Stochastic Programming in Gas Storage and Gas Portfolio ... · in Gas Storage and Gas Portfolio Management ÖGOR-Workshop, September 23rd, 2010 Dr. Georg Ostermaier. ... – Determinsistic

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

2

• 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

3

– 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

4

– 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

6

• 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

7

€\

Example: Storage Operation under consideration of future spot price uncertaintyStochastic vs. Deterministic optimization

01.10.200601.10.2006

01.10.2007

365 stochastische Optimierungen

8

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

9

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



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





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

23

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



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


VV33minmin <= <= VV33

S1S1 <= 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



. . .. . .



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

16

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)

24

The parameters of volatility (standard deviation), mean reversion and correlation define the generation of the scenario tree

Set up of gas procurement portfolios (storages)

25

Set up of gas procurement portfolios (contracts)

26

Definition of min/max power of storages and contracts,min/max volume of storages

27

Definition of Forward-Portfolios (Prices, already closed positions)

28

Definition of liquidity and market depth of day-ahead gas market

29

• 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

32

Sensitivity of P&L distribution with different trading strategies

33

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)

35

Gas Storage Valuation based on Scenario Tree(overall profit over 30 years)

36

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

Documents

Stochastic Programming in Gas Storage and Gas Portfolio ... · in Gas Storage and Gas Portfolio Management ÖGOR-Workshop, September 23rd, 2010 Dr. Georg Ostermaier. ... – Determinsistic