(I1)Respond to peak demand (I2)Capacity factor (I3)LCOE (I4)Service life (I5)Land use (I6)GHG emissions (I7)Social acceptance (I8)Social risks (I9)Environmental risks (I10)Large hydropower80 [80-95]5 [4-5]45 [35-56]0.047 [0.044-0.049]50 [30-80]160 [2-750]18.5 [2-75]1 [1-2]5 [4-5]5 [4-5]Small hydropower80 [70-90]1 [1-2]45 [34-56]0.06 [0.02-0.10]35 [30-60]0.5 [.003-1]5 [0.3-13]3 [2-4]3 [2-4]3 [2-4]Solar PV15 [15-18]1 [1-2]18 [14-18]0.062 [0.045-.068]25 [20-35]20 [1-64]48 [9-300]5 [4-5]1 [1-2]2 [2-3]Onshore wind power35 [30-40]1 [1-2]25 [20-32]0.05 [0.033-0.083]25 [20-25]100 [72-137]11 [7-124]5 [4-5]2 [1-2]1 [1-2]Bioenergy50 [35-60]3 [3-4]64 [40-85]0.057 [0.04-0.09]20 [20-30]900 [72-2200]230 [14-1000]4 [3-5]3 [2-4]4 [3-5]An element Xij in Eq. (2.1) represents the value of the ith alternative Ai, for the jth indicator Ij.
b. The normalization of decision matrix as presented by rij is shown below(2.2)
c. The rij is now multiplied to respective weights (Wj) of indicators to obtain the matrix vij as calculated in Eq. (2.3).(2.3)Wj in Eq. (2.3) is the corresponding weight of the indicator
d. The best solution (V+) and worst solutions (V−) is then calculated as shown in below equations:(2.4)(2.5)J is the set of beneficiary indicators and j’ is of non-beneficiary indicators.
e. The distance of separation of each alternative from the best and the worst solution is calculated as:(2.6)(2.7)
f. The corresponding closeness of the alternative Aij from the best solution is calculated as:(2.8)
g. Finally, according the value of Ri in descending order the alternatives are ranked.
2.4.3.2 The Fuzzy-TOPSIS
The fuzzy-TOPSIS, a well-known MCDM method, was developed by [87]. Fuzzy-TOPSIS is extensively used in the various fields for ranking where decision makers give judgement in the form of crisp value especially for qualitative indicators. In qualitative approaches, the decisions are usually expressed in the form of linguistic terms (e.g., low, high, very high), and due to the possibility of vagueness in a human decision, uncertainty is always associated with the result. Fuzzy combined with the MCDM method has proven to be useful in handling qualitative indicators with associated uncertainties [53, 54].
Before elaborating the steps of fuzzy-TOPSIS. The basic theory of fuzzy is explained as follows:
(i) The membership function for a triangular fuzzy number à given by (a1, b1, c1) is defined as:(2.9)
(ii) The distance between two triangular fuzzy numbers à = (a1, b1, c1) and is given by Eq. (2.10):(2.10)
(iii) The multiplication of fuzzy triangular numbers is given by Eq. (2.11):(2.11)
(iv) The addition of fuzzy triangular numbers is given by Eq. (2.12):(2.12)
Based on the basic fuzzy theory explained above. The fuzzy-TOPSIS method steps are described as below:
1 (a) Establish a decision matrix choosing the linguistic values for alternatives (i = 1, 2, … m) with respect to indicator (j = 1, 2,…n).
2 (b) Choose the suitable linguistic variables for the weights of the indicators.
3 (c) The fuzzy linguistic value are in range of [0, 1]; thus, normalization is not required.
4 (d) The weighted decision matrix is calculated.
5 (e) Calculate the positive ideal (V+) and negative ideal solutions (V-) as below:J is the set of beneficiary indicators and J’ is of non-beneficiary indicators(2.13)(2.14)
6 (f) The distance of each alternative from V+ and V- is calculated as:(2.15)(2.16)
7 (g) Determine the corresponding closeness to the best solution as below:(2.17)
8 (h) Finally, according the value of Ri in descending order the alternatives are ranked.
2.5 Ranking of RE Technologies
2.5.1 The TOPSIS
As per methodology described in Section 2.5.3, Table 2.3 is input decision matrix. Following the steps (b) – (f) as mentioned in Section 2.5.3, the results obtained are presented as Table 2.4.
2.5.2 The Fuzzy-TOPSIS
To address the uncertainties associated with qualitative data fuzzy-TOPSIS is used for ranking. Table 2.3 will be the input decision matrix for fuzzy-TOPSIS. Since indicator’s ratings are not in the form of fuzzy linguistic variables, normalization in the range of [0, 1] is done using equations (2.18) and (2.19), and presented as Table 2.5.
(i) For the beneficial indicators:(2.18)
(ii) For the non-beneficial indicators:(2.19)
In the next step, decision matrix with a fuzzy linguistic variable has to be established using triangular membership functions defined as shown in Figure 2.2.
Table 2.5 is now transformed to Table 2.6 as described with an example. If the numeric rating of an indicator value is 0.50, the equivalent fuzzy linguistic value then will be “Medium”. Table 2.6 now is converted to Table 2.7 the fuzzy decision matrix using Figure 2.2. Table 2.7 presents the fuzzy decision matrix. As per step (e) of Section 2.4, V+ and V- are defined as Vj+ = (0,0,0) and Vj- = (1,1,1) for beneficial indicators and Vj- = (0, 0, 0) and Vj+
