Introduction
Existing models and methods
Theoretical analysis
Estimation of hourly direct normal irradiance
Parametric (broadband) models
Equation | Description | ||
---|---|---|---|
\(I_{{{\text{DNI}},{\text{FR }}}} = I_{\text{oN}} \tau_{\text{bulk}}^{{m_{\text{e}} }}\) | (1) | Fu and Rich model | [9] |
\(I_{{{\text{DNI}},{\text{ASH}}}} = A {\text{exp}}\left[ {\frac{ - B}{{\cos \theta_{\text{z}} }}} \right]\) | (2) | ASHRAE model | |
\(I_{{{\text{DNI}},{\text{HLJ}}}} = I_{\text{oN}} \tau_{\text{aa}}\) \(\tau_{\text{aa}} = a_{\text{aa}} + b_{\text{aa}} {\text{exp}}\left[ { - \frac{{c_{\text{aa}} }}{{\cos \theta_{\text{z}} }}} \right]\) | (3) | HLJ model | [22] |
\(I_{{{\text{DNI}},{\text{KUM}}}} = 0.56 I_{\text{oN}} [\exp ( - 0.65 m_{\text{air}} ) + { \exp }( - 0.095 m_{{{\text{air}},{\text{KUM}}}} )]\) \(m_{\text{air}} = m_{\text{r}} p/p_{\text{o}}\) \(m_{\text{r}} = \{ [1229 + \left( {614 \cos \theta_{\text{z}} } \right)^{2} ]^{0.5} - 614 \cos \theta_{\text{z}} \}\) | (4) | Kumer model | [9] |
\(I_{{{\text{DNI}},{\text{HS}}1}} = I_{\text{oN}} \exp ( - m_{\text{air}} \sigma T_{\text{LTF}} )\) \(\sigma = 1/\left( {6.62960 + 1.7513m_{\text{air}} - 0.1202m_{\text{air}}^{2} + 0.0065m_{\text{air}}^{3} - 0.00013m_{\text{air}}^{4} } \right)\) \(m_{\text{air}} = m_{\text{r}} p/p_{\text{o}}\) \(m_{\text{r}} = 1/\cos \theta_{\text{z}}\) \(T_{\text{LTF}}\): Linke turbidity factor [22] | (5) | Heliosat-1 model | [9] |
\(I_{{{\text{DNI}},{\text{ESRA}}}} = I_{\text{oN}} \exp ( - m_{\text{air}} \sigma T_{\text{LTF}} )\) \(m_{\text{air}} = m_{\text{r}} p/p_{\text{o}}\) \(m_{\text{r}} = 35/[(1224 {\text{cos}}^{2} \theta_{\text{z}} ) + 1]^{0.5}\) | (6) | ESRA model | [9] |
\(I_{{{\text{DNI}},{\text{Bird}}}} = 0.9662 I_{\text{oN}} \tau_{\text{total}}\) \(\tau_{\text{total}} = \tau_{\text{rt}} \tau_{\text{ot}} \tau_{\text{gt}} \tau_{\text{wt}} \tau_{\text{at}}\) \(\tau_{\text{rt}} = { \exp }\left[ { - 0.0903 m_{\text{air}}^{0.84} \left( {1 + m_{\text{air}} - m_{\text{air}}^{1.01} } \right)} \right]\) \(\tau_{\text{ot}} = 1 - [0.1611U_{3} (1 + 139.48U_{3} )^{ - 0.3035} - 0.002715U_{3} \left( {1 + 0.044U_{3} + 0.0003U_{3}^{2} )^{ - 1} } \right]\)\(\tau_{\text{gt}} = \exp ( - 0.0127m_{\text{air}}^{0.26} )\) \(\tau_{\text{wt}} = 1 - 2.4959U_{1} [1 + 79.034U_{1} )^{0.6828} + 6.385U_{1} ]^{ - 1}\) \(\tau_{\text{at}} = { \exp }\left[ { - L_{\text{ao}}^{0.873} \left( {1 + L_{\text{ao}} - L_{\text{ao}}^{0.7808} } \right)m_{\text{air}}^{0.9108} } \right]\) \(m_{\text{air}} = m_{\text{r}} p/p_{\text{o}}\) \(m_{\text{r}} = 35/[(1224 {\text{cos}}^{2} \theta_{\text{z}} ) + 1]^{0.5}\) \(L_{\text{ao}} = f\left( {\beta_{1} , \beta_{2} } \right)\) | (7) | Bird model | |
