The research of optimal control based on wave prediction in wave energy conversion

Bin ZHAO, Li YANG, Xiao CUI

(Institute of 0ceanographic Instrmentation，Qilu University of Technology (Shandong Academy of Sciences), Qingdao 266100, China)

Abstract: As a kind of energy, wave energy in ocean is a renewable energy with the characteristics of widespread, abundant and highly energy flux density. But it’s hard to extract energy from incident wave which is random, unstable with big variance of power input. Wave energy converter (WEC) is a device designed to capture the wave power. Recently much more attention is paid to oscillate WEC (point absorber wave energy capture device) for its flexible, low cost and easy deployment. In particular, the efficiency of oscillating WEC can be significantly increased by motion control. Some investigations show that an optimal control strategy can be designed to alter the oscillator dynamics. As a result, the efficient energy conversion will occur over a wide range of wave conditions. This point of view is also verified by some practical application such as latching control. Study also shows that short term wave prediction is essential for reactive control. However, the fact of prediction time horizon is still unclear. In this paper, the prediction effect is described mathematically based on hydrodynamic theory. Furthermore, a simulation study is also performed to show the increasing of energy capture by the effect of wave prediction, and the problem of this study is analyzed. Finally, an economical prediction time horizon is proposed considering costs and benefits and a more insightful understanding of WEC optimal control is obtained by this paper.

Key words: Wave energy converter (WEC), Optimal control, Prediction time

1 Introduction

Ocean wave power attracted considerable interests in recent years for its representation of a sustainable and renewable energy. Wave energy converter (WEC) is the device to capture wave power. Recently，much more attention is paid to oscillate WEC (point absorber wave energy capture device) for its flexible, low cost and easy deployment. In particular, the efficiency of oscillating WEC can be significantly increased by motion control. Control strategy can be designed to alter the oscillator’s dynamics such that the efficient energy conversion occurs over a wide range of wave conditions[1]. The main purpose of control is to regulate the dynamics of the system, which is named as intrinsic impedance[2], to be turned for maximum energy absorption at different peak frequencies corresponding to different incoming wave spectra[3]. Many literatures present the frequency model and the controller design method in frequency domain[4], including experiment in regular wave[5] and some theoretical analysis in irregular wav[6-7]. But real time controller can only be achieved in time domain and the main difficulty is that the transformation into the time domain will result in non-causal transfer functions, so that the conditions for maximum power absorption can be obtained only if future motion of the device, or of the future incident wave profile, are known[8]. But the essential problem needs to be solved firstly is to determine the prediction time horizon, which is still unclear. This paper tries to answer two fundamental questions that have to be clarified in WEC controller design. One is how long for the prediction time is meaningful to the WEC optimal control. The other is, to what extent the wave profile prediction improved the performance of WEC.

In section 2, the dynamic motion of a WEC is obtained based on hydrodynamic theory. A mathematical model is built according to the prior hydrodynamic analysis. In this section, the characteristics of wave profile prediction are presented by a mathematical description. In section 4, to illustrate the function of wave prediction and give a more insightful understanding of WEC optimal control, a simulation study is also conducted based on practical WEC parameters. The simulation results show that energy absorption is improved about 5 times by the optimal control and the reasonable prediction time horizon is about 10 seconds.

2 Mathematical model of a WEC

For simplicity, we only consider an oscillating WEC in one mode such as the heave mode. Also, assuming that amplitudes of oscillation are sufficiently small to make linear wave theory applicable. That is to say, the dimension of WEC is small enough compared with wave length. These two assumptions are frequently applied in most of the study of WEC motion control[9]. Considering most of the WEC buoy equipped with mooring system and it will cause additional nonlinear dragging force, we neglect these forces for simplicity.

As shown in Fig.1, a semi-submerged cylindrical WEC constrained in heaven is considered. A spring and mass acceleration are applied to the heaving WEC as many articles studied. Furthermore, a linear mass-spring-damping system is employed to represent the PTO force.

Fig.1 Schematic of the heaving buoy WEC

According to the Newton’s second law the governing equation of motion for a simple oscillating WEC is shown in Eq. (1).

fe-fr-fs-fpto=ma

(1)

Herefeis the excitation force caused by incident wave.fris the force due to radiation of waves,fsis the restoring force consists of buoyancy force and gravity force.fptois the external force provided by the PTO system, stands for the weight of WEC and is its acceleration.

Since the system is linear and there is only a single degree of freedom, based on Froude-Krylov theory the excitation force amplitude,Fe(ω) is proportional to the wave amplitudeH(ω).

