Where techno- meets -economics

 

Annual Energy Production. Where techno- meets -economics.  

But what are the devilish details along the way to coming up with one number that describes the power capture of a wave energy converter? And how does this fit into a conversation about commercial development? And what red flags 🚩🚩🚩should we be on the lookout for when trying to make sense of published numbers on wave energy converter performance? 

  

Back in the good old days we had 'capture width ratio'.  

This was the ideal performance metric for desk-based PhD students, or for competing device developers, as it muddied the waters of a sensible economic comparison.  Capture width expresses the power captured by a wave energy converter (WEC) in terms of the width of the incident wave that has been absorbed. When we divide this by a characteristic device width then we get the 'capture width ratio', often thought of as a hydrodynamic efficiency.  

There are some awesome mathematical results for the theoretical upper limit of capture width: it depends only on the wavelength and the degree of freedom of power extraction: surge and pitch devices have twice the upper limit of heave devices. This can be useful for spotting suspicious performance claims by device developers🚩 To do this, you would find the capture width as a function of each frequency component, and check whether it exceeded the upper limit at any point. Alternatively capture width could be calculated for a specific sea-state, and this can be used for spotting outliers 🚩in performance claims (e.g. in Babarit's database comparing 156 published performance results). In general though, various types of murkiness hide behind this metric, including: 

• This upper limit is based on the assumption of 'optimal reactive control'. There are several ways to hack 🚩 this assumption without it being obvious or even intentional. For example you can get closer to this upper limit by applying reactive control in the context of various assumptions, such as neglecting PTO losses, using regular waves, using irregular waves with a narrow-banded single-peaked spectrum that you have knowledge of, or running a linear simulation without checking that the results are wildly out-with the bounds of linearity.  
• There was no common definition of the waves: use of regular waves was common to compare to the theoretical upper limit, but this is a poor proxy for how much energy could be captured in 'irregular' sea waves.  
• The term 'hydrodynamic' points to the first stage of the wave-to-wire process: where the power is calculated from the force and velocity on the hydrodynamic body; so the behaviour and losses of the power train are not included. This is one way to get closer to our theoretical 'optimal reactive control', because this upper limit applies to the hydrodynamic power. The assumptions associated with this upper limit are often transferred to modelling: so something to watch out for 🚩is that much of the early publications on wave energy do not model power train losses. 
• The 'width' parameter is a poor proxy for capital costs and tricky to apply systematically.  Babarit's database addresses this, but for a long time irrelevant comparisons between devices were made.  

Recipe for AEP  

Eventually there was a need to standardise things, so there is now much less wriggle room when describing WEC performance. Most of this standardisation is around how we calculate a projected annual energy production (AEP).  

The ingredients for annual energy production are availability, the device power matrix and the resource joint probability distribution.  Both the incident wave power at a particular site (the resource), and the power generated by a particular device, are plotted against a characteristic wave height and period. The product of the two matrices is summed, availability is factored in, or sometimes not, and that's your estimated AEP.   

Ingredient 1: Availability 

Availability is the percentage of time that the device is in a fit state to operate. Availability is often not factored in during early stages of development, but becomes increasingly significant once expensive kit hits the waves.  

Availability is difficult to assess at pre-commercial levels, as teething problems are being ironed out and learning-by-doing is to be expected. However, some design choices have a big impact on availability: particularly the ability to disconnect and tow away in short weather windows using affordable vessels.  
 

Ingredient 2: Wave resource 

The necessity of standardisation has introduced a lot of abstraction and additional terminology, and it is useful to be aware of how much real information is thrown away to allow us to converge on this common approach, which can be used in sea trails, tank tests and simulations. Let us follow the journey of a dataset collected by a waverider buoy or a bottom-mounted acoustic doppler profiler, to see how this becomes a resource joint probability distribution.   

• Our dataset starts off as a continuous recording over at least a year. The first step is to 'clean' the data by flagging bad data points or interruptions in logged data.   
• We are interested in the 'wave elevation' - the vertical motion of the surface for a fixed point on the map. This time series is extracted from the cleaned data. Here we are throwing away the horizontal component.  Generally this is not needed, as standard hydrodynamic models relate the wave excitation force to the surface elevation, even for surge devices where the horizontal component is driving the motion. 
• Typically we also discard the directional content of the waves. This is not suitable for devices designed to work in waves from a particular direction. 🚩 
• The wave elevation time series is split into successive wave records, typically 0.5 - 2 hours long. This length depends on how fast the sea state is changing. We need each record to be 'statistically stationary': a sliding window applied to the record should show there are no trends in the statistical characteristics. 
• Fourier analysis is carried out on each record to give a spectrum, from which spectral moments are calculated.  
• A pair of spectral moments are used to place each record on a 'scatter plot'. The characteristic wave height is Hm0, and there is some variation in the choice of the characteristic period. Two common parameters calculated from moments are Tz and Te.  The entire dataset is represented as a scatter plot using these characteristics as axes. Each dot on the plot represents one record.  
• The scatter plot is then discretised by grouping the dots into 'bins': squares defined by a range of periods and a range of wave heights. The result is a grid with a tally of the number of wave records in each square. Dividing everything by the total number of wave records in the dataset gives us the resource joint probability distribution. Each square contains the percentage of time for which that 'sea state' was measured.  
• Although the data measured represents average weather, typically over one or two years, it is used as an indication of the (longer term) climate at that location.  

