1 Introduction

Electricity is of vital importance to every region as an essential energy resource in people’s daily life. Prediction of electricity load from one day to one week, namely short-term load forecasting, has extremely vital significance for the development of the whole national economy. However, with the development of economy, the structure of power system becomes more and more complicated and the features of power load have more obvious changes such as nonlinear, time-varying and uncertainty. It is difficult to establish an appropriate mathematical model to clearly express the relationship between the load and the variables affecting the load. Therefore, an accurate forecasting method is particularly indispensable.

Nowadays, there is a tendency that an accumulating number of scholars try to analyze time series prediction in various fields. Paper [1] proposed a directed weighted complex network from time series. Multivariate weighted complex network analysis was proposed in paper [2] for characterizing nonlinear dynamic behavior in two-phase flow. Paper [3,4,5] are other two well performed examples about time series analysis. Since the 1960s, it should come as no surprise to learn that an amassing number of researchers began to study load forecasting. For example, linear multiple regression models of electrical energy consumption in Delhi for different seasons have been developed in paper [6, 7] presented a novel approach for short term load forecasting using fuzzy neural networks. The commonly used prediction methods include grey model, the traditional mathematical statistical model, artificial intelligence approach (spring up in the 1990s), combination model and hybrid model. Grey model, a part of grey system theory, was first proposed by Chinese scholar professor JL. Deng [8], in March 1982. Grey forecasting models are amongst the latest prediction methods [9]. For example, paper [10] used an optimized grey model to forecast the annual electricity consumption of Turkey. In paper [11], a grey correlation contest modeling was used for Short-term power load forecasting. The traditional mathematical statistical model, described by existing mathematical expressions, is an approach combined the mathematical theories with the practical problems such as the Kalman filtering model [12], ARMA (Autoregressive Moving Average) model [13, 14], ARIMA (Auto-Regressive Integrated Moving Average) model [15], the linear extrapolated method [16], the dynamic regression model [17], the GARCH (Generalized Auto-Regressive Conditional Heteroskedastic) model [18] and the time series analysis method [19], etc. Artificial intelligence approach is a branch of computer science. With the gradual application of the artificial intelligence technology in the time series forecasting, people have proposed many kinds of forecasting method based on artificial intelligence such as knowledge based expert system [20], SVR (Support Vector Regression) [21], CMAC (Cerebellar Model Articulation Controller) [22] and ANN (Artificial neural networks) [23, 24].

In order to enhance the forecasting performance, emphases have been laid on combined models or hybrid models. The key of combined model is how to determine the weighting coefficient of every individual model. A series of researches which solved the problem have been made in recent years. A combined model based on data pre-analysis was proposed for electrical load forecasting and cuckoo search algorithm was applied to optimize the weight coefficients [25]. A combined model has been developed for electric load forecasting and adaptive particle swarm optimization (APSO) algorithm was used to determine the weight coefficients allocated to each individual model [26]. Hybrid model has also been put into several different models and makes full use of the information of every model. However, each prediction model, different from the combined model, is only a process of the whole hybrid model and the project is completed in the most effective order. A hybrid model based on wavelet transform combined with ARIMA and GARCH models was proposed in day-ahead electricity price forecasting [13]. A new hybrid evolutionary-adaptive methodology, called HEA, was proposed for electricity prices forecasting [27].

In this paper, a hybrid model combined MEC and WNN based on rolling forecast and EEND for electricity load forecasting was proposed. The main contents of this paper are explained in detail as follows.

First, a concept named the minimum cycle decomposition (MCD) that raw data were grouped according to the minimum cycle (MCD) was proposed. When conducting the pre-processing of the raw data, how to divide the load type is a particular important problem. Through analyzing the published research in short-term load forecasting we could find that the load data are usually divided into different data type to predict. For example, a week was divided into two types [26], working days (Monday to Friday) and rest days (Saturday and Sunday), or into seven types [28] which take every day of a week as a type. All of these methods are not flexible, so this paper proposed a new flexible load classification model which took cycle of load data into account. This approach believes that the same observation point in different period of load have similarities, therefore load data could be divided into different types according to the cycle of load data. The periodic time series may be based on several years, quarter, month, week and day, so this classification method is more flexible.

Second, this paper presented the WNN \(+\) MEC model as a forecast engine to cover nonlinear pattern. Compared with traditional BP network, the promotion of wavelet theory leads to the inborn advantage of WNN. Firstly, the WNN has a strong ability to approximate nonlinear function and to extract implicit function. Secondly, its convergence speed is faster than BPNN. Finally, the generalization capability of WNN and memory ability of nonlinear function are also better than BPNN. Wavelet neural network is a combination of neural network and wavelet analysis. In 1988, affine discrete wavelet network model was proposed by Pati and Krishnaprasad [29]. The concept of wavelet neural network was formally put forward by Zhang and Benveniste [30]. The basic idea of which are that the activation function Sigmoid function is replaced by positioned wavelet function and the connection between wavelet transform and network coefficient is established by affine transformation. However, the proposal of wavelet neural network is relatively late, fruitful results have been achieved. For example, WNN was used to improve the accuracy of the short-term load forecasting in paper [31]. A fault prognosis architecture consisting dynamic wavelet neural networks had been developed [31]. A new load forecasting (LF) approach using bacterial foraging technique (BFT) trained WNN was proposed in paper [32].

