Control and Shape Optimization of Wave Energy Converters

In some applications, the simultaneous optimization of both the system design and the control is crucial for optimizing the system operations. Examples of such applications include the design and control of offshore wind turbines, and ocean wave energy converters. The latter application is discussed in some detail in this chapter. The results presented in this chapter were first published in reference [6].

One of the challenges in wave energy harvesting is the motion control. There have been significant developments for different control methods for WECs. Most studies on the control of one-degree-of-freedom heaving wave energy converters adopt a linear dynamic model—the Cummins’ equation—which can be written as:

where z is the heave displacement, m is the buoy mass, к is the hydrostatic stiffness due to buoyancy, a is the added mass, Fex is the excitation force, и is the control force, Bv is a viscous damping coefficient, and hr is the radiation impulse response function (radiation kernel). The radiation term is called radiation force, Fr, and the buoyancy stiffness term is called the hydrostatic force.

There are multiple sources of possible nonlinearities in the WEC dynamic model though.For example, if the buoy shape is not a vertical cylinder near the water surface, then the hydrostatic force will be nonlinear [60]. The hydrodynamic forces can also be nonlinear in the case of large motion. Control strategies that aim at maximizing the harvested energy usually increase the motion amplitude and, hence, increase the impact of these nonlinearities.

In the case of using nonlinear control, it is possible that the motion of the buoy grows large enough to make the buoy almost fully submerged or almost fully in the air. In such cases, the linear hydrodynamic model becomes invalid, and modeling of nonlinear hydrodynamics becomes inevitable. In this section, we examine the impact of having nonlinear terms in the equation of motion whether they appear due to nonlinear hydrodynamics, nonlinear hydrostatics, nonlinear damping, and/or nonlinear control forces. Toward that goal, the control force is here assumed in the form of a summation of two quantities:

where 14 is the linear part of the control and йс is the nonlinear control part. The harvested power can be expressed as:

The nonlinear control part is assumed in the form:

where aCj and pCj are constant coefficients and Nc and Mc are the number of nonlinear terms that determine the order of control forces.

The hydrodynamic and hydrostatic (hydro) forces along with all other opti- mizable nonlinear forces are referred to as the system nonlinearities. The system nonlinearities and the control (also nonlinear) are here optimized simultaneously. For the sake of control design, it is convenient to express the optimizable system nonlinearities as a series function as follows:

where fs is the nonlinear force, aSi and f5Si are constant coefficients, Vi; Ns and Ms are the number of nonlinear terms that determine the order of the nonlinear forces. Equation (10.5) is written intuitively; consider for example the Proportional-Derivative (PD) controls which are widely used in linear systems. In a PD control, the proportional part is constructed as linear term in the state, and the derivative term is constructed as a linear term in the state derivative. The proportional term is a stiffness term since it has spring-like effect, which means this part of the force does not add/remove energy on average. The derivative term, however, is a damper-like term, and it continuously adds/removes power. One might think of nonlinear stiffness or damping terms, as discussed in details in several references such as [85]. The first term in fs represents a nonlinear stiffness force, and the second term contains a nonlinear damping force. Note that all j3Vy are always negative coefficients, and hence the second term is always a damping term (energy flow is always from the water to the device). Optimizing the system nonlinearities means in this case finding the optimal coefficients aSi, V/ = 1 ■ ■ -Ns, and pSi, Vi = 1 • ■ Ms. Similarly, optimizing the control means finding the optimal ac., Vi = 1 ■ ■ -Nc and Д., Vi = 1 ■ • Me. Once iic and fs are optimized, the WEC system (e.g., the buoy shape) is designed so that the WEC nonlinear force matches the optimized nonlinear force fs. This last step of designing a WEC system to generate a prescribed nonlinear force is not addressed in this chapter; the focus of this chapter is on the optimization of fs and Cic.

The equation of motion of the system then is:

The equation of motion, Eq. (10.6), is derived assuming that the buoy does not leave the water nor gets fully submerged in the water. In the case of nonlinear WECs presented in this chapter, the motion of the buoy may grow large and these two cases should not be excluded. Hence the model in Eq. (10.6) is modified as follows. Consider the coordinates defined in Fig. 10.1, a range |z| < zs is defined in which the model in Eq. (10.6) is considered valid. The limit zs is selected based on the buoy dimensions and the wave height. When |z| > zs, there are two possible cases. The first case is when (z > 0), that is the buoy is (or very close to being) fully submerged under water. The second case is when (z < 0), that is the buoy is (or very close to being) totally out of the water. In these two cases, the dynamic model in Eq. (10.6) is not valid, and an approximate dynamic model is defined as follows:

Case 1: (z > 0) The linear stiffness term becomes a constant khf2. The nonlinear stiffness force will also be saturated, that is fs = X^=2 +

Л^'1 Ay |z7>/£«(z). The excitation force is assumed to remain the same as in Eq. (2.26).

Case 2: (z < 0) The buoy is out of the water so there is no buoyancy force on it, meaning that kz. = —mg, where g is the gravitational acceleration. There is no excitation force acting on the WEC; and there is no linear damping term in Eq. (10.6). Also the nonlinear hydro force vanishes, that is fs = 0. The equation of motion reduces to m'z = mg + Ui + йс.

The harvested power P(t) is expressed as:

Buoy coordinate system

Figure 10.1: Buoy coordinate system.

In the analysis conducted in this study, 14 is assumed a damping force as follows:

where В is the linear control force damping coefficient; В is negative. The shape and control coefficients will be optimized so as to maximize the harvested energy. The optimization problem can be formulated as follows:

where E is the energy harvested over a period T, which can be expressed as:

The design variables in this optimization problem are Ns, Ms, Nc, Mc, aSi, jiSj, V i = 1 ,...,NS and j = 1 act, and Д.,, V k= 1 ,...,NC and / = 1 The

overall number of variables is variable since Ns, Ms, Nc, and Mc are variables, and hence this is a VSDS optimization problem. This problem can be solved using HGGA, SCGA, or DSMPGA. A numerical case study is presented below.

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