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土木工程专业裂缝宽度容许值: allowable value of crack width使最优化: optimized次最优化: suboptimization主梁截面: girder section主梁: girder|main beam|king post桥主梁: bridge girder单墩: single pier结构优化设计: optimal structure designing多跨连续梁: continuous beam on many supports裂缝crackcrevice刚构桥: rigid frame bridge刚度比: ratio of rigidity|stiffness ratio等截面粱: uniform beam|uniform cross-section beam 桥梁工程: bridgeworks|LUSAS FEA|Bridge Engineering桥梁工程师: Bridge SE预应力混凝土: prestressed concrete|prestre edconcrete 预应力混凝土梁: prestressed concrete beam预应力混凝土管: prestressed concrete pipe最小配筋率minimum steel ratio轴向拉力, 轴向拉伸: axial tension英语重点词汇承台: bearing platform|cushioncap|pile caps桩承台: pile cap|platformonpiles低桩承台: low pile cap拱桥: hump bridge|arch bridge|arched bridge强度: intensity|Strength|Density刚强度: stiffness|stiffne|westbank stiffness箍筋: stirrup|reinforcement stirrup|hooping预应力元件: prestressed element等效荷载: equivalent load等效荷载原理: principle of equivalent loads模型matrixmodelmouldpattern承载能力极限状态: ultimate limit states正常使用极限状态: serviceability limit state 弹性: elasticity|Flexibility|stretch平截面假定: plane cross-section assumption抗拉强度intensity of tensiontensile strength安全系数safety factor标准值: standard value,|reference value作用标准值: characteristic value of an action重力标准值: gravity standard设计值: design value|value|designed value作用设计值: design value of an action荷载设计值: design value of a load可靠度: Reliability|degree of reliability不可靠度: Unreliability高可靠度: High Reliability几何特征: geometrical characteristic塑性plastic natureplasticity应力图: stress diagram|stress pattern压应力: compressive stress|compression stress配筋率: reinforcement ratio纵向配筋率: longitudinal steel ratio有限元分析: FEA|finite element analysis (FEA)|ABAQUS有限元法: finite element method线性有限元法: Linear Finite Element Method裂缝控制: crack control控制裂缝钢筋: crack-control reinforcement应力集中: stress concentration主拉应力: principal tensile stress非线性nonlinearity非线性振动: nonlinear vibration弯矩: bending moment|flexural moment|kN-m弯矩图: bending moment diagram|moment curve弯矩中心: center of moments|momentcenter剪力: shearing force|shear force|shear剪力墙: shear wall|shearing wall|shear panel弹性模量elasticity modulus剪力图: shear diagram|shearing force diagram剪力和弯矩图: Shear and Moment Diagrams剪力墙结构: shear wall structure轴力: shaft force|axial force框架结构frame construction板单元: plate unit曲率curvature材料力学mechanics of materials结构力学: Structural Mechanics|theory of structures 弯曲刚度: bending stiffness|flexural rigidity截面弯曲刚度: flexural rigidity of section弯曲刚度，抗弯劲度: bending stiffness钢管混凝土结构: encased structures极限荷载: ultimate load极限荷载设计: limit load design|ultimate load design 板壳力学: Plate Mechanic主钢筋: main reinforcement|Main Reinforcing Steel 钢筋混凝土的主钢筋: main bar悬臂梁: cantilever beam|cantilever|outrigger悬链线: Catenary,|catenary wire|chainetteribbed stiffener加劲肋: stiffening rib|stiffener|ribbed stiffener短加劲肋: short stiffener支承加劲肋: bearing stiffener技术标准technology standard水文: Hydrology招标invite public bidding连续梁: continuous beam|through beam多跨连续梁: continuous beam on many supports wind resistance抗风: Withstand Wind |wind resistance基础的basal初步设计predesignpreliminary plan技术设计: technical design|technical project施工图设计: construction documents design基础foundationbasebasis 结构形式: Type of construction|form of structure屋顶结构形式: roof form地震earthquake地震活动: Seismic activity|seismic motion耐久性: durability|permanence|endurance耐久性试验: endurance test|life test|durability test短暂状况: transient situation偶然状况: accidental situation永久作用: permanent action永久作用标准值: characteristic value of permanent action可变作用: variable action可变作用标准值: characteristic value of variable action可变光阑作用: iris action偶然作用: accidental action作用效应偶然组合: accidental combination for action effects作用代表值: representative value of an action作用标准值: characteristic value of an action地震作用标准值: characteristic value of earthquake action可变作用标准值: characteristic value of variable action作用频遇值Frequent value of an action安全等级: safety class|Security Level|safeclass设计基准期: design reference period作用效应: effects of actions|effect of an action作用效应设计值Design value of an action effect分项系数: partial safety factor|partial factor作用分项系数: partial safety factor for action抗力分项系数: partial safety factor for resistance作用效应组合: combination for action effects结构重要性系数Coefficient for importance of a structure桥涵桥涵跟桥梁比较类似，主要区别在于:单孔跨径小于5m或多孔跨径之和小于8m的为桥涵,大于这个标准的为桥梁水力: hydraulic power|water power|water stress跨度span人行道sidewalk无压力: stress-free净高clear height矩形rectangle无铰拱: arch without articulation|fixed end arch荷载load荷载强度: loading intensity|loading inte ity荷载系数: load factor|loading coefficient桥头堡bridgeheadbridge tower美观pleasing to the eyebeautifulartistic经济的economicaloecumenicaleconomic适用be applicable防水waterproof剪切模量: shear modulus|rigidity modulus|GXY剪切强度: shear strength|shearing strength|Fe-Fe扭转剪切强度: torsional shear strength剪切破坏: shear failure|shear fracture|shear damage 纯剪切破坏: complete shear failure局部剪切破坏: local shear failure永久冻土: permafrost|perennial frost土的侧压力: earth lateral pressure收缩shrinkpull backcontract徐变: creep摩擦系数: coefficient of friction|friction factor风荷载: wind load|wind loading风荷载标准值: characteristi cvalue of windload 风荷载体型系数: shape factor of windload温度作用: temperature action支座: support|bearing|carrier 外支座: outer support|outersu ort代表值: central value|representative value结构自重: self-weightstructure|dead load最不利分布: Least favorable distribution,抗震antiknockquake-proofearthquake proofing constructionearthquake-resistanceearthquake proof钢结构steel structure钢结构设计: Design Of Steel Structure钢结构设计规范: Code for design of steel structures 混凝土结构设计规范: Code for design of concrete structures预应力混凝土结构设计软件: PREC温度梯度: temperature gradient|thermal gradient动力系数: dynamic coefficient制动力系数: Braking force coefficient动力学kineticsdynamicsdyn内摩擦角: angle of internal friction有效内摩擦角: effective angle of internal friction主效应main effect主效应: Main effect,主效应模型: Main effect model超静定的: hyperstatic超静定结构: statically indeterminate structure静定: statically determinate静定梁: statically determinate beam附属设备: accessories|accessory equipment稳定系数: coefficient of stabilizationearth pressure at rest静土压力: earthpressureatrest挡土墙retaining wallabamurus主动土压力: active earth pressure被动土压力: passive earth pressure土层soil horizon土层剖面: