Chain derailment from front crank. Fine on large ring won't hold on smaller inner ring Second hand bike. GTZR3000

Shimano 105 9 speedshifter
9 speed rear group shimano
9 speed tiagra rear derailleur 

The bike runs fine on large front ring, shifting is perfect.

When I change for an incline to the smaller front ring and apply pedal presure the chain comes straignt off. On a stand without pressure resistance on the drive train, everything moves perfectly. Back on the road and again and pedal pressure sends the chain off.

Can anyone advise if they are aware of potential reason why?
I could be caused because the chain is not running a straight line from the rear cogs to the front chain rings. If you put the bike on the stand and follow the chain from the top of the cogs to the chainrings. You will see if it follows a straight line. It might be slight off so that it won't jump off when on the stand but it does when there is pressure. 

Another reason could the rear cogs wable. If it so, you will need to change the cogs.