Kalmen estimator as a robust solution for output ...

IEEJ TRANSACTIONS ON ELECTRICAL AND ELECTRONIC ENGINEERING IEEJ Trans 2015 Published online in Wiley Online Library (wileyonlinelibrary.com). DOI:10.1002/tee.22098

Paper Kalmen Estimator as a Robust Solution for Output Power Maximization of Wave Energy Converter Ahmed M. Kassem∗a , Non-member Ahmad H. Besheer∗∗,† , Non-member Ibrahem E. Atawi∗∗∗ , Non-member In this paper, the application of linear quadratic Gaussian (LQG) control for a buoy-type point absorber of a wave energy converter (PA-WEC) system is investigated. The proposed wave energy conversion is considered as a two-body system, which is taut-anchored to the sea floor using three cables. The main goal of this study is to extract the maximum available power from the ocean wave. This is accomplished via determining the optimal value of the force exerted on the power take-off (PTO) system taking in account the physical constraints on the position and velocity. First, the reduced nonlinear dynamical model of the WEC system is obtained. The nonlinearity in the mooring force is replaced by a linear law to yield the state space linear model of the system. Then, the standard Kalman filter technique is employed to estimate the full states of the system. Based on the LQG control approach, the optimal PTO force is computed at which the maximum output power can be easily harvested. The computational burden is minimized to a great extent by computing the optimal state feedback gains and the Kalman state space model offline. The feasibility of the proposed control approach in extracting the optimal power of the ocean wave is validated via the simulation example even under different values of the mooring constant and without violating the system limitation. © 2015 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc. Keywords: wave energy converter, two-body point absorber model, linear quadratic Gaussian control, Kalman estimator

Received 16 April 2014; Revised 11 July 2014

1. Introduction Fueled by increasing global energy demands, exhaustion of energy resources, and heavy environmental impacts (ozone layer depletion, global warming, climate change, etc.), the need for alternatives to the large coal- and oil-fired power plants has become a mandatory requirement. Renewable energy is heavily involved in such a direction, and in particular photovoltaic and wind turbines have proven to be feasible solutions [1–3]. Among the factors that contribute to the success of renewable energy technologies are the arrival of the new power device technologies, new circuit topologies, and novel control strategies. One of the most promising—though under development—renewable technologies that has been attracting massive research interest in very recent years is the ocean wave energy. Ocean waves are an enormous, unharnessed energy resource, and the possibility for debriefing energy from waves is considerable. It is estimated that the potential worldwide of wave energy is in the range 8000–80 000 TWh/year, which is equivalent to 1–10 TW [4]. According to [5], up to 15% of current UK electricity demand could be met by wave energy; when combined with tidal stream generation, up to 20% of the UK demand could be met. Approximately, more than half the world annual energy consumption—which is estimated to a

Correspondence to: Ahmed M. Kassem. E-mail: [email protected]

* Electrical Engineering Department, Faculty of Engineering, Sohag University, Sohag, Egypt ** Environmental Studies and Research Institute, University of Sadat City, Six District - Sadat City 32897, Egypt † Current address: Center for Sustainable Technology, University of Ulster, Jordanstown, Northern Ireland BT370QB, UK *** Electrical Engineering Department, Faculty of engineering, Tabuk University, Tabuk, Saudi Arabia

be 148 000 TWh in 2008—can be supplied by wave energy systems. Wave energy converters are new power devices that are designed to convert the energy from the motion of ocean waves into usable power. Recently, numerous concepts and distinction of technological approaches for wave energy conversion have been patented in USA, Japan, and Europe [6]. Oscillating body, oscillating water column, and overtopping are the most wellestablished techniques among these. In this work, the modeling of a point absorber—which is a subclass of the oscillating body—is studied. A point absorber is a relatively small equipment that captures power from all directions at one point by at least one floating body which is moved by the waves [7]. Different forms of point absorbers are found in the literature, but, in general, it can be described as relatively small, linear, damped oscillators excited by ocean waves. Incident waves force the mass element of the wave energy converter (WEC) equipment into motion, but the motion is resisted by some power take-off (PTO) machinery [8,9]. Then beneficial power can be generated and transferred to the shore. Point absorber (PA) devices are considered to be deployed in the commercial phase in arrays of many units, which are known as wave farms [10]. To extract energy from the waves, control techniques that improve the capacity of PAs are a must for such equipment; this may include, for example, latching and phase control; however, they suffer from poor performance [11–13]. A point absorber must be compelled to adjust its behavior to the wave climate, which may vary significantly at a given site. To optimize the energy conversion, several controllers are applied, such as reaction force control and real-time control [14,15]. These studies assume a heaving point absorber with semisubmerged single body, and mooring forces are not taken into consideration. Related to these assumptions, it was shown in Ref. [8] that the velocity and hence the absorbed power from waves extracted by

