  Cell-Free Massive multiple-input multiple-output (MIMO) and reconfigurable
intelligent surface (RIS) are two promising technologies for application to
beyond-5G networks. This paper considers Cell-Free Massive MIMO systems with
the assistance of an RIS for enhancing the system performance under the
presence of spatial correlation among the scattering elements of the RIS.
Distributed maximum-ratio processing is considered at the access points (APs).
We introduce an aggregated channel estimation approach that provides sufficient
information for data processing with the main benefit of reducing the overhead
required for channel estimation. The considered system is studied by using
asymptotic analysis which lets the number of APs and/or the number of RIS
elements grow large. A lower bound for the channel capacity is obtained for a
finite number of APs and scattering elements of the RIS, and closed-form
expressions for the uplink and downlink ergodic net throughput are formulated
in terms of only the channel statistics. Based on the obtained analytical
frameworks, we unveil the impact of channel correlation, the number of RIS
elements, and the pilot contamination on the net throughput of each user. In
addition, a simple control scheme for controlling the configuration of the
scattering elements of the RIS is proposed, which is shown to increase the
channel estimation quality, and, hence, the system performance. Numerical
results demonstrate the effectiveness of our system design and performance
analysis. In particular, the performance benefits of using RISs in Cell-Free
Massive MIMO systems are confirmed, especially if the direct links between the
APs and the users are of insufficient quality with a high probability.
