# Assessment and Day-Ahead Forecasting of Hourly Solar Radiation in Medellín, Colombia

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}. This shows that Medellín, Colombia, has a substantial availability of the solar resource that can be a complementary source of energy during the dry season periods. In the case of the Markov model, the estimates exhibited typical root mean squared errors between ~80 W/m

^{2}and ~170 W/m

^{2}(~50%–~110%) under overcast conditions, and ~57 W/m

^{2}to ~171 W/m

^{2}(~16%–~38%) for clear sky conditions. In general, the proposed model had a performance comparable with the WRF model, while presenting a computationally inexpensive alternative to forecast hourly solar radiation one day in advance. The Markov model is presented as an alternative to estimate time series that can be used in energy markets by agents and power-system operators to deal with the uncertainty of solar power plants.

## 1. Introduction

- We performed an initial assessment of the intra-day and daily variability of the solar resource in Medellín, Colombia, which is a piece of information that was not available in the site of study and that is necessary for determining the intra-day generation potentials.
- We proposed a statistical model based on Markov chains for forecasting the hourly GHI series for one day-ahead lead time, with low computational costs. We also used an NWP model (WRF) and a persistence-based Markov model as a benchmarking for the proposed Markov model.
- We evaluated the performance of the Markov model at estimating the hourly GHI and daily clearness coefficient considering different cloud covers in a tropical climate region.
- The performance of the Markov model was also evaluated under local atypical and synoptic atypical cloudy conditions.
- The method used for the formulation and training of the Markov model can be extended to other locations with different climatological conditions.
- The magnitude of the errors obtained with the Markov model are comparable to the errors obtained with other models identified in the literature.

## 2. Data and Methods

#### 2.1. Pyranometer Data

^{2}. The pyranometer considered (hereafter SIATA station) is located in Medellín, Colombia, at 6.2593° latitude and −75.5887° longitude (Figure 1). The SIATA pyranometers are inspected monthly and are calibrated according to the ISO 9847:1992. For the hourly values, the expected uncertainty is of 3% of the true value of radiation. Measurements are provided at a one-minute time resolution.

#### 2.2. Clearness Coefficient Estimation

^{2}. The zenith angle can be estimated at the same resolution of the GHI measurements.

#### 2.3. Discrete Markov Chain Model

#### 2.4. Construction of the Markov Transition Matrices

#### 2.4.1. First-Degree Markov Transition Matrix (MTM) for ${k}_{s}$

- ${f}_{ij}$: Number of times the variable passes from $i$ to $j$.
- ${T}_{i}$: Number of times the variable departs from $i$.

- (1)
- Extract all the hourly GHI values from the days of the dataset that have a ${k}_{d}=z$. Keep the information regarding the hour and day of the year that corresponds to each hourly datum.
- (2)
- Calculate the corresponding ${k}_{s}$ values using Equations (1)–(4), and discretize them using Table 1.
- (3)
- Using the discrete ${k}_{s}$ data, calculate the corresponding ${f}_{ij}$ and ${T}_{i}$ values for all ${k}_{s}$ discrete values. The $i$ and $j$ values are iterated over all the states of ${k}_{s}$.
- (4)
- Calculate the ${P}_{ij}$ probabilities with Equation (7), and position them in the MTM at the corresponding ($i,j$) position.

#### 2.4.2. Second-Degree MTM for ${k}_{d}$

#### 2.5. Simulations of Global Horizontal Irradiance (GHI) Using a Markov Chains Model

- (1)
- Transform the estimates $\left\{{k}_{s,6},{k}_{s,7},\dots ,{k}_{s,18}\right\}$ from states (1 to 20) to extinction percentages (0.0–1.0). This is done by first assuming that for a state $i$ of ${k}_{s}$, the continuous values inside the corresponding interval (see Table 1), follow a uniform distribution such that ${k}_{s}~\mathcal{U}\left(\left(i-1\right)\times 0.05,i\times 0.05\right)$. Then, using a pseudo-random process a continuous value of ${k}_{s}$ is picked from the given distribution.
- (2)
- Once a continuous value of ${k}_{s}$ is obtained and knowing the hour and the day for which it was estimated, the total hourly clearness coefficient, ${k}_{t}$ must be calculated again in order to include the deterministic effects. This is done by using Equation (4) as:$${k}_{t,est}={k}_{s,est}\times 1.031\mathrm{exp}\left(\frac{-1.4}{0.9+\frac{9.4}{A.M}}\right)+0.1$$
- (3)
- Now, given that the extraterrestrial radiation can be modeled as a deterministic variable, the Equation for the total clearness coefficient, ${k}_{t}$, can be used now to estimate the corresponding GHI value as:$$GH{I}_{est}={k}_{t}\times {I}_{ext}$$

#### 2.6. The Weather Research and Forecasting (WRF) Model

#### 2.7. Persistence-Markov Model

#### 2.8. Error Metrics

^{2}), the normalized root mean square error (nRMSE; in %), the mean bias error (MBE; in W/m

^{2}), and the normalized mean bias error (nMBE; in %), as follows:

- N corresponds to the number of hourly GHI values during the day.
- $GH{I}_{est,i}$ is the estimated average GHI value for the hour $i$.
- $GH{I}_{meas,i}$ is the measured average GHI value for the hour $i$.
- ${\overline{GHI}}_{meas}$ is the daily mean of the hourly GHI measured values.

