Temporal Fluctuations Scaling Analysis: Power Law of Ramp Rate’s Variance for PV Power Output

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.

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 (${}^{\circ }$)	55.491	55.506	−158.084	−61.517
Latitude (${}^{\circ }$)	−21.34	−21.269	21.3120	16.217
Elevation (m)	75	550	11	6

Table 1.

Description of sites under study.

Figure 1.

Geographical location of measurement sites under study: Oahu, Fouillole campus, tampon and Saint-Pierre.

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 $+/-3.0\%$ for the daily sun of GHI. Measurements are provided by LARGE laboratory from Université des Antilles on a 1 second basis data.

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 $+/-3.0\%$ for the daily sun of GHI. Measurements provided by PIMENT are available on a 10 min basis and two years of data.

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 ${\mathit{Kt}}^{\ast }$ (ratio of measured GHI to theoretical clear sky GHI) defined as Eq. (1).

$${\mathit{Kt}}^{\ast }=\frac{{\mathit{GHI}}_m}{{\mathit{GHI}}_{\mathit{\text{clear}}}}$$E1

where $\mathit{GHI}$ is the Global Horizontal Irradiance, index $m$ refers to the measured GHI and index $\mathit{\text{clear}}$ refers to theoretical clear sky irradiance.

In order to better consider variability for a time scale, we investigate in the temporal increment for a given time scale $\mathrm{\Delta }t$. The temporal increment of ${\mathit{Kt}}^{\ast }$ corresponding to the selected time scale $\mathrm{\Delta }t$ is noted $\mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)$ and is defined Eq.(2) such as in [5]. A sequence of $\mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)$ for each measurement sites with $\mathrm{\Delta }t=20\mathit{min}$ from original ${\mathit{Kt}}^{\ast }$ time series at 10 min, is presented in Figure 2.

Figure 2.

Signals of $\mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)$ for each measurement sites with $\mathrm{\Delta }t=20\mathit{min}$ from original time series of ${\mathit{Kt}}^{\ast }$ at $10$ min.

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

$$\mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)=K^{\ast }\left(t+\mathrm{\Delta }t\right)-K^{\ast }\left(t\right)$$E2

Figure 3 presents ${\mathit{Kt}}^{\ast }$ time series for 1 week sequence obtained from 1 week GHI data at 10 minutes time scales and Figure 2 presents $\mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)$ time series for five other days sequences.

Figure 3.

(a) 1 week GHI data and b) the corresponding ${\mathit{Kt}}^{\ast }$ signal obtained for Fouillole site. The latter signal exhibits the high variability of solar flux.

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:

$$\mathit{\text{Nominal variability}}=\sigma (\mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)=\sqrt{\mathit{Var}\left[\right(\mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)]}$$E3

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 $\sigma \left(\mathit{Kt}\right)$ [5, 16]. Nominal variability $\sigma (\mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)$ is a metric such a measure of the variability of the dimensionless clear sky index ${\mathit{Kt}}^{\ast }$.

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 $\sigma =\sqrt{\frac{1}{N}{\sum }_{i=1}^N{\left(x_i\left(t\right)-<x>\right)}^2}$ of a signal $x\left(t\right)$ and its mean value $<x>=\frac{1}{N}{\sum }_{i=1}^N{x}_i\left(t\right)$ estimated over a sequence of length N of the considered signal x(t) defined as in [19] and described by equation Eq.(4):

$${\sigma }_{\mathrm{\Delta }t}=C_0<x{>}^{\lambda }$$E4

with $<.>$ defining the statistical average, and $\mathrm{\Delta }t$ is the increment corresponding to the time scales explored, $C_0$ is a constant and $\lambda$ the Taylor exponent. The Taylor law is therefore a power law and a scaling relationship between the standard deviation of a phenomenon and its mean value.

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 $\mid \mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)\mid$. The metric $\mid \mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)\mid$ exposes directly the fluctuations’ magnitude. Thus, we verify a scaling relationship between the nominal variability ${\sigma }_{\mid \mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)\mid }$ and the mean value ${\mu }_{\mid \mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)\mid }$. Therefore, the process of intradaily variability irradiance will obey power Taylor law if the equation Eq. (5) is verified:

$$\sigma \left(|\mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)|\right)=C_0\mu ({\left|\mathrm{\Delta }{K}^{\ast }(t,\mathrm{\Delta }t)\right|}^{\lambda }$$E5

with $\lambda$ for a given time scale $\mathrm{\Delta }t$.

