S TOCHASTIC O PTIMIZATION OF A G AS P LANT
WITH S TORAGE TAKING INTO ACCOUNT
TAKE - OR -PAY R ESTRICTIONS
Nils Löhndorf
David Wozabal
Vienna University of Economics and Business
International Conference on Stochastic Programming. July, 2013
Nils Löhndorf, David Wozabal
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
O UTLINE
Problem: plant and storage management under uncertainty
Modeling approach: Markov decision process
Assumptions
Problem setup
Solution Method: Approximate Dual Dynamic Programming
Algorithm
Theoretical results
Case Study
Stochastic process
Numerical results & convergence
Value of a stochastic solution
Conclusion
Nils Löhndorf, David Wozabal
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
P ROBLEM D ESCRIPTION
P ROBLEM
Operator of a gas fired power station and a gas storage wants to
maximize expected revenue from trading on electricity and gas
markets. Gas can either be bought from the spot market or from a
take-or-pay contract.
Nils Löhndorf, David Wozabal
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
P ROBLEM D ESCRIPTION
P ROBLEM
Operator of a gas fired power station and a gas storage wants to
maximize expected revenue from trading on electricity and gas
markets. Gas can either be bought from the spot market or from a
take-or-pay contract.
Medium term: one year planning horizon
Uncertainty: gas & electricity price
Optimal dispatch is a stochastic-dynamic decision problem
Decisions
Gas contract management (hourly)
Power generation (hourly)
Storage operation (hourly)
Gas market (daily)
Contract: fixed price and take-or-pay restriction
Nils Löhndorf, David Wozabal
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
D ECOMPOSITION STRATEGY
Intraday decision problem
 Optimize bidding on markets
 Uncertain electricity prices
 Stochastic programming
Interday decision process




Optimize storage levels and contract
Uncertain long term gas price
State transition as Markov process
Dynamic programming
Pritchard, Philpott & Neame (2005)
Nils Löhndorf, David Wozabal
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
I NTER -DAY AND I NTRA -DAY
I NTER - DAY P ROBLEM
Interday Problem: Markov decision process (daily resolution)
Resource state Rt : Continuous storage, contract and plant state.
Environmental state St : Not influenced by decisions
Factors influencing gas and electricity prices.
Discrete Markov process
Modeled by scenario lattice.
Nils Löhndorf, David Wozabal
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
I NTER -DAY AND I NTRA -DAY
I NTER - DAY P ROBLEM
Interday Problem: Markov decision process (daily resolution)
Resource state Rt : Continuous storage, contract and plant state.
Environmental state St : Not influenced by decisions
Factors influencing gas and electricity prices.
Discrete Markov process
Modeled by scenario lattice.
I NTRA - DAY P ROBLEM
Intraday Problem: Stochastic program
Environmental state of the MDP → electricity price scenarios
Hourly day ahead bidding on electricity and gas markets
Operational decisions
Value functions from the MDP
Nils Löhndorf, David Wozabal
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
D ISCRETIZATION : T REE VS L ATTICE
Scenariotree
Scenariolattice
31 Nodes, 16 Scenarios
15 Nodes, 120 Scenarios
State-time-history graph
State-time graph
Nils Löhndorf, David Wozabal
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
ADDP OVERVIEW
I TERATIVE STRATEGY FOR CONSTRUCTING VALUE FUNCTIONS
Forward pass
sample path (s0 , . . . , sT ) on the lattice
solve intraday problems, generating a sequence (r0 , . . . , rT ).
Backward pass: generate supergradients of the value functions Vt
at the nodes st and the points rt .
Approximate value functions by maximum of the supergradients.
P ROPOSITION
1 For convex problems, approximate value functions
are upper bounds for real value functions
convergence to true value functions
2
If problems on the nodes are linear → finite convergence.
