Decentralized Nonlinear Control of Wind Turbine

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 23, NO. 2, MAY 2008
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Decentralized Nonlinear Control of Wind Turbine With Doubly Fed Induction Generator
Feng Wu, Xiao-Ping Zhang, Senior Member, IEEE, Ping Ju, Member, IEEE, and Michael J. H. Sterling
Abstract—In this paper, a novel nonlinear control design method for the wind turbine (WT) with doubly fed induction generator (DFIG) using differential geometry theory is proposed. The WT with DFIG is represented by a third-order model where electromagnetic transients of the stator are neglected. Then the model is exactly linearized using the coordinate transformation in differential geometry theory. Since the relative degree of the model of the WT with DFIG is equal to the order of the original system model, the linearized model can be transformed into the Bronovsky norm form. The linear quadratic regulator (LQR) design method is introduced to design the optimal control for the linearized system, and the nonlinear control for the WT with DFIG obtained by the inverse coordinate transformation only depends on the parameters of the WT with DFIG. Hence, the nonlinear control can be considered as a decentralized control, which is a desirable feature for a WT with DFIG. To illustrate the effectiveness of the nonlinear controller proposed, simulations on a single machine in?nite bus (SMIB) system and a four-machine system are performed. Compared to the conventional PI control, the proposed nonlinear control can improve the transient stability of the power systems and enhance the system damping. Index Terms—Control systems, differential geometry, doubly fed induction generator, nonlinear systems, transient stability, wind power generation.
I. INTRODUCTION
W
ITH the growing integration of wind energy into power grids, the impact of wind generators on power system stability, especially in the aspect of transient stability, is of increasing concern. In the development of wind turbine (WT) techniques, WT with doubly fed induction generator (DFIG) is becoming the dominant type of WT system used in wind farms [1], because it has the characteristics of low investment and ?exible control. A decoupling control technique for the active power and reactive power proposed in [2] has been widely used in control of the WT with DFIG [3]–[5]. The decoupling control technique was based on conventional PID controllers. PID controllers are well accepted in the engineering ?eld for their reliability and simplicity. With these features, the PID control has been widely used in excitation control and steam valve control of power systems. However, the parameters of the PID controllers are usually tuned with the approximately linearized model. In this situation,
Manuscript received June 12, 2007; revised October 14, 2007. Paper no. TPWRS-00433-2007. F. Wu, X.-P. Zhang, and M. J. H. Sterling are with the Department of Electronic, Electrical, and Computer Engineering, University of Birmingham, Birmingham B15 2TT, U.K. (e-mail: wfcn77@https://www.360docs.net/doc/d614387090.html,; X.P.Zhang@bham.ac. uk). P. Ju is with the College of Electrical Engineering, HoHai University, Nanjing 210098, China (e-mail: pju@https://www.360docs.net/doc/d614387090.html,). Digital Object Identi?er 10.1109/TPWRS.2008.920073
the dynamic control performance may not be guaranteed during the transients of the WT with DFIG, especially under large disturbances in power systems. Hence, new methods for the design of controllers for the WT with DFIG system should be investigated in order to achieve better control performance in terms of transient stability of power systems. Differential geometry is a branch of mathematics that studies the geometry of curves, surfaces, and manifolds. During the past decades, differential geometry has been proven to be successful in the study of the nonlinear system. Based on the branch of the exact linearization via feedback, the nonlinear control design method using differential geometry theory has been developed [6], [7]. In the design, the model of the system to be controlled is exactly linearized by coordinate transformation using differential geometry theory. Then the optimal control law is obtained for the linearized system using linear optimal control principles. Finally the nonlinear control law is derived from the linear optimal control law by inverse coordinate transformation using differential geometry theory. In the past, nonlinear control design methods using differential geometry theory have been used in power system stability control problems. A nonlinear control design method using differential geometry theory has been applied successfully in excitation control design of synchronous generators for multimachine systems [8] where both small signal stability and transient stability of the power systems were improved. A nonlinear rocontrol apbust control based on the combination of the proach and the exact linearization approach using differential geometry theory has been presented in [9] for excitation control of synchronous generators. The robustness of the controller to the uncertainty of the network structure and parameters, and to the disturbance was improved, and the transient stability control of power systems was enhanced. A nonlinear coordinated control of excitation and governor systems for a hydraulic power plant using differential geometry theory has been proposed in [10], and better performances of the hydraulic generator under large disturbance were achieved. A nonlinear control design method using differential geometry theory has been used in the dc link voltage control of PWM ac/dc converters [11] where the dc voltage was regulated effectively. A nonlinear control for PWM back to back converters using differential geometry theory, which incorporates the power balance of the input and output of the converters, the current transients of the load, has been proposed in [12] while the dynamics of mechanical motion of the load were not considered. The responses of the voltage to the load change were improved and the capacitance of the dc electrolytic capacitor was considerably reduced. The control objective in both [11] and [12] was to regulate the dc voltage. A feedback control
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Dynamic Equations: (1)
(2)
Fig. 1. Equivalent circuit of DFIG.
