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

Authors: Zhen Zhang; Yanhong Liu
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

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.


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## Body

## 1. 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 [1–3]. 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. [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 pth 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 Section2. 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 Figure1 [20], where rs, xs, rr, xr, rm, xm, and U1 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, s is the slip, and I is the stator current.Figure 1
The simplified equivalent circuit of the asynchronous generators.From Figure1, we have(1)I2=U12rs+rr/s2+xs+xr2.The active power consumption in the rotor windings can be formulated as(2)ΔPE=U12rrrs+rr/s2+xs+xr2.The electromagnetic power and the mechanical power of the system are(3)PE=U12rmrm2+xm2+U12rs+rr/srs+rr/s2+xs+xr2,(4)Pe=ΔPE+PE=U12rmrm2+xm2+U12rr+rs+rr/srs+rr/s2+xs+xr2, respectively. Writing the parameters ware as per unit values, we have Te=Pe and(5)Te=U12rmrm2+xm2+U12rr+rs+rr/srs+rr/s2+xs+xr2.Ignoring the high-end items, the Taylor series expansion ofTe at s0 is(6)Te=Te0+dTedsΔs.AssumingR=dTe/ds, we have(7)R=-U12rrrs+rr/s2+xs+xr2s2rs+rr/s2+xs+xr22+2U12rrrr+rs+rr/srs+rr/ss2rs+rr/s2+xs+xr22s=s0.


### 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 Figure2.Figure 2
The mass model of wind turbine with asynchronous generators.Without consideration of the random excitation, the determinate shaft model of the wind turbines can be written as(8)Ttdωtdt=Tw-Tsh,Tgdωrdt=Tsh-Te,dθwdt=ωt-ωrω0,with(9)Tsh=Kθw+Ddθwdt,where ωt is the angle speed of the wind turbine, ω0 is the synchronous angle speed, ωr is the angle speed of the asynchronous generator, Tt is the rotary inertia of the wind turbine, Tg is the rotary inertia of the asynchronous generator, Tw, Tsh, Te are output mechanical torque, shaft torque, and generator electromagnetic torque of wind turbine, respectively, θw is the shaft twist angle, K is the shaft stiffness coefficient, and D 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(10)Ttdωtdt=Tw-Tsh+σ1Wt,Tgdωrdt=Tsh-Te+σ2Wt,dθwdt=ωt-ωrω0, where σ1 and σ2 are the random excitation intensities and W(t) is the standard Wiener process.Substituting (9) into (10), we have the stochastic dynamic model of the asynchronous wind turbine generators as follows:(11)Ttdωtdt=Tw-Kθw-Dωt-ωrω0+σ1Wt,Tgdωrdt=Kθw+Dωt-ωrω0-Te+σ2Wt.


### 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 ofD.Considering the random small excitation and denotingωt=ω0+Δωt,ωr=ω0+Δωr, we have(12)dΔωtdt=1TtTw-Kθw-DΔωt-Δωrω0+σ1TtWt,dΔωrdt=1TgKθw+DΔωt-Δωrω0-Te+σ2TgWt.Noting thatωr=(1-s)ω0 and Δωr=-sω0, we can obtain(13)dΔωtdt=1TtTw-Kθw-DΔωt+sω0+σ1TtWt,dsdt=-1Tgω0Kθw+DΔωt+sω0-Te-σ2Tgω0Wt.When the asynchronous wind turbine generators are working at a stable operating point, we have(14)Tw=Kθw=Te0.Lets=s0+Δs; we can obtain(15)dΔωtdt=-Dω0TtΔωt-Dω0TtΔs-Dω0Tts0+σ1TtWt,dΔsdt=-DTgΔωt-Dω0-RTgω0Δs-DTgs0-σ2Tgω0Wt.So the stochastic state equation of power systems with asynchronous wind turbine generators can be written as(16)dxt=Axtdt+Gdt+A′dBt, where dB(t)=W(t)dt, xt=[ΔωtΔs]T,(17)A=-Dω0Tt-Dω0Tt-DTg-Dω0-RTgω0,G=-Dω0Tts0-DTgs0,A′=σ1Tt-σ2Tgω0.For power systems, the damping coefficientD is always uncertain with some upper and lower bound. Denoting D∈[D_,D¯], where D_ is the lower bound and D¯ is the upper bound of D, respectively, we have A∈[A_,A¯], where(18)A_=-D¯ω0Tt-D¯ω0Tt-D¯Tg-D¯ω0-RTgω0,A¯=-D_ω0Tt-D_ω0Tt-D_Tg-D_ω0-RTgω0. Let(19)A∗=12A_+A¯,A~=12A¯-A_, so we have A=A∗+ΔA and ΔA∈(-A~,A~). The stochastic dynamic interval model of the power systems with asynchronous wind turbine generators can be formulated as(20)dxt=A∗+ΔAxtdt+Gdt+A′dBt.


