研究控制非线性动力学模型

Study on Nonlinear Dynamical Model and Control Strategy of Transient Process in Hydropower Stationwith Francis turbineHaiyan Bao , Jiandong Yang, Liang FuState Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University No.8 Donghu South Road, Wuchang District, Wuhan 430072, ChinaHaiyan_8931@Abstract —The transient process in conduits of hydropowerstations is a very complicated dynamic procedure coupled withfluid, machines, electricity. In this paper, a whole nonlinear dynamical model of transient process in hydropower station with Francis turbine has been developed, and the control strategies of each transient process are studied. The nonlinear characteristics of hydraulic turbine and the elastic water hammer effect of pressure water supply conduit are considered in the model. The developed model is accurate enough to represent and simulate each transient process of the plant and may enable a plant operator to carry out economical, convenient study for the static stability and transient stability of the hydropower station under a wide range of transient processes. In addition, the literature takes a hydropower station as engineering case to simulate the transient processes of hydro-generator units ’ start-up, load variation, full load rejection from the grid and emergency stop. And the results of simulation are very satisfied.Keywords-hydraulic transients; nonlinear mathematical model; numerical simulation; control strategyI.I NTRODUCTIONH ydropower is an important and vital renewable energy resource, which converts energy in flowing water into electricity. Generally, a hydro-generator unit has many different operating conditions, and any operating condition changes will result in different hydraulic transients. The calculation of hydraulic transient is a key link for the safety and reliability of units and hydraulic installations. Traditionally, the objective of hydraulic transient calculation is to predict three primary regulation guaranteed parameters including the maximum dynamic pressure in the spiral case, the maximum rising ratio of rotating speed and the draft tube minimum pressure, consequently to ensure safety operation of hydropower station. H owever, with the development of hydroelectric construction and technology in China, the content of hydraulic transient calculation is continuously beingenriched, it already not only include calculation of regulationguaranteed parameters, but also include calculation and research of stabilization and dynamic quality [1]. In conventional hydropower stations, there are a series of hydraulic transient processes, such as start-up, load variation, full load rejection from the grid, and emergency stop, wherepower and frequency regulations may always be needed [2]. In order to design suitable control law, stabilize the nonlinear systems, solve many existing control problems, reduce operating costs and energy losses, and improve guarantee security and safety of equipments and plants, it is necessary to develop a whole nonlinear dynamical model that is accurate enough to represent and simulate each transient process of the plant. The developed model may enable a control system designer or a plant operator to carry out accurate, economical, convenient study for the static stability and transient stability of the hydropower station under a wide range of operational modes and nonlinear process conditions, and to design the suitable control strategy, so as to improve stability of hydro-generator units.The literature review carried out in this work finds some published research works. In [3], a new kind of start-up rule is proposed, by using this rule the contradiction between fast start-up and smooth start-up is eliminated; In [4], it analyses the adjusting mode of power adjustment in digital electric-hydraulic governor, and how to realize power adjustment; In [5], the transient performance index of hydro-generator unit in a full load rejection are studied. owever, in the aforementioned published research works, the effect of hydraulic turbine characteristics and the elasticity of conduit walls on the transient process are neglected . In addition, a whole nonlinear dynamical model that can simulate each transient process of the plant isn’t developed in predecessors’ research works. In china, some large-scale hydropower stations often use the complex arrangement nowadays, moreover, the hydraulic conduits are getting longer, and its nonlinearity is very obvious. Therefore, it is very important and necessary to develop a whole nonlinear dynamical model for the complex hydropower system.II.M ATHEMATICAL M ODELS For developing the whole nonlinear mathematical model, the hydropower plant system is decomposed into decoupled dynamical modules as illustrated in Fig. 1, and a mathematical model for each module is developed. 