On a near-optimal control approach for a wave energy converter in irregular waves

Applied Ocean Research 46 (2014) 79–93

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On a near-optimal control approach for a wave energy converter in irregular waves Umesh A. Korde ∗ Department of Mechanical Engineering, South Dakota School of Mines and Technology, Rapid City, SD 57701, USA

a r t i c l e

i n f o

Article history: Received 15 June 2013 Received in revised form 1 February 2014 Accepted 9 February 2014 Keywords: Wave energy conversion Real time control Irregular waves Impulse response functions Up-wave surface elevation

a b s t r a c t Real-time smooth reactive control and optimal damping of wave energy converters in irregular waves is difficult in part because the radiation impulse response function is real and causal, which constrains the frequency-dependent added mass and radiation damping according to the Kramers–Kronig relations. Optimal control for maximum energy conversion requires independent synthesis of the impulse response functions corresponding to these two quantities. Since both are non-causal (one being odd and other even), full cancellation of reactive forces and matching of radiation damping requires knowledge or estimation of device velocity into the future. To address this difficulty and the non-causality of the exciting force impulse response function, this paper investigates the use of propagating-wave surface elevation up-wave of the device to synthesize the necessary forces. Long-crested waves are assumed, and the approach is based on the formulations of Naito and Nakamura [2] and Falnes [22]. A predominantly heaving submerged device comprised of three vertically stacked discs driving a linear power take-off is studied. The overall formulation leads to smooth control that is near-optimal, given the approximations involved in the time-shifting of the non-causal impulse response functions and the consequent up-wave distances at which wave surface elevation is required. Absorbed power performance with the nearoptimal approach is compared with two other cases, (i) when single-frequency tuning is used based on non-real time adjustment of the reactive and resistive loads to maximize conversion at the spectral peak frequency, and (ii) when no control is applied with damping set to a constant value. Simulation results for wave spectra over a range of energy periods and significant wave heights are compared for the three situations studied. While practical implementation presents engineering challenges, in terms of timeaveraged absorbed power, unconstrained near-optimal control is found to perform significantly better than single-frequency tuning in the spectra with longer energy periods (>10 s for the present device), and somewhat better in the spectra with shorter energy periods (here ≤10 s). © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Wave energy conversion technology has developed rapidly in recent years, and many devices have matured to the point of full-scale ocean operation [1]. Despite the inherent advantages of greater energy density and less variability, wave energy often proves less cost effective than wind and solar power. Active control of the hydrodynamic response of wave energy devices can potentially enable over 3-5 fold increase in overall efficiency, allowing fewer, structurally efficient, smaller devices to meet the required energy generation targets. This could make wave energy cost competitive in many locations and justify the added expense of providing large reactive forces and short term on-board energy storage. As pointed out in the literature [2], hydrodynamic control for optimal energy conversion in irregular waves presents

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fundamental physical difficulties. These arise in part from the fact that a body oscillating in waves also forces the free surface to make its own waves. The radiation impulse response function is causal, which implies that the frequency-dependent added mass and radiation damping satisfy the Kramers–Kronig relations. Optimal control for maximum energy conversion requires independent synthesis of the time-domain impulse response kernels for these two quantities. Both are non-causal, since the ‘impedance kernel’ corresponding to added mass must be an odd function, and that corresponding to radiation damping an even function. Therefore, real-time generation of the control force requires knowledge or estimation of device velocity into the future. Compromise solutions using velocity estimation based on time-series analysis of past velocities were reported several years ago [3,4]. An alternative time-domain latching type control approach using real-time application of non-reactive clutching and declutching forces has been investigated since the mid-seventies by many authors [5–10], etc. Recently, the use of a high-pressure hydraulic power take-off was studied for a heaving buoy type device to optimize the converted hydraulic power in the time

80

U.A. Korde / Applied Ocean Research 46 (2014) 79–93

˛, ˇ ı(t) (· ; t) (ω) (∞) (ω) ω cd D Fa (t) Ff (t) Fl (t) Fr (t) hf (t) hr (t) Hs h (t) h (t) kh m Pf (t) Pw (t) Pnc (t) R S(ω) tc Te tf U

v(t) vm (t) vo (t) vact (t) vnc (t) voC (t) vopt (t) xA xB xR

2 parameters in the 2-parameter spectrum representation Dirac delta function located at t = 0 wave surface elevation at point · and time t frequency dependent radiation damping for the 3disc body infinite-frequency added mass of the three disc body in heave frequency dependent added mass of the 3-disc body (total added mass − infinite-frequency added mass) angular frequency of waves, oscillation damping coefficient representing viscous friction and other dissipation, assumed constant power take-off damping in constant damping case instantaneous reactive force applied in nearoptimal control exciting force in heave on the 3-disc body instantaneous loading force applied in near-optimal control actuator applied force on the 3-disc body impulse response kernel for exciting force radiation impulse response kernel significant wave height inverse Fourier transform of radiation damping (ω) inverse Fourier transform of frequency-dependent added mass (ω) times ω steady stiffness in the hydraulic power take-off in-air mass of the 3-disc body instantaneous power absorbed through single (peak) frequency tuning control instantaneous power absorbed through nearoptimal control instantaneous power absorbed through constant damping without control device radius power spectral density of incoming waves time duration beyond which hr (t) is truncated to zero energy period time duration beyond which hf (t) is truncated to zero wind speed used in power spectrum calculation instantaneous heave velocity of the 3-disc body instantaneous heave velocity under singlefrequency tuning control instantaneous device heave velocity under nearoptimal control actual instantaneous velocity under near-optimal control instantaneous heave velocity under constant damping and without control desired near-optimal velocity of a ‘virtual’ identical device at an up-wave station instantaneous optimal device velocity in heave leading to maximum power absorption distance d up-wave of device device location relative to origin distance dR up-wave of device

domain [11]. For a small, tubular oscillating water column device, a ‘non-predictive’ phase control strategy was considered more recently [12], with the understanding that the radiation impedance was small. Frequency domain ‘complex-conjugate

