Which graph partitioning algorithm can solve this problem?
Your problem is $\text{NP}$-hard. To see why, consider the version of the 3-Partition problem where we're given multiset $ X = \{ x_1, x_2, \ldots, x_{3n}\}$ of $3n$ nonnegative integers, and we need to decide whether $X$ can be partitioned into three disjoint multisets with the same sum each. Clearly, an instance $X = \{ x_1, x_2, \ldots, x_{3n}\}$ of the 3-Partition problem can be translated in polynomial time into a complete graph $G = \langle V, E \rangle$, where $V = \{ v_i\}_{i\in [3n]}$  and for each $v_i$, we have that the weight $w(v_i) = x_i$. That is, we view the integers as $3n$ vertices, and every two vertices are connected with an edge. Obviously, a balanced 3-clustering of $G$ corresponds to an appropriate 3-partition of the multiset $X$.