Site Selection Evaluation of Pumped Storage Power Station Based on Multi-Energy Complementary Perspective: A Case Study in China

Abstract

Pumped storage power stations (PSPSs, hereafter) have garnered significant attention due to their critical roles in peak regulation and frequency modulation, contributing to the advancement of global new energy and power systems. Site selection of power stations is the key to successful operation. In this paper, a new site selection index system and evaluation model covering hydrogeology, construction, social economy, and energy grid are proposed to meet the multi-energy complementary needs of new energy sources. The index system was constructed by the literature review and Delphi method, the subjective and objective weights were calculated by the G1 method and Gini weighting method, and the combined weights were obtained by modifying the G1 method based on the Gini coefficient. The VIKOR method was used to evaluate the pre-selected sites, determine the best scheme, and verify the stability of the results. The results of the case study show that the Centian station site in Guangdong Province is the most promising. This study provides decision support for the construction of pumped storage power plants and has important significance for the development of clean energy and new power systems.

1. Introduction

In the face of the increasingly serious problem of global warming, adjusting the power supply model, implementing the energy revolution, and building a new energy power system have become the universal and only decisions in the world. Globally, nations are embracing innovative approaches, reevaluating their energy sector trajectories, and choosing renewable sources for alternatives.

Multi-energy complementarity refers to an energy use method that adopts a variety of energy sources to supplement each other under varying resource scenarios and by different energy users, thereby resolving the conflict between energy and consumer demands, judiciously conserving and exploiting natural resources, and concurrently achieving beneficial environmental outcomes [1]. At present, many countries and local governments attach importance to the development of multi-energy complementary systems and formulate policies and measures to safeguard them. The UK is deeply analyzing the multi-energy complementary strategy to ensure energy security, stability, and sustainable development [2]. As early as the beginning of the 21st century, the United States started the construction of a multi-energy complementary system and introduced laws and regulations to promote the collaborative planning and construction of multiple energy sources [3]. China advocates the construction of an energy Internet, which uses advanced technologies such as power electronics, information and communication, big data analysis, and intelligent management of cloud platforms to connect distributed energy sources, storage devices, and various loads to form an energy network with two-way energy flow and efficient sharing, so as to optimize resource allocation and promote the sustainable development of the energy industry [4].

PSPSs store excess electricity during off-peak hours (pumping to the reservoir) and release water to generate electricity during peak hours to achieve energy conversion and efficient utilization (Figure 1) [5]. It is the core of the new energy power system, which not only undertakes peak regulation, frequency modulation, and emergency backup but also optimizes the power structure of the grid, promotes the complementary and joint dispatch of clean energy, and enhances the synergistic effect with new energy. In recent years, China’s pumped storage capacity construction scale has accelerated. According to industry statistics, as of February 2023, 67 PSPS projects have been approved across China, with a total investment amount of about 611.6 billion yuan. Among them, as of February 2023, eight power stations have been approved, featuring an overall capacity of 9.495 million kW and an investment of about 68.5 billion yuan, and the installed capacity has exceeded that of 2021 [6]. Figure 2 shows the general development of China’s PSPS installed capacity. The installed capacity has grown rapidly since 2010 and has been growing rapidly year by year [7]. At present, the installed capacity of China’s PSPSs has not yet met the actual demand, and the rapid development of wind power and optoelectronics has put forward more urgent requirements for the supporting construction of PSPSs. Therefore, it is particularly important to accelerate the construction of pumped storage power plants to balance and optimize the energy structure. In this process, the location of the power station has become the core link. However, the current site selection work mainly relies on qualitative analysis, and the application of quantitative evaluation is relatively small, which makes the site selection process easily affected by subjective factors, and it is difficult to make an accurate and objective evaluation. In addition, in the site evaluation, the consideration of new energy multi-energy complementary factors is also relatively scarce, and there is a lack of scientific and systematic guidance programs.

This study develops an integrated site selection evaluation model that combines subjective–objective weighting with behavioral decision theory from the perspective of multi-energy complementarity in renewable energy systems. The proposed framework significantly enhances both the scientific rigor and adaptive capacity of site selection methodologies, thereby facilitating the systematic development of future pumped storage power stations. The key points are as follows: (1) establish a site selection index system that meets the requirements of multi-energy complementarity; (2) use the Gini coefficient to modify the G1 method to give weight and improve its scientific nature; (3) improve the traditional VIKOR method to build a new multi-attribute decision model; and (4) take Guangzhou Province as an example to verify the validity of the model and optimize it to ensure accuracy and reliability.

The rest of this article is as follows: “Literature Review” discusses the research progress of multi-energy complementary PSPSs. The “Site selection evaluation index system” section establishes the relevant evaluation system. The “Methodology” section discusses the main research methods. “Case Studies” introduce sensitivities and comparative analysis cases. The “Discussion” section analyzes the case process and results. The “Conclusion” section summarizes the views and achievements and looks forward to the future research directions.

2. Literature Review

2.1. Development Status of PSPS Based on Multi-Energy Complementarity

In the wake of escalating worldwide efforts in sustainable energy advancement, the creation of an adaptable, versatile multi-energy hybrid system focusing on pumped storage is increasingly acknowledged by governments and specialists globally. The strategy successfully reduces unpredictability and instability inherent in generating renewable energy. In their study, P Sun et al. [8] optimized the selection of pumped storage units, combined with economic indicators and return on investment evaluation, and provided a reference for unit selection and joint operation. Li X et al. [9] established the scheduling model of the wind–light–water–thermal storage hybrid energy system (HESWPHTP), proposed the operation mode of HESWPHTP, and evaluated its value. Gao R et al. [10] combined the energy storage technology of an abandoned mine reservoir with wind and solar power generation to optimize electric energy allocation. Wang J et al. [11] proposed a pumped storage–hydrogen storage system based on deep learning and intelligent optimization to minimize power fluctuations and maximize economic benefits. Zhang et al. [12] discussed the complementary application of pumped storage power plants and wind power generation. Salimi et al. [13] studied the safety constraints of pumped storage power plants under high wind energy penetration and proposed a scenario-based modeling method to reduce market costs. The above research fully proves that PSPS and wind and other new energy grid operations can greatly enhance the security and dependability of the power source and effectively solve the problem of absorption in the development of clean energy, so the new PSPS should fully consider the requirements of joint operation with nearby clean energy power stations.

2.2. PSPS Site Selection Research

The location of PSPSs is the key to planning, affecting the layout, scale, investment and benefits. The site selection should consider hydrogeology, engineering construction, social economy, clean energy, power grid development, and other factors, and it needs a scientific evaluation model. Scholars have conducted in-depth studies on this issue: Liyan Ji et al. [14] introduced circular elimination mechanism to improve a PSPS location selection model. Nzotcha et al. [15] proposed a multi-criteria decision-making method to select the best location for a PSPS. Deng et al. [16] evaluated PSPSs by assigning weights based on game theory combinations to make the results more reliable. Wu et al. [17] selected the most ideal PSPS sites in Zhejiang Province by using a multi-criteria decision-making technique. Zhao et al. [18] discussed PSPS site selection criteria and strategies in detail to ensure accurate and scientific site selection.

Based on the research, we find that the pumped storage multi-energy complementary system is an important part of the future power system, and site selection should consider the energy structure and grid optimization. The existing selection methods and indicators lead to different conclusions, so it is necessary to build a more scientific and reasonable evaluation system. This paper innovatively puts forward comprehensive evaluation indicators and scientific evaluation models, explores new site selection schemes, improves accuracy and efficiency, and supports the development of power systems.

3. Establish the Evaluation Index System of Site Selection

3.1. PSPS Site Selection Evaluation Index Summary

Selecting optimal sites for large-scale power projects requires regions with stable natural conditions, robust infrastructure, strong government/community support, and high investment returns. Establishing such locations demands a scientifically rigorous evaluation framework. Our synthesis of global research on power project siting (Table 1) reveals three key patterns: (1) Core Criteria: Economic viability, environmental impact, and social acceptability form the fundamental evaluation triad, addressing both direct project benefits and broader sustainability. (2) Technical Drivers: Technology maturity (feasibility) and resource availability (long-term sustainability) are consistently prioritized in site assessments. (3) Contextual Factors: Transportation, safety, services, and risk profiles, while project-specific, significantly influence suitability for certain regions.

3.2. Index System Construction

Based on the results of existing scholars, this study aims to establish an objective and comprehensive site selection evaluation system for PSPSs. In order to ensure accuracy and completeness, we combined the literature review method and the Delphi method: first, key indicators were selected through literature review and then optimized according to actual needs; then, we asked the experts for their opinions and improved the system through multiple rounds of feedback. The final system has both theoretical support and practical wisdom, enhancing objectivity and comprehensiveness. The development process is shown in Figure 3.

A panel of 10 experts in pumped storage and large-scale project evaluation (Table 2) was convened to develop a rigorous PSPS site selection framework.

After in-depth expert analysis and discussion, we carefully examined and adjusted the risk factors initially identified by the literature review method. By deleting redundant items, combining similar items, and supplementing related indicators in key fields such as energy and power, we finally established a comprehensive and accurate PSPP location evaluation index system. This system covers hydrogeological factors, engineering construction factors, social and economic factors, and energy grid factors—four aspects and a total of 21 specific evaluation indicators. Quantitative indicators are involved in the operation based on their numerical values, and qualitative indicators are used for preliminary quantitative classification according to five language value levels: L = {good, relatively good, medium, relatively poor, poor} = {1,0.75, 0.5, 0.25, 0}. A detailed list of indicators is shown in Figure 4 and described and categorized in detail in Table 3.

