  A normal $k$-edge-coloring of a cubic graph is an edge-coloring with $k$
colors having the additional property that when looking at the set of colors
assigned to any edge $e$ and the four edges adjacent it, we have either exactly
five distinct colors or exactly three distinct colors. We denote by
$\chi'_{N}(G)$ the smallest $k$, for which $G$ admits a normal
$k$-edge-coloring. Normal $k$-edge-colorings were introduced by Jaeger in order
to study his well-known Petersen Coloring Conjecture. More precisely, it is
known that proving $\chi'_{N}(G)\leq 5$ for every bridgeless cubic graph is
equivalent to proving Petersen Coloring Conjecture and then, among others,
Cycle Double Cover Conjecture and Berge-Fulkerson Conjecture. Considering the
larger class of all simple cubic graphs (not necessarily bridgeless), some
interesting questions naturally arise. For instance, there exist simple cubic
graphs, not bridgeless, with $\chi'_{N}(G)=7$. On the other hand, the known
best general upper bound for $\chi'_{N}(G)$ was $9$. Here, we improve it by
proving that $\chi'_{N}(G)\leq7$ for any simple cubic graph $G$, which is best
possible. We obtain this result by proving the existence of specific no-where
zero $\mathbb{Z}_2^2$-flows in $4$-edge-connected graphs.
