*

#  Battery Operations in Electricity Markets:
Strategic Behavior and Distortions

Jerry Anunrojwong Columbia University, Graduate School of Business. Email:
<EMAIL_ADDRESS>Santiago R. Balseiro Columbia University,
Graduate School of Business. Email<EMAIL_ADDRESS>Omar Besbes  Columbia
University, Graduate School of Business. Email<EMAIL_ADDRESS>Bolun Xu
Columbia University, Earth and Environmental Engineering. Email:
<EMAIL_ADDRESS>
###### Abstract

Electric power systems are undergoing a major transformation as they integrate
intermittent renewable energy sources, and batteries to smooth out variations
in renewable energy production. As privately-owned batteries grow from their
role as marginal “price-takers” to significant players in the market, a
natural question arises: How do batteries operate in electricity markets, and
how does the strategic behavior of decentralized batteries distort decisions
compared to centralized batteries?

We propose an analytically tractable model that captures salient features of
the highly complex electricity market. We derive in closed form the resulting
battery behavior and generation cost in three operating regimes: (i) no
battery, (ii) centralized battery, and (ii) decentralized profit-maximizing
battery. We establish that a decentralized battery distorts its discharge
decisions in three ways. First, there is quantity withholding, i.e.,
discharging less than centrally optimal. Second, there is a shift in
participation from day-ahead to real-time, i.e., postponing some of its
discharge from day-ahead to real-time. Third, there is reduction in real-time
responsiveness, or discharging less in response to smoothing real-time demand
than centrally optimal. We quantify each of the three forms of distortions in
terms of market fundamentals. To illustrate our results, we calibrate our
model to Los Angeles and Houston and show that the loss from incentive
misalignment could be consequential.

Keywords: energy storage, batteries, electricity markets, economic
withholding, market power, renewable energy, duck curve, two-stage settlement,
day-ahead, real-time.

###  1 Introduction

Climate change is the defining issue of our time (IPCC, 2023), and countries
and regions have pledged to reduce their carbon emissions through
international agreements and carbon neutrality pledges. For example, the
United States and the European Union planned to reach net zero by 2050. The
power sector in the United States is one of the largest emitting sectors,
accounting for around 30 percent of total U.S. emissions (CBO, 2022).

To achieve this goal, the electric power systems are currently undergoing a
major transformation by incorporating renewable energy resources, such as
solar and wind. However, the availability of these renewable resources depends
on exogenous factors that cannot be controlled. Because supply of power has to
equal demand at all times on the electric grid, the system operator has to
compensate for the real-time variability in one of two ways. The traditional
way is to call up fast-responding “peaker” plants, but these plants are both
expensive and have high emissions. Alternatively, the system can have enough
energy storage resources, such as batteries, to smooth out fluctuations and
variations in energy production and consumption over the course of a day. As
costs fall and incentive schemes for renewables and batteries are enacted,
battery storage capacity has been growing rapidly since 2021. California and
Texas have emerged as front runners in the deployment of battery storage, with
8.6 GW and 4.1 GW of battery, respectively, as of April 2024, while other
states are only starting to deploy batteries at scale (EIA, 2024b).111To put
these numbers in context, California and Texas electricity demands on a
typical day are around 20–40 GW and 40–70 GW, respectively, so California’s
batteries are already a substantial fraction of demand. In deregulated
electricity markets such as California and Texas, these grid-scale batteries,
like other generation assets, are often privately owned by profit-driven
investors. As batteries grow from their previous role as marginal “price-
takers” to significant players in the market, a natural question arises:

> How do batteries operate in electricity markets, and how does the strategic
> behavior of decentralized batteries distort decisions compared to
> centralized batteries?

System operators have already observed strategic battery behaviors leading to
negative outcomes. The Australian Energy Regulator reported strategic behavior
from its (relatively small) 100MW/150MWh battery during tight market
conditions on March 16-17, 2023 (AER, 2023; Parkinson, 2023). After a
generator outage (March 16) and a change in forecast price (March 17), the
battery rebid its capacity from the price floor up to $10,000/MWh and
$15,000/MWh, respectively, setting the price. The report concludes, “This
short-term strategic rebidding to capitalise on market conditions had the
effect of exacerbating high prices. Again, while this behaviour may not be a
breach of the rules, the ability of these participants to increase price
through these rebidding strategies highlights the market power that
participants may be able to exercise at certain times.” Batteries are also
strategic in “normal” market conditions. California’s special report on
battery storage (CAISO, 2023) suggests that batteries avoid being scheduled in
day-ahead on average, preferring to participate in real-time markets. This
strategic shift from day-ahead to real-time means the system operator has to
commit additional more expensive generators in day-ahead.

Figure 1: Los Angeles’ “duck curve,” hourly mean net demand by year,
2019–2023. The net demand is the energy demand minus renewable production.
(Source: CAISO)

To understand the role of batteries, we must first understand the challenges
from the mismatch between renewable energy production and demand. As an
illustrative example, in Figure 1, we depict for the years 2019-2023 and for
Los Angeles, the average hourly net load (or net demand), defined as the
energy demand minus renewable production, which needs to be covered by
conventional generators. The net demand admits a peak around 7–8PM when solar
wanes and people come back from work, while during the off-peak around noon
the net load is very low and has been getting lower every year due to
increasing solar capacity. This leads to an increasingly steep ramp period
during which energy generation needs to be increased quickly, and only a
subset of expensive and emissions-heavy generators can do the task, negating
part of the benefits of renewables. Due to its shape, this curve is often
referred to as the “duck curve.”

Batteries are a natural complement to renewables in the electrical grid
because they can charge during off-peak when energy is plentiful and price is
low, and discharge during peak when energy is scarce and price is high. In
doing so, batteries smooth out the demand by arbitraging between the off-peak
and the peak periods during the day and make a profit from the price
difference. As a price taker, such a battery is straightforwardly beneficial,
but as discussed earlier, batteries are now a substantial fraction of demand
in most regions in California and other markets are about to follow, and
strategic behavior and market power become questions of interest.

Electricity markets are especially susceptible to the exercise of market power
because demand and supply must be exactly equal at every location at any given
time. Even though there are many grid-scale batteries in California, each can
resemble a local monopoly in its region, because the transmission line
infrastructure limits the amount of energy that can be transferred across
regions. California regulators acknowledged that increasing volatility from
renewables further exarcerbates the market fragmentation problem and approved
a $7.3 billion plan to build additional transmission capacity (St. John,
2023). The fragmented nature of the market is evident from the fact that
wholesale electricity prices are very different across locations in
California. For example, on May 27, 2024 at noon (off-peak), the real-time
“base” price is $4/MWh but the congestion prices in some regions of California
were as high as $120/MWh.

Starting from the deregulation of electricity markets in the 1990s, and
learning from painful historical lessons along the way,222Most notoriously,
the 2000–2001 California electricity crisis caused by flawed market design and
market manipulation by energy companies, mainly Enron (Weare, 2003). a large
literature on detecting and mitigating market power has developed, and all US
short-term wholesale markets have adopted various forms of market power
mitigation procedures (Graf et al., 2021). These measures primarily target
conventional generators, because until recently the only feasible storage
technology at scale was pumped hydro, which were relatively small. However,
grid-scale battery systems are now projected to rise rapidly, and they are
crucial for integrating renewables into the grid. These developments bring the
questions around the potential for battery market power back to the forefront
of electricity market design.

Batteries also pose an additional challenge to regulators more used to
monitoring market power from generators. The cost bids of conventional
generators are largely determined by known physical and operational
constraints, which allows the system operators to ensure that generators’ bids
are “reasonable” most of the time based on such characteristics. The bids of
batteries, however, are determined by (their predictions of) opportunity costs
and not just physical marginal costs like generators. The questions of what
form of strategic behavior a battery might take, and how it impacts system
performance, are therefore crucial questions.

####  1.1 Summary of Main Contributions

At a high level, this paper identifies an important question – market power of
batteries – and proposes a tractable model that is rich enough to analyze how
batteries behave strategically in a two-settlement market, isolate the effects
at play, and quantify their impact on system cost.

##### Modeling Contribution.

We view the formulation of our model as one of our main contributions, because
our model is fully microfounded and rich enough to incorporate salient
institutional features yet simple enough to solve in closed form and isolate
the main forces at play. In particular, we capture the two-settlement
structure that is common in most markets, the duck curve demand trend (peak
and off-peak), demand stochasticity and autocorrelation, and heterogeneity in
generator ramp speeds. We can directly see from solutions to the model how
each of these factors impact the types of strategic behaviors of batteries,
and how they impact the resulting cost, under different operating regimes.

##### Main Features of the Model.

We model a two-settlement centralized market with two periods in a day, a peak
period and an off-peak period. The day-ahead market clears for each period at
the beginning of the day with a day-ahead demand forecast. Then for each
period, the real-time market clears after the demand for that period is
realized. We assume that the demands in two periods are random and correlated,
to investigate how batteries, as fast responders, react to demand
stochasticity and temporal dependence. This form of modeling allows us to
disentangle the two different kinds of demand-smoothing done by batteries: (i)
intertemporal demand smoothing, or transferring predictable components of
demand from peak to off-peak period, and (ii) smoothing unpredictable
components of demand, by being responsive in real-time, discharging more when
demand is unusually high, and vice versa, to reduce the cost impact of
deviations from forecast.

We also assume that the battery’s charge and discharge behavior has price
impact. In other words, the price is endogenously determined.

Lastly, we microfound the model by assuming an exogenous set of generators.
Only a subset of generators have fast enough ramping time to participate in
real-time, and other generators can only participate in day-ahead. Given the
supply curves of “slow” and “fast” generators, the prices in both day-ahead
and real-time are set at a point where the supply from the generators equals
the exogenous consumer demand net battery charge/discharge.

##### Battery Behavior.

We analyze the behavior of a large battery in two regimes: the centralized
regime, in which the system operator can control the battery charge/discharge
to minimize system cost, and the decentralized regime, in which the battery
independently makes charge/discharge decisions to maximize its profit. We find
that in the centralized case, the battery perfectly smoothes the predictable
demand in day-ahead, and unpredictable demand in real-time. Therefore, there
is no economic withholding of any kind under this ideal baseline.

In the decentralized case, we show that the strategic battery distorts its
discharge decisions relative to the centrally optimal in three ways. First is
quantity withholding: expected total battery discharge is less than centrally
optimal. Second is the shift from day-ahead to real-time: expected real-time
battery discharge is positive, whereas it is zero in the centralized case, as
the centrally optimal battery only responds to mean-zero demand fluctuations
in real time. In other words, the battery “hides” part of its capacity in day-
ahead, reducing its day-ahead participation, making the day-ahead planning
more costly. Third is reduction in real-time responsiveness: the strategic
battery discharges less to smooth real-time demand fluctuation, i.e., to
reduce the cost impact of real-time deviations from forecast.

The first two types of distortion (quantity withholding and shift from day-
ahead to real-time) are about arbitrage across time from peak to off-peak, and
the relative weight between them depends on the generation composition. If
most generators are fast (and can participate in real-time), then the shift
from day-ahead to real-time dominates. If most generators are slow, then
quantity withholding dominates. We elaborate on this in Section 5 (cf. Table
1).