© 2015 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

A. M. KASSEM, A. H. BESHEER, AND I. E. ATAWI Fm

z1 Float

PTO z2

Spar

z2

Tank

Fc

l′ δ

Sea floor

(a)

α

(b)

(a)

Fig. 1. (a) L10 wave energy converter and (b) schematic diagram

the presented model is restricted to up and down motion. The assumptions to derive the equations of motion for the two bodies of a PA in the time domain are as follows [4–8,16]: 1. The motion forming is based on linear wave theory. 2. The presented PA-dependent parameters are constant. 3. During operation, it is assumed that the spar is completely submerged. 4. The buoyancy spar force can be neglected. 5. The cable is considered to behave as a spring. 6. Velocity cross-coupled radiation terms based on damping can be neglected. 7. The convolution terms that normally appear in the radiation force equations are neglected.

2.1. PA-WEC nonlinear model In this paper, a reduced nonlinear two-body model for the proposed PA-WEC is used. This model is a reduced form of the extended model in Ref. [19]. Using Newton’s second law, the dynamic motion equations of the buoy system is given in (1) and (2), where z1 denotes the position of the buoy, and z2 is the position of the spar.

2. Proposed WEC System Dynamics

3.5 0.76

1.1 7.03

(1)

−FPTO + Fe2 − Fr2 + Fm − Fr21 = m2 z¨2

(2)

Fr1 = A1 z¨1 + b1 z˙1 Fh1 =

2 gρπ rbuoy z1

(3) = K1 z1

Fr12 = A12 z¨2 Fr2 = A22 z¨2 + b2 z˙2 Fr21 = A21 z¨1

(4) (5) (6) (7)

In this paper, a taut-moored buoy with three cables is considered. The three mooring cables are configured like a tripod, which makes an angle ‘α’ with respect to the sea floor, as shown in Fig. 2(a) [9]. According to Hook’s law, the total mooring force Fc in the direction of the cables for all three cables can be written as  Fc = −3K (l  − l ) = −3K ( l 2 + z22 − 2lz2 cos(90◦ + α) − l ) (8)

Table I. Dimensions of the L10 WEC

Diameter (m) Height (m)

FPTO + Fe1 − Fr1 − Fh1 − Fr12 = m1 z¨1

The radiation force Fri , the hydrodynamic force Fh1 , the coupling radiation force Fr12,21 and the mooring Fm force are given in (3)–(4):

In this section, a semisubmerged cylindrical PA constrained to move in heave only is considered. Figure 1 shows the oscillating type of WEC (L10 point absorber) which is developed at Oregon State University and its schematic diagram [20,21]. The L10 PA is a direct-drive WEC and mainly consists of two bodies: a buoy floating on the surface of the ocean, and a damping body which includes a spar and a ballast tank. Table I shows the dimension of the L10 point absorber used in this paper. To convert the relative motion into usable energy, these two bodies are connected together through the PTO generation system. In this type of WEC, the kinetic energy due to the relative motion of the two bodies is converted directly into electrical power using a linear generator without converting kinetic energy into mechanical power as in hydraulic PTO systems. Also to dampen the spar’s motion, the spar is moored to the sea floor. Generally, a PA moves in six degrees of freedom, but the absorbed energy is produced only from the up and down motion (heave motion) in most cases. Therefore,

Spar

(b)

