  We study the energy levels and transport properties of an extended Kitaev
chain with a phase gradient. It is demonstrated that the hopping phase
difference can effectively induce the generation of Majorana bound states,
which are located at the non-endpoint sites of the chain. The number and
positions of the non-endpoint Majorana bound states can be modulated by the
hopping phase difference and initial hopping phase, respectively. In addition,
we propose a protocol to realize topological braiding operation by exchanging
the positions of two Majorana bound states in the extended Kitaev ring.
Furthermore, we also implement the braiding of any two of the multiple Majorana
bound states in the extended Kitaev double rings.
