# Nonlinear Dynamics of a PI Hydroturbine Governing System with Double Delays

Authors: Hongwei Luo; Jiangang Zhang; Wenju Du; Jiarong Lu; Xinlei An
Journal: Journal of Control Science and Engineering
 (2017)

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## Abstract

A PI hydroturbine governing system with saturation and double delays is generated in small perturbation. The nonlinear dynamic behavior of the system is investigated. More precisely, at first, we analyze the stability and Hopf bifurcation of the PI hydroturbine governing system with double delays under the four different cases. Corresponding stability theorem and Hopf bifurcation theorem of the system are obtained at equilibrium points. And then the stability of periodic solution and the direction of the Hopf bifurcation are illustrated by using the normal form method and center manifold theorem. We find out that the stability and direction of the Hopf bifurcation are determined by three parameters. The results have great realistic significance to guarantee the power system frequency stability and improve the stability of the hydropower system. At last, some numerical examples are given to verify the correctness of the theoretical results.


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## Body

## 1. Introduction

It is well known that the hydroturbine system is the nonminimum phase system and is operated in a complex condition, constituted by controller and governor. The parameters of the hydroturbine system would change significantly under different operating conditions. According to specific goals, controlled system is researched within a given range. There exist two regulators: PID regulator (Proportion Integration Differentiation regulator) and the soft type feedback regulator (proportional-integral regulator), respectively [1]. Although the characteristics of the PID regulator are simple structure and adaptable and easily adjusting parameters, the regulation law of PID is not efficient. PI regulator is a closed-loop system with phase lag, which can reach steady state by using parameter setting. The PI regulator possesses the properties of the optimal regulator and good robustness. Furthermore, the PI regulator is easy to use in the field, similar to the parameter setting method of the PID regulator. At present, the PI regulator plays a vital role in maintaining the stability of electrical systems and is widely used in China [2, 3]. On account of lacking systematic management, it is a challenge to maintain the stability of a large hydroelectric station [4–6]. Many efforts are focused on constructing different mathematical models of the hydroturbine governing system and analyzing the stability and the bifurcation phenomena [7–16]; for example, Ling and Tao [13] analyzed the stability and the bifurcation phenomena of a proportional-integral- (PI-) (controller) type speed hydroturbine governing system with saturation.Over the past one decade, many researchers have paid great attention to analyzing dynamic characteristic when the parameters of the hydroturbine systems are changed. For instance, using PI controller, Silva et al. [17] have revealed the problem of stabilizing of a first-order plant with time delay and obtained the stabilizing PI gain values. Shu and Pi [18] introduced a PID neural network (PIDNN) with control time delay and gave examples of analysis. Strah et al. [19] designed a speed and active power controller of hydroturbine units; some controller parameters were obtained. Li and Zhou [20] developed a gravitational search algorithm (GSA), which was applied to parameter identification of the hydraulic turbine governing system (HTGS), and analyzed the stability of the power system. Jiang et al. [5] proposed a deterministic chaotic mutation evolutionary programming (DCMEP) method to efficiently optimize the PID parameters of the hydroturbine governing systems. Utilizing a maximum peak resonance specification method, a new PID controller for automatic generation control (AGC) of hydroturbine power systems was presented by Khodabakhshian and Hooshmand [21]. On the basis of necessary and sufficient condition, Liu et al. [22] proposed a new method to analyze the stability of automatic generation control (AGC) systems with commensurate delays. Zhang et al. [23] analyzed a PID-type load frequency control (LFC) scheme by using delay-dependent robust method. Based on state space equations, Chen et al. [24] studied the nonlinear dynamical behaviors of a novel hydroturbine mode with the effect of the surge tank. Xu et al. [25] proposed a Hamiltonian model of the hydroturbine governing system, which included fractional item and time-lag, and explored the effect of the fractional item and the time-lag on the dynamic variables of the hydroturbine governing system. Wang et al. [26] studied a novel fractional-order Francis hydroturbine governing system with time delay and verified the effects of the fractional item and time delay on the system by the principle of statistical physics, respectively. The stability and Hopf bifurcation of a Goodwin model with four different delays were investigated by Zhang et al. [27]. Zhang et al. [28] analyzed a hydroturbine governing system in the process of load rejection transient and got the stable regions of the hydroturbine governing system by means of numerical simulations.However, the existence of a Hopf bifurcation is rarely reported in proportional-integral (PI) type hydroturbine generating system with time delay. In the paper, we generate a PI hydroturbine governing system with saturation and double delays. The nonlinear dynamic behavior of the system is analyzed. The scope of some parameter values is obtained to maintain the stability of the system, which has great realistic significance in a small hydropower station.The basic structure of the rest paper is as follows. In Section2, we present a new PI hydroturbine governing system, which is affected by the speed control delay of the generator and the displacement-control delay of the servomotor. In Section 3, the stability of equilibrium points and Hopf bifurcation for PI hydroturbine governing system are investigated in the four different delay cases, respectively. The stability and direction of the Hopf bifurcation are illustrated in Section 4. Numerical simulations are given to support our theory by Matlab software in Section 5. Finally, a brief discussion is given in Section 6.


