Description of sites under study.

## Abstract

The characterization of irradiance variability needs tools to describe and quantify variability at different time scales in order to optimally integrate PV onto electrical grids. Recently in the literature, a metric called nominal variability defines the intradaily variability by the ramp rate’s variance. Here we will concentrate on the quantification of this parameter at different short time scales for tropical measurement sites which particularly exhibit high irradiance variability due to complex microclimatic context. By analogy with Taylor law performed on several complex processes, an analysis of temporal fluctuations scaling properties is proposed. The results showed that the process of intradaily variability obeys Taylor’s power law for every short time scales and several insolation conditions. The Taylor power law for simulated PV power output has been verified for very short time scale (30s sampled data) and short time scale (10 min sampled data). The exponent λ presents values between 0.5 and 0.8. Consequently, the results showed a consistency of Taylor power law for simulated PV power output. These results are a statistical perspective in solar energy area and introduce intradaily variability PV power output which are key properties of this characterization, enabling its high penetration.

### Keywords

- nominal variability
- power Taylor law
- intradaily variability
- temporal fluctuations scaling
- PV power output

## 1. Introduction

Solar energy is an environmental process composed of a stochastic component, source of this intermittent nature and a deterministic component depending on solar geometry and time/location parameters. The stochastic component is complex to define due to significant fluctuations, particularly at intradaily time scales or short time scales. This component is the result of several factors of clouds motion and weather systems and is the main source of limited penetration onto electrical grids of systems exploiting solar energy such as photovoltaic panels (PV systems). Recently in literature, irradiance short-term variability attracted the interest of many studies. Indeed, the variability of irradiance particularly at short time scales is a very complex process that needs tools to characterize it to optimally integrate it onto electrical grids. The dynamic of fluctuations remains a challenging parameter to define. Several works defined this dynamic by metrics. In [1], a scoring method, termed an Intra-Hour Variability Score (IHVS), quantified variability characteristics into a single metric which represents an hour of irradiance. In [2], the one-minute intra-hourly solar variability based upon hourly inputs has defined four metrics characterizing intra-hourly variability, such as the standard deviation of the global irradiance clear sky index, the mean index change from one-time interval to the next, the maximum and the standard deviation of the latter. Other metrics defining intradaily of irradiance are described in the literature such as VI index (variability index) with the daily clear sky index in [3], the daily probability of persistence (POPD) in [4], the nominal variability which is the ramp rate standard deviation calculated from the change in the clear sky index developed in [5], MAD metric which is defined by the median absolute deviation of the change in the clear sky index in [6].

Analysis of variability was also applied to PV power output such as [7] who defined a frequency domain of PV output variability analysis, or [8] describing the frequency of a given fluctuation from PV power output for a certain day by an analytic model and [9] which demonstrated rapid ramps observed in point measurements would be smoothed by large PV plants and the aggregation of multiple PV plants with [5] who completed and strengthened this result. In [10], a quantitative metric called the Daily Aggregate Ramp Rate (DARR) is proposed to quantify, categorize, and compare daily variability from power output, across multiple sites.

In this chapter, we examine a temporal scaling fluctuation modeling namely power Taylor law applied to irradiance intradaily variability and PV power output intradaily variability. The influence of parameters such as increment, data sampling on this modeling is also assessed in order to reinforce the quantification and characterization of this complex process which is ramp rate’s variance. This study is a supplementary results to works about intradaily variability quantification but also showed evidence to the universality of power Taylor for environmental complex processes.

## 2. Data set for the study

### 2.1 Context of study

In this work, the sites under study are located in tropical islands (Guadeloupe, La Reunion and Hawaii). These exhibit high variability irradiance due to a large diversity of microclimates. This complex process evolves on different time and spatial scales. Table 1 summarizes the description of sites under study and Figure 1 presents the geographical location of measurement sites under study. Measurements are available on a basis of two years of data. The study of temporal irradiance fluctuations scaling is therefore analyzed for different locations.

