stochastic programming in gas storage and gas portfolio ... · in gas storage and gas portfolio...
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Stochastic Programmingin Gas Storage and Gas Portfolio Management
ÖGOR-Workshop, September 23rd, 2010
Dr. Georg Ostermaier
Agenda
• Optimization tasks in gas storage and gas portfolio management
• Scenario Tree based Stochastic Programmingapplied to Gas Portfolio Management
• Stochastic Processes and Scenario Tree generation
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• Technical, contractual and gas market constraints in gas portfolio management
• Exemplary Results
• Conclusions
Optimization tasks in gas portfolio management
• Objectives: – Cost minimal supply of retail demand by optimal utilizaiton of gas supply sources and storages
– Dimensioning of assets (gas storages, flexible contracts)
• Supply sources:– Forward market products (Months, Quarters, Years)
– Spot market products (Daily)
– Balancing market
– Flexible supply contracts
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– Flexible supply contracts
– Gas storages
• Methods:– Deterministic Optimization and Simulation
– Stochastic Optimization
• Challenges:– Mathematical modelling of gas markets and assets with all contractual and technical constraints
– Future uncertainties of market prices and retail demand
Efficiency of gas procurement portfolios
• Forward market products + flexible supply contract :– Uncetainty: future retail demand
– Structuring of procurement by monthly average of retail demand
– Structuring of procurement by mathematical optimiaztion of forward products
• Forward market products + flexible supply contract + spotmarket:
– Additional uncertainty: future uncertainty of spot market prices
– Determinsistic optimization with daily price forward curve of gas spot market
Effi
cien
cy
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– Determinsistic optimization with daily price forward curve of gas spot marketprices and gas retail demand
– Stochastic optimizaiton with scenario tree of gas spot market prices and gas retail demand
• Forward market products + flexible supply contract + spotmarket + gas storage:
– Determinsistic optimization with daily price forward curve of gas spot marketprices and gas retail demand
– Stochastic optimizaiton with scenario tree of gas spot market prices and gas retail demand
Effi
cien
cy
Not path dependent:• Today‘s exercise of an option has no impact on future optionality• Examples: monthly strip of options, coal fired plant with „unlimited“ fuel• Adequate valuation technique: Monte-Carlo-Simulation (= „one path scenario
simulation“)
Valuation and exercise of flexibilities in the energy industry
Path dependent:
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• Today‘s exercise of an option does have impact on future optionality• Examples: Swing Options, Pumped storage, CCGT with limited fuel supply
contract, Gas Storage• Path dependency is:
„When I fill the storage today, I cannot inject tomorrow anymore.“• Adequate valuation technique: Tree based stochastic optimization, Least Square
Monte Carlo methods
Path dependent:
Least Square Monte Carlo:
• Advantage: Generation of price scenarios can use a variety of price processes and can represent daily granularity
• Disadvantage:Technical and contractual constraints of the storage are difficult to implement because of the growth of the state space
Comparison of valuation approaches
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• Advantage:Technical and contractual constraints can be modeled precisely
• Disadvantage:Scenario tree cannot branch on a daily basis, scenario generation is limited to scaled daily price forward curves in periods of the tree.
Tree based Stochastic Optimization:
- daily price forward curves for TTF (every tenth day)- historic spot prices (red)
\MW
hDiscrepancy between forward and day ahead prices
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€\
Example: Storage Operation under consideration of future spot price uncertaintyStochastic vs. Deterministic optimization
01.10.200601.10.2006
01.10.2007
365 stochastische Optimierungen
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01.10.2006
02.10.2006
02.10.2006
01.10.2007
365 deterministische Optimierungen
MW
hStorage volume trajectories
100% 100% profitprofit73.85% 73.85% profitprofit73.2% 73.2% profitprofit
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MW
h
Analysis Analysis ofof profitsprofits / / costscosts zzs,ns,n derivedderived fromfrom decisionsdecisions(u(u11, ... , , ... , uuSS)) forfor different different priceprice processesprocesses (p(p11, ... , , ... , ppNN))
pp11 ⇒⇒ zzs,1s,1
pp22 ⇒⇒ zzs,2s,2
pp33 ⇒⇒ zzs,3s,3
pp44 ⇒⇒ zzs,4s,4
Concept of Simulation
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Drawbacks:Drawbacks:•• Different first stage Different first stage
decisionsdecisions with different with different
price processes price processes pp11, ... , p, ... , pNN
•• Anticipativity of decision makingAnticipativity of decision making
pp44 ⇒⇒ zzs,4s,4
pp55 ⇒⇒ zzs,5s,5
pp66 ⇒⇒ zzs,6s,6
......
