A Fuzzy Model for Daily Solar Irradiation

The algorithm outlined below has been designed for the computation of daily global solar energy and is a slightly modified variant of the one reported in (Tulcan- Paulescu and Paulescu 2007). The model is conducted with two input linguistic variable: daily amplitude of air temperature Atj = tmax, j – tmi„t j and Julian day j. The output variable is the clearness index kT = Hj/He^ j (Liu and Jordan 1960) where Hj represents the daily solar irradiation in the j day while Hej is its extraterrestrial value.

In fuzzy modeling practice the number of attributes associated to a linguistic variable increase with increasing of input/output data spreading. Because of relative higher than medium scattering of At to kT, these variables are characterized by eight attributes. It is emphasized that the attributes T and K often are described in terms
as LOW, MEDIUM or HIGH. For a simplified notation the numeral subscription i is more suitable, assign with ascending i attributes ranging from VERY LOW to VERY HIGH. For Julian day linguistic variable only two attributes have been considered, WINTER (W) and SUMMER (S)

All the input membership functions are plotted in Fig. 7.6, where the notation for every attribute is specified.

The membership functions for Atj (i = 1 …8) attributes are triangular:

The coefficients ai, bi and Ci have the signification depicted in Fig. 7.6. The mem­bership function of attributes T1 and T8 are saturated towards zero (mAtд = 1 if At < c1) and infinite (mAt 8 = 1 if At > c8), respectively. The factor | fits the algo­rithm to the territory. As was introduced in Eq. (7.18) it compresses or expands the membership functions associated to At attributes to overlay the specific At range in a given location. A recipe for the computation of | as function of yearly mean of air temperature Tand yearly mean of daily air temperature amplitude At is reported in (Tulcan-Paulescu and Paulescu 2007):

I (AT T) = 0.00413? – 0.964? + 1.078T – 0.00565?AT-

-0.023TAT+ 0.009476ЇЛ? + 0.495AT – 0.0468A? + -0.002223A? – 3.581 (7.19)

Equation (7.9 a, b) is not applicable in every location; the condition to use it in a given location characterized by the pair (At, t) is 0.6 < | (At, t) < 1.4.

The results from our study can be regarded as a starting point for future devel­opments of increasing the generality level of temperature based models. The model universality and versatility is determined by the way in which the factor | can be related to the local meteo-climate.

The role of the Julian day linguistic variable is to enhance model prediction in cold season, when the irradiation models accuracy decays. Thus it is allowed to enable specific rules for days characterized with WINTER attribute. On the other hand, everyone knows from routine observations that some spring or autumn days are sometimes closer to the summer one and other times to the winter ones; this behavior is well accounted for by the trapezoidal membership functions of Julian day attributes:

(7.20 a, b)

The membership functions of Ki attributes are fixed as triangular, symmetric and equidistant:

(7.21)

We underline that the potential users can not tune the output membership functions, as they have no measurements of the daily solar irradiation. They apply the numeri­cal algorithm exactly for obtaining these data. The coefficients a,, b, and c, with the signification from Fig. 7.6, are depicted in Fig. 7.7 where kT, j membership function are displayed.

Table 7.1 Input/output associative rules of the fuzzy algorithm

Rule#

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

At

T1

T2

T3

T4

T5

T6

T7

T8

T1

T2

T3

T4

T5

T6

T7

T8

j

S

S

S

S

S

S

S

S

W

W

W

W

W

W

W

W

kT

K1

K2

K3

K4

K5

K6

K7

K8

K1

K3

K4

K5

K5

K5

K6

K7

The input/output mapping of the fuzzy system is presented in Table 7.1. Every rule is encompassed in a column meaning a fuzzy implication in Eq. (7.15). By example the rule #7 is reading:

IF At IS T7 AND j IS S THEN kT IS K7 (7.22)

Thus the rules are expressed closer to the human thinking if we bear in mind that the attributes notation with numeral subscript replaces words. As a matter of fact, the rule Eq. (7.22) has to be understood as: If daily temperature amplitude is high in a summer day then also the clearness index is high, with the assumption that HIGH is associated to T7 and K7 attributes.

With the input/output mapping listed as a matrix in Table 7.1, the fuzzy algorithm is ready for use. A handling example of fuzzy model application and its implemen­tation in a computer program are presented in the following.

Updated: August 4, 2015 — 12:39 am