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9СОЛНЕЧНАЯ ЭНЕРГЕТИКА
SOLAR ENERGY
Статья поступила в редакцию 22.02.10. Ред. рег. № 726 The article has entered in publishing office 22.02.10. Ed. reg. No. 726
УДК 620.91
СРАВНЕНИЕ ПРИМЕНЕНИЯ ФУНКЦИЙ ФУРЬЕ И ГАУССА ДЛЯ МОДЕЛИРОВАНИЯ СОЛНЕЧНОЙ РАДИАЦИИ В ТОГО
К.А. Амоу, С. Оуро-Джобо, К. Напо
Лаборатория солнечной энергетики, Кафедра возобновляемых источников энергии UNESCO (L.E.S) Факультет естественных наук (CUER) Университета Ломе-Того BP: 1515, Ломе-Того, тел.: (228) 2255094; e-mail: mapkamou@yahoo.fr
Заключение совета рецензентов: 10.03.10 Заключение совета экспертов: 15.03.10 Принято к публикации: 20.03.10
Применение функции Фурье для моделирования солнечного излучения в Того позволяет получить все важные гармоники для корректировки моделируемых значений по сравнению с измеренными значениями. Эта функция дает хорошие результаты при сравнении моделируемого среднемесячного глобального солнечного излучения с измеренными значениями. Функция Гаусса позволяет нам изначально получить часовые значения воздействия излучения, а затем дневные глобальные значения. Сравнение результатов этих двух моделей позволяет сделать вывод о том, что результаты метода, использующего функцию Гаусса, лучше при относительном колебании порядка 1,7% от глобальных значений для трех городов. Однако обе модели подходят для моделирования солнечного излучения в Того.
Ключевые слова: моделирование, солнечное излучение, среднемесячный, анализ Фурье, гармоники, часовые значения, функция Гаусса.
COMPARISON BETWEEN THE USE OF FOURIER AND GAUSS FUNCTIONS TO SIMULATE THE SOLAR IRRADIATION IN TOGO
K.A. Amou, S.Ouro-Djobo, K.Napo
Laboratoire sur l'Energie Solaire(L.E.S), Chaire Unesco sur les Energies Renouvelables(CUER), Faculté des Sciences, Université de Lomé-Togo BP: 1515, Tel: 2255094, e-mail: mapkamou@yahoo.fr
Referred: 10.03.10 Expertise: 15.03.10 Accepted: 20.03.10
The use of the function of Fourier to simulate the solar irradiation in Togo allows us to obtain harmonics all important in the adjustment of the simulated values compared to the measured values. This function gives good results when we compare the monthly average of global solar irradiation simulated to the measured values. The function of Gauss enables us to initially obtain hourly values of sunning and then the daily global values. The comparison of the results of these two methods enabled us to conclude that the results of the method using the function of Gauss is better with a relative variation of approximately 1.7% on the global values for the three cities. However the two models are valid to simulate the solar irradiation in Togo.
Key words: simulation, solar irradiation, monthly average, Fourier analysis, harmonics, hourly values, Gaussian function.
Organization: Université de Lomé, Faculté des Sciences, Chaire de l'UNESCO sur les Energies Renouvelables, Laboratoire sur l'Energie Solaire.
Education: Bachelor of Science degree in physics in 2004, master's degree in material science in 2006 at the Université de Lomé. I am preparing my Phd degree.
Experience: Scientific research project, member of Laboratoire sur l'Energie Solaire of Université de Lomé, ATER (Attaché Temporaire pour l'Enseignement et la Recherche) of Université de Lomé since January 2008.
Main range of scientific interests: renewable energy, solar energy, optoelectronic thin films and its applications.
Apélété Amou
S. Ouro-Djobo
Organization: Université de Lomé, Faculté des Sciences, Chaire de l'UNESCO sur les Energies Renouvelables, Laboratoire sur l'Energie Solaire.
Education: Bachelor degree in Electronic in 1994 at the University of Lille, master's degree in Systems and Electronic option in 1995 at the University Clermont-Ferrand II, PhD degree in 2001 at the University of Nantes.
Experience: Lecturer at University of Lomé since 2002, Maîtres Assistant since 2005 and member of Laboratoire sur l'Energie Solaire of Université de Lomé. Working on the characterization of Togo solar irradiation project, drying Agricultural products and thin films grown by chemical bath deposition for photovoltaic applications.
Main range of scientific interests: renewable energy, solar energy, product drying and optoelectronic thin films and its applications.
