High resolution global irradiance time series are needed for accurate simulations of photovoltaic (PV) systems, since the typical volatile PV power output induced by fast irradiance changes cannot be simulated properly with commonly available hourly averages of global irradiance. We present a twostep algorithm that is capable of synthesizing oneminute global irradiance time series based on hourly averaged datasets. The algorithm is initialized by deriving characteristic transition probability matrices (TPM) for different weather conditions (cloudless, broken clouds and overcast) from a large number of high resolution measurements. Once initialized, the algorithm is locationindependent and capable of synthesizing oneminute values based on hourly averaged global irradiance of any desired location. The oneminute time series are derived by discretetime Markov chains based on a TPM that matches the weather condition of the input dataset. Oneminute time series generated with the presented algorithm are compared with measured high resolution data and show a better agreement compared to two existing synthesizing algorithms in terms of temporal variability and characteristic frequency distributions of global irradiance and clearness index values. A comparison based on measurements performed in Lindenberg, Germany, and Carpentras, France, shows a reduction of the frequency distribution root mean square errors of more than 60% compared to the two existing synthesizing algorithms.
The efficiency of PV modules depends mainly on the irradiance, amongst other secondary effects such as module temperature [
For the understanding of the dynamic interaction of PV generator, storage systems, loads, and grids on a worldwide scale, oneminute data series of high quality in terms of realistic variability and frequency distributions are a key factor. Simulating those systems with hourly averaged values neglects significant behavior patterns like short time power enhancements [
To illustrate the importance of oneminute data for the simulation of PV systems, a 1 kWp PV example system with PV generator, DC/AC inverter, and grid is analyzed at the location of HTW Berlin, Germany. DC/AC inverters are used in gridconnected PV systems as power processing interface between the PV power source (DC) and the electric grid (AC). The output power is very sensitive to the temporal variability of the solar radiation which is highest during broken clouds.
In some important markets (e.g., Germany), PV systems can be affected by grid connection restrictions that define the maximum AC power output of the inverter as a percentage of the installed PV power on the DC side, where the usual limit is around 70% [
Power output of a 1 kWp PV system at HTW Berlin, Germany, on April 01, 2012, measured oneminute values (grey) and hourly averaged values (blue). The yield losses due to maximum power clipping (output power is cut above 700 W) are calculated.
Following Vanicek et al. in his contribution on the energy yield losses as a function of inverter dimensioning [
Yearly energy yield losses of a 1 kWp PV system at HTW Berlin, Germany, for various inverter sizing factors (the relation between installed PV power on the DC side and nominal AC inverter output) and a maximum power clipping value of 70% (output power is cut at 70% of the installed DC power). Using hourly averaged values for the simulation of PV systems leads to a significant underestimation of the yearly yield losses. With hourly averaged values (grey), the total energy loss is at 1.3% while the more precise simulation with oneminute values (blue) returns a total energy loss of 3.9%. In addition, this figure illustrates that the optimal inverter sizing factor (here, 143%) for systems with maximum power clipping is the reciprocal of the clipping value (70%).
While there exist several commercial providers and free sources of meteorological data in a resolution of one hour (e.g., Meteotest, SolarGIS, and TMY), covering nearly the whole earth, the availability of measured irradiance data with a resolution of less than an hour is very limited. This limited availability leads to the necessity to synthesize oneminute time series from hourly averaged data.
Several algorithms were developed in the past in order to synthesize oneminute global irradiance datasets with realistic variability and frequency distributions from hourly averaged datasets. The supposedly most established algorithms were developed by Aguiar and CollaresPereira [
The contribution of Aguiar and CollaresPereira was originally designed for the generation of hourly averaged time series with daily averages as input. It is based on the modeling of probability densities as Gaussian functions that depend on the clearness index
Other important contributions to this topic were provided by Assunção et al. [
However, current algorithms only insufficiently withstand the validation against measurement values, since they underestimate irradiance enhancements caused by broken clouds, overestimate mid irradiance values, and provide oneminute time series with a variability that is too high.
Therefore, we developed an improved algorithm capable of synthesizing oneminute global irradiance time series based on hourly averaged global irradiance. The algorithm takes three different weather conditions (cloudless, broken clouds and overcast) into consideration. We show that the improved algorithm exceeds the performance of the Aguiar and the Skartveit algorithm in terms of temporal variability and characteristic frequency distributions for the calculation of shortterm global irradiance at two exemplary PV installation locations.
The new algorithm consists of two parts. The first part comprises a data preparation process that categorizes the input dataset and produces transition probability matrices (TPM) for three weather conditions: cloudless, broken clouds and overcast. The preparation process has to be executed only once.
The input dataset used for the initialization consists of global irradiance measurements conducted by the Baseline Surface Radiation Network (BSRN), featuring more than 50 locations all over the world with up to 20 years of measurements. The BSRN database is updated continuously with new measurement data; in this study we used a snapshot of May 2013. A subset of these data, oneminute global irradiance measurements performed in Lindenberg, Germany (2005), and Carpentras, France (2001), is used for the model validation.
The second part is the synthesis process for oneminute time series from hourly averaged time series. The required input of this process only consists of the prepared set of TPM and the hourly averaged time series of global irradiance that is to be disaggregated. The core of the process is based on Markov chains [
The central idea in both parts of the new algorithm is the classification of weather situations by the temporal feature of the clearness index. In the first part, the preparation process, the BSRN dataset is split into three individual datasets corresponding to three weather conditions: cloudless, broken clouds and overcast. Each subset is then processed separately and transformed into a transition probability matrix. In the second part, the synthesis process, each daily dataset of the hourly averaged input values is categorized as well and processed according to their weather category. As a consequence, the main process steps of the new algorithm are only depending on those weather categories, in disregard of specific location information.
This leads to the advantage that the algorithm can be applied to hourly averaged datasets of arbitrary locations. Furthermore, the only required input is the hourly averaged datasets, once the TPM are created. Hence, the new algorithm combines aspects of existing work on this subject with a universally applicable method for the synthesis of oneminute time series from hourly averaged values.
The determination of predominant weather conditions is needed in both steps of the presented algorithm. The weather conditions are determined by the calculation of the clearness index
The predominant weather condition on a particular day results in a characteristic temporal pattern of
Table
Overview of the three weather classes and their detection conditions.
Weather class  Condition 