4. Methodology

The site selection for pumped storage power stations is inherently complex and uncertain, necessitating careful consideration of the decision-makers’ psychological influences. Conventional real-number evaluations prove inadequate in this context. Therefore, this study conducts a dual analysis of the proposed model from both theoretical rationality and practical applicability perspectives after completing site ranking. First, the combined weighting method preserves the structural clarity of the G1 method while incorporating the Gini coefficient for objective weighting. This approach significantly enhances both the objectivity and stability of indicator weight calculations, effectively compensating for potential consistency deviations arising from subjective ranking processes [38]. Second, the integration of prospect theory enables the model to authentically capture decision-makers’ psychological responses to risks and gains, thereby overcoming the limitations of the “perfectly rational agent” assumption inherent in traditional expected utility theory [39]. Particularly in high-stakes decision scenarios involving massive investments and environmental uncertainties—characteristic of pumped storage power station site selection—our methodology demonstrates superior explanatory power for real-world situations. In summary, the proposed model not only improves ranking stability but also enhances the credibility and behavioral rationality of decision support. It exhibits strong potential for both theoretical extension and practical application. The logical framework is shown in Figure 5.

4.1. Calculate the Weights of Evaluation Indicators

4.1.1. Calculation of Subjective Weight by G1 Method

Guo Yajun introduced the order relation analysis method (G1 method), a subjective weighting technique, in his work, Comprehensive Evaluation Theory Methods and Expansion. The primary benefit of this method over the analytic hierarchy process (AHP) lies in the uniformity of indicator rankings by specialists, ensuring the consistency of the judgment matrix within the analytic hierarchy [40].

(1). Indicator Sequencing

Unlike AHP, the G1 method requires experts to rank indicators by importance using Equation (1) but eliminates the need for pairwise comparison matrices. For an evaluation system with m indicators $x_1,x_2,\dots ,x_m$, their ordinal relationship is expressed as

(1)$x_1>x_2>\dots >x_m$

(2). Adjacent indicator comparison

After establishing the indicator ranking (Equation (1)), only the relative importance between adjacent indicators needs to be assessed. For example, with three indicators x_i > x_j > x_k, only x_i:x_j and x_j:x_k comparisons are required, eliminating the need for the x_i:x_k comparison. This approach minimizes comparisons while preserving consistency and expert-friendly simplicity. These features give the G1 method distinct advantages over more complex weighting approaches.

(3). Subjective weight calculation using G1 method

According to the ranking of the importance of evaluation indicators given by experts, the rational assignment $r_k$ of adjacent indicators can calculate the subjective weight of evaluation indicators. The calculation method primarily involves determining the weight of the last indicator based on all the rational assigned values. Then, by considering the ratio of the importance between adjacent indicators, the weights of all indicators are calculated step by step by moving up the hierarchy. Based on the previous assumption, the G1 weight $w_m$ of the mth index in the order is as follows in Equation (2):

(2)$w_m=(1+{\sum }_{k=2}^m{\prod }_{i=1}^m{r}_i{)}^{-1}$

According to weight $w_m$, the weights of m − 1, m − 2$,\dots ,$3, 2 indicators can be calculated as shown in Equation (3):

(3)$w_{k-1}=w_k·r_k, k=m,m-1,\dots , 3, 2$

where $w_{k-1}$ is the (k − 1)th indicator under the criteria layer for the G1 method weight of this criteria layer; $w_k$ is the G1 method weight of the kth indicator at this criterion level; $r_k$ is a rational assignment given by experts.

4.1.2. Determination of Objective Weight by Gini Coefficient

Li Gang et al. proposed the Gini coefficient weighting method (Gini weighting method) and conducted a comparative study of the Gini weighting method and the entropy weighting method. They demonstrated that the Gini weighting method is a highly applicable and stable objective weighting method. A major benefit of the Gini coefficient weighting technique is that it does not need to standardize the original index data and can directly calculate the index weights based on the original data [41]. Here are the steps to compute:

Suppose that there are now m evaluation indicators, and in the decision-making of n evaluation objects, $Y_{ki}$ represents the i data of the k evaluation indicator, and ${\mathsf{\mu }}_k$ represents the mean of all data for the kth evaluation metric of the n evaluation objects. Then, the Gini coefficient value $G_k$ of the kth index is shown in Equation (4):

(4)$G_k={\sum }_{i=1}^n{\sum }_{j=1}^n\left|Y_{ki}-Y_{kj}\right|/2n^2{\mathsf{\mu }}_k$

The Gini coefficient value $G_k$ of all m evaluation indicators can be calculated through Equation (4). The Gini coefficient value $G_k$ of m evaluation indicators can be normalized to obtain the Gini coefficient weight of evaluation indicators, as shown in Equation (5):

(5)$g_k=G_k/({\sum }_{i=1}^m{G}_i)$

$g_k$—Gini coefficient weight of the kth indicator; $G_k$—Gini coefficient value of the kth indicator.

4.1.3. Combinatorial Weighting Calculation Based on Modified Gini Coefficient G1 Method

Due to the inherent shortcomings of both subjective and objective weighting methods, this paper explores a combined weighting approach that corrects the deficiencies of subjective weighting with the advantages of objective weighting—specifically, the combined weighting method based on the Gini coefficient adjustment of the G1 method, so as to make the weights of indicators more scientific. Here are the steps to take:

(1). The importance ranking of evaluation indicators is given by experts.

(2). Calculate the importance ratio of adjacent indicators based on the weight of the Gini coefficient. Through Equations (6) and (7), the Gini coefficient weights of the evaluation indicators can be obtained, fully reflecting the data information of the indicators. Then, by comparing the Gini coefficient weights of the two adjacent indicators, the ratio $r_k$ of the importance degree of the adjacent indicators $x_{k-1}$ and $x_k$ is determined. Details are shown in Equation (8).

(6)$G_k={\sum }_{i=1}^n{\sum }_{j=1}^n\left|Y_{ki}-Y_{KJ}\right|/2n^2{\mathsf{\mu }}_k$

(7)$g_k=G_k/\left({\sum }_{i=1}^m{G}_i\right)$

(8)$r_k=g_{k-1}/g_k$

In order to prevent situations such as the importance ratio of adjacent indicators being less than 1 or the weight ratio of adjacent indicators being too large, Equation (8) is improved here:

(9)$r_k=\left\{\begin{array}{c}min\left\{2,{\displaystyle \raisebox{1ex}{$g_{k-1}$}\!\left/ \!\raisebox{-1ex}{$g_k$}\right.}\right\},g_{k-1}>g_k\hfill \\ 1,g_{k-1}<g_k\hfill \end{array}\right.$

(3). Determine the combined weights. Using the idea of the G1 method to calculate the combined weight, the combined weight of the last ranked mth indicator is

(10) $w_m=(1+{\sum }_{k=2}^m{\prod }_{i=k}^m{r}_i{)}^{-1}$

Therefore, the combined weight of m − 1,m − 2$,\dots ,$3, 2 indicators is

(11)$w_{k-1}=r_k{w}_k, k=m,m-1,\dots , 3, 2$

4.2. PSPS Site Selection Evaluation Model Calculation Steps

Suppose that a PSPS site selection evaluation problem includes m sites to be evaluated and n evaluation indicators. Let $Y=\left\{Y_1,Y_2,\dots ,Y_m\right\}$, $A=\left\{A_1,A_2,\dots ,A_n\right\}$ represent the supplier set and the indicator set, respectively, and the indicator weight is unknown, represented by a vector $W={(w_1,w_2,\dots ,w_n)}^T,w_j\in [0,1],{\sum }_{j=1}^n{w}_j=1$. Let $G={\left(h_{ij}\right)}_{m\times n}$ be the hesitancy fuzzy decision matrix, where $h_{ij}(i=1,2,\dots ,m;j=1,2,\dots ,n)$ is a hesitancy fuzzy number, expressing the degree to which supplier $Y_i$ satisfies the index $A_j$, expressed as $h_{ij}=H\left\{{\mathsf{\gamma }}_{ij}^1,{\mathsf{\gamma }}_{ij}^2,\dots ,{\mathsf{\gamma }}_{ij}^{l_{ij}}\right\}$, $\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} l_{ij}$ is the number of elements in the hesitancy fuzzy number.

Step 1. Apply the Gini coefficient modified G1 method for indicator weighting.

Step 2. Construct a standardized hesitation fuzzy decision matrix.

The elements in the hesitancy fuzzy decision matrix $G={\left(h_{ij}\right)}_{m\times n}$ are arranged in increasing order. For hesitancy fuzzy numbers with different numbers of elements, the formula $\overline{\mathsf{\gamma }}=\mathsf{\xi }\mathsf{\gamma }+(1-\mathsf{\xi })\mathsf{\gamma }″$ is applied to add elements to the same hesitancy fuzzy number with fewer elements, where $\mathsf{\gamma }=min\left\{{\mathsf{\gamma }}^k|k=1,2,\dots ,l\right\},\mathsf{\gamma }″=max\left\{{\mathsf{\gamma }}^k|k=1,2,\dots ,l\right\}$ and l are the number of elements in the hesitancy fuzzy number with the largest number of elements.

