The nice thing about external power supplies is that you can make them be exactly like a battery frozen in time.
This is an old, old topic, but I think maybe many people here haven't seen it.
Every voltage source can be modelled as a perfect voltage source driving some kind of imperfection network between it and the output pins of the real device.
The word "modelled" means "you cannot tell the difference between the real thing and the model if you did the model right."
If you take the open circuit voltage of a battery at some particular state of (dis)charge, that's the open circuit voltage. If you then load the battery with some external resistance, the battery voltage sags a bit. That sag is caused by the current flowing through the internal resistance. If you load it with both a DC load and an AC load, the battery voltage will vary both in DC voltage but perhaps also in phase compared to the load current. That is, the battery may have some internal inductance and capacitance too.
So if you measure a battery carefully when it's just at the right amount of depleted, you can design a power supply with the same open circuit voltage, the same internal resistance, and the same AC/frequency phase shift if you want to go that far. Generally voltage and series resistance is enough.
There is an article at GEO for how to do this if you want to experiment. See
http://geofex.com/Article_Folders/pedal ... edalbd.htm