# Application of utilizability computation methods to Europe and Africa

Maria Joao Carvalho, Bernard Bourges

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Maria Joao Carvalho, Bernard Bourges. Application of utilizability computation methods to Europe and Africa. Intersol Eighty Five, Pergamon, pp.2439-2443, 1986, 9780080331775. 10.1016/B978-0-08-033177-5.50459-2. hal-00949145

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
APPLICATION OF UTILIZABILITY COMPUTATION
METHODS TO EUROPE AND AFRICA

+

*

Maria JOAO CARVALHO and Bernard BOURGES

+ Laboratorio Nacional de Engenharia e Tecnologia Industrial, Lisboa

* Centre d'Energétique, Ecole des Mines de Paris

ABSTRACT

Some methods for computing solar energy utilizability are applied to European and African locations and compared with actual values obtained from long-term measurements on horizontal and south-oriented tilted planes. Most of analyzed methods give acceptable results (within 20%) when the actual monthly average daily irradiation on the considered surface is known. When this value is not known (i.e. computed from monthly mean on a horiz. plane), errors on the monthly available energy above a threshold irradiance may be in the range 30-40%.

KEYWORDS

Solar radiation; utilizability; tilted planes; design methods.

INTRODUCTION

Utilizability has now become an usual meteorological input for a number of simplified design methods of solar systems (thermal active, passive, photovoltaic). Various methods for the computation of solar energy utilizability have been published in the former years (1-6) and reviewed recently (7). Some characteristics are general to these methods: most of them are derived from North American radiation data; solar radiation on tilted planes is computed through the usual and so-called "isotropic" assumption for diffuse sky radiation. It is now possible to question both assumptions, as radiation data have become available for other climates (e.g. Europe and Africa) and for tilted planes. Objectives of the present study were, first, to apply the selected Utilizability methods to some sample of representative meteorological stations and, second, to compare results with values directly obtained by analyzing hourly figures of measured solar radiation, on a horizontal plane and on tilted planes.

UTILIZABILITY AND AVAILABLE ENERGY

Solar system design can be done by simplified methods which need, as input for the calculation of solar fraction, the long-term energy delivery of the system at different thresholds. These computations are based on the concept of
"Utilizability" defined as $\theta(I_c) = \int_{\Delta t} (I - I_c)^+ dt / nH_T$

The Utilizability $\theta(I_c)$ is the fraction of solar radiation $nH_T$ incident upon a collector's aperture during a certain period $\Delta t$ (month or year), that lies above the specific threshold irradiance $I_c$. The product $\theta(I_c) \cdot nH_T$ is sometimes called the "available solar energy above the threshold $I_c$" (for the period).

## UTILIZABILITY CALCULATION METHODS

Various methods have been published for utilizability computation and reviewed recently (7). Six of these methods have been considered in the present work (1-6). The first four methods only need as climatological input the mean daily global radiation value on a horizontal plane, $H_H$, and on the collector plane, $H_T$ (This last value may be also computed from $H_H$ with method (8), if it is not known). They have been derived as fits to North-American data vs the "clearness index" $K_T = H_H/H_o$ which is the parameter used to indicate the dependence of $\theta$ with climate. The two last methods require an additional parameter $I_{MAX}$. maximal irradiance on the considered plane.

Collares-Pereira/Rabl method (1) includes in the same step computations of $\theta$ and $H_T$ from $H_H$; it is also suitable for concentrating collectors. Clark's method (2) is the only hourly-based utilizability method. Evans method (3) proposes three different correlations of $\theta$ suitable for flat-plate solar collectors: only one of them is reported here, after transformation into function of $I_c$ and $K_T'$ (related to $K_T$)

$$ \theta = 1 - 1.07 \cdot 10^{-3} \cdot I_c / K_T' + 3.02 \cdot 10^{-7} \cdot (I_c / K_T')^2 $$

Gordon and Zarmi's method (5) is an analytical approach to the Utilizability concept, with the semi-empirical form for flat-plate collector

$$ \theta(I_c) = (1 - I_c / I_{MAX})^{1/\nu} $$

with $1/\nu = I_{MAX} \cdot t_D / H_T$ and $t_D$ is the time between sunrise and sunset. In method (6), "utilizability" is derived from Cumulative Frequency Curves (CFC) of solar irradiance incident on the collector. CFC's are defined as curves giving the time $nw$ (mean daily number of hours) during which the solar radiation is above some threshold $I_c$. Correlations have been developed from a Principal Components Factor Analysis to give the Coefficients of a 5TH order polynomial CFC fitting from the parameter $\nu = H_T / (I_{MAX} \cdot t_D)$.

