In this paper, an optimal control scheme for wind turbine output torque and power regulation under the influence of wind disturbances is presented. The system considered is a dynamic mechanical-based model with pitch and generator torque actuators for controlling the pitch and generator torque. The performance of linear matrix inequality (LMI) formalism of linear quadratic regulator (LQR); linear quadratic regulator with integral action (LQRI) and model predictive control (MPC) were compared in response to a step change in wind disturbance. It is shown by Matlab simulation that the LQRI outperformed both LQR and MPC controllers.
Wind energy is one of the primitive sources of energy. The kinetic energy from the wind can be converted to mechanical energy using wind energy conversion systems. Wind energy conversion systems include: wind mill, wind pump and the modern-day wind turbine. The first practical windmill, called Sistan mill was developed in the 7th century in Iran. According to [
Depending on the axis of rotation, wind turbines can be differentiated into two categories: Horizontal axis wind turbine (HAWT) and vertical axis wind turbine (VAWT) (see
Wind turbine systems can be classified into three basic system configurations typically used on wind farms depending on the type of generators used [
tion cost and highly complex control system. Dynamic modeling issues and control methods of wind turbine systems are presented in [
In this paper, we contribute to the further development of control for a class of wind turbine systems by designing improved output torque and power regulation is face of wind disturbances. Several distinct control design approaches are considered:
• Linear quadratic control (LQR) design, for which we establish an LMI formalism yielding a static state-feedback gain.
• Linear quadratic control with Integral action (LQRI), for which we enhance the LQR design by adding an integral term.
• Model predictive control (MPC), for which we establish an LMI formalism yielding a static state-feedback gain.
•H2 control design, for which we establish an LMI formalism based on the H2-norm condition and yielding a dynamic-feedback structure and
• H ∞ control design, for which we establish an LMI formalism based on the H ∞ -norm condition and yielding a dynamic-feedback structure.
All the design procedures are casted into the format of feasibility problem over linear matrix inequalities (LMIs). By this way, effective computational methods are established yielding guaranteed quality solution. Simulation studies are performed for all the approaches yielding good performance results.
Several wind turbine models exist depending on the objective of the modeling. Some models focus mainly on fault detection and identification while other models are control-oriented. Regardless of the objective, the main subsystems usually considered are: aerodynamics, drive train, generator and the grid (for grid connected systems).
The aerodynamics of the system is defined by nonlinear algebraic equations. The aerodynamic power of the turbine is expressed as [
P = 1 2 ρ π R b 2 V w 3 C p ( λ , β )
C p = 0.73 ( 151 λ i − 0.53 β − 0.002 β 2.14 − 13.2 ) e − 18.4 λ i
λ i = ( 1 λ − 0.02 β − 0.003 β 3 + 1 ) − 1 (1)
where ρ represents the air density, R b is the blade radius, V w is the wind velocity and the power coefficient C p is the power coefficient is a highly nonlinear function of λ .
The dynamics of the drive train can be represented with six-mass, three-mass, two-mass and one mass models. The six-mass model is considered most detailed and represents accurately the non-linear behavior of the drive-train. In the six-mass model, six different inertias are defined: the three different blade inertias, hub, gearbox, generator inertia. The three-mass model is can be obtained from the six-mass model by lumping the weights of the three blades and the hub together. In the two-mass model (2), one mass is used to represent the low speed turbine and the high speed generator. The connecting shaft is then modeled as a spring and damper. A simplified single-mass drive train model (3) can be obtained by removing the shaft stiffness and mutual damping from the two-mass model. A single-inertia is thus used to represent the whole system.
ω ˙ r = Q r K r − K s J r ϕ Δ − D s J r ω r + D s J r N g ω g
ω ˙ g = Q g J g − K s J g N g ϕ Δ − D s J r N g ω r + D s J r N g 2 ω g
ϕ ˙ Δ = ω r − ω g N g (2)
ω ˙ m = − f J ω m − T e m + T m (3)
The most common electrical generators connected studied in literature are the Permanent Magnet Synchronous Generator (PMSG) and the Doubly-Fed Induction Generator (DFIG). The nonlinear model of a DFIG-based wind turbine [
ϕ ˙ r d = − R r σ L r ϕ r d + R r L m σ L r L s ϕ s d + ω r ϕ r q + V r d (4)
ϕ ˙ r q = − R r σ L r ϕ r q + R r L m σ L r L s ϕ s q − ω r ϕ r d + V r q (5)
ϕ ˙ s d = − R s σ L s ϕ s d + R s L m σ L r L s ϕ r d + ω s ϕ s d + V s d (6)
ϕ ˙ s q = − R s σ L s ϕ s q + R s L m σ L r L s ϕ r q − ω s ϕ s d + V s q (7)
The dynamic nonlinear model of a PMSG-based wind turbine is described by [
i ˙ s d = − R s L s d i s d + L s q L s d ω d q i s q + V s d L s d (8)
i ˙ s q = − R s L s d i s q − L s q L s d ω d q i s d + ω d q ϕ r d + V s q L s d (9)
Various combinations of the above dynamic models of each subsystem are possible. Models described in literature may be characterized as mechanical- based or electrical-based models. The mechanical based models are more detailed in the model of the aerodynamics and drive train. These models are used for designing pitch and torque actuator controllers [
The linearized dynamic model of the system is defined by (10)-(11). The operating parameters of the system were obtained at 18 m/s of wind velocity. The model consist of a two-mass dive drain, a pitch and generator torque actuator.