978-1-4244-2487-0/09/$25.00 ©2009 IEEEFigure 1. Hydropower plant dynamical systemA.Modelling of Hydraulic ConduitIn hydraulic transient calculation of hydropower stations, one-dimensional continuity equation and momentum equation will be applied to the unsteady flow in conduit [6]. These equations constitute a system of two equations of partial derivatives – of the hyperbolic type. An exact integration of the equations is however very complicated, analytical solutions are rare. Nevertheless, there exist different numerical techniques making solutions possible. The method of characteristics is often and efficiently used to obtain the solutions of the equations. The method is reasonably convenient to deal with complicated boundary conditions. In the system of hydropower station, there are some boundary conditions (such as turbine) that are nonlinear themselves. Therefore, adopting the method of characteristics, a system of complex nonlinear equations will be avoided, so as to make the solution process simplify greatly. By using the method of characteristics, put the system of the equations of partial derivatives into a system of equations of ordinary derivatives. And under the condition of fixed time step, the equations of ordinary derivatives will be put into two linear algebraic equations with two unknown functions, discharge P Q and water head P H , along the curves of the characteristics, the positive characteristic +C and the negative characteristic −C . The simple forms of the two equations can be written as [7]:+C :P QP CP P H C Q Q .−=, (1)−C :P QM CM P H C Q Q .+=. (2)where CP Q ,QP C ,CM Q ,QM C are known coefficients that are related to the velocity at the forward time step, the water headat the forward time step and the geometrical sizes of conduits. B.Modelling of Hydraulic TurbineThe type of hydraulic turbine is various, in this paper, the mathematical model of Francis turbine will be developed. It is well known that the characteristics of hydraulic turbine are usually represented by the synthetic characteristic curves. In order to develop the mathematical model of hydraulic turbine, it is necessary to carry on data processing for the synthetic characteristic curves. Using the special-purpose software, translate the synthetic characteristic curves into discharge characteristic curves and torque characteristic curvesthat respectively take the specific discharge 1Q ′, the specific torque 1M ′as the ordinate, take the specific speed 1n ′as the abscissa, and take the wicket gate servomotor stroke y as parameter. During the calculation of hydraulic transient process, by using the wicket gate servomotor stroke and water head at the time level, the operating point will be found at discharge characteristic curves and torque characteristic curves. Therefore, the mathematical model of the characteristic of hydraulic turbine may be written as:()y n f Q ,11′=′, (3) ()y n f M ,11′=′. (4) owever, there are still no detailed and precisemathematical models that can describe 1Q ′ and 1M ′; therefore, 1Q ′ and 1M ′are usually obtained by interpolation method. The discharge equation of every type hydraulic turbine may be written as follows:H D Q Q P 211′=. (5) where 1D is the turbine-specific diameter, H is the availablehydraulic head of turbine, the specific discharge 1Q ′ can be obtained using (3).And the specific speed 1n ′in (3) can be shown by following equationH D n n /11=′. (6) where nis the rotational speed of hydro-generator unit, and the computational method of nwill be discussed in next segment.According to the theory of hydraulic turbine, the mechanical torque of hydraulic turbine t M can be written as follows:H D M M t 311′=. (7)where the specific torque 1M ′ can be obtained using (4). The mechanical torque t Min terms of the hydraulic turbine mechanical power P may be written as follows:n P M t ʌ/30=. (8)The mathematical model of hydraulic turbine should be coupled with the characteristic curves, +C and −C (as shown in(1) and (2)), of the upper and downstream conduits linked with the hydraulic turbine. The schematic diagram of turbine boundary is shown in Fig. 2.Figure 2. Schematic diagram of turbine boundaryC.Modelling of Synchronous GeneratorThe synchronous generator converts the mechanical power of the hydraulic turbine to electrical power at a specific voltage and frequency. The electric dynamics have very short time constants compared to hydrodynamics and can be ignored [8]. The mechanical equations of a rotating machine are based on the swing equation of the rotating inertia. Constant shaft speed for a given machine is maintained when there is equilibrium between the mechanical shaft and braking electrical torques. Any imbalance between the torques will cause the acceleration or deceleration of the machine according to the laws of motion of a rotating body. The swing equation may be written as follows:g t M M dtdn−=30Jπ. (9)where J is inertia torque of the machine,g M is braking electrical torque.D.Modelling of Turbine GovernorThe governor is important assistant equipment of the hydroelectric generator, it cooperates with computer monitoring system to complete hydro-generator units’ start-up, load variation, full load rejection from the grid, emergency stop and so on.As for control mode of modern turbine governor, there are commonly speed regulation, power control mode and opening control mode. And now, the parallel connection PID controller has been widely used in hydropower process control. The schematic diagram of PID controller is expressed as Fig. 3 andFig.4.f cfcY cP gP Figure 3.Schematic diagram of PID controller (a)gff cY cP gP Figure 4. Schematic diagram of PID controller (b)Contrasting the two Figs, power control mode and opening control mode in each Fig are different. In Fig. 3, the power control mode and opening control mode adopt PI control law, whereas that adopts “Integral + feedforward