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Journal of Energy Markets Volume 6/Number 2, Summer 2013 (317)

An equilibrium analysis of third-party access

to natural gas storage

Alan Holland

Department of Computer Science, University College Cork, College Rd, Co.

Cork, Ireland; email: [email protected]

Christopher Walsh

Keelvar Systems, Rubicon Centre, CIT Campus, Bishopstown, Cork, Ireland;

email: [email protected]

(Received November 11, 2011; accepted November 26, 2012)

Underground natural gas storage facilities are vital to the operation of energy

networks.Giventhe inelasticnature ofthe supplycapacityand thehighvolatility of

demand, storage acts as a buffer that facilitates physical delivery at peak demand

periods and also smooths the price of gas. Operational efficiency is important

for the overall cost-effectiveness of gas and electricity networks and is therefore

deserving of study. We study the effects of competition when firms share space,

the injection/withdrawal of resources and game-theoretic behavior. We present a

game-theoretic model in which strategic interaction occurs between firms due to

the interdependencies that arise given the uniform pressure level in the store. In

our model, firms are traders that seek to lock in the intrinsic value of storage at

the outset of the term by buying and selling forward contracts subject to physical

flow constraints. We conduct an empirical analysis using fictitious play and asetup representative of actual facilities and observable forward curve prices. Our

results indicate that large losses can occur during the withdrawal season when

the forward curve has a specific shape. This motivates the need for alternative

economic mechanisms to counteract strategic manipulation.

1 INTRODUCTION

Natural gas is a voluminous commodity that forms an integral part of energy networks

in terms of consumption for domestic and industrial heating as well as combustion

for electricity generation. It is more environmentally friendly than other fossil fuels

because it releases less CO2 than oil, coal or peat-based fuels per unit of electric-

ity generated. The significance of natural gas in Western countries in particular is

increasing for several reasons, including the following.

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4 A. Holland and C. Walsh

Oil production is projected to decrease far more rapidly than natural gas pro-

duction in the coming decade.

A proliferation of wind-powered electricity generation increases demand for

gas-fired generators that are more responsive and offer a shorter ramp-up time.

Technological advances have improved the accessibility of larger and more

distant gas producing fields, thus increasing supply and prompting investment

in more gas-fired generators.

A gas storage facility is limited in terms of space and also the rate at which goods

can be injected or withdrawn from the store. This is sometimes referred to as ratch-

eted injection and withdrawal. The maximum rates of injection/withdrawal typically

depend on the total inventory level within the storage facility. The majority of stores

consist of partially depleted gas fields that have undergone a lengthy conversion pro-

cess that can cost hundreds of millions of dollars. For example, the Rough Field in

the North Sea has the ability to deliver approximately 455 GWh (1.5 billion cubic

feet) of gas per day and a total storage capacity of 30 TWh (100 billion cubic feet) of

natural gas at pressures of over 200 bar. It is currently the largest gas storage facility

in the United Kingdom. It can supply approximately 10% of the current UK peak day

demand and is therefore an important supply buffer.

The inelastic nature of supply patterns offers value in storage because shippers can

buy gas at low prices when demand is low (for example, during summer). As the

demand peaks in winter, the price of natural gas increases, and gas can be released

from storage in order to avoid payment of high spot prices. A key driver of gas storagevalue is the spread between summer and winter prices. Other key reasons for the high

value of storage include

a shortage of geologically suitable sites for storing gas at high pressure,

a gradual exhaustion of gas fields in close proximity to areas of heavy con-

sumption and thus a requirement to transport gas over longer distances, and

increased political uncertainty over transportation pipelines and supply capa-

bility.

Given the enormous value attached to gas storage facilities, energy regulators in most

developed countries have determined that they should not be monopolized and access

should be provided on an open market basis. This entails a situation in which firms are

assigned rights to flow and physical space capacity. The storage owner is faced with

Journal of Energy Markets Volume 6/Number 2, Summer 2013


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Third-party access to natural gas storage 5

a nontrivial decision regarding the choice of an appropriate policy of access rights to

unused flow or space capacity.

We look at a typical setting wherein a storage manager sells these resources so that

each agent holds the same ratio of injection, withdrawal and space resources. During

the term of a storage contract, the agents can then suggest which actions the store

operator should take on their behalf. The deliverability rates depend on the physical

attributes of the store as wellas the pressure.This pressure, however, crucially depends

on other firms decisions. Hence, the strategic interaction is induced by the inability

to partition underground storage among leaseholders.

