Evaluation and strategic optimization of public institution energy efficiency: a hybrid framework combining CRITIC weighting and tripartite evolutionary game

1 Introduction

Driven by the rapid urbanization in China, the number of public institutions and their associated floor area have witnessed continuous expansion, leading to a sustained surge in energy demand. As major energy consumers with large operational scales and significant social demonstration effects, public institutions play a critical role in promoting economy-wide energy conservation, carbon emission reduction, and green sustainable development (Xu, 2022). Recognizing this importance, in 2025, the National Administration of Government Offices of China issued the “Work Plan for Energy and Resource Conservation in Public Institutions”, which calls for nationwide action by public institutions to reduce energy consumption and carbon emissions in alignment with the country’s carbon peaking and carbon neutrality goals (Zou, 2025). Indeed, strategic perspectives on China’s dual carbon path emphasize that adhering to the “conservation and efficiency priority” is the fundamental premise for the green transition, identifying energy efficiency improvement as a key driver for ensuring energy security and reducing emissions (Du, 2023). However, as reforms in public institutions deepen and the energy system undergoes a rapid transformation toward low-carbon alternatives, energy management has become increasingly complex (Li and Zeng, 2025). The continuous expansion of building stock and evolving energy use patterns have made system regulation more challenging. Concurrently, existing evaluation standards, management mechanisms, and technical solutions are often inadequate to support green and low-carbon development requirements, making the establishment of a refined, intelligent, and systematic energy governance framework a pressing necessity (Chu et al., 2019).

In response to the high energy consumption of public buildings, extensive research has been conducted in the fields of energy-saving technologies and evaluation methodologies. Energy retrofitting efforts have primarily focused on optimizing high-energy-consuming systems such as HVAC and lighting (Zhou et al., 2024). For instance (Yang, 2019), report that energy-saving measures, such as replacing traditional steam-driven lithium bromide chillers with high-efficiency air-cooled heat pump units and implementing smart LED lighting systems, have led to over 20% energy savings in public building retrofits in Qingdao. Additionally (Jin et al., 2024), introduced an air-conditioning load potential assessment method under thermal comfort constraints, providing insights for energy-efficient air-conditioning retrofits. Other studies have explored innovative technical solutions, such as Model Predictive Control (MPC) methods to optimize HVAC operation (Li et al., 2020), and the integration of simulation models to assess the impact of occupant behavior on energy performance (Huang et al., 2022). Complementing the focus on technological and physical retrofits, research into human-building interaction suggests a parallel pathway for energy conservation. For example (Zaidan et al., 2021), indicate that adjusting occupant behavior and engagement in office environments alone can yield energy savings of 9%–18%, highlighting the significant, technology-independent potential of behavioral interventions.

In terms of energy efficiency evaluation, various multi-dimensional indicator systems and methodologies have been proposed, including DEA models for integrated energy systems (Zhao et al., 2022), multi-dimensional comfort frameworks for building performance (Zhong et al., 2025; Xue et al., 2023), and performance evaluation models for public institutions in cold regions (Liu et al., 2025). Furthermore (D’Agostino et al., 2017), proposed a comprehensive energy efficiency evaluation approach for public building electricity users and applied a TOPSIS–RSR model to rank energy efficiency across multiple users. However, existing research often overlooks the unique characteristics of public institutions, such as their management structures, fiscal dependencies, and energy use patterns. Previous studies tend to focus mainly on isolated technical dimensions like energy consumption and carbon emissions, without considering a comprehensive, systemic evaluation approach (Lazaro and Baba, 2024). As pointed out by (Родригес et al., 2023), energy-saving evaluation for the public sector remains at an early stage, with a shortage of scientifically unified quantitative tools and systematic methodologies. The absence of such comprehensive evaluation approaches limits the accurate identification of energy-saving potential and undermines the effectiveness of energy efficiency improvement initiatives in public institutions.

Furthermore, energy efficiency improvement in public institutions inherently involves strategic interactions among multiple stakeholders, including government authorities, energy service companies (ESCOs), and public institutions themselves. Given that evolutionary game theory (EGT) is well suited for analyzing strategy interactions among boundedly rational agents, therefore it has emerged as an effective analytical tool in this context (Cao et al., 2023). In recent years, evolutionary game models have been widely applied to multi-agent coordination problems in energy management, environmental governance, and technological innovation (Chen et al., 2025). developed a payoff game model for rooftop photovoltaic development involving residential, commercial-industrial, and public-sector entities (schools and government buildings), and proposed region-specific policy recommendations for improving market mechanisms (Yang et al., 2025). constructed a stochastic evolutionary game model among wind, photovoltaic, and pumped-storage power producers, revealing coalition formation mechanisms and evolutionary stability conditions (Xie and Liu, 2022). found that decision-making behaviors among stakeholders are interdependent and that the effectiveness of government supervision policies depends on both project profitability and public willingness to participate (Yang et al., 2020). introduced “whistleblowers” as an additional game player and demonstrated that well-designed incentive and penalty mechanisms can significantly alter corporate strategies (Ren et al., 2020). developed a tripartite evolutionary game model involving government, energy service companies, and users, showing that government subsidies and ESCO pricing strategies are critical determinants of system evolution. These studies confirm the effectiveness of evolutionary game theory in capturing stakeholder interactions, strategy selection, and dynamic evolution processes, providing a solid theoretical and methodological foundation for its application to energy efficiency improvement in public institutions.

In summary, although substantial progress has been made in both energy retrofit evaluation methods and evolutionary game applications, systematic studies that integrate energy efficiency evaluation with multi-stakeholder game analysis for public institutions remain limited. To bridge this gap, this study constructs an energy efficiency evaluation system for public institutions utilizing the CRITIC weighting method and develops a performance prediction model to accurately identify energy-saving potential and prioritize retrofit measures. Furthermore, a tripartite evolutionary game model involving the government, public institutions, and Integrated Energy Service Providers (IESPs) is established to investigate decision-making behaviors and interaction mechanisms in energy efficiency improvement projects. The overall research framework is illustrated in Figure 1.

