Three Duopoly Game-Theoretic Models for the Smart Grid Demand Response Management Problem

Abstract

Demand response management (DRM) significantly influences the prospective advancement of electricity smart grids. This paper introduces three distinct game-theoretic duopoly models for the smart grid demand response management problem. It delineates several rational assumptions regarding the model variables, functions, and parameters. The first model adopts a Cournot duopoly form, offering a unique closed-form equilibrium solution. The second model adopts a Stackelberg duopoly structure, also providing a unique closed-form equilibrium solution. Following a comparison of the economic viability of the two model equilibria and an assessment of their sensitivity to parametric changes, the paper proposes a third model with a Cartel structure and discusses its advantages over the earlier models. Finally, the paper examines how demand forecasting affects the equilibrium quantities and pricing solutions of each model.

1. Introduction

Smart grids (Figure 1) are essential for maintaining the safe, efficient, and reliable functioning of electricity networks, significantly contributing to a reduction in power losses. Despite their critical importance, modern smart grids encounter numerous economic and technical challenges in their quest to provide energy securely and affordably to consumers. Some of the most pressing issues include conducting load-flow analysis, managing scheduling, and controlling the electric energy system. To achieve optimal operation and effective planning of the power system, it is vital to develop and implement an accurate predictive model.

Game theory serves as a conceptual and analytical framework designed to explore intricate interactions among rational and independent players. This field is deeply rooted in mathematical tools and can be divided into two main branches: cooperative and non-cooperative. Non-cooperative game theory centers on the analysis of strategic decision-making by independent agents, who often have partially or entirely conflicting interests. This branch is particularly relevant in applications like real-time demand-side management. Game theory, in general, serves as a robust framework for developing strategies that meet specific performance requirements while accounting for realistic assumptions.

The Cournot game model involves economic agents formulating their strategic decisions simultaneously, aiming to maximize their respective utilities. Numerous authors, such as Cui et al. [1], Belhaiza et al. [2,3], Goudarzi et al. [4], Rajasekharan et al. [5], Mansouri et al. [6], and Zareein et al. [7] have proposed game-theoretic models for smart grid DRM, where agents make simultaneous decisions.

The Stackelberg game model involves a leader and a follower, both seeking to maximize their respective utilities while being aware of their economic positions. Several authors, including Shakrina and Margossian [8], AlShehri et al. [9], Ma et al. [10], Ozcan et al. [11], and Apostolopoulos et al. [12], have proposed game-theoretic models for smart grid DRM where agents make sequential decisions.

1.1. Importance of Load Forecasting in Smart Grids

Load forecasting has become a rapidly growing area of interest and research within the realm of smart grids. A variety of forecasting techniques have been introduced to predict power system loads, with many of these methods showing significant success through the use of mathematical models. The primary goal of these techniques is to reduce the estimation errors between the predicted energy demands and the actual measured future values. Many researchers have contributed to the development of load forecasting models for smart grids. In particular, Amjady and Keynia [13], Hippert et al. [14], Macedo et al. [15], Baliyan et al. [16], and Raza and Khosravi [17] and many others, have focused on artificial intelligence techniques for demand load forecasting in smart grids. Along these lines, this paper explores how demand forecasting influences the Nash equilibrium quantities and pricing solutions of each proposed model.

Figure 1. Smart grid sample design.

Figure 1. Smart grid sample design.

1.2. Motivation and Novelties of Our Approach

Building upon the existing research literature, this paper introduces three distinct game-theoretic duopoly models for smart grid demand response:

• A Cournot duopoly model where two companies seek their energy supply Nash equilibrium strategies simultaneously.
• A Stackelberg leader–follower duopoly model where two companies seek their energy supply Nash equilibrium strategies sequentially.
• A duopoly collusion model where the two companies form a cartel and seek their energy supply Nash equilibrium strategies.

The paper establishes a series of rational assumptions regarding the variables, functions, and parameters of each game model before seeking their closed-form equilibrium solutions. Additionally, it compares the results of the three duopoly models and evaluates their sensitivity to parametric changes.

Moving forward, Section 2 reviews some of the existing literature on Cournot and Stackelberg game models for smart grid DRM. Section 3 provides detailed insights into the three different game-theoretic duopoly models. It not only presents and compares their diverse results, but also discusses the sensitivity of the equilibrium strategies concerning parametric variations. Section 4 delves into the effects of forecasted demand variation on the Nash equilibrium quantities and pricing solutions of each model. Section 5 summarizes and concludes the paper.

2. Literature Review of Game-Theoretic Models for DRM

The energy management issue within smart grids has garnered attention from various angles due to its significance in attaining stability, reliability, and cost-effectiveness for both electricity distribution grids and individual consumers in their residences. This section reviews some of the most recent Cournot, Stackelberg, and data-driven game models proposed for smart grid DRM.