## 3. Results

#### 3.1. Characterization of the GHI

^{2}to 643 W/m

^{2}. The daytime values are skewed, with higher frequencies of the lowest radiation values, especially in the 0–100 W/m

^{2}range.

^{2}. April 2016, May 2016 and late November 2016 are the periods where more GHI variability is observed, with April 2016 exhibiting several days with solar radiation below 300 W/m

^{2}at any time of the day.

^{2}, suggesting a similar annual behavior of the GHI. However, measurements for both sites (Sevilla and Medellín) exhibit seasonal changes with rather different amplitudes. The 2016–2017 annual GHI series obtained from the SIATA pyranometer (Figure 5) shows a rather uniform pattern when compared to the larger contrasts between summer and winter seasons at higher latitudes that would alter significantly the daylight hours and the intensity of incoming radiation (see [64]). In spite of lacking the strong seasonal effects of middle latitudes, the SIATA pyranometer data suggests two periods of reduced solar radiation: April to early May 2016, and late October to November 2016. During November 2016, the time interval when solar irradiance exceeds 200 W/m

^{2}is reduced from ~10 h to ~7.5 h.

^{2}, commonly used in power systems and energy budget. August 2016 and February 2017 have the highest values of mean daily incoming energy and the lowest average of hourly CV. On the other hand, wet season months (i.e., April, May, October, November and December 2016) show the highest variability with CV values between 43% and 50% of the hourly GHI averages. April and November 2016 are the months with the highest hourly variability values (46% and 50%, respectively). Consistent with the mean daily values of solar energy, the mean ${k}_{d}$ per month show that during the months of August 2016, January 2017 and February 2017, there was an overall low level of cloudiness, hence the high ${k}_{d}$ values.

#### 3.2. Clearness Coefficient

#### 3.3. Clearness Coefficient and GHI Estimates

#### 3.3.1. Daily Clearness Coefficient Estimates

#### 3.3.2. Hourly Estimates of GHI

^{2}from the outer domain (d01) with respect the inner domain (d02). In the case of the inner domain, d02, the WRF estimates consistently underestimated the measured GHI series.

^{2}, which is lower than the RMSE value for the Persistence-Markov model but higher than the mean RMSE produced by the WRF experiment. This indicates a lower performance at estimating the hourly GHI for this day compared to the WRF experiment but an improvement with respect the persistence-based model. For 23 December 2017, the Markov model, the persistence-Markov model and the WRF experiments presented a similar performance at estimating the hourly GHI, all of them exhibiting nRMSE values of 47%, 45% and 46%, respectively. For the case of 6 June 2017, the performance of the Markov model increases with respect to the performance of the WRF experiments at estimating the GHI, however, the persistence-Markov model presents an improvement with respect the proposed Markov model. For the case of 24 November 2017, it can be seen that the proposed Markov model has a better performance than the persistence-Markov mode and the WRF model, presenting a lower RMSE than the other two models. This shows that for this particular day, neither the persistence-based model nor the WRF model were capable of simulating the correct state of the ${k}_{d}$, while the proposed Markov model did. Finally, for the case of 19 August 2017, the WRF presents a better performance at reproducing the effects of the larger scale event over the region of study, while the proposed Markov model and the persistence-Markov model fail to reproduce these effects over the GHI and thus, result in estimations with high RMSE values.

^{2}.

#### 3.4. Daily Forecasts of GHI for the Validation Period of May 2017–May 2018

## 4. Concluding Remarks

#### 4.1. Solar Assessment

^{2}, typically surpassing the 800 W/m

^{2}between 11 and 13LT. Additionally, the measured data showed that the average daily solar energy that reached the surface in the region during March 2016 to February 2017 was of ~5 kWh/m

^{2}, which corresponds to 5 h of equivalent peak sun hours (PSH). The hourly measurements showed that this value would increase to an average of 5.5 PSH during the dry season months. This resource availability is important since it is comparable or even larger than what has been observed in regions that are referents regarding the use of solar radiation for energy production, like California, USA, with average values ranging from 3.8 to 6.3 PSH [68], or cities in Germany, with average PSH values that can range from 2.9 PSH to 3.6 PSH [69].

#### 4.2. Two-Part Markov Chain Model

^{2}to ~171 W/m

^{2}(~16%–~38%). For broken cloud sky conditions, the RMSE values ranged from ~149 W/m

^{2}to ~250 W/m

^{2}(~32%–~81%) while for overcast sky conditions, RMSE values ranged from ~80 W/m

^{2}to ~170 W/m

^{2}(~50%–~110%). In general, these results are in agreement with the errors found in similar studies regarding the hourly GHI forecasting for one day-ahead [56,73,74]. Although not frequently, the Markov model managed to produce GHI hourly estimates with very low RMSEs, even for highly overcast conditions. These values ranged from ~30 W/m

