Investment in electricity capacity under fuel cost uncertainty: Dual-fuel and a mix of single-fuel technologies

Energy Policy 126 (2019) 518–532

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Investment in electricity capacity under fuel cost uncertainty: Dual-fuel and a mix of single-fuel technologies

T

Nurit Gala, Irena Milsteinb, , Asher Tishlerc, C.K. Wood ⁎

a

Public Utility Authority – Electricity, Jerusalem, Israel Holon Institute of Technology, Holon, Israel c Faculty of Management, Tel Aviv University, Tel Aviv, Israel d Education University of Hong Kong, Hong Kong b

ARTICLE INFO

ABSTRACT

JEL codes: D24 D43 L11 L94 L95

We study the effect of the price and price volatility of natural gas on investment in electricity capacity in two technology scenarios: (1) dual-fuel units that use natural gas and diesel; and (2) a mix of single-fuel plants that use coal or natural gas. We develop a two-stage (capacity and operation) model and derive analytical solutions for both scenarios. Based on the observed log-normal distribution of the natural gas price, we show that optimal capacity investment increases moderately with natural gas price volatility, countering a commonly held view that fuel cost uncertainty tends to discourage capacity investment. Thus, higher volatility of the natural gas price tends to reduce the 'missing money' problem. We use Texas data to show that higher gas price volatility implies higher profits and consumer surplus in the first scenario, even when the per MWh diesel cost is much higher than the expected value of the per MWh gas cost. In the second scenario, firms invest only in gas capacity, unless the per MWh coal cost is significantly below the expected per MWh gas cost, thus explaining the popularity of gas generation.

Keywords: Natural gas price volatility Investment under uncertainty Capacity mix Dual-fuel plants Electricity price behavior

1. Introduction Since the 1980s, many countries have reformed their electricity sectors, so as to attract private investment, improve cost efficiency and lower retail prices (Newbery, 2005; Joskow, 2006). Unlike capacity investment by a regulated integrated monopoly, new plants are constructed by profit-maximizing independent power producers (IPPs) operating in a restructured market with wholesale competition. These IPPs face fuel cost risks, unlike the monopoly that could pass its fluctuating fuel costs on entirely to its retail customers under cost of service ratemaking. Further, electricity prices and sales are set by the interactions among market participants, exacerbating the IPPs’ profit risks (Gatti, 2008; Gal et al., 2017). This paper studies electricity capacity investment in competitive markets under fuel cost uncertainty, thus responding to Professor Paul Joskow’s insightful observation that: “Revenue adequacy has emerged as a problem in many organized wholesale electricity markets and has been of growing concern in liberalized electricity markets in the U.S. and Europe. The revenue adequacy or 'missing money' problem arises when the expected net revenues from sales of energy and ancillary services at market prices provide inadequate incentives for merchant

⁎

investors in new generating capacity or equivalent demand-side resources to invest in sufficient new capacity to match administrative reliability criteria at the system and individual load serving entity levels” (Joskow, 2013a; p. v). We show that capacity investment increases with fuel cost volatility, countering a commonly held view that investment tends to decline when fuel costs become more volatile. This view comes from the intuition that higher fuel cost risks may adversely affect a new plant’s expected profitability, thus reducing the value of the marginal capacity units. Power plants fueled by natural gas (“gas plants” hereafter) have been accounting for a large share of new capacity expansion in the last decade due to declining gas prices and the relatively low emissions and short construction periods of these plants (MIT, 2011; Joskow, 2013b). However, IPPs face large fuel cost risks because: (a) natural gas constitutes about 80% of their variable costs; and (b) the volatility of the price of natural gas is larger than that of coal and oil (Geman and Ohana, 2009; Graves and Levine, 2010; Smead, 2010; Alterman, 2012; Gal et al., 2017). This raises a substantive research question: what is the effect of gas price volatility on electricity capacity investment? Answering this question requires modeling an IPP’s investment problem under fuel cost uncertainty, an endeavor complicated by the reality of

Corresponding author. E-mail addresses: [email protected] (N. Gal), [email protected] (I. Milstein), [email protected] (A. Tishler), [email protected] (C.K. Woo).

https://doi.org/10.1016/j.enpol.2018.10.040

Available online 11 December 2018 0301-4215/ © 2018 Elsevier Ltd. All rights reserved.

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N. Gal et al.

heterogeneous generation technologies. To wit, a power plant may have the dual-fuel capability of using natural gas and diesel, implying that the plant’s fuel consumption decision responds to the relative fuel prices. Moreover, an IPP may build a mix of single-fuel plants, which may be fueled by natural gas or coal. For clarity, we refer to the case of dual-fuel plants as the first technology scenario and to the case of a mix of single-fuel plants as the second technology scenario. While reflecting real-world situations, these two scenarios under fuel cost uncertainty have not been thoroughly explored in the extant literature of electricity capacity investment. Our exploration herein is an extension of Gal et al. (2017), which finds that gas capacity investment tends to moderately increase with gas price volatility. This paper makes the following contributions. First, it develops an analytical model of a Cournot electricity market to assess the effect of fuel cost uncertainty on dual-fuel capacity investment. To the best of our knowledge, this model and its findings are new, lending support to an IPP’s incentive to hedge against excessively high fuel costs. Second, it shows that in the second technology scenario, an IPP will invest in coal capacity only if the cost of coal generation is substantially below the expected cost of gas generation. This finding explains the popularity of gas generation in new plant construction. Moreover, the market share of coal capacity tends to decline as gas price volatility increases. This seemingly counter-intuitive result has a reasonable explanation. An increase in gas price volatility tends to increase the IPP’s expected profit from gas generation, thanks to the observed log-normal distribution of the gas price, which implies a higher probability of low gas costs when gas prices become more volatile. Finally, we show that the more volatile the gas prices, the higher the market’s total capacity, leading to lower electricity prices and higher consumer surplus. Thus, a government-sponsored natural gas price stabilization policy is unnecessary for promoting generation capacity investment. Moreover, the 'missing money' that plagues deregulated electricity markets, especially those with vast renewable generation development, becomes less of a problem (Spees et al., 2013; Woo et al., 2016). Milstein and Tishler (2011, 2015), for example, show that electricity prices tend to spike higher and more often in deregulated electricity markets with high share of renewables and, consequently, possibly yield higher profits for gas-fired generators, which may in turn reduce the 'missing money' problem. The paper proceeds as follows. Section 2 discusses the evolution of fossil fuel prices and their volatilities. Section 3 briefly reviews the literature on generation capacity expansion. Section 4 develops a Cournot model for investment in dual-fuel plants. Section 5 presents a Cournot model for investment in coal and gas plants. Section 6 illustrates the models’ empirics. Section 7 concludes.