soil profile土层剖面特性: soil-profile characteristics密度densitythickness宽度width净距: clear distance|gabarit|Clearance钢筋强度标准值: characteristic value of strength of steel bar钢材强度标准值: characteristic value of strength of steel折减系数: reduction factor|discount coefficient强度折减系数: strength reduction factor线性linearity线性代数linear algebra位移displacement位移角: angle of displacement|angle of slip应变量: dependent variable|strain capacityuniform stress均布应力: uniform stress非均布应力: non-uniform stress均布荷载: uniformly distributed load集中荷载: concentrated load|point load可变集中荷载: variable concentrated load法向集中荷载: normal point load影响线: influence line反力影响线: influence line for reaction影响线方程: equation of the influence line车辆荷载: car load|vehicular load|traffic load计算跨径: calculated span重力加速度: acceleration of gravity膨胀系数: coefficient of expansion|expansivity术语termterminology恒载: dead load|deadloading|permanent load活载: live load楼面活载: floor live load概率分布: probability distribution 联合概率分布: Joint probability distribution,边缘概率分布: Marginal probability distribution,拱腹: soffit|intrados|arch soffit三铰拱: three hinged arch土木工程系: Department of Civil Engineering土木工程师协会: ICE土木工程师协会: Institute of Civil Engineers作用准永久值: quasi－permanentvalueofanaction 直径diameter验算: checking|check calculation验算公式: check formula变形验算: deformation analysis建筑材料tignum刚度rigidityseveritystiffness单元: cell|Unit|module节点node位移方程式: strain displacement equation三维three dimensional 3d插值: Interpolation|interpolate|Spline插值法: interpolation|method of interpolation轴对称axial symmetryrotational symetryaxisymmetric(al)应变矩阵strain matrix应变矩阵: strain matrix单元应变矩阵: element strain matrix应力应变矩阵: stress-strainmatrix阻尼矩阵: damping matrix|daraf|damped matrix 弹性系数矩阵: elastic coefficient matrix雅可比矩阵: Jacobi matrix|jacobian matrix刚度矩阵: stiffness matrix|rigidity matrix质量矩阵: mass matrix|ma matrix节点力: nodal forces等效节点力: equivalent nodal force节点荷载: joint load|nodal loads节点荷载: joint load|nodal loads一致节点荷载: consistent nodal load应力矩阵: stress matrix挠度: deflection|flexivity|flexure转角: corners|intersection angle|rotor angle单元刚度矩阵: element stiffness matrix边界条件: boundary condition|edge conditions疲劳强度: fatigue strength|endurance strength抗疲劳强度: fatigue resistance工程局: construction bureau沉井基础: open caisson foundation水泥cement水泥砂浆cement mortar石膏: Gypsum|plaster|Plaster of Paris简支梁: simply supported beam|simple beam简支梁桥: simple supported girder bridge平衡条件: equilibrium condition|balance condition约束条件: constraint condition|constraint数值解: numerical solution|arithmeticsolution力法: force method|brute force method位移法: displacement method|di lacement method力矩分配法: moment distribution method|moment diagram理论力学: Theoretical Mechanics弹性力学: Theory of Elastic Mechanics结构动力学: Structural Dynamics|Clough高等结构动力学: Advancd Dynamics of Structures测量学: surveying|metrology|geodesy道路工程: road works|highway construction铁路工程: railway engineering|rairoad engineering隧道: Tunnels|subway|underpass轨道: orbit|track|trajectory砂子: sand抗压强度pressive strength焊接技术: Welding Engineering Technology (WET)断裂力学: Fracture Mechanics|fracturing mechanics基础工程: foundation engineering|foundation works 地质学: geology|die Geologie, opl.|geognosy岩土力学: rock mechanics|rock-soil mechanics工程力学: engineering mechanics轴线axes拱脚: arch springing|abutment|spring木桥: timber bridge|wodden bridge|Woodbridge枕木sleeper crosstie残余应力: residual stress|remaining stress 复合应力: combined stress|compound stress初始应力: initial stress|primary stress屈服极限: yield limit|minimum yield|yield strength疲劳屈服极限: fatigue yield limit应力幅值: stress amplitude冲击韧性: impact toughness|Impelling strength反弯点: knick point|pointofcontraflexure桁架: truss|tru|Girder网架结构: space truss structure|grid structure锚孔: anchor eye大跨度: High-span柱: column|pillar|Clmn. Coloumn常微分方程: Ordinary Differentical Equations|ODE|ODEs增大系数: enhancementcoefficient浮桥flying bridge raft bridgepontoon bridge pontoonfloat bridge浮桥: pontoon bridge|pontoon|floating bridge轮渡: Ferry|Ferries|ferry boat钢桥: steel bridge立面图: elevation|elevation drawing|profile背立面图: back elevation平面图: plan|plan view|planar graph泥石流: debris flow|rollsteinfluten|mud-rock flow大型泥石流: macrosolifluction滑坡泥石流: landslide模板: template|die plate, front board|formwork沉降: settlement|sedimentation|subside沉降缝: settlement joint伸缩缝: expansion joint路灯street lamp排水系统: drainage system|sewerage system泄水管: drain pipe|Scupper Pipe|tap pipe土力学: soil mechanics|Bodenmechanik高等土力学: Advanced Soil Mechanics扩展(扩大)基础: spread foundation桩基础: pile foundation|pile footing|Pile砂桩基础: sand pile foundation群桩基础: multi-column pier foundation沉箱基础caisson foundation沉箱基础: caisson foundation|laying foundation管状沉箱基础: cylinder caisson foundation气压沉箱基础: pneumatic caisson foundation桩承台: pile cap|platformonpiles桩: pile|pile group|pale灌注桩: cast-in-place pile|cast in place管灌注桩: driven cast-in-place pile灌注混凝土基础: cast-in-place concrete foundation 承台结构: suspended deck structure工作机理working mechanism铆钉: rivet|rivet riv|clinch bolt卵石: cobble|gravel|pebble钢筋混凝土结构: reinforced concrete structure预应力混凝土结构: prestressed concrete structure软化: softening|mollification|malacia强化: reinforcement|consolidate|intensification固体力学: solid mechanics|механика твердого тела 虚功原理: principle of virtual work偏心距: eccentricity|throw of eccentric偏心距增大系数: amplified coefficient of eccentricity 强度准则: strength criterion变形: Deformation|Transforms|deform工程建设: engineering construction石油工程建设: Petroleum Engineering Construction 偏心受压: eccentric compression偏心受压构件: eccentric compression member弹性支承: elastomeric bearing|yielding support temperature load温度荷载: temperature load施工控制: construction control经纬仪theodolite transit instrument夹具jig tongs clamp切线: tangent|Tangent line,|tangential line水平角: horizontal angle|inclination高程index elevation height altitude沼泽marsh swamp glade水准仪water level公寓apartment砂浆mortar sand pulp骨料skeletal material aggregate骨料级配: aggregate grading|aggregate gradation碱性的: alkalic|basic|alkalescent耐碱性的: alkali-proof风洞试验: wind tunnel test先张法: pre-tensioning|pretensioning method配合比设计: mix design|design of mix proportion 和易性: workability渗透性osmosis penetrability水泥浆: grout|cement slurry|cement paste对称的symmetrical symmetric(al)扭转reverseturn around (an undesirable situation)扭转应力: torsion stress|warping