integrated planning of biomass inventory and energy production - … · 2014. 8. 27. · integrated...

Integrated planning of biomass inventoryand energy production

Marco Chiarandini1 Niels Kjeldsen1,2 Napoleão Nepomuceno3

1Department of Mathematics and Computer Science, University of Southern Denmark2Model development, DONG Energy Thermal Power A/S

3Universidade de Fortaleza, Programa de Pós-Graduação em Informática Aplicada, Fortaleza, Brazil

July 5th, 2012

Investment evaluationMathematical modelBenders decompositionResultsOutline

1. Production planning and investment evaluationChanging fuel: From coal to wood pellets

2. Mathematical modelMixed integer linear programming model

3. Benders decompositionBenders optimality cutsHandling multiple scenarios

4. Results

M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 2





4. Results


Investment evaluationMathematical modelBenders decompositionResultsDanish energy system

(source Energinet.dk)

Weekly demand profile

200 250 300

01000

3000

5000

Hours

MW

Demand

Remaining

Wind


Investment evaluationMathematical modelBenders decompositionResultsEnergy production

I Uncontrollable:I Wind powerI Solar power

I ControllableI Thermal units:

Providing heat to the local heating areaI Connections to neighboring countries

I Other sources:I SmartGridI Electric cars


Investment evaluationMathematical modelBenders decompositionResultsOverview of Avedøre power plant


Investment evaluationMathematical modelBenders decompositionResultsWood pellet storage at Avedøre


Investment evaluationMathematical modelBenders decompositionResultsFuel delivery processes

Logistics differences

Coal logistics

Wood pellets logistics


Investment evaluationMathematical modelBenders decompositionResultsBiomass contracts

Uniform contract

hours

0 730 1460 2190 2920 3650 4380 5110 5840 6570 7300 8030 8760

Seasonal contract

hours

0 730 1460 2190 2920 3650 4380 5110 5840 6570 7300 8030 8760


Investment evaluationMathematical modelBenders decompositionResultsTwo stage stochastic approach

I Biomass contracts must be decided a year ahead.

I But future demand, prices and exact delivery times are unknown uncertainty.

Two stage stochastic approach (look-ahead policy):

I First stage: long term decisions on biomass contracts might yield:I Running out of fuel (underflow)I Running out of storage space (overflow)

I Second stage: optimize when uncertainty is revealedI Production of electricity and heat.I Foreign trade (only electricity).I Using an alternative (fossil) fuelI Redirection of deliveries.






4. Results


Investment evaluationMathematical modelBenders decompositionResultsMixed integer linear programming model

Several scenarios for future uncertainty

Objective function (minimize):I Cost of biomass contractsI Use of fossil fuelI Foreign tradeI Over/under production (slack/surplus demand)

Constraints:I Electricity and heat demandI Power plant production (including trade with neighboring countries)I Biomass fuel levels and redirection of deliveries


Investment evaluationMathematical modelBenders decompositionResultsConstraints

Electricity and heat balance

Electricity

t − 1 t

pk,i,t−1,s

Dpt−1,s

p t−1,s

p t−1,

s

p en,t−1,s

pk,i,t,s

Dpt,s

p t,s

p t,s

p en,t−1,s

Heat

t − 1 tqah,t−2,s

qk,i,t−1,s

Dqh,t−1,s

qah,t−1,s

q h,t−

1,s

q h,t−

1,s

qk,i,t,s

Dqh,t,s

qah,t,s

q h,t,s

q h,t,s



Biomass fuel level constraints

time

fuellevel

capacity

f2,t,s = f2,t−1,s − τ · u2,i,t,s + ∑j∈J Aj,t,s · xj,i,t,s



Modeling power plant production

Cogeneration power plant

ρmax

ρmin p = ςbq

q

p

p = ρmax − ςvq

p = ρmin − ςvq


Investment evaluationMathematical modelBenders decompositionResultsThe full model






4. Results


Investment evaluationMathematical modelBenders decompositionResultsBenders Decomposition

We consider the MILP with complicating y -variables, which are the biomasscontracts:

minx,y

z = cT x + f T y

Ax + By ≥ by ∈ Yx ≥ 0

or emphasizing the two stage approach:

miny∈Y

[f T y +min

x≥0

(cT x |Ax ≥ b − By

)]


Investment evaluationMathematical modelBenders decompositionResultsBenders optimality cuts

Given a specific set of biomass contracts y the dual of the inner problem is:

maxu

f T y + (b − By)Tu

ATu ≤ cu ≥ 0

The solution u to the dual problem gives a lower bound to the originalproblem.The lower bound is valid for all biomass contracts y and the generalizationgives:

Benders optimality cut

z ≥ f T y + (b − By)Tu.


Investment evaluationMathematical modelBenders decompositionResultsHandling multiple scenarios

Block angular structureVariables

Con

straints

Biomasscontracts

One yearscenarios

Benders optimality cuts for multiplescenarios

miny∈Y

f T y +1|S |∑

s∈S

zs

s.t. zs ≥ (bs − Bsy)Tusk∀s ∈ S , k = 1 . . .K






4. Results


Investment evaluationMathematical modelBenders decompositionResultsTime to optimality

Empirical cumulative distribution function of the time to completion of therun for the four models

1e+02 5e+02 5e+03 5e+04

0.0

0.2

0.4

0.6

0.8

1.0

Time to optimum

ecdf

integralrelaxedbranch_boundbranch_cut


Investment evaluationMathematical modelBenders decompositionResultsOptimality gap

cost

8−04−2008−04−1008−06−2008−06−1008−12−2008−12−1008−24−2008−24−1004−04−2004−04−1004−06−2004−06−1004−12−2004−12−1004−24−2004−24−1001−04−2001−04−1001−06−2001−06−1001−12−2001−12−1001−24−2001−24−100

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Investment evaluationMathematical modelBenders decompositionResultsExploitation of computational resources

Average number of CPUs used during a run for varying number of scenarios(x-axis) and number of contracts and step size (strip text in the panels).

scenarios

n.cp

us

2

4

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integral relaxed branch_bound branch_cut● ● ● ●


Investment evaluationMathematical modelBenders decompositionResultsConclusions

I Biomass logistics complicates long term planning

I Relaxing some of the binary variables does not impact significantly thetotal cost assessment

I It is important to consider several scenarios and flexible contracts

I Benders relaxation does not improve solution timesbut it is able to exploit computational resources potential improvement by primal heuristics + more aggressive cuts

(future work)


Investment evaluationMathematical modelBenders decompositionResults

Thank you for your attention!


Production planning and investment evaluationChanging fuel: From coal to wood pellets

Mathematical modelMixed integer linear programming model

Benders decompositionBenders optimality cutsHandling multiple scenarios

Results

integrated planning of biomass inventory and energy production - … · 2014. 8. 27. · integrated...

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