Long-term energy output estimation for photovoltaic energy systems using synthetic solar irradiation data

Energy 28 (2003) 479–493 www.elsevier.com/locate/energy

Long-term energy output estimation for photovoltaic energy systems using synthetic solar irradiation data A.N. Celik ∗ Mechanical Engineering Department, School of Engineering and Architecture, Mustafa Kemal University, 31024 Antakya, Hatay, Turkey Received 11 May 2001

Abstract A general methodology is presented to estimate the monthly average daily energy output from photovoltaic energy systems. Energy output is estimated from synthetically generated solar radiation data. The synthetic solar radiation data are generated based on the cumulative frequency distribution of the daily clearness index, given as a function of the monthly clearness index. Two sets of synthetic solar irradiation data are generated: 3- and 4-day months. In the 3-day month, each month is represented by 3 days and in the 4-day month, by 4 days. The 3- and 4-day solar irradiation data are synthetically generated for each month and the corresponding energy outputs are calculated. A total of 8-year long measured hourly solar irradiation data, from five different locations in the world, is used to validate the new model. The monthly energy output values calculated from the synthetic solar irradiation data are compared to those calculated from the measured hour-by-hour data. It is shown that when the measured solar radiation data do not exist for a particular location or reduced data set is advantageous, the energy output from photovoltaic converters could be correctly calculated.  2003 Elsevier Science Ltd. All rights reserved.

1. Introduction Energy output estimation for a photovoltaic energy system (PVES) for a period of time is the initial step in designing a stand-alone PVES. The accuracy of the initial energy output estimation will be the determining factor in whether or not the energy system is optimally sized. In systems where a poor estimation of energy output occurs, either over or under-sizing of PVESs is inevitable. On the other hand, as Haas et al. [1] state, the technical reliability of photovoltaic systems

∗

Tel.: +90-532-227-7353; fax: +90-326-618-4932. E-mail address: [email protected] (A.N. Celik).

0360-5442/03/$ - see front matter  2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0360-5442(02)00140-8

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Nomenclature A photovoltaic array area AM relative air mass AM0 1.5 regression coefficients for EPTC An regression coefficients for ENRA Bn regression coefficients for EMAT Cn COM compressed weather data E daily total energy production, Wh EFG-Si edge-defined-film-fed growth silicon module EMAT energy rating at maximum ambient temperature ENRA energy ratings algorithm EPTC energy at photovoltaic test conditions GEN weather generator algorithm G solar irradiance, W/m2 G0 1000 W/m2 h solar irradiance, W/m2 H daily solar irradiation, Wh/m2 Ht daily solar irradiation at day t, Wh/m2 Ht+1 daily solar irradiation at day t+1, Wh/m2 H0 daily extraterrestrial solar irradiation, Wh/m2 ¯ H monthly average daily solar irradiation, Wh/m2 ¯ H0 monthly average daily extraterrestrial solar irradiation, Wh/m2 I hourly solar irradiation, Wh/m2 ¯KT monthly clearness index daily clearness index KT LLP loss of load probability LTD long-term data n number of days in a month N number of days in COM P power output, W PVES photovoltaic energy system ratio of hourly to daily solar irradiation rt SF solar fraction Tamb ambient temperature, °C Tmax maximum ambient temperature, °C TMY typical meteorological year TRY test reference year WS wind speed at 10 m above ground, m/s h overall system efficiency for photovoltaic modules n cell temperature 25 °C n0 w hour angle sunset hour angle ωs