2.1. Cournot Models for Smart Grids

Cui et al. [1] employed a game involving n players to simulate energy price competition, where the pricing decisions made by one player impact the payoffs of others. They introduced two models: the first explored Nash equilibrium solutions, demonstrating uniqueness under certain assumptions, while the second model incorporated two key factors into the energy price function—energy generation cost and the homeowner’s response to changes in energy usage. The study revealed that adjusting the price function did not enhance expected profits for any company attempting to do so.

Belhaiza et al. [2,3] presented single and multi-period game-theoretic models for demand response management (DRM) focusing on a $0-1$ mixed-integer linear programming approach to generate Pareto non-dominated extreme Nash equilibria.

A game-theory-based demand response program was introduced by Goudarzi et al. [4], integrating concepts from both incentive-based and price-based demand response programs, with a focus on the residential, commercial, and industrial sectors. The program explores three pricing strategies: fixed pricing, time-of-use pricing (applied to both utility and customer sides), and real-time pricing, as well as various combinations of these strategies. The study includes three case studies conducted across seven different scenarios. The findings demonstrate that the developed multi-criteria, security-constrained, game-theory-based demand response program can create a win–win situation for both the utility and the customers.

A cooperative game-theoretic model was developed by Rajasekharan et al. [5] for smart grids within local neighborhood households equipped with energy storage systems. In this model, users collaborate to exchange stored or supplied energy, aiming to minimize consumption costs by taking advantage of pricing disparities. The study presented a straightforward economic market scenario where two users exchanged two goods, illustrating how trading prices could self-regulate, thereby encouraging users to lower their costs.

A three-layer, risk-averse, game-theoretic strategy was proposed by Mansouri et al. [6] to coordinate smart buildings and electric vehicle fleets with micro-grid scheduling. The approach begins with the development of a demand response program tailored for smart buildings, followed by the coordinated scheduling of both smart buildings and electric vehicle fleets. Afterward, micro-grid operators receive power exchange data from the smart buildings, allowing them to adjust their scheduling accordingly. Finally, the scheduling of micro-grids and distribution systems (DS) is executed using cooperative game theory.

A three-layer framework was also introduced by Zareein et al. [7]. The process begins with an algorithm that manages demand response within each microgrid, utilizing load models in smart buildings to determine the load consumption for each consumer. Next, each microgrid sets its selling price for consumers and determines the amount of energy required from the utility grid to maximize profit. Finally, the microgrid cluster agent determines the equilibrium point between the microgrids.

2.2. Stackelberg Models for Smart Grids

Drawing inspiration from the Stackelberg game, Shakrina and Margossian [8] proposed a smart grid game model that simulates a trading model between the network operator (the leader) and the customers (the followers). The operator focuses on optimizing generation costs and offers market prices, while the customers optimize their consumption behavior in response. The model is designed as a one-leader, N-follower iterative game, where the optimization problems are addressed using both deterministic and stochastic optimization techniques.

A game-theoretic approach was proposed by AlShehri et al. [9] to address the multi-period demand response problem by modeling the interaction between companies and consumers through a Stackelberg game. In this setup, companies act as leaders and consumers as followers. The approach uncovered a unique equilibrium that allows companies to maximize their revenues while enabling consumers to optimize their utilities.

A real-time pricing scheme was introduced by Ma et al. [10] for demand response management, involving one energy provider and multiple energy hub operators. The scheme employs a Stackelberg game approach to model interactions between the provider and consumers. A distributed algorithm was proposed to determine the Stackelberg equilibrium, which maximizes the utilities of the game participants.

A Stackelberg game was also introduced by Ozcan et al. [11] involving a coordination agent and participating households to manage the overall load consumption within a residential neighborhood. The coordination agent’s objective is to set a price vector that keeps the neighborhood’s total load consumption close to a specified target value.

In a similar vein, a two-stage game-theoretic optimization framework was formulated by Apostolopoulos [12]. In the first stage, the optimal electricity consumption for customers is calculated, while in the second stage, the optimal pricing for power companies is determined. The results from the demand response management problem are then used by the learning system to accumulate knowledge and make informed decisions on the best selection of power companies.

2.3. Data-Driven Game Models for Smart Grids

Nguyen et al. [18] investigated a smart power system with dynamic energy requests from users. Energy prices, updated by the provider using consumption data, prompted users to charge batteries during low-demand periods and discharge during high-demand periods. The study focused on minimizing the power system’s peak-to-average ratio (PAR) through a game-theoretical approach. Their algorithm, tested via simulation, demonstrated simultaneous minimization of PAR and total energy costs.

Stephens et al. [19] proposed a game-theoretic model using predictive control (MPC) for real-time demand-side management (DSM) data. Assuming perfect forecasting information, their MPC-based algorithm aimed to identify subgame perfect equilibrium strategies for distributed generation and storage. This approach effectively mitigated forecasting errors, particularly when facing errors exceeding 10%, leading to significant electricity cost savings and a reduction in the peak-to-average demand ratio (PAR).

Wen et al. [20] proposed a dynamic pricing model within a district-scale smart grid with high photovoltaic power penetration. The existence of a Nash equilibrium for the proposed model is demonstrated. The game theory model is also employed to investigate the interaction between thre energy service provider and electricity consumers.

