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2017 Maynooth University Wave Energy Workshop

Control of the PTO system of OWCs:

feedback vs model predictive control

Joao C. C. Henriques and A. F. O. Falcao

[email protected]

[email protected]

2017 Maynooth University Wave Energy Workshop / Control of the PTO system of OWCs: feedback vs model predictive control 1/31

Outline of the presentation

PART 1

Generator feedback control: Mutriku test case

Power take-off system: biradial vs Wells turbine

Mathematical model

Generator feedback control law: computation and sensitivity analysis

PART 2

Model predictive latching control: Spar-buoy OWC test case

Power take-off system: biradial turbine equipped with an HSSV

Changes to mathematical model

A discrete control algorithm based on Pontryagin’s Maximum Principle

A new continuous control method based on the Discontinuous GalerkinMethod

Major conclusions


PART1 - Mutriku power plant test case


Hydrodynamic model

Hydrodynamic model developed by Wanan Sheng

High-order method, 2192 variables, 300 frequencies


Installed Mutriku power take-off

Wells turbine with biplane rotor without guide vanes

Rotor diameter: 0.75 m. Generator rated power: 18.5 kW


Biradial turbine to be installed at Mutriku within OPERA H2020 Project

Biradial turbine with fixed guide vanes (Kymaner/IST patent WO/2011/102746)

Rotor diameter: 0.50 m. Generator rated power: 30.0 kW

To be installed at Mutriku in mid-April 2017 and in the Oceantec spar-buoy,deployed at BIMEP site, in Sept. 2017 (OPERA H2020 Project)


8 6 4 2 0 2 4 6 8

Ψ [− ]

0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

[−]

Π

η

Φ

Turbine dimensionless curves

The performance characteristics of a turbine in dimensionless form

Ψ =p

ρinΩ2 d2

Φ =mturb

ρinΩ d3

Π =Pturb

ρinΩ3 d5

η =Π

ΨΦ

d - rotor diameter

For Re > 106 and Ma < 0.3 the dimensionless curves Φ, Π and η are independentof Re and Ma

⇒ mturb (p,Ω, ρin) = ρinΩ d3 Φ(Ψ), Pturb (p,Ω, ρin) = ρinΩ3 d5 Π(Ψ)


Mathematical model

OWC motion is modelled as a rigid piston (LOWC λwaves)

(m2 + A∞22) x2 = −ρwgS2 x2︸︷︷︸buoyancy

−R22︸︷︷︸radiation

+Fd2︸︷︷︸diffraction

−patm S2 p∗︸︷︷︸

air chamber

Air chamber pressure (see [1, 2])

p∗ = −γ (p∗ + 1)Vc

Vc− γ (p∗ + 1)(γ−1)/γ mturb

ρVc

Turbine/generator set dynamics

I Ω∗ = (Tturb − Tgen)Ω−1max =

(ρinΩ

2d5 Π(Ψ) − Tgen

)Ω−1

max

Dimensionless variables

p∗ =p − patm

patmand Ω∗ = Ω/Ωmax

O [x2] ≈ O [p∗] ≈ O [Ω∗] ≈ 1 ⇒ Similar orders of magnitude for errors


Generator feedback control law


If the turbine operates at the best efficiency point ηmax

Pturb ≈ ρatm d5 Π( Ψ(ηmax) )︸︷︷︸const

Ω3 = constΩ3

the generator should follow the turbine in time-average

Feedback control law

Pgen = min(aΩb, P rated

gen

)Tgen = min

(aΩb−1,

P ratedgen

Ω

)

The control law is clipped at generator rated power



Constants “a” and “b” are functions of the WEC hydrodynamics and turbinegeometry but not of sea-state

Usually 2.4 < b < 3.6

To compute “a” and “b” we use a constant rotational speed model

Ω∗ = 0 ⇒ very large inertia

for the Mutriku wave climate (14 sea states)

Further details can be found in [1, 2]



Wells turbineBiradial turbine

1000 2000 3000 4000 5000 6000 7000Ω [rpm]

0

10

20

30

40

50

Ptu

rb[k

W]

Pgen = aΩb [W]

sea states

1000 2000 3000 4000 5000 6000 7000Ω [rpm]

0

10

20

30

40

50

Ptu

rb[k

W]

