PSV is independent of system failure. What about pilot operated safety valves?
Pilot Operated Safety Valves
Pilot operated safety valves are also mechanical device. Instead of spring, it utilizes system operating pressure itself to hold the valve seat in its closed position.
At top and bottom of valve seat, there is same pressure during normal operation. So, there is no resultant force on seat. As soon as pressure rises above set point, the top section above valve seat releases pressure to atmosphere or header (very small amount).
So, now there is difference in upstream and downstream pressure of valve seat. Therefore, it opens as a +ve upstream pressure.
Again, read API-RP 520. Read free literatures on web about “pilot operated valve”. Anderson Greenwood’s website provided very good articles on it.