# Market Mechanisms for Low-Carbon Electricity Investments: A Game-Theoretical
Analysis

Dongwei Zhao, Sarah Coyle, Apurba Sakti, and Audun Botterud This work is
supported by Equinor ASA.Dongwei Zhao is with the MIT Energy Initiative and
the Laboratory for Information and Decision Systems, Massachusetts Institute
of Technology, Cambridge, MA 02139 (e-mail: zhaodw@mit.edu). Sarah Coyle is
with the Department of System Design & Management, Massachusetts Institute of
Technology, Cambridge, MA 02139 (e-mail: sarahcoyle@chevron.com). Apurba Sakti
is with the MIT Energy Initiative, Massachusetts Institute of Technology,
Cambridge, MA 02139 (e-mail: sakti@mit.edu). Audun Botterud is with the
Laboratory for Information and Decision Systems, Massachusetts Institute of
Technology, Cambridge, MA 02139 (e-mail: audunb@mit.edu).

###### Abstract

Electricity markets are transforming from the dominance of conventional energy
resources (CERs), e.g., fossil fuels, to low-carbon energy resources (LERs),
e.g., renewables and energy storage. This work examines market mechanisms to
incentivize LER investments, while ensuring adequate market revenues for
investors, guiding investors’ strategic investments towards social optimum,
and protecting consumers from scarcity prices. To reduce the impact of
excessive scarcity prices, we present a new market mechanism, which consists
of a Penalty payment for lost load, a supply Incentive, and an energy price
Uplift (PIU). We establish a game-theoretical framework to analyze market
equilibrium. We prove that one Nash equilibrium under the penalty payment and
supply incentive can reach the social optimum given quadratic supply costs of
CERs. Although the price uplift can ensure adequate revenues, the resulting
system cost deviates from the social optimum while the gap decreases as more
CERs retire. Furthermore, under the traditional marginal-cost pricing (MCP)
mechanism, investors may withhold investments to cause scarcity prices, but
such behavior is absent under the PIU mechanism. Simulation results show that
the PIU mechanism can reduce consumers’ costs by over 30% compared with the
MCP mechanism by reducing excessive revenues of low-cost CERs from scarcity
prices.

###### Index Terms:

Mechanism design, Market competition, Renewable energy, Energy storage, Game
theory, Nash equilibrium

##  Nomenclature

###  -A Sets and Indexes

$\mathcal{D}$

Set of operational horizons (days) in the investment horizon with total
$D_{a}$ days, indexed by $d$

$\mathcal{I}$

Set of LER investors, indexed by $i$

$\mathcal{I}^{\text{RE}}$

Set of VRE investors

$\mathcal{I}^{\text{ES}}$

Set of ES investors

$\mathcal{K}$

Set of types of LER investors, indexed by $k$

$\mathcal{T}$

Set of hours in an operational horizon, indexed by $t$

$\mathcal{Z}_{i}^{o}$

Self-constraint set of LER $i$’s resource operations

$\tilde{\mathcal{Z}}_{i}^{o}$

Self-constraint set of LER $i$’s resource operations including lost load

$\mathcal{Z}_{i}^{c}$

Joint-constraint set of LER $i$’s resource operations associated with other
investors

$\tilde{\mathcal{Z}}_{i}^{c}$

Joint-constraint set of LER $i$’s resource operations associated with other
investors including lost load

$\Omega$

Set of scenarios, indexed by $\omega$

###  -B Parameters

$a^{\omega}{[t]}$

Quadratic coefficient of CER supply curve

$b^{\omega}{[t]}$

Linear coefficient of CER supply curve

$c^{\omega}{[t]}$

No-load cost of CER supply curve

$c_{i}^{S}$

Energy-capacity cost of ES

$c_{i}^{P}$

Power-capacity cost of ES

$c_{i}^{X}$

Power-capacity cost of VRE

$c_{i}^{ch/dis}$

Charge/Discharge cost of ES

$D^{\omega}{[t]}$

System demand

$\bar{p}^{cv}$

Initial capacity of CERs

$\underline{U}_{i}/\overline{U}_{i}$

Lower/Upper bound of duration of ES

VOLL

Value of lost load

$\rho^{\omega}$

Probability of scenario $\omega$

$\gamma$

Ratio of remaining capacities of CERs to the initial capacities

$\nu_{i}^{\omega}{[t]}$

Generation output factor of VRE $i$

$\eta_{i}^{c/d}$

Charge/Discharge efficiency of ES

$\Delta\pi^{\omega}{[t]}$

Energy-price uplift

###  -C Decision Variables

$A_{i}^{\omega}{[t]}$

Net output of LERs

$\tilde{A}_{i}^{\omega}{[t]}$

Net output of LERs including lost load

$e^{\omega}{[t]}$

State-of-charge of ES

$f_{i}$

Profit of LERs

$\tilde{f}_{i}$

Profit of LERs including lost-load cost

$\tilde{\tilde{f}}_{i}$

Profit of LERs including lost-load cost and supply incentive

$g^{\omega}{[t]}$

Supply cost of CERs

$p^{ch/dis,\omega}{[t]}$

Charge/ Discharge of ES

$p^{cv,\omega}{[t]}$

Output of CERs

$p^{sh,\omega}{[t]}$

Load shedding in the market

$P_{i}$

Power capacity of ES

$S_{i}$

Energy capacity of ES

$X_{i}$

Power Capacity of VRE

$\bm{z}_{i}$

LER decision vector

$\tilde{\bm{z}_{i}}$

LER decision vectors including the lost load

${\pi}^{\omega}{[t]}$

General notation of market prices

$\hat{\pi}^{\omega}{[t]}$

Market price in the PIU mechanism without price uplift

$\tilde{\pi}^{\omega}{[t]}$

Market price in the PIU mechanism with price uplift

##  I Introduction

The threat of climate change is accelerating the global transition of
electricity supply from conventional energy resources (CERs), e.g., fossil
fuels, to low-carbon energy resources (LERs), e.g., variable renewable energy
(VRE) and energy storage (ES). Many countries and states have set zero-carbon
targets for electricity supplies. For example, California aims to achieve the
target of 100% zero-carbon electricity by 2045 [1]. From 2010 to 2020, the
globally installed solar PV capacity increased from about 40 GW to 760 GW,
while wind energy grew from about 200 GW to 740 GWh [2]. Due to the variable
and uncertain nature of VRE, ES is becoming a crucial flexible resource to
support VRE integration. The global stationary ES market is estimated to grow
from $15.2$ GWh in 2019 to $222.7$ GWh in 2035 [3]. In this work, we will
focus on VRE (wind and solar energy) and ES for LERs.

The increasing integration of LERs is transforming the electricity market,
which is possibly on a path towards a 100%-LER market in the future. Such
transition, however, is bringing new challenges, e.g., more frequent scarcity
prices [4] and inadequate revenues for LER investments [5]. Due to the
uncertain and variable nature of VRE, it is economically difficult to
guarantee adequate supply. This may lead to an increase in the frequency of
scarcity price spikes and thus face consumers’ backlash due to political and
social reasons [6]. Meanwhile, as more CER generators retire, ensuring
adequate revenues for the LER investment may be more difficult. Since VRE has
zero marginal cost, the marginal-cost-based market prices are expected to fall
with increasing VRE[7, 4]. Furthermore, the retirement of CERs may require
much larger capacities of ES and VRE to fill the gap due to supply
variability. Thus, the high capital costs, e.g., of ES [8], combined with
large capacity needs, will shift the cost structure in the system towards high
investment and low operating costs, possibly reducing revenues under current
market designs.

This prospect of volatile prices and diminishing revenues can influence the
strategic behaviors of LER investors. In the deregulated market, where
investors make independent investment and operation decisions, they will make
decisions strategically to pursue profits. Since most electricity markets are
imperfectly competitive [6], the diminishing market returns may discourage
profit-seeking investors from investing in LERs, or encourage them to withhold
the investment and supply to increase prices [9]. Under the decarbonization
target, the strategic behaviors of VRE and ES may incur a higher frequency of
supply shortages and price spikes. Although most markets perform market power
mitigation to reduce the strategic behavior of suppliers in the short term
[10], it is hard to monitor and regulate investments in the long term.
Besides, compared with CERs’ certain outputs and certain marginal costs, VRE
generation is uncertain. ES marginal cost is not straightforward and can be
dependent on the opportunity cost. It is still an open problem for the market
monitor how to mitigate the market power of VRE and ES.

Market mechanism design under an increasing penetration of LERs still poses a
number of open questions, e.g., how to set the price in zero-carbon markets
[11], and how to incentivize ES operations and investments into the market
[12]. In this paper, we do not focus on a 100%-LER market, but rather on the
transition from a CER-dominated mix to 100%-LER. We study ways to adjust the
CER-based market pricing mechanism to incentivize investors’ strategic
investments in LERs and to address the following key question:

  * •

Under the increasing percentage of LERs in electricity markets, how can we
ensure adequate revenues for LER investors, guide their investments towards
the social optimum, and protect consumers from excessive prices?

Notably, our work focuses on the long-term planning of LERs in the energy-only
electricity market. We do not consider the capacity market or capacity
requirements for energy resources. Our model is based on joint optimization
over the long-term investment horizon and short-term operations.

###  I-A Related work

There is extensive literature on incentive mechanisms for encouraging
investments in VRE and ES in electricity markets. In terms of market
mechanisms, the approaches are centered around the energy market, capacity
market, and supportive financial tools. (i) For the energy market, besides the
short-term marginal-cost pricing, extensive studies proposed long-term
enhancement. For example, long-term bilateral contracts (e.g., power purchase
agreements) [13], long-term auction-based energy markets [14], and pricing
mechanisms based on long-run marginal costs [15]. (ii) Besides energy markets,
most of the US electricity markets have established the long-term capacity
market or resource adequacy requirements [16], which provides remuneration for
resource investments. Approaches (e.g., effective load carrying capability)
have been proposed to effectively characterize capacity credits for VRE and ES
[17, 18, 19]. (iii) Financial tools have been discussed to hedge the risk of
lost load or high prices and meanwhile provide adequate incentives for the VRE
and ES investors, e.g., through reliability options [20] [21] and insurance
contracts [22][23]. Finally, in addition to the market mechanisms, government
policies also play an important role in incentivizing the deployment of VRE
and ES [11] [24]. For example, a renewable portfolio standard (RPS) is a
regulatory mandate to increase electricity supply from renewable energy.
Eligible resources that satisfy specific RPS obligations receive credits.
Production and investment tax credits incentivize the expansion of clean
resources by offering tax credits to investors. Carbon pricing usually imposes
a direct tax on carbon emissions or implements a carbon cap-and-trade
framework for resources, which benefits clean energy resources.

In this work, we study how to adjust marginal-cost-based pricing mechanisms to
incentivize the investment of LERs in energy-only markets. The marginal-cost
pricing (MCP) mechanism has been widely adopted in current electricity markets
and extensively studied in the literature. Earlier works showed that the MCP
mechanism can achieve short-term and long-term market efficiency under convex
assumptions [25, 26, 27]. However, the MCP mechanism can cause high scarcity
prices, which may create consumers’ concerns [6]. Some markets enforce a price
cap, e.g., 2000$/ MWh in CAISO, but such a price cap is still very high for
consumers and additionally contributes to the missing money problem [28]. In
contrast, our work proposes adjustments to the traditional MCP mechanism by
reducing the impact of excessive scarcity prices and meanwhile ensuring
adequate revenues for investors. Furthermore, in the traditional MCP
mechanism, consumers bear the cost of the lost load. Our work lets investors
pay this cost by designing a penalty payment. Such a penalty payment works
similarly to the contract for price difference [29][13] or insurance [30] by
reducing the risk for consumers. We analyze the market equilibrium under
imperfect competition under such a penalty payment, which is not captured in
[29, 13, 30].

The market equilibrium under the MCP mechanism is mostly analyzed under
perfect competition [25, 26, 27], which assumes market participants as price-
takers. However, considering imperfect competition, a finite number of market
participants can behave strategically to influence the market price. It is
highly challenging to analyze and compute the equilibrium under the MCP
mechanism under imperfect competition, which usually requires solving
equilibrium programs with equilibrium constraints (EPEC) for multiple players
[31, 32, 33]. A pure-strategy Nash equilibrium of EPEC generally may not exist
[34]. In this work, we characterize a special pure-strategy Nash equilibrium
for the traditional MCP mechanism to demonstrate that strategic LER investors
may withhold investments to obtain more frequent scarcity prices.

We establish an imperfect competition model of LERs based on the Cournot model
where firms compete in quantity [35, 36, 37]. The classic Cournot model [35,
36, 37] focused on theoretical analysis with simplified assumptions on firms.
The works [38, 39, 40, 41, 42, 43, 44, 45] applied Cournot competition in
electricity markets to study suppliers’ strategic behaviors. However, none of
the prior studies include market competition in the joint investment of VRE
and ES, while our work will capture it. The studies [38, 39, 40, 41, 42] were
established based on assumed demand curves, which cannot reflect the impact of
zero-marginal-cost VRE. Furthermore, electricity demand is usually not elastic
whose demand curve is difficult to quantify. In contrast, our work focuses on
inelastic demand. We utilize real market data to construct the supply curve,
in an attempt to reflect the impact of VRE on market revenues and costs.

###  I-B Our contributions

Expanding on the literature above, we summarize the key contributions of this
work in the following.

Strategic LER investments under imperfect competition: To the best of our
knowledge, our work is the first to study strategic investments of LERs,
including VRE and ES, under imperfect competition through a game-theoretical
framework. We use market data from the California Independent System Operator
(CAISO) to approximate the system marginal cost curve. We analyze the pure-
strategy Nash equilibrium of VRE and ES investors’ investment and operation
under the conventional MCP mechanism and the proposed PIU mechanism.

PIU mechanism: We present a novel mechanism consisting of a Penalty payment
for lost load, a supply Incentive, and a price Uplift (PIU), to LER investors.
Under the assumption of quadratic supply cost of CERs, we prove that one Nash
equilibrium coincides with the social optimum under perfect competition in the
P-mechanism, and under imperfect competition in the PI-mechanism. To solve the
resulting profit loss under the retirement of CERs, we introduce a price
uplift to ensure adequate revenues for LER investors. The PIU mechanism will
incentivize more LER investment than the social optimum. However, the system-
cost gap between the PIU mechanism and the social optimum decreases as more
CERs retire.

Comparison between the PIU and MCP mechanisms: The traditional MCP mechanism
can induce the social optimum under perfect competition. However, under
imperfect competition, we prove that market participants may withhold
investments and intentionally cause scarcity prices for consumers, which will
not occur in the PIU mechanism. Simulation results show that the PIU mechanism
can reduce consumers’ cost by over 30% than the traditional MCP mechanism even
under perfect competition, by reducing the excessive profit of low-cost CERs
under scarcity prices.

The rest of this paper is organized as follows. In Section II, we present the
models of the LER investors and system operator. In Section III, we formulate
a game-theoretical framework for market competition. Then, we analyze the Nash
equilibrium for the traditional MCP mechanism in Section IV. We introduce a
new PIU mechanism and analyze the Nash equilibrium in Section V. In Section
VI, we conduct simulations to demonstrate the equilibrium results under the
PIU mechanism. Finally, we conclude in Section VII.

##  II System model

We consider a set of LER investors in the market denoted by
$\mathcal{I}=\mathcal{I}^{\text{RE}}\bigcup\mathcal{I}^{\text{ES}}$, which
contains two subsets $\mathcal{I}^{\text{RE}}$ and $\mathcal{I}^{\text{ES}}$
corresponding to two classes of LERs: VRE and ES, respectively. The VRE can be
solar energy or wind energy. Without loss of generality, We assume that each
investor has only one type of resources. One type means the same class of
energy resources and the same technology parameters.