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is one of the most important criteria for a broader acceptance and dissemination of this technology. Avoiding over and under-sized systems and designing techno-economically optimum systems may possibly make such renewable energy systems competitive and a better alternative as compared to conventional energy systems for electricity production. Another utilisation of energy output estimation is the simplified performance models. Many simplified performance algorithms have been developed for PVESs. The simplified algorithms predict the long-term performance by means of mathematical models, thus eliminating the use of hour-by-hour simulation procedure. In such simplified models, the produced energy by the modules is the input with some model parameters and the system performance (mostly in terms of loss of load probability, LLP, or solar fraction, SF) is the output. Some examples of such simplified models for PVES can be seen in [2–5]. In the present study, a general methodology is presented to estimate the monthly average daily energy output from PVESs. Energy output is estimated from synthetically generated solar radiation data. The synthetic solar radiation data are generated based on the cumulative frequency distribution of the daily clearness index (KT), given as a function of the monthly clearness index (K¯ T). Two sets of synthetic solar irradiation data are generated: 3- and 4-day months. In the 3day month, each month is represented by 3 days and in the 4-day month, by 4 days. The 3- and 4-day solar irradiation data are synthetically generated for each month and the corresponding energy outputs are calculated. The validity of the proposed method is demonstrated by comparing the energy output results from the synthetically generated data with those from the measured hour-by-hour solar radiation data, using a total of 8-year long data, from five different locations in the world. 2. Literature survey The correct estimation of energy output for a PVES normally requires measured data for a period of time and an energy model. However, the solar radiation data to be used in the absence of measured data should be a realistic substitution to the measured data, representing some major characteristics of the measured data. On the other hand, the energy model to be used should be able to provide accurate energy output when compared to the measured energy production. Missing either the real or realistic solar radiation data or a realistic energy model would lead to an inaccurate energy estimation for PVES. In the present paper, synthesised solar radiation data are used instead of measured ‘long-term data’ (LTD), using a validated energy model and the quality of the synthetically generated data is studied. Neither the development of an energy model nor the study of the accuracy of existing energy models is the objective in this paper. Different types of synthetic solar radiation data and their use in energy and system performance calculations and some energy models are reviewed next. 2.1. Synthetic solar radiation data The ‘typical meteorological year’ (TMY), or otherwise known as the ‘test reference year’ (TRY), is one of the most common synthetic weather data sequences used in solar system energy and performance calculations. TRY is a term used in Europe while TMY is a term mainly used in US. TMY of hourly weather data usually consists of 12 months of hourly data. Each month

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is selected from a long-term weather data as being the best representative of that particular month or is generated from several years of weather data which would yield the same statistics (such as the average solar irradiation and K¯ T) as those of several years data. TMY or TRY of hourly weather data has been developed and commonly used in performance and energy output calculations for many locations (see Refs. [2,6,7]). Klein and Beckman presented a method in Ref. [2] for estimating LLP of stand-alone photovoltaic systems using such solar irradiation data. Bergauer-Culver and Jager [6] generated a TRY, on the basis of 6 year of LTD for the town of Leonding and Mount Loser in Austrian Alps. The power output in Ref. [6] is calculated from its short circuit current which linearly depends on the irradiance, open circuit voltage (which is a linear function of the temperature) in the maximal power point. The photovoltaic panels installed in Leonding and at Mount Loser have a rated output power of 10 kW each. The energy output calculated using the generated TRY was compared with the actual energy output of the plants for 6 years (1990–1995). It was concluded in Ref. [6] that the mean deviations of the complete energy yield of the different years to the predicted yields using the new TRYs are 2.4% for Mount Loser and 3.6% for Leonding. Other than TMY and TRY, other techniques in generation of the solar radiation in particular, wind speed and temperature data have been the objective of several studies. Among these studies, Knight et al. [8] presented techniques for the generation of hourly solar irradiation and ambient temperature data, as well as suggestions for humidity and wind speed. The weather generator algorithm (GEN) developed in Ref. [8] requires the input of the monthly average solar radiation value and generates the hourly solar radiation values based on the cumulative frequency distributions of KT. Gordon and Reddy developed a solar irradiation generator on an hourly, [9], and on a daily basis [10]. Another alternative to the methods discussed above is the ‘compressed weather data’ (COM). COM is generated based on the same theory as the GEN, but attempts to represent the important statistics of an entire month of days with N selected days, where N is less than the actual number of days in the month. A question posed in Ref. [11] arising concerning the COMs for performance simulations of solar systems is the number of days, N, required and the sequence of them. It is clear that as the number of days decreases, the accuracy of the COM in representing the actual weather data will change. The researchers who used such synthetic data preferred a 4-day month, for examples see Gansler et al. [11] and Morgan [12]. Morgan generated 4-day month synthetic solar irradiation data for a year. He then compared the monthly specific energy resource using both the generated data and LTD and found that the synthesised data underestimated the resource available by some 13% for this particular data. Even though there seems to be an agreement on the number of compressed days as 4 days a month among the researchers [11–14], there is, however, no neat explanation for choosing this particular number of days. Therefore, 1 and 2day months should further be investigated and compared to the 3- and 4-day months. The biases introduced by synthetic solar radiation data have also been investigated for systems other than PVESs. Both Gansler et al. [11] and Bourges and Kadi [15] analysed four different synthetic solar radiation data files for the solar water heaters and compared their system performances and energy outputs to those using LTD.

