A hybrid prediction model for photovoltaic power generation based on information entropy

IET Generation, Transmission & DistributionVolume 15, Issue 3 p. 436-455 ORIGINAL RESEARCH PAPEROpen Access A hybrid prediction model for photovoltaic power generation based on information entropy Shiping Pu, School of Automation, Central South University, Changsha, ChinaSearch for more papers by this authorZhiyong Li, Corresponding Author lizy@mail.csu.edu.cn School of Automation, Central South University, Changsha, China Hunan Xiangjiang Artificial Intelligence Academy, Central South University, Changsha, China Correspondence Zhiyong Li, School of Automation, Central South University, 932 South Lushan Road, Changsha, China. Email: lizy@mail.csu.edu.cnSearch for more papers by this authorHui Wan, School of Automation, Central South University, Changsha, China Hunan Xiangjiang Artificial Intelligence Academy, Central South University, Changsha, ChinaSearch for more papers by this authorYougen Chen, School of Automation, Central South University, Changsha, ChinaSearch for more papers by this author Shiping Pu, School of Automation, Central South University, Changsha, ChinaSearch for more papers by this authorZhiyong Li, Corresponding Author lizy@mail.csu.edu.cn School of Automation, Central South University, Changsha, China Hunan Xiangjiang Artificial Intelligence Academy, Central South University, Changsha, China Correspondence Zhiyong Li, School of Automation, Central South University, 932 South Lushan Road, Changsha, China. Email: lizy@mail.csu.edu.cnSearch for more papers by this authorHui Wan, School of Automation, Central South University, Changsha, China Hunan Xiangjiang Artificial Intelligence Academy, Central South University, Changsha, ChinaSearch for more papers by this authorYougen Chen, School of Automation, Central South University, Changsha, ChinaSearch for more papers by this author First published: 16 December 2020 https://doi.org/10.1049/gtd2.12032Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinked InRedditWechat Abstract Photovoltaic power is affected by various random and coupled meteorological factors, and its changing trend implies the non-linear effects of these factors. According to the quantitative analysis results, a statistical prediction model is proposed to accurately predict the power, which is of great significance to the safe and efficient use of solar energy. In this study, the authors first use grey relation analysis to select four main meteorological factors affecting photovoltaic power. Further, they combine grey relation analysis with information entropy and apply grey relation entropy to similar day analysis. On this basis, they take grey relation analysis to optimise extreme learning machine model to establish the grey relation analysis-extreme learning machine model, while taking similar day analysis to optimise firefly algorithm to establish the similar day analysis-firefly algorithm. By combining the two sub-models with information entropy, a hybrid prediction model for photovoltaic power generation based on information entropy is proposed. The experimental results show that in various weather conditions, the values of mean absolute percentage error, root mean square error and standard deviation of error are 2.8425%, 2.5675 and 2.2642, respectively. Therefore, the proposed hybrid model has superior prediction performance. Nomenclature GRA Grey relation analysis SDA Similar day analysis ELM Extreme learning machine FA Firefly algorithm GRA-ELM Grey relation analysis-extreme learning machine SDA-FA Similar day analysis-firefly algorithm IE Information entropy MAPE Mean absolute percentage error RMSE Root mean square error nRMSE Normalized root mean square error SDE Standard deviation of error MARS Multivariate adaptive regression splines MBE Mean bias error NN Neural network NWP Numerical weather prediction GFS Global Forecasting System MM5 Meso-scale Model 5 BP Back propagation SOM Self-organizing map GCPV Grid-connected photovoltaic plants SARIMA Seasonal auto-regressive integrated moving average method SVM Support vector machine GPR Gaussian process regression ESE EMD-SCA-ELM WPS Wavelet-PSO-SVM GAS Genetic algorithm-based support vector machine DKASC Desert Knowledge Australia Solar Centre 1 INTRODUCTION Nowadays, due to the challenges of climate change and energy crisis, more and more attention is paid to using renewable energy in the world. Solar energy has the unique advantages of rich resources, no air pollution and wide distribution. According to Renewables 2020 Global Status Report, by the end of 2019, the solar photovoltaic global capacity has reached 627 GW, an increase of 