Besides the new additions of VRE and ES, there are also existing CERs, mainly
fossil thermal generations, in the market.111For conventional generators,
first, we assume that in the future they will gradually retire in the market
and there will be no new investment due to decarbonization targets. Therefore,
we do not consider the strategic long-term investment behavior of CERs.
Second, CER generators are dispatchable with controllable outputs. Current
electricity markets have sophisticated mechanisms to mitigate the market power
of CER short-term supply. Thus, we assume no strategic short-term withholding
from CERs. However, CERs may strategically decide how much capacity they want
to retire, which may impact the market equilibrium outcome. Our current model
does not capture retirement decisions, which we leave as future work. We
assume that the system operator will schedule these existing CERs, and ensure
the demand and supply balance. During periods of scarcity, the system operator
can also leave some demand unserved, as we assume that the demand is inelastic
without demand response. Besides, our work assumes no congestion in the
network and captures key insights into market mechanisms. We partially
generalize our model to incorporate networks using DC power flow, where some
key results will hold as shown in Appendix.VIII [46].

Next, we will introduce the timescale of the investors’ decision-making.

###  II-A Timescale of decision-making

Figure 1 illustrates two timescales of decision making. At the beginning of an
investment horizon $\mathcal{D}=\\{1,2,...,D_{a}\\}$ of $D_{a}$ days (e.g., 10
years), each investor $i\in\mathcal{I}$ decides the investment decision
$\bm{X}_{i}$ of VRE or ES.222 We focus on the one-time investment decisions
and capital costs of LERs to capture key insights from long-term planning. We
do not consider the multiple sequential investment decisions during the
investment horizon and we leave it to future work.

The investment horizon is divided into a number of operational horizons, i.e.,
each day $d\in\mathcal{D}$ corresponds to one operational horizon, which is
further divided into multiple time slots $\mathcal{T}=\\{1,2,...,T\\}$ (e.g.,
24 hours). The system demand is denoted by $D^{d}[t]$ for hour $t$ of day $d$.
For day $d\in\mathcal{D}$, each investor $i$ schedules the operation decision
over time slots $\mathcal{T}$, i.e.,
$\bm{p}_{i}^{d}=({p}_{i}^{d}[t],t\in\mathcal{T})$. We denote by a vector
$\bm{z}_{i}=\Big{(}\bm{X}_{i},(\bm{p}_{i}^{d},d\in\mathcal{D})\Big{)}$ all the
decision variables of investor $i$ over the two timescales, which we will
elaborate later. The system operator will schedule the CER operation
$\bm{p}^{cv,d}=({p}_{i}^{cv,d}[t],t\in\mathcal{T})$ and the shed load
$\bm{p}^{sh,d}=({p}^{sh,d}[t],t\in\mathcal{T})$ to maintain the power balance.

To facilitate the formulation using stochastic programming, we use historical
data to construct a scenario set $\Omega$ that represents all the days in the
investment horizon. Specifically, for any day $d\in\mathcal{D}$ in the
operation, scenario $\omega\in\Omega$ occurs with a probability
$\rho^{\omega}$. We use scenarios to model the discrete distributions of daily
demand, operational cost, and renewable energy outputs. We calculate the
expected revenues and operation costs for investors on a daily basis and scale
the long-term capital cost into one day. For example, we use three-year data
from CAISO to construct over 1000 scenarios for daily market operation
profiles. This can effectively capture multiple periods of operations within
the investment horizon. Since we model the operation by scenarios, we
substitute $d\in\mathcal{D}$ by $\omega\in\Omega$ for the related decisions
and parameters.

Figure 1: Timescales.

###  II-B Model of VRE and ES investors, and system operator

We will introduce the decision variables, constraints, and costs for the VRE
and ES investors, and the system operator.

####  II-B1 VRE investor

For the VRE investor $i\in\mathcal{I}^{\text{RE}}$, the investment decision
$\bm{X}_{i}$ is the invested capacity $X_{i}$. The operational decision
$\bm{p}_{i}$ includes the market supply
$\bm{p}_{i}^{mk}=({p}_{i}^{mk,\omega}[t],\forall
t\in\mathcal{T},\forall\omega\in\Omega)$ and the curtailment
$\bm{p}_{i}^{cur}=({p}_{i}^{cur,\omega}[t],\forall
t\in\mathcal{T},\forall\omega\in\Omega)$ for each hour $t$ of scenario
$\omega$.

The operational constraints for investor $i\in\mathcal{I}^{\text{RE}}$ satisfy

$\displaystyle
p_{i}^{mk,\omega}[t]+p_{i}^{cur,\omega}[t]=\nu_{i}^{\omega}[t]\cdot
X_{i},\forall t\in\mathcal{T},\forall\omega\in{\Omega},$ (1a) $\displaystyle
X_{i}\geq 0,p_{i}^{mk,\omega}[t]\geq 0,p_{i}^{cur,\omega}[t]\geq 0,\forall
t\in\mathcal{T},\forall\omega\in{\Omega}.$ (1b)

Here the parameter $\nu_{i}^{\omega}[t]$ denotes the capacity factor for time
$t\in\mathcal{T}$ of scenario $\omega\in\Omega$. Note that such capacity
factors can be quite different between solar and wind energy. Equation (1a)
ensures the power balance of VRE. Constraint (1b) gives non-negative
variables. We denote the constraint set constructed by (1a)-(1b) by
$\mathcal{Z}_{i}^{o}$, $\forall i\in\mathcal{I}^{\text{RE}}$.

In terms of the costs, the variable operation cost of VRE is zero
$C_{i}^{opr}(\bm{z}_{i})=0$. The capital cost is
$C_{i}^{inv}(\bm{z}_{i})=\kappa_{i}\cdot c_{i}^{X}X_{i}$, where $c_{i}^{X}$
denotes the unit capacity cost of the wind or solar energy. The factor
$\kappa_{i}$ scales the capital cost over the investment horizon into one day
for stochastic programming [45].

####  II-B2 Storage investor

For the ES investor $i\in\mathcal{I}^{\text{ES}}$, the investment decision
$\bm{X}_{i}$ includes the invested energy capacity and power capacity
$(S_{i},P_{i})$. The operation decision $\bm{p}_{i}$ includes the charge and
discharge profiles for each hour $t$ of scenario $\omega$, i.e.,
$\bm{p}_{i}^{ch}=({p}_{i}^{ch,\omega}[t],\forall
t\in\mathcal{T},\forall\omega\in\Omega)$ and
$\bm{p}_{i}^{dis}=({p}_{i}^{dis,\omega}[t],t\in\mathcal{T},\forall\omega\in\Omega)$,
as well as the state-of-charge (SOC) in storage by
$\bm{e}_{i}^{\omega}=\\{e_{i}^{\omega}[t],~{}\forall t\in
0\bigcup\mathcal{T},\forall\omega\in\Omega\\}$.

The operation constraints for investor $i\in\mathcal{I}^{\text{RE}}$ satisfy

$\displaystyle 0\leq p_{i}^{dis,\omega}[t]\leq P_{i},\forall
t\in\mathcal{T},\forall\omega\in\Omega,$ (2a) $\displaystyle 0\leq
p_{i}^{ch,\omega}[t]\leq P_{i},\forall
t\in\mathcal{T},\forall\omega\in\Omega,$ (2b) $\displaystyle
e_{i}^{\omega}[t]=e_{i}^{\omega}[t-1]+\eta_{i}^{c}p_{i}^{ch,\omega}[t]-p_{i}^{dis,\omega}[t]/\eta_{i}^{d},\forall
t\in\mathcal{T},\forall\omega\in\Omega,$ (2c) $\displaystyle 0\leq
e_{i}^{\omega}[t]\leq S_{i},\forall
t\in\mathcal{T}^{\prime},\forall\omega\in\Omega,$ (2d) $\displaystyle
e_{i}^{\omega}[0]=e_{i}^{\omega}[T],\forall\omega\in\Omega,$ (2e)
$\displaystyle\underline{U}_{i}\leq\frac{S_{i}}{P_{i}}\leq\overline{U}_{i}.$
(2f)

We elaborate the constraints next. First, constraints (2a)-(2b) are the power-
capacity limits for charge and discharge, respectively. Second, constraints
(2c)-(2e) are the constraints for the SOC in the storage. Constraint (2c) is
the SOC change with time due to the charge and discharge operation, where
$\eta_{i}^{c}$ and $\eta_{i}^{d}$ denote the charge and discharge efficiency,
respectively. Equation (2d) ensures the SOC bounded by the storage capacity.
Equation (2e) makes the initial SOC equal to the final level, which decouples
the storage operation across different days. Finally, constraint (2f) enforces
a general duration bound on the investment alternatives of the energy capacity
and power capacity, where $\underline{U}_{i}$/$\overline{U}_{i}$ are the
lower/upper bounds.333We omit the constraint of non-simultaneous charge and
discharge $p_{i}^{dis,\omega}[t]\cdot p_{i}^{ch,\omega}[t]=0$ to maintain the
convexity of the model. In simulations, simultaneous charge and discharge does
not occur. We denote the strategy set constructed by (2a)-(2f) by
$\mathcal{Z}_{i}^{o}$, $\forall i\in\mathcal{I}^{\text{ES}}$.

In terms of the costs, the investment cost includes the capital costs for both
energy and power capacity
$C_{i}^{inv}(\bm{z}_{i})=\kappa_{i}\left(c_{i}^{S}S_{i}+c_{i}^{P}P_{i}\right)$,
where $c_{i}^{S}$ and $c_{i}^{P}$ denote the unit costs for energy and power
capacities, respectively. For the operation cost incurred by charge and
discharge, we adopt a linear model and calculate the expected cost by
$C_{i}^{opr}(\bm{z}_{i})$ on a daily basis.

$\displaystyle
C_{i}^{opr}(\bm{z}_{i})=\mathbb{E}_{\omega\in\Omega}\Big{[}\sum_{t\in\mathcal{T}}\left(c_{i}^{ch}{p}_{i}^{ch,\omega}[t]+c_{i}^{dis}{p}_{i}^{dis,\omega}[t]\right)\Big{]},$
(3)

where $c_{i}^{ch}$ and $c_{i}^{dis}$ denote the unit costs for charge and
discharge, respectively. This can model the O&M cost as well as the potential
degradation cost.

####  II-B3 System operator

The system operator schedules the generation of CERs $p^{cv,\omega}[t]$ and
the lost load $p^{sh,\omega}[t]$.

To introduce the system constraints, we first let $A_{i}^{\omega}[t]$ be
investor $i$’s net market supply. For the VRE investor
$i\in\mathcal{I}^{\text{RE}}$, $A_{i}^{\omega}[t]\triangleq
p_{i}^{mk,\omega}[t]$. For the storage investor $i\in\mathcal{I}^{\text{ES}}$,
$A_{i}^{\omega}[t]\triangleq-p_{i}^{ch,\omega}[t]+p_{i}^{dis,\omega}[t]$. We
list the system constraints.

$\displaystyle\sum_{i\in\mathcal{I}}A_{i}^{\omega}[t]+p^{cv,\omega}[t]+p^{sh,\omega}[t]=D^{\omega}[t],\forall
t\in\mathcal{T},\forall\omega\in{\Omega},$ (4) $\displaystyle 0\leq
p^{cv,\omega}[t]\leq\gamma\cdot\bar{p}^{cv},\forall
t\in\mathcal{T},\forall\omega\in{\Omega},$ (5) $\displaystyle 0\leq
p^{sh,\omega}[t],\forall t\in\mathcal{T},\forall\omega\in\Omega.$ (6)

Here constraint (4) ensures power balance. Constraint (5) gives the bound on
the capacity of available CERs. The parameter $\gamma\in[0,1]$ adjusts the
retirement amount of the current CERs. For example, $\gamma=0.4$ means the
retirement of 60% CERs.

In terms of the costs, we calculate the expected cost of lost load based on
the value of lost load (VOLL):

$\displaystyle
G^{sh}(\bm{p}^{sh})=\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\text{VOLL}\cdot
p^{sh,\omega}[t].$ (7)

For the operation cost of CERs, we denote it as a function
$g^{cv,\omega}[t](\cdot)$ in $p^{cv,\omega}[t]$ for hour $t$ of scenario
$\omega$. The overall expected cost is $G^{cv}(\bm{p}^{cv})$.

$\displaystyle
G^{cv}(\bm{p}^{cv})=\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}g^{cv,\omega}[t]({p}^{cv,\omega}[t]).$
(8)

We assume that the function $g^{cv,\omega}[t](\cdot)$ is convex, i.e., the
marginal cost $\frac{dg^{cv,\omega}[t]}{dp^{cv,\omega}[t]}$ is non-decreasing,
and $\frac{dg^{cv,\omega}[t]}{dp^{cv,\omega}[t]}<\text{VOLL}$. Next, we will
introduce a social-optimum benchmark.

###  II-C Social-optimum benchmark: System cost minimization

We consider a social-optimum benchmark where the system operator decides the
investment and operation of all the LER investors. The system operator aims to
minimize the system cost, which includes the investment cost and operation
cost of new LERs, the operation cost of the existing CERs, and the cost of
system lost load. Since we consider inelastic demand, the social optimum is
equivalent to the least system cost.

Problem SO: System Cost Minimization

$\displaystyle\min$
$\displaystyle\sum_{i\in\mathcal{I}}C_{i}^{inv}(\bm{X}_{i})+\sum_{i\in\mathcal{I}}C_{i}^{opr}(\bm{p}_{i})+G^{cv}(\bm{p}^{cv})+G^{sh}(\bm{p}^{sh})$
s.t.
$\displaystyle\eqref{eq:const_ren_a}-\eqref{eq:const_ren_b},\eqref{eq:ch}-\eqref{eq:duration},\forall
i\in\mathcal{I},$
$\displaystyle\eqref{eq:const_balance},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{dual}:~{}\lambda^{b,\omega}[t]$
$\displaystyle\eqref{eq:const_convention},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{dual}:~{}\underline{\mu}^{cv,\omega}[t],~{}\overline{\mu}^{cv,\omega}[t]$
$\displaystyle\eqref{eq:const_loss},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{dual}:~{}\underline{\mu}^{sh,\omega}[t]$
var: $\displaystyle(\bm{X}_{i},\bm{p}_{i},\forall
i\in\mathcal{I}),\bm{p}^{cv},\bm{p}^{sh}.$

Problem SO is convex. We denote its optimal solution as
$\bm{z}^{\dagger}=(\bm{X}_{i}^{\dagger},\bm{p}_{i}^{\dagger},\forall
i\in\mathcal{I}),~{}\bm{p}^{cv\dagger}$, and $\bm{p}^{sh\dagger}$. We denote
the dual variables corresponding to the system constraints (4)-(6) as
$\lambda^{b,\omega}[t]$,
$(\underline{\mu}^{cv,\omega}[t],~{}\overline{\mu}^{cv,\omega}[t])$, and
$\underline{\mu}^{sh,\omega}[t]$, respectively.

##  III Deregulated market: Game theory model

We now focus on a deregulated market where each new LER investor can decide
the invested capacity and market supply by themselves. Each LER investor aims
to maximize its profit. The system operator will schedule CERs and shed load
over the operation horizons to ensure power balance.

The market energy price is the key to investors’ market revenues. Note that
the market price is affected by all the investors’ decisions. Thus, we model
the market price of time $t$ of scenario $\omega$ as a function of all the
investors’ decision variables, i.e.,
${\pi}^{\omega}[t](\bm{z}_{1},\bm{z}_{2}\ldots\bm{z}_{I})$. Next, we introduce
the revenue, profit, and constraint formulations for the VRE and ES investors,
based on which we establish a game-theoretical model among these competing
investors.

###  III-A Revenue and profits

For investor $i$, the market revenue $R_{i}(\bm{z}_{i},\bm{z}_{-i})$ comes
from the net supply ${A}_{i}^{\omega}[t]$ on the market.

$\displaystyle
R_{i}(\bm{z}_{i},\bm{z}_{-i})=\mathbb{E}_{\omega\in\Omega}\Big{[}\sum_{t\in\mathcal{T}}{A}_{i}^{\omega}[t]\cdot\pi^{\omega}[t](\bm{z}_{i},\bm{z}_{-i})\Big{]},$
(9)

where $\bm{z}_{-i}$ denotes all the investors’ decision variables other than
investor $i$. The profit $f_{i}(\bm{z}_{i},\bm{z}_{-i})$ is the revenue minus
the investment and operation cost, which is coupled with others.

