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Diagnostic Metrics for The Adequate Development ofEfficient-Market Baseload Natural Gas Storage Capacity

By Ernesto Guzman∗

This paper develops four metrics to diagnose the adequate de-velopment of baseload natural gas storage capacity anticipated foran efficient market. They are based on the market equilibrium ofbaseload natural gas storage operations under perfect competitionand monopoly environments. These storage operations are simu-lated using intertemporal choice models. The four diagnostic met-rics can help FERC monitor potential and unintended deterrenteffects of their own regulatory policies on midstream infrastruc-ture development. In addition, the seasonal market price predictedby my model helps explain the shape of the Henry Hub forwardcurve. This paper fills a current gap in the economic literature ofcommodity storage by addressing natural gas demand seasonalityin structural models.

The Federal Energy Regulatory Commission (FERC) faces conflicting facul-ties granted by the Natural Gas Act (NGA) and the Federal Power Act (FPA)concerning the development of interstate natural gas midstream infrastructure –namely, pipelines, storage, and LNG facilities. On one hand, under the NGA,FERC has jurisdiction over the siting and abandonment of midstream infrastruc-ture and reviews applications for its construction and operation (FERC, 2016).In addition, FERC has authority over the ratemaking policies of midstream ser-vices – namely, transport and storage – which are the sources of return on capitalinvestment. On the other hand, under the FPA, FERC has the responsibility topromote the development of robust, reliable, and secure midstream infrastruc-ture as this is critical to ensure that natural gas supply can reach market areas(FERC, 2014). These conflicting faculties require FERC to check that their ownregulatory policies on midstream infrastructure development and rate schedulesare not acting as potential deterrents of efficient-market development. In particu-lar, FERC has attempted to check on the adequate efficient-market development1

of baseload natural gas storage, which helps moderate prices over the seasons.As shown in Wright and Harvey (2006), FERC empirically evaluates seasonal

storage sufficiency based on historical trends of inventory levels, storage capacity,and marginal benefit of baseload storage.2 To gain insights, FERC contrasts

∗ Guzman: Colorado School of Mines, Golden, CO, [email protected] this paper, adequate or sufficient development refers to that anticipated for an efficient market.2FERC estimates the marginal benefit of baseload storage as the difference between the natural gas

price averages realized during withdrawal and injection periods. This estimate is not accurate and worksas a market signal proxy only. The proper estimate is shown in expression (44) and discussed in theparagraph immediately following it.

1

2 SEPTEMBER 2017

these domestic trends against those from European countries. This comparableanalysis approach, however, is significantly affected by the heterogeneous nature ofindustry and market environments in different countries. Fundamentally, storagecapacity adequacy cannot be completely assessed without an identification of thedrivers behind these trends and the proper quantification of their effects.

To monitor the effect of regulatory policies on storage development, it is neces-sary to have diagnostic metrics for the assessment of storage capacity adequacy(or lack thereof). Unfortunately, such diagnostic metrics are not currently foundin the literature. Having them could help policy makers determine whether bil-lions of dollars in storage capital investments must be incentivized over the years.This research looks to fill this gap by formulating them. In addition, this researchfills a current gap in the economic literature of commodity storage by addressingnatural gas demand seasonality in structural models.

In this research, intertemporal choice models are built using an optimal con-trol approach. The models solve for baseload natural gas storage operationshandling seasonal demand in one year under two market environments – per-fect competition and monopoly – and varying levels of binding storage capacity.Model outputs capture how natural gas market prices, inventories, consumption,and storage flows, all as endogenous results, respond to these changing scenar-ios. Four diagnostic metrics are developed using the parametric results of thetheoretical storage operations. These diagnostic metrics are based on (1) marketequilibrium of storage capacity investment, (2) maximum seasonal price spread,(3) correlation between price and consumption, and (4) correlation between priceand inventories.

In this paper, Section I begins identifying the uses of natural gas storage. Sec-tion II contains a discussion on the scarce literature available on the subject ofinterest. Section III introduces the base of my intertemporal choice models. Sec-tion IV tailors the base model to a scenario absent of storage operations. SectionsV and VI formulate the models for non-binding and binding storage capacity op-erations, respectively, and present the main results and findings. Section VIIcompares qualitatively the market responses predicted by the competitive modelagainst those observed in the market. Section VIII develops the four metrics usedto diagnose the adequate development of baseload natural gas storage capacityanticipated for an efficient market. Last, Section IX concludes.

I. Natural Gas Storage

Generally, natural gas storage has three uses: (1) to enhance reliability of sup-plies, (2) to substitute transmission services temporarily, and (3) to match deliv-eries with fluctuating seasonal, daily, or hourly demand.3 The first two uses areoriented to optimize the engineering design of natural gas networks in tandemwith operational arrangements. The first two uses are of interest in engineering

3The first and third uses are identified by Kidnay, Parrish and McCartney (2011).

DIAGNOSTIC METRICS FOR NATURAL GAS STORAGE CAPACITY 3

design but are not relevant to the economic aspects addressed in this research.Under the third use, storage can be either baseload or peak load. Baseload storageprovides a large volume for a long steady injection period (typically 5-7 months),followed by a shorter withdrawal period (typically 3-5 months).4 In the U.S., gasinjection and gas withdrawal typically occur during the Apr-Oct and Nov-Marperiods, respectively. In contrast to baseload storage, peak-load storage balancesgas volumes over a short time frame of hours or days. Peak-load storage alsomeets unanticipated and sudden demand increases. Natural gas generally re-quires voluminous storage unless pressurized or liquefied. Large quantities of gasfor baseload make use of underground facilities that operate at elevated pressures.Most of these facilities are constructed in depleted oil and gas fields, salt caverns,and aquifers. According to EIA (2008), there are about 400 underground storagefacilities in the U.S.: 326 depleted natural gas or oil fields, 43 aquifers, and 31salt caverns. Because baseload storage drives the bulk volume of storage capacity,this type of storage is the one of interest in this research.

