Effects of Topography and Geography on Solar Diffuse Fraction Modeling in Taiwan

Abstract

A correlation model for the diffuse fraction was recently developed on the basis of a data set obtained in the western part of the Taiwanese mainland. However, it is widely agreed that no existing diffuse fraction correlation model is applicable to all geographical regions and climatic conditions, which is a viewpoint stated from a macro perspective. This study re-justifies this viewpoint through the consideration of a rather small geographical region: Taiwan. The topographic profile of the Taiwanese mainland primarily comprises the high-rise Central Mountain Ranges running from north–northeast to south–southwest, which separate the mainland into eastern and western parts. Furthermore, there are a number of small, remote islands around the Taiwanese mainland. The humidity over the sky dome of these small islands, carried from the moist sea (or ocean) air, is usually greater than that of the Taiwanese mainland. This results in different diffuse fraction patterns between these two geographical regions due to the climatic factor of atmospheric constituents. Two diffuse fraction correlation models for Taiwan were developed using in situ data sets for the eastern part of the Taiwanese mainland and an island in the Penghu archipelago, respectively. In particular, one case considered the topographic effect on modeling the diffuse fraction in Taiwan, while the other considered the geographical effect. Statistical assessments indicate that each correlation model developed in the present study performed better than the previous one developed using the in situ data set for the western part of the Taiwanese mainland, with both applied to the specific site where the data set was used for the model’s development. This work demonstrates the need to consider the effects of topography and geography when modeling the diffuse fraction in Taiwan.

1. Introduction

Information about solar radiation on the Earth’s surface is an important parameter, as it controls a remarkable variety of factors necessary for life such as the atmospheric environment [1], terrestrial climate [2], and terrestrial ecosystems [3]. Fluctuations in solar radiation intensity occur due to the geometry of the Earth relative to the sun [4], changes in atmospheric constituents [5], and variations in the coverage of clouds [6]. Therefore, cloud coverage and atmospheric constituents are two factors that play significant roles in determining the solar radiation properties at a given site. These two factors vary in space and time and lead to the observed variability in solar global and diffuse radiation, even in small geographical regions such as Taiwan (with a total area of 36,006 ${\mathrm{km}}^2$) [7,8].

While imported fossil fuels have been the main energy source in Taiwan for years, the use of renewable energy has received an increasing amount of public support. In addition, the use of renewable energy helps to decarbonize the energy system which, in Taiwan, is still heavily dependent on burning fossil fuels. Solar energy is one of two key renewable energy resources in Taiwan, with the other being wind energy. The harnessing of solar energy has witnessed significant advancements in recent years. Advances in the use of solar energy are not limited to solar panel efficiency improvements or energy storage technologies; they also extend to the way in which solar radiation data are analyzed and used.

Global solar radiation ($I_{global}$) in the sky comprises beam radiation ($I_{beam}$) and diffuse radiation ($I_{diffuse}$):

$$I_{global}=I_{beam}+I_{diffuse}$$

(1)

Beam radiation data are a prerequisite for energy assessments related to concentrating solar applications, which are sometimes applied to large-scale systems; nevertheless, there is a lack of beam and diffuse radiation information in daily reports from all weather stations of the Central Weather Bureau (CWB) of the Republic of China (Taiwan). Very recently, a database of global solar radiation for a typical meteorological year (TMY) has been established, derived from 30 weather stations of the CWB across Taiwan (Figure 1); most of these stations are located on the Taiwanese mainland, with a few stations on various remote islands or islets (hereafter called islands for simplicity) [7]. Figure 1 shows that mountains and hills account for nearly two-thirds of the Taiwanese mainland. Five mountain ranges (hereafter named the Central Mountain Ranges) run from north–northeast to south–southwest on the Taiwanese mainland, separating the mainland into eastern and western parts. The land slopes gently to broad plains/basins to the Taiwan Strait in the west. The precipitous mountains in the Central Mountain Ranges have more than 200 peaks with elevations of more than 3000 m, which descend to the Pacific Ocean in the east. A study by Hsieh et al. [7] showed that there is a significant topographic effect on global radiation at the weather stations located between the eastern and western parts of the Taiwanese mainland and that there is a significant geographic effect on global radiation at the weather stations located between the mainland and either the island in the Pacific Ocean or the island in the Taiwan Strait. Monitoring of the eastern and western parts of the Taiwanese mainland and one remote island in the Penghu archipelago by Chang et al. [8] also revealed significant seasonal variations in the diffuse fractions observed at these three sites due to topographical and geographical effects.

Beam radiation can be directly measured using a pyrheliometer, which uses a collimated detector to measure solar radiation at normal incidence on a small portion of the sky around the sun’s disc through the use of a sun-tracking device. In contrast to the relatively complicated measurement approach for beam radiation, diffuse radiation can be measured with two sets of pyranometers without the need for a sun-tracking device. One pyranometer measures the global radiation, while the other is equipped with a shadow band to measure the diffuse radiation. Beam radiation can be calculated using Equation (1) when the global and diffuse radiation values are readily obtained. Accordingly, there are more measured diffuse radiation data than directly measured beam radiation data in the literature. Although there exist several world maps of solar radiation, they are not detailed enough to be used for the determination of the available solar energy in small regions. This situation has prompted the development of correlation models to provide radiation estimates for areas where ground-based measurements are not carried out. Considering that there are no available diffuse radiation data in the daily reports from all CWB weather stations, an empirical approach employing meteorological data and regression techniques can be used to estimate the diffuse radiation data for Taiwan. The beam radiation is then calculated, using Equation (1), from the estimated diffuse radiation data together with the measured global radiation data for each CWB weather station.

