fda@es.aau.dk Department of Electronic Systems, November 29, 211
Agenda
Motivation Main challenges for the application of wind turbine control: Known parameter-dependencies (gain-scheduling); Unknown parameter variations (robustness); Faults (reconfiguration); 3 Simple implementation. uctured Linear Parameter Varying 3 w(k) u(k) Wind turbine y(k) LPV controller ˆθ op (t) ˆθ f (t) Wind speed estimator Fault diagnosis system g. 1: Block diagram of the controller structures. The black boxes are common to e LPV controllers, while the red dashed box illustrates the fault diagnosis system
Motivation Linear parameter-varying (LPV) modeling and control for practical wind turbine control problems. where minimize T z w (θ, α, K) i,2 K K K represents a structural constraint on the controller; 4 θ is a vector of time-varying scheduling parameters; α is a vector of uncertain parameters.
Nominal 6 Christoffer Sloth, Jakob Stoustrup β ref(t) Pitch System V w(t) V (t) q(t) β(t) Q g,ref(t) Q a(t) Ω g(t) Generator P g(t) Aerodynamics Drive train & Converter T a(t) Ω r(t) Q g(t) Tower Fig. 3: Sub-model-level block diagram of a variable-speed variable-pitch WT. BEM aerodynamics (static); f T W(r,t) Fore-aft V (t)(1 a) tower translation Rotorplane (first bending ϕ f Q mode); Flexible two-mass drive-train; Second-order pitch system; β rω r(t)(1 + a ) First order torque delay. ϕ α Chord line 5
Aerodynamics Linearized torque and thrust equations, Q a (t) Q + Q a Q a V ˆV (t) + ˆΩr Ω r (t) + Q a β ˆβ(t) T a (t) T + T a V ˆV (t) + T a Ω r ˆΩ r (t) + T a β ˆβ(t) 6 x 16 5 x 15 1 x 17 Qa/ Ωr [knm/rad/s] 2 4 Qa/ V [knm/rad] 4 3 2 Qa/ β [knm/rad] 1 2 6 5 1 15 2 V [m/s] 1 5 1 15 2 V [m/s] 3 5 1 15 2 V [m/s] 2 x 15 4.5 x 14 x 16 Ta/ Ωr [kn/rad/s] 2 Ta/ V [knm/m/s] 4 3.5 3 2.5 Ta/ β [knm/rad].5 1 1.5 2 4 5 1 15 2 V [m/s] 2 5 1 15 2 V [m/s] 2.5 5 1 15 2 V [m/s]
Aerodynamic Uncertainty Simplification of the aerodynamic phenomena: Blade Element Momentum (BEM) codes; Neglected dynamics (e.g. dynamic inflow); Deviations from the operating points. Q β (θ, α) := Q β + f l (α), f l (α) := a l + b l α where a l, b l characterizes the additive uncertainty for the l-th aerodynamic gain and α is an uncertainty parameter. Λ = {α : α α α}. 7 Q β [Nm/rad] x 17 1 2 3 α = α = α α = α 4 12 14 16 18 2 22 24 26 V [m/s]
Faults Failures that gradually change system s dynamics: Bias and proportional error in sensors: pitch angle, generator speed; Offset of the generated torque due to an offset in the internal power converter control loops; Reduction in conversion efficiency; Altered dynamics of pitch system (Pressure drop, pump wear, high air content in the oil); A comprehensive list of wind turbine faults is given in Sloth et al, 29, Odgaard and Stoustrup, 29, Adegas et al, 211. 8
Faults Example: Pitch system Damping ratio and natural frequency from their nominal values ζ and ω n, to their faulty values ζ f and ω n,f. Convex combination of the vertices of the parameter sets, β(t) = -2ζ(θ f )ω n (θ f ) β(t) ω 2 n(θ f )β(t) + ω 2 n(θ f )β ref (t) ω 2 n(θ f ) = (1 θ f )ω 2 n, + θ f ω 2 n,lp -2ζ(θ f )ω n (θ f ) = -2(1 θ f )ζ ω n, 2θ f ζ lp ω n,lp where θ f [, 1] is an scheduled indicator for the fault. 9 Pitch angle, β [ ] 1.8.6.4.2.5 1 1.5 2
