Real-time Constrained Nonlinear Optimization for Maximum Power Take-off of a Wave Energy Converter

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1 Real-time Constrained Nonlinear Optimization for Maximum Power Take-off of a Wave Energy Converter Thomas Bewley 23 May 2014 Southern California Optimization Day

2 Summary 1 Introduction 2 Nonlinear Model Predictive Control 3 Simulations 4 Conclusions

3 Summary 1 Introduction 2 Nonlinear Model Predictive Control 3 Simulations 4 Conclusions

4 Wave Energy Converters (WECs) Pros Green and emission-free Abundant and widely available Reliable and predictable Cons High installation and maintenance costs Impact on the marine ecosystem

5 Dynamic model of a point-absorber wave energy converter Dynamic equations ma(t) + r v(t) + k p(t) = F R (t) + F D (t) + u(t) + F E (t) Remarks m and r and the mass and viscous dissipation of the device and mooring system k is the hydrodynamic stiffness due to to the buoyancy force

6 Dynamic model of a point-absorber wave energy converter Dynamic equations ma(t) + r v(t) + k p(t) = F R (t) + F D (t) + u(t) + F E (t) Radiation Force Remarks m and r and the mass and viscous dissipation of the device and mooring system k is the hydrodynamic stiffness due to to the buoyancy force

7 Dynamic model of a point-absorber wave energy converter Dynamic equations ma(t) + r v(t) + k p(t) = F R (t) + F D (t) + u(t) + F E (t) Nonlinear Drag Force Remarks m and r and the mass and viscous dissipation of the device and mooring system k is the hydrodynamic stiffness due to to the buoyancy force

8 Dynamic model of a point-absorber wave energy converter Dynamic equations ma(t) + r v(t) + k p(t) = F R (t) + F D (t) + u(t) + F E (t) Machinery Force Remarks m and r and the mass and viscous dissipation of the device and mooring system k is the hydrodynamic stiffness due to to the buoyancy force

9 Dynamic model of a point-absorber wave energy converter Dynamic equations ma(t) + r v(t) + k p(t) = F R (t) + F D (t) + u(t) + F E (t) Excitation Force Remarks m and r and the mass and viscous dissipation of the device and mooring system k is the hydrodynamic stiffness due to to the buoyancy force

10 Hydrodynamic forces t F R (t) = m a(t) k R (t τ)v(τ)dτ = m a(t) F r (t) F D (t) = 1 2 C D ρ S (v(t)) 2 sgn(v(t)) F E (t) = + k E (t τ)η(τ)dτ Remarks F D (t) is a nonlinear and non-smooth function of the device velocity m is the added mass, k R (t) is the causal radiation kernel k E (t) is the noncausal excitation kernel, η(t) is the wave elevation at the device location

11 State-space model State-space realization of F r (t) ż(t) = A p z(t) + B p v(t) F r (t) = C p z(t) + D P v(t) State-space WEC model ẋ = Ax(t) + f D (x(t)) + Bu(t) + E F E (t) Remarks u(t) is the control variable F E (t) is supposed to be known in advance or predicted through estimation techniques

12 Summary 1 Introduction 2 Nonlinear Model Predictive Control 3 Simulations 4 Conclusions

13 MPC formulation The goal is optimizing the WEC power take-off by maximizing the energy absorption over the interval [t 0, t 0 + T]: maxe a = max t0 +T t 0 v(t)u(t)dt subject to mechanical and dynamics constraints. This reduces to the solution of the following nonlinear optimization problem: min 1 2 t0 +T t 0 subject to x(t 0 ) = x 0 x T S T v u + ut S v xdt ẋ(t) = Ax(t) + f D (x(t)) + Bu(t) + E F E (t), t [t 0, t 0 + T] d(x(t), u(t)) 0, t [t 0, t 0 + T]

14 Problem discretization The time interval is divided into N shooting intervals. In each interval, the following conditions must be imposed: u k (t) = u k, t [t k, t k+1 ] F E k (t) = F E k, t [t k, t k+1 ] ẋ k (t) = Ax k (t) + f D (x k (t)) + Bu k + E F E k (t), t [t k, t k+1 ] x k (t k+1 ) = x k+1 d(x k, u k ) 0 The nonlinear dynamic equations are discretized using an explicit fourth-order Runge-Kutta scheme