(3) technique using differential geometry theory has been proposed in the control of the full scale back to back converters of a synchronous generator based wind generation system [13] while the electromechanical dynamics of the wind generation system have not been considered. The control objective in [13] was to improve the fault-ride through ability of the wind generation system. With the continuous integration of wind generation systems into power systems, it will be of great importance to propose novel control methods in improving the transient stability of power systems. As far as we know, transient stability control of power systems with the electromechanical dynamics of wind generation systems using differential geometry theory has not been investigated yet. In this paper, a nonlinear control method for a WT with DFIG using differential geometry theory will be investigated in the transient stability control of power systems where the electromechanical dynamics of DFIG are considered. Transient stability control performance of the nonlinear control will be examined in terms of critical clearing time (CCT). The paper is arranged as follows. In Section II, a third-order model of the WT with DFIG for stability studies is presented at ?rst. A nonlinear control design method using differential geometry theory is introduced in Section III, and then in Section IV, the method is applied to the control of the WT with DFIG where the WT with DFIG is integrated into the power system. Based on the relative degree analysis, the nonlinear model of the DFIG is exactly linearized, and the linear optimal control design method is applied to obtain the control law. In Section V, simulations are performed on a single machine in?nite bus system (SMIB) and a four-machine system to verify the effectiveness of the proposed nonlinear controller for the WT with DFIG in the transient stability control, while conclusions are drawn in Section VI. where
Electrical Equations: (4) (5) (6) (7) where is the rotor slip; is the total inertia constant of the is the output active power of the turbine and the generator; stator of the DFIG; is the mechanical power of the WT; is the stator self-inductance; is the rotor self-inductance; is the mutual inductance; is the synchronous angle speed; is the stator reactance; is the stator leakage reactance; is the rotor leakage reactance; is the stator transient reactance; and are the and axis voltages behind the transient is the rotor circuit time constant; reactance, respectively; and are the and axis stator currents, respectively; and are the and axis stator terminal voltages, respectively; and are the and axis rotor voltages, respectively; is the reactive power of the stator of the DFIG. Equation (1)–(3) shows the third-order DFIG model and the voltage equations and the ?ux linkage equations of the DFIG are based on the motor convention. In the case of transient stability studies, it is common to reduce the ?fth-order model to such a third-order model [17], [18]. From (2) and (3), the model of the WT with DFIG is a system reference frame. The inputs are with two inputs in the and , respectively. III. NONLINEAR CONTROL DESIGN METHOD A. Exact Linearization In the conventional linear control design method, the model of the nonlinear system is linearized near a chosen operating point using the ?rst-order Taylor’s series approximation. The accuracy of the approximate linearized model is acceptable only when the system is operating near the point chosen during design. Under large disturbances, the system operating point may be far away from the design point chosen. In this situation,
II. MODEL OF WT WITH DFIG The WT system consists of a drive train and DFIG. The drive train of the WT system is represented with a one-mass model [14], while the DFIG is modeled by a second order model where the electromagnetic transients of the stator are neglected [15]–[18]. The equivalent circuit of DFIG is shown in Fig. 1. Hence, the model of the WT with DFIG used in this paper is a third-order model as follows:
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WU et al.: DECENTRALIZED NONLINEAR CONTROL OF WIND TURBINE WITH DOUBLY FED INDUCTION GENERATOR
