Abstract
Natural-gas and electric power systems and their corresponding markets have evolved over time independently. However, both systems are increasingly interdependent since combined cycle gas turbines that use natural gas to produce electricity increasingly couple them together. Therefore, suitable analysis techniques are most needed to comprehend the consequences on market outcomes of an increasing level of integration of both systems. There is a vast literature on integrated natural-gas and electric power markets assuming that the two markets are operated centrally by a single operator. This assumption is often untrue in the real world, which necessitates developing models for these interdependent yet independent markets. In this vein, this paper addresses the gap in the literature and provides analytical Nash-Cournot equilibrium models to represent the joint operation of natural-gas and electric power markets with the assumption that the market participants in each market make their own decisions independently seeking the maximum profits, as often is the case in the real world. We develop an analytical equilibrium model and apply the Karush-Kuhn-Tucker (KKT) approach to obtain Nash-Cournot equilibria for the interdependent natural-gas and electric power markets. We use a double-duopoly case to study the interaction of both markets and to derive insightful analytical results. Moreover, we derive closed-form analytical expressions for spot-market equilibria in both natural-gas and electric power markets, which are relevant and of practical significance for decision makers. We complement the double-duopoly study with a detailed sensitivity analysis.
THE natural-gas systems and electric power systems were created independently and have evolved over time as two separate infrastructures with limited or no coordination [
The literature on the analysis of natural-gas and electric power markets individually is diverse. However, the literature on the coordinated analysis of the integrated natural-gas and electric power markets as well as the tools to comprehend their interdependency is still limited [
The approach of the first cluster, which can be found in the works of [
In this paper, we adopt the the approach of the second cluster, which can be found in [
Equilibrium models have been used in the literature to analyze the non-integrated natural-gas [
1) We develop an analytical equilibrium model to analyze the interdependency of the natural-gas and electric power markets. Such analytical models based on classical optimization theory allow characterizing the optimality of the outcomes as well.
2) We use a Nash-Cournot equilibrium model to derive relevant equilibria between natural-gas and electric power markets when there is limited coordination or information exchange between the two markets, as often is the case in the real world.
3) Our approach allows us to derive closed-form analytical expressions for spot-market equilibria in both natural-gas and electric power markets, which allows policy-makers and regulators to examine the potential outcomes of the proposed policies and regulations quickly without the usual complexity of other models.
4) We use a double-duopoly case along with detailed sensitivity analyses to investigate the interaction of both markets and derive insightful analytical results.
The remainder of this paper is organized as follows. Section II presents the proposed market model and shows some of its structural properties. The model is demonstrated via stylized analysis and numerical examples in Section III. Section IV draws the conclusions.
In this section, we present the proposed Nash-Cournot equilibrium model of the natural-gas and electric power markets.
We model the natural-gas market as consisting of a set of producing firms. The natural-gas producers compete in natural-gas spot market to serve two groups of natural-gas consumers, i.e., electric power firms who convert natural gas to electric power and other natural-gas consumers. The natural-gas producing firms are modeled as being quantity-setting competitors. This means that the firms simultaneously determine their production levels in the spot market, after which the market price is adjusted to clear the market, i.e., the demand is exactly equal to the supply. Let and , measured in cf, denote the production levels of natural-gas producing firm being sold to the electric power sector and other natural-gas consumers, respectively, in the spot market. We assume that the firms have quadratic production costs. Thus, the cost to firm of producing and in a spot market is:
(1) |
where and are the non-negative coefficients of the quadratic cost function of natural-gas producing firm . We assume that the demand in the spot market changes linearly with the price. Thus, the natural-gas spot price, which is measured in $/Ccf, is given by:
(2) |
where and are the non-negative coefficients of the linear price-demand function in natural-gas spot market; and is the total natural-gas production of all producing firms in the natural-gas market. We assume hereafter that is sufficiently large compared with for all firms to produce a strictly positive amount of energy in the spot market.
The natural-gas spot market consists of the firms determining their production levels to maximize their profits, meaning that the production level of firm is derived from the following profit-maximization problem:
(3) |
s.t.
(4) |
where and are the profit and the maximum production capacity of firm , respectively. We also exclude the non-negativity constraint because we assume that the market price is sufficiently high for all firms to produce a strictly positive quantity.
We apply the Karush-Kuhn-Tucker (KKT) optimality conditions [
(5) |
(6) |
(7) |
where is the Lagrange multiplier associated with the capacity constraint (4). To derive a more refined version of (5) and (6), we calculate equivalent terms for the partial derivatives in these equations. Considering (2), we can obtain:
(8) |
(9) |
Note that and are zero in general unless firm has reached its maximum capacity, in which they take the value of -1. Therefore, we can obtain:
(10) |
(11) |
where is the set of natural-gas producing firms working at the maximum capacity in the spot market.
Replacing (10) and (11) in (8) and (9) renders:
(12) |
(13) |
