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Status: round two; last active here December 2010

Wind Power Curve Model

(formally: Reconcile Wind Farm output with BoM wind data)

ABSTRACT

We want to simulate the power that would be generated from a wind farm placed in the vicinity of a wind speed data source, usually a Bureau of Meteorology (BoM) Station. To establish methodology for achieving this, we use real wind farm output data in conjunction with wind speed data from nearby BoM stations.

We start by developing a Wind Power Curve that reasonably estimates wind farm power output based on sampled wind speed data. We then examine in some detail how well this curve works, before introducing further adjustments to the methodology (in particular dealing with the Diurnal Bias). We finish by examining and recording how well the methodology works, as a benchmark for future development.

The Round One work from earlier in 2010 has plots showing the correspondence between five SA wind farms and nearly BoM Station wind speed data. For each pairing, there are two aspects shown: (i) plots showing the data comparison for one month at 30 min resolution, and (ii) a snap from Google Earth showing the relative locations of the wind farm and BoM Station.

INTRODUCTION

BoM weather stations are not generally positioned in the locations where wind farm builders might choose. For example, BoM wind data may be associated with a town or airstrip, which has most likely protected from the wind to some extent by the local geography. In contrast, a wind tower would most likely be located on the top of a hill, or a ridge running perpendicular to the prevailing wind. Also, BoM wind measurements are taken at 10m above ground level, whereas the wind turbine hub is usually at around 80m, and it is well understood that the wind speed increases with height.

SO, we do not try to calculate wind farm output in a direct, physical, way from the wind speed data, but rather we will look at how the two compare for a number of cases where we have wind farm generation data and a nearby (or not so nearby) BoM station. We will establish a heuristic relationship that is good enough for the purpose of building renewable scenarios with wind farms in different places.

Wind farm power output can be related to wind speed via a response curve that we call a Wind Power Curve (WPC). This is a heuristic curve, and there is an important distinction to maintain: a "power curve" as might be given by a wind turbine manufacturer describes the instantaneous power output that can be expected from that turbine when presented with a given wind speed; whereas, we are working to produce quite a different 'power curve'. Here the wind speed data is a proxy in space for what a nearby turbine might see; it is also a sample in time, a single measurement every half or one hour.

What this amounts to is a balancing act. We use the wind speed data, primarily, to describe the variability rather than the magnitude of the wind resource. That is, we will more-or-less impose a capacity factor on a modelled wind farm, and the Wind Power Curve will in effect scale the wind speeds to achieve this. Blindly applied this methodology can take any (ludicrously inappropriate) wind data and produce a respectable power output for a hypothetical wind farm. It is necessary to keep our wits about us in avoiding any and all magic puddings.

THE DATA

Has a page of its own, Pairings of Wind Speed Data with Wind Farms, from where both the wind speed and wind farm generation data can be downloaded. What follows here is just an outline.

The realities of the data provide two divisions, which together induce a split into four 'tiers' of data, as below. First, the 2009 reference period does not work for all the wind farms as for some we only have data from ~mid 2009 (and these cases use 1st Aug 2009 to 31st Jul 2010 instead); second, some of the BoM data is reliably half-hour data, and some is, or is best treated as, hourly data.

  Tier One: Half hourly BoM, 2009 reference

  case  Wind Farm     Wind-speed data
 ----------------------------------------------------
   #1.  WPWF          BoM 22046 EDITHBURGH
   #2.  STARHLWF      BoM 23875 SECOND VALLEY FOREST AWS       Tier Two: Half hourly BoM, 1 Aug 2009 to 31 July 2010 reference 
   #3.  CNUNDAWF *    BoM 26021 MT GAMBIER AIRPORT *
   #4.  LKBONNY1      BoM 26021 MT GAMBIER AIRPORT *             CAPTL_WF *     BoM 70014 CANBERRA AIRPORT
   #5.  CHALLHWF      BoM 79101 BEN NEVIS                        WAUBRAWF       BoM 89002 BALLARAT AERODROME
   #6.  YAMBUKWF      BoM 90175 PORT FAIRY AWS
   #7.  WOOLNTH1      BoM 91245 CAPE GRIM BAPS


  Tier Three: Hourly BoM, 2009 reference                       Tier Four: Hourly BoM, 1 Aug 2009 to 31 July 2010 reference 

        MTMILLAR       BoM 18116 CLEVE AERODROME                 CAPTL_WF *     BoM 69132 BRAIDWOOD RACECOURSE AWS
        CATHROCK       BoM 18217 CUMMINS AERO                    CULLRGWF       BoM 70330 GOULBURN AIRPORT AWS
        CNUNDAWF *     BoM 26091 COONAWARRA

(*) Used in more than one pairing

In what follows (i.e. in the comments for now) we are using only the Tier One data thus far.

ANALYSIS

The full development of the Wind Power Curve (WPC) model remains in the comments for now [Nov 2010]. What follows is an outline that is being incrementally developed.

Part I - The Wind Power Curve

After some consideration, and building on comment #6, we adopted the following initial form for the Wind Power Curve (see #12):

Note that mostly the wind speed sits in the left half of this power curve.

