**4. Power System with DFIG**

In this section, comparative studies of the calculated and estimated Floquet multipliers for two cases are performed. Starting with the mathematical modeling of a power system with DFIG, the application of the suggested method in test power systems and power system with a complicated DFIG model are studied.

The modeling of a DFIG is not as simple as a general synchronous generator. The state matrix of the system becomes larger, considering not only the characteristics of controllers but also the complexity of the machine. The specified model of the DFIG is provided with network equations in the appendix of [16]. In this paper, Floquet multiplier have been estimated under even more complex system. That is, DFIG model of [16] is added to the three-bus system with induction motor which have been constructed at [15]. The specified model of this system is provided at Appendix A.

Figure 5 shows a three-bus power system network with the DFIG installed at bus 3. Time series voltage data in rms were obtained at the load bus. The Floquet multiplier was estimated for the results of the time domain simulation of *VL*. The bifurcation diagram of power grid with DFIG is represented in Figure 6, which was determined by the mathematical model of the system. The mathematical model for the system containing aerodynamics and electric power system dynamics are given as follows:

**Figure 5.** three-bus power system network with doubly fed induction generator (DFIG).

**Figure 6.** A bifurcation diagram of three-bus power system with DFIG (wind speed 12 m/s).

In Figure 6, from the left Hopf bifurcation point 8.669 MVar to right Hopf bifurcation point 11.33 MVar, periodic solutions exist where the amplitudes of the orbit are large in the middle of the range. Before estimating the Floquet multiplier in the power system with DFIG, the sample three-bus power grid was examined.

Similar to the previous case, the three-bus power grid in [14–16] showed two Hopf bifurcation points. Periodic orbit can be observed in parameter *Q*1 from 10.946 MVar to 11.407 MVar in Figure 7. Time domain simulations were performed in between these two bifurcation points to check nonlinear oscillatory behaviors. The Floquet multiplier was calculated by the monodromy matrix of the system and the corresponding Floquet multiplier was estimated for comparison with the critical multiplier.

**Figure 7.** A bifurcation diagram of sample power system.

### *4.1. Calculated Floquet Multiplier in Sample Power System*

In Figure 8, four modes of the Floquet multiplier are calculated and the trajectories of these modes are described. Mode 1 is always zero, while the other modes move inside the unit circle. When the parameter increases at 10.946 MVar, the critical multiplier in mode 4 stays on the right side of the unit circle until 10.8728 MVar; then it moves to the left side of the circle with a slight change in parameter. This is the point where the periodic solution of the system loses stability. Without stopping, the critical multiplier moves to the left along the real axis approximately −30, and then changes direction and heads to the unit circle. The value when the trajectory meets the unit circle is 11.3874 MVar and moves to the right side of the circle with a small change at 11.3887 MVar.

**Figure 8.** Trajectory of Floquet multiplier for three-bus power system.

### *4.2. Estimated Floquet Multiplier in Sample Power System*

Figure 9a,b shows the rms voltage values at the load bus (left-top), phase portraits of the generator angle and load voltage (left-bottom), and Floquet multiplier estimation (right) for specific parameter values. Stable oscillatory behavior is observed in Figure 9a. The local minima of the rms voltage marked in red circles appear to be steady. The phase portraits on the left-bottom show that the trajectory of the solution is in the attractor. On the right side of the picture, the points marked in red crosses are the values related to *V*[*k* + 1]/*V*[*k*] in Equation (15). The estimated value was determined from the average of these values.

The left-bottom of Figure 9b shows that the periodic solution is still inside the attractor, but the orbit is inconsistent. Figure 9b is obviously an unstable periodic solution such that ratio *V*[*k* + <sup>1</sup>]/*V*[*k*], which are marked in red crosses, are scattered from 0.96 to 1.18. The average of these points is 1.002, marked in blue circles. To summarize the sample cases for Figure 9a ( *Q*1 = 10.931 MVar), the estimated and calculated values are 1 and 1.005, respectively. For Figure 9b ( *Q*1 = 11.378 MVar), the estimated and calculated values are 1.0023 and −2.317, respectively.

**Figure 9.** Floquet multiplier estimation for rms value of load voltage.

To apply Equation (15), at least two local minima are required. We set a three-cycle time interval for the ordinary differential equation (ODE) tool from MATLAB (R2014a, Mathworks, Netic, MA, USA). For the three-cycle time series data of all parameter ranges, the Floquet multipliers were estimated using Equation (15). In Figure 10a, the overall values are not as large as the moduli of the calculated Floquet multipliers. The points marked with blue dots are the points which are near unity, while the points marked with red dots are the points when the deterministic Floquet multiplier goes outside the unit circle. The estimated Floquet multiplier has unity value when the deterministic Floquet multiplier calculated from monodromy matrix is stable (unity), as shown in Figure 10a. Likewise, the estimated Floquet multiplier is greater than unity when the moduli of the deterministic Floquet multiplier are unstable (greater than one), as shown in Figure 10b. The correlation coefficient between two results Figure 10a,b for partial interval are 0.824.

Comparing the two approaches shows that the estimated Floquet multiplier using Equation (15) gave similar information on a short-term signal. The boundary of stability was less likely to be observed in the estimated Floquet multiplier, while the actual calculated Floquet multiplier changed its sign upon losing stability.

**Figure 10.** Floquet multiplier verification for the sample power system.

### *4.3. Comparison between Estimated Floquet Multiplier and Calculated Floquet Multiplier*

Figure 11a shows an estimated (data-driven) Floquet multiplier and Figure 11b shows the calculated (model-based) Floquet multiplier of test power system with DFIG. For a higher-order system, the estimated Floquet multiplier values remained near unity when these were calculated with the monotomy matrix inside the unit circle. When the calculated Floquet multipliers were about to lose their stability, the estimated Floquet multiplier values stayed at approximately 1 from 10.993 to 11.064 MVar. Then, the estimated Floquet multipliers increased in the positive direction although this was not as large as the calculated values. Likewise, the calculated Floquet multipliers dramatically decreased in the negative direction. Once the estimated Floquet multiplier lost stability, it also lost its increment tendency, i.e., stiff growth was observed from 10.604 to 10.635 MVar, 10.781 to 10.807 MVar, and 10.973 to 10.993 MVar. The correlation coefficient between two results Figure 11a,b for partial interval are 0.758. Specifically, the proposed method accurately estimated the Floquet multiplier as long as the oscillation behavior was purely periodical.

**Figure 11.** Floquet multiplier verification for the power system with wind generation.
