Solar Zoning Maps of Algeria Based on Sunshine Duration Data and Kriging Method

Solar energy is employed in photovoltaic systems for electricity generation or thermal systems (solar water heaters) for hot water production. It is a growing industry, especially in the housing sector. This energy is unlimited and abundant [1]. It's ideal for decentralized manufacturing since it doesn't need a significant distribution infrastructure. Its fluctuation in distance and time depends on astronomical and climatic conditions.

The solar field is a data set that illustrates the variations in solar radiation across time and space. All future studies of solar energy will build upon this foundational study. It's a simulation tool for determining how large a solar array should be and how best to arrange its panels to maximize energy production. Solar energy potential at a particular location is an essential metric for designers of systems that convert solar energy into electricity (photovoltaic modules) or provide hot water (plane concentration collectors) [2].

On the national territory, meteorological stations gather various meteorological data and a limited amount of radiometric data. To do this, it is required to use models designed to predict the principal components of solar energy impacting the ground. The reconstitution of solar radiation on the ground provides solar energy system designers with accurate, exact, and simply usable sun irradiation data sequences for the sizing and installation of solar energy systems on any territory. In addition, insolation statistics are crucial for both the design and development of these systems as well as the assessment of their performance.

There are two sources of information for determining the quantity of solar radiation: measurements of radiation at meteorological and radiometric stations and theoretical estimates. Although the first approach is the most direct and precise, it is not always feasible due to factors such as the low density of meteorological stations and their often large physical distance from one another across countries. The alternative technique employs models for radiation estimation based on a set of input factors. Several research papers have been done to construct estimation models, ranging from semi empirical models [3, 4] to artificial intelligence [5, 6] models.

Algeria possesses significant solar potential with a substantial solar deposit, experiencing over 2500 hours of sunlight annually across most of its region [7]. Despite favorable geographical, climatic, and meteorological characteristics, the utilization of solar energy has yet to make significant progress and this market is essentially virgin. For a nation like Algeria, it's important to quantify the potential of this form of energy so we can determine the most effective ways to use it in our energy strategy.

In Algeria, only the meteorological stations of Bouzareah, Ghardaia, Oran and Tamanrasset give solar radiation measurements. Moreover, their distribution is not enough to cover all the Algerian territory [8].

Hence, we propose in this paper, a complete study of the sunshine fraction distribution over Algeria based on the analysis of 56 meteorological stations data from 1992 to 2002 that monitor the sunlight hours on a regular basis. The main objective is to categorize the territory into zones which help to determine the number of sunshine hours and solar irradiation amount at any location even if the meteorological stations is no installed. In addition, using kriging interpolation method, which can be used in the field of solar irradiation to estimate solar radiation quantities at unmeasured places.

Kriging contributes to the creation of a smooth and continuous surface of solar irradiance by taking into account the spatial correlation of solar radiation data. This data is useful for evaluating possible solar energy resources, optimizing solar panel location, and constructing energy-efficient solar power systems as well as constructing solar zoning maps.

Our objective is to develop solar zoning map, which will provide a graphical representation of the similar zones for better understanding of the same solar regions over Algeria.

The remainder of the paper is structured as follows: Section 2 will show us where exactly in Algeria all the radiometers are located. In Section 3, we'll look at how the Kriging method is used in the making of zoning maps. Discussion on final thoughts follows.

Our main objective is to obtain solar radiation statistics for each location in Algeria However, obtaining complete data sets at every desired location due to practical restrictions. As a result, interpolation is critical to the visualization and interpretation of 2D data. New data points are created using interpolation, which is a technique for creating new data points within a given collection of data points. Linear and nonlinear interpolation techniques as well as more complex approaches such as principal analysis components and kriging have been proposed in the literature. In what follows, kriging interpolation method is used to categories the zoning maps for Algeria and distinguish similar zones with the same energetic properties.