##### Generation Cost Comparison.

We compare the generation cost in the three regimes. Centralized cost is lower
than decentralized cost, which is lower still than no-battery cost. Although
there is incentive misalignment, having an independent battery is still better
than not having one. We quantify the incentive misalignment by the ratio
between the cost reduction achieved by a centrally controlled versus profit-
maximizing battery, which we call the Price of Anarchy (PoA). We prove that
PoA is between $9/8$ and $4/3$, and PoA is decreasing (i.e. the incentive
alignment is better) in the share of fast generators and the steepness of the
duck curve.

##### Numerical Illustration.

We illustrate our results by calibrating our model with real data in two
regions: Los Angeles and Houston. We find that, with a single monopoly
battery, market power could lead to a significant increase in generation cost,
and all three types of distortion can be significant.

####  1.2 Related Work

Our paper is related to several streams of literature.

##### Sequential Markets and Market Power

Our work is most closely related to the literature on market power in
sequential markets, starting with the seminal work of Allaz and Vila (1993).
The latter considers producers (“generators”) rather than batteries, but some
of their insights transfer to our setting. Just like in Allaz and Vila (1993),
producers use the forward (“day-ahead”) market even under complete certainty
and perfect foresight because the forward market changes the marginal revenue
on the spot (“real-time”) market. Under market power, the forward market
improves both producer profit and social welfare, because the producer can use
the day-ahead market to reduce withholding. Ito and Reguant (2016) extends the
static Cournot game of Bushnell et al. (2008) to sequential markets, and
quantifies market power with limits to arbitrage in the Iberian electricity
market. Borenstein et al. (2008) and Saravia (2003) also consider market power
and arbitrage in electricity markets in California and New York, respectively.
You et al. (2019) considers a fixed strategic load that can allocate to day-
ahead or real-time. While these works focus on (perfect or imperfect)
arbitrage between day-ahead and real-time by generators or purely financial
“virtual” bidders and only assume one time period, our work assumes that
generators are nonstrategic and focuses also on the arbitrage between peak and
off-peak time periods by batteries.

##### Renewable Energy Operations.

There is a vast literature on renewable energy in operations; for surveys, see
e.g. Agrawal and Yücel (2021); Parker et al. (2019); Sunar and Swaminathan
(2022). Here, we highlight modeling works that are related to batteries,
sequential markets, or market power. Sioshansi (2010) observes that price
smoothing by batteries create welfare gains but the incentives may not be
properly aligned for centrally optimal storage use. Sioshansi (2014) shows
that the introduction of storage always increases welfare when generators are
nonstrategic, but it can reduce welfare when generators are strategic.
However, Sioshansi (2014) assumes that the demand is deterministic and clears
in one stage, whereas we highlight the role that demand stochasticity and
sequential market clearing play in different forms of distortion in battery
behavior. Peura and Bunn (2021) uses a game-theoretic model to analyze how
intermittent renewable production affect electricity prices in the presence of
a forward market. While they do not consider batteries, the use of forward
markets to improve welfare and reduce market power as in Allaz and Vila (1993)
is related to our work. Acemoglu et al. (2017), Genc and Reynolds (2019) and
Bahn et al. (2021) consider the impact of ownership models (similar to our
centralized versus decentralized regimes) on competition and market power in
renewables without batteries. Kaps et al. (2023) and Peng et al. (2021)
develop models of joint investment in renewables, conventional generators, and
storage. Wu et al. (2023) and Qi et al. (2015) consider investment in storage
in different locations, but do not consider incentives. Zhou et al. (2016)
analyzes storage operations and energy disposal in the presence of negative
electricity prices.

There is also a nascent line of work on smoothing demand or shaving the peak
beyond the use of batteries. Agrawal and Yücel (2022) analyzes the design of
demand response programs, paying consumers to reduce consumption when the grid
is under stress. Fattahi et al. (2023, 2024) analyze the use of direct load
control contracts to smooth demand. Gao et al. (2024) studies different ways
of aggregating distributed energy resources. Electric vehicles can also be
used to shave the peak, although such uses are currently limited (Wu et al.,
2022; Perakis and Thayaparan, 2023).

##### Market Power in Electricity Markets.

There is a large empirical literature in economics measuring market power in
electricity markets; see (Kellogg and Reguant, 2021, Section 4.2) for a
survey, and Graf et al. (2021) for market power mitigation mechanisms. This
literature is mostly focused on market power of generators; the exceptions are
Karaduman (2023) and Butters et al. (2023). Karaduman (2023) does not model
sequential market clearing and focuses more on numerically computing a dynamic
equilibrium between a monopoly storage and conventional generators calibrated
to the South Australian electricity market. Like us, he documents the
discrepancy between private and social incentives. In contrast, Butters et al.
(2023) assumes that batteries do not have market power and focuses more on the
impact of different incentive schemes on investments.

##### Battery Market Power in Power Systems

There is a body of work in the power systems literature that study the market
power of a monopoly battery with price impact (Mohsenian-Rad, 2016; Bjørndal
et al., 2023; Hartwig and Kockar, 2016; Huang et al., 2018; Schill and
Kemfert, 2011). Other works design algorithms to maximize battery profit over
time (Tómasson et al., 2020; Ward and Staffell, 2018; Cruise et al., 2019).
Whereas these papers propose detailed models in the form of large-scale
mathematical programs that are numerically solved, our work gives a stylized
model that can be directly analyzed and solved in closed form. The two lines
of work are complementary; black box models can incorporate more details,
while stylized models give sharp analytical insights and economic intuition.
In particular, it is generally understood from this literature that batteries
can strategically withhold capacity, but our work clarifies different forms of
withholding and how they depend on market fundamentals.

###  2 Model

##### Two-Settlement Market.

We consider a two-settlement centralized market clearing that is used by all
wholesale electricity markets in the United States. The day-ahead (DA) market
clears before the day begins, setting a price in each time period such that
the amount of supply from all generators below the price equals the mean
demand for that period. In other words, we assume that the system operator’s
day-ahead demand forecast is unbiased. Then, during the day, the real-time
(RT) market clears the incremental real-time demand (henceforth just “RT
demand”), which is the difference between the realized demand and the pre-
committed DA demand. The RT demand can either be positive (higher demand than
expected) or negative (lower demand than expected). If the RT demand is
positive (resp. negative), a subset of generators that are fast enough to
adjust in real-time are called on to increase (resp. decrease) production (cf.
the generators section below). The battery makes the charge/discharge decision
and amount after that period’s actual demand is realized, affecting the net
demand for that period (cf. the battery participation section below).

##### Generators.

We model two types of generators: generators that have a slow ramp speed can
only participate in DA (henceforth “slow generators”), whereas generators with
a fast ramp speed can participate in both DA and RT (henceforth “fast
generators”). We assume that we have a continuum of infinitesimally small
generators, and each generator is specified by the cost. All conventional
generators are assumed to be non-strategic, that is generators bid their true
marginal cost. Each generator also follows the market operator’s dispatch
instructions on whether they produce. This is an intentional modeling choice
to capture the shape of the supply curve and how it determines the clearing
price and generation cost, while abstracting away non-convex elements such as
start-up and no-load costs which we instead capture in a stylized way via the
dichotomy between slow and fast generators. This model of generators is
adapted from You et al. (2019). (While thinking of generators as a continuum
is convenient, it is not necessary: we get the same result with a finite set
of generators with zero start-up and no-load costs whose combined cost
functions correspond to the supply curve.)

Let $G_{s}(\lambda)$, respectively $G_{f}(\lambda)$, be the mass of slow,
respectively fast generators, with cost less or equal than $\lambda$. The cost
distributions $G_{s}(\cdot)$ and $G_{f}(\cdot)$ of slow and fast generators
are primitives of the model, assumed to be strictly increasing.

##### Demand Process.

There are two time periods in a day. Period 1 is the peak period, and period 2
is the off-peak period. Period 1 demand $D_{1}$ and period 2 demand $D_{2}$
are both random variables with a known joint distribution
$(D_{1},D_{2})\sim\pi$.

For $t\in\\{1,2\\}$, let $\mu_{t}\equiv\mathbb{E}[D_{t}]$ and
$\sigma_{t}^{2}\equiv\mathbb{E}[(D_{t}-\mu_{t})^{2}]$ be the period-$t$
marginal mean and variance, respectively. We will now define two notions of
correlation that will later be important.

For each realization of $D_{1}$, we define the conditional mean and variance
of $D_{2}$ as

$\displaystyle\mu_{2|D_{1}}\equiv\mathbb{E}[D_{2}|D_{1}]\,,\quad\sigma_{2|D_{1}}^{2}\equiv\mathrm{Var}(D_{2}|D_{1}).$
(1)

Note that when $D_{1}$ is a random variable, so are $\mu_{2|D_{1}}$ and
$\sigma_{2|D_{1}}^{2}$. By the law of total variance, we have
$\sigma_{2}^{2}=\mathrm{Var}(D_{2})=\mathbb{E}(\mathrm{Var}(D_{2}|D_{1}))+\mathrm{Var}(\mathbb{E}(D_{2}|D_{1}))$.

We define the sequential correlation $\rho_{s}$ to be such that

$\displaystyle\mathrm{Var}(\mathbb{E}(D_{2}|D_{1}))=\mathbb{E}[(\mu_{2|D_{1}}-\mu_{2})^{2}]=\rho_{s}^{2}\sigma_{2}^{2}\,,\quad\mathbb{E}[\sigma_{2|D_{1}}^{2}]=(1-\rho_{s}^{2})\sigma_{2}^{2}$
(2)

and $\rho_{s}^{2}\in[0,1]$. Note that if $D_{2}$ and $D_{1}$ are independent,
then
$\mathrm{Var}(\mathbb{E}(D_{2}|D_{1}))=\mathrm{Var}(\mathbb{E}(D_{2}))=0$, so
$\rho_{s}=0$, whereas if $D_{2}$ is completely determined by $D_{1}$, then
$\mathrm{Var}(\mathbb{E}(D_{2}|D_{1}))=\mathrm{Var}(D_{2})=\sigma_{2}^{2}$, so
$\rho_{s}=1$. Therefore, $\rho_{s}$ is a “reasonable” normalized measure of
dependence between $D_{2}$ and $D_{1}$.

Note that $\rho_{s}$ does not need to be the same as the standard Pearson
correlation $\rho$ defined by

$\displaystyle\rho\equiv\frac{\mathbb{E}[(D_{1}-\mu_{1})(D_{2}-\mu_{2})]}{\sigma_{1}\sigma_{2}}.$
(3)

However, if $(D_{1},D_{2})$ are jointly normally distributed, then
$\rho_{s}=\rho$. We will not make the normal assumption through most of our
results; when we do, we will be explicit, and the normal assumption is made
only for convenience or to highlight certain insights.

Recall that this demand process describes net demand, so uncertainties in the
net demand comes from both consumer demand and renewable production, and they
are both exogenous. In particular, we assume that electricity demand is
perfectly inelastic. This is well-supported by empirical evidence (Joskow,
2006). We model the demand process this way because we want to capture the
fact that markets with high renewable and battery penetration such as Los
Angeles exhibit a duck curve (cf. Figure 1): the price is highest in the
evenings when solar generation wanes (the sun sets) and demand peaks as people
come home from work, and the price is lowest in midday when solar production
peaks. Period 1 and period 2 are meant to correspond to the peak (including
the steep ramp leading to the peak) and the off-peak of the duck curve.
Demands in both periods have a significant amount of variability, and they are
also highly correlated, hence our modeling choice.

