# Study on the Power-Frequency Waves Distribution Characteristics for Half-Wavelength Transmission Lines Based on the Frequency-Length Factor

Authors: Chang Chen; Xiaoyang Ma; Honggeng Yang; Weikang Wang; Yilu Liu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

To analyze the distribution characteristics of voltage and current along half-wavelength transmission lines (HWTLs) in the cases with or without short circuit in the steady state, the method based on the frequency-length factor (FLF) for lossy lines is proposed. Firstly, according to the pole condition of the FLF, the distribution characteristics of power-frequency waves along HWTLs are analyzed. Then, the comprehensive effects of the system parameters and fault resistance are explored, revealing the mechanism of the power-frequency resonance caused by nonmetallic short circuit. Meanwhile, unbalanced short-circuit fault is studied by exploiting additional impedance. The results show that the distribution of the maximum value of power-frequency resonance voltage is related to the system parameters but not to the fault impedance. When a HWTL is short circuited at 2640 km∼2930 km, the resonance voltage can reach to 21 p.u. In relation to symmetrical short circuit, the resonance voltage appears at 1469 km from the short-circuit point, while the position moves towards the short-circuit point with the increase of additional impedance in asymmetrical short-circuit conditions. Additionally, the model theoretically proves that the power-frequency overvoltage induced by short circuit does not appear on a line whose length is less than 1469 km. Finally, cases are studied on PSCAD to verify the accuracy of the model.


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## Body

## 1. Introduction

Half-wavelength power transmission refers to the long-distance three-phase AC transmission technology. The electrical distance of a half-wave length transmission line (HWTLs) is close to half of the power-frequency wavelength (3000 km for 50 Hz systems and 2600 km for 60 Hz systems) [1–4]. Compared with HVDC lines [5–7] or highly compensated AC lines [8], half-wavelength power transmission technology has an advantage with no requirement of compensation, which may be the most economical solution for ultra-long distance power transmission [9–11]. Besides, HWTLs have strong transmission capacity and excellent voltage stability. Therefore, many relative research studies have been carried out by several countries including China and Russia [12, 13]. For example, in order to deliver large hydropower to load centers, Brazil has operated detailed research and simulation calculation on half-wavelength transmission technology [14, 15]. Korea investigates the feasibility of transmitting the water electricity from Siberia by means of HWTLs [16]. Developing clean energy has become an unavoidable path for China and even the whole world [17, 18]. To deliver western wind power to the eastern load center, a lot of research on HWTLs has been carried out in China recently. Due to the development of global energy interconnection in recent years, half-wavelength power transmission has again attracted wide attention worldwide [19, 20].The distribution of power-frequency voltage and current along the transmission line is the premise to ensure the safety of the electrical equipment and the stable operation of the system. At present, copious research studies have been conducted to discuss the characteristics for HWTLs in the steady state [21–24]. Meanwhile, some investigations have also been carried out for the distribution of the power-frequency voltage and current on HWTLs after short circuit. However, most of these analyses either rely on simulation or neglect line loss, making them unable to give a deep insight into the mechanism [24]. Zhang et al. [25] proposed an approach to find out the location feature of power frequency along HWTLs after short circuit. However, it focused on the symmetrical fault and neglected the fault resistance. Besides, it also probed for the characteristics for other long-distance transmission lines. But the results were concluded by software simulation, lacking theoretical proof. On the research foundation above, the asymmetric short-circuit fault was analyzed in [26]. However, rather than considering the comprehensive influence of various parameters on transmission characteristics, each impact parameter was only analyzed as a single variable. In brief, the mechanism of the comprehensive influence of the parameters, i.e., system parameters and fault condition, on distribution characteristics is not clear and visible, which need a further excavation.Based on the frequency-length factor (FLF), the model of power-frequency voltage and current distribution along lines considering line loss is deduced in this paper. Based on the pole distribution condition of the FLF, the distribution characteristics of steady-state power-frequency voltage and current on HWTLs are firstly analyzed. In this part, the effect of the active power and power factor on the distribution characteristics are obtained. Then, the power-frequency resonance mechanism after short-circuit fault in the steady state is also discussed. The proposed model considers both the system operating parameters and the fault conditions, making it possible to understand the mechanism of power-frequency resonance. Additionally, the characteristics of power-frequency resonance for other long-distance transmission lines are also proved theoretically. Comparison between the calculation model and the simulation results on PSCAD and other reliable software is made to verify the accuracy of the model and the theoretical conclusions.The current paper is structured as follows. Section2 proposes the calculation models of the power-frequency waves (voltage and current) distribution on lossy lines based on the FLF. Section 3 discusses the distribution characteristics of steady-state power-frequency waves without the fault. By using the similar approach as Section 2, Section 4 proposes the models for the distribution of steady-state power-frequency waves on HWTLs with short-circuit fault occurring. Meanwhile, the distribution features are obtained. Section 5 gives a theoretical verification of other long-distance transmission lines, while Section 6 provides several case studies and discussions, and the final section concludes.


## 2. Transmission Model Based on the Frequency-Length Factor Theory

### 2.1. Frequency-Length Factor

The simplified structure and model of ultra-high voltage power systems with a long-distance AC transmission line are shown in Figure1. ZL is the load impedance, S=P + iQ is the apparent power of the load, and P and Q are the active power and reactive power, respectively.Figure 1
The simplified model of UHVAC power systems: (a) structure; (b) model.
(a)
(b)The line is divided into two distributed-parameter models connected in series to obtain the distribution characteristics of voltage and current on it, as shown in Figure1(b). The two-port network of a transmission line can be expressed as UM,IMT=C−1US,IST, where US and IS are the voltage and current of the sending end (node S), respectively. UM and IM are the voltage and current of the measuring point (Node M) which is at a distance of l km from node S. C is the T-parameter matrix of the transmission line, of which the inverse matrix is(1)C−1=chγl−ZCshγl−shγlZCchγl,where γ=α+iβ is the propagation constant, α and β are the attenuation constant and the phase shift constant of the line, respectively, and ZC=RC+iXC is the characteristic impedance.Equation (1) contains hyperbolic functions with complex variables γl=αl+iβl, where Dl=αl and μl=βl reflect the change of amplitude and phase in transmission characteristics, respectively. μl is called the frequency-length factor (FLF), which is related to the product of frequency and line length [27].The hyperbolic function in equation (1) can be expressed by the FLF as(2)chγl=chDlcosμl+ishDlsinμl,shγl=shDlcosμl+ichDlsinμl.Equation (1) is expanded into a trigonometric function of the FLF in equation (2). Since it is technically simple to obtain the pole distribution of trigonometric functions, the transmission characteristics can be analyzed with the aid of the pole distribution feature of the FLF.


### 2.2. Equivalent Transmission Model for Voltage Transmission

In Figure1, the relationship between the voltage and the current of node M is(3)UM=IM×ZM.Z
M is the equivalent impedance seen from node M to node R (the receiving end), which can be simplified as(4)ZM=ZCn⋅shγl2+chγl2shγl2+n⋅chγl2,where n=Zc/ZL, l2=L−l is the length between node M and R, and L is the total length of the line.SinceZL=V2/S∗, it can be obtained that n=a−ib, a = ZCP/V2, b = ZCQ/V2 = a · tan θ, and n=a2+b2=ZCS/V2. V is the power-frequency voltage. θ is the power factor angle, which is positive when the load is inductive and capacitive otherwise.Substituting equation (3)∼(4) into equation (1), the voltage transfer function between the input source at node S and output response at node M can be written as(5)Hu=UMUS=1chγl+ZC/ZMshγl.Simplifying equation (5) by equation (4), there is(6)Hu=n⋅shγl2+chγl2n⋅shγL+chγL.It can be seen that the structure of the numerator and denominator in equation (6) are similar to each other. Therefore, we construct an intermediate function, fuμx=n⋅shγx+chγx, which can be simplified by equation (2) as(7)fuμx=chDx+ashDxcosμx+bchDxsinμx+ishDx+achDxsinμx−bshDxcosμx,where μx=βx. By introducing an auxiliary angle φ, the magnitude of fuμx can be expressed as(8)fuμx=fuμx=Ausin2μx+φ+Bμx,where Au=sgnb⋅A. sgnb is the signal function of b, and there is sgnb=sgnθ. sgnb=1 when the load is inductive (θ>0). And sgnb=−1 when θ<0. sgnb=0 when the load is a pure resistance. A=1−n22/4+b2, while Bμx and φ are(9)Bμx=1+n22×ch2Dx+ash2Dx,φ=arctan1−n22b,φ in equation (9) is π/2 and −π/2 when b⟶0 and ∞, respectively. Otherwise, −π/2<φ<π/2.The magnitude ofHu is numerically equal to the per unit value of voltage. That is, UM∗=Hup.u.. Therefore, substituting equation (8) into (6), it can be obtained that(10)UM∗=Ausin2μl2+φ+Bμl2Ausin2μL+φ+BμL=fuμl2fuμL.


### 2.3. Equivalent Transmission Model for Current Transmission

The current transfer function at any point on a long-distance transmission line can also be derived by the similar method as that inSection 2.1, as follows:(11)Hi=IMUS=1ZC⋅shγl2+n⋅chγl2n⋅shγL+chγL.Similarly, constructing an intermediate functionfiμx=shγx+n⋅chγx. Then, simplifying the magnitude of fiμx, the current on the line can be expressed as(12)IM=USZC⋅Aisin2μl2+φ+Bμl2Ausin2μL+φ+BμL=U1ZC⋅fiμl2fuμL,where Ai=−sgnb1−n22/4+b2=−Au.


### 2.4. The Distribution Characteristics of FLF

Equations (10) and (12) simplify the transmission coefficients into trigonometric functions verifying with the FLF. While Bμxalso contains the product of frequency and length, it changes little at the power frequency f0. Therefore, Bμx is considered as a parameter that has no effect on the pole distribution of the FLF.The pole conditions offuμx and fiμx can be expressed as follows:(13)μx=−φ2−π2+Nπ,(14)μx=−φ2+π2+Nπ,where N is the integer. When θ>0, there is Au>0; thus, equations (13) and (14) are the minimum and maximum conditions of fuμx, respectively. Meanwhile, since Ai=−Au<0, they are the maximum and minimum conditions of fiμx, respectively. The pole distribution conditions are contrary to the above when θ<0. Moreover, when θ=0, there is Au=Ai=0. Accordingly, the transmission of the power-frequency voltage and current is no longer distributed as trigonometric functions of the FLF.Combining with the position (in numerator or denominator) offuμx or fiμx in the fractional functions, the extreme conditions of UM∗ and IM can be discussed. For example, when the load is inductive, the analysis of the extreme value of voltage distribution on a line is translated to a problem of figuring out the polar condition of fuμx as the numerator of UM∗. Therefore, equation (14) is the maximum condition of UM∗, while equation (13) is its minimum condition. When discussing the influence of line length (L) on voltage transmission, the pole condition of fuμx as the denominator of UM∗ is aimed to be found out. Therefore, equations (13) and (14) are the maximum and minimum conditions, respectively.


