(1) Characterization of quantitative indicator

Uncertainties exist in the market environment like on-grid price, electricity sales, power supply cost, power sales price, etc., so each indicator is also uncertain. The uncertainty of each quantitative indicator is described in this paper using the interval number method. Set the interval of xij is [tijL,tijU], where tijL and tijU are lower and upper limits of xij.

(2) Characterization of qualitative indicators

This paper quantifies qualitative indicators using triangular fuzzy numbers [14,35]. Table 1 shows the fuzzy language and numbers.

Suppose that K experts are assigned to participate in scoring. The final evaluation results of all experts on the j-th index of the i-th project are as follows:

(1){bijL=1K∑k=1KbijLkbijM=1K∑k=1KbijMkbijU=1K∑k=1KbijUkwhere bijL, bijM and bijU are respectively the fuzzy attribute values of the j-th indicator of the i-th project, and the superscript k indicates the k-th expert.

This section transforms the above indicator values to [−1, 1] interval based on the RGPBIT operator. The main method is: when the indicator value is better than the average level, assign the index value as [0, 1]; When the index value is less than the average value, assign the index value as [−1, 0].

(1) RGPBIT of quantitative indicators

Let:

(2)Tj¯=12m∑i=1m(tijL+tijU),i=1,2,…,m;j=1,2,…,n

Among them, for benefit type indicators, it is:

(3){yijL=tijL−Tj¯|Tj¯|yijU=tijU−Tj¯|Tj¯|

For cost type indicators, it is:

(4){yijL=Tj¯−tijU|Tj¯|yijU=Tj¯−tijL|Tj¯|where yijL and yijU are the lower limit and upper limit of the j-th indicator of the i-th PGCP after linear transformation respectively. Since the values of yijL and yijU may be greater than 1 or less than −1, to keep their values in the interval [−1, 1], the following normalization formula is adopted:

(5){eijL=yijLmaxj{|yijL|,|yijU|}eijU=yijUmaxj{|yijL|,|yijU|}where eijL and eijU are upper and lower limits of the j-th indicator of the i-th PGCP after RGPBIT.

(2) RGPBIT of qualitative indicators

Let:

(6)vj=13m∑i=1m(bijL+bijM+tijU),i=1,2,…,m;j=1,2,…,n

Among them, for benefit type indicators, it is:

(7){eijL=bijL−vjmaxj{bijU∗−vj,vj−bijL∗}eijM=bijM−vjmaxj{bijU∗−vj,vj−bijL∗}eijU=bijU−vjmaxj{bijU∗−vj,vj−bijL∗}

For cost type indicators, it is:

(8){eijL=vj−bijUmaxj{bijU∗−vj,vj−bijL∗}eijM=bijM−vjmaxj{bijU∗−vj,vj−bijL∗}eijU=vj−bijLmaxj{bijU∗−vj,vj−bijL∗}where bijU∗=max1≤i≤m{bijU}, bijL∗=max1≤i≤m{bijL}, j=1,2,…,n.

Through the RGPBIT operator, the evaluation matrix A=(xij)m×n composed of original data is normalized to a normalized matrix E=(eij)m×n with an indicator interval of [−1, 1].

The best and worst schemes are given below for quantitative and qualitative indicators, respectively.

(1) The best and worst schemes of quantitative indicators (interval indicator)

Define:

(9)ej+=max{eijL+eijU2,1≤i≤m},j=1,2,…,n

The corresponding attribute value is recorded as [eijL(+),eijU(+)], and ej+ is the best scheme point of interval indicator.

Define:

(10)ej−=min{eijL+eijU2,1≤i≤m},j=1,2,…,n

The corresponding attribute value is recorded as [eijL(−),eijU(−)], and ej− is the worst scheme point of the interval indicator. (2)

The best and worst schemes of qualitative indicators (fuzzy indicator)

Define:

(11){eijL(+)=max1≤i≤m{eijL}eijM(+)=max1≤i≤m{eijM}eijU(+)=max1≤i≤m{eijU},j=1,2,…,n

The corresponding fuzzy attribute value is recorded as (eijL(+),eijM(+),eijU(+)), and ej+ is the best scheme point of fuzzy indicator.

