Variable speed control of wind turbines using nonlinear and adaptive algorithms

Transcription

1 Journal of Wind Engineering and Industrial Aerodynamics 85 (2000) 293}308 Variable speed control of wind turbines using nonlinear and adaptive algorithms Y.D. Song*, B. Dhinakaran, X.Y. Bao Department of Electrical Engineering, North Carolina A&T State University, Greensboro, NC7411, USA Department of Electrical Engineering, University of Virginia, Charlottesville, VA 22903, USA Abstract Automatic control represents one of the most important factors responsible for the e$ciency and reliability of wind power conversion systems. Thus far, only a few isolated applications of conventional control to wind power systems have appeared in the literature. To make wind power generation truly cost-e!ective and reliable, advanced control techniques are imperative. In this paper we develop a control scheme for wind turbine using nonlinear and adaptive control theory. Based on both mechanical and electrical dynamics, nonlinear and adaptive control algorithms are derived to on-line adjust the excitation winding voltage of the generator. This method is shown to be able to achieve smooth and asymptotic rotor speed tracking, as justi"ed by both analysis and computer simulation Elsevier Science Ltd. All rights reserved. Keywords: Wind turbine; Variable speed control; Adaptive control; Nonlinear control; Stability 1. Introduction Several key areas of research in control of wind turbines have been identi"ed during the recent workshop held at Santa Clara University [1]. Of particular interest to wind power industry is the development of innovative control algorithms for smoother and more ezcient operation of wind power generation systems. Traditionally, most wind turbines operate at "xed speeds except when starting and stopping [2]. Fixed-speed operation means that the maximum coe$cient of performance is available only at a particular wind speed. A low coe$cient of performance is observed for all other * Corresponding author. address: songyd@ncat.edu (Y.D. Song) /00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S ( 9 9 )

2 294 Y.D. Song et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 293}308 wind speeds, which reduces the energy output below that which might be expected from variable speed operation [3}13]. As noted by several researchers (e.g. Refs. [3,10,11]), to e!ectively extract wind power while at the same time maintaining safe operation, the wind turbine should be driven according to the following three fundamental modes associated with wind speed, maximum allowable rotor speed and rated power, i.e., Mode 1 * operating at variable speed/optimum tip-speed ratio: u )u)u, Mode 2 * operating at constant speed/variable tip-speed ratio: u )u)u, Mode 3 * operating at variable speed/constant power: u )u)u, which are illustrated in Fig. 1, where u is the cut-in wind speed, u denotes the wind speed at which the maximum allowable rotor speed is reached, u is the rated wind speed and u is the furling wind speed at which the turbine needs to be shut down for protection. It is seen that if high-power e$ciency is to be achieved at lower wind speeds, the rotor speed of the wind turbine must be adjusted continuously against wind speed. A common practice in addressing the control problem of wind turbines is to use linearization approach. This method allows the linear system theory to be applied in control design and analysis. However, due to the stochastic operating conditions and Fig. 1. The operation modes of turbine speed.

3 Y.D. Song et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 293} the inevitable uncertainties inherent in the system, such a control method comes at the price of poor system performance and low reliability [11}13]. In this work we present a method for variable speed control of wind turbines. The objective is to make the rotor speed track the desired speed that is speci"ed according to the three fundamental operating modes as described earlier. This is achieved by auto-adjusting the excitation winding voltage of the generator through the developed nonlinear and adaptive control algorithms. Such a control scheme leads to more energy output without involving additional mechanical complexity to the system. Test of the proposed method based on a two-bladed horizontal axis wind turbine similar to DOE MOD-0 is conducted. Several operating conditions are simulated and satisfactory results are obtained. 2. System modeling The power extraction of wind turbine is a function of three main factors: the wind power available, the power curve of the machine and the ability of the machine to respond to wind #uctuation. The expression for power produced by the wind is given by [1}3] P (u)" C (λ, β)ρπru, (1) where ρ is air density, R is radius of rotor, u is wind speed, C denotes power coe$cient of wind turbine, λ is the tip-speed ratio and β represents pitch angle. Note that the tip}speed ratio is de"ned as λ" Rω u, (2) where ω is the rotor speed. It is seen that if the rotor speed is kept constant, then any change in the wind speed will change the tip-speed ratio, leading to the change of power coe$cient C as well as the generated power out of the wind turbine. If, however, the rotor speed is adjusted according to the wind speed variation, then the tip-speed ratio can be maintained at an optimal point, which could yield maximum power output from the system. From Eqs. (1) and (2) we can see that where P (ω)"k ω, (3) k " 1 2 C R ρπ λ. For a typical wind power generation system, the following simpli"ed block diagram (Fig. 2) is used to illustrate the fundamental work principle. We see that such a system primarily consists of an aeroturbine, which converts wind energy into mechanical energy, a gearbox, which serves to increase the speed and decrease the torque and

