## Nonlinear Problems: Mathematical Modeling, Analyzing, and Computing for Finance

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

**Academic Editor:**Fenghua Wen

#### 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 is, in whichis assumed to follow geometric Brownian process,is the number of investors, and. Whilehas 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: whereis power sale price,is demand parameter, andis the total power demand.is the effects of relevant factors except sale price on power demand, and we assume that whereandis 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 whereis cost of power producer,is power quantity of producer,, andis total number of producers in the power market.are cost coefficient.

Supposings that the range ability of power producer’s biding function is the same with that of its marginal cost function, there is whereis bidding strategy parameter, that is, rational power produceradjustto 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 producer; thus whereis the transaction price of competitive power. Then, we get the total demand powerwith (1) and substitute it into the equation above; there is

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 proportionof CFDs and CFDs’ power price, there is wheredenotes the proportion of transmission cost in sale price. Substitute (7) into (6); there is Substitute the equation above into (4); there is Then, power producer’s profit function can be described by

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 asand setas the cost of per unit renewable energy. Solving the first order condition ofin (10), there is

Solving (11), we get where

Iterating (12), we get

Substituting this equation into (12), we get in equilibrium bidding condition. And substituting into (7) and (9), we can get the equilibrium power quantityof oligopoly power producers; that is, where,, and.

Obviously, oligopoly power producer’s power quantity should satisfy, andis the largest of power quantity determined by its capacity, so the equilibrium power quantity should satisfy where, and .

In market equilibrium, we can get an equation as follows from (7) and (1):

If given price cap, there is

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

##### 2.2. Oligopoly Power Producer’s Capacity Investment Model

With classical real option theory, power producer’s investment value function is as follows whereis risk-free rate anddenotes expectation. Unfolding the equation above with Ito lemma, there is

The result of this differential equation is where, anddenotes the number of’s intervals divided by boundary point, and in (19).denotes the value function of power producerin interval. And and are functions ofto be determined.is a particular solution in intervaland generally denoted by

Obviously, substituting (22) into (21) we can get the analytical solution of value functionwith the largest power quantityfor power producer. Owing to the piecewise function, the value functionis also piecewise. But in the boundary, value functionshould be continuous and smooth. If given boundary points as, there is

Substitute (21) into the equation above, there is Furthermore, there iswhenandwhen, in whichdenotes the rightmost interval ofdivided by(Dangl [18], Nagel and Rammerstorfer, [22]). Combined with (21), there is

Applying (24) and (25), we can get, 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 term. So the valueof the option to wait satisfies following equation:

To solve the above equation, there is whereis also a function ofjust like. The investment threshold should satisfy value matching and smooth pasting; so there is whereis power producer’s investment and is a function of. We would like to consider the investment function as follows: where are parameters and is a variable of investment policy. Solving (28) and eliminating, we can get the functional relationship between optimal investment thresholdand largest power quantity. Furthermore, for rational oligopoly power producer, when choosing maximum power quantity, there is

Combining (28) and (30), we can get the investment thresholdand optimal capacityfor power producer. 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 surplusin the power bidding equilibrium; there is

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

Therefore, the aim of regulation institute can be simply described by whereis the moment to start unit operation for power producer,is the examining period of regulator institute,is separately upper limit of average unit load and floor level of economic operation when power system is safely operated, andis 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)When, there is where, , ; , , and .(2)When , there is where,, , , , and .(3)When, there is where and .

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 threshold and the optimal investment capacity.

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

It is quite clear that the analytic solution of investment threshold and optimal investment capacitycannot be obtained, so we simulate these numerically in the following research. We assumed that market parameters are, , and, cost parameters are, and capacity investment parameters are.

If given, Figure 1 shows the curve of oligopoly power producer’s investment value function whenand.

**(a)**

**(b)**

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 when is higher thanbut decreasing whenis lower than. The third one is that investment value is increasing with increasing price cap. And the forth one is that investment value is increasing withbut decreasing after it reaches the highest point and then invariant in Figure 1(a). While in Figure 1(b) wheninvestment value is increasing first, then variant, and at last decrease with increasing, and the maximum value is increasing with increasing, and the investment value will be always increasing withwhen price cap is ignored. As shown in Figure 1, numberof power producer, price cap, and the rateof 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 capon 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; 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,holds; Grenadier did not consider price cap. The third, power producer’s optimal investing capacity is decreasing with increasing number of power producers; that is,. The forth, oligopoly power producer’s investment threshold and optimal capacity are increasing with increasing price cap.

**(a)**

**(b)**

Power producer’s investment threshold and capacity curve are shown in Figure 3 when. 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)**

##### 3.3. Optimal Capacity Investment Policy Analysis

Simplifying (33), there is where

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 capand the rateof CFDs whenresponding to Figures 2 and 3.

**(a)**

**(b)**

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 satisfies, then we can seewhen,when,when, andwhen. 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)**

From Figure 6, we can see that the rate of CFDsshould be lower thanwhen we adopt the standard, andis different in different cases of, for example,when,when, andwhen. 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 when. Similarly, if reducing specified unit load rate standard, the corresponding rateof CFDs will decrease, the weighting access grid power price will increase step by step, and market efficiency will descend.

**(a)**

**(b)**

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 price. Consider thatmeans that all the power is access grid through market competition, andmeans that all the power is regulated by regulation institute, sodenotes that part of power is traded through market competition and the rest is traded through regulation. Based on the real condition, we assume that, 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 policyin three cases which are.

**(a)**

**(b)**

**(c)**

**(d)**

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 0.850.25 and adjusting subsidymay 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)**

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 byin 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)**

#### 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.

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#### Copyright

Copyright © 2013 Xinhua Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.