control’ approaches comprising adjustable reactive loading for selective tuning to changing wave spectra have been studied since the mid-seventies [13–15], etc. Such an approach was tested recently on the Wavestar device in Denmark [19]. A method based on a coupled fuzzy logic–robust controller was used for short term tuning with incoming-wave prediction [17]. A stochastic control approach based on past information only was recently investigated and found to produce good performance relative to optimal time-domain control [18]. ‘Hard’ displacement and force constraints on the primary converter make it necessary but challenging to find a practical optimal control strategy in the time domain, and a Kalman-filter based predictor was used to derive a time-domain constrained optimal controller for a heaving buoy device [19]. The effect of device geometry on the ‘prediction horizon’ for real-time control was studied with a view to velocity/exciting force prediction [20]. Some recently proposed time-domain control approaches were evaluated in [21]. This paper adopts the approach discussed in [2] and later in [22] to infer the necessary velocity values from free surface elevation information ‘up-wave’ of the device (i.e. to the left of the device for waves propagating from left to right). For up-wave distances on the order of 1–2 km, a deterministic understanding of wave propagation may be valid (especially where a predominant wave direction is present) [22–24], etc. A single-mode device with unconstrained oscillations is used here so as to enable greater insight into potential difficulties with such control. The primary energy converter is a submerged body comprised of three vertically stacked discs. Power absorption and control are by means of a double-acting hydraulic cylinder here, but a linear electric generator/motor may also be used instead. It was found in an earlier work [25] that the exciting force and hydrodynamic coefficient variations became flatter and more ‘broad-band’ with increasing depth, leading to narrower inverse Fourier transforms in the time domain. As discussed later, this is advantageous because the nature of these functions determines the up-wave distance at which the wave surface elevation is needed, as shorter distances imply greater validity of the present approach and more economical measurement system hardware in a practical implementation. Performance of the control based on up-wave surface elevation is compared with single-frequency tuning, where a reactive load component cancelling the inherent + frequencydependent reactance and a resistive component matching the radiation damping at the peak frequency is used. The performance is also compared with a case where no control is used and constant damping is applied. Section 2 following this introduction discusses the overall formulation and the rationale behind the method. Section 3 describes the purpose of the up-wave surface elevation measurements and the corresponding distances. Section 4 derives the impulse response functions leading to device response under control. The manner in which up-wave surface elevation records are generated for the present simulations is discussed in Section 5. Simulation procedure for a number of 2-parameter spectra is described in Section 6. Aspects related to practical implementation of the present approach are considered in Section 7. Results for the three cases are discussed in Section 8, and the main findings are summarized in Section 9. 2. Formulation The device used in this work consists of three submerged discs of radius 8 m and stacked at a spacing of 2 m on a single central axis (Fig. 1). With this stack configuration heave motion is expected to predominate. A double-acting hydraulic cylinder reacting against a deeply submerged tension-leg moored platform utilizes the disc heave for energy generation. Single-mode conversion is assumed.

U.A. Korde / Applied Ocean Research 46 (2014) 79–93

81

Fig. 1. Side-view drawing of the device being investigated here (dimensions in m).

This submerged device is used here because results for exciting force and hydrodynamic coefficients are readily available using the method of [26], and because it was found that deeper submergence allowed shorter lengths of time beyond which the radiation and exciting force kernels tended to zero [25]. The equation of motion for heave velocity in an irregular wave train is,





∞

[m + (∞)] v˙ + cd v +

t

hr ()v(t − )d + kh

v()d = Ff + Fr −∞

0

∞

−iωt

hr (t)e

dt = iω(ω) + (ω)



∞

Ff (t) =

hf ()(xB ; t − )d

(2)

With respect to (xB ; t), hf is not causal, and hf (t) = / 0, t < 0. For maximum energy absorption, the reactive terms in Eq. (1) need to be externally forced out, and the rate of energy radiation from the device needs to be equaled by the rate of energy absorption by the power take-off. In some cases, such as with the present hydraulic power take-off, the same mechanism could be used for reactive force cancellation and energy absorption. Thus, for maximum power absorption, the force Fr (t) needs to be



∞

0

2 (ω) cos ωtdω = − 



∞

ω(ω) sin ωtdω





hr (t) sin ωtdt 0



(4)

∞

(7)

Thus,

hr (t) cos ωtdt

1 (ω) = − ω

where the first two terms contain the frequency-independent inertia and stiffness effects only, and can be derived in real time using velocity and acceleration measurements at the current instant without the need for prediction. Fc (t) is the part that needs to provide the correct power absorption rate and cancel the remaining reactive effects. Fc (t) is the inverse Fourier transform of Fc (iω) where Fc (iω) = −[(ω) + cd ]v(iω) + iω(ω)v(iω)

∞

0

(6)

(3)

0

and it follows that (ω) =

v()d + Fc (t) −∞

hr (t) is real-valued, and its causality means that hr (t) = 0, t < 0. Therefore, ω(ω) must be an odd function and (ω) must be an even function of ω. Furthermore, the full inverse Fourier transform of iω(ω) being odd and real and that of (ω) being even and real, the two must be equal to each other for t ≥ 0, and must be equal and sign-opposite for t < 0. Consequently,



t

Fr (t) = [m + (∞)]v˙ + kh

−∞

2 hr (t) = 

(5)

−∞

(1)

A constant cd is used to represent viscous and other dissipative losses, following [22]. Here, the convolution integral on the left represents the radiation force in heave, due to the waves created by device oscillation near the free surface. The radiation impulse response function/kernel hr (t) is causal, because of which only the present and past velocity of the device affect the force at the present. Causality of hr (t) constrains the real and imaginary parts of the Fourier transform to satisfy the Kramers–Kronig relations. In,



Important to note however is that the full inverse Fourier transforms of iω(ω) and (ω) are respectively, odd and even functions and therefore non-causal or acausal. The exciting force Ff (t) may be written in terms of the wave surface elevation  at the device centroid xB as