This curve has five sections, with the first and last (red) being zero up to a cut-in (starting) wind speed, W_s, and after a cut-out speed, W_e. The second section (green) is a rising cubic, motivated by the fact that energy density in wind goes with the cube of the wind speed. The middle section (magenta) exists above a threshold P_t, with associated 'transition' wind speed W_t, and is modelled as a quadratic forced to join the previous curve at X2 and have its maximum at X3 (corresponding to wind speed W_m). Note that without such a 'dome' curve at the top, the power estimate would spend too much time at full power. Finally, a gradual application of the cutout (cyan section) is provided via a liner drop from X4 (defined symmetrically from X3 and X2) to zero power at wind speed W_e.

This was the initial exploratory approach, which after some use (see #13, #18) was quickly simplified (#20, #23) in the following ways:
- set W_s to zero, leaving the cubic to impose the flat start;
- drop the linear 'cut-out' (cyan) section, allowing the quadratic to provide the drop to zero;
- force the transition from cubic to quadratic at X₂ to be C1 continuous.

The end result is a WPC with a single parameter, W_t (and with P_t set at 0.4); that is, the shape of the WPC is set, and the parameter W_t acts to stretch or compress the WPC along the wind-speed axis. Note that W_t typically ends up being approximately the median wind speed; note also that, with P_t as set, the peak of the WPC, W_m, is close to twice W_t.

Part II - Applying the WPC

The Wind Power Curve (WPC) is applied to data in one of two ways. In the development work here we fit the WPC as a model, seeking parameterisation that minimises the difference between the simulated and actual wind farm output: for now we use a simple sum of squares of the individual data point differences as the Objective function for this fitting work. This is standard work, requiring an initial 'guess' at parameters, and we use the fminsearch function in Matlab to do the minimisation work.

In general use, when converting wind-speed data into simulated wind farm output, an imposed Capacity Factor (CF) is used (generally 35%) and the parameter W_t adjusted as needed to achieve this CF. See the latter part of #12 for some further detail.

Part III - systematic biases, seasonal and diurnal

Work on diurnal and seasonal biases proceeds from #30.

Part IV - Scripts and Demonstration

Here are the Matlab scripts that I use to perform the wind-speed to simulated wind-farm-output transformation. Care and understanding is needed in applying the corrections - be careful, and consider not applying if not understood. The seasonal correction simply requires having the DPpD (Data Points per day) and DYst (Day of Year start) parameters correctly set; the diurnal correction requires having the DCp (phase) parameter adjusted for local midday compared to the reference case here (at longitude 137.7 E) -- see #38. In any case, these corrections are 'first pass' and in some cases (especially for the diurnal correction) might introduce more systematic bias than they remove.

For a demonstration of application, see: Simulate Wind Farms on Eyre Peninsula from BoM wind speed data. For their main application, see: Simulated Wind Farms from Broome to Cooktown.

[note: have not provided scripting for the fitting of wind-speed data against known wind farm output; will provide this in time / as needed]

Part V - Some Sensitivity and Benchmarks

As in the Remarks below, we hold off on a full sensitivity analysis for now. However, some basic analysis of sensitivity is in order, and this also records some benchmarks for the data fitting.

When fitting the model to known wind-farm output, the Objective function is simply the sum of squares of the differences of the (normalised) simulated and observed generation values. For the current Tier One data (as above) we examine the value of the objective function: in the absence of both time-of-day (diurnal bias), and time-of-year (seasonal bias) corrections, then in terms of each correction individually (including a look at parameter sensitivity), and finally with both corrections in the form we are proceeding with.

The parameters examined below are:
S_m is the magnitude of the seasonal correction [default 0.1] (see #34)
S_p is the phase of the seasonal correction [default 30]
D_m is the magnitude of the diurnal correction [default 7] (see #33)

  Case              #1      #2     #3     #4      #5     #6     #7
 ---------------------------------------------------------------------

  No corrections:  472    1235   1013    876    1478    566     397

  Just season:     415    1139    906    795    1449    526     397
   Sm @ 0.05       431    1176    949    825    1457    536     386
   Sm @ 0.15       425    1115    878    778    1452    531     433
   Sp  +60         466    1215    990    863    1445    570     440
   Sp  -60         465    1187    949    828    1502    547     397

  Just Diurnal:    420    1244    919    809    1677    579     405
   Dm -> 3         438    1234    966    841    1566    565     399
   Dm -> 5         425    1238    939    822    1622    570     401
   Dm -> 9         421    1252    905    799    1729    591     409

  Both:            353    1141    806    723    1646    534     405

For our reference case, the final Objective value of 353 corresponds to ~1 unit per day, which for 48 time points (ignoring any missing data) is ~0.02 per data point average for the squared difference, which gives a blunt average of 14% difference between the simulation and the known wind farm output. In contrast, case #4 works through to blunt average difference of 20%.