Thirdly, it regretfully suggests that few researchers use MEC algorithm for WNN parameters optimization problem in short-term load forecasting. Therefore, this paper proposed an approach, based on rolling forecast, which could select the best weight, shift factor and scalability factor of wavelet neural network by the means of MEC. There is a very close relationship between the precision of artificial neural network and the selection of ANN’s parameters such as weights. To improve the accuracy, some optimal algorithm techniques were applied, such as PSO (Particle Swarm Optimization) [33, 34], APSO (Adaptive Particle Swarm Optimization) [26], DE (Differential Evolution) [35], GA (Genetic Algorithms) [36], BFT (Bacterial Foraging Technique) [32] and high-order Markov chain model [37]. Evolutionary computation algorithm (EC), the most famous optimal algorithm, was also used to tune the connection weights and the parameters of dilation and translation in the WNN [38]. There are a lot of famous evolutionary computation such as genetic algorithm (GA), evolution strategy (ES) and evolutionary programming (EP). However, some problems and defects still exist, for example, early-ripe, the complex parameters control, slow convergence rate and the high computation costs. In order to solve those problems, mind evolutionary computation (MEC) was proposed by Sun et al. [39]. It is inspired by the process of human mind evolution and inherited architecture and conceptual framework from GA that include group, individual and environment, at the same time, proposed new conceptions such as subgroup, bulletin board, the similar-taxis and the dissimilation.

Finally, in this paper, two historical load data from Queensland and New South Wales respectively were used. In order to evaluate the validity and accuracy of the methods proposed in this paper, two simulation experiment which experimental data from NEM (National Electricity Market) were employed. The NEM, the Australian wholesale electricity market and the associated synchronous electricity transmission grid, began operation on 13 December 1998 and operations are currently based in five interconnected regions—Queensland, New South Wales, Tasmania, Victoria and South Australia [40].

The organization of the rest of this paper is as follows: In Sect. 2 the theory and formula of the pre-process tools are provided. Section 3 introduces the MEC \(+\) WNN model. Section 4 introduces the hybrid model. Simulation results and analysis are provided in Sect. 5. Finally, the conclusions of this paper are given in Sect. 6.

2 Description of the Per-process Tools

All the per-process methods are introduced in this section, including the difference and ensemble empirical mode decomposition.

2.1 The Difference

The difference is used to eliminate correlation by subtracting through item by item. Difference algorithm denoted by backward-shift algorithm B, difference operator \(\nabla \) and the order number d.

d Order difference:

$$\begin{aligned} \nabla ^{d}X_t =( 1 -B )^{d}X_t \end{aligned}$$
(1)

The d order difference operator \(\nabla ^{d}\):

$$\begin{aligned} \nabla ^{d}=( {1-B} )^{d}=1-C_d^1 B+C_d^2 B^{2}+\cdots +({-1} )^{d-1}C_d^{d-1} B^{d-1}+({-1})^{d}B^{d} \end{aligned}$$
(2)

where, \(C_d^k ={d!}/{k!({d-k} )!}\) .

2.2 Ensemble Empirical Mode Decomposition

In order to solve that “one of the main drawbacks of EMD (Empirical Mode Decomposition) is mode mixing [41]”, EEMD (Ensemble Empirical Mode Decomposition) was proposed by Wu and Huang [42].

Process of EEMD as follows:

  1. (1)

    Add the white Gaussian noise to signal x(t),

    $$\begin{aligned} x_i (t)=x(t)+\omega _i (t) \end{aligned}$$
    (3)

    where, \(\omega _i (t)\) is the ith added white noise signal.

  2. (2)

    The series \(x_i (t)\) is decomposed into multiple IMF (Intrinsic mode functions) modes \(IMF_{ij} (t)\) by the standard EMD.

  3. (3)

    Repeat Step (1) and Step (2), add the new white noise sequence each repetition.

  4. (4)

    Calculate the average of all IMFs \(IMF_{ij} (t)\),

    $$\begin{aligned} IMF_j (t)=1/N\sum \nolimits _{i=1}^N {I_{ij} (t)} \end{aligned}$$
    (4)

    where, \(IMF_j (t)\) and N are the first j IMFs and the times of adding white nose, respectively.