^{2}to ~80 W/m

^{2}(~20%–~50%). Even though these values are not typical among the estimates obtained in this work, they indicate that the model has the potential to produce series that are closer to the measured series for overcast condition than what is typically found in other works [8,50,51,73,75,76,77]. This improvement can be achieved by a further characterization of the hourly GHI series that could return typical intervals in which the hourly GHI usually lies for each hour of the day. Based on these intervals, the simulated series can be evaluated after they are produced and discarded if necessary. Additionally, because of how the Markov model was trained in this study, the hourly simulations of GHI did not take into account the dependency on the time of the day and the atmospheric mechanisms that could affect cloudiness. As a way of correcting this, the transition probabilities stored inside the Markov transition matrices could be obtained considering the time of the day.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Krakauer, N.; Cohan, D. Interannual variability and seasonal predictability of wind and solar resources. Resources
**2017**, 6, 29. [Google Scholar] [CrossRef] - Salgueiro, V.; Costa, M.J.; Silva, A.M.; Bortoli, D. Effects of clouds on the surface shortwave radiation at a rural inland mid-latitude site. Atmos. Res.
**2016**, 178, 95–101. [Google Scholar] [CrossRef] - Vignola, F.; Grover, C.; Lemon, N.; McMahan, A. Building a bankable solar radiation dataset. Sol. Energy
**2012**, 86, 2218–2229. [Google Scholar] [CrossRef] - CREG. Resolución No. 30 de febrero de 2018; CREG: Bogotá, Colombia, 2018; p. 27.
- Energías Renovables. Available online: http://www1.upme.gov.co/Paginas/Energias-renovables.aspx (accessed on 22 July 2019).
- Hidroituango, el Megaproyecto de Ingeniería en Colombia, Tiene Otro Problema: Un Socavón a 40 Metros CNN. Available online: https://cnnespanol.cnn.com/2019/01/14/hidroituango-el-megaproyecto-de-ingenieria-en-colombia-tiene-otro-problema-un-socavon-a-40-metros/#0 (accessed on 22 July 2019).
- Al-Kayiem, H.; Mohammad, S. Potential of renewable energy resources with an emphasis on solar power in Iraq: An outlook. Resources
**2019**, 8, 42. [Google Scholar] [CrossRef] - Lauret, P.; Diagne, M.; David, M. A neural network post-processing approach to improving NWP solar radiation forecasts. Energy Procedia
**2014**, 57, 1044–1052. [Google Scholar] [CrossRef] - Ngoko, B.O.; Sugihara, H.; Funaki, T. Synthetic generation of high temporal resolution solar radiation data using Markov models. Sol. Energy
**2014**, 103, 160–170. [Google Scholar] [CrossRef] - Skamarock, W.C.; Klemp, J.B.; Dudhi, J.; Gill, D.O.; Barker, D.M.; Duda, M.G.; Huang, X.-Y.; Wang, W.; Powers, J.G. A Description of the Advanced Research WRF Version 3; NCAR: Boulder, Colorado, USA, 2008. [Google Scholar]
- Perez, R.; Lorenz, E.; Pelland, S.; Beauharnois, M.; Van Knowe, G.; Hemker, K.; Heinemann, D.; Remund, J.; Müller, S.C.; Traunmüller, W.; et al. Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe. Sol. Energy
**2013**, 94, 305–326. [Google Scholar] [CrossRef] - Perez, R.; Kivalov, S.; Schlemmer, J.; Hemker, K.; Renné, D.; Hoff, T.E. Validation of short and medium term operational solar radiation forecasts in the US. Sol. Energy
**2010**, 84, 2161–2172. [Google Scholar] [CrossRef] - Doorga, J.R.S.; Rughooputh, S.D.D.V.; Boojhawon, R. Modelling the global solar radiation climate of Mauritius using regression techniques. Renew. Energy
**2019**, 131, 861–878. [Google Scholar] [CrossRef] - Ibrahim, S.; Daut, I.; Irwan, Y.M.; Irwanto, M.; Gomesh, N.; Farhana, Z. Linear regression model in estimating solar radiation in perlis. Energy Procedia
**2012**, 18, 1402–1412. [Google Scholar] [CrossRef] - Gairaa, K.; Bakelli, Y. A Comparative study of some regression models to estimate the global solar radiation on a horizontal surface from sunshine duration and meteorological parameters for Ghardaïa Site, Algeria. ISRN Renew. Energy
**2013**, 2013, 1–11. [Google Scholar] [CrossRef] - Paoli, C.; Voyant, C.; Muselli, M.; Nivet, M.L. Forecasting of preprocessed daily solar radiation time series using neural networks. Sol. Energy
**2010**, 84, 2146–2160. [Google