  Cell-free Massive MIMO (multiple-input multiple-output) is a promising
distributed network architecture for 5G-and-beyond systems. It guarantees
ubiquitous coverage at high spectral efficiency (SE) by leveraging signal
co-processing at multiple access points (APs), aggressive spatial user
multiplexing and extraordinary macro-diversity gain.
  In this study, we propose two distributed precoding schemes, referred to as
\textit{local partial zero-forcing} (PZF) and \textit{local protective partial
zero-forcing} (PPZF), that further improve the spectral efficiency by providing
an adaptable trade-off between interference cancelation and boosting of the
desired signal, with no additional front-hauling overhead, and implementable by
APs with very few antennas.
  We derive closed-form expressions for the achievable SE under the assumption
of independent Rayleigh fading channel, channel estimation error and pilot
contamination. PZF and PPZF can substantially outperform maximum ratio
transmission and zero-forcing, and their performance is comparable to that
achieved by regularized zero-forcing (RZF), which is a benchmark in the
downlink. Importantly, these closed-form expressions can be employed to devise
optimal (long-term) power control strategies that are also suitable for RZF,
whose closed-form expression for the SE is not available.
