Uncertainty in wave energy resour...
Uncertainty in wave energy resource assessment. Part 1: Historic data Edward B.L. Mackay a,*, AbuBakr S. Bahaj a, Peter G. Challenor b a Sustainable Energy Research Group, School of Civil Engineering and the Environment, University of Southampton, Highfield, Southampton SO17 1BJ, UK b Ocean Observing and Climate Group, National Oceanography Centre, Southampton SO14 3ZH, UK a r t i c l e i n f o Article history: Received 16 September 2009 Accepted 16 October 2009 Available online 12 November 2009 Keywords: Wave energy resource Numerical wave model Hindcast Calibration Uncertainty a b s t r a c t The uncertainty in estimates of the energy yield from a wave energy converter (WEC) is considered. The study is presented in two articles. This first article deals with the accuracy of the historic data and the second article considers the uncertainty which arises from variability in the wave climate. Estimates of the historic resource for a specific site are usually calculated from wave model data calibrated against in- situ measurements. Both the calibration of model data and estimation of confidence bounds are made difficult by the complex structure of errors in model data. Errors in parameters from wave models exhibit non-linear dependence on multiple factors, seasonal and interannual changes in bias and short-term temporal correlation. An example is given using two hindcasts for the European Marine Energy Centre in Orkney. Before calibration, estimates of the long-term mean WEC power from the two hindcasts differ by around 20%. The difference is reduced to 5% after calibration. The short-term temporal evolution of errors in WEC power is represented using ARMA models. It is shown that this is sufficient to model the long- term uncertainty in estimated WEC yield from one hindcast. However, seasonal and interannual changes in model biases in the other hindcast cause the uncertainty in estimated long-term WEC yield to exceed that predicted by the ARMA model. �� 2009 Elsevier Ltd. All rights reserved. 1. Introduction Before a wave farm is installed developers and planners need to have an estimate of the energy that will be produced over the expected life time. Like other sources of renewable energy, ocean waves are a variable resource, impossible to predict precisely. This increases the risk associated with the development of a wave energy farm, since the upfront cost of a project is large and the return is variable and imprecisely known. It is therefore necessary to estimate the expected yield from the wave farm, the variability in power production and confidence bounds on these estimates. The uncertainty in estimates of the electrical power which will be produced by a wave farm can be split into three categories: 1 Uncertainty in future wave conditions. 2 Uncertainty in conversion from wave energy to electrical energy. 3 Uncertainty in availability of machines. This paper will focus on the uncertainty in future wave condi- tions but it is worth making some notes on the other sources of uncertainty as well. The electrical energy produced by a wave energy converter (WEC) in a given sea state is dependent on the full directional wave spectrum. However, for the purposes of esti- mating the yield it is useful to describe the response in terms of a small number of parameters. Most device manufacturers specify the power produced by a WEC in a ���power matrix��� in terms of the significant wave height, Hs, and energy period, Te. The power matrices are usually calculated from numerical simulation of the WEC using theoretical spectral shapes such as Pierson-Moskowitz (PM) or JONSWAP (see e.g. ), and are validated using a combi- nation of scale-model tank tests and sea trials with prototype devices. For real wave spectra there will be some deviation in the spectral shape and directional distribution from theoretical forms, which will result in differences in the power produced from the values specified in the power matrix. Parameterising the WEC response solely in terms of Hs and Te leads to uncertainties in estimates of power for a given sea state, but at this stage of the industry, where few experimental or hard data exist, it is a necessary approximation. On the whole, the effect of parameterisation is less important for higher Hs, since spectra tend toward standard shapes in larger seas. Kerbiriou et al.  have shown that partitioning directional spectra into separate sea states improves accuracy compared to using a simple parametric repre- sentation of the whole spectrum when estimating the performance * Corresponding author. Fax: ��44(0) 7861 383995 E-mail address: email@example.com (E.B.L. Mackay). Contents lists available at ScienceDirect Renewable Energy journal homepage: www.elsevier.com/locate/renene 0960-1481/$ ��� see front matter �� 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2009.10.026 Renewable Energy 35 (2010) 1792���1808
of the SEAREV device. Once further data has been gathered on the effects of varying spectral shapes and directional distributions, this can be factored into the estimated energy yield in a probabilistic manner. This is discussed further in Section 6. Interactions between WECs within arrays will cause differences in the power absorbed compared to an isolated device. This intro- duces a further level of uncertainty into the conversion from wave energy to electrical energy. There has been a considerable amount of workon thetheoreticalaspectsof WECinteraction effects (see  for a brief overview). Millar et al.  take a slightly different approach in order to examine the impact of an offshore wave farm on the shoreline wave climate, but their method could be applied to model array losses. They use the spectral wave model SWAN  to examine the effect of a generic WEC removing energy at various points in the wave field. This paper will focus on modelling the uncertainty in the predicted yield of a single device. Array losses can be factored into calculations when more precise information is available. The third category of uncertainty mentioned above is perhaps the most difficult to quantify. Mechanical failures are inherently unpredictable in a new technology. As operational experience is gained maintenance requirements will be better understood and it will be possible to estimate the availability of machines. At present it is difficult to put a realistic figure on this type of uncertainty. The aim of this paper is to estimate the uncertainty in predicted energy yield resulting from uncertainty in future wave conditions. Estimates of future wave conditions are based on historic condi- tions. The accuracy is limited by the accuracy of the historic data and the variability in the resource. Due to the length and complexity of the analysis, the work will be presented in two articles. This first article deals with uncertainty in the historic data and the second article  deals with uncertainty in the future wave conditions resulting from variability in the resource. 