A commodity’s annual price volatility is commonly based on its daily percentage price changes over a 1-year period (Roesser, 2009).2 In the case of natural gas, the price distribution has been found to be lognormal, which is characterized by its mean μ and variance 2 (Pilipovic, 1997; Eydeland and Wolyniec, 2003; Gal et al., 2017, Fig. 2). Hence, we make the assumption that the random natural gas prices are log-normally distributed; this is notwithstanding that our models can be solved for other distribution functions of the gas prices, and some of our analytical results (e.g., Propositions 1, 2 and 3, and Corollary 2 part (a)) hold for any distribution function of the gas prices. Natural gas price volatility is caused mainly by transportation constraints and storage limitations (Eydeland and Wolyniec, 2003). Transportation of natural gas is limited by pipeline capacity and/or liquefied natural gas (LNG) capacity. Natural gas storage is limited to depleted reservoirs, aquifers, salt formations or LNG tanks. Natural gas prices surge when natural gas production is disrupted or when demand exceeds pipeline capacity or LNG gasification capability (Alterman, 2012). Volatility is further amplified by low inventory levels (Geman and Ohana, 2009). Fig. 2 reports natural gas price spikes in New England during 2007–2016. While the Algonquin price generally follows the Henry Hub price, it exhibits winter spikes due to extreme weather and local pipeline capacity constraints. Hence, we reasonably postulate that an IPP in this region anticipates highly volatile natural gas prices when making its capacity investment decision. 3. Literature review 3.1. Dual-fuel capability Dual-fuel capability is estimated to increase the annual capacity cost of gas generation by ~10%, reflecting the additional costs for oil storage facilities, 3–5 days of fuel inventory, adapting the combustion turbines to burn liquid fuels and reduce NOx emissions from burning these fuels (Silve and Noël, 2010; Newell et al., 2014). It enhances an IPP’s risk management by mitigating: (a) the regional risk of gas supply interruptions due to pipeline capacity constraints (e.g., PJM), political disruptions (e.g., Eastern Europe), or military threats (e.g., Israel); and (b) the financial risk of natural gas price spikes due to supply interruptions or demand surges. The share of natural gas fired plants with dual-fuel capability was 19% in Europe in 2001 (Glachant et al., 2013, p. 26) and 30% in the USA in 2016 (EIA, 2016b; Table 4.11). Several recent studies, sponsored by the US department of Energy focused on the challenges rising from the increasing use of natural gas as a primary fuel in the US electricity sector (Shahidehpour and Li, 2014; NERC, 2016; DOE, 2017). The conclusion of these studies is that dual-fuel capability and gas storage can mitigate the increasing dependence of the electricity sector on natural gas. The increasing rates of extreme weather events in the US resulted in new incentives for generators to invest in dual fuelcapability. For example, PJM new capacity payment scheme enhances power plants to invest in dual fuel-capability due to pipeline constrains and natural gas price spikes (PJM, 2015). PJM concludes that although this new policy will increase capacity payments, it will decrease overall

2. Fossil fuel prices and their volatility Natural gas, unlike oil and coal, is traded regionally, due chiefly to the limitations on its transportation. In some regions such as the USA and parts of Europe, natural gas prices are determined in competitive spot markets. In other regions, they may link to oil prices (e.g., Eastern Europe). Spot prices (US$/MMBtu) of fossil fuels reflect interactions between market demands and supplies. Fig. 1 depicts the price history of Australian coal, Brent oil, Henry Hub natural gas and German border Russian gas over the past 25 years. Up to the year 2000, all four fuel prices were relatively stable and moved in tandem. Since 2001, the figure shows large price volatilities and divergences. According to the World Bank (2017), this divergence in fuel prices will likely continue during the next decade.1

(footnote continued) MMBtu in Europe and the USA, respectively, and coal prices will fall to about $2/MMBtu. 2 Eydeland and Wolyniec (2003) define the annualized volatility, σ, as follows (page 82, eq. (3.9)): . n

(

logP

logP

n

logP

logP

)

2

i i 1 i i 1 = n 1 i=1 , where {Pi} denotes the i=1 ti ti 1 n ti ti 1 time series of historical prices observed at times ti, i = 0,...,n, and ti ti 1 are year fractions. A year fraction equals the length of the interval, in days, between two observations, divided by 365 or by 250 (when only trading days are accounted for), respectively. 1

1 According to the World Bank (2017), during the next decade, crude oil prices will likely reach 11$/MMBtu, natural gas prices will be $4 to $7 per

519

1

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Fig. 1. History of fossil fuel prices. Source: IMF – monthly commodity prices.

investment opportunities with uncertain cash flows. Frayer (2001) applies the same theory to evaluate various generation alternatives, finding high option values for flexible peak load generators such as gas turbines. Using the geometric Brownian motion to characterize uncertainty, Gahungu and Smeers (2009) develop a real options capacity expansion model for power generation in a competitive market with heterogeneous technologies that differ in operation and investment costs. Using a real options approach, Chen and Tseng (2011) search for the optimal investment timing when a coal-fired plant owner considers introducing clean technologies in the face of two possible policies, tradeable permits or carbon tax. They demonstrate that higher levels of volatility in the permit prices are likely to induce suppliers to take early actions to hedge against carbon risks. Finally, one may develop the efficient frontier that minimizes a resource portfolio’s cost variance for a given level of cost expectation (Awerbuch and Yang, 2007; Zhu and Fan, 2010; Arnesano et al., 2012). As these financial approaches assume price-taking market participants, they do not characterize investment decisions made by IPPs operating in a Cournot market environment that realistically reflects wholesale electricity trading.

Fig. 2. Natural gas price ($/MMBtu) history of Henry Hub and Algonquin Citygate. Source: https://www.eia.gov/todayinenergy/detail.php?id=29032.

retail price by limiting wholesale prices during extreme weather events. This policy was adopted by NE-ISO and was approved by FERC (FERC order 148, 2014). Kulatilaka (1993) employed real options theory to assess fuel switching as a hedge for high fuel cost risk. He shows that the option value of the dual-fuel capability in the industrial sector increases with natural gas price volatility. Newell et al. (2014) indicate that firms in areas with repeated gas supply interruptions will have the incentive to invest in dual-fuel capability. Sieminski (2013) describes gas supply interruptions in the New England region and discusses various solutions, including dual-fuel capability, demand response, pipeline expansion, and LNG peak contracts. Silve and Noël (2010) study the role of dual-fuel capability in Bulgaria’s electricity sector, suggesting that diesel backup can be an economic alternative to gas pipeline capacity expansion, as evidenced by its common use in other countries with unstable gas supply (e.g., Singapore, UK and Spain).

3.3. Summary remark IPPs operating in a Cournot market likely make capacity decisions that incorporate their exposure to natural gas price spikes and gas supply interruptions. As Gal et al. (2017) have already studied the use of call options for fuel cost risk management, the rest of this paper analyzes optimal capacity investment in the two technology scenarios, yielding new insights in capacity investment under the fuel cost uncertainty that is clearly present in real-world situations. 4. The effect of natural gas price volatility on investment in dualfuel plants

3.2. A mix of single-fuel generation technologies

This section develops and solves an analytical model of a Cournot electricity market in which IPPs build and operate dual-fuel plants.3

Numerous studies derive and assess the optimal single-fuel capacity choice of a competitive firm under fuel price certainty (e.g., Andersson and Bergman, 1995; Fehr and Harbord, 1997; Borenstein and Bushnell, 1999; Tishler et al., 2008). There are also studies that analyze a competitive market’s conventional generation capacity mix (e.g., Murphy and Smeers, 2005; Joskow, 2006; Fabra et al., 2006, 2011; Milstein and Tishler, 2012; Bajo-Buenestado, 2017), which is found to depend on the presence of renewable resources (Larson et al., 2003; Pindyck, 2004; Diakoulaki and Karangelis, 2007; De Jonghe et al., 2011; Milstein and Tishler, 2015). Financial theory offers another line of inquiry. Pindyck (1991) and Dixit and Pindyck (1994) use the real options theory to value

3

Eurostat (2017) reports that the number of main electricity companies participating in most European electricity markets is usually less than 5, due chiefly to significant entry costs, various regulatory restrictions on entry, economies of scale in fossil-fired electricity generation, ability to obtain project financing at reasonable costs, and credibility as a counter party in a large bilateral transaction (e.g., a multi-year power purchase agreement). The small number of dominant market players indicates that these players are likely to possess some market power. Using Cournot market to assess electricity markets is therefore a common assumption in the electricity market reform literature (e.g., Wolfram, 1999; Borenstein et al., 2002; Murphy and Smeers, 2005; 520

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parameters are µt and

Each plant uses natural gas as its main fuel and diesel as its secondary fuel. The price of natural gas is assumed to be a log-normally distributed random variable and the price of diesel a known constant.4 Our two-stage model captures the lead time in capacity construction and the daily fluctuations of the fuel costs. In the first stage, only the probability distribution function of the daily natural gas price is known. Profit-seeking IPPs maximize their expected profits by determining the amount of the dual-fuel capacity to be constructed. In the second stage, once the daily price of natural gas becomes known, each IPP decides how much electricity to generate up to the capacity built in the first stage, which, in turn, determines the equilibrium market price. Solved recursively, our two-stage model assumes a Cournot market with N identical IPPs (or firms) operating daily over a period of T days. The market’s inverse demand function on day t = 1, …, T is assumed to be:

Pt = a

ln

=

T

+

T

Ki + ctmin Qit ,

et al. (2017), we assume that ln

( ) ct 1

(3)

2 ct t

=

c0 ct

ct

2 t

e

1 2

(c )

ln c0t

µt

2

2 ct t

g (ct ).

