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2564 Advances in Environmental Biology, 6(9): 2564-2573, 2012 ISSN 1995-0756 This is a refereed journal and all articles are professionally screened and reviewed ORIGINAL ARTICLE LQR Optimum Control for Wind Energy Conversion system 1 E. Nekooei, 2S.M. Mousavi Badjani, 2M.R. Alizadeh Pahlavani 1 Mapna Power Plants Construction & Development Co., Mapna-MD1, 15 Africa Ave., Tehran 1518614411, Iran. 2 Malek-Ashtar University of Technology (MUT), Shabanlo St., Lavizan, Tehran, Iran. E. Nekooei, S.M. Mousavi Badjani, M.R. Alizadeh Pahlavani; LQR Optimum Control for Wind Energy Conversion system ABSTRACT In this paper, a grid-connected wind energy converter system including a Matrix Converter (MC), a Squirrel-Cage Induction Generator (SCIG), a gear box and a horizontal axes wind turbine is proposed. An overall dynamical model for the proposed wind turbine system is considered. The dynamic behaviour of the model is investigated by simulating the response of the overall model to step changes in selected input variables. Moreover, a linear model of the system is developed at a typical operating point, and a LQR optimum control is designed for the system. The nonlinear and linear governing equations of the system are implemented using MATLAB. The obtained results show that the settling time of closed loop system using LQR step response is 0.002 sec., steady state error is zero and overshoot is zero too, which is highly acceptable for performance goals of this system. Key words: Wind turbine, MC, LQR Introduction Wind is one of the most abundant renewable sources of the energy in the nature. The economical and environmental advantages offered by wind energy are the most important reasons why electrical systems based on wind energy are receiving widespread global attention [1]. The SCIG is a lowcost and robust machine and the industry has a lot of experience in dealing with it. In addition, the induction generator has high reliability and the right mechanical properties for wind turbines, such as slip and a certain degree of overload capability. These are the main reasons for choosing this type of generator in the proposal system [2]. As a power electronic converter, a MC is used. MC interfaces the SCIG with the grid and implements shaft speed control. In the proposed system, the MC input is connected to the grid and the generator terminals are connected to the MC output. MC is highly controllable and allows independent control of the output voltage magnitude, frequency, and phase angle, as well as input power factor. In this paper first, MC dynamical model is described. And in sec.IV dynamical model of wind turbine system is proposed. Linearized model described in sec.V and LQR design is proposed in sec.VI and at last the simulation results are taken in sec.VII. Wind Energy Conversion System(WECS): Wind energy conversion system composed of wind turbine blades, a SCIG, a MC and corresponding control system. Fig.1 shows the block diagram of basic components of WECS. The functional objective of this system is converting the wind kinetic energy into electric power and injecting this electric power into a utility grid. Fig. 1: Block diagram of a WECS. Corresponding Author E. Nekooei, Mapna Power Plants Construction & Development Co., Mapna-MD1, 15 Africa Ave., Tehran 1518614411, Iran. E-mail: [email protected] 2565 Adv. Environ. Biol., 6(9): 2564-2573, 2012 MC Structure and Dynamic Model: The structure of a MC is shown in Fig.2 and the parameters of the circuit are defined in Table1 [3-5]. The MC connects the three phase ac voltages on input side to the three phase voltages on output side by 3x3 matrixes of bi-directional switches. Fig. 2: A WECS circuit including MC. Table 1: Parameters of the MC circuit. Parameters Descriptions Three phase MC input voltages and currents Three phase MC output voltages and currents Three phase input voltages sources Three phase output voltages sources Input source resistance per phase Input source plus filter inductance per phase Input filter capacitor per phase per phase Output source resistance per phase Output source inductance Switch between phase and For deriving of dynamical model first the switching method should be chosen. Switching function is as (1): Where is the on period of the switches in a switching sequence time. The following two restrictions