## AVAILABLE DATA AND METHODOLOGY

Solar radiation data used in the present analysis are summarized in Table 1. Three to seven years of solar global radiation measured hourly values have been collected for horizontal planes or south-oriented tilted planes (without ground-reflected diffuse radiation).

<table><caption>Table 1: Solar Radiation Data Used in the Present Analysis</caption><thead><tr><th>Station</th><th>Country</th><th>Tilt</th><th>Latitude</th><th>Longitude</th><th>Altitude</th><th>Period</th></tr></thead><tbody><tr><td>Nice</td><td>France</td><td>0°</td><td>43.7°N</td><td>7.2°E</td><td>100 m</td><td>1970-75</td></tr><tr><td>Millau</td><td>France</td><td>8°</td><td>44.1°N</td><td>3.0°E</td><td>720 m</td><td>1970-75</td></tr><tr><td>Carpentras</td><td>France</td><td>0°</td><td>44.1°N</td><td>5.1°E</td><td>99 m</td><td>1970-75</td></tr><tr><td></td><td></td><td>45°-90°</td><td></td><td></td><td></td><td>1979-81</td></tr><tr><td>Lisboa</td><td>Portugal</td><td>0°</td><td>38.7°N</td><td>9.1°W</td><td>77 m</td><td>1975-77</td></tr><tr><td>Trappes</td><td>France</td><td>0°</td><td>48.8°N</td><td>2.0°E</td><td>167 m</td><td>1970-75</td></tr><tr><td></td><td></td><td>45°-90°</td><td></td><td></td><td></td><td>1979-81</td></tr><tr><td>Kilkenny</td><td>Ireland</td><td>0°</td><td>52.7°N</td><td>7.3°W</td><td>64 m</td><td>1969-75</td></tr><tr><td>Maputo</td><td>Mozambique</td><td>0°</td><td>25.9°S</td><td>32.6°E</td><td>44 m</td><td>1965-69</td></tr></tbody></table>
Available data may be combined in three groups:

Carpentras, Millau, Lisboa)

3. Tropical Africa (Maputo)

1. Southern Europe (Nice,

2. Northern Europe (Trappes, Kilkenny)

The considered quantity, $\theta.H_T$, is not Utilizability, but the monthly available energy above given thresholds, $\theta.H_T$. That makes possible the comparison of methods for which precise definition of "utilizability" is not exactly homogeneous (1-2) with the others. Available energy above various critical intensities is computed, first from these original measured data for thresholds $I_c = 0, 100, 200$ to $600 \text{ W/m}^2$ and, secondly, using various utilizability methods (only taking, as an input, the mean monthly sum of global irradiation on a horizontal plane and, if available, on the tilted plane). For a given location, plane and period (one month or the whole year) different comparison criteria have been selected: Mean bias error (all thresholds) and Root mean square error (RMSE) (or Standard-deviation of errors) (all thresholds). Both these parameters are given as relative values (divided by the mean available energy for all thresholds from 0, 100 to $600 \text{ W.m}^{-2}$). The relative error on the available energy is also given for one specific threshold ($200 \text{ W.m}^{-2}$).

## RESULTS FOR HORIZONTAL PLANES

Results are quite homogeneous within each group. Four methods (1,2,4,6) give very satisfactory results in group 1 (Southern Europe): Mean yearly bias error for all thresholds within 2%; relative errors for monthly available energy within 3 to 8% at low thresholds (Tables 2-3); largest errors in winter. Algorithms (3) and (5) show important discrepancies (10 to 20% in average; large errors in winter, up to 30% for (3)). For the third group (Maputo), all methods seem acceptable (with the exception of (3) in winter), but with systematic underestimate. For group 2 (characterized by more cloudy conditions and small values of clearness index $K_T$), method (6) gives good results; methods (1-2) offer a satisfactory accuracy, although with some systematic overestimate (within 10% at low thresholds). Methods (3,5) show larger errors.

