The derivative value of SG investment is a derivative system related to power grid investment, and specifically refers to the current and potential value derived from the original assets, which is reflected in the security, environmental protection, economy, and technological progressiveness of the power grid. According to the evolution of energy and power system in complex environments, the value derivative chain and mechanism of power grid investment in stimulating industrial development, promoting technological progress, and renewable consumption, thus supporting dual carbon targets are significant, which is the specific derivative value brought by SG investment. Generally speaking, the SG investment derivative value criteria should be applied to government regulators, power industry associations, and relevant business departments of power grid enterprises. For each application scenario, the following SG investment derivative value criteria should be followed: 1)

Support the strategic layout of power grid enterprises and power grid construction. Under the dual carbon goal orientation, SG investment derivative value evaluation can guide the productivity layout, planning and construction of relevant business departments of power grid enterprises, optimize the power grid investment structure and mode to meet the infrastructure needs for the low-carbon transformation of electricity, and carry out the value evaluation of investment projects and apply them in power grid investment, to provide support for the sustainable and stable operation and development of enterprises.

The construction of indicators is the top priority in the field of evaluation. Its suitability not only determines the scope of application of the evaluation, but also affects the feasible value of the evaluation. In fact, the influencing factors of the derivative value of SG investment are complex. “Dual carbon” goal puts forward new requirements for energy transformation. In addition, it is necessary to ensure the calculability, subjectivity and objectivity of the evaluation indicators. Therefore, considering the connotation of SG investment derivative value under the dual carbon target, this paper draws on relevant literature [8,16,18] and measures the derivative value of SG investment from three aspects: power grid investment performance, operation and maintenance performance, and sustainable development performance. The index framework is shown in Fig. 1.

According to Fig. 1, the power grid investment performance index is the intuitive embodiment of the derivative value of SG investment, reflecting the direct economic and physical benefits of SG investment, including revenue rate per unit investment, voltage qualification rate per unit investment, reserve rate per unit investment, increased load per unit investment and system overload reduction per unit investment. The operation and maintenance performance index indicates the reliability guarantee generated by SG investment, including user failure self-healing rate, equipment failure rate, and time delay rate of information transmission. The sustainable development performance index represents the development potential of SG investment, including renewable energy investment ratio, electric vehicle investment ratio, and primary energy efficiency. The constructed SG investment derivative value evaluation index system takes into account the multiple goals of SG development under the dual carbon goal, that is, SG investment needs to enhance the strong level and power supply guarantee ability of the power grid system while obtaining appropriate economic benefits, so that the power grid can withstand the impact of large-scale renewable energy uncertainty, and thus ensure the large-scale consumption of renewable energy from the infrastructure level. The connotation and calculation method of each secondary indicator are shown in Table 1, and the names of relevant parameters are shown in Table 2.

The framework of the SG investment derivative value evaluation model proposed in this paper is shown in Fig. 2. Based on the evaluation index system shown in Fig. 1, the hybrid MCDM model includes two parts:

Part 1: Indicator weighting. A weighting model integrating subjective and objective weighting methods (called AEW and LBWA method) is developed in this paper based on the constructed evaluation index system. Firstly, based on all indicators’ values of multiple alternatives, the anti-entropy values of each indicator is calculated, and then the anti-entropy values are normalized to obtain the objective weights. Secondly, invite experts to judge the importance of each indicator, layer the indicators based on the judgment results and construct an LBWA judgment vector, and then calculate the influence function of each indicator. Based on this, calculate the subjective weight based on LBWA. Finally, an optimization model is constructed by minimizing the heterogeneity between objective and subjective weights, and the integrated weights can be obtained to evaluate the derivative value of SG investment.

Part 2: MCDM. The constructed indicators are quantitative and this paper aims to evaluate the derivative value of multiple SG investments, so an MCDM model based on the MARCOS technique is proposed. Firstly, based on the values of multiple alternatives on all indicators, a normalized evaluation matrix is constructed, and then combined with the weight results, a weighted decision matrix is constructed. Secondly, according to the weighted decision matrix, the utility functions of each alternative for anti-ideal and ideal solutions are calculated as the distance. Finally, calculate the comprehensive utility function of each alternative, and rank them accordingly.