3. Three Game-Theoretic Duopoly Models for DRM

In our proposed smart grid duopoly models, two energy producing companies determine their best responses’ production $q_1$ and $q_2$ based on the expected base demand $\tilde{d}$. Each company market’s price is a function of the quantities produced. Each company aims at maximizing its own profit.

Assumption 1.

We assume that for company 1, the price writes as follows:

$$p_1=a_1-b_1(q_1+q_2-\tilde{d}),$$

and for company 2 the price is written as:

$$p_2=a_2-b_2(q_1+q_2-\tilde{d}),$$

with pricing elasticities $b_1,b_2>0$.

The rationale behind this assumption on the form of pricing functions is that if the two companies decide not to totally meet the expected demand, the price would slightly increase. Meanwhile, it would decrease if they provide a total energy exceeding the demand.

Assumption 2.

We assume that for company 1 the production cost is $C_1=c_1{q}_1$ and for company 2 the production cost it is $C_2=c_2{q}_2$.

The rationale behind this assumption is that the total energy production cost is linear in terms of the total quantity produced.

Hence, the profit for company 1 is:

$${\pi }_1(q_1,q_2)=p_1{q}_1-c_1{q}_1$$

and the profit for company 2 is:

$${\pi }_2\left(q_1{,}_2\right)=p_2{q}_2-c_2{q}_2.$$

Assumption 3.

We assume that for company 1, we have $a_1>c_1$, and for company 2 we have $a_2>c_2$.

The reasoning behind this assumption is rooted in the necessity for energy base prices $a_1$ and $a_2$ to exceed $c_1$ and $c_2$, respectively. This ensures the economic viability of any energy production activity.

3.1. Cournot Duopoly Model

Under the proposed Cournot duopoly model, the two companies would simultaneously determine their best production responses $q_1^{*}$ and $q_2^{*}$. The Nash equilibrium market prices $p_1^{*}$ and $p_2^{*}$ are obtained using first order and second order derivatives, respectively.

We have:

$${\displaystyle \frac{d{\pi }_1}{d{q}_1}}=a_1-b_1{q}_1-b_1(q_1+q_2-\tilde{d})-c_1$$

and

$${\displaystyle \frac{d{\pi }_2}{d{q}_2}}=a_2-b_2{q}_2-b_2(q_1+q_2-\tilde{d})-c_2.$$

We then solve the system of linear equations: ${\displaystyle \frac{d{\pi }_1}{d{q}_1}}=0$ and ${\displaystyle \frac{d{\pi }_2}{d{q}_2}}=0.$

The simultaneous solutions here yield:

$$q_1^{*}={\displaystyle \frac{b_1(c_2-a_2)+2b_2(a_1-c_1)+\tilde{d}{b}_1{b}_2}{3b_1{b}_2}}$$

and

$$q_2^{*}={\displaystyle \frac{b_2(c_1-a_1)+2b_1(a_2-c_2)+\tilde{d}{b}_1{b}_2}{3b_1{b}_2}}.$$

The total quantity supplied by the two companies would then be:

$$Q^{*}=q_1^{*}+q_2^{*}={\displaystyle \frac{b_1(a_2-c_2)+b_2(a_1-c_1)+2\tilde{d}{b}_1{b}_2}{3b_1{b}_2}}.$$

One can notice that Assumptions 1–3 do not guarantee that $q_1^{*}$ and $q_2^{*}$ are both strictly positive. Hence, the pricing parameters $a_1$, $b_1$, $a_2$, and $b_2$ of the two companies must adhere to the condition of Assumption 4.

Assumption 4.

We assume that the Cournot model parameters satisfy the following conditions:

$${\displaystyle \frac{b_1(a_2-c_2)+2b_2(c_1-a_1)}{b_1{b}_2}}\le \tilde{d}.$$

and

$${\displaystyle \frac{2b_1(c_2-a_2)+b_2(a_1-c_1)}{b_1{b}_2}}\le \tilde{d}.$$

In addition, the second derivatives ${\displaystyle \frac{d^2{\pi }_1}{d{q}_1^2}}=-2b_1$ and ${\displaystyle \frac{d^2{\pi }_2}{d{q}_2^2}}=-2b_2$ are both negative, which guarantees that $q_1^{*}$ and $q_2^{*}$ provide the absolute maximum values of ${\pi }_1$ and ${\pi }_2$, respectively.

Proposition 1.