stress容许扭转应力: allowable twisting stress扭转角: angle of torsion|angle of twist夯实回填土: tamped backfill|tamped/compacted backfill圆锥贯入仪: cone penetrometer水化（作用）: hydration水化热: heat of hydration|heat of hydratation振捣器: vibrating tamper|vibrorammer|vibrator板振捣器: slab vibrator破裂fracture burst结合力: binding force|Adhesion|cohesion碎石gravel gravely脆性brittleness脆性材料: brittleness material|brittle material脆性破坏: brittle failure|brittle fracture素混凝土: plain concrete素混凝土结构: plain concrete construction含水量liquid water content钢筋: Reinforcement|bar tendon主钢筋: main reinforcement|Main Reinforcing Steel钢筋条: reinforcement bar|steel bar极限抗拉应力: ultimate tensile strength极限抗拉强度: ultimate tensile strength|UTS混凝土板: concrete slab预制混凝土板: precast concrete plank锚固: anchoring|anchorage|Anchor锚具: anchorage|anchorage device|ground tackle削弱weaken埋置: embedding|elutriator|imbedment预应力钢筋: prestressed reinforcement回弹: resilience|spring back|rebound有说服力的: persuasive|convincing|convictive形心centre of figurecentre of formcentroid重心center of gravity(n) core; main part惯性矩: moment of inertia极惯性矩: polar moment of inertia质心centroid center of mass回转半径: radius of gyration|turning radius容许应力: allowable stress|permissible stress排架: shelving|bent frame|bent桩排架: pile bent纵梁longeron carling横梁: beam|cross beam|transverse beam缆索cable thick rope阻尼damping刚架: rigid frame|frame|stiffframe缀板batten plate缀板: batten plate|stay plate|batte latebatten plate缀板: batten plate|stay plate|batte late上部缀板: upper stay plate推力: thrust|Push|Push Power槽钢channel steel特征值: Eigenvalue,|characteristic value冷拔钢丝: cold drawn wire自振频率: natural frequency of vibration自振周期: natural period of vibration土壤加固工程: soil stabilization works结构加固工程: structural fortification应力分析: stress analysis|stress distribution结构分析: structural analysis|ETABS NL结构稳定性: structural stability结构工程: Structural Engineering|structural works 认可标准: recognized standard|approved standard 官方认可标准: officially recognized standard,再循环: recycle|recirculation|recycling快硬水泥: rapid hardening cement|ferrocrete曲率半径: radius of curvature|curve radius|ρ刚性系数: coefficient of rigidity乡郊地区: rural area饱和saturation饱和密度: saturated density|Saturation density脚手架staging scaffold falsework立体剖面图: sectional axonometric drawing结构控制: structural control收缩量: Shrinkage|amount of shrinkage间距space between 钢管steel tube工字钢桩: steel H pile钢绞线: Steel Strand|Steel Stranded Wire|strand群震: swarm earthquake系统误差: systematic error|fixed error|system error最大剪应力: maximum shear|maximum shearing stress最大剪应变: maximum shear strain千斤顶: jack|lifting jack|Wheeljack地震系数: seismic coefficient|seismic factor。
一、介绍Jacobi矩阵Jacobi矩阵，又称雅可比矩阵，是数学中的一个重要概念，在许多领域中都有着广泛的应用。
Jacobi矩阵是一个矩阵，它由向量值函数的偏导数组成。
在数学分析中，Jacobi矩阵被用来描述多变量函数的导数，它能够提供有关函数在某一点处的局部线性逼近。
在优化算法、控制理论、微分方程数值解等领域中，Jacobi矩阵也经常被用来进行求解和分析。
二、Jacobi矩阵的稀疏模式稀疏模式表示的是Jacobi矩阵中大部分元素都为零的特点。
在实际的应用中，许多Jacobi矩阵都具有稀疏性，这意味着矩阵中只有少量的非零元素。
稀疏模式的Jacobi矩阵具有较低的存储成本和计算成本，因此在大规模的数值计算和数据处理中具有重要的作用。
对于稀疏模式的Jacobi矩阵，我们通常希望能够高效地求解和分析其特征和性质。
三、用MATLAB语言求解Jacobi矩阵的稀疏模式MATLAB是一种强大的数值计算软件，它提供了丰富的工具和函数来进行矩阵运算、线性代数计算、数值分析等操作。
在MATLAB中，我们可以利用其丰富的工具箱和函数来求解Jacobi矩阵的稀疏模式。
下面，我们将介绍一种常用的方法，来利用MATLAB对Jacobi矩阵的稀疏模式进行求解。
1. 导入稀疏矩阵我们需要将Jacobi矩阵表示成稀疏矩阵的形式。
在MATLAB中，我们可以使用稀疏矩阵的数据结构来存储Jacobi矩阵的稀疏模式。
通过使用稀疏矩阵，我们可以高效地存储和处理Jacobi矩阵的数据，节约内存并提高计算效率。
2. 使用MATLAB函数进行求解一旦Jacobi矩阵被表示成稀疏模式的稀疏矩阵，我们便可以利用MATLAB中丰富的线性代数函数来进行求解。
MATLAB提供了许多高效的函数，如“eigs”、“svds”等，这些函数可以帮助我们对稀疏矩阵进行特征值分解、奇异值分解等操作。
通过这些函数，我们可以快速求解Jacobi矩阵的特征和性质，对其稀疏模式进行分析和研究。
三重积分球坐标变换公式雅可比行列式英文版The Jacobian determinant is a key concept in multivariable calculus, particularly when dealing with transformations between coordinate systems. In the context of triple integrals, the Jacobian determinant plays a crucial role in converting integrals from one coordinate system to another.Consider a triple integral in spherical coordinates, where the integrand is a function f(r, θ, φ) and the volume element is given by dV = r^2 sin(φ) dr dθ dφ. To convert this integral to Cartesian coordinates, we need to calculate the Jacobian determinant of the transformation from spherical to Cartesian coordinates.The Jacobian determinant for the transformation from spherical coordinates (r, θ, φ) to Cartesian coordinates (x, y, z) is given by the formula:J = |∂(x, y, z)/∂(r, θ, φ)|To calculate this determinant, we need to find the partial derivatives of x, y, and z with respect to r, θ, and φ. These partial derivatives can be expressed in terms of the unit vectors in each coordinate system:∂x/∂r = sin(φ) cos(θ)∂x/∂θ = -r sin(φ) sin(θ)∂x/∂φ = r cos(φ) cos(θ)∂y/∂r = sin(φ) sin(θ)∂y/∂θ = r sin(φ) cos(θ)∂y/∂φ = r cos(φ) sin(θ)∂z/∂r = cos(φ)∂z/∂θ = 0∂z/∂φ = -r sin(φ)By calculating the determinant of the Jacobian matrix formed by these partial derivatives, we can obtain the Jacobian determinant J. This determinant will then be used to convert the volume element dV = r^2 sin(φ) dr dθ dφ in spherical coordinates to the corresponding volume element in Cartesian coordinates.In summary, the Jacobian determinant is a powerful tool in transforming integrals between different coordinate systems, allowing us to work with a variety of mathematical problems in a more convenient and efficient manner.完整中文翻译雅可比行列式是多元微积分中的一个关键概念，特别是在处理坐标系之间的变换时。
雅可比矩阵求导雅可比矩阵是一种表示向量函数与向量导数之间关系的矩阵形式。
在机器学习、优化、控制等领域中，雅可比矩阵被广泛应用。
在本文中，我们将介绍如何求解基本的雅可比矩阵，以及如何对雅可比矩阵进行导数求解。
1. 雅可比矩阵的定义雅可比矩阵（Jacobian Matrix）是指一个由一个向量函数的一阶偏导数组成的矩阵。
具体来说，如果有一个n维实向量函数$f: \mathbb{R}^n \rightarrow \mathbb{R}^m$，即f将n维向量映射到m维向量。
假设$f$的分量函数为$f_i$，那么它的雅可比矩阵$J_f$为一个$m \times n$的矩阵，其中第$i$行第$j$列的元素为$\frac{\partialf_i}{\partial x_j}$，即：$$J_f(\mathbf{x})=\begin{bmatrix}\frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots & \frac{\partial f_1}{\partial x_n} \\\frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_2}{\partial x_n} \\\vdots & \vdots & \ddots & \vdots \\\frac{\partial f_m}{\partial x_1} & \frac{\partial f_m}{\partial x_2} & \cdots & \frac{\partial f_m}{\partial x_n} \\\end{bmatrix}$$我们可以将雅可比矩阵看做是一个函数$f$在一个给定点$\mathbf{x}$处的线性近似，即：$$f(\mathbf{x}+\Delta\mathbf{x})\approxf(\mathbf{x})+J_f(\mathbf{x})\Delta\mathbf{x}$$其中，$\Delta\mathbf{x}$为一个足够小的向量，可以看做是从点$\mathbf{x}$处的微小偏移。
拉格朗日乘子法和KKT条件的定义及选取原因拉格朗日乘子法（Lagrange Multiplier Method）是一种用于求解有等式约束的无约束极值问题的方法。