## 2. Model Description

We study a PI type hydroturbine governing system with saturation and time delay. The structure of the hydroturbine governing system is shown in Figure1.Figure 1
Structure of the hydroturbine governing system.The transfer function of soft feedback regulator is(1)GPIs=KP+KIs.We use an approximate linearization approach for the hydroturbine governing system, which is a first-order mathematical model. Moreover the system is set in small perturbation. Therefore, we have(2)Gts=ey1-eqyeh/ey-eqhTωs1+eqhTωs,Gss=1Tas+eg-ex,where Gt(s) is used in a nonelastic water column model [29, 30]. Gs(s) is the transfer function from the hydroturbine moment to the speed of the generator. eqy is servomotor stroke transfer coefficient of the flow rate. ey is servomotor stroke transfer coefficient of the turbine torque. eh is the transfer coefficient of the torque on the water head of a turbine. eqh is the transfer coefficient of the flow rate on the water head of a turbine. Tω is the water inertia time constant of a pressure guide-water system. s is the strength of the elastic water hammer effect. Ta is the sum of the machine starting time and load time constants. eg is the load self-regulation factor. ex is the transfer coefficient of a speed on the turbine torque. e is equal to eqyeh/ey-eqh.If we neglect the load perturbations, then we obtain the state space equations as(3)x˙1=x2+-eyeeqhTayx˙2=-eneqhTωTax1-Ta+eneqhTωeqhTωTax2+-eyeqhTωTa+Ta+eneqhTωeyeeqh2TωTa2y,where en is equal to eg-ex.When the nonlinear part can be shown by the nonlinear functiony=N(x3-KPx1), then we have the nonlinear state equation, which is a closed-loop system; we define x3 as a state variable,(4)x˙3=KPC˙+KIC-x1,where KP is the proportional component; KI is the integral component.Next, whenC=0, by combining (3) and (4), we have the state space equations of the PI hydroturbine governing system with saturation as(5)x˙1t=x2t+b1Nx3t-KPx1tx˙2t=-a0x1t-a1x2t+b0-a1b1Nx3t-KPx1tx˙3t=-KIx1t,where a1=Ta+eneqhTω/eqhTωTa, a0=en/eqhTωTa, b1=-eye/eqhTa, b0=-ey/eqhTωTa, e=eqyeh/ey-eqh, and N is the constant of nonlinear part and equal to 1.Although Ling and Tao [13] have investigated the existence and direction of the Hopf bifurcation, for the PI hydroturbine governing system, speed control delays and the displacement delays of the servomotor are never considered in the previous studies. In the paper, we consider the dynamics of the system (5) with two different delays. Therefore, we have the following PI hydroturbine governing system as(6)x˙1t=x2t+b1Nx3t-KPx1t-τ1x˙2t=-a0x1t-a1x2t+b0-a1b1Nx3t-τ2-KPx1tx˙3t=-KIx1t,where τ1 is the speed control delay of the generator. τ2 is the displacement-control delay of the servomotor.