22.46% compared with 2018. Nowadays, the output power of large-scale photovoltaic plants has exceeded 500 MW, such as Gujarat Solar Park in India and Topaz Solar Farm in the United States. However, because of the influence of the earth's rotation and revolution, the radiation received by photovoltaic plants presents periodic changes. At the same time, owing to the daily changes of meteorological factors (such as weather temperature, weather relative humidity and wind speed), the output power has great intermittence, volatility and randomness. When a photovoltaic power generation system is incorporated into the power grid, it will harm the reasonable formulation of the power generation plan due to the uncontrollability. Moreover, it will cause an impact on the generating units, reduce the stability of the power system, increase the operation loss, and bring challenges to the security, scheduling and economic aspects of the power system [1]. The improvement of prediction accuracy of photovoltaic power can provide a reliable basis for the optimal scheduling of power system and reduce the impact on energy market management when a large-scale photovoltaic power system is injected into the grid. Furthermore, accurate forecasting can better match the market commitment with the actual photovoltaic production, reduce the energy imbalance cost, improve the utilization of solar energy resources, and increase the return on investment of photovoltaic plants. Therefore, the establishment of a photovoltaic power generation prediction model with high accuracy and stability has become a major issue that has to be broken through. In terms of characteristics and performance, photovoltaic power prediction methods can be classified into physical methods and statistical methods. Physical methods are mainly aimed at solar irradiance prediction and cloud detection. They consist of three sub-models: numerical weather prediction (NWP), sky imagery and satellite-imaging models [2]. NWP models are essentially based on the numerical integration of coupled differential equations describing the radiation transport mechanisms [3]. These models are used widely to predict atmospheric state up to 15-days ahead. Meanwhile, they are biased by less than 50 W/m2 for measured clear conditions [4]. In [5], a short-term forecasting system for hourly electrical energy production is proposed. The system includes a global NWP module corresponding to the Global Forecasting System (GFS) model, a meso-scale NWP module corresponding to the Meso-scale Model 5 model (MM5) and an energy forecasting model based on artificial intelligence. The forecasting horizon ranges from 1 to 39 h, covering all of the following day. With forecasting horizons from 16 to 39 h, the RMSE error represents 11.79% of the rated power of the PV plant. Sky imager is a digital camera that is applied for clouds detection, cloud height measurement above ground, and cloud motion determination. Researchers can use sky imager to obtain sky images and recognize cloud. On the basis of establishing cloud motion vectors, they can predict cloud cover, irradiance and photovoltaic power. In [6], a solar irradiance prediction methodology for all sky conditions using the TSI-880 model is presented. The results indicate the average normalized root mean square error (nRMSE) value of beam irradiance, diffuse irradiance and global irradiance is 25.44% in cloudless skies, 11.6% in partially cloudy skies and 11.17% in overcast skies. Satellite imaging is another model to obtain sky image data. The satellite-based sensors can take visible and infrared images to determine cloud pattern. In [7], a satellite-derived, ground data model is developed, which improves the intra-day solar prediction and neural network (NN) model. The results indicate the model increases the forecast time scales from 15.47% to 22.17% for Co-Pozo Izquierdo station and 25.15–34.09% for C1-Las Palmas station. The disadvantage of physical methods is the need for accurate topographic maps, coordinate maps, photovoltaic array power curves and other relevant data. At the same time, the local anti-interference ability is poor and the overall robustness is weak [8]. The advantage of physical methods is that they do not require a large amount of historical data, save the calculation time, and enhance the prediction ability when the weather conditions change. Statistical prediction is a direct way to forecast the photovoltaic power by considering some key influencing factors. Statistical methods usually analyse historical data to obtain the prediction trend. Further, they take the changing trend as a basis for photovoltaic power generation prediction. The