L6 Backtracking And Search
算法设计与分析(王晓东)
a b a b
（2）方法重载：Java允许方法重载，即允许定义有不同签名的同名方法。
上述方法ab可重载为：
public static double ab(double a, double b) { return (a+b+Math.abs(a-b))/2.0; } 12
4.异常
1.3 描述算法
6
1.2 表达算法的抽象机制
2.抽象数据类型
抽象数据类型是算法的一个数据模型连同定义在该模型上 并作为算法构件的一组运算。
抽象数据类型带给算法设计的好处有：
（1）算法顶层设计与底层实现分离； （2）算法设计与数据结构设计隔开，允许数据结构自由选择； （3）数据模型和该模型上的运算统一在ADT中，便于空间和时间耗费的折衷； （4）用抽象数据类型表述的算法具有很好的可维护性； （5）算法自然呈现模块化； （6）为自顶向下逐步求精和模块化提供有效途径和工具； （7）算法结构清晰，层次分明，便于算法正确性的证明和复杂性的分析。
中国计算机学会 “21世纪大学本科计算机专业系列教材”
算法设计与分析
王晓东 编著
1
主要内容介绍
• • • • • • 第1章 第2章 第3章 第4章 第5章 第6章 算法引论 递归与分治策略 动态规划 贪心算法 回溯法 分支限界法
Unit+6+Nurturing+Nature+Understanding+ide课文知识点逐句讲解
so/
did so 这样做 that 引导定语从句，修饰care care 关注，关爱，关心 deserve 值得，应受，应得
译文： 我很们自很豪 自我豪们我建们了建我了们世界“上不可“能不”可能建的成的”的铁铁路路，， 过并程且中我我们们带也着对关环注境这给样予做了，应环有境的应关得注到。这种关注
Language points
_S_i_t_ti_n_g__(sit) back in my seat, I can’t quite believe that I’m about to travel along the railway __th_a_t_/_ many foreign experts claimed was “ impossible”.which quite =very 十分，非常 引导定从，修饰railway be about to do sth. 将要做，正要做 在从句中做claim的宾语 be about to do sth. ... when正要做某事，这时 claim 宣称，认领
15. be _l_o_c_a_t_ed___ in/at位于，坐落于
16.thin air 稀薄的空气
17. _ch__a_n_g_a_b_le___ weather多变的天气
18. present 导致，带来，引发
Thanks for your attention
massive 巨大的 wind v.蜿蜒，迂回，缠绕 wind one’s way蜿蜒前行 particularly=_i_n_ particular 尤其，特别 roof 屋顶
译文： 当我旅行穿过“世界屋脊”时，“一条条巨龙翻山 越岭”这句歌词似乎特别生动。
1.quite =very 十分，非常 2.be about _to__d_o_ ...w_h_e_n__ 正要干某事...这时 3.claim 宣称，认领 4.have been doing 一直做 5.race v. 快速移动，快速运转,比赛
宽度优先搜索详解
宽度优先搜索详解宽度优先搜索（Breadth First Search， BFS）是一种用来遍历或搜索图形或树数据结构的算法。
该算法以广度为优先，从根节点开始，依次访问同层节点，直到遍历完整个图形或树。
本文将详细介绍宽度优先搜索的原理、应用场景以及实现方法。
一、原理解析宽度优先搜索主要基于队列数据结构实现，其具体流程如下：1. 将根节点（起始节点）放入队列中；2. 当队列不为空时，执行以下步骤：a. 取出队首元素进行访问；b. 将当前节点的所有相邻未访问过的节点加入队列；c. 标记当前节点为已访问；3. 重复步骤2，直到队列为空。
宽度优先搜索的核心思想是在同一层级的节点访问完之后才会继续访问下一层级的节点，确保了先广度后深度的遍历顺序。
二、应用场景宽度优先搜索在图形和树等数据结构中有广泛的应用。
以下是一些常见的应用场景：1. 最短路径问题：当图中每条边的权重相等时，宽度优先搜索可以用来求解起点到终点的最短路径。
2. 连通性问题：宽度优先搜索可以用来判断两个节点之间是否存在路径联通。
3. 键值搜索：对于带有层次结构的数据，如树结构或图像中的像素布局，宽度优先搜索可以帮助我们在最短时间内找到目标节点。
4. 社交网络分析：在社交网络中，宽度优先搜索可以用来寻找两个人之间的熟人关系链，或者寻找某个人的最近邻居。
5. 游戏路径搜索：在一些游戏中，如迷宫游戏或棋盘游戏，宽度优先搜索可以用来寻找到达目标位置的最短路径。
三、实现方法以下是宽度优先搜索的一种实现方法（以无向图为例）：```pythonfrom collections import dequedef bfs(graph, start):visited = set() # 用于记录已访问的节点queue = deque([start]) # 使用双端队列作为辅助数据结构visited.add(start) # 将起始节点标记为已访问while queue:node = queue.popleft() # 取出队首节点print(node) # 访问节点的操作for neighbor in graph[node]: # 遍历当前节点的相邻节点if neighbor not in visited:queue.append(neighbor) # 将未访问过的节点加入队列visited.add(neighbor) # 标记为已访问```上述代码中，`graph`表示无向图的邻接表表示，`start`表示起始节点。
6自由度机械手的算法
6自由度机械手的算法介绍6自由度机械手是一种具有6个自由度的机械臂，可以在空间中完成复杂的运动任务。
为了实现机械手的精确控制和运动规划，需要使用一系列算法来实现。
本文将探讨6自由度机械手的算法，包括逆运动学、正运动学、轨迹规划等。
逆运动学逆运动学是指已知机械手末端位置和姿态，计算出各个关节角度的过程。
对于6自由度机械手而言，逆运动学问题是一个复杂的数学问题。
以下是逆运动学算法的基本步骤：1.确定机械手的DH参数，包括关节长度、关节偏移、关节旋转角度等。
2.根据机械手的DH参数，构建正运动学方程，即末端位置和关节角度的关系。
3.根据末端位置和姿态，求解正运动学方程，得到关节角度的解。
4.对于多解的情况，选择最优解，例如使关节角度变化最小或满足特定约束条件的解。
正运动学正运动学是指已知机械手各个关节角度，计算出末端位置和姿态的过程。
对于6自由度机械手而言，正运动学问题相对简单，可以通过矩阵变换来实现。
以下是正运动学算法的基本步骤：1.确定机械手的DH参数。
2.根据机械手的DH参数，构建正运动学方程，即关节角度和末端位置的关系。
3.根据关节角度，求解正运动学方程，得到末端位置的解。
轨迹规划轨迹规划是指在给定起始位置和目标位置的情况下，确定机械手的运动路径和速度的过程。
对于6自由度机械手而言，轨迹规划需要考虑运动的平滑性和避免碰撞等因素。
以下是轨迹规划算法的基本步骤：1.确定起始位置和目标位置。
2.根据起始位置和目标位置，计算出机械手的途径点和运动方向。
3.根据途径点和运动方向，生成平滑的运动路径。
4.考虑机械手的运动速度和加速度，生成合适的速度曲线。
5.考虑碰撞检测，避免机械手和其他物体的碰撞。
动力学建模动力学建模是指根据机械手的结构和参数，建立机械手的运动学和动力学模型的过程。
对于6自由度机械手而言，动力学建模需要考虑关节间的耦合效应和惯性等因素。
以下是动力学建模的基本步骤：1.确定机械手的质量、惯性等参数。
prefix beam search原理
prefix beam search原理
Prefix Beam Search是一种改进的搜索算法，用于在语言模型中寻找最优输出序列。
其核心原理在于减少Beam Search中候选序列的重复部分，从而提高搜索效率。
首先，了解Beam Search的基本概念是必要的。
Beam Search是一种启发式搜索算法，用于在图形模型（如隐马尔可夫模型和条件随机场）中寻找最优序列。
它通过在每个时间步保留最有前景的n个部分序列（即beam宽度），而不是穷举所有可能的序列，从而在合理的时间内找到近似最优解。
然而，当使用Beam Search与语言模型结合时，会遇到一个问题：在搜索过程中，不同的候选序列可能会有很多相同的前缀部分。
这意味着算法在多个序列上重复计算相同部分的得分，造成了不必要的计算浪费。
为了解决这个问题，Prefix Beam Search被提出。
这种算法的核心思想是将共有的前缀部分从各个候选序列中分离出来，只保留差异化的部分进行扩展和评分。
这样，算法就不需要对每个候选序列从头开始计算，而是从共享的前缀开始，只对剩余部分进行搜索和评分。
这种方法显著减少了计算量，尤其是在处理长序列时更为有效。
Prefix Beam Search通过避免重复计算共有前缀，提高了搜索效率，特别是在处理大规模语言模型时，这种优化可以大幅度减少计算资源的消耗，同时保持了搜索质量。
动态对等的最佳策略