它基于拉格朗日函数的构建，通过引入拉格朗日乘子将等式约束转化为无约束问题，进而求解极值点。
首先，对于有等式约束的问题，我们可以将其建模为以下形式：$$\begin{align*}\text{min} \quad &f(x) \\\text{s.t.} \quad &g(x) = 0\end{align*}$$其中，$f(x)$为目标函数，$x$为自变量，$g(x)$为等式约束。
为了处理这类问题，我们构建拉格朗日函数：$$L(x, \lambda) = f(x) + \lambda g(x)$$其中，$\lambda$为拉格朗日乘子。
接下来，我们求解拉格朗日函数的极值。
首先对$x$求偏导，得到：$$\frac{\partial L}{\partial x} = \frac{\partialf(x)}{\partial x} + \lambda \frac{\partial g(x)}{\partial x} = 0$$然后对$\lambda$求偏导，得到：$$\frac{\partial L}{\partial \lambda} = g(x) = 0$$最后，我们得到一组方程：$$\frac{\partial f(x)}{\partial x} + \lambda \frac{\partial g(x)}{\partial x} = 0$$$$g(x)=0$$这组方程被称为KKT条件（Karush-Kuhn-Tucker conditions）。
KKT 条件是拉格朗日乘子法的核心。
KKT条件定义了一组必要条件，使得函数$f(x)$在满足等式约束$g(x)=0$的前提下可以达到极值。
KKT条件包含了以下几个要素：1. 雅可比矩阵（Jacobian Matrix）：雅可比矩阵是目标函数$f(x)$对自变量$x$的偏导数组成的矩阵，表示了目标函数的梯度。
自动化英语专业英语词汇表文章摘要：本文介绍了自动化英语专业的一些常用的英语词汇，包括自动化技术、控制理论、系统工程、人工智能、模糊逻辑等方面的专业术语。
本文按照字母顺序，将这些词汇分为26个表格，每个表格包含了以相应字母开头的词汇及其中文释义。
本文旨在帮助自动化专业的学习者和从业者掌握和使用这些专业英语词汇，提高他们的英语水平和专业素养。
A英文中文acceleration transducer加速度传感器acceptance testing验收测试accessibility可及性accumulated error累积误差AC-DC-AC frequency converter交-直-交变频器AC (alternating current) electric drive交流电子传动active attitude stabilization主动姿态稳定actuator驱动器，执行机构adaline线性适应元adaptation layer适应层adaptive telemeter system适应遥测系统adjoint operator伴随算子admissible error容许误差aggregation matrix集结矩阵AHP (analytic hierarchy process)层次分析法amplifying element放大环节analog-digital conversion模数转换annunciator信号器antenna pointing control天线指向控制anti-integral windup抗积分饱卷aperiodic decomposition非周期分解a posteriori estimate后验估计approximate reasoning近似推理a priori estimate先验估计articulated robot关节型机器人assignment problem配置问题，分配问题associative memory model联想记忆模型associatron联想机asymptotic stability渐进稳定性attained pose drift实际位姿漂移B英文中文attitude acquisition姿态捕获AOCS (attritude and orbit control system)姿态轨道控制系统attitude angular velocity姿态角速度attitude disturbance姿态扰动attitude maneuver姿态机动attractor吸引子augment ability可扩充性augmented system增广系统automatic manual station自动-手动操作器automaton自动机autonomous system自治系统backlash characteristics间隙特性base coordinate system基座坐标系Bayes classifier贝叶斯分类器bearing alignment方位对准bellows pressure gauge波纹管压力表benefit-cost analysis收益成本分析bilinear system双线性系统biocybernetics生物控制论biological feedback system生物反馈系统C英文中文calibration校准，定标canonical form标准形式canonical realization标准实现capacity coefficient容量系数cascade control级联控制causal system因果系统cell单元，元胞cellular automaton元胞自动机central processing unit (CPU)中央处理器certainty factor确信因子characteristic equation特征方程characteristic function特征函数characteristic polynomial特征多项式characteristic root特征根英文中文charge-coupled device (CCD)电荷耦合器件chaotic system混沌系统check valve单向阀，止回阀chattering phenomenon颤振现象closed-loop control system闭环控制系统closed-loop gain闭环增益cluster analysis聚类分析coefficient of variation变异系数cogging torque齿槽转矩，卡齿转矩cognitive map认知图，认知地图coherency matrix相干矩阵collocation method配点法，配置法combinatorial optimization problem组合优化问题common mode rejection ratio (CMRR)共模抑制比，共模抑制率commutation circuit换相电路，换向电路commutator motor换向电动机D英文中文damping coefficient阻尼系数damping ratio阻尼比data acquisition system (DAS)数据采集系统data fusion数据融合dead zone死区decision analysis决策分析decision feedback equalizer (DFE)决策反馈均衡器decision making决策，决策制定decision support system (DSS)决策支持系统decision table决策表decision tree决策树decentralized control system分散控制系统decoupling control解耦控制defuzzification去模糊化，反模糊化delay element延时环节，滞后环节delta robot德尔塔机器人demodulation解调，检波density function密度函数，概率密度函数derivative action微分作用，微分动作design matrix设计矩阵E英文中文eigenvalue特征值，本征值eigenvector特征向量，本征向量elastic element弹性环节electric drive电子传动electric potential电势electro-hydraulic servo system电液伺服系统electro-mechanical coupling system电机耦合系统electro-pneumatic servo system电气伺服系统electronic governor电子调速器encoder编码器，编码装置end effector末端执行器，末端效应器entropy熵equivalent circuit等效电路error analysis误差分析error bound误差界，误差限error signal误差信号estimation theory估计理论Euclidean distance欧几里得距离，欧氏距离Euler angle欧拉角Euler equation欧拉方程F英文中文factor analysis因子分析factorization method因子法，因式分解法feedback反馈，反馈作用feedback control反馈控制feedback linearization反馈线性化feedforward前馈，前馈作用feedforward control前馈控制field effect transistor (FET)场效应晶体管filter滤波器，滤波环节finite automaton有限自动机finite difference method有限差分法finite element method (FEM)有限元法finite impulse response (FIR) filter有限冲激响应滤波器first-order system一阶系统fixed-point iteration method不动点迭代法flag register标志寄存器flip-flop circuit触发器电路floating-point number浮点数flow chart流程图，流程表fluid power system流体动力系统G英文中文gain增益gain margin增益裕度Galerkin method伽辽金法game theory博弈论Gauss elimination method高斯消元法Gauss-Jordan method高斯-约当法Gauss-Markov process高斯-马尔可夫过程Gauss-Seidel iteration method高斯-赛德尔迭代法genetic algorithm (GA)遗传算法gradient method梯度法，梯度下降法graph theory图论gravity gradient stabilization重力梯度稳定gray code格雷码，反向码gray level灰度，灰阶grid search method网格搜索法ground station地面站，地面控制站guidance system制导系统，导航系统gyroscope陀螺仪，陀螺仪器H英文中文H∞ control H无穷控制Hamiltonian function哈密顿函数harmonic analysis谐波分析harmonic oscillator谐振子，谐振环节Hartley transform哈特利变换Hebb learning rule赫布学习规则Heisenberg uncertainty principle海森堡不确定性原理hidden layer隐层，隐含层hidden Markov model (HMM)隐马尔可夫模型hierarchical control system分层控制系统high-pass filter高通滤波器Hilbert transform希尔伯特变换Hopfield network霍普菲尔德网络hysteresis滞后，迟滞，磁滞I英文中文identification识别，辨识identity matrix单位矩阵，恒等矩阵image processing图像处理impulse response冲激响应impulse response function冲激响应函数inadmissible control不可接受控制incremental encoder增量式编码器indefinite integral不定积分index of controllability可控性指标index of observability可观测性指标induction motor感应电动机inertial navigation system (INS)惯性导航系统inference engine推理引擎，推理机inference rule推理规则infinite impulse response (IIR) filter无限冲激响应滤波器information entropy信息熵information theory信息论input-output linearization输入输出线性化input-output model输入输出模型input-output stability输入输出稳定性J英文中文Jacobian matrix雅可比矩阵jerk加加速度，冲击joint coordinate system关节坐标系joint space关节空间Joule's law焦耳定律jump resonance跳跃共振K英文中文Kalman filter卡尔曼滤波器Karhunen-Loeve transform卡尔胡南-洛维变换kernel function核函数，核心函数kinematic chain运动链，运动链条kinematic equation运动方程，运动学方程kinematic pair运动副，运动对kinematics运动学kinetic energy动能L英文中文Lagrange equation拉格朗日方程Lagrange multiplier拉格朗日乘子Laplace transform拉普拉斯变换Laplacian operator拉普拉斯算子laser激光，激光器latent root潜根，隐根latent vector潜向量，隐向量learning rate学习率，学习速度least squares