Under Assumptions 1–4, the Cournot model Nash equilibrium prices would be:

$$p_1^{*}={\displaystyle \frac{b_1(c_2-a_2)+b_2(2a_1+c_1)+\tilde{d}{b}_1{b}_2}{3b_2}}$$

and

$$p_2^{*}={\displaystyle \frac{b_2(c_1-a_1)+b_1(2a_2+c_2)+\tilde{d}{b}_1{b}_2}{3b_1}}.$$

The differentiation with respect to the pricing elasticities $b_1$ and $b_2$ yields Corollaries 1 and 2, as we observe that:

$${\displaystyle \frac{d{q}_1^{*}}{d{b}_1}}=2{\displaystyle \frac{c_1-a_1}{3b_1^2}}<0,{\displaystyle \frac{d{q}_2^{*}}{d{b}_1}}={\displaystyle \frac{a_1-c_1}{3b_1^2}}>0,$$

and

$${\displaystyle \frac{d{q}_1^{*}}{d{b}_2}}={\displaystyle \frac{a_2-c_2}{3b_2^2}}>0,{\displaystyle \frac{d{q}_2^{*}}{d{b}_2}}=2{\displaystyle \frac{c_2-a_2}{3b_2^2}}<0.$$

Consequently, Corollary 1 generalizes the impact of an increase in the pricing elasticity of any given company on its energy market share, as well as on the market share of the other company.

Corollary 1.

Any increase in the pricing elasticity $b_i$ of company i ($i=1,2$) results in a decrease in its share of the total energy market at Cournot model equilibrium, with a simultaneous increase in the share of the other company $j\ne i$.

We also observe that the differentiation of the Cournot model equilibrium prices with respect to the pricing elasticities yields:

$${\displaystyle \frac{d{p}_2^{*}}{d{b}_1}}={\displaystyle \frac{b_2(a_1-c_1)}{3b_1^2}}>0,$$

$$\mathrm{and}{\displaystyle \frac{d{p}_1^{*}}{d{b}_1}}={\displaystyle \frac{c_2-a_2+b_2\tilde{d}}{3b_2}}\ge 0\mathrm{if}{b}_2\ge {\displaystyle \frac{a_2-c_2}{\tilde{d}}},$$

and

$${\displaystyle \frac{d{p}_1^{*}}{d{b}_2}}={\displaystyle \frac{b_1(a_2-c_2)}{3b_2^2}}>0$$

$$\mathrm{and}{\displaystyle \frac{d{p}_2^{*}}{d{b}_2}}={\displaystyle \frac{c_1-a_1+b_1\tilde{d}}{3b_1}}\ge 0\mathrm{if}{b}_1\ge {\displaystyle \frac{a_1-c_1}{\tilde{d}}}.$$

Consequently, Corollary 2 automatically follows.

Corollary 2.

At the Cournot model equilibrium, any increase in the pricing elasticity $b_i$ leads to an increase in the price of company i ($i=1,2$), potentially causing either a decrease or an increase in the price of company $j\ne i$.

In addition, the differentiation with respect to the unit production costs $c_1$ and $c_2$ yields Corollary 3, as we have:

$${\displaystyle \frac{d{q}_1^{*}}{d{c}_1}}=-{\displaystyle \frac{2}{3b_1}}<0,{\displaystyle \frac{d{q}_2^{*}}{d{c}_1}}={\displaystyle \frac{1}{3b_1}}>0,$$

and

$${\displaystyle \frac{d{q}_1^{*}}{d{c}_2}}={\displaystyle \frac{1}{3b_2}}>0,{\displaystyle \frac{d{q}_2^{*}}{d{c}_2}}=-{\displaystyle \frac{2}{3b_2}}<0.$$

Corollary 3.

Any increase in the unit production cost $c_i$ of company i ($i=1,2$) results in a decrease in its share of the total energy market at the Cournot model equilibrium, with a simultaneous increase in the share of company $j\ne i$.

We also observe that the differentiation of the Cournot model equilibrium prices with respect to the unit production costs yields:

$${\displaystyle \frac{d{p}_1^{*}}{d{c}_1}}={\displaystyle \frac{d{p}_2^{*}}{d{c}_2}}={\displaystyle \frac{1}{3}}>0,$$

$${\displaystyle \frac{d{p}_1^{*}}{d{c}_2}}={\displaystyle \frac{b_1}{3b_2}}>0\mathrm{and}{\displaystyle \frac{d{p}_2^{*}}{d{c}_1}}={\displaystyle \frac{b_2}{3b_1}}>0.$$

Corollary 4.

An increase in the unit production costs $c_1$ or $c_2$ results in an increase of the Cournot model equilibrium prices.

To conclude the sensitivity analysis of the Cournot model equilibrium, we observe that the differentiation with respect to the base prices $a_1$ and $a_2$ yields:

$${\displaystyle \frac{d{q}_1^{*}}{d{a}_1}}={\displaystyle \frac{2}{3b_1}}>0,{\displaystyle \frac{d{q}_2^{*}}{d{a}_1}}=-{\displaystyle \frac{1}{3b_1}}<0,$$

and

$${\displaystyle \frac{d{q}_1^{*}}{d{a}_2}}=-{\displaystyle \frac{1}{3b_2}}<0,{\displaystyle \frac{d{q}_2^{*}}{d{a}_2}}={\displaystyle \frac{2}{3b_2}}>0.$$

Corollary 5.

