Introduction

Rapid economic growth in developing countries has highlighted the dependence on energy sources. Meanwhile, burning fossil fuels can have adverse environmental consequences, leading to air pollution and contamination of soil and water resources. It also leads to increased carbon dioxide emissions and eventually to global warming, which is particularly important in developing countries, including Iran. Rising sea levels, melting glaciers, sharper droughts and increasing heat waves, powerful storms and floods, changing ecosystems, growing and spreading agricultural pest populations and diseases and reduced food security are some of the most important sequels of climate change, accelerated by the use of fossil fuels. These threats have united governments towards taking management decisions regarding reduction and control of fossil fuel consumption and carbon dioxide emissions, which are enforced through the annual Climate Change Conference.

Rs is one of the main sources of clean, renewable and accessible energy and can be an alternative to fossil fuels. Particularly, there is tremendous potential in such a country as Iran, with an arid to semi-arid climate and more than 13 h of sunshine per day in some areas. Rs is the primary driving force in agriculture and impacts many key processes such as photosynthesis, evapotranspiration and irrigation scheduling. It is also very useful from an industrial standpoint, playing a role in the design of solar panels and urban facilities and in photovoltaic power generation.

Rs can be directly measured at meteorological stations by pyranometers. In many developing countries, however, not all meteorological stations are equipped with these instruments, and incidentally they are costly to maintain. Therefore, either there are no recorded data for this parameter or the available data are not wholly credible. In recent years, accordingly, researchers have tried to estimate this parameter by various methods. Empirical models and artificial intelligence (AI) models are some of the main tools for Rs estimation.

Empirical models

Empirical models can be divided into four categories: temperature-based (i.e. Hargreaves-Samani (1982); Annandale et al. (2002)), sunshine-based (Angstrom-Prescott (1940); Feng et al. (2018a, 2018b)), day of the year-based (Quej et al. (2017); Zang et al. (2018)) and hybrid models (Wu et al. (2007); Jahani et al. (2017)).

Adaramola (2012) investigated the performance of seven empirical models in estimating monthly Rs in Nigeria and showed that Angstrom-Prescott model exhibited the best performance (RMSE of 0.257 kWh.m−2.day−1). Quej et al. (2017) studied the performance of five empirical day of the year-based models (four existing, and one proposed model) and reported RMSEs ranging between 0.975 MJ.m−2.day−1 (proposed model) and 2.197 MJ.m−2.day−1 (sine wave model). Jamil and Akhtar (2017) estimated monthly mean diffuse Rs using 16 proposed models categorised into two groups in a humid subtropical climatic region of India. RMSEs for the first group ranged from 1.29 to 1.47 MJ.m−2.day−1, and those of the second group from 1.29 to 1.47 MJ.m−2.day−1. Performance of two proposed models based on sunshine duration and relative humidity and 14 existing models in estimating daily Rs in Turkey was studied by Yildirim et al. (2018). Qin et al. (2018) used MODIS atmospheric and land products as well as daily meteorological data recorded at 837 meteorological stations in China as inputs of Yang’s hybrid model (YHM), an efficient physically based model (EPP), an hourly solar radiation model (HSRM) and a neural network model (ANNM) for estimation of solar radiation, and reported YHM as providing better estimates compared to EPP, ANNM and HSRM, with mean daily values ​​of 2.414, 2.535, 2.855 and 3.645 MJ.m−2.day−1, respectively. Performance of 97 available models in literature and 5 newly established models for estimating diffuse Rs at 17 stations over China was evaluated by Wang et al. (2019). Their results showed that the proposed model with clearness index and relative sunshine duration as inputs, produced the highest accuracy. The results showed superiority of proposed models (with RMSEs of 0.947 MJ.m−2 at Adana station, 1.086 MJ.m−2 at Göksun station, 1.074 MJ.m−2 at Tarsus station) over existing models. Mohammadi and Moazenzadeh (2021) evaluated the performance of existing and proposed empirical models in estimating daily Rs at 13 weather stations of Peru. According to the RMSEs, the worst and best results are achieved at San Martin station (RMSE = 509 J.cm−2.day−1) and Tacna station (RMSE = 223 J.cm−2.day−1), respectively.

AI-based models

Cao and Cao (2006) improved the accuracy of daily Rs estimates by combining wavelet analysis with a back propagation training algorithm, with the error rate reduced from 2.82 to approximately 0.72 MJ.m−2.day−1. Various combinations of day of the year, temperature and relative humidity were used by Rehman and Mohandes (2008) as input variables of ANN models for estimation Rs in Saudi Arabia, with MAPEs of 4.49 and 11.8 for different scenarios. Using main meteorological parameters, evaporation and soil temperature, Asl et al. (2011) estimated daily Rs in Dezful, Iran, by multi-layer perceptron (MLP) neural networks with an MAPE of 6.08. Wang et al. (2016a, 2016b) evaluated the performance of three neural network models including generalised regression neural network (GRNN), MLP and radial basis neural network (RBNN) in estimating daily Rs. The results of their study showed that MLP outperformed the other models, with RMSEs ranging from 1.94 to 3.27 MJ.m−2.day−1.