Almost all correlation models have been developed to estimate the solar diffuse fraction (d), which is defined below, instead of solar diffuse radiation, using predictors for global radiation and other meteorological factors:

$$d=\raisebox{1ex}{$I_{diffuse}$}\!\left/ \!\raisebox{-1ex}{$I_{global}$}\right.$$

(2)

Kambezidis et al. [6] recently developed a universal methodology to estimate the upper and lower limits for the diffuse fraction. Such information can be applied to classify the sky condition into clear, intermediate, or overcast at any site in the world. Nevertheless, it is generally agreed that no existing correlation models are applicable to all geographical regions and climatic conditions [9,10,11,12,13,14,15]. For example, Badescu et al. [9] performed sensitivity tests using 54 correlation models, with the input being testing data from two weather stations in Romania. They claimed that no model was ranked the best for all sets of input data. This is because each of the correlation models was derived for a specific site using the meteorological data for that site, which was not one of the two tested sites in Romania. Despotovic et al. [11] conducted similar tests using 50 correlation models but on a global scale, using local long-term meteorological data from 267 different sites around the world. They concluded that there was no general diffuse fraction model that was applicable to any site in the world. Berrizbeitia et al. [14] pooled the hourly global and diffuse radiation data from 19 different sites worldwide to obtain three latitude-dependent regression models relating the month-averaged hourly diffuse ratio to the clearness index, with each showing a high relationship between these two variables. However, they confessed that it was not possible to obtain a unique regression curve for all sites in the latitude group under the study. Hung [12] evaluated 21 correlation models using meteorological data for Taipei, which is located in northern Taiwan. The model comparison results in the study indicated that the model developed by Kuo et al. [16] using the training data from a site in southern Taiwan ranked first of all the tested models, as the coefficients of Kou’s model were locally tuned using data measured in Taiwan.

To overcome the site-dependent restriction in the development of correlation models, Every et al. [13] developed a model capable of covering all Australian climate conditions by means of the Köppen–Geiger climate classification system [17]. Although this model can, in general, better match all the local diffuse fraction estimates within the Australian territory, it sacrifices the accuracy of diffuse fraction estimation for any specific climate condition (or region) of interest in Australia. In contrast to the use of the Köppen–Geiger climate classification system for modeling, Li et al. [18] adopted four weather types—defined in terms of the hourly clearness index—as the classification basis for modeling. They combined five classical diffuse fraction models collected from the literature and individually determined the weights for each model according to the weather type. The training data set was the TMY data for global and diffuse radiation in Beijing. The approach performed better than any selected classical model in tests using the sub-typical year’s radiation data as the testing data. However, the applicability of this approach to combinations of different diffuse fraction models available in the literature and to different climate conditions remains to be further validated.

Very recently, Lin et al. [15] developed three correlation models that use TMY data for global and diffuse radiation, which were measured at the Kuei-Jen campus of National Cheng Kung University, Tainan (22°56′ N, 121°16′ E), in the western part of the Taiwanese mainland (see Figure 1). Of the three developed correlation models in their study, the piecewise linear multiple predictor correlation model—which is a function of the hourly and daily sky clearness indices ($k_t$ and $K_T$, respectively), the persistence of the global radiation level ($\psi$), the solar altitude angle ($\alpha$, in rad), and the apparent solar time (AST, in h), as given below—performed better than the modified Boland–Ridley–Lauret-type model [19], which also uses multiple predictors, and the modified Liu–Jordan-type model [20], which uses a single predictor (i.e., $k_t$).

$$\left\{\begin{array}{c}d=1.0 0\le k_t<0.2412\\ d=\min \left(1.0,1.2207-0.6179k_t-0.1993\psi \right) 0.2412\le k_t<0.4091\\ d=\min (1.0, 1.6215-1.5379k_t+0.1486\alpha +0.006AST-0.3186K_T-0.4051\psi \\ 0.4091\le k_t<0.7222\\ d=0.9718-0.3805K_T-0.7169\psi k_t\ge 0.7222\end{array}\right.$$

(3)

with

$$k_t=\raisebox{1ex}{$I_{global}$}\!\left/ \!\raisebox{-1ex}{$I_0$}\right.$$

(4)

$$K_T=\raisebox{1ex}{$\sum _{i=1}^{24}{I}_{global, i}$}\!\left/ \!\raisebox{-1ex}{${\sum }_{i=1}^{24}{I}_{0,i}$}\right.$$

(5)

$$\psi =\left\{\begin{array}{c}(k_{t-1}+k_{t+1})/2 \\ k_{t+1} for t=sunrise \\ k_{i-1} for t=sunset \end{array}\right.$$

(6)

where $I_0$ is the hourly extraterrestrial global horizontal radiation for a one-hour period between the hour angles ${\omega }_1$ and ${\omega }_2$, expressed as [21]:

$$I_0=\frac{12\times 3600}{\mathsf{\pi }}{G}_{sc}\left(1+0.033 cos\frac{360 n}{365}\right)\left[cos\varphi cos\delta \left(sin{\omega }_2-sin{\omega }_1\right)+\frac{\pi ({\omega }_2-{\omega }_1)}{180}sin\varphi sin\delta \right]$$

(7)

where $G_{sc}$ is the solar constant, n represents the $n$th day of the year, $\varphi$ is the latitude, $\mathsf{\delta }$ is the sun declination angle, and $\omega$ is the hour angle (in degrees).