Overcast 

Cloudless 

Broken clouds  Otherwise 
Example results of the weather category detection algorithm based on
Visualization of the classification of weather conditions by
For each class that represents a specific weather situation, matrices of transition probabilities (TPM) are created. The TPM contain information on how probable the switch is from one specific
Excerpt of an example TPM for broken clouds weather condition. For each

 

0  0.01  0.02  0.03  0.04  0.05  0.06  0.07  0.08  0.09  0.1 


0  0  0  0  0  0  0  0  0  0  0  0  
0.01  0  0.82927  0.17073  0  0  0  0  0  0  0  0  
0.02  0  0.10345  0.72414  0.17241  0  0  0  0  0  0  0  
0.03  0  0  0.06897  0.76724  0.15517  0  0.00862  0  0  0  0  
0.04  0  0  0.00709  0.12057  0.70922  0.14894  0.01418  0  0  0  0  
0.05  0  0  0  0  0.07004  0.75875  0.15564  0.01167  0.00389  0  0  
0.06  0  0  0  0  0.01136  0.14773  0.64394  0.14773  0.03788  0.00758  0.00379  
0.07  0  0  0  0  0  0.0084  0.19328  0.53361  0.2395  0.02521  0  
0.08  0  0  0  0  0.0059  0  0.0236  0.17109  0.57817  0.17404  0.03245  
0.09  0  0  0  0  0  0  0  0.03378  0.19932  0.46959  0.21284  
0.1  0  0  0  0  0  0  0  0  0.02067  0.17571  0.5323  