Publications: 10 articles and communications.
Kossi Napo
Organization: Université de Lomé, Faculté des Sciences, Chaire de l'UNESCO sur les Energies Renouvelables, Laboratoire sur l'Energie Solaire.
Education: Master degree in physic in 1982 at University of Abidjan (Côte d'Ivoire), DEA in material sciences in 1983 at University of Nantes, First PhD degree in 1986 at university of Nantes, Second PhD degree in 1998 at University of Nantes and Université de Lomé.
Experience: Lecturer at Université de Lomé since 2003, Person in charge of PhD training program in material science at Université de Lomé, National coordinator of UNITWIN of UNESCO at Université de Lomé, Senior associate of ICTP (Italy), member of scientific committee of Université de Lomé.
Main range of scientific interests: renewable energy, solar energy, thin film research, optoelectronic and electronic thin films applications.
Publication: 20 papers in international and national scientific journal, 10 oral communications and 5 posters.
Presentation of the two methods
Analysis of Fourier
In order to design any solar energy system and to study its performance characteristics, information on the availability of solar radiation is required. But solar radiation data are not easily available because of not being able to afford the measuring equipments and techniques involved. Therefore, it is necessary to develop methods to estimate the solar radiation on the basis of the more readily available meteorological data. By their nature, meteorological data are time series data and one effective way of studying periodic data is by Fourier analysis [1]. Fourier's analysis is a method that breaks a series into independent components called harmonics. The harmonics represent the important features of the particular series [2] because they are independent, they are additive. Usually, the first few harmonics are enough to explain the major features of any series [3-8]. In the study of the station for Seeb, Sutanate of Oman, Atsu Dorvlo shows that the annual series dominates the rest of the harmonics. Fagbenle and Karayiannis [5] show the same thing in their study of 28 stations in Nigeria. Fourier's analytical method is used by Selcedo to obtain simulation of solar radiation data of Spain.
In this paper, we use first Fourier's analysis method to model solar radiation data for three towns of Togo.
Methodology The method used in this paper is the same used by Atsu Dorvlo [9]. G(t) is a linear combination of trigonometric functions, a Fourier series of function. This can be written in the form:
^ s ~ ^n/2 . Г2nm ) N/2П Г2%m
G (t) = Go + Y a sin I-1 l + Y ß cosj-1
w L-i m=1 m I p l m=1 m I p
(1)
G(t) = Go + Y:=i^m cos j^t + Фm J, (2)
where t is the time, p is the period, N is the number of the months (N = 12), the mth amplitude
Rm
and the mt phase angle
Pm = tan-1 (-am / ßm ) .
(3)
(4)
So Gi, ..., Gn are particular monthly averaged daily global irradiation of a station, then the least squares estimates for the Fourier coefficients are am and Pm are:
2vn . 2nm . N
am = nXsin-p, m=-i; (5)
2 v N 2nm . N
Pm = NX,=iGcos-p, m = ^.-y-i; (6)
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«N/2 = 0;
в N/2 = N S í.c-im
(7)
(8)
=
tan-1 (-«m / Pm ) , SiPm > 0
tan-1 (-«m / Pm )-П, SiPm < 0, «m > 0. (9)
tan-1 (-«m / Pm )-П SiPm < 0, «m ^ 0
And
Фm =--SiPm = 0, am > 0
Ф,» = SiPm = 0, am < 0. Фm = arbitraire siPm = 0, am = 0
(10)
P (t ) = -
Тл/2п
exp-
(t - la )(l - lb )
2a2
(11)
where the unknown parameter c for every month is determined by the resolution of (1) for ta = tb = 12 or 13 according to whether the maximum of sunning is obtained at 12:00 pm, 1:00 pm, or between 12:00 am and 1:00 pm i.e.
