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The {\sc Plane Diameter Completion} problem asks, given a plane graph $G$ anda positive integer $d$, if it is a spanning subgraph of a plane graph $H$ thathas diameter at most $d$. We examine two variants of this problem where theinput comes with another parameter $k$. In the first variant, called BPDC, $k$upper bounds the total number of edges to be added and in the second, calledBFPDC, $k$ upper bounds the number of additional edges per face. We prove thatboth problems are {\sf NP}-complete, the first even for 3-connected graphs offace-degree at most 4 and the second even when $k=1$ on 3-connected graphs offace-degree at most 5. In this paper we give parameterized algorithms for bothproblems that run in $O(n^{3})+2^{2^{O((kd)^2\log d)}}\cdot n$ steps.

Variants of Plane Diameter Completion