In the process of the above, parameters N need to satisfy the following equation:

$$\begin{aligned} \varepsilon _n =\varepsilon /{\sqrt{N}} \end{aligned}$$
(5)

where, \(\varepsilon \) is the range of the white noise sequence and \(\varepsilon _n\) is the standard deviation between origin signal and the final result.

The non-noise signal \(\hat{{x}}(t)\) can be obtained by:

$$\begin{aligned} \hat{{x}}(t)=\sum _{j=1}^m {IMF_j (t)} +R_m (t) \end{aligned}$$
(6)

where, m is the number of \(IMF_j (t)\)s and \(R_m (t)\) is the residue.

3 Proposed MEC \(+\) WNN Based on Rolling Forecast

This section introduces the explicit theory of the hybrid model wavelet neural network optimized by mind evolutionary computation based on rolling forecast. Three parameters the best weights, shift factor and scalability factor are optimized by the mind evolutionary computation. Section 3.1 states the theory of the wavelet neural network. The mind evolutionary computation is introduced in Sect. 3.2. The hybrid model MEC \(+\) WNN based on rolling forecast is presented in Sect. 3.3.

3.1 Mind Evolutionary Computation

Mind evolutionary computation (MEC) is inspired by the process of human mind evolution. It inherited architecture and conceptual framework of GA including group, individual and environment. At the same time, new conceptions such as the subgroup, the bulletin board, the similar-taxis and the dissimilation were proposed.

The population of MEC consist of several groups surviving around the environment. Those groups are divided into superior groups and temporary groups according to their evolutionary action during an evolution. Each group owns a local billboard and a set of individuals. Every individual is given a score. The score is the main information to guide the evolution. In order to trace the local and global competition, MEC provides two kinds of billboards (a local one and a global one) to record the evolutionary information drawn by the knowledge abstractor [43].

The details of the MEC are shown as follows [43]:

  • Step 1 Initialization of individuals

    Generate a certain scale individual randomly within the solution space. Then, according to the scores, seek out some superior individuals with the highest scores and temporary individuals.

  • Step 2 Initialization of groups

    Get some superior groups and temporary groups by generating some new individuals with every superior individual and temporary individual as the center, respectively.

  • Step 3 Similar-taxis and local competition

    Within each subgroup perform similar-taxis operation until the subgroup is mature. Then calculate each individual scoring, and set the subgroups score equal to the superior individual score.

  • Step 4 Dissimilation and global competition

    After every subgroup matured, publish each subgroup score on the global bulletin board. Then perform dissimilation operation between subgroups to complete the process of replacing and abandoning between superior groups and temporary groups and the process of releasing some individuals.

  • Step 5 If meet the termination conditions, the global superior individual and its score are obtained. If not, generate new subgroups under the condition of that the number of temporary subgroups is constant and perform the Step 3.

3.2 Wavelet Neural Network

Wavelet neural network (WNN) is a combination of wavelet analysis and neural network. It follows the topology of BP neural network including the forward signal propagation and error back propagation. However, the activation function of WNN is the mother-wavelet function (MWF) but not sigmoid function.

We assume that \(x_1 ,x_2 ,\ldots ,x_k \) and \(y_1 ,y_2 ,\ldots ,y_m\) are input parameters and output parameters of WNN, respectively; \(\omega _{ij}\) and \(\omega _{jk} \) are weights of the interconnections.

The output of the hidden layer (second layer) can be obtained by:

$$\begin{aligned} h(j)=h_j \left( {{\left( {\sum _{i=1}^k {\omega _{ij} x_i -b_j } } \right) }\Big /{a_j }} \right) ,j=1,2,\ldots ,l \end{aligned}$$
(7)

where, h(j) is the jth neuron of hidden layer, k and l are the number of the input layer (first layer) and the hidden layer neurons, \(\omega _{ij}\) is the connection weights of the first layer and second layer, \(b_j\) and \(a_j \) are the shift factor and the contraction-expansion factor, respectively, \(h_j \) is the mother-wavelet function.

The output of the output layer (third layer) can be obtained by:

$$\begin{aligned} y(k)=\sum _{i=1}^l {\omega _{ik} h(i ),k=1,2,\ldots ,m} , \end{aligned}$$
(8)

where, \(\omega _{jk} \) is the connection weights of second layer and third layer, m is the number of the output layer neurons.

Gradient correction method is employed for updating the parameters of mother-wavelet function and the weights of interconnections.

3.3 WNN Optimized by MEC

Based on rolling forecast, this paper came up with using the MEC to select best weights, shift factor and scalability factor of wavelet neural network. A flowchart of MEC for parameters selection of WNN based on rolling forecast is shown in Fig. 1. Neural network learning result is influenced by initial value of parameters. To select the best network weights, shift factor and scalability factor are of great service to a successful wavelet neural network. Using MEC, employed the fitness function instead of gradient descent method, can obtain the global optimal solution even on polymorphism and discontinuous function. Additionally, rolling forecast can improve accuracy of the forecasting by combine characteristics of long-term data and of recent data, which is emphasized by system.