Considering that there are generally two types of indicators—benefit type and cost type, $\overline{h_{ij}}=h_{ij}$ for benefit type and $\overline{h_{ij}}=h_{ij}^c$ for cost type—the formula $h_{ij}^c=H\left\{1-{\mathsf{\gamma }}_{ij}^1,1-{\mathsf{\gamma }}_{ij}^2,\dots ,1-{\mathsf{\gamma }}_{ij}^{l_{ij}}\right\}$ is applied to normalize the decision matrix, and a standardized hesitant fuzzy decision matrix $\overline{G}={\left(\overline{h_{ij}}\right)}_{m\times n}$ is obtained.

Step 3. Comprehensive foreground value calculation.

According to the normalization matrix $\overline{G}={\left(\overline{h_{ij}}\right)}_{m\times n}$, find the median $h_{jk}$ as the decision reference point to determine the prospect value function $\mathrm{v}\left(\overline{{\mathrm{h}}_{\mathrm{i}\mathrm{j}\mathrm{k}}}\right)$:

(12)$v(\overline{h_{ijk}})=\left\{\begin{array}{c}{(d_E(\overline{h_{ijk}},h_{jk}))}^{\mathsf{\alpha }} , \overline{h_{ijk}}\ge h_{jk}\\ -\mathsf{\lambda }{(d_E(\overline{h_{ijk}},h_{jk}))}^{\mathsf{\beta }} , \overline{h_{ijk}}<h_{jk}\end{array}\right.$

(13)$d_E(h_1,h_2)=\sqrt{{\sum }_{\mathsf{\lambda }=1}^l({\mathsf{\gamma }}_1^{\mathsf{\lambda }}-{\mathsf{\gamma }}_2^{\mathsf{\lambda }}{)}^2/l}$

where k is the median number; and $d_E(\overline{h_{ijk}},h_{jk})$ is determined by Equation (8). $\mathsf{\alpha }$ and $\mathsf{\beta }$ describe the sensitivity of decision-makers to gains and losses, $\mathsf{\alpha }>0$, $\mathsf{\beta }<1$; $\mathsf{\lambda }$ reflects that decision-makers are more sensitive to losses than gains, $\mathsf{\lambda }>1$.

The decision weight function of gain and loss under hesitancy fuzzy environment is expressed as

(14)$\left\{\begin{array}{c}\mathsf{\pi }(p_j{)}^{+}={\displaystyle \frac{p_j^{\mathsf{\theta }}}{[p_j^{\mathsf{\theta }}+(1-p_j{)}^{\mathsf{\theta }}{]}^{1/\mathsf{\theta }}}},\overline{{\mathrm{h}}_{\mathrm{i}\mathrm{j}\mathrm{k}}}\ge {\mathrm{h}}_{\mathrm{j}\mathrm{k}}\\ \mathsf{\pi }(p_j{)}^{-}={\displaystyle \frac{p_j^{\mathsf{\delta }}}{[p_j^{\mathsf{\delta }}+(1-p_j{)}^{\mathsf{\delta }}{]}^{1/\mathsf{\delta }}}},\overline{{\mathrm{h}}_{\mathrm{i}\mathrm{j}\mathrm{k}}}<{\mathrm{h}}_{\mathrm{j}\mathrm{k}}\end{array}\right.$

where $p_j$ is the index weight given by experts; and $\mathsf{\theta }$ and $\mathsf{\delta }$ are the income preference degree and risk avoidance degree of decision-makers, respectively, and are the different risk preference coefficients of decision-makers in the state of profit and loss. Related parameters are set to $\mathsf{\lambda }=2.25$, $\mathsf{\alpha }=\mathsf{\beta }=0.88$, $\mathsf{\theta }=0.61$, and $\mathsf{\delta }=0.69$. The empirical study by Tversky et al. [42] demonstrated that $\mathsf{\lambda }=2.25$ represents the median value of loss aversion coefficients in human decision-making. The meta-analysis by Zhang et al. [43] revealed that $\mathsf{\alpha }=\mathsf{\beta }=0.88$ constitutes typical parameters for diminishing sensitivity to both gains and losses in energy infrastructure decisions. Meanwhile, $\mathsf{\theta }=0.61$ and $\mathsf{\delta }=0.69$ were calibrated through Delphi expert surveys.

Thus, the combined prospect value can be determined as

(15)$v(h_{ij})={\sum }_{\overline{v(h_{ijk}})\ge 0}\mathsf{\pi }{(p_j)}^{+}v(\overline{h_{ijk}})+{\sum }_{\overline{v(h_{ijk}})<0}\mathsf{\pi }{(p_j)}^{-}v(\overline{h_{ijk}})$

Then, transform the multi-attribute hesitant fuzzy evaluation matrix into a multi-attribute comprehensive prospect matrix, which is denoted as $V={(h_{ij})}_{m\times n}$.

Step 4. Use the VIKOR method to sort the schemes:

(1). According to the comprehensive prospect matrix, the positive ideal solution $f^{*}$ and the negative ideal solution $f^{-}$ of each index are calculated:

$f^{*}=\left\{max{v}_i(h_{i1}),max{v}_i(h_{i2}),\dots ,max{v}_i(h_{in})\right\}$

(16)$f^{-}=\left\{min{v}_i(h_{i1}),min{v}_i(h_{i2}),\dots ,min{v}_i(h_{in})\right\}$

(2). Calculate the group benefit value $S_i$ and individual regret value $R_i$ of each evaluation object:

$S_i=\sum _{j=1}^n{w}_j(f^{*}-v(h_{ij}))/(f^{*}-f^{-})$

(17)$R_i={max}_j\left\{w_j(f^{*}-v(h_{ij}))/(f^{*}-f^{-})\right\}$

In the formula, $S_i$ represents the group benefit of the evaluation object, and the smaller $S_i$ is, the larger the group benefit; $R_i$ represents individual regret. The smaller $R_i$ is, the smaller the individual regret. $w_j$ is the attribute weight.

(3). Determine the interest ratio Qi of each evaluation object:

(18)$Q_i=\mathsf{\phi }{\displaystyle \frac{S_i-S_i^{-}}{S_i^{*}-S_i^{-}}}+(1-\mathsf{\phi }){\displaystyle \frac{R_i-R_i^{-}}{R_i^{*}-R_i^{-}}}$

where $S_i^{*}={max}_i{S}_i$, $S_i^{-}={min}_i{S}_i$, $R_i^{*}={max}_i{R}_i$, $R_i^{-}={min}_i{R}_i$, $S_i^{*}$ is the maximum utility of the group, $R_i^{*}$ is the minimum regret of the individual, and $\mathsf{\phi }$ represents the decision preference of the decision-maker.

(4). Determine the ranking of alternatives and compromise:

According to $S_i$, $R_i$, $Q_i$ in order from small to large, the decision scheme is sorted, and the priority of the object is evaluated. If the following two conditions are met, the order is made according to the size of $Q_i$. The smaller the value of $Q_i$, the better the scheme to be decided. If condition ② cannot be met, then $Y_1$ and $Y_2$ are both compromise solutions; if the scheme ranked first and several other schemes do not satisfy condition ① but only satisfy condition ②, then it can be determined that the overall evaluation of the schemes that do not satisfy condition ① is optimal.

① Acceptable dominance criteria: $Q(Y_2)-Q(Y_1)\ge DQ,DQ=1/(n-1)$;

where $Y_1$ is the optimal evaluation object in $Q_i$ sort, and $Y_2$ is the second-optimal evaluation object in $Q_i$ sort.

② Acceptable stability criteria: $Y_1$ is the preceding object of $S_i$ or $R_i$.

5. Case Studies

Guangdong’s social and economic prosperity has driven the growth of electricity demand, and its future electricity demand mainly depends on new energy and nuclear power. In recent years, the construction of PSPSs in Guangdong has been remarkable, forming a multi-source complementary system of coal power, nuclear power, west–east power transmission, gas power, hydropower, pumped storage, and wind power. However, the diversified power supply structure leads to an increase in the peak–valley difference in the power grid, and an increase in the proportion of nuclear power, western power, and new energy generation increases the difficulty of peak regulation. The power supply reliability standard upgrade requires more emphasis on peak capacity. Therefore, reasonable planning of peak power supply is the key, with PSPS as a flexible adjustment means to help new energy consumption and storage; achieving the “double carbon goal” is crucial. Guangdong needs to urgently evaluate and optimize the construction conditions of pumped storage stations, which is related to the optimization of power structure, energy transformation, and the realization of the “double carbon goal”.

Through a thorough analysis of the geography, water resources, electricity demand, and distribution of renewable energy in Guangdong Province, and based on specific tasks such as field surveys, environmental assessments, and feasibility studies, this study selected five pre-selected sites: Zhongdong (Y1), Centian (Y2), Meixu Phase II (Y3), Langjiang (Y4), and Shuiyuan Mountain (Y5). Figure 6 and Table 4 show information about these five sites. To ensure methodological transparency and reproducibility, all parameter values for site evaluation (including hydrogeological characteristics, investment costs, and environmental impacts) were rigorously sourced from (1) project technical documentation, (2) publicly available reports, and (3) expert evaluations. Quantitative parameters (e.g., annual seepage volume, regulating storage capacity) were obtained directly from engineering records, while qualitative indicators (e.g., topographic–geological conditions, transportation accessibility) were scored through Delphi-based consensus among three senior hydropower specialists. The comprehensive dataset (Appendix A) provides validated evidence for optimal site selection.

