Short-term load forecasting for power systems based on Kalman filtering

First, the Kalman filter algorithm is introduced, and a set of recursive calculation formulas is given. Then, this algorithm is applied to short-term load forecasting, and the algorithm is improved according to the characteristics of load forecasting itself. The two algorithms are used to perform actual load forecasting calculations and obtain relatively accurate forecasting results.
Keywords: Kalman filter; power load forecasting; forecasting model

Power system short-term load forecast by Kalman filter

Li Ming-gan, Sun Jian-li, Liu Pei

(Huazhong University of Science & Technology, Wuhan 430074, China)

Abstract: Kalman filter algorithm is introduced in this paper firstly, and a set of formula is presented. Then Kalman filter model is used in short-term load forecast. And the model is revised by a approach aims at the specialty of load forecast. Application of these algorithms gains preferably results.
Key words: Kalman filter; power load forecast; forecast model

0 Introduction
Short-term load forecasting is a very important part of power system operation and dispatching. It is the premise of safe and economical operation of power grid and the basis of dispatching and arranging start-up and shutdown plans. It is very important for automatic control of power grid dispatching. Its prediction accuracy directly affects the economic benefits of power system. With the deepening of power reform and the further opening of power market, high-quality short-term load forecasting becomes more and more important and urgent.
There are many methods for load forecasting. Traditional methods include regression analysis[1] and least square method[2]. These methods have relatively simple algorithms and mature technologies. However, because their models are too simple, it is difficult to reflect new information about load changes during power system operation into the models, so the prediction accuracy is not satisfactory. In recent years, people have been committed to applying new theories and methods to load forecasting and have made great progress. Chaos model method[3], neural network (RBF)[4], fuzzy neural network method[5], expert system method[6], etc. have been proposed. These methods have achieved better prediction results than traditional methods.
Kalman filter (KF) was proposed by Kalman in 1960. It describes the filter by using the state space model of a linear stochastic system composed of a state equation and an observation equation. It also uses the recursive property of the state equation and the linear unbiased minimum mean square error estimation criterion to make the best estimate of the state variables of the filter using a recursive algorithm, thereby obtaining the best estimate of the useful signal after removing the noise. Kalman filter theory not only has a filter model, but also a predictor model [7]. By estimating the model parameters, the observation sequence can be predicted. Therefore, Kalman filter is suitable for short-term load forecasting.
There have been studies on the use of Kalman filter for short-term load forecasting abroad. In 1998, M. Huelsemann, MD Seiser and others discussed the use of Kalman filter and autocorrelation for load forecasting [8], and proposed the idea of ​​using Kalman filter for load forecasting, which achieved a theoretical breakthrough. Subsequently, Ngan, HW and others also discussed this method [9] and made some progress. In China, although there are predictions of other aspects of Kalman filter, the use of Kalman filter for load forecasting is still in its infancy. The author uses Kalman filter theory to establish a short-term load forecasting model for power systems, and uses relevant data such as load data and meteorological data in historical data to predict short-term load.

1 Introduction to Kalman filter model
Consider a linear discrete time system:

Where x is an n-dimensional state variable, Φ(t+1,t) is an n×n state transfer matrix, B(t) is an n×r input noise transfer matrix, w(t) is a p-dimensional input noise; y(t) is an m-dimensional measurement vector, H(t) is an m×n-dimensional measurement matrix, and v(t) is an m-dimensional measurement noise.

Where E represents the mathematical expectation, the p×p-order input noise covariance matrix Q(t) is symmetric positive definite; the m×m-order measurement noise covariance matrix R(g) is symmetric positive definite.
Assume that the initial state x(0) is independent of w(t) and v(t), and its statistical characteristics are known to be:






For detailed derivation process, see [7].