Decision support tools for storage traders employ sophisticated techniques bor-

rowed from computer science, mathematics and particle physics in an effort to deter-

mine optimal trading and operational policies (Hull 2003; Holland 2008). We assume

that each firm is rational and utility maximizing, thus facilitating a game-theoretic

analysis.This paper is structured as follows. We first present a model of the gas storage

problem. We then focus upon an empirical study of expected losses given realistic

values for the parameters of a storage facility and prices for natural gas forward

contracts. We conclude with a brief discussion of the implications of this work and

how inefficiencies may be resolved.

2 RELATED WORK

A growing literature attempts to quantify the inefficiency of Nash equilibrium prob-

lems in noncooperative games in general. The fact that the system is not fully efficient

is well known and was first quantified by Papadimitriou and Koutsoupias (1999).Subsequently, Roughgarden and Tardos (2002) applied this idea to the network equi-

librium problem in transportation with separable cost functions. In other work, Correa

et al (2005) derived bounds on the inefficiency of Nash equilibria for situations in

which the equilibrium costs are within reasonable limits of uncongested settings.

These bounds help to explain empirical observations in vehicular traffic networks and

related congestion games.

3 STORAGE GAME

We present the normal form representation of a stage game that ensues when one

considers the flow capacities to be allocated on a use-it-or-lose-it basis. Therefore,

there is contention for resources at each interval because when the number of firms

sharing either injection or withdrawal resources increases, the flow capacity available

to each firm decreases.

Research Paper www.risk.net/journal


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FIGURE 1 Congestion game.

Inject

Withdraw

t0 t1 tj

Do nothing

tj1

Do nothingDo nothingDo nothing

WithdrawWithdraw

Inject Inject

A congestion model (I, X, J, .Ai /i2I, .cj;x/j;x2JX) is defined as follows.

I D f1 ; : : : ; I g denotes the set of players.

J X D f1 ; : : : ; J g f1 ; : : : ; Nxg denotes the set of feasible inventory levels

for each period.

For all i 2 I, Ai denotes the set of strategies for agent i , where ai 2 Ai is the

profile of inject/withdrawal actions for firm i over all periods.

For all j; x 2 J X, cj;x 2 Rn denotes the vector of costs, where cj;x.k/ is

related to the cost of inventory in period j , given aggregate inventoryP

i2I xi .

The game associated with this model has a set of strategies, .Ai /i2I, and a cost

function defined as follows. Let A D i2I be the set of all possible deterministic

strategy profiles. For any a 2 A and for any j; x 2 J X, let invj;x.a/ be the

aggregate inventory. Figure 1 represents the set of strategies for an agent over all

periods. We define the overall cost function for i as a function of aggregate inventory:

ui .a/ DX

j2ai

cj.invj;x.a//:

Note that inventory is summated across all firms to determine the congestion levels

at nodes. This cost function is nonseparable. In contrast to transport congestion,

paths through the network indicate inventory levels. Therefore, the function invj;x.a/

depends upon the inventory levels of other firms in period j to determine the cost of

injection or withdrawal.

Figure 2 on the facing page presents an example of realistic flow rates as a function

of aggregate inventory. Unfortunately, the nature of the injection profile indicates that

the cost function is not necessarily convex or differentiable.



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FIGURE 2 Sample flow rates from an underground gas storage reservoir.

60

40

20

0

20

40

60

80

0 20 40 60 80 100

F

lowrate(GWh/day)

Percentage full

Withdrawal capacity (GWh/day)

Injection capacity (GWh/day)

4 SOCIAL PLANNERS MODEL

The allocation problem is a mixed integer program for optimizing the injection/

withdrawal resources across all firms. Binary variables are required because the flowconstraints are activated at discrete intervals that depend upon aggregate inventory.

We thus use indicator variables to determine the active ratchets to constrain injection

and withdrawal.

At interval 0, the objective is to maximize the sum of expected rewards across I

agents over J intervals in the storage contract. We consider two traders and and a

single market for monthly forward contracts for gas that can be used to lock in the

intrinsic value of storage, sj, for all j 2 f1 ; : : : ; J g. A social optimum is to maximize

the sum of payoffs to all agents:

max

IX

iD1

JX

jD0

.sji C inj/.inj

ji C / C .s

ji C wit/.wit

ji

NC /; (4.1)

where inj and wit are the marginal costs of injection and withdrawal, respectively.