Reunion | Hawaii | Guadeloupe | ||
---|---|---|---|---|

Saint-pierre | Tampon | Kalaeloa Oahu | Pointe-à-pitre (Fouillole) | |

Data provider | PIMENT | PIMENT | NREL | LARGE |

GHI measurement time step | 10 min | 10 min | 3 seconds | 1 second |

Period of record | 2 years | 2 years | 2 years | 2 years |

Longitude ( | 55.491 | 55.506 | −158.084 | −61.517 |

Latitude ( | −21.34 | −21.269 | 21.3120 | 16.217 |

Elevation (m) | 75 | 550 | 11 | 6 |

#### 2.1.1 Case of Oahu

Kalaeloa Oahu is located in a tropical zone, at the West of the Hawaii island. This station is affected by clouds formation during summer due to the trade winds effect and are generated by the local topography (located inland with medium orography with an elevation of about 11 m). This dataset is provided on the NREL (National Renewable Energy Laboratory) website. The procedure of data acquisition is described on the website. GHI is measured by using a LICOR LI-200 Pyranometer mounted on an Irradiance Inc. Rotating Shadowband Radiometer (RSR). RSR mounted on the ground and the LI-200 sensor height is approx. The uncorrected value is for testing and troubleshooting purposes only. Voltage is measured across a 100 Ohm precision resistor in parallel to the sensor output.

#### 2.1.2 Case of Fouillole

Fouillole site is located at the campus of the French West Indies University situated in the West of Grande-Terre island in coastal topography and also located in an urban area. This context generates a complex microclimatic context. The clouds are generated by land/sea contrast and the local topography (elevation lower than 10 m, Table 1). Data are measured by a pyranometer CM22 from Kipp and Zonen whose response time is less than one second. The precision of pyranometer is

#### 2.1.3 Case of Saint-Pierre and Le tampon

Concerning Reunion island, two locations at the West of the island are under our study: Saint-Pierre which is a coastal site, and Le Tampon an inland site. According to [11, 12], these two sites exhibit very different sky conditions. Concerning Le Tampon, the inland site orographic clouds are mainly generated by the local topography. This site is located in a mountainous orography (elevation about 550 m) in an urban zone. It presents higher variability irradiance than Saint-Pierre site which is in a climate tropical ocean with an urban coastal topography. The irradiance data is measured with a secondary standard pyranometer CMP11 from Kipp and Zonen. The precision of the pyranometer is

### 2.2 Data preprocessing

The profile of GHI that is due to solar geometry is predictable by several models [13, 14, 15]. In our study, we will focus on intra-daily variability induced by cloud mass passage that is stochastic in nature [5].

In order to study this variability component, the solar-geometry effects must be first removed. The parameter usually considered in the solar energy area is the clear sky index

where

In order to better consider variability for a time scale, we investigate in the temporal increment for a given time scale

This change is often referred to as the ramp rate [5].

Figure 3 presents

Recently in the literature [5, 6], a metric is defined to characterize the intradaily variability of the change in the clear sky index over the considered day i.e. the ramp rate’s variance, or its square root. This metric is the ramp rate standard deviation called nominal variability defined by this equation:

This metric can clearly distinguish two extremum cases of insolation conditions, namely perfectly clear conditions (i.e., no variability) and heavily overcast conditions (i.e., again, no variability), contrary to other metric proposed such as

## 3. Taylor power law, a statistical perspective in solar energy

### 3.1 Definition of the Taylor power law

Many fields exhibit complex process such as biology, ecology and, engineering sciences. The analysis of these complex process exhibited the universality of the Taylor power law defined by [17] by a scaling relationship more precisely described as” temporal fluctuation scaling” [18]. The Taylor power law (or temporal fluctuations scaling), is a scaling relationship between the standard deviation

with

### 3.2 Taylor law in solar energy data

Solar energy is a complex process. Particularly for insular context, this energy resource exhibits high fluctuations at all temporal and spatial short time scales. The analysis of the stochastic nature of this resource is in growing in the literature and have shown evidence of scaling properties despite its complexity [3, 5, 6, 20]. In this paper, an analysis of scaling properties of irradiance fluctuations is proposed. By analogy with Taylor law performed on several complex processes, we investigate in the study of Taylor power law performed on the intradaily variability of irradiance field, specifically on the

with

The four sites previously mentioned, characterized by tropical insular context hence exhibiting high variability, were chosen to test the consistency of this temporal fluctuation scaling method.

