Question:
You are given the full text of a scientific research paper.

    Summarize the study’s **policy implications**—how the research findings can inform policy or decision-making. Your response should:
    - Interpret what the results suggest for governments, regulators, or institutions
    - Highlight potential actions, strategies, or decisions informed by the study
    - Use policy-appropriate language without speculation beyond the study’s conclusions
    - Be clear, practical, and informative

    Write a short paragraph for the “Policy Implications” section of a policy brief.

    Scientific Research Paper:
    Articles
https://doi.org/10.1038/s41560-019-0476-1
1Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, USA. 2Federal Energy Regulatory Commission,
Washington, DC, USA. *e-mail: jacobmays@cornell.edu
S
ince the inception of liberalized electricity markets in the
late 1990s, merchant investment into new generation capac-
ity in the United States has been dominated by gas-fired units.
The simplest explanation for this trend points to marked improve-
ments in the efficiency of combined-cycle plants along with, more
recently, low natural gas prices brought about by the shale revo-
lution. Without attempting to compare the magnitude of the two
effects, we propose a second explanation: distortions brought about
by capacity markets in the absence of more complete risk trading.
Three alternative paradigms govern investments in generation
capacity: vertical integration, energy-only markets and installed
capacity (ICAP) markets1. In traditional vertically integrated sys-
tems, investors are guaranteed a rate of return by a regulatory pro-
cess, which effectively shifts most risk to ratepayers and creates
the incentive for excess investment. One goal behind deregulation
was to encourage a more efficient allocation of this risk2. The theo-
retically ideal competitive market provides the basis for a second
paradigm, energy-only markets. Under the energy-only design,
generators earn all their revenue through the sale of energy and
ancillary services. Prices rise substantially during times of scarcity,
which allows generators to recover their fixed costs. In practice, a
lack of price-responsive demand, combined with inconsistencies
between the market clearing process and actions taken by opera-
tors on the grounds of reliability, hampered the formation of effi-
cient scarcity prices3,4. An administrative response is to introduce an
operating reserves demand curve that reflects the probability that
the system operator will need to take emergency actions (for exam-
ple, voltage reductions or rolling blackouts) to prevent a cascad-
ing failure5. As the probability of such actions approaches one, the
price grows to an estimated value of the lost load, which is typically
two orders of magnitude larger than average prices (for example,
US$9,000 MWh–1 for the Electric Reliability Council of Texas). With
the exception of the Electric Reliability Council of Texas, wholesale
market operators in the United States opted for the third paradigm.
Under the ICAP design, load-serving entities are required to either
supply or procure an administratively determined level of capacity.
Energy prices in these markets are capped at a much lower level
(for example, US$2,000 MWh–1 or lower), and revenue from the
sale of energy and ancillary services is supplemented by payments
for capacity. The intent of these capacity payments is to balance the
‘missing money’ that results from the cap and other price-suppress-
ing actions to restore the outcomes that would be achieved in an
ideal energy-only market6–8.
We are not the first to suggest that capacity markets bring about
a distortion of the technology mix. Indeed, this assertion is com-
mon enough that the overview in Cramton et al.7 deems it ‘a deep
confusion concerning capacity markets’ that ‘takes many forms and
contains contradictory views’. In this context, our contribution lies
in formalizing a mechanism by which distortion could arise.
To illustrate this possibility, we developed a stochastic equilib-
rium model of the type first described in Ehrenmann and Smeers9,
as well as an algorithm to solve large-scale instances of the problem.
The fuel neutrality of capacity markets relative to an ideal energy-
only market relies on the assumption embedded in the classic opti-
mization framework that the cost of capital is exogenous to the
market design. The model in this article relaxes that assumption,
which enables us to take into account the stabilizing effect that the
introduction of capacity mechanisms has on generator revenues.
Our numerical examples describe a simple system with three gener-
ation technologies: a baseload technology with high capital costs but
low operating costs (for example, nuclear), a renewable technology
with a similar cost profile but variable output (for example, wind)
and a peaker technology with low capital costs but high operating
costs (for example, gas). Results on these test systems suggest that
an increased reliance on existing capacity market structures without
the emergence of other forms of risk trading is likely to work against
technologies with low operating costs, a category that includes all of
the most scalable forms of low-carbon generation.
Resource adequacy and risk trading
Separate from restoring the missing money required for proper
investment incentives, advocates for capacity markets point to the
benefits that the ICAP design provides in reducing risk. By replac-
ing highly volatile scarcity prices with a more consistent monthly
or annual payment, capacity mechanisms provide a financial hedge
for both generators and loads. In the traditional analysis, capacity
Asymmetric risk and fuel neutrality in electricity
capacity markets
Jacob Mays   1,2*, David P. Morton   1 and Richard P. O’Neill2
In many liberalized electricity markets, power generators can receive payments for maintaining capacity through capacity
markets. These payments help stabilize generator revenues, making investment in capacity more attractive for risk-averse
investors when other outlets for risk trading are limited. Here we develop a heuristic algorithm to solve large-scale stochastic
equilibrium models describing a competitive market with incomplete risk trading. Introduction of a capacity mechanism has
an asymmetric effect on the risk profile of different generation technologies, tilting the resource mix towards those with lower
fixed costs and higher operating costs. One implication of this result is that current market structures may be ill-suited to
financing low-carbon resources, the most scalable of which have high fixed costs and near-zero operating costs. Development
of new risk trading mechanisms to replace or complement current capacity obligations could lead to more efficient outcomes.
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payments have the impact of a call option with a strike price equal
to the price cap for the market2,7,10. Loads make an upfront payment
to generators, in exchange for which the effective price they see in
the spot market does not exceed the price cap. As energy markets
do not produce an accurate underlying price in times of scarcity,
real-world capacity markets do not precisely replicate an option.
We discuss some of the implementation challenges this causes in
Supplementary Note 1. In the ideal case, however, incentives on
the margin are maintained: loads must pay high spot prices if they
consume more than they have procured in advance, and generators
have to buy back their position if they are unable to perform.
It has long been acknowledged that risk aversion could lead to a