δ x ˙ = A δ x + B 1 δ u i n + B 2 δ v i n
y = C δ x (10)
A = [ A 1 0 0 A 2 ] , B 1 = [ B 11 0 0 B 22 ] ,
A 1 = [ 1 J r ∂ Q r ∂ ω r | ω ¯ r D s J r N g − k s J r D s J g N g D s J g N g 2 − k s J g N g 1 1 N g 0 ] , A 2 = [ − 1 τ g 0 0 0 0 1 0 − ω n 2 − 2 ζ ω n ] ,
B 12 = [ 0 0 1 τ g 0 0 ω n 2 ] , B 11 = [ 0 1 J r ∂ Q r ∂ θ | θ ¯ 1 J g 0 0 0 ]
B 2 = [ B 21 0 ] , B 21 = [ 1 J r ∂ Q r ∂ v | v ¯ 0 0 ]
C = [ 0 Q ¯ g 0 Ω ¯ g 0 0 1 0 0 0 0 0 ] (11)
where x = [ ω r , ω g , ϕ , Q g , β 1 , β 2 ] , u = [ Q g , β , Q g r e f , β r e f ] , ω r is the rotor/ mechanical angular speed; ω g is the generator angular speed; ϕ is the torsional speed; Q g is the generator torque; β 1 is the pitch displacement; β 2 is the pitch velocity. Letting B = [ B 1 B 2 ] , u = [ δ u i n δ v i n ] , we cast the wind-turbine system into the compact form:
x ˙ = A x ( t ) + B u (t)
y = C x ( t ) , (12)
Typical numerical values of the model parameters are presented in
Now we look at the optimal control design and express all the design methods as feasibility problems over linear matrix inequalities (LMIs).
With focus on the linear-quadratic regulator (LQR) design, the associated quadratic cost function [
J = ∫ 0 ∞ [ y t ( t ) Q y ( t ) + u t ( t ) R u ( t ) ] d t (13)
where 0 < Q , 0 < R are output error and control weighting matrices. Initially, we present an LMI-based formulation to the LQ control of system (12) while minimizing the quadratic cost (13). We proceed to determine a linear optimal control u = L x that achieves this goal.
Assumption 1. There exists a Lyapunov functional V ( x ) which has the properties:
• V ( x ) = x t K x , K > 0
• There exists γ + > 0 such that x 0 t K x 0 ≤ γ +
• V ˙ ( x ) ≤ − [ x t C t Q C x + u t R u ]
Considering system (12) with linear control u = L x . The following theorem provides an LMI-based LQR design:
| Parameter | Value |
|---|---|
| 5000 KW | |
| 12.1 rpm | |
| 867.637 × 106 N∙m/rad | |
| 6.215 × 106 N∙m/(rad/s) | |
| 534.116 kg/m2 | |
| 3.8768 × 107 kg/m2 | |
| 4.3792 × 107 kg/m2 | |
| 63 m | |
| 1 | |
| 0.1 s | |
| 0.88 rad/s | |
| 0.9 |
Theorem 1 System (12) with the LQR control u = L x is asymptotically stable and J ∞ ≤ V ( x o ) . Given matrices Q > 0 , R > 0 . If there exist matrices S , Y such that
min γ + , Y , S γ + subject to
[ ( A Y + B S ) + ( A Y + B S ) t Y C t Q Y L t R • − Q 0 • • − R ] < 0 (14)
[ γ + x o t • Y ] ≥ 0 (15)
has a feasible solution, then LQR gain matrix is L = S Y − 1 .
