Wind power scenario generation

Geoffrey PritchardUniversity of Auckland

by regression clustering

Scenarios for stochastic optimization

• Uncertain problem data represented by a probability distribution.

• For computational tractability, need a finite discrete distribution, i.e. a collection of scenarios.

Make decision here

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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 2hrTararua/Te Apiti 28/5/2004-31/3/2010

Change in wind power over next 2hrTararua/Te Apiti 28/5/2004-31/3/2010

Change in wind power: 7 discrete scenarios

Each scenario is a function of the present wind power x.

Change in wind power over next 2hrTararua/Te Apiti 28/5/2004-31/3/2010

Change in wind power: 7 discrete scenarios

Each scenario is a function of the present wind power x.

• Have data xi and yi for i=1,…n

x

y

Scenarios by quantile regression

• Have data xi and yi for i=1,…n

• Want scenarios for y, given x.

x

y

Scenarios by quantile regression

Scenarios by quantile regression

• Have data xi and yi for i=1,…n

• Want scenarios for y, given x.

• Quantile regression: choose scenario sk() to

minimize i k( yi – sk(xi) )

for a suitable loss function k().

x

y

Quantile regression fitting

• For a scenario at quantile , is the loss function

Scenarios as functions

• Choose each scenario to be linear on a feature space:

sk(x) = j jkbj(x)

• Typically bj() are basis functions (e.g. cubic splines).

• The quantile regression problem is then a linear program.

Change in wind power over next 2hrTararua/Te Apiti 28/5/2004-31/3/2010

Change in wind power: 7 discrete scenarios

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

… choose scenarios to minimize expected distance of a random point to the nearest scenario. (Wasserstein approximation.)

Robust to general stochastic optimization problems.

Given one probability distribution …

Scenarios for conditional distributions

• Have data xi and yi for i=1,…n

• Want scenarios for y, given x.

x

y

Scenarios for conditional distributions

• Have data xi and yi for i=1,…n

• Want scenarios for y, given x.

• Wasserstein:

minimize i mink | yi – sk(xi) |

over scenarios sk() chosen from some function space.

x

y

Scenarios as functions

• Choose each scenario to be linear on a feature space:

sk(x) = j jkbj(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

i | 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)

to minimize

i | yi – sk(i)(xi) |,

until convergence (cf. k-means clustering algorithm).

For univariate y, a median regression problem

Example: wind powerExample: wind power, next 2 hours

Scenario probabilities

Each scenario gets a probability: that of the part of the distribution closest to it.

Given one probability distribution …

• 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(j jkbj(x))

where jk are to be found.

Conditional scenario probabilities

Wind: scenarios and probabilities

9%

7%

3%

90%

33%

70% 41%

26%

21%

Wind: scenarios and probabilities

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