Literature Review

The BC model was tested in many different locations and papers. It is often proposed as alternative method from the sunshine based models², that were largely applied in the past. Goodin et Mendoza (1999) [4] evaluated the model considering cities in Kansas, while Chen et al. (2011) [5] compared a support vector machine (SVM) approach with the empirical model. They found that SVM using maximum and minimum temperatures as input and polynomial kernel functions outperform empirical models. Phakamas at al. (2013) [6] applied it for cities in Thailand. A more recent application, from Gomez et al. (2021) [7], uses the Bristow and Campbell and Hargreaves and Samani models in Tropical and Mountainous environments; in the work it is also proposed a different specification using a logistic regression.

Bristow and Campbell’s Model

The BC model specifies a non linear relation between the daily air temperature range (Eq. 4.1) and the global horizontal irradiance (GHI). \[\tag{4.1} \Delta T = \frac{T_t^{max} - T_t^{min} }{2}\] The daily transmittance coefficient is defined as the ratio between the horizontal radiation that reach the ground (GHI) and the extraterrestrial one. Considering a day \(t\), the transmittance coefficient, denoted as \(\hat{T}_t\) is computed as: \[\tag{4.2} \hat{T}_t = \frac{GHI_t}{H_{o,t}}\] The Bristow and Campbell relation is defined between \(\hat{T}_t\) and \(\Delta T\) as: \[\tag{4.3} \hat{T}_t = A \cdot (1-e^{-B \Delta T_t^{C}})\] The parameters \(A\), \(B\), and \(C\) in the equation (Eq. 4.3) represent three empirical coefficients. \(A\) can be interpreted as the maximum value that \(\hat{T}_t\) can assume, while \(B\) and \(C\) determine how fast the maximum value \(A\) is reached as \(\Delta T\) increase. In order to consider the seasonality the authors suggest to vary the parameter \(B\) on a monthly basis. Denoting as \(u\) a generic month, the parameter \(B_u\) is computed with the following relation: \[\tag{4.4} B_u = \beta_0 \cdot e^{-\beta_1 \Delta \overline{T_u}}\] In this way a the Global Horizontal Irradiance can be expressed as: \[\tag{4.5} GHI_t = \hat{T}_t \cdot H_{o,n}\] The empirical relation between \(\Delta T\) and \(\hat{T}_t\) for Bologna is shown in Figure @ref(fig:PlotBristowRelation). The points are divided by season: this highlights that Summer days are less disperced around the black line, while the Autumn and Winter’s ones are more variable. In the Table @ref(tab:BCParameters) the coefficients obtained for the BC model in different locations are reported. Being the parameter \(B\) dependent on the monthly mean of the temperature range’s, we have reported the mean values for the coefficient by distinguishing between Summer and Winter months.

Parameters and Residual Errors (Bristow and Campbell Model)

Place	A	\(\overline{B}_{summer}\)	\(\overline{B}_{winter}\)	C	\(MSE(\hat{T}_t)\)	\(MSE(GHI)\)
Amsterdam	0.774	0.018	0.014	2.353	0.189	1.291
Berlino	0.773	0.015	0.009	2.177	0.161	1.004
Parigi	0.777	0.014	0.009	2.204	0.156	1.010
Oslo	0.771	0.017	0.011	2.156	0.177	0.884
Bologna	0.766	0.011	0.008	2.282	0.142	1.030
Rimini	0.787	0.013	0.010	2.322	0.176	1.291
Catania	0.764	0.010	0.007	2.452	0.139	1.090
Roma	0.768	0.012	0.009	2.435	0.148	1.123

