Short-circuit current 1 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.l 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.l Effects:1. Interruption of power supply.2. Destruction of system components.3. Development of unacceptable mechanical and thermal stresses in electrical operational equipment.l 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.l 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.l 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.l 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 characteristic number.l Peak short circuit current:The largest possible momentary value of the short circuit occurring.l Steady state short circuit current:Effective value of the initial symmetrical short circuit current remaining after the decay of all transient phenomena.l DC aperiodic component:Average value of the upper and lower envelope curve of the short circuit current, which slowly decays to zero.l Symmetrical breaking current:Effective value of the short circuit current which flows through the contact switch at the time of the first contact separation.l 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.l 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.l Voltage factor:Ratio between the equivalent voltage source and the network voltage Un,divided by 3.l Equivalent electrical circuit:Model for the description of the network by an equivalent circuit.l Far-from-generator short circuit:The value of the symmetrical AC periodic component remains essentiallyconstant.l 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.l Positive-sequence short circuit impedance:The impedance of the positive-sequence system as seen from the position of the short circuit.l Negative-sequence short circuit impedance:The impedance of the negative-sequence system as seen from the position of the short circuit.l Zero-sequence short circuit impedanceThe impedance of the zero-sequence system as seen from the position of the short circuit. Three times the value of the neutral point to ground impedance occurs here.l Short circuit impedance:Impedance required for calculation of the short circuit currents at the position of the short circuipt.1.2 Short circuit path in the positive-sequence systemFor the same external conductor voltages, a three-pole short circuit allows three currents of the same magnitude to develop between the three conductors. It is therefor only necessary to consider one conductor in further calculations. Depending on the distance from the position of the short circuit from the generator, here it is necessary to consider near-to-generator andfar-from-generator short circuits separately.For far-from-generator and near-to-generator short circuits, the short circuit path can be represented by a mesh diagram with AC voltage source, reactances X and resistances R (Figure 1.2). Here, X and R replace all components such as cables,conductors, transformers, generators and motors.Fig. 1.2: Equivalent circuit of the short circuit