Integrated planning of biomass inventory and energy production Marco - - PowerPoint PPT Presentation

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Integrated planning of biomass inventory and energy production Marco - - PowerPoint PPT Presentation

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Integrated planning of biomass inventory and energy production Marco Chiarandini 1 Niels Kjeldsen 1 , 2 Napoleo Nepomuceno 3 1 Department of Mathematics and Computer Science, University of Southern Denmark 2 Model development, DONG Energy Thermal

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Integrated planning of biomass inventory and energy production

Marco Chiarandini1 Niels Kjeldsen1,2 Napoleão Nepomuceno3

1Department of Mathematics and Computer Science, University of Southern Denmark 2Model development, DONG Energy Thermal Power A/S 3Universidade de Fortaleza, Programa de Pós-Graduação em Informática Aplicada, Fortaleza, Brazil

July 5th, 2012

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Investment evaluation Mathematical model Benders decomposition Results

Outline

  • 1. Production planning and investment evaluation

Changing fuel: From coal to wood pellets

  • 2. Mathematical model

Mixed integer linear programming model

  • 3. Benders decomposition

Benders optimality cuts Handling multiple scenarios

  • 4. Results
  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 2

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SLIDE 3

Investment evaluation Mathematical model Benders decomposition Results

Outline

  • 1. Production planning and investment evaluation

Changing fuel: From coal to wood pellets

  • 2. Mathematical model

Mixed integer linear programming model

  • 3. Benders decomposition

Benders optimality cuts Handling multiple scenarios

  • 4. Results
  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 3

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Investment evaluation Mathematical model Benders decomposition Results

Danish energy system

(source Energinet.dk)

  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 4

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Investment evaluation Mathematical model Benders decomposition Results

Danish energy system

(source Energinet.dk)

Weekly demand profile

200 250 300 1000 3000 5000 Hours MW

Demand Remaining Wind

  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 4

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Investment evaluation Mathematical model Benders decomposition Results

Energy production

◮ Uncontrollable:

◮ Wind power ◮ Solar power

◮ Controllable

◮ Thermal units:

Providing heat to the local heating area

◮ Connections to neighboring countries

◮ Other sources:

◮ SmartGrid ◮ Electric cars

  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 5

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Investment evaluation Mathematical model Benders decomposition Results

Overview of Avedøre power plant

  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 7

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Investment evaluation Mathematical model Benders decomposition Results

Wood pellet storage at Avedøre

  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 8

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Investment evaluation Mathematical model Benders decomposition Results

Fuel delivery processes

Logistics differences

Coal logistics Wood pellets logistics

  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 9

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Investment evaluation Mathematical model Benders decomposition Results

Biomass contracts

Uniform contract

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

Seasonal contract

hours 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

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Investment evaluation Mathematical model Benders decomposition Results

Two stage stochastic approach

◮ Biomass contracts must be decided a year ahead. ◮ But future demand, prices and exact delivery times are unknown

uncertainty. Two stage stochastic approach (look-ahead policy):

◮ First stage: long term decisions on biomass contracts might yield:

◮ Running out of fuel (underflow) ◮ Running out of storage space (overflow)

◮ Second stage: optimize when uncertainty is revealed

◮ Production of electricity and heat. ◮ Foreign trade (only electricity). ◮ Using an alternative (fossil) fuel ◮ Redirection of deliveries.

  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 11

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Investment evaluation Mathematical model Benders decomposition Results

Outline

  • 1. Production planning and investment evaluation

Changing fuel: From coal to wood pellets

  • 2. Mathematical model

Mixed integer linear programming model

  • 3. Benders decomposition

Benders optimality cuts Handling multiple scenarios

  • 4. Results
  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 12

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Investment evaluation Mathematical model Benders decomposition Results

Mixed integer linear programming model

Several scenarios for future uncertainty Objective function (minimize):

◮ Cost of biomass contracts ◮ Use of fossil fuel ◮ Foreign trade ◮ Over/under production (slack/surplus demand)

Constraints:

◮ Electricity and heat demand ◮ Power plant production (including trade with neighboring countries) ◮ Biomass fuel levels and redirection of deliveries

  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 14

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Investment evaluation Mathematical model Benders decomposition Results

