*3.2. Floquet Multiplier Estimation*

In Section 2.3, the matrix *∂P*(*q*<sup>∗</sup>)/*∂q* is provided by the monodromy matrix with restriction to the *n* − 1-dimensional hyperspace Ω. The *<sup>P</sup>*(*q*) can be expressed by Taylor series in the vicinity of the fixed point *q*<sup>∗</sup>. Assuming that periodic solution is stable, higher-order term can be negligible. Recalling that *<sup>P</sup>*(*q*<sup>∗</sup>) = *q*∗ at the fixed point, Equation (10) is linearized as

$$P(q) - q^\* = [\frac{\partial P(q^\*)}{\partial q}](q - q^\*)\tag{11}$$

For the specific vector *q*1, the corresponding critical eigenvalue *μ*1 is calculated using the linear approximated equation

$$P(q\_1) - q^\* = \mu\_1 (q\_1 - q^\*) \tag{12}$$

If we have series data *x*, the relationship between the mapping *P* of point *q* and point *q* can be followed by sequence *x*[*k*] and *x*[*k* + 1]. Hence, the computational form gives Equation (11) by

$$\mathbf{x}[k+1] - \mathbf{x}^\* = \mathfrak{A}(\mathbf{x}[k] - \mathbf{x}^\*) \tag{13}$$

Equation (12) is a typical form of linear regression, where *μ*ˆ is a constant to be determined by series data *x*.

For application to time series data, the equation should be modified to a scalar form. In that case, the calculated Floquet multiplier might not be the same as the actual value since the scalar value cannot reflect the direction. However, there will be negligible differences when selecting the acceptable component of the state variable. Using engineering judgment, it will be natural to select the component with the largest variance among the state variables *x*.

$$s[k+1] - s^\* = \pounds\_F(s[k] - s^\*) \tag{14}$$

The estimated Floquet multiplier *μ*ˆ *F* can be easily computed with Equation (10). The estimated value, however, is not the exact value of the calculated Floquet multiplier because it is estimated and projected in one dimension, regardless of the number of original system state variables it contains. But as long as the signal is periodic, the estimated Floquet multiplier gives information on stability even if we choose single measurement as a state variable.

For application to time series data, Equation (13) can be expressed after changing scalar *s* to voltage term *V*

$$\hat{\mu}\_{F,series} = \frac{1}{N} \sum\_{k=1}^{N} \frac{V[k+1] - (1 - \mu)V^\*}{V[k]} \tag{15}$$

Ironically, the linearized form of Equation (13) contradicts the stable orbits when *s*[*k* + 1] ≈ *s*[*k*] ≈ *<sup>s</sup>*<sup>∗</sup>. Hence, the *μ* term of Equation (14) should be 1 by applying the stable orbit condition.

$$\hat{\mu}\_{F,series} = \frac{1}{N} \sum\_{k=1}^{N} \frac{V[k+1]}{V[k]} \tag{16}$$

Then, the final estimation Equation (15) further simplifies the problem such that the estimated Floquet multiplier is only the average of the ratio of the previous and current local minima of the voltage at all peak data. In other words, we can set the fixed-point value to zero. It can be guaranteed that *μ*ˆ *F*,*series* = 1 is a sufficient condition for stable orbit. Using final Equation (15), other unstable cases are demonstrated by two examples of power systems in Section 4.
