Oligopoly Power Producer’s Capacity Investment Model with Contracts for Differences

Abstract

This paper presents a three-level oligopoly power producer’s capacity investment game model, whose first level considers optimal regulation policy, and second-level models producer’s capacity investment strategy based on the analysis of power producer’s equilibrium biding strategy with capacity and price cap constraints at third level. We solve the model with backward induction and simulate the symmetric case. Precisely, we examine the effect of the number of oligopoly power producers, price cap, and contracts for differences (CFDs) on the unit load and power sale price and explore the optimal investment policy based on the maximization of discounted social welfare. For the proportion of power in CFDs being very big and power supply being relatively nervous in Chinese power market, we discuss the effect of power capacity investment subsidies and CFDs power price on power supply and demand, whose results indicate that reducing the proportion of CFDs’ power in the power producer’s access grid power is an effective way to alleviate the tension in power supply and demand, and the current renewable energy policy can neither necessarily ease the tension condition of power supply nor can it necessarily promote the construction of renewable power generation units.

1. Introduction

Before market-oriented reforms of the Chinese electric power industry, two types of power pricing policy, repaying capital with interests pricing and operation period pricing, were implemented in turn. Therefore, there was no risk of power capacity investment for investor, and the investment strategy was ignored. But in a mature power market, power capacity investment is a market behavior regulated by government, and power producer faces lots of uncertain factors such as market demand and relevant policy; how to avoid the risk is important for investors [1, 2]. Meanwhile, power capacity investment has several characteristics such as expensive sunk cost, investment irreversible, and long construction period, and excess or inadequate investment will both bring great economic losses. On the other hand, as a fundamental industry, stable power supply is the premise of the national economic development; otherwise it will bring immeasurable economic and social loss. Hence, to explore the power market environment of oligopoly power producer’s investment threshold, the optimal investment timing, and capacity choice, the corresponding invested amount is very significant.

Murphy and Smeers [3] considered the capacity investment decision as an optimization problem based on mathematical programming, whose idea was net present value (NPV) greater than zero, but Dixit and Pindyck [4] have shown that real option analysis leads to better investment decisions than traditional NPV analysis when the investor has the opportunity to postpone his investment. With uncertainty and irreversibility of power capacity investment, Botterud and Korpås [5], Bøckman et al. [6], Wang and Min [7], and Baofeng et al. [8] employed the real option to analyze, respectively, the investment timing of generation capacity, but capacity choice or competitors’ strategy are ignored. Other related studies can be found in similar literature [9–14].

Grenadier [15] has proved that investment strategy is not optimal if competitors were ignored. Options game is a combination of real option and game theory, in which competitors’ strategy and uncertainty are both considered. Nuo and Fushuai [16] considered that load growth, assumed to follow geometric Brownian process, was a major source of uncertainty of generation capacity investment and discussed the investment strategy in the case of duopoly investors, but it is very difficult to expand the investment model to the case of three or more than three investors, which is different from power investment practice.

The available literature of option game has shown that there is no general method to solve the model when three or more than three compete investment, and analytical solution is almost impossible (Bouis et al.) [17], which increases the difficulty to employ option game theory to solve practical investment decision problems. Meanwhile, the expressions of market demand and return of investment are implicit; Bouis et al. assumed that the return of investment is𝑌(𝑡)𝐷_𝑘, in which𝑌(𝑡)is assumed to follow geometric Brownian process,𝑘is the number of investors, and𝐷₁>𝐷₂>⋯>𝐷_∞. While𝐷has no specific expression, this is different from general works of literature that make a decision based on business profit derived from market demand function, and this also constrains further promotion and application of option game model such as empirical analysis and analog simulation. Furthermore, present option game works of literature mostly only consider investment threshold, and few of them discuss the optimal amount of investment corresponding to the threshold, which is not conform to the investment practices. So investors need to consider uncertainty factors in power market like demand and competitors, and they also should consider policy restriction like price cap and simultaneously choose the optimal capacity investment.

Dangl (1999) [18] developed theory and methodology for investment and capacity choice under uncertain demand, in which only complete monopoly market was considered. Aguerrevere [19] analyzed the effects of competitive interactions on investment decisions, and Roques and Savva [20] studied the impact of price cap regulation on the level and timing of investment in an oligopolistic industry facing stochastic demand, but all of them did not consider the choice of optimal capacity investment. Combined with actual condition of Chinese power industry, we use these ideas to develop an oligopoly power producer’s capacity investment model and analyze the optimal investment strategy by numerical simulation. And the results show that several factors including the number of oligopoly power producer, price cap, and the proportion of CFDs do have impacts on the unit load and sale price. And if given exogenous specified unit load rate and price level accepted by consumers, we can find the optimal investment policy such as higher price cap and moderate number of power producer based on the maximization of discounted social welfare. For the proportion of CFDs in Chinese power market being very big and power supply being relatively nervous, we discuss the impacts of power capacity investment subsidies and CFDs’ power price on power supply and demand. The results indicate that reducing the rate of CFDs in the power producer’s access power is an effective way to alleviate the tension in power supply and demand, and the current renewable energy policy can neither necessarily ease the nervous power supply nor can it necessarily promote the construction of renewable power generation units.

This paper is structured as follows. Section 2 presents our model of investment timing and capacity choice. Section 3 analyzes and simulates symmetric oligopoly investment model, and optimal oligopoly power producers, price cap, and contracts for differences are determined. Chinese power market is discussed in Section 4. Section 5 presents a summary of our main findings.