Fe(ω)=W(ω)H(ω)

(2)

J. Falnes[10] suggests that the transfer functionW(ω) is dependent on the body’s shape and the wave frequency, which can be calculated by Eq. (3).

(3)

Wheregis the gravity andρis the density of sea water. An empirical formula derived by A. Hulme[11] gives the added mass and damping coefficientsB(ω) associated with the periodic motions of a floating hemisphere, which shows that:

(4)

As a result, the excitation force can be obtained by Eq. (2) to Eq. (4). Actually, in the coming section we will find thatFe(ω) can be cancelled as deriving the maximum energy capture condition. But the obtaining of excitation force is still very important because it is a prerequisite for determining the physical parameters of WEC.

The radiation forcefrrepresents the body in motion creating waves and hence wave forces. It is decomposed into damping force and inertial force related with buoy velocity and acceleration respectively. The added massm∞and time retardation functionk(t-τ) are the main reasons of WEC memory effect[13]. This is given by Eq. (5).

(5)

(6)

The dynamic buoyancy forcefscan be obtained from the following formula:

fs(t)=mg-ρgv=ρgawx(t)

(7)

Whereawis the water plane area of buoy andx(t) he displacement of buoy in heave.

Considering Eq. (5) and Eq. (7), eq. (1) can be expanded and rearranged as:

(8)

Taking the Fourier Transform of Eq. (8) andρgawis replaced byc:

Fpto(ω)=Fe(ω)-iωmU(ω)-

(9)

Eq. (9) will be utilized for the optimal control problem which will be shown in the next section.

3 Optimal control problems for WEC

In the early theoretical studies of Falnes[14], concept of impedance is applied to study the mechanic properties of oscillation buoy and it shows its superior characteristic. Taking the impedance theory, the excitation forceFe(ω)) and PTO forceFpto(ω) can be rewritten as [Z(ω)+Zpto(ω)]U(ω) andZpto(ω)U(ω). Since PTO system can always be represented by an inertial, damping and spring system, the PTO impedance can be written as:

(10)

The power absorbed by WEC is:

W=|Zpto(ω)|U2(ω)=

(11)

|Zpto(ω)|=|Z(ω)|

(12)

For a vector can be described in complex domain, the imaginary part implies that some components of the vector altered in direction. This causes the asynchronous between excitation force and velocity of WEC. Specifically, Eq. (8) shows that damping force is in phase with the velocity. The spring force has a phase lag of π/2 and the inertial force has a phase lead of π/2 related to the velocity. In the optimal control, PTO force is controlled intentionally to neutralize the inertial and spring parts in excitation force. At the same time, considering Eq. (12) the most reasonable result is:

Zpto(ω)=Z*(ω)

(13)

(14)

As retardation functionK(ω) is defined as:

K(ω)=B(ω)+iω[M(ω)-m∞)

(15)

So

(16)

To perform the controller design, time-domain equation must be acquired. It is obvious to link this to Inverse Fourier Transform. Given that PTO force must be real,K*(ω)=K(-ω), soF-1{K*(ω)}=k(-t).

(17)

Convolution ofk(-t) andu(t) in Eq. (17) can be rewritten in integral form:

(18)

Then substituting a variableτ-t=t-τ, and noting that the dummy variable could stand forτitself.

(19)

Eq. (19) includes an integral of variableτfrom 0 totand termu(2t-τ) which indicates that the ideal control force at time 0 needs some information of future velocity, which isu(2t). Considering the memory effect in solving convolution ofk(t) andu(t), a conclusion can be drawn that for an ideal optimal control of the WEC in timet, not only the floating body velocity of [0→t] but also [t→2t] are required in controller. Specifically, the inverse of retardation functionk(-t) is responsible for this requiring a prediction time. However, in practice, it’s impossible to store too much memory data and to estimate too much future data. As a result, the essential time horizons of future should be investigated and this is the working in section 4.

4 Available data for simulation

Table 1 shows the parameters applied in simulation. As described in Fig.1, the WEC buoy is assumed as a vertical cylindrical body. Its radius isaand its height isl, with an extended hemisphere in its lower end to reduce viscous effects. The wave data is generated by Pierson-Moskowitz spectrum forHsandTp.

Table 1 Parameters applied in simulation

The corresponding wave elevation is presented in Fig.2. The excitation force experienced by buoy in 200 seconds is calculated by Eq. (2) to Eq. (4) and shown in Fig.3. Since the buoy is small compared to the incoming wavelength, diffracted force is neglected and the excitation force is simply equal to the Froude-Krylov force.

