A Tax-Subsidy Scheme for Efficient Investment in Renewable Generation Capacity
3ptaCorresponding author. KTH Royal Institute of Technology, Stockholm,
Sweden. E-mail<EMAIL_ADDRESS>3ptbKTH Royal Institute of Technology,
Stockholm, Sweden. E-mail<EMAIL_ADDRESS>3ptcKTH Royal Institute of
Technology, Stockholm, Sweden. E-mail<EMAIL_ADDRESS>3ptdKTH Royal Institute
of Technology, Stockholm, Sweden. E-mail<EMAIL_ADDRESS>
The impact of energy production significantly affects system sustainability,
which has enabled a shift towards renewable energy sources. Thus, producer
behavior is crucial in electricity markets to achieve sustainability goals. In
this paper, we address two key challenges comprising electricity markets and
generation investment. Firstly, electricity markets typically are operated
with competitive market clearing and merit-order dispatch, which neglects
negative externalities from pollution. A Pigouvian tax is proposed in order to
investigate the impacts of these externalities on electricity prices and
resolve this issue. Secondly, renewable energy sources entail low operational
costs, which result in lower system prices and reduced profits for producers.
Furthermore, producers face high investment costs when moving into renewable
energy resources, which leads to strategic investment decisions. In order to
mitigate this strategic behavior, subsidies are proposed equal to producers’
contribution to consumer surplus. These subsidies incentivize producers to
decrease prices and increase consumer surplus, so, producers would be
motivated to invest in socially optimal generation capacity. Finally, we
demonstrate that implementing the proposed tax and subsidy does not increase
the regulator’s information burden. Renewable energy, electricity generation
capacity, market power, environmental externalities, incentives. D43, D62,
H21, H23, L11, L13, O21, Q41

##  1 Introduction

###  1.1 Motivation

Electricity generation significantly contributes to environmental pollution
Ritchie et al., (2020) and according to the measurements, power plants emit
substantial carbon by consuming fossil fuels. However, global environmental
sustainability goals endeavor to utilize cleaner electricity generation, which
includes those by the United Nations (United Nations,, 2022). For example, in
Britain, the Office of Gas and Electricity Markets (Ofgem) aims for zero
pollution by 2050 (Ofgem,, 2022). Indeed, British power systems are shifting
toward renewable sources (WWEA,, 2018), which reflects broader European
Commission advocacy for market mechanisms to reduce emissions (European
Commission,, 2019). We face a context where most governments have introduced
auctions for Contract-for-Differences (CfDs), Power Purchase Agreements
(PPAs), and capacity payments to ensure efficient investment in renewable
generation capacity. The CfD contracts are used to hedge the risk for
producers against the volatility of carbon prices in the future, as
implemented by California’s cap-and-trade program (Bruninx and Ovaere,, 2022).
On the other hand, PPAs are contracts between an Independent System Operator
(ISO) and an off-taker, which ensure a profitable electricity price for
renewable energy resources for a period of 3 to 5 years. This type of hedge
contract is implemented in some EU countries such as Spain, Germany, and
France (Mesa-Jiménez et al.,, 2023). Finally, capacity payment is another
approach to support clean energy and to leverage sustainable development. In
this scheme, the system operator pays money to each producer based on their
capacity, even if they do not produce electricity; this is implemented by the
UK and Sweden, for instance (Kozlova et al.,, 2023).

Besides these market policies, carbon markets and emission regulations have
also been introduced to accelerate the energy transition in one form or
another. Carbon futures could hedge the risk for participants through this
market. However, given the fact that negative externalities do not have a
straightforward relation to the output level of generation, it might be
challenging to design an optimal hedge contract that would hedge the risk of
participants against the volatility of carbon prices in the carbon market. For
this reason, regulators proposed carbon taxes or carbon prices to penalize
outputs, leading to that renewable resources enjoy several forms of subsidy
support. These new carbon markets and emission regulations, through different
tax-subsidy schemes, motivate reforms in spot-market design. According to the
above motivation, there have been some endeavors among scholars to address
this issue using different approaches.

###  1.2 Literature review

The Independent System Operator (ISO) manages technical aspects and performs
competitive market clearing in liberalized electricity markets. Based on
supply and demand curves, generation prices are determined from merit-order
dispatch considering power-system constraints (Biggar and Hesamzadeh,, 2014).
Constraints regarding negative externalities are not considered in generation
prices in a competitive market. There are two types of policies in order to
address externalities, particularly carbon emissions: cap-and-trade and carbon
tax. These policies aim to reduce carbon emissions. Currently, the carbon
futures market is responsible for more than 90% of total carbon trade,
indicating that participants use this market to hedge their risks (Xi et al.,,
2023). On the other hand, some countries, such as China, have a successful
carbon trade market and are planning to implement a carbon tax scheme (Li et
al.,, 2017). This decision might result from the inefficiency of the cap-and-
trade mechanism in the long-term operation of the electricity market,
particularly in terms of capacity investment. It is shown that if we impose
some price control in cross-border trade, it could easily lead to a reduction
in the investment of renewable energy resources. (Carlos Muñoz et al.,, 2024)
tried to combine a price control mechanism with carbon policies, and they
showed that a carbon tax with a price control mechanism may lead to an
increase in investment in renewable energy resources without the
aforementioned negative impact of the price control mechanism. However,
implementing a carbon tax poses challenges because the total pollution levels
do not directly correlate with individual generation levels, which makes it
difficult to allocate the exact amount of these negative externalities. In
order to overcome these challenges, some carbon tax schemes have been
developed. For instance, carbon taxes (Maryniak et al.,, 2019) or carbon
allowances (Limpaitoon et al.,, 2011; Kunsch et al.,, 2004) are used
separately and can be traded in exclusive markets. ISOs optimize the market
outcomes considering wind or solar generation units EWEA, (2022) or hydro
units in systems with hydro reservoirs (Rand,, 2018). Thus, ISOs could address
externalities as the part of market clearing mechanism by imposing a Pigouvian
tax on producers, effectively resolving this issue.

The ISO can impose taxes on non-renewable energy sources with higher levels of
pollution, which make these energy resources more expensive. This could
resolve the lack of incentive for producers to invest in renewable energy
sources. But, renewable energy sources consume free energy sources such as
wind and solar to produce electricity, leading to decrease market clearing
prices. As a result of reducing profits for producers, some producers
therefore might withhold generation capacity or falsely declare higher costs
to decrease generation, thereby increasing electricity prices and overall
profits (Joskow and Tirole,, 2005) and (Hesamzadeh and Biggar,, 2012). For
example, (Fogelberg and Lazarczyk,, 2019) proposed an algorithm to identify
whether a generator or producer is exercising withholding of generation
levels. Indeed, their algorithm shows whether they reduce their capacity for
economic incentives or technical issues through their validation within the
Swedish electricity market. Authors in (Moiseeva et al.,, 2017) discuss the
generation flexibility and the potential for generators to exercise market
power using their ramp-rate capabilities. In the European Union (European
Commission,, 2019), there are policies in place to force producers to generate
at their full capacity and declare their true costs, which protects consumers
from higher prices. Besides, price caps have also been proposed to mitigate
strategic behavior (Vogelsang and Finsinger,, 1979). (Downward,, 2010) showed
that a carbon price could lead to increased carbon emissions in a small power
system comprising two nodes and two generators. Thus, the scheme of carbon
charges imposed on each producer or generator is very substantial because it
can easily lead to a system with a higher pollution level. Besides, (Kim and
Chang,, 1993) considers the problem of pollution and strategic behavior
together by modeling pollution as a decreasing function of pollution volume.
(Varawala et al.,, 2023) proposed an incentive framework to prevent strategic
behavior related to pollution while considering transmission lines and nodal
pricing in their mechanism.

Similar to strategic generation behavior, the system would suffer from
strategic investment for the same reason and with the same effect, leading to
an increase in electricity prices (Fogelberg and Lazarczyk,, 2019) and (Tohidi
et al.,, 2016). Indeed, shifting from traditional fossil-based power plants
towards renewable energy plants requires high investment costs, which
disincentivizes producers from adopting this generation technology.
Consequently, strategic investments would occur in generation capacity. While
strategic generation behavior can be prevented through policy measures,
strategic investments cannot be prevented by similar policies. Therefore, the
best solution is to turn to incentives. Some researchers have proposed a
market mechanism with subsidies for monopolistic producers to incentivize them
to maximize social welfare without detailing their cost functions (Loeb and
Magat,, 1979). For example, (Milstein et al.,, 2024) proposed a subsidy scheme
based on their two-stage formulation for the investment cost of batteries in
order to ensure that investors invest in the right mixture and amount of
generation in the right locations, which maximizes the social welfare of
participants in the long term. They showed how this subsidy could change the
mixture of generation into carbon-free electricity systems through their
simulation of the Israeli power system. Moreover, some scholars have extended
this approach by using locational spot prices to determine consumer utilities
in electricity markets. However, the entire surplus as a subsidy poses funding
challenges in practice (Gans and King,, 2000). Besides, (Holmberg and
Tangerås,, 2023) showed that allocating the right strategic reserves in the
electricity market, compared to the capacity mechanism, especially for
renewable energy resources, could incentivize producers to invest in carbon-
free energy resources, such as those in Sweden. However, as discussed by
(Gilmore et al.,, 2023), the power system needs to have some conventional
generators in order to meet its demand at all times in the long term. As a
result, they suggested a mixture of renewable energy resources with fuel-based
technology, such as hydrogen, in their optimal mixture of generators. This
mixture would align with zero carbon emissions if hydrogen is generated by
renewable energy resources, which is called green hydrogen.

###  1.3 Contributions

At this research background, the current paper contributes to the relevant
literature as follows: First, it proposes a tax-subsidy scheme for producers
based on their marginal contribution to consumer surplus. The proposed scheme
induces social-welfare decisions and it considers discrete generation
technologies and costs111This is inspired by the HRGV mechanism proposed in
(Hesamzadeh et al.,, 2018) and studied in (Khastieva et al.,, 2020). Second,
the introduced tax-subsidy scheme addresses pollution externalities and
strategic investment in its proposed formulation. It also respects the
operational limits of power systems. Third, a detailed analytical example and
a set of comprehensive numerical experiments are provided to carefully explain
the operation of the tax-subsidy scheme.