suboptimal investment in generation capacity9,11,12. However, prior
studies do not demonstrate any consistent technological shift. In
energy-only markets, peaking units face substantial risk due to
their dependence on the occurrence of high prices in a small num-
ber of hours. At the same time, several authors emphasized the
natural hedge enjoyed by gas generators, which often set the price
of electricity13–16. The ability to trade risk is a key element of this
discussion. With complete risk trading, along with the standard
assumptions of competitive markets, an equilibrium would achieve
the (risk-aware) social optimum17–20. The connection between equi-
librium and optimum implies that the impact of risk aversion on the
generation mix can be in any direction depending on the types of
risk that most concern market participants. This theoretical result,
however, depends on the presence of ‘complete’ markets, which
include a security that corresponds to every possible future state of
the world. In this context, our focus is less on incompleteness than
on asymmetry: the introduction of a capacity market amounts to a
trade against a subset of future world states that peaking plants are
particularly concerned about. In the PJM interconnection (PJM),
for example, a unit with a marginal cost of US$10 MWh–1 could
expect to earn 17% of its operating profits from the capacity market,
whereas a unit with a US$100 MWh–1 marginal cost could expect
90% (Supplementary Note 2). Along these lines, our results suggest
a different interpretation of the natural hedge of gas. By itself, the
hedge does not confer an advantage. It is only when combined with
a capacity mechanism, which removes risk related to the frequency
of scarcity events, that the hedge becomes an advantage.
The size of this advantage depends on the availability of other
outlets for risk trading. In practice, the evidence suggests that mar-
kets in risk are far from complete21. A natural resolution to the
problem occurs if investors in generation are able to sign long-term
contracts with loads, who are exposed to price risk in the oppo-
site direction. With the support of state and corporate procurement
programmes, a robust market has developed for renewable genera-
tors that seek long-term power purchase agreements22. In areas with
competitive retail markets, however, contracts rarely exceed two
years outside these programs23. Small retailers are unable to com-
mit to long-term contracts without risking customer defection in
periods when wholesale prices drop below the contracted rate11.
The phenomenon of insufficient demand-side interest in hedging
is seen as characteristic to commodity markets beyond electricity24,
especially in the presence of transaction costs25.
If they are unable to secure a power purchase agreement, mer-
chant generators are often able to hedge risk on timescales of five
to seven years through contracts with financial traders able to bear
some of the risk26. However, these bilateral deals cannot match the
liquidity of centralized capacity markets, which leads some to pro-
pose formalized exchanges27,28. Owing to the proprietary nature of
these arrangements, it is a challenge to assess the liquidity in this
market. Nevertheless, the available evidence suggests some endo-
geneity between cash flows and the cost of capital. Among pub-
licly available ratings, debt issued for projects in the energy-only
Electric Reliability Council of Texas market receive ratings of B
and CCC, whereas projects in PJM typically receive a higher BB
rating29. Separate evidence comes from renewable support schemes
in Europe. Although some programmes (for example, feed-in tariffs)
lead to predictable revenue streams, others (for example, quotas)
result in payments that vary with market conditions. The relative
ease of financing stable revenues can lead to a greater deployment
for the same amount of government support30,31.
Even if the size of the effect is small at present, insufficient risk
trading may have large consequences for long-term market design.
In general, long-run models predict that the incorporation of solar
and wind will lead to lower average electricity prices but greater vol-
atility, which prompted many to consider how capacity mechanisms
may need to evolve to guarantee resource adequacy1,32–35.
Two-generator example
To illustrate the impact that incomplete markets in risk may have
on the capacity mix that arise in equilibrium, we developed two
simple test systems. The first features two technologies: a baseload
resource with a high capital cost but a low and certain operating
cost, and a peaking resource with a low capital cost but a high and
uncertain operating cost. The second example, developed in the
next section, adds a renewable resource with variable availability.
In the context of current US markets, these three technologies can
be thought of as existing nuclear plants, many of which are con-
sidering retirement in the near term, and new gas plants and wind
farms, which are being constructed at a rapid rate. With that said,
the parameters were chosen to best illustrate the impact of risk trad-
ing, rather than to recreate or predict market outcomes. In keep-
ing with US market design principles, the focus is on economic
efficiency, with no explicit environmental goals within the market.
Support mechanisms for carbon-free generation are assumed to be
reflected through the investment and energy costs associated with
each technology. Unlike most US markets, we did not implement a
mandatory capacity market: resource decisions are driven directly
by the value of load expressed by consumers. Instead, we mim-
icked a capacity market through a call option with a strike price of
US$1,000 MWh–1. If the cost of capital were exogenous to the mar-
ket design, this would have no effect on the resulting capacity mix7,8.
We also considered a future settling at US$50 MWh–1.
For the experiments, the nominal demand is based on the load
duration curve of PJM in 2017, during which demand ranged from
58 to 146 GW. The first example focuses on two generation technol-
ogies, baseload and peaker, and two sources of uncertainty, demand
and the fuel cost of the peaker. For both technologies, 90% of ICAP
is available in every time block. Fuel prices and demand shifters are
modelled as random variables with 100 equally likely year-long sce-
narios for the dispatch problem. We considered two contracts: a call
option with a strike price of US$1,000 MWh–1 and a future settling
at US$50 MWh–1. The probabilistic model is discussed in Methods,
and additional details on the scenario assumptions are given in
Supplementary Note 2 and Supplementary Tables 1 and 2.
To characterize the risk attitude of market participants, we used
a weighted sum of the expected value of surplus, with weight β, and
conditional value at risk (CVaR) of the α-level tail of surplus, with
weight 1 − β. CVaR is a coherent risk measure that lends itself to
inclusion in a mathematical optimization model36,37. Lower values
of α and β imply a greater risk aversion, with either α = 1 or β = 1
implying a risk-neutral agent. In real-world contexts, market opera-
tors have limited ability to ascertain or influence the risk preferences
of market participants. Accordingly, our results do not focus on the
effect of risk aversion itself, but instead on the availability of risk
trades, a phenomenon that market operators and regulators have
a greater potential to address. Along these lines, our priority when
choosing α and β was not necessarily to match the risk preferences
of real-world market participants, but to offer a clear illustration of
the impact of incomplete markets. Tests with a number of values for
these parameters yielded the same directional results.
