WIND energy is a renewable energy source. It has

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1 or Standalone Wind Energy Conversion Systems Hoa M. Nguyen, Member, IAENG and D. S. Naidu. Abstract This paper presents a direct uzzy adaptive control or standalone Wind Energy Conversion Systems (WECS) with Permanent Magnet Synchronous Generators (PMSG). The problem o maximizing power conversion rom intermittent wind o time-varying, highly nonlinear WECS is dealt with by an adaptive control algorithm. The adaptation is designed based on the Lyapunov theory and carried out by the uzzy logic technique. Comparison between the proposed method and the eedback linearization method is shown by numerical simulations veriying the eectiveness o the suggested adaptive control scheme. Index Terms wind energy conversion system, permanent magnet synchronous generators, uzzy logic, nonlinear adaptive control. I. INTRODUCTION WIND energy is a renewable energy source. It has developed signiicantly in the last ew decades and now has the astest growth among other renewable energy sources []. Wind energy is harvested by WECS or wind turbines. These wind energy converters can be classiied as gridconnected or standalone depending on their connections to utility grids or local grids. Nowadays most WECS are connected to utility grids. However there is still demand or standalone WECS which provide electrical power or remote areas where utility grids cannot reach. To guarantee continuous energy supply, standalone WECS are combined with other energy sources such as battery storage systems, solar energy systems, diesel generators, etc. resulting in Hybrid Wind Energy Systems (HWES). Due to the presence o other energy sources, the control o HWES is quite dierent rom that o grid-connected WECS. Beside attempting to capture as much power as possible, controllers need to ensure constant power low to local loads. A number o control strategies has been proposed or HWES in [2], [3], [4]. Most o these works ocus on the interaction between WECS and storage systems. Meanwhile, authors in [] deal with the control o maximizing the power conversion o WECS in HWES based on the nonlinear eedback linearization control method. The adaptive control is an attractive control which provides controllers ability to learn as systems and/or environments change [6]. This eature is particularly useul or WECS which are immersed in highly stochastic wind surroundings. Dierent adaptive structures were presented to WECS in [7] [2]. Basically these works study adaptive Proportional Integral Derivative (PID) control using uzzy logic [7], [8] Manuscript received July 2, 22; revised July 3, 22. Hoa M. Nguyen is with the Department o Electrical Engineering, Idaho State University, Pocatello, ID, 832 USA nguyhoa@isu.edu. D. S. Naidu is with the Department o Electrical Engineering, School o Engineering, Idaho State University, Pocatello, ID, 832 USA naiduds@isu.edu. or neural networks [9] [2] to tune PID parameters. Our paper ocuses on an adaptive control scheme based on the input-output linearization control where the adaptive rule is designed based on the Lyapunov analysis. The chosen HWES is the same as in []. The results o the adaptive control will be compared to that o the eedback linearization control in [] proving advantages o the proposed control. The paper is organized as ollows. Section II describes the HWES and the standalone PMSG-based WECS nonlinear model. The adaptive control design is presented in Section III ollowed by its application to the standalone WECS in Section IV. Based on simulation results shown in Section V, some discussions and conclusions are drawn in Section VI. II. STANDALONE WIND ENERGY CONVERSION SYSTEM MODELING A HWES includes a WECS interacting with another source o energy as shown in Fig.. Due to the output power rom the WECS luctuating according to wind changes, other energy sources such as battery or solar systems or diesel generators must be added to ensure a constant power supply to the local grid. Maximum power conversion o the WECS is obtained by adjusting the generator speed ω g as wind speed V changes. This is achieved by modiying the