Research on an Investment Decision Model of Waste Incineration Power under Demand Guarantee Policies

Abstract

In order to encourage social capital to sustainably enter waste incineration power generation projects, policy-makers propose demand guarantee policies to ensure the fundamental interests of social capital. Nowadays, demand guarantee policies in China are artificially set based on industry experience and similar biomass power generation projects but lack theoretical support, thus bringing pitfalls to sustainable development. To address this issue, this paper constructs a decision model under the Real Option Approach to obtain investment triggers and guarantee level. Under lower, upper and bidirectional demand guarantee policies, this paper compares three Real Option models considering uncertain factors. The results show that demand guarantee policies usually have an excess guarantee phenomenon that affects long-term interests, and the lower demand guarantee policy can most effectively promote social capital to invest. Policy-makers can choose appropriate policies based on their demands or adjust existing guarantee policies to avoid the excessive guarantees phenomenon and attract social capital to invest in waste incineration.

1. Introduction

1.1. Background and Motivation

As the global climate is rapidly changing, carbon neutrality and emission reduction have attracted much attention. With the rapid development of China’s economy, the waste industry is facing extremely high pressure regarding waste treatment. Improper disposal methods can lead to significant waste pollution. The Guiding Opinions on Accelerating the Construction of Urban Environmental Infrastructure clearly states that, by 2025, the capacity for classified collection and transportation of domestic waste will reach around 700,000 tons per day [1]. Therefore, the capacity of urban domestic waste to be treated is considerable in the foreseeable future and, if disposed rationally, waste pollution can be reduced tremendously.

The mainstream waste treatment methods include green incineration and sanitary landfill [2]. As shown in Figure 1, due to the environmental pollution caused by sanitary landfill, China has vigorously promoted the transformation and development of domestic waste treatment from sanitary landfill to green incineration, effectively improving the quality of urban and rural living environment as well as promoting sustainable economic and social development. According to data from the National Bureau of Statistics, the harmless treatment capacity of domestic waste incineration for electricity generation in China reached 18,000 tons in 2022, accounting for 72% of the harmless treatment capacity, and will continue to increase in the next decade. Thus, more effective and accurate decision making of waste incineration power is imminent and facilitates the development of the waste industry.

The impressive demand for harmless treatment of waste has laid a solid foundation for vigorous development of power generation projects for waste incineration in China. However, it should be noted that such projects are still confronted with many uncertain factors and high operational risks, and the thought that making government departments take sole responsibility for investment and operation is irrational [3]. It is necessary to apply government guarantee policies [4] (such as demand guarantee policies and income guarantee policies [5,6]) to attract social capital for investment and achieve high-speed development in waste incineration. To encourage social capital to invest in waste incineration power generation projects, relevant departments usually provide demand guarantee policies.

In order to compare and analyze the differences in commonly applied demand guarantee policies at present, this paper considers three options for the demand guarantee policy of waste incineration power generation projects to assist decision-makers in achieving sustainable development with balance between social capital and relevant departments:

• Lower demand guarantee policy: When the actual capacity of waste incineration is lower than the set value, relevant departments compensate the project company or allow price adjustments.
• Upper demand guarantee policy: When the actual capacity of waste incineration exceeds the set value, relevant departments share the excess profit.
• Bidirectional demand guarantee policy: When the actual capacity of waste incineration is lower than the set value, relevant departments compensate the project company or allow price adjustments; when it exceeds a certain set value, relevant departments share the excess profit.

In fact, as the revenue of a waste incineration power generation project is entirely determined by demand, a demand guarantee policy can be converted into a revenue guarantee policy. The minimum volume of waste in the waste incineration power generation project can be converted into the minimum income guarantee through the revenue generated from each ton of garbage treatment.

1.2. Literature Review

In the context of Carbon Neutrality and Carbon Peak, investing in biomass power generation can effectively generate green electricity. As a typical project for biomass power generation, waste incineration power generation projects can be harmless, reduced and make use of resourceful household waste, which has certain investment value [7]. With the extremely increasing demand for waste incineration power generation, social capital is encouraged to participate in related projects. The waste incineration power generation project, which is jointly invested and operated by social capital and the public sector (or relevant departments), is essentially a typical Public-Private-Partnership (PPP) project [8].

In order to encourage social capital to actively and sustainably participate in the construction and operation of PPP projects, the public sector usually proposes certain guarantee policies to help share some part of the financing and operational risks of social capital. Oxley et al. [9] believed that various forms of guarantee policies can be used to achieve sustainable development, such as government payments, tax exemptions, low interest loans, and cross subsidies during the development process. Wu et al. [10,11] designed the optimal construction cost guarantee and the optimal operation risk compensation, that is, the minimum income guarantee, and then explored the impact of government compensation on investment triggers and timing. Wang et al. [12] pointed out that the public sector can bear the demand risk of PPP projects by providing compensation through income guarantee, but there was a problem of transitional guarantee. Gao et al. [13] proposed a demand compensation mechanism, which determined the compensation capacity based on the product of the demand compensated by government and the price of social capital charges, establishing a reasonable and effective compensation mechanism. Gao et al. [14] ensured reasonable investment returns by designing government fees or feasibility gap subsidies. Zheng et al. [15] selected a tax compensation guarantee—which means that the government compensates for social capital using financial subsidies derived from taxation—to explore the optimal investment decisions and the impact of government compensation under public cultural PPP projects. Chen et al. [16] studied the impact of three compensation methods, namely, investment subsidies, income subsidies, and demand subsidies, on government cash flow. Shi et al. [17] incorporated the charge mechanism for a minimum traffic guarantee (MTG) into consideration and proposed a method to choose the optimal concession contract variables with a paid MTG.

At present, the waste incineration industry usually adopts a policy of ensuring the minimum capacity of waste treatment set by relevant departments. For example, in the franchise agreement of the waste incineration power generation BOT project, the government and the project company agree on the minimum daily average treatment capacity standard and provide corresponding waste treatment fees [18]. The investment in waste incineration power generation projects is uncertain, irreversible, and deferrable, which allows investors to wait for the appropriate investment triggers and choose the optimal investment opportunity [19]. The Real Option Approach can be applied to solve the problem during decision making caused by the delayed investment characteristics [20] and can effectively reduce the risk impact of uncertain factors such as electricity price fluctuations, uncertain operating returns, random waste treatment capacity, and changes in operating costs on waste incineration power generation projects for sustainable development.