\(I_{{{\text{DNI}},{\text{Hoyt}}}} = I_{\text{o}} \left( {1 - \mathop \sum \limits_{i = 1}^{5} a_{i} } \right) \tau_{\text{as}} \tau_{\text{r}}\) \(m_{\text{air}} = m_{\text{r}} p/p_{\text{o}}\) \(m_{\text{r}} = 35/[(1224 {\text{cos}}^{2} \theta_{\text{z}} ) + 1]^{0.5}\) \(m_{r} = [\cos \theta_{z} + 0.15 (93.885 - \theta_{z} )^{ - 1.253} ]^{ - 1}\) \(a_{1}\), \(a_{2}\), \(a_{3}\), \(a_{4}\),\(a_{5} = f\left( {U_{1} ,U_{3} ,m_{\text{r}} ,m_{\text{a}} ,\tau_{\text{ot}} ,\tau_{\text{as}} } \right)\) | (8) | Hoyt (Iqbal B) model | |
\(I_{{{\text{DNI}},{\text{MET}}}} = 0.9751 I_{\text{oN}} \tau_{\text{total}}\) All transmittances \((\tau_{\text{total}} )\) are similar to Bird model except aerosol transmittance, \(\tau_{\text{at}} = { \exp }\left( { - m_{\text{air}} L_{\text{ao}} } \right)\) \(m_{\text{air}} = m_{\text{r}} p/p_{\text{o}}\) \(m_{\text{r}} = 35/[(1224 {\text{cos}}^{2} \theta_{\text{z}} ) + 1]^{0.5}\) | (9) | METSTAT model | [9] |
\(I_{{{\text{DNI}},{\text{CSR}}}} = C_{\text{CSR}} I_{\text{oN}} \tau_{\text{total}}\) \(C_{\text{CSR}} = \left[ {50 + \left| {{ \cos }\left( {\frac{{N_{j} }}{325}} \right)} \right|} \right]/49.25\) All transmittances \((\tau_{\text{total}} )\) are similar to Bird model except aerosol transmittance, \(\tau_{\text{at}} = { \exp }\left( { - m_{\text{air}} L_{\text{ao}} } \right)\) \(m_{\text{air}} = m_{\text{r}} p/p_{\text{o}}\) \(m_{\text{r}} = 35/[(1224 {\text{cos}}^{2} \theta_{\text{z}} ) + 1]^{0.5}\) | (10) | CSR model | [22] |
\(I_{{{\text{DNI}},{\text{IqbalC}}}} = 0.9751 I_{\text{oN}}\)\(\tau_{\text{total}}\) All transmittances (\(\tau_{\text{total}}\)) are similar to Bird model | (11) | Iqbal model C | [10] |
\(I_{{{\text{DNI}},{\text{MIqbalC}}}} = 0.9751 I_{\text{oN}} \tau_{\text{total}}\) \(\tau_{\text{at}} = \left( {0.12445\beta_{1} - 0.0162} \right) + \left( {1.003 - 0.125\beta_{2} } \right){ \exp }\left[ { - m_{\text{air}} \beta_{1} \left( {1.089 \beta_{2} + 0.5123} \right)} \right]\)\(\tau_{\text{w}} = 1 - 2.4959U_{1} [1 + 79.034U_{1} )^{0.6828} + 6.385U_{1} ]^{ - 1}\) \(U_{1} = W^{{\prime }} m_{\text{r}}\) \(W^{{\prime }} = 0.1\exp \left( {2.2572 + 0.05454 T_{\text{dew}} } \right) = {\text{Won'sequation}}\) \(m_{\text{air}} = m_{\text{r}} p/p_{\text{o}}\) \(m_{\text{r}} = 35/[(1224 {\text{cos}}^{2} \theta_{\text{z}} ) + 1]^{0.5}\) | (12) | Modified Iqbal model C | [22] |
\(I_{{{\text{DNI}},{\text{AWB}}}} = I_{\text{o}} \left( {\tau_{\text{md}} - a_{\text{w}} } \right) \tau_{\text{at}}\) \(\tau_{\text{md}} = 1.041 - 0.16 [m_{\text{r}} (949 \times 10^{ - 6} p + 0.051 )]^{0.5}\) \(a_{\text{w}} = 0.077(U_{1} m_{\text{air}} )^{0.3}\) \(U_{1} = W m_{\text{r}}\) \(W = W^{{\prime }} \left( {\frac{p}{{p_{\text{o}} }}} \right)^{0.75} (T_{\text{o}} /T_{\text{amb}} )^{0.5}\) \(W^{{\prime }} = 0.1\exp \left( {2.2572 + 0.05454 T_{\text{dew}} } \right) = {\text{Won'sequation}}\) \(m_{\text{air}} = m_{\text{r}} p/p_{\text{o}}\) \(m_{\text{r}} = 35/[(1224 {\text{cos}}^{2} \theta_{\text{z}} ) + 1]^{0.5}\) | (13) | Atwater and Ball model. (The model can be used for clear and cloudy sky) | |