Pgen = aΩb [W]

sea states

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

0.2

0.4

0.6

0.8

Π

η

Φ

0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.0

0.2

0.4

0.6

0.8

Π ×100

η

Φ


3.0

3.2

3.4

3.6

3.8

4.0b

sea state 3

0.27 kW

0.37 kW

0.46 kW

0.56 kW

0.65 kW

0.75 kW

0.85 kW

0.85 kW

0.94 kW

0.94 kW

1.04 kW

1.04 kW

sea state 7

2.00 kW

2.78 kW

3.57 kW

4.36 kW

5.15 kW

5.93 kW

6.72 kW

6.72 kW

7.51 kW

7.51 kW

8.30 kW

8.30 kW

0.1 0.2 0.3 0.5 1.0 2.0 4.0

a ×10 4

3.0

3.2

3.4

3.6

3.8

b

sea state 11

5.54 kW

7.48 kW

9.42 kW

11.36 kW

13.30 kW

15.24 kW

17.18 kW

19.12 kW

21.07 kW

21.07 kW

0.1 0.2 0.3 0.5 1.0 2.0 4.0

a ×10 4

Annual average

1.06 kW

1.47 kW

1.87 kW2.28 kW

2.69 kW

3.10 kW

3.50 kW

3.91 kW

3.91 kW

4.32 kW

4.32 kW

Biradial turbine power output sensitivity to “a” and “b”


3.0

3.1

3.2

3.3

3.4

3.5

3.6b

sea state 3 0.03 kW

0.10 kW0.16 kW

0.22 kW

0.29 kW0.35 kW

0.42 kW

0.42 kW

0.48 kW

0.48 kW

0.54 kW

0.54 kW

0.61 kW

0.61 kW

sea state 7

0.48 kW

1.45 kW

2.42 kW

3.39 kW

4.36 kW

4.36 kW

5.33 kW

5.33 kW

6.30 kW

6.30 kW

7.27 kW

7.27 kW

8.23 kW

8.23 kW

9.20 kW

9.20 kW

1 2 3 4 5 6 7

a ×10 5

3.0

3.1

3.2

3.3

3.4

3.5

b

sea state 11

0.96 kW

2.87 kW

4.78 kW

6.69 kW

8.60 kW

10.51 kW

12.42 kW

14.33 kW

14.33 kW

16.25 kW

16.25 kW

18.16 kW

18.16 kW

1 2 3 4 5 6 7

a ×10 5

Annual average

0.22 kW0.67 kW1.11 kW

1.56 kW

2.00 kW

2.45 kW

2.45 kW

2.89 kW

2.89 kW

3.34 kW

3.34 kW

3.78 kW

3.78 kW

4.23 kW

4.23 kW

Wells turbine power output sensitivity to “a” and “b”


Comparison time-averaged turbine output power and rotational speed

Biradial turbine Wells turbine

6 8 10 12 14 16

Tp [s]

0.5

1.0

1.5

2.0

2.5

3.0

Hs

[m]

12 3 4 5

67

8

9

10

11

12

13 14

0.93 kW

0.93 kW

2.79 kW

2.79 kW

4.65 kW

4.65 kW

6.51 kW

6.51 kW

8.36 kW

8.36 kW

10.22 kW

10.22 kW

12.08 kW

12.08 kW

13.94 kW

13.94 kW

15.8

0 k

W

17.66 kW

6 8 10 12 14 16

Tp [s]

0.5

1.0

1.5

2.0

2.5

3.0

Hs

[m]

12 3 4 5

67

8

9

10

11

12

13 14

1.39 kW4.12 kW

6.85 kW

9.58 kW

12.31 kW

15.04 kW

17.78 kW

20.51 kW

23.24 kW

25.97 kW

Tur

bine

pow

er

Gen. law clipping

Rot

atio

nal s

peed

6 8 10 12 14 16

Tp [s]

0.5

1.0

1.5

2.0

2.5

3.0

Hs

[m]

12 3 4 5

67

8

9

10

11

12

13 14

65 rad/s

168 rad/s

168 rad/s

272 rad/s

272 rad/s

376 rad/s

376 rad/s

479 rad

/s

479 rad/s

583 rad/s

687

rad/

s

791 rad/s

894 rad/s

998 rad/s

6 8 10 12 14 16

Tp [s]

0.5

1.0

1.5

2.0

2.5

3.0

Hs

[m]

12 3 4 5

67

8

9

10

11

12

13 14

56 rad/s

101 rad/s

146 rad/s

191 rad/s

237 rad

/s

282 rad/s

327 rad/s

372 rad/s

417 rad/s

462 rad/s


PART 2 - Optimal control of the turbine High-Speed Stop Valve (Latching)

rotor

HSSV

The HSSV valve of a biradial turbine with 0.5 m rotor diameter HSSV movie

Latching with the HSSV will be tested at Mutriku, June-July 2017 (OPERA)

HSSV will operate as safety valve in Oceantec MARMOK-A-5, October 2017(OPERA). Buoy: 5 m diameter, 42 m in length and 80 tonnes weight.


