---|---|---|---|---|---|
12266.7 | 0.0621 | 571.9 | 484.4 | 2.9745 | 5.8418 | 0.0359 |
12230.4 | 0.0530 | 573.9 | 486.0 | 2.9983 | 5.6954 | 0.0360 |
12135.5 | 0.0455 | 532.7 | 451.8 | 2.8276 | 6.6546 | 0.0334 |
12846.0 | 0.0676 | 568.0 | 501.2 | 3.1217 | 5.2926 | 0.0357 |
12266.7 | 0.0620 | 551.5 | 459.0 | 2.8142 | 5.8748 | 0.0346 |
Pareto front of TV-RMOP model for enrollment.
We select 12135.5 and 0.0455 as optimal parameters to forecast the enrollment from 1990 to 1992. For this initial and ratio, the universe of discourse is defined as
The forecasting results for
Year | Actual enrollment | [ |
[ |
TV-RMOP model |
---|---|---|---|---|
1990 | 19328 | 18685 | 18970 | 19271 |
1991 | 19337 | 19138 | 19306 | 19271 |
1992 | 18876 | 19176 | 19315 | 19271 |
The measures for
Year | Actual enrollment | [ |
Initial | Ratio | RMSE | MAE | MAPE | MAP | THEIL |
---|---|---|---|---|---|---|---|---|---|
12135.5 | 0.0455 | 233.5 | 172.7 | 0.9098 | 2.79 | 0.01212 |
Long term load forecasting is needed for power supply and delivery system. We use twenty years (1990~2010) load of Shanghai region in China for forecast (data from the 2011 Shanghai statistical yearbook). The load is shown in Table
Load in Shanghai region.
Year | Load (hundred million Kwh) |
---|---|
1990 | 264.74 |
1991 | 288.78 |
1992 | 317.38 |
1993 | 345.86 |
1994 | 337.3 |
1995 | 403.27 |
1996 | 430.4 |
1997 | 454.26 |
1998 | 482.94 |
1999 | 501.2 |
2000 | 559.42 |
2001 | 592.99 |
2002 | 645.71 |
2003 | 745.97 |
2004 | 821.44 |
2005 | 921.97 |
2006 | 990.15 |
2007 | 1072.38 |
2008 | 1138.22 |
2009 | 1153.38 |
2010 | 1295.87 |
The minimal value of
The initialization of genetic algorithm parameters is set as follows. The fraction of population on non-dominated front is 0.1, the number of individuals is 50, maximum number of generations allowed is 100, and termination tolerance on fitness function value is
Pareto solutions of load.
Initial | Ratio | RMSE | MAE | MAPE | MAP | THEIL |
---|---|---|---|---|---|---|
257.68 | 0.2317 | 45.28 | 34.47 | 5.8185 | 12.184 | 0.0719 |
250.80 | 0.2400 | 44.06 | 34.49 | 5.9488 | 13.627 | 0.0699 |
256.51 | 0.2329 | 45.22 | 34.54 | 5.8476 | 12.448 | 0.0718 |
250.80 | 0.2400 | 44.06 | 34.49 | 5.9488 | 13.627 | 0.0699 |
252.48 | 0.2380 | 44.24 | 34.30 | 5.8942 | 13.273 | 0.0702 |
Pareto front of TV-RMOP model for load forecasting.
Select 250.8 and 0.24 as optimal parameters to forecast the loads from 2008 to 2010. Tables
The forecasting results for
Year | Actual load | TV-RMOP model |
---|---|---|
2008 | 1138.22 | 1021.1 |
2009 | 1153.38 | 1266.2 |
2010 | 1295.87 | 1266.2 |
The measures for
Initial | Ratio | RMSE | MAE | MAPE | MAP | THEIL |
---|---|---|---|---|---|---|
250.8 | 0.24 | 95.43 | 86.53 | 7.45 | 10.29 | 0.07966 |
Fuzzy time series forecasting method are getting quite popular in recent years. Most studies focus on the forecasting rules and methods of defuzzification. How to determine the lengths of intervals is less studied. The decision on what the lengths will be is important for forecasting accuracy.
In this study, a new model TV-RMOP is proposed, which formulates the forecasting as a multiobjective optimization problem and solve it by the use of genetic algorithms. The optimal lengths of intervals can be determined in the proposed model. The model is tested for forecasting the enrollment of University of Alabama. The experimental results show that the TV-RMOP model is more accurate than existing models. Finally, we apply this model for long-term load forecasting of Shanghai Region with satisfactory result.
The authors would like to thank the anonymous reviewers in MPE for helpful suggestions and correction. This work was conducted by using of the MATLAB software and the authors declared that it has no conflict of interests to this work. The work was financially supported by Shanghai Municipal Nature Science Foundation under Grant 10ZR1401400.