apply to the duty cycle: (1) The (2) constraint applies to the switching function: (2) This constraint comes from this fact that MC is supplied from a voltage source and generally feeds an inductive load. Connecting more than one input phase to one output phase causes a short circuit between two input sources. Also, disconnecting all input phases from an output phase causes an open circuit. Therefore 27 allowable combinations from 512 states can exit. For transfer all quantity to one side and consider the same frequency for all quantity, we define a duty cycle as (3): (3) By considering the duty cycles of the switches, the matrix D (modulation matrix) is as follows: So, the output voltages and input currents of the matrix converter can be written as: (4) Where and are a set of sinusoidal input voltages and input currents. The transformation matrix D should satisfy the following three requirements: (a) Restrictions on duty cycles 2566 Adv. Environ. Biol., 6(9): 2564-2573, 2012 (b) Sinusoidal output voltages with controllable frequency, magnitude and phase angle. (c) Sinusoidal input current, (d) Desired input displacement power factor. The following low- frequency transformation matrices are two basic solutions that individually satisfy the requirements (a), (b) and (c) (5) Where : Output frequency of the MC, : output voltage angle of the MC and is MC output to input voltage gain. The combination of the two transformation matrices provide controllability of input displacement power factor. (6) In (6), is a constant parameter varying between 0 and 1. All equations transferred to qdo frame because of investigating of dynamical behaviour of the model is simple. so, By KVL and KCL laws, the state space qdo model of matrix converter system can be written as Eq. (7): (7) 2567 Adv. Environ. Biol., 6(9): 2564-2573, 2012 Dynamical Model of Wind Turbine System: The power coefficient analytically according to: is approximated The mechanical power and torque on the wind turbine shaft are given by 8, 9 equations respectively [6-17]: (11) (8) (9) Where: : Mechanical power extracted from turbine rotor, : Mechanical torque extracted from turbine rotor, : Area covered by the rotor, : Performance coefficient, : Air density, : Tip speed ratio, : Rotor blade pitch angle, : Angular speed of turbine shaft, : Wind velocity, : Turbine rotor radius. The power coefficient is related to tip speed ration and blade pitch angle and changes with different values of the pitch angle, but the efficiency and also for maximum power is obtained for point tracking. In this paper, it is assumed that the rotor pitch angle is fixed and equal to zero. The mechanical model of turbine can be considered as Fig.3 the driven train converts the aerodynamic torque on the low-speed shaft to the torque on the high-speed shaft. The dynamic of the driven train are described by the following three differential equations: (12) (13) The blade tip speed ratio is defined as Eq. (10): (10) (14) In the above equations: And Fig. 3: Mechanical model of wind turbine shaft 2568 Adv. Environ. Biol., C(): CC-CC, 2012 A commonly-used induction machine model is based on the flux linkages. The dynamic equivalent circuit of the induction machine in qdo frame is illustrated in Fig.4 Fig. 4: qdo equivalent circuit of an induction machine. In order to model the gearbox, it is only needed to consider that the gearbox torque can simply be transferred to the low speed shaft by a multiplication. So, for gearbox of Fig.1, one can be write: (15) So, the overall model of the system in state space qdo frame can be written as follow: (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) 2569 Adv. Environ. Biol., C(): CC-CC, 2012 Where: , , Linearized model performance evaluation: Table2. Simulation results are shown in simulation results section. In order to evaluate the linearization error at an operating point, the linearized model responses to step changes in some inputs are compared with those of nonlinear model. The operating point is shown in LQR Design: First, an integrator is added to the wind turbine system, the augmented system is shown in Fig. 5 Table 2: State variables at the operating point for linearization. -8.45 2.5 1632.3 -2.16 26.45 109.5 1591.1 378.5 0.004 9.45 1632.5 Fig. 5: Augmented system. The wind turbine model has the input vector , the state vector contains eleven , the output