$\displaystyle
f_{i}(\bm{z}_{i},\bm{z}_{-i})=R_{i}(\bm{z}_{i},\bm{z}_{-i})-C_{i}^{inv}(\bm{z}_{i})-C_{i}^{opr}(\bm{z}_{i}).$
(10)

###  III-B Strategy set: constraints

There are two parts of each investor’s constraint. First, each investor’s
operation decision is constrained by its resource
$\bm{z}_{i}\in\mathcal{Z}_{i}^{o}$. Second, investors’ decision sets are
coupled due to the system constraints (4)-(6), which leads to the following
constraint on the total supply $\sum_{i\in\mathcal{I}}A_{i}^{\omega}[t]$ of
all the investors.

$\displaystyle
D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}\leq\sum_{i\in\mathcal{I}}A_{i}^{\omega}[t]\leq
D^{\omega}[t],\forall t\in\mathcal{T},\forall\omega\in{\Omega}.$ (11)

Given other investors’ decisions $\bm{z}_{-i}$, we denote the coupled strategy
set of investor $i$ from (11) by $\mathcal{Z}_{i}^{c}(\bm{z}_{-i})$. Overall,
the constraint set of investor $i$ is
$\mathcal{Z}_{i}(\bm{z}_{-i})\triangleq\mathcal{Z}_{i}^{o}\bigcup\mathcal{Z}_{i}^{c}(\bm{z}_{-i})$.

###  III-C Game-theoretical model

Now we formulate a competition game $G$ among investors to model investors’
strategic behaviors.

  * •

Players: Investor $i\in\mathcal{I}$

  * •

Strategy: $\bm{z}_{i}\in\mathcal{Z}_{i}(\bm{z}_{-i})$ of investor $i$

  * •

Profit: $f_{i}(\bm{z}_{i},\bm{z}_{-i})$ of investor $i$

We aim to find the Nash equilibrium of the game. If the players’ strategy sets
are also coupled, the Nash equilibrium is usually called a generalized Nash
equilibrium, which we define for the competition game $G$ in Definition 1
[47].

######  Definition 1 (Nash equilibrium).

In the game $G=\langle\mathcal{I},(\mathcal{Z}_{i}),(f_{i})\rangle$, the
strategy profile
$\bm{z}^{*}\in\Pi_{i\in\mathcal{I}}\mathcal{Z}_{i}(\bm{z}_{-i}^{*})$ is a
generalized pure strategy Nash equilibrium if for every investor
$i\in\mathcal{I}$, $f_{i}(\bm{z}_{i}^{*},\bm{z}_{-i}^{*})\geq
f_{i}(\bm{z}_{i},\bm{z}_{-i}^{*})$ for any
$\bm{z}_{i}\in\mathcal{Z}_{i}(\bm{z}_{-i}^{*})$.

The above definition states that the Nash equilibrium is a state where every
investor’s decision maximizes its profit given other investors’ decisions. The
Nash equilibrium will serve as the market equilibrium for analysis in this
paper.

We make the following assumptions for this game-theoretical model. First,
investors make decisions simultaneously, and they do not observe others’
decisions when making their own. Second, we consider a complete-information
setting, i.e., each investor knows the market price function and the
technology parameters of all the other investors. Third, every investor is
rational and will make decisions following the Nash equilibrium. Otherwise,
some investors can always change their strategies in an attempt to pursue
higher profits.

Different market mechanisms lead to different market price functions
${\pi}^{\omega}[t](\bm{z})$. Next, in Section IV, we discuss the traditional
MCP mechanism, which can lead to scarcity prices during supply shortages. In
Section V, we propose a new mechanism without scarcity prices and analyze Nash
equilibrium. We present mathematical proofs in appendices [46].

##  IV MCP mechanism

The market price is set based on the shadow price $\lambda^{b,\omega}[t]$
corresponding to the power balance constraint (4) in Problem SO given
investors’ decisions of investment and operation, i.e.,

$\displaystyle\pi^{\lambda,\omega}[t](\bm{z})=\frac{\lambda^{b,\omega}[t]}{\rho^{\omega}},$
(12)

where we modify the shadow price by the probability $\rho^{\omega}$ due to the
probability coefficient for revenues.444 The reason for this price
modification is that in Problem SO, we optimize the expected system cost based
on scenario probabilities. According to the KKT conditions of Problem SO, we
show the following stationary condition associated with the CER operation
variable $p^{cv,\omega}[t]$:
$\frac{dg^{cv,\omega}[t]}{dp^{cv,\omega}[t]}-\frac{\lambda^{b,\omega}[t]}{\rho^{\omega}}+\frac{\overline{\mu}^{cv,\omega}[t]}{\rho^{\omega}}-\frac{\underline{\mu}^{cv,\omega}[t]}{\rho^{\omega}}=0.$
Given each hour $t$ of each scenario $\omega$, the CER actual marginal cost
(or roughly system marginal cost) is set up based on
$\frac{\lambda^{b,\omega}[t]}{\rho^{\omega}}$. Therefore, we use
$\frac{\lambda^{b,\omega}[t]}{\rho^{\omega}}$ as the market-clearing price.

Problem SO is a joint optimization problem for the investment and operation of
energy resources. The shadow price (12) can reflect investment capital costs.
In Proposition 1 later, we prove that the shadow price (12) can incentivize
investors to invest and operate their resources reaching the long-term social-
optimum outcome.

However, in the energy-only market, if we fix the invested capacity for energy
resources, conventional short-term shadow prices from the economics dispatch
problem can only reflect short-term operation costs and thus may not provide
proper market signals for LER investment [48].555If we fix investors’
investment decisions and re-run Problem SO, the dual variable
$\lambda^{b,\omega}[t]$ can have multiple solutions, one of which coincides
with the dual variable computed when investors’ investment decisions are
variables [48]. Thus, the market signal is missing when we only focus on
short-term operations under given LER capacities. Short-term shadow prices may
also take account of the long-term investment cost if new mechanisms are
introduced. The obtained shadow price (12) actually can elicit this special
shadow price to reach the long-term social optimum.

In this work, we focus on the long-term planning of LERs. We will directly
utilize the market price (12) for the MCP mechanism as it gives nice market
signals to the LER investment. It is not this paper’s focus on how to
practically incorporate investment factors into short-term shadow prices,
which we leave as future work. Next, we analyze the Nash equilibrium under
perfect and imperfect competition based on the price (12), respectively.

###  IV-A Market equilibrium under perfect competition

In Proposition 1, we prove that the social-optimum result is one Nash
equilibrium under perfect competition (i.e., investors are price takers),
which achieves zero surplus for investors.

######  Proposition 1 (Perfect competition under the shadow price).

Given the market price (12) , under perfect competition, the social-optimum
solution $\bm{z}^{\dagger}$ is one Nash equilibrium, at which each investor’s
profit is zero, i.e., $f_{i}(z^{\dagger})=0,\forall i\in\mathcal{I}$.

Although the MCP mechanism can achieve the social optimum under perfect
competition, the resulting high scarcity prices typically face criticism from
political and consumer groups [6], and the social optimum may not be achieved
under imperfect competition.

###  IV-B Market equilibrium under imperfect competition

With a finite number of investors, the perfect competition assumption is not
realistic as investors’ decisions will impact the market price. It is
challenging to analyze the pure-strategy Nash equilibrium under the MCP
mechanism.

We focus on a case of homogeneous VRE investors and characterize a Nash
equilibrium that can always achieve a market price at VOLL in Proposition 2.
The impact of storage charge and discharge can be complicated, but the VRE
supply will simply reduce the market price. When the supply from LERs
satisfies
$\sum_{i\in\mathcal{I}}A_{i}^{\omega}[t]<D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}$,
some demand is not served after utilizing the maximum capacity of CERs, which
leads to the price at VOLL. When
$D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}<\sum_{i\in\mathcal{I}}A_{i}^{\omega}[t]$,
the market price is bounded by the marginal cost of the CERs. The market price
is not unique at
$\sum_{i\in\mathcal{I}}A_{i}^{\omega}[t]=D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}$,
which we will avoid in the analysis.

######  Proposition 2.

Assume $N$ homogeneous VRE investors and $D^{\omega}[t]-\gamma\bar{p}^{cv}>0$
for any time $t$ of scenario $\omega$. Given an $\epsilon^{\omega}[t]>0$, we
consider the following optimization problem.

$\displaystyle\max~{}$
$\displaystyle\sum_{i\in\mathcal{I}}f_{i}(\bm{z}_{i},\bm{z}_{-i})$ s.t.
$\displaystyle\pi^{\omega}[t](\bm{z})=VOLL,$
$\displaystyle\sum_{i\in\mathcal{I}}A_{i}^{\omega}[t]\leq
D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}-\epsilon^{\omega}[t],\forall
t\in\mathcal{T},\forall\omega\in{\Omega},$
$\displaystyle\eqref{eq:const_ren_a}-\eqref{eq:const_ren_b}.$

If VOLL satisfies (13) for any time $t$ of scenario $\omega$

$\displaystyle
VOLL\geq\left(1+\frac{N\cdot\gamma\bar{p}^{cv}}{D^{\omega}[t]-\gamma\bar{p}^{cv}}\right)\cdot\frac{dg^{cv,\omega}[t]}{dp^{cv,\omega}[t]}\Big{|}_{\gamma\cdot\bar{p}^{cv}},$
(13)

then the solution to the above optimization problem approaches a Nash
equilibrium as $\epsilon^{\omega}[t]\rightarrow 0$.

If all the CERs retire, i.e., $\gamma\rightarrow 0$, (13) will be
$\text{VOLL}>\frac{dg^{cv,\omega}[t]}{dp^{cv,\omega}[t]}\Big{|}_{\gamma\cdot\bar{p}^{cv}}$,
which is easily satisfied. This implies that more retirement of the CERs makes
it easier for the investors to exercise market power. If
$N\rightarrow+\infty$, the condition (13) will require
$\text{VOLL}\rightarrow+\infty$, which implies that exercising market power is
harder with more investors in the market. Next, we propose a new mechanism in
an attempt to address the shortcomings of high scarcity prices.

##  V PIU mechanism

In the new mechanism, we let the market price be bounded by the marginal cost
of the CERs. Even if there is unserved load in the market, the market price is
still set by the marginal cost of the CERs at the maximum output.
Specifically, we have the price with LER supply as follows.

$\displaystyle\hat{\pi}^{\omega}[t](\bm{z})=\left\\{\begin{aligned}
&\frac{dg^{cv,\omega}[t]}{dp^{cv,\omega}[t]}\Bigg{|}_{\gamma\cdot\bar{p}^{cv}},~{}\text{if}~{}\sum_{i\in\mathcal{I}}A_{i}^{\omega}[t]<D^{\omega}[t]-\gamma\cdot\bar{p}^{cv};\\\
&\frac{dg^{cv,\omega}[t]}{dp^{cv,\omega}[t]},~{}~{}~{}~{}~{}\text{if}~{}D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}\\\
&\hskip 94.72192pt\leq\sum_{i\in\mathcal{I}}A_{i}^{\omega}[t]\leq
D^{\omega}[t].\end{aligned}\right.$ (14)

Figure 2: Market price with LER supply without scarce prices.

We illustrate the price $\hat{\pi}^{\omega}[t](\bm{z})$ in Fig.2, which
decreases with the supply from LERs. The price $\hat{\pi}^{\omega}[t](\bm{z})$
is bounded by the marginal cost of CERs. This price bound shares some
similarities with the existing practice that ISOs set a fixed value of price
cap. We clarify some key differences in the following. First, the existing
price cap is still very high for consumers, e.g., 5001$/MWh in ERCOT and
2000$/MWh in CAISO. This price cap can be much higher than the maximum
marginal cost of CERs, especially when we see retirements of high-cost CERs
under decarbonization. Such a high price cap may lead to strategic behaviors
of investors, as we show in Proposition 2. Second, the existing price cap is
fixed in the long term. It can be challenging to choose a proper cap to
reflect the value of lost load while protecting consumers. In contrast, our
work sets the price cap at the maximum marginal cost of CERs, which will vary
when more CERs retire. Finally, the ISOs usually set a price cap lower than
the actual values of lost load. Although such price caps may protect
consumers, they can cause missing money problems for investors to recover
investment costs. Our pricing restraints will work together with supply
incentives and energy price lift to ensure revenues for suppliers as shown
later.

However, such a price bound cannot reflect the value of lost load in the
system in the resulting market prices. To solve this problem, we design a
penalty payment for investors based on the lost load.

###  V-A Penalty payment

The system cost includes the cost of lost load. After we remove the scarcity
price, if investors’ profits can not reflect the cost of lost load, the market
equilibrium cannot eliminate the gap to the social-optimum benchmark. Imposing
penalty payments for the lost load on investors can well bridge this gap and
motivate investors to invest in more capacity. Next, we introduce the model of
the penalty payment (P-mechanism). We discuss the Nash equilibrium under the
penalty payment and the price setting $\hat{\pi}^{\omega}[t]$.

####  V-A1 Model of penalty payment

Under the penalty payment, each investor $i$ decides an additional
contribution item in the lost load $p_{i}^{sh,\omega}[t]\geq 0$ at each hour
$t$ of scenario $\omega$. The system operator ensures that all the investors’
total lost load satisfies the demand balance in the market as in the
following.

$\displaystyle\sum_{i\in\mathcal{I}}A_{i}^{\omega}[t]+p^{cv,\omega}[t]+\sum_{i\in\mathcal{I}}p_{i}^{sh,\omega}[t]=D^{\omega}[t],$
$\displaystyle\hskip 129.16626pt\forall
t\in\mathcal{T},\forall\omega\in{\Omega},$ (15) $\displaystyle
p_{i}^{sh,\omega}[t]\geq 0,\forall i\in\mathcal{I},\forall
t\in\mathcal{T},\forall\omega\in\Omega.$ (16)

We let the new market supply of investor $i$ be
$\tilde{A}_{i}^{\omega}[t]=A_{i}^{\omega}[t]+p_{i}^{sh,\omega}[t]$ since the
lost load also balances the demand. We assume that the system operator will
also pay for $p_{i}^{sh,\omega}[t]$ at the market price to each investor $i$.
We denote $\tilde{\bm{z}}_{i}=\left(\bm{z}_{i},\bm{p}_{i}^{sh}\right)$, where
$\bm{p}_{i}^{sh}=(p_{i}^{sh,\omega}[t],\forall
t\in\mathcal{T},\forall\omega\in\Omega)$.

We update the strategy set, market price function, and profit formulation
based on the new variables $\tilde{A}_{i}^{\omega}[t]$ and
$\tilde{\bm{z}}_{i}$. First, the constraint (11) is changed to

$\displaystyle
D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}\leq\sum_{i\in\mathcal{I}}\tilde{A}_{i}^{\omega}[t]\leq
D^{\omega}[t],\forall t\in\mathcal{T},\forall\omega\in{\Omega}.$ (17)

Based on the new constraint (17), we update the coupled strategy set
$\mathcal{Z}_{i}^{c}(\bm{z}_{-i})$ to
$\tilde{\mathcal{Z}}_{i}^{c}(\tilde{\bm{z}}_{-i})$. We add a new constraint
$p_{i}^{sh,\omega}[t]\geq 0$ in the strategy set $\mathcal{Z}_{i}^{o}$, which
is now changed to $\tilde{\mathcal{Z}}_{i}^{o}$. Overall, we denote the
strategy set for investor $i$ as
$\tilde{\bm{z}}_{i}\in\tilde{\mathcal{Z}}_{i}(\tilde{\bm{z}}_{-i})\triangleq\tilde{\mathcal{Z}}_{i}^{o}\bigcup\tilde{\mathcal{Z}}_{i}^{c}(\tilde{\bm{z}}_{-i})$.
Second, we update the market price function in (14) by directly replacing
${A}_{i}^{\omega}[t]$ with $\tilde{A}_{i}^{\omega}[t]$ as the lost load will
not impact the pricing. Finally, the investor pays the additional penalty
payment for the lost load $p_{i}^{sh,\omega}[t]$, which leads to the following
new profit $\tilde{f}_{i}(\tilde{\bm{z}}_{i},\tilde{\bm{z}}_{-i})$.

