Stochastic Programming in 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 Programming applied to Gas Portfolio Management Stochastic Processes and Scenario Tree generation Technical, contractual and gas market constraints in gas portfolio management Exemplary Results Conclusions 2
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 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 3
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 + spot market: Additional uncertainty: future uncertainty of spot market prices Determinsistic optimization with daily price forward curve of gas spot market prices and gas retail demand Stochastic optimizaiton with scenario tree of gas spot market prices and gas retail demand Forward market products + flexible supply contract + spot market + gas storage: Determinsistic optimization with daily price forward curve of gas spot market prices and gas retail demand Stochastic optimizaiton with scenario tree of gas spot market prices and gas retail demand Effici iency 4
Valuation and exercise of flexibilities in the energy industry 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 ) Path dependent: 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 5
Comparison of valuation approaches 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 Tree based Stochastic Optimization: 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. 6
Discrepancy between forward and day ahead prices - daily price forward curves for TTF (every tenth day) - historic spot prices (red) \ \MWh 7
Example: Storage Operation under consideration of future spot price uncertainty Stochastic vs. Deterministic optimization 01.10.2006 01.10.2006 02.10.2006 365 stochastische Optimierungen 01.10.2007 01.10.2006 02.10.2006 365 deterministische Optimierungen 01.10.2007 8
Storage volume trajectories 100% profit 73.85% profit 73.2% profit MWh 9
Concept of Simulation Analysis of profits / costs z s,n derived from decisions (u 1,..., u S ) for different price processes (p 1,..., p N ) Drawbacks: Different first stage decisions with different price processes p 1,..., p N Anticipativity of decision making t p 1 z s,1 p 2 z s,2 p 3 z s,3 p 4 z s,4 p 5 z s,5 p 6 z s,6... p N z s,n 10
Concept of Stochastic Optimization Unique optimal decision in every node with respect to all possible future developments (1,1) Scenario tree (binary process) (1) Non-anticipativity of the decisions (1,2) (2,1) (1,1,1) Szenario 1 (1,1,2) Szenario 2 (1,2,1) Szenario 3 (1,2,2) Szenario 4 (2,1,1) Szenario 5 (2) (2,2) (2,1,2) Szenario 6 (2,2,1) Szenario 7 t (2,2,2) Szenario 8 11
Storage balance equations in the scenario tree Shadow price of volume is derived from dual variable of volume balance equation of first node Storage Balance equation of period 1, scenario 1 V S1 1 + W S1 1 I S1 1 S1 = V 0-3 -2-1 Scenario tree Path dependent PnL P(Scenario1) P(Scenario 2) Storage Balance equation of period 0, scenario 1 V 0 + W 0 - I 0 = V -1 Value-at-Risk (10%) 0 0.05 0 0.15 0.1 0.25 0.2 0.3 0.4 0.35 0.45 Constant! Initial Volume V -1 Storage Balance equation of period 1, scenario 2 V S2 1 + W S2 1 I S2 1 = V 0 2 1 V. Volume W: Withdrawal I : Injection t 0 t 1 t 2 t 3 3 Min/max Volume constraints for each path of the tree 12
Storage balance equations in the scenario tree Shadow price of volume is derived from dual variable of volume balance equation of first node Storage Balance equation of period 1, scenario 1 V S1 1 + W S1 1 I S1 1 S1 = V 0 V 3 min <= V S1 3 <= V max 3 min <= V 3 min <= V S2 3 <= V max 3 min <= V 3 min <= V S3 3 <= V max 3 min <= Storage Balance equation of period 0, scenario 1 V 0 + W 0 - I 0 = V -1... Constant! Initial Volume V -1 V. Volume W: Withdrawal I : Injection Storage Balance equation of period 1, scenario 2 V S2 1 + W S2 1 I S2 1 = V 0 t 0 t 1 t 2 t 3 V 3 min <= V S8 3 <= V max 3 min <= max 13
Multi-dimensional scenario tree generation Day Ahead price (Market 1) Gas demand Day Ahead price (Market 2) Storage outages FX rate Oil Price indices All trees share the same branching structure but hold different values, according to expectation, volatility, correlations and other parameters of the underlying stochastic processes 14