The rest of this paper is organized as follows. Section 2 presents the construction of the energy efficiency evaluation system and the performance prediction model. Section 3 describes the development of the tripartite evolutionary game model. Section 4 applies the proposed framework to a case study to demonstrate the process and effectiveness of the proposed models. Section 5 provides the results and discussion.

2 Energy efficiency evaluation system and improvement effect prediction model

2.1 Energy efficiency evaluation indicator system for public institutions

Based on national standards for energy efficiency evaluation and management of public institutions, relevant literature (Song et al., 2023; Zhao et al., 2018; Wang J. et al., 2023), and survey data obtained from competent authorities, a comprehensive energy efficiency evaluation indicator system tailored to public institutions is established in this study. As shown in Table 1, the proposed evaluation system consists of four dimensions with a total of ten core indicators: (i) energy and resource consumption (C1), (ii) advanced guidance (C2), (iii) operational management (C3), and (iv) scenario-specific characteristics (C4). The energy and resource consumption dimension focuses on controlling overall energy use intensity and includes three cost-type indicators: per capita comprehensive energy consumption, energy consumption per unit floor area, and per capita water consumption. Higher values of these indicators indicate lower energy efficiency performance.

TABLE 1. Comprehensive energy efficiency assessment index system for public institutions.

△ denotes a benefit-type indicator (higher value indicates higher comprehensive energy efficiency); ▽ denotes a cost-type indicator (higher value indicates lower comprehensive energy efficiency).

The advanced guidance dimension is designed to reflect low-carbon transition objectives and technological leadership. It incorporates carbon emissions per unit floor area as a cost-type indicator, along with benefit-type indicators such as the utilization rate of renewable energy and green electricity, and the return on investment of energy efficiency retrofit projects. These indicators collectively capture both environmental performance and economic effectiveness of low-carbon technologies. The operational management dimension evaluates management effectiveness through two benefit-type indicators: the maturity of energy management systems and their implementation, and the compliance rate of energy efficiency diagnostics for energy-consuming equipment. These indicators emphasize the role of institutional governance and standardized management practices in improving energy performance.

To account for heterogeneity across different types of public institutions, scenario-specific indicators are introduced. For educational institutions, indicators include campus green space coverage and the elasticity coefficient of energy consumption between academic semesters and vacation periods. Government offices are evaluated using the penetration rate of new energy vehicles in official fleets and an office energy efficiency optimization index. Medical institutions incorporate indicators such as the energy efficiency ratio of key medical equipment and the standby energy loss rate for emergency energy systems. For public venues, indicators include the baseline energy load ratio during idle periods and the thermal energy recovery rate. By integrating differentiated scenario-based indicators, the proposed system captures unique operational characteristics such as periodic energy use in educational facilities, emergency response requirements in hospitals, and intermittent operation patterns in public venues. Overall, the evaluation system achieves a multidimensional integration of energy intensity, carbon emissions, management performance, and economic efficiency. All indicators are explicitly classified as either benefit criteria or cost criteria, providing a clear directional basis for subsequent indicator weighting and comprehensive energy efficiency assessment.

2.2 CRITIC-based indicator weighting method

The key in multi-criteria evaluation lies in determining the weights of each evaluation indicator. The accuracy and objectivity of the weights significantly affect the credibility of the evaluation results. The entropy weighting method calculates weights based on the variability of each indicator and is unaffected by subjective factors, offering a straightforward calculation process. In contrast, the CRITIC (Criteria Importance Through Intercriteria Correlation) method determines weights by considering both the contrast intensity and the conflict between indicators. Given that the indicators in the energy efficiency evaluation system for public institutions not only exhibit variability but also some degree of inter-correlation, the CRITIC method is employed in this study for indicator weighting. The specific calculation steps are as follows: where y i j represents the jth energy efficiency evaluation indicator for the ith energy retrofit option. where d i j is the normalized value of the jth indicator for the ith retrofit option, and max y j and min y j are the maximum and minimum values of the jth indicator across all alternatives. where z ¯ j is the mean value of the jth normalized indicator, and σ j represents the standard deviation of the jth indicator. A higher standard deviation σ j indicates a stronger ability of the indicator to distinguish between alternatives. where d ¯ p and d ¯ q are the normalized values of the pth and qth indicators for the ith option, and d i q and d i q are the mean values of the pth and qth indicators. Generally, r p q ≤ 1, and the closer r p q is to 1, the higher the correlation between the two indicators. where σ j is the standard deviation of the jth evaluation indicator, and r p j is the correlation coefficient between the pth and jth indicators. c j is the quantification of index conflict. where w j represents the weight of the jth indicator, and g j is the information content of the jth indicator.

• Construct the energy efficiency evaluation matrix for public institutions. Let the number of alternative energy retrofit schemes be denoted by v, and the number of selected evaluation indicators be denoted by n, where n = 10 in this study. The evaluation matrix is formulated as shown in Equation 1:

Y = y i j v × n = y 11 y 12 ⋯ y 1 n y 21 y 22 ⋯ y 2 n ⋮ ⋮ ⋱ ⋮ y v 1 y v 2 ⋯ y v n (1)

• Standardize the original evaluation data matrix. Due to differences in the units of the evaluation indicators, the data must be normalized before applying the CRITIC method to compute the weights, as expressed in Equation 2.

d i j = y i j − min ⁡ y j max ⁡ y j − min ⁡ y j benefit − type indicator max ⁡ y j − y i j max ⁡ y j − min ⁡ y j cost − type indicator (2)

• Compute the comparative strength of the indicators. The comparative strength of each indicator reflects its ability to distinguish between different alternatives, and is determined based on the standard deviation of each indicator, as expressed in Equation 3.