The variables injji 2 1;0 and wit

ji 2 0; 1 indicate the consumption of injection

and withdrawal resources in each period. The inventory of each firm at every interval



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is xji > 0:

jX

jjD0

.injjji C / C .wit

jji

NC / > 0; 8 j 2 f0 ; : : : ; J g;

For all i 2 f1 ; : : : ; I g, where C and NC are the maximum injection and withdrawal

amounts, respectively. We employ a shared space policy so that the cumulative inven-

tory across all firms does not exceed Nx during each period j :

IX

iD1

jX

jjD0

.injji C / C .wit

ji

NC / > Nx C

IX

iD1

xki ; 8 j 2 f0 ; : : : ; J g;

Standard monthly forward contracts for natural gas stipulate that the commodity is

delivered at uniform flow rates across all days in that month. This, in turn, causes flow

capacity to become a stepwise function of aggregate inventory. The injectability and

deliverability rates are restricted by the tightest ratchet constraints encountered during

the time interval, as seen in Figure 3 on the facing page. By inspecting the aggregate

inventory of the store in the next interval, it is possible to establish the feasibility of

the cumulative action set across all agents I.

4.1 Injection constraints

The injection ratchet constraints indicate the decreasing rate of injection as inventory

increases. With two ratchet points at inventory levels r1 and r2, we partition the

inventory-dependent aggregate flow capacity into three regions:

ifPIiD1 x

jC1i 6 r1 Nx, thenP

IiD1 jinj

ji j < jc1j,

ifPI

iD1 xjC1i > r1 Nx and

PIiD1 x

jC1i 6 r2 Nx, then

PIiD1 jinj

ji j < jc2j,

ifPI

iD1 xjC1i > r2 Nx, then

PIiD1 jinj

ji j < jc3j,

where r1 < r2 and jc1j > jc2j > jc3j. We use binary indicator variables Ri;j to

activate or deactivate relevant flow constraints associated with the specific aggregate

inventory ranges:

IX

iD1

xjC1i .1 R1;j/M 6 r1 Nx;

IX

iD1

injji C .1 R1;j/M > c1;

for all j 2 f0 ; : : : ; J g. Constraints that apply to decreased flow rates as aggregate

inventory increases are defined similarly.



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FIGURE 3 Ratchet points for injection constraints (negative flow) and withdrawal (positive

flow).

100

50

0

50

100

0 20 40 60 80 100

Percen

tageflowcapacity

Percentage full

Withdrawal capacity (% of maximum)

Injection capacity (% of maximum)

We also impose constraints to ensure that only a single ratchet is active at each

interval.

R1;j C R2;j C R3;j D 1; 8j 2 f0 ; : : : ; J g:

The withdrawal constraints follow a similar pattern, with outflow becoming more

constrained as inventory decreases.

5 EQUILIBRIUM ANALYSIS

Fictitious play is a game-theoretic learning convention that informs the expected

pattern of learning for a selfish player (Brown 1951). It assumes that opponents are

playing set strategies that involve mixing. Players examine the frequency of their

opponents actions and best respond in the expectation that their opponents mixed

strategy is fixed. In situations when an opponents strategy is conditional upon their

own perception of fellow players, this learning mechanism may be flawed.

Tennenholtz and Zohar (2009) described a learning equilibrium involving co-

operative players that can lead to a socially optimal outcome. However, the equi-

librium strategy relies on a learn-or-punish approach that is only enforceable in an



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infinitely repeated game. A storage game involves a finite horizon in which defecting

players can leave after the storage term expires. Furthermore, storage leaseholders are

generally competing firms in gas markets and are forbidden to enter into agreements

on trading strategies by the competition rules in most countries.

In this setup we present simplifying assumptions regarding the nonseparability of

the cost function. In each iteration of fictitious play we assume that this function is

separable and evolves between iterations. We lose guarantees on convergence even

though they are present in each stage game given the existence of a potential function

when costs are separable. This is a justifiable relaxation given that gas storage owners

can provide reasonable estimates of pressure levels at various times of the year. We

thus assume that the injection and withdrawal capacity of the compressors are known

at each interval. The cost functions for the facilities at each point are therefore also

known, but each agent is unsure of the number of firms accessing the resource, hence

the contention for flow resources.

5.1 Welfare losses

We consider a symmetric two-agent setting in which each agent has an identical view

of price evolution during the remainder of the storage contract. We use a learning

approach to find an equilibrium in which the agents respond to each other given

their expectation of their opponents actions. We let i denote the identity of the firm

imposing an externality on the other:

max

JX

jD0

.sji C inj/.inj

j;bri C / C .s

ji C wit/.wit

j;bri

NC /: (5.1)

When the agent providing the best response seeks to change their respective flow

plan, they will be restricted by all the aforementioned physical constraints. Addition-

ally, these actions will be constrained by the fixed agents legacy actions.