Electric Power Systems Research 84 (2012) 58–64Contents lists available at SciVerse ScienceDirectElectric Power SystemsResearchj o u r n a l h o m e p a g e :w w w.e l s e v i e r.c o m /l o c a t e /e p srDynamic equivalence by an optimal strategyJuan M.Ramirez a ,∗,Blanca V.Hernández a ,Rosa Elvira Correa b ,1a Centro de Investigación y de Estudios Avanzados del I.P.N.,Av del Bosque 1145.Col El Bajio,Zapopan,Jal.,45019,MexicobUniversidad Nacional de Colombia –Sede Medellín.Facultad de Minas.Carrera 80#65-223Bloque M8,114,Medellín,Colombiaa r t i c l ei n f oArticle history:Received 21May 2011Received in revised form 26September 2011Accepted 29September 2011Keywords:Power system dynamics Equivalent circuitPhasor measurement unit Stability analysisPower system stabilitya b s t r a c tDue to the curse of dimensionality,dynamic equivalence remains a computational tool that helps to analyze large amount of power systems’information.In this paper,a robust dynamic equivalence is proposed to reduce the computational burden and time consuming that the transient stability studies of large power systems represent.The technique is based on a multi-objective optimal formulation solved by a genetic algorithm.A simplification of the Mexican interconnected power system is tested.An index is used to assess the proximity between simulations carried out using the full and the reduced model.Likewise,it is assumed the use of information stemming from power measurements units (PMUs),which gives certainty to such information,and gives rise to better estimates.© 2011 Elsevier B.V. All rights reserved.1.IntroductionOne way to speed up the dynamic studies of currently inter-connected power systems without significant loss of accuracy is to reduce the size of the system model by means of dynamic equiva-lents.The dynamic equivalent is a simplified dynamic model used to replace an uninterested part,known as an external part,of a power system model.This replacement aims to reduce the dimension of the original model while the part of interest remains unchanged [1–6].The phrases “Internal system”(IS)and “external system”(ES)are used in this paper to describe the area in question,and the remaining regions,respectively.Boundary buses and tie lines can be defined in each IS or ES.It is usually intended to perform detailed studies in the IS.However,the ES is important to the extent where it affects IS analyses.The equivalent does not alter the transient behavior of the part of the system that is of concern and greatly reduces the dimen-sion of the network,reducing computational time and effort [4,7,8].The dynamic equivalent also can meet the accuracy in engineering,achieving effective,rapid and precise stability analysis and security controls for large-scale power system [4,8].However,the determi-∗Corresponding author.Tel.:+523337773600.E-mail addresses:jramirez@gdl.cinvestav.mx (J.M.Ramirez),bhernande@gdl.cinvestav.mx (B.V.Hernández),elvira.correa@ (R.E.Correa).1Tel.:+57314255140.nation of dynamic equivalents may also be a time consuming task,even if performed off-line.Moreover,several dynamic equivalents may be required to represent different operating conditions of the same system.Therefore,it is important to have computational tools that automate the procedure to evaluate the dynamic equivalent [7].Ordinarily,dynamic equivalents can be constructed follow-ing two distinct approaches:(i)reduction approach,and (ii)identification approach.The reduction approach is based on an elimination/aggregation of some components of the existing model [4,5,9].The two mostly found in the literature are known as modal reduction [6,10]and coherency based aggregation [2,11,12].The identification approach is based on either parametric or non-parametric identification [13,14].In this approach,the dynamic equivalent is determined from online measurements by adjusting an assumed model until its response matches with measurements.Concerning the capability of the model,the dynamic equivalent obtained from the reduction approach is considerably more reli-able and accurate than those set up by the identification approach,because it is determined from an exact model rather than an approximation based on measurements.However,the reduction-based equivalent requires a complete set of modeling data (e.g.model,parameters,and operating status)which is rarely avail-able in practice,in particular the generators’dynamic parameters [5,13,15,16].On the other hand,due to the lack of complete system data,and/or frequently variations of the parameters with time,the importance of estimation methods is revealed noticeably.Especially,on-line model correction aids for employing adaptive0378-7796/$–see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.epsr.2011.09.023J.M.Ramirez