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Using risk parameters α = β = 0.5 for all the market partici-
pants, the socially optimal mix (that is, the complete trading case)
includes a baseload capacity of 89.7 GW and a peaking capacity of
77.6 GW. We tested cases with no trading, each contract individu-
ally, and both contracts available, and compared the results to this
social optimum. Table 1 shows the capacity mix that arises in each
case, the financial trades made by investors in each technology and
the decrease in consumer surplus relative to the complete trading
case. The Methods section describes the associated mathematical
model, solution algorithm and metric measuring proximity of the
algorithm’s solution to an equilibrium.
Incomplete trading leads to a lower ICAP and significant degra-
dation of surplus. To achieve an equilibrium close to the complete
trading ideal, both contracts must be available.
The most striking result in Table 1 occurs in the Option Only
case, in which the baseload technology is absent from the equilib-
rium mix. This observation is particularly important because, as
discussed above, the options contract is designed to have the same
financial impact as the capacity markets currently utilized or under
discussion in many real-world markets. A hint as to the mechanism
at work is provided by the trades made in the Both Contracts case.
The peaker technology prefers to sell options, selling 78.4 GW of
options contracts against only 1.2 GW of futures. The baseload tech-
nology exhibits the opposite preference, selling 80.7 GW of futures
but no options. Tables 2 and 3 show the reason for these trading
preferences. Table 2 shows the payouts for the futures contract in
the Both Contracts case. The 50 bold cells in the table represent
the worst-case realizations of demand and fuel cost for the basel-
oad technology in the No Trading case. As α = 0.5 in this example,
these 50 scenarios are the worst-case scenarios identified by the
CVaR calculation for the baseload investors. Analogously, Table 3
shows the payouts for the option contract in the Both Contracts case
along with the worst-case combinations of demand and fuel costs
for the peaker technology. The difference in the bold cells between
the two tables reflects the risk exposure of the two technologies:
although investors in the peaking technology are only concerned
about lower-than-expected demand, investors in the baseload tech-
nology are concerned about both lower-than-expected demand and
lower-than-expected fuel cost. As we discuss in Methods section,
employing CVaR is equivalent to modifying the nominal probability
distribution under a risk-neutral objective function. In this numeri-
cal example, the modified probability mass function for each agent
corresponds to placing probability 3
200
I
 on the bold scenarios and
1
200
I
 on the regular scenarios in the respective tables (in place of the
nominal 1
100
I
 on all scenarios).
To improve their overall risk profile, investors in each technol-
ogy choose to sell contracts with payouts that align with favourable
underlying outcomes. Higher payouts for the futures contract cor-
respond to non-bold cells in Table 2, whereas higher payouts for the
options contract correspond to non-bold cells in Table 3. In cases
for which only one trade is available, the equilibrium shifts towards
the resource with a risk profile that is better balanced by that trade.
Accordingly, moving from the No Trading case to the Future Only
case increases the baseload capacity from 76.1 to 89.2 GW. The
effect is more drastic in moving from the No Trading case to the
Option Only case, which brings the peaking capacity from 88.9 to
166.7 GW and eliminates the baseload technology from the mix.
An additional interesting feature of Table 1 is that, under the
Option Only case, investors in peaking technology sell a volume of
options that exceeds the ICAP. As the assumed availability is 0.9,
it is clear that these units cannot physically back up the financial
trade. Real-world capacity mechanisms typically require physical
assets. From a liquidity standpoint, a related issue is that the two-
stage model developed in this article enables generators to hedge
over their entire operating life, whereas real-world capacity com-
mitments are often much shorter (for example, one year in PJM).
It is a mistake to object that a short delivery period eliminates
the financial hedging property of capacity markets: more important
than the cash flow for the agreed period is the volatility of future
cash flows. The option payouts in Table 3, which reflect the distri-
bution of annual revenues during scarcity periods that might be
expected in an energy-only setting, range from US$0 to 138 kW–1 yr–1,
with an average of US$41 kW–1 yr–1 and a coefficient of variation of
1.12. The presence of a capacity market reduces this volatility sub-
stantially. In PJM, for example, capacity market prices have ranged
from US$28 to 60 kW–1 yr–1 in the six years since the introduction
of Capacity Performance38, with an average of US$47 kW yr–1 and a
coefficient of variation of 0.24. That said, both the physical back-up
requirement and the short-term nature of real-world commitments
make the level of hedging achieved in Table 1 unlikely. Similar to
the strategy in de Maere d’Aertrycke et al.39, we mimicked the lack
of liquidity by limiting the allowed volume of options trades to a
percentage of ICAP for each technology and recalculated the equi-
librium mix. As shown in Fig. 1, the amount of baseload capacity in
equilibrium declines as liquidity in the options market increases, to
reach zero when trade volumes equal 60% of ICAP.
Three-generator example
In the second example, we added a variable generation resource with
four scenarios to govern its availability. The availability scenarios for
the variable technology are distinguished in how they are coupled with
the load duration curve. Although each scenario has an average avail-
ability of 37.5% over time, the four are temporally correlated with a
fixed load in the following manner: strongly positive, weakly positive,
weakly negative and strongly negative. Availability for the baseload
and peaker generators remains at 0.9. Scenarios for fuel and demand
are retained from the two-generator example, which results in a total
of 400 equally likely year-long second-stage scenarios. Investment cost
and risk parameters were updated to ensure that all three technologies
are represented in both the socially optimal mix and the No Trading
equilibrium. Additional details on these assumptions are given in
Supplementary Note 4. In this example, the socially optimal mix
Table 1 | Equilibria in two-generator example
No trading
Option
only
Future
only
Both
contracts
Capacity (GW)
 Baseload
76.1
0.0
89.2
89.6
 Peaker
88.9
166.7
76.8
77.7
 Total
165.1
166.7
166.0
167.3
Trade volume (GW)
 Baseload
 Future
80.3
80.7
 Option
0.0
0.0
 Peaker
 Future
31.3
1.2
 Option
167.1
78.4
Proximity to
equilibrium (%)
0.000
0.000
0.000
0.002
Change in surplus
(US$ billion yr–1)
−12.3
−6.8
−2.1
−0.0
Four financial trading cases are shown in order of increasing surplus. The No Trading case leads
to underinvestment in capacity. The capacity mix tilts towards the peaker technology in the
Option Only case and towards the baseload technology in the Future Only case. A mix close to
the complete trading ideal is achieved in the Both Contracts case. Risk parameters α = β = 0.5 are
assumed for the consumer and both generators.
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includes a baseload capacity of 31.3 GW, peaking capacity of 104.9 GW
and variable capacity of 139.2 GW.