2. Model

Generally, power demand function is difficult to be truly described because power demand is influenced by many factors, but the empirical analysis of Pindyck [21] shows that the linear function as follows can simulate real power demand condition to a certain degree: 𝑝(𝑡)=𝜃(𝑡)−𝜉⋅𝑄(𝑡), (1) where𝑝(𝑡)is power sale price,𝜉>0is demand parameter, and𝑄(𝑡)is the total power demand.𝜃(𝑡)is the effects of relevant factors except sale price on power demand, and we assume that 𝑑𝜃=𝜔𝜃𝑑𝑡+𝜎𝜃𝑑𝑧, (2) where𝜔,𝜎>0and𝑑𝑧is standard Wiener process.

2.1. Oligopoly Power Market Equilibrium Analysis with Capacity Constraint and Price Cap

We assume that the cost function of power producers is a quadratic function of power quantity; there is 𝐶𝑞_𝑖=𝛼_𝑖+𝛽_𝑖𝑞_𝑖+0.5𝛾_𝑖𝑞²_𝑖, (3) where𝐶is cost of power producer,𝑞_𝑖is power quantity of producer𝑖,𝑖=1,2,…,𝑛, and𝑛is total number of producers in the power market.𝛼,𝛽,𝛾≥0are cost coefficient.

Supposings that the range ability of power producer𝑖’s biding function is the same with that of its marginal cost function, there is 𝑏_𝑖=𝜙_𝑖+𝛾_𝑖𝑞_𝑖, (4) where𝜙_𝑖>0is bidding strategy parameter, that is, rational power producer𝑖adjust𝜙_𝑖to choose its biding strategy to get higher access grid power and maximum profit. If capacity constraints are ignored, we get total power supply by the sum of (4) for each power producer𝑖; thus ^𝑛_𝑖=1𝑞_𝑖=𝑝₂^𝑛_𝑖=11𝛾_𝑖−^𝑛_𝑖=1𝜙_𝑖𝛾_𝑖, (5) where𝑝₂is the transaction price of competitive power. Then, we get the total demand power[𝜃(𝑡)−𝑝]/𝜉with (1) and substitute it into the equation above; there is 𝜃(𝑡)−𝑝(𝑡)𝜉=𝑝₂^𝑛_𝑖=11𝛾_𝑖−^𝑛_𝑖=1𝜙_𝑖𝛾_𝑖. (6)

In China, power sale price includes average access grid price, transmission cost, and reasonable profit ignoring taxes. And access grid power can be divided into two kinds including competitive power and uncompetitive power. In this paper, competitive power is assumed to be determined by (4), and uncompetitive power is called contract power which means power producer makes a deal with grid corporation to assure the contract power’s price and quantity before power bidding under the leading of regulation institutions. This is actually a transitory stage of power market reform which will be accomplished with lower proportion of CFDs or lower difference between CFDs’ power price and market price. If given the proportion𝜂of CFDs and CFDs’ power price𝑝₁, there is 𝑝(𝑡)=(1+𝑘)𝜂𝑝₁+1−𝜂𝑝₂, (7) where0<𝑘<1denotes the proportion of transmission cost in sale price. Substitute (7) into (6); there is 𝑝₂=𝜃(𝑡)−(1+𝑘)𝜂𝑝₁+𝜉∑^𝑛_𝑖=1𝜙_𝑖/𝛾_𝑖(1+𝑘)1−𝜂+𝜉∑^𝑛_𝑖=11/𝛾_𝑖. (8) Substitute the equation above into (4); there is 𝑞_𝑖=1𝛾_𝑖𝜃(𝑡)−(1+𝑘)𝜂𝑝₁+𝜉∑^𝑛_𝑖=1𝜙_𝑖/𝛾_𝑖(1+𝑘)1−𝜂+𝜉∑^𝑛_𝑖=11/𝛾_𝑖−𝜙_𝑖. (9) Then, power producer’s profit function can be described by 𝜋_𝑖𝜙_𝑖=1−𝜂𝑝₂𝑞_𝑖+𝜂𝑝₁𝑞_𝑖−𝐶𝑞_𝑖−𝜏𝜌𝑞_𝑖. (10)

In the lower carbon economic environment, Chinese regulator forcibly sets a certain proportion of renewable energy in power producer’s access grid power, and we set this proportion as𝜏and set𝜌as the cost of per unit renewable energy. Solving the first order condition of𝜙_𝑖in (10), there is 𝜉1−𝜂𝑞_𝑖=1−𝜂𝑝₂+𝜂𝑝₁−𝛽_𝑖−𝛾_𝑖𝑞_𝑖×()−𝜏𝜌1+𝑘1−𝜂+𝜉^𝑛_𝑗≠𝑖1𝛾_𝑗. (11)

Solving (11), we get 𝜙^∗_𝑖=𝐴_𝑖1−𝐴_𝑖2𝜃−(1+𝑘)𝜂𝑝₁+𝜉∑^𝑛_𝑖=1𝜙^∗_𝑖/𝛾_𝑖𝐴_𝑖1(1+𝑘)1−𝜂+𝜉∑^𝑛_𝑖=11/𝛾_𝑖−𝐴_𝑖3𝐴_𝑖1, (12) where 𝐴_𝑖1=𝜉1−𝜂𝛾_𝑖+(1+𝑘)1−𝜂+𝜉^𝑛_𝑗≠𝑖1𝛾_𝑗,𝐴_𝑖2=1−𝜂(1+𝑘)1−𝜂+𝜉^𝑛_𝑗≠𝑖1𝛾_𝑗,𝐴_𝑖3=𝜂𝑝₁−𝛽_𝑖−𝜏𝜌(1+𝑘)1−𝜂+𝜉^𝑛_𝑗≠𝑖1𝛾_𝑗. (13)