\(I_{{{\text{DNI}},{\text{DH}}}} = I_{\text{o}} \left( {\tau_{\text{o}} \tau_{\text{rt}} - \alpha_{\text{w}} } \right) \tau_{\text{A}}\) \(\tau_{\text{o}} = \left\{ {\left[ {\frac{{\left( {1 - 0.02118X_{\text{o}} } \right)}}{{\left( {1 + 0.042X_{\text{o}} + 0.000323X_{\text{o}}^{2} } \right)}}} \right] - [(1.082X_{\text{o}} )/(1 + 138.6X_{\text{o}} )^{0.805} ] - [(0.0658X_{\text{o}} )/(1 + (103.6X_{\text{o}} ))^{3} } \right\}\)\(\alpha_{\text{w}} = 2.9X_{\text{w}} /[(1 + 141.5X_{\text{w}} )^{0.635} + 5.925 X_{\text{w}} )]\) \(\tau_{\text{A}} = \left( {0.12445\alpha - 0.0162} \right) + \left( {1.003 - 0.125\alpha } \right) {\text{exp}}\left[ { - \beta m_{\text{air}} \left( {1.089\alpha + 0.5123} \right)} \right]\) \(X_{\text{o}} = U_{3} m_{\text{r}}\) \(X_{\text{w}} = U_{1} m_{\text{r}}\) \(m_{\text{air}} = m_{\text{r}} p/p_{\text{o}}\) \(m_{\text{r}} = 35/[(1224 {\text{cos}}^{2} \theta_{z} ) + 1]^{0.5}\) | (14) | Davies and Hay model | [12] |
\(I_{{{\text{DNI}},{\text{DPP}}}} = 950.2 \left\{ {1 - \exp \left[ { - 0.075 \left( {90{^\circ } - \theta_{\text{z}} } \right)} \right]} \right\}\) | (15) | Daneshyar–Paltridge–Proctor (DPP) model | [8] |
\(I_{{{\text{DNI}},{\text{Meinel}}}} = I_{\text{oN}} 0.7^{{m_{\text{air}}^{0.678} }}\) \(m_{\text{air}} = m_{\text{r}} p/p_{\text{o}}\) \(m_{\text{r}} = 1/\cos \theta_{\text{z}}\) | (16) | Meinel model | [8] |
\(I_{{{\text{DNI}},{\text{Laue}}}} = I_{\text{oN}} \left[ {\left( {1 - 0.14 L} \right) 0.7^{{m_{\text{air}}^{0.678} }} + 0.14 L} \right]\) \(m_{\text{air}} = m_{\text{r}} p/p_{\text{o}}\) \(m_{\text{r}} = 1/\cos \theta_{\text{z}}\) | (17) | Laue model | [8] |
\(I_{{{\text{DNI}},{\text{Haw}}}} = 1098 \cos \theta_{\text{z}} \exp [ - 0.057/\cos \theta_{\text{z}} ]\) | (18) | Haurwitz model | [86] |
\(I_{{{\text{DNI}},{\text{BD}}}} = 0.70 I_{\text{oN}} \cos \theta_{\text{z}}\) | (19) | Berger and Duffie model | [86] |
\(I_{{{\text{DNI}},{\text{ABCG}}}} = 951.39 (\cos \theta_{\text{z}} )^{1.15}\) | (20) | Adnot, Bourges, Campana and Gicquel model | [86] |
\(I_{{{\text{DNI}},{\text{KC}}}} = 910 \cos \theta_{\text{z}} - 30\) | (21) | Kasten and Czeplak model | [86] |
\(I_{{{\text{DNI}},{\text{RS}}}} = 1159.24 \{ (\cos \theta_{\text{z}} )^{1.179} \exp [ - 0.0019 (90 - \theta_{\text{z}} )]\}\) | (22) | Robledo and Sole model | [86] |
Equation | Parameters name | Parameters type | ||
---|---|---|---|---|
\(\cos \theta_{\text{z}} = \sin L \sin \theta_{\delta } + \cos L \cos \theta_{\delta } \cos \theta_{\text{h}}\) | (23) | Solar zenith angle | Astronomical | [6] |
\(\theta_{\delta } = 23.45\sin \left[ {\frac{360}{365}(284 + N_{j} )} \right]\) | (24) | Declination angle | Astronomical | [6] |
\(\theta_{\text{h}} = 15{^\circ }\left( {{\text{ST}} - 12} \right)\) | (25) | Solar angle | Astronomical | [6] |
\({\text{ST}} = {\text{SDT}} + 4\left( {L_{\text{st}} - L_{\text{loc}} } \right) + E\) \(E\, = \,229.2 \, (75\, \times \,10^{ - 6} + 186 \times 10^{ - 6} \sin B - 0.032207 \sin B - 0.014615 \sin 2B - 0.04089 \sin 2B)\) \(B = (N_{j} - 1)\frac{360}{365}\) | (26) | Solar time Time equation | Astronomical | [22] |