II. Literature Review

While the economic literature on commodity storage is ample, my researchobjective has no direct precedent in this literature. That is, the formulation ofdiagnostic metrics to assess the adequate development of baseload natural gasstorage capacity anticipated for an efficient market has not been attempted un-til now (as best as I know). In addition, the related literature and my researchdiverge on the market features of interest. While the related literature mainlyfocuses on supply shocks (e.g. in agricultural commodities) and demand shocks(e.g. in oil and gas), my research focuses on the deterministic oscillation of sea-sonal demand (in natural gas). Because the seasonal component of natural gasdemand is dominant and mostly deterministic, my research on adequate baseloadnatural gas storage capacities ignores market shocks.5 Regarding seasonal de-mand as a market feature of interest is rarely seen in the literature though. Thus,just like my research objective, my research model has no direct precedent either.In fact, less than a handful of works can be found in the economic literature ofcommodity storage addressing seasonality in some form using structural models.Pyatt (1978) is perhaps the earliest reference in this rare literature. Althoughmy model formulation differs from Pyatt’s in numerous aspects,6 Pyatt identifies

4Natural gas is primarily used for heating and power generation. Greatest demand is seen in thewinter when heating demand peaks. Demand can be significant in the summer when power generationpeaks.

5“The main effect of storage on seasonality can be seen under certainty.” (Williams and Wright, 1991)6To begin, the intertemporal choices in Pyatt’s model are concerned with minimizing costs via pro-

duction scheduling. On the other hand, the intertemporal choices in my model are concerned withmaximizing profits via storage operations. Given these operational policies, Pyatt’s model accounts forthe costs of production while my model assumes them away as implied by a perfectly inelastic supply. Inaddition, given our distinct research objectives, Pyatt’s model constrains production capacity while mymodel constrains storage capacity. Last but not least, Pyatt’s method of solution is based on the Eulercondition, which is a limiting and dated mathematical technique relative to the contemporary maximum

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three stages in production scheduling – namely, (a) increasing, (b) decreasing,and (c) zero stocks, which are echoed in the set of four stages I later identify.After Pyatt (1978), interest in characterizing seasonal demand fades away.

Williams and Wright (1991) is a classic textbook on structural models of com-petitive equilibrium for storers subject to random shocks. Williams and Wrightfocus mostly on agricultural commodities. Although not characteristic of agricul-tural commodities, they briefly cover the subject of demand seasonality. Theirnumerical model results show that seasonality in price and production is signifi-cantly reduced (almost flattened) when storage modulates the seasonal demandof a commodity. This observation is reflected by my parametric solution for priceunder perfect competition shown later in Table 1.

Schroder-Amundsen (1991) develops a linearly integrated natural gas modelincluding capacity-constrained production, storage, and distribution where de-mand fluctuates seasonally. The purpose of this model is to determine the effectsof storage facilities on market equilibrium prices (peak and off-peak), both withinthe cycle of a year and over several years. The overlapping capacity constraintsin this model, however, mask the individual binding effects across the gas supplychain. In addition, the numerical results cannot be easily used to discern theeffects of individual capacity constraints. Given these modeling complexities, hisanalyses of the model output are limited and fall short of general insights.

Urıa and Williams (2007) build an elaborate econometric model based on astructural setup to investigate whether the net injection profiles of natural gasstorage facilities in California over the period 2002-2006 are influenced by in-tertemporal price signals from the NYMEX futures market. They validate thisinvestigation and find that the timing and magnitude of the net injection profilesvary largely across facilities, due to the differences in the regulatory and physicalconstraints. Although these findings may inform operational decisions, they donot provide structural insights that can help determine the degree of adequatecapacity offered by the natural gas network in place.

Beyond the four aforementioned references, no other significant work was foundin the economic literature of commodity storage addressing natural gas seasonalityin structural models. The research that continues should contribute to fill thisgap in the literature.

III. Model Introduction

In my model, an economic agent operates natural gas storage over time with agoal that is specific to the economic environment embodied by the agent. Underperfect competition, the agent is a social planner whose goal is to maximize socialwelfare.7 Under monopoly, the agent is a monopolist whose goal is to maximize

principle used by my method of solution. Consequently, my research results provide insights that aredifferent from Pyatt’s.

7First and second welfare theorems imply that a competitive equilibrium is a social optimum andvice versa. Also, according to Williams and Wright (1991), rational competitive storers are collectively


profits. The agent (or storage operator) simply chooses how much flow of naturalgas to inject into or to withdraw from storage over time to meet the correspond-ing goal. Given the intertemporal choices faced by the agent in this dynamicoptimization, the model is formulated as an optimal control problem where thechoice variable is storage flow u, the state variable is natural gas inventory N ,and the goal changes with the nature of the agent (or market environment). Un-der my sign convention, storage flow u is positive when (purchased) natural gasis injected into storage and negative when (sold) natural gas is withdrawn fromstorage.

The planning horizon for intertemporal choices is one year, which capturesone complete seasonal cycle. Over the course of the year, the agent uses baseloadstorage to modulate supply to the seasonal demand. Producers supply the marketwith a constant flow Q0. This means that supply flow is perfectly inelastic andconstant, which is a fair assumption of producer behavior over the one-year periodevaluated. Arguments supporting this assumption are later introduced in SectionIII.B. No inventory is carried over from one year to the next. Therefore, eachcycle is completely independent from all others. Because seasonal demand isanticipated in the short term, the model is deterministic for operational purposes.

A. Seasonal Demand

Seasonal (inverse) demand is represented by the linear form,

(1) Pt(QDt ) = A · St −B ·QDt .

where price Pt is a function of quantity demanded QDt . Subscript t is a contin-uous time index over planning horizon T .St is a T -periodic function that oscillates around 1.0. A is the reservation price

when St = 1. Reservation price (A · St) changes over the seasonal cycle andshifts the demand curve up and down. B is a constant (inverse) demand slope.Seasonality factor St is represented by the sinusoidal form

(2) St = 1− a · sin bt.

Amplitude a determines the magnitude of the seasonal peak and trough factorsto be (1 + a) and (1− a), respectively. a ∈ [0, 1] implies St ∈ [0, 2]. b is the factornormalizing the planning horizon over one seasonal cycle and is simply

(3) b =2π

T.