## 2.1. Frequency-Length Factor

The simplified structure and model of ultra-high voltage power systems with a long-distance AC transmission line are shown in Figure1. ZL is the load impedance, S=P + iQ is the apparent power of the load, and P and Q are the active power and reactive power, respectively.Figure 1
The simplified model of UHVAC power systems: (a) structure; (b) model.
(a)
(b)The line is divided into two distributed-parameter models connected in series to obtain the distribution characteristics of voltage and current on it, as shown in Figure1(b). The two-port network of a transmission line can be expressed as UM,IMT=C−1US,IST, where US and IS are the voltage and current of the sending end (node S), respectively. UM and IM are the voltage and current of the measuring point (Node M) which is at a distance of l km from node S. C is the T-parameter matrix of the transmission line, of which the inverse matrix is(1)C−1=chγl−ZCshγl−shγlZCchγl,where γ=α+iβ is the propagation constant, α and β are the attenuation constant and the phase shift constant of the line, respectively, and ZC=RC+iXC is the characteristic impedance.Equation (1) contains hyperbolic functions with complex variables γl=αl+iβl, where Dl=αl and μl=βl reflect the change of amplitude and phase in transmission characteristics, respectively. μl is called the frequency-length factor (FLF), which is related to the product of frequency and line length [27].The hyperbolic function in equation (1) can be expressed by the FLF as(2)chγl=chDlcosμl+ishDlsinμl,shγl=shDlcosμl+ichDlsinμl.Equation (1) is expanded into a trigonometric function of the FLF in equation (2). Since it is technically simple to obtain the pole distribution of trigonometric functions, the transmission characteristics can be analyzed with the aid of the pole distribution feature of the FLF.


## 2.2. Equivalent Transmission Model for Voltage Transmission

In Figure1, the relationship between the voltage and the current of node M is(3)UM=IM×ZM.Z
M is the equivalent impedance seen from node M to node R (the receiving end), which can be simplified as(4)ZM=ZCn⋅shγl2+chγl2shγl2+n⋅chγl2,where n=Zc/ZL, l2=L−l is the length between node M and R, and L is the total length of the line.SinceZL=V2/S∗, it can be obtained that n=a−ib, a = ZCP/V2, b = ZCQ/V2 = a · tan θ, and n=a2+b2=ZCS/V2. V is the power-frequency voltage. θ is the power factor angle, which is positive when the load is inductive and capacitive otherwise.Substituting equation (3)∼(4) into equation (1), the voltage transfer function between the input source at node S and output response at node M can be written as(5)Hu=UMUS=1chγl+ZC/ZMshγl.Simplifying equation (5) by equation (4), there is(6)Hu=n⋅shγl2+chγl2n⋅shγL+chγL.It can be seen that the structure of the numerator and denominator in equation (6) are similar to each other. Therefore, we construct an intermediate function, fuμx=n⋅shγx+chγx, which can be simplified by equation (2) as(7)fuμx=chDx+ashDxcosμx+bchDxsinμx+ishDx+achDxsinμx−bshDxcosμx,where μx=βx. By introducing an auxiliary angle φ, the magnitude of fuμx can be expressed as(8)fuμx=fuμx=Ausin2μx+φ+Bμx,where Au=sgnb⋅A. sgnb is the signal function of b, and there is sgnb=sgnθ. sgnb=1 when the load is inductive (θ>0). And sgnb=−1 when θ<0. sgnb=0 when the load is a pure resistance. A=1−n22/4+b2, while Bμx and φ are(9)Bμx=1+n22×ch2Dx+ash2Dx,φ=arctan1−n22b,φ in equation (9) is π/2 and −π/2 when b⟶0 and ∞, respectively. Otherwise, −π/2<φ<π/2.The magnitude ofHu is numerically equal to the per unit value of voltage. That is, UM∗=Hup.u.. Therefore, substituting equation (8) into (6), it can be obtained that(10)UM∗=Ausin2μl2+φ+Bμl2Ausin2μL+φ+BμL=fuμl2fuμL.


## 2.3. Equivalent Transmission Model for Current Transmission

The current transfer function at any point on a long-distance transmission line can also be derived by the similar method as that inSection 2.1, as follows:(11)Hi=IMUS=1ZC⋅shγl2+n⋅chγl2n⋅shγL+chγL.Similarly, constructing an intermediate functionfiμx=shγx+n⋅chγx. Then, simplifying the magnitude of fiμx, the current on the line can be expressed as(12)IM=USZC⋅Aisin2μl2+φ+Bμl2Ausin2μL+φ+BμL=U1ZC⋅fiμl2fuμL,where Ai=−sgnb1−n22/4+b2=−Au.


## 2.4. The Distribution Characteristics of FLF

Equations (10) and (12) simplify the transmission coefficients into trigonometric functions verifying with the FLF. While Bμxalso contains the product of frequency and length, it changes little at the power frequency f0. Therefore, Bμx is considered as a parameter that has no effect on the pole distribution of the FLF.The pole conditions offuμx and fiμx can be expressed as follows:(13)μx=−φ2−π2+Nπ,(14)μx=−φ2+π2+Nπ,where N is the integer. When θ>0, there is Au>0; thus, equations (13) and (14) are the minimum and maximum conditions of fuμx, respectively. Meanwhile, since Ai=−Au<0, they are the maximum and minimum conditions of fiμx, respectively. The pole distribution conditions are contrary to the above when θ<0. Moreover, when θ=0, there is Au=Ai=0. Accordingly, the transmission of the power-frequency voltage and current is no longer distributed as trigonometric functions of the FLF.Combining with the position (in numerator or denominator) offuμx or fiμx in the fractional functions, the extreme conditions of UM∗ and IM can be discussed. For example, when the load is inductive, the analysis of the extreme value of voltage distribution on a line is translated to a problem of figuring out the polar condition of fuμx as the numerator of UM∗. Therefore, equation (14) is the maximum condition of UM∗, while equation (13) is its minimum condition. When discussing the influence of line length (L) on voltage transmission, the pole condition of fuμx as the denominator of UM∗ is aimed to be found out. Therefore, equations (13) and (14) are the maximum and minimum conditions, respectively.


## 3. Distribution Characteristics of Power-Frequency Current and Voltage on HWTLs in Steady State

### 3.1. Feature of Voltage Distribution

According to the pole distribution condition of the FLF, the maximum voltage on the line (Um∗) and the distance (lm) between Um∗ and node S under various operation modes can be obtained. Moreover, the characteristics of Um∗ and lm variating with load conditions can be analyzed, as shown in Figure 2(a). The following is the concrete analysis approach.Figure 2
Distribution characteristics of maximum power-frequency wave along the line: (a) voltage; (b) current.
(a)
(b)


#### 3.1.1.θ=0

Assume the load power isP=kPn, where Pn is the natural power of the line. When θ=0, there is b=0, thus n=k. Accordingly, Au=n2−1/2 and Bμx=n2+1⋅ch2Dx/2+ash2Dx.Whenn=1, which means that the natural power is transmitted on the line. Thus, Au=0, andBμx=e2Dx. Therefore, the distribution function of the power-frequency voltage on the line can be expressed as UM∗=e−2αl=e−2Dl. That is to say that the voltage distribution on the line attenuated slowly by the natural exponent.Whenn>1, there is Au>0 and φ=−90∘. From the extreme condition equation (14), it can be obtained that Um∗ occurs at μ=π/4, which is the midpoint of the HWTL. Meanwhile, a increases with P, thus Um∗ also increases with P.Whenn<1, there is φ=90°. So, Um∗ appears at the node S, while the minimum value is at the midpoint of the line.


#### 3.1.2.θ>0

Equation (14) is the condition for the occurrence of the maximum voltage on the line. Combining with the range of φ, some conclusions can be obtained as follows.Whenn>1, it can be obtained that λ0/4<lm<3/8λ0. With the increase of P, the parameter a increases, thus decreasing φ and resulting in a bigger lm. That is to say that Um∗ move towards node S with the increase of P. Moreover, because the parameter A increases with a, Um∗ also becomes bigger with P. On the other hand, b decreases with the increase of the load power factor cosθ. Considering that the cosθ of a UHV system is generally greater than 0.9, φ decreases with cosθ, resulting in a smaller lm. Meanwhile, because b decreases with A, Um∗ also decreases with A.Whenn<1, it can be obtained in the similar way that lm=3/8λ0. Additionally, lm increases with P and cosθ, while Um∗ increases P but decreases with cosθ. When n=1, there is lm=3/8λ0.In summary, when the line transmits inductive reactive power,Um∗ appears in the second half of the line. It increases with P, but decreases with cosθ.


#### 3.1.3.θ<0

Similarly, according to equation (13), when n>1, λ0/8<lm<λ0/4. lm increases with P and cosθ. When n<1, there is 0<lm<λ0/8. lm increases with P, but decreases with cosθ. When n=1, lm=λ0/8. In brief, when the load is capacitive, Um∗ appears in the first half of the line, and the relationship between Um∗ and P or cosθ are the same as the condition of θ>0.


### 3.2. Feature of Current Distribution

Using the similar method as that in Section3.1, the impact of operation modes on the current distribution along the line is discussed. The distribution of the maximum current on the line is shown in Figure 2(b). In particular, when θ=0 and k=1, the current transmission function is IM=U1/ZCe−2Dl. Otherwise, it is a trigonometric function as (12).


### 3.3. Amplitude Feature

Based on the analysis above,Um∗ can be expressed as(15)Um∗=Au+Bμl2Ausinφ+BμL.It can be obtained thatAusinφ=1−n2/2. In addition, chx is expended to the Taylor series as ch2DL=1+x2/2!+x4/4!⋯=1+οx for convenience. Accordingly, when there is a Um∗ on the line, it can be simplified as(16)Um∗=n1+δ,where δ=n2+1⋅ο2DL/2.Similarly,Im can also be expressed as(17)Im=U1ZCn1+δ.It can be seen thatUm∗ and Im are both related to n. Um∗ and Im increases with P and Q, which means they decrease with cosθ.