(4)

Under Cournot competition, each of the N firms determines its daily optimal electricity production, Qit , to maximize its operating profit, it , subject to the capacity built in stage 1.8 Hence, the optimization problem of firm i on day t is:

max Qit s. t.

it

= (Pt

Qit

Ki

(5)

ctmin) Qit

The properties of the equilibrium solution are summarized in Proposition 1 below. For brevity and readability, the proofs of all propositions and corollaries presented below are available in the Appendix A. Proposition 1 Q Let c Ki a 2bKi bQ it , where Q it j i jt . Then, equilibrium quantity and price at stage 2 are given by one of the following two cases: Case A: When c Ki c d ,

(2)

where Ki = firm i’s MW capacity; = capacity cost ($/MW-year) of gas generation; = incremental capacity cost ($/MW-year) due to dual-fuel capability; and ctmin min {ct , c d } = per MWh fuel cost ($/MWh) under least-cost dispatch = minimum of the per MWh cost of natural gas, ct , and the per MWh cost of diesel, c d . We require ctmin < a to realistically reflect that some electricity consumption, however small, for critical end-use services (e.g., lighting and refrigeration), will still occur when fuel prices are excessively high. Since a unit’s fuel cost is the unit’s heat rate times the applicable fuel price, its fuel cost volatility is its heat rate times the fuel price volatility. Let c0 denote the initial per MWh cost of natural gas at the beginning of the planning period. Based on Pilipovic (1997), Geman (2005) and Gal ct

2 ct t )

4.1. Stage 2 solution: energy market

= total electricity generated where a > 0 and b > 0; Qt = by the N firms on day t; and Qit = electricity generated by the i-th firm on day t.5 Following Newbery (1998), Besanko and Dorazelski (2004), Murphy and Smeers (2005), Tishler et al. (2008), and Gal et al. (2017), we assume linear amortization of the annual capacity cost.6 Hence, the cost function of electricity generation of firm i on day t is: i)

ct | µt , c0

g

N Q i = 1 it

C (Qit ,

ct ~lnN (µt , c0

2 ct t

and

(1)

bQt ,

ct ~N µt , c0

2 7 ct t :

Ki , 1 (a 2b 1 ( a 2b

Qit* =

a Pt*

is normally distributed with mean

ct

bQ it ), ifc Ki

cd

bQ it ),

bKi

1 (a 2 1 (a 2

=

if 0 ct < c Ki

bQ it ,

ifc d

ct < c d ct <

(6a)

if 0 ct < c Ki

+ ct

bQ it ), if c Ki

+ cd

bQ it ),

if c d

ct < c d ct <

(6b)

Case B: When c Ki > , Qit* = K i and Pt* = a bKi bQ it for any value of ct . Note that c Ki is the per MWh gas cost ceiling below which the firm will generate at full capacity. When c Ki > c d (i.e., the per MWh diesel cost is relatively low), c Ki > ctmin for any realization of ct and, therefore, the firm will use gas or diesel to generate at full capacity, depending on whether ctmin is equal to ct or c d . When c Ki c d , the diesel fuel cost is relatively high. If ct c Ki , the firm will use gas to generate at full capacity; otherwise it will produce at less than full capacity, using gas when the per MWh gas cost is smaller than the per MWh diesel cost, ct < c d , and diesel when ct c d .

cd

μ and variance c2t . We use ct to measure the volatility of the per MWh cost of natural gas on day t. As ct has a log-normal distribution, its c0

(footnote continued) Tishler et al., 2008). 4 The diesel price assumption is made because it greatly simplifies and sharpens the analytical results. We have tried numerical simulations to determine that relaxing this assumption does not materially change the findings presented below. 5 We acknowledge that it is possible to develop the model with one demand function for the day (or peak) period and another for the night (or off-peak) period. However, a multiple-demand specification precludes a closed form solution of the game’s first stage, rendering some of the analytical results impossible to prove. To be sure, one can use numerical methods to obtain more detailed results based on two (or more) demand functions. The nature and magnitude of such results, however, are very similar to those with only one demand function, as shown by Tishler et al. (2008) and Milstein and Tishler (2015). As the primary focus of this study is to assess the role of natural gas price volatility that already requires a fairly complicated modeling effort, our choice of a single demand function aims to yield a mathematically tractable model with closed form solutions. 6 There are several ways to formulate capacity costs in our model. In reality, the choice of the financing period and the stream of debt payments for the capacity are related to the lifetime of the investment. We chose to abstract from the optimal financing decision by each investor and assume a linear amortization of the debt over the lifetime of the project.

4.2. Stage 1 solution: optimal capacity The optimization problem of firm i is: T

max Ki

e t=1

rt

E [ it ]

+ T

· Ki

(7)

The expected operating profit of firm i on day t is: 7

2 c t

Assuming risk-neutral investment implies µt = rt 2t , where r is the riskless interest rate (see Lee et al., 2010 on this issue). 8 Stage 2 is repeated, independently, on each of the T days in the planning horizon (e.g., T = 365 for a 1-year horizon). Note that it is straightforward to solve the model for 8760 h or 17,520 half-hours of the year. 521

Energy Policy 126 (2019) 518–532

N. Gal et al. c Ki 0

E [ it ] =

+ +

[a cd c Ki cd

bKi 1 [a 4b 1 [a 4b

bQ

bQ bQ

by natural gas (to be denoted by G), and the other fueled by coal (to be denoted by C). The two generation types differ in their generation and capacity costs. As in Section 4, we assume that the per MWh natural gas cost, ct , follows a log-normal distribution and the per MWh coal cost, c C , is a known constant.10 The annual per MW capacity cost of coal generation is assumed to exceed the annual per MW capacity cost of gas generation.11 The market’s inverse demand function is assumed to be:

ct ] Ki g (ct ) dct

it

ct ]2 g (ct ) dct

it

2

c d] g (ct ) dct

it

(8a)

for Case A in Proposition 1, and cd 0

E [ it ] =

+

[a cd

bKi [a

bQ bKi

ct ] Ki g (ct ) dct

it

bQ

it

c d] Ki g (ct ) dct

Pt = a

(8b)

+ = total MWh generated by where a > 0, b > 0; Qt = the N firms on day t; QitG = MWh generated by firm i on day t using technology G; and QitC = MWh generated by firm i on day t using technology C. We continue to employ a linear amortization of the annual capacity cost. Thus, the costs of electricity generation by firm i on day t are: N i=1

for Case B in Proposition 1. The properties of the solution of the first stage are summarized in Proposition 2 below. Proposition 2 The optimal capacity K * is given by the following two cases (which correspond to the two cases of Proposition 1): c

T

d

e r (1

e rT ) · e r

Case A: When t = 1 e rt · 0 (c d ct ) g (ct ) dct 1 optimal capacity is given by the following implicit form: T

e t=1

rt ·

a b (N + 1) K * 0

(a

b (N +1) K *

= 0.