SO, we see:
#1. All good
#2. Seasonal correction good (prefer stronger correction); diurnal correction mildly unhelpful
#3. Seasonal correction good (prefer stronger correction); diurnal correction good
#4. Seasonal correction good (prefer stronger correction); diurnal correction good
#5. Seasonal correction flat; diurnal correction bad
#6. Seasonal correction good; diurnal correction unhelpful / untuned
#7. Seasonal correction flat (prefer weaker correction); diurnal correction mildly unhelpful

For #5, the Ben Nevis BoM station ... been here before. The station is ~20km to NNW of the Challicum Hills Wind Farm, and at ~865m altitude appears to be the high point of the region (compared to ~450m for the WF). BUT, this BoM station is in the forest, and surrounded by trees to 10m.

REMARKS

We have developed here a Wind Power Curve (WPC) model that transforms BoM wind speed data into simulated wind farm output. This method does not assess how suitable the vicinity of the BoM station really is for a wind farm, but rather takes the variability information in the wind-speed data, along with an imposed capacity factor, and transforms this into a time series of simulated output in a way that is consistent with real wind farms.

The transformation process has two parts. First we apply time-of-day and time-of-year adjustments to the wind-speed data; second we apply the WPC to this adjusted wind speed data, using a single 'stretching' parameter (W_t) in order to achieve the specified capacity factor.

In establishing this procedure we examined cases where output from existing wind farms could be compared with simulated data based on nearby BoM wind speed measurements. The goodness-of-fit, measured as a sum of squares of differences, gives values that are not as low as we might like. A case in point is the reference (#1) discussed above, where the objective value of 353 breaks down to give a blunt average of 14% difference between simulated and observed wind farm output. In contrast, physical inspection of the traces shows a very reasonable correspondence.

The sum of squares of differences approach may be acting in a deficient way as these traces undergo some steep transitions, and so small 'horizontal' shifts can give large differences in the 'vertical' positioning of the simulated and observed data. A better approach may be to quantify 'difference' as the area between the two traces around each data point. There are also questions about time resolution that it would be ideal to explore: e.g. how does model performance change for hourly compared to half hourly data? And, can we usefully use three hourly data?

Now, however, we proceed first into testbed analysis on simulating Eyre Peninsula wind farms, and then onto examining spatial smoothing of wind power at a national level (The Broome to Cooktown Challenge). Developing the analysis pipeline in this way allows us to establish specifically what end point 'answers' need confidence intervals, at which point we can proceed here again with a clear direction and purpose.

Finally, it is to be noted that the work here involved substantive contributions for an anonymous commenter 'dashpool'; kudos and thanks to dashpool for engaging in the Open Science aspect - and helping provide a good example of how valuable contributions from individuals outside the core OzEA group can be in developing the work.

6 OzEA_AWBW0006

dashpool Subject: figures for diurnal variation, overall 5 stations result Date: 2010-08-29 (at 03:38:41)

The power curve I used was roughly based on typical turbine power curves, with zero power below a cut in speed, fairly rapid increase up to a plateau, and a cut out ~8 times higher that cut in at high speeds.

Because the wind varies somewhat over a half hour with gusts and so on, and the various turbines experience somewhat different winds, the actual power curve for the wind farm won't be exactly the single-turbine constant wind power curve. The 'curve' on the plateau is necessary to match the real data reasonably well, otherwise the modelled wind farm spends too much time at exactly full power compared to the real farm.



It may be more straightforward to justify just using the usual manufacturers power curve: I have tried a couple of different power curves, and the results averaged over several farms aren't too sensitive to the choice of models.

The diurnal power curves for the 5 wind farms, versus the modelled wind farms, with capacity factor of 0.3 assumed for the modelled farms:




There is a strong tendency for the model to overestimate diurnal variation. I haven't been very successful trying to correct for this with a location independent curve. Getting the real wind shear information may be necessary.

The average power of the real and modelled wind farms has a similar distribution:

which is reassuring (the five farms were all normalised to 100MW to avoid the larger farms dominating the mean).

It seems like it would be useful to extend this to all the wind farms where we have nearby data (beyond SA), because this would be a strong verification that we can get reasonable statistical information about the total power of the wind farm network. Getting the distribution correct would be enough for most purposes, but it would be good to get the time variation correct to look at storage and ramping of thermal plant.

8 OzEA_AWBW0008

francis Subject: Power Curve exploration; Edithburgh BoM station and Wattle Point Wind Farm Date: 2010-09-09 (at 22:28:33)

Following on from dashpool above, I'm having an exploratory go at transforming the Wind Speed data from Edithburgh BoM station into Wind Farm output, using Wattle Point Wind Farm (2009) as the reference. The wind speeds distribute - as shown - in a way that I now recognise as typical for these data, with a tail up to ~15 m/s. Have 2009 data (354 days up to 20th Dec), at 30 min, for BoM wind speed and WPWF output (16992 points).



Trying to make a scatter plot of the correspondence between BoM Wind and WPWF output is hopeless - to much data and too much spread in the data. So, have binned according to wind speed, and plotted as:



Up to around 9 m/s the bins are 1000+ data points each; the next two bins are ~250 data points each, and the final 4 higher speed bins have 130 data points each. These smaller bins at the end allow a look at what happens here. The blue bars show simply calculated means with +/- std error bars. The distributions are not normal, so I've also plotted the medians (black crosses).

