Two-stage joint equilibrium model of electricity market with tradable green certificates

Abstract

Renewable portfolio standard (RPS) with tradable green certificate (TGC) scheme has important influences on the market equilibrium outcomes and generation firms’ strategic behaviors. The main objective of this paper is to investigate that under the RPS with TGC scheme, who and how to exercise the market power, and to what extent market powers are exercised in the electricity wholesale and TGC markets. This is achieved by firstly proposing a two-stage joint equilibrium model based on the oligopolistic competition equilibrium theory. The model is then formulated as an equilibrium problem with equilibrium constraints (EPEC) by using the backward induction method, which is further solved by the nonlinear complementarity approach. Finally, simulation results show that renewable firms tend to withhold some of TGCs to raise the TGC prices when the RPS is relatively low, otherwise they choose to cut down their electricity output to reduce the volume of TGC and raise the TGC price. Moreover, facing the increasing TGC price, fossil fuel firms tend to withhold their electricity output to decrease the demand of TGCs and lower the TGC price. This study has meaningful implications for design of the electricity markets with TGC market.

Keywords

Equilibrium analysis renewable portfolio standard tradable green certificate Cournot competition supply function competition

Introduction

Owing to the ongoing concerns regarding energy crisis, sustainable development and smart grid, renewable energy such as wind, solar and hydroelectricity, receives increasing attention around the world (Du et al., 2017; Kolhe, 2012). Compared with conventional fossil energy, the main characteristics of these energies are environment-friendly and renewable. For these reasons, many countries set specific targets for the renewable energy development. In Europe, France, Germany and Britain put forward the goals that the share of renewable energy should reach 23%, 18% and 15% of the total energy consumption by 2020, respectively (Tamas et al., 2010). In Asia, Japan claims that the renewable electricity must meet 20% of its power demand by 2020 (Wang et al., 2014), and the latest long-term target of China’s wind power development is that the wind power cumulative installed capacity is expected to reach 200 GW, 400 GW, and 1000 GW in 2020, 2030, and 2050, respectively (Lu et al., 2015).

Aiming to achieve the expected targets, many support policies are proposed to bolster the renewable energy development. Currently, the common support policies of renewable energy can be roughly divided into two categories. One is not market-based, such as Feed-in Tariff (FIT) (Shahrouz and Alma, 2014; Tamas et al., 2010; Wang et al., 2014), Carbon Tax (CT) (Pereira et al., 2016; Shahrouz and Alma, 2014). The advantage of these policies is easy to implement, but may have poor efficiency. The other is based on market mechanism, such as Emission Trading Scheme (ETS) (Ju et al., 2016; Limpatioon et al., 2014; Wang et al., 2014) and Renewable Portfolio Standard (RPS) with Tradable Green Certification (TGC) scheme (Ding et al., 2013; Farooq et al., 2013; Ju et al., 2016; Nielsen and Jeppesen, 2003; Shahrouz and Alma, 2014; Tamas et al., 2010; Wang et al., 2014). Especially in recent years, the fast development of renewable generation technology makes its generation cost decrease sharply, participation of the renewable generation firms in the electricity market is becoming an overwhelming trend. Under this background, the market-based support policies will be more and more popular.

RPS and TGC scheme is an advanced market-based support scheme to promote the renewable energy in power industry. The RPS policy requires the electricity producers, retailers or consumers to derive a certain percent of electricity generation/consumption from renewable resources. To meet the RPS requirement, apart from self-generation, these RPS obligation undertakers also can procure renewable power via bilateral contracts or purchase the TGCs from the TGC market. In particular, TGC is an efficient and widely-used tool to meet RPS requirement for the RPS obligation undertakers. TGC divides the electricity generated by units into two parts: the physical electricity that can be traded in the electricity market, and the associated green certificates that can be traded in the TGC market. Our paper considers a TGC market. In this TGC market, the sellers are the generation firms of renewable source, and the buyers are the RPS obligation undertakers; the market clearing price of TGC is determined by the interplay of the demand and supply of TGCs.

Many researchers have been devoted to studying the impact of RPS with TGC on the electricity market. Mozumder and Marathe (2004) discussed the benefits of integrated TGC market. Gillenwater (2008a,b) explained the challenges of using RPS with TGC scheme to offset pollution emissions in the electricity market. Shahrouz and Alma (2014), Ju et al. (2016) and Wang et al. (2014) compared the RPS with TGC scheme and other renewable support policies, and have shown that RPS with TGC scheme is an appropriate policy when a market view policy is applied, which can effectively increase the share of renewable energy power and lead to renewable resource diversity. Using UK data, Tamas et al. (2010) compared FIT and TGC in a formal model; the results show that social welfare under TGC is consistently higher than FIT for a wide range of values of the parameters. Langniss and Wiser (2003) described the design of the Texas RPS and offered an early assessment of this program; the results show that an RPS can effectively spur renewables development and encourage competition among renewable energy producers.

Owing to the fact that the current electricity wholesale markets are more akin to oligopolistic markets, generation firms with large market shares have the ability to manipulate the market price through their strategic bidding behaviors, which can seriously affect the overall efficiency and reliability of the electricity market (Helman, 2006; Li et al., 2011; Wang et al., 2004; Yu et al., 2010). The similar situation might also happen in the TGC market. Moreover, because of the interplay between electricity wholesale and TGC markets, the market power in the TGC market can spillover to the wholesale market, and further impact the generation firms’ bidding strategies in the wholesale market. Therefore, it is necessary to study the interactive influence between electricity wholesale and TGC markets, which is helpful to the design and operation of electricity markets with TGC market.

