Devices and methods for operating a diaphragm compressor system provide high output pressure and high throughput. In some embodiments, modular diaphragm compressors are stacked with a clamping mechanism pressing the compressor modules together. In embodiments, multiple stacks are provided as stages of a pressurization process. In embodiments, a main stage valve controls one or more pressure circuits for one or more hydraulic actuators of compressor modules. In embodiments, orifices configured for damping are incorporated to control actuator piston movement within a compressor module.