current path inthe positive-sequence systemThe following differential equation can be used to describe the short circuit processwhere w is the phase angle at the point in time of the short circuit. This assume that the current before S closes (short circuit) is zero. The inhomogeneous first order differential equation can be solved by determining the homogeneous solution ik and a particular solution ik.The homogeneous solution, with the time constant g = L/R, solution yields:For the particular solution, we obtain: The total short circuit current is composed of both components:The phase angle of the short circuit current (short circuit angle) is then, in accordance with the above equation, For the far-from-generator short circuit, the short circuit current is therefore made up of a constant AC periodic component and the decaying DC aperiodic component. From the simplified calculations, we can now reach the following conclusions: l The short circuit current always has a decaying DC aperiodic component in addition to the stationary AC periodic component.l The magnitude of the short circuit current depends on the operating angle of the current. It reaches a maximum at c = 90 (purely inductive load). This case serves as the basis for further calculations.l .The short circuit current is always inductive.1.4 Methods of short circuit calculationThe equivalent voltage source will be introduced here as the only effective voltage of the generators or network inputs for the calculation of short circuit currents. The internal voltages of generators or network inputs are short circuited, and at the position of the short circuit (fault position) the value ( is used as the only effective voltage (Figure 1.4).l The voltage factor c 5 considers (Table 1.1):l The different voltage values, depending on time and positionl The step changes of the transformer switchl That the loads and capacitances in the calculation of the equivalent voltage source can be neglectedl The subtransient behavior of generators and motorsl This method assumes the following conditions:l The passive loads and conductor capacitances can be neglectedl The step setting of the transformers do not have to be consideredl The excitation of the generators do not have to be consideredl The time and position dependence of the previous load (loading state) of the network does not have to be considered Fig. 1.4: Network circuit with equivalent voltage sourcea) three-phase network, b) equivalent circuit in positive sequence system1.4.2 Superposition methodThe superposition method is an exact method for the calculation of the short circuit currents. The method consists of three steps. The voltage ratios and the loading condition of the network must be known before the occurrence of the short circuit. In the first step the currents, voltages and the internal voltages for steady-state operation before onset of the short circuit are calculated (Figure 1.5b). The calculation considers the impedances, power supply