Define:

(12){eijL(−)=min1≤i≤m{eijL}eijM(−)=min1≤i≤m{eijM}eijU(−)=min1≤i≤m{eijU},j=1,2,…,n

The corresponding fuzzy attribute value is recorded as (eijL(−),eijM(−),eijU(−)), and ej− is the worst scheme point of fuzzy indicator.

The best scheme point set and the worst scheme point set of each indicator constitute the attribute values of the best scheme and the worst scheme respectively. The vectors composed of the best attribute values and the worst attribute values are as follows:

The best attribute value vector:

(13)e+=(e1+,e2+,…,en+)

In the formula, if ej+ is the interval indicator, ej+=[eijL(+),eijU(+)], and if ej+ is the fuzzy indicator, ej+=(eijL(+),eijM(+),eijU(+)).

The worst attribute value vector:

(14)e−=(e1−,e2−,…,en−)

In the formula, if ej− is the interval indicator, ej−=[eijL(−),eijU(−)] and if ej− is the fuzzy indicator, ej−=(eijL(−),eijM(−),eijU(−)).

The closeness of each indicator to the best and worst scheme points reflects the advantages and disadvantages of the indicator. Based on this, this paper uses the gray correlation method [36,37] to characterize the closeness of each indicator to the best and worst scheme points, and constructs the decision-making matrix of alternatives.

If the index set of the i-th PGCP is si=(ei1,ei2,…,ein), e+=(e1+,e2+,…,en+) and e−=(e1−,e2−,…,en−) are the best scheme and the worst scheme, respectively, then the correlation coefficients ξ of the project indicator set with the best and worst scheme points are:

(15){ξij+=miniminj|eij−ej+|+ρmaximaxj|eij−ej+||eij−ej+|+ρmaximaxj|eij−ej+|ξij+=miniminj|eij−ej−|+ρmaximaxj|eij−ej−||eij−ej−|+ρmaximaxj|eij−ej−|,i=1,2,…,m;j=1,2,…,nwhere ρ is the resolution coefficient, and ρ∈(0,1). In this paper, ρ=0.5.

If eij is a quantitative indicator, then

(16){|eij−ej+|=12(eijL−eijL(+))2+(eijU−eijU(+))2|eij−ej−|=12(eijL−eijL(−))2+(eijU−eijU(−))2

If eij is a qualitative indicator, then

(17){|eij−ej+|=13(eijL−eijL(+))2+(eijM−eijM(+))2+(eijU−eijU(+))2|eij−ej−|=13(eijL−eijL(−))2+(eijM−eijM(−))2+(eijU−eijU(−))2

The positive correlation coefficient matrix ξ+=(ξij+)m×n and negative correlation coefficient matrix ξ−=(ξij−)m×n can be obtained accordingly.

In decision-making, the value function [39] is the value formed by subjective feelings. The commonly used value function is as follows:

(19)v(x)={xα,x≥0−θ(−x)β,x<0where parameter α and β respectively represent the concavity and convexity of the regional value power function of benefits and losses [40], reflecting the sensitivity of decision-makers to benefits and losses, and generally α,β≤1. θ reflects the degree of the decision-makers aversion to loss, and generally θ>1.

Generally, the decision-making weight [41] is determined according to Eq. (20):

(20)w(p)=pγ(pγ+(1−p)γ)1/γwhere γ represents the decision-maker’s risk attitudes towards benefits and losses.

The prospect theory shows that when facing the decision, the decision-maker will judge the benefits and losses of the decision according to the reference point value, so selecting the reference point is crucial to the results of the decision. In this paper, the best scheme and the worst scheme based on RGPBIT are used as reference points to judge the benefits and losses of decision results. Specifically, when the best scheme is taken as a reference point, the indicator values of the alternative are inferior to the best scheme, so the decision-maker is faced with losses, and at this time the decision-maker pursues risks. When the worst scheme is taken as a reference point, the indicator values of the alternative are superior to the worst scheme, so the decision-maker is faced with benefits, and the decision-maker is risk-averse at this point.

