【導(dǎo)讀】流輸電系統(tǒng)設(shè)備的優(yōu)化配置問(wèn)題的有效性。這篇論文也談及了在文章中易于被忽視的概念和細(xì)節(jié)，因?yàn)?。?yōu)化算法的選擇在很大程度依賴于這些概念和細(xì)節(jié)。柔性交流輸電系統(tǒng)的優(yōu)化配置概念還處在一個(gè)相對(duì)早期的調(diào)查階段。個(gè)被普遍接受的方法，許多研究人員都聲稱他們的方法比其他人的方法更好。模大小以及操作條件。這篇論文為經(jīng)典算法和準(zhǔn)啟發(fā)式算法的性能比較提供了一個(gè)共同的背景。證實(shí)比其它的準(zhǔn)啟發(fā)式技術(shù)更加有效。一項(xiàng)基于穩(wěn)態(tài)和經(jīng)濟(jì)指標(biāo)的簡(jiǎn)單且符合實(shí)際的優(yōu)。于局部最優(yōu)與全局最優(yōu)問(wèn)題的探討。的主要算法是奔德斯分解算法，分支定界法和格莫瑞流割算法。性整數(shù)方程需要考慮到目標(biāo)函數(shù)和約束條件的使用。是，依賴系統(tǒng)參數(shù)的問(wèn)題的規(guī)模和非凸性是可能引起收斂問(wèn)題的關(guān)鍵。大多數(shù)情況下，目標(biāo)函數(shù)。結(jié)果是梯度下降法往往受到局部極值的限制。點(diǎn)不同的是，這個(gè)主題跟應(yīng)用到動(dòng)力系統(tǒng)中的不同目標(biāo)函數(shù)中的值并沒(méi)有關(guān)系。

　　

【正文】 ecision 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 number of units: 1≤ M ≤ 5. ? The size of each unit: 0≤ ηp ≤ 250 MVA. ? The desired voltage profile requires N additional restrictions defined as: ?? kV ?? ,?? k????1,2,..., N? (6) Each solution that does not satisfy the above constraints is considered infeasible. V. OPTIMIZATION ALGORITHMS For the optimal allocation of multiple FACTS units in a 45 bus system , three algorithms are fully developed and pared: Bender’s deposition, Bamp。B, and Enhanced PSO. A. Benders’ Deposition This method separates two sets of decisions that are made into two consecutive sta ges. In the first stage of the decision making , some of the constraints are delayed to reduce the plexity of the original (master) problem. In the second stage , some of the parameters that influence the decision, whose values were originally uncertain, are known and fixed after the first decision vector is found . Thus the secondary problem is reduced in plexity and in the number of variables [21], [22]. In the case of the STATCOM allocation problem, the master problem considers the decision vector in (5) that can be naturally separated into one subvector for selecting optimal locations and another subvector for choosing optimal sizes. The separation of the constraints can be stated as follows: ? First stage: sizes of the STATCOM units bee delayed constraints , thus the reactive power limits for these devices are relaxed in the solution of the power flow. The voltage reference is set to . for each STATCOM controller. The objective function corresponds to the voltage deviation metric defined in (1). ? Second stage: with the locations of the devices determined, the set of constraints is limited to those related to the maximum size of each unit . The objective function incl udes the voltage deviation metric and the cost metric as in (4). B. Branch and Bound Bamp。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