The rest of the paper is organized as follows. In Section 2, we present our
model of the power system including buses, transmission lines, generators and
pollution. Then, we compare the socially optimal spot market generation
outcomes, which include consideration of pollution, to the competitive spot
market clearing. Finally, in this section, we derive the tax part of the
scheme which aligns the competitive market clearing with the socially optimal
outcome. In Section 3, we model how investment in generation capacity would
affect the social welfare created in the spot market and compare the
producers’ profit maximizing investment decisions to what is socially optimal.
To align the producers’ profit maximization with the social welfare
maximization, we derive the subsidy part of the scheme. In Section 4, we
present the properties of the tax and subsidy scheme and in Section 5 and
Section 6, we present an analytical example and case study for illustrating
the strength of the proposed scheme. Finally, we conclude in Section 7.

##  2 Generation in wholesale power markets

###  2.1 Socially optimal spot market generation

We consider a wholesale electricity market over a power system with a set
$\mathcal{N}$ of buses connected to each other by a set $\mathcal{L}$ of
transmission lines. The market participants comprise a set $\mathcal{I}$ of
producers forming an oligopoly, along with a large set of consumers modeled as
the aggregated utility functions. Each producer has a set $\mathcal{J}$ of
generating technologies, where units employing each technology may be present
at multiple buses. We use indices $n$, $l$, $i$, and $j$ to refer to
individual buses, lines, producers, and technologies, respectively. For
simplicity, these variables vary over positive integer numbers (e.g.,
$n\in\mathbb{N},1\leq n\leq|\mathcal{N}|$). Additionally, we consider
dynamically varying constraints and the social welfare maximization problem in
the electricity spot market over a planning horizon of $T$ dispatch intervals.
We use index t to represent a single dispatch interval, where
$t\in\mathbb{N},t\leq T$.

Additionally, we demonstrate the generated power output of producer $j$ using
technology j at bus n during time interval $t$ as $q_{tinj}$ and the
corresponding generation capacity as $k_{inj}$. We assume that the generation
capacity remains constant throughout the planning horizon. Specifically, for
renewable energy sources like wind power plants, there exists an _availability
factor_ denoted by $A_{tinj}$, which represents the portion of available
capacity during time interval $t$ based on factors such as wind speed. Thus,
we define the _relative generation_ by $r_{tinj}$ as the fraction of available
capacity utilized for generation. Thus, the output power of these technologies
would be represented as (1).

$q_{tinj}=r_{tinj}A_{tinj}k_{inj}\leftrightarrow\omega_{tinj},\forall
t\in\mathbb{N},t\leq T,n\in\mathcal{N},i\in\mathcal{I},j\in\mathcal{J}$ (1)

where $\omega_{tinj}$ is the associated Lagrangian multiplier. The generation
output must be positive and less than its corresponding capacity, which is
represented by (2).

$0\leq r_{tinj}\leq 1\leftrightarrow(\mu_{tinj},\nu_{tinj}),\forall
t\in\mathbb{N},t\leq T,n\in\mathcal{N},i\in\mathcal{I},j\in\mathcal{J}$ (2)

Where $\mu_{tinj}$ and $\nu_{tinj}$ are the associated Lagrangian multipliers,
each generation technology may also have its ramping limit denoted by
$R_{inj}$. This limit constrains the relative output change between two
neighboring time intervals. We would represent this limit as (3), where
$\rho_{tinj}$ and $\sigma_{tinj}$ are two Lagrangian multipliers associated
with the upper and lower limits of the ramping rate.

$-R_{inj}\leq r_{tinj}-r_{t-1inj}\leq
R_{inj}\leftrightarrow(\rho_{tinj},\sigma_{tinj}),\forall t\in\mathbb{N},2\leq
t\leq T,n\in\mathcal{N},i\in\mathcal{I},j\in\mathcal{J}.$ (3)

Basically, power generation is associated with the cost and pollution for the
producers related to the technology they employed. The generation cost
function and the amount of pollution created by power output at bus $n$ for
producer $i$ with technology $j$ at output level $q_{tinj}$ are represented as
$C_{inj}\left(q_{tinj}\right)$ and $x_{tinj}\left(q_{tinj}\right)$,
respectively. We assume that these two functions are static over the planning
horizon, non-negative, non-decreasing, and convex with respect to $q_{tinj}$.
As a result of these assumptions, these functions must be piece-wise first and
second differentiable in $q_{tinj}$. They exhibit only jump discontinuities.
Also, environmental pollution results in a negative externality, which is a
function of the total pollution at each bus. The negative externality function
at bus $n$ is denoted by222In this paper, we adopt the convention that
omitting any index implies a summation over that index e.g.
$x_{tin}=\sum_{j\in\mathcal{J}}x_{tinj}$. $E_{n}\left(x_{tn}\right)$ and is
assumed to be static over the planning horizon. $E_{n}\left(x_{tn}\right)$ is
non-negative, non-decreasing and convex in $x_{tn}$. Since
$E_{n}\left(x_{tn}\right)$ is non-decreasing and convex, it must be piece-wise
first and second differentiable in $x_{tn}$ with only jump discontinuities.

The power consumption in interval $t$ at bus $n$ is represented as $d_{tn}$,
where it is expected to be positive. We denote this representation as
constraint (4). Correspondingly, the utility generated by a consumer in
interval $t$ at bus $n$ through consumption $d_{tn}$ is denoted by
$\tilde{U}_{tn}\left(d_{tn}\right)$. Notably,
$\tilde{U}_{tn}\left(d_{tn}\right)$ is non-decreasing and concave with respect
to $d_{tn}$. Consequently, $\tilde{U}_{tn}\left(d_{tn}\right)$ can be modeled
as a piece-wise first and second differentiable function in $d_{tn}$,
exhibiting only jump discontinuities.

$0\leq d_{tn}\leq D_{tn}\leftrightarrow(\alpha_{tn},\beta_{tn}),\;\forall
t\in\mathbb{N},t\leq T,\forall n\in\mathcal{N}.$ (4)

According to the power balance equation, the sum of power generated by
producers is equal to the sum of power consumed by consumers in each time
interval $t$. We represent this relationship as equation (5).

$\sum_{n\in\mathcal{N}}d_{tn}-\sum_{n\in\mathcal{N}}q_{tn}=0\leftrightarrow(\lambda_{t}),\;\forall
t\in\mathbb{N},t\leq T.$ (5)

For transmission line $l$ and bus $n$, we determine the flow of the line by
considering the power generated by producers and the power consumed by
consumers, using the power transfer distribution factor matrix ($H_{ln}$) with
a linear expression (Hesamzadeh et al.,, 2014). The flow limitation of line
$l$ during time interval $t$ is denoted by equation (6).

$-F_{l}\leq\sum_{n\in\mathcal{N}}H_{ln}\left(d_{tn}-q_{tn}\right)\leq
F_{l}\leftrightarrow(\tau_{tl},\zeta_{tl}),\forall
t\in\mathbb{N},l\in\mathcal{L}.$ (6)

In this paper, we represent vectors by square brackets with the running index
as a subscript instead of using the set builder notation, e.g.
$\left[q_{tinj}\right]_{tinj}:=\left(q_{tinj}|t\in\mathbb{N},t\leq
T,n\in\mathcal{N},i\in\mathcal{I},j\in\mathcal{J}\right)$. Also, we assume
that the running index runs over the entire set unless a condition is
specified. By considering above constraints and parameters, and we could write
social welfare maximization problem over a planning horizon of $T$ as

$W\left(\left[k_{inj}\right]_{inj}\right)=\\\
\max_{\left[q_{tinj},r_{tinj}\right]_{tinj},\left[d_{tn}\right]_{tn}}\sum_{t=1}^{T}\sum_{n}\left(\tilde{U}_{tn}\left(d_{tn}\right)-\sum_{ij}C_{inj}\left(q_{tinj}\right)-E_{n}\left(\sum_{ij}x_{tinj}\left(q_{tinj}\right)\right)\right)$
(7)

subject to the constraints (1) to (6). In the following, we are trying to
remove dependency of social welfare maximization problem in respect to
$d_{tn}$. First, we may separate the demand $d_{tn}$ out from the social
welfare maximization problem by defining an _aggregate_ consumer utility as

$U_{t}\left(\left[q_{tn}\right]_{tn}\right)=\max_{\left[d_{tn}\right]_{tn}}\sum_{n}\tilde{U}_{tn}\left(d_{tn}\right)\;\forall
t\in\mathbb{N},t\leq T$ (8)

subject to the constraints (4) to (6). According to constraint (5), it is
clear that $\partial d_{t}/\partial q_{tn}=1$,
$\tilde{U}_{tn}\left(d_{tn}\right)$ is non-decreasing and concave in $d_{tn}$,
and $\partial d_{tn^{\prime}}/\partial q_{tn}\geq 0$, thus,
$U_{t}\left(\left[q_{tn}\right]_{tn}\right)$ is non-decreasing and concave
function in $q_{tn}$. By similar deduction, we may deduce that
$U_{t}\left(\left[q_{tn}\right]_{tn}\right)$ would be a piece-wise first and
second differentiable function in $q_{tn}$ with only jump discontinuities.
Also, in order to overcome the jump discontinuities and to allow the
derivative of $\partial U_{t}\left(\left[q_{tn}\right]_{tn}\right)/\partial
q_{tn}$ with respect to $q_{tn^{\prime}}$ to exist everywhere, we have only
considered its right-hand derivative at every $\left[q_{tn}\right]_{tn}$.
Moreover, we will follow this as a convention for all piece-wise
differentiable functions.

By using an _aggregate_ utility function with respect to $q_{tn}$ in (7), the
social welfare maximization problem would be

$W\left(\left[k_{inj}\right]_{inj}\right)=\max_{\left[q_{tinj},r_{tinj}\right]_{tinj}}\sum_{t=1}^{T}\left(U_{t}\left(\left[q_{tn}\right]_{tn}\right)-\sum_{n}\left(\sum_{ij}C_{inj}\left(q_{tinj}\right)+E_{n}\left(x_{tn}\right)\right)\right)$
(9)

subject to the constraints (1) to (3) which do not depend on $d_{tn}$ anymore.

According to the Karush-Kuhn-Tucker (KKT) conditions for the above social
welfare maximization problem, we obtain (10) and (11) equation, which relate
to $r_{tinj}$ and $q_{tinj}$, respectively. In (11), we use the superscript
$*$ to represent the _optimal_ values of the optimization variables in (9) and
the KKT multipliers of the constraints (1) to (3).