commonly used statistical prediction methods are multi-factor line regression method [9], time series forecasting method [10] and Markov chain method [11]. In [9], a non-linear regression model known as multivariate adaptive regression splines (MARS) is proposed to forecast the solar power of a grid-connected 2.1 kW PV system. The comparative experiments show the MARS model can provide more reliable forecast performance. In [12], the hourly photovoltaic power generation in Chile is estimated by using multiple linear regression models. The results show the RMSE value and mean bias error (MBE) value of the proposed model are reduced by 11% and 10% compared with the comparison model. Statistical prediction also includes artificial intelligence prediction. The commonly used artificial intelligence prediction mainly includes back propagation (BP) neural network, support vector machine (SVM) and artificial neural network [13]. In [14], a photovoltaic prediction model based on artificial neural network is proposed, which predicts the photovoltaic power according to solar radiation intensity, relative air humidity and temperature training self-organizing map (SOM). In [15], a BP neural network model is proposed to forecast the output power of a PV system located in Ashland at 24-hour-ahead. The best performance can be obtained with the BP neural network structure of 28-20-11, while the MAPE error is within 8%. Different from physical methods, statistical methods have no requirements for the geographical location and construction parameters of photovoltaic power stations. Therefore, compared with physical methods, they have the advantages of simple modelling and strong universality. However, a large amount of historical data and meteorological data are needed for model training to ensure the prediction accuracy. Besides, the accuracy of statistical methods depends on the data quality to a large extent. Although data screening and false data elimination can improve the prediction accuracy, they extend the calculation time, so the statistical methods are not suitable for ultra-short-term photovoltaic prediction. Since photovoltaic power generation is affected by a variety of dynamically changing environmental factors, all influencing factors should be scientifically analysed according to actual working conditions. At the same time, a single prediction model cannot meet the requirements, so it is particularly important to develop an accurate hybrid model based on different single models. In [16], a hybrid model to forecast the power produced by small-scale grid-connected photovoltaic plants (GCPV) is introduced, which combines the seasonal auto-regressive integrated moving average method (SARIMA) and SVM. The experimental results indicate that the developed hybrid model performs better than both SARIMA and SVM. In [17], a hybrid short-term prediction model combining improved K-means clustering, GRA and Elman neural network is proposed. The results show that the average daily RMSE, MAPE and R2 of the model are 4.3210 kW, 2.8366% and 0.9953, respectively. In [18], a short-term combination forecasting model for PV power based on similar days and cross entropy theory is proposed. Compared with combination models based on the sum of squared errors and correlation coefficient, the model has lower prediction error and better prediction performance. There is uncertainty in the results of photovoltaic power prediction models, and the degree of distortion is expressed by their internal information entropy. The higher the information entropy, the lower the credibility. Therefore, to reduce the information entropy of the system and improve the prediction accuracy, this work takes GRA to optimise the ELM model to establish GRA-ELM, while taking SDA to optimise the FA model to establish SDA-FA. On this basis, GRA-ELM and SDA-FA are integrated to establish a hybrid prediction model based on information entropy. Compared with existing single prediction models and hybrid prediction models in accuracy and stability, the superiority of the model is clearly verified. The innovations and contributions of this work are as follows: GRA is applied to select four main influencing factors from seven meteorological indicators as the input of ELM, so as to establish the GRA-ELM model; SDA is used to select similar days from historical days by introducing grey correlation entropy, further, the photovoltaic power of similar days is taken as the input of FA to establish the SDA-FA model; A hybrid prediction model for photovoltaic power generation based on information entropy is proposed, and its prediction accuracy and stability are compared, further, its running