Equilibrium theory based on oligopolistic competition is a powerful tool to study the generation firms’ strategic behaviors and market power problems, in which the Cournot and supply function (SF) equilibrium models are two popular methods for electricity market equilibrium analysis (An et al., 2015; Li et al., 2011; Wang et al., 2004). More recently, some researchers committed to studying the market power associated with renewable energy in the TGC market, which were not enough. Jensen and Skytte (2002) proposed a static equilibrium model including TGC market to study the interplay of the electricity and TGC markets, but the TGC market was perfect competition and the strategic behaviors of generation firms in the TGC market were not considered. Amundsen and Bergmen (2012) considered the imperfect competition in the TGC market. It is assumed that the consumers and retailers need to undertake RPS obligation and show that the producers with abundant renewable resource may exercise the market power using TGC market by holding back TGCs. Zhou and Tamas (2010) developed an equilibrium model to examine the market power problem when the conventional firm was merged with the renewable firm. The model assumed that both the electricity wholesale and TGC markets were imperfect, but only one renewable firm was considered, the TGC market was monopolistic. Tanaka and Chen (2013) developed a dominant firm-competitive fringe model to account for market power and showed that market power could have a significant impact on the TGC and electricity prices. In particular, when a nonrenewable generator was a dominant firm, the nonrenewable firm had a strong incentive to lower the TGC price for reducing TGC costs. But this competition model was not widely used in the study of electricity market.

Our paper proposes a two-stage joint model based on the oligopolistic competition equilibrium theory to address the market power problems associated with renewable energy in the oligopolistic electricity wholesale and TGC markets. The main contributions of this paper are summarized as follows:

A two-stage game model is proposed to analyze generation firms’ strategic behaviors in electricity wholesale and TGC markets, where the two markets are assumed to be imperfect. In the first stage, the renewable generation firms compete a la supply function competition mode to sell the TGCs in the TGC market. Then, in the second stage, all generation firms compete a la Cournot competition mode to sell their electricity in the wholesale market. All the generation firms are obligated to undertake RPS requirement, and the renewable firms are the sellers of TGC market.

The backward induction method is utilized to solve this complicated joint equilibrium model, by which, the model is reformulated as an equilibrium problem with equilibrium constraints (EPEC), and the nonlinear complementarity approach is used to solve this EPEC problem.

The strategic behaviors of generation firms in the two markets are examined. Simulation results show that the introduction of TGC market complicates the generation firms’ strategic behaviors. The renewable firms can exercise the market power not only by holding back some of TGCs, but also by cutting back the electricity output of wholesale market, which way is chosen depending on the RPS value. Facing the different TGC prices, the fossil fuel firms also may exercise market power in the electricity wholesale market.

The rest of this paper is organized as follows: the second section formulates the strategic electricity wholesale and TGC market competition. The third section gives the solution method of the joint equilibrium model. The fourth section presents numerical examples to verify the effectiveness of the method. The fifth section concludes the paper.

The two-stage joint model

Assumption

Figure 1 shows the market architecture. It is assumed that there are N strategic fossil fuel firms (G_f,₁, G_f,₂,…, G_f,_N) and R strategic renewable firms (G_r,₁, G_r,₂,…, G_r,_R) competing in the electricity wholesale market. All these generation firms are rational and risk neutral. The paper supposes that the environmental regulation requires that all generation firms are subject to a minimum share of their supply from renewable energy. The generation firms fulfill the mandatory quota by a TGC market, in which the sellers of TGCs are the renewable firms, the buyers are the fossil fuel firms. Let $α$ denote mandatory RPS requirement, $α \in (0, 1)$ . Assume that One MW green energy gives rise to one unit TGC.

Figure 1.

Market architecture.

A stylised joint model is developed to formulate a two-stage game of an electricity wholesale market and a TGC market. In the first stage, the renewable generation firms sell the excess TGCs in the TGC market. Then, in the second stage, all generation firms sell their electricity in the wholesale market. The affine supply function competition model is used to simulate competitive bidding for the TGC market, and the electricity wholesale market is formulated with the Cournot competition model.

The cost function of fossil fuel firms takes a form of quadratic function as follows (Li et al, 2015; Tanaka and Chen, 2013; Xiao et al., 2016)

c_{f, i} (q_{f, i}) = 0.5 c_{f, i} q_{f, i}^{2} + b_{f, i} q_{f, i} + a_{f, i}, i = 1, 2, \dots, N

(1)

where $c_{f, i}$ , $b_{f, i}$ , $a_{f, i}$ are the non-negative generation cost parameters of fossil fuel firm i, $q_{f, i}$ is the electricity output of fossil fuel firm i’s in the wholesale market.

Unlike the fossil fuel energy generations, renewable energy generations have no fuel cost. Because of the intermittence and randomness, renewable energy generation needs costs, such as operation and maintenance cost, spinning reserve cost. Following the assumption of many existing work (Amundsen and Bergmen, 2012; Tanaka and Chen, 2013; Xiao et al., 2016; Zhou and Tamas, 2010) the cost function of renewable firms takes the following form of quadratic function

c_{r, j} (q_{r, j}) = 0.5 c_{r, j} q_{r, j}^{2} + b_{r, j} q_{r, j} + a_{r, j}, j = 1, 2, \dots, R

(2)

where $c_{r, j}$ , $b_{r, j}$ , $a_{r, j}$ are the non-negative generation cost parameters of renewable firm j, $q_{r, j}$ is the electricity output of renewable firm j’s in the wholesale market.

Assume that the renewable firm j takes bid form of linear supply function in the TGC market

q_{j}^{TGC} = x_{j} + d_{j} p^{TGC}, j = 1, 2, \dots, R

(3)

where $d_{j}$ is the slope of linear supply function of renewable firm j, $x_{j}$ is the intercept of linear supply function of renewable firm j. $q_{j}^{TGC}$ is the actual traded TGC volume of renewable firm j in the TGC market, $p^{TGC}$ is the TGC price at time t. In this paper, x_j is chosen as the strategic variable of renewable firm j in the TGC market, and d_j is assumed to be fixed.

