Injection/Withdrawal Scheduling for
Natural Gas Storage Facilities
Alan Holland
Cork Constraint Computation Centre
Department of Computer Science
University College Cork, Ireland
[email protected]
ABSTRACT
enabling gas to be stored within the reservoir, thousands
of feet underground or under the seabed and withdrawn to
meet peaks in demand. These facilities do not supply gas
directly to end users (domestic and industrial users) but
acts as a storage facility for gas shippers and suppliers, allowing gas to be fed into a transmission system at times
of peak demand (e.g. winter) or withdrawn from the grid
and re-injected into the reservoir at times of low demand
(e.g. summer). The movement of gas either into or out of
the reservoir is based on “nominations” made by gas shippers as a result of demands placed on them by their end
customers. These facilities have various deliverability rates
depending on their size and physical attributes.
We examine the problem faced by owners of gas storage
contracts of how to inject and withdraw natural gas in an
optimal manner so that gas is injected when prices are lowCategories and Subject Descriptors
est and withdrawn when prices are high. Storage contracts
G.1.2 [Approximation]: Linear approximation; I.2.8 [Artificial are typically of twelve months duration and the storage opIntelligence]: Problem Solving, Control Methods, and Search.; erators must be informed at the outset of each day whether
they should inject or withdraw gas on that day. Gas prices
I.6.8 [Types of Simulation]: Monte Carlo
exhibit a noticeable seasonality each year. We focus upon
the northern European market where prices drop in the sumGeneral Terms
mer as consumption for heating purposes decreases and rise
Experimentation , economics, algorithms.
in the winter as temperatures drop. We model gas prices using a stochastic process and determine the expected-profit
Keywords
maximizing injection/withdrawal for an energy trader who
wishes to decide on injection or withdrawal on a daily basis.
Natural gas storage, scheduling, optimization.
The theory of real options is based on the realization that
many business decisions have properties similar to those of
1. INTRODUCTION
derivative contracts used in financial markets. A natural gas
Our work focuses on gas storage facilities that consist of
well can be thought of as a series of call options on the price
partially depleted gas fields such as the Rough field in the
of natural gas, where the strike or exercise price is the toSouthern North Sea, 18 miles from the east coast of Yorktal operating and opportunity costs of producing gas [4], if
shire. There is a shortage of gas storage facilities worldwide
we ignore operating characteristics. In markets where there
that has contributed to an increase in their value [3]. There
exists a liquid secondary derivatives market, valuable inis, therefore, a greater incentive for optimizing its utilizaformation for the valuation of such “real options” can be
tion given the rising costs in storage contracts. These storreadily garnered. Derivative prices can be used to deterage facilities were generally originally developed to produce
mine the market’s opinion on the probability distribution of
natural gas. Fields can be converted to storage facilities,
future prices. By operating a gas storage facility in the way
that maximizes the expected cash flow with respect to the
market’s view of future uncertainties and its risk tolerances
for those uncertainties, one can subsequently maximize the
Permission to make digital or hard copies of all or part of this work for
market value of the facility itself.
Control decisions for gas storage facilities are made in the
face of extreme uncertainty over future natural gas prices on
world markets. We examine the problem faced by owners of
storage contracts of how to manage the injection/withdrawal
schedule of gas, given past price behavior and a predictive
model of future prices. Real options theory provides a framework for making such decisions. We describe the theory
behind our model and a software application that seeks to
optimize the expected value of the storage facility, given capacity and deliverability constraints, via Monte-Carlo simulation. Our approach also allows us to determine an upper
bound on the expected valuation of the remaining storage
facility contract and the gas stored therein.
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ACM SAC ’07, March 11–15, 2007, Seoul, Korea.
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2.
GAS STORAGE
Difficulties arise when operating characteristics and extreme price fluctuations are included in a pricing model [1].
The exotic nature of storage facilities and gas prices requires
the development of complex methodologies both from the
theoretical as well as the numerical perspective. The operating characteristics of a real storage facility pose a theoretical challenge due to the nature of the opportunity cost
structure. When gas is withdrawn from storage the opportunity to release that gas in the future ceases to exist. Also,
when gas is released the deliverability of the remaining gas
in storage decreases because of the drop in pressure. Similarly, when gas is injected into storage both the amount and
the rate of future gas injections are decreased. The opportunity costs and thus the exercise price varies nonlinearly
with the amount of gas in the reservoir [2].