Our response model uses the following principles to estimate feasibility. When a

responding firm alters a plan, the fixed firms actions may become infeasible due to

an insufficiency of deliverability or injectability in the following period. We assume

that minimal curtailments are imposed on the allocation of flow in order to preserve

feasibility. We omit the full description of the mixed integer programming model for

this response optimization model due to space constraints.

6 EXPERIMENTAL SETUP

In this section, we discuss the choice of store parameters, ratchet profiles, transmission

charges, initial flow allocation and the number of intervals that were used in each stage

of the empirical analysis. In the best response analysis we consider a two-agent model



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TABLE 1 Store specifications.

Store Nx NC C

1 1000 400 300

2 1000 500 300

3 1000 600 300

TABLE 2 Ratchet profiles for each store.

Store Type Inventory Withdrawal Injection

1 I > 0.5 6 0.8

I > 0.8 6 0.5

W 6 0.1 6 0.5

W 6 0.5 6 0.75

2 I > 0.5 6 0.8

I > 0.8 6 0.5

W 6 0.1 6 0.3

W 6 0.45 6 0.6

3 I > 0.5 6 0.8

I > 0.8 6 0.5

W 6 0.05 6 0.25

W 6 0.35 6 0.35

where both agents have an identical view on price evolution over the remainder of

the storage contract.

The store parameters were selected arbitrarily with max inventory Nx D 1000 for all

experiments and the maximum injectability and deliverability (C and NC, respectively)

varied across each of the three stores used in this analysis. The parameters in Table 1

reflect realistic ratchet profiles for actual gas storage facilities.

It was useful from a computational viewpoint to consider just two ratchets of each

type (ie, two injection and two withdrawal ratchets). See Figure 4 on the next page

for a graphical depiction of the feasible action set given these ratchets.

In order for the ratchet model described previously to function optimally and to

eliminate anypotential redundancy betweenconstraints, ratchet points must be spaced

such that no two ratchets are encountered in a single interval. If this were not the case,

an agent could bypass an entire flow region in a single interval with the related flow

constraint never coming into effect.



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FIGURE 4 Ratchet profiles.

1.0

0.5

0

0.5

1.0

0 0.2 0.4 0.6 0.8 1.0

Proportionalflowrate

Proportional inventory

Withdrawal capacity

Injection capacity

-1.0

0.5

0

0.5

1.0

0 0.2 0.4 0.6 0.8 1.0



1.0

0.5

0

0.5

1.0

0 0.2 0.4 0.6 0.8 1.0





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TABLE 3 Transmission charges.

Experiment wit inj

1 0.1 0.2

2 0.5 0.6

3 1.0 1.1

With a store separated into n C 1 distinct flow regions by n number of ratchets, the

minimum number of intervals required to fill the store from empty or, alternatively,

empty the store when full is nC1. When employing two injection ratchets, a minimum

of three days of injection at the maximum permissable capacity is required to fill the

store. Similarly, a further three days of withdrawal at the maximum permissable

capacity is required to empty the store.

An additional interval was included in order to allow withdrawal flow throttlingsuch that firms can better avail of peak spreads by deliberately withdrawing less

at earlier intervals and more at later intervals. It is also commonplace in storage

sites for maximum withdrawal rates to far outstrip maximum injection rates. This

traditionally leads to agents gradually building up reserves over a long period of time

with subsequent extract occurring over a shorter period of time. It is for this purpose

that the number of intervals required to fill the store was increased to five.

The model used in this empirical analysis therefore has nine intervals, which, in

the event of a suitable spread in prices, permits five intervals to fill the store and a

further four to empty it again.

6.1 Transmission chargesFor each store, we analyze three different sets of transmission charges: low, medium

and high. Here we set injection costs above the cost of withdrawal, in keeping with

typical storage conventions, as seen in Table 3. When these charges are said to be low,

inj and wit are set to 0.2 and 0.1, respectively. When they are said to be medium,

these values are set to 0.6 and 0.5. When they are said to be high, we use 1.1 and 1.0.

6.2 Experimental setup

We assume a single injection season wherein agents inject gas until the store is full

because prices are low.Agentsthen choose to switch to a do nothing or withdrawal

mode in the next interval. Our analysis is thus focused upon how firms can time their

withdrawals to avail of high deliverability earlier in a trade-off against the maximal

price.