In addition to the option and futures contracts used in the pre-
vious example, we define a unit contingent contract that matches
the availability of the variable generator40. The interaction between
availability and demand in determining the occurrence of scar-
city conditions means it is difficult to determine in advance which
combinations of random variable realizations will represent the
worst-case scenarios for market participants. However, the intent
is to structure one contract that matches the risk profile of each
technology: futures to the baseload technology, options to the peak-
ing technology and unit contingent to the variable technology. The
equilibrium results are split into two tables, oriented in order of
increasing surplus. Table 4 shows the capacity mix that arises when
at most one trade is available. Comparing the bold cells within each
row, it can be seen that introducing any of the trades individually
results in more capacity of the corresponding technology. The
market share of the baseload unit collapses after the introduction of the
unit contingent contract. As in the two-generator example, intro-
duction of the option contract (analogous to current capacity mar-
kets) results in a complete exit of the baseload technology.
Table 5 shows the results for the cases in which two or three
contracts are available. Here the bold cells demonstrate the effect
of introducing the third contract. In all three pairwise comparisons,
the addition of the third contract shifts the capacity mix towards the
corresponding technology.
Although the directional pattern holds when moving from one to
two available trades, the magnitudes are in some instances weaker.
In the most extreme example, the Future case and the Future + Unit
case result in precisely the same equilibrium. With that capacity
mix, the worst-case outcomes for the consumer involve a low output
for the variable technology. Accordingly, the consumer is reluctant
to purchase unit contingent contracts and none are traded.
Table 2 | Alignment of futures payout to risk faced by baseload technology
Demand
Fuel cost
1
2
3
4
5
6
7
8
9
10
 1
−225
−194
−162
−131
−107
−77
−47
−16
14
45
 2
−221
−189
−157
−125
−99
−68
−37
−5
26
57
 3
−218
−185
−151
−118
−92
−60
−28
4
37
69
 4
−214
−179
−145
−111
−85
−52
−19
14
48
81
 5
−192
−161
−130
−99
−75
−43
−9
26
60
94
 6
−148
−116
−84
−52
−28
4
35
66
97
128
 7
−119
−84
−48
−12
19
52
84
116
148
180
 8
−101
−65
−30
6
41
69
105
140
176
211
 9
−73
−38
−3
32
67
97
132
168
203
239
 10
−41
−5
31
67
102
132
168
203
238
273
Each cell corresponds to one combination of demand (increasing by row) and fuel cost (increasing by column). The values show the US dollar payout per kilowatt of a futures contract in that demand and
fuel cost scenario in the Both Contracts case. Bold indicates the 50 worst-case scenarios for the baseload technology in the No Trading case. Alignment between the payout structure and risk profile leads
investors in baseload generation to prefer futures to options when both are available.
Table 3 | Alignment of option payout to risk faced by peaker
technology
Demand
Fuel cost
1
2
3
4
5
6
7
8
9
10
 1
0
0
0
0
0
0
0
0
0
0
 2
0
0
0
0
0
0
0
0
0
0
 3
0
0
0
0
0
0
0
0
0
0
 4
0
0
0
0
0
0
0
0
0
0
 5
14
10
6
2
0
0
0
0
0
0
 6
51
47
43
39
36
35
31
27
23
19
 7
76
75
74
73
72
71
67
63
59
55
 8
89
87
85
83
81
81
80
79
78
77
 9
111
108
105
102
99
99
97
95
93
91
 10
138
135
132
129
126
126
123
120
117
114
Each cell corresponds to one combination of demand (increasing by row) and fuel cost (increasing
by column). The values show the US dollar payout per kilowatt of an options contract in that
demand and fuel cost scenario in the Both Contracts case. Bold indicates the 50 worst-case
scenarios for the peaker technology in the No Trading case. Alignment between the payout
structure and risk profile leads investors in peaking generation to prefer options to futures when
both are available.
0
20
40
60
80
100
0
40
80
120
160
200
Baseload
Total
Maximum option trade volume (% of installed capacity)
Equilbrium capacity (GW)
Fig. 1 | Decline of baseload capacity with increased options trading. Total
ICAP grows slightly with the availability of options contracts. As these
contracts are better suited to the risk profile of the peaker technology,
baseload capacity falls to zero.
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The trades that emerge in the All Contracts case echo the results
of the two-generator example. Each technology prefers contracts of
the trade adapted to its risk profile: investors in the baseload tech-
nology sell 38.5 GW of futures, investors in the peaker sell 96.7 GW
of options and investors in the variable technology sell 91.8 GW of
unit contingent contracts. Unlike the two-generator example, in
which the Both Contracts case reached an equilibrium close to the
social optimum, there remains a US$0.6 billion yr–1 gap between the
All Contracts case and the complete trading solution. Additional
contracts are required to bridge this divide.
Conclusion
This article argues that the introduction of a capacity market will
tilt the technology mix that arises in equilibrium towards resources
with higher operating costs. The syllogism proceeds as follows. The
financial impact of a capacity mechanism is to replace volatile scar-
city prices with more regular revenues. The higher a unit’s operating
costs, the more its cost recovery in an energy-only market relies on
scarcity prices. Therefore, the introduction of a capacity market has
a stronger impact on the risk profile of technologies with higher
operating costs.
The magnitude of this effect in real-world markets is difficult
to evaluate due to its dependence on the risk tolerance of market
participants, their evaluation of potential future scenarios and the
availability of other outlets for risk trading. However, this article’s
numerical experiments reveal the potential for substantial shifts,
as both examples exhibit a significant increase in capacity of the
peaking technology after the introduction of an options contract.
The struggles of US nuclear units, which have been unable to find
counterparties for long-term contracts without state support despite
their potential role as a hedge against future increases in natural
gas and carbon prices, suggests that the mechanism demonstrated
in this article may be at work in current markets. This result is
particularly important in the context of efforts to reduce carbon
Table 4 | Equilibria in the three-generator example for cases
with 0 and 1 contract
No trading
Unit only
Option
only
Future
only
Capacity (GW)
 Baseload
38.7
1.9
0.0
70.9
 Peaker
97.6
121.3
131.0
79.9
 Variable
118.1
195.1
167.7
48.9
Trade volume (GW)
 Baseload
 Future
63.8
 Option
0.0
 Unit
1.9
 Peaker
 Future
52.3
 Option
132.0
 Unit
35.8
 Variable
 Future
17.4
 Option
8.1
 Unit
195.1
Proximity to
equilibrium (%)
0.012
0.003
0.001
0.004
Change in surplus
(US$ billion yr–1)
−17.6
−13.1
−6.0
−3.6
Four financial trading cases are shown in order of increasing surplus. Bold cells facilitate pairwise
comparison within each row. Compared to the No Trading case, introducing a single contract leads
to more capacity of the corresponding technology.
Table 5 | Equilibria in the three-generator example for cases with 2 and 3 contracts
Future + Unit
Option + Unit
Option + Future
All Contracts
Capacity (GW)
 Baseload
70.9
0.0
68.2
42.5
 Peaker
79.9
126.9
81.3
96.6
 Variable
48.9
195.7
56.8
120.4
Trade volume (GW)
 Baseload
 Future
63.8
61.4
38.5
 Option
0.0
0.0
1.5
 Unit
0.0
0.0
0.0
 Peaker
 Future
52.3
10.2
0.9
 Option
138.1
67.9
96.7
 Unit
0.0
14.3
0.0
 Variable
 Future
17.4
21.5
12.8
 Option
2.3
0.0
0.0
 Unit
0.0
199.1
91.8
Proximity to equilibrium (%)
0.006%
0.003%
0.144%
0.004%