Fig. 2. (a) Mooring configuration with three cables. (b) Force derivation used diagram

a PA can be maximized using the matching impedance of the linearized equivalent electrical circuit. Over the course of the last four decades, advanced control techniques have been applied successfully to the WEC systems to maximize the extracted power and improve the system performance. They include fuzzy logic control (FL) [16], robust fuzzy logic control (RFL) [17], artificial neural networks control [18], and nonlinear model predictive control (NMPC) [19]. The main idea beyond these recent methods is to maximize the power output of the WEC by explicitly regulating the PTO force via a feedback and/or predicted feedback control system with appropriate constraints. In this paper, a two-body point absorber with a mooring WEC is studied. The closed-loop wave energy system is conceived to generate maximum available power from the waves of the ocean. The design and development of the controller for the proposed system is presented under the linear quadratic Gaussian (LQG) approach. First, the nonlinear model of the proposed PA-WEC is obtained and linearized around an operating point. The excitation forces are al so calculated from the water surface wave elevation. Then, the optimal PTO force based on LQG control is obtained. The proposed PA-WEC with the proposed controller has been tested through a wave elevation change for different values of the mooring constant. Simulation results show the feasibility and effectiveness of the proposed LQG control in maximizing the generated power from the proposed wave energy system.

Buoy

l

α

The vertical mooring force Fm (the effective component of Fc on the spar) for all three cables in total is then Fm = −3Fc sin(α) 2

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POWER MAXIMIZATION OF A WAVE ENERGY CONVERTER

⎡

Table II. System parameters of the L10 WEC Variable

Value

m1 m2 A11 A12 A22

2625.3 2650.4 8866.7 361.99 361.99

Variable

Value

A21 b1 b2 k1

361.99 kg 5000 N/(m/s) 50 000 N/(m/s) 96 743 N/m

kg kg kg kg kg

⎤

A12 ⎥ ⎢− ⎥ ⎢ E2 = ⎢ me1 (m2 +A2 ) ⎥ , 0 ⎦ ⎣

2 K1 = gρπ rbouy

1 me2

3. LQG Control Strategy In this study, the LQG approach is employed to optimize the PTO in the WEC system. The LQG is a method of the state and output feedback controller technique for designing optimal dynamic regulators. It has the following advantages: 1. It enables the trading off of regulation performance and control effort. 2. It takes into consideration the system disturbance and measurement noise. Generally, the LQG controller consists of an optimal state feedback gain F and a Kalman state estimator. The optimal feedback gain is calculated such that the following performance index is minimized:

From Fig. 2(b), the total vertical mooring force Fm can be rewritten as Fm = −3Fc sin(α)   ◦ 2 2 = −3K sin(α) l + z2 − 2lz2 cos(90 + α) − l

0

(10)

The different parameters in the above equations are defined in the Nomenclature section. In the above nonlinear model (1)–(4), it is assumed that the inertia and the damping terms are the dominating terms for the radiation forces, and the nonlinearity is found in the mooring force equation (10). Table II shows the L10 PA model parameters.

∝ (x T Qx + u T Ru)dt

H =

(14)

0

u = −Fx

2.2. PA-WEC linear model The linear model of the proposed PA-WEC can be obtained by linearizing the nonlinear term in (10). It is expected that linearization around z2 = 0 yields good results, since z2  l . Since the displacements in ropes are generally much smaller than the rope lengths, these changes can be neglected and the angle ‘α’ can be assumed to be constant. The linearization gives

The feedback control law can be written as in (15) [22]. u = −F [z1

◦

(z2 =0)

= −3K (sin α)2 z2 = −Km z2

(11)

This holds for z2 > 0. The same result with the opposite sign follows for z2 < 0. Even for large displacements such as 0.5 m, the difference between the nonlinear and the linear mooring force calculation is only 0.04% for a cable length of 170 m. Hence, the linear case is a very good approximation, and in such a case the nonlinear mooring force will be replaced by the following linear one: Fm(linear) = −Km z2

(13)

where u = FPTO ,

v = Fe1 ,

z˙1 0

1

0

⎢ ⎢ A=⎢ ⎣

− m1 e1 0

k

− m1 e1 0

b

A12Km me1 (m2 +A2 )

A21K 1 me2 (m1 +A1 )

A21b 1 me2 (m1 +A1 )