Iterating (12), we get ^𝑛_𝑖=1𝜙_𝑖𝛾_𝑖=𝜃−(1+𝑘)𝜂𝑝₁^𝑛_𝑖=1𝐴_𝑖1−𝐴_𝑖2𝐴_𝑖1𝛾_𝑖−(1+𝑘)1−𝜂+𝜉^𝑛_𝑖=11𝛾_𝑖^𝑛_𝑖=1𝐴_𝑖3𝐴_𝑖1𝛾_𝑖×(1+𝑘)1−𝜂+𝜉^𝑛_𝑖=11𝛾_𝑖−𝜉^𝑛_𝑖=1𝐴_𝑖1−𝐴_𝑖2𝐴_𝑖1𝛾_𝑖⁻¹. (14)

Substituting this equation into (12), we get𝜙^∗_𝑖 in equilibrium bidding condition. And substituting𝜙^∗_𝑖 into (7) and (9), we can get the equilibrium power quantity𝑞^∗_𝑖of oligopoly power producers; that is, 𝑞^∗_𝑖=1𝐴_𝑖1𝛾_𝑖𝐴_𝑖2𝜃−(1+𝑘)𝜂𝑝₁𝑀−𝜉Σ₁−𝜉𝐴_𝑖2Σ₂𝑀−𝜉Σ₁+𝐴_𝑖3, (15) where𝑀=(1+𝑘)(1−𝜂)+𝜉∑^𝑛_𝑖=11/𝛾_𝑖,Σ₁=∑^𝑛_𝑖=1(𝐴_𝑖1−𝐴_𝑖2)/𝐴_𝑖1𝛾_𝑖, andΣ₂=∑^𝑛_𝑖=1𝐴_𝑖3/𝐴_𝑖1𝛾_𝑖.

Obviously, oligopoly power producer’s power quantity should satisfy0≤𝑞_𝑖≤𝑚_𝑖, and𝑚_𝑖is the largest of power quantity determined by its capacity, so the equilibrium power quantity should satisfy 𝑞^∗_𝑖=0if𝜃≤𝜁_𝑖1,𝐴_𝑖2𝐴_𝑖1𝛾_𝑖𝜃−(1+𝑘)𝜂𝑝₁−𝜉Σ₂𝑀−𝜉Σ₁+𝐴_𝑖3𝐴_𝑖2if𝜁_𝑖1<𝜃≤𝜁_𝑖2𝑚_𝑖,𝑚_𝑖if𝜁_𝑖2𝑚_𝑖<𝜃, (16) where𝜁_𝑖1=(1+𝑘)𝜂𝑝₁+𝜉Σ₂−(𝐴_𝑖3/𝐴_𝑖2)(𝑀−𝜉Σ₁),  and  𝜁_𝑖2(𝑚_𝑖)=(1+𝑘)𝜂𝑝₁+𝜉Σ₂+(𝑀−𝜉Σ₁)((𝐴_𝑖1𝛾_𝑖𝑚_𝑖−𝐴_𝑖3)/𝐴_𝑖2).

In market equilibrium, we can get an equation as follows from (7) and (1): 𝑝^∗₂=𝜃−𝜉∑^𝑛_𝑖=1𝑞^∗_𝑖(−1+𝑘)1−𝜂𝜂𝑝₁.1−𝜂 (17)

If given price cap𝑝, there is 𝑝^∗₂=𝜃−𝜉∑^𝑛_𝑖=1𝑞^∗_𝑖()−1+𝑘1−𝜂𝜂𝑝₁1−𝜂if𝜃−𝜉∑^𝑛_𝑖=1𝑞^∗_𝑖(𝜃)<(1+𝑘)1−𝜂𝑝+(1+𝑘)𝜂𝑝₁,𝑝if𝜃−𝜉∑^𝑛_𝑖=1𝑞^∗_𝑖(𝜃)≥(1+𝑘)1−𝜂𝑝+(1+𝑘)𝜂𝑝₁. (18)

Apparently, power producer’s equilibrium power quantity function and competitive equilibrium price function are piecewise function with𝜃. Substituting these into (10), we can get oligopoly power producer’s profit function𝜋(𝜃)with𝜃.

2.2. Oligopoly Power Producer’s Capacity Investment Model

With classical real option theory, power producer𝑖’s investment value function is as follows 𝑉_𝑖𝜃,𝑚_𝑖=𝜋_𝑖𝜃,𝑚_𝑖+𝑒^{−𝑟𝑑𝑡}𝐸𝑉_𝑖𝜃,𝑚_𝑖+𝑑𝑉_𝑖𝜃,𝑚_𝑖, (19) where𝑟is risk-free rate and𝐸(⋅)denotes expectation. Unfolding the equation above with Ito lemma, there is 12𝜎²𝜃²𝜕²𝑉_𝑖𝜃,𝑚_𝑖𝜕𝜃²+𝜔𝜃𝜕𝑉_𝑖𝜃,𝑚_𝑖𝜕𝜃−𝑟𝑉_𝑖𝜃,𝑚_𝑖+𝜋_𝑖𝜃,𝑚_𝑖=0. (20)

The result of this differential equation is 𝑉^𝑗_𝑖𝜃,𝑚_𝑖=𝐴^𝑗₁𝑚_𝑖𝜃^𝜆1+𝐴^𝑗₂𝑚_𝑖𝜃^𝜆2+𝑉^𝑗_𝑖𝜃,𝑚_𝑖, (21) where𝜆_1,2=1/2−𝜔/𝜎²±(𝜔/𝜎²−1/2)²+2𝑟/𝜎²,𝑗=1,2,…,𝐽, and𝐽denotes the number of𝜃’s intervals divided by boundary point𝜁_𝑖1,𝜁_𝑖2, and  𝜁_𝑖3in (19).𝑉^𝑗_𝑖(𝜃,𝑚_𝑖)denotes the value function of power producer𝑖in interval𝑗. And𝐴^𝑗₁(𝑚_𝑖)  and  𝐴^𝑗₂(𝑚_𝑖)are functions of𝑚_𝑖to be determined.𝑉^𝑗_𝑖(𝜃,𝑚_𝑖)is a particular solution in interval𝑗and generally denoted by 𝑉^𝑗_𝑖𝜃,𝑚_𝑖=𝐸^∞₀𝜋^𝑗_𝑖𝜃,𝑚_𝑖𝑒^−𝑟𝑡𝑑𝑡. (22)