$A_{tinj}k_{tinj}\omega_{tinj}^{*}=-\mu_{tinj}^{*}+\nu_{tinj}^{*}-\rho_{tinj}^{*}+\sigma_{tinj}^{*}+\rho_{t+1inj}^{*}-\sigma_{t+1inj}^{*},\\\
\;\forall t\in\mathbb{N},t\leq
T,n\in\mathcal{N},i\in\mathcal{I},j\in\mathcal{J}$ (10)

$\left.\frac{\partial C_{inj}}{\partial
q_{tinj}}\right|_{\left[q_{tinj}\right]_{tinj}=\left[q_{tinj}^{*}\right]_{tinj}}+\omega_{tinj}^{*}=\left.\frac{\partial
U_{t}}{\partial
q_{tinj}}\right|_{\left[q_{tinj}\right]_{tinj}=\left[q_{tinj}^{*}\right]_{tinj}}\rule{-17.07182pt}{0.0pt}-\left.\frac{\partial
E_{n}}{\partial x_{tinj}}\frac{\partial x_{tinj}}{\partial
q_{tinj}}\right|_{\left[q_{tinj}\right]_{tinj}=\left[q_{tinj}^{*}\right]_{tinj}},\\\
\forall t\in\mathbb{N},t\leq
T,n\in\mathcal{N},i\in\mathcal{I},j\in\mathcal{J}$ (11)

It is clear that $r_{tinj}$ and $q_{tinj}$ will depend on
$\left[k_{inj}\right]_{inj}$ and consequently,
$W\left(\left[k_{inj}\right]_{inj}\right)$ therefore depends on
$\left[k_{inj}\right]_{inj}$.

###  2.2 Competitive spot market clearing

The electricity market is managed by the ISO, which determine the market price
at each location and time interval. According to (Varawala et al.,, 2023), the
ISO faces information gaps, particularly regarding the specifics of individual
producers’ technology and pollution outputs that are
$\left[q_{tinj}\right]_{j}$ and $\left[x_{tinj}\right]_{j}$, respectively.
This leads to difficulties in accurately assessing the environmental impact.
_Competitive_ generation levels and pricing are derived from maximizing social
welfare, excluding the externality term. We represent these optimal values by
$q_{tn}^{\dagger}$ and $\omega_{tinj}^{\dagger}$, which obtain from (11) by
eliminating $\frac{\partial E_{n}}{\partial x_{tinj}}\frac{\partial
x_{tn}}{\partial q_{tinj}}$ term. The current paper consistently positions
producer costs on the left side of the equation (11) for clarity. The
generation price at a given location and time interval is determined by the
equation (11) minus $\frac{\partial E_{n}}{\partial x_{tn}}\frac{\partial
x_{tinj}}{\partial q_{tinj}}$ term, which is negative externalities term.
Thus, we obtain price in interval $t$ at bus $n$ by

$P_{tn}^{\dagger}\left(\left[k_{inj}\right]_{inj}\right)=\left.\frac{\partial
U_{t}\left(\left[q_{tn}\right]_{tn}\right)}{\partial
q_{tn}}\right|_{\left[q_{tn}\right]_{tn}=\left[q_{tn}^{\dagger}\right]_{tn}},\forall
t\in\mathbb{N},t\leq T,n\in\mathcal{N}.$ (12)

Since $U_{t}\left(\left[q_{tn}\right]_{tn}\right)$ is non-decreasing in
$q_{tn}$, $P_{tn}^{\dagger}\left(\left[k_{inj}\right]_{inj}\right)$ is also
non-negative. Furthermore, the ISO depends on producers to clearly declare
their capacity and cost functions at bus $n$. Unlike (Varawala et al.,, 2023),
we assume producers cannot act strategically and must truthfully declare these
values, operating as price-takers. Hence, the price in (12) is not influenced
by $\left[q_{tn}^{\dagger}\right]_{n}$. It is clear that the maximum profits
of producer $i$ over the planning horizon of $T$ dispatch intervals in a
Cournot equilibrium is

$\left[\overline{\overline{Y}}_{i}\left(\left[k_{inj}\right]_{inj}\right)\right]_{i}=\max_{\left[q_{tinj},r_{tinj}\right]_{tnj}}\sum_{t=1}^{T}\sum_{n}\left(P_{tn}^{\dagger}\left(\left[k_{inj}\right]_{inj}\right)q_{tinj}-\sum_{j}C_{inj}\left(q_{tinj}\right)\right)$
(13)

subject to the constraints (1) to (3). Therefore, by Karush-Kuhn-Tucker (KKT)
conditions, the producers’ profit maximizing would be written as

$\left.\frac{\partial C_{inj}}{\partial
q_{tinj}}\right|_{q_{tinj}=q_{tinj}^{\dagger}}+\omega_{tinj}^{\dagger}=P_{tn}^{\dagger}\left(\left[k_{inj}\right]_{inj}\right),\forall
t\in\mathbb{N},t\leq T,n\in\mathcal{N},i\in\mathcal{I},j\in\mathcal{J}.$ (14)

which differs from the social welfare maximizing condition (11) except it
lacks the non-negative term $\frac{\partial E_{n}}{\partial
x_{tn}}\frac{\partial x_{tinj}}{\partial q_{tinj}}$, as designed in
$P_{tn}^{\dagger}$. Consequently, as $\frac{\partial C_{inj}}{\partial
q_{tinj}}$ increases with $q_{tinj}$, competitive generation levels are
generally higher or equal to optimal levels, i.e. $q_{tinj}^{\dagger}\geq
q_{tinj}^{*}\;\forall t\in\mathbb{N},t\leq
T,n\in\mathcal{N},i\in\mathcal{I},j\in\mathcal{J}$.

As discussed earlier, $C_{inj}\left(q_{tinj}\right)$ is a convex, piece-wise
second differentiable function in $q_{tinj}$. We consider $k_{inj}$ is large
enough number. Under this condition and considering constraints (1) to (3),
the value of $\mu_{tinj}^{\dagger}$ remains independent of $k_{inj}$ and other
KKT multipliers are zero. So, $q_{tinj}^{\dagger}$ is also independent of
$k_{inj}$. On the other hand, as $k_{inj}$ decreases and becomes small enough,
$q_{tinj}^{\dagger}$ varies linearly respect to $k_{inj}$. So,
$q_{tinj}^{\dagger}$ is continuous and piece-wise differentiable in $k_{inj}$.
Notably, Since the objective function in (13) is continuous and piece-wise
differentiable in $q_{tinj}$, the KKT multipliers of binding constraints and
$\omega_{tinj}^{\dagger}$ from (10) are continuous and piece-wise
differentiable in $k_{inj}$. Thus,
$P_{tn}^{\dagger}\left(\left[k_{inj}\right]_{inj}\right)$ is piece-wise
differentiable in $k_{inj}$.

###  2.3 Proposed tax scheme

In this section, we propose a tax scheme in spot market, which maximizes the
social welfare. Note that we use $*$ in the superscript to denote optimal
values of the optimization variables in the profit maximization under the tax
scheme. Similar to the (12), we can deduce that the price of generation in
interval $t$ at bus $n$ would be modified to (15) for the social welfare
maximizing with generation levels $\left[q_{tinj}^{*}\right]_{tinj}$.

$P_{tn}^{*}\left(\left[k_{inj}\right]_{inj}\right)=\left.\frac{\partial
U_{t}\left(\left[q_{tn}\right]_{tn}\right)}{\partial
q_{tn}}\right|_{\left[q_{tn}\right]_{tn}=\left[q_{tn}^{*}\right]_{tn}},\forall
t\in\mathbb{N},t\leq T,n\in\mathcal{N}$ (15)

It is clear that $P_{tn}^{*}\left(\left[k_{inj}\right]_{inj}\right)$ in this
scheme is non-negative. We assume that the tax imposed in interval $t$ on
producer $i$ is represented by $\phi^{ti}$. Consequently, the maximum profits
of producer $i$ over the planning horizon of $T$ dispatch intervals,
$\left[Y_{i}\right]_{i}\left(\left[k_{inj}\right]_{inj}\right)$, would
decrease by the amount of tax charged as

$\max_{\left[q_{tinj},r_{tinj}\right]_{tnj}}\sum_{t=1}^{T}\sum_{n}\left(P_{tn}^{*}\left(\left[k_{inj}\right]_{inj}\right)q_{tin}-\sum_{j}C_{inj}\left(q_{tinj}\right)-\phi_{ti}\right)$
(16)

subject to the constraints (1) to (3). Similarly, the producers’ profit
maximizing condition is

$\left.\left(\frac{\partial C_{inj}}{\partial
q_{tinj}}+\frac{\partial\phi_{ti}}{\partial
q_{tinj}}\right)\right|_{\left[q_{tinj}\right]_{tinj}=\left[q_{tinj}^{*}\right]_{tinj}}\rule{-19.91684pt}{0.0pt}+\omega_{tinj}^{*}=P_{tn}^{*}\left(\left[k_{inj}\right]_{inj}\right),\forall
t\in\mathbb{N},t\leq T,n\in\mathcal{N},i\in\mathcal{I},j\in\mathcal{J}.$ (17)

By comparing (11) and (17), we easily obtain that the optimal tax would be
feasible in the following equation

$\frac{\partial\phi_{ti}}{\partial q_{tinj}}=\frac{\partial E_{n}}{\partial
x_{tn}}\frac{\partial x_{tinj}}{\partial q_{tinj}},\;\forall
t\in\mathbb{N},t\leq T,n\in\mathcal{N},i\in\mathcal{I},j\in\mathcal{J}.$ (18)

By integrating (18) over the box starting from zero for all $q_{ti}$ to the
$\left[q_{ti}\right]_{ti}$, the producer $i$’s tax is

$\phi_{ti}-\left.\phi_{ti}\right|_{\left[q_{tinj}\right]_{nj}=\left[0\right]_{nj}}=\sum_{n}\left(E_{n}\left(x_{tn}\right)-E_{n}\left(x_{tn}-x_{tin}\right)\right),\forall
t\in\mathbb{N},t\leq T,i\in\mathcal{I}.$ (19)

It is worth mentioning that the producer $i$’s tax in interval $t$ is its
marginal contribution to the negative externality due to pollution, i.e., it
is a so-called _Pigouvian tax_ (Sandmo,, 2016). By adding the tax to the
producers’ profit maximizing condition, it is changed into (17).

