This paper employs chaos theory into power load forecasting. Lyapunov exponents on chaos theory are calculated to judge whether it is a chaotic system. Delay time and embedding dimension are calculated to reconstruct the phase space and determine the structure of artificial neural network (ANN). Improved back propagation (BP) algorithm based on genetic algorithm (GA) is used to train and forecast. Finally, this paper uses the load data of Shaanxi province power grid of China to complete the short-term load forecasting. The results show that the model in this paper is more effective than classical standard BP neural network model.

Chaos theory is the important component of the nonlinear science [

Short-term power load forecasting is a multidimensional nonlinear system. It is easy to get the load time series in power system. But these data are nonlinear and difficult to establish a matched mathematical model to forecast the next-hour load. Recently, more and more nonlinear time series forecasting models based on chaos theory [

There are many models to be adopted into power system load forecasting. They can generally be summarized as follows: time series model, regression model, expert system model, grey theory model, and fuzzy logic model. But according to chaotic characters of the load time series, ANN [

Many intrinsic deficiencies of ANN are still in existence. The structure is difficult to confirm. The blindness that initial weights are chosen results into slow convergence speed and easily falling into local minimum. However GA [

Power system loads are a set of time series. Chaos theory can analyze chaotic characteristics of time series and reveal the sequence itself of the objective regularities to avoid the predicted human subjectivity and improve the accuracy and credibility of load forecasting.

At present, phase-space delay coordinate reconstitution method is employed to analyze chaotic characteristics of time series. Generally, the dimension is very great even infinite. In fact, phase-space delay coordinate reconstitution method can expand the given time series to three-dimensional and even higher-dimensional space, and the information which exposed sufficiently from time series can be classified and extracted.

Phase-space reconstitution theory is the basis for chaotic time series forecasting. Packard and Takens proposed the technology of phase-space reconstitution for chaotic time series

Takens proved that phase-space reconstitution through selecting an appropriate delay time

Chaos is characterized by extreme sensitivity to movement on the initial conditions. Lyapunov exponents quantify the exponential divergence of initially close state-space trajectories and estimate the amount of chaos in a system. When the largest Lyapunov exponent [

M. T. Rosenstin, J. J. Collins, and G. J. Deluca proposed an approach of small data sets. This method is more reliable, with a smaller calculation and easier to operate than others. So the largest Lyapunov exponent is computed by this approach. The process is as follows.

For

Figure out correlation dimension

Reconstruct the phase-space based on

Find out nearest neighbor

Calculate the distance after

Compute by the follow formula:

Kolmogorov theory supposes if

That is to say, there is a three-layer network of which hidden-layer function is

Chaos neural network in Figure

Chaos neural network base model.

In order to get the better weights and thresholds and determine the structure of BP neural network, GA can be drawn in to improve the forecasting accuracy of chaotic time series.

There are some deficiencies of BP neural network, such as a lower pace, being easy to local minimum, and the uncertainty structure. But GA can overcome these and improve network performance and convergence rate and optimize chaos neural network further.

The methods and steps to achieve genetic algorithm and optimize chaos neural network are the following.

After analyzing the chaotic characters of the time series, we get a data set of the weights and thresholds that are unknown. These data are encoded and made as an individual. Several individuals can constitute the initial population.

Suppose

Calculate the fitness of every individual

Crossover operation is used to enhance the global search ability of GA. After selecting operation, randomly two individuals are selected to match but avoid choosing the individuals of the same gene. Randomly select a cross point of each individual, and according to the probability

Mutation operation is mainly used to enhance local search ability of GA. After crossover operation, randomly select a mutation point to change the code according to the probability

In order to overcome precocity of GA, the improvement is necessary to operate. The parent and the offspring individuals are looked as a whole to form

For the new population, repeat the previous operation until the relative error between two iteration operations meets the accuracy requirements. Error formula is the following formula:

On the basis of the above, GA is just like the diagram in Figure

Genetic algorithm flow diagram.

There are some power data from 0:00 at 1/1/2009 to 23:00 at 3/30/2009 in Shaanxi province power grid of China to use as the sample data for short-term power load forecasting. Then the data can be established into the time series

The load time series.

The time series are analyzed by chaos theory. Then there are some parameters to determine. Delay time

Since

The graphics of the three parameters.