To get a power curve (red) I decided to follow the medians; I also decided it should top out at 1 (i.e. full power). This is an exploration -- the curve is just a curve that I reckon worth trying out.

SO, with this power curve it was straightforward to transform the wind data into simulated Wind Farm output. Plotting this and comparing to the actual WPWF gives what appears to be surprisingly good correspondence (looking in ~20 day chunks). I also converted to total daily output and again visually compared (overlaid) with the reference data -- some differences, but overall it looked good. To look more quantitatively at the differences between simulated and reference, I considered the individual differences ('residuals' if you like), and these are shown below for the 30 minute data, and also summed into 3 hr blocks and 24 hr blocks.



We see that some of the differences are large (up to around one half of the full capacity), and that overall the simulated data is underestimating the power output. Following dashpool's lead I looked at the differences in terms of the time of day, as follows:



[14 Nov 2010] As in #30 below, it turns out that above plot is in error. There is a strong diurnal bias, but what is shown here is the seasonal bias.

A clear diurnal pattern is observed. Here I was more inclined to follow the means (red curve), although this is just a curve for looking at. Note that this diurnal pattern is close to a sinusoid with a peak at 3am and maximal average underestimation of power at 3pm (AEST).

Also, I had a look at how the differences behaved in relation the BoM Wind Speed:



Again the spreads are large and the distributions far from normal. It is interesting that the medians are not so bad, but the means show a clear underestimation of power at the lower and mid-level Wind Speeds, with overestimation at the high end. It seems that in setting up the power curve it might have been better to have followed the means, and perhaps held the curve down a little at the high end.

For now I'm happy to sit and muse on this. It could be useful to develop and apply a moderately complex model, including the use of random components drawn from appropriate distributions, in order to more closely simulate the Wind Farm output. However, I'm more inclined to stick with a power curve for now. I'd like to understand the basis of the diurnal behaviour before getting too gung-ho incorporating it. I think the big question now is whether a single power curve can usefully be applied to data from disparate BoM stations.

10 OzEA_AWBW0010

dashpool Subject: More comparisons, correlations. Date: 2010-09-18 (at 23:42:27)

I extended the histogram analysis to the 9 wind farms where the 2009 data looks stationary (WPWF STARHLWF MTMILLAR CATHROCK LKBONNY1 WOOLNTH1 YAMBUKWF CHALLHWF CNUNDAWF), by simulating model wind farms at the 40 weather stations, and pairing the actual wind farms with the model farm with the highest correlation.

This produces some surprises: e.g., 'CAPE JAFFA' gives better correlation with LKBONNY1 than 'MOUNT GAMBIER AERO'.

It would be helpful to have 2010 data for the weather stations, and the latest AEMO data to do this for the full set of wind farms.

I am sticking with a fixed capacity factor for the modelled farms (0.3): I think trying to directly calculate it from the weather station data is problematic, given how sensitive average power at hub height is going to be to mean wind speeds.

Again, the power histograms are very similar for the modelled and actual wind farms, which is a hopeful sign. This distribution is a little tighter with these extra wind farms.



Another important feature of the real wind farms that we wish to reproduce is the correlation between farms as a function of distance.



Here we see that the modelled and actual wind farms show a similar pattern of decorrelation with increasing distance: beyond about 1000km separation the correlation is effectively zero. For this case I used all the wind farms, with the last 270 days of the AEMO data (where all the wind farm data looks OK).

All of this together suggests to me that we can get a pretty good idea of the power output distribution of the proposed set of turbines using a simple power curve analysis, and a guess at the likely capacity factors. I show the power output for equal sized wind farms at each of the weather stations:



But some things are not so straightforward: i.e. diurnal cycle.

12 OzEA_AWBW0012

Francis Subject: General form of the OzEA power curve Date: 2010-11-01 (at 20:28:39)

[Will be posting here every day until our Power Curve approach is sorted out.]

Analysis will continue to focus on the Edithburgh BoM station wind data and Wattle Point Wind Farm pair (see comment #8 above); in this case the wind as seen by the BoM station is a good proxy for that seen by the wind farm. This is not the case with all of the pairings. To demonstrate, consider the pair STARHLWF (Starfish Hill Wind Farm) and the wind speed data from our associated BoM station (Second Valley Forest AWS):



[14 Nov 2010] As in #30 below, it turns out that right panel above is in error. There is a strong diurnal bias, but what is shown here is the seasonal bias.

In the left panel the red 'power curve' is simply a curve that I have constructed 'by eye' to follow the mean values (blue crosses); that the wind data is not a good proxy for the wind farm is apparent from the 'power curve' topping out at around 0.6 (should be 1.0). Note, however, that even with this crappy correspondence, and having continued the analysis as in #8 above, the diurnal bias is again clearly seen in right panel.

Before proceeding to give the general form of the power curve that we will proceed with, there is an important distinction to make clear: a "power curve" as might be given by a wind turbine manufacturer describes the instantaneous power output that can be expected from that turbine when presented with a given wind speed, whereas we are working to produce quite a different 'power curve'. In our case the wind speed data is different in two distinct ways: it is a proxy in space for what a nearby turbine might see (for starters, BoM wind speed is at 10m compared to a typical hub hight of perhaps 80m); it is also a proxy in time representing an average over half or one hour. We seek a 'power curve' that does a good job at estimating wind farm power output based on sampled wind speed data.