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2.2. Existing energy output estimation models for PVES Energy models for photovoltaic converters have been dealt with by a number of researchers. The energy models vary from simple to complex with varying degrees of accuracy. The basic form of the energy models is E ⫽ A × h × h.

(1)

However, h goes through a substantial modification according to the objective of energy models for different module types. For example Durisch et al. [16] give the following five-parameter correlation for the polycrystalline silicon module h ⫽ P1(G / G0)P2 ⫹ P3(G / G0) [1 ⫹ P4(n / n0) ⫹ P5(AM / AM0)].

(2)

Similar correlations were also developed for the monocrystalline silicon modules and their accuracies were tested in Ref. [16]. In the following three models, the function of h is transferred to some correlation coefficients (An, Bn and Cn). First model is the ‘energy at photovoltaic test conditions’ (EPTC) [17]. The power output of EPTC is calculated by P ⫽ A1h ⫹ A2h2 ⫹ A3hTamb ⫹ A4h(WS).

(3)

When Eq. (3) is multiplied by the number of sunshine hours during a specified time period we obtain the energy output for that time period. The second model is the ‘energy ratings’ (ENRA) [18], which is given by P ⫽ B1h ⫹ B2h2 ⫹ B3h ln h.

(4)

Eq. (4) is then integrated over a standard reference day to determine the energy rating where integration of H is given as a function of five more parameters. In developing EPTC and ENRA only irradiance above 500 W/m2 was used. The third model is the ‘energy rating at maximum ambient temperature’ (EMAT) [19]. Since EMAT uses total daily irradiation, it yields the energy output of the module directly, based on the following equation: E ⫽ C1H ⫹ C2HT⫺2 max ⫹ C3Tmax.

(5)

The EPTC, ENRA and EMAT methods were compared in Ref. [19], for three different cases (clear day, cloudy day and the whole month), and for three different module types (EFG-Si, monocrystalline silicon and polycrystalline silicon). The clear day comparison shows that, for three types of photovoltaic cells, EMAT predicts energy very close to the measured energy for all modules, within 0.7%. EPTC over-predicts energy by at least 4% for all modules while ENRA over-predicts energy for all modules by less than 10%. For the cloudy day, EMAT predicts energy within 5–2% for different modules. ENRA over-predicts energy by more than 60% for all modules. This is due to the reference day’s irradiation (1000 W/m2), which is much greater than that of the cloudy day. Thus, if the reference day’s conditions are not close to the operating conditions, ENRA is not a good predictor of module energy output. The comparison based on the whole month shows that EMAT predicts energy within 0.07% for all modules, while EPTC and ENRA within 3 and 20%, respectively. The models require a total of 12 parameters for each module type (three parameters for each season).