## 3.1. Feature of Voltage Distribution

According to the pole distribution condition of the FLF, the maximum voltage on the line (Um∗) and the distance (lm) between Um∗ and node S under various operation modes can be obtained. Moreover, the characteristics of Um∗ and lm variating with load conditions can be analyzed, as shown in Figure 2(a). The following is the concrete analysis approach.Figure 2
Distribution characteristics of maximum power-frequency wave along the line: (a) voltage; (b) current.
(a)
(b)


### 3.1.1.θ=0

Assume the load power isP=kPn, where Pn is the natural power of the line. When θ=0, there is b=0, thus n=k. Accordingly, Au=n2−1/2 and Bμx=n2+1⋅ch2Dx/2+ash2Dx.Whenn=1, which means that the natural power is transmitted on the line. Thus, Au=0, andBμx=e2Dx. Therefore, the distribution function of the power-frequency voltage on the line can be expressed as UM∗=e−2αl=e−2Dl. That is to say that the voltage distribution on the line attenuated slowly by the natural exponent.Whenn>1, there is Au>0 and φ=−90∘. From the extreme condition equation (14), it can be obtained that Um∗ occurs at μ=π/4, which is the midpoint of the HWTL. Meanwhile, a increases with P, thus Um∗ also increases with P.Whenn<1, there is φ=90°. So, Um∗ appears at the node S, while the minimum value is at the midpoint of the line.


### 3.1.2.θ>0

Equation (14) is the condition for the occurrence of the maximum voltage on the line. Combining with the range of φ, some conclusions can be obtained as follows.Whenn>1, it can be obtained that λ0/4<lm<3/8λ0. With the increase of P, the parameter a increases, thus decreasing φ and resulting in a bigger lm. That is to say that Um∗ move towards node S with the increase of P. Moreover, because the parameter A increases with a, Um∗ also becomes bigger with P. On the other hand, b decreases with the increase of the load power factor cosθ. Considering that the cosθ of a UHV system is generally greater than 0.9, φ decreases with cosθ, resulting in a smaller lm. Meanwhile, because b decreases with A, Um∗ also decreases with A.Whenn<1, it can be obtained in the similar way that lm=3/8λ0. Additionally, lm increases with P and cosθ, while Um∗ increases P but decreases with cosθ. When n=1, there is lm=3/8λ0.In summary, when the line transmits inductive reactive power,Um∗ appears in the second half of the line. It increases with P, but decreases with cosθ.


### 3.1.3.θ<0

Similarly, according to equation (13), when n>1, λ0/8<lm<λ0/4. lm increases with P and cosθ. When n<1, there is 0<lm<λ0/8. lm increases with P, but decreases with cosθ. When n=1, lm=λ0/8. In brief, when the load is capacitive, Um∗ appears in the first half of the line, and the relationship between Um∗ and P or cosθ are the same as the condition of θ>0.


## 3.1.1.θ=0

Assume the load power isP=kPn, where Pn is the natural power of the line. When θ=0, there is b=0, thus n=k. Accordingly, Au=n2−1/2 and Bμx=n2+1⋅ch2Dx/2+ash2Dx.Whenn=1, which means that the natural power is transmitted on the line. Thus, Au=0, andBμx=e2Dx. Therefore, the distribution function of the power-frequency voltage on the line can be expressed as UM∗=e−2αl=e−2Dl. That is to say that the voltage distribution on the line attenuated slowly by the natural exponent.Whenn>1, there is Au>0 and φ=−90∘. From the extreme condition equation (14), it can be obtained that Um∗ occurs at μ=π/4, which is the midpoint of the HWTL. Meanwhile, a increases with P, thus Um∗ also increases with P.Whenn<1, there is φ=90°. So, Um∗ appears at the node S, while the minimum value is at the midpoint of the line.


## 3.1.2.θ>0

Equation (14) is the condition for the occurrence of the maximum voltage on the line. Combining with the range of φ, some conclusions can be obtained as follows.Whenn>1, it can be obtained that λ0/4<lm<3/8λ0. With the increase of P, the parameter a increases, thus decreasing φ and resulting in a bigger lm. That is to say that Um∗ move towards node S with the increase of P. Moreover, because the parameter A increases with a, Um∗ also becomes bigger with P. On the other hand, b decreases with the increase of the load power factor cosθ. Considering that the cosθ of a UHV system is generally greater than 0.9, φ decreases with cosθ, resulting in a smaller lm. Meanwhile, because b decreases with A, Um∗ also decreases with A.Whenn<1, it can be obtained in the similar way that lm=3/8λ0. Additionally, lm increases with P and cosθ, while Um∗ increases P but decreases with cosθ. When n=1, there is lm=3/8λ0.In summary, when the line transmits inductive reactive power,Um∗ appears in the second half of the line. It increases with P, but decreases with cosθ.


## 3.1.3.θ<0

Similarly, according to equation (13), when n>1, λ0/8<lm<λ0/4. lm increases with P and cosθ. When n<1, there is 0<lm<λ0/8. lm increases with P, but decreases with cosθ. When n=1, lm=λ0/8. In brief, when the load is capacitive, Um∗ appears in the first half of the line, and the relationship between Um∗ and P or cosθ are the same as the condition of θ>0.


## 3.2. Feature of Current Distribution

Using the similar method as that in Section3.1, the impact of operation modes on the current distribution along the line is discussed. The distribution of the maximum current on the line is shown in Figure 2(b). In particular, when θ=0 and k=1, the current transmission function is IM=U1/ZCe−2Dl. Otherwise, it is a trigonometric function as (12).


## 3.3. Amplitude Feature

Based on the analysis above,Um∗ can be expressed as(15)Um∗=Au+Bμl2Ausinφ+BμL.It can be obtained thatAusinφ=1−n2/2. In addition, chx is expended to the Taylor series as ch2DL=1+x2/2!+x4/4!⋯=1+οx for convenience. Accordingly, when there is a Um∗ on the line, it can be simplified as(16)Um∗=n1+δ,where δ=n2+1⋅ο2DL/2.Similarly,Im can also be expressed as(17)Im=U1ZCn1+δ.It can be seen thatUm∗ and Im are both related to n. Um∗ and Im increases with P and Q, which means they decrease with cosθ.


## 4. Characteristics of Steady-State Power-Frequency Resonance for HWTLs after Short Circuit

Figure3 shows the system equivalent circuit when the fault point (node F) is short circuited. Taking node F as the demarcation point, the system is divided into the models of the left side and right side. Since the calculation models of both sides are the same, it is reasonable to analyze the left-side model merely.Figure 3
Diagram of three-phase short circuit of the research system.The one-side model of the system in Figure3 is similar to that shown in Figure 1(a). Therefore, equation (3)∼(12) can also be derived in short-circuit analysis, where L and l2 have different definitions in short-circuit analysis. L represents the distance between node F and S, and l2 is the length between node F and M.There is a local steady-state maximum voltage and short-circuit current on the line after the fault, which isl2m kilometers away from node F. In addition, the local maximum voltage and current vary with node F. The fault point that makes them reach the maximum value (Ufm∗ and Ifm) is called “the most serious short-circuit point,” which is at a distance of Lm from node S.


### 4.1. Calculation Models

Considering the effect of system parameters and fault conditions on the distribution of power-frequency voltage simultaneously, there is(18)USUG=ZSZS+ZG,(19)ZS=ZCn⋅shγL+chγLshγL+n⋅chγL,where n=ZC/Zf. ZG is the total impedance of the sending system with generators and transformers included. Zf represents different fault conditions. It is the fault resistance for symmetric short-circuit analysis, or the additional impedance to simulate asymmetric short circuit.Combining equations (6), (18), and (19), it can be obtained that(20)UMf∗=UMUG=shγl2+n⋅chγl2n+mshγL+nm+1chγL,where m=ZG/ZC.Similarly, the short-circuit current can be obtained as(21)If1=UGZf⋅1n+mshγL+nm+1chγL.SinceZG=iXG is mainly a inductance in practical engineering, it can be assumed that m=id and d>0. Using the similar method as that in Section 1.2, it can be obtained that(22)UMf∗=fuμl2gμL,where fuμl2 is the molecular partial of equation (10):(23)gμL=Agsin2μL+φg+BμLg,Ag=n2−1d2−1+4bd22+n2d−bd2+b−d2,BμLg=ad2+1sh2DL+n2+1d2+12ch2DL,φg=arctan1−d21−n2+4bd2n2d−bd2+b−d.In the similar way, the short-circuit current can be simplified to(24)If1=UGZf⋅1gμL.


### 4.2. Characteristics of Symmetrical Short-Circuit Fault

In this section, the comprehensive effect of the sending system parameter and the fault condition are represented by Zf.Zf=Rf on transmission characteristics is analyzed for symmetrical grounding short circuit.Given that the real part ofZC is usually much larger than that of the imaginary part in engineering, it can be considered that a>0 and b is positive but very small, which is neglected in analysis. Meanwhile, according to the parameters of the UHVAC systems in actual demonstration project [28, 29], the system impedance under small and large operation mode are ZG = i0.07 p.u. and ZG = i0.002 p.u., respectively. The base voltage and base capacity of the system are Ub = 1050 kV and Sb = 1000 MVA, respectively. In addition, for the reason that the fault resistance of UHV systems is generally less than 50 Ω after three-phase short circuit [30], it can be considered that d<1, a>1, and d≪a. Therefore, it can be obtained that(25)Au=a2−12,Bμl2=a2+12ch2Dl2+ash2Dl2,φ=−π2,Ag=d2+1a2−12,BμLg=d2+1a2+12ch2DL+ash2DL,φg=arctand2−12d.By the simplification above, the distribution characteristics of power-frequency voltage and short-circuit current on the line after the fault are expressed by functions containing parametersa and d. This can be used to analyze the comprehensive influence of both fault resistance Rf and system impedance ZG on short-circuit characteristics intuitively.


#### 4.2.1. Location Characteristics of the Maximum Short-Circuit Voltage along the Line

The condition for the maximum value of the calculation model equation (22) is μl2m=π/2+Nπ. It can be obtained that l2m=λ0/4 when λ0/4<L<λ0/2, while l2m=L when L<λ0/4. That is to say, the position that the short-circuit voltage reaches maximum on the line is independent of the sending system parameters and the fault resistance. It is at the distance of λ0/4 from node F when some point of the second half of the line is short-circuited. Otherwise, it is at node S.