## 3. Stability Analysis and Hopf Bifurcation

Usually, it is not easy to find out its accurate solutions. It is sufficient to research the stability ofE∗=(x1∗,x2∗,x3∗); we only consider E0=(0,0,0). At equilibrium point E0, the Jacobi matrix of the system (6) is(7)-b1N·KP·e-λτ11b1Na1b1-b0N·KP-a0-a1b0-a1b1N·e-λτ2-KI00,Then the associated characteristic equation is(8)λ3+A1λ2+A2λ+A3+A4λ2+A5λe-λτ1+A6e-λτ2=0,where (9)A1=a1,A2=NKPb0+NKIb1+a0-NKPa1b1A3=NKIa1b1,A4=NKPb1,A5=NKPa1b1,A6=-NKIa1b1+NKIb0.Next, we investigate the distribution of the roots of (8) with different delay values for τ1 and τ2.It is apparent that (8) assumes the following form when τ1=0, τ2=0.(10)λ3+λ2A1+A4+λA2+A5+A3+A6=0.Furthermore, we propose the Routh-Hurwitz stability criterion; a corresponding certificate shall be found [31].Lemma 1 (see [32]).
The polynomialL(λ)=λ3+p1λ2+p2λ+p3 with real coefficients has all roots with negative real parts if and only if the numbers p1,p2,p3 are positive and the inequality p1p2>p3 is satisfied.According to Lemma1, all roots of (10) have negative real parts if and only if(H1)
A1+A4>0,A2+A5>0,A3+A6>0 and (A1+A4)(A2+A5)>A3+A6.So E∗=(x1∗,x2∗,x3∗) is locally asymptotically stable when (H1) holds.Case 1 (τ=τ1=τ2≠0).
We rewrite (8) as follows:(11)λ3+A1λ2+A2λ+A3+A4λ2+A5λ+A6e-λτ=0.If λ=iω is a root of (11), then we have(12)ω10+ω8C1+ω6C2+ω4C3+ω2C4+C5=0,where (13)C1=2B1B2+B62-B14B12,C2=2B1B3-2B12B4+B22+2B6B7B12,C4=B32-2B4B5+2B7B8B12,C3=2B2B3-2B12B5-B42+2B6B8+B72B12,C5=-B52+B82B12.Let z=ω2, we define(14)hz=z5+z4C1+z3C2+z2C3+zC4+C5.
From the definition ofh(z), we can obtain h(+∞)=+∞; thus we can get that (14) has at least one positive root if h(0)=C5<0 holds.
We assume that (14) has five positive roots, which are defined by z1,z2,z3,z4, and z5. Then, (12) has five positive roots ωk=zk,k=1,2,3,4,5.
If we denote(15)τki=1ωkarccos⁡ω4B6+ω2B7+B8ω4B12+ω2B4+B5+2iπ,i=0,1,2,…,k=1,2,3,4,5,then ±iωk is a pair of purely imaginary roots of (8) with τ=τk(i).Lemma 2.
Defineτ0=τk(0)=mink∈{1,2,3}⁡{τk(0)} and ω0=ωk0. Let λ(τ)=ξ(τ)+iω(τ) be the root of (8) near τ=τ0, satisfying ξ(τ0)=0,ω(τ0)=ω0; the following transversality condition holds: dRe⁡λ(τ)/dττ=τk(i)≠0.By applying Lemma2 to (11), we can obtain the following theorem.Theorem 3.
SupposeH2=PRQR+P1Q1≠0 holds; with the increasement of delay variable τ from zero, there is a value of τ0, such that the positive equilibrium point E∗ is locally asymptotically stable for τ∈[0,τ0) and unstable for τ>τ0. Moreover, system (6) occurs with a Hopf bifurcation at E∗ when τ=τ0.Case 2 (τ1≠0,τ2=0).
We rewrite (8) as follows:(16)λ3+A1λ2+A2λ+A3+A6+A4λ2+A5λe-λτ1=0.Let λ=iω be a root of (16), then we have(17)z4+z3D1+z2D2+zD3+D4=0,where(18)D1=A12A42+A52-A44-2A2A42A42,D2=-2A42A52+A12A52+A22A42-2A2A52-2A1A3A42-2A1A42A6A42,D3=-A54+A32A42+A42A62+A22A52+2A3A42A6-2A1A3A52-2A1A52A6A42,D4=A3A5+A5A62A42.If we denote(19)τki=1ωkarccos⁡ω3A5-A1A4+ωA3A4-A2A5+A4A6ω3A42+ωA52+2iπ,i=0,1,2,…,k=1,2,3,4,then ±iωk is a pair of purely imaginary roots of (8) with τ1=τk(i).Lemma 4.