Based on existing literature, the Real Option Approach has been widely used in the waste incineration power generation industry. Tolis et al. [21] conducted a comparative study on investment in combined power generation based on urban solid waste, focusing on the evolution of its economic performance over time, applying the Real Option Approach to compare different options for waste energy recovery—incineration, gasification, and landfill biogas extraction. Incineration has been proven to be more attractive than other harmless waste treatment methods, mainly due to its higher power generation efficiency, lower investment costs, and lower emission rates. Taking the Philippines as an example, Agaton et al. [22] adopted the Real Option Approach under uncertain conditions to compare the option value, waiting value, and the optimal timing for technology conversion from landfill to waste and then to energy. After considering energy production and investment costs, the optimization results found that incineration is the best technology choice for sustainable development. Boomsma et al. [23] adopted the Real Option Approach to analyze investment timing and capacity choice for renewable energy projects under different support schemes. The relevant literature indicates that, for PPP projects with social capital participation, compensation or guarantee policies can be harmoniously integrated into the Real Option theory. Marzouk et al. [24] believed that a minimum-flow guarantee was a suitable risk mitigation strategy and developed a new stochastic model using the real option theory in order to simulate the wastewater inflow to the wastewater treatment plant under PPP projects. Hu et al. [25] proposed a Real Option model to evaluate the value of flexibility in the Waste to Energy (WTE) project based on incineration technology. The proposed model reflected the real decision-making process and better supported subsidy design through considering flexibility and uncertainty, applying to the WTE project based on incineration technology in China. Kim et al. [26] used a framework based on real options to reasonably quantify the capacity of government subsidies required for private entities to implement clean energy power generation systems. The proposed framework is expected to assist energy policy-makers in selecting appropriate levels of government subsidies, thereby effectively encouraging private entities to install clean energy systems. Wang et al. [27] established a Real Option model with policy benefit to evaluate biomass power generation investment in China. In this process, uncertainties in the purchase price of domestic waste, government incentives and technology improvements were considered. The research results showed that direct investment in biomass power generation was not ideal without government incentives in China.

At present, the vast majority of literature chose to assign artificially set compensation or guarantee policies to parameter values in the Real Option model, but they lacked numerical solutions to derive the optimal compensation or guarantee level from systematic models, resulting in a lack of persuasiveness in the pricing of artificially set guarantee policies and sustainable pressure for social capital or relevant departments.

1.3. Contribution and Innovation

The existing research has laid the foundation for the application of the Real Option Approach in the decision making in the waste incineration power industry. However, compared with related research, the contribution and innovations of this study are summarized as follows:

(1)

Most research only paid attention to the decision making of waste incineration power generation projects and conveniently considered guarantee policies decision conditions without offering accurate guarantee level, leading to long-term economic pressures for social capital or relevant departments. Thus, this paper considers and compares three typical demand guarantee policies in the investment decision model and points out the respective advantages of policies to assist policy-makers in formulating relevant policies for long-term development.

(2)

The artificially set guarantee quantity indicators in traditional methods will have a subtle long-term economic impact on social capital or relevant departments. In order to analyze demand guarantee policies comprehensively, this paper does not treat policies as variables or parameters like other literature does, but rather as the research object of the Real Option model to obtain the optimal demand guarantee level.

Thus, based on the Real Option Approach proposed by Dixit and Pindyck [19], this paper conducts an in-depth analysis on the decision-making research of waste incineration power generation projects in combination with different demand guarantee policies. An empirical case study is conducted, taking a municipal household waste incineration licensed power generation project as an example, and sensitivity analysis is conducted on subsidy coefficients and grid electricity prices. From theoretical and empirical evidence, the investment decision model under the Real Option Approach can reprice the minimum capacity of waste treatment for household waste incineration licensed power generation projects in China.

This study is presented in four sections. Section 2 introduces the assumptions of waste incineration power generation projects and calculates the closed solutions for demand guarantee level, option value and optimal investment timing. Section 3 presents the Real Option model that assesses the feasibility of the methodology in real-world applications and reveals the results. Section 4 presents the conclusions made in the study, as well as limitations and further research.

2. Methodology

2.1. Assumption

Without changing the essence of the problem, the following assumptions are made:

(1) The total investment cost of the waste incineration power generation project is $I$, fully funded by social capital. And the average cost of waste combustion per ton is $C$ CNY, which includes labor, debt service, and daily operation and maintenance costs.

(2) The investment income ($V_t$) of the waste incineration power generation project consists of the waste treatment fee and the electricity sales income brought by the online power generation after the waste treatment. The waste treatment capacity of tons per phase is assumed to be $Q_t$, and an average waste treatment fee is $n$ CNY per ton. Each ton of waste can be converted into $\eta$ kWh, and the online electricity price of waste is $P$ CNY per kWh. Therefore, we have $V_t=\left(n+\eta P\right)Q_t$. In addition, the subsidy coefficient from demand guarantee policy is $\phi \left(0\le \phi <1\right)$, i.e., the subsidy share is $\phi V$ CNY.

(3) The capacity of waste incineration ($Q_t$) under the waste incineration power generation project follows the geometric Brownian motion:

$$d{Q}_t=\mu Q_{t}dt+\sigma Q_{t}d{W}_t$$

(1)

where $\mu$ and $\sigma$ are respectively drift term and standard deviation, $W_t$ is increment of standard Wiener process, and $t=1,2,\dots$ is the time period. The cash flows generated during the cycle of the waste incineration power generation project are discounted at a discount rate $r$. Order $\delta =r-\mu$ represents the opportunity cost of holding investment options for project construction. However, according to the relationship between operating income and waste incineration in hypothesis (2), $V_t$ is also consistent with the geometric Brownian motion.

(4) The waste incineration power generation project is risk-neutral and does not consider the construction time. After investing in the project, it can immediately enter into production and operation. The project company can make decision at $t=0$, the earliest possible time.

2.2. Real Option Model under Lower Demand Guarantee Policy

Under the lower demand guarantee policy, when the actual capacity of waste incineration ($Q_t$) is lower than the set value ($\underset{\_}{Q}$), relevant departments compensate the project company or allow price adjustments. The set value ($\underset{\_}{Q}$) is the lower reflecting wall in the stochastic process of the waste incineration capacity. According to the assumption of the minimum capacity of waste, the lower demand guarantee is a correction to the stochastic process of operating income, namely, the geometric Brownian motion. Simply put, operating income will not decrease significantly under the lower demand guarantee policy. The lower reflection wall with corresponding operating income is $\underset{\_}{V}=\left(n+\eta P\right)\underset{\_}{Q}$.

