# Solar GHI Ensemble Prediction Based on a Meteorological Model and Method Kalman Filter

Authors: Yuanyuan Liu
Journal: Advances in Meteorology
 (2022)

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## Abstract

The intensity of light emanating from sun is determined by using a meteorological version and is altered with the numerical version, and the forecast accuracy is improved in advance by using Kalman Filter. As the accuracy of the version output related to its specific position is often questionable, group prediction constituting three members is suggested and agreed upon measurement. Also, this ensemble prediction provides an estimation of the solar global horizontal irradiance uncertainty (i.e., coverage rate of the prediction interval), which can be useful to provide flexible energy production forecasts. This article displays how the method Kalman filter could be used as an error correction way to alter the predicted irradiance value. The Kalman filter ameliorates the prediction of solar global horizontal irradiance as well as its interval. As the empirical coverage rate increases and closes to the nominal coverage rate, the interval size reduces.


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## Body

## 1. Introduction

Having been installed and connected to power grids, a large amount of photovoltaic (PV) systems with their prognostication in power generation or the prediction of the solar irradiance have been used in handling the electric grids more efficiently to make the energy provision steadier. With the help of dependable and conforming prediction data [1–9], system operators ameliorate the electrical management to balance the users and power generation [10–12]. Also, with the development of the forecasting models, support is offered in an efficient way for electricity suppliers in managing, dispatching, and planning power plants [13]. By applying a meteorological model, the authors have advanced a system in ensemble prediction of solar global horizontal irradiance [14], and not only the irradiance but also its prediction interval has been forecasted [15–17]. The interval corresponds to the reliability of the forecasting and will be also useful for power grid management. With the use of model WRF (the Weather Research and Forecasting), Liu et al. proposed solar global horizontal irradiance and verified its forecasting accuracy [14]; the simulated WRF irradiance included RMSEs, which imply root mean square errors [18–26] as well as positive biases. In order to improve solar global horizontal irradiance forecasting accuracy, the meteorological calculation used in a post-\processing technique is introduced [27–30]; and, the result in the amelioration on the forecasting of the prediction interval is also discussed in this work [31–34].Different approaches for processing in post-period, with examples of MOS (model output statistics) [35–39] and Kalman filter, are put into use in weather forecast in a frequent way for the sake of modifying the anticipatory output in gaining possession by NWP models which stand for numerical weather prediction. In 1995, Homleid depicted a model Kaman Filter to estimate temperature forecasting corrections in day time [34]. In 2011, De Carvalho et al. [35] applied the model Kalman filter for correcting the mean temperature errors which are assessed by the model PRECIS, a system developed by the English Hadley Centre used for climate predictions in regional area. In 2008, Monache et al. [36] estimated systematic errors in surface ozone forecasts. Galanis et al. in 2006 [37] as well as Cassola and Burlando in 2012 [38] put the method Kalman filtering procedure into removing bias in forecasting for wind speed. Pelland et al. in 2013 [39] as well as Diagne et al. in 2014 [40] used the model Kalman filter to remove bias in solar irradiance prediction. In 2018, Rincón with his teammates [41] made a combination with model Kalman filter and MOS for bias correction for the sake of improving the entire predicted values in solar irradiance. The creation of correction function method on linear regression traditionally depends mainly on MOS. According to Kalman [42] and Homleid [34], the theory of the Kalman filter offers equations in modifying the assessments in a recursive way through observing.The probabilistic forecasting of solar global horizontal irradiance can have an important impact on the management of the power grid with many PV systems and its risk analysis. For this case, the method Kalman filter is used as a posttreatment technique to rectify the system flaws of prediction of solar gleam and, on the other hand, influences the development on anticipating the gauge interval employing a group anticipating approach also inspected.


## 2. Meteorological Model and Solar Global Horizontal Irradiance Forecasting

### 2.1. Model WRF Configuration

As is shown previously in the research work [14], the performance with the WRF model, which means Weather Research and Forecasting by Skamarock et al. 2008 [43], was supervised from July 2013 to June 2014 for the prediction of inexhaustible energy resources, for example photovoltaics or wind power generations [44–47]. The model WRF is also applied in an extensive way, for the reason that it can be used to simulate the weather in reality with the technique of high resolution. The calculation back drop of meteorology with WRF are summed up in Table 1.Table 1
Configuration of the WRF model.
Input dataJMA GSM-JP (3-hourly, 0.2° × 0.25°) NCEP GFS (3-hourly, 0.5° × 0.5°)Vertical layer50 levels (surface to 100 hPa)Nesting2-way nestingDomainDomain 1, D01 (18 km, 120 × 110 grids), domain 2, D02 (6 km, 130 × 130 grids), and domain 3, D03 (2 km, 160 × 229 grids)Physical optionsMellor-Yamada-Janjic PBL parameterization Betts-Miller-Janjic cumulus parameterization WSM 6-class graupel scheme Dudhia short wave radiation scheme RRTM long wave radiation scheme (domain 1 and domain 2 only) Noah land surface modelThe simulation was driven based on the analysis of data in a global scope for the GSM-JP standing for area of Global Spectral Model for Japan. The spatial resolution of the analyses was 0.2° × 0.25°, while the temporal resolution was 3 h. The WRF exemplary composition constitute three consecutive installed realms with 18 kilometers of grid spacing, which is the grid in coarser style showed in domain 1 [48]. Also, the 6 kilometers of grid spacing will be showed in domain 2 and 2 kilometers of grid spacing is in domain 3, which is the best grid, accompanied with 50 vertical levels. Estimates which correspond to the most inside domain grid points (domain 3) enclosing the stations of experimental radiometric were put into use in the assessment.Physical parameterizations in a wide range are enclosed in the model WRF, allowing setting the WRF model to depict the physical processes in a better way which are based on resolution, model domain, application, and location. Under such circumstances, various problems were chosen to pursue. Shimada et al. [49] supervised an assessment study of the achievement of distinct problems of the exemplary in the identical researching field.Thus, the model WRF was functioned with the MYJ planetary boundary layer scheme [50], the Betts-Miller-Janjic cumulus scheme [51], WSM 6 microphysical scheme [52], the Noah land surface model, the scheme of Dudhia for shortwave radiation scheme [53], and the RRTM scheme for long-wave radiation scheme [54]. With regard to every day, a 3-day (72 h) ahead prediction was fulfilled, which started at 12 h Universal Coordinated Time (UTC).