loop” control law in Fig. 4.When the turbine governor carry out speed regulation, frequency deviation between the frequency set point and the present frequency come into parallel proportional (P) function, integral (I) function and differential (D) function , then get across mechanical servo system. The derivative equation of speed regulation can be written as follows:¸¸¹·¨¨©§++−=+++++++x dt dx T dt x d T T yb dtdyT b T b b T dtyd T T b T T b T T b dt y d T T T b d n d p y p d p t d y d p y d t n d p y n d p 22233)()(. (9)where x is the speed relative deviation given by rcn n n x −=;p b ,t b ,d T ,n T are governor parameters.Usually, turbine governor is in power control mode tocontrol load variation of operating units. H owever, opening control mode is only a supplement control mode of power control mode. Therefore, the literature mainly analyses power control mode, the derivative equation of the control mode in Fig. 3 and Fig. 4 can be written as follows respectively:()()g c p g c d p d t y d t P P b dt P P d T b dt dyT b dt y d T T b −+−=+22, (10) ()d t P P T b b P t y dt dy T t g c d t p c y ³−⋅⋅+Δ=+01)(. (11) where c P is power set point; g P is present power. Except the aforementioned mathematical models, the mathematical models of other boundary conditions (such as surge tank, bifurcated pipe) in hydropower station system are also developed. As limited by the scope of this paper, the expressions of the mathematical models are omitted. The omitted models and the aforementioned models form the whole nonlinear dynamical model which can be solved by timedomain method that is a kind of numerical methods. On thebasis of the nonlinear dynamical model, Section III will study the different control strategies of turbine governor, in order to carry out optimum control for every hydraulic transient. III.H YDROPOWER P ROCESS C ONTROL S TRATEGY According to the regulation and control of turbine governorfor hydroelectric generator, there are three steady states of hydroelectric generator: normal operation, on-load operation,shut-down waiting state. The schematic diagram of conversionconnection of the three steady states is shown in Fig. 5.Figure 5. Schematic diagram of conversion connectionThe literature takes a hydropower station as engineering case to simulate hydro-generator units’ start-up, load variation, full load rejection from the grid and emergency stop transient processes. Conduit parameters and essence data of the hydropower station are shown as follows: nominal power600=r P MW, nominal speed 7.166=r n rad/min, nominal discharge 5.232=r Q m 3/s, nominal head 0.288=r H m 3/s, the length of the conduit 166601=L m, the length of the penstock 83.2332=L m, the area of upstream throttled surge tank 3.349=A m, the water inertia time constant 40.1=w T s, the unit inertia time constant 361.9=a T s.A.Start-up Control StrategyFast start-up and synchronization depends directly upon a good start-up algorithm. Moreover, the start-up procedure is closely linked with the network reliability, dynamic performance and economic operation. Open loop start-up rule,closed loop start-up rule and “open loop + closing loop” start-up rule are the most common start-up rules. The lecture mainlysimulates “open loop + closed loop” start-up procedure. Theprinciple of the start-up rule is shown as follows: after a start-up command is issued to the turbine governor under shut-down waiting state, the gate starts to open at maximum speed until a start-up gate opening, the gate opening stays constant until thedifference between the present frequency and the frequency setpoint 40HZ attains certain values. Then PID regulation routineis put in to regulate the gate opening and make the frequency follow the frequency set point, the frequency will reposefully attain 50H Z. When the frequency deviations, as well as the phase and voltage deviation between the unit and the network reach certain values, the generating unit can be synchronized to the network. During the start-up process, the generating unit isn’t connecting with the network, so braking electrical torque is zero, i.e.0=g M . At the phase of open loop start-up, the opening rule of the wicket gate servomotor stroke y is known.And at the phase of closed loop start-up, the value of y can beobtained using the equation of speed regulation, as shown in(9). The turbine governor parameters under no-load condition can be estimated by the value of w T and a T [9], let t b =0.3,p b =0, d T =7s, n T =0.6s. And let start-up gate 0y =0.1, wicket opening time k T =50s, to simulate start-up transient process. The simulation results are shown in Fig.6.Figure 6. Simulation results of start-up transient process The results of simulation have shown that start-up process of the hydropower station is fast and smooth. The unit speed follows its own set point very good, and the unit speed has good dynamic response, short response time, small overshoot, and fast attenuation. Therefore, the “open loop + closed loop” start-up rule is a perfect and simple start-up control mode.B.Load Control StrategyThe generating unit is connected with the network during load variation. There are two main control mode, speed regulation and power control mode, to control load variation. 