After the evaluation index is constructed, the index weighting method is the key to the quantitative analysis of the derivative value of SG investment. To meet the multiple development goals of the future power grid and implement the action plan of the “Dual Carbon” goal, the index weight needs to take into account the nature of the index itself and the difference of the original data. Therefore, this paper proposes a combined weighting method based on AEW and LBWA, which ensures the reliability of index weight results through the fusion of subjective and objective ideas.

(1) Objective weighting method based on AEW

In this paper, the AEW method is used to determine the objective weight of the index. Entropy is a concept used to measure the disorder degree of a system in a thermodynamic system. When there are m states in the system, and the probability of each state is pj (j=1,2,…,m), then entropy is:

(1)h=−∑j=1mpjlnpjwhere 0≤pj≤1 and ∑jpj=1.

For AEW, assuming that there are m objects and n indicators for one MCDM issue, the index value can be expressed as xij (i=1,2,…,n, j=1,2,…,m), and the corresponding decision matrix can be written as X=[xij]n×m. Therefore, the anti-entropy value of each index can be expressed as:

(2)hi=−∑j=1mrijln⁡(1−rij)where rij=xij/∑jxij represents the dimensionless value of the index value. By normalizing the anti-entropy value of the index, the AEW weight of the index (w1i) can be obtained:

(3)w1i=hi/∑ihi

(2) Subjective weighting method based on LBWA

LBWA method is a subjective weighting method proposed by Serbian scholars Žižović et al. in 2019 [40]. Based on defining the most important indicators among the indicator system, the LBWA method uses it as a benchmark to layer other indicators according to the degree of importance, and then sort the importance of indicators within the layer and make adjacent comparisons and judgments. This method can effectively deal with the inconsistency of ranking caused by a large number of indicators through the index importance stratification, and simplify the index importance comparison process through the index importance ranking within the layer and adjacent comparison judgment. In recent years, the LBWA method has attracted the attention of relevant scholars and has been applied to many MCDM fields, such as renewable energy alternative evaluation, and offshore wind farm location decisions [41,42]. In this paper, the basic steps of using the LBWA method to determine the subjective weight of indicators are as follows:

Step 1: determine the optimal index. According to expert opinions, determine the index with the greatest importance in the index set C={C1,C2,…,Cn} and define it as the optimal index CB.

Step 2: index stratification. The remaining indicators except the optimal indicator are divided into different layers according to the importance of the indicators. The hierarchical basis is as follows:

S1 Layer: among the remaining indicators, indicators with the same importance as the optimal indicator or indicators less than two times (excluding two times) the importance of the optimal indicator are divided into this layer.

S2 Layer: among the remaining indicators, the indicators that are two to three times less important than the optimal indicator (excluding three times) are divided into this layer.

Sk Layer: among the remaining indicators, indicators that are k times to k+1 times less important than the optimal indicator (excluding k+1 times) are divided into this layer.

Through the above indicator stratification, it can have a rough definition of the importance of indicators. Assuming that for any indicator Ci, its importance can be expressed as S(Ci), then for the divided layer Sk (k=1,2,…,K), there is:

(4)S=S1∪S2∪⋯∪SK

And for any p,q∈{1,2,…,K}, if p≠q, then Sp∩Sq=∅.

Step 3: hierarchical judgment of index importance. In any layer Sk={Ck1,Ck2,…,CkNk}, define the optimal index CkB in the layer, and then compare the importance of other indexes in the layer with the optimal index, and the result is recorded as Ikik. Ikik indicates the importance of the k-th index compared with the optimal index of the k-th layer. The greater the importance of the k-th index, the smaller the value of Ikik, and Ikik is defined as an integer in the interval [0,r]. In particular, when k=1, there is C1B=CB and I1B=0. r is a constant determined by the hierarchical results of index importance, that is:

(5)r=max{|S1|,|S2|,…,|SK|}where |Sk| represents the number of indicators in the set Sk.

Step 4: determine the LBWA elasticity coefficient r0. The constant r represents the maximum difference in the judgment of the importance of indicators in the layer. According to the value of r, the elasticity coefficient of LBWA r0 is defined to meet r0>r. According to Žižović et al. [40], and Torkayesh et al. [43], it can be r0=r+1.

Step 5: calculate the influence function f(Ckik) of each index, and the formula is as follows:

(6)f(Ckik)=r0k×r0+Ikikwhere Ckik represents the ik-th index in the k-th layer.