Augmented System Discrete-time LPV system obtained by discretization (Bilinear) of continous-time counterpart, x(k + 1) = A(θ, α)x(k) + B w (θ, α)w(k) + B u (θ, α)u(k) z(k) = C z (θ, α)x(k) + D zw (θ, α)w(k) + D zu (θ, α)u(k) y(k) = C y (θ, α)x(k) + D yw (θ, α)w(k). Affine in scalar functions ρ i (θ) known as basis functions and θ f,m. A B w B u C z D zw D zu + A B w B u C z D zw D zu (ρ i (θ) + f i (α)) C y D yw i C y D yw i + A B w B u C z D zw D zu θ f,m, i = 1,..., n ρ, m = 1,..., n θf. m C y D yw m 1
Augmented System The aerodynamic gains are natural candidates for ρ i (θ), ρ 1 (θ) := 1 Q a J r Ω ρ 2 (θ) := 1 Q a J r V ρ 3 (θ) := 1 Q a J r β ρ 4 (θ) := 1 M t T a Ω ρ 5 (θ) := 1 M t T a V ρ 6 (θ) := 1 M t T a β where the division by J r and M t is adopted to improve numerical conditioning. 11
LPV Controller The LPV controller has the form, x c (k + 1) = A c (θ)x c (k) + B c (θ)y(k) u(k) = C c (θ)x c (k) + D c (θ)y(k), Controller matrices are continuous functions of θ with similar type of dependence, A c (θ) = A c, + B c (θ) = B c, + C c (θ) = C c, + D c (θ) = D c, + n θ n θ n θ n θ n θf ρ i (θ)a c,i + θ f,i A c,nρ+i, n θf ρ i (θ)b c,i + θ f,i B c,nρ+i, n θf ρ i (θ)c c,i + θ f,i C c,nρ+i, n θf ρ i (θ)d c,i + θ f,i D c,nρ+i. 12
Closed-Loop LPV System The controller matrices can be represented in a compact way, [ ] Dc (θ) C K(θ) := c (θ). B c (θ) A c (θ) The interconnection of system and controller leads to the following closed-loop LPV system denoted S cl, S cl : x cl (k + 1) = A(θ, α, K(θ))x cl (k) + B(θ, α, K(θ))w(k) z(k) = C(θ, α, K(θ))x cl (k) + D(θ, α, K(θ))w(k). θ ranges over a hyperrectangle denoted Θ, 13 Θ = { θ : θ i θ i θ i, i = 1,..., n θ }. Rate of variation θ = θ(k + 1) θ(k) belongs to a hypercube denoted V, V = { θ : θ i v i, i = 1,..., n θ }.
Induced L 2 -gain Proposition (L 2 -gain) If there exist K(θ), P(θ, α) = P(θ, α) T and Q(θ) satisfying, r 2 P(θ + θ, α) A(θ, α, K(θ))Q(θ) B(θ, α, K(θ)) P(θ, α) + Q(θ) T + Q(θ) T C(θ, α, K(θ)) T γi D(θ, α, K(θ)) T γi with r = 1, (θ, θ, α) Θ V Λ, then the system S cl is exponentially stable and T zw (θ, α) 2 < γ. >
Lyapunov and Slack Matrices The Lyapunov and slack variables are here defined affine functions of the basis functions, P(θ, α) = P + Q(θ) = Q + n θ n θ n θf (ρ i (θ) + f i (α)) P i + θ f,i P nρ+i ρ i (θ)q i + θ f,i Q nρ+i The Lyapunov function at θ + := θ + θ can be described as, P(θ +, α) = P + n θ n θf ( ρi (θ + ) + f i (α) ) n θf ( ) P i + θ + f,i P nθ +i. Conveniently, the basis functions at θ + are approximated by a linear function of ρ(θ) and θ, ρ i (θ + ) := ρ i (θ) + ρ i(θ) θ, θ 15
Iteration Scheme Sequence of LMI problems: Q(θ) {j} = P(θ) {j 1} ; Gridded parameter space subset denoted Θ g Θ. LMI checked at Θ g Vert(V) Vert(Λ) at each iteration; Minimization of performance level γ; Feasibility phase creates a convergent sequence r j that tries to tend 1. r γ γ 2 1.5 1.5 1 2 3 4 5 6 7 8 9 1 11 15 1 5 1 2 3 4 5 6 7 8 9 1 11 5 4.5 4 3.5 3 2.5 Feasibility Cost Optimization 2 3 4 5 6 7 8 9 1 11 Iteration (j) 16