$\displaystyle\tilde{f}_{i}(\tilde{\bm{z}}_{i},\tilde{\bm{z}}_{-i})=f_{i}(\tilde{\bm{z}}_{i},\tilde{\bm{z}}_{-i})-\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\text{VOLL}\cdot
p_{i}^{sh,\omega}[t].$ (18)

Accordingly, we update the game-theoretical model $G$.

####  V-A2 Analysis of the Nash equilibrium

To demonstrate the key insights, we analyze the Nash equilibrium under the
assumption of a quadratic supply cost of CERs, i.e., at hour $t$ of scenario
$\omega$,

$\displaystyle
g^{cv,\omega}[t](p)=\frac{1}{2}a^{\omega}[t]\cdot{p}^{2}+b^{\omega}[t]\cdot{p}+c^{\omega}[t],~{}a^{\omega}[t]>0.$
(19)

The analytical results in the rest of Section V are based on the quadratic-
cost assumption. In Section VI, we use CAISO data to characterize such
quadratic supply-cost functions. We leave a more general cost function as
future work.

Computing Nash equilibrium: We characterize one Nash equilibrium for the
updated competition game $G$ considering the penalty payment, which is
computed in Proposition 3.

######  Proposition 3.

One Nash equilibrium coincides with the optimal solution to the following
problem, where
$\Phi^{\omega}(\tilde{\bm{z}}_{i},\tilde{\bm{z}}_{j})=\tilde{A}_{i}^{\omega}\cdot\tilde{A}_{j}^{\omega}.$

$\displaystyle\max_{\bm{z}}~{}$
$\displaystyle\sum_{i\in\mathcal{I}}\tilde{f}_{i}(\tilde{\bm{z}})+\mathbb{E}_{\omega\in\Omega}\Big{[}\sum_{t\in\mathcal{T}}\sum_{1\leq
i<j\leq
I}a^{\omega}[t]\cdot\Phi^{\omega}(\tilde{\bm{z}_{i}},\tilde{\bm{z}}_{j})\Big{]},$
(20) s.t.
$\displaystyle\tilde{\bm{z}}_{i}\in\tilde{\mathcal{Z}}_{i}^{o},\forall
i\in\mathcal{I},~{}\eqref{eq:const_balancemodify_new}.$

In the objective (20), the first part
$\sum_{i\in\mathcal{I}}\tilde{f}_{i}(\tilde{\bm{z}})$ is the total profit of
all the investors. The second part with
$\Phi^{\omega}(\tilde{\bm{z}}_{i},\tilde{\bm{z}}_{j})$ is an additional item
indicating the impact of strategic interactions among investors.

The objective function (20) does not have a direct physical or economic
interpretation. We characterize this objective function based on mathematical
results to compute Nash equilibrium. One approach to prove Proposition 3 is
based on the potential game theory [49] [50], which states that if there
exists a potential function for this game, the optimum of the potential
function is one Nash equilibrium. Our game-theoretical model is in nature a
Cournot competition model. Based on the linear supply function, we can
characterize a potential function, i.e., the objective (20). A stylized
Cournot model with a characterized potential function is given in [49]. Our
previous work applies the result in [49] to the storage competition problem
[45], and this work applies it to the mechanism design for the joint
investment of VRE and ES. We show the details of the potential game theory in
Appendix IV [46]. Another perspective of proving Proposition 3 is to establish
the KKT condition equivalence between the game model and the centralized
optimization problem in Proposition 3. In Appendix IV, we show that the KKT
conditions of the centralized optimization problem in Proposition 3 suffice to
satisfy the KKT conditions for all the investors at the Nash equilibrium.

The optimization problem in Proposition 3 is quadratic programming, which can
be efficiently solved. Note that our work proposes a penalty payment for
investors and analyzes the equilibrium result. There can be multiple
equilibria of the lost load allocation among investors. How to design the
rules to distribute the lost load to the individual investors is beyond the
scope of this work but deserves future investigations.

Discussion on the Nash equilibrium set: The Nash equilibrium computed in
Proposition 3 may not give all the pure-strategy Nash equilibria. If we do not
have the coupled strategy constraint (17), the set of all the pure-strategy
Nash equilibria will exactly coincide with the solutions in Proposition 3
based on the equivalent KKT conditions. However, with the coupled strategy
constraint (17), the solutions in Proposition 3 can only give a subset of all
the Nash equilibria. The reason is that the optimization problem in
Proposition 3 will generate a uniform dual variable for (17). However, when
each investor optimizes its profit under the constraint (17) in the game
model, the dual variables wrt. (17) can be different among investors. Thus, to
ensure that Proposition 3 gives all the equilibria, we may need additional
constraints to make the dual variables of (17) identical among investors. How
to introduce and implement such additional constraints for investors is beyond
this paper’s scope. We will just focus on the Nash equilibrium from
Proposition 3 in the rest of the paper.

Perfect competition: We prove in Proposition 4 that under perfect competition,
the Nash equilibrium will coincide with the social optimum.

######  Proposition 4.

Considering a set of $K$ types of investors $\mathcal{K}$. Each type $k$ has
$N^{k}$ investors. As $N^{k}\rightarrow+\infty$ for any $k$, the Nash
equilibrium approaches the social-optimum solution.

Proof sketch: The objective function in Proposition 3 for computing Nash
equilibrium is equivalent to

$\displaystyle\max~{}C^{0}-\text{System
cost}-\mathbb{E}_{\omega}\sum_{i\in\mathcal{I}}\sum_{t\in\mathcal{T}}\frac{1}{2}\cdot
a_{n}^{\omega}[t]\cdot(\tilde{A}_{i}^{\omega}[t])^{2},$ (21)

where $C^{0}$ is some constant value. Here we can see that the gap between the
Nash equilibrium and the social-optimum benchmark comes from the item
$\mathbb{E}_{\omega}\sum_{i\in\mathcal{I}}\sum_{t\in\mathcal{T}}\frac{1}{2}\cdot
a_{n}^{\omega}[t]\cdot(\tilde{A}_{i}^{\omega}[t])^{2}$. Note that the total
invested capacity of energy resources or the total market supply is upper-
bounded in the market, which implies
$\sum_{i\in\mathcal{I}^{k}}\mid\tilde{A}_{i}^{\omega}[t]\mid\leq\textit{(Certain
upper bound)}$ for any type $k\in\mathcal{K}$. As the number of each investor
type approaches infinity, the item
$\sum_{i\in\mathcal{I}}\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\frac{1}{2}\cdot
a_{n}^{\omega}[t]\cdot(\tilde{A}_{i}^{\omega}[t])^{2}$ will approach zero due
to the quadratic formulation.

Although the Nash equilibrium coincides with the social optimum under perfect
competition, a finite number of investors lead to imperfect competition, where
investors can withhold capacities to increase market prices and get higher
profits.

###  V-B Supply incentive

Under the finite number of investors, as shown in (21), the gap between the
Nash equilibrium and the social-optimum benchmark comes from the item
$\sum_{i\in\mathcal{I}}\sum_{t\in\mathcal{T}}\frac{1}{2}\cdot
a_{n}^{\omega}[t]\cdot(\tilde{A}_{i}^{\omega}[t])^{2}$. To bridge the gap, we
design a variable supply incentive for each hour of each scenario based on the
quadratic values of supply $\tilde{A}_{i}^{\omega}[t]$ for each investor. We
present the new profit $\tilde{\tilde{f}}_{i}$ of investor $i$ as follows.666
The design of penalty payments and supply incentives is to theoretically
ensure that the Nash equilibrium approaches the social-optimum outcome under
certain conditions. If we change the value of supply incentives, there is no
such theoretical guarantee. Besides, arbitrarily increasing the supply
incentive can intuitively promote investment in LERs and mitigate investors’
strategic behaviors. However, if we simply adjust the quadratic coefficient by
a factor $\xi$, i.e., supply incentive $\frac{1}{2}\cdot\xi\cdot
a^{\omega}[t]\cdot(\tilde{A}_{i}^{\omega}[t])^{2}$, we will have a non-convex
structure of the investor’s profit function when $\xi\geq 3$. The existence of
the pure-strategy Nash equilibrium can not be ensured. The discussion on the
design of other structures of supply incentives is beyond the current game-
theoretical framework. We leave it as future work.

$\displaystyle\tilde{\tilde{f}}_{i}(\tilde{\bm{z}}_{i},\tilde{\bm{z}}_{-i})=\tilde{f}_{i}(\tilde{\bm{z}}_{i},\tilde{\bm{z}}_{-i})+\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\frac{1}{2}a^{\omega}[t](\tilde{A}_{i}^{\omega}[t])^{2}.$
(22)

With such a supply incentive, the objective function (21) for computing Nash
equilibrium is changed to

$\displaystyle\max~{}C^{0}-\text{System cost},$ (23)

which is exactly equivalent to the objective of the social-optimum benchmark.
Therefore, under the penalty payment and supply incentive (PI-mechanism), the
Nash equilibrium can reach the social optimum, as concluded in Proposition 5.

######  Proposition 5.

Under the PI mechanism, one Nash equilibrium coincides with the social-optimum
solution.

Although the penalty payment and supply incentive can make the Nash
equilibrium coincide with the social optimum, the investors may face profit
loss. As more CERs retire in the market, i.e., $\gamma$ goes down, the maximum
market price
$\frac{dg^{cv,\omega}[t]}{dp^{cv,\omega}[t]}\Big{|}_{\gamma\cdot\bar{p}^{cv}}$
decreases, which can incur negative profits to investors due to the penalty
payment.

Figure 3: Equilibrium under the MCP and PIU mechanisms vs. social-optimum
benchmark.

###  V-C Energy-price uplift

To ensure adequate revenue for investors, we propose an uplift of energy
prices $\Delta\pi^{\omega}[t]\geq 0$:

$\displaystyle\tilde{\pi}^{\omega}[t](\bm{z})=\hat{\pi}^{\omega}[t](\bm{z})+\Delta\pi^{\omega}[t].$
(24)

which is added to the market price (14) as above.

The energy-price uplift serves as the basic rate for market prices, which
means suppliers will get additional price uplift $\Delta\pi^{\omega}[t]$
besides the market rate $\hat{\pi}^{\omega}[t](\bm{z}).$ This can provide
additional revenues for suppliers. The uplift $\Delta\pi^{\omega}[t]$ is set
as parameters in this work, which can be flexibly adjusted at any time $t$ and
scenario $\omega$.

Adding the price uplift increases investors’ profits, but makes the Nash
equilibrium deviate from the social optimum. Under the penalty payment, supply
incentive, and price uplift (PIU-mechanism), Proposition 6 shows that the Nash
equilibrium coincides with the solution to a modified Problem SO.

######  Proposition 6.

Under the PIU mechanism, one Nash equilibrium is the optimum solution to
Problem SO when we change the linear cost of the CERs from $b^{\omega}[t]$ in
(19) to $b^{\omega}[t]+\Delta\pi^{\omega}[t]$.

Proposition 6 shows that the price uplift will equivalently increase the cost
of the CERs. This will reduce the supply from the CERs and incentivize more
investment and supply from LERs. Furthermore, the increasing retirement of the
CERs will lessen the impact of the CERs’ cost, which implies the Nash
equilibrium will be closer to the social optimum. The uplift can be
interpreted as a long-term incentive in practice, such as premium fees of
insurance. How to design and implement the uplift is an interesting problem
for future research. We later show some simulation results on the uplift in
Section VI.

In Fig.3, we summarize the equilibrium under the MCP and PIU mechanisms
compared with the SO benchmark. Compared with MCP, the advantages of the PIU
mechanism are further summarized below: (i) Under imperfect competition, the
PIU mechanism can prevent the strategic behaviors of investors from incurring
scarcity prices. As shown in Proposition 2, when VOLL is higher than a
threshold, the Nash equilibrium under MCP is that investors will withhold
capacity and supply to incur scarcity prices. However, such strategic
behaviors will not occur under the PIU mechanism. (ii) Under perfect
competition, the proposed PIU mechanism can significantly reduce the energy
cost of consumers and reduce excessive revenues of low-cost CERs compared with
MCP. We demonstrate it in detail in Section VI.E.

##  VI Numerical studies

We conduct numerical studies using CAISO data. We first introduce the
simulation setup, especially how we use the market data to approximate the
supply cost of CERs. Then, we discuss the equilibrium results under the P, PI,
and PIU mechanisms, respectively. In Proposition 2, we have shown that the
traditional MCP mechanism may cause manipulated scarcity prices and zero
surplus for consumers under imperfect competition. In simulations, we will
further compare the PIU mechanism with the MCP mechanism under perfect
competition, which shows that the PIU mechanism can still achieve a higher
consumer surplus.

###  VI-A Simulation setup

####  VI-A1 Data-driven system cost model

We assume a quadratic supply-cost function for CERs and introduce how to
utilize the historical data from the CAISO to characterize it.

We consider the historical market price as the system marginal cost, and use
this to approximate the marginal cost of CERs. Specifically, we examine the
three-year market data from 2019 to 2021 in CAISO, where each day corresponds
to one scenario $\omega$ and is divided into 24 hours $\mathcal{T}$. We
utilize the day-ahead prices $\pi_{0}^{\omega}[t]$, day-ahead forecast system
demand $D_{0}^{\omega}[t]$, and system VRE of solar and wind energy
$R_{0}^{\omega}[t]$ [51]. Recall that the marginal cost of the CERs is
represented as $a^{\omega}[t]\cdot p^{cv,\omega}[t]+b^{\omega}[t]$ based on
(19). Then, the net demand $D^{\omega}[t]=D_{0}^{\omega}[t]-R_{0}^{\omega}[t]$
is assumed to be served by the existing CERs only in the historical data,
which leads to the following expression for the historical market price
$\pi_{0}^{\omega}[t]$, i.e.,

$\displaystyle\pi_{0}^{\omega}[t]=a^{\omega}[t]\cdot(D_{0}^{\omega}[t]-R_{0}^{\omega}[t])+b^{\omega}[t].$
(25)

Next, we will analyze the historical data to characterize the quadratic
coefficient $a^{\omega}[t]$ based on linear regression. Once we obtain
$a^{\omega}[t]$, we compute $b^{\omega}[t]$ according to (25). Note that the
fixed cost $c^{\omega}[t]$ in (19) will not impact the social-optimum solution
or Nash equilibrium result; so, we set it to zero.

Figure 4: (a) Price as a function of net demand for all the hours in each
month of 2019, 2020, and 2021; (b) Linear approximation between the price and
net demand for each month of 2019, 2020, and 2021.

To obtain $a^{\omega}[t]$, in Fig.4(a), we plot the data of day-ahead prices
$\pi_{o}^{\omega}[t]$ and net demand $D_{0}^{\omega}[t]-R_{0}^{\omega}[t]$ of
all the hours with prices below 250$/MWh in each month from January to
December in the years 2019 (red points), 2020 (blue points), and 2021 (green
points), respectively.777We consider the prices below 250$/MWh only due to the
following reasons. Prices over 250$/MWh only occur in 0.33% of all the hours.
Some extremely high prices (e.g., over 600$/MWh) occur due to some rare events
like the Feb 2021 Texas outage, which have a limited impact on the market
revenues in the long run. Second, one limitation of our linear model is that
it cannot accurately model very high prices. A piece-wise linear price
function can be more accurate to model the high prices but it also
significantly increases the complexity of the analysis and the computational
effort to find the Nash equilibrium. We leave this extension as future work.
Fig.4 (a) indicates that the linear model provides a reasonable approximation
for the relationship between price and net demand. In Fig.4(b), we use linear
regression to characterize the slope $a^{\omega}[t]$ of supply functions for
each month from January to December in the year 2019 (red curves), 2020 (blue
curves), and 2021 (green curves), respectively.888There were abnormal prices
in February 2021 due to the Texas outage. We exclude this month’s data since
the linear model does not have a good approximation. Note that we cluster the
scenarios based on each month of each year and have a uniform $a^{\omega}[t]$
for each cluster. For the linear coefficient $b^{\omega}[t]$, it can be
different for each hour of each scenario according to (25).