Step 6: calculate the index weight. According to f(Ckik), the weight of the optimal index can be expressed as:

(7)w2B=1∑k=1K∑ik=1Nkf(Ckik)

Further, the weights of other indicators are:

(8)wCkik=f(Ckik)×w2B

So far, the subjective weights of all indicators under the LBWA method are obtained, which are recorded as w2i.

(3) Subjective and objective integrated weighting method

The fundamental principle of integrating weights is to minimize the heterogeneity between the objective weights and the subjective weights [44]. Thus, the subjective and objective integrated weights can be obtained by solving the following optimization problems:

(9)min∑i=1n[(w0i−w1i)2×(w0i−w2i)2]s.t.∑i=1nw0i=1where w0i is the normalized subjective and objective integrated weight of the i-th index.

In this paper, the evaluation object of SG investment derivative value is the power grid of multiple regions, so it can be studied through the attribute integration method with the ranking function. TOPSIS method is a common multi-alternative ranking method. The traditional TOPSIS takes the relative closeness degree as the basis for alternative ranking, while the relative closeness degree of TOPSIS only considers the relative distance between the alternative and the ideal solution, and is prone to the vertical problem [45]. Based on this, this paper adopts the evaluation method named measurement of alternatives and ranking according to compromise solution (MARCOS), which is developed in 2020 [46]. The specific steps of MARCOS are as follows:

Step 1: Construct the initial decision matrix. Set a multi-criteria model with n indicators and m alternatives. According to the actual date of each alternative, the evaluation matrix is obtained.

Step 2: Construct the extended initial matrix X=[xij]m×n. The ideal solution (AI) and anti-ideal solution (AAI) are defined to extend the initial matrix. The ideal solution AI has the best characteristics, and the anti-ideal solution AAI has the worst. (10)X=c1c2⋯cnAAIA1A2⋯AmAI(xaa1xaa2x11x12x21x22⋯⋯xm1xm2xai1xai1⋯xaan⋯x1n⋯x2n⋯⋯⋯xmn⋯xai1)

According to the attribute of indicators, the values of AAI and AI are:

(11)AAI=minixij,ifj∈B andmaxixij,ifj∈C

(12)AI=maxixij,ifj∈B andminixij,ifj∈Cwhere B represents the benefit type indicator, and C represents the cost type indicator.

Step 3: Calculate the normalized decision matrix N=[nij]m×n. The normalized decision matrix is obtained by normalizing the initial matrix:

(13)nij=xijxai,if j∈B

(14)nij=xaixij,if j∈C

Step 4: Determine the weighted normalized decision matrix V=[vij]m×n. It is obtained by multiplying the normalized decision matrix and weight vector:

(15)vij=nij×wj

Step 5: Calculate the utility degree of the alternative Ki relative to the anti-ideal and ideal solutions:

(16)Ki−=SiSaai

(17)Ki+=SiSaiwhere S(i=1,2,…,m) represents the sum of elements in the weighting matrix V, which is calculated by:

(18)Si=∑j=1nvij

Step 6: Determine the utility function f(Ki) of the alternative by the following formula:

(19)f(Ki)=Ki++Ki−1+1−f(Ki+)f(Ki+)+1−f(Ki−)f(Ki−)where f(Ki−) is the utility function of the alternative for the anti-ideal solution, and f(Ki+) is the utility function of the alternative for the ideal solution. Their expressions are:

(20)f(Ki+)=Ki−Ki++Ki−

(21)f(Ki−)=Ki+Ki++Ki−

Step 7: Rank the alternatives according to the utility function value in Eq. (19). The final realization takes the alternative with the highest utility function value as the ideal alternative.

(1) AEW weighting results

The basic information of the examined 7 regional SGs is listed in the Appendix (Table A1). Collect the original data of each regional SG on each index (the descriptive statistical results are listed in Table 3), calculate the anti-entropy value of each grid on each index, as shown in Table 4, and then calculate the anti-entropy weight of each index according to Eqs. (2) and (3), as shown in Fig. 3.

It can be seen in Fig. 3 that the c31 has the biggest AEW weight, followed by c32 and c12. According to the principle of AEW, it can be explained that the performance differences of the 7 regional SGs to be evaluated are the greatest in the above three indicators, that is, different SGs shows significant differences in the promotion effect on the emerging source (renewable energy) and loads (electric vehicles), as well as the support effect on system stability. Therefore, from a differentiation-driven perspective, SG investment derivative value should focus on these indicators with significant differences.