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3. System setting and simulation procedure The same system setting as the experimental system, installed at Tal-y-Bont (TyB, a remote site near Cardiff, UK), described by Celik [14] is assumed for the system investigated in the present paper. The test facility was operational for over 8 years from 1990 –1998. The number of the components used and the battery controllers and control strategies applied have changed during these years in the experimental site. It finally consisted of 20 amorphous silicon PV panels (each of 0.3 m2 area and 10 W nominal power at 1 kW/m2), three wind turbine machines (two of them at 50 W and one at 170 W rated power at 10 m/s wind speed), one lead-acid battery with under and over-voltage controllers, an auxiliary lead-acid battery, and a load. The energy models for the PV modules and the wind generator were validated with respect to solar irradiation and wind speed, which are explained in detail in Refs. [20] and [21]. Weather data measurements were carried out in the test facility also. 4. Analysis of the solar radiation data used The latitudes and longitudes for the locations considered in the present paper (Cardiff, Canberra, Davos, Athens, and Ankara) are given in Table 1. The monthly average daily solar irradiation values for the same locations are also presented in Table 1, calculated from the LTD on horizontal surface. Cardiff has the least amount of solar irradiation level among the locations studied, contributed especially in the winter months. The yearly average daily solar irradiation level for Cardiff Table 1 Monthly average daily solar irradiation (H) values in kWh/m2 calculated from LTD on horizontal surface for different locations Cardiff 1991 Latitude Longitude

51.30N 3.13W

January February March April May June July August September October November December Average

0.65 1.28 2.00 3.64 4.45 4.34 4.63 4.55 3.29 1.47 0.63 0.47 2.62

Cardiff 1994

0.78 1.00 2.09 3.81 3.78 5.32 5.03 3.92 2.91 1.91 0.86 0.65 2.67

Cardiff 1995

0.55 0.97 2.47 3.74 5.09 5.56 4.86 4.77 2.76 1.61 0.80 0.45 2.80

Cardiff 1996

0.48 1.39 1.57 3.57 4.31 5.47 5.15 4.23 2.91 1.33 0.88 0.52 2.65

Canberra 1993

Davos 1996

Athens 1995

Ankara 1995

35.18S 149.08E

46.48N 9.50E

38.00N 23.44E

39.55N 32.50E

6.88 8.35 5.37 4.13 2.62 1.80 2.57 3.78 4.19 7.10 7.71 8.00 5.21

2.12 3.25 4.70 5.80 5.78 5.74 5.81 5.19 4.31 3.20 2.32 1.79 4.17

1.67 3.81 4.23 5.24 5.69 5.69 5.86 5.73 5.77 3.08 2.78 2.00 4.30

2.21 2.57 4.35 3.44 4.43 5.35 4.95 5.41 5.05 3.63 2.41 1.62 3.79

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differs only slightly from one year to the next. Canberra having the smallest latitude receives most of solar irradiation. Davos has a fairly clear sky with a relatively high yearly average solar irradiation level of 4.17 kWh/m2/day. The monthly clearness indices (K¯ T) calculated from the following equation are given in Table 2: ¯ /H ¯ 0. (6) K¯ T ⫽ H ¯ 0 required for the calculation of Eq. (6) is calculated using the well-known equation The H given by Duffie and Beckman [22]. It is seen from Table 2 that, due to low level of irradiation, the K¯ Ts for Cardiff are the lowest among the locations analysed, the yearly average values varying from 0.36 to 0.38. Davos has the highest yearly average K¯ T value of 0.62, with monthly values as high as 0.72. The maximum K¯ T value of 0.77 seen in Table 2 occurs in February in Canberra. 5. Synthetical generation of solar radiation data In the present paper the 3- and 4-day long of hourly solar irradiation data are generated for each month, adding up to a total of 36 and 48 days in a year, respectively. The method to generate COM solar irradiation data mentioned earlier consists of estimating hourly values from the monthly average data, following the steps given next. The hourly solar irradiation data are generated in two steps. First, the daily clearness indices (KT) are obtained from the cumulative distribution of daily clearness index, given as a function of monthly clearness index (K¯ T). This cumulative distribution can either be derived from LTD or alternatively the cumulative distribution functions, developed and assumed location independent (general) by Bendt et al. [23] and Hollands and Huget [24], can be used. Bendt et al. [23] analysed 20 years of data from 90 US locations and developed expression for the KT distribution based on these data, while Hollands and Huget [24] developed an implicit expression for the Liu and Jordan distributions presented Table 2 The monthly clearness indices (K¯ T) calculated from LTD for different locations Cardiff 1991 January February March April May June July August September October November December Average