Fig.2 The profile of wave elevation

Fig.3 The profile of excitation force

4.1 Simulation Scheme

Fig.4 represents the scheme of simulation. The system model is developed using MATLAB/Simulink. In detail, the whole simulation is running under Simulink environment and the convolution algorithm, in which the kernel of calculation of radiation force is realized in M-files. As for PM spectrum wave and excitation force, it is also realized in M-files.

Fig.4 The scheme of simulation

4.2 Simulation result

To demonstrate the ability of power absorption and force properties of optimal control, the model is run either with optimal control or without control. Fig.5 reveals that in the irregular sea state, the ability of power absorption by WEC is weaken for the phase inconsistency of wave excitation force and WEC’s velocity. This phase difference is caused by the hydrodynamic characteristic of WEC’s motion in the sea wave. In Fig.6, it is noticeable that under optimal control the velocity of WEC is almost in phase with the excitation force, somewhat like the resonance phenomenon in the mechanical vibration system.

Fig.5 Phase of normalized excitation force and WEC velocity without active control

Fig.6 Phase of normalized excitation force and WEC velocity with active control

According to the previous research of the optimal wave power capture, complete absorption occurs when radiated wave cancel oncoming transmitted waves both in amplitude and phase[15]. Fig 7 demonstrates this cancelling effect by radiated wave, which is generated by radiation force. It can be noticed that the phase of radiated wave (its profile can be represented by radiation force) is in counter-phase (180° out of phase) with the oncoming wave. The amplitude inconsistency of two curves is partly for the reason why the complete energy absorption for the point absorber must be involved in heave and surge motion, and here only heave motion is concerned in simulation. According to Eq. (12) the optimal impedance implies conditions of both in amplitude and phase. The amplitude condition is achieved by adjusting PTO damping equal to the intrinsic damping. The phase condition is achieved by adjusting PTO reactance to cancel the intrinsic reactance.

Fig.7 Phases of normalized excitation force and radiation force

Fig.8 and Fig.9 show the power captured by WEC with or without active control. Energy can be obtained by power integral. It is notable that the average power absorption of WEC can be improved about 5 times under active control. The other thing to be noted is that the maximum instantaneous power is 10 times of average power under the active control. This would be the main problem need to be solved in WEC active control for the very high ration of maximum and average power absorption, which always leads to big trouble in full-scale WEC equipment design.

Fig.8 Power (instantaneous and average) and energy captured by WEC without active control

Fig.9 Power (instantaneous and average) and energy captured by WEC with active control

As explained in the last part of section 2, the equation of optimal controller needs the information of body velocity [t→2t]. If the lower limit of integral is replaced by (t-Δt), the prediction time of body velocity can be changed to Δt. That means termsu(2t) is replaced byu(t+Δt) and only a small horizon of future information of body velocity is needed to know. Fig.10 gives the comparison of full-time prediction and reduced time premonition. For a clearer description, only the range from 30 to 40 seconds are selected to show below.

Fig.10 Damping force with different time horizon premonition, full time, 10 seconds and 1 second respectively

Fig.11 is the error profile under different prediction time horizons. It is clearly that the error decrease remarkably when the time horizon extends from 1 second to 10 seconds. The impact of prediction time included on deviation is shown in Fig.12. which indicates that only including a few seconds in the prediction is seen to give bad estimations of the full premonition. When the time horizon lies at one second, the maximum deviation is about 65% yet no less than 15% when extends to 5 seconds and 10% is reached with 9 seconds. If the time span prolongs to 50 seconds, the maximum deviation is only about 1%, very close to the full-time premonition. In practice however, it is costly and hardly to achieve the body motion velocity prediction with the time span of 50 seconds.

Fig.11 Error with different time horizon premonition, full time, 10 seconds and 1 second respectively

Fig.12 The deviation of different prediction time horizon

5 Conclusions

A mathematical model of point absorbed WEC is setup based on hydrodynamic equations. To achieve maximum power absorption, the optimal PTO control force is derived, which reveals that the prediction of body velocity is needed in the controller design. Simulations using MATLAB/Simulink are conducted based on the mathematical model and P-M spectrum. The numerical results show that optimal control strategy has a big effect on the power absorption. This is the reason why by PTO force control radiated wave cancels the oncoming wave both in amplitude and phase. Further investigation of time prediction horizon impacting on control error is also conducted. A reasonable time predication span is suggested by balancing error and computational costs.

The optimal control described in the paper shows its promising future at present at early stages of its development. Concerning the short-term wave predication algorithms is optimizing and the great performance improvement on power capture, the optimal control of WEC needs to be given further attention.