Obviously, substituting (22) into (21) we can get the analytical solution of value function𝑉_𝑖with the largest power quantity𝑚_𝑖for power producer𝑖. Owing to the piecewise function𝜋_𝑖(𝜃,𝑚_𝑖), the value function𝑉_𝑖is also piecewise. But in the boundary, value function𝑉^𝑗_𝑖(𝜃,𝑚_𝑖)should be continuous and smooth. If given boundary points as𝜁_𝑖𝑗(𝑖=1,2,…,𝑛,𝑗=1,2,…,𝐽), there is 𝑉_𝑖𝜁⁻_𝑖𝑗,𝑚_𝑖=𝑉_𝑖𝜁⁺_𝑖𝑗,𝑚_𝑖,𝜕𝑉_𝑖𝜁⁻_𝑖𝑗,𝑚_𝑖=𝜕𝜃𝜕𝑉_𝑖𝜁⁺_𝑖𝑗,𝑚_𝑖.𝜕𝜃 (23)

Substitute (21) into the equation above, there is 𝐴₁^𝑗+1(𝑚)=𝐴^𝑗₁(𝑚)+𝜆₂𝑉^𝑗𝜁_𝑗,𝑚−𝑉^𝑗+1𝜁_𝑗,𝑚−𝜁_𝑗𝜕𝑉^𝑗𝜕𝜃_{𝜃=𝜁𝑗}−𝜕𝑉^𝑗+1𝜕𝜃_{𝜃=𝜁𝑗}×𝜆₂−𝜆₁𝜁_𝑗^𝜆1−1𝐴₂^𝑗+1(𝑚)=𝐴^𝑗₂(𝑚)−𝜆₁𝑉^𝑗𝜁_𝑗,𝑚−𝑉^𝑗+1𝜁_𝑗,𝑚−𝜁_𝑗𝜕𝑉^𝑗𝜕𝜃_{𝜃=𝜁𝑗}−𝜕𝑉^𝑗+1𝜕𝜃_{𝜃=𝜁𝑗}×𝜆₂−𝜆₁𝜁_𝑗^𝜆2−1. (24) Furthermore, there is𝑉_𝑖(0,𝑚_𝑖)=0when𝜃=0andlim_𝜃→+∞𝑉_𝑖(𝜃,𝑚_𝑖)=𝑉_𝑖^𝐽when𝜃→+∞, in which𝐽denotes the rightmost interval of𝜃divided by𝜁_𝑖𝑗(Dangl [18], Nagel and Rammerstorfer, [22]). Combined with (21), there is 𝐴¹₂=0,𝐴^𝐽₁=0. (25)

Applying (24) and (25), we can get𝐴^𝑗₁(𝑚_𝑖),𝐴^𝑗₂(𝑚_𝑖), and piecewise investment value function of (21). Power producers’ capacity investment strategy is actually an option to invest or to wait. Assuming power suppliers are myopic based on Cournot game theory; that is, power producer thinks competitors will not invest in term𝑑𝑡→0. So the value𝐹_𝑖(𝜃,𝑚_𝑖)of the option to wait satisfies following equation: 12𝜎²𝜃²𝜕²𝐹_𝑖𝜃,𝑚_𝑖𝜕𝜃+𝜔𝜃𝜕𝐹_𝑖𝜃,𝑚_𝑖𝜕𝜃−𝑟𝐹_𝑖𝜃,𝑚_𝑖=0. (26)

To solve the above equation, there is 𝐹_𝑖𝜃,𝑚_𝑖=𝐷_𝑖𝑚_𝑖𝜃^𝜆1, (27) where𝐷_𝑖(𝑚_𝑖)is also a function of𝑚_𝑖just like𝐴^𝑗₁(𝑚_𝑖)and𝐴^𝑗₂(𝑚_𝑖). The investment threshold should satisfy value matching and smooth pasting; so there is 𝐹_𝑖𝜃^∗,𝑚_𝑖=𝑉_𝑖𝜃^∗,𝑚_𝑖−𝐼𝑚_𝑖,𝜕𝐹_𝑖𝜃^∗,𝑚_𝑖=𝜕𝜃𝜕𝑉_𝑖𝜃^∗,𝑚_𝑖,𝜕𝜃 (28) where𝐼(𝑚_𝑖)is power producer’s investment and is a function of𝑚_𝑖. We would like to consider the investment function as follows: 𝐼𝑚_𝑖=𝑏𝑚_𝑖^𝜍+𝜇(𝑡)𝑚_𝑖, (29) where 0<𝜍≤1,𝑏>0 are parameters and𝜇(𝑡) is a variable of investment policy. Solving (28) and eliminating𝐷_𝑖(𝑚_𝑖), we can get the functional relationship between optimal investment threshold𝜃^∗and largest power quantity𝑚_𝑖. Furthermore, for rational oligopoly power producer, when choosing maximum power quantity, there is 𝜕𝑉_𝑖𝜃,𝑚_𝑖𝜕𝑚_𝑖=𝑑𝐼𝑚_𝑖𝑑𝑚_𝑖. (30)

Combining (28) and (30), we can get the investment threshold𝜃^∗and optimal capacity𝑚^∗_𝑖for power producer𝑖. Apparently, power producer’s investment strategy is determined by policy factors such as price cap and contract power, so it is very important to examine relevant policy factors for equilibrium of power capacity investment market.

