A global annual optimum tilt angle model for photovoltaic generation to use in the absence of local meteorological data

Highlights

• Model to estimate the annual optimum tilt angle for any location worldwide.

• Search based model based on polynomial regression, using data from 14468 sites.

• Studied influence variables on optimum tilt: hemisphere, diffuse fraction and albedo.

• The energy loss when the optimum tilt angle is misestimated increases with latitude.

• An estimation of a 0.2 annual albedo is only accurate for absolute latitudes <60°.

Abstract

This manuscript proposes a series of global models to estimate optimum annual tilt angle (${\beta }_{opt}$) as a function of local variables (latitude, diffuse fraction and albedo) based on the hourly irradiance data of 14,468 sites spread across the globe from the One Building database. As a result, these models can be used for any location in the absence of local meteorological data. First, a polynomial regression model, applicable worldwide, is proposed to estimate ${\beta }_{opt}$ as a function of latitude. This model fits the global data considered with a 2% RMSE error. Average energy losses are estimated to be 1% for a 10° variation from ${\beta }_{opt}$. A variation of 40° with respect to ${\beta }_{opt}$, implies a 12–18% energy loss depending on latitude. In addition, if only latitude is considered to estimate ${\beta }_{opt}$, different expressions should be used for latitudes $>50°$ depending on the hemisphere. These variations are a result of the influence of diffuse irradiance on ${\beta }_{opt}$, due to the fact that sites with higher amounts of diffuse irradiance have a lower ${\beta }_{opt}$. Secondly, a polynomial surface regression model to estimate ${\beta }_{opt}$ as a function of latitude and the annual diffuse fraction is proposed improving the results, reaching a 0.7% RMSE error. Thirdly, a simplified polynomial surface regression model to estimate ${\beta }_{opt}$ as a function of latitude and albedo (without the influence of the diffuse fraction) is proposed, and finally a model gathering all three variables under study (latitude, annual diffuse fraction and albedo) to calculate the optimum tilt angle is presented.

Introduction

The performance of photovoltaic (PV) modules is highly influenced by the orientation (an azimuth or horizontal angle with respect to a reference) and tilt angle (the vertical angle with respect to the ground) for a given location. A key objective when installing a photovoltaic array (PVA) is to achieve the maximum energy output in a time period. To achieve that, a regular solution is to use tracking systems, though, in many applications, the tilt and azimuth angles are fixed or periodically and manually adjusted. Placing the PVA facing the equator at an optimum tilt angle ${\beta }_{opt}$, and to correct the tilt periodically is a common practice. For this purpose, the optimum tilt should be determined at any latitude, on any day, or any period of the year. For grid-connected PV plants, the optimum orientation is computed according to site-specific weather data e.g., for the typical meteorological year (TMY) or satellite data. However, designers for medium and small PV systems, mainly in developing countries, often lack either the site-specific data or the calculation tools to benefit from it. These situations could clearly benefit from a simple and universal equation that accurately estimates the optimum tilt angle as a function of known local variables.

Technical literature shows various methods to express the optimum tilt function considering different criteria and based on latitude, time, declination, or other local characteristics, [20]. There are many studies that analyze the angle of inclination that optimizes the capture of the annual irradiation of the PV panels. To calculate ${\beta }_{opt}$, different optimization criteria can be considered; maximizing the total incident solar energy on the surface, either in extraterrestrial or in atmospheric conditions, is the most applied criterion [43]. Other optimization techniques applied include the Artificial Neural Network [42]; Genetic Algorithms [8,44] or Particle-Swarm Optimization techniques [25].

To compute the irradiance on the plane of a PVA, in order to calculate ${\beta }_{opt}$ at a given location, the direct normal irradiance (DNI) and diffuse radiation data of a TMY type year are required. Because it is not normally easy to obtain this kind of data, available measured global horizontal irradiance (GHI) data are used, and separation methods are usually applied to determine its DNI and diffuse horizontal irradiance (DHI) components. Separation or decomposition models are typically based on the correlation between the clearness index and the diffuse fraction. The clearness index is defined as the ratio of the GHI to the extraterrestrial horizontal irradiance while the diffuse fraction is the ratio of the DHI to GHI. The relationship between the diffuse fraction and the clearness index, originally proposed by Ref. [28]; is independent of the cloud type. Models including a stability index parameter (intended to account for the effect of variable/in-homogeneous clouds) in their formulation were found to perform better than the simple Liu and Jordan based models formulation [19].

