Nonlinear Dynamics of a PI Hydroturbine Governing System with Double Delays
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.
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 . 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  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.  have revealed the problem of stabilizing of a first-order plant with time delay and obtained the stabilizing PI gain values. Shu and Pi  introduced a PID neural network (PIDNN) with control time delay and gave examples of analysis. Strah et al.  designed a speed and active power controller of hydroturbine units; some controller parameters were obtained. Li and Zhou  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.  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 . On the basis of necessary and sufficient condition, Liu et al.  proposed a new method to analyze the stability of automatic generation control (AGC) systems with commensurate delays. Zhang et al.  analyzed a PID-type load frequency control (LFC) scheme by using delay-dependent robust method. Based on state space equations, Chen et al.  studied the nonlinear dynamical behaviors of a novel hydroturbine mode with the effect of the surge tank. Xu et al.  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.  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. . Zhang et al.  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 Section 2, 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 Figure 1.
The transfer function of soft feedback regulator is
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 havewhere is used in a nonelastic water column model [29, 30]. is the transfer function from the hydroturbine moment to the speed of the generator. is servomotor stroke transfer coefficient of the flow rate. is servomotor stroke transfer coefficient of the turbine torque. is the transfer coefficient of the torque on the water head of a turbine. is the transfer coefficient of the flow rate on the water head of a turbine. is the water inertia time constant of a pressure guide-water system. is the strength of the elastic water hammer effect. is the sum of the machine starting time and load time constants. is the load self-regulation factor. is the transfer coefficient of a speed on the turbine torque. is equal to .
If we neglect the load perturbations, then we obtain the state space equations aswhere is equal to .
When the nonlinear part can be shown by the nonlinear function , then we have the nonlinear state equation, which is a closed-loop system; we define as a state variable,where is the proportional component; is the integral component.
Next, when , by combining (3) and (4), we have the state space equations of the PI hydroturbine governing system with saturation aswhere , , , , , and is the constant of nonlinear part and equal to 1.
Although Ling and Tao  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 aswhere is the speed control delay of the generator. 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 of ; we only consider . At equilibrium point , the Jacobi matrix of the system (6) is
Then the associated characteristic equation iswhere
Next, we investigate the distribution of the roots of (8) with different delay values for and .
Lemma 1 (see ). The polynomial with real coefficients has all roots with negative real parts if and only if the numbers are positive and the inequality is satisfied.
Case 1 (). We rewrite (8) as follows:If is a root of (11), then we havewhere Let , we defineFrom the definition of , we can obtain ; thus we can get that (14) has at least one positive root if holds.
We assume that (14) has five positive roots, which are defined by , and . Then, (12) has five positive roots
If we denote then is a pair of purely imaginary roots of (8) with .
Lemma 2. Define and . Let be the root of (8) near , satisfying ; the following transversality condition holds: .
Theorem 3. Suppose holds; with the increasement of delay variable from zero, there is a value of , such that the positive equilibrium point is locally asymptotically stable for and unstable for . Moreover, system (6) occurs with a Hopf bifurcation at when .
Lemma 4. Define and . Let be the root of (16) near , satisfying and ; then , , and have the same sign.
Next, we have to look for the conditions required for (17) to have at least one positive root.
Theorem 5. For (20), the following result holds.(i)If and , then the zero solution of system (6) is asymptotically stable for (ii)If and , then of system (6) is asymptotically stable for , and it is unstable when (iii)If all the conditions stated in (ii) and are satisfied, system (6) occurs with a Hopf bifurcation at when
Case 3 (). We rewrite (8) as follows:By letting be the root of (21), we haveThe roots of (22) are considered by Lemma 1. Without losing generality, we suppose that (22) has three positive roots, which are defined by , and .
If we denotethen is a pair of purely imaginary roots of (8) with .
Define and . Let be the root of (21) near , satisfying and , then the following transversality condition holds.
Lemma 6. Suppose that and , then , , and have the same sign.