I hope you are impressed with just how different the idea of a sea-state is from what's actually happening in the sea.  

Ingredient 3: Power matrix 

The third component needed to estimate annual energy is the device power matrix, and it is interesting to see how the joint probability distribution is used to calculate this in the absence of sea trial data.  

The starting point is an assumption that all the wave records in a particular bin can be described by a generalised spectrum with a characteristic height and period that are in the centre of each grid square. 🚩The wave power transport of this generalised spectrum is unlikely to be representative of the wave records in that bin for the simple reason that wave power is proportional to the square of the wave height.  

The spectra used in simulations or wave tanks (e.g. JONSWAP) tend to be single peaked, which are not the most common spectral shape in recorded wave data.  It is easier to approach our theoretical limit with a single-peaked spectrum than a bimodal spectrum, so this is an abstraction which can widen the gap between anticipated power production and results from sea trials.  

Here is a typical process for generating a device power matrix from simulation or wave tank data:  

• In tank tests it might be too costly to run all the sea states, so several are selected across a range of heights and periods, and the results are later interpolated. For example, the Wave Energy Prize (WEP) used six different period/height combinations and the WECCCOMP (wave energy control competition) used three. 
• JONSWAPs are typically defined by the peak period, so when the power matrix is generated using a generalised spectrum rather than sea trial results, it is common to use Tp as the period index. 
• Some standardised spectra have a third parameter to describe 'peakiness' and this isn't always disclosed. For JONSWAP this is g, and the smaller the value, the peakier the spectrum. It is worth noting how peaky the spectra are, as a good control algorithm can extract more power in a peakier spectrum. This is why the WECCOMP included sea states with two different values of g.   
• Each sea-state is represented by a JONSWAP spectrum with height and period characteristics from the centre of its grid. The spectrum is used to synthesise wave elevation time series. In a tank this would be Froude –scaled down to a scale appropriate to the tank and WEC model. The phase of each frequency component is assigned a quasi-random number. This allows us to either use the exact same sequence of waves to compare variations (for example, control policies), or to generate different instances of the same sea state. In wave tanks, calibrating a particular run of waves is additional work, so it is standard practice to select one set of phases, and then generate a time series equivalent to two hours of the sea state: long enough for it to be statistically representative and for those rare high waves to appear.  
• The wave elevation time series is a repeating sequence that can be run in a loop. This gives us time to ramp up the experiment, then select a block of data that is as long as the repeating sequence for our analysis. In hydrodynamic simulations we would measure the 'hydrodynamic power' over this selected block of data: this is the product of the velocity of and PTO force on the hydrodynamic body. Typically in tank tests we would measure the 'mechanical power' entering the PTO, for example the velocity and force of the PTO mechanism. 
• It is quite common to see simulation results averaged at this point and used to populate a power matrix. However, for a more accurate reflection of what to expect in sea trials, you would apply a model of the power train to this time series to estimate the electrical power generated. 
• In tank tests it is necessary to Froude-scale the results back up to full-scale, and often to interpolate results to fully populate the power matrix so that its resolution matches that of the resource joint probability distribution. The power is interpolated linearly with respect to period and with the square of the wave height.  

 The devil in the detail: power train modelling 

There is no standardised process for the power train modelling, and it is not uncommon to see results where this is not included. For example, Babarit's study includes performance data from devices tested at sea, in tanks and in simulation, so sometimes the power train losses are included, and sometimes not. 

In particular, it is worth flagging 🚩🚩that the combination of ignoring PTO losses and allowing reactive control will give much higher power capture than could be realised in the real world. This does not allow a level playing field for comparing concepts with and without reactive control. This is one oversight of the otherwise well-designed WEP. 

🚩One of the difficulties lies in an ambiguity in terminology, particularly when applied to different types of WEC or power train. Take the term 'PTO' for example: does it refer to the first step in a wave-to-wire model, where the device's motion is damped, perhaps by (a) a hydraulic system that pumps fluid into an accumulator, or (b) an electric generator that pumps reciprocating electrical power into a capacitor bank? Or does it refer to the final step in this process, where (a) the hydraulic accumulator drives a uni-directional electric generator, or (b) the capacitor bank feeds power electronics that output grid-quality electricity? For most people, the term PTO is associated with an electrical generator. In the two examples above, the generator is doing different jobs, so talking about electrical power isn't enough to make the distinction. The first step of capturing energy will involve energy losses. So for case (b) the input mechanical power at the PTO (e.g. generator torque times speed) will be greater than the electrical power (voltage times current), and these losses can be defined using an efficiency curve (usually provided by the generator supplier).  The process of smoothing and rectifying the captured power involves more losses. These two steps are different in nature; in the first step, reactive control will result in bidirectional power, whereas in the last step, you are always generating power, unless something has gone drastically wrong.   