It is well agreed that the values of solar resources such as global radiation, beam radiation, and diffuse fraction vary from one location to another. This variation—besides local meteorological conditions—depends on the geographical location such as those observed in previous studies in Taiwan [7,8]. To determine the most suitable correlation models for the diffuse fraction to account for topographic and geographic effects, instead of a unique universal correlation model for the entire terrain of Taiwan, the correlation model described by Equation (3)—which was developed using in situ data sets for global and diffuse radiation measured at a site in the western part of the Taiwanese mainland—was re-modelled using two other in situ data sets. One was measured at the Taitung campus of National Open University, Taitung (22°45′ N, 121°07′ E), in the eastern part of the Taiwanese mainland (see Figure 1), while the other was measured at Penghu University of Science and Technology (23°35′ N, 119°35′ E) on an island of the Penghu archipelago in the Taiwan Strait (see Figure 1). The re-modeling approaches were the same as those used for the development of Equation (3) [15].

2. Experimental Method

2.1. Determination of TMY

The data set that was used to model the diffuse fraction correlation is based on the TMY data of the measured global radiation at a specific site. A TMY includes 12 typical meteorological months (TMMs), as determined by the TMY 3 method [22], for each of the 12 calendar months using a long-term database. The TMM selection procedure was the same as that used in the study of Hsieh et al. [7]. The selection procedure begins with the generation of the specified daily weather parameters. Each set of daily weather parameters is stored by month and is used to establish 12 long-term cumulative distribution functions (CDFs) that cover the entire measurement period. The value for each calendar month is then determined. The short- and long-term CDFs for each weather parameter x are written as the function $S_n\left(x\right)$:

$$S_n\left(x\right)=\left\{\begin{array}{c}0 \mathrm{for} x\le x_1 \\ (k-0.5)/n \mathrm{for} x_k\le x\le x_{k+1} \\ 1 \mathrm{for} x\ge x_{n }\end{array}\right.$$

(8)

where n is the total number of elements, and k is the ranked order number. Candidate monthly CDFs are compared to long-term CDFs using the Finkelstein–Schafer (FS) statistics [23], defined as:

$$\mathrm{F}\mathrm{S}=\frac{1}{m}\sum _{k=1}^m{\sigma }_k$$

(9)

where ${\sigma }_k$ is the absolute difference between the long-term CDF and the candidate month CDF at $x_k$, and m is the number of daily readings over the long-term period for the candidate month. The lower the FS value for a month, the closer that month is to a typical month for a specific weather parameter. In general, some weather parameters may be more important than others; therefore, a weighted sum (WS) of the FS statistics is calculated as follows:

$$\mathrm{W}\mathrm{S}=\sum _{j=1}^J{\omega }_j{\left(\mathrm{F}\mathrm{S}\right)}_j$$

(10)

with

$$\sum _{j=1}^J{\omega }_j=1$$

(11)

where ${\omega }_j$ is the weighting factor for the jth weather parameter, and J is the total number of weather parameters. The weighting factors for the weather parameters in Equation (10) are assigned according to the impact of each weather parameter on the performance of a specific solar energy application. The study by Hsieh et al. [7] showed that the full weight for global radiation can be reasonably accepted in Equation (10) to calculate the TMY for non-concentrating solar applications and, as such, was adopted in the study.

Five candidate months with the lowest WS values were selected. The final selection of each TMM from the five candidate months was performed using the root-mean-square difference (RMSD) [7,15,24,25,26], defined as:

$${\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{D}}_k={\left[\sum _{\mathcal{l}=1}^{N_k}{\left(x_{k\mathcal{l}}-{\overline{x}}_{\mathcal{l}}\right)}^2/N_k\right]}^{1/2}$$

(12)

where the indices k and $\mathcal{l}$ denote the year and hour of the day, respectively, $N_k$ is the number of hours in the kth year, and ${\overline{x}}_{\mathcal{l}}$ is the average value for the long-term data for the ${\mathcal{l}}^{th}$ hour of the day. The month with the smallest RMSD value was selected as the TMM.

2.2. Experimental Setup and Quality Check of Data

Two stations were established to measure the global and diffuse radiation on a remote island (at Penghu University of Science and Technology on the Penghu archipelago) and in eastern Taiwan (at National Open University in Taitung County). Each station used two sets of pyranometers (Model SMP11 of Kipp & Zonen, Delft, The Netherlands); one was used for the measurement of global irradiance, while the other was equipped with a shadow band stand (Model SBS of Eppley Laboratory) for the measurement of diffuse irradiance. More information on the operation and correction to compensate for blocked diffuse irradiance by the shadow band is given in the recent study by this research group [15] and is not repeated here. The sampling rate for the pyranometers was 1 Hz. The missing data for individual seconds or minutes—which mostly occurred during transmission—were recreated by interpolating a cubic spline, as described in another recent study by this research group [7].