0.27 0.33 0.33 0.42 0.42 0.38 0.42 0.49 0.48 0.33 0.24 0.24 0.36

Cardiff 1994 0.33 0.26 0.34 0.44 0.35 0.46 0.46 0.42 0.42 0.43 0.33 0.33 0.38

Cardiff 1995 0.23 0.25 0.40 0.43 0.48 0.48 0.44 0.51 0.40 0.36 0.30 0.23 0.38

Cardiff 1996 0.20 0.36 0.26 0.41 0.41 0.47 0.47 0.45 0.42 0.30 0.33 0.26 0.36

Canberra 1993 0.57 0.77 0.60 0.61 0.51 0.41 0.54 0.62 0.51 0.70 0.66 0.65 0.60

Davos 1996 0.70 0.72 0.70 0.64 0.53 0.50 0.52 0.54 0.58 0.63 0.70 0.68 0.62

Athens 1995

Ankara 1995

0.36 0.64 0.54 0.54 0.51 0.49 0.52 0.56 0.68 0.48 0.57 0.48 0.53

0.52 0.46 0.57 0.36 0.40 0.46 0.44 0.54 0.61 0.59 0.53 0.43 0.49

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in Ref. [25]. Knowing K¯ T for a particular month, N days (3 and 4 days in the present study) are selected from the cumulative frequency distribution curve or are calculated using either of the functions mentioned above. Each day would then have a KT, with a condition that the average of NKTs be equal to K¯ T. Knight et al. [8] illustrated the generation of a 5-day month hourly solar irradiation data, obtaining the KTs from the daily cumulative distributions function using the correlation given in Ref. [23]. The fractional time axis is divided into five equal regions and the corresponding KT values of these midpoints of the regions are found. The same approach used in Ref. [8] to obtain KTs is shown in Fig. 1 for a 4-day month. The corresponding KTs are 0.29, 0.43, 0.52 and 0.62, with the average value of 0.46. For this particular month, this average value is also equal to the actual K¯ T. However, if the average value of the 4-day month was different than the actual value, KTs of 4-day month should have been modified to obtain the exact value of the actual K¯ T. The sequence of KTs is another important point. Knight et al. [8] order them such that the lag one autocorrelation (i.e. persistence of solar irradiation, which refers to the dependence of today’s solar irradiation on the solar irradiation of preceding days and is given by Eq. (7)) is approximately equal to the actual monthly average persistence.

冘

n⫺1

f⫽

t⫽1

¯ )(Ht+1⫺H ¯) (Ht⫺H

冘

(7)

n⫺1

¯) (Ht⫺H

2

t⫽1

The first step is the most difficult part of the whole procedure. This is complex, time consuming and could be a deterrent for non-expert people. This complex approach can be abandoned for simplicity. Instead of deriving or calculating the monthly cumulative frequency distribution curves, predetermined coefficients have been used in the present paper to obtain KTs, with the

Fig. 1. Obtaining KT values from the cumulative frequency distribution curve for a 4-day month.

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condition that the average of nKTs be equal to K¯ T. The predetermined coefficient values given in Table 3 are used to calculate the KTs in the present paper. The predetermined coefficients and corresponding KTs for an example month in which K¯ T is 0.46 are presented in Table 3 for the 3and 4-day months. For the 4-day month, it is observed that the Kt values obtained from the cumulative frequency curves are very close to those obtained from the predetermined coefficients. In the second step, the hourly solar irradiation data are generated for each day (for each KT calculated above), using the following equations developed by Collares-Pereira and Rabl [26]: rt ⫽

冋

册

π I cos w⫺cos ws ⫽ (a ⫹ b cos w) , H 24 sin ws⫺(πws / 180) cos ws

(8a)

where a ⫽ 0.409 ⫹ 0.501 sin(ws⫺60),

(8b)

b ⫽ 0.6609 ⫹ 0.4764 sin(ws⫺60),

(8c)

KT ⫽ H / H0.

