# SVR with Hybrid Chaotic Immune Algorithm for Seasonal Load Demand Forecasting

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology of the SSVRCIA Model

#### 2.1. Support Vector Regression (SVR) Model

**w**($w\in {\Re}^{{n}_{h}}$) and b ($b\in \Re $) are adjustable. SVR method aims at minimizing the empirical risk as Equation 2:

**y**by using the ε-insensitive loss function. C is a parameter to trade off these two terms. Training errors above $\epsilon $ are denoted as ξ

_{i}, whereas training errors below-ε are denoted as ξ

_{i}.

**w**in Equation 1 is obtained from:

_{1}and a

_{2}: $K({x}_{i},{x}_{j})={({a}_{1}{x}_{i}{x}_{j}+{a}_{2})}^{d}$. Till now, it has been hard to determine the type of kernel functions for specific data patterns [53,54]. However, the Gaussian RBF kernel is not only easier to implement, but also capable of nonlinearly mapping the training data into an infinite dimensional space, thus, it is suitable to deal with nonlinear relationship problems. Therefore, the Gaussian RBF kernel function is specified in this study.

#### 2.2. Chaotic Immune Algorithm (CIA) in Selecting Parameters of the SVR Model

_{k},Max

_{k}) into chaotic variable ${x}_{k}^{(i)}$ located in the interval (0,1).

_{k}denotes the SVR forecasting errors obtained by the antibody k. The similarity between antibodies is expressed as in Equation 11:

_{ij}denotes the difference between the two SVR forecasting errors obtained by the antibodies inside (existed) and outside (will be entering) the memory cell.

_{k}are considered to be potential candidates for entering the memory cell. However, the potential antibody candidates with Ab

_{ij}values exceeding a certain threshold are not qualified to enter the memory cell. In this investigation, the threshold value is set to 0.9.

_{c}) is set as 0.5. Finally, the three crossover parameters are decoded into a decimal format.

_{k},Max

_{k}) are mapped to chaotic variable interval [0,1] to form the crossover chaotic variable space ${\widehat{x}}_{k}^{(i)},\text{}k=C,\text{}\sigma \text{,}\epsilon $, as Equation 12:

_{max}is the maximum evolutional generation of the population. Then, the ith chaotic variable ${x}_{k}^{(i)}$ is summed up to ${\widehat{x}}_{k}^{(i)}$ and the chaotic mutation variable are also mapped to interval [0,1] as in Equation 13:

_{k},Max

_{k}) by definite probability of mutation (p

_{m}), thus completing a mutative operation:

_{i}is the actual value at period i; ${\widehat{f}}_{i}$ denotes is the forecasting value at period i.

#### 2.3. Seasonal Adjustment

**Table 1.**Monthly electric load in Northeastern China (from January 2004 to April 2009). Units: hundred million kW/h.

Time | Electric Load | Time | Electric Load | Time | Electric Load |
---|---|---|---|---|---|

January 2004 | 129.08 | November 2005 | 150.84 | September 2007 | 175.41 |

February 2004 | 127.24 | December 2005 | 165.27 | October 2007 | 179.64 |

March 2004 | 136.95 | January 2006 | 155.31 | November 2007 | 188.89 |

April 2004 | 125.34 | February 2006 | 138.5 | December 2007 | 197.62 |

May 2004 | 126.86 | March 2006 | 133.27 | January 2008 | 200.35 |

June 2004 | 129.34 | April 2006 | 151.41 | February 2008 | 169.24 |

July 2004 | 131.91 | May 2006 | 155.63 | March 2008 | 196.97 |

August 2004 | 136.22 | June 2006 | 155.7 | April 2008 | 186.15 |

September 2004 | 131.56 | July 2006 | 162.98 | May 2008 | 188.485 |

October 2004 | 134.62 | August 2006 | 163.41 | June 2008 | 190.82 |

November 2004 | 144.62 | September 2006 | 157.57 | July 2008 | 196.53 |

December 2004 | 154.62 | October 2006 | 160.15 | August 2008 | 197.67 |

January 2005 | 151.48 | November 2006 | 168.13 | September 2008 | 183.77 |

February 2005 | 126.74 | December 2006 | 180.71 | October 2008 | 181.07 |

March 2005 | 148.57 | January 2007 | 179.94 | November 2008 | 180.56 |

April 2005 | 136.6 | February 2007 | 147.29 | December 2008 | 189.03 |

May 2005 | 138.83 | March 2007 | 172.45 | January 2009 | 182.07 |

June 2005 | 136.6 | April 2007 | 169.98 | February 2009 | 167.35 |

July 2005 | 146.21 | May 2007 | 173.21 | March 2009 | 189.3 |

August 2005 | 146.09 | June 2007 | 177.43 | April 2009 | 175.84 |

September 2005 | 140.04 | July 2007 | 184.29 | ||

October 2005 | 142.02 | August 2007 | 183.53 |

## 3. A Numerical Results

#### 3.1. Data Set

Data Sets | SVRCIA and SSVRCIA Models | TF-ε-SVR-SA Model (Wang et al., 2009) |
---|---|---|