An increase in the base price $a_i$ ($i=1,2$) of the company results in an increase in the Cournot model equilibrium market share for company i and a decrease in the market share for company $j\ne i$.

Example 1.

Let us consider that companies 1 and 2 have pricing and cost parameters $a_1=32$, $a_2=30$, $b_1=4$, $b_2=3$, and $c_1=10$, $c_2=8$, with an expected base demand of $\tilde{d}=6$. For the Cournot game model, the Nash equilibrium quantities are $q_1^{*}=3.22$, $q_2^{*}=5.05$, and the prices are $p_1^{*}=22.88$, and $p_2^{*}=23.16$. In this scenario, the Cournot model equilibrium profits are ${\pi }_1^{*}=41.53$ and ${\pi }_2^{*}=76.67$, respectively.

3.2. Stackelberg Duopoly Model

In the proposed Stackelberg duopoly model, the two companies sequentially determine their best response quantities, denoted as $q_1^{**}$ and $q_2^{**}$. Company 1 assumes the role of the “leader”, while Company 2 plays the role of the “follower”. The Nash equilibrium market prices $p_1^{**}$ and $p_2^{**}$ are derived using backward induction, considering the first and second-order derivatives. Backward induction is commonly applied in extensive games with perfect information, solving the game backward based on the assumption of common knowledge rationality.

We start first by solving:

$${\displaystyle \frac{d{\pi }_2}{d{q}_2}}=a_2-b_2{q}_2-b_2(q_1+q_2-\tilde{d})-c_2=0.$$

We get

$$q_2^{**}={\displaystyle \frac{a_2+b_2(d-q_1)-c_2}{2b_2}}.$$

This is the best response of company 2 to any level of production set by company 1. We plug $q_2^{**}$ into ${\pi }_1\left(q_1\right)$ and then solve the equation ${\displaystyle \frac{d{\pi }_1}{d{q}_1}}=0$. We obtain:

$$q_1^{**}={\displaystyle \frac{2b_2(a_1-c_1)+b_1(c_2-a_2)+\tilde{d}{b}_1{b}_2}{2b_1{b}_2}},$$

and then

$$q_2^{**}={\displaystyle \frac{3b_1(a_2-c_2)+2b_2(c_1-a_1)+\tilde{d}{b}_1{b}_2}{4b_1{b}_2}}.$$

The total quantity supplied by the two companies would then be: $Q^{**}=q_1^{**}+q_2^{**}$

$$={\displaystyle \frac{b_1(a_2-c_2)+2b_2(a_1-c_1)+3\tilde{d}{b}_1{b}_2}{4b_1{b}_2}}.$$

One can observe again that Assumption 4 ensures that $q_1^{**}$ only is strictly positive. Therefore, the pricing parameters $a_1$, $b_1$, $a_2$, and $b_2$ of the two companies must satisfy the condition of Assumption 5 to ensure that $q_2^{**}$ is also positive.

Assumption 5.

We assume that the Stackelberg model parameters satisfy the condition:

$${\displaystyle \frac{3b_1(c_2-a_2)+2b_2(a_1-c_1)}{b_1{b}_2}}\le \tilde{d}.$$

Proposition 2.

Under Assumptions 1–5, the Stackelberg model Nash equilibrium prices would be:

$$p_1^{**}={\displaystyle \frac{b_1(c_2-a_2)+2b_2(a_1+c_1)+\tilde{d}{b}_1{b}_2}{4b_2}}$$

and

$$p_2^{**}={\displaystyle \frac{2b_2(c_1-a_1)+b_1(3a_2+c_2)+\tilde{d}{b}_1{b}_2}{4b_1}}.$$

Corollary 6 promptly ensues as we observe that the differentiation of the Stackelberg model equilibrium quantities with respect to the pricing elasticities yields:

$${\displaystyle \frac{d{q}_1^{**}}{d{b}_1}}={\displaystyle \frac{c_1-a_1}{b_1^2}}<0,{\displaystyle \frac{d{q}_2^{**}}{d{b}_1}}={\displaystyle \frac{a_1-c_1}{2b_1^2}}>0,$$

and,

$${\displaystyle \frac{d{q}_1^{**}}{d{b}_2}}={\displaystyle \frac{a_2-c_2}{2b_2^2}}>0,{\displaystyle \frac{d{q}_2^{**}}{d{b}_2}}=3{\displaystyle \frac{c_2-a_2}{4b_2^2}}<0.$$

Corollary 6.

An increase in the pricing elasticity $b_i$ of company i ($i=1,2$) leads to a reduction in its share of the total energy market at Stackelberg model equilibrium, with a simultaneous increase in the share of company $j\ne i$.