Change in surplus (US$ billion yr–1)
−3.6
−1.0
−0.9
−0.6
Four financial trading cases are shown in order of increasing surplus. Bold cells facilitate pairwise comparison within each row. Compared to the All Contracts case, removing a single contract leads to a
lower capacity of the corresponding technology.
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emissions. The majority of energy in low-carbon systems is likely to
be provided by some combination of hydroelectric, nuclear, wind
and solar resources, all of which are characterized by high capital
costs and low operating costs. Accordingly, capacity markets as cur-
rently structured may work against efforts to decarbonize.
To achieve an efficient capacity mix requires the ability to share
risk. Integrated resource planning, which can be made versa-
tile enough to incorporate whatever sources of risk are viewed to
be most salient, is one way to do so. A second, modelled in this
article, is an energy-only market accompanied by the robust trade
of instruments adapted to the risk profile of customers and vari-
ous technologies. Neither of these strategies is without challenges.
Capacity markets can offer a partial solution to the problem of risk
sharing. The results of this study, however, reveal the consequences
that choices made in the design of these markets may have for the
resource mix that arises in liberalized electricity systems.
Methods
Modelling framework. Here we formulated a multiple-agent variant of a two-stage
stochastic program for capacity expansion. In the first stage, risk-averse investment
and financial trading decisions are made by a number of agents. Spot market prices
are determined through a perfectly competitive dispatch over time blocks, t 2 T
I
,
performed by the system operator in the second stage. The ultimate source of all
generator revenue in the model is energy sales; financial contracts settle based on
the prices realized in the second stage. Instead of writing the optimality conditions
for each of the subproblems and reformulating as a complementarity problem9,41,42,
we anticipate the algorithm that we describe later in this section and focus on the
problems solved by each agent.
Our modelling framework builds most directly on the approaches of de Maere
d’Aertrycke and co-workers41,]42, which describe a risk-averse capacity equilibrium
with incomplete markets. We make two modifications of note, which we
describe in detail after introducing the model. Incomplete trading has significant
ramifications, both computationally and conceptually. The computational issue
is that, instead of a comparatively simple optimization problem, the problem
is formulated using a construct called a multiple optimization problem with
equilibrium constraints43. Although modest-sized instances can be solved by the
PATH solver44, numerical tests on larger instances can fail to converge. To help
address this issue, we proposed a heuristic decomposition algorithm similar
in spirit to that of Höschle et al.45, which allows the identification of equilibria
through a series of convex quadratic programs. Although the test systems used
in this article are kept simple, this decomposition approach allows an easier
extension to a richer set of technologies and risks. The conceptual concern is that
the uniqueness of the equilibrium solutions cannot, in general, be guaranteed. We
revisit this topic after introducing the algorithm. The optimization models are
implemented in AMPL46 and solved using CPLEX47.
In the three-generator example, uncertainty comes from three elements.
The random variable D+, with scenarios indexed by s 2 S
I
, reflects an upward
demand shift in all the time blocks. The random variables Agt, with scenarios
indexed by r 2 R
I
, denotes the availability of each generator in a given time
block. The random variables CEN
g
I
 and D−, with scenarios indexed by f 2 F
I
,
comprise generator fuel costs as well as the downward demand shift perfectly
correlated with higher prices. These three sources of randomness are assumed
to be independent and to have finite support. In the two-generator example,
randomness is modelled in the same way except that it does not include
the random variables Agt. Each market participant, a 2 A
I
, uses a coherent
risk measure ρa in its decision problem. As shown in Artzner et al.36, this
corresponds to choosing the worst-case distribution in a risk set, which takes
the form of a convex subset of probability measures on F ´ R ´ S
I
. The risk
measures we specifically chose are convex combinations of expected value
and CVaR of the surplus (Philpott et al.18 and de Maere d’Aertrycke et al.39 give
additional examples of this construction). The CVaR portion of the calculation
identifies the worst (100α) percentage of second-stage outcomes for the market
participant in question, and calculates the average surplus only among those
scenarios. Thus, starting from a shared nominal distribution, each participant
performs a risk-averse optimization that implicitly assigns a higher probability
to the combinations of fuel, availability and demand scenarios most harmful
to their interest. The difference in expected profit when evaluated using this
adjusted distribution versus the nominal distribution corresponds to the risk
premium required by investors. This endogenous representation of the risk
premium stands in contrast to the classical capacity expansion framework, in
which investment cost is annualized according to an exogenous cost of capital.
Dispatch. The economic dispatch (ED) problem was formulated as a convex
quadratic program, the outcome of which depends on the realization of the
random variables that govern demand, availability and fuel costs. The problem
splits operations into simple time blocks of varying length, which reflect the
different levels of load seen throughout the year without explicitly representing
operational considerations, such as unit commitment decisions, which in general
preclude equilibria, or ramping constraints, which can exacerbate the issue
of multiple equilibria. In general, the inclusion of these constraints leads to a
greater need for capacity, but their implications for the resource mix are not well
understood48.
Notation.
Sets:
g 2 G
I
, set of all generation technologies
t 2 T
I
, set of time blocks
f 2 F
I
, set of scenarios for fuel prices
r 2 R
I
, set of scenarios for generator availability profiles
Dfix
t
I
, set of scenarios for demand shifters
Parameters:
B, value of non-price-responsive load (US$ MWh–1)
Lt, length of time block t (h)
Dfix
t
I
, baseline level of non-price-responsive load in time block t (MW)
Dres
t
I
, amount of price-responsive demand, which bids at a value declining
linearly from B to 0 (MW)
Scenario-specific parameters (that is, realizations of random variables):
Agrt, availability of generator g in time block t under availability profile scenario
r (%)
D
f
I
, marginal cost for generator g under fuel price scenario f (US$ MWh–1)
D
f
I
, downward shift in demand in all time blocks under fuel price scenario f
(MW)
dfix
t
I
, upward shift in demand in all time blocks under demand shifter scenario
s (MW)
Variables:
ygt, production by technology g in time block t (MW)
dfix
t
I
, amount of non-price-responsive demand cleared in time block t (MW)
dfix
t
I
, amount of responsive demand cleared in time block t (MW)
Capacity expansion decision (fixed in dispatch model):
xg, quantity installed of generation technology g (MW)
Formulation. The ED problem, with Hfrs denoting the total surplus given scenarios f
for fuel price, r for generator availability and s for demand, is stated as:
ðEDÞfrs
Hfrs ¼
maximize
y;d
X
t2T
LtB dfix
t
þ dres
t
 ðdres
t Þ2=ð2Dres
t Þ