For example, a seasonal demand with a change of up to 50% over/under the annual

equivalent to a benevolent planner.

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average is represented for a 12-month planning horizon with a = 0.5, b = 2π12 = π

6

and, thus, St = 1− 0.5 · sin(π6 · t

). Figure 1 illustrates the seasonality factor (St)

for this numerical example. Figure 1 also illustrates factor (1− St), or (a · sin bt),which is the deviation from one unit. In Figure 1, the symmetry of the sinusoidalform implies that the low-demand season lasts as long as the high-demand season.In the actual U.S. seasonal cycle, however, the number of low-demand months(Apr-Oct) outnumbers the number of high-demand months (Nov-Mar). Althoughthe symmetric form in the model diverges from the asymmetric form of the actualseasonal cycle, the symmetric approach facilitates the derivation of a simplerparametric solution that can provide greater insights.

Figure 1. Seasonality factor.

B. Constant Supply

Natural gas production from wells in place, either as main or associated product,generally does not respond to short-term price changes because production isset to optimize ultimate hydrocarbon recovery. Deviating from the technically-optimal production path can be detrimental to the formation and to ultimatehydrocarbon recovery. Anderson, Kellog and Salant (2014) show empirically that“crude oil production from existing wells in Texas does not respond to current orexpected oil prices.” Under this argument, natural gas supply can be assumed tobe perfectly inelastic over the short-term horizon of the model (one year).

Natural gas production for most mature wells is approximately steady. There-fore, if new wells added to a region over the course of a year represent a small


fraction relative to mature wells already in place then regional production can beassumed to be steady in a year. Under this condition, natural gas supply can beassumed to be not just perfectly inelastic but also equal to a constant Q0 overthe planning horizon. This constant and perfectly inelastic supply assumption isused in the model to ease the analytical derivations that follows.

C. Supply and Demand Equilibrium

Figure 2 illustrates the supply and demand diagram over the course of thedynamic optimization faced by the storage operator. As indicated before, naturalgas supply is assumed to be perfectly inelastic and equal to constant Q0 over time.Supply is represented by the vertical bar in the diagram. On the other hand,natural gas demand is assumed to be linear, downward sloping, and cyclicallyshifting over time as illustrated by the four-arrow sequence in Figure 2.8 Overthe seasonal cycle, the agent can inject a fraction of Q0 into storage or withdrawfrom storage and add to Q0. Storage flow u is positive when injected into storageand negative when withdrawn from storage. To fulfill market clearing, quantitydemanded QDt is simply equal to the difference between constant supply Q0 andstorage flow u. This expression is also known as the accounting identity,

(4) QDt = Q0 − u.

Figure 2 shows storage flow u as injected (I) during the low-demand season andwithdrawn (W) during the high-demand season. In Figure 2, storage flow u can beobserved graphically as a deviation from constant supplyQ0 because u = Q0−QDt .Market equilibrium price realized over time is simply

(5) Pt(QDt)

= Pt (Q0 − u) = A · St −B · (Q0 − u) .

D. Profit and Welfare

Ignoring costs of injection and withdrawal, costs of storage, and interest rates,9

the profit at instant t for the operating agent is

(6) πt = −u · Pt(QDt)

= −u · Pt (Q0 − u) .

Integrating πt over the planning horizon yields total profits,

8The four arrows show that reservation price decreases from A down to (1 − a) · A over the firstquarter of the time horizon, increases back to A over the second quarter, further increases to (1 + a) ·Aover the third quarter, and decreases back to A over the fourth quarter.

9Interest rates are ignored in this model (assumed to be zero) given the short time horizon of theone-year operation. Consequently, the discount factor is simply one over the course of a year.

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Figure 2. Seasonal demand, perfectly inelastic supply, and storage flow.

(7) π =

∫ T

0πt dt =

∫ T

0−u · Pt (Q0 − u) dt.

Welfare at instant t is reflected by the consumer benefit at instant t measured bythe area underneath the demand curve up to quantity demanded QDt as

(8) Wt =QDt2·[A · St + Pt

(QDt)].

After substituting QDt (4) and Pt (5), welfare can be expressed as a function of(Q0 − u).

(9) Wt = A · St · (Q0 − u)− B

2· (Q0 − u)2


Integrating Wt (9) over the planning horizon yields total welfare.

(10) W =

∫ T

0Wt dt

Next, Section IV tailors this base model to a scenario absent of storage op-erations. In progressive complexity, Sections V and VI setup the deterministicoptimal control problem for non-binding and binding storage capacity operations,respectively.10 Both sections present the market equilibrium of storage operationsunder monopoly and perfect competition. This brings the total number of modelsin this paper up to five. The parametric solutions from these five models includemarket equilibrium price, consumption, profit, welfare, storage flow, inventories,and minimum non-binding storage capacity. These five parametric-solution setsprovide the grounds to develop the sought-after diagnostic metrics.

IV. No Storage Operation

When there is no storage operation (NSO), quantity demanded QDt is simplyequal to constant supply Q0 and market equilibrium price PNSOt is

(11) PNSOt = Pt (Q0)

where

(12) Pt (Q0) = A · St −B ·Q0.

Because there is no storage operation, there can neither be inventories (Nt = 0 ∀t)nor profits (πt = 0 ∀t). Welfare at instant t is simply the area underneath thedemand curve as it fluctuates over time bound on the right hand side by constantsupply Q0. Total welfare under no storage operation WNSO can be found afterreplacing QDt with Q0 in (8) and integrating as follows.

WNSO =

∫ T

0Wt dt =

∫ T

0

Q0

2[A · St + Pt (Q0)] dt =

∫ T

0

Q0

2[2A · St −B ·Q0] dt

Integrating the last expression above and substituting∫ T

0 St dt = T ,

(13) WNSO = T ·Q0

[A− B

2·Q0

].

The expressions found in this section are later referenced in Sections V and VI

10Storage capacity is binding when inventory level Nt is bound by nominal capacity N at some timet. Storage capacity is non-binding when inventory level Nt is unbound for any time t.

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where storage operations are in place (non-binding and binding, respectively).