According to the Real Option theory, if the operation income process is strictly downward without the influence of the lower reflecting wall, then the net present value function ($F$) of the income in the waste incineration power generation project at time $t$ is:

$$F=\frac{\left(1+\phi \right)V}{\delta }-\frac{C}{r}$$

(2)

where $\delta$ is the opportunity cost of holding investment options when the project construction is waiting. $\delta$ is also the difference between the risk-free interest rate and the expected growth rate under the lower demand guarantee policy. When the current reflection wall exists, the project value is corrected on $F$.

In order to quantify this correction, we start from the initial waste incineration volume $Q>\underset{\_}{Q}$; in a sufficiently subtle time interval, $dt$, the operating income is basically above the lower reflecting wall, so the following differential equation is obtained by using Itō's lemma and the Bellman equation:

$$\frac{1}{2}{\sigma }^2{V}^2{F}^{″}\left(V\right)+\left(r-\delta \right)V{F}^{\prime }\left(V\right)-rF\left(V\right)+V=0$$

(3)

The general solution is:

$$F\left(V\right)=D{V}^{{\beta }_1}+\frac{\left(1+\phi \right)V}{\delta }-\frac{C}{r}$$

(4)

where $D$ is the undetermined coefficient and ${\beta }_1$ is the negative root of the following equation:

$$\frac{1}{2}{\sigma }^2\beta \left(\beta -1\right)+\left(r-\delta \right)\beta -r=0$$

(5)

At this point, $\left[\frac{\left(1+\phi \right)V}{\delta }-\frac{C}{r}\right]$ is the option value when there is no lower reflection wall of operating income, and $\left[D{P}^{{\beta }_1}\right]$ is a correction to the lower limit of operating income, so $D$ should be a positive number. To determine the undetermined coefficient $D$, we start from a point extremely close to the lower reflection wall, and the price will almost certainly rise in the next subtle time interval. To exclude the existence of arbitrage behavior, the slope of the value function near this point must be zero. Then follows

$$F^{\prime }\left(\underset{\_}{V}\right)\equiv {\beta }_{1}D{\underset{\_}{V}}^{{\beta }_1-1}+\frac{1+\phi }{\delta }=0$$

(6)

We obtain

$$D=-\frac{{\left(1+\phi \right)\underset{\_}{V}}^{{1-\beta }_1}}{{\beta }_1\delta }>0$$

(7)

It has been proven $D$ to be a positive number, indicating that the lower reflection wall has removed some potential for a decrease in operating income, so the correction of option value is an upward trend. Then we bring $D$ into the general solution and obtain:

$$F\left(V\right)=-\frac{{\left(1+\phi \right)\underset{\_}{V}}^{{1-\beta }_1}}{{\beta }_1\delta }{V}^{{\beta }_1}+\frac{\left(1+\phi \right)V}{\delta }-\frac{C}{r}$$

(8)

The critical value of the lower reflection wall set by relevant departments must be such that the option value on the reflection wall is equal to its investment cost, indicating that social capital will not lose money under the lower demand guarantee policy. Dynamic zero condition occurs when the profit shortage condition is met. We achieve:

$$\underset{\_}{V}=\frac{{\beta }_1}{{\beta }_1-1}\frac{\delta }{1+\phi }\left(\frac{C}{r}+I\right)$$

(9)

Considering the situation where there is no demand guarantee, the real option value held by the social capital preparing to enter the waste incineration power generation project meets the following requirement:

$$f\left(V\right)=E{V}^{{\beta }_2}$$

(10)

where $E$ is an undetermined parameter and ${\beta }_2>1$ is the positive root of Equation (5). At the investment threshold ($V^{\mathrm{*}}$) with and without demand guarantee, the value matching condition and smooth pasting condition should satisfy:

$$\left\{\begin{array}{c}E{V^{*}}^{{\beta }_2}=F{V^{*}}^{{\beta }_1}+\frac{{\left(1+\phi \right)V}^{*}}{\delta }+\frac{C}{r}-I\\ {\beta }_{2}E{V^{*}}^{\left({\beta }_2-1\right)}={\beta }_{1}F{V^{*}}^{\left({\beta }_1-1\right)}+\frac{1}{\delta }\end{array}\right.$$

(11)

It should also satisfy the boundary of $f\left(0\right)=0$. Thus, we achieve:

$$V^{*}=\sqrt[{\beta }_1]{\frac{I-C/r}{1-\left(1+\phi \right){\beta }_1}·\frac{1+\phi }{{\beta }_1\delta }·{\underset{\_}{V}}^{1-{\beta }_1}}$$

(12)

There is also an investment threshold ($Q^{*}$) for the capacity of waste incineration:

$$Q^{*}=\sqrt[{\beta }_1]{\frac{I-C/r}{1-\left(1+\phi \right){\beta }_1}·\frac{1+\phi }{{\beta }_1\delta }·{\left(\left(n+\eta P\right)\underset{\_}{Q}\right)}^{1-{\beta }_1}}/\left(n+\eta P\right)$$

(13)

2.3. Real Option Model under the Upper Demand Guarantee Policy

The demand guarantee corresponding to the upper reflection wall is the upper demand guarantee policy, which means that, when the incineration capacity of waste exceeds a certain set value $\overline{Q}$, social capital needs to return the income generated by the excess capacity to the relevant departments. $\overline{Q}$ is the upper reflection wall in the stochastic process of the waste incineration capacity, which further indicates that there is an upper limit to the revenue of social capital, and the corresponding lower reflection wall of operating income is $\overline{V}=\left(n+\eta P\right)\overline{Q}$. The general solution form of the net present value function ($F\left(V\right)$) of the income from investment in the waste incineration power generation project is still:

$$F\left(V\right)=G{V}^{{\beta }_2}+\frac{\left(1+\phi \right)V}{\delta }-\frac{C}{r}$$

(14)

