# Sandia PV Array Performance Model

The Sandia PV Array Performance Model (SAPM) defines five points on the IV curve.  These points are shown in the figure below.

The SAPM defines the primary points ($$L_{sc}$$, $$I_{mp}$$, $$V_{oc}$$, and $$V_{mp}$$) with the following equations:

$$I_{sc}=I_{sc0}\times f_{1} \left ( \frac{E_{b}f_{2}+f_{d}E_{d}}{E_{0}} \right )\times \left ( 1+\alpha _{Isc}\left ( T_{c}-T_{0} \right ) \right )$$           (eq. 1)

$$I_{mp}=I_{mp0}\left ( C_{0}E_{e}+C_{1}{E_{e}}^{2} \right )\left ( 1+\alpha _{Imp}\left ( T_{c}-T_{0} \right )\right )$$           (eq. 2)

$$V_{oc}=V_{oc0}+N_{s}\delta \ln \left ( E_{e} \right )+\beta _{Voc}\left ( T_{c}-T_{0} \right )$$(eq. 3)

$$V_{mp}=V_{mp0}+C_{2}N_{s}\delta \ln \left ( E_{e} \right )+C_{3}N_{s}\left ( \delta \ln \left ( E_{e} \right ) \right )^{2}+\beta _{Vmp}\left ( T_{c} -T_{0}\right )$$            (eq. 4)

## Functions:

$$f_1$$ is a 4th order polynomial function of absolute air mass, $$AM_a$$, and is called the air mass modifier.  It is defined as:

$$f_{1}\left ( AM_{a} \right )=a_{0}+a_{1}AM_{a}+a_{2}\left ( AM_{a} \right )^{2}+a_{3}\left ( AM_{a} \right )^{3}+a_{4}\left ( AM_{a} \right )^{4}$$

where $$a$$ is the vector of coefficients that are determined from module testing.

$$f_2$$ is a 5th order polynomial function of angle of incidence, $$AOI$$, and is called the angle of incidence modifier. it is defined as:

$$f_{2}\left ( AOI \right )=b_{0}+b_{1}AOI+b_{2}\left ( AOI \right )^{2}+b_{3}\left ( AOI \right )^{3}+b_{4}\left ( AOI \right )^{4}+b_{5}\left ( AOI \right )^{5}$$

where $$b$$ is the vector of coefficients that are determined from module testing.

$$E_e$$ is the “effective irradiance”.  It is defined as:

$$E_{e}=\frac{I_{sc}}{I_{sc0}\left \{ 1+\alpha _{Isc}\left ( T_{c}-T_{0} \right ) \right \}}$$, where:

• $$I_{sc0}$$ is the short circuit current at reference conditions.
• $$I_{sc}$$ can be calculated from (eq. 1) above.

$$\delta$$ is a function of $$T_{c}$$ defined as:   $$\delta =\frac{n\times k\left ( T_{c}+273.15 \right )}{q}$$, where:

• $$n$$ is an empirically determined ‘diode factor’,
• $$k$$ is Boltzmann’s constant ($$1.38066\times 10^{-23}J/K$$),
• $$q$$ is the elementary charge constant ($$1.60218\times 10^{-19}coulomb$$)

$$\beta_{Voc}$$ is a function of effective irradiance,$$E_e$$ , defined as: $$\beta _{Voc}=\beta _{Voc0}+m_{\beta Voc}\left ( 1-E_{e} \right )$$, where:

• $$\beta_{Voc0}$$ is the temperature coefficient for module open circuit voltage at irradiance conditions of $$1000 W/m^2$$ .
• $$m_{\beta Voc}$$ is a coefficient describing the irradiance dependence for the open circuit voltage temperature coefficient (typically equals zero)

## Parameters:

• $$E_{b}$$ is beam irradiance on the plane of array
• $$E_{d}$$ is the diffuse irradiance on the plane of array
• $$E_{0}$$ is reference solar irradiance ($$1000 W/m^2$$)
• $$T_{c}$$ is cell temperature ($$^{\circ}C$$)
• $$T_{0}$$ is reference cell temperature ($$25^{\circ}C$$)
• $$f_{d}$$ is the fraction of the diffuse light that is used by the module.  For typical flat plate modules $$f_{d}$$ is usually assumed to be equal to 1.  For concentrators the value can be smaller than 1.
• $$\alpha_{lsc}$$ is the normalized temperature coefficient for short circuit current. Units are $$1/^{\circ}C$$
• $$\alpha_{lmp}$$ is the normalized temperature coefficient for maximum power current. Units are $$1/^{\circ}C$$
• $$N_{s}$$ is the number of cells in series
• $$C$$ is a vector of coefficients determined by module testing using a method developed at Sandia.