## 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 Figure1 [20], where rs, xs, rr, xr, rm, xm, and U1 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, s is the slip, and I is the stator current.Figure 1
The simplified equivalent circuit of the asynchronous generators.From Figure1, we have(1)I2=U12rs+rr/s2+xs+xr2.The active power consumption in the rotor windings can be formulated as(2)ΔPE=U12rrrs+rr/s2+xs+xr2.The electromagnetic power and the mechanical power of the system are(3)PE=U12rmrm2+xm2+U12rs+rr/srs+rr/s2+xs+xr2,(4)Pe=ΔPE+PE=U12rmrm2+xm2+U12rr+rs+rr/srs+rr/s2+xs+xr2, respectively. Writing the parameters ware as per unit values, we have Te=Pe and(5)Te=U12rmrm2+xm2+U12rr+rs+rr/srs+rr/s2+xs+xr2.Ignoring the high-end items, the Taylor series expansion ofTe at s0 is(6)Te=Te0+dTedsΔs.AssumingR=dTe/ds, we have(7)R=-U12rrrs+rr/s2+xs+xr2s2rs+rr/s2+xs+xr22+2U12rrrr+rs+rr/srs+rr/ss2rs+rr/s2+xs+xr22s=s0.


## 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 Figure2.Figure 2
The mass model of wind turbine with asynchronous generators.Without consideration of the random excitation, the determinate shaft model of the wind turbines can be written as(8)Ttdωtdt=Tw-Tsh,Tgdωrdt=Tsh-Te,dθwdt=ωt-ωrω0,with(9)Tsh=Kθw+Ddθwdt,where ωt is the angle speed of the wind turbine, ω0 is the synchronous angle speed, ωr is the angle speed of the asynchronous generator, Tt is the rotary inertia of the wind turbine, Tg is the rotary inertia of the asynchronous generator, Tw, Tsh, Te are output mechanical torque, shaft torque, and generator electromagnetic torque of wind turbine, respectively, θw is the shaft twist angle, K is the shaft stiffness coefficient, and D 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(10)Ttdωtdt=Tw-Tsh+σ1Wt,Tgdωrdt=Tsh-Te+σ2Wt,dθwdt=ωt-ωrω0, where σ1 and σ2 are the random excitation intensities and W(t) is the standard Wiener process.Substituting (9) into (10), we have the stochastic dynamic model of the asynchronous wind turbine generators as follows:(11)Ttdωtdt=Tw-Kθw-Dωt-ωrω0+σ1Wt,Tgdωrdt=Kθw+Dωt-ωrω0-Te+σ2Wt.