##### Battery Participation.

There is a single battery. Before the day, the battery decides on the DA
discharge amount $z_{1}^{DA}$ and $z_{2}^{DA}$ in period 1 and period 2 day-
ahead. Then the real-time scenario materializes: the period 1 demand $D_{1}$
is realized, then depending on the demand realization, the battery decides on
the RT discharge amount $z_{1}^{RT}(D_{1})$. After period 2 demand $D_{2}$ is
realized, the battery RT discharge amount in period 2 is
$z_{2}^{RT}(D_{1},D_{2})$. Note that the battery charging is represented by
negative discharge $z$.

The battery is assumed to have no state-of-charge constraint, no operating
cost, and no efficiency losses. While our model features only one day (with
multiple time periods in a day), we should interpret the model as a day in
steady state. Equivalently, we have the same day that happens day after day.
The battery cannot produce its own energy, so the total energy charged must
equal the total energy discharged (plus the cycle inefficiency loss, which we
assume to be zero).

Furthermore, we impose the condition that the net discharge throughout the day
is zero for each demand realization: $z_{1}^{DA}+z_{2}^{DA}=0$ and
$z_{1}^{RT}(D_{1})+z_{2}^{RT}(D_{1},D_{2})=0$ for every $(D_{1},D_{2})$ in the
support. This assumption is made because (i) batteries cannot produce energy,
only shift it across time, and (ii) batteries overwhelmingly arbitrage between
peak and off-peak periods within a day rather than between days.

The fact that batteries typically have negligible net daily discharge, in
agreement with (ii) above, is evident from the data. For each year in
2021–2023, we can calculate the mean absolute daily discharge over the year,
which captures the average net daily position of batteries over the year. This
net position is 1.0%, 2.9%, and 1.1% of total battery capacity in 2021, 2022,
and 2023. The daily charge cycle of batteries is by design: less than 7% of
installed storage have duration exceeding 4 hours (Denholm et al., 2023).

Note that while we model the battery as choosing a quantity in each scenario,
the quantity discharged in practice depends on the specific market framework,
which broadly falls into two categories. First is self-scheduling: the battery
can decide the quantity to discharge in each period, which is the same as our
model. Second is economic bidding, where the battery bids the charge and
discharge curves as price-quantity pairs, and the quantity charged/discharged
is determined from market clearing conditions. Given that the battery is the
only strategic player in the environment, the battery can choose the bid
curves to achieve any desired quantity level.

To conclude, the battery’s decision variables are
$z_{1}^{DA},z_{2}^{DA},z_{1}^{RT}(D_{1}),z_{2}^{RT}(D_{1},D_{2})$ subject to
constraints $z_{1}^{DA}+z_{2}^{DA}=0$ and
$z_{1}^{RT}(D_{1})+z_{2}^{RT}(D_{1},D_{2})=0$ for all $(D_{1},D_{2})$ in the
support. Note that once we decide the discharge amount $z_{1}^{RT}(D_{1})$
during peak period 1, then the discharge amount in off-peak period 2 is
determined: during each off-peak, the battery charges back to full to be ready
for discharge during peak the next day.

##### Net Demand and Price Formation Process.

The DA demands in period 1 and period 2 are taken to be the mean $\mu_{1}$ and
$\mu_{2}$ of $D_{1}$ and $D_{2}$, from the system operator’s unbiased demand
forecast. With the battery discharge, the DA net demand in period 1 and 2 are
$d_{1}^{DA}=\mu_{1}-z_{1}^{DA}$ and
$d_{1}^{DA}=\mu_{2}-z_{2}^{DA}=\mu_{2}+z_{1}^{DA}$. The RT net demand in
period 1 and 2 are $d_{1}^{RT}(D_{1})=D_{1}-\mu_{1}-z_{1}^{RT}(D_{1})$ and
$d_{2}^{RT}(D_{1},D_{2})=D_{2}-\mu_{2}-z_{2}^{RT}(D_{1},D_{2})=D_{2}-\mu_{2}+z_{1}^{RT}(D_{1})$.
Note that the RT net demand is the incremental demand, i.e., the adjustment to
the quantity cleared in day-ahead. (We slightly abuse the terminology here.
The traditional definition of net demand is system demand minus renewable
production, which is covered by conventional generators and batteries; this
corresponds to $D_{1}$ and $D_{2}$ in the demand process section earlier. The
“net demands” $d_{1}^{DA},d_{2}^{DA},d_{1}^{RT},d_{2}^{RT}$ in this section
are covered by conventional generators only.)

In each time $t$, the DA price $\lambda_{t}^{DA}$ is set at the market
clearing price, that is, the price such that the energy produced by generators
with costs below the price exactly equals the net demand:

$\displaystyle G_{s}(\lambda_{t}^{DA})+G_{f}(\lambda_{t}^{DA})=d_{t}^{DA}.$
(4)

In RT, the price $\lambda_{t}^{RT}$ is set such that the total energy produced
equals the net demand (DA demand plus incremental RT demand). However, slow
generators with total energy output $G_{s}(\lambda_{t}^{DA})$ can no longer be
adjusted in real-time, so the system operator sets the price so that the RT
generators adjust their output to match the realized net demand:

$\displaystyle
G_{s}(\lambda_{t}^{DA})+G_{f}(\lambda_{t}^{RT})=d_{t}^{DA}+d_{t}^{RT}\,.$ (5)

Equations (4) and (5) relate the net DA and RT demands $d_{t}^{DA},d_{t}^{RT}$
to the DA and RT prices $\lambda_{t}^{DA},\lambda_{t}^{RT}$. (Note that
$d_{t}^{RT}$ and $\lambda_{t}^{RT}$ depend on $D=(D_{1},D_{2})$ which we omit
for brevity.) We can invert these to get prices in terms of net demands.

We note that if RT demand is zero ($d_{t}^{RT}=0$, no adjustment to demand),
then DA and RT prices are equal: $\lambda_{t}^{DA}=\lambda_{t}^{RT}$. If RT
demand is positive (resp. negative), then the RT price is higher (resp. lower)
than the DA price.

##### Generation Cost.

As the price formation process suggests, the slow generators are cleared in
DA: they produce if and only if their costs are below $\lambda_{t}^{DA}$. The
fast generators are cleared in RT: they produce if and only if their costs are
below $\lambda_{t}^{RT}$. The total generation cost follows from integrating
the mass of generators with cost less than the corresponding clearing price,
$\lambda_{t}^{DA}$ for slow generators and $\lambda_{t}^{RT}$ for fast
generators. Therefore, the total generation cost is given by

$\displaystyle\sum_{t=1}^{2}\left(\int_{\lambda\leq\lambda_{t}^{DA}}\lambda
dG_{s}(\lambda)+\mathbb{E}_{D}\left[\int_{\lambda\leq\lambda_{t}^{RT}}\lambda
dG_{f}(\lambda)\right]\right)\,,$

where the expectation is taken over the random demand.

Throughout, we consider the system cost to be generation cost from
conventional generators. Because consumers are assumed to be price inelastic,
maximizing welfare is equivalent to minimizing generation cost. This also
matches the prevailing objective of independent system operators: the “unit
commitment” and “economic dispatch” procedures in day-ahead and real-time both
minimize generation cost (Kirschen and Strbac, 2018; Cretì and Fontini, 2019).

##### Battery Operation Models.

We will compare three operating regimes.

  * •

A first benchmark system is one without batteries.

  * •

Centralized participation: In this system, the battery is directly controlled
by the system operator and makes charge/discharge decisions to minimize
generation cost.

  * •

Decentralized participation: in this system, the battery is an independent
entity that makes charge/discharge decisions to maximize its own profit.

Note that the DA and RT prices are endogenously determined by the battery’s
charge/discharge decisions, as outlined earlier as part of the price formation
process.

##### Day-Ahead and Real-Time Supply Curves

We have the relationships between DA and RT prices and demands in (4) and (5),
which depend on $G_{s}$ and $G_{f}$. $G_{s}$ and $G_{f}$ are demand functions
for slow and fast generators, respectively, so we have a flexible way to
define the relationship between demand and price via the specification of
$G_{s}$ and $G_{f}$.

We assume that at each price $\lambda$, a fraction $k_{f}$ of generators are
fast, and $k_{s}=1-k_{f}$ are slow. In other words, at any price point, there
are fast generators that can adjust their production up and down. This
assumption reflects the operating characteristics of the generators
themselves: coal and nuclear plants are “slow,” whereas natural gas and hydro
plants are “fast.” This assumption is also an implicit model of the system
operator’s reserve requirement in day-ahead scheduling, which ensure this
property by committing some fast-responding generators in day-ahead (even when
they are relatively expensive) for reliability. Let
$G(\lambda)=G_{s}(\lambda)+G_{f}(\lambda)$ be the total supply function, then
$G_{s}(\lambda)=k_{s}G(\lambda)$ and $G_{f}=k_{f}G(\lambda)$. Equations (4)
and (5) imply

$\displaystyle\lambda_{t}^{DA}$ $\displaystyle=G^{-1}\left(d_{t}^{DA}\right)$
(6) $\displaystyle\lambda_{t}^{RT}$
$\displaystyle=G^{-1}\left(d_{t}^{DA}+\frac{1}{k_{f}}d_{t}^{RT}\right).$ (7)

Note that while $G(\cdot)$ describes a supply curve that maps price to
quantity, $G^{-1}(\cdot)$ also describes a supply curve, mapping quantity to
price. We assume that the supply curve is linear:

$\displaystyle G^{-1}(x)=\alpha+\beta x,$ (8)

where $\alpha,\beta\geq 0$ are known constants. The parameter $\alpha$ is the
“intercept” (minimum marginal cost for conventional generators), and the
parameter $\beta$ is the “slope.” The linear supply curve assumption is
commonly made in the literature, e.g., Sioshansi (2010, 2014); Ito and Reguant
(2016), and we also make this assumption primarily for parsimony. We thus have
price-demand relationships of the form
$\lambda_{t}^{DA}=\alpha+\beta^{DA}d_{t}^{DA},\lambda_{t}^{RT}=\lambda_{t}^{DA}+\beta^{RT}d_{t}^{RT}$
with $\beta^{DA}=\beta,\beta^{RT}=\beta/k_{f}$. These price-demand
relationships are similar to those derived in (You et al., 2019, Equations (5)
and (8)). In this case, we can interpret $\beta^{DA}$ and $\beta^{RT}$ as
price elasticities of day-ahead and real-time demand, respectively.333Strictly
speaking, they are not price elasticities, because price elasticities are
conventionally defined as the ratio of one percentage (relative) change
against another percentage change, whereas our coefficient is the slope, or
the ratio of the absolute price change against the absolute demand change. The
intuition governing both is similar.

###  3 No Battery Baseline

In the next three sections, we will characterize the optimal battery behavior
and the corresponding generation cost in three regimes: no battery (§3),
centralized cost-minimizing battery (§4), and decentralized profit-maximizing
battery (§5). Our results hold for any given distribution $\pi$ over
$(D_{1},D_{2})$.