Pereira & Pinto (1991), Philpott & Guan (2008), Philpott & de Matos (2012), Löhndorf, W.,
Minner (2013)
Nils Löhndorf, David Wozabal
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
VALUE F UNCTION A PPROXIMATION
V(St,R)
Value function
R
Nils Löhndorf, David Wozabal
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
VALUE F UNCTION A PPROXIMATION
V(St,R)
Value function
supporting hyperplane at Rn
Rn
Nils Löhndorf, David Wozabal
R
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
S OLUTION S TRATEGY
Nils Löhndorf, David Wozabal
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
S OLUTION S TRATEGY
Scenario reduction
Initial solution
Test for convergence
Construct value function
for every lattice node
Simulate solution with
current value function
Remove redundant hyperplanes
Löhndorf, W., Minner. Optimal Bidding of Electricity Storage using Approximate Dynamic
Programming. Operations Research (forthcoming).
Nils Löhndorf, David Wozabal
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
C ASE S TUDY: S ETUP & F LOWS
Gas flow
Power flow
Nils Löhndorf, David Wozabal
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
C ASE S TUDY: PARAMETERS
Gas Plant
Efficiency
Maximum output
Ramping costs
50%
800 MW
EUR 30 / MWh
Gas Storage
Size
330,000 MWh
Initial level
75,000 MWh
Maximum change per hour 200 MWh
Contract
Volume
Take-or-pay limit
Penalty
Nils Löhndorf, David Wozabal
4,100,000 MWh
3,750,000 MWh
20 EUR / MWh
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
C ASE S TUDY: S TOCHASTIC P ROCESSES
Scenario lattice
 Resource variables
Day of the year
 Renewables
 Temperature
 Gas price
 Renewables
 Remove seasonal trends
 clustering Residuals
Renewables
Temperature
Holidays
Gas price
 Gas Price: Mean-Reverting GBM
Environmental
State
State dependent spot prices
 2x24 linear models
 Interactions up to second order
 Model selection: Stepwise backward
forward regression
Spot price
 Price scenarios: NIG distributions
Hour of the day
Nils Löhndorf, David Wozabal
Weekday
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
C ASE S TUDY: S IMULATED P RICES
Weekly Electricity Prices (EUR/MWh)
55
50
45
40
35
01.2013
03.2013
04.2013
06.2013
08.2013
09.2013
11.2013
09.2013
11.2013
Daily Gas Prices (EUR/MWh)
32
30
28
26
24
22
01.2013
03.2013
04.2013
Nils Löhndorf, David Wozabal
06.2013
08.2013
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
C ASE S TUDY: I NTRA -DAY P ROBLEM
P ROBLEM F ORMULATION
Objective Function
Profit from market transactions
Ramping costs of the gas plant
Penalty from take-or-pay contract
Salvage value of gas (price at the end of horizon)
Constraints
Technical constrains of plant, storage, and pipelines
Market balance
Contract
Gas market
Bid-ask-spread at the gas market
Contract price greater than average market price
Lattice
365 Stages
Nodes per stage: 10, 100, 1000
Equivalent scenario tree has 6.7 × 10833 end-nodes
Nils Löhndorf, David Wozabal
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
C ASE S TUDY: R ESULTS
Simulated deterministic benchmarks
Clairvoyant: solves deterministic problem knowing the future
Rolling deterministic: for every stage
compute average values for random variables given current state;
solve deterministic problem with averages;
simulate transition to next state.
Nils Löhndorf, David Wozabal
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
C ASE S TUDY: R ESULTS
Simulated deterministic benchmarks
Clairvoyant: solves deterministic problem knowing the future
Rolling deterministic: for every stage
compute average values for random variables given current state;
solve deterministic problem with averages;
simulate transition to next state.
Nils Löhndorf, David Wozabal
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
C ASE S TUDY: S TORAGE P OLICY
Nils Löhndorf, David Wozabal
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
C ASE S TUDY: C ONTRACT P OLICY
Nils Löhndorf, David Wozabal
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract
C ONCLUSIONS
ADDP solves the problem over 365 stages in acceptable time
Value of stochastic solution is around 14%
Complicated stochastic processes require a lot of branching
Consistent results from 1000 nodes per stage onwards
Scenario trees can not be used for this problem
Comparison of policies
Contract management differs between stochastic and
deterministic policy
Deterministic solutions yield more extreme results
Future Research
Dynamic future price models
Lattice generation
Cut elimination & value function updates
Nils Löhndorf, David Wozabal
Optimization of a Gas Plant with Storage and a Take-or-Pay Contract