strategic use of storage: the impact of carbon policy ... · ﬀ by market power, carbon policy,...

Introduction Mathematical Formulation Comparative Statics Numerical Examples Conclusions

Strategic Use of Storage: The Impact of

Carbon Policy, Resource Availability, and

Technology Efficiency on a

Renewable-Thermal Power System

Sebastien Debiaa Pierre-Olivier Pineaua Afzal S. Siddiquib

aHEC Montreal

bUniversity College London, Stockholm University, HEC Montreal, and Aalto University,

e-mail address: [email protected]

15 February 2019


Table of Contents

1 Introduction

2 Mathematical Formulation

3 Comparative Statics

4 Numerical Examples

5 Conclusions


Introduction


Motivation

Is Energy Storage the Answer?


Motivation

Role of Flexible Generation

Marginal costs: e20-22/MWh (lignite-coal), e30/MWh

(CCGT), e45/MWh (gas)

Daily prices: e24.1/MWh and e43.5/MWh

*Residual load = buy volume - wind - solar


Motivation

“Texas Utility Oncor Wants to Invest $5.2B in Storage: Can It

Get Approval?,” Greentech Media, 10 November 2014


Related Work

Equilibrium Analysis of Storage

Crampes and Moreaux (2001) show theoretically how market

power may lower water value and even render it negative

Bushnell (2003) finds empirical support via a complementarity

analysis of the California hydro-thermal system

Restructuring and sustainable transition enhance hydropower

producers’ incentives to shift generation between peak and

off-peak periods (Mathiesen et al., 2013)

Sioshansi (2010) analyses the welfare impacts of storage use

via a stylised partial-equilibrium model with perfectly

competitive electricity generation

Sioshansi (2014) demonstrates when storage can reduce social

welfare

In the context of transmission capacity, Downward et al.

(2010) examine how a carbon tax may actually increase

emissions in the presence of market power


Complementarity 101

Evolving Paradigms


Complementarity 101

Playing Games

Optimisation Problem constrained by

Optimisation Problems (OPcOP)

Equilibrium Constraints

Mathematical Program with Equilibrium

Constraints (MPEC)

Equilibrium Constraints


Contribution

Research Objective and Findings

How are storage operations, producer profit, and social welfare

affected by market power, carbon policy, and device efficiency?

Bushnell (2003)’s main findings about more off-peak RE

production and a negative marginal value of RE storage undermarket power hold for only a low level of the carbon tax

Moreover, a carbon tax may actually increase peak thermal

production under perfect competition

Greater storage efficiency monotonically decreases marginalvalue of RE storage under market power but has anon-monotonic effect under perfect competition

RE producer profits from a ceteris paribus increase in the

efficiency of an inefficient device but a “turning point” when the

device is so efficient that the peak price is actually depressed

Social welfare increases monotonically with device efficiency

Conflict between private and social incentives can be partially

resolved through a carbon tax


Mathematical Formulation


Assumptions

Assumptions

Two periods, j = 1, 2, that correspond to off- and on-peak,

respectively

No uncertainty or transmission constraints

Equilibrium model with a (i) storage-enabled renewable-energy

(RE) and (ii) thermal producer

Pj(xj + yj) = Aj − (xj + yj) [in $/MWh], where

A2 > A1 > 0

RE production incurs no explicit cost, but available stock of

RE capacity is D ≥ 0 [in MWh] and x1 + Fx2 = D, where

F ≥ 1

Thermal cost function 12Cy2

j [in $/MWh], where C > 0 [in

$/MWh2] with emission rate R > 0 [in t/MWh] and carbon

tax T [in $/t]

For interior solutions, require (i) A2 > FA1 and (ii)

C > 3 − 2√2 ≈ 0.17


Problem Formulation and KKT Conditions

RE Producer

RE producer’s problem is to maximise its profit over the twoperiods consisting of revenue from sales subject to its resourceavailability

Maximisexj≥0 [A1 − (x1 + y1)]x1 + [A2 − (x2 + y2)]x2(1)

s.t. x1 + Fx2 = D : µ (2)

µ [in $/MWh] is the dual variable on the RE resource

constraint representing the marginal value of RE storage

Since (1)–(2) is a convex optimisation problem, it may bereplaced by its Karush-Kuhn-Tucker (KKT) conditions foroptimality:

0 ≤ x1 ⊥ − [A1 − (2x1 + y1)] + µ ≥ 0 (3)

0 ≤ x2 ⊥ − [A2 − (2x2 + y2)] + Fµ ≥ 0 (4)

µ free D − x1 − Fx2 = 0 (5)