Case B: When optimal capacity is:

a cd K* = + b (N + 1)

T e rt · t=1

d

cd 0

(c d

ct ) g (ct ) dct <

(c d

ct ) g (ct ) dct

b (N +1)·

e r (1 e rT ) · 1 e r

+ T

, the

+ T

T e rt t=1

(10)

Proposition 2 implies that when the per MWh diesel cost is relatively high (Case A), the optimal capacity with dual-fuel capability is lower than the optimal capacity without it.9 However, when the per MWh diesel cost is relatively low (Case B), the optimal capacity with dual-fuel capability may be smaller than, equal to or larger than that of the optimal single-fuel capacity. Corollary 1 below summarizes the effect of the volatility of ct on the equilibrium solution. Corollary 1. An increase in

ct

C

T

(QitG

QitC )

K iG + ct QitG ,

(12a)

K iC + c CQitC ,

(12b)

where K iG (K iC ) is the i-th firm’s natural gas (coal) capacity (in MW), and G ( C ) is the capacity cost ($/MW-year) of the natural gas (coal) technology. We also assume G < C and c C < a . Following Milstein and Tishler (2012), we employ a two-stage model. In the first stage, each firm decides how much capacity to build of each technology (gas and coal), given the distribution function of the per MWh gas cost, g (ct ). In the second stage, the spot natural gas cost, ct , becomes known and the firms decide how much electricity to generate. The model is solved backwards, with the second stage repeating T independent times.

(9) c T e rt · 0 t=1

T

C (QitC , K iC ) =

+ T

ct ) g (ct ) dct

G

C (QitG , K iG ) =

, the

+ T

(11)

bQt , t =1, …, T

5.1. Stage 2 solution: energy market The objective of the i-th firm in stage 2 is to maximize its operating profit, it , subject to the installed capacity constraints of K iG and K iC :

max

QitG,

s. t.

causes:

QitC

QitG

it

= (Pt

K iG ,

ct )· QitG + (Pt

QitC

K iC ,

c C )·QitC QitG

0, QitC

(13)

0

The Karush-Kuhn-Tucker (KKT) conditions are:

(a) An increase in the optimal capacity. (b) An increase in the expected operating profits at the optimal firststage solution on day t. (c) An increase in the expected consumer surplus at the optimal firststage solution on day t. (d) A decline in the expected electricity price at the optimal first-stage solution on day t.

a

2bQitG

bQGit 2bQitC

bQCit

ct

a

2bQitG

bQGit 2bQitC

bQCit

cC

µG ·QitG = 0,

K iG

QitG 0, 0,

The properties of the log-normal distribution drive Corollary 1. When ct increases, so does the probability of low ct . Thus, higher volatility of the natural gas cost induces higher capacity investment, so as to benefit from the higher probability of low per MWh gas costs. This, in turn, leads to lower electricity prices and higher consumer surplus.

µC · QitC = 0,

KiC

QitC 0,

G ·(K G i

QitG 0,

G

C

(14)

+ µG = 0,

+ µC = 0,

QitG ) = 0,

QitC 0,

C ·(K C i

G

0,

QitC ) = 0, C

0,

µG

µC 0,

where QGit

QG QCit QC G and C are the dual variables of j i jt , j i jt , the capacity constraints, and µG and µC are the dual variables of the non-negativity constraints. For given values of market demand and generation costs, the optimal quantities are equal to the capacities (i.e., QitG * = KiG and QitC * = K iC ), provided that the capacities built in stage 1 are relatively small and the capacity costs are relatively high (Tirole, 1988). In other words, a producer will never install in the first stage a capacity which

5. Technology mix in competitive markets with uncertain generation cost This section models a Cournot electricity market with N identical IPPs. Each IPP can invest in two types of single-fuel plants: one fueled

10 The annualized price volatility of natural gas in 2014 is 96%, while that of Australian coal is only 8% (see Gal et al., 2017, Fig. 1). Hence, our assumption that the price of coal is constant, which is made for expositional simplicity, is a good approximation of reality. 11 For simplicity, we ignore other dissimilarities between these two technologies, such as emissions, for example.

9 This follows from direct comparison of Eq. (9) with Eq. (13) in Gal et al. (2017). The first-order conditions differ only in capacity cost of the dual-fuel capability, / T , which is added in the case of dual-fuel capability. As a result, the optimal capacity is lower in response to the higher capacity cost.

522

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N. Gal et al.

will be left idle in the second stage (Bulow et al., 1985). Note, however, that in our model the per MWh natural gas cost, ct , is a random variable. When c C = ct , generation using coal and gas is at full capacity. Since c C is a known constant, generation using coal will always (for any ct ) be at full capacity, K iC . Gas generation, however, depends on the realized value of ct , ranging from zero to full capacity, K iG . The equilibrium solution of stage 2 which satisfies (14) is:

(K iG, (QitG *,

QitC *)

(

=

1 [a 2b

KiC ), bQGit

ct

T

e

a b (N + 1) K C

a b (N + 1) K C

ifc KiG+ KiC

µ1· K G = 0,

K iC , KiC ),

(0,

ifc KiC

where

2bK iG 2bK iG

a

bQGit 2bK iC

bQCit , bQGit

c KiC

a

2bK iC

bQGit

bQCit ,

bQCit .

Scenario

C

c +

max

e

KiG, KiC

E(

G it |K i ,

G

KiC )

T

t=1

C

K iG

G it |K i ,

K iC ) =

0

T

ct ) QitG + (Pt

[(Pt

K iC

,

(16)

c C ) QitC ] g (ct ) dct .

max

e

KiG, KiC t = 1

c K G +K C

rt

i

0

i

(a

bK iG

bQGit

bK iC

c K G+ K C i

0

1 (a 4b

cK C

i c K G+ K C i i

cK C

e (17)

bQGit

bQCit

bK iC

bQCit

bK iC

(a

cK C

bQGit

T

T

K iG 0,

s. t.

e

rt ·

e

rt ·

t=1

T

e

t=1

e bQ Cit

rt ·

t=1

c C ) K iC g (ct ) dct

e

1

e e

rT ) r

·

G

T

T e rt · t=1

. G e r (1 e rT ) ·T 1 e r

cC + C / T T e rt · 0 t=1

<

T rt ·

e

cC + C / T

t=1

cC

[ct

C / T ] g (c ) dc , t t

0

[a

b (N + 1)(K G * + K C *)

ct ] g (ct ) dct

a b (N + 1) K C *

[ct

a + b (N + 1) K C *] g (ct ) dct =

0

c C ) g (ct ) dct

(ct

e r (1 e 1 e

rT ) r

·

cC + C / T T e rt · 0 t=1

C

G

T

[c C +

.

C /T

ct ] g (ct ) dct >

cC + C / T

t=1

=

b (N + 1)(K G + K C )

cC

[ct

C / T ] g (c ) dc , t t

K C * = 0 , and K G * is given by the following implicit form: rt ·

a b (N + 1) K G * 0

e r (1 e 1 e

rT ) r

·

G

T

[a

b (N + 1) K G *

ct ] g (ct ) dct

.

Proposition 3 implies that only one of three scenarios is possible for any given values of the variable and capacity costs of technology C. When these costs are low relative to the expected variable and known capacity costs of technology G (scenario (i)), the firm will build only coal capacity. When the costs of both technologies are comparable (scenario (ii)), the firm builds a mix of coal and gas capacities. When the variable and capacity costs of technology C are high relative to the expected costs of technology G (scenario (iii)), the firms will build only gas capacity. Corollary 2 below characterizes the optimal first-stage

ct ] g (ct ) dct (19)

r (1

G

T

T

e

[a

(ii). When ct ] g (ct ) dct

G e r (1 e rT ) ·T + 1 e r

K iC 0

0

t=1

T

b (N + 1)

Scenario (iii). When

ct )2g (ct ) dct+

K iC

a b (N + 1)(K G + K C )

rt ·

e r (1 e rT ) · 1 e r

C

cC

a

T

c C ) KiG g (ct ) dct +

Assuming a symmetric solution (i.e., K1G = …=K NG = K G and = …=KNC = K C ), the Karush-Kuhn-Tucker (KKT) conditions are:

K1C

0,

ct ] g (ct ) dct ,

a b (N + 1)(K G *+ K C *)

T

C

K iG

µ2

T

i

G

0,

and

i

+

µ1

e r (1 e rT ) G · , 1 e r T

=

(18)

bQGit

rt ·

t=1

c C )· K iC g (ct ) dct

(ct

i

c K G+ K C i

bKiG

(a

i

0,

the optimal solution is given by the following implicit form: T

bQCit

ct ) K iG g (ct ) dct+

KC

0,

When

C /T

e r (1 e rT ) G · + 1 e r T

Inserting (15) into (17), the optimization problem (16) becomes: T

[c C +

Scenario [c C + C / T

where expectations are taken over ct (t = 1, …, T ). The i-th firm’s expected operating profit on day t is:

E(

/T

(i).