### 2.2. Study Area and Observed Data

The central region of Japan is the study field for the solar global horizontal irradiance prediction; the research area includes Aichi, Nagano, Mie, Shizuoka, and Gifu Prefectures. Results of the present research have been assessed with the observation data which are collected at 61 radiometric observation stations displayed in Figure1. The main reasons why these stations are chosen is because they are located near to regions where the solar power plants are running or under construction. The solar global horizontal irradiance (GHI) data were collected by EKO pyranometer MS–402 in these stations. In this research, to get data in 30 minutes, measurements in 10 seconds were averaged. Moreover, since researches were operated in one field where the commercial electric power grid locates, the GHI averaged data for GHI from all 61 stations of Figure 1 are used for the assessment. The data at night, which refer to time period from sunset in the previous day to sunrise in the intraday time, are excluded from the assessment [55–57].Figure 1
Study region and distribution of the radiometric stations with global irradiance measurements (dark stars).


### 2.3. Forecasted Results with WRF

With the method Kalman filter as a posttreatment technique of solar global horizontal irradiance, calculation by model WRF is correlated with the checked information. As is explained before, the solar GHI averaged values over 61 stations in the Figure1 and with 30-minute frequency is used for the checking. The data at night are not used for the evaluation.Figure1 shows correlation charts between the forecasted solar global horizontal irradiance and the observation from the beginning in July 2013 to the end of June in 2014. Such statistical error indexes as biases, root mean square errors, and correlation coefficients, which are abbreviated as Bias, RMSE, and CORR respectively, are listed in Table 2. The WRF computes 72-h ahead forecasting in one operation, which includes the intraday Mi,0, the next-day Mi,1 and the 2-day ahead Mi,2 prediction. Subscript i indicates operation day of the forecasting. The labels “Mi,j” only refers to the simulated GHI with model WRF, and “Mi,j−KF” refers to the result which is adjusted with method Kalman filter. The outcomes are studied separately for Mi,0, Mi,1, and Mi,2 forecasting in this figure and this table. For the same target day, the ensemble average value is computed as a mean of the members, which consist of Mi,0, Mi−1,1, and Mi−2,2, standing for the very day forecasting in the operation day i.Table 2
Statistical indexes for solar GHI different forecasts.
Forecasting method and memberBias [W/m2](Relative bias)RMSE [W/m2](Relative RMSE)CORR [−]Mi,049.7(28.8%)102.9(59.6%)0.950Mi,0−KF−2.4(−1.4%)79.1(45.8%)0.951Mi,151.4(29.7%)118.9(68.9%)0.929Mi,1−KF−6.0(-3.5%)94.6(54.7%)0.930Mi,250.8(29.4%)129.8(75.2%)0.911Mi,2−KF−8.5(−4.9%)106.6(61.7)0.911Ensemble mean50.6(29.3%)104.9(60.7%)0.946Ensemble mean−KF−5.6(−3.2%)81.8(47.4%)0.947Persistent0.3(0.15%)191.7(110.9%)0.718


## 2.1. Model WRF Configuration

As is shown previously in the research work [14], the performance with the WRF model, which means Weather Research and Forecasting by Skamarock et al. 2008 [43], was supervised from July 2013 to June 2014 for the prediction of inexhaustible energy resources, for example photovoltaics or wind power generations [44–47]. The model WRF is also applied in an extensive way, for the reason that it can be used to simulate the weather in reality with the technique of high resolution. The calculation back drop of meteorology with WRF are summed up in Table 1.Table 1
Configuration of the WRF model.
Input dataJMA GSM-JP (3-hourly, 0.2° × 0.25°) NCEP GFS (3-hourly, 0.5° × 0.5°)Vertical layer50 levels (surface to 100 hPa)Nesting2-way nestingDomainDomain 1, D01 (18 km, 120 × 110 grids), domain 2, D02 (6 km, 130 × 130 grids), and domain 3, D03 (2 km, 160 × 229 grids)Physical optionsMellor-Yamada-Janjic PBL parameterization Betts-Miller-Janjic cumulus parameterization WSM 6-class graupel scheme Dudhia short wave radiation scheme RRTM long wave radiation scheme (domain 1 and domain 2 only) Noah land surface modelThe simulation was driven based on the analysis of data in a global scope for the GSM-JP standing for area of Global Spectral Model for Japan. The spatial resolution of the analyses was 0.2° × 0.25°, while the temporal resolution was 3 h. The WRF exemplary composition constitute three consecutive installed realms with 18 kilometers of grid spacing, which is the grid in coarser style showed in domain 1 [48]. Also, the 6 kilometers of grid spacing will be showed in domain 2 and 2 kilometers of grid spacing is in domain 3, which is the best grid, accompanied with 50 vertical levels. Estimates which correspond to the most inside domain grid points (domain 3) enclosing the stations of experimental radiometric were put into use in the assessment.Physical parameterizations in a wide range are enclosed in the model WRF, allowing setting the WRF model to depict the physical processes in a better way which are based on resolution, model domain, application, and location. Under such circumstances, various problems were chosen to pursue. Shimada et al. [49] supervised an assessment study of the achievement of distinct problems of the exemplary in the identical researching field.Thus, the model WRF was functioned with the MYJ planetary boundary layer scheme [50], the Betts-Miller-Janjic cumulus scheme [51], WSM 6 microphysical scheme [52], the Noah land surface model, the scheme of Dudhia for shortwave radiation scheme [53], and the RRTM scheme for long-wave radiation scheme [54]. With regard to every day, a 3-day (72 h) ahead prediction was fulfilled, which started at 12 h Universal Coordinated Time (UTC).


## 2.2. Study Area and Observed Data

The central region of Japan is the study field for the solar global horizontal irradiance prediction; the research area includes Aichi, Nagano, Mie, Shizuoka, and Gifu Prefectures. Results of the present research have been assessed with the observation data which are collected at 61 radiometric observation stations displayed in Figure1. The main reasons why these stations are chosen is because they are located near to regions where the solar power plants are running or under construction. The solar global horizontal irradiance (GHI) data were collected by EKO pyranometer MS–402 in these stations. In this research, to get data in 30 minutes, measurements in 10 seconds were averaged. Moreover, since researches were operated in one field where the commercial electric power grid locates, the GHI averaged data for GHI from all 61 stations of Figure 1 are used for the assessment. The data at night, which refer to time period from sunset in the previous day to sunrise in the intraday time, are excluded from the assessment [55–57].Figure 1
Study region and distribution of the radiometric stations with global irradiance measurements (dark stars).