\(I_{\text{oN}} = I_{\text{o}} \left[ {1 + 0.033\cos \left( {\frac{{360N_{j} }}{365}} \right)} \right]\) | (27) | Extraterrestrial radiation measured on the plane normal to the radiation | Astronomical | [6] |
\(m_{\text{e}} = \exp \left( { - 0.000118\;{\text{h}} - 1638 \times 10^{ - 9} {\text{h}}^{2} } \right)/\cos \theta_{\text{z}}\) | (28) | Air mass corrected for elevation | Atmospheric | [22] |
\(m_{\text{r}} = \{ [1229 + (614 \cos \theta_{\text{z}} )^{2} ]^{0.5} - 614 \cos \theta_{\text{z}} \}\) | (29) | A specific air mass | Atmospheric | [22] |
\(m_{\text{air}} = m_{\text{r}} p/p_{\text{o}}\) | (30) | Air mass at actual pressure | Atmospheric | [9] |
\(m_{\text{r}} = 1/\cos \theta_{\text{z}}\) | (31) | Air mass at standard pressure | Atmospheric | [9] |
\(m_{{{\text{air}},{\text{MIqbalC}}}} = m_{\text{r}} \exp \left( { - 0.001184 {\text{h}}} \right)\) | (32) | Actual air mass value depends on altitude and relative air mass at standard pressure | Atmospheric | [22] |
\(m_{\text{r}} = 35/[(1224 {\text{cos}}^{2} \theta_{\text{z}} ) + 1]^{0.5}\) | (33) | Air mass at standard pressure | Atmospheric | [12] |
\(m_{\text{r}} = [\cos \theta_{\text{z}} + 0.15 (93.885 - \theta_{\text{z}} )^{ - 1.253} ]^{ - 1}\) | (34) | Air mass at standard pressure | Atmospheric | [22] |
Cloud cover model (CRM)
Equation | Description | ||
---|---|---|---|
\(I_{{{\text{G}}_{\text{cs}} }} = A\sin \theta_{\alpha } - B\) \(\sin \theta_{\alpha } = \cos \theta_{z} = \sin L \sin \theta_{\delta } + \cos L \cos \theta_{\delta } \cos \theta_{h}\) \(\sin \theta_{\text{h}} = \frac{{\sin \theta_{\alpha } - \sin \theta_{\delta } \sin L}}{{\cos \theta_{\delta } \cos L}}\) A, B: empirical coefficients | (35) | Hourly global solar radiation on a horizontal surface under cloudless sky | [30] |
\(I_{{{\text{G}}_{\text{cc}} }} = I_{{{\text{G}}_{\text{cs}} }} \left[ {1 - C\left( {\frac{\text{N}}{8}} \right)^{D} } \right]\) \(N = {\text{cloud cover}}\; ({\text{Oktas}});\quad [0({\text{clear}}\;{\text{sky}}) - 8 ( {\text{completely}}\;{\text{overcast}}\;{\text{sky)}}]\) C, D: empirical coefficients | (36) | Hourly global solar radiation on a horizontal surface under cloud cover condition | |
\(I_{\text{d}} = I_{{{\text{G}}_{\text{cc}} }} \left[ {0.3 - 0.7\left( {\frac{\text{N}}{8}} \right)^{2} } \right]\) | (37) | Hourly diffuse radiation on a horizontal surface | [30] |
\(I_{{{\text{DNI}},{\text{KC}}}} = \left( {I_{{{\text{G}}_{\text{cc}} }} - I_{\text{d}} } \right)/\cos \theta_{\text{z}}\) | (38) | Direct normal irradiance (DNI) under different sky conditions |
A hierarchical calculation methodology
Estimation of monthly average hourly direct solar irradiance from daily data
Daily global solar radiation (decomposition models)
Equation | Description | ||
---|---|---|---|