####  VI-A2 Technology parameters

We consider three types of LERs: solar energy, wind energy, and energy
storage, and focus on cost parameters for 2020. Specifically, we consider PV
solar energy and onshore wind energy with capacity costs of 885$/kW and 1355
$/kW, respectively, and a lifetime of 25 years [52]. For storage, we use Li-
ion batteries (LFP), with energy-capacity cost 385 $/kWh, power-capacity cost
85 $/kWh, roundtrip efficiency 0.88, and a lifetime of 10 years [8]. In the
later sensitivity analysis, we reduce the capital cost below the 2020 baseline
for illustration. We set the VOLL at 3500$/MWh and vary it in some cases.
Furthermore, we present all the results of profits and revenues in the values
scaled to one day.

###  VI-B P-mechanism: Imperfect competition

With the penalty payment only, we show that imperfect competition can lead to
the equilibrium system cost much higher than the social-optimum results
because the investors withhold investment. We also find that the increased
market competition due to more investors can change the capacity share of
different resources in the market.

Figure 5: P-mechanism: (a) System cost with the capital cost reduction of new
investments; (b) Total invested capacity of solar, wind, and storage with the
number of investors of each resource.

In Fig.5, we focus on the P-mechanism. In Fig.5(a), we present the system
costs as the capital cost of the LERs decreases. The blue solid curve shows
the result under the competition of three investors, i.e., one solar-energy
investor, one wind-energy investor, and one storage investor, while the blue
dotted curve shows the competition among fifteen investors, i.e., five
investors of each LER type. The red curve presents the social-optimum result.
Fig.5(b) shows the total installed power capacity of each LER type in the
market when the investors’ number of each type increases: the blue solid curve
for solar energy, red for wind energy, and green for energy storage. The
dotted curves are the social-optimum (SO) results. To better illustrate the
impact of investors’ number, in Fig.5(b), we reduce the capital cost by 80%,
40%, and 20% for energy storage, solar energy, and wind energy, respectively.

In Fig.5(a), the system cost under the 15-investor case is always close to the
social-optimum result. However, the system cost under the 3-investor case can
be much higher than the social-optimum result, especially if the capital cost
is significantly reduced (by over 60%). This reflects the ability of investors
to withhold the invested capacity to support a higher market price, which is
not beneficial to social welfare. In Fig.5(b), with more investors competing,
the total capacity of each resource type will increase. The increasing number
of investors will also change the market position of different resources. When
there is only one investor for each type, the invested capacity of wind energy
(red solid curve) is larger than other types. However, as the number
increases, the capacity of solar energy will surpass wind energy. As the
number of each type goes to infinity, the results approach the social optimum,
where wind-energy capacity will be the lowest (red dotted curve).

###  VI-C PI-mechanism: Impact of CER retirements

Although the PI-mechanism can achieve social optimum, when the CERs retire,
the investors may face profit loss due to the decreasing market price and
increasing demand for LERs. We also find that in CAISO, a low CER retirement
ratio results in more wind energy investments, while a high ratio incentivizes
more energy storage paired with solar energy.

In Fig.6, we focus on the PI-mechanism where the investment results coincide
with the social optimum. For illustration, we reduce the capacity costs of
solar energy, wind energy, and storage by 50%, respectively. We present the
equilibrium profits of three investors in Fig.6(a) and capacities in Fig.6(b)
as the CERs retire. We show the blue curve for solar energy, red curve for
wind energy, and green curve for storage (both power and energy capacity).

In Fig.6(a), when the retirement ratio of CERs is low, e.g., smaller than 10%,
all the investors have non-negative profits (zero for storage but small
positive values for wind and solar energy). However, when the retirement ratio
increases, the profits of the three investors turn negative. Note that if
investors do not face the penalty payment of the lost load, the lowest profits
are always zero since investors can choose not to invest. However, the penalty
payment may result in a negative profit for the investor. In Fig.6(b), the
capacities of LERs increase significantly to fill the supply gap under the
retirement of the CERs. The optimal amount of investment trades off the costs
of the LER investments and system lost load. Furthermore, as shown in
Fig.6(b), a lower or medium retirement ratio of the CERs, e.g., smaller than
40%, results in more wind energy investments (in red) than solar energy (in
blue) and storage (in green), while a high retirement ratio gives more solar
energy paired with storage than wind energy.

Figure 6: PI-mechanism: (a) Profit; (b) Capacity; As a function of CER
retirement ratio under PI mechanism and 3 investors.

###  VI-D PIU mechanism: Profit and system cost with price uplift

The price uplift can flexibly adjust the investors’ profits and prevent profit
loss. Although the price uplift will lead to a higher system cost than the
social optimum, a higher penetration level of LERs can mitigate this
deviation.

Figure 7: PIU-mechanism: (a) Total profit; (b) System cost; Both are with the
energy price uplift under the different retirement ratios.

In Fig.7, we focus on the PIU mechanism. As the price uplift (assumed uniform
for hours and scenarios) increases, we present the total profit of three
investors in Fig.7(a) and the system cost in Fig.7(b), where the blue, red,
green curves refer to retirement ratios of 30%, 50%, and 70%, respectively.
Considering future cost reduction, we reduce the capacity costs of solar
energy, wind energy, and storage by 30%, respectively.

In Fig.7(a), when the price uplift is zero, the total profit is negative. As
the uplift increases, the total profit will turn positive. The break-even
price uplift increases with the retirement ratio of the CERs, e.g., about
24$/MWh under 30% retirement (in blue) and about 90 $/ MWh under 70%
retirement (in green). In Fig.7(b), when the uplift is zero, the system cost
is equal to the social optimum. As the uplift increases, the system cost
increases and deviates from the social optimum. However, such deviation is
lessened with more retirement of the CERs. Under 70% retirement, the system
cost rises only slightly with the uplift, while the system cost increases
significantly under 30% retirement. The reason is that the uplift is
equivalent to an increase in the operating cost of the CERs. This impact
dampens when there are fewer CERs in the system.

###  VI-E Economic surplus comparisons between the PIU mechanism and
traditional MCP mechanism

We show that the PIU mechanism can achieve a lower consumer cost than the MCP
mechanism, by reducing the excessive surplus of CERs. We summarize a surplus
comparison table for all the market participants in Appendix.I[46].

####  VI-E1 Impact of energy price uplift

We show how the energy price uplift will impact the surplus of participants
under the PIU and MCP mechanisms. In Fig.8, we compare the profits and costs
of different entities under the PIU and MCP mechanisms. In Proposition 2, we
have shown that the MCP mechanism may cause manipulated scarcity prices under
imperfect competition. We now focus on a comparison under perfect competition.
In Fig.8(a)-(d), we present the LER profit, consumer cost, CER profit, and
system cost, respectively, as the price uplift varies. The blue and red curves
show the PIU and MCP mechanisms, respectively. The solid curves show 30%
retirement of CERs and the dashed curves show 70% retirement. Considering
future scenarios, we reduce the capacity cost of solar energy, wind energy,
and storage by 30% in this example.

In Fig.8(a), the LER profit under the MCP mechanism is just zero. The profit
under the PIU mechanism increases with the price uplift, and we denote the
break-even points as ${BE}^{30\%}$ and ${BE}^{70\%}$ for the 30% and 70% CER
retirement, respectively. In Fig.8(b), as the uplift increases, the consumer
cost under the PIU mechanism increases, but it is over 30% lower than the MCP
mechanism at the break-even point ${BE}^{70\%}$. The reason is that the CERs
have a much higher profit under the MCP mechanism as shown in Fig.8(c), which
increases the consumer cost. Comparing the red curves of Fig.8(c), the CER
profit under 30% retirement is much lower than that under 70% retirement. The
high ratio of CER retirement creates more frequent scarcity prices and lets
the remaining low-cost CERs get excessive profits under MCP mechanism, which
is absent in the PIU mechanism. In Fig.8(d), the system cost under the PIU
mechanism is higher than the social-optimum under the MCP mechanism, but the
gap at the break-even point ${BE}^{70\%}$ is less than 7%, when the CER-
retirement ratio is high at 70%.

Figure 8: PIU vs MCP: (a) LER profit; (b) Consumer cost; (c) CER profit; (d)
System cost, as functions of price uplift.

####  VI-E2 Impact of VOLL

We show how the parameter VOLL will impact the surplus of participants under
the PIU and MCP mechanisms. In Fig.9, we set a 70% retirement ratio of CERs.
We vary the values of VOLL from 500$/MWh to 9500$MWh. For the PIU mechanism,
we choose the energy-price uplift that achieves break-even profit for LER
investors as shown in Fig.9a. Recall that under the MCP mechanism, the
equilibrium profit of LER investors is always zero.

As the value of VOLL increases, we compare consumer costs, CER profits, and
system costs under the break-even price uplift in Fig.9b-9d. As shown in
Fig.9b, the consumer cost increases with VOLL. Under the PIU mechanism, the
consumer cost is always much lower than the MCP mechanism, even under an
unrealistic low VOLL of 500$/MWh. Also, the PIU mechanism can reduce the
excessive profits for CERs compared with MCP as shown in Fig.9c. Although the
system cost under the PIU mechanism is higher than the MCP mechanism in
Fig.9d, the gap is not significant compared with the reduction of the consumer
cost.

Figure 9:  PIU vs MCP: (a) Consumer cost; (b) CER profit; (d) System cost, as
functions of VOLL.

##  VII Conclusion

In this paper, we establish a game-theoretical framework for the competition
among LER investors, to investigate strategic market interactions, incentive
mechanisms, and efficiency in future electricity markets. We present a PIU
mechanism for LERs investors, which involves a penalty payment for lost load,
a supply incentive, and an energy price uplift. With a quadratic supply cost
function of CERs, the penalty payment leads to the resulting Nash equilibrium
coinciding with the social optimum under perfect competition. The supply
incentive achieves the social optimum under imperfect competition by
encouraging investors to supply more energy, while the price uplift aims at
ensuring adequate revenues for investors. The traditional MCP mechanism can
achieve the social optimum under perfect competition. However, under imperfect
competition, investors can withhold investment to trigger more frequent
scarcity prices, which will not happen under the PIU mechanism. The PIU
mechanism can also reduce consumers’ cost, by reducing the excess-surplus of
low-cost CER suppliers under the MCP mechanism.

Although the proposed PIU mechanism shows nice properties, it still exhibits
limitations that require attention in future research. First, it cannot ensure
social optimum and revenue adequacy simultaneously. The energy-price uplift
can ensure revenue adequacy for investors but make the system cost deviate
from the social optimum. However, this deviation can be mitigated when more
CERs retire in the market. Second, the analysis of Nash equilibrium in this
work relies on the assumption of a quadratic supply curve, which selects one
equilibrium among potential multiple solutions. The impact of different supply
curves and multiple Nash equilibria should be further investigated. In future
work, we will include demand response in the market and study how consumers
may respond to alternative mechanism designs. We will also study the
practicality of implementing elements of the PIU mechanism in future
electricity markets, e.g., through long-term insurance contracts where
premiums can work as the energy price uplift and reimbursement can serve as
the penalty payment.

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|  Dongwei Zhao (M’21) received the B.S. degree from Zhejiang University in
2015, and the Ph.D. degree from The Chinese University of Hong Kong in 2021.
He is currently a postdoctoral associate at MIT Energy Initiative and
Laboratory for Information & Decision Systems, Massachusetts Institute of
Technology. His main research interests are in the optimization, game theory,
and mechanism design of power and energy systems. More information at
https://sites.google.com/view/joris-zhao
---|---
|  Sarah Coyle received a B.S. in Geological Sciences from the University of
Texas at Austin in 2012, a M.S. in Geophysics from the University of Texas at
Austin in 2014, and a M.S. in System Design and Management from Massachusetts
Institute of Technology in 2022. She is currently working as the Product
Manager for Asset Development and Portfolio Optimization at Chevron.
---|---
|  Apurba Sakti received his Ph.D. in Engineering and Public Policy from
Carnegie Mellon University in 2013. He then pursued a postdoctoral position
with the MIT Energy Initiative before being promoted to a Research Scientist.
At MIT, Sakti’s research focused on different techno-economic aspects of
energy storage systems, which included the development of improved analytics
for low-carbon energy systems to help understand evolving market dynamics.
Since 2022, Sakti is an advisor with the data science team at Albemarle
Corporation.
---|---
|  Audun Botterud Audun Botterud is a Principal Research Scientist in
Laboratory for Information and Decision Systems (LIDS) at MIT, where he leads
the Energy Analytics Group. He has a co-appointment in the Energy Systems and
Infrastructure Analysis Division at Argonne National Laboratory. His research
interests include power systems, electricity markets, renewable energy, and
energy storage. Audun holds a M.Sc. (Industrial Engineering) and a Ph.D.
(Electrical Power Engineering), both from the Norwegian University of Science
and Technology. He was previously with SINTEF Energy Research in Trondheim,
Norway.
---|---

TABLE I: Comparison of market participants’ surplus |  PIU mechanism |  MCP mechanism
---|---|---
LERs |  Regulated by the uplift |  0
CERs |  Lower (by simulation) |  Higher (by simulation)
Consumers |  Higher (by simulation) |  Lower (by simulation)
System operator |  Non-negative under perfect competition |  0
Social welfare |  • $\Delta\bm{\pi}=\bm{0}$: Social optimum; • $\Delta\bm{\pi}>\bm{0}$: It deviates from social optimum and incentivizes more LER investment but the gap is reduced as CERs retire. |  Social optimum

## Appendix.I: Surplus comparison between the PIU mechanism and MCP mechanism

We have demonstrated the shortcomings of the MCP mechanism in terms of the
high scarce prices, and supply withhold under imperfection competition.
However, the MCP mechanism can achieve the social optimum under perfect
competition, while the PIU mechanism cannot. In this appendix, we will further
compare the surplus of various market participants between the PIU mechanism
and MCP mechanism under perfect competition.

The surplus of the LER investors is the profit that we have derived. The
surplus of the CERs is the market revenue deducting the operation cost, i.e.,

$\displaystyle\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\pi^{\omega}[t](\bm{z})\cdot
p^{cv,\omega}[t]-G^{cv}(\bm{p}^{cv}).$ (26)

Consumers’ surplus is the value of system served demand minus the energy
payment

$\displaystyle\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\left(\text{VOLL}-\pi^{\omega}[t](\bm{z})\right)\cdot(D^{\omega}[t]-p^{sh}[t]).$
(27)

Here we use consumers’ surplus instead of cost to be consistent with other
participants’ surplus. The system operator’s surplus under the MCP mechanism
is zero since the money flow is only between the consumers and suppliers. In
the PIU mechanism, the surplus is the total penalty payment charged from the
investors minus the supply incentive, i.e.,

$\displaystyle\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\sum_{i\in\mathcal{I}}\left(\Big{(}\text{VOLL}-\pi^{\omega}[t](\bm{z})\Big{)}\cdot
p_{i}^{sh}[t]-\frac{1}{2}a^{\omega}[t](\tilde{A}_{i}^{\omega}[t])^{2}\right).$
(28)

Under perfect competition, i.e., a sufficiently large number of investors, the
total supply incentive goes to zero because no supply incentive is needed. In
this case, the operator’s surplus is non-negative and can be distributed to
consumers.

We summarize the surplus comparison in Table I under perfect competition.

## Appendix.II: Proof of Proposition 1

Proof: We prove that the social-optimum solution is one Nash equilibrium by
KKT conditions in Part 1. We show the zero profit of investors under the
social optimum by the Lagrange dual problem in Part 2.

### Part 1

Given the decisions of the other investors and the system operator, any
investor $i$ maximizes its own profit.