− mkm e2 ⎡

⎡

z˙2 ] ,

z2

⎤

0

A12 ⎢ 1 + ⎥ ⎢ me1 (m2 +A2 ) ⎥ B = ⎢ me1 ⎥, 0 ⎣ ⎦ A21 1 − m − m (m +A ) e2

e2

1

1

0

w = Fe2 ⎤ 0 A12b ⎥ 2 me1 (m2 +A2 ) ⎥ ⎥ 1 ⎦ b2 −m e2 ⎤

0 1 ⎢ ⎢ me1 E1 = ⎢ 0 ⎣ A − m (m21+A e2

1

(15)

(16)

where L is the Kalman gain, which can be determined by knowing the system noise and measurement covariance matrices Qn and Rn , respectively. However, the accuracy of the filter’s performance depends mainly upon the accuracy of these covariance matrices. On the other hand, the matrices A and B containing the wave energy converter system parameters are not required to be very accurate because of the inherent feedback nature of the system. The Kalman filter performs best for linear systems. The optimal state feedback gains and the Kalman state space model are calculated offline, which results in great savings in computation. On this basis, the implementation of the proposed controller becomes easier and the hardware is reduced to a minimum.

Using (12) instead of (10) and after performing the necessary manipulation on (1)–(7), the PA-WEC linear state space model used to design a robust control based on linear quadratic Gaussian approach is presented by

x = [z1 ⎡

z˙2 ]

p xˆ = (A − BF − LC )xˆ + Ly

(12)

x˙ = Ax + Bu + E1 v + E2 w

z2

where Q and R are positive-definite or semidefinite Hermittian or real symmetric matrices. The optimal state feedback controller is not implementable without full state measurement. In our case, the position and the velocity of the buoy are chosen to be the output measured signal. The linear acceleration signal of the buoy is measured using an accelerometer, and then the acceleration signal is integrated twice to get both the position and the velocity signals. To avoid DC offset gain resulting from integration, a circuit that includes low- and high-pass filters is used. The proposed Kalman filter in this paper is used to drive the state estimated vector such that u = −F xˆ remains optimal for the output feedback problem [22]. 

xˆ = z˙ˆ 1 zˆ1 zˆ2 z˙ˆ 2

Fm(linear) = (Fm + Fm z2 )z2 =0

   z2 − l cos(90 + α)  = −3K sin(α)  z2   ◦ 2 2 l + z2 − 2lz2 cos(90 + α) 

z˙1

4. PA-WEC System Simulation Using LQG Control 4.1. Closed-loop control The block diagram of the proposed PA-WEC system with the proposed LQG controller is shown in Fig. 3. The estimated states values are superscripted with a hat in the diagram. The LQG controller contains the Kalman state estimator in addition to optimal state feedback gains. The Kalman estimator uses both the position and the velocity of the buoy as measured states, which, by using only one sensor as an

⎥ ⎥ ⎥, ⎦

1)

3

IEEJ Trans (2015)

A. M. KASSEM, A. H. BESHEER, AND I. E. ATAWI

Disturbance forces Fe1, Fe2

Relative position and wave elevation (m) 3

WEC system

u = FPTO

Measurable states Z1, Z1

0 –1.5

LQG-controller

–3

Optimal gains “F ”

Wave elev. Relative. pos.

1.5

Optimization & calculation

X

Kalamn estimator

1

0

10

20

30

40 50 60 Generation force (N)

70

80

90

100

0

10

20

30

40 50 60 Relative velocity (m/s)

70

80

90

100

0

10

20

30

40 50 60 70 Generation power (KW)

80

90

100

0

10

20

30

80

90

100

x 106

0 –1

Fig. 3. Block diagram of the wave energy control with the proposed LQG controller

3 0

accelerometer to estimate all the full states of the system, include the position and the velocity of the spar. These states are multiplied by the corresponding optimal gains to produce the control signals necessary to generate the optimal signal of PTO. The proposed closed-loop system is simulated using the MATLAB/Simulink software package. The parameters of the system used in the simulation procedure are shown in Tables I and II. The noise and measurement covariance matrices are set as follows:

–3

200 0 –200

Also, the values of Q and R matrices, which are necessary to calculate the optimal feedback gains, are set to be

10

20

30

40

50 60 Time (s)