Problem Formulation and KKT Conditions

Thermal Producer

Thermal producer’s problem is to maximise its total profit,which consists of sales revenue, generation cost, and CO2

taxes:

Maximiseyj≥0 [A1 − (x1 + y1)] y1 + [A2 − (x2 + y2)] y2

−1

2C

(y21 + y2

2

)− TR (y1 + y2) (6)

Since (6) is a convex optimisation problem, it may be replacedby its KKT conditions for optimality:

0 ≤ y1 ⊥ − [A1 − (2y1 + x1)] + Cy1 + TR ≥ 0 (7)

0 ≤ y2 ⊥ − [A2 − (2y2 + x2)] + Cy2 + TR ≥ 0 (8)


Interior Solutions

Perfect Competition

xPC1 =

F (FA1 − A2) + D + F (F − 1)RT/C

1 + F 2(9)

xPC2 =

A2 − FA1 + FD − (F − 1)RT/C

1 + F 2(10)

yPC1 =

A1 + FA2 − D − F (F − 1)RT/C −(1 + F 2

)RT

(1 + F 2) (C + 1)(11)

yPC2 =

F (A1 + FA2 − D) + (F − 1)RT/C −(1 + F 2

)RT

(1 + F 2) (C + 1)(12)

µPC =C (A1 + FA2 − D) + (F + 1)RT

(1 + F 2) (C + 1)(13)


Interior Solutions

Cournot Oligopoly

xCO1 =

F (FA1 − A2) (C + 1) + F (F − 1)RT

(1 + F 2) (2C + 3)+

D

1 + F 2(14)

xCO2 =

(A2 − FA1) (C + 1) − (F − 1)RT

(1 + F 2) (2C + 3)+

FD

1 + F 2(15)

yCO1 =

(2C + 3)A1 + F 2 (C + 2)A1 + F (C + 1)A2 − (2C + 3)D

(1 + F 2) (C + 2) (2C + 3)

−[2C + 3 − F + F 2 (2C + 4)

]RT

(1 + F 2) (C + 2) (2C + 3)(16)

yCO2 =

F (C + 1)A1 + (C + 2)A2 + F 2 (2C + 3)A2 − F (2C + 3)D

(1 + F 2) (C + 2) (2C + 3)

−[2C + 4 − F + F 2 (2C + 3)

]RT

(1 + F 2) (C + 2) (2C + 3)(17)

µCO =(C + 1)A1 + F (C + 1)A2 − (2C + 3)D + (F + 1)RT

(1 + F 2) (C + 2)(18)


Equilibrium Characterisation

Interior Solution Definitions

Definition

The interior solution set for perfect competition is defined as

SPC={(x, y)

PC ∈ R4+∣∣ xPC

j > 0, yPCj > 0, j = 1, 2

}(19)

Definition

The interior solution set for Cournot oligopoly with µ < 0 is defined as

SCOµ =

{(x, y)

COµ ∈ R4+

∣∣ µCO< 0, x

COj > 0, y

COj > 0, j = 1, 2

}(20)



Perfect Competition



Perfect Competition

Lemma

SPC is non-empty if and only if

(i) D > F

(A2 − FA1 − (F − 1)

RT

C

)if RT ≤

C(A2 − FA1)

F − 1(21a)

(ii) D >−A2 + FA1 + (F − 1)RT/C

Fif RT >

C(A2 − FA1)

F − 1(21b)

(iii) D < A1 + FA2 − F (F − 1)RT

C− (1 + F

2)RT (21c)

(iv) RT < min

A1,A2

F +F−1C

. (21d)

Furthermore, µPC > 0 is in SPC .



Cournot Oligopoly



Cournot Oligopoly

Lemma

SCOµ is non-empty if and only if

(i) D >(C + 1)(A1 + FA2) + RT (1 + F )

2C + 3(22a)

(ii) D <A1(C + 2)(1 + F2) + (A1 + FA2)(C + 1) − RT

[(2C + 3)(1 + F2) + F (F − 1)

]2C + 3

(22b)

(iii) RT <A1

2. (22c)


Comparative Statics


Impact of Carbon Policy

Main Results

Proposition

∂xPC1

∂T> 0,

∂xPC2

∂T< 0,

∣∣∣∣∂xPC1

∂T

∣∣∣∣ > ∣∣∣∣∂xPC2

∂T

∣∣∣∣ ,∂yPC

1

∂T< 0,

∂yPC2

∂T> 0 if C <

F − 1

1 + F 2,

∂µPC

∂T> 0.