V. Non-binding Storage Capacity Operations

When storage capacity is not binding, the optimal control problem setup forstorage operations under perfect competition is a social planner maximizing wel-fare W (10), or

(14) maxu

∫ T

0Wt (u) dt,

and under monopoly is a singular private agent maximizing profits π (7), or

(15) maxu

∫ T

0πt (u) dt,

each subject to the equation of motion11 and terminal conditions12

(16) N = u, N (0) = 0, N (T ) = 0

where storage flow u is the control variable and inventory N is the state variable.Table 1 shows the parametric solutions under the two market environments

evaluated – monopoly and perfect competition13 – numbered from (17) through(30).

Descriptions of the results in Table 1 are now in order. In the first line, Q0

bounds must be met to ensure interior solutions. Qb is a constant referred to as thebase flow for optimal monopoly storage. Storage flow u∗ (17) checks the intuitionthat flow would be stored when seasonal demand is low

(t < T

2 → St < 1 and u∗ > 0)

and withdrawn when seasonal demand is high(t > T

2 → St > 1 and u∗ < 0).

Market equilibrium storage flow under perfect competition u∗∗ doubles its monopolycounterpart u∗ as shown in (18). Under the price line, a new term called neu-tral price P0 is introduced. P0 is the initial price reached before any seasonalfluctuation begins, expressed as

(31) P0 (Q0) = A−BQ0.

In theory, introducing monopoly storage operations to the market reduces pricefluctuation by a half. On the other hand, introducing perfectly competitive stor-age operations flattens out price completely to a constant – see (22). For inventory,both N∗

t (25) and N∗∗t (26) reach a maximum when t = T/2. These maximum

11Over time, storage flow u changes inventory level N according to the expression N = u.12Inventory is set to zero at both ends of the planning horizon as no inventory is carried over cycles.13One star and two stars as superscripts of the variables denote market equilibrium results under

monopoly and perfect competition environments, respectively. When the superscript is omitted, thevariable becomes a placeholder for both environments – still under market equilibrium.


levels are called the minimum non-binding storage capacities, denoted as SC∗ un-der monopoly and SC∗∗ under perfect competition. Both SC∗ (23) and SC∗∗ (24)are proportional to seasonal variation (SC ∝ a) and reservation price (SC ∝ A)and inversely proportional to the linear (inverse) demand slope

(SC ∝ 1

B

). The

minimum non-binding storage capacity under perfect competition SC∗∗ doublesits monopoly counterpart SC∗ as shown in (24). Economic profits achieved bya monopolist from storage operations are shown in π∗ (27). As anticipated, noeconomic profits are achieved under perfect competition as shown in πs (28).

The welfare increase from no-storage operation to non-binding monopoly stor-age operation can be estimated as the difference (WM −WNSO), or (29) minus(13).

(32) WM −WNSO =3

4BT ·Q2

b

Analogously, the welfare increase from no-storage operation to perfectly compet-itive storage operation can be estimated as the difference (W ∗∗

s −WNSO), or (30)minus (13).

(33) W ∗∗s −WNSO = BT ·Q2

b

Comparing (32) and (33) shows that the welfare increase from no-storage opera-tion to monopoly operation (32) is three quarters of that realized from no-storageoperation to perfectly competitive storage operation (33), or

(34) WM −WNSO =3

4(W ∗∗

s −WNSO) .

Equations (33) and (27) are later referenced in Section VI.C.

VI. Binding Storage Capacity Operations

Binding storage capacity adds to the optimal control problem setup introducedin Section V two inventory (N) constraints: an upper constraint

(N ≤ N

)and

a lower constraint (N ≥ 0), where N is the nominal storage capacity and theminimum inventory level is zero. Table 2 shows the parametric solutions under thetwo market environments evaluated – monopoly and perfect competition. SectionVI.A outlines a solution procedure to use Table 2. Section VI.B introduces anumerical example to illustrate the results graphically. Section VI.C describes anapplied procedure to find the marginal benefit of storage capacity (MBSC). Last,Section VI.D illustrates the long-term market equilibrium of storage capacityinvestment, which is the procedural base of the first metric introduced in SectionVIII.

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In Table 2, four stages are identified over the planning horizon for both perfectlycompetitive and monopoly operations when storage capacity is binding. Theyare (A) stage of injection, (B) stage of binding storage capacity, (C) stage ofwithdrawal, and (D) stage of stockout.14 Four time delimiters, τ1 through τ4,define the bounds of each stage.

Table 2 shows the equilibrium solutions by stage found for endogenous price P ,storage flow u (choice variable), inventory N (state variable), and shadow value ofinventory θUt . To apply these solutions, the four unknowns (τ1, τ2, τ3, τ4) needto be solved first. As shown later in Section VI.A, they can be solved on the basisof normalized storage capacity NSC (≡ N/SC), which is simply nominal storagecapacity N normalized by the minimum non-binding storage capacity SC.15 Be-cause they are a function of NSC, time delimiters (τ1, τ2, τ3, τ4) turn out to bethe same under both perfect competition and monopoly when normalized storagecapacities are the same – when NSC∗ = NSC∗∗.

In Table 2, price in stages B and D are the same under both operational en-vironments and equal to Pt (Q0), which is the price realized as if there were nostorage operation – see (11). On the other hand, in stages A and C, while pricebecomes flat under perfect competition

(P I , PW

), price changes with storage

flow under monopoly. Table 2 shows that price under monopoly is the averageof price under perfect competition and price under no storage operation. Forboth operations, there is no storage flow (u∗ = 0 and u∗∗ = 0) in stages B and D.On the other hand, storage flow is injected (positive) in stage A and withdrawn(negative) in stage C.16 Storage flow under monopoly is half of storage flow underperfect competition (u∗ = u∗∗/2). For both operations, there is no inventory instage D (ND = 0) and storage capacity binds in stage B

(NB = N

).17 For both

operations, the shadow value of storage capacity θUt is equal to the rate of changein price realized when there is no storage operation,

(35) P (Q0) = −aAb cos bt,

and is only present in stage B. When NSC∗ = NSC∗∗, the shadow values ofstorage capacity are equal in value under both operations

(θU∗t = θU∗∗

t

).