## 2.3. Forecasted Results with WRF

With the method Kalman filter as a posttreatment technique of solar global horizontal irradiance, calculation by model WRF is correlated with the checked information. As is explained before, the solar GHI averaged values over 61 stations in the Figure1 and with 30-minute frequency is used for the checking. The data at night are not used for the evaluation.Figure1 shows correlation charts between the forecasted solar global horizontal irradiance and the observation from the beginning in July 2013 to the end of June in 2014. Such statistical error indexes as biases, root mean square errors, and correlation coefficients, which are abbreviated as Bias, RMSE, and CORR respectively, are listed in Table 2. The WRF computes 72-h ahead forecasting in one operation, which includes the intraday Mi,0, the next-day Mi,1 and the 2-day ahead Mi,2 prediction. Subscript i indicates operation day of the forecasting. The labels “Mi,j” only refers to the simulated GHI with model WRF, and “Mi,j−KF” refers to the result which is adjusted with method Kalman filter. The outcomes are studied separately for Mi,0, Mi,1, and Mi,2 forecasting in this figure and this table. For the same target day, the ensemble average value is computed as a mean of the members, which consist of Mi,0, Mi−1,1, and Mi−2,2, standing for the very day forecasting in the operation day i.Table 2
Statistical indexes for solar GHI different forecasts.
Forecasting method and memberBias [W/m2](Relative bias)RMSE [W/m2](Relative RMSE)CORR [−]Mi,049.7(28.8%)102.9(59.6%)0.950Mi,0−KF−2.4(−1.4%)79.1(45.8%)0.951Mi,151.4(29.7%)118.9(68.9%)0.929Mi,1−KF−6.0(-3.5%)94.6(54.7%)0.930Mi,250.8(29.4%)129.8(75.2%)0.911Mi,2−KF−8.5(−4.9%)106.6(61.7)0.911Ensemble mean50.6(29.3%)104.9(60.7%)0.946Ensemble mean−KF−5.6(−3.2%)81.8(47.4%)0.947Persistent0.3(0.15%)191.7(110.9%)0.718


## 3. Kalman Filter for Improvement of WRF Forecasting

### 3.1. Method Kalman Filter

Designed to collect a signal in an efficient way from data in noisy environment that consists of statistical noise and other inexact ones, Kalman filter method is therefore expected to show a more exact outcome by using training data [40, 58, 59]. This approach is ordinarily employed to rectify systems flaws of weather forecast further and has been ratified as one of establishment approach of meteorology forecast supervision; such meteorological parameters as amount of precipitation, temperature, and probability of precipitation are studied based on this method, according to Kunitsugu [60]. The application of Kalman filter in the meteorological field is different from other fields. The correction coefficient of the observation and prediction error of solar irradiance are obtained, not physical quantities. Compared with other methods, Kalman filter is not so complex and is suitable for the solar GHI forecasting corrections. Meanwhile, the Kalman filter has not been widely applied in this field before. Also, the results of Kalman filter could be realized by MATLAB efficiently and conveniently.By and large, previous values before the timek have an effect on the arbitrary instantaneous k time value to some extent. The method Kalman filter follows a trail of the previous data as the state of the system and renews it with the observation value. The Kalman filter method usually contains observation equations and state equations.(1)zk=Hxk+vk,(2)xk=Axk−1+wk.On the basis of state equation, equation (1), it is assumed in Kalman filter that the state at time k is generated from the previous one at time k−1. In equation (2), xk refers to a state variable, A refers to a state transition matrix, and wk refers to a state transition noise. On the basis of observation equation, equation (1), the observation data zk is computed by utilizing the state variable xk. H refers to a state-to-observation matrix and vk means the noisy data in the observation. It is assumed that noise plays a key role in method Kalman filter. The two white noises wk and vk are compliant with Gaussian zero mean.Figure2 displays the calculation flow in Kalman filter algorithm which is described as equations, referring to Kim and Huh [61]. The first step is to set the initial values, which include the error covariance matrix P0, an internal variable initial value, and evaluated state variable x^0. Then, the estimated state variable x^k at time k is evaluated and updated with the observation data zk by following the algorithm.Figure 2
Algorithm of Kalman filter.In this case, we make use of the method Kalman filter to correct the unfairness of the predicted solar global horizontal irradiance in accordance with the system flaws of prediction. Predicting the future bias by using a linear relationship between the estimated previous bias present and the forecasting error, Kalman filter is such a mathematical approach [49]. After the estimation of current bias, to ameliorate the forecasting, it is detached from the solar global horizontal irradiance which is forecasted with model WRF. It is assumed that, the simulated solar global horizontal irradiance with model Kalman filter is assessed from forecasted GHI based upon model WRF according to the following equation:(3)I^WRF=aIWRF+b,where a and b are correction coefficients. On this assumption, the state variable is defined as xk=abT. Also, the observation equations and the state corresponding to equations (1) and (2) can be inferred in the following equations.(4)zk=Hxk+vk,=IWRF1abk+vk,(5)xk=Axk−1+wk,=1001abk−1+wk,where zk=I^WRF. With regard to the function of Kalman filter, vk in the equation (4) and its relevant covariance matrix R, and wk in the equation (5) and its relevant covariance matrix Q in the equations Figure 2 are essential impact factors. To perform different WRF simulations, various matrices R and Q are input to gain their appropriate values. Figure 3 displays the solar GHI’s RMSE during the intraday forecasting under different matrices R and Q, which reduce the predicted RMSE. According to Figure 3, the adequate rate between the impact factors R and Q could be found out. From this result, we gain that R is equal to 1×1011⋅5 and Q is equal to 1×1001001.Figure 3
RMSE of the solar GHI intraday forecasting under the different values of covariance matricesR and Q.The time series of the correction coefficientsa and b in the equation (3) is displayed in Figure 4. Compared with the solar GHI’s order about 100 or 1,000 W/m2, b can be ignored. From this figure, b increases over time, and a varies by season. In the filter, the two coefficients vary with time, working to regulate the solar global horizontal irradiance which is predicted with Model WRF.Figure 4
Time series of correction coefficients in the method Kalman filter for the intraday forecasting during the period from July 2013 to July 2014.


