Complex conjugate control without the equations

Many people would like to know about complex conjugate control of wave energy converters, and the problems with its implementation, without wading through masses of equations. Ok, lets not kid ourselves, only a few people want to know this, but I cater to all tastes, including highly refined ones.

While I won't use equations directly, I will need to make reference to a well-known mathematical model, if you're an engineer, that is. Different mathematical models suit different purposes. For explaining how complex conjugate control works, a simple linear mass-spring-damper model, with excitation at one frequency only, is a good place to start. If you need to brush up on this model, there are some reminders here.

The natural frequency of the system (f_n) is where the response amplitude is highest, which happens to be the only frequency where the response phase is zero (with respect to the excitation phase). A system that is being excited at its natural frequency is described as resonant.

The basic idea behind complex conjugate control is to have a small (= low cost) absorber that has a natural frequency higher than the power-rich frequency components of a typical wave spectrum, and to then force the system into resonance by doing clever things with the power take off (PTO) force.

What resonance means for a wave energy absorber

With a natural frequency higher than the excitation frequency, a small absorber will not be resonant. We noted that at resonance there is zero phase difference between the excitation and the response. Here the response is the velocity of the absorber. The phase is zero because, at the resonant frequency, restoring (spring) force and the mass-acceleration (do not call this a force in front of a physicist) cancel each other out. This leaves an equation with excitation force on the one side and velocity on the other. Hence the velocity and excitation have the same phase.

We also know from the theory of optimum wave absorption [Evans1976] [Mei1976] [Newman1976], that for this simple mass-spring-damper, we need velocity to be in phase with the excitation to get optimum absorption. As I have just shown, this is equivalent to saying we need resonance.

The theory behind tricking the system into resonance

Normally, when you think of extracting energy from a linear mass-spring-damper system, you'd damp the velocity. The PTO force would then be in phase with the velocity. We can't use PTO damping to change the natural frequency of the system, which depends on mass and spring only. However, if you are able to use the PTO system to apply a force that has some damping, and also cancels out the mass-acceleration and restoring forces at that frequency, then the system acts like it is in resonance. 

Does this idea of tricking the system into resonance work?

Let's consider what this method entails and how it is applied. We can think of the required PTO force as having two components: one in phase with velocity, that is extracting power, and another ±90° out of phase with velocity, that stores and releases power. During each wave cycle (period), the system would alternate between opposing the motion to extract wave power (generating), and driving the motion (motoring). When the power is averaged over the cycle, you still get more average power than by generating only (a force in phase with velocity). Here are the requirements for such a system:

• The PTO system must be able to switch between operation as a generator and as a motor.
• For complete absorption, the PTO system would need to be 100% efficient at all stages of the power cycle. Once again, physics spoils our fun! Nevertheless, it is conceivable that some of the advantages of this method could be attained with a high efficiency PTO system. For this to happen, the extra wave power extracted by operating at resonant conditions must be more than the power lost in the drive train.
• In such a system the peaks of instantaneous power would be much greater than the average power. The same could be said for a damping-only PTO, but to a lesser degree. This would have an impact on the size of the PTO plant, and hence the mass, volume, hydrodynamic design, and cost of the project.
• Sea waves are made up of a spectrum of frequencies. This method only allows us optimise the average power extracted from one frequency. At other frequencies the system does not look resonant; the waves radiated at these frequencies are not the right phase to cancel out (capture) these frequency components of the incident wave.
• The biggest problem with this method is that this first step in the theory does not consider motion constraints (a popular research topic now). Basically, to generate big waves with a small body, it needs to make big motions. The smaller the body, the smaller the motion at which linear theory is no longer valid or useful. For example, it is not useful if our linear model tells us our 5m deep heaving buoy oscillates with an amplitude of 50m. Anything which stops the motion required to radiate waves of the optimum phase and amplitude (extra spring to move the system out of resonance, generator losses, end-stops, gravity, etc) will stop us achieving our goal of efficient power absorption.

It is clear that the laws of physics will confound our goal of achieving 100% absorption using reactive control. The jury is still out on whether some advantage could be gained by applying these principles in some way. For example, the technique called 'latching' is a non-linear implementation of complex conjugate control (also known as reactive control). My personal opinion is that complex conjugate control holds the most potential benefit for devices that are far from resonance, which in practice means small, and that implementation problems are more severe for smaller devices. 

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