\(r_{\text{t}} = \frac{{\bar{I}_{\text{G}} }}{{\bar{H}_{\text{G}} }} = \frac{\pi }{24} \left( {a + b\cos \theta_{\text{h}} } \right) \left[ {\frac{{\cos \theta_{\text{h}} - \cos \theta_{\text{hs}} }}{{\sin \theta_{\text{hs}} - \left( {\frac{{\pi \theta_{\text{hs}} }}{180}} \right)\cos \theta_{\text{hs}} }}} \right]\) \(\theta_{\text{hs}} = \cos^{ - 1} \left[ { - \tan L \cdot \tan \theta_{\delta } } \right]\) \(\theta_{\text{h}} = \pm 0.25 \left( {{\text{number}}\;{\text{of}}\;{\text{minutes}}\;{\text{from}}\;{\text{local}}\;{\text{solar}}\;{\text{noon}}} \right)\) \(a = 0.4090 + 0.5016\sin (\theta_{\text{hs}} - 60)\) \(b = 0.6609 - 0.4767\sin (\theta_{\text{hs}} - 60)\) | (39) | Collares-Pereira and Rabl correlation (ratio of monthly average hourly global irradiance to monthly average daily global irradiance) | [6] |
\(r_{\text{d}} = \frac{{\bar{I}_{\text{d}} }}{{H_{\text{d}} }} = \frac{\pi }{24} \left[ {\frac{{\cos \theta_{\text{h}} - \cos \theta_{\text{hs}} }}{{\sin \theta_{\text{hs}} - \left( {\frac{{\pi \theta_{\text{hs}} }}{180}} \right)\cos \theta_{\text{hs}} }}} \right]\) | (40) | Liu and Jordan correlation (ratio of monthly average hourly diffuse irradiance to monthly average daily diffuse irradiance) | [6] |
\(\bar{I}_{\text{DNI,H}} = \bar{I}_{\text{G}} - \bar{I}_{\text{d}}\) | (41) | Monthly average hourly direct solar irradiance on a horizontal surface | |
\(\bar{I}_{\text{DNI}} = \bar{I}_{{{\text{DNI}},{\text{H}}}} /\cos \theta_{\text{z}}\) | (42) | Monthly average hourly direct solar irradiance |
Angstrom–Prescott correlation
Equation | Description | ||
---|---|---|---|
\(\frac{{\bar{H}_{\text{G}} }}{{\bar{H}_{\text{o}} }} = a + b\frac{{\bar{S}}}{{\bar{S}_{\text{o}} }}\) | (43) | Linear model | [60] |
\(\frac{{\bar{H}_{\text{G}} }}{{\bar{H}_{\text{o}} }} = a + b \frac{{\bar{S}}}{{\bar{S}_{\text{o}} }} + c \left( {\frac{{\bar{S}}}{{\bar{S}_{\text{o}} }}} \right)^{2}\) | (44) | Quadratic model | [47] |
\(\frac{{\bar{H}_{\text{G}} }}{{\bar{H}_{\text{o}} }} = a + b \frac{{\bar{S}}}{{\bar{S}_{\text{o}} }} + c T + d R\) | (45) | Multi-parameters model | [47] |
\(\frac{{\bar{H}_{\text{G}} }}{{\bar{H}_{\text{o}} }} = a + b\cos L + c H + d \frac{{\bar{S}}}{{\bar{S}_{\text{o}} }} + e T + f R\) | (46) | Gopinathan’s model | [74] |
\(\bar{K}_{\text{T}} = \frac{{\bar{H}_{\text{G}} }}{{\bar{H}_{\text{o}} }}\) | (47) | Monthly mean clearness index | [6] |
\(H_{\text{o}} = \frac{24}{\pi } H_{\text{sc}} \left[ {1 + 0.033\cos \left( {\frac{{360N_{j} }}{365}} \right)} \right]\left[ {\cos L \cos \theta_{\delta } \sin \theta_{\text{hs}} + \frac{\pi }{180} \theta_{\text{hs}} \sin L \sin \theta_{\delta } } \right]\) | (48) | Monthly average daily extraterrestrial solar irradiance on a horizontal surface | [6] |
\(S_{\text{o}} = 2 \theta_{\text{hs}} /15\) | (49) | Maximum possible monthly average daily length (h) | [6] |
Empirical models
Equation | Description | ||
---|---|---|---|