Considering (12) and (13) together with the KKT conditions (5)-(7), we derive closed-form analytical expressions for the total production capacity of firm in the spot market and the spot market clearing price in (14) and (15), respectively.
(14) |
(15) |
where denotes the value of the variables at the obtained equilibrium.
Similar to the natural-gas market, we model the electric power market as consisting of a set of electric power firms. The electric power firms compete in the spot market to serve the electricity consumers. The electric power firms are assumed to have two ways of generating electricity, i.e., converting natural gas to electricity and converting non-natural-gas energy sources to electricity. The electric power firms, modeled as quantity-setting competitors, simultaneously determine their generation levels in the spot market, after which the market price adjusts to clear the market.
Let denote the natural gas being converted to electricity by electric power firm , which is measured in cf, at a conversion rate of and let denote the generation level, which is measured in MW, of electric power generated by firm from converting non-natural-gas energy sources to electricity. We assume that the firms have quadratic generation costs. Thus, the cost of firm for generating electricity from non-natural-gas energy sources and converting natural gas to electricity in a spot market is:
(16) |
where and are the non-negative coefficients of the quadratic cost function of electric power firm . We assume that the electricity demand in the spot market changes linearly with the price. Thus, the spot electricity price, which is measured in $/MW, is given by:
(17) |
where and are the non-negative coefficients of the linear price-demand function in electric power spot market; and and are the sets of generation levels of all electric power firms from natural-gas and non-natural-gas energy sources, respectively.
We note that (16) and (17) couple together the natural-gas and electric power markets. Furthermore, the total natural gas consumed by the firms in the electric power market to generate electric power should match the total natural gas sold by the firms in the natural-gas market to the firms in the electric power market. Thus, we can obtain (18), which is used not only to check for the consistency of an equilibrium but also to differentiate the total natural gas sold to firms in the electric power market and other natural-gas consumers.
(18) |
The electric power spot market consists of the firms determining their generation decisions to maximize their profits, meaning that the generation decision of firm is derived from the following profit-maximization problem:
(19) |
s.t.
(20) |
(21) |
where , , and are the profit, the maximum natural-gas-based generating capacity, and the maximum non-natural-gas-based generating capacity of the firm , respectively. We also exclude the non-negativity constraint because we assume that the market price is sufficiently high for all firms to produce a strictly positive quantity.
We apply the KKT optimality conditions to the problem (19)-(21) and derive the following KKT conditions, which are both necessary and sufficient for a global optimum [
(22) |
(23) |
(24) |
(25) |
where and are the Lagrange multipliers associated with the capacity constraints (20) and (21), respectively. To derive a more refined version of (22) and (23), we calculate equivalent terms for the partial derivative terms in these equations. Note that the two energy sources of generating electric power, i.e., natural-gas and non-natural-gas energy sources, are independent, we can obtain:
(26) |
Considering (17) and (26), we can obtain:
(27) |
(28) |
Therefore, (22) and (23) can be rewritten as:
(29) |
(30) |
Formulae (24), (25), (29), and (30) provide the solution of the problem in the electric power market and yield analytical expressions for the generation levels of the firms in the electric power market. Considering (24) and (25) may or may not be binding for each electric power firm yields the following four cases to obtain the generation portfolio of each electric power firm as a function of the market clearing price. We further analyze these formulae considering the following four cases.
1) Case 1: and . Considering (25) and (30), we can obtain the generation level of electric power from natural gas for firm as:
(31) |
(32) |
2) Case 2: and . Considering (24) and (29), we can obtain the generation level of electric power from non-natural-gas energy sources for firm as:
(33) |
(34) |
3) Case 3: and . Considering (24), (25), (29), and (30), we can obtain:
(35) |
(36) |
Equations (
(37) |
4) Case 4: and . The total electric power generation of firm can be obtained as:
(38) |
To calculate the market clearing price, the total electric power generation of the firms derived from (32), (34), (37), (38) are plugged in (17). Note that each firm may fall into any of the four aforementioned cases in the spot market, which results in potential equilibria in the electric power market, where denotes the number of electric power firms. Similarly, the natural-gas market yields potential equilibria. Therefore, we may obtain potential equilibria for the integrated natural-gas and electric power markets.
We illustrate the proposed equilibrium model and solution methodology with a stylized case study. This analysis considers a duopoly in the natural-gas market and a duopoly in the electric power market with the values for the parameters of the model. This analysis is illustrative because we can present the interactions between the two markets and highlight how the equilibria are obtained.
In this subsection, we analyze a stylized double-duopoly case in which there are two producers in the natural-gas market and two electric power firms in the electric power market. The power equilibrium outcomes (14), (15), coupling condition (18), and natural-gas equilibrium outcomes (17), (32), (34), (37), and (38) are used.
Given that there is at least one consistent equilibrium in the natural-gas market, we may examine the existence of consistent equilibria in the electric power market. Note that there exist 16 potential equilibria for each consistent equilibrium of the natural-gas market, i.e., 32 equilibria in total.
As illustrated in Tables
We conclude our analysis of the stylized double-duopoly case with a number of sensitivity analyses.