Three types of AI-based models including ANFIS-GP, ANFIS-SC and M5Tree were evaluated for estimating Rs at 21 stations over China by Wang et al. (2016a, 2016b) and a general tendency to under-estimation of high radiation values in some stations is reported. An experimental model named “Iqbal” and four AI-based models, including extreme learning machine (ELM), back-propagation neural networks optimised by genetic algorithm (GANN), random forests (RF) and GRNN were compared by Feng et al. (2017) in terms of estimating diffuse Rs. The results showed that GANN at Beijing station (RRMSE = 17.1%) and Zhengzhou station (RRMSE = 13.4%) had the best performance. Halabi et al. (2018) assessed the performance of four models, including ANFIS and its hybrids with three optimisation algorithms — particle swamp optimisation (PSO), genetic algorithm (GA) and differential evolution (DE) — in estimating monthly Rs in Malaysia, concluding that ANFIS-PSO (RMSE = 0.3065) has outperformed other models. Performance of back propagation (BP) and radial basis function (RBF) models and a new hybrid model, “ensemble empirical mode decomposition and self-organising map-back propagation hybrid neural networks” (EEMD-SOM-BP) was studied by Lan et al. (2018) in estimating seasonal Rs in China, with the proposed model leading to the best results in spring, summer, autumn and winter with RMSEs of 137.85, 123.58, 72.84 and 135.42 W.m−2, respectively. Among different models applied for modeling daily photosynthetically active radiation (Feng et al. (2018a, 2018b)), genetic model outperformed with lowest RMSE (0.5 MJ.m−2.day−1). Kuhe et al. (2019) evaluated the performance of three ANN models including feed-forward back-propagation neural network (FFNN), radial basis function network (RBFN) and GRNN in estimating monthly Rs in Makurdi, Nigeria. All models had acceptable accuracies, with R2 = 0.998 and MSE = 0.0142 (MJ. m−2.day−1) on average. Performance of SVM-FA, Copula-based nonlinear quantile regression (CNQR) and empirical models in estimating daily diffuse Rs at four stations in China was studied by Liu et al. (2020). Their findings showed that CNQR and SVM-FA were much better than empirical models. SVM-FA results were slightly better than those of CNQR, resulting in a 0.67% decrease in MABE.

Given the vast surface area of Iran and its arid and semi-arid climate, and since many parts of the country receive long hours of sunshine, the present study aimed at (i) evaluating the effect of input parameters (geographical and meteorological) on estimating Rs, (ii) proposing a novel strategy based on multi-model approach by combining either support vector machine (SVM) or SVM boosted by the cuckoo search algorithm (SVM-CSA) with multi-layer perceptron and (iii) discussing the generalisation capability of proposed models using local and external analysis. Considering a data-fusion approach via multi-layer perceptron model for combining the advantages of empirical and AI models for estimating Rs and discussing the potential of proposed models in three different intervals are the main novelty aspects of the present study. Meteorological data and measured Rs values recorded at seven weather stations in Iran over 2010–2019 were used for this purpose.

Materials and methods

Study area

With an area of 1,648,000 km2, Iran extends between longitudes 44° and 63° east and between latitudes 25° and 40° north (Fig. 1). The country’s climate is generally arid or semi-arid (Aghelpour et al., 2019), with maximum sunshine hours of about 14 h per day in some regions, which is a high figure. Iran characterised by hot summers and cold winters and different type of climates, from arid (most part of Iran) to humid region (north of Iran). The average of annual precipitation is about 250 mm. Iran is a developing country and inevitably utilises various energy sources. However, fossil fuel consumption can cause irreparable damage to the environment in the long term. Studies on clean and renewable sources of energy such as Rs, and its proper estimation, can therefore be of great help in reducing the country’s dependence on fossil fuels. In the present study, geographical and meteorological parameters recorded at seven weather stations in Iran were used for estimating daily Rs. Statistical indices of the meteorological parameters are given in Table 1.

Fig. 1
figure 1

Iran (the study area), neighbouring countries and locations of weather stations

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Table 1 Geographical locations of the stations and statistical indices of meteorological parameters used in this study
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Rs estimation

In the present study, two different structures were used for Rs estimation, flowcharts of which are shown in Fig. 2. In structure 1 (local analysis), data from each of the seven stations were divided into two sets (training and testing), and performance of each method at each station was evaluated separately. In structure 2 (external analysis), data from four stations (Ahvaz, Isfahan, Kermanshah, and Mashhad) were used for the training, and the data from three other stations (Bandar Abbas, Kerman, and Tabriz) were used for testing. In fact, in structure 2 an attempt was made to examine the generalisation capability of the models through rendering the data of the two sets (training and testing set) dissimilar, from two perspectives, including the distance between stations and climatic differences between stations. In this study, inputs of Rs simulator models were divided into two categories, geographical and meteorological inputs, so that in addition to evaluating model performance, the effect of input type could also be examined. A summary of model input data can be seen in Table 2. The quality of meteorological data (inputs) and solar radiation (output) was checked before applying in the modeling processes. For this aim, outliers and missing data were found and time series without noise data were considered for modeling.

Fig. 2
figure 2

Two structures used for estimating daily Rs. Structure 1: dataset (n) from each station is divided into training and testing sets. Structure 2: datasets from four stations are used for training of the models, and the results are validated separately against data from each of the other three stations

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Table 2 Rs estimator models and their inputs categorised as geographical or meteorological parameters
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Empirical models

In this study, performance of empirical models based on temperature (6 models) or sunshine hours (2 models) as well as 3 hybrid models in estimating Rs over 2010–2019 was studied based on meteorological data recorded at seven weather stations (Table 3).