14These four stages are the result of superposing the Karush-Kuhn-Tucker (KKT) conditions intro-duced by the pair of upper and lower storage capacity constraints.

15NSC is defined as NSC∗ ≡ N/SC∗ under monopoly and as NSC∗∗ ≡ N/SC∗∗ under perfectcompetition – See Equation (36). SC∗ and SC∗∗ are defined in (23) and (24), respectively.

16This is why Stages A and C are called the stage of injection and stage of withdrawal, respectively.17This is why Stages B and D are called the stage of binding capacity and stage of stockout, respectively.


A. Solution Procedure

To apply the solutions shown in Table 2, the four unknowns (τ1, τ2, τ3, τ4)need to be solved first. To begin, normalize the nominal storage capacity N using

(36) NSC ≡ N/SC

where SC is equal to SC∗ (23) under monopoly and SC∗∗ (24) under perfectcompetition. Time delimiter τ1 (∈ [0, T4 ]) can then be solved iteratively from

(37) g (τ1) = NSC

where

(38) g (τ1) ≡ cos bτ1 − b(T

4− τ1

)sin bτ1.

To find the remaining three time delimiters, the resulting τ1 can be substitutedinto the following equations.

(39) τ2 =T

2− τ1

(40) τ3 =T

2+ τ1

(41) τ4 = T − τ1

Supplementing the expressions in Table 2, use (1), (4), and (35) to solve pricesPt(QDt), quantity demanded QDt , and the shadow value of storage capacity θUt ,

respectively. To solve inventories, use the inventory cumulative function f (τi, t),

(42) f (τi, t) ≡ −aA

bB[b (t− τi) (sin bτi ) + (cos bt− cos bτi )] .

The relationship between prices in stages A and C for both environments is givenby

(43) P0 (Q0)− Pτ1 (Q0) = Pτ3 (Q0)− P0 (Q0) .

For both operations, the marginal benefit of storage capacity (MBSC) can be

14 SEPTEMBER 2017

solved as a price difference

(44) MBSC = Pτ3 (Q0)− Pτ2 (Q0) ,

or as a function of τ1

(45) MBSC (τ1) = 2aA · sin bτ1 .

MBSC, expressed as the price difference (44), supports the intuition that asocial planner would arbitrage any price difference away through storage oper-ations as long as storage capacity is not binding. Consequently, under perfectcompetition and non-binding storage capacity, price becomes flat.

B. Numerical Example

Applying the procedure previously described, storage operations when normal-ized storage capacity is 25% (NSC = 25%) are solved using the model inputvalues shown in Table 3. The results of the simulation are illustrated in Figure3 for perfect competition and Figure 4 for monopoly. Figure 3 and Figure 4show the resulting profiles for (a) price, (b) inventory, (c) storage flow, and (d)shadow value of storage capacity. These profiles (25% NSC) are contrasted againstthose obtained when there is no storage (0% NSC) and when storage capacity isnot binding (100% NSC).18 The four vertical dashed lines are time delimiters(τ1, τ2, τ3, τ4) separating stages A, B, C, and D. The qualitative descriptiongiven before for the solutions in Table 2 can be visually validated in Figure 3and Figure 4, respectively. The profile of quantity demanded

(QDt), which is not

shown in these figures, is simply an upside-down mirror image of storage flowprofile (c) where the horizontal axis is Qo rather than zero – as QDt = Q0 − u.

C. Finding the Marginal Benefit of Storage Capacity (MBSC)

As shown in Figure 3(d) and Figure 4(d), the profile of the shadow value ofcapacity θUt is the same under both perfect competition and monopoly giventhe same level of NSC. Thus, the MBSC for different levels of NSC can befound as the area underneath the profiles of θUt from either perfect competitionor monopoly environments. Alternatively, a rating table for τ1 can be setup fordiscrete intervals in the domain τ1 ∈ [0, T4 ] as shown in Table 4. For each τ1,NSC is calculated as NSC = g (τ1) using (38). MBSC is calculated using (45).Nominal storage capacity N is estimated as a function of NSC under monopoly

18The solution profiles under no storage operations (0% NSC) are estimated using the expressionsintroduced in Section IV. The solution profiles under non-binding storage capacity operations (100%NSC) are estimated using the expressions introduced in Table 1.


as

(46) NM

= NSC · SC∗

and under perfect competition as

(47) NS

= NSC · SC∗∗,

where SC∗ and SC∗∗ are defined in (23) and (24), respectively.

Table 4 shows the results while Figure 5 illustrates the MBSC curves for bothstorage operations. The area underneath the MBSC curve for social planner isthe welfare added by storage operations under perfect competition. Accordingly,this area can be calculated as W ∗∗

s −WNSO = $150 using (33). On the other hand,the area underneath the MBSC curve for monopolist is the profit generated bystorage operations under monopoly. Accordingly, this area can be calculated asπ∗ = $75 using (27).

D. Market Equilibrium of Storage Capacity Investment

In a static investment model, the intersect of the (annual) marginal benefitof storage capacity (MBSC) and the (annual) marginal cost of storage capacity(MCSC) would determine the market equilibrium storage capacities under bothoperational environments. For instance, assuming a constant MCSC equal to 2 inFigure 5 results in market equilibrium storage capacities of approximately 8 and16 NG units under monopoly and perfect competition, respectively: SC∗

EQ = 8and SC∗∗

EQ = 16.

MCSC can decrease with storage capacity due to economies of scale availableon an individual site basis. However, the right geological conditions are not alwaysavailable for natural gas storage and there can be sites more suitable for storagethan others. On this basis, MCSC can increase with storage capacity as thebest sites for storage are built up first. As MCSC can increase and decrease dueto these factors, assuming MCSC to be flat (constant) on average is within therealm of possibilities and, thus, a fair initial qualitative assumption. In this case,when MCSC is constant, the market equilibrium storage capacities are related bythe expression SC∗∗

EQ = 2 ·SC∗EQ just as minimum non-binding storage capacities

are related by the analogous expression SC∗∗ = 2 · SC∗.