### 3.2. Forecasting Results with WRF and Kalman Filter

Figure5 shows correlation diagrams of the observed GHI and the forecasted data. Ensemble means shown in Figures 5(g) and 5(h) are irradiances averaged in the same target day. Also, the persistent model as a reference one is displayed in Figure 5(i). Table 2 enlists scientific mistakes indices of weather prediction. Both of the results in CORRs and RMSEs based on model WRF forecasts are superior to those which are acquired by the persistent model. The validity of the forecasting with model WRF are uncovered in this result. But the bias value of model WRF forecasts is larger than that of the persistent model, which is zero in theory.Figure 5
Correlation diagrams of the observed GHI and the forecasted data for the intraday prediction, next-day prediction, 2-day ahead prediction, and ensemble mean with the meteorological model WRF only (Mi,0, Mi,1, Mi,2, and ensemble mean), with WRF and posttreatment Kalman filter (Mi,0−KF, Mi,1−KF, Mi,2−KF, and ensemble mean−KF), and the persistent prediction. (a) Mi,0 prediction, (b) Mi,0−KF prediction, (c) Mi,1 prediction, (d) Mi,1−KF prediction, (e) Mi,2 prediction, (f) Mi,2−KF prediction, (g) ensemble mean, (h) ensemble mean −KF, and (i) persistent.
(a)(b)(c)(d)(e)(f)(g)(h)(i)Biases ofMi,0, Mi,1 and Mi,2 forecasting by using model WRF are 49.7 W/m2, 51.4 W/m2, and 50.8 W/m2, respectively. According to these outcomes, a notable positive bias is found in the solar GHI forecasting with model WRF, which signifies that the solar GHI with WRF model is overestimated [62–65].As found in this figure and this table, the bias values are decreased clearly by utilizing the method Kalman filter; for instance, with the help of method Kalman filter, the bias ofMi,0 forecasting turns from 49.7 W/m2 to −2.4 W/m2. By comparing with the solar irradiance mean intensity, the biases could be practically negligible. With the use of method Kalman filter, the two indices RMSEs and biases decrease while CORRs raise as is shown in Table 2. Those relative RMSEs and biases in this table are normalized with their mean values. These relative indices are also improved by applying Kalman filter. The function of the method Kalman filter to the forecast precision is proved with the amelioration of these indices.The correction coefficienta of the Kalman filter in Figure 4 is nearly 0.88, and smaller than unity. The method Kalman filter decreases the solar global horizontal irradiance intensity which is predicted with model WRF and rectifies the overestimation of model WRF output with the value a.The intraday forecasting of daily operation is associated and stated in the form of time series in Figure6. The observed solar global horizontal irradiance is also plotted in this figure. The irradiance which is forecasted with model WRF seems a little greater than the measured one, but after the application of Kalman filter to the WRF’s results, the evaluated irradiance reduces and approaches the observation data. The method Kalman filter has an effect on the amelioration of the solar global horizontal irradiance forecast shown in this figure.Figure 6
Results of intraday prediction of the solar GHI (global horizontal irradiance) by using WRF onlyMi,0, based on WRF and method Kalman filter Mi,0−KF and observed one during the period from July 11 to 20 in 2013.


## 3.1. Method Kalman Filter

Designed to collect a signal in an efficient way from data in noisy environment that consists of statistical noise and other inexact ones, Kalman filter method is therefore expected to show a more exact outcome by using training data [40, 58, 59]. This approach is ordinarily employed to rectify systems flaws of weather forecast further and has been ratified as one of establishment approach of meteorology forecast supervision; such meteorological parameters as amount of precipitation, temperature, and probability of precipitation are studied based on this method, according to Kunitsugu [60]. The application of Kalman filter in the meteorological field is different from other fields. The correction coefficient of the observation and prediction error of solar irradiance are obtained, not physical quantities. Compared with other methods, Kalman filter is not so complex and is suitable for the solar GHI forecasting corrections. Meanwhile, the Kalman filter has not been widely applied in this field before. Also, the results of Kalman filter could be realized by MATLAB efficiently and conveniently.By and large, previous values before the timek have an effect on the arbitrary instantaneous k time value to some extent. The method Kalman filter follows a trail of the previous data as the state of the system and renews it with the observation value. The Kalman filter method usually contains observation equations and state equations.(1)zk=Hxk+vk,(2)xk=Axk−1+wk.On the basis of state equation, equation (1), it is assumed in Kalman filter that the state at time k is generated from the previous one at time k−1. In equation (2), xk refers to a state variable, A refers to a state transition matrix, and wk refers to a state transition noise. On the basis of observation equation, equation (1), the observation data zk is computed by utilizing the state variable xk. H refers to a state-to-observation matrix and vk means the noisy data in the observation. It is assumed that noise plays a key role in method Kalman filter. The two white noises wk and vk are compliant with Gaussian zero mean.Figure2 displays the calculation flow in Kalman filter algorithm which is described as equations, referring to Kim and Huh [61]. The first step is to set the initial values, which include the error covariance matrix P0, an internal variable initial value, and evaluated state variable x^0. Then, the estimated state variable x^k at time k is evaluated and updated with the observation data zk by following the algorithm.Figure 2
Algorithm of Kalman filter.In this case, we make use of the method Kalman filter to correct the unfairness of the predicted solar global horizontal irradiance in accordance with the system flaws of prediction. Predicting the future bias by using a linear relationship between the estimated previous bias present and the forecasting error, Kalman filter is such a mathematical approach [49]. After the estimation of current bias, to ameliorate the forecasting, it is detached from the solar global horizontal irradiance which is forecasted with model WRF. It is assumed that, the simulated solar global horizontal irradiance with model Kalman filter is assessed from forecasted GHI based upon model WRF according to the following equation:(3)I^WRF=aIWRF+b,where a and b are correction coefficients. On this assumption, the state variable is defined as xk=abT. Also, the observation equations and the state corresponding to equations (1) and (2) can be inferred in the following equations.(4)zk=Hxk+vk,=IWRF1abk+vk,(5)xk=Axk−1+wk,=1001abk−1+wk,where zk=I^WRF. With regard to the function of Kalman filter, vk in the equation (4) and its relevant covariance matrix R, and wk in the equation (5) and its relevant covariance matrix Q in the equations Figure 2 are essential impact factors. To perform different WRF simulations, various matrices R and Q are input to gain their appropriate values. Figure 3 displays the solar GHI’s RMSE during the intraday forecasting under different matrices R and Q, which reduce the predicted RMSE. According to Figure 3, the adequate rate between the impact factors R and Q could be found out. From this result, we gain that R is equal to 1×1011⋅5 and Q is equal to 1×1001001.Figure 3
RMSE of the solar GHI intraday forecasting under the different values of covariance matricesR and Q.The time series of the correction coefficientsa and b in the equation (3) is displayed in Figure 4. Compared with the solar GHI’s order about 100 or 1,000 W/m2, b can be ignored. From this figure, b increases over time, and a varies by season. In the filter, the two coefficients vary with time, working to regulate the solar global horizontal irradiance which is predicted with Model WRF.Figure 4
Time series of correction coefficients in the method Kalman filter for the intraday forecasting during the period from July 2013 to July 2014.