equivalent load at the generator terminal via power electronics converters. The equivalent standalone WECS is depicted in Fig. 2 where R L and L L are the equivalent load resistance and inductance, respectively. The equivalent load resistance is considered the control input or the control system. The dynamic model o the standalone WECS is obtained by combining the aerodynamics, drive train dynamics and generator dynamics. Note that the power electronics dynamics is ignored because it is much aster than the other dynamics. The aerodynamics converts wind lows into aerodynamic torque and mechanical power given respectively as T r = 2 ρπr3 V 2 C Q (λ, β), () P r = 2 ρπr2 V 3 C P (λ, β), (2) where T r is the aerodynamic torque, ρ is the air density, R is the radius o the wind rotor swept area, V is the wind speed, C Q (λ, β) is the torque coeicient, P r is the mechanical power, and C P (λ, β) is the power coeicient. It is seen that both torque and power coeicient are unctions o the so-called tip-speed ratio λ and pitch angle o the wind rotor blades β. The tip-speed ratio is deined as the ratio between the speed at the tip o blades and the wind speed, which is given as λ = ω rr V, (3)

2 Fig.. Hybrid Wind Energy System Power Coeicient Tip speed Ratio Fig. 3. Power Coeicient Curve Fig. 2. Standalone WECS where ω r is the wind rotor rotational speed. However or the optimal power conversion purpose the torque coeicient in () is only dependent on the tip-speed ratio and can be approximated as the ollowing sixth-order polynomial unction o the tip-speed ratio [] C Q (λ) = a 6 λ 6 + a λ + a 4 λ 4 + a 3 λ 3 + a 2 λ 2 +a λ + a. (4) The power coeicient C P (λ) has its maximum value at the so-called optimal tip-speed ratio λ as illustrated in Fig. 3. Thereore, to maximize the power conversion, the WECS must operate at the optimal tip-speed ratio. However, when the wind speed changes, the tip-speed ratio is perturbed away rom the optimal value as seen rom (3). To maintain the optimal tip-speed ratio, the wind rotor speed ω r must be adjusted by the control system. The standalone PMSG model in the direct and quadrature (d,q) rame is given as [] d dt i d = R s + R L i d + p(l q L L ) i q ω g, () L d + L L L d + L L d dt i q = R s + R L i q p(l d + L L ) i d ω g + pφ m ω g, (6) T g = pφ m i q, (7) where i d and i q are the d- and q-components o the stator currents respectively; L d and L q are the d- and q-components o the stator inductances respectively; R s is the stator resistance; R L is the equivalent load resistance; p is the number o pole pairs; Φ m is the linkage lux; ω g is the high-speed or generator speed; and T g is the generator electromagnetic torque. The drive train system consists o a low-speed shat connected to a high-speed shat through a gearbox which is a rotational speed multiplier. The drive train dynamics can be represented by a rigid model as dω g J h = η dt i T r T g, (8) where J h is the equivalent inertia transormed into the highspeed side, η and i are the gearbox eiciency and speed ratio respectively, T r is the aerodynamic torque, and T g is the generator electromagnetic torque. Deining x = [x x 2 x 3 ] T = [i d i q ω g ] T as the state variable vector, u = R L as the control input, and y = ω g is the system output. Combining (), (3), and ()-(8) gives a nonlinear state space model o the standalone PMSG-based WECS: where and ẋ ẋ 2 ẋ 3 } {{ } ẋ = (x) 2 (x) 3 (x) + g (x) g 2 (x) g 3 (x) u, (9) } {{ } } {{ } (x) g(x) y = h(x), () (x) = R s L d + L L x + p(l q L L ) L d + L L x 2 x 3, () 2 (x) = R s x 2 p(l d + L L ) x x 3 + pφ m x 3, (2) 3 (x) = ηρπr3 V 2 C Q (x 3, V ) pφ m x 2, (3) 2iJ h J h g (x) = x, L d + L L (4) g 2 (x) = x 2, () g 3 (x) =, (6) h(x) = x 3. (7) Note that the wind rotor speed ω r is multiplied i times ater going through the gearbox, thereore the generator speed ω g is i times larger than the wind rotor speed. Consequently the torque coeicient C Q (x 3, V ) in (3) can be expressed as C Q (x 3, V ) = a 6 ( Rx3 +a 3 ( Rx3 +a ( Rx3 ) 6 ( ) ( ) 4 Rx3 Rx3 + a + a 4 ) 3 ( ) 2 Rx3 + a 2 ) + a. (8)