$\displaystyle\max~{}$ $\displaystyle
f_{i}(\bm{z}_{i},\bm{z}_{-i})=R_{i}(\bm{z}_{i},\bm{z}_{-i})-C_{i}^{inv}(\bm{z}_{i})-C_{i}^{opr}(\bm{z}_{i})$
s.t.
$\displaystyle\bm{z}_{i}\in\mathcal{Z}_{i}^{o}\bigcup\mathcal{Z}_{i}^{c}(\bm{z}_{-i}),$
var: $\displaystyle\bm{z}_{i}.$

The market price is set at $\lambda^{\omega}[t]/\rho^{\omega}$, which is not
affected by any investor’s decision under perfect competition. We have the
following equivalent problem given $\bm{z}_{-i}$, $\bm{p}^{cv}$ and
$\bm{p}^{sh}$.

$\displaystyle\min~{}$
$\displaystyle-\sum_{t\in\mathcal{T}}{A}_{i}^{\omega}[t]\cdot\lambda^{b,\omega}[t]+C_{i}^{inv}(\bm{z}_{i})+C_{i}^{opr}(\bm{z}_{i})$
s.t.
$\displaystyle~{}\bm{z}_{i}\in\mathcal{Z}_{i}^{o}\bigcup\mathcal{Z}_{i}^{c}(\bm{z}_{-i}).$
var: $\displaystyle~{}\bm{z}_{i}.$ (29)

Since the above problem is convex, the optimal profit solution is equivalent
to the solutions derived by the KKT conditions. It is straightforward that the
KKT conditions of the above problem are the necessary condition for the KKT
conditions of Problem SO. Note that we can additionally add the constraint
(11) (which models the set $\mathcal{Z}_{i}^{c}(\bm{z}_{-i})$) to Problem SO.
Therefore, under the social-optimum solution to Problem SO, any investor
achieves the optimal profit and will not deviate unilaterally. ∎

### Part 2

We prove the zero profit of investors at the social optimum based on the
Lagrange dual problem of Problem SO, which is formulated as follows.

$\displaystyle\max_{\bm{\lambda}^{b},\bm{\mu}}~{}~{}\min_{\bm{z},\bm{p}^{cv},\bm{p}^{sh}}~{}$
$\displaystyle\sum_{i\in\mathcal{I}}-f_{i}(\bm{\lambda}^{b},\bm{z_{i}})$
$\displaystyle+\sum_{t\in\mathcal{T}}\sum_{\omega\in\Omega}\Big{(}\big{(}-\lambda^{b,\omega}[t]-\underline{\mu}^{cv,\omega}[t]+\overline{\mu}^{cv,\omega}[t]\big{)}\cdot
p^{cv,\omega}[t]$ $\displaystyle\hskip
77.49976pt-\overline{\mu}^{cv,\omega}[t]\cdot\bar{p}^{cv}\Big{)}$
$\displaystyle+\sum_{t\in\mathcal{T}}\sum_{\omega\in\Omega}\Big{(}-\lambda^{b,\omega}[t]-\underline{\mu}^{sh,\omega}[t]\Big{)}\cdot
p^{sh,\omega}[t]$
$\displaystyle+\sum_{t\in\mathcal{T}}\sum_{\omega\in\Omega}\lambda^{b,\omega}[t]\cdot
D^{\omega}[t]+G^{cv}(\bm{p}^{cv})+G^{sh}(\bm{p}^{sh})$
$\displaystyle\text{s.t.}~{}\eqref{eq:const_ren_a}-\eqref{eq:const_ren_b},\eqref{eq:ch}-\eqref{eq:duration},\forall
i\in\mathcal{I},$

where we let

$\displaystyle-
f_{i}(\bm{\lambda}^{b},\bm{z_{i}})=C_{i}^{inv}(\bm{X}_{i})+C_{i}^{op}(\bm{p}_{i})-\sum_{t\in\mathcal{T}}\sum_{\omega\in\Omega}\lambda^{b,\omega}[t]\cdot
A_{i}^{\omega}[t].$

Note that
$\sum_{t\in\mathcal{T}}\sum_{\omega\in\Omega}\lambda^{b,\omega}[t]A_{i}^{\omega}[t]$
is equivalent to
$\sum_{t\in\mathcal{T}}\mathbb{E}_{\omega}\left[\pi^{\lambda,\omega}[t]A_{i}^{\omega}[t]\right]$
based on (12), which means that $f_{i}(\bm{\lambda}^{b},\bm{z_{i}})$ is the
profit of investor $i$. We denote the optimal solution above by
$\bm{\lambda}^{b*},\bm{\mu}^{*},\bm{z}^{*},\bm{p}^{cv*}$, and $\bm{p}^{sh*}$.
Given $\bm{\lambda}^{b*}$ and $\bm{\mu}^{*}$, the solution
$\bm{z}^{*},\bm{p}^{cv*}$, and $\bm{p}^{sh*}$ minimizes the objective function
and the solution is equivalent to the social optimum in Problem SO due to
strong duality.

We now prove the zero profit of investors at the social optimum by
contradiction. (i) Suppose $-f_{i}(\bm{\lambda}^{b*},\bm{z}_{i}^{*})>0$ at the
social optimum. Given $\bm{\lambda}^{b*}$ and $\bm{\mu}^{*}$, we can always
have a new solution $\bm{z}_{i}^{\prime}=\bm{0}$ satisfying the constraints
such that $-f_{i}(\bm{\lambda}^{b*},\bm{z}_{i}^{\prime})=0$, which will reduce
the objective value and contradicts the optimal solution. (ii) Suppose that we
have $-\sum_{i\in\mathcal{I}}f_{i}(\bm{\lambda}^{b*},\bm{z}_{i}^{*})<0$ at the
social optimum. Given $\bm{\lambda}^{b*}$ and $\bm{\mu}^{*}$, we can always
have a new solution $\bm{z}_{i}^{\prime}=\alpha\cdot\bm{z}_{i}^{*},\alpha>1$
satisfying the affine constraints such that
$-\sum_{i\in\mathcal{I}}f_{i}(\bm{\lambda}^{b*},\bm{z}_{i}^{\prime})<-\sum_{i\in\mathcal{I}}f_{i}(\bm{\lambda}^{b*},\bm{z}_{i}^{*})$,
which will reduce the objective value and contradicts the optimal solution.
Thus, we have $f_{i}(\bm{\lambda}^{b*},\bm{z}_{i}^{*})=0$ at the social
optimum. ∎

## Appendix.III: Proof of Proposition 2

Proof: We prove Proposition 2 based on the symmetric solutions for the
homogeneous VRE investors and near Nash equilibrium.

To begin with, We define $\alpha-$Nash equilibrium as follows.

######  Definition 2 ($\alpha-$Nash equilibrium).

In the competition game
$G=\langle\mathcal{I},(\mathcal{Z}_{i}),(f_{i})\rangle$, the strategy profile
$\bm{z}^{*}\in\Pi_{i\in\mathcal{I}}\mathcal{Z}_{i}(\bm{z}_{-i}^{*})$ is an
$\alpha-$Nash equilibrium if for every investor $i\in\mathcal{I}$,
$f_{i}(\bm{z}_{i}^{*},\bm{z}_{-i}^{*})\geq
f_{i}(\bm{z}_{i},\bm{z}_{-i}^{*})-\alpha$ for any
$\bm{z}_{i}\in\mathcal{Z}_{i}(\bm{z}_{-i}^{*})$.

Furthermore, the optimization problem in Proposition 2 is concave, so we
consider the symmetric solutions among investors. We denote each investor’s
solution as $\bm{z}_{i}^{*}=\bm{z}_{u}^{*}$. Next, we will prove the Nash
equilibrium results by discussing $\sum_{i\in\mathcal{I}}A_{i}^{\omega}[t]$ in
the ranges $\sum_{i\in\mathcal{I}}A_{i}^{\omega}[t]\leq
D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}-\epsilon^{\omega}[t]$,
$D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}-\epsilon^{\omega}[t]\leq\sum_{i\in\mathcal{I}}A_{i}^{\omega}[t]\leq
D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}$, and
$D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}<\sum_{i\in\mathcal{I}}A_{i}^{\omega}[t]\leq
D^{\omega}[t]$, respectively.

### Step 1

We define a constraint $\mathcal{M}_{1}^{\omega}[t]$ for $A_{i}^{\omega}[t]$
as $A_{i}^{\omega}[t]+\sum_{j\in\mathcal{I}\setminus i}A_{j}^{\omega*}[t]\leq
D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}-\epsilon^{\omega}[t]$ for some time $t$
of scenario $\omega$. We show that any investor $i$ will not change its
strategy unilaterally to any $\bm{z}_{i}$ that satisfies
$\mathcal{M}_{1}^{\omega}[t]$ for some time $t$ of scenario $\omega$.

Note that the market price is fixed at $\pi=\text{VOLL}$ in this constraint
range, any investor $i$ who changes his strategy unilaterally within this
constraint will not impact other investors’ profits. Meanwhile, the
optimization problem in Proposition 2 maximizes all the investors’ total
profit, which implies that any investor $i$’s unilateral strategy change will
not increase its profit within this constraint $\mathcal{M}_{1}^{\omega}[t]$,
i.e.,

$\displaystyle f_{i}(\bm{z}_{i}^{*},\bm{z}_{-i}^{*})\geq
f_{i}(\bm{z}_{i},\bm{z}_{-i}^{*}),$ (30)

for any $\bm{z}_{i}$ that satisfies $\mathcal{M}_{1}^{\omega}[t]$.

### Step 2

We define a constraint $\mathcal{M}_{2}^{\omega}[t]$ for $A_{i}^{\omega}[t]$
as $D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}-\epsilon^{\omega}[t]\leq
A_{i}^{\omega}[t]+\sum_{j\in\mathcal{I}\setminus i}A_{j}^{\omega*}[t]\leq
D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}$. We will show that when any investor
$i$ changes its strategy unilaterally from $\bm{z}^{*}$ to any
$\bm{z}_{i}^{\dagger}$ such that $\mathcal{M}_{1}^{\omega_{a}}[t_{a}]$ holds
for some time $t_{a}$ of scenario $\omega_{a}$ and
$\mathcal{M}_{2}^{\omega_{b}}[t_{b}]$ holds for some time $t_{b}$ of scenario
$\omega_{b}$, this investor’s profit increase will be bounded above by a
function of ${\epsilon}^{\omega}[t]$.

Specifically, we consider a benchmark strategy $\bm{z}_{i}^{\prime}$ for
$\bm{z}_{i}^{\dagger}$, i.e., investor $i$ changes its strategy unilaterally
from $\bm{z}_{i}^{*}$ to $\bm{z}_{i}^{\prime}$ such that
$D^{\omega_{b}}[t_{b}]-\gamma\cdot\bar{p}^{cv}-\epsilon^{\omega_{b}}[t_{b}]=A_{i}^{\omega_{b}^{\prime}}[t_{b}]+\sum_{j\in\mathcal{I}\setminus
i}A_{j}^{\omega_{b}*}[t_{b}]$ for time $t_{b}$ of scenario $\omega_{b}$.
Meanwhile we let $\bm{z}_{i}^{\prime}=\bm{z}_{i}^{\dagger}$, except at time
$t_{b}$ of scenario $\omega_{b}$. Since $\bm{z}_{i}^{\dagger}$ satisfies
$\mathcal{M}_{1}^{\omega_{a}}[t_{a}]$ and
$\mathcal{M}_{2}^{\omega_{b}}[t_{b}]$, we easily have

$\displaystyle R_{i}^{\omega}(\bm{z}_{i}^{\prime},\bm{z}_{-i}^{*})[t]\geq
R_{i}^{\omega}(\bm{z}_{i}^{\dagger},\bm{z}_{-i}^{*})[t]-\epsilon^{\omega}[t]\cdot\text{VOLL},$
(31)

i.e., the expected daily revenue

$\displaystyle R_{i}(\bm{z}_{i}^{\prime},\bm{z}_{-i}^{*})\geq
R_{i}(\bm{z}_{i}^{\dagger},\bm{z}_{-i}^{*})-\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\epsilon^{\omega}[t]\cdot\text{VOLL},$
(32)

Note that when $A_{i}^{\omega}+\sum_{j\in\mathcal{I}\setminus
i}A_{j}^{\omega*}[t]=D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}$, (31) is always
satisfied for any market price $\pi^{\omega}[t]\leq\text{VOLL}$.

Since $A_{i}^{\omega_{b}^{\prime}}[t_{b}]\leq
A_{i}^{\omega_{b}\dagger}[t_{b}]$, the corresponding invested capacity
$X_{i}^{\prime}$ of investor $i$ under $\bm{z}_{i}^{\prime}$ is no greater
than the capacity $X_{i}^{\dagger}$ under $\bm{z}_{i}^{\dagger}$, which makes
the expected daily profit satisfy

$\displaystyle f_{i}(\bm{z}_{i}^{\prime},\bm{z}_{-i}^{*})\geq
f_{i}(\bm{z}_{i}^{\dagger},\bm{z}_{-i}^{*})-\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\epsilon^{\omega}[t]\cdot\text{VOLL},$
(33)

which leads to

$\displaystyle f_{i}(\bm{z}_{i}^{*},\bm{z}_{-i}^{*})\geq
f_{i}(\bm{z}_{i}^{\dagger},\bm{z}_{-i}^{*})-\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\epsilon^{\omega}[t]\cdot\text{VOLL}$
(34)

based on (30). Thus, we have $\bm{z}^{*}$ is
$\big{(}\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\epsilon^{\omega}[t]\big{)}-$near
Nash equilibrium under the constraint set
$\sum_{i\in\mathcal{I}}A_{i}^{\omega}[t]\leq
D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}$.

### Step 3

We define a constraint $\mathcal{M}_{3}^{\omega}[t]$ for $A_{i}^{\omega}[t]$
as
$D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}<A_{i}^{\omega}[t]+\sum_{j\in\mathcal{I}\setminus
i}A_{j}^{\omega*}[t]\leq D^{\omega}[t]$. We will characterize a sufficient
condition such that $\bm{z}^{*}$ is
$\big{(}\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\epsilon^{\omega}[t]\big{)}-$near
Nash equilibrium globally.

To characterize such a sufficient condition, we suppose that any investor $i$
changes its strategy unilaterally to any $\bm{z}_{i}^{\ddagger}$ such that
$\mathcal{M}_{1}^{\omega_{a}}[t_{a}]$ holds for some time $t_{a}$ of scenario
$\omega_{a}$, $\mathcal{M}_{2}^{\omega_{b}}[t_{b}]$ holds for some time
$t_{b}$ of scenario $\omega_{b}$, and $\mathcal{M}_{3}^{\omega_{c}}[t_{c}]$
holds for some time $t_{c}$ of scenario $\omega_{c}$. Then, we compare it with
a benchmark case: investor $i$ changes its strategy unilaterally to
$z_{i}^{\prime\prime}$ that satisfies
$A_{i}^{\omega_{c}^{\prime\prime}}[t_{c}]+\sum_{j\in\mathcal{I}\setminus
i}A_{j}^{\omega_{c}*}[t_{c}]=D^{\omega_{c}}[t_{c}]-\gamma\cdot\bar{p}^{cv}$,
$\mathcal{M}_{2}^{\omega_{b}}[t_{b}]$, and
$\mathcal{M}_{1}^{\omega_{a}}[t_{a}]$. We let
$\pi^{\omega_{c}}(\bm{z}_{i}^{\prime\prime},z_{-i}^{*})[t_{c}]=\text{VOLL}$
because (31) is always satisfied for any market price
$\pi^{\omega}[t]\leq\text{VOLL}$. We let
$\bm{z}_{i}^{\prime\prime}=\bm{z}_{i}^{\ddagger}$, except at time $t_{c}$ of
$\omega_{c}$.