∞

Fc (t) = −cd v(t) −

∞

h ()v(t − )d + −∞

h ()v(t − )d −∞

(8)

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U.A. Korde / Applied Ocean Research 46 (2014) 79–93

Substitution of Eqs. (6) and (8) into Eq. (1) results in v(t) ≡ vopt (t) with







∞

cd vopt (t) +

2

h ()vopt (t − )d

= Ff (t)

(9)

−∞

∞

[cd ı() + h ()]vopt (t − )d = Ff (t)

2

(10)

−∞

where h and h are the full inverse Fourier transforms of (ω) and iω(ω) respectively, and h (t) =

1 2

1 h (t) = 2



∞

(ω)eiωt dω

−∞ ∞

(11) iω(ω)eiωt dω

−∞





tc

Fc (t) ≈ −cd v(t) −



tc

cd vo (t) +

h ()vo (t − )d

= Ff (t)

(14)

where now v(t) is replaced by vo (t), which is the device velocity under near-optimal control as enabled by Fr (t) based on the approximate Fc (t). 3. Use of up wave surface elevation As mentioned previously, a deterministic analysis of propagation in the predominant wave direction may be valid over distances on the order of 1–2 km. The surface elevation for a uni-directional wave propagating from left to right over x > 0 is described as

h ()v(t − )d

(12)

−tc



1 (xB ; t) = 2

(15)

∞

(xB ; ω)eiωt dω

(16)

−∞

The complex amplitude A(iω) contains both amplitude and phase information [22]. Ignoring any viscous attenuation effects, if xA is a point to the left of, i.e. up-wave of xB , so that xB − xA = d, then for the frequency ω and wave number k(ω), (xB ; iω) = e−ik(ω)d (xA ; iω)

(17)

The time-domain relation between the surface elevations  at x = xA and x = xB is generally not causal [2,22], given the theoretical presence of zero-frequency or infinite-period waves. However, only a range of frequencies is relevant in practical wave spectra, and though longer water waves travel faster, a maximum velocity limit  is reached in finite water depth h. Past this limit (= gh), group velocity equals phase velocity, propagation becomes nondispersive, and the relation between  at x = xA and x = xB becomes causal.2 This fact is used to define the time-shifted impulse response functions below and to determine the location of the surface elevation used in control. The exciting force Ff (iω) in the frequency domain can be written as



∞

Ff (iω) =

Ff (t)e−iωt dt

(18)

−∞

In order to approximate Fc (t) as in Eq. (12), estimates of velocity up to t → tc are required, in addition to a memory of values up to tc into the past. Eq. (5) also has an integral with limits from −∞ to ∞, as hf (t) = / 0, t < 0, so that (xB , t) must in theory be known/recorded from t→ − ∞ and known or predicted up to t→ ∞ into the future. Again, in practice, hf (t) → 0, t < − tf1 and hf (t) → 0, t > tf2 , where tf1 and tf2 may be equal and set to tf (alternatively, tf = max [tf1 , tf2 ] may be used) and Ff (t) may be approximated as, tf

hf ()(xB ; t − )d

where

tc

h ()v(t − )d + −tc

Ff (t) =

2



(xB ; iω) = A(iω)e−ik(ω)xB

h (t) and h (t) are even and odd functions respectively as argued / 0, and h (t) = / 0 when before, and hence are non-causal, or h (t) = t < 0. For this reason, negative values of  must be admitted in the two convolution integrals in Eq. (8). This means that the synthesis of the two integrals in Eq. (8) ideally requires future velocity values up to v(t + ∞) as → − ∞, in addition, of course to past velocity values down to v(t − ∞) as → ∞. Therefore in theory the controller must have infinite memory for past velocity and must know or be able to predict velocity infinitely long into the future. In practice, hr (t) → 0, t > tc where tc is several seconds long, implying that once a motion is over tc seconds old it stops significantly affecting the present radiation force. Now h (t) = h (− t) = hr (t)/2, t > 0 with h (0) = hr (0)/2. Further, h (t) = − h (− t) = hr (t)/2, t > 0 and h (0+ ) = hr (0)/2. Thus, even though h (t) and h (t) are non-causal, both become vanishingly small beyond time tc in either direction. Therefore, only velocity values up to time tc into the past and velocity values time tc into the future will contribute significantly to the integrals in Eq. (8). These should clearly be included in the integrals, however. The error in the control force then (relative to the optimum) will depend on the order of magnitude at which hr (t) is truncated for the purpose of determining tc . Consequently, the control force in Eq. (8) can only be near-optimal in a practical application, and





−tc

Equivalently,



Finally, with Fc (t) approximated as in Eq. (12), the device dynamics under near-optimal control will reduce to,

(13)

−tf

Thus, in addition to estimates of velocity v(t) a time duration tc into the future, also required are estimates of free surface elevation (xB ; t) a time duration tf further into the future in order to compute the exciting force Ff if this is to be used in the velocity estimation.1

and Eq. (5) transforms to, Ff (iω) = Hf (iω)(xB ; iω) where



∞

Hf (iω) =

hf (t)e−iωt dt

(19)

(20)

−∞

is the complex frequency response function for the exciting force. Following arguments similar to [22], one can construct a version hfd (t) of hf (t) that is right-shifted on the time axis by tf seconds so that, hfd (t) = hf (t − tf )

(21)

Since hf is truncated to zero for t < − tf , hfd is approximately causal, or hfd (t) ≈ 0, t < 0. In the frequency domain, Hfd (iω) = Hf (iω)e−ik(ω)d

(22)

1

In harmonic waves, such estimates are straightforward to obtain, and hence such control is relatively easy to implement, as demonstrated by Salter et al. and Budal and Falnes in the seventies, and there is no need for a time domain treatment. In strictly monochromatic waves at a known frequency, Eq. (1) becomes a differential equation for that frequency with coefficients constant at that frequency.

2 In case h→ ∞, a finite depth hs may be used for which the exciting and radiation force impulse response functions very closely approach those for infinite depth [22].