### 3.3 Criterion of the temporal limit of Δ t

The time increment

## 4. Verification of the existence of power Taylor law

### 4.1 Verification of the existence of power Taylor law for very short time scales dataset

The existence of power Taylor law is first verified for sampled data at very short time scales, i.e. 3 s, for the whole of dataset (2 years). This time scale of data sampling is available for Fouillole and Oahu measurement sites. The increment

3 s | 0.78 | 0.84 | 10mn | 0.62 | 0.68 |

15 s | 0.74 | 0.74 | 20mn | 0.57 | 0.68 |

30s | 0.71 | 0.70 | 30mn | 0.55 | 0.73 |

45 s | 0.70 | 0.69 | 40mn | 0.54 | 0.76 |

1mn | 0.68 | 0.68 | 50mn | 0.55 | 0.79 |

2mn | 0.65 | 0.67 | 1 h | 0.55 | 0.83 |

4mn | 0.62 | 0.68 | 2 h | 0.56 | 0.85 |

6mn | 0.60 | 0.69 | 3 h | 0.61 | 1.02 |

8mn | 0.58 | 0.68 | 4 h | 0.68 | 1.10 |

This power Taylor law verifies that:

### 4.2 Verification of the existence of power Taylor law for short time scales dataset

The temporal fluctuation scaling is analyzed by assessing power Taylor law on 10 min sampled data which is available on the whole of sites under study (Tampon, Saint-Pierre, Fouillole, Oahu).

The result of this analysis for

10 min | 0.70 | 0.80 | 0.41 | 0.63 |

20 min | 0.71 | 0.84 | 0.36 | 0.63 |

30 min | 0.72 | 0.89 | 0.33 | 0.64 |

40 min | 0.71 | 0.93 | 0.31 | 0.67 |

50 min | 0.72 | 0.97 | 0.29 | 0.71 |

1 h | 0.72 | 1.02 | 0.26 | 0.74 |

2 h | 0.74 | 1.16 | 0.30 | 0.81 |

3 h | 0.77 | 1.14 | 0.36 | 0.76 |

4 h | 0.78 | 1.12 | 0.31 | 0.68 |

The same analysis is also done for sampled data at 30s, 1 min, 2 min, 5 min and showed the consistency of power Taylor law for intradaily variability irradiance process. The results are presented in Section 5, Figure 7 for the study of the evolution of

## 5. Illustration of λ as a function of temporal parameters

This analysis allows assessing if there is a dependence between

### 5.1 Evolution of coefficient λ as a function of increment Δ t parameter conditioned to data sampling

The first study assessing the evolution of

The profile of evolution of coefficients

### 5.2 Verification of the Taylor law stationarity

The evolution of coefficients

## 6. Temporal fluctuations scaling analysis for PV power output

### 6.1 Characteristics of photovoltaics panels and PV power output modeling

In order to verify the consistency of Taylor power law for PV power output (power production from photovoltaic panel), the PV power output time series is simulated and obtained by a theoretical model for a first approach. The PV power output modeling is calculated by the following equation Eq. (7) such described in [24]. We have chosen arbitrarily a classic panel of monocrystalline technology for the simulation. The characteristics of the photovoltaic panel are described in Table 4. The required parameters for this modeling are the number of panels set at 1, the panel area, and the panel’s efficiency according to the theoretical model equation Eq. (7).

Technology | Monocrystalline |
---|---|

Nominal power | |

Voltage for maximal power | |

Current for maximal power | |

Voltage of open circuit | |

Current of short circuit | |

Dimension of photovoltaic cells (mm) | |

Number of cells | |

PV module dimension (mm) | |

Panels’ efficiency |

where, GHI is the measured irradiance in

The data in Table 4 are based on measurements under the standards conditions SRC (Standard Reporting Conditions, knowledge also: STC or Standard Test Conditions) which: an illumination of 1

The aim here is to obtain an output power profile to evaluate the existence of the Taylor power law. Considering the transfer function between the GHI and the power output of the panel, one should expect the same results found for irradiance. We decided on a first approach to verify Taylor’s law on simulated data which should be a good approximation of the real case. To reinforce this study in perspective, we will need real data from PV power output. An example of a sequence of PV power output time series is presented in Figure 10.