A Power Curve General Form
After some consideration, and building on comment #6, I am adopting the following form:



Note that mostly the wind speed sits in the left half of this power curve.

The curve has five sections, with the first and last (red) being zero up to a cut-in (starting) wind speed, W_s, and after a cut-out speed, W_e. The second section (green) is a rising cubic, motivated by the fact that energy density in wind goes with the cube of the wind speed. The middle section (magenta) exists above a threshold P_t (which is held at 0.8 of full power for now) and model as a quadratic forced to join the previous curve at X2 and have its maximum at X3 (corresponding to wind speed W_m). Note that without such a curve the power estimate would spend too much time at full power. Finally, a gradual application of the cutout (cyan section) is provided via a liner drop from X4 (defined symmetrically from X3 and X2) to zero power at wind speed W_e.

So, given the functional forms as above, the power curve is defined by the parameters P_t, W_s, W_t, W_m and W_e. To simplify matters we proceed by focusing on W_m alone (but will examine the others later): we set P_t = 0.8 (as noted above); we set W_s = 1/5 W_m; we set W_t = 3/4 W_m; and we set W_e = 8/7 x_4 = 8/7 (2W_m - W_t). It is this parameterisation that is shown in the figure above.

We set W_m, for given wind data (i.e. a given BoM station), so as to give a desired capacity factor. For now we work with Capacity Factor at 35%. Applying this to the Wattle Point Wind Farm gives:



The analysis has two solutions for a 35% capacity factor: W_m ~ 10.5 and W_m ~ 4. The latter solution is non-physical (or at least non-sensical), corresponding to a Wind Farm that gives its maximum output at low wind speeds. Similarly, it would be wrong-headed to read this analysis as suggesting that capacity factors of 0.6 are possible (other than perversely; i.e. by choking the turbine at the higher wind speeds so as to convert, for example, a 2 MW turbine with a CF of 35% into a 0.2 MW turbine with a CF of 60%).

more tomorrow...

13 OzEA_AWBW0013

francis Subject: Power curve estimation by model fitting Date: 2010-11-03 (at 00:57:16)

Following on from #12, the job today has been to use a modelling approach to parameterise The General Curve (all 5 parameters) by 'fitting' to data (more details below). I take the Edithburgh BoM station wind data along with the Wattle Point Wind Farm generation data, and also an initial guess at the power curve parameterisation, and use the differences between the modelled and actual wind farm output to construct an Objective Function (simply summing the squares of these difference for now). After scripting up a little Matlab machinery [available on request] to wrap around the 'fminsearch' function, the fitting converges to give:



Note that the highest value in this wind data is 17.5 m/s, and so the 'cutoff' section of the model fit is meaningless / irrelevant. The bulk of the wind data exists below 10 m/s. The values of the model fit are: P_t = 0.31, W_s = 0.003, W_t = 5.66, W_m = 14.25, W_e = 28.8

Some more details
The model fitting proceeds as follows:
1. Take the associated wind speed and wind farm output data; normalise the output data (in this case divide by 91 MW); set up the data as two rows in a table (16992 long in this case); remove all columns that include a missing data value.
2. Parameterise an initial guess at the power curve (shown above in figure); transform parameters into a suitable form for the model fitting, which here means taking logs to enforce the zero-lower-limit (after first enforcing that W_t > W_s (etc) by subtraction).
3. Set up the objective function: this takes the current power curve, the wind speed data, and the wind farm output data; the first of which is applied to the second in order to get the simulated estimate of output, which is then subtracted from the actual output data to get differences. The differences are squared and summed to give the value of the Objective Function.
4. With appropriate tolerances, set fminsearch running on the problem.

In this case the search converged easily, and a little experimentation with the initial guess convinced me this is the solution we want (aside from the cutoff section of the curve flapping around in a data free zone on the right).


How good is it?
As for #8 above we look at the differences and the diurnal pattern:



The differences are more centred than previously (which is not unexpected given the use of medians in the preliminary look given in #8), otherwise there is no obvious change. And for the diurnal bias, we again see the same pattern.

[14 Nov 2010] As in #30 below, it turns out that above right panel is in error. There is a strong diurnal bias, but what is shown here is the seasonal bias.

What the diurnal plot also indicates is that the +/- 1 SD for the differences span around 30% of the power output range, and across the various time points this exists from around -30% to +20%. Correcting for the diurnal pattern would go some way to pulling in the extremes, especially at the -30 end.

Concluding Remarks
We how have a rational and well defined method for estimating a power curve for cases where we have both wind farm output data and wind speed data that is a reasonable proxy for what the wind turbines see. The individual 30min timepoint estimates remain rough around the edges; the diurnal bias is the obvious starting point for obtaining improvement.

However, will sign off here for today. Tomorrow I expect to go back to data, and see about getting the BoM pairing data for all 13 of the Wind Farms neat and tidy and ready for ensemble power curve analysis.