${\beta }_2$ is the positive root of $\frac{1}{2}{\sigma }^2\beta \left(\beta -1\right)+\left(r-\delta \right)\beta -r=0$. The undetermined parameter $G<0$. $\left[\frac{\left(1+\phi \right)V}{\delta }-\frac{C}{r}\right]$ is the option value when there is no upper reflection wall of operating income, and $\left[G{P}^{{\beta }_1}\right]$ is a correction to the upper limit of operating income, so $G$ should be a negative number. Then we have:

$$F^{\prime }\left(\overline{V}\right)\equiv {\beta }_{2}G{\overline{V}}^{{\beta }_2-1}+\frac{1+\phi }{\delta }=0$$

(15)

This condition holds for any reflective wall during the diffusion process. The solution is:

$$G=-\frac{{\left(1+\phi \right)\overline{V}}^{{1-\beta }_2}}{{\beta }_2\delta }<0$$

(16)

Furthermore,

$$F\left(V\right)=-\frac{{\left(1+\phi \right)\overline{V}}^{{1-\beta }_2}}{{\beta }_2\delta }{V}^{{\beta }_2}+\frac{\left(1+\phi \right)V}{\delta }-\frac{C}{r}$$

(17)

To find the critical value of the upper investment wall, we use dynamic zero conditions. It is known that the critical value of the upper reflection wall ($\overline{V}$) set by the relevant departments must be such that, when social capital enters the critical point ($\overline{V}$), the project value on the reflection wall ($F\left(V\right)$) is equal to the project value near the critical point ($\left[I+\overline{V}/\delta \right]$). In this way, we obtain:

$$\overline{V}=\frac{\delta {\beta }_2}{\left(1+\phi \right)\left({\beta }_2-1\right)-{\beta }_2}\left(\frac{C}{r}+I\right)$$

(18)

Meanwhile, the value matching condition and smooth pasting condition should satisfy:

$$\left\{\begin{array}{c}E{V^{*}}^{{\beta }_2}=J{V^{*}}^{{\beta }_2}+\frac{{\left(1+\phi \right)V}^{*}}{\delta }+\frac{C}{r}-I\\ {\beta }_{2}E{V^{*}}^{\left({\beta }_2-1\right)}={\beta }_{2}J{V^{*}}^{\left({\beta }_2-1\right)}+\frac{1}{\delta }\end{array}\right.$$

(19)

$E$ and $J$ are undetermined parameters. And the boundary of $f\left(0\right)=0$ is satisfied. Thus, we obtain:

$$V^{*}=\frac{{\beta }_2}{{\beta }_2-1}\frac{\delta }{\phi +1}\left(I-\frac{C}{r}\right)$$

(20)

And the investment threshold ($Q^{*}$) for the capacity of waste incineration is as follows:

$$Q^{*}=\frac{{\beta }_2}{{\beta }_2-1}\frac{\delta }{\phi +1}\left(I-\frac{C}{r}\right)/\left(n+\eta P\right)$$

(21)

Note that the investment threshold in the case of an upper guarantee demand is consistent with that in the case of no demand guarantee. This is because the upper reflection wall only affects the size of revenue but does not affect the timing of entering project options. Simply put, as long as the random variable of operating income or waste incineration capacity reaches a certain threshold and returns, social capital will execute the project.

2.4. Real Option Model under the Bidirectional Demand Guarantee Policy

A more reasonable and effective scenario is for the public sector to promise compensation to investors when the actual capacity of household waste incineration is less than the set value ($\underset{\_}{Q}$). At the same time, when the actual capacity of waste incineration is higher than a certain set value ($\overline{Q}$), investors also need to return the income generated by the higher part to the relevant departments. This policy is called the bidirectional demand guarantee policy. The set value $\underset{\_}{Q}$ is the lower reflection wall in the stochastic process of operating income, and the set value $\overline{Q}$ is the upper reflection one. According to the assumption of operating income ($V=\left(n+\eta P\right)Q$), bidirectional demand guarantee is a two-way correction to the stochastic process of operating income, namely, geometric Brownian motion. Simply put, through the government guarantee policy, operating income will not decrease very low because of the lower reflection wall at some locations, but at the same time, it will not rise very high, due to the upper reflection wall. When the lower reflection wall does not work, the enterprise value is:

$$v_1(V)=\frac{(1+\phi )V}{\delta }-\frac{C}{r}+A_1{V}^{{\beta }_1}+A_2{V}^{{\beta }_2}$$

(22)

When the lower reflection wall works, the enterprise value is:

$$v_2(V)=\frac{\underset{\_}{V}-C}{r}+B_1{V}^{{\beta }_1}+B_2{V}^{{\beta }_2}$$

(23)

When the upper reflection wall works, the enterprise value is:

$$v_3(V)=\frac{\overline{V}-C}{r}+C_1{V}^{{\beta }_1}+C_2{V}^{{\beta }_2}$$

(24)

Specifically, there are valuable matching conditions and smooth adhesion conditions on the critical value of the reflection wall:

$$\genfrac{}{}{0pt}{}{\begin{array}{c}{v}_1(\underset{\_}{V})=aI , v_1^{\prime }(\underset{\_}{V})=0\\ v_2(\underset{\_}{V})=aI , v_2^{\prime }(\underset{\_}{V})=0\end{array}}{\begin{array}{c}{v}_2(\overline{V})=aI+\frac{\overline{V}}{\delta } , v_2^{\prime }(\overline{V})=0\\ v_3(\overline{V})=aI+\frac{\overline{V}}{\delta } , v_3^{\prime }(\overline{V})=0\end{array}}$$

(25)

There are eight equations to determine two reflection walls $\overline{V}$ and $\underset{\_}{V}$, along with six constants $A_1$, $A_2$, $B_1$, $B_2$, $C_1$ and $C_2$. We obtain:

$$\underset{\_}{V}=\left[arI+\frac{1-{\beta }_2}{1+{\beta }_1}r+\frac{{\beta }_2-{\beta }_1}{1+{\beta }_1}C\right]/\left(1+\phi \right)$$

(26)

and

$$\overline{V}=\left[\frac{\left(1+{\beta }_1\right)\left(aI-C/r\right)}{\left(1+{\beta }_1\right)/r+\left({\beta }_1-1\right)/\delta }\right]/\left(1+\phi \right)$$

(27)

