Optimization of the solar panel tilt angle for various seasonal periods

The design of photovoltaic (PV) plants should provide the maximum use of solar radia-tion for electricity generation. The quantity of produced energy per radiation collecting sur-face unit depends on several factors: the radiation profiles in time, the effectiveness of PV modules, the angles of solar panel tilt and orientation, the operating temperature and the tem-perature characteristics of PV cells, the characteristics of inverters and other plant compo-nents. The most easily changed parameters, which haven’t constructional restrictions, are the inclination and the azimuth orientation of solar panels. Numerous articles devoted to the op-timization of inclination and orientation of fixed solar panels angles were published to date. These works can be divided into those in which the optimal angles are determined by direct measurement of energy generation by PV modules [1-5] and those that based mainly on vari-ous statistical relations and theoretical models [6-14]. The first approach has the advantage of direct experimental determination of the tilt angle for a given location, but such experiments are usually curried out for a short time in comparison with the multi-year intervals of meteo-rological measurements. In earliest studies within the second approach the optimum orienta-tion is usually suggested to be south-facing and tilt angle β is assumed depends only on the local latitude L, for example βopt=(L+15º) ±15º [6], βopt= L±15º [7] (plus for winter season, minus for summer one). It is clearly discrepancies and the difference in the choice of tilt angle 15 degrees can lead to the difference of year yield estimations of about 5% for PV plants in the mid-latitudes (see Figure 1 below). Numerous works are devoted to determination the relation , which was tested, for example, for various regions in China [8]. The more general expression and monthly values of the correlation coefficients were determined for cities located at different latitudes from Stockholm to Tripoli [9]. 

The one of simplest theoretical methods for determining the optimal solar panel angles is based on calculation the extraterrestrial radiation on a tilted surface, neglecting the influence of the atmosphere [10]. However, to take into account the atmospheric and local conditions it is necessary to rely on the local terrestrial measurements. In number of works the daily amount of radiation averaged for each month is used for the optimal angle calculation [11 - 13]. More detailed calculations are based on a hourly irradiance data which are now available for any geographical place from NASA website or from local meteorological sources. The use of these hourly data allows more reasonably determine the optimal tilt and azimuth angles.

The radiation measurements are performed usually for a horizontal surface and to determine the radiation components (direct, diffuse and reflected) on tilted surface the particular model of solar radiation is necessary. The direct radiation at any given time can be easily determined by a horizontal radiation using a simple geometrical relations. A more difficult problem is determining of diffuse and reflected components which are calculated usually with a help of various radiation models [14]. These models can be divided on isotropic models (uniform irradiance from the sky dome) and anisotropic ones (non-uniform irradiance). In works [8, 11-12, 15-16] one can see the examples of optimal tilt angle calculations based on isotropic models and in works [13, 16-18] - on anisotropic models.
The method of optimal angles calculation can be represented by purely numerical solutions [12, 13, 17, 18], analytical equations for optimal angles [9, 11, 15], algorithms with using of neural networks [15], computer simulations with programs as TRNSYS [19].

Because the solar radiation varies with season and time of a day under unpredictable weather conditions the systematic long-term data measurements can be regarded as most reli-able and accurate radiation data for using in optimal tilt and azimuth angles determination. The optimization method based of a data of this kind has been developed in this work. This method can be applied to any geographical place for any seasonal period with knowledge of the long-term radiation data. The initial point for our computing is the measured hourly global and diffuse irradiance on a horizontal surface and albedo also. In developed method the analytical formulas or equations for optimum angles using a number of well-known dif-fuse radiation models [20-24] are derived. As example we present the calculation results for a number of regions in Ukraine for various seasonal periods, including summer, winter and all year.

RESULTS OF OPTIMIZATION
To calculate the optimal tilt angle of solar panels the following cities of Ukraine were se-lected: Kiev, Odessa, Zaporozhye, Kherson and Uzhgorod. The significant distance between this cities and various solar radiation conditions result to substantial differences of the opti-mum tilt angles. Table 1 shows the results of calculating by the analytical method for isotropic and anisotropic radiation models. The parameters P, Q1, Q2 and others entering to equations for optimal angle were calculated on the basis of the long-term experimental radiation data [27] obtained by averaging for each month of hourly direct, diffuse and reflected irradiance components.

These formulas enable us to find the optimum angle for all values of the azimuth angle of the receiving surface. Table 1 shows the values for solar panels oriented to the south. Evidently the bigger angles are typical for the winter months and the lower ones - for periods with summer months. The differences between the data obtained by various models are 9 - 11 degrees for winter months, 1 - 2 degrees for summer months and 3 - 10 degrees for all year period. The differencies of "all year" data for various models is most significant for Tran-scarpatia region (Uzhgorod). Reindl's model gives the biggest angles for this region due to significant influence of the horizon brightening term in diffuse irradiance component.
In general all anisotropic models except Klucher's model result to bigger tilt angles than isotropic one for all considered regions. The lowest differencies of optimal angles are observed between Hay and Hay-Davies models (< 1 deg) for all regions.
To make sure, as far as the optimum orientation is to the south, we calculated in the framework of the Hay-Davies model the average daily amount of radiation depending on the angle for various azimuth angles near the line due south (Figure 1). According to the calculation for Odessa region the more preferred azimuth is 3 degree (to the East). The daily energy gain is 1 Wh/m2 only compared with the south facing panels.

Etot beta

FIG.1. The dependence of average daily amount of radiation per unit area on the tilt an-gle at different panel orientation. Odessa region; yearly period; radiation model of Hay-Davies

As can be seen from Table. 1, the optimum tilt angles for the winter and summer months' differ more than doubled. The conditions for radiation collecting can be slightly improve by installation the optimal angle at each month. The energy gain due to this adjustment will be less than 3% for considered regions.
Since this work is devoted to determining the optimum inclination, the natural question is, what will be the energy losses when the tilt angle is not optimal. Figure 2 shows the energy losses as a function of angle for different periods for Odessa region.

LossEtot beta
FIG. 2. Average daily yield losses as function of the tilt angle deviation from the opti-mum value. Odessa region; yearly period; radiation model of Hay-Davies

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Source: A. Gaevskii