σ j = 1 v − 1 ∑ i = 1 n z i j − z ¯ j 2 (3)

• Calculate the correlation coefficients between the energy efficiency evaluation indicators. The correlation coefficient between indicators is calculated using the product-moment method, as shown in Equation 4.

r p q = ∑ i = 1 v d i p − d ¯ p d i q − d ¯ q ∑ i = 1 v d i p − d ¯ p 2 · ∑ i = 1 v d i q − d ¯ q 2 (4)

• Calculate the amount of information contained in each indicator. The information content of each indicator is quantified based on both its standard deviation and its correlation with other indicators. The corresponding conflict and information-content measures are given by Equations 5, 6, respectively.

c j = ∑ p = 1 n 1 − r p j (5)

g j = σ j × c j (6)

• Calculate the weights of the indicators. Finally, the weights of each indicator are computed by normalizing the information content of all indicators, as shown in Equation 7.

w j = g j ∑ j = 1 n g k (7)

2.3 Integrated energy system modeling for public institutions

The integrated energy system (IES) for public institutions is a system with multiple energy inputs and outputs, and the specific architecture of a typical system is shown in Figure 2. In order to improve the accuracy and efficiency of the model, it is decomposed into a number of unit groups using a hierarchical modeling approach. where O U T s , h GB , heat is the thermal power output of the gas boiler at state s and time h, I N s , h GB , G is the gas power input, and η GB , heat is the constant thermal efficiency of the gas boiler. η GB , heat is the thermal efficiency of the gas boiler, which is a fixed value. I N s , h GB , G is the gas power input, k represents the conversion factor between the amount of gas and energy, and V indicates the input/output gas volume. where δ s , h represents the solar irradiance received by the solar cell components; s pvs is the area occupied by a single solar cell unit; n pvs is the number of PV units; and η pvs is the photovoltaic conversion efficiency. The number of solar cell modules n pvs is determined by the available roof area. where C O P WSHP is the coefficient of performance (COP) of the water-source heat pump, with superscripts “heat” and “cool” denoting heating and cooling modes, respectively.

• Gas boiler. Gas boiler is a heating device primarily based on thermal conversion, which generates heat by consuming natural gas to meet thermal energy demand. Its mathematical model is given by Equations 8, 9:

O U T s , h GB , heat = I N s , h GB , G · η GB , heat (8)

I N s , h GB , G = k s , h GB , G · V s , h GB , G (9)

• Photovoltaic System. Photovoltaic (PV) systems are typically installed on building rooftops. The electricity generation of the PV system is calculated using Equation 10 based on the received solar irradiance:

O U T s , h pvs , ele = δ s , h · s pvs · n pvs · η s , h pvs (10)

• Water-Source Heat Pump. The water-source heat pump is a type of heat pump system that uses water as a heat source for cooling or heating. It cannot perform both heating and cooling simultaneously. Typically, in winter, surface water is used as a heat source for heating, while in summer, surface water is used as a heat sink for cooling. The water-source heat pump offers energy-saving, clean, and economical advantages. The specific model of the water-source heat pump is as follows the heating and cooling models of the water-source heat pump are given by Equations 11, 12, respectively:

O U T s , h WSHP , heat = I N s , h WSHP , ele · C O P WSHP , heat (11)

O U T s , h WSHP , cool = I N s , h WSHP , ele · C O P WSHP , cool (12)

In practice, the construction scale of the water-source heat pump n WSHP is limited by the available area in the region, as expressed in Equation 13:where n WSHP represents the number of installed water-source heat pumps, and s WSHP is the area occupied by a single unit of water-source heat pump.

n WSHP · s WSHP ≤ A WSHP (13)

• Combined Cooling, Heating, and Power (CCHP) System. The CCHP system described in this paper mainly consists of a gas internal combustion engine, a lithium bromide absorption chiller, and a waste heat boiler. The gas engine generates electricity, the lithium bromide chiller provides cooling in summer and heating in winter, and the waste heat boiler absorbs high-temperature waste heat to generate steam. The CCHP system achieves efficient energy utilization through cascade energy recovery, offering high overall energy efficiency.

To facilitate mathematical modeling, individual system components are abstracted as a unified entity. The system’s energy conversion characteristics are described by Equations 14–17. It is worth noting that the target campus employs a single-network piping architecture operating under a strict seasonal switchover regime (space heating in winter and cooling in summer). Therefore, simultaneous heating and cooling is mutually exclusive, which is mathematically characterized by Equation 18. The corresponding operational power, gas input, and steam-output relationships are further given by Equations 19–21: where O U T represents the output energy, with superscripts “ele”, “cool”, “heat” and “stm” denoting electricity, cooling, heating, and steam, respectively; and I N represents input energy, with superscript “G” indicating gas input. η is the energy conversion efficiency, P is the operational power, Δ t is the duration of the time period, V is the input/output gas volume, with superscripts “G” and “stm” referring to gas and steam, respectively; and k is the conversion factor between gas volume and energy. where C O P ERU is the coefficient of performance of the electric cooling unit, which is a fixed value. where η vehicle , ele is the energy conversion efficiency, with the subscript “Vehicle” referring to electric vehicles. where D is the user load demand, and I N grid , ele is the electricity purchased from the grid. The equality constraint represents the nodal electrical energy balance, where total generation from CCHP, PVS, and grid imports equals total electrical demand plus consumption by all electric-driven equipment (WSHP, ERU, CP). The inequality constraints represent the thermal and cooling energy balances, where total supply from various devices must meet or exceed the respective demands and internal consumptions, allowing for potential energy surplus or storage charging. where C E represents CO₂ emissions, C S represents CO₂ sequestration, and C E F is the carbon emission factor. where P k is the output power of equipment k, with superscripts “min” and “max” denoting the minimum and maximum output power, respectively, and S E k represents the storage capacity of energy storage equipment k. where w j represents the weight of the jth indicator, and I j is the normalized value of the jth indicator.