At the investment threshold ($V^{*}$) under the two states of bilateral demand guarantee and no demand guarantee, the value matching condition and smooth pasting condition should be met:

$$\left\{\begin{array}{c}E{V^{*}}^{{\beta }_2}=B_1{V^{*}}^{{\beta }_1}+B_2{V^{*}}^{{\beta }_2}+\frac{{\left(1+\phi \right)V}^{*}}{\delta }+\frac{C}{r}-I\\ {\beta }_{2}E{V^{*}}^{\left({\beta }_2-1\right)}={\beta }_1{B}_1{V^{*}}^{\left({\beta }_1-1\right)}+{\beta }_2{B}_2{V^{*}}^{\left({\beta }_2-1\right)}+\frac{1}{\delta }\end{array}\right.$$

(28)

$E$, $B_1$ and $B_2$ are undetermined parameters. And the boundary of $f\left(0\right)=0$ is satisfied.

$$V^{*}=\sqrt[{\beta }_1]{\frac{I-C/r}{K\left(1-\left(1+\phi \right){\beta }_1\right)}}$$

(29)

where $K=\frac{\frac{1+{\beta }_2}{1+{\beta }_1}\frac{C}{r}+\frac{1-{\beta }_2}{1-{\beta }_1}}{\frac{{\beta }_2-{\beta }_1}{{\beta }_2}{\left[rI+\frac{1-{\beta }_2}{1-{\beta }_1}C-\frac{1-{\beta }_2}{1+{\beta }_1}r\right]}^{{\beta }_1}}$. and the investment threshold ($Q^{*}$) for the capacity of waste incineration:

$$Q^{*}=\sqrt[{\beta }_1]{\frac{I-C/r}{K\left(1-\left(1+\phi \right){\beta }_1\right)}}/\left(n+\eta P\right)$$

(30)

2.5. Real Option Value

The waste incineration power generation project is essentially a financial product in which social capital holds deferred options. The optimal execution time ($\tau$) for different demand guarantee policies is:

$$\tau =inf\left(t\ge 0|V_t>V^{*}\right)$$

(31)

The value function of investment options held by social capital is as follows:

$$F\left(V_t\right)=\left\{\begin{array}{c}\left(\frac{\left(1+\phi \right)V^{*}}{r-\mu }-I-\frac{C}{r}\right){\left(\frac{V_t}{V^{*}}\right)}^{{\beta }_2} ,V_t\le V^{*}\\ \frac{\left(1+\phi \right)V^{*}}{r-\mu }-I-\frac{C}{r} , V_t>V^{*}\end{array}\right.$$

(32)

3. Case Study

A certain city signed a domestic waste incineration franchise power generation project agreement in 2015 and began construction of the first phase in May 2016. It was officially put into operation in May 2018. The second phase of the project started construction in August 2020 and commenced operation in December 2021. Both phases of the project refer to the practices of the waste treatment industry and have set a minimum capacity of waste treatment. The project stipulates that the minimum capacity of waste in the first, second and third year of commercial operation is 70% of the design scale, and the minimum capacity of waste in the fourth and subsequent years is 90% of the design scale.

However, based on the local waste collection and transportation situation, waste growth trends and industry forecasts, there are doubts about whether the establishment of the minimum capacity of waste treatment is reasonable and brings the highest benefits. Therefore, this article proposes demand guarantee policies and studies the impact of different demand guarantee policies on project investment thresholds, timing and real option values, developing more reasonable and effective guarantee policies for relevant departments.

3.1. Parameter Settings

3.1.1. Estimation of Parameters Related to Waste Incineration Volume

This paper selected the data of the annual treatment capacity of domestic waste in the city that had undergone harmless treatment by incineration from 2006 to 2021 from the Wind database. According to Section 2, we assume that the treatment capacity of waste incineration conforms to the geometric Brownian motion and the maximum likelihood estimation method is used to estimate the drift rate ($\mu$) and fluctuation rate ($\sigma$) of the treatment capacity of domestic waste in the city. The experimental results show that the drift rate ($\mu$) of the domestic waste treatment volume in a certain city is 0.0355, with a confidence interval of $\left[0.0089,0.0620\right]$. The fluctuation rate ($\sigma$) of the domestic waste treatment volume in a certain city is 0.2317, with a confidence interval of $\left[0.2042,0.2855\right]$. We also select the domestic waste incineration treatment capacity of 257,100 tons in 2021 in this city, of which 80% is treated by the domestic waste incineration power generation project. Therefore, the initial value of the waste incineration treatment capacity ($Q_0$) is 2,064,800 tons.

3.1.2. Estimation of Other Parameters

(1) Waste power grid electricity price

The province where the certain city is located has approved online electricity consumption for waste incineration power generation in accordance with the Notice on Improving the Price Policy for Garbage Incineration Power Generation issued by the National Development and Reform Commission. The online electricity price is 0.65 CNY/kWh (including tax), and the remaining online electricity consumption is executed according to the same type of coal-fired power generation units in the province. The Implementation Plan for Improving the Construction and Operation of Biomass Power Generation Projects (NDRC Energy [2020] No. 1421) stipulates that all biomass power generation projects that have been approved but have not yet started or have been newly approved within the plan will be allocated and determined through competitive means for grid electricity prices. The household waste incineration concession project studied in this paper has been completed and put into operation, so the electricity price for waste incineration in this project is 0.65 CNY/ten million hours (including tax).

(2) Electricity generation per ton of waste

A similar municipal household waste incineration power plant was connected to the grid for power generation, with an annual treatment capacity of 219,000 tons of household waste and an annual power generation capacity of approximately 83.4 million kilowatt hours in 2020. Therefore, the power generation capacity per ton of waste in the case study is approximately 380.82 kWh/ton.

(3) Waste treatment fee per ton

The social capital can earn an income of 50 CNY/ton (including tax) for the treatment fee per ton of household waste incineration according to a similar waste incineration power generation projects.

(4) Subsidy for waste incineration power generation

According to the Notice on Improving the Price Policy for Garbage to Energy Generation issued by the National Development and Reform Commission (FGJG [2012] No. 801), the local provincial power grid bears 0.1 CNY/10 million hours for household waste to energy generation projects, the capacity of electricity generated through waste treatment is converted into online electricity. And according to the Notice on Reducing the Additional Subsidy Fund for Renewable Energy Tariff of Environmental Illegal Waste to Energy Generation Projects (CJ [2020] No. 801) and the Implementation Plan for Improving the Construction and Operation of Biomass Power Generation Projects (FG Energy [2020] No. 1421), the subsidy funds for environmental illegal waste to energy generation projects can be reduced, and the project subsidy funds are jointly borne by the central and local governments. This waste incineration power generation concession project disposes of domestic sludge, and the subsidy standard for sludge electricity price is 50 CNY/ton. Therefore, the calculated subsidy coefficient ($\phi$) for household waste incineration power generation is 0.168.