Defineτ10=τk(0)=mink∈{1,2,3}⁡{τk(0)} and ω0=ωk0. Let λ(τ)=ξ(τ)+iω(τ) be the root of (16) near τ1=τ10, satisfying ξ(τ10)=0,ω(τ10)=ω0,zk=ωk2 and h′(zk)≠0; then dRe⁡λ(τ10)/dτ1τ1=τk(i)≠0, dRe⁡λ(τ10)/dτ1, and h′(zk) have the same sign.Next, we have to look for the conditions required for (17) to have at least one positive root.We denote(20)hz=z4+z3D1+z2D2+zD3+D4.By applying Lemma1 to (17), we obtain the following theorem.Theorem 5.
For (20), the following result holds.(i)
IfD4>0 and Δ=D1D2D3-D4D12-D32≤0, then the zero solution of system (6) is asymptotically stable for τ1>0.(ii)
IfD4<0 and Δ=D1D2D3-D4D12-D32>0, then E∗=(x1∗,x2∗,x3∗) of system (6) is asymptotically stable for τ1∈0,τ10, and it is unstable when τ>τ10.(iii)
If all the conditions stated in (ii) andh′(z)≠0 are satisfied, system (6) occurs with a Hopf bifurcation at E∗=(x1∗,x2∗,x3∗) when τ1=τk(i)(i=0,1,2,…).Case 3 (τ1=0,τ2≠0).
We rewrite (8) as follows:(21)λ3+A1λ2+A2λ+A3+A6e-λτ2=0.By letting λ=iω be the root of (21), we have(22)z3+E1z2+E2z+E3=0.
The roots of (22) are considered by Lemma 1. Without losing generality, we suppose that (22) has three positive roots, which are defined by z1,z2, and z3.
If we denote(23)τki=1ωkarccos⁡A1ω2-A3A6+2iπ,i=0,1,2,…,k=1,2,3,then ±iωk is a pair of purely imaginary roots of (8) with τ2=τk(i).Defineτ20=τk(0)=mink∈{1,2,3}⁡{τk(0)} and ω0=ωk0. Let λ(τ)=ξ(τ)+iω(τ) be the root of (21) near τ2=τ20, satisfying ξ(τ10)=0 and ω(τ10)=ω0, then the following transversality condition holds.Lemma 6.
Suppose thatzk=ωk2 and h′(zk)≠0, then dRe⁡λ(τ20)/dτ2τ1=τk(i)≠0, dRe⁡λ(τ20)/dτ2τ1=τk(i), and h′(zk) have the same sign.The proof is similar to that of Lemma4, so we ignore the proofs. By applying Lemma 6 to (21), we have the following theorem.Theorem 7.
Asτ2 increases from zero, there exists a critical value τ20, such that E∗ is locally asymptotically stable for τ1∈0,τ20 and unstable when τ2>τ20. Moreover, system (6) occurs a Hopf bifurcation at E∗ when τ2=τ20.Case 4 (τ1≠τ2,τ1>0, and τ2>0).
We consider (8) with τ1 in its stability range. Regarding τ2 as a parameter, without losing generality, we consider system (6) in Case 2. Let ωi(ω>0) be a root of (8); then we obtain(24)F1ω+F2ωsin⁡ωτ1+F3cos⁡ωτ1=0,where(25)F1ω=ω6+ω4A12+A42-2A2+ω2A22+A52-2A1A3+A32-A62,F2ω=-2ω5A4+ω32A2A4-2A1A5+2ωA3A5,F3ω=-2ω5A4+2ω4A1A4+2ω3A2A4-2ω2A3A4.
Therefore, utilizing the general Hopf bifurcation theorem for functional differential equations (FDEs) as given in Hale [33], we obtain the following results for system (6).Theorem 8.
Suppose that (24) has at least finite positive roots, PRQR+PIQI≠0 and τ1∈0,τ10, then the positive equilibrium point E∗ of system (6) is locally asymptotically stable for τ2∈0,τ2∗. For E∗, system (6) undergoes a Hopf bifurcation when τ2=τ2∗. System (6) has a branch of periodic solutions bifurcating from the zero solution near τ2=τ2∗.


