Chaotic Signatures and Global Solar Radiation model estimate over Nigeria, a Tropical region

1 In a tropical region like Nigeria, accurate estimation and chaotic signatures of global solar 2 radiation (Rs) are essential to the design of solar energy utilization systems in PV technology 3 companies and one conservative energy source required in developing drying devices in today’s 4 mechanized Agriculture. The Rs model is a function of solar declination angle, temperature 5 difference, and relative humidity. In this paper, the daily re-analyzed atmospheric data obtained 6 from the archive of ERA-Interim was used to estimate the nonlinear Global Solar radiation model 7 and investigated chaotic signatures across the tropical climatic regions of Nigeria. The well-known 8 statistical tools were used to analyze the chosen meteorological parameters and the correlation 9 was found to be perfect, close with low values of RMSE across the selected regions over Nigeria. 10 For proper modeling and prediction of the underlying dynamics, the extensive chaotic measures of 11 phase space reconstruction using recurrence plots and recurrence quantification analyses are also 12 presented, analyzed and discussed with the appropriate choice of embedded dimension, m, and 13 time delay τ . 14

Since the nonlinear approach to atmospheric convection using Lorenz's model (Lorenz 1963), vari-34 ous atmospheric situations have been applied to the field of science and engineering. These numerous 35 scientific disciplines help in providing the information that is needed in forecasting the weather con-36 dition. That is, the complexity of the global solar radiation and the inherent irregularities occurrence 37 in space can be identified to be chaotic or hyperchaotic based on the availability of climatological data 38 which has been enriched with the theory of nonlinear dynamics. It is worth noting that large time-39 series data is very vital in providing more insight into the internal activities of the atmosphere (Kantz temperature difference air temperature have been proposed (Sarker and Sifat 2016). However, the 3 estimated model in this paper expressed the relationship between clearness index, relative humidity, 4 and temperature difference. We also employed the chaotic quantifiers, RP and RQA, to investigate 5 the daily and seasoning variations in the meteorological parameters for the four basic climatic regions 6 in Nigeria, namely; the Coastal region, Guinea savannah region, Midland region and Sahel savannah 7 region. This paper presents a nonlinear global solar radiation model estimate and chaotic signatures 8 from available meteorological data. Section 2 comprises of the study area, data analysis, and statisti-9 cal performance evaluation. The mathematical analyses of the used chaotic quantifiers are discussed 10 in section 3, while section 4 concludes the paper. The re-analyzed atmospheric data from four climatic regions of Nigeria were obtained from the archive 14 of the ERA-Interim database. Ten years of daily data covering the period of 2006 to 2015, for so-15 lar radiation, relative humidity, and temperature differences were collected and analyzed for this 16 research. Table 1 shows the selected geographical locations includes Kano (Sahel savannah region),

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South-Western station, Akure (Coastal region), Ilorin (Midland region) and Yola (Guinea Savannah).  The collected data were used in global solar nonlinear model estimation, given as: which can be expressed as, where clearness index, R s = R/R a , is the solar ratio which depicts the ratio of the global solar ra- The sunset hour angle, ω s , is given by:

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Fit measured is very important in comparing and assessing models with the aid of statistical tools. 6 The statistical tools have been used in recent years extensively to evaluate different models ranging 7 from linear, quadratic, third-degree, logarithmic and exponential models after comparing and assess- good prediction and weak if close to zero. That is, R 2 =1 is an indication of a perfect match between 20 the predicted and the observed values. The expression for R 2 is: The RMSE is however defined as: while SSE can also be expressed as: It is worth noting that the RMSE is always positive while a zero value is ideal. In the Eqs.  In nonlinear science, a PSR can be used in estimating the characteristic properties of a natural system 2 such as global solar radiation. It requires decision making regarding the size of the space, the value of 3 the time shifts between the coordinates, and another important-although often overlooked-aspect, 4 that is, which one or which combination of observable(s), if several of them are available will be used In accordance to Takens theorem (Takens 1981), the PSR is defined as where, x 1 , x 2 , x 3 ....., x n−1 , x n denotes the chaotic time series, embedding dimension, m, time 26 delay, τ and N is the number of samples after reconstruction.

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In the same vein, Fraser and Swinney (1986) also proposed a method of AMI to determine the 28 delay, and the best choice for the delay is where the AMI has its smallest local minimum. However, 29 the delay,τ has to be carefully chosen due to linear dependence between the subsequent vectors which 30 impregnated from random errors and low measurement precision. The AMI is expressed as: where P ψ,φ (τ ) is the joint probability that u i = ψ and u i+τ = φ. P ψ and P φ are the probability 33 that u i has the value ψ andφ, respectively. Also, the generalized mutual information for higher di- On the other hand, one of the chaotic quantifiers necessary in the study of the nonlinear behavior Lyapunov exponent λ can be expressed as It is worth noting that entropy is reciprocal to Lyapunov exponent and it is mostly used in both 15 physics and information theory to describe the amount of uncertainty or information inherent in an  The parameter q itself is not a measure of the complexity of the system, but measures the degree 21 of non-extensivity of the system: q → 1 corresponds to the standard extensive Boltzmann-Gibbs 22 statistics which generalizes the Boltzmann-Gibbs theory. Also, it is the time variations of the Tsallis 23 entropy for a given q(S q ) that quantify the dynamic changes in the complexity of the system, that is, 24 lower values of S q characterize the portions of the signal with lower complexity (Fraser 1989). The 25 Tsallis entropy S q is calculated using and the entropic index q for systems A and B, respectivley, characterizes the degree of nonaddi-27 tivity reflected in the following pseudo-additivity rule: where p i is the probabilities associated with the microscopic configurations, W is their total 29 number, q is a real number, and k is Boltzmann's constant.