Proposition

∂xCO1

∂T> 0,

∂xCO2

∂T< 0,

∣∣∣∣∂xCO1

∂T

∣∣∣∣ > ∣∣∣∣∂xCO2

∂T

∣∣∣∣ ,∂yCO

1

∂T<

∂yCO2

∂T< 0,

∂µCO

∂T> 0.

Proposition

Off-peak RE generation under CO exceeds that under PC only if

T < (A2−FA1)(C+2)CR(F−1)(C+3)

.



Decomposition


Impact of RE Availability

Main Results

Proposition

∂xPC2

∂D>

∂xPC1

∂D> 0,

∂yPC2

∂D<

∂yPC1

∂D< 0,

∂µPC

∂D< 0.

Proposition

∂xCO2

∂D>

∂xCO1

∂D> 0,

∂yCO2

∂D<

∂yCO1

∂D< 0,

∂µCO

∂D< 0.


Impact of RE Storage Efficiency

Main Results

Assuming T = 0 and any D ∈ SPC ∩ SCOµ

Proposition

∂xPC1

∂F< 0,

∂yPC1

∂F> 0,

∂µPC

∂F> 0 if F <

(D − A1) +√

A22 + (D − A1)2

A2

∂xPC2

∂F< 0,

∂yPC2

∂F> 0.

Proposition

∂xCO1

∂F< 0,

∂xCO2

∂F< 0,

∂yCO1

∂F> 0,

∂yCO2

∂F> 0,

∂µCO

∂F> 0.


Numerical Examples


Data for Numerical Examples

Data for Numerical Examples

Parameter Value

A1 125

A2 250

D 250

C 0.18

F 1.80

R 1.0

T 20



Marginal Value of RE Storage

0 5 10 15 20 25-20

-10

0

10

20

30



RE Production

0 5 10 15 20 2540

60

80

100

120



Thermal Production

0 5 10 15 20 250

20

40

60

80

100

120



Producer Profits

0 5 10 15 20 250

2000

4000

6000

8000

10000

12000

14000



Carbon Emissions and Social Welfare

0 5 10 15 20 2560

80

100

120

140

160

180

200

0 5 10 15 20 253.3

3.4

3.5

3.6

3.7

3.8104



Marginal Value of RE Storage

220 240 260 280 300-30

-20

-10

0

10

20

30



RE Production

220 240 260 280 30040

60

80

100

120

140



Thermal Production

220 240 260 280 3000

20

40

60

80

100

120

140



Producer Profits

220 240 260 280 3000

2000

4000

6000

8000

10000

12000

14000



Carbon Emissions and Social Welfare

220 240 260 280 30060

80

100

120

140

160

220 240 260 280 3003.3

3.4

3.5

3.6

3.7

3.8104



Marginal Value of RE Storage (PC)

1 1.2 1.4 1.6 1.8 2

10

15

20

25



Marginal Value of RE Storage (CO)

1 1.2 1.4 1.6 1.8 2-100

-80

-60

-40

-20

0



RE Production (T = 0)

1 1.2 1.4 1.6 1.8 20

50

100

150

200



RE Production (T = 20)

1 1.2 1.4 1.6 1.8 250

100

150

200



Thermal Production (T = 0)

1 1.2 1.4 1.6 1.8 20

20

40

60

80

100

120

140



Thermal Production (T = 20)

1 1.2 1.4 1.6 1.8 20

20

40

60

80

100

120

140



Producer Profits (T = 0)

1 1.2 1.4 1.6 1.8 20

2000

4000

6000

8000

10000

12000



Producer Profits (T = 20)

1 1.2 1.4 1.6 1.8 20

2000

4000

6000

8000

10000

12000

14000



Carbon Emissions and Social Welfare (T = 20)

1 1.2 1.4 1.6 1.8 220

40

60

80

100

120

140

1 1.2 1.4 1.6 1.8 23.3

3.4

3.5

3.6

3.7

3.8

3.9104


Conclusions


Conclusions

Summary

Transformation of power sector with higher RE penetrationentices more storage operations

Examine the impacts of carbon policy, RE availability, and

storage efficiency on market equilibria

Carbon tax may reverse results from the literature about the

role of market power in shifting RE deployment and the marginal

value of RE storage

With relatively efficient thermal production, carbon taxation

may even increase thermal output in peak period under perfect

competition

More RE availability benefits the RE producer and welfare at the

expense of the thermal producer

Higher device efficiency is more desirable for society but may be

resisted by RE producers

A carbon tax aligns private and social incentives more effectively

Future work: storage investment, transmission constraints,

intermittent RE output

strategic use of storage: the impact of carbon policy ... · ﬀ by market power, carbon policy,...

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