The conceptual algorithm outlines a four-step process for achieving the optimal market structure.
It should be noted that each cascading optimization in the study calculates various factors such as nodal price, nodal demand, nodal supply, nodal consumer surplus, nodal producer surplus, and power flow for each market configuration. However, for the purpose of our analysis, we primarily focus on the Residual Supply Index (RSI) calculation for each market setting. Specifically, we select the highest RSI value among all possible market settings as our main point of emphasis. This algorithm’s overarching goal is to achieve a competitive market structure that maximizes welfare while addressing any issues with generation capacity and reserve margin. Policymakers and market designers can follow this four-step process.
The methodology used in this study is a single-shot game methodology in which generating firms submit fixed supply functions to the ISO while taking into account their competitors’ bid functions. This is used for a single bidding period, and once all firms have submitted bids, the ISO clears the market, resulting in a market-clearing price. This price is determined by the ISO based on the market’s generation and transmission network structure, as well as the balance of supply and demand for electricity. The ISO pays the bus nodal price to all generating firms based on the amount of electricity sold to the pool in this model, and consumers pay the ISO the electricity price based on the active power load received.
3.1. Nodal Pricing and DC Power Transmission System
The concept of nodal pricing, as established by [
34,
35], is a fundamental theory used to determine optimal electricity prices that achieve welfare maximization under specific constraints. In this research, we perform a nodal pricing analysis based on Optimal Power Flow (OPF) and a marginal cost calculation at each node. The authors of ref. [
36] incorporated transmission constraints and optimized electricity prices as a dual value in the program’s implementation of nodal pricing in the England and Wales market. The authors of ref. [
27] expanded on this implementation by accounting for transmission losses in the optimal power flow (OPF) in order to apply nodal pricing in the same market. However, in this study, transmission losses are assumed to be minimal, and the load flow formulation is approximated into the DC load flow. The equilibrium structure is then used to calculate the Nash equilibrium for specific bid functions and the electricity network [
37].
Kirchhoff’s law is a fundamental principle that governs power flow in power systems. The vector sum of the currents at any node in a circuit must be zero. This law governs all electrical circuits, including power grids. Kirchhoff’s law can be restated in terms of power flow in a grid as the vector sum of a node’s input and output being equal to the electricity injection at that node. This means that the power flow through any node in the system must be balanced, with the power flowing in equaling the power flowing out. Kirchhoff’s law must be followed to ensure the stability and reliability of the power system, as any deviation from this principle can cause power flow imbalances and potential power outages. To ensure the accuracy and stability of our power flow calculations, we use Kirchhoff’s law as a fundamental principle in our bottom-up electricity market model in this study.
We assume that bus
i is the sending node and
j is the receiving node, connected by transmission line
ij.
is the voltage magnitude,
is the voltage angle, and
is the phase angle.
and
, respectively, are conductance and susceptance of transmission line
ij. The
AC power flow from bus
i to
j is defined as
:
If is positive, then the power is flowing from bus i to bus j, and vice versa.
Using the
AC power flow equation, the
DC power flow
assumes that voltage magnitudes bus
i and
j equal to 1 p.u. (per unit) and
, which implies that
and
for a normal operating condition in the power system.
In power system theory, the susceptance
is the imaginary part of admittance
and could be denoted as
where
is the reactance of the transmission lines. Thus, the
DC power flow could be stated as:
We assume a power setting with
m transmission lines and
n nodes.
is a vector of reactance (
m ×
m).
is a vector of
DC power flow (
m × 1). M is the node–branch incidence matrix for the angle phase vector matrix (
n − 1 ×
m) excluding the reference node (slack bus), i.e., the node with phase angle is zero.
P is the power injection matrix (
n − 1 × 1). B is the susceptance matrix. PTDF is the power transfer distribution factor. The DC power flow vector equation could be expressed in a matrix form as PTDF vector multiplies the power injection vector:
The power injection at node n is calculated using the DC power flow assumption as the difference between power generation production and consumer demand . As a result, the transmission line’s power flow can be represented as a linear function of PTDF and .
3.2. Supply and Demand
The GenCo generates electricity based on the actual cost of the generator, while the consumer provides demand functions that represent the amount of electricity consumed. The demand function is the derivative of the benefit function, while the marginal cost function is the derivative of the total cost function. The consumer utility function is an inverse quadratic function as follows:
The load demand coefficient and active load demand at node i are represented by , , and , respectively. The coefficient is a positive value, and indicates the number of consumers.
The demand function’s slope is negative, and it follows a linear form in the inverse.