The excerpt of a TPM shown in Table
Since the TPM are created using real weather data in oneminute resolution, each measured irradiance within a given time interval leaves a fingerprint in a TPM. Hence, the spatial and temporal validity of the algorithm is increasing with the number of input datasets. As of May 2013, the BSRN comprises more than 6900 irradiance measurement months distributed globally, which is equal to more than 200 000 measurement days in oneminute resolution that leave their fingerprint in the TPM. The influence of the number of input data on the synthesis quality is referred to in Results section as well.
To generate oneminute values from hourly averaged sequences of the global irradiance, the weather condition of the day in question is detected at first. Depending on the weather condition the correspondent
The actual generation of the oneminute values is conducted with the help of the socalled discretetime Markov chains (DTMC). DTMC is a statebased process for the modelling of realworld events. In the first order, the process is memoryless, so that the next state only depends on the current state [
To determine the successor
In the following section the new algorithm is validated with measurement data and compared to the algorithms by Aguiar and Skartveit. The result comparison is conducted for two exemplary datasets of one year at two different locations: Lindenberg, Germany, 2005, and Carpentras, France, 2001. Both datasets are taken from the BSRN database. To avoid selfreference in the presented results, the creation process of the TPM excludes all measurement data of the respective location.
First, the results are presented on the basis of diurnal courses to assess the temporal variability, afterwards in the form of frequency distributions. In addition we provide a table with comparative uncertainties.
When assessing the temporal variability of synthetized oneminute values, the results for days with broken clouds and overcast skies are more important, since the simulation of sunny days is not difficult. In Figure
Lindenberg, May 14, 2005. The temporal course of the measured global irradiance (a) on a day that was rated as a day with overcast sky is compared to values generated by the new algorithm (b), the algorithm by Aguiar (c), and the algorithm by Skartveit (d). The mean variability, that is, the mean irradiance change from one minute to the next, of the measured irradiance of 7.0 W/m² shows good congruence with the new algorithm (8.2 W/m²), while the usage of the algorithms by Aguiar and Skartveit leads to higher variability values of 13.1 W/m² and 16.6 W/m².
Although the exact occurrence of irradiance peaks in the modelled time series may differ from the measured time series, the variability of the values modelled with the new algorithm agrees with measured values to a very high degree. The mean variability of irradiance changes from one minute to the next is 7.0 W/m² for measured time series, whereas it is 8.2 W/m² for the data modelled with the new algorithm in the example dataset of Figure
A more complete picture of the variability of solar irradiance can be obtained by analyzing the frequency of its gradients over a whole year. The gradients, in this case the absolute difference of the irradiance values of one minute to the next for the measured data and model data, are calculated and transferred into frequency plots. Figure
Frequency of gradients of the global irradiance in Lindenberg, Germany (2005). The model quality in lower gradient ranges of up to 10 W/m² is similar in all models. In the range of 10 to 100 W/m², significant deviations can be detected for the models of Aguiar and Skartveit (grey), whereas the new method (blue) shows good congruence. For gradient values of more than 100 W/m², the model of Aguiar underestimates the frequency significantly, while the new method and the method of Skartveit feature similar frequency values compared to the measurement data (black).
Frequency of gradients of the global irradiance in Carpentras, France (2001). As in Figure
In both cases, the frequency distribution of the data modelled by the algorithms of Aguiar and Skartveit, respectively, shows significant overestimations for the gradient range from 10 to 100 W/m², while the new algorithm is able to produce irradiance values that feature a similar frequency distribution in this range. For gradients of less than 10 W/m² the data modelled by all algorithms show similar deviations from the measured data. For gradients of more than 100 W/m², the new algorithm and the approach of Skartveit display similar quality, whereas the algorithm of Aguiar shows significant underestimations for both locations.
For the transfer into deviation indicators, the deviations of the modelled data from the measured ones for each irradiance value are squared, weighed by its frequency, and summed up. The frequency weight
Root mean square errors (RMSE) of the frequency distributions of irradiance gradients of modelled data versus measurement in W/m². The new model is able to produce significantly smaller values of RMSE than the models of Aguiar and Skartveit for both locations, Lindenberg, 2005, and Carpentras, 2001.
Model  Lindenberg  Carpentras 