Scholar] [CrossRef] - Ji, W.; Chee, K.C. Prediction of hourly solar radiation using a novel hybrid model of ARMA and TDNN. Sol. Energy
**2011**, 85, 808–817. [Google Scholar] [CrossRef] - David, M.; Ramahatana, F.; Trombe, P.J.; Lauret, P. Probabilistic forecasting of the solar irradiance with recursive ARMA and GARCH models. Sol. Energy
**2016**, 133, 55–72. [Google Scholar] [CrossRef] - Notton, G.; Voyant, C. Forecasting of intermittent solar energy resource. In Advanced in Renewable Energies and Power Technologies; Yahyaoui, I., Ed.; Elsevier Science: Amsterdam, The Netherlands, 2018; pp. 78–114. ISBN 9780128129593. [Google Scholar]
- Aguiar, R.; Collares-Pereira, M. TAG: A time-dependent, autoregressive, Gaussian model for generating synthetic hourly radiation. Sol. Energy
**1992**, 49, 167–174. [Google Scholar] [CrossRef] - Bouabdallah, A.; Olivier, J.C.; Bourguet, S.; Machmoum, M.; Schaeffer, E. Safe sizing methodology applied to a standalone photovoltaic system. Renew. Energy
**2015**, 80, 266–274. [Google Scholar] [CrossRef] - Poggi, P. Stochastic study of hourly total solar radiation in. Int. J. Climatol.
**2000**, 1860, 1843–1860. [Google Scholar] [CrossRef] - Hocaoglu, F.O.; Serttas, F. A novel hybrid (Mycielski-Markov) model for hourly solar radiation forecasting. Renew. Energy
**2017**, 108, 635–643. [Google Scholar] [CrossRef] - Shakya, A.; Michael, S.; Saunders, C.; Armstrong, D.; Pandey, P.; Chalise, S.; Tonkoski, R. Solar irradiance forecasting in remote microgrids using markov switching model. IEEE Trans. Sustain. Energy
**2017**, 8, 895–905. [Google Scholar] [CrossRef] - Li, S.; Ma, H.; Li, W. Typical solar radiation year construction using k-means clustering and discrete-time Markov chain. Appl. Energy
**2017**, 205, 720–731. [Google Scholar] [CrossRef] - Munkhammar, J.; Widén, J. A spatiotemporal Markov-chain mixture distribution model of the clear-sky index. Sol. Energy
**2019**, 179, 398–409. [Google Scholar] [CrossRef] - Hocaoĝlu, F.O. Stochastic approach for daily solar radiation modeling. Sol. Energy
**2011**, 85, 278–287. [Google Scholar] [CrossRef] - Eugster, W. Mountain Meteorology: Fundamentals and Applications. Mt. Res. Dev.
**2001**, 21, 200–201. [Google Scholar] [CrossRef] - Bedoya-Soto, J.M.; Aristizábal, E.; Carmona, A.M.; Poveda, G. Seasonal shift of the diurnal cycle of rainfall over medellin’s valley, central andes of Colombia (1998–2005). Front. Earth Sci.
**2019**, 7, 92. [Google Scholar] [CrossRef] - Poveda, G.; Mesa, O.J.; Waylen, P.R. Nonlinear Forecasting of River Flows in Colombia Based Upon ENSO and Its Associated Economic Value for Hydropower Generation; Springer: Dordrecht, The Netherlands, 2003; pp. 351–371. [Google Scholar]
- Poveda, G.; Waylen, P.R.; Pulwarty, R.S. Annual and inter-annual variability of the present climate in northern South America and southern Mesoamerica. Palaeogeogr. Palaeoclimatol. Palaeoecol.
**2006**, 234, 3–27. [Google Scholar] [CrossRef] - Poveda, G. La hidroclimatología de Colombia: una síntesis desde la escala inter-decadal hasta la escala diura por ciencias de la tierra. Rev. Acad. Colomb. Cienc. Exactas, Físicas y Nat.
**2004**, 28, 201–222. [Google Scholar] - Badescu, V. Modeling Solar Radiation at the Earth’s Surface; Springer: Berlin Heidelberg, Germany, 2008; p. 53. ISBN 978-3-540-77454-9. [Google Scholar]
- Kasten, F. A new table and approximate formula for relative optical air mass. Archiv für Meteorologie, Geophysik und Bioklimatologie
**1966**, 14, 206–223. [Google Scholar] [CrossRef] - Kasten, F.; Young, A.T. Revised optical air mass tables and approximation formula. Appl. Opt.
**2000**, 28, 4735–4738. [Google Scholar] [CrossRef] - Liu, B.Y.H.; Jordan, R.C. The interrelationship and characteristic distribution of direct, diffuse and total solar radiation. Sol. Energy
**1960**, 4, 1–19. [Google Scholar] [CrossRef] - Thompson, G.; Tewari, M.; Ikeda, K.; Tessendorf, S.; Weeks, C.; Otkin, J.; Kong, F. Explicitly-coupled cloud physics and radiation parameterizations and subsequent evaluation in WRF high-resolution convective forecasts. Atmos. Res.
**2016**, 168, 92–104. [Google Scholar] [CrossRef] - Iqbal, M. An Introduction to Solar Radiation; Elsevier Inc.: Vancouver, BC, Canda, 1983; ISBN 978-0-12-373750-2. [Google Scholar]
- Perez, R.; Ineichen, P.; Seals, R.; Zelenka, A. Making full use of the clearness index for parameterizing hourly insolation conditions. Sol. Energy