Fig. 1
figure 1

Flowchart of MEC \(+\) WNN based on rolling forecast

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The details of the MEC \(+\) WNN based on rolling forecast are shown as follows:

  • Step 1 Input time series \(X=({x_1 ,} x_2 ,\ldots ,{x_{n-1} } )\) and determine the training set and testing set.

  • Step 2 Determine the structure of the wavelet neural network including number of nodes in each layer.

  • Step 3 According to the structure of WNN, the training input vector and output vector are obtained by rolling mechanism. Assuming that t is the input layer of the WNN and the output layer is one. Thus, the last t (\(t<n-1)\) load data are used to forecast the next one load data, and \((n-1)-t\) training samples are obtained, which shown in the Table 1, \(\hat{{x}}_k\) is the predicted value of \(x_k\) by WNN.

  • Step 4 Optimize weights, shift factor and scalability factor of wavelet neural network. The specific process of optimized is described below

    1. (1)

      Initialization. Map the solution space to coding space, of which every code corresponding to a solution (individual) of the problem is formed by weights, shift factor and scalability factor.

    2. (2)

      Fitness evaluation. For each subgroup, calculate and record the fitness function values, and then evaluate its fitness. In this paper, the fitness function is defined as the mean square error

    3. (3)

      Update weights, shift factor and scalability factor according to fitness evaluation results.

    4. (4)

      Execute similar-taxis and dissimilation task, and then update all subgroups.

    5. (5)

      Termination. Circulating until the stop criterion is satisfied and outputting the best individual.

  • Step 5 Training wavelet neural network.

  • Step 6 Testing the trained network. Set the forecast horizon h. obtained the testing input vector and output vector by rolling mechanism in \(h<t\) and \(h\ge t\), shown in the Tables 2 and 3, respectively.

  • Step 7 Output the predicted results and calculate the accuracy.

Table 1 Training input vector and output vector obtained by rolling mechanism
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Table 2 The testing input vector and out vector obtained by rolling mechanism in \(h<t\)
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Table 3 The testing input vector and output vector obtained by rolling mechanism in \(h\ge t\)
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4 Proposed the Hybrid Model Combined the Minimum Cycle Decomposition for Load Forecasting

The purpose of this section is to describe a hybrid model combined the minimum cycle decomposition (HMM model) for load forecasting.

First of all, a concept named the minimum cycle decomposition (MCD) was proposed. According to the periodicity of time series, MCD puts forward that when the original series are large enough, the original series are divided into more subseries and each subsequence fitting a sub-model, respectively. Secondly, we should take some data processing methods to separate unsteady property from the series to make time series become stable. From the statistical sense, a time series are a sequence of numbers arranged according to chronological order. One of the important characteristics of time series is periodicity which is used to accurately describe the time sequence, and predicted its development trend. And then, most time series are un-stationary. Therefore, in this paper one order difference was used to remove the nonstationary factors. Besides, some uncertain factors also can introduce noises, which lead to inaccurate results. So it requires special treatment in electric load forecasting. EEMD, which has many advantages discussed in Sect. 2.2, was used to address the problem. Above all, the WNN \(+\) MAC model was used on the process data series to acquire the final forecasting results after data processing.

The diagrammatic figure of HMM model proposed in this paper is shown in Fig. 1. the hybrid algorithm is described in more details as follow.

Part 1 The minimum cycle decomposition

Time series \(A=\{ {a_1 ,a_2 ,\ldots ,a_s ,a_{s+1} ,\ldots a_{ns} } \}\) presents the similarity made over s time intervals, so the sequence has the cycle characteristics and the minimum cycle is s. Reshape A to \({A}'=({X_1^T },X_2^T ,\ldots ,{X_S^T})\), which is a matrix with cycle s as column and cycle point as row, where subsequence \(X_i =({a_i ,a_{s+i} ,a_{2s+i} ,\ldots ,a_{(n-1)s+i}})\).

Fig. 2
figure 2

Diagrammatic figure of the HMM model proposed in this paper

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Part 2 Fitting Sub-model

Sub-model of each subsequence \(X_i ({i=1,2,\ldots ,s} )\) pertaining to \({A}'=({X_1^T },X_2^T ,\ldots , {X_S^T } )\) is established, respectively. The details of the processing of every sub-model are shown as follows:

Step 1 Data pre-process, on the one hand, can make the sequence’s characteristic more obvious helping to choose the appropriate model; on the other hand, is also to meet the requirement of the model. According to Fig. 2, firstly, get the first order differential operator \(X_i^1\), by one order difference with \(X_i^0\), which god by rewriting \(X_i\). Secondly, in order to obtain the noiseless signal \(X_i^2 \) to employ EEMD with \(X_i^1\) to get m IMFs \(b_1 ,b_2 ,\ldots ,b_m \) and one residue \(R_m \).