VII. Comparing Predicted and Observed Market Responses

This section compares qualitatively the shapes of the market responses pre-dicted by my intertemporal choice model against those observed in the U.S. mar-ket. Inventories and prices are chosen as the responses of interest because they areat the core of the diagnostic metrics to be developed. Beginning with inventories,

16 SEPTEMBER 2017

Figure 619 shows the U.S. working gas in underground storage over the two yearsprior to February 2017 contrasted against the 5-year maximum and minimum.

Figure 6 illustrates a bell shape for inventories, which is consistent with that il-lustrated in Figure 3(b) under perfect competition and Figure 4(b) under monopoly.Without additional information, shape alone does not indicate the degree of mar-ket competitiveness. One shape feature, however, that can hypothetically provideinformation is the top of the bell. In theory, a flat top flags storage capacity con-straints regardless of competitive environment – see stage B in Figure 3(b) andFigure 4(b). Nonetheless, the examination of a flat top may be justified at aregional level but not at a national level. Case in point, a round top in Figure6 should not be misinterpreted as the absence of storage capacity constraintsbecause nationwide inventory levels will not necessarily show possible regionalconstraints.

Continuing with prices, Figure 720 illustrates two forward curves for HenryHub gas starting at two different times. The first one captures the momentwhen oil was at its most recent prominent peak (6/20/2014) when WTI was$107.26. The second one captures the subsequent dramatic fall off ending ata trough (2/14/2015) when WTI was $48.48. Forward prices rather than spotprices are illustrated to remove stochastic noise from the chart. Forward pricesare the expected spot prices in the future under risk-neutral probability. The twocurves show a clear growth trend and a seasonal component fluctuating around it.The historical WTI timing of the two curves highlights that while forward pricelevels for natural gas can change substantially over time their growth trend andseasonal components do not necessarily change because of it. This means thatthe diagnostic metric to be formulated on the maximum seasonal price spread(in Section VIII.B) is not influenced by changes in levels, which makes it morereliable and robust.

After the growth trend is removed, the shape of the remaining seasonal com-ponent can be evaluated in two sections. As illustrated in Figure 7, the firstsection contains stages D, A, and B while the second section contains stage C.The first section resembles the first half of the price profile illustrated for a com-petitive environment with some degree of storage capacity constraint (25% NSC)in Figure 3(a). The second section resembles the second half of the price profileillustrated for a competitive environment completely constrained (0% NSC) inthe same figure. These two sections are different not only in shape but also inlength. The first section is longer than the second one. Both differences in shapeand length can be explained by the actual asymmetry of the seasonal pattern:the pace of demand change is not the same for every season. As low-demandmonths outnumber high-demand months, injection stage A is widespread allow-ing a continuous injection that modulates and flattens an otherwise natural gas

19Source: EIA Weekly Natural Gas Storage Report for the week ending February 10, 2017,http://ir.eia.gov/ngs/ngs.html.

20Source: FERC Staff Report.


price drop. On the other hand, peak-demand (or withdrawal) stage C is shortand intense,21 which puts significant stress on the overall system capacity beyondstorage. During withdrawal stage C, other system components beyond storage(transmission) could constrain the system and prevent the anticipated price mod-ulation. Releasing this specific system constraint (in withdrawal stage C) maynot be economically feasible under a maximum seasonal price spread that is lessthan 50 cents (See Figure 7). The effects of asymmetric seasonal patterns couldbe evaluated, as follow-up research, by feeding the actual seasonal patterns to anumerical model of the Henry Hub region and others of interest while introduc-ing capacity limits for other system elements besides storage. Although numericalmodels are necessary to address market nuances, the brief qualitative analysis in-troduced shows the significant probing power of the seasonal shapes predicted bymy model.

VIII. Diagnostic Metrics for Natural Gas Storage Capacity

Four diagnostic metrics are now developed using the parametric results foundfor the theoretical non-binding and binding storage capacity operations undermonopoly and perfect competition. The four-diagnostic metrics are based on(1) market equilibrium of storage capacity investment, (2) maximum seasonalprice spread, (3) correlation between price and consumption, and (4) correlationbetween price and inventories. The first metric addresses directly the level ofadequate development. The second metric reads the response from the market inthe form of the maximum seasonal price spread to indirectly determine whetheradequate development is in effect. Metrics (3) and (4) complement the resultsfrom metrics (1) and (2). Metrics (3) and (4) are used to quantify the relativedegree of market power that might be present in different regions.

The use of these diagnostic metrics requires that the model assumptions intro-duced in Section III be valid. In addition, when the natural gas supply Q0 inthe region of interest is constant and inelastic throughout the year, Q0 must bewithin the bounds of an interior solution as indicated in Table 1. In the U.S.,regional demand parameters A and B can be derived from the National EnergyModeling System (NEMS) maintained by the Energy Information Administration(EIA). Marginal cost of storage capacity (MCSC) can be found from regional en-gineering records. If seasonal demand amplitude a is not available directly fromdemand models, it can be initially approximated to the seasonal amplitude of thesum of Cooling Degree Days (CDDs) and Heating Degree Days (HDDs) in theregion22 or be more accurately derived using econometric models.

21The low injection and high withdrawal rates of the seasonal cycle are consistent with the magnitudeof the upward and downward slopes illustrated in Figure 6, respectively.

22CDDs and HDDs measure the energy demand for cooling in warm days and heating in cold days,respectively. CDDs and HDDs are the main drivers of the natural gas demand seasonal pattern.

18 SEPTEMBER 2017

A. Market Equilibrium of Storage Capacity

The first diagnostic metric is based on the theoretical market equilibrium ofstorage capacity (SCEQ) itself. Actual storage capacity (ASC)23 can be comparedagainst the SC∗∗

EQ under perfect competition and SC∗EQ under monopoly. These

two can be determined as a function of the marginal cost of storage capacity(MCSC) using the capital investment model introduced in Section VI.D. IfASC < SC∗

EQ then ASC is theoretically insufficient and further investigationof potential physical constraints in the natural gas infrastructure or regulatorydeterrents would be required. This level of storage adequacy can be qualified witha red flag. If SC∗

EQ ≤ ASC < SC∗∗EQ then ASC theoretically falls in a gray area

between monopoly and perfect competition. In this case, further investigation ofthe nature of the market may be required to determine whether market poweris a factor holding up storage capacity. This level can be qualified with a yellowflag. Last, if SC∗∗

EQ ≤ ASC then ASC is theoretically sufficient. This level canbe qualified with a green flag. Table 5 summarizes the evaluation criteria of thefirst diagnostic metric.