X
t2T
X
g2G
LtCEN
fg ygt
ð1Þ
subject to :
dfix
t
þ dres
t
þ Dþ
s  D
f ¼
X
g2G
ygt
8t 2 T
ð2Þ
0≤ygt ≤Agrtxg
8g 2 G; t 2 T
ð3Þ
0≤dfix
t ≤Dfix
t
8t 2 T
ð4Þ
0≤dres
t ≤Dres
t
8t 2 T
ð5Þ
Equation (1) includes a linear term for the value of non-price-responsive load, a
quadratic expression for the value of price-responsive load and a linear term for
the cost of producing power. The responsive load is the first departure from the
model in de Maere d’Aertrycke et al.41. Its presence does not carry any philosophical
significance: it is introduced for computational convenience, to help avoid the
inevitability of multiple optimal dual solutions in the linear version of the model
discussed in Abada et al.42. The power balance constraint in equation (2) splits
demand into four segments: dfix
t
I
, dres
t
I
, D
f
I
 and D
f
I
. The second, more meaningful
departure from de Maere d’Aertrycke et al.41 is the absence of the terms D
f
I
and D
f
I

from the objective function. This choice implies that consumers do not experience
any gain or loss of surplus as a result of these exogenous shocks to demand, which has
important consequences when comparing our numerical results to those of de Maere
d’Aertrycke et al.39, which are based on the model in de Maere d’Aertrycke et al.41.
Conceptually, the question is which events in the sample space F ´ R ´ S
I
 should
customers be most concerned about. Upward shocks to demand could come
from, for example, a higher-than-expected economic growth that leads to more
productive uses of electricity. In this case, consumers might not be concerned about
high-demand scenarios, as shortages would be balanced by an increase in surplus
from consumption. An upward shock could also come, however, from an increased
use of air conditioning due to higher summer temperatures, which consumers
might experience as neutral or even a loss. By excluding these shocks, we ensure
that consumers concentrate on the high-demand scenarios in which shortages
occur more frequently. This definition of consumer surplus appears subsequently
in the consumer model (CON). The optimal values of the ED model across all the
scenarios define the social surplus (SOC) used in the complete trading model.
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Contracts. In conjunction with the capacity investment decisions made in the
first stage, investors can sign contracts with consumers that settle based on the
energy price in the second stage. Assume a small set of contracts ηk
frs
I
 is available to
trade. Let λfrst be the price of energy in time block t given fuel price f, profile r and
demand shifter s, that is, the dual variable that corresponds to the power balance
constraint in equation (2) divided by the length Lt, and let ηk
frs
I
 represent the payout
of contract k 2 K
I
 given this scenario. A call option gives the purchaser the right to
buy a unit of energy for a predetermined price if the spot price exceeds that level.
Thus, if contract k indexes a call option that covers all the time blocks with strike
price λk, we can calculate:
ηk
frs ¼
X
t2T
Lt maxf0; λfrst  λkg
ð6Þ
If k instead indexes a futures contract that covers all time blocks at price λk, the
payout is calculated as:
ηk
frs ¼
X
t2T
Ltðλfrst  λkÞ
ð7Þ
We also define a unit contingent contract that tracks the availability of generator g,
a construct often used for variable technologies. When k indexes a unit contingent
contract for generator g, the payout is calculated as:
ηk
frs ¼
X
t2T
AgrtLtðλfrst  λkÞ
ð8Þ
Market participants. Each market participant solves a convex quadratic program,
choosing a quantity of financial contracts that maximizes its risk-adjusted surplus.
Generator capacities, outputs of the dispatch model for each scenario and the
prices and payouts of contracts are fixed parameters in these models. We define
notation here that can be shared between the consumer and generator models, that
is, among all market participants.
Notation.
Sets:
a 2 A
I
, set of market participants (generators and consumers)
vk
a;vk
a
I
, set of contracts
Parameters:
αa, tail probability at which CVaR is evaluated by market participant a,
0 < αa ≤ 1
βa, weight given to the expected value in the risk measure for market
participant a, 0 ≤ βa ≤ 1
γ, weight on the term that penalizes imbalance in the contract volumes between
market participants
vk
a;vk
a
I
, minimum and maximum volume of contract k to be purchased or sold by
market participant a (MW)
pfrs, nominal probability of scenario (f,r,s)
ηk
frs
I
, investment cost for generator g annualized at risk-free rate (US$ MW–1)
Provisional parameters (that is, values calculated by other agents):
λfrst, price of energy in time block t under scenario (f,r,s) (US$ MWh–1)
πfgrst, operating profit for technology g in block t under scenario (f,r,s)
(US$ MW–1)
ηk
frs
I
, payout of contract k under scenario (f,r,s) (US$ MW–1)
ϕk, price of contract k incurred in the first stage (US$ MW–1)
Variables:
vk
a
I
, volume of contract k purchased or sold by market participant a (MW)
VaRa, value at risk for market participant a (US$)
ua
frs
I
, surplus for market participant a under scenario (f,r,s) (US$)
c 2 A
I
, auxiliary variable used in the calculation of value at risk (VaR) (US$)
Consumer model. We distinguished consumer and generator agents by using c 2 A
I