The way to get around variations in terminology is to look at the equation for power. If it is force times velocity then this is the mechanical or hydrodynamic power: losses in the first step have not yet been included. It is also helpful to note that in simulation work this mechanical power is often referred to as the 'power', e.g. in a well-known numerical benchmarking. And it is very common in laboratory work (e.g. WECCOMP) to consider the losses in the first stage of capture, but not the losses associated with smoothing and rectification. As soon as a developer is grid connected, then neglecting any of these losses is not an option.  

Sometimes the electrical power matrix is estimated by multiplying the mechanical power matrix by a single value of efficiency.  While this is a quick approach, there are advantages to modelling power train losses using the time series. This includes the possibility of representing:  

• Real world limits on the first stages of the PTO, e.g. maximum power, speed or torque, or rate of change in demand torque.   
• Efficiency of the first stage of the PTO as a function of speed and torque, which is significant for reciprocating motion, where the speed drops to zero twice with every passing wave.  
• Limits on storage capacity in the smoothing stage of the power train.  
• Real world limits on the final stages of the power train, e.g. limits on any of the inputs as well as the maximum (rated) power.  
• Non-linear behaviour specific to the power train used, such as backlash, hydraulic transmission, or intermediate storage. 
• In the case of  🚩 reactive power, the sign of the efficiency depends on whether the PTO is damping or driving the buoy motion (e.g. WECCOMP ensured that PTO efficiency was correctly applied for bi-directional power flow).  

Is is common to see simulation results where the maximum power is clipped; in reality this would be achieved with a lower torque demand, and this would have an impact on the hydrodyamic response. If possible, it is better to include limits in the simulation rather than the post-processing.

How is Annual Energy Production used? 

The way AEP is collected and used typically depends on the stage of development: 

• TRL8-9: In devices backed up by sea trials, electrical power fed into the grid could be used to estimate LCOE to build a case for funding.  Typically the annual energy production would encompass 'availability', power train losses, and transmission losses up to the grid connection point. The wave conditions would be measured at the same time, so the performance could be generalised as an electrical power matrix, allowing LCOE projections for other sites or for proposed improvements.
• TRL 7-8: Often the generated power in smaller scale sea trials isn't fed into the grid, but used to charge a battery or dumped into a heating element. So the measured fluctuating power would include losses due to the first stage of power capture, but not due to power smoothing. To use the measured electrical power matrix to estimate AEP for LCOE, conversion losses and availability would need to be estimated and factored in.
• TRL6-8 To win support for sea trials, typically annual energy would be estimated from tank tests, and often combined with power-train bench tests. There isn't a standard approach to how availability and power train losses are factored in. There can often be surprising differences in estimated power between tank tests and sea trials. 🚩 Richard Yemm recalled that the first Pelamis trials generated half the power and experienced twice the loads than expected, citing 'non-linearity' and variability in component behaviour as the causes. 
• TRL 3-5: For laboratory research, mechanical power (speed times load) measured in tank trials might be used alongside proxies for capital costs to assess economic viability. For example, both the WEP (for device concept) and the WECCCOMP (control policies) penalised behaviour associated with high costs: low PTO load ratios, and forces or displacements that exceed limits. The WEP used the volume of the WEC as a proxy for capital cost, but did not include a proxy for the cost of resisting mooring forces. Often availability is ignored. There is growing consensus about the importance of representing power train losses, but no standard implementation, e.g. WECCCOMP assumed 10% electrical machine losses, while these weren't modelled in the WEP.  
• TRL2-4: The previously mentioned papers (numerical benchmarking and the database comparison) are examples of a common practice in simulation studies, where power train losses and availability aren't modelled. In both these studies, power matrices and resource joint probability distributions were used, and the results were then expressed as the mean capture width ratio for a given wave resource. 

Take away message 

There are now de facto standards for calculating annual energy production, which greatly aids sensible commercial conversations. But (a) there is so much simplification that high uncertainty is built in, and (b) there is still also enough variation in methodology that it is essential to read the small print. The 🚩red 🚩flags 🚩 above indicate some of the things to look out for. This is surely not an exhaustive list.   

 

Credits 

This blog was written as a first step in generating material for the AIM-WEC project, at the University of Heriot Watt. 

Image credit: Allstarzombie55

References:

Spectral moment table from coastalmonitoring.org 

WECCCOMP 

Wave Energy Prize 

Richard Yemm's Peaks and Troughs talk 

Babarit's database comparing published WEC performance 

A numerical benchmarking by Babarit et al comparing simulated results