O U T s , h CCHP , ele = I N s , h CCHP , G · η CCHP , ele (14)

O U T s , h CCHP , cool = I N s , h CCHP , G · η CCHP , cool (15)

O U T s , h CCHP , heat = I N s , h CCHP , G · η CCHP , heat (16)

O U T s , h CCHP , stm = I N s , h CCHP , G · η CCHP , stm (17)

O U T s , h CCHP , cool · O U T s , h CCHP , heat = 0 (18)

P s , h CCHP · Δ t s , h = O U T s , h CCHP , ele (19)

I N s , h CCHP , G = k CCHP , G · V s , h CCHP , G (20)

O U T s , h CCHP , stm = k CCHP , stm · V s , h CCHP , stm (21)

• Electric Cooling Unit. An electric cooling unit converts input electrical energy into cooling output to meet cooling demand. The mathematical model for the electric cooling unit is given by Equation 22:

O U T S , h ERU , cool = I N S , h ERU , ele · C O P ERU (22)

• Charging pile Constraints. Charging piles (CP) are supporting infrastructure for electric vehicles (EVs), with their input side connected directly to the power grid and the output side providing interfaces for charging EVs. The charging-pile model is expressed by Equations 23, 24:

O U T s , h CP , ele = I N s , h CP , ele (23)

I N s , h vehicle , ele = O U T s , h CP , ele · η vehicle , ele (24)

• Energy Flow Balance Constraint. The system involves three types of energy flows: electricity, heat, and cooling. Energy flow balance must be maintained between the supply and demand sides, as described by Equations 25–27.

O U T s , h CCHP , ele + O U T s , h PVS , ele + I N s , h grid , ele = D s , h ele + I N s , h WSHP , ele + I N s , h ERU , ele + I N s , h ISAC , ele + I N s , h CP , ele (25)

O U T s , h CCHP , heat + O U T s , h WSHP , heat + O U T s , h GB , heat + O U T s , h HSWT , heat ≥ D s , h heat + I N s , h HSWT , heat (26)

O U T s , h CCHP , cool + O U T s , h WSHP , cool + O U T s , h ERU , cool + O U T s , h ISAC , cool ≥ D s , h cool + I N s , h ISAC , cool (27)

• Carbon Emission Constraints. The carbon emissions of the integrated energy system for public institutions studied here primarily consist of three components: (1) the carbon emissions from external energy inputs, such as electricity purchased from the public grid; (2) the carbon emissions generated internally by the system, such as those from the gas internal combustion engine in the gas boiler or CCHP system when converting gas into energy; and (3) the carbon sequestration function of green spaces, which absorb and fix CO₂, thereby reducing system carbon emissions. Carbon emissions are calculated as follows:

C E = ∑ s ∑ h I N s , h grid , ele · C E F ele + V s , h G · C E F G − C S (28)

• Equipment Operation Constraints. All equipment within the integrated energy system of the public institution must satisfy the basic operational constraints given by Equations 29, 30:

P k , s , h min ≤ P k , s , h ≤ P k , s , h max (29)

S E k , s , h min ≤ S E k , s , h ≤ S E k , s , h max (30)

• 10. Objective Function. The optimization goal of the model is to comprehensively enhance the energy efficiency of the public institution while meeting its energy demand. This goal is reflected in the maximization of the institution’s comprehensive energy efficiency index (CEI). The optimization objective is formulated in Equation 31:

max CEI = max ∑ j = 1 n w j I j (31)

It should be noted that the model architecture and key parameters are derived from our previous work (Wang et al., 2026). Given that case-specific parameter settings vary considerably, direct comparisons of our results with those of similar studies are inherently difficult. Nevertheless, the core IES model has been well validated in our earlier research (Hu et al., 2025; Jing, 2022). We have also established a benchmark scenario for reference. Although the model possesses broad applicability, the inherent variability of parameters precludes direct comparison or cross-validation of results across different cases.

3 Tripartite evolutionary game model

In the process of enhancing energy efficiency within public institutions, the ability of the government, integrated energy service providers (IESPs), and public institutions to interact effectively and establish stable, mutually beneficial partnerships is critical to the project’s sustained success. Given that evolutionary game theory offers a robust framework for analyzing strategic interactions and dynamic evolutionary processes among multiple agents in complex environments, this study constructs a tripartite evolutionary game model. This model serves to elucidate the decision-making processes and reciprocal interaction mechanisms of these diverse stakeholders in the context of energy efficiency projects.

3.1 Game model assumptions

Implementing energy efficiency retrofits in public institutions under the framework of the “Carbon Inclusion” mechanism involves a complex game among three distinct entities: the government, integrated energy service providers (IESPs), and public institutions. To construct an effective game model, it is necessary to clarify the benefit and cost structures for each stakeholder under different strategic choices, while fully accounting for the pivotal impact of the carbon trading market mechanism on the returns of all parties.

3.1.1 The government

The government aims to maximize environmental improvement and social benefits. In energy efficiency improvement projects, it assumes the roles of policy formulation, supervision, and incentivization. Generally, the government chooses between two strategies: “ Implement Policy” or “ No Policy.”

When the government chooses to implement incentive policies, it incurs management costs G r related to regulatory standardization and provides subsidies G s to IESPs that implement retrofits. If public institutions do not undertake energy efficiency retrofits, the government must also bear environmental governance costs G e However, regardless of whether the government implements incentive policies, as long as public institutions undertake energy efficiency retrofits, the government obtains social benefits G t derived from resource conservation and environmental improvement. The G t is calculated by Equation 32, where R env represents environmental improvement benefits and R res represents resource conservation benefits. Furthermore, when the government implements incentive policies but public institutions do not undertake retrofits, the government still accrues social benefits G g arising from the enhancement of energy usage comfort. For the purpose of theoretically analyzing the transient unstable nodes in the evolutionary process, a simplifying assumption is made that G g < G e, considering current environmental governance pressures (Yang and Li, 2023).

G t = R env + R res (32)

3.1.2 Integrated energy service providers (IESPs)

IESPs are market entities oriented toward economic efficiency. In energy efficiency projects, they choose between “ Implement Project” or “ No Project.”