et al./Electric Power Systems Research84 (2012) 58–6459Fig.1.190-buses46-generators power system.controllers,power system stabilizers(PSS)or transient stability assessment.The capability of such methods has become serious rival of the old conventional methods(e.g.the coherency[11,12] and the modal[6,10]approaches).The equivalent estimation meth-ods have spread,because it can be estimated founded on data measured only on the boundary nodes between the study system and the external system.This way,without any need of informa-tion from the external system,estimation process tries to estimate a reduced order linear model,which is replaced for the external part.Evidently,estimation methods can be used,in presence of perfect data of the network as well to compute the equivalent by simulation and/or model order reduction[15].Sophisticated techniques have become interesting subject for researchers to solve identification problems since90s.For example, to obtain a dynamic equivalent of an external subsystem,an opti-mization problem has been solved by the Levenberg–Marquardt algorithm[17].Artificial neural networks(ANN)are the most prevalent method between these techniques because of its high inherent ability for modeling nonlinear systems,including power system dynamic equivalents[15,18–25].Power system real time control for security and stability has pro-moted the study of on-line dynamic equivalent technologies,which progresses in two directions.One is to improve the original off-line method.The mainstream approach is to obtain equivalent model based on typical operation modes and adjust equivalent parame-ters according to real time collected information[4].Distributed coordinate calculation based on real time information exchanging makes it possible to realize on-line equivalence of multi-area inter-connected power system in power market environment[4].Ourari et al.[26]developed the slow coherency theory based on the struc-ture preservation technique,and integrated dynamic equivalence into power system simulator Hypersim,verifying the feasibility of on-line computation from both computing time and accuracy[27].Prospects of phasor measurement technique based on global positioning system(GPS)applied in transient stability control of power system are introduced in Ref.[4].Using real data collected by phasor measurement unit(PMU),with the aid of GPS and high-speed communication network,online dynamic equivalent of interconnected power grid may be achieved[4].In this paper,the dynamic equivalence problem is formulated by two objective functions.An evolutionary optimization method based on genetic algorithms is used to solve the problem.2.PropositionThe main objective of this paper is the external system’s model order reduction of an electrical grid,preserving only the frontier nodes.That is,those nodes of the external system directly linked to nodes of the study system.At such frontier nodes,fictitious gen-erators are allocated.The external boundary is defined by the user. Basically,it is composed by a set of buses,which connect the exter-nal areas to the study system.There is not restriction about this set. Different operating conditions are taken into account.work reductionAfirst condition for an equivalence strategy is the steady state preservation on the reduced grid;this means basically a precise60J.M.Ramirez et al./Electric Power Systems Research 84 (2012) 58–64Fig.2.Proposed strategy’s flowchart.voltage calculation.In this paper,all nodes of the external system are eliminated,except the frontier nodes.By a load flow study,the complex power that should inject some fictitious generators at such nodes can be calculated.The nodal balance equation yields,j ∈Jp ij +Pg i +Pl i =0,∀i ∈I(1)where I is the set of frontier nodes;J is the set of nodes linked directly to the i th frontier node;p ij is the active power flowing from the i th to j th node;Pg i is the generation at the i th node;Pl i is the load at the i th node.Thus,the voltages for the reduced model become equal to those of the full one.For studies where unbalanced conditions are important,a similar procedure could be followed for the negative and zero equivalent sequences calculation.2.2.Studied systemThe power system shown in Fig.1depicts a reduced version of the Mexican interconnected power system.It encompasses 7regional