I was wondering whether there is a Euler–Maclaurin formula of sorts for expressions such as $$ \sum_{x \in [a,b]^d\cap \mathbb{Z}^d} f(x) - \int_{[a,b]^d}f(x) $$ where $d\ge 2$ is an integer, $a,b \in \mathbb{R}$ and $f:\mathbb{R}^d \longrightarrow \mathbb{R}$ is a smooth function in $[a,b]^d$. I am particularly interested in such expansion with a control of the error term.

I appreciate any reference or suggestions.