## 3.2. Forecasting Results with WRF and Kalman Filter

Figure5 shows correlation diagrams of the observed GHI and the forecasted data. Ensemble means shown in Figures 5(g) and 5(h) are irradiances averaged in the same target day. Also, the persistent model as a reference one is displayed in Figure 5(i). Table 2 enlists scientific mistakes indices of weather prediction. Both of the results in CORRs and RMSEs based on model WRF forecasts are superior to those which are acquired by the persistent model. The validity of the forecasting with model WRF are uncovered in this result. But the bias value of model WRF forecasts is larger than that of the persistent model, which is zero in theory.Figure 5
Correlation diagrams of the observed GHI and the forecasted data for the intraday prediction, next-day prediction, 2-day ahead prediction, and ensemble mean with the meteorological model WRF only (Mi,0, Mi,1, Mi,2, and ensemble mean), with WRF and posttreatment Kalman filter (Mi,0−KF, Mi,1−KF, Mi,2−KF, and ensemble mean−KF), and the persistent prediction. (a) Mi,0 prediction, (b) Mi,0−KF prediction, (c) Mi,1 prediction, (d) Mi,1−KF prediction, (e) Mi,2 prediction, (f) Mi,2−KF prediction, (g) ensemble mean, (h) ensemble mean −KF, and (i) persistent.
(a)(b)(c)(d)(e)(f)(g)(h)(i)Biases ofMi,0, Mi,1 and Mi,2 forecasting by using model WRF are 49.7 W/m2, 51.4 W/m2, and 50.8 W/m2, respectively. According to these outcomes, a notable positive bias is found in the solar GHI forecasting with model WRF, which signifies that the solar GHI with WRF model is overestimated [62–65].As found in this figure and this table, the bias values are decreased clearly by utilizing the method Kalman filter; for instance, with the help of method Kalman filter, the bias ofMi,0 forecasting turns from 49.7 W/m2 to −2.4 W/m2. By comparing with the solar irradiance mean intensity, the biases could be practically negligible. With the use of method Kalman filter, the two indices RMSEs and biases decrease while CORRs raise as is shown in Table 2. Those relative RMSEs and biases in this table are normalized with their mean values. These relative indices are also improved by applying Kalman filter. The function of the method Kalman filter to the forecast precision is proved with the amelioration of these indices.The correction coefficienta of the Kalman filter in Figure 4 is nearly 0.88, and smaller than unity. The method Kalman filter decreases the solar global horizontal irradiance intensity which is predicted with model WRF and rectifies the overestimation of model WRF output with the value a.The intraday forecasting of daily operation is associated and stated in the form of time series in Figure6. The observed solar global horizontal irradiance is also plotted in this figure. The irradiance which is forecasted with model WRF seems a little greater than the measured one, but after the application of Kalman filter to the WRF’s results, the evaluated irradiance reduces and approaches the observation data. The method Kalman filter has an effect on the amelioration of the solar global horizontal irradiance forecast shown in this figure.Figure 6
Results of intraday prediction of the solar GHI (global horizontal irradiance) by using WRF onlyMi,0, based on WRF and method Kalman filter Mi,0−KF and observed one during the period from July 11 to 20 in 2013.


## 4. Solar Irradiance Ensemble Prediction

### 4.1. Ensemble Prediction Method and Forecasting Conditions

Everyday group prediction of solar global horizontal irradiance is controlled with WRF, similar to previous work proposed by Liu et al. [14], with the exception of the application of method Kalman filter, which is a postprocessor for the WRF model. We use similar calculations and circumstances as the latter case in order to contrast [66–71]. The detailed process of the ensemble prediction is stated in the previous research [14]; therefore, we give only the outlines here [72].The lagged averaged forecast (LAF) which is made up of members with distinct original situation and time is used for group prediction. Figure7 shows the chart of the cycle of operation for the ensemble prediction of solar GHI. Mi,j−KF indicate the members of the ensemble prediction based on WRF output and Kalman filter, in which i refers to the running day and amounts to the day of the termination of calculation, and j represents the scope of the forecasting day. Hence, j=0 refers to the result of the intraday; j=1 refers to the result of the next-day; and j=2 refers to the results of the 2-day ahead forecasting from the running day. As the figure displayed, GHI of the intraday forecasting in the running day i is predicted based on three members: Mi,0−KF, Mi−1,1−KF, and Mi−2,2−KF. GHI of the next-day forecasting in the running day i is predicted based on two members: Mi,1−KF and Mi−1,2−KF. GHI of the 2-day ahead forecasting in the running day i is predicted based on one member: Mi,2−KF.Figure 7
Cycle of operation for the solar GHI ensemble prediction based upon model WRF and method Kalman filter.The spread of the forecasting members is also defined with the following equation:(6)Spreadi,j=1N−j∑k=0N−j−1GHIi−k,j+k−GHI¯2,where j equals to 0, 1, and 2; N represents the number of forecasting day in one running, and it is equal to 3 in this work; N−j means the member number of ensemble prediction; GHI¯ is the averaged GHI value of the ensemble members; and GHIi−k,j+k refers to the predicted GHI value of the member Mi−k,j+k. This spread serves as the evaluation factors of uncertainty of the forecasting [14].


### 4.2. Solar Global Horizontal Irradiance Ensemble Prediction

The certainty of the outcome pointed out for solar global horizontal irradiance prediction is examined in Figure5 and Table 2. It is found that Bias, RMSE, and CORR, the statistical error indices of Mi,0 forecasting, are the optimal in those figure and table. In general, in the ensemble prediction method, the ensemble average is selected as the predicted value, brief noted as ensemble mean−KF in this research. Nevertheless, Mi,0−KF, which represents the intraday forecasting value based on WRF and Kalman filter, is used as the predicted value in this work, for the reason that the precision of the intraday forecasting is optimum.


### 4.3. Evaluation Index of Ensemble Prediction

Figure8 displays the link in between the distribution within a day prediction and its prediction flaws from the examination throughout the time from September 1 to November 30 2013. Also, the top and bottom peripheries in the prediction interval whose coverage rates are 50 percent, 80 percent, 90 percent, and 95 percent respectively are linked with lines in this chart. These lines are assessed during per 10 W/m2 in the target spread on the one hand and, on the other hand, are assessed negative and positive errors, respectively. The plotted data gather around the original point of the chart. Meanwhile, the forecasting error values deviate in a wide way with the target spread getting large as figured with the periphery lines of the extents.Figure 8
Relationship chart between the ensemble spread and solar GHI’s forecasting error for the intraday prediction by using model WRF onlyMi,0, based on WRF and method Kalman filter Mi,0−KF during the period from September 1 to November 30 in 2013.The periphery lines in the prediction extents are coarse, for the reason that three is the limitation for the ensemble member with the target spread containing random error, which results in the limitation range of the computing environment in this project. The increasement of ensemble members makes contribution to the amelioration of the interval evaluation [73–76].The distribution of scatter diagrams and the periphery lines ofMi,0, which is the calculated outcomes with model WRF, deflects to the positive side, because WRF has the tendency to overestimate the solar global horizontal irradiance as explained before. The pure WRF forecasting is symmetric with respect to the horizontal line 30 W/m2. On the other hand, the plots and the periphery lines, with an exception of the lines whose coverage rate is 95 percent, distribute in a symmetrical way relative to the axis of abscissas. This is the result of method Kalman filter, which corrects the overestimating calculation of model WRF.