B. Maximum Seasonal Price Spread

The second diagnostic metric is based on the Maximum Seasonal Price Spread(MSPS). The benchmarks for this metric are the theoretical MSPSs under(1) no storage operation, (2) perfect competition, and (3) monopoly. The stepsto obtain these three MSPSs are now described. First, time delimiter τ1 isestimated as half of the time span when storage capacity is binding.24 Next,MBSC (τ1) is estimated using (45) with τ1 as input. According to (44), MSPSis equal to MBSC under perfect competition (subscript PC ), or

(48) MSPSPC = MBSCPC (τ1) .

MSPSPC is the lower benchmark. MSPS under no storage operation (subscriptNSO) is25

(49) MSPSNSO = 2aA.

23In a region with self-contained operations, the identification of actual storage capacity is straightfor-ward as it is all enclosed within the region. On the other hand, an interconnected region would requirestorage allocation. A given region could be served partially by local storage and partially by otherregions’ storage.

24In theory, storage capacity binds in stage B. The time span of stage B is defined by (τ3 − τ2). Inturn, (τ3 − τ2) = 2τ1 and consequently τ1 = (τ3 − τ2) /2.

25MSPSNSO = maxPt (Q0) − minPt (Q0) = [A (1 + a) −BQ0] − [A (1 − a) −BQ0] = 2aA.


MSPSNSO is the upper benchmark. MSPS under monopoly (subscript MP) isthe average of the upper and lower benchmarks, or26

(50) MSPSMP = aA (1 + sin bτ1) .

The actual MSPS (abbr. APS) can be calculated as the difference between themaximum and minimum weekly-running average prices in a year. IfMBSC (τ1) <APS < aA (1 + sin bτ1) then APS falls in a gray area where storage capacity maybe limited by market power. The closer APS is to MBSC (τ1) from above themore competitive the market environment is. If APS > aA (1 + sin bτ1) thenstorage capacity may be held back by potential physical constraints in the natu-ral gas infrastructure or regulatory deterrents. Table 6 summarizes the evaluationcriteria.

C. Correlation Metrics

The two diagnostic metrics that follows can only be used to assess the degree ofmarket power in storage operations. The third diagnostic metric is based on thecorrelation in the short term (one year) between price and natural gas consump-tion. As illustrated in Figure 3(a) and (c),27 price is flat when quantity demandedchanges and vice versa. Thus, under perfect competition, the correlation betweenprice and quantity demanded is theoretically zero

[corr

(Pt, Q

Dt

)= 0]. On the

other hand, as seen in Figure 4(a) and (c), price and quantity demanded changesimultaneously in stages A and C. Thus, under monopoly, the same correlation istheoretically greater than zero

[corr

(Pt, Q

Dt

)> 0].

The fourth diagnostic metric is based on the correlation in the short term (oneyear) between price and inventories. As illustrated in Figure 3(a) and (b), price isflat when inventory changes and vice versa. Thus, under perfect competition, thecorrelation between price and inventories is theoretically zero [corr (Pt, Nt) = 0].On the other hand, as seen in Figure 4(a) and (b), price and inventory changesimultaneously in stages A and C. Thus, under monopoly, the same correlation istheoretically different from zero [|corr (Pt, Nt)| > 0].

The third and fourth diagnostic metrics flag market power in storage operationswhen

(51) corr(Pt, Q

Dt

)> 0

or

(52) |corr (Pt, Nt)| > 0.

Out of the four metrics, the first one provides the most complete diagnosis of

26MSPSMP is the average of (48) and (49). MSPSMP = (MBSC (τ1) + 2aA) /2 = aA (1 + sin bτ1).27Note that quantity demanded is simply an upside-down mirror image of the storage flow profile.

20 SEPTEMBER 2017

storage capacity adequacy. The second metric follows in order. Unlike the firsttwo, the third and fourth metrics can only assess the degree of market power instorage operations. Thus, the last two metrics complement the first two. Thepotential increase in welfare achieved by improving market efficiency can be mea-sured using welfare expressions (32) and (33).

IX. Conclusions

This paper developed four diagnostic metrics to evaluate the adequate efficient-market development of natural gas baseload storage capacity based on economicfundamentals. The four diagnostic metrics are based on (1) market equilibrium ofstorage capacity investment, (2) maximum seasonal price spread, (3) correlationbetween price and consumption, and (4) correlation between price and inventories.The first metric addresses directly the level of adequate development. The secondmetric reads the maximum seasonal price spread to indirectly determine whetheradequate development is in effect. The last two metrics complement the firsttwo metrics. The last two metrics quantify the relative degree of market powerthat might be present in different regions, which might be preventing efficient-market development. Market regulatory agencies like FERC can make use ofthese metrics.

In a separate extension of this paper, seasonal pattern uncertainty (which canbe magnified by climate change) is addressed via the market equilibrium of capitalinvestment and accordingly adjusted in the diagnostic metrics. Future researchcan adjust my proposed diagnostic metrics to account for the effects of asymmetricseasonal patterns. Based on my model insights, I posited that withdrawal ortransmission constraints would explain the asymmetric shape of the Henry Hubforward curve. Future research can test this hypothesis by calibrating my modelnumerically while accounting for storage, injection/withdrawal, and transmissioncapacities. Besides Henry Hub, other markets should also be evaluated.

REFERENCES

Anderson, Soren T, Ryan Kellog, and Stephen W. Salant. 2014.“Hotelling Under Pressure.” NBER Working Paper Series, Paper No.20280(JULY).

EIA. 2008. “About US Natural Gas Pipelines - Transporting Natural Gas.” Web-site.