Proof. By Assumption (1), and using control u = L x in system (12), the inequality of the derivative of the Lyapunov functional is expressed as
x t [ K ( A + B L ) + ( A + B L ) t K t ] x ≤ − x t [ C t Q C + L t R L ] x (16)
It is evident that (16) is satisfied if there exists L and K such that
[ K ( A + B L ) + ( A + B L ) t K t ] − [ C t Q C + L t R L ] ≤ 0 (17)
Simple computations on (13) in view of Assumption (1) yields J ∞ ≤ V ( x 0 ) . By minimizing the upper bound γ + on the cost x 0 t K x 0 , we obtain
min γ + , K , L γ + subject to (17) (18)
To convexify the above problem, we first express (17) as
Φ = K ( A + B L ) + ( A + B L ) t K t
Π = [ Φ C t Q L t R • − Q 0 • • − R ] ≤ 0 (19)
Pre- and post-multiply (19) by diag { Y , I , I } and using Y = K − 1 , S = L K − 1 it yields (14). Additionally, the inequality bound of the Lyapunov functional can be expressed as
[ γ + x 0 t • K − 1 ] ≥ 0 (20)
which can be manipulated to yield (15). When a feasible solution is attained, we get L = S Y − 1 , K = Y − 1 as desired.
A modified formulation of the LQR (13) is considered with an additional term in the cost representing the integral of the deviation of the output from its initial state z ( t ) = ∫ 0 t y ( τ ) d τ . This formulation will be referred to as the LQRI.
J ˜ = ∫ 0 ∞ ( y ( τ ) t y ( τ ) + ρ u ( τ ) t u ( τ ) + σ z ( τ ) t z ( τ ) ) d τ (21)
Treating z ˙ = C x as additional state variable, we define η = [ x z ] t . The augmented system becomes
η ˙ = [ A 0 C 0 ] η + [ B 0 ] u = A ˜ η + B ˜ u
y ¯ = [ C 0 ] η = C ˜ η (22)
and hence we can re-write (21) as:
J ˜ = ∫ 0 ∞ [ η t ( τ ) Q ˜ η ( τ ) + u ( τ ) t R ˜ u ( τ ) ] d τ
Q ˜ = [ C t C 0 0 σ I ] , R ˜ = ρ I , ρ , σ > 0 (23)
Next, we present an LMI-based formulation to the LQI control of system (22) while minimizing the quadratic cost (23). We proceed to determine a linear optimal control u = L ˜ x that achieves this goal.
Assumption 2. There exists a Lyapunov functional V ˜ ( x ) which has the properties:
• V ˜ ( η ) = η t K ¯ η , K ¯ > 0 ,
• There exists γ + > 0 such that η o t K ¯ η o ≤ γ +
• V ˜ ˙ ( η ) ≤ − [ η t Q ˜ η + u t R ˜ u ]
The following theorem provides an LMI-based LQRI design for system (22) with linear control u = L ˜ x :
Theorem 2 System (22) with the LQRI control u = L ˜ x is asymptotically stable and J ∞ ≤ V ˜ ( η 0 ) . Given matrices Q ˜ > 0 , R ˜ > 0 . If there exist matrices S ˜ , Y ˜ such that
min γ + , Y ¯ , S ¯ γ + subject to
[ ( A ˜ Y ¯ + B ˜ S ¯ ) + ( A ˜ Y ¯ + B ˜ S ¯ ) t Y ˜ Q ˜ Y ¯ L ˜ t R ˜ • − Q ˜ 0 • • − R ˜ ] < 0 (24)
[ γ + x o t • Y ¯ ] ≥ 0 (25)
has a feasible solution, then LQRI gain matrix is L ˜ = S ¯ Y ¯ − 1 .
Proof. By Assumption (2), and using control u = L ˜ x in system (22), the inequality of the derivative of the Lyapunov functional is expressed as
x t [ K ¯ ( A ˜ + B ˜ L ˜ ) + ( A ˜ + B ˜ L ˜ ) t K ¯ t ] x ≤ − x t [ Q ˜ + L ˜ t R ˜ L ˜ ] x (26)
It is evident that (26) is satisfied if there exists L ˜ and K ¯ such that
K ¯ ( A ˜ + A ˜ L ˜ ) + ( A ˜ + A ˜ L ˜ ) t K ¯ t − [ Q ˜ + L ˜ t R ˜ L ˜ ] ≤ 0 (27)
Simple computations on (13) in view of Assumption (1) yields J ∞ ≤ V ˜ ( x 0 ) . By minimizing the upper bound γ + on the cost x 0 t K x 0 , we obtain
min γ + K ¯ , L ˜ γ + subject to (27) (28)
To convexify the above problem, we first express (27) as
Φ = K ¯ ( A ˜ + B ˜ L ˜ ) + ( A ˜ + B ˜ L ˜ ) t K ¯ t
Π = [ Φ Q ˜ L ˜ t R • − Q ˜ 0 • • − R ˜ ] ≤ 0 (29)
Pre- and post-multiply (29) by diag { Y ¯ , I , I } and using Y ¯ = K ¯ − 1 , S ¯ = L K ¯ − 1 it yields (24). Additionally, the inequality bound of the Lyapunov functional can be expressed as
[ γ + x o t • K ¯ − 1 ] ≥ 0 (30)
which can be manipulated to yield (25). When a feasible solution is attained, we get L ˜ = S ¯ Y ¯ − 1 , K ¯ = Y ¯ − 1 as desired.