### 4.4. Solar Global Horizontal Irradiance Forecasting with Prediction Interval

Every day solar global horizontal irradiance ensemble predictioning is performed, and the solar lustrous and its prognosis interval are forethought. Figure9 shows the irradiance forecasted with WRF only, and Figure 10 displays the irradiance with model WRF and method Kalman filter. In these charts, the calculated outcomes of the intraday forecasting data are linked up and shown during the time series between December 1 and December 10 2013. By respectively using black full lines and red dot lines in these charts, the data of the predicted irradiance and the data of observed ones are plotted. As a result of the overestimation of model WRF, the outcome of irradiance which is predicted with model WRF is greater than the observed data in Figure 9. On the other hand, the forecasting results with WRF and Kalman filter is suited to the observed data as indicated in Figure 10.Figure 9
Time series of solar GHI intraday prediction by using model WRF onlyMi,0 and its 50%, 80%, 90%, and 95% prediction interval during the period from December 1 to 10 in 2013. (a) Mi,0 prediction and ensemble spread, (b) Mi,0 prediction and 50% prediction interval, (c) Mi,0 prediction and 80% prediction interval, (d) Mi,0 prediction and 90% prediction interval, and (e) Mi,0 prediction and 95% prediction interval.
(a)(b)(c)(d)(e)Figure 10
Time series of solar GHI intraday predictionMi,0−KF based on model WRF and method Kalman filter, and its 50%, 80%, 90%, and 95% prediction interval during the period from December 1 to 10 in 2013. (a) Mi,0−KF prediction and ensemble spread, (b) Mi,0−KF prediction and 50% prediction interval, (c) Mi,0−KF prediction and 80% prediction interval, (d) Mi,0−KF prediction and 90% prediction interval, and (e) Mi,0−KF prediction and 95% prediction interval.
(a)(b)(c)(d)(e)The ensemble spread of the members with model WRF and based on model WRF and method Kalman filter is also displayed by using dashed lines in green in Figures9(a) and 10(a). The target spread value alters with time; for instance, the target spread on December 6 is not small, while the target spread on December 3 is close to zero in these charts. As the ensemble spread becomes smaller, the predicted GHI values come closer to the observed one.The top and bottom peripheries in the prediction interval are displayed by using two dashed lines in blue which situate from Figures9(b)–9(e) and from Figures 10(b)–10(e), whose coverage rates are respectively 50 percent, 80 percent, 90 percent, and 95 percent. For the sake of evaluating the peripheries of the predicted intervals, the top and bottom peripheries in the prediction interval in Figure 8(a) or 8(b) are assessed based on the given rates of nominal coverage and the target spread in Figures 9(a) or 10(a) at first. And, defined as the prediction interval, they are drawn based on the forecasted irradiance. In these charts, accompanied with the decrease of target spread, the interval size, which can be found from the data on December third, is small. On the contrary, the interval size becomes large when the target spread increases, just like the data on December 6. As is displayed in these charts, the rate of nominal coverage gets large compared with the interval size in proportional way.Table3 displays the rate of daily empirical coverage, which refers to the ratio of the actual data number of observed data fallen in the estimated interval, to the total data number within the effective sunshine time, for the intraday solar GHI ensemble prediction during the period from December 1 to 10 in 2013. Mi,0 represents the predicted result based on model WRF, and Mi,0−KF represents the predicted result based on model WRF and method Kalman filter in this table. It is revealed that the rate of empirical coverage is in proportion to the rate of nominal coverage, which is greater than the rate of daily empirical coverage of Mi,0 and approached by the rate of daily empirical coverage of Mi,0−KF.Table 3
Rates of empirical coverage for the intraday solar GHI prediction with regard to the prediction interval of the ensemble prediction during the period from December 1 to 10th 2013.
Rate of nominal coverage for prediction interval (%)Forecasting method and memberRate of empirical coverage for prediction interval [%]12/0112/0212/0312/0412/0512/0612/0712/0812/0912/10Ave50Mi,031.610.531.673.726.326.321.126.336.863.234.7Mi,0−KF57.952.642.157.957.947.463.157.936.868.454.280Mi,063.226.352.689.536.857.936.836.873.794.756.8Mi,0−KF73.778.978.989.584.268.484.289.563.194.780.590Mi,073.742.157.994.747.457.957.947.489.594.766.3Mi,0−KF84.278.994.794.794.789.594.794.778.994.790.095Mi,084.247.463.294.752.694.778.947.489.594.774.7Mi,0−KF94.784.294.794.794.794.794.794.789.594.793.2During the period from December 1 to 10 2013, Table4 shows the daily prediction interval sizes for the intraday solar GHI ensemble prediction. For the size computation, the data at night are refused. It is revealed that the sizes get in proportion to the rates of nominal coverage in this table. The sizes of Mi,0−KF forecasting is smaller than the ones of Mi,0 forecasting.Table 4
Prediction interval sizes for the intraday solar GHI prediction with regard to the prediction interval of the ensemble prediction during the period from December 1 to 10 in 2013.
Rate of nominal coverage for prediction interval (%)Prediction method and memberPrediction interval size [W/m2]12/0112/0212/0312/0412/0512/0612/0712/0812/0912/10Ave50Mi,079.965.465.4112.965.4120.982.965.490.8104.785.4Mi,0−KF76.755.055.0107.954.9109.778.354.991.698.978.380Mi,0155.6111.8111.8231.9111.8232.1164.5111.7196.7208.0163.6Mi,0−KF147.299.699.6212.299.5196.9155.899.4188.4190.6148.990Mi,0217.0144.5144.5316.4144.4304.6233.3144.3287.1284.4222.1Mi,0−KF208.6127.1126.9328.0126.9290.3223.1126.8272.8279.0210.995Mi,0274.8171.9171.9427.6171.8408.2320.7171.8381.3378.4287.9Mi,0−KF266.8163.5163.3431.1163.3398.9288.6163.2328.8338.1270.6To improve the WRF calculation, solar global horizontal irradiance via Kalman filter is used. Applying it to the ensemble prediction brings about the size of the prognosis interval gradually little and the factual analysis quota rises and closes to the nominal one. It signifies that the amelioration of the forecasting provides a better assessment for prediction interval.Furthermore, monthly empirical coverage rates and prediction interval sizes from October 2013 to June 2014 are illustrated in Figures11 and 12. With the application of Kalman filter in improving the solar GHI forecasting, the prediction interval sizes get small, while the rates of empirical coverage add and become near the nominal rates in the long period. It means that the improvement in the forecasting works well for the improvement in the prediction interval estimation.Figure 11
Association diagram between monthly rate of empirical coverage and monthly rate of nominal coverage for the intraday solar GHI prediction with regard to the prediction interval of the ensemble prediction during the period from October 2013 to June 2014.Figure 12
Monthly sizes of prediction interval for the intraday solar GHI prediction with regard to the prediction interval of the ensemble prediction during the period from October 2013 to June 2014.