variables, the output vector is and the tracing error reference is defined as: , so, the state space model of the augmented system as follows: (29) 2570 Adv. Environ. Biol., C(): CC-CC, 2012 Fig. 6 shows the closed loop system with regulator state-feedback. The closed loop state space model is as Eq. 30: Fig. 6: Closed loop system with state feedback. (30) The aim of the controller design is to determine the gain of the controller, to make the augmented system stable and result in a proper undershoot/overshoot and settling time of the closed loop time domain responses. The cost function used in the design of an LQR controller, is defined as: (31) ~ ~ ~ For the closed loop feedback system if ( A, B ) is controllable and (Q, A) is observable, then the control gain can be calculated from the formula: (32) Where P is a unique symmetric positive semi- definite solution of the algebraic riccati equation: (33) (34) Simulation Results: Parameters of the system under study are shown in Table 3. Fig.7, 8 shows the system response to a step decrease of 1 HZ in the MC output frequency Table 3: Parameters of the system under study. Grid MC 1 for linearized and nonlinear models. In the next simulation the system response to a 10% step changes in wind velocity for linearized and nonlinear model are shown. Wind turbine Induction Generator 2571 Adv. Environ. Biol., C(): CC-CC, 2012 Active Output Pow er [p.u] 3 Nonlinear Linear 2 1 0 0.2 0.4 0.6 0.8 Time[sec] 1 1.2 Fig. 7: Active output power response to 1 frequency decrease in 0.3 sec. Reactive Output pow er [p.u] 1 Nonlinear Linear 0.5 0 0.2 1.2 1 0.8 0.6 Time[sec] 0.4 Fig. 8: Reactive output power response to 1 frequency decrease in 0.3sec. As Fig.7, 8 shows with 1 frequency decrease, we have approximately 2 p.u increase in active and reactive power. Active Output Pow er [p.u] 1 Nonlinear Linear 0.8 0.6 0.4 0.4 0.6 0.8 Time[sec] 1 1.2 Reactive Output Pow er [p.u] 0.42 Nonlinear Linear 0.4 0.38 0.2 0.4 0.6 0.8 Time[sec] 1 1.2 Fig. 9: Active and reactive power response to 10% decrease of wind velocity in 0.5 sec. And also Fig.10 shows active power response to 50% decrease in parameter a. Active Output Pow er [p.u] 2 1 Nonlinear 0 Linear -1 0.4 0.6 0.8 Time[sec] 1 Fig.10: Active power response to 50% decrease of parameter a in 0.7 sec. 1.2 2572 Adv. Environ. Biol., C(): CC-CC, 2012 The simulation results reveal that the linear model is an accurate approximation of the nonlinear model. By tuning Q, R, we can achieve an appropriate controller for the system. For this system Q and R matrix and also the active and reactive power step responses are as follows: Step Response of Active and Reactive Powers [p.u] 1.5 1 0.5 0 0 0.05 0.1 0.15 0.2 0.15 0.2 Time[sec] 30 [p.u] 20 10 0 -10 0 0.05 0.1 Time[sec] Fig. 11: System (linearized model) with LQR controller: step response of the active and reactive powers. As Fig.11 shows the settling time is less than 0.002sec. 2. Conclusions In this paper, an overall dynamical model of wind energy conversion system is proposed. A linear model of it is derived and evaluation of this linear model investigated with nonlinear model. A closed loop feedback controller, based on the linear model, is designed. The open loop system is locally stable around the operating point, and the controller design objective is to track a class of desired active and reactive powers delivered to the grid. A state feedback controller is adopted for the WECS. In the state feedback controller, since the goal is to achieve zero steady state error for the family of step reference signals, first an integrator is added after the system. The combination of the system with the integrator forms the augmented system. Then, the state feedback controller is designed based on the LQR method. 3. 4. 5. 6. References 7. 1. Surugiu, L. and I. Paraschivoiu, "Environmental, social, and economical aspects of windenergy," AIAA-2000-3008 Environmental, Social and Economical Aspects of Wind Energy, 35th Intersociety, 2: 1167-1174. Torrey, D.A., 1993. "Variable-reluctance generators in wind-energy systems, " Rec. in IEEE PESC'93, Power Electronics Specialists Conf., pp: 561-567. Kazerani, M., 1995. Dynamic Matrix Converter Theory Development and Application, Ph.D. thesis, Department of Electrical Eng., McGill University, Canada. Ooi, B.T. and M. Kazerani, 1994. 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