Before we proceed, we first derive an expression for generation cost in terms
of demands, which is used in all regimes we consider. Define the modified DA
and RT demands as $\tilde{d}_{t}^{DA}=d_{t}^{DA}$,
$\tilde{d}_{t}^{RT}=d_{t}^{DA}+d_{t}^{RT}/k_{f}$. (Both $d_{t}^{RT}$ and
$\tilde{d}_{t}^{RT}$ can depend on $D=(D_{1},D_{2})$, but we sometimes omit it
for brevity.) The generation cost is then given by

$\displaystyle\sum_{t=1}^{2}k_{s}\left(\alpha\tilde{d}_{t}^{DA}+\beta\frac{(\tilde{d}_{t}^{DA})^{2}}{2}\right)+k_{f}\mathbb{E}_{D}\left[\alpha\tilde{d}_{t}^{RT}+\beta\frac{(\tilde{d}_{t}^{RT})^{2}}{2}\right].$
(9)

In the no-battery case, the generation cost is computed from (9) with
$\tilde{d}_{1}^{DA}=\mu_{1},\tilde{d}_{2}^{DA}=\mu_{2},\tilde{d}_{1}^{RT}(D_{1})=\mu_{1}+(D_{1}-\mu_{1})/k_{f}$
and $\tilde{d}_{2}^{DA}(D_{1},D_{2})=\mu_{2}+(D_{2}-\mu_{2})/k_{f}$. The proof
is given in Appendix A.

######  Theorem 1 (No Battery).

The generation cost under no battery Cost(NB) is given by

$\displaystyle\textnormal{Cost(NB)}=\alpha(\mu_{1}+\mu_{2})+\beta\left(\frac{\mu_{1}^{2}+\mu_{2}^{2}}{2}+\frac{\sigma_{1}^{2}+\sigma_{2}^{2}}{2k_{f}}\right).$

This no-battery generation cost is a baseline to which we compare the other
two regimes, centralized and decentralized. The generation cost depends only
on the marginal mean $\mu_{t}$ and marginal variance $\sigma_{t}^{2}$, and not
on the actual distribution beyond these moments, or the correlation of demands
from the two periods. Correlation does not matter because there is no decision
making (i.e., battery) linking the two periods, and the market clears in each
period independently. The generation cost depends on the variance because the
generation cost is quadratic in demand. Variability in demand therefore leads
to higher cost. The generation cost is also decreasing in $k_{f}$, because a
larger fraction of fast generators means that more generators can buffer the
real-time variability of demand. In other words, with more fast generators,
less expensive fast generators around the day-ahead point are enough to
satisfy the real-time incremental demand, and the system does not need to use
more expensive generators further away. This is also why $k_{f}$ only appears
in conjunction with the variance terms and not the mean terms: if the demands
are deterministic $\sigma_{1}=\sigma_{2}=0$, then the cost no longer depends
on $k_{f}$.

###  4 Battery Operation: Centralized Solution

We now consider the case when there is a battery, and the system operator can
directly control the battery to achieve the system goal, namely, minimize
generation cost. The decision variables are the DA and RT discharges
$z_{1}^{DA}$ and $z_{1}^{RT}(D_{1})$ for each realization of period-1 demand
$D_{1}$, and the system operator solves

$\displaystyle\min_{z_{1}^{DA},z_{1}^{RT}(\cdot)}$
$\displaystyle\sum_{t=1}^{2}\left[k_{s}\left(\alpha\tilde{d}_{t}^{DA}+\beta\frac{(\tilde{d}_{t}^{DA})^{2}}{2}\right)+k_{f}\mathbb{E}_{D}\left(\alpha\tilde{d}_{t}^{RT}+\beta\frac{(\tilde{d}_{t}^{RT})^{2}}{2}\right)\right],$

where

$\displaystyle\tilde{d}_{1}^{DA}=d_{1}^{DA}=\mu_{1}-z_{1}^{DA}$
$\displaystyle\tilde{d}_{2}^{DA}=d_{2}^{DA}=\mu_{2}+z_{1}^{DA}$
$\displaystyle\tilde{d}_{1}^{RT}=d_{1}^{DA}+\frac{d_{1}^{RT}}{k_{f}}=\mu_{1}-z_{1}^{DA}+\frac{D_{1}-\mu_{1}-z_{1}^{RT}(D_{1})}{k_{f}}$
$\displaystyle\tilde{d}_{2}^{RT}=d_{2}^{DA}+\frac{d_{2}^{RT}}{k_{f}}=\mu_{2}+z_{1}^{DA}+\frac{D_{2}-\mu_{2}+z_{1}^{RT}(D_{1})}{k_{f}}.$

The following theorem gives the optimal battery decisions and the
corresponding generation cost.

######  Theorem 2 (Centralized Battery).

The centralized battery discharge decisions are given by

$\displaystyle z_{1}^{DA}$ $\displaystyle=\frac{1}{2}(\mu_{1}-\mu_{2}),$
$\displaystyle z_{1}^{RT}(D_{1})$
$\displaystyle=\frac{1}{2}(D_{1}-\mu_{1})-\frac{1}{2}(\mu_{2|D_{1}}-\mu_{2}).$

The corresponding system generation cost Cost(CN) is

$\displaystyle\textnormal{Cost(CN)}=\alpha(\mu_{1}+\mu_{2})+\frac{\beta}{4}(\mu_{1}+\mu_{2})^{2}+\frac{\beta}{4k_{f}}\left[\sigma_{1}^{2}+(2-\rho_{s}^{2})\sigma_{2}^{2}+2\rho\sigma_{1}\sigma_{2}\right].$

We can show that the centralized cost minimization problem is a convex
quadratic optimization problem, whose unique optimal solution can be found
from first-order conditions. The proof is given in Appendix A.

The DA demands in the two periods are $\mu_{1}$ and $\mu_{2}$ respectively,
and the DA discharge is $z_{1}^{DA}=(\mu_{1}-\mu_{2})/2$, so the net demands
of the two periods are equalized: $\mu_{1}-z_{1}^{DA}=\mu_{2}+z_{2}^{DA}$. The
RT smoothing is more subtle, as we have to make the discharge decision
$z_{1}^{RT}(D_{1})$ knowing the realization of period 1 demand $D_{1}$ but not
the period 2 demand $D_{2}$. If we replace the unknown period 2 demand with
its conditional mean $\mu_{2|D_{1}}$, then the incremental RT demands of the
two periods are $D_{1}-\mu_{1}$ and $\mu_{2|D_{1}}-\mu_{2}$, so the RT
discharge
$z_{1}^{RT}(D_{1})=\frac{1}{2}(D_{1}-\mu_{1})-\frac{1}{2}(\mu_{2|D_{1}}-\mu_{2})$
is set such that the net RT demands of the two periods are equalized:
$(D_{1}-\mu_{1})-z_{1}^{RT}(D_{1})=(\mu_{2|D_{1}}-\mu_{2})+z_{1}^{RT}(D_{1})$.

Therefore, batteries improve social welfare by “smoothing” demand, in the
sense that batteries shift demand from peak, where it is scarce and expensive,
to off-peak, where it is plentiful and cheap, making the net demand profile
over the day more equalized and smooth. Theorem 2 shows that in a centrally
optimal world, the battery “perfectly” smooths demand between periods (peak
versus off-peak) in DA stage, and perfectly smooths the “residual” random
demand within each period in RT stage, in the sense that the net demands after
battery between two periods are equal in both day-ahead and real-time. In
other words, the battery does its “best” and does not withhold its capacity.

An alternative way to view $z_{1}^{RT}(D_{1})$ is that it smooths the real-
time component of demand followed by a correlation correction. The incremental
demand in period 1 is $D_{1}-\mu_{1}$ so the battery shifts half of it at
$\frac{1}{2}(D_{1}-\mu_{1})$ to period 2. The extra term
$-\frac{1}{2}(\mu_{2|D_{1}}-\mu_{2})$ captures the effect of the demand
dependence of period 2 that influences the decision in period 1. On one end of
the spectrum, if $D_{1}$ and $D_{2}$ are independent, then
$\mu_{2|D_{1}}=\mu_{2}$, so this term is zero, in agreement with the intuition
that if demands are independent, then the future should not influence the
current period’s decision. On the other end, if $D_{1}=D_{2}$ always, then
$\mu_{2|D_{1}}=D_{1}$, and $\mu_{2|D_{1}}-\mu_{2}=D_{1}-\mu_{1}$, so the
correlation correction term exactly cancels out the main term, and the battery
discharges zero. This is also in agreement with the intuition that if the two
periods are always the same, then there is no smoothing for the battery to do.

The correlation correction intuition also shows that the real-time battery
discharge does depend on the correlation between two demand periods, albeit
implicitly, and that higher demand correlation reduces battery discharge.

The perfect smoothing of demand also implies that the prices in two periods
are equal for both day-ahead and real-time, so battery profit is zero. This is
centrally optimal but clearly not aligned with the goal of battery profit. If
the battery is instead operated by an independent profit maximizer, then the
battery will notice that the social optimum discharges “too much” and
withholds some of its discharge. This is the source of incentive misalignment
that we will quantify in §5.

We now discuss the generation cost. Generation cost depends on the
distribution $\pi$ of $(D_{1},D_{2})$ through marginal means $\mu_{t}$,
marginal variances $\sigma_{t}^{2}$, the sequential correlation $\rho_{s}$,
and Pearson correlation $\rho$. The cost depends on the variances for the same
reason as the no-battery case: because the cost is convex, demand variability
increases cost. The cost is also decreasing in $k_{f}$, and $k_{f}$ appears
only in the variance terms for the same reason: more fast generators buffer
the impact of demand variability. However, whereas the no-battery cost does
not depend on the correlation, the centralized battery cost depends on the
correlation because the battery shifts demand across different periods.

###  5 Battery Operation: Decentralized Solution

We now consider the regime when there is an independently operated battery
that chooses its discharge/charge decisions $z_{1}^{DA},z_{1}^{RT}(D_{1})$ to
maximize its profit. The battery solves

$\displaystyle\max_{z_{1}^{DA},z_{1}^{RT}(\cdot)}\lambda_{1}^{DA}z_{1}^{DA}+\lambda_{2}^{DA}z_{2}^{DA}+\mathbb{E}_{D}\left[\lambda_{1}^{RT}z_{1}^{RT}+\lambda_{2}^{RT}z_{2}^{RT}\right],$

where the DA and RT prices are given by (6), (7), and the supply curve is
given by (8). We derive the optimal battery behavior and the corresponding
cost in the following theorem.

######  Theorem 3 (Decentralized Battery).

The decentralized battery discharge decisions are given by

$\displaystyle z_{1}^{DA}$
$\displaystyle=\frac{(2-k_{f})}{2(4-k_{f})}(\mu_{1}-\mu_{2}),$ $\displaystyle
z_{1}^{RT}(D_{1})$
$\displaystyle=\frac{k_{f}}{2(4-k_{f})}(\mu_{1}-\mu_{2})+\frac{1}{4}(D_{1}-\mu_{1})-\frac{1}{4}(\mu_{2|D_{1}}-\mu_{2}).$

The corresponding generation cost Cost(DCN) is

$\displaystyle\textnormal{Cost(DCN)}=$ $\displaystyle\alpha(\mu_{1}+\mu_{2})$
$\displaystyle+\beta\Bigg{\\{}\frac{(20-11k_{f}+k_{f}^{2})}{4(4-k_{f})^{2}}(\mu_{1}^{2}+\mu_{2}^{2})+\frac{(12-5k_{f}+k_{f}^{2})}{2(4-k_{f})^{2}}\mu_{1}\mu_{2}+\frac{5\sigma_{1}^{2}+(8-3\rho_{s}^{2})\sigma_{2}^{2}+6\rho\sigma_{1}\sigma_{2}}{16k_{f}}\Bigg{\\}}.$

We can show that the profit maximization problem is also a convex quadratic
optimization problem, whose unique solution can be found from first-order
conditions. The proof is given in Appendix A.

