storing power: on the importance of market structure

Storing Power: on the Importance of Market Structure

David Andrés-Cerezo1 and Natalia Fabra2

1EUI, 2UC3M

January 27, 2020

D.Andrés-N.Fabra Storing Power: on the Importance of Market Structure January 27, 2020

The “Duck curve”


Introduction

Storage should play a key role in the energy transition:By making a more efficient use of existing resources (e.g. excess renewable production).By providing energy when renewables are not available.By reducing the need to invest in (polluting) back-up generation capacity.

Goals:Analyze whether the private and social incentives for investing in storage are aligned.Understand decentralized storage operation and its impact on market outcomes.Asses how this depends on market structure.


Storing technologies and market structure

Different types of storage facilities...

Figure: Pumped hydro Figure: Grid-scale batteries Figure: Electric vehicle fleet

...that imply different horizontal and vertical market structures:Competitive vs. strategic storage.Competitive vs. strategic productionDifferent ownership structures ⇒ stand-alone vs. vertically integrated storage firms.


Relevant questions

1 Investment in storage capacity .Will it be socially optimal?Does it depend on market structure?

2 Productive efficiency and prices.What are the effects of strategic behavior and the ownership structure?

3 Do storage facilities confer market power? Do they mitigate market power in generation?


Related literature

Storage and commodity speculation.McLaren (1999) Williams& Wright (1991); Mitraille, Thille (2014); etc...

Natural resource extraction.Hotelling (1931); Salant (1976); etc...

Electricity storage.Schmalensee (2019); Ambec&Crampes (2018); Karaduman (2020); Crampes&Trochet(2019); Helm&Mier (2018); García& Stacchetti (2001); etc...


Outline

1 Model set-up

2 First Best

3 Market solutionSecond BestCompetitive storageIndependent storage monopolistVertically integrated firm

4 Welfare comparison


Modelling set-up

DemandPrice- inelastic demand θ; consumers’ valuation v.θ is distributed according to a symmetric G (θ) in

[θ, θ

].

θ can be interpreted as demand net of renewables.Known at the production stage → Focus on seasonal variation.

GenerationExisting assets allow to produce with costs C̃(Q) increasing and convex.

StorageStorage capacity K (in MWh); Investment cost C(K) increasing and convex.qB(θ), qS(θ) : quantities bought (B) and sold (S) by the storage facility.


Demand process


Outline

1 Model set-up

2 First Best




First Best Problem

Welfare is gross consumer surplus minus production and investment costs:

maxqB(θ),qS(θ),K

W =

∫ θ

θvθdG (θ)−

∫ θ

θC̃(θ − qS(θ) + qB(θ)

)dG (θ)− C (K)

s.t. (λ) :

∫ θ

θqB(θ)dG (θ) ≥

∫ θ

θqS(θ)dG (θ)

(µ) :

∫ θ

θqB(θ)dG (θ) ≤ K


First Best

θ

θ θ̄

θ

q(θ), p(θ)

Figure: First Best


First Best

θ

θ θ̄

qFBS

qFBB

θFB2

θFB2

θFB1

θFB1

MC(θ)

θ

q(θ), p(θ)

Figure: First Best


First Best

θ

θ θ̄

qFBS

qFBB

θFB2

θFB2

θFB1

θFB1

MC(θ)

µFBC′(K) =

θ

q(θ), p(θ)

Figure: First Best


First Best

Optimal storage management:Store when demand is low and release when demand is high.Equalization of marginal costs within storing and releasing regions.Minimization of total costs of production.

Optimal investment in storage:Marginal benefit ⇒ Marginal cost saving from storing one more unit of output.No full marginal cost equalization.

First Best


Outline

1 Model set-up

2 First Best




Horizontal market structure: Production

Existing assets are owned by:a dominant firm (α share), with costs C̃D(q) = q2

2α ·a competitive fringe (1 − α share) with costs C̃F(q) = q2

2(1−α) ·α ∈ (0, 1)

Fringe produces qF = (1 − α) p(θ),

Dominant firm faces an elastic residual demandD(θ) = θ − qS(θ) + qB − (1 − α) p(θ)(θ).