Latching control and air compressibility

OWCs vs oscillating body WECs

Latching on OWCs not so effective as in the case of an oscillating-bodyWECs

air compressibility has a spring effect

closing the HSSV does not stop the relative motion between the OWCand buoy

Decreases the forces resulting from latching in comparison with solid-bodybreaking

the valve surface area subject to the chamber pressure is a smallfraction of the area of the OWC free surface

no impact forces - air compressibility spring effect

Removes the constraint of latching having to coincide with an instant of zerorelative velocity between the floater and the OWC


Changes to the mathematical model of Mutriku

Spar-buoy OWC - two body system constrained to oscillate in heave

Buoy: (m1 + A∞11) x1 + A∞12 x2 = −ρwgS1 x1 − R11 − R12 + Fd1 +patm S2 p∗

OWC: A∞21 x1 + (m2 + A∞22) x2 = −ρwgS2 x2︸︷︷︸buoyancy

−R21 − R22︸︷︷︸radiation

+Fd2︸︷︷︸diffraction

−patm S2 p∗︸︷︷︸

air chamber

HSSV simulation requires a new definition of Ψ

Ψ = uΨ = up∗pat

ρinΩ2 d2

where u ∈ 0,1 is the discrete control (close/open)

System of equations written as a 1st-order ODE

x = f(t,x,u) (1)

x are called the states


Optimal control and the Pontriagyn Maximum Principle

Optimal control

Maximization of a performance index based on the control u of the HSSV

max

( ∫T0

L (x,u) dt

)

Example: L (x,u) =Pturb(t,x,u)

ρatmΩ3maxd

5

Using the Pontriagyn Maximum Principle (PMP) the optimal problem isrecast as

max

∫T0

L (x, u) dt︸︷︷︸performance index

−

∫Tf

0

λT (x − f(t,x,u)) dt︸︷︷︸constrained to system dynamics

λ ′s are called adjoint variables


Solution of the Pontriagyn Maximum Principle

The PMP shows that:

along the optimal control path (x,u) the Hamiltonian function H

H (t,x, u,λ) = λT f (t,x,u) + L (x, u) ,

is maximum for the optimal input u subjected to:

x = f(t,x,u)

x(0) = x0

⇒ forward solution

λ = −∇xf(t,x,u)Tλ−∇xL(x, u)T

λ(Tf ) = 0

⇒ backward solution

for t ∈ [ 0,Tf ]

non-causal control - Fd (t) required to compute f for t ∈ [ 0,Tf ]

For complete details see the two classical books of Luenberger [5] andBryson & Ho [6]


Receding Horizon control in a real-time framework


Hardware-in-the-loop tests at Tecnalia test rig

See [3, 4] for further details



0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

L∗ c

[−]

Hs = 2 m

Te [s]

Hs = 4 m

∆TRH = 24 s

∆TRH = 16 s

∆TRH = 10 s

NL (No latching)

8 9 10 11 12 13 14 15 16

Te [s]

0

20

40

60

80

100

120

ε∗ c=(

L∗ c−

L∗ c| N

L

)

/L∗ c| N

L[%

]

8 9 10 11 12 13 14 15 16

Te [s]



3.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

P∗ tu

rb[−

]

Time [s]

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

η t[−

]

Time [s]

-0.4

-0.2

0.0

0.2

p∗

[−]

Time [s]close

open

u[−

]

∆ TRH = 24 s

∆ TRH = 10 s

No latching


Lessons learned from the Tecnalia hardware-in-the-loop tests

To improve stability, all state variables must have the same order of magnitude

The algorithm sometimes opens/closes the HSSV intermittently during shortperiods (less than 1.5 seconds)

The discrete nature of the numerical solution of the PMP problem implies a2nd-order accuracy

A new challenge in the HSSV control

The sub-optimal problem of open/closing the HSSV only at the end ofeach (non-infinitesimal) time step, ∆t = 0.1 ∼ 0.2 s

No solution found in the searched bibliography!

We start the quest for a new solution method...