These facts, coupled with the complicated nature of gas
prices, have serious implications for numerical valuation and
control. There are three common numerical techniques used
in option pricing: Monte-Carlo simulation, binomial/trinomial
trees, numerical partial differential equation techniques. MonteCarlo simulation is the most flexible approach because it can
handle a wide range of underlying uncertainties. However,
it is not ideal for handling problems for which an optimal
exercise strategy needs to be determined exactly, and in particular when that strategy may be non-trivial. Although
imperfect, because of the inaccuracies, this approach is very
popular because it is computationally tractable and accuracy can be improved by allowing more simulations. Price
spikes should be an integral part of any gas market model
and Monte-Carlo simulations are the most robust means of
replicating such price behavior.
Given a Monte-Carlo price simulation, we inspect the generated prices and optimize the injection/withdrawal sequences
retrospectively. For each simulation we have an anticipated
gas price for some day in the future. There remains the problem of deciding the optimal injection/withdrawal schedule
for this simulated price movement. The trader has a choice
of three actions for each day of the year, inject, withdraw or
do nothing. In the northern hemisphere the critical times of
the year are Spring and Autumn.
Energy traders in gas supply companies make decisions
on a daily basis. They must also bear in mind the duration
of their storage contract in this analysis. Storage operators
adopt a “use it or lose it” policy with regard to gas that
remains in storage after the expiry of the contract. It is
therefore ideal to deplete the store at the end of a contract.
The following integer linear program formulation represents
the optimization problem:
max
N
X
pi (Wi dWi − dIi ),
(1)
i=1
where pi is the price on day i, Wi is the max withdrawal
amount on day i, Ii is the max injection amount on day
i, dWi is the decision variable on whether to inject or not,
dIi is whether to withdraw on day i or not. Injection and
withdrawal are mutually exclusive decisions, therefore,
dIk + dWk ≤ 1, ∀k = 1 . . . N.
(2)
Also, we cannot have a negative amount of gas in storage,
so the following capacity constraints apply:
IN V +
N
X
dIk Ik + dWk Wk ≥ 0, ∀j = 1 . . . N,
(3)
k=j
where IN V is the amount currently stored in the facility.
Similarly, we cannot exceed our maximum capacity:
IN V +
N
X
dIk Ik + dWk Wk ≤ M AXCAP , ∀j = 1 . . . N,
k=j
(4)
where M AXCAP is the maximum storage capacity as agreed
in the contract. In this model there are 2N variables and 2N
constraints, where N is the number of days remaining in the
contract. To aid computability, we relax the integrality constraints on the injection and withdrawal decision variables.
This facilitates solving of the model in polynomial time. We
find that in practice, less than 2% of the solutions exhibit
fractional solutions. This indicates that our approximate
solution technique does not seriously affect our results.
This optimization problem is solved many times, once for
each simulated price process. The results of dWi and dIi
are then averaged in order to determine what the best decision for that day i is likely to be. Given that gas suppliers can decide daily on their injection policy, only dI0 and
dW0 are required to know what needs to be done for that
day. Nevertheless, energy traders can see the probability
of certain strategies being optimal for future days given the
status quo. This helps with budgeting and planning for the
gas supplier. We conducted experiments to determine the
scalability of this approach. In a worst case situation, where
N = 365, over 300 simulations can be solved in five minutes.
This is ample time for energy traders who are not pressed
to make daily injection/withdrawal decisions at short notice. In practice, the software tool is required in Autumn
and Spring and contracts begin in April. Therefore, it is
principally used when N < 220, which allows the problems
to be solved much faster. Our results and discussions with
energy traders indicate that the do-nothing policy is often
overlooked by traders and in fact should be adopted more
frequently. This tool helps to illustrate how in certain circumstances, a policy of inaction is best.
3. CONCLUSION
Stochastic optimization of gas storage facilities enables
gas suppliers to schedule injection and withdrawal over the
duration of a storage contract in a manner that maximizes
expected profitability. We utilized a mean-reverting price
model that incorporates diffusion and jump components and
presented an ILP formulation of the optimization problem
and found that the linear relaxation can solve 300 simulations, given a contract of 365 days, in just under five
minutes. We showed that in practice fractional solutions
have a minor impact on solution accuracy.
4.
REFERENCES
[1] Hyungsok Ahn, Albina Danilova, and Glen Swindle.
Storing arb. Wilmott, 1, 2002.
[2] Mike Ludkovski. Optimal switching with application to
energy tolling agreements. PhD thesis, Princeton
University, 2005.
[3] Ken Silverstein. More storage may be key to managing
natural gas prices. PowerMarketers Industry
Publications, October 2004.
[4] Matt Thompson, Matt Davison, and Henning
Rasmussen. Natural gas storage valuation and
optimization: A real options application. preprint.