Therefore, using the prospect theory and the correlation coefficient of every indicator with the best/worst scheme, the prospect value of each indicator will be calculated.

The attribute value of the j-th indicator of the i-th PGCP is eij. If the reference point is ej−, the decision-maker is faced with benefits. If the reference point is ej+, the decision-maker will face losses. So the benefits and losses value function of eij is:

(21){v+(eij)=(1−ξij−),eij≥ej−v−(eij)=−θ[−(ξij+−1)]β,eij<ej+

In the formula, v+(eij) and v−(eij) are respectively the positive and negative prospect values brought by benefits and losses.

The positive and negative prospect value matrices of the project indicators can be obtained from the benefits and losses value functions, which are v+=(vij+)m×n and v−=(vij−)m×n, respectively.

According to Eq. (20), the indicator decision-making weights of the scheme benefits and losses are:

(22){w+(pj)=pjγ+[pjγ++(1−pj)γ+]1/γ+w−(pj)=pjγ−[pjγ−+(1−pj)γ−]1/γ−where γ+ and γ− respectively represent the risk attitude of decision-makers towards benefits and losses, and also reflect the bending degree of decision-making weight function.

The comprehensive prospect value Vi of the project si is the sum of positive and negative prospect values and their corresponding decision-making weights, that is:

(23)Vi=∑j=1nv+(eij)w+(pj)+∑j=1nv−(eij)w−(pj)

When performing MCDM issues, in order to make each alternative’s comprehensive prospect value larger, the optimal weight of each index is obtained through optimization methods. The optimization model is as follows:

(24a)maxV=∑i=1m∑j=1nv+(eij)w+(pj)+∑i=1m∑j=1nv−(eij)w−(pj)

(24b)s.t.{∑j=1npj=1pj≥0aj≤pj≤bj≤1i=1,2,…,m;j=1,2,…,n

In the formula, a and b are lower and upper limits for each indicator weight. The optimal weight p∗=[p1∗,p2∗,…,pn∗] can be obtained through the optimization method, then the comprehensive optimal prospect value of si is:

(25)Vi=∑j=1nv+(eij)w+(p∗)+∑j=1nv−(eij)w−(p∗)

The comprehensive optimal prospect values of various PGCPs are gained from Eq. (25), and the projects can be ranked from large to small based on the comprehensive optimal prospect values.

In this section, the typical IEEE-RTS79 node system [42] is applied for a case study to verify the applicability of the developed model. As a result of regional load growth, transmission lines and transformers need to be planned. The information on planned PGCPs (1, 2, 3, 4) is as follows. The marketization rate is set to be 0.3. Then, the project cost of the transmission line for one km is 1.4 million yuan and the project cost of the transformer is 400000 yuan/MVA [43]. The service life of the equipment is 18 years.

PGCP 1: There is an increased load of 136 MW on the bus 6. Therefore, it is proposed to construct a new 25.76 km transmission line, which connects node 6 and node 10. Besides, an 80 MVA transformer is added at the same time. The investment fee is 68.06 million yuan and the operation cost for the project is 5 million yuan/year.

PGCP 2: There is a changed load at bus 4, increasing from 74 to 285 MW. A new 44 km transmission line is proposed to be built from node 4 to node 9, and the transformer capacity is increased from 240 to 300 MVA. Then, the investment fee is 85.6 million yuan and the operation cost for the project is 5.9 million yuan/year.

PGCP 3: There is a changed load at bus 5, increasing from 71 to 300 MW. A new 36 km transmission line is proposed to be built from node 1 to node 5, and the transformer capacity is increased from 120 to 320 MVA. Then, the investment fee is 130.4 million yuan and the operation cost for the project is 5.2 million yuan/year.

PGCP 4: There is a changed load at bus 8, increasing from 171 to 270 MV, a new 26 km transmission line is proposed to be built between nodes 7 and 8, and the transformer capacity is increased from 200 to 300 MVA. The investment fee is 76.4 million yuan and the operation cost for the project is 5 million yuan/year.