As in the centralized case, the discharge has a “non-random” component
smoothing predictable demand fluctuations between time periods, and a “random”
component (a factor of $D_{1}-\mu_{1}$ and $\mu_{2|D_{1}}-\mu_{2}$) smoothing
the demand fluctuation within each period. We discuss each of these components
in turn.

##### Non-random component of discharge.

Let $\Delta\mu\equiv\mu_{1}-\mu_{2}$ be the difference in mean net demands of
the two periods, i.e., the steepness of the duck curve. The total expected
discharge in period 1,
$z_{1}^{DA}+\mathbb{E}_{D}[z_{1}^{RT}]=\frac{1}{(4-k_{f})}\Delta\mu$ is
strictly less than the centrally optimal discharge $\frac{1}{2}\Delta\mu$. We
call this distortion quantity withholding. This form of distortion is familiar
from the standard account of market power. As we observed in §4, centrally
optimal battery discharges “too much,” perfectly smoothing demand resulting in
zero profit. The independent battery exercises market power by withholding
capacity, resulting in less total discharge. Quantity withholding increases
generation cost because it makes peak demand higher and off-peak demand lower,
which nets out to higher cost because cost is quadratic in demand. Note that
quantity withholding occurs even without demand randomness. We quantify the
extent of quantity withholding by computing one minus the ratio between the
decentralized and centralized expected battery discharge, which we define as

$\displaystyle\text{quantity withholding}\equiv
1-\frac{(z_{1}^{DA})_{DCN}+(\mathbb{E}_{D}[z_{1}^{RT}])_{DCN}}{(z_{1}^{DA})_{CN}+(\mathbb{E}_{D}[z_{1}^{RT}])_{CN}}=\frac{2-k_{f}}{4-k_{f}}.$
(10)

Note that quantity withholding is defined such that if the decentralized
battery (DCN) discharges as much as the centralized battery (CN), then
withholding will be zero, whereas if the decentralized battery has zero
discharge, then withholding will be one. Therefore, our definition captures
the percentage of quantity withholding. Our computation shows that quantity
withholding $(2-k_{f})/(4-k_{f})$ is decreasing in $k_{f}$. Intuitively, this
is because more fast generators mean that the battery can discharge more with
less price impact, so the battery needs to withhold less to maximize profit.
The quantity withholding percentage is $1/2=50\%$ when generators are mostly
slow ($k_{f}\approx 0$) and is $1/3\approx 33.3\%$ when generators are mostly
fast ($k_{f}\approx 1$).

Furthermore, the battery shifts a positive amount of the expected discharge
$\frac{k_{f}}{2(4-k_{f})}\Delta\mu$ to real time. In contrast, the centrally
optimal battery has zero real-time discharge in expectation. We call this
distortion the shift from day-ahead to real-time. Once the monopolist battery
already commits the cleared quantity in the first (day-ahead) market, the
battery’s marginal revenue changes, enabling the battery to discharge more in
total. In other words, the shift to real-time, while undesirable in itself,
enables the battery to do less quantity withholding. This intuition is similar
to how a monopolist sells to a population of nonstrategic consumers over two
stages: with a higher-price in the first stage to capture higher-value
consumers, and a lower price in the second stage. The quantity sold is split
over two stages, but the total quantity sold is higher than a single-stage
monopoly quantity. This effect is analogous to the forward market equilibrium
in Allaz and Vila (1993), but we are the first to analyze this effect for
batteries in electricity markets. A more informal (and perhaps more intuitive)
way to think about this effect is this: because the two markets clear
separately each with exogenous demand, the battery splits its quantity into
two markets to reduce the quantity in each market and thus reduce the adverse
price impact, which is increasing in each market’s quantity.

Notably, This shift to real-time is a structural consequence of sequential
market clearing by itself: it still exists even without different elasticities
in two markets or demand randomness. Nevertheless, the relative elasticities
of the two markets, as determined by the share of fast generators $k_{f}$,
determines the extent of the shift to real-time. We quantify the extent of
shift from day-ahead to real-time by computing the share of expected discharge
in real-time as a fraction of total discharge, and compare this share between
decentralized and centralized regime. In other words, we define shift from
day-ahead to real-time as

$\displaystyle\textnormal{shift from day-ahead to real-
time}\equiv\frac{(\mathbb{E}_{D}[z_{1}^{RT}])_{DCN}}{(z_{1}^{DA})_{DCN}+(\mathbb{E}_{D}[z_{1}^{RT}])_{DCN}}=\frac{k_{f}}{2}$
(11)

Therefore, the shift from day-ahead to real-time is increasing in $k_{f}$. The
shift percentage is 0% when generators are mostly slow ($k_{f}\approx 0$) and
50% when generators are mostly fast ($k_{f}\approx 1$). Intuitively, more fast
generators mean that the real-time price impact is less so real-time
participation is more attractive and the battery discharges more in real time.
If (almost) all generators are slow, then the price impact is so large that it
is not worth discharging in real time at all. Instead, the battery exercises
market power by quantity withholding; we have seen earlier that quantity
withholding is highest in this slow generator case.

The upshot of our discussion is that the battery strategically distorts the
discharge via quantity withholding and shift from day-ahead to real-time,
independent of demand randomness. Both types of distortion increase generation
cost and the relative weight of each type depends on generator composition.
More fast (resp. slow) generators mean more shift from day-ahead to real-time
(resp. quantity withholding). We summarize the expected discharge in day-
ahead, real-time, and total, as well as the extent of quantity withholding and
shift from day-ahead to real-time in Table 1.

regime |  | generator
---
composition
| DA
---
discharge
| RT
---
discharge
| total
---
discharge
| quantity
---
withholding
| shift from
---
DA to RT
decentralized |  | slow gen.
---
dominate
($k_{f}\approx 0$)
$\Delta\mu/4$ | $0$ | $\Delta\mu/4$ | $50\%$ | $0\%$
| fast gen.
---
dominate
($k_{f}\approx 1$)
$\Delta\mu/6$ | $\Delta\mu/6$ | $\Delta\mu/3$ | $33.3\%$ | $50\%$
centralized |  | centrally
---
optimal
$\Delta\mu/2$ | $0$ | $\Delta\mu/2$ | $0\%$ | $0\%$
Table 1: Non-random component of battery discharge as a function of generation
composition.

##### Random component of discharge.

The random component of $z_{1}^{RT}(D_{1})$ is
$\frac{1}{4}(D_{1}-\mu_{1})-\frac{1}{4}(\mu_{2|D_{1}}-\mu_{2})$, which is half
of the centrally optimal perfect smoothing. We define the one minus the ratio
of the random component of decentralized versus centralized discharge as
reduction in real-time responsiveness:

$\displaystyle\textnormal{reduction in real-time responsiveness}\equiv
1-\frac{\frac{1}{4}(D_{1}-\mu_{1})-\frac{1}{4}(\mu_{2|D_{1}}-\mu_{2})}{\frac{1}{2}(D_{1}-\mu_{1})-\frac{1}{2}(\mu_{2|D_{1}}-\mu_{2})}=\frac{1}{2}$
(12)

Intuitively, the reduction in real-time responsiveness can also be viewed as a
form of battery exercising market power via quantity withholding, but on the
real-time mean-zero component of demand. This reduction in real-time
responsiveness is always exactly 50% on both the realized demand and the
correlation correction components. Just as we argued in the centralized case,
the term $(\mu_{2|D_{1}}-\mu_{2})$ can be viewed as “correlation correction”
because it is zero when $D_{1}$ and $D_{2}$ are independent, and it exactly
cancels out the main term when $D_{1}=D_{2}$.

####  5.1 Comparing Generation Costs Across Different Regimes

We have derived battery discharge decisions and the corresponding generation
costs under three regimes: no battery (Theorem 1), centralized battery
(Theorem 2), and decentralized battery (Theorem 3). We can now compare
generation costs (which are the system operator’s objectives) between the
three regimes, which we denote by Cost(NB), Cost(CN), and Cost(DCN),
respectively.

We define the price of anarchy (PoA) as the relative cost reduction from the
no-battery default of the centralized versus decentralized battery:

$\displaystyle\textnormal{PoA}\equiv\frac{\textnormal{Cost(NB)}-\textnormal{Cost(CN)}}{\textnormal{Cost(NB)}-\textnormal{Cost(DCN)}}.$
(PoA)

While not obvious, having an independent battery always yields a cost
reduction relative to the no-battery default (cf. Theorem 4). Given this,
(PoA) is well-defined, and $\textnormal{PoA}\geq 1$.

The PoA metric captures the fact that there is a part of the generation cost
that is “unavoidable” in the sense that even the centrally controlled battery
cannot avoid it. Therefore, the PoA is defined to compare the relative cost
reduction relative to the no-battery benchmark, which is the status quo before
the introduction of battery.

Throughout this subsection, we will assume that $(D_{1},D_{2})$ is jointly
normal with correlation $\rho$; this is to highlight the qualitative
dependence of different cost metrics on market fundamentals and to enable us
to meaningfully discuss the correlation $\rho$.

######  Theorem 4 (Cost Comparisons).

  1. (a)

$\textnormal{Cost(NB)}\geq\textnormal{Cost(DCN)}$ and
$\textnormal{Cost(DCN)}\geq\textnormal{Cost(CN)}$. Both inequalities become
equalities if and only if $\mu_{1}=\mu_{2}$ and $\sigma_{1}=\rho\sigma_{2}$.

  2. (b)

$\textnormal{PoA}\in[9/8,4/3]$. The lower bound is achieved when $k_{f}=1$ and
$\sigma_{1}=\rho\sigma_{2}$. The upper bound is achieved when $k_{f}=0$. PoA
is decreasing in $|\mu_{1}-\mu_{2}|$, increasing in
$|\sigma_{1}-\rho\sigma_{2}|$, and decreasing in $k_{f}$.

The result that $\textnormal{Cost(NB)}\geq\textnormal{Cost(DCN)}$ is in the
spirit of Sioshansi (2014), while
$\textnormal{Cost(DCN)}\geq\textnormal{Cost(CN)}$ is necessarily true by
definition. Taken together, these cost comparisons show that the three costs
are ranked, so the PoA metric is well-defined and meaningful. We also derive a
lower bound of $9/8$ and an upper bound of $4/3$ on PoA, and both bounds are
the best possible. In other words, the incentive misalignment can increase the
generation cost from $12.5\%$ to $33.3\%$ relative to the no-battery
benchmark. Both bounds are attainable and are independent of the market
parameters. In particular, the existence of an absolute lower bound strictly
away from 1 means that in any market, there is always incentive misalignment
(increasing cost by at least $12.5\%$).