For a time period t (1h), the inverse demand function of electricity wholesale market can generally be expressed by the following linear function

p = A - ξ Q = A - ξ (\sum_{i = 1}^{N} q_{f, i} + \sum_{j = 1}^{R} q_{r, j})

(4)

where $A$ , $ξ$ are the non-negative coefficients of inverse demand function, $Q$ is the total electricity demand of the wholesale market, p is the electricity price at time t. The transmission constraints are not considered in this paper.

Joint equilibrium model

Electricity wholesale market competition equilibrium model

Under the assumption of Cournot-type competition, each rational generation firm maximizes its profit by setting its electricity output.

The profit of each fossil fuel firm is equal to the revenue for selling its electricity in the wholesale market, minus its generation costs and the payment for buying TGCs from the TGC market to fulfill its RPS requirement. Thus, the fossil fuel firm i’s optimization problem in the wholesale market can be described as

\underset{q_{f, i}}{M a x} π_{f, i} = [A - ξ (\sum_{n = 1}^{N} q_{f, n} + \sum_{m = 1}^{R} q_{r, m})] q_{f, i} - c_{f, i} (q_{f, i}) - α p^{T G C} q_{f, i}

(5)

s . t . q_{f, i}^{min} \leq q_{f, i} \leq q_{f, i}^{max}, i = 1, 2, \dots, N

(6)

where $π_{f, i}$ is the fossil fuel firm i’s profit, $q_{f, i}^{min}$ , $q_{f, i}^{max}$ are the minimum and maximum output limits of fossil fuel firm i in the wholesale market.

The profit of each renewable firm is equal to the revenue for selling its electricity in the wholesale market plus the revenue for selling the excess TGCs in the TGC market, minus its generation costs. Thus, the renewable firm j’s optimization problem in the wholesale market can be described as

\underset{q_{r, j}}{M a x} π_{r, j} = [A - ξ (\sum_{n = 1}^{N} q_{f, n} + \sum_{m = 1}^{R} q_{r, m})] q_{r, j} - c_{r, j} (q_{r, j}) + p^{T G C} q_{j}^{T G C}

(7)

s . t . q_{r, j}^{min} \leq q_{r, j} \leq q_{r, j}^{max}, j = 1, 2, \dots, R

(8)

where $π_{r, j}$ is the renewable firm j’s profit, $q_{r, i}^{min}$ , $q_{r, i}^{max}$ are the minimum and maximum output limits of renewable firm j in the wholesale market.

The Cournot equilibrium model for the wholesale market can be obtained by gathering N+R generation firms’ optimization problems expressed by (5)–(8).

TGC market competition equilibrium model

For renewable firms, apart from selling the green energy in the wholesale market, they also can sell the surplus renewable energy in the TGC market in form of the green certificates to obtain the extra revenue.

Under the assumption of affine SF-type competition as in (3), each rational renewable firm aims to maximize its profit by choosing its strategic variables $x_{j}$ , $j = 1, 2, \dots, R$ in the TGC market. Thus, the renewable firm j’s optimization problem in the TGC market can be described as

\underset{x_{j}}{M a x} π_{r, j} = [A - ξ (\sum_{n = 1}^{N} q_{f, n} + \sum_{m = 1}^{R} q_{r, m})] q_{r, j} - c_{r, j} (q_{r, j}) + p^{T G C} q_{j}^{T G C}

(9)

s . t . 0 \leq q_{j}^{TGC} \leq (1 - α) q_{r, j}

(10)

\sum_{j = 1}^{R} q_{j}^{TGC} = α \sum_{i = 1}^{N} q_{f, i}

(11)

q_{j}^{TGC} = x_{j} + d_{j} p^{TGC}, j = 1, 2, \dots, R

(12)

where constraint (10) means that the actual traded TGC volume of renewable firm j cannot be greater than the tradable TGC volume of itself, constraint (11) is the balance constraint of supply and demand in the TGC market.

The SF equilibrium model for the TGC market can be obtained by gathering R renewable firms’ optimization problems expressed by (9)–(12).

Solution method

The TGC and wholesale markets interplay. It can be seen from (5) –(8), when decision-making in the wholesale market, the generation firms should consider the impact of TGC price and volume $(q_{j}^{TGC}, p^{TGC})$ (j=1,2,…,R). And from (15)–(18), it can be observed, to solve the equilibrium model of the TGC market, the impact of all firms’ strategies $(q_{f, i}, q_{r, j})$ (i=1,2,…,N; j=1,2,…,R) in the wholesale market should be incorporated too. Moreover, the explicit function between $(q_{f, i}, q_{r, j}, p)$ (i=1,2,…,N; j=1,2,…,R) and $(q_{j}^{TGC}, p^{TGC})$ (j=1,2,…,R) is hard to be obtained, but the implicit function is definitely incorporated in KKT conditions of wholesale market optimization problems. In this paper, the backward induction method (Willems, 2005) is used to solve this two-stage model. By this method, we can derive the Nash Equilibrium in the second stage of the game as a function of the TGCs sold and TGC price in the first stage firstly, this function can be described by the following KKT conditions of wholesale market optimization problems

(q_{f, i}, \forall i) : A - ξ (\sum_{n = 1}^{N} q_{f, n} + \sum_{m = 1}^{R} q_{r, m}) - ξ q_{f, i} - c_{f, i} q_{f, i} - b_{f, i} - α \cdot p^{T G C} + ρ_{f, i}^{-} - ρ_{f, i}^{+} = 0

(13)

(ρ_{f, i}^{-}, \forall i) : 0 \leq ρ_{f, i}^{-} ⊥ q_{f, i} - q_{f, i}^{min} \geq 0