3.1 Ordinary kriging method

Geostatisticians utilize the standard Kriging approach [13] to estimate the values of an invisible random field by analyzing surrounding data. Delhomme [14] provides an in-depth explanation of kriging and its applications. Linear kriging can be classified as either basic (mean is known), ordinary (mean is unknown but remains constant), or universal, depending on how the mean value is stated (mean is an unknown linear combination of known functions). Regular kriging is the best linear unbiased estimator, and as such is the primary emphasis of this study. Linearity in the data allows for fairness in the estimates provided by ordinary kriging, which is why the approach strives to maintain a mean residual of zero.

The goal of the ordinary kriging technique is to provide an approximation for the missing data by utilizing the existing data The use of ordinary Kriging in solar zoning maps was justified due to the spatial correlation of solar irradiance, enabling accurate estimation of values at unsampled locations. Its unbiased estimation minimized errors and ensured reliable solar energy resource assessment. The method's flexibility with uneven data distribution and ability to incorporate elevation effects further enhanced the accuracy of the zoning maps [8].

The summary of the ordinary Kriging method is given as follows,

Let the data be sampled at N locations (x₁, ..., x_N), and the corresponding values be (v₁, ..., v_N). The value $\hat{v}_j$ at an unknown location

$\widetilde{v}_j=\sum_{i=1}^N w_i v_i$       (4)

Here, $\tilde{v}_j$ is the estimate, and let $\tilde{v}_j$ be the actual value (unknown) at $\hat{x}_j$. To find the weights, the values $v_j$ and $\tilde{v}_j$) are assumed to be stationary random functions:

$E\left[v_i\right]=E\left[\hat{v}_i\right]=E(v)$     (5)

Optimal weights $w_i$ are calculated assuming that the estimation is unbiased. The condition needed for unbiased estimator is:

$\sum_{i=1}^N w_i=1$        (6)

Let the residue be r_j:

_{$r_j=\widetilde{v}_j-\hat{v}_j$      (7)}

Therefore, the residual variance is given by:

$\operatorname{Var}\left(r_j\right)=\operatorname{Cov}\left\{\tilde{v}_j \tilde{v}_j\right\}-2 \operatorname{Cov}\left\{\tilde{v}_j \hat{v}_j\right\}+\operatorname{COV}\left\{\hat{v}_j \hat{v}_j\right\}$    (8)

The first term can further be simplified as follows:

$\begin{gathered}\operatorname{Cov}\left\{\tilde{v}_j \tilde{v}_j\right\}=\operatorname{Var}\left(\tilde{v}_j\right)=\operatorname{Var}\left\{\sum_{i=1}^N w_i v_i\right\} \\ =\sum_{i=1}^N \sum_{k=1}^N w_i w_k \operatorname{Cov}\left\{v_i v_k\right\}=\sum_{i=1}^N \sum_{k=1}^N w_i w_k \hat{C}_{i k}\end{gathered}$        (9)

The second term can be written as:

$\operatorname{Cov}\left\{\tilde{v}_j \hat{v}_j\right\}=\operatorname{Cov}\left\{\left(\sum_{i=1}^N w_i v_i\right) \hat{v}_j\right\}=\sum_{i=1}^N w_i \operatorname{Cov}\left\{v_i \hat{v}_j\right\}=\sum_{i=1}^N w_i \hat{C}_{i 0}$       (10)

Finally, the third term can be expressed as:

$\operatorname{Cov}\left\{\hat{v}_j \hat{v}_j\right\}=\hat{C}_{00}$     (11)

Substituting from Eqns. (9)-(11) in Eq. (8),

$\operatorname{Var}\left(r_j\right)=\sum_{i=1}^N \sum_{k=1}^N w_i w_k \hat{C}_{i k}-2 \sum_{i=1}^N w_i \hat{C}_{i 0}+\hat{C}_{00}$      (12)

For ordinary Kriging, it is required to find w by minimizing Var (r_j) with respect to w subject to the constraint Eq. (12). This can be written as the minimization of the penalized cost function,

$J(w)=\sum_{i=1}^N \sum_{k=1}^N w_i w_k \hat{C}_{i k}-2 \sum_{i=1}^N w_i \hat{C}_{i 0}+\hat{C}_{00}+2 \lambda\left(\sum_{i=1}^N w_i-1\right)$ (13)  

Taking derivatives of J with respect to w and λ,

$\frac{\partial J}{\partial w_i}=\sum_{k=1}^N w_k \hat{C}_{i k}-2 \hat{C}_{i 0}+2 \lambda$    (14)

$\frac{\partial J}{\partial \lambda}=2\left(\sum_{i=1}^N w_i-1\right)$     (15)

2λ is the Lagrange multiplier. setting this to zero, we get the following system to solve, to obtain the weights w and λ.