Step 2 Fitting MEC \(+\) WNN model for the noiseless signal \(X_i^2\) to forecast the result \({\textit{predict}}\_{\textit{output}}\), whose horizon is h.

Step 3 Obtain \(X_i \)’s prediction result \(Y_i \) by inverse one order difference.

Part 3 Combination of the final results

Obtain the time series \({A}'\)’s prediction results \(\hat{{Y}}=( {Y_1^T },Y_2^T ,\ldots ,{Y_S^T })\) which is also the A’s prediction results by combining all of \(Y_i \).

5 Example and Results Analysis

The main purpose of this section is to simulate two experiments of electricity load forecasting using the proposed method.

5.1 Evaluation Measurements of the Algorithm Performance

Forecasting accuracy is closely following that of forecasting error. The greater the error, the accuracy is low, on the contrary, the smaller the error, accuracy is high. To evaluate the performance of the hybrid model proposed in this paper, four criterions are used, including the relative absolute error, the mean absolute percentage error (MAPE), the mean absolute error (MAE), the mean square error (MSE) and grey relation analysis (GRA).

Relative absolute error at point i, MAPE, MAE and MSE can be given by:

$$\begin{aligned} Rae(i)= & {} \left| {{x_i -\hat{{x}}_i }/{x_i }} \right| \end{aligned}$$
(9)
$$\begin{aligned} MAPE= & {} \frac{1}{n}\sum _{i=1}^n {\left| {{x_i -\hat{{x}}_i }/{x_i }} \right| } \times 100\%\end{aligned}$$
(10)
$$\begin{aligned} MAE= & {} {1}/n\sum _{i=1}^n {\left| {x_i -\hat{{x}}_i } \right| }\end{aligned}$$
(11)
$$\begin{aligned} MSE= & {} {1}/n\sum _{i=1}^n {({x_i -\hat{{x}}_i } )^{2}} \end{aligned}$$
(12)

GRA [44] is a method to analysis the relation among system factors proposed by grey system theory. This method is used to compare the fitting degree between active curve and forecast curves of different prediction models. When the curve’s relation degree is big, the performance of the corresponding prediction model is good and the fitting error is little.

Suppose you have a reference sequence \(y_0 \) and m forecast sequences \(y_i (i=1,2,\ldots ,m)\),

$$\begin{aligned}&y_0 =\left\{ {y_0 (1),y_0 (2),\ldots ,y_0 (n)} \right\} \end{aligned}$$
(13)
$$\begin{aligned}&y_i =\left\{ {y_i (1),y_i (2),\ldots ,y_i (n)} \right\} \end{aligned}$$
(14)

Then the GRA \(\xi _i (k )\) of \(y_0 \) and \(y_i \) is defined as follow:

$$\begin{aligned} \xi _i (k)=\frac{\mathop {\min }\limits _i \mathop {\min }\limits _k \left| {y_0 (k)-y_i (k)} \right| +\rho \mathop {\max }\limits _i \mathop {\max }\limits _k \left| {y_0 (k)-y_i (k)} \right| }{\left| {y_0 (k)-y_i (k)} \right| +\rho \mathop {\max }\limits _i \mathop {\max }\limits _k \left| {y_0 (k)-y_i (k)} \right| } \end{aligned}$$
(15)

where k is an integer in terms of n; resolution coefficient \(\rho \) is a number between 0 and 1, usually set \(\rho =0.5\).

Relation degree \(r_i \) of the curve \(y_i \) and the reference curve \(y_0\) is defined as:

$$\begin{aligned} r_i =\hbox {1}/n\sum _{k=1}^n {\xi _i (k)} \end{aligned}$$
(16)

5.2 Comparison Algorithm

To evaluate the performance of the HMM model proposed in this paper, five forecasting models (BP-1, BP-2, WNN, DEEPLSSVM and DEESGRNN) were adopted to realize the series of electricity load data predictions.

To compare results and to test the effectiveness of MCD proposed in this paper, two comparison BP models, BP-1 and BP-2, were employed. The reason is that wavelet neural network, which adopted the concept and framework of BP neural network, is the combination and interpenetrative of wavelet analysis and neural network. BP-1 divided the original series into seven groups according to the day of week (the second minimum cycle). These seven groups were Monday, Tuesday, and Wednesday and so on. BP-2 divided the original series into 48 subseries by half hour of day according to MCD and each subsequence fitted a BP Neural Network respectively. To compared the performance between BP and WNN in practice and to evaluate the effectually of the data pre-processing and the effectiveness of MEC, the WNNM forecasting model, whose data and structure were the same as the BP-2, was also employed.