speed and convergence effect are analysed; The proposed hybrid model is compared with existing single prediction models and hybrid prediction models in terms of MAPE, RMSE and SDE to verify its superior prediction performance [19]. The single prediction models include BP, SVM and Gaussian process regression (GPR), while the hybrid prediction models include EMD-SCA-ELM (ESE) [20], wavelet-PSO-SVM (WPS) [21] and genetic algorithm-based support vector machine (GAS). The rest of the work is as follows: Sections 2 and 3 introduce the construction of GRA-ELM model and SDA-FA model, respectively. A hybrid prediction model based on information entropy is proposed in Chapter 4. Chapter 5 tests the hybrid prediction model and makes a comparative analysis. Chapter 6 summarizes the whole paper and puts forward research objectives of the next stage. 2 GRA-ELM 2.1 Meteorological factors sorting based on GRA As the photovoltaic power generation prediction is a time-delay, non-linear, multi-parameter process, affected by weather temperature, weather relative humidity, wind speed and other factors, it is difficult to establish an accurate mathematical model. Therefore, it is necessary to use GRA to analyse and rank multiple meteorological factors according to the historical data, thus establishing the basis for the subsequent photovoltaic prediction models. In GRA, the major and minor factors leading to the development of the system are determined by analysing the relationship between factors. For the photovoltaic prediction, this work takes the photovoltaic power as the reference data column and uses X0 to represent it. X 0 = [ x 0 ( 1 ) , x 0 ( 2 ) , x 0 ( 3 ) , … , x 0 ( j ) … , x 0 ( k ) ] T where k represents the selected historical data quantity. The larger the historical data quantity, the smaller the random error, and the more accurate the analysis results are. The photovoltaic power prediction results are affected by the random disturbance of many factors in the natural environment. We chose Desert Knowledge Australia Solar Centre (DKASC) as the experimental power station and analysed the meteorological data from 04.01.2016 to 04.01.2018. Take the data from 04.01.2016 to 10.31.2017 as training samples, while taking the data from 11.01.2017 to 04.01.2018 as test samples [22]. The research period of the recorded data is 08:00–17:00. The output power and meteorological indicators are measured every 5 min. Since sensor failure will cause errors in the recorded data, pre-processing must be done to exclude all error values in the original data to ensure the reliability of the analysis results. To describe the characteristics of the natural environment in more detail, this work selects seven indicators affecting the photovoltaic power generation as the influence factor array (expressed by X1, X2, X3, X4, X5, X6 and X7, respectively). The seven indicators are clearly explained in Table 1. In GRA, with the continuous change of weather conditions, the coupling relationship between the photovoltaic power and meteorological factors will also change. Therefore, the sorting results of meteorological factors need to be refreshed dynamically every 30 min. In this work, to highlight the effectiveness of GRA, we select a rainy day (09.30.2017 14:00–14:25, Rainy) with strong randomness and unpredictability as the training sample to describe the specific process of GRA. The values of seven indicators are shown in Table 2. TABLE 1. Seven indicators and values Definition Data range Average value Unit X0 Photovoltaic power [−1.42,229.10] 47.31 kW X1 Weather temperature [1.08,51.26] 22.21 ℃ X2 Weather relative humidity [0.01,99.99] 37.90 % X3 Global horizontal radiation [0.29,2015.18] 260.89 W/m2 X4 Diffuse horizontal radiation [0.36,707.58] 53.93 W/m2 X5 Weather daily rainfall [0.00,32.60] 0.34 mm X6 Wind speed [0.31,36.94] 6.24 m/s X7 Wind direction [0.41,353.92] 236.26 degrees TABLE 2. Values of meteorological factors (09.30.2017 Rainy) Meteorological factors Time X0 X1 X2 X3 X4 X5 X6 X7 14:00 21.99 13.36 91.19 113.80 111.56 6.99 53.07 334.32 14:05 26.04 13.41 95.27 137.54 133.73 7.19 40.51 294.82 14:10 24.71 13.42 94.21 130.98 128.27 7.19 34.94 277.63 14:15 31.94 13.42 91.52 163.99 159.06 7.19 35.96 286.45 14:20 46.28 13.57 91.05 240.32 234.29 7.19 37.40 288.90 14:25 47.58 13.81 90.91 260.89 254.84 7.19 41.82 302.11 In GRA, it is difficult to directly compare and get accurate results due to the different dimensions of meteorological factors. Therefore, to ensure the unity in quantity, the above data must be dimensionless: x ∗ i j = x i j 1 k ∑ j = 1 k x i j , i = 0 , 1 , … , m , j = 1 , 2 , … , k , (1)where m represents the number of meteorological