P(12 or 13) =
1
ал/2П
then
a =
P(12 or 13)л/2п '
(12)
(13)
Gaussian Function The average values of the hourly global solar irradiation measured on the horizontal ground are very useful for the multiple applications of solar energy. These data are available for certain places only and for most of the time it should be estimated, from the ideal models which are developed from the measured values of a place. The first attempt to simulate the hourly solar irradiation from certain available data was made by Whillier and Hottel [10] then taken again by Whillier
[11] which used the ratio of the hourly global solar irradiation on the daily total radiation according to the time angle of the sun to obtain curves. Liu and Jordan
[12] tried in their turn to find a relation between the length of the day and this ratio. Thereafter they used the approaches of Whillier for good extrapolation of the experimental curves. Recently, Jain [13] put forward the idea to use the normal distribution P(t), the monthly average of solar irradiation and the hourly global values to obtain a model which gives good performance. From the results of Jain, we noticed that in certain cases, the curves corresponding to the data calculated from the formula of Jain are very close to the experimental curves obtained from the measured data if the maximum of the sunning in the course of the day is obtained at 12 o'clock. For the days when this maximum is at 1:00 pm or between 12:00 am and 1:00 pm, there is a great shift between the experimental and theoretical curves if we apply the formula of Jain [14]. This is why we propose a substitute of the formula of Jain, thus allowing to obtain a better result. By this new formula, we take into account the moment when the maximum is reached for each month differently.
Methodology The method consists in calculating the ratio of the daily high value of solar irradiation and the other values of hourly solar irradiation from the following normal distribution:
After having determined the various values of c for the various months, we use the equation (11) to calculate the theoretical values of the ratio hourly/daily for all solar irradiation data from 6:00 a.m. to 7:00 pm whose center varies between 12:00 pm and 1:00 pm. From the raw data of the total sunshine duration for the three cities since 2002 up to 2007, and the ratio of sunshine duration /maximum possible sunshine duration of the day were calculated. These calculations enabled us to determine for each month of the year, the hour to which the maximum is reached in the course of the day. In order to minimize the errors, we individually take into account this maximum for each month whereas those which preceded me fixed the maximum at 12 pm. To know the value of c in a place where the measured values are not available, a linear correlation between c and the maximum possible of daily sunshine duration for Atakpamé and Mango, and with the monthly average sunshine duration of Lomé is used to calculate c then to obtain the values of solar irradiation.
Data of studies The data used to do this work are obtained from the pyranometers (sensor of global solar irradiation LI-1400) established on the three sites which are Lomé, Atakpamé and Mango in Togo. These power stations measure the total sunning of the day. We present the typical results of 2003, 2004 and 2007 for Lomé 2002, 2003, 2004 and 2007 for Atakpamé then 2004, 2005 and 2007 for Mango. For the model using the Gaussian function, all the data are taken into account since 2002 up to 2007 for the three stations.
Results and Analysis
Method of Fourier By applying the method of analysis of Fourier for the various years taken separately, we obtained the values of the amplitudes and the phases of angle allowing us to simulate the data of sunning. The various parameters of the first three harmonics are presented in Table 1 [9].
1
Таблица 1
Оценки ежегодных гармоник солнечного излучения для Того
Table 1
Estimates of solar irradiation annual harmonics for Togo
Stations Years Annual average 1st harmonic 2nd harmonic 3rd harmonic
amplitude phase amplitude phase amplitude phase
Lomé 2003 4610,2 75,247 -2,023 532,223 2,300 120,144 2,440
Lomé 2004 444l,l 13l,S94 -0,370 47S,703 2,214 139,312 1,127
Lomé 2007 4026,5 S9,345 -1,111 706,714 1,7S6 155,919 0,295
Atakpamé 2002 5005,3 553,304 -1,620 602,193 1,505 20S,S64 3,059
Atakpamé 2003 4794,5 349,769 -1,361 615,050 1,726 332,221 2,SSS
Atakpamé 2004 4774,l 2S9,745 -1,333 544,907 1,945 330,992 3,167
Atakpamé 2007 4l0l,S 356,23S -0,917 739,337 1,499 144,702 2,S32
Mango 2004 5591,4 361,300 -1,223 367,733 2,496 321,42S 3,077
Mango 2005 550S,4 671,2S7 -1,605 469,213 1,776 1S2,01S 3,101
Mango 2007 5393,0 S66,365 -1,2S6 366,331 2,170 431,772 -3,0S0
The analysis of the estimated values shows that the estimated average values correspond exactly to the measured averages. By comparing the amplitudes of the various harmonics, we note that they all are important and make it possible to obtain values very close to the measured values. The third harmonic is thus not negligible but has an importance in the adjustment of the values obtained contrary to work of Dorvlo Atsu [9] which stopped with the second harmonic. From the various harmonics, we obtain the annual function using three harmonics to simulate a global solar irradiation of the three cities: Lomé:
H(t) = 436 i,5 +100,83cos I y - i,168 I +
Tit I I nt
+572,547 cos I — + 2,100 J + 13S,45Scos I y + i,2S7
Atakpamé:
nt
H(t) = 4S20,6 + 3S7,264 cos ^y - i, 30SJ + +625,372cos I y +1,669 I + 254,195 cos I П- +1,415 J ;
Mango:
nt
H (t) = 5497,6 + 632,984 cos - 1,371j +
+401,092cos| y + 2,147j + 311,739cos ^y +1,033 j.