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the control capability of PID controllers may be greatly weakened in terms of large disturbance transient stability control. The exact linearization method linearizes the nonlinear model in the global state space or in a large enough region of state space by choosing suitable coordinate transformation. Designing the nonlinear controller based on the exact linearized model, the robust control performance can be guaranteed at any point where the system would operate. In order to simplify the introduction, without loss of generality, a system with two inputs and two outputs is taken as an example to illustrate the principle of the exact linearization. The two inputs and two outputs system can be written as
is a nonlinear vector function, which new state variables; is called a diffeomorphism between two coordinate spaces, satis?es
(10) B. Nonlinear Control Design , When relative degrees of the nonlinear system, equals the order of the model , (9) can be transformed into the Bronovsky norm form as follows:
(8) is the vector of the state variables; is the order where , is the th of the nonlinear system; is the number of control variables or the control variable; output variables; and are the n-dimensional vector is the th output variable; ?elds in the state space; is the scalar function of . as in [7], the origChoosing the similar mapping inal system in (8) can be transformed into the following form: (11) From the th and th equations in (12), we have
(12) where we see the equation at the bottom of the page. With (10) and (12), the corresponding control for the original system can be derived as follows:
(13) where
(9) ( ) to where is the relative degree from output function the nonlinear system; is the relative degree from output function to the nonlinear system; denotes the Lie with respect to ; is the vector of the derivative of
Applying linear optimal principles [19] to the system described by (11), the linear optimal control law can be obtained as follows: (14)
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where is the optimal vector of the control variables for the is the optimal feedback gain matrix and linearized system; is given by
(15) can be found by solving the following Riccati algewhere braic equation:
It can be seen that the system (17) is a two-input two-output nonlinear system. The relative degree of the model of the WT with DFIG can be calculated as follows. For when when then For when .
(16) Substituting (14) into (13), the control law for the original system can be obtained. IV. NONLINEAR CONTROL DESIGN FOR THE WT WITH DFIG A. Model Exact Linearization of the WT With DFIG The model of WT with DFIG, (1)–(3), is written as a standard af?ne nonlinear system as follows:
then Hence, the relative degree of the model of the WT with DFIG, , is 3, which equals to the order of the model. Now we choose the mapping for the coordinate transformation as follows: (21a)
(21b) (21c) Since the mechanical dynamics of the WT system is much slower than the electromagnetic dynamics of the WT system, it is reasonable to assume that the input mechanical power is constant during the transients of the DFIG under a system disturbance. Applying the mapping relationships in (21), the model of the WT with DFIG can be exactly linearized as follows: (22a) (22b) (22c) Substituting (2) and (3) into (22), we have (23a)
(17) where
Choosing the output variables: (18) (19) where is the terminal voltage of the WT with DFIG; is the control reference of the terminal voltage of the WT with DFIG, and the control reference is normally chosen as 1 p.u. reference frame with the Aligning the axis of the becomes zero, while direction of the terminal voltage, is equal to the magnitude of the terminal voltage. Based on (6) and (19), we have (23c) (23b)
B. Nonlinear Control Design for the WT With DFIG Equation (23) can be written as (24a) (24b) (24c)
(20)
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According to the linear quadratic regulator (LQR) method in the linear control theory [19], the optimal control law of the system shown in (24) can be obtained by
Fig. 2. Con?guration of the SMIB system.