U.A. Korde / Applied Ocean Research 46 (2014) 79–93

where d = vmax tf ,

and

vmax =



If the device were a distance dc up-wave of the location xB , (23)

gh

vmax is the fastest wave speed (for the longest wave) in the spectrum at a roughly constant water depth h [22]. From Eq. (17), (xA ; iω) = eik(ω)d (xB ; iω)

(24)

Ff (iω) = Hf (iω)(xB ; iω) ≈ Hfd (iω)(xA ; iω)

(25)

Taking the inverse Fourier transform of Eq. (25),



tf

Ff (t) ≈

hf ()(xB ; t − )d = −tf

hfd ()(xA ; t − )d

(34)

Defining xR as a station dc up-wave of xA (= xB − d), such that xR = xA − dc , (xA ; iω) = e−ik(ω)dc (xR ; iω)

(35)

Hfd (iω)(xR ; iω)

voC (iω) =

(36)

2[cd + (ω)]

In the time domain, this can be expressed as



2tf

(26)

0

2tR

voC (t) =

hoC ()(xR ; t − )d

where 1 hoC (t) = 2



∞

−∞

Hfd (iω) 2[cd + (ω)]

(27)

hc (t) = h (t − tc )

Since h and h are approximately zero for |t| > tc , hc and hc are approximately causal, i.e. ≈0, t < 0. In light of the above discussion, a distance dc can be defined in the up-wave direction such that



gh

(28)

(38)

with

vmax =



(39)

gh

Generation of the near-optimal Fc (t) therefore requires use of the surface elevation information a distance dR up-wave of the device. 4. Response under control The inverse Fourier transform of Eq. (33) can be expressed as,



2tf

vo (t) =

In the frequency domain, this defines two functions,

eiωt dω

is causal to the approximation implicit here, and tR = tf + tc . Further, xR = xB − d − dc = xB − dR , dR = vmax (tf + tc ) = vmax tR ,

hc (t) = h (t − tc )

with vmax =

(37)

0

(xA ; t) is the surface elevation at a station xA up-wave of the device. The present and past values of (xA ; t) from 0 to 2tf are required for a good approximation of Ff (t) at the present instant. The near-optimal control force to be synthesized according to Eq. (12) has two integrals with non-causal impulse response kernels h and h . Analogously to hfd , the time-shifted versions of h and h can be found as

dc = vmax tc ,

voC (iω) = vo (iω)eik(ω)dc

implying that,

where d is as determined in Eq. (23). Therefore,



83

ho ()(xA ; t − )d

(40)

0

c (iω) = (ω)e−ik(ω)dc

(29)

c (iω) = (ω)e−ik(ω)dc

where ho (t) =

With the waves propagating from left to right, were the device placed a station dc up-wave of its current location xB at a distance xC , where xC = xB − dc , then subject to response linearity, its velocity in the frequency domain would be related to the current device velocity according to,

vc (iω) = eik(ω)dc v(iω)

(30)

In light of Eqs. (29) and (30), the control force Fc (iω) in the frequency domain can be expressed as, Fc (iω) = −c (iω)eik(ω)dc v(iω) − cd v(iω) + iωc (iω)eik(ω)dc v(iω) = −c (iω)vc (iω) − cd v(iω) + iωc (iω)vc (iω)

1 2



∞

−∞

Hfd (iω)

eiωt dω

2[(ω) + cd ]

Thus, ho (t) with respect to (xA ; t) is seen to be equal to hoC (t) with respect to (xR ; t), (xR = xA − dc ), confirming that here the velocity values at t for the ‘virtual’ device dc up-wave provide the future velocity values at (t + tc ) at the device location xB , with dc = vmax tc , vmax = gh. ho (t) for the present device is shown in Section 8. In a response simulation in the time domain according to Eq. (1) or in experimental testing, it is straightforward, knowing the physical parameters of the device and its infinite-frequency added mass in heave to construct the causal part of the control force as,

 (31)

(41)

t

Fs (t) = [m + (∞)]v˙ (t) + kh

v()d

(42)

−∞

which in the time domain takes the form,



Fc (t) = −



2tc

cd vc (t) +

 +

Application of this force requires real time measurement of the device acceleration and displacement. With Fs applied, the device velocity in the frequency domain is,



hc ()vc (t − )d 0

2tc

hc ()vc (t − )d

(32)

0

where the two impulse response kernels hc and hc are causal to the degree of approximation implied in Eq. (12). The velocity under near-optimal control in the frequency domain can be expressed as,

vo (iω) =

Hfd (iω)(xA ; iω) 2[cd + (ω)]

(33)

[iω(ω) + (ω) + cd ]v(iω) = Ff (iω) + Fc (iω)

(43)

Fc (iω) may be constructed based on the second of equations (31). Using the desired vo (iω) and voC (iω) to generate Fc (iω), Fc (iω) =

iωc (iω)Hfd (iω) 2[cd + (ω)] −

cd Hfd (iω) 2[cd + (ω)]

(xR ; iω) −

(xA ; iω)

c (iω)Hfd (iω) 2[cd + (ω)]

(xR ; iω) (44)

84

U.A. Korde / Applied Ocean Research 46 (2014) 79–93

Substitution of Eq. (44) into (43) and some algebra leads to v(iω) ≡ vact (iω), [cd + 2(ω)]Hfd (iω)

vact (iω) =

2[cd + (ω)][iω(ω) + (ω) + cd ]

(xA ; iω)

[iωc (ω) − c (ω)]Hfd (iω)

+

2[(ω) + cd ][iω(ω) + (ω) + cd ]

(xR ; iω)

(45)

Introducing the impulse response functions hA (t) and hR (t)



1 2

hA (t) =



1 2

hR (t) =

∞

[cd + 2(ω)]Hfd (iω) 2[cd + (ω)][iω(ω) + (ω) + cd ]

−∞ ∞

[iωc (ω) − c (ω)]Hfd (iω) 2[(ω) + cd ][iω(ω) + (ω) + cd ]