3 It is observed rom (8) that the dynamical model (9)-() is highly nonlinear. III. ADAPTIVE CONTROL METHOD Consider a Single Input Single Output (SISO) nonlinear system deined in the region D x R n as ẋ = (x) + g(x)u, (9) y = h(x), (2) where x R n is the state vector, u R is the control input, y R is the system output, (x) R n and g(x) R n are smooth vector ields, and h(x) R is a scalar smooth unction. It is assumed the nonlinear system (9)-(2) has a relative degree r (r < n) at x D x and internal dynamics is stable, taking derivatives o the output y with respect to time up to r times gives y (r) = L r h(x) + L g L r h(x) u, (2) }{{} where L r h(x) is the Lie derivative o h(x) along the direction o the vector ield (x) up to r times, L g L r h(x) is the Lie derivative o L r h(x) a long the direction o the vector ield g(x). Deining α(x) = L r h(x) and β(x) = L gl r h(x) and rewriting (2) yields y (r) = α(x) + β(x)u, (22) Assumption : The unction β(x) is positive. It means that < β(x) < with x D x. Choosing the state eedback linearization control: u (x) = ( α(x) + v), (23) β(x), then the input-output nonlinear relation (22) becomes linear as A. Fuzzy Approximation Strategy y (r) = v. (24) The linear input-output relation (24) can be designed or stabilization or tracking with any linear methods. It is apparent that the ideal eedback linearization control (23) is only applicable i α(x) and β(x) are known. However, there are uncertainties imposed on α(x) and β(x) in practice, thereore the control (23) is not accurate. In this case, adaptive algorithms were proposed to realize the ideal control u (x) by using an approximate nonlinear unction û(x), called direct adaptive control, or by using approximate ˆα(x) and ˆβ(x) o nonlinear unctions α(x) and β(x), called indirect adaptive control [3], [4]. This paper presents the direct adaptive control using the uzzy approximation. The control problem is to drive the output y track a reerence signal y m (y m is a smooth unction). The eedback linearized control input v in (24) is deined as v = y (r) m + ē s + γe s, (2) where γ is a positive constant, ē s and e s are deined as e s = e (r ) o + k e (r 2) o k r e o, (26) ē s = ė s o = k e (r ) o k r ė o, (27) e o = y m y, (28) where e s is the tracking error, e o is the output error. Coeicients k i are chosen such that ollowing polynomial is Hurwitz E(s) = s r + k s r k r 2 s + k r, (29) The ideal control input in (23) is approximated by a uzzy system as û(x) = θ T u ξ u (x), (3) where θ u is a parameter vector including values o singleton membership unction at consequent propositions o the uzzy system rule base, ξ u (x) is a uzzy regressive vector. θ u is updated online such that û(x) approaches u (x). The optimal parameter vector is θ u = arg min θ D θ { sup θ T u ξ u (x) u x D x }. (3) Because u (x) is approximated by a uzzy system possessing a inite number o rules, there exists an unavoidable structural error δ u (x). Thereore the actual ideal control u (x) is u (x) = θ ut ξ u (x) + δ u (x). (32) The dierence between the approximate control û(x) and ideal control u (x) is where û(x) u (x) = θ T u ξ u (x) δ u (x), (33) θ u = θ u θ u (34) is the approximation error. Assumption 2: The adaptive control is chosen such that the structural error is bounded ( δu (x) δu ) with x Dx and the upper bound δ u is known. Due to the presence o the structural error, an additional supervisory control u s is added to guarantee the closed-loop stability. Thereore the real control is B. Adaptive Law Design u = û + u s. (3) The adaptive law is designed based on an Lyapunov analysis is presented here. Adding and subtracting β(x)u (x) into (22) gives y (r) = α(x) + β(x)u (x) + β(x) [u(x) u (x)], = v + β(x) [u(x) u (x)]. (36) Combining (2), (28), (33), (3), and (36) gives o = y m (r) y (r), o = y m (r) v β(x) [u(x) u (x)], o = ē s γe s β(x) [û(x) + u s u (x)], o = ē s γe s β(x) θ u T ξ u (x) + β(x)δ u (x) β(x)u s. (37) Combining (27) and (37) yields the error dynamic as ė s + γe s = β(x) θ T u ξ u (x) + β(x)δ u (x) β(x)u s. (38) Considering a positive semideinite quadratic Lyapunov unction: V = 2β(x) e2 s + 2 θ T u Q u θu, (39)