$\left.\frac{\partial C_{inj}}{\partial
q_{tinj}}\right|_{q_{tinj}=q_{tinj}^{*}}+\omega_{tinj}^{*}+\left.\frac{\partial
E_{n}}{\partial x_{tn}}\frac{\partial
x_{tinj}}{q_{tinj}}\right|_{q_{tinj}=q_{tinj}^{*}}=P_{tn}^{*}\left(\left[k_{inj}\right]_{inj}\right),\\\
\;\forall t\in\mathbb{N},t\leq
T,n\in\mathcal{N},i\in\mathcal{I},j\in\mathcal{J}.$ (20)

Observe that the term $\frac{\partial E_{n}}{\partial x_{tn}}\frac{\partial
x_{tinj}}{\partial q_{tinj}}$ is now on the left hand side of the equation
since, due to the tax, it would be a cost to the generator. While it may
appear that only producers are paying the tax, higher prices also reduce
consumer surplus because producers control the taxed pollution source which
they pay.

##  3 Generation capacity investment in wholesale power markets

###  3.1 Socially optimal generation capacity investment

In this section, we try to formulate the problem of socially optimal
generation capacity investment. We suppose that the existing capacity at bus
$n$ of producer $i$’s technology $j$ is $K_{inj}$ and the incremental capacity
in generation is $\Delta k_{inj}$. It is clear that $\Delta k_{inj}$ must be
positive as

$\Delta k_{inj}\geq 0\leftrightarrow\tau_{inj}\;\forall
n\in\mathcal{N},i\in\mathcal{I},j\in\mathcal{J}$ (21)

and also the capacity at bus $n$ of producer $i$’s technology $j$ would be the
sum of the existing and the increase as

$k_{inj}=K_{inj}+\Delta k_{inj}.$ (22)

Increasing generation capacity incurs an investment cost linked to the
technology used. We denoted it as $\mathfrak{C}_{inj}\left(\Delta
k_{inj}\right)$, which associates to the investment cost at bus $n$ for
producer $i$ using technology $j$. Note that, this function is non-negative,
non-decreasing, and convex in $\Delta k_{inj}$. This cost function should be
piece-wise first and second differentiable with jump discontinuities. Now by
considering social welfare maximization over the investment timescale, it
involves summing the social welfare generated in spot market intervals minus
investment costs, it could be shown as

$\mathfrak{W}=\max_{\left[\Delta
k_{inj}\right]_{inj}}W\left(\left[k_{inj}\right]_{inj}\right)-\sum_{nij}\mathfrak{C}_{inj}\left(\Delta
k_{inj}\right)$ (23)

subject to the constraint defined by equation (21) and the relationship
provided in equation (22). Accordingly, given the modified price in (15), the
equation representing the condition for maximizing social welfare is given by
(24).

$\left.\frac{\partial\mathfrak{C}_{inj}}{\partial\Delta
k_{inj}}\right|_{\Delta k_{inj}=\Delta
k_{inj}^{*}}+\tau_{inj}^{*}=\left.\frac{\partial W}{\partial
k_{inj}}\right|_{\left[k_{inj}\right]_{i}=\left[k_{inj}^{*}\right]_{i}}$ (24)

which results in

$\left.\left(\sum_{t=1}^{T}\left.\left(\frac{\partial C_{inj}}{\partial
q_{tinj}}+\frac{\partial E_{n}}{\partial x_{tn}}\frac{\partial
x_{tinj}}{\partial
q_{tinj}}\right)\right|_{\left[q_{tinj}\right]_{inj}=\left[q_{tinj}^{*}\right]_{inj}}\frac{\partial
q_{tinj}^{*}}{\partial
k_{inj}}+\frac{\partial\mathfrak{C}_{inj}}{\partial\Delta
k_{inj}}\right)\right|_{\left[\Delta k_{inj}\right]_{i}=\left[\Delta
k_{inj}^{*}\right]_{i}}+\tau_{inj}^{*}\\\
=\sum_{t=1}^{T}\left.P_{tn}^{*}\frac{\partial q_{tinj}^{*}}{\partial
k_{inj}}\right|_{\left[\Delta k_{inj}\right]_{inj}=\left[\Delta
k_{inj}^{*}\right]_{inj}},\forall
n\in\mathcal{N},i\in\mathcal{I},j\in\mathcal{J}.\vspace{-0.3cm}$ (25)

###  3.2 Strategic generation capacity investment

Producers may manipulate spot market prices by declaring higher generation
costs or withholding capacity, which leads to reduced generation levels and
increased prices (Varawala et al.,, 2023), (Fogelberg and Lazarczyk,, 2019).
In Section 2.2, we assumed producers do not engage in such behavior, as a
result generation level would be competitive generation levels. This
assumption is reasonable due to policies preventing strategic behavior in
critical infrastructures like power systems (European Union,, 2022). However,
while the modified price with considering pollution in (15) does not depend on
$\left[q_{tn}^{\dagger}\right]_{tn}$, it can be changed by
$\left[k_{inj}\right]_{inj}$. In a Cournot equilibrium, the maximum profit is
achieved by all producers by (26).

$\left[\overline{\mathcal{Y}}_{i}\right]_{i}=\left[\max_{\left[\Delta
k_{inj}\right]_{nj}}{\overline{Y}}_{i}\left(\left[k_{inj}\right]_{inj}\right)-\sum_{nj}\mathfrak{C}_{inj}\left(\Delta
k_{inj}\right)\right]_{i}$ (26)

subject to the constraint (21) given (22). Thus , from (16) given the tax in
(19) the producers’ profit-maximizing condition over the investment period in
a Cournot equilibrium is

$\left.\frac{\partial\mathfrak{C}_{inj}}{\partial\Delta
k_{inj}}\right|_{\Delta k_{inj}=\Delta
k_{inj}^{\\#}}+\tau_{inj}^{\\#}=\left.\frac{\partial Y_{i}}{\partial
k_{inj}}\right|_{\left[k_{inj}\right]_{i}=\left[k_{inj}^{\\#}\right]_{i}}$
(27)

which results in

$\left.\left(\sum_{t=1}^{T}\left.\left(\frac{\partial C_{inj}}{\partial
q_{tinj}}+\frac{\partial E_{n}}{\partial x_{tn}}\frac{\partial
x_{tinj}}{\partial
q_{tinj}}\right)\right|_{\left[q_{tinj}\right]_{inj}=\left[q_{tinj}^{*}\right]_{inj}}\frac{\partial
q_{tinj}^{*}}{\partial
k_{inj}}+\frac{\partial\mathfrak{C}_{inj}}{\partial\Delta
k_{inj}}\right)\right|_{\left[\Delta k_{inj}\right]_{i}=\left[\Delta
k_{inj}^{\\#}\right]_{i}}+\tau_{inj}^{\\#}\\\
=\sum_{t=1}^{T}\left.\left(\sum_{n^{\prime}}\frac{\partial
P_{tn^{\prime}}^{*}}{\partial
k_{inj}}q_{tin^{\prime}}^{*}+P_{tn}^{*}\frac{\partial q_{tinj}^{*}}{\partial
k_{inj}}\right)\right|_{\left[\Delta k_{inj}\right]_{inj}=\left[\Delta
k_{inj}^{\\#}\right]_{inj}},\forall
n\in\mathcal{N},i\in\mathcal{I},j\in\mathcal{J}.$ (28)

The $\\#$ superscript denotes the _strategic_ values of optimization variables
in producers’ profit maximization (26) and the KKT multiplier of the
investment constraint (21). Although the social welfare maximization condition
is (25), this condition includes the additional term
$\sum_{n^{\prime}}\frac{\partial{P_{t}n^{\prime}}^{*}}{\partial
k_{inj}}q_{tn^{\prime}}^{*}$ compared to optimal investment problem. Hence,
strategic capacity increases differ from optimal ones, i.e. $\Delta
k_{inj}^{\\#}\neq\Delta k_{inj}^{*}\;\forall
n\in\mathcal{N},i\in\mathcal{I},j\in\mathcal{J}$.

###  3.3 Proposed subsidy scheme

In this section, we propose a subsidy scheme aligning producers’ profit
maximization with social welfare over the investment period, with subsidies
provided in the spot market. Denoting the subsidy to producer $i$ in interval
$t$ as $\chi^{ti}$, the maximum profits of all producers in a Cournot
equilibrium increase by the subsidy amount. Producers’ maximum profit over the
investment period becomes the same maximization problem with greater objective
function in comparison with problem proposed in Section 2.3. Consequently, the
producers’ profit-maximizing condition over the investment period in a Cournot
equilibrium is:

$\left.\left(\sum_{t=1}^{T}\left.\left(\frac{\partial C_{inj}}{\partial
q_{tinj}}+\frac{\partial E_{n}}{\partial x_{tn}}\frac{\partial
x_{tinj}}{\partial q_{tinj}}-\frac{\partial\chi_{ti}}{\partial
q_{tinj}}\right)\right|_{\left[q_{tinj}\right]_{inj}=\left[q_{tinj}^{*}\right]_{inj}}\frac{\partial
q_{tinj}^{*}}{\partial
k_{inj}}+\frac{\partial\mathfrak{C}_{inj}}{\partial\Delta
k_{inj}}\right)\right|_{\left[\Delta k_{inj}\right]_{i}=\left[\Delta
k_{inj}^{*}\right]_{i}}+\tau_{inj}^{*}\\\
=\sum_{t=1}^{T}\left.\left(\sum_{n^{\prime}}\frac{\partial
P_{tn^{\prime}}^{*}}{\partial
k_{inj}}q_{tin^{\prime}}^{*}+P_{tn}^{*}\frac{\partial q_{tinj}^{*}}{\partial
k_{inj}}\right)\right|_{\left[\Delta k_{inj}\right]_{inj}=\left[\Delta
k_{inj}^{*}\right]_{inj}},\forall
n\in\mathcal{N},i\in\mathcal{I},j\in\mathcal{J}.$ (29)

Here, the $*$ superscript denotes expected profit maximization under the
subsidy scheme. With the same logic as tax scheme, comparing (29) to the
social welfare maximization condition (25), we design the subsidy so that the
conditions align with the following condition.

