1 Introduction

Half-wavelength alternating current transmission(HWACT) is an ultra-long distance three-phase AC power transmission technology with an electrical distance near to half-wavelength at the system power frequency. In 50Hz system, the distance is 3000km, while it is 2500km in 60Hz system. The advantages of HWACT are infinite theoretical maximum transmission capacity and no additional reactive power compensation. Hence it is suitable to serve as the long-distance backbone frame in the future Global Energy Interconnection.

The concept of HWACT is initially proposed by the experts from Soviet Union in 1940s [1]. After some following exploratory research in this field [2-4], the principal transmission characteristics and some correlative key technical problems, such as corona loss and overvoltage [5, 6], have been solved out. But the research on HWACT has not been continuously and deeply carried out due to the limited demand and lower technical level at that time.

四要全面推进节水型社会建设，突出抓好工业、农业、城市生活等领域的节水防污，搞好废污水处理回用、雨水集蓄利用、海水直接利用与淡化利用。

（1）根据水功能区水质达标管理要求，以水功能区为功能控制单元、流域为统筹管理单元，实施分阶段排污总量控制，重点提高不达标水功能区内污水处理能力与质量，加大超采水功能区限排力度，有序提升水功能区达标水平。

In the new century, the ultra-long distance, large capacity transmission has attracted global attention because of the serious energy shortage problem. Many researchers in different countries focus on HWACT once again. In Brazil, HWACT is used as the backup scheme to transmit large scale hydropower from Amazon River basin to the load centers [7, 8]. And in China, the scheme of HWACT has also been proposed to transmit the coal and hydro energy to the east coast load centers [9-16]. In particular,the State Grid Corporation of China proposed the idea of building Global Energy Interconnection in July 2014. The UHV power grid serves as the backbone to mainly deliver clean energy and achieve a globally interconnected smart grid. Long distance UHV transmission lines are needed for the long distance power transmission or exchange.Therefore, the study on HWACT to satisfy the requirement of large capacity long distance point-to-point power transmission has broader significance and practical value.

The fault characteristics of an HWACT line can be different from those of a conventional transmission line.It is very important for the construction and operation of HWACT to analyze its fault features and corresponding protection technology. References [13-15] all deal with the analysis of fault for HWACT line, but the studies are limited to overvoltage along the line. Reference [12] gives a simple analysis of the fault characteristics for HWACT line, but not further.

本文提出的时变转移概率IMM-RDCKF算法，充分利用了量测方程中只有部分状态变量是非线性的特点，只对非线性状态变量采样，不仅减小了计算量，而且继承了CKF算法滤波精度高的优点，有效克服了常规IMM-CKF算法计算量大的缺点；同时通过优化代码，消除了计算过程的数值计算误差积累，保障协方差矩阵在滤波过程中始终保持对称和正定，提高了算法稳定性；通过实时修正模型转移概率，使与目标运动匹配的模型概率增大，加快模型切换速度，最终提高了精度。因此，本文提出的时变转移概率IMM-RDCKF算法在计算效率、跟踪精度、鲁棒性方面都比常规交互多模算法强，是一种工程应用价值比较高的机动目标跟踪算法。

When a symmetrical short-circuit fault occurs, the three-phase system can be regarded as a single-phase system in order to simplify the fault analysis. The symmetrical fault analysis is the basis of the asymmetrical fault analysis. Therefore, the steady-state voltage and current characteristics of the bus bar and fault point and the steady-state overvoltage distribution along the line will be analyzed in this paper when a three-phase symmetrical short-circuit fault occurs on an HWACT line.

2 HWACT system model

Only single phase should be considered for a symmetrical fault. And when the fault resistance is 0, the right part of the fault point has no influence on the voltage and current of bus m. According to the hyperbolic function solution of the uniform transmission line equations, the voltage and current of the bus m and bus n are related to the voltage and current of the fault point f,

Fig. 1 Model of an HWACT system

The analysis of fault characteristics of an HWACT line will be done by using the HWACT system model shown in Fig. 1. The line mn is a 1000kV HWACT line, and the rest parts of the system are equivalent to two voltage sources with impedances. The phase angle difference between two equivalent voltage sources of the bus m and bus n is larger than 180°. The power frequency is 50Hz. The line parameters in sequence quantities are shown as Table 1.

Table 1 Parameters of HWACT Line

Parameters Positive sequence Zero sequence Resistance(Ω/km) 0.00805 0.205 Inductance (mH/km) 0.825 2.375 Capacitance(nF/km) 14.0 9.03

According to the line parameters, the length of line mn can be calculated as follows, where λ is the wavelength with the power frequency f=50Hz. L and C are the line positive sequence inductance and capacitance per unit length respectively.