The possible withdrawal periods represent the months from November to February.

This period is when gas prices are at their highest and most volatile. By taking the



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FIGURE 5 Losses in equilibrium for store 1.

Welfarelossesatequilibrium

0 10 20 30 40 50

November price

0

10

20

30

40

50

Januaryprice

0

2

4

6

8

10

12

December price to represent the peak gas price at 50 we study the effect of varying

prices at either side of this peak value, ie, November and January contracts, where

sji 2 0;: : : ;50 for all j 2 f6 ; : : : ; 9g. In order to reduce the curse of dimensionality

we set the price at each of the last two intervals equal to one another s8i D s9i . The

resulting price profile for each agent looks like s D 0;0;0;0;0;s7;50;s8; s9.The best response is carried out in each iteration by successive agents when the

social optimum is taken as a starting point. This process is repeated until both agents

cannot unilaterally increase their own payoff and thus an equilibrium is reached. It

is not possible to guarantee convergence but our empirical results indicate that in a

setting with two agents, convergence was invariably observed.

6.3 Results

Our experimental results indicate that the ratio of prices within the forward curve

has an enormous impact upon the expected returns from the withdrawal and sale of

gas in storage. Figure 5 shows that a precipitous fall in value is observed when the

November price is slightly higher than the January price. The reasoning behind this

result is straightforward.An agent has an incentive to shift withdrawal from January to

November but also wishes to be patient so that inventories are high during December,

when prices peak. However, given that deliverability in December is reduced when



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and curtail injection. The same selfish incentive to take injectability when available

results in harm to social welfare.

6.5 Magnified impact on storage contract value

It is important to consider that when a 1% reduction in expected returns from the sale

of gas from a storage contract is observed, the value of storage is reduced by a much

larger percentage. For example, wintersummer spreads for gas prices in the United

Kingdom are less than 10% of gas prices. There is, therefore, a loss in value of more

than 10% for a storage contract.

7 CONCLUSION

We described a model of a natural gasstorage game that employs a congestion model to

simulate a contest for injection and flow resources in successive periods. We identifiedrelated results for the price of anarchy in congestion games that are applicable to this

setting. We also described an empirical study in which the players adopt a fictitious

play approach to a game in which the cost functions are learned in successive periods.

For a specific instance, we showed that this converges to an equilibrium.

The problem of strategic manipulation in natural gas storage is significant even

when theprice of anarchyis small in terms of percentage. Theenormous value attached

to these resources and their critical importance in our energy networks demands that

an effort be initiated to recover equilibrium losses due to competition. For example,

approximately US$30bn of gas is stored in North America annually. This paper pro-

vides a step toward calculating the extent of such losses and motivating the design of

more sustainable mechanisms for welfare maximization.

In future work, we aim to identify more specific theoretical results pertaining to

upper bounds on welfare losses for individual stores. Furthermore, we aim to develop

a more complete understanding of the precise losses due to competition in gas stores

given specific market data and physical descriptions of stores.

REFERENCES

Brown, G. W. (1951). Iterative solutions of games by fictitious play. In Activity Analysis of

Production and Allocation. Wiley.

Correa, J. R., Schulz, A. S., and Stier-Moses, N.E. (2005). On the inefficiency of equilibria

in congestion games. In Integer Programming and Combinatorial Optimization, Jnger,M., and Kaibel, V. (eds).Lecture Notes in Computer Science,Volume 3509, pp. 171177.

Springer.

Holland, A.(2008). A decision support tool for energy storage optimization. In Proceedings

of 20th IEEE International Conference on Tools with Artificial Intelligence (ICTAI 2008).



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Hull, J. C. (2003). Options, Futures and Other Derivatives. Prentice Hall, Englewood Cliffs,

NJ.

Papadimitriou, C., and Koutsoupias, E. (1999). Worst-case equilibria. In STACS 99: Pro-

ceedings of the 16th Annual Conference on Theoretical Aspects of Computer Science,pp. 404413.

Roughgarden,T., andTardos,E.(2002).Bounding the inefficiencyof equilibria in nonatomic

congestion games. Journal of the ACM 49(2), 236259.

Tennenholtz,M., andZohar, A.(2009).Learning equilibria in repeated congestiongames. In

Proceedings of the 8th International Conference on Autonomous Agents and Multiagent

Systems, Volume 1, pp. 233240. International Foundation for Autonomous Agents and

Multiagent Systems, Richland, SC.



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