(5) Waste power generation cost

According to the social capital winning a domestic waste incineration power generation concession project in another city in 2019, the cost per ton ($C$) of waste power generation includes personnel costs, financing costs, daily operations (transportation, facilities, equipment maintenance and all other daily expenses), and the winning fee is 128 CNY/ton.

(6) Design scale of waste treatment capacity

The total planned construction scale ($S$) of this household waste incineration concession power generation project is 1400 tons/day, which is divided into two phases. The first phase provides a waste treatment scale of 800 tons/day, and the second phase provides a waste treatment scale of 600 tons/day.

(7) Discount rate

The domestic waste incineration licensed power generation project uses the 10-year treasury bond and its interest rate to calculate the risk-free interest rate. With reference to the industry literature [28], the discount rate ($r$) is 3.7%.

(8) Investment cost

All financing for this household waste incineration concession project is completed by the social capital. When the project company cannot successfully complete the financing, the social capital will solve it through shareholder loans, supplementary guarantees and other channels. The investment cost ($I$) of this project is as high as CNY 417 million.

To sum up, we set the parameters in the Real Option model as shown in

Table 1. Parameter Table of Domestic Waste Incineration Power Generation Concession Project.

Parameter	Symbol	Unit	Value
Parameters related to waste incineration volume
Drift rate	$\mu$	-	0.0355
Fluctuation rate	$\sigma$	-	0.2317
Initial capacity	$Q_0$	$\times {10}^4\mathrm{t}$	20.648
Other parameters
Waste power grid electricity price	$P$	CNY/KWh	0.65
Electricity generation of waste	$\eta$	KWh/t	380.82
Waste treatment fee	$n$	CNY/t	50
Subsidy coefficient	$\phi$	%	0.168
Generation cost	$C$	CNY/t	128
Discount rate	$r$	%	3.7
Design scale	$S$	t	1400
Investment cost	$I$	$\times {10}^8$CNY	4.17

.

3.2. Analysis on Different Demand Guarantee Policies

3.2.1. Investment Analysis under the Lower Demand Guarantee Policy

The parameter estimation results of the geometric Brownian motion are substituted into the investment decision-making model of the domestic waste incineration licensed power generation project selected in this paper, and the operating income and investment threshold results of waste incineration treatment capacity are obtained as shown in Figure 2. The investment threshold of waste incineration capacity and the investment threshold of operating income are reduced in proportion, and we select a 90% confidence interval for the volatility and drift rate of operating income in line with the geometric Brownian motion. The upper and lower limits of the investment threshold are displayed with the estimated expectation of the investment threshold. The results show that the investment threshold of operating income of domestic waste incineration power generation projects in this paper is around $6\times {10}^7$ CNY per year, and the investment threshold of waste incineration capacity is at $2\times {10}^5$ tons per year. That is about 548 tons per day.

Specifically, the project stipulates that the minimum waste guarantee rate for the first, second and third years of commercial operation is 70% of the design scale, and the minimum waste guarantee rate for the fourth and subsequent years is 90% of the design scale. To more intuitively observe the relationship between the minimum waste guarantee rate and investment threshold, we assumed that the minimum waste guarantee rate remains unchanged during the operation period. Based on this, we find a negative correlation between the minimum waste guarantee rate and the operational investment threshold. A higher minimum waste guarantee rate means that the operational risks that the social capital needs to bear are reduced. However, as relevant departments set minimum waste guarantee rates artificially and rely on operating experience of similar projects in the industry, there is a possibility of missing the optimal minimum waste guarantee rates.

3.2.2. Comparative Analysis under Different Policies

In order to find the optimal guarantee capacity level of waste under different demand guarantee policies, three scenario investment analyses were conducted on the reflective walls. The results of real option value, reflective wall, operating income, investment threshold and optimal investment timing for waste incineration treatment capacity were obtained as shown in

Table 2. Investment threshold and option value of domestic waste incineration power generation concession projects with different demand guarantees.

	Lower Demand Guarantee Policy	Upper Demand Guarantee Policy	Bidirectional Demand Guarantee Policy
Option value	$4.5626\times {10}^{10}$	$5.5394\times {10}^{10}$	$4.8502\times {10}^{10}$
Lower reflection wall of operation revenue	257,350	-	234,640
Lower reflection wall of waste treatment capacity	790	-	833
Upper reflection wall of operation revenue	-	324,010	264,340
Upper reflection wall of waste treatment capacity	-	1090	890
Investment trigger of revenue	162,000	196,370	172,120
Investment trigger of capacity	545	661	580
Demand guarantee rate	61%	73%	64%
Investment timing	0	4.3856	0.7032

. The data indicate that, under the lower demand guarantee policy, the project can be carried out when the daily waste incineration treatment capacity reaches 545 tons per day. Since 545 is less than the initial waste incineration treatment capacity, it reflects that the optimal investment decision is to invest immediately. Under the premise of the upper demand guarantee policy, project options can be executed when the waste incineration treatment capacity is 661 tons per day, and the optimal execution time is approximately 4.4 years after the initial period. Under the policy of ensuring the demand for bidirectional reflective walls, the investment threshold for waste incineration treatment capacity reaches 580 tons per day, and the optimal execution time is around 0.7 years from the initial period.

Obviously, the demand guarantee policy with the lower reflection wall provides the highest guarantee level and shares some part of risks for social capital, making the investment threshold and optimal investment timing the lowest one. The guarantee of the demand for the lower reflection wall purely has a limitation on the operation revenue of social capital, setting an upper limit for operating income and collecting partial benefits. Therefore, the investment threshold for social capital in the case of the upper reflection wall is the highest and the optimal investment timing is the latest. Although the dual reflection wall provides a certain income guarantee for social capital, it still needs to share the benefits with the relevant departments. Therefore, the investment threshold and optimal investment timing under the bidirectional demand guarantee policy are moderate.