(14)

(ρ_{f, i}^{+}, \forall i) : 0 \leq ρ_{f, i}^{+} ⊥ q_{f, i}^{max} - q_{f, i} \geq 0

(15)

(q_{r, j}, \forall j) : A - ξ (\sum_{n = 1}^{N} q_{f, n} + \sum_{m = 1}^{R} q_{r, m}) - ξ q_{r, j} - c_{r, j} q_{r, j} - b_{r, j} + ρ_{r, j}^{-} - ρ_{r, j}^{+} = 0

(16)

(ρ_{r, j}^{-}, \forall i) : 0 \leq ρ_{r, j}^{-} ⊥ q_{r, j} - q_{r, j}^{min} \geq 0

(17)

(ρ_{r, j}^{+}, \forall i) : 0 \leq ρ_{r, j}^{+} ⊥ q_{r, j}^{max} - q_{r, j} \geq 0

(18)

where $ρ_{f, i}^{-}$ and $ρ_{f, i}^{+}$ are the Lagrange multipliers for the minimum and maximum output limits of fossil fuel firm i, respectively, i=1,2,…,N. $ρ_{r, j}^{-}$ and $ρ_{r, j}^{+}$ are the Lagrange multipliers for the minimum and maximum output limits of renewable firm j, respectively, j=1,2,…,R.

After deriving the second stage equilibrium conditions, put these KKT conditions as the constraints of TGC market optimization problems, thus, the two-stage joint equilibrium model can be formulated as an EPEC problem, in which each renewable firm’s profit maximization problem in the TGC market is a mathematical program with the wholesale market equilibrium constraints (MPEC)

\begin{matrix} \underset{x_{j}}{Max} π_{r, j} = [A - ξ (\sum_{n = 1}^{N} q_{f, n} + \sum_{m = 1}^{R} q_{r, m})] q_{r, j} - c_{r, j} (q_{r, j}) \\ + \frac{α \sum_{n = 1}^{N} q_{f, n} - \sum_{m = 1}^{R} x_{m}}{\sum_{m = 1}^{R} d_{m}} \cdot (x_{j} + d_{j} \cdot \frac{α \sum_{n = 1}^{N} q_{f, n} - \sum_{m = 1}^{R} x_{m}}{\sum_{m = 1}^{R} d_{m}}) \end{matrix}

(19)

s . t . 0 \leq x_{j} + d_{j} \cdot \frac{α \sum_{n = 1}^{N} q_{f, n} - \sum_{m = 1}^{R} x_{m}}{\sum_{m = 1}^{R} d_{m}} \leq (1 - α) q_{r, j}

(20)

KKT conditions described by equations (13) - (18)

(21)

The detail of the backward induction method is shown in Appendix A. Because of the presence of complementarity conditions in the form of $u \geq 0, v \geq 0, u \cdot v = 0$ in the wholesale and TGC markets equilibrium conditions, the nonlinear complementarity approach is used in this paper (Wang et al., 2004). The solution to this EPEC problem should satisfy the first-order optimality (KKT) conditions for each renewable firm’s optimization problem expressed by (19)–(21). Giving the RPS requirement $α$ , aggregating all the KKT conditions of all renewable firms’ optimization problem in the TGC market will yields the equilibrium outcomes of the two markets, simultaneously (see Appendix B).

Numerical examples

Data assumption

Suppose that there are four fossil fuel firms (G_f,₁, G_f,₂, G_f,₃, G_f,₄) and two renewable firms (G_r,₁, G_r,₂) competing in the wholesale market. The data of fossil fuel firms’ cost parameters are taken from the literature of Xiao et al. (2016). Following the assumption of most existing work, the marginal costs of renewable firms are assumed to be higher than those of fossil fuel firms. Specifically, we assume that the intercept and slope of the marginal costs for renewable firms are greater than those for fossil fuel firms (Tanaka and Chen, 2013). The two renewable firms are assumed to be symmetrical. All generation firms’ cost parameters are listed in Table.1. Assume that $A = 100$ , $ξ = 0.05$ .

Table 1.

Parameters of generation firms.

Firms	c($/(MW)²h)	b($/MWh)	q^max(MW)
G_f,₁	0.011	25.265	400
G_f,₂	0.076	23.588	250
G_f,₃	0.073	29.037	200
G_f,₄	0.082	33.971	120
G_r,₁	0.095	37.889	250
G_r,₂	0.095	37.889	250

Main result analysis

To gain insights into the interaction between the electricity wholesale market and the TGC market, two cases are considered in this section.

Case 1 considers a Cournot competition wholesale market with a perfect competition TGC market. In this case, all firms are TGC price-takers, no firm can exercise market power in the TGC market. The tradable TGC volume shall be greater than or equal to the net demand of TGCs, the constraint for TGCs is thereby given by

(1 - α) \sum_{m = 1}^{R} q_{r, m} \geq α \sum_{n = 1}^{N} q_{f, n}

(22)

The dual variable to this constraint is the TGC price $p^{TGC}$ . If there are excess TGCs, the TGC price $p^{TGC}$ will be zero; in contrast, if the demand for TGCs equals to the supply, the TGC price $p^{TGC}$ is positive. This is consistent with the literatures of Jensen and Skytte (2002) and Tanaka and Chen (2013). This case serves as a reference for comparison purpose.

Case 2 considers a Cournot competition wholesale market with a SF competition TGC market. This case is the main focus of this paper.

Impact of RPS on renewable firms’ bidding strategies

Take renewable firm G_r,₁ for example, Figure 2 shows the impact of RPS on electricity output of G_r,₁ in two cases, Figure 3 shows the impact of RPS on actual traded TGC volume of G_r,₁ in two cases, and Figure 4 shows the impact of RPS on actual traded TGC volume and tradable TGC volume of G_r,₁ in case 2.