Table 3 Empirical models used for estimating Rs
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Support vector machine (SVM)

SVM was introduced by Vapnik (1995) as a supervised learning algorithm and a statistical learning technique which can be used for solving classification, regression and forecasting problems. SVM employs kernel functions to transform the data from input space into a higher dimensional feature space in order to simplify classification problems. In addition, the “ε” insensitive loss function enables SVM to solve nonlinear regression problems. For non-separable classes, where an exact separating hyperplane cannot be found, the input space is mapped to a higher-dimensional feature space using nonlinear functions called feature functions (∅) and kernels, thus enabling SVM to form nonlinear boundaries and model highly complex problems (Raghavendra and Deka, 2014). SVM has been used for simulating evaporation (Goyal et al., 2014; Moazenzadeh et al., 2018), reference evapotranspiration (Kisi and Cimen, 2009; Mohammadi and Mehdizadeh, 2020), estimation of lake water level fluctuations (Cimen and Kisi, 2009), streamflow simulation (Mohammadi et al., 2021), drought forecasting (Deo et al., 2017), soil temperature (Moazenzadeh and Mohammadi, 2019), velocity prediction (Ebtehaj and Bonakdari, 2016), prediction of discharge coefficient and depth around bridge piers (Sharafi et al., 2016; Azimi et al., 2019) and soil saturated hydraulic conductivity (Kashani et al., 2020) estimation.

Cuckoo search algorithm (CSA)

This algorithm is based on the following assumptions:

  1. a.

    Each cuckoo randomly selects a nest and lays a single egg in it.

  2. b.

    The nests with the highest quality of eggs (that is, solutions to the problem) are carried over to the next generation.

  3. c.

    The number of the available nests is fixed. Original owners of the nests are able to recognise cuckoo eggs with a probability Pa \(\in\) [0, 1].

Upon learning that an alien egg is laid in its nest, the host bird will either discard that egg or desert the nest and build a new one. The last assumption can be approximated by a fraction Pa of the n nests being replaced by new nests (with new random solutions at new locations). For maximisation problems, the fitness of a solution can be proportional to the objective function. For details and background information about CSA, see Yang and Deb (2009) and Gandomi et al. (2013). In the present study, CSA was used to determine the best parameters and weights for SVM in the form of hybrid models. Previous studies have confirmed that such coupled CSA optimisation approach via AI models can produce a capable hybrid model (Liu and Fu 2014; He et al., 2018; Puspaningrum et al., 2020). Figure 3 shows the modelling flowchart for the base SVM and SVM coupled with CSA (SVM-CSA) as used in this study.

Fig. 3
figure 3

Flowchart of the hybrid model (SVM-CSA) used in the present study

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Multi-model approach

A new approach based on the multi-model concept was developed, tested and compared with the empirical and AI models for estimation of daily Rs. In this approach, outputs of empirical and SVM/SVM-CSA models under the best scenarios were used as inputs to the multi-model, structure of which was based on MLP. Multi-model outputs were named SVM-MLP and SVM-CSA-MLP. Figure 4 outlines the multi-model approach employed in our study.

Fig. 4
figure 4

Flowchart of the multi-model strategy used for estimating Rs

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Evaluation indices

Performance of the developed Rs estimator models was evaluated using three statistical indices including RMSE, mean absolute percentage error (MAPE) and relative root mean square error (RRMSE), calculated using Eqs. 1, 2, 3, respectively. Ertekin and Yaldiz (2000) proposed the following categories for rating models according to accuracy: A model is excellent if its RRMSE is below 10%, good if 10% < RRMSE < 20%, fair if 20% < RRMSE < 30%, and poor if the RRMSE is higher than 30%.

$$RMSE={\left[{\sum }_{i=1}^{n}{\left({Rs}_{(i,obs)}-{Rs}_{\left(i,est\right)}\right)}^{2}\div n\right]}^{0.5}$$
(1)
$$MAPE=\left[\left({\sum }_{i=1}^{n}\left|\left({Rs}_{\left(i,est\right)}-{Rs}_{(i,obs)}\right)\div{Rs}_{(i,obs)}\right|\right)\div n\right]$$
(2)
$$RRMSE=\left[RMSE\div\left(\frac{1}{n}{\sum }_{i=1}^{n}{Rs}_{(i,obs)}\right)\right]$$
(3)

where \({\text{Rs}}_{\mathrm{i},\text{obs}}\) and \({\text{Rs}}_{\mathrm{i},\text{est}}\) are observed and estimated solar radiation values and n refers to the number of data points.

Results

Local performance of the models (structure 1)

For this structure, we divided the data from each station into two parts (training and testing). Results of the best empirical model, SVM, SVM-CSA and the multi-model approach under the best scenarios at each station are plotted in Fig. 5.

