  Model order reduction (MOR) is often applied to spatially-discretized partial
differential equations to reduce their order and hence decrease computational
complexity. A reduced system can be obtained, e.g., by time-limited balanced
truncation, a method that aims to construct an accurate reduced order model on
a given finite time interval $[0, T]$. This particular balancing related MOR
technique is studied in this paper. An $L^2_T$-error bound based on the
truncated time-limited singular values is proved and is the main result of this
paper. The derived error bound converges (as $T\rightarrow \infty$) to the
well-known $\mathcal H_\infty$-error bound of unrestricted balanced truncation,
a scheme that is used to construct a good reduced system on the entire time
line. The techniques within the proofs of this paper can also be applied to
unrestricted balanced truncation so that a relatively short time domain proof
of the $\mathcal H_\infty$-error bound is found here.
