Pure damper PTO force

[Hoda-Markieh]

Hello,

I have a WEC design for a simple surge-only device, and I'm trying to add a PTO force that acts purely as a damper. My goal is to optimize the damping coefficient to achieve the best average mechanical power output from this damper PTO, without involving constraints or electrical power conversion for now.

I’ve tested three methods, and only the first one (where I define the damping manually) gives a reasonable power output. Here’s a brief overview of the methods:

1. Defining the controller as resistive and adding the force to the PTO object

(Code attached for all methods). In this case, I get position, velocity, and force signals in the time domain that follow the wave elevation.
Question: Is this behavior normal since it’s a linear case, and I’m not optimizing the control state vector? What could be wrong with this method?

    def resistive_controller(pto, wec, x_wec, x_opt, waves, nsubsteps=1):
        v = pto.velocity(wec, x_wec, x_opt, waves, nsubsteps)
        return -c_damp * v
    
    kinematics = np.eye(box_body.nb_dofs)
    
    #PTO with the resistive controller
    pto = wot.pto.PTO(ndof=ndof, 
                      kinematics=kinematics,
                      controller=resistive_controller,
                      impedance=None,
                      loss=None,
                      names="PTO")

    # Add PTO force to WEC
    f_add = {'PTO': pto.force_on_wec}

2. Using a D-controller that behaves like a damper

Here, I get an average power of zero because both positive and negative values cancel each other out, and there’s very limited motion.
Question: Does using the built-in controllers affect the mechanical power calculation? Should I expect a more significant motion?

# PTO with the PID controller 
    pto = wot.pto.PTO(ndof=ndof, 
                  kinematics=kinematics,
                  controller=lambda pto, wec, x_wec, x_opt, waves, nsubsteps: controller_pid(
                      pto, wec, x_wec, x_opt, waves, nsubsteps,
                      proportional=False,  
                      integral=False,     
                      derivative=True,     
                      saturation=None),    
                  impedance=None,
                  loss=None,
                  names="PTO")

    # Add PTO force to WEC
    f_add = {'PTO': pto.force_on_wec}

3. Using an unstructured controller with a damper impedance for the PTO object

My understanding is that impedance is generally used for mechanical-to-electrical conversion. However, since it relates the PTO force to its velocity, can I just use the damping ratio to calculate the force? I encounter a singular matrix error when I set the voltage component to zero.
Question: How can I handle the PTO force and voltage in this case if I don’t need the conversion?

    Z = np.zeros((2*ndof, 2*ndof, nfreq), dtype=complex)

    for k, w in enumerate(freqs):
        if w > 0:
            Z[0,1,k] = c_damp / (1j*w)   
            Z[1,0,k] =  c_damp / (1j*w)  
        else:
            Z[:,:,k] = 0.0

    impedance = xr.DataArray(
    Z,
    dims=('radiating_dof','influenced_dof','omega'),
    coords={'omega': freqs}
    )

Any insights or suggestions would be greatly appreciated!

[cmichelenstrofer]

• Method 1 should work
• Method 2 should also work, but you want a proportional controller (P=velocity, D=acceleration, I=position).You also don't need the lambda function, it should simply be controller=controller_pid(...)
• Method 3 is not the right approach since you are not introducing a new dynamic states (e.g. generator voltage/current)

Use method 1 if you want to specify the damping coefficient. Use method 2 if you want to optimize the damping coefficient. (you could also use method 1 and define c_damp=x_opt[0] and it would be equivalent to method 2. )

[Hoda-Markieh]

Hello, thank you for your response!

I have a question, I'm modeling a 1x1x0.1 m screen facing a regular wave, using 6 DOF. When I use the second method with an unstructured or None controller, I get a very unrealistic dynamic response, with extremely large velocity and position values.

Why could this be happening? Could it be related to the lack of damping or the way the controller is defined?

I've attached the response plot for reference.

Thanks again!

[cmichelenstrofer]

When you use the unstructured controller, the optimizer finds the optimal PTO force timeseries. In your case this PTO force seems very large which would be driving the very large and fast dynamics. For the linear problem, you can get the analytical solution using frequency domain analysis. Since you don't include the PTO dynamics (e.g., a PTO impedance matrix) the optimal solution should be Z_pto = Z_i* (PTO impedance equal complex conjugate of intrinsic impedance). I would recommend starting there, and comparing these results to linear theory