2. Summary of approach In order to get a reasonable estimate of the long-term mean and interannual variability in the power produced by a WEC at a specific site a long record of wave conditions is required. It is rare that at a site of a proposed wave farm there will be an existing long-term dataset. In absence of a long record for the site of interest, an approach similar to the Measure-Correlate-Predict (MCP) method used by the wind energy industry can be applied. In the MCP procedure short-term measurements recorded at the site of a proposed development (the predictor site) are correlated with concurrent measurements taken at a nearby reference site for which long-term data exists. This calibration is then applied to the historic data at the reference site to estimate the historic climate at the predictor site. The US and Canada have an extensive network of offshore wave buoys which can be used as long-term reference datasets. Recent assessments of wave energy potential from these buoys are given in [7, 8]. In Europe there are fewer offshore buoys. Halliday and Douglas  have presented a survey of the long-term wave data available in UK waters. They note that there is relatively little in-situ data available for the most energetic locations and that it would aid wave energy development if coverage was increased in these areas. Where there are no long-term measurements to use as refer- ence datasets, some authors have proposed the use of data from numerical wave models as a long-term reference [10���12]. Mollison  proposed that offshore data from ocean-scale models could be used as the boundary conditions of a smaller scale shallow-water wave model, which is used to estimate the wave conditions at the site of interest. Since wave model data are estimates rather than measurements, Mollison  suggests that the model data should be calibrated against nearby buoy measurements before use. Bar- stow et al.  take a similar approach, but use satellite altimeter measurements to calibrate the offshore wave model data, before using it to drive a nearshore model. This approach is now common in wave energy resource studies. Pitt  compared estimates of wave power from the UK Met Office wave model to estimates from buoy measurements at the location of the proposed Wave Hub site in South West Britain. Several recent studies [13���15] have used data from the nearshore model SWAN  with boundary conditions from the WAM model  to estimate the nearshore wave resource. However, the issue of uncertainty in energy yield predictions, necessary for the economic assessment of a wave energy project, has not yet been addressed. This is in part because until recently the industry has not required such detailed calculations. With the first full scale devices being deployed at present and rapid expansion of the wave energy industry foreseen over the next decade, the problem of making accurate yield predictions with quantified uncertainty needs to be considered. As mentioned earlier, the uncertainty in the historic data is discussed in this first article and the second article  considers uncertainty arising from the vari- ability of the resource. The uncertainty in the estimate of the historic WEC yield is examined using two hindcasts for the European Marine Energy Centre (EMEC) in Orkney. The Pelamis wave energy converter is used as an example. The power matrix for an early version of the Pelamis WEC is given in Table 1. It will be shown later on that over Table 1 Power matrix for an early version of the Pelamis WEC, values in kW. Te [s] 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 0.5 Idle Idle Idle Idle Idle Idle Idle Idle Idle Idle Idle Idle Idle Idle Idle Idle Idle 1.0 Idle 22 29 34 37 38 38 37 35 32 29 26 23 21 Idle Idle Idle 1.5 32 50 65 76 83 86 86 83 78 72 65 59 53 47 42 37 33 2.0 57 88 115 136 148 153 152 147 138 127 116 104 93 83 74 66 59 2.5 89 138 180 212 231 238 238 230 216 199 181 163 146 130 116 103 92 3.0 129 198 260 305 332 340 332 315 292 266 240 219 210 188 167 149 132 3.5 270 354 415 438 440 424 404 377 362 326 292 260 230 215 202 180 4.0 462 502 540 546 530 499 475 429 384 366 339 301 267 237 213 Hs [m] 4.5 554 635 642 648 628 590 562 528 473 432 382 356 338 300 266 5.0 739 726 731 707 687 670 607 557 521 472 417 369 348 328 5.5 750 750 750 750 750 737 667 658 586 520 496 446 395 355 6.0 750 750 750 750 750 750 711 633 619 558 512 470 415 6.5 750 750 750 750 750 750 750 743 658 621 579 512 481 7.0 750 750 750 750 750 750 750 750 676 613 584 525 7.5 750 750 750 750 750 750 750 750 686 622 593 8.0 750 750 750 750 750 750 750 750 690 625 E.B.L. 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an 8-year period covered by both datasets, the mean power produced by the Pelamis calculated from each hindcast differs by over 20%. It is clear that one or both models produce significantly biased estimates. In this article the calibration of wave model data is discussed and a method is proposed to calculate confidence bounds for estimates of WEC yield from calibrated model data. The paper begins with a description of the errors in model data in Section 3. In Section 4 techniques for estimating model errors are discussed. In Section 5 these techniques are applied to calibrate the hindcasts for the EMEC site. In Section 6 the estimation of confi- dence bounds on the estimate of energy yield from the calibrated hindcasts is discussed, and in Section 7 the two calibrated hindcasts are compared in more detail. In Section 8 the results are discussed and conclusions are presented. 