$\left.\frac{\partial\chi_{ti}}{\partial
q_{tinj}}\right|_{\left[q_{tinj}\right]_{inj}=\left[q_{tinj}^{*}\right]_{inj}}\frac{\partial
q_{tinj}^{*}}{\partial k_{inj}}=-\sum_{n^{\prime}}\frac{\partial
P_{tn^{\prime}}^{*}}{\partial k_{inj}}q_{tin^{\prime}}^{*},\forall
t\in\mathbb{N},t\leq T,n\in\mathcal{N},i\in\mathcal{I},j\in\mathcal{J}.$ (30)

The subsidy must account for the additional term in comparison to (15):

$\left.\frac{\partial\chi_{ti}}{\partial
q_{tinj}}\right|_{\left[q_{tinj}\right]_{inj}=\left[q_{tinj}^{*}\right]_{inj}}\frac{\partial
q_{tinj}^{*}}{\partial k_{inj}}=\sum_{n^{\prime}}\left(\frac{\partial
U_{t}}{\partial q_{tn^{\prime}}}\frac{\partial q_{tin^{\prime}j}^{*}}{\partial
k_{inj}}-\frac{\partial\left(P_{tn^{\prime}}^{*}q_{tin^{\prime}j}^{*}\right)}{\partial
k_{inj}}\right),\\\ \;\forall t\in\mathbb{N},t\leq
T,n\in\mathcal{N},i\in\mathcal{I},j\in\mathcal{J}.$ (31)

Upon integration with respect to $k_{inj}$, iterating over $j$ values, and
finally, over $n$ values, we obtain:

$\chi_{ti}-\left.\chi_{ti}\right|_{\left[q_{tinj}\right]_{nj}=\left[0\right]_{nj}}=\sum_{n}\left(U_{t}\left(\left[q_{tn}\right]_{n}\right)-U_{t}\left(\left[q_{tn}-q_{tin}\right]_{n}\right)-P_{tn}^{*}q_{tin}\right)\;\forall
t\in\mathbb{N},t\leq T,i\in\mathcal{I}.$ (32)

Notice that the subsidy in interval $t$ equals the producer’s marginal
contribution to consumer surplus. From updated
$\left[Y_{i}\right]_{i}\left(\left[k_{inj}\right]_{inj}\right)$ with subsidy,
the producers’ maximum profit, including the tax and subsidy is:

$\max_{\left[q_{tinj},r_{tinj}\right]_{tnj}}\sum_{t=1}^{T}\left(U_{t}\left(\left[q_{tn}\right]_{n}\right)-U_{t}\left(\left[q_{tn}-q_{tin}\right]_{n}\right)\right.\\\
\left.-\sum_{n}\left(\sum_{j}C_{inj}\left(q_{tinj}\right)+E_{n}\left(x_{tn}\right)-E_{n}\left(x_{tn}-x_{tin}\right)\right)-\left.\phi_{ti}\right|_{\left[q_{tinj}\right]_{nj}=\left[0\right]_{nj}}+\left.\chi_{ti}\right|_{\left[q_{tinj}\right]_{nj}=\left[0\right]_{nj}}\right)$
(33)

subject to constraints (1) to (3). Notice that all terms in producer $i$’s
decisions $\left[q_{tinj}\right]_{nj}$ are contained in the maximum social
welfare problem in (9). This ensures that producers maximize social welfare
under this scheme. In comparison with (Varawala et al.,, 2023), we do not
allow false declaration of generation cost function. Yet, both our tax and
subsidy schemes are equivalent to theirs. This prevents the potential for
producers to exercise strategic behavior through capacity withholding.

##  4 Properties of the proposed scheme

Based on
$\left.\phi^{ti}\right|_{\left[q_{nj}^{ti}\right]{nj}=\left[0\right]{nj}}$ and
$\left.\chi^{ti}\right|_{\left[q_{nj}^{ti}\right]{nj}=\left[0\right]{nj}}$,
the proposed scheme exhibits the following properties:

  * •

Individually Rational:

If the social welfare in interval $t$ does not decrease upon producer $i$’s
participation, then the proposed scheme could ensure the individual
rationality of producers. Producer $i$’s revenue would be adequate if:

$\left.\left(\chi_{ti}-\phi_{ti}\right)\right|_{\left[q_{tinj}\right]_{nj}=\left[0\right]_{nj}}\geq
U_{t}\left(\left[q_{tn}-q_{tin}\right]_{n}\right)-U_{t}\left(\left[q_{tn}\right]_{n}\right)\\\
\rule{-11.38092pt}{0.0pt}+\sum_{n}\left(\sum_{j}C_{inj}\left(q_{tinj}\right)+E_{n}\left(x_{tn}\right)-E_{n}\left(x_{tn}-x_{tin}\right)\right).$
(34)

Choosing
$\left.\left(\chi_{ti}-\phi_{ti}\right)\right|_{\left[q_{tinj}\right]{nj}=\left[0\right]{nj}}\geq
0$ satisfies this condition for all possible values of
$\left[q_{tinj}\right]_{inj}$ because of
$U_{t}\left(\left[q_{tn}\right]_{n}\right)$, $E_{n}\left(x_{tn}\right)$, and
$x_{tn}\left(\left[q_{tinj}\right]_{tn}\right)$ are non-decreasing and
$C_{inj}\left(q_{tinj}\right)$ is non-negative in term of their variables.

  * •

Robust to Information Asymmetry:

The scheme is designed to depend only on aggregate values $q_{tin}$ and
$x_{tin}$, i.e.
$\left.\phi_{ti}\right|_{\left[q_{tinj}\right]_{nj}=\left[0\right]_{nj}}$ and
$\left.\chi_{ti}\right|_{\left[q_{tinj}\right]_{nj}=\left[0\right]_{nj}}$,
which makes the proposed scheme to be robust to information asymmetry.

  * •

Non-discriminatory:

If

$\left.\phi_{ti}\right|_{\left[q_{tinj}\right]{nj}=\left[0\right]{nj}}\equiv\left.\phi_{t}\right|_{\left[q_{tinj}\right]{nj}=\left[0\right]{nj}}$
$\left.\chi_{ti}\right|_{\left[q_{tinj}\right]{nj}=\left[0\right]{nj}}\equiv\left.\chi_{t}\right|_{\left[q_{tinj}\right]{nj}=\left[0\right]{nj}}$

for every producer $i$, the functional forms for $\phi_{ti}$ and $\chi_{ti}$
would be the same for every producer. We may also choose
$\left.\phi_{ti}\right|_{\left[q_{tinj}\right]_{nj}=\left[0\right]_{nj}}$ and
$\left.\chi_{ti}\right|_{\left[q_{tinj}\right]_{nj}=\left[0\right]_{nj}}$ to
satisfy all the two conditions above simultaneously.

  * •

Incentive Compatible with Respect to Direct Costs:

The proposed scheme ensures that producers declare their costs and parameters
accurately, e.g. generation and investment costs, generation capacity, ramping
limits, and availability factor. Also, this scheme aligns profit maximization
with social welfare maximization, which leads to a burden on the data utilized
by the ISO. It is worth mentioning that ramp limits, $\rho_{tinj}$ and
$\sigma_{tinj}$, would affect producers’ cost declarations and pollution
externality, $E_{n}\left(x_{tn}\right)$, via a tax, also change profit
maximization problem (20). On the other hand, decreasing pollution externality
effects can increase the probability of manipulating pollution levels. Thus,
ISOs must measure pollution levels in this condition. Additionally, if the ISO
incorporates pollution in the market clearing price (20), producers would
strategically increase $E_{n}\left(x_{tn}\right)$ or manipulate the price (20)
and profits (13), which differs from the goal of maximizing social welfare.

  * •

Not Budget Balanced in General:

If the scheme results in a net subsidy in any interval $t$, i.e.,
$\sum_{i\in\mathcal{I}}\left(\chi_{ti}-\phi_{ti}\right)\geq 0$, the ISO may
suffer from funding issues, that it can be resolved by adjusting fixed fees
for market participants. Additionally, for every producer $i$,
$\left.\left(\phi_{ti}-\chi_{ti}\right)\right|_{\left[q_{tinj}\right]_{nj}=\left[0\right]_{nj}}$
can be considered as a fixed fee. However, according to the (34), we need to
set it to a reasonable number in order for the scheme to have individual
rationality.

  * •

Producers’ Profits Independent of Price:

From (33), producers’ profits under the scheme are independent of the price,
which makes it compatible with price caps and overcoming their drawbacks.

##  5 Analytical Example

In this section, we demonstrate the developed theory through a practical
example. First of all, we examine the spot market timescale, comparing optimal
spot market generation with competitive market clearing. We consider a
simplified scenario of a one-bus system, denoted as $\mathcal{N}=\\{1\\}$,
with two producers $\mathcal{I}=\\{1,2\\}$ and one technology
$\mathcal{J}=\\{1\\}$ over a planning horizon of $T=3$. Since there are no
transmission lines in a one-bus system, therefore, there are no constraints on
power flow capacities (6). Table 1 shows the generation cost function,
pollution factors, and existing generation capacities for each producer.
Firstly, we neglected investment in generation capacity, which means the
generation capacities are maintained at their existing levels. Additionally,
we set $A_{ti11}=1,R_{i11}=1,\;\forall t\in\\{1,2,3\\},i\in\\{1,2\\}$ to
simplify generation capacity constraints and to eliminate ramping constraints.
By this assumption, the relative generation $r_{ti11}$ is eliminated by
merging constraints (1) and (2) into $0\leq q_{ti11}\leq k_{i11}\;\forall
t\in\\{1,2,3\\},i\in\\{1,2\\}$. Moreover, the negative externality function
that represents pollution is expressed as $E_{1}\left(x_{t1}\right)=x_{t1}$.
Finally, consumer utility is defined as
$\tilde{U}_{t1}\left(dV\right)=c_{t}d_{t1}-d_{t1}^{2}/2\;\forall
t\in\\{1,2,3\\}$, where $c_{1}=6,c_{2}=12,c_{3}=20$.

Table 1: Generation Cost Function, Pollution, Generation Capacity and Investment Cost Function $i$ | $C_{i11}\left(q_{ti11}\right)$ | $x_{ti11}\left(q_{ti11}\right)$ | $K_{i11}$ | $\mathfrak{C}_{i11}\left(\Delta k_{i11}\right)$
---|---|---|---|---
|  | $\forall t\in\\{1,2,3\\}$ |  |
1 | $2q_{t111}$ | $4q_{t111}$ | 4 | $9\Delta k_{111}$
2 | $4q_{t211}$ | 0 | 3 | $9\Delta k_{211}$

From the power balance constraint (5), $d_{t1}=q_{t1}\geq 0,\;\forall
t\in\\{1,2,3\\}$. Thus, the aggregate consumer utility would be
$U_{t1}\left(q_{t1}\right)=\tilde{U}_{t1}\left(q_{t1}\right),\;\forall
t\in\\{1,2,3\\}$.

Figure 1: Optimal Spot Market Generation

(a) Interval $t=1$

(b) Interval $t=2$

(c) Interval $t=3$

Figure 1 represents the optimal spot market generation for every interval. It
is clear that the marginal utility is $c_{t}-q_{t1}\;\forall t\in\\{1,2,3\\}$,
which represents a solid blue line in all the figures.