systems,with a generation capacity of 54GW in 2004and an annual consumption level of 183.3TWh in 2005.The transmis-sion grid comprises a large 400/230kV system stretching from the southern border with Central America to its northern interconnec-tions with the US.The grids at the north and south of the country are long and sparsely interconnected transmission paths.The major load centers are concentrated on large metropolitan areas,mainly Mexico City in the central system,Guadalajara City in the western system,and Monterrey City in the northeastern system.The subsystem on the right of the dotted line is considered as the system under study.Thus,the subsystem on the left is the exter-nal one.There are five frontier nodes (86,140,142,148and 188)and six frontier lines (86–184,140–141,142–143,148–143(2)and 188–187).Thus,the equivalent electrical grid has five fictitious generators at nodes 86,140,142,148and 188.Transient stabil-ity models are employed for generators,equipped with a static excitation system;its formulation is described as follows,dıdt=ω−ω0(2)dωdt=1T j [Tm −Te −D (ω−ω0)](3)dE q dt=1T d 0 −E g −(x d −x d )i d +E fd(4)dE ddt =1T d 0−E d +(x q −xq )i q (5)dE fd dt=1T A−E fd +K A (V ref +V s −|V t |) (6)where ı(rad)and ω(rad/s)represent the rotor angular positionand angular velocity;E d (pu)and E q(pu)are the internal transient voltages of the synchronous generator;E fd (pu)is the excitationvoltage;i d (pu)and i q (pu)are the d -and q -axis currents;T d 0(s)and T q 0(s)are the d -and q -open-circuit transient time constants;x’d (pu)and x q(pu)are the d -and q -transient reactances;Tm (pu)and Te (pu)are the mechanical and electromagnetic nominal torque;Tj is the moment of inertia;D is the damping factor;K A and T A (s)are the system excitation gain and time constant;V ref is the voltage reference;V t is the terminal voltage;V s is the PSS’s output (if installed).The corresponding parameters are selected as typical [28].2.3.FormulationGiven some steady state operating point (#CASES )the following objective functions are defined,min f =[f 1f 2]f 1=#CASESop =1w 1opNg intk =1ωk ori (t )−ωkequiv (x,t )2(7)f 2=#CASESop =1w 2opNg intk =1Pe kori (t )−Pe k equiv (x,t )2(8)subject to:0=Ngen eqj =1H j −ni =1i ∈LH i(9)where ωk ori is the time behavior of the angular velocity of those generators in the original system,that will be preserved (Ng int),J.M.Ramirez et al./Electric Power Systems Research84 (2012) 58–6461Fig.3.Fitness assignment of NSGA-II in the two-objective space.Fig.4.Case1:from top to bottom(i)angular position37(referred to slack);(ii)angular speed28;(iii)electrical torque41,after a three-phase fault at bus172.62J.M.Ramirez et al./Electric Power Systems Research 84 (2012) 58–64Fig.5.Case 3:from top to bottom (i)angular position 40(referred to slack);(ii)angular speed 34;(iii)electrical torque 39,after a three-phase fault at bus 144.after a disturbance within the internal area;ωk equiv is the time behavior of the angular velocity of those generators in the equiv-alent system,after the same disturbance within the internal area;Pe k ori is the time behavior of the electrical power of those gen-erators of the original system within the internal area;Pe k equiv is the time behavior of the electrical power of those generators in the equivalent system within the internal area;H k is the k th genera-tor’s inertia;L is the set of generators that belong to the external system;Ngen eq is the number of equivalent generators [16,25].The set of voltages S ={V i ,V j ,...,V k |complex voltages stemming from PMUs }has been included in the solution.The main challenge in a multi-objective optimization environ-ment is to minimize the distance of the generated solutions to the Pareto set and to maximize the diversity of the developed Pareto set.A good Pareto set may be obtained by appropriate guiding of the search process through careful design of reproduction oper-ators and fitness assignment strategies.To obtain diversification special care has to be taken in the selection process.Special care is also to be taken to prevent non-dominated solutions from being lost.Elitism addresses the problem of losing good solutions dur-ing the optimization process.In this paper,the NSGA-II algorithm (Non dominated Sorting Genetic Algorithm-II)[29–31]is used to solve the formulation.The algorithm NSGA-II has demonstrated to exhibit a well