The bounds and comparative statics of PoA follow from the following argument.
The cost reductions of both centralized and decentralized regimes have two
components: the contribution from the differences in means, and from the
differences in variance, as shown by

$\displaystyle\textnormal{Cost(NB)}-\textnormal{Cost(CN)}$
$\displaystyle=\beta\frac{1}{4}(\mu_{1}-\mu_{2})^{2}+\frac{\beta}{4k_{f}}(\sigma_{1}-\rho\sigma_{2})^{2}$
$\displaystyle\textnormal{Cost(NB)}-\textnormal{Cost(DCN)}$
$\displaystyle=\beta\frac{(12-5k_{f}+k_{f}^{2})}{4(4-k_{f})^{2}}(\mu_{1}-\mu_{2})^{2}+\frac{3\beta}{16k_{f}}(\sigma_{1}-\rho\sigma_{2})^{2}$

The variance component of the two cost gaps are always a factor of $4/3$ from
each other, whereas the mean component of the two cost gaps are a factor of
$\frac{12-5k_{f}+k_{f}^{2}}{(4-k_{f})^{2}}$ from each other. This factor
reaches a minimum of $9/8$ at $k_{f}=1$ and a maximum of $4/3$ at $k_{f}=0$.
Therefore, the ratio PoA is between $9/8$ and $4/3$. Because the mean factor
component is less than the variance factor component of $4/3$, if
$|\mu_{1}-\mu_{2}|$ increases, the mean component becomes more important and
the PoA decreases, whereas if $|\sigma_{1}-\rho\sigma_{2}|$ increases, the
variance component becomes more important and PoA decreases. Lastly, because
the mean factor is decreasing in $k_{f}$, the ratio PoA is also decreasing in
$k_{f}$.

Because PoA is decreasing in $|\mu_{1}-\mu_{2}|$, as renewable energy
(especially solar) increases, widening the gap between peak mean net demand
$\mu_{1}$ and off-peak mean net demand $\mu_{2}$, incentive alignment
improves! A more severe duck curve increases the cost gaps of both centralized
and decentralized regimes, but the decentralized cost reduction grows at a
faster rate. Moreover, the fact that PoA is decreasing in $k_{f}$ means that
more fast generators improve incentive alignment. Intuitively, this is because
non-strategic fast generators are ready to “step in” and cushion the price
impact of real-time battery withholding. Lastly, because PoA is increasing in
$|\sigma_{1}-\rho\sigma_{2}|$, it is not necessarily monotonic in the
correlation $\rho$. On the one hand, if $\sigma_{1}\leq\sigma_{2}$ then PoA is
always increasing in $\rho$. On the other hand, if $\sigma_{1}\geq\sigma_{2}$
then PoA is decreasing in $\rho$ when $\rho\leq\sigma_{1}/\sigma_{2}$ and
increasing in $\rho$ when $\rho\geq\sigma_{1}/\sigma_{2}$.

###  6 Numerical Illustrations

In this section, we illustrate the battery behavior and incentive misalignment
numerically. To anchor ideas, and for illustration purposes, we use data from
Los Angeles and Houston. We choose these regions as examples of markets that
are well on the way in terms of renewable and battery adoption, and markets
that are in transition, respectively. We also observe local monopoly effects
in these regions. For example, in Los Angeles, as of April 2024, AES is the
biggest player with 4 batteries with total capacity of 355 MW, the second
biggest is VESI with two batteries, 20 MW each, and the rest are very small
batteries with total capacity 27.2 MW (EIA, 2024a).

We emphasize that the results we present here are illustrative of the forces
at play, but also do not account for how submarkets are connected in the
electricity market. A full network analysis is beyond the scope of the current
work.

Figure 2: Mean net demand for each hour of the day in 2023 in Los Angeles and
Houston, and the corresponding peak and off-peak hours.

We calibrate the supply curve parameters $\alpha,\beta$ from public market
data. From $\lambda_{t}^{DA}=\alpha+\beta d_{t}^{DA}$, we estimate $\alpha$
and $\beta$ by linear regression of the day-ahead price against day-ahead
demand, with L1 loss to be robust to outliers. California reports this data by
region on its web portal, while Texas provides historical data upon
request.444Because the price of anarchy and different types of withholding
depend only on $k_{f}$ and the distribution of $(D_{1},D_{2})$ but not on
$\alpha,\beta$, strictly speaking we do not need to estimate these parameters
for our numerical illustration.

We estimate $k_{f}$ from the share of energy produced from “fast” sources. In
2023, the share of energy produced from conventional generators are the
following: in California, natural gas 56.5%, large hydro 10.2%, nuclear 4.9%
(CAISO, 2023); in Texas, combined-cycle natural gas 37%, natural gas 8%,
nuclear 9%, coal 14% (ERCOT, 2023). If we assume that natural gas and hydro
are fast, while nuclear and coal are slow, then we have $k_{f}=0.93$ for Los
Angeles and $k_{f}=0.66$ for Houston.

In contrast to the generator parameters $\alpha$, $\beta$, $k_{f}$ which can
be assumed to be constant throughout the year, the net demand has a
significant seasonality component. We assume that the peak and off-peak net
demands are jointly normal, and calibrate the marginal means
$\mu_{1},\mu_{2}$, marginal variances $\sigma_{1}^{2},\sigma_{2}^{2}$, and the
correlation $\rho$ for each quarter of 2023. Each quarter corresponds roughly
to a season: first quarter (Jan-Mar) is winter; second quarter (Apr-Jun) is
spring; third quarter (Jul-Sep) is summer, and fourth quarter (Oct-Dec) is
fall. Both Los Angeles and Houston typically have mild winters, and
electricity usage is highest in the summer due to air conditioning. For Los
Angeles, we take the peak hour to be 19:00-20:00 and the off-peak hour to be
12:00-13:00. For Houston, we take the peak hour to be 19:00-20:00 and the off-
peak hour to be 09:00-10:00 (cf. Figure 2).

For each quarter, we calculate the price of anarchy. In Los Angeles, we have
1.15 for all four quarters. In Houston, we have Q1: 1.26, Q2: 1.24, Q3: 1.22,
Q4: 1.26 with an average of 1.25. Therefore, even though the demand has high
seasonality, the price of anarchy is fairly stable across seasons. The Price
of Anarchy is a measure of incentive misalignment in terms of generation cost:
15% for Los Angeles and 25% in Houston, assuming a single large battery. (We
subtract one when we represent the PoA as percentages for convenience.) The
drivers of increased cost are quantity withholding, shift of participation
from day-ahead to real-time, and reduction in real-time responsiveness, as
computed in (10), (11), and (12) which depends only on $k_{f}$. The results
are summarized in Table 2. We can see that, if the battery achieves local
monopoly in a region, the price of anarchy as well as all three types of
distortion are practically significant.

|  | distortion types
---|---|---
| PoA |  quantity withholding |  shift from DA to RT |  | reduction in
---
RT responsiveness
Los Angeles | 15% | 35% | 47% | 50%
Houston | 25% | 40% | 33% | 50%
Table 2: Incentive misalignment in generation cost (Price of Anarchy) and in
withholding in quantity, time, and responsiveness in Los Angeles and Houston.

###  7 Conclusion

We formulate and solve an analytical model of market power of batteries in
electricity markets. We find that profit-maximizing batteries strategically
distort their decisions by quantity withholding, shifting participation from
day-ahead to real-time, and reducing real-time responsiveness, and quantify
the extent of each form of distortion. Quantity withholding ranges from 33.3%
to 50%; the shift from day-ahead to real-time ranges from 0% to 50%, and
reduction in real-time responsiveness is always a constant at 50%. The larger
the share of fast generators, the more batteries shift to real-time rather
than withhold in quantity, and vice versa. Battery distortion due to incentive
misalignment leads to an increase in generation cost between 12.5% and 33.3%,
and the misalignment is largest in relative terms when generators are slow,
and and the duck curve is shallow. Numerical illustrations with Los Angeles
and Houston data suggest that, if a battery achieves local monopoly, these
effects could be practically significant.

There are many avenues for future work. Our model is intentionally
parsimonious to most clearly highlight the main drivers of battery incentive
misalignment. Our model can be used as a framework to understand the role of
constraints such as battery inefficiency and physical operating costs. Given
our model’s focus on daily cycles of the market, taking market participants as
fixed, it naturally fits with the system operator’s goal of ensuring proper
market functioning. However, it can also be used as a building block to
understand higher-level decisions such as investment in battery capacity and
government incentives. Our model considers each region separately, which can
be a good first-order approximation for highly fragmented markets. For
moderately fragmented markets, the network structure and locational marginal
pricing market clearing should be modeled explicitly, and the question of
market power over a network is worth investigating. Our model also assumes
that the environment is probabilistically known. The energy market is a data-
rich environment with many predictable patterns, and our model captures
essential parts of that. Arguably, however, battery behavior is also partly
shaped by uncertainty and robustness considerations. For example, if the
market price occasionally spikes, then part of battery withholding behavior
might simply be contingency preparation rather than market power.
Understanding the role of price and system forecast, Bayesian and non-Bayesian
uncertainty, and distinguishing between strategic behavior and standard
operating procedures is an important future direction.

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Electronic Companion:
Battery Operations in Electricity Markets:
Strategic Behavior and Distortions
Jerry Anunrojwong,111Columbia University, Graduate School of Business. Email:
<EMAIL_ADDRESS>Santiago R. Balseiro,222Columbia University,
Graduate School of Business. Email<EMAIL_ADDRESS>Omar
Besbes,333Columbia University, Graduate School of Business. Email:
<EMAIL_ADDRESS>. Bolun Xu444Columbia University, Earth and Environmental
Engineering. Email<EMAIL_ADDRESS>.

## Appendix

###  Appendix A Proofs for Section 3

######  Proposition A-1.

We have

$\displaystyle\mathbb{E}[(\mu_{2|D_{1}}-\mu_{2})^{2}]$
$\displaystyle=\rho_{s}^{2}\sigma_{2}^{2}$
$\displaystyle\mathbb{E}[\sigma_{2|D_{1}}^{2}]$
$\displaystyle=(1-\rho_{s}^{2})\sigma_{2}^{2}$
$\displaystyle\mathbb{E}[D_{2}-\mu_{2|D_{1}}]$ $\displaystyle=0$
$\displaystyle\mathbb{E}[(D_{2}-\mu_{2|D_{1}})(D_{1}-\mu_{1})]$
$\displaystyle=0$
$\displaystyle\mathbb{E}[(D_{2}-\mu_{2|D_{1}})(D_{2}-\mu_{2})]$
$\displaystyle=(1-\rho_{s}^{2})\sigma_{2}^{2}$
$\displaystyle\mathbb{E}[(D_{1}-\mu_{1})(\mu_{2|D_{1}}-\mu_{2})]$
$\displaystyle=\rho\sigma_{1}\sigma_{2}$
$\displaystyle\mathbb{E}[(D_{2}-\mu_{2})(\mu_{2|D_{1}}-\mu_{2})]$
$\displaystyle=\rho_{s}^{2}\sigma_{2}^{2}$

Also, for each constant $c$, we have

$\displaystyle\mathbb{E}[(D_{2}-c)^{2}|d_{1}]=\sigma_{2|d_{1}}^{2}+(\mu_{2|d_{1}}-c)^{2}$