## 4.1. Ensemble Prediction Method and Forecasting Conditions

Everyday group prediction of solar global horizontal irradiance is controlled with WRF, similar to previous work proposed by Liu et al. [14], with the exception of the application of method Kalman filter, which is a postprocessor for the WRF model. We use similar calculations and circumstances as the latter case in order to contrast [66–71]. The detailed process of the ensemble prediction is stated in the previous research [14]; therefore, we give only the outlines here [72].The lagged averaged forecast (LAF) which is made up of members with distinct original situation and time is used for group prediction. Figure7 shows the chart of the cycle of operation for the ensemble prediction of solar GHI. Mi,j−KF indicate the members of the ensemble prediction based on WRF output and Kalman filter, in which i refers to the running day and amounts to the day of the termination of calculation, and j represents the scope of the forecasting day. Hence, j=0 refers to the result of the intraday; j=1 refers to the result of the next-day; and j=2 refers to the results of the 2-day ahead forecasting from the running day. As the figure displayed, GHI of the intraday forecasting in the running day i is predicted based on three members: Mi,0−KF, Mi−1,1−KF, and Mi−2,2−KF. GHI of the next-day forecasting in the running day i is predicted based on two members: Mi,1−KF and Mi−1,2−KF. GHI of the 2-day ahead forecasting in the running day i is predicted based on one member: Mi,2−KF.Figure 7
Cycle of operation for the solar GHI ensemble prediction based upon model WRF and method Kalman filter.The spread of the forecasting members is also defined with the following equation:(6)Spreadi,j=1N−j∑k=0N−j−1GHIi−k,j+k−GHI¯2,where j equals to 0, 1, and 2; N represents the number of forecasting day in one running, and it is equal to 3 in this work; N−j means the member number of ensemble prediction; GHI¯ is the averaged GHI value of the ensemble members; and GHIi−k,j+k refers to the predicted GHI value of the member Mi−k,j+k. This spread serves as the evaluation factors of uncertainty of the forecasting [14].


## 4.2. Solar Global Horizontal Irradiance Ensemble Prediction

The certainty of the outcome pointed out for solar global horizontal irradiance prediction is examined in Figure5 and Table 2. It is found that Bias, RMSE, and CORR, the statistical error indices of Mi,0 forecasting, are the optimal in those figure and table. In general, in the ensemble prediction method, the ensemble average is selected as the predicted value, brief noted as ensemble mean−KF in this research. Nevertheless, Mi,0−KF, which represents the intraday forecasting value based on WRF and Kalman filter, is used as the predicted value in this work, for the reason that the precision of the intraday forecasting is optimum.


## 4.3. Evaluation Index of Ensemble Prediction

Figure8 displays the link in between the distribution within a day prediction and its prediction flaws from the examination throughout the time from September 1 to November 30 2013. Also, the top and bottom peripheries in the prediction interval whose coverage rates are 50 percent, 80 percent, 90 percent, and 95 percent respectively are linked with lines in this chart. These lines are assessed during per 10 W/m2 in the target spread on the one hand and, on the other hand, are assessed negative and positive errors, respectively. The plotted data gather around the original point of the chart. Meanwhile, the forecasting error values deviate in a wide way with the target spread getting large as figured with the periphery lines of the extents.Figure 8
Relationship chart between the ensemble spread and solar GHI’s forecasting error for the intraday prediction by using model WRF onlyMi,0, based on WRF and method Kalman filter Mi,0−KF during the period from September 1 to November 30 in 2013.The periphery lines in the prediction extents are coarse, for the reason that three is the limitation for the ensemble member with the target spread containing random error, which results in the limitation range of the computing environment in this project. The increasement of ensemble members makes contribution to the amelioration of the interval evaluation [73–76].The distribution of scatter diagrams and the periphery lines ofMi,0, which is the calculated outcomes with model WRF, deflects to the positive side, because WRF has the tendency to overestimate the solar global horizontal irradiance as explained before. The pure WRF forecasting is symmetric with respect to the horizontal line 30 W/m2. On the other hand, the plots and the periphery lines, with an exception of the lines whose coverage rate is 95 percent, distribute in a symmetrical way relative to the axis of abscissas. This is the result of method Kalman filter, which corrects the overestimating calculation of model WRF.