2.3. Optimal Capacity Investment Policy

For regulation institute, the purpose of regulating power capacity investment includes two aspects, one is to assure power system’s safe operation and adequate power supply and the other one is to accomplish economic operation of power unit; that is, the vacancy rate of capacity is at a low level and the power price can be accepted by consumers. With (1), we can get consumer surplus𝑢(𝜃_𝑡,𝑝,𝜂,𝑝₁,𝜇)in the power bidding equilibrium; there is 𝑢𝜃_𝑡,𝑝,𝜂,𝑝₁=𝜃,𝜇0.5_𝑡−(1+𝑘)𝜂𝑝₁+1−𝜂𝑝₂𝜃_𝑡²𝜉. (31)

From economic theory, the total society welfare is the sum of consumer surplus and the profit from all the power producers, while grid companies should be nonprofitable in mature power market, so we did not consider grid companies’ profit; then there is 𝑤𝜃_𝑡,𝑝,𝜂,𝑝₁,𝜇=𝑢𝜃_𝑡,𝑝,𝜂,𝑝₁,𝜇+^𝑛_𝑖=1𝜋_𝑖𝜃_𝑡,𝑝,𝜂,𝑝₁,𝜇. (32)

Therefore, the aim of regulation institute can be simply described by 𝑊=max_{𝑛,𝑝,𝜂,𝑝1,𝜇}^𝑇_𝑇0𝑒^−𝑟𝑡𝑤𝜃_𝑡,𝑝,𝜂,𝑝₁,𝜇𝑑𝑡s.t.𝜓_𝑖=𝑞_𝑖𝜃_𝑡,𝑝,𝜂,𝑝₁,𝜇𝑚_𝑖𝜃_𝑡,𝑝,𝜂,𝑝₁∈𝜓,𝜇^𝐿,𝜓_𝐻𝑝=(1+𝑘)𝜂𝑝₁+1−𝜂𝑝₂×𝜃_𝑡,𝑝,𝜂,𝑝₁∈𝑝,𝜇_𝐿,𝑝_𝐻, (33) where𝑇₀is the moment to start unit operation for power producer,𝑇is the examining period of regulator institute,𝜓_𝐿,𝜓_𝐻is separately upper limit of average unit load and floor level of economic operation when power system is safely operated, and𝑝_𝐿,𝑝_𝐻is separately the price cap accepted by consumers and price floor accepted by power producers deemed by regulator.

Obviously, (18), (28), (30), and (33) separately build bidding equilibrium, investment equilibrium, and equilibrium of maximum society welfare. Based on game theory, firstly regulation institute implement relevant policy, then power producers choose their capacity investment strategy and bid after starting unit operation. Apparently, this process is a typical dynamic game, so we would like to use backward induction to solve these equilibrium models, that is, substituting the result of (18) into (28) and (30) to get power producer’s capacity investment strategy, which is substituted into (33) to get optimal result with inequality constraints. But it is difficult to solve (33) analytically, so we will simulate it numerically in the following research.

3. Model Analysis and Numerical Simulation

3.1. Oligopoly Power Producer’s Investment Model in Symmetrical Case

In symmetrical oligopolistic power capacity investment market, all the power producers’ cost parameters are the same, so we substitute (16) and (18) into (10); the profit function of each power producer is obtained as the following three cases.(1)When𝑝>(𝐴₁𝛾𝑚−𝐴₃)/𝐴₂, there is 𝜋(𝜃)=0𝜃≤𝜁₁𝐻₁₁𝜃−(1+𝑘)𝜂𝑝₁−𝜉Σ₂𝑀−𝜉Σ₁²+𝐻₁₂𝜃−(1+𝑘)𝜂𝑝₁−𝜉Σ₂𝑀−𝜉Σ₁+𝐻₁₃𝜁₁<𝜃≤𝜁₂𝑚(𝜃−𝑛𝜉𝑚)−𝑚1+𝑘𝛽+𝜏𝜌−0.5𝛾𝑚²−𝛼𝜁₂<𝜃≤𝜁₃1−𝜂𝑝+𝜂𝑝₁−𝛽−𝜏𝜌𝑚−0.5𝛾𝑚²−𝛼𝜁₃<𝜃, (34) where𝜁₁=(1+𝑘)𝜂𝑝₁+𝜉Σ₂−(𝑀−𝜉Σ₁)𝐴₃/𝐴₂, 𝜁₂=(1+𝑘)𝜂𝑝₁+𝜉Σ₂+(𝑀−𝜉Σ₁)(𝐴₁𝛾𝑚−𝐴₃)/𝐴₂, 𝜁₃=𝑛𝜉𝑚+(1+𝑘)[(1−𝜂)𝑝+𝜂𝑝₁]; 𝐻₁₂=[𝐴₃(1−𝜂)+𝐴₂(𝜂𝑝₁−𝛽−𝜏𝜌)−𝐴₂𝐴₃/𝐴₁]/(𝐴₁𝛾), 𝐻₁₁=𝐴₂(1−𝜂−0.5𝐴₂/𝐴₁)/(𝐴₁𝛾),  and  𝐻₁₃=𝐴₃[(𝜂𝑝₁−𝛽−𝜏𝜌)−0.5𝐴₃/𝐴₁]/(𝐴₁𝛾)−𝛼.(2)When −𝐴₃/𝐴₂<𝑝<(𝐴₁𝛾𝑚−𝐴₃)/𝐴₂, there is 𝜋(𝜃)=0𝜃≤𝜁₁𝐻₁₁𝜃−(1+𝑘)𝜂𝑝₁−𝜉Σ₂𝑀−𝜉Σ₁²+𝐻₁₂𝜃−(1+𝑘)𝜂𝑝₁−𝜉Σ₂𝑀−𝜉Σ₁+𝐻₁₃𝜁₁<𝜃≤𝜁₂𝐻₂₁𝜃−(1+𝑘)𝜂𝑝₁−𝜉Σ₂𝑀−𝜉Σ₁²+𝐻₂₂𝜃−(1+𝑘)𝜂𝑝₁−𝜉Σ₂𝑀−𝜉Σ₁+𝐻₂₃𝜁₂<𝜃≤𝜁₃1−𝜂𝑝+𝜂𝑝₁−𝛽_𝑖−𝜏𝜌𝑚−0.5𝛾_𝑖𝑚²−𝛼𝜁₃<𝜃, (35) where𝜁₁=(1+𝑘)𝜂𝑝₁+𝜉Σ₂−(𝑀−𝜉Σ₁)𝐴₃/𝐴₂,𝜁₂=(1+𝑘)𝜂𝑝₁+𝜉Σ₂+𝑝[𝑀−𝜉Σ₁], 𝜁₃=(1+𝑘)𝜂𝑝₁+𝜉Σ₂+(𝑀−𝜉Σ₁)(𝐴₁𝛾𝑚−𝐴₃)/𝐴₂, 𝐻₂₁=−0.5𝐴²₂/(𝐴²₁𝛾), 𝐻₂₂=(𝐴₂/𝐴₁𝛾)[(1−𝜂)𝑝+𝜂𝑝₁−𝛽−𝜏𝜌−(𝐴₃/𝐴₁)],  and  𝐻₂₃=(𝐴₃/𝐴₁𝛾)[(1−𝜂)𝑝+𝜂𝑝₁−𝛽_𝑖−𝜏𝜌−(𝐴₃/2𝐴₁)]−𝛼.(3)When𝑝≤−𝐴₃/𝐴₂, there is 𝜋(𝜃)=0𝜃≤𝜁₁𝐻₂₁𝜃−(1+𝑘)𝜂𝑝₁−𝜉Σ₂𝑀−𝜉Σ₁²+𝐻₂₂𝜃−(1+𝑘)𝜂𝑝₁−𝜉Σ₂𝑀−𝜉Σ₁+𝐻₂₃𝜁₁<𝜃≤𝜁₂1−𝜂𝑝+𝜂𝑝₁−𝛽_𝑖−𝜏𝜌𝑚−0.5𝛾_𝑖𝑚²−𝛼𝜁₂<𝜃, (36) where𝜁₁=(1+𝑘)𝜂𝑝₁+𝜉Σ₂−(𝑀−𝜉Σ₁)𝐴₃/𝐴₂  and  𝜁₂=(1+𝑘)𝜂𝑝₁+𝜉Σ₂+(𝑀−𝜉Σ₁)(𝐴₁𝛾𝑚− 𝐴₃)/𝐴₂.