and g 2 A
I
 in place of the generic agent a, and we assume a single consumer; that
is, in terms of our notation, g 2 G
I
. Given the assumption of perfect competition,
the resulting decisions are equivalent to those of a larger number of small, identical
load serving entities. Defining ρa to be the risk measure associated with market
participant a, the consumer’s problem is stated as follows:
ðCONÞ
ρc ¼
maximize
vc;uc;ucþ;VaRc
ð1  βcÞ
VaRc  1=αc
X
f 2F
X
r2R
X
s2S
pfrsucþ
frs
0
@
1
A
þβc
X
f 2F
X
r2R
X
s2S
pfrsuc
frs
0
@
1
A  γ=2
X
k2K
X
a2A
vk
a

!2
ð9Þ
subject to
uc
frs ¼
X
k2K
vk
c ϕk  ηk
frs


þ
X
t2T
LtB dfix
t
þ dres
t
 ðdres
t Þ2=ð2Dres
t Þ



X
t2T
Ltλfrst dfix
t
þ dres
t
þ Dþ
s  D
f


8f 2 F; r 2 R; s 2 S
ð10Þ
VaRc  uc
frs ≤ucþ
frs
8f 2 F; r 2 R; s 2 S
ð11Þ
0≤ucþ
frs
8f 2 F; r 2 R; s 2 S
ð12Þ
vk
c ≤vk
c ≤vk
c
8k 2 K
ð13Þ
The consumer maximizes a convex combination of CVaR and expected value
of surplus, subtracting a proximal term that penalizes imbalances between the
contracts bought and sold by the market participants. In equilibrium, this third
term of equation (9) must equal zero. Although similar in appearance, the inclusion
of this proximal term does not technically yield an augmented Lagrangian, because
the market clearing condition for financial contracts is not part of the consumer’s
true objective function. Constraint (10) calculates the surplus for the consumer
in every scenario that results from the purchase of contracts in the first stage
and energy in the second. Constraints (11) and (12) dictate the value of auxiliary
variables used in the CVaR calculation. Constraint (13) dictates a minimum and
maximum volume of each contract. Although we discuss specific choices of these
parameters in Supplementary Note 3, an algorithmic advantage of including these
constraints on trading is to guarantee that all the subproblems are bounded. Note
that vc
k is a decision variable in model (CON), but the contract volumes for the
generators, which appear in the objective function’s final term, enter model (CON)
as provisional parameters.
Generator model. We modelled a single investor in each generation technology
g 2 G
I
. As in the case of the consumer, investment decisions can equivalently be
represented as a large number of identical firms. The investor’s problem is stated as:
ðGENÞg
ρg ¼
maximize
vg;ug;ugþ;VaRg
ð1  βgÞ
VaRg  1=αg
X
f 2F
X
r2R
X
s2S
pfrsugþ
frs
0
@
1
A
þβg
X
f 2F
X
r2R
X
s2S
pfrsug
frs
0
@
1
A  γ=2
X
k2K
X
a2A
vk
a

!2
ð14Þ
subject to
ug
frs ¼ CINV
g
xg
X
k2K
vk
g ϕk  ηk
frs


þ
X
t2T
πfgrstxg
8f 2 F; r 2 R; s 2 S
ð15Þ
VaRg  ug
frs ≤ugþ
frs
8f 2 F; r 2 R; s 2 S
ð16Þ
0≤ugþ
frs
8f 2 F; r 2 R; s 2 S
ð17Þ
vk
g ≤vk
g ≤vk
g
8k 2 K
ð18Þ
The consumer and generator models differ only in the calculation of scenario
surplus in equations (10) and (15). For simplicity, we employed the operating profit
πfgrst, calculated as the dual variable to the maximum generation constraint in the
dispatch model, equation (3), multiplied by availability Agrt.
Decomposition approach. With the assumption of perfect competition, investors
in technology g will build new capacity until the risk measure ρg = 0 in equilibrium.
Thus, the challenge is to identify values for the ICAP x and contract prices ϕ that
balance the market in all financial contracts and result in zero risk-adjusted profit
for all generation technologies, given the prices and operating profits that arise
from the dispatch model. We refer to this equilibrium problem as (EQ).
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954

