Contract decisions analysis of shared savings energy performance contracting based on Stackelberg game theory

: Energy Performance Contracting is a contractual arrangement between energy users and Energy Service Companies (ESCOs) and is currently the main mechanism for implementing energy-saving retrofitting measures in existing buildings. The paper centers around the decision-making conundrum pertaining to the energy-saving sharing percentage and initial project investment within the realm of shared savings Energy Performance Contracting. By formulating a Stackelberg game-based decision-making game model, we examine the optimal contract decisions of both the energy user and the ESCO. The results of numerical experiments demonstrate that this method yields significant advantages for both energy users and ESCOs. Additionally, we observed that the employment of more sophisticated energy-saving technologies by the ESCO and a higher share of investment by the energy users result in superior energy-saving efficiency.


INTRODUCTION
Energy conservation and emission reduction are essential issues that China and the world will pay attention to in the future. There are at least 40% of global energy consumed and more than 30% of greenhouse gas emissions produced by buildings1. The traditional model of energy-saving retrofitting in buildings is to implement energy-saving efficiency projects by energy users as independent individuals. However, it is difficult for energy users to retrofit buildings alone due to some barriers [2,3]. First, Energy users do not have professional knowledge of energy-saving retrofit, which leads to the unreasonable selection of energy-saving equipment and formulation of energy-saving plans for buildings, resulting in secondary waste of energy.
Second, the traditional model of energy-saving retrofitting in buildings is not supported by the financial market but still requires energy users to finance the purchase of expensive energy-efficiency equipment, making it difficult for energy users to finance. Energy Performance Contracting (EPC) represents a contractual arrangement between Energy Service Companies (ESCOs) and energy users, offering an effective solution to overcome these barriers related to energy performance4. The difference between the traditional energy retrofitting model and EPC is shown in Figure. 1. However, energy users and ESCOs in EPC depend on each other and constrain each other, so how to help the decision-making of energy users and ESCOs is an important issue to increase their active participation in EPC projects. The utilization of EPC, as an advanced energy management strategy and a market-driven mechanism for energy conservation, allows for the recovery of expenses associated with implementing, operating, and Based on the above analysis, this paper will establish a Stackelberg game analysis model to help energy users and ESCOs to allocate their benefits. The remaining sections of this paper are organized as follows: Section II analyzes the decision-making process between energy users and ESCOs, describes the main symbols and their definitions, and establishes the Stackelberg game analysis model between energy users and ESCOs under Shared Savings Model; Section III is numerical simulations of the model. for both sides of the game and the government. The conclusions of this paper are given in Section IV.

Problem Description
In EPC projects, the energy user has ownership of the energy retrofit project and can decide whether to implement the energy management contract or not and occupies the dominant position in the EPC project. The dependency and constraints between the energy users and the ESCOs involve the energy retrofitting period and the project operation period, in which the game process between the two parties and the benefits and costs during the project operation period are shown in Figures  T , according to the contract. In determining its optimal investment, the ESCO is bound by its own maximum investment amount TE I , as is the EU, which has a maximum investment amount TO I .This is because the EPC project investment is large and the ESCO needs to obtain financing from a financial institution, which will provide a loan based on the ESCO's past creditworthiness.  Figure. 3. T is the remaining design working life of the retrofit project. It is the amount of time that the project can be used for its intended purpose without major repairs under conventional design, construction, use, and maintenance after the retrofit.

Hypothesis and Description of Symbols
For better description and modeling, the following assumptions are made in this paper: Hypothesis 1. The annual energy cost of the energy user is stable, i.e., the annual energy cost is ep . Hypothesis 2. The percentage of energy savings after retrofitting is a concave function of increasing input capital, i.e. the function ( ) S I satisfies / 0, dS dI 

Model
Based on the specified parameters and underlying assumptions, the mathematical programming model can be formulated using the following equation.
s.t. 0 1, s.t. , Equation (2) comprises three terms. The first term represents the total energy-saving benefits obtained by the energy user throughout the remaining design working life of the buildings. These benefits are discounted to the present value of the year when the energy-saving retrofit of the energy-using building is completed. The second term represents the present value of O&M costs during the non-contract period, and the third term represents the investment costs of the energy user. Constraint (3) constrains the benefit allocation ratio. Equation (4) also consists of three terms. The first term represents the present value of benefits gained by the ESCO during the contract period, discounted to the year when the energy-saving retrofit is completed. The second term represents the present value of O&M costs during the contract period, and the third term represents the investment costs of the ESCO. Constraints (5) and (6) limit the investment amount of energy users and ESCOs to their respective investment ceilings. Constraints (7) and (8) ensure that the energy savings in the final year of O&M exceed the O&M costs. Constraint (9) imposes a non-negative constraint on investment costs.