Substituting (34), (35), and (36) into (22) separately, we can get particular solution of investment value function. Then to substitute this result into (21), we can get power producer’s investment value function. Furthermore, substituting the function into (28) and (30) we can get the investment threshold𝜃^∗ and the optimal investment capacity𝑚^∗.

3.2. Numerical Simulation Analysis of Symmetrical Oligopoly Power Capacity Investment Model

It is quite clear that the analytic solution of investment threshold𝜃^∗ and optimal investment capacity𝑚^∗cannot be obtained, so we simulate these numerically in the following research. We assumed that market parameters are𝑟=0.06,𝜔=0.02,  𝜎=0.05, and𝜉=0.02, cost parameters are𝛼=0,𝛽=2,and𝛾=0.025, and capacity investment parameters are𝑏=12and𝜍=0.85.

If given𝑝₁=4,𝑚=80, Figure 1 shows the curve of oligopoly power producer’s investment value function when𝑛=1,3,5and𝑝=3.5,5.0.

(a)

(b)

Figure 1 

Investment value curve when𝑝=3.5,5.0.

We can see several phenomena from Figure 1: firstly, investment value is decreasing with increasing number of power producer. Secondly, investment value is increasing with increasing rate of CFDs’ power when𝑝₁ is higher than𝑝but decreasing when𝑝₁is lower than𝑝. The third one is that investment value is increasing with increasing price cap. And the forth one is that investment value is increasing with𝜃but decreasing after it reaches the highest point and then invariant in Figure 1(a). While in Figure 1(b) when𝑝<𝑝₁investment value is increasing first, then variant, and at last decrease with increasing𝜃, and the maximum value is increasing with increasing𝑝, and the investment value will be always increasing with𝜃when price cap is ignored. As shown in Figure 1, number𝑛of power producer, price cap𝑝, and the rate𝜂of CFDs have remarkable effects on capacity investment value, and we will analyze the effect in following research.

The effect of oligopoly power producer’s price cap𝑝on investment threshold and optimal capacity is shown in Figure 2. First, when price cap is very small, power producers will not invest and delay investing infinitely because their weighting access grid power price is small and their profit is small or even negative. Second, when price cap is small, there is𝜕𝜃^∗/𝜕𝑛>0; that is, the larger the number of power produceris, the longer they delay to invest. And this result is opposite to the conclusions of classical real option works of literature such as Grenadier [23], but further analysis shows that when price cap is large enough,𝜕𝜃^∗/𝜕𝑛<0holds; Grenadier did not consider price cap. The third, power producer’s optimal investing capacity is decreasing with increasing number of power producers; that is,𝜕𝑚^∗/𝜕𝑛<0. The forth, oligopoly power producer’s investment threshold and optimal capacity are increasing with increasing price cap.

(a)

(b)

Figure 2 

Investment threshold and optimal capacity. Parameters:𝑘=0.2,𝜏=0.25,𝜌=1.0,𝜂=0.5,𝜇=0,and𝑝1=4.0.

