Consider an off-grid PV system with storage that powers an incandescent bulb load using an inverter. The area of the PV array is 1.7[m2] and the conversion efficiency of the modules is 20% . The charge controller has no conversion losses, but the MPPT algorithm is 94.1% efficient. The system is equipped with a 12[V] battery with an internal resistance of 100[mΩ] .
At a certain moment, the irradiance on the modules is 500[W⋅m−2] and the load consumes 100[W] . At that operating point the conversion efficiency of the inverter is 92.6% . What is the operating voltage of the battery?
To find the operating voltage of the battery, we need to consider the power flow in the system. We can break down the power flow into different stages and calculate the losses at each stage.
1. Power generated by the PV array:
Since the area of the PV array is 1.7 m^2 and the irradiance on the modules is 500 W/m^2, the power generated by the PV array can be calculated as follows:
Power_generated = Area_of_PV_array * Irradiance = 1.7 m^2 * 500 W/m^2 = 850 W
2. Power at the output of the MPPT algorithm:
The MPPT algorithm is 94.1% efficient, so the power at the output of the MPPT algorithm is:
Power_MPPT_output = Power_generated * MPPT_efficiency = 850 W * 0.941 = 799.85 W
3. Power after considering inverter efficiency:
The inverter has an efficiency of 92.6%, so the power at the output of the inverter is:
Power_inverter_output = Power_MPPT_output * Inverter_efficiency = 799.85 W * 0.926 = 740.6841 W
4. Power consumed by the load:
The load consumes 100 W of power.
5. Power available for battery charging:
The power available for battery charging is the difference between the power output of the inverter and the power consumed by the load:
Power_available_for_charging = Power_inverter_output - Power_load = 740.6841 W - 100 W = 640.84 W
Now, let's calculate the operating current of the battery. We know the power available for charging, but we need to convert it into current using Ohm's law.
6. Calculation of battery charging current:
The formula for power in terms of current and voltage is:
Power = Current^2 * Resistance
We can rearrange this formula to find the charging current:
Current = sqrt(Power / Resistance)
The internal resistance of the battery is given as 100 mΩ, which is equivalent to 0.1 Ω.
Using this information, we can calculate the charging current:
Charging_current = sqrt(Power_available_for_charging / Internal_resistance)
Charging_current = sqrt(640.84 W / 0.1 Ω) = sqrt(6408.4 A) = 80 A
Now, we have the operating current of the battery, but we still need to find the operating voltage.
7. Calculation of operating voltage:
The operating voltage can be found by dividing the power available for charging by the charging current:
Operating_voltage = Power_available_for_charging / Charging_current
Operating_voltage = 640.84 W / 80 A = 8 V
Therefore, the operating voltage of the battery in this off-grid PV system is 8 V.