4 where Q u is a positive deinite weighting matrix. Taking the derivative o V with respect to time, with the observation rom (34) that θu = θ u, yields V = β(x) e sė s 2β 2 (x) e2 s + θ u T Q u θu. (4) Substituting (38) into (4) produces V = e s β(x) [ γe s β(x) θ T u ξ u (x) + β(x)δ u (x) β(x)u s ] 2β 2 (x) e2 s + θ T u Q u θu, V = γe2 s β(x) e su s + e s δ u (x) + θ u T ( ) Qu θu ξ(x)e s 2β 2 (x) e2 s. (4) Choosing the adaptive law: and substituting (42) into (4) gives θ u = Q u ξ u (x)e s, (42) V = γe2 s β(x) e su s + e s δ u (x) 2β 2 (x) e2 s, (43) V γe2 s β(x) e su s + ( δu es (x) + es ). 2β 2 (44) (x) Assumption 3: There exist positive lower bound and upper bound o β(x). It means < β β(x) β. Assumption 4: The velocity o β(x) is bounded. It means βv. Combining (44) with assumptions 3 and 4 yields V γe2 s β e su s + es ( δu + β v 2β 2 es ). (4) Choosing the supervisory control u s as u s = ( δu + β v ) es 2β 2 sgn(e s ), (46) and substituting (46) into (4) gives V γe2 s β. (47) Note that e s sgn(e s ) = es. It is seen rom (39) and (47), the positive semideinite Lyapunov unction V has its negative semideinite derivative V, thereore the closed-loop adaptive system is stable []. IV. ADAPTIVE CONTROL DESIGN FOR THE STANDALONE WECS In this section the adaptive control method presented in Section III is applied to the standalone nonlinear PMSGbased WECS given in (9) and (). The control objective is to track the optimal generator speed reerence ωg in order to maintain the optimal tip-speed ratio λ as the wind speed V changes. Unlike the eedback linearization design in [] where authors simpliied the sixth-order polynomial torque coeicient in (4) by using an approximate secondorder polynomial which captures only the steady-state region, Fig. 4. Gaussian Fuzzy Sets o the Linguistic Variable x 3 Fig.. Structure our study uses the sixth-order polynomial torque coeicient which captures all operating regions. The uzzy model û(x) used to approximate the control u (x) is chosen as Takagi-Sugeno model which has inerence rules in the orm I x is A i and x 2 is A 2i and x 3 is A 3i then û(x) = θ ui, where x, x 2, and x 3 are state variables; A i, A 2i, and A 3i are uzzy sets o linguistic variables describing x, x 2, and x 3 respectively at the i th rule. The orms and number o uzzy sets or linguistic variables are chosen based on trial and error basis. In this system uzzy set orms were chosen as Gaussian and there are ive uzzy sets or each linguistic variable as shown in Fig. 4. Consequently there are total 2 rules. The approximate uzzy system in (3) is where û(x) = θ T u ξ u (x), θ u = [ θ u θ u2...θ u2 ] T, (48) ξ u (x) = [ ξ u (x) ξ u2 (x)...ξ u2 (x) ] T, (49) µ Ai (x ).µ A2i (x 2 ).µ A3i (x 3 ) ξ ui = 2 i= µ A i (x ).µ A2i (x 2 ).µ A3i (x 3 ), () where ξ ui is the regressor at the i th rule. The direct uzzy adaptive control structure is shown in Fig. where the adaptive controller is given in (3) with the adaptive law given in (42), and the supervisory controller is given in (46). V. SIMULATION RESULTS Simulations were carried out with a 3KW standalone PMSG-based WECS which has the optimal power coeicient

5 TABLE I SIMULATION DATA Wind Rotor Drive Train PMSG R = 2. m i = 7 p = 3, R s = 3.3 Ω ρ =.2 kg/m 3 η = L d =.46 H J h =.2 kg.m 2 L q =.46 H L s =.8 H Φ m =.4382 W b Load Resistance (Ohm) Wind Speed (m/s) Fig Time(s) 3 2 Simulation Wind Speed Proile Fig. 8. Tracking Error Square Control Inputs Speed (rad/s) 2 Optimal Generator Speed Fig Tracking Errors Fig Output Reerence Tracking C Pmax =.478 and the optimal tip-speed ratio λ = 7. Other system parameters are given Table I. Lower and upper bounds in assumption 2, 3, and 4 are δ u =., β =, and β v = 3. The nonlinear PMSG-based WECS has the relative degree r = 2, so parameters or the error dynamics are chosen as γ = and k =. The control input is set bounded as < u. The stochastic wind proile is shown in Fig. 6. Control perormances o both Direct Fuzzy Adaptive Control (DFAC) proposed in this paper and (FLC) proposed in [] are compared in parallel. Regarding the output tracking perormance, Fig. 7 and 8 show both DFAC and FLC track the output reerence Power Coeicient Fig Optimal Power Coeicient Maintenance satisactorily. However the DFAC provides better tracking than the FLC does as seen rom the Fig. 9 which indicates tracking errors.