factors and takes 7, and k takes 6. The dimensionless matrix is obtained in Table 3. TABLE 3. Dimensionless matrix of Meteorological factors Meteorological factors (Asterisk represents dimensionless data) X 0 ∗ X 1 ∗ X 2 ∗ X 3 ∗ X 4 ∗ X 5 ∗ X 6 ∗ X 7 ∗ 0.6646 0.9898 0.9873 0.6518 0.6551 0.9767 1.3066 1.1242 0.7869 0.9935 1.0315 0.7878 0.7853 1.0047 0.9974 0.9914 0.7468 0.9942 1.0200 0.7502 0.7532 1.0047 0.8602 0.9336 0.9652 0.9942 0.9909 0.9393 0.9340 1.0047 0.8854 0.9633 1.3986 1.0053 0.9858 1.3765 1.3758 1.0047 0.9208 0.9715 1.4379 1.0231 0.9843 1.4943 1.4965 1.0047 1.0296 1.0159 After the dimensionless reference data column X 0 ∗ and dimensionless meteorological factor data column X i ∗ are determined, the correlation coefficient between meteorological factors and photovoltaic power is defined as ξ i j = min i min j x 0 ∗ j − x i ∗ j + ρ max i max j x 0 ∗ j − x i ∗ j x 0 ∗ j − x i ∗ j + ρ max i max j x 0 ∗ j − x i ∗ j , (2)where ρ is the resolution coefficient and takes value in [0,1]. The correlation coefficient matrix is shown in Table 4. TABLE 4. Correlation coefficient matrix of meteorological factors ξ1 ξ2 ξ3 ξ4 ξ5 ξ6 ξ7 0.4981 0.5001 0.9643 0.9740 0.5085 0.3343 0.4124 0.6101 0.5691 1.0000 0.9978 0.5974 0.6056 0.6126 0.5663 0.5417 0.9923 0.9832 0.5561 0.7410 0.6339 0.9197 0.9285 0.9279 0.9140 0.8929 0.8031 0.9969 0.4507 0.4387 0.9382 0.9363 0.4503 0.4030 0.4303 0.4375 0.4156 0.8529 0.8480 0.4268 0.4414 0.4332 Finally, the meteorological factors are analysed according to Table 4, and the average value of correlation coefficients is calculated and defined as correlation degree r i = 1 k ∑ j = 1 k ξ i j , (3)where r i ∈ ( 0 , 1 ) , it indicates that any meteorological factor is not strictly independent of photovoltaic power, nor is it the only determining factor. If l , j ∈ { 1 , 2 , … , m } and r l ≥ r j , it means that the correlation of X l is stronger than X j . Similarly, select 09.17.2017 (10:30–10:55, Sunny) and 10.20.2017 (11:00–11:25, Cloudy) as training samples. The correlation degrees of all meteorological factors are summarized and shown in Table 5. The correlation degree between the system reference column and each meteorological factor column is sorted in descending order ( X 1 > X 2 > X 7 > X 3 > X 6 > X 4 > X 5 , Sunny) ( X 3 > X 1 > X 6 > X 7 > X 2 > X 4 > X 5 , Cloudy) ( X 3 > X 4 > X 7 > X 1 > X 5 > X 2 > X 6 , Rainy). The results are as follows: The four meteorological factors mainly affecting photovoltaic power on sunny days are weather temperature, weather relative humidity, wind direction and global horizontal radiation; on cloudy days, they are global horizontal radiation, weather temperature, wind speed and wind direction; on rainy days, they are global horizontal radiation, diffuse horizontal radiation, wind direction and weather temperature. TABLE 5. Correlation degree of meteorological factors Training samples X1 X2 X3 X4 X5 X6 X7 09.17.2017 10:30–10:55(Sunny) 0.9895 0.9381 0.7924 0.7454 0.3385 0.7860 0.8584 10.20.2017 11:00–11:25(Cloudy) 0.9372 0.9190 0.9418 0.8798 0.3365 0.9370 0.9358 09.30.2017 14:00–14:25(Rainy) 0.5804 0.5656 0.9460 0.9422 0.5720 0.5547 0.5865 Comprehensive correlation degree 0.8876 0.8520 0.8611 0.8200 0.3892 0.7707 0.8169 When we cannot judge the weather type of the forecast day through the weather forecast, in order to reasonably evaluate the impact of each meteorological factor on photovoltaic power, this work introduces p1, p2 and p3 as the preference coefficients of sunny days, cloudy days and rainy days, respectively, and assumes ∑ i = 1 3 p i = 1 . On this basis, the comprehensive correlation degree is defined as c j = ∑ i = 1 3 p i X j i , (4)where j = 1 , 2 , … , 7 , X j i represents the correlation degree of meteorological factor j in weather condition i. According to the recorded data from 04.01.2016 to 04.01.2018, we found that in 730 samples, 399 samples are sunny days, accounting for 54.66%; 171 samples are cloudy days, accounting for 23.42%; 160 samples are rainy days, accounting for 21.92%. Therefore, p1, p2 and p3 can be set to 0.5466, 0.2342 and 0.2192, respectively. Calculate the comprehensive correlation degree between each meteorological factor and photovoltaic power as shown in Table 5. Sort them in descending order ( X 1 > X 3 > X 2 > X 4 > X 7 > X 6 > X 5 ). We can conclude that according to the recorded data from 04.01.2016 to 04.01.2018, weather temperature, global horizontal radiation, weather relative humidity and diffuse horizontal radiation can be selected as the four main meteorological factors. According to the calculation results in Table 5, the correlation degree histograms are drawn in Figure 1. In summary, we can analyse two