method最小二乘法Lebesgue integral勒贝格积分Legendre polynomial勒让德多项式Lennard-Jones potential莱纳德-琼斯势level set method水平集方法Liapunov equation李雅普诺夫方程Liapunov function李雅普诺夫函数Liapunov stability李雅普诺夫稳定性limit cycle极限环，极限圈linear programming线性规划linear quadratic regulator (LQR)线性二次型调节器linear system线性系统M英文中文machine learning机器学习machine vision机器视觉magnetic circuit磁路，磁电路英文中文magnetic flux磁通量magnetic levitation磁悬浮magnetization curve磁化曲线magnetoresistance磁阻，磁阻效应manipulability可操作性，可操纵性manipulator操纵器，机械手Markov chain马尔可夫链Markov decision process (MDP)马尔可夫决策过程Markov property马尔可夫性质mass matrix质量矩阵master-slave control system主从控制系统matrix inversion lemma矩阵求逆引理maximum likelihood estimation (MLE)最大似然估计mean square error (MSE)均方误差measurement noise测量噪声，观测噪声mechanical impedance机械阻抗membership function隶属函数N英文中文natural frequency固有频率，自然频率natural language processing (NLP)自然语言处理navigation导航，航行negative feedback负反馈，负反馈作用neural network神经网络neuron神经元，神经细胞Newton method牛顿法，牛顿迭代法Newton-Raphson method牛顿-拉夫逊法noise噪声，噪音nonlinear programming非线性规划nonlinear system非线性系统norm范数，模，标准normal distribution正态分布，高斯分布notch filter凹槽滤波器，陷波滤波器null space零空间，核空间O英文中文observability可观测性英文中文observer观测器，观察器optimal control最优控制optimal estimation最优估计optimal filter最优滤波器optimization优化，最优化orthogonal matrix正交矩阵oscillation振荡，振动output feedback输出反馈output regulation输出调节P英文中文parallel connection并联，并联连接parameter estimation参数估计parity bit奇偶校验位partial differential equation (PDE)偏微分方程passive attitude stabilization被动姿态稳定pattern recognition模式识别PD (proportional-derivative) control比例-微分控制peak value峰值，峰值幅度perceptron感知器，感知机performance index性能指标，性能函数period周期，周期时间periodic signal周期信号phase angle相角，相位角phase margin相位裕度phase plane analysis相平面分析phase portrait相轨迹，相图像PID (proportional-integral-derivative) control比例-积分-微分控制piezoelectric effect压电效应pitch angle俯仰角pixel像素，像元Q英文中文quadratic programming二次规划quantization量化，量子化quantum computer量子计算机quantum control量子控制英文中文queueing theory排队论quiescent point静态工作点，静止点R英文中文radial basis function (RBF) network径向基函数网络radiation pressure辐射压random variable随机变量random walk随机游走range范围，区间，距离rank秩，等级rate of change变化率，变化速率rational function有理函数Rayleigh quotient瑞利商real-time control system实时控制系统recursive algorithm递归算法recursive estimation递归估计reference input参考输入，期望输入reference model参考模型，期望模型reinforcement learning强化学习relay control system继电器控制系统reliability可靠性，可信度remote control system遥控系统，远程控制系统residual error残差误差，残余误差resonance frequency共振频率S英文中文sampling采样，取样sampling frequency采样频率sampling theorem采样定理saturation饱和，饱和度scalar product标量积，点积scaling factor缩放因子，比例系数Schmitt trigger施密特触发器Schur complement舒尔补second-order system二阶系统self-learning自学习，自我学习self-organizing map (SOM)自组织映射sensitivity灵敏度，敏感性sensitivity analysis灵敏度分析，敏感性分析sensor传感器，感应器sensor fusion传感器融合servo amplifier伺服放大器servo motor伺服电机，伺服马达servo valve伺服阀，伺服阀门set point设定值，给定值settling time定常时间，稳定时间T英文中文tabu search禁忌搜索，禁忌表搜索Taylor series泰勒级数，泰勒展开式teleoperation遥操作，远程操作temperature sensor温度传感器terminal终端，端子testability可测试性，可检测性thermal noise热噪声，热噪音thermocouple热电偶，热偶threshold阈值，门槛time constant时间常数time delay时延，延时time domain时域time-invariant system时不变系统time-optimal control时间最优控制time series analysis时间序列分析toggle switch拨动开关，切换开关tolerance analysis公差分析torque sensor扭矩传感器transfer function传递函数，迁移函数transient response瞬态响应U英文中文uncertainty不确定性，不确定度underdamped system欠阻尼系统undershoot低于量，低于值unit impulse function单位冲激函数unit step function单位阶跃函数unstable equilibrium point不稳定平衡点unsupervised learning无监督学习upper bound上界，上限utility function效用函数，效益函数V英文中文variable structure control变结构控制variance方差，变异vector product向量积，叉积velocity sensor速度传感器verification验证，校验virtual reality虚拟现实viscosity粘度，黏度vision sensor视觉传感器voltage电压，电位差voltage-controlled oscillator (VCO)电压控制振荡器W英文中文wavelet transform小波变换weighting function加权函数Wiener filter维纳滤波器Wiener process维纳过程work envelope工作空间，工作范围worst-case analysis最坏情况分析X英文中文XOR (exclusive OR) gate异或门，异或逻辑门Y英文中文yaw angle偏航角Z英文中文Z transform Z变换zero-order hold (ZOH)零阶保持器zero-order system零阶系统zero-pole cancellation零极点抵消。
2012年第05期吉林省教育学院学报No.05，2012第28卷JOURNAL OF EDUCATIONAL INSTITUTE OF JILIN PROVINCEVol .28（总293期）Total No .293收稿日期：2012—03—05作者简介：张明洪（1966—），男，湖北枝江人，三峡旅游职业技术学院，讲师，研究方向：计算机教育、休闲服务与管理的教学与研究。
浅论如何使用MATLAB 作张量运算张明洪（三峡旅游职业技术学院，湖北宜昌443100）摘要：本文介绍并分析了如何使用MATLAB 作张量的创建以及缩并、乘积、求导等运算的方法和步骤。
关键词：MATLAB ；张量；张量创建；张量运算中图分类号：O183文献标识码：A文章编号：1671—1580（2012）05—0054—02一、引言张量作为物理或几何的具体对象，充分反映了这些现象的物理和几何属性，是这些现象的一种数学抽象，在分析力学、固体力学、流体力学、几何学、电磁场理论和相对论等方面有着广泛的应用。
张量（tensor ）是几何与代数中的基本概念之一，从代数角度讲，张量是数量、向量、矩阵的自然推广，在为n空间中的N 阶张量有n N个分量，下面是n =2时的张量示意图：T（T 1，T 2）标量（阶N =0）矢量（阶N =1）T 11T 12T 21T ()22矩阵（阶N =2）张量（阶N =3）可见，零阶张量可用一个数表示，一阶张量可用一行数组表示，二阶张量可用矩阵表格表示，三阶张量可用“立体矩阵”表示，更高阶的张量不能用图形表示，正因为如此，关于张量的推演计算有时会很复杂繁琐。
利用MATLAB 可以使复杂繁琐的推演计算变得简单方便。
由于难以见到相关的文献，在此作简要的介绍，以方便读者学习。
二、张量运算函数命令MATLAB 是通过调用MAPLE 的张量包（ten-sor ）进行运算的，格式为：＞＞maple （‘函数名’），或者借用procread 指令把整段MAPLE 程序送往MAPLE 计算。
雅格比行列式
摘要：
1.雅格比行列式的定义
2.雅格比行列式的性质
3.雅格比行列式的计算方法
4.雅格比行列式在数学中的应用
正文：
雅格比行列式（Jacobian matrix）是一个数学概念，它用于描述一个可微函数在给定点处的偏导数。
具体来说，对于一个函数$f: mathbb{R}^n ightarrow mathbb{R}^m$，在点$x in mathbb{R}^n$处的雅格比行列式是一个$m times n$的矩阵，其元素为：
$$D_{ij} = frac{partial f_i}{partial x_j}(x)$$
其中$f_i$表示函数$f$的第$i$个分量。
雅格比行列式的性质包括：
1.行列式中的每个元素都是雅格比行列式的雅可比行列式。
也就是说，如果我们有一个$p times q$的雅格比行列式，那么它的每个元素都是一个$p times q$的雅可比行列式。
2.雅格比行列式的行列式值等于函数$f$在点$x$处的梯度向量的范数。
梯度向量是函数在该点处的局部最速上升方向。
3.雅格比行列式的行列式值可以用函数$f$在点$x$处的海塞矩阵的行列式来表示。
海塞矩阵是函数$f$在点$x$处的梯度向量和一个标量的乘积。
4.如果函数$f$在点$x$处可微并且连续，那么雅格比行列式在点$x$处是
连续的。
计算雅格比行列式的方法有多种，其中最常用的是高斯消元法。
在实际应用中，雅格比行列式常用于求解微分方程、优化问题和最优化问题。
雅可比矩阵(Jacobi方法)Jacobi 方法Jacobi方法是求对称矩阵的全部特征值以及相应的特征向量的一种方法，它是基于以下两个结论1) 任何实对称矩阵A可以通过正交相似变换成对角型，即存在正交矩阵Q,使得Q T AQ = diag(λ1 ,λ2,…,λn) (3.1)其中λi(i=1,2,…,n)是A的特征值，Q中各列为相应的特征向量。
2) 在正交相似变换下,矩阵元素的平方和不变。
即设A=(aij )n×n,Q交矩阵,记B=Q T AQ=(bij )n×n, 则Jacobi方法的基本思想是通过一次正交变换,将A中的一对非零的非对角化成零并且使得非对角元素的平方和减小。
反复进行上述过程,使变换后的矩阵的非对角元素的平方和趋于零，从而使该矩阵近似为对角矩阵，得到全部特征值和特征向量。
1 矩阵的旋转变换设A为n阶实对称矩阵，考虑矩阵易见 Vij(φ)是正交矩阵, 记注意到B=VijA的第i,j行元素以及的第i,j列元素为可得≠0，取φ使得则有如果aij对A(1)重复上述的过程，可得A(2) ,这样继续下去, 得到一个矩阵序列{A(k) }。
可以证明，虽然这种变换不一定能使矩阵中非对角元素零元素的个数单调增加，但可以保证非对角元素的平方和递减，我们以A与A(1)为例进行讨论。
设由式(3.4)可得这表明,在上述旋转变换下,非对角元素的平方和严格单调递减,因而由(3.2)可知，对角元素的平方和单调增加。
2. Jacobi方法通过一系列旋转变换将A变成A(k+1) ,求得A的全部特征值与特征向量的方法称为Jacobi方法。