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 $\mathrm{\Delta }t$

The time increment $\mathrm{\Delta }t$ or resolution of the irradiance data is a parameter that affects the magnitude. Moreover, the increment affects the length of daily data sampling. This is an important parameter to consider. In order to justify the choice of $\mathrm{\Delta }t$ threshold for our study, the Pearson coefficient is assessed between $\mathit{log}(\sigma \left(|\mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)|\right)$ and $\mathit{log}\left(\mu \left(|\mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)|\right)\right)$ as a function of $\mathrm{\Delta }t$. The threshold of $\mathrm{\Delta }t$ is considered as being the value of $\mathrm{\Delta }t$ when the Pearson coefficient is lower than 0.6. The Pearson coefficient is analyzed for several data sampling and for each site under the study. The results are exposed in Figure 4. According to the Figure 4, the Pearson coefficient is lower than 0.6 from $\mathrm{\Delta }t=3h$ for Tampon site but increases for $\mathrm{\Delta }t=4h$. We consequently considered that the threshold of $\mathrm{\Delta }t$ for our study would be $\mathrm{\Delta }t=4h$ which corresponds to a differentiation on a quasi half day since the daylight sequences are from 7 am to 5 pm (10 hours). Moreover, taking account of the length of daily data for $\mathrm{\Delta }t$ higher than 4 hours, we can deduce that is not representative for our study. Indeed, an average or a standard deviation on a small length of time series is not representative for our study and can give absurd results for our analysis of intradaily variability.

Figure 4.

Evolution of Pearson coefficient between $\mathit{log}\left(\sigma \left(|\mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)|\right)\right)$ and $\mathit{log}(\mu \left(|\mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t|\right)\right)$ as a function of $\mathrm{\Delta }t$ conditionned to the data sampling.

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 $\mathrm{\Delta }t$ ranges from 3 s until to the limit of 4 h as previously mentioned. The normalization of $\sigma \left(|\mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)|\right)$ by $C_0$ was done to remove the influence of specificities due to locations. Here, the goal is to highlight the existence of Taylor’s law on irradiance data. Figure 5 illustrates the evolution of the variance as a function of the average in log–log scale for an example of time scale $\mathrm{\Delta }t=30s$. We observe the existence of a power law between $\sigma \left(|\mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)|\right)$ an $\mu \left(|\mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)|\right)$ represented by weighted least squares function [21, 22] in log–log scale plot. This power law is in accordance with the Taylor power law. The results showed that the process of intradaily variability irradiance obeys power Taylor law for sampled 3 s dataset and each $\mathrm{\Delta }t$ from $\mathrm{\Delta }t=3s$ to $\mathrm{\Delta }t=4h$Table 2 presents the values of $\lambda$ as a function of time scales $\mathrm{\Delta }t$ for the two sites. The exponent $\lambda$ of power Taylor law varies between 0.4 and 0.8 for $\mathrm{\Delta }t=30s$ to $\mathrm{\Delta }t=4h$. The majority of values are higher than 0.5. According to [23], Taylor’s fluctuation scaling results from the ubiquitous second law of thermodynamics called the maximum entropy principle and the number of states, a concept borrowed from physics.

Figure 5.

Evolution of $\frac{\sigma }{C_0}$ versus the mean $\mu$ for several $\mathrm{\Delta }t=30s$ with $3s$ sampled data.

$\mathrm{\Delta }t$	${\lambda }_{\mathit{\text{Fouillole}}}$	${\lambda }_{\mathit{\text{Oahu}}}$	$\mathrm{\Delta }t$	${\lambda }_{\mathit{\text{Fouillole}}}$	${\lambda }_{\mathit{\text{Oahu}}}$
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

Table 2.

Table of evolution of $\lambda$ as a function of $\mathrm{\Delta }t$.

This power Taylor law verifies that:

$$\frac{\sigma \left(|\mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)|\right)}{C_0}=\mu {\left(|\mathrm{\Delta }{K}^{\ast }\left(t,\mathrm{\Delta }t\right)|\right)}^{\lambda }$$E6

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 $\mathrm{\Delta }t=1h$ over all measurements sites is presented in Figure 6. In Table 3, the results are described for each $\mathrm{\Delta }t$ and each measurement sites. Two comments can be given. Firstly, the results showed the consistency of power Taylor law for 10 min sampled data for every increment ($\mathrm{\Delta }t=10\mathit{min}$ from $\mathrm{\Delta }t=4h$). The result for $\mathrm{\Delta }t=1h$ and each measurements site is illustrated in Figure 6. Secondly, the $\lambda$ coefficients exhibit a different trend from a site to another with particularity for Saint-Pierre site presenting the lowest values of this coefficient ($\lambda$ lower than 0.5 for each $\mathrm{\Delta }t$). This can highlight a particularity of factors governing the process of intradaily variability irradiance at this measurement site clearly different from the other locations.