−∞

eiωt dω (46) eiωt dω



vact (t) in the time domain can be evaluated as





2tf

vact (t) =

hR ()(xR ; t − )d

0

(47)

0

The impulse response functions hA and hR for the present device are discussed in Section 8. In a physical experiment their role is performed by the device itself. If the present simulations work correctly, vact (t) must be nearly identical to vo (t). In a practical implementation any difference between vact and vo arising from disturbances could be corrected using negative feedback based on the difference (and its derivative and integral if needed). However, a controller leading to maximum power absorption will still need voC up-wave (or future-velocity estimates such as based on time-series models) as, in a sense, a ‘feedforward’ input. The part of the control force Fc responsible for power absorption can be expressed in the frequency domain as, Fl (iω) = −

c (iω)Hfd (iω) 2[cd + (ω)]

(xR ; iω) −

cd Hfd (iω) 2[cd + (ω)]

so that, in the time domain,





2tR

Fl (t) = −

(48)

2tf

hl1 ()(xR ; t − )d − 0

(xA ; iω)

hl2 ()(xA ; t − )d (49) 0

The impulse response functions hl1 and hl2 are plotted in Section 8, and can be found as follows. hl1 (t) =

1 2

1 hl2 (t) = 2



∞

−∞ ∞ −∞

c (ω)Hfd (iω) 2[cd + (ω)] cd Hfd (iω) 2[cd + (ω)]

iωc (iω)Hfd (iω) 2[cd + (ω)]

e

(xR ; iω)

(51)

2tR

Fa (t) =

ha ()(xR ; t − )d

(52)

0

where ha (t) =

1 2



∞

−∞

U=

 4 1/4 g 5

⇒˛=

ˇ

ωe

 gH 2 s

2U

(55)

ˇ

The wave amplitude A(ω) for a chosen frequency is found as



A(ω) =

S(ω) 2

(56)

which is consistent with the inverse Fourier transformation procedure used here for computing (x ; t). Thus, the surface elevation at xR is generated using 1 (xR ; t) = 2



∞

A(ω)e−i[k(ω)xR −ωt+(ω)] dω

(57)

−∞

where (ω) is the random phase angle mentioned above. The surface elevation at xA is computed as 1 2



∞

A(ω)e−i[k(ω)xA −ωt+(ω)] dω

(58)

−∞

6. Simulations

dω

and in the time domain as,



(54)

(50) iωt

The reactive part of Fc (t) can be written in the frequency domain as, Fa (iω) =

4

With ˇ = 0.74, for a given significant wave height Hs and energy period Te = 2/ωe ,

(xA ; t) =

eiωt dω



g ˛g 2 exp −ˇ Uω ω5

S(ω) =

2tf

hA ()(xA ; t − )d +

transducers, wave rider buoys, or accelerometers on very small surface floats. Despite their power requirements non-contact sensing techniques such as radar based systems may be worth considering on a long-term basis. In case significant reflections from the device or shore exist, measurements at least two additional points in the predominant propagation direction would be needed for separation of the incident and reflected wave components at xA and xR . In the present simulations, the surface elevation  at xR is generated numerically for Pierson-Moskowitz type 2-parameter spectra over a range of energy periods and significant wave heights. The elevation at xA here is also generated numerically using the known coordinate xA . Phase angle for each wave component is generated using a random-number generator between [0, 2] (but see [11]). Based on [27], we use the following 2-parameter spectrum,

iωc (iω)Hfd (iω) 2[cd + (ω)]

eiωt dω

(53)

The impulse response function ha for the present device is shown in Section 8. 5. Wave surface elevation at xR and xA In a practical implementation, the instantaneous wave surface elevation at xR and xA could be measured using submerged pressure

Time domain calculations were performed for the present submerged 3-disc device with disc radius R of 8 m and disc vertical spacing of 2 m. Submergence depth of the topmost disc was 4 m, to prevent loss of static submergence up to wave amplitudes approaching 2 m. The hydraulic power take-off was assumed to be capable of generating the large control forces required in swelldominated irregular waves. Energy storage was also assumed to be available near the device, so that the required reactive power could be supplied. The exciting force and hydrodynamic coefficients in heave were computed using an integral equation solution code [28] based on the method of [26]. These results were verified numerically using the code HYDRAN based on [29–31]. The effect of hydrodynamic interaction among the discs on the exciting force and hydrodynamic coefficients was ignored, and their respective added masses were summed to get the total added mass and radiation damping values were added to obtain the total radiation damping for the device. The exciting force contributions of the three discs were added by summing the real and imaginary parts separately and then finding the amplitude and phase. A uniform disc thickness of 0.35 m was used for each disc, and a steady stiffness

U.A. Korde / Applied Ocean Research 46 (2014) 79–93

kh = 10 × 103 kN/m was used for the hydraulic power take-off. Further cd = 500 Ns/m was also assumed. A water depth of h = 225 m was used in this work. A program developed within Matlab was used to simulate the velocity response in a number of wave spectra at different Hs and Te . The standard Discrete Fourier transformation routine in Matlab was used here for the various inverse Fourier transform calculations. For a spectrum with a chosen Hs –Te combination, the response was computed over 10 min. vact (t) and vo (t) were compared over this duration. For the control forces applied, vact (t) was plotted alongside Ff (t) to test whether velocity was synchronous with exciting force. Also computed was the average power absorbed over the 10-min interval, and this was compared later with the incident power at the given Hs and Te . As mentioned, the device response and absorbed power were also found when (i) single-frequency tuning control was used, where the forces were tuned to cancel the reactive terms and match the radiation damping at the spectral peak frequency (‘peak-frequency tuning’); and when (ii) just constant linear damping was applied with no active control. In the case with single-frequency tuning, the device velocity in heave could be expressed in frequency domain as, Hfd (iω)(xA ; iω)

vm (iω) =

(59)

iω[(ω) − o ] + [(ω) + o + 2cd ]

where o is the frequency-dependent added mass at the peak frequency in the spectrum, and o is the frequency-dependent radiation damping for the device at the peak frequency. No measurement of (xA ; t) is required for this case in practice, and only periodic updating of the incoming-wave spectrum is needed (as implemented in [19]). The simulation here uses it simply as a means to evaluate the time domain velocity vm (t) within the existing formulation. Thus, in the time domain,