Power producer’s investment threshold and capacity curve are shown in Figure 3 when𝑛=3. We can get several results that oligopoly power producer will preemptive in investing but decrease the investing amount with increasing rate of CFDs’ power. And CFDs’ power price has little effect on oligopoly power producer’s investment strategy when the rate of CFDs is small, but it becames much more important with increasing rate of CFDs. The higher the CFD’s price is, the longer the power producers delay investing.

(a)

(b)

Figure 3 

Investment threshold and optimal capacity. Parameters:𝑛=3,𝑘=0.2,𝜏=0.25,𝜌=1.0,𝑝=20,and𝜇=0.

3.3. Optimal Capacity Investment Policy Analysis

Simplifying (33), there is 𝑊=max_{𝑛,𝑝,𝜂,𝑝1,𝜇}𝐿₁𝐸₁+𝐿₂𝐸₂+𝐿₃𝑟, (37) where 𝐿₁=𝑛𝐻₁₁+𝑛²𝜉𝐴²₂/2𝐴²₁𝛾²𝑀−𝜉Σ₁²,𝐿₂=𝑛𝐻₁₂+𝑛²𝜉𝐴₂𝐴₃/𝐴²₁𝛾²𝑀−𝜉Σ₁,𝐿₃=𝑛𝐻₁₃+𝑛²𝜉𝐴²₃2𝐴²₁𝛾²,𝐸₁=𝜃^∗2𝑟−2𝜔−𝜎²1−𝑒^{−(𝑟−2𝜔−𝜎2)𝑇}−2(1+𝑘)𝜂𝑝₁+𝜉Σ₂𝜃^∗𝑟−𝜔1−𝑒^{−(𝑟−𝜔)𝑇}+(1+𝑘)𝜂𝑝₁+𝜉Σ₂²𝑟1−𝑒^−𝑟𝑇,𝐸₂=𝜃^∗𝑟−𝜔1−𝑒^{−(𝑟−𝜔)𝑇}−(1+𝑘)𝜂𝑝₁+𝜉Σ₂𝑟1−𝑒^−𝑟𝑇. (38)

Substituting parameters in Figures 2 and 3 into (37), we can get responding total society welfare. And Figure 4 shows the curve of total society welfare with price cap𝑝and the rate𝜂of CFDs when𝑇=5responding to Figures 2 and 3.

(a)

(b)

Figure 4 

Discounted society welfare curve.

We can clearly see that the total society welfare is increasing with increasing price cap and increasing number of power producers but decreasing with increasing rate of CFDs. So regulation organizations should choose higher price cap, larger number of power producer, and lower rate of CFDs to maximize the total society welfare under the constraint of unit load and sale power price.

Figure 5 shows that the power system will be safely operated with proper number of power producer and high price cap. If unit load satisfiesΨ∈[50%,75%], then we can see𝑝∈(3.125,5.95)when𝑛=1,𝑝∈(5.22,17.25)when𝑛=2,𝑝∈(12.35,+∞)when𝑛=3, and𝑝∈(34.95,+∞)when𝑛=4. What is more, with the increasing number of power producers, the weighting access grid power price is decreasing and the unit load rate is increasing because power producer’s profit rate is low when weighting access grid power price is low, so they need higher unit load to ensure their total profit. Above all, sale price is decreasing with increasing number of power producers and is increasing with increasing price cap, so assuring these two aspects is very important for regulators and the whole society.

(a)

(b)

Figure 5 

Power unit load and sale power price corresponding to Figure 2.

From Figure 6, we can see that the rate of CFDs𝜂should be lower than𝜂^∗when we adopt the standardΨ^∗∈[50%,80%], and𝜂^∗is different in different cases of𝑝₁, for example,𝜂^∗≈0.625when𝑝₁=3,𝜂^∗≈0.645when𝑝₁=4, and𝜂^∗≈0.675when𝑝₁=5. If considering weighting access grid power price corresponding to investment threshold in the meantime, it is clear that the access grid power price is lowest and the market efficiency is highest when𝜂=𝜂^∗. Similarly, if reducing specified unit load rate standard, the corresponding rate𝜂^∗of CFDs will decrease, the weighting access grid power price will increase step by step, and market efficiency will descend.

(a)

(b)

Figure 6 

Power unit load and sale power price corresponding to Figure 3.

The above analysis results show that if given the ceiling and floor of unit load and weighting access grid power price exogenously, we can get the optimal price cap, the number of power producers, the contract power’s rate, and price by (37).

4. Simple Discussion of Power Capacity Investment in China

Chinese power industry has been in market reforming for 30 years, while the original administrative monopoly has been broken. Some part of power industry has been brought in market mechanism, especially the experiment exercised in northeast power market of China from 2004 to 2005, although it did not realize the expected aim, it basically built the fundamental framework of Chinese competitive power market. At present, the real condition of Chinese power market is that generally power supply is nervous, and most of power is traded at the price set by government regulation institutes, but little is traded through market regulation and do not form a market clearing mechanism. The problem about power capacity in China is complicated; simply, we will discuss two aspects: one is the reason of nervous power supply and relevant suggestion, and the other one is the impact of renewable energy policy on power capacity investment.

The above oligopoly power producer’s investment model and analysis framework can be used to examine Chinese oligopoly power producer’s investment strategy and market behavior. We can regard the access grid power price set by government regulators as CFDs and its price as CFDs’ power price𝑝₁. Consider that𝜂=0means that all the power is access grid through market competition, and𝜂=1means that all the power is regulated by regulation institute, so0<𝜂<1denotes that part of power is traded through market competition and the rest is traded through regulation. Based on the real condition, we assume that𝑛=5,𝜂=0.85, and other parameters are invariant; then we can know that oligopoly power producer lack the positivity to invest in capacity, and their unit load surpasses the specified standard ceiling even close to 100%. This is unacceptable in power system’s safely operating, so power shortage is unavoidable.