4 296 Y.D. Song et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 293}308 Fig. 2. Schematic diagram of wind power system. a generator to convert mechanical energy into electrical energy. Driving by the input wind torque ¹, the rotor of the wind turbine runs at the speed ω. The transmission output torque ¹ is then fed to the generator, which produces a shaft torque of ¹ at generator angular velocity of ω. Note that the rotor speed and generator speed are not the same in general, due to the use of the gearbox. The dynamics of the system can be characterized by the following equations: ¹!¹"J ω #B ω#k θ, (4) ¹!¹ "J ω B ω #K θ, (5) ¹ ω "¹ω, (6) where B, K, B, K are the friction- and torsion-related constants, ¹, ¹, ¹, ¹ the shaft torque seen at turbine end, generator end, before and after gear box, J, J the moment of inertia of the turbine and the generator, and ω, ω the angular velocity of the shaft at turbine end and generator end. γ the gear ratio is de"ned as γ" ω ω. (7) Upon using Eqs. (6) and (7), we can combine Eqs. (4) and (5) to get Jω #Bω#Kθ"¹!γ¹ or equivalently Jω #Bω#Kθ" P ω!γ P ω (8)

5 Y.D. Song et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 293} with J"J #γj, B"B #γb, K"K #γk, where P denotes the wind power given by Eq. (3) and P represents the electric power generated by the system. It is well known that P is related to the excitation current of the generator via [5] P "K ( ω c(i ). (9) where K ( is a machine-related constant, c(.) is the #ux in the generating system, and I is the "eld current. In this work we consider the case that the system is operating over the nonlinear but nonsaturation magnetic range. This can be ensured by suitable pre-compensation. The exciter dynamics of the system is governed by IQ #I R "u, (10) where is the inductance of the circuit, I the "eld current, R the resistance of the rotor "eld, and u the "eld voltage. In designing the control scheme for the system, both the rotor dynamics (8) and the exciter dynamics (10) must be considered in order to achieve good control performance. We shall address this issue in detail in next section. 3. Control design The rotor speed of the wind turbine is controlled through the adjustment of excitation winding voltage, as shown in Fig. 3. The main idea behind this method is to control the reaction torque (power) of the generator via changing the winding voltage, so that the rotor speed is correspondingly adjusted. As such, the control problem can be stated as follows: Design a control voltage (< ) such that the rotor speed (ω) of the wind turbine closely tracks the desired speed (ωh) given according to the three operational modes. It is assumed that ωh, ω H, ωk H are bounded Nonlinear Variable Speed Control For later development, we rewrite Eqs. (8) and (10) as and ω " a!ac(i ) IQ "bu!ai, (11a) (11b)

6 298 Y.D. Song et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 293}308 Fig. 3. Wind power system with "eld excitation. where a " k J, a "! B J, a "!K J, "ω, "ω, " ω dτ, A" γk ( J, a" R and b"1. To design the tracking controller, let us de"ne the rotor tracking error as e"ω!ωh. (12) It then follows that e " a!ac(i )!ω H. (13) Now, we need to design a control scheme such that the tracking error e converges to zero. To this end, we express Eq. (13) as e "!k e#z, (14) where z " a #k e!ac(i )!ω H (15)

7 and k '0 is a design constant. It is interesting to note that if z as de"ned in Eq. (15) is made to approach zero as tpr, then ep0 astpr. Thus our attention now is focused on making z P0. First we take derivative of Eq. (15) w.r.t. time to get z " c a ω!a IQ ω I #k e!ωk H (16) Substituting for ω, e and IQ in the above equation gives z "F!b u, (17) where F " a ω #k a!ac(i ) #Aa c I!k ω H!ωK H, I b "Ab c I. (18) If we design the control voltage u so that u " 1 b (F #k z ), (19) where k '0 is a design constant, we obtain z "!k z (20) from which we readily have that z tends to zero as time increases. To summarize, we have the following result. Theorem 1. Consider the wind power generation system given by Eq. (8) with the xeld exciter as shown in Eq. (10). If the xeld voltage is adjusted according to Eq. (19), in which z is calculated by Eq. (15) and F is generated by Eq. (18), then the rotor speed is ensured to track the desired speed asymptotically. Proof. The result can be justi"ed by considering the Lyapunov function candidate <"0.5e#0.5z, which leads to Y.D. Song et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 293} !1 <Q "! e z k 0 k e z )!min(k, k ) e z. Regarding the control scheme, the following convergence result can be established. Theorem 2. The convergent rate of the tracking error with the proposed control can be found as e, where ν"min(k, k ).