Articles
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Algorithm 1 shows the solution approach that we employ. The algorithm
chooses a capacity mix, finds the implied dispatch solutions, identifies the prices
that balance the markets for financial contracts (to tolerance ε) and then updates
the capacity mix based on profitability. With capacity decisions made in the outer
loop, one interpretation of the algorithm is that it simulates the process of entry
and exit in competitive markets. The use of a proximal term in the objective
function for the market participants, as well as the sequential updating, invites
comparison to the alternating direction method of multipliers49,50, an avenue
pursued on a model similar to ours in Höschle et al.45. As in Höschle et al.45,
convergence is not guaranteed, but was achieved in all the numerical tests after
limited experimentation with different step sizes σ and μ. We compare our
approach to that of Höschle et al.45 in Supplementary Note 5.
Algorithm 1.
Input: An instance of (EQ) defined by models (ED), (CON), and (GEN).
Output: Near-equilibrium solution to (EQ)
  Define σ, μ, δ, ε > 0; let ρa ¼ 0 : 8a 2 A
I
; initialize x, ϕ
  loop
    λfrst; πfgst; ηk
frs : 8ðf ; r; sÞ 2 F ´ R ´ S; k 2 K
I
    solve (ED)frs; update λfrst; πfgst; ηk
frs : 8ðf ; r; sÞ 2 F ´ R ´ S; k 2 K
I
    solve (CON)
    solve maxk2K j P
a2A
vk
aj>ε
I
    while ϕk  ϕk þ σ P
a2A
vk
a
8k 2 K
I
 do
      ϕk  ϕk þ σ P
a2A
vk
a
8k 2 K
I
      solve (CON)
      solve ðGENÞg
8g 2 G
I
    end while
    if maxg2G jρgj<δ
I
 then
      return x and ϕ
    end if
  end loop
The potential for multiple equilibria poses a challenge for the interpretation of
the numerical results51. In an effort to avoid this possibility, we omit intertemporal
constraints from model (ED) and maintain a constant merit order through all the
scenarios. Under these assumptions and with no financial trading, uniqueness can
be shown for the model in Abada et al.42. In both numerical examples, capacity is
initialized at the same, socially optimal starting point for all the cases within each
example, with the goal to avoid a spurious result. Contract prices are initialized at
their expected payouts using the nominal probability distribution. Starting from
alternative points in ad hoc tests did not uncover alternative equilibria.
Complete trading. For comparison, we also constructed a model that assumes
complete trading. In this setting, an equilibrium can be identified through a
risk-averse optimization problem without the need for equilibrium constraints,
using the intersection of the risk sets of all market participants17–20. This can be
represented as a single large-scale optimization problem that comprises investment
decisions and the dispatch model for each scenario. To be concise, we employed
as a variable Hfrs to indicate the surplus that arises from the dispatch using the
chosen amount of capacity x. Replacing this variable with the objective function
from (ED)frs and inserting the constraints in (ED)frs for each scenario recovers the
complete model. With subscript i denoting societal risk preferences, the complete
trading model is written as:
ðSOCÞ
ρi ¼
maximize
x;y;d;H;ui;uiþ;VaRi
ð1  βiÞ VaRi  1=αi
X
f 2F
X
r2R
X
s2S
pfrsuiþ
frs
2
4
3
5
þβi
X
f 2F
X
r2R
X
s2S
pfrsui
frs
2
4
3
5
ð19Þ
subject to
ui
frs ¼
X
g2G
CINV
g
xg þ Hfrs
8f 2 F; r 2 R; s 2 S
ð20Þ
VaRi  ui
frs ≤uiþ
frs
8f 2 F; r 2 R; s 2 S
ð21Þ
0≤uiþ
frs
8f 2 F; r 2 R; s 2 S
ð22Þ
Convergence. As is characteristic of the alternating direction method of
multipliers, the algorithm exhibits slow convergence near the equilibrium. For
cases in which the termination criterion δ in Algorithm 1 is not met, we report the
solution found at iteration 50,000 of the outer loop. As a measure of proximity to
equilibrium, we compute the value:
maxfg2G:xg > 0g
ρg
CINV
g

ð23Þ
for each case, where ρg is the risk measure calculated by problem (GEN)g.
Data availability
The code and data used for numerical tests in this study are available in a public
repository (https://doi.org/10.5281/zenodo.3242844).
Received: 24 January 2019; Accepted: 6 September 2019;
Published online: 28 October 2019
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Acknowledgements
This research was supported, in part, by Northwestern University’s Center for
Optimization and Statistical Learning. The views expressed in this article are not
necessarily those of the Federal Energy Regulatory Commission.
Author contributions
J.M. planned and performed the analysis. D.P.M. reviewed the manuscript, in particular
with reference to the models and algorithm. R.P.O. reviewed the manuscript, in
particular with reference to the electricity market design.
Competing interests
The authors declare no competing interests.
Additional information
Supplementary information is available for this paper at https://doi.org/10.1038/
s41560-019-0476-1.
Correspondence and requests for materials should be addressed to J.M.
Reprints and permissions information is available at www.nature.com/reprints.
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© The Author(s), under exclusive licence to Springer Nature Limited 2019
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    Policy Implications:

Answer: • Mechanisms designed to guarantee sufficient electricity supply may have unintended consequences on resource mix.• Implementing capacity markets can shift the resource mix away from baseload and variable resources like nuclear and renewables, toward peaking technologies like oil and gas.• Regions relying on competitive markets for resource adequacy should work to ensure robust markets for risk-trading instruments.• Regions with explicit capacity requirements should consider a range of risks when designing those requirements.