To ensure the quality of the data for model development and comparison, the measured data for the global and diffuse irradiances were examined using two flagging procedures, as listed in

Table 1. Lower and upper limits of the extremely rare intervals for flagging the radiation quantities (excerpted from [27]).

Parameter	$\mathbf{Lower} \mathbf{Bound} (\mathbf{W}/{\mathbf{m}}^2$)	$\mathbf{Upper} \mathbf{Bound} (\mathbf{W}/{\mathbf{m}}^2$)
$G_g$	−2	$1.2G_{sc}{\mu }^{1.2}+50$
$G_d$	−2	$0.75G_{sc}{\mu }^{1.2}+30$

where $\mu =cos{\theta }_z$.

and

Table 2. Condition for test of across-quantity intervals for radiation quantities (excerpted from [27]).

$d<1.05 \mathrm{for} G_g>50 \mathrm{W}/{\mathrm{m}}^2 \mathrm{and} {\theta }_z<{75}^{°}$

$d<1.10 \mathrm{for} G_g>50 \mathrm{W}/{\mathrm{m}}^2 \mathrm{and} {75}^{°}<{\theta }_z<{93}^{°}$

[27]. These are used by the Baseline Surface Radiation Network [28] and were used in a previous study by this research group [15]. Measured data that complied with the quality checks of Table 1 and Table 2 were considered acceptable data for individual seconds and transformed to an hourly value for model development and comparison.

3. Methodology for Model Development and Performance Test

3.1. Collection of Data Set for Regression Modeling

A TMY for the global radiation data was first generated at a given station. All diffuse radiation data for 12 TMMs were used as the training data set to develop the correlation model for the diffuse fraction at the station. The raw diffuse radiation data were examined through the quality checks in Table 1 and Table 2 in order to eliminate poor-quality data. The approved data were then used in the following regression analysis.

3.2. Regression Model Analysis

A statistical study using regression analysis of the filtered hourly $d-k_t$ data for the piecewise linear multiple predictors model, in terms of the same predictors as in Equation (3), was performed as follows. Following the model of Equation (3), four segments were established, where each segment was established using the library “Segmented” in the R software (Version R 4.2.3) [29].

Developing the multiple predictor correlation model involved identifying the predictors that were significantly correlated to the diffuse fraction using an iterative stepwise regression with a backward elimination rule [30]. The backward elimination rule starts with all predictors and then discards the least statistically significant predictor one by one until the remaining predictors in the model are all statistically significant. To determine the multiple collinearity between the five predictors, a p-value test [31] and the coefficient of determination ($R^2$), as defined below, were used to determine the significance of the selected predictors.

$$R^2=1-\left[\sum _{i=1}^n{\left(d_{est,i}-d_{mea,i}\right)}^2/\sum _{i=1}^n{\left(d_{mea,i}-{\overline{d}}_{mea}\right)}^2\right]$$

(13)

$${\overline{d}}_{mea}=\frac{1}{n}\sum _{i=1}^n{d}_{mea,i}$$

(14)

A p-value of less than 0.005 indicates strong significance, a p-value between 0.005 and 0.05 indicates a moderate level of significance, and a p-value greater than 0.05 denotes that the result is not significant. The coefficient of determination ($R^2$, bounded in [0, 1]) indicates the proportion of the variance for a dependent variable that is expressed by a single or multiple independent variables in a regression model. The greater the $R^2$ value, the better the fit between the observed and the predicted data.

After determination of the significant indicators in all the segments was completed, the regression submodel for each segment in terms of the determined significant indicators was formulated. Finally, a combination of all the regression submodels for the four segments yielded the proposed piecewise linear multiple predictors model for the diffuse fraction at the station.

3.3. Test of Model Performance

Five common statistical indicators were used, including the root-mean-square error (RMSE), the mean absolute error (MAE), the mean absolute percentage error (MAPE), the standard deviation (SD), and the coefficient of determination ($R^2$), which are, respectively, defined by Equations (15)–(18) and Equation (13).

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}={\left[\frac{1}{n}\sum _{i=1}^n{\left(d_{bias,i}\right)}^2\right]}^{1/2}$$

(15)

$$\mathrm{M}\mathrm{A}\mathrm{E}=\frac{1}{n}$$

(16)

$$\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}=\frac{1}{n}\left|\raisebox{1ex}{$d_{bias,i}$}\!\left/ \!\raisebox{-1ex}{$d_{mea,i}$}\right.\right|\times 100\%$$

(17)

$$\mathrm{S}\mathrm{D}={\left[\frac{1}{n-1}\sum _{i=1}^n{{(d}_{bias,i}-{\overline{d}}_{bias})}^2\right]}^{1/2}$$

(18)

where

$$d_{bias,i}=d_{mea,i}-d_{est,i}$$

(19)

$${\overline{d}}_{bias}=\frac{1}{n}\sum _{i=1}^n{d}_{bias,i}$$

(20)

