Short-circuit current

1 Terms and Definitions

The 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 characteristic number.

● Peak short circuit current:

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 impedance of the positive-sequence system as seen from the position of the short circuit.

● Negative-sequence short circuit impedance:

The impedance of the negative-sequence system as seen from the position of the short circuit.

● Zero-sequence short circuit impedance

The 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.

● Short circuit impedance:

Impedance required for calculation of the short circuit currents at the position of the short circuit.

1.2 Short circuit path in the positive-sequence system

For 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 in

the positive-sequence system

The following differential equation can be used to describe the short circuit process

where 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 i²k.

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:

● The short circuit current always has a decaying DC aperiodic component in addition to the stationary AC periodic component.

● 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.

● .The short circuit current is always inductive.

1.4 Methods of short circuit calculation

The 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).

● The voltage factor c [5] considers (Table 1.1):

● The different voltage values, depending on time and position

● The step changes of the transformer switch

● That the loads and capacitances in the calculation of the equivalent voltage source can be neglected

● The subtransient behavior of generators and motors

● This method assumes the following conditions:

● The passive loads and conductor capacitances can be neglected

● The step setting of the transformers do not have to be considered

● The excitation of the generators do not have to be considered

● 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 source

a) three-phase network, b) equivalent circuit in positive sequence system

1.4.2 Superposition method

The 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 method

a) undisturbed operation, b) operating voltage at the fault location, c) superposition of a) and b)

1.4.3 Transient calculation

With 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 variables

There 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:

● Voltage U in V

● Current 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, MVA

2.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, X¢d, X²d 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。

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

+ =, (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i²k求解。

=+ (1.2)

齐次解有一个时间常量=,方程式为：

= (1.3)

对于特解，我们得出：

= (1.4)

总短路电流由两部分构成：

= (1.5)

根据以上方程，单相短路电流的短路角为：

== (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

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

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

● 单位为VA的视在功率S

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, MVA

2. 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’’都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。

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