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When a three-phase short-circuit fault occurs at the point f on the HWACT line, the fault characteristics will be analyzed without considering the fault resistance.

3 Fault characteristics of bus bar

The steady-state voltage and current characteristics will be analyzed when the symmetrical faults occur on an HWACT line in the cases of different fault distances.

where the parameters Zc and γ are the surge impedance and propagation constant of the HWACT line. x is the distance from the fault point f to the bus m. and are the voltage and current of the bus m respectively.represents the voltage of the fault point f and its value is 0. is the current which flows into the fault point f from the bus m.

The boundary condition at the bus m is as follows,

whereis the equivalent voltage source out of the bus m and Zm is the equivalent impedance.

Analyzing (2) (3), the voltage of the bus m can be obtained as follows.

对于表现特别优异，确实能力强的，我会跟该学生私下交流，鼓励学生参与班级管理。参与到班级管理中，既能够成为班级其他同学的榜样，又有助于让自身更上一层楼。通过交流，确认学生意愿之后，直接在班级宣布该学生的班干部工作岗位。

For simplicity, the resistances in the system will be neglected. Therefore,

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where Lm is the inductance in Zm. Evidently, k is real.

In the cases of longer transmission lines, when a symmetrical fault occurs, by setting the fault point as a reference point, the voltage RMS along the line at bus m side varies as sinusoidal positive half cycle periodically with the distance between fault point and measuring point increasing. So does the voltage RMS along the line at bus n side. However, if the equivalent impedances of the bus m and bus n are not equal, the sinusoidal peaks of the two sides are different.

The extreme point exists in the interval (λ/4, λ/2)and it is related to the line parameters and the equivalent impedance out of the analyzed bus.

Fig. 2 Voltage RMS of bus m calculated with the change of the fault distance

The voltage RMS (Root Mean Square) of bus m is calculated with the change of the fault distance mf, as shown in Fig. 2. And it can be also regarded as |k| with the change of x. From Fig. 2, we can see that the voltage RMS of bus m increases from 0 to +∞ (extreme point) and then decreases to 0 with the fault distance mf increasing.

For the current of the bus m, according to (3),

Evidently, k’ is pure imaginary. When x=λ/4, k’=0. And the extreme point of k’ is the same as that of k.

基因表达实验结果显示，该基因在植物的花被片、花丝、花柱等花部组织中表达量较高，而在其他部位几乎不表达，表现出一定的组织特异性，研究结果与前人[6]对其他百合品种该基因的表达分析结果一致。LhsorMYB12在花蕾发育过程中表达量逐渐升高，花蕾发育至12 cm时表达量最高，之后开始降低，郁金香的TfMYB1基因在花朵发育过程中的表达水平呈先上升后下降的趋势，与LhsorMYB12具有相似的表达模式[38]。这种基因表达量的变化与花被片的花色性状变化一致，提示LhsorMYB12的表达与 ‘索邦’花青素苷的积累呈正相关。

The current RMS of bus m is calculated with the change of the fault distance mf, as shown in Fig. 3. And it can be also regarded as |k’| with the change of x. From Fig. 3, we can see that the current RMS of bus m firstly decreases with the fault distance mf increasing. It decreases to 0 at the middle point of the line. After that, it increases to +∞ (extreme point) and then decreases.

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In the cases of longer transmission lines, when the symmetrical faults occur, there is a half-wavelength periodic extension phenomenon for the voltage and current RMS of bus m with the change of the fault distance mf. The extreme points are:

Fig. 3 Current RMS of bus m calculated with the change of the fault distance

When a symmetrical fault occurs on a full-wavelength line, the voltage RMS of bus m is calculated with the change of the fault distance mf, as shown in Fig. 4.We can see that the voltage RMS of bus m displays a periodic change and the change period is the halfwavelength.

Fig. 4 Voltage RMS of bus m calculated with the change of the fault distance when a symmetrical fault occurs on a fullwavelength line

4 Fault characteristics of fault point

The current of the fault point is the sum of and,which are respectively the currents that flow into the fault point f from the bus m and bus n. That is

According to (2), when a three-phase short-circuit fault occurs on the HWACT line at point f, the overvoltage distribution along the line is as follows,

The currents that flow into the fault point f from the bus m can be obtained according to (2) (3),

The voltage RMS of fault point is calculated for different fault distances under different fault resistances, as shown in Fig. 9. The curves of different colors in the figure represent different fault resistances. It can be seen from the figure that when the fault resistance is 0, the voltage of fault point is 0. And when the fault resistance is infinite,the voltage curve of fault point is the same with the voltage distribution curve along the line. When the fault resistance exists, the voltage of fault point is fixed at two pointsAnd it will not be influenced by the fault resistance.

