Short-circuit current1 Terms and DefinitionsThe following terms and definitions correspond largely to those defined in IEC 60 909. Refer to this standard for all terms not used in this book.The terms short circuit and ground fault describe faults in the isolation of operational equipment which occur when live parts are shunted out as a result.• Causes:1. Overtemperatures due to excessively high overcurrents.2. Disruptive discharges due to overvoltages.3. Arcing due to moisture together with impure air, especially on insulators.• Effects:1. Interruption of power supply.2. Destruction of system components.3. Development of unacceptable mechanical and thermal stresses in electrical operational equipment.• Short circuit:According to IEC 60 909, a short circuit is the accidental or intentional conductive connection through a relatively low resistance or impedance between two or more points of a circuit which are normally at different potentials.• Short circuit current:According to IEC 60 909, a short circuit current results from a short circuit in an electrical network.It is necessary to differentiate here between the short circuit current at the position of the short circuit and the transferred short circuit currents in the network branches.• Initial symmetrical short circuit current:This is the effective value of the symmetrical short circuit current at the moment at which the short circuit arises, when the short circuit impedance has its value from the time zero.• Initial symmetrical short circuit apparent power:The short circuit power represents a fictitious parameter. During the planning of networks, the short circuit power is a suitable characteristicnumber.• Peak short circuit curre nt:The largest possible momentary value of the short circuit occurring.• Steady state short circuit current:Effective value of the initial symmetrical short circuit current remaining after the decay of all transient phenomena.• DC aperiodic component:Average value of the upper and lower envelope curve of the short circuit current, which slowly decays to zero.•Symmetrical breaking current:Effective value of the short circuit current which flows through the contact switch at the time of the first contact separation.• Equivalent voltage source:The voltage at the position of the short circuit, which is transferred to the positive-sequence system as the only effective voltage and is used for the calculation of the short circuit currents.• Superposition method:The superposition method considers the previous load of the network before the occurrence of the short circuit. It is necessary to know the load flow and the setting of the transformer step switch.• Voltage factor:Ratio between the equivalent voltage source and the network voltage Un,divided by 3.• Equivalent electrical circuit:Model for the description of the network by an equivalent circuit.• Far-from-generator short circuit:The value of the symmetrical AC periodic component remains essentially constant.• Near-to-generator short circuit:The value of the symmetrical AC periodic component does not remain constant. The synchronous machine first delivers an initial symmetrical short circuit current which is larger than twice the rated current of the synchronous machine.• Positive-sequence short circuit impedance:The impeda nee of the positive-seque nee system as see n from the positi on of the short circuit.• Negative-seque nee short circuit impeda nee:The impeda nee of the n egative-seque nee system as see n from the positi on of the short circuit.