## 4. Stability and Direction of the Hopf Bifurcation

In the above section, we have studied that PI hydroturbine governing model (6) with double delays undergoes a Hopf bifurcation for τ=τk(i)(i=0,1,2,…,k=1,2,…). In this section, we assume that system (6) undergoes a Hopf bifurcation at τ=τ0(i)(i=0,1,2,…). Utilizing the normal form theory and the center manifold reduction, the stability, the direction, and the bifurcation of the periodic solutions are determined.For convenience sake, we suppose thatτ0 is less than τ2∗, where τ2∗∈0,τ10. Let u(t)=(x1(t),x2(t),x3(t))T∈R3,τ1=τ0+μ, in which μ∈R, then we transform system (6) into functional differential equations (FDEs) in C=C-1,0,R3 as(26)x˙t=Lμxt+fμ,xt,where Lμ:C→R3,f:R×C→R3 are given, respectively. We obtain(27)Lμxt=τ0+μ01c1c2c30c400ϕ10ϕ20ϕ30+τ0+μc50000c6000ϕ1-1ϕ2-1ϕ3-1,fu,ϕ=τ0+μc5ϕ1-1c6ϕ3-10,where c1=b1N,c2=a1b1NKP-b0NKP-a0,c3=-a1,c4=-KI,c6=(b0-a1b1)N,c5=-b1NKP and ϕ=ϕ1,ϕ2,ϕ3T∈C-1,0,R3.Utilizing the Riesz representation theorem, there is a3×3 matrix-valued function ηθ,μ of bounded variation for θ∈-1,0, such that(28)Lμϕ=∫-10dηθ,0ϕθ,whereϕ∈C-1,0,R3.Indeed, we may take(29)ηθ,μ=τ0+μ01c1c2c30c400δθ-τ0+μc50000c6000δθ+1,where δ is the Dirac delta function.Forϕ∈C-1-1,0,R3, we define(30)A∗ψs=-dψsds,s∈0,1∫-10dηTt,0ψ-t,s=0and a bilinear inner product:(31)ψs,ϕθ=ψ-0ϕ0-∫-10∫ξ=0θψ-ξ-θdηθϕξdξ,where ηθ=ηθ,0; A0 and A∗ are adjoint operators. From the above analysis, we obtain that qθ and q∗s are eigenvectors of A and A∗ corresponding to iω0τk and -iω0τk, respectively. Suppose that qθ=1,v1,v2Teiω0τkθ is the eigenvector of A0 corresponding to iω0τk; then A0qθ=iω0τ0qθ. It follows from the definition of A0, Lμϕ, and ηθ,μ that(32)qθ=1,v1,v2Teiω0τkθ=q0eiω0τkθ,and similarly, by definition of A∗,(33)q∗θ=D1,v1∗,v2∗Teiω0τkθ=q∗0eiω0τkθ.Through a simple calculation, we