Training data | December 2004–July 2007 | December 2004–September 2008 |

Validation data | August 2007–September 2008 | |

Testing data | October 2008–April 2009 | October 2008–April 2009 |

#### 3.2. SSVRCIA Electric Load Forecasting Model

Population Size (${\mathit{p}}_{\mathit{s}\mathit{i}\mathit{z}\mathit{e}}$) | Maximal Generation (${\mathit{q}}_{\mathbf{max}}$) | Probability of Crossover (${\mathit{p}}_{\mathit{c}}$) | The Annealing Operation Parameter ($\mathit{\delta}$) | Probability of Mutation (${\mathit{p}}_{\mathit{m}}$) |
---|---|---|---|---|

200 | 500 | 0.5 | 0.9 | 0.1 |

Nos. of Input Data | Parameters | MAPE of Testing (%) | ||
---|---|---|---|---|

σ | C | ε | ||

5 | 14.744 | 347.33 | 1.8570 | 4.1953 |

10 | 9.9515 | 90.244 | 0.1459 | 3.638 |

15 | 109.06 | 7298.3 | 11.953 | 3.897 |

20 | 48.030 | 8399.7 | 14.372 | 3.514 |

25 | 30.262 | 4767.3 | 22.114 | 3.0411 |

Time Point (Month) | Seasonal Index | Time Point (Month) | Seasonal Index |
---|---|---|---|

January | 1.0153 | July | 1.0663 |

February | 0.9089 | August | 1.0615 |

March | 1.0126 | September | 1.0076 |

April | 0.9853 | October | 0.9734 |

May | 1.0187 | November | 1.0247 |

June | 1.0225 | December | 1.0614 |

**Table 6.**Forecasting results of the ARIMA, TF-ε-SVR-SA, SVRCIA, and SSVRCIA models (units: hundred million kW/h).

Time Point (Month) | Actual | ARIMA(1,1,1) | TF-ε-SVR-SA | SVRCIA | SSVRCIA |
---|---|---|---|---|---|