From the parameters of Table 1, the simulated and measured monthly average sunning were compared (Fig. 1, 2, 3).
Рис. 1. Сравнение измеренных и cмоделированных значений среднемесячного дневного глобального солнечного излучения для г. Ломе в 2003, 2004 и 2007 гг. Fig. 1. Comparison between measured and simulated values of monthly average of daily global solar irradiation
for Lomé in 2003, 2004 and 2007
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Рис. 2. Сравнение измеренных и cмоделированных значений среднемесячного дневного глобального солнечного излучения для г. Атакпаме в 2002, 2003, 2004 и 2007 г. Fig. 2. Comparison between measured and simulated values of monthly average of daily global solar irradiation
for Atakpame in 2002, 2003, 2004 and 2007
Рис. 3. Сравнение измеренных и смоделированных значений среднемесячного дневного глобального солнечного излучения для г. Манго в 2004, 2005 и 2007 гг. Fig. 3. Comparison between measured and simulated values of monthly average of daily global solar irradiation
for Mango in 2004, 2005 and 2007
Анализ погрешностей теоретических значений Error analysis on the theoretical values
Таблица 2 Table 2
Months Lomé Atakpamé Mango
H measured H simulated R.V.(%) H measured H simulated R.V.(%) H measured H simulated R.V.(%)
January 3598,4 3671,0 0,7 4395,4 4269,3 1,3 5550,8 5249,3 3,0
February 4277,7 4129,2 1,5 4865,2 4609,5 2,6 6121,9 5751,2 3,7
March 4733,5 4941,8 2,1 5448,7 5511,6 0,6 6407,0 6603,4 2,0
April 4980,2 5043,4 0,6 5267,7 5740,5 4,7 5994,1 6537,3 5,4
May 4634,1 4458,5 1,8 5412,9 5172,9 2,4 5883,5 5614,2 2,7
June 3801,2 3984,1 1,8 4969,1 4581,9 3,9 5526,7 4988,7 5,4
July 4076,4 3906,9 1,7 4059,0 4232,7 1,7 4682,6 4944,9 2,6
August 3924,1 4026,7 1,0 3647,7 4014,8 3,7 4434,8 4880,4 4,5
September 4451,7 4359,1 0,9 4554,0 4251,9 3,0 5037,5 4829,2 2,1
October 4790,9 4824,6 0,3 5269,4 5039,8 2,3 5795,4 5259,0 5,4
November 4869,4 4831,5 0,4 5267,1 5485,1 2,2 5389,1 5744,7 3,6
December 4200,2 4160,9 0,4 4691,0 4936,9 2,5 5148,2 5569,1 4,2
Average 4361,5 4361,5 1,1 4820,6 4820,6 2,6 5497,6 5497,6 3,7
The simulated and measured monthly average of solar irradiation was compared and the error analysis on the values simulated for the three stations. These results are presented in Table 2.
These results show that we obtained theoretical values with a maximum relative variation of 5.4% for Mango, 4.7% for Atakpamé and 2.1% for Lomé. These results are very interesting and show that this method is valid for the stations of Togo. The representative curves assembling a comparison between the simulated and measured values enable us to notice that the maxima are reached in March and in October whereas the minimum is obtained in August for Mango. For Atakpamé and Lomé, the maxima are obtained between March and April and in October whereas the minimum is in August.
Gaussian Function As shown in Fig. 4-21, there is a good correspondence between the theoretical and
experimental curves for the majority of the months. From the values day laborers, the monthly averages are calculated and error analyzes on the theoretical values are weak (lower than 2%) for every month. The results of error analyzes on the monthly averages of the theoretical sunning are presented in the table below just as curves. We thus minimized the errors in order to be closer to the experimental results by considering separately the time on which the maximum is obtained, that was not the case in the precedent work carried out by other authors. The correspondence between the theoretical and experimental curve is very remarkable between 7:00 am and 5:00 pm. The majority of the relative variations are observed at the sunrise and sunset. These results are better than those obtained by Banna et al. [15] when they used this same method to simulate the time sunning in Togo. They had considered that at any time of the year, the maximum is obtained in the day at 12 am which was an approximation of the reality.