(25) (26) are the optimal feedback gain coef?cients; where is the initial mechanical power, which equals to the active power output of the DFIG stator at the initial condition. From (25) and (26), it can be seen that the active power and the terminal voltage are controlled independently via and , respectively. Choosing the weighting matrix in the Riccati equation as identity matrix, the optimal feedback coef?cients can be ob. tained: Using (26), the nonlinear control law for the WT with DFIG is given by
V. DYNAMIC SIMULATIONS The nonlinear control law, as given in (27), was implemented in MATLAB while the parameters of the WT with DFIG is given in the Appendix. Simulations were performed on an SMIB system and a four-machine system to verify the effectiveness and control performances of the nonlinear controller in improving the transient stability, the system damping and the fault ride-through capability of the WT with DFIG. A. Simulations on SMIB System The con?guration of the SMIB system is illustrated in Fig. 2. The SMIB system consists of an in?nite bus, a wind farm (WF) and an induction motor (IM) load, the details can be found in SimPowerSystems of MATLAB v7.1. To make the comparison more accurately, particle swarm optimization was applied to optimize the parameters of the PI controllers of the WT with DFIG. The optimization aims to shift all the eigenvalues as far to the left of the left hand side of the complex plane as possible, so as to achieve better damping characteristic. The details of the optimization and the optimized parameters can be found in [20]. The speed of the wind in the WF is kept constant at 12 m/s. The WF comprises six WTs. A three phase ground fault with , and the ground resistance of 10 was applied at B2 at s. Under this disturbance, the the fault was cleared at terminal voltage of the WT with DFIG dropped to 0.8 p.u. The simulations were carried out to compare the damping of the WT with DFIG using the PI controller, optimized PI controller and nonlinear controller. The details about the PI controller can be found in the SimPowerSystems Demo of MATLAB v7.1. The dynamics of the active power, reactive power, terminal voltage and the dc link voltage of the WT with DFIG using the nonlinear controller, PI controllers and optimized PI controllers are illustrated in Fig. 3. It can be seen that applying the nonlinear control, the system is stable and the damping of the WT system has been improved. The oscillations of the terminal voltage decayed very quickly after the fault was cleared. The peak value of the dc link voltage is 1400 V, which is lower than 1450 V with the PI control used. The improvement is bene?cial to the operation of the converters since this will reduce the stress to the converters. During the transient process, the minimum active power of the WT with DFIG with the proposed nonlinear controller is 0 MW, while the minimum active power of the WT with DFIG using the MW where the WT absorbed conventional PI controller is the active power. This reveals that the WT with DFIG using the conventional PI controller may lead to a worse transient stability problem of the power system. It should also be pointed out that under the large disturbance, the terminal voltage decreased signi?cantly, the WT with DFIG generated more reactive power to
(27a)
(27b) From (27), it can be seen that the nonlinear control law is only related to its own parameters. Hence, the nonlinear control can be considered as a decentralized control, which is a desirable feature of the controller for a WT with DFIG. Under disturbances, the active power and the terminal voltage of the WT with DFIG system may deviate far from the normal and , operating values, which may cause the rotor voltage calculated by (27), exceed the voltage capacity of the feedback and should be limited. In this paper, converter. Hence, for the WT with DFIG, if the calculated rotor voltage exceeds and are scaled proportionally as follows: If the limit, , then set and . The relationship between the active powers of the rotor and stator of the DFIG is implicitly described by the electrical equations of the DFIG. The sum of the rotor reactive power, the stator reactive power and the reactive power exchange between the DFIG and the grid is balanced by the reactive power generated by the dc link. The nonlinear controller proposed in (27) is used to regulate the voltage injections of the rotor side converter to control the slip and the stator terminal voltage to their control settings. While the conventional PI controller is used to regulate the voltage injections of the grid side converter to control the voltage of the dc link and the reactive power exchange between the DFIG and the grid to their control settings.