## RESULTS FOR TILTED PLANES

Two types of comparison have been made: first, using only as input the mean monthly irradiation on a horizontal surface; the mean monthly irradiation on the tilted plane is computed by $\theta$, except for method (1). Second, the right value on the tilted plane is introduced, except for (1). Bourges method (6) has not been tested.

In the first case, errors are important, due to a systematical underestimate

<figure><img src="image_2.png" /><figcaption>Fig. 1: Available energy $\theta.H_T$ (MJ/m$^2$/day) above threshold irradiance $I_c$.</figcaption></figure>
<table><caption>Table 2: Computation of Available Solar Energy above Threshold Radiation Intensities from 0 to 600 W.m⁻² (Horizontal plane) using different methods. This table gives, for each method, the mean relative bias error and the monthly root mean square error (between brackets) (in % of the average Available Energy).</caption><thead><tr><th>Station</th><th>Col. Per<br>/1/</th><th>Clark<br>/2/</th><th>Evans<br>/3/</th><th>Klein<br>/4/</th><th>Gordon<br>/5/</th><th>Bourges<br>/6/</th></tr></thead><tbody><tr><td>Nice</td><td>-1.6<br>(2.3)</td><td>0.9<br>(2.6)</td><td>8.4<br>(6.8)</td><td>1.1<br>(2.7)</td><td>-5.5<br>(5.2)</td><td>-0.9<br>(1.8)</td></tr><tr><td>Millau</td><td>-1.8<br>(2.4)</td><td>0.2<br>(1.7)</td><td>5.6<br>(6.5)</td><td>1.2<br>(2.7)</td><td>-6.2<br>(5.2)</td><td>-0.8<br>(1.3)</td></tr><tr><td>Carpentras</td><td>-1.9<br>(2.3)</td><td>0.5<br>(1.6)</td><td>8.4<br>(6.6)</td><td>0.5<br>(2.6)</td><td>-5.1<br>(5.0)</td><td>-0.2<br>(1.9)</td></tr><tr><td>Lisboa</td><td>-0.3<br>(1.7)</td><td>2.2<br>(2.5)</td><td>8.6<br>(7.2)</td><td>1.9<br>(2.0)</td><td>-2.8<br>(2.6)</td><td>1.7<br>(1.7)</td></tr><tr><td>Trappes</td><td>3.3<br>(6.5)</td><td>4.3<br>(4.3)</td><td>7.2<br>(7.0)</td><td>6.9<br>(6.6)</td><td>-6.0<br>(6.2)</td><td>0.3<br>(1.7)</td></tr><tr><td>Kilkenny</td><td>4.2<br>(6.1)</td><td>4.5<br>(4.7)</td><td>10.6<br>(9.5)</td><td>7.2<br>(7.3)</td><td>-4.0<br>(4.5)</td><td>2.7<br>(3.2)</td></tr><tr><td>Maputo</td><td>-4.8<br>(2.8)</td><td>-2.1<br>(2.8)</td><td>1.7<br>(7.3)</td><td>-2.5<br>(2.8)</td><td>-6.7<br>(4.6)</td><td>-2.7<br>(2.4)</td></tr></tbody></table>

<table><caption>Table 3: Relative errors (%) on the computed value of available energy above the threshold irradiance $I_c = 200 \text{ W.m}^{-2}$.</caption><thead><tr><th>Station</th><th>Period</th><th>/1/</th><th>/2/</th><th>/3/</th><th>/4/</th><th>/5/</th><th>/6/</th></tr></thead><tbody><tr><td rowspan="3">Nice</td><td>Year</td><td>-2.3</td><td>-0.1</td><td>7.3</td><td>0.6</td><td>-5.7</td><td>-0.6</td></tr><tr><td>June</td><td>-1.9</td><td>-1.3</td><td>3.7</td><td>0.9</td><td>-5.6</td><td>-1.9</td></tr><tr><td>Dec.</td><td>-7.7</td><td>-3.1</td><td>32.1</td><td>-4.7</td><td>-17.6</td><td>-3.5</td></tr><tr><td rowspan="2">Kilkenny</td><td>Year</td><td>1.3</td><td>5.7</td><td>15.1</td><td>8.4</td><td>-4.9</td><td>2.7</td></tr><tr><td>June</td><td>1.3</td><td>3.9</td><td>10.6</td><td>7.7</td><td>-3.3</td><td>1.7</td></tr><tr><td rowspan="3">Maputo</td><td>Year</td><td>-5.8</td><td>-3.7</td><td>-0.3</td><td>-3.0</td><td>-7.1</td><td>-3.5</td></tr><tr><td>June</td><td>-6.9</td><td>-5.0</td><td>9.0</td><td>-6.2</td><td>-8.7</td><td>-3.2</td></tr><tr><td>Dec.</td><td>-4.6</td><td>-3.3</td><td>-3.4</td><td>-1.4</td><td>-6.1</td><td>-3.4</td></tr></tbody></table>