Figure 2.

Impact of RPS on electricity output of G_r,₁ in two cases.

Figure 3.

Impact of RPS on actual traded TGC volume of G_r,₁ in two cases.

Figure 4.

Impact of RPS on actual traded TGC volume and tradable TGC volume of G_r,₁ in case 2.

For a RPS value, the more electricity of renewable firms selling in the wholesale market means they have more tradable TGC volume that can be sold in the TGC market. Figure 2 shows that, in case 2 when the RPS is below 20%, the output of G_r,₁ in the wholesale market is apparently higher than the output in case 1, but the actual traded TGC volume is less than the volume in case 1, as shown in Figure 3. This implies that, although renewable firms have more TGCs to sell in case 2, they do not choose to get more revenues by selling more TGC volume in the TGC market. On the contrary, they choose to withhold the actual traded TGC volume intentionally, the aim is to decrease the supply of TGCs, thus to raise the TGC price and improve the profit.

As the RPS increases, the TGC demand of renewable firms themselves increases as well. As shown in Figure 4, when the RPS is above 20%, the actual traded TGC volume of G_r,₁ is exactly equal to the tradable TGC volume. This indicates that, when the RPS is greater than 20%, G_r,₁ sells all tradable TGC volume, and has no more TGCs to withhold in the TGC market. But this does not mean there is no strategic behaviour of generation firms.

Figure 2 illustrates that, when the RPS is above 20%, the output of G_r,₁ in the wholesale market in case 2 is less than the output in case 1. This suggests that, compared to case 1, G_r,₁ cuts back its output in the wholesale market. Instead of withholding traded TGC volume in the TGC market, G_r,₁ turns to the wholesale market, by withholding the electricity output to further reduce the tradable TGC volume, so as to depress the supply of TGC, drive up the TGC price and increase the profit.

In order to fully measure the magnitude of market power in two cases, we introduce the price cost margin index (PCM index) k, which can reflect the ability and the degree of generation firms to raise market prices. Consider that renewable firms bid both in the wholesale and TGC markets, k_r,_j is defined as follows

k_{r, j} = \frac{p + p^{TGC} - m c_{r, j}}{m c_{r, j}}, j = 1, 2, \cdot \cdot \cdot, R

(23)

where $m c_{r, j}$ is the marginal cost of renewable firm j.

Take renewable firm G_r,₁ for example, Table 2 shows the impact of RPS on the PCM index k_r,₁ in two cases. It can be noted that the value of k_r,₁ in case 1 is always less than that in case 2, which means the market power of renewable firms in case 2 is larger than that in case 1. Moreover, as RPS increases, the market power of renewable firms is greater.

Table 2.

Impact of RPS on the PCM index.

RPS (%)	k_r,₁		k_f,₁
	case 1	case 2	case1	case 2
5	0.0931	0.1896	0.4601	0.4644
10	0.0931	0.2807	0.4601	0.4788
15	0.0931	0.3602	0.4601	0.5010
20	0.1172	0.5287	0.4661	0.5297
25	0.1646	0.7752	0.4884	0.6478
30	0.2165	0.9565	0.5245	0.7595
35	0.2718	1.0527	0.5744	0.8652
40	0.3292	1.1017	0.6376	0.9655
45	0.3879	1.1231	0.7133	1.0607
50	0.4469	1.1280	0.8000	1.1511

Impact of RPS on fossil fuel firms’ bidding strategies

Because the fossil fuel firms only bid in the wholesale market, the PCM index k_f,_i is written as

k_{f, i} = \frac{p - m c_{f, i}}{m c_{f, i}}, i = 1, 2, \cdot \cdot \cdot, N

(24)

where $m c_{f, i}$ is the marginal cost of fossil fuel firm i.

Take fossil fuel firm G_f,₁ for example, Table 2 shows the impact of RPS on the PCM index k_f,₁ in two cases, and Figure 5 shows the impact of RPS on G_f,₁’s electricity output in two cases. It can be seen that, in case 2, the value of k_f,₁ is also greater than that in case 1, which demonstrates that fossil fuel firms also have larger market power in case 2. The reason is that, facing increasing TGC price, strategic fossil fuel firms choose to reduce more output in the wholesale market to exercise market power (see Figure 5). On the one hand, these behaviours can raise the electricity price and improve their wholesale market profit. On the other hand, by this way, the demand of TGCs is reduced, thus the TGC price can be lowered and the cost of purchasing TGCs is reduced. As such, the market power indexes of fossil fuel firms are similar to that of renewable firms, and k_f,₁ increases with increase of RPS.

Figure 5.

Impact of RPS on electricity output of G_f,₁ in two cases.

Impact of RPS on TGC and electricity prices

Table 3 shows the impact of RPS on TGC and electricity prices in two cases. It is found that, the electricity and TGC prices are greatly driven up in case 2 because of the strategic behaviors of generation firms. The reason why the TGC prices are zeros when the RPS is blow 20% in case 1 is that, in the perfect TGC market, the renewable firms have no opportunities to exercise market power in the TGC market when RPS is relatively small, but the fossil fuel firms can exercise market power by cutting back the wholesale electricity output, which decreases the demand of TGCs and depress the TGC price even to zero. Figure 6 summarizes the ways to exercise market power of generation firms in detail in case 2.

Table 3.

Impact of RPS on TGC and electricity prices.

RPS (%)	p^TGC ($/MWh)		p ($/MWh)
	case1	case2	case1	case2
5	0	4.4291	50.3167	50.3767
10	0	8.5711	50.3167	50.7059
15	0	12.1979	50.3167	51.2096
20	2.8074	18.3497	50.1109	51.8519
25	7.7992	32.0679	50.1208	54.4139
30	12.7715	41.5472	50.4871	56.7171
35	17.6183	47.0959	51.2020	58.7987
40	22.2407	50.3353	52.2434	60.6892
45	26.5528	52.1429	53.5764	62.4139
50	30.4870	53.0292	55.1568	63.9936

Figure 6.