K G * = 0, and K C * =

The firm sets the capacity mix at stage 1 to maximize the sum of its expected daily profits. Only g (ct ) is known at this stage. Hence, its optimization problem is: rt

C

0

5.2. Stage 1 solution: optimal capacities

T

KG

Proposition 3 There exists a unique equilibrium solution in the first stage, given by one of the following three scenarios:

a and Eq. (15) states that if the per MWh gas cost, ct , is low on day t, the firm uses coal and gas to produce at full capacity. If ct is high on day t (i.e., ct > c C ), the firm decreases its gas generation below full capacity. If ct is very high, the firm generates electricity employing only the coal units. c KiG + KiC

µ 2 ·K C = 0,

where µ1 and µ 2 are the dual variables for the non-negativity constraints. Proposition 3 summarizes the properties of the first-stage equilibrium solution.

ct < (15)

c KiG

e r (1 e rT ) C · + µ2 1 e r T

c C ] g (ct ) dct

= 0,

ct < c KiC

)

KiC ,

b (N + 1) K C

[a

c C ] g (ct ) dct +

c C ) g (ct ) dct+

(ct

a b (N + 1)(K G + K C )

b (N + 1)(K G + K C )

[a

0

t=1

if 0 ct < c KiG + KiC bQ Cit ]

a b (N + 1)(K G + K C )

rt ·

+ µ1 = 0, 523

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solution. Corollary 2. (a) The optimal overall capacities under scenarios (ii) and (iii) of Proposition 3 are the same, whereas the optimal capacity under scenario (i) is higher. (b) When the IPP operates under scenario (ii), an increase in the volatility of ct increases the optimal overall capacity.12 Corollary 2 implies that the same overall capacity level is optimal whenever the IPPs decide to build natural gas plants. Capacity mix, however, may vary in this case. In addition, higher gas cost volatility induces larger capacity when constructing (some) gas plants is optimal. The latter result is due to the log-normal distribution of the per MWh gas cost, similar to the finding for a single (gas) technology (Gal et al., 2017) and that for dual-fuel plants (Corollary 1 above). Fig. 3. Industry capacity as a function of the per MWh diesel-fuel cost.

6. Illustration of the models’ empirics

Fig. 3 presents the industry capacity as a function of the per MWh cost of diesel, c d , for two levels of gas price volatility: ct = 40% and 100%. The optimal first-stage solution corresponds to Case B of Proposition 2 for c d = 10, 20, 30, 40 $/MWh when ct = 40%; and for c d = 10, 20, 30 $/MWh when ct = 100%. The optimal capacities for the other values of c d correspond to Case A of Proposition 2. Fig. 3 shows that the optimal industry capacity increases when volatility rises. However, it decreases, albeit at a decreasing rate, as the per MWh dualfuel (diesel) cost rises. Figs. 4–7 present the average electricity price, industry production, consumer surplus and profits as functions of the per MWh cost of diesel. These figures are based on simulations entailing the following steps. First, we compute the per MWh cost of gas for each day in the year for a given value of the annualized volatility (40% or 100%) and repeat this process 100 times. For a given optimal capacity, we compute the electricity price, industry production, profits and consumer surplus for each day of the year for all 100 repetitions. The values in Figs. 4–7 are the averages across days and repetitions. The average electricity price increases (Fig. 4), industry production decreases (Fig. 5) and consumer surplus declines (Fig. 6) when c d rises or when ct declines, illustrating the results of Corollary 1. Fig. 7 compares industry profits when the IPPs use dual-fuel capability or only natural gas capability for ct = 40% (the graph on the left) and for ct = 100% (the graph on the right). It shows that industry profits decrease as c d rises or ct declines. In addition, dual-fuel capability has an advantage in terms of profits over a single-fuel capability for small per MWh costs of diesel. This advantage is more pronounced for higher values of the gas price volatility.

This section uses data from Texas to illustrate the two models’ empirics. Our regional choice is motivated by the state’s robust electricity market housing a number of major competitors (Harvey, 2015). The primary generation sources in Texas in 2015 were natural gas (53%) and coal (27%), and the installed capacity was 117,144 MW. The average hourly electricity consumption in 2015 was 44,787 MWh, at an average retail price of $87 per MWh.13 We assume that the price elasticity of demand equals -0.25.14 Thus, the inverse demand function for Texas in 2015 is given by: Pt = 435 0.0078 Qt . We also assume that the number of electricity producers in the market is five (N = 5).15 As in Gal et al. (2017), we set the gas capacity cost to $285/MWday, based on a $1023/kW total installed cost, a 6% WACC, 20 years of depreciation and an annual fixed cost of $15.37/kW (EIA, 2013, Table 1). The initial per MWh natural gas fuel cost, c0 , is $29.5/MWh, based on an average Henry Hub natural gas price of $3.3/MMBtu in 2015, a constant heat rate of 7.88 MMBtu/MWh, and variable non-fuel cost of $3.5/MWh (EIA, 2015, Table 1). The annualized price volatility of natural gas ranges from 20% to 120%, the value for 2014 being 96% (Gal et al., 2017). We also assume a riskless annual interest rate of 4%. 6.1. Technology scenario 1: dual-fuel capability We assume that the dual-fuel plant is a CCGT plant with the capability of burning diesel,16 achieved through a 10% increase in capacity cost. Therefore, the cost of the marginal capacity unit with a dual-fuel capability is T + T = 1.1 × $285/24 = $13.07 per MW-hour. We also assume that the diesel fuel cost, c d , ranges from $10/MWh to $100/ MWh.17

6.2. Technology scenario 2: capacity mix of two single-fuel generation types

12

The model’s outcomes under scenario (i), which describes a market with a single (coal) technology, do not depend on the volatility of ct . Scenario (iii) describes a market with a single (gas) technology, which is analyzed in Gal et al. (2017). 13 U.S. Energy Information Administration, Texas Data: https:// www.eia.gov/electricity/state/Texas. 14 Table 1 in Lijesen (2007) presents a summary of short term price estimates of electricity price elasticities, with values ranging from -0.04 to -0.89. Borenstein and Holland (2005) employ a value of -0.1 and Borenstein (2005) uses elasticity values between -0.025 and -0.5 for their assessments of the California electricity market. Solving the model for other values of price elasticity (-0.1 or -0.05, say) yields similar trends and conclusions. 15 Solving the model for larger N yields similar trends and conclusions. 16 Coal-fired plants with dual-fuel capabilities are less common and much less efficient compared to dual-fuel gas-fired plants that can also use diesel fuel. 17 A diesel fuel cost of $80/MWh is equivalent to a price of $72/barrel, a heat rate of 6.4 MMBtu/MWh, and 5.8 MMBtu per barrel (i.e., a price of $12.4 per

C

This section assumes a coal capacity cost of T = $26.72/MW‐hour, based on a $2290/kW total installation cost, a 6% WACC, 20 years of depreciation and $34.4/kW/year fixed cost (Tidball et al., 2010). We use three per MWh coal costs: $5/MWh, $15/MWh and $25/MWh, based on the assumptions of (a) 19.4 MMBtu per ton of coal; (b) coal prices of $1.7/ton, $18.3/ton and $35/ton; and (c) a heat rate of 11.4 MMBtu/MWh and a variable cost of $4.0/MWh (Tidball et al., 2010). Fig. 8 presents industry capacity as a function of the annualized price volatility of natural gas, ct , for the three per MWh costs of coal, c C . When c C is low, at $5/MWh, the optimal first-stage solution is given by scenario (i) of Proposition 3, in which the IPPs build only coal (footnote continued) MMBtu of diesel oil). 524

