Wind turbines are subject to periodic loads that result in a time-varying 'trim' condition. Linearizing the nonlinear turbine dynamics around this trim condition yields a periodic, linear time-varying (PLTV) system. A linear time-invariant (LTI) approximation is typically obtained in two steps. First, the multi-blade coordinate transformation is applied to the PLTV system to obtain a weakly periodic system. Second, the state matrices of the weakly periodic system are averaged over one period to obtain an LTI approximation. This paper presents an alternative approach to construct optimal LTI approximations using convex optimization. It is also shown that the multi-blade coordinate transformation followed by averaging is equivalent to a special case of the proposed convex optimization procedure. The proposed approach is demonstrated on a linearized model of a utility-scale turbine.