These five statistical indicators were used to test the performance of the models for this study. Lower values of the RMSE, MAE, MAPE, and SD indicate more accurate predictions using a regression model, while a higher $R^2$ value indicates a better fit between the predicted and the measured data. However, it frequently occurs that not all statistical indicators identify a specific model as being the most accurate. The global performance indicator (GPI) [32], defined below, is a combined statistical indicator that was used for statistical analyses of the solar diffuse fraction [7,33] and of global solar radiation sunshine models [34].

$${GPI}_i=\sum _{j=1}^5{\omega }_j\left({\overline{y}}_j-y_{ij}\right)$$

(21)

where ${\overline{y}}_j$ is the median (i.e., 50th percentile) of the scaled values of the jth statistical indicator, $y_{ij}$ is the scaled value of the jth statistical indicator for the ith model, and ${\omega }_j$ is the weighting factor for the jth statistical indicator, which takes a value of 1 for all statistical indicators except for the coefficient of determination, for which it takes a value of −1, as its value-judgment trend is opposite to those of the other four indicators. The greater the value of the GPI, the more accurate the model.

4. Results and Discussion

4.1. Correlation Model for the Diffuse Fraction for Remote Island

A TMY for the global radiation data measured at Penghu University of Science and Technology in the period between April 2015 and December 2018 (3 years and 9 months in total) was generated and is presented in

Table 3. List of the TMMs for the stations on a remote island of the Penghu archipelago and in Taitung (on the eastern side of the Taiwanese mainland).

Station	Jan.	Feb.	Mar.	Apr.	May	Jun.	Jul.	Aug.	Sep.	Oct.	Nov.	Dec.
Penghu archipelago	2018	2018	2017	2017	2016	2016	2017	2016	2018	2015	2016	2018
Taitung	2018	2017	2016	2017	2016	2017	2017	2016	2018	2017	2016	2016

. All the diffuse radiation data for the 12 TMMs shown in Table 3 were used as the training data set to develop the correlation model for the diffuse fraction on the remote island. A total of 37% of the raw data were eliminated after examination according to the quality checks described in Table 1 and Table 2. Details of the raw data and the data that complied with the quality checks are provided in Figure 2.

A regression analysis of the filtered hourly $d-k_t$ data for the piecewise linear multiple predictors model, in terms of the same five predictors in Equation (3), was performed as follows. Following the model of Equation (3), four segments were established using the library “Segmented” in the R software [29]. Three break points were determined at 0.2247 (between the first and second segments), 0.3863 (between the second and third segments), and 0.7641 (between the third and fourth segments).

Figure 2 shows that some of the filtered data for the diffuse fraction (d) had values that were slightly greater than unity, which is theoretically unrealistic due to the relationships in Equations (1) and (2), particularly in the low $k_t$ subrange. This is also true for the measurements from National Open University in Taitung County in the eastern part of the Taiwanese mainland (see Figure 3). The greater values (d > 1) using a pyranometer with a shadow band stand were attributed to solar irradiance that was reflected from the ground onto the pyranometer through the bottom side of the shadow band under a very cloudy sky or soon after dawn or near nightfall [15,35]. However, following the method of our previous study [15], all of the filtered data—including the data for which the d values exceeded unity, as shown in Figure 2 and Figure 3—were used to develop the correlation models.

As can be observed from Figure 2, almost all the d values for the filtered data in the first segment ($0\le k_t<0.2247$) were slightly greater than 1.0, which is the upper bound for d defined by Equation (2) together with Equation (1). Thus, the upper bound for d was set for the first segment as d = 1.0. For the second segment ($0.2247\le k_t<0.3863$), the backward elimination rule [30] was employed. The first stepwise regression used the five predictors $k_t, K_T, \alpha$, AST, and $\psi$, as shown in

Table 4. Summary of the p-values and $R^2$ for the (a) second, (b) third, and (c) fourth segments of the piecewise multiple predictor regression process for the data set from Penghu University of Science and Technology.

(a) Second segment $0.2247\le k_t<0.3863$
Predictors	$k_t$	${{\rm K}}_T$	$\alpha$	AST	$\psi$	$R^2$
$k_t, {{\rm K}}_T, \alpha , AST, \psi$	<0.001	0.023	0.389	0.159	0.333	0.341
$k_t, {{\rm K}}_T, AST, \psi$	<0.001	0.032	-	0.164	0.234	0.340
$k_t, {{\rm K}}_T, AST$	<0.001	<0.001	-	0.155	-	0.337
$k_t, {{\rm K}}_T$	<0.001	<0.001	-	-	-	0.333
(b) Third segment $0.3863\le k_t<0.7641$
Predictors	$k_t$	${{\rm K}}_T$	$\alpha$	AST	$\psi$	$R^2$
$k_t, {{\rm K}}_T, \alpha , AST, \psi$	<0.001	<0.001	<0.001	0.160	<0.001	0.864
$k_t, {{\rm K}}_T, \alpha , \psi$	<0.001	<0.001	<0.001	-	<0.001	0.864
(c) Fourth segment $0.7641\le k_t\le 1.0$
Predictors	$k_t$	${{\rm K}}_T$	$\alpha$	AST	$\psi$	$R^2$
$k_t, {{\rm K}}_T, \alpha , AST, \psi$	0.255	0.340	<0.001	0.068	0.009	0.682
$k_t, \alpha , AST, \psi$	0.265	-	<0.001	0.052	<0.001	0.679
$\alpha , AST, \psi$	-	-	<0.001	0.053	<0.001	0.654
$\alpha , \psi$	-	-	<0.001	-	<0.001	0.650