8 300 Y.D. Song et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 293}308 Proof. In fact, from Eqs. (14) and (20), we have e(t)"e(0)exp#exp expz (τ)dτ and z (t) " z (0)e. Therefore, it can be veri"ed that exp#<(0)t exp if k "k, e(t)"e(0) e(0) exp# z (0) (exp!exp) otherwise, k!k implying the convergent rate of the tracking error is at least ν"min(k, k ). Remark. The proposed control scheme can be realized by simply specifying the design parameters k '0 and k '0. There is no ad hoc or trial and error process involved. It should be mentioned that the control scheme is based on the assumption that all the system parameters are available during system operation. New control algorithms are needed if unknown parameters are involved. We address this issue in next section Adaptive Variable Speed Control In practice, those parameters such as k, B and K may not known precisely. For this reason, we now consider the control problem in which the parameters a are unknown. Let a( be the estimate of a and a be the estimate error de"ned by a "a!a(. Motivated by the backstepping method [6], we rewrite the error dynamic equation as e "!k e#z # where z " a, (21) a(!ac(i )!ω H#k e. (22) To derive the control scheme, we take derivative of the above equation and substituting for ω, e and IQ to get z "a( # a( ω #k (!k e#a #z )!A c bu #Aa c I #a( ω H!ωK H I I ω "F!b u # a( ω #k (a ), (23)

9 Y.D. Song et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 293} where F "a( # a( ω #k (!k e#z ) #Aa c I!a( ω H!ωK H, (24) I ω b "Ab c. (25) I Now if the control voltage is adjusted by u " 1 b (k z #F ), (26) the error dynamics become z "!k z # a( ω #k (a ). (27) Consider the Lyapunov function candidate <" 1 2 e#1 2 z #1 2 a. (28) Di!erentiating V along trajectory (27) leads to <Q "ee #z z!a a( "e(!k e#z #a ) #z!k z # a( "!k e!k z #ez #a Mω #k (a )!a a( (29) e #z a( ω #k!a( (30) If the estimate parameters a( are updated via a( "e # a( ω #k z, (31) then <Q "!k e!k z#ez 1 "(ez )!k 0!k e z 4!λ (k, k )(e#z )40. (33) Therefore, we have e3, z 3 and a (i.e., a( ) 3. (32)

10 302 Y.D. Song et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 293}308 From (22) we see that the boundedness of z, e and a( ensures that a( is bounded. Thus Therefore a 3., which, in view of Eq. (22), leads 3 to e By Barbalat Lemma [6], we can conclude that both e and z converge to 3. zero asymptotically and a( remain bounded. Thus the following result is established. Theorem 3. Consider the wind power generation system given by Eq. (8) with the xeld exciter as shown in Eq. (10). If the xeld voltage is adjusted according to Eq. (26), where z is generated by Eq. (22) and a( are updated by Eq. (31), then the rotor speed tracks the desired speed asymptotically. Remark. The power coe$cient k represents one of the most di$cult parameters to be obtained because of its dependence on operating-point. The control scheme developed here does not rely on the precise value of k, which could prove useful in practice. 4. Simulation study The simulation study was performed to verify the e!ectiveness of the proposed control algorithms. The following system parameters are considered. R "0.02Ω, "0.001H, < "480 <, J"16Kg!m, P"8, γ"37.5, f"60hz, k "3, B"52, K"52, k "1.7. ( The control parameters are chosen as k "2, k "54, k "54. The "rst simulation conducted was the tracking of the following desired trajectory ωh"2#sin(t) (rad/s). The tracking pro"le is shown in Fig. 4. The tracking error and the control voltage are depicted in Figs. 5 and 6, respectively. Fig. 7 shows the estimate parameters. It is seen that both nonlinear and adaptive control schemes lead to good tracking performance. The second simulation is based on a more practical situation. Namely we consider a two-bladed horizontal-axis wind turbine similar to DOE MOD-0 [4] as shown in Fig. 8. The speci"cation of the wind turbine is given in Table 1. The rotor speed (ω) is to be adjusted to follow the following desired trajectory: u(k)(u ωh"0,, (u(k)!s ) X (1#sin π 2 d, u(k)(u, X, u(k)(u, (u(k)!s ) X (1!sin π 2 d, u(k)(u, 0, u(k)'u,

11 Y.D. Song et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 293} Fig. 4. Tracking process with nonlinear and adaptive control. Fig. 5. Tracking error. where s " u #u 2 s " u #u 2, d " u!u, 2, d " u!u, 2 u "21.3 m/s, X "4.1 rad/s.