(2) LBWA weighting results

In order to calculate the LBWA weight of the index system, 9 experts from government agencies, power grid enterprises and research institutions are invited to determine the optimal index in the index system, and conduct index stratification and judgment. According to experts’ opinions, the renewable energy investment ratio (c31) is the optimal index. The remaining indicators can be divided into three layers according to their importance relative to the optimal indicator:

S1={c31,c11,c21,c33}

S2={c12,c14,c22,c23,c32}

S3={c13,c15}

According to Eq. (5), there is:

r=max{|S1|,|S2|,|S3|}=5, r0=r+1=6

Then, the importance judgment results of all indicators relative to the optimal indicator are obtained according to expert opinions, that is:

S1 Layer: I31=0, I11=3, I21=2, I33=4;

S2 Layer: I12=3, I14=5, I22=4, I23=1, I32=2;

S3 Layer: I13=1, I15=2.

Based on Eq. (6), the influence function of each index is calculated as:

S1 Layer: f31=61×6+0=1, f11=61×6+3=23, f21=61×6+2=34, f33=61×6+4=35;

S2 Layer: f12=62×6+3=25, f14=62×6+5=617, f22=62×6+4=38, f23=62×6+1=613, f32=62×6+2=37;

S3 Layer: f13=63×6+1=619, f15=63×6+2=310.

Furthermore, according to Eq. (7), the weight of the optimal index can be calculated as:

w31=1∑f=15.6505=0.1770

Finally, the weight results of other indicators can be obtained according to Eq. (8). To sum up, the LBWA weights of the SG investment derivative value evaluation index system are shown in Fig. 4. It can be seen from Fig. 4 that the indicator with the highest weight is c31, followed by c21 and c11, indicating that from the perspective of the indicator meanings under the dual carbon goal, the promotion of renewable energy investment, the improvement of self reliability, and reasonable revenues are the key points that need to be paid attention to in the derivative value of SG investment. It should be noted that there are differences in the weight results between AEW and LBWA. This is because the two weighting methods have different starting points. AEW only considers the differences in indicator data, while LBWA only considers the meaning of the indicator itself. Both weight results have certain limitations, so it is necessary to combine the two weight results to improve the reliability of the weight results.

(3) Combined weighting results

On the basis of determining AEW weight and LBWA weight, MATLAB programming is used to solve Eq. (9), and the subjective and objective integrated weight is obtained:

w0=[0.1069,0.0790,0.0686,0.0720,0.0663,0.1160,0.0758,0.0846,0.1480,0.0837,0.0991]

It can be seen that the four indicators with the largest weight are renewable energy investment ratio (c31), user failure self-healing rate (c21), revenue rate per unit investment (c11), and primary energy efficiency (c33), with weights of 0.1480, 0.1160, 0.1069 and 0.0991, respectively. Among them, two indicators reflect the sustainable development performance of SG, one indicator reflects the investment performance of SG, and one indicator reflects the operation and maintenance performance of SG, indicating that for SG investment, the sustainable development value can better reflect its derivative value. Specifically, under the “dual carbon” goal, the scale of renewable energy is increasing. On the one hand, the SG investment is required to support renewable energy access to the power grid to a greater extent; on the other hand, the SG is required to have stronger fault self-healing ability (an important direction of the development of power grid intelligence), while maintaining a good investment revenue and ensuring a stable source of investment. Therefore, the future development of SG should pay more attention to the promotion of renewable energy consumption, and further improve the level of intelligence and pay attention to investment benefits.

According to Eqs. (11)~(14), standardize the original data, so as to process the index data into the form of a positive correlation with the evaluation goal. Then, the standardized decision matrix can be obtained, as shown in Table 5.

Combined with the index weighting results, the weighted normalized decision matrix is calculated, as shown in Table 6.

According to Table 6, the ideal and negative ideal solutions are obtained as follows:

AI=(0.1069,0.0790,0.0686,0.0720,0.0663,0.1160,0.0758,0.0846,0.1480,0.0837,0.0991)

AAI=(0.0805,0.0383,0.0431,0.0533,0.0464,0.0846,0.0368,0.0485,0.0331,0.0368,0.0883)

According to the weighted normalized decision matrix, the compromise values of MARCOS for each alternative relative to the ideal and negative ideal solutions are determined, and the utility function results of each SG relative to the ideal and negative ideal solutions are calculated by using Eqs. (16)~(21), so as to judge the derivative value of SG investment. The evaluation results of the derivative value of SG investment based on MARCOS are shown in Table 7.