6 The Tip speed Ratio Fig Optimal Tip-speed Ratio Maintenance Regarding the power conversion eiciency, two important actors representing the optimal power extraction are the power coeicient maintenance and tip-speed ratio maintenance under wind speed luctuations. Fig. and prove the DFAC better than the FLC in optimizing the power conversion. It is obviously observed rom those igures that the DFAC stays constantly steady at the optimal power coeicient and tip-speed ratio values ater the transient time. Meanwhile, the FLC keeps oscillating around optimal values. [] I. Munteanu, A. Bratcu, N. Cutuluslis, and E. Ceanga, Optimal Control o Wind Energy Systems: Toward a Global Approach. Springer, 28. [6] K. Astrom and T. Hagglund, Adaptive Control. Dover Publications Inc., Mineola, N.Y., U.S., 28, second Edition. [7] W. Dinghui, X. Lili, and J. Zhicheng, Fuzzy adaptive control or wind energy conversion system based on model reerence, in Control and Decision Conerence, 29. CCDC 9. Chinese, Jun. 29, pp [8] J. Qi and Y. Liu, PID control in adjustable-pitch wind turbine system based on uzzy control, in Industrial Mechatronics and Automation (ICIMA), 2 2nd International Conerence on, vol. 2, May 2, pp [9] X. Yao, H. Wen, Y. Deng, and Z. Zhang, Research on rotor excitation neural network PID control o variable speed constant requency wind turbine, in Electrical Machines and Systems, 27. ICEMS. International Conerence on, Oct. 27, pp [] X. Yao, X. Su, and L. Tian, Wind turbine control strategy at lower wind velocity based on neural network PID control, in Intelligent Systems and Applications, 29. ISA 29. International Workshop on, May. 29, pp.. [], Pitch angle control o variable pitch wind turbines based on neural network PID, in Industrial Electronics and Applications, 29. ICIEA 29. 4th IEEE Conerence on, May. 29, pp [2] Z. Xing, Q. Li, X. Su, and H. Guo, Application o BP neural network or wind turbines, in Intelligent Computation Technology and Automation, 29. ICICTA 9. Second International Conerence on, vol., Oct. 29, pp [3] L.-X. Wang, A Course in Fuzzy Systems and Control. Prentice-Hall. Inc., Upper Saddle River, New Jersey, U.S., 996. [4] H. T. Huynh, Intelligent Control Systems. Ho Chi Minh National University Press, 26. [] H. Khalil, Nonlinear Systems, 2 nd Edition. Prentice-Hall. Inc., Upper Saddle River, New Jersey, U.S., 996. VI. DISCUSSION AND CONCLUSION The simulation results show that the proposed DFAC is very good in dealing with the time-varying, nonlinear nature o WECS. The DFAC was also proven more eective than the FLC regarding the control perormance and power capture. However, the design o DFAC requires certain assumptions to be met as stated in Section III which are not always satisied in practice. For example, it is hard to ind lower and upper bounds o the nonlinear unction β(x) and structural error δ u (x). These values were ound based on the system physical eatures combined with the trial and error method in this paper. Another important note is that both DFAC and FLC require all states to be accessible. As we know that system states are not available sometime and/or somewhere; consequently the DFAC and FLC are not applicable in those situations. Thereore, an extension o this study would be constructing a nonlinear observer to estimate unavailable states and integrating the nonlinear observer into the DFAC scheme. REFERENCES [] M. Stiebler, Wind Energy Systems or Electric Power Generation. Springer-Verlag, 28. [2] C. Abbey, K. Strunz, and G. Joos, A knowledge-based approach or control o two-level energy storage or wind energy systems, Energy Conversion, IEEE Transactions on, vol. 24, no. 2, pp , June 29. [3] S. Teleke, M. Baran, A. Huang, S. Bhattacharya, and L. Anderson, Control strategies or battery energy storage or wind arm dispatching, Energy Conversion, IEEE Transactions on, vol. 24, no. 3, pp , Sept. 29. [4] S. Teleke, M. Baran, S. Bhattacharya, and A. Huang, Optimal control o battery energy storage or wind arm dispatching, Energy Conversion, IEEE Transactions on, vol. 2, no. 3, pp , Sept. 2.