A stochastic approach for the daily temperature range

In its original formulation, the BC model and its variants consider the daily temperature range as given, therefore its usage is limited only to the explanation of the relation between the two variables and it is not possible to forecast the value of the solar irradiance in the future. However if we immagine that \(\Delta T\) is stochastic and that is possible to specify an SDE to reproduce its dynamics, we could make a further step in the applications of the model. In order to model the daily temperature range we could use the OU-process specified in (Eq. 3.1). However, despite this model works almost perfectly if applied to the mean temperature, when it is used for daily radiation, gives poor results³. An alternative that we have found is to specify two different equations, both based on the same model, but fitted separately for the maximum and the minimum temperature. Once the two variables \(T_t^{max}\) and \(T_t^{min}\) are defined, we can derive two more variables whose value is implied: the mean temperature \(T_t\) and the daily temperature range \(\Delta T_t\). The equations specified for the maximum \(T_t^{max}\) and the minimum \(T_t^{min}\) temperature are given by: \[\tag{4.6} T_t^{max} = (T_s^{max} - T_t^{max, m}) e^{-\theta^{max} (t-s)} + T_t^{max,m} + \int_{s}^{t} \sigma_u^{max} e^{-\theta^{max} (t-u)} dW_u^{max}\] \[\tag{4.7} T_t^{min} = (T_s^{min} - T_t^{min, m}) e^{-\theta^{min} (t-s)} + T_t^{min,m} + \int_{s}^{t} \sigma_u^{min} e^{-\theta^{min} (t-u)} dW_u^{min}\] Finally in the following system we represent the two implied values for the mean temperature \(T_t\) and daily range temperature variation \(\Delta T\): \[\begin{cases} T_t = \frac{1}{2} \bigl( T_t^{max} + T_t^{min} \bigl) \\ \Delta T_t = T_t^{max} - T_t^{min} \end{cases}\] The time series considered for the estimation are from 2005-01-01 up to 2020-01-01. As done for direct and diffuse radiation, a correlation coefficient between \(T_t^{min}\) and \(T_t^{min}\) was estimated, in order to take an eventual correlation into account in the simulations. In Table @ref(tab:TemperatureParameters) the mean reversion parameters \(\theta\), the errors of the models and the kurtosis of the residuals \(K\) are reported. Both the maximum and the minimum leads to a kurtosis that is greater than three denoting a more pointed distribution with respect to the normal one.

Parameters, Mse and Kurtosis of the Residuals (Models 4.6 and 4.7)

Place	\(\theta_{max}\)	\(\theta_{min}\)	\(\rho\)	\(Mse_{max}\)	\(Mse_{min}\)	\(Mse_{\Delta T}\)	\(K_{max}\)	\(K_{min}\)
Amsterdam	0.244	0.189	0.424	2.195	1.800	2.191	3.323	3.683
Berlino	0.193	0.211	0.401	2.624	2.234	2.707	3.694	3.580
Oslo	0.247	0.225	0.402	2.337	2.185	2.398	3.592	3.589
Parigi	0.257	0.294	0.366	2.375	2.204	2.601	3.209	2.861
Bologna	0.212	0.507	0.171	2.232	2.094	2.799	3.805	3.189
Catania	0.293	0.361	0.211	1.994	1.436	2.215	3.846	3.187
Rimini	0.253	0.317	0.324	1.994	1.839	2.236	3.819	3.266
Roma	0.251	0.363	0.241	1.585	1.713	2.018	4.184	3.252

In Figure @ref(fig:FittedMaxMin) we show the fitted data for the maximum and minimum temperature together with the implied time series of the temperature range and the mean temperature for the year 2019. As we can see, modeling the two variables separately lightly affect the goodness of fit of \(T_t\) and \(\Delta T\), however the model performs the best for the maximum temperature, while less well for the minimum.

In general, the implied values for \(\Delta T\) and \(T_t\) are very close to the real ones. Once established that the models work fine, we would like to compare the performance of the Bristow and Campbell model considering two distinct inputs:

1. the realized \(\Delta T\). (Real) obtained by the data.
2. the fitted \(\Delta T\). (Implied) obtained implying a value of \(\Delta T\) using the values from the models (Eq. 4.6) and (Eq. 4.7).

In the Figure @ref(fig:SimulatedMaxMin) the empirical value of the GHI (Realized) and the two fitted values are compared.

The results obtained with the direct model Empirical, are obviously much better than the Fitted ones. The daily temperature range is again confirmed to be an optimal proxy that can be used to explain the variation of the Global Horizontal Irradiance.

Simulated Trajectories

In order to validate the model under the assumption of a stochastic \(\Delta T\), we have decided to simulate five trajectories for the maximum and minimum for each considered city. Then the simulated data will be used together with the fitted BC models to obtain a simulation of the Global Horizontal Irradiance. A possible trajectory for the GHI was computed for each city, the simulated paths do not perform the best, especially for North european cities. However the simulations for Catania and Rome are very close to the real dynamics. In conclusion, applying a stochastic approach on Bristow and Campbell model leads to good results for cities at lower latitudes while it does not perform the best in cities with higher latitudes. In Figure @ref(fig:BCSimulated1),@ref(fig:BCSimulated2), @ref(fig:BCSimulated3) and @ref(fig:BCSimulated4) a possible trajectory for each city is shown. We note that this approach leads to better results for Italian cities while worse ones for northern European ones, while with the model specified in chapter 3 the results are reversed.