feeders and node loads of the active elements. In the second step the voltage applied to the fault location before the occurrence of the short circuit and the current distribution at the fault location are determined with a negative sign (Figure 1.5c). This voltage source is the only voltage source in the network. The internal voltages are short-circuited. In the third step both conditions are superimposed. We then obtain zero voltage at the fault location. The superposition of the currents also leads to the value zero. The disadvantage of this method is that the steady-state condition must be specified. The data for the network (effective and reactive power, node voltages and the step settings 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.Fig. 1.5: Principle of the superposition methoda) undisturbed operation, b) operating voltage at the fault location, c) superposition of a) and b)1.4.3 Transient calculationWith the transient method the individual operating equipment and, as a result, the entire network are represented by a system of differential equations. The calculation is very tedious. The method with the equivalent voltage source is a simplification relative to the other methods. Since 1988, it has been standardized internationally in IEC 60 909. The calculation is independent of a current operational state. In thisbook, we will therefore deal with and discuss the method with the equivalent voltage source.1.5 Calculating with reference variablesThere are several methods for performing short circuit calculations with absolute and reference impedance values. A few are summarized here and examples are calculated for comparison. To define the relative values, there are two possible reference variables.For the characterization of electrotechnical relationships we require the four parameters:l Voltage U in Vl Current I in Al Impedance Z in Wl 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 this system are UB and SB.Example: The reactances of a synchronous machine Xd, Xd, Xd are given 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.短路電流1 術(shù)語和定義以下術(shù)語和定義對應(yīng)IEC 標(biāo)準(zhǔn)60 909。 未出現(xiàn)在本書中的術(shù)語可以在該標(biāo)準(zhǔn)中查詢。短路和接地故障主要是操作設(shè)備的帶電部分被分流而導(dǎo)致絕緣損壞的結(jié)果。l 原因1. 溫度過高導(dǎo)致強烈的過電流；2. 火花放電導(dǎo)致過電壓3. 由于水分和污穢空氣混雜導(dǎo)致的電弧作用，特別是在絕緣體上。l 后果：1. 供電中斷2. 系統(tǒng)部件癱瘓3. 在電氣操作設(shè)備中產(chǎn)生不可接受的機械力和熱應(yīng)力。l 短路：根據(jù)IEC 60 909，短路是經(jīng)歷一段相對低電阻或在兩個或更多不同電位之間的電阻間意外或故意的導(dǎo)電連接。l 短路電流：根據(jù)IEC 60 909，短路電流是在電力網(wǎng)絡(luò)中短路的結(jié)果。在這里有必要區(qū)分在短路過程中產(chǎn)生的短路電流和在網(wǎng)絡(luò)分支中的轉(zhuǎn)移電流。l 初始對稱短路電流：這是在短路出現(xiàn)瞬間，短路阻抗從零開始變化時的對稱短路電流的有效值。l 初始對稱短路視在功率：短路功率代表了一個虛構(gòu)的參數(shù)。在網(wǎng)絡(luò)規(guī)劃中，短路功率是一個合適的典型參數(shù)。l 最大短路電流:短路時可能的最大電流瞬時值。l 穩(wěn)態(tài)短路電流：初始對稱短路電流在暫態(tài)過程中衰減完畢之后的電流有效值。l 短路電流非周期分量：慢慢衰減的短路電流上下包絡(luò)曲線的平均值。l 對稱斷路電流：聯(lián)絡(luò)開關(guān)第一次接觸分離時流過短路電流的有效值。l 等效電壓源：被轉(zhuǎn)移到正序系統(tǒng)作為唯一有效的短路位置的電壓，并且主要用于短路電流的計算。l 疊加方法：疊加法考慮到在發(fā)生短路前網(wǎng)絡(luò)的負(fù)荷情況。因此很必要知道負(fù)荷留了和變壓器開關(guān)的設(shè)定。