12 304 Y.D. Song et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 293}308 Fig. 6. Control voltage. Fig. 7. Estimate parameters. Note that X (rad/s) is speci"ed according to the allowable rotor speed (r/m of the wind turbine (Figs. 9 and 10). The values for u, u, u are given in Table 1 and u is so chosen to give a smooth shutdown pro"le. It is observed that for both low and high wind speed, the proposed control is able to achieve smooth and precise asymptotic speed tracking. All the internal signals are bounded. We have tested a number of operating points and similar results are obtained (Figs. 11}13).

13 Y.D. Song et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 293} Fig. 8. General view of MOD-0 wind turbine. Fig. 9. Rotor Speed Tracking with Nonlinear Control. 5. Conclusion Variable speed operation of wind turbine is necessary to increase power generation e$ciency. A nonlinear wind turbine control method is explored in this paper. This method is based on the regulation of excitation winding voltage of the generator.

14 306 Y.D. Song et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 293}308 Fig. 10. Rotor Speed Tracking with Adaptive Control. Fig. 11. Tracking error. Table 1 Allowable rotor speed 40 r/m Generator output power 100 kw Optimal coe$cient of performance C Cut-in wind speed u 4.3 m/s Rated wind speed u 7.7 m/s Furling wind speed u 17.9 m/s Rotor diameter 37.5 m Hub height 30 m Coning angle 73 E!ective swept area 1072 m Weight of blades 2090 kg Generator voltage 480 V

15 Y.D. Song et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 293} Fig. 12. Control voltage. Fig. 13. Estimate parameters. Based on both mechanical and electrical dynamics, nonlinear and adaptive control algorithms are derived. Analysis and simulation show that the proposed method is able to achieve smooth and satisfactory rotor speed tracking. In addition, we have carried out a comparison study between the proposed method and the tradition constant speed control method. We found that under the same operating conditions the proposed method is able to gain more power if the wind turbine is operated at variable speed mode by the proposed scheme.

16 308 Y.D. Song et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 293}308 Acknowledgements This work was partially supported by the U.S. National Renewable Energy Laboratory under subcontract REP No. RCX References [1] Mike Robinson, Paul Veers, Wind Turbine Control Workshop, Santa Clara University, Santa Clara, CA, June, [2] S.A. Salle, D. Reardon, W.E. Leithead, M.J. Grimble, Review of wind turbine control, Int. J. Control 52 (6) (1990) 1295}1310. [3] E. Muljadi, C.P. Butter"eld, P. Migliore, Variable speed operation of generators with rotor-speed feedback in wind power applications, Fifteenth ASME Wind Energy Symposium, Houston, Texas, [4] G.L. Johnson, Wind Power Systems, Prentice-Hall Inc., Englewood Cli!s, NJ, [5] A.R. Bergen, Power System Analysis, Prentice-Hall, Englewood Cli!s, NJ, [6] I. Kanellakopoulos, P.V. Kokotovic, A.S. Morse, Systematic design of adaptive controllers for feedback linearizable systems, IEEE Trans. Automat. Control 36 (1991). [7] T. Thirnger, J. Linders, Control of variable speed of a "xed-pitch wind turbine operating in a wide speed range, IEEE Trans. Energy Conversion 8 (3) (1993) 520}526. [8] W.E. Leithead, Dependence of performance of variable speed wind turbine on the turbulence, dynamics and control, IEE proc. 137 (6) (1990). [9] M. Steinbuch, O.H. Bosqra, Optimal output feedback of a wind energy conversion system, Power systems modeling and control applications, in: A.J. Calvaer (Ed.), IFAC Proceeding series, No. 9, [10] D.J. Leith, W.E. Leithead, Implementation of wind turbine controllers, Int. J. Control 66 (3) (1997) 349}380. [11] A. McIver, D.G. Holmes, P. Freere, Optimal control of a variable speed wind turbine under dynamic wind conditions, IAS'96 Conference Record of the IEEE Industry Applications Conference Thirty- First IAS Annual Meeting, [12] Donald S. Zinger, Eduard Muljadi, Andrew Miller, A simple control scheme for variable speed wind turbines, IAS'96 Conference Record of the IEEE Industry Applications Conference 31st IAS Annual Meeting, [13] Gregor E. Van Baars, Peter M.M. Bongers, Wind turbine control design and implementation based on experimental models, Proceedings of the 31st Conference on Decision and Control, Tucson, Arizona- December, 1992.