2tf

vm (t) =

hfc ()(xA ; t − )d

(60)

0

where



1 2

hfc (t) =

∞

−∞

Hfd (iω) iω[(ω) − o ] + [(ω) + o + 2cd ]

eiωt dω

(61)

This impulse response function for the 3-disc heaving device is shown in Section 8. For the case with constant damping D and no active control, the velocity in the time domain is evaluated using,



2tf

vnc (t) =

hnc ()(xA ; t − )d

(62)

0

where hnc (t) =

1 2



∞

−∞

Hfd (iω) iω[m + (∞) + (ω)] − i(1/ω)k + [(ω) + cd + D]

Section 8 plots the impulse response function hnc for the present device. The incident power in kW incident over the device diameter 2R for a given Hs and Te combination was found using Pinc = 0.49Hs2 Te (2R)

(64)

The average power absorbed in the three cases could be found over a simulation duration of [0, T] seconds as follows. Pw =

1 T



T

[Fl (t) + cd vact (t)]vact (t)dt

(65)

0

for near optimal control. The instantaneous reactive power to be supplied to provide this control can be found as, Pa =

1 T



T

Fa (t)vact (t)dt 0

(66)

85

For single-frequency tuning, Pf =

1 T



T

[o + cd ]v2m (t)dt

(67)

0

With no control and constant damping, Pnc =

1 T



T

Dv2nc (t)dt

(68)

0

The average capture efficiency in the three situations above was found by dividing the average absorbed powers above by the average incident power in Eq. (64). The simulation used discrete-time approximations for the various continuous-time expressions in this treatment. The sampling period was linked to the calculation time window (and frequency range). 7. Practical considerations Device velocities are left unconstrained in this work. Increasingly larger velocity maxima may result under near-optimal control in spectra with longer energy periods. The corresponding displacement amplitudes may be greater than can be supported by the power take-off hardware, and in addition may cause surfacing of the discs. Velocity constraints will therefore be required in practice, limiting the amount of conversion efficiency gains enabled by this control approach. Large deflections/velocities also challenge the small-amplitude approximation, and the present linear theory based approach may no longer be sufficient. The formulation may then need to be amended to handle large-amplitude nonlinearities. These issues are not examined further here, but it is noted that the average efficiency results in the next section for Te ≥ 15 s would likely be revised downward once velocity/displacement constraints are introduced. If wave rider buoys or submerged pressure transducers are used for up-wave surface elevation measurement, this information will need to be transmitted to the device using telemetry or acoustic modem hardware. Computational requirements for direct wave profile measurement using wave rider buoys or submerged pressure transducers should be modest. The right-shifted impulse response functions here used can be synthesized on finite impulse response (FIR) digital filters. Evaluation of each convolution integral will likely involve about 103 products and sums at each sampling interval. With the sampling intervals on the order of 100 ms, practical implementation may be feasible using a modern Digital Signal Processor (DSP) or other types of rapidly developing embedded parallel processors.

eiωt dω(63)

The current formulation assumes long-crested waves, for which there exists a well-defined up-wave direction. For small enough wave amplitudes, short-crested seas could be considered as a linear superposition of waves approaching from a sector of directions. A small number of point sensors (wave rider buoys or submerged pressure transducers) distributed over an arc could be used, with Eqs. (49) and (52) being amended to include all measurements. This and the issues touched on below may be addressed in more detail in a future publication. As already mentioned, large amounts of energy need to flow between the power take-off and the primary converter in order to provide near optimal velocity. These amounts increase as the spectral energy period exceeds the natural resonant period of the device. For instance, the natural resonant period for the device under present conditions is ∼6.5 s, so considerably high reactive forces

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U.A. Korde / Applied Ocean Research 46 (2014) 79–93

Fig. 2. Exciting force magnitude for the 3-disc body in heave.

and energy may be required for near-optimal velocity in many realistic swell-dominated spectra. Consequently, near-optimal control needs a large energy storage unit on board and/or the ability to exchange energy with the grid until a sufficient amount can be stored. Secondly, any dissipative losses in the power take-off will remove energy from the system, and reduce the overall conversion efficiency [32,19]. Actuator inertia may also be large and needs to be considered in the overall formulation. The results discussed below should therefore be seen as nearoptimal from a mainly hydrodynamic control point of view, and an additional step comprising important practical design along with a similarly detailed simulation should be considered in further work. 8. Discussion of results Device hydrodynamics is important for smooth control (as distinct from ‘stop and release’ type latching control) as the control forces to be applied by the actuator are strongly dependent on the exciting force and hydrodynamic coefficient variations. Smaller radiation damping implies that large velocities must be produced in order to generate large control forces and achieve large power absorption. Large added mass on the other hand results in a large reactive force and large instantaneous reactive power with the consequent need for a large amount of stored energy. Figs. 2 and 3 plot the exciting force magnitude and phase respectively for the 3-disc device. The peak is seen to be at kR ∼ 0.7 or ω ∼ 0.93 rad/s, which translates into a wave period of 6.8 s. The shorter waves in spectra with energy periods roughly ≥10 s can influence the device response considerably, without contributing much energy to the spectrum. Figs. 4 and 5 show the radiation damping and added mass variations. Greater submergence of the discs leads to a smaller radiation force overall. However, the infinite-frequency added mass is present regardless of proximity to the free surface and places a lower limit on the minimum reactive forces needed. Greater submergence also causes flatter frequency variations as noted in [25] which implies narrower impulse response kernels for the radiation and exciting forces, resulting in shorter distances for up-wave surface elevation measurement. Figs. 6 and 7 plot the impulse response kernels just mentioned. It is seen that hr (and consequently h and h ) →0 beyond t → 20.0 s, and hf → 0 beyond |t| → 15 s, and these values may be used, respectively, as tc and tf here for the determination of distances dc and d

