Theoretically in transformers, will the secondary voltage increase or decrease (with increasing load current and a lagging load power factor) and why?
Primary winding driven by simple AC sinusoidal voltage
Assuming there are no tuned resonances occuring (due to distributed winding capacitances), if you apply a lagging load (inductive) to the secondary, you can expect the secondary voltage to drop. It might drop a really small amount if the inductive load is many henries or it might drop a lot if the load takes lots of current.
The equivalent circuit of a transformer should confirm my answer: -