###### Proof of Proposition A-1.

The first two equations hold by definition of $\rho_{s}$. For the third
equation,
$bE[D_{2}-\mu_{2|D_{1}}]=\mathbb{E}[\mathbb{E}[D_{2}-\mu_{2|D_{1}}|D_{1}]]=\mathbb{E}[\mu_{2|D_{1}}-\mu_{2|D_{1}}]=0$.
For the fourth equation,
$\mathbb{E}[(D_{2}-\mu_{2|D_{1}})(D_{1}-\mu_{1})]=\mathbb{E}[\mathbb{E}[(D_{2}-\mu_{2|D_{1}})(D_{1}-\mu_{1})|D_{1}]]=\mathbb{E}[(D_{1}-\mu_{1})(\mu_{2|D_{1}}-\mu_{2|D_{1}})]=0$.
For the fifth equation,

$\displaystyle\mathbb{E}[(D_{2}-\mu_{2|D_{1}})(D_{2}-\mu_{2})]$
$\displaystyle=\mathbb{E}[(D_{2}-\mu_{2|D_{1}})^{2}]+\mathbb{E}[(D_{2}-\mu_{2|D_{1}})(\mu_{2|D_{1}}-\mu_{2})]$
$\displaystyle=\mathbb{E}[\mathbb{E}[(D_{2}-\mu_{2|D_{1}})^{2}|D_{1}]]+\mathbb{E}[\mathbb{E}[(D_{2}-\mu_{2|D_{1}})(\mu_{2|D_{1}}-\mu_{2})|D_{1}]]$
$\displaystyle=\mathbb{E}[\sigma_{2|D_{1}}^{2}]+\mathbb{E}[(\mu_{2|D_{1}}-\mu_{2|D_{1}})(\mu_{2|D_{1}}-\mu_{2})]$
$\displaystyle=(1-\rho_{s}^{2})\sigma_{2}^{2}+0=(1-\rho_{s}^{2})\sigma_{2}^{2}$

For the sixth equation,
$\mathbb{E}[(D_{1}-\mu_{1})(\mu_{2|D_{1}}-\mu_{2})]=\mathbb{E}[\mathbb{E}[(D_{1}-\mu_{1})(D_{2}-\mu_{2})|D_{1}]]=\mathbb{E}[(D_{1}-\mu_{1})(D_{2}-\mu_{2})]=\rho\sigma_{1}\sigma_{2}$.
For the seventh equation,
$\mathbb{E}[(D_{2}-\mu_{2})(\mu_{2|D_{1}}-\mu_{2})]=\mathbb{E}[\mathbb{E}[(D_{2}-\mu_{2})(\mu_{2|D_{1}}-\mu_{2})|D_{1}]]=\mathbb{E}[(\mu_{2|D_{1}}-\mu_{2})^{2}]=\rho_{s}^{2}\sigma_{2}^{2}$

Lastly, we have

$\displaystyle\mathbb{E}[(D_{2}-c)^{2}|d_{1}]$
$\displaystyle=\mathbb{E}[(D_{2}-\mu_{2|d_{1}}+\mu_{2|d_{1}}-c)^{2}|d_{1}]$
$\displaystyle=\mathbb{E}[(D_{2}-\mu_{2|d_{1}})^{2}|d_{1}]+(\mu_{2|d_{1}}-c)^{2}+2(\mu_{2|d_{1}}-c)\mathbb{E}[(D_{2}-\mu_{2|d_{1}}|d_{1}]$
$\displaystyle=\sigma_{2|d_{1}}^{2}+(\mu_{2|d_{1}}-c)^{2}+2(\mu_{2|d_{1}}-c)\cdot
0$ $\displaystyle=\sigma_{2|d_{1}}^{2}+(\mu_{2|d_{1}}-c)^{2}$

∎

###### Proof of Theorem 1.

For $t\in\\{1,2\\}$, we compute

$\displaystyle\mathbb{E}[(\tilde{d}_{t}^{RT})]$
$\displaystyle=\mathbb{E}\left[\mu_{t}+\frac{D_{t}-\mu_{t}}{k_{f}}\right]=\mu_{t}$
$\displaystyle\mathbb{E}[(\tilde{d}_{t}^{RT})^{2}]$
$\displaystyle=\mathbb{E}\left[\left(\mu_{t}+\frac{D_{t}-\mu_{t}}{k_{f}}\right)^{2}\right]=\mu_{t}^{2}+\frac{2\mu_{t}}{k_{f}}\mathbb{E}[(D_{t}-\mu_{t})]+\frac{1}{k_{f}^{2}}\mathbb{E}[(D_{t}-\mu_{t})^{2}]=\mu_{t}^{2}+\frac{\sigma_{t}^{2}}{k_{f}^{2}}$

Therefore, the generation cost is

$\displaystyle\sum_{t=1}^{2}k_{s}\left[\alpha\mu_{t}+\frac{\beta}{2}\mu_{t}^{2}\right]+k_{f}\left[\alpha\mu_{t}+\frac{\beta}{2}\left(\mu_{t}^{2}+\frac{\sigma_{t}^{2}}{k_{f}^{2}}\right)\right]$

which simplifies to the given expression.

∎

###  Appendix B Proofs for Section 4

###### Proof of Theorem 2.

Generation cost is

$\displaystyle\alpha(\mu_{1}+\mu_{2})$
$\displaystyle+k_{s}\left[\frac{\beta}{2}\left[(\mu_{1}-z_{1}^{DA})^{2}+(\mu_{2}+z_{1}^{DA})^{2}\right]\right]$
$\displaystyle+k_{f}\mathbb{E}_{D_{1},D_{2}}\Bigg{\\{}\frac{\beta}{2}\left[\left(\mu_{1}-z_{1}^{DA}+\frac{D_{1}-\mu_{1}-z_{1}^{RT}(D_{1})}{k_{f}}\right)^{2}+\left(\mu_{2}+z_{1}^{DA}+\frac{D_{2}-\mu_{2}+z_{1}^{RT}(D_{1})}{k_{f}}\right)^{2}\right]\Bigg{\\}}$

We first note that the generation cost is strictly convex (quadratic) in the
decision variables. So, the global minimum is achieved where first-order
conditions hold with equality.

For each fixed $D_{1}=d_{1}$, we take the derivative w.r.t
$z_{1}^{RT}(d_{1})$. By the law of iterated expectations, the expectation
$\mathbb{E}_{D_{1},D_{2}}$ can be viewed as taking expectation
$\mathbb{E}_{D_{1}}$ followed by the conditional expectation
$\mathbb{E}_{D_{2}|D_{1}}$, by focusing on $z_{1}^{RT}(d_{1})$ we fix the
value $D_{1}=d_{1}$ while the inner expectation becomes an expectation over
$D_{2}\sim\pi(\cdot|D_{1}=d_{1})$. We therefore get

$\displaystyle\mathbb{E}_{D_{2}\sim\pi(\cdot|D_{1}=d_{1})}\Bigg{\\{}\beta\left[-\frac{1}{k_{f}}\left(\mu_{1}-z_{1}^{DA}+\frac{d_{1}-\mu_{1}-z_{1}^{RT}(d_{1})}{k_{f}}\right)+\frac{1}{k_{f}}\left(\mu_{2}+z_{1}^{DA}+\frac{D_{2}-\mu_{2}+z_{1}^{RT}(D_{1})}{k_{f}}\right)\right]\Bigg{\\}}=0.$

The expression is linear in $D_{2}$, and the expectation is
$\mathbb{E}[D_{2}|D_{1}=d_{1}]=\mu_{2|d_{1}}$. Therefore,

$\displaystyle-
k_{f}(\mu_{1}-\mu_{2})+2k_{f}z_{1}^{DA}-(d_{1}-\mu_{1})+(\mu_{2|d_{1}}-\mu_{2})+2z_{1}^{RT}(d_{1})=0$

This allows us to write $z_{1}^{RT}(d_{1})$ in terms of $z_{1}^{DA}$:

$\displaystyle
z_{1}^{RT}(d_{1})=-k_{f}z_{1}^{DA}+\frac{k_{f}}{2}(\mu_{1}-\mu_{2})+\frac{1}{2}(d_{1}-\mu_{1})-\frac{1}{2}(\mu_{2|d_{1}}-\mu_{2})$

In particular, this implies

$\displaystyle\mathbb{E}[z_{1}^{RT}(D_{1})]=-k_{f}z_{1}^{DA}+\frac{k_{f}}{2}(\mu_{1}-\mu_{2})$

Taking the derivative w.r.t. $z_{1}^{DA}$ gives

$\displaystyle
k_{s}\Bigg{[}\beta\left[-(\mu_{1}-z_{1}^{DA})+(\mu_{2}+z_{1}^{DA})\right]\Bigg{]}$
$\displaystyle+k_{f}\mathbb{E}_{D_{1},D_{2}}\Bigg{\\{}\beta\left[-\left(\mu_{1}-z_{1}^{DA}+\frac{D_{1}-\mu_{1}-z_{1}^{RT}(D_{1})}{k_{f}}\right)+\left(\mu_{2}+z_{1}^{DA}+\frac{D_{2}-\mu_{2}+z_{1}^{RT}(D_{1})}{k_{f}}\right)\right]\Bigg{\\}}=0$

or

$\displaystyle(\mu_{2}-\mu_{1}+2z_{1}^{DA})+\mathbb{E}\left[2z_{1}^{RT}(D_{1})\right]=0$

Using the expression for $\mathbb{E}[z_{1}^{RT}(D_{1})]$ derived earlier, we
get

$\displaystyle z_{1}^{DA}$ $\displaystyle=\frac{\mu_{1}-\mu_{2}}{2}$
$\displaystyle z_{1}^{RT}(d_{1})$
$\displaystyle=\frac{1}{2}(d_{1}-\mu_{1})-\frac{1}{2}(\mu_{2|d_{1}}-\mu_{2}).$

We now compute the generation cost. We have

$\displaystyle\left(\mu_{1}-z_{1}^{DA}+\frac{D_{1}-\mu_{1}-z_{1}^{RT}(D_{1})}{k_{f}}\right)^{2}$
$\displaystyle=\frac{1}{k_{f}^{2}}\left(k_{f}\mu_{1}+(D_{1}-\mu_{1})-k_{f}z_{1}^{DA}-z_{1}^{RT}(D_{1})\right)^{2}$
$\displaystyle=\frac{1}{4k_{f}^{2}}\left[k_{f}(\mu_{1}+\mu_{2})+(D_{1}-\mu_{1})+(\mu_{2|D_{1}}-\mu_{2})\right]^{2}$

Using Proposition A-1, we can evaluate

$\displaystyle\mathbb{E}\left[\left(\mu_{1}-z_{1}^{DA}+\frac{D_{1}-\mu_{1}-z_{1}^{RT}(D_{1})}{k_{f}}\right)^{2}\right]=\frac{1}{4k_{f}^{2}}\left[k_{f}^{2}(\mu_{1}+\mu_{2})^{2}+\sigma_{1}^{2}+\rho_{s}^{2}\sigma_{2}^{2}+2\rho\sigma_{1}\sigma_{2}\right]$