#### 4.2.2. Distribution Characteristics of the Most Serious Short Circuit

(1) Location Feature. As can be seen from equation (25), φg is only related to d but not to a. That is, the location of the most serious fault point is only depended on the sending system parameters but not on the fault resistance. In addition, d increases with XG, thus φg increases, resulting in the decrease of Lm. That is, the most serious fault point move towards the midpoint of the line from the receiving end.According to the pole distribution characteristics ofgμL, it can be obtained that Lm=3/4−φg/2πλ0/2. Due to −π/2<φg<π/2, there is λ0/4<Lm<λ0/2. That is to say, the point causing the most serious three-phase short circuit is at the second half of a HWTL.(2) Amplitude Feature. By using the same method as that in Section 3.3, the most serious short-circuit voltage Ufm∗ and short-circuit current Ifm are simplified as(26)Ufm∗=1d2+1a2+δλ0/41+δLmf,Ifm=UGZCad2+11+δLmf,where δλ0/4=a2+1/2⋅οDλ0/4 and δLmf=a2+1/2⋅οDLmf.WhenXG increases from a very small value, δLmf is the dominant variation, which decreases with XG. Therefore, Ifm increases with XG. After that, d becomes the major variation, which increases with XG, resulting in the decrease of Ufm∗ and Ifm. In brief, Ufm∗ and Ifm increase with XG first, but then decrease. On the other hand, when Rf becomes larger, Ifm and Ufm∗ decrease because of the decrease of a.


### 4.3. Characteristics of Asymmetrical Short-Circuit Fault

Asymmetrical short-circuit can be simulated by changing the additional impedanceZf=iXf [31], where b>0 and a=0. Thus, it can be obtained that(27)Au=b2+12,Bμl2=Auch2Dl2,φ=arctan12b−b2,Ag=b2+1d2+12,BμLg=AuGch2DL,φg=arctan12d+1/b−1d+1/b.


#### 4.3.1. Location Characteristics of the Maximum Short-Circuit Voltage on the Line

It can be seen from equation (27) that φ is related to b but independent of d. That is, the position that the short-circuit voltage reaches maximum is only related to the additional impedance Zf but not to the sending system parameters. When Zf decreases, b increases but φ decreases, thus causing a larger l2m. In other words, the distance between the position of the maximum short-circuit voltage and the short-circuit point decreases with Zf.Based on the pole distribution characteristics offuμl2, there is l2m=1/4−φ/2πλ0/2. Therefore, it can be obtained that 0<l2m<λ0/4 when λ0/4<L<λ0/2, while l2m=L when L<λ0/4.


#### 4.3.2. Location Characteristics of the Most Serious Fault

It can be seen from equation (27) that φg is related to both d and b. In addition, the increase of XG (or Zf) enlarges d (or reduces b). Hence, φg becomes larger, resulting in a bigger Lm. In other words, the distance between the most-serious fault point and node S moves towards the midpoint of the line from node R with the increase of XG and Zf.It can be obtained in the same way as (1) in Section4.2.2 that λ0/4<Lm<λ0/2. That is, the most serious point of three-phase short circuit occurs in the second half of the HWTL. The power-frequency voltage and short-circuit current along the line reach to maximum when such point is short circuited.


## 4.1. Calculation Models

Considering the effect of system parameters and fault conditions on the distribution of power-frequency voltage simultaneously, there is(18)USUG=ZSZS+ZG,(19)ZS=ZCn⋅shγL+chγLshγL+n⋅chγL,where n=ZC/Zf. ZG is the total impedance of the sending system with generators and transformers included. Zf represents different fault conditions. It is the fault resistance for symmetric short-circuit analysis, or the additional impedance to simulate asymmetric short circuit.Combining equations (6), (18), and (19), it can be obtained that(20)UMf∗=UMUG=shγl2+n⋅chγl2n+mshγL+nm+1chγL,where m=ZG/ZC.Similarly, the short-circuit current can be obtained as(21)If1=UGZf⋅1n+mshγL+nm+1chγL.SinceZG=iXG is mainly a inductance in practical engineering, it can be assumed that m=id and d>0. Using the similar method as that in Section 1.2, it can be obtained that(22)UMf∗=fuμl2gμL,where fuμl2 is the molecular partial of equation (10):(23)gμL=Agsin2μL+φg+BμLg,Ag=n2−1d2−1+4bd22+n2d−bd2+b−d2,BμLg=ad2+1sh2DL+n2+1d2+12ch2DL,φg=arctan1−d21−n2+4bd2n2d−bd2+b−d.In the similar way, the short-circuit current can be simplified to(24)If1=UGZf⋅1gμL.


## 4.2. Characteristics of Symmetrical Short-Circuit Fault

In this section, the comprehensive effect of the sending system parameter and the fault condition are represented by Zf.Zf=Rf on transmission characteristics is analyzed for symmetrical grounding short circuit.Given that the real part ofZC is usually much larger than that of the imaginary part in engineering, it can be considered that a>0 and b is positive but very small, which is neglected in analysis. Meanwhile, according to the parameters of the UHVAC systems in actual demonstration project [28, 29], the system impedance under small and large operation mode are ZG = i0.07 p.u. and ZG = i0.002 p.u., respectively. The base voltage and base capacity of the system are Ub = 1050 kV and Sb = 1000 MVA, respectively. In addition, for the reason that the fault resistance of UHV systems is generally less than 50 Ω after three-phase short circuit [30], it can be considered that d<1, a>1, and d≪a. Therefore, it can be obtained that(25)Au=a2−12,Bμl2=a2+12ch2Dl2+ash2Dl2,φ=−π2,Ag=d2+1a2−12,BμLg=d2+1a2+12ch2DL+ash2DL,φg=arctand2−12d.By the simplification above, the distribution characteristics of power-frequency voltage and short-circuit current on the line after the fault are expressed by functions containing parametersa and d. This can be used to analyze the comprehensive influence of both fault resistance Rf and system impedance ZG on short-circuit characteristics intuitively.


### 4.2.1. Location Characteristics of the Maximum Short-Circuit Voltage along the Line

The condition for the maximum value of the calculation model equation (22) is μl2m=π/2+Nπ. It can be obtained that l2m=λ0/4 when λ0/4<L<λ0/2, while l2m=L when L<λ0/4. That is to say, the position that the short-circuit voltage reaches maximum on the line is independent of the sending system parameters and the fault resistance. It is at the distance of λ0/4 from node F when some point of the second half of the line is short-circuited. Otherwise, it is at node S.


### 4.2.2. Distribution Characteristics of the Most Serious Short Circuit

(1) Location Feature. As can be seen from equation (25), φg is only related to d but not to a. That is, the location of the most serious fault point is only depended on the sending system parameters but not on the fault resistance. In addition, d increases with XG, thus φg increases, resulting in the decrease of Lm. That is, the most serious fault point move towards the midpoint of the line from the receiving end.According to the pole distribution characteristics ofgμL, it can be obtained that Lm=3/4−φg/2πλ0/2. Due to −π/2<φg<π/2, there is λ0/4<Lm<λ0/2. That is to say, the point causing the most serious three-phase short circuit is at the second half of a HWTL.(2) Amplitude Feature. By using the same method as that in Section 3.3, the most serious short-circuit voltage Ufm∗ and short-circuit current Ifm are simplified as(26)Ufm∗=1d2+1a2+δλ0/41+δLmf,Ifm=UGZCad2+11+δLmf,where δλ0/4=a2+1/2⋅οDλ0/4 and δLmf=a2+1/2⋅οDLmf.WhenXG increases from a very small value, δLmf is the dominant variation, which decreases with XG. Therefore, Ifm increases with XG. After that, d becomes the major variation, which increases with XG, resulting in the decrease of Ufm∗ and Ifm. In brief, Ufm∗ and Ifm increase with XG first, but then decrease. On the other hand, when Rf becomes larger, Ifm and Ufm∗ decrease because of the decrease of a.


## 4.2.1. Location Characteristics of the Maximum Short-Circuit Voltage along the Line

The condition for the maximum value of the calculation model equation (22) is μl2m=π/2+Nπ. It can be obtained that l2m=λ0/4 when λ0/4<L<λ0/2, while l2m=L when L<λ0/4. That is to say, the position that the short-circuit voltage reaches maximum on the line is independent of the sending system parameters and the fault resistance. It is at the distance of λ0/4 from node F when some point of the second half of the line is short-circuited. Otherwise, it is at node S.


## 4.2.2. Distribution Characteristics of the Most Serious Short Circuit

(1) Location Feature. As can be seen from equation (25), φg is only related to d but not to a. That is, the location of the most serious fault point is only depended on the sending system parameters but not on the fault resistance. In addition, d increases with XG, thus φg increases, resulting in the decrease of Lm. That is, the most serious fault point move towards the midpoint of the line from the receiving end.According to the pole distribution characteristics ofgμL, it can be obtained that Lm=3/4−φg/2πλ0/2. Due to −π/2<φg<π/2, there is λ0/4<Lm<λ0/2. That is to say, the point causing the most serious three-phase short circuit is at the second half of a HWTL.(2) Amplitude Feature. By using the same method as that in Section 3.3, the most serious short-circuit voltage Ufm∗ and short-circuit current Ifm are simplified as(26)Ufm∗=1d2+1a2+δλ0/41+δLmf,Ifm=UGZCad2+11+δLmf,where δλ0/4=a2+1/2⋅οDλ0/4 and δLmf=a2+1/2⋅οDLmf.WhenXG increases from a very small value, δLmf is the dominant variation, which decreases with XG. Therefore, Ifm increases with XG. After that, d becomes the major variation, which increases with XG, resulting in the decrease of Ufm∗ and Ifm. In brief, Ufm∗ and Ifm increase with XG first, but then decrease. On the other hand, when Rf becomes larger, Ifm and Ufm∗ decrease because of the decrease of a.


## 4.3. Characteristics of Asymmetrical Short-Circuit Fault

Asymmetrical short-circuit can be simulated by changing the additional impedanceZf=iXf [31], where b>0 and a=0. Thus, it can be obtained that(27)Au=b2+12,Bμl2=Auch2Dl2,φ=arctan12b−b2,Ag=b2+1d2+12,BμLg=AuGch2DL,φg=arctan12d+1/b−1d+1/b.