## 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 ofD.Considering the random small excitation and denotingωt=ω0+Δωt,ωr=ω0+Δωr, we have(12)dΔωtdt=1TtTw-Kθw-DΔωt-Δωrω0+σ1TtWt,dΔωrdt=1TgKθw+DΔωt-Δωrω0-Te+σ2TgWt.Noting thatωr=(1-s)ω0 and Δωr=-sω0, we can obtain(13)dΔωtdt=1TtTw-Kθw-DΔωt+sω0+σ1TtWt,dsdt=-1Tgω0Kθw+DΔωt+sω0-Te-σ2Tgω0Wt.When the asynchronous wind turbine generators are working at a stable operating point, we have(14)Tw=Kθw=Te0.Lets=s0+Δs; we can obtain(15)dΔωtdt=-Dω0TtΔωt-Dω0TtΔs-Dω0Tts0+σ1TtWt,dΔsdt=-DTgΔωt-Dω0-RTgω0Δs-DTgs0-σ2Tgω0Wt.So the stochastic state equation of power systems with asynchronous wind turbine generators can be written as(16)dxt=Axtdt+Gdt+A′dBt, where dB(t)=W(t)dt, xt=[ΔωtΔs]T,(17)A=-Dω0Tt-Dω0Tt-DTg-Dω0-RTgω0,G=-Dω0Tts0-DTgs0,A′=σ1Tt-σ2Tgω0.For power systems, the damping coefficientD is always uncertain with some upper and lower bound. Denoting D∈[D_,D¯], where D_ is the lower bound and D¯ is the upper bound of D, respectively, we have A∈[A_,A¯], where(18)A_=-D¯ω0Tt-D¯ω0Tt-D¯Tg-D¯ω0-RTgω0,A¯=-D_ω0Tt-D_ω0Tt-D_Tg-D_ω0-RTgω0. Let(19)A∗=12A_+A¯,A~=12A¯-A_, so we have A=A∗+ΔA and ΔA∈(-A~,A~). The stochastic dynamic interval model of the power systems with asynchronous wind turbine generators can be formulated as(20)dxt=A∗+ΔAxtdt+Gdt+A′dBt.