Aguiar  8131  8541 
Skartveit  4758  5112 
New  2787  3218 
Frequency distributions of measured global irradiance for Lindenberg (Germany, 2005) against values generated by different algorithms. Mid values are slightly overestimated, and high values are underestimated by the existing models (grey dotted), resulting in RMSE of 0.530% for Aguiar and 0.684% for Skartveit. The modelling quality of the new method (blue) does not overestimate mid irradiance values and shows only little underestimation of high irradiance values. The new RMSE can be reduced to 0.210%.
Frequency distributions of measured global irradiance for Carpentras (France, 2001) against values generated by different algorithms. For locations with higher yearly global irradiation, the high irradiance peak grows. Mid values are slightly overestimated, and high values are underestimated by the existing models (grey dotted), resulting in RMSE of 0.549% for Aguiar and 0.575% for Skartveit. The modelling quality of the new method (blue) does not overestimate mid irradiance values and shows only little underestimation of high irradiance values. The new RMSE can be reduced to 0.237%.
Frequency distributions of the clearness index
Frequency distributions of the clearness index
For the generation of those figures, measured oneminute values were averaged to hourly means, which were then fractionized again using the new improved algorithm as well as the approaches of Aguiar and Skartveit. The figures show how often a specific irradiance value occurs in a year. The maximum at high irradiance values represents clear sky situations, while the second maximum at lower values is evoked by skies covered by clouds. Hence, the maximum at high irradiance values is considerably more pronounced at sunnier locations than at locations with very variable weather.
It can be seen that the new algorithm is reproducing the frequency distributions of the global irradiance much better than the conventional approaches. Mid irradiance values are not overestimated, and a good modelling quality is present at high irradiance values. However, very high irradiances above 1100 W/m² are slightly overestimated.
If those frequency distributions are looked at in the form of the clearness index
These visual impressions give an indication, but an analysis of the uncertainty can be used for quantitative assessment. Table
Root mean square error (RMSE) values comparing the frequency distributions of the existing and the new algorithms with measured data. Smaller values of RMSE denote better congruence of the frequency distributions of modelled oneminute values with measured values.
Model  RMSE of irradiance in %  RMSE of 


Lindenberg  Carpentras  Lindenberg  Carpentras  
Aguiar  0.530  0.549  596  801 
Skartveit  0.684  0.575  862  962 
New  0.210  0.237  207  248 
To analyze the influence of the amount of input data for the TPM on the synthesis quality of the algorithm, the creation process of the TPM is varied as follows.
First, the algorithm is processed three times with its original setup, which includes all TPM except the ones from the respective location, to estimate the influence of the random Markov number generator on the RMSE range. Second, only TPM of the respective location are used. In a third iteration, the only measurement values included in the creation process are taken from BSRN stations that are located in the same climate zone as per the definition of Köppen [
The synthesis of oneminute irradiance values is now repeated with all varied TPM. The RMSE values are determined according to the previous chapter. Table
Comparison of synthesis quality of the new algorithm as a function of input data for the locations of Lindenberg, Germany, 2005, and Carpentras, France, 2001.
Variation  RMSE of irradiance in %  RMSE of 


Lindenberg  Carpentras  Lindenberg  Carpentras  
All except own (1)  0.210  0.237  207  248 
All except own (2)  0.244  0.174  210  273 
All except own (3)  0.235  0.217  204  239 
Own TPM only  0.232  0.199  202  279 

0.193  —  315  — 
All TPM  0.253  0.186  204  254 
The repetition of the synthesis process with the original setup (all TPM except own 1–3) demonstrates the RMSE range that can be expected due to the random nature of the Markov number generator. The interesting aspect of the various TPM modifications (own TPM only,
An improved method for synthesizing oneminute time series of global irradiance has been presented that was developed on the basis of a large worldwide measurement dataset. It combines the advantages of conventional algorithms and adds new elements like the differentiation of weather conditions. It could be demonstrated that with the new approach it is possible to synthesize oneminute values of high statistical quality and realistic temporal variability. The independence on the location has been shown for selected cases. Such an independence would allow synthesizing oneminute time series for any location.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors acknowledge support by Deutsche Forschungsgemeinschaft and Open Access Publishing Fund of Leibniz Universität Hannover.