5.1. The Gini Coefficient Modified G1 Method Was Applied to Assign Weights to Indicators

(1). First, 10 experts were given a ranking according to their previous engineering experience; then, through organizing online discussion among experts, each expert explained his reasons for such ranking, and other experts put forward questions and opinions; finally, after three rounds of discussion, 10 experts jointly gave the final index ranking through continuous adjustment, as follows: $C_{22}, C_{23},C_{45},C_{12},C_{21},C_{41},C_{34},C_{35},C_{13},C_{25},C_{24},C_{46},C_{33},C_{15},C_{14},C_{31}{C}_{32},C_{11},C_{44},C_{43},C_{42}$.

(2). The ratio $r_k$ of the importance of adjacent indicators based on the weight of the Gini coefficient was calculated by Equations (6) and (7) and included in column 2 of Table 5.

(3). According to Equation (8), the combined weight $w_k$ is listed in column 3 of Table 5.

(4). Inter-indicator Correlation Validation

To ensure the scientific validity of weight allocation, we conducted a comprehensive correlation analysis among the 21 evaluation indicators (Appendix B). Key findings reveal the following: (i) strong positive correlations between C12 (topographic-geological conditions) and C13 (water quality conditions) ($\mathsf{\rho }$ = 0.905), demonstrating the direct influence of geological stability on water quality, and between C25 (adjustable storage capacity) and C35 (emission reduction effect) ($\mathsf{\rho }$ = 0.971), confirming the significant contribution of storage regulation to emission mitigation; (ii) notable negative correlations, including C15 (seismic intensity) versus C33 (new energy efficiency) ($\mathsf{\rho }$ = −0.954), indicating challenges in renewable energy integration in high seismic risk zones, and C21 (investment per kilowatt) versus C34 (power reliability) ($\mathsf{\rho }$ = −0.739), reflecting the cost–reliability trade-off; and (iii) orthogonal relationships, such as C11 (water quality) versus C21 (investment per kilowatt) ($\mathsf{\rho }$ = 0.015) and weak correlation between C44 (construction conditions) and C45 (traffic conditions) ($\mathsf{\rho }$ = 0.297), validating their independence for separate weighting. These results systematically justify the robustness of our weighting system against multicollinearity while highlighting inherent synergies and conflicts among critical indicators.

5.2. The Prospective Theory and VIKOR Method Were Used to Evaluate and Sort the Pre-Selected Sites

Following the methods described in the Methodology section, the selection was made as follows:

Step 1: Taking “hydrogeological factors” as an example, the method of this paper is used to calculate. When choosing a site, the risk of each index has three states: high, medium, and low, and the probability is 0.3, 0.5, and 0.2, respectively. The P1 = 0.3 (high), P2 = 0.5 (medium), P3 = 0.2 (low) risk probabilities align with Guangdong’s 2022–2023 flood frequency data (30% high-risk months, 20% low-risk) [44]. Experts scored the five indicators of C1 using the standardized 9-point scale hesitant fuzzy scoring method. The original scoring data are shown in Table 6. Experts use a 9-point scale method to score five indicators. Assuming expert risk neutrality, $\mathsf{\xi }$ = 0.5; the standardized matrix obtained after processing is shown in Table 7.

Step 2: Determination of comprehensive prospect matrix. First, find the median as the decision reference point $h_{jk}$. Then, by calculating the foreground value function matrix from Equation (15), Table 8 can be obtained. Then, by calculating the decision weight from Equation (16), we can obtain $\mathsf{\pi }{(p1)}^{+}=0.3185,\mathsf{\pi }{(p2)}^{+}=0.4206,\mathsf{\pi }{(p3)}^{+}=0.2608,\mathsf{\pi }{(p1)}^{-}=0.3276,\mathsf{\pi }{(p2)}^{-}=0.4540,\mathsf{\pi }{(p3)}^{-}=0.2570$. Finally, the comprehensive prospect value matrix of all indicators can be obtained from Equation (17), as shown in Table 9.

Step 3: Sort the alternatives according to the VIKOR method. If the decision preference of the decision-maker $\mathsf{\phi }$ = 0.5, the group benefit value $S_i$, individual regret value $R_i$, and comprehensive value $Q_i$ can be calculated from Equations (14) and (15). The findings are displayed in Table 10.

From Table 10, the comprehensive evaluation value $Q_i$ ranks the schemes as follows: Y2 > Y1 > Y34 > Y5 > Y3. According to the VIKOR method’s decision rules, the scheme ranked first by $S_i$ is also Y2, meeting condition ②: acceptable stability criterion; from the formula Q2 − Q1 = 0.2960 > 0.05, it can be concluded that the results also satisfy condition ①: the acceptable advantage criterion, and thus, an ideal scheme with Y2 as acceptable is obtained.

6. Sensitivity Analysis and Comparative Analysis

For additional confirmation of the framework’s practicality and efficacy, a sensitivity analysis was conducted and juxtaposed with VIKOR’s technique and the enhanced TOPSIS method already in use.

6.1. Sensitivity Analysis

(1). In the case application, coefficient selection affects the decision result. To balance benefits and regrets, ${\mathsf{\phi }}_i$ = 0.5 is chosen as a compromise. However, experts have different preferences and will choose different compromise coefficients, resulting in changes in scheme ranking. Therefore, in this paper, ${\mathsf{\phi }}_i\in [0,1]$ with 0.1 step is taken for 11 times to obtain 11 groups of the comprehensive evaluation value $Q_i$ (Table 11). The stability of the model is investigated through a sensitivity analysis.

The sensitivity analysis results shown in Figure 7 and Figure 8 are obtained from Table 11:

The ranking of the four alternative plans will basically remain unchanged under different values of ${\mathsf{\phi }}_i$. When the value range of ${\mathsf{\phi }}_i$ is [0, 0.876], the order is Y2 $>$ Y1 $>$ Y4 $>$ Y5 $>$ Y3. Currently, the decision-maker’s focus is primarily on the personal value of regret, overlooking the collective advantage. When the value of ${\mathsf{\phi }}_i$ falls within the range (0.876, 1], meaning that as the decision-maker increasingly considers group benefits and individual regret gradually decreases, the resulting ranking is Y2 $>$ Y1 $>$ Y4 $>$ Y3 $>$ Y5. With ${\mathsf{\phi }}_i$ = 0.876 as the boundary, only two sorting results are produced in the end, which is mainly reflected in the difference between the Y3 and Y5 sorting results. Moreover, under 11 perturbations of the ${\mathsf{\phi }}_i$ value, nine times (81.8%) of the scheme are sorted by Y2 > Y1 > Y4 > Y5 > Y3, so it can be concluded that the model is not sensitive to perturbations of the ${\mathsf{\phi }}_i$ value.

(2). In the case study section, we adopted the risk-neutral assumption for experts by setting $\mathsf{\xi }$ = 0.5 as the baseline. To systematically evaluate the parameter’s influence, we conducted sensitivity tests with $\mathsf{\xi }$ ranging across {0, 0.25, 0.5, 0.75, 1} at intervals of 0.25, covering both extreme and intermediate values. The test results are presented in Table 12. Key findings reveal the following: Lower $\mathsf{\xi }$ values (minimization-oriented completion) elevate the ranking of Y3 (low-risk alternative). Higher $\mathsf{\xi }$ values (maximization-oriented completion) favor Y5 (high-return alternative). Stability threshold: Ranking outcomes remain completely consistent when $\mathsf{\xi }$ ∈ [0.4, 0.6], empirically validating the rationality of the default $\mathsf{\xi }$ = 0.5 setting.

(3). Sensitivity Analysis of Prospect Theory Parameters

To systematically evaluate the impact of behavioral parameters on model outcomes, we conducted perturbation analyses on (i) loss aversion coefficient $\mathsf{\lambda }$, (ii) gain sensitivity $\mathsf{\alpha }$, and (iii) loss sensitivity $\mathsf{\beta }$. Each analysis held other parameters constant while varying the target parameter, with the subsequent recording of comprehensive evaluation index (Q) variations and alternative rankings.

First, $\mathsf{\lambda }$ was varied within [1.0, 3.0] at 0.5 intervals ($\mathsf{\alpha }=\mathsf{\beta }$ fixed at 0.88). As Table 13 demonstrates, Y2 consistently maintained optimal ranking despite minor fluctuations in Y3/Y5 positions, confirming $\mathsf{\lambda }$’s limited effect on optimal solution selection but notable influence on intermediate alternatives’ scores—highlighting its critical role in loss preference modulation.

Subsequent $\mathsf{\alpha }$ and $\mathsf{\beta }$ perturbations within [0.5, 1.0] (Table 14 and Table 15) revealed marginal Q-value variations without altering the overall ranking structure, designating them as secondary sensitivity parameters.

Table 13, Table 14 and Table 15 demonstrate that perturbations in $\mathsf{\lambda }$ exert more substantial influence on the model’s comprehensive prospect values. Notably, when $\mathsf{\lambda }$ exceeds 2.5, ranking fluctuations emerge for specific sites (e.g., Y3 and Y5), which aligns with $\mathsf{\lambda }$’s established role in governing loss-averse behavior within prospect theory. In contrast, variations in $\mathsf{\alpha }$ and $\mathsf{\beta }$—serving as curvature adjustment parameters for the gain/loss value functions—exhibit minimal impact on ranking outcomes, as evidenced by consistently stable rankings and the persistent optimal solution (Y2). These findings validate the parameter importance hierarchy in prospect theory as $\mathsf{\lambda }$ > $\mathsf{\alpha }$ ≈ $\mathsf{\beta }$, confirming that our adopted parameter configuration ($\mathsf{\lambda }$ = 2.25, $\mathsf{\alpha }$ = $\mathsf{\beta }$ = 0.88) maintains both structural representativeness and robustness.

