Use of Plane of Array Measurements

Where operational PV plants are analysed, it is typical to have measurements of global POA irradiance $G_\text{poa}$ rather than horizontal global $G_{\text{h}}$ and diffuse $G_\text{dif}$ irradiances. The SolarFarmer calculation chain starts with $G_\text{h}$ and $G_\text{dif}$, so it is necessary to convert $G_\text{poa}$ as a first step. This involves first transposing the measured irradiance to the horizontal plane using the Hay model, then decomposing it to estimate the diffuse component using the Erbs model.

It may also become necessary to remove near shading effects included in the POA measurements, but this is not considered at present.

The modelled global POA irradiance based on horizontal global and diffuse irradiance is:

$$G_\text{poa} = G_\text{dir,poa} + G_\text{dif,poa} + G_\text{r,poa}$$

We can relate these components to the concurrent horizontal irradiance components using the following from previously:

$$G_\text{dir,poa} = G_\text{dir} \frac{cos \left( \theta \right)}{\cos \left(\theta_\text{z} \right)}$$

$$G_\text{r,poa} = \frac{1}{2} \rho G_\text{h} \left(1 - \cos \left( \beta \right) \right)$$

$$G_\text{dif,poa} = G_\text{dif} \left\lbrack \left( \frac{G_\text{dir}}{G_{0} \left( n \right)} \right) \left( \frac{a}{\cos \theta_\text{z}} \right) + \left( \frac{1 + \cos\beta}{2} \right) \left( 1 - \frac{G_\text{dir}}{G_{0} \left( n \right)} \right) \right\rbrack$$

with $$a = \max(0,\ \cos\theta)$$

Where $G_0$ is the hourly extra-terrestrial irradiation (Solar Constant) on a normal plane as described in Extra-Terrestrial Beam Irradiation.

As described in the Erbs model, the Erbs decomposition model defines the Diffuse Fraction ($G_\text{dif} / {G_\text{h}}$ - ratio of the hourly horizontal diffuse to global irradiance) as a function of Clearness Index $k_\text{T}$. This function has different forms depending on the value of $k_\text{T}$ and we will denote it as $f\left(k_\text{T}\right)$ here. This allows us to formulate $G_\text{poa} $ as a function of the diffuse fraction, $G_\text{dif} / {G_\text{h}}$.

$$G_{dir,poa} = k_\text{T} G_0 \cos \theta \left(1 - f \left( k_\text{T} \right) \right)$$

For consistency with the usual transposition for the direct irradiance, set $G_\text{dir,poa}$ to zero for $\theta_\text{z} > 88^\circ$, see Direct Irradiance.

$$G_\text{r,poa} = \frac{1}{2} \rho k_\text{T} G_0 \cos \theta_\text{z} \left(1 - \cos \beta \right)$$

To avoid the reflected irradiance being negative, set $G_\text{r,poa}$ to zero for $\theta_\text{z} \geq 90^\circ$.

$$G_\text{dif,poa} = f \left( k_\text{T} \right) k_\text{T} G_0 \cos \theta_\text{z} \left\lbrack k_\text{T} \left( 1 - f \left( k_\text{T} \right) \right) \frac{a}{\cos\theta_{z}} + \left( \frac{1 + \cos\beta}{2} \right) \left( 1 - k_\text{T} \left(1 - f \left( k_\text{T} \right) \right) \right) \right\rbrack$$

To avoid the sky diffuse irradiance being negative, set $G_\text{dif,poa}$ to zero for $\theta_\text{z}\geq 90^\circ$. It should be noted that this gives different results than the Hay model because this means that the diffuse part is neglected for cases with $\theta_\text{z}\geq 90^\circ$ but there is no way to consider this as the cosine of the zenith angle is used as a multiplier in this formulation of the sky diffuse irradiance.

Our target when solving this for $k_\text{T}$ is that modelled POA is equal to the measured POA (within a tolerance).

Marion [13] suggests a numerical method but it has been found that there are problems with the convergence of the result. The secant method has been tried which works well in most cases. $\cos \theta$ is set to zero for $\theta > 89^\circ$ when solving for $k_\text{T}$. It has been found that for a few cases where $\cos \theta < 0$ (sun behind the rack) there is no root found so that the value for $k_\text{T}$ is used that gives the smallest error in POA.

Once $k_\text{T}$ is found, the global horizontal and diffuse horizontal irradiances are calculated as:

$$G_\text{h} = k_\text{T} G_0 \cos \theta_\text{z}$$

$$G_\text{dif} = f \left( k_\text{T} \right) G_\text{h}$$