**1990**, 45, 111–114. [Google Scholar] [CrossRef] - Palomo, E. Hourly solar radiation time series as first-order Markov chains. In Proceedings of the Actes du International Solar Energy Society Solar World Congress, Kobe, Japan, September 1989; pp. 2146–2150. [Google Scholar]
- Munkhammar, J.; van der Meer, D.; Widén, J. Probabilistic forecasting of high-resolution clear-sky index time-series using a Markov-chain mixture distribution model. Sol. Energy
**2019**, 184, 688–695. [Google Scholar] [CrossRef] - Nguyen, B.T.; Pryor, T.L. A computer model to estimate solar radiation in Vietnam. Renew. Energy
**1996**, 9, 1274–1278. [Google Scholar] [CrossRef] - Aguiar, R.J.; Collares-Pereira, M.; Conde, J.P. Simple procedure for generating sequences of daily radiation values using a library of Markov transition matrices. Sol. Energy
**1988**, 40, 269–279. [Google Scholar] [CrossRef] - Graham, V.A.; Hollands, K.G.T. A method to generate synthetic hourly solar radiation globally. Sol. Energy
**1990**, 44, 333–341. [Google Scholar] [CrossRef] - Hollands, K.G.T.; Huget, R.G. A probability density function for the clearness index, with applications. Sol. Energy
**1983**, 30, 195–209. [Google Scholar] [CrossRef] - Diagne, M.; David, M.; Lauret, P.; Boland, J.; Schmutz, N. Review of solar irradiance forecasting methods and a proposition for small-scale insular grids. Renew. Sustain. Energy Rev.
**2013**, 27, 65–76. [Google Scholar] [CrossRef] [Green Version] - Hoyos, C.D.; Zapata, M.H. Análisis del Impacto de la Interacción Suelo-Atmósfera en las Condiciones Meteorológicas del Valle de Aburrá Utilizando el Modelo WRF; Universidad Nacional de Colombia: Bogotá, Colombia, 2015. [Google Scholar]
- Diagne, M.; David, M.; Boland, J.; Schmutz, N.; Lauret, P. Post-processing of solar irradiance forecasts from WRF Model at Reunion Island. Energy Procedia
**2014**, 57, 1364–1373. [Google Scholar] [CrossRef] - Incecik, S.; Sakarya, S.; Tilev, S.; Kahraman, A.; Aksoy, B.; Caliskan, E.; Topcu, S.; Kahya, C.; Odman, M.T. Evaluation of WRF parameterizations for global horizontal irradiation forecasts: A study for Turkey. Atmósfera
**2019**, 32, 143–158. [Google Scholar] [CrossRef] [Green Version] - Zempila, M.M.; Giannaros, T.M.; Bais, A.; Melas, D.; Kazantzidis, A. Evaluation of WRF shortwave radiation parameterizations in predicting Global Horizontal Irradiance in Greece. Renew. Energy
**2016**, 86, 831–840. [Google Scholar] [CrossRef] - Lara-Fanego, V.; Ruiz-Arias, J.A.; Pozo-Vázquez, D.; Santos-Alamillos, F.J.; Tovar-Pescador, J. Evaluation of the WRF model solar irradiance forecasts in Andalusia (southern Spain). Sol. Energy
**2012**, 86, 2200–2217. [Google Scholar] [CrossRef] - Jimenez, P.A.; Hacker, J.P.; Dudhia, J.; Haupt, S.E.; Ruiz-Arias, J.A.; Gueymard, C.A.; Thompson, G.; Eidhammer, T.; Deng, A. WRF-SOLAR: Description and clear-sky assessment of an augmented NWP model for solar power prediction. Bull. Am. Meteorol. Soc.
**2016**, 97, 1249–1264. [Google Scholar] [CrossRef] - Arbizu-Barrena, C.; Ruiz-Arias, J.A.; Rodríguez-Benítez, F.J.; Pozo-Vázquez, D.; Tovar-Pescador, J. Short-term solar radiation forecasting by advecting and diffusing MSG cloud index. Sol. Energy
**2017**, 155, 1092–1103. [Google Scholar] [CrossRef] - Ruiz-Arias, J.A.; Arbizu-Barrena, C.; Santos-Alamillos, F.J.; Tovar-Pescador, J.; Pozo-Vázquez, D. Assessing the surface solar radiation budget in the WRF model: A spatiotemporal analysis of the bias and its causes. Mon. Weather Rev.
**2016**, 144, 703–711. [Google Scholar] [CrossRef] - De Meij, A.; Vinuesa, J.F.; Maupas, V. GHI calculation sensitivity on microphysics, land- and cumulus parameterization in WRF over the Reunion Island. Atmos. Res.
**2018**, 204, 12–20. [Google Scholar] [CrossRef] - Verbois, H.; Rusydi, A.; Thiery, A. Probabilistic forecasting of day-ahead solar irradiance using quantile gradient boosting. Sol. Energy
**2018**, 173, 313–327. [Google Scholar] [CrossRef] - Janjić, Z.I. The step-mountain Eta coordinate model: Further developments of the convection, viscous sublayer, and turbulence closure schemes. Mon. Weather Rev.