6.2. Comparative Analysis

To validate the feasibility and superiority of our proposed method, we conducted systematic comparisons with three classical decision-making approaches: VIKOR, GRA-TOPSIS, and prospect theory-integrated TOPSIS. The implementation details are specified as follows:

(1). VIKOR method

The standard VIKOR calculation process is adopted to perform linear normalization processing on the positive indicators:

$x_{ij}^{\prime }={\displaystyle \frac{x_{ij}-x_j^{min}}{{x_j^{max}-x}_j^{min}}}$

Subsequently, we computed the group utility measure ($S_i$), individual regret measure ($R_i$), and compromise index ($Q_i$), with the decision-maker’s preference coefficient set to $\mathrm{v}$ = 0.5. All weightings were derived through the Gini coefficient-adjusted G1 method to ensure consistent comparison baselines.

(2). GRA-TOPSIS method:

First, perform Min-Max normalization processing on the index matrix; then, calculate the grey correlation degree between each scheme and the ideal solution, as well as the negative ideal solution; then, the relative closeness degree value in the classical TOPSIS is replaced by the grey correlation degree, thereby obtaining the ranking result:

$\mathsf{\gamma }(x_0,x_i)={\displaystyle \frac{{∆}_{min}+{\mathsf{\zeta }∆}_{max}}{{∆}_{ij}+{\mathsf{\zeta }∆}_{max}}}$

Although this method improves the fuzziness problem of traditional TOPSIS, it still ignores the differences in decision preference behaviors.

(3). Prospect Theory-TOPSIS Method:

This approach first normalizes the decision matrix, then applies prospect theory’s value function to psychologically transform gains and losses before conducting TOPSIS proximity calculations. Its key advantage lies in better capturing real human decision-making under risk and uncertainty.

All comparative methods utilized identical decision matrix and indicator weights to ensure fair benchmarking. As shown in Table 16 and Figure 9, all four methods produced fully consistent rankings (Y2 > Y1 > Y4 > Y5 > Y3) at the numerical level. However, our method’s superiority extends beyond ranking consistency—its theoretical framework demonstrates stronger behavioral interpretability and decision elasticity, effectively overcoming traditional methods’ reliance on the “perfectly rational agent” assumption.

To statistically validate ranking consistency, we computed Spearman’s rank correlation coefficients between methods ($\mathsf{\rho }$ = 1.000, p < 0.001), confirming complete ordinal agreement and further evidencing our model’s validity and robustness.

7. Conclusions

The precise site selection of pumped storage power stations plays a pivotal role in achieving the “dual carbon” goals. This study establishes a comprehensive decision-making evaluation framework that not only guides the location selection of pumped storage stations but also extends its applicability to new energy power plant siting and industrial layout optimization. Our research systematically reviewed existing studies on pumped storage power station site selection, through which we identified and refined critical evaluation indicators. By developing an integrated weighting method that combines the Gini coefficient with the G1 method, we have enhanced the scientific rigor of indicator weighting. Furthermore, the novel decision-making approach incorporating hesitant fuzzy sets, prospect theory, and the VIKOR method has optimized the decision-making process to better reflect real-world conditions. Comparative analysis with alternative methods confirms the effectiveness of our proposed approach as an improved version of the VIKOR method, offering innovative solutions to complex siting problems.

Three key findings emerge from this research. First, site-specific indicators such as average water head (C22), adjustable storage capacity (C23), and height-to-distance ratio (C21) are crucial determinants of a pumped storage station’s operational capacity. Second, the synergistic effects between pumped storage stations and new energy power plants significantly enhance grid stability and reliability, underscoring the importance of energy grid considerations in site selection for achieving carbon neutrality objectives. Third, hydrological–geological and socio-economic factors, including social security and ecological protection indicators, must be incorporated into the evaluation system to ensure sustainable development and local community advancement.

Despite these contributions, we acknowledge certain limitations. The relatively recent development of pumped storage projects in China means that practical experience remains limited, and the site evaluation system requires further refinement. Additionally, the reliance on expert scoring for data collection introduces subjective elements. Future research should focus on acquiring more comprehensive and objective datasets while developing quantitative analytical methods to minimize subjectivity and enhance the robustness of the conclusions.

Author Contributions

Conceptualization, Y.X.; Methodology, Y.X.; Software, Y.X.; Validation, Y.X. and H.Z.; Formal Analysis, Y.X.; Investigation, Y.X.; Resources, Y.X. and H.Z.; Data Curation, Y.X.; Writing–Original Draft Preparation, Y.X.; Writing—Review and Editing, Y.X. and H.Z.; Visualization, H.Z.; Supervision, H.Z.; Project Administration, Y.X. and H.Z. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Conflicts of Interest

The authors declare no conflicts of interest; this research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1 Schematic diagram of PSPS.

Figure 2 Annual development of installed capacity of PSPS.

Figure 3 Development process of PSPS site selection evaluation index system.

Figure 4 Evaluation index system of PSPS.

Figure 5 The logical framework.

Figure 6 Location of evaluation station.

Figure 7 Changes in comprehensive evaluation value $Q_i$.

Figure 8 Ranking change of comprehensive value $Q_i$.

Figure 9 Comparative analysis.

Table 1

Relevant indicators of site selection evaluation.

Factor	[19]	[20]	[21]	[22]	[23]	[24]	[25]	[26]	[27]	[28]	[29]	Sum
Social	√	√	√	√	√	√		√	√	√		9
Service										√		1
Economic	√		√	√	√	√	√	√	√		√	9
Resource		√	√			√	√	√	√	√	√	8
Environmental	√	√		√	√	√	√	√	√	√	√	10
Technical	√		√	√		√		√				5
Safety			√					√		√		2
Traffic		√				√					√	3
Support Condition		√										1
Risk					√							1
Climate								√				1

Table 2

Basic information for each expert.

No	Work Unit	Position	Research Focus	Relevance to PSPS Site Selection
1	North China Electric Power University	Professor	Power system planning, renewable energy integration	Provides theoretical support for hydrological/geological and grid compatibility indicators
2	Shandong University	Professor	Environmental engineering, social impact assessment of energy projects	Leads optimization of environmental and socio-economic indicators
3	Shandong Electric Power Survey and Design Institute	Senior engineer	Power engineering geological survey, topographic mapping	Validates technical aspects of hydrological/geological and construction indicators
4	Shandong Electric Power Survey and Design Institute	Senior engineer	Electrical design, grid connection solutions	Refines energy/grid factors
5	China Shandong Electric Power Construction Group	Project manager	Large-scale power infrastructure project management, construction risk assessment	Optimizes engineering/construction indicators and risk management
6	China 11th Water Conservancy and Hydropower Engineering Bureau	Project manager	Hydraulic engineering construction, geological adaptability analysis	Verifies practicality of hydrological/geological indicators
7	Institute of Water Conservancy and Hydropower Planning	Senior engineer	Regional water resource planning, ecological impact assessment	Supplements cross-cutting indicators
8	China Electric Power Planning and Design Institute	Senior engineer	Power system stability analysis, energy storage project planning	Strengthens energy/grid factors
9	China International Engineering Consulting Corporation	Professional consultant	Life-cycle assessment of energy projects, policy compliance	Ensures comprehensiveness of socio-economic factors
10	Hebei Electric Power Company of State Grid	Dispatching specialist	Grid operation/dispatch, grid-connected energy storage experience	Provides operational insights to optimize grid compatibility and dispatch flexibility

Table 3

Indicator description and index.

Evaluation Indicators	Indicator Description
Water quality condition	Evaluating the plant’s water source quality primarily hinges on the extent of water contamination and the level of dissolved oxygen present, crucial for the efficient functioning and upkeep of energy storage and power generation machinery [30].
Topographic geological condition	When PSPS are located, the geological conditions need to be ideal, which affects construction, costs, and long-term operation. Site surveys to determine optimal geological conditions are essential [31].
Annual water seepage loss	Annual water loss due to leakage within the factory area.
Seismic intensity	Reflect the intensity of earthquake impact on the surface and engineering buildings [14].
Average annual rainfall	When PSPSs are built in arid areas, water is insufficient; when they are built in places with heavy rainfall, although the water source is stable, the flood control requirements are increased [32].
Height-to-distance ratio	The height-to-distance ratio describes the ratio between the horizontal distance between the inlet/outlet of the upper reservoir and the inlet/outlet of the lower reservoir, and the average total head of the power station [33].
Mean head	At present, generally, the economic head of PSPSs is 330–600 m, and the highest is not more than 700 m. A higher water head will increase the complexity and cost of equipment; a lower water head requires a larger flow rate, resulting in an increase in unit cost [18].
Adjustable storage capacity	Adjustable storage capacity refers to the effective capacity of PSPSs that can store or release water by adjusting the amount of water in and out during operation [14].
Traffic condition	The site is located with excellent transportation facilities, which helps lay a solid foundation for the construction of the power station. The evaluation parameters mainly include the weighted average distance (km) to highways/railways and the heavy-load vehicle passability index (1–5 scale) [34].
Construction and installation conditions	The construction and installation conditions mainly refer to the construction site conditions (hydropower layout, plant layout), technical difficulty, and the difficulty of equipment installation. Quantification is usually based on terrain adaptability index (slope stability, excavation difficulty), equipment transport accessibility (road class, bridge capacity), and modular construction feasibility (%) [34].
Land expropriation	The cost of acquiring land for the construction of PSPSs [30].
Relocation and resettlement population	The number of people relocated due to the construction of PSPSs [35].
Distance from the city	The distance from the site to the city [35].
Investment per kilowatt	When constructing a power station facility, this is the investment cost required per kilowatt of installed capacity [35].
Investment per unit of energy storage	The investment cost per unit of storage capacity required when constructing storage facilities [35].
Distance to new energy base	Feasibility study on the complementary operation of PSPSs and a clean energy base [36].
Optimize the power supply structure	The ability of PSPSs to improve the electricity supply structure. Quantified as the percentage reduction in fossil-fuel-based generation capacity displaced by PSPS-flexible renewable integration over a 5-year horizon [30].
Power reliability and stability	This is usually measured by two sub-indicators: the System Average Interruption Frequency Index (SAIFI, occurrences/year) and the Voltage Deviation Rate (%) from nominal levels during peak loads [14].
New energy efficiency	The ratio (%) of actual electricity output to maximum theoretical output when integrating renewable energy sources, calculated as $\mathsf{\eta }=({\displaystyle \frac{E_{actual}}{E_{theoretical}}})\times 100\%$.
Distance from substation	If there is a substation near the project site, it can reduce unnecessary power loss along the transmission line, and good power transmission conditions can greatly reduce the cost of embedded cables [34].
Annual emission reduction effect	The power station can reduce pollutant emissions within a year by complementing clean energy [37].