l 電壓因素：等效電壓源和網(wǎng)絡(luò)電壓之間除以三的比例。l 等效電路：網(wǎng)絡(luò)描述的模型采用等效電路。l 遠(yuǎn)離發(fā)電機短路：對稱交流周期分量維持原本不變的量的短路形式。l 靠近發(fā)電機短路：對稱交流周期分量不保持不變的值的短路。同步機首先會產(chǎn)生一個大于兩倍額定電流的初始對稱短路電流。l 正序短路阻抗：短路位置正序系統(tǒng)的短路阻抗。l 負(fù)序短路阻抗：短路位置負(fù)序系統(tǒng)的短路阻抗。l 零序短路阻抗：短路位置零序系統(tǒng)的短路阻抗。就是中性點到短路位置阻抗的三倍。l 短路阻抗：計算短路位置的短路電流所需要的阻抗。1.2 正序系統(tǒng)的短路路徑：對于相同的外部導(dǎo)體電壓，三相短路允許同一數(shù)量級的三相電流在三相導(dǎo)體中發(fā)展。所以在進一步計算中只需要考慮一相導(dǎo)體的情況。根據(jù)短路位置到發(fā)電機的距離，這里有必要將遠(yuǎn)離發(fā)電機短路和靠近發(fā)電機短路這兩種情況分開考慮。對于遠(yuǎn)離和靠近發(fā)電機短路的情況，短路路徑可以用一個有交流電壓源，電抗X，電阻R構(gòu)成的網(wǎng)絡(luò)圖表示。（圖1.2）這里X和R替代所有的原件，如電纜，導(dǎo)體，變壓器，發(fā)電機和電機。圖1.2 短路電流路徑在正序系統(tǒng)中的等效電路下面的微分方程可以用來描述短路過程 ikRk+Lkdikdt =usin(t+), (1.1)是短路點的相位角。這是假設(shè)電流在S關(guān)閉（短路）之前是零。非線性一階微分方程可以通過決定齊次解ik和特解ik求解。 ik=ik+ik- (1.2)齊次解有一個時間常量=LR,方程式為：ik=-u(R2+X2)etgsin(-k) (1.3)對于特解，我們得出：ik=-u(R2+X2)sin(t+-k) (1.4)總短路電流由兩部分構(gòu)成：ik =-u(R2+X2)sin(t+-k)- etgsin(-k) (1.5)根據(jù)以上方程，單相短路電流的短路角為：k=-v=tan-1XR (1.6)對于遠(yuǎn)離發(fā)電機形式的短路，短路電流是由一個不變的交流周期分量和一個衰減的直流非周期分量構(gòu)成。從簡化計算，我們可以得出以下結(jié)論:l 短路電流總是由一個固定的交流周期分量和一個衰減的直流非周期分量構(gòu)成；l 短路電流的大小取決于電流的工作角，最大值為90（純電感負(fù)載）。這種情況作為進一步計算的基礎(chǔ)。l 短路電流都是感應(yīng)的。1.4 短路電流計算方法:三相系統(tǒng)中的短路電流有三種計算方法：1.在故障位置計算等效電壓源；2.疊加法確定負(fù)載流量情況；3.瞬態(tài)計算。1.4.1 等效電壓源:這里的等效電壓源主要作為發(fā)電機或投入電網(wǎng)的短路電流計算的唯一有效電源。發(fā)電機和投入電網(wǎng)的內(nèi)部電壓是短路的，短路地點（故障位置）的值就作為唯一有效電壓。（圖1.4）l 電壓因素考慮35（表1.1）l 不同電壓值取決于時間和地點的不同l 變壓器開關(guān)的階躍變化l 等效電壓源的計算中負(fù)荷和容量都可忽略不計l 發(fā)電機和電機的起始狀態(tài)該方法假設(shè)以下條件：l 被動負(fù)荷和導(dǎo)體容量可以忽略不計l 變壓器的步驟設(shè)定可以不需考慮l 發(fā)電機的激勵不需要考慮l 前負(fù)荷的負(fù)荷狀態(tài)的時間和位置可以忽略不計圖1.4 具有等效電壓源的網(wǎng)絡(luò)電路 a)三相系統(tǒng) b)正序系統(tǒng)的等效電路表1.1 根據(jù)E DIN IEC 73/89/CDV (VDE 0102, Part 100):1997-08的電壓因素網(wǎng)絡(luò)電壓 電壓因素 Un 最大短路電流 最小短路電流 Cmax Cmin低壓 1.05 0.95100V至1000V（IEC38 表1） 1.103中壓 1.10 1.001KV至35KV高壓大于35KVCmaxUn不能超過網(wǎng)絡(luò)中操作設(shè)備的最大工作電壓Um在低壓網(wǎng)絡(luò)有+6%的公差在低壓網(wǎng)絡(luò)有+10%的公差1.4.2 疊加法:疊加法是一種精確的計算短路電流的方法。該方法包含三個步驟。在短路發(fā)生前變壓器的變壓比和網(wǎng)絡(luò)負(fù)荷條件必須已知。在第一階段，穩(wěn)態(tài)允許下的電流，電壓，穩(wěn)態(tài)電壓在短路前都要先計算出來(圖1.5b)。計算考慮了阻抗，電源和有源元件的節(jié)點負(fù)荷。第二步，在短路前故障位置處的電壓和電流分配要加以符號確定（圖1.5c）。電壓源是網(wǎng)絡(luò)中唯一的。內(nèi)部電壓時短路的。在第三階段兩種狀態(tài)會疊加起來。我們最終在故障點獲得零序電壓。電流的疊加同樣也會導(dǎo)致零值。這種方法的缺點是穩(wěn)定狀態(tài)必須被指定。網(wǎng)絡(luò)參數(shù)（有功功率和無功功率，節(jié)點電壓和變壓器的步驟設(shè)定）往往是很難決定的。問題同時出現(xiàn)，這將導(dǎo)致工作狀態(tài)時出現(xiàn)最大的短路電流。圖1.5 疊加法原理 a) 穩(wěn)定系統(tǒng) b) 故障處運行電壓 c) a和b的疊加1.4.3 瞬態(tài)計算:用瞬態(tài)方法計算每個設(shè)備時，整個網(wǎng)絡(luò)被一系列的微分方程所表示。計算過程非常乏味。擁有等效電壓源的這種方法相比其他方法比較簡單。自1988年以來已經(jīng)在IEC 60 909中被國際化規(guī)范。運算和當(dāng)前運行狀態(tài)是獨立的。所以在這本書中我們將解決和討論有等效電壓源的這種方法。1.5 參考變量的計算:有很多方法根據(jù)絕對的和參考抗值進行短路計算。一些在這兒已經(jīng)總結(jié)出來，還有實例加以比較。為了定義相對值，有兩種可能的參考變量。為表征電工關(guān)系，我們要求四個參數(shù)：l 單位為V的電壓Ul 單位為A的電流Il 單位為KW的阻抗Zl 單位為VA的視在功率S以下三種方法可以用來計算短路電流：1. 歐姆系統(tǒng)：單位：kV, kA, V, MVA2. 標(biāo)幺值系統(tǒng)：這種方法主要用于發(fā)電機系統(tǒng)：所有四個參數(shù)，u,i,還有s都被給定了值（單位=1）。參考容量是100MV。系統(tǒng)的兩個參考變量分別是UB和SB。例如：同步機的電抗Xd,XdXd都以標(biāo)幺值的形式被給定，或者乘以100%以百分比標(biāo)幺值形式給定。3. %/MVA系統(tǒng)這種方法特別適用于短路阻抗的快速測定。作為正式單位用%加以補充。