FERC. 2014. “Strategic Plan: FY2014-2018.” FERC, Washington, DC.

FERC. 2016. “FERC’s Official Website - www.ferc.gov.”

Kidnay, Arthur, William Parrish, and Daniel McCartney. 2011. Funda-mentals of Natural Gas Processing. . 2nd Edition ed., Boca Raton, FL:CRCPress. Taylor & Francis Group.


Pyatt, Graham. 1978. “Marginal Costs, Prices, and Storage.” The EconomicJournal, 88: 749 – 762.

Schroder-Amundsen, Erik. 1991. “Seasonal Fluctuations of Demand and Op-timal Inventories of a Non-Renewable Resource Such as Natural Gas.” Re-sources and Energy, 13(3): 285–306.

Urıa, Rocıo, and Jeffrey Williams. 2007. “The Supply of Storage for NaturalGas in California.” The Energy Journal, 28(3): 31–50.

Williams, Jeffrey C., and Brian D. Wright. 1991. Storage and CommodityMarkets. . 1st ed., Cambridge:Cambridge University Press.

Wright, Jeff, and Steve Harvey. 2006. “Storage Overview.” FERC.

22 SEPTEMBER 2017

Table 1—Market equilibrium solutions to non-binding storage capacity operations.

MarketEquilibrium

Solutions

Storage operationsunder monopoly

Storage operationsunder perfectcompetition

Q0 bounds forinterior solution

Qb ≤ Q0 ≤(AB −Qb

)where Qb = aA

2B

2Qb ≤ Q0 ≤ AB

where Qb = aA2B

Storage flow (17) u∗ = Qb · (sin bt) (18) u∗∗ = 2u∗

Quantitydemanded

(19) QD∗t = Q0 − u∗ (20) QD∗∗

t = Q0 − u∗∗

Price (21) P ∗t = P0 (Q0)−B · u∗ (22) P ∗∗

t = P0 (Q0)

Minimumnon-binding

storage capacity(23) SC∗ =

aA

bB(24) SC∗∗ = 2 · SC∗

Inventory levels (25) N∗t =

SC∗

2(1 − cos bt) (26) N∗∗

t =SC∗∗

2(1 − cos bt)

Profit (27) π∗ =BQ2

b

2T (28) πs = 0

Welfare (29) WM = WNSO +3

4BT ·Q2

b(30) W ∗∗

s = WNSO +BT ·Q2b

Restatedequations

P0 (Q0) = A−B ·Q0 from (31), andWNSO = T ·Q0

[A− B

2 ·Q0

]from (13)


Table 2—Binding operation solutions by stage under perfect competition and monopoly.

Variable Stage A Stage B Stage C Stage D

t ∈ (τ1, τ2) ∈ [τ2, τ3] ∈ (τ3, τ4)∈ [τ4, T ]∧

[0, τ1]

θU∗t , θU∗∗

t 0 P (Q0) 0 0

Perfect Competition

P ∗∗ P I [= Pτ1 (Q0)] Pt (Q0) PW [= Pτ3 (Q0)] Pt (Q0)

u∗∗ 1B

[P I − Pt (Q0)

]0 1

B

[PW − Pt (Q0)

]0

N∗∗ f (τ1, t) N N + f (τ3, t) 0

Monopoly

P ∗ 12

[P I + Pt (Q0)

]Pt (Q0) 1

2

[PW + Pt (Q0)

]Pt (Q0)

u∗ 12u

∗∗ 0 12u

∗∗ 0

N∗ 12f (τ1, t) N N + 1

2f (τ3, t) 0

Table 3—Model input parameters for numerical example.

Parameter NameParameter

ValueParameter Units

Time horizon, T 12 Months

Seasonal amplitude, a 0.5 Unitless

Reservation price, A 20 Price unit

Demand slope, B 2 Price unit over flow unit

Constant supply, Q0 5 Natural gas flow unit

Initial inventory, N (0) 0 Natural gas volume unit

24 SEPTEMBER 2017

Figure 3. Storage operation profiles with 25% NSC under perfect competition.


Figure 4. Storage operation profiles with 25% NSC under monopoly.

26 SEPTEMBER 2017

Table 4—Marginal benefit of storage capacity (MBSC) under perfect competition and

monopoly.

τ1 (months) NSC (fraction) MBSC($/NG)Nominal storage capacity,N (NG Unit)

PerfectCompetition

Monopoly

0.0 1.00 0.0 19.10 9.55

0.5 0.63 5.2 11.98 5.99

1.0 0.34 10.0 6.54 3.27

1.5 0.15 14.1 2.90 1.45

2.0 0.05 17.3 0.89 0.44

2.5 0.01 19.3 0.11 0.06

3.0 0.00 20.0 0.00 0.00

Figure 5. Marginal benefit of storage capacity (MBSC) under perfect competition and

monopoly.


Figure 6. U.S. working gas in underground storage (2015-2016).

Table 5—Comparing actual versus theoretical storage capacity as diagnostic metric.

Where actual storage

capacity (ASC) falls

Storage Capacity Ade-

quacy (Qualification)Potential issues

ASC < SC∗EQ Insufficient (Red flag)

Physical constraints in the nat-

ural gas infrastructure or regu-

latory deterrents.

SC∗EQ ≤ ASC ≤ SC∗∗

EQ

May be subject to market

power (Yellow flag)

Market power or any of the

above.

SC∗∗EQ < ASC Sufficient (Green flag)

Excessive investment that may

not be recovered.

28 SEPTEMBER 2017

Figure 7. Henry Hub forward curve at a recent peak and trough of oil price.

Table 6—Comparing actual versus theoretical seasonal price spreads as diagnostic metric.

Where actual seasonal

price spread (APS) falls

Storage Capacity Ade-

quacy (Qualification)Potential issues

APS > aA (1 + sin bτ1) Insufficient (Red flag)

Physical constraints in the nat-

ural gas infrastructure or regu-

latory deterrents.

MBSC (τ1) ≤ APS ≤aA (1 + sin bτ1)

May be subject to market

power (Yellow flag)

Market power or any of the

above.

APS < MBSC (τ1) N.A. Inconsistent estimates.

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