performance;it is reliable and easy to handle.It uses elitism and a crowded comparison operator that keeps diver-sity without specifying any additional parameters.Pragmatically,it is also an efficient algorithm that has shown better results to solve optimization problems with multi-objective functions in a series of benchmark problems [31,32].There are some other meth-ods that may be used.For instance,it is possible to use at least two population-based non-Pareto evolutionary algorithms (EA)and two Pareto-based EAs:the Vector Evaluated Genetic Algorithm (VEGA)[36],an EA incorporating weighted-sum aggregation [33],the Niched Pareto Genetic Algorithm [34,35],and the Nondom-inated Sorting Genetic Algorithm (NSGA)[29–31];all but VEGA use fitness sharing to maintain a population distributed along the Pareto-optimal front.3.ResultsIn this case,the decision variables,x ,are eight parametersper each equivalent generator:{x d ,x d ,x q ,x q ,T d 0,T q 0,H,D }.In this paper,for five equivalent generators,there are 40parameters to be estimated.Likewise,in this case,a random change in the load of all buses gives rise to the transient behavior.A normal distribution with zero mean is utilized to generate the increment (decrement)in all buses.The variation is limited to a maximum of 50%.The disturbance lasts for 0.12s and then it is eliminated;the studied time is 2.0s.To attain more precise equivalence for severe operating conditions,this improvement could require load variations greater than 50%.However,this bound was used in all cases.Fig.2depicts a flowchart of the followed strategy to calculate an optimal solution.In this paper,three operating points are taken into account:(i)Case 1,the nominal case [37];(ii)Case 2,an increment of 40%in load and generation;(iii)Case 3,a decrement of 30%in load and gen-eration.To account for each operating condition into the objective functions,the same weighted factors have been utilized (w i =1/3),Eqs.(7)–(8).Table 1summarizes the estimated parameters for five equiv-alent generators,according to the two objective functions.TheJ.M.Ramirez et al./Electric Power Systems Research84 (2012) 58–6463 Table1Parameters of the equivalent under a maximum of50%in load variation.G equiv1G equiv2G equiv3G equiv4G equiv5f1f2f1f2f1f2f1f2f1f2x d0.1080.104 2.140 2.100 1.920 1.9100.1590.1790.3240.362x d0.1130.1130.8190.8690.3960.3960.08760.132 1.900 1.950T d010.7010.7039.6039.6018.8018.7011.5011.5011.7011.70x q0.7100.7500.5300.5190.2880.308 2.470 2.4600.7050.803x q0.3950.3850.8740.8050.9010.9260.9220.9530.8820.884T q0 4.950 4.82033.3033.40 4.280 4.10016.9016.9012.8012.90H 4.610 4.61022.2722.2747.2647.2669.2369.2333.9333.93D18.0318.40535.6536.6239.2239.141.9042.00705.1705.2 electromechanical modes associated to generators of the internalsystem are closely preserved.These generators arefictitious andbasically are useful to preserve some of the main interarea modesbetween the internal area and the external one[16].Thus,in order to avoid the identification of the equivalent gener-ators’parameters based on a specific disturbance,in this paper theuse of random changes in all the load buses is used.This will giverise to parameters valid for different fault locations.The allowedchange in the load(in this paper,50%)will result in a slight varia-tion of transient reactances.Further studies are required to assesssensitivities.Fig.3shows a typical Pareto front for this application.The NSGA-II runs on a Matlab platform and the convergence lastsfor3.25h for a population of200individuals and20generations.It is assumed that phasor measurement units(PMUs)areinstalled at specific buses(188,140,142,148,and86in the externalsystem,and141,143,145,and182in the internal system),whichbasically correspond to the frontier nodes.Likewise,it is assumedthat the precise voltages are known in these buses every time.Bythe inclusion of the PMUs,there is a noticeable improvement in thevoltages’information at the buses near them,due to the fact that itis assumed the PMUs’high precision.In this paper,the simulation results obtained by the full andthe reduced system are compared by a closeness measure,themean squared error(MSE).The goal of a signalfidelity measureis to compare two signals by providing a quantitative score thatdescribes the degree of similarity/fidelity or,conversely,the levelof error/distortion between ually,it is assumed that oneof