Fig. 5
figure 5

Measured versus estimated radiation values under the best empirical model, the best scenarios of SVM and SVM-CSA (with either geographical or meteorological inputs) and the multi-model approach at each station (local analysis: structure 1)

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Ahvaz station

The use of meteorological data as model inputs greatly improved the accuracy of radiation estimates, to the extent that RMSEs of the SVM10M and SVM10M-CSA models (best scenarios with meteorological inputs) were approximately 49% and 60% lower than those of SVM1G and SVM1G-CSA (best scenarios with geographical inputs), respectively. Coupling the CSA to the SVM model was effective and reduced RMSE from 153.21 to 114.22. Garj-Garj model exhibited the best performance among the empirical models, with RMSE = 166.97 J.cm−2.day−1. Examination of the results showed that the RMSE of the multi-model approach has been 4 and 2% lower compared to the base SVM and SVM-CSA, respectively. However, the difference is insignificant and the use of the multi-model approach is not recommended.

Bandar Abbas station

The results, under the best scenarios, of SVM, SVM-CSA, the best empirical model and the multi-model approach are represented in Fig. 5, indicating that the use of meteorological rather than geographical parameters as inputs of SVM and SVM-CSA has reduced RMSEs by about 21 and 28 per cent, respectively (Table 4). Among the empirical models, Garj-Garj (RMSE = 220.24 J.cm−2.day−1) outperformed SVM, but its performance was inferior to SVM-CSA with meteorological parameters. Also, the SVM8M-CSA scenario reduced estimation error by 31% compared to SVM8M, which highlights the importance of coupling the CSA to the base SVM model.

Table 4 Performance evaluation indices for the best empirical model, SVM, SVM-CSA, and the multi-model approach for structure 1 (local analysis). G and M denote geographical and meteorological inputs, respectively
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Isfahan station

Using meteorological rather than geographical parameters has greatly improved the performance of SVM and SVM-CSA, reducing their estimation errors by about 56 and 47%, respectively. Comparing the outputs of SVM (RMSE = 125.31 J.cm−2.day−1), SVM-CSA (RMSE = 125.86 J.cm−2.day−1) and Garj-Garj empirical model (RMSE = 125.87 J.cm−2.day−1) shows that application of the CSA to the base SVM model has failed to improve the results, and the empirical Garj-Garj model is recommended according to its ease of use compared to the more complex AI models. Performance of 42 different SVM structures in estimating daily Rs in Ghardaia, Algeria, was studied by Belaid and Mellit (2016). Daily RMSE values for the four selected structures of SVM ranged between 2.777 and 2.807 MJ.m−2, whereas for the MLP model, the range of RMSEs increased to 2.788–3.047.

Kerman station

At this station, Abdalla’s model outperformed all other empirical models, with an error rate similar to that of SVM. However, application of the CSA to the base SVM model reduced estimation error by 15%. Examination of the results also shows that application of the multi-model approach to outputs of SVM and SVM-CSA has only slightly improved the results, reducing RMSEs by about 7 and 2%, respectively.

Kermanshah station

Coupling the CSA to SVM base model has improved radiation estimates, with the SVM9-M-CSA scenario reducing RMSE by about 30% and 38% compared to the best empirical model (third model of Jahani et al.) and the best SVM scenario (SVM9-M), respectively. Application of the multi-model approach to SVM outputs reduced the RMSE by 9%. Kim et al. (2018) evaluated the performance of single soft computing models, including MLP, SVM, ANFIS and MARS (multivariate adaptive regression spline) in estimating daily Rs. With various inputs, the best performance was obtained for SVM (RMSE = 4.399) and MARS (RMSE = 4.207) at Big Bend and Incheon stations, respectively.

Mashhad station

The results showed that replacing geographical parameters with meteorological parameters as inputs of SVM and SVM-CSA reduced RMSEs by 45 and 25%, respectively. Both the base SVM and SVM coupled with CSA effectively estimated Rs, with RMSEs reduced by about 11% and 37%, respectively, compared to the best empirical model (Abdalla’s model).

Tabriz station

The results indicate that although the use of SVM model has failed to improve radiation estimation in comparison with the best empirical model (the first model of Jahani et al., RMSE = 366.88 J.cm−2.day−1), SVM coupled with CSA has reduced the RMSE by 14% compared to the best empirical model. Application of the multi-model approach to outputs of the best scenarios of SVM and SVM-CSA was effective, reducing error rates by 7% and 16%, respectively.

External performance of the models (structure 2)

We defined this structure in order to examine the possibility of generalising the performance of radiation estimator models at stations which had not played a role in model training process. For this purpose, radiation data from four stations (Ahvaz, Isfahan, Kermanshah and Mashhad) were used to train the models, and performance of the models was tested on radiation data from the other three stations (Bandar Abbas, Kerman and Tabriz). Figure 6 depicts the results of the best empirical model, SVM, SVM-CSA and the multi-model approach under the best scenarios.

Fig. 6
figure 6

Measured versus estimated radiation values under the best empirical model, the best scenarios of SVM and SVM-CSA (with either geographical or meteorological inputs) and the multi-model approach at each station (external analysis: structure 2)

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Bandar Abbas station

According to the results, although SVM has not led to lower estimation errors in comparison with the best empirical model (Abdalla model), application of CSA has reduced the error rate by 10%. The results also show that application of the multi-model approach to outputs of SVM and SVM-CSA has failed to improve radiation estimates.

