# Solar Module Electrical Model

An electrical model for the solar module will allow us to determine the true operating conditions imposed by the battery and regulator, and the loss of efficiency that occurs. In this study the efficiency of the solar power system is studies for different types of regulators.
1. A regulator that simply connects the solar module to the battery, so that the voltage output of the module is fixed at the battery voltage.
2. A Maximum Power Point Tracking (MPPT) regulator that follows the Maximum Power Point of the solar module, transforming the module power output to a voltage level that matches the battery. Such MPPT regulators can generally only achieve about 92%-97% efficiency at best in the transformation process (some top range ones claim as much as 99%), so some of the gain achieved by these regulators will be lost. A good MPPT regulator can also boost low module voltages up to the battery voltage, thus allowing charging to continue under very low light levels.
The scenario is that of a solar module located at Adelaide, South Australia on the shortest day of the year. This is the time when the amount of sunlight is at its minimum and is just the time that maximum efficiency is desired from the system.

The system diagram for this model is as follows: As a solar cell develops a voltage across a p-n junction in response to incident light, a simple model can be devised that consists of a current source (dependent on the incident illumination) and a parallel diode. Internal
resistance can be significant for some cells, but we shall ignore this for the moment. Thus:

I = Isc - I0 (eV/Vk-1)

where I is the module current delivered to the battery,
Isc is the light dependent current, equal to the short circuit current in the case of zero series resistance, and I0 is a diode reverse bias current. V is the voltage across the cell and Vk is another diode characteristic parameter proportional to temperature. We can obtain the two diode parameters by curve fitting to three points on the characteristic V-I curve that is usually published for a solar module. The three points that are normally specified are the short circuit current, open circuit voltage, and maximum power point voltage and current.

If internal series resistance
Rs of a cell is considered then the voltage V must be replaced by V+IRs which is the voltage across the ideal diode. Thus we get:

I = Isc - I0 (e(V+IRs)/Vk-1)

The total voltage across a module is then the sum of the individual series cell voltages. Note that V and Rs refer to an individual cell. For a module which has total series resistance Rsm and voltage Vm it is necessary to divide by the number of cells in series Ns.

As an example, the BP module BP3125, a 125W specified module, has the following datasheet values at an incident sunlight power of
1000W/m² and at 25oC:
• Short circuit current 8.02A
• Maximum power point current 7.23A
• Maximum power point voltage 17.3V
• Open circuit voltage 22.1V
• Number of cells in series Ns
By fitting the simpler model to these values we get I0 = 0.185mA, and Vk = 0.058V. The latter is double that of the theoretical 0.026V for an ideal pn junction, however we will assume that Vk has the usual linear temperature proportionality and that neither characteristic parameter depends on light intensity. A plot of the V-I curve for this model is shown below. This demonstrates the main features of the characteristic curve of the data sheets. If series resistance were important, the curve would show a reduced slope to the right beyond the maximum power point (the output voltage is reduced by IR losses). Parallel resistance would show increased negative slope of the upper part of the curve (current is reduced by diversion through the resistance as voltage increases).

The model of the solar module as described above is combined with the model for the insolation of the module:
1. for the simple regulator, limiting the output of the solar module to the battery voltage and determining the current.
2. for the MPPT regulator, determining the module's maximum power point and converting the voltage/current at that point to match the battery voltage at 100% efficiency.
The battery model is that of an uncharged battery that has its voltage always fixed at 12V (although this can vary between about 11V and 14V depending on the state of charge). The program used to study the model prints out the ampere-hours that would be put into a 12V uncharged battery by the above described solar module over the shortest day of the year at the latitude of Adelaide. These are:
• If the module followed the sun and the panel output voltage was fixed:: 22.5 AH
• If the module followed the sun and the panel output voltage was at the MPP: 28.3 AH
• If the module were fixed to north and the panel output voltage was fixed: 19.6 AH
• If the module were fixed to north and the panel output voltage was at the MPP: 24.4 AH
Thus there is a loss of 16% through having a fixed module over a sun-following module, and a further 25% through having a non-MPPT regulator between the module and the battery. With a possible loss of 10% through inefficiencies in the regulator, an MPPT regulator will allow a gain of only 12% over a simple regulator (such regulators should be quite low loss). This additional cost of such a regulator has to be balanced against the cost of a larger solar module. It is quite likely that the additional cost of sun-following modules and high efficiency MPPT regulators will not offset the simple strategy of purchasing a larger module, or even just using reflective surfaces to enhance collected solar energy.

To model a cloudy day, the sunlight intensity was reduced by a factor of 10. The results showed that an MPP regulator provided very little gain over a simple regulator, as the module output was close to 12V most of the time. This conclusion may vary with other modules whose output may be less than 12V in certain conditions, and a good MPP regulator could provide a small advantage.

A number of runs of the model under various conditions did not show MPP regulator gains greater than 30%. Some manufacturers claim over 40% gain for their products. The gain will increase as the open-circuit module voltage increases above the battery voltage.

First created 26 June 2007