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annualized price volatility of natural gas: the higher ct , the higher the share of gas-fired CCGTs in the market’s total capacity. For example, the share of CCGT is only 2.3% when ct = 40%. It equals 66.7% when C ct = 120% . When c is high, at $25/MWh, as it was during the last decade, and as the World Bank (2017) expects it to be during the next decade,18 the optimal first-stage solution is given by scenario (iii) of Proposition 3. The IPPs will only build gas capacity. Total capacity in this case rises when the annualized price volatility of natural gas rises. Confirming Corollary 2, Fig. 8 demonstrates that (a) overall industry capacity is highest when the IPPs choose to build only coal capacity (which happens when the price of coal is sufficiently low), and (b) for any level of gas price volatility, the industry’s overall capacity is the same when the IPPs employ mixed or gas-only capacity, though the share of natural gas in the overall capacity increases with the gas price volatility. Fig. 9 presents the overall industry capacity as a function of the per MWh cost of coal, c C , for ct = 100%. When the per MWh cost of coal is sufficiently high, c C 16 , the optimal first-stage solution is to build only gas capacity. When the coal cost is sufficiently low, c C 9, the optimal first-stage solution is to build only coal capacity. Only in the narrow range of 9
Fig. 4. Average electricity price as a function of the per MWh diesel-fuel cost.

Fig. 5. Industry production as a function of the per MWh diesel-fuel cost.

18 A future price of $2/MMBtu, which is expected by World Bank (2017), implies a fuel cost of $24/MWh (using 10.1 MMBtu per MWh). 19 The narrow range of coal prices in which the coal technology is viable (0 < c C <16) becomes even more narrow when the regulator imposes a tax (penalty) on CO2 emissions. These emissions are 53 and 93 kg of CO2 per MMBtu for natural gas and coal, respectively (EIA, 2016a; Table A.3). If we suppose that the heat rates of natural gas and coal are 7.6 and 10.1 MMBtu per MWh, respectively (EIA, 2015; Table 8.2), CO2 emissions in kg per MWh, are 7.6 × 53 = 402.8 and 10.1 × 97 = 979.7, for natural gas and coal, respectively. Thus, a CO2 tax of $30 per ton, say, implies an increase in the total fuel cost to the IPP, in $/MWh, of 402.8 × 0.03 = 12.08 and 979.7 × 0.03 = 29.39, for natural gas and coal, respectively. In fact, it is straightforward to show that coal-fired generators may not be built at all if the tax on CO2 is sufficiently high. 20 The per MWh gas cost was computed for each day of the year for three values of the annualized volatility: 80%, 100%, and 120%. 21 The increase in the per MWh cost of coal always increases the equilibrium electricity price when coal is used for electricity production (scenarios (i) and (ii)). Here, we chose to present the results of simulations in which the evolution of the natural gas price is relatively smooth and does not deviate too much (more than 70%, say) from the current price of natural gas. When the natural gas price becomes very small or very large over time, which is possible with the log-normal distribution, we may obtain very small or very large average electricity prices (and, consequently, extreme values of industry production, industry profit and consumer surplus). Hence, it is possible, in extreme situations, to find that the average electricity price declines (under scenario (ii), for example) when the annualized volatility of the price of natural gas declines.

Fig. 6. Consumer surplus as a function of the per MWh diesel-fuel cost.

capacity that remains unchanged when ct increases. When c C is moderate, at $15/MWh, the optimal first-stage solution is given by scenario (ii) of Proposition 3 (with the exception of ct = 20%), where the IPPs build both coal and gas plants. In this case, the overall industry capacity increases when the annualized price volatility of natural gas rises (see Corollary 2 above). In addition, the capacity mix depends on the 525

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Fig. 7. Industry profits as a function of the per MWh diesel-fuel cost.

Fig. 8. Industry capacity as a function of the annualized volatility of the per MWh cost of gas. Legend: The share of CCGT in overall capacity is depicted by the striped areas of the bars. The share of coal technology is depicted by the white areas of the bars.

Fig. 10. Average electricity prices as a function of the per MWh cost of coal. 382

50

Annualized volatility = 80% 43.04

42.82

42.80

42.80

42.80

Annualized volatility = 100%

42.80

Industry production (1000 MWh)

Industry capacity (1000 MW)

43.25 40

30

20

10

Annualized volatility = 120%

378.88 378

374

369.66

370

369.10 368.35 0

5

7

9

11

13

15

369.66 369.04 367.94

366

17

5

The per MWh cost of coal ($/MWh)

15

25

The per MWh cost of coal ($/MWh)

Fig. 9. Industry capacity as a function of the per MWh cost of coal ( ct = 100%). Legend: The share of CCGT in overall capacity is depicted by the striped areas of the bars. The share of coal technology is depicted by the white areas of the bars.

Fig. 11. Industry production as a function of the per MWh cost of coal.

profits, and consumer welfare in a Cournot electricity market with dualfuel plants or a mix of single-fuel capacities. Based on the empirically reasonable assumption that the per MWh gas cost is log-normally distributed, we derive closed-form solutions and analytically assess their properties for each technology scenario. Optimal capacity investment in gas technology increases when gas price volatility rises. This occurs because rising gas price volatility increases the probability of low gas prices, which, in turn, increases the IPP’s marginal expected profits. Consequently, the market’s total installed capacity increases, thereby

7. Conclusion and policy implications In the last two decades, natural gas prices in competitive wholesale markets have become more volatile than the prices of other fossil fuels. Large gas price spikes occasionally occur, exposing IPPs to the risk of high fuel costs and low profits. This paper studies the effect of gas price and gas price volatility on capacity investment, production, prices, 526

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N. Gal et al. 25.8

25.47

Industry profits ($ Billion)

25.4

electricity market with identical firms, thus abstaining from the reality of heterogeneous power plants with different heat rates, and the possibility of using renewables. Note, however, that the overall capacity in the two models that we present here is similar (for similar fuel prices and cost and demand parameters), and that the nature of the distribution of the shares of coal and natural gas technologies is similar to those of other studies (Bajo-Buenestado, 2017; Milstein and Tishler, 2012). Second, we employ here a single demand function, while in reality electricity markets are better characterized by two or more different demand schedules over the day (base and peak demands, say).22 Third, we do not model an electric grid’s planning reserve margin, which is often administratively set by a regulator to achieve a predetermined reliability target such as loss of load expectation of 1-dayin-ten-years. The resulting reserve margin is ~15% of the forecast system peak demand for Texas and California. The grid also has a daily reserve margin of ~6% of the daily peak system demand to ensure safe and reliable operation (see, for example, NERC, 2018). One can model such reserve requirements by scaling up the demand function, which is not expected to materially change our model’s qualitative results. Fourth, the optimal solution when investment in capacity can occur only in a lumpy manner may differ from ours portrayed herein, particularly when the electricity market size is small. The reality of lumpy capacity investment is likely to have a minimal effect on the optimal solution of our model for large electricity markets such as those in Texas, California and most European countries. Finally, large-scale development renewable energy will likely accelerate in electricity markets, thanks to the international commitments made in 2015 at the Paris Climate Change Summit. In the future we intend to extend our models to include renewable generation technologies.23

Annualized volatility = 80% Annualized volatility = 100% Annualized volatility = 120%

25.0

24.6

24.48

24.2

24.24

24.15 24.01

24.03 23.93

23.8 5

15

25

The per MWh cost of coal ($/MWh)

Fig. 12. Industry profits as a function of the per MWh cost of coal.