2 Kalman filter load forecasting model
The same moment of several consecutive days is used as a set of time series to predict the next load value at that moment. Usually, the load value can be divided into several parts: the

basic load at that moment, Lpk(t) is the load value at the same moment of the previous day, LTk(t) is the temperature at that moment, Vk(t) is the error, HNk(t), HPk(t), HTk(t) are all parameter matrices. Since the value at a certain moment is predicted, all the quantities in the formula are one-dimensional.
In order to facilitate the application of Kalman filter theory for state prediction, the following transformation is made:

In the formula: yk(t) is the observed value, Hk(t) is the observation matrix, Φk(t) is the state transfer matrix, and Wk(t) is the state error. Since the state variable in this article is the temperature at the same moment of several consecutive days, it can be regarded as a slowly changing state in short-term load forecasting, so Φk(t)=I can be set, I is the unit matrix.

2.1 Improved model
In the actual prediction process, the temperature forecast value at the prediction time can generally be provided, or the temperature value of other points can be obtained by interpolation through the forecast values ​​of several points. This paper proposes a prediction value correction method, that is, a load prediction method that adds a temperature correction value on the basis of this prediction value.
Suppose the Kalman filter prediction value of the load at the (t+1)th day to be predicted is yk(t+1|t), the state estimation value at this moment is LTK(t+1|t), and the temperature forecast value at the prediction moment is Tk. The state estimation value is the best estimate of the state of the system at the next moment obtained by the Kalman filter through the historical load, and the new state value of the system obtained by the forecast reflects the future state of the system. Therefore, their combination can allow the prediction model to obtain more information and thus obtain a more accurate prediction value. In order to use this information, the predicted value can be corrected, that is, a correction coefficient is multiplied before [Tk-LTK(t+1|t)], that is:

Where: bk is the correction coefficient, which can be obtained through experiments, and yk(t+1) is the corrected load forecast value at this moment.

3 Short-term load prediction example
Kalman filtering and the improved model are used to calculate the power load prediction example in Wuhan. In practice, it is difficult to accurately grasp the initial state xk(0|0) and Pk(0|0). However, since the Kalman filter continuously uses new information to correct the state during the recursive process, when the filtering time is sufficiently long, the influence of the initial state value xk(0|0) on xk(t+1|t) will decay to near zero, and the influence of the initial covariance matrix Pk(0|0) on the filter estimation covariance matrix Pk(t+1|t) will also decay to zero. Therefore, the initial conditions of the filter can be approximately determined.
In each recursive operation, the predicted value xk(t+1|t) must be calculated first, and then the variance of the prediction error Pk(t+1|t) is calculated based on the predicted value, and the Kalman gain Kk(t+1) is calculated by the optimal filtering rule. After the error compensation of the Kalman gain, the optimal filtering value xk(t+1) is obtained, and then the load prediction value is calculated by the prediction equation.
The temperature parameters are obtained from historical data before the day to be predicted and filtered and estimated, while the temperature of the day to be predicted is obtained from the temperature forecast of the day.
Introducing error indicators:
Relative error:

The Kalman filter model and the improved model are used to make actual predictions of the power load in Wuhan area. The prediction results and errors of a random day are shown in the following figures (1), (2), (3), and (4).

Figure 1 is a comparison between the Kalman filter prediction value and the actual value for 24 hours of the day, Figure 2 is the prediction error of the Kalman filter at 24 points, and its average absolute relative error is 3.43%. Figure 3 is a comparison between the load value calculated by the improved algorithm for 24 hours of the day and the actual value, and Figure 4 is the relative error of the 24 points of the improved algorithm, and its average absolute relative error is 2.94%. It can be seen that the improved algorithm is effective.

4 Summary
This paper uses the Kalman filter theory to establish a short-term load forecasting model and conduct short-term load forecasting. The feasibility of the Kalman filter model prediction is verified by examples. At the same time, according to the characteristics of load forecasting, the prediction accuracy is improved by improving the Kalman filter algorithm.
Since the Kalman filter continuously uses new information to correct the state estimate during the recursive process, the Kalman filter is asymptotically stable. When the time series is long enough, the state value of the initial state and the influence of the covariance matrix on the estimate will decay to zero. Therefore, the Kalman filter model can continuously update the state information and obtain a relatively accurate estimate. This method can be used not only for short-term prediction, but also for ultra-short-term load prediction.