## 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.Definition 1 (see [8]).
System (20) is interval p-moment stable, if(21)limt→∞Extp<ε, where ε>0 is a constant. Especially, when p=1 and p=2, the system is interval mean stable and interval mean square stable, respectively.Lemma 2 (see [21]).
LetZ be n×n real symmetric matrix and x be n-dimensional column vector; we have(22)λminZxTx≤xTZx≤λmaxZxTx,where λmax(Z) and λmin(Z) are the maximum and minimum eigenvalue of Z, respectively.Lemma 3 (see [22]).
Consider the system(23)dxt=fx,tdt. If there exists a continuous positive definite function V with infinitesimal upper bound and infinity lower bound such that(24)LV=∂V∂t+∑i=1nfix,t∂V∂xi<0, system (23) is asymptotically stable.Theorem 4.
Suppose there exists a reversible matrixQ such that P=QJQ-1; we have(25)eP≤QQ-1eαB1+Bτ-1+Bη+1+⋯+Bk+2eατBτcos⁡βτ+⋯+eαηBηcos⁡βη︸η-τ+1/2, where (26)λiP=α1,…,ατ-1,ατ+βτj,ατ-βτj,…,αη+αηj,αη-βηj,αη+1,…,αk,α=max⁡α1,…,αk,i=1,…,τ,…,η,…,k≤n,Bi=1112!⋯1ri-1!⋱⋱⋱⋮⋱⋱⋱⋱⋱12!⋱11,ri is the order of the same roots, and η-τ+1 is an even number.Proof.
The Taylor expansion ofφ(x) at λi(P) can be written as(27)φx=φλiP+φ′λiPx-λiP+12!φ′′λiPx-λiP2+⋯+1n!φnλiPx-λiPn+⋯. Assume matrix P is similar to a Jordan matrix J; that is,(28)P~J=J1⋱Jk, where i=1,2,…,k≤n,(29)Ji=λiP1λiP⋱⋱1λiPri×ri=λiPI+H, with H=Ji-λi(A)I, (30)H=010⋱⋱10,H2=0010⋱1⋱00,⋮Hri-1=0ri×ri. Substituting Ji=λi(P)I+H into (27) gives (31)φJi=φλiPI+φ′λiPJi-λiPI+12!φ′′λiPJi-λiPI2+⋯+1n!φnλiPJi-λiPIn+⋯=φλiPI+φ′λiPH+12!φ′′λiPH2+⋯+1n!φnλiPHn+⋯. Let φ(x)=ex and we have(32)eJi=eλiPI+H+12!H2+⋯=eλiPBi, where(33)Bi=1112!⋯1ri-1!⋱⋱⋱⋮⋱⋱⋱⋱⋱12!⋱11.
If there exists a matrixQ such that P=QJQ-1, we have(34)eP=QeJQ-1≤QeJQ-1=QQ-1eλ1B1+⋯+eλkBk.
Assumeλ1,…,λτ-1,λη+1,…,λk are real roots, λτ,…,λη are plurals, and λτ,λτ+1 is a pair of conjugate complex roots of P. It is obvious that λη-1,λη is a pair of conjugate complex roots of P and η-τ+1 is an even number. Let(35)λτ=ατ+βτj,λτ+1=ατ-βτj; then we have(36)eλτBτ=eατ+βτjBτ=eατeβτjBτ,eλτ+1Bτ+1=eατ-βτjBτ+1=eατe-βτjBτ+1. According to the structure of Bi, we have Bτ=Bτ+1 and (37)eλτBτ+eλτ+1Bτ+1=eατBτeβτj+e-βτj=eατBτcos⁡βτ+jsin⁡βτ+cos⁡-βτ+jsin⁡-βτ=2eατBτcos⁡βτ. So eP can be written as (38)eP=QeJQ-1≤QQ-1eλ1B1+⋯+eλτ-1Bτ-1+2eατBττcos⁡β+⋯+2eαη-1Bη-1cos⁡βη-1+eλη+1Bη+1+⋯+eλkBk≤QQ-1eαB1+Bτ-1+Bη+1+⋯+Bk+2eατBτcos⁡βτ+⋯+eαηBηcos⁡βη︸η-τ+1/2; thus the proof is complete.Deduction 1.
For the two order power systems withP=A, we have(39)eA≤1.618QQ-1eα, if A has two same real roots,(40)eA≤2QQ-1eα, if A has two different real roots, and(41)eA≤2cos⁡βQQ-1eα, if A has a pair of conjugate complex roots, where α=max{Re⁡(λi(A))},i=1,2.Proof.
SupposeA has two same real roots; then α=λ1=λ2 and(42)A~J=α1α=λI+H, where(43)I=1001,H=0100,H2=0,B1=1101,B1=1.618. According to Theorem 4, we have (44)eA=QeJQ-1≤QQ-1eJ≤QQ-1eαB=1.618QQ-1eα.
IfA has two real roots with λ1≠λ2, choosing α=max⁡{λ1,λ2}, then we have(45)A~J=λ1λ2. There exists a matrix Q such that A=QJQ-1. So we have (46)eA=QeJQ-1≤QQ-1eJ≤QQ-1eλ1+eλ2≤2QQ-1eλ.
IfA has a pair of conjugate complex roots, denote λi(A)=α±βj and we have (47)eA=QeJQ-1≤QQ-1eJ≤QQ-1eα+βj+eα-βj=QQ-1eαcos⁡β+jsin⁡β+cos⁡-β+jsin⁡-β=2cos⁡βQQ-1eα; thus the proof is complete.Theorem 5.
System (20) is interval mean stable if(48)λmaxA_+A¯+A_+A¯T+2A¯-A_<0, where λmax[(A_+A¯)+(A_+A¯)T] is the maximum eigenvalue of [(A_+A¯)+(A_+A¯)T], and(1)
ifA has two same real roots, we have(49)limt→∞Ext≤1.618QQ-12G2+A′2-2α;(2)
