7 Stakeholders’ Desires 96 [ ( X ⊗Y ) ⊕ ( X ⊗Y ) ⊕... ⊕ ( X Y ) ] 1 = 1 i1 2 i2 Nst i N for i=1,…, Nsd (7.3) st N U i ⊗ st with: U i importance degree of desire i from the viewpoint of transmission planners Nst number of stakeholder groups Fuzzy arithmetic operations are defined using α-cuts of fuzzy intervals [74]-[75]. If Xj and Yij are substituted by triangular fuzzy numbers i.e. Xj=(aj, bj, cj) and Yij=( oij, pij, qij), U i is approximated by (li, mi, ni) [27]. where: 1 Nst li = ∑ N st j= 1 1 o Nst mi = ∑ N st j= 1 1 Nst ni = ∑ N st j= 1 ij q p ij a ij c j b j , i=1,…, Nsd (7.4) j , i=1,…, Nsd (7.5) , i=1,…, Nsd (7.6) Appropriateness degree of plan k versus combination of all decision criteria is equal to weighted mean of k N i , i.e.: k k k [ ( U ⊗ N ) ⊕ ( U ⊗ N ) ⊕... ⊕ ( U N ) ] 1 = 1 1 2 2 Nd N for k=1, …, Np (7.7) N k Fap ⊗ sd sd with: k F ap fuzzy appropriateness index of plan k versus combination of all decision criteria Np with considering importance degree of stakeholders in decision making number of expansion plans 7.4.2 Selecting the Final Plan The expansion plan which has the greatest fuzzy appropriateness index is the optimal plan. As chapter 6, convex combination of left and right integral value [27], [75]-[76], centroid indices [77], and extended centroid index [77] are used for ranking fuzzy numbers.

7 Stakeholders’ Desires 97 7.5 Case Study: IEEE 30 Bus Test System The proposed approach is applied to IEEE 30 bus test system [62], [64]. Figure 5.1 shows the single line diagram of IEEE 30 bus system. Data of generators, loads, and transmission lines are given in tables 5.2-5.4. Consider the single scenario case which was described in section 5.3.1. PDFs of LMPs were computed for the peak load of planning horizon of existing network in section 5.3.1. If between each two buses that have average LMP difference greater than $5/MWhr a new transmission line suggested as expansion candidate, we have 89 decision alternatives including alternative “do nothing”. Stakeholders and their desires are weighted according to tables 7.2 and 7.3. Importance degrees of decision criteria (desires) from the viewpoint of transmission planners ( U i ) are obtained by aggregating tables 7.2 and 7.3. Table 7.4 shows the importance degrees of decision criteria from the viewpoint of transmission planners. In this table the importance degree of each criterion is a triangular fuzzy number. Appropriateness degrees of expansion plans versus competition, reliability, flexibility of operation, network charge, and environmental impacts are computed using the criteria described in section 7.2. Average congestion cost is used to measure the competition. Columns 3-7 of table 7.5 show the appropriateness degrees of expansion plans versus different decision criteria. Fuzzy appropriateness index ( F ) was computed by aggregating importance degrees of decision criteria from the viewpoint of transmission planners (table 7.4) and appropriateness degrees of expansion plans versus decision criteria (columns 3-7 of table 7.5). Fuzzy appropriateness indices are shown in column 8 of table 7.5. Fuzzy appropriateness indices were ranked using different methods. Convex combination of right and left integral values with α=0.5 is shown in column 9 of table 7.5. All ranking method show that plan 3 i.e. line 1-10 has the greatest fuzzy appropriateness index and is selected as optimal plan. If the capacity of this line be greater than 325 MW, then the probability of violating its limit is less than one percent. Table 7.4 - Importance degrees of decision criteria form viewpoint of transmission planners Desire Importance degree Competition (0.1125, 0.3125, 0.5375) Reliability (0.1250, 0.3875, 0.6625) Flexibility of Operation (0.0750, 0.2875, 0.6000) Network Charge (0.1125, 0.3250, 0.6250) Environmental Impacts (0.0375, 0.1250, 0.3250) k ap