When an IESP chooses to implement the project and the public institution chooses to “undertake energy efficiency improvement,” the ISP receives government subsidies G s and generates revenue E e by participating in carbon trading through project-based certified emission reductions (CERs). Simultaneously, the IESP bears project promotion and operation costs E c and pays incentives E i to the participating public institution.

If the IESP chooses to “implement” but the public institution chooses “ No Retrofit,” the cooperation fails to materialize; consequently, the IESP neither bears project operation costs nor receives subsidies, maintaining only its original energy revenue E s 2. It is worth noting that IESPs obtain energy revenue regardless of whether they choose to implement the project. This revenue is contingent upon whether the public institution implements the retrofit: energy revenue after the retrofit is denoted as E s 1, and before the retrofit as E s 2. Since energy efficiency retrofits reduce total energy consumption, the energy sales revenue decreases, implying E s 2 > E s 1.

3.1.3 Public institutions

As energy end-users and the source of carbon emissions, public institutions aim to minimize energy expenditures. They choose between ” Undertake Retrofit” or “ No Retrofit.”

When a public institution chooses to ” Undertake Retrofit,” its energy cost is E s 1, and it receives additional incentives, denoted as E i, for participating in the project. When the public institution chooses No Retrofit,” its energy cost is E s 2. Upon undertaking the retrofit, the public institution gains energy-saving benefits C e, calculated as Equation 33. Additionally, it may need to bear the “retrofit cost” associated with implementing the energy efficiency project.

C e = E s 2 − E s 1 (33)

3.1.4 Coupling the physical and economic models

To explicitly couple the integrated energy system model with the tripartite evolutionary game model, the core economic parameters in the payoff matrix are directly derived from the physical energy flows and carbon emission constraints.

First, the public institution’s energy-saving benefit C e, is determined by the difference in energy procurement costs before E s 2 and after E s 1 the retrofit. These costs are calculated based on the grid electricity input I N grid , ele and natural gas consumption V s , h G necessary to maintain the energy flow balance. The corresponding energy procurement cost is calculated using Equation 34:where P e and P g represent the time-of-use unit prices for grid electricity and natural gas, respectively.

E s = P e ∑ s ∑ h I N s , h grid , ele + P g ∑ s ∑ h V s , h G (34)

Second, the revenue from carbon trading, E e, which incentivizes the IESP, is defined by the total verified carbon emission reduction. Utilizing the carbon emissions C E evaluated in Equation 28. The carbon trading revenue is thus formulated in Equation 35:where Q c represents the carbon emission reduction generated after the implementation of the project, and P c denotes the unit carbon price in the carbon trading market. By introducing these equations, the strategic decisions in the evolutionary game are strictly bounded by the physical operations of the proposed integrated energy system.

E e = P c · Q c = P c · C E b a s e − C E (35)

3.1.5 Model framework

Let x represent the probability that the government chooses the “Implement Policy” strategy, and 1 − x represent the probability that it does not. Let y represent the probability that the IESP chooses the “Implement Project” strategy, and 1 − y represent the probability that it does not. Finally, let z represent the probability that the public institution chooses the “Undertake Retrofit” strategy, and 1 − z represent the probability that it does not.

Based on the assumptions and parameter definitions outlined above, the logical framework regarding the stakeholders in the tripartite evolutionary game for public institution energy efficiency improvement projects payoff matrix for the game is presented in Table 2.

TABLE 2. Tripartite game payoff matrix for public institution energy efficiency projects under carbon inclusive mechanisms.

The tripartite payoff matrix is constructed by systematically mapping the conditional costs and benefits defined in the assumptions to each of the eight possible strategy combinations. For any given profile, an entity’s net expected payoff is calculated algebraically by summing its baseline revenues and conditional incentives, and subtracting its operational and opportunity costs.

For instance, in the ideal cooperative scenario where all parties opt for active participation, the Government gains environmental and resource benefits G t while bearing the costs of policy regulation G r and IESP subsidies G s. Concurrently, the IESP receives the government subsidy G s, post-retrofit energy revenue E s 1, and carbon trading revenue E e, offset by project implementation costs E c and user incentives E i. Finally, the Public Institution acquires energy-saving benefits C e and IESP incentives E i, minus the retrofit capital cost C c.

3.2 Expected payoff functions and replicator dynamic equations

Based on the payoff matrix, the expected payoff functions for the three stakeholders are derived. By applying evolutionary game theory, the replicator dynamic equations for the agents under different strategic scenarios are calculated. The average expected payoff functions for the government, the IESP, and the public institution are denoted as U 1, U 2, and U 3, respectively. The corresponding replicator dynamic equations are represented by F x, F y, and F z, respectively.

The average expected payoff function for the government is given by Equation 36:where U 12 and U 11 denote the expected payoffs for the government choosing to implement or not implement the incentive policy, respectively, and U 1 represents the government’s average expected payoff.

U 1 = x U 11 + 1 − x U 12 (36)

The replicator dynamic equation for the government choosing to implement the policy is given by Equation 37:

F x = d x d t = x U 11 − U 1 = x 1 − x − G r − y G s − 1 − z G e + 1 − z G g (37)

The average expected payoff for the integrated energy service provider is given by Equation 38:where U 21 and U 22 denote the expected payoffs for the integrated energy service provider (IESP) choosing to implement or not implement the energy efficiency project, respectively, and U 2 represents the IESP’s average expected payoff.

U 2 = y U 21 + 1 − y U 22 (38)

The replicator dynamic equation for the integrated energy service provider is given by Equation 39:

F y = d y d t = y U 21 − U 2 = z y 1 − y E e − E i − E c (39)

The average expected payoff for the public institution is given by Equation 40:where U 31 and U 32 denote the expected payoffs for the public institution choosing to undertake or not undertake the energy efficiency retrofit, respectively, and U 3 represents the institution’s average expected payoff.