Independent storage: Dominant firm

Maximize profits over the residual demand:

maxp(θ)

πD = p (θ)D(θ)− [D(θ)]2

2α

Optimal prices:

p(θ) = θ − qS(θ) + qB(θ)

1 − α2

Constant mark-up equal to α.Distorted market shares:

Dominant produces α/(1 + α) < α.Fringe produces 1/(1 + α) > 1 − α


Outline

1 Model set-up

2 First Best




Second Best

Maximize total welfare taking production decisions as given:

maxqB(θ),qS(θ),K

W =

∫ θ

θvθdG (θ)−

∫ θ

θ

(q2D

2α +q2

F2(1 − α)

)dG (θ)− C (K)

subject to the storage constraints and taking as given that:

qD =α

1 + α

[θ − qS(θ) + qB(θ)

]qF =

11 + α

[θ − qS(θ) + qB(θ)

]


Second Best

θ θ̄

θ

θSB1 θSB

2

θ

p(θ),MC(θ)

Figure: Second Best


Second Best

θ θ̄

θ

θSB1 θSB

2

WMC(θ)

θ

p(θ),MC(θ)

Figure: Second Best


Second Best

θ θ̄

θ

WMC(θ)

θSB1 θSB

2

C′(K) = µSB

θ

p(θ),MC(θ)

Figure: Second Best


Second Best

Optimal storage management:Similar to first best.Equalization of industry (weighted) marginal costs within storing and releasing regions.Weighted marginal cost: sum of the product of each firm’s market share and marginal cost.

Optimal investment in storage:Marginal benefit ⇒ Marginal cost saving from adding one unit of storage.Market power amplifies differences in industry marginal costs.

Second Best


Outline

1 Model set-up

2 First Best




Problem of competitive storage firms

Perfect competition:1 Large set of small owners (e.g. electric cars).2 Take generation prices as given.3 Free entry in the market ⇒ zero-profit condition.

Storage firms maximize:

maxqS(θ),qB(θ)

ΠS =

∫ θ

θp (θ)

[qS(θ)− qB(θ)

]g (θ) dθ − C(K)

subject to the storage constraints and the zero-profit condition.


Competitive storage

θ

θ θ̄θC2θC

1

θ

q(θ), p(θ)


Competitive storage

θ

θ θ̄θC2θC

1

p(θ)

θ

q(θ), p(θ)


Competitive storage

θ

θ θ̄θC2θC

1

p(θ)

C(K)/K = µC

θ

q(θ), p(θ)


Competitive storage

Optimal storage management:Storage operators exploit arbitrage opportunities.Prices (not marginal costs) equalized within storage and releasing regions.

Equilibrium investment in storage:Marginal value of storage capacity equals price differential that an extra unit of capacityallows to arbitrage.Market power in the product market amplifies arbitrage profits.

Competitive


First Best vs. Second Best vs. Competitive

θ θ̄

θ

θ1 θ2

MC(θ)

µFB

WMC(θ)

µSB

p(θ)

µC

θ


First Best vs. Second Best vs. Competitive

Under competitive storage with free-entry in the market, there is over-investment andover-utilization of storage ⇒ KC > KSB > KFB.

Price differential higher than marginal cost savings.

θ2 − θ11 − α2︸︷︷︸

µC

>(θ2 − θ1)(1 + α− α2)

(1 − α2)(1 + α)︸︷︷︸µSB

> θ2 − θ1︸︷︷︸µFB

Cost convexity → Higher infra-marginal profits → C(K)/K < C′(K)

KSB > KFB → Storage mitigates market power by reducing residual demand at highdemand levels.


Outline

1 Model set-up

2 First Best




Storage Monopolist

The storage monopolist internalizes the effects of its decisions on market prices:

maxqS(θ),qB(θ),K

ΠS =

∫ θ

θ

θ − qS(θ) + qB(θ)

1 − α2[qS(θ)− qB(θ)

]g (θ) dθ − C(K)

subject to the storage constraints.


Storage monopolist

θ θ̄

θ

q(θ)

θM1 θM

2

θ

q(θ), p(θ)


Storage monopolist

θ θ̄

θ

q(θ)

θM1 θM

2

p(θ)

θ

q(θ), p(θ)


Storage monopolist

θ θ̄

θ

q(θ)

θM1 θM

2

p(θ)

MRS

MCSC′(K) = µM

θ

q(θ), p(θ)


Storage monopolist

Optimal storage management:Less production smoothing ⇒ Storage monopolist avoids a strong price increase (decrease)when it buys (sells).