The values of reflective walls under different demand guarantee policies are presented in Table 2. The lower reflective wall under the lower demand guarantee policy has a waste incineration treatment capacity of 790 tons per day, the upper reflective wall under the upper demand guarantee policy has a capacity of 1090 tons per day, and the upper and lower reflective walls under the bidirectional demand guarantee policy are 890 tons and 833 tons per day, respectively. The pricing of the lower reflection wall under the bidirectional demand guarantee is slightly higher than that under the lower demand guarantee, while the pricing of the upper reflection wall under the bidirectional demand guarantee is lower than that under the upper demand guarantee. This is because the bidirectional demand guarantee ensures a portion of the operating income of social capital, but at the same time limits the excessively high operating income of social capital. Therefore, the pricing of the reflective wall under bidirectional conditions is between that of reflective walls under unidirectional conditions, sacrificing some operating income in exchange for a certain guarantee of income.

The current or previously implemented demand guarantee policies have the excessive guarantee phenomenon due to artificial setting. The guarantee policy directly affects the operational benefits and costs of social capital, thereby affecting the optimal execution conditions of waste incineration power generation concession projects. It can be seen from Table 2 that, based on the design scale of 900 tons per day of waste treatment capacity under the first phase of the project, the waste guarantee rates of the waste incineration power generation concession project studied in this article under the policies of lower, upper and bidirectional demand guarantee are 61%, 73% and 64% respectively, which are far less than the 90% set artificially or through the industry experience. Therefore, the current or previously implemented demand guarantee policies have an excess guarantee rate for waste, which causes the relevant departments to bear excessive operational risks. According to the specific parameter settings of the waste incineration power generation concession project, the artificially set minimum guarantee rate of waste can be adjusted appropriately to avoid the excessive guarantee phenomenon and help relevant departments and social capital to reasonably share operational risks for sustainable development.

3.3. Sensitivity Analysis

3.3.1. Sensitivity Analysis of Electricity Prices

Since October 2022, many provinces have successively introduced or planned to introduce new policies for biomass power generation grid electricity prices in waste incineration power generation projects. For example, provinces such as Shandong, Guizhou and Zhejiang require waste incineration grid electricity prices to be compared to the benchmark prices for coal-fired power generation during the same period, and form grid electricity prices through competitive allocation in the later stage. The 0.65 CNY/kWh grid electricity price in the case study is the sum of the benchmark grid electricity price for desulfurization coal-fired units, provincial grid subsidy of 0.1 CNY/kWh and national subsidy.

However, the Notice on Improving the Price Policy for Garbage to Energy Generation (FGJG [2012] No. 801) states that waste to energy generation projects using household waste as raw materials will first be settled by converting the incoming waste treatment capacity into grid electricity. Each ton of household waste will be converted into grid electricity of 280 kWh, and the remaining excess electricity will be subject to the grid electricity price of similar coal-fired power generation units in the local area. Thus, the newly introduced policy of biomass power grid electricity price has resulted in the electricity price in this model not necessarily being guaranteed at 0.65 CNY/kWh. Therefore, in the sensitivity analysis of electricity prices, the changes in investment thresholds for waste incineration power generation in Shandong, Guizhou, Zhejiang and Ningxia province were compared under different electricity prices, and the following evaluations were obtained.

(1) The grid electricity price is directly proportional to the investment threshold under various demand guarantee policies.

After selecting different data for the grid electricity price of waste power generation as shown in

Table 3. Numerical data of electricity price sensitivity analysis.

Parameters	Cases
	Shandong	Guizhou	Zhejiang	Ningxia
Waste power grid electricity price (CNY/kWh)	0.4949	0.4515	0.5153	0.3595
Lower demand guarantee policy
Investment trigger of waste treatment capacity (t)	680	731	659	868
Investment opportunity (year)	5.188	7.225	4.305	12.060
Upper demand guarantee policy
Investment trigger of waste treatment capacity (t)	825	886	799	1052
Investment opportunity (year)	10.628	12.638	9.7267	17.4756
Bidirectional demand guarantee policy
Investment trigger of waste treatment capacity (t)	723	777	700	922
Investment opportunity (year)	6.9111	8.9402	6.005	13.76

, we find that, the higher the grid electricity price of waste power generation, the higher the income of the household waste incineration power generation concession project, and the lower the investment threshold for social capital to enter such projects, thus leading to advanced optimal investment timing. By comparing and analyzing the setting of the waste power grid electricity price for the biomass power generation policies between Guizhou Province and Ningxia Province, Guizhou Province provides a grid electricity price of 0.4515 CNY/kWh, which is only about 0.1 CNY/kWh higher than the grid electricity price provided by Ningxia Province. However, the difference of 0.1 CNY/kWh greatly influences the investment threshold and optimal investment timing of domestic waste incineration power generation concession projects. Under the three demand guarantee policies, the investment threshold for waste treatment in Guizhou has been reduced by about 100 tons per day, and the optimal investment timing in Guizhou has been advanced by about five years. The advance in investment timing is extremely beneficial for investors and society to promote the sustainable development of waste-to-energy in power generation.

(2) The investment trigger under the lower demand guarantee policy is the lowest, as that under the upper demand guarantee policy is the highest.

Comparing the data of Shandong, Guizhou, Zhejiang and Ningxia Province,

Table 4. Numerical data of subsidy coefficient sensitivity analysis.

Parameters	Case
	Waste incineration	Agroforestry Biology	Animal Manure and Straw	Biogas
Subsidy (CNY/KWh)	0.1313	0.25	0.1991	0.156
Subsidy coefficient	0.168	0.32	0.2548	0.1997
Lower demand guarantee policy
Lower reflection wall (t)	790	415	521	665
Investment trigger of waste treatment capacity (t)	545	286	359	458
Investment timing (year)	0	0	0	0
Upper demand guarantee policy
Upper reflection wall (t)	1090	572	719	917
Investment trigger of waste treatment capacity (t)	661	347	436	556
Investment timing (year)	4.3856	0	0	0
Bidirectional demand guarantee policy
Lower reflection wall (t)	833	437	549	710
Upper reflection wall (t)	890	467	587	749
Investment trigger of waste treatment capacity (t)	580	357	448	572
Investment timing (year)	5.1839	0	0	0

indicates that the properties of the three demand guarantee policies after adjusting the grid electricity price for waste power generation are consistent with those obtained in the Case Study. The demand with the lower reflection wall ensures that social capital shares operational risks and reduces operating costs without being shared benefits with relevant departments, having the lowest investment threshold and the earliest investment opportunity. The upper demand guarantee policy not only fails to share risks but shares benefits with the relevant departments, which exacerbates the operational pressure of social capital and has the highest investment threshold and the latest investment timing. The bidirectional demand guarantee policy allows social capital to share benefits with relevant departments but also provides a certain level of income guarantee for social capital. Therefore, the investment threshold and timing are between the guarantee of demand for unidirectional reflective walls.