Contract Decision Calculation Example
Referring to some of the data from Liu et al. 15, this paper assumes that the Energy Service Company E undertakes an energy-saving retrofit project belonging to the Energy user O. The initial energy consumption costs of the retrofit project are 20,000 per year, and the remaining design working life is 15 years. The ratio of the investment from the energy user is 0.3. The discount rate is 5%. The O&M costs function of the energysaving retrofit project in year t is 2 2 200 t + . The ESCO analyzes the energy-saving ratio function based on past historical data to be . The maximum investment that can be undertaken by the energy user and the energy efficiency service company is 1500 and 3500 respectively.
The model was implemented using Matlab R2020b with the parameters mentioned above, and the results are shown in Table 2.
The data in Table II shows that the energy user proposes an optimal energy efficiency sharing ratio of 26.23%, and the optimal initial investment amount of the ESCO is 1,668. It means that the project requires an initial investment of 2,383, of which the energy user needs to provide 715 and the ESCO needs to provide 1,668. The ESCO uses 2,383 to implement building energy-saving efficiency retrofits for the energy user's energy-using building, is responsible for the building's O&M during the contract period, and pays for the O&M costs. After the end of the contract period until the end of the remaining design working life of the project, the energy user will be responsible for the operation and maintenance of the project. The overall benefit of the project is 70,240. The total benefits received by the energy user in the project are 63,305, and the profit achievable by the ESCO during the contract period is 6,935.

The Impacts of the Parameters
The sophistication of energy-saving technologies. The results in Table III for the two main contract parameters for the different " sophistication of energy-saving technologies of the ESCO" for b are as follows. The optimal sharing percentage of the energysavings determined by the energy user tends to increase with b . The optimal initial project investment determined by the ESCO increases with b . The results of these two variables are determined by the dynamic game process between the energy user and the ESCO. After these two variables are determined, ESCO's profit follows the same trend as b . Both the profit of the energy user and the total project benefit increase with the increase of b . From the data in Table III, it also follows that the ratio of total project profit to initial investment at each ESCO technology level also increases as b increases. The above analysis shows that the increase in the level of the ESCO's capability advancement improves the profit margin per unit investment, brings higher energy efficiency and socio-economic benefits, and has a positive effect on the development of the EPC mechanism.  Table IV for different "investment sharing ratios" for  are as follows. The initial investment decided by the ESCOs is opposite trend to  , and the benefit sharing ratio decided by the energy user is also opposite trend to  . This is because the ESCOs have to get a higher share percentage of the energysavings by increasing their own investment. After these two variables are determined, the ESCO profit decreases with the increase of  , and the profit of the energy user and the total project profit increase with the increase of  . From the above analysis, it is clear that energy user with a higher share of investment can generate higher energy-saving benefit themselves and higher total project revenues. Such energy users are more likely to participate in EPC projects for energy-saving efficiency retrofits. ESCOs with high financing pressure should also give priority to such energy users to relieve investment pressure and generate greater social energy efficiency benefits.

CONCLUSIONS
Energy Performance Contracting can improve the success rate of energy-saving retrofit projects of energy users, which is of great significance to the development of energy-saving in China. This paper focuses on the decision making of the contract in Shared Savings Model, where the decision variable of the energy user is the sharing percentage of the annual energy-savings and the decision variable of the ESCO is its own investment amount. This paper further establishes the Stackelberg game analysis model to analyze the decision-making process of both sides of the game, and obtains the following conclusions: (1) The improvement in the level of ESCO energy-saving technology can improve the profitability of the project's unit investment, increase the overall revenue, and bring higher energy efficiency and socio-economic benefits. It has a positive effect on the development of the EPC mechanism.
(2) ESCOs with high financing pressure may prefer to choose energy users that decide to share a higher ratio of investment. Such action can help ESCOs to relieve financing pressure and also increase the total return of EPC projects.