The computation of ${\beta }_{opt}$ requires obtaining the irradiance components on the tilted panel from DNI and DHI data. While the beam component can be computed from a simple geometric relationship with the DNI, the other two components (diffuse and albedo radiation) must be estimated from models and empirical correlations [6,15]. Moreover, an adequate transposition model must be selected in order to account for all three components at the PVA tilt angle. In the valuable review paper by Ref. [50]; the performance of 26 transposition models was evaluated and compared. Although no universal model is found, some are recommended; according to the linear ranking results on nRMSE, the top families of models are the four Perez models [[37], [38], [39],48], the Muneer model [33]; and the Hay-Davies model [12].

Many studies that calculate ${\beta }_{opt}$ from a specific location or region using the Perez model or another are available. Most of them have been reviewed in Ref. [11,20]. However, there are few studies that have sufficient radiation data to give a worldwide generalized expression of the ${\beta }_{opt}$ as a function of latitude [22,41] or as a function of latitude and other local variables. Variables that can influence the calculation of ${\beta }_{opt}$ in addition to latitude are, according to Ref. [27]: the longitude, the sign of latitude, the altitude of the location, temperature, humidity, diffuse fraction, or albedo radiation among others [18]. developed an analytical model of the optimum tilt angle as a function of the latitude, clearness index and day number based on long-term solar data in Iran [9]. showed that, for the US, the local clearness index could be used as a relationship factor between latitude and ${\beta }_{opt}$.

The few expressions that model the annual optimum tilt angle as a function of latitude obtained by a search-based approach use databases where there are very few data locations for each latitude. In those cases, variations in $\beta opt$ of up to 10% are exposed. This supports the fact that for the same latitude, the optimal angle depends on other local variables. Other studies, [5,22]; also support this idea.

The ground type on which the photovoltaic installation is located affects the proportion of radiation reflected on it. Models typically consider an albedo (ρ) default value of 0.2 across sites, based on [29]; this value is generally used all throughout the literature [36]. However, the evolution and use of anisotropic models allows for the consideration of the influence of albedo, which has been proven to be important. If the effect of ground albedo is ignored, errors can be observed, for example, in the peak of daily irradiance if the default value is used instead of the real one [43]. In Ref. [4]; effective albedo values for 22 commonly occurring surface materials in photovoltaic applications are provided. Furthermore, high-tilt-angle ground-mounted PV systems may be impacted by albedo irradiation [1,26].

Firstly, this paper proposes a simple, global, worldwide model to estimate ${\beta }_{opt}$ as a function of latitude. The optimization criterion used to compute ${\beta }_{opt}$ is the one maximizing the total incident solar energy on the surface, either in extraterrestrial or in atmospheric conditions, [43]. The [37] model is the selected transposition model. However, a small comparison between the results obtained for ${\beta }_{opt}$ using the Perez and Hay-Davies model is found in subsection 3.3. The model is developed by using carefully selected and treated hourly radiation data from 14,468 sites around the globe from the One Building database (subsection 2.1). Given the amount of data used to develop the model, its applicability is beyond any other existing ${\beta }_{opt}$ model.

Throughout this manuscript, the variables influencing ${\beta }_{opt}$ perceived as most relevant are considered: the sign of the latitude, relative diffuse radiation, and albedo. Any differences in ${\beta }_{opt}$ for sites in the northern versus southern hemisphere are studied and discussed in section 3. The influence of diffuse irradiance on ${\beta }_{opt}$ is quantified in section 4. A simple worldwide model for ${\beta }_{opt}$ as a function of latitude and the diffuse fraction is also proposed.

The concept of albedo, its variability, and estimated values are discussed in section 5. In addition, a model quantifying the influence of albedo on ${\beta }_{opt}$ is also proposed. The relevance of accurately estimating ${\beta }_{opt}$ in terms of energy loss is carefully studied for all latitudes. The impact of correctly estimating variables such as the diffuse fraction or albedo for a correct energy assessment of a site is also considered throughout the paper.

All variables under study (latitude, annual diffuse fraction and albedo) are used in section 6 to obtain a regression model that allows to compute the optimum tilt angle.