Kerman station

At this station, application of meteorological rather than geographical parameters greatly improved Rs estimates, to the extent that RMSEs were reduced by 49 and 48% for SVM and SVM-CSA, respectively. Although the best empirical model (Garj-Garj) slightly outperformed the best scenario of SVM, coupling the CSA to the base SVM reduced RMSEs by 29% and 18% compared to SVM and Garj-Garj models, respectively. Application of the multi-model approach to both SVM and SVM-CSA led to better radiation estimates; but this improvement was negligible in both cases, especially for SVM-CSA.

Tabriz station

Application of CSA to the base SVM model has reduced the error rate by 25% compared to the best empirical model, testifying to the importance of coupling the optimisation algorithm to SVM. Unlike other stations, application of the multi-model approach to outputs of both SVM and SVM-CSA was effective, reducing error rates by about 24% (SVM-MLP) and 27% (SVM-CSA-MLP), respectively.

Discussion

Local performance of the models (structure 1)

Results of the best scenarios — in terms of under- or over-estimation of radiation amounts — and model accuracies, respectively, over the three discussed intervals (first, middle and last third of measured radiation values) are depicted in Figs. 7 and 8. Statistical indices including RMSE, MAPE and RRMSE are also listed in Table 4.

Fig. 7
figure 7

Under-estimated (yellow circles) and over-estimated (purple circles) radiation values under the best model at each station. Vertical lines divide measured radiation data points (ranked by magnitude) into three numerically equal groups (local analysis: structure 1)

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Fig. 8
figure 8

RMSEs in the first, middle and last third intervals of estimated radiation amounts under the best model, in under- and over-estimation sets; at each station (local analysis: structure 1)

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Ahvaz station

Apart from SVM-G and SVM-G-CSA, whose performances were “good” according to their RRMSE indices, performance of all other models was excellent, indicating the reliable performance of Garj-Garj model, superiority of meteorological variables over geographical variables and the importance of coupling the CSA to the base SVM model (Table 4).

Hassan et al. (2017) evaluated the performance of three different machine-learning algorithms as well as a proposed algorithm titled “decision trees” in estimating Rs in Cairo, Egypt. Among day of the year-based models, the proposed decision trees model exhibited the best performance (RMSE = 2.0489). Zang et al. (2018) studied the performance of 14 empirical models, five ANFIS-based models and GPR and SVR models, in estimating daily Rs in China. RMSEs varied from 1.39 to 3.065 MJ.m−2.day−1 for the empirical models, from 1.379 to 2.976 for SVR, from 1.287 to 2.711 for GPR and between 1.203 and 2.721 for ANFIS-based models. Application of the multi-model approach to the SVM-CSA (SVM-CSA-MLP) led to more accurate results in over-estimation set. The results showed that SVM-CSA-MLP has estimated Rs with the lowest error rates in under- and over-estimation sets in the first third interval (RMSE = 103.86) and the last third interval (RMSE = 41.64), respectively.

Bandar Abbas station

The results presented in Table 4 show that except SVM-M-CSA and SVM-CSA-MLP, which have RRMSEs in the “excellent” category and highlight the role of coupling the optimisation algorithm to the base SVM model and the importance of employing the multi-model approach in radiation estimation, respectively; other models are “good” with RRMSEs ranging between 10 and 20%. Performance of two empirical models as well as eight existing hybrid models and four proposed hybrid models in China was evaluated by Fan et al. (2018). According to their findings, Bahel model was recommended for cases where only sunshine hours data were available. Sanz et al. (2018) investigated the performance of four models including extreme learning machine (ELM), SVR, multiple linear regression (MLR) and multivariate adaptive regression spline (MARS), either alone or coupled with two optimisation algorithms, CRO (coral reefs optimisation) and GGA (grouping genetic algorithm), in Australia. The four base models had error rates (MJ.m−2) varying from 4.224 (ELM) to 4.289 (MARS), whereas coupling GGA and CRO algorithms to them led to error rates ranging from 4.246 (GGA-ELM) to 4.533 (GGA-MLR) and from 4.21 (CRO-ELM) to 4.468 (CRO-SVR), respectively.

As shown in Fig. 8, the best performance of all models in under- and over-estimation sets is obtained in the first third interval (measured radiation values below 1675 J.cm−2.day−1) and the last third interval (measured radiation values above 2241 J.cm−2.day−1), respectively. In under-estimation set, SVM-CSA-MLP in the first third (RMSE = 102.84 J.cm−2.day−1) and SVM-G in the last third (RMSE = 375.25 J.cm−2.day−1) had the best and the poorest performance, respectively. The best and poorest models in over-estimation set were 12.23 (last third, the best empirical model) and 412.88 (first third, SVM-G-CSA), respectively. Similar to what happened to most models at this station in the last third interval, Wang et al. (2016a, 2016b) showed in a study on 12 Chinese stations that ANN models under-estimated high radiation amounts at some stations.

Isfahan station

With the exception of SVM-G-CSA, which has performed better in over-estimation set with an RMSE of 190.97 J.cm−2.day−1, accuracy of the other models has been higher in under-estimation set. Similar to the previous two stations, although all models have performed best in under- and over-estimation sets in the first third (measured radiation values below 1675 J.cm−2.day−1) and the last third (measured radiation values above 2415 J.cm−2.day−1), respectively, the difference between estimation errors in the first third (under-estimation set) and the last third (over-estimation set) is much less compared to the previous two stations. In under-estimation set, SVM-M-CSA in the first third interval (RMSE = 96.66 J.cm−2.day−1) and SVM-G in the last third interval (RMSE = 231.8 J.cm−2.day−1) demonstrated the best and the poorest performance, respectively. For the over-estimation set, corresponding values were 76.06 (last third, the SVM-MLP model) and 425.31 (first third, the SVM-G model), respectively.