Calculate delay time by mutual information

Calculating embedding dimension by Cao method

Calculating the largest Lyapunov exponent by wolf method

When it got the parameters, the phase space can be reconstructed in Figure

Phase-space reconstitution.

Figure

After completing phase-space reconstitution, the data that come from 0:00 at 3/31/2009 to 23:00 at 3/31/2009 in Shaanxi province power grid of china are used as the testing sample. The structure of BP neural network is employed 4-9-1. Relative error and root-mean-square relative error are used as the final evaluating indicators:

The results are in Table

Comparison of the forecasting results and errors by the different methods.

Time (h) | Real load (MW) | This paper’s method | Standard BP network method | ||

Results (MW) | Error (%) | Results (MW) | Error (%) | ||

0 | 6416.69 | 6482.81 | 1.03 | 6554.61 | 2.15 |

1 | 6111.52 | 6185.50 | 1.21 | 6329.72 | 3.57 |

2 | 6044.52 | 6098.35 | 0.89 | 6206.50 | 2.68 |

3 | 5998.76 | 6041.46 | 0.71 | 6031.26 | 0.54 |

4 | 5813.06 | 5833.42 | 0.35 | 6073.57 | 4.48 |

5 | 5945.27 | 5852.57 | −1.56 | 5823.41 | −2.05 |

6 | 6195.15 | 6163.60 | −0.51 | 6075.02 | −1.94 |

7 | 6863.36 | 7001.31 | 2.01 | 6956.71 | 1.36 |

8 | 7232.71 | 7347.72 | 1.59 | 7457.66 | 3.11 |

9 | 7781.65 | 7879.76 | 1.26 | 7949.74 | 2.16 |

10 | 7847.12 | 7910.78 | 0.81 | 7974.20 | 1.62 |

11 | 8000.54 | 8154.24 | 1.92 | 8151.80 | 1.89 |

12 | 7756.46 | 7882.10 | 1.62 | 7614.51 | −1.83 |

13 | 7154.43 | 7055.72 | −1.38 | 6967.08 | −2.62 |

14 | 7340.18 | 7263.15 | −1.05 | 7077.44 | −3.58 |

15 | 7467.57 | 7588.56 | 1.62 | 7344.41 | −1.65 |

16 | 7513.92 | 7577.81 | 0.85 | 7586.82 | 0.97 |

17 | 7856.26 | 7963.15 | 1.36 | 7955.23 | 1.26 |

18 | 7862.16 | 7871.63 | 0.12 | 7831.54 | −0.39 |

19 | 7891.19 | 7914.19 | 0.29 | 7776.06 | −1.46 |

20 | 8487.14 | 8406.51 | −0.95 | 8617.81 | 1.54 |

21 | 8180.10 | 8220.22 | 0.49 | 8362.52 | 2.23 |

22 | 7476.11 | 7570.30 | 1.26 | 7682.50 | 2.76 |

23 | 6416.69 | 6347.42 | −1.08 | 6632.33 | 3.36 |

RSM | // | // | 1.080 | // | 2.133 |

Table

Standard BP neural network method just uses the historical load and the uncertain structure of BP neural network to forecast the next-hour load. Added the deficiencies of BP, the results could not be very accurate and the speed not so quick. But the method in this paper takes the chaotic characters of the power load time series, appropriately determining the structure of input layer and hidden layer in BP network into account. So this can make the network training well and improve the prediction accuracy further obviously.

Above all, Figure

The different methods forecasting comparison.

The results of the different methods

The errors of the different methods

As can be seen from the graphics in Figure

This paper employs the chaos theory into eclectic system load forecasting. From Lyapunov exponents to phase-space reconstitution, it tells that power load time series are nonlinear and chaotic and have all the characters of chaos. The chaotic phase diagram also shows there is a chaos attractor. So chaos theory introduced into power load forecasting can describe the nonlinear dynamic behavior of the system and get the accuracy and the precision improved greatly.

Meanwhile, BP neural network under being improved by GA based on chaos theory is used for forecasting to improve the accuracy and the training rate further. In the last, applied into Shaanxi province power grid short-term load forecast of China and compared with the standard BP neural network model, the method mentioned in this paper gets more accurate results and more efficient training rate. So this way can bring a broad application prospects in power load forecasting.