Dynamic Dependency Structures in Short-term Solar Power Forecasts

Dynamic Dependency Structures in Short-term Solar Power Forecasts

Jethro Browell, Bri-Mathias Hodge* & Tarek Elgindy*

University of Strathclyde, Glasgow

*NREL, Golden, CO

[email protected] +44 (0) 141 444 7297

Quarterly Forecasting Forum, University of Lancaster, 16 March 2018

A little about me…

Live and work in Glasgow, Scotland Married to Naomi

2011-15: PhD “Spatio-termporal Prediction of Wind Fields”

2015-Present: Postdoc, University of Strathclyde

Research interests: Energy Forecasting and decision- making; wind, solar, load and price forecasting

methodology…

• From 1880 to 2012, the average global temperature increased by 0.85°C.

• Oceans have warmed, the amounts of snow and ice have diminished and the sea level has risen.

– From 1901 to 2010, the global average sea level rose by 19 cm

– The sea ice extent in the Arctic has shrunk in every successive decade since 1979

Motivation

Changing Climate

• It is likely that we will see a 1–2°C

increase in global temperature by 2100 above the 1990 level

• Most aspects of climate change will persist for many centuries, even if emissions are stopped.

The UN Intergovernmental Panel on Climate Change, Fifth Assessment Report, 2013 https://xkcd.com/1732/

Then…

• Reasonably Predictable Demand

• Controllable Generation

• Cheap Storage (solid fuels, pump hydro)

Motivation

A Changing Energy System

Now, and increasingly…

• Flexible and Active Demand

• Variable Generation with Limited Predictability

• Shortage of cheap storage (pump hydro, batteries still expensive)

• Electrification of heat & transport

0 6 12 18 24

What

• Power output of solar generator depends on:

– Resource (Solar Irradiance [Wm^-2]) – Physical parameters of generator:

(area, orientation, efficiency…) – Single generator, regional

aggregation?

• Uncertainty?

– Instantaneous/averaging periods?

– Spatial and temporal dependency?

– Interval/distribution/trajectories?

Solar Power Forecasting

What, Why and How

Why

• Power system operation requires planning and re-planning from days ahead to real-time

• Participation in electricity markets

• “Optimising” other operations

Diagram Credit: Jeffrey Brownson

Solar Power Forecasting

What, Why and How

Decision Category Application Forecast Type How*

Symmetric Penalties for Forecast Errors

Forecast Evaluation Marketing

Situational Awareness

Deterministic Regression, Physical Model

Asymmetric Penalties for Forecast Errors

Trading

Power System Operation Density

Parametric

Non-parametric (quantile regression, KDE, analog ensemble…)

Multi-temporal Effects

Power System Operation Storage Operation

Scheduling Tasks

Trajectories/

Scenarios

Multi-dimensional density forecast Ensemble NWP

AnEn + Schaake Shuffle

*For more than a few hours ahead, post-processing NWP is necessary. For shorter horizons, time series methods are usually preferable.

Solar Power Forecasting

What, Why and How

Decision Category Application Forecast Type How*

Symmetric Penalties for Forecast Errors

Forecast Evaluation Marketing

Situational Awareness

Deterministic Regression, Physical Model

Asymmetric Penalties for Forecast Errors

Trading

Power System Operation Density

Parametric

Non-parametric (quantile regression, KDE, analog ensemble…)

Multi-temporal Effects

Power System Operation Storage Operation

Scheduling Tasks

Trajectories/

Scenarios

Multi-dimensional density forecast Ensemble NWP

AnEn + Schaake Shuffle

*For more than a few hours ahead, post-processing NWP is necessary. For shorter horizons, time series methods are usually preferable.

Scenario Forecasting

Model Chain

Numerical Weather Prediction

• Irradiance, cloud cover, etc…

• 0-48 hours ahead

Density Forecasts

• Non-parametric (QR)

• Parametric Tails

Multi-Dim. Density Forecasts

• Gaussian Copula

• Covariance structure describes temporal dependency

Physical Model

Statistical Models Trained on Historic Data

Numerical Weather Prediction

Input Data

• Outputs from physical model of the atmosphere

– Irradiance, cloud cover at multiple heights, and temperature

– Provide by the European Centre for Medium Range Weather Forecasting

• Historically, irradiance hasn’t been a high

“priority” parameter, but the rapid growth in solar capacity has change this!