70

80

90

100

Fig. 4. Time series of a specific simulation of the wave energy conversion system with the proposed LQG control and Km = 100 000

R = 3.08 − 66860

With optimization Without optimization

0

Q = diag(200 10 10 10000)

13930

70

80 60 40 20 0

Qn = diag(4, 4), Rn = 1

F = [−10580

40 50 60 Energy gained (KJ)

− 527510]

Relative position and wave elevation (m)

4.2. Simulation results

Computer simulations were carried out to validate the LQG approach and its performance regarding the optimal signal of PTO. The two-body wave energy conversion dynamic model takes the wave surface elevation as input and solves for the velocity and position of the buoy, spar, and plate. A wave data sample collected by the NDBC offshore buoy [23] is used to generate a time series for the wave elevation. This time series is used as the input signal to calculate the excitation forces. The wave data have a significant height of 3.6 m, an energy period of 9.4 s, a dominant period of 10.4 s, and a power of 86 kW per meter crest length. The simulations reported in Figs 3–5 involved a PA excited by ocean waves, where the peak-to-peak wave height is 3.6 m and the up-crossing period is 100 s. These figures show the WEC system dynamic response of the relative position, generation force, relative velocity, optimized generation PTO, and the energy gained with and without the proposed optimization technique for the specified wave elevation and at different values of the mooring constant (in this study, the case of nonoptimized generation power is related to the system without the controller). It is obvious from these figures that the generated power depends on the value of the mooring constant. Also, the proposed optimization method based on LQG approach keeps both the relative position and velocity values within their limits of ±1.4 m and ±1.6 m/s, respectively. The average generated output power of the proposed WEC with and without optimization is listed in Table III. It is seen that the values in both cases decrease with the increase of the mooring constant. Hence it is clear that the average generated power can be greatly enhanced using the proposed optimization method compared with the case without optimization for different mooring constants. Table IV provides a brief comparison between the proposed technique in this paper and other approaches previously presented

3

Wave elev. Relat. pos.

1.5 0 –1.5 –3

10

20

30

40 50 60 Generation force (N)

70

80

90

100

0

10

20

30

40 50 60 Relative velocity (m/s)

70

80

90

100

0

10

20

30

40 50 60 70 Generation power (KW)

80

90

100

–200 0

10

20

30

80

90

100

1

0 x

106

0 –1 3 0 –3 200 0 40 50 60 Energy gained (KJ)

70

80 60

With optimization Without optimization

40 20 0 0

10

20

30

40

50 60 Time (s)

70

80

90

100

Fig. 5. Time series of a specific simulation of the wave energy conversion system with the proposed LQG control and Km = 250 000 4

IEEJ Trans (2015)

POWER MAXIMIZATION OF A WAVE ENERGY CONVERTER

Table III. Average generated output power (kW)

Relative position and wave elevation (m) 3

Km

With optimization

Without optimization

31 28.2 25

10.5 4 3.6

100 000 250 000 350 000

0 –1.5 –3

Table IV. Comparison with fuzzy and robust fuzzy logic control

Proposed LQG controller FL controller [16] RFL controller [17]

IWH

AF

Control law

3.6 m 1.5 m 1.1 m

65–85% 15–25% 32–80%

Very simple Moderate Complicated

Wave elev. Relative pos.

1.5

1

0

10

20

30

40 50 60 Generation force (N)

70

80

90

100

0

10

20

30

40 50 60 Relative velocity (m/s)

70

80

90

100

0

10

20

30

40 50 60 70 Generation power (KW)

80

90

100

0

10

20

30

80

90

100

x

106

0 –1 3 0 –3

in the literature [16,17] to show the effectiveness of the adopted technique. Our work is different from those in Refs [16,18,19] in the following ways:

200 0 –200

1. A very simple and straightforward control law is used when compared to the more sophisticated controller structure in Refs [16,18,19]. 2. The resulting optimization problem in Ref [19] for controlling the same PA-WEC model as used in our paper is very large for solving online within a small step time, which may lead to a real-time applicability problem that can be avoided using the proposed LQG framework in this paper. 3. The NMPC proposed in [19] requires the solution of a nonlinear problem. In general, these problems are nonconvex, so it cannot be ensured that a global optimum can be found. Additionally, the solution can be computationally expensive. 4. The quality of the generated power is enhanced using the proposed LQG technique compared to that in [19] (see e.g. Figs 4–6).