15 Nonlinear Constrained Programming (1/2) The discretized problem now appears as min 1 2 N 1 k=0 subject to x T k ST v u k + u T k S vx k x 0 = x 0 s x k + t b i K RK i x k+1 = 0, k [0, N 1] i=1 ( ( ) i 1 = A x k + t a ij K RK i 1 j )+f D x k + t a ij K RK j + Bu k + E F E k K RK i j=1 d(x k, u k ) 0, k [0, N 1] j=1

16 Nonlinear Constrained Programming (2/2) Define: w = [u T x T ] T The optimization problem becomes: minj(w) = min 1 2 wt Hw subject to C(w) + Λ x 0 = 0 D(w) 0 where: H = [ 0 ] S v S v T 0 The bar sign accounts for the discretization of the time integral

17 Sequential Quadratic Programming (SQP) Define an initial guess for the optimization variable and the Lagrange multipliers (w 0, λ 0, µ 0 ) At each iteration k, starting from (w k, λ k, µ k ), solve the following quadratic programming problem (QP): min w,λ,µ subject to 1 2 wt B k w + b T k w C wk w + C wk = 0 D wk w + D wk 0 where b k = J(w) wk is the gradient of the cost function and B k is the Hessian of the associated Lagrangian function: B k = H i λ i 2 C i wk j µ j 2 D j wk Perform the update (w k+1, λ k+1, µ k+1 ) = (w k + α w, λ, µ ) Repeat until convergence

18 Implementation features Gradients can be computed analytically or numerically. Many numerical approaches are available: Automatic Differentiation Adjoint Gradient Finite Differences Complex Derivative For the WEC implementation all gradients are computed analytically. The Hessian of the Lagrangian is calculated analytically as well A full step implementation, i.e. α = 1, is considered The Hessian of the cost function is indefinite Inequality constraints involve motion constraints of the device and saturation constraints of the actuator: u min u k u max p min S p x k p max v min S v x k v max

19 Implementation features Gradients can be computed analytically or numerically. Many numerical approaches are available: Automatic Differentiation Adjoint Gradient Finite Differences Complex Derivative For the WEC implementation all gradients are computed analytically. The Hessian of the Lagrangian is calculated analytically as well A full step implementation, i.e. α = 1, is considered The Hessian of the cost function is indefinite Inequality constraints involve motion constraints of the device and saturation constraints of the actuator: u min u k u max p min S p x k p max v min S v x k v max

20 Implementation features Gradients can be computed analytically or numerically. Many numerical approaches are available: Automatic Differentiation Adjoint Gradient Finite Differences Complex Derivative For the WEC implementation all gradients are computed analytically. The Hessian of the Lagrangian is calculated analytically as well A full step implementation, i.e. α = 1, is considered The Hessian of the cost function is indefinite Inequality constraints involve motion constraints of the device and saturation constraints of the actuator: u min u k u max p min S p x k p max v min S v x k v max

21 Implementation features Gradients can be computed analytically or numerically. Many numerical approaches are available: Automatic Differentiation Adjoint Gradient Finite Differences Complex Derivative For the WEC implementation all gradients are computed analytically. The Hessian of the Lagrangian is calculated analytically as well A full step implementation, i.e. α = 1, is considered The Hessian of the cost function is indefinite Inequality constraints involve motion constraints of the device and saturation constraints of the actuator: u min u k u max p min S p x k p max v min S v x k v max

22 Handling the drag term The calculation of the gradient of the equality constraints requires differentiability of the dynamic equations. The drag force, due to the sgn function is non-smooth: F D (t) = 1 2 C D ρ S (v(t)) 2 sgn(v(t)) A smooth approximation of the drag term is: F D (t) = 1 2 C D ρ S (v(t)) 2 tanh(k v(t)) where K is a parameter governing the degree of smoothness. The dynamic equations are now twice differentiable

23 Handling the drag term The calculation of the gradient of the equality constraints requires differentiability of the dynamic equations. The drag force, due to the sgn function is non-smooth: F D (t) = 1 2 C D ρ S (v(t)) 2 sgn(v(t)) A smooth approximation of the drag term is: F D (t) = 1 2 C D ρ S (v(t)) 2 tanh(k v(t)) where K is a parameter governing the degree of smoothness. The dynamic equations are now twice differentiable

24 Condensed SQP Considerations The KKT matrix arising from the QP problem has a very sparse nature This sparsity can be exploited by projecting the cost function to the null space of the equality constraints This allows to remove the dependent variable x, which accounts for the state perturbation, and solve an optimization problem in u only This reduces the optimization space and produces a dense KKT matrix