18 OzEA_AWBW0018

francis Subject: Some preliminary analysis of the power curve model for seven Wind Farm / BoM Wind Speed Data pairs Date: 2010-11-06 (at 20:14:25)

Some preliminary analysis of the power curve model for seven Wind Farm / BoM Wind Speed Data pairs.

Have fitted the Power Curve Model as in #18 to the following data:

	Wind Farm	BoM Station
A.	WPWF	BoM 22046 EDITHBURGH
B.	STARHLWF	BoM 23875 SECOND VALLEY FOREST AWS
C.	CNUNDAWF	BoM 26021 MT GAMBIER AIRPORT
D.	LKBONNY1	BoM 26021 MT GAMBIER AIRPORT
E.	CHALLHWF	BoM 79 101 BEN NEVIS
F.	YAMBUKWF	BoM 90175 PORT FAIRY AWS
G.	WOOLNTH1	BoM 91245 CAPE GRIM BAPS

And get the following fits:

	Obj.	Pt	Ws	Wt	Wm	We	CF obs.	CF mod.
A.	514.8976	0.3041	0.0780	5.6677	14.3156	23.7486	0.3361	0.3307
B.	1034	0.5011	0.0188	6.3864	big	big	0.2959	0.2533
C.	826. 9342	0.0882	0.0000	2.4762	19.5432	54.8557	0.2949	0.2887
D.	701.5299	0.0774	0.0000	2.4967	22.3476	113.7592	0.2593	0.2541
E.	943.7424	0.2728	0.1043	3.7328	38.2592	73.1930	0.2990	0.2953
F.	462.6351	0.1050	0.0000	3.4541	23.0798	52.7411	0.2773	0.2760
G.	398.9159	0.4638	0.2975	9.9817	26.9763	55.7840	0.3988	0.4005

The following plot pairs show (left) the modelled Wind Power Curve along with the data, and (right) the Power Output distribution for the actual Wind Farm as compared with that estimated from the wind data using the modelled Wind Power Curve.

















There are lots of observations and comments that can be made here, but for now this is just a progress report. More as it happens.

Notes:
* The 2009 reference period has turned out to be problematic as some of the Wind Farm data only kicks in from July 2009. Will treat these cases separately later.
* The Capacity Factors (observed and modelled) match up surprisingly well - there is no constraint applied here.

23 OzEA_AWBW0023

francis Subject: a step forward on the Wind Power Curve model Date: 2010-11-09 (at 03:51:37)

Following on from #18 above, here I first simplify the power curve (see #12), and then for comparision redo the data analysis and plots from #18 above.

Have set W_s to zero (now rely on the cubic to provide the initial dead response), and have set P_t at 0.4 (after some experimentation, and in order to provide the desired 'stiffness'). Also, the parameters W_t and W_m are now degenerate due to fixing the cubic and the parabola to be C1 continuous at the join point (i.e. x=W_t, y=P_t). What this all means is that the Wind Power Curve is for now defined and fitted using one parameter, P_t.

Also, the discreteness of the BoM wind speed data (mostly 0.5 increments) was making analysis difficult. Since this discreetness means that a data value of x readily translates to x +/- 0.2, I add a small random component to this wind data (0.15 * randn) before applying the power curve. Also, for further smoothing, the resultant generation values are similarly perturbed at the 1% level before plotting the power curves below. None-the-less, I remain suspicious that the kernel density filter used to make these curves, from the observed and estimated generation data, may be mucking up a little in the first part (but these curves are much easier to examine and compare than histograms).

In #18 I gave the observed and modelled Capacity Factors, and - as since discussed - these are not so meaningful in this context. However... in what follows the CFs are not given, and do not match as nicely as did above (the modelled CFs are low - this will make more sense shortly). In what follows, instead of the CFs, I give a more direct measure of the low end; given is the percentage of the time that power output is less that 5% and less than 10% of capacity, for comparison between the actual Wind Farm data, and that estimated from the wind speed using the power curve. In most cases the Wind Power Curve is over-estimating the amount of time that output is very low. Note that this will (at least somewhat) resolve when we stop fitting the wind speed against the wind farm data, and simply apply the Wind Power Curve (WPC) model to wind speed data, as at the end of #12 above (but this is two steps away from here - still to include a correction for the diurnal bias).

So, here is the re-analysis (of #18) using the current simplified WPC model:

Using the same seven pairings as in #18 above:

	Wind Farm	BoM Station
A.	WPWF	BoM 22046 EDITHBURGH
B.	STARHLWF	BoM 23875 SECOND VALLEY FOREST AWS
C.	CNUNDAWF	BoM 26021 MT GAMBIER AIRPORT
D.	LKBONNY1	BoM 26021 MT GAMBIER AIRPORT
E.	CHALLHWF	BoM 79 101 BEN NEVIS
F.	YAMBUKWF	BoM 90175 PORT FAIRY AWS
G.	WOOLNTH1	BoM 91245 CAPE GRIM BAPS

Get the following fits:

	Obj.	Wt	Wm	gen<5%	mod<5%	gen<10%	mod<10%
A.	520.3	6.4	12.9	20.5	13.4	30.8	26.2
B.	1180.7	6.9	13.7	25.4	27.9	35.7	42.0
C.	980.8	6.1	12.3	26.1	29.1	36.5	44.0
D.	837.5	6.6	13.3	27.5	33.7	38.8	49.6
E.	1413.8	7.0	14.1	22.0	28.4	32.3	44.6
F.	577.6	7.2	14.4	21.4	24.5	32.9	41.2
G.	418.8	10.1	20.1	16.4	14.4	24.0	23.3

As shown in the following plot pairs: (left) the modelled Wind Power Curve along with the data, and (right) the Power Output distribution for the actual Wind Farm as compared with that estimated from the wind data using the modelled Wind Power Curve.
