计算过程如下1)令k=0, A(k) =A2) 求整数i,j, 使得3) 计算旋转矩阵4) 计算A(k+1)5) 计算6) 若E(A(k+1))<ε, 则为特征值，Q T = (V(0) V(1)…V(k+1))T的各列为相应的特征向量；否则，k+1=>k返回2,重复上述过程。
雅可比矩阵和行列式-概述说明以及解释1.引言1.1 概述雅可比矩阵和行列式是线性代数中的两个重要概念，它们在数学和物理学等领域中具有广泛的应用。
雅可比矩阵是由一组向量的偏导数组成的方阵，而行列式则是一个矩阵的一个标量值。
雅可比矩阵在数学和工程领域中有着广泛的应用。
它可以用来描述多变量函数的导数，从而在优化和控制理论中起到关键作用。
雅可比矩阵还可以用来解决线性方程组、求解非线性方程和最小二乘法等问题。
在机器学习和人工智能领域，雅可比矩阵常常用于计算梯度和求解优化问题。
行列式是线性代数中另一个重要的概念。
它是一个方阵的一个标量值，常用来描述线性变换对空间的拉伸和旋转效果。
行列式的值可以告诉我们方阵的特征，比如它是否可逆或奇异。
行列式也可以用来解决线性方程组的问题，判断线性相关性和计算向量的体积。
本文将从定义、性质、计算方法和应用领域四个方面介绍雅可比矩阵和行列式。
首先，我们将给出雅可比矩阵和行列式的数学定义，为读者提供清晰的概念框架。
然后，我们将详细讨论它们的性质，包括可逆性、特征值和特征向量等。
接下来，我们将介绍计算雅可比矩阵和行列式的方法，包括手工计算和数值计算。
最后，我们将探讨雅可比矩阵和行列式在各个领域的应用，包括优化、控制理论、机器学习等。
通过对雅可比矩阵和行列式的全面讨论，本文旨在帮助读者深入理解它们的概念和应用。
这将为读者在数学和工程领域的学习和研究提供基础，并鼓励读者进一步探索相关领域的知识。
在本文的结论部分，我们将总结主要观点，并展望未来对雅可比矩阵和行列式的研究方向。
最后，我们还将提供一些建议进一步阅读的参考资料，以便读者深入学习和了解这一领域的更多内容。
1.2 文章结构文章结构部分的内容可以描述整篇文章的组织和内容分布。
以下是可以使用的示例内容：在本篇文章中，我们将讨论雅可比矩阵和行列式的相关概念、性质、计算方法和应用领域。
文章主要分为四个部分。
第一部分是引言部分。
我们将概述本文的主题，介绍雅可比矩阵和行列式在数学和应用领域的重要性。
ICSET2008Abstract—A method based on “partial Jacobian matrix” for assessing the voltage support ability of integrated wind farms in power system is proposed. The focused nodes are regarded as injection node. And then the “partial Jacobian matrix”, which only includes nodes in the studied region, is obtained by eliminating the elements relevant to all other nodes. The sensitivities of point of interconnection (POI) voltage on local reactive power injection and reactive power injection of wind farm are derived. The proposed method can be used to determine the appropriate reactive compensation location and capacity by comparison of related sensitivities. Numerical results on the actual system with integrated wind farms shows that the method proposed in this paper is effective, feasible and adaptive.Index Terms— Power systems, wind power, partial Jacobian matrix, sensitivity, voltage support.I.I NTRODUCTIONS one of the most feasible renewable energy, windpower will be developed more widely because of the sustained developing strategy. Therefore, its impact on power system [1]−[4], operation characteristic [5]−[7] and control method in different operation modes as in [8]−[11] are more and more paid attention to.Wind generator can be divided into two types, namely fixed-speed and variable-speed. Fixed-speed wind generator is used in earlier wind farms. The lack of voltage regulation ability makes it need to absorb reactive power from grid with active power injected into grid, maybe leading voltage sag [6],[12]−[13]. Traditional solution is to install reactive power compensation devices at wind farm bus or POI to compensate the reactive power command, as in [12]. Reference [14] has studied the impact on transient voltage stability in wind farm by installing static var compensator(SVC) at wind farm bus. Then capacitors Manuscript received May 7, 2008. This work was supported in part by Key Project of the National Eleventh-Five Year Research Program of China under Grant 2008BAA14B04. Full names of authors are preferred in the author field, but are not required. Put a space between authors' initials.Ning Chen is with the State Grid Electric Power Research Institute, sub-branch of State Grid Corporation of China, Nanjing 210003, Jiangsu Province, China, phone:86-25-83098031; fax:86-25-83098025; e-mail: chenning8375@.Lingzhi Zhu is with the State Grid Electric Power Research Institute, sub-branch of State Grid Corporation of China, Nanjing 210003, Jiangsu Province, China, e-mail: zhulingzhi@.Wei Wang is with the State Grid Electric Power Research Institute, sub-branch of State Grid Corporation of China, Nanjing 210003, Jiangsu Province, China, e-mail: wangwei@.Xiaodong Zhu is with the State Grid Electric Power Research Institute, sub-branch of State Grid Corporation of China, Nanjing 210003, Jiangsu Province, China, e-mail: zhuxiaodong@. control rules at POI was studied by genetic algorithm in [15]. Also reactive power compensation technology for improving low voltage ride through (LVRT) ability of wind generator is proposed in [16]−[17]. However, with wind power installation increasing, the voltage problem will become more serious.Doubly-fed induction generator (DFIG) has gradually become the main stream in new wind farm. DFIG can realize variable-speed constant-frequency control and decoupling of active and reactive power, with its stator interconnected into grid and AC excitation control by rotor side converter, as in [8]−[9]. DFIG can take part in voltage control in integrating region by absorbing or generating reactive power according to system operation mode and control strategy. Many scholars have made a lot of research on voltage stability and control problems caused by wind farm integrated into grid. A method of calculating wind generator reactive power limit based on state model of DFIG is proposed in [18]. A control and distribution strategy for local reactive power compensation by DFIG is proposed in [19]. However, for a large-scale wind farm with reactive power generating ability, problems such as how to assess its voltage support ability and choose the best location to compensate reactive power need to be studied further. For the characteristic of wind power application in China, namely many large-scale wind farms integrated into grid, researches on these problems are significant to construction layout and operation of wind power.This paper will research the above problems using static analysis method and give a method of assessing voltage support ability in wind power integrating region. The concept of “partial Jacobian matrix” is described in Section II ,the method of assessing voltage support ability in Sectio Q ċ, and the numerical analysis in Section IV. Finally, the conclusions are drawn in Section V.II.C ONCEPT OF P ARTIAL J ACOBIAN M ATRIX Assumed that there is power injected into every node in system, system Jacobian matrix will include sensitivities information of voltage on power injection of all nodes. But in actual power system, there are only partial nodes with load or generator connected to, so that the analysis method based on traditional Jacobian matrix can not avoid