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B. Fault Ride-Through Capability Analysis The drop in the terminal voltage of WT with DFIG is increasing with the decreasing of the ground resistance of the three phase to ground fault. Smaller is the ground resistance, larger is the drop in the terminal voltage, and the disturbance is more serious. In order to study the fault ride-through capability of the WT with DFIG with the optimized PI controller and the nonlinear controller, simulations were performed under the fault, which is as same as the fault used in Section V-A, with the ground resistance decreasing step by step. When the ground resistance of the fault was decreased to 2.3 , the WT with DFIG with the optimized PI controller was tripped because of the over-voltage in the dc link in the feed-back converters, while the WT with DFIG with the nonlinear controller can still operate and is kept connected with the power system. The dynamics of the WT with DFIG with the optimized PI control and the nonlinear control under the fault with ground resistance of 2.3 are shown in Fig. 4. It can been seen that under the disturbance, the terminal voltage of the dc link of the WT with DFIG using the optimized PI control increased quickly for the accumulation of the active power in the capacitor, and reached the maximum dc voltage limit of 1900 V, and subsequently the dc over-voltage protection was triggered to trip the WT while the output active power and reactive power of the WT with DFIG with the optimized PI controller dropped to zero. However, under the disturbance, the WT with DFIG with the nonlinear control generated more reactive power to support the terminal voltage so that the generator was able to output more active power, hence less active power accumulated in the capacitor and the dc power of the dc link did not reach the dc voltage limit. In this situation, the dc over-voltage was not trigged to trip the WT where the WT was able to recover to the initial equilibrium point. Hence, the WT with DIFG with nonlinear control kept the WT being connected with the grid and recovered to the initial operation point. The simulations were performed with the ground resistance decreased further. When the ground resistance decreased to 2.1 , the WT with DFIG with the nonlinear controller tripped from the power system. It can be concluded that the fault ride-through capability of the WT with DFIG can be improved with the nonlinear controller proposed. C. Simulations on Multimachine System The four-machine system used in the simulations is shown in Fig. 5. The system was divided into two areas, in each of which there were two machines. To investigate the impact of the WT with different controllers such as the optimized PI controller and nonlinear controller on the transient stability performance of the power system, the four-machine system was modi?ed by replacing generator G3 with a wind farm (WF), which is the same as the WF in Fig. 2. The power ?ow of the four-machine system is also shown in Fig. 5. The parameters of the generators, transformers and lines of the four-machine system can be found in SimPowerSystems of MATLAB v7.1. Case 1: A three phase to ground fault was applied at bus 7 . The CCT was calculated under different conditions at such as the original four-machine system, the modi?ed four-machine system with the WT using the PI control (PI), optimized PI
Fig. 3. Dynamics of the WT with DFIG.
restore the terminal voltage, and after the fault, the output reactive power of the WT with DFIG with the nonlinear controller was higher than the initial output. After the fault was cleared, the system became stable at a new equilibrium where the reactive power output at this point is higher than that at the initial equilibrium before the system fault. It can also be concluded that under the large disturbance the improvement in the damping of the WT with DFIG with the optimized PI controllers is not as large as that under the small disturbance [20]. For the convenience of demonstration, only WTs with the optimized PI controller and nonlinear controller are compared in the subsequently part of the paper.
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TABLE I CCT OF THE SIMULATED SYSTEM UNDER DIFFERENT CONDITIONS
From Table I, it can be seen that in comparison to the original four-machine system, the multimachine system with the WT with DFIG using either the PI controller, or the optimized PI control, or the nonlinear controller can actually enhance the transient stability margin of the system in terms of CCT. Furthermore, for the four-machine systems with the WT with DFIG, the CCT using the nonlinear controller is higher than those using the PI controller and the optimized PI controller. The transient stability control bene?t of the four-machine system with the WT using the nonlinear controller is signi?cant. Case 2: A three phase to ground fault was applied in the middle of the transmission line 1 at , and the fault was cleared after 0.2s. From (27), it can be seen that there are only some simple mathematical operations in the nonlinear control law. Hence, the computational requirement of the nonlinear control is comparable to that of the PI control. The simulation program for the multimachine was run on a Pentium 4 2.20-GHz PC with 1 GB of RAM. It takes 35 s to simulate 45 s dynamics of the multimachine when the nonlinear control is applied. When the PI control is applied, it takes 45 s to complete the simulation. Dynamics of the terminal voltage, dc link voltage, and the output active power of the WT with DFIG are shown in Fig. 6(a)–(c), respectively. It can be seen that using the nonlinear controller, the damping of the WT with DFIG was improved and the oscillations decayed very quickly. The damping of G4, which is located in the same area as the WF, was signi?cantly affected. The dynamics of the terminal voltage of G4 is shown in Fig. 6(d). It can also be seen that the damping of the terminal voltage of G4 was much improved with the proposed nonlinear controller.
Fig. 4. Fault ride-through capability of WT with DFIG.