Kerman station

According to the results from Isfahan and Kerman stations (Table 4) and in confirmation of high RMSEs for SVM-G and SVM-G-CSA, RRMSEs of these two models are above 10% compared to the other models, placing them in “good” category; and this denotes the subordinate role of geographical parameters in estimating radiation, even in case of using an AI model solely or its coupling with the CSA. This finding is important in that employing the AI model will not necessarily lead to a proper estimation of radiation, and care must be taken when selecting model inputs. Meenal and Slavakumar (2018) studied the performance of 16 empirical models, 16 different structures of the SVM, and 3 structures of the ANN, at four stations in India. According to the results, the lowest RMSEs for empirical models were 0.638, 1.15 and 0.744 MJ.m−2.day−1, for sunshine-based, temperature-based and hybrid models, respectively, and the lowest RMSEs for ANN and SVM were 0.581 and 0.42 MJ.m−2.day−1, respectively. Zou et al. (2017) introduced improved forms of two empirical models (Bristow-Campbell’s and Yang’s hybrid model) and an ANFIS-based model for estimating daily Rs in China. Their results indicated that ANFIS had the lowest RMSEs and MAEs, ranging from 0.59 to 1.6 and from 0.42 to 1.21 MJ.m−2.day−1, respectively.

The lowest and highest differences in error rates between under- and over-estimation sets were 7.38 and 164.72 J.cm−2.day−1 for Abdalla’s empirical model and SVM-G-CSA, respectively (Fig. 8). In under-estimation set, except for SVM-M-CSA and SVM-CSA-MLP which had their lowest error rates in the middle third interval (measured radiation values above 1887 and below 2590 J.cm−2.day−1), the best performance of the other models occurred in the first third interval (measured radiation values below 1887 J.cm−2.day−1). In over-estimation set, all models performed noticeably better in the last third compared to the other two intervals. According to the results, the minimum and maximum estimation errors were 69.44 J.cm−2.day−1 (last third of over-estimation set, SVM-M model) and 540.06 J.cm−2.day−1 (first third of over-estimation set, SVM-G model).

Kermanshah station

Aside from SVM-M-CSA and SVM-CSA-MLP, which demonstrated a slightly better performance in over-estimation set (RMSE = 161.64 and RMSE = 161.47, respectively), all other models exhibited their best performance in under-estimation set. In under-estimation set, all seven models exhibited their best performance in the first third interval (measured radiation values below 1352 J.cm−2.day−1), with the lowest and highest differences in error rates being ΔRMSE = 36.11 J.cm−2.day−1 (between the first and the last third intervals, SVM-CSA-MLP model) and ΔRMSE = 185.56 J.cm−2.day−1 (between the first and the last third intervals, the best empirical model), respectively (Fig. 7). The minimum and maximum estimation errors in over-estimation set were RMSE = 76.34 and RMSE = 523.65 J.cm−2.day−1, for the last third interval of SVM-M-CSA and the first third interval of SVM-G, respectively.

Mashhad station

According to the results presented in Table 4, only RRMSEs of SVM-M-CSA and SVM-CSA-MLP are below 10% (“excellent”), which, in confirmation of RMSEs, point to the importance of using meteorological parameters, application of the optimisation algorithm and employment of the multi-model approach in estimating Rs at Kermanshah and Mashhad stations. Using air temperature as the sole input data, Feng et al. (2019) reported RMSEs in the ranges (3.309–3.375), (3.834–4.021), (3.379–3.406) and (3.811–4.053) MJ.m−2 for empirical models and (2.814–3.103), (3.715–3.939), (3.35–3.491) and (3.54–3.866) for machine learning models at Turpan, Yinchuan, Dunhuang and Xilingol stations in China, respectively. Performance of an empirical model and four AI models in estimating daily Rs in Zhengzhou region, China, was investigated by Xue and Zhou (2019). According to RMSE values, which ranged from 0.7524 to 1.9632 MJ.m−2.day−1 for the 5 models, PSO-LSSVM and the empirical model had the best and the poorest performance, respectively.

Overall, all the seven models performed better in under-estimation compared to over-estimation set; and the lowest and highest differences in error rates between the two sets were observed for the SVM-CSA-MLP model (ΔRMSE = 29.06 J.cm−2.day−1) and the SVM-G model (ΔRMSE = 277.76 J.cm−2.day−1), respectively. The best performances of all models in under- and over-estimation sets were obtained in the first third interval (measured radiation values below 1518 J.cm−2.day−1) and the last third interval (measured radiation values above 2466 J.cm−2.day−1), respectively (Fig. 8). Abdalla’s empirical model in the first third of under-estimation set (RMSE = 28.39 J.cm−2.day−1) and SVM-G in the first third of over-estimation set (RMSE = 573.39 J.cm−2.day−1) exhibited the best and the poorest performance in estimating solar radiation, respectively.