Structure of the overall mathematical model Shadow price of gas is derived from dual variable of gas balance equation of first node 15
Workflow for the valuation of gas storages / gas portfolios Stochastic impacts: Gas spot prices (i.e. TTF) Gas demand (optionally) Oil price index, FX-Rate (optionally) Historic demand Estimation of volatility/mean reversion (Demand) Stochastic optimization: 1) Scenario trees Historic TTF-spot price Estimation of volatility/mean reversion (Spot prices) (Spot prices and demand) 2) max Objective function Distributions Historc TTF- spot price Calculation of DPFC (Fleten s model) Actual price forward curve 3) Solving the math. model Solver (CPLEX ) Constraints Mean of P&L- Distribution = Value of the Storage 16
70000 60000 50000 40000 30000 20000 10000 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Structure of the mathematical optimization kernel Energy [ MWh / 2 weeks ] Scenario trees scen. tree lower appr. expected load real load scen. tree upper appr Analytical model formulation Periods Stochastic multistage (mixed integer) program Solver (CPLEX ) LP-Solver (Simplex, Barrier) MIP-Solver (Branch-and-Bound) Decisions and P&L Distribution 17
Spot price process: Pilipovic Spot price process for the gas day ahead market: Mean reversion Sigma Price Spot price process: Equilibrium price process: Uncorrelated Brownian motions Sigma Trend 18
Stochastic processes of multiple uncertainties Spot price processes for the Gas-Day-Ahead-Market, Gas Demand, oil price and FX rate: Spot price process: Long term price process: Gas demand: Oil price: FX rate: 19 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: Uncorrelated Brownian Motions
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 branches In each node of the tree, the DPFC is scaled to a certain scenario of the DPFC 12/5/2006 01/22/2007 09/02/2007 12/4/2006 12/25/2006 03/19/2007 06/11/2007 1 day 4 weeks 8 weeks 12 weeks 12 weeks 2 weks DPFC /MWh 20
Scenarios as deviations from DPFC 1. Scenarios of the logarithmic spot price x p for each period of the tree x 0 2. Scenarios of exp(x p 0.5 Var(x p )) exp (x ½ Var(x)) 1 3. Scenarios of DPFC t exp(x p 0.5 Var(x p )) DPFC exp (x ½ Var(x)) DPFC 21
Generation of scenario trees 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 Approximation of the multidimensional standard normal 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 22
Definition of the planning horizon Valuation date: Today Start/End Date: Defines the planning horizon The user can define the number and dates of the scenario tree branching steps 23
Definition of the stochastic spot price process The basis for the spot price scenario generation is a Daily price forward curvecurve) The parameters of volatility (standard deviation), mean reversion and correlation define the generation of the scenario tree 24
25 Set up of gas procurement portfolios (storages)
26 Set up of gas procurement portfolios (contracts)
27 Definition of min/max power of storages and contracts, min/max volume of storages
28 Definition of Forward-Portfolios (Prices, already closed positions)
29 Definition of liquidity and market depth of dayahead gas market
Modeling of gas storages 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 Shrinkage of volume over time Stochastic outages / interruption of transport capacity 30
Modeling of gas markets / traded products 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 Representation of already closed positions 31
Representation of characteristics of flexible supply contracts 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 32
Sensitivity of P&L distribution with different trading strategies Expected profits increase with high share of spot trading 33
Example of gas storage valuation results 34
Gas Storage Valuation based on Least Square Monte Carlo (overall profit over 30 years) 35
Gas Storage Valuation based on Scenario Tree (overall profit over 30 years) 36
37 Sensitivities with short term spot price volatility
38 Sensitivities with short term spot price volatility
39 Sensitivities with long term spot price volatility
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 Representation of uncertainty in the valuation and operation of gas procurement portfolios leads to lower expected procurement costs 40