Predictability of WEC yield

6.4.1 Distribution of multi-year mean values

By combining the time series model for the NAO (equation 6.2) with the relationship between the NAO and Pelamis power we can generate sequences of annual anomalies in Pelamis power of arbitrary length. An example 500 year simulation is shown in Figure 6.12 with 5 and 20 year moving averages. There are extended periods when the 20 year moving average is above or below zero, for over 50 years at times. It should be

reiterated though, that this is a stationary process, but one that has long-range dependence.

Figure 6.12. 5 and 20 year moving averages of simulated anomalies in annual mean Pelamis power from FAR(1) model.

A measure of the predictability of the wave resource can be obtained from the

distribution of differences in average power between consecutive periods. This tells us the uncertainty in the future resource, if the estimate is based on the historic resource.

To do this we generate a long series (1 million years in this case) of power anomalies as described above. The series is then divided into 5, 10, or 20 year blocks and the

difference in mean power between each block is calculated. The differences were found to be normally distributed, with standard deviations of 17.5 kW, 13.3 kW and 10.3 kW between consecutive 5, 10 and 20 year mean values respectively.

How does this compare to the assumption that annual power anomalies are uncorrelated noise? The standard deviation of the annual anomalies is 28.6 kW, so under the

assumption that interannual variability is a white noise process the standard deviations of differences between mean power over consecutive 5, 10 and 20 year periods are 18.0 kW, 12.8 kW and 9.0 kW respectively. So the variance in this case decreases slightly faster than for the FAR(1) model. The difference is small since the FAR(1) model for NAO variability is dominated by irregular fluctuations with only a small level of long-range dependence. Moreover, the influence of the residuals in equation 6.1 is of roughly the same magnitude as that from the NAO: the residual standard deviation is 15.4 kW compared to the value for the slope of 14.4 kW / hPa and the NAO index varying between ±4 hPa. This means that a lot of the structure of the modelled NAO variability will be masked by the random noise term in 6.1. Moreover the assumption that ε in 6.1 is a white noise process may not be accurate and there may be a small amount of temporal correlation which we were not able to detect from the relatively short record.

A similar finding was made by Pitt (2006a). In order to study the variability of the wave power climate at the Wave Hub site in South West England, Pitt (2006a) constructed a predictor based on an ‘Index of Westerlies’ over the North Atlantic, analogous to the NAO index. Despite the reasonable correlation (0.79) between monthly mean wave power and the Index of Westerlies, there was a large uncertainty in the resulting predictor and he concluded that the use of a longer hindcast is likely to give more accurate results.

This uncertainty in our NAO-based predictor may explain why the observed difference of 33 kW in mean power levels between 1960-1980 and 1980-2000 exceeds that expected from the FAR(1) model. A difference of 33 kW between mean power over consecutive 20 year periods corresponds to the 99.93 percentile of the distribution of differences from the FAR(1) model. From this we can conclude one of two things:

either that the combined use of equations 6.1 and 6.2 is not an appropriate model for annual anomalies in WEC power levels at this location, or that a statistically significant change in wave climate occurred between the 1960s and 1990s. There is a large

uncertainty related to both conclusions. As noted before detection and attribution of human influence on the NAO and wave climate is made difficult by the short length of climate record available and the differing results between climate models on the effects of GHG forcing. Nevertheless, the evidence is building to suggest that the observed increase in the NAO index and corresponding increase in North Atlantic wave heights in the latter part of the twentieth century are the result of increasing greenhouse gas concentrations (Gillett et al, 2003; Wang et al, 2006b).

With regard to the appropriateness of the stochastic model, fitting a ‘long-range’

dependence model to a series of only 100 years results in quite a high uncertainty in the maximum likelihood estimate of the fractional difference parameter. This uncertainty in the choice of model for the NAO is reflected in the fact that there is still no consensus in the appropriate choice of model (Stephenson et al, 2000; Mills, 2004; Barbosa et al, 2006). Moreover, fitting a stochastic model to the NAO record to explain ‘natural’

variability may not be appropriate since the record may already contain some effect of anthropogenic forcing.

Finally, Woolf et al (2002) have observed that the residual anomalies in winter wave heights in the north-east North Atlantic, once the NAO signal has been removed, are correlated with the “East Atlantic pattern” in atmospheric pressure (see e.g. Rogers, 1990; Zveryaev, 1999). This may mean that our assumption of independent residuals in equation 6.1 is not strictly valid. However there is little literature on the temporal variability of the East Atlantic pattern, so this will not be investigated further here.

6.4.2 The effect of uncertainty in historic data

In Chapter 5 we saw that over an 8 year period the mean Pelamis power calculated from the two calibrated hindcasts differed by 10 kW. The differences in annual mean power over the 8 years concurrent period had a standard deviation of 9 kW, with a maximum of 20 kW and a minimum of -5 kW. Is difficult to say which hindcast these differences resulted from, or if the hindcast data used in this chapter has similar behaviour, but we can make some general remarks on the effect that any errors in the historic data would have on the predictability of WEC yield.

The first observation to make is that a bias in the estimate of the historic resource will result in a bias in the mean value of the predicted future resource. Whether this is a constant bias each year or the net effect of zero-mean random errors in the annual mean (as might arise from the use of altimeter data), the effect on the mean predicted resource will be the same. However, whether the errors in the historic data are predominantly bias errors or random errors will effect the estimated distribution of possible future resource. The observed distribution of annual power anomalies is the convolution of the true distribution of annual power anomalies with the distribution of errors in the

estimate of annual mean power. This will result in a positive bias in the estimate of the standard deviation of power anomalies. If we assume that the errors in annual power have a standard deviation of 8 kW, as for the hindcasts in Chapter 5, and the true annual anomalies have a standard deviation of 28 kW, as found above, then the observed annual anomalies would have a standard deviation of 29 kW - only a small increase.

