ons sddp workshop, august 17, 2011 slide 1 of 50 andy philpott electric power optimization centre...

ONS SDDP Workshop, August 17, 2011 Slide 1 of 50

Andy PhilpottElectric Power Optimization Centre (EPOC)

University of Auckland(www.epoc.org.nz)

joint work with

Anes Dallagi, Emmanuel Gallet, Ziming Guan, Vitor de Matos

Recent work on DOASA


Dynamic Outer Approximation Sampling Algorithm

• EPOC version of SDDP with some differences• Version 1.0 (P. and Guan, 2008)

– Written in AMPL/Cplex– Very flexible– Used in NZ dairy production/inventory problems– Takes 8 hours for 200 cuts on NZEM problem

• Version 2.0 (P. and de Matos, 2010) – Written in C++/Cplex with NZEM focus– Adaptive dynamic risk aversion– Takes 8 hours for 5000 cuts on NZEM problem

DOASA


Notation for DOASA


SDDP (PSR) versus DOASA

Hydro-thermal scheduling

SDDP (NZ model) DOASA

Fixed sample of N openingsin each stage. Solves all.

Fixed sample of N openings in each stage. Solves all.

Fixed sample of forward pass scenarios (50 or 200)

Resamples forward pass scenarios (1 at a time)

High fidelity physical model Low fidelity physical model

Loose convergence criterion Stricter convergence criterion

Risk models (None for NZ) Risk model (Markov chain)


Overview of this talk

This talk should be about optimization…

• A Markov Chain inflow model• Risk modelling example in DOASA• River chain optimization

DOASA

My next talk(?) is about benchmarking electricity markets using SDDP.


Part 1

Markov chains and risk aversion

(joint work with Vitor de Matos, UFSC)

ONS SDDP Workshop, August 17, 2011 Slide 7 of 50http://www.med.govt.nz/

57%

20%

7%

11%4%1%

HYDRO

GAS

COAL

GEOTHERMAL

WIND

OTHER

Electricity sector by energy supply in 2009


New Zealand electricity mix


9 reservoir model

MAN

HAW

WKO

Experiments in NZ system


Benmore inflows over 1981-1985

Inflow modelling

Source: [Harte and Thomson, 2007]


• DOASA model assumes stagewise independence

• SDDP models use PAR(p) models.

• NZ reservoir inflows display regime jumps.

• Can model this using “Hidden Markov

models” ( [Baum et al, 1966])

Markov-chain model


Hidden Markov model with 2 climate states

1 2 3 4 5 6

p11 p26

DRYWET

INFLOWS


Hidden Markov model with AR1 (Buckle, Haugh, Thomson, 2004)

Yt is log of inflowsSt a Markov Chain with 4 statesZt is an AR1 process


Hidden Markov model with AR1 Benmore inflows in-sample test

Source: [Harte and Thomson, 2007]


Markov Model with 2 climate states

1 2 3 4 5 6

p11 p26

DRYWET

WET INFLOWS DRY INFLOWS

Aim: test if we can optimize with Markov states


Transition matrix P

q 1-q 1-p p

P =


Markov-chain DOASA This gives a scenario tree


• Climate state for each island in New Zealand (W or D)

• State space is (WW, DW, WD, DD).• Assume state is known.• Sampled inflows are drawn from historical record

corresponding to climate state e.g. WW.• Record a set of cutting planes for each state.• Report experiments with a 4-state model:

– (WW, DW, WD, DD).

Markov-chain model for experiments


Markov-chain SDDP

P is a transition matrix for S climate states, each with inflows ti

(c.f. Mo et al 2001)


Ruszczynzki/Shapiro risk measure construction


Coherent risk measure constructionTwo-stage version


Multi-stage version (single Markov state)Coherent risk measure construction


State-dependent risk aversionWe can choose lambda according to Markov state

t+1(i) = 0.25, i=1, 0.75, i=2.


State-dependent risk aversion“4 Lambdas” model in experiments


Experiments

Reservoir inflow samples drawn from 1970-2005 inflow dataEach case solved with 4000 cutsSimulated with 4000 Markov Chain scenarios for 2006 inflows

Nine reservoir model (+ four Markov states)


Average storage trajectoriesExperiments


ExperimentsFuel and shortage cost in 200 most expensive scenarios


ExperimentsFuel and shortage cost in 200 least expensive scenarios


ExperimentsNumber of minzone violations


ExperimentsExpected cost compared with least expensive policy


Part 2

Mid-term scheduling of river chains(joint work with Anes Dallagi and Emmanuel Gallet at EDF)


What is the problem?

Mid-term scheduling of river chains

• EDF mid-term model gives system marginal price scenarios from decomposition model.

• Given price scenarios and uncertain inflows how should we schedule each river chain over 12 months?

• Test SDDP against a reservoir aggregation heuristic


A parallel system of three reservoirs

Case study 1


A cascade system of four reservoirs

Case study 2


• weekly stages t=1,2,…,52• no head effects• linear turbine curves• reservoir bounds are 0 and capacity• full plant availability• known price sequence, 21 per stage• stagewise independent inflows• 41 inflow outcomes per stage

Case studiesInitial assumptions


Revenue maximization modelMid-term scheduling of river chains


DOASA stage problem SP(x,(t))Outer approximation using cutting planes

Θt+1

Reservoir storage, x(t+1)

V(x,(t)) =


xi0xi1 xi2

i0+i0 xi1

xi3

i0

i1

Heuristic uses reduction to single reservoirsConvert water values into one-dimensional cuts


Upper bound from DOASA with 100 iterations Results for parallel system

430

435

440

445

450

455

460

0 10 20 30 40 50 60 70 80 90 100


Difference in value DOASA

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.300 -0.200 -0.100 0.000 0.100 0.200 0.300

Difference in value DOASA - Heuristic policyResults for parallel system


Upper bound from DOASA with 100 iterations Results cascade system

715

720

725

730

735

740

745

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96


Results: cascade system

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1 0 1 2 3 4

Difference in value DOASA - Heuristic policy


• weekly stages t=1,2,…,52• include head effects• nonlinear production functions• reservoir bounds are 0 and capacity• full plant availability• known price sequence, 21 per stage• stagewise independent inflows• 41 inflow outcomes per stage

Case studiesNew assumptions


Modelling head effectsPiecewise linear production functions vary with volume


Modelling head effectsA major problem for DOASA?

• For cutting plane method we need the future cost to be a convex function of reservoir volume.

• So the marginal value of more water is decreasing with volume.

• With head effect water is more efficiently used the more we have, so marginal value of water might increase, losing convexity.

• We assume that in the worst case, head effects make the marginal value of water constant at high reservoir levels.

• If this is not true then we have essentially convexified C at high values of x.


Modelling head effectsConvexification

• Assume that the slopes of the production functions increase linearly with reservoir volume, so

energy = volume.flow• In the stage problem, the marginal value of

increasing reservoir volume at the start of the week is from the future cost savings (as before) plus the marginal extra revenue we get in the current stage from more efficient generation.

• So we add a term p(t)..E[h()] to the marginal water value at volume x.


Modelling head effects: cascade systemDifference in value: DOASA - Heuristic policy


Modelling head effects: casade systemTop reservoir volume - Heuristic policy


Modelling head effects: casade systemTop reservoir volume - DOASA policy


FIM

ons sddp workshop, august 17, 2011 slide 1 of 50 andy philpott electric power optimization centre...

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