Cell-free massive multiple-input multiple-output (MIMO) systems, leveraging tight cooperation among wireless access points, exhibit remarkable signal enhancement and interference suppression capabilities, demonstrating significant performance advantages over traditional cellular networks. This paper investigates the performance and deployment optimization of a user-centric scalable cell-free massive MIMO system with imperfect channel information over correlated Rayleigh fading channels. Based on the large-dimensional random matrix theory, this paper presents the deterministic equivalent of the ergodic sum rate for this system when applying the local partial minimum mean square error (LP-MMSE) precoding method, along with its derivative with respect to the channel correlation matrix. Furthermore, utilizing the derivative of the ergodic sum rate, this paper designs a Barzilai-Borwein based gradient descent method to improve system deployment. Simulation experiments demonstrate that under various parameter settings and large-scale antenna configurations, the deterministic equivalent of the ergodic sum rate accurately approximates the Monte Carlo ergodic sum rate of the system. Furthermore, the deployment optimization algorithm effectively enhances the ergodic sum rate of this system by optimizing the positions of access points.