The transformer must be powered continuously to supply power to the valve internal electronics, not through the PID.
The PID output connects in place of the thermostat as per drawing.
My buttterfly valves are air or hand operated not motor driven.
I tried a 2Nm Belimo actuator on my 2" Definox brand butterflies, did not close completely.
The Belimo high speed actuator part #LMQ24A 24VDC.
Ebay is a good source for Belimos.
Purchased three slow Belimos a month ago...
You can purchase the 1/2" SS 9-24 VDC 3-wire valves for $26.00 shipped.
I am currently modefining two for an additional $ 3.00 to meet my needs.
My pneumatic valves I can move in less than 1 second.
The butterflies I have don't require more than 4 Nm torque.
This hobby can drive you...