October 2008 | 181.07 | 192.9316 | 184.5035 | 179.0276 | 174.2737 |

November 2008 | 180.56 | 191.127 | 190.3608 | 179.4118 | 183.8444 |

December 2008 | 189.03 | 189.9155 | 202.9795 | 179.7946 | 190.8367 |

January 2009 | 182.07 | 191.9947 | 195.7532 | 180.1759 | 182.9343 |

February 2009 | 167.35 | 189.9398 | 167.5795 | 180.5557 | 164.1062 |

March 2009 | 189.30 | 183.9876 | 185.9358 | 180.9341 | 183.2106 |

April 2009 | 175.84 | 189.3480 | 180.1648 | 178.1036 | 175.4833 |

MAPE (%) | 6.044 | 3.799 | 3.041 | 1.766 |

Compared Models | Wilcoxon Signed-Rank Test | |
---|---|---|

α = 0.025 W = 2 | α = 0.05 W = 3 | |

SSVRCIA vs. ARIMA(1,1,1) | 1 * | 1 * |

SSVRCIA vs. TF-ε-SVR-SA | 0 * | 0 * |

SSVRCIA vs. SVRCIA | 2 * | 2 * |

## 4. Conclusions

## Acknowledgements

## References

- Fan, S.; Chen, L. Short-term load forecasting based on an adaptive hybrid method. IEEE Trans. Power Syst.
**2006**, 21, 392–401. [Google Scholar] [CrossRef] - Morimoto, R.; Hope, C. The impact of electricity supply on economic growth in Sri Lanka. Energy Econ.
**2004**, 26, 77–85. [Google Scholar] [CrossRef] - Box, G.E.P.; Jenkins, G.M. Time Series Analysis, Forecasting and Control; Holden-Day Press: San Francisco, CA, USA, 1970. [Google Scholar]
- Chen, J.F.; Wang, W.M.; Huang, C.M. Analysis of an adaptive time-series autoregressive moving-average (ARMA) model for short-term load forecasting. Electr. Power Syst. Res.
**1995**, 34, 187–196. [Google Scholar] [CrossRef] - Vemuri, S.; Hill, D.; Balasubramanian, R. Load forecasting using stochastic models. In Proceedings of the 8th Power Industrial Computing Application Conference, Minneapolis, MN, USA, 1973; pp. 31–37.
- Christianse, W.R. Short term load forecasting using general exponential smoothing. IEEE Trans. Power Apparatus Syst.
**1971**, 90, 900–911. [Google Scholar] [CrossRef] - Park, J.H.; Park, Y.M.; Lee, K.Y. Composite modeling for adaptive short-term load forecasting. IEEE Trans. Power Syst.
**1991**, 6, 450–457. [Google Scholar] [CrossRef] - Brown, R.G. Introduction to Random Signal Analysis and Kalman Filtering; John Wiley & Sons Inc. Press: New York, NY, USA, 1983. [Google Scholar]
- Gelb, A. Applied Optimal Estimation; The MIT Press: Cambridge, MA, USA, 1974. [Google Scholar]
- Moghram, I.; Rahman, S. Analysis and evaluation of five short-term load forecasting techniques. IEEE Trans. Power Syst.
**1989**, 4, 1484–1491. [Google Scholar] [CrossRef] - Asbury, C. Weather load model for electric demand energy forecasting. IEEE Trans. Power Apparatus Syst.
**1975**, 94, 1111–1116. [Google Scholar] [CrossRef] - Papalexopoulos, A.D.; Hesterberg, T.C. A regression-based approach to short-term system load forecasting. IEEE Trans. Power Syst.
**1990**, 5, 1535–1547. [Google Scholar] [CrossRef] - Soliman, S.A.; Persaud, S.; El-Nagar, K.; El-Hawary, M.E. Application of least absolute value parameter estimation based on linear programming to short-term load forecasting. Int. J. Electr. Power Energy Syst.
**1997**, 19, 209–216. [Google Scholar] [CrossRef] - Rahman, S.; Bhatnagar, R. An expert system based algorithm for short-term load forecasting. IEEE Trans. Power Syst.
**1998**, 3, 392–399. [Google Scholar] [CrossRef] - Chiu, C.C.; Kao, L.J.; Cook, D.F. Combining a Neural Network with a rule-based expert system approach for short-term power load forecasting in Taiwan. Expert Syst. Appl.
**1997**, 13, 299–305. [Google Scholar] [CrossRef] - Rahman, S.; Hazim, O. A generalized knowledge-based short-term load- forecasting technique. IEEE Trans. Power Syst.
**1993**, 8, 508–514. [Google Scholar] [CrossRef] - Park, D.C.; El-Sharkawi, M.A.; Marks, R.J., II; Atlas, L.E.; Damborg, M.J. Electric load forecasting using an artificial neural network. IEEE Trans. Power Syst.
**1991**, 6, 442–449. [Google Scholar] [CrossRef] - Novak, B. Superfast autoconfiguring artificial neural networks and their application to power systems. Electr. Power Syst. Res.
**1995**, 35, 11–16. [Google Scholar] [CrossRef] - Darbellay, G.A.; Slama, M. Forecasting the short-term demand for electricity—do neural networks stand a better chance? Int. J. Forecast.
**2000**, 16, 71–83. [Google Scholar] [CrossRef] - Abdel-Aal, R.E. Short-term hourly load forecasting using abductive networks. IEEE Trans. Power Syst.
**2004**, 19, 164–173. [Google Scholar] [CrossRef] - Hsu, C.C.; Chen, C.Y. Regional load forecasting in Taiwan—application of artificial neural networks. Energy Convers. Manag.
**2003**, 44, 1941–1949. [Google Scholar] [CrossRef] - Suykens, J.A.K. Nonlinear modelling and support vector machines. In Proceedings of IEEE Instrumentation and Measurement Technology Conference, Budapest, Hungary, 2001; pp. 287–294.
- Tay, F.E.H.; Cao, L.J. Application of support vector machines in financial time series forecasting. Omega