The results of above model analysis are consistent with present power supply condition in China. For efficient power supply being directly related to national economic development and people’s daily lives, government regulators generally propose inspiring policy to encourage power producer to increase capacity investment to mitigate nervous power supply. Based on the above power capacity investment model, there are simple and direct inspiring policies that give power producer subsidy of capacity investment or increasing the price of CFDs, and Figure 7 shows the curve of power producer’s investment strategy and market behavior with variant investing policy𝜇in three cases which are𝜂=0.25,𝑝₁=4;𝜂=0.85,𝑝₁=4;𝜂=0.85,𝑝₁=6.

(a)

(b)

(c)

(d)

Figure 7 

Power producer’s investment strategy and its market behavior. Parameters:𝑛=5,𝑘=0.2,𝜏=0.25,𝜌=1.0,and𝑝=20.

It is easy to see in the figure that increasing CFDs’ power price or reducing the rate of CFDs can delay investment timing of power producers to a certain extent, but once they invest they will increase capacity investment and reduce unit load to a certain degree, while further analysis indicates that decreasing unit load caused by increasing CFDs’ power price is for the reason that increasing sale price restrains power demand. And reducing the rate of CFDs can increase unit capacity through power market competition, reduce unit load and sale price and increase total society welfare, so this is more effective in alleviating nervous power supply than increasing CFDs’ power price. What is more, if regulation institutes subsidize capacity investment directly, it can reduce investment threshold and bring more capacity investment, but it may reduce unit load only when subsidy reaches a certain degree; that is, small subsidy cannot improve power supply condition, and the effect of this direct subsidy on sale price is not obvious and depends on other factors. From the above, if reducing the rate of CFDs such as 0.85→0.25 and adjusting subsidy𝜇may adjust power producer’s investing behavior and marketable impact effectively, so we can improve unit load rate to alleviate nervous power supply by reducing the rate of CFDs and increasing investment subsidy.

Furthermore, we would like to consider the effect of regulators’ subsidy to power capacity investment on power producer’s actual investment. As shown in Figure 8, policy subsidy increases total power capacity investment but decreases power producer’s actual investment, while reducing rate of CFDs can encourage power producers increasing their capacity investment.

(a)

(b)

Figure 8 

The effect of policy subsidy on net investment of producer.

Therefore, reducing rate of CFDs is a simpler and more efficient way to alleviate nervous power supply in China comparing to increase CFDs price and subsidy capacity investment directly; that is, reducing rate of CFDs can increase competitive to encourage power producers to increase unit capacity, reduce unit load to supply power adequately, and reduce sale price to enhance society welfare and make better use of limit power resources.

Nowadays, Chinese renewable energy policies in power generation mainly are investing and tax policies supporting wind power generation and photovoltaic power, and compelling policy for nonrenewable energy power companies such as coal-fired plant to set a certain proportion of renewable energy power in access grid power. In the short term, nonrenewable energy power companies can only buy renewable energy “index” from renewable energy power companies to satisfy the rule, which is expressed by𝜏,𝜌in above model. Obviously, this policy would increase cost of nonrenewable energy power companies and income of renewable energy power companies.

Figure 9 shows the effect of renewable energy policies on investment strategy and market price of nonrenewable energy companies. Apparently, power producer’s capacity investment will be postponed by implement of these policies and will increase once they invest and also will increase market sale price to a certain degree. For power units mainly being coal-fired units in China, it will not alleviate or even aggravate nervous power supply to implement the above renewable energy policies. Similarly, the result of investing strategy of renewable energy power companies indicates that the above policies can reduce their investment threshold and also their investing capacity. Therefore, present renewable energy policies will not necessarily be efficient in alleviating nervous power supply and popularizing constructions of renewable energy unit.

(a)

(b)

Figure 9 

The effect of renewable energy policies on investment strategy. Parameters:𝜂=0.85,𝑝1=4.0,and𝑝=20.0.

5. Conclusion

Power industry is a basic industry, and adequate power supply is significant for national economic development and people’s everyday lives. In power market environment, power producers face great risk in investing power capacity, so regulation institute needs to consider this and establish optimal investment policy to promote efficient power supply. To discuss oligopoly power producer’s investment strategy and relevant investment policy, this paper presents a three-layer investment game model, which begins with relevant policy setup by regulation institutions, then considers oligopoly power producer’s investment strategy, and analyzes power producer’s biding strategy with capacity and price cap constraint. Based on the backward induction thought in dynamic game theory, we solve and simulate the model in symmetric case and examine the effect of oligopoly power producer’s number, price cap, the proportion of contracts for differences (CFDs) and CFDs’ power price on the unit load, and power sale price, and we try to find the optimal investment policy based on the maximization of discounted social welfare.

For the proportion of power in CFDs in Chinese power market being very big and power supply being relatively nervous, we discuss the effect of power capacity investment subsidies and CFDs power price on power supply and demand. The results indicate that reducing the proportion of CFDs’ power in the power producer’s access grid power is an effective way to alleviate the tension in power supply and demand, and the current renewable energy policy can neither necessarily ease the tension condition of power supply nor can it necessarily promote the construction of renewable power generation units.

Combined with actual condition of power capacity investment, the above model expands Dangl (2009) [18] and Roques and Savva (2009) [20], analyzes Chinese actual power capacity investment, and proposes several suggestions for alleviating nervous power supply. But the model is based on the symmetric of power investment market and can also be used in an asymmetrical condition which is not discussed owing to its complexity. Moreover, this paper did not consider technical factors such as power grid block and also did not consider time to build and policy changes; if these factors are included in this model, it will expands the conclusions of this paper.

Acknowledgments

This work was supported by National Nature Science Foundation of China under 71271033 and 70971012, Education Department Talent Support of New Century under NCET-11-0978, and Scientific Research Fund of Hunan Provincial Education Department under Grant 13K057.