3.3.2. Sensitivity Analysis of Subsidy

There are also differences in subsidy policies for different types of biomass power generation projects, but the conclusion remains unchanged. As a typical representative of biomass power generation, domestic waste incineration power generation has numerical differences from the subsidy policies received by other biomass power generation projects. For example, in the Notice on Issues Related to the 2023 Biomass Power Generation Tariff Subsidy issued in December 2022, the subsidy capacity for each biomass power generation project that has expired as a national subsidy has been clearly pointed out. In this section, we compare and analyze the subsidy for household waste incineration power generation with that of other types of biomass subsidies and convert the subsidy capacity for different biomass power generation into a subsidy coefficient, as shown in Table 4. We calculate the impact of different subsidy coefficients on the reflection wall, investment threshold and timing of biomass power generation concession projects in the case of agricultural, forestry and biological natural gas power generation concession projects, and the following evaluation is obtained.

(1) There is a negative correlation between subsidy coefficient and reflection wall.

There is a negative correlation between the subsidy coefficient and the reflection wall, that is, the higher the subsidy coefficient, the lower both the upper and lower reflection walls. When other parameters remain unchanged, the higher the subsidy coefficient, the greater the capacity of social capital received from national or provincial subsidies, and the lower the return guarantee provided by relevant departments to achieve balance, that is, the lower the reflection wall. On the other hand, when the subsidy coefficient increases, social capital can receive more investment subsidies. In order to balance social capital, the corresponding operating benefits shared by social capital and relevant departments during operation are higher, resulting in lower upper and lower reflection walls.

(2) There is a negative correlation between subsidy coefficient and investment threshold.

The higher the subsidy coefficient, the lower the investment and financing pressure and cost of social capital, and the easier it is for social capital to enter the investment of biomass power generation concession projects, thus lowering the corresponding investment threshold. When the investment threshold in some cases is lower than the initial waste treatment capacity, it means that the project should be executed ahead of time. However, the biomass power project is risk-neutral, meaning that social capital can invest at the earliest time. Therefore, when the optimal investment opportunity is negative, the optimal investment opportunity is at $\mathrm{t}=0$, recommending immediate investment.

4. Discussion

Based on the Real Option model and case study above, we obtain the following main conclusions and promote discussion based on these conclusions:

(1) The lack of persuasive and reliable demand guarantee level in the waste incineration power industry results in the excess guarantee phenomenon and the possibility of missing the optimal waste guarantee rate.

(2) There is a positive correlation between the grid electricity price of waste power generation and the investment threshold under all three demand guarantee policies.

(3) There is a negative correlation between the subsidy coefficient and the reflection wall and investment threshold.

(4) The investment triggers under lower, bidirectional and upper demand policy gradually increase, the investment timing is gradually delaying.

In industries such as waste incineration and even biomass power generation, there is an urgent need for capital investment to support long-term sustainable new energy development. In order to attract social capital, relevant departments have proposed a series of compensation policies, but currently there is no systematic model analysis of the most common demand guarantee policies to prove whether current policy formulation can most contribute to the sustainable development of waste incineration power generation.

For the lower demand guarantee policy in other industries, Song et al. [29] suggested that an increased lower demand guarantee level could significantly reduce social welfare. Especially, relevant departments need to guarantee all unrealized demand and assume full demand risk. On the other hand, an increased lower demand guarantee level allows for the social capital to receive a high revenue without risks and reduces the social capital to improve demand. The conclusion of this paper that the lower demand guarantee in the waste incineration power industry is too high and the existence of the excess guarantee phenomenon indicates that the pitfalls brought by the lower guarantee policy is also common in waste incineration power industry. With the help of the Real Option theory, we offer an advanced Real Option model to calculate the accurate demand guarantee level. In this way, we believe that the decision-making model can help the industry to set systematic planning for demand guarantee policy.

Another way to compensate for the deficiencies of the lower demand guarantee policy is to shift to another guarantee policy. Zhao et al. [30] compared the lower demand guarantee with the flexible demand guarantee policy and proved that social capital’s optimal effort decreases with respect to the guarantee level under the lower demand guarantee policy. Moreover, the bidirectional demand guarantee policy can not only decrease the financial risk of relevant departments, but also provide social capital with a disincentive to increase demand based on our results. Furthermore, the upper demand guarantee policy is equally valuable under the situation where the industry develops too rapidly to be controlled. The upper demand guarantee policy can encourage relevant departments to control the speed of the new energy industry with high-quality technology, thereby achieving stable and sustainable development.

5. Conclusions

Waste incineration helps to achieve great potential and long-term development in the harmless, reduced and resourceful treatment of waste. In order to encourage social capital to enter waste incineration power generation projects sustainably, decision-makers propose demand guarantee policies to ensure the fundamental interests of social capital. This paper provides a systematic Real Option model for the waste incineration power generation industry to achieve investment trigger, investment timing and option value of projects under the lower, upper and bidirectional demand policies.

The current guarantee policy for the waste incineration power industry usually results in the excess guarantee phenomenon. Although policy-makers desire to guide social capital into the industry with the help of the compensation share, the excess guarantee phenomenon generally brings relevant departments financial pressure. The recommendation that policy-makers reduce demand guarantee level from the perspective of the Real Option model is necessary for the relevant departments to achieve sustainable development with the most economic benefits. Policy-makers can choose appropriate demand guarantee policies based on the requirements of project investment trigger and timing, thereby promoting the entry of social capital into project investment for sustainable development.

One limitation of this study is the assumption of uncertainty. We simply assume that only the amount of waste incineration is stochastic under the geometric Brownian motion. However, uncertainty is possible for other elements, such as the price of electricity, investment cost and discount rate. Other stochastic processes are also significant to study. In addition, we limit to acquire project value under the perspective of economic effects and ignore the environmental effects for waste incineration power projects.

Thus, further studies can investigate several uncertainties together to achieve a more accurate Real Option model in waste incineration power projects. In addition, the stochastic processes of uncertainties are not limited to the geometric Brownian motion, but mean regression process, Poisson process and so on. Furthermore, as the waste incineration power project is a typical green energy project, further study must be conducted on the carbon emission trading mechanism to calculate investment triggers and timing in the perspective of environmental effects.