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A comparison in this work shows the complex link between the Lyapunov exponent of the solar 31 radiation and that of its Tsallis entropy. This is based on the fact that Tsallis entropy has been linked 32 to a significant degree of response to the edge of chaos and chaotic regime dynamical systems due to In term of N × N matrix, the RPs can be defined as: The ε i is a predefined cut-off distance, . is a norm (i.e Euclidean norm) and Θ(x) is the Heaviside where ε is the threshold, P ε (l) is the histogram of the length l of the diagonal structures. For 37 l min = 1, then the determinism (DET) is equal to recurrence rate (RR). Periodic signals (e.g., sine 38 waves) will give very long diagonal lines. Chaotic signals (e.g. Henon attractor) will give very short 39 diagonal lines, and stochastic signals (e.g., random numbers) will give no diagonal lines (Shannon 40 1948). This has been used to quantify how deterministic a system is (Webber and Zbilut 1994).    Fig.(1). We also consider the 3D-Phase portrait, which reveals the relationship between the three 2 meteorological parameters as plotted in Fig. (2). The phase space construction which represents the 3 state of any real-world systems taking into consideration the dynamics emanating from the set of its 4 state variable is plotted in Fig.(  Also, the typical plots for the false nearest neighbors against embedded dimension(m) and the AMI 26 against delay, τ were then plotted as shown in Fig.5(a). The choice of embedded dimension, m=7, 27 and delay time, τ = 10, is essential for phase PSR in this work to avoid over embedded (Fraser and   The number of FNN plotted against the embedded dimension depicts the variation of solar ratio and 31 the temperature difference in each station which is very high and similar compared to the low values 32 of the relative humidity in each station (see Fig.5(b)). The PSR preserves relevant geometrical and  The positive maximum Lyapunov exponent can be observed for solar radiation in Midland, Coastal 37 and Guinnea Savannah regions (see Fig.6(a)), while Fig.6(b) shows higher positive values of Lyapunov 38 exponent through-out the Sahel Savannah region. The direct relationship between Lyapunov expo-39 nent and Tsallis entropy has been perfectly displayed in Fig. 7(a) for Coastal region and Fig.7(b) for 40 Sahel savannah zone, respectively. Also, the direct relationship and indirect relationship among the  (see Fig. 8). Coastal regions reveal the presence of checkerboard structures around the diagonal lines. 1 For instance, high insolation has been noticed at days < 100, 300 ≤days≤ 500 and 700 ≤days≤ 900 2 for the dry seasons (see Fig.8(a)). Conversely, in Fig. 8(b), relative humidity shows inverse trend 3 during these insolation periods of the year, but high during the wet season of the year. Similar events 4 can also be observed in Kano station (see Fig.9(a-b)), however, more solar radiation is observed dur-5 ing this period of the year which can be a result of harmattan and the evidence of chaos from the 6 oscillatory nature of the system. Fig.9(b)shows little rain throughout the year.  year before rising again at the return of the dry season (i.e. November and December). In contrast, 21 the Sahel Savannah region experience very long but one dry season months for the year, usually from 22 November of one year to May of the following year (see Fig. 10(b)). The chaotic signatures that 23 were discovered through the aforementioned oscillatory nature of both solar radiation and relative 24 humidity clearly show evidence of chaos, that is, higher (dry/wet) and lower (wet/dry) chaoticity, 25 respectively during the season months of the year. The complex signatures across the regions also 26 confirm the direct relationship of a temperature difference to solar radiation for the period under 27 study. However, daily or monthly variation in solar irradiance which is a function of the aforemen-28 tioned meteorological parameters may be attributed to the effect of the intertropical discontinuity, 29 which affects the atmospheric stability during these periods. The results also indicate the availability 30 of solar radiation throughout the whole year, which is an associated factor for the tropical regions.    In general, when the meteorological parameters of a natural system are chaotic, it implies that the 11 Lmax, determinism and entropy are all low which depicts that the system is far from equilibrium. for each region. However, the availability of solar irradiance in all selected zones has been discovered 31 with the highest irradiance observed in Sahel savanna zone, which shows that the zone exhibit more 32 complex solar irradiance than the other selected regions. Other regions such as Coastal, Midland 33 and Guinea savannah regions were suitable for Agricultural purposes during the wet season as well 34 as solar energy capture through-out the year. Therefore, with the implementation of accurate and 35 efficient prediction, we will be able to identify which regions is/are suitable for optimal capture of 36 solar radiation for human use and energy source designing. The data that support the findings of this study are available from the corresponding author upon 40 reasonable request and can also be obtained from the archive of ERA-Interim.

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Daily global solar radiation modeling using data-driven techniques and empirical equations