  Let $k\geqslant 3$ and let $A=\{0=a_{0}<a_{1}<\cdots<a_{k-1}\}$ with
$\gcd(A)=1$. Let $2^{\wedge}A$ denote the set of all sums of two distinct
elements of $A$. Freiman and Lev conjectured that the cardinality of
$2^{\wedge}A$ is greater than or equal to $a_{k-1}+k-2$ if $a_{k-1}\leqslant
2k-5$ and greater than or equal to $3k-7$ otherwise [V.F. Lev, Restricted set
addition in groups, I. The classical setting, J. London Math. Soc. 62(2000),
27-40]. Up to now, Freiman-Lev conjecture is still open for all
$a_{k-1}\geqslant 2k-2$. In this paper, we prove the Freiman-Lev conjecture is
true for $a_{k-1}\geqslant 2k-2$ and $a_{k-2}<2k-4$. Moreover, we determine
some structures of $A$ when $|2^{\wedge}A|=3k-7$.