1)Speed Regulation RoutineWhen the generating unit is connected with small or isolated network, or connected with large network and operating under speed regulation routine, the speed regulation routine is put in to control load variation. Let p b=0.01, tb=0.3, d T=7s, n T=0.6s, to simulate the transient process of -20% step change in load. The simulation results are shown inFig.7.Figure 7. Simulation of -20% step change in load under speed regulation The results of simulation have shown: after -20% step change in load is detected, the unit speed rises quickly with the speed overshoot only once, then the unit speed stabilizes at normal speed newly; while the power decrease quickly, finally stabilizes at power set point with a very small overshoot. The response curves in Fig.7 have small oscillation that is caused by the fluctuation in upstream surge tank after step change in load.2)Power Control ModeWhen the generating unit is connected with large network, and is controlled by automatic generation control system (AGC), the power control mode is put in to control load variation. Because the generating unit is connected with large network, we can think that the speed is normal speed and stays constant during load variation. Let p b=0.04, t b=0.3, d T=7s, to simulate the transient process of -20% step change in load.The simulation result is shown in Fig.8.Figure 8. Simulation of -20% step change in load under power control modeAs shown in Fig.8, the two parallel center lines are ±5% [9] power set point. The solid is the simulation result under “Integral + feedforward loop” control law, and the dashed is the simulation result under PI control law. Under “Integral + feedforward loop” control law, the power decrease quickly, and stabilizes in bandwidth of ±5% power set point with a very small reverse overshoot at t=18.4s. However, in contrast to the previous control law, the variation velocity of the power is slower under PI control law. Although there is no overshoot, the system response under PI control law is not very satisfactory; because the regulating time which is 112s is too long. Therefore, the “Integral + feedforward loop” control law is superior to the PI control law.C.Control Strategy of Emergency Stop and Full LoadRejection1)Emergency StopWhile the unit is on normal operation, the unit suddenly rejects full load that is caused by the fatal accident of the unit itself, then wicket gate closes rapidly, finally brings on emergency stop. During emergency stop, the generating unit has broken away the network, so braking electrical torque is zero, the wicket gate closes at maximum speed. The simulation results of emergency stop are shown in Fig.9.Figure 9. Simulation results of emergency stop transient processThe gate opening and speed curves in Fig.9 show: at the time of emergency stop, the wicket gate closes at maximum speed; at the same time, the speed rapidly rises firstly, and then the speed descends slowly.2)Full Load Rejection from the NetworkIf the unit suddenly rejects full load that is caused by other accidents, such as oil switch tripping, the wicket gate will close until no-load gate opening, and braking electrical torque is zero. Let p b=0.01ˈt b=0.3ǃd T=7sǃn T=0.6s, to simulate the transient process of full load rejection. The simulation resultsare shown in Fig.10.Figure 10. Simulation results of full load rejection from networkAfter full load rejection from the network, the wicket gate closes at maximum speed, and the speed rises rapidly, then the turbine governor traces and regulates the speed, in order to make the speed stabilizes at normal speed, finally the unit is on no-load operation. The simulations of the full load rejection have shown that both the dynamic response of speed with overshoot being≥3% only once and the regulating time satisfy the desire of specification [10].IV.C ONCLUSIONIn this paper, a whole nonlinear dynamical model of transient process in hydropower station with Francis turbine has been developed. The models vary in complexity and are meant to be used for the study of hydro turbine governing system problems of different types. And these nonlinear characteristics of hydraulic turbine and the elastic water hammer effect of pressure water supply conduit are considered in the modelling.The developed model is used to simulate different types of transient processes, such as start-up, load variation, full load rejection from the grid, and emergency stop, and to study the different control strategies of turbine governor. It could be found that the hydro-generator unit has good stability and dynamic performances under each transient process with optimal control strategy. The results of simulation and control strategies are very satisfied, it shows that the developed model is accurate enough to enable a control system designer or a plant operator to carry out accurate, economical, convenient study for the static stability and transient stability of the hydropower station, and the model is valuable.R EFERENCES[1]L. Q. Ye, “H ydropower Process Control – Theory, Application &Development,” Huazhong University of Science and Technology Press, China, 2002.[2]J. S. Chang, “Transients of H ydraulic Machine Installations,” H igherEducation Press, China, 2005.[3]H. B. Zhang, J. C. Xie, S. B. Jiao, “Study on optimum start_up rule forhydroelectric generating units,” Journal of H ydraulic Engineering, vol.35, no. 3, pp. 53-59, 2004.[4]S. P. Wei, Y. J. Wang, P. 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