Also, the marginal costs including the pollution damage for producers $i=1$
and $i=2$ are $6+\omega_{t111}^{*}$ and $4+\omega_{t211}^{*}$ ($\forall
t\in\\{1,2,3\\}$), respectively. Considering costs within each generator’s
capacity limits, i.e., $0<q_{ti11}<k_{i11}$, the associated KKT multipliers
are $\omega_{ti11}^{*}=0;\forall t\in\\{1,2,3\\},i\in\\{1,2\\}$. According to
the social welfare maximization in (9), which leads to a least-cost dispatch,
producer $i=2$ is prioritized for generation due to lower costs, with $i=1$
generating only if $i=2$ has reached maximum capacity. The marginal cost
curves for $i=2$ and $i=1$ exhibit jumps at $q_{t1}=k_{211}$ and
$q_{t1}=k_{211}+k_{111}$, respectively, which characterized by the KKT
multipliers $\omega_{t211}^{*}$ and $\omega_{t111}^{*}$. Pollution damage is
feasible to depict individually for each producer since the marginal pollution
damage $\partial E_{1}/\partial x_{t1}$ is independent of $\partial x_{t1}$,
which is represented by the green shaded region. Optimal generation,
determined at the intersection of the marginal cost and utility curves, gives
maximum social welfare, depicted by the lightly shaded region. The detailed
information comprising total generation, pollution damage, and maximum social
welfare in each interval is summarized in Table 2.

Table 2: Comparison of Optimal Spot Market Generation and Competitive Market Clearing (SW - Maximum Social Welfare) | $t$ | $q_{t111}$ | $q_{t211}$ | $q_{t11}$ | $P_{t1}$ | $E_{1}\left(x_{t1}\right)$ | SW | $\phi_{t1}-\left.\phi_{t1}\right|_{q_{t111}=0}$
---|---|---|---|---|---|---|---|---
| 1 | 0 | 2 | 2 | 4 | 0 | 2 | 0
Optimal | 2 | 3 | 3 | 6 | 6 | 12 | 24 | 12
| 3 | 4 | 3 | 7 | 13 | 16 | 79.5 | 16
| 1 | 4 | 0 | 4 | 2 | 16 | -8 | -
Competitive | 2 | 4 | 3 | 7 | 5 | 16 | 23.5 | -
| 3 | 4 | 3 | 7 | 13 | 16 | 79.5 | -
Figure 2: Competitive Spot Market Clearing

(a) Interval $t=1$

(b) Interval $t=2$

(c) Interval $t=3$

Figure 2 illustrates competitive market clearing where pollution damage costs
are ignored, consequently, dispatch relies only on marginal generation cost.
For producers $i=1$ and $i=2$, marginal costs are $\frac{\partial
C_{i11}}{\partial q_{ti11}}+\omega_{ti11}^{\dagger},\;\forall
i\in\\{1,2\\},\forall t\in\\{1,2,3\\}$. Thus, producer $i=1$ is prioritized in
this case which determines the marginal cost curve for the system and it is as
shown in the figures with the dotted line representing producer $i=1$ where
the jump at $q_{t1}=k_{111}$ from 2 to 4 is characterized by the KKT
multiplier $\omega_{t111}^{\dagger}$ and the dashed line representing producer
$i=2$ where the jump at $q_{t1}=k_{211}$ is characterized by the KKT
multiplier $\omega_{t211}^{\dagger}$. Moreover, pollution damage is shaded
darkly in each time step. Similar to the optimal market, optimal generation is
determined at the intersection of marginal cost and utility curves. The social
welfare is in the lightly shaded region, with price $P_{t1}^{\dagger}$ as a
solid line. It is worth mentioning that social welfare is negative for $t=1$
and therefore cannot be represented. The total generation, prices, pollution
damage, and social welfare are summarized in Table 2.

The competitive generation and pollution damage are not less than optimal:
$q_{t11}^{\dagger}\geq q_{t11}^{*}$ and
$E_{1}\left(x_{t1}^{\dagger}\right)\geq E_{1}\left(x_{t1}^{*}\right)\;\forall
t\in\\{1,2,3\\}$. However, the competitive social welfare is not greater than
optimal. The tax levied on producers $i=1$ and $i=2$ to align their profit
maximization with the social welfare optimization is obtained by (19) and they
are represented in Table 2, which is the same as the pollution damage in
Figure 1. Also, the price following the tax from (15) aligns with optimal spot
market generation, labeled as _Optimal_. Price $P_{t1}^{*}$ is solid-lined in
Figure 1 and the price is not lower than the one under competitive market
clearing.

According to the information presented in Table 1, both producers have the
same marginal investment cost and are linear functions. Since producer $i=2$
has a lower marginal cost including pollution damage, only this producer
increases generation capacity, i.e. $k_{111}=K_{111}=4$.

Figure 3: Optimal Generation Capacity Investment

(a) Interval $t=1$

(b) Interval $t=2$

(c) Interval $t=3$

In order to compute the optimal generation capacity investment, we first
represent the maximum social welfare from the spot market as a function of
generation capacity $k_{111}$. Social welfare is the area between marginal
utility and marginal cost curves. Figure 1 and Table 3 provide generation
levels for each producer and interval prices.

Table 3: Generation Levels and Price as Functions of the Generation Capacity $k_{11}^{1}$ | $q_{t111}$ | $q_{t211}$ | $q_{t11}$ | $P_{t1}^{*}\left(K_{111},k_{211}\right)$
---|---|---|---|---
$t=1$ | 0 | 2 | 2 | 4
$t=2$ | $6-k_{211}$ | $k_{211}$ | 6 | 6
$t=3$ | 4 | $k_{211}$ | $4+k_{211}$ | $16-k_{211}$

With $\Delta k_{211}$ small enough, the marginal utility curve intersects the
marginal cost curve as in Figure 1. So, the total social welfare in the spot
market is $W\left(K_{111},k_{211}\right)=68+14\times k_{211}-0.5\times
k_{211}^{2}$. The optimal increase in generation capacity from the social
welfare maximization condition in (24) is $\Delta k_{211}^{*}=2$.

Figure 3 represents market clearing prices from optimal investment in
capacity, which indicates marginal utility, marginal cost, social welfare, and
price. The striped region below the price line represents producer $i=2$’s
profit from (16) after tax. Table 4 tabulates each producer’s generation,
price from (15), tax for producer $i=1$, maximum profit of producer $i=2$, and
maximum social welfare in the spot market.

Table 4: Comparison of Optimal and Strategic Generation Investment (PS2 \- Maximum Profit of Producer $i=2$, SW - Maximum Social Welfare in Spot Market) | $t$ | $q_{t111}$ | $q_{t211}$ | $q_{t11}$ | $P_{t1}^{*}\left(K_{111},k_{211}\right)$ | $\phi_{t1}$ | PS2 | SW | $\chi_{t1}$
---|---|---|---|---|---|---|---|---|---
| 1 | 0 | 2 | 2 | 4 | 0 | 0 | 2 | 2
Optimal | 2 | 1 | 5 | 6 | 6 | 4 | 10 | 28 | 12.5
| 3 | 4 | 5 | 9 | 11 | 16 | 10 | 95.5 | 12.5
| 1 | 0 | 2 | 2 | 4 | 0 | 0 | 2 | -
Strategic | 2 | 3 | 3 | 6 | 6 | 12 | 12 | 24 | -
| 3 | 4 | 3 | 7 | 13 | 16 | 16 | 79.5 | -

Now, we are going to determine the strategic generation investment by producer
$i=2$. Firstly, we represent producer 2’s profit as a function of generation
capacity $k_{211}$, which is the area between the price line and segments of
the marginal cost curve for producer $i=2$. Table 3 provides generation levels
for each producer and interval prices. The total profit in the spot market for
producer $i=2$ and the profit maximization condition are

$\displaystyle Y_{2}\left(K_{111},k_{211}\right)$ $\displaystyle=14\times
k_{211}-k_{211}^{2}.$ (35) $9+\tau_{111}^{\\#}=14-2\times\left(K_{211}+\Delta
k_{211}\right).$ (36)

Given $\Delta k_{211}\geq 0$ and convex optimization, strategic increase in
generation capacity is $\Delta k_{211}^{\\#}=0$ with KKT multiplier
$\tau_{111}^{\\#}=-1$. Strategically, an increase in generation is not equal
to optimal increase, i.e. $\Delta k_{211}^{\\#}\neq\Delta k_{211}^{*}$. Table
4 lists each producer’s generation, price from (15), tax for producer $i=1$,
maximum profit of producer $i=2$, and maximum social welfare in the spot
market. The subsidy granted to producer $i=2$ is

$\chi_{t1}\left(q_{t211}\right)=\begin{cases}2+\left.\chi_{t1}\right|_{q_{1211}=0}&\text{if}\;t=1,\\\
0.5\left({k_{211}^{*}}\right)^{2}+\left.\chi_{t1}\right|_{q_{2211}=0}&\text{if}\;t=2,\\\
0.5\left({k_{211}^{*}}\right)^{2}+\left.\chi_{t1}\right|_{q_{3211}=0}&\text{if}\;t=3.\\\
\end{cases}$ (37)

With the subsidy amount based on corresponding generation levels, the optimal
capacity investment occurs. Table 4 indicates producer $i=2$’s variable
subsidy portion $\chi_{t2}-\left.\chi_{t2}\right|_{q_{t211}=0}$ for each
interval.

##  6 Numerical Experiments

In this section, we evaluate the effectiveness of the proposed tax-subsidy
scheme on the IEEE 24-bus test system, which has 33 generators and 38 lines.
The system comprises ten producers utilizing six types of technologies
including renewable, hydro-power, nuclear, coal, fuel oil, and gas, which are
distinguished by their cost function values. Before investment, the total
installed generation capacity and load consumption in the first year are equal
to 3405 MW and 2850 MW, respectively. Also, the cost functions for generators
and utility functions for loads are linearized using piece-wise linear
functions consisting of ten segments each. Similarly, the externality cost
function is assumed to be linear, with the coefficient of the externality
function for each technology type comprised of renewable, hydro-power,
nuclear, coal, fuel oil, and gas are assigned to be 0, 0, 0, 90, 95, and 110
($\$$/MWh), respectively. The reason for choosing these externality
coefficients is to adjust the net cost functions of each technology in the
following increasing order: renewable, hydro-power, nuclear, coal, gas, and
fuel oil. These numbers show the contribution of each producer in a specific
area to pollution costs. These costs, imposed on each producer, might be
significant because it could change the net merit order list in the spot
market. Additionally, the coefficient of utility function and power
consumption are assumed to increase by 4 and 2.5 percentage per time step,
respectively. Furthermore, the rated power limits of lines are reduced to 70
percentage of their nominal limits in order to experience line congestion in
our simulations. Also, each generator is assigned to each of the ten producers
in such a way that each producer has different technologies and the total
capacity of generators for each producer ranges from 6$\%$ to 14$\%$ of the
total installed capacity of generators. It is worth mentioning that the annual
investment cost for each technology is obtained based on its lifetime and
total investment costs, which include both fixed and variable investment costs
(EIA,, 2024). We assume that the investment cost per MW is the same for each
technology. Finally, the simulation is conducted over 20 time steps.