the signals is a pristine original,while the other is distorted orcontaminated by errors[38].Suppose that z={z i|i=1,2,...,N}and y={y i|i=1,2,...,N}aretwofinite-length,discrete signals,where N is the number of signalsamples and z i and y i are the values of the i th samples in z and y,respectively.The MSE is defined by,MSE(z,y)=1NNi=1(z i−y i)2(10)Figs.4–5illustrate the transient behavior of some representa-tive signals after a three-phase fault at buses172(Fig.4)for theCase1;and144(Fig.5)for the Case3.Bus39is selected as theslack bus.Values in Table2show the corresponding MSE valuesfor the twelve generators of the internal system for each operat-ing case.Such values indicate a close relationship between the fulland reduced signals’behavior.In order to improve the equivalenceof a specific operating point,it is possible to weight it differentlythrough the factors w1–3,Eqs.(7)–(8).Table2shows the MSE’s val-ues when Case2has a higher weighting than Case1and Case3(w2=2/3,w1=w3=1/6).These values indicate that a closer agree-ment is attained between signals with the full and the reducedmodel.It is emphasized that the equivalence’s improvement couldrequire load variations greater than50%for off-nominal operatingconditions.4.ConclusionsUndoubtedly,the power system equivalents’calculationremains a useful strategy to handle the large amount of data,cal-culations,information and time,which represent the transientstability studies of modern power grids.The proposed approachis founded on a multi-objective formulation,solved by a geneticalgorithm,where the objective functions weight independentlyeach operating condition taken into account.The use of informationstemming from PMUs helps to improve the estimated equivalentgenerators’parameters.Results indicate that the strategy is ableto closely preserve the oscillating modes associated to the inter-nal system’s generators,under different operating conditions.Thatis due to the preservation of the machines’inertia.The use ofan index to measure the proximity between the signal’s behav-ior after a three-phase fault,indicates that good agreement isattained.In this paper,the same weighting factors have been usedto assess different operating conditions into the objective func-tions.However,depending on requirements,these factors can bemodified.The equivalence based on an optimal formulation assuresTable2MSE for Case2(Three-Phase Fault at Bus168).Angular position Angular speed Electrical powerw k=1/3w2=2/3,w1=w3=1/6w k=1/3w2=2/3,w1=w3=1/6w k=1/3w2=2/3,w1=w3=1/6 Gen39 2.13E−01 1.54E−01 6.36E−05 6.90E−059.40E−02 6.91E−02Gen28 1.79E−019.39E−02 3.43E−05 2.75E−05 4.09E−05 2.50E−05Gen29 1.12E−01 4.96E−02 4.84E−05 2.29E−059.25E−04 5.38E−04Gen30 1.01E−01 4.27E−02 3.31E−05 1.62E−059.75E−04 4.52E−04Gen32 2.80E−01 1.02E−01 4.34E−05 2.69E−05 4.26E−04 5.16E−04Gen33 2.97E−01 1.09E−01 4.81E−05 3.29E−059.85E−04 1.15E−04Gen34 1.70E−01 1.04E−01 4.03E−05 4.09E−059.19E−03 6.17E−03Gen37 1.68E−01 1.08E−01 5.23E−05 5.39E−05 1.12E−028.48E−03Gen38 1.53E−019.54E−02 4.47E−05 4.60E−05 1.29E−02 1.01E−02Gen40 1.91E−01 1.15E−01 5.27E−05 5.14E−05 2.26E−03 1.06E−03Gen41 1.93E−01 1.18E−01 5.52E−05 5.39E−05 2.50E−04 1.16E−04Gen42 1.68E−01 1.07E−01 4.17E−05 4.28E−05 5.23E−05 3.02E−0564J.M.Ramirez et al./Electric Power Systems Research84 (2012) 58–64proximity between the full and the reduced models.Closer prox-imity is reached if more stringent convergence’s parameters are defined,as well as additional objective functions,as line’s power flows,are included.References[1]P.Nagendra,S.H.nee Dey,S.Paul,An innovative technique to evaluate networkequivalent for voltage stability assessment in a widespread sub-grid system, Electr.Power Energy Syst.(33)(2011)737–744.[2]A.M.Miah,Study of a coherency-based simple dynamic equivalent for transientstability assessment,IET Gener.Transm.Distrib.5(4)(2011)405–416.[3]E.J.S.Pires de Souza,Stable equivalent models with minimum phase transferfunctions in the dynamic aggregation of voltage regulators,Electr.Power Syst.Res.81(2011)599–607.[4]Z.Y.Duan Yao,Z.Buhan,L.Junfang,W.Kai,Study of coherency-based dynamicequivalent method for power system with HVDC,in:Proceedings of the2010 International Conference on Electrical and Control Engineering,2010.[5]T.Singhavilai,O.Anaya-Lara,K.L.Lo,Identification of the dynamic equiva-lent of a power system,in:44th International Universities’Power Engineering Conference,2009.