(8d)

Collares-Pereira and Rabl [26] developed equations to generate the hourly solar irradiation data, representing the Liu and Jordan’s general curves. Liu and Jordan [25] had previously shown that rt can be represented as general curves for a number of stations. Further research has shown that the curves are satisfactory for most of mid-latitude locations [22], but not for latitudes higher than 60°N. The generated hourly solar irradiation values for the 4-day month are shown in Fig. 2 for an example year. The 3- and 4-day months of hourly synthetic solar irradiation data are generated in the same way for the available 8-year long data. 6. Energy output estimation The quality of the synthetically generated solar irradiation data is evaluated here. The monthly average daily energy outputs calculated from the synthetic data are compared to those calculated from the measured LTD. Fig. 3 provides the monthly average daily energy outputs from the LTD and the 3- and 4-day months synthetic data for Cardiff 1991. As Fig. 3 shows, both the 3- and Table 3 The predetermined coefficients and corresponding KTs for an example month for the 3- and 4-day months 3-Day month Day

Coefficient

1 2 3 4

0.7 1.0 1.3

4-Day month KT

0.32 0.46 0.60 –

Coefficient

0.6 0.9 1.1 1.4

KT

0.28 0.41 0.51 0.64

KT by cumulative frequency curve 0.29 0.43 0.52 0.62

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Fig. 2. Daily solar irradiation profiles calculated from KTs for the 4-day month for Cardiff 1991.

Fig. 3. Monthly average daily photovoltaic energy outputs calculated from LTD versus those calculated from the 3and 4-day months synthetic data for Cardiff 1991.

4-day months synthetic data provide very close energy output figures to the LTD. The maximum daily energy output from the photovoltaic system is in July with a value of 93.56 Wh/m2. For the same month, the monthly average daily energy outputs calculated from the 3- and 4-day months of synthetic data are 93.21 and 93.20 Wh/m2, respectively. The minimum daily energy output is observed in December, in which the monthly average daily energy output is only 7.98 Wh/m2. Considering a daily energy demand of 360 Wh, the size of photovoltaic converter required to meet the load is 45 m2. However, this was only 3.85 m2 in July, which is the best solar month. Therefore, if a photovoltaic system was sized for the worst month, it would become largely oversized for the rest of the year. This in turn makes the system too costly and non-optimal in terms of techno-economics. Therefore, if a high level of reliability (90% or over) is required, an alternative system should be considered. For example, hybrid photovoltaic-wind energy system could be an option depending on the wind regime in the location. The errors in estimation of the monthly average energy outputs calculated from the 3- and 4-day months synthetic solar irradiation data,

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when compared to those calculated from the LTD, are shown in Fig. 4 for Cardiff 1991. The maximum errors observed for both the 3- and 4-day months correspond to the months in which the solar irradiation level is quite low, which are January and December: for these months, a deviation of 5% from the actual energy output occurs. An overall trend is also observed that the errors tend to decrease in the summer months and increase again in spring and autumn, and winter most of all. The only exception is February in which the minimum deviation occurs, 0.1% both for the 3- and 4-day months. Duffie and Beckman [22] state that the method used in the present paper is adequate for individual days, with the best results for clear days and increasingly uncertain results as the daily total irradiation decreased. This explains the increasing deviations for rather cloudy months. For most of the months, the synthetic data underestimate the monthly average daily energy output. Only in 3 months, August, September and October, overestimation occurs. Overall, the average deviation is 0.5% both for the 3- and 4-day months of synthetic solar irradiation data for Cardiff 1991. The errors in estimation of the monthly specific photovoltaic energy as calculated from the 3day month synthetic solar irradiation data are represented in Fig. 5 for different locations. Among the locations studied only Canberra is in the southern hemisphere. Therefore, the months are converted from the southern hemisphere to equivalents for the northern hemisphere. For the 3day month, the trend, observed previously in the analysis of Cardiff 1991 data, is also obvious for other locations studied. The errors decrease in the summer months and increase again in spring, autumn and winter months. On the monthly basis, the minimum error occurs in July both for Canberra and Athens in which the energy output is estimated without an error by the 3-day synthetic data. The maximum deviation occurs in January for Cardiff 1996 with a value of 7.8%. Taking the absolute values of the monthly deviations, the minimum deviation value of all locations occurs in July, 0.23%. This minimum value is followed by June and August with 0.31 and 0.44%, respectively. The maximum error value including all locations is observed in January, 3.3%, which is followed by December with 2.6%. It is noted from Fig. 5 that, the monthly average daily energy is underestimated in January, March, April, May, and July at all locations studied without an

Fig. 4. Errors in estimation of the monthly specific photovoltaic energy calculated from the 3- and 4-day months synthetic solar irradiation data for Cardiff 1991.