$\left(\begin{array}{cccc}\hat{C}_{11} & \cdots & \hat{C}_{1 N} & 1 \\ \vdots & \ddots & \vdots & \vdots \\ \hat{C}_{N 1} & \cdots & \hat{C}_{N N} & 1 \\ 1 & \cdots & 1 & 0\end{array}\right)\left(\begin{array}{c}w_1 \\ \vdots \\ w_N \\ \lambda\end{array}\right)=\left(\begin{array}{c}\hat{C}_{10} \\ \vdots \\ \hat{C}_{N 0} \\ 1\end{array}\right)$   (16)

Substituting Eq. (16) into Eq. (12), we can derive the Kriging variance,

$\operatorname{Var}\left(r_j\right)=\hat{C}_{00}-\sum_{i=1}^N w_i \hat{C}_{i 0}-\lambda$   (17)

Eq. (16) needs to be solved at each unknown location $\hat{x}_j$.

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Figure 5. Sunshine zoning maps for 12 months for Algeria using ordinary Kriging method

• Covariance function

The amount of covariance depends on the Euclidean distance between spatial points. The correlation length L has different values, which mean we are dealing with a global pseudo Euclidean distance. The used function can be expressed as,

$\hat{C}_{i j}=\sigma^{\prime \prime 2}\left(1+\frac{\left\|x_i-x_j\right\|}{L}\right) \exp \left(-\frac{\left\|x_i-x_j\right\|}{L}\right)$   (18)

where, σ’’ is the standard deviation, and there is an assumption that all the random variables have the same standard deviation. The value of σ’’ does not have an effect on the weights w and the interpolation result, but it does have a direct effect on the Kriging variance, i.e., the uncertainty of the interpolation result.

Figure 5 shows the average amount of sunshine in Algeria for each month of the year, based on the results of the simulation. The database for the simulation is the number of hours of sunshine at all 56 stations from 1992 to 2002. The maps are made with Surfer software.

According to colors distinction obtained from the solar zoning maps, we can tell that Algeria is made up of six energetic zones:

- Zone 1 includes the coast and the high plateaus in the east (stations 16, 17, 18, 19, 20, 27, 28, 29, 30, 31). It has a low amount of sunshine, and the amount of sunshine doesn't change much during the year.

- Zone 2 has the central coastal areas (stations 9, 10, 11, 12, 13, 14, 24, 26). It has a small amount of sunshine, but it's a little higher than zone 1, and it's spread out in a very different way throughout the year. The sun gives off more energy in the summer than in the winter.

- Zone 3: It has the coast, the high plateaus in the west, and the central parts of the interior (stations 1, 2, 3, 4, 5, 6, 7, 8, 15, 21, 22, 23, 25, 32). Compared to zones 1 and 2, it has a lot of sunshine, but it's spread out in different ways at different times of the year.

- Zone 4 includes the northern part of the Sahara (stations 33, 34, 35, 36, 37, 38, 39, 40, 41, 42). It has a high percentage of sunny days, with only a small difference between the summer and winter months.

- Zone 5: It's the middle of the Sahara (stations 43, 44, 45, 46, 47, 48, 49, 50, 55, 56). Compared to other zones, it has the most amount of sunshine. It is given out the same way every day of the year.

- Zone 6: It includes the southern part of the Sahara (Hoggar and Tassili, Station Numbers: 51, 52, 53, 54). The seasons don't have the effect they used to. Compared to the summer months, there is a lot of sunshine in the winter.