Applications of Stochastic Programming in the Energy Industry

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Applications of Stochastic Programming in the
Energy Industry

• Chonawee Supatgiat
• Research Group
• Enron Corp.
• INFORMS Houston Chapter Meeting
• August 2, 2001

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Outline

• Stochastic Program
• Colombia Hydro-thermal System
• Model
• Solution techniques
• Nested Decomposition
• Abridged Nested Decomposition
• Example result
• Fuel Inventory and Electric Generation
• Model
• Solution techniques
• Bender Decomposition
• Lagrangian Relaxation
• Example result

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Stochastic Program

• Mathematical program where some of the data
  incorporated into the objective or constraints is
  uncertain
• Recourse program some decisions or recourse
  actions can be taken after uncertainty is
  disclosed

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Two-Stage Stochastic Linear Program
Stage 2
Stage 1
x
y
x
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Extensive Form
y(x1)
x1
x
x2
y(x2)
x3
y(x3)
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Multi-Stage Stochastic Linear Program

• QN1(xN) 0, for all xN

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Application 1Colombia Long-Term Power
Planning(joint work with John Birge,
Northwestern University)
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Power System
Area 2
Area 1

• Colombia System
• 8 areas
• 47 hydro units
• 70 thermal units
• 28 fuel types
• hydro generation 50

Area 3
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Problem

• Central Dispatch Planning Problem
• Colombia government need to determine capacity
  payment
• Socially optimal dispatch planning of Colombia
  hydro-thermal generating units
• Decision in each period, water release (hydro)
  and generation (thermal), and export/import power
  flow between areas
• Stochastic water inflow in each location

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Model for Colombia Power

• Objective
• minimize total generation costs outage penalty
  alert water level penalty
• subject to the operating constraints
• meet load constraints
• water balance constraints
• thermal capacity constraints
• hydro maximum/minimum flow constraints
• export/import capacity constraints
• minimum/maximum reservoir level

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Inflow Scenario Tree Serial Correlation
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Multi-Stage Stochastic Linear Program

• QN1(xN) 0, for all xN

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Solution Technique Nested Decomposition

• Basic Idea
• Solve each subproblem separately
• xt is passed forward to the Stage t1 subproblem
• Function Qt is passed backward to Stage t-1
  subproblem
• Updating x and functions Q until converge

Q3(x2,?)
Q2 (x2,?)
Q3(x2,?)
x2
x1
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Feasibility when Passing Forward

• If xt from Stage t subproblem makes a subproblem
  at Stage t1 infeasible
• Stage t1 subproblem sends a message to Stage t
  that this xt is a bad solution

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Nested Decomposition

• Forward Pass
• Starting at the root node and proceeding forward
  through the scenario tree, solve each node
  subproblem
• Add feasibility cuts as infeasibilities arise
• Backward Pass
• Starting in top node of Stage t N-1, use
  optimal dual values in descendant Stage t1 nodes
  to construct new optimality cut. Repeat for all
  nodes in Stage t, resolve all Stage t nodes, then
  t t-1.
• Repeat until converge

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Nested Decomposition (ND) v.s. Dynamic
Programming (DP)

• DP
• starting from the last nodes and evaluating Q for
  all possible values of x
• move backward when get complete information of Q
• ND
• evaluating Q only for one value of x in each
  iteration
• move forward to generate new value of x

Q
Qt(xt-1)
x
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Abridged Nested Decomposition

• Incorporates sampling into the general framework
  of Nested Decomposition
• Assumes
• relatively complete recourse a feasible solution
  exists for every feasible solution in the
  previous stage
• serial independence the stochastic data in each
  stage is independent of the realized values in
  the previous stages
• Samples both the subproblems to solve and the
  solutions to continue from in the forward pass

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Abridged Nested Decomposition
Forward Pass 1. Solve root node
subproblem 2. Sample Stage 2 subproblems and
solve selected subset 3. Sample Stage 2
subproblem solutions and branch in Stage 3 only
from selected subset (i.e., nodes 1 and 2)

