Cooperative game-based method to determine the weights of load forecasting solution incorporated with various algorithms

: Ultra-short-term load forecasting is one of the core techniques for distribution networks operation. Since individual load in distribution networks is small and with strong randomness, single prediction model cannot guarantee the forecasting accuracy and combined forecasting solution is usually used. In such forecasting scheme, how to weight the different forecasting model is a challenge. By introducing the concept of cooperative game, the weight of every algorithm can be determined according to the error distribution. The proposed method has been tested for a real distribution networks and the results show this scheme has good prediction precision, and can control the error distribution at the in ﬂ ection points.


Introduction
In real practice, ultra-short-term load forecasting is one of the main sources of pseudo-measurement for distribution state estimation (DSE) [1,2]. In distribution networks, the measurements are mainly from automatic meter reading system (AMR). However, the failures or time delay of AMR may lead to data missing. Therefore, load forecasting is needed to generate pseudomeasurement to mend these missing data for DSE. The accuracy of pseudo-measurement will significantly affect the results of DSE.
Till now, many practical prediction models have been proposed. For the published works, most predictive models are single prediction methods, and the core idea is the process of making predictions of the future based on past and present data and most commonly by analysis of trends [3,4]. However, the trend of distribution network load is influenced by many random factors, which leads to a very large load random fluctuation. Therefore, a single model cannot guarantee its prediction accuracy. The combination forecasting model, incorporated with various forecasting algorithms, can significantly improve the prediction precision [5]. Such combination model overcomes the defects of the single prediction model, and brings more choices [6,7].
Since the weights of different algorithms in the combination prediction model influence the prediction accuracy, how to optimise these parameter is a key problem [8,9].
These methods can be roughly divided into two categories: the first one is to formulate a optimisation problem with certain constraints, so as to obtain the weighting coefficient of the combined forecasting method; for the other idea, the weight coefficient of each individual forecasting model is selected inversely proportional to its variance of the prediction error. Since different time series have different dimensions, the sum of squares of the absolute errors is not comparable, so the criterion of the sum of squares of errors may not accurately reflect the validity of the prediction method, the results of the first class of weight solving methods may not be optimal in this situation. Furthermore, the computation burden of this method is very heavy. For the second type schemes, a simple weight calculation formula cannot guarantee the weight positive.
In this paper, the game theory is used to determine the weights of various prediction algorithms in the combination model. Each individual forecasting model is regarded as the cooperative player in the combination forecasting model, the sum of square error of combination forecasting is considered as a total payoff, then the weights to each single forecasting model are allocated using the Shapley value method. The forecasting results for a real distribution network show that this method has good prediction accuracy and can control the error distribution.
The remainder of this paper is organised as follows. In Section 2, the characteristics of a Chinese city's load series are explained. Section 3 introduces short-term load forecasting using combination forecasting model. Section 4 introduces the Shapley value. Section 5 presents a cooperative game method to determine the weights of combination forecasting model. The numerical test results are given in Section 6, and follows conclusions in Section 7.

Load series
In this section, the load series of a China's urban distribution networks is analysed (named city A load series). The load series consist of all 15 min observations in 2006, the data of the first 3 months was used to determine the model parameters, and the remaining months of data is used to evaluate the post-sample forecast accuracy. The data series are shown in Fig. 1, the load data of distribution network changes significantly, and the data of some days are absent and abnormal. This situation will affect the prediction accuracy and error distribution, and it is the main problem to be solved in this paper. Fig. 2 shows the load profile data of a typical week in city A. This figure emphasises the differences between the load patterns of different days within a week. Fig. 3 shows the load series of typical week and weekend days in city A. These two series both possess strong information seasonality with lower demand in the weekend day. This indicates that the load on weekends and holidays are influenced by various factors and a single model cannot tackle all issues. Therefore, the combination of multiple single models is a better option to capture various load characteristics.