Table 4

Information about the site.

Site Name	Location	Brief Introduction
Zhongdong	Huidong County, Huizhou City	The water source of Zhongdong PSPS in Huizhou is sufficient and the average water head is large, and the proximity to the new energy power station is conducive to the power transmission of the power station.
Centian	Dongyuan County, Heyuan City	Centian PSPS lower reservoir has a large drainage area and belongs to a large-scale PSPS. The area where the power station is located belongs to the coverage area of the Guangdong power grid, which can be continuously full for 7 h.
Meixu Phase II	Wuhua County, Meizhou City	The basin area of the lower reservoir of the second phase of Meixu Phase II PSPS is also very large, but the static investment is smaller than that of Centian Power Station, and the economic advantage is large.
Langjiang	Guangning County, Zhaoqing City	Langjiang PSPS is located in the Huanglian Mountain range, the terrain between the upper and lower reservoirs is different, and the rainfall is greater in the south.
Shuiyuan Mountain	Xinxing County, Yunfu City	The water supply of Shuiyuan Mountain PSPS is sufficient and the distance from the city is close, which is conducive to improving the transmission efficiency. However, the ecological red line is involved in the region, which is not advantageous in investment.

Table 5

${\mathrm{r}}_{\mathrm{k}}$ and ${\mathrm{w}}_{\mathrm{k}}$ values of evaluation indicators.

Evaluation Index	${\mathit{r}}_{\mathit{k}}$	${\mathit{w}}_{\mathit{k}}$
C11	2	0.0016
C12	1.0552	0.1512
C13	2	0.0190
C14	1	0.0031
C15	2	0.0031
C21	1.9931	0.0758
C22	\	0.2154
C23	1.3508	0.1519
C24	1.5228	0.0125
C25	1	0.0190
C31	1	0.0031
C32	1	0.0031
C33	2	0.0062
C34	2	0.0379
C35	1	0.0379
C41	1	0.0758
C42	1	0.0012
C43	1.3516	0.0012
C44	1	0.0016
C45	1	0.1595
C46	1	0.0125

Table 6

Initial scoring matrix.

Risk Probability	Option	C11	C12	C13	C14	C15
P1 = 0.3	Y1	$\left\{0.6,0.7,0.8\right\}$	$\left\{0.4,0.6,0.7\right\}$	$\left\{0.4,0.5,0.7\right\}$	$\left\{0.6,0.7,0.9\right\}$	$\left\{0.6,0.7,0.8\right\}$
	Y2	$\left\{0.6,0.9\right\}$	$\left\{0.6,0.8\right\}$	$\left\{0.2,0.3,0.4\right\}$	$\left\{0.4,0.5\right\}$	$\left\{0.7,0.8\right\}$
	Y3	$\left\{0.4,0.5,0.7\right\}$	$\left\{0.6,0.7,0.8\right\}$	$\left\{0.2,0.5\right\}$	$\left\{0.1,0.2,0.4\right\}$	$\left\{0.2,0.5,0.6\right\}$
	Y4	$\left\{0.2,0.3,0.5\right\}$	$\left\{0.3,0.5,0.7\right\}$	$\left\{0.6,0.7\right\}$	$\left\{0.3,0.5,0.6\right\}$	$\left\{0.5,0.7,0.8\right\}$
	Y5	$\left\{0.4,0.7\right\}$	$\left\{0.1,0.3,0.4\right\}$	$\left\{0.1,0.3,0.5\right\}$	$\left\{0.6,0.7\right\}$	$\left\{0.6,0.7\right\}$
P2 = 0.5	Y1	$\left\{0.7,0.8\right\}$	$\left\{0.5,0.6,0.7\right\}$	$\left\{0.4,0.5,0.6\right\}$	$\left\{0.6,0.7,0.8\right\}$	$\left\{0.5,0.7,0.8\right\}$
	Y2	$\left\{0.6,0.8,0.9\right\}$	$\left\{0.7,0.8\right\}$	$\left\{0.1,0.2,0.3\right\}$	$\left\{0.3,0.4,0.5\right\}$	$\left\{0.6,0.7\right\}$
	Y3	$\left\{0.5,0.6,0.7\right\}$	$\left\{0.6,0.7,0.8\right\}$	$\left\{0.3,0.4\right\}$	$\left\{0.1,0.2,0.3\right\}$	$\left\{0.4,0.5,0.6\right\}$
	Y4	$\left\{0.3,0.4,0.5\right\}$	$\left\{0.4,0.5,0.6\right\}$	$\left\{0.6,0.8\right\}$	$\left\{0.3,0.4,0.5\right\}$	$\left\{0.7,0.8\right\}$
	Y5	$\left\{0.5,0.6\right\}$	$\left\{0.2,0.3,0.5\right\}$	$\left\{0.5,0.7,0.8\right\}$	$\left\{0.5,0.7\right\}$	$\left\{0.5,0.6,0.7\right\}$
P3 = 0.2	Y1	$\left\{0.4,0.6,0.7\right\}$	$\left\{0.3,0.5,0.7\right\}$	$\left\{0.3,0.5,0.6\right\}$	$\left\{0.5,0.6,0.7\right\}$	$\left\{0.5,0.6\right\}$
	Y2	$\left\{0.5,0.7,0.8\right\}$	$\left\{0.5,0.6,0.7\right\}$	$\left\{0.3,0.4,0.5\right\}$	$\left\{0.5,0.6\right\}$	$\left\{0.4,0.5,0.6\right\}$
	Y3	$\left\{0.6,0.7\right\}$	$\left\{0.6,0.7\right\}$	$\left\{0.5,0.6\right\}$	$\left\{0.3,0.4,0.5\right\}$	$\left\{0.4,0.7\right\}$
	Y4	$\left\{0.4,0.5,0.6\right\}$	$\left\{0.4,0.5,0.7\right\}$	$\left\{0.4,0.5\right\}$	$\left\{0.4,0.7\right\}$	$\left\{0.6,0.7,0.8\right\}$
	Y5	$\left\{0.4,0.5,0.6\right\}$	$\left\{0.3,0.4,0.6\right\}$	$\left\{0.4,0.5,0.7\right\}$	$\left\{0.4,0.5,0.6\right\}$	$\left\{0.5,0.6,0.7\right\}$

Table 7

Standardization matrix.