Remark 1. labelremA The results of Theorems 1-2 establish improved tools for control designer since on one hand it affords the computational effectiveness in determining the feedback gain matrix and the associated cost. On the other hand, it gives an opportunity to compromise between design efficiency and performance.
The model predictive controller performs a discrete-time optimization of of the form [
x ( k + 1 ) = A x ( k ) + B u (k)
y ( k ) = C x ( k ) (31)
min J = x t ( k + n p ) S x ( k + n p ) + ∑ i = 0 n p − 1 x ( k + i ) t Q x ( k + i ) + u ( k + i ) t R u ( k + i ) (32)
where x k ∈ ℜ n , u k ∈ ℜ m , y k ∈ ℜ p , A ∈ ℜ n × n , B ∈ ℜ n × m and C ∈ ℜ q × n . It is desired to synthesize a controller of the form:
x c ( k + 1 ) = A c x ( k ) + B c u (k)
y c ( k ) = C c x c ( k ) (33)
that stabilizes the closed loop system. The closed loop system can be represented as:
ζ ( k + 1 ) = A ¯ ζ ( k ) + B ¯ u (k)
u ( k ) = K ζ ( k ) , y ( k ) = C ¯ ζ ( k ) , B ¯ = [ B 0 ]
ζ = [ x ( k ) x c ( k ) ] , A ¯ = [ A 0 B c A c ]
C ¯ = [ C 0 ] , K ¯ = [ 0 C c ] (34)
We now consider infinite-time horizon formulation of (32) with R > 0 , 0 ≤ Q ¯ = diag [ Q 0 ] :
J = ∑ i = 0 ∞ ζ ( k + i ) t Q ¯ ζ ( k + i ) + u ( k + i ) t R ¯ u ( k + i ) (35)
Again, we define the quadratic Lyapunov function, V ( x ) = ζ T P + ζ , P + > 0 such that
V ( x ( k + i + 1 ) ) − V ( x ( k + i ) ) < − ζ ( k + i ) T Q ¯ ζ ( k + i ) − u ( k + i ) T R u ( k + i ) (36)
The problem is thus cast into the LMI form:
max γ + − 1 s .t .
[ Y I I X ] − γ − 1 ϕ > 0
[ Y I Γ Z 0 0 • X A Δ 0 0 • • Π Ω 0 0 • • • X L T X Q 1 / 2 • • • • R − 1 0 • • • • • 0 ] > 0
[ γ − 1 u max 2 I 0 L • Y I • • X ] > 0 , [ Y I A T C T • X ( A X + C B L ) T • • γ − 1 y max 2 I ] > 0 (37)
γ = Y A + F C , Δ = A X + B L , Π = Y − Q , Ω = I − Q X When a feasible solution to (37) is obtained as V = ( I − Y X ) ( U T ) − 1 , C c = L ( U T ) − 1 , B c = V − 1 F and A c = V − 1 Z ( U T ) − 1 .
Remark 2. labelremD The controllers designed in the foregoing sections enjoy the following features:
• The feedback gains are computed from the feasible solution of convex optimization problems over linear matrix inequalities for which the LMI-solver of Matlab provides an effective software support.
• The realization of the feedback gains is simple and readily programmed in microprocessor ships.
• The developed designs are easily reproducible, repeated and fine-tuned whenever needed.
In this section, we report on the results obtained from wind turbine emulation available at the Distributed Control Research Lab (DCRL). We consider data from the National Renewable Energy Laboratory (NREL) 5 MW offshore wind turbine [
The ensuing results are depicted in Figures 3-7, from which it is evident that the LQRI outperforms the other controllers when the system is subjected to a step change in wind velocity.
Three control algorithms: LQR, LQRI and MPC have been compared for two
different wind turbine models. For system 1, the LQRI was better at rejecting wind velocity disturbances. The same responses were observed in system 2 for both LQR and LQRI controllers with both performing better than the MPC controllers.
We thank the Editor and the referee for their comments. This work is supported by the deanship for scientific research (DSR) at KFUPM through distinguished professorship award project no. IN 141003.
Mahmoud, M.S. and Oyedeji, M.O. (2017) Optimal Control of Wind Turbines under Islanded Operation. Intelligent Control and Automation, 8, 1-14. https://doi.org/10.4236/ica.2017.81001