ppNN ⇒⇒ zzs,Ns,N
tt
UniqueUnique optimal decision in every node optimal decision in every node with respect to all possible future developmentswith respect to all possible future developments
(1)(1)
(1,1)(1,1)
(1,2)(1,2)
(1,1,1)(1,1,1)
(1,1,2)(1,1,2)
(1,2,1)(1,2,1)
Scenario tree(binary process)
Szenario 1Szenario 1
Szenario 2Szenario 2
Szenario 3Szenario 3
Concept of Stochastic Optimization
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Non-anticipativityof the decisions
(2)(2)
(2,1)(2,1)
(2,2)(2,2)
(1,2,2)(1,2,2)
(2,1,1)(2,1,1)
(2,1,2)(2,1,2)
(2,2,1)(2,2,1)
(2,2,2)(2,2,2)tt Szenario 8Szenario 8
Szenario 5Szenario 5
Szenario 4Szenario 4
Szenario 6Szenario 6
Szenario 7Szenario 7
0.45
-3-2
-1
Shadow price of volume is derived from dual variable of volume balance equation of first node
P(Scenario 2)P(Scenario 2)
P(Scenario1)P(Scenario1)
Value-at-Risk (10%)
Storage balance equations in the scenario tree
Scenario treePath dependent PnL
VV11S1S1 + + WW11
S1S1 –– II11S1S1 = = VV00
Storage Balance equation of period 1, scenario 1
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0 0.05
0.10.15
0.20.25
0.3
0.350.4
0.4501
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tt00
Value-at-Risk (10%)
tt11 tt22 tt33
VV00 + + WW00 -- II0 0 = V= V--1 1
VV11S2S2+ + WW11
S2S2 –– II11S2S2 = = VV00
V . Volume
W: Withdrawal
I : Injection
Constant!Initial Volume VV--11
Storage Balance equation of period 1, scenario 2
Storage Balance equation of period 0, scenario 1
Min/max Volume constraintsfor each path of the tree
Shadow price of volume is derived from dual variable of volume balance equation of first node
Storage balance equations in the scenario tree
VV11S1S1 + + WW11
S1S1 –– II11S1S1 = = VV00
Storage Balance equation of period 1, scenario 1
VV33minmin <= <= VV33
S1S1 <= V<= V33maxmax
VV33minmin <= <= VV33
S2S2 <= V<= V33maxmax
VV33minmin <= <= VV33
S3S3 <= V<= V33maxmax
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tt00 tt11 tt22 tt33
VV00 + + WW00 -- II0 0 = V= V--1 1
VV11S2S2+ + WW11
S2S2 –– II11S2S2 = = VV00
V . Volume
W: Withdrawal
I : Injection
Constant!Initial Volume VV--11
Storage Balance equation of period 1, scenario 2
Storage Balance equation of period 0, scenario 1
. . .. . .
VV33minmin <= <= VV33
S8S8 <= V<= V33maxmax
Day Ahead price (Market 1)
Day Ahead price (Market 2)
Storage outages
Multi-dimensional scenario tree generation
Gas demand
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Oil Price indices
FX rate All trees share the same branching structure but hold different values, according to expectation, volatility, correlations and other parameters of the underlying stochastic processes
Structure of the overall mathematical model
Shadow price of gas is derived from dual variable of gas balance equation of first node
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Workflow for the valuation of gas storages / gas portfolios
Stochastic optimization:
Estimation of volatility/mean reversion
(Demand)
Stochastic impacts:Gas spot prices (i.e. TTF)Gas demand (optionally)Oil price index, FX-Rate (optionally)
Historic demand
1) Scenario trees
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Calculation of DPFC
(Fleten‘s model)
Historic TTF-spot price
HistorcTTF- spot price
Distributions
Estimation of volatility/mean reversion
(Spot prices)
1) Scenario trees(Spot prices and demand)
2) max Objective function
Constraints
Solver (CPLEX™)
3) Solving the math. model
Actual price forward curve
Mean of P&L-Distribution =Value of the
Storage
Solver (CPLEX™)
0
10000
20000
30000
40000
50000
60000
70000
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Periods
Energ
y [ M
Wh / 2 w
eeks ]
scen. treelower appr.