### 4.3.1. Location Characteristics of the Maximum Short-Circuit Voltage on the Line

It can be seen from equation (27) that φ is related to b but independent of d. That is, the position that the short-circuit voltage reaches maximum is only related to the additional impedance Zf but not to the sending system parameters. When Zf decreases, b increases but φ decreases, thus causing a larger l2m. In other words, the distance between the position of the maximum short-circuit voltage and the short-circuit point decreases with Zf.Based on the pole distribution characteristics offuμl2, there is l2m=1/4−φ/2πλ0/2. Therefore, it can be obtained that 0<l2m<λ0/4 when λ0/4<L<λ0/2, while l2m=L when L<λ0/4.


### 4.3.2. Location Characteristics of the Most Serious Fault

It can be seen from equation (27) that φg is related to both d and b. In addition, the increase of XG (or Zf) enlarges d (or reduces b). Hence, φg becomes larger, resulting in a bigger Lm. In other words, the distance between the most-serious fault point and node S moves towards the midpoint of the line from node R with the increase of XG and Zf.It can be obtained in the same way as (1) in Section4.2.2 that λ0/4<Lm<λ0/2. That is, the most serious point of three-phase short circuit occurs in the second half of the HWTL. The power-frequency voltage and short-circuit current along the line reach to maximum when such point is short circuited.


## 4.3.1. Location Characteristics of the Maximum Short-Circuit Voltage on the Line

It can be seen from equation (27) that φ is related to b but independent of d. That is, the position that the short-circuit voltage reaches maximum is only related to the additional impedance Zf but not to the sending system parameters. When Zf decreases, b increases but φ decreases, thus causing a larger l2m. In other words, the distance between the position of the maximum short-circuit voltage and the short-circuit point decreases with Zf.Based on the pole distribution characteristics offuμl2, there is l2m=1/4−φ/2πλ0/2. Therefore, it can be obtained that 0<l2m<λ0/4 when λ0/4<L<λ0/2, while l2m=L when L<λ0/4.


## 4.3.2. Location Characteristics of the Most Serious Fault

It can be seen from equation (27) that φg is related to both d and b. In addition, the increase of XG (or Zf) enlarges d (or reduces b). Hence, φg becomes larger, resulting in a bigger Lm. In other words, the distance between the most-serious fault point and node S moves towards the midpoint of the line from node R with the increase of XG and Zf.It can be obtained in the same way as (1) in Section4.2.2 that λ0/4<Lm<λ0/2. That is, the most serious point of three-phase short circuit occurs in the second half of the HWTL. The power-frequency voltage and short-circuit current along the line reach to maximum when such point is short circuited.


## 5. Characteristics of Steady-State Power-Frequency Resonance for Long-Distance Transmission Lines after Fault

From the abovementioned analysis, it can be seen that there is0<μL<π/2 when 0<L<λ0/4. According to the pole distribution condition of fuμl2, the maximum voltage Ufm∗ along the line appears at node S, that is, l2=L and μl2=μL. Hence, the most-serious short-circuit voltage can be expressed as Ufm∗=fuμL/gμL.In order to analyzeUfm∗, construct a function as FμL=gμL2−fuμL2. By calculating the derivative of FμL, it can be obtained that FμL increases monotonously or increases first but then decreases. Therefore, it reaches minimum when μL=0 or μL=λ0/4. Since F0, Fπ/2, fuμL, and gμL are all larger than 0, it can be easily obtained that gμL>fuμL. Therefore, there is Ufm∗<1. That is to say, for a line with length less than λ0/4, power-frequency resonance cannot appear along the line after short circuit, and there is no overvoltage.


## 6. Case Study and Discussions

### 6.1. The Distribution Characteristic of Voltage and Current on HWTLs

A point-to-point power system with a HWTL is built in PSCAD, as shown in Figure1, where the traveling wave Bergeron model of the transmission line is chosen. This model is based on the distributed LC parameters and lumped resistance. It represents the inductance and capacitance of the π segments in a distributed way rather than lumped parameters and uses the lumped resistance to reflect line loss. Therefore, the Bergeron model in PSCAD can accurately simulate transmission lines at the power frequency f0 and is suitable for the cases related to the power flow at f0 [32]. The studied HWTL is an existing transmission line of the UHV power system in China, where the parameters are calculated and shown in Table 1.Table 1
The information of an UHVAC transmission line.
Type of wire
8 × LGJ-500/35 (with 400 mm between conductors in the bundle)
Type of the ground wire
JLB20A-170
Information of conductor arrangement
Distance from suspension point of upper conductor to ground: 65 m
Distance from suspension point of lower conductor to ground: 45 m
Vertical distance between suspension points of the ground wire and upper conductor: 12 m
Horizontal distance between upper conductor and central line: 0
Horizontal distance between lower conductor and central line: 16
Horizontal distance between ground wire and central line: 14
Conductor sag: 18.5 m
Ground wire sag: 12.5 m
Per unit length parameters
R
0 = 0.0081 Ω/km, X0 = 0.2631 Ω/km, C0 = 0.01383 μF/km,Pn = 4063 MW, λ0/2 = 2938 kmFigure4 shows the results of maximum power-frequency voltage, current, and their distribution characteristics under different operation modes. The error between the results of simulation and the proposed model calculation is tiny, which is less than 0.15%. That is to say that the method presented in this paper can accurately analyze the distribution characteristics of power-frequency voltage and current in the steady state. It can be seen from Figure 4 that the distribution characteristics of Um∗ and Im are consistent with Figure 2 and the conclusions in Section 3.3. When θ = 0 (cos θ = 1), Um∗ appears at the sending bus when a power of less than 4063 MW is transmitted or at the midpoint of the line (l = 1469 km) when the transmitted power is more than 4063 MW. The position of Im is opposite to that of Um∗. It occurs at l = 1469 km when P < 4063 MW or at the sending bus when P > 4063 MW. Moreover, Um∗ increases with P, while Im decreases with it. Given that the voltage on the line in an actual UHVAC power system cannot exceed 1.2 p.u. for the safety operation [33], it is suggested that the active power of the UHV system with HWTLs should be less than 1.25Pn = 5080 MW.Figure 4
Distribution of power-frequency waves along the HWTL in the steady state under different operation conditions: (a)θ = 0. (b) θ = 0. (c) θ > 0. (d) θ > 0. (e) θ < 0. (f) θ < 0.
(a)
(b)
(c)
(d)
(e)
(f)However, there is cosθ < 1 in practical engineering in general. Meanwhile, it has been obtained that both Um∗ and Im decrease with cos θ. That is, the decrease of the power factor raises the risk of overvoltage. Therefore, to avoid power-frequency overvoltage, it is necessary to set a limit on the power factor according to the power load, as shown in Figure 5. For example, cos θ is needed to be controlled above 0.86 when P = 0.9Pn = 3660 MW. Hence, the operating modes with P < 3660 MW and cos θ > 0.86 are safety for the system.Figure 5
The safe operating modes and overvoltage operating modes.Generally, 0.85 is considered as the minimum acceptable power factor for AC transmission. Therefore, the safe operating modes are shown in the shadow area of Figure5. It can be seen that keeping the power factor larger than 0.98 and the active power less than 5000 MW can eliminate the power-frequency voltage. However, it is challenge for a UHV power system to keep such a large power factor. Thus, the improvement of the power factor for UHV systems is significantly important for HWTLs to put into practice.