**2001**, 29, 309–317. [Google Scholar] [CrossRef] - Huang, W.; Nakamori, Y.; Wang, S.Y. Forecasting stock market movement direction with support vector machine. Comput. Oper. Res.
**2005**, 32, 2513–2522. [Google Scholar] [CrossRef] - Hung, W.M.; Hong, W.C. Application of SVR with improved ant colony optimization algorithms in exchange rate forecasting. Control Cybern.
**2009**, 38, 863–891. [Google Scholar] - Pai, P.F.; Lin, C.S. A hybrid ARIMA and support vector machines model in stock price forecasting. Omega
**2005**, 33, 497–505. [Google Scholar] [CrossRef] - Pai, P.F.; Lin, C.S.; Hong, W.C.; Chen, C.T. A hybrid support vector machine regression for exchange rate prediction. Int. J. Inf. Manag. Sci.
**2006**, 17, 19–32. [Google Scholar] - Hong, W.C.; Dong, Y.; Chen, L.Y.; Wei, S.Y. SVR with hybrid chaotic genetic algorithms for tourism demand forecasting. Appl. Soft Comput.
**2011**, 11, 1881–1890. [Google Scholar] [CrossRef] - Pai, P.F.; Hong, W.C. An improved neural network model in forecasting arrivals. Ann. Tourism Res.
**2005**, 32, 1138–1141. [Google Scholar] [CrossRef] - Pai, P.F.; Hong, W.C. Software reliability forecasting by support vector machines with simulated annealing algorithms. J. Syst. Softw.
**2006**, 79, 747–755. [Google Scholar] [CrossRef] - Hong, W.C.; Pai, P.F. Predicting engine reliability by support vector machines. Int. J. Adv. Manuf. Technol.
**2006**, 28, 154–161. [Google Scholar] [CrossRef] - Hong, W.C. Rainfall forecasting by technological machine learning models. Appl. Math. Comput.
**2008**, 200, 41–57. [Google Scholar] [CrossRef] - Hong, W.C.; Pai, P.F. Potential assessment of the support vector regression technique in rainfall forecasting. Water Resour. Manag.
**2007**, 21, 495–513. [Google Scholar] [CrossRef] - Wang, W.; Xu, Z.; Lu, J.W. Three improved neural network models for air quality forecasting. Eng. Comput.
**2003**, 20, 192–210. [Google Scholar] [CrossRef] - Mohandes, M.A.; Halawani, T.O.; Rehman, S.; Hussain, A.A. Support vector machines for wind speed prediction. Renew Energy
**2004**, 29, 939–947. [Google Scholar] [CrossRef] - Hong, W.C. Hybrid evolutionary algorithms in a SVR-based electric load forecasting model. Int. J. Electr. Power Energy Syst.
**2009**, 31, 409–417. [Google Scholar] [CrossRef] - Hong, W.C. Chaotic particle swarm optimization algorithm in a support vector regression electric load forecasting model. Energy Convers. Manag.
**2009**, 50, 105–117. [Google Scholar] [CrossRef] - Hong, W.C. Electric load forecasting by support vector model. Appl. Math. Modell.
**2009**, 33, 2444–2454. [Google Scholar] [CrossRef] - Pai, P.F.; Hong, W.C. Support vector machines with simulated annealing algorithms in electricity load forecasting. Energy Convers. Manag.
**2005**, 46, 2669–2688. [Google Scholar] [CrossRef] - Pai, P.F.; Hong, W.C. Forecasting regional electric load based on recurrent support vector machines with genetic algorithms. Electr. Power Syst. Res.
**2005**, 74, 417–425. [Google Scholar] [CrossRef] - Hong, W.C. Application of chaotic ant swarm optimization in electric load forecasting. Energy Policy
**2010**, 38, 5830–5839. [Google Scholar] [CrossRef] - Mori, K.; Tsukiyama, M.; Fukuda, T. Immune algorithm with searching diversity and its application to resource allocation problem. Trans. Inst. Electr. Eng. Jpn.
**1993**, 113-C, 872–878. [Google Scholar] - Prakash, A.; Khilwani, N.; Tiwari, M.K.; Cohen, Y. Modified immune algorithm for job selection and operation allocation problem in flexible manufacturing system. Adv. Eng. Softw.
**2008**, 39, 219–232. [Google Scholar] [CrossRef] - Wang, L.; Zheng, D.Z.; Lin, Q.S. Survey on chaotic optimization methods. Comput. Technol. Autom.
**2001**, 20, 1–5. [Google Scholar] - Li, B.; Jiang, W. Optimizing complex functions by chaos search. Cybern. Syst.
