Wind power scenario generation
by regression clustering
Geoffrey Pritchard
University of Auckland
Scenarios for stochastic optimization
Make
decision
here
?
• Uncertain problem data represented by a probability
distribution.
• For computational tractability, need a finite discrete
distribution, i.e. a collection of scenarios.
Power system applications
• Wind power generation, 2 hours from now.
• Inflow to hydroelectric reservoir, over the next week.
Typical problems solved repeatedly:
– Need a procedure to generate scenarios for many
problem instances, not just one.
Situation-dependent uncertainty
• Scenarios represent the conditional distribution of the
variable(s) of interest, given some known information x.
• Different problem instances have different x.
Change in wind power over next 2hr
Tararua/Te Apiti 28/5/2004-31/3/2010
Change
in wind
power:
7 discrete
scenarios
Change
in wind
power
over next
2hr
Tararua/Te
Apiti
28/5/2004-31/3/2010
Each
scenario is
a function
of the present wind power x.
Change
in wind
power:
7 discrete
scenarios
Change
in wind
power
over next
2hr
Tararua/Te
Apiti
28/5/2004-31/3/2010
Each
scenario is
a function
of the present wind power x.
Scenarios by quantile regression
• Have data xi and yi for i=1,…n
y
x
Scenarios by quantile regression
• Have data xi and yi for i=1,…n
• Want scenarios for y, given x.
y
x
Scenarios by quantile regression
• Have data xi and yi for i=1,…n
• Want scenarios for y, given x.
y
• Quantile regression:
choose scenario sk() to
minimize Si rk( yi – sk(xi) )
for a suitable loss function
rk().
x
Quantile regression fitting
• For a scenario at quantile t (0 < t < 1) ,
rt is the loss function
t-1
t
Scenarios as functions
• Choose each scenario to be linear on a feature space:
sk(x) = Sj bjkbj(x)
• Typically bj() are basis functions (e.g. cubic splines).
• The quantile regression problem is then a linear program.
Change
in wind
power:
7 discrete
scenarios
Change
in wind
power
over next
2hr
Tararua/Te
Apiti
28/5/2004-31/3/2010
Equally likely
scenarios,
modelled
by quantiles 1/14, 3/14, … 13/14.
Quantile regression: pros and cons
• Each scenario has its own model. Scenario models are fitted
separately.
• Fitting is computationally easy.
• Scenarios have fixed probabilities. Events with low probability
but high importance may be left out.
Another way to choose scenarios
Given one probability distribution …
… choose scenarios to minimize expected distance of a random
point to the nearest scenario. (Wasserstein approximation.)
Robust to general stochastic optimization problems.
Scenarios for conditional distributions
• Have data xi and yi for i=1,…n
• Want scenarios for y, given x.
y
x
Scenarios for conditional distributions
• Have data xi and yi for i=1,…n
• Want scenarios for y, given x.
y
• Wasserstein:
minimize Si mink | yi – sk(xi) |
over scenarios sk() chosen
from some function space.
x
Scenarios as functions
• Choose each scenario to be linear on a feature space:
sk(x) = Sj bjkbj(x)
• Typically bj() are basis functions (e.g. cubic splines).
• The Wasserstein approximation problem is then a MILP with
SOS1 constraints (not that that helps).
Algorithm: clustering regression
Let each observation (xi,yi) be assigned to a scenario k(i).
Choose alternately
• the functions sk
• the assignments k(i)
to minimize
Si | yi – sk(i)(xi) |,
until convergence (cf. k-means clustering algorithm).
Clustering regression
Let each observation (xi,yi) be assigned to a scenario k(i).
Choose alternately
• the functions sk
• the assignments k(i)
For univariate y, a
median regression
problem
to minimize
Si | yi – sk(i)(xi) |,
until convergence (cf. k-means clustering algorithm).
Example:
wind power
Example:
wind power,
next 2 hours
Scenario probabilities
Given one probability distribution …
Each scenario gets a probability: that of the part of the
distribution closest to it.
Conditional scenario probabilities
• Probability pk(x) of scenario k
must reflect the local density
of observations (xi , yi) near
(x, sk(x)).
• Multinomial logistic regression:
probabilities proportional to
exp(Sj gjkbj(x))
where gjk are to be found.
Wind: scenarios and probabilities
Wind: scenarios and probabilities
9%
26%
21%
70%
41%
90%
7%
33%
3%
The End
Wind power 2hr from now: lowest scenario,
conditional on present power/wind direction
Wind power 2hr from now: lowest scenario,
conditional on present power/wind direction