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.
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 [1–3]. 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 [4–6].
In recent years, a lot of researches have been done on the stability analysis and feedback control of stochastic systems; see [7–10] and the references therein. For power systems under stochastic excitations, Humberto et al.  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 . 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 , 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 . 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 . 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.
2. Stochastic Dynamic Model of Power Systems with Asynchronous Wind Turbine Generators
2.1. 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 , where , , , , , , and 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 respectively. Writing the parameters ware as per unit values, we have and
Ignoring the high-end items, the Taylor series expansion of at is
Assuming , we have
2.2. Stochastic Model of Asynchronous Wind Turbine Generators
The asynchronous wind power generation systems is mainly composed of the wind turbine, gear box, and asynchronous generator, as shown in Figure 2.
Without consideration of the random excitation, the determinate shaft model of the wind turbines can be written aswithwhere is the angle speed of the wind turbine, 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, , , 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 and are the random excitation intensities and is the standard Wiener process.
2.3. 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 , we have
Noting that and , we can obtain
When the asynchronous wind turbine generators are working at a stable operating point, we have
Let ; 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 and . The stochastic dynamic interval model of the power systems with asynchronous wind turbine generators can be formulated as
3. 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 ). Let be real symmetric matrix and be -dimensional column vector; we havewhere and are the maximum and minimum eigenvalue of , respectively.
Theorem 4. Suppose there exists a reversible matrix such that ; we have where is the order of the same roots, and 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 , with , Substituting into (27) gives Let and we have whereIf there exists a matrix such that , we haveAssume are real roots, are plurals, and is a pair of conjugate complex roots of . It is obvious that is a pair of conjugate complex roots of and is an even number. Let then we have According to the structure of , we have and So can be written as thus the proof is complete.
Deduction 1. For the two order power systems with , we have if has two same real roots, if has two different real roots, and if has a pair of conjugate complex roots, where .
Proof. Suppose has two same real roots; then and where According to Theorem 4, we have If has two real roots with , choosing , then we have There exists a matrix such that . So we have If has a pair of conjugate complex roots, denote and we have thus the proof is complete.
Theorem 5. System (20) is interval mean stable if where is the maximum eigenvalue of , and(1) if has two same real roots, we have(2) if has two different real roots, we have(3) if has a pair of conjugate complex roots, we have
Proof. Consider the following system: where is the same as the system matrix in (20).
Choosing a Lyapunov function , there exist positive constants and such that and has the infinitesimal upper bound and the infinite upper bound. Moreover, we have where is a constant. Let and ; we have Thus and system (52) is stable under condition (48). Furthermore, we have .
Now consider the stochastic power system (20). The solution process of (20) can be expressed as  Taking expectation of the both sides, we haveNoticing 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 getAccording to Deduction 1, if has two same real roots, choose and we have Direct calculation shows thatSubstituting (60)-(61) into (59) givesAccording to Cauchy-Schwarz inequality and noticing that we haveSimilarly, if has two different real roots, we have and if has a pair of conjugate complex roots, we get So there exists such that and system (20) is interval mean stable. Thus the proof is complete.
Theorem 6. System (20) is interval mean square stable, if Moreover,(1) if has two the same real roots, we have(2) if has two different real roots, we have(3) if has a pair of conjugate complex roots, we get
Proof. The proof is similar to Theorem 5 and is omitted here.
Remark 7. Consider -machine stochastic power systems described byThe system can be reformulated as where is state matrix, is coefficient matrix, is gain matrix, and is 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.
In this section, we simulate the power system to verify the effectiveness of the proposed results. The parameters of the system are chosen as , = 10 p.u., = 2.5 p.u., = 1 p.u., = 0.08 p.u., = 0.003 p.u., = 0.125 p.u., = 0.004 p.u., = 0.05 p.u., = 2.5 p.u., and = 10.52 p.u. and we can get ,
Choosing , we can get so . According to Theorems 5 and 6, the theoretical bounds of mean and mean square error with different and are shown in Table 1 when . The response of the system states with different and is shown in Figures 3–5. The bounds of mean and mean square error with different and are shown in Table 2, where and denote the absolute maximum values of stable states and , respectively.
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.
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  and feedback controller design  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.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This work is supported by the National Natural Science Foundation of China (61473265, 61603345, and 61773351) and the Innovation Research Team of Science & Technology of Henan Province (17IRTSTHN013).
Y. Zhou, Y. Li, W. Liu, D. Yu, Z. Li, and J. Liu, “The stochastic response surface method for small-signal stability study of power system with probabilistic uncertainties in correlated photovoltaic and loads,” IEEE Transactions on Power Systems, vol. 32, no. 6, pp. 4551–4559, 2017.View at: Publisher Site | Google Scholar
V. Humberto, K. Woifgang, V. Luis et al., “Stability region and radius in electric power systems under sustained random perturbations,” International Journal of Electrical Power & Energy Systems, vol. 73, pp. 725–733, 2015.View at: Google Scholar
S. Q. Bu, W. Du, H. F. Wang, Z. Chen, L. Y. Xiao, and H. F. Li, “Probabilistic analysis of small-signal stability of large-scale power systems as affected by penetration of wind generation,” IEEE Transactions on Power Systems, vol. 27, no. 2, pp. 762–770, 2012.View at: Publisher Site | Google Scholar
B. Wang and Q. X. Zhu, “Stability analysis of Markov switched stochastic differential equations with both stable and unstable subsystems,” Systems & Control Letters, vol. 105, pp. 55–61, 2017.View at: Google Scholar
V. Humberto, V. Luis, K. Woifgang et al., “Stability of linear stochastic systems via Lyapunov exponents and applications to power systems,” Applied Mathematics & Computation, vol. 218, no. 22, pp. 11021–11032, 2012.View at: Google Scholar
Q. X. Zhu and Q. Y. Zhang, “pth moment exponential stabilisation of hybrid stochastic differential equations by feedback controls based on discrete-time state observations with a time delay,” IET Control Theory & Applications, vol. 11, no. 12, pp. 1992–2003, 2017.View at: Google Scholar
Z. H. Lu, W. X. Zhao, and D. Xie, “P-moment stability of power system under small Gauss type random excitation,” Chaos Solitons Fractals the Interdisciplinary Journal of Nonlinear Science Nonequilibrium Complex Phenomena, vol. 81, pp. 30–37, 2015.View at: Google Scholar
Z. N. Wei, Q. Yan, G. Q. Sun et al., “Power system interval linear state estimation considering network parameter uncertainty,” Journal of the Acm, vol. 2, no. 2, pp. 99–110, 2015.View at: Google Scholar
G. Q. Li. Power System, Power System Trans.ient Analysis Inequality, Water Power Press, Beijing, China, 1985.
S. G. Wang, M. X. Wu, and Z. Z. Jia, Matrix Inequality, Science Press, Beijing, China, 2006, Matrix Inequality, Science Press.
Z. E. Ma and Y. C. Zhou, Qualitative and Stability for Ordinary Differential Equations, Science Press, Beijing, China, 2005.
H. Wang and Q. X. Zhu, “Global Stabilization of Stochastic Nonlinear Systems via C1 and C8 Controllers,” IEEE Transactions on Automatic Control, vol. 62, no. 11, pp. 5880–5887, 2017.View at: Google Scholar