Ways to exercise market power of generation firms in case 2.

Sensitivity analysis

With development of generation technology, the generation cost of renewable drops continuously. In this section, we focus on examining the impact of the renewable firms’ generation cost reduction on the market equilibrium outcomes and the strategic behaviors in case 2. We alter the intercept c_r,_j of marginal cost for renewable firms from 38 to 36, 34; other parameters are kept unchanged.

Table 4 shows the impact of the renewable firms’ generation cost reduction on the TGC and electricity prices. It can be noted that, the TGC and electricity prices can be lowered with generation cost reduction of renewable firms. Especially for the TGC price, when the RPS is relatively high, the effect is more obvious. The reason is that, aiming to increase profit, the renewable firms with high generation cost have larger market power in the wholesale market to raise the TGC price. When the RPS is relatively small, the TGC price is nearly invariable. The reason is, when the RPS is relatively low, the impact of RPS on the fossil fuel firms’ output in the wholesale market is relatively small (see Figure 8), which makes the actual demand of TGC change less, so that the TGC price changes slightly.

Table 4.

Impact of renewable firms’ generation cost reduction on TGC and electricity prices.

RPS (%)	p^TGC ($/MWh)			p ($/MWh)
	Intercept 34	Intercept 36	Intercept 38	Intercept 34	Intercept 36	Intercept 38
5	4.4395	4.4341	4.4288	49.6117	50.0052	50.3986
10	8.5913	8.5809	8.5705	49.9417	50.3347	50.7278
15	12.2268	12.2119	12.1971	50.4466	50.8390	51.2314
20	15.2113	17.1928	18.9695	51.0723	51.4640	51.9380
25	17.5076	24.2307	32.5284	51.7635	53.0275	54.4954
30	29.0295	35.4669	41.9044	54.0070	55.4007	56.7944
35	37.0855	42.2335	47.3816	56.2189	57.5456	58.8723
40	42.1476	46.3583	50.5689	58.2278	59.4937	60.7595
45	45.3284	48.8329	52.3374	60.0605	61.2708	62.4811
50	47.2781	50.2358	53.1934	61.7391	62.8986	64.0580

Table 5 shows that the impact of renewable firms’ generation cost reduction on PCM index of generation firm G_r,₁ and G_f,1. It can be seen that, the k_r,₁ of renewable firm G_r,₁ with relatively high generation cost is greater, which means that the renewable firms with relatively high generation cost will have more market power in the two markets. The k_f,₁ of fossil fuel firm G_f,1 is relatively larger when the generation cost of renewable firms is relatively high. The reason is that, when the generation cost of renewable firms is relatively high, the renewable firms have more market power to raise the TGC price. To reduce the TGC purchasing cost, the fossil fuel firms will exercise more market power to depress the TGC price.

Table 5.

Impact of renewable firms’ generation cost reduction on k_r,₁ and k_f,_1.

RPS (%)	k_r,₁			k_f,₁
	Intercept 34	Intercept 36	Intercept 38	Intercept 34	Intercept 36	Intercept 38
5	0.1233	0.1567	0.1916	0.7332	0.7436	0.7540
10	0.2110	0.2461	0.2827	0.7474	0.7577	0.7681
15	0.2879	0.3243	0.3624	0.7691	0.7794	0.7897
20	0.3507	0.4297	0.5080	0.7961	0.8064	0.8202
25	0.3986	0.5753	0.7871	0.8262	0.8746	0.9323
30	0.6283	0.7958	0.9660	0.9251	0.9800	1.0353
35	0.7790	0.9186	1.0606	1.0246	1.0773	1.1303
40	0.8670	0.9867	1.1085	1.1168	1.1674	1.2182
45	0.9174	1.0223	1.1291	1.2025	1.2511	1.2998
50	0.9446	1.0380	1.1333	1.2822	1.3289	1.3758

Figure 7 and Figure 8 show the impact of renewable firms’ generation cost reduction on the outputs of generation firm G_r,₁ and G_f,1 in the wholesale market, respectively. It can be seen from Figure 7 that the generation cost reduction makes the renewable firm more competitive in the wholesale market, so that its electricity output increases obviously. The reason why this effect is less obvious when the RPS is above 20% is that, when the RPS is relatively large, the renewable firm with high generation cost has more market power (see Table 5). From section 4.2.1, when the RPS is above 20%, the way to exercise market power of renewable firms is to cut back their electricity output in the wholesale market, so that the renewable firms with relatively high generation cost will cut back more electricity output to exercise market power.

Figure 7.

Impact of renewable firms’ generation cost reduction on electricity output of G_r,_1.

Figure 8.

Impact of renewable firms’ generation cost reduction on electricity output of G_f,_1.

The electricity output of fossil fuel firms not only is affected by renewable firms’ generation cost reduction, but also affected by the TGC price. When the RPS is relatively low, the TGC price change slightly, the renewable firms’ generation cost reduction is the main factor to affect the electricity output of fossil fuel firms. Under this case, the lower the renewable firms’ generation cost is, the more electricity they sell in the wholesale market, so that the fossil fuel firms sell less electricity in the wholesale market. With the RPS increases, the market power of renewable firms increases, thus the renewable firm with relatively high generation cost has more market power to raise the TGC price. In order to reduce the TGC purchasing cost, the fossil fuel firms will choose to cut down more wholesale electricity output, thus to reduce the demand of TGCs and lower the TGC price.