Additionally, the market share of company 1, as the energy market leader, exhibits a higher sensitivity to variations in its pricing elasticity $b_1$ compared with the energy market Cournot equilibrium, as expressed by:

$${\displaystyle \frac{d{q}_1^{**}}{d{b}_1}}={\displaystyle \frac{c_1-a_1}{b_1^2}}<{\displaystyle \frac{d{q}_1^{*}}{d{b}_1}}=2{\displaystyle \frac{c_1-a_1}{3b_1^2}}.$$

We also observe that the differentiation of the Stackelberg model equilibrium prices with respect to the pricing elasticities yields:

$${\displaystyle \frac{d{p}_2^{**}}{d{b}_1}}={\displaystyle \frac{b_2(a_1-c_1)}{2b_1^2}}>0,$$

$$\mathrm{and}{\displaystyle \frac{d{p}_1^{**}}{d{b}_1}}={\displaystyle \frac{c_2-a_2+b_2\tilde{d}}{4b_2}}\ge 0\mathrm{if}{b}_2\ge {\displaystyle \frac{a_2-c_2}{\tilde{d}}},$$

with

$${\displaystyle \frac{d{p}_1^{**}}{d{b}_2}}={\displaystyle \frac{b_1(a_2-c_2)}{4b_2^2}}>0$$

$$\mathrm{and}{\displaystyle \frac{d{p}_2^{**}}{d{b}_2}}={\displaystyle \frac{2(c_1-a_1)+b_1\tilde{d}}{4b_1}}\ge 0\mathrm{if}{b}_1\ge {\displaystyle \frac{2(a_1-c_1)}{\tilde{d}}}.$$

Corollary 7.

As the pricing elasticity $b_i$ of company i ($i=1,2$) increases at the Stackelberg model equilibrium, company i would increase its price, while the price of company $j\ne i$ may either increase or decrease.

In addition, the differentiation with respect to the unit production costs $c_1$ and $c_2$ yields Corollary 8, as we have the following:

$${\displaystyle \frac{d{q}_1^{**}}{d{c}_1}}=-{\displaystyle \frac{1}{b_1}}<0,{\displaystyle \frac{d{q}_2^{**}}{d{c}_1}}={\displaystyle \frac{1}{2b_1}}>0,$$

and

$${\displaystyle \frac{d{q}_1^{**}}{d{c}_2}}={\displaystyle \frac{1}{2b_2}}>0,{\displaystyle \frac{d{q}_2^{**}}{d{c}_2}}=-{\displaystyle \frac{3}{4b_2}}<0.$$

Corollary 8.

Any increase in the unit production cost $c_i$ of company i ($i=1,2$) results in a decrease in its share of the total energy market at the Stackelberg model equilibrium, with a simultaneous increase in the share of company $j\ne i$.

We also observe that the differentiation of the Cournot model equilibrium prices with respect to the unit production costs yields:

$${\displaystyle \frac{d{p}_1^{**}}{d{c}_1}}={\displaystyle \frac{1}{2}}>0,{\displaystyle \frac{d{p}_2^{**}}{d{c}_2}}={\displaystyle \frac{1}{4}}>0,$$

$${\displaystyle \frac{d{p}_1^{**}}{d{c}_2}}={\displaystyle \frac{b_1}{4b_2}}>0,\mathrm{and}{\displaystyle \frac{d{p}_2^{**}}{d{c}_1}}={\displaystyle \frac{b_2}{2b_1}}>0.$$

Corollary 9.

An increase in the unit production costs $c_1$ or $c_2$ results in an increase in the Stackelberg model equilibrium prices.

To conclude the sensitivity analysis of the Stackelberg model equilibrium, we observe that the differentiation with respect to the base prices $a_1$ and $a_2$ yields:

$${\displaystyle \frac{d{q}_1^{**}}{d{a}_1}}={\displaystyle \frac{1}{b_1}}>0,{\displaystyle \frac{d{q}_2^{**}}{d{a}_1}}=-{\displaystyle \frac{1}{2b_1}}<0,$$

and

$${\displaystyle \frac{d{q}_1^{**}}{d{a}_2}}=-{\displaystyle \frac{1}{2b_2}}<0,{\displaystyle \frac{d{q}_2^{**}}{d{a}_2}}={\displaystyle \frac{3}{4b_2}}>0.$$

Corollary 10.

An increase in the base price $a_i$ ($i=1,2$) of company results in an increase in the Stackelberg model equilibrium market share for company i and a decrease in the market share for company $j\ne i$.

Table 1. Summary of the sensitivity analysis results.

Param.	${\mathit{q}}_1^{*}$	${\mathit{q}}_2^{*}$	${\mathit{p}}_1^{*}$	${\mathit{p}}_2^{*}$	${\mathit{q}}_1^{**}$	${\mathit{q}}_2^{**}$	${\mathit{p}}_1^{**}$	${\mathit{p}}_2^{**}$
$b_1\nearrow$	↘	↗	→	↗	↘	↗	→	↗
$b_2\nearrow$	↗	↘	↗	→	↗	↘	↗	→
$c_1\nearrow$	↘	↗	↗	↗	↘	↗	↗	↗
$c_2\nearrow$	↗	↘	↗	↗	↗	↘	↗	↗
$a_1\nearrow$	↗	↘	↗	↘	↗	↘	↗	↘
$a_2\nearrow$	↘	↗	↘	↗	↘	↗	↘	↗

summarizes the sensitivity analysis corollaries on Cournot and Stackelberg models’ equilibria quantities and prices.