Each cell records a p-value except the cells in the last column, which record the $R^2$ values. In red, representing the largest value of the predictors in the row.

a. The p-value for the $\alpha$ predictor (0.389 > 0.05) was greatest (marked in red color, Table 4a), and so $\alpha$ was discarded. The next stepwise regression used the four remaining predictors: $k_t, K_T,$ AST, and $\psi$. The greatest p-value was for the $\psi$ predictor (0.234 > 0.05, marked in red color, Table 4a), which was thus also discarded. This process was repeated until there were only two predictors—namely, $k_t$ and $K_T$ (see Table 4a)—for which the calculated p-values were all less than 0.001 and, so, were highly significant. The corresponding $R^2$ value was 0.333 (see Table 4a). The regression sub-model for the second segment was thus formulated using two predictors: $k_t$ and $K_T$. A stepwise regression process was applied to the third segment ($0.3863\le k_t<0.7641)$ starting with the five predictors; the calculated p and $R^2$ values are listed in Table 4b. The p values for $k_t, K_T, \alpha$, and $\psi$ using the stepwise regression for these four predictors (after discarding AST, for which the p value was 0.16 > 0.05 in the first backward elimination process; see Table 4b for details) were all less 0.001, indicating high significance; the corresponding $R^2$ value was 0.864. Thus, the regression sub-model for the third segment was formulated using these four predictors. A stepwise regression process was applied to the last (fourth) segment ($0.7641\le k_t\le 1.0)$ using the five predictors; the calculated p and $R^2$ values are listed in Table 4c. After discarding the predictors $K_T, k_t,$ and AST, for which the p-values were greater than 0.05 in the first three backward elimination process steps (see Table 4c for details), the p-values for $\alpha$ and $\psi$ using the stepwise regression for only these two predictors were less than 0.001 with a corresponding $R^2$ value of 0.650. Thus, the regression sub-model for the fourth segment was formulated using $\alpha$ and $\psi$ as predictors.

The proposed piecewise linear model for these five predictors through regression analysis in each segment is expressed as follows:

$$\left\{\begin{array}{c}d=1.0 0\le k_t<0.2247\\ d=\min \left(1.0,1.1588-0.4836k_t-0.0557K_T\right) 0.2247\le k_t<0.3863\\ d=\min (1.0, 1.7653-1.6041k_t-0.4417K_T+ 0.3863\le k_t\le 0.7641\\ 0.1012\alpha -0.2239\psi ) \\ d=0.9886-0.2061\alpha -0.7357\psi 0.7641\le k_t\le 1.0\end{array}\right.$$

(22)

The “minimum function” operator was imposed upon the regression sub-formulae for the second and third segments to prevent the predictions for d from exceeding the theoretical upper bound of unity. A graphical comparison of the diffuse fractions predicted using Equation (22) and the filtered data versus the sky clearness index ($k_t$) is presented in Figure 2, which shows that the predictions for d matched the filtered data well but did not cover the dispersion range for the measurement database for d. This also indicates that the “minimum function” operator had to be imposed upon the regression sub-formulae for the second and third segments in order to prevent some d predictions around $k_t\simeq 0.4$ from exceeding the upper bound of unity.

4.2. Correlation Model for the Diffuse Fraction for the Eastern Part of the Taiwanese Mainland

A TMY for the global radiation data measured at National Open University in Taitung County from January 2016 to December 2018 (3 years) was generated, as presented in Table 3. All the diffuse radiation data for the 12 TMMs in Table 3 were used for the TMY as the training data set used to develop the correlation model for the diffuse fraction for the eastern part of the Taiwanese mainland. Using the same data analysis method as in the previous subsection, 43% of the raw data were eliminated after the quality checks detailed in Table 1 and Table 2. The raw and filtered data are shown in Figure 3.

Using the model of Equation (3), four segments were established using the library “Segmented” in the R software [29]. The three break points were determined at 0.2876 (between the first and second segments), 0.4366 (between the second and third segments), and 0.7548 (between the third and fourth segments). Figure 3 shows that almost all of the filtered d data had a value that was slightly greater than unity in the first segment and, so, the upper bound for d (i.e., d = 1.0) was set for the first segment ($0\le k_t<0.2876)$. The stepwise regression process initially using the five predictors $k_t, K_T, \alpha , AST,$ and $\psi$ was repeatedly applied for the second ($0.2876\le k_t<0.4566$), third ($0.4366\le k_t<0.7548$), and fourth $(0.7548\le k_t\le 1$) segments; the calculated p and $R^2$ values are, respectively, listed in

Table 5. Summary of the p and $R^2$ values for the (a) second, (b) third, and (c) fourth segments of the piecewise multiple predictor regression process for the data set from National Open University in Taitung County.