expectedload
real load
scen. treeupper appr
Scenario treesAnalytical
modelformulation
Stochasticmultistage
Structure of the mathematical optimization kernel
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Solver (CPLEX™)
LP-Solver(Simplex, Barrier)
MIP-Solver(Branch-and-Bound)
Decisions and P&L Distribution
multistage(mixed integer)
program
Spot price process: Pilipovic
Spot price process:
Mean reversion Sigma Price
Spot price process for the gas day ahead market:
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UncorrelatedBrownian motions
Spot price process:
Equilibrium price process:
Sigma Trend
Stochastic processes of multiple uncertainties
Spot price process:
Long term price process:
Spot price processes for the Gas-Day-Ahead-Market, Gas Demand, oil price and FX rate:
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Uncorrelated Brownian Motions
process:
Sigma long term price:Sigma short term price:Sigma demand:Sigma oil price:Alpha (Mean Reversion) short term price:Alpha (Mean Reversion) gas demand:Alpha (Mean Reversion) oil price:Correlation between gas demand and short term gas price:
Gas demand:
Oil price:
FX rate:
12/4/2006
12/5/2006
12/25/2006
01/22/2007
03/19/2007 06/11/2007
09/02/2007
Division of the planning horizon
• Problem: exponential growth of the problem with the number of stages in the tree („curse of dimensionality)
• Solution: Division of the planning horizon into a limited number of periods at which the tree branchesIn each node of the tree, the DPFC is scaled to a certain scenario of the DPFC
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1 day
2 weks
4 weeks 8 weeks 12 weeks 12 weeks
DPFC€/MWh
Scenarios as deviations from DPFC
1. Scenarios of the logarithmic spot price xp for each period of the tree
2. Scenarios of exp(xp – 0.5 Var(xp))
0
x
exp (x – ½ Var(x))
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3. Scenarios of DPFCt exp(xp – 0.5 Var(xp))DPFCexp (x – ½ Var(x))
1
exp (x – ½ Var(x))
DPFC
• Estimation of stochastic process parameters– Volatilities, Mean Reversions
– Correlations
• Analytic Integration of stochastic differencial equations in order to define the moments of the distributions (Var/Covar-Matrix) at every branching step of the scenario tree
Generation of scenario trees
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every branching step of the scenario tree
• Approximation of the multidimensional standardnormal distribution with multinomial distributions
• Transformation of the approximation of the standard normal distribution with the moments of the desired distribution at every step t of the scenario tree
Definition of the planning horizon
Valuation date:Today
Start/End Date:Defines the planning horizon
The user can define the
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The user can define the number and dates of the scenario tree branching steps
Definition of the stochastic spot price process
The basis for the spot price scenario generation is a Daily price forward curveCurve)
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The parameters of volatility (standard deviation), mean reversion and correlation define the generation of the scenario tree
Set up of gas procurement portfolios (storages)
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Set up of gas procurement portfolios (contracts)
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Definition of min/max power of storages and contracts,min/max volume of storages
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Definition of Forward-Portfolios (Prices, already closed positions)
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Definition of liquidity and market depth of day-ahead gas market
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• Time series of min/max injection, withdrawal and volume
• Non linear constraints of maximum injection/withdrawal dependent on volume (piecewise linear approximations or stair curves ->integer modelling)
• Total depletion only at a maximum number of days
• Time series of injection/withdrawal costs
• Time series of injection/withdrawal losses
Modeling of gas storages
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• Time series of injection/withdrawal losses
• Shrinkage of volume over time
• Stochastic outages / interruption of transport capacity
• Joint optimization of forward, spot (and balance market) products
• Maximum day ahead trading limits (market depth)
• Maximum forward trading limits
• Price influence of large traded positions (market elasticity)
• Lot size of traded products as integer numbers
• Availability of standard and non-standard products
Modeling of gas markets / traded products
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• Representation of already closed positions
• Daily, monthly, quarterly, seasonal and yearly min/max quantities
• Discounts after withdrawal of specific quantities
• Oil price indexed and FX-rate dependent price
• Time lags in oil price indexation
Representation of characteristics of flexible supply contracts
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Sensitivity of P&L distribution with different trading strategies
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Expected profits increase with high share of spot trading
Example of gas storage valuation results
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Gas Storage Valuation based on Least Square Monte Carlo(overall profit over 30 years)
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Gas Storage Valuation based on Scenario Tree(overall profit over 30 years)
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Sensitivities with short term spot price volatility
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Sensitivities with short term spot price volatility
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Sensitivities with long term spot price volatility
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Conclusions
• Valuation of gas storages and gas procurement portfolios requires methods that consider path dependency in the decision process
• Tree based Stochastic Optimization is an approach to represent uncertainty and at the same time complex technical and contractual constraints in the portfolio
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• Representation of uncertainty in the valuation and operation of gas procurement portfolios leads to lower expected procurement costs