**1994**, 122, 927–945. [Google Scholar] [CrossRef] [Green Version] - Thompson, G.; Field, P.R.; Rasmussen, R.M.; Hall, W.D. Explicit forecasts of winter precipitation using an improved bulk microphysics scheme. Part II: Implementation of a new snow parameterization. Mon. Weather Rev.
**2008**, 136, 5095–5115. [Google Scholar] [CrossRef] - Thompson, G.; Eidhammer, T. A Study of aerosol impacts on clouds and precipitation development in a large winter cyclone. J. Atmos. Sci.
**2014**, 71, 3636–3658. [Google Scholar] [CrossRef] - Morrison, H.; Thompson, G.; Tatarskii, V. Impact of cloud microphysics on the development of trailing stratiform precipitation in a simulated squall line: Comparison of one- and two-moment schemes. Mon. Weather Rev.
**2009**, 137, 991–1007. [Google Scholar] [CrossRef] [Green Version] - Grell, G.A.; Dévényi, D. A generalized approach to parameterizing convection combining ensemble and data assimilation techniques. Geophys. Res. Lett.
**2002**, 29, 38-1–38-4. [Google Scholar] [CrossRef] [Green Version] - Deng, A.; Gaudet, B.; Dudhia, J.; Alapaty, K. Implementation and evaluation of a new shallow convection scheme in WRF. In Proceedings of the 26th Conference on Weather Analysis and Forecasting/22nd Conference on Numerical Weather Prediction, Atlanta, GA, USA, 2–6 February 2014; pp. 2–6. [Google Scholar]
- Kain, J.S. The kain–fritsch convective parameterization: An update. J. Appl. Meteorol.
**2004**, 43, 170–181. [Google Scholar] [CrossRef] [Green Version] - Moreno-Tejera, S.; Silva-Pérez, M.A.; Lillo-Bravo, I.; Ramírez-Santigosa, L. Solar resource assessment in Seville, Spain. Statistical characterisation of solar radiation at different time resolutions. Sol. Energy
**2016**, 132, 430–441. [Google Scholar] [CrossRef] - Bowman, K.P.; Fowler, M.D. The diurnal cycle of precipitation in tropical cyclones. J. Clim.
**2015**, 28, 5325–5334. [Google Scholar] [CrossRef] - Liou, K.-N. An Introduction to Atmospheric Radiation, 2nd ed.; Elsevier Science: Amsterdam, The Netherlands, 2002; ISBN 0124514510. [Google Scholar]
- US Department of Commerce, NOAA, N.W.S. Hurricane Harvey Info. Available online: https://www.weather.gov/hgx/hurricaneharvey (accessed on 22 September 2019).
- Roberts, B.J. Solar Maps | Geospatial Data Science | NREL. Available online: https://www.nrel.gov/gis/solar.html (accessed on 13 June 2019).
- HotSpot Energy Solar Sun Hours | Average Daily Solar Insolation | Europe. Available online: https://www.hotspotenergy.com/DC-air-conditioner/europe-solar-hours.php (accessed on 13 June 2019).
- World Bank. Colombia Recent Economic Developments in Infrastructure II; World Bank: Washington, DC, USA, 2004. [Google Scholar]
- Capacidad Efectiva Por Tipo de Generación. Available online: http://paratec.xm.com.co/paratec/SitePages/generacion.aspx?q=capacidad (accessed on 22 July 2019).
- Stockdale, T.; Balmaseda, M.; Ferranti, L. The 2015/2016 El Niño and Beyond. Available online: https://www.ecmwf.int/en/newsletter/151/meteorology/2015-2016-el-nino-and-beyond (accessed on 22 October 2019).
- Aryaputera, A.W.; Yang, D.; Walsh, W.M. Day-ahead solar irradiance forecasting in a tropical environment. J. Sol. Energy Eng.
**2015**, 137, 051009. [Google Scholar] [CrossRef] - Verbois, H.; Huva, R.; Rusydi, A.; Walsh, W. Solar irradiance forecasting in the tropics using numerical weather prediction and statistical learning. Sol. Energy
**2018**, 162, 265–277. [Google Scholar] [CrossRef] - Husein, M.; Chung, I.Y. Day-ahead solar irradiance forecasting for microgrids using a long short-term memory recurrent neural network: A deep learning approach. Energies
**2019**, 12, 1856. [Google Scholar] [CrossRef] [Green Version] - Lan, H.; Yin, H.; Hong, Y.Y.; Wen, S.; Yu, D.C.; Cheng, P. Day-ahead spatio-temporal forecasting of solar irradiation along a navigation route. Appl. Energy
**2018**, 211, 15–27. [Google Scholar] [CrossRef] - Lan, H.; Zhang, C.; Hong, Y.Y.; He, Y.; Wen, S. Day-ahead spatiotemporal solar irradiation forecasting using frequency-based hybrid principal component analysis and neural network. Appl. Energy
**2019**, 247, 389–402. [Google Scholar] [CrossRef]