### 6.2. Analysis of Short Circuit Fault

#### 6.2.1. Symmetrical Short Circuit

In order to analyze the power-frequency resonance characteristics after the three-phase short-circuit fault, the half-wavelength transmission system is built in the electromagnetic transient simulation program PSCAD, as shown in Figure3. The distributed Bergeron model is also used here as a reference due to its correctness for the calculation of the conditions related to power frequency.In this case, the HWTL is short to ground with no resistance (Rf = 10−19 Ω), low resistance (Rf = 5 Ω), and high resistance (Rf = 50 Ω). The condition of Rf = 100 Ω is also analyzed as a supplement. Figure 6 shows the simulation results of the maximum voltage UMf∗ and short-circuit current IMf on the line with different fault locations L. Among these, except for ZG within the general range in practice (i0.002 Ω∼i0.07 Ω), ZG = i0.2 Ω and ZG = i1 Ω are also analyzed to find out the characteristics as a whole.Figure 6
Characteristics of the maximum voltage and current after the short-circuit fault with different fault locations. (a) Voltage; (b) current.
(a)
(b)It can be seen from Figure6 that when ZG increases from i0.002 Ω to i0.07 Ω, both Ufm∗ and Ifm increase. However, they drop with ZG when ZG > i0. 07 Ω. When the system runs in the minimum operating mode (ZG = i0.07 Ω), Ufm∗ reaches the maximum value of 21 p.u. and Ifm reaches to 90 kA. Therefore, to limit the power-frequency voltage resonance and the short-circuit current, the system shall be operated in the mode as large as possible.In addition,Ufm∗ and Ifm decrease significantly with the fault resistance Rf. On the contrary, Lm is independent on Rf. It only decreases with ZG. Even so, Lm is always larger than λ0/4 = 1469 km, regardless of the value of Rf. Moreover, it is worth mentioning that the most serious fault overvoltages are accompanied by the maximum short-circuit currents. In other words, the most serious overvoltages and maximum short-circuit currents for HWTLs occur at the same time. This characteristic makes the short-circuit consequence of HWTLs much more serious.To conclude, the most serious short-circuit point is mainly related toRf rather than ZG. Considering the range of ZG and Rf in practice, it is at the distance of about 2653 km∼2929 km from the sending bus. In addition, Ufm∗ and Ifm can reach about 4∼21 p.u. and 16∼90 kA, respectively. The potential overvoltages and maximum short-circuit currents are far larger than the standard. Both of them need further analysis for the specific suppress strategies.Considering the actual operation of the system (i0.002 < ZG < i0.07 Ω and Rf < 50 Ω), the voltage distribution on a HWTL after the most serious short circuit are analyzed, as shown in Figure 7. In addition, the comparison between the simulation and calculation results are shown in Table 2.Figure 7
The voltage distribution along lines after the most serious short-circuit fault.Table 2
The simulation and calculation results of the voltage and current distribution characteristics under different fault conditions for the symmetric short circuit.
Z
G (p.u.)
i0.002
i0.02
i0.07
R
f (Ω)
10−19
5
50
10−19
5
50
10−19
5
50
U
fm
∗
p
.
u
.
Simulation
20.97
14.71
3.89
21.44
14.92
4
22.1
15.47
4
Calculation
19.89
14.17
3.72
20.21
14.57
3.81
20.9
14.88
3.88
I
fm
kA
Simulation
85.21
59.73
15.73
87.11
60.59
16.2
89.8
61.03
16.2
Calculation
83.65
57.92
14.94
85.25
58.34
15.13
86.4
59.58
15.3
I
fm
kA
Simulation
2929
2929
2926
2854
2854
2853
2653
2653
2650
Calculation
2929
2925
2924
2848
2848
2846
2650
2649
2646
L
m
km
Simulation
1469
1469
1466
1469
1469
1468
1469
1469
1466
Calculation
1469
1469
1469
1469
1469
1469
1469
1469
1469
l
m
km
Simulation
1460
1460
1460
1385
1385
1385
1184
1184
1184
Calculation
1460
1456
1455
1379
1379
1377
1181
1180
1177To confirm the results above, the following are some supplemental cases, as shown in Table3. Case A is simulated with the single-phase distributed parameter line model in DigSILENT/Power factory, while Case B uses the three-phase distributed parameter line model in MATLAB/Simulink. In addition, Case C is simulated in PSCAD, and the data on the steady state is collected. In this case, the frequency dependent model of the transmission line in PSCAD is used. This model is one of the most accurate model of the transmission line models, which is able to represent all the frequency-dependent effects of transmission lines.Table 3
The results of the voltage and current distribution in different cases.
Z
G (p.u.)
i0.002
i0.02
i0.07
R
f (Ω)
10−19
5
50
10−19
5
50
10−19
5
50
U
fm
∗
p
.
u
.
Case A
20.92
14.70
3.84
21.39
14.88
3.96
22.0
15.42
3.98
Case B
20.9
14.72
3.88
21.25
14.88
4.02
21.9
15.29
3.96
Case C
21.27
14. 95
3.96
21.58
15.24
4
22.43
15.87
4.
I
fm
kA
Case A
85.16
59.65
15.62
87.11
60.53
16.12
89.69
60.93
16.13
Case B
85.13
59.68
15.64
87.02
60.51
16.19
89.61
60.86
16.1
Case C
85.11
58.88
15.63
87.19
60.66
16.25
89.76
61.13
16.22
I
fm
kA
Case A
2929
2929
2926
2854
2854
2853
2653
2653
2650
Case B
2929
2928
2926
2855
2855
2853
2652
2651
2650
Case C
2930
2930
2927
2858
2857
2854
2655
2655
2650
L
m
km
Case A
1469
1469
1466
1469
1469
1468
1469
1469
1466
Case B
1468
1468
1466
1469
1468
1468
1469
1468
1466
Case C
1469
1469
1466
1469
1469
1465
1470
1470
1462
l
m
km
Case A
1460
1460
1460
1385
1385
1385
1184
1184
1184
Case B
1461
1460
1460
1386
1387
1385
1183
1183
1184
Case C
1461
1461
1461
1389
1388
1389
1185
1185
1188It can be seen that the calculation results in Table2 are fairly close to that shown in Case A and B. That is, the distributed Bergeron model can represent the distributed-parameter characteristic of the line at power frequency precisely, both in the aspect of single phase and three phase. Moreover, the calculation results in Table 2 also have a good accuracy compared with Case C. It shows that the Bergeron model in PSCAD can describe the feature of lines in the steady state. However, the difference between the calculation results and Case C are larger than the difference between calculation results and Case A or B. The reason is that there can be a slight difference between the actual parameter in the tower line model and the calculated parameters that is used in the distributed PI models.From the abovementioned analyses, it can be concluded that the distributed Bergeron model is accurate enough to describe the feature of transmission lines in the steady state after the short-circuit fault.


#### 6.2.2. Asymmetrical Short Circuit

In this case, the effect of the sending system parameters and the additional impedance on the position characteristics (Lm and l2m) for asymmetric short circuit are discussed. The PSCAD simulation results and the calculation results are shown in Table 4.It can be obtained that the position of the most-serious fault point moves towards the midpoint of the line with the increase of the impedance of the sending system and the additional impedance. Meanwhile, the position of the maximum short-circuit voltage along the line moves from 1469 km to the short-circuit point with the increase ofZf. That is, the distance between the overvoltage caused by the frequency-power resonance and the sending bus is about 1184 km∼1460 km. Comparing with Tables 3 and 4, it can be seen that lm in Table 4 is approximately the same as the cases of the symmetric short circuit. In other words, it can be considered that asymmetric short circuit does not affect the positions of the most-serious overvoltage on the line. The position characteristics of the asymmetric short circuit can be analyzed by the same way as the three-phase short circuit fault.Table 4
The simulation and calculation results of the voltage and current distribution for asymmetric short circuit.
Z
G (p.u.)
i0.002
i0.02
i0.07
Zf (p.u.)
0.001
0.01
0.1
0.001
0.01
0.1
0.001
0.01
0.1
L
m
km
Simulation
2925
2887
2535
2850
2812
2460
2649
2611
2259
Calculation
2920
2872
2528
2841
2802
2455
2645
2605
2253
l
2
m
km
Simulation
1465
1427
1075
1465
1427
1075
1465
1427
1075
Calculation
1465
1425
1074
1465
1425
1069
1465
1425
1069
l
m
km
Simulation
1460
1460
1460
1385
1385
1385
1184
1184
1184
Calculation
1455
1453
1452
1376
1373
1374
1180
1176
1172The simulation results are consistent with the theoretical results of the proposed method. That is, the proposed algorithm can analyze the position characteristics of the most-serious asymmetric short-circuit fault. The error of the method is less than 1%.


#### 6.2.3. Voltage Distribution Characteristics of Different Length Lines

To investigate the voltage distribution characteristics for different lines, transmission lines with length of 200 km, 500 km, 800 km, 1469 km (λ0/4), 2100 km, and 2700 km are discussed here. Figure 8 shows both the simulation and calculation results of the voltage distribution along these lines after the most-serious short circuit, where the receiving end is assumed as an infinite-bus system.Figure 8
The voltage distribution along lines after the most-serious short-circuit fault for different lines (Rf=10−19Ω). (a) L = 2700km; (b) L=2100km; (c) L=1469km; (d) L=800km; (e) L=500km; (f) L=200km.
(a)
(b)
(c)
(d)
(e)
(f)It can be obtained that, for transmission lines longer than 1469 km, power-frequency resonance overvoltage can occur at a distance of about 1469 km from the short-circuit point. However, for lines with length less than 1469 km, there is no overvoltage along the line after short circuit.That is to say, for any lines that is shorter than one-quarter of the power-frequency wave, there is no need to worry about the issue of power-frequency overvoltage after the short circuit. However, for other lines, certain strategies are indispensable.


## 6.1. The Distribution Characteristic of Voltage and Current on HWTLs

A point-to-point power system with a HWTL is built in PSCAD, as shown in Figure1, where the traveling wave Bergeron model of the transmission line is chosen. This model is based on the distributed LC parameters and lumped resistance. It represents the inductance and capacitance of the π segments in a distributed way rather than lumped parameters and uses the lumped resistance to reflect line loss. Therefore, the Bergeron model in PSCAD can accurately simulate transmission lines at the power frequency f0 and is suitable for the cases related to the power flow at f0 [32]. The studied HWTL is an existing transmission line of the UHV power system in China, where the parameters are calculated and shown in Table 1.Table 1
The information of an UHVAC transmission line.
Type of wire
8 × LGJ-500/35 (with 400 mm between conductors in the bundle)
Type of the ground wire
JLB20A-170
Information of conductor arrangement
Distance from suspension point of upper conductor to ground: 65 m
Distance from suspension point of lower conductor to ground: 45 m
Vertical distance between suspension points of the ground wire and upper conductor: 12 m
Horizontal distance between upper conductor and central line: 0
Horizontal distance between lower conductor and central line: 16
Horizontal distance between ground wire and central line: 14
Conductor sag: 18.5 m
Ground wire sag: 12.5 m
Per unit length parameters
R
0 = 0.0081 Ω/km, X0 = 0.2631 Ω/km, C0 = 0.01383 μF/km,Pn = 4063 MW, λ0/2 = 2938 kmFigure4 shows the results of maximum power-frequency voltage, current, and their distribution characteristics under different operation modes. The error between the results of simulation and the proposed model calculation is tiny, which is less than 0.15%. That is to say that the method presented in this paper can accurately analyze the distribution characteristics of power-frequency voltage and current in the steady state. It can be seen from Figure 4 that the distribution characteristics of Um∗ and Im are consistent with Figure 2 and the conclusions in Section 3.3. When θ = 0 (cos θ = 1), Um∗ appears at the sending bus when a power of less than 4063 MW is transmitted or at the midpoint of the line (l = 1469 km) when the transmitted power is more than 4063 MW. The position of Im is opposite to that of Um∗. It occurs at l = 1469 km when P < 4063 MW or at the sending bus when P > 4063 MW. Moreover, Um∗ increases with P, while Im decreases with it. Given that the voltage on the line in an actual UHVAC power system cannot exceed 1.2 p.u. for the safety operation [33], it is suggested that the active power of the UHV system with HWTLs should be less than 1.25Pn = 5080 MW.Figure 4
Distribution of power-frequency waves along the HWTL in the steady state under different operation conditions: (a)θ = 0. (b) θ = 0. (c) θ > 0. (d) θ > 0. (e) θ < 0. (f) θ < 0.
(a)
(b)
(c)
(d)
(e)
(f)However, there is cosθ < 1 in practical engineering in general. Meanwhile, it has been obtained that both Um∗ and Im decrease with cos θ. That is, the decrease of the power factor raises the risk of overvoltage. Therefore, to avoid power-frequency overvoltage, it is necessary to set a limit on the power factor according to the power load, as shown in Figure 5. For example, cos θ is needed to be controlled above 0.86 when P = 0.9Pn = 3660 MW. Hence, the operating modes with P < 3660 MW and cos θ > 0.86 are safety for the system.Figure 5
The safe operating modes and overvoltage operating modes.Generally, 0.85 is considered as the minimum acceptable power factor for AC transmission. Therefore, the safe operating modes are shown in the shadow area of Figure5. It can be seen that keeping the power factor larger than 0.98 and the active power less than 5000 MW can eliminate the power-frequency voltage. However, it is challenge for a UHV power system to keep such a large power factor. Thus, the improvement of the power factor for UHV systems is significantly important for HWTLs to put into practice.