Total generation cost
consists of fixed (
) and variable cost
The bid or marginal cost function
of a generating firm is represented as a linear function because a constant marginal cost does not fully capture the true cost of generation in the power sector. We defined
as the intercept of the marginal cost, while
was the slope of the marginal cost which reflected the true cost of the Genco.
Consumer surplus is defined as the net benefit of consumers. The total consumer surplus is computed by adding the individual consumer surpluses based on the price. The consumer surplus in each region is determined by a variety of factors, including the structure of the power network, the location of generators and consumers, and transmission constraints. Let
represent consumer i’s electricity demand at price
. The consumer surplus for the inverse linear demand function is expressed as follows:
The producer surplus is the profit earned by a generator from selling electricity to the power pool and can be expressed as
.
3.4. Cournot Equilibrium Determination
We assume each node has a unique inverse linear demand function and the total demand function is defined as:
where
Q is the total demand for a particular interconnected power system and
P is the Cournot equilibrium price.
In a competitive electricity market, generating firms make decisions on the amount of electricity they wish to produce and sell, based on market conditions and their own costs. This output decision, in turn, affects the market price level through the inverse demand relationship. In a single-shot game, firms assume that their rivals’ outputs are constant and use this information to calculate their own profit function, which is the difference between their benefit function and total cost function. The role of the independent system operator (ISO) is to ensure that this profit function is maximized for all players, taking into account the constraints imposed by the power network. In a Cournot equilibrium, the profit function for each firm (
i) is given by a specific mathematical equation that takes into account the output decisions and costs of all firms in the market.
The above profit function is in quadratic form and concave. Hence, the derivative solution for ISO profit maximization is easy to calculate.
Taking the function’s derivative with respect to each node and setting it to zero yields a matrix equation with the variable subject to the total demand and cost function coefficients. The array’s size is determined by the number of participants in the electricity network.
When the derivative function is set to zero in each node, a matrix equation with
as the subject variable and subject to the total demand and cost function coefficient is obtained. The array’s size is determined by the number of participants in the electricity network.
The FOC profit maximization has diagonal function with and ; therefore, the coefficient of the matrix is positive and will give a unique solution for each .
The object variable
is the Cournot best response function in the network and could be defined as
In a case where the generating firms are symmetric, i.e., produce electricity in a uniform marginal cost, and coefficients are equivalent for all firms.
The power system is composed of numerous subnetworks in the actual power system topology, with each node representing a single load serving entity (LSE) and one or more power plant technologies. Each power plant technology, such as the base, intermediate, and peaking power plants, has a distinct linear marginal cost that reflects its distinct generation characteristics. The power system’s power plant technology mix can be classified based on the power plant’s ability to ramp-up and ramp-down to adjust to fluctuations in LSE aggregate electricity demand. Peak-load power plants have high ramping rates, low fuel costs, and relatively short construction times, whereas base-load power plants have low ramping rates, low fuel costs, and relatively short construction times.
When a GenCo operates multiple power plants, it behaves similarly to a monopolist with multiple plants, with the marginal cost of each plant determining the cost of generating electricity. When two power plants merge, their marginal cost functions are added horizontally. The efficiency constant used in the merger process is 1, which means that the GenCo’s efficiency before and after the merger is the same ().
To describe the system with two power plants that have a linear marginal cost and , the combining marginal cost is , where and . The merger of the power plants results in changes to both the original marginal costs and the supplier capacity k where . Note that for uniform power plant capacity, the combined generation capacity is the maximum supplier capacity .
The application of forward contract in this study is following the work by the authors of [
23]. The producers could supply electricity in the forward market by determining a definite quantity as contract coverage, and then bid in the spot market. The system operator clears the market according to the Cournot game. Based on [
23], we assume two players
and
with inverse demand function
. The cost functions are
and
with forward contracts
and
, respectively. The profit functions in the Cournot equilibrium for
and
with a constant marginal cost and forward sales are
and
. Solving FOC for both equations above leads to reaction function
. The quantity’s Nash equilibriums are
and
, while the Cournot price is
.
The authors of ref. [
21] conducted a forward contract study using the RSI and proving that the RSI is suitable for a competition analysis and has an ability to determine the potential of generators to raise prices considering contracts and non-price-responsive supplies. Incorporating the forward contract, the RSI formula is
, where
is the total installed capacity,
Q is the total demand equilibrium, and
is the relevant capacity (capacity—forward contract). Based on this equation, the forward contract reduces the GenCo’s relevant capacity. Therefore, the contract mitigates the market power by increasing the residual supply faced by the market.