It can be seen that among the seven regional SGs, SG3 has the largest utility value, indicating that the derivative value of the grid investment is the largest; SG7 has the smallest utility value, so its investment derivative value is the smallest. In terms of the performance of secondary indicators, among the four secondary indicators with the largest weight, SG3 performs best in c11, c21 and c31, and second best in c33. In comparison, SG7 performs the worst in c21, c31 and c33, and the third worst in c11.

Therefore, when SG investment performs poorly in promoting renewable energy consumption and improving primary energy efficiency, its investment will significantly reduce the driving force for future sustainable development, making grid investment lack derivative value. In addition, SG investment also needs to be able to improve its fault resistance, which is reflected in the fact that the power grid can achieve self-healing through intelligent ability in the event of a fault. Finally, on the basis of promoting the consumption of renewable energy and improving its own fault repair ability, SG investment also needs to pay attention to the economy of investment, which is an important guarantee to ensure that the power grid has an adequate and stable source of investment funds.

In order to verify the effectiveness of the constructed SG investment derivative value evaluation model, three comparative models are designed to carry out the ranking consistency test and sample separation test. The set comparison model is shown in Table 8, in which Model 1 is the model proposed in this paper.

(1) Ranking consistency test

Ranking consistency is an important index to reflect the robustness of MCDM methods. According to references [47,48], this paper constructs the following ranking consistency index:

(22)rS=1−6×∑i=1N(xi−yi)2N×(N2−1)

(23)rW=1−6×∑i=1N(xi−yi)2[(N−xi+1)+(N−yi+1)]N×(N3+N2−N−1)

(24)WS=1−∑i=1N(2−xi|xi−yi|max{|xi−1|,|xi−N|})where rS, rW and WS are Spearman ranking correlation coefficient, weighted Spearman ranking correlation coefficient, and WS ranking correlation coefficient, respectively. xi and yi are two ranking results, and N is the number of alternatives in the ranking. The larger the correlation coefficient is, the closer the two ranking results are.

Based on the basic data of the above-mentioned 7 SGs on various indicators, the ranking results of the 7 SG investment derivative value evaluations under each comparison model are obtained, and then the ranking consistency of the three comparison models relative to Model 1 is calculated. The results are shown in Figs. 5 and 6.

According to Fig. 6, the values of the three comparison models on the three ranking consistency indicators are greater than 0.8, and the values of the other ranking consistency indicators are greater than 0.85 except for the WS of Model 4, indicating that the ranking results of the comparison model are highly consistent with those of the model proposed in this paper, which further verifies that the model in this paper has high robustness and the model ranking results are relatively reliable.

(2) Sample separation test

Sample separation test is an important means to judge the effectiveness of the ranking results of the MCDM model. According to references [49,50], the following four indicators for the sample separation test are set, named standard deviation (σ), relative range (θ), coefficient of variation (ϑ) and sensitivity (η):

(25)σ=∑j=1m(δj−δ¯)2m

(26)θ=δj,max−δj,minδ¯×100%

(27)ϑ=σ/δ¯

(28)η=δj,max−δj,secδj,maxwhere δj is the MARCOS comprehensive utility value of each alternative, δ¯ is the average comprehensive utility value, m represents the number of entities and m=7 in this section. δj,max and δj,min are the maximum and minimum values of the comprehensive utility value, respectively, and δj,sec is the second largest value in δj. The larger the above four indicators, the better the sample separation degree of the MCDM method used.

Similarly, this section uses the three comparison models in Tables 7 and 8 SGs to test the sample separation, and the results are shown in Table 9. It can be seen from Table 9 that the four sample separation indexes of Model 1 are greater than those of the other three models, indicating that compared with the comparison model, the hybrid MCMD model proposed in this paper has better performance in sample differentiation and can better reflect the differences in the derivative value of different SG investments. Therefore, the SG investment derivative value evaluation model based on AEW-LBWA weighting and MARCOS proposed in this paper can better improve the decision-making efficiency on the premise of ensuring robustness.