Internal resistance

Basic resistance

When using a battery, there is an intrinsic resistance (supposed identical for both charging and discharging), which affects the battery voltage by 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 may be modified in the advanced parameters. However new default values will only be effective when defining a new battery. For attributing it to an existing battery, you have to check the corresponding default box in the battery parameters.Then you will have to save your battery.

Internal resistance vs Temperature

The Lead Acid batteries internal resistance is supposed independent of the temperature.

For the Li-Ion batteries, PVsyst applies the Arrhenius relationship (Ref. Lundgren 2016¹), which has been checked against many experimental discharge curves. It was observed that the law was giving very good satisfaction for the modeling purpose 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 catch the increase of voltage at the end of charge. Its real shape has little impact on the battery balance calculation as the concerned SOC range between the disconnecting threshold an 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 !