\(\frac{{\bar{H}_{\text{d}} }}{{\bar{H}_{\text{G}} }} = 1.39 - 4.027 \left( {\frac{{\bar{H}_{\text{G}} }}{{\bar{H}_{\text{o}} }}} \right) + 5.5310\left( {\frac{{\bar{H}_{\text{G}} }}{{\bar{H}_{\text{o}} }}} \right)^{2} - 3.108\left( {\frac{{\bar{H}_{\text{G}} }}{{\bar{H}_{\text{o}} }}} \right)^{3}\) | (50) | Liu and Jordan model | [82] |
\(\frac{{\bar{H}_{\text{d}} }}{{\bar{H}_{\text{G}} }} = 1.2547 - 1.2547 \left( {\frac{{\bar{S}}}{{\bar{S}_{\text{o}} }}} \right)\) | (51) | Iqbal model | [82] |
\(\frac{{\bar{H}_{\text{d}} }}{{\bar{H}_{\text{G}} }} = 1.194 - 0.838\left( {\frac{{\bar{H}_{\text{G}} }}{{\bar{H}_{\text{o}} }}} \right) - 0.0446\left( {\frac{{\bar{S}}}{{\bar{S}_{\text{o}} }}} \right)\) | (52) | Gopinathan model | [82] |
\(\frac{{\bar{H}_{\text{d}} }}{{\bar{H}_{\text{G}} }} = 0.775 + 0.00606 \left( {\theta_{\text{hs}} - 90} \right) - \left[ {0.505 + 0.00455 \left( {\theta_{\text{hs}} - 90} \right)} \right]{\text{cos }}\left( {115 \left( {\frac{{\bar{H}_{\text{G}} }}{{\bar{H}_{\text{o}} }}} \right) - 103} \right)\) | (53) | Collares-Pereira and Rabl | [6] |
A hierarchical calculation methodology
Site description and data collection
Statistical methods of model evaluation
Equation | Description | |
---|---|---|
\({\text{MBE}} = \frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {I_{\text{cal}} - I_{\text{meas}} } \right)}}{n}\) | (54) | Mean bias error |
\({\text{RMSE}} = \sqrt {\frac{{\mathop \sum \nolimits_{i = 1}^{n} (I_{\text{cal}} - I_{\text{meas}} )^{2} }}{n}}\) | (55) | Root mean square error |
\({\text{MPAPE}} = \frac{100}{n} \mathop \sum \limits_{i = 1}^{n} \frac{{\left( {I_{\text{cal}} - I_{\text{meas}} } \right)}}{{I_{\text{meas}} }}\) | (56) | Absolute percent error |
\(R^{2} = 1 - \frac{{\mathop \sum \nolimits_{i = 1}^{n} (I_{\text{cal}} - I_{\text{meas}} )^{2} }}{{\mathop \sum \nolimits_{i = 1}^{n} (I_{\text{meas}} - I_{\text{meas,avg}} )^{2} }}\) | (57) | Coefficient of determination |
\(t_{\text{stat}} = \left[ {\frac{{\left( {n - 1} \right){\text{MBE}}^{2} }}{{{\text{RMSE}}^{2} - {\text{MBE}}^{2} }}} \right]^{1/2}\) | (58) | t Statistic method |
\(e{\text{\% }} = \frac{{I_{\text{cal}} - I_{\text{meas}} }}{{I_{\text{meas}} }}\) | (59) | Percentage error |
Results and discussion
Month | Hour | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Daily | |
January | 0.23 | 0.34 | 0.40 | 0.45 | 0.48 | 0.50 | 0.50 | 0.49 | 0.47 | 0.42 | 0.32 | 0.462 | |||
February | 0.24 | 0.34 | 0.41 | 0.45 | 0.48 | 0.51 | 0.52 | 0.52 | 0.50 | 0.46 | 0.37 | 0.476 | |||
March | 0.03 | 0.23 | 0.34 | 0.40 | 0.46 | 0.50 | 0.52 | 0.54 | 0.55 | 0.55 | 0.50 | 0.42 | 0.478 | ||
April | 0.16 | 0.29 | 0.37 | 0.44 | 0.49 | 0.53 | 0.55 | 0.58 | 0.58 | 0.57 | 0.54 | 0.46 | 0.33 | 0.502 | |
May | 0.02 | 0.21 | 0.31 | 0.40 | 0.46 | 0.50 | 0.54 | 0.57 | 0.59 | 0.60 | 0.60 | 0.58 | 0.51 | 0.517 | |
June | 0.05 | 0.24 | 0.33 | 0.45 | 0.51 | 0.57 | 0.61 | 0.65 | 0.65 | 0.65 | 0.64 | 0.62 | 0.56 | 0.44 | 0.568 |
July | 0.01 | 0.23 | 0.34 | 0.46 | 0.54 | 0.59 | 0.61 | 0.65 | 0.65 | 0.65 | 0.65 | 0.61 | 0.56 | 0.44 | 0.582 |