Next we look at a correction for the diurnal bias - as suggested by dashpool.

30 OzEA_AWBW0030

francis Subject: a dog ate my homework - diurnal bias Date: 2010-11-12 (at 02:34:50)

Started the day (the day before yesterday) thinking the Diurnal Bias would be easy to sort out.. It's taken a while to unwrap.

It appears that previous plots of diurnal bias (#8, #12, #13) are in error. Second time this year that I've mucked up when turning vectors in matrices. Let's start again, and with WPWF vs Edithburgh BoM station for 2009 as the reference. The two plots below show first the diurnal bias, and second the average bias over the course of the year.

It appears that what I presented before was a seasonal bias (2nd / right panel), which I reckon is derivative to the daily (diurnal) one. And I reckon the diurnal bias has a lot to do with sunshine - and in the future we may look at this explicitly. For now, it is the plot on the left that we focus on. The dashed magenta curve is a shifted and stretched instance of a y = exp( -t^2 ) form, which I'll return to shortly. First, I want to subtract in the wind speed dimension, as in the following 3D surface plot:



looking in from slightly to the left we see an exploded version of the first plot above. While there are some issues showing up at the high end, this is enough for me to be happy (for now) applying a simple time-of-day correction to wind speed. This is done by reducing the wind speed slightly across the bulge, and (for starters) I do this as:

w(t) = w(t).[1 - D_m exp( -((t-D_p)/D_s)² ) ]

where w(t) is the wind speed for a time of day [0,24), and D_m, D_p and D_s do the stretching and shifting. For example, in the above (dashed magenta) case: D_m ~ 0.15, D_p ~ 13.5, and D_s ~ 4 (plus the other translation and inversion to get the alignment shown).

Using this, again for the WPWF vs Edithburgh 2009 reference case, gives a reduction in the Objective from 471 to 428, which is pleasing. The parameters for this fit come out as:

P_t = 6.13
D_m = 0.107
D_p = 14.7
D_s = 4.9

And looking again at what diurnal bias is left, gives:



It seems obvious enough (to me here and now) that the form chosen above (i.e. y = exp( t^2 ) ) is too 'peaky', and that something much the same basic shape but with a flatter profile would be better. Nothing jumps to mind, but maybe over the next couple of days someone will come up with a better idea. Could ... grind out a synthetic one, but that might muck with empirical. Let's put that game in the bag for ron.

So, we could go on and on with this diurnal (and seasonal) bias - BUT, there can be another time. Probably in January. For now need to bank the insights and get on with the BtCC. I'm backing off for a few days, and will decide later on what the appropriate sign off is here (please express a view), but will default towards simply grinding out what we have.

33 OzEA_AWBW0033

francis Subject: Diurnal Bias corrected! Date: 2010-11-16 (at 02:20:12)

The plots showing Diurnal Bias in #30 certainly demonstrate its presence; however, given that our approach to correction is through adjustments to the wind speed data, what we really need to see is what adjustments to wind speed would give the desired correction (perhaps "because of the nonlinear process of putting the wind speeds through the power curve").

That is, at each data point we have a wind speed value, w(t), which we plug through the Wind Power Curve (WPC) to translate into simulated WF output, S(t), which we compare with the observed WF output, G(t). What is needed is to ask What corrected windspeed, w'(t), would give S(t) = G(t), and then to consider the difference w'(t) - w(t) in our plots (rather than the difference S(t) - G(t) used above). In order to perform this analysis we consider only data points where 0 < w(t) < W_m, thus providing for an inverse function to the WPC (which I call the PWC).

SO, with PWC as the inverse of WPC for w < Wm; just a matter of getting all the scripting details right...and we get the desired view of the diurnal bias in terms of (average) wind speed correction (continuing with WPWF vs Edithburgh BoM station for 2009 as the reference):



It turns out to be about the same shape as before, but we can clearly see that an -average- correction to wind speed of ~1 m/s across the diurnal bulge is needed. As before we make a 3D surface plot so that we can get past blunt averages and see if the corrections are preferentially needed at particular wind speeds:



And indeed we see that the correction is preferentially required at the lower wind speeds. The low end of the WPC is clearly sensitive to the time of day, and a BoM wind speed of 4 m/s means something quite different at 1 in the afternoon to 1 in the morning. So, in order to apply this correction, I look at the maximum correction vs. wind speed (i.e. the underneath 'spine' of the 3D plot):



So, rather than the magnitude of the correction scaling with the wind speed as previously, we do the exact opposite and scale as the inverse of the wind speed. I have also adopted dashpool's truncated cosine form for the time slice, thus arriving at the following form for the correction:

w'(t) = w(t) - D_m/ w(t) * [ max( 0, cos( D_s ( t - D_p ))) ]

And have a starting guess for the parameters in this form as: D_m ~ 6, D_s ~ 0.26, and D_p ~ 13. Proceeding with this as the form and starting parameters for the diurnal correction, obtain a model fit:

P_t = 6.05,
D_m = 8.073,
D_p = 13.8,
D_s = 0.289,
Obj. = 415.96

Does it work?
Using this correction and seeing what is left (as in #30) gives:



Which I call a win.
Still a bit to do here with checks against other data etc, but it seems we now have all the bits for the WPC model.