Density Forecasts

GBM + Tails

• Quantile Regression:

– Proportion of Clear Sky Irradiance – Trees: Gradient Boosting Machine – Features engineered from NWP

• Tails

– Lower bound: 0, Upper bound: ?

– Clear sky models are imperfect + lensing – Exponential Tails form last quantile to

boundary

𝐹෠_𝑡+ℎ(𝑥_𝑡+ℎ)

Density Forecasts

Evaluation

• Are the density forecast reliable?

𝑈 = ෠𝐹_𝑖(𝑥_𝑖)

Gaussian Copula

Temporal Dependency

• High-dimensional Gaussian Copula

• Marginal comprise the predictive distribution for each forecast horizon, 0-48 hours ahead

• Temporal dependency characterised by covariance matrix, 𝑅

𝐺 𝑥෠ _𝑡+1, … , 𝑥_𝑡+𝐻 = Φ_𝑅 Φ⁻¹ 𝐹෠_𝑡+1(𝑥_𝑡+1) , Φ⁻¹ 𝐹෠_𝑡+2(𝑥_𝑡+2) , … , Φ⁻¹ 𝐹෠_𝑡+1(𝑥_𝑡+𝐻)

Φ: Standard Normal CDF Φ_𝑅: Multivariate Normal with covariance matrix 𝑅

𝐹෠_𝑡+ℎ: Predictive CDF

Gaussian Copula

Empirical Covariance Matrix

Temporal Structure

• Width of diagonal strip indicates strength of correlation with

neighbouring forecast errors

• Strong diurnal effects

Gaussian Copula

Parametric Covariance Matrix

Temporal Dependency

Different Day Types

Different Sources of Error:

• Clear Day: Clear sky estimate (aerosol content etc)

• Partially Cloudy: Time and duration of clear/cloudy spells

• Cloudy Day: Irradiance penetrating cloud layer(s)

Gaussian Copula

Parametric Covariance Matrix

Δℎ

Covariance CloudyPartialClear

Cloudy Partial

Clear

Gaussian Copula

Parametric Covariance Matrix

Δℎ

Covariance

Gaussian Copula

Sampling

…and transform through the marginals:

𝒙^(𝑘) = 𝑥_𝑡+1^(𝑘), … , 𝑥_𝑡+𝐻^(𝑘) = ෠𝐹_𝑡+1⁻¹ Φ (𝑧₁^(𝑘)) , … , ෠𝐹_𝑡+𝐻⁻¹ Φ (𝑧_𝐻^(𝑘)) Draw 𝑘 samples 𝑧₁^(𝑘), … , 𝑧_𝐻^(𝑘) from Φ_𝑅…

Forecast Evaluation

Scoring Rules

Multivariate Energy Score

• Compares observation to all scenarios

• Strictly Proper

• Discrimination weighted towards marginals, not dependency

𝑆_𝑒𝑛 𝐺, 𝒚 = 𝐸෠ _𝐺 𝑿 − 𝒚 − 1

2𝐸_𝐺 𝑿 − 𝑿′

Variogram Score

• Compares variogram of observation and scenarios

• Proper, but not strictly (agnostic to level changes)

• Discriminates well between covariance structures

𝑆_𝛾𝑝 𝐺, 𝒚 = ෍෠

𝑖,𝑗

𝑦_𝑖 − 𝑦_𝑗 ^𝑝 − 𝑥_𝑖 − 𝑥_𝑗 ^{𝑝 2}

Forecast Evaluation

Scoring Rules

Covariance Matrix MV Energy Score

Variogram Score

Identity 419.0 27348

Empirical, Static 411.5 27062

Empirical, Dynamic 412.3 27127

Parametric, Static 411.9 27147

Parametric, Dynamic 411.6 27087

Forecast Evaluation

Scoring Rules

Covariance Matrix MV Energy Score

Variogram Score

Identity 419.0 27348

Empirical, Static 411.5 27062

Empirical, Dynamic 412.3 27127

Parametric, Static 411.9 27147

Parametric, Dynamic 411.6 27087

Summary

• Temporal dependency structures in solar irradiance are dynamic

• By conditioning models on underlying processes we aught to be able to capture some of these effects…

• Much still to be done:

• Model development

• Comparison to Ensemble NWP and other methods

• Forecast evaluation – what’s is the value in reality?

Thanks!

Papers and more available at www.jethrobrowell.com