prediction scenarios separately. (1) If we have accurately judged the weather type of the forecast day (such as sunny, cloudy or rainy), we can select the corresponding main meteorological factors; (2) If we cannot judge the weather type of the forecast day, we can use the comprehensive correlation degree for photovoltaic forecasting, which is slightly less accurate than the first prediction scenario. It can be seen that GRA can provide a reliable basis for the selection of meteorological information. FIGURE 1Open in figure viewerPowerPoint Correlation degree histograms of meteorological factors 2.2 Construction of GRA-ELM model In this chapter, ELM machine learning algorithm is combined with GRA to predict photovoltaic power. Take the four main meteorological factors in GRA as the input of ELM, to establish GRA-ELM model. ELM is a single hidden layer feedforward neural network learning algorithm. The traditional neural network learning algorithm (such as BP neural network algorithm) needs a lot of parameters in the training model, and it is easy to fall into the local optimal solution. By contrast, ELM only needs to set the number of neurons in the hidden layer, and the algorithm does not need to iterate the hidden layer. The output weight is determined by calculating the generalized inverse of the output matrix of the hidden layer. ELM not only guarantees the good generalization performance of the network, but also greatly improves the learning speed of the feedforward neural network. It has the advantages of high training speed, and can effectively overcome the inherent defects of BP neural network and learning vector quantization neural network. There are seven meteorological factors affecting the photovoltaic power generation, and each factor has a certain coupling with the output power. For different prediction time, the coupling strength of each meteorological factor is different. Take the photovoltaic power before the prediction time as the training sample and use GRA to select four main influencing factors (such as weather temperature, global horizontal radiation, weather relative humidity and diffuse horizontal radiation). Further, take the above factors as the input layer node of ELM and the photovoltaic power as the output layer node, thereby establishing GRA-ELM photovoltaic power prediction model as shown in Figure 2. FIGURE 2Open in figure viewerPowerPoint GRA-ELM photovoltaic power prediction model GRA-ELM needs a large number of samples to non-dimensionalize them. The samples are classified into training samples and test samples. Suppose the training samples are [ x i , y i ] , the number of neurons in the hidden layer is k, and the infinitely differentiable excitation function is g ( x ) ( g ( x ) can be sigmoid, sine or radial basis function). The output model can be calculated as follows: O j = ∑ i = 1 k β i g W i X j + b i , (5)where X j represents the input matrix, which is composed of main meteorological factors (weather temperature X1, global horizontal radiation X2, weather relative humidity X3 and diffuse horizontal radiation X4); O j represents the output matrix, which is the predicted value of photovoltaic power; b represents the bias matrix of hidden layer neurons, b = [ b 1 , b 2 , · · · , b N ] . N is the number of training samples. In (5), W represents the connection weight matrix between the input layer and the hidden layer W = w i j N × k = w 11 w 12 … w 1 k w 21 w 22 … w 2 k ⋮ ⋮ ⋱ ⋮ w N 1 w N 2 … w N k . (6) In (5), β represents the connection weight matrix between the hidden layer and the output layer β = β i j N × k = β 11 β 12 … β 1 k β 21 β 22 … β 2 k ⋮ ⋮ ⋱ ⋮ β N 1 β N 2 … β N k . (7) The purpose of training GRA-ELM model is to make the prediction error close to 0, that is lim N → ∞ ∥ ∑ i = 1 N β i g W i X j + b i − T j ∥ = 0 , (8)where T j represents the target value of GRA-ELM, which is the actual value of photovoltaic power. In other words, there are W i , b i and β i satisfying (9) and (10) ∑ i = 1 N β i g W i X j + b i = T j , (9) H β = T , (10)where H = g w 1 x 1 + d 1 g w 2 x 1 + d 2 … g w k x 1 + d k g w 1 x 2 + d 1 g w 2 x 2 + d 2 … g w k x 2 + d k ⋮ ⋮ ⋱ ⋮ g w 1 x N + d 1 g w 2 x N + d 2 … g w k x N + d k . (11) According to (10), the training of a single hidden layer neural network is transformed into calculating the least square norm solution of the weight matrix as follows: β ̂ = H + T (12)where H + is the Moore–Penrose generalized inverse matrix of H. To study the influence of the hidden layer excitation function on the prediction accuracy of the model, considering the availability and validity of the data, this work selects the data from 09.30.2017 to 10.31.2017 as the t