Рис. 4. Сравнение временных графиков, смоделированных и измеренных в январе в г. Ломе Fig. 4. Comparison between time curves simulated and measured in January in Lomé
0,160
0,140
0,120
5 0,100
"п
с 0,080
<z
cl 0,060
0,040
0,020
0,000
-я- P{t)
1
1 31
•о га о о
zi о
ZI IM
ZI
zi to
ZI
ra
Hours
Рис. 6. Сравнение временных графиков, смоделированных и измеренных в апреле в г. Ломе Fig. 6. Comparison between time curves simulated and measured in April in Lomé
Рис. 5. Сравнение временных графиков, смоделированных и измеренных в феврале в г. Ломе Fig. 5. Comparison between time curves simulated and measured in February in Lomé
Рис. 7. Сравнение временных графиков, смоделированных и измеренных в августе в г. Ломе Fig. 7. Comparison between time curves simulated and measured in August in Lomé
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Рис. 8. Сравнение временных графиков, cмоделированных и измеренных в октябре в г. Ломе Fig. 8. Comparison between time curves simulated and measured in October in Lomé
Рис. 11. Сравнение временных графиков, cмоделированных и измеренных в феврале в г. Атакпаме Fig. 11. Comparison between time curves simulated and measured in February in Atakpame
Рис. 9. Сравнение временных графиков, смоделированных и измеренных в ноябре в г. Ломе Fig. 9. Comparison between time curves simulated and measured in November in Lomé
Рис. 12. Сравнение временных графиков, смоделированных и измеренных в марте в г. Атакпаме Fig. 12. Comparison between time curves simulated and measured in March in Atakpame
Рис. 10. Сравнение временных графиков, смоделированных и измеренных в январе в г. Атакпаме Fig. 10. Comparison between time curves simulated and measured in January in Atakpame
Рис. 13. Сравнение временных графиков, cмоделированных и измеренных в апреле в г. Атакпаме Fig. 13. Comparison between time curves simulated and measured in April in Atakpame
Рис. 14. Сравнение временных графиков, смоделированных и измеренных в сентябре в г. Атакпаме Fig. 14. Comparison between time curves simulated and measured in September in Atakpame
Рис. 17. Сравнение временных графиков, смоделированных и измеренных в феврале в г. Манго Fig. 17. Comparison between time curves simulated and measured in February in Mango
Рис. 15. Сравнение временных графиков, смоделированных и измеренных в декабре в г. Атакпаме Fig. 15. Comparison between time curves simulated and measured in December in Atakpame
Рис. 18. Сравнение временных графиков, cмоделированных и измеренных в марте в г. Манго Fig. 18. Comparison between time curves simulated and measured in March in Mango
Рис. 16. Сравнение временных графиков, cмоделированных и измеренных в январе в г. Манго Fig. 16. Comparison between time curves simulated and measured in January in Mango
Рис. 19. Сравнение временных графиков, cмоделированных и измеренных в июле в г. Манго Fig. 19. Comparison between time curves simulated and measured in July in Mango
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Рис. 20. Сравнение временных графиков, cмоделированных и измеренных в сентябре в г. Манго Fig. 20. Comparison between time curves simulated and measured in September in Mango
Рис. 21. Сравнение временных графиков, cмоделированных и измеренных в ноябре в г. Манго Fig. 21. Comparison between time curves simulated and measured in November in Mango
Анализ погрешностей теоретических значений для модели функции Гаусса Error analysis on the theoretical values for the Gaussian model
Таблица 3 Table 3
Month Lomé Atakpamé Mango
H measured H calculated R.V.% H measured H calculated R.V.% H measured H calculated R.V.%
January 3 7S7,5 3 764,S 0,60 4406,3 4463,3 1,29 5479,7 5531,4 0,94
February 4 330,4 4 311,5 0,44 5039,4 5096,0 1,12 5979,4 6027,1 0,S0
March 4 996,5 4 957,9 0,77 53S4,9 5439,9 1,02 6125,2 6164,3 0,64
April 4 964,S 5 007,2 0,S6 52S9,4 5233,1 1,07 5S90,2 5SS7,S 0,04
May 4 732,6 4 747,3 0,31 5321,5 5239,1 1,55 5450,4 53S2,6 1,24
June 3 904,0 3 S37,3 1,71 5032,0 4942,0 1,79 4927,S 4S50,5 1,57
July 4 346,1 4 2S3,3 1,44 40SS,9 4039,3 1,21 4564,5 44S9,4 1,65
August 4 1S2,5 4 135,0 1,13 3764,2 37S6,0 0,5S 4552,3 4492,0 1,32
September 4 621,6 4 574,3 1,02 4564,1 4575,9 0,26 522S,9 5253,1 0,46
October 4 943,4 4 9S5,1 0,S4 52S6,4 5215,5 1,34 5937,S 5S54,4 1,40
November 4 975,2 5 020,4 0,91 5216,6 5155,5 1,17 5512,7 5461,0 0,94
December 4 324,6 4 370,7 1,07 4604,0 4650,9 1,02 5310,2 535S,5 0,91
Average 4 509,1 4 499,6 0,21 4S33,2 4S19,7 0,24 5413,3 5396,0 0,37
We obtained theoretical values very close to experimental values. From the theoretical Pt values we calculated the daily total sunning then monthly averages. A comparison between the theoretical values obtained by this model and the experimental data are calculated with the made errors. All these results are gathered in Table 3.