RiskProbability	Option	C11	C12	C13	C14	C15
P1 = 0.3	Y1	$\left\{0.6,0.7,0.8\right\}$	$\left\{0.4,0.6,0.7\right\}$	$\left\{0.3,0.5,0.6\right\}$	$\left\{0.1,0.3,0.4\right\}$	$\left\{0.6,0.7,0.8\right\}$
	Y2	$\left\{0.6,0.75,0.9\right\}$	$\left\{0.6,0.7,0.8\right\}$	$\left\{0.6,0.7,0.8\right\}$	$\left\{0.5,0.55,0.6\right\}$	$\left\{0.7,0.75,0.8\right\}$
	Y3	$\left\{0.4,0.5,0.7\right\}$	$\left\{0.6,0.7,0.8\right\}$	$\left\{0.5,0.75,0.8\right\}$	$\left\{0.6,0.8,0.9\right\}$	$\left\{0.2,0.5,0.6\right\}$
	Y4	$\left\{0.2,0.3,0.5\right\}$	$\left\{0.3,0.5,0.7\right\}$	$\left\{0.3,0.35,0.4\right\}$	$\left\{0.4,0.5,0.7\right\}$	$\left\{0.5,0.7,0.8\right\}$
	Y5	$\left\{0.4,0.55,0.7\right\}$	$\left\{0.1,0.3,0.4\right\}$	$\left\{0.1,0.3,0.5\right\}$	$\left\{0.3,0.35,0.4\right\}$	$\left\{0.6,0.65,0.7\right\}$
P2 = 0.5	Y1	$\left\{0.7,0.75,0.8\right\}$	$\left\{0.5,0.6,0.7\right\}$	$\left\{0.4,0.5,0.6\right\}$	$\left\{0.2,0.3,0.4\right\}$	$\left\{0.5,0.7,0.8\right\}$
	Y2	$\left\{0.6,0.8,0.9\right\}$	$\left\{0.7,0.75,0.8\right\}$	$\left\{0.7,0.8,0.9\right\}$	$\left\{0.5,0.6,0.7\right\}$	$\left\{0.6,0.65,0.7\right\}$
	Y3	$\left\{0.5,0.6,0.7\right\}$	$\left\{0.6,0.7,0.8\right\}$	$\left\{0.6,0.65,0.7\right\}$	$\left\{0.7,0.8,0.9\right\}$	$\left\{0.4,0.5,0.6\right\}$
	Y4	$\left\{0.3,0.4,0.5\right\}$	$\left\{0.4,0.5,0.6\right\}$	$\left\{0.2,0.3,0.4\right\}$	$\left\{0.5,0.6,0.7\right\}$	$\left\{0.7,0.75,0.8\right\}$
	Y5	$\left\{0.5,0.55,0.6\right\}$	$\left\{0.2,0.3,0.5\right\}$	$\left\{0.2,0.3,0.5\right\}$	$\left\{0.3,0.4,0.5\right\}$	$\left\{0.5,0.6,0.7\right\}$
P3 = 0.2	Y1	$\left\{0.4,0.6,0.7\right\}$	$\left\{0.3,0.5,0.7\right\}$	$\left\{0.4,0.5,0.7\right\}$	$\left\{0.3,0.4,0.57\right\}$	$\left\{0.5,0.55,0.6\right\}$
	Y2	$\left\{0.5,0.7,0.8\right\}$	$\left\{0.5,0.6,0.7\right\}$	$\left\{0.5,0.6,0.7\right\}$	$\left\{0.4,0.45,0.5\right\}$	$\left\{0.4,0.5,0.6\right\}$
	Y3	$\left\{0.6,0.65,0.7\right\}$	$\left\{0.6,0.65,0.7\right\}$	$\left\{0.4,0.45,0.5\right\}$	$\left\{0.5,0.6,0.7\right\}$	$\left\{0.4,0.55,0.7\right\}$
	Y4	$\left\{0.4,0.5,0.6\right\}$	$\left\{0.4,0.5,0.7\right\}$	$\left\{0.5,0.55,0.6\right\}$	$\left\{0.3,0.45,0.6\right\}$	$\left\{0.6,0.7,0.8\right\}$
	Y5	$\left\{0.4,0.5,0.6\right\}$	$\left\{0.3,0.4,0.6\right\}$	$\left\{0.3,0.5,0.6\right\}$	$\left\{0.4,0.5,0.6\right\}$	$\left\{0.5,0.6,0.7\right\}$

Table 8

Foreground value function matrix of some indicators.

Status	Option	C11	C12	C13	C14	C15
P1	Y1	0.1944	0.0000	0.0000	−0.6363	0.0000
	Y2	0.2426	0.1788	0.2828	0.0897	0.0897
	Y3	−0.0994	0.1788	0.2617	0.3168	−0.7405
	Y4	−0.5887	−0.2481	−0.4097	0.0813	−0.1829
	Y5	0.0000	−0.7799	−0.4810	−0.4373	−0.2018
P2	Y1	0.2138	0.0000	0.0000	−0.7799	0.0897
	Y2	0.2358	0.1944	0.3466	0.0000	0.0813
	Y3	0.0442	0.1318	0.1944	0.2426	−0.3457
	Y4	−0.5093	−0.2966	−0.5459	0.0000	0.1788
	Y5	−0.1829	−0.7127	−0.4810	−0.5459	−0.0994
P3	Y1	0.0000	−0.1829	0.0813	−0.2613	−0.1829
	Y2	0.1318	0.1103	0.1318	−0.1829	−0.2613
	Y3	0.1537	0.1821	−0.2018	0.1537	−0.1829
	Y4	−0.2481	0.0000	0.0897	−0.1829	0.1537
	Y5	−0.2481	−0.2966	−0.1829	0.0442	0.0442

Table 9

Comprehensive foreground value matrix.

	Y1	Y2	Y3	Y4	Y5
C11	0.1944	0.2426	−0.0994	−0.5887	0.0000
C12	0.0000	0.1788	0.1788	−0.2481	−0.7799
C13	0.0000	0.2828	0.2617	−0.4097	−0.4810
C14	−0.6363	0.0897	0.3168	0.0813	−0.4373
C15	0.0000	0.0897	−0.7405	−0.1829	−0.2018
C21	−0.4810	0.2138	−0.6363	0.1318	0.0000
C22	0.3168	−0.2613	−0.4024	0.0000	0.2617
C23	−0.6363	0.0813	0.1318	0.0000	−0.7127
C24	−0.1829	−0.5459	0.1162	0.0000	0.1788
C25	0.0000	0.2138	0.0813	−1.0046	−0.8632
C31	−0.5459	−0.6363	0.2987	0.0000	0.1788
C32	0.0000	−0.9273	0.2426	−0.2966	0.1537
C33	0.0000	−0.8770	−0.5842	0.1650	0.3168
C34	0.2828	−0.1829	0.4177	−0.4810	0.0000
C35	0.2596	−0.2481	0.2828	−0.3457	−0.2481
C41	0.1788	0.0000	0.0813	−0.8632	−0.6363
C42	0.1318	0.0000	0.0897	−0.6682	−0.4024
C43	0.1103	0.0000	−0.6363	−0.3714	0.0442
C44	0.4546	0.0813	0.0813	−0.1829	−0.2481
C45	−0.1829	0.2264	−0.4024	0.0000	0.0813
C46	0.3836	−0.4810	0.0000	0.1103	−0.2481

Table 10

Evaluation results and rank of the alternatives.

	Y1	Y2	Y3	Y4	Y5
$S_i$	0.3448	0.2644	0.5318	0.4857	0.5504
$R_i$	0.1310	0.0929	0.2154	0.1319	0.1595
$Q_i$	0.2960	0.0000	0.9675	0.5462	0.7718
Rank	2	1	5	3	4

Table 11

Ranking of different $\mathsf{\phi }$.

	Q1	Q2	Q3	Q4	Q5	Rank
0	0.3108	0.0000	1.0000	0.3185	0.5435	21453
0.1	0.3079	0.0000	0.9935	0.3640	0.5892	21453
0.2	0.3049	0.0000	0.9870	0.4096	0.6348	21453
0.3	0.3019	0.0000	0.9805	0.4551	0.6805	21453
0.4	0.2990	0.0000	0.9740	0.5006	0.7261	21453
0.5	0.2960	0.0000	0.9675	0.5462	0.7718	21453
0.6	0.2930	0.0000	0.9610	0.5917	0.8174	21453
0.7	0.2901	0.0000	0.9545	0.6373	0.8631	21453
0.8	0.2871	0.0000	0.9480	0.6828	0.9087	21453
0.9	0.2841	0.0000	0.9415	0.7284	0.9544	21435
1	0.2812	0.0000	0.9350	0.7739	1.0000	21435

Table 12

Sorting under $\mathsf{\xi }$ changes.

$\mathit{\xi }$	Rank
0	Y2 > Y1 > Y4 > Y3 > Y5
0.25	Y2 > Y1 > Y4 > Y3 > Y5 (When $\mathsf{\xi }$ = 0.25, Y3/Y5 reverses.)
0.5	Y2 > Y1 > Y4 > Y5 > Y3
0.75	Y2 > Y1 > Y4 > Y5 > Y3
1	Y2 > Y1 > Y4 > Y5 > Y3

Table 13

Sorting changes under $\mathsf{\lambda }$ perturbation.

$\mathbf{\lambda }$ Value	Y1	Y2	Y3	Y4	Y5	Rank
1.0	0.322	0.000	0.970	0.510	0.740	Y2 > Y1 > Y4 > Y5 > Y3
1.500	0.310	0.000	0.987	0.455	0.589	Y2 > Y1 > Y4 > Y5 > Y3
2.0	0.298	0.000	0.961	0.592	0.817	Y2 > Y1 > Y4 > Y5 > Y3
2.25	0.296	0.000	0.967	0.546	0.772	Y2 > Y1 > Y4 > Y5 > Y3
2.5	0.284	0.000	0.948	0.729	0.909	Y2 > Y1 > Y4 > Y3 > Y5
3.0	0.281	0.000	0.935	0.774	0.974	Y2 > Y1 > Y4 > Y3 > Y5

Table 14

Sorting changes under $\mathsf{\alpha }$ perturbation.

$\mathbf{\alpha }$	Y1	Y2	Y3	Y4	Y5	Rank
0.50	0.298	0.000	0.967	0.546	0.772	Y2 > Y1 > Y4 > Y5 > Y3
0.60	0.296	0.000	0.962	0.550	0.768	Y2 > Y1 > Y4 > Y5 > Y3
0.70	0.293	0.000	0.958	0.563	0.763	Y2 > Y1 > Y4 > Y5 > Y3
0.80	0.291	0.000	0.954	0.575	0.758	Y2 > Y1 > Y4 > Y5 > Y3
0.88	0.296	0.000	0.967	0.546	0.772	Y2 > Y1 > Y4 > Y5 > Y3
0.90	0.289	0.000	0.950	0.585	0.750	Y2 > Y1 > Y4 > Y5 > Y3

Table 15

Sorting changes under $\mathsf{\beta }$ perturbation.