Storage monopolist equalizes marginal revenues (costs) when selling (buying).No price-equalization.

Optimal investment in storage:Marginal value of storage capacity equals the difference between MR and MC that anextra unit of capacity allows to arbitrage.Increasing in α

Monopolist


Infra-utilization of storage capacity

θ θ̄E(θ)

q(θ)

θ

q(θ)



θ θ̄E(θ)

q(θ)

KFB

KFB

qFB(θ)

θ

q(θ)



θ θ̄E(θ)

q(θ)

KFB

KFB

qFB(θ)

KM

KM qM(θ)

θ

q(θ)


Second Best & Second Best vs. Storage monopolist

Second Best: The storage monopolist under-invests → KM < KSB.1 Market power in storage ⇒ lower storage utilization.

First Best: The storage monopolist under-invests (over-invest) if α < α̂ (α < α̂), withα ∈ (0, 1) → KM ⋛ KSB.

1 Market power in storage ⇒ lower storage utilization.2 Market power in generation ⇒ arbitrage profits higher than at the first best.


Outline

1 Model set-up

2 First Best




Vertically Integrated Firm

Dominant firm vertically integrated with storage monopolist.

maxp(θ),qB(θ),qS(θ)

πS =

∫ θ̄

θ

[p(θ)D (p; θ)− [D (p; θ)− qS(θ) + qB(θ)]2

2α

]g (θ) dθ,

subject to the storage constraints.

Higher residual demand (firm controls storage) → D (p; θ) = θ − (1 − α)p(θ).

Storage facilities ⇒ Help the dominant producer smooth its production over time.


Vertically integrated firm

θ θ̄

θ

q(θ)

θI1 θI

2

θ

q(θ), p(θ)



θ θ̄

θ

q(θ)

θI1 θI

2

MCD(θ)

θ

q(θ), p(θ)



θ θ̄

θ

q(θ)

θI1 θI

2

MCD(θ)

p(θ)

θ

q(θ), p(θ)



θ θ̄

θ

q(θ)

θI1 θI

2

MCD(θ)

p(θ)

C′(K) = µI

θ

q(θ), p(θ)



Optimal storage management:Vertically integrated firm uses storage to smooth own production.Under-utilization of given storage capacity with respect to first best.

Optimal investment in storage:Marginal value of storage capacity equals own marginal cost savings.Investment decreases in α.

Integrated


First Best vs. Vertically integrated firm

In a market with a vertically integrated dominant firm, there is under-investment instorage, KI < KFB < KSB.

In contrast to previous cases, KI is decreasing in α.Efficiency gains from higher α dominate larger arbitrage opportunities

The under-investment problem is aggravated with respect to the case of an independentmonopolist, KI < KM


Outline

1 Model set-up

2 First Best




Consumer’s surplus

Consumer’s surplus only depends on the price profile (i.e. weighted average price)

CS = vθ −∫ θ̄

θp(θ)θg(θ)dθ.

Market power in generation increases the price level.Market power in storage increases the variance of the market price.

The ranking of consumer surplus across market structures is

CSFB > CSC ≥ CSSB > CSM > CSI > CSNS.


Consumer’s surplus

Consumer’s surplus only depends on the price profile (i.e. weighted average price)

CS = vθ −∫ θ̄

θp(θ)θg(θ)dθ.

Market power in generation increases the price level.Market power in storage increases the variance of the market price.The ranking of consumer surplus across market structures is

CSFB > CSC ≥ CSSB > CSM > CSI > CSNS.


Price profile: no capacity restrictions

pC(θ)E(θ)1−α2

E(θ)

E(θ) pFB(θ)

θ

pNS(θ)

pM(θ)

pI(θ)

θ̄

θ

p(θ)


Total welfare

Total welfare is just a function of the total costs of production.Market power creates static & dynamic productive inefficiencies:

Generation (static) ⇒ Distorted market shares.Storage (dynamic) ⇒ Lower storage usage, production not flatenned.Aggravated with vertical integration ⇒ Fringe absorbs demand variations.

The ranking of total welfare across market structures is

TWFB > TWSB > TWC > TWM > TWI > TWNS.