Discontinuous Galerkin Method

The computational domain is discretized in small time-elements

where

State, adjoint and control variables approximated by a set of Legendrepolynomials, pj (t) such that x =

∑j pj (t)xj in each time element

The numerical solution is continuous within the time elements andallowed to be discontinuous across element boundaries

Solution of the sub-optimal control problem - compute u that maximizes theHamiltonian H in a integral sense

maxu∈ 0,1

∫ te+1

te

H(t,x, u,λ) dt



In the DG Finite Element Method, the original problem

x − f(t, x, u) = 0

is replaced by a weak formulation∫Ie

vh

(dxhdt

− f

)︸︷︷︸state equations

dt + vh(te) [xh(t+e ) − x(t−e )]︸︷︷︸

weakly enforced BC

= 0

vh is the so-called test function in the FEM framework

Applying an affine transformation from t ∈ Ie to τ ∈ [−1,1]∫ 1

−1

vhdxhdτ

dτ+ vh(−1) xh(−1+) =∆t

2

∫ 1

−1

vh f dτ+ vh(−1) x(−1−) (2)

we write (2) as linear system A xh = b(t,x,u) in each element



To improve the convergence rate, a regularization term was introduced

L(x,u) =1

T

∫T0

(Pturb(t,x,u)

ρatmΩ3maxd

5+ ε (1 − u)2

)dt

ε is a small constant

0 2 4 6 8 10 12 14 16 18 200.0075

0.0080

0.0085

0.0090

0.0095

0.0100

0.0105

0.0110

0.0115

0.0120

∆ t = 0.001 s

∆ t = 0.005 s

∆ t = 0.010 s

∆ t = 0.050 s

∆ t = 0.100 s

∆ t = 0.200 s

iteration, n

Power without optimal control

0 2 4 6 8 10 12 14 16 18 20iteration, n

0.0075

0.0080

0.0085

0.0090

0.0095

0.0100

0.0105

0.0110

0.0115

0.0120

∆ t = 0.001 s

∆ t = 0.005 s

∆ t = 0.010 s

∆ t = 0.050 s

∆ t = 0.100 s

∆ t = 0.200 sPower without optimal control

ε = 0.0001 ε = 0.05



close

open

u[−

]

t ime, t [s]0.4

0.2

0.0

0.2

0.4

p∗

[−]

ε = 0.0001

ε = 0.05

no control


Major conclusions

Receding horizon optimal latching algorithm can be used to improve the OWCspar-buoy capture width

Receding horizon time interval between 10 to 24 s

The Discontinuous-Galerkin solves the HSSV open/closing intermittency problemof the discrete optimal control

Probably the greatest advantage of the OWC technology

Simple control of the available power to PTO system by using a HSSV

HSSV can be used for latching and control of the available power to theturbine and generator

The proposed Discontinuous-Galerkin method is an efficient alternative to thewell know Pseudo-Spectral Methods

High-order accuracy, mesh and polynomial (h-p) refinement, local nature ofthe solution, simple parallelization (real-time applications)


Acknowledgements

The work was funded by

Portuguese Foundation for Science and Technology (FCT) through IDMEC

LAETA Pest-OE/EME/LA0022

WAVEBUOY project, Wave-powered oceanographic buoy for long termdeployment, PTDC/MAR-TEC/0914/2014

European Union’s Horizon 2020 research and innovation programme undergrant agreement No 654444 (OPERA Project)

The first author was supported by FCT researcher grant No. IF/01457/2014


References

[1] A. F. O. Falcao, J. C. C. Henriques, Oscillating-water-column wave energy converters and airturbines: A review, Renewable Energy 85 (2016) 1391 – 1424.

doi:10.1016/j.renene.2015.07.086.

[2] J. C. C. Henriques, J. C. C. Portillo, L. M. C. Gato, R. P. F. Gomes, D. N. Ferreira, A. F. O.Falcao, Design of oscillating-water-column wave energy converters with an application toself-powered sensor buoys, Energy 112 (2016) 852 – 867.

doi:10.1016/j.energy.2016.06.054.

[3] J. C. C. Henriques, L. M. C. Gato, A. F. O. Falcao, E. Robles, F.-X. Fay, Latching control of afloating oscillating-water-column wave energy converter, Renewable Energy 90 (2016) 229 – 241.

doi:10.1016/j.renene.2015.12.065.

[4] J. C. C. Henriques, L. M. C. Gato, J. M. Lemos, R. P. F. Gomes, A. F. O. Falcao, Peak-powercontrol of a grid-integrated oscillating water column wave energy converter, Energy 109 (2016)378 – 390.

doi:10.1016/j.energy.2016.04.098.

[5] D. G. Luenberger, Introduction to dynamic systems : theory, models, and applications, J. Wiley& Sons, New York, Chichester, Brisbane, 1979.

[6] A. E. Bryson, Y.-C. Ho, Applied optimal control: optimization, estimation, and control, BlaisdellPublishing Company, Waltham, 1969.


http://dx.doi.org/10.1016/j.renene.2015.07.086

http://dx.doi.org/10.1016/j.energy.2016.06.054

http://dx.doi.org/10.1016/j.renene.2015.12.065

http://dx.doi.org/10.1016/j.energy.2016.04.098

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