3. Errors in wave model data Understanding the features of errors in model data is vital both for calibration purposes and for the calculation of the uncertainty of derived wave energy statistics. Error sources can be viewed as either internal or external to the model. The internal sources of error are the formulation of model physics or ���sources terms��� and the numerical resolution, while external errors refer to errors in the input data, primarily the wind field. An in depth review of the present state of the art and limiting factors in wave modelling is given in . In this section we are concerned with a description rather than diagnosis of model errors. Modelled wave spectra can be considered an estimate of the average conditions over the grid spacing and time step used in the model. Typically, global or oceanic scale wave models will be run with a grid spacing somewhere between 0.5 and 3 (about 50���300 km) with a time step of 3 or 6 h. Measured data are obtained over a smaller scale, with data from buoys representing a point average over time (between 20 min and 1 h) and altimeter data representing an instantaneous spatial average over an area of 5���10 km in diameter. The spatial and temporal variability of wave conditions will therefore result in differences between measure- ments and modelled data, even if both are perfectly accurate. The larger scales over which wave models estimate conditions result in time series of model data appearing smoother than those from in-situ measurements. It can also lead to small intense pressure systems being subject to some smoothing, resulting in systematic underestimation of peak wind speeds and hence peak wave heights . The calibration of wave model data involves estimating the mean error under a given set of conditions. Modelling the random errors is necessary for estimating confidence bounds. It can be difficult to distinguish between the mean and random model errors, since the error at a given location is the integrated effect of mean (predominantly internal) and random (predominantly external) errors over the whole wave field. Both the mean and random components will have a complex dependence on the actual wave conditions. For instance the bias in a model estimate of Hs may have a dependence on the actual Hs, period, spectral shape, swell age, etc. Due to the way that errors occur in wave models and propagate through the model domain, the biases are non- stationary with location and with time. Janssen  presents a particularly clear illustration of the non-stationary biases in spectra from the European Centre for Medium Range Weather Forecasts (ECMWF) WAM model. A plot of the bias in spectral energy binned by frequency shows that the model tends to over- predict energy at lower frequencies in the Northern Hemisphere summer and much less in the winter time. Moreover, the magni- tude of this bias and its dependence on both frequency and time of year changes from year to year. He notes that the main reasons for the changing biases are that large swells generated in the Southern Ocean in the Southern Hemisphere winter time are not well modelled due to the formulation of the dissipation source term and unresolved islands and atolls. This goes to show that it is difficult to define and adjust for a ���mean error component��� since varying conditions lead to varying amounts of internal and external errors occurring and aggregating over the model domain. Therefore errors in wind seas and young swells can be expected to have different characteristics to older swells that have propagated further, increasing uncertainties. A further reason for non-stationary biases in model data is changes made to the models themselves. This is more of an issue for archived data from operational models than for hindcasts. However, despite the fact that hindcasts are run with a constant model setup, the quality of the input wind fields and assimilated wave data may be varying. As well as biases changing with time and location, the random error will persist in both time and location. For instance models will tend to over or under predict the intensity of an entire storm, which leads to correlation of errors up to a few days. Additionally errors in various parameters can be correlated. At high sea states, since wave spectra tend toward standard PM or JONSWAP type forms, an overestimate in model Hs will result in an overestimate of period as well. This correlation of errors between parameters means that one needs to be careful when calibrating model data, since adjusting model parameters independently may lead to changes in the shape of their joint distribution. Finally, we note that modelled data may be subject to temporal offsets, with the model predicting that a storm arrives slightly early or late. This type of error is sometimes referred to as a ���jitter error���. Jitter errors are not so important when calculating long-term mean statistics from modelled data, but are important for validation purposes where concurrent modelled and measured data are compared. To summarise, the main features of the errors in model data are: The bias and variance of modelled parameters may depend on multiple factors such as Hs, Te, swell age, etc. The bias and variance of the modelled parameters may be non- stationary with both time and location. Errors in parameters exhibit short-term auto-correlation. There may also be correlation of errors between parameters, e.g. errors in Hs and Te may be correlated. There may be temporal offsets or ���jitter errors��� in modelled parameters. 4. Techniques for estimating model errors 4.1. Published studies Model errors are estimated by comparing collocated modelled and measured parameters, where the measurements are usually from wave buoys or satellite altimeters. Various techniques have been proposed to determine the errors. If there is reason to believe that the model bias may be a linear function of a model parameter then linear regression can be used [20,21] to test for non-linear- ities the bias and standard deviation of model data, can be plotted against integrated buoy parameters such as Hs and Tp  or the bias can be calculated in discrete frequency bands [19,22]. A more sophisticated approached was implemented by Caires and Sterl , in which corrections were estimated using a non-parametric method, based on analogues in a learning dataset. Alternatively, if three or more concurrent datasets are available then a multiple collocation technique can be used to explicitly E.B.L. Mackay et al. / Renewable Energy 35 (2010) 1792���1808 1794