Figure 6.

Evolution of $\frac{\sigma }{C_0}$ versus the mean $\mu$ for $\mathrm{\Delta }t=1h$ for $10\mathit{min}$ sampled data.

$\mathrm{\Delta }t$	${\lambda }_{\mathit{\text{Fouillole}}}$	${\lambda }_{\mathit{\text{Oahu}}}$	${\lambda }_{\mathit{\text{Saint}}-\mathit{\text{pierre}}}$	${\lambda }_{\mathit{\text{Tampon}}}$
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

Table 3.

Table of evolution of $\lambda$ as a function of $\mathrm{\Delta }t$.

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 $\lambda$ as a function of data sampling.

Figure 7.

Evolution of $\lambda$ as a function of $\mathrm{\Delta }t$ conditioned to the data sampling.

5. Illustration of $\lambda$ as a function of temporal parameters

This analysis allows assessing if there is a dependence between $\lambda$ and temporal parameters such as the increment $\mathrm{\Delta }t$ and the time scale data sampling, conditioned to the measurement sites.

5.1 Evolution of coefficient $\lambda$ as a function of increment $\mathrm{\Delta }t$ parameter conditioned to data sampling

The first study assessing the evolution of $\lambda$ as a function of $\mathrm{\Delta }t$ showed that $\lambda$ coefficients increase until to about $\mathrm{\Delta }t=2h$ excepted for Saint-Pierre site where $\lambda$ coefficients decrease until to $\mathrm{\Delta }t=1h$. Moreover, it is observed that $\lambda$ coefficients become quasi constant from $\mathrm{\Delta }t=2h$ (Figure 8). The time increment is a parameter that affects the magnitude, hence we can suppose that from $\mathrm{\Delta }t=2h$ until to $\mathrm{\Delta }t=4h$ the consistency of $\lambda$ coefficient characterizes ramp rate as being quasi invariant. Moreover, the increment affecting the length of data sampling can alter the accuracy of $\lambda$ value.

Figure 8.

Evolution of $\lambda$ as a function of $\mathrm{\Delta }t$.

The profile of evolution of coefficients $\lambda$ as a function of $\mathrm{\Delta }t$ does not vary a lot from a time scale of sampling to an other (Figure 7). This highlights the non dependence of coefficients $\lambda$ to time scale of data sampling. Consequently, there is a consistency of evolution trend of $\lambda$ as a function of $\mathrm{\Delta }t$, in particular, an averaged trend of $\lambda$ whatever the data sampling available but specific to a site. Thus, 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 may be useful for inefficient forecasting model at very short time scale for example Numerical Weather prediction (NWP) models such as in [6].

5.2 Verification of the Taylor law stationarity

The evolution of coefficients $\lambda$ as a function of $\mathrm{\Delta }t$ is assessed for several years available in our data set for Fouillole and Oahu. This coefficient $\lambda$ is computed for a database of two years. This analysis is performed for several data sampling. Figure 9 represents the results. From a year to another, the profile is substantially the same. This highlights the yearly stationarity of the evolution of $\lambda$ as a function of $\mathrm{\Delta }t$. We can deduce that for an analysis of $\lambda$, the user needs only one year data set to generalize his results. Nevertheless, to support this result and extend this analysis, more available years data are needed.

Figure 9.

Evolution of $\lambda$ as a function of $\mathrm{\Delta }t$ conditioned to the data sampling.

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	$P_c=185W_c$
Voltage for maximal power	$V_{\mathit{MPP}}=33.7V$
Current for maximal power	$I_{\mathit{MPP}}=5.49A$
Voltage of open circuit	$V_{\mathit{oc}}=40V$
Current of short circuit	$I_{\mathit{sc}}=5.8A$
Dimension of photovoltaic cells (mm) $A_P$	$125\ast 125$
Number of cells	$5\ast 10\left(50\right)$
PV module dimension (mm)	$1600\ast 900\ast 35$
Panels’ efficiency	${\eta }_P=15\%$

Table 4.

Characteristics of photovoltaic panel parameters.

$$P_s=N_P\ast \mathit{GHI}\ast A_P\ast {\eta }_P$$E7

where, GHI is the measured irradiance in $W.m^{-2}$, $A_P$ is the panel area, $N_P$ is the number of panels, and ${\eta }_P$ is the panels’ efficiency.

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 $\mathit{kW}/m^2$ (1 sun) to a spectrum AM 1.5; a temperature of cell of 25°C.

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.

Figure 10.

