Stochastic Small Signal Interval Stability of Power Systems with Asynchronous Wind Turbine Generators

The stochastic dynamic interval model of power systems with asynchronous wind turbine generators is established with consideration of the interval uncertain parameters and random small excitation disturbances. The conditions of interval mean stability and interval mean square stability of the power systems are proposed. The relationship between the bounds of the mean (mean square) error and the parameter interval range of the systems is discussed. Finally, we simulate the power systems to demonstrate the effectiveness of the proposed results.


Introduction
With the rapid development of wind power and other emerging new energy and grid, the random excitation has a significant impact on the power systems stability and power quality [1][2][3].In addition, the system parameters are generally uncertain because of the effects of the temperature, humidity, and other environmental conditions and the restriction of the information collection technology.It is of great importance to consider the influence of the stochastic excitation and uncertain parameters in the stability analysis of power systems [4][5][6].
In recent years, a lot of researches have been done on the stability analysis and feedback control of stochastic systems; see [7][8][9][10] and the references therein.For power systems under stochastic excitations, Humberto et al. [11] discussed the almost sure asymptotic stability of linear stochastic systems with bounded Markov diffusion process perturbation by the means of Lyapunov exponents and applied the theoretical results to one machine infinite bus electric power systems.Ma et al. established a continuous Markov power system model with multiple operating conditions considering the stochastic characteristic of wind speed and proposed a Lyapunov function based method for the stability analysis of the system [4].Robust stochastic stability of power systems under stochastic excitation and random perturbations was discussed in [12,13].Noticing that the mean and mean square stability [14,15] are of great importance for power systems, the small signal mean stability and mean square stability of power systems under random excitation were investigated by some researchers.In [2], Zhang et al. put forward a stochastic differential equation model for power system under Gauss type random excitation and proposed some conditions for the mean stability and mean square stability of the system.Yuan et al. further investigated the the steady-state expectation and covariance of the system state variables of the power systems with asynchronous wind turbines [16].For a general case of mean stability and mean square stability, Lu et al. discussed the th stochastic stability and dynamic characteristics of power systems under small Gauss type random excitations [17].For power systems whose network parameters are being known to be within certain bounds, under the assumption of with mixed phasor and conventional power measurements, some state estimation methods were put forward based on the interval linear system model [18,19].So far, there have been no researches carried out for the stochastic mean stability and mean square stability analysis of power systems with both consideration of random excitation disturbances and interval parameter uncertainties.
In this paper, we discuss the stochastic small signal stability of power systems with wind turbines and parameter uncertainties.First, the stochastic interval dynamic model of the power systems is established with consideration of the interval parameter uncertainties and small random excitations.Then, we propose some results for the mean stability and mean square stability by analyzing the eigenvalues of the system matrix.The relationship between the bounds of the mean and the mean square error and the small random excitation and interval uncertain parameters is also discussed.Finally, we simulate a single machine infinite bus power system with asynchronous wind turbine Generator to verify the effectiveness of the proposed method.The paper is organized as follows.First, the stochastic dynamic interval model of power system is established considering interval uncertain parameters under small random excitation in Section 2. The interval mean stability and interval mean square stability of the system are given in Section 3. Section 4 gives the simulation of power systems.Finally, conclusions are drawn in the last section.

Dynamic Model of Asynchronous Generators.
Ignoring the electromagnetic transient process of asynchronous generator, the simplified equivalent circuit of the asynchronous generator can be shown in Figure 1 [20], where   ,   ,   ,   ,   ,   , and  1 are the stator resistance, the stator reactance, the rotor resistance, the rotor reactance, the excitation resistance, the excitation reactance, and the voltage of the asynchronous generator, respectively,  is the slip, and  is the stator current.
From Figure 1, we have The active power consumption in the rotor windings can be formulated as The electromagnetic power and the mechanical power of the system are Wind turbine

Gear box
Asynchronous generator respectively.Writing the parameters ware as per unit values, we have   =   and Ignoring the high-end items, the Taylor series expansion of   at  0 is Assuming  =   /, we have with where   is the angle speed of the wind turbine,  0 is the synchronous angle speed,   is the angle speed of the asynchronous generator,   is the rotary inertia of the wind turbine,   is the rotary inertia of the asynchronous generator,   ,  sh ,   are output mechanical torque, shaft torque, and generator electromagnetic torque of wind turbine, respectively,   is the shaft twist angle,  is the shaft stiffness coefficient, and  is the damping coefficient.
Considering the power fluctuations of the wind turbine and the asynchronous generator, which can be regarded as Gaussian random small excitation in a short time [9,10], the stochastic dynamic model of the asynchronous wind turbine generators can be formulated as where  1 and  2 are the random excitation intensities and () is the standard Wiener process.Substituting ( 9) into (10), we have the stochastic dynamic model of the asynchronous wind turbine generators as follows:

Stochastic Interval Model of Power Systems with Wind
Turbines.In this section, the stochastic dynamic interval model of power systems with asynchronous wind turbine generators is established with consideration of the parameter uncertainty of .
Considering the random small excitation and denoting   =  0 + Δ  ,   =  0 + Δ  , we have Noting that   = (1−) 0 and Δ  = − 0 , we can obtain When the asynchronous wind turbine generators are working at a stable operating point, we have Let  =  0 + Δ; we can obtain So the stochastic state equation of power systems with asynchronous wind turbine generators can be written as where For power systems, the damping coefficient  is always uncertain with some upper and lower bound.Denoting  ∈ [, ], where  is the lower bound and  is the upper bound of , respectively, we have  ∈ [, ], where Let so we have  =  * + Δ and Δ ∈ (− Ã, Ã).The stochastic dynamic interval model of the power systems with asynchronous wind turbine generators can be formulated as

Stochastic Small Signal Interval Stability Analysis of the Power Systems
In this section, we analyze the interval mean stability and interval mean square stability of the power systems and establish the relationship between the bounds of the mean and the mean square error and the system uncertain parameter and random excitation.
Lemma 2 (see [21]).Let  be  ×  real symmetric matrix and  be -dimensional column vector; we have where  max () and  min () are the maximum and minimum eigenvalue of , respectively.
Lemma 3 (see [22]).Consider the system If there exists a continuous positive definite function  with infinitesimal upper bound and infinity lower bound such that system ( 23) is asymptotically stable. where is the order of the same roots, and −+1 is an even number.
Proof.The Taylor expansion of () at   () can be written as Assume matrix  is similar to a Jordan matrix ; that is, where  = 1, 2, . . .,  ≤ , with . . .
Proof.Suppose  has two same real roots; then  =  1 =  2 and where 2 = 0, (51) Proof.Consider the following system: where  is the same as the system matrix in (20).Choosing a Lyapunov function  =   , there exist positive constants  1 and  2 such that Thus  < 0 and system (52) is stable under condition (48).Furthermore, we have  < 0. Now consider the stochastic power system (20).The solution process of ( 20) can be expressed as [8]  () =    ( 0 ) + ∫ (56) Taking expectation of the both sides, we have Noticing that the expectation of the nonrandom variable is equal to itself and the expectation of the random variable is equal to 0, we have Substituting (58) into (57), we can get According to Deduction 1, if  has two same real roots, choose  4 = 1.618‖‖‖ −1 ‖ and we have Direct calculation shows that Substituting ( 60)-( 61) into (59) gives According to Cauchy-Schwarz inequality (73) The system can be reformulated as where () is 2 × 1 state matrix, Â is 2 × 2 coefficient matrix, Ĝ is 2 × 1 gain matrix, and Â is 2 × 1 random excitation matrix.From the proof procedure of Theorems 5 and 6, we can see that similar results can be obtained for the interval mean and mean square stability of multimachine stochastic power systems.

Simulation
In this section, we simulate the power system to verify the effectiveness of the proposed results.The parameters of the system are chosen as  ∈ [ From Figure 3, we can see that the system states can converge fast to the stable operation under small excitation and the system is stable.When the random excitation intensity       increases, we can see from Figure 4 that the fluctuations of the system states increase compared to Figure 3.As shown in Figure 5, when we select a random excitation intensity largely, the system states fluctuate seriously and the system is unstable.

Conclusions
The paper established a stochastic interval dynamic model and analyzed the interval mean stability and interval mean square stability of power systems with asynchronous wind turbine generators.The relationship between the bounds of the mean and the mean square error of the system and the parameter interval range was discussed under random excitation.Theoretical analysis and simulation show that, for stochastic power systems with interval parameter uncertainties, the bounds of the mean and the mean square error are directly related to the random excitation.The power system will be unstable if the random excitation is large enough.Noticing that the mean square exponential stability analysis [23] and feedback controller design [24] for stochastic power systems are of great importance, in the future, we will carry out researches on the mean square exponential stability analysis of power systems with asynchronous wind turbine generators and design feedback controllers to improve the stochastic stability of the system under large random excitations.

Figure 1 :
Figure 1: The simplified equivalent circuit of the asynchronous generators.

Figure 2 :
Figure 2: The mass model of wind turbine with asynchronous generators.

Figure 3 :
Figure 3: Response of system with small random excitation.

Figure 4 :Figure 5 :
Figure 4: Response of system with large random excitation.

Table 1 :
The theoretical bounds of mean and mean square error.

Table 2 :
The simulation bounds of mean and mean square error.