Fig. 3. Exciting force phase for the 3-disc body in heave.

and the synthesis of the near-optimal control forces. Fig. 8 plots the impulse response kernel for the constant damping case [Eq. (63)]. Considerable ringing is noticed in the response, perhaps partly due to the smallness of the damping terms relative to the rest. Fig. 9 plots the impulse response with single-frequency tuning [Eq. (61)]. As expected, it is causal with respect to the surface elevation at x = xA required for Ff (t) determination. hfc need not be used in practice for the single-frequency tuning control if the power take-off is simply continually re-tuned to different frequencies based on measurements of the incoming spectra at a point in the vicinity of the device. This process only requires a look-up table with known frequency distributions of the added mass and radiation damping along with the terms comprising the fully-causal Fs (t). Re-tuning may be carried out at 10–15 min intervals (e.g. as in [19]). Larger in magnitude is the kernel ho [Eq. (41)] used to compute the near-optimal velocity vo (t) at the current instant (Fig. 10). Surface elevation from x = xA is used for finding the vo (t) at the current instant. ho is used in these simulations simply to obtain vo for comparison with the vact resulting from the applied control forces.

Fig. 4. Radiation damping for the 3-disc body in heave.

U.A. Korde / Applied Ocean Research 46 (2014) 79–93

Fig. 5. Added mass for the 3-disc body in heave.

87

Fig. 7. Exciting force impulse response kernel for the 3-disc body in heave.

Figs. 11 and 12 show the impulse response kernels hA and hR [Eq. (46)]. Both are seen to be nearly causal with respect to the surface elevation information they each receive [(xA ; t) for hA and (xR ; t) for hR ]. The impulse response functions hl1 , hl2 , and ha (t) are shown in Figs. 13–15 are also seen to be almost causal. These kernels have a long memory, however, which is likely due to the highly oscillatory phase variation introduced by the e−ik(ω)d (and like) operators. It is recalled that for the left → right propagating waves here this substitutes for the Fourier transform of the timeshift operator subject to the arguments in Section 3, [2] and [22]. The greater the up-wave distance, the more sensitive the phase angle becomes to k. As mentioned, this work uses the standard Discrete Fourier transformation in Matlab, but alternative approaches including a purely time-domain implementation with nested convolutions need to be considered. Calculations were carried out for the 2-parameter spectra over a range of energy periods and significant wave heights. For Te = 11 s and Hs = 1 m, the wave profile synthesized at xR = 356 m (1644 m up-wave) is shown in Fig. 16. This and the wave profile at xA are here generated using Eqs. (57) Fig. 8. Impulse response function hnc without control and with constant damping.

Fig. 6. Radiation impulse response for the 3-disc body in heave.

Fig. 9. Impulse response function hfc with single-frequency tuning.

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U.A. Korde / Applied Ocean Research 46 (2014) 79–93

Fig. 10. Impulse response function ho leading to near-optimal velocity.

Fig. 13. Impulse response function hl1 used in evaluating control force.

and (58) respectively. Fig. 17 shows the velocity vact based on hA and hR . This matches the desired vo very closely once the control force is fully synthesized. The control force component Fc as derived from hl1 and hl2 [with (xR ; t)] is shown in Fig. 18. Fig. 19 shows the absorbed power under near-optimal control over the calculation range. While the maximum exceeds 120 kW, the average over 600 s is about 4.7 kW. Small amounts of power are returned due to slight errors in the velocity–force combination. This behaviour was also noted in [2] and could arise at least in part from an incomplete synthesis of hl1 and hl2 , in part due to errors in the inverse Fourier transformation and in part due to the combined effect of total simulation time, frequency range, and sampling time. Fig. 20 plots the instantaneous reactive power needed to apply the reactive part Fa (t) of the control force. The instantaneous magnitudes are seen to be very high, although the average over 600 s in the case shown is about 50 W. In practice, this would equal the power lost in the actuator, which has been ignored here. Therefore the value found here is a result of numerical inaccuracies in the synthesis of the control force relative to velocity. Nevertheless, this result shows Fig. 11. Impulse response function hA used in evaluating device velocity.

Fig. 12. Impulse response function hR used in evaluating device velocity.

Fig. 14. Impulse response function hl2 used in evaluating control force.

U.A. Korde / Applied Ocean Research 46 (2014) 79–93

Fig. 15. Impulse response function ha used in evaluating control force.

89

Fig. 17. Velocity resulting from forces and desired velocity under near-optimal control (Te = 11 s, Hs = 1 m).

that a large instantaneous power magnitude needs to be provided by the on-board machinery including any energy storage system. As may be expected, this magnitude is greater in spectra with longer energy periods. The power absorbed with single-frequency tuning is shown in Fig. 21. Because of the fact that only pre-selected positive definite values for damping are used in this case, negative power values are not produced in this calculation. While a maximum power over 47 kW is achieved during the calculation window, the average is 2.87 kW. The power absorption without any control and constant damping (Fig. 22) is smaller, with the average absorbed power about 318 W in this case. Control is seen to lead to considerable efficiency improvement. Finally, the heave exciting force and device heave velocity are shown in Fig. 23. The velocity seems to be synchronous with the force for a large part. Some differences exist, which may have resulted from numerical errors including incomplete synthesis of

Fig. 18. Control force Fc applied on the device under near-optimal control (Te = 11 s, Hs = 1 m).

the terms used to compute Ff (t) for this purpose and errors in the inverse Fourier transformations. A direct all time-domain approach may improve the quality of the match in this figure. The next set of figures represents a wider perspective on the device behaviour with control. Fig. 24–26 plot the average capture efficiency3 variation with energy period for 3 significant wave heights. Averages over the 600 s computations above are used here. All three plots show near-optimal control producing close to 20–50% average energy absorption efficiency over a Te range between 15 s and 20 s. The improvement over peak-frequency tuning increases with increasing energy period. Peak-frequency tuning produces a marked improvement over constant damping with no control, but its time-averaged efficiency maximum is found

Fig. 16. Wave surface elevation (xR ; t) in a Pierson–Moskowitz spectrum (simulation) (Te = 11 s, Hs = 1 m).