[6]J.M.Undrill,A.E.Turner,Construction of power system electromechanicalequivalents by modal analysis,IEEE Trans.PAS90(1971)2049–2059.[7]A.B.Almeida,R.R.Rui,J.G.C.da Silva,A software tool for the determinationof dynamic equivalents of power systems,in:Symposium-Bulk Power System Dynamics and Control–VIII(IREP),2010.[8]A.Akhavein,M.F.Firuzabad,R.Billinton,D.Farokhzad,Review of reductiontechniques in the determination of composite system adequacy equivalents, Electr.Power Syst.Res.80(2010)1385–1393.[9]U.D.Annakkage,N.C.Nair,A.M.Gole,V.Dinavahi,T.Noda,G.Hassan,A.Monti,Dynamic system equivalents:a survey of available techniques,in:IEEE PES General Meeting,2009.[10]B.Marinescu,B.Mallem,L.Rouco,Large-scale power system dynamic equiva-lents based on standard and border synchrony,IEEE Trans.Power Syst.25(4) (2010)1873–1882.[11]R.Podmore,A.Germond,Development of dynamic equivalents for transientstability studies,Final Report on EPRI Project RP763(1977).[12]E.J.S.Pires de Souza,Identification of coherent generators considering the elec-trical proximity for drastic dynamic equivalents,Electr.Power Syst.Res.78 (2008)1169–1174.[13]P.Ju,L.Q.Ni,F.Wu,Dynamic equivalents of power systems with onlinemeasurements.Part1:theory,IEE Proc.Gener.Transm.Distrib.151(2004) 175–178.[14]W.W.Price,D.N.Ewart,E.M.Gulachenski,R.F.Silva,Dynamic equivalents fromon-line measurements,IEEE Trans.PAS94(1975)1349–1357.[15]G.H.Shakouri,R.R.Hamid,Identification of a continuous time nonlinear statespace model for the external power system dynamic equivalent by neural net-works,Electr.Power Energy Syst.31(2009)334–344.[16]J.M.Ramirez,Obtaining dynamic equivalents through the minimizationof a lineflows function,Int.J.Electr.Power Energy Syst.21(1999) 365–373.[17]J.M.Ramirez,R.Garcia-Valle,An optimal power system model order reductiontechnique,Electr.Power Energy Syst.26(2004)493–500.[18]E.de Tuglie,L.Guida,F.Torelli,D.Lucarella,M.Pozzi,G.Vimercati,Identificationof dynamic voltage–current power system equivalents through artificial neural networks,Bulk power system dynamics and control–VI,2004.[19]A.M.Azmy,I.Erlich,Identification of dynamic equivalents for distributionpower networks using recurrent ANNs,in:IEEE Power System Conference and Exposition,2004,pp.348–353.[20]P.Sowa,A.M.Azmy,I.Erlich,Dynamic equivalents for calculation of powersystem restoration,in:APE’04Wisla,2004,pp.7–9.[21]A.H.M.A.Rahim,A.J.Al-Ramadhan,Dynamic equivalent of external power sys-tem and its parameter estimation through artificial neural networks,Electr.Power Energy Syst.24(2002)113–120.[22]A.M.Stankovic, A.T.Saric,osevic,Identification of nonparametricdynamic power system equivalents with artificial neural networks,IEEE Trans.Power Syst.18(4)(2003)1478–1486.[23]A.M.Stankovic,A.T.Saric,Transient power system analysis with measurement-based gray box and hybrid dynamic equivalents,IEEE Trans.Power Syst.19(1) (2004)455–462.[24]O.Yucra Lino,Robust recurrent neural network-based dynamic equivalencingin power system,in:IEEE Power System Conference and Exposition,2004,pp.1067–1068.[25]J.M.Ramirez,V.Benitez,Dynamic equivalents by RHONN,Electr.Power Com-pon.Syst.35(2007)377–391.[26]M.L.Ourari,L.A.Dessaint,V.Q.Do,Dynamic equivalent modeling of large powersystems using structure preservation technique,IEEE Trans.Power Syst.21(3) (2006)1284–1295.[27]M.L.Ourari,L.A.Dessaint,V.Q.Do,Integration of dynamic equivalents in hyper-sim power system simulator,in:IEEE PES General Meeting,2007,pp.1–6. [28]P.M.Anderson,A.A.Fouad,Power System Control and Stability.Appendix D,The Iowa State University Press,1977.[29]K.Deb,Evolutionary algorithms for multicriterion optimization in engineer-ing design,in:Evolutionary Algorithms in Engineering and Computer Science, 1999,pp.135–161(Chapter8).[30]N.Srinivas,K.Deb,Multiple objective optimization using nondominated sort-ing in genetic algorithms,put.2(3)(1994)221–248.[31]K.Deb,A.Pratap,S.Agarwal,T.Meyarivan,A fast and elitist multi-objectivegenetic algorithm:NSGA-II,IEEE put.6(2)(2002)182–197.[32]K.Deb,Multi-Objective Optimization using Evolutionary Algorithms,1st ed.,John Wiley&Sons(ASIA),Pte Ltd.,Singapore,2001.[33]P.Hajela,C.-Y.Lin,Genetic search strategies in multicriterion optimal design,Struct.Optim.4(1992)99–107.[34]Jeffrey Horn,N.Nafpliotis,Multiobjective optimization using the niched paretogenetic algorithm,IlliGAL Report93005,Illinois Genetic Algorithms Laboratory, University of Illinois,Urbana,Champaign,July,1993.