SERIAL CORRELATION ... andOVERCONFIDENCE

Sandia-EPRI 2016 PV Systems Symposium5th PV Performance Modeling Workshop

Mark HandschySolarRetina, LLCMay 9, 2016

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CONTEXT: MODEL PURPOSE

• Such models are used to ... determine the future value of PV generation projects (expressed as the predicted energy yield).... Greater confidence in the accuracy of performance models will lead to lower financing costs and an increase in the number of projects that are built.

• Historical data is typically used to predict output of proposed systems.

— https://pvpmc.sandia.gov/

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P50 ESTIMATION3

time

prediction error

(here, the mean of the 20 pre-construction annual resource averages)

PREDICTION ERROR DISTRIBUTION4

assumptions:• resource values are statistically independent• resource values are normally distributed• variability is the same pre- and post-construction

definition:

then: • t has Student’s t-distribution with n1 + n2 – 2 degrees of

freedom

2121

212

222

11

21

)2()]()1()1[(

nnnnnnnn

XXt

−++−+−

−≡

σσ

Long Records

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SUNSHINE DURATION: CAMPBELL-STOKES6

ARMAGH OBSERVATORY7

daily sunshine hours01/01/1881–12/31/2014

48,942 days (134 y)140 bad: 0.3%

BLUE HILL OBSERVATORY8

monthly sunshine hours01/1886–12/2015

1560 months (130 y)

www.bluehill.org

KTH (STOCKHOLM)9

Moll-Gorczynski pyranometer (ca. 1936)

monthly avg. irradiance01/1923–12/2013

1092 months (91 y)

data from: Global Energy Balance Archive

Results

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TIME SERIES11

avg. irradiance (W/m2)avg. daily sunshine (h)

Armagh Blue Hill Stockholm

randomly permuted versions of time series

P50 ESTIMATION

Procedure:• “predict” average of each decade, using...

• mean of previous decade as estimator

• 12 cases for 130-y records: Armagh, Blue Hill; 8 cases for 90-y record: Stockholm

• difference means; divide by “σ ,” giving tOutputs:

• distribution of t-statistic

• distribution of # of years out 10 exceeding P50Control:• same outputs for randomly permuted inputs

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P50 ERROR DISTRIBUTIONS: t-STATISTIC13

Armagh Blue Hill Stockholm

randomly permuted version of time series

theory 1.4 1.4 1.4interquartile

ranges fort-statistic

chronological 1.4 2.9 4.2

random 1.0 1.3 0.8

chrono/theory 1.0 2.1 3.1

energy error spreads are larger than expected

P50 EXCEEDANCE DISTRIBUTIONS (# years/10)14

Armagh Blue Hill Stockholm

randomly permuted version of time series

theory 2.0 2.0 2.0interquartile

ranges for P50 exceedance

chronological 2.5 4.0 7.5

random 2.5 1.5 2.5

chrono/theory 1.3 2.0 3.8

exceedance spreads are larger than expected

INTERANNUAL VARIABILITY (σ𝒿𝒿/μ)15

Armagh Blue Hill Stockholm

record length 𝒿𝒿 (y)

IAV

CONCLUSIONS

• assumption of statistical independence is NOT warranted:

• resource levels exhibit year-to-year correlation over decades

• IAV appears to grow with record length, out to 130-y limit of available records

• assumption of independence will lead to material understatement of risk:– 2–3× understatement of energy production prediction interval – 2–3× understatement of exceedance-range prediction interval

(based on interquartile ranges).

THANK YOU

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