34 OzEA_AWBW0034

francis Subject: Seasonal Bias Date: 2010-11-16 (at 22:53:44)

I've been facing down the clock trying to deal with the seasonal bias, and the clock won. Or has it...

Let's proceed first with how this started, and we'll get the minor win by the end.

I was largely on the wrong track in trying to capture the seasonal bias as a tweak to the diurnal bias correction. Didn't really work out. It's time to move on here, although this will tick along and we'll return later. Other things need a first pass - a break here, then a fresh start, will likely help in any case.

So, to recap, we can fit the Wind Power Curve model to a reference case (the Wattle Point WF vs Edithburgh BoM station for 2009), and apply the resultant Wind Power Curve (WPC) back onto the wind data; giving simulated wind farm output, which can be compared against the actual wind farm output in order to look for systematic biases and the general quality of the transformation. Two particular aspects to examine are how the fitting works on average over the course of the day (365 days to average), and then how the the fitting works over the course of the year (broken into 30 chunks of 292 hours = ~12 days each). It is these two aspects that are shown in the first two plots in #30 above: the diurnal bias is clear as a bell, while a systematic seasonal bias exists but is somewhat messy.

Correction for the diurnal bias is described in #33 above. The correction is made predominately at the lower end of the wind speeds, and involves subtracting from the BoM wind speed data during the day to give a 'corrected' value for use in other calculations. We have not thus far worked to explore the physical basis of this diurnal effect; however, some comments can be made. We know that sheer forces at the ground, and up through 'layers' of the air, reduce wind speed close to the ground compared to higher up (and we have the BoM measurements at 10m compared to wind turbine hubs at ~80m). We suppose that the daytime sunshine 'loosens' up these shear forces, probably through simple warming of the lower air 'layers' (thus making them comparatively less viscous), although more complex fluid dynamics are not doubt involved at some level.

Now, the seasonal bias...

My thinking had been that the different lengths of the days through the year, compounded with seasonal sunshine strength, made the diurnal correction so far established something of a blunt instrument. Exploring this, and having the magnitude of the diurnal correction itself vary sinusoidally over the course of the year, did produce a modest improvement in the fit, but did almost nothing to flatten out this systematic seasonal variation.

SO, we put aside the diurnal bias, and look just at the seasonal as a phenomenon in its right. My guess at the physical basis is, as above, temperature. That is, the daily-weekly-monthly average temperature over the year is taken as varying sinusoidally. Some plotting and experimenting indicates that a straight multiplicative correction to the wind speed is appropriate, at least for wind speeds up to ~W_t (~6 m/s here). Since the focus is on the low end, this will do for now. Again, a simple manual parameter examination shows that the sinusoidal correction function wants to peak at the end of January (good!), and have a magnitude of around 0.1 giving a reduction in the Objective function from 471 to 415. The minima is shallow, and any fit in this general vicinity is nearly as good (+/- 30 days on the phase, and 0.06 - 0.15 for the magnitude). Good enough for me, here and now.

Hence, the first correction we make is this adjustment to the wind speed data:

w(t) => w(t)*(1 - 0.1*cos( 2*pi/365*(t/48 - 30) ))

Where in this case t indexes the data in half hour blocks from January 1 (hence t/48). Applying this transformation, the seasonal bias reduces to:



Comparing this to the uncorrected version in #30, it is clear that the correction is working. What other consequences it might have remain to be established.

Finally, it will be interesting to look at this Seasonal Bias, and its correction, in terms of daily temperature data. Another job for Ron.

35 OzEA_AWBW0035

francis Subject: Both biases corrected Date: 2010-11-17 (at 00:27:32)

Back to Diurnal Bias after the correction for Seasonal Bias as in #34 above.

So, with the seasonal correction applied, look again at the diurnal bias (to check it has not been unduly altered), and also test what happens when diurnal correction (as in #33 above) is applied:



No complaints here.
And the Obj. has now further reduced from 415 to 353.

What I want to check next is that the power distribution (as in #23) continues to match with the generation data:



Here the known wind farm data has 30.7% of values at less than 10% capacity, compared with 28.4% in the simulated version. Comparison with #23 above is favourable (a minor improvement).


So, that is our basic model for converting BoM wind speed data to simulated wind farm output. Still have to go through examination of other pairings, and otherwise lay out the robustness characteristics a bit more (and do a major rewrite the head post), but I'm out of the loop for at least 24 hr now.

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