In the indicator system in Fig. 1, PGCP early planning ability, PGCP participation team management and control ability, PGCP bidding control ability, technical innovation level of the project, and investment risk level are qualitative indicators, which are quantified by fuzzy methods according to expert scoring. For the rest quantitative indicators, the calculation of the network coordination indicator is the same as that in reference [44], and calculating the load loss rate is used to determine the reliability of the power supply. For indicators such as reduction of line loss rate, the power flow with/without the planning project is calculated in a given system, and then the indicator values are obtained accordingly. For the increase in electricity sales, based on the economic development situation, the gray prediction model is employed to predict the electricity demand, and the increased electricity amount is the increased electricity sold. For the investment income ratio index, the investment amount and income are calculated respectively by combining the life-cycle and uncertainty methods [45], and the ratio of the two is the investment income ratio. The calculation process and results of each indicator are shown in Appendix A, and the results of each indicator for investment ranking of PGCPs are obtained, as shown in Table 2.

(1) Results of RGPBIT

In the indicator system in Fig. 1, only the investment risk level is the cost type indicator, and other indicators are benefit type indicators. Firstly, the RGPBIT operator is carried out for the indicators, and the normalized matrix of the transformed indicator value belonging to [−1, 1] is obtained. The best and worst scheme points are obtained accordingly, and the correlation coefficient matrices of the project indicators and the ideal points are obtained. The results are shown in Table 3.

(2) Positive and negative prospect value matrices

The value function and decision-making weight parameters used in the prospect theory calculation are [40]: α=0.88, β=0.88, γ+=0.61, γ−=0.68. Substitute the correlation coefficient into the loss function and value function, to obtain the positive and negative prospect value matrices of all PGCPs, as shown in Table 4.

(3) Optimal decision-making weight and comprehensive prospect value

The optimal decision-making weight is calculated using Eq. (24), as shown in Table 5.

Further, according to Eq. (25), the comprehensive prospect values of PGCPs 1 to 4 are respectively 0.1554, −1.8594, −0.7078, and −0.5704, so the investment ranking of the four PGCPs should be V1≻V4≻V3≻V2. Among the 4 indicators with the highest optimal weights shown in Table 5, V1 performs best in c13, c14 and c22, and is only slightly lower than V4 in c11. That is, V1 has better network coordination, the lowest investment risk level, the highest power supply reliability, and the largest increase of electricity sales per unit power grid investment. On the contrary, V2 performs the worst in c11 and c22, and the second worst in c13 and c14. According to the ranking results, it can be found that the technical and benefits indicators, as the intuitive expression of the technical feasibility and economic sustainability of the PGCP, are the key considerations in the investment decision-making of PGCP. At the same time, management factors such as project importance and project maturity will play an important role in supporting decision-making for PGCPs’ investment ranking.

To further verify the effectiveness of the proposed model, three comparison models are designed in this section. The model information and corresponding evaluation results are shown in Table 6.

Based on the calculation results in Table 6, this section further tests the robustness and effectiveness of the proposed model. Among them, the robustness is analyzed by the ranking consistency test [46,47], and the effectiveness of the model is analyzed by the sample separation test [9,48]. The following indicators are constructed:

(26)rs=1−6×∑i=1N(xi−yi)2N×(N2−1)

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

(28)WS=1−∑i=1N(2−xi|xi−yi|max{|xi−1|,|xi−N|})where rs, rw and WS are the Spearman ranking correlation coefficient, weighted Spearman ranking correlation coefficient, and WS ranking correlation coefficient. xi and yi are two ranking results, and N is the alternatives’ total number. These three indicators reflect the similarity between the two ranking results. Larger indicator value reveals higher similarity between the ranking results. In this paper, if the rankings of the constructed model and the comparison model have high similarity, it indicates that the model in this paper is robust.

(29)σ=∑i=1m(Vi−V¯)2m

(30)θ=Vi,max−Vi,minV¯×100%

(31)ϑ=σV¯where σ, θ and ϑ are standard deviation, relative range, and coefficient of variation respectively, reflecting the degree of sample separation of the model. V¯ is the mean value of the comprehensive prospect value of the scheme, and Vi,max and Vi,min are maximum and minimum values of comprehensive prospect values. The larger the three indicators, the greater the difference between the comprehensive prospect values of different schemes, that is, the better the sample separation ability ofthe model.