August | 0.20 | 0.36 | 0.49 | 0.57 | 0.61 | 0.63 | 0.64 | 0.66 | 0.64 | 0.64 | 0.61 | 0.53 | 0.39 | 0.590 | |
September | 0.14 | 0.33 | 0.45 | 0.51 | 0.57 | 0.59 | 0.62 | 0.62 | 0.62 | 0.61 | 0.56 | 0.46 | 0.558 | ||
October | 0.27 | 0.39 | 0.47 | 0.52 | 0.56 | 0.58 | 0.59 | 0.60 | 0.57 | 0.50 | 0.37 | 0.522 | |||
November | 0.21 | 0.34 | 0.42 | 0.48 | 0.51 | 0.54 | 0.54 | 0.54 | 0.50 | 0.43 | 0.30 | 0.479 | |||
December | 0.13 | 0.31 | 0.40 | 0.45 | 0.49 | 0.51 | 0.52 | 0.51 | 0.49 | 0.42 | 0.29 | 0.462 |
Month | T (°K) | RH% | \(\frac{{\bar{S}}}{{\bar{S}_{\text{o}} }}\) |
---|---|---|---|
January | 16 | 62 | 0.194 |
February | 18.7 | 60 | 0.283 |
March | 23.1 | 54 | 0.349 |
April | 26.8 | 54 | 0.451 |
May | 29.6 | 57 | 0.476 |
June | 33.2 | 54 | 0.573 |
July | 35 | 50 | 0.603 |
August | 35.2 | 49 | 0.676 |
September | 31.8 | 53 | 0.605 |
October | 27.6 | 53 | 0.528 |
November | 22.2 | 54 | 0.351 |
December | 17.5 | 59 | 0.278 |
Equation | Description | MBE | RMSE | MAPE | e % | R2 |
---|---|---|---|---|---|---|
\(\frac{{\bar{H}_{\text{G}} }}{{\bar{H}_{\text{o}} }} = 0.3841 + 0.2946 \frac{{\bar{S}}}{{\bar{S}_{\text{o}} }}\) | Linear model | − 0.11 | 0.17 | − 3.4 | 3.7 | 0.98 |
\(\frac{{\bar{H}_{\text{G}} }}{{\bar{H}_{\text{o}} }} = 0.4656 - 0.1235 \frac{{\bar{S}}}{{\bar{S}_{\text{o}} }} + 0.4767\left( {\frac{{\bar{S}}}{{\bar{S}_{\text{o}} }}} \right)^{2}\) | Quadratic model | − 0.11 | 0.18 | − 3.3 | 4.2 | 0.98 |
\(\frac{{\bar{H}_{\text{G}} }}{{\bar{H}_{\text{o}} }} = 0.235 + 0.179 \frac{{\bar{S}}}{{\bar{S}_{\text{o}} }} + 0.0036 T + 0.0019 R\) | Multi-parameters model | − 0.10 | 0.17 | − 3.4 | 3.8 | 0.98 |
\(\frac{{\bar{H}_{\text{G}} }}{{\bar{H}_{\text{o}} }} = 0.801 - 0.378\cos L + 0.0128 H + 0.316 \frac{{\bar{S}}}{{\bar{S}_{\text{o}} }} - 1.214 \times 10^{ - 3} T - 1.049 \times 10^{ - 3} R\) | Gopinathan’s model | − 0.02 | 0.14 | − 1.4 | 2.8 | 0.99 |
Conclusions
- Based on the preliminary assessment for the potential of solar energy for the selected location by performing the comprehensive analysis. The San Antonio region in Texas is unequivocally amenable to harnessing solar energy as the prime source of energy by utilizing concentrating and non-concentrating solar energy systems because the analysis of the monthly average hourly clearness index through the classification of the clearness index level shows that more than 80% of the days can be defined as either sunny (\(k_{\text{t}}\) > 0.5) or partly cloudy (0.3 ≤ \(k_{\text{t}}\) ≤ 0.5) and less than 20% of the days are classified as cloudy (\(k_{\text{t}}\) < 0.3).
- Based on five statistical indictors, most estimated values of hourly direct normal irradiance for 22 parametric models are in favorable agreement with the measured values for all the months of the year.