U 3 = z U 31 + 1 − z U 32 (40)

The replicator dynamic equation for the public institution is given by Equation 41:

F z = d z d t = z 1 − z C e + E i y − C c (41)

3.3 Stability analysis of the replicator dynamic equations

Drawing upon the replicator dynamic equations formulated for the public institution energy efficiency game under the Carbon Inclusion mechanism, we derive the following three-dimensional replicator dynamic system, denoted as D. The Jacobian matrix and the three-dimensional replicator dynamic system are given by Equations 42, 43, respectively: where J a c o denotes the Jacobian matrix of the three-dimensional tripartite evolutionary dynamic system. This matrix is composed of the partial derivatives of the replicator dynamic equations F x, F y, and F z with respect to the strategic probabilities of the three stakeholders x, y, and z.

J a c o = ∂ F x ∂ x ∂ F x ∂ y ∂ F x ∂ z ∂ F y ∂ x ∂ F y ∂ y ∂ F y ∂ z ∂ F z ∂ x ∂ F z ∂ y ∂ F z ∂ z (42)

D = F x = d x d t = x U 11 − U 1 F y = d y d t = y U 21 − U 2 F z = d z d t = z U 31 − U 3 (43)

By equating the replicator dynamic functions of the government, IESPs, and public institutions to zero, we identify the equilibrium points of the tripartite evolutionary game. A Nash equilibrium is achieved when the directional evolution of the system stabilizes. We select eight pure strategy equilibrium points for stability analysis: E 1 0 , 0 , 0 through E 8 1 , 1 , 1. In accordance with Lyapunov’s first method (Lyapunov, 1992), the Jacobian matrix of the tripartite evolutionary game is constructed.

Substituting the equilibrium points E 1 ∼ E 8 into the Jacobian matrix J a c o allows for the calculation of eigenvalues. An equilibrium point is classified as an Evolutionarily Stable Strategy (ESS) only if the real parts of all eigenvalues are negative. The stability analysis results are summarized in Table 3.

TABLE 3. Stability of game equilibrium points for energy efficiency improvement projects in public institutions under the carbon inclusive mechanism.

The results indicate that points E 1 , E 3 , E 5 , E 6 , E 7 and E 8 are unstable. However, points E 2 and E 4 represent strategies that may stabilize under certain conditions. These stable evolutionary strategies and the associated stakeholder behaviors are analyzed as follows:

Scenario E 2 (0,0,1): under the conditions C C < C e and E e < E i + E c, public institutions proactively choose “Undertake Retrofit.”. However, the probability of IESPs implementing such projects falls to zero, as the revenue derived from carbon trading on emission reductions is insufficient to offset implementation costs. Concurrently, as government subsidies fail to yield the expected outcomes, the likelihood of the government maintaining incentive policies also diminishes to zero.

This analysis elucidates the system’s potential evolutionary paths. Subsequent sections will utilize a case study to parameterize the model, thereby verifying the validity of the stability analysis and exploring the critical factors governing the stability of this dynamic system.

4 Case study

We select an education institution in Fujian, China as the case study for this educational institution. The campus occupies an area of 600 acres, with approximately 30,000 faculty and students. It consists of four primary functional zones: teaching buildings, dormitories, laboratories, and dining halls. The current energy system of the campus utilizes a decentralized energy supply system. The municipal grid serves as the main energy supply source, accounting for 85% of total energy consumption, while natural gas and solar energy play a supplementary role. The current energy system of the case institution is shown in Figure 3.

A typical day method is employed to classify the regional load, with five typical days defined as follows: summer semester days (May-July, September-October) as s1, summer vacation typical day (August) as s2, transition season semester days (March, April, November) as s3, winter semester days (December-January) as s4, and winter vacation typical day (February) as s5. Figure 4 shows the hourly electricity, cooling, heating, and electric vehicle charging loads for each typical day.

4.1 Optimization of energy efficiency retrofit measures and multi-dimensional benefit evaluation

Building upon the IES model established in the previous section, this study investigates the energy efficiency evaluation and selection of energy-saving retrofit measures for the educational institution The objective is to identify the optimal method and system operation strategy to maximize comprehensive energy efficiency under coordinated supply and demand planning. Furthermore, a priority ranking of energy efficiency improvement solutions is provided.

The Mixed-Integer Nonlinear Programming (MINLP) model developed for this study is implemented using GAMS platform, and the model is solved using the LINDO optimization solver. Seven energy-saving retrofit schemes (S1-S7) are introduced, based on the campus’s resource availability and the current energy system status.

• S1: Multiple retrofit measures, optimized for the highest comprehensive energy efficiency index.
• S2: CCHP-forced scenario. The CCHP system is mandated to operate at no less than 95% of its maximum installable capacity, which is limited by site availability to approximately 2.0 MW. All other technologies remain subject to optimization.
• S3: PV-forced scenario. The photovoltaic system is mandated to operate at no less than 95% of its maximum installable capacity, constrained by available rooftop area to roughly 3.4 MW. Remaining equipment is optimized accordingly.
• S4: WSHP-forced scenario. The photovoltaic system is mandated to operate at no less than 95% of its maximum installable capacity, constrained by available rooftop area to roughly 3.4 MW. Remaining equipment is optimized accordingly.
• S5: Management-forced scenario. An energy management system is deployed across all energy-consuming floor areas of the campus.
• S6: Envelope retrofit scenario. Full retrofit of the building envelope.
• S7: Equipment replacement scenario. Chillers and other energy-intensive equipment are replaced with high-efficiency units.

The evaluation of each scheme is conducted based on the comprehensive evaluation indicator system, with indicator weights determined using the CRITIC method. Figure 5 presents the CRITIC weighting results.

