Internal resistance

Basic resistance

Batteries have intrinsic resistance (assumed identical for both charging and discharging) that causes a voltage drop:

\(Vbatt = Voc_{Batt} + ResInt * IBatt\)

where:

• Voc_Batt = open circuit voltage, it may be seen as a virtual intrinsic voltage,
• Vbattery = either charging or discharging voltage, at the terminals of the battery,
• IBatt = either charging or discharging current (positive or negative).

The internal resistance is rarely mentioned on the datasheets. PVsyst provides a default value based on a reasonable voltage drop at a given current.

• For lead-acid batteries, this default is 40 mV at C10.
• For Li-Ion, this is much less, 16 mV at C10. This low value allows to use Li-Ion batteries at very high charge/discharge rates without prohibitive heating.

NB: These values can be modified in advanced parameters. However, new default values apply only when defining a new battery. To apply the default to an existing battery, check the corresponding box in the battery parameters, then save the battery.

Internal resistance vs Temperature

Lead-acid batteries internal resistance is assumed independent of temperature.

For Li-ion batteries, PVsyst applies the Arrhenius relationship (Ref. Lundgren 2016¹), which has been validated against many experimental discharge curves. This relationship provides very good agreement for modeling purposes in PVsyst.

\(ResInt (T°C) = ResInt (T°CRef) * e^{((Activation Energy/8.315) / (1/Tbatt[K])-1/(TRef[K]))}\)

where:

• ResInt = internal resistance
• Tbatt[k] and TRef[K] are temperatures expressed in Kelvin (T[K] = T[°C] + 273.16)
• Activation Energy = activation energy for the underlying thermally activated mechanism; usual values are 30 to 40 kJ/mol.

Li-Ion batteries: internal resistance vs SOC

End of charge

The exponential increase of the Internal resistance at the end of charge is responsible for the sudden voltage increase at high SOC. This indicates the charging limit for disconnecting the controller.

We define an empirical exponential correction factor, applied above a SOC = 90%, and so that the value at SOC = 1 is around 15: \(ResCorr = 1 + A * e^{-B * (1-SOC)}\)

The battery voltage becomes, above SOC = 0.9: \(VBattery (SOC, Temp) = Voc_{battery} (SOC) + ResCorr * ResInt (Temp) * IBattery\)

PVsyst applies this empirical correction factor to account for the voltage increase at the end of charge. Its exact shape has little impact on the battery balance calculation since the relevant SOC range between the disconnection threshold and 0.9 is small.

Deep discharge

An equivalent resistance correction is applied to the deep discharge, when SOC is lower than 15%. It is evident for everybody that the battery voltage drops quickly when using it at very low SOC !