Problem description
For a distribution network, the load data is very complex and random, so the prediction precision of a single model cannot meet the requirements in most cases. Bate and Granger [5] also pointed out: (i) Even for a very poor prediction method, if it contains some independent information, it can improve the prediction ability of the system when it combined with a good prediction method. (ii) Single prediction method has its unique value in different criteria of the data processing.
A combination model, incorporated with different forecasting methods in an appropriate way, can provide comprehensive information and improve the prediction accuracy. The basic form of combination forecasting model can be formulated as where k i is the weight of the ith prediction algorithm, f i the ith predictive value, and m the number of prediction algorithm. The formula to optimise the weights is The above model is a quadratic problem, and the existing method can find the optimal solution, but it cannot control the error distribution. For example, we should guarantee the prediction accuracy of the inflection points instead of minimising the overall sum square of the prediction errors. Where v(C ) denotes the variance of the combined forecasting from the coalitional of C.
We use the Shapley value (in the coalition, the proceeds to which the ith participant should be assigned): An allocation is individually rational and efficient if  The combination forecasting model combines the results of different models, and the essence of the integrated model is the weight of each single method in the final prediction of the proportion of the results. Then if there are some single methods in the combination no contribution in the combined model, we can remove it first. So that model does not participate in the process of combination forecasting, which is the formation of the basic idea of forecasting.
, then the contribution of these two models is the same, one of which can be removed.
. v(N ), then this model causes loss in combination forecasting, it is removed.
The determination strategy of weights for combination forecasting model is: Step 1: Assume that the absolute error produced by each single forecasting model at time t is e t (i) = y t − f it . We calculate the variance of a single model: Step 2: Assume that the total return in the combined forecast is: Var(e t (i)) (5) Step 3: Applying the Shapley value assign to each forecasting algorithm, it can fairly and fully reflect the contribution of each prediction model to the total prediction: Step 4: Calculate the weight of combination forecasting (we used variance analysis method [17]): Obviously, k i satisfy condition (2). The flowchart of this solution can be described in Fig. 4: 6 Numerical test

Performance of single prediction model
Exponential smoothing (ES) and trend extrapolation methods (TE) are widely used in ultra-short-term load forecasting. In this paper, we mainly focus on these two methods and naive method [18], and now very efficient Holt-Winter [19] model to verify the practicality of this paper method. The weights of these method are limited between 0 and 1. In order to calculate the parameters, we use OPTI packagenon-linear least square algorithm method. To compare the improvement of combination solution, the accuracy of each single prediction model is listed in Table 1. Table 1 shows these methods all have reasonable prediction precisions.

Comparison of the weight determining methods
Besides the proposed method, variance analysis method [17] and NLS method of OPTI Software Package are also used to solve the weights from comparison. Table 2 shows that the error distribution of the proposed method is smaller than the traditional method, so it has the best prediction performance. Although the average error of this method is bigger than that of the NLS method, but its error distribution is better than the NLS method. This is because the NLS method is to minimise the sum square of error, but the proposed method is to control error distribution.
From Table 3, we can see accuracy of the proposed method is significantly better than the other two methods in holidays.  shows that most of the blue lines are covered by red lines and black lines; this indicates error distribution of this paper is significantly better than the other two methods. Furthermore, Fig. 6 shows the errors at the inflection points for these two methods. In Fig. 6, it can be seen that the proposed method has a much high precision than the NLS method at the inflection points. This is the most important advantage of the proposed method.
Assuming the prediction error distributions are subject to a normal distribution, Table 4 lists the interval width of the predictions results under a given confidence level of 90%. It shows that NLS and proposed method have the smallest width which indicates that the combination model has the best performance. NLS has smaller width than the proposed method, but the proposed method has high prediction precision at the inflection points. The large prediction error at the inflection points may lead to more conservative dispatch strategies. Therefore, the proposed method is more useful for real practice.

Conclusions
This paper proposes a weight determination method for combined forecasting model in a distribution network. This method can improve the prediction accuracy and control error distribution. The proposed method is especially suitable to forecast the loads of small values and strong randomness. Moreover, it usually has a good accuracy at the inflection points where many other published methods may produce notable forecasting errors.