(a) Second segment $0.2876\le k_t<0.4366$
Predictors	$k_t$	${{\rm K}}_T$	$\alpha$	AST	$\psi$	$R^2$
$k_t, {{\rm K}}_T, \alpha , AST, \psi$	<0.001	0.013	0.059	0.632	0.003	0.395
$k_t, {{\rm K}}_T, \alpha , \psi$	<0.001	0.006	0.078	-	0.004	0.395
$k_t, {{\rm K}}_T, \psi$	<0.001	0.003	-	-	0.008	0.389
(b) Third segment $0.4366\le k_t<0.7548$
Predictors	$k_t$	${{\rm K}}_T$	$\alpha$	AST	$\psi$	$R^2$
$k_t, {{\rm K}}_T, \alpha , AST, \psi$	<0.001	<0.001	<0.001	0.060	<0.001	0.863
$k_t, {{\rm K}}_T, \alpha , \psi$	<0.001	<0.001	<0.001	-	<0.001	0.861
(c) Fourth segment $0.7548\le k_t\le 1$
Predictors	$k_t$	${{\rm K}}_T$	$\alpha$	AST	$\psi$	$R^2$
$k_t, {{\rm K}}_T, \alpha , AST, \psi$	<0.001	<0.001	0.110	0.259	<0.001	0.720
$k_t, {{\rm K}}_T, \alpha , \psi$	<0.001	<0.001	0.138	-	<0.001	0.717
$k_t, {{\rm K}}_T, \psi$	<0.001	<0.001	-	-	<0.001	0.710

Each cell records a p-value except the cells in the last column, which record the $R^2$ values. In red, represnting the largest value of the predictors in the row.

a–c. The results in Table 5 show that the regression sub-models for the second, third, and fourth segments were formulated using the three predictors $k_t, K_T, \mathrm{a}\mathrm{n}\mathrm{d} \psi$; the four predictors $k_t, K_T, \alpha , \mathrm{a}\mathrm{n}\mathrm{d} \psi$; and the three predictors $k_t, K_T, \mathrm{a}\mathrm{n}\mathrm{d} \psi$, respectively. The proposed piecewise linear model using the five predictors through regression analysis in each segment is expressed as follows:

$$\left\{\begin{array}{c}d=1.0 0\le k_t<0.2876\\ d=\min \left(1.0,1.3527-0.8494k_t-0.1469K_T-0.1304\psi \right) 0.2876\le k_t<0.4366\\ d=\min (1.0, 1.8375-1.7883k_t-0.4991K_T+ 0.4366\le k_t<0.7548\\ 0.1691\alpha -0.2669\psi ) \\ d=-0.2284-1.3803k_t-0.4153K_T-0.5520\psi 0.7548\le k_t\le 1\end{array}\right.$$

(23)

A graphical comparison of the diffuse prediction using Equation (23) and the filtered data versus $k_t$ is presented in Figure 3, which shows that the predictions for d matched the filtered data well but, again, they did not cover the entire dispersion range of the measurement database for d.

4.3. Performance Assessments for the Developed Models

To examine the superiority of the two correlation models developed for the remote island—Equation (22), accounting for the geographical effect—and the eastern part of the Taiwanese mainland—Equation (23), accounting for the topographic effect—the performance of the correlation model—Equation (3), which was developed using a training data set generated for the western part of the Taiwanese mainland [7]—was compared with that of Equations (22) and (23).

To compare the performance of the two tested models using Equations (3) and (22) at Penghu University of Science and Technology on a remote island of the Penghu archipelago, the test data for 33 months (45 months in total, excluding the 12 TMMs; from April 2015 to December 2018) that passed the quality check were used as an independent data set. The five performance indicators for each correlation model for each year were calculated, and the results for each correlation model are listed in

Table 6. Overall results for MAE, RSME, MAPE, SD, and $R^2$ for the two correlation models, Equations (3) and (22), that were developed using data for (a) the western part of the Taiwanese mainland and (b) the remote islands of the Penghu archipelago.

	(a) Equation (3)					(b) Equation (22)
Year	MAE	RSME	MAPE (%)	SD	${\mathit{R}}^2$	MAE	RSME	MAPE (%)	SD	${\mathit{R}}^2$
2015	0.064	0.086	15.92	0.085	0.923	0.065	0.086	18.21	0.085	0.923
2016	0.065	0.086	13.74	0.081	0.925	0.060	0.081	14.14	0.081	0.933
2017	0.064	0.085	14.60	0.081	0.934	0.061	0.080	15.78	0.080	0.941
2018	0.068	0.092	16.61	0.089	0.903	0.065	0.088	17.06	0.088	0.910

. As can be observed from Table 6, it is hard to judge which correlation model performed better compared to the other on the basis of the individual results obtained with each of the five statistical indicators. The GPI for each year was calculated using Equation (21) together with the results shown in Table 6 for each model. The GPI values for all the examined years and models are listed in

Table 7. GPI results for the two correlation models using Equations (3) and (22).