Figure 9 and Figure 10 show the impact of the renewable firms’ generation cost reduction on the profits of G_r,₁ in the two markets. It can be seen that, as the generation cost of renewable firms reduces, the renewable firms’ profit transfers from the TGC market to the wholesale market. The renewable firms with relatively low generation cost are more inclined to give up gaining additional profit by exercising the market power, and turn to obtaining more profit by selling more electricity in the wholesale market. From this viewpoint, the generation cost reduction of renewable firms can mitigate the market power of renewable firms and has positive effects on the market operation efficiency.

Figure 9.

Impact of renewable firms’ generation cost reduction on wholesale market profit of G_r,_1.

Figure 10.

Impact of renewable firms’ generation cost reduction on TGC market profit of G_r,_1.

Conclusions

The market-based renewable support policies engage more attention in the electricity market. The introduction of these support policies may exert an important influence on the electricity market. In this paper, we focus on the impact of RPS with TGC scheme on the electricity market. The main contributions are as follows:

A two-stage game model is proposed to analyze generation firms’ strategic behaviors in oligopolistic electricity wholesale market with an imperfect TGC market. In the first stage, the renewable generation firms compete a la supply function competition mode to sell the TGCs in the TGC market. Then, in the second stage, all generation firms compete a la Cournot mode to sell their electricity in the wholesale market. The model is reformulated as an EPEC problem by the backward induction method and solved by the nonlinear complementarity approach.

The strategic behaviors of generation firms in the two markets are examined. Simulation results show that the introduction of TGC market complicates the generation firms’ strategic behaviors. With the oligopolistic TGC market, renewable firms will exercise market power in two markets by different ways. When the RPS is relatively low, renewable firms tend to hold back some TGCs in the TGC market to raise the TGC price; when the RPS is relatively high, renewable firms choose to cut down their output in the electricity wholesale market, so as to reduce the supply of TGCs, raise both the TGC and electricity prices, and improve the profits. Facing increasing TGC prices, strategic fossil fuel firms choose to reduce their output in the wholesale market, so as to reduce the demand of TGC, bring down TGC price and TGCs’ purchasing cost.

Sensitivity analysis shows that the generation cost reduction of renewable firms can lower both TGC and electricity prices, and alleviates the market power of generation firms in two markets, which has a positive impact on the market operation efficiency.

Footnotes

Appendix A

Appendix B

KKT conditions of all renewable firms’ optimization problems in the TGC market

(B1)

(ρ_{f, i}^{-}, \forall i) : ω_{1 ij} + ω_{2 ij} [1 - \frac{ρ_{f, i}^{-}}{\sqrt{ρ_{f, i}^{- 2} + {(q_{f, i} - q_{f, i}^{min})}^{2}}}] = 0

(B2)

(ρ_{f, i}^{+}, \forall i) : - ω_{1 ij} + ω_{3 ij} [1 - \frac{ρ_{f, i}^{+}}{\sqrt{ρ_{f, i}^{+ 2} + {(q_{f, i}^{max} - q_{f, i})}^{2}}}] = 0

(B3)

\begin{matrix} (q_{f, i}, \forall i) : - ξ q_{r, j} + \frac{α}{\sum_{m = 1}^{R} d_{m}} (x_{j} + d_{j} \cdot \frac{α \sum_{n = 1}^{N} q_{f, n} - \sum_{m = 1}^{R} x_{m}}{\sum_{m = 1}^{R} d_{m}}) + \frac{α \sum_{n = 1}^{N} q_{f, n} - \sum_{m = 1}^{R} x_{m}}{\sum_{m = 1}^{R} d_{m}} \cdot \frac{α d_{j}}{\sum_{m = 1}^{R} d_{m}} \\ + (μ_{j}^{-} - μ_{j}^{+}) \cdot \frac{α d_{j}}{\sum_{m = 1}^{R} d_{m}} + (\sum_{n = 1}^{N} ω_{1 nj} + \sum_{m = 1}^{R} υ_{1 mj}) (- ξ) + ω_{1 ij} (- ξ - c_{f, i} - \frac{α^{2}}{\sum_{m = 1}^{R} d_{m}}) \\ + ω_{2 ij} (1 - \frac{q_{f, i} - q_{f, i}^{min}}{\sqrt{ρ_{f, i}^{- 2} + {(q_{f, i} - q_{f, i}^{min})}^{2}}}) + ω_{3 ij} (- 1 + \frac{q_{f, i}^{max} - q_{f, i}}{\sqrt{ρ_{f, i}^{+ 2} + {(q_{f, i}^{max} - q_{f, i})}^{2}}}) = 0 \end{matrix}

(B4)

(ρ_{r, k}^{-}, \forall k) : υ_{1 kj} + υ_{2 kj} [1 - \frac{ρ_{r, k}^{-}}{\sqrt{ρ_{r, k}^{- 2} + {(q_{r, k} - q_{r, k}^{min})}^{2}}}] = 0

(B5)

(ρ_{r, k}^{+}, \forall k) : - υ_{1 kj} + υ_{3 kj} [1 - \frac{ρ_{r, k}^{+}}{\sqrt{ρ_{r, k}^{+ 2} + {(q_{r, k}^{max} - q_{r, k})}^{2}}}] = 0

(B6)

\begin{matrix} (q_{r, k}, \forall k, k \neq j) : - ξ q_{r, j} + (\sum_{n = 1}^{N} ω_{1 nj} + \sum_{m = 1}^{R} υ_{1 mj}) (- ξ) \\ + υ_{1 kj} (- ξ - c_{r, k}) \\ + υ_{2 kj} (1 - \frac{q_{r, k} - q_{r, k}^{min}}{\sqrt{ρ_{r, k}^{- 2} + {(q_{r, k} - q_{r, k}^{min})}^{2}}}) \\ + υ_{3 kj} (- 1 + \frac{q_{r, k}^{max} - q_{r, k}}{\sqrt{ρ_{r, k}^{+ 2} + {(q_{r, k}^{max} - q_{r, k})}^{2}}}) = 0 \end{matrix}