The effect of random errors in the historic wave data on the stochastic model described in Section 6.3 is slightly more subtle. The uncertainty in the estimate of the slope in equation 6.1 results both from errors in the wave data and from sampling effects from having only 50 years of data. The standard deviation of the residuals in equation 6.1 is

15.4 kW, so even if the standard deviation in the estimates of annual power is 8 kW (which is probably an overestimate), then the true residual standard deviation would be 13.2 kW. Therefore finite sampling effects are likely to be the main source of

uncertainty in the estimated relationship between power anomalies and the NAO in this case.

So we can conclude that the main effect of errors in the estimate of the historic resource is to bias the estimate of the mean future resource and that the effect on the estimate of interannual variability is small. In relation to a ±20-30 kW uncertainty in the predicted mean power over a 20 year period, a 10 kW bias is still significant, but whether it is worth investing further time and money to improve estimates of the historic resource will depend on the sensitivity of the economics to uncertainty in the predicted yield.

6.4.3 Sensitivity to climate change

We investigate the sensitivity of WEC yield at this location to climate change through the link with the NAO. The output of climate models has been used to examine the effect of increased atmospheric concentrations of GHGs on the NAO by several authors (e.g. Gillett et al, 2003; Osborn, 2004; Terray et al, 2004; Kuzmina et al 2005;

Stephenson et al, 2006; Pinto et al, 2007). Results have varied depending on the climate model used. An estimate of the uncertainty in the model representation of climate physics can be made by using a multi-model approach (Collins et al, 2006; Tebaldi and Knutti, 2007).

Stephenson et al (2006) have used the output from 18 AOGCMs to investigate the response of wintertime NAO to increasing concentrations of atmospheric carbon dioxide (CO₂). They examine the model simulations of the NAO over 80 year periods with both constant forcing and transient forcing at 1% per year increasing CO2, (other possible anthropogenic forcing, such as changes in the concentration of other

greenhouse gases, aerosols, or ozone are not included). Of the 18 models compared, 15 appeared to be able simulate the main features of NAO, but there was much model-dependent variation in how the models simulated the amplitudes, spatial patterns, and future trends of the NAO. Of the 15 models able to simulate the NAO pattern, 13

(2006) estimate the NAO sensitivity at 0.0061 ± 0.007 hPa per %CO₂ and note that this result is relatively robust to the exclusion of the models without the NAO dipole. They note that the true confidence interval is likely to be larger than this since the model sensitivities are neither independent nor normally distributed, but since even this interval includes zero, the null hypothesis that there is no effect of CO2 increase on the NAO index, cannot be rejected at the 5% level of significance. Since the combined results from 18 models cannot be used to reject the no-effect hypothesis, caution should be used when drawing conclusions from studies using smaller subsets of models (e.g.

Gillett et al, 2003; Osborn, 2004; Terray et al, 2004; Kuzmina et al 2005; Pinto et al, 2007).

The results of Stephenson et al (2006) are for the response of the winter (Dec-Feb) index, so in order to use them to investigate the effect of climate change on wave energy, we must assume that they apply equally to the annual index that we have used.

Since the highest power levels are in winter anyway, this should capture most of the response.

At present, atmospheric concentrations of CO2 are rising at around 1.9ppm per year or approximately 0.5% per year, with current (February 2009) levels around 387ppm. This gives an increase of around 10% in a 20 year period. So even if we assume that the NAO will respond with the most extreme sensitivity of the models reported by Stephenson et al (2006), of 0.04 hPa per %CO2 (estimated by the ECHAM4 model), then in a 20 year period the NAO could increase by 0.4 hPa. So under our model for the relationship of Pelamis power with the NAO, equation 6.1, this would result in an increase of 5.7 kW. Assuming this happens gradually over the 20 year period, this would result in a net increase of around 2.9 kW over the entire period. Considering that we calculated the standard deviation of the differences between mean power levels over consecutive 20 year periods as 10.3 kW, this increased power caused by the increasing mean value of the NAO would be undetectable amongst the ‘natural’ variability. This implies that changes in WEC power levels related to anthropogenic emissions of GHGs will probably not effect the predictability of the future resource based on historic

estimates, since the change in the NAO over timescales we are interested in is much less than the ‘natural’ noise level. However, whether the increase in wave power between the 1960s and 1990s was natural variability or not is debatable.

This is, of course, based on many assumptions which are difficult to justify. Firstly, the response of the NAO to increased levels of CO₂ is highly uncertain and will almost certainly not be a linear increase over time. Future CO2 emissions are also uncertain.

Moreover, we are assuming that the response of wave power to the NAO will remain the same in a changed NAO state. Wang and Swail (2006) give a similar analysis of the uncertainty in predictions of future wave conditions and note that the uncertainty due to differences among the emissions scenarios is much smaller than that due to differences between predictions from different climate models.

Since there are such high uncertainties in predictions of future climate, it may be more useful to note the effect on WEC yield under various NAO change scenarios. If the mean value of the NAO index increases by 1 standard deviation (about 1.5 hPa for the index used here), then the mean Pelamis power at the location in question here would increase by 21.6 kW (5.6%). A mean increase of 2 standard deviations in the NAO index would result in a mean increase of 43.2 kW (11.2%). Since the NAO has an affect on storm track, frequency and intensity it may be naïve to assume that residuals ε in equation 6.1 will not be affected by a change in the NAO state. Nevertheless, the NAO accounts for about 70% of the interannual variability in Pelamis power at this site, so we can certainly expect some response to changes in the mean NAO state.