Conclusions

The market does not provide adequate investment incentives in storage capacity.Market power in generation leads to over-investment.Market power in storage to under-investment.

Vertical integration between storage and generation yields the most inefficient outcome.Texas regulator: utilities are not permitted to use storage.

Storage reduces the ability to exercise market power in generation, conditional on beingindependently owned.

Storage capacity auctions.Solve investment problem, although inefficient storage operation.


Ways forward

Introduce other cases (e.g. load-owned storage).

Introduce stochastic demand and/or production.

Empirical simulation for the Spanish market.


First Best (cont)

Optimal storage management:For given K, storage decisions are

qFBB (θ) = max

{θFB

1 − θ, 0}

and qFBS (θ) = max

{θ − θFB

2 , 0}

whereθFB

1 = E [θ]− µ

2 ≤ θFB2 = E [θ] +

µ

2 ,

and µ = µFB(K) is the unique solution to∫ θFB1 (µ)

θ

[θFB

1 (µ)− θ]

g(θ)dθ = K.

Optimal investment in storage:∂W∂K = 0 ⇒ µ

(KFB

)= C′

(KFB

)⇒ θFB

2 − θFB1 = C′

(KFB

)Back D.Andrés-N.Fabra Storing Power: on the Importance of Market Structure January 27, 2020

Second Best (cont)

Optimal storage management:For given K, storage decisions are

qSBB (θ) = max

{θSB

1 − θ, 0}

and qSBS (θ) = max

{θ − θSB

2 , 0}

whereθSB

1 = E [θ]− µ

2(1 − α2)(1 + α)

1 + α− α2 ≤ θSB2 = E [θ] +

µ

2(1 − α2)(1 + α)

1 + α− α2 ,

and µ = µFB(K) is the unique solution to∫ θSB1 (µ)

θ

[θSB

1 (µ)− θ]

g(θ)dθ = K.

Optimal investment in storage: Back

∂W∂K = 0 ⇒ µ

(KSB

)= C′

(KSB

)⇒ (θSB

2 − θSB1 )

1 + α− α2

(1 − α2)(1 + α)= C′

(KSB

)D.Andrés-N.Fabra Storing Power: on the Importance of Market Structure January 27, 2020

Competitive storage (cont)

Optimal storage management:For given K, the equilibrium storage decisions are

qCB(θ) = max

{θC

1 − θ, 0}

and qCS(θ) = max

{θ − θC

2 , 0}

whereθC

1 = E [θ]−µ(1 − α2)

2 ≤ θC2 = E [θ] +

µ(1 − α2)

2 ,

with µ = µC(K) implicitly defined by:∫ θC1 (µ)

θ

[θC

1 (µ)− θ]

g(θ)dθ = K.

Investment in storage:

µC(K) = (θC2 − θC

1 )/(1 − α2) = C (K) /K < C′(K).

Back D.Andrés-N.Fabra Storing Power: on the Importance of Market Structure January 27, 2020

Storage Monopolist (cont)


qMB (θ) = max {(θ1 − θ) /2, 0} and qM

S (θ) = max {(θ − θ2) /2, 0} ,

whereθM

1 = E [θ]− µ(1 − α2)/2 ≤ θM2 = E [θ] + µ(1 − α2)/2,

with µ = µM(K) is the unique solution to∫ θM

1 (µ)θ

θM1 (µ)−θ

2 g(θ)dθ = KOptimal investment in storage:

C′(K) = µM(K) = (θM2 − θM

1 )/(1 − α2)

Back


Vertically Integrated Firm (cont)


qIB(θ) = max

{(θI

1 − θ)/2, 0

}and qI

S(θ) = max{(

θ − θI2

)/2, 0

},

whereθI

1 = E [θ]− µ(1 + α)/2 ≤ θI2 = E [θ] + µ(1 + α)/2,

with µ = µI(K) is the unique solution to∫ θI1(µ)

θ

θI1 (µ)− θ

2 g(θ)dθ = K.

Optimal investment in storage:

C′(K) = µI(K) = (θI2 − θI

1)/ (1 + α) .

BackD.Andrés-N.Fabra Storing Power: on the Importance of Market Structure January 27, 2020

storing power: on the importance of market structure

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