can obtain(34)v1=-ω0c5e-iω0τk+ω02+c1c4iω0,v2=c3c5e-ω0iτk-c2-ω02-ω0c3+c5e-ω0iτkic6e-ω0iτk-c1c3+c1ω0i,v1∗=c1c4+ω02-ω0c5e-ω0iτkic2ω0i-c4c6e-ω0iτk,v2∗=c2+ω02-c3c5e-ω0iτk-ω0c3+c5ω0e-ω0iτkiω0i+c3c4.From (31), one has(35)q∗s,qθ=q-∗0q0-∫-10∫ξ=0θq-∗ξ-θdηθqξdξ=D-1,v-1∗,v-2∗1,v1,v2T-∫-10∫ξ=0θD-1,v-1∗,v-2∗e-ω0iτkξ-θdηθ1,v1,v2Teω0iτkξdξ=D-1+v-1∗v1+v-2∗v2+τkc5+c6v2∗e-ω0τki.Thus we can choose D- as(36)D-=11+v-1∗v1+v-2∗v2+τkc5+c6v2∗e-ω0τki,such that q∗s,qθ=1 and q∗s,q-θ=0.Next, through the use of the same notations in Hassard et al. [34], we can calculate the coordinates describing the center manifold C0 at μ=0. Therefore, we have the following:(37)g20=2D-τkc5+c6v2∗e-iω0τk+1+c3,g11=2D-τkc5+c6v-2∗v22Re⁡eiω0τk+1+c1,g02=2D-τkc5+c6v2∗eiω0τk+1+c3,g21=D-τkc5W201-1+W111-1+c6W203-1+W113-1,where (38)W20θ=ig20ω0τkq0eiω0τkθ+ig-023ω0τkq-0e-iω0τkθ+E1e2iω0τkθ,W11θ=-ig11ω0τkq0eiω0τkθ+ig-11ω0τkq-0e-iω0τkθ+E2,E1=2c5e-2iω0τk1c1+2iω0c2+2iω0c3+2iω0c6e-2iω0τkc4+2iω000-1c5e-iω0τkc6e-iω0τk0,E2=2c51c1c2c3c6c400-1c5Re⁡e-iω0τkc6Re⁡e-iω0τk0.Thus, we get the following quantities:(39)c10=i2ω0τkg20g11-2g112-13g022+12g21,μ2=-Re⁡c10Re⁡λ′0,T2=-Im⁡c10+μ2Im⁡λ′0ω0τk,β2=2Re⁡c10.From the above analysis, we have the theorem as follows.Theorem 9.
Whenτk is equal to τ0i, the stability and direction of the Hopf bifurcation for system (6) are confirmed by the parameters μ2,T2, and β2.(1)
μ2 determines the direction of the Hopf bifurcation: if μ2>0 and τk>τ00, then the Hopf bifurcation is supercritical; if μ2>0 and τk>τ00, then he Hopf bifurcation is subcritical. In both cases, the bifurcating periodic solutions of system (6) exist.(2)
T2 determines the period of the bifurcating periodic solutions: if T2>0, the period increases; else the period decreases.(3)
β2 determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) for β2<0 (β2>0).