For $\bm{z}_{i}^{\prime\prime}$, at time $t_{c}$ of scenario $\omega_{c}$,
investor $i$’s revenue satisfies

$\displaystyle
R_{i}^{\omega_{c}}(\bm{z}_{i}^{\prime\prime},\bm{z}_{-i}^{*})[t_{c}]$
$\displaystyle=\text{VOLL}\cdot A_{i}^{\omega_{c}^{\prime\prime}}[t_{c}]$
$\displaystyle=\text{VOLL}\cdot(D^{\omega_{c}}[t_{c}]-\gamma\cdot\bar{p}^{cv}-(N-1)A_{u}^{\omega_{c}*}[t_{c}]).$
(35)

For $\bm{z}_{i}^{\ddagger}$, at time $t_{c}$ of scenario $\omega_{c}$,
investor $i$’s revenue satisfies

$\displaystyle
R_{i}^{\omega_{c}}(\bm{z}_{i}^{\ddagger},\bm{z}_{-i}^{*})[t_{c}]$
$\displaystyle~{}\leq\big{(}D^{\omega_{c}}[t_{c}]-(N-1)A_{u}^{\omega_{c}*}[t_{c}]\big{)}\cdot\frac{dg^{cv,\omega_{c}}[t_{c}]}{dp^{cv,\omega_{c}}[t_{c}]}\Big{|}_{\gamma\cdot\bar{p}^{cv}}.$
(36)

To let
$R_{i}^{\omega_{c}}(\bm{z}_{i}^{\prime\prime},\bm{z}_{-i}^{*})[t_{c}]\geq
R_{i}^{\omega_{c}}(\bm{z}_{i}^{\ddagger},\bm{z}_{-i}^{*})[t_{c}]$, it suffices
to show

$\displaystyle\text{VOLL}\cdot(D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}-(N-1)A_{u}^{\omega*})$
$\displaystyle~{}~{}~{}~{}~{}~{}\geq\big{(}D^{\omega}[t]-(N-1)A_{u}^{\omega*}\big{)}\cdot\frac{dg^{cv,\omega}[t]}{dp^{cv,\omega}[t]}\Big{|}_{\gamma\cdot\bar{p}^{cv}},$
(37)

for all the time $t$ of scenario $\omega$. Note that
$\text{VOLL}\geq\frac{dg^{cv,\omega}[t]}{dp^{cv,\omega}[t]}\Big{|}_{\gamma\cdot\bar{p}^{cv}}$
and $A_{u}^{\omega*}[t]<\frac{D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}}{N}$. To
satisfy (37), it suffices to show

$\displaystyle\text{VOLL}\cdot(D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}-(N-1)\frac{D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}}{N})$
$\displaystyle\geq\Big{(}D^{\omega}[t]-(N-1)\frac{D^{\omega}[t]-\gamma\cdot\bar{p}^{cv}}{N}\Big{)}\cdot\frac{dg^{cv,\omega}[t]}{dp^{cv,\omega}[t]}\Big{|}_{\gamma\cdot\bar{p}^{cv}},$
(38)

i.e.,

$\displaystyle\text{VOLL}\geq\left(1+\frac{N\cdot\gamma\bar{p}}{D^{\omega}[t]-\gamma\bar{p}}\right)\cdot\frac{dg^{cv,\omega}[t]}{dp^{cv,\omega}[t]}\Big{|}_{\gamma\cdot\bar{p}^{cv}}.$
(39)

Therefore, as long as (39) is satisfied, we have
$R_{i}^{\omega_{c}}(\bm{z}_{i}^{\prime\prime},\bm{z}_{-i}^{*})[t_{c}]\geq
R_{i}^{\omega_{c}}(\bm{z}_{i}^{\ddagger},\bm{z}_{-i}^{*})[t_{c}]$ , which
easily makes the the expected daily revenue $R$ satisfy

$\displaystyle R_{i}(\bm{z}_{i}^{\prime\prime},\bm{z}_{-i}^{*})\geq
R_{i}(\bm{z}_{i}^{\ddagger},\bm{z}_{-i}^{*}).$ (40)

Since $A_{i}^{\omega_{c}^{\prime\prime}}[t_{c}]\leq
A_{i}^{\omega_{c}\ddagger}[t_{c}]$, the corresponding invested capacity
$X_{i}^{\prime\prime}$ of investor $i$ under $\bm{z}_{i}^{\prime\prime}$ is no
greater than the capacity $X_{i}^{\ddagger}$ under $\bm{z}_{i}^{\ddagger}$,
which makes the expected daily profit also satisfy

$\displaystyle f_{i}(\bm{z}_{i}^{\prime\prime},\bm{z}_{-i}^{*})\geq
f_{i}(\bm{z}_{i}^{\ddagger},\bm{z}_{-i}^{*}).$ (41)

Note that here $\bm{z}_{i}^{\prime\prime}$ is one special case of
$\bm{z}_{i}^{\dagger}$ in Step 2. Based on (34), we have

$\displaystyle f_{i}(\bm{z}_{i}^{*},\bm{z}_{-i}^{*})\geq
f_{i}(\bm{z}_{i}^{\ddagger},\bm{z}_{-i}^{*})-\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\epsilon^{\omega}[t]\cdot\text{VOLL},$
(42)

for any $\bm{z}_{i}^{\ddagger}$ such that
$\mathcal{M}_{1}(A_{i}^{\omega_{a}}[t_{a}])$ holds for some time $t_{a}$ of
scenario $\omega_{a}$, $\mathcal{M}_{2}(A_{i}^{\omega_{b}}[t_{b}])$ holds for
some time $t_{b}$ of scenario $\omega_{b}$, and
$\mathcal{M}_{3}(A_{i}^{\omega_{c}}[t_{c}])$ holds for some time $t_{c}$ of
scenario $\omega_{c}$.

Overall, we have $\bm{z}^{*}$ is
$\big{(}\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\epsilon^{\omega}[t]\big{)}-$near
Nash equilibrium globally, which approaches a pure-strategy Nash equilibrium
as $\epsilon^{\omega}[t]\rightarrow 0$. ∎

## Appendix.IV: Proof of Proposition 3

###  A.Proof based on potential game theory

We can prove Proposition 3 from the perspective of the potential game theory,
which we define as follows. Let
$H=\langle\mathcal{N},(\mathcal{Z}_{i}),(h_{i})\rangle$ be any strategic game
and $\mathcal{Z}=\times_{i\in\mathcal{N}}\mathcal{Z}_{i}$ be the collection of
all players’ strategy profiles. We have the following result for potential
games [49] [50].

######  Definition 3 (Potential game).

A function $\Phi:\mathcal{Z}\rightarrow\mathbb{R}$ is called an (exact)
potential function for the game $H$ if for each $i\in\mathcal{N}$ and all
$\bm{z}_{-i}\in\mathcal{Z}_{-i}$,

$\Phi(\bm{z}^{\prime}_{i},\bm{z}_{-i})-\Phi(\bm{z}_{i},\bm{z}_{-i})=h_{i}(\bm{z}^{\prime}_{i},\bm{z}_{-i})-h_{i}(\bm{z}_{i},\bm{z}_{-i}),$

for all $\bm{z}^{\prime}_{i},\bm{z}_{i}\in\mathcal{Z}_{i}$. The game $H$ is
called an (exact) potential game if it admits a potential function.

######  Proposition 7 (Pure strategy Nash equilibrium).

If the potential function $\Phi$ has a maximum point $\bm{z}^{*}$ in
$\mathcal{Z}$, then the game $H$ has a pure-strategy Nash equilibrium that
coincides with $\bm{z}^{*}$.

The above proposition shows that if we can find a potential function for this
game, the optimum of the potential function is one Nash equilibrium. Our game-
theoretical model is in nature a Cournot competition model. Based on the
linear supply functions, we can characterize a potential function, i.e., the
objective (20) in Proposition 3. A stylized Cournot model with a characterized
potential function is given in [49]. Our previous work [45] extends it to the
storage competition problem [45], and this work extends to the mechanism
design for the joint investment of renewable energy and energy storage.

### B.Proof based on KKT conditions

At the Nash equilibrium of the game-theoretical model, each investor’s
decision is the best response given other investors’ decisions. We show that
the KKT conditions of the optimization problem in Proposition 3 suffice to
satisfy the investors’ KKT condition set when each investor optimizes its own
profit given other investors’ decisions.

For each investor $i$, at the Nash equilibrium, each investor decides its
decision $\bm{z}_{i}$ to maximize the profit given other investors’ decisions
$\bm{z}_{-i}$, i.e.,

$\displaystyle\max~{}$ $\displaystyle\tilde{f}_{i}(\tilde{\bm{z}})$ s.t.
$\displaystyle\tilde{\bm{z}}_{i}\in\tilde{\mathcal{Z}}_{i}^{o}\bigcup\tilde{\mathcal{Z}}_{i}^{c}(\tilde{\bm{z}}_{-i})$

Based on the equation (15), we have
$p^{cv,\omega}[t]=D^{\omega}[t]-\sum_{i\in\mathcal{I}}\tilde{A}_{i}^{\omega}[t],\forall
t\in\mathcal{T},\forall\omega\in{\Omega}$, which leads to

$\displaystyle\tilde{f}_{i}(\tilde{\bm{z}})$
$\displaystyle=\sum_{t\in\mathcal{T}}\mathbb{E}_{\omega}\left[\frac{dg^{cv,\omega}[t]}{dp^{cv,\omega}[t]}\cdot\tilde{A}_{i}^{\omega}[t]\right]-C_{i}^{inv}(\tilde{\bm{z}}_{i})-C_{i}^{opr}(\tilde{\bm{z}}_{i})$
$\displaystyle=\sum_{t\in\mathcal{T}}\mathbb{E}_{\omega}\left[a^{\omega}[t](D^{\omega}[t]-\sum_{i\in\mathcal{I}}\tilde{A}_{i}^{\omega}[t])\cdot\tilde{A}_{i}^{\omega}[t]+b^{\omega}[t]\cdot\tilde{A}_{i}^{\omega}[t]\right]$
$\displaystyle\hskip 77.49976pt-
C_{i}^{inv}(\tilde{\bm{z}}_{i})-C_{i}^{opr}(\tilde{\bm{z}}_{i}).$ (43)

Investors’ profits are coupled with $\tilde{\bm{A}}^{\omega}[t]$ directly so
we examine
$\frac{d\tilde{f}_{i}(\tilde{\bm{z}})}{d\tilde{A}_{i}^{\omega}[t]}$:

$\displaystyle\frac{d\tilde{f}_{i}(\tilde{\bm{z}})}{d\tilde{A}_{i}^{\omega}[t]}$
$\displaystyle=a^{\omega}[t]\Big{(}D^{\omega}[t]-\sum_{i\in\mathcal{I}}\tilde{A}_{i}^{\omega}[t]\Big{)}-a^{\omega}[t]\cdot\tilde{A}_{i}^{\omega}[t]+b^{\omega}[t]$
$\displaystyle\hskip
81.8053pt-\frac{C_{i}^{inv}(\bm{z}_{i})+C_{i}^{opr}(\bm{z}_{i})}{d\tilde{A}_{i}^{\omega}[t]}.$
(44)

Considering the objective function of the optimization problem in Proposition
3, we easily verify

$\displaystyle\frac{d\left(\sum_{i}\tilde{f}_{i}(\tilde{\bm{z}})+\mathbb{E}_{\omega\in\Omega}\Big{[}\sum_{t\in\mathcal{T}}\sum_{1\leq
i<j\leq
I}a^{\omega}[t]\cdot\Phi^{\omega}(\tilde{\bm{z}_{i}},\tilde{\bm{z}}_{j})\Big{]}\right)}{d\tilde{A}_{i}^{\omega}[t]}$
(45)
$\displaystyle=\frac{d\tilde{f}_{i}(\tilde{\bm{z}})}{d\tilde{A}_{i}^{\omega}[t]}.$
(46)

Note that the joint constraints of all the investors in the game-theoretical
model coincide with the constraints of the optimization problem in Proposition
3. It is straightforward that the KKT conditions of the optimization problem
in Proposition 3 suffice to satisfy the investors’ KKT condition set when each
investor optimizes its own profit given other investors’ decisions in the
game-theoretical model. ∎

## Appendix.V: Proof of Proposition 4

Proof: We prove it based on the Nash equilibrium derived in Proposition 3 and
symmetric solutions for the homogeneous investors.

First, based on (43), the objective function of the optimization problems in
Proposition 3 can be written into

$\displaystyle
F(\tilde{\bm{z}})\triangleq\sum_{i\in\mathcal{I}}\tilde{f}_{i}(\tilde{\bm{z}})+\mathbb{E}_{\omega\in\Omega}\Big{[}\sum_{t\in\mathcal{T}}\sum_{1\leq
i<j\leq
I}a^{\omega}[t]\cdot\Phi^{\omega}(\tilde{\bm{z}_{i}},\tilde{\bm{z}}_{j})\Big{]}$
$\displaystyle=\sum_{t\in\mathcal{T}}\mathbb{E}_{\omega}\Big{[}a^{\omega}[t]\cdot
D^{\omega}[t]\cdot\sum_{i\in\mathcal{I}}\tilde{A}_{i}^{\omega}[t]+b^{\omega}[t]\cdot\sum_{i\in\mathcal{I}}\tilde{A}_{i}^{\omega}[t]\Big{]}$
$\displaystyle-\sum_{t\in\mathcal{T}}\mathbb{E}_{\omega}\Big{[}\frac{1}{2}a^{\omega}[t]\Big{(}(\sum_{i\in\mathcal{I}}\tilde{A}_{i}^{\omega}[t])^{2}+\sum_{i\in\mathcal{I}}(\tilde{A}_{i}^{\omega}[t])^{2}\Big{)}\Big{]}$
$\displaystyle-\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\sum_{i\in\mathcal{I}}\text{VOLL}\cdot
p_{i}^{sh,\omega}[t]-\sum_{i\in\mathcal{I}}C_{i}^{inv}(\tilde{\bm{z}}_{i})-\sum_{i\in\mathcal{I}}C_{i}^{opr}(\tilde{\bm{z}}_{i}).$

Then, based on the convexity of the optimization problem in Proposition 3, we
assume symmetric solutions for investors in the same type, i.e., any investor
$i$ in type $k$ has $\tilde{A}_{i}^{\omega}[t]=\tilde{A}_{k}^{\omega}[t]$,
which leads to

$\displaystyle\sum_{i\in\mathcal{I}}\tilde{A}_{i}^{\omega}[t]=\sum_{k\in\mathcal{K}}N^{k}\tilde{A}_{k}^{\omega}[t].$
(47)

We let
$N^{k}\tilde{A}_{k}^{\omega}[t]\triangleq\tilde{A}_{k}^{\omega^{\prime}}[t]$
and $p^{sh,\omega}[t]\triangleq\sum_{i\in\mathcal{I}}p_{i}^{sh,\omega}[t]$ and
have

$\displaystyle~{}~{}~{}~{}F(\tilde{\bm{z}})$
$\displaystyle=\sum_{t\in\mathcal{T}}\mathbb{E}_{\omega}\Big{[}a^{\omega}[t]\cdot
D^{\omega}[t]\cdot\sum_{k\in\mathcal{K}}\tilde{A}_{k}^{\omega^{\prime}}[t]+b^{\omega}[t]\cdot\sum_{k\in\mathcal{K}}\tilde{A}_{k}^{\omega^{\prime}}[t]\Big{]}$
$\displaystyle-\sum_{t\in\mathcal{T}}\mathbb{E}_{\omega}\Big{[}\frac{1}{2}a^{\omega}[t]\Big{(}(\sum_{k\in\mathcal{K}}\tilde{A}_{k}^{\omega^{\prime}}[t])^{2}+\sum_{k\in\mathcal{K}}\frac{(\tilde{A}_{k}^{\omega^{\prime}}[t])^{2}}{N^{k}}\Big{)}\Big{]}$
$\displaystyle-\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\text{VOLL}\cdot
p^{sh,\omega}[t]-\sum_{i\in\mathcal{I}}C_{i}^{inv}(\tilde{\bm{z}}_{i})-\sum_{i\in\mathcal{I}}C_{i}^{opr}(\tilde{\bm{z}}_{i}).$

Now, we consider the case when $N^{k}\rightarrow+\infty$. Note that the
optimal value $F(\tilde{\bm{z}})$ is lower bounded by the special solution of
no investment, which implies that the solutions of the invested capacity of
any resource will be bounded above. Thus, $\tilde{A}_{k}^{\omega^{\prime}}[t]$
is bounded above. As $N^{k}\rightarrow+\infty$ for any $k\in\mathcal{K}$, we
have
$\sum_{k\in\mathcal{K}}\frac{(\tilde{A}_{k}^{\omega^{\prime}}[t])^{2}}{N^{k}}\rightarrow
0$, which leads to