Time series of GHI and the theoretical ouput PV power corresponding.

The stochastic component of PV power output is obtained by removing the solar-geometry effects. Similarly to the clear sky index ${\mathit{Kt}}^{\ast }$ (ratio of measured GHI to theoretical clear sky GHI) defined as Eq. (1), the detrending of PV output is described by the equation Eq. (8).

$$P\ast =\frac{P_m}{P_{\mathit{\text{clear}}}}$$E8

where $P_m$ is the PV output estimated from measured irradiance for index $m$ and index $\mathit{\text{clear}}$ refers to PV output estimated from theoretical clear sky irradiance. For this analysis, we used the data from Fouillole measurement site. In [5], a metric called power variability is defined by $\sigma \left(\mathrm{\Delta }{P}_{\mathrm{\Delta }t}\right)$. As the previous study, we define the metric of the ramp rate standard deviation from PV output power by this equation Eq. (9).

$$\sigma (\mathrm{\Delta }{P}^{\ast }\left(t,\mathrm{\Delta }t\right)=\sqrt{\mathit{Var}\left[\mathrm{\Delta }{P}^{\ast }\left(t,\mathrm{\Delta }t\right)\right]}$$E9

By analogy with power Taylor law performed for irradiance, we verify a scaling relationship between ${\sigma }_{\mid \mathrm{\Delta }{P}^{\ast }\left(t,\mathrm{\Delta }t\right)\mid }$ and the mean value ${\mu }_{\mid P^{\ast }\left(t,\mathrm{\Delta }t\right)\mid }$ for several increments from $\mathrm{\Delta }t$ (Figure 11).

Figure 11.

Evolution of $\frac{\sigma }{C_0}$ versus the mean $\mu$ for $\mathrm{\Delta }t=2\mathit{min}$ for $30s$ sampled data and $\mathrm{\Delta }t=30\mathit{min}$ for $10\mathit{min}$ sampled data.

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):

Figure 12.

Evolution of $\lambda$ as a function of $\mathrm{\Delta }t$ conditioned to the data sampling from GHI data and from estimated PV power ouput data.

$$\sigma \left(|\mathrm{\Delta }{P}^{\ast }\left(t,\mathrm{\Delta }t\right)|\right)=C_0{\mu }_{\mid \mathrm{\Delta }{P}^{\ast }\left(t,\mathrm{\Delta }t\right)\mid }^{\lambda }$$E10

The $\lambda$ power coefficients show significant similarities both in the values and in the evolution profile of the $\lambda$ as a function of increment $\mathrm{\Delta }t$ between power PV field and irradiance field. We can deduce from this first approach that irradiance data are sufficient to model the ramp rate standard deviation of PV ooutput by power Taylor law without having to use $P^{\ast }$. Hence, the modeling ramp rate standard deviation of PV output can be described by this following equation:

$$\sigma \mid \mathrm{\Delta }{P}^{\ast }\left(t,\mathrm{\Delta }t\right)\mid =C_0\mu (\mid \mathrm{\Delta }{P}^{\ast }{\left(t,\mathrm{\Delta }t\right)}^{\lambda }$$E11

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 $\lambda$ exponent has shown that the user needs only one year data set to generalize his results. Nevertheless, to support this result and extend this analysis, more available years of data are needed.

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 $\sigma \left(\mathrm{\Delta }|P^{\ast }\left(t,\mathrm{\Delta }t\right)\right)$ even for very short time scales. Hence, this study allowed to model this metric by a power law based on $\mu \left(\mathrm{\Delta }{P}^{\ast }\left(t,\mathrm{\Delta }t\right)\right)$. The exponent seems to depend on the location of sites. This can be due to different factors causing cloudy formation specific to a site which is the source of the ramp rate’s variance particularities. The invariance of $\lambda$ evolution profile as a function of $\mathrm{\Delta }t$ conditioned to the sampling of data highlighting the possibility to approximately model ramp rate variance at high frequency from lower frequency data. The stationarity of $\lambda$ evolution profile as a function of $\mathrm{\Delta }t$ from a year to another showed that the user does not need a long dataset to establish this power law describing ramp rate’s variance. Moreover, the study showed evidence that this modeling also applies to PV power output. For all increments $\mathrm{\Delta }t$ of this study from 30s sampled and 10 min sampled data, the exponent values of Taylor power law $\lambda$ are between 0.5 and 0.8. The results of these works are a statistical perspective in PV power output area and introduce the multifractility analysis of intradaily variability PV power output which is a prerequisite of this characterization, enabling its high penetration.

Conflict of interest

The authors declare no conflict of interest.

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.