influence caused by nodes without power injection. Reference [20] proposed the concept of “injection node reduced Jacobian matrix”, in which all nodes are divided into “injection node” with power injection and “connection node” without powerPartial Jacobian Matrix Based Method for Assessing Reactive Power/Voltage Support Ability of Wind FarmN. Chen, L. Z. Zhu, W. Wang and X. D. ZhuAinjection. Then the influence on analysis result caused bynonexistent injected power can be avoided by eliminating “connection node”. Based on this, the concept of “partial Jacobian matrix” is proposed, in which load nodes, generator nodes and nodes with reactive power compensation devices are regarded as “injection node” andall other nodes as “connection node”. By further reducing order of Jacobian matrix, “partial Jacobian matrix” including sensitivities information of voltage on node injected power is derived. The detailed process is describedbelow.In power flow calculation, common modified equation isgenerally defined as X J Y Δ⋅=Δ (1) Where, »¼º«¬ª=QV Q șPV P șJ J J J J is system Jacobian Matrix in polar coordinate,θ∂∂=P P șJ ,V P ∂∂=PV J ,θ∂∂=Q Q șJ andV Q ∂∂=QV J are sub-matrixes of J .T][Q P Y ΔΔ=Δis injected power vector,T ][V șX ΔΔ=Δis state variablesvector including node voltage and phase angle. In case that ΔY is divided into two parts as ΔY =[ΔY I ΔY N ]T ,ΔX will be ΔX =[ΔX I ΔX N ]Ttoo. ΔY I and ΔX I are injected power vector and state variables vector of “injection node”, ΔY N and ΔX N are injected power vector and statevariables vector of “connection node”. Accordingly, J isdivided into four parts, the modified equation is described as »¼º«¬ªΔΔ⋅»¼º«¬ª=»¼º«¬ªΔΔN I NN NI IN II N I X X J J J J Y Y (2) Assumed that outer network voltage and injected power of “connection node” are constant, the actual change of injected power can approximatively be regarded as ΔY I . The modified equation becomes ¯®Δ⋅+Δ⋅=Δ⋅+Δ⋅=ΔNNN I NI N IN I II I 0X J X J X J X J Y (3) Generally speaking, J NN is nonsingular, so ΔX N can be eliminated by matrix transformation. Then ΔY I can be described as I R I I NI 1NN IN II I )(X J X J J J J Y Δ⋅=Δ⋅⋅⋅−=Δ− (4)Where,NI 1NNIN II R I J J J J J ⋅⋅−=− is partial Jacobian matrix,»¼º«¬ª′′′′=QVQ șPVPșRI J J J J J ,ΔY I and ΔX I are described as T ][I II Q P Y ΔΔ=Δand T ][I II V șX ΔΔ=Δ.It is seen from above that for a n-dimension system, if the dimension of injected power vector of studied region is less than n, the relation between injected power and state variables can be derived by reducing order of Jacobian matrix.Further, considering the relativity of injected active power and reactive power in every node, the relation can bedescribed as ΔQ =K ×ΔP (5) In equation (5), »»»¼º«««¬ª=m k k K 0000001%,ii i dP dQ k =.Then equation (4) can be described as »¼º«¬ªΔΔ⋅»¼º«¬ª′′′′=»¼º«¬ªΔ⋅ΔI I QV Q șPV PșI I V șJ J J J P K P (6) Usually, J ′P θis nonsingular, the below equation is derived from equation (6) as I PV1P șQ șQV I 1P șQ ș)()(V J J J J P J J K Δ⋅′⋅′⋅′−′=Δ⋅′⋅′−−− (7)Further, if )(1P șQ ș−′⋅′−J J K is nonsingular, the “partial Jacobian matrix” considering the relation between active power and reactive power can be derived as)()(PV1P șQ șQV 11P șQ șSI J J J J J J K J ′⋅′⋅′−′′⋅′−=−−− (8) Equation (8) reflects that the relation between nodevoltage magnitude and injected active power when injectedactive power variation is proportional to that of reactivepower. Because injected active power and reactive powerare relative, the analysis of voltage stability by J SI will be more exact and according with the fact. III.A SSESSING M ETHOD A.Method Introduction As wind farm always locates at the end of transmissionlines and its output power change only influences operation status in the region with it integrated into, we choose windpower integrated region as research object. An equivalentsystem with integrated wind farms is shown in Fig 1. The equivalent system contains m wind farms which are interconnected into infinite electric network through n POIs. In the system, wind farms and POIs are selected as “injection node” and all other nodes selected as “connection node”.The relation between injected power and state variable can be derived by 2(m+n) dimension “partial Jacobian matrix” J SI , which is described as[][]TT C W C W SI C W C W ,,,,,,V V șșJ ǻQ ǻQ ǻP ǻP ΔΔΔΔ⋅= (9)Where, ΔV W and ΔθW are voltage and phase angle vectorsof wind farms, ΔV C and ΔθC are voltage and phase angleFarm 1Farm mvectors of POIs, ΔP W and ΔQ W are injected active power and reactive power vectors of wind farms, ΔP C and ΔQ C are injected active power and reactive power vectors of POIs. Inverse matrix of J SI and equation (9) are described as»»»¼º«««¬ª=++++−)(2),(21),(2)(2,11,11SI n m n m n m n m A A A A "#%#"J (10)[][]T T C W C W 1SI C W C W ,,,,,,Q Q P P J V V șșΔΔΔΔ⋅=ΔΔΔΔ− (11)Then voltage magnitude change of i th POI can becalculated according to¦+=++++++Δ⋅+Δ⋅=Δn m j j j n m i n m j ji n m i Q A P AV 1,2,2)( (12)If active power output of wind farm is ignored, the sensitivity of voltage of i th POI on reactive power of j th wind farm can be calculated according toj n m i n m j i A Q V S ++++=ΔΔ=,21 (13)If injected active power of POI is not considered, the sensitivity of voltage of i th POI on local reactive load or compensation can be calculated according toi n m i n m iiA Q V S ++++=ΔΔ=2,22 (14)The best reactive power compensating location can be chosen according to equation (13) and (14). If S 1>S 2at i th POI,it indicates that reactive power of wind farm influences voltage of i th POI more. So regulating reactive power of wind farm or compensating reactive power at wind farm bus will be more effective. By contrary, it indicates that local reactive power influences voltage of i th POI more. So compensating reactive power at i th POI will be more effective.If we only consider the influence of output power of j th wind farm on the voltage of i th POI, the voltage magnitude change of i th POI can be calculated according to)(,21,2j j n m i n m mj j ji n m i Q A P AV Δ⋅+Δ⋅=Δ++++=++¦ (15)Where, ΔQ j ∈[Q j,min −Q j,0ˈQ j,max −Q j,0]ˈQ j,min