The stochastic component of PV power output is obtained by removing the solar-geometry effects. Similarly to the clear sky index

where

By analogy with power Taylor law performed for irradiance, we verify a scaling relationship between

### 6.2 Power Taylor law consistency for PV ouput area

The power Taylor law for PV power output has been verified for very short time scale (30s sampled data) and short time scale (10 min sampled data). The results showed a consistency of Taylor power law for PV area output (Figure 12) which is an expected result due to the relation between irradiance and PV power output modeling. Therefore, there is no changing of the inherent cause of variability. The results have shown evidence for the existence of temporal fluctuation scaling for PV power output data. Hence, the ramp rate standard deviation of power PV can be modelized by this equation Eq. (10):

The

Consequently, theoretically, the user does not need to have available PV output data set to characterize ramp rate standard deviation of PV output. To reinforce this study in perspective, we will need real data from PV output.

## 7. Discussion

The development of installed photovoltaic (PV) power increases problems related to the underlying variability of PV power production. Characterizing the underlying spatiotemporal volatility of solar radiation is a key ingredient to the successful outlining and stable operation of future power grids [25]. In literature, scientifics attention and studies related to the understanding of weather-induced PV power output variability are in full development.

Each time scale interval of solar generation is associated with a specific problem of load management challenges. In [5], a characterization of how solar energy’s resource variability impacts energy systems and a definition of the temporal or the spatial scales context are given. In our study context that concerns very short time scales fluctuations, voltage control issues are a specific problem [5, 26]. This observation implies an understanding of the ramp’s variance at very short time scales.

As PV power variability is mainly determined by irradiance variability, irradiance variability quantifications are essential to the successful outlining and stable operation of future power grids [27]. Variability in irradiance itself as interesting as variability in irradiance increments. Indeed, irradiance increments are transitions from one point in time to another, namely ramp rates. Irradiance variability and irradiance increments impact the system PV differently. Irradiance variability mainly impacts a PV system’s yield and the proper dimensioning of energy storage, while increment variability affects power quality as well as the maintenance of the generation load balance [25]. Therefore, our study is firstly focused on increment variability in irradiance. Then, this analysis is applied to PV power output time series.

The works in this article bring a complementary understanding of underlying variability. The results highlighted a new modelization of ramp rate’s variance of irradiance and PV power output based on the fluctuations’ magnitude from Taylor power law. This model makes it possible to extrapolate the resulting variability of PV power output. Moreover, a synthetic time series data at high frequency which are not commonly available would be produced from lower frequency by using nominal variability modeling from power Taylor law. This new model fills a gap in temporal scales. This may be useful for inefficient PV power output and irradiance forecasting model at very short time scale for example Numerical Weather prediction (NWP) models.

Analysis of

## 8. Conclusion

This chapter presented a characterization of the irradiance and PV power output intradaily variability describing a temporal fluctuation scaling. By analogy of environmental complex process, the works have demonstrated that power Taylor law is verified for the ramp rate’s variance of irradiance named nominal variability, namely the standard deviation of the changes in the clear sky index

## Thanks

The authors thank Laboratory PIMENT (Laboratoire de Physique et Ingénierie Mathématique pour l’Energie et l’environnement) from University of La Réunion, for providing ground measurements databases at locations Tampon and Saint-Pierre and NREL (National Renewable Energy Laboratory) website for providing ground measurements databases at locations Oahu.