VI. CONCLUSION In this paper a nonlinear controller for a WT with DFIG based on differential geometry theory has been proposed where the electromechanical dynamics have been considered. Based on the relative degree analysis, the linearization via state feedback design method has been employed in the nonlinear control design for the WT with DFIG. The nonlinear control law obtained only depends on the own parameters of the WT with DFIG. Hence, the nonlinear control can be considered as a decentralized control, which is a desirable feature for a WT with DFIG. The dynamic simulation results on the SMIB system have shown that the damping and the fault ride-through capability have been improved using the proposed nonlinear controller. The results of CCT analysis and simulations on the four-machine system have demonstrated that both the transient
Fig. 5. Con?guration of the four-machine system.
control (OPI) and nonlinear control (NC). The results are shown in Table I.
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Wind Turbine: DFIG: pu, Converter:
s, pu, pu
s, pu,
pu/rad pu
pu REFERENCES
Fig. 6. Dynamics of the four-machine system.
stability margin and the system damping have been improved using the nonlinear control proposed. The transient stability control bene?t with the nonlinear control is signi?cant.
[1] P. B. Eriksen, T. Ackermann, H. Abildgaard, P. Smith, W. Winter, and J. M. Rodriguez Garcia, “System operation with high wind penetration,” IEEE Power Energy Mag., vol. 3, no. 6, pp. 65–74, Nov. 2005. [2] M. Yamamoto and O. Motoyoshi, “Active and reactive power control for doubly-fed wound rotor induction generator,” IEEE Trans. Power Electron., vol. 6, no. 4, pp. 624–629, Oct. 1991. [3] R. Pena, J. C. Clare, and G. M. Asher, “Doubly fed induction generator using back-to-back PWM converters and its application to variablespeed wind-energy generation,” Proc. Inst. Elect. Eng., Elect. Power, vol. 143, no. 3, pp. 231–241, May 1996. [4] F. M. Hughes, O. Anaya-Lara, N. Jenkins, and G. Strbac, “Control of DFIG-based wind generation for power network support,” IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1958–1966, Nov. 2005. [5] J. B. Ekanayake, L. Holdsworth, X. G. Wu, and N. Jenkins, “Dynamic modeling of doubly fed induction generator wind turbine,” IEEE Trans. Power Syst., vol. 18, no. 2, pp. 803–809, May 2003. [6] D. Cheng, T. J. Tarn, and A. Isidori, “Global external linearization of nonlinear system via feedback,” IEEE Trans. Autom. Control, vol. 30, no. 8, pp. 808–811, Aug. 1985. [7] Q. Lu, Y. Sun, and S. Mei, Nonlinear Control System And Power System Dynamics. Norwell, MA: Kluwer, 2001. [8] G. Guo, Y. Wang, and D. J. Hill, “Nonlinear output stabilization control for multimachine power system,” IEEE Trans. Circuits Syst., vol. 47, no. 1, pt. Part I, pp. 46–53, Jan. 2000. [9] C. Sun, Z. Zhao, Y. Sun, and Q. Lu, “Design of nonlinear robust excitation control for multimachine power systems,” Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 143, no. 3, pp. 253–257, May 1996. [10] M. J. Jin, W. Hu, F. Liu, S. W. Mei, and Q. Lu, “Nonlinear co-ordinated control of excitation and governor for hydraulic power plant,” Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 152, no. 4, pp. 544–548, Jul. 2005. [11] J. Jung, S. Lim, and K. Nam, “A feedback linearizing control scheme for a PWM converter-inverter having a very small DC-Link capacitor,” IEEE Trans. Ind. Appl., vol. 35, no. 5, pp. 1124–1131, Sep. 1999. [12] D. C. Lee, G. M. Lee, and K. D. Lee, “DC-bus voltage control of threephase AC/DC PWM converters using feedback linearization,” IEEE Trans. Ind. Appl., vol. 36, no. 3, pp. 826–833, May 2000. [13] A. Mullane, G. Lightbody, and R. Yacamini, “Wind-turbine fault ridethrough enhancement,” IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1929–1937, Nov. 2005. [14] Y. Lei, A. Mullane, G. Lightbody, and R. Yacamini, “Modeling of the wind turbine with a doubly