From the scatter plot of comparative strength versus conflict, it can be observed that the advanced guidance indicators (C2 group) exhibit relatively high information content and independence. Specifically, the utilization rates of renewable energy and green electricity (C22) and the carbon emissions per unit floor area (C21) both have the highest comparative strength and conflict, indicating that these two indicators provide the most differentiated and independent information for the scheme evaluation. The energy efficiency diagnosis compliance rate (C32) and the energy consumption per unit floor area (C12) also exhibit strong evaluation characteristics. In contrast, the campus green space construction indicator (C41) shows relatively low comparative strength, indicating its limited ability to differentiate between various schemes. This analysis suggests that, in the energy efficiency evaluation of educational institutions, technical indicators in the advanced guidance dimension play a stronger role in supporting decision-making for scheme selection. Figure 5B presents a pie chart of indicator weight distribution, visually reflecting the relative importance of each indicator in the comprehensive evaluation. The larger the weight of an indicator, the greater its decision-making value in the selection of energy efficiency improvement schemes.

By calculating the weights of each indicator using the CRITIC method and combining them with the performance scores of each scheme, the comprehensive evaluation scores for the schemes are obtained, as shown in Table 4. The results indicate that Scheme 1 (a combination of various retrofit measures) has the highest comprehensive score (0.907), followed by Scheme 3 (installation of PV systems, 0.822) and Scheme 7 (replacement of energy-consuming equipment, 0.797). This suggests that, in the energy efficiency improvement process for the campus, comprehensive retrofit schemes offer the best overall benefits, while technology-constrained cmeasures such as equipment replacement and the application of renewable energy technologies (e.g., PV and water-source heat pumps) are prioritized.

TABLE 4. Evaluation results of various renovation plan indicators.

Figure 6 illustrates the performance characteristics of the seven energy efficiency improvement schemes across four major indicator categories (C1-C4). The analysis shows that Schemes 1, 3, and 7 perform well in the energy and resource consumption indicators (C1), with scores of 0.262, 0.210, and 0.244, respectively. Schemes 3 and 4 score highly in the advanced guidance indicators (C2), with scores of 0.304 and 0.261, respectively, indicating strong investment returns and green, low-carbon performance. Most schemes show relatively low scores in the characteristic evaluation indicators (C4), suggesting that there is still room for improvement in campus green space construction and energy consumption elasticity. Overall, while technology-constrained schemes may optimize certain indicators, they are unlikely to deliver comprehensive improvements in energy efficiency. Based on the case analysis, it is recommended that the institution prioritize comprehensive retrofit schemes, such as replacing key energy-consuming equipment and installing renewable energy systems.

To validate the robustness of the proposed model, we compared its rankings with those obtained from two alternative methods: TOPSIS and the entropy weight method. The results are presented in Table 5, all three methods consistently identify S1 as the optimal scenario and S7 and S3 among the top three, confirming the robustness of the proposed evaluation framework despite minor variations in the ranking of intermediate scenarios.

TABLE 5. Evaluation results of different assessment methods.

Table 6 presents the main performance indicators for the current energy system and the comprehensive retrofit scheme S1 over a 20-year operational period. The results indicate that the performance of Scheme S1 is superior to the current energy system across all key indicators. Compared to the current system, Scheme S1 reduces the net present cost by 18.4%, decreases annual carbon emissions by 25.2%, lowers annual primary energy consumption by 28.1%, and reduces the annual electricity purchase from the grid by 34.4%, significantly decreasing the institution’s dependence on the public grid.

TABLE 6. Performance comparison of the original energy system and S1.

4.2 Design of policy response mechanism based on tripartite evolutionary game

To verify the validity of the evolutionary game analysis presented in the preceding chapters, and to investigate the influence of the stakeholders’ initial willingness and key parameters on strategic evolutionary paths within public institution energy efficiency projects under the Carbon Inclusion mechanism, this section employs the energy efficiency retrofit model for educational institutions described earlier. Relevant parameters are assigned specific values, and MATLAB is utilized to conduct a numerical analysis of the case study.

Drawing upon current domestic cases of energy efficiency retrofits in public institutions (Tang and Wang, 2024), the Carbon Inclusion mechanism (Fan and Chen, 2024), fluctuations in carbon trading prices (Fudan Carbon Price Index in June 2025 - Fudan University Sustainable Development Research Center, 2025), and analyses of complex variables such as government subsidies (Han, 2024) and perceived benefits (Wang X.-P. et al., 2023) found in relevant literature, the initial values for the evolutionary game simulation are established as shown in Table 7. All revenues and costs are calculated on an annual basis.

TABLE 7. Parameter values for the energy efficiency improvement case study of public institution.

The stability analysis of equilibrium points indicates that this game system presents three evolutionary equilibrium scenarios. Due to fluctuations in carbon market prices, the revenue derived from carbon trading based on emission reductions may vary significantly during the project implementation period. Consequently, initial values for carbon trading revenue are set according to different carbon trading price scenarios.

According to the stability conditions derived in Section 3.3, the autonomous participation of the IESP relies strictly on the condition E e > E i + E c. Using the operational parameters E c = 150, E i = 30 and the verified carbon emission reduction (Q c = 2.525), the exact carbon trading price threshold required to trigger participation in the IESP is calculated to be 71.29 RMB/ton; this serves as the subsidy-free baseline.

To explicitly verify the validity of this tipping point and explore the system’s evolutionary trajectories, the subsequent numerical simulations will conduct a comparative sensitivity analysis by setting the carbon trading prices at two representative levels: 70 RMB/ton (below the threshold, yielding E e = 177) and 80 RMB/ton (above the threshold, yielding E e = 202).

Figure 7 displays the computational results of the replicator dynamic system over 50 evolutionary time steps under the current initial conditions. The data indicates that the system stabilizes at only two evolutionary equilibrium points: E 2 0 , 0 , 1 and E 4 0 , 1 , 1.

In the scenario where the carbon trading price is 70 RMB/ton, the government opts not to enforce energy efficiency incentive policies. Similarly, the IESP declines to implement the project, as the carbon trading revenue fails to offset the necessary investment costs. Conversely, the educational institution elects to autonomously pursue the energy efficiency retrofit, as the realized benefits outweigh the modification costs.