3 It is recalled that capture efficiency is here the ratio of the absorbed power and the power incident over a width equal to the disc diameter.

90

U.A. Korde / Applied Ocean Research 46 (2014) 79–93

Fig. 19. Instantaneous power absorbed under near-optimal control (Te = 11 s, Hs = 1 m).

Fig. 20. Instantaneous reactive power supplied under near-optimal control (Te = 11 s, Hs = 1 m).

to be less than 20%. At the lower energy periods, perhaps because of the nature of the exciting force frequency variation described earlier near-optimal control is more sensitive to velocity-force phase errors resulting in greater power losses, so that the overall performance is only slightly better than with single-frequency tuning. Depending on the power requirement of the wave profile measurement system, it may be advantageous to use single-frequency tuning control in the Te ∼ 5–10 s as range for this device. Because of assumed linearity, power production is correspondingly greater for larger Hs values, though efficiencies remain approximately the same.4 Figs. 27–29 more directly show the dependence of absorbed power on Hs for different energy periods Te . The difference in

4 Due to the randomness of the phases associated with various amplitudes in a spectrum, successive runs of the same procedure may produce slightly different results.

Fig. 21. Instantaneous power absorbed under single-frequency tuning control (Te = 11 s, Hs = 1 m).

Fig. 22. Instantaneous power absorbed with no control and constant damping (Te = 11 s, Hs = 1 m).

the average absorbed power widens with increasing Hs and Te , suggesting that near-optimal control can significantly enhance system productivity in wave climates with longer energy periods for this device.5 Finally, additional simulations should be performed in bi-modal spectra, where real-time near-optimal control may be particularly advantageous. However, in a practical situation, a combination of near-optimal and single-frequency tuning control appears desirable. In any case, control of one type or another seems necessary for the present submerged device. The forces involved in control are large, and may approach magnitudes comparable to the exciting force itself. The large forces represent a design challenge for the power take-off/actuator. The forces are smoothly varying in the absence of velocity or displacement constraints, and may remain so if such constraints were implemented by adding a steady-damping contribution in the

5

Velocity constraints will likely limit this increase in practice.

U.A. Korde / Applied Ocean Research 46 (2014) 79–93

91

Fig. 26. Average-power capture efficiency as a function of energy period (Hs = 3 m).

Fig. 23. Exciting force and velocity vact under near-optimal control (Te = 11 s, Hs = 1 m).

Fig. 27. Average absorbed power as a function of significant wave height (Te = 13 s).

Fig. 24. Average-power capture efficiency as a function of energy period (Hs = 1 m).

control forces. Energy storage on board or near the device also appears necessary and may prove to be a challenge depending on the duration over which control is to be maintained. Wave elevation measurements may be obtained with point sensors such as wave rider buoys and submerged pressure transducers. In addition,

Fig. 25. Average-power capture efficiency as a function of energy period (Hs = 2 m).

remote (i.e. non-contact) sensing techniques such as X-band radars (adapted for surface elevation measurement) may be worth considering, due to the possibility of obtaining concurrent measurements at xR and xA from a single image, and providing any corrections to the current propagation model.

Fig. 28. Average absorbed power as a function of significant wave height (Te = 15 s).

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U.A. Korde / Applied Ocean Research 46 (2014) 79–93

energy from swell-dominated wave spectra. Possible ways to measure up-wave surface elevation may be to use wave rider buoys, submerged pressure transducers, or a radar type remote sensing techniques designed to provide this information in real time. Acknowledgements I am grateful to Larry and Linda Pearson and the SDSM&T Foundation for their support through the Pearson endowment. I am also grateful to the reviewers of this paper for their helpful comments. References

Fig. 29. Average absorbed power as a function of significant wave height (Te = 17 s).

9. Conclusion The main goal of this study has been to investigate a close approximation to real-time optimal control of a wave energy converter. Wave surface elevation up-wave of the device was used as discussed in the formulations of [2] and [22]. Control was applied to cancel out the reactive forces acting on the device and to match the wave energy radiation rate in irregular waves. In the approach followed here, the predominant wave direction is thought to be from left to right over x > 0, and wave surface elevation is measured at an up-wave distance determined using (i) the time shift required to make the inverse Fourier transform of the radiation damping approximately causal, (ii) the time shift needed to make the inverse Fourier transform of the exciting force variation approximately causal, and (iii) the maximum wave propagation speed at the water depth of operation past which propagation becomes non-dispersive. A submerged device comprised of three vertically stacked discs was used, since greater submergence was recently found to require shorter up-wave distances at which wave surface elevation was needed. The problem of evaluating the control force and the resulting device velocity was formulated using a combination of derived impulse response functions operating on the free surface elevation at two up-wave locations. The velocity under control (termed ‘near-optimal’ due to the necessary approximations implied in (i), (ii), and (iii) above) and the control force were computed over time values ranging from 0 to 600 s. The wave profile at the two locations was here synthesized numerically for a number of wave spectra. The absorbed power was computed at each instant and averaged over the simulation duration. Over a range of energy periods and significant wave heights in 2-parameter wave spectra, results were compared with two other conditions: (i) when single-frequency tuning was used to set the reactive force and power take-off damping to provide reactive force cancellation and radiation damping match at the peak frequency in a spectrum (‘peak-frequency tuning’), and (ii) when no control was used, with damping maintained at a constant value. Device velocity was unconstrained in this work, but constraints may be applied in practice by adding a constant-damping contribution to the control force. This may be programmed to increase with the approaching significant wave heights once they exceed a known energy capture limit. Given the large forces involved in near-optimal control, energy storage on-board or near the device may be required. Such storage would be particularly important for small devices if they are efficiently to convert large amounts of

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