of the mean monthly irradiation (3 to 17%, the biggest in winter, see Tab. 4). For low thresholds values ($I_c = 200 \text{ W.m}^{-2}$), errors are within 25% for method /2/, within 30% for /1/. All other methods show errors within 50% or larger, particularly for vertical planes (Table 5).

When the correct average irradiation value is introduced, results are better (Tab. 5): e.g. for /2/ the mean bias error for all thresholds is reduced by 6 to 11%, but monthly errors for a given threshold ($I_c=200$) are still important (10% at 45°; 20% to 25% for vert. surfaces). Methods (3-5) cannot be applied for vert. planes. Method (5) seems to be more reliable for sunny climates.

## CONCLUSIONS

As a general result, all methods have significant errors when the mean monthly irradiation value on the considered plane is not known: within 30% for reasonably low thresholds, with systematical underestimation for methods /1/ and /2/; still larger for other methods. When this value is known, available energy may be computed on a more accurate way, but nonnegligible errors still remain (within 20%, with the most general, and also the most complicate, method /2/). Method (3) (developped for inclined surfaces with tilt angle close to some optimum) and methods (4) and (5) are acceptable for tilt-angles close to
the latitude, but are not quite reliable for other angles. Some additional work seems to be necessary in order to get a simple method for computing the mean monthly irradiation on a tilted plane and simple empirical utilizability formulas of a general use, for any south-oriented surface.

REFERENCES

1. M. Collares-Pereira and A. Rabl, A simple procedure for predicting long-term average performance of nonconcentrating and of concentrating solar collectors. Solar Energy, 23, 235 (1979).

2. D.R.Clark, S.A.Klein and W.A.Beckman, Algorithm for evaluating hourly radiation utilizability function. ASME J. of Sol. Energy Eng., 103, 281 (1983).

3. D.L.Evans, T.T Rule and B.D.Wood, A new look at long-term collector performance and utilizability. Solar Energy, 28, 13 (1982).

4. J.C.Theilacker and S.A.Klein, Improvements in the utilizability relationships. Presented at the AS/ISES Meeting, Phoenix, 271 (1980).

5. J.M.Gordon and Y.Zarmi, The utilizability function: 1. Theoretical development of a new approach. and 2. Validation of theory against data-based correlations. Solar Energy, 31, 529 and 537 (1983).

6. B.Bourges and F.Lasnier, Statistical distribution of solar radiation: cumulative frequency curves. Preparation of a european data set. CEC-Report ESF-008-F (1983).

7. S.A.Klein and W.A.Beckman, Review of solar radiation utilizability. ASME Journal of Sol. Energy Eng., 106, 393 (1984).

8. S.A.Klein and J.C.Theilacker, An algorithm for calculating monthly average radiation on inclined surface. ASME J. of Sol. Energy Eng., 103, 29 (1981).

<table><caption>Table 4: Relative error (%) on the computed value of the mean daily irradiation on tilted planes.</caption><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="4">/1/</th><th colspan="4">/8/</th></tr><tr><th>Year</th><th>June</th><th>Marc</th><th>Dec.</th><th>Year</th><th>June</th><th>Marc</th><th>Dec.</th></tr></thead><tbody><tr><td>Trappes</td><td>45°</td><td>-7.2</td><td>-2.9</td><td>-6.0</td><td>-8.4</td><td>-6.5</td><td>-2.4</td><td>-4.8</td><td>-8.5</td></tr><tr><td>Trappes</td><td>90°</td><td>-9.5</td><td>-7.8</td><td>-3.9</td><td>-7.7</td><td>-7.4</td><td>-1.4</td><td>-2.6</td><td>-7.9</td></tr><tr><td>Carpent.</td><td>45°</td><td>-8.9</td><td>-5.6</td><td>-7.9</td><td>-14.6</td><td>-8.2</td><td>-4.9</td><td>-7.0</td><td>-14.3</td></tr><tr><td>Carpent.</td><td>90°</td><td>-12.7</td><td>-13.9</td><td>-7.3</td><td>-16.9</td><td>-10.8</td><td>-6.8</td><td>-6.2</td><td>-16.6</td></tr></tbody></table>

