This paper explores some variations of a hier-archical control framework that has been recently proposed. This framework is dedicated to the control of a network of interconnected subsystems such as those describing cryogenic processes or power plants. Recent studies have shown that handling constraints and non-linearities could challenge the real-time feasibility of the approach. This paper investigates and combines two successful directions, namely the use of truncated fast gradient and deep-neural-network-based controller modeling, to reduce the computational time of the most critical subsystem. It is also shown that by doing so, the control update period can be significantly reduced and the closed-loop performance is greatly improved. This paper can therefore be seen as a concrete implementation and validation of some key ideas in the design of real-time distributed NMPCs. All concepts are validated using the realistic and challenging example of a real cryogenic refrigerator.