[35]J.Horn,N.Nafpliotis,D.E.Goldberg,A niched pareto genetic algorithm formultiobjective optimization,in:Proceedings of the First IEEE Conference on Evolutionary Computation,IEEE World Congress on Computational Computa-tion,vol.1,82–87,Piscataway,NJ,IEEE Service Center,1994.[36]J.D.Schafer,Multiple objective optimization with vector evaluated geneticalgorithms,in:J.J.Grefenstette(Ed.),Proceedings of an International Confer-ence on Genetic Algorithms and their Applications,1985,pp.93–100.[37]A.R.Messina,J.M.Ramirez,J.M.Canedo,An investigation on the use of powersystem stabilizers for damping inter-area oscillations in longitudinal power systems,IEEE Trans.Power Syst.13(2)(1998)552–559.[38]W.Zhou,A.C.Bovik,Mean squared error:love it or leave it?A new look at signalfidelity measures,IEEE Signal Process.Mag.26(1)(2009)98–117.。
整数规划 孙小玲 课件2
Example. Consider the following 0-1 knapsack problem: max 8x1 + 11x2 + 6x3 + 4x4 s.t. 5x1 + 7x2 + 4x3 + 3x4 ≤ 14 x ∈ {0, 1}4 .
5/1
6/1
7/1
8/1
Solving Integer Knapsack by B&B
◮ ◮ ◮ ◮
◮
◮
◮
◮
Best First Depth-First Hybrid Strategies Best Estimate
◮
◮
One way to minimize overall solution time is to try to minimize the size of the search tree. We can achieve this by choosing the subproblem with the best bound (lowest lower bound if we are minimizing). A candidate node is said to be critical if its lower bound is also a lower bound of the original IP. Every critical node will be processed no matter what the search order Best first is guaranteed to examine only critical nodes, thereby minimizing the size of the search tree. Drawbacks of Best First
机器人路径规划算法
机器人路径规划算法机器人路径规划算法是指通过特定的计算方法，使机器人能够在给定的环境中找到最佳的路径，并实现有效的移动。
这是机器人技术中非常关键的一部分，对于保证机器人的安全和高效执行任务具有重要意义。
本文将介绍几种常见的机器人路径规划算法，并对其原理和应用进行探讨。
一、迷宫走迷宫算法迷宫走迷宫算法是一种基本的路径规划算法，它常被用于处理简单的二维迷宫问题。
该算法通过在迷宫中搜索，寻找到从起点到终点的最短路径。
其基本思想是采用图的遍历算法，如深度优先搜索（DFS）或广度优先搜索（BFS）等。
通过递归或队列等数据结构的应用，寻找到路径的同时保证了搜索的效率。
二、A*算法A*算法是一种启发式搜索算法，广泛应用于机器人路径规划中。
该算法通过评估每个节点的代价函数来寻找最佳路径，其中包括从起点到当前节点的实际代价（表示为g(n)）和从当前节点到目标节点的估计代价（表示为h(n)）。
在搜索过程中，A*算法综合考虑了这两个代价，选择总代价最小的节点进行扩展搜索，直到找到终点。
三、Dijkstra算法Dijkstra算法是一种最短路径算法，常用于有向或无向加权图的路径规划。
在机器人路径规划中，该算法可以用来解决从起点到目标点的最短路径问题。
Dijkstra算法的基本思想是，通过计算起点到每个节点的实际代价，并逐步扩展搜索，直到找到目标节点，同时记录下到达每个节点的最佳路径。
四、RRT算法RRT（Rapidly-exploring Random Tree）是一种适用于高维空间下的快速探索算法，常用于机器人路径规划中的避障问题。
RRT算法通过随机生成节点，并根据一定的规则连接节点，逐步生成一棵树结构，直到完成路径搜索。
该算法具有较强的鲁棒性和快速性，适用于复杂环境下的路径规划。
以上介绍了几种常见的机器人路径规划算法，它们在不同的场景和问题中具有广泛的应用。
在实际应用中，需要根据具体的环境和需求选择合适的算法，并对其进行适当的改进和优化，以实现更好的路径规划效果。
C语言经典算法
C语言经典算法1.河内之塔说明河内之塔(Towers of Hanoi)是法国人M.Claus(Lucas)于1883年从泰国带至法国的，河内为越战时北越的首都，即现在的胡志明市；1883年法国数学家Edouard Lucas曾提及这个故事，据说创世纪时Benares有一座波罗教塔，是由三支钻石棒（Pag）所支撑，开始时神在第一根棒上放置64个由上至下依由小至大排列的金盘（Disc），并命令僧侣将所有的金盘从第一根石棒移至第三根石棒，且搬运过程中遵守大盘子在小盘子之下的原则，若每日仅搬一个盘子，则当盘子全数搬运完毕之时，此塔将毁损，而也就是世界末日来临之时。
解法如果柱子标为ABC，要由A搬至C，在只有一个盘子时，就将它直接搬至C，当有两个盘子，就将B当作辅助柱。
如果盘数超过2个，将第三个以下的盘子遮起来，就很简单了，每次处理两个盘子，也就是：A->B、A ->C、B->C这三个步骤，而被遮住的部份，其实就是进入程式的递回处理。
事实上，若有n个盘子，则移动完毕所需之次数为2^n - 1，所以当盘数为64时，则所需次数为：264- 1 = 18446744073709551615为5.05390248594782e+16年，也就是约5000世纪，如果对这数字没什幺概念，就假设每秒钟搬一个盘子好了，也要约5850亿年左右。
#include <stdio.h>void hanoi(int n, char A, char B, char C) {if(n == 1) {printf("Move sheet %d from %c to %c\n", n, A, C);}else {hanoi(n-1, A, C, B);printf("Move sheet %d from %c to %c\n", n, A, C);hanoi(n-1, B, A, C);}}int main() {int n;printf("请输入盘数：");scanf("%d", &n);hanoi(n, 'A', 'B', 'C');return 0;}2.Algorithm Gossip: 费式数列说明Fibonacci为1200年代的欧洲数学家，在他的着作中曾经提到：「若有一只免子每个月生一只小免子，一个月后小免子也开始生产。
最优路径问题的启发式搜索算法
最优路径问题的启发式搜索算法启发式搜索算法是一种常用的优化算法，广泛应用于求解最优路径问题。
最优路径问题是指在给定的图中，寻找两个节点之间的最短路径或最优路径。
启发式搜索算法通过引入启发函数，对搜索过程进行优化，以提高搜索效率。
一、问题描述最优路径问题可以用图表示，图由节点和边组成。
节点表示位置或状态，边表示两个节点之间的关系或连接。
给定一个起始节点和目标节点，最优路径问题的目标是找到从起始节点到目标节点的最短路径或最优路径。
二、传统的搜索算法传统的搜索算法包括深度优先搜索（DFS）和广度优先搜索（BFS）。
DFS从起始节点开始，沿着每条可能的路径一直搜索到目标节点或无法继续搜索为止。
BFS则按层次遍历的方式，先搜索起始节点所有邻接节点，然后依次搜索这些邻接节点的邻接节点，直到找到目标节点。
传统的搜索算法存在效率低下的问题。
DFS通常能够找到一条路径，但该路径未必是最短路径或最优路径。
而BFS虽然能够找到最短路径，但在搜索过程中需要存储和遍历大量的节点，导致计算成本高。
三、启发式搜索算法启发式搜索算法引入了启发函数，用于评估搜索过程中每个节点的价值或成本。
启发函数通常根据问题的特定性质和经验进行设计，可以根据启发函数的值对节点进行评估和排序。
基于启发函数的评估，启发式搜索算法能够优先考虑具有更高潜在价值的节点，提高搜索效率。
最著名的启发式搜索算法之一是A*算法。
A*算法综合考虑了两个因素：从起始节点到当前节点的实际路径成本（表示为g值），以及从当前节点到目标节点的预估路径成本（表示为h值）。
A*算法通过计算启发函数值f = g + h，来评估节点的价值，从而选择具有最小f值的节点进行搜索。
A*算法在搜索过程中通过维护一个优先队列，不断扩展距离起始节点较好的节点，直到找到目标节点或搜索完成。
四、应用实例启发式搜索算法在许多领域都有应用，其中最著名的例子之一就是在计算机游戏中的路径规划。
在游戏中，启发式搜索算法能够帮助角色或NPC找到最短路径或最优路径，以避开障碍物或敌人。