## 4.4. Solar Global Horizontal Irradiance Forecasting with Prediction Interval

Every day solar global horizontal irradiance ensemble predictioning is performed, and the solar lustrous and its prognosis interval are forethought. Figure9 shows the irradiance forecasted with WRF only, and Figure 10 displays the irradiance with model WRF and method Kalman filter. In these charts, the calculated outcomes of the intraday forecasting data are linked up and shown during the time series between December 1 and December 10 2013. By respectively using black full lines and red dot lines in these charts, the data of the predicted irradiance and the data of observed ones are plotted. As a result of the overestimation of model WRF, the outcome of irradiance which is predicted with model WRF is greater than the observed data in Figure 9. On the other hand, the forecasting results with WRF and Kalman filter is suited to the observed data as indicated in Figure 10.Figure 9
Time series of solar GHI intraday prediction by using model WRF onlyMi,0 and its 50%, 80%, 90%, and 95% prediction interval during the period from December 1 to 10 in 2013. (a) Mi,0 prediction and ensemble spread, (b) Mi,0 prediction and 50% prediction interval, (c) Mi,0 prediction and 80% prediction interval, (d) Mi,0 prediction and 90% prediction interval, and (e) Mi,0 prediction and 95% prediction interval.
(a)(b)(c)(d)(e)Figure 10
Time series of solar GHI intraday predictionMi,0−KF based on model WRF and method Kalman filter, and its 50%, 80%, 90%, and 95% prediction interval during the period from December 1 to 10 in 2013. (a) Mi,0−KF prediction and ensemble spread, (b) Mi,0−KF prediction and 50% prediction interval, (c) Mi,0−KF prediction and 80% prediction interval, (d) Mi,0−KF prediction and 90% prediction interval, and (e) Mi,0−KF prediction and 95% prediction interval.
(a)(b)(c)(d)(e)The ensemble spread of the members with model WRF and based on model WRF and method Kalman filter is also displayed by using dashed lines in green in Figures9(a) and 10(a). The target spread value alters with time; for instance, the target spread on December 6 is not small, while the target spread on December 3 is close to zero in these charts. As the ensemble spread becomes smaller, the predicted GHI values come closer to the observed one.The top and bottom peripheries in the prediction interval are displayed by using two dashed lines in blue which situate from Figures9(b)–9(e) and from Figures 10(b)–10(e), whose coverage rates are respectively 50 percent, 80 percent, 90 percent, and 95 percent. For the sake of evaluating the peripheries of the predicted intervals, the top and bottom peripheries in the prediction interval in Figure 8(a) or 8(b) are assessed based on the given rates of nominal coverage and the target spread in Figures 9(a) or 10(a) at first. And, defined as the prediction interval, they are drawn based on the forecasted irradiance. In these charts, accompanied with the decrease of target spread, the interval size, which can be found from the data on December third, is small. On the contrary, the interval size becomes large when the target spread increases, just like the data on December 6. As is displayed in these charts, the rate of nominal coverage gets large compared with the interval size in proportional way.Table3 displays the rate of daily empirical coverage, which refers to the ratio of the actual data number of observed data fallen in the estimated interval, to the total data number within the effective sunshine time, for the intraday solar GHI ensemble prediction during the period from December 1 to 10 in 2013. Mi,0 represents the predicted result based on model WRF, and Mi,0−KF represents the predicted result based on model WRF and method Kalman filter in this table. It is revealed that the rate of empirical coverage is in proportion to the rate of nominal coverage, which is greater than the rate of daily empirical coverage of Mi,0 and approached by the rate of daily empirical coverage of Mi,0−KF.Table 3
Rates of empirical coverage for the intraday solar GHI prediction with regard to the prediction interval of the ensemble prediction during the period from December 1 to 10th 2013.
Rate of nominal coverage for prediction interval (%)Forecasting method and memberRate of empirical coverage for prediction interval [%]12/0112/0212/0312/0412/0512/0612/0712/0812/0912/10Ave50Mi,031.610.531.673.726.326.321.126.336.863.234.7Mi,0−KF57.952.642.157.957.947.463.157.936.868.454.280Mi,063.226.352.689.536.857.936.836.873.794.756.8Mi,0−KF73.778.978.989.584.268.484.289.563.194.780.590Mi,073.742.157.994.747.457.957.947.489.594.766.3Mi,0−KF84.278.994.794.794.789.594.794.778.994.790.095Mi,084.247.463.294.752.694.778.947.489.594.774.7Mi,0−KF94.784.294.794.794.794.794.794.789.594.793.2During the period from December 1 to 10 2013, Table4 shows the daily prediction interval sizes for the intraday solar GHI ensemble prediction. For the size computation, the data at night are refused. It is revealed that the sizes get in proportion to the rates of nominal coverage in this table. The sizes of Mi,0−KF forecasting is smaller than the ones of Mi,0 forecasting.Table 4
Prediction interval sizes for the intraday solar GHI prediction with regard to the prediction interval of the ensemble prediction during the period from December 1 to 10 in 2013.
Rate of nominal coverage for prediction interval (%)Prediction method and memberPrediction interval size [W/m2]12/0112/0212/0312/0412/0512/0612/0712/0812/0912/10Ave50Mi,079.965.465.4112.965.4120.982.965.490.8104.785.4Mi,0−KF76.755.055.0107.954.9109.778.354.991.698.978.380Mi,0155.6111.8111.8231.9111.8232.1164.5111.7196.7208.0163.6Mi,0−KF147.299.699.6212.299.5196.9155.899.4188.4190.6148.990Mi,0217.0144.5144.5316.4144.4304.6233.3144.3287.1284.4222.1Mi,0−KF208.6127.1126.9328.0126.9290.3223.1126.8272.8279.0210.995Mi,0274.8171.9171.9427.6171.8408.2320.7171.8381.3378.4287.9Mi,0−KF266.8163.5163.3431.1163.3398.9288.6163.2328.8338.1270.6To improve the WRF calculation, solar global horizontal irradiance via Kalman filter is used. Applying it to the ensemble prediction brings about the size of the prognosis interval gradually little and the factual analysis quota rises and closes to the nominal one. It signifies that the amelioration of the forecasting provides a better assessment for prediction interval.Furthermore, monthly empirical coverage rates and prediction interval sizes from October 2013 to June 2014 are illustrated in Figures11 and 12. With the application of Kalman filter in improving the solar GHI forecasting, the prediction interval sizes get small, while the rates of empirical coverage add and become near the nominal rates in the long period. It means that the improvement in the forecasting works well for the improvement in the prediction interval estimation.Figure 11
Association diagram between monthly rate of empirical coverage and monthly rate of nominal coverage for the intraday solar GHI prediction with regard to the prediction interval of the ensemble prediction during the period from October 2013 to June 2014.Figure 12
Monthly sizes of prediction interval for the intraday solar GHI prediction with regard to the prediction interval of the ensemble prediction during the period from October 2013 to June 2014.


## 5. Conclusion

This report portraits how solar global horizontal irradiance is accomplished with weather forecast model WRF and collecting data approach Kalman filter. The Kalman filter get rid of aberration of the irradiance calculated with WRF and enhance prognosticating. Also, the ensemble prediction is applied to the forecasting of the solar global horizontal irradiance as well as its prediction interval. The spread of the ensemble forecasting is calculated as a parameter corresponding to the unreliability of the forecasting. And the relation between the spread and the forecasting error is discussed to evaluate the prediction interval. With the prediction interval sizes reducing, its rate of empirical coverage raises and comes close to the nominal ones by applying the method Kalman filter. In the ensemble prediction, the amelioration through putting the method Kalman filter into use does not fit for the forecasting simply. Meanwhile, it is also suited for the prediction interval assessment. The probabilistic forecasting of solar irradiance presented in this article will be used to predict photovoltaic system generation for electric power grid management and its risk analysis. Compared with other methods, the first highlight of this article is the application of ensemble prediction method to achieve the probabilistic forecasting of solar irradiance. The second highlight is that the Kalman filter is applied into the prediction accuracy of solar irradiance, which is an innovation in solar irradiance prediction research, making the results more exact and reliable.Further work is to improve the solar GHI prediction by using some interpretable machine learning tools to quantify the parameter importance. The work also seeks to achieve uncertainty quantification of the solar GHI prediction by transferred recurrent neural network for battery calendar health prognostics of energy transportation systems. Furthermore, it is also of great interest to compare and analyze the evaluation of the photovoltaic system installation impact to an electric power grid.



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*Source: 1523198-2022-06-10.xml*