Constraints

Electricity and heat balance

Electricity t − 1 t

pk,i,t−1,s Dp

t−1,s

pt−1,s pt−1,s pe

n,t−1,s

pk,i,t,s Dp

t,s

pt,s p

t,s

pe

n,t−1,s

Heat t − 1 t

qa

h,t−2,s

qk,i,t−1,s Dq

h,t−1,s

qa

h,t−1,s

qh,t−1,s qh,t−1,s qk,i,t,s Dq

h,t,s

qa

h,t,s

qh,t,s q

h,t,s

  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 15

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Investment evaluation Mathematical model Benders decomposition Results

Constraints

Biomass fuel level constraints

time fuel level 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

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Investment evaluation Mathematical model Benders decomposition Results

Constraints

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

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Investment evaluation Mathematical model Benders decomposition Results

The full model

  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 18

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Investment evaluation Mathematical model Benders decomposition Results

Outline

  • 1. Production planning and investment evaluation

Changing fuel: From coal to wood pellets

  • 2. Mathematical model

Mixed integer linear programming model

  • 3. Benders decomposition

Benders optimality cuts Handling multiple scenarios

  • 4. Results
  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 19

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Investment evaluation Mathematical model Benders decomposition Results

Benders Decomposition

We consider the MILP with complicating y-variables, which are the biomass contracts: min

x,y

z = cTx + f Ty Ax + By ≥ b y ∈ Y x ≥ 0

  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 20

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Investment evaluation Mathematical model Benders decomposition Results

Benders Decomposition

We consider the MILP with complicating y-variables, which are the biomass contracts: min

x,y

z = cTx + f Ty Ax + By ≥ b y ∈ Y x ≥ 0

  • r emphasizing the two stage approach:

min

y∈Y

  • f Ty + min

x≥0

  • cTx|Ax ≥ b − By
  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 20

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Investment evaluation Mathematical model Benders decomposition Results

Benders optimality cuts

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

u

f Ty + (b − By)Tu ATu ≤ c u ≥ 0 The solution u to the dual problem gives a lower bound to the original problem. The lower bound is valid for all biomass contracts y and the generalization gives: Benders optimality cut z ≥ f Ty + (b − By)Tu.

  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 22

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Investment evaluation Mathematical model Benders decomposition Results

Handling multiple scenarios

Block angular structure

Variables Constraints Biomass contracts One year scenarios

Benders optimality cuts for multiple scenarios min

y∈Y

f Ty + 1 |S|

  • s∈S

zs s.t. zs ≥ (bs − Bsy)Tus

k

∀s ∈ S, k = 1 . . . K

  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 24

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Investment evaluation Mathematical model Benders decomposition Results

Outline

  • 1. Production planning and investment evaluation

Changing fuel: From coal to wood pellets

  • 2. Mathematical model

Mixed integer linear programming model

  • 3. Benders decomposition

Benders optimality cuts Handling multiple scenarios

  • 4. Results
  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 25

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Investment evaluation Mathematical model Benders decomposition Results

Time to optimality

Empirical cumulative distribution function of the time to completion of the run 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 integral relaxed branch_bound branch_cut

  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 26

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Investment evaluation Mathematical model Benders decomposition Results

Optimality gap

cost

8−04−200 8−04−100 8−06−200 8−06−100 8−12−200 8−12−100 8−24−200 8−24−100 4−04−200 4−04−100 4−06−200 4−06−100 4−12−200 4−12−100 4−24−200 4−24−100 1−04−200 1−04−100 1−06−200 1−06−100 1−12−200 1−12−100 1−24−200 1−24−100 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

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Investment evaluation Mathematical model Benders decomposition Results

Exploitation 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.cpus

2 4 6

  • 24

100

2 4 6 8

  • 12

100

  • 6

100

2 4 6 8

  • 4

100

2 4 6 8

  • 24

200

  • 12

200

2 4 6 8

  • 6

200

2 4 6

  • 4

200 integral relaxed branch_bound branch_cut

  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

.::. Integrated planning of biomass inventory and energy production 29

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Investment evaluation Mathematical model Benders decomposition Results

Conclusions

◮ Biomass logistics complicates long term planning ◮ Relaxing some of the binary variables does not impact significantly the

total cost assessment

◮ It is important to consider several scenarios and flexible contracts ◮ Benders relaxation does not improve solution times

but 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

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Investment evaluation Mathematical model Benders decomposition Results

Thank you for your attention!

  • M. Chiarandini, N. Kjeldsen, N. Nepomuceno

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