<table><caption>Table 5: Relative errors (%) on the computed value of available energy above the threshold irradiance $I_c = 200 \text{ W.m}^{-2}$ on tilted planes, according to the type of input data: $H_M$ (monthly average of daily global irradiation on a horizontal plane) or $H_T$ (idem on tilted plane).</caption><thead><tr><th rowspan="2">Station</th><th rowspan="2">Tilt Period</th><th rowspan="2"></th><th colspan="5">Input data:</th><th colspan="4">Input data:</th><th colspan="4">Ht</th></tr><tr><th>/1/</th><th>/2/</th><th>/3/</th><th>/4/</th><th>/5/</th><th>/2/</th><th>/3/</th><th>/4/</th><th>/5/</th></tr></thead><tbody><tr><td rowspan="3">Trappes</td><td rowspan="3">45°</td><td>Year</td><td>-10.4</td><td>-8.6</td><td>-2.4</td><td>-13.2</td><td>-11.5</td><td>-2.4</td><td>4.3</td><td>-7.3</td><td>-5.5</td></tr><tr><td>June</td><td>4.6</td><td>6.3</td><td>12.7</td><td>6.3</td><td>10.0</td><td>8.9</td><td>15.5</td><td>9.0</td><td>12.8</td></tr><tr><td>Dec.</td><td>-0.4</td><td>-5.2</td><td>-6.8</td><td>-34.6</td><td>-49.2</td><td>3.5</td><td>1.9</td><td>-28.5</td><td>-45.4</td></tr><tr><td rowspan="3">Trappes</td><td rowspan="3">90°</td><td>Year</td><td>-17.3</td><td>-20.6</td><td>5.1</td><td>-31.8</td><td>-20.5</td><td>-13.9</td><td>13.9</td><td>-26.3</td><td>-13.8</td></tr><tr><td>June</td><td>10.3</td><td>8.4</td><td>78.3</td><td>11.4</td><td>59.7</td><td>9.9</td><td>80.7</td><td>13.0</td><td>62.0</td></tr><tr><td>Dec.</td><td>-0.5</td><td>-15.8</td><td>-4.3</td><td>-49.0</td><td>-58.0</td><td>-8.6</td><td>3.8</td><td>-44.6</td><td>-54.7</td></tr><tr><td rowspan="3">Carpent.</td><td rowspan="3">45°</td><td>Year</td><td>-13.8</td><td>-12.1</td><td>-6.3</td><td>-15.1</td><td>-8.5</td><td>-4.4</td><td>1.9</td><td>-7.6</td><td>-0.5</td></tr><tr><td>June</td><td>-6.3</td><td>-6.5</td><td>0.9</td><td>-10.8</td><td>1.2</td><td>-1.6</td><td>6.1</td><td>-6.2</td><td>6.4</td></tr><tr><td>Dec.</td><td>-25.4</td><td>-20.5</td><td>-17.8</td><td>-26.5</td><td>-20.7</td><td>-7.2</td><td>-4.1</td><td>-14.3</td><td>-7.5</td></tr><tr><td rowspan="3">Carpent.</td><td rowspan="3">90°</td><td>Year</td><td>-23.0</td><td>-20.9</td><td>2.0</td><td>-30.6</td><td>-9.1</td><td>-10.8</td><td>-14.5</td><td>-21.6</td><td>2.2</td></tr><tr><td>June</td><td>-22.9</td><td>-15.1</td><td>59.7</td><td>-46.4</td><td>53.2</td><td>-8.9</td><td>71.3</td><td>-42.5</td><td>64.3</td></tr><tr><td>Dec.</td><td>-29.8</td><td>-25.3</td><td>-18.4</td><td>-32.7</td><td>-25.3</td><td>-10.4</td><td>-2.3</td><td>-19.3</td><td>-10.4</td></tr></tbody></table>