and Q j,max are upper and lower limit of j th wind farm, Q j,0is initial reactive power output of j th wind farm.As the impact on voltage of POI is generally decided by reactive power output of wind farm, the voltage support ability index of all wind farms on i th POI can be calculated according to¦+++=++Δ⋅=Δnm n m j j j i n m i Q A V 21,2 (16)The voltage support ability index of j th wind farm on i th POI can be calculated according toj n m j n m i n m i Q A V ++++++Δ⋅=Δ,2 (17)putation Process(1)Initial load flow calculation,(2)Calculating initial system Jacobian matrix,(3)Finding elements related to “injection nodes” in initial Jacobian matrix,(4)Calculating “partial Jacobian matrix” and inverting it, (5)Calculating the sensitivities of voltage of POI on local reactive power and reactive power of wind farm,(6)Calculating the voltage support ability index according to equation (15) or (17),(7)Comparing S 1 with S 2 to decide the best reactive power compensation location.IV.N UMERICAL A NALYSISA.Test System IntroductionThe test system is an actual grid with integrated wind farms. A part region with three wind farms integrated into is used to be analyzed and testify the proposed method. The equivalent system is shown in Fig.2. Three wind farms interconnect to the mains by two POIs. Each wind farm consists of 134 wind generators, in which maximum active power of single generator is 1.5 MW and its power factor ranges from -0.95 to 0.95.B.Voltage Support Ability AssessmentAssumed that all wind farms whose initial power factor are all 1 operate in constant power factor mode, the voltage support ability indices of wind farm are calculated and testified by actual power flow. The voltage support ability indices of single wind farm action on POI and actual power flow in normal operation mode are shown in TABLE I. The voltage support ability indices of multi-wind farms action together on POI and actual power flow in normal operation mode are shown in TABLE II.wind farm1wind farm2wind farm3˖ Fig. 2 Test system.From TABLE I and TABLE II, it is known that thevoltages calculated by the proposed sensitivities indices of single wind farm action or multi-wind farm action together are both basically consistent with actual power flow. For single wind farm action, the largest error is 0.03%. For multi-wind farm action, the largest error is 0.12%.Summarizing the above analysis, the proposed method can assess the voltage support ability of reactive power regulation of wind farm. The operation mode adaptability is analyzed as below. Firstly, four modes are defined as follow: Mode1: normal operation,Mode2: line broke between POI1 and POI2,Mode3: load of POI1 reduced to forty percents of the initial value,Mode4: power output and loads all reduced to fifty percents of the initial value.Then the voltage support ability of wind farm in different operation mode is calculated. The result of wind farm 1 is shown in TABLE III. Results of else wind farm are similar with wind farm 1.From TABLE III, it is known that the largest sensitivity index error is 5% when some transmission lines broke. So proposed sensitivity indices have good adaptability in different operation modes. Besides, the fact that the voltages calculated by the proposed sensitivity indices are still basically consistent with actual power flow further proves that the proposed method is efficient.C.Choice of Reactive Compensation Location Sensitivity indices S 1and S 2will change with power factor changing. Sensitivity curves of voltage of POI on local reactive power regulation are shown in Fig.3. And sensitivity curves of voltage of POI on reactive power regulation of wind farm are shown in Fig.4. In the condition that power factor is 0.99 0r -0.99 and reactive power regulation is 0.35p.u., the voltage changes of POI caused by regulating reactive power of wind farm and compensating reactive power at POI are shown in TABLE IV.In Fig3, it is shown that the point whose S 1 equals to S 2 is selected as the critical point for reactive power compensation in some given power factor. If S 1>S 2,compensating reactive power at wind farm bus will be more efficient. By contrary, compensating reactive power at POI will be more efficient.power factors e n s i t i v i t yFig. 4 Sensitivity curve of voltage of POI on reactive power regulation of wind farm.power factors e n s i t i v i t y Fig. 3 Sensitivity curve of voltage of POI on local reactive power regulation.power compensation location is feasible and efficient and the reactive power compensation can be calculated by proposed sensitivity indices and voltage command.V.C ONCLUSIONA partial Jacobian matrix based method for assessing reactive power/voltage support ability of wind farm is proposed in this paper. This method can calculate sensitivity of voltage of POI on reactive power of wind farm and local reactive power deviation. Further, the voltage support ability of wind farm is given. Besides, the proposed sensitivity indices can be applied on choosing the best reactive power compensating location, which is significant to construction layout and operation of wind power. Test system analysis shows that:a. The voltage support ability can be calculated according to proposed sensitivity S1 reflecting voltage change of POI caused by reactive power of wind farm,b. The proposed sensitivity S1 is adaptive and suitable for engineering application,c. The sensitivity indices S1and S2can reflect voltage support ability of reactive power injection of different nodes. By comparing S1 with S2, it is helpful to choose proper reactive power compensating location.R EFERENCES[1]J. P. E. Hochheimer, “Wind generation integration &operation-technical challenges/issues”, presented at IEEE PowerEngineering Society General Meeting, Montreal, Canada, June 18-22,2006.[2] A. M. Azmy, I. Erlich, “Impact of distributed generation on thestability of electrical power system”, presented at IEEE PowerEngineering Society General Meeting, San Francisco, CA, UnitedStates, June 12-16, 2005.[3]I. Erlich, K. Rensch, F. 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