If the fault point is determined, andare constant values. By setting the fault point as a reference point, the voltage RMS along the line at bus m side varies as sine with the distance between fault point and measuring point increasing. The peak value of the sine is related toSimilarly, the voltage RMS along the line at bus n side also varies as sine and the peak value is related to.

When either of the sines in Am and An is 0, the current of the fault point will be infinite. The extreme points are:

The two extreme points exist in the interval (0, λ/4)and interval (λ/4, λ/2) respectively. The extreme point in the interval (0, λ/4) is related to the line parameters and the equivalent impedance of the bus n. And the one in the interval (λ/4, λ/2) is related to the line parameters and the equivalent impedance of the bus m.

Fig. 5 Current RMS of fault point calculated with the change of the fault distance

The current RMS of fault point is calculated with the change of the fault distance mf, as shown in Fig. 5. We can see that the current RMS of fault point firstly increases to infinite (first extreme point) and then decreases (not to 0) with the fault distance mf increasing. Subsequently, it increases to infinite again (second extreme point) and then decreases. In particular, if the equivalent impedances of the bus m and bus n are equal, the current RMS of fault point will be symmetric with respect to λ/4.

In the cases of longer transmission lines, when the symmetrical faults occur, the impedance coefficient Am remains the same while An changes. If the line length is l,

Here, the extreme points are:

Fig. 6 Current RMS of fault point calculated with the change of the fault distance when a symmetrical fault occurs on a line with the length of 1.25λ

When a symmetrical fault occurs on a line with the length of 1.25λ, the current RMS of fault point is calculated with the change of the fault distance mf, as shown in Fig. 6.We can see that the current RMS of fault point displays a periodic change and the change period is the halfwavelength.

5 Overvoltage distribution along the line

where Am and An are the impedance coefficients.

where x is the distance from the fault point f to the bus m. represents the voltage of the point from which the distance is y to the bus m. If the resistances in the system are neglected,

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In the same way,

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Fig. 7 Voltage RMS along the line calculated for different fault distances

The voltage RMS along the line is calculated for different fault distances, as shown in Fig. 7. Here, the curves of different colors mean different distances between fault point and bus m.

The value of k will be discussed while the fault distance x changes from 0 to λ/2. With x increasing, the cotangent decreases from +∞ to -∞ and k increases. When x=0,the cotangent equals to +∞ and k=0. When x=λ/4, the cotangent equals to 0 and k=1. When the denominator in(5) decreases to 0, an extreme point appears and here k increases to +∞. And after that, k increases from -∞. When x=λ, the cotangent equals to -∞ and k=0. The extreme point is:

When a symmetrical fault occurs on a full-wavelength line, the voltage RMS along the line is calculated for different fault distances, as shown in Fig. 8. It shows the same result to the analysis above.

Fig. 8 Voltage RMS along the line calculated for different fault distances when a symmetrical fault occurs on a fullwavelength line

6 Fault characteristics considering fault resistance

When there is a fault resistance, the voltage at the fault point f is no longer 0,

After ignoring the resistors in the equivalent impedances, Am and An are sinusoidal pure imaginary.

Fig. 9 Voltage RMS of fault point calculated for different fault distances

If the resistances in the system are neglected,

7 Conclusions

The fault characteristics of the HWACT line are quite different from those of the conventional transmission line. In this paper, the steady-state voltage and current characteristics of the bus bar and fault point and the steadystate overvoltage distribution along the line are analyzed when a three-phase symmetrical short-circuit fault occurs on an HWACT line. The conclusions are summarized as follows.

1) When a three-phase symmetrical short-circuit fault occurs in the HWACT line, if there is no fault resistance,the steady-state voltage and current RMS at the bus have an extreme point between the line middle point and the opposite bus bar. If the resistances in the system are neglected, the extreme value is infinite. And the current at the middle point is zero.

2) The steady-state current RMS at the fault point has two extreme points, which locate at the two sides of the line middle point. When the equivalent impedances on both sides are equal, the variation of the RMS current at the fault point is symmetrical about the line middle point.

3) By setting the fault point as a reference point, the voltage RMS along the line varies as sine. The peak values of the sine on both sides are related to the corresponding equivalent impedances.

4) For longer distance transmission system, the threephase short-circuit fault characteristics will display a periodic change and the change period is the half-wavelength.

Acknowledgements

This work was supported by National Key Research and Development Program of China (2016YFB0900100).

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