## 5. Numerical Example

Suppose the hydroturbine governing system which is set works under certain operating conditions, the parameters are as follows:N=1,a0=0.667,a1=2.522,b0=0.311,b1=-0.342. Utilizing the function dde23, numerical calculations have been performed as follows.In Case1, system (6) has two same time delays τ, initial value x0=(0.05,0.05,0.05),KP=4.5, and KI=-1.326; we can obtain the Hopf bifurcation value τ0=0.35 s. When τ<τ0=0.35 s, at the equilibrium E∗=(x1∗,x2∗,x3∗), system (6) is asymptotically stable, and it is unstable when τ>τ0 are shown in Figures 2(b) and 2(d), respectively.Figure 2
Time waveforms and phase portrait of the PI hydroturbine governing system. (a) Time waveforms withKP=4.5,KI=-1.326, and τ<0.35 s. (b) The phase portrait with KP=4.5, KI=-1.326, and τ=0.15 s. (c) Time waveforms of xi(t) with KP=1.1, KI=-3.6326, and τ≥0.35 s. (d) The phase portrait with KP=1.5 and τ=0.4 s.
(a)
(b)
(c)
(d)In Case2, when initial value x0=(0.05,0.05,0.05), KI<1.7326,KP=0.5, and τ1>0, the corresponding oscillation curves of xi(t) are shown in Figure 3(a). System (6) is asymptotically stable at the equilibrium point E0=0,0,0. We can compute the Hopf bifurcation value τ10=0.8322 s. When KI≥1.7326, τ2=0, the equilibrium point E∗=(x1∗,x2∗,x3∗) of system (6) is asymptotically stable for τ1∈0,τ10, and it is unstable when τ1>τ10 are shown in Figures 4(b) and 4(d), respectively.Figure 3
Time waveforms and phase portrait of the PI hydroturbine governing system with the initial valuex0=(0.05,0.05,0.05),KI<1.7326, and τ2=0. (a) Time waveforms of xi(t). (b) The phase portrait with τ1>0.
(a)
(b)Figure 4
Time waveforms and phase portrait of the PI hydroturbine governing system. (a) Time waveforms with0<τ1<τ10. (b) The phase portrait with τ1=0.75s<τ10. (c) Time waveforms for τ1=0.92s>τ10. (d) The phase portrait with τ1=0.84s>τ10.
(a)
(b)
(c)
(d)In Case3, when initial value x0=(0.05,0.05,0.05),KP=5,KI=-1.2, and τ1=0, we obtain the Hopf bifurcation value τ20=0.455 s. System (6) is asymptotically stable for τ1∈0,τ20, and it is unstable when τ2 is more than τ20 at the equilibrium point E∗=(x1∗,x2∗,x3∗), which are shown in Figures 5(b) and 5(d), respectively.Figure 5
The corresponding oscillation curves and phase portrait of the PI hydroturbine governing system. (a) The corresponding oscillation curves forKP=5 and τ2∈0,τ20. (b) The phase portrait with KP=5 and τ2=0.35s∈0,τ20. (c) The corresponding oscillation curves with KP=1.5 and τ2≥τ20. (d) The phase portrait with KP=1.5 and τ2=0.46s≥τ20.
(a)
(b)
(c)
(d)In Case4, when initial value x0=(0.05,0.05,0.05),KI=-1.5326,τ10=0.8322s, and τ2∗=0.5655s. The phase portraits are obtained from Theorem 8 in Figures 6(b) and 6(d). When KP=5, at Hopf point E∗=(x1∗,x2∗,x3∗) system (6) is stable and for each τ2<τ20, but close to τ20 there is a stable periodic orbit near the asymptotically stable equilibrium point E∗=(x1∗,x2∗,x3∗). When τ2=τ2∗ and τ1=τ10, system (6) has a transversal Hopf point at E∗=(x1∗,x2∗,x3∗) (see Figure 6(d)).Figure 6
The corresponding oscillation curves and phase portrait of the PI hydroturbine governing system. (a) The corresponding oscillation curves withKP=5 and τ2∈0,τ20. (b) The phase portrait with KP=5 and τ2∈0,τ20. (c) The corresponding oscillation curves with KP=1.2,τ1=0.01s, and τ2≥τ2∗. (d) Hopf bifurcation occurs for KP=1.2,τ1=0.01s, and τ2=τ2∗.
(a)
(b)
(c)
(d)


## 6. Conclusions

In the paper, we establish a PI hydroturbine governing system with saturation and double delays. In the case of positive equilibrium pointE∗, the stability of the PI hydroturbine governing system is discussed when the values of the speed control delay and the displacement delay of the servomotor is equal to zero and greater than zero, respectively. The results show that the PI hydroturbine governing system may have unexpected limit cycle oscillation when the delay parameters meet certain conditions. We obtain the scope of three parameters, which determine the stability of periodic solution, the direction of Hopf bifurcation, and the cycle of periodic solutions, respectively. Finally, the theoretical results are validated via the numerical simulation. In addition, a novel approach is proposed to analysis dynamic characteristics of the PI hydroturbine governing system with double delays.Our work illustrates that the oscillation can be effectively controlled by decreasing speed control delay and setting up the high efficiency of PI controller parameters. A time response device is designed to offset the speed control delay in the hydroturbine governing system. Utilizing data analysis method, the accuracy of servomotor displacement can be improved. The research provides theoretical guidance for hydropower station in maintaining the stability of the hydropower system. In the future work, the model of the PI hydroturbine governing system will be constituted by new materials. The rich nonlinear dynamic characteristics of the system will be analyzed accurately by the theory of fractional order. These methods and results will provide new ideas to the research of the stability of the hydropower station.



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*Source: 2796090-2017-09-12.xml*