**1. Introduction**

Over the past few decades, power systems have experienced significant changes regarding the amount of power consumed as well as the complexity of the network. Voltage instabilities tend to occur in power systems that are heavily loaded or in faulty condition. There are various indicators of voltage instability, which are determined by the generator, load dynamics, and network structure [1]. Nonlinear oscillatory behaviors are usually noticeable even before the voltage collapse occurs. These oscillatory behaviors can be observed with high-resolution devices. After detecting nonlinear oscillatory behavior, the type of oscillation needs to be determined to check whether the condition will be harmful to the power system.

Nonlinear oscillatory behaviors are an intrinsic characteristic of power systems caused by the structure of the system under a specific condition. To analyze nonlinear behavior, system topology and dynamics are necessary to describe the system mathematically. Then, mathematical expressions of the system, such as state equations or system Jacobian matrix, can provide detailed information on the current status of the system. In power systems, practical issues such as special nonlinear oscillatory behavior can be revealed by time series data measurements. However, without the input of other measurement data or conditions, it is hard to assess the system state since the information about the system is limited to local measurements. Fortunately, studies related to limited time series data applications have been conducted using mathematical models.

Based on the mathematical modeling of nonlinear system local stability, calculation of the maximal Lyapunov exponent in time series data was proposed by Wolf et al. [2]. Other researchers improved the calculation efficiency of the maximal Lyapunov exponent in time series data [3,4]. The maximal Lyapunov exponent was also modified for applications in time series data on power systems [5,6]. The maximal Lyapunov exponent is a stability index that is strongly connected to the fluctuation of data. If the maximum Lyapunov exponent of the system is negative (positive), then the nearby system trajectories will converge (diverge) toward each other [7]. In the results of [5], the maximal Lyapunov exponent in time series data was reasonable compared to the original signal of the bus voltage after the fault was cleared. In other words, all the simulations in [5] involved increased oscillation or oscillation with large amplitudes of approximately 0.4 p.u. However, in reality, the maximal Lyapunov exponent in time series data is difficult to apply since it strongly depends on the initial values and data size. For example, when the bus voltage oscillates for a long time interval, the maximal Lyapunov exponent will be different for the initial values and data size selected. When oscillation occurs, the maximal Lyapunov exponent also fluctuates, regardless of whether it is positive or negative, which makes it hard to determine system stability. Therefore, the characteristics or stability of the oscillatory behavior of nonlinear systems differ from local stability, especially for time series data. There are some certain approaches (real-time or near-to-real-time) in large amounts of literature to detect local stability such as positively or negatively damped oscillations (for example, Voltage stability indices or maximum Lyapunov exponent [5,6]). However, there are few existing solutions or approaches in power systems or other applicable engineering field to detect uncertain response as marginal stability (mathematically defined but not practically). However, still marginal stability issues such as sustained oscillation or forced oscillation could lead some damages or instability to power system [8]. So, this paper focuses on the feature of periodic stability of the power system dynamics.

Knowing the type of oscillatory behavior is important to predict how the oscillatory behavior of the system will change. In practical applications, filter-based approaches are often used to detect and classify the oscillation [9]. For mathematically based applications, the stability of nonlinear oscillatory behaviors is determined with a Floquet multiplier. A Floquet multiplier is the eigenvalue of a matrix that gives orbital stability for the periodic solution of the system. A solution is stable when all the calculated moduli of the eigenvalues in the monodromy matrix are below the unity. A monodromy matrix is a matrix that influences whether the periodic solution decays or grows for the initial perturbation [7]. In classical texts, a geometric concept called a Poincaré map has been introduced to discuss some behaviors of periodic solutions in terms of a Floquet multiplier [7]. One study [10] introduced a method for nonlinear time series analysis that provided some guidelines on constructing a Poincaré map from data-based signals. However, the method reduced the phase space dimensionality one at a time to turn the continuous time flow into a discrete time map. It notes that the method is the intersection count and not simply proportional to the original time *t* of the flow. The number of intersections that are counted might be very small, since the parameter is over specific value or interval of unstable region (the chaos), the system may collapse. Therefore, surfaces should be carefully selected or else they will not contain enough information on the signal.

From the perspective of biomechanical engineering, orbital stability is defined using estimated Floquet multipliers from measured data of physical rotation and orbital movement [11–13]. The study by [11] provided a method that generated accurate estimates with noisy experimental data. However, there is no generalized method to choose the proper Poincaré section. Moreover, the verification of the estimated Floquet multiplier against the calculated Floquet multiplier in an actual system model is required.

In this paper, the selection of Poincaré sections for time series dimensions is proposed. Then, the Floquet multiplier is estimated for a one-dimensional Poincaré section. The estimation technique is based on linear regression applied not only to a simple three-bus system but a complex mathematical model of a power system with a wind generator modeled as a doubly fed induction generator (DFIG).

The paper starts with an introduction of the stability of a periodic solution. Then, mathematical concepts of the Poincaré section and Floquet multiplier will be expanded to the proposed method. Comparison of the proposed method and calculated Floquet multipliers will be performed in a test power grid that is integrated with DFIG. Three noteworthy summaries are provided followed by the conclusion.