**1998**, 29, 409–419. [Google Scholar] [CrossRef] - Ohya, M. Complexities and their applications to characterization of chaos. Int. J. Theor. Phys.
**1998**, 37, 495–505. [Google Scholar] [CrossRef] - Lorenz, E.N. Deterministic nonperiodic flow. J. Atmos. Sci.
**1963**, 20, 130–141. [Google Scholar] [CrossRef] - Dagum, E.B. Modelling, forecasting and seasonally adjusting economic time series with the X-11 ARIMA method. J. R. Stat. Soc. Series D
**1978**, 27, 203–216. [Google Scholar] - Kenny, P.B.; Durbin, J. Local trend estimation and seasonal adjustment of economic and social time series. J. R. Stat. Soc. Series A
**1982**, 145, 1–41. [Google Scholar] [CrossRef] - Wang, J.; Zhu, W.; Zhang, W.; Sun, D. A trend fixed on firstly and seasonal adjustment model combined with the ε-SVR for short-term forecasting of electricity demand. Energy Policy
**2009**, 37, 4901–4909. [Google Scholar] [CrossRef] - Xiao, Z.; Ye, S.J.; Zhong, B.; Sun, C.X. BP neural network with rough set for short term load forecasting. Expert Syst. Appl.
**2009**, 36, 273–279. [Google Scholar] [CrossRef] - Vapnik, V. The Nature of Statistical Learning Theory; Springer-Verlag Press: New York, NY, USA, 1995. [Google Scholar]
- Amari, S.; Wu, S. Improving support vector machine classifiers by modifying kernel functions. Neural Netw.
**1999**, 12, 783–789. [Google Scholar] [CrossRef] [PubMed] - Vojislav, K. Learning and Soft Computing—Support Vector Machines, Neural Networks and Fuzzy Logic Models; The MIT Press: Cambridge, MT, USA, 2001. [Google Scholar]
- Pan, H.; Wang, L.; Liu, B. Chaotic annealing with hypothesis test for function optimization in noisy environments. Chaos Solitons Fractals
**2008**, 35, 888–894. [Google Scholar] [CrossRef] - Zuo, X.Q.; Fan, Y.S. A chaos search immune algorithm with its application to neuro-fuzzy controller design. Chaos Solitons Fractals
**2006**, 30, 94–109. [Google Scholar] [CrossRef] - Liu, B.; Wang, L.; Jin, Y.H.; Tang, F.; Huang, D.X. Improved particle swam optimization combined with chaos. Chaos Solitons Fractals
**2005**, 25, 1261–1271. [Google Scholar] [CrossRef] - Yang, D.; Li, G.; Cheng, G. On the efficiency of chaos optimization algorithms for global optimization. Chaos Solitons Fractals
**2007**, 34, 1366–1375. [Google Scholar] [CrossRef] - Li, L.; Yang, Y.; Peng, H.; Wang, X. Parameters identification of chaotic systems via chaotic ant swarm. Chaos Solitons Fractals
**2006**, 28, 1204–1211. [Google Scholar] [CrossRef] - Tavazoei, M.S.; Haeri, M. Comparison of different one-dimensional maps as chaotic search pattern in chaos optimization algorithms. Appl. Math. Comput.
**2007**, 187, 1076–1085. [Google Scholar] [CrossRef] - Coelho, L.D.S.; Mariani, V.C. Chaotic artificial immune approach applied to economic dispatch of electric energy using thermal units. Chaos Solitons Fractals
**2009**, 40, 2376–2383. [Google Scholar] [CrossRef] - Wang, J.; Wang, Y.; Zhang, C.; Du, W.; Zhou, C.; Liang, Y. Parameter selection of support vector regression based on a novel chaotic immune algorithm. In Proceedings of the 4th International Conference on Innovative Computing, Information and Control, City, Country, 2009; pp. 652–655.
- Deo, R.; Hurvich, C. Forecasting realized volatility using a long-memory stochastic volatility model: estimation, prediction and seasonal adjustment. J. Econometrics
**2006**, 131, 29–58. [Google Scholar] [CrossRef] - Azadeh, A.; Ghaderi, S.F. Annual electricity consumption forecasting by neural network in high energy consuming industrial sectors. Energy Convers. Manag.
**2008**, 49, 2272–2278. [Google Scholar] [CrossRef]

© 2011 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Hong, W.-C.; Dong, Y.; Lai, C.-Y.; Chen, L.-Y.; Wei, S.-Y.
SVR with Hybrid Chaotic Immune Algorithm for Seasonal Load Demand Forecasting. *Energies* **2011**, *4*, 960-977.
https://doi.org/10.3390/en4060960

**AMA Style**

Hong W-C, Dong Y, Lai C-Y, Chen L-Y, Wei S-Y.
SVR with Hybrid Chaotic Immune Algorithm for Seasonal Load Demand Forecasting. *Energies*. 2011; 4(6):960-977.
https://doi.org/10.3390/en4060960

**Chicago/Turabian Style**

Hong, Wei-Chiang, Yucheng Dong, Chien-Yuan Lai, Li-Yueh Chen, and Shih-Yung Wei.
2011. "SVR with Hybrid Chaotic Immune Algorithm for Seasonal Load Demand Forecasting" *Energies* 4, no. 6: 960-977.
https://doi.org/10.3390/en4060960