According to the calculation results of the four models, the ranking consistency and sample separation index of each model can be calculated, as shown in Tables 7 and 8.

According to Tables 7 and 8, on the one hand, the ranking consistency test results show that the ranking results of Model 3 and Model 4 are identical to that of Model 1, and the ranking consistency coefficients are all 1. The ranking results of Model 2 and Model 1 are different, but the best and worst schemes are the same. Therefore, the ranking results of the three comparison models are highly consistent with those of Model 1, indicating that the constructed model is robust. On the other hand, the sample separation test results show that the three separation index values (or the absolute values of the index) of Model 1 are significantly greater than the values of the three contrast models, indicating that the constructed model can more clearly distinguish the sample differences, thus improving the decision-making efficiency. Next, this section further compares and analyzes the results of the four models.

(1) Comparison and analysis with traditional prospect theory model

This paper adopts the prospect theory method improved by RGPBIT operator. Compared with the traditional prospect theory method, RGPBIT combines reward and punishment, overcoming the shortcomings of the existing method of “only rewarding but not punishing”.

The results based solely on prospect theory are shown in the second row of Table 6. It can be seen that the ranking result based on the comprehensive prospect value of the traditional prospect theory model is V1≻V3≻V4≻V2. The difference from the proposed RGPBIT prospect theory model lies in PGPCs 3 and 4, and the numerical range of the comprehensive prospect value based solely on prospect theory is smaller than the proposed method in this paper.

Table 9 represents the prospect values of each PGPC indicator for the two methods (Model 1 and Model 2). It can be seen that under the effect of “punishing the good”, the prospect values of network coordination, power supply reliability, and investment income ratio of PGCP 3 are obviously smaller than those based on traditional prospect theory. However, only one indicator named power supply reliability of PGCP 4 is significantly less than the result based on traditional prospect theory. Therefore, it can be seen that the proposed RGPBIT prospect theory model in this paper not only considers the “rewarding the good”, but also takes into account the “punishing the bad”. Compared with the traditional prospect theory results based on “rewarding the good”, the investment ranking results of PGCPs may be different.

(2) Comparative analysis with utility theory model

The investment ranking results of PGCPs based on utility theory are further given in this paper, as shown in the third row of Table 6. The utility theory takes the decision-maker as a rational person to make rational judgments on various risks in decision-making problems, and then make decisions [49,50]. The common point of this method with the proposed RGPBIT prospect theory is that decision-makers risk preference is taken into account when making investment decisions. The difference is that in RGPBIT prospect theory, decision-makers have different risk attitudes towards benefits and losses, while the utility theory judges whether the decision-maker is risk-seeking, risk aversion, or risk neutral according to the expected value calculated from the utility function. Generally, in most cases, the results of prospect theory are consistent with those of utility theory [51,52]. It can be seen from Table 6 that the ranking of four PGCPs based on comprehensive utility values is V1≻V4≻V3≻V2, which is consistent with the results of the RGPBIT prospect theory, which shows the feasibility of the methods proposed in this paper.

(3) Comparison and analysis with TOPSIS model

According to TOPSIS, the positive ideal solution is the optimal solution, and its attribute values all reach the optimal value. The negative ideal solution is the worst solution with all indicator values reaching the worst. This method compares each alternative with the positive and the negative ideal solutions, and the alternative closest to the positive one and farthest from the negative one is the optimal solution [53,54].

The idea of the TOPSIS method is the same as that of the prospect theory method. The difference is that the prospect theory uses the value function and decision-making weight to reflect the decision-maker’s risk attitude towards benefits and losses, to determine the comprehensive prospect value. The ranking method of TOPSIS uses the distance degree from the ideal solution. It is noted from Table 6 that the rankings of TOPSIS are consistent with that of the proposed RGPBIT prospect theory model in this paper, both of which are V1≻V4≻V3≻V2.