Numerical Example Fault-Tolerant PI LPV Control for ^ ^ w= v 2 s+2 s+ u= ^ Q g ref Classical Design kdt 2 dt dts(1+s ) 2 s+2 s+ dt dt dt Q g g G p(s, ) k( p ) g ^ q (s+z) 1 s k 2 (s + z 1)(s + 2 ) 2 z 17 k( i y Weighting Functions k( q LPV Controller k 3 (s+z) 3 (s+p) 3
Numerical Example Fault-Tolerant PI LPV Control for Considering G p augmented with the integrator filter as the plant for synthesis purposes, the LPV controller structure reduces to a parameter-dependent static output feedback of the form, 6 K(θ) = D c, + ρ i (θ)d c,i + θ f D c,7, D c,n := [ D p,n D i,n D q,n ], n =, 1,..., 7. Initial K(θ) based on analytical pole placement (tower fore aft mode neglected). ( 2ξ Ω ω Ω Jr + N 2 ) Q g g J g Ng + ρ 1 (θ) Ω g k p (θ) =, N g ρ 3 (θ) ( ) ( k i (θ) = ω2 Ω 1 + ξ 2 Ω Jr + Ng 2 ) J g N g ρ 3 (θ) The tower feedback gain of the initial controller is k q (θ) =, meaning no active tower damping. 18
Numerical Example Fault-Tolerant PI LPV Control for γ {j} k {j} p k {j} i 4 2.8.6.4.4.3.2.1 19 k {j} q.5 1 2 3 4 5 6 7 8 9 j (Iterations) Figure: Evolution of performance level γ and controller gains k p, k i, k q during the iterative LMI synthesis. Controller gains computed at θ op = 15 m/s, θ f =.
Numerical Example Fault-Tolerant PI LPV Control for PI and Tower Feedback Gains kp 1.2 1.8.6 ki.5.4.3.2.4.2 1.8.6 θf.4.2 2 V 15 1.1 1.8.6.4 θf.2 2 V 15 1 2.12.1 kq.8.6.4 1.8.6 θf.4.2 2 V 15 1
Numerical Example Fault-Tolerant PI LPV Control for vhub [m/s] 26 24 22 2 18 16 14 12 1 8 1 2 3 4 5 6 (d) Ωr [rad/s] 1.64 1.62 1.6 1.58 1.56 1.54 1.52 1.5 1.48 Pole Placement LPV PI w/ tower damping 1 2 3 4 5 6 (e).4.35 Pole Placement LPV PI w/ tower damping.15.1 21.3.5 q [m]. q [rad/s].2.5.15.1.1 1 2 3 4 5 6 Pole Placement LPV PI w/ tower damping 1 2 3 4 5 6 (f) (g)
Numerical Example Fault-Tolerant PI LPV Control for β [deg] 22 2 18 16 14 12 1 8 2 Pole Placement LPV PI w/ tower damping 1 (h) (i) β [deg/s] 2 Pole Placement LPV PI w/ tower damping 3 1 2 3 4 5 6 1 2 3 4 5 6 Pg [kw] 2.2 x 16 Pole Placement LPV PI w/ tower damping 2.1 2 1.99 1.98 1.97 1.96 1.95 1 22 1.94 1 2 3 4 5 6 (j)
Numerical Example Fault-Tolerant PI LPV Control for vhub [m/s] ωr [rad/s] 26 24 22 2 18 16 14 12 1 8 1 2 3 4 5 6 1.7 1.65 1.6 1.55 Pg [kw ] θf [ ] 1.9.8.7.6.5.4.3.2.1 5 1 15 2 3 35 4 45 5 55 6 2.1 x 16 2.5 2 1.95 23 1.5 1.9 PI LPV PI LPV (Fault Tolerant) PI LPV PI LPV (Fault Tolerant) 5 1 15 2 3 35 4 45 5 55 6 1.85 5 1 15 2 3 35 4 45 5 55 6
Numerical Example Fault-Tolerant PI LPV Control for.5.45.4.3.4.35.2.1 q [m].3..2 q [rad/s].1.2.15.1.5 PI LPV PI LPV (Fault Tolerant).3.4.5 PI LPV PI LPV (Fault Tolerant) 5 1 15 2 3 35 4 45 5 55 6.6 5 1 15 2 3 35 4 45 5 55 6 22 2 8 6 PI LPV PI LPV (Fault Tolerant) 24 18 4 16 2 β [deg] 14 12 β [deg/s] 2 1 4 8 PI LPV PI LPV (Fault Tolerant) 6 6 5 1 15 2 3 35 4 45 5 55 6 8 5 1 15 2 3 35 4 45 5 55 6
Thank you! Work sponsored by the Danish Ministry of Science, Technology and Innovation under Project CASED (Concurrent Aeroservoelastic Analysis and Design of Wind ).