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Fig. 5. Errors in estimation of the monthly specific photovoltaic energy calculated from the 3-day month synthetic solar irradiation data for different locations.

exception. While the energy output is overestimated in September and October at all locations, it is overestimated in August and November in majority of the locations by the 3-day month synthetic data. Fig. 6 shows the errors in estimation of the monthly specific energy calculated from the 4-day month synthetic solar irradiation data for varying locations. The monthly energy output figures from the 4-day month synthetic data are quite similar to those from the 3-day month. On the monthly basis, the minimum deviation occurs in June and July for Davos in which the energy output is estimated without an error. For the 4-day month synthetic data the maximum deviation occurs in January for Cardiff 1996, 8.0%. Considering the absolute values of the monthly errors, the average error value of all locations is observed in July, 0.24%. June follows this minimum

Fig. 6. Errors in estimation of the monthly specific photovoltaic energy calculated from the 4-day month synthetic solar irradiation data for different locations.

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value with 0.31% and this is in turn followed by August, 0.43%. The maximum error including all locations is 3.3% in January, which is followed by December with 2.5%. It is noted from Fig. 6 that the monthly average daily energy is underestimated in January, March, April, May, and July at all locations. While the energy output is overestimated in September and October at all locations, it is overestimated in August and November in majority of the locations by the 4-day month synthetic data. The errors in calculating the yearly average daily energy outputs are presented in Table 4 for various locations. The 3- and 4-day months underestimate the energy output in six of the locations, while the energy outputs are overestimated in two locations. The minimum error in calculating the yearly average daily energy output is by the 3-day month synthetic data with a 0.03% underestimation for Athens. For the 4-day month synthetic data, the minimum error is observed to be a 0.07% underestimation for Athens and a 0.07% overestimation for Ankara. The maximum errors in estimation of the yearly average daily energy outputs are 1.38 and 1.36% underestimation by the 3- and 4-day months synthetic data, respectively, both occurring in Cardiff 1991. 7. Conclusions The energy output estimation using synthetically generated solar irradiation data has been the subject of this paper. The monthly average energy outputs were calculated from the synthetic irradiation data and compared to those calculated from LTD for a total of 8-year long data. The most important findings that arose from this study can be summarised as follows: 앫 The novel approach developed in the present paper, the introduction of predetermined coefficients, can be successfully used to obtain KTs, instead of deriving or calculating them from the cumulative frequency distribution curves. 앫 As far as the 3- and 4-day months compressed data are concerned, the number of compressed days has no effect on the quality of the energy output prediction for PVESs. 앫 The error in energy output estimation using both the 3- and 4-day months synthetic data is Table 4 Yearly average values of errors in estimation of the monthly average daily photovoltaic energy from the 3- and 4-day months synthetic data for different locations 3-Day month error (%) Cardiff 1991 Cardiff 1994 Cardiff 1995 Cardiff 1996 Canberra 1993 Davos 1996 Athens 1995 Ankara 1995 Average

⫺1.18 ⫺0.65 ⫺1.38 ⫺1.26 ⫺0.50 0.20 ⫺0.03 0.09 0.66

4-Day month error (%) ⫺1.18 ⫺0.63 ⫺1.36 ⫺1.28 ⫺0.52 0.17 ⫺0.07 0.07 0.66

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relatively larger in the winter months. So, a possible further study can investigate the reason and develop new techniques to reduce this relatively larger errors in the winter months. 앫 Overall small error value of 0.66%, both for the 3- and 4-day months, shows that the synthetic data generated in the present paper estimate the energy output with a very high level of accuracy.

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