Tabriz station

In confirmation of the higher RMSEs compared to the other stations, performance of the best empirical model, SVM-CSA and SVM-CSA-MLP were “good” (10–20%) and that of the other models were “fair” (20–30%), according to the RRMSE. These findings are indicative of the fine performance of the first model of Jahani et al. and the noticeable superiority of SVM-CSA over the stand-alone SVM model. A critical review of the various types of solar radiation estimator models was undertaken by Zhang et al. (2017). Their results showed that RMSEs have been ranging from 1.11 to 4.5 MJ.m−2 for sunshine-based models, from 2.05 to 4.7 MJ.m−2 for non-sunshine-based models, and from 1.24 to 4.2 MJ.m−2 for ANN models. Ghimire et al. (2019) compared the performance of various models at 5 sites in Australia and reported that ANN (with a mean RMSE of (1.715–2.27) MJ.m−2.day−1) has outperformed the other models (RMSE = 2.14–5.9). Hou et al. (2018) compared the performance of ELM integrated with variable forgetting factor (FOS-ELM) and classical ELM in estimating Rs in Burkina Faso. The results showed that FOS-ELM has reduced RMSE and MAE by (68.8–79.8)% compared to ELM.

The smallest differences in error rates between under- and over-estimation sets were observed for the first model of Jahani et al. (ΔRMSE = 43.09 J.cm−2.day−1) and the SVM-G-CSA model (ΔRMSE = 86.8 J.cm−2.day−1), respectively. In under-estimation set, all seven models exhibited their best performance in the first third interval, and error rates were rising from the first third to the last third interval (Fig. 7). Accordingly, the minimum and maximum error rates in under-estimation set were 163.99 J.cm−2.day−1 (first third interval, SVM-G-CSA model) and 725.28 J.cm−2.day−1 (last third interval, SVM-G). Unlike under-estimation set, the lowest error rates in over-estimation set did not occur in a single interval: SVM-M, SVM-M-CSA and SVM-MLP exhibited their best performance in the first third interval (measurement radiation values below 1386 J.cm−2.day−1); SVM-G performed best in the middle third interval (1386 ≤ Rs ≤ 2431); and the best performance of SVM-G-CSA, the first model of Jahani et al., and SVM-CSA-MLP was obtained in the last third interval (measured radiation values above 2431 J.cm−2.day−1). In over-estimation set, the lowest and highest error rates were approximately 106.85 (last third interval, the first model of Jahani et al.) and 497.04 (last third interval, SVM-G-CSA), respectively.

External performance of the models (structure 2)

Results of the best scenarios — in terms of under- or over-estimation of radiation amounts — and model accuracies, respectively, over the three discussed intervals (first, middle, and last third of measured radiation values) are shown in Figs. 9 and 10. Statistical indices including RMSE, MAPE and RRMSE are also listed in Table 5.

Fig. 9
figure 9

Under-estimated (yellow circles) and over-estimated (purple circles) radiation values under the best model at each station. Vertical lines divide measured radiation data points (ranked by magnitude) into three numerically equal groups (structure 2)

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Fig. 10
figure 10

RMSEs in the first, middle and last third intervals of estimated radiation amounts under the best model, in under- and over-estimation sets, at each station (structure 2)

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Table 5 Performance evaluation indices for the best empirical model, SVM, SVM-CSA, and the multi-model approach for structure 2 (external analysis). G and M denote geographical and meteorological inputs, respectively
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Bandar Abbas station

The ranges of RRMSE variations are given in Table 5, showing that all models have had a “good” performance; the lowest and highest RRMSEs (11.39 and 19.28%) are those of SVM-G-CSA and SVM-G, respectively. This finding underlines the importance of coupling the CSA with the base SVM and is confirmed by comparison of RMSEs of these two models. Marzo et al. (2017) evaluated the performance of the ANN in estimating daily Rs in desert areas, using data from several stations in Chile to train the network and data from other stations to validate the results (RRMSD = 6.6%). Also, data from two other stations in Chile and four in Israel, South Africa, Saudi Arabia and Australia were used to test the generalisation capability of the proposed model to other desert regions of the world, with RRMSD values ranging from 8.1% (one of the stations in Chile) to 22.9% (the Australian station).

Overall, of the seven models studied, only SVM-M-CSA and SVM-CSA-MLP performed better in over-estimation set compared to the under-estimation set. However, the difference in error rates between under- and over-estimation sets for Abdalla’s empirical model, SVM-M and SVM-MLP were about 7, 23 and 31 J.cm−2.day−1, respectively, which are negligible and indicative of comparable performance of these models in the two sets. In under- and over-estimation sets, the best performance of all seven models occurred in the first third interval (measured radiation values below 1646 J.cm−2.day−1) and the last third interval (measured values above 2171 J.cm−2.day−1), respectively. The minimum and maximum estimation errors in under-estimation set were 126.51 and 315.02 J.cm−2.day−1 for SVM-G-CSA and SVM-G models, respectively; whereas corresponding values in over-estimation set were 77.94 and 675.34 for the SVM-M-CSA and SVM-G models, respectively, indicating that the differences in performance of all models between the one-third intervals have been greater in over-estimation set.