• Zero-seque nee short circuit impeda neeThe impeda nee of the zero-seque nee system as see n from the positi on of the short circuit. Three times the value of the neutral point to ground impeda nee occurs here.• Short circuit impeda nee:Impeda nee required for calculati on of the short circuit curre nts at the positi on of the short circuit.1・2 Short circuit path in the positive-sequence systemFor the same exter nal con ductor voltages, a three-pole short circuit allows three curre nts of the same magn itude to develop betwee n the three eon ductors. It is therefor only n ecessary to eon sider one eon ductor in further calculati ons. Depe nding on the dista nee from the positi on of the short circuit from the gen erator, here it is n ecessary to con sider n ear-to-ge nerator an dfar-from-ge nerator short circuits separately.For far-from-generator and near-to-generator short circuits, the short circuit path can be represe nted by a mesh diagram with AC voltage source, reacta nces X and resista nces R (Figure 1.2). Here, X and R replace all comp onents such as cables,c on ductors, tra nsformers, gen erators and motors.Fig. 1.2: Equivale nt circuit of the short circuit curre nt path inthe positive-seque nee systemThe following differential equation can be used to describe the short circuitprocesswhere w is the phase angle at the point in time of the short circuit. This assume that the curre nt before S closes (short circuit) is zero. The in homoge neous first order differe ntial equati on can be solved by determ ining the homogeneous solution ik and a particular solution i2k.The homogeneous solution, with the time constant g = L/R, solution yields:For the particular solution, we obtain:* -』显:牡j站伽+串一叫-The total short circuit curre nt is composed of both comp onen ts:fL = y 話 M打 询伽+申_吨一占 鈕妙一巩厂The phase an gle of the short circuit curre nt (short circuit an gle) is the n, in accorda nee with the above equati on,% = — v = arctan 菱.For the far-from-ge nerator short circuit, the short circuit curre nt is therefore made up of a con sta nt AC periodic comp onent and the decay ing DC aperiodic component. From the simplified calculations, we can now reach the following con clusi ons:• The short circuit curre nt always has a decay ing DC aperiodiccomp onent in additi on to the stati on ary AC periodic comp onent.• The magn itude of the short circuit curre nt depe nds on the operat ingan gle of the curre nt. It reaches a maximum at c = 90 (purely in ductive load). This case serves as the basis for furthercalculati ons.• .The short circuit current is always inductive.1.4 Methods of short circuit calculationThe equivale nt voltage source will be in troduced here as the only effective voltage of the gen erators or n etwork in puts for the calculati on of short circuit curre nts. The in ter nal voltages of gen erators or n etwork in puts are short circuited, and at the positi on of the short circuit (fault positi on) the value ( is used as the only effective voltage (Figure 1.4).• The voltage factor c [5] con siders (Table 1.1):• The different voltage values, depending on time and position• The step cha nges of the tra nsformer switch• That the loads and capacita nces in the calculati on of the equivale nt voltage source can be n eglected• The subtra nsie nt behavior of gen erators and motors• This method assumes the followi ng con diti ons:• The passive loads and con ductor capacita nces can be n eglected• The step setting of the tra nsformers do not have to be con sidered• The excitatio n of the gen erators do not have to be con sidered• The time and positi on depe ndence of the previous load (load ing state) of the n etwork does not have to be con sideredJL// /z/rFault positionFig. 