Fig. 1 Production level of producers in natural-gas market as a function of .

Fig. 2 Market clearing price in natural-gas market as a function of .
Figures

Fig. 3 Market clearing price in electric power market as a function of .

Fig. 4 Generation level of electric power firm as a function of .
In general, the increase in causes a decrease in the market clearing price as well as the total generation level of the electric power firms. If a firm is not operating at the maximum capacity, it lowers its generation level as the market becomes less attractive with the increase of . As a case in point, the dotted-line arrows in Figs.

Fig. 5 Generation level of electric power firm as a function of .
Next, we analyze how a change in the cost parameters of an electric power firm, namely in this case study, influences the equilibria in electric power market. Figures

Fig. 6 Generation level of electric power firm as a function of .

Fig. 7 Generation level of electric power firm as a function of .
Intuitively, one expects that an increase in , i.e., an increase in the generation cost of electric power firm , causes a decrease in the generation level of electric power firm and potentially an increase in the generation level of electric power firm , if electric power firm is not operating at the maximum capacity. The arrows show a set of equilibria that are aligned with this intuition. We use different line formats for the arrows and lines to draw attention to some equilibria but use the same line format for the arrows and lines in related figures to illustrate the behavior of the equilibria with respect to the changes in the parameter values of the sensitivity analysis. The equilibria highlighted by the red line, however, counter this intuition. Albeit the available capacity, the generation level of electric power firm 2 does not change while the generation level of electric power firm 1 increases. Such equilibria and market behavior stem from the profit-maximization nature of these firms. A higher profit may be earned occasionally, as it is the case for electric power firm , if a lower price helps a firm sell more of its product. Such market complexities necessitate developing analytical models such as the one proposed in this paper to shed light on the behavior of the market participants, and more importantly assist the stakeholders and decision-makers of the interdependent natural-gas and electric power markets to make more informed decisions when pursuing new policies, regulations, market incentives, and others.
Finally, we conduct a sensitivity analysis to evaluate the impact of the natural gas to power conversion factor on the total generation level as well as the market clearing price of the two electric power firms. Figures

Fig. 8 Total generation level of electric power firm as a function of .

Fig. 9 Total generation level of electric power firm as a function of .

Fig. 10 Market clearing price in electric power market as a function of .
A higher conversion factor makes the natural gas a more competitive energy source in the electric power market and enables the electric power firms to supply more electricity to their customers. Furthermore, Figs.

Fig. 11 Natural gas purchased by electric power firm as a function of .

Fig. 12 Natural-gas purchased by electric power firm as a function of .
The proposed model allows the derivation of analytical expressions that characterize the multi-firm equilibria in the interdependent natural-gas and electric power markets. Such analytical expressions may be utilized to provide insights on the outcomes and characteristics of operation decisions. Hence, we derive these analytical expressions for the double-duopoly case, as discussed in the previous section.
As shown in the figures throughout the paper, we find the cases in which a set of parameter values may yield multiple equilibria. As is common in non-cooperative games, these equilibria introduce trade-offs in terms of which market agents, i.e., firms and consumers, benefit from one equilibrium relative to another. On the other hand, another set of parameter values may not yield consistent equilibria, which suggests that only mixed-strategy equilibria which are not found by the proposed model and which are beyond the scope of our work may exist. Therefore, there is no way to guarantee the existence of convergence to one consistent equilibrium, and there is no way to prove that a particular market equilibrium has been achieved until the spot markets clear in both markets.
Further, we discuss that one of the key advantages of the proposed model is that it allows identifying general trends and interactions in both natural-gas and electric power markets, which helps comprehend the general behavior of the two markets. As a case in point, any of the beneficiaries of the proposed model may need to investigate how to decrease the interdependency of the two markets and intuitively may expect that higher efficiency converters, i.e., higher values of , will increase the attractiveness of natural gas for electric power firms, drive up the consumption of natural gas in electric power market, and make the two markets more interdependent. In such scenarios, the proposed model can be utilized to test the hypothesis and provide insightful results quickly. In this specific scenario, the proposed model is utilized to test this hypothesis. As shown in Figs.
In future, we would like to expand on the findings of this research, by carrying out detailed numerical analysis using the actual data of real-world natural-gas and electric power markets, considering capacity additions in the proposed model, and using a supply-function equilibrium model instead of the Cournot model.
We propose a Nash-Cournot equilibrium model pertaining for interdependent natural-gas and electric power markets. This model allows us to comprehend the impact on market outcomes with an increasing integration of the natural gas and electric power markets. The general analytical results derived based on this model are useful to inform the decision-making processes of both regulators and operators. In a simple double-duopoly case, we identify a large number of equilibria, which reveals the inherent complexity of the considered problem. We illustrate the evolution of such equilibria via sensitivity analysis, which renders some counter-intuitive market behavior outcomes.
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