25 Condensed SQP Looking at the structure of C wk : I + t B C wk = I I + t s i b i x0 K i RK wk I I + t B I + t s i b i xn 1 K i RK wk I Through an appropriate permutation matrix P, it is possible to rewrite the gradient as P C wk = [ L I ] This allows to rewrite the equality constraint in the QP iteration as: x = L u + P C wk Substituting this relationship everywhere gives the reduced dense QP problem: min 1 2 ut H u u subject to d u (u) 0 H u is now positive definite, hence convergence is ensured

26 Condensed SQP Looking at the structure of C wk : I + t B C wk = I I + t s i b i x0 K i RK wk I I + t B I + t s i b i xn 1 K i RK wk I Through an appropriate permutation matrix P, it is possible to rewrite the gradient as P C wk = [ L I ] This allows to rewrite the equality constraint in the QP iteration as: x = L u + P C wk Substituting this relationship everywhere gives the reduced dense QP problem: min 1 2 ut H u u subject to d u (u) 0 H u is now positive definite, hence convergence is ensured

27 Condensed SQP Looking at the structure of C wk : I + t B C wk = I I + t s i b i x0 K i RK wk I I + t B I + t s i b i xn 1 K i RK wk I Through an appropriate permutation matrix P, it is possible to rewrite the gradient as P C wk = [ L I ] This allows to rewrite the equality constraint in the QP iteration as: x = L u + P C wk Substituting this relationship everywhere gives the reduced dense QP problem: min 1 2 ut H u u subject to d u (u) 0 H u is now positive definite, hence convergence is ensured

28 Summary 1 Introduction 2 Nonlinear Model Predictive Control 3 Simulations 4 Conclusions

29 Results for sine wave with H m0 = 2m, T 0 = 8s, T = 10s (1/3) Absorbed power x 10 Machinery Force P a [kw] 1000 F m [N] t [s] p [m] WEC Position t [s] v [m/s] t [s] WEC Velocity t [s] Constraints are u 10 7 N, p 3m/s, v 5m/s

30 Results for sine wave with H m0 = 2m, T 0 = 8s, T = 10s (2/3) Absorbed power x 10 Machinery Force P a [kw] 500 F m [N] p [m] t [s] WEC Position t [s] v [m/s] t [s] WEC Velocity t [s] Constraints are u 10 7 N, p 1m/s, v 5m/s

31 Results for sine wave with H m0 = 2m, T 0 = 8s, T = 10s (3/3) 1000 Absorbed power 1 6 x 10 Machinery Force P a [kw] F m [N] p [m] t [s] WEC Position t [s] v [m/s] t [s] WEC Velocity t [s] Constraints are u 10 6 N, p 1m/s, v 5m/s

32 Results for JONSWAP sea spectrum with H m0 = 2m, T 0 = 8s, T = 10s 2 Wave elevation η [m] t [s] Constraints are u 10 6 N, p 1m/s, v 5m/s

33 Results for JONSWAP sea spectrum with H m0 = 2m, T 0 = 8s, T = 10s 1000 Absorbed power 1 6 x 10 Machinery Force P a [kw] F m [N] p [m] t [s] WEC Position t [s] v [m/s] t [s] WEC Velocity t [s] Constraints are u 10 6 N, p 1m/s, v 5m/s

34 Budal s diagram for sine waves with H m0 = 2m, T = 10s Legend: RL = Resistive Loading, ACC = Approximate Complex-Conjugate Control, AVT = Approximate Optimal Velocity Tracking, MPC = Model Predictive Control, PML = Peak-Matching Latching Control, PMC = Peak-Matching Clutching Control

35 Summary 1 Introduction 2 Nonlinear Model Predictive Control 3 Simulations 4 Conclusions

36 Conclusions Direct multiple shooting provides a fast way of solving constrained MPC problems involving nonlinear systems The condensing step further accelerates the solution of the QP problem Nonlinear MPC applied to the maximization of power take-off of a wave energy converter allows to approach the theoretical limit while outperforming other techniques This approach can easily be extended to any other WEC configuration and constraints

Constrained optimization: direct methods (cont.)

Constrained optimization: direct methods (cont.) Jussi Hakanen Post-doctoral researcher jussi.hakanen@jyu.fi Direct methods Also known as methods of feasible directions Idea in a point x h, generate a