ifA has two different real roots, we have(50)limt→∞Ext≤2QQ-12G2+A′2-2α;(3)
ifA has a pair of conjugate complex roots, we have(51)limt→∞Ext≤2cos⁡βQQ-12G2+A′2-2α.Proof.
Consider the following system:(52)dxt=Axtdt, where A is the same as the system matrix in (20).
Choosing a Lyapunov functionV=xTx, there exist positive constants c1 and c2 such that(53)c1x2≤V=xTx≤c2x2, and V has the infinitesimal upper bound and the infinite upper bound. Moreover, we have (54)LV=2xTAx=2xTA∗x+2xTΔAx=xTA_+A¯+A_+A¯Tx+2xTΔAx≤λmaxA_+A¯+A_+A¯TxTx+2xTΔAx≤λmaxA_+A¯+A_+A¯TxTx+c3xT2+1c3ΔAx2=λmaxA_+A¯+A_+A¯TxTx+c3xTx+1c3xTΔATΔAx≤λmaxA_+A¯+A_+A¯TxTx+c3xTx+1c3ΔA2xTx≤λmaxA_+A¯+A_+A¯TxTx+c3+ΔA2c1c3xTx≤λmaxA_+A¯+A_+A¯TxTx+c3+A~2c1c3xTx=λmaxA_+A¯+A_+A¯TxTx+c3+A¯-A_24c1c3xTx, where c3>0 is a constant. Let c3=1/2(A¯-A_) and c1=1; we have(55)LV≤λmaxA_+A¯+A_+A¯T+2A¯-A_xTx. Thus LV<0 and system (52) is stable under condition (48). Furthermore, we have α<0.
Now consider the stochastic power system (20). The solution process of (20) can be expressed as [8](56)xt=eAtxt0+∫0teAt-ϕGdϕ+∫0teAt-ϕA′dBϕ. Taking expectation of the both sides, we have(57)ExTtxt=EeAtx0TeAtx0+EeAtx0T∫0teAt-ϕGdϕ+EeAtx0T∫0teAt-ϕA′dBϕ+E∫0teAt-ϕGdϕTeAtx0+E∫0teAt-ϕGdϕT∫0teAt-ϕGdϕ+E∫0teAt-ϕGdϕT∫0teAt-ϕA′dBϕ+E∫0teAt-ϕA′dBϕTeAtx0+E∫0teAt-ϕA′dBϕT∫0teAt-ϕGdϕ+E∫0teAt-ϕA′dBϕT∫0teAt-ϕA′dBϕ.
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(58)EeAtx0TeAtx0=eAtx02,EeAtx0T∫0teAt-ϕGdϕ=eAtx0T∫0teAt-ϕGdϕ,E∫0teAt-ϕGdϕTeAtx0=∫0tGTeATt-ϕdϕeAtx0,E∫0teAt-ϕGdϕT∫0teAt-ϕGdϕ=∫0tGTeATt-ϕdϕ∫0teAt-ϕGdϕ=∫0teAt-ϕGdϕ2,E∫0teAt-ϕA′dBϕT∫0teAt-ϕA′dBϕ=∫0tA′TeATt-ϕeAt-ϕA′dϕ=∫0teAt-ϕA′2dϕ,EeAtx0T∫0teAt-ϕA′dBϕ=0,E∫0teAt-ϕGdϕT∫0teAt-ϕA′dBϕ=0,E∫0teAt-ϕA′dBϕTeAtx0=0,E∫0teAt-ϕA′dBϕT∫0teAt-ϕGdϕ=0. Substituting (58) into (57), we can get(59)ExTtxt=eAtx02+eAtx0T∫0teAt-ϕGdϕ+∫0tGTeATt-ϕdϕeAtx0+∫0teAt-ϕGdϕ2+∫0teAt-ϕA′2dϕ.
According to Deduction1, if A has two same real roots, choose c4=1.618QQ-1 and we have(60)eAtx02≤c42e2αtx02. Direct calculation shows that(61)2eAtx0T∫0teAt-ϕGdϕ≤eAtx0TeAtx0+∫0tGTeATt-ϕdϕ∫0teAt-ϕGdϕ=eAtx02+∫0teAt-ϕGdϕ2≤eAtx02+∫0teAt-ϕ2G2dϕ≤c42e2αtx02+G22αe2αt-1,∫0tGTeATt-ϕdϕeAtx0≤12c42e2αtx02+G22αe2αt-1,∫0teAt-ϕGdϕ2≤c42G22αe2αt-1,∫0teAt-ϕA′2dϕ≤c42A′22αe2αt-1.
Substituting (60)-(61) into (59) gives(62)ExTtxt≤c422e2αtx02+G2α+A′22αe2αt-1.
According to Cauchy-Schwarz inequality(63)Ext2≤Ext2=ExTtxt, and noticing that(64)Ext≤ExTtxt, we have(65)limt→∞Ext≤1.618QQ-12G2+A′2-2α.
Similarly, ifA has two different real roots, we have(66)limt→∞Ext≤2QQ-12G2+A′2-2α, and if A has a pair of conjugate complex roots, we get(67)limt→∞Ext≤2cos⁡βQQ-12G2+A′2-2α. So there exists ε1>0 such that(68)limt→∞Ext<ε1, and system (20) is interval mean stable. Thus the proof is complete.Theorem 6.
System (20) is interval mean square stable, if(69)λmaxA_+A¯+A_+A¯T+2A¯-A_<0. Moreover,(1)
ifA has two the same real roots, we have(70)limt→∞Ext2≤1.309Q2Q-122G2+A′2-α;(2)
ifA has two different real roots, we have(71)limt→∞Ext2≤2Q2Q-122G2+A′2-α;(3)
ifA has a pair of conjugate complex roots, we get(72)limt→∞Ext2≤2cos2⁡βQ2Q-122G2+A′2-α.Proof.
The proof is similar to Theorem5 and is omitted here.Remark 7.
Considern-machine stochastic power systems described by(73)Ttidωtidt=Twi-Kiθwi-Diωti-ωriω0+σ1iWt,Tgidωridt=Kiθwi+Diωti-ωriω0-Tei+σ2iWt.
The system can be reformulated as(74)dXt=A^Xtdt+G^dt+A^′dBt, where X(t) is 2n×1 state matrix, A^ is 2n×2n coefficient matrix, G^ is 2n×1 gain matrix, and A^′ is 2n×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.