As the utility functions are linear, we can transform the optimization problem
related to utility function maximization with constraints (4) to (6), into a
set of linear equations. This set of equations consists of the primal and dual
constraints with linearized complementary slackness equations. We can
determine the optimal values for load consumption and the Lagrangian
multipliers corresponding to constraints (4) to (6) by solving this set of
equations, which ultimately leads to have price of each node in power system.
Similarly, the optimization problem linked to maximizing profit via investment
capacity can be converted into a set of optimality conditions, which enables
us to derive the amount of capacity investment for the strategic investment
case study. The aforementioned formulation is described in more detail in the
Appendix. Finally, we implement the proposed model in Python along with the
Gurobi solver.

Figure 4a shows the average nodal price and social welfare for both optimal
and competitive spot market models within twenty-time steps. As discussed in
Section 2, we have established that the optimal spot market price surpasses
that of the competitive spot market, and likewise, the optimal social welfare
is greater than under the competitive model. Our results reveal that in the
optimal model, total social welfare experiences a 0.23$\%$ increase compared
to the competitive model. Moreover, Figure 4c shows the tax levied on the two
producers under the proposed tax-subsidy scheme as a function of time. The
figure shows that as the load increases, the tax levied on the two producers
changes very differently. The reason for the difference is that producer 2
operates two nuclear units and one coal unit, while producer 4 operates two
renewable units and one fuel oil unit. Given the higher externality cost
coefficient for the fuel oil unit compared to the coal unit, producer 4
experiences a higher tax changes over time compared to producer 2.

Figure 4: Comparison of Optimal and Competitive Spot Market

(a) Average market price

(b) Social welfare

(c) Proposed tax for two producers

Figure 5a illustrates the average nodal electricity price and social welfare
for both the optimal and strategic investment models. As predicted in Section
3, the electricity price obtained from the optimal investment model is higher
than that under the strategic investment model. Also, the annual social
welfare in the optimal investment scenario is higher than in the strategic
investment one. Our analysis shows that under optimal investment, social
welfare is 4.36$\%$ higher compared to strategic investment. The total
generation investment in the optimal and strategic models are 256.57MW and
1937.31MW, respectively. Also, Figure 5c displays the profit of producer 10 at
each time step. It is clear that the profit of producer 10 increases with the
increases of loads in the optimal investment scenario. However, in strategic
investment, the profit does not exhibit a similar increase with load
consumption due to the strategic game dynamics inherent in the strategic
investment model. Additionally, our findings indicate that the total profit of
all producers in the optimal investment scenario is greater than the strategic
investment by 13.30$\%$.

Figure 5: Comparison of Optimal and Strategic Investments

(a) Average market price

(b) Social welfare

(c) Profit of two producer

Finally, Figure 6 illustrates the subsidy paid to producers 2 and 4 at each
time step in the optimal investment model. As depicted in the chart, the
subsidy value increases with some fluctuation over time. As we observed
previously, the profit of producers without our proposed subsidy does not
exhibit a meaningful trend. With this subsidy, the profit of each producer
might increase with the increment of loads. Thus, the profit of each producer
would increase in the final optimal investment framework.

Figure 6: Optimal Subsidy is Paid to two Sample Producers

##  7 Conclusion

We identified two main challenges about the growing penetration of renewable
energy sources in electricity markets. Firstly, the current competitive market
mechanisms cannot consider pollution externalities accurately. We proposed a
Pigouvian tax on producers as the function of their externalities that the tax
aligns producers’ profit maximization with the social welfare maximization
problem. This tax leads to encouraging preferences for renewable energy
sources in market clearing processes. Secondly, there is a lack of motivation
for producers to invest in renewable due to their lower operational costs
resulting in reduced electricity prices and ultimately reducing the total
profits of producers. To tackle this, we proposed subsidies to producers
equivalent to their marginal contribution to consumer surplus. This approach
ensures that their profits remain unaffected by electricity prices, and
ultimately incentivizes them to social-welfare maximizing investment in
renewable energy capacity. Crucially, these proposed tax and subsidy
mechanisms can be calculated using existing data within the ISO.

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## Appendix

In this section, we developed our proposed model for the numerical experiments
for linear utility, generation cost, and externality functions. These
functions are achievable by approximating exact functions with their piece-
wise linear approximations.

Firstly, we develop the mathematical formulation of the competitive
electricity market. The competitive electricity market problem is formulated
as (A1)-(A5), where (A1) represents the total social welfare of participants
without externalities. In this equation, $P_{d,t}^{D}$ and $P_{g,t}^{G}$
represent the consumption level of demand $d$ at time step $t$ and the
generation level of generator $g$ at time step $t$, respectively.
Additionally, $b_{d,t}^{D}$ and $b_{g,t}^{G}$ are the coefficients of the
demand utility function and the generation cost function, respectively. It is
worth mentioning that we assume $\Delta P_{g}^{G\max}$ is equal to zero for
all $g\in\mathcal{N}_{G}$ in the competitive and optimal markets, which means
the investment level is considered to be zero. This variable will be used in
the optimal and strategic capacity investment, which leads to an increase in
the capacity of generation units in the planning horizon. (A2) and (A3)
represent the limitations of generation output level and consumption level and
their associated Lagrange multipliers, respectively. Indeed, the maximum level
of consumption for demand $d$ at time step $t$ and the generation capacity of
generator $g$ at time step $t$ are represented by $P_{d,t}^{D\max}$ and
$P_{g}^{G\max}$, respectively. The power balance equation and its associated
Lagrange multiplier, i.e., electricity price, is formulated as (A4), and
finally, the line flow limits are shown by (A5) where $H_{g,l}^{G}$ and
$H_{d,l}^{D}$ are the Power Transfer Distribution Factors (PTDFs) for
generation and demand, respectively. Finally, $F_{l}^{\max}$ depicts the rated
power of line $l$.

$W\left(\Delta
P_{g}^{G\max}\right)=\max\sum_{t\in\mathbb{N}}\left(\sum_{d\in\mathcal{N}_{D}}b_{d,t}^{D}P_{d,t}^{D}-\sum_{g\in\mathcal{N}_{G}}b_{g,t}^{G}P_{g,t}^{G}\right)$
(A1)

subject to

$0\leq P_{g,t}^{G}\leq P_{g}^{G\max}+\Delta
P_{g}^{G\max}\leftrightarrow(\mu_{g,t}^{G\min},\mu_{g,t}^{G\max}),\forall
g\in\mathcal{N}_{G},\forall t\in\mathbb{N},$ (A2) $0\leq P_{d,t}^{D}\leq
P_{d,t}^{D\max}\leftrightarrow(\mu_{d,t}^{D\min},\mu_{d,t}^{D\max}),\forall
d\in\mathcal{N}_{D},\forall t\in\mathbb{N},$ (A3)
$\sum_{d\in\mathcal{N}_{D}}P_{d,t}^{D}=\sum_{g\in\mathcal{N}_{G}}P_{g,t}^{G}\leftrightarrow(\lambda_{t}),\forall
t\in\mathbb{N},$ (A4)
$-F_{l}^{\max}\leq\sum_{g\in\mathcal{N}_{G}}H_{g,l}^{G}P_{g,t}^{G}-\sum_{d\in\mathcal{N}_{D}}H_{d,l}^{D}P_{d,t}^{D}\leq
F_{l}^{\max}\leftrightarrow(\mu_{l,t}^{L\min},\mu_{l,t}^{L\max}),\forall
l\in\mathcal{N}_{L},\forall t\in\mathbb{N}.$ (A5)

On the other hand, the optimal market problem is represented with the
objective function formulated as (A6) subject to (A2)-(A5), which considers
negative externalities through the piece-wise linear externality functions
with coefficient $e_{g,t}^{G}$ for the generator $g$ at time step $t$.

$W\left(\Delta
P_{g}^{G\max}\right)=\max\sum_{t\in\mathbb{N}}\left(\sum_{d\in\mathcal{N}_{D}}b_{d,t}^{D}P_{d,t}^{D}-\sum_{g\in\mathcal{N}_{G}}\left(b_{g,t}^{G}+e_{g,t}^{G}\right)P_{g,t}^{G}\right)$
(A6)

Now, we need to develop a formulation to compute the total consumer utility,
which is used for determining the electricity price by another optimization
problem or a set of equations. This problem is formulated with the objective
function as (A7) subject to (A3)-(A5).

$U_{t}^{PP}=\max\sum_{d}b_{d,t}^{D}P_{d,t}^{D}$ (A7)

The dual problem of maximizing total consumer utility is as follows:

$U_{t}^{DP}=\min\sum_{d,t}P_{d,t}^{D\max}\mu_{d,t}^{D\max}+\left(\sum_{g\in\mathcal{N}_{G}}P_{g,t}^{G}\right)\lambda_{t}-\sum_{l\in\mathcal{N}_{L}}\left(F_{l}^{\max}+\sum_{g\in\mathcal{N}_{G}}H_{g,l}^{G}P_{g,t}^{G}\right)\mu_{l,t}^{L\max}$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\sum_{l\in\mathcal{N}_{L}}\left(F_{l}^{\max}-\sum_{g\in\mathcal{N}_{G}}H_{g,l}^{G}P_{g,t}^{G}\right)\mu_{l,t}^{L\min}$
(A9)

subject to

$\mu_{d,t}^{D\min}+\mu_{d,t}^{D\max}+\lambda_{t}-\sum_{l\in\mathcal{N}_{L}}H_{d,l}^{D}\left(\mu_{l,t}^{L\min},\mu_{l,t}^{L\max}\right)=b_{d,t}^{D}\leftrightarrow(P_{d,t}^{D}),\forall
d\in\mathcal{N}_{D},\forall t\in\mathbb{N},$ (A10) $\mu_{d,t}^{D\min}\leq
0,\mu_{d,t}^{D\max}\geq 0,\forall d\in\mathcal{N}_{D},\forall t\in\mathbb{N},$
(A11) $\mu_{l,t}^{L\min}\leq 0,\mu_{l,t}^{L\max}\geq 0,\forall
l\in\mathcal{N}_{L},\forall t\in\mathbb{N}.$ (A12)

If we assume $P_{g,t}^{G}$ is a parameter, then the dual problem is a Linear
Programming (LP) problem, and $U_{t}^{DP}$ and $U_{t}^{PP}$ are equal at each
time step. Thus, if we consider the equations of the primal problem of
maximizing total consumer utility, i.e., (A3)-(A5), with the equations of the
dual problem, i.e., (A10)-(A12), plus the zero duality gap equations, i.e.,
$U_{t}^{PP}=U_{t}^{DP}$, we would have a set of equations leading to a
solution for maximizing total consumer utility with its Lagrange multiplier.
Based on the fact that $P_{g,t}^{G}$ is a variable in our problem, this set of
equations is non-linear. This system of equations can be linearized by
removing the strong-duality equations and considering the linearized
complementary slackness condition with binary variables as defined by
(A10)-(A12), where $\gamma$ is an arbitrary positive number which is greater
than one. In practice, $\gamma$ equal to two has an appropriate performance
with respect to getting a solution in a reasonable time.