## References

- 1.
Lenox C, Nelson L. Variability Comparison of Large-Scale Photovoltaic Systems across Diverse Geographic Climates. In: 25th European Photovoltaic Solar Energy Conference, 2010,Valencia, Spain. - 2.
Perez R, Kivalov S, Schlemmer J, Hemker K, Hoff T. Parameterization of site-specific short-term irradiance variability. Solar Energy. 2011;85: 1343-1353 - 3.
Stein JS, Hansen CW, Reno MJ. The Variability Index: A New and Novel Metric for Quantifying Irradiance and PV Output Variability. In: World Renewable Energy Forum, 2012, Denver, Colorado. - 4.
Kang BO, Tam KS. A new characterization and classification method for daily sky conditions based on ground-based solar irradiance measurement data. Solar Energy. 2013; 94: 102–118 - 5.
Perez R, David M, Hoff TE, Jamaly M, Kivalov S, Kleissl J, Lauret P and Perez M. Spatial and temporal Variability of solar energy. Foundations and Trends® in Renewable Energy. 2016;1(1):1-44 - 6.
Lauret P, Perez R, Mazorra Aguiar L, Tapachès E, Diagne HM, David M. Characterization of the intraday variability regime of solar irradiation of climatically distinct locations. Solar Energy. 2016; 125: 99-110 - 7.
Hansen T and Phillip D. Utility solar generation valuation methods. Tecnichal report. - 8.
Marcos J, Marroyo L, Lorenzo E, Alvira D, Izco E. Power output fluctuations in large scale PV plants: One year observations with one second resolution and a derived analytic model. Prog. Photovolt. Res. Appl. 2011; 19: 218–227 - 9.
Mills A, Ahlstrom M, Brower M, Ellis A, George R, Hoff T, Kroposki B, Lenox C, Nicholas M, Stein J, Wan YW. Understanding Variability and Uncertainty of Photovoltaics for Integration with the Electric Power System. IEEE Power and Energy Magazine. 2011; 9(3) - 10.
Van Haaren R, Morjaria M, Fthenakis V. Empirical assessment of short-term variability from utility-scale solar PV plants. Prog. Photovol. Res. Appl. 2012;22: 548–559 - 11.
Lauret P, Voyant C, Soubdhan T, David M, Poggi P. A benchmarking of machine learning techniques for solar radiation forecasting in an insular context. Solar Energy. 2015; 112: 446-457. - 12.
Diagne HM. Gestion intelligente du réseau électrique Réunionnais. Prévision de la ressource solaire en milieu insulaire. PhD thesis, Université de La Réunion; 2015. - 13.
Bird RE, Hulstrom RL. Simplified the Clear Sky Model for Direct and Diffuse Insolation on Horizontal Surfaces. In: Technical Report No. 1981; Solar Energy Research Institute. - 14.
Ineichen P. Comparison of eight clear sky broadband models against 16 independent data banks. Solar Energy. 2006; 80: 468–478 - 15.
Kasten F. Parametrisierung der globaslstrahlung durch bedekungsgrad undtrubungsfaktor. Ann Meteorol 1984;20:49-50. - 16.
Skartveit A, Olseth JA. The probability density of autocorrelation of short-term global and beam irradiance. Solar Energy. 1992; 46(9):477-488 - 17.
Taylor LR. Aggregation, variance and the mean. Nature. 1961; 189: 732-735 - 18.
Eisler Z, Kertész J. Scaling theory of temporal correlations and size-dependent fluctuations in the traded value of stocks. Physical Review E. 2006; 73(040109) - 19.
Calif R, Schmitt FG. Modeling of atmospheric wind speed sequence using a lognormal continuous stochastic equation. Journal of Wind Engineering and Industrial Aerodynamics. 2012; 109:1-8 - 20.
Lave M, Reno MJ, Broderick, RJ. Characterizing local high-frequency solar variability and its impact to distribution studies. Solar Energy. 2015; 118: 327-337. - 21.
Street JO, Carroll RJ, Ruppert D. a note on computing robust regression estimates via iteratively reweighted least squares. The American Statistician. 1988; 42: 152-154. - 22.
Draper NR, Smith H. Applied regression analysis, 3rd edition: Wiley series in probability and statistics; 1998. - 23.
Fronczak A, Fronczak P. Origins of Taylor’s power law for fluctuation scaling in complex systems. Physical Review E. 2010; 81(6). - 24.
Khaled U, Eltamaly A M, Beroual A. Optimal power flow using particle swarm optimization of renewable hybrid distributed generation. Energies. 2017; 10 (1013) - 25.
Lohmann, Gerald M. Irradiance Variability quantification and small-scale averaging in space and time: A short review. Atmosphere. 2018;9(7):264 - 26.
Widen J, Wäckelgård E, Paatero J, Lund P. Impacts of distributed photovoltaics on network voltages: Stochastic simulations of three Swedish low-voltage distribution grids. Electric Power Systems Research. 2010;80(12):1562-1571 - 27.
Mills A, Ahlstrom M, Brower M, Ellis A, George R, Hoff T, Kroposki B, Lenox C, Miller N, Milligan M, Stein J, Wan Y-h. Dark Shadows. IEEE Power and Energy Magazine. 2011;9(3):33-41