The above three models do not data preprocessing. For the method BP-1, we used 3 days’ data to forecast the next one day’s data. That means the BP-1 contains 144(48*3) input nodes and 48 output nodes. For the method BP-2, the structure is set to 7-9-1, and prediction horizon is set to 7. In addition, the iteration number of two BP methods is 50; the learning rater parameter is 0.1; and the resolution level is 0.0004. And, the structure of every subsequence’s WNNM model is set to 7-9-1, and prediction horizon is set to 7. The filter type of every WNN is mother-wavelet function Morlet; the learning rater parameter of shift factor and scalability factor is 0.0001, and of weight is 0.001; the iteration number is 50.

To evaluate the effectually of the hybrid model HMM, two another hybrid method referred to as DEEPLSSVM and DEESGRNN were used. For DEEPLSSVM and DEESGRNN models, data grouping and preprocessing were the same as the HMM model.

DEEPLSSVM model combines the LSSVM \(+\) PSO with the MCD, the EEMD and one order difference. Every LSSVM (Least Squares Support Vector Machine) models consisting of 7 inputs, 7 outputs and the RBF (the radial basis function) kernel function was adopted for the kernel type of LSSVM. This model uses PSO for two parameters optimization, regularization parameter c and kernel parameter g. The fitness function of PSO is defined as the MAPE.

DEESGRNN model combines the GRNN \(+\) SA with the MCD, the EEMD and one order difference. Every GRNN (Generalized Regression Neural Network) model consisting of an inputting layer, a pattern layer, a summation layer and an outputting layer was adopted. Set the input layer nodes to be 7 and the number of output layer nodes to be 7. The SA (simulated annealing algorithm) with the default parameter and 30 annealing chain length was used to find the optimal smoothing parameter.

5.3 The Simulation Data

The electric load data in QLD Electricity Market and NSW Electricity Market was employed to evaluate the performances of the proposed model. A total of 77376 history load observations of the QLD electricity markets Australian and the NSW electricity markets Australian, respectively, were collected every half hour of 1612 days from 1st January 2011 to 31st May 2015. There are 48 electricity load data in 1 day. The training sat was started in 1st January 2011 and ended in 24th May, a total of 77040 data of 1605 days. 336 electricity load data from 25th May 2015 to 31st May 2015 was the test set.

5.4 The Simulation and Results Analysis of the Two Series

The forecasting horizon is weekly. Set the origin series cycle \(s=48\). Minimum sampling period of electric load data is 48. Meanwhile, on the same half hour of the day in several different days, people’s lives and production are very similar. According to MCD, training data decomposed into 48 subsequences, each of which denotes the same point sample of different days. In other words, the data have been divided into the different half hour of the day. 48 sub-model corresponding to 48 subseries needed to be simulated. Set the prediction interval of each model \(h=7\). In this way, 336(48*7) prediction results are obtained from 48 sub-models.

Then the structure of WNN is set to 7-9-1. Seven input layer nodes are seven power load points before the predicted point and one output layer nodes is the predicted point. In addition, the filter type of every WNN is mother-wavelet function Morlet; the learning rater parameter of shift factor and scalability factor is 0.0001, and of weights is 0.001; the iteration number is 50.

When MEC is used to optimize the parameters of WNN, the individual code word length is 90 including 63(9*7) weights of input layer and hidden layer, 9(1*9) weights of hidden layer and output layer, 9 shift factor and 9 scalability factor. In addition, the iteration number is 50; the maximum number of individual of each group \({\textit{popsize}}\) is 200; the number of superior group \({\textit{bestsize}}\) is 5, and of temporary group \({\textit{tempsize}}\) is 5; and the number of individual of every subgroup can be obtained by \(SG={popsize}/{({bestsize+temsize})}\).

Fig. 3
figure 3

The line chart of the subseries original data one and the first-order difference operator for QLD Electricity Market started in 1st January 2011 and ended in 24th May, a total of 77040 data of 1605 days

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Fig. 4
figure 4

The IMFs and residue of the subseries one’ first-order difference operator generated in noise eliminating process by EEMD for QLD Electricity Market started in 1st January 2011 and ended in 24th May, a total of 77040 data of 1605 days

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Fig. 5
figure 5

The line chart of the subseries original data one and the first-order difference operator for NSW Electricity Market started in 1st January 2011 and ended in 24th May, a total of 77040 data of 1605 days

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Fig. 6
figure 6

The IMFs and residue of the subseries one’ first-order difference operator generated in noise eliminating process by EEMD for NSW Electricity Market started in 1st January 2011 and ended in 24th May, a total of 77040 data of 1605 days

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Before training each sub-model, the training sequence required preprocessing. Firstly, Figs. 3 and 5 are the line chart of the subseries original data and the first-order difference operator of two real cases, QLD Electricity Market and NSW Electricity Market. By comparing the original data with the first-order difference operator, it can be observing that the first-order difference operator became more stability. Secondly, the ratio of the standard deviation of the added noise for the EEMD is 0.1 and the ensemble number for the EEMD is 50. Figs. 4 and 6 shown all IMFs and residue of the first-order difference operator generated in noise eliminating process by EEMD. As can be seen from Figs. 4 and 6, the periodic characteristic of first-order difference operator is very obvious. In addition, decomposed signal consists of the high frequency components in the short cycle, the middle frequency components in the middle cycle and the low frequency components in long period, because of that the periodic time series may be based on several years, quarter, month, week and day.