Model	2015	2016	2017	2018
Equation (22)	2.06	0.66	0.74	2.21
Equation (3)	0.15	−1.80	−1.20	0.28

. The GPI comparison results in Table 7 show that the developed correlation model—namely, Equation (22)—performed better for the remote island of the Penghu archipelago than the model developed using the data set measured at the station in the western part of the Taiwanese mainland. The reason leading to this comparison difference is as follows. The Penghu archipelago is located in the Taiwan Strait (see Figure 1). The valley effect—which occurs during the Northeast Monsoon season in the strait—results in windy and foggy weather in the Penghu archipelago. In particular, the moist sea air over the sky dome of the small island for the whole year leads easily to higher humidity on the remote island than on the Taiwanese mainland. The high humidity in the atmosphere weakens the intensity of sun rays passing through it, thus reducing the beam irradiance that is incident on the ground. Thus, the diffuse fraction pattern was certainly different from that observed on the Taiwanese mainland, which was considered as a geographic effect on the diffuse fraction in this study.

The performance of the two models using Equations (3) and (23) was compared for the data set collected at National Open University in Taitung County in the eastern part of the Taiwanese mainland. The test data for 24 months from 2016 to 2018, excluding the 12 TMMs, which passed the quality check were used as an independent data set. The five performance indicators for each correlation model for each year were calculated, and the results for each correlation model are listed in

Table 8. Overall results for MAE, RSME, MAPE, SD, and $R^2$ for the two correlation models, Equations (3) and (23), that were developed using data for (a) the western part and (b) the eastern part of the Taiwanese mainland, respectively.

	(a) Equation (3)					(b) Equation (23)
Year	MAE	RSME	MAPE (%)	SD	${\mathit{R}}^2$	MAE	RSME	MAPE (%)	SD	${\mathit{R}}^2$
2016	0.071	0.091	20.58	0.091	0.935	0.067	0.090	17.56	0.090	0.936
2017	0.066	0.088	19.02	0.088	0.939	0.059	0.083	14.60	0.083	0.946
2018	0.065	0.089	16.35	0.089	0.934	0.063	0.089	14.53	0.088	0.934

. The GPI for each year was calculated using Equation (21) together with the results shown in Table 8 for each model, and the GPI values for all the examined years and models are listed in

Table 9. GPI results for the two correlation models using Equations (3) and (23).

Model	2016	2017	2018
Equation (23)	0.97	2.12	1.62
Equation (3)	−0.47	−0.52	0.61

. The GPI comparison results in Table 9 show that the developed model, namely, Equation (23), performed better for the eastern part of the Taiwanese mainland than the model developed using data obtained from a station located in the western part of the Taiwanese mainland. As mentioned above, the topographic effect on the diffuse fraction for the Taiwanese mainland mainly stems from the Northeast Monsoon [8]. During the monsoon season (November–March), moist ocean air is blown southwest-bound and is lifted upward by the Central Mountain Ranges, which are almost parallel to the eastern coastline. Therefore, cloudy weather and orographic rain occur frequently in the windward side of the mountains, resulting in high diffuse fraction values in the eastern part of the Taiwanese mainland in this season. In contrast, the effect of the Northeast Monsoon becomes insignificant to the leeward of the Central Mountain Ranges (i.e., the west part of the Taiwanese mainland). The diffuse fraction value is, thus, lower than that in the eastern part in the monsoon season.

5. Conclusions

Taiwan, with a total area of 36,006 ${\mathrm{k}\mathrm{m}}^2$, is a rather small geographical region that consists of a mainland (the Taiwanese mainland) and a number of remote islands (see Figure 1). The Central Mountain Ranges, running north–northeast to south–southwest on the Taiwanese mainland, separate the mainland into two (east and west) parts with specific topographies. The interaction between the high-rise Central Mountain Ranges and the Northeast Monsoon from November to March (i.e., the Northeast Monsoon season) results in remarkable variations in the coverage of clouds between the windward (the eastern part of the Taiwanese mainland) and leeward (the western part of the Taiwanese mainland) sides of the Central Mountain Ranges. Thus, the two piecewise multiple prediction correlation models for the diffuse fraction—namely, Equations (3) and (23)—which were developed using in situ data measured in the eastern and western parts of the Taiwanese mainland, respectively, presented different patterns, which can mainly be attributed to the climatic factor of cloud coverage, thus indicating the effect of topography on the modeling process. The humidity over the sky dome of a small remote island, due to the moist sea air, is usually higher than that in the Taiwanese mainland. The different patterns between the two piecewise multiple prediction correlation models for the diffuse fraction—namely, Equations (3) and (22)—which were developed using in situ data measured in the two different geographic regions—the western part (inland) of the Taiwanese mainland and a remote island of the Penghu archipelago located in the west side of Taiwan mainland, respectively—can mainly be attributed to the climatic factor of atmospheric constituents, indicating the effect of geography on the modeling process. The three piecewise multiple prediction correlation models for the diffuse fraction, accounting for the topographic and geographic effects in Taiwan developed in either our previous study [7] or the present study—that is, Equations (3), (23), and (22)—are, respectively, applicable for the estimation of the diffuse fraction in the western and eastern parts of the Taiwanese mainland as well as on the remote islands around Taiwan. It was shown that such an approach can generate better estimates for the diffuse fraction than simply using the unique correlation model developed in the previous study [15] (i.e., Equation (3)), taken as the national model for the entirety of Taiwan.