**Figure 1.**Location of the site of study and the pyranometer used. (

**a**) Location of Colombia in South America. (

**b**) Location of Medellín in Colombia. Color shades represents topography.

**Figure 2.**Simulation process of the normalized clearness coefficient, ${k}_{s}$ using the two-part Markov model proposed in this work.

**Figure 3.**Simulation process of the normalized clearness coefficient, ${k}_{s}$ using the persistence-Markov model proposed in this work.

**Figure 4.**Histogram of hourly global horizontal irradiance (GHI) from the SIATA station recorded during the period 1 March 2016–28 February 2017.

**Figure 5.**Hourly GHI values from SIATA station for all days of the period March 2016–February 2017. The color scale represents the intensity of solar radiation in W/m

^{2}. White gaps correspond to missing values.

**Figure 6.**Daytime violin plots of GHI during the period March 2016–February 2017. Three hour segments are considered: 6–10 LT, 10–14 LT and 14–18 LT. The colored surfaces of each segment correspond to the empirical probability density function (PDF) of the GHI data. The black lines inside each distribution correspond to the boxplots of the GHI values and the white dots correspond to the median values of the distributions.

**Figure 7.**Empirical PDFs of ${k}_{s}$ for each month between March 2016 and February 2017, according to SIATA records. The y-axis values can be larger than 1.0 since this is a distribution plot, thus meaningful information can only be obtained from its integral over an interval rather than from a single point. This integral is equal or less to 1.

**Figure 8.**Empirical PDFs for four types of days with ${k}_{d}$ in different percentages of daily mean extraterrestrial radiation reaching the surface: (

**a**) 25%–30%, (

**b**) 50%–55%, (

**c**) 60%–65%, and (

**d**) 70%–75%.

**Figure 9.**Bar plots of the ${k}_{d}$ estimates Distributions of the daily clearness coefficient estimated by WRF, the Markov model and the persistence model. (

**a**) ${k}_{d}$ estimates distributions for 1 September 2017, measured ${k}_{d}$ = 0.72. (

**b**) ${k}_{d}$ estimates distributions 23 December 2017. Measured ${k}_{d}$ = 0.54. (

**c**) ${k}_{d}$ estimates distributions 6 June 2017. Measured ${k}_{d}$ = 0.48. (

**d**) ${k}_{d}$ estimates distributions for 24 November 2017 measured ${k}_{d}$ = 0.39. (

**e**) ${k}_{d}$ distributions for 19 August 2017, measured ${k}_{d}$ = 0.18. The black lines correspond to the measured ${k}_{d}$ values.

**Figure 10.**Hourly distributions of the GHI estimates produced by the Markov model and the WRF model for five particular days (Table 2). (

**a**) Corresponds to 1 September 2017, with a ${k}_{d}$ = 0.72. (

**b**) Corresponds to 23 December 2017, with a measured ${k}_{d}$ = 0.54. (

**c**) Corresponds to 6 June 2017, with a measured ${k}_{d}$ = 0.48. (

**d**) Corresponds to 24th November 2017, with a measured ${k}_{d}$ = 0.39. (

**e**) Corresponds to 19 August 2017, with a measured ${k}_{d}$ = 0.18. (

**f**) Also corresponds to 19 August 2017, but in this case the estimated ${k}_{d}$ was forced to be equal to the measured ${k}_{d}$ = 0.18. The black dashed lines correspond to the measured hourly GHI values for each day.

**Figure 11.**Frequency histogram of the $\Delta {k}_{d}$ of the estimates performed during the period May 2017–May 2018.

**Figure 12.**(

**a**) Distributions of the root mean square error (RMSE) vs $\Delta {k}_{d}$ for the period May 2017–May 2018. (

**b**) As in (

**a**) but for nRMSE. Orange boxes correspond to the wet season estimates. Blue boxes correspond to the dry season estimates.

Discrete State | Continuous interval | Discrete State | Continuous interval | Discrete State | Continuous interval | Discrete State | Continuous interval |
---|---|---|---|---|---|---|---|

State 1 | 0–0.05 | State 6 | 0.25–0.3 | State 11 | 0.5–0.55 | State 16 | 0.75–0.8 |

State 2 | 0.05–0.1 | State 7 | 0.3–0.35 | State 12 | 0.55–0.6 | State 17 | 0.8–0.85 |

State 3 | 0.1–0.15 | State 8 | 0.35–0.4 | State 13 | 0.6–0.65 | State 18 | 0.85–0.9 |

State 4 | 0.15–0.2 | State 9 | 0.4–0.45 | State 14 | 0.65–0.7 | State 19 | 0.9–0.95 |

State 5 | 0.2–0.25 | State 10 | 0.45–0.5 | State 15 | 0.7–0.75 | State 20 | 0.95–1 |

**Table 2.**Particular days considered to analyze the performance of the clearness coefficient simulations obtained with the Markov chains-based model and the weather research and forecasting (WRF) model.