Theorem 7. As increases from zero, there exists a critical value , such that is locally asymptotically stable for and unstable when . Moreover, system (6) occurs a Hopf bifurcation at when
Case 4 (, and ). We consider (8) with in its stability range. Regarding as a parameter, without losing generality, we consider system (6) in Case . Let be a root of (8); then we obtainwhereTherefore, utilizing the general Hopf bifurcation theorem for functional differential equations (FDEs) as given in Hale , we obtain the following results for system (6).
Theorem 8. Suppose that (24) has at least finite positive roots, and , then the positive equilibrium point of system (6) is locally asymptotically stable for . For , system (6) undergoes a Hopf bifurcation when . System (6) has a branch of periodic solutions bifurcating from the zero solution near .
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 . In this section, we assume that system (6) undergoes a Hopf bifurcation at . 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 that is less than , where . Let , in which , then we transform system (6) into functional differential equations (FDEs) in aswhere are given, respectively. We obtainwhere and
Utilizing the Riesz representation theorem, there is a matrix-valued function of bounded variation for , such that
Indeed, we may takewhere is the Dirac delta function.
For , we defineand a bilinear inner product:where ; and are adjoint operators. From the above analysis, we obtain that and are eigenvectors of and corresponding to and , respectively. Suppose that is the eigenvector of corresponding to ; then . It follows from the definition of , , and thatand similarly, by definition of ,
Through a simple calculation, we can obtainFrom (31), one hasThus we can choose assuch that and
Next, through the use of the same notations in Hassard et al. , we can calculate the coordinates describing the center manifold at . Therefore, we have the following:where Thus, we get the following quantities:From the above analysis, we have the theorem as follows.
Theorem 9. When is equal to , the stability and direction of the Hopf bifurcation for system (6) are confirmed by the parameters , and .(1) determines the direction of the Hopf bifurcation: if and , then the Hopf bifurcation is supercritical; if and , then he Hopf bifurcation is subcritical. In both cases, the bifurcating periodic solutions of system (6) exist.(2) determines the period of the bifurcating periodic solutions: if , the period increases; else the period decreases.(3) determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) for ().
5. Numerical Example
Suppose the hydroturbine governing system which is set works under certain operating conditions, the parameters are as follows: . Utilizing the function dde23, numerical calculations have been performed as follows.
In Case , system (6) has two same time delays , initial value , and ; we can obtain the Hopf bifurcation value s. When s, at the equilibrium , system (6) is asymptotically stable, and it is unstable when are shown in Figures 2(b) and 2(d), respectively.
In Case , when initial value , , and , the corresponding oscillation curves of are shown in Figure 3(a). System (6) is asymptotically stable at the equilibrium point . We can compute the Hopf bifurcation value s. When , , the equilibrium point of system (6) is asymptotically stable for , and it is unstable when are shown in Figures 4(b) and 4(d), respectively.
In Case , when initial value , and , we obtain the Hopf bifurcation value s. System (6) is asymptotically stable for , and it is unstable when is more than at the equilibrium point , which are shown in Figures 5(b) and 5(d), respectively.
In Case , when initial value , and . The phase portraits are obtained from Theorem 8 in Figures 6(b) and 6(d). When , at Hopf point system (6) is stable and for each , but close to there is a stable periodic orbit near the asymptotically stable equilibrium point . When and , system (6) has a transversal Hopf point at (see Figure 6(d)).
In the paper, we establish a PI hydroturbine governing system with saturation and double delays. In the case of positive equilibrium point , 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.
Stability Analysis and Hopf Bifurcation
According to time delays and , we analyze the nonlinear dynamic behavior of system (6) under the four different cases.
Case A.1 (). If is a root of (11), then we haveFrom (A.1), we can obtainwhere Then we have (12). Let ; we haveTo analyze the existence and distribution of roots of (A.4), we have (14) and the following:
From the above analysis, we have Lemma 2. The proof is as follows.
From the above analysis, we have Lemma 4. The proof is as follows.
Proof. Taking the derivative of with respect to in (16), we haveIn addition, we haveFor simplification of analysis, we define as and as . From (A.15)-(A.16) and (A.12), we can obtainwhere . Thus we have