The realistic(practical) voltage source is modeled as an ideal voltage source plus a series resistor R in the above figure. In an ideal voltage source whatever the current load is the voltage would remain the same.
Apparently in a practical voltage source when the load current increases the imaginary series resistor will cause a higher voltage drop.
I think the wire resistance between the source and the load is not the issue here, since we are talking about the voltage change accross the power-supply terminals.
My question is then: What is this series resistor in a power-supply? What causes it to exist? Is that the change in Thevenin equivalent resistor?