This table shows that the value of the monthly average of global solar irradiation is about 4500 W/m2 for the town of Lomé, 4800 W/m2 for Atakpamé and 5400 W/m2 approximately for Mango when we take into
account all the period of measurement. These results show that we obtained theoretical values with a maximum relative variation of 1.7% for Mango, 1.8% for Atakpamé and 1.7% for Lomé. These results are very interesting and show that this method is also valid to obtain simulated values of the total sunning for the stations of Togo. Thus whatever the position of the measuring site, errors are weak (Table 4).
Table 4
Geographical positions of the stations
Таблица 4
Географическое местоположение станций
The last part consists in finding a relation binding the parameter c and the monthly average of the daily
sunshine duration which is easily measurable by the majority of the weather stations and with little means. The sunshine durations were used for Lomé but by lack of available data for the two other cities, the maximum possible sunshine duration was used for Atakpamé and Mango. The results obtained are shown on Fig. 22, 23 and 24.
Stations Latitude Longitude Altitude
Lomé 06°10' Northern 01°15' East 19,6 m
Atakpamé 07°35' Northern 01°07' East 499,66 m
Mango 10°22' Northern 00°28' East 144,7 m
Рис. 22. Кривая соотношения а в соответствии с N для Ломе Fig. 22. Curve of fitage of а according to N for Lomé
Рис. 23. Кривая соотношения о в соответствии с No для Атакпаме Fig. 23. Curve of fitage of о according to No for Atakpame
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Рис. 24. Кривая соотношения а в соответствии с No для Манго Fig. 24. Curve of fitage of а according to No for Mango
The results of the correlations making it possible to calculate the values of c knowing the values of means sunshine duration or its maximum arise as follows:
For Lomé:
a = 0,0346N5 - 0,9751N4 + 10,901N3 -- 60,325N2 +164,95N -175,14 ;
R2 = 0,96.
For Atakpamé
a = 0,784N03 - 28,02N02 + 333,6N0 -1322 ; R2 = 0,93.
For Mango
a = 0,253N03 + 9,133N02 - 109,5N0 + 439,6;
R2 = 0,99.
The results show that, theoretical curves correspond well with the experimental curves except for some shifts observed at the sunrise and at the sunset. This is due to the fact that the exponential function is cancelled only when t tends towards less infinite whereas this is not physically realizable in our case. For the Gaussian method, obtaining correlation between a and the sunshine duration with R2 very high will make it possible to simulate the hourly global solar irradiation in Togo constantly. The errors made by the use of the function of Fourier on the global values are higher than for the Gaussian method but lower than 5%.
Conclusion
The results obtained from these two models show that the two methods give good performances. In the case of the Gaussian method, we obtain a new method of simulation which allows us to have best result. In the case of the function of Fourier, the theoretical curve
corresponds well to the experimental curves and it seems the averages values of experimental ones. The errors made on the experimental values are weaker for the Gaussian method than for the analysis of Fourier. The second method enables us to obtain simulated values as soon as the sunshine duration is available. The two methods are thus valid to simulate the sunning in Togo.
List
H: Monthly average of global daily solar irradiation
No: monthly average of maximum possible sunshine duration
N: monthly average of sunshine duration on the ground
R2: coefficient of correlation
Rt: the ratio of global hourly sunning and the daily measured global sunning
P(t): the ratio of global hourly sunning and the daily simulated global sunning
g: Parameter of standard deviation of the normal distribution
R.V.: Relative Variation
References
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