  Reconfigurable intelligent surface (RIS) can be employed in a cell-free
system to create favorable propagation conditions from base stations (BSs) to
users via configurable elements. However, prior works on RIS-aided cell-free
system designs mainly rely on the instantaneous channel state information
(CSI), which may incur substantial overhead due to extremely high dimensions of
estimated channels. To mitigate this issue, a low-complexity algorithm via the
two-timescale transmission protocol is proposed in this paper, where the joint
beamforming at BSs and RISs is facilitated via alternating optimization
framework to maximize the average weighted sum-rate. Specifically, the passive
beamformers at RISs are optimized through the statistical CSI, and the transmit
beamformers at BSs are based on the instantaneous CSI of effective channels. In
this manner, a closed-form expression for the achievable weighted sum-rate is
derived, which enables the evaluation of the impact of key parameters on system
performance. To gain more insights, a special case without line-of-sight (LoS)
components is further investigated, where a power gain on the order of
$\mathcal{O}(M)$ is achieved, with $M$ being the BS antennas number. Numerical
results validate the tightness of our derived analytical expression and show
the fast convergence of the proposed algorithm. Findings illustrate that the
performance of the proposed algorithm with two-timescale CSI is comparable to
that with instantaneous CSI in low or moderate SNR regime. The impact of key
system parameters such as the number of RIS elements, CSI settings and Rician
factor is also evaluated. Moreover, the remarkable advantages from the adoption
of the cell-free paradigm and the deployment of RISs are demonstrated
intuitively.