Example 2.

As in Example 1, let us assume that $a_1=32$, $a_2=30$, $b_1=4$, $b_2=3$, and $c_1=10$, $c_2=8$, with an expected base demand of $\tilde{d}=6$. For the Stackelberg game model, the Nash equilibrium quantities are $q_1^{**}=4.83$ and $q_2^{**}=4.25$, with prices $p_1^{**}=19.67$ and $p_2^{**}=20.75$. The Stackelberg model equilibrium profit for company 1 is ${\pi }_1^{*}=46.72$, and for company 2, it is ${\pi }_2^{*}=54.19$. It is noteworthy that despite company 1 supplying 50% more energy at the Stackelberg model equilibrium, its profit only increases by 12.5% compared with the Cournot model equilibrium. Furthermore, despite the higher total energy supplied to the market at the Stackelberg model equilibrium, $q_1^{**}+q_2^{**}=9.08$, compared with the Cournot model equilibrium with $q_1^{*}+q_2^{*}=8.27$, the combined profit for both companies decreases from $118.21$ to $100.91$.

3.3. Cartel Model

The example above underscores the potential advantage for the two companies to collaborate and jointly make decisions as a cartel. In such a scenario, their combined profit can be expressed as follows:

$$\pi (q_1,q_2)={\pi }_1(q_1,q_2)+{\pi }_2(q_1,q_2).$$

Setting $\nabla \pi =0$ results in the following system of equations:

$$\begin{array}{cc}2b_1{q}_1+(b_1+b_2)q_2& =b_1\tilde{d}+a_1-c_1,\\ (b_1+b_2)q_1+2b_2{q}_2& =b_2\tilde{d}+a_2-c_2.\end{array}$$

We obtain:

$$q_1^{***}={\displaystyle \frac{2b_2(c_1-a_1)+(b_1+b_2)(a_2-c_2)+\tilde{d}{b}_2(b_2-b_1)}{{(b_1-b_2)}^2}},$$

and

$$q_2^{***}={\displaystyle \frac{2b_1(c_2-a_2)+(b_1+b_2)(a_1-c_1)+\tilde{d}{b}_1(b_1-b_2)}{{(b_1-b_2)}^2}}.$$

The total quantity supplied by the cartel would then be:

$$Q^{***}=q_1^{***}+q_2^{***}=\tilde{d}+{\displaystyle \frac{(a_1-c_1)+(a_2-c_2)}{b_1-b_2}}.$$

Assumptions 1–5 do not guarantee that both $q_1^{***}$ and $q_1^{***}$ are strictly positive. Thus, the pricing parameters $a_1$, $b_1$, $a_2$, and $b_2$ of the two companies must adhere to the condition stipulated in Assumption 6 to ensure that $q_1^{***}$ and $q_2^{***}$ remain positive.

Assumption 6.

We assume that the Cartel model parameters satisfy the conditions:

$${\displaystyle \frac{2b_2(a_1-c_1)+(b_1+b_2)(c_2-a_2)}{b_2}}\le (b_2-b_1)\tilde{d}$$

and

$${\displaystyle \frac{2b_1(a_2-c_2)+(b_1+b_2)(c_1-a_1)}{b_1}}\le (b_1-b_2)\tilde{d}.$$

Proposition 3.

Under Assumptions 1–3 and Assumption 6, the Cartel model Nash equilibrium prices would be:

$$p_1^{***}={\displaystyle \frac{b_1(c_1-c_2)+a_2{b}_1-a_1{b}_2}{b_1-b_2}}$$

and

$$p_2^{***}={\displaystyle \frac{b_2(c_1-c_2)+a_2{b}_1-a_1{b}_2}{b_1-b_2}}$$

Example 3.

In our previous example, the total quantity provided by the cartel would then be $Q^{***}=(q_1^{***}+q_2^{***})=4+2=6$. The energy prices set by the cartel would be $p_1^{***}=32$ and $p_2^{***}=30$. The total profit of the cartel is 132, with company 1 receiving 88 and company 2 receiving 44. This total profit surpasses that of any other configuration. Despite producing less energy, the cartel ensures that prices remain high, resulting in increased profit.

Figure 2, Figure 3 and Figure 4 illustrate the variation the individual and total quantities provided by the two energy companies through the three Examples 1–3.

4. Discussion

This discussion expands upon the sensitivity analysis concerning the smart grid’s forecasted base demand within the context of the three proposed game models. First of all, the differentiation of the Cournot model Nash equilibrium quantities and prices with respect to the base expected demand $\tilde{d}$ yields:

$${\displaystyle \frac{d{q}_1^{*}}{d\tilde{d}}}={\displaystyle \frac{1}{3}}>0{\displaystyle \frac{d{q}_2^{*}}{d\tilde{d}}}={\displaystyle \frac{1}{3}}>0,$$

and

$${\displaystyle \frac{d{p}_1^{*}}{d\tilde{d}}}={\displaystyle \frac{b_1}{3}}>0,{\displaystyle \frac{d{p}_2^{*}}{d\tilde{d}}}={\displaystyle \frac{b_2}{3}}>0.$$

Corollary 11.