22 Accounting for baseload and peak capacities can occur through using different demand functions for the off-peak and for the peak periods. The demand during the peak period is higher than the demand during the off-peak period. This setup appears in Milstein and Tishler (2015). Using different demand functions for the peak and off-peak periods in this study will not change the nature of our results, but will likely yield higher electricity generation during the peak period and a smaller one during the off-peak period. Total capacity will be determined by the peak period. In addition, price spikes will likely be higher and more frequent during the peak period. The shares of the coal (base) and natural gas (peaking) technologies depend on the relative fuel prices, and on the parameters of the demand and cost functions. That said, the solution of these shares in Milstein and Tishler (2012), which employed stochastic demand function, is similar to that in the technology mix of this study. Furthermore, the overall capacity is fairly similar (for similar values of the parameters) in the two models in this study (dual-fuel, technology mix). 23 There are two possible ways to model renewable energy (wind, or PV, say) in an optimal capacity and operation model. First, one may treat renewable energy development as exogenous. The introduction of new renewables often requires subsidies and/or major regulatory restrictions/interventions such as the allocation of the renewable capacities and exogenous pricing of the renewables (Bhagwat et al., 2017). In this case, one can model the net demand (= actual market demand - renewable energy output). As the marginal fuel cost of PV or wind is zero, it will always be dispatched whenever available. Hence, the analysis and results of our paper will remain the same. Alternatively, one may treat the capacity of the renewable technology as endogenous (Milstein and Tishler, 2015). Comparing the model of Milstein and Tishler (2015), in which the price of natural gas is deterministic, to our technology mix model suggests the nature of the results of our model will not change materially. In Milstein and Tishler (2015), natural-gas-fired generation retains a large share of the market’s total capacity. Hence, higher volatility of the natural gas price tends to increase natural-gas-fired generation’s capacity and the market’s overall capacity. In summary, the preceding discussion indicates that the quantitative solution of our model can be changed by the market share of renewables; this is notwithstanding that the qualitative nature of our results, and the conclusion that higher gas price volatility increases capacity, will remain valid.

Fig. 13. Consumer surplus as a function of the per MWh cost of coal.

reducing electricity prices, raising generation output and improving consumer welfare. For the first technology scenario of dual-fuel generation, we find that consumer surplus may be increased (reduced) when dual-fuel capability is used and the diesel price is low (high). However, IPPs’ expected profit may decline due to the additional cost incurred to obtain dual-fuel capability. The dual-fuel capability is more advantageous for both consumers and producers when the per MWh diesel cost is low or when a high diesel cost is accompanied by a high gas volatility. For the second technology scenario of a capacity mix of single-fuel coal and gas generation plants, we find that an IPP adds coal capacity into its capacity mix only if the coal generation cost is significantly lower than the expected gas generation cost. Indeed, we observe that during the last several years, gas plants account for most of the conventional generation’s new capacity in the world and it seems reasonable that this trend will continue in the future. Moreover, the share of coal capacity in the capacity mix tends to decline as the volatility of natural gas increases. The policy implication of our findings is that a government should not intervene to reduce the price volatility of a well-functioning competitive natural gas spot market because such an action can have the unintended consequence of discouraging capacity investment, thereby raising electricity prices, and reducing consumer surplus. Nevertheless, the 'missing money' problem may still remain in an electricity market, necessitating the continued use of capacity markets to encourage generation capacity expansion. We would be remiss had we ignored some of the limitations of the models presented herein. First, our models characterize a simplified 527

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Acknowledgements

C.K. Woo's research was supported by a research grant from the Faculty of Liberal Arts and Social Sciences of the Education University of Hong Kong.

We thank two referees for very helpful comments and suggestions. Appendix A 7.1. Proof of Proposition 1 The first-order condition of (5) requires:

a

where Q

it

andQit* =

1 (a 2b

j

(A1)

ctmin = 0,

b·(2Qit + Q it )

Q Q* = i jt . The optimal solution of (5) is it

Ki if

ctmin

a 2bKi

bQ it ; otherwise, it is (A1), i.e. Qit* =

1 (a 2b

ctmin

bQ it ) .

Let c Ki a 2bKi bQ it . Then, there are two possible cases: c Ki c d (case A) and c Ki > c d (case B). For each case we can build the sequence of non-overlapping intervals of ct such that the value of ct is monotonically increasing. 1 Case A. If ct c Ki , then ctmin = ct and Qit* = Ki . If ct > c Ki but ct c d , then ctmin = ct and Qit* = 2b (a ct bQ it ) . If ct > c d , then ctmin = c d

c d bQ it ) . The equilibrium price is obtained by inserting the equilibrium quantity into (1) for each interval of ct . Case B. If ct c d , then ctmin = ct and Qit* = Ki . If ct > c d , then ctmin = c d and Qit* = Ki . The equilibrium price is obtained by inserting the equilibrium quantity into (1) for each interval of ct .

7.2. Proof of Proposition 2 Case A. Inserting (8a) into (7) and differentiation yield the first-order condition for optimal capacity: T

c Ki

rt ·

e

[a

0

t=1

2bKi

bQ

+ T

ct ] g (ct ) dct

it

= 0.

(A2)

Assuming symmetry of all N producers (i.e., K1 = …=KN = K ), we obtain (9). Case B. Inserting (8b) into (7) and differentiation yield the first-order condition for optimal Ki : T

cd

rt ·

e

[a

0

t=1

2bKi

bQ

ct ] g (ct ) dct +

it

cd

[a

2bKi bQ

+ T

c d] g (ct ) dct

it

=0

(A3)

Assuming symmetry of all N producers (i.e., K1 = …=KN = K ) and rearranging yield: T

cd

rt ·

e

0

t=1

[a

b (N + 1) K

ct ] g (ct ) dct +

T

=

e

rt ·

e

rt ·

[a

t=1 T

= t=1

cd

b (N + 1) K ]

cd

(c d

0

0

ct ) g (ct ) dct +

ct g (ct ) dct

cd

e r (1 e 1 e

[a

rT ) r

c d] g (ct ) dct

[a b (N + 1) K

cd

g (ct ) dct

+ T

b (N + 1) K ]

cd

cd

+ T

+ T

=0

(A4)

We obtain the optimal solution, K *, from (A4): T t=1

K* =

rt ·

e

cd 0

(c d

ct ) g (ct ) dct +

T e rt t=1

b (N + 1)·

1 a b (N +1)

+ T

cd

.

(A5)

We also obtain the condition for case B from (A4): T

c K * > cd

e t=1

rt ·

cd 0

(c d

ct ) g (ct ) dct <

e r (1 e 1 e

rT ) r

·

+ T

.

(A6)

7.3. Proof of Corollary 1 (a) Case A. The proof is identical to that of Proposition 1 in Gal et al. (2017), since the capacity cost (the only part distinguishing (9) from Eq. (13) in Gal et al., 2017) does not depend on ct . Case B. Differentiation of (10) with respect to ct yields:

K*

=

ct

T

1 b (N + 1)·

T e rt t=1

·

e ct t = 1

cd

rt 0

(c d

ct ) g (ct ) dct .

(A7)

Using the same logic as in the proof of Proposition 1 in Gal et al. (2017), it is straightforward to show that That is,

K* ct

>0 .

(b) Case A. Expected operating profits on day t in (8a) at the optimal first-stage solution are: 528

ct

c T e rt 0 t=1

d

(c d

ct ) g (ct ) dct > 0 .

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N. Gal et al.

E[

| K *] =

it

0 cd 1 cK* b

+ where c K * a

E[

it

| K *]

cK*

a ct 2 g (ct ) dct N+1

ct ]· K *g (ct ) dct +

1 b

cd

2

a cd N+1

g (ct ) dct ,

(A8)

bNK *. The chain rule implies: K* it | K *] · . K* ct

E[

=

ct

From part (a), we have

| K *] = K*

E[

bNK *

[a

cK*

it

E [ it | K *]

Therefore,

ct

ct

(A9)

>0 . Differentiation of (A8) with respect to K * yields

b (N + 1) K *

[a

0

K*

cK*

ct ] g (ct ) dct =

0

(c K *

ct ) g (ct ) dct > 0.

(A10)

>0 .

