**(1) Dimension reduction by projection**

The proposed method is targeted to use local phasor measurement unit data from power systems. Thus, estimating the Floquet multiplier is strictly limited to time series data in one-dimensional space. Figure 13a,b shows the three-dimensional intersection in the Poincaré section and their projections onto one-dimensional line and chaotic behavior, respectively. Figure 13a is a typical limit cycle oscillation. Clearly, no distortions occurred. Moreover, when the measured data showed period

doubling, the projection of two intersections onto the three-dimensional hyperplane Ω may start to reveal unintended differences.

**Figure 13.** Projection of three-dimensional Poincaré section into load voltage axis.

Figure 13b shows the results of chaotic behavior. For the sample three-bus power system, distortion occurred for a step of projection. Initially, the three-dimensional intersections were scattered on the space. The projection onto a two-dimensional plane gave different values. Then, the projection onto a line, which was the result for time series data, yielded totally distorted values compared to the original intersection.

The power system with wind generator is a 10-dimensional system. The Poincaré surface would be nine-dimensional, thus projections will be carried out eight times. Then, ignoring the other axes, only the local minima of time series data would give the estimated value of the Floquet multiplier. Therefore, it is natural that the estimated Floquet multiplier is less than the calculated Floquet multiplier.