## 4. Simulation

In this section, we simulate the power system to verify the effectiveness of the proposed results. The parameters of the system are chosen asD∈[1.35p.u.,1.65p.u.], Tt = 10 p.u., Tg = 2.5 p.u., ω0 = 1 p.u., s0 = 0.08 p.u., rs = 0.003 p.u., xs = 0.125 p.u., rr = 0.004 p.u., xr = 0.05 p.u., xm = 2.5 p.u., and U1 = 10.52 p.u. and we can get R=-7.23,(75)A_=-0.165-0.165-0.66-3.552,A¯=-0.135-0.135-0.54-3.432,λmaxA_+A¯+A_+A¯T+2A¯-A_=-0.129<0.According to Theorems5 and 6, we know that the system is interval mean stability and interval mean square stability.ChoosingD=1.45p.u.∈[1.35p.u.,1.65p.u.], we can get(76)A=-0.145-0.145-0.58-3.472,J=-0.119900-0.34971,Q=0.98540.0432-0.17050.9991,Q-1=1.0073-0.04360.17190.9935,G=-0.0116-0.0464T,A′=0.1σ1-0.4σ2T,G2=0.0022875,A′2=0.01σ12+0.16σ22,Q=1.0620,Q-1=1.0705,λ1A=-0.1199,λ2A=-0.34971, so α=-0.1199. According to Theorems 5 and 6, the theoretical bounds of mean and mean square error with different σ1 and σ2 are shown in Table 1 when t→∞. The response of the system states with different σ1 and σ2 is shown in Figures 3–5. The bounds of mean and mean square error with different σ1 and σ2 are shown in Table 2, where x1tmax and x2tmax denote the absolute maximum values of stable states Δωt and Δs, respectively.Table 1
The theoretical bounds of mean and mean square error.
σ
1
σ
2
E
x
t
E
x
t
2
0.002
0.002
0.313327
0.098173
0.003
0.003
0.313376
0.098204
0.004
0.004
0.313416
0.098230
0.005
0.005
0.313347
0.098263
0.020
0.020
0.315643
0.099631
0.030
0.030
0.318520
0.101454
0.040
0.040
0.322503
0.104008
0.050
0.050
0.327553
0.107291
0.500
0.500
0.456192
0.208111
1.000
1.000
0.928481
0.862078Table 2
The simulation bounds of mean and mean square error.
Corresponding figure
σ
1
σ
2
x
1
t
m
a
x
x
2
t
m
a
x
E
x
(
t
)
E
x
(
t
)
2
(
a
1
)
0.002
0.002
0.080190
0.000792
0.080212
0.006431
(a2)
0.003
0.003
0.080490
0.001105
0.080498
0.006480
(a3)
0.004
0.004
0.080980
0.001229
0.080994
0.006560
(a4)
0.005
0.005
0.081290
0.002478
0.081328
0.006614
(a5)
0.020
0.020
0.082080
0.006803
0.082359
0.006783
(a6)
0.030
0.030
0.086460
0.009684
0.087000
0.007569
(a7)
0.040
0.040
0.087030
0.021220
0.089582
0.008025
(a8)
0.050
0.050
0.093510
0.026570
0.097211
0.009450
(a9)
0.500
0.500
0.193300
0.265300
0.328251
0.107749
(a10)
1.000
1.000
0.377300
0.435100
0.575905
0.331667Figure 3
Response of system with small random excitation.Figure 4
Response of system with large random excitation.Figure 5
Response of system with larger random excitation.From Figure3, 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.


## 5. 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.



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*Source: 7617394-2018-01-11.xml*