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
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 3
Investment evaluationMathematical modelBenders decompositionResultsDanish energy system
(source Energinet.dk)
Weekly demand profile
200 250 300
01000
3000
5000
Hours
MW
Demand
Remaining
Wind
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 4
Investment evaluationMathematical modelBenders decompositionResultsDanish energy system
(source Energinet.dk)
Weekly demand profile
200 250 300
01000
3000
5000
Hours
MW
Demand
Remaining
Wind
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 4
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
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 5
Investment evaluationMathematical modelBenders decompositionResultsOverview of Avedøre power plant
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 7
Investment evaluationMathematical modelBenders decompositionResultsWood pellet storage at Avedøre
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 8
Investment evaluationMathematical modelBenders decompositionResultsFuel delivery processes
Logistics differences
Coal logistics
Wood pellets logistics
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 9
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
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 10
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.
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 11
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 12
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
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 14
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
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 15
Investment evaluationMathematical modelBenders decompositionResultsConstraints
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
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 16
Investment evaluationMathematical modelBenders decompositionResultsConstraints
Modeling power plant production
Cogeneration power plant
ρmax
ρmin p = ςbq
q
p
p = ρmax − ςvq
p = ρmin − ςvq
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 17
Investment evaluationMathematical modelBenders decompositionResultsThe full model
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 18
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 19
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
)]
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 20
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
)]
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 20
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.
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 22
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
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 24
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 25
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
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 26
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
10^8 10^9 10^10 10^11
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
integral
10^8 10^9 10^10 10^11
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
relaxed
10^8 10^9 10^10 10^11
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
branch_bound
10^8 10^9 10^10 10^11
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
branch_cut
ub lb
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 28
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
6
●●
●●
●
●
●
●
●
●
●
●
24100
2 4 6 8
●
●
●
●
●
●●
●●
●
●
●
12100
●
●
●
●
● ●
●
●●
●
●●
6100
2 4 6 8
●
●●
● ●●
●
●●
●
●
●
4100
2 4 6 8
●
●●
●
●
●
●
●
●
●●
●
24200
●
●
●
●
●
●
●
●
●
●●
●
12200
2 4 6 8
●
●
●
●●
●
●
●
●
●
● ●
6200
2
4
6●
●●
●
●
●●
● ●
●
●
●
4200
integral relaxed branch_bound branch_cut● ● ● ●
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 29
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)
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 30
Investment evaluationMathematical modelBenders decompositionResults
Thank you for your attention!
M. Chiarandini, N. Kjeldsen, N. Nepomuceno .::. Integrated planning of biomass inventory and energy production 31
Production planning and investment evaluationChanging fuel: From coal to wood pellets
Mathematical modelMixed integer linear programming model
Benders decompositionBenders optimality cutsHandling multiple scenarios
Results