$\mathbf{\beta }$	Y1	Y2	Y3	Y4	Y5	Rank
0.50	0.296	0.000	0.967	0.546	0.772	Y2 > Y1 > Y4 > Y5 > Y3
0.60	0.294	0.000	0.969	0.541	0.776	Y2 > Y1 > Y4 > Y5 > Y3
0.70	0.291	0.000	0.965	0.538	0.781	Y2 > Y1 > Y4 > Y5 > Y3
0.80	0.289	0.000	0.963	0.535	0.786	Y2 > Y1 > Y4 > Y5 > Y3
0.88	0.296	0.000	0.967	0.546	0.772	Y2 > Y1 > Y4 > Y5 > Y3
0.90	0.288	0.000	0.961	0.531	0.790	Y2 > Y1 > Y4 > Y5 > Y3

Table 16

Ranking results of the four methods.

	Y1	Y2	Y3	Y4	Y5
The method used in this article	0.2960	0.0000	0.9675	0.5462	0.7718
Rank	2	1	5	3	4
VIKOR method	0.2393	0.0000	0.7143	0.6287	0.6904
Rank	2	1	5	3	4
GRA-TOPSIS decision	0.6097	0.6542	0.4763	0.5223	0.4917
Rank	2	1	5	3	4
Prospect theory improvesTOPSIS method	0.5932	0.6310	0.4663	0.5271	0.5049
Rank	2	1	5	3	4

Appendix A

Table A1

The relevant information value of the indicator.

	Y1	Y2	Y3	Y4	Y5
Water quality condition	good	good	relatively good	relatively poor	medium
Topographic geological condition	relatively good	good	good	medium	relatively poor
Annual water seepage loss (/10,000 cubic meters)	73	58	64	80	79
Seismic intensity (/degree)	6	4	3	4	5
Average annual rainfall (/mm)	2004.3	1978.7	1665.2	2108.9	1883.9
Height-to-distance ratio	6.0	4.1	7.3	5.6	4.5
Mean head(/m)	699.6	485	400.5	493	558.3
Adjustable storage capacity (/10,000 cubic meters)	735	785	1008	774.9	660
Traffic condition	medium	relatively poor	relatively good	medium	relatively good
Construction and installation conditions	relatively good	good	good	relatively poor	relatively poor
Land expropriation (/hm2)	344.03	372.64	40.04	339	326.44
Relocation and resettlement population (/person)	200	778	0	240	71
Distance from the city (/km)	153	180	175	128	81
Investment per kilowatt (yuan/kw)	5033	5229	3424	5713	5222
Investment per unit of energy storage (yuan/kwh)	706	741	231	935	733
Distance to new energy base (/km)	70	140	90	170	160
Optimize the power supply structure	good	relatively good	medium	relatively poor	medium
Power reliability and stability	relatively good	good	relatively poor	medium	relatively good
New energy efficiency	good	medium	relatively good	relatively poor	relatively poor
Distance from substation (/km)	36	17	77	41	36
Annual emission reduction effect (ten thousand tons/year)	122.6	23.1	47.1	90	39.5

Appendix B

Table A2

Index correlation matrix.

	C11	C12	C13	C14	C15	C21	C22	C23
C11	1.0000	0.7059	0.7449	−0.2354	−0.2182	0.0151	0.2877	0.1515
C12	0.7059	1.0000	0.9050	0.4372	−0.3488	−0.3470	−0.3392	0.7234
C13	0.7449	0.9050	1.0000	0.4442	−0.4068	−0.0103	−0.4035	0.5508
C14	−0.2354	0.4372	0.4442	1.0000	−0.4857	−0.3106	−0.9885	0.7808
C15	−0.2182	−0.3488	−0.4068	−0.4857	1.0000	0.5213	0.4935	−0.6744
C21	0.0151	−0.3470	−0.0103	−0.3106	0.5213	1.0000	0.2084	−0.7568
C22	0.2877	−0.3392	−0.4035	−0.9885	0.4935	0.2084	1.0000	−0.6980
C23	0.1515	0.7234	0.5508	0.7808	−0.6744	−0.7568	−0.6980	1.0000
C24	−0.4125	−0.4125	−0.4431	0.1048	−0.6386	−0.5173	−0.1443	0.2077
C25	0.8344	0.9609	0.9365	0.3181	−0.4778	−0.2854	−0.2411	0.6443
C31	0.0185	0.4387	0.3036	0.653	−0.8687	−0.8263	−0.6050	0.8998
C32	−0.4285	−0.4091	−0.5732	−0.0319	−0.4616	−0.6652	0.0316	0.1949
C33	−0.7866	−0.9566	−0.9787	−0.4118	0.4713	0.1927	0.3485	−0.6422
C34	0.3201	0.5650	0.4842	0.5048	−0.9339	−0.7396	−0.4580	0.8456
C35	0.4318	0.7480	0.6135	0.5306	−0.8189	−0.7584	−0.4530	0.9145
C41	0.0011	−0.4666	−0.3701	−0.7488	0.8702	0.7518	0.7099	−0.9186
C42	0.9081	0.4036	0.4030	−0.6154	0.0288	0.0863	0.6618	−0.1771
C43	0.4372	−0.0673	0.1923	−0.5385	0.5183	0.8799	0.4840	−0.7093
C44	0.7647	0.6176	0.3965	−0.2691	−0.2831	−0.5583	0.3792	0.3693
C45	0.2065	−0.1960	0.0124	−0.5183	0.7385	0.9057	0.4685	−0.7795
C46	−0.0439	−0.1613	−0.5152	−0.5837	0.4616	−0.4011	0.6764	−0.1844
	C24	C25	C31	C32	C33	C34	C35	C41
C11	−0.4125	0.8344	0.0185	−0.4258	−0.7866	0.3201	0.4318	0.0011
C12	−0.4125	0.9609	0.4387	−0.4091	−0.9566	0.5650	0.7480	−0.4666
C13	−0.4431	0.9365	0.3036	−0.5732	−0.9787	0.4842	0.6135	−0.3701
C14	0.1048	0.3181	0.6530	−0.0319	−0.4118	0.5048	0.5306	−0.7488
C15	−0.6386	−0.4778	−0.8687	−0.4616	0.4713	−0.9339	−0.8189	0.8702
C21	−0.5173	−0.2854	−0.8263	−0.6652	0.1927	−0.7396	−0.7584	0.7518
C22	−0.1443	−0.2411	−0.6050	0.0316	0.3485	−0.4580	−0.4530	0.7099
C23	0.2077	0.6443	0.8998	0.1949	−0.6422	0.8456	0.9145	−0.9186
C24	1.0000	−0.3152	0.6043	0.9408	0.3589	0.5144	0.2953	−0.5484
C25	−0.3152	1.0000	0.4390	−0.3613	−0.9872	0.6342	0.7722	−0.4577
C31	0.6043	0.4390	1.0000	0.5544	−0.4155	0.9454	0.9019	−0.9893
C32	0.9408	−0.3613	0.5544	1.0000	0.4435	0.4373	0.2658	−0.4647
C33	0.3589	−0.9872	−0.4155	0.4435	1.0000	−0.5977	−0.7331	0.4570
C34	0.5144	0.6342	0.9454	0.4373	−0.5977	1.0000	0.9640	−0.9272
C35	0.2953	0.7722	0.9019	0.2658	−0.7331	0.9640	1.0000	−0.8881
C41	−0.5484	−0.4577	−0.9893	−0.4647	0.4570	−0.9272	−0.8881	1.0000
C42	−0.3669	0.5494	−0.2336	−0.3022	−0.4630	0.0667	0.1532	0.2967
C43	−0.6814	0.0289	−0.8336	−0.7539	−0.0672	−0.6302	−0.5811	0.7985
C44	−0.0458	0.6827	0.3273	0.1056	−0.5579	0.5189	0.6237	−0.2350
C45	−0.7444	−0.1715	−0.9449	−0.7647	0.1239	−0.8255	−0.7563	0.9049
C46	−0.0070	−0.2418	−0.1806	0.3236	0.3734	−0.2271	−0.1474	0.3022
	C42	C43	C44	C45	C46
C11	0.9081	0.4372	0.7647	0.2065	−0.0439
C12	0.4036	−0.0673	0.6176	−0.1960	−0.1613
C13	0.4030	0.1923	0.3965	0.0124	−0.5152
C14	−0.6154	−0.5385	−0.2691	−0.5183	−0.5837
C15	0.0288	0.5183	−0.2831	0.7385	0.4616
C21	0.0863	0.8799	−0.5583	0.9057	−0.4011
C22	0.6618	0.4840	0.3792	0.4685	0.6764
C23	−0.1771	−0.7093	0.3693	−0.7795	−0.1844
C24	−0.3669	−0.6814	−0.0458	−0.7444	−0.0070
C25	0.5494	0.0289	0.6827	−0.1715	−0.2418
C31	−0.2336	−0.8336	0.3273	−0.9449	−0.1806
C32	−0.3022	−0.7539	0.1056	−0.7647	0.3236
C33	−0.4630	−0.0672	−0.5579	0.1239	0.3734
C34	0.0667	−0.6302	0.5189	−0.8255	−0.2271
C35	0.1532	−0.5811	0.6237	−0.7563	−0.1474
C41	0.2967	0.7985	−0.2350	0.9049	0.3022
C42	1.0000	0.5385	0.7736	0.3482	0.2652
C43	0.5385	1.0000	−0.1009	0.9525	−0.1728
C44	0.7736	−0.1009	1.0000	−0.2677	0.4618
C45	0.3482	0.9525	−0.2677	1.0000	−0.0564
C46	0.2652	−0.1728	0.4618	−0.0564	1.0000

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