An increase in the expected demand $\tilde{d}$ results in an equal increase in the Cournot model equilibrium quantities for the two companies.

Secondly, the differentiation of the Stackelberg model Nash equilibrium quantities and prices with respect to the expected base demand $\tilde{d}$ yields the following:

$${\displaystyle \frac{d{q}_1^{**}}{d\tilde{d}}}={\displaystyle \frac{1}{2}}>0{\displaystyle \frac{d{q}_2^{**}}{d\tilde{d}}}={\displaystyle \frac{1}{4}}>0,$$

and

$${\displaystyle \frac{d{p}_1^{**}}{d\tilde{d}}}={\displaystyle \frac{b_1}{4}}>0,{\displaystyle \frac{d{p}_2^{**}}{d\tilde{d}}}={\displaystyle \frac{b_2}{4}}>0.$$

Corollary 12.

An increase in the expected demand $\tilde{d}$ results in an increase in the Stackelberg model equilibrium quantities for the two companies.

We can already observe that both the Cournot and Stackelberg model equilibrium prices demonstrate a linear sensitivity concerning the forecasted base demand variation. However, it is noteworthy that the Cournot model equilibrium prices exhibit a higher level of sensitivity.

Finally, the differentiation of the Cartel model equilibrium quantities and prices with respect to the expected base demand $\tilde{d}$ yields:

$${\displaystyle \frac{d{q}_1^{***}}{d\tilde{d}}}=-{\displaystyle \frac{b_2}{b_1-b_2}}{\displaystyle \frac{d{q}_2^{***}}{d\tilde{d}}}={\displaystyle \frac{b_1}{b_1-b_2}},$$

and

$${\displaystyle \frac{d{p}_1^{***}}{d\tilde{d}}}=0,{\displaystyle \frac{d{p}_2^{***}}{d\tilde{d}}}=0.$$

Corollary 13.

An increase in the expected demand $\tilde{d}$ leads to opposite fluctuations in the equilibrium quantities of the two companies in the Cartel model.

Conversely, it is evident that the equilibrium market prices in the Cartel model remain unchanged despite variations in the expected base demand $\tilde{d}$. This observation underscores the economic soundness of the Cartel model, as it effectively stabilizes energy prices.

Figure 5, Figure 6 and Figure 7 depict the alterations in the optimal quantities of supplied energy $q_1^{*}$, $q_2^{*}$, $q_1^{**}$, $q_2^{**}$, $q_1^{***}$, and $q_2^{***}$, respectively, as the forecasted base demand $\tilde{d}$ fluctuates across 100 consecutive periodic predictions that adhere to Assumptions 4–6. Considering the parameter values outlined in Examples 1–3 and Assumption 6, we have ${\displaystyle \frac{22}{4}}\le \tilde{d}\le {\displaystyle \frac{22}{3}}$.

5. Conclusions

Demand response management (DRM) plays a pivotal role in shaping the future development of electricity smart grids. This paper introduced three distinct game-theoretic duopoly models aimed at addressing the smart grid demand response management problem. It outlines several rational assumptions concerning the variables, functions, and parameters of the models. The first model takes on a Cournot duopoly format, offering a unique closed-form equilibrium solution. Similarly, the second model follows a Stackelberg duopoly structure, also presenting a unique closed-form equilibrium solution. After comparing the economic viability of the two model equilibria and evaluating their sensitivity to parametric changes, the paper proposes a third model structured as a Cartel and discusses its advantages over the preceding models. Subsequently, the paper delves into how demand forecasting impacts the equilibrium quantities and pricing solutions of each model.

The Cartel model demonstrates its effectiveness in stabilizing energy prices. Moreover, it yields a total profit surpassing that of any other model. Despite producing less energy, the cartel ensures that prices remain high, thereby leading to increased profits.

Although the Cartel model offers valuable theoretical insights into how energy providers might collude to maximize their profits, it may fall short when considering the impact of regulatory frameworks designed to prevent this kind of collusion. Furthermore, it is important to note that preventing such collusive behavior requires robust governance structures and highly effective regulatory enforcement agencies. The ability to deter collusion is not merely a matter of having regulations in place; it demands strong institutional capacity, stringent oversight, and the consistent application of penalties. Without a firm commitment to enforcing these regulations, even the most well-designed policies may fail to curb anti-competitive practices. Therefore, the success of regulatory efforts hinges on both the strength of governance and the efficacy of enforcement mechanisms.

The three models presented in this paper lay the groundwork for a wide range of potential extensions, including different types of pricing and cost functions. Furthermore, these models could be expanded to account for constraints such as production, distribution, and storage capacities. They also offer the flexibility to incorporate various duopoly or oligopoly market structures.