Table 4 Every sub-model’s performance evaluation results of MAPE, MAE, MSE using HMM model for load data from 25th May, 2015 to 31st May, 2015 (336 load data)
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Table 4 and Fig. 7 are every sub-model’ s performance evaluation results of MAPE, MAE, MSE using HMM model for load data from 25th May, 2015 to 31st May, 2015 (336 load data). From Table 4, for QLD Electricity Market, all MAPE are less than 2.1%, among them, the minimum MAPE is 0.42%, the maximum MAPE is 2.06% and the average MAPE is 0.96%; and for NSW Electricity Market, all MAPE are less than 3.30%, among them, the minimum MAPE is 0.53%, the maximum MAPE is 3.03% and the average MAPE is 1.82%. The figures lead us to the conclusion that very satisfactory results and high accuracy are obtained using the HMM model proposed in this paper. In addition, as can be seen from Fig. 7, the two curves shown the fluctuation of every sub-model’ s performance evaluation results for two electricity market. We can clearly find that there is obvious difference between two curves. For QLD Electricity Market, the overall trend is down after rising firs, while, for NSW Electricity Market is first rise after falling. That is to say, for QLD Electricity Market, MAPE increases about from 0:30 to 12:30 and decreases about from 14:30 to 24:00. However, it is interesting that the situation for NSW Electricity Market is just the opposite. Unfortunately, the causes of this phenomenon still need further research.

Fig. 7
figure 7

Every sub-model’ s performance evaluation results of MAPE using HMM model for load data from 25th May, 2015 to 31st May, 2015 (336 load data)

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5.5 Analysis of Comparison

Figures 8 and 9 list the curves of above six model’s results series and one real data series. From Figs. 8 and 9, it is apparent that the curve of HMM model proposed in this paper is closer to real data curve, which means that the HMM model shows best performance compared with other five contrasted models BP-1, BP-2 and WNNM. Furthermore, the experimental result that the performance of three forecasting models, BP-2, WNNM and HMM model, are better than BP-1 model demonstrates the effectiveness of decomposition time-series according to minimum period (period \(=\) 48).

Fig. 8
figure 8

Forecasted electricity load result for QLD: the solid line-black show the actual load signal, the dashed line-blue, dash-dotted line-red lines, solid line-aquamarine and solid line-yellow show the forecasted load signals with four models, respectively. (electricity load: MW). (Color figure online)

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Fig. 9
figure 9

Forecasted electricity load result for NSW: the solid line-black show the actual load signal, the dashed line-blue, dash-dotted line-red lines, solid line-aquamarine and solid line-yellow show the forecasted load signals with four models, respectively. (electricity load: MW). (Color figure online)

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Relative absolute error is used to estimate the performance of the six forecasting models. Figures 10 and 11 have shown the relative absolute error space of four forecasting models in QLD and NSW Electricity Market data, respectively. Error space is a very complex surface in the three-dimensional space on which the height of each point corresponds to a relative absolute error value; the x-axis and y-axis are the half hour of day and the day of week, respectively. From the above two figures, we can draw the following conclusions. Firstly, dividing the series into 48 subseries shows the good performance than dividing the series into seven groups. For short-term load forecasting, the error range [\(-3, +3\%\)] is always considered as a standard to measure forecasting results [45], the range is also used to compare four methods as follows [35]. For QLD Electricity Market shown in Fig. 10, we can easily observe that HMM model just has less than 5% forecasting result points exceed the range in total 336 result points. For DEEPLSSVM and DEESGRNN models, there are about 10% result points larger than 3%. In the WNNM model and BP-2 model, there are about 25% forecasting result points larger than 3%. However, the BP-1 model forecasting errors are almost 80% exceed the range. Secondly, the values of the data pro-process and of the parameter optimization algorithm MEC are indicated by phenomenon that the proposed HMM algorithm has good potential than WNNM model. Thirdly, rolling forecast mechanism may lead to the error transfer and amplification by using forecast results to further forecasting. From the error distribution of three models BP-2, WNNM and HMM model, it can be obviously found that the relative absolute error significantly increased about on the sixth and the seventh day of a week. This phenomenon mainly stems from the fact that these three models are adopted rolling forecast mechanism. In conclusion, the performance of the HMM model is better than other five contrasted models. Similar phenomenon on NSW Electricity Market data have been obtained as shown in Fig. 11, although all error distributions, among which the error of the HMM method is just about 80% in the range, are not as good as on QLD Electricity Market data.