(B7)

\begin{matrix} (q_{r, k}, \forall k, k = j) : A - ξ (\sum_{n = 1}^{N} q_{f, n} + \sum_{m = 1}^{R} q_{r, m}) - ξ q_{r, j} - c_{r, j} q_{r, j} \\ - b_{r, j} + μ_{j}^{+} (1 - α) \\ + (\sum_{n = 1}^{N} ω_{1 nj} + \sum_{m = 1}^{R} υ_{1 mj}) (- ξ) + υ_{1 jj} (- ξ - c_{r, j}) \\ + υ_{2 jj} (1 - \frac{q_{r, j} - q_{r, j}^{min}}{\sqrt{ρ_{r, j}^{- 2} + {(q_{r, j} - q_{r, j}^{min})}^{2}}}) \\ + υ_{3 jj} (- 1 + \frac{q_{r, j}^{max} - q_{r, j}}{\sqrt{ρ_{r, j}^{+ 2} + {(q_{r, j}^{max} - q_{r, j})}^{2}}}) = 0 \end{matrix}

(B8)

\begin{matrix} (x_{j}, \forall j) : \frac{- 1}{\sum_{m = 1}^{R} d_{m}} (x_{j} + d_{j} \cdot \frac{α \sum_{n = 1}^{N} q_{f, n} - \sum_{m = 1}^{R} x_{m}}{\sum_{m = 1}^{R} d_{m}}) \\ + \frac{α \sum_{n = 1}^{N} q_{f, n} - \sum_{m = 1}^{R} x_{m}}{\sum_{m = 1}^{R} d_{m}} \cdot (1 - \frac{d_{j}}{\sum_{m = 1}^{R} d_{m}}) \\ + \sum_{n = 1}^{N} ω_{1 nj} (\frac{α}{\sum_{m = 1}^{R} d_{m}}) + (μ_{j}^{-} - μ_{j}^{+}) \cdot (1 - \frac{d_{j}}{\sum_{m = 1}^{R} d_{m}}) = 0 \end{matrix}

(B9)

(μ_{j}^{-}, \forall j) : 0 \leq μ_{j}^{-} ⊥ (x_{j} + d_{j} \cdot \frac{α \sum_{n = 1}^{N} q_{f, n} - \sum_{m = 1}^{R} x_{m}}{\sum_{m = 1}^{R} d_{m}}) \geq 0

(B10)

(μ_{j}^{+}, \forall j) : 0 \leq μ_{j}^{+} ⊥ [(1 - α) q_{r, j} - (x_{j} + d_{j} \cdot \frac{α \sum_{n = 1}^{N} q_{f, n} - \sum_{m = 1}^{R} x_{m}}{\sum_{m = 1}^{R} d_{m}})] \geq 0

(B11)

\begin{matrix} (ω_{1 ij}, \forall i, j) : A - ξ (\sum_{n = 1}^{N} q_{f, n} + \sum_{m = 1}^{R} q_{r, m}) - ξ q_{f, i} - c_{f, i} q_{f, i} \\ - b_{f, i} - α \cdot \frac{α \sum_{n = 1}^{N} q_{f, n} - \sum_{m = 1}^{R} x_{m}}{\sum_{m = 1}^{R} d_{m}} \\ + ρ_{f, i}^{-} - ρ_{f, i}^{+} = 0 \end{matrix}

(B12)

(ω_{2 ij}, \forall i, j) : ρ_{f, i}^{-} + (q_{f, i} - q_{f, i}^{min}) - \sqrt{ρ_{f, i}^{- 2} + {(q_{f, i} - q_{f, i}^{min})}^{2}} = 0

(B13)

(ω_{3 ij}, \forall i, j) : ρ_{f, i}^{+} + (q_{f, i}^{max} - q_{f, i}) - \sqrt{ρ_{f, i}^{+ 2} + {(q_{f, i}^{max} - q_{f, i})}^{2}} = 0

(B14)

(υ_{1 kj}, \forall k, j) : A - ξ (\sum_{n = 1}^{N} q_{f, n} + \sum_{m = 1}^{R} q_{r, m}) - ξ q_{r, k} - c_{r, k} q_{r, k} - b_{r, k} + ρ_{r, k}^{-} - ρ_{r, k}^{+} = 0

(B15)

(υ_{2 kj}, \forall k, j) : ρ_{r, k}^{-} + (q_{r, k} - q_{r, k}^{min}) - \sqrt{ρ_{r, k}^{- 2} + {(q_{r, k} - q_{r, k}^{min})}^{2}} = 0

(B16)

(υ_{3 kj}, \forall k, j) : ρ_{r, k}^{+} + (q_{r, k}^{max} - q_{r, k}) - \sqrt{ρ_{r, k}^{+ 2} + {(q_{r, k}^{max} - q_{r, k})}^{2}} = 0

where, k=1,2,…,R; $μ_{j}^{-}$ , $μ_{j}^{+}$ are Lagrange multipliers for minimum and maximum traded TGCs limits of renewable firm j, respectively; $ω_{1 ij}$ , $ω_{2 ij}$ , $ω_{3 ij}$ , $υ_{1 kj}$ , $υ_{2 kj}$ , $υ_{3 kj}$ are Lagrange multipliers for KKT conditions (13)–(18), respectively, which are converted by the nonlinear complementarity function $ψ (u, v) = u + v - \sqrt{u^{2} + v^{2}}$ (Wang et al., 2004).

Declaration of conflicting interest

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Research grants from National Natural Science Foundation of China (No. 61633016) and Natural Science Foundation of Shanghai (No. 14ZR1415300, 15JC1401900) are acknowledged.

ORCID iD

Dajun Du

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