Case B. The expected operating profits on day t in (8b) at the optimal first-stage solution are given by

E[

it

cd

| K *] =

bNK *

[a

0

ct ] K *g (ct ) dct +

cd

[a

c d]· K *g (ct ) dct .

bNK *

(A11)

Differentiation of (A11) with respect to K * yields

E[ cd 0

cd

it | K *] = K*

[c K *

b (N + 1) K *

[a

0

ct ] g (ct ) dct +

As long as c K * >

cd

[c K *

ct ] g (ct ) dct +

cd

[a

b (N + 1) K * c d] g (ct ) dct =

c d] g (ct ) dct .

(A12)

holds for case B, (A12) is positive. Applying the chain rule and the result of part (a), i.e.,

cd

(c) Generally, the expected consumer surplus on day t is given by

E [CSt ] =

1 (a 2

0

K* ct

>0 , yields

E [ it | K *] ct

>0 .

Pt ) Qt g (ct ) dct .

(A13)

Case A. Inserting the equilibrium solution into (A13), we get the expected consumer surplus at the optimal first-stage solution on day t:

1 b (NK *)2g (ct ) dct + 2

cK*

E [CSt | K *] =

0

cd cK*

1 N 2 (a ct )2 g (ct ) dct + 2b (N + 1)2

cd

1 N 2 (a c d ) 2 · g (ct ) dct . 2b (N + 1)2

(A14)

Applying the chain rule, we get:

E [CSt | K *]

E [CSt | K *] K * · . K* ct

=

ct

From part (a), we have

E [CSt | K *] = K*

cK* 0

ct

(A15)

>0 . Differentiation of (A14) with respect to K * yields

bN 2K *g (ct ) dct >0.

E [CSt | K *]

Therefore,

K*

ct

(A16)

>0 .

Case B. Inserting the equilibrium solution into (A13), we get the expected consumer surplus at the optimal first-stage solution on day t:

E [CSt | K *] =

1 b (NK *)2g (ct ) dct . 2

0

(A17)

Differentiation of [A.17] with respect to K * yields E [CSt | K *] ct

E [CSt | K *] K*

=

bN 2K *> 0

. Applying the chain rule and the result of part (a), i.e.,

>0 .

K* ct

>0 , yields

(d) Generally, the expected price on day t is given by

E [Pt ] =

0

Pt g (ct ) dct .

(A18)

Case A. Inserting the equilibrium solution into (A18), we get the expected price at the optimal first-stage solution on day t: cK*

E [Pt | K *] =

0

(a

bNK *) g (ct ) dct+

cd cK*

a + Nct g (ct ) dct + N +1

c

d

a + Nc d g (ct ) dct . N +1

(A19)

Applying the chain rule, we get:

E [Pt | K *]

=

ct

E [Pt | K *] K * · . K* ct

From part (a), we have

E [Pt | K *] = K* Therefore,

cK* 0

ct

(A20)

>0 . Differentiation of (A19) with respect to K * yields

( bN ) g (ct ) dct < 0.

E [Pt | K *] ct

K*

(A21)

<0 .

Case B. Inserting the equilibrium solution into (A18), we get the expected price of the optimal first-stage solution on day t: 529

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N. Gal et al.

E [Pt | K *] =

bNK *) g (ct ) dct .

(a

0

(A22) E [Pt | K *] K*

Differentiation of (A22) with respect to Ki * yields E [Pt | K *] ct

bN < 0 . Applying the chain rule and the result of part (a), i.e.,

=

<0 .

K* ct

>0 , yields

7.4. Proof of Proposition 3

K G * and K C *, the optimal capacities of the i-th firm, may be either interior or corner solutions. That is, we have to examine whether the KKT conditions, given by (19), are satisfied simultaneously at each one of the three possible solutions (we do not analyze the trivial solution, i.e., K G * = 0 and K C * = 0 ). Scenario (i). K G * = 0 and K C * 0 , i.e., µ 2 = 0 . K C * and µ1 are given by: K C*

=

C

cC

a

T

,

b (N + 1)

(A23)

and T

e r (1 e rT ) G · 1 e r T

µ1 =

e

cC + C / T

rt ·

0

t=1

[c C +

C /T

ct ] g (ct ) dct .

Thus, this point is optimal iff 1) K C * 0 , i.e., a c C + Scenario (ii). K G * and K C * are both interior solutions, T rt ·

e t=1

a b (N + 1)(K G *+ K C *)

b (N + 1)(K G * + K C *)

[a

0

(A24)

G e r (1 e rT ) , and 2) µ1 0 , i.e. 1 e r · T T i.e., = 0 , µ1 = 0 , and µ 2 = 0 . K G *

C

cC + C / T T e rt · 0 t=1 K C*

and

[c C + C /T are derived from:

ct ]·

e r (1 e rT ) G · , 1 e r T

g (ct ) dct =

ct ] g (ct ) dct .

(A25)

and T

e

rt ·

a b (N + 1) K C *

t=1

a + b (N + 1) K C *] g (ct ) dct

[ct

T

=

rt ·

e t=1

e r (1 e 1 e

c C ) g (ct ) dct

(ct

0

rT ) r

·

C

G

T

.

(A26)

The RHS of (A26) may be rewritten as follows: T

e

rt ·

t=1

0

C

cC

ct

T

T C

c +

C

/T

cC

[ct

T

e r (1 e rT ) G · = 1 e r T

g (ct ) dct +

C / T ] g (c ) dc t t

e

cC + C / T

rt ·

0

t=1

e

[c C +

C

C

c + T e rt · 0 t=1

/T

[c C +

C /T

C

c +

C

/T

cC

[ct

Scenario (iii). K G * T

e t=1

rt ·

a b (N + 1) K G * 0

cC + C / T T e rt · 0 t=1

<

[c C +

C /T

ct ] g (ct ) dct , and 2) K C *> 0 , i.e.

ct ] g (ct ) dct

T t=1

ct ]·

G e r (1 e rT ) ·T 1 e r

Thus, this point is optimal iff 1) K G *> 0 , i.e.,

rt ·

C /T

e r (1 e rT ) G · . 1 e r T

g (ct ) dct +

e

rt ·

t=1

C / T ] g (c ) dc t t

+

e r (1 e rT ) G · . 1 e r T

0 and K C * = 0 , i.e., µ1 = 0 . K G * and µ 2 are solved from:

[a

b (N + 1) K G *

e r (1 e 1 e

ct ] g (ct ) dct =

rT ) r

·

G

T

,

(A27)

and

µ2 =

e r (1 e 1 e

rT ) r

·

C

G

T

T

e

rt ·

t=1

0

(ct

Thus, this point is optimal iff µ 2 0 , i.e.

g (ct ) dct >

T e rt · c C + C / T t=1

[ct

cC

c C ) g (ct ) dct . C

c + T e rt · 0 t=1 C / T ] g (c ) dc t t

+

(A28)

C

/T

[c C +

C /T

G e r (1 e rT ) ·T . 1 e r

530

ct ]·

Energy Policy 126 (2019) 518–532

N. Gal et al.

7.5. Proof of Corollary 2 (a)

K C*

under

scenario

(i)

G a b (N + 1) K C * T e r (1 e rT ) ·T e rt · 0 [a t=1 1 e r C* a b ( N + 1) K T e rt · 0 [a b (N + 1) K C * t=1 K G * + K C * under scenario (ii).

is

given

b (N + 1) K C *

ct ] g (ct ) dct

by

(A23).

ct ] g (ct ) dct ,

G e r (1 e rT ) ·T . 1 e r

Substituting which

(A23) should

into be

the

RHS positive.

of

(A24)

yields

That

is,

Therefore, it follows from (A25) that K C * under scenario (i) is at least

Direct comparison of (A25) and (A27) shows that K G * + K C * under scenario (ii) and K G * under scenario (iii) are equal.

(b) The overall capacity under scenario (ii) is given by (A25). Thus, the proof is identical to the proof of Proposition 1 in Gal et al. (2017), since Eq. (13) in Gal et al. is identical to (A25) for overall capacity.

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