A distributed port-Hamiltonian (DPH) system is a framework of boundary controls based on energy for partial differential equations (PDEs). A DPH system is defined in terms of a Stokes-Dirac structure that implies boundary integrability in the sense of Stokes' theorem. Because of the boundary integrability, the energy flows of the systems distributed on a domain can be transformed into energy flows across the boundary of the domain. In general, there might exist boundary nonintegrable energy flows in PDEs. Such energy flows can be modeled by an extended representation of DPH systems, called a distributed energy flow. This paper shows that DPH systems with distributed energy flows can be transformed into standard DPH systems without distributed energy flows by adding extra DPH systems. As a result, we can apply boundary controls and boundary interconnections based on energy to a larger class of PDEs. We call this transformation the boundary completion of DPH systems. This paper derives the necessary and sufficient conditions for such boundary completions.