$\displaystyle~{}~{}~{}~{}F(\tilde{\bm{z}})$ (48)
$\displaystyle=\sum_{t\in\mathcal{T}}\mathbb{E}_{\omega}\Big{[}a^{\omega}[t]\cdot
D^{\omega}[t]\cdot\sum_{k\in\mathcal{K}}\tilde{A}_{k}^{\omega^{\prime}}[t]+b^{\omega}[t]\cdot\sum_{k\in\mathcal{K}}\tilde{A}_{k}^{\omega^{\prime}}[t]\Big{]}$
$\displaystyle-\sum_{t\in\mathcal{T}}\mathbb{E}_{\omega}\Big{[}\frac{1}{2}a^{\omega}[t]\Big{(}\sum_{k\in\mathcal{K}}\tilde{A}_{k}^{\omega^{\prime}}[t]\Big{)}^{2}\Big{]}$
$\displaystyle-\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\text{VOLL}\cdot
p^{sh,\omega}[t]-\sum_{i\in\mathcal{I}}C_{i}^{inv}(\tilde{\bm{z}}_{i})-\sum_{i\in\mathcal{I}}C_{i}^{opr}(\tilde{\bm{z}}_{i}).$

Since
$\sum_{k\in\mathcal{K}}\tilde{A}_{k}^{\omega^{\prime}}[t]=D^{\omega}[t]-p^{cv,\omega}[t]$,
we have

$\displaystyle~{}~{}~{}~{}F(\tilde{\bm{z}})$
$\displaystyle=\sum_{t\in\mathcal{T}}\mathbb{E}_{\omega}\Big{[}\frac{1}{2}a^{\omega}[t]\cdot
D^{\omega}[t]\cdot D^{\omega}[t]+b^{\omega}[t]\cdot D^{\omega}[t]\Big{]}$
$\displaystyle-\sum_{t\in\mathcal{T}}\mathbb{E}_{\omega}\Big{[}\frac{1}{2}a^{\omega}[t](p^{cv,\omega}[t])^{2}+b^{\omega}[t]\cdot
p^{cv,\omega}[t]\Big{]}$
$\displaystyle-\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\text{VOLL}\cdot
p^{sh,\omega}[t]-\sum_{i}C_{i}^{inv}(\tilde{\bm{z}}_{i})-\sum_{i}C_{i}^{opr}(\tilde{\bm{z}}_{i}).$

Thus, $\max F(\tilde{\bm{z}})$ is equivalent to

$\displaystyle\min$
$\displaystyle\sum_{t\in\mathcal{T}}\mathbb{E}_{\omega}\Big{[}\frac{1}{2}a^{\omega}[t](p^{cv,\omega}[t])^{2}+b^{\omega}[t]\cdot
p^{cv,\omega}[t]\Big{]}$
$\displaystyle+\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\text{VOLL}\cdot
p^{sh,\omega}[t]+\sum_{i}C_{i}^{inv}(\tilde{\bm{z}}_{i})+\sum_{i}C_{i}^{opr}(\tilde{\bm{z}}_{i}),$

which is the system cost in Problem SO. The Nash equilibrium derived in
Proposition approaches the social optimum as $N^{k}\rightarrow+\infty$ for any
$k$.∎

## Appendix.VI: Proof of Proposition 5

Proof: As implied in (48) in Appendix.V, if the part
$-\sum_{t\in\mathcal{T}}\mathbb{E}_{\omega}\Big{[}\frac{1}{2}a^{\omega}[t]\sum_{i\in\mathcal{I}}(\tilde{A}_{i}^{\omega}[t])^{2}\Big{]}$
in $F(\tilde{\bm{z}})$ is eliminated, the Nash equilibrium derived in
Proposition 3 will coincide with the social optimum. Thus, we design the
additional supply incentive
$\frac{1}{2}a^{\omega}[t](\tilde{A}_{i}^{\omega}[t])^{2}$ for each time
$t\in\mathcal{T}$ and scenario $\omega\in\Omega$ added to investor $i$’s
profit. ∎

## Appendix.VII: Proof of Proposition 6

Proof: The market price is set by the marginal cost of CERs:

$\displaystyle\frac{dg^{cv,\omega}[t]}{dp^{cv,\omega}[t]}=a^{\omega}[t]\cdot
p^{cv,\omega}[t]+b^{\omega}[t]$ (49)

Adding the price uplift is equivalent to changing $b^{\omega}[t]$ to
$b^{\omega}[t]+\Delta\pi^{\omega}[t]$, which will be reflected in the
corresponding social-optimum result. ∎

## Appendix.VIII: Generalization of transmission network

Our current work assumes no congestion in the network. We focus on the key
insights from the PIU mechanism involving the penalty payment, supply
incentive, and energy price uplift. However, we can partially generalize our
model to incorporate networks with DC power flow, where some key results will
still hold. Next, we will first present the system constraints based on the DC
power flow model. Then, we show how to generalize the game-theoretical model
and the PIU mechanism considering networks.

For the transmission network, we denote by $\Lambda$ the set of transmission
lines, indexed by $(n,m)$, where $n/m$ is sending/receiving end of a
transmission line. We denote by $\mathcal{N}$ the set of grid buses, indexed
by $n$, where $(\cdot)_{n}$ denotes the mapping of $(\cdot)$ into the set of
buses. The notation $\delta_{n}^{\omega}[t]$ denotes voltage angles and
$x_{n,m}$ is transmission line $(n,m)$ reactance. Without loss of generality,
we assume that each LER investor is located on only one bus with only one
resource in the network.

###  -A System constraints and costs

We now update the system constraints and costs based on the DC power flow
model.

For the system constraints, we have

$\displaystyle\sum_{i\in\mathcal{I}_{n}}A_{i}^{\omega}[t]+p_{n}^{cv,\omega}[t]+p_{n}^{sh,\omega}[t]-p_{n}^{inj,\omega}[t]=D_{n}^{\omega}[t],$
$\displaystyle\hskip 86.11084pt\forall n\in\mathcal{N},\forall
t\in\mathcal{T},\forall\omega\in{\Omega},$ (50) $\displaystyle 0\leq
p_{n}^{cv,\omega}[t]\leq\gamma\cdot\bar{p}_{n}^{cv},\forall
n\in\mathcal{N},\forall t\in\mathcal{T},\forall\omega\in{\Omega},$ (51)
$\displaystyle 0\leq p_{n}^{sh,\omega}[t],\forall
t\in\mathcal{T},\forall\omega\in\Omega,$ (52)
$\displaystyle-\overline{F}_{n,m}\leqslant
p_{n}^{inj,\omega}[t]\leqslant\overline{F}_{n,m},\forall
n\in\mathcal{N},\forall(n,m)\in\Lambda,\forall t\in\mathcal{T},$ (53)
$\displaystyle
p_{n}^{inj,\omega}[t]=\sum_{m:(n,m)\in\Lambda}\frac{\delta_{n}^{\omega}[t]-\delta_{m}^{\omega}[t]}{x_{n,m}},\forall
n\in\mathcal{N}.$ (54)

Compared with the initial model with no network, we have additional
constraints (53)-(54) for the transmission power flow. Note that the
additional constraints are linear.

The system costs include the operation costs of lost load and CERs, as well as
the investment and operation costs of VRE and ES. We calculate the expected
cost of lost load for all the buses based on the value of lost load (VOLL):
$G^{sh}(\bm{p}^{sh})=\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\sum_{n\in\mathcal{N}}\text{VOLL}\cdot
p_{n}^{sh,\omega}[t].$ For the operation cost of CERs, we assume a convex
supply-cost function $g_{n}^{cv,\omega}[t](\cdot)$ for CERs at each bus
$n\in\mathcal{N}$. The total expected cost is
$G^{cv}(\bm{p}^{cv})=\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\sum_{n\in\mathcal{N}}g_{n}^{cv,\omega}[t]({p}_{n}^{cv,\omega}[t]).$
The investment and operation costs of VRE and ES are calculated following the
same way as the initial no-network model.

###  -B Social-optimum benchmark

We can easily generalize the social-optimum benchmark based on the system cost
minimization problem. Compared with the case of no network, we replace the
initial system constraints (4)-(6) of the manuscript with the new system
constraints (50)-(54). The system cost is updated accordingly.

###  -C Game-theoretical model

We now generalize the game-theoretical model. We introduce the system operator
as the player in the game. The models for LER investors maintain the same
structure as the initial model of no network.

Besides investors, we assume that the system operator is also a player in the
game, which aims to minimize the system cost given LER investors’ decisions.
Compared with the initial model that does not consider networks, we need the
system operator as an additional player to optimize transmission power flow.
Specifically, the system operator decides the system variables, denoted as
$\bm{z}_{s}$, which includes the CER supply $\bm{p}^{cv}$, the lost load
$\bm{p}^{sh}$, the voltage angle $\bm{\delta}$, and transmission power flow
$\bm{p}^{inj}$. The constraint set of the system operator is denoted as
$\mathcal{Z}_{s}$ based on system constraints (50)-(54). Note that
$\mathcal{Z}_{s}$ can be written as $\mathcal{Z}_{s}(\bm{z}_{-s})$ that is
dependent on all the LER investors’ decisions
$\bm{z}_{-s}=(\bm{z}_{1},\ldots,\bm{z}_{I})$.

We set up the game-theoretical model in the following.

  * •

Players: Investors $i\in\mathcal{I}$ and system operator $s$

  * •

Strategy:

    * –

investor $i\in\mathcal{I}$: $\bm{z}_{i}\in\mathcal{Z}_{i}(\bm{z}_{-i})$

    * –

system operator: $\bm{z}_{s}\in\mathcal{Z}_{s}(\bm{z}_{-s})$

  * •

Payoff:

    * –

Investor $i$: maximize its profit

$\max_{\bm{z}_{i}\in\mathcal{Z}_{i}(\bm{z}_{-i})}f_{i}(\bm{z}_{i},\bm{z}_{-i})$

    * –

System operator $i$: minimize the system cost (given investors’ decisions)

$\min_{\bm{z}_{s}\in\mathcal{Z}_{s}(\bm{z}_{-s})}G^{cv}(\bm{p}^{cv})+G^{sh}(\bm{p}^{sh})$

Next, we generalize the PIU mechanism based on the established game-
theoretical model and analyze the Nash equilibrium.

###  -D PIU mechanism

We will first generalize the price function for each bus. Then, we generalize
the penalty payment, supply incentive, and energy price uplift accordingly
considering the transmission network.

We have the price function for each bus.

$\displaystyle\hat{\pi}_{n}^{\omega}[t](\bm{z})=\left\\{\begin{aligned}
&\frac{dg_{n}^{cv,\omega}[t]}{dp_{n}^{cv,\omega}[t]}\Bigg{|}_{\gamma\cdot\bar{p}_{n}^{cv}},~{}\text{if}~{}\sum_{i\in\mathcal{I}_{n}}A_{i}^{\omega}[t]<D_{n}^{\omega}[t]\\\
&\hskip 94.72192pt+p_{n}^{inj,\omega}[t]-\gamma\cdot\bar{p}_{n}^{cv};\\\
&\frac{dg_{n}^{cv,\omega}[t]}{dp_{n}^{cv,\omega}[t]},~{}~{}~{}~{}~{}\text{if}~{}D_{n}^{\omega}[t]+p_{n}^{inj,\omega}[t]-\gamma\cdot\bar{p}_{n}^{cv}\\\
&\hskip 94.72192pt\leq\sum_{i\in\mathcal{I}_{n}}A_{i}^{\omega}[t]\leq
D_{n}^{\omega}[t].\end{aligned}\right.$

####  -D1 Penalty payment

As we do for the case of no network, we let each investor bear a penalty on
the allocated lost load $p_{i}^{sh,\omega}$. The lost load satisfies the power
balance as follows.

$\displaystyle\sum_{i\in\mathcal{I}_{n}}A_{i}^{\omega}[t]+p_{n}^{cv,\omega}[t]+\sum_{i\in\mathcal{I}_{n}}p_{i}^{sh,\omega}[t]-p_{n}^{inj,\omega}=D_{n}^{\omega}[t],$
$\displaystyle\hskip 86.11084pt\forall n\in\mathcal{N},\forall
t\in\mathcal{T},\forall\omega\in{\Omega}$ (55) $\displaystyle
p_{i}^{sh,\omega}[t]\geq 0,\forall i\in\mathcal{I},\forall
t\in\mathcal{T},\forall\omega\in\Omega.$ (56)

We let $\tilde{A}_{i}^{\omega}[t]={A}_{i}^{\omega}[t]+p_{i}^{sh,\omega}[t]$
and $\tilde{\bm{z}}_{i}=\left(\bm{z}_{i},\bm{p}_{i}^{sh}\right)$. The profit
of investor $i$ is modified to $\tilde{f}_{i}$.

$\displaystyle\tilde{f}_{i}(\tilde{\bm{z}}_{i},\tilde{\bm{z}}_{-i})=f_{i}(\tilde{\bm{z}}_{i},\tilde{\bm{z}}_{-i})-\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\text{VOLL}\cdot
p_{i}^{sh,\omega}[t].$ (57)

We will characterize the Nash equilibrium based on Proposition 8, which is a
modification of Proposition 3.

######  Proposition 8.

One Nash equilibrium coincides with the optimal solution to the following
problem.

$\displaystyle\min$
$\displaystyle\sum_{n\in\mathcal{N}}\sum_{i\in\mathcal{I}_{n}}\Big{(}C_{i}^{inv}(\bm{X}_{i})+C_{i}^{op}(\bm{p}_{i})+G_{i}^{sh}(\bm{p}_{i}^{sh})\Big{)}$
$\displaystyle+\sum_{n\in\mathcal{N}}G_{n}^{cv}(\bm{p}_{n}^{cv})+\sum_{n\in\mathcal{N}}\sum_{i\in\mathcal{I}_{n}}\sum_{t\in\mathcal{T}}\frac{1}{2}\cdot
a_{n}^{\omega}[t]\cdot(\tilde{A}_{i}^{\omega}[t])^{2}$ s.t.
$\displaystyle\tilde{\bm{z}}_{i}\in\tilde{\mathcal{Z}}_{i}^{o},$
$\displaystyle\text{system
constraints}~{}\eqref{eq:app_const_balance_net}-\eqref{eq:app_trans}.$

Proposition 4 in the main text still holds, i.e., the Nash equilibrium
coincides with the social optimum under perfect competition when we have the
penalty mechanism.

####  -D2 Supply incentive

As we do for the case of no network, the supply incentive still has the
structure $\frac{1}{2}a^{\omega}[t].(\tilde{A}_{i}^{\omega}[t])^{2}$. The
profit of investor $i$ is

$\displaystyle\tilde{\tilde{f}}_{i}(\tilde{\bm{z}}_{i},\tilde{\bm{z}}_{-i})=\tilde{f}_{i}(\tilde{\bm{z}}_{i},\tilde{\bm{z}}_{-i})+\mathbb{E}_{\omega}\sum_{t\in\mathcal{T}}\frac{1}{2}a^{\omega}[t](\tilde{A}_{i}^{\omega}[t])^{2}.$
(58)

Proposition 5 still holds, i.e., under imperfect competition, one Nash
equilibrium coincides with the social-optimum solution when we have the
penalty payment and supply incentive.

####  -D3 Energy price uplift

We give an uplift $\Delta\pi_{n}^{\omega}[t]$ for the price of each bus.

$\displaystyle\tilde{\pi}_{n}^{\omega}[t](\bm{z})=\hat{\pi}_{n}^{\omega}[t](\bm{z})+\Delta\pi_{n}^{\omega}[t].$
(59)

Proposition 6 of the main text still holds, where we replace $b^{\omega}[t]$
and $\pi^{\omega}[t]$ with $b_{n}^{\omega}[t]$ and $\pi_{n}^{\omega}[t]$.
Still, the proposition shows that the price uplift will equivalently increase
the cost of the CERs.

Note that we only partially generalize our results for the transmission
network. The propositions not mentioned here may not be extended.