Date | Daily Clearness Coefficient (k_{d}) | Category |
---|---|---|

1 September 2017 | 0.72 | Clear sky |

23 December 2017 | 0.54 | Broken clouds |

5 June 2017 | 0.48 | Cloudy |

24 November 2017 | 0.39 | Very cloudy (local conditions) |

19 August 2017 | 0.18 | Very cloudy (Synoptic conditions) |

**Table 3.**Experiments scheme configurations. The Deng scheme is also referred as Deng’s mass-flux-scheme.

Experiments | Cumulus | Shallow Convection | Microphysics | |||
---|---|---|---|---|---|---|

d01 | d02 | d01 | d02 | d01 | d02 | |

KF-TE | Kain-Fritsch [63] | Off | Off | Off | Thompson and Eidhammer | Thompson and Eidhammer |

KF-TE-02 | Kain-Fritsch | Kain-Fritsch | Off | Off | Thompson and Eidhammer | Thompson and Eidhammer |

KF-MO | Kain-Fritsch | Off | Off | Off | Morrison | Morrison |

GR-TE | Grell 3D | Off | Off | Off | Thompson and Eidhammer | Thompson and Eidhammer |

GR-TE-DE | Grell 3D | Off | Deng | Off | Thompson and Eidhammer | Thompson and Eidhammer |

GR-MO-DE | Grell 3D | Off | Deng | Off | Morrison | Morrison |

**Table 4.**Statistical summary of the hourly GHI from the SIATA station recorded during the period 1 March 2016 to 28 February 2017.

Total Size of the Sample with Hourly Resolution = 4658 | Percentiles (W/m^{2}) | |||||||
---|---|---|---|---|---|---|---|---|

Mean (W/m^{2)} | Median (W/m^{2}) | Standard Deviation (W/m^{2}) | IQR (W/m^{2}) | 0% | 25% | 50% | 75% | 100% |

385 | 332 | 315 | 553 | 0 | 90 | 332 | 643 | 1217 |

**Table 5.**Monthly mean hourly coefficient of variation (CV) and monthly mean daily radiation (in energy units of kilowatts-hour) from SIATA station during the period March 2016–February 2017.

Month | Hourly Mean CV (%) | Month Mean Daily Solar Energy (kWh/m^{2}) | $\mathbf{Mean}{\mathit{k}}_{\mathit{d}}$ |
---|---|---|---|

March 2016 | 39.7 | 5.4 | 0.52 |

April 2016 | 46.2 | 4.7 | 0.45 |

May 2016 | 40.7 | 4.8 | 0.48 |

June 2016 | 29.9 | 5.6 | 0.56 |

July 2016 | 33.5 | 5.6 | 0.55 |

August 2016 | 31 | 6 | 0.58 |

September 2016 | 37.2 | 5.4 | 0.52 |

October 2016 | 45.7 | 5.0 | 0.50 |

November 2016 | 49.7 | 4.0 | 0.43 |

December 2016 | 43 | 4.6 | 0.51 |

January 2017 | 36.3 | 5.3 | 0.57 |

February 2017 | 32.9 | 5.9 | 0.60 |

**Table 6.**Summary of hourly GHI estimates produced by the Markov model and by the WRF experiment, GR-MO-DE.

Simulated day | Markov | Persistence-Markov | WRF: GR-MO-DE |
---|---|---|---|

RMSE (Median) | RMSE (Median) | RMSE | |

01/09/2017 | 144 W/m^{2} (26%) | 190 W/m^{2} (33%) | 116 W/m^{2} (21%) |

k_{d} = 0.72 | |||

23/12/2017 | 177 W/m^{2} (47%) | 171 W/m^{2} (45%) | 174 W/m^{2} (46%) |

k_{d} = 0.54 | |||

05/06/2017 | 165.1 W/m^{2} (44%) | 158 W/m^{2} (43%) | 204 W/m^{2} (55%) |

k_{d} = 0.48 | |||

24/11/2017 | 209 W/m^{2} (76%) | 233 W/m^{2} (85%) | 288 W/m^{2} (104%) |

k_{d} = 0.39 | |||

19/08/2017 | 415.2 W/m^{2} (288.4%) | 372 W/m^{2} (258%) | 133 W/m^{2} (92%) |

k_{d} = 0.18 | |||

19/08/2017 | 97.5 W/m^{2} (68%) | 372 W/m^{2} (258%) | 133 W/m^{2} (92%) |

k_{d} = 0.18 | |||

(k_{d} corrected) |

Error Metric | Markov | Persistence-Markov | WRF:GR-MO-DE |
---|---|---|---|

Median RMSE (W/m^{2}) | 177 | 190 | 174 |

Error Metric | Markov | Persistence-Markov |
---|---|---|

Median RMSE (W/m^{2}) | 214 | 217 |

Median MBE (W/m^{2}) | 33.9 | 7.21 |

$\Delta {k}_{d}$ | 0.076 | 0.003 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Urrego-Ortiz, J.; Martínez, J.A.; Arias, P.A.; Jaramillo-Duque, Á.
Assessment and Day-Ahead Forecasting of Hourly Solar Radiation in Medellín, Colombia. *Energies* **2019**, *12*, 4402.
https://doi.org/10.3390/en12224402

**AMA Style**

Urrego-Ortiz J, Martínez JA, Arias PA, Jaramillo-Duque Á.
Assessment and Day-Ahead Forecasting of Hourly Solar Radiation in Medellín, Colombia. *Energies*. 2019; 12(22):4402.
https://doi.org/10.3390/en12224402

**Chicago/Turabian Style**

Urrego-Ortiz, Julián, J. Alejandro Martínez, Paola A. Arias, and Álvaro Jaramillo-Duque.
2019. "Assessment and Day-Ahead Forecasting of Hourly Solar Radiation in Medellín, Colombia" *Energies* 12, no. 22: 4402.
https://doi.org/10.3390/en12224402