Fig. 10
figure 10

The space of forecasting error distribution with different model for QLD

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Fig. 11
figure 11

The space of forecasting error distribution with different model for NSW

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Fig. 12
figure 12

Bar graph of MAPE, MAE, MSE and R using different model for QLD

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But, when comparing the BP-2 with WNNM from Figs. 8, 9, 10 and 11, it can be clearly seen that the result curves of the two models in the Figs. 8 and 9 are very close, and the error distributions of the two models in the Figs. 10 and 11 are also very similar. In order to compare the performance of four models, the MAPE, MAE, MSE and R (GRA) of the four methods are employed. The MAPE, MAE, MSE and R of each model for QLD Electricity Market load data are shown in the Fig. 12 and Table 5, while, Fig. 13 and Table 6 list that values for NSW Electricity Market load data. We can draw the following conclusions.

Table 5 Performance evaluation results of MAPE, MAE, MSE and R using different model for QLD
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Fig. 13
figure 13

Bar graph of MAPE, MAE, MSE and R using different model for NSW

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Table 6 Performance evaluation results of MAPE, MAE, MSE and R using different model for NSW
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First of all, from Fig. 12 and Table 5, MAPE, MAE, MSE and R of HMM model on QLD load data are 0.97%, 57.01, 5997.73 MW and 0.91, respectively. The lowest values of MAPE, MAE and MSE and the highest value of R indicate that the HMM model has the best forecast accuracy and high curve fitting degree. Those values are seen in WNNM model are 1.78%, 104.50, 1.8867.2 MW and 0.82, respectively. BP-2, by contrast, the MAPE, MAE and MSE increase 0.97%, 46.94 and 12254.31 MW, although R value just reduce 0.014.

And then, those values of BP-1 model are 6.81%, 407.13, 245764 MW and 0.644, respectively. The worst values of all these indicate that BP-1 makes the poorest performance among all models. This phenomenon displays visually that the advantages, that divided the load data into different data type according to half hour of day when forecasting, is verified.

What’s more, the following phenomenon shows that BP neural network and wavelet neural network also can be a good fitting of nonlinear functions, but wavelet neural network have better accuracy and faster convergence speed. It is very interesting that, compare with BP-2 model in this simulation, WNNM model’s R also just increased 0.039, which is very low compare with 0.09(0.06) obtained from HMM model and WNNM for QLD(NSW) load data.

Last but not the least, the model with lower fitting error does not have high relation. When comparing the results of WNNM with DEEPLSSVM, form Fig. 12 and Table 5, it can be seen that the WNNM model has same value of MAPE with GEEPLSSVM and higher value of R. This phenomenon shows that fitting performance of the WNNM model is better than the DEEPLSSVM model when they have the same fitting error, since the relation of WNNM model is higher.

In the same way, from Fig. 13 and Table 6, that values obtained from NSW load data shown follow as: for HMM model are 1.82%, 150.44, 40940.22 MW and 0.88, respectively, for WNNM model those are 2.87%, 240.07, 100994 MW and 0.83, respectively; for BP-2 those are 3.94%, 334.87, 178652.5 MW and 0.79, respectively; and for BP-1 those are 6.17%, 526.01, 449072 MW and 0.69, respectively.

6 Conclusion

In this paper, a concept named the minimum cycle decomposition (MCD) and a new hybrid model combining MEC and WNNM based on rolling forecast and EEMD for electricity load forecasting were proposed. The proposed HMM model consists three parts including the decomposition of load series according to MCD, fitting sub-model by the hybrid model and the combination of final result. To evaluate the performance of the proposed HMM model, five compared models and five evaluation measurements are employed. The proposed forecasting model is simulated on real the QLD and NSW electricity markets. The result shows that the average MAPE of the HMM model is 0.97 and 1.82% which is lower than the existing hybrid model named MFES proposed by Zhao et al. [46]. It had reduced MAPE of MFES by 29.99 and 29.14%. The results of comparison demonstrate the excellent performance of the HMM model. The two experiment results show that the proposed HMM model has great performance than other compared model including accuracy and applicability. To sum up, firstly, wavelet neural network with better stability compared with BP neural network, and with high relation compared with DEEPLSSVM and DEEAGRNN, is more suitable for electricity load forecasting. Secondly, the MCD is very effective for improving load forecasting accuracy. Thirdly, according to these values, the advantages of dividing the load data into different data type by half hour of day to predict are verified. Finally, the wavelet neural network has stability performance than BPNN. In a nutshell, the proposed HMM model outperforms other models. With higher accuracy, it is a promising tool in the future.