fed induction generator for grid integration studies,” IEEE Trans. Energy Convers., vol. 21, no. 1, pp. 257–264, Mar. 2006. [15] F. Mei and B. C. Pal, “Modelling and small-signal analysis of a grid connected doubly-fed induction generator,” in Proc. IEEE Power Eng. Soc. General Meeting 2005, San Francisco, CA, 2005. [16] P. Kundur, Power System Stability and Control.. New York: McGraw-Hill, 1994, pp. 279–306. [17] J. B. Ekanayake, L. Holdsworth, and N. Jenkins, “Comparison of 5th order and 3rd order machine models for doubly fed induction generator (DFIG) wind turbines,” Elect. Power Syst. Res., vol. 67, no. 3, pp. 207–215, Dec. 2003. [18] P. Ledesma and J. Usaola, “Doubly fed induction generator model for transient stability analysis,” IEEE Trans. Energy Convers., vol. 20, no. 2, pp. 388–397, Jun. 2005. [19] H. Kwakernaak and R. Sivan, Linear Optimal Control System. New York: Wiley-Interscience, 1972. [20] F. Wu, X. P. Zhang, K. Godfrey, and P. Ju, “Small signal stability analysis and optimal control of a wind turbine with doubly fed induction generator,” Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 1, no. 5, pp. 751–760, Sep. 2007. Feng Wu received the B.Eng. and M.Sc. degrees from HoHai University, Nanjing, China, in 1998 and 2002, respectively. He is now pursuing the Ph.D. degree at the University of Birmingham, Birmingham, U.K. His research interest is wind turbine modeling and control.
APPENDIX PARAMETERS OF WIND TURBINE SYSTEM Per Unit System: MW, V
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Xiao-Ping Zhang (M’95–SM’06) received the B.Eng., M.Sc. and Ph.D. degrees in electrical engineering from Southeast University, Nanjing, China, in 1988, 1990, 1993, respectively. He worked at NARI, Ministry of Electric Power, China, on EMS/DMS advanced application software research and development between 1993 and 1998. He was visiting the University of Manchester Institute of Science and Technology, Manchester, U.K., from 1998 to 1999. He was a Lecturer and then an Associate Professor at the University of Warwick, Coventry, U.K., until early 2007. Currently, he is a Reader at the University of Birmingham, Birmingham, UK. He is also Director of the Institute for Energy Research and Policy. He is a coauthor of the monograph “Flexible AC Transmission Systems: Modelling and Control” published in 2006. Dr. Zhang is a member of CIGRE. He was an Alexander-von-Humboldt Research Fellow with the University of Dortmund, Dortmund, Germany, from 1999 to 2000.
He is now Professor of Electrical Engineering in the College of Electrical Engineering at HoHai University, Nanjing. Dr. Ju was an Alexander-von-Humboldt Fellow at the University of Dortmund, Dortmund, Germany.
Ping Ju (M’95) received the B.Eng. and M.Sc. degrees in electrical engineering from Southeast University, Nanjing, China, in 1982 and 1985, respectively. In 1988, he received the Ph.D. degree in electrical engineering from Zhejiang University, Hangzhou, China.
Michael J. H. Sterling received the B.Eng. degree with First Class Honours in electronic and electrical engineering at the University of Shef?eld, Shef?eld, U.K., and the Ph.D. degree in computer control in 1971. In 1971, he joined the Department of Control Engineering, University of Shef?eld, as a Lecturer, being promoted to Senior Lecturer in 1978. In 1980, he was appointed as Professor of engineering at the University of Durham, Durham, U.K. In 1989, he was appointed Vice-Chancellor and Principal of Brunel University, London, U.K. He is currently Vice-Chancellor at the University of Birmingham, Birmingham, U.K. Prof. Sterling is a Fellow of the Institute of Measurement and Control, the IEE, and the Royal Academy of Engineering. Professor Sterling has served on the IEE Council, is a former Chairman of the Russell Group, and is also a past President of the IEE and the Institute of Measurement and Control.
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