## 6.2. Analysis of Short Circuit Fault

### 6.2.1. Symmetrical Short Circuit

In order to analyze the power-frequency resonance characteristics after the three-phase short-circuit fault, the half-wavelength transmission system is built in the electromagnetic transient simulation program PSCAD, as shown in Figure3. The distributed Bergeron model is also used here as a reference due to its correctness for the calculation of the conditions related to power frequency.In this case, the HWTL is short to ground with no resistance (Rf = 10−19 Ω), low resistance (Rf = 5 Ω), and high resistance (Rf = 50 Ω). The condition of Rf = 100 Ω is also analyzed as a supplement. Figure 6 shows the simulation results of the maximum voltage UMf∗ and short-circuit current IMf on the line with different fault locations L. Among these, except for ZG within the general range in practice (i0.002 Ω∼i0.07 Ω), ZG = i0.2 Ω and ZG = i1 Ω are also analyzed to find out the characteristics as a whole.Figure 6
Characteristics of the maximum voltage and current after the short-circuit fault with different fault locations. (a) Voltage; (b) current.
(a)
(b)It can be seen from Figure6 that when ZG increases from i0.002 Ω to i0.07 Ω, both Ufm∗ and Ifm increase. However, they drop with ZG when ZG > i0. 07 Ω. When the system runs in the minimum operating mode (ZG = i0.07 Ω), Ufm∗ reaches the maximum value of 21 p.u. and Ifm reaches to 90 kA. Therefore, to limit the power-frequency voltage resonance and the short-circuit current, the system shall be operated in the mode as large as possible.In addition,Ufm∗ and Ifm decrease significantly with the fault resistance Rf. On the contrary, Lm is independent on Rf. It only decreases with ZG. Even so, Lm is always larger than λ0/4 = 1469 km, regardless of the value of Rf. Moreover, it is worth mentioning that the most serious fault overvoltages are accompanied by the maximum short-circuit currents. In other words, the most serious overvoltages and maximum short-circuit currents for HWTLs occur at the same time. This characteristic makes the short-circuit consequence of HWTLs much more serious.To conclude, the most serious short-circuit point is mainly related toRf rather than ZG. Considering the range of ZG and Rf in practice, it is at the distance of about 2653 km∼2929 km from the sending bus. In addition, Ufm∗ and Ifm can reach about 4∼21 p.u. and 16∼90 kA, respectively. The potential overvoltages and maximum short-circuit currents are far larger than the standard. Both of them need further analysis for the specific suppress strategies.Considering the actual operation of the system (i0.002 < ZG < i0.07 Ω and Rf < 50 Ω), the voltage distribution on a HWTL after the most serious short circuit are analyzed, as shown in Figure 7. In addition, the comparison between the simulation and calculation results are shown in Table 2.Figure 7
The voltage distribution along lines after the most serious short-circuit fault.Table 2
The simulation and calculation results of the voltage and current distribution characteristics under different fault conditions for the symmetric short circuit.
Z
G (p.u.)
i0.002
i0.02
i0.07
R
f (Ω)
10−19
5
50
10−19
5
50
10−19
5
50
U
fm
∗
p
.
u
.
Simulation
20.97
14.71
3.89
21.44
14.92
4
22.1
15.47
4
Calculation
19.89
14.17
3.72
20.21
14.57
3.81
20.9
14.88
3.88
I
fm
kA
Simulation
85.21
59.73
15.73
87.11
60.59
16.2
89.8
61.03
16.2
Calculation
83.65
57.92
14.94
85.25
58.34
15.13
86.4
59.58
15.3
I
fm
kA
Simulation
2929
2929
2926
2854
2854
2853
2653
2653
2650
Calculation
2929
2925
2924
2848
2848
2846
2650
2649
2646
L
m
km
Simulation
1469
1469
1466
1469
1469
1468
1469
1469
1466
Calculation
1469
1469
1469
1469
1469
1469
1469
1469
1469
l
m
km
Simulation
1460
1460
1460
1385
1385
1385
1184
1184
1184
Calculation
1460
1456
1455
1379
1379
1377
1181
1180
1177To confirm the results above, the following are some supplemental cases, as shown in Table3. Case A is simulated with the single-phase distributed parameter line model in DigSILENT/Power factory, while Case B uses the three-phase distributed parameter line model in MATLAB/Simulink. In addition, Case C is simulated in PSCAD, and the data on the steady state is collected. In this case, the frequency dependent model of the transmission line in PSCAD is used. This model is one of the most accurate model of the transmission line models, which is able to represent all the frequency-dependent effects of transmission lines.Table 3
The results of the voltage and current distribution in different cases.
Z
G (p.u.)
i0.002
i0.02
i0.07
R
f (Ω)
10−19
5
50
10−19
5
50
10−19
5
50
U
fm
∗
p
.
u
.
Case A
20.92
14.70
3.84
21.39
14.88
3.96
22.0
15.42
3.98
Case B
20.9
14.72
3.88
21.25
14.88
4.02
21.9
15.29
3.96
Case C
21.27
14. 95
3.96
21.58
15.24
4
22.43
15.87
4.
I
fm
kA
Case A
85.16
59.65
15.62
87.11
60.53
16.12
89.69
60.93
16.13
Case B
85.13
59.68
15.64
87.02
60.51
16.19
89.61
60.86
16.1
Case C
85.11
58.88
15.63
87.19
60.66
16.25
89.76
61.13
16.22
I
fm
kA
Case A
2929
2929
2926
2854
2854
2853
2653
2653
2650
Case B
2929
2928
2926
2855
2855
2853
2652
2651
2650
Case C
2930
2930
2927
2858
2857
2854
2655
2655
2650
L
m
km
Case A
1469
1469
1466
1469
1469
1468
1469
1469
1466
Case B
1468
1468
1466
1469
1468
1468
1469
1468
1466
Case C
1469
1469
1466
1469
1469
1465
1470
1470
1462
l
m
km
Case A
1460
1460
1460
1385
1385
1385
1184
1184
1184
Case B
1461
1460
1460
1386
1387
1385
1183
1183
1184
Case C
1461
1461
1461
1389
1388
1389
1185
1185
1188It can be seen that the calculation results in Table2 are fairly close to that shown in Case A and B. That is, the distributed Bergeron model can represent the distributed-parameter characteristic of the line at power frequency precisely, both in the aspect of single phase and three phase. Moreover, the calculation results in Table 2 also have a good accuracy compared with Case C. It shows that the Bergeron model in PSCAD can describe the feature of lines in the steady state. However, the difference between the calculation results and Case C are larger than the difference between calculation results and Case A or B. The reason is that there can be a slight difference between the actual parameter in the tower line model and the calculated parameters that is used in the distributed PI models.From the abovementioned analyses, it can be concluded that the distributed Bergeron model is accurate enough to describe the feature of transmission lines in the steady state after the short-circuit fault.


### 6.2.2. Asymmetrical Short Circuit

In this case, the effect of the sending system parameters and the additional impedance on the position characteristics (Lm and l2m) for asymmetric short circuit are discussed. The PSCAD simulation results and the calculation results are shown in Table 4.It can be obtained that the position of the most-serious fault point moves towards the midpoint of the line with the increase of the impedance of the sending system and the additional impedance. Meanwhile, the position of the maximum short-circuit voltage along the line moves from 1469 km to the short-circuit point with the increase ofZf. That is, the distance between the overvoltage caused by the frequency-power resonance and the sending bus is about 1184 km∼1460 km. Comparing with Tables 3 and 4, it can be seen that lm in Table 4 is approximately the same as the cases of the symmetric short circuit. In other words, it can be considered that asymmetric short circuit does not affect the positions of the most-serious overvoltage on the line. The position characteristics of the asymmetric short circuit can be analyzed by the same way as the three-phase short circuit fault.Table 4
The simulation and calculation results of the voltage and current distribution for asymmetric short circuit.
Z
G (p.u.)
i0.002
i0.02
i0.07
Zf (p.u.)
0.001
0.01
0.1
0.001
0.01
0.1
0.001
0.01
0.1
L
m
km
Simulation
2925
2887
2535
2850
2812
2460
2649
2611
2259
Calculation
2920
2872
2528
2841
2802
2455
2645
2605
2253
l
2
m
km
Simulation
1465
1427
1075
1465
1427
1075
1465
1427
1075
Calculation
1465
1425
1074
1465
1425
1069
1465
1425
1069
l
m
km
Simulation
1460
1460
1460
1385
1385
1385
1184
1184
1184
Calculation
1455
1453
1452
1376
1373
1374
1180
1176
1172The simulation results are consistent with the theoretical results of the proposed method. That is, the proposed algorithm can analyze the position characteristics of the most-serious asymmetric short-circuit fault. The error of the method is less than 1%.


### 6.2.3. Voltage Distribution Characteristics of Different Length Lines

To investigate the voltage distribution characteristics for different lines, transmission lines with length of 200 km, 500 km, 800 km, 1469 km (λ0/4), 2100 km, and 2700 km are discussed here. Figure 8 shows both the simulation and calculation results of the voltage distribution along these lines after the most-serious short circuit, where the receiving end is assumed as an infinite-bus system.Figure 8
The voltage distribution along lines after the most-serious short-circuit fault for different lines (Rf=10−19Ω). (a) L = 2700km; (b) L=2100km; (c) L=1469km; (d) L=800km; (e) L=500km; (f) L=200km.
(a)
(b)
(c)
(d)
(e)
(f)It can be obtained that, for transmission lines longer than 1469 km, power-frequency resonance overvoltage can occur at a distance of about 1469 km from the short-circuit point. However, for lines with length less than 1469 km, there is no overvoltage along the line after short circuit.That is to say, for any lines that is shorter than one-quarter of the power-frequency wave, there is no need to worry about the issue of power-frequency overvoltage after the short circuit. However, for other lines, certain strategies are indispensable.