$P_{d,t}^{D}\leq\gamma P_{d,t}^{D\max}\left(1-z_{d,t}^{D\min}\right),\forall
d\in\mathcal{N}_{D},\forall t\in\mathbb{N},$ (A13)
$\mu_{d,t}^{D\min}\geq-\gamma b_{d,t}^{D}z_{d,t}^{D\min},\forall
d\in\mathcal{N}_{D},\forall t\in\mathbb{N},$ (A14) $P_{d,t}^{D}\geq
P_{d,t}^{D\max}-\gamma P_{d,t}^{D\max}\left(1-z_{d,t}^{D\max}\right),\forall
d\in\mathcal{N}_{D},\forall t\in\mathbb{N},$ (A15)
$\mu_{d,t}^{D\max}\leq\gamma b_{d,t}^{D}z_{d,t}^{D\max},\forall
d\in\mathcal{N}_{D},\forall t\in\mathbb{N},$ (A16)
$\sum_{g\in\mathcal{N}_{G}}H_{g,l}^{G}P_{g,t}^{G}-\sum_{d\in\mathcal{N}_{D}}H_{d,l}^{D}P_{d,t}^{D}\leq-
F_{l}^{\max}+\left(\gamma+1\right)F_{l}^{\max}\left(1-z_{l,t}^{L\min}\right),\forall
l\in\mathcal{N}_{L},\forall t\in\mathbb{N},$ (A17)
$\mu_{l,t}^{L\min}\geq-\gamma\max_{d\in\mathcal{N}_{D}}\left(b_{d,t}^{D}\right)z_{l,t}^{L\min},\forall
l\in\mathcal{N}_{L},\forall t\in\mathbb{N},$ (A18)
$\sum_{g\in\mathcal{N}_{G}}H_{g,l}^{G}P_{g,t}^{G}-\sum_{d\in\mathcal{N}_{D}}H_{d,l}^{D}P_{d,t}^{D}\geq
F_{l}^{\max}-\left(\gamma+1\right)F_{l}^{\max}\left(1-z_{l,t}^{L\max}\right),\forall
l\in\mathcal{N}_{L},\forall t\in\mathbb{N},$ (A19)
$\mu_{l,t}^{L\max}\leq\gamma\max_{d\in\mathcal{N}_{D}}\left(b_{d,t}^{D}\right)z_{l,t}^{L\max},\forall
l\in\mathcal{N}_{L},\forall t\in\mathbb{N},$ (A20)
$z_{d,t}^{D\min},z_{d,t}^{D\max}\in\\{0,1\\},\forall
d\in\mathcal{N}_{D},\forall t\in\mathbb{N},$ (A21)
$z_{l,t}^{L\min},z_{l,t}^{L\max}\in\\{0,1\\},\forall
l\in\mathcal{N}_{L},\forall t\in\mathbb{N}.$ (A22)

By adding (A10)-(A22) to the competitive electricity markets or optimal
electricity markets, we can obtain a Mixed Integer Linear Programming (MILP)
problem, which enables us to directly compute the Lagrange multiplier of
competitive electricity markets or optimal electricity markets. According to
the value of the Lagrange multiplier, the electricity price at each node, each
generator node, and each demand node is represented by (A23), (A24), and
(A25), respectively. Moreover, $H_{n,l}^{N}$ shows the PTDF associated with
the power changes in bus $n$.

$Pr_{n,t}^{N}=\lambda_{t}-\sum_{l\in\mathcal{N}_{L}}H_{n,l}^{N}\left(\mu_{l,t}^{L\min}+\mu_{l,t}^{L\max}\right),\forall
n\in\mathcal{N}_{N},\forall t\in\mathbb{N},\\\ $ (A23)
$Pr_{g,t}^{G}=\lambda_{t}-\sum_{l\in\mathcal{N}_{L}}H_{g,l}^{G}\left(\mu_{l,t}^{L\min}+\mu_{l,t}^{L\max}\right),\forall
g\in\mathcal{N}_{G},\forall t\in\mathbb{N},$ (A24)
$Pr_{d,t}^{D}=\lambda_{t}-\sum_{l\in\mathcal{N}_{L}}H_{d,l}^{D}\left(\mu_{l,t}^{L\min}+\mu_{l,t}^{L\max}\right),\forall
d\in\mathcal{N}_{D},\forall t\in\mathbb{N}.$ (A25)

Now, the capacity investment problem, i.e., optimal investment and strategic
investment, will be presented using the model we developed earlier. The
optimal capacity investment problem is represented by maximizing the objective
functions as shown by (A26) subject to (A2)-(A5), (A10)-(A22), and (A27). In
(A27), a limit on the capacity investment for generator $g$ is shown by
$P_{g}^{\Delta G\max}$.

$\sum_{t\in\mathbb{N}}\left(\sum_{d\in\mathcal{N}_{D}}b_{d,t}^{D}P_{d,t}^{D}-\sum_{g\in\mathcal{N}_{G}}\left(b_{g,t}^{G}+e_{g,t}^{G}\right)P_{g,t}^{G}\right)-\sum_{g\in\mathcal{N}_{G}}c_{g}^{G\text{cap}}\Delta
P_{g}^{G\max}$ (A26)

$0\leq\Delta P_{g}^{G\max}\leq P_{g}^{\Delta
G\max}\leftrightarrow(\mu_{g}^{\Delta G\min},\mu_{g}^{\Delta G\max}),\forall
g\in\mathcal{N}_{G}$ (A27)

On the other hand, the strategic investment problem is a bi-level optimization
problem with maximizing the objective function represented by (A1) subject to
(A26), (A2)-(A5), (A10)-(A22), and (A27)-(A28).

$\Delta P_{g}^{G\max}=\underset{{0\leq\Delta P_{g}^{G\max}\leq P_{g}^{\Delta
G\max}}}{\text{argmax}}\left[\sum_{t\in\mathbb{N}}\left(Pr_{g,t}^{G}-b_{g,t}^{G}-e_{g,t}^{G}\right)P_{g,t}^{G}-c_{g}^{G\text{cap}}\Delta
P_{g}^{G\max}\right],\forall g\in\mathcal{N}_{G}.$ (A28)

In order to have a single-level problem, we need to transform the optimization
problem with the objective function (A28) and constraints (A2) and (A27) into
a system of equations using a similar approach as before. Because
$Pr_{g,t}^{G}$ and $P_{g,t}^{G}$ are not decision variables for the problem of
maximizing the profit of generators, they are determined after knowing the
amount of capacity investment by solving the optimal markets problem. Thus,
the dual problem of maximizing the profit of generators is represented by
minimizing (A29) subject to (A30)-(A32).

$\min\sum_{t\in\mathbb{N}}\left(P_{g}^{G\max}-P_{g,t}^{G}\right)\mu_{g,t}^{G\max}+P_{g}^{\Delta
G\max}\mu_{g}^{\Delta G\max}$ (A29) $\mu_{g}^{\Delta G\min}+\mu_{g}^{\Delta
G\max}-\sum_{t\in\mathbb{N}}\mu_{g,t}^{G\max}=-c_{g}^{G\text{cap}}\leftrightarrow(\Delta
P_{g}^{G\max}),\forall g\in\mathcal{N}_{G},$ (A30) $\mu_{g}^{\Delta G\min}\leq
0,\mu_{g}^{\Delta G\max}\geq 0,\forall g\in\mathcal{N}_{G},$ (A31)
$\mu_{g,t}^{G\max}\geq 0,\forall g\in\mathcal{N}_{G},\forall t\in\mathbb{N}.$
(A32)

We could similarly derive a system of equations that gives us the solution for
maximizing the profit of generators in capacity investment by considering the
linearized complementary slackness equations represented by (A33) to (A38). As
a result of these linearizations, the strategic investment problem has the
objective function represented by (A1) subject to constraints comprising
(A26), (A2)-(A5), (A10)-(A22), (A27), and (A30)-(A38).

$\Delta P_{g}^{G\max}\leq\gamma P_{g}^{G\max}\left(1-z_{d,t}^{\Delta
G\min}\right),\forall g\in\mathcal{N}_{G},$ (A33) $\Delta
P_{g}^{G\max}\leq\gamma P_{g}^{G\max}\left(1-z_{d,t}^{\Delta
G\min}\right),\forall g\in\mathcal{N}_{G},$ (A34) $mu_{g}^{\Delta
G\min}\geq-\gamma c_{g}^{G\text{cap}}z_{d,t}^{\Delta G\min},\forall
g\in\mathcal{N}_{G},$ (A35) $\mu_{g}^{\Delta G\max}\leq\gamma
c_{g}^{G\text{cap}}z_{d,t}^{\Delta G\max},\forall g\in\mathcal{N}_{G},$ (A36)
$P_{g,t}^{G}-P_{g}^{G\max}-\Delta P_{g}^{G\max}\geq-\gamma
P_{g}^{G\max}\left(1-z_{g,t}^{G\max}\right),\forall
g\in\mathcal{N}_{G},\forall t\in\mathbb{N},$ (A37)
$\mu_{g,t}^{G\max}\leq\gamma c_{g}^{G\text{cap}}z_{d,t}^{G\max},\forall
g\in\mathcal{N}_{G},\forall t\in\mathbb{N}.$ (A38)