1.4: Network circuit with equivale nt voltage source a) three-phase n etwork, b) equivale nt circuit in positive seque nee system益 竝 FVoltage farter r: fbf orLhe largest shortcitclul cutretil1Lrnaxthe smallest short cltclljL curre tiL■An-inlrw vdLage加0.95JOO V s lOOO VfBC 3E. Table 1|1.101Mc-diLLm1.10l.M> lkV to55 kV]ligh vol tiged 35脚1| cmax U.H mitsL notei^eed Lh.e hjghe^L Uffl fetr 心peral蚯nal«eqLiipmenl lh lh.e tie2) for low voltage nefWDtks wtha tolerance of +&%1) for lew wltagp netwmtkg 垃th a tcleranjn1 cFj-I%1.4.2 Superposition methodThe superposition method is an exact method for the calculation of the short circuit curre nts. The method con sists of three steps. The voltage ratios and the load in g con diti on of the n etwork must be known before the occurre nce of the short circuit. In the first step the curre nts, voltages and the in ter nal voltages for steady-state operati on before on set of the short circuit are calculated (Figure 1.5b). The calculati on con siders the impeda nces, power supply feeders and node loads of the active eleme nts. In the sec ond step the voltage applied to the fault locati on before the occurre nce of the short circuit and the curre nt distributi on at the fault locati on are determ ined with a n egative sig n (Figure 1.5c). This voltage source is the on ly voltage source in the n etwork. The internal voltages are short-circuited. In the third step both conditions are superimposed. We the n obta in zero voltage at the fault locati on. The superpositi on of the curre nts also leads to the value zero. The disadva ntage of this method is that the steady-state condition must be specified. The data for the n etwork (effective and reactive power, node voltages and the step se ttings of the transformers) are often difficult to determine. The question also arises, which operating state leads to the greatest short circuit current. Figure 1.5 illustrates the procedure for the superposition method. r* i i b) __ >PowerP内QnelwurkL:'irQ IPuwer nelwurkFig. 1.5: Pri nciple of the superpositi on methoda) undisturbed operation, b) operating voltage at the fault locati on, c) superpositi on of a) and b)1.4.3 Transient calculationWith the tra nsie nt method the in dividual operat ing equipme nt and, as a result, the entire network are represented by a system of differential equations. The calculatio n is very tedious. The method with the equivale nt voltage source is a simplification relative to the other methods. Since 1988, it has been sta ndardized in ter natio nally in IEC 60 909. The calculatio n is in depe ndent of a current operational state. In thisbook, we will therefore deal with and discuss the method with the equivale nt voltage source.1.5 Calculating with reference variablesThere are several methods for performing short circuit calculations with absolute and refere nce impeda nce values. A few are summarized here and examples are calculated for comparis on. To defi ne the relative values, there are two possible refere nce variables.For the characterizati on of electrotech ni cal relati on ships we require the four parameters:• Voltage U in V• Curre nt I in A• Impedance Z in W• Apparent power S in VA.Three methods