60 40 20 0 –20

40 50 60 Energy gained (KJ)

70

With optimization Without optimization

0

10

20

30

40

50 60 Time (s)

70

80

90

100

Fig. 6. Time series of a specific simulation of the wave energy conversion system with the proposed LQG control and Km = 350 000 values of the mooring constant. The obtained results indicated that average generated power is maximized compared with the uncontrolled case at different values of the mooring constant. Also the results showed that optimum power could be obtained keeping the physical system limitations as positions and velocities.

Finally, we can make the following general remarks:

Nomenclature

• We have tailored the LQG framework, which is successfully

z1 , z2

used as an optimization method in the feedback control theory, to address the problem of maximizing energy harvesting from ocean wave energy. • We have proved the suitability of such optimization methodology to a highly nonlinear application like the two-body PA with mooring WEC. • The optimized output feedback control under Kalmen estimator approach can facilitate the extraction of maximum power with high amplification factor even under high exogenous disturbance. • The adopted controller has the effect of enhancing the energy output while curtailing the deflection of the buoy motion to the predefined limits.

z˙1 , z˙2 m1 Fe1 Fr1 FPTO A1,2 b1,2 g ρ rbuoy α Km K

5. Conclusion

IWH

In this paper, a control scheme based on LQG approach was used to maximize the energy capture of a wave-energy PA system. The nonlinear reduced model and the state space linear model of the WEC system were obtained. Different values of linear mooring forces were assumed. Under such circumstances, the proposed LQG control successfully demonstrated the ability to extract and maximize the generated power while keeping the PA states and the generator force within their limits. In order to evaluate the performance, the WEC system with proposed LQG control was tested for different values of wave elevation and also for different

AF

Position of the buoy and the position of the spar, respectively. Velocity of the buoy and the velocity of the spar, respectively. Mass of the buoy. Excitation force induced by the incoming waves. Radiation force induced of the buoy. Force produced by the power take-off system. Added mass Viscous damping factors Acceleration of gravity. Density of seawater. Radius of the buoy. Angle between the mooring cable and the sea floor. Mooring constant Cable stiffness Incident wave height And (peak-to-peak disturbance amplitude And) Amplification factor References

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Ahmed M. Kassem (Non-member) received the B.Sc. degree from Assiut University, Egypt, in 1991, the M.Sc. degree from Eindhoven Technical University, The Netherlands, in1999, and the Ph.D. degree from Minia University, Minia, Egypt, in 2006, all in Electrical Engineering. He is currently an Associate Professor of electrical power engineering in the Department of Electrical Engineering, Faculty of Engineering, Sohag University, Egypt. His research interests include voltage stability analysis, power systems operation and control, renewable energy, electric machines control, and intelligent control applications. Ahmad H. Besheer (Non-member) received the Ph.D. degree in Electrical Power and Machines from Cairo University, Egypt, in 2006. He is currently an Associate Professor with the Environmental Studies and Research Institute, University of Sadat City, Egypt. His research interests include optimization techniques, fuzzy system, and renewable energy. Dr. Besheer is senior member of the IEEE, and a member of the Electric Vehicles Community, Technical Community, the Smart Grid Community, the IEEE Technical community, IEEE Industrial Electronics Society (IEEE IES), IEEE Computational Intelligence Society (IEEE CIS), and IEEE CIS Social Media Subcommittee (2012). Also, he is a member of the Editorial Board of the International Journal in Research Studies for Science, Engineering and Technology (IJRSSET). Ibrahem E. Atawi (Non-member) received the B.S. degree in Electrical Engineering from King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, in 2005, dual M.S. degrees in Electrical Engineering and Engineering Management from Florida Institute of Technology, Melbourne, FL, USA, in 2008, and the Ph.D. degree in Electrical Engineering from the University of Pittsburgh, Pittsburgh, PA, in 2013. Since 2013, he has been an Assistant Professor with the Electrical Engineering Department and Vice Dean of the Collage of Engineering, University of Tabuk, Tabuk, Saudi Arabia.

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