• 4. For each selected Stage t-1 subproblem
  solution, sample Stage t subproblems and solve
  selected subset
• 5. Sample Stage t subproblem solutions and branch
  in Stage t1 only from selected subset

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Abridged Nested Decomposition
Backward Pass Starting in first branching node
of Stage t N-1, solve all Stage t1 descendant
nodes and construct new optimality cut for all
stage t subproblems. Repeat for all sampled
nodes in Stage t, then repeat for t t - 1
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NDUM and CPLEX v. No. of Scenarios
19.4 hrs
2.8 hrs
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Example Results (selected plants)
MWh
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Example Hydro Generation
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Example Thermal Generation
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Example Dual Prices
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Example Max MW (selected plants)
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Application 2Energy Marketer with Gas Storage
and Generators (joint work with Samer Takriti
and Lilian Wu, IBM Research)
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Coordinating Fuel Inventory with Electricity
Generation
Energy marketer
Gas-Turbine generation plants
Natural gas market
Gas storage
buy
fuel
sell
gas demand
sell power
Power market
Gas customers
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Properties of Gas Turbine Generators

• Minimum up time
• Minimum down time
• Start up cost
• Quadratic gas consumption
• gas consumption a b (generation) c
  (generation)2
• Operating level
• Sell power at market (bid) price

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Gas Storage

• Buy gas at market ask price, sell gas at market
  bid price
• Storage holding cost
• Inject/withdraw fees
• Inject/withdraw limits
• Storage capacity
• Gas loss

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Randomness and Decisions

• Have multiple forecasts for natural gas demand,
  natural gas prices, electricity prices
• Observe spot gas (bid/ask) prices, spot
  electricity (bid) price, and current gas demand
• State current gas storage level, status of the
  electric generators
• Decide on the amount to buy/sell natural gas and
  the electricity generation

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Scenario Tree
p, g, d
p, g, d
p, g, d
p, g, d
p, g, d
p, g, d
p, g, d
p, g, d
p, g, d
p, g, d
p, g, d
p, g, d
p, g, d
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Full Model

• Large stochastic mixed integer program
• Max total discounted expected future revenue
  from gas and power selling minus future operation
  costs from gas storage and generators and gas
  buying cost
• S.t.
• Minimum up/down time constraints (integer)
• Min/max power generation levels (conditional)
• Gas to power conversion equation (integer
  possibly non-linear)
• Gas inventory balance constraints
• Gas storage capacity
• Gas injection/withdraw capacity

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Bender Decomposition

• Max total discounted expected future revenue
  from gas and power selling minus future operation
  costs from gas storage and generators and gas
  buying cost
• S.t.

Minimum up/down time constraints
(integer) Min/max power generation levels
(conditional) Gas to power conversion equation
(integer non-linear) Gas inventory balance
constraints Gas storage capacity Gas
injection/withdraw capacity
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Bender Decomposition

• First Stage Unit commitment problem stochastic
  integer program
• Second Stage Inventory problem stochastic
  linear program

integer
conditional
Integer non-linear
Bender cuts
To be solved by simple LP
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Unit Commitment Problem

• Lagrangian Relaxation of the Bender cuts
• Max L(l), l gt 0

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Lagrangian Relaxation of Unit Commitment Problem

• Max Si Li(l)pbl
• l gt 0
• where

Vector of avg. gas price of generator i in each
node
Individual generator problem
to be solved by stochastic DP
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Solution Technique Summary

• Pure BB
• Use OSL BB to solve the full problem in one shot
• Bender BB
• Decompose into two stages
• Solve first stage by BB and second stage by OSL
  LP
• Bender Lagrangian
• Decompose into two stages
• Relax the Bender cuts and decompose the first
  stage. Solve its sub-problems by DP
• Solve the second stage by OSL LP

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Numbers of Benders cuts needed are between 7-38
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