【正文】 B is a classical approach to search for an optimal solution by evaluating only a subset of the total possible solutions. The main steps in the algorithm are [18], [21], [22]: ? Branching: the set of feasible solutions is partitioned into simpler subsets. At each iteration, one of the promising subsets is chosen and an effort is made to find the best feasible solution within it. ? Bounding: the algorithm proceeds to find upper and lower bounds for the optimal objective value. There is only one upper bound u at each stage, which corresponds to the lowest among the objective values of all the feasible solutions that have appeared so far. ? Pruning: if at certain a stage, one of the subsets has a lower bound which is greater than the current upper bound, then the algorithm prunes (discards) that set. Branching, bounding and pruning are repeated until the optimal solution is found. For this particular problem, the objective function is defined as in (4). The branching strategy corresponds to the “depthfirst search”: for each subset of feasible locations,branching is performed by dividing progressively the STATCOM size intervals into smaller subintervals. The bounding and pruning strategies help to narrow the search by discarding as many subintervals as possible until the optimal value, for the particular subset of feasible locations, is the next stage another subset of feasible locations is chosen, and the process is repeated until the set of all feasible locations is covered. C. Enhanced PSO 1) Canonical PSO formulation The PSO algorithm considers that each particle represents a potential solution to the problem, thus the particles are defined as the decision vector in (5). The quality of the solution, that allows the best position for each particle and the swarm to be determin ed, is assessed using the fitness function defined in (4). At each iteration, t, the position of each particle is determined by [23], [24]: ? ? ? ? ? ?tvtxtx iii ??? ??? 1 (7) The velocity of each particle is determined by both the individual and group experien ces: ? ? ? ? ? ?? ? ? ?? ?1.. .11 2211 ???????????? txprctxprctvwtv ijiiiii (8) where w1 is a positive number between 0 and 1, c1 and c2 are the cognitive and social acceleration constants respectively, r1 and r2 are random numbers with uniform distr ibution in the range of [0, 1] . Finally, pi is the individual best position found by the corresponding particle and pg is the global best position found by the entires warm. To avoid the divergence of the swarm, a maximum velocity for each dimension of the problem hyperspace is defined ( maxv ). Additionally, since integer variables are in cluded in the optimization problem , the IntegerPSO version is used , where the par ticle’s position is rounded off to the nearest integer [24]. The PSO parameters used in this study are [1]: Parameter Optimal value Inertia constant (wi) Linear decrease ( to ) Individual acceleration constant (c1) Social acceleration constant (c2) Vmax for bus location 9 Vmax for STATCOM size 50 2) EnhancedPSO For this particular application, the canonical PSO 。 therefore an exhaustive manual search can be performed to find the global optimum, (ii) the problem has a reduced, scattered and non convex feasible region , and (iii) only a steady state criterion is considered to avoid possible discrepancies if a transient nalysis was also to be included [20]. A. Objective Function Two goals are considered: (i) to minimize voltage deviations in the system and (ii) to minimize the cost. Thus, two metrics J1 and J2 are defined as in (1) and (3). ? ?? ?? N kVJ 1 21 (1) where J1 is the voltage deviation metric, kV is the . value of the voltage at bus k and N is the total number of buses. The total cost function, Ctotal, consists of two ponents: a fixed cost per unit that is installed in the system and a variable cost that is a linear function of each unit size: ?????? Mp pvft ot al CMCMC 1)( ? (2) where M is the number of units to be allocated, Cf is the fixed cost per unit, Cv is the cost per MVA, and??p is the size in MVA of unit p. Since Cf Cv, it is convenient to normalize each term of the cost function prior to its inclusion in the objective function: M V AaxMMMCCMCMCJMppvMppvff_1m a xm a xm a x1m a x2 ???????? ????????? (3) where J2 is the cost metric, maxM is the maximum number of STATCOM units to be allocated, and? max? ?is the maximum size in MVA of each STATCOM unit. The multiobjective optimization problem can now be defined using the weighted sum of both metrics J1 and J2 to create the overall objective function J shown in (4) 2211 JJJ ???? ?? (4) The weight for each metric is adjusted to reflect the relative importance of each goal. In this case, considering the maximum magnitudes of J1 and J2, it is decided to assign values of? 1? = 1 and? 2? = , such that both metrics have equal importance. B. Decision Variables The decision variables are the location of the STATCOM units and their sizes. These variables can be arranged in a vector as: ? ?MMix ???? .. .11? (5) where? p? , p=1...M, is the location (bus number)of STATCOM unit p. All pon ents of the decision vector are integer numbers, thus Mi Z2?? ??C. Constraints There are several constraints in this problem regarding the characteristics of the power system and the desired voltage profile. Each constraint represents a limit in the search space, which in this particular case corresponds to: Generator buses are omitted from the search process since they have voltage regulators to regulate the voltage. ? Bus numbers are limited to {1, 2,…, N}. ? Only one unit can be connected at each bus. ? The