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This paper shows that conic programming is an effective tool to solve robust power system stabilizer (PSS) design problems, namely coordinated gain tuning and coordinated phase and gain tuning. Design robustness is achieved by simultaneously considering several operating scenarios. The method is implemented through a sequence of conic programming runs that define a multivariable root locus along which the eigenvalues move. Specifically, the eigenvalues corresponding to the unstable and poorly damped modes are moved to a conic sector in the left half of the s-plane, whereas the eigenvalues corresponding to the well damped modes are constrained to stay within the boundaries of this conic sector. At each step of the solution, the PSS design parameters are restricted in a trust-region such that the computation of the eigenvalue shift based on the residue method holds valid. The proposed method is demonstrated on a 68-bus test system with nine different operating conditions. Comparisons are carried out between conic programming implementations for PSS coordinated gain tuning and for simultaneous tuning of gain and phase characteristics.