- Some simple parametric models that have a few parameters (less than three geographic and astronomical parameters) such as Meinel and Laue have shown a good fit accuracy for most months during the year with the values of R2 are in the range of 0.93–0.99. While some values were not consistent perfectly with the measured data.
- More sophisticated (complex) parametric models such as Bird, Iqbal C, METSTAT, Modified Iqbal C, CSR, Atwater–Ball, ESRA, Hoyt (Iqbal B), Heliosat-1, Davies–Hay and Iqbal A models have shown more accuracy in estimating DNI values during winter months (October–March) with the values R2 are in the range of 0.87–0.99 than summer months (April–September) with the values of R2 are in the range of 0.33–0.96.
- The significant influence of cloud amount on reducing the intensity of global solar radiation, specifically DNI, was studied by using the cloud-cover radiation model (CRM) and the cloud amount indicator in Oktas, ranging from 0 to 8. For illustrate, the global solar radiation intensity has been attenuated from 765 W/m2 (0 Oktas, clear sky) to 213 W/m2 (8 Oktas, overcast sky). While the amount of diffuse irradiance increases in the atmosphere with growing the cloud amount until reaching zero under the overcast sky.
- The estimated values of the monthly average daily global solar radiation on a horizontal surface obtaining from four formulations of the Angstrom–Prescott correlation, which were developed through regression analysis to determine their local coefficients, show a good agreement with measured data from different databases with the values of R2 are in the range of 0.98–0.99.
- The validation of four selected empirical models was performed by comparing their estimated values of monthly average daily diffuse solar irradiance against the measured data. Clearly, the estimated values, which were obtained from three models including Collares-Pereira and Rabl, Liu and Jordan, Gopinathan models, are in good agreement with the measured data with the values of R2 are ranging from 0.94 to 0.98 except for Iqbal model that shows less consent with measured data with the R2 value is 0.65.
- The estimated values of monthly average hourly direct solar irradiance on a horizontal surface, which were calculated to attain monthly average DNI values through utilizing the Angstrom–Prescott correlation (linear model), the empirical model (Liu and Jordan model), two decomposition models, and zenith angle, showed a relative agreement (R2 = 0.82) with the measured data because some used models require obtaining locally fitted coefficients.
- It is obvious that the proposed methodologies have offered a reasonably good estimation for the hourly solar radiation values and they can be implemented for other locations around the world by creating new locally fitted coefficients for empirical and regression correlations. However, it is worth noting that the estimated solar data (by solar radiation modeling) can never substitute the measured solar data (by measurement equipment).