Kerman station

Comparison of RRMSEs (Table 5) shows that Garj-Garj model, SVM-M-CSA and the multi-model approach have had an “excellent” performance, with RRMSEs below 10%. Hassan et al. (2016) assessed the generalisation capability of their proposed empirical models with application on the ten separate stations and showed that RMSEs were in the range (0.7035–2.447).

From the viewpoint of model performance over different one-third intervals, all models have estimated radiation amounts with the lowest error rates in the first third interval (measured radiation values below 1928 J.cm−2.day−1) in the under-estimation set and in the last third interval (measured radiation values above 2652 J.cm−2.day−1) in the over-estimation set (Figs. 9 and 10). According to the results, only SVM-G-CSA and SVM-M have had lower error rates in under-estimation set. All other models have estimated radiation values more accurately in over-estimation set, although the differences in error rates between the two sets are negligible for SVM-G and SVM-MLP (approximately 3.2 and 1.6 J.cm−2.day−1, respectively), indicating consistent performance of these models in under- and over-estimation sets. In under-estimation set, SVM-M-CSA and SVM-CSA-MLP were the most accurate models, with RMSEs of 131.27 and 133.26 J.cm−2.day−1, respectively (both in the first third interval); and SVM-G had the lowest accuracy with RMSE = 500.71 J.cm−2.day−1 (in the last third interval). In over-estimation set, SVM-G (RMSE = 56.38 J.cm−2.day−1 in the last third interval) and SVM-G-CSA (RMSE = 567.5 J.cm−2.day−1 in the first third interval) had the highest and lowest accuracies in estimating Rs, respectively. Maximal variations in estimation error over the three one-third intervals occurred in over-estimation set and for the SVM-G model, from first interval to middle interval (46% error reduction) and from middle interval to last interval (81% error reduction), and for the SVM-G-CSA model, first to middle interval (42% error reduction) and middle to last interval (62% error reduction).

Tabriz station

According to the results (Table 5), the most challenging attempt at generalisation of the models to stations with no role in model training has been the one at Tabriz station. Although SVM-G (a “poor” model) and SVM-M, SVM-MLP and the first model of Jahani et al. (“fair”) did not perform well in estimating radiation at this station, SVM-CSA with meteorological or geographical inputs had “good” RRMSEs of 16.24 and 18.5%, respectively. The results show that the multi-model approach has been effective and efficient in generalising radiation estimation at a station that has played no role in its training, with RRMSE = 11.91%, which is very close to the borderline value separating “excellent” and “good” categories (10%). Almost all models have performed noticeably better in under-estimation set compared to over-estimation set, with the minimum and maximum differences in error rates between the two sets being 49.39 J.cm−2.day−1 (the first model of Jahani et al.) and 314.17 J.cm−2.day−1 (SVM-G model). In under-estimation set, the lowest and highest error rates were 28.93 J.cm−2.day−1 (SVM-G, first third interval) and 420.69 J.cm−2.day−1 (the first model of Jahani et al., last third interval), respectively. An important point worth mentioning about the under-estimation set is the relatively low error rates in the first third interval, with all models except the first model of Jahani et al. having estimation errors below 90 J.cm−2.day−1. In over-estimation set, there is almost no consistency in error rates between the three intervals. The lowest errors rates of SVM-G, SVM-G-CSA and the first model of Jahani et al. occurred in the last third interval; whereas for SVM-M, SVM-M-CSA, SVM-MLP and SVM-CSA-MLP the lowest error rates occurred in the first third interval. SVM-G-CSA (RMSE = 160.4), SVM-CSA-MLP (RMSE = 164.74) and the first model of Jahani et al. (RMSE = 165.99) exhibited the best performance, and SVM-G (RMSE = 666.17) had the poorest performance in over-estimation set.

Conclusions

In the current research, we evaluated the performance of empirical models, two AI models (SVM and SVM-CSA), and the novel “multi-model” approach, in estimating daily Rs values at seven Iranian meteorological stations over 2010–2019. For the first structure, model performances were examined separately at each station and in training and testing sets (local analysis). For the second structure (external analysis), an attempt was made to examine the generalisation capability of the models by separating the data used for model training (Ahvaz, Isfahan, Kermanshah and Mashhad stations) from those used for testing (Bandar Abbas, Kerman and Tabriz stations). The results showed that overall, meteorological parameters have played a more effective role in estimating radiation compared to geographical parameters. Considering the atmospheric conditions including energy transferring and sunshine duration, is one of the main advantages of meteorological compared to geographical parameters. SVM-CSA significantly improved radiation estimates at all stations except Isfahan, where Garj-Garj empirical model performed equally well and was comparable to SVM-CSA. All models except SVM-G at Bandar Abbas station, Garj-Garj model, SVM-M-CSA and the multi-model approach at Kerman station and SVM-CSA-MLP at Tabriz station could be effectively and efficiently generalised to stations that played no role in training those models (second structure). Although the multi-model approach demonstrated a much better performance under both structures and at most stations compared with empirical models and the base SVM, it is not preferable to the SVM-CSA model given that its superiority over SVM-CSA is negligible at most stations (except Tabriz) on the one hand, and it requires longer and more complex computations on the other hand. Using meteorological as well as geographical inputs and considering the ability of multi-model approach beside AI-based models are the main advantages of the proposed models in the present study. It appears that further studies at climatically diverse stations are needed before recommending the use of the multi-model approach for radiation estimation.