## 6.2.1. Symmetrical Short Circuit

In order to analyze the power-frequency resonance characteristics after the three-phase short-circuit fault, the half-wavelength transmission system is built in the electromagnetic transient simulation program PSCAD, as shown in Figure3. The distributed Bergeron model is also used here as a reference due to its correctness for the calculation of the conditions related to power frequency.In this case, the HWTL is short to ground with no resistance (Rf = 10−19 Ω), low resistance (Rf = 5 Ω), and high resistance (Rf = 50 Ω). The condition of Rf = 100 Ω is also analyzed as a supplement. Figure 6 shows the simulation results of the maximum voltage UMf∗ and short-circuit current IMf on the line with different fault locations L. Among these, except for ZG within the general range in practice (i0.002 Ω∼i0.07 Ω), ZG = i0.2 Ω and ZG = i1 Ω are also analyzed to find out the characteristics as a whole.Figure 6
Characteristics of the maximum voltage and current after the short-circuit fault with different fault locations. (a) Voltage; (b) current.
(a)
(b)It can be seen from Figure6 that when ZG increases from i0.002 Ω to i0.07 Ω, both Ufm∗ and Ifm increase. However, they drop with ZG when ZG > i0. 07 Ω. When the system runs in the minimum operating mode (ZG = i0.07 Ω), Ufm∗ reaches the maximum value of 21 p.u. and Ifm reaches to 90 kA. Therefore, to limit the power-frequency voltage resonance and the short-circuit current, the system shall be operated in the mode as large as possible.In addition,Ufm∗ and Ifm decrease significantly with the fault resistance Rf. On the contrary, Lm is independent on Rf. It only decreases with ZG. Even so, Lm is always larger than λ0/4 = 1469 km, regardless of the value of Rf. Moreover, it is worth mentioning that the most serious fault overvoltages are accompanied by the maximum short-circuit currents. In other words, the most serious overvoltages and maximum short-circuit currents for HWTLs occur at the same time. This characteristic makes the short-circuit consequence of HWTLs much more serious.To conclude, the most serious short-circuit point is mainly related toRf rather than ZG. Considering the range of ZG and Rf in practice, it is at the distance of about 2653 km∼2929 km from the sending bus. In addition, Ufm∗ and Ifm can reach about 4∼21 p.u. and 16∼90 kA, respectively. The potential overvoltages and maximum short-circuit currents are far larger than the standard. Both of them need further analysis for the specific suppress strategies.Considering the actual operation of the system (i0.002 < ZG < i0.07 Ω and Rf < 50 Ω), the voltage distribution on a HWTL after the most serious short circuit are analyzed, as shown in Figure 7. In addition, the comparison between the simulation and calculation results are shown in Table 2.Figure 7
The voltage distribution along lines after the most serious short-circuit fault.Table 2
The simulation and calculation results of the voltage and current distribution characteristics under different fault conditions for the symmetric short circuit.
Z
G (p.u.)
i0.002
i0.02
i0.07
R
f (Ω)
10−19
5
50
10−19
5
50
10−19
5
50
U
fm
∗
p
.
u
.
Simulation
20.97
14.71
3.89
21.44
14.92
4
22.1
15.47
4
Calculation
19.89
14.17
3.72
20.21
14.57
3.81
20.9
14.88
3.88
I
fm
kA
Simulation
85.21
59.73
15.73
87.11
60.59
16.2
89.8
61.03
16.2
Calculation
83.65
57.92
14.94
85.25
58.34
15.13
86.4
59.58
15.3
I
fm
kA
Simulation
2929
2929
2926
2854
2854
2853
2653
2653
2650
Calculation
2929
2925
2924
2848
2848
2846
2650
2649
2646
L
m
km
Simulation
1469
1469
1466
1469
1469
1468
1469
1469
1466
Calculation
1469
1469
1469
1469
1469
1469
1469
1469
1469
l
m
km
Simulation
1460
1460
1460
1385
1385
1385
1184
1184
1184
Calculation
1460
1456
1455
1379
1379
1377
1181
1180
1177To confirm the results above, the following are some supplemental cases, as shown in Table3. Case A is simulated with the single-phase distributed parameter line model in DigSILENT/Power factory, while Case B uses the three-phase distributed parameter line model in MATLAB/Simulink. In addition, Case C is simulated in PSCAD, and the data on the steady state is collected. In this case, the frequency dependent model of the transmission line in PSCAD is used. This model is one of the most accurate model of the transmission line models, which is able to represent all the frequency-dependent effects of transmission lines.Table 3
The results of the voltage and current distribution in different cases.
Z
G (p.u.)
i0.002
i0.02
i0.07
R
f (Ω)
10−19
5
50
10−19
5
50
10−19
5
50
U
fm
∗
p
.
u
.
Case A
20.92
14.70
3.84
21.39
14.88
3.96
22.0
15.42
3.98
Case B
20.9
14.72
3.88
21.25
14.88
4.02
21.9
15.29
3.96
Case C
21.27
14. 95
3.96
21.58
15.24
4
22.43
15.87
4.
I
fm
kA
Case A
85.16
59.65
15.62
87.11
60.53
16.12
89.69
60.93
16.13
Case B
85.13
59.68
15.64
87.02
60.51
16.19
89.61
60.86
16.1
Case C
85.11
58.88
15.63
87.19
60.66
16.25
89.76
61.13
16.22
I
fm
kA
Case A
2929
2929
2926
2854
2854
2853
2653
2653
2650
Case B
2929
2928
2926
2855
2855
2853
2652
2651
2650
Case C
2930
2930
2927
2858
2857
2854
2655
2655
2650
L
m
km
Case A
1469
1469
1466
1469
1469
1468
1469
1469
1466
Case B
1468
1468
1466
1469
1468
1468
1469
1468
1466
Case C
1469
1469
1466
1469
1469
1465
1470
1470
1462
l
m
km
Case A
1460
1460
1460
1385
1385
1385
1184
1184
1184
Case B
1461
1460
1460
1386
1387
1385
1183
1183
1184
Case C
1461
1461
1461
1389
1388
1389
1185
1185
1188It can be seen that the calculation results in Table2 are fairly close to that shown in Case A and B. That is, the distributed Bergeron model can represent the distributed-parameter characteristic of the line at power frequency precisely, both in the aspect of single phase and three phase. Moreover, the calculation results in Table 2 also have a good accuracy compared with Case C. It shows that the Bergeron model in PSCAD can describe the feature of lines in the steady state. However, the difference between the calculation results and Case C are larger than the difference between calculation results and Case A or B. The reason is that there can be a slight difference between the actual parameter in the tower line model and the calculated parameters that is used in the distributed PI models.From the abovementioned analyses, it can be concluded that the distributed Bergeron model is accurate enough to describe the feature of transmission lines in the steady state after the short-circuit fault.


## 6.2.2. Asymmetrical Short Circuit

In this case, the effect of the sending system parameters and the additional impedance on the position characteristics (Lm and l2m) for asymmetric short circuit are discussed. The PSCAD simulation results and the calculation results are shown in Table 4.It can be obtained that the position of the most-serious fault point moves towards the midpoint of the line with the increase of the impedance of the sending system and the additional impedance. Meanwhile, the position of the maximum short-circuit voltage along the line moves from 1469 km to the short-circuit point with the increase ofZf. That is, the distance between the overvoltage caused by the frequency-power resonance and the sending bus is about 1184 km∼1460 km. Comparing with Tables 3 and 4, it can be seen that lm in Table 4 is approximately the same as the cases of the symmetric short circuit. In other words, it can be considered that asymmetric short circuit does not affect the positions of the most-serious overvoltage on the line. The position characteristics of the asymmetric short circuit can be analyzed by the same way as the three-phase short circuit fault.Table 4
The simulation and calculation results of the voltage and current distribution for asymmetric short circuit.
Z
G (p.u.)
i0.002
i0.02
i0.07
Zf (p.u.)
0.001
0.01
0.1
0.001
0.01
0.1
0.001
0.01
0.1
L
m
km
Simulation
2925
2887
2535
2850
2812
2460
2649
2611
2259
Calculation
2920
2872
2528
2841
2802
2455
2645
2605
2253
l
2
m
km
Simulation
1465
1427
1075
1465
1427
1075
1465
1427
1075
Calculation
1465
1425
1074
1465
1425
1069
1465
1425
1069
l
m
km
Simulation
1460
1460
1460
1385
1385
1385
1184
1184
1184
Calculation
1455
1453
1452
1376
1373
1374
1180
1176
1172The simulation results are consistent with the theoretical results of the proposed method. That is, the proposed algorithm can analyze the position characteristics of the most-serious asymmetric short-circuit fault. The error of the method is less than 1%.


## 6.2.3. Voltage Distribution Characteristics of Different Length Lines

To investigate the voltage distribution characteristics for different lines, transmission lines with length of 200 km, 500 km, 800 km, 1469 km (λ0/4), 2100 km, and 2700 km are discussed here. Figure 8 shows both the simulation and calculation results of the voltage distribution along these lines after the most-serious short circuit, where the receiving end is assumed as an infinite-bus system.Figure 8
The voltage distribution along lines after the most-serious short-circuit fault for different lines (Rf=10−19Ω). (a) L = 2700km; (b) L=2100km; (c) L=1469km; (d) L=800km; (e) L=500km; (f) L=200km.
(a)
(b)
(c)
(d)
(e)
(f)It can be obtained that, for transmission lines longer than 1469 km, power-frequency resonance overvoltage can occur at a distance of about 1469 km from the short-circuit point. However, for lines with length less than 1469 km, there is no overvoltage along the line after short circuit.That is to say, for any lines that is shorter than one-quarter of the power-frequency wave, there is no need to worry about the issue of power-frequency overvoltage after the short circuit. However, for other lines, certain strategies are indispensable.


## 7. Conclusions

Based on the pole distribution characteristics of the FLF, the distribution of steady-state power-frequency voltage and current along HWTLs and the resonance mechanism after the fault are analyzed in this paper. The contributions are as follows:(1)
The influence of load power and power factor on the distribution of steady-state power-frequency voltage and current along the line under the steady state is analyzed. It is proved theoretically that the increase of load power or the decrease of the power factor will increase the maximum voltage and current along the line. To reduce the risk of power-frequency overvoltage for a HWTL in practice, it is necessary to keep the load power less than 1.2 times of natural power.(2)
The method reveals the mechanism of steady-state power-frequency resonance for the symmetrical short circuit, which considers both the system parameters and fault condition simultaneously. The results show that, when a HWTL is short circuited at a distance of 2640 km∼2930 km from the sending bus, there can be an overvoltage that reaches up to 21 p.u. caused by power-frequency resonance. Meanwhile, the overvoltage occurs at a distance of 1469 km from the fault point.(3)
The fault point that leads to the most serious asymmetric short circuit moves towards the midpoint of the line with the increase of the sending system impedance or additional impedance. And the position of the maximum short-circuit voltage on the line moves from 1469 km to the short circuit points only with the increase of the additional impedance.(4)
Based on the algorithm, it is proved theoretically that for lines with length less than 1469 km, there is no steady-state power-frequency resonance and overvoltage in theory, no matter what the system and fault conditions are.(5)
Based on the proposed method, the position and amplitude of the maximum power-frequency voltage and current of the HWTL can be analyzed accurately. The simulation results verify its high applicability and accuracy.



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*Source: 3497625-2020-02-27.xml*