can be used to calculate the short circuit current:1. The Ohm system: Units: kV, kA, V, MVA2. The pu system:This method is used predominantly for electrical machines; all four parameters u, i, z and s are given as per unit (unit = 1). The reference value is 100 MVA. The two reference variables for thissystem are UB and SB.Example: The reactances of a synchronous machi ne Xd, Xcd, X2d are give n in pu or in % pu, multiplied by 100 %.3. The %/MVA system:This system is especially well suited for the fast determination of short circuit impedances. As formal unit only the % symbol is add.短路电流1 术语和定义以下术语和定义对应 IEC 标准 60 909。
未出现在本书中的术语可以在该 标准中查询短路和接地故障主要是操作设备的带电部分被分流而导致绝缘损坏的结果 •原因1. 温度过高导致强烈的过电流；2. 火花放电导致过电压3. 由于水分和污秽空气混杂导致的电弧作用，特别是在绝缘体上• 后果：1. 供电中断2. 系统部件瘫痪3. 在电气操作设备中产生不可接受的机械力和热应力• 短路：根据IEC 60 909，短路是经历一段相对低电阻或在两个或更多不同电位 之间的电阻间意外或故意的导电连接• 短路电流：根据IEC 60 909，短路电流是在电力网络中短路的结果在这里有必要 区分在短路过程中产生的短路电流和在网络分支中的转移电流• 初始对称短路电流： 这是在短路出现瞬间，短路阻抗从零开始变化时的对称短路电流的有效 值• 初始对称短路视在功率： 短路功率代表了一个虚构的参数在网络规划中，短路功率是一个合适 的典型参数• 最大短路电流: 短路时可能的最大电流瞬时值• 稳态短路电流： 初始对称短路电流在暂态过程中衰减完毕之后的电流有效值• 短路电流非周期分量： 慢慢衰减的短路电流上下包络曲线的平均值• 对称断路电流： 联络开关第一次接触分离时流过短路电流的有效值。
• 等效电压源：被转移到正序系统作为唯一有效的短路位置的电压，并且主要用于短路 电流的计算• 叠加方法：叠加法考虑到在发生短路前网络的负荷情况因此很必要知道负荷留了 和变压器开关的设定• 电压因素： 等效电压源和网络电压之间除以三的比例• 等效电路： 网络描述的模型采用等效电路• 远离发电机短路： 对称交流周期分量维持原本不变的量的短路形式• 靠近发电机短路： 对称交流周期分量不保持不变的值的短路同步机首先会产生一个大于 两倍额定电流的初始对称短路电流• 正序短路阻抗：短路位置正序系统的短路阻抗• 负序短路阻抗：短路位置负序系统的短路阻抗• 零序短路阻抗： 短路位置零序系统的短路阻抗就是中性点到短路位置阻抗的三倍• 短路阻抗：计算短路位置的短路电流所需要的阻抗1.2 正序系统的短路路径：对于相同的外部导体电压，三相短路允许同一数量级的三相电流在三相导体 中发展所以在进一步计算中只需要考虑一相导体的情况根据短路位置到发电 机的距离，这里有必要将远离发电机短路和靠近发电机短路这两种情况分开考 虑对于远离和靠近发电机短路的情况，短路路径可以用一个有交流电压源，电 抗X,电阻R构成的网络图表示。
图1.2）这里X和R替代所有的原件，如电 缆，导体，变压器，发电机和电机■Rk~i Xk 厂 让ik图1.2短路电流路径在正序系统中的等效电路下面的微分方程可以用来描述短路过程-X 讥+7才-- , (1.1)「是短路点的相位角这是假设电流在S关闭(短路)之前是零非线性一 阶微分方程可以通过决定齐次解ik和特解Pk求解m (1.2)齐次解有一个时间常量、=亍，方程式为：t■：<= (1.3)对于特解，我们得出：■=二二三—一二 (1-4)总短路电流由两部分构成：t■:< -二二［J _ 壬二 1 — (1.5)根据以上方程，单相短路电流的短路角为：二=7 一(1.6)对于远离发电机形式的短路，短路电流是由一个不变的交流周期分量和一 个衰减的直流非周期分量构成从简化计算，我们可以得出以下结论：短路电流总是由一个固定的交流周期分量和一个衰减的直流非周期分量 构成；• 短路电流的大小取决于电流的工作角，最大值为90° （纯电感负载）这 种情况作为进一步计算的基础• 短路电流都是感应的1.4 短路电流计算方法:三相系统中的短路电流有三种计算方法：1. 在故障位置计算等效电压源；2. 叠加法确定负载流量情况；3. 瞬态计算。
1.4.1 等效电压源:这里的等效电压源主要作为发电机或投入电网的短路电流计算的唯一有效电 源发电机和投入电网的内部电压是短路的，短路地点（故障位置）的值就作为 唯一有效电压 （图1.4 ）• 电压因素考虑3[5] （表1.1）• 不同电压值取决于时间和地点的不同• 变压器开关的阶跃变化• 等效电压源的计算中负荷和容量都可忽略不计• 发电机和电机的起始状态该方法假设以下条件：• 被动负荷和导体容量可以忽略不计• 变压器的步骤设定可以不需考虑• 发电机的激励不需要考虑• 前负荷的负荷状态的时间和位置可以忽略不计QMVLVUk故障位置Xc Rc Xt Rt Xl RlIk''图1・4具有等效电压源的网络电路a)三相系统b)正序系统的等效电路表 1.1 根据E DIN IEC 73/89/CDV (VDE 0102, Part 100):1997-08的电压因素网络电压电压因素Un最大短路电流最小短路电流CmaxCmi n低压1.050.95100 V 至 1000V（IEC38 表1）l,10a中压1.101.001KV至 35KV高压大于35KVCmaxU n不能超过网络中操作设备的最大工作电压Um在低压网络有+6%的公差在低压网络有+10%的公差1.4・2 叠加法：叠加法是一种精确的计算短路电流的方法。
该方法包含三个步骤在短路发 生前变压器的变压比和网络负荷条件必须已知在第一阶段，稳态允许下的电流, 电压，稳态电压在短路前都要先计算出来（图1.5b）计算考虑了阻抗，电源和有 源元件的节点负荷第二步，在短路前故障位置处的电压和电流分配要加以符号 确定（图1.5c）电压源是网络中唯一的内部电压时短路的在第三阶段两种 状态会叠加起来我们最终在故障点获得零序电压电流的叠加同样也会导致零 值这种方法的缺点是稳定状态必须被指定网络参数（有功功率和无功功率， 节点电压和变压器的步骤设定）往往是很难决定的问题同时出现，这将导致工 作状态时出现最大的短路电流图1..5叠加法原理a）稳定系统b）故障处运行电压c） a和b的叠加1.4.3 瞬态计算:用瞬态方法计算每个设备时，整个网络被一系列的微分方程所表示计算过 程非常乏味拥有等效电压源的这种方法相比其他方法比较简单自1988年以 来已经在IEC 60 909中被国际化规范运算和当前运行状态是独立的所以在 这本书中我们将解决和讨论有等效电压源的这种方法1.5 参考变量的计算:有很多方法根据绝对的和参考抗值进行短路计算一些在这儿已经总结出 来，还有实例加以比较。
为了定义相对值，有两种可能的参考变量为表征电工关系，我们要求四个参数：•单位为V的电压U•单位为A的电流I•单位为KW的阻抗Z以下三种方法可以用来计算短路电流：1. 欧姆系统：单位：kV, kA, V, MVA2. 标幺值系统：这种方法主要用于发电机系统：所有四个参数，u,i,还有s都被给定了值（单位=1）参考容量是100MV系统的两个参考变量 分别是UB和SB例如：同步机的电抗Xd,Xd'Xd'P以标幺值的形式被 给定，或者乘以100%以百分比标幺值形式给定3. %/MVA系统 这种方法特别适用于短路阻抗的快速测定作为正式单位用%加以补充。