Similarly,

$\displaystyle\mathbb{E}\left[\left(\mu_{2}+z_{1}^{DA}+\frac{D_{2}-\mu_{2}+z_{1}^{RT}(D_{1})}{k_{f}}\right)^{2}\right]$
$\displaystyle=\frac{1}{4k_{f}^{2}}\mathbb{E}\left[\left(2(D_{2}-\mu_{2|D_{1}})+(\mu_{2|D_{1}}-\mu_{2})+(D_{1}-\mu_{1})+k_{f}(\mu_{1}+\mu_{2})\right)^{2}\right]$
$\displaystyle=\frac{1}{4k_{f}^{2}}\mathbb{E}\left[4\sigma_{2|D_{1}}^{2}+\left((\mu_{2|D_{1}}-\mu_{2})+(D_{1}-\mu_{1})+k_{f}(\mu_{1}+\mu_{2})\right)^{2}\right]$
$\displaystyle=\frac{1}{4k_{f}^{2}}\Big{[}4(1-\rho_{s}^{2})\sigma_{2}^{2}+\rho_{s}^{2}\sigma_{2}^{2}+\sigma_{1}^{2}+k_{f}^{2}(\mu_{1}+\mu_{2})^{2}+2\rho\sigma_{1}\sigma_{2}\Big{]}$

So

$\displaystyle\mathbb{E}\left[\left(\mu_{1}-z_{1}^{DA}+\frac{D_{1}-\mu_{1}-z_{1}^{RT}(D_{1})}{k_{f}}\right)^{2}\right]+\mathbb{E}\left[\left(\mu_{2}+z_{1}^{DA}+\frac{D_{2}-\mu_{2}+z_{1}^{RT}(D_{1})}{k_{f}}\right)^{2}\right]$
$\displaystyle=\frac{1}{2k_{f}^{2}}\left[\sigma_{1}^{2}+(2-\rho_{s}^{2})\sigma_{2}^{2}+k_{f}^{2}(\mu_{1}+\mu_{2})^{2}+2\rho\sigma_{1}\sigma_{2}\right].$

We also have

$\displaystyle(\mu_{1}-z_{1}^{DA})^{2}+(\mu_{2}+z_{1}^{DA})^{2}=\left(\frac{\mu_{1}+\mu_{2}}{2}\right)^{2}+\left(\frac{\mu_{1}+\mu_{2}}{2}\right)^{2}=\frac{1}{2}(\mu_{1}+\mu_{2})^{2}$

Therefore, the total generation cost is

$\displaystyle\alpha(\mu_{1}+\mu_{2})+k_{s}\left[\frac{\beta}{4}(\mu_{1}+\mu_{2})^{2}\right]+k_{f}\left[\frac{\beta}{2}\frac{1}{2k_{f}^{2}}\left[\sigma_{1}^{2}+(2-\rho_{s}^{2})\sigma_{2}^{2}+k_{f}(\mu_{1}+\mu_{2})^{2}+2\rho\sigma_{1}\sigma_{2}\right]\right]$

which simplifies to the generation cost expression in the theorem. ∎

###  Appendix C Proofs for Section 5

###### Proof of Theorem 3.

The prices are given by

$\displaystyle\lambda_{1}^{DA}$ $\displaystyle=\alpha+\beta d_{1}^{DA}$
$\displaystyle\lambda_{2}^{DA}$ $\displaystyle=\alpha+\beta d_{2}^{DA}$
$\displaystyle\lambda_{1}^{RT}$
$\displaystyle=\alpha+\beta\left(d_{1}^{DA}+\frac{d_{1}^{RT}}{k_{f}}\right)$
$\displaystyle\lambda_{2}^{RT}$
$\displaystyle=\alpha+\beta\left(d_{2}^{DA}+\frac{d_{2}^{RT}}{k_{f}}\right)$

with

$\displaystyle d_{1}^{DA}$ $\displaystyle=\mu_{1}-z_{1}^{DA}$ $\displaystyle
d_{2}^{DA}$ $\displaystyle=\mu_{2}+z_{1}^{DA}$ $\displaystyle d_{1}^{RT}$
$\displaystyle=D_{1}-\mu_{1}-z_{1}^{RT}(D_{1})$ $\displaystyle d_{2}^{RT}$
$\displaystyle=D_{2}-\mu_{2}+z_{1}^{RT}(D_{1})$

The battery maximizes profit:

$\displaystyle\Pi=(\lambda_{1}^{DA}-\lambda_{2}^{DA})z_{1}^{DA}+\mathbb{E}\left[(\lambda_{1}^{RT}-\lambda_{2}^{RT})z_{1}^{RT}(D_{1})\right]$

We can write

$\displaystyle\Pi$
$\displaystyle=\beta(\mu_{1}-\mu_{2}-2z_{1}^{DA})z_{1}^{DA}$
$\displaystyle+\mathbb{E}\left[\beta\left(\mu_{1}-\mu_{2}-2z_{1}^{DA}+\frac{((D_{1}-\mu_{1})-(D_{2}-\mu_{2})-2z_{1}^{RT}(D_{1}))}{k_{f}}\right)z_{1}^{RT}(D_{1})\right]$

Taking derivative w.r.t. $z_{1}^{RT}(d_{1})$ for a given fixed $d_{1}$ gives,
for each $d_{1}$,

$\displaystyle\mathbb{E}_{D_{2}\sim\pi(\cdot|d_{1})}\Bigg{[}\beta\left(\mu_{1}-\mu_{2}-2z_{1}^{DA}+\frac{((d_{1}-\mu_{1})-(D_{2}-\mu_{2})-4z_{1}^{RT}(d_{1}))}{k_{f}}\right)\Bigg{]}=0$

This reduces to

$\displaystyle
z_{1}^{RT}(d_{1})=\frac{1}{4}(d_{1}-\mu_{1})-\frac{1}{4}(\mu_{2|d_{1}}-\mu_{2})+\frac{k_{f}}{4}(\mu_{1}-\mu_{2}-2z_{1}^{DA})$

In particular,

$\displaystyle\mathbb{E}[z_{1}^{RT}(D_{1})]=\frac{k_{f}}{4}(\mu_{1}-\mu_{2}-2\bar{z}_{1}^{DA})$

Now we take derivative w.r.t $z_{1}^{DA}$:

$\displaystyle\beta(\mu_{1}-\mu_{2}-4z_{1}^{DA})+\mathbb{E}\left[\beta(-2)z_{1}^{RT}(D_{1})\right]=0$

Using the expression for $\mathbb{E}[z_{1}^{RT}(D_{1})]$ derived earlier gives

$\displaystyle\bar{z}_{1}^{DA}$
$\displaystyle=\frac{(2-k_{f})}{2(4-k_{f})}(\mu_{1}-\mu_{2})$
$\displaystyle\bar{z}_{1}^{RT}(d_{1})$
$\displaystyle=\frac{1}{4}(d_{1}-\mu_{1})-\frac{1}{4}(\mu_{2|d_{1}}-\mu_{2})+\frac{k_{f}}{2(4-k_{f})}(\mu_{1}-\mu_{2})$

Now we compute the generation cost. The demands are given by

$\displaystyle d_{1}^{DA}$
$\displaystyle=\mu_{1}-z_{1}^{DA}=\frac{(6-k_{f})}{2(4-k_{f})}\mu_{1}+\frac{(2-k_{f})}{2(4-k_{f})}\mu_{2}$
$\displaystyle d_{2}^{DA}$
$\displaystyle=\mu_{2}+z_{1}^{DA}=\frac{(2-k_{f})}{2(4-k_{f})}\mu_{1}+\frac{(6-k_{f})}{2(4-k_{f})}\mu_{2}$
$\displaystyle d_{1}^{RT}$
$\displaystyle=\frac{3}{4}(d_{1}-\mu_{1})+\frac{1}{4}(\mu_{2|d_{1}}-\mu_{2})-\frac{k_{f}}{2(4-k_{f})}(\mu_{1}-\mu_{2})$
$\displaystyle d_{2}^{RT}$
$\displaystyle=(d_{2}-\mu_{2})+\frac{1}{4}(d_{1}-\mu_{1})-\frac{1}{4}(\mu_{2|d_{1}}-\mu_{2})+\frac{k_{f}}{2(4-k_{f})}(\mu_{1}-\mu_{2})$

We calculate the modified real-time demand

$\displaystyle\tilde{d}_{1}^{RT}$
$\displaystyle=d_{1}^{DA}+\frac{d_{1}^{RT}}{k_{f}}=\frac{3}{4k_{f}}(d_{1}-\mu_{1})+\frac{1}{4k_{f}}(\mu_{2|d_{1}}-\mu_{2})+\frac{(5-k_{f})}{2(4-k_{f})}\mu_{1}+\frac{(3-k_{f})}{2(4-k_{f})}\mu_{2}$
$\displaystyle\tilde{d}_{2}^{RT}$
$\displaystyle=d_{2}^{DA}+\frac{d_{2}^{RT}}{k_{f}}=\frac{1}{k_{f}}(d_{2}-\mu_{2})+\frac{1}{4k_{f}}(d_{1}-\mu_{1})-\frac{1}{4k_{f}}(\mu_{2|d_{1}}-\mu_{2})+\frac{(3-k_{f})}{2(4-k_{f})}\mu_{1}+\frac{(5-k_{f})}{2(4-k_{f})}\mu_{2}$

We calculate, using Proposition A-1:

$\displaystyle\mathbb{E}[(\tilde{d}_{1}^{RT})^{2}]$
$\displaystyle=\mathbb{E}\left(\frac{3}{4k_{f}}(D_{1}-\mu_{1})+\frac{1}{4k_{f}}(\mu_{2|D_{1}}-\mu_{2})+\frac{(5-k_{f})}{2(4-k_{f})}\mu_{1}+\frac{(3-k_{f})}{2(4-k_{f})}\mu_{2}\right)^{2}$
$\displaystyle=\frac{9\sigma_{1}^{2}+\rho_{s}^{2}\sigma_{2}^{2}+6\rho\sigma_{1}\sigma_{2}}{16k_{f}^{2}}+\left(\frac{(5-k_{f})}{2(4-k_{f})}\mu_{1}+\frac{(3-k_{f})}{2(4-k_{f})}\mu_{2}\right)^{2}$
$\displaystyle\mathbb{E}[(\tilde{d}_{2}^{RT})^{2}]$
$\displaystyle=\mathbb{E}\left(\frac{4(D_{2}-\mu_{2})+(D_{1}-\mu_{1})-(\mu_{2|D_{1}}-\mu_{2})}{4k_{f}}+\frac{(3-k_{f})}{2(4-k_{f})}\mu_{1}+\frac{(5-k_{f})}{2(4-k_{f})}\mu_{2}\right)^{2}$
$\displaystyle=\frac{\sigma_{1}^{2}+(16-7\rho_{s}^{2})\sigma_{2}^{2}+6\rho\sigma_{1}\sigma_{2}}{16k_{f}^{2}}+\left(\frac{(3-k_{f})}{2(4-k_{f})}\mu_{1}+\frac{(5-k_{f})}{2(4-k_{f})}\mu_{2}\right)^{2}$

The generation cost is

$\displaystyle\alpha(\mu_{1}+\mu_{2})+k_{s}\Bigg{[}\frac{\beta}{2}\left[(d_{1}^{DA})^{2}+(d_{2}^{DA})^{2}\right]\Bigg{]}+k_{f}\mathbb{E}\Bigg{[}\frac{\beta}{2}\left[(\tilde{d}_{1}^{RT})^{2}+(\tilde{d}_{2}^{RT})^{2}\right]\Bigg{]}$

Using the expressions we have previously computed, this generation cost
simplifies to the one given in the theorem. ∎