In the scenario where the carbon trading price rises to 80 RMB/ton, the government remains passive regarding incentive policies. In contrast, the IESP actively elects to implement the energy efficiency project and offers incentives to the participating public institution. Driven by these external incentives, the public institution reciprocates by actively selecting to undertake the energy efficiency retrofit.

To elucidate how distinct initial strategies influence the system’s evolutionary trajectory, this study performed a sensitivity analysis on the stakeholders’ initial willingness, utilizing the parameters associated with equilibrium point x , y , z. As illustrated in Figure 8, the initial probabilities for the government and the integrated energy service provider (IESP) were categorized into low (0.2,0.2,0.2), medium (0.4,0.4,0.4), and high (0.6,0.6,0.6) levels.

Simulation results demonstrate that irrespective of the government’s initial probability, its strategy invariably converges to 0, signifying a decision not to implement incentive policies. Regarding the IESP, while initial willingness does not alter the final stable strategy of “implementing the energy efficiency project,” it significantly impacts the velocity and trajectory of evolution. Higher initial willingness correlates with faster convergence to the stable state; conversely, at lower levels of initial willingness, evolutionary speed exhibits an initial increase followed by a decline. The evolutionary speed of public institutions is determined by the collective initial willingness of all three stakeholders: higher aggregate initial willingness accelerates the public institution’s stabilization.

To analyze the effects of specific variables, key model parameters were assigned varying values: Government Subsidies ( G s = {50, 100, 150}), IESP Incentives (E i = {15, 30, 45}), and Perceived Retrofit Costs (C c = {1,600, 2000, 2,400}). Figure 9 depicts how variations in these factors influence the evolutionary process, where x, y, and z denote the strategic probabilities of the government, the IESP, and the public institution, respectively.

The findings indicate that variations in the magnitude of government subsidies and IESP incentives exert no significant influence on the evolutionary speed of the stakeholders’ strategies during the stabilization process. This suggests that within the proposed framework, these financial levers function primarily as guiding and auxiliary mechanisms. Notably, however, IESPs must balance these incentives against practical constraints, particularly operational costs, when designing service packages.

Further analysis highlights a critical pattern in the cost sensitivity of public institutions. While the perceived cost of retrofitting does not alter the system’s final equilibrium state within a specific range, it dramatically affects the rate of evolution. Specifically, lower perceived costs accelerate convergence. However, should the perceived cost exceed the expected energy-saving benefits, the decision-making rationale rapidly shifts toward non-participation, driving the system to stabilize at a non-cooperative equilibrium characterized by withdrawal from the energy efficiency project.

5 Conclusion

This study proposes a public institution energy efficiency improvement strategy framework based on the CRITIC weighting method and a tripartite evolutionary game. By constructing a four-dimensional evaluation system covering energy resource consumption, advanced guidance, operational management efficiency, and scenario characteristics, and integrating the government-energy service provider-public institution tripartite game mechanism, the study systematically analyzes the energy efficiency improvement pathways under multi-agent interactions. An empirical study is conducted using a higher education institution in Fujian Province as a case study. The main conclusions are as follows.

• CRITIC-based Indicator Weighting: The results of the CRITIC-based weighting method indicate that the advanced guidance indicators, such as the utilization rate of renewable energy (C22) and carbon emissions per unit area (C21), carry the highest decision-making weights (0.258 and 0.234, respectively). This suggests that low-carbon technologies play a pivotal role in enhancing the energy efficiency of educational institutions. Evaluation of the retrofit solutions reveals that the comprehensive retrofit solution provides the best overall benefit, reducing net present costs by 18.4%, annual carbon emissions by 25.2%, and annual electricity consumption by 34.4%. Among individual retrofit measures, the replacement of energy-consuming equipment and the installation of photovoltaic systems are the most highly recommended options.
• Evolutionary Game Results: The results of the evolutionary game analysis show that under the carbon inclusive mechanism, government subsidy policies have a significant effect in the early stages of public institution energy efficiency projects. These policies not only effectively increase the market participation willingness of IESPs but also enhance the likelihood of public institutions engaging in energy efficiency retrofitting. However, as the market mechanism evolves and participants develop, government subsidies will gradually phase out from the project operation.
• Energy Service Providers’ Role: Comprehensive IESPs need to establish dynamic carbon pricing response and incentive adjustment mechanisms. By promoting models such as energy management outsourcing, they can offer integrated solutions that lower the technical barriers for public institutions’ energy efficiency retrofitting. The participation behavior of public institutions in decision-making is influenced by both the economic viability of the projects and environmental awareness. It is recommended to strengthen the promotion of quantified energy-saving benefits and environmental education to increase institutional participation and engagement in energy efficiency improvements.

Specifically, rather than applying uniform policies, policymakers should utilize this case-based calculation methodology to determine specific “dynamic policy triggers” for different institutions. Subsidies should be activated when the carbon price falls below the institution-specific mark and systematically phased out once the market price exceeds it, thereby minimizing redundant fiscal expenditures.

Finally, limitations of this study lie in two main aspects. First, while the proposed evaluation and optimization framework possesses conceptual transferability, the empirical results and specific indicator weights are currently based on a single university case study. As scenario-specific indicators for other institution types have not yet been empirically tested, future research should apply this framework to hospitals, government agencies, and public venues to demonstrate and validate its broader external validity. Second, regarding the analytical methodology, the deterministic nature of the tripartite evolutionary game model may not fully capture the significant stochastic volatility inherent in real-world energy and carbon trading markets. Future research should also consider integrating stochastic evolutionary game frameworks, potentially by introducing Gaussian white noise into dynamic pricing parameters, to further enhance the practical robustness of the equilibrium analysis.Competing Interests

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fenrg.2